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International Journal of Engineering, Science and Technology

Vol. 2, No. 2, 2010

International Journal of Engineering, Science and Technology (IJEST)

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International Journal of Engineering, Science and Technology (IJEST) Editor: S.A. Oke, PhD, Department of Mechanical Engineering, University of Lagos, Nigeria E-mail : [email protected] Associate Editor (Electrical Engineering): S.N. Singh, PhD, Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur-208016, India EDITORIAL BOARD MEMBERS Kyoji Kamemoto (Japan) M. Abdus Sobhan (Bangladesh) Sri Niwas Singh (India) Shashank Thakre (India) Jun Wu (USA) Jian Lu (USA) Raphael Jingura (Zimbabwe) V. Sivasubramanian (India) Shaw Voon Wong (Malaysia) S. Karthikeyan (Sultanate of Oman)

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A.M. Rawani (India) Saurabh Mukherjee (India) P. Dhavachelvan (India) A. Bandyopadhyay (India) Velusamy. Sivasubramanian (India) S. Vinodh (India) Víctor Hugo Hinojosa Mateus (Chile)

International Journal of Engineering, Science and Technology (IJEST) REVIEWERS The following reviewers have greatly helped us in reviewing our manuscripts and have brought such submissions to high quality levels. We are indebted to them. Marcus Bengtsson (Sweden) Kit Fai Pun (West Indies) Peter Koh (Australia) Erhan Kutanoglu (USA) Jayant Kumar Singh (India) Maneesh Singh (Norway) RRK Sharma (India) Jamil Abdo (Oman) Agnes S. Budu (Ghana) Yuan-Ching Lin (Taiwan)

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Marisa Viera (Argentina) Shiguo Jia (China) S. Devasenapati Babu (India) Rajeeb Dey (India) Subrata Kumar Ghosh (India) Timothy Payne (Australia) Diwakar Tiwari (India) Mustafa Soylak (Turkey) Jerzy Merkisz (Poland) Md Fahim Ansari (India)

Jiun-Hung Lin (Taiwan) Tzong-Ru Lee (Taiwan) Subir Kumar Sarkar (India) Kee-hung Lai (Hong Kong) Jochen Smuda (Switzerland) Roland Hischier (China) Ahmed Abu-Siada (Australia) Hamzah Abdul Rahman (Jordan) Chih-Huang Weng (Taiwan) Yenming Chen (Taiwan)

Dinesh Verma (USA) Devanandham Henry (USA) M. Habibnejad Korayem (Iran) Radu Radescu (Romania) Hsin-Hung Wu (Taiwan) Amy Trappey (Taiwan) A.B. Stevels (Netherlands) Liang-Hsuan Chen (Taiwan) Richard Hischier (Switzerland) Shyi-Chyi Cheng (Taiwan)

Andrea Gerson (Australia) Ingrid Bouwer Utne (Norway) Maruf Hossain (Bangladesh) Enso Ikonen (Finland) Kwai-Sang Chin (Hong Kong) Jiunn – I Shieh (Taiwan) Hung-Yan Gu (Taiwan) Pengwei (David) Du (US) Min-Shiang Hwang (Taiwan) Ekata Mehul (India)

Shashidhar Kudari (India) Khim Ling Sim (USA) Rong-Jyue Fang (Taiwan) Chandan Guria (India) Rafael Prikladnicki (Brazil) Juraj Kralik (Slovak) Indika Perera (Sri Lanka) R K Srivastava (India) Ramakrishnan Ramanathan (UK) Suresh Premil Kumar (India)

Fernando Casanova García (Colombia) J. Ashayeri (The Netherlands) Siddhartha Kumar Khaitan (USA) Jim Austin (UK) Rafael Prikladnicki (Brazil) V. Balakrishnan (India) P. Dhasarathan (India) R. Venckatesh (India) Sarmila Sahoo (India) Avi Rasooly (USA)

Barbara Bigliardi (Italy) Huiling Wu (China) Ahmet N. Eraslan (Turkey)

Ryoichi Chiba (Japan) P.K.Dutta (India) Kimon Antonopoulos (Greece)

Josefa Mula (Spain) Amiya Ku.Rath (India) Fabio Leao (Brazil)

Francisco Jesus Fernandez Morales (Spain)

Serial No. Title of paper Authors Page No.

1 Parametric optimization of CNC end milling using entropy measurement technique combined with grey-Taguchi method

Sanjit Moshat, Saurav Datta, Asish Bandyopadhyay and Pradip Kumar Pal 1-12

2 Minimization of sink mark defects in injection molding process – Taguchi approach D. Mathivanan, M. Nouby and R. Vidhya 13-22

3 A price based automatic generation control using unscheduled interchange price signals in Indian electricity system

Saurabh Chanana, Ashwani Kumar 23-30

4 Propagation of S-waves in a non-homogeneous anisotropic incompressible and initially stressed medium

S. Gupta, S. Kundu1, A.K. Verma and R. Verma 31-42

5 Explicit solutions of the Rand Equation A. Huber 43-53

6 Cycle multiplicity of total graph of Cn, Pn, and K1,n M.M. Akbar Ali, S. Panayappan 54-58

7 A hybridized K-means clustering approach for high dimensional dataset Rajashree Dash, Debahuti Mishra, Amiya Kumar Rath, Milu Acharya 59-66

8 Effect of magnetic field on the blood flow in artery having multiple stenosis: a numerical study Gaurav Varshney, V.K. Katiyar, Sushil Kumar 67-82

9 Effect of training algorithms on neural networks aided pavement diagnosis Kasthurirangan Gopalakrishnan 83-92

10 Permanent magnet synchronous motor dynamic modeling with genetic algorithm performance improvement

Adel El Shahat, Hamed El Shewy 93-106

11 Waves, conservation laws and symmetries of a third-order nonlinear evolution equation A. Huber 107-116

12 Magnetic field effect on a three-dimensional mixed convective flow with mass transfer along an infinite vertical porous plate

N. Ahmed 117-135

13 Five-phase induction motor drive for weak and remote grid system Shaikh Moinoddin, 2Atif Iqbal* and 3Elmahdi M. Elsherif 136-154

14 Influencing parameters on performance of a mantle heat exchanger for a solar water heater - a simulation study

G. Naga Malleshwara Rao, K. Hema Chandra Reddy, M. Sreenivasa Reddy 155-164

15 Evaluation of thermal characteristics of oscillating combustion J. Govardhan, G.V.S. Rao 165-173

16 Persistence and stability of a two prey one predator system T. K. Kar and Ashim Batabyal 174-190

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Vol. 2, No. 2, 2010, pp. 1-12

INTERNATIONAL JOURNAL OF

ENGINEERING, SCIENCE AND TECHNOLOGY

www.ijest-ng.com

© 2010 MultiCraft Limited. All rights reserved

Parametric optimization of CNC end milling using entropy measurement technique combined with grey-Taguchi method

1Sanjit Moshat, 2*Saurav Datta, 3Asish Bandyopadhyay and 4Pradip Kumar Pal

1, 3, 4Department of Mechanical Engineering, Jadavpur University, Kolkata- 700032, West Bengal, INDIA

2Department of Mechanical Engineering, National Institute of Technology (NIT), Rourkela, Orissa-769008, INDIA *Corresponding Author (e-mail: [email protected], Saurav Datta)

Abstract End milling is the most important milling operation, widely used in most of the manufacturing industries due to its capability of producing complex geometric surfaces with reasonable accuracy and surface finish. However, with the inventions of CNC milling machine, the flexibility has been adopted along with versatility in end milling process. In order to build up a bridge between quality and productivity and to achieve the same in an economic way, the present study highlights optimization of CNC end milling process parameters to provide good surface finish and high material removal rate (MRR). The surface finish of the machined surface has been identified as quality attribute whereas MRR has been treated as performance index directly related to productivity. Attempt has been made to optimize quality and productivity in a manner that these multi-criterions could be fulfilled simultaneously up to the expected level. This has invited a multi-objective optimization problem which has been solved by Taguchi method coupled with grey relational analysis. Depending on relative importance, the priority weights of individual quality and performance attributes have been estimated by entropy measurement technique. Multi-objectives related to quality and productivity has been accumulated to evaluate an equivalent single quality index (called grey relational grade); which has been optimized finally by Taguchi method. Application feasibility of the aforesaid optimization technique has been illustrated in this reporting. Keywords: CNC end milling, surface finish, material removal rate (MRR), entropy measurement technique, Taguchi method

1. Introduction Among different types of milling processes, end milling is one of the most vital and common metal cutting operations used for machining parts because of its capability to remove materials at faster rate with a reasonably good surface quality. Also, it is capable of producing a variety of configurations using milling cutter. In recent times, computer numerically controlled (CNC) machine tools have been adopted to make the milling process fully automated. It provides greater improvements in productivity, increases the quality of the machined parts and requires less operator input. For these reasons, CNC end milling process has been recently proved to be very versatile and useful machining operation in most of the modern manufacturing industries. Only the implementation of automation in end milling process is not the last achievement. It is also necessary to improve the machining process and machining performances continuously for effective machining and also for the fulfillment of requirements of the industries. Surface roughness is a key factor in the machining process while considering machining performance and that is why in many cases, industries are looking for maintaining the good surface quality of the machined parts. Surface roughness is a measure of the technological quality of a product and a factor that greatly influences manufacturing cost and quality. It describes the geometry of the machined surface and combined with the surface texture, it can play an important role on the operational characteristics of the part. It also influences several functional attributes of a part, such as light reflection, heat transmission, coating characteristics, surface friction, fatigue resistance etc. However, the mechanism behind the formation of surface roughness is very dynamic, complicated and process dependent; therefore it is very difficult to calculate its value through analytical formulae. Various theoretical models that have been proposed are not accurate enough and apply only to a limited range of processes and cutting

Moshat et al. / International Journal of Engineering, Science and Technology, Vol. 2, No.2, 2010, pp. 1-12

2

conditions. Therefore, machine operators usually use “trial and error” approaches to set-up milling machine cutting conditions in order to achieve the desired surface roughness. Obviously, the “trial and error” method is not effective and efficient. Also, this method is very much time consuming. Therefore, the dynamic nature and widespread usage of milling operations in practice have raised a need for seeking a systematic approach that can help to set-up milling operations and also to help in achieving the desired surface roughness quality. On the other hand, material removal rate (MRR), which indicates processing time of the work piece, is another important factor that greatly influences production rate and cost. It is also necessary to study the material removal rate along with surface roughness in CNC end milling process. So, as a whole, there is a need for a tool that will allow the evaluation of the surface roughness and material removal rate (MRR) before the machining of the part and which, at the same time, can be easily used in the production-floor environment for contributing to the minimization of required time and cost and the production of desired surface quality. Both the surface roughness and material removal rate greatly vary with the change of cutting process parameters. That is why proper selection of cutting process parameters is also essential along with its prediction to obtain good surface finish (lower Ra value) and higher material removal rate in CNC end milling process. 2. Prior state of art and motivation of the present work Alauddin et al. (1995) developed two mathematical polynomial models in end milling of 190 BHN steel considering the factors: cutting speed, feed and axial depth of cut. Response surface contours were constructed based on these models. Baek et al. (1997) suggested that both static and dynamic variables of the milling process should be included in the surface roughness model. In the research made by Fuh and Wu (1995), Analysis of variance (ANOVA) method was adopted in order to determine the most influential parameters concerning surface quality. It was shown that the tool nose radius and feed influence surface roughness. Lee et al. (2001) developed simulation algorithms to predict the machined surface in end milling. Tsai et al. (1999) used neural networks for in-process prediction of surface roughness in milling operations. They had considered spindle speed, feed, depth of cut, and vibration “intensity” per revolution as parameters which would influence surface roughness. Benardos and Vosniakos (2002) used a neural network modeling approach for the prediction of surface roughness (Ra) in CNC face milling. Ozcelik and Bayramoglu (2006) developed a statistical model for surface roughness estimation in a high-speed flat end milling process under wet cutting conditions, using machining variables such as spindle speed, feed rate, depth of cut, and step over. Back et al. (2001) studied the effects of the insert run-out errors and the variation of the feed rate on the surface roughness and the dimensional accuracy in a face-milling operation using a surface roughness model. Alauddin et al. (1997) presented a study of the development of mathematical models for tool life in end milling steel (190 BHN) using high-speed steel slot drills under dry conditions. Yang and Chen (2001) demonstrated a systematic procedure of using Taguchi parameter design in process control of individual milling machines. The Taguchi parameter design had been done in order to identify the optimum surface roughness performance with a particular combination of cutting parameters in an end-milling operation. Ghani et al. (2004) applied Taguchi optimization methodology to optimize cutting parameters in end milling while machining hardened steel with TiN coated carbide insert tool under semi-finishing and finishing conditions of high speed cutting considering the milling parameters: cutting speed, feed rate and depth of cut. From the analysis of the result, the optimal combination of process parameters for low resultant cutting force and good surface finish was determined. Mansour and Abdalla (2002) developed a surface roughness model for the end milling EN32M (a semi-free cutting carbon casehardening steel with improved merchantability). Reddy et al. (2006) investigated the role of solid lubricant assisted machining with graphite and molybdenum disulphide lubricants on surface quality, cutting forces and specific energy while machining AISI 1045 steel using cutting tools of different tool geometry (radial rake angle and nose radius). Colak et al. (2007) used genetic expression programming method for predicting surface roughness of milling surface under varying cutting parameters. Cutting speed, feed and depth of cut of end milling operations were collected for predicting surface roughness. A linear equation was predicted for surface roughness estimation in this study. Oktem et al. (2005) focused on the development of an effective methodology to determine the optimum cutting conditions leading to minimum surface roughness in milling of mold surfaces by coupling Response Surface Methodology (RSM) with a developed genetic algorithm (GA). RSM was utilized to create an efficient analytical model for surface roughness in terms of cutting parameters: feed, cutting speed, axial depth of cut, radial depth of cut and machining tolerance. Fuh and Chang (1997) presented a dimensional-accuracy model for the peripheral milling of aluminum alloys under dry and down-milling conditions. Bayoumi, Kopac and Krajnik (2007) had presented the robust design of flank milling parameters dealing with the optimization of the cutting forces, milled surface roughness and the material removal rate (MRR) in the machining of an Al-alloy casting plate for injection moulds. In end milling, surface finish and material removal rate are two important aspects, which require attention and both are to be controlled precisely, because these two factors greatly influence machining performances. In modern industry, one of the trends is to manufacture low cost, high quality products in short time. Automated and flexible manufacturing systems are employed for that purpose. CNC machines are considered most suitable in flexible manufacturing system. Above all, CNC milling machine is very


3

useful for both its flexibility and versatility. These machines are capable of achieving reasonable accuracy and surface finish. Processing time is also very low as compared to some of the conventional machining process. Surface roughness is a measure of the technological quality of a product and a factor that greatly influences manufacturing cost and quality. On the other hand, material removal rate (MRR), which indicates processing time of the work piece, is another important factor that greatly influences production rate and cost. So, there is a need for a tool that will allow the evaluation of the surface roughness and material removal rate (MRR) value before the machining of the part and which, at the same time, can easily be used in the production-floor environment contributing to the minimization of required time and cost and the production of desired surface quality. Both the surface roughness and material removal rate greatly vary with the change of cutting process parameters. That is why proper selection of process parameters is also essential along with the prediction of the surface finish (lower Ra value) and material removal rate in CNC end milling process. Literature depicts that previous investigators focused on various aspects of modeling and simulation of surface roughness in various milling operations. Predictive models were developed to represent correlations among surface roughness and various milling parameters. Compared to surface roughness, study on MRR has been found done to a limited extent. Parametric optimization of milling process responses have been addressed too. But it has been observed that optimization has highlighted a single objective function. In a multi-response process, it may happen that optimization of a single response may cause severe quality loss for rest of the responses. Therefore, for solving a multi-criteria optimization problem; it is advised to convert individual objectives into an equivalent single objective function. In doing so, individual response weights have to be assigned depending on their relative importance. It seems that there is no specific guideline for assignment of individual response weights. It entirely depends on the decision maker. This may change the optimal setting in the case where optimization methodology that is being adopted is sensitive. In view of the above consideration the present study has been dealt with application of grey-Taguchi method for optimization of surface roughness and MRR in end milling operation; with adaptation to entropy measurement technique to evaluate priority value (weight) of individual responses; which is based on statistical analysis of the experimental data. 3. Experimental setup and procedure: Details of experimental plan are given below.

a) Checking and preparing the Vertical CNC milling machine system ready for performing the machining operation. b) Preparing rectangular aluminum plates of size 95 mm × 75 mm × 10 mm in shaping machine for performing CNC end

milling. c) Calculating weight of each plate by the high precision digital balance meter before machining. d) Creating CNC part programs for tool paths with specific commands using different levels of spindle speed, feed and depth

of cut and then performing end milling operation. e) Calculating weight of each machined plate again by the digital balance meter. f) Measuring surface roughness and surface profile with the help of a portable stylus-type profilometer, Talysurf (Taylor

Hobson, Surtronic 3+, UK) The present experimental work seeks to evaluate the optimal result for selection of spindle speed (S), feed rate (f) and depth of cut (d) in order to achieve good surface roughness (Ra value) and high material removal rate (MRR) during the CNC end milling process. Table 1 represents selected process control parameters used during the experiments. These parameters have been allowed to vary in three different levels.

Table 1: Process control parameters and their limits

Coded levels Spindle speed, S (rpm)

Feed rate, f (mm/min)

Depth of cut, d (mm)

-1 300 30 0.2 0 450 50 0.5

+1 600 70 0.8 The values of coded and actual value of each parameter used in this work are listed in Table 1. The experimental design matrix adopted as per Taguchi’s L9 Orthogonal Array (OA) design. The coded number for variables, used in Table 1 is obtained from the following transformation equations:

Spindle speed: SSSA

Δ−

= 0


4

Feed rate: f

ffBΔ−

= 0

Depth of cut: dddC

Δ−

= 0

Here A, B and C are the coded values of the variables S, f and d respectively; S0, f0 and d0 are the values of spindle speed, feed rate and depth of cut at zero level; ΔS, Δf and Δd are the units or intervals of variation in S, f and d respectively. The machine used for the milling is a ‘DYNA V4.5’ CNC vertical milling machine having the control system SINUMERIK 802 D with a vertical milling head. For generating the milled surfaces, CNC part programs for tool paths have been created with specific commands. The photograph of DYNA V4.5 CNC milling machine has been shown in Figure 1. The compressed coolant servo-cut has been used as cutting environment.

Figure 1: CNC vertical milling machine used for experiment

Commercially available CVD coated carbide tools have been used. The tools used are flat end mill cutters produced by WIDIA (EM-TiAlN). The tools are coated with TiAlN coating. For each material a new cutter of same specification has been used. The details of the end milling cutter are given below: Cutter diameter = 8 mm Overall length = 108 mm Fluted length = 38 mm Helix angle = 300

Hardness = 1570 HV Density = 14.5 g/cc Transverse rupture strength =3800 N/mm2 The test work pieces are made of Aluminum of size 95 mm x 75 mm x 10mm rectangular plate. Different plates of same dimension and material are used for each experimental run. Material properties are given below: Density: 2600-2800 kg/m3 Melting Point: 660 °C Elastic Modulus: 70-79 GPa Tensile Strength: 230-570 MPa Yield Strength: 215-505 MPa Roughness measurement has been done using a portable stylus-type profilometer, Talysurf (Taylor Hobson, Surtronic 3+, UK).

Material removal rate (MRR) has been calculated from the difference of weight of work piece before and after experiment.


5

3 1.mini f

s

W WMRR mm

tρ−−

=

Here, Wi is the initial weight of work piece in g; Wf is the final weight of work piece in g; t is the machining time in minutes; ρs is the density of aluminum (2.7 x 10-3 g/mm3). The weight of the work piece has been measured in a high precision digital balance meter (Model: DHD – 200 Macro single pan DIGITAL reading electrically operated analytical balance made by Dhona Instruments), which can measure up to the accuracy of 10-4 g and thus eliminates the possibility of large error while calculating material removal rate (MRR) in CNC end milling process. 4. Taguchi method Taguchi’s philosophy, developed by Dr. Genichi Taguchi, is an efficient tool for the design of high quality manufacturing system. It is a method based on Orthogonal Array (OA) experiments, which provides much-reduced variance for the experiment resulting optimum setting of process control parameters. Orthogonal Array provides a set of well-balanced experimental settings (with less number of experimental runs), and Taguchi’s Signal-to-Noise ratios (S/N), which are logarithmic functions of desired output; serve as objective functions in the optimization process. This technique helps in data analysis and prediction of optimum results. In order to evaluate optimal parameter settings, Taguchi method uses a statistical measure of performance called signal-to-noise ratio. The S/N ratio takes both the mean and the variability into account. The S/N ratio is the ratio of the mean (Signal) to the standard deviation (Noise). The ratio depends on the quality characteristics of the product/process to be optimized. The standard S/N ratios generally used are as follows: - Nominal-is-Best (NB), Lower-the-Better (LB) and Higher-the-Better (HB). The optimal setting is the parameter combination, which has the highest S/N ratio, [Mahapatra and Chaturvedi (2009), Moshat et al. (2010)]. 5. Grey relational analysis In grey relational analysis, experimental data i.e. measured features of quality characteristics of the product are first normalized ranging from zero to one. This process is known as grey relational generation. Next, based on normalized experimental data, grey relational coefficient is calculated to represent the correlation between the desired and actual experimental data. Then overall grey relational grade is determined by averaging the grey relational coefficient corresponding to selected responses. The overall performance characteristic of the multiple response process depends on the calculated grey relational grade. This approach converts a multiple- response- process optimization problem into a single response optimization situation; the single objective function is the overall grey relational grade. The optimal parametric combination is then evaluated by maximizing the overall grey relational grade. In grey relational generation, the normalized data corresponding to lower-the-better (LB) criterion can be expressed as:

)(min)(max

)()(max)(

kykykyky

kxii

iii −

−= (1)

For higher-the-better (HB) criterion, the normalized data can be expressed as:

)(min)(max)(min)(

)(kyky

kykykx

ii

iii −

−= (2)

Here )(kxi is the value after the grey relational generation, )(min kyi is the smallest value of )(kyi for the kth response,

and )(max kyi is the largest value of )(kyi for the kth response. An ideal sequence is )(0 kx for the responses. The purpose

of grey relational grade is to reveal the degrees of relation between the sequences say, 0[ ( ) ( ), 1, 2,3.......,9]ix k and x k i = .

The grey relational coefficient )(kiξ can be calculated as

max0

maxmin

)()(

Δ+ΔΔ+Δ

=ψψ

ξk

ki

i (3)


6

Here )()(00 kxkx ii −=Δ = difference of the absolute value )(0 kx and )(kxi ; ψ is the distinguishing

coefficient 10 ≤≤ψ ; )()(0minmin

min kxjkxkij −∀∈∀=Δ = the smallest value of i0Δ ; and

)()(0maxmax

max kxjkxkij −∀∈∀=Δ = largest value of i0Δ . After averaging the grey relational coefficients, the grey

relational grade iγ can be computed as:

∑=

=n

kii k

n 1

)(1 ξγ (4)

Here n = number of process responses. The higher value of grey relational grade corresponds to intense relational degree between the reference sequence )(0 kx and the given sequence )(kxi . The reference sequence )(0 kx represents the best process sequence. Therefore, higher grey relational grade means that the corresponding parameter combination is closer to the optimal. In the aforesaid study, it has been assumed that all quality features are equally important. But in practical case, it may not be so. Depending on the area of application, different response may have different preference and thereby, different tolerance limit. For example, the surface roughness and the MRR; both may be or may not be of equal importance. It depends on the decision maker. Therefore, different weightages have to be assigned to different responses. If different priority weightages have been assigned to different responses, the equation for calculating overall grey relational grade becomes:

∑

∑

=

== n

kk

n

kik

i

w

kw

1

1)(ξ

γ (5)

Here, iγ is the overall grey relational grade for ith experiment. ( )i kξ is the grey relational coefficient of kth response in ith

experiment and kw is the weightage assigned to the kth response. Justification of selecting grey based Taguchi method has been explained in the work by Nandi (2009). 6. Entropy measurement technique In information theory, entropy is a measure of how disordered a system is. As applying the concept of entropy to weight measurement, an attribute with a large entropy means it has a great diversity to responses so the attribute has more significant influence to the response. Recently, entropy measurement method is used to decide the weights in grey relational analysis. According to the definition proposed by Wen, Chang, and You (1998), the mapping function :[0,1] [0,1]if → used in entropy

should satisfy three conditions: (1) (0) 0if = (2) ( ) (1 )i if x f x= − and (3) ( )if x is monotonic increasing in the

range (0,0.5)x∈ . Thus, the following function ( )ew x can be used as the mapping function in entropy measure.

(1 )( ) . (1 ) 1x xew x x e x e−= + − − (6)

The maximum value of this function occurs at 0.5x = , and the value is 0.5 1 0.6487e − = . In order to get the mapping result in the range[0,1] , Wen et al. (1998) defined new entropy:

0.51

1 ( )( 1)

m

e ii

W w xe =

≡− ∑ (7)

Assume there is a sequence (1), (2),..................., ( )i i i ir r r n∈ = . Where, ( )ir j is the grey relational coefficient. Note that

1, 2,.............., ; 1, 2,.............., .i m j n= = erimentsofnumberTotalm exp= and responsesofnumberTotaln = The steps for weight calculation are as follows (Wen, 2004; Datta et al., 2009).


7

(a) Calculation of the sum of the grey relational coefficient in all sequences for each quality characteristic

1( ), 1, 2,............, .

m

j ii

D r j j n=

= =∑ (8)

(b) Evaluation of the normalized coefficient

0.5

1 1( 1) 0.6487

ke m m

= =− × ×

(9)

(c) Calculation of the entropy of each quality characteristics

1

( ) , 1, 2,................, .m

ij e

i j

r je k w j nD=

⎛ ⎞= =⎜ ⎟⎜ ⎟

⎝ ⎠∑ (10)

Here, (1 )( ) . (1 ) 1x x

ew x x e x e−= + − − (d) Calculation of the sum of entropy

1

n

jj

E e=

=∑ (11)

(e) Calculation of the weight of each quality characteristic:

1

[1 ]1 .1 .[1 ]

jj n

jj

ew

n E en E=

−=

− −−∑

, here, 1, 2,................, .j n= (12)

7. Optimization of surface roughness and MRR in end milling process Experimental data of Ra value and MRR (material removal rate) corresponding to L9 OA design of experiment has been tabulated below (Table 2). Experimental data have been normalized first (grey relational generation), shown in Table 3. For surface roughness (LB) and for MRR (HB) criterion has been used (equation (a) and (b) respectively).

Table 2: Experimental design and collected response data

Parametric combination (Design of experiment) Response features Sl. No. Spindle Speed

(rpm) Feed rate (mm/min)

Depth of Cut (mm)

Ra (μm)

MRR (mm3/min)

1 300 30 0.2 1.167 62.500 2 300 50 0.5 1.544 229.743 3 300 70 0.8 1.720 428.803 4 450 30 0.5 0.9696 133.076 5 450 50 0.8 1.541 304.770 6 450 70 0.2 1.210 148.200 7 600 30 0.8 1.250 172.811 8 600 50 0.2 1.000 116.400 9 600 70 0.5 1.44 338.680


8

Table 3: Normalized response values (grey relational generation) and quality loss estimates Response values (normalized) Quality loss estimates Sl. No. Ra MRR Ra MRR

Ideal sequence 1.0000 1.0000 0.0000 0.0000 1 0.7369 0.0000 0.2631 1.0000 2 0.2345 0.4566 0.7655 0.5434 3 0.0000 1.0000 1.0000 0.0000 4 1.0000 0.1927 0.0000 0.8073 5 0.2385 0.6614 0.7615 0.3386 6 0.6796 0.2340 0.3204 0.7660 7 0.6263 0.3011 0.3737 0.6989 8 0.9595 0.1471 0.0405 0.8529 9 0.3731 0.7540 0.6269 0.2460

Table 3 also represents quality loss estimates of individual responses. Grey relational coefficients of individual responses have been calculated using equation (c), and tabulated in Table 4. From the knowledge of literature distinguishing coefficient has been selected as 0.5. Table 5 represents overall grey relational grade (using equation (e)) for three sets of response weightages. In case 1, equal priority has been given to both roughness value and MRR. In case 2, 75% importance has been given to roughness value; and rest 25% for MRR. Similarly in case 3, roughness is assigned a weightage of 0.25 and for MRR priority weight is 0.75.

Table 4: Grey relational coefficients of individual responses Response values (normalized) Sl. No. Ra MRR

1 0.6552 0.3333 2 0.3951 0.4792 3 0.3333 1.0000 4 1.0000 0.3825 5 0.3964 0.5962 6 0.6095 0.3949 7 0.5723 0.4170 8 0.9251 0.3696 9 0.4437 0.6702

Table 5: Overall grey relational grade and predicted optimal setting

Case 1 Case 2 Case 3 Sl. No.

1 2 0.5w w= = 1 20.75, 0.25w w= = 1 20.25, 0.75w w= = 1 0.4943 0.5747 0.4138 2 0.4372 0.4161 0.4582 3 0.6667 0.5000 0.8333 4 0.6912 0.8456 0.5369 5 0.4963 0.4463 0.5463 6 0.5022 0.5558 0.4486 7 0.4947 0.5335 0.4559 8 0.6473 0.7862 0.5084 9 0.5570 0.5003 0.6136

Optimal setting A3 B3 C2 A2 B1 C1 A1 B3 C3 Rank of factors 2(A), 1(B), 3(C) 3(A), 2(B), 1(C) 3(A), 1(B), 23(C)

Optimal factor setting for the aforesaid three case studies have been shown in Table 5. These have been evaluated from Figure 2, 3 and 4 respectively. It has been observed that the optimal factorial combination is sensitive to the individual response weights. But assignment of these weights highly depends on the decision maker. Therefore, it seems necessary to propose a tool which can estimate these response weights mathematically so as to avoid variation of optimal setting due to various setting of response priorities defined by the decision maker. In view of the above fact, the study proposes application of entropy measurement technique for systematic calculation and strategic evaluation of individual response weights. The evaluation procedure is given in detail.


9

Figure 2: S/N Ratio plot for evaluation of optimal setting (Case 1)

Figure 3: S/N Ratio plot for evaluation of optimal setting (Case 2)

Figure 4: S/N Ratio plot for evaluation of optimal setting (Case 3) The sum of grey relational coefficients , 1, 2jD j = for both surface roughness and MRR, have been calculated using equation (h). These are shown in Table 6. The value of the normalized coefficient has been calculated using equation (i). In the present case,

9m = . Calculated value of the normalized coefficient becomes 0.1713k = .


10

Table 6: Calculation of jD (Sum of grey relational coefficients)

Sum of grey relational coefficients of each responses Surface roughness MRR

5.3306 4.6429

Table 7: Calculation of ( )i

j

r jD

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

( )i

j

r jD

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Sl. No.

Surface roughness MRR 1 0.1229 0.0718 2 0.0741 0.1032 3 0.0625 0.2154 4 0.1876 0.0824 5 0.0744 0.1284 6 0.1143 0.0851 7 0.1074 0.0898 8 0.1735 0.0796 9 0.0832 0.1443

The values of ( )i

j

r jD

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

and( )i

ej

r jk wD

⎛ ⎞× ⎜ ⎟⎜ ⎟

⎝ ⎠ for surface roughness and MRR have been furnished in Table 7 and 8, respectively.

Entropy of the responses has been calculated using equation (j); the values have been furnished in Table 9. The sum of entropy 0.7966E = has been calculated using equation (k). The individual response weights (Table 9) have been calculated using

equation (l). It has been found that both surface roughness and MRR are equally important 1 2 0.5w w= = . It corresponds to Case 1. The overall grey relational grade has been calculated using equation (e), shown in Table 5 earlier. Thus, multi-criteria optimization problem has been transformed into a single objective optimization problem using the combination of Taguchi approach and grey relational analyses. Higher is the value of grey relational grade, the corresponding factor combination is said to be close to the optimal. The optimal factor combination as evaluated using (Figure 2) is A3 B3 C2. After evaluating the optimal setting, confirmatory test has been conducted. It showed satisfactory results. At optimal setting, experimentally obtained value of overall grey relational grade became more compared to Taguchi’s prediction.

Table 8: Calculation of ( )i

ej

r jk wD

⎛ ⎞× ⎜ ⎟⎜ ⎟

⎝ ⎠

( )ie

j

r jk wD

⎛ ⎞× ⎜ ⎟⎜ ⎟

⎝ ⎠ Sl. No.

Surface roughness MRR 1 0.0492 0.0306 2 0.0316 0.0424 3 0.0270 0.0763 4 0.0690 0.0347 5 0.0316 0.0510 6 0.0463 0.0357 7 0.0438 0.0375 8 0.0650 0.0337 9 0.0350 0.0562


11

Table 9: Calculation of je (entropy of each quality indexes)

Entropy of each responses Surface roughness MRR

0.3985 0.3981

Table 10: Calculation of jw (weightage value of each quality characteristics)

Response weights Priority weight of surface roughness 1( )w Priority weight MRR 2( )w

0.50 0.50 8. Conclusions The foregoing study deals with multi-criteria optimization of CNC end milling by applying grey based Taguchi method. Application of grey relation theory is recommended to convert multiple objectives into a single objective function (overall grey relational grade) to facilitate exploration of Taguchi method. Traditional Taguchi method cannot solve a multi-objective optimization problem. That is why in the present work Taguchi method has been integrated with grey relation theory. Entropy measurement technique has been proposed to calculate individual response weights according to their relative priority. The integrated technique adapted in the present study may be efficiently applied for continuous quality improvement and off-line quality control of process not only CNC end milling but also any other machining operations where multiple objectives (responses) come under consideration. In the said research it has been assumed that all responses are independent i.e. uncorrelated. But in practice this assumption may deviate. How to tackle this situation seems really a challenging task. There exists enough scope to continue research in this particular direction. Moreover, interaction effect of process parameters may also be considered in future work. References Alauddin M., El Baradie M.A. and Hashmi M.S.J., 1995. Computer aided analysis of a surface roughness model for end milling,

Journal of Material Processing Technology, Vol. 55, pp. 123-127. Dae Kyun Baek, Tae Jo Ko and Hee Sool Kim, 1997. A dynamic surface roughness model for face milling, Precision Engineering,

Vol. 20, pp. 171-178. Fuh Kuang-Hua and Wu Chiu-Fu, 1995. A proposed statistical model for surface quality prediction in end milling of Al alloy,

International Journal of Machine Tools & Manufacturing, Vol. 35, No. 8, pp. 1187-1200. Ki Yong Lee, Myeong Chang Kang, Yung Ho Jeong, Deuk Woo Lee and Jeong Suk Kim, 2001. Simulation of surface roughness

and profile in high-speed end milling, Journal of Material Processing Technology, Vol. 113, pp. 410-415. Tsai Yu-Hsuan, Chen Joseph C. and Lou Shi-Jer, 1999. An in-process surface recognition system based on neural networks in end

milling cutting operations, International Journal of Machine Tools & Manufacturing, Vol. 39, pp. 583-605. Benardos P.G. and Vosniakos G.C., 2002, Prediction of surface roughness in CNC face milling using neural networks and

Taguchi’s design of experiments, Robotics and Computer Integrated Manufacturing, Vol. 18, pp. 343 - 354. Ozcelik B. and Bayramoglu M., 2006, The statistical modeling of surface roughness in high-speed flat end milling, International

Journal of Machine Tools & Manufacturing, Vol. 46, pp. 1395-1402. Back D.K., Ko T.J. and Kim H.S., 2001, Optimization of feed rate in a face milling operation using a surface roughness model,

International Journal of Machine Tools & Manufacturing, Vol. 41, pp. 451-462. Alauddin M., El Baradie M.A. and Hashmi M.S.J., 1997. Prediction of tool life in end milling by response surface methodology,

Journal of Material Processing Technology, Vol. 71, pp. 456-465. Yang L. John and Chen C. Joseph, 2001. A systematic approach for identifying optimum surface roughness performance in end-

milling operations, Journal of Industrial Technology, Vol. 17, pp.1-8. Ghani J.A., Choudhury I.A. and Hassan H.H., 2004. Application of Taguchi method in the optimization of end milling parameters,

Journal of Material Processing Technology, Vol. 145, pp. 84-92. Mansour A. and Abdalla H., 2002. Surface roughness model for end milling: a semi-free cutting carbon casehardening steel

(EN32) in dry condition, Journal of Material Processing Technology, Vol. 124, pp. 183-191. Reddy N., Kumar S. and Rao P. Venkateswara, 2006. Experimental investigation to study the effect of solid lubricants on cutting

forces and surface quality in end milling, International Journal of Machine Tools & Manufacturing, Vol. 46, pp. 189-198. Colak O., Kurbanoglu C. and Kayacan M.C., 2007. Milling surface roughness prediction using evolutionary programming

methods, Materials and Design, Vol. 28, pp. 657-666. Oktem H., Erzurumlu T. and Kurtaran H., 2005. Application of response surface methodology in the optimization of cutting

conditions for surface roughness, Journal of Material Processing Technology, Vol. 170, pp. 11-16.


12

Lou M.S., Chen J.C. and Li C.M., 1999. Surface roughness prediction technique for CNC end-milling, Journal of Industrial Technology, Vol. 15, No. 1, pp. 1-6.

Fuh K.H. and Chang H.Y., 1997. An accuracy model for the peripheral milling of aluminum alloys using response surface design, Journal of Material Processing Technology, Vol. 72, pp. 42-47.

Kopac J. and Krajnik P., 2007. Robust design of flank milling parameters based on grey-Taguchi method, Journal of Material Processing Technology, pp. 1-4.

Wen K.L., Chang T.C. and You M.L., 1998. The grey entropy and its application in welding analysis, IEEE International Conference on Systems, Man, and Cybernetics, Vol. 2, pp. 1842-1844.

Wen K.L., 2004. Grey system: Modeling and prediction, Yang’s Scientific Press, Tucson. Datta S., Nandi G. and Bandyopadhyay A, 2009. Application of entropy measurement technique in grey based Taguchi method for

solution of correlated multiple response optimization problems: A case study in welding, Journal of Manufacturing Systems, (in press) DOI: 10.1016/j.jmsy.2009.08.001.

Mahapatra S.S. and Chaturvedi, V., 2009, Modeling and analysis of abrasive wear performance of composites using Taguchi approach, International Journal of Engineering, Science and Technology, Vol. 1, No. 1, pp. 123-135.

Nandi G., 2009, Development of robust methodology for multi-response optimization and off-line quality control in submerged arc welding, Ph. D. Thesis, Jadavpur University, Kolkata.

Moshat S., Datta S., Bandyopadhyay A. and Pal P.K., 2010, Optimization of CNC end milling process parameters using PCA based Taguchi method, International Journal of Engineering, Science and Technology, Vol. 2, No. 1, pp. 92-102.

Biographical notes Mr. Sanjit Moshat is an Ex. P. G. scholar of Department of Mechanical Engineering, Jadavpur University, Kolkata, India. He did his M. E. in Production Engineering from Jadavpur University in the year 2007. Dr. Saurav Datta (*) is an Assistant Professor in the Department of Mechanical Engineering, National Institute of Technology Rourkela India. He obtained his BME (Hons), and PhD degrees in Mechanical Engineering from Jadavpur University. He has more than 6 years of experience in teaching and research. His current area of research includes Welding Technology, Manufacturing Processes, Multi-Criteria Decision-Making, Quality Engineering, optimization & Simulation modeling. Dr. Asish Bandyopadhyay is a Professor of the Department of Mechanical Engineering, Jadavpur University, Kolkata, India. He has been serving at the university for the last 23 years. He completed his undergraduate and postgraduate studies in Mechanical Engineering and was awarded PhD (Engg) degree from Jadavpur University. Before joining Jadavpur University, he served in heavy industries in India for approximately six years. His fields of interest include manufacturing and machining technology and heat transfer. Dr. Pradip Kumar Pal is presently a Professor of the Department of Mechanical Engineering, Jadavpur University, India. He obtained his BME (Hons), MME and PhD degrees in Mechanical Engineering from Jadavpur University. He has been involved with teaching and research since 1985 at said university. His teaching and research areas include manufacturing science, machine tool vibration and welding technology. Received December 2009 Accepted January 2010 Final acceptance in revised form February 2010

MultiCraft


Vol. 2, No. 2, 2010, pp. 23-30



www.ijest-ng.com


A price based automatic generation control using unscheduled interchange price signals in Indian electricity system

Saurabh Chanana*, Ashwani Kumar

Department of Electrical Engineering, National Institute of Technology Kurukshetra, INDIA *Corresponding Author: e-mail: [email protected], Tel +91-1744-233401, Fax.+91-1744-238050

Abstract The Availability Based Tariff (ABT) mechanism has been introduced in Indian system mainly to ensure grid security and to deal with grid indiscipline prevailing in the system prior to its introduction. Unscheduled Interchange (UI) charge - one of the components of ABT, acts a mechanism for regulating the grid frequency. At the same time, this mechanism offers opportunity to participants to exchange as and when available surplus energy at a price determined by prevailing frequency conditions. Although the underlying principle on which UI mechanism of ABT operates is quite different from the conventional load frequency control mechanism, it can still be viewed as a price based secondary generation control mechanism. Presently, the generators are responding to price signals manually. In this paper, a model for price based automatic generation control is presented. A modified control scheme is proposed which will prevent unintended unscheduled interchanges among the participants. The proposed scheme is verified by simulating it on a model of isolated area system having four generators. It has been shown here that such control mechanism, if adopted by all generating stations, can improve the control of frequency and bring down the UI obligation of participants. Keywords: Availability based tariff, generation control error, price based automatic generation control, unscheduled interchange price. 1. Introduction India is one of the fastest growing economies in the world. Providing adequate electricity supply to fuel high growth rates has been one of the major challenges before the country. Despite of steady increase in power generation capacity over last few decades, demand growth has far outstripped supply growth. This has led to a continued energy shortage over the years. Even at present, the energy shortage and peak demand shortage are estimated to be 9.3% and 12.6% respectively (CEA, 2009). The power deficit situation in the country was further aggravated by problems like fuel supply bottlenecks, high transmission and distribution losses, poor financial health of public utilities and a volatile electric grid with wide and rapid frequency fluctuations (Gupta and Sathaye, 2009; Yadav et al., 2005). Government of India has taken some path breaking initiatives during the last decade to overcome these problems. EA 2003 was enacted to introduce competition and efficiency in Indian power sector. The act envisages de-licensing of generation, unbundling of vertically integrated public utilities, introducing open access and allowing private participation in transmission and distribution sector (The Gazette of India, Extraordinary, 2003). Another important step has been introduction of ABT mechanism for bulk power transactions. The ABT mechanism has been introduced mainly to ensure grid security and deal with grid indiscipline prevailing in the system prior to its introduction (Bhushan, 2005).

Before the introduction of ABT scheme, grid operators in India faced a major problem in the form of grid indiscipline. A glaring symptom of which was wide and rapid fluctuations in grid frequency from below 48.0 Hz to above 52.0 Hz on daily basis. Abnormally low frequency during peak load hours was caused by inadequate generation capacity and attempts of meeting consumer loads in excess of available generation by public distribution utilities. High frequency during off-peak hours was the result of generation stations not backing down adequately when the consumer demand came down. The root cause of this problem was the then prevailing single part tariff structure for bulk power supply which disregarded the withdrawal pattern, deviation from schedule, system condition etc.

Chanana and Kumar / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 23-30

24

ABT is a three-part tariff scheme. First part being a fixed component is linked to the availability of generating stations, second part is a variable component linked to the energy charges for scheduled interchange and third part is a frequency dependent component linked with the unscheduled interchange. In the given generation shortage scenario of Indian power system, the third component of ABT – the UI charge acts as a mechanism for regulating the grid frequency. At the same time, this mechanism offers opportunity to participants to exchange as and when available surplus energy at a price determined by prevailing frequency conditions (Bhushan et al., 2004). Although the underlying principle on which UI mechanism of ABT operates is quite different from the conventional load frequency control mechanism, it can still be viewed as a price based secondary generation control mechanism. Currently, the nature of this control is manual as generators see the price signal and respond to it by increasing or decreasing their output manually. An automatic generation control scheme based in UI price has been presented by Tyagi and Srivastava (2004). This paper attempts further investigation on a price based automatic generation control (PBAGC) in Indian power system and how it can be modeled accurately. A modified control scheme is proposed which will prevent unintended unscheduled interchanges among the participants if PBAGC is introduced. It has been shown here that such mechanism, if adopted by all generating stations, can improve the control of frequency and bring the UI price down. 2. Structure of Indian Power Sector Before we give details of the constituents of ABT scheme, it is necessary to explain the backdrop in which this scheme was introduced. The structure of ownership of generation and transmission facilities in India is based on the federal structure of the country. In each of the states in the region, most of the generation is owned by the State Electricity Boards (SEBs), which are vertically integrated utilities with generation, transmission and distribution under their control. Many of the SEBs are now unbundled into separate generation, transmission and distribution corporations. The Union Government (also called the Central Government) own generation utilities that supply bulk power to the SEBs based on allocations to the states made by the Union Government. These utilities are known as Central Generating Stations (CGSs) and currently supply around 34% of the total generation to the system. Their share in generation capacity is projected to increase to 39% by 2012. The transmission grid is divided into five regional grids Northern, Western, Southern, Eastern and North-Eastern. All grids except Southern are now inter-connected via synchronous links. A Regional Load Dispatch Center (RLDC) in each of the region coordinates the daily scheduling process for dispatch of centrally generated power. Inter-regional exchanges are coordinated through National Load Dispatch Centre (IEGC, 2006). After collecting the availability of power from CGS for the next day (in 96 slots of 15 min each), RLDC allocates this power to respective SEBs as per their percentage share in CGS pool. SEBs then carry out an exercise to see how best they can meet the load of their consumers over the day, from their own generation stations along with their entitlement in CGS. They submit their requisitions to the RLDC, which then decides the dispatch schedule for CGS and withdrawal schedule for the SEBs. 3. Availability Based Tariff To deal with the problems faced by grid operators a new tariff scheme: Availability Based Tariff was introduced in July 2002. ABT comprises of three components: (a) Capacity Charge (b) Energy Charge (c) Unscheduled Inter-change (UI) Charge. 3.1 Capacity Charge: This component represents the fixed cost and is linked to the availability of the plant, i.e., its capability to deliver MWs on a day-by-day basis. The total amount payable to the generating company over a year towards the fixed cost would depend on the average availability of the plant over the year. In case the average actually achieved over the year is higher than the specified norm for plant availability, the generating company would get a higher payment. In case the average availability achieved is lower, the payment will be lower. Hence, the scheme is named Availability Based Tariff. 3.2 Energy Charge: This component of ABT comprises of the variable cost, i.e. the fuel cost of the power plant for generating energy as per given schedule for the day. Therefore, this energy charge is not according to the actual generation but only for scheduled generation. 3.3 Unscheduled Interchange Charge: In case there are deviations from schedule, this third component of ABT comes into picture. Deviations from schedule are determined in 15-minute time blocks through special metering. They are priced according to the system condition prevailing at that time. If the frequency is above 50 Hz (nominal frequency in Indian System), UI rate will be small and if it is below 50 Hz, it will be high. As long as the actual generation/withdrawal is according to the given schedule, the third component of ABT is zero. In case of over-drawal (withdrawal in excess of schedule), beneficiary has to pay UI charge according to the frequency dependent rate specified. Beside promoting competition, efficiency and economy and leading to more economically viable power scenario, ABT has been able to pave way for high quality power with more reliability and availability through enhanced grid discipline.


25

• By giving incentives for enhancing the output capability of power plants, it enables more consumer load to be met during peak hours.

• By separating fixed charges based on availability from variable charges, backing down during off peak hours no longer results in financial loss to generating stations. Therefore, earlier incentive for not backing down and raising system frequency during off-peak hours no longer exists.

• By charging separately for unscheduled interchanges, the problem of over-drawal during peak load condition, resulting in lowering of frequency, has been controlled. UI rate is high during the low frequency condition, which discourages the over-drawal of power.

Apart from these intended benefits, the ABT mechanism has provided a vast scope of unscheduled interchange of as and when available surplus energy in the grid. 4. Modeling of Frequency-UI Price Block The shape of UI price vs. frequency curve has been a subject of much debate among the sector participants. Regular modifications have been done in the shape of UI curve since it was introduced in 2000. The modifications have been ordered by Central Electricity Regulatory Commission (CERC), so as to meet the stated objectives of ABT mechanism. Initially, the frequency range in which UI prices vary was set between 49.0 and 50.5 Hz (CERC, 2000). In 2009, CERC has come up latest regulations (CERC, 2009) which set the frequency range between 49.2 and 50.3 Hz. UI Price varies inversely with frequency. In the original regulations, the minimum price was zero INR/kWh at 50.5 Hz and maximum price limit was 4.80 INR/kWh at 49.0 Hz. In the 2009 regulations, minimum price is zero INR/kWh at 50.3 Hz and maximum price is 7.35 INR/kWh at 49.2 Hz. This curve is shown in Figure 1. The maximum price cap is set by CERC to accommodate the highest price generation in the system. Now, CERC has adopted a process whereby the max UI price cap is to be reviewed six monthly. Additionally, there are kinks (or dual slopes) in the UI curve. In the March 2009 UI curve shown in Figure 1, the price varies from 7.35 INR/kWh at 49.2 Hz to 4.80 INR/kWh at 49.5 Hz in steps of 0.17 INR/kWh per 0.02 Hz and from 4.80 INR/kWh at 49.2 Hz to zero INR/kWh at 50.3 Hz, in steps of 0.12 INR/kWh per 0.02 Hz.

49 49.2 49.5 50 50.3 50.50

1

2

3

4

5

6

7

8

Frequency (in Hz)

UI C

harg

e( in

INR

/kW

h)

Figure 1. UI Price vs. Frequency Chart (March, 2009) We use a frequency to price conversion block for modeling the price responsive generation in this paper. Although the UI price varies in discreet steps of 0.02 Hz, we assume continuous variation of frequency for this work. The block is implemented in SIMULINK using embedded MATLAB function block. The code for the block is given in Figure 2.

if f <= 49.2

ρ = 7350 elseif f <= 49.5

ρ = 4800 + 8500 * (49.5 - f) elseif f <= 50.3

ρ = 6000* (50.3 - f) else ρ = 0 end end

end

Figure 2. Frequency to Price conversion block


26

5. Proposed Generation Control Model As a result of successful implementation of UI mechanism in Indian electricity sector, a UI price signal is always available in real-time. This price signal fluctuates with grid frequency and can easily be accessed by any participant (generator or load) connected to the grid. This section explains a price based generation control mechanism, which can be adopted by generators by utilizing the UI price signal. Indian Electricity Grid Code (IEGC) stipulates a Free Governor Mode of Operation (FGMO) for all generation units connected to the regional grid. According to IEGC, the droop of the governors should be set between 3 to 6 percent (IEGC, 2006). The FGMO is a primary level control, which acts as a first line of defense against sudden frequency rise\fall. With only FGMO acting, the system deviates from nominal frequency and generation units deviate from their respective schedules, in response to changes in load. Conventionally, a secondary level control (LFC or AGC) is recommended to bring frequency back to nominal value. Implementation of a successful secondary control mechanism is not possible in India under present circumstances, as there is a generation shortage. However, the UI mechanism of ABT itself is meant to provide a secondary level control. The generation units are expected to respond to the UI price signal in real-time. Although the wide fluctuations in frequency have been tamed through implementation of UI mechanism, the frequency profile is still not as smooth as desirable. The Generation units respond to the UI price signal manually, often resulting in a delayed response. This causes frequency to fluctuate rapidly. Sometimes, the generation units may not act or may act out of merit. This results in a higher UI price than expected.

1 1+sTg

1 1+sTt

1 R

Frequencyto Price

Calculate Marginal

Cost

Kis

f0

∆f

ρ ∆Pg

Pg0

γ

GCE

Figure 3. Price based automatic generation control This section describes a PBAGC that will ensure that generators respond to the UI price automatically and in a desirable manner. The desirable properties of this controller are:

• It should ensure that frequency control is as smooth as possible • It should ensure that UI price is minimum possible. The basic principle of this control is illustrated in Figure 3. Each generator individually monitors the UI price ρ and compares

with its marginal cost γ. It derives an error signal, which is the difference of current UI price and its own marginal cost. This error signal, which can be termed as generation control error (GCE), is fed to an integral controller. A positive GCE indicates that the generator will profit by increasing generation level. A negative GCE indicates that Generator will profit by decreasing the generation level. Since under ABT, the payments received by generators for UI are separate from the payments for SI, the generators earn profit in both cases. 6. Simulation Study and Results The control scheme described in the previous section is similar to one suggested by Tyagi and Srivastava (2004). This scheme has a shortcoming. It runs into problem due to fixed nature of UI curve. Under current regulations (CERC, 2009), UI price is pegged at 1800 INR/MWh at 50 Hz frequency. This means that if everyone (Generators and Loads) adhere to the schedule, the frequency should be 50 Hz and UI price 1800 INR/MWh. However, at 1800 INR/MWh UI price some generators get an error signal causing them to deviate from their schedule. This may even cause the frequency to deviate from nominal value. This outcome is undesirable, as it results in UI among generators even when load is as per schedule. To illustrate our point, we simulated an isolated area system having a capacity of 5000 MW supplied by four generating stations. The relevant data of isolated area and generating stations is given in Table 1 and Table 2 respectively. All models are created using MATLAB/SIMULINK. Let us consider three scenarios:


27

Table 1. Area Data Capacity 5000 MW

H 25000 MWs D 100 MW/Hz f 0 50 Hz

Table 2. Generator Data

Generator 1 Generator 2 Generator 3 Generator 4 Capacity (MW) 1500 1500 1000 1000

bi (INR/MWh) 800 1000 1600 2000 Cost Coefficients ci (INR/MWh2) 0.3 0.3 0.4 0.4

Table 3. Generation Schedule (in MW)

Generator 1 Generator 2 Generator 3 Generator 4 Scenario 1 1500 1333.33 250 0 Scenario 2 1500 1500 83.33 0 Scenario 3 1500 1500 250 0

Scenario 1: The generators are scheduled in merit order and load level results in system marginal cost of 1800 INR/MWh. This scenario represents the only case in which price based generation control works successfully. The scheduled generation of this scenario is given in first row of Table 3. For this scenario, the generation is scheduled so that the overall system marginal cost is 1800 INR/MWh. The load level does not change during the simulation. This means that none of the generators will get any error signal. The outcome of simulation is shown in Figure 4. It is observed that there is no impact on either frequency/UI price or scheduled generation. Scenario 2: Load level results in system marginal cost of 1800 INR/MWh but generators are not scheduled in merit order. The scheduled generation for this scenario is given in the second row of Table 3. In this case, the generation is not scheduled as per merit order. Even if the load level is kept same, the generators scheduled out of merit will get error signal and will reschedule. The results of simulation are shown in Figure 5. Although the frequency and UI price are restored, the generation of second and third generator is rescheduled. Scenario 3: Load level results in a system marginal cost higher that 1800 INR/MWh. The scheduled generation for this scenario is given in third row of Table 3. Until now, the load has been kept at a level so that the system marginal cost comes out to be 1800 INR/MWh. However, this would rarely be the case. The results of simulation are shown in Figure 6. The outcome of simulation shows that even when there is no load change, neither the nominal frequency / UI price nor the scheduled generation can be maintained.

0 20 40 60 80 10049.99

49.995

50

50.005

Freq

uenc

y (in

Hz)

0 20 40 60 80 1001740

1760

1780

1800

1820

1840

1860

UI P

rice

(in IN

R/M

Wh)

0 20 40 60 80 100-200

-100

0

100

200

ΔP

g (in

MW

)

Time in s

ΔPg1 ΔPg2 ΔPg3 ΔPg4

Figure 4. Scenario 1

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Therefore, there is a need to improve the proposed control mechanism so that no action is taken by generators if all the loads and other generators stick to their respective schedules. Instead of computing GCE for each generator in a simple manner as shown in Figure 3, a new algorithm for computing GCE is proposed. Flowchart of modified control scheme is shown in Figure 7. This control scheme ensures that schedules are maintained even if there is no load change, and suitable action is taken if there is a load change. The basic approach of the proposed control scheme is that it deals with generators with marginal cost higher than 1800 INR/MWh, and those with marginal cost less than equal to 1800 INR/MWh in a different manner. For the generators whose marginal cost is greater than 1800 INR/MWh, the next step is to see if UI price is higher than its marginal cost, than GCE is set at ρ minus γ. In case UI price is less than 1800 INR/MWh, GCE is set at ρ -1800 and in case UI price is in between the marginal cost of generator and 1800, GCE is set at zero. This ensures that GCE would be positive if UI price is greater than the marginal cost of generator and negative if UI price is less than 1800 INR/MWh and would be zero otherwise. Similarly, for generators having marginal cost less than 1800 INR/MWh, GCE would be positive if UI price is greater than 1800 INR/MWh and negative if UI price is less than the marginal cost of generator and would be zero otherwise.

Read γ, ρ

GCE = ρ - γ

If γ >1800

GCE = ρ - 1800 GCE = 0

If ρ > γ

IF ρ < 1800

IF ρ >1800

IF ρ < γ

Y

Y

Y

Y

Y

N

N

N N

Figure 7. Flowchart for calculating GCE

To verify the outcome of proposed control, we apply this scheme to all three scenarios described before. The generation schedule or frequency does not deviate in any of the three cases. The result of application of proposed control on third scenario is given in Figure 8. We further test the operation of proposed control in Scenario 3 under application of a step load change of 100 MW. The results are shown in Figure 9. In this case, Generator 1 and 2 are loaded to their maximum capacity, hence cannot absorb any load increase. Only Generator 3 loaded at 250 MW and Generator 4 loaded at zero MW have a capacity to increase generation. As soon as load increase is applied, the frequency falls and consequently UI price rises, reaching around 2400 INR/MWh. Since, the current marginal cost of Generator 3 is 1800 INR/MWh and that of Generator 4 is 2000 INR/MWh, both generators get a positive GCE. Generator 3 and 4 take corrective action and UI price finally settles at 1875 INR/MWh. At this UI price, Generator 3 should absorb all the load change and Generator 4 should get a GCE equal to zero. We observe that Generator 3 has not completely absorbed the load increment. Out of 100 MW, around five MW is taken by Generator 4. This small generation by Generator 4 is the result of primary action (FGMO) of Generator 4. The final value of UI price is 1875 INR/MWh, which corresponds to 49.9875 Hz. The frequency settles at a value less than nominal, hence some generation of Generators 3 and 4 will be due to their FGMO or primary control action. Ultimately, we can conclude that proposed price based control, unlike a conventional secondary, may not restore frequency to nominal value. However, if compared to only FGMO, it reduces the frequency error considerably and minimizes the UIs taking place in the system.


29

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load change)

A step load change of -100 MW is applied to check the performance of the proposed control under load decrement. The results are shown in Figure 10. A rise in frequency is observed and consequently the UI price falls to around 1450 INR/MWh. Only Generators 1, 2, and 3 are capable of reducing their outputs. At scheduled load, marginal cost of Generators 1, 2, 3 are 1700, 1900 and 1800 INR/MWh respectively. All generators including Generator 1, whose marginal cost is less than 1800 INR/MWh, get a negative GCE. As a result of corrective action taken by these generators, the UI price rises to 1800 INR/MWh and frequency is restored to 50 Hz. Since the generators were loaded out of merit initially, there is sufficient capacity available with Generator 2, who has the highest marginal cost, to absorb reduction of 100 MW. Had the generators been loaded in merit order, a load reduction of 100 MW would have certainly resulted in small frequency rise. We have seen that the proposed control scheme gives satisfactory results in terms of frequency control and UI reduction for a variety of operational conditions. 7. Conclusions This paper discusses UI mechanism of ABT as a price based secondary control, which is manual in nature. It presents a price based automatic generation control that can be implemented in Indian system. It is shown that this control, if not suitably modified, will result in undesirable UIs. A modified controller is proposed which is able to deal with fixed nature of UI curve. The working of modified controller is verified through simulation of various scenarios and it is shown that it is able to avoid undesirable UIs. The successful operation of modified PBAGC is further tested by applying a step load increase and a step load decrease. The results of simulation show that the control is successful in bringing down the frequency deviation. The final frequency under this control may not settle to nominal value, but to a value equivalent to system marginal cost (on UI price vs. frequency curve) while serving the load. Implementation of proposed control on all central and state generating stations will not only result in better control of frequency, but merit order dispatch of generation can also be ensured at the same time. The UI obligations of participants can be drastically reduced through this mechanism. In this paper, modified PBAGC is applied to an isolated area test system. In future, the impact of such mechanism on frequency, UIs and tie-line exchanges can be observed by taking multi-area systems. Nomenclature f Frequency (Hz) ρ Unscheduled interchange price (INR/MWh) γ Marginal cost (INR/MWh) bi First order cost coefficient (INR/MWh) ci Second order cost coefficient (INR/MWh2) H Inertia constant of isolated area (MW-s) D Load damping of isolated area (MW/Hz) ABT Availability Based Tariff


30

AGC Automatic Generation Control CERC Central Electricity Regulatory Commission CGS Central Generating Station FGMO Free Governor Mode of Operation GCE Generation Control Error IEGC Indian Electricity Grid Code LFC Load Frequency Control PBAGC Price Based Automatic Generation Control RLDC Regional Load Dispatch Center SEB State Electricity Board UI Unscheduled Interchange References Bhushan B., Roy A. and Pentayya P., 2004. The Indian Medicine. Proc. of IEEE PES General Meeting, Denver, USA. Vol. 2, pp.

2236-2239. Bhushan B., 2005. ABC of ABT: A primer on availability tariff. Available online: http://www.nrldc.org/docs/abc_abt.pdf. Central Electricity Authority, 2009. Load Generation Balance Report 2009-10. CEA, New Delhi, India. Available online:

http://cea.nic.in/god/gmd/lgbr_report.pdf Central Electricity Regulatory Commission, 2000. ABT Order. CERC, New Delhi, India. Available online:

http://www.cercind.gov.in/orders/2-1999GOIABT040100.pdf Central Electricity Regulatory Commission, 2009. Unscheduled interchange charges and related matters regulation, 2009. CERC,

New Delhi, India. Available online: http://www.cerc.gov.in. Gupta A.P. and Sathaye J., 2009. Electrifying India. IEEE Power and Energy Magazine. Vol. 7, No. 5, pp. 53-61. Indian Electricity Grid Code, 2006. Available online: http://cercind.gov.in/Regulations/Indian-Electricity-Grid-Code-2006.pdf. Roy A., Khaparde S.A., Pentayya P. and Pushpa S., 2005. Operating Strategies for Generation Deficient Power System. Proc.

IEEE PES General Meeting, San Francisco, USA. Vol. 3, pp. 2738-2745. The Gazette of India, Extraordinary, 2003. The Electricity Act, 2003 Part II Section 3 Sub-section (ii) June 10, 2003. Ministry of

Power, Government of India, New Delhi., Available online: http://www.powermin.nic.in/acts_notification/electricity_act2003 /pdf/The%20Electricity%20Act_2003.pdf

Tyagi B. and Srivastava S.C., 2004. A mathematical framework for frequency linked availability based tariff mechanism in India. National Power Systems Conference, IIT Madras, Chennai, India, Vol. 1, pp. 516-521.

Yadav R.G., Roy A., Khaparde S.A., Pentayya P., 2005. India’s fast-growing power sector. IEEE Power and Energy Magazine. Vol. 3, No. 4, pp. 39-48.

Biographical notes Saurabh Chanana received bachelors and master degree in technology from National Institute of Technology Kurukshetra, India in 1996 and 2002, respectively. Currently, he is assistant professor in the Electrical Engineering, National Institute of Technology Kurukshetra, India. His research interests include power system restructuring, power system optimization & control, demand side management and FACTS applications. He is a life member of ISTE (India) and a member of IEEE. Ashwani Kumar did his bachelors of technology in Electrical Engineering from G. B. Pant University of Agriculture and Technology, Pant Nagar in 1988 and master’s degree from Punjab University Chandigarh in 1994 in honors. He obtained his Ph.D. from IIT Kanpur, India in Oct. 2003. He has his interests in power system restructuring, FACTS applications, power systems dynamics, distributed generation, demand side management, and price forecasting. He is member of IEEE, life member of Indian Society of Technical Education, (ISTE), and life member of Institution of Engineers (IEI) India. Presently, he is working as associate professor in the Department of Electrical Engineering, National Institute of Technology, Kurukshetra, India. Received December 2009 Accepted February 2010 Final acceptance in revised form March 2010

MultiCraft


Vol. 2, No. 2, 2010, pp. 13-22



www.ijest-ng.com


Minimization of sink mark defects in injection molding process – Taguchi approach

D. Mathivanan1*, M. Nouby2 and R. Vidhya3

1Director of CAE Infotech, Chennai-600020, INDIA

2AU/FRG Institute for CAD/CAM, Anna University, Chennai-600025, INDIA 3Institute of Remote Sensing, Anna University, Chennai-600025, INDIA E-mail: [email protected] (Mathivanan D.); *Corresponding author

Abstract Optimal setting up of injection molding process variables plays a very important role in controlling the quality of the injection molded products. It is all the most important to control attribute defects like sink marks. Sink marks are basically a “designed in” problem and hence it is to be attended during designs stages. Owing to certain conditions and constraints, sometimes, it is rather ignored during design stages and it is expected to be handled by molders with only instruction to ‘do the best’. Handling of numerous processing variables to control defects is a mammoth task that costs time, effort and money. This paper presents a simple and efficient way to study the influence of injection molding variables on sink marks using Taguchi approach. Using the Taguchi approach, optimal parameter settings and the respective sink depth were arrived. The sink depth based on the validation trials was compared with the predicted sink depth and they are found to be in good agreement. The results demonstrate the ability of this approach to predict sink depth for various combination of processing variables with in the design space. Keywords: Sink mark, plastic injection molding, Taguchi optimization, process optimization, attribute defects in injection molding

1. Introduction Injection molding is one of the major net shape forming processes for thermoplastic polymers. Over 30% of all the plastic parts manufactured are by injection molding. Injection molding is ideally suited for manufacturing large quantities of mass produced plastic parts of complex shapes and sizes. In the injection molding process, hot melt of plastic is forced into a cold empty cavity of desired shape called mold. Then, the hot melt is allowed to solidify. Solidified net shape product is ejected out of the mold upon opening. Although the process is simple, prediction of final part quality is a complex phenomenon due to the numerous processing variables. Common defects in injection molding process can be classified in to two ways. They are:

1. Dimensional related 2. Attribute related

Dimensional related defects can be controlled by correcting the mold dimensions. But, attribute related defects are generally dependent on the processing parameters. Some of the common attribute related defects are splay marks, sink marks, voids, weld/meld lines, poor surface finish, air traps, burn marks etc. Of all attribute defects, sink marks are considered to be perennial. Sink marks can be defined as ‘an unwanted depression or dimple on the surface of molding due to localized shrinkage’. The sink marks can be minimized by optimizing the process parameter settings. The process parameter settings were traditionally based on operator’s experiences. A great deal of research is being carried out to understand, identify critical factors and possibly to optimize the molding process. Most of the work carried out in the last decade was based on: theoretical, computer aided engineering based simulation models and practical experimental trials (Kazmer, 1997). Shi and Gupta (1998) tried to predict sink mark depths using localized shrinkage

Mathivanan et al. / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 13-22

14

analysis through finite element methods. They also tried to establish approximate empirical equations based on the rib geometry, packing time and packing pressure (Shi and Gupta, 1999). But, other parameters like melt temperature, mold temperature, etc were not considered. Predicted Sink mark depths were observed to be smaller than the actual. Dan Tursi and Bistany (2000) attempted to study the effect of tooling factors like kind of mold material, gate type in addition to some processing parameters. In their study, barrel temperature was considered instead of melt temperature. It was observed that, gate design did not significantly contribute to sink marks but choice of mold material did significantly influence sink marks. Iyer and Ramani (2002) in an attempt to study the use of an alternate high thermal conductivity mold material, sink mark defect was taken up as quality control parameter. Using finite element method, an attempt was made to a study sink marks. It was observed that thermal conductivity of the mold material does influence sink marks. DOE has been widely used by various researchers for optimization of injection molding process to control defects and improve quality. Patel and Mallick (1998) applied DOE for defect reduction in injection molding. Sink index was included as one of the quality indicators for investigation as part of their study. Processing variables like melt temperature, injection time, ejection temperature, fill/pack switch over, pack time, injection rate and coolant temperature were considered. Effect of mold temperature, rib-to-wall ratio and rib distance from feed point were ignored. Erzurumlu and Ozcelik (2006) used Taguchi technique to minimize warpage and the sink index. In their study, certain processing variables like mold temperature, melt temperature, packing pressure, rib cross section and rib layout (orientation) were considered. Shen et.al. (2007) made an investigation on effect of molding variables on sink mark index using Taguchi’s fractional factorial design methodology. Shen et.al. considered melt temperature, mold temperature, injection time, pack time, distance between gate and rib and global increase of thickness. Mathivanan and Parthasarathy (2009a, 2009b) reported comprehensive modeling of sink marks using DOE based regressions.

The detailed literature survey indicates the following:

1. Though comprehensive studies on the effects of molding variables on sink marks do exist, a simple to use method for the molders is still required on the same lines. An approach like Taguchi method by applying a comprehensive approach as proposed (Mathivanan and Parthasarathy, 2009a, 2009b) will be of quick use to the molders.

2. Most of the Taguchi based studies used sink mark index or sink index as the parameter. It is an indirect measure for the sink marks. The sink index is an indication of the potential shrinkage due to a hot core. However, whether or not the shrinkage would result in sink mark depends on geometry characteristics (MPI user guide, MoldFlow). Hence, need for a study on sink using sink mark depth as direct response does arise.

Hence the present work was aimed at:

1. Conducting a comprehensive study on effects of variables on sinks using sink mark depth as direct response. 2. Bringing out an easy to use methodology like Taguchi, suitable for molders as well as designers, for control of sink

marks.

Conducting comprehensive study on injection molding process using conventional practical approach is very expensive and also time consuming. With the advent of CAE technology, numerical simulation of injection molding process, comparatively less expensive and quicker trial runs can be experimented virtually (Mathivanan and Parthasarathy, 2008). Hence, in this research, it is proposed to employ Taguchi’s design of experiments in combination with computer aided engineering (CAE) based simulated experimental data for investigation. 2. Materials and methods Different steps involved in the methodology are as follows:

1. Design of simple and scalable generic model 2. Selection of processing variables and their levels 3. Initial screening Taguchi’s experiments, data collection and analysis 4. Arriving at critical variables based on initial screening 5. Additional expanded Taguchi’s experiments for minimization of sink marks

2.1. Design of simple, scalable and generic model and machine selection: A simple and scalable disc part (Figure 1) was prepared using Pro/Engineer. The model base wall was fixed at 3mm. The model was constructed in such a way that, it had 3 different rib thicknesses (1.5mm,1.95mm and 2.4mm) having rib-to-wall ratios of 50%, 65% and 80%. The ribs were located at three levels from the feed point (15mm, 40mm and 65mm). Distance between ribs (25mm) was calculated to maintain a minimum distance of 10 times the maximum thickness of rib (2.4mm). This is required to isolate effects of neighborhood ribs on sink mark.

2.1.1 Molding material and machine: Commercially available, amorphous, Cycolac AR ABS Co-Polymer from GE Plastics and a generic injection molding machine with 7000T clamping tonnage capable of applying 180 MPa injection pressure were selected for the study. Properties of the Cycolac resin are given in Table -1.


15

2.1.2 Taguchi methodology: Taguchi techniques were developed by Dr. Genichi Taguchi. Taguchi developed the foundations of robust design and validated its basic philosophies by applying them in the development of many products (Phadke, 1989). Taguchi method can be used for optimization methodology that improves the quality of existing products and processes and simultaneously reduces their costs very rapidly, with minimum engineering resources and development man-hours. It achieves this objective by making the product or process performance "insensitive" to variations in factors such as materials, manufacturing equipment, workmanship and operating conditions. It also makes the product or process robust and therefore it is called as robust design.

Table-1 Properties of cycolac AR ABS Material Properties of the Material

Commercial product name Cycolac AR

Solid Density (g/cm3) 1.0541

Melt Density (g/cm3) 0.94383

MoldFlow Viscosity index VI(240)0234

Recommended Mold Temperature °C 60

Recommended Melt Temperature °C 240

Material Characteristics Amorphous

Ejection Temperature °C 108

Modulus of Elasticity Mpa 2240

Poisson ratio 0.392

Shear Modulus 804.6

Thermal Conductivity W/m-C° 0.27 @ 2408C

All man-made machines or set-up are classified as engineering systems according to Taguchi. Engineering systems can be classified in to two categories: 1. Static and 2. Dynamic. Dynamic system has signal factors (input from the end user) in addition to control and noise factors, whereas in static system signal factors are not present. Optimization of injection molding process is a static system (Refer Figure 2). Figure 2 is called the P-diagram. The ‘P’ means process or product according to Taguchi.

Figure 1. Disc part


16

Injection Molding Process

Control factors -Controllable processing variables based on pressure, temperature, time etc

Noise factors - Fixed processing parameters, Mold parameters, ambient conditions, human factor etc.

Output Quality, Quantity

Figure 2. P-diagram of injection moulding process

Taguchi views design of any system as a three phase program: 1. System design, 2. Parameter design and 3. Tolerance design. Genesis of new idea, concepts, processes etc., due to technological advancements, comes under system design. Technological advantage gained by a new system design can be lost quickly when competitors produce the same idea in a more uniform manner. Hence, as a holistic approach, one needs to incorporate parameter design as well as tolerance design. Parameter design improves product/process uniformity and can be used to cost savings at no cost. This means that certain parameters are set to make the performance less sensitive to causes of variations. Tolerance design phase improves quality at a minimal cost (Ross, 1996). Few recent successful attempts using Taguchi’s approach for modelling and analysis of abrasive wear performance of composites and parameter optimization of end milling can be seen from Mahapatra and Chaturvedi (2009) and Sanjit et.al. (2010). In this present work, parameter design is utilized to arrive at the optimum levels of process parameters for minimization of sink depth/mark during manufacturing. According to Taguchi, two major tools are employed to achieve any quality goal or any robust design. They are: 1. Signal -to- Noise ratio (S/N ratio), which measures quality and 2. Orthogonal arrays, which are used to study many parameters simultaneously (Phadke, 1989) Taguchi uses the S/N ratio to measure quality characteristic deviating from the desired value. The S/N ratio characteristics can be divided into three categories: the-nominal-the-best, the smaller-the-better, and the-larger-the-better when the quality characteristic is continuous (Ross, 1996). Since, the objective of this study was to minimize the sink mark depth; smaller-the-better quality characteristic was employed. Two orthogonal arrays (OA) were used for experiments. One OA is used for initial screening of processing variables and the other to arrive at optimal process conditions.

2.1.3 Experimental set-up: In order to mold a component on the injection molding machine, a proper mold based on good mold design is required. Mold design basically involves designing of feed system to feed the material from the machine nozzle into the mold cavity, cooling systems to solidify the product after injection and clamping system to keep the mold closed during pressurized injection. Feed system consists of sprue, runner and gate. Cooling system consists of cooling channels and it should be capable of maintaining the required mold temperature. For the present study, Tapered central sprue (4mm diameter) feed point, Disc type runner (4mm) and diaphragm gate (1mm) were designed to have uniform flow based on standard mould design guide lines. Twelve diameter cooling channels were designed for efficient maintenance of mold temperature. The 3D Model, made using Pro/E, was exported to computer aided Simulation tool (in this study MoldFlow was used). Mid plane finite element model was created by meshing the 3D model with 1684 linear triangular elements. Average aspect ratio of the mesh was found to be 1.528. Mesh was thoroughly checked to eliminate mesh related errors. Feed system and cooling channels were created as designed earlier. This set-up was used for conducting trials. Meshed models are shown in Figure 3.


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Figure 3. Meshed model of the disc part 2.2 Selection of processing variables and their levels: Based on the detailed literature survey, extensive discussion with molders and through one initial trial, the following processing parameters were considered for the initial screening experiments (Table -2).

Table: 2 Initial screening parameters and their levels Levels

Number Coded Parameters Uncoded Parameters

Low (1) High (2) 1 A Melt Temperature (°C) 220 260 2 B Mould Temperature (°C) 40 80 3 C Injection Time (sec) 1.20 1.80 4 D Packing Time (sec) 8 12 5 E Packing Pressure (MPa) 23 29 6 F Rib-to-wall Ratio (%) 50 80

7 G Rib Distance from gate (mm) 15 65

3. Injection molding experiments 3.1 Initial screening Taguchi’s experiments, data collection and analysis: Taguchi L8 screening experiments were conducted to identify the “most significant” input variables by ranking with respect to their relative impact on the sink mark. Table -3 shows the Taguchi’s array for L8 experimental runs.


18

The S/N ratio η is given by:

η = -10log (MSD) (1)

Where MSD is the mean-square deviation for the output characteristic. MSD for the smaller-the-better quality characteristic is calculated by the following equation,

⎥⎦

⎤⎢⎣

⎡= ∑

=

n

iiYN

MSD1

21 (2)

Where Yi is the sink mark depth for the ith test, n denotes the number of tests and N is the total number of data points. The function ‘-log’ is a monotonically decreasing one, it means that we should maximize the S/N value. The S/N values were calculated using equations (1) and (2). Table -4 shows the response table for S/N ratios using smaller-the-better approach.

Table-3 Taguchi L8 Array

Table -4 Response table for S/N ratios using smaller-the-better

Level Melt temperature

Mold Temperature

Injection Time

Packing Time

Packing Pressure

Rib-to-Wall ratio

Rib Distance

1 25.72 27.55 26.58 26.42 26.66 28.13 29.96 2 27.73 25.90 26.87 27.03 26.80 25.33 23.49

Delta 2.01 1.64 0.30 0.61 0.14 2.80 6.47 Rank 3 4 6 5 7 2 1

3.2 Parameters selection for follow-up experiments: It was found that, rib distance made significant contribution in the formation of sinks followed by Rib-to-Wall ratio, melt temperature, mold temperature, packing pressure, packing time and injection time. After this initial screening and ranking, it was decided to treat injection time and pack pressure as fixed parameters. The Injection time was fixed at 1.2sec. The pack pressure was fixed at 26 MPa. These decisions were taken under the consideration of overall quality and economics in mind. Maintaining higher pack pressure requires additional power and cost. Packing a part with higher pressure normally leads to higher residual stress and it was not desirable. Though the ranking for pack time was the lowest, it was included in the follow-up experiments to study its impact.

3.3 Taguchi L27 follow-up experiments, data collection and analysis: During the follow-up experiments for minimization, processing variables were considered at three levels. Table – 5 shows the variables and its levels considered for the follow-up experiments.

Run A B C D E F G

1 1 1 1 1 1 1 1

2 1 1 1 2 2 2 2

3 1 2 2 1 1 2 2

4 1 2 2 2 2 1 1

5 2 1 2 1 2 1 2

6 2 1 2 2 1 2 1

7 2 2 1 1 2 2 1

8 2 2 1 2 1 1 2


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Table – 5 Follow-up experiment variables and its levels Levels

Number Coded Parameters Uncoded Parameters

Low (1) Mid (2) High (3) 1 A Melt Temperature (°C) 220 240 260 2 B Mold Temperature (°C) 40 60 1.80 3 D Packing Time (sec) 8 10 12 4 F Rib-to-wall Ratio (%) 50 65 80

5 G Rib Distance from gate (mm) 15 40 65

Twenty seven experiments were conducted and all the sink mark data points were collected. Collected data points were analyzed using the “smaller-the-better approach”. The S/N ratios were calculated using equations (1) and (2). Response table for signal to noise ratio was constructed (Table – 6). Main effects plot for S/N ratio is shown in Figure 4.

Table -6 S/N ratio table for follow-up experiments

Experimental Run

Melt Temperature

(A)

Mold Temperature

(B)

Pack Time (D)

Rib-to-wall Ratio

(F)

Rib Distance

(G)

Sink Depth In mm

S/N ratio

1 220 40 8 50 15 0.025460 31.883003 2 220 40 10 65 40 0.056458 24.965459 3 220 40 12 80 65 0.088900 21.021965 4 220 60 8 65 65 0.088617 21.049620 5 220 60 10 80 15 0.036564 28.739021 6 220 60 12 50 40 0.053858 25.374947 7 220 80 8 80 40 0.065347 23.695474 8 220 80 10 50 65 0.083950 21.519555 9 220 80 12 65 15 0.038355 28.323470

10 240 40 8 50 15 0.020054 33.955980 11 240 40 10 65 40 0.043997 27.131460 12 240 40 12 80 65 0.074100 22.603636 13 240 60 8 65 65 0.072478 22.795864 14 240 60 10 80 15 0.030481 30.319444 15 240 60 12 50 40 0.042833 27.364450 16 240 80 8 80 40 0.058235 24.696393 17 240 80 10 50 65 0.068565 23.277988 18 240 80 12 65 15 0.032409 29.786687 19 260 40 8 50 15 0.022367 33.007923 20 260 40 10 65 40 0.037175 28.595051 21 260 40 12 80 65 0.063400 23.958215 22 260 60 8 65 65 0.061962 24.157519 23 260 60 10 80 15 0.031746 29.966247 24 260 60 12 50 40 0.035834 28.914167 25 260 80 8 80 40 0.062701 24.054469 26 260 80 10 50 65 0.060334 24.388815 27 260 80 12 65 15 0.033013 29.626353

4. Results and discussion

From the Table – 6 and from main effects plot for S/N ratio (Figure 4), it is observed that, rib distance from the feed point is a most influential variable on sink. This factor needs to be considered while designing the part as well as during mold design. If the feed points cannot be provided near a rib, flow leaders can be designed in to the component. This could be an important input to product designers.


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Figure 4. Main effects plot

The prime objective of this study was to find optimum level for each of the variables and to arrive at a combination of these factors that could result in minimum sink. From figure. 4, it can be observed that A3-B1-D2-F1-G1 is the optimum combination for minimum sink depth. Similarly, A1-B3-D1-F3-G3 is the combination for maximum sink depth. These combinations were not included in the experimental runs. Hence, additional two confirmation experiments were run at both combinations. The results are shown in Table 7.

Table -7 Verification experimental results

Validation Run A B D F G S/N

ratio

Predicted Sink

Depth

Measured Sink

Depth

% Deviation

1 260 40 10 50 15 33.5847 0.0164416 0.0154293 6.2% 2 220 80 8 80 65 20.2139 0.0939485 0.0897736 4.4%

As is seen from the Table 7, the difference or the variation between the predicted and measured sink depth is well below 10%. It shows the adequacy of the approach in prediction of the sink depth. Authors have also continued the research with regression analysis and further analyses on the prediction and minimization of the sink marks. Those analyses and findings are not included in this work, as they have been performed in a different study. 5. Conclusion Manipulation of numerous processing variables of the injection moulding process to control defects is a mammoth task that costs time, effort and money. This paper presents a simple and efficient way to study the influence of injection molding variables on sink marks using Taguchi approach. Application of Taguchi approach also helps in arriving at optimal parameter settings. The sink depth through the validation trials based on the optimal parameters and the predicted sink depth using Taguchi’s approach for the same settings are found to be in good agreement. The results show the ability of this approach to predict sink depth for various combination of processing variables with in the design space. It is observed that, increased distance of rib from the feed point seems to produce deeper sinks. This could be an important input to product designers for designing alternatives or to give effective and corrective solutions. This methodology can also be applied while designing parts. Though this study was meant for the sink marks, it can be extended to other defects and also for improving overall quality.

260240220

0.07

0.06

0.05

0.04

0.03

806040 12108

806550

0.07

0.06

0.05

0.04

0.03

654015

Melt Temperature

Mea

n of

Mea

ns

Mold Temperature Pack Time

Rib-to-Wall ratio Rib Distance


21

Nomenclature

% - Percent °C - Degree centigrade η - The S/N ratio 3D - Three Dimension ABS - Acrylonitrile Butadiene Styrene ANN - Artificial Neural Network ANOVA - Analysis of Variance A - Melt Temperature in °C B - Mold Temperature in °C C - Injection Time in sec CAD - Computer Aided Design CAE - Computer Aided Engineering D - Packing time in sec DOE - Design of Experiments E - Packing Pressure in MPa F - Rib-to-wall Ratio in % FE - Finite Element FEA - Finite Element Analysis G - Rib Distance from gate in mm mm - millimetres MPa - Mega Pascal MSD - Mean square deviation N - Total number of data points sec - Seconds

References Erzurumlu T. and Ozcelik B. 2006. Minimization of warpage and sink index in injection-molded thermoplastic parts using Taguchi

optimization method. Materials and Design, Vol. 27, pp. 853–861. Iyer N. and Ramani K. 2002. A study of localized shrinkage in injection molding with high thermal conductivity molds. Journal of

Injection Molding Technology, Vol. 6, No.2, pp. 73–90. Kazmer D., 1997. The foundation of intelligent process control for injection molding. Journal of Injection Molding Technology,

Vol. 1, No.1, pp. 44–56. Mahapatra S.S. and Chaturvedi V. 2009, Modelling and analysis of abrasive wear performance of composites using Taguchi

approach. International Journal of Engineering, Science and Technology, Vol. 1, No. 1, 2009, pp. 123-135 Mathivanan D. and Parthasarathy N.S. 2008. Sink mark prediction and optimization – a review. Society of Plastics Engineers

ANTEC 2008, Milwaukee, USA Mathivanan D. and Parthasarathy N.S. 2009a. Prediction of sink depths using non-linear modeling of injection molding variables.

International Journal of Advanced Manufacturing Technology, Springerlink, Vol. 43, pp. 654–663. Mathivanan D. and Parthasarathy N.S. 2009b. Sink-mark minimization in injection molding through response surface regression

modeling and genetic algorithm. International Journal of Advanced Manufacturing Technology, Springerlink, Vol. 45, pp. 867–874.

Moldflow MPI user guide. 2005. Moldflow, Waltham, MA, US. Moshat S., Datta S., Bandyopadhyay A. and Pal P.K. 2010. Parametric optimization of CNC end milling using entropy

measurement technique combined with grey-Taguchi method, International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 1-12

Patel S.A. and Mallick P.K. 1998. Development of a methodology for defect reduction in injection molding using process simulations Journal of Injection Molding Technology, Vol. 2, No. 4, pp176-191.

Phadke M.S.. 1989. Quality Engineering Using Robust Design, Prentice Hall, NJ,US Ross P.J.. 1996. Taguchi Techniques for Quality Engineering. McGraw-Hill, NY, US Shen, C., Wang, L., Cao W. and Qian L. 2007. Investigation of the effect of molding variables on sink marks of plastic injection

molded parts using Taguchi’s DOE technique. Polymer -Plastics Technology and Engineering, Vol. 46, No. 3, pp. 219-225. Shi L. and Gupta M. 1998. A localized shrinkage analysis for predicting sink marks in injection-molded plastic parts. Journal of

Injection Molding Technology, Vol. 2, No. 4, pp149-155. Shi L. and Gupta M. 1999. An approximate prediction of sink mark depth in rib-reinforced plastic parts by empirical equations.

Journal of Injection Molding Technology, Vol. 3, No. 1, pp 1-10.


22

Tursi D. and Bistany S. P. 2000. Process and tooling factors affecting sink marks for amorphous and crystalline resins. Journal of Injection Molding Technology, Vol. 4, No. 3, pp.114-119.

Biographical notes Dr. D. Mathivanan is a Practicing Engineer in the field of Mechanical Engineering. He obtained his B.E. (Mechanical) from GCT, Coimbatore, India in the year 1990. He received his M.E. (Manufacturing) and Ph.D. from Anna University, Chennai, India. He has international, national journals and conference papers to his credit. His research areas are CAD, CAM, CAE and Plastics. M. Nouby is a Research Scholar in the Department of Mechanical Engineering, Anna University, India since 2007. He received his B.Sc. and M.S. degrees from Department of Automotive and Tractor Engineering, Minia University, Egypt in 1999 and 2003 respectively. Prior to joining the faculty at Anna University, he worked as an Assistant Lecturer in Minia University from 2003 to 2007. His research interests are in the areas of Vehicle Dynamics, Finite Element Methods, Automotive Design, Noise and Vibration. Dr. R. Vidhya is an Assistant Professor in the Institute of Remote Sensing, Department of Civil Engineering of Anna University. She has many international and national journals and conference papers to her credit. She is currently dealing with numerous projects sponsored by Department of Space, Government of India. Her current research interests are Climate change, Object identification, GIS, RS and Optimization techniques. Received December 2009 Accepted January 2010 Final acceptance in revised form February 2010

MultiCraft


Vol. 2, No. 2, 2010, pp. 31-42



www.ijest-ng.com


Propagation of S-waves in a non-homogeneous anisotropic incompressible and initially stressed medium

S. Gupta1*, S. Kundu1, A.K. Verma2 and R. Verma3

1*Department of Applied Mathematics, I.S.M. Dhanbad, INDIA

2 Department of Mathematics, Hampton University, Hampton, VA, USA 3 Department of Mathematics, Norfolk State University, Norfolk, VA, USA

*Corresponding Author: e-mail: [email protected], Tel +91-326-2235464

Abstract In this paper we have studied the propagation of shear waves in a non-homogeneous anisotropic incompressible and initially stressed medium. Analytical analysis reveals that the velocities of the shear waves depend upon the direction of propagation, the anisotropy, the non-homogeneity of the medium and the initial stress. Numerical computation shows that the presence of initial compressive stress in the medium reduces the velocity of propagation whereas, the tensile stress increases it. It is found that the variation in parameters associated with anisotropy and non-homogeneity of the medium directly affects the velocity of the wave. The velocity of wave also depends on the inclination of the direction of its propagation. An increase in the inclination angle decreases the velocity in the beginning and takes a minimum value before increasing. Keywords: Shear waves; anisotropic; stress; non-homogeneity; half-spaces.

1. Introduction The term “Initial stress” is meant by stresses developed in a medium before it is being used for study. The earth is an initially stressed medium, Due to presence of external loading, slow process of creep and gravitational field, considerable amount of stresses (called pre-stresses or initial stresses) remain naturally present in the layers. These stresses may have significant influence on elastic waves produced by earthquake or explosions and also in the stability of the medium. The propagation of surface waves is well documented in the literature (e.g., Achenbach (1973), Bath (1968), Biot (1965), Ewing (1957)). Biot (1940) formulated the dynamical equations for pre-stressed elastic medium and discussed the influence of pre-stresses on the propagation of elastic waves in a body. The problem of finite deformations of an elastic body and the effect of high initial stress on wave propagation were discussed in a series of investigations by Kappus (1939), Murnaghan (1951) and others. Qian et al. (2004) have investigated the effect of initial stress on the propagation behavior of SH-waves in multilayered piezoelectric composite structures. Chattopadhyay et al. (2009) have studied the propagation of shear waves in an internal magnetoelastic monoclinic stratum sandwiched between two semi-infinite isotropic elastic media and with a rectangular irregularity in lower interface. Chattopadhyay et al. (2010) have also investigated the propagation of shear waves in a monoclinic layer with an irregularity lying between two isotropic semi infinite elastic medium. The effect of inhomogeneous initial stress on Love wave propagation in layered magneto-electro-elastic structures have been studied by Zhang et al. (2008). Sharma (2005) has demonstrated the effect of initial stress on the propagation of plane waves in a general anisotropic poro-elastic medium. The Edge wave propagation in an incompressible anisotropic initially stressed plate of finite thickness has been studied by Dey et al. (2009). Addy et al. (2005) have studied Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence of the temperature field. Liu et al. (2008) have demonstrated the propagation characteristics of converted refracted wave and its application in static correction of converted wave. Moczo et al. (2007) provided mathematical modeling of seismic wave propagation using the Finite-Difference time-domain method. Huber (2010) has explained the physical meaning of a nonlinear evolution equation of the fourth order relating to locally and non-locally supercritical waves in his work. Duan1 et al. (2006, 2007)

Gupta et al. / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 31-42

32

have investigated heterogeneous fault stresses from previous earthquakes and the effect on dynamics of parallel strike-slip faults and non-uniform pre-stress from prior earthquakes and the effect on dynamics of branched fault systems. Zhou and Chen (2005) have studied the influence of seismic cyclic loading history on small strain shear modulus of saturated sands. Sharma et al. (2007) discussed about the wave velocities in a pre-stressed anisotropic elastic medium. Selim et al. (2006) have demonstrated the propagation and attenuation of seismic body waves in dissipative medium under initial and couple stresses. Seismology is the study of progressive elastic wave. But most of this studies and investigations do not include very important factor viz, the influence of initial stress, anisotropy and non-homogeneity present in the body. In this paper, an attempt has been made to show the effect of initial stress, the anisotropy and non-homogeneity of the medium on the propagation of shear wave. 2. Solution of the problem Most materials behave as incompressible media and their influence on seismic waves are very high. (The velocities of longitudinal waves in them are very high) The varieties of hard rocks present in the earth are also, almost incompressible. Due to the factors like external pressure, slow process of creep, difference in temperature, manufacturing processes, nitriding, pointing etc., the medium stay under high stresses. These stresses are regarded as initial stresses. Owing to the variation of elastic properties and the presence of these initial stresses, the medium becomes anisotropic as well. We consider an unbounded incompressible anisotropic medium under initial stresses S11 and S22 along the x-, y- directions respectively. When the medium is slightly disturbed, the incremental stresses s11, s12 and s22 are developed and the equations of motion given by Biot (1965) are

2

11 122 ,s s w uP

x y y tρ∂ ∂ ∂ ∂

+ − =∂ ∂ ∂ ∂

(1)

2

12 222 .s s w vP

x y x tρ∂ ∂ ∂ ∂

+ − =∂ ∂ ∂ ∂

(2)

where P = S22 – S11, 12⎛ ⎞∂ ∂

= −⎜ ⎟∂ ∂⎝ ⎠

v uwx y

, and ρ represents the density of the medium. Also jis , are incremental stresses, (u, v)

are incremental deformations, w is the rotational component about the z-axis. The incremental stress-strain relations for an incompressible medium may be taken as

11 , 22 122 2 and 2 .xx yy xys s N e s s N e s Q e− = − = = (3)

where 11 22

2+

=S S

s , jie are incremental strain components and N and Q are the rigidities of the medium.

The incompressibility condition 0,xx yye e+ = is satisfied by:

andu vy xϕ ϕ∂ ∂

= − =∂ ∂

. (4)

Using eqs. (3) and (4) in eqs. (1) and (2), we obtain

2 2 2

2 2 222 2

s P P uN Q Qx y x y t

ϕ ϕ ρ⎡ ⎤∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞− − − + + =⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

, (5)

2 2 2

2 2 222 2

s P P vN Q Qy x y x t

ϕ ϕ ρ⎡ ⎤∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞+ − − + − =⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

. (6)

We assume the non-homogeneity as

( )( )( )

0

0

0

1

1

1

Q Q ay

N N by

cyρ ρ

= + ⎫⎪

= + ⎬⎪= + ⎭

(7)

where N0 and Q0 are rigidities and 0ρ is the density in homogeneous isotropic medium. Eliminating s from eqs. (5) and (6) we get


33

( ) ( ) ( )

( ) ( )

( )

4 4

0 0 04 2 2

4 3 3

0 0 0 04 2 3

4 4 3

0 02 2 2 2 2

1 4 1 2 12

1 4 2 22

1 .

PQ ay N by Q ayx x y

PQ ay N b Q a Q ay x y y

cy cx t y t y t

ϕ ϕ

ϕ ϕ ϕ

ϕ ϕ ϕρ ρ

∂ ∂⎡ ⎤+ − + + − +⎡ ⎤⎣ ⎦⎢ ⎥ ∂ ∂ ∂⎣ ⎦∂ ∂ ∂⎡ ⎤+ + + + − +⎢ ⎥ ∂ ∂ ∂ ∂⎣ ⎦

⎛ ⎞∂ ∂ ∂= + + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

(8)

3. Solution of the problem For propagation of sinusoidal waves in any arbitrary direction we take the solution of eq.(8) as

( ) ( )1 2 1, , ik p x p y c tx y t Aeϕ + −= . (9)

where 1p and 2p are cosine of the angles made by the direction of propagation with the x- and y- axes, and 1c and k are the velocity of propagation and the wave number respectively. Using eq.(9) in eq.(8) and equating real and imaginary parts separately, one gets

( ) ( ) ( )

24 2 2011 1 2

0 0

42

0

21 1 2 1 11 2

1 ,2

Nc Pay p by ay p pcy Q Q

Pay pQ

β⎧⎛ ⎞ ⎡ ⎤⎛ ⎞ ⎪= + − + + − +⎨⎜ ⎟ ⎢ ⎥⎜ ⎟ + ⎪⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎩

⎫⎛ ⎞ ⎪+ + + ⎬⎜ ⎟⎪⎝ ⎠ ⎭

(10)

and

2

2 2011 2

0

22 2Nc b a ap pQ c c cβ

⎛ ⎞⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

. (11)

where

12

0

0

Qβ

ρ⎛ ⎞

= ⎜ ⎟⎝ ⎠

, the velocity of shear wave in homogeneous isotropic medium. Eq. (10) gives the velocity of propagation of

shear wave and eq. (11) gives the damping.

Equation (10) shows that the velocity depends on the anisotropy factor 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

, the initial stress factor 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

and also on the

direction of propagation denoted by ( )1 2,p p . 4. Particular cases: Following cases have been discussed to gain more insight information from eq.(10) and eq.(11): Case I: When 0a → , i.e., rigidity along vertical direction is constant, eq. (10) reduces to

( ) ( )

24 2 2 4011 1 2 2

0 0 0

21 1 2 1 1 11 2 2

Nc P Pp by p p pcy Q Q Qβ

⎧ ⎫⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎪ ⎪= − + + − + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ + ⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎩ ⎭. (12)

The velocity of wave along x-direction ( )1 2 1 111, 0,p p c c= = = , is obtained as

( )

2 211

0

112 1PcQ cy

β⎛ ⎞

= −⎜ ⎟ +⎝ ⎠. (13)

This shows that velocity of wave along x-direction depends on initial stress. If the medium is free from initial stress, i.e when 0P → and 0c → , the velocity of wave is given by 11c β= .


34

Similarly the velocity of propagation along y-direction ( )1 2 1 220, 1,p p c c= = = , is obtained as

( )

2 222

0

112 1PcQ cy

β⎛ ⎞

= +⎜ ⎟ +⎝ ⎠, (14)

It is interesting to note that( )

2 222 11

20

11

c c PQ cyβ

−=

+, a function of initial stress and density.

It is also observed that if 0P > , the effect of initial stresses on the body is compressive along x-direction and which reduces the velocity of shear wave along x-direction while tensile stress increases the velocity of shear wave, where as along y-direction shear wave velocity shows the reverse effect. Case II: When 0b → , i.e., rigidity along horizontal direction is constant, eq. (10) reduces to

( ) ( ) ( )

( )

24 2 2011 1 2

0 0

42

0

21 1 2 11 2

1 .2

Nc Pay p ay p pcy Q Q

Pay pQ

β⎧⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎪= + − + − +⎨⎢ ⎥ ⎢ ⎥⎜ ⎟ + ⎪⎝ ⎠ ⎣ ⎦ ⎣ ⎦⎩

⎫⎡ ⎤ ⎪+ + + ⎬⎢ ⎥⎪⎣ ⎦ ⎭

(15)

The velocity of wave along x- direction ( )1 2 1 111, 0,p p c c= = = , is given by

( ) ( )2 211

0

112 1Pc ayQ cy

β⎡ ⎤

= + −⎢ ⎥ +⎣ ⎦, (16)

which depends on the depth y and the wave is dispersive. The velocity along y-direction is

( ) ( )2 222

0

112 1Pc ayQ cy

β⎡ ⎤

= + +⎢ ⎥ +⎣ ⎦, (17)

In case of 0P > , the velocity along y-direction may increase considerably at a distance y from free surface and the wave is dispersive. Case III: When 0a → , 0b → , i.e., the rigidity along horizontal direction is constant but density is linearly varying with depth, eq. (10) transforms to

( )

24 2 2 4011 1 2 2

0 0 0

21 1 2 1 11 2 2

Nc P Pp p p pcy Q Q Qβ

⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎪ ⎪= − + − + +⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ + ⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭. (18)

In the absence of initial stress the velocity of wave is

( )

22 2011 2

0

1 1 4 11

Nc p pcy Qβ

⎡ ⎤⎛ ⎞⎛ ⎞= − −⎢ ⎥⎜ ⎟⎜ ⎟ +⎝ ⎠ ⎝ ⎠⎣ ⎦

. (19)

This shows that velocity( )

221 1

ccy

β=

+, in x-direction ( )1 2 1 111, 0,p p c c= = = , and in

y -direction ( )1 2 1 220, 1,p p c c= = = it does not depend on anisotropy. However, in other directions the anisotropy affects the velocity. For 0 0N Q= , i.e., for isotropic medium with variable density the wave velocity is

( )

2 2 21 1 2

1c p p

cyβ⎛ ⎞

=⎜ ⎟ +⎝ ⎠, (20)

which depends on the direction of propagation. Case IV : In the absence of initial stress i.e. 0P → , eq. (10) gives the velocity of wave as


35

( ) ( ) ( ) ( ) ( )

24 2 2 4011 1 2 2

0

1 1 2 2 1 1 11

Nc ay p by ay p p ay pcy Qβ

⎧ ⎫⎡ ⎤⎛ ⎞ ⎪ ⎪= + + + − + + +⎨ ⎬⎢ ⎥⎜ ⎟ + ⎪ ⎪⎝ ⎠ ⎣ ⎦⎩ ⎭. (21)

which for 0 0N Q= , takes the form

( ) ( ) ( ) ( ) ( )

24 2 2 411 1 2 2

1 1 2 2 1 1 1 .1

c ay p by ay p p ay pcyβ

⎛ ⎞= + + + − + + +⎡ ⎤⎜ ⎟ ⎣ ⎦+⎝ ⎠

(22)

Along y-direction ( )1 2 1 220, 1,p p c c= = = ,

2

22 11

c aycyβ

⎛ ⎞ +=⎜ ⎟ +⎝ ⎠

. (23)

Along x-direction ( )1 2 1 111, 0,p p c c= = = ,

2

11 11

c aycyβ

⎛ ⎞ +=⎜ ⎟ +⎝ ⎠

. (24)

The wave is dispersive and velocities are same in two direction x and y. Case V : When 0a → , i.e., rigidity along vertical direction is constant, eq. (11) transforms to

2

2011

0

22 Nc b pQ cβ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠, (25)

this shows that velocity of shear wave is always damped. The velocity of wave along x-direction ( )1 2 1 111, 0,p p c c= = = , is obtained as

2

011

0

22 Nc bQ cβ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠, (26)

this shows that actual wave velocity in x-direction is damped by 0

0

22

N bQ c

⎛ ⎞⎜ ⎟⎝ ⎠

, whereas, no damping takes place along y-direction

( )1 2 1 220, 1,p p c c= = = . Case VI : When 0b → , i.e. rigidity along horizontal direction is constant, eq. (11) reduces to

2

2 211 22 2c a ap p

c cβ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠. (27)

The velocity of wave along x- direction ( )1 2 1 111, 0,p p c c= = = , is given by

2

11 2c acβ

⎛ ⎞ ⎛ ⎞= −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

. (28)

Existence of negative sign shows that damping does not takes place along x-direction for 0b → , whereas damping of magnitude 2ac

⎛ ⎞⎜ ⎟⎝ ⎠

, takes place along y-direction.

Case VII: When 0a → , 0b → , i.e. rigidity along horizontal direction is constant but density varying linearly with depth, eq.

(11) gives 2

1 0cβ

⎛ ⎞=⎜ ⎟

⎝ ⎠, i.e. no damping takes place for 0a → , 0b → .


36

5. Numerical calculation To get numerical information on the velocity of shear wave in the non-homogeneous initially stressed medium, the equation (10) is non-dimensionalized as

( )2

4 2 2011 1 2

0 0

42

0

21 1 2 1 121

1 .2

Nc a P aby p by by p pc b Q Q bbyb

a Pby pb Q

β⎧⎛ ⎞ ⎡ ⎤⎛ ⎞ ⎪ ⎛ ⎞= + − + + − +⎨⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎛ ⎞ ⎝ ⎠⎪⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎩+⎜ ⎟

⎝ ⎠⎫⎛ ⎞ ⎪+ + + ⎬⎜ ⎟⎪⎝ ⎠ ⎭

(29)

The numerical values of 1cβ

has been calculated for different values of ab

, 0

0

NQ

, 1p , 2p and02

PQ

, and the results are

presented in Figures 1 through 6. Figure-1 gives the variation in velocities of shear wave in the direction of θ = 300 and 600 with x-axis at different depth and

different values of density parametercb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.7, 0.8, and 0.9 for ab

= 4.0, 02

PQ

= 0.5, and 0

0

NQ

= 2.5. The velocity of the wave

increases as depth increases.

Figure 1a (θ = 300): by vs. 1cβ

for v1:cb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.7, v2: cb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.8, v3: cb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.9.


37

Figure 1b (θ = 600): by vs. 1cβ

for v1:cb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.7, v2: cb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.8, v3: cb

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.9.

Figure. 2 gives the variation in velocities of shear wave in the direction of θ = 300 and 600 with x-axis at different depth and

different values of ab

when cb

= 0.8, 02

PQ

= 0.5, and 0

0

NQ

= 2.5. The velocity of the wave increases as depth increases.


for v1: ab

⎛ ⎞⎜ ⎟⎝ ⎠

= 3.0, v2: ab

⎛ ⎞⎜ ⎟⎝ ⎠

= 3.5, v3: ab

⎛ ⎞⎜ ⎟⎝ ⎠

= 4.0.


38


for v1: ab

⎛ ⎞⎜ ⎟⎝ ⎠

= 3.0, v2: ab

⎛ ⎞⎜ ⎟⎝ ⎠

= 3.5, v3: ab

⎛ ⎞⎜ ⎟⎝ ⎠

= 4.0.

Figure. 3 gives the variation in velocities of shear wave in the direction of θ = 300 and 600 with x-axis at different depth and

different values of 0

0

NQ

when cb

= 0.8, 02

PQ

= 0.5, and ab

= 4.0. Figure.3 gives the information of variation of velocity for

different values of anisotropic factor and reflects that with the increase in the values of 0

0

NQ

, the velocity of shear wave

increases.


for v1: 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 2.0, v2: 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 2.5, v3: 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 3.0.


39


for v1: 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 2.0, v2: 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 2.5, v3: 0

0

NQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 3.0.

Fig.4 gives the variation in velocities of shear wave in the direction of θ = 300 and 600 with x-axis at different depth and different

values initial stress parameter 02

PQ

when cb

= 0.8, 0

0

NQ

= 2.5, and ab

= 4.0. The velocity of the wave increases as depth

increases.


for v1: 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

= -0.8, v2: 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.0, v3: 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.8.


40


for v1: 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

= -0.8, v2: 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.0, v3: 02

PQ

⎛ ⎞⎜ ⎟⎝ ⎠

= 0.8.

Figure.5 gives the velocity of shear wave in an anisotropic initially stressed homogeneous medium for different values of initial

stress parameter 02

PQ

and 0

0

NQ

with 02

PQ

= 0.5, cb

= 0.8, 0

0

NQ

= 2.5, andab

= 4.0.

Figure 5: by vs. 1cβ

for v1: θ = 300, v2: θ = 600

Figure.6 shows the variation of velocity of shear wave with respect to direction of propagation in non-homogeneous anisotropic

initially stressed medium with 02

PQ

= 0.5, cb

= 0.8, 0

0

NQ

= 2.5, ab

= 4.0, and by = 2.0;


41

Figure 6: θ (in degrees) vs. 1cβ

.

6. Conclusion From equation (10) it is concluded that: 1. When rigidity along vertical direction is constant then shear wave velocity is influenced by initial stress. We have derived the

velocity of wave in both x and y direction and we have seen both depend on initial stress. Compressive initial stress reduces the velocity of shear wave along x-direction while tensile stress increases. Shear wave velocity shows the reverse effect along y-direction.

2. If rigidity along horizontal direction is constant then shear wave velocity exists and we have obtained the velocity equation in both x and y direction. In case 0P > , the velocity along x-direction may decrease considerably and the velocity along y-direction may increase considerably.

3. When the rigidity along horizontal direction is constant but density is linearly varying with depth then also shear wave velocity is still influenced by initial stress and in absence of initial stress the velocity also exist. The velocity of wave in x-direction and y-direction does not depend on anisotropy. However, in other directions the anisotropy affects the velocity. We have also observed for isotropic medium with variable density the velocity depends on the direction of propagation.

4. In the absence of initial stress the shear wave velocity is still available. It also exists in isotropic medium. But in this case velocities are same in two direction x and y.

From equation (11) it is concluded that: 1. If rigidity along vertical direction is constant then the velocity of shear wave is always damped. We have obtained the velocity

equation of wave along x-direction and it is also damped whereas no damping takes place along y-direction. 2. When rigidity along horizontal direction is constant then the velocity of shear wave is damped and in this case the damping

does not take place along x-direction whereas damping takes place along y-direction. 3. When rigidity along horizontal direction is constant but density varying linearly with depth then no damping takes place. Thus it seen that the anisotropy, non-homogeneity , the initial stresses , the direction of propagation and the depth (in case of non-homogeneous medium) have considerable effect in the velocity of propagation of shear wave and attracts the attention of earth scientists in their work. References Achenbach, J.D. 1973. Wave propagation in elastic solids, North Holland Publishing Comp., New York. Addy, S. K. and Chakraborty ,N. R., 2005. Rayleigh waves in a viscoelastic half-space under initial hydrostatic stress in presence

of the temperature field. International Journal of Mathematics and Mathematical Sciences, Vol. 24, pp. 3883–3894 Bath, M. A., 1968. Mathematical Aspects of Seismology, Elsevier Publishing Comp., New York.


42

Biot, M.A., 1940. The influence of initial stress on elastic waves, Journal of App. Phy., Vol.2, p. 522 Biot, M.A., 1965. Mechanics of incremental deformations, John Wiley and Sons Inc, New York. Chattopadhyay, A., Gupta, S., Singh, A.K. and Sahu, S.A., 2009. Propagation of shear waves in an irregular magnetoelastic

monoclinic layer sandwiched between two isotropic half-spaces, International Journal of Engineering, Science and Technology, Vol. 1, No. 1, pp. 228-244.

Chattopadhyay, A., Gupta, S., Sharma, V.K. and Kumari, P., 2010. Effects of irregularity and anisotropy on the propagation of shear waves, International Journal of Engineering, Science and Technology, Vol. 2, No. 1, pp. 116-126.

Duan1, B. and Oglesby, D.D., 2006. Heterogeneous fault stresses from previous earthquakes and the effect on dynamics of parallel strike-slip faults. Journal of Geophysical Research, Vol. 111, B05309.

Duan1, B. and Oglesby, D.D., 2007. Nonuniform prestress from prior earthquakes and the effect on dynamics of branched fault systems, Journal of Geophysical Research, Vol. 112, B05308.

Dey, S. and De, P. K. 2009. Edge wave propagation in an incompressible anisotropic initially stressed plate of finite thickness, International Journal of Computational Cognition, Vol. 7, No. 3, pp. 55-60.

Ewing, W. M., Jardetzky, W. S. and Press, F. 1957. Elastic waves in layered media, McGraw Hill Book Comp., New York. Huber, A., 2010. The physical meaning of a nonlinear evolution equation of the fourth order relating to locally and non-locally

supercritical waves, International Journal of Engineering, Science and Technology, Vol. 2, No. 1, pp. 70-79. Kappus, R. 1939. Zeitschr. A ngew Math.Mech, Vol. 19, No. 5, p. 27. Liu, Y. and Wei , X. C. 2008. Propagation characteristics of converted refracted wave and its application in static correction of

converted wave, Science in China Series D: Earth Sciences, Vol. 51, No. 2, pp. 226-232. Murnaghan, F. D. 1951. Finite Deformation of an Elastic Solids, John Willey and Sons, New York. Moczo, P., Robertsson, J.O.A., and Eisner, L., 2007. The Finite-difference time-domain method for modeling of seismic wave

propagation, Advances in Geophysics, Vol. 48, pp. 421-516. Qian ,Z., Jin, F., Kishimoto. K., and Wang, Z., 2004. Effect of initial stress on the propagation behavior of SH-waves in

multilayered piezoelectric composite structures, Sensors and Actuators A: Physical, Vol. 112, No. 2-3, pp. 368-375. Selim. M.M. and Ahmed. M.K., 2006. Propagation and attenuation of seismic body waves in dissipative medium under initial and

couple stresses, Applied Mathematics and Computation, Vol. 182, No. 2, pp. 1064-1074. Sharma M. D. 2005. Effect of initial stress on the propagation of plane waves in a general anisotropic poroelastic medium, Journal

of Geophysical Research, Vol. 110, No. B11, pp. B11307.1-B11307.14 Sharma M. D. and Garg N., 2006. Wave velocities in a pre-stressed anisotropic elastic medium, Journal of Earth System Science,

Vol. 115, No. 2, pp. 257-265. Zhang, J., Shen, Y.P. and Du J.K., 2008. The effect of inhomogeneous initial stress on Love wave propagation in layered

magneto-electro-elastic structures, Smart Mater. Struct. Vol. 17, No. 025026 (9pp) Zhou, Y. and Chen, Y., 2005. Influence of seismic cyclic loading history on small strain shear modulus of saturated sands. Soil

Dynamics and Earthquake Engineering, Vol. 25, No. 5, pp. 341-353. Biographical notes Dr. Shishir Gupta is an Associate Professor in the Department of Applied mathematics, Indian School of Mines, Dhanbad. A Gold Medalist from Ranchi University, he has had a brilliant career. He has more than 21 years of teaching experience at undergraduate and postgraduate levels in Indian School of Mines, Dhanbad. He possesses experience of guiding students of MPhil and PhD. He has published more than 45 papers in International/National journals/Proceedings. He has served as reviewer in renowned International/ National books and journals. He has also carried out several sponsored research projects. Mr. Santimoy Kundu is a Junior Research Fellow (JRF) in the Department of Applied Mathematics, Indian School of Mines (ISM), Dhanbad, Jharkhand, India. He is pursuing his PhD under the supervision Dr. Shishir Gupta of ISM in the field of Theoretical Seismology. He did his B.Sc. from University of Calcutta, Kolkata in 2001 and then switched to ISM to earn M.Sc. in Maths & Computing (2004) and MPhil in Applied Mathematics (2006). A.K. Verma is an Endowed University Professor of Mathematics at Hampton University, Hampton, Virginia, USA. In his teaching caereer of over twenty-five years he has authored/co-authored research papers in fluid mechanics, controlled thermo-nuclear fusion and education technology. Dr. Verma had served as director and consultant for state and federal projects, and reviewed federal proposals. He has earned several accolades at institution, regional, state and national levels for his quality teaching and effective use of technology in instruction. R. Verma is an Associate Professor in the Department of Mathematics, Norfolk State University, Norfolk, Virginia. She earned her Ph.D. from Indian Institute of Technology, Kharagpur. Dr. Verma has taught mathematics in traditional way and online to technical and non-technical students for over twenty-years. Her research focuses on the use of integral equations in crack problems of elasticity. She has several publications in this field in international journals. Dr. Verma has also been involved in training school teachers to incorporate new teaching methods using technology. Received December 2009 Accepted March 2010 Final acceptance in revised form April 2010

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Vol. 2, No. 2, 2010, pp. 43-53



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Explicit solutions of the Rand Equation

A. Huber 1,2

1 Address constantly: Prottesweg 2a, A-8062 Kumberg, AUSTRIA

2 Institute of Theoretical Physics – Computational Physics, Technical University Graz, Petersgasse 16, A-8010 Graz, AUSTRIA E-mail: [email protected]

Abstract In this paper the meaning of a nonlinear partial differential equation (nPDE) of the third-order is shown to the first time. The equation is known as the ‘Rand Equation’ and belongs to a class of less studied nPDEs. Both the explicit physical meaning as well as the behaviour is not known until now. Therefore we believe it is indispensable to study this evolution equation in detail. We perform a classical Lie Group analysis to analyze the point symmetries. By using a similarity reduction we are able to deduce more classes of solutions of general character. Special nonlinear transformations are given in a most general form. In addition, we also study Lie’s non-classical case relating to potential and generalized symmetries. Both the potential and approximate symmetries are discussed to the first time leading to new results. So we expect a better understanding and concrete physical as well as technical application in future. Keywords: Nonlinear partial differential equations, evolution equations, symmetries, similarity solutions, Rand Equation. PACS-Code: 02.30Jr, 02.20Qs, 02.30Hq. AMS-Code: 35L05, 35Q53, 14H05.

1. Introduction - outline of the problem The scaled nPDE in (1+1) dimension under consideration is given by:

3

32

3

3xuu

tu

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

, ),( txuu = , ),(3 ∞−∞∈ Ru , 0>t , (1)

where the function ),( txu is related to a variation of a physical quantity depending upon the positive time t . We seek for classes of solutions for which ),( txFu = , where ∈F R3 and ⊂D R2 is an open set and further we exclude

0),(:~)),(,(: ≠∈= txuDtxuD . Suitable classes of solutions are Iu∈ an interval so that DI ⊆ and →Iu : R 2 . Note: We suppress the item ‘classes’, so that ‘classes of solutions’ are simply ‘solutions’. Firstly, taking a look at eq.(1) concluding the following: Unlike classical evolution equations (e.g. the Korteweg de Vries equation and many others (Whitham, 1974; Drazin and Johnson, 1989; Eilenberger, 1983; Ablowitz and Clakson, 1991; Dodd et al., 1988; Huber, 2010)), where the nonlinear part is counterbalanced by a linear part and therefore responsible for stable waves. Here, on the contrary, we have no such balance. That means we cannot expect a priori classical wave motion where the steepening effect is counterbalanced by some linear parts since both the l.h.s. and the r.h.s. of eq.(1) are nonlinear. In other words, e.g. the effect of beach wave breaking cannot occur. This leads to the assumption that other types of waves might appear (if we assume that the eq.(1) at least admits wave solutions). This is the main task of the given paper whereby we are interested in questions about the meaning, validity and existence of solutions. Another question of importance is how we can associate a concrete physical and/or technical application with the eq.(1).

Huber / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 43-53

44

1.2. Classical symmetry analysis - algebraic group properties We take up now the developments given in (Ibragimov, 1994a; Olver, 1986; Bluman and Kumei, 1989, Gaeta, 1994) omitting all technical details. To use symmetry groups in any application we first deduce the symmetries of eq.(1). The result is a well-defined system of eight linear homogeneous PDEs (describing the point symmetries) for the infinitesimals

),( uxii ξ=ξ and ),( uxii φ=φ . These constitute the so-called determining equations for the symmetries of eq.(1) generated by Fréchet’s derivative (Ibragimov, 1994a; Huber, 2007a,b; Huber, 2008a,b; Huber, 2009):

02

2221 =

∂φ∂

=∂ξ∂

=∂ξ∂

=∂ξ∂

uxuu, (1.1)

039 2

3

31

31 =

∂∂φ∂

+∂ξ∂

−∂ξ∂

uxxt, (1.2)

09 3

3=

∂φ∂

−∂φ∂

xt, (1.3)

021

22

=∂ξ∂

−∂∂φ∂

xux, (1.4)

03 12 =∂ξ∂

−∂ξ∂

xt. (1.5)

Solving the above given set of equations (1.1) to (1.5) we derive at the infinitesimals:

.),(),(3

1

422

431

txFtxuktkk

xkk

+=φ+=ξ+=ξ

(1.6)

The result shows that the symmetry group of eq.(1) constitutes an infinite four-dimensional point group where the group parameters are denoted by ik , 3,2,1i = . The infinite part of the group is generated by the function ),( txF whereby the latter

function has to satisfy the linear third-order equation: 09 =− xxxt FF . The arbitrary function ),( txF does not satisfy any further equation(s). So, in what follows we have the freedom to set the function ),( txF equal to zero (or individually otherwise). Eq.(1) admits the four-dimensional Lie algebra L of its classical infinitesimal point symmetries related to the following vector fields: uxttx uVxtVVV ∂=∂+∂=∂=∂= 4321 ,3,, . (1.7)

This group of four vector fields contains translations in time and space so that λ+→λ+→ xxtt ',' holds for 21 V,V

and the associated differential operators 3V and 4V are related to dilatation operations. The symmetry vector fields form a Lie algebra L by: [ ] [ ] [ ] 334224343242 ],[,3,,,,3, VVVVVVVVVVVV −=−=== . (1.8)

For this four-dimensional Lie algebra the commutator table for the iV is a (4 x 4)- table whose

th)j,i( entry expresses the Lie Bracket [ ]ji VV , given in (1.8). The table is skew-symmetric and the diagonal elements vanish.

The coefficient kjiC ,, is the coefficient of iV of the th)j,i( entry of Table 1 and the related structure constants can be read from Table 1: 1,3,1,3 3,3,42,2,43,4,32,4,2 ==−=−= CCCC . (1.9)


45

Table 1 The commutator table of the Rand Equation

1V 2V 3V 4V

1V 0 0 0 0

2V 0 0 0 23V−

3V 0 0 0 3V−

4V 0 23V 3V 0

Theorem: The Lie algebra of eq.(1) is solvable. Proof: A Lie algebra L is called solvable if 0)( =nV for some n > 0. It can be shown that L is reducible to 0)4( =V starting by

the ideal )4()1( ....,, VV since the algebra is four-dimensional.

Other useful algebraic group properties are mentioned: Eq.(1) has the Casimir operator by 1V , the group order is four containing 15 subgroups. These subgroups are important below to perform a similarity reduction deducing suitable solutions. The metric ( 44⊗ Cartanian tensor ) satisfies:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

120

00

L

MOM

L

ijg , with det(g) = 0 , (1.10)

and, since the condition 0)gdet( = holds, the given algebra is therefore degenerate and commutative. Note: Alternatively one

can write with eq.(1.9) ∑=

=n

ki

kmi

ilkim ccg

1,.

2. Similarity solutions Let us now discuss the most important three similarity solutions for special subgroups. If we set the group parameters 132 == kk and 041 == kk , the following linear ODE of the third-order results:

093

3=

ζ+

ζ ddS

dSd

, S : R × R → R , ∈ζ R, ∞≤ζ≤∞− , 0)(:~),(: ≠ζ∈ζ= SDSD . (2)

The similarity variable ζ together with the relevant transformation (Case H) reads as uSxt =ζ=− , which is closely related to the case of traveling waves. Following Peanos’ theorem we expect that at least solutions exist (locally in this sense) and secondly solutions are unique on the entire real axis. We calculate a superposition of harmonic wave trains by

[ ] [ ] ζ+ζ+=ζ 3sin3cos31)( 213 CCCS , (2.1)

where the iC , 3,2,1=i are arbitrary constants of integration. A compact written form gives

( ) ( ) ( )( )∑

∞

=

++

+Γζ−−+−ζ

+=ζ0

211

)23(31192

34)(

k

kkkk

kkS , (2.2)

where (.)Γ means the gamma function. At this stage a question is of interest: What physical meaning can we associate with this solution, eq.(2.1) or otherwise, represent this solution a solitary wave? If so, the following condition must hold: 0→S as ∞→ζ . It is shown that ∞→S as

∞→ζ , that means no solitary motion is possible. So we have periodic wave trains on the entire real axis which is seen in Figure 1.


46

-2 -1 1 2

-1.5

-1

-0.5

0.5

1

1.5

Figure 1 A planar sketch of the periodic wave train solution eq.(2.1) by using different values of the integration constants

iC , 3,2,1=i . We chose 11 <<− iC for the domain of the constants and the periodic wave trains are stable. The further behaviour strongly depends on the choice of the constant 3C . If we set 03 =C the function vanishes R∈∀ζ \

412 /,/ ππ− , otherwise, if ±∈ NC3 , especially 13 =C the function vanishes ∈ζ∀ ℜ \ ( )⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ±−− 7321

31 arccos ,

that is numerically )5,08,0( i+−≈ .

If we consider another choice for the group parameters, i.e. 143 == kk and 021 == kk (corresponding to Case C) the

transformation is ζ=+ 3)1( xt

, Su = and the following linear homogeneous ODE of the third-order with non-constant

coefficients result:

0)320(369 2

22

3

33 =

ζ+ζ+

ζζ+

ζζ

dSd

dSd

dSd

, S : R × R → R , ∈ζ R, ∞≤ζ≤∞− . (2.3)

This is solved explicitly by introducing the transformation )(' ζ= pS to get a second-order equation

0)320('36''9 23 =+ζ+ζ+ζ ppp , 0)(:~),(: ≠ζ∈ζ= pDpD (2.4) where the prime means the derivation ζdd / . This is solved by Bessel functions of broken order:

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ζζ+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ζζ=ζ − 3

232)( 3/13

23/13

1 JCJCp , 0≠ζ . (2.5)

Finally we get the solution function after integrating once to

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−⎥

⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−=

ζζπ

ΓζΓζ

ζ31

35

34

32

4

323931

34

32

31

35

323 2

92

3

61

3 ;,;F)/(C

;,;F)/(

CC)(S q

pqp . (2.6)

Here (.)Γ means the gamma function and );;;( 21 zbbaFqp is the generalized hypergeometric function. In our case we

explicitly have the function [ ]ζ;.....,;...., 1121 qp bbaaF .

For further considerations we are interested in the asymptotic case ∞→ζ . By using the asymptotic behaviour of the hypergeometric function we have

[ ]⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ζ+×ζ

ΓπΓΓ

ζ ζ⎟⎠⎞

⎜⎝⎛ +−− 11

)(2)()(

~;.....,;...., 221

21

1

2111

21

211

Oeabb

bbaaFbba

qp . (2.7)

The calculation for the real part leads to the asymptotic representation which is shown in Figure 2:


47

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ζζ

ΓπΓΓ

ζ −

31cos

)3/1(2)3/4()3/2(~)( 6/7S , as ∞→ζ . (2.8)

Figure 2 A planar sketch of the asymptotic solution function eq.(2.8) with 1321 === CCC . The asymptotic behaviour

is clearly seen. The point 0=ζ is the irregularity of the gamma function and the function decreases rapidly for ∞→ζ . Here, for relevant applications only the real part is considered.

Note: The solution function eq.(2.5) can be expressed in terms of Airy functions. Imagine the fact that Bessel functions of order 1/3 are expressible so that we have alternatively by putting together

⎥⎦

⎤⎢⎣

⎡

ζ−

ζ=ζ 3

3/4

6/5 33)( Aip , 0>ζ . (2.9)

Both expressions are similar and do not describe any periodic or traveling wave motions. Finally, the Case N is of interest if we use the following choice for the group parameters:

1321 === kkk and 04 =k . The transformation reads as ζ=− xt and Sue x =− . This choice represents the case of traveling waves and we have to solve the following nODE of the third-order:

0123 322

22

3

3=−

ζ+

ζ−

ζS

ddSS

dSdS

dSd

, S : R × R → R , ∈ζ R, ∞≤ζ≤∞− . (2.10)

This equation cannot be solved explicitly so we decided to perform a power series representation up to order five:

( ) ( )( ) [ ]5420

21100

31

20

30

2210 88

81312

61)( ζ+ζ−−+ζ+−+ζ+ζ+=ζ OaaaaaaaaaaaaS , (2.11)

with arbitrary chosen coefficient ia , 2,1,0=i whereby this polynomial solution is continuously differentiable on the entire real axis. A graphical overview is given in Figure 3.

-4 -2 0 2 4-6

-4

-2

0

2

4

6

Figure 3 The behaviour of the solution function eq.(2.11) for the similarity function S. This polynomial solution was

generated by the choice of the coefficients ia for the values 11 1 <<− a and the same for 2a and 3a .

5 10 15 20x

0.01

0.02

0.03

0.04

0.05

0.06 S


48

If we transform by ζ=− xt and Sue x =− we show a sequence of solution surfaces depending on the independent variable x and t considering special values for the parameter λ in Figure 4. This shows that a traveling wave solution does not appear. If we introduce the notation )(ζP for the polynomial part of eq.(2.11), one can write the complete solution )(),( ζ= Petxu x in short and comparing with the animations given, we conclude:

-20

2

x

-2

0

2t

-4000-20000

u

-20

2

x

-2

0

2t

-20

2x -2

0

2

t-3´ 106-2´ 106

-1´ 1060

u

-20

2x

-20

2x -2

0

2

t-3´ 1010-2´ 1010-1´ 1010

0u

-20

2x

Figure 4 An animation of different solution surfaces of eq.(2.11), left: 2=λ , middle: 10=λ , right: 10−=λ , all 1=ia .

The exponential part influences the solution in the sense of a damping effect whereby the exponential part either does not decrease or increase. So, this part covers a domain of saturation. After we have discussed all similarity cases of relevance we finish this chapter to proceed further with the analysis. For completeness, in Table 2 we show all relevant nonlinear transformations by considering special values of the group parameters.

Table 2 Symmetry calculation and nonlinear transformations for the Rand Equation Case Choice of the group parameters Transformation for ζ Transformation for S

A 1,0 4321 ==== kkkk ζ=−1tx Su =

B 1,0 3421 ==== kkkk ζ=t Su =

C 1,0 4321 ==== kkkk ζ=+ −3)1( xt Su =

D 1,0 2431 ==== kkkk ζ=x Su =

E 1,0 4231 ==== kkkk ζ=+ −33)31( xt Su =

F 1,0 4321 ==== kkkk ζ=++ −3)1(3)31( xt Su =

G 1,0 1432 ==== kkkk non-solvable non-solvable

H 1,0 3241 ==== kkkk ζ=− xt Su =

I 1,0 4132 ==== kkkk ζ=−3xt Sxu =−1

J 1,0 3142 ==== kkkk ζ=t Seu x =−

K 1,0 4312 ==== kkkk ζ=+ −3)1( xt Sxu =+ −1)1(

L 1,0 2143 ==== kkkk ζ=x Su =

M 1,0 4213 ==== kkkk ζ=+ −33)31( xt Sxu =−1

N 1,0 3214 ==== kkkk ζ=− xt Seu x =−

O 14321 ==== kkkk ζ=++ −3)1(3)31( xt Sxu =+ −1)1( 3. Analysis by the dominant balance method Again, consider the Rand Equation, eq.(1). If we introduce the similarity ‘ansatz’ )(),( ξ= ftxu , tx λ−=ξ , we derive the following nODE of the third-order:


49

033

3

32 =⎟⎟

⎠

⎞⎜⎜⎝

⎛ξ

−ξ d

dfd

fdf , )(ξ= ff , 0''','',',0)(:~),(: ≠≠ζ∈ζ= ffffDfD . (3)

Let the domain 131 RRRDD~ ×⊆×= . We seek proper solutions on the interval I , DI ∈ and 2RI:f → . Unfortunately, this nODE cannot be solved analytically in a closed form. Therefore, we apply the Dominant Balance Method in order to generate new solutions. Generally, from eq.(3) follows that 32 '3''' fff = and 0'''00''' 22 =∨=→= ffff . This is not possible since we require both the existence of the function and their derivation. By balancing we have to treat the following cases considering a two-term balance:

Case(i): 02 ≈xxxff requiring that 23 /)'(3'' fff >> otherwise the condition holds: 0)'(30 32 <→> fff xxx from

eq.(3) for a suitable balance.

Case(ii): 0)'(3 3 ≈− f requiring that 0)'( 3 =f with the condition 00)'( 23 <→> xxxfff .

The ODE for Case(i) is solved explicitly by 3321~)( ξ+ξ+ξ cccf .

For proper of solutions (e.g. if )R ±∈ζ it is seen that this solution contradicts the given inequalities. So we conclude that the polynomial of the third-order represents a consistent balance solution and therefore we have 3

321 )()(~),( txctxcctxu λ−+λ−+ , 0≠λ , (3.1)

as proper solutions as before with suitable chosen coefficients ic , but 02 ≠c , 03 ≠c .

For practical calculations we perform a series representation of the nODE, eq.(3) with arbitrary constants ia , 2,1,0=i up to order four:

( ) [ ]54

30

2120

213

20

312

210 43

2)( ξ+ξ

−+ξ+ξ+ξ+=ξ O

aaaaa

aa

aaaf . (3.2)

For this series solution we give a graphical overview in Figure 5 by using suitable chosen values for the parameters ia .

-6 -4 -2 0 2 4 6-300

-200

-100

0

100

200

300

Figure 5 A planar plot of the series solution, eq.(3.2) generated with the choice of the parameters for all ia so that the

domain of the parameters is given by 20 << ia , the curves are all symmetrically to the x-axis. 4. The non-classical case I: Potential symmetries For more technical details we refer to (Olver, 1986; Bluman and Kumei, 1989, Gaeta, 1994, Huber, 2009) respectively. For the Rand Equation, eq.(1) we found the following: The equation admits only one possible potential system, 1Ψ consisting of two

relations. The systems can be formulated for the dependent variable 1V and can be treated in their derivations w.r.t. the independent variables ),( tx denoted by subscripts:

021

21 =⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+xu

tVuu xx , 0

29 2

1 =+∂∂ u

xV

. (4)

Unlike other evolution equations having two or more potential systems, here we are confronted with an unexpected case: Calculating the infinitesimals we see that no new potential symmetry occurs:


50

( ) 1212521

5425

31

361

3

Vkkukk

tkkxk

k

+=ϕ−=ϕ

+=ξ+=ξ . (4.1)

It is of interest to compare with the classical case, eq.(1.6). The dimension of the group increases (we have a five-dimensional group) and also an infinite group generated by the function ),( txu is seen. Here no new potential symmetry could observe. 5. The non-classical case II: Generalized symmetries We find it advisable to mention some basic notes. It is obvious from Lie theory that point symmetries are a subset of generalized symmetries (Ibragimov, 1985; 1994c). The determination of the characteristics for the general case follows by a similar algorithm as in the case of point transformations (PT) in the classical case. Classical symmetries of a (n)PDE (assumed to be in a general form 0=Δ ) are PT which guarantee the invariance of the solution space and so, PT are created by infinitesimal transformations. The determining equations for the characteristics αGS are consequences of the relation

00=Δυ =ΔGSprr

, (5)

where GSpr υr denotes the prolongation of the vector field GSυ and ‘GS’ means generalize symmetry. The main difference however is the fact that in general the characteristics depend on derivatives of an infinite order. If the order is equal to identity we arrive at the so-called contact transformations. By increasing the order of derivatives 1>n we shall find higher order GS. In case of the Rand Equation, eq.(1) we found GS depending on the first derivative: ( ) xtx ukukuuutxGS 121 ,,,, += . (5.1) This symmetry also changes from the symmetries given in (1.6), (4.1). Here we are confronted with a two-dimensional finite group of transformations where the second part xu ∂∂ / is related to dilatation operations. For the case 2=n by assuming second partial derivatives we further found ( ) xtx ukukuuutxGS 122 ,,,, += . (5.2) as a quite similar result. 6. Approximate symmetries In this section we follow (Ibragimov, 1985, 1994c; Huber, 2009) respectively and our intension is to present new results without referring too much theory. However, some remarks will be indicated. Definition: Approximate symmetries: We assume that ,..)( 21xxx = are independent coordinates of functions which are analytic in their arguments. Let us further assume that ε is a small parameter on which the functions additionally

depend. We will denote the involved infinitesimal small functions of order 1+ε p by ),( εθ xp , where

0≤p and p is a positive constant. This condition is expressed by pp Ox ][),( ε=εθ . In addition an equivalent representation of

this condition can be written by

pp x

ε

εΘ

→ε

),(lim

0. (6)

Let f and g be analytic functions in x . We define an approximation of order ,p gf ≈ by the relation

pOzgzf ][),(),( ε+ε=ε (6.1) for some fixed value of 0≤p . This definition is the basis of all calculations we will carry out in the following. Let us introduce ε as a small parameter measuring the influence of the nonlinear term of the eq.(1) so that we can write

xxxt uuu 223 =ε . First order approximate symmetries follow by


51

.)1()1(

14

32

21

ε+=φε+=ξε+=ξ

kkkk

(6.2)

Here we have another unexpected situation comparing with the symmetries as above. The order remains equal (four-dimensional) toward the classical case but the dimension is finite. The generating vector fields containing the perturbation parameter reads as txut VVVV ∂=∂ε+=∂ε+=∂ε= 4321 ,)1(,)1(, , (6.3) and the associated coefficients of these vector fields are given by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0,1,00,11,0,0,0,,0 ε+ε+ε . (6.4) In total we have four possible combinations for the vector fields. Possible reductions can be calculated by combining several sub-groups, that is nml VVV ⊗⊗ with 4,3,2,1,, =nml . We now restrict the analysis to the most important case, the case of traveling waves. This case arises by calculating the combination 43 VVVl ⊗⊗ which gives the traveling-wave transformation for the similarity

variable xt −=ζ and Su = for the similarity function once again. The relating linear ODE of the third-order is similar as in the classical case, eq.(2), however the difference is the occurrence of a linear part:

093

3=+

ζε S

dSd

, S : R × R → R , ∈ζ R, ∞≤ζ≤∞− , 0≠ε , (6.5)

which is solved explicitly by [ ] [ ] [ ]ζ−+ζ−+ζ−−=ζ 3/23/1

33/2

23/2

1 3)1(exp)3(exp)3(exp)( CCCS , (6.6)

where 21 ,CC and 3C are arbitrary constants of integration.

Considering special values of the constants, say, 1321 === CCC the real part of the solution, eq.(6.6) is written as

[ ] [ ] [ ]ζ+ζ+ζ−=ζ 6/12

33/22

13/2 3cos)3(exp2)3(exp)(S . (6.7) The function has a finite value at 0=ζ and it is further proven that 3)(lim

0=ζ

→ζS holds.

The limiting behaviour is ∞=ζ∞+→ζ

)(lim S but vanishes for −∞→ζ , that is 0)(lim =ζ∞+→ζ

S . Since the second derivative

vanishes as 0=ζ , takes positive real-valued as 1−=ζ and negative real-valued as 1=ζ , one can conclude that the solution is not stable at least in the domain 11 <ξ<− . In addition a compact written form of eq.(6.7) follows by the representation

∑∞

=

+ ζ−=ζ

0

321

)!3(3)1()(

k

kkk

kS . (6.8)

In Figure 6 we give a graphical overview of the behaviour of the real-valued function, eq.(6.7) by considering special values of the integration constants.

-2 0 2 4

-200

0

200

400

Figure 6 A planar plot of the real-valued solution, eq.(6.7) by using different values of the integration constants iC ;

that is 50 1 << C , 91 2 <<− C , 53 3 <<− C . The solution is unstable in the domain 11 <ξ<− .


52

Marked maxima and minima could observe on the real positive domain and both the maxima and minima lay on a vertical straight line. Consider the unusual behaviour in the range near the origin recognizing the instability. 7. Conclusion The present paper represents a valuable contribution to the understanding of a rarely studied evolution equation, the so called Rand Equation. Unlike many other evolution equations well known in this field, properties and behaviour of the Rand Equation are not available. This paper however is suitable to extend the spectrum of knowledge to get a deeper insight in the solution manifold. As a first step we performed a classical Lie Group analysis to study the point symmetries. Applying the procedure of similarity reduction several cases of interest were shown especially the traveling wave reduction leading to periodic wave trains. Due to complexity of the underlying nODEs asymptotic solutions were given as well as some series representations using in practical calculations were shown. A complete symmetry table containing nonlinear transformations was performed. By using the Dominant Balance Method further similar solutions could derive. Secondly, the non-classical cases were studied. We show how one can derive potential as well as generalized symmetries. Here, interestingly the nPDE does not admit potential symmetries. Otherwise, the nPDE behave in a same kind to other evolution equations relating to generalize symmetries. By increasing the order (remains two-dimensional finite) the symmetry does not change and physically speaking the group transformation correlates with dilatation operations. As a last fact of interest approximate symmetries were studied in order to show how one can calculate new solutions. The special case of traveling waves leads to unstable solutions at least in a certain domain. We also did not found any kinds of solitons, neither line solitons nor loop or cusp solitons and actually, the highly nonlinear eq.(1) admits therefore no physical description correlating to known processes. Further studies in future done by the author are necessary, especially concerning the following: We shall stress some theoretical basic questions such like the complete integrability. If the eq.(1) can be integrated completely we further can show the existence of a related Bäcklund system. Another question of interest is the affiliation to a known hierarchy also closely related to the integrability. Further it is of interest to know if the eq.(1) possesses the Painlevé property. If so, one can proof that the highly nonlinear eq.(1) can be solved in principle by the Inverse Scattering Method. Otherwise the Painlevé algorithm is suitable to solve the eq.(1) on the singular manifold. At this stage we can expect the limitation of calculating intensions, since, due to the complexity of the eq.(1), future theoretically derivations might fail. References Ablowitz M., Clarkson P., 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press Bluman G., Kumei S., 1989. Symmetries and Differential Equations, Springer Dodd R., Eilbeck J., Gibbon J., Morris H., 1988. Solitons and Nonlinear Wave Equations, Academic Press Drazin P., Johnson R., 1989. Solitons: An Introduction, Cambridge University Press Eilenberger G., 1983. Solitons, Springer-Verlag, Berlin, p.140 Gaeta G., 1994. Nonlinear Symmetries and Nonlinear Equations, Kluwer, Acad. Press Huber A., 2010. The physical meaning of a nonlinear evolution equation of the fourth order relating to locally and non-locally

supercritical waves, Int. J. Engineering, Science and Technology, Vol.2, No.1, p.70 Huber A., 2007a. On class of solutions of a nonlinear partial differential equation of sixth order generated by a classical and non-

classical symmetry analysis, Int. J. Differential Equations and Dynamical Systems, Vol.15, Nos.1&2, p.27 Huber A., 2007b. A note on class of traveling wave solutions of a nonlinear third-order system Generated by Lie’s approach,

Chaos, Solitons and Fractals, Vol. 32, No. 4, p.1357 Huber A., 2008a. A note on class of solitary-like solutions of the Tzitzéica-equation generated by a similarity reduction, Physica D

237, p.1079 Huber A., 2008b. A note on new solitary and similarity class of solutions of a fourth-order nonlinear evolution equation, Appl.

Math. and Comp. 202, p.787 Huber A., 2009. The Cavalcante -Tenenblat equation – Does the equation admit a physical significance?, Appl. Math. and Comp.

212, p.14 Ibragimov N., 1985. Transformation Groups Applied to Mathematical Physics, Reidel Publ., Dortrecht Ibragimov N., 1994a. Lie Group Analysis, Vol. III, CRC Press, Inc. Ibragimov N., 1994b. Lie Group Analysis, Vol. II, CRC Press, Inc. Ibragimov N., 1994c. Sophus Lie and harmony in mathematical physics, on the 150th anniversary of his birth, Math. Intel. 16, p.20. Klein F., 1918. Über Differentialgesetzte für die Erhaltung von Impuls und Energie in der EinsteinschenGravitationstheorie,

Nachr. Ges. Wiss. Göttingen Math. Phys. Vol. 2, p.171 Noether E., 1971. Invariante Variationsprobleme, Transport Theory Stat. Phys. Vol. 1, p.186 Olver P., 1986. Applications of Lie Groups to Differential Equations, Springer Whitham G.B., 1974. Linear and Nonlinear Waves, J. Wiley, New York.


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Biographical notes Dipl.-Ing. Dr. techn. Huber Alfred is a distinguished lecturer at the Institute of Theoretical Physics – Computational Physics at the Technical University Graz, Austria following his habilitation treatise. He did his diploma thesis titled ‘Systematic in the physics of elementary particles focusing the quarkonium states’ in the field of elementary particle physics at the former Institute of Nuclear Physics at the Technical University Graz, Austria. He completed his scientific education with the doctoral programme of technical sciences at the Institute of Chemical Technology of Inorganic Compounds at the Technical University Graz, Austria subject to nuclear solid state physics and advanced electrochemistry. Thesis titled ‘Synthesis and characterization of doped γ-manganese dioxides’. Also the author has a learnt vocation for a chemical assistant at the Research Centre of Electron Microscopy at the former Technical High School Graz, Austria. He is the author of 27 articles which have appeared in world-wide renowned scientific journals. His research interests are nonlinear partial differential equations (nPDE) of higher order with applications especially in physics and chemistry. The author developed several new algebraic procedures for solving nPDE. Special interests are further given in classical and non-classical symmetry methods, nonlinear transformations and the application of nonlinear methods in describing electrochemical interfaces, nonlinear wave propagation and further nonlinear topics of advanced character. Received February 2010 Accepted April 2009 Final acceptance in revised form April 2010

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A hybridized K-means clustering approach for high dimensional dataset

Rajashree Dash1, Debahuti Mishra*, Amiya Kumar Rath2, Milu Acharya3

1,*,3Institute of Technical Education and Research, Bhubaneswar, Orissa, INDIA

2 Director, College of Engineering Bhubaneswar, Orissa, INDIA *Corresponding Authors’ e-mails: [email protected], [email protected] ,[email protected], [email protected]

Abstract Due to incredible growth of high dimensional dataset, conventional data base querying methods are inadequate to extract useful information, so researchers nowadays is forced to develop new techniques to meet the raised requirements. Such large expression data gives rise to a number of new computational challenges not only due to the increase in number of data objects but also due to the increase in number of features/attributes. Hence, to improve the efficiency and accuracy of mining task on high dimensional data, the data must be preprocessed by an efficient dimensionality reduction method. Recently cluster analysis is a popularly used data analysis method in number of areas. K-means is a well known partitioning based clustering technique that attempts to find a user specified number of clusters represented by their centroids. But its output is quite sensitive to initial positions of cluster centers. Again, the number of distance calculations increases exponentially with the increase of the dimensionality of the data. Hence, in this paper we proposed to use the Principal Component Analysis (PCA) method as a first phase for K-means clustering which will simplify the analysis and visualization of multi dimensional data set. Here also, we have proposed a new method to find the initial centroids to make the algorithm more effective and efficient. By comparing the result of original and new approach, it was found that the results obtained are more accurate, easy to understand and above all the time taken to process the data was substantially reduced. Keywords: Cluster analysis, K-means Algorithm, Dimensionality Reduction, Principal Component Analysis, Hybridized K-means algorithm

1. Introduction Data mining is a convenient way of extracting patterns, which represents knowledge implicitly stored in large data sets and focuses on issues relating to their feasibility, usefulness, effectiveness and scalability. It can be viewed as an essential step in the process of knowledge discovery. Data are normally preprocessed through data cleaning, data integration, data selection, and data transformation and prepared for the mining task. Data mining can be performed on various types of databases and information repositories, but the kind of patterns to be found are specified by various data mining functionalities like class/concept description, association, correlation analysis, classification, prediction, cluster analysis etc. Cluster analysis is one of the major data analysis methods widely used for many practical applications in emerging areas. Clustering is the process of finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups. A good clustering method will produce high quality clusters with high intra-cluster similarity and low inter-cluster similarity. The quality of a clustering result depends on both the similarity measure used by the method and its implementation and also by its ability to discover some or all of the hidden patterns. A good survey on clustering methods is found in Xu et al. (2005). K-means is a commonly used partitioning based clustering technique that tries to find a user specified number of clusters (k), which are represented by their centroids, by minimizing the square error function. Although K-means is simple and can be used for a wide variety of data types, it is quite sensitive to initial positions of cluster centers. There are two simple approaches to cluster center initialization i.e. either to select the initial values randomly, or to choose the first k samples of the data points. As an alternative, different sets of initial values are chosen (out of the data points) and the set, which is closest to optimal, is chosen. However, testing different initial sets is considered impracticable criteria, especially for large number of clusters Ismail et al (1989). Therefore, different methods have been proposed in literature by Pena et al. (1999). Again the computational complexity

Dash et al. / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 59-66

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of original K-means algorithm is very high, especially for large data sets. In addition the number of distance calculations increases exponentially with the increase of the dimensionality of the data. When the dimensionality increases usually, only a small number of dimensions are relevant to certain clusters, but data in the irrelevant dimensions may produce much noise and mask the real clusters to be discovered. Moreover when dimensionality increases, data usually become increasingly sparse, due to which data points located at different dimensions can be considered as all equally distanced and the distance measure, which, essentially for cluster analysis, becomes meaningless. Hence, attribute reduction or dimensionality reduction is an essential data-preprocessing task for cluster analysis of datasets having a large no. of features/attributes.

Dimensionality reduction specified by Maaten et al. (2007) and Davy et al. (2007) is the transformation of high-dimensional data into a meaningful representation of reduced dimensionality that corresponds to the intrinsic dimensionality of the data. It falls into two categories i.e. Feature Selection (FS) and Feature Reduction (FR). Feature Selection algorithm aims at finding out a subset of the most representative features according to some objective function in discrete space. The algorithms of FS are always greedy. Thus, they sometimes cannot even find the optimal solution in the discrete space. Feature Extraction/ Feature Reduction algorithms aim to extract features by projecting the original high-dimensional data into a lower-dimensional space through algebraic transformations. It finds the optimal solution of a problem in a continuous space, but the computational complexity is more comparative to feature selection algorithm. Various types of feature reduction methods have been developed. PCA is a commonly used feature reduction method in terms of minimizing the reconstruction error. Traditional K-means algorithm for cluster analysis developed for low dimensional data, often do not work well for high dimensional data like microarray gene expression data and the results may not be accurate most of the time due to noise and outliers associated with original data. Also the computational complexity increases rapidly as the dimension increases. Hence, to improve the efficiency, we proposed a method to apply PCA on original data set, so that the correlated variables exist in the original dataset would be transformed to possibly uncorrelated variables, which are reduced in size. Before applying PCA the dataset needs to be normalized, so that any attribute with larger domain will not dominate attributes with smaller domain. The resulting reduced data set obtained from the application of PCA will be applied to a K-means clustering algorithm. Here also we have proposed a new method to find the initial centroids to make the algorithm more effective and efficient. The main advantage of this approach stems from the fact that this framework is able to obtain better clustering with reduced complexity and also provides better accuracy and efficiency for high dimensional datasets. Section 1 of the paper deals with the introductory concepts of clustering, K-means clustering, its limitations, the need of dimensionality reduction for clustering and the goal of the paper. Some recent related works and other preliminaries on K-means algorithm, dimensionality reduction methods and some concepts of PCA have been discussed in section 2. Section 3 describes our new proposed algorithm for K-means clustering. Section 4 describes our approach in various steps with experimental activities and corresponding result discussion followed by conclusion in Section 5.

2. Related Work

Several attempts were made by researchers to improve the effectiveness and efficiency of the K-means algorithm. Yuan et al. (2004) proposed a systematic method for finding the initial centroids. However, Yuan’s method does not suggest any improvement to the time complexity of the K-means algorithm. Belal et al. (2005) proposed a new method for cluster initialization based on finding a set of medians extracted from a dimension with maximum variance. Zoubi et al. (2008) proposed a new strategy to accelerate K-means clustering by avoiding unnecessary distance calculations through the partial distance logic. Fahim et al. (2009) proposed a method to select a good initial solution by partitioning dataset into blocks and applying K-means to each block. But here the time complexity is slightly more. . Though the above algorithms can help finding good initial centers for some extent, they are quite complex and some use the K-means algorithm as part of their algorithms, which still need to use the random method for cluster center initialization. Deelers et al. (2007) has proposed an enhancing K-means algorithm based on the data partitioning algorithm used for color quantization. The algorithm performs data partitioning along the data axis with the highest variance. Nazeer et al. (2009) proposed an enhanced K-means algorithm, which combines a systematic method for finding initial centroids and an efficient way for assigning data points to cluster. This method ensures the entire process of clustering in O (n2) time without sacrificing the accuracy of clusters. Similarly Xu et al. (2009) specify a novel initialization scheme to select initial cluster centers based on reverse nearest neighbor search. But all the above methods do not work well for high dimensional data sets. Yeung et al. (2000) presented an empirical study on principal component analysis for clustering gene expression data. But here the initial centroids are chosen randomly. Chao et al. (2005) also proposed a method for dimension reduction for microarray data analysis using Locally Linear Embedding. 2.1 K-means Clustering Algorithm The K-means algorithm is one of the partitioning based, nonhierarchical clustering methods. Given a set of numeric objects X and an integer number k, the K-means algorithm searches for a partition of X into k clusters that minimizes the within groups sum of squared errors. The K-means algorithm starts by initializing the k cluster centers. The input data points are then allocated to one of the existing clusters according to the square of the Euclidean distance from the clusters, choosing the closest. The mean (centroid) of each cluster is then computed so as to update the cluster center. This update occurs as a result of the change in the


61

membership of each cluster. The processes of re-assigning the input vectors and the update of the cluster centers is repeated until no more change in the value of any of the cluster centers. The steps of the K-means algorithm are written below:

1. Initialization: choose randomly K input vectors (data points) to initialize the clusters. 2. Nearest-neighbor search: for each input vector, find the cluster center that is closest, and assign that input vector to the

corresponding cluster. 3. Mean update: update the cluster centers in each cluster using the mean (centroid) of the input vectors assigned to that

cluster. 4. Stopping rule: repeat steps 2 and 3 until no more change in the value of the means.

2.2 Principal Component Analysis (PCA) Principal Component Analysis by Valarmathie et al. (2009) and Yan et al. (2006) is an unsupervised Feature Reduction method for projecting high dimensional data into a new lower dimensional representation of the data that describes as much of the variance in the data as possible with minimum reconstruction error. PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. It transforms a number of possibly correlated variables into a smaller number of uncorrelated variables called principal components. Hence, PCA is a statistical technique for determining key variables in a high dimensional data set that explain the differences in the observations and can be used to simplify the analysis and visualization of high dimensional data set, without much loss of information. 2.3 Principal Component (PC) Technically, a principal component can be defined as a linear combination of optimally weighted observed variables which maximize the variance of the linear combination and which have zero covariance with the previous PCs. The first component extracted in a principal component analysis accounts for a maximal amount of total variance in the observed variables. The second component extracted will account for a maximal amount of variance in the data set that was not accounted for by the first component and it will be uncorrelated with the first component. The remaining components that are extracted in the analysis display the same two characteristics: each component accounts for a maximal amount of variance in the observed variables that was not accounted for by the preceding components, and is uncorrelated with all of the preceding components. When the principal component analysis will complete, the resulting components will display varying degrees of correlation with the observed variables, but are completely uncorrelated with one another.

PCs are calculated using the Eigen value decomposition of a data covariance matrix/ correlation matrix or singular value decomposition of a data matrix, usually after mean centering the data for each attribute. Covariance matrix is preferred when the variances of variables are very high compared to correlation. It would be better to choose the type of correlation when the variables are of different types. Similarly the SVD method is used for numerical accuracy. 2.4 Elimination Methods of Unnecessary PCs The transformation of the dataset to the new principal component axis produces the number of PCs equivalent to the no. of original variables. But for many datasets, the 1st several PCs explain the most of the variances, so the rest can be eliminated with minimal loss of information. The various criteria used to determine how many PCs should be retained for the interpretation is as follows:

Using Scree Diagram plots the variances in percentage corresponding to the PCs, which will automatically eliminate the PCs with very low variances.

Fixing a threshold value of variance, so that PCs having variance more than the given threshold value will be retained rejecting others.

Eliminate PCs whose Eigen values are smaller than a fraction of the mean Eigen value. 3. Proposed Hybridized K-means Clustering Algorithm As original K-means clustering algorithm often does not work well for high dimension, hence, to improve the efficiency, we proposed to apply PCA on original data set, to obtain a reduced dataset containing possibly uncorrelated variables. Then the resulting reduced data set will be applied to the K-means clustering algorithm to determine the precise no. of clusters. As quality of the final clusters heavily depends on the selection of the initial centroids, here we proposed a new method to choose such data objects as initial centroids whose squared Euclidian distance is maximum among all the data objects, to make the algorithm more effective and efficient.


62

The proposed model is illustrated in Figure 1. The steps of the hybridized k-means clustering algorithm are as follows. Input: X = d1, d2,……..,dn // set of n data items. K // Number of desired clusters. An array Cen [ ] having size k initially being empty. Output: A set of k clusters // Phase-1: Apply PCA to reduce the dimension of the data set

1. Organize the dataset in a matrix X. 2. Normalize the data set using Z-score. 3. Calculate the singular value decomposition of the data matrix. 4. Calculate the variance using the diagonal elements of D. 5. Sort variances in decreasing order. 6. Choose the p principal components from V with largest variances. 7. Form the transformation matrix W consisting of those p PCs. 8. Find the reduced projected dataset Y in a new coordinate axis by applying W to X.

//Phase-2: Find the initial centroids

9. Set m=1. 10. Compute the distance between each data points in the set Y. 11. Choose the two data points yi and yj such that distance (yi, yj ) is maximum. 12. Cen[m] = yi ; Cen[m+1] = yj ; m=m+2 ; 13. Remove the two objects yi , yj from Y. 14. While (m <= k)

1. Find the distance of each object in Y to Cen[i], for i = 1 to m-1. 2. Find the average of all the distances to the centroid for each object in Y. 3. Choose the data object yo having maximum average distance from previous centroids. 4. Cen[m] = yo ; m = m+1; 5. Remove the object yo from Y.

// Phase-3: Apply the K-means clustering with the initial centroids given in array Cen.

15. For each data point, in set Y, find the nearest cluster center from list Cen that is closest and assign that data point to the corresponding cluster.

16. Update the cluster centers in each cluster using the mean of the data points, which are assigned to that cluster. 17. Repeat the steps 15 and 16 until there are no more changes in the values of the centroids.

4. Experimental Activities and Result Discussion Initially, we evaluated the proposed algorithm on a synthetic dataset with 15 data objects having 10 attributes as shown in table 1. Then three datasets, Pima Indian Diabetes data set, Breast Cancer data set and SPECTF Heart data set, taken from the UCI

TUDVX =

Normalize the data set using Z-score

method.

Apply the SVD method of PCA to get

PCs.

Eliminate the unnecessary

PCs.

Get the cluster index of each object of the

reduced dataset by k-means clustering.

Derive the initial centroids for k-

means clustering.

Find the reduced projected data set

using reduced PCs.

Figure 1: A Hybridized model for K-means Clustering


63

machine learning repository are used for testing the accuracy and efficiency of the hybridized algorithm. Here the Sum of Squared Error (SSE), representing distances between data points and their cluster centers have used to measure the clustering quality. Among two solutions for a given dataset, the smaller the value of SSE and higher the accuracy, the better the solution. Step 1: Normalizing the original data set Using the Normalization process, the initial data values are scaled so as to fall within a small-specified range. An attribute value V of an attribute A is normalized to V’ using Z-Score as follows: V’=(V-mean(A))/std(A) It performs two things i.e. data centering, which reduces the square mean error of approximating the input data and data scaling, which standardizes the variables to have unit variance before the analysis takes place. This normalization prevents certain features to dominate the analysis because of their large numerical values.

Table 1: The original data matrix X with 15 data objects having 10 attribute values.

Figure 2: Plotting of data along with the normalized data.

Step 2: Calculating the PCs using Singular Value Decomposition of the normalized data matrix Applying the steps given in phase 1 of the new proposed algorithm, the no. of PCs obtained is same with the no. of original variables. To eliminate the weaker components from this PC set we have calculated the corresponding variance, percentage of variance and cumulative variances in percentage, which is shown in Table 2. Then we have considered the PCs having variances less than the mean variance, ignoring the others. The reduced PCs are shown in Table 3. Step 3: Finding the reduced data set using the reduced PCs The transformation matrix with reduced PCs is formed and this transformation matrix is applied to the normalized data set to produce the new reduced projected dataset, which can be used for further data analysis. The reduced data set is shown in Table 4. We have also applied the PCA on three biological dataset and the reduced no. of attributes obtained for each dataset is shown in Figure 3. Step 4: Comparison of efficiency and accuracy of the original k-means clustering and proposed algorithm. The clustering results shown in Figure 4 and 5 by applying the standard k-means clustering to the original synthetic dataset and the proposed method to the reduced dataset are approximately same, but the time taken for clustering will be reduced due to less number of attributes. Again we compared the clustering results obtained by the k-means algorithm using random initial centers and initial centers derived by the proposed algorithm over 4 datasets with original dimension and with reduced dimension based on the sum of squared error distances (SSE), which is shown in Figure 6 and 7. The clustering results of k-means using random initial centers are the average results over 10 runs since each run gives different results. The SSE value obtained and the time taken in ms for 4 datasets with original k-means and new proposed algorithm is given in Table 5.

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

Data1 1 5 1 1 1 2 1 3 1 1 Data2 2 5 4 4 5 7 10 3 2 1 Data3 3 3 1 1 1 2 2 3 1 1 Data4 4 6 8 8 1 3 4 3 7 1 Data5 5 4 1 1 3 2 1 3 1 1 Data6 6 8 10 10 8 7 10 9 7 1 Data7 7 1 1 1 1 2 10 3 1 1 Data8 8 2 1 2 1 2 1 3 1 1 Data9 9 2 1 1 1 2 1 1 1 5

Data10 10 4 2 1 1 2 1 2 1 1 Data11 11 1 1 1 1 1 1 3 1 1 Data12 12 2 1 1 1 2 1 2 1 1 Data13 13 2 1 1 1 2 1 2 1 1 Data14 14 5 3 3 3 2 3 4 4 1 Data15 15 1 1 1 1 2 3 3 1 1


64

Table 2: The variances, variances in percentage, cumulative Table 3: Reduced PCs having variance

variances in percentage corresponding to the PCs . greater than the mean variance.

Table 4: The reduced dataset containing 3 attributes

Figure 3: Plotting of original and reduced no. of attributes for Synthetic, Pima Indian Diabetes, Breast Cancer and SPECTF Heart datasets.

Figure 4: Clustering with original dataset by Figure 5: Clustering with reduced dataset by standard K-means algorithm. proposed algorithm.

Variances Variances in %

Cumulative variances

in% pc1 6.210578 62.10578 62.10578 pc2 1.054022 10.54022 72.646 pc3 1.016014 10.16014 82.80614 pc4 0.86546 8.654603 91.46075 pc5 0.455458 4.554576 96.01532 pc6 0.24649 2.464901 98.48022 pc7 0.108508 1.085079 99.5653 pc8 0.030248 0.302483 99.86779 pc9 0.010731 0.10731 99.9751

pc10 0.00249 0.024904 100

Pc1 Pc2 Pc3 0.161104 -0.70692 0.222666 -0.34575 0.06143 0.104589 -0.37811 -0.12852 0.225285 -0.37785 -0.12565 0.21828 -0.34923 0.061232 -0.09679 -0.34712 0.275694 -0.1149 -0.28668 0.21192 -0.29648 -0.34062 -0.24545 -0.14089 -0.34594 -0.26379 0.318984 0.091787 0.457921 0.780392

v1 v2 v3 data1 0.536985 1.045863 -0.56448 data2 -2.76212 1.914677 -1.15055 data3 0.858597 0.73036 -0.64746 data4 -2.78991 -0.43292 1.310152 data5 0.502757 0.444459 -0.51029 data6 -7.19228 -0.525 0.259885 data7 0.691562 0.513194 -1.21072 data8 1.149898 -0.19297 -0.28827 data9 2.060598 1.744708 2.866126

data10 1.08491 -0.31268 -0.00678 data11 1.749453 -0.8052 -0.20218 data12 1.620462 -0.6419 -0.08667 data13 1.656486 -0.79997 -0.03688 data14 -0.70786 -1.51675 0.500656 data15 1.540447 -1.16586 -0.23254


65

Table 5: SSE values obtained and time taken in ms with original k-means and new proposed algorithm

Dataset No of Instances

Original K-means Algorithm

SSE Time taken(ms)

Proposed Algorithm

SSE Time taken(ms) Synthetic 15 608.446 78 47.8 65

Pima Indian Diabetes

50 59255 158 182.39 122

Breast Cancer 80 29253 167 165.94 131 SPECTF Heart 40 97075 145 996.8 112

The above results show that the new algorithm provides better SSE values for all the cases. Hence, in this regard it increases the efficiency of the original k-means algorithm. The accuracy of clustering determined by comparing the clusters obtained by the experiments with the available clusters for three data sets in UCI data set is shown in Figure 8. In all the cases the proposed algorithm provides better accuracy compared to the original k-means algorithm.

Figure 8: Clustering accuracy of PID, Breast cancer, SPECTF Heart datasets.

5. Conclusion In this paper a hybridized K-means algorithm has been proposed which combines the steps of dimensionality reduction through PCA, a novel initialization approach of cluster centers and the steps of assigning data points to appropriate clusters. Using the proposed algorithm a given data set was partitioned in to k clusters in such a way that the sum of the total clustering errors for all clusters was reduced as much as possible while inter distances between clusters are maintained to be as large as possible. The experimental results show that the proposed algorithm provides better efficiency and accuracy comparison to original k-means algorithm with reduced time. Though the proposed method gave better quality results in all cases, over random initialization methods, still there is a limitation associated with this, i.e. the number of clusters (k) is required to be given as input. Again the method to find the initial centroids may not be reliable for vary large dataset. Evolving some statistical methods to compute the value of k, depending on the data distribution is suggested for future research. Methods for refining the computation of initial centroids are worth investigating.

Figure 6: SSE results on synthetic, PID, Breast cancer, SPECTF heart datasets with original dimension.

Figure 7: SSE results on synthetic, PID, Breast cancer, SPECTF heart datasets with reduced dimension.


66

References Belal M. and Daoud A., 2005. A new algorithm for cluster initialization, World Academy of Science, Engineering and Technology,

Vol. 4, pp. 74-76. Chao Shi and Chen Lihui, 2005. Feature dimension reduction for microarray data analysis using locally linear embedding, 3rd Asia

Pacific Bioinformatics Conference, pp. 211-217. Davy Michael and Luz Saturnino, 2007. Dimensionality reduction for active learning with nearest neighbour classifier in text

categorization problems, Sixth International Conference on Machine Learning and Applications, pp. 292-297. Deelers S. and Auwatanamongkol S., 2007. Enhancing K-means algorithm with initial cluster centers derived from data

partitioning along the data axis with the highest variance, International Journal of Computer Science, Vol. 2, No. 4, pp. 247-252.

Fahim A. M., Salem A. M., Torkey F. A., Saake G. and Ramadan M. A., 2009. An efficient k-means with good initial starting points, Georgian Electronic Scientific Journal: Computer Science and Telecommunications, Vol. 2, No. 19, pp. 47-57.

Ismail M. and Kamel M., 1989. Multidimensional data clustering utilization hybrid search strategies, Pattern Recognition, Vol. 22, No. 1, pp.75-89.

Maaten L.J.P., Postma E.O. and Herik H.J. van den, 2007. Dimensionality reduction: A comparative review”, Tech. rep. University of Maastricht.

Nazeer K. A. Abdul and Sebastian M.P., 2009. Improving the accuracy and efficiency of the k-means clustering algorithm, Proceedings of the World Congress on Engineering, Vol. 1, pp. 308-312.

Pena J. M., Lozano J. A. and Larranaga P., 1999. An empirical comparison of four initialization methods for the k-means algorithm, Pattern Recognition Letters, Vol. 20, No. 10, pp. 1027-1040.

Valarmathie P., Srinath M. and Dinakaran K., 2009. An increased performance of clustering high dimensional data through dimensionality reduction technique, Journal of Theoretical and Applied Information Technology, Vol. 13, pp. 271-273.

Xu R. and Wunsch D., 2005. Survey of clustering algorithms, IEEE Trans. Neural Networks, Vol. 16, No. 3, pp. 645-678. Xu Junling, Xu Baowen, Zhang Weifeng, Zhang Wei and Hou Jun, 2009. Stable initialization scheme for K-means clustering,

Wuhan University Journal of National Sciences, Vol. 14, No. 1, pp. 24-28. Yan Jun, Zhang Benyu, Liu Ning, Yan Shuicheng, Cheng Qiansheng, Fan Weiguo, Yang Qiang, Xi Wensi, and Chen Zheng,

2006. Effective and efficient dimensionality reduction for large-scale and streaming data preprocessing, IEEE transactions on Knowledge and Data Engineering, Vol. 18, No. 3, pp. 320-333.

Yeung Ka Yee and Ruzzo Walter L., 2000. An empirical study on principal component analysis for clustering gene expression Data”,Tech. Report, University of Washington.

Yuan F., Meng Z. H, Zhang H. X and Dong C. R, 2004. A new algorithm to get the initial centroids, Proc. of the 3rd International Conference on Machine Learning and Cybernetics, pp. 1191–1193.

Zhang Z., Zhang J. and Xue H., 2008. Improved K-means clustering algorithm, Proceedings of the Congress on Image and Signal Processing, Vol. 5, No. 5, pp. 162-172.

Zoubi M. B. Al., Hudaib A., Huneiti A. and Hammo B., 2008. New efficient strategy to accelerate k-means clustering algorithm”, American Journal of Applied Sciences, Vol. 5, No. 9, pp. 1247-1250.

Biographical notes Rajashree Dash has completed her B.Tech in Computer Sc. & Engineering from KIIT University. Now she is perusing her M.Tech in Computer Sc. & Engg at Institute of Technical Education & Research (ITER) under Siksh `O` Anusandhan University, Bhubaneswar. Her research areas include Data mining, Computer Graphics etc. Debahuti Mishra is an Assistant Professor and research scholar in the department of Computer Sc. & Engg, Institute of Technical Education & Research (ITER) under Siksh `O` Anusandhan University, Bhubaneswar. She received her Masters degree from KIIT University, Bhubaneswar. Her research areas include Data mining, Bio-informatics Software Engineering, Soft computing . She is an author of a book Aotumata Theory and Computation by Sun India Publication (2008). Dr.Amiya Kumar Rath obtained Ph.D in Computer Science in the year 2005 from Utkal University for the work in the field of Embedded system. Presently working with College of Engineering Bhubaneswar (CEB) as Professor of Computer Science & Engg. Cum Director (A&R) and is actively engaged in conducting Academic, Research and development programs in the field of Computer Science and IT Engg. Contributed more than 30 research level papers to many national and International journals. and conferences Besides this, published 4 books by reputed publishers. Having research interests include Embedded System, Adhoc Network,Sensor Network ,Power Minimization, Biclustering, Evolutionary Computation and Data Mining. Dr. Milu Acharya obtained her Ph.D at Utkal University. She is a Professor in Department of Computer Applications at Institute of Technical Education and Research (ITER), Bhubaneswar. She has contributed more than 20 research level papers to many national and International journals and conferences Besides this, published 3 books by reputed publishers. Her research interests include Biclustering,Data Mining , Evaluation of Integrals of analytic Functions , Numerical Analysis , Complex Analysis , Simulation and Decision Theory. Received February 2010 Accepted March 2010 Final acceptance in revised form April 2010

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Cycle multiplicity of total graph of Cn, Pn, and K1,n

M.M. Akbar Ali1*, S. Panayappan2

1*Department of Mathematics,Sri Shakthi Institute of Engineering and Technology, Coimbatore, Tamil Nadu, INDIA

2 Department of Mathematics,Government Arts College,Coimbatore, Tamil Nadu, INDIA *Corresponding Author: e-mail: [email protected], Tel +91-9994279835.

Abstract Cycle multiplicity of a graph G is the maximum number of edge disjoint cycles in G. In this paper, we find the cycle multiplicity of total graph of cycles Cn, paths Pn, and star graph K1,n respectively. Keywords: cycle multiplicity, total graph, cycle, path, star graph. 1. Introduction Line partition number (Chatrand et al., 1971) of a graph G is the minimum number of subsets into which the edge-set of G can be partitioned so that the subgraph induced by each subset has property P. Dual to this concept of line partition number of graph is the maximum number of subsets into which the edge -set of G can be partitioned such that the subgraph induced by each subset does not have the property P. Define the property P such that a graph G has the property P if G contains no subgraph which is homeomorphic from the complete graph K3. Now the line partition number and dual line partition number corresponding to the property P is referred to as arboricity and cycle multiplicity of G respectively. Equivalently the cycle multiplicity is the maximum number of line disjoint subgraphs contained in G so that each subgraph is not acyclic. This number is called the cycle multiplicity of G denoted by CM(G). The formula for cycle multiplicity of a complete and complete bipartite graph is given in (Chatrand et al., 1971). In (Simões Pereira, 1972), the author found an upper bound for the line and middle graph of any graph. Also he proved that the bound becomes the formula for line and total graph of any forest.

We consider finite, simple, undirected graph G(V(G), E(G)) where V(G) and E(G) represent vertex set and edge set of G respectively. For any real number r, [r] and r⎡ ⎤⎢ ⎥ denote the largest integer not exceeding r and the least integer not less than r, respectively. The other notations and terminology used in this paper can be found in (Harary, 1969).

Let G be a graph with vertex set V(G) and edge set E(G). The total graph (Michalak, 1981) of G, denoted by T(G) is defined as follows. The vertex set of T(G) is V(G)∪ E(G). Two vertices x, y in the vertex set of T(G) are adjacent in T(G) in case one of the following holds: (i) x, y are in V(G) and x is adjacent to y in G. (ii) x, y are in E(G) and x, y are adjacent in G (iii) x is in V(G), y is in E(G), and x, y are incident in G

2. Cycle multiplicity of total graph of Cn It is obvious that cycle multiplicity of any cycle is one. We obtain a formula to find the cycle multiplicity of the total graph of a

cycle.

Theorem 2.1

Cycle multiplicity of Total Graph of n-Cycle, 1)]([ += nCTCM n

Akbar Ali and Panayappan / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 54-58

55

Proof

Figure 1. n-cycle and its Total Graph

Let V(Cn) = v1, v2,…….,vn and E(Cn) = e1, e2,……,en, in which ei = vivi+1. By the definition of total graph, V[T(Cn)]= v1, v2,……,vn∪ e1,e2,……,en and E[T(Cn)] = eiei+1 / ( ni ≤≤1 -1) ∪ ene1 ∪ vivi+1 / ni ≤≤1 -1∪ vnv1 ∪ eivi+1 / ni ≤≤1 -1∪ env1∪ viei / ni ≤≤1 . The cycles of T(Cn) are Ci = eivi+1ei+1 ( ni ≤≤1 -1), Cn = ene1v1,

iiii evvC 1|

+= ( ni ≤≤1 -1), nnn evvC 1| = . Let Cn+1 = v1v2,………,.vnv1, Cn+2 = e1e2……..,ene1, Cn+3 = v1e1v2e2v3,……..,vnenv1.

Now we collect set of line disjoint cycles, C1 = Ci / ( ni ≤≤1 -1)∪ Cn∪ Cn+1, C2 = 11/ | −≤≤ niCi ∪ |nC ∪

Cn+2, C 3 = Cn+1, Cn+2, Cn+3 . Clearly Ci ( )31 ≤≤ i is a set of line disjoint cycles in T(Cn) and |C1| = |C2| = n+1. Since n ≥ 3,

|C1| or |C2| ≥ |C3| and either C1 or C2 contains maximum number of line disjoint cycles of T(Cn) and hence 1)]([ += nCTCM n . 3. Cycle multiplicity of total graph of Pn As Pn does not contain any cycle, its cycle multiplicity is zero. In the following theorem we states a formula to find the maximum

number of line disjoint cycles in the total graph of a path.

Theorem 3.1

Cycle multiplicity of total graph of path, nPTCM n =)]([

.Proof

Figure 2. Path and its Total Graph

Let V(Pn) = v1, v2,……..,vn+1 and E(Cn) = e1, e2,……,en, By the definition of total graph, V[T(Pn)] = V(Pn)∪ E(Pn), E[T(Pn)] =viei / ( ni ≤≤1 ) ∪ eivi+1 / ni ≤≤1 ∪ vivi+1 / ni ≤≤1 ∪ eiei+1 / ni ≤≤1 -1 The cycles of T(Pn) are


56

Ci = vivi+1ei ( ni ≤≤1 ) and |iC = eiei+1vi+1 ( ni ≤≤1 -1) . Let C1 = Ci /( ni ≤≤1 ) and C2 = 11/ | −≤≤ niCi . The cycles

in the set Ci ( i =1,2) are line disjoint cycles of T(Pn). Also |C 2| < |C 1| = n and hence nPTCM n =)]([ 4. Cycle multiplicity of total graph of K1, n Since the star graphs are acyclic its cycle multiplicity is zero. We find a formula for the cycle multiplicity of total graph of a star graph Theorem 4.1

Cycle multiplicity of total graph of Km,n,

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦⎤

⎢⎣⎡ +

⎥⎦

⎤⎢⎣

⎡ +

=

evenisnifnn

oddisnifnn

KTCM n

6)4(

65

)]([

2

,1

Proof

Figure 3. Star graph and its Total Graph

Let V(K1,n) = v1, v2,…….,vn and E(K1,n) = e1, e2,……,en, By the definition of total graph, we have

V[T(K1,n)] = v∪ ei / ( ni ≤≤1 )∪ vi / ( ni ≤≤1 ), in which the vertices e1, e2,……,en induces a cliques of order n (say

Kn). Also the vertex v is adjacent with vi ( ni ≤≤1 ).

Case (i)

If n is odd

We collect the set of line disjoint cycles of T(K1,n) as below.

C1 = veiei+1v / (i = 1, 3,……..,n-2), C2 = veiei+1v / i = 2, 4,……..,n-1, C3 = set of line disjoint cycles in the clique Kn .

C4 = veiviv / ( ni ≤≤1 ) , Clearly |C1| = |C2| = 2

1−n.

To prove |C3| = ⎥⎦

⎤⎢⎣

⎡ −6

2 nn, i.e., we have to prove the number of line disjoint cycles in C3 is ⎥

⎦

⎤⎢⎣

⎡ −6

2 nn if n is odd. If n =1, the

number of line disjoint cycles in the clique K1 is zero and ⎥⎦

⎤⎢⎣

⎡ −6

2 nn = 0 for n =1. Similarly if n =3, then the number of line


57

disjoint cycles in the clique K3 is one and ⎥⎦

⎤⎢⎣

⎡ −6

2 nn = 1 for n =2. Therefore |C3 | = ⎥

⎦

⎤⎢⎣

⎡ −6

2 nn if n = 1, 3. Assume that the result

is true for m = 2k-1 for some k. i.e., |C3| = ⎥⎦

⎤⎢⎣

⎡ +−3

132 2 kk, i.e., Number of line disjoint cycles in Km = |C3| = ⎥

⎦

⎤⎢⎣

⎡ +−3

132 2 kk.

Now consider the clique Kn where n = 2k + 1. Consider Kn-2 = Kn – e2k, e2k+1 = K2k-1 = Km. Number of line disjoint cycles in Kn-2

is ⎥⎦

⎤⎢⎣

⎡ +−3

132 2 kk. Also the number of line disjoint cycles is decreased by ⎥⎦

⎤⎢⎣⎡ −

314k

. Therefore |C3| in Kn is

⎥⎦

⎤⎢⎣

⎡ +−3

132 2 kk+ ⎥⎦

⎤⎢⎣⎡ −

314k

. i.e., |C3| = ⎥⎦

⎤⎢⎣

⎡ +3

2 2 kk, i.e. |C 3| = ⎥

⎦

⎤⎢⎣

⎡ −6

2 nn, where n = 2k+1. Since n is odd there exist

no edges in the clique which are left out in the extraction of line disjoint cycles. Since 2≥n , ≤−2

2n⎥⎦

⎤⎢⎣

⎡ −6

2 nn. Therefore |C1|

= |C2| ≤ |C3| . The cycles in C 3 and C 4 are line disjoint. Therefore maximum number of line disjoint cycles in T(K1, n),

CM[T(K1,n)] = |C3| + |C4| = ⎥⎦

⎤⎢⎣

⎡ −6

2 nn+ n = ⎥

⎦

⎤⎢⎣

⎡ +6

52 nn if n is odd.

Case (ii)

If n is even

In this case we collect the set of line disjoint cycles as below.

C1 = veiei+1v / i = 1, 3,……..,n-1), C2 = veiei+1v / i= 2, 4,……..,n-2, clearly |C1| = 2n

and |C2| = 2

2−n C3 = set of line disjoint

cycles in Kn . C4 = veiviv / ( ni ≤≤1 ) , We prove |C 3 | = 6

)2( −nn. Maximum number of line disjoint cycles are extracted

from Kn using the following steps.

Step 1:

Extract the line disjoint cycles ci = eiei+1ei+2ei ( i = 1, 3, 5,…….n-1). Clearly c1, c2,…………,cn-1 are line disjoint cycles.

Thus we got 2n

line disjoint cycles.

Step 2:

Delete the edges ei⎟⎠⎞

⎜⎝⎛ +ine

2

(i = 1, 2,………,2n

) from Kn.

Step 3:

Extract ⎥⎦

⎤⎢⎣

⎡ −6

52 nnline disjoint 3–cycles from Kn – ei

⎟⎠⎞

⎜⎝⎛ +ine

2

(i= 1, 2,………, 2n

) .


58

Therefore 2n

+ 6

52 nn − = ⎥⎦

⎤⎢⎣⎡ −

6)2(nn

. Since ei⎟⎠⎞

⎜⎝⎛ +ine

2

(i = 1, 2,……… ,2n

) are mutually non adjacent edges in T(K1, n). Let C 5

= v ei⎟⎠⎞

⎜⎝⎛ +ine

2

v / (i = 1, 2,……… ,2n

). The cycles in C 5 are line disjoint. The cycles in C 3 and C 5 are line disjoint and also the

cycles in C 3 and C 4 are line disjoint . Since |C 5 | ≤ |C 4 |. Therefore maximum number of line disjoint cycles in T(K1, n) ,

CM[T(K1,n)] = |C 3 | + |C 4 | = ⎥⎦⎤

⎢⎣⎡ −

6)2(nn

+ n = ⎥⎦⎤

⎢⎣⎡ +

6)4(nn

.

Acknowledgement The present version of this paper owes much to the referees for their comments, suggestions and remarks that have resulted in the improvement of this paper. References Bondy J.A. and Murty U.S.R. 1976. Graph theory with Applications. London: MacMillan Michalak D., 1981. On middle and total graphs with coarseness number equal 1, Springer Verlag Graph Theory, Lagow

proceedings, Berlin Heidelberg, New York, Tokyo, 139 -150. Harary F., 1969. Graph Theory, Narosa Publishing home. Chatrand G., Geller D. and Hedetniemi S., 1971. Graphs with forbidden Subgraphs, Journal of Combinatorial Theory, Vol. 10, pp.

12-41 Simões Pereira J.M.S., 1972. A note on the cycle multiplicity of line-graphs and total graphs, Journal of Combinatorial Theory,

Vol. 12, No. 2, pp. 194-200. Biographical notes M.M. Akbar Ali. is working as an Assistant Professor in the Department of Mathematics, Sri Shakthi Institute of Engineering and Technology, Coimbatore, India. He received his M.Phil degree in Graph Theory from Bharathiar University in 2004 and his M.Sc from Government Arts College, Coimbatore and currently pursuing PhD from Bharathiar University, Coimbatore. He has Published 16 papers in National and International Journals/Proceedings. He has participated in 15 National and International Conferences and presented more than 10 papers. Won the Best Paper Award at the International Conference on Mathematics and Computer Science (ICMCS 2007), March 1-3, 2007, Loyola College, Chennai, India. Area of interest includes Graph Colorings and Decomposition of graphs. Dr. S. Panayappan has 25 years of research experience and completed 5 research projects. He is presently serving as Principal investigator of University Grant Commission's Major project at Government Arts College Coimbatore. 45 M.Phil and 5 PhD scholars have been awarded the degree under his supervision and he is guiding 8 PhD scholars at present. He has Published 25 papers in National and International Journals/Proceedings. He has organized 3 conferences, including one graph theory conference. Area of interest includes Graph theory and Operator Theory. Received February 2010 Accepted March 2010 Final acceptance in revised form April 2010

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A hybridized K-means clustering approach for high dimensional dataset

Rajashree Dash1, Debahuti Mishra*, Amiya Kumar Rath2, Milu Acharya3

1,*,3Institute of Technical Education and Research, Bhubaneswar, Orissa, INDIA

2 Director, College of Engineering Bhubaneswar, Orissa, INDIA *Corresponding Authors’ e-mails: [email protected], [email protected] ,[email protected], [email protected]

Abstract Due to incredible growth of high dimensional dataset, conventional data base querying methods are inadequate to extract useful information, so researchers nowadays is forced to develop new techniques to meet the raised requirements. Such large expression data gives rise to a number of new computational challenges not only due to the increase in number of data objects but also due to the increase in number of features/attributes. Hence, to improve the efficiency and accuracy of mining task on high dimensional data, the data must be preprocessed by an efficient dimensionality reduction method. Recently cluster analysis is a popularly used data analysis method in number of areas. K-means is a well known partitioning based clustering technique that attempts to find a user specified number of clusters represented by their centroids. But its output is quite sensitive to initial positions of cluster centers. Again, the number of distance calculations increases exponentially with the increase of the dimensionality of the data. Hence, in this paper we proposed to use the Principal Component Analysis (PCA) method as a first phase for K-means clustering which will simplify the analysis and visualization of multi dimensional data set. Here also, we have proposed a new method to find the initial centroids to make the algorithm more effective and efficient. By comparing the result of original and new approach, it was found that the results obtained are more accurate, easy to understand and above all the time taken to process the data was substantially reduced. Keywords: Cluster analysis, K-means Algorithm, Dimensionality Reduction, Principal Component Analysis, Hybridized K-means algorithm

1. Introduction Data mining is a convenient way of extracting patterns, which represents knowledge implicitly stored in large data sets and focuses on issues relating to their feasibility, usefulness, effectiveness and scalability. It can be viewed as an essential step in the process of knowledge discovery. Data are normally preprocessed through data cleaning, data integration, data selection, and data transformation and prepared for the mining task. Data mining can be performed on various types of databases and information repositories, but the kind of patterns to be found are specified by various data mining functionalities like class/concept description, association, correlation analysis, classification, prediction, cluster analysis etc. Cluster analysis is one of the major data analysis methods widely used for many practical applications in emerging areas. Clustering is the process of finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups. A good clustering method will produce high quality clusters with high intra-cluster similarity and low inter-cluster similarity. The quality of a clustering result depends on both the similarity measure used by the method and its implementation and also by its ability to discover some or all of the hidden patterns. A good survey on clustering methods is found in Xu et al. (2005). K-means is a commonly used partitioning based clustering technique that tries to find a user specified number of clusters (k), which are represented by their centroids, by minimizing the square error function. Although K-means is simple and can be used for a wide variety of data types, it is quite sensitive to initial positions of cluster centers. There are two simple approaches to cluster center initialization i.e. either to select the initial values randomly, or to choose the first k samples of the data points. As an alternative, different sets of initial values are chosen (out of the data points) and the set, which is closest to optimal, is chosen. However, testing different initial sets is considered impracticable criteria, especially for large number of clusters Ismail et al (1989). Therefore, different methods have been proposed in literature by Pena et al. (1999). Again the computational complexity


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of original K-means algorithm is very high, especially for large data sets. In addition the number of distance calculations increases exponentially with the increase of the dimensionality of the data. When the dimensionality increases usually, only a small number of dimensions are relevant to certain clusters, but data in the irrelevant dimensions may produce much noise and mask the real clusters to be discovered. Moreover when dimensionality increases, data usually become increasingly sparse, due to which data points located at different dimensions can be considered as all equally distanced and the distance measure, which, essentially for cluster analysis, becomes meaningless. Hence, attribute reduction or dimensionality reduction is an essential data-preprocessing task for cluster analysis of datasets having a large no. of features/attributes.

Dimensionality reduction specified by Maaten et al. (2007) and Davy et al. (2007) is the transformation of high-dimensional data into a meaningful representation of reduced dimensionality that corresponds to the intrinsic dimensionality of the data. It falls into two categories i.e. Feature Selection (FS) and Feature Reduction (FR). Feature Selection algorithm aims at finding out a subset of the most representative features according to some objective function in discrete space. The algorithms of FS are always greedy. Thus, they sometimes cannot even find the optimal solution in the discrete space. Feature Extraction/ Feature Reduction algorithms aim to extract features by projecting the original high-dimensional data into a lower-dimensional space through algebraic transformations. It finds the optimal solution of a problem in a continuous space, but the computational complexity is more comparative to feature selection algorithm. Various types of feature reduction methods have been developed. PCA is a commonly used feature reduction method in terms of minimizing the reconstruction error. Traditional K-means algorithm for cluster analysis developed for low dimensional data, often do not work well for high dimensional data like microarray gene expression data and the results may not be accurate most of the time due to noise and outliers associated with original data. Also the computational complexity increases rapidly as the dimension increases. Hence, to improve the efficiency, we proposed a method to apply PCA on original data set, so that the correlated variables exist in the original dataset would be transformed to possibly uncorrelated variables, which are reduced in size. Before applying PCA the dataset needs to be normalized, so that any attribute with larger domain will not dominate attributes with smaller domain. The resulting reduced data set obtained from the application of PCA will be applied to a K-means clustering algorithm. Here also we have proposed a new method to find the initial centroids to make the algorithm more effective and efficient. The main advantage of this approach stems from the fact that this framework is able to obtain better clustering with reduced complexity and also provides better accuracy and efficiency for high dimensional datasets. Section 1 of the paper deals with the introductory concepts of clustering, K-means clustering, its limitations, the need of dimensionality reduction for clustering and the goal of the paper. Some recent related works and other preliminaries on K-means algorithm, dimensionality reduction methods and some concepts of PCA have been discussed in section 2. Section 3 describes our new proposed algorithm for K-means clustering. Section 4 describes our approach in various steps with experimental activities and corresponding result discussion followed by conclusion in Section 5.

2. Related Work

Several attempts were made by researchers to improve the effectiveness and efficiency of the K-means algorithm. Yuan et al. (2004) proposed a systematic method for finding the initial centroids. However, Yuan’s method does not suggest any improvement to the time complexity of the K-means algorithm. Belal et al. (2005) proposed a new method for cluster initialization based on finding a set of medians extracted from a dimension with maximum variance. Zoubi et al. (2008) proposed a new strategy to accelerate K-means clustering by avoiding unnecessary distance calculations through the partial distance logic. Fahim et al. (2009) proposed a method to select a good initial solution by partitioning dataset into blocks and applying K-means to each block. But here the time complexity is slightly more. . Though the above algorithms can help finding good initial centers for some extent, they are quite complex and some use the K-means algorithm as part of their algorithms, which still need to use the random method for cluster center initialization. Deelers et al. (2007) has proposed an enhancing K-means algorithm based on the data partitioning algorithm used for color quantization. The algorithm performs data partitioning along the data axis with the highest variance. Nazeer et al. (2009) proposed an enhanced K-means algorithm, which combines a systematic method for finding initial centroids and an efficient way for assigning data points to cluster. This method ensures the entire process of clustering in O (n2) time without sacrificing the accuracy of clusters. Similarly Xu et al. (2009) specify a novel initialization scheme to select initial cluster centers based on reverse nearest neighbor search. But all the above methods do not work well for high dimensional data sets. Yeung et al. (2000) presented an empirical study on principal component analysis for clustering gene expression data. But here the initial centroids are chosen randomly. Chao et al. (2005) also proposed a method for dimension reduction for microarray data analysis using Locally Linear Embedding. 2.1 K-means Clustering Algorithm The K-means algorithm is one of the partitioning based, nonhierarchical clustering methods. Given a set of numeric objects X and an integer number k, the K-means algorithm searches for a partition of X into k clusters that minimizes the within groups sum of squared errors. The K-means algorithm starts by initializing the k cluster centers. The input data points are then allocated to one of the existing clusters according to the square of the Euclidean distance from the clusters, choosing the closest. The mean (centroid) of each cluster is then computed so as to update the cluster center. This update occurs as a result of the change in the


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membership of each cluster. The processes of re-assigning the input vectors and the update of the cluster centers is repeated until no more change in the value of any of the cluster centers. The steps of the K-means algorithm are written below:

1. Initialization: choose randomly K input vectors (data points) to initialize the clusters. 2. Nearest-neighbor search: for each input vector, find the cluster center that is closest, and assign that input vector to the

corresponding cluster. 3. Mean update: update the cluster centers in each cluster using the mean (centroid) of the input vectors assigned to that

cluster. 4. Stopping rule: repeat steps 2 and 3 until no more change in the value of the means.

2.2 Principal Component Analysis (PCA) Principal Component Analysis by Valarmathie et al. (2009) and Yan et al. (2006) is an unsupervised Feature Reduction method for projecting high dimensional data into a new lower dimensional representation of the data that describes as much of the variance in the data as possible with minimum reconstruction error. PCA is mathematically defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. It transforms a number of possibly correlated variables into a smaller number of uncorrelated variables called principal components. Hence, PCA is a statistical technique for determining key variables in a high dimensional data set that explain the differences in the observations and can be used to simplify the analysis and visualization of high dimensional data set, without much loss of information. 2.3 Principal Component (PC) Technically, a principal component can be defined as a linear combination of optimally weighted observed variables which maximize the variance of the linear combination and which have zero covariance with the previous PCs. The first component extracted in a principal component analysis accounts for a maximal amount of total variance in the observed variables. The second component extracted will account for a maximal amount of variance in the data set that was not accounted for by the first component and it will be uncorrelated with the first component. The remaining components that are extracted in the analysis display the same two characteristics: each component accounts for a maximal amount of variance in the observed variables that was not accounted for by the preceding components, and is uncorrelated with all of the preceding components. When the principal component analysis will complete, the resulting components will display varying degrees of correlation with the observed variables, but are completely uncorrelated with one another.

PCs are calculated using the Eigen value decomposition of a data covariance matrix/ correlation matrix or singular value decomposition of a data matrix, usually after mean centering the data for each attribute. Covariance matrix is preferred when the variances of variables are very high compared to correlation. It would be better to choose the type of correlation when the variables are of different types. Similarly the SVD method is used for numerical accuracy. 2.4 Elimination Methods of Unnecessary PCs The transformation of the dataset to the new principal component axis produces the number of PCs equivalent to the no. of original variables. But for many datasets, the 1st several PCs explain the most of the variances, so the rest can be eliminated with minimal loss of information. The various criteria used to determine how many PCs should be retained for the interpretation is as follows:

Using Scree Diagram plots the variances in percentage corresponding to the PCs, which will automatically eliminate the PCs with very low variances.

Fixing a threshold value of variance, so that PCs having variance more than the given threshold value will be retained rejecting others.

Eliminate PCs whose Eigen values are smaller than a fraction of the mean Eigen value. 3. Proposed Hybridized K-means Clustering Algorithm As original K-means clustering algorithm often does not work well for high dimension, hence, to improve the efficiency, we proposed to apply PCA on original data set, to obtain a reduced dataset containing possibly uncorrelated variables. Then the resulting reduced data set will be applied to the K-means clustering algorithm to determine the precise no. of clusters. As quality of the final clusters heavily depends on the selection of the initial centroids, here we proposed a new method to choose such data objects as initial centroids whose squared Euclidian distance is maximum among all the data objects, to make the algorithm more effective and efficient.


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The proposed model is illustrated in Figure 1. The steps of the hybridized k-means clustering algorithm are as follows. Input: X = d1, d2,……..,dn // set of n data items. K // Number of desired clusters. An array Cen [ ] having size k initially being empty. Output: A set of k clusters // Phase-1: Apply PCA to reduce the dimension of the data set

1. Organize the dataset in a matrix X. 2. Normalize the data set using Z-score. 3. Calculate the singular value decomposition of the data matrix. 4. Calculate the variance using the diagonal elements of D. 5. Sort variances in decreasing order. 6. Choose the p principal components from V with largest variances. 7. Form the transformation matrix W consisting of those p PCs. 8. Find the reduced projected dataset Y in a new coordinate axis by applying W to X.

//Phase-2: Find the initial centroids

9. Set m=1. 10. Compute the distance between each data points in the set Y. 11. Choose the two data points yi and yj such that distance (yi, yj ) is maximum. 12. Cen[m] = yi ; Cen[m+1] = yj ; m=m+2 ; 13. Remove the two objects yi , yj from Y. 14. While (m <= k)

1. Find the distance of each object in Y to Cen[i], for i = 1 to m-1. 2. Find the average of all the distances to the centroid for each object in Y. 3. Choose the data object yo having maximum average distance from previous centroids. 4. Cen[m] = yo ; m = m+1; 5. Remove the object yo from Y.

// Phase-3: Apply the K-means clustering with the initial centroids given in array Cen.

15. For each data point, in set Y, find the nearest cluster center from list Cen that is closest and assign that data point to the corresponding cluster.

16. Update the cluster centers in each cluster using the mean of the data points, which are assigned to that cluster. 17. Repeat the steps 15 and 16 until there are no more changes in the values of the centroids.

4. Experimental Activities and Result Discussion Initially, we evaluated the proposed algorithm on a synthetic dataset with 15 data objects having 10 attributes as shown in table 1. Then three datasets, Pima Indian Diabetes data set, Breast Cancer data set and SPECTF Heart data set, taken from the UCI

TUDVX =

Normalize the data set using Z-score

method.

Apply the SVD method of PCA to get

PCs.

Eliminate the unnecessary

PCs.

Get the cluster index of each object of the

reduced dataset by k-means clustering.

Derive the initial centroids for k-

means clustering.

Find the reduced projected data set

using reduced PCs.

Figure 1: A Hybridized model for K-means Clustering


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machine learning repository are used for testing the accuracy and efficiency of the hybridized algorithm. Here the Sum of Squared Error (SSE), representing distances between data points and their cluster centers have used to measure the clustering quality. Among two solutions for a given dataset, the smaller the value of SSE and higher the accuracy, the better the solution. Step 1: Normalizing the original data set Using the Normalization process, the initial data values are scaled so as to fall within a small-specified range. An attribute value V of an attribute A is normalized to V’ using Z-Score as follows: V’=(V-mean(A))/std(A) It performs two things i.e. data centering, which reduces the square mean error of approximating the input data and data scaling, which standardizes the variables to have unit variance before the analysis takes place. This normalization prevents certain features to dominate the analysis because of their large numerical values.

Table 1: The original data matrix X with 15 data objects having 10 attribute values.

Figure 2: Plotting of data along with the normalized data.

Step 2: Calculating the PCs using Singular Value Decomposition of the normalized data matrix Applying the steps given in phase 1 of the new proposed algorithm, the no. of PCs obtained is same with the no. of original variables. To eliminate the weaker components from this PC set we have calculated the corresponding variance, percentage of variance and cumulative variances in percentage, which is shown in Table 2. Then we have considered the PCs having variances less than the mean variance, ignoring the others. The reduced PCs are shown in Table 3. Step 3: Finding the reduced data set using the reduced PCs The transformation matrix with reduced PCs is formed and this transformation matrix is applied to the normalized data set to produce the new reduced projected dataset, which can be used for further data analysis. The reduced data set is shown in Table 4. We have also applied the PCA on three biological dataset and the reduced no. of attributes obtained for each dataset is shown in Figure 3. Step 4: Comparison of efficiency and accuracy of the original k-means clustering and proposed algorithm. The clustering results shown in Figure 4 and 5 by applying the standard k-means clustering to the original synthetic dataset and the proposed method to the reduced dataset are approximately same, but the time taken for clustering will be reduced due to less number of attributes. Again we compared the clustering results obtained by the k-means algorithm using random initial centers and initial centers derived by the proposed algorithm over 4 datasets with original dimension and with reduced dimension based on the sum of squared error distances (SSE), which is shown in Figure 6 and 7. The clustering results of k-means using random initial centers are the average results over 10 runs since each run gives different results. The SSE value obtained and the time taken in ms for 4 datasets with original k-means and new proposed algorithm is given in Table 5.

v1 v2 v3 v4 v5 v6 v7 v8 v9 v10

Data1 1 5 1 1 1 2 1 3 1 1 Data2 2 5 4 4 5 7 10 3 2 1 Data3 3 3 1 1 1 2 2 3 1 1 Data4 4 6 8 8 1 3 4 3 7 1 Data5 5 4 1 1 3 2 1 3 1 1 Data6 6 8 10 10 8 7 10 9 7 1 Data7 7 1 1 1 1 2 10 3 1 1 Data8 8 2 1 2 1 2 1 3 1 1 Data9 9 2 1 1 1 2 1 1 1 5

Data10 10 4 2 1 1 2 1 2 1 1 Data11 11 1 1 1 1 1 1 3 1 1 Data12 12 2 1 1 1 2 1 2 1 1 Data13 13 2 1 1 1 2 1 2 1 1 Data14 14 5 3 3 3 2 3 4 4 1 Data15 15 1 1 1 1 2 3 3 1 1


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Table 2: The variances, variances in percentage, cumulative Table 3: Reduced PCs having variance

variances in percentage corresponding to the PCs . greater than the mean variance.

Table 4: The reduced dataset containing 3 attributes

Figure 3: Plotting of original and reduced no. of attributes for Synthetic, Pima Indian Diabetes, Breast Cancer and SPECTF Heart datasets.

Figure 4: Clustering with original dataset by Figure 5: Clustering with reduced dataset by standard K-means algorithm. proposed algorithm.

Variances Variances in %

Cumulative variances

in% pc1 6.210578 62.10578 62.10578 pc2 1.054022 10.54022 72.646 pc3 1.016014 10.16014 82.80614 pc4 0.86546 8.654603 91.46075 pc5 0.455458 4.554576 96.01532 pc6 0.24649 2.464901 98.48022 pc7 0.108508 1.085079 99.5653 pc8 0.030248 0.302483 99.86779 pc9 0.010731 0.10731 99.9751

pc10 0.00249 0.024904 100

Pc1 Pc2 Pc3 0.161104 -0.70692 0.222666 -0.34575 0.06143 0.104589 -0.37811 -0.12852 0.225285 -0.37785 -0.12565 0.21828 -0.34923 0.061232 -0.09679 -0.34712 0.275694 -0.1149 -0.28668 0.21192 -0.29648 -0.34062 -0.24545 -0.14089 -0.34594 -0.26379 0.318984 0.091787 0.457921 0.780392

v1 v2 v3 data1 0.536985 1.045863 -0.56448 data2 -2.76212 1.914677 -1.15055 data3 0.858597 0.73036 -0.64746 data4 -2.78991 -0.43292 1.310152 data5 0.502757 0.444459 -0.51029 data6 -7.19228 -0.525 0.259885 data7 0.691562 0.513194 -1.21072 data8 1.149898 -0.19297 -0.28827 data9 2.060598 1.744708 2.866126

data10 1.08491 -0.31268 -0.00678 data11 1.749453 -0.8052 -0.20218 data12 1.620462 -0.6419 -0.08667 data13 1.656486 -0.79997 -0.03688 data14 -0.70786 -1.51675 0.500656 data15 1.540447 -1.16586 -0.23254


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Table 5: SSE values obtained and time taken in ms with original k-means and new proposed algorithm

Dataset No of Instances

Original K-means Algorithm

SSE Time taken(ms)

Proposed Algorithm

SSE Time taken(ms) Synthetic 15 608.446 78 47.8 65

Pima Indian Diabetes

50 59255 158 182.39 122

Breast Cancer 80 29253 167 165.94 131 SPECTF Heart 40 97075 145 996.8 112

The above results show that the new algorithm provides better SSE values for all the cases. Hence, in this regard it increases the efficiency of the original k-means algorithm. The accuracy of clustering determined by comparing the clusters obtained by the experiments with the available clusters for three data sets in UCI data set is shown in Figure 8. In all the cases the proposed algorithm provides better accuracy compared to the original k-means algorithm.

Figure 8: Clustering accuracy of PID, Breast cancer, SPECTF Heart datasets.

5. Conclusion In this paper a hybridized K-means algorithm has been proposed which combines the steps of dimensionality reduction through PCA, a novel initialization approach of cluster centers and the steps of assigning data points to appropriate clusters. Using the proposed algorithm a given data set was partitioned in to k clusters in such a way that the sum of the total clustering errors for all clusters was reduced as much as possible while inter distances between clusters are maintained to be as large as possible. The experimental results show that the proposed algorithm provides better efficiency and accuracy comparison to original k-means algorithm with reduced time. Though the proposed method gave better quality results in all cases, over random initialization methods, still there is a limitation associated with this, i.e. the number of clusters (k) is required to be given as input. Again the method to find the initial centroids may not be reliable for vary large dataset. Evolving some statistical methods to compute the value of k, depending on the data distribution is suggested for future research. Methods for refining the computation of initial centroids are worth investigating.

Figure 6: SSE results on synthetic, PID, Breast cancer, SPECTF heart datasets with original dimension.

Figure 7: SSE results on synthetic, PID, Breast cancer, SPECTF heart datasets with reduced dimension.


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References Belal M. and Daoud A., 2005. A new algorithm for cluster initialization, World Academy of Science, Engineering and Technology,

Vol. 4, pp. 74-76. Chao Shi and Chen Lihui, 2005. Feature dimension reduction for microarray data analysis using locally linear embedding, 3rd Asia

Pacific Bioinformatics Conference, pp. 211-217. Davy Michael and Luz Saturnino, 2007. Dimensionality reduction for active learning with nearest neighbour classifier in text

categorization problems, Sixth International Conference on Machine Learning and Applications, pp. 292-297. Deelers S. and Auwatanamongkol S., 2007. Enhancing K-means algorithm with initial cluster centers derived from data

partitioning along the data axis with the highest variance, International Journal of Computer Science, Vol. 2, No. 4, pp. 247-252.

Fahim A. M., Salem A. M., Torkey F. A., Saake G. and Ramadan M. A., 2009. An efficient k-means with good initial starting points, Georgian Electronic Scientific Journal: Computer Science and Telecommunications, Vol. 2, No. 19, pp. 47-57.

Ismail M. and Kamel M., 1989. Multidimensional data clustering utilization hybrid search strategies, Pattern Recognition, Vol. 22, No. 1, pp.75-89.

Maaten L.J.P., Postma E.O. and Herik H.J. van den, 2007. Dimensionality reduction: A comparative review”, Tech. rep. University of Maastricht.

Nazeer K. A. Abdul and Sebastian M.P., 2009. Improving the accuracy and efficiency of the k-means clustering algorithm, Proceedings of the World Congress on Engineering, Vol. 1, pp. 308-312.

Pena J. M., Lozano J. A. and Larranaga P., 1999. An empirical comparison of four initialization methods for the k-means algorithm, Pattern Recognition Letters, Vol. 20, No. 10, pp. 1027-1040.

Valarmathie P., Srinath M. and Dinakaran K., 2009. An increased performance of clustering high dimensional data through dimensionality reduction technique, Journal of Theoretical and Applied Information Technology, Vol. 13, pp. 271-273.

Xu R. and Wunsch D., 2005. Survey of clustering algorithms, IEEE Trans. Neural Networks, Vol. 16, No. 3, pp. 645-678. Xu Junling, Xu Baowen, Zhang Weifeng, Zhang Wei and Hou Jun, 2009. Stable initialization scheme for K-means clustering,

Wuhan University Journal of National Sciences, Vol. 14, No. 1, pp. 24-28. Yan Jun, Zhang Benyu, Liu Ning, Yan Shuicheng, Cheng Qiansheng, Fan Weiguo, Yang Qiang, Xi Wensi, and Chen Zheng,

2006. Effective and efficient dimensionality reduction for large-scale and streaming data preprocessing, IEEE transactions on Knowledge and Data Engineering, Vol. 18, No. 3, pp. 320-333.

Yeung Ka Yee and Ruzzo Walter L., 2000. An empirical study on principal component analysis for clustering gene expression Data”,Tech. Report, University of Washington.

Yuan F., Meng Z. H, Zhang H. X and Dong C. R, 2004. A new algorithm to get the initial centroids, Proc. of the 3rd International Conference on Machine Learning and Cybernetics, pp. 1191–1193.

Zhang Z., Zhang J. and Xue H., 2008. Improved K-means clustering algorithm, Proceedings of the Congress on Image and Signal Processing, Vol. 5, No. 5, pp. 162-172.

Zoubi M. B. Al., Hudaib A., Huneiti A. and Hammo B., 2008. New efficient strategy to accelerate k-means clustering algorithm”, American Journal of Applied Sciences, Vol. 5, No. 9, pp. 1247-1250.

Biographical notes Rajashree Dash has completed her B.Tech in Computer Sc. & Engineering from KIIT University. Now she is perusing her M.Tech in Computer Sc. & Engg at Institute of Technical Education & Research (ITER) under Siksh `O` Anusandhan University, Bhubaneswar. Her research areas include Data mining, Computer Graphics etc. Debahuti Mishra is an Assistant Professor and research scholar in the department of Computer Sc. & Engg, Institute of Technical Education & Research (ITER) under Siksh `O` Anusandhan University, Bhubaneswar. She received her Masters degree from KIIT University, Bhubaneswar. Her research areas include Data mining, Bio-informatics Software Engineering, Soft computing . She is an author of a book Aotumata Theory and Computation by Sun India Publication (2008). Dr.Amiya Kumar Rath obtained Ph.D in Computer Science in the year 2005 from Utkal University for the work in the field of Embedded system. Presently working with College of Engineering Bhubaneswar (CEB) as Professor of Computer Science & Engg. Cum Director (A&R) and is actively engaged in conducting Academic, Research and development programs in the field of Computer Science and IT Engg. Contributed more than 30 research level papers to many national and International journals. and conferences Besides this, published 4 books by reputed publishers. Having research interests include Embedded System, Adhoc Network,Sensor Network ,Power Minimization, Biclustering, Evolutionary Computation and Data Mining. Dr. Milu Acharya obtained her Ph.D at Utkal University. She is a Professor in Department of Computer Applications at Institute of Technical Education and Research (ITER), Bhubaneswar. She has contributed more than 20 research level papers to many national and International journals and conferences Besides this, published 3 books by reputed publishers. Her research interests include Biclustering,Data Mining , Evaluation of Integrals of analytic Functions , Numerical Analysis , Complex Analysis , Simulation and Decision Theory. Received February 2010 Accepted March 2010 Final acceptance in revised form April 2010

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Effect of magnetic field on the blood flow in artery having multiple stenosis: a numerical study

Gaurav Varshney1*, V.K. Katiyar2, Sushil Kumar3

1*Department of Mathematics, Government Degree College Karanprayag,Chamoli, INDIA

2 Department of Mathematics,Indian Institute of Technology Roorkee, INDIA Department of Mathematics,S.V. National Institute of Technology Surat, INDIA

*Corresponding Author: e-mail: [email protected], Tel +91-9456717427, Fax.+91-1363-244129

Abstract In the present study a mathematical model for the blood flow in stenosed artery in the presence of magnetic field is proposed. The laminar, incompressible, fully developed, non-Newtonian flow of blood in an artery having multiple stenosis is numerically studied under the action of transverse magnetic field. The governing equations are transformed by using a radial transformation and the numerical results are obtained using a finite difference technique. Effect of overlapping stenosis and externally applied magnetic field in the blood flow is discussed with the help of graph. All the flow characteristics are found to be affected by the presence of multiple stenosis and exposure of magnetic field of different intensities. Keywords: Blood flow, Magnetic Field, Overlapping Stenosis, generalized Power Law, Finite Difference Method 1. Introduction Many cardiovascular diseases, particularly atherosclerosis, have been found to be responsible for deaths in both developed and developing countries. The study of blood flow through a stenotic artery is very important because the nature of blood movement and mechanical behaviour of vessel walls are causes of many cardiovascular diseases. Most of the authors (Bali and Awasthi, 2007; Chakravarty, 1987; Deplano and Siouffi, 1987; Haldar, 1985; Liu et al., 2004; Mandal, 2005; Shalman et al., 2002) studied the pulsatile blood flow in the artery having single mild stenosis. The multiple stenosis is commonly found in the femoral and pulmonary arteries, so the problem of blood flow becomes more acute in the presence of overlapping stenosis (Chakravarty and Mandal, 1994; Ismail et al., 2007). The study of blood rheology and blood flow has several objectives such as not only understanding health and disease but also in essence, what kind of fluid it is (Buchanan et al., 2000, Hernan and Gonzalez, 2007, Ismail et al., 2007, Mandal, 2005). Some authors in the area of blood flow feel that blood can be assumed to be Newtonian in nature especially in large blood vessels such as the aorta (Katiyar and Vasarajappa, 2002; Liepsch, 2002; Liu et al., 2004; Prakash et al., 2004; Tashtoush and Magableh, 2008; Tzirtzilakis, 2005). In fact blood is a suspension of cells in plasma. The plasma which is a solution of proteins, electrolyte and other substances, is an incompressible virtually Newtonian fluid. From biomechanical point of view, blood is considered as an intelligent fluid, probably the most one in the nature, capable of adapting itself in a great extent in order to provide nutrients to the organs. It is well known that blood behaves differently when flowing in large vessels, in which Newtonian behaviour is expected and in medium and small vessels where non-Newtonian effects appear (Buchanan et al., 2000; Chakravarty and Mandal, 1994; Deplano and Siouffi, 1987; Ismail et al., 2007; Mandal, 2005). Blood can be regarded as magnetic fluid, in which red blood cells are magnetic in nature. Liquid carriers in the blood contain the magnetic suspension of the particle (Tzirtzilakis, 2005). Human body experiences magnetic fields of moderate to high intensity in many situations of day to day life. In recent times, many medical diagnostic devices especially those used in diagnosing cardiovascular disease make use of magnetic fields. It is known from the magneto-hydrodynamics that when a stationary, transverse magnetic field is applied externally to a moving electrically conducting fluid, electrical currents are induced in the fluid. The interaction between these induced currents and the

Varshney et al. / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 67-82

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applied magnetic field produces a body forces (known as the Lorentz force) which tends to retard the movement of blood (Sud and Sekhon, 1989) Chester (1957) analyzed the interaction of the magnetic field with the electric current to propose the pressure of body forces in stokes problem for the discussion of the motion of the fluids in the classical hydrodynamics. Low frequency, low intensity magnetic energy has been employed for treating chronic pain secondary to tissue ischemia and slow healing and non healing ulcers with satisfactory to excellent results. This type of energy appears to affect biological process, not through heat production but through electrically induced changes in the environment of cells within the organism. Jauchem (1997) studied the effects of low frequency electromagnetic energy on the peripheral blood circulation and concluded that low frequency, low intensity magnetic field increased blood flow in the great majority. Hypertension is one flight mechanics presumably exposed to radio frequency radiation at a level of 38 times above the permissible exposure limit. There are a number of emerging technologies involving the use of Electro-Magnetic frequencies including new types of cellular telephones, magnetically levitated trains and superconducting magnetic energy storage. The possible effects of these particular Electro-Magnetic frequencies on health have not been studied directly. Magnetic Resonance Imaging is a tool to study the blood flow phenomena in which magnetic field of large intensity is applied on the body. Although existing guidelines on Magnetic Resonance Imaging magnetic fields have been adequate to preclude any known biological problem to date, the Magnetic Resonance Imaging industry would like to have greater flexibility in developing future designs. Simunic et al. used a model to simulate exposure of the human torso to switched magnetic field that would be present during Magnetic Resonance Imaging (Jauchem, 1997) Villoresi et al. (1994) proposed that when magnetic field exceed 0.05 T, it leads to real dangers connected with development of heart fibrillation and further irreversible changes. He reported a large decrease in myocardial infarction rates on Saturday, Sundays and public holidays and remarked that there were greater man made magnetic disturbances on workdays than during weekends. Myocardial infarctions could be triggered by man made magnetic fields. Time varying electromagnetic field produced by electrical currents are used to treat non-unified bone fractures. With regards to sympathetic function, time varying electromagnetic fields can modify electrical activity in the brain. This change in electrical activity of the central nervous system can inhibit muscle sympathetic nerve activity and increase skin sympathetic nerve activity (Kinouchi et al., 1996; Kuipers et al., 2007). Magnetic devices sold to patients commonly utilize static magnetic fields generated by permanent magnets and not time varying electromagnetic fields. Like time varying electromagnetic field, there is some evidence to suggest that static magnetic fields alter autonomic function in human. A 2.0 T static magnetic field can increase cardiac cycle length, which may be caused by changes within the sinus node. A 0.4 T static magnetic field can alter skin blood flow in humans, possibly caused by alteration in calcium dynamics (Bali and Awasthi, 2007; Chester, 1957; Gmitrov, 2007; Jauchem, 1997; McKay et al., 2007). Kuipers et al. (2007) studied the influence of static magnetic fields on cardiovascular and sympathetic function at rest and during physiological stress and also investigated the influence of static magnetic field on pain perception during noxious stimuli. The biological effects of Magnetic fields have often been linked to nitric oxide (NO), which is responsible for the changes in vessel diameter following magnetic field exposure. Recently magnetic fields have been shown to have positive effects on numerous human systems. For instance, it is documented that magnetic field exposure can provide analgesia, decrease healing time for fractures, increase the speed of nerve regeneration, act as a treatment for depression and provide other medical benefits (McKay et al., 2007). Magnetic force therapy could be useful for the reperfusion of ischemic tissue or during sepsis. When blood flow to a tissue becomes blocked or reduced, necrosis will eventually occur. Local exposure of a magnetic field could potentially result in blood vessel relaxation and increased blood flow. The effects of magnetism on blood vessels and the cardiovascular system are very interesting. There is still no experiment that shows the effect of quite magnetic field on blood circulation. In recent years some studies (Katiyar and Basavarajappa, 2002; Kinouchi et al., 1996; Sud and Sekhon, 1989; Tashtoush and Magableh, 2008; Tzirtzilakis, 2005) have been reported on the analysis of blood flow through single arteries in the presence of externally applied magnetic field. However there are very few studies focusing on the effect of magnetic field in the stenotic artery. Considering the influence of magnetic field on the stenotic artery, in this study, we look at the effect of transverse magnetic field and multi-stenosis on the blood flow in blood vessel. It is assumed that the arterial segment is cylindrical tube with time dependent multi-stenosis and the flowing blood is characterized by generalized Power-law model. Governing equations are solved by using suitable finite difference method. The effect of externally applied magnetic field on velocity, flow rate, flow resistance and wall shear stress is studied 2. Formulation of the problem 2.1 Mathematical model It is well known that blood acts as an electrically conducting fluid. When solid material as Fe is moving in a magnetic field, it experiences an electromotive force and as a result an electric current may flow. If we apply a magnetic field on an electrically


69

conducting fluid like blood, an electromagnetic force will be produced due to the interaction of current with magnetic field. The electromotive force is proportional to the speed of motion and the magnetic flux intensity B (Tashtoush and Magableh, 2008) Maxwells relations are performed as follows: The current density J is expressed by

( )J E v B= σ + × (1) Where E is the electric field intensity, σ is the electrical conductivity, B is the magnetic flux intensity and v is the velocity vector. In the momentum equation, the electromagnetic force Fm is included and is defined as

( )mF J B v B B= × = σ × × (2) The Navier Stokes equations for blood flow including a Lorentz force is

( ) 2v v v v B B p vt

∂⎛ ⎞ρ + ⋅∇ = σ × × −∇ +μ∇⎜ ⎟∂⎝ ⎠ (3)

Where ρ is the density, μ is viscosity and p is the pressure. The artery having stenosis is considered as a cylindrical elastic tube of the circular cross section containing an incompressible non-Newtonion fluid. The blood flow is modeled to be laminar, unsteady, two-dimensional, axi-symmetric and fully developed. Blood is characterized by generalized Power-law model. The flow is considered to take place under the influence of externally applied magnetic field in axial direction. Under these assumptions, the governing equations may be written in the cylindrical coordinates system (r, z, θ) as Equation of continuity

r r zv v v0

r r z∂ ∂

+ + =∂ ∂

(4)

Equation of axial momentum

( )2

z z zr z rz zz z

v v v 1 p 1 1 Bv v r ( ) vt r z z r r z

∂ ∂ ∂ ∂ ∂ ∂ σ⎡ ⎤+ + = − σ + σ −⎢ ⎥∂ ∂ ∂ ρ ∂ ρ ∂ ∂ ρ⎣ ⎦ (5)

Equation of radial momentum

( ) ( )r r rr z rr rz

v v v 1 p 1 1v v rt r z r r r z

∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤+ + = − − σ + σ⎢ ⎥∂ ∂ ∂ ρ ∂ ρ ∂ ∂⎣ ⎦ (6)

Where the relationship between the shear stress and the shear rate in case of two dimensional motions are given by (Gerrard and Taylor, 1977)

n 11m ( · )2

−⎡ ⎤⎢ ⎥σ = − γ γ γ⎢ ⎥⎣ ⎦

& & (7)

With 22 2 2

r r z r zv v v v v1 ( · ) 22 r r z z r

⎧ ⎫ ⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪γ γ = + + + +⎨ ⎬ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭& &

Here σ denotes the stress tensor, γ& , the symmetric rate of deformation tensor, m and n are the respective consistency and fluid behavior parameters

n 11/ 22 2 2 2r r z r r

zzv v v v v

2 mr r z z r

−⎧ ⎫⎡ ⎤∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞σ = − + + + +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦

⎩ ⎭

zvz

∂⎛ ⎞⎜ ⎟∂⎝ ⎠

,

n 11/ 22 2 2 2r r z r z z r

rzv v v v v v v

mr r z z r r z

−⎧ ⎫⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞σ = − + + + + +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦

⎩ ⎭

and n 11/ 22 2 2 2

r r z r z rrr

v v v v v v2 m

r r z z r r

−⎧ ⎫⎡ ⎤∂ ∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞σ = − + + + +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦

⎩ ⎭

(8)

Here ( )zv r, z, t and ( )rv r, z, t represents the axial and the radial velocity components respectly p is the pressure and ρ is the


70

density of blood. The pressure gradient p / z∂ ∂ appearing in equation (5) , is given by

0 1p A A Cos t, t 0.z∂

− = + ω >∂

(9)

Where 0A → constant amplitude of the pressure gradient

1A → amplitude of the pulsatile component giving rise to systolic and diastolic pressure and p2 fω = π , where pf is the pulse frequency . 2.2 Boundary conditions (1) There is no radial flow along the axis of the artery and the axial velocity gradient of the streaming blood may be assumed to

be equal to zero i.e., there is no shear rate of fluid along the axis.

( )rv r, z, t 0= , zv (r,z, t) 0r

∂=

∂and rz 0σ = at r 0= (10)

(2) The velocity on the artery wall is taken as

( )rRv r, z, t t

∂=∂

and ( )zv r, z, t 0= at ( )r R z, t= (11)

(3) It is assumed that no flow takes places when the system is at rest. ( )rv r, z, t 0= and ( )zv r, z, t 0= at t 0= (12)

2.3 Geometry of stenosed artery The multiple stenosis is commonly found in the femoral and pulmonary arteries. The geometry of the arterial segment having multiple stenosis is mathematically given by (Chakarvarty and Mandal, 1994)

( ) ( ) ( ) ( ) ( )2 3 43 21

1

11 47 1 3la 1 a (t) z d l z d l z d l z d , d z dR z, t 32 48 3 2

a.a (t), ;otherwise

⎧ ⎫⎛ ⎞⎡ ⎤− − − − + − − − ≤ ≤ +⎪ ⎪⎜ ⎟⎢ ⎥= ⎣ ⎦⎨ ⎬⎝ ⎠⎪ ⎪⎩ ⎭

(13) where a is the unconstricted radius of stenosed artery, d is the location of stenosis and l is the length of the stenosis. The time variant parameter is given by ( ) –b t

1a t 1 – b Cos( t – 1) e ω= ω ; p2 fω = π . 3. Solution procedure 3.1 Transformation of the governing equations

Let us introduce a radial coordinate transformation rx

R(z, t)⎡ ⎤

=⎢ ⎥⎣ ⎦

, which has the effect of immobilizing the vessel wall in the

transformed coordinate x . Using this transformation, equations (4-8) together with boundary conditions (10-12) take the following form : -

r r z zv v v v1 x R 0r x xR z R x z∂ ∂ ∂ ∂

+ + − =∂ ∂ ∂ ∂

(14)

z r z zz z

xz zz xzxz z

v v v vx R x R 1 pv vt R t R R z x z z

1 1 1 x R MvxR R x z R x z

∂ ∂ ∂∂ ∂ ∂⎡ ⎤= − + − − −⎢ ⎥∂ ∂ ∂ ∂ ∂ ρ ∂⎣ ⎦∂σ ∂σ ∂σ ∂⎡ ⎤σ + + − −⎢ ⎥ρ ∂ ∂ ∂ ∂⎣ ⎦

(15)

Where 2BM σ

=ρ

Normal stress


71

n 11/ 22 2 2 2r r z z r r 2

zz

z z

v v v v v v v1 x R x R 12 mR x xR z R z x z R z r x

x vx Rz R z x

−⎧ ⎫⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞σ = − + + − + − +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦

⎩ ⎭∂ ∂∂⎛ ⎞−⎜ ⎟∂ ∂ ∂⎝ ⎠

(16)

Shear stress

n 11/ 22 2 2 2r r z z r r z

xz

z r z

v v v v v v v1 x R x R 1mR x xR z R z x z R z x r x

v v v1 x R ,R z z R z x

−⎧ ⎫⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞σ = − + + − + − +⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦

⎩ ⎭∂ ∂ ∂∂⎛ ⎞+ −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(17)

where n is the fluid behavior index parameter 3.2 Transformation of the Boundary Conditions

( )rv x, z, t 0= , zv (x,z, t) 0r

∂=

∂and xz 0σ = an x 0= (18)

( )rRv x, z, t t

∂=∂

and ( )zv x, z, t 0= on x 1= and (19)

( )rv x, z, t 0= and ( )zv x, z, t 0= at t 0= . (20) In order to get the radial velocity component, ( )rv x, z, t , we have to consider the equation (4) as

2r z zr

v v v Rx v xR x · 0x z x z

∂ ∂ ∂ ∂+ + − =

∂ ∂ ∂ ∂ (21)

Now, integrating the equation (21) with respect to x from the limits 0 x→ , we get (Mandal, 2005):

( ) ( )2r z

R Rv x, z, t x v z xz t

∂ ∂⎡ ⎤= − −⎢ ⎥∂ ∂⎣ ⎦ (22)

3.3 Discretization of the axial velocity component, ( )zv x, z, t

The discretization of axial velocity ( )zv x, z, t is written as ( )z j i kv x , z , t or ( )kz i, jv

We define

jx j. x; j 0, 1, 2 . N= Δ = … where Nx 1.0=

iz i. z; i 0, 1, 2 . M = Δ = …

( )Kt K 1 t; K 1, 2, .= − Δ = … The Finite difference scheme is used to solve the governing transformed equation by using central difference approximations for all the spatial derivatives in the following manner:

( ) ( )( )

x

K Kz zi, j 1 i, j 1z

z f

v vv Vx 2 x

+ −−∂= =

∂ Δ and

( ) ( )( )

z

K Kz zi 1, j i j, jZ

z f

v vv vx 2 z

+ −−∂= =

∂ Δ (23)

and the time derivatives are approximated by

( ) ( )K 1 Kz zi, j i, jz

v vvt 2 t

+ −∂=

∂ Δ (24)


72

Similarly, the derivatives for r zz xzv , and σ σ are

( ) ( )( )

x

K Kr ri, j 1 i, j 1r

r f

v vv vx 2 x

+ −−∂= =

∂ Δ and

( ) ( )( )

z

K Kr ri 1, j i 1, jr

r fz

v vv v2 z

+ −−∂= =

∂ Δ (25)

( ) ( )( )

x

K Kzz zzi, j 1 i, j 1zz

zz fx 2 x+ −σ − σ∂σ

= = σ∂ Δ

,

( ) ( )( )

z

K Kzz zzi 1, j i 1 , jzz

zz fz 2 z+ −σ − σ∂σ

= = σ∂ Δ

,

and ( ) ( )

( )x

K Kxz xzi, j 1 i, j 1xz

xz fx 2 x+ −σ − σ∂σ

= = σ∂ Δ

(26)

Using above discritization techniques equation (22-26) for axial velocity, equation (15) may be transformed to following difference equations:

( ) ( )( )

( )KK K

rK 1 K Ki, jj jz z zi, j i, j i, jk k K

i it t i

vx xR Rv V t . v .t zR R R

+⎧⎡ ⎤∂ ∂⎪ ⎛ ⎞ ⎛ ⎞⎢ ⎥= + Δ − +⎨ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

( )( ) ( ) ( )( ) ( )K 1

KK K Kz z z xzfx i, j tz i, jK Ki, j i, j

i i i

1 p 1 1 1v v vz x R R

+ ⎡∂⎛ ⎞− − − σ +⎢⎜ ⎟ρ ∂ ρ⎝ ⎠ ⎢⎣( ) ( )K K

xz x zz zi, j i, jf f⎡ σ ⎤ + ⎡ σ ⎤ −⎣ ⎦ ⎣ ⎦

( ) ( )K

K Kixz x z i, jK i, j

ij

x Rf M vZR

⎫⎤∂ ⎪⎛ ⎞⎡ σ ⎤ −⎥ ⎬⎜ ⎟⎣ ⎦ ∂⎝ ⎠ ⎥ ⎪⎦ ⎭ (27)

The equation (16) and (17) has their discritized form as:

( ) ( )( )( )

2 KK rK i, j

zz ri, j fxK Ki, ji j i

v12 m vR x R

⎛ ⎞⎧ ⎡⎛ ⎞⎪ ⎜ ⎟σ = +⎢⎜ ⎟⎨ ⎜ ⎟ ⎜ ⎟⎢⎪ ⎝ ⎠⎣⎩ ⎝ ⎠

+

( )( ) ( )( )z x

2KK Kjz zf fKi, j i, jii

x Rv vzR

⎛ ⎞∂⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠ +

( )( ) ( )( ) ( )( )z x

2 1/2 n 1KK KKjr z x zf fK ki, ji, j i, jii j

x R 1v v f VzR R

−⎛ ⎞∂ ⎞ ⎤ ⎫⎛ ⎞⎜ ⎟− + ⎬⎜ ⎟ ⎟ ⎥⎜ ⎟∂⎝ ⎠ ⎠ ⎦ ⎭⎝ ⎠

( )( ) ( )( )z z

kK Kjz zf fKi, j i, jii

x Rv vZR

⎛ ⎞∂⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠ (28)

and

( ) ( )( ) ( )x

2K2K rK i, j

xz ri, j fk ki, ji j i

v12 m vR x R

⎧ ⎡ ⎤⎛ ⎞⎛ ⎞⎪ ⎢ ⎥⎜ ⎟σ = − +⎜ ⎟⎨ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎪ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎩

+

( )( ) ( )( )z x

2KK Kjz zf fKi, j i, jii

x Rv vzR

⎛ ⎞∂⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠ +


73

( )( ) ( )( ) ( )( ) )z x x

j

KK K 1/ 2 n 12j Kx r z i, jf f fK Ki, j i jii i

x R 1v v vzR R

−⎛ ⎞∂ ⎫⎛ ⎞ ⎤− +⎜ ⎟ ⎬⎜ ⎟ ⎦⎜ ⎟∂⎝ ⎠ ⎭⎝ ⎠

( )( ) ( )( ) ( )( )z x x

KK K Kjr x zf f fK Ki, j i, j i, jij i

x R 1v v vzR R

⎛ ⎞∂⎛ ⎞⎜ ⎟− +⎜ ⎟⎜ ⎟∂⎝ ⎠⎝ ⎠ (29)

The boundary conditions in discretized form are as follows:

( )Kr i, jv 0= , ( ) ( )K K

z zi,1 i, 2v v ,= ( )Kxz i, 1 0,σ = ( )K

z i, N 1v + =0, ( )K

Kr i, N 1

i

Rvt+

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

and

( ) ( )1 1r zi, j i, jv 0 , v 0= = (30)

The radial velocity can be calculated from the equation (22) and its discretized form is

( ) ( )K K

K 1 K 1 2r j z ji, j i, j

i i

R Rv x v 2 xz t

+ +⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞ ⎡ ⎤= − −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎣ ⎦∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

The flow rate can be obtained by R

z0Q 2 rv dr= π∫

by using the radial coordinate transformation, the discretized form of flow rate Q is given by

( ) ( )2 1 KK K

i i z ji, j0Q 2 R x v dx= π ∫ (31)

The Resistance of flow λ can be calculated from | L( p / z) |

Q∂ ∂

λ =

Its discretized form is given by K

Ki K

i

| L( p / z) |Q

∂ ∂λ = (32)

The wall share stress is defined as

z xw

v vx z

∂ ∂⎛ ⎞τ = μ +⎜ ⎟∂ ∂⎝ ⎠

In the discretized from equation (41) can be written as

( ) ( )( ) ( )( ) ( )( )z x

K KK KKK jw z r z rf fK Ki i, ji, j i, j i ii i x 1

x1 R Rv v f v cos arctanZ zR R

=

⎡ ⎤ ⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞τ = μ + − ×⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ (33)

4. Result and discussion Numerical computations have been carried out using the following parameter values (Chakravarty, 1987, Ismail et al., 2007; Katiyar and Vasarajappa, 2002; Mandal, 2005).

03 2 2

p 0 1 0 m

a 0.08cm,L 5cm, l 1.6cm,d 1.3cm,b 0.1,m 0.1735P, 0.035P,n 0.639,

1.06g cm ,f 1.2Hz,A 10g cm s ,A 0.2A , 0.4*a, 2.4*pi,

x 0.02, z 0.1 and t 0.00001

− −

= = = = = = μ = =

ρ = = = = τ = ω =

Δ = Δ = Δ =

The result appeared to converge with an accuracy of the order ~ 10-5 when the time step was chosen to be 0.00001tΔ = . Figure 1 shows the geometry of the moving arterial wall of the time variant multiple stenosis for different taper angles. It is clear that the radius of artery increases during systole and as the diastolic phase start, the radius of the artery decreases.


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Figure 1. Radius of constricted Artery at different time The complete axial and radial velocity profile in the artery geometry is shown in Figures 2 and 3, respectively. In order to analyze the flow field insensibly along the arterial segment under study, Figure 4 exhibits the axial velocity profiles at various axial locations for M=2 and t =0.3sec. The axial velocity profile is parabolic at the upstream (z=10mm), the area without stenosis, while a flattering trend is followed at the converging section (z=15mm) and subsequently it becomes much blunter at the locations (z=24mm and z=28mm), the critical height of the stenosis than at the entry. The velocity appears to be enhanced at the diverging section (z=35mm) and finally at the end of the stenosis, the axial velocity profile gets back again into parabolic patterns. The Figure 5 shows the radial velocity profile at various locations for t = 0.3 sec. The radial velocity component varying radially for different axial position, which is found to be increasing from zero on the axis with positive values as one moves away from it and finally to attain some finite value on the wall surface. Radial velocity is higher in magnitude in the area without stenosis and the presence of stenosis slows down radial velocity to a considerable extent.

Figure 2. Axial velocity profile at t=0.3 sec in stenosed artery

Figure 3. Radial velocity profile at t=0.3 sec in stenosed artery


75

Figure 4. Axial velocity profiles for different axial positions at t= 0.3s (Tm=0.4a, M=2, d=13mm, l0=16mm)

Figure 5. Radial velocity profiles for different axial positions at t= 0.3s (Tm=0.4a, d=13mm, M=2, l0=16mm)


76

Figure 6 shows the variation of the axial velocity profile at z = 23mm for different time periods. The flow profiles are directly responsible for the pulsatile pressure gradient produced by the pumping action of heart. Whenever the time shifted increases from 0.1 sec to 0.3 sec, the curve shifts towards the origin. At t = 0.6 sec again the axial velocity increases because of the diastole. It is also clear that the streaming blood is much higher than the non-Newtonian values. It is because of the fact that if flowing blood is treated as Newtonian, it has a high shear rate flow, which increases the axial velocity profile. Thus the non-Newtonian characteristics of the flowing blood affect the axial velocity profile. The results agrees qualitatively with “Chaturani and Palanisamy (1991) and Sud and Sekhon (1985)” with their research on stenotic blood flow treated as Newtonian fluid.

Figure 6. Axial velocity profiles for different times at z=23mm (Tm=0.4a, d=13mm, M=2, l0=16mm)

In Figure 7, the radial velocity profile at different time period is shown. The radial velocity profile assume a positive value during systole i.e. at t = 0.1sec and 0.3sec. and negative during diastole i.e. at t = 0.6sec and 0.75sec. This is due to the fact that in systolic phase the heart muscle has fully contracted and blood is squeezed out and during diastole, the heart relaxes and becomes full of blood. Effect of magnetic field intensity on the axial velocity at the specific region z=23 mm and time t=0.3sec is shown in Figure 8. The higher values of magnetic field intensity reduce the axial velocity in a larger extent. This is due to the fact that as magnetic field applied on the body, the Laurentz force opposes the flow of blood and hence reduces its velocity. The variation of fluid acceleration F at time t = 0.6 sec at the location z = 23mm, where the artery get maximum narrowing is shown in Figure 9. The acceleration is maximum on the axis and attains some finite value on the wall. By increasing the magnetic field intensity, the fluid acceleration decreases rapidly. Blood is accelerated more for the cases without magnetic field and the magnetic field exposure slows down the fluid acceleration. The variation of flow rate at the location z = 23mm over a period of nearly four cardiac cycle for various magnetic field intensity is illustrated in Figure 10. The pulsatile nature of the flow rate has been found in all cases thought the time scale. One may note the flow rate for an artery without applied magnetic field having higher magnitude than the flow rate for artery with applied magnetic field. It is concluded that the presence of magnetic field affect the flow rate to considerable extent. Figure 11 illustrates the variations of the flow rate Q with time t for different values of the frequency and amplitude of pressure gradient for the magnetic field intensity M=2 at z=23mm. It is clear that by increasing the pulse frequency fp , the back flow increases. The magnitude of flow rate increases with increasing 0A with positive values i.e. no back flow occurs. Thus the frequency as well as the amplitude of the pressure gradient affects the rate of flow to a considerable extent in the presence of magnetic field.


77

Figure 7. Radial velocity profiles for different times at z=23mm (Tm=0.4a, d=13mm, M=2, l0=16mm)

The variations of the flow resistance with time for different magnetic field are shown in Figure 12. The resistance of flow gives the reverse trend of the flow rate. This is because as the flow rate increases, resistance decreases. The values of resistance are higher for the artery without having applied magnetic field than the artery having the effects of applied magnetic field.

Figure 8. Axial velocity profiles for different magnetic field at z=23mm (Tm=0.4a, d=13mm, l0=16mm)


78

The variation of the time dependent wall shear stress at a specific location of z = 23mm for various magnetic field intensity has been shown in Figure 13. The stress yields all time higher values for an artery without applied magnetic field than the artery having applied magnetic field. The values are negative by direction Figure 14 illustrates the variation of wall shear stress .for several values of the frequency and the amplitude of pressure gradient. The wall shear stress becomes more negative for small value of pf and an early separation can be noticed for large value of pf .

The wall shear stress is pulsatile in nature for all cases. The flow separation decreases by increasing the value of 0A .

Figure 9. Radial variation of fluid acceleration at t=0.6 sec. ( a0=100mm/s2

, z=23mm, Tm=0.4a, d=13mm, l0=16mm)

Figure 10. variation of the rate of flow with time at z=23mm (Tm=0.4a, d=13mm, l0=16mm)


79

Figure 11. Variation of the rate of flow with time for different fb and a0 at z=23mm (fp=1.2 Hz,, a0=100mm/s2

, M=2)

Figure 12. Variation of the resistance of flow with time for different magnetic field at z=23mm (fp=1.2 Hz, a0=100mm/s2)


80

Figure 13. Variation of the wall shear stress with time at z=23mm (Tm=0.4a, d=13mm, l0=16mm)

Figure 14. Variation of the wall shear stress with time for different fp and a0 at z=28mm (Tm=0.4a, d=13mm, l0=16mm, M=2)


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5. Conclusion This analysis presented numerical results for an unsteady blood flow in an artery with multiple stenosis, using the generalized power law model of blood viscosity under the influence of applied magnetic field. In this paper we used the time dependent radius of the artery having multiple stenosis, which is an important factor considered in this paper. The advantage of this study is that here we calculated effect of magnetic field on various fluid parameters like blood velocity, flow rate, wall shear stress, flow resistance and flow acceleration with the presence of multiple stenosis in the artery. All the flow characteristics are found to be affected by the influence of applied magnetic field as well as presence of multiple stenosis. Various physiological problems due to the back flow and low shear stress can be caused by the sufficiently large applied magnetic field. The present study would be helpful to analyze the effects of magnetic field on the human by various medical therapies like MRI, Catheter insertion and use of many devices e.g. cellular phone. Acknowledgement One of the authors Gaurav Varshney is thankful to Dr. M. S. Rautela, Principal, Government Degree College Karanprayag, Chamoli, India for his guidance, support and providing all facilities in the College for the preparation of this Manuscript. References Bali, R. and Awasthi, U. 2007. Effect of magnetic field on the resistance to blood flow through stenotic artery. Applied Mathematics and

Computation, Vol. 188, pp. 1635-1641. Buchanan Jr., J. R., Kleinstreuer, C. and Corner J. K. 2000. Rheological effects on pulsatile hemodynamics in a stenosed tube. Computers &

Fluids, Vol. 29, pp. 695-724. Chakravarty, S. 1987. Effects of stenosis on the flow behaviour of blood in an artery. International Journal of Engineering Science, Vol. 25, No.

8, pp. 1003-1016. Chakravarty, S. and Datta, A. 1992. Pulsatile blood flow in a porous stenotic artery. Mathl. Comput. Modelling, Vol. 16, No. 2, pp. 35-54. Chakravarty, S. and Mandal, P. K. 1994. Mathematical modeling of blood flow through an overlapping arterial stenosis. Mathl. Comput.

Modelling, Vol. 19, No. 1, pp. 59-70. Chester, W. 1957. The effect of a magnetic field on Stokes flow in a conducting fluid. J. Fluid Mech,.Vol. 3, pp. 304-308. Deplano, V. and Siouffi, M. 1999. Experimental and Numerical study of pulsatile flow through a tapered artery with stenosis. Journal of

Biomechanics, Vol. 32, pp. 1081-1090. Gerrard, J. H. and Taylor, L. A. 1977. Mathematical model representing blood flow in arteries. Med. & Biol. Eng. and Comput., Vol. 15, pp.

611-617. Gmitrov, J. 2007. Static magnetic field effect on the arterial baroreflex-mediated control of microcirculation: implications for cardiovascular

effects due to environmental magnetic fields. Radiat. Environ. Biophys., Vol. 46, pp. 281-290. Haldar, K. 1985. Effects of the shape of stenosis on the resistance to blood flow through an artery. Bulletin of Mathematical Biology, Vol. 47, No.

4, pp. 545-550. Hernan, A. and Gonzalez, R. 2007. Numerical implementation of viscoelastic blood flow in a simplified arterial geometry. Medical Engineering

and Physics, Vol. 29, pp. 491-496. Ismail, Z., Abdullah, I., Mustapha, N. and Amin, N. 2007. A power-law model of blood flow through a tapered overlapping stenosed artery.

Applied Mathematics and Computation, Vol. 195, No. 2, pp. 669-680. Jauchem, J.R. 1997. Exposure to extremely-low-frequency electromagnetic fields and radiofrequency radiation: cardiovascular effects in humans,

Review Int Arch Occup Environ Health, Vol. 70, pp. 9-21. Kaazempur – Mofrad, M., Wada, S., Myers, J. G. and Ethier, C. R. 2005. Mass transport and fluid flow in stenotic arteries: axisymmetric and

asymmetric models. International Journal of Heat and Mass Transfer, Vol. 48, pp. 4510-4517. Katiyar, V. K. and Basavarajappa, K.S. 2002. Blood flow in the cardiovsular system in the presence of magnetic field. International Journal of

Applied Science and Computations, Vol. 9, No. 3, pp. 118-127. Khanafer, K. M., Gadhoke, P., Berguer, R. and Bull, J. L. 2006. Modeling pulsatile flow in aortic aneurysms. Biorheology, Vol. 43, pp. 661-679. Kinouchi, Y., Yamaguchi, H., and Tenforde, T.S. 1996. Theoretical analysis of magnetic field interactions with aortic blood flow.

Bioelectromagnetics, Vol. 17, pp. 21-32. Kuipers, N. T., Sauder, C.L. and Ray, C. A. 2007. Influence of static magnetic fields on pain perception and sympathetic nerve activity in

humans. J. Appl. Physiol., Vol. 102, pp. 1410-1415. Kumar, A., Varshney, C.L. and Sharma, G.C. Performance modeling and analysis of blood flow in elastic arteries. Applied Mathematics and

Mechanics, Vol. 26, No. 3, pp. 345-354. Liepsch, D. 2002. An introduction to biofluid mechanics – basic models and applications. Journal of Biomechanics, Vol. 35, pp. 415-435. Liu, G. T., Wang, X. J., Ai, B. Q. and Liu, L. G. 2004. Numerical study of pulsatile flow through a tapered artery with stenosis. Chinese Journal

of Physics, Vol. 42, No. 4-I, pp. 401-409. Mandal, P. K. 2005. An unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. International Journal of Non-

Linear Mechanics, Vol. 40, pp. 151-164. McKay, J.C., Prato, F.S. and Thomas, A.W. 2007. A literature review: the effects of magnetic field exposure on blood flow and blood vessels in

the microvasculature. Bioelectromagnetics, Vol. 28, pp. 81-98.


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Prakash, J., Makinde, O. D. and Ogulu, A. 2004. Magnetic effect on oscillary blood flow in a constricted tube. Botswana Journal of Technology April , pp. 45-50.

Shalman. E., Rosenfeld, M., Dgany, E. and Einav, S. 2002. Numerical modeling of the flow in stenosed coronary artery: the relationship between main hemodynamic parameters. Computers in Biology and Medicine, Vol. 32, pp. 329-344.

Sud, V. K. and Sekhon, G. S. 1989. Blood flow through the human arterial system in the presence of a steady magnetic field. Phys. Med. Biol. Vol. 34, No. 7, pp. 795-805.

Tashtoush, B. and Magableh, A. 2008. Magnetic field effect on heat transfer and fluid flow charecteristics of blood flow in multi-stenotic arteries. Heat and Mass Transfer, Vol. 44, No. 3, pp. 297-304.

Tzirtzilakis, E. E. 2005. A mathematical model for blood flow in magnetic field. Physics of Fluids, Vol. 17:077103, pp. 1-15. Villoresi, G., Kopytenko, Y. A.., Ptitsyna, N. G., Tyasto, M. I., Kopytenko, E. A, Iucci, N. and Voronov, P. M. 1994. The influence of

geomagnetic storms and man made magnetic field disturbances on the incidence of myocardial infarction in St. Petersburg (Russia). Phys. Med. , Vol. 10, pp. 107-117.

Biographical notes Gaurav Varshney received Ph.D. from Indian Institute of Technology Roorkee, India in 2009. He is Assistant Professor in the Department of Mathematics, Government Degree College Karanprayag, Chamoli, India. He has published 5 research papers in referred international journals and presented more than 15 research article in National and International Conferences. His area of research includes Bio-fluid Dynamics, Mathematical Modeling, Respiratory Mechanics, Numerical Solutions of ODE and PDE. Dr. V. K. Katiyar is a Professor in the Department of Mathematics, Indian Institute of Technology Roorkee, India. He has more than 30 years of experience in teaching and research. His current area of research includes Biomechanics, Mathematical Modeling, Respiratory Mechanics, Heat and Mass Transfer, Industrial Mathematics. He has published more than One hundred papers in referred international journals. He has also presented more than one hundred research articles in national and international conferences and organized many National and International Conferences. Sushil Kumar received M.Sc. and Ph.D. from Indian Institute of Technology Roorkee, India in 2001 and 2007 respectively. He is Assistant Professor in the Department of Mathematics, S. V. National Institute of Technology Surat, India. He has published 5 research papers in referred international journals. His area of research includes Heat and Mass Transfer, Computational Fluid Dynamics, Mathematical Modeling, Numerical Solutions of ODE and PDE. Received March 2010 Accepted April 2010 Final acceptance in revised form April 2010

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Effect of training algorithms on neural networks aided pavement diagnosis

Kasthurirangan Gopalakrishnan

Department of Civil, Construction, and Environmental Engineering, 354 Town Engineering Bldg., Iowa State University, Ames, IA 50011-3232, USA e-mail: [email protected], Phone: (515) 294-3044 Fax: (515) 294-8216

Abstract Routine pavement maintenance necessitates present structural diagnosis and condition evaluation of pavements by employing non-destructive test equipment such as the Falling Weight Deflectometer (FWD). FWD testing of pavements involves measuring time-domain surface deflections resulting from applied impulse loading on the pavement. Through inverse analysis of FWD deflection data, the stiffness parameters of the individual pavement layers are, in general, determined using iterative optimization routines. In recent years, Neural Networks (NN) aided inverse analysis has emerged as a successful alternative for predicting pavement layer moduli from FWD deflection data in real-time. Especially, the use of Finite Element (FE) based pavement modeling results for training the NN aided inverse analysis is considered to be accurate in realistically characterizing the non-linear stress-sensitive response of underlying pavement layers in real-time. Efficient NN learning algorithms have been developed and proposed to determine the weights of the network, according to the data of the computational task to be performed. In this paper, the effect of training algorithms on the NN aided inversion process is analyzed and discussed. Keywords: Neural networks; Non-destructive testing; Inverse analysis; Finite element; Flexible pavement.

1. Introduction Roads deteriorate over time due to accumulated damage from vehicular traffic loading as well as environmental effects. Damage assessment is routinely carried out by engineers from public works, road authorities, or other public bodies that build and maintain the road system. According to recent statistics (ASCE 2009), poor road conditions cost U.S. motorists more than billions of dollars a year in repairs and operating costs. Thus, to improve the performance and serviceability of pavements, to reduce the cost to taxpayers, and to schedule proactive pavement repair and maintenance activities, consistent, cost-effective, and accurate monitoring of pavement is necessary. A major portion of the road networks in the United States are reported to be composed of flexible pavement (Highway Statistics 2000). Flexible pavements are multi-layered, heterogeneous structures that are designed to “flex” under repeated traffic loading. A typical flexible pavement structure consists of the surface course (typically Hot-Mix Asphalt) at the top, underlying base and subbase (optional) courses (typically unbound granular material), and a subgrade (typically soil) at the bottom. In the field, Non-Destructive Testing (NDT) of in-service pavements using a Falling Weight Deflectometer (FWD) equipment is carried out to measure the deflection response of the pavement structure to applied dynamic load that simulates a moving wheel (see Figure 1). Using FWD test results, back-calculation of in-situ pavement layer moduli from measured deflections (inverse analysis) is carried out. The backcalculated pavement layer moduli are indicators of pavement layer condition as well as necessary inputs for mechanistic pavement structural analysis and thus remaining life calculations (Lytton, 1989; Ullidtz and Coetzee, 1995). Most of the conventional commercial backcalculation programs involve Multi-Layer Elastic Theory (MLET) in their forward calculation routines and assume that pavement materials are linear-elastic, homogenous, and isotropic resulting in unrealistic backcalculated pavement layer moduli. Several research studies have shown that pavement geomaterials used in the underlying pavement layers exhibit non-linear, stress-sensitive behavior under repeated traffic loading. Unbound aggregates used in pavement base/sub-base course exhibit stress-hardening and fine-grained soils show stress-softening-type behavior (Brown and Pappin, 1981; Thompson and Elliot, 1985; Garg et al., 1998). Research studies have shown that non-linear analysis using Finite Element (FE) based approach increases the precision of the forward model (Gopalakrishnan et al., 2010).

Gopalakrishnan / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 83-92

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Figure 1. Non-destructive testing of pavements using Falling Weight Deflectometer (FWD)

Recent research studies have focused on the development of Neural Networks (NN) based flexible pavement analysis models trained using FE solutions database to predict critical pavement responses and layer moduli (Ceylan, Guclu, Bayrak, and Gopalakrishnan 2007; Gopalakrishnan and Thompson 2004). The objective of this paper is to study the effect of different training algorithms on the performance of FE-NN backcalculation models which has not been done in previous studies. 2. Neural networks aided inversion of pavement surface deflections NNs are parallel connectionist structures constructed to simulate the working network of neurons in the human brain. They attempt to achieve superior performance via dense interconnection of non-linear computational elements operating in parallel and arranged in a pattern reminiscent of a biological neural network. The perceptrons or processing elements and interconnections are the two primary elements which make up a neural network. A single perceptron is mathematically represented as follows (Hicks and Monismith 1971):

⎟⎠

⎞⎜⎝

⎛−== ∑

=

n

ijijijk bwxvy

1)( ϕϕ

(1) where xi is input signal, wij is synaptic weight, bj is bias value, vj is activation potential, φ() is activation function, yk output signal, n is the number of neurons for previous layer, and k is the index of processing neuron. Multilayer perceptrons (MLPs), frequently referred as multi-layer feedforward neural networks, consist of an input layer, one or more hidden layer, and an output layer and they have numerous applications in the engineering domain (Murali et al., 2010). Learning in a MLP is an unconstrained optimization problem, which is subject to the minimization of a global error function depending on the synaptic weights of the network. For a given training data consisting of input-output vectors, values of synaptic weights in a MLP are iteratively updated by a learning algorithm to approximate the target behavior. This update process is usually performed by backpropagating the error signal layer by layer and adapting synaptic weights with respect to the magnitude of error signal (Goktepe et al., 2004). Rumelhart et al., (1986) presented the first backpropagation learning algorithm (referred to as gradient-descent backpropagation) for use with MLP structures, which is typically employed in the development of FE-NN backcalculation models. In the backpropagation algorithm, the error energy used for monitoring the progress toward convergence is the generalized value of all errors that is calculated by the least-squares formulation and represented by a Mean Squared Error (MSE) as follows (Haykin 1999):

( )∑∑

=

−=P M

kkk yd

MPMSE

1 1

21

(2) where yk are the actual outputs and dk are the desired outputs; M is the number of neurons in the output layer and P represents the total number of training patterns. The objective of this research is to study the effect of various training algorithms on the prediction performance of FE-NN backcalculation models.


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A FE-NN backcalculation procedure was developed and implemented in MATLAB® to approximate the FWD backcalculation function. Using the synthetic training and testing dataset generated using a 2-D axisymmetric pavement FE program considering the nonlinear stress-dependent geomaterial characterization, the NN was trained to learn the relation between the synthetically generated surface deflection basins (inputs) and the pavement layer moduli (outputs). A generic three-layer flexible pavement structure consisting of Asphalt Concrete (AC) surface layer, unbound aggregate base layer, and subgrade layer was modeled using the FE program. The top surface AC layer was characterized as a linear elastic material with Young’s Modulus, EAC, and Poisson ratio, ν. The K-θ model (Hicks and Monismith 1971)was used as the stress-dependent resilient modulus model for the unbound aggregate layer.

n

R KE θ= (3) where ER is resilient modulus, θ = σ1 + σ2 + σ3 = σ1 + 2σ3 = bulk stress, and K and n are multiple regression constants obtained from repeated load triaxial test data on granular materials. Previous research studies have shown that K and n model parameters can be correlated to characterize the non-linear stress dependent behavior with only one model parameter. Fine-grained subgrade soils were modeled using the commonly used bi-linear resilient modulus model (Thompson and Elliot 1985):

diddidRiR

diddidRiRforKEEforKEE

σσσσσσσσ

>−+=<−+=

).().(

2

1

(4) where ERi is the breakpoint resilient modulus, σd is the breakpoint deviator stress (σd = σ1 - σ3), σdi is the breakpoint deviator stress, and K1 and K2 are statistically determined coefficients from laboratory tests. As indicated by Thompson and Elliot (1985), the value of the resilient modulus at the breakpoint in the bilinear curve, ERi, can be used to classify fine-grained soils as being soft, medium or stiff. The ERi is the main input for subgrade soils in ILLI-PAVE. The bilinear model parameters were set to default values. Also, the Asphalt Institute’s Thickness Design Manual MS-1 recommends ERi as the subgrade modulus input for ELP analysis. Thus, asphalt concrete modulus, EAC, granular base K-θ model parameter K, and the subgrade break-point resilient moduli, ERi, were used as the layer stiffness inputs for all the FE runs. The 40-kN wheel load was applied as a uniform pressure of 552 kPa over a circular area of radius 150 mm simulating the FWD loading. A comprehensive FE synthetic database was generated by varying the AC layer thickness (in the range of 75 to 700 mm), aggregate base layer thickness (in the range of 100 to 550 mm), EAC (in the range of 6.9 to 41.5 GPa), K (in the range of 21 to 82 MPa), and ERi (in the range of 7 to 105 MPa) for NN training and testing. A total of 3,000 and 1,000 independent datasets, respectively, was used for training and testing the FE-NN backcalculation model. Two hidden layers were found to be sufficient in solving a problem of this size. Therefore the architecture was reduced to a four-layer feedforward network. Before discussing the effect of different training algorithms on FE-NN aided inversion of pavement deflection data for prediction of non-linear pavement layer stiffnesses, a brief review of the different training algorithms is presented first. 3. Neural networks training algorithms The ability to ‘learn’ the mapping between inputs and outputs is one of the main advantages that make the NNs so attractive. Efficient learning algorithms have been developed and proposed to determine the weights of the network, according to the data of the computational task to be performed. The learning ability of the NNs makes them suitable for unknown and non-linear problem structures such as pattern recognition, medical diagnosis, time series prediction, and others. Considerable research has been carried out to accelerate the convergence of learning algorithms which can be broadly classified into two categories: (1) development of ad-hoc heuristic techniques which include such ideas as varying the learning rate, using momentum and rescaling variables; (2) development of standard numerical optimization techniques. The three types of numerical optimization techniques commonly used for NN training include the conjugate gradient algorithms, quasi-Newton algorithms, and the Levenberg-Marquardt algorithm (Hagan and Menhaj 1994). Although numerous training algorithms appear in recent neural network literature, it is difficult to know which algorithm works best, in terms of convergence speed and accuracy, for a given problem. A number of factors, including the complexity of the problem, the number of datasets used in training, the number of weights and biases in the network, the error goal, and whether the NN is used for function approximation or classification, etc., seem to have influence (Coskun and Yildrim 2003). The following sub-sections briefly describe the various NN training algorithms considered in this study. 3.1Gradient Descent Backpropagation (GD) The gradient descent backpropagation training algorithm is based on minimizing the mean square error between the network’s output and the desired output. Once the network’s error has decreased to the specified threshold level, the network is said to have converged and is considered to be trained. The GD backpropagation algorithm updates synaptic weights and biases along the negative gradient of the error energy function (MATLAB Toolbox, User’s Guide, 2010).


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3.2 Gradient Descent with Momentum Backpropagation (GDM) This is a batch steepest descent training algorithm that often provides faster convergence. Momentum allows a network to respond not only to the local gradient, but also to recent trends in the error surface. Acting like a low-pass filter, momentum allows the network to ignore small features in the error surface. Without momentum a network may get stuck in a shallow local minimum. With momentum a network can slide through such a minimum (MATLAB Toolbox, User’s Guide, 2010). 3.3 Gradient Descent with Adaptive Learning Rate Backpropagation (GDA) In the standard steepest descent (GD) backpropagation algorithm, the learning rate parameter is held constant throughout training. However, the performance of the algorithm is very sensitive to the proper setting of the learning rate parameter. For this reason, the GDA algorithm was developed to allow the learning rate parameter to be adaptive to keep the learning step size as large as possible while keeping learning stable. In GDA algorithm, the optimal value of the learning rate parameter changes with the gradient’s trajectory on the error surface (MATLAB Toolbox, User’s Guide, 2010). 3.4 Gradient Descent with Momentum and Adaptive Learning Rate Backpropagation (GDX) The GDX training algorithm combines adaptive learning rate with momentum training. It is similar to GDA algorithm except that it has a momentum coefficient as an additional training parameter. Thus, the weight vector update is carried out the same way as in GDM except that a varying learning rate is used as in GDA (MATLAB Toolbox, User’s Guide, 2010). 3.5 Resilient Backpropagation (RP) To overcome the inherent disadvantages of pure gradient-descent, the RP training algorithm performs a local adaptation of the weight-updates according to the behavior of the error function. In contrast to other adaptive techniques, the effect of the RP adaptation process is not affected by the size of the derivative, but only by the temporal behavior of its sign leading to an efficient and transparent adaptation process. Resilient Backpropagation is generally much faster than the standard steepest descent algorithm although it requires only a modest increase in memory requirements (MATLAB Toolbox, User’s Guide, 2010). 3.6 Conjugate Gradient Backpropagation with Fletcher-Reeves Update (CGF) The basic backpropagation algorithm adjusts the weights in the steepest descent direction (negative of the gradient), the direction in which the cost function is decreasing most rapidly. Although the function decreases most rapidly along the negative of the gradient, this does not necessarily produce the fastest convergence. In the conjugate gradient algorithms a search is performed along conjugate directions, which produces generally faster convergence than steepest descent directions (MATLAB Toolbox, User’s Guide, 2010). All the conjugate gradient algorithms start out by searching in the steepest descent direction (negative of the gradient) on the first iteration. A line search is then performed to determine the optimal distance to move along the current search direction. Then the next search direction is determined so that it is conjugate to previous search directions. The general procedure for determining the new search direction is to combine the new steepest descent direction with the previous search direction. The various versions of the conjugate gradient algorithm are distinguished by the manner in which the constant associated with the determination of new search direction is computed. For the Fletcher-Reeves update, this constant is computed as the ratio of the norm squared of the current gradient to the norm squared of the previous gradient (MATLAB Toolbox, User’s Guide, 2010). 3.7 Conjugate Gradient Backpropagation with Polak-Ribiére Update (CGP) For the Polak-Ribiére update, the constant associated with the determination of new search direction is computed as the inner product of the previous change in the gradient with the current gradient divided by the norm squared of the previous gradient. The storage requirements for Polak-Ribiére (four vectors) are slightly larger than for Fletcher-Reeves (three vectors) (MATLAB Toolbox, User’s Guide, 2010). 3.8 Conjugate Gradient Backpropagation with Powell-Beale Restarts (CGB) For all conjugate gradient algorithms, the search direction is periodically reset to the negative of the gradient. The standard reset point occurs when the number of iterations is equal to the number of network parameters (weights and biases), but there are other reset methods that can improve the efficiency of training. One such reset method is the Powell-Beale Restarts. This technique restarts if there is very little orthogonality left between the current gradient and the previous gradient. The storage requirements for the Powell-Beale algorithm (six vectors) are slightly larger than for Polak-Ribiére (four vectors) (MATLAB Toolbox, User’s Guide, 2010).


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3.9 Scaled Conjugate Gradient Backpropagation (SCG) The three conjugate gradient algorithms discussed so far require a line search at each iteration, which is computationally expensive, since it requires that the network response to all training inputs be computed several times for each search. The SCG training algorithm was developed to avoid this time-consuming line search, thus significantly reducing the number of computations performed in each iteration, although it may require more iterations to converge than the other conjugate gradient algorithms. The storage requirements for the SCG algorithm are about the same as those of CGF (MATLAB Toolbox, User’s Guide, 2010). 3.10 BFGS Quasi-Newton Backpropagation (BFGS) Newton's method is an alternative to the conjugate gradient methods for fast optimization. The Broyden–Fletcher–Golfarb–Shanno (BFGS) algorithm is one of the most popular of the quasi-Newton algorithms (Haykin 1999). The basic step of Newton's method is to form the Hessian Matrix (second derivatives). This method often converges faster than conjugate gradient methods but it is complex and expensive to compute the Hessian Matrix for feedforward neural networks. For smaller networks, however, BFGS can be an efficient training function (MATLAB Toolbox, User’s Guide, 2010). 3.11 One Step Secant Backpropagation (OSS) Since the BFGS algorithm requires more storage and computation in each iteration than the conjugate gradient algorithms, there is need for a secant approximation with smaller storage and computation requirements. The OSS training algorithm requires less storage and computation per epoch than the BFGS algorithm. It requires slightly more storage and computation per epoch than the conjugate gradient algorithms. Thus, the OSS method can be considered a compromise between full quasi-Newton algorithms and conjugate gradient algorithms (MATLAB Toolbox, User’s Guide, 2010). 3.12 Levenberg-Marquardt Backpropagation (LM) The LM second-order numerical optimization technique combines the advantages of Gauss–Newton and steepest descent algorithms. While this method has better convergence properties than the conventional backpropagation method, it requires O(N2) storage and calculations of order O(N2) where N is the total number of weights in an MLP backpropagation. The LM training algorithm is considered to be very efficient when training networks which have up to a few hundred weights. Although the computational requirements are much higher for each iteration of the LM training algorithm, this is more than made up for by the increased efficiency. This is especially true when high precision is required (MATLAB Toolbox, User’s Guide, 2010). 3.13 Bayesian Regularization Backpropagation (BR) The BR training algorithm is considered as one of the best approaches to overcome the over-fitting tendencies of NNs so that their prediction accuracies for unseen data can be further enhanced. This approach minimizes the over-fitting problem by taking into account the goodness-of-fit as well as the network architecture. The BR network training function updates the weight and bias values according to Levenberg-Marquardt optimization. It minimizes a combination of squared errors and weights, and then determines the correct combination so as to produce a network that generalizes well. This process is called Bayesian regularization (MATLAB Toolbox, User’s Guide, 2010). 4. Effect of Training Algorithms on NN Aided Inversion of Deflection Data Twelve different training algorithms discussed in the previous section were considered in evaluating their effect on the prediction performance of FE-NN based inversion of pavement deflection data. The network architecture was set to 8 input neurons (six surface deflections at equal radial offsets, and AC surface and base layer thicknesses), two hidden layers with 60 hidden neurons each and one output layer (layer modulus). The determination of optimum number of hidden neurons and hidden layers for the pavement modulus backcalculation problem considered in this study is discussed elsewhere (Ceylan, Guclu, Bayrak, and Gopalakrishnan 2007). Two different architectures, one for AC modulus, EAC, and one for subgrade break-point non-linear resilient modulus, ERi, were used. To enable easy comparison, the size of the networks and the learning parameters were held constant in studying the effect of different training algorithms on prediction accuracies of FE-NN inversion models. The effect of gradient-descent backpropagation training algorithms (GD, GDM, GDA, GDX, and RP) on the training performance of FE-NN EAC prediction model is captured in Figure 2. While training the networks, a set of validation vectors were used to stop training early if further training on the primary vectors will hurt generalization to the validation vectors. The effect of conjugate gradient training algorithms (CGF, CGP, CGB, and SCG) on the training performances of FE-NN EAC prediction model is displayed in Figure 3. Similar results showing the effect of other numerical optimization algorithms (BFGS, OSS, LM, and BR) on the training performance of FE-NN EAC prediction model are shown in Figure 4. It is seen that the learning curves of gradient-descent backpropagation algorithms and conjugate gradient training algorithms have different trends.


88

Figure 2. Effect of gradient-descent backpropagation learning algorithms on training performance of FE-NN AC modulus

prediction model.

Figure 3. Effect of conjugate gradient backpropagation learning algorithms on training performance of FE-NN AC modulus

prediction model


89

Figure 4. Effect of quasi-Newton and Levenberg-Marquardt backpropagation learning algorithms on training performance of FE-

NN AC modulus prediction model

Table 1. R2 values with Various NN Training Algorithms Algorithm Modulus Training Testing

EAC 0.804 0.795 GD ERi 0.829 0.822 EAC 0.067 0.131

GDM ERi 0.115 0.078 EAC 0.631 0.637

GDA ERi 0.696 0.699 EAC 0.824 0.815

GDX ERi 0.839 0.829 EAC 0.978 0.972

RP ERi 0.229 0.248 EAC 0.992 0.990

CGF ERi 0.978 0.977 EAC 0.864 0.861

CGP ERi 0.973 0.973 EAC 0.978 0.975

CGB ERi 0.985 0.984 EAC 0.915 0.906

SCG ERi 0.963 0.964 EAC 0.995 0.994

BFGS ERi 0.987 0.987 EAC 0.893 0.892

OSS ERi 0.980 0.980 EAC 0.999 0.998

LM ERi 0.999 0.998 EAC 1.000 1.000

BR ERi 0.998 0.999

In Figure 2-4, the training performances of neural networks are displayed in terms of Mean Squared Error (MSE) for normalized input-output values. Table 1 summarizes the prediction performance results using different training algorithms in terms of coefficient of determination (R2) values obtained with the training and testing datasets.


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Both in terms of training and testing, the LM algorithm seems to provide the best performance. It was also one of the fastest algorithms in terms of speed of convergence. This result is also in agreement with the results of a previous research study which evaluated the role of learning algorithms in NN based backcalculation models trained with elastic layered pavement analysis program generated synthetic datasets (Goktepe, Agar, and Lav 2006). Using the Bayesian Regularization (BR) approach, the performance is further enhanced although over-fitting tendency of the network for this problem is likely to be minimal considering the relatively large number of datasets used in training the network. The performance of FE-NN model trained with quasi-Newton BFGS algorithm is almost as good (with respect to the coefficient of determination) as with the LM algorithm, although the BFGS algorithm took considerably longer time for training. The prediction performance of the FE-NN models using LM and BFGS training algorithms on the 1,000 independent test datasets is displayed in Figs. 5 and 6. These results indicate that the LM training algorithm could be used quite reliably with backpropagation NN for predicting pavement layer modulus from FWD surface deflection basins. In terms of Average Absolute Errors (AAEs), the NN-based backcalculation models trained with LM learning algorithm successfully predicted pavement layer moduli values with an overall AAE value of less than 1.5%. The adoption of NN-based approach can result in both a drastic reduction in computation time and a simplification of the backcalculation approach from the viewpoint of a pavement designer/analyst. Rapid prediction ability of the NN models, capable of analyzing 100,000 FWD deflection profiles in less than a second, provide a tremendous advantage to the pavement engineers by allowing them to nondestructively assess the condition of the transportation infrastructure system in real time, including when the FWD testing is conducted in the field.

Figure 5. Neural networks AC and non-linear subgrade moduli prediction accuracies using Lavenberg-Marquardt (LM) training

algorithm.


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Figure 6. Neural networks AC and non-linear subgrade moduli prediction accuracies using BFGS training algorithm.

5. Conclusion In the field, Non-Destructive Testing (NDT) of in-service pavements is routinely carried out using a Falling Weight Deflectometer (FWD) equipment to measure the surface deflection response of the pavement structure to applied dynamic load that simulates a moving wheel. Then, through an inversion procedure, referred to as backcalculation, the pavement layer stiffness properties are determined from the pavement deflection data using parameter identification routines. In recent years, hybrid Finite Element-Neural Networks (FE-NN) aided inverse analysis has emerged as a successful alternative for predicting non-linear pavement layer moduli from FWD deflection data. The effect of different training algorithms on the performance of FE-NN backcalculation models has been presented in this paper. The Lavenberg-Marquardt algorithm with Bayesian Regularization (BR) procedure was found to be the most successful algorithm for non-linear pavement layer moduli backcalculation using hybrid FE-NN inversion models. Elimination of the seed layer moduli selection step, combined with the integration of NN-based direct backcalculation approach, can be invaluable for the state and federal agencies for rapidly analyzing a large number of pavement deflection basins needed for routine pavement evaluation for both project-specific and network-level FWD testing. Research studies have shown that to successfully backcalculate the pavement layer stiffness, or to predict the critical pavement responses (maximum stresses, strains and deflections), accurate layer thickness information is needed, especially at the FWD testing points. Future research efforts should focus on conducting sensitivity studies to determine the effect of pavement layer thickness on pavement performance data using the mechanistic-empirical based pavement design concepts. This will help to determine how much tolerance can be accommodated in assessing the pavement thickness by the means of NDT techniques and devices.


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References ASCE. 2009 Report Card for America’s Infrastructure. Available:

http://www.infrastructurereportcard.org/sites/default/files/RC2009_full_report.pdf (accessed on Feb. 1, 2009) (URL). Brown S. F. and Pappin J. W., 1981. Analysis of pavements with granular bases,” Transportation Research Record 810, pp. 17-23. Ceylan H., Guclu A., Bayrak M. B. and Gopalakrishnan K., 2007. Nondestructive evaluation of Iowa pavements-Phase I, CTRE,

Iowa State University, Ames, IA, CTRE Project 04-177, Dec. Coskun N. and Yildrim T., 2003. The effects of training algorithms in MLP network on image classification, in Proc. Int. Joint

Conf. on Neural Networks, Vol. 2, pp. 1223-1226. Garg N.E., Tutumluer E. and Thompson M. R., 1998. Structural modelling concepts for the design of airport pavements for heavy

aircraft, in Proc. Fifth International Conf. on the Bearing Capacity of Roads and Airfields, Trondheim, Norway. Goktepe A.B., Agar E., and Lav A. H., 2006. Role of learning algorithm in neural network-based backcalculation of flexible pavements, Technical Notes, Journal of Computing in Civil Engineering, Vol. 20, No. 5, pp. 370-373. Gopalakrishnan K. and Thompson M.R., 2004. Backcalculation of airport flexible pavement non-linear moduli using artificial

neural networks, in Proc. 17th International FLAIRS Conference, Miami Beach, Florida, May. Gopalakrishnan, K., Ceylan, H. and Attoh-Okine, N. O. (Eds.), 2010. Intelligent and Soft Computing in Infrastructure Systems:

Recent Advances, Springer, Inc., Germany. Haykin S., 1999. Neural networks: A comprehensive foundation, Prentice-Hall, Inc., NJ, USA. Hicks R. G. and Monismith C. L., 1971. Factors influencing the resilient properties of granular materials, Transportation Research

Record no. 345, TRB, pp. 15-31. Hagan M. T. and Menhaj M. B., 1994. Training feedforward networks with the Marquardt algorithm, IEEE Transactions on

Neural Networks, Vol. 5, No. 6, pp. 989-993. Lytton R.L., 1989. Backcalculation of layer moduli, state of the art, In: NDT of pavements and backcalculation of moduli, A. J.

Bush and G. Y. Baladi, editors., Vol. 1, ASTM Special Technical Publication (STP) 1026, pp. 7–38. MATLAB Neural Network ToolboxTM. User’s Guide. Available: http://www.mathworks.com/access/helpdesk/help/toolbox/nnet/

(accessed on Mar 3, 2010) (URL). Murali, R. V., Puri, A. B., and Prabhakaran, G., 2010. Artificial neural networks based predictive model for worker assignment

into virtual cells, International Journal of Engineering, Science and Technology, Vol. 2, No. 1, 2010, pp. 163-174. Office of Highway Policy Information. 2004. Highway Statistics 2000, Federal Highway Administration, Washington, DC. Rumelhart D.E., Hinton G. E., and Williams R. J., 1986. Learning internal representation by error propogation in Rumelhart, D.E.

eds, Parallel Distributed Processing, MIT Press, Cambridge, MA, pp. 318-362. Thompson M. R. and Elliot R. P., 1985. ILLI-PAVE based response algorithms for design of conventional flexible pavements,

Transportation Research Record 1043, pp.50-57. Ullidtz P. and Coetzee N. F., 1995. Analytical procedures in nondestructive testing pavement evaluation, Transportation Research

Record 1482, pp. 61-66. Biographical notes Dr. Kasthurirangan Gopalakrishnan is a Research Faculty in the Department of Civil, Construction and Environmental Engineering (CCEE) at Iowa State University, Ames, Iowa and is also affiliated with the Ames Lab, US Department of Energy and the Iowa Bioeconomy Institute. He obtained his Ph.D from University of Illinois at Urbana-Champaign in 2004 and M.S. from Louisiana State University in 2000. He is also a recipient of the Federal Highway Administration (FHWA) Dwight D. Eisenhower Graduate Research Fellowship. He has more than 100 peer-reviewed publications in the form of journal articles, articles in conference proceedings, and book chapters. He is also the lead editor of the following Springer book publications: (1) Intelligent and Soft Computing in Infrastructure Systems Engineering: Recent Advances; (2) Sustainable and Resilient Critical Infrastructure Systems: Simulation, Modeling, and Intelligent Engineering. His research interests include green energy and technology in civil infrastructure, sustainable systems engineering, nanotechnology and MEMS in transportation structures, knowledge discovery and intelligent data mining paradigms for civil engineering informatics. Received April 2010 Accepted April 2010 Final acceptance in revised form April 2010

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Vol. 2, No. 2, 2010, pp. 93-106

INTERNATIONAL JOURNAL OF ENGINEERING, SCIENCE AND TECHNOLOGY

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2010 MultiCraft Limited. All rights reserved

Permanent magnet synchronous motor dynamic modeling with genetic algorithm performance improvement

Adel El Shahat1*, Hamed El Shewy2

1*Department of Electrical and Computer Engineering, Ohio State University, USA

2Department of Electrical Power and Machines Engineering, Zagazig University, EGYPT *Corresponding Authors: (e.mails: [email protected]; [email protected])

Abstract This paper proposes dynamic modeling simulation for ac Surface Permanent Magnet Synchronous Motor (SPMSM) with the aid of MATLAB – Simulink environment. The proposed model would be used in many applications such as automotive, mechatronics, green energy applications, and machine drives. The modeling procedures are described and simulation results are presented. The validity of this dynamic model here is verified. Then, two genetic algorithm trials are presented to improve SPMSM performance. Maximum torque per ampere genetic algorithm function with maximum efficiency constrained is illustrated. Also, genetic algorithm maximum efficiency function constrained by GA maximum power factor is proposed. Simulations are implemented using MATLAB with its genetic algorithm toolbox. Finally, the required voltage to drive the motor at the desired improved characteristics is deduced for each case. All various characteristics are well depicted in the form of comparisons with such ones from original characteristics at rated voltage. Keywords- Permanent Magnet, Synchronous Motor, Simulink, Genetic Algorithm, optimization and MATLAB

1. Introduction Permanent magnet (PM) motor drives have been a topic of interest for the last twenty years due to its suitability for many topics like in automotive, mechatronics, green energy applications, and machine drives. Pillay and Krishnan (1988, 1989) presented PM motor drives and classified them into two types such as permanent magnet synchronous motor drives (PMSM) and brushless dc motor (BDCM) drives. Morimoto et al. (1994), in their paper, aimed to improve efficiency in permanent magnet (PM) synchronous motor drives. The paper of Wijenayake and Schmidt (1997) described the development of a two-axis circuit model for permanent magnet synchronous motor (PMSM) by taking machine magnetic parameter variations and core loss into account. Jang-Mok and Seung-Ki (1997) proposed a novel flux-weakening scheme for an Interior Permanent Magnet Synchronous Motor (IPMSM). Bose (2002) presented different types of synchronous motors and compared them to induction motors. Mademlis and Margaris, (2002) presented an efficiency optimization method for vector-controlled interior permanent-magnet synchronous motor drive. Jian-Xin, et al. (2004) applied a modular control approach to a permanent magnet synchronous motor (PMSM) speed control. Onoda and Emadi (2004) had developed a modeling tool to study automotive systems using the power electronics simulator (PSIM) software. Genetic algorithm is based on natural evolution. As a result, much of the terminology is drawn from biology and evolution (Goldberg, 1989). Genetic algorithm is started by randomly generating a population of strings, representing the encoded parameters. The strings are then evaluated to obtain a quantitative measure of how well they perform as possible problem solution. Genetic operators are crossover and mutation. Crossover produces one pair of output strings for a given pair of input string. Mutation is a unary operator which takes a binary string as its input and outputs a binary string that is almost identical to the input string except at most at a single bit (Rudolph, 1994). Since Holland (1995) presented the GA as a computer algorithm, a wide range of applications of GA has appeared in various scientific areas, and GA has been proved powerful enough to solve the complicated problems, especially the optimal design problems. Some of the possible methods are to fix the number of generations and to use the best individual of all generations as the optimum result; to fix the time elapsed and to select the optimum similarly; or to let the entire population converge in to an average fitness with some error margin (Cvetkovski et al., 1998). El Shahat et al.

El Shahat and El Shewy / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 93-106

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(2010) used the genetic algorithm in an efficient manner in high speed PM synchronous motor flywheel design aspects. Finally, noticeable efforts are introduced in the topic of PM machines modeling and improving the machine performance using artificial intelligence like neural network, genetic algorithm, etc (El Shahat and El Shewy, 2009a,b,d,e, 2010; El Shahat et al., 2010; El Shewy et al., 2008). 2. Dynamic Modeling of PMSM Figure 1 presents equivalent circuit of PMSM in d-q axis to be used in both dynamic equations of PMSM, and static characteristics.

Figure1. PMSM Equivalent Circuit

The two axes PMSM stator windings can be considered to have equal turns per phase. The rotor flux can be assumed to be concentrated along the d axis while there is zero flux along the q axis. Further, it is assumed that the machine core losses are negligible. Also, rotor flux is assumed to be constant. Variations in rotor temperature alter the magnet flux, but its variation with time is considered to be negligible. A dynamic model of PMSM can be illustrated as follow (Krishnan, 2006):

vq = rs iq + ρ ( λq ) + ωr λd (1)

vd = rs id + ρ ( λd ) – ωr λq (2)

λq = Lq iq (3)

λd = Ld id + λaf (4)

where ωr : Electrical velocity of the rotor; λaf : The flux linkage due to the rotor magnets linking the stator; vd, vq : d, q voltages; λd, λq : d, q flux ρ (λaf) = 0, λaf = Lm ifr; ρ : Operator The electromagnetic torque is given by:

Te=23 )(

2 dqqd iiP

λλ − =23

))((2 dqqdqm iiLLiP

−+λ (5)

The electromechanical power

Pem = ωrm Te = 23ωr ( λd iq – λq id ) (6)

ωr = 2P

ωrm (7)

where P: Poles No; ωrm: Rotor Mechanical velocity The general mechanical equation for the motor is:

Te = Tl + Td + B ωrm + J ρ ωrm (8)

where B: Viscous friction’s coefficient; J: Inertia of the shaft and the load system; Td : Dry friction; Tl: Load torque 3. Dynamic Modeling & Simulation Results This dynamic simulation of PMSM is done with the aid of SIMULINK in MATLAB package.


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Figure2. PMSM Dynamic Model Block

Figure 2 presents PMSM block in which, voltage and load torque are considered as inputs, with the speed and current as outputs. In this model some assumptions included: 1) Saturation is neglected. 2) The induced EMF is sinusoidal. 3) Eddy currents and hysteresis losses are negligible. 4) There are no field current dynamics. 5) All motor parameters are assumed constant. 6) Leakage inductances are zero.

Ns1

current in q -axis

u(1)/Lq

current in d -axis

f(u)

Te Calculation

f(u)

Sine Wave 3

Sine Wave 2

Sine Wave 1

Poles / 2

-K-

Mechanical

f(u)

Load Torque

T_L

Integrator 1

1s

Integrator

1s

Gain 4

-K-

Gain

B

From 3-phase to d -q-

In1

V_d

V_q

Flux in q -axis

V_q

i_q

Lambda_d

Omega_r

Lambda_q

Flux in d -axis

V_d

i_d

Lambda_q

Omega_r

Lambda_d

Constant 1

T_d

Constant

J

...

Figure 3. PM Synchronous Motor Model

Figure 3, introduces the PMSM more detailed model with the aid of Simulink, and its details are described below. Some simulation performance characteristics of this model are presented in the following figures using scope; with the prescribed inputs as in Figure 2.


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Figure 4. Synchronous Speed with Time

Figure 5. Electromagnetic Torque with Time

Figure 6. Angle Delta with Time

The following figures deal with the frequency variations under v/f constant pattern to show the synchronous speed response of this model as a simple check. From figures 7 to 10, it is clear that; each synchronous speed values are equal to 120 f / P.


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Figure 7. Synchronous Speed at frequency = 50 Hz




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Figure 10. Synchronous Speed at frequency = 20 Hz 4. Genetic algorithm The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generations, the population "evolves" toward an optimal solution. We can apply the genetic algorithm to solve a variety of optimization problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, non - differentiable, stochastic, or highly nonlinear (Conn et al., 1991, 1997; El Shahat et al., 2009c, 2010). The genetic algorithm uses three main types of rules at each step to create the next generation from the current population: • Selection rules select the individuals, called parents, which contribute to the population at the next generation. • Crossover rules combine two parents to form children for the next generation. • Mutation rules apply random changes to individual parents to form children. The genetic algorithm differs from a classical, derivative-based, optimization algorithm in two main ways, as summarized in the following: Classical Algorithm: generates a single point at each iteration in which the sequence of points approaches an optimal solution, also it selects the next point in the sequence by a deterministic computation. Genetic Algorithm: generates a population of points at each iteration in which the best point in the population approaches an optimal solution, moreover it selects the next population by computation which uses random number generators. The motor characteristics equations, with the terminal voltage as optimizing variable (x1) are used; then the desired function (Matlab m-file) is formulated. After that, the Genetic technique is used to maximize the function. This optimizing variable is constrained or bounded by 0: 200. The aim of these examples is to deduce the required voltage to drive the motor at desired phenomena. A. MATLAB GA with Nonlinear Constraint Description Solver To use the genetic algorithm at the command line (the same results could be drawn from GUI), call the genetic algorithm function ga with the syntax [x fval] = ga (@fitnessfun, nvars, options) Where • @fitnessfun is a handle to the fitness function. • nvars is the number of independent variables for the fitness function. • Options are structure containing options for the genetic algorithm. If we do not pass in this argument, ga uses its default options. The results are given by • x — Point at which the final value is attained • fval — Final value of the fitness function Using the function ga is convenient if we want to


99

• Return results directly to the MATLAB workspace • Run the genetic algorithm multiple times with different options, by calling ga from an M-file.

The genetic algorithm uses the Augmented Lagrangian Genetic Algorithm (ALGA) to solve nonlinear constraint problems. The optimization problem solved by the ALGA algorithm is min x f (x), such that

ci (x) ≤ 0, i = 1 ….. m (9) ceqi (x) = 0, i = m+1 … mt A . x ≤ b Aeq . x = beq lb ≤ x ≤ ub,

Where: c (x) represents the nonlinear inequality constraints, ceq (x) represents the equality constraints, m is the number of nonlinear inequality constraints, and mt is the total number of nonlinear constraints. The Augmented Lagrangian Genetic Algorithm (ALGA) attempts to solve a nonlinear optimization problem with nonlinear constraints, linear constraints, and bounds. In this approach, bounds and linear constraints are handled separately from nonlinear constraints. A sub problem is formulated by combining the fitness function and nonlinear constraint function using the Lagrangian and the penalty parameters. A sequence of such optimization problems are approximately minimized using the genetic algorithm such that the linear constraints and bounds are satisfied. A sub - problem formulation is defined as

∑∑∑+=+==

++−−=Θmt

mii

mt

miii

m

iiiii xcxcxcssxfsx

1

2

11

)10()(2

)())(log()(),,,(ρλλρλ

Where the components λi of the vector (λ) are nonnegative and are known as Lagrange multiplier estimates. The elements si of the vector (s) are non – negative shifts, and ρ is the positive penalty parameter. The algorithm begins by using an initial value for the penalty parameter (Initial Penalty). The genetic algorithm minimizes a sequence of the sub – problem, which is an approximation of the original problem. When the sub - problem is minimized to a required accuracy and satisfies feasibility conditions, the Lagrangian estimates are updated. Otherwise, the penalty parameter is increased by a penalty factor (Penalty Factor). This results in a new sub – problem formulation and minimization problem. These steps are repeated until the stopping criteria are met. For a complete description of the algorithm, see the references (Conn et al., 1991, 1997).

Our genetic trials use the following prescribed terminologies:

Population type: Double Vector with Populations size = 20 Creation function, Initial population, Initial Score, and Initial range: Default Fitness scaling: Rank Selection function: Stochastic uniform Reproduction; Elite Count: Default (3), Crossover fraction: Default (0.8) Mutation function: Adaptive feasible (due to its benefits) Crossover function: Scattered Migration; Direction: Forward, Fraction: Default (0.2), Interval: Default (20) Stopping criteria (Defaults): Generations: 100, Time limit: Inf., Fitness limit: Inf., Stall generations: 50, Stall time limit: Inf., Function Tolerance: 1e-6, nonlinear constraint tolerance: 1e-6 B. Maximum Torque per Ampere GA with Maximum Efficiency Constrained Efficient example is introduced here, by maximizing the torque per ampere ratio with the voltage as optimizing variable, and with bounds as previous. This is done with maximum efficiency function as constraint. By using the motor characteristics equations, the maximum torque per ampere ratio function (Matlab m-file) is formulated. The maximum efficiency constraint function is implemented using Matlab (m-file) too. The aim of this example is to deduce the required voltage to drive the motor at maximum torque per ampere with the described constraints. Maximum torque per ampere (Function MTAC) as Objective function to be maximized is presented in the following depending on previous relations. The parameters in figure 11 are defined as the following: δ: load angle, α : torque angle, φ: power factor angle. (optimizing variable x(1): terminal voltage (Vs)) .


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Figure11. Phasor diagram of the PMSM Function MTAC = f (x)

Vq = x(1) cos δ (11)

Vd = – x(1) sin δ (12) Vd and Vq are d and q axis stator voltages.

Iq = (Vq / Rs – C2 ( λaf + (Ld / Rs) Vd)) / C3 (13)

Id = Vd / Rs + C2 Lq Iq (14) C2=(ωr / Rs). C3=(1 + C2

2 Lq Ld). Id and Iq are d and q axis stator currents (neglect core loss).

Is = (Iq2 + Id

2)1/2 (15) Is : Stator current value.

α = tan– 1( Iq / Id ) (16)

The mutual flux linkage ( λm ), is the resultant of the rotor flux linkages and stator flux linkages. It is then given as

λm = ( λq2 + λd

2 )1/2. (Wb – Turn) (17)

The core loss, stray loss …etc are negligible, and so the copper loss Pcu is in Eq. (19).

Pcu = 3 Is2 Rs (18)

Pin , Pout are the input and output power. p.f = cos φ = cos ( π/2 + δ – α ) (19)

Pin = 3 x(1) Is p.f = (3/2) (Vd Id + Vq Iq) (20)

Pout = Pin – Pcu (21)

The efficiency η = Pout / Pin (22)

Torque per ampere output function (MTAC) = Te/I s (23)

Function constraints: This optimizing variable (x(1)) is bounded by [0 200]. The maximum efficiency function is used here as nonlinear constraint as follows:


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Function [c,ceq] = f (x) Using the machine’ characteristics relations in previous function

z = Efficiency = Pout / Pin c = [ ] ceq = [ z – 0.99999 ] The following figures present the various characteristics obtained from this function in comparable with such ones at rated terminal voltage. This is to show how much saving in energy.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Electromagnetic Torque (N.m)

Torque Per Ampere Ratio (N.m/Amp.)

Original Torque per Amp. RatioMaximum Torque per Amp. Ratio

Figure12. Maximum torque per ampere in both two cases

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9


Stator C

urrent (A

mp.)

At Max. Torque per Amp.At Original Case

Figure13. Stator current value in both two cases


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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000


Input P

ower (W

att)

At Max. Torque per Amp.At Original Case

Figure14. Input power in both two cases

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1


Efficiency

At Max. Torque per Amp. At Original Case

Figure15. Efficiency in both two cases

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

140


Required Voltage (V

olt)

Required Voltage 4th degree Eqn

Figure16. Required voltage to drive motor at maximum T per A

The relation between the electromagnetic torque (x-axis) and required voltage (y-axis) to drive the motor at maximum torque per ampere could be deduced using the curve fitting facility in Matlab as in equation (24).

y = 0.03*x^4 - 0.1*x^3 + 0.23*x^2 + 5*x + 1.2e+002 (24)


103

From this trial’s figures; it is clear that, the performance characteristics like torque per ampere ratio, stator current, losses, input power and efficiency are highly improved. C. GA maximum efficiency constrained by GA maximum power factor This example improves motor performance by maximizing the efficiency with the voltage also as optimizing variable, and the same bounds as the previous example. This is done with GA maximum power factor as constraint also. By using the motor characteristics equations, the maximum efficiency function (Matlab m-file) is formulated. The maximum power factor constraint function is implemented using Matlab (m-file) too, and then using the Genetic technique. The aim of this example is to deduce the required voltage to drive the motor at maximum efficiency as possible with the described constraints. Maximum efficiency (Function MEFC) as objective function to be maximized is presented in the following depending on previous illustrated relations. (optimizing variable x(1): terminal Voltage (Vs)) . Function MEFC = f (x)

The same previous machine relations

MEFC = Efficiency = Pout/Pin

Function constraints: This optimizing variable (x(1)) is bounded by [0 200]. The maximum power factor function is used here as nonlinear constraint as follows: Function [c,ceq] = f (x) Using the machine characteristics relations in previous functions

zz = Power factor = cos ( π/2 + δ – α )

c = [ ] ceq = [ zz – 0.99999 ] The following figures present the various characteristics obtained from this function in comparison with such ones from original characteristics at rated voltage. This is to show how much performance improvement could be obtained using this example.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1


Efficiency

Original EfficiencyMax. Efficiency

Figure17. Efficiency in the two cases


104

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8


Stator C

urrent (A

mp.)

At Max. EfficiencyAt Org. Condition

Figure18. Stator current in the two cases

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000


Input Pow

er (W

att)

At Max. EfficiencyAt Orig. Condition

Figure19. Input power in the two cases

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120


Required Voltage (Volt)

Required Voltage 5th degree Eqn

Figure20. Required voltage to drive motor at GA maximum efficiency

By the same way, the relation between the electromagnetic torque (x-axis) and required voltage (y-axis) to drive the motor at maximum efficiency could be deduced using the curve fitting facility in Matlab as in equation (25).

y = - 0.916*x^5 + 4.97*x^4 - 9.85*x^3 + 8.58*x^2 - 3.04*x + 118 (25)


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The resulting figures in this trial is very satisfied for overall characteristics especially in efficiency, stator current, losses, and input power. 5. Conclusions This paper addresses new simple dynamic SPMSM modeling to can be used in many topics like in automotive applications, mechatronics, green energy applications, machine drives, etc. This simulation is done with the aid of MATLAB – Simulink to facilitate a good method for machine dynamic behavior prediction for the previous applications. The modeling procedures are described and simulation results are presented. This dynamic model is developed by coupling electrical equations and mechanical equations of the PMSM. Also, there are two proposed trials for performance improvement of PM synchronous motor using genetic algorithm. This idea is done by implementing two genetic algorithm functions with different constraints, same optimizing variable bounds and the same optimizing variable which is the voltage. The last one about GA maximum efficiency constrained by GA maximum power factor has the most powerful effect on all various machine characteristics. Second rank for performance improvement is in maximum torque per ampere GA with maximum efficiency constrained. All functions and simulations are implemented using Matlab environment with the aid of genetic algorithm toolbox. For each trial; the required voltage relation to drive the motor at the desired improved performance characteristics; with the aid of curve fitting facility in Matlab is presented. These algebraic equations may be used directly later without doing the genetic algorithm work each time. All various characteristics obtained are well depicted in the form of comparisons with such ones from original characteristics at rated voltage.

Appendix SPMSM parameters: Lq = 0.0115 H; Ld = 0.0115 H; λaf = 0.283; P = 4; Rs = 6.8 Ω; Vs = 200 V; Ns = 2000 rpm; B = 0.0005416; J = 0.0000144; Td = 0.1698 Acknowledgment I would like to thank Ms. Shaza M. Abd Al Menem for her effort in this research editing References

Bose B. K. 2002: Modern power electronics and AC drives. Prentice Hall, New Jersey. Conn A.R., Gould N. I. M., and Toint Ph. L. 1991. A globally convergent augmented lagrangian algorithm for optimization with

general constraints and simple bounds, SIAM Journal on Numerical Analysis, Vol. 28, No. 2, pp. 545–572. Conn A.R., Gould N. I. M., and Toint Ph. L. 1997. A globally convergent augmented lagrangian barrier algorithm for optimization

with general inequality constraints and simple bounds. Mathematics of Computation, Vol. 66, No. 217, pp. 261-288. Cvetkovski G., Petkovska L., Cundev M. and Gair S. 1998. Mathematical model of a permanent magnet axial field synchronous

motor for GA optimisation, in Proc. ICEM'98, Vol. 2/3, pp. 1172-1177. El Shahat A., and El Shewy H., 2009a. PM synchronous motor control strategies with their neural network regression functions, Journal of Electrical Systems, Vol. 5, No. 4, Dec, Paper No. 6.

El Shahat A., and El Shewy H., 2009b. PM synchronous motor genetic algorithm performance improvement for renewable energy applications, MDGEN11, International Conference on Millennium Development Goals (MDG): Role of ICT and other technologies, December 27 – 29, in Chennai, India.

El Shahat A., and El Shewy H., 2009c. Permanent magnet machines reliability appraisals and fault types review, EP-126, 13th International Conference on Aerospace Science & Aviation Technology, May 26 – 28, ASAT 2009 – Military Technical College, Cairo, Egypt.

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Holland H., 1995. Adaptation in Natural and Artificial Systems, Univ. of Michigan Press, Ann Arbor.. Jang-Mok K. and Seung-Ki S., 1997. Speed control of interior permanent magnet synchronous motor drive for the flux weakening

operation, IEEE Transactions on Industry Applications, Vol. 33, pp. 43-48. Jian-Xin X., Panda S. K., Ya-Jun P., Tong Heng L., and Lam B. H., 2004. A modular control scheme for PMSM speed control

with pulsating torque minimization, IEEE Transactions on Industrial Electronics, Vol. 51, pp. 526-536. Krishnan R., 2006. Electric Motor Drives: Modeling, Analysis & Control, Prentice Hall, Upper Saddle River, New Jersey. Mademlis C. and Margaris N., 2002. Loss minimization in vector-controlled interior permanent-magnet synchronous motor drives, IEEE Transactions on Industrial Electronics, Vol. 49, pp. 1344-1347.

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Pillay P. and Krishnan R., 1989. Modeling, simulation, and analysis of permanent-magnet motor drives. I. The permanent-magnet synchronous motor drive, IEEE Transactions on Industry Applications, Vol. 25, pp. 265-273.

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Wijenayake H.A. and Schmidt P. B., 1997. Modeling and analysis of permanent magnet synchronous motor by taking saturation and core loss into account, Proceedings 1997 International Conference on Power Electronics and Drive System, 26-29 May 1997, Vol. 2, pp. 530-534.

Biographical notes Adel El Shahat is a Research Scientist, ECE Dept., Mechatronics-Green Energy Lab, The Ohio State University, USA. His interests are: Electric Machines, Artificial Intelligence, Renewable Energy, Power System, Control Systems, PV cells, Power Electronics, and Smart Grids. He has more than 30 papers between journals and refereed conferences, and 15 abstracts and posters with one accepted book chapter. Member of IEEE, IEEE Computer Society, ASEE, IAENG, IACSIT, EES, WASET and ARISE. He gains honors and recognitions from OSU, USA 2009, Suez Canal University, Egypt, 2006, ARE 2000, and EES, 1999, Egypt. Hamed El Shewy currently is a Professor of Electrical Machines in the Electrical Power and Machines Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt. He was the previous head of the same department. His research interests include Electric Machines, Power Systems, Power Electronics, and Electric Drives.

Received March 2010 Accepted April 2010 Final acceptance in revised form April 2010

MultiCraft


Vol. 2, No. 2, 2010, pp. 107-116



www.ijest-ng.com


Waves, conservation laws and symmetries of a third-order nonlinear evolution equation

A. Huber 1

1 Address constantly: Prottesweg 2a, A-8062 Kumberg, AUSTRIA

Abstract In this paper a less studied nonlinear partial differential equation of the third-order is under consideration. Important properties concerning advanced character such like conservation laws and the equation of continuity are given. Characteristic wave properties such like dispersion relations and both the group and phase velocities are derived explicitly. In addition, we discuss the non-classical case relating to potential symmetries for the first time. Further, for practical applications in several domains of sciences we discuss in detail approximate symmetries. Finally, as a further new contribution we deduce new generalized symmetries of lower order. Some important notes relating to future intensions are given. Key Words: Nonlinear partial differential equations, evolution equations, symmetries. PACS-Code: 02.30Jr, 02.20Qs, 02.30Hq, 03.40Kf. AMS-Classification: 35K55, 35D35.

1. Introduction, outline the problem Progresses in recent years in the study and analysis of nPDEs have made significant contributions to the understanding of many physical systems. Modelling of physical systems often leads to nonlinear evolution equations of the general form [ ]uKut = , where [ ]uK is a locally defined function (or a nonlinear operator in general) of the function u and its x -derivatives. Well-known evolution equations describing physical phenomena could found in several domains of applied sciences. We restrict the pool of equations to ‘classical’ nPDEs, such like the Korteweg de Vries Equation (Drazin and Johnson, 1989; Witham, 1974; Eilenberger, 1983) and its varieties, the cylindrical KdV (Drazin and Johnson, 1989) and the generalized KdV (Eilenberger, 1983) modeling the propagation of weakly nonlinear waves in dispersive media. Otherwise a well known variety of the KdV is known for a long time, the so-called modified KdV equation (Ablowitz and Clakson, 1991; Dodd et al., 1988), describing nonlinear acoustic waves in anharmonic lattices (Zabusty, 1967 and Alfvén waves in a collisionless plasma (Kakutani, 1969). Here we concentrate our intensions to a less studied unnamed variety of the mKdV Equation (Fung and Au, 1984; Au and Fung, 1984), which differs from the mKdV Equation by a first local-derivative term whereby this term changes basically the equation’s property:

066 23

3=

∂∂

λ+∂∂

−∂

∂+

∂∂

xu

xuu

xu

tu , (1)

with ),( txuu = , ),(3 ∞−∞∈Cu , 0,, ≠tx uuu , Rtx ∈),( , +∈ Rt , 0>t and λ is a non-vanishing parameter. We assume that the function ),( txu acts as the amplitude and is suitable therefore to describe wave propagation depending upon time t in the sense of an evolution equation in which the steepening effect of the nonlinear term is counterbalanced by the (linear) dispersion term(s).

Huber / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 107-116 108

2. Physical properties concerning wave motion Up till now no direct physical applications are known and this is the crucial purpose of study in this paper. In the following without any loss of generality we can set the parameter 1=λ . Normally, the addition of an odd derivative term leads to the fact that the dispersion relation is real for real k , the wave vector. To see this, we assume the linear eq.(1) and introduce the ‘ansatz’ [ ])(exp),( xktiAtxu −ω∝ into eq.(1) to derive the dispersion

relation )6()( 2kkk −=ω=ω for 1=A .

This relation differs from the mKdV that is 3)( kk =ω=ω . We observe that a linear part of the wave vector is overlaid. Further, we deduce two characteristic velocities: The phase velocity pc and the group velocity gc respectively by:

)6(/ 2kkc p −=ω= , kkddcg 2/ −=ω= . (2.1)

gc remains negative for all +∈ Rk and takes positive for all −∈ Rk .

Both velocities tends to ∞→),( gp cc as ∞→k . That means that all waves of large wave numbers (small wavelengths)

propagate in the negative x -direction for all +∈ Rk ; if they exist anyway (similar to the KdV equation). Introducing a velocity field )(xυ and the amplitude field )(xu we deduce the equation of continuity

)3(2)(,0)( 2 −+=υ=υ+ uu

uuuu xx

xt , (2.2)

where )(xu and )(xυ are sufficiently smooth functions. This equation can be linearized for small perturbations about the equilibrium state 0uu = and 0υ=υ so that we can introduce ),(~

0 txuuu += and ),(~0 txυ+υ=υ .

Then the continuity equation (2.2) reduces to a first-order equation 0~~0 =υ+ xt uu with solutions )(~

0 txfu υ−= representing traveling waves. Theorem I: A general equation of the form

0=∂∂

+∂∂

xX

tT (2.3)

is called a conservation law where T and X are known as the density and the flux. If both T and X are integrable on ),( ∞−∞ so that X tends to a constant as ∞→x , then (2.3) can be integrated to

0=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∫∞

∞−

xdTtd

d or equivalently ∫∞

∞−

= .constxdT , (2.4)

where the latter integral is called a constant of motion. This leads to the following Theorem II: The nPDE (1) admits three constants of motion: (i) the conservation of mass, (ii) the conservation of the horizontal momentum and (iii) the conservation of energy so that

∫∞

∞−

=→++∂∂

=∂∂ ).()62( 3 iconstdxuuuu

xu

t xx , (2.5)

∫∞

∞−

=→⎟⎠⎞

⎜⎝⎛ −+

∂∂

=⎟⎠⎞

⎜⎝⎛

∂∂ ).(

21

21

23

21 2242 iiconstdxuuuuu

xu

t xxx , (2.6)

∫∞

∞−

=⎟⎠⎞

⎜⎝⎛ +→++

∂∂

=⎟⎠⎞

⎜⎝⎛ +

∂∂ ).(

41)66(

41 424 iiiconstdxuuuuuu

xuu

t xxxxxxx . (2.7)

All conservation laws are proven by a direct calculation ® . 3. Algebraic group properties (the classical case) In this section we use the classical Lie group analysis in order to derive new classes of solutions otherwise we are interested in the algebraic group behaviour of the nPDE (1). Hint: In what follows we suppress the item ‘classes’; so ‘classes of solutions’ are simply solutions. We take up now the developments given in (Ibragimov, 1984; Olver, 1986; Bluman and Kumei, 1989; Gaeta, 1994; Huber, 2008; Huber, 2009) omitting all technical details.


To use symmetry groups in any application, we first deduce the symmetries of eq.(1). The result is a system of eight linear homogeneous PDEs for the infinitesimals ),( uxii ξ=ξ and ),( uxii φ=φ :

02

2221 =

∂

φ∂=

∂ξ∂

=∂ξ∂

=∂ξ∂

uxuu, (3.1)

03666612 2

3

31

32221211 =

∂∂

φ∂+

∂

ξ∂−

∂ξ∂

+∂ξ∂

+∂ξ∂

−∂ξ∂

−∂ξ∂

−φuxtt

utx

uxt

u , (3.2)

03 12 =∂ξ∂

−∂ξ∂

xt, (3.3)

021

22=

∂

ξ∂−

∂∂φ∂

xux, (3.4)

066 3

32 =

∂φ∂

+∂φ∂

+∂φ∂

+∂φ∂

xxu

xt. (3.5)

The infinitesimals are given by solving the above set of equations (3.1) to (3.5) leading to

),(

3)12(

3

312

321

txuktkk

txkk

−=φ+=ξ

++=ξ . (3.6)

The result shows that the symmetry group of eq.(1) constitutes a finite three-dimensional point group containing translations in the independent variables and scaling transformations. In (3.6) the group parameters are denoted by ik , 3,2,1=i . Eq.(1) admits the three-dimensional Lie algebra L of its classical infinitesimal point symmetries related to the following vector fields uxtxt uxttVVV ∂−∂++∂=∂=∂= )12(3,, 321 . (3.7) These vector fields form a Lie algebra L by: [ ] 2131 123, VVVV +−= , [ ] 232 , VVV = , [ ] 2113 123, VVVV −−= , [ ] 223 , VVV −= . (3.8) For this three-dimensional Lie algebra the commutator table for iV is a )33( ⊗ table whose

thji ),( entry expresses the Lie Bracket [ ]ji VV , given in (3.8). The table is skew-symmetric and the diagonal elements all vanish.

The coefficient kjiC ,, is the coefficient of iV of the thji ),( entry of Tab.1 and the related structure constants can be easily calculated from Tab.1 to give: 1,12,3,1,13,3 2,2,32,1,31,1,32,3,22,3,11,3,1 ===−=−=−= CCCCCC . (3.9) Tab.1 Commutator table for the Lie algebra V of the nPDE (1).

1V 2V 3V

1V 0 0 21 123 VV −−

2V 0 0 2V−

3V 21 123 VV + 2V 0

Theorem III: The Lie algebra of eq.(1) is solvable. Proof: A Lie algebra L is called solvable if 0)( =nV for some 0>n . )1(V is an ideal containing 321 ,, VVV , )2(V is an ideal

containing , 21 VV ; this can be reduced to 0)3( =V . These subgroups are important later to perform a similarity reduction deducing new solutions. The metric ( 33⊗ Cartanian tensor) satisfies:


⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

120

00

L

MOM

L

jig with det(g) = 0 . (3.10)

Since the condition 0)det( =g holds the given algebra is degenerate. Notes: Other useful algebraic group properties are worth to mention: Eq.(1) has no Casimir operator, the group order is three containing seven subgroups.

Alternatively, one can write eq.(3.10) with eq.(3.9) ∑=

=n

ki

kmi

ilkim ccg

1,

.

3.1 The derivation of similarity solutions Let us now discuss the most important similarity solutions for special subgroups to derive new solutions where we restrict the analysis to the most important cases. Case A: If we set the group parameters 11 =k and 03 =k , the similarity variable and the relevant transformation reads as

0=ζ−x and uS = . The related nODE of the third-order is

066 23

3=

ζ+

ζ+

ζ ddS

ddSS

dSd , ∈S ℜ3 ( ∞−∞, ), ),( ∞−∞∈ζ . (3.1.1)

Case B: (the case of traveling waves): Here we assume the parameters to be 121 == kk and 03 =k respectively. This choice means the traveling wave transformation by ζ=− xt and uS = . The nODE of the third-order is calculated to

056 23

3=

ζ+

ζ+

ζ ddS

ddSS

dSd , ∈S ℜ3 ( ∞−∞, ), ),( ∞−∞∈ζ (3.1.2)

which differs from eq.(3.1.1) by a constant factor only Remark: For the given nODEs we both assume the existence of solutions and moreover, functions of the r.h.s. are continuously differentiable following a Lipschitz condition in the considered domain. Also note that we enlarge the domain to ensure the possibility of complex-valued solutions. 4. The non-classical case I – Potential symmetries For more technical details we refer to Ibragimov (1984), Olver (1986), Bluman and Kumei (1989), Gaeta (1994) and Huber (2009). For the nPDE (1) we found the following: The equation admits two possible potential systems 1Ψ and 2Ψ . Both systems can be formulated for two dependent variables iV , 2,1=i and both variables are treated in their derivations w.r.t. the independent variables and are denoted by subscripts: Potential system 1Ψ Potential system 2Ψ

0

026

1

2

213

=∂∂

+

=∂∂

+∂∂

−+

xVu

xu

tV

uu (4)

02

021

233

22

2

22

242

=∂∂

+

=∂

∂+

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−+

xVu

xuu

tV

xuuu

(4.1)

The given systems are related to new symmetries which differ from the symmetry group (3.6) completely: In opposite to the symmetry group (3.6), here, we are confronted with a finite four- dimensional PT; that means that the difference exists in the dimension of the group as well as the number of the elements: Symmetry 1 Symmetry 2

12

41

422

431

3)12(

kuk

tkkxtkk

=φ−=φ

+=ξ++=ξ

(4.2)

2412

41

422

431

3)12(

Vkkuk

tkkxtkk

−=φ−=φ

+=ξ++=ξ

(4.3)

If we examine the infinitesimals upon the dependence of variables starting with capital letter ,V we realize that these infinitesimals are independent of the new potential variables. Comparing this behavior with known results (e.g. the KdV Equation) we see that both systems do not contribute to potential symmetries (further examples are also well-known; e.g. the nonlinear Reaction Diffusion Equation, the cylindrical Korteweg de


Vries Equation and the Burgers Equation) and tells us that potential symmetries are rare symmetries but can occur in connection with some equations. 5. Approximate symmetries In this section we follow Huber (2009), Ibragimov (1985) and Ibragimov (1994), respectively and our intension is to deduce new results without referring too much theory. Let us introduce ε as a small parameter 10 <<ε< determining the strength of the nonlinearity of eq.(1) so that we can write without loss of generality 1=λ :

066 23

3=

∂∂

+∂∂

ε−∂

∂+

∂∂

xu

xuu

xu

tu , ∈u ℜ3 ( ∞−∞, ), ∞<<−∞ x , 0>t (5)

to ensure the complete solution-manifold. Then, approximate symmetries follow by

,)(3

)(3

43

4

483

76322

775

3311

ukkuk

kktkk

xkkk

xktkk

+ε+−=φ

+ε++=ξ

⎟⎠

⎞⎜⎝

⎛++ε+++=ξ

(5.1)

representing an eight-dimensional approximate symmetry group in the first-order approximation. The generating vector fields for this model read:

.,

34,,

,,3

43

,,

8765

4321

xxtut

uxuttx

VxttVVV

uVxtutVVV

∂ε=∂⎥⎦

⎤⎢⎣

⎡ε⎟⎠⎞

⎜⎝⎛ ++∂ε=∂ε=∂ε=

∂ε=∂⎟⎠⎞

⎜⎝⎛ ++∂⎟

⎠⎞

⎜⎝⎛−+∂=∂=∂=

(5.2)

Possible reductions can be calculated by combining several sub-groups of (5.2). For the present calculations we choose three cases of interest: Case A: Combining 521 VVV ⊗⊗ the transformation and the defining equation for S are:

01

=ζ−ε+

−xt , Su = , 0')65()1(''' 22 =ε+ε−ε++ SSS , )(ζ= SS , ζ= ddSS /' . (5.3)

Case B: Combining 621 VVV ⊗⊗ one derives at:

0)1( =ζ−ε−− xt , Su = , 0'))1(65(''')1( 23 =ε+ε++ε− SSS . (5.4) Case C: Combining 651 VVV ⊗⊗ leads to the defining equation for )(ζS :

01

=ζ−ε+

ε−

xt , Su = , 0')651()1(''' 2223 =ε+ε+−ε++ε SSS , 0≠ε . (5.5)

The result is a similarity representation of the solution(s) linearly depending upon the perturbation parameter ε and also in second and fourth-order dependence. Note that for the given nODEs the same assumptions as in Chapter 3.1 have been made. 6. The non-classical case II: General symmetries (GS) We find it advisable mentioning some basic notes. It is obvious from Lie theory that point symmetries are a subset of generalized symmetries, Abramowitz and Stegun (1972) as well as Noether (1971) and Klein (1918). The determination of the characteristics for the general case follows by a similar algorithm as in the case of point transformations (PT) in the classical case. Classical symmetries of a (n)PDE (assumed to be in a general form 0=Δ ) are PT which guarantee the invariance of the solution space and so, PT are created by infinitesimal transformations. The determining equations for the characteristics αGS are consequences of the relation 00=Δυ =ΔGSpr

r, (6)

where GSpr υr

denotes prolongation of the vector field GSυ and ‘GS’ means generalize symmetry. The main difference however is the fact that in general the characteristics depend upon derivatives of an infinite order. If the order is equal to identity we arrive at the so-called contact transformations. By increasing the order of derivatives 1>n we shall find higher order GS. In case of eq.(1) we found GS of the first order depending on the first derivative: ( ) xtx ukuuutxGS 11 ,,,, = . (6.1) This symmetry also changes from the symmetries given in (3.6), (4.3) and (5.1). Here we are confronted with a one-


dimensional finite group of transformations where the second part xu ∂∂ / is related to scaling and/or stretching operations (more precisely dilatations). For the case 2=n by assuming second partial derivatives we found ( ) xxtttxxx ukuuuuutxGS 12 ,,,,,, = (6.2) as a quite similar result. At this stage we remark that higher cases are difficult to deal with. 7. Analysis and results Now we use (3.1.1) and (3.1.2) respectively to derive new solutions. An analogues equation can be obtained if we investigate solutions for which ),( txFu = , ,)(3 DCF ∈ 3RD∈ is a domain. Introducing the frame of reference

)(),( ξ=Utxu , tx α−=ξ , ∈ξ ℜ1, R∈α \ 0 into eq.(1) leads to 0''6'6''' 2 =α−++ UUUUU and the prime means derivation w.r.t. ξ . Due to the similar structure of the latter nODE it is sufficient to consider (3.1.1) and it can be shown that the traveling wave reduction results into eq.(3.1.2) exactly. We summarize the analytical properties of eq.(3.1.1) resulting in a polynomial of the fourth-order in Tab.2. Thus we have to treat four cases depending upon the choice of the integration constants iK leading to new solutions: For the Case A, 121 == KK appropriate numerical standard procedures are necessary. Case B: 11 =K and 02 =K leads to a sine amplitude with a pure imaginary modulus:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−−

++−+⎟

⎠⎞

⎜⎝⎛ ++−= kxsnxu ,

28171)222(2

))22(24(2121022111)(

3/23/13/2

3/23/13/2

3/2 , (7)

with )30722602784( 3/23/1573

12

12

1 −+−=k ; that is numerically ik 4362,0= . The constant factor under the root sign is approximately 73,0≈ and the first factor takes 5,2≈ . Using a complex modulus transformation Abramowitz and Stegun (1972) we convert the function eq.(7) numerically into a real-valued function so that we have finally a pure local dependence [ ]kxsdxu ,8,03,2)( −= with the real modulus 9166,0=k . (7.1) A graphical plot for different values of the modulus shows Fig.1. In case of 1=k the function degenerates to the hyperbolic sine function as usual. For practical calculations a trigonometric series (Erdelyi, 1981), is sometimes useful so that one can write in a more convenient form:

∑∞

=−

− π−

+−

π=

112 2

8,0)12(sin

1)1(

'6,4)(

21

nn

nn

Kx

nq

qKkk

xu with the nome )([ K/'Kexpq π−= , (7.2)

valid in every strip of the form τπ<π Im)2/Im( 21Kx and K is the complete elliptic integral of the first kind.

Tab.2 Algebraic properties of the eq.(3.1.1) after converting. All zeros of the polynomial of the fourth-order )(uP .

The relating nODE of the first-order becomes )(226

024

ζ−ζ=+++∫ uuu

du and 0ζ is an arbitrary

constant of integration.

Case Integration

constants iK Polynomial )(uP Zeros iu of )(uP

A 121 == KK 226 24 +++ uuu iu 56,018,02,1 ±−= , iu 40,218,04,3 ±=

B 11 =K , 02 =K uuu 26 24 ++ 01 =u , 3/23/12 22 −=u , 3/2

3/1

4,3 2

)31(2321 iiu

±+±−=

C 01 =K , 12 =K 26 24 ++ uu 732,1 −±= iu , 734,3 +±= iu

D 021 == KK 24 6uu + 02,1 =u , 64,3 iu ±=


A formal power expansion up to order three yields 432 ]1[)1(11,0)1(18,0)1(36,128,2)( −+−+−−−+= xOxxxxu . (7.3)

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

0 0.5 1 1.5 2 2.5 30

1

2

3

4

Fig.1 Left: Some solution curves of the function (7) for different values of the modulus: 9166,0=k (solid line), 5,0=k (short dotted line), 1=k (dotted line) - this case degenerates to the sinh function. Right: The undisturbed solution eq.(7.1), (solid line) by comparison with the second-order approximation calculated from the power expansion (7.3) respectively, (dotted line). This can be compared with the exact solution in Fig.2 and therefore we conclude that the expansion is valid in the domain 23,0 << x . One can also make use of the Weierstrassian expansion [14]. Case C: 01 =K and 12 =K leads to complex-valued solutions for ∈x ⊆+, 0>x :

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −−= k

xisnAixu ,

2)517(

4)( with

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++

+=

172395174

172395

20A . (7.4)

Note that the modulus is 1>k , that is ( ) 2255,34/175421 2/1 =+=k . To proceed further one has to use suitable transformations so that 10 << k holds. Refering to [19] we calculate

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −= ',

2)517(

)( kx

tnAxu , (7.5)

where the new modulus 'k is given by 9954,01' 2 =−= kk . We also note that for 1<x , 0≠x the function (6.6) takes real value. The development of this function in the complex plane shows Fig.2 by using different initial values and Fig.3 shows the real-valued function considering the assumption 1<x .

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 60

1

2

3

4

5

6

-4 -2 0 2 4 6 8

-4

-2

0

2

4

6

8

Fig.2. The behavior of the branch lines near a zero of the function (7.5) in the complex plane with 2≈A . Different values for the complete integral of the first kind are used: Right: The complete integral is assumed to 2=K , middle: 4=K and left: 6=K .


0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

Fig.3 The real-valued tn-function for different values for the modulus and 2≈A ; solid line: 9954,0'=k , dotted line: 5,0'=k and short dotted line: 1'=k . Note: The tn-function degenerates to the sinh function. Case D: 021 == KK :

))62(cosh)62(sinh(6

)))1(6(sinh))1(6((12

6

12)(6262

)1(6

xxc

xxCosh

ee

exux

x

−+

+++=

−=

+

, (7.6)

where c means ( ))62(cosh)62(sinh − and the constant 0x is assumed to be the identity. For this solution we calculate a closed-form analytical expression in terms of infinite series

k

k

k

k

xk

ec

xk

e

xukk

kk

∑

∑∞

=

++

∞

=

++

+

=

0

6120

612

!32

!32

)(22

3

22

, (7.7)

where the convergence ∈∀ x ⊆ can be proven by using the d’Alembertian criterion immediately. For large values of the argument, say ∞→x the following asymptotic formula holds:

32

3

6

2

6 11)6(

)36(36(361)6(

)6(6126

12~)( ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

+

−++⎟

⎠⎞

⎜⎝⎛

+

−+

+ xO

xcecc

xc

cc

exu . (7.8)

It is proven that the following limiting behaviour holds: 0

1)(lim→

≈x

xu and ∞±→=

xxu 0)(lim ; the first and the second derivation are finite

at the point 0=x . It is remarkable that the analysis of eq.(3.1.2) also leads to similar results depending upon the choice of the integration constants. Although eq.(3.1.2) admits the case of traveling motion which is concern to the appropriate similarity variable and no classical wave propagation is observed. Finally, we discuss the equations relating to approximate symmetries eq.(5.3), (5.4) and (5.5). The first and the second equations lead to a linear ODE of the third-order by setting 0=ε , that is 0'5''' =+ SS and the prime means derivation w.r.t. ζ . An analytical solution of the first-order approximation is therefore:

][5sin5cos5

1)( 321 ε++ζ+ζ=ζ OcccS , (7.9)

where the ic are arbitrary constants. This case covers the traveling wave solution for 0)1/( =ζ−ε+− xt if we set 0=ε . For 0→ζ the function(s) takes a finite value but remains indefinite as ±∞→ζ . For solution eq.(6.5) a closed-form analytical

expression can be obtained:

∑∞

=

+−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+Γ−

+−

ζ=ζ0

21

2

)22()5(

)!2(5)1()(

k

kkkk

kkS , (7.10)

where the convergence is also proven immediately. A graphical overview for the traveling motion represents Fig.4. To analyze eq.(5.5) we expand about the fourth-order term by setting 1=ε and a solution of the fourth-order approximation is given: [ ] 43222 )3016()611(21)( ε+εζ++εζ++ε+−=ζ OS , (7.11)


leading to a quadratic dependence. We show some solution curves by using different initial conditions in Fig.4. Here, the traveling motion as well as the quadratic dependence is remarkable.

-3 -2 -1 0 1 2 3

-0.50

0.51

1.52

-3 -2 -1 0 1 2 30

100

200

300

Fig.4 Left: Solutions of eq.(7.9) representing traveling waves by using different values of the integration constant ic , especially 121 == cc and 13 =c (solid line), 121 == cc , 03 =c (dotted line); the perturbation parameter is assumed to be 0=ε . Right: The quadratic dependence of eq.(7.11) in fourth-order approximation. 8. Conclusion remarks – main propositions - outlook In this paper a less studied nPDE of the third-order is under consideration. Let us emphasise in brief the results of the analysis. Usually by introducing special similarity variables, say )(),( ξ=Utxu with tx α−=ξ , one would expect traveling motion as a result. For this special nPDE we did not found such solutions in the general case. However, by choosing suitable values of the constants involved the traveling behaviour results. In addition the nPDE admits conservation laws derived for the first time similar to the KdV and analogues equations. The derived conservation laws are connected directly with physical measurable quantities like mass, the horizontal momentum and the energy. The dispersion relation, the group and the phase velocity as further physically important quantities are in agreement with many other evolution equations. It is important to point out that we apply a classical group analysis to generate new solutions for the first time. The non-classical case, also performed for the first time leads to the expected traveling wave result. A further important contribution shows that the nPDE (1) does not allow potential symmetries (similar to the KdV). It is known that similarity ‘ansätze’ of the form tx α−=ξ does not guarantee the existence of physically important wave propagation; the nPDE (1) is a notable example for this behavior. We also show the existence of approximate and generalized symmetries to the first time.Finally, it is seen that the nPDE(1) does not belong to the hierarchy of the KdV Equation. Naturally, the next step is to prove the integrability by assuming that the nPDE(1) can be written in form of an equation of motion, the so-called Lax equation [ ]BLut ,= , where L means the Schrödinger operator (as a first assumption) which commutate with the commutator B . If so, we then can show that the nPDE(1) is integrable completely possessing infinitely many laws of conservation and has a related Bäcklund system. References Drazin P., Johnson R., 1989. Solitons: An Introduction, Cambridge University Press. Witham G., 1974. Linear and Nonlinear Waves, Wiley, N.Y, p.577. Eilenberger G., 1983. Solitons, Springer Verlag, Berlin, p.140. Ablowitz M., Clarkson P., 1991. Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press. Dodd R., Eilbeck J., Gibbon J., Morris H., 1988. Solitons and Nonlinear Wave Equations, Academic Press. Zabusky N., 1967. A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, Proc. Symp. on

Nonlinear Partial Differential Equations (ed. by W. Ames), Academic Press, Boston, p.223. Kakutani T., Ono H., 1969. Weak nonlinear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Jpn. 31, p.1246. Fung P. C. W., Au C., 1984. A series of a new analytical solution to the nonlinear equation 066 2 =λ+++ xxxxxt yyyyy , J. Math.

Phys. Vol. 25, No. 5, p.1370. Au C., Fung P. C. W., 1984. A KdV soliton propagating with varying velocity, J. Math. Phys. 25, p.1364. Ibragimov N., 1994. Lie Group Analysis, Vol. III, CRC Press, Inc. Olver P., 1986. Applications of Lie Groups to Differential Equations, Springer. Bluman G., Kumei S., 1989. Symmetries and Differential Equations, Springer Gaeta G., 1994. Nonlinear Symmetries and Nonlinear Equations, Kluwer, Acad. Press. Huber A., 2008. A note on new solitary and similarity class of solutions of a fourth order nonlinear evolution equation, Appl.

Math. and Comp., Vol. 202, p.787


Huber A., 2009. The Cavalcante-Tenenblat equation – Does the equation admit a physical significance? Appl. Math. and Comp. 212, p.14.

Ibragimov N., 1985. Transformation Groups Applied to Mathematical Physics, Reidel Publ., Dortrecht. Ibragimov N., 1994. Sophus Lie and harmony in mathematical physics on the 150th anniversary of his birth, Math. Intel. 16, p.20. Erdelyi A., 1981. Higher Transcendent Functions, N. Y., Krieger. Abramowitz M., Stegun I., 1972. Handbook of Mathematical Functions, Tenth Printing. Noether E., 1971. Transport Theory Stat. Phys. Vol. 1, p.186. Klein F., 1918. Über Differentialgesetzte für die Erhaltung von Impuls und Energie in der Einsteinschen Gravitationstheorie,

Nachr. Ges. Wiss. Göttingen Math. Phys.2, p.171. Biographical notes Dipl.-Ing. Dr. techn. Huber Alfred is a distinguished lecturer at the Institute of Theoretical Physics – Computational Physics at the Technical University Graz, Austria following his habilitation treatise. He did his diploma thesis titled ‘Systematic in the physics of elementary particles focusing the quarkonium states’ in the field of elementary particle physics at the former Institute of Nuclear Physics at the Technical University Graz, Austria. He completed his scientific education with the doctoral programme of technical sciences at the Institute of Chemical Technology of Inorganic Compounds at the Technical University Graz, Austria subject to nuclear solid state physics and advanced electrochemistry. Thesis titled ‘Synthesis and characterization of doped γ-manganese dioxides’. Also the author has a learnt vocation for a chemical assistant at the Research Centre of Electron Microscopy at the former Technical High School Graz, Austria. He is the author of 27 articles which have appeared in world-wide renowned scientific journals. His research interests are nonlinear partial differential equations (nPDE) of higher order with applications especially in physics and chemistry. The author developed several new algebraic procedures for solving nPDE. Special interests are further given in classical and non-classical symmetry methods, nonlinear transformations and the application of nonlinear methods in describing electrochemical interfaces, nonlinear wave propagation and further nonlinear topics of advanced character. Received March 2010 Accepted April 2010 Final acceptance in revised form April 2010

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Vol. 2, No. 2, 2010, pp. 117-135



www.ijest-ng.com © 2010 MultiCraft Limited. All rights reserved

Magnetic field effect on a three-dimensional mixed convective flow with

mass transfer along an infinite vertical porous plate

N. Ahmed

Department of Mathematics, Gauhati University, Guwahati -781014 , Assam, INDIA E-mail: [email protected]

Abstract An analytical solution to the problem of the MHD free and forced convection three dimensional flow of an incompressible viscous electrically conducting fluid with mass transfer along a vertical porous plate with transverse sinusoidal suction velocity is presented. A uniform magnetic field is assumed to be applied transversely to the direction of the free stream. The expressions for skin friction at the plate in the direction of the main flow and the rate of heat transfer and mass transfer from the plate to the fluid are obtained in non-dimensional form. The amplitudes of the perturbed parts of these fields and the skin friction at the plate are presented in graphs and the effects of different physical parameters like Hartmann number M, Reynolds number R and the Schmidt number S on these fields are discussed and the results obtained are physically interpreted. Keywords: Viscous , incompressible, electrically conducting, sinusoidal suction 1. Introduction Many natural phenomena and technological problems are susceptible to MHD analysis. Geophysics encounters MHD characteristics in the interactions of conducting fluids and magnetic fields. Engineers employ MHD principle, in the design of heat exchangers pumps and flow meters, in space vehicle propulsion, thermal protection, braking, control and re-entry, in creating novel power generating systems etc. From technological point of view, MHD convection flow problems are also very significant in the fields of stellar and planetary magnetospheres, aeronautics, chemical engineering and electronics. Model studies of the above phenomena of MHD convection have been made by many. Some of them are Sanyal and Bhattacharya (1992), Ferraro and Plumpton (1966), Cramer and Pai (1973) and Nikodijevic et al. (2009). On the other hand, along with free convection currents, caused by the temperature difference, the flow is also affected by the difference in concentrations on material constitutions. Many investigators have studied the phenomena of MHD free convection and mass transfer flow of whom the names of Acharya et al. (2000), Bejan and Khair (1985), Babu and Prasad Rao (2006), Raptis and Kafousias (1982), Singh and Singh (2000) as well as Singh et al. (2007), etc. are worth mentioning. Investigations of the problems of laminar flow control are being done by many researchers due to its importance in the field of aeronautical engineering in view of its application to reduce drag and hence the vehicle power requirement by a substantial amount. The development of this subject has been compiled by (Lachman 1961). Theoretical and experimental investigations indicate that the transition from laminar to turbulent flow which causes the drag co-efficient to increase may be prevented by suction of the fluid, by the application of transverse magnetic field and by heat and mass transfer from the boundary layer to the wall. To obtain any desired reduction in the drag by increasing suction alone is uneconomical as the energy consumptions of the suction pumps will be more. Therefore the method of “cooling of the wall” in controlling the laminar flow together with the application of suction has become more useful and hence received the attention of many workers. The effect of the flow caused by the periodic suction velocity perpendicular to the main flow when the difference between the wall temperature and free stream temperature gives rise to buoyancy force in the direction of the free stream on heat transfer characteristics has been investigated by Singh et al. (1978). Ahmed and Sarma (1997) have extended the work of Singh et al. (1978) to the case when the medium is porous. Gupta and Johari (2001) have analyzed the effects of magnetic field on the three-dimensional forced flow of an incompressible viscous fluid past a porous plate. Singh and Sharma (2001) have studied the effect

Ahmed / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 117-135 118

of the porosity of the porous medium on the three-dimensional Couette flow and heat transfer. The same authors (Singh and Sharma, 2002) have also studied the effect of the periodic permeability on the free convective flow of a viscous incompressible fluid through a highly porous medium. The effect of transverse sinusoidal injection velocity distribution on the three dimensional free convective Couette flow of a viscous incompressible fluid in slip flow regime under the influence of heat source has been studied by Jain and Gupta (2006). Ahmed et al. (2006) have obtained an analytical solution to the problem of the three-dimensional free convective flow of an incompressible viscous fluid past a porous vertical plate with the transverse sinusoidal suction velocity taking into account the presence of species concentration. Singh (1991) has studied the effect of a uniform transverse magnetic field on the free convection flow of an electrically conducting viscous incompressible fluid past an infinite vertical porous plate with sinusoidal suction velocity and uniform free stream. As the present author is aware till now no attempt has been made to study the effect of a transverse magnetic field on a mixed convective flow of an incompressible viscous electrically conducting fluid with mass transfer along a vertical porous plate with transverse sinusoidal suction velocity taking into account the effect of Ohmic and viscous dissipations together. Such an attempt has been made in the present work. Though the flow geometry of the present work and that of the work of Singh (1991) are common, yet this paper is not a routine extension of the paper (Singh, 1991). Both papers differ in several aspects such as the forms of suction velocities and the combinations of dimensionless substitutions.

2. Basic equations The equations governing the steady motion of an incompressible viscous electrically conducting fluid in presence of a magnetic field are – the equation of continuity (law of conservation of mass) :

div q =0 (1) the Gauss’s law of magnetism (law of conservation of magnetic flux) :

div 0=B (2) the momentum equation (law of conservation of momentum) :

( ) gqBJpqq +∇+×

+∇−=∇ 21. υρρ

(3)

the energy equation (law of conservation of energy) :

( )[ ]σ

φρ2

2. JTkTqCp ++∇=∇ (4)

the species continuity equation (law of conservation of species) : ( ) CDCq 2. ∇=∇ (5) the Ohm’s law ( Current density and electric field relation) :

[ ]BqEJ ×+= 0σ (6) All physical quantities are defined in the Nomenclature. We now consider the steady free and forced convection flow of an incompressible viscous electrically conducting fluid taking into account the species concentration past a vertical porous plate with transverse sinusoidal suction velocity as mentioned earlier by making the following assumptions.

(i) All the fluid properties except the density in the buoyancy force term are constants. (ii) A magnetic field of uniform strength B0 is applied transversely to the direction of the free stream. (iii) The magnetic Reynolds number Rm is small so that the induced magnetic field can be neglected. (iv) The level of species concentration in the fluid is very low so that the Soret and Dufour effects can be neglected. (v) wT T ∞> and wC C∞>

We introduce a coordinate system ( )zyx ,, with X-axis vertically upwards along the plate, Y-axis perpendicular to it and directed into the fluid region and Z-axis along the width of the plate.

Let wkvjuiq ˆˆˆ ++= be the fluid velocity at the point ( )zyx ,, and jBB ˆ0= be the applied magnetic field.

The transverse sinusoidal suction velocity is taken as follows:

( ) ⎥⎦

⎤⎢⎣

⎡+−=

LzCosVzvw

πε10 (7)


which consists of basic steady distribution 0

V with superimposed weak distribution 0zV Cos

Lπε confined in the boundary layer

only. Here negative sign indicates that the direction of the suction velocity is towards the plate. . This suction velocity ( )wv z is

applied transversely to the plate and weak distribution 0zV Cos

Lπε will have no role in the outer edge of the boundary layer. Due

to application of suction at the surface, the fluid particles at the edge of the boundary layer will have a tendency to get displaced towards the plate surface. Therefore 0v V→ − at y →∞ . This phenomenon is clearly supported by the equation of continuity.

The velocity ,temperature and concentration fields are independent of x , because an asymptotic flow has been considered but the flow itself is three dimensional due to cross flow.

Fig. 1 The flow configuration

Z

∞→

→

TT

Uu

∞→CC

p p∞→

∞→y

0=u

YO

0Vv −→

0→w

g

0=w

wvv =

jBB ˆ0=

u

v

w

j

wC C=

wT T=

k

X i


With the foregoing assumptions and under the usual boundary layer and Boussinesq approximations, the equations (1), (3), (4) and (5) reduce to:

Equation of continuity 0=∂∂

+∂∂

zw

yv

(8)

x-component of momentum equation 2 2

202 2

u u p u uv w g B uy z y zx

ρ ρ μ σ⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂

+ = − − + + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂∂⎝ ⎠ ⎝ ⎠ (9)

At the outer edge of the boundary layer the parallel component u U= , the free stream velocity. Since there is no large velocity gradient here, the viscous term in the equation (9) vanishes for small μ and hence for the outer flow, we have

200

pg B U

xρ σ∞∞

∂= − − −

∂ (10)

It is emphasized by (Schlichting 1950) that in case of hot vertical plate, the pressure in each horizontal plane is equal to the gravitational pressure. That is p p∞= . Hence (10) reduces to

200 p g B U

xρ σ∞

∂= − − −

∂ (11)

By eliminating the pressure term from the equations (9) and (11), we obtain

( ) ( )2 2

202 2

u u u uv w g B U uy z y z

ρ ρ ρ μ σ∞

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂+ = − + + + −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

(12)

The Boussinesq approximation gives

( ) ( )T T C Cρ ρ ρ β ρ β∞ ∞∞ ∞ ∞− = − + − (13)

On using (13) in the equation (12) and noting that ρ∞ is approximately equal to ρ , we obtain the momentum equations as follows:

x -component: ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+−+−=∂∂

+∂∂

∞∞ 2

2

2

2

)(zu

yuCCgTTg

zuw

yuv υββ ( )uUB

−+ρ

σ 20 (14)

y -component: ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−=∂∂

+∂∂

2

2

2

21zv

yv

yp

zvw

yvv υ

ρ (15)

z -component:zww

ywv

∂∂

+∂∂

ρσ

υρ

wBzw

yw

zp 2

02

2

2

21−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−= (16)

Energy equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+∂∂

2

2

2

2

zT

yT

zTw

yTv α

222

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

+⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+yw

zv

zu

yu

Cp

υ⎥⎥⎦

⎤

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+22

2zw

yv

( )2

2 20

p

B U u wC

σρ

⎡ ⎤+ − +⎢ ⎥⎣ ⎦ (17)

Species continuity equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=∂∂

+∂∂

2

2

2

2

zC

yCD

zCw

yCv

(18) The symbols involved have been defined in the Nomenclature. The relevant boundary conditions are:


:0=y ,0=u ,wvv = 0=w , ,wTT = wCC = , (19)

:y →∞ ,Uu = ,0Vv −= ,0=w ,∞= TT ∞= CC , p p∞= (20) We introduce the following non-dimensional quantities:

,Lyy = ,

Lzz =

0Vuu = , ,

0Vvv =

0

,UUV

= 0V

ww =

,∞

∞

−−

=TTTT

w

θ w

C CC C

φ ∞

∞

−=

−, Pr υ

α= , Sc

Dυ

= , ( )

20

wLg T TGr

V

β ∞−= , 2

⎟⎠⎞

⎜⎝⎛

=

L

ppυρ

( )

20

wLg C CGm

V

β ∞−= , ( )∞−

=TTC

VEwp

20 , 2

0

20

VBMρ

υσ= , 0Re V L

υ= , 2

⎟⎠⎞

⎜⎝⎛

= ∞∞

L

ppυρ

The non-dimensional forms of (8), (14), (15), (16), (17) and (18) are:

0∂ ∂+ =

∂ ∂v wy z

(21)

( )2 2

2 2

1u u u uv w Gr Gm MR e U uy z R e y z

θ φ⎡ ⎤∂ ∂ ∂ ∂

+ = + + + + −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ (22)

2 2

2 2 2

1 1 ⎡ ⎤∂ ∂ ∂ ∂ ∂+ = − + +⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦

v v p v vv wy z Re y Re y z

(23)

2 2

2 2 2

1 1 ⎛ ⎞∂ ∂ ∂ ∂ ∂+ = − + + −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

w w p w wv w MRe wy z Re z Re y z

(24)

2 22 22 2

2 2

1 2θ θ θ θ ⎡ ⎤ ⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎛ ⎞ ⎛ ⎞+ = − + + + + +⎢ ⎥ ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎪ ⎪⎣ ⎦ ⎩ ⎭

E u u E v wv wy z PrRe y z Re y z Re y z

( ) 2

2 2⎛ ⎞∂ ∂+ + + − +⎜ ⎟∂ ∂⎝ ⎠

E v w MERe U u wRe z y

(25)

2 2

2 2

1e

φ φ φ φ⎛ ⎞∂ ∂ ∂ ∂+ = +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

v wy z ScR y y

(26)

With relevant boundary conditions: :0=y ,0=u ( ),1 zCosv πε+−= 0=w ,1=θ 1=φ , (27)

:∞→y ,Uu = ,1−=v ,0=w ,0=θ 0=φ , p p∞= (28)

3. Method of solution We assume the solutions of the equations (21) to (26) to be of the form: ( ) ( ) ( )2

10 0, εε ++= zyuyuu (29)

( ) ( ) ( )210 0, εε ++= zyvyvv (30)

( ) ( ) ( )210 0, εε ++= zywyww (31)

( ) ( ) ( )210 0, εε ++= zypypp (32)

( ) ( ) ( )210 0, εεθθθ ++= zyy (33)

( ) ( ) ( )210 0, εεφφφ ++= zyy (34)


with ,0 ∞= pp 00 =w

Substituting these in the equations (21) to (26) and equating the co-efficient of same degree terms and neglecting ,2ε we get the following sets of the differential equations. Zeroth-order equations:

00 =dydv

(35)

( )2

0 00 0 0 02

1du d uv Gr Gm MRe U udy Re dy

θ φ= + + + − (36)

( )2

22 20 00 0 0 02

1 2d d E Ev v u MERe U udy PrRe dy Re Reθ θ ′ ′= + + + − (37)

20 0

0 21d dv

dy Sc Re dyφ φ

= (38)

First order equations:

011 =∂∂

+∂∂

zw

yv

(39) 2 2

01 1 11 1 1 12 2

1uu u uv Gr Gm MReuy y Re y z

θ φ⎛ ⎞∂∂ ∂ ∂

− + = + + + −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (40)

2 21 1 1 1

2 2 2

1 1v p v vy Re y Re y z

⎛ ⎞∂ ∂ ∂ ∂− = − + +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

(41)

2 20 01 1 1 1

1 2 2

1 2d du duEvy dy PrRe y z Re dy dy

θθ θ θ⎡ ⎤∂ ∂ ∂− + = + +⎢ ⎥∂ ∂ ∂⎣ ⎦

( )0 10 1

4 2dv vE MERe U u uRe dy y

∂+ + −

∂ (42)

2 21 1 1 1

12 2 2

1 1w p w w MRewy Re z Re y z

⎡ ⎤∂ ∂ ∂ ∂− = − + + −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

(43)

2 21 1 10

1 2 2

1dvy dy ScRe y zφ φ φφ ⎛ ⎞∂ ∂ ∂

− + = +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ (44)

With conditions: :0=y ,00 =u ,10 −=v ,10 =θ 10 =φ , ,01 =u ,1 zCosv π−=

,01 =w ,01 =θ 01 =φ (45)

:∞→y ,0 Uu = ,10 −=v ,00 =θ 01 =φ , ,01 =u

,01 =v ,01 =w 01 =p ,01 =θ 01 =φ (46) The solutions of the equations (35) and (38) subject to boundary conditions (45) and (46) are respectively

10 −=v (47) Re

0Sc yeφ −= (48)

In order to solve the coupled equations (36) and (37) under the above boundary conditions, we note that E<1 for all the incompressible fluids and it is assumed that the solutions of these equations to be of the form:

( ) =yu0 ( ) ( ) ( )20100 0 EyEuyu ++ (49)

( ) =y0θ ( ) ( ) ( )20100 0 EyEy ++ θθ (50)

Substituting from (49) and (50) in the equations (36) and (37) and equating the co-efficients of the same degree terms and neglecting the term of 0(E2), the following differential equations with corresponding boundary conditions are derived. 2 2

00 00 00 00ScReyu Reu MRe u MRe U Gr Re GmReeθ −′′ ′+ − = − − − (51)


201 01 01 01u Reu MRe u Grθ′′ ′+ − = − (52)

00 00 0PrReθ θ′′ ′+ = (53)

( )22 201 01 00 00PrRe Pru MPrRe U uθ θ′′ ′ ′+ = − − − (54)

Subject to the boundary conditions:

:0=y ,000 =u ,001 =u ,100 =θ 001 =θ (55)

:∞→y ,00 Uu = ,001 =u ,000 =θ 001 =θ (56) The solutions of these equations under the boundary conditions (55) and (56) are as follows:

00PrReyeθ −= (57)

00 1 2 3Pr Re y ScRe y Re yu U A e A e A e λ− − −= + + + (58)

2 2 201 0 1 2 3

Pr Rey Pr Rey ScRey ReyE e E e E e E e λθ − − − −= + + +

( ) ( ) ( )4 5 6

Re Pr Sc y Re Sc y Re Pr yE e E e E eλ λ− + − + − ++ + + (59)

2 201 0 1 2 3

Re y Pr Re y Pr Re y ScRe yu Gr F e F e F e F eλ− − − −⎡= − − −⎣

( )

( ) ( )

24 5

6 7

Re Pr Sc yRey

Re Sc y Pr Re y

F e F e

F e F e

λ

λ λ

− +−

− + − +

⎤− −⎥

− − ⎥⎦ (60)

where

2411 M++

=λ ,( )1 2

GrARe Pr Pr M

=− −

,( )2 2

GmARe Sc Sc M

=− −

, UAAA −−−= 213 ,2

21

1AB = ,

22

2 4 2A ScB

Sc Pr=

−,

23

3 4 2AB

Prλ

λ=

−, 6B

( )1 32A A Pr

Prλ λ=

+, 4B

( )1 22A A PrSc

Sc Pr Sc=

+, 5B

( )( )2 32A A Sc

Sc Sc Prλ

λ λ=

+ + −2

11 2 2,

2AD

Re Pr= ,

( )22

2 22 2AD

ScRe Sc Pr=

−,

( )23

3 22 2AD

Re Prλ λ=

−, 4D

( )1 2

22A A

ReSc Pr Sc=

+, 5D

( )( )2 3

22A A

Re Sc Sc Prλ λ=

+ + −, 6D

( )1 3

22A A

Re Prλ λ=

+, 2

1 1 1,E MPr Re D PrB= − − 22 2 2E PrB MPr Re D= − −

23 3 3E PrB MPr Re D= − − , 2

4 4 4E PrB MPrRe D= − − , 25 5 5E PrB MPrRe D= − − , 2

6 6 6E PrB MPr Re D= − − ,

( )6543210 EEEEEEE +++++−= , 1F( )

02 2

ERe Pr Pr M

=− −

, 2F( )

12 24 2Pr

EPr M Re

=− −

, 4F

( )3

2 24 2E

M Reλ λ=

− −, 5F

( ) ( ) 4

22

ERe Pr Sc Pr Sc M

=+ − + −

, 3F

22 24 2

ERe Sc Sc M

=− −

6F( ) ( )

52 2

ESc Sc M Reλ λ

=+ − + −

,( ) ( )

67 2 2

EFPr Pr M Reλ λ

=+ − + −

, ∑=

=7

10

k

FkF

4. Cross flow solution We shall first consider the equations (39), (41) and (42) for ( )zyv ,1 , ( )zyw ,1 and ( )1 ,p y z which are independent of main

flow component 1u , temperature field 1θ and concentration field 1φ .The suction velocity.


( )zCosvw πε+−= 1 consists of a basic uniform distribution –1 with superimposed weak sinusoidal distribution zCosπε .

Hence the velocity components pandwv, are also separated into mean and small sinusoidal components 111, pandwv . We

assume 111, pandwv to be of the following form:

( ) zCosyvv ππ 111 −= (61)

( ) zSinyvw π111 ′= (62) ( )2

1 11p Re p y Cos zπ= (63) On substitution of (61), (62) and (63), the equation (39) is satisfied and the equations (41) and (42) reduce to the following differential equations:

2 1111 11 11

Re pv Rev vπ

π′

′′ ′+ − = − (64)

( )2 211 11 11 11v Rev MRe v Re pπ π′′′ ′′ ′+ − + = − (65)

The relevant boundary conditions for these equations are:

:0=y ,111 π=v 011 =′v (66)

:∞→y ,011 =v 011 =′v (67) The solution of the equations (64) and (65) subject to the boundary conditions (66) and (67) is

( )[ ]yy eev ξξ ξξξξπ

−− −−

=1

11 (68)

Where 2 2 24

2Re Reλ λ πξ + +

= ,2 2 24

2Re Reλ λ πξ + +

=1 1 4 ,

2Mλ − +

= 1 1 4

2Mλ + +

=

Hence the solutions for the velocity components 1v , 1w and pressure 1p are as follows:

1v ( )[ ] zCosee yy πξξξξ

ξξ −− −−

=1

(69)

1w ( )[ ] zSinee yy πξξπ

ξξ ξξ −− −−

= (70)

( )1 1 12y yReP e eξ ξξξ ξ ξ

π ξ ξ− −⎡ ⎤= −⎣ ⎦−

(71)

Where, 22 2

1 MRe Reξ π ξ ξ= + + − , 2 2 21 MRe Reξ π ξ ξ= + + −

5. Solutions for flow, concentration, and temperature field We shall now consider the equations (40), (43) and (44). The solutions for the velocity component u, concentration field φ and

temperature field θ are also separated into mean and sinusoidal components 111 ,, φθu . To reduce the partial differential equations

(40), (43) and (44) into ordinary differential equations, we consider for the following forms for 111 ,, φθu .

( ) zCosyuu π111 = (72)

( ) zCosy πθθ 111 = (73)

( ) zCosy πφφ 111 = (74)

Using the expressions for 1111 ,,, φθuv in (40), (43), (44), we get the following ordinary differential equations

( )2 211 11 11u Reu MRe uπ′′ ′+ − + 11 0 11 11Rev u Gr Re Gm Reπ θ φ′= − − − (75)


211 11 11PrReθ θ π θ′′ ′+ − ( )2

11 0 0 11 0 112 2PrRev EPr u u EMRe Pr U u uπ θ ′ ′ ′= − − + − (76) 2

11 11 11ScReφ φ π φ′′ ′+ − 11 0ScRe vπ φ′= − (77) with the boundary conditions

:0=y 0,0,0 111111 === φθu (78)

0,0,0: 111111 ===∞→ φθuy (79) The solution of the equation (77) subject to the boundary conditions (78) and (79) is

( ) ( )11 0 1 2

ScRe yScRe yayH e H e H e ξξφ − +− +−= + + (80) Where

2 2 242

ScRe Sc Rea

π+ += ,

( )( )2 2

1 2 2

Sc ReHSc Reξ

ξ ξ ξ ξ π=

− + −

( )( )2 2

2 2 2

Sc ReHScRe

ξξ ξ ξ ξ π

−=

− + −, ( )210 HHH +−=

Now in order to solve the coupled equations (75) and (76), the solutions of these two equations are assumed to be of the form:

( ) ( ) ( ) ( )21011 0 EyEfyfyu ++= (81)

( ) ( ) ( ) ( )21011 0 EyEyy ++= ψψθ (82)

Substituting these in the equations (75) and (76) and equating the coefficients of similar terms and neglects E2, we get the following differential equations: ( )2 2

0 0 0f Ref MRe fπ′′ ′+ − + 11 00 0 11Rev u GrRe GmReπ ψ φ′= − − − (83)

( )2 21 1 1f Ref MRe fπ′′ ′+ − + 1 11 01GrRe Rev uψ π ′= − − (84)

20 0 0PrReψ ψ π ψ′′ ′+ − 11 00PrRevπ θ ′= − (85)

21 1 1PrReψ ψ π ψ′′ ′+ − ( )2

00 0 00 02 2Pru f MRe Pr U u f′ ′= − + − 11 01PrRevπ θ ′− (86) with boundary conditions:

:0=y ,00 =f ,01 =f ,00 =ψ 01 =ψ (87)

:∞→y ,00 =f ,01 =f ,00 =ψ 01 =ψ (88) The equations (83) to (86) are solved subject to the boundary conditions (87) and (88), but not presented here for the

sake of brevity.

6. Skin friction and heat and mass flux.

The non-dimensional skin-friction in the direction of the free stream at the wall y = 0 is given by

Uvyu

y

0

0

ρ

μτ =

⎥⎦

⎤∂∂

−

= 0

1

y

uRe y =

⎤∂= − ⎥∂ ⎦

= ( ) ( )0 111 0 0u u Cos zRe

ε π′ ′⎡ ⎤− +⎣ ⎦

( )0 1 , , , , , ,Q Pr Sc Re Gr Gm E M Cos zτ ε π= + (89)

Where ( )11

1

0uQ

Re′

= − , ( )0

0

0uRe

τ′

= −

The heat flux from the plate to the fluid in terms of Nusselt number is given by

( )0 0 0

1

p w y y

k TNuy Pr Re yv C T T

θρ ∞ = =

⎛ ⎞ ⎤∂ ∂= − = −⎜ ⎟ ⎥∂ ∂− ⎝ ⎠ ⎦

( )0 2 , , , , , ,Nu Q Pr Re M E Gr Gm Sc Cos zε π= + (90)


where ( ) ( )110 0 2

01 0 ,Nu QPrRe Pr Re

θθ

′′= − = −

The mass flux at the wall y = 0 in terms of Sherwood number Sh is given by

( )0 0w y

D CShyV C C∞ =

⎛ ⎞∂= − ⎜ ⎟∂− ⎝ ⎠ 0

1

yScRe yφ

=

⎛ ⎞− ∂= ⎜ ⎟∂⎝ ⎠

( ) ( )0 111 0 0 Cos z

ScReφ εφ π− ′ ′⎡ ⎤= +⎣ ⎦

( )31 , ,Q Sc Re M Cos zε π= + (91)

where ( )11

3

0Q

ScReφ′

= −

7. Discussion of results In order to get the physical insight into the problem, the numerical values for 21,, QQτ and 3Q which are respectively the skin friction and the amplitudes of the first order skin friction, first order Nusselt number and first order Sherwood number at the plate are obtained for different values of the physical parameters involved in the problem and these are demonstrated in graphs. The investigation is restricted to Prandtl number Pr equal to 0.7 which corresponds to air . The value of each of G and Gm has been chosen as 10. The Schmidt number S are chosen in such a way that they represent the diffusing chemical species of common interest in air and water (for example S = 0.24 for H2, S = .60 for H2O, S = 0.78 for NH3 and S = 1 for CO2). That is in the present investigation the air is considered as the diffusing medium (solvent) and H2, H2O, NH3 and CO2 as diffusing species (solutes). The free stream velocity U is taken to be equal to 1 and E is selected to be 0.05. The values of the other physical parameters namely M and Re are chosen arbitrarily. Figures 2 and 3 depict the variation of the skin friction τ at the plate under the influences Re, M, and Sc. It is observed from Figure 2 that an increase in M leads to a decrease in the magnitude τ of the skin friction. That is there is a reduction in the viscous drag (shearing stress) on the plate due to the application of the transverse magnetic field. This result has a good agreement with the physical realities. Because the application of a transverse magnetic field to a flow of an electrically conducting fluid has a retarding effect to the fluid motion and hence the increase of the velocity gradient in the direction normal to the plate is prevented due to imposition of magnetic field for which the sharing stress at the plate is reduced. It is inferred from Figure 3 that an increase in Schmidt number results in a decrease in τ . That is the mass diffusion causes the viscous drag on the plate to increase and it clearly supports the physical situation as the mass diffusion accelerates the fluid motion for which the velocity gradient at the plate increases causing growth in drag on the plate. It is also seen from these two figures that τ , the magnitude of the skin friction at the plate becomes very large for small Reynolds number whereas there is a fall in τ

for large R . In other words the frictional force on the plate becomes high for high viscosity. This result is clearly supported by the Newton’s law of viscosity. There is a clear indication from the Figures 2 and 3 the magnetic field as well as mass diffusion ceases to affect the sharing stress at the plate for low viscosity or large suction. The change of behaviour of 1Q , the amplitude of the first order skin friction at the plate against the Reynolds number R under the

effects of M and S is presented in Figures 4 and 5. . Figure 4 shows that the magnitude of 1Q decreases due to application of the

magnetic field. It is seen that 1Q is negative for small and moderate values of R and it takes its positive values for large R. That is the direction of the first order shearing stress at z=0 for large R is opposite to that of the first order skin friction at z=0 for small and moderate R. Fig.5 clearly shows that an increase in the Schmidt number S leads to a decrease in the magnitude of 1Q . There is

an indication from figure 5 that the magnitude of 1Q first decreases as R increases for small R and then it slowly and steadily increases as R. The variation of the amplitude Q2 of the first order Nusselt number 1Nu is demonstrated in Figure 6. It is observed from Figure 6 that Q2 decreases under effect of the magnetic field. The same figure also shows that Q2 is diminished by the frictional property of the fluid. In other words the rate of first order heat transfer (for z=0) from the plate to the fluid drops due to application of the transverse magnetic field or due to small suction..


τ

M = 1

M = 0

M = 3

M = 5 M = 7

Re

Fig .2: Skin friction τ versus Re for Pr = 0.7, Sc = 0.60, Gr=10, G m= 10, U = 1, E = 0.05

-70

-60

-50

-40

-30

-20

-10

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


-45

-40

-35

-30

-25

-20

-15

-10

-5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Re

τ

S= 0.22

S= 0.60

S= 0.78

S= 1

Fig. 3: Skin frictionτ versus Re for Pr =0.7, M = 1, Gr=10, Gm = 10, U = 1, E = 0.05


The effects of the Reynolds number R, the Hartmann number M and the Schmidt number S on Q3, the amplitude of the first order Sherwood number are shown in figures 7 and 8. These two figures show that there is a steady fall in

3Q when M and R are

increased whereas 3Q steadily increases for increasing Schmidt number.

Fig.4: The amplitude Q1 of the first order skin friction 1τ versus Re for P =0.7, S = 0.60, G=10, Gm = 10, U = 1, E = 0.05

-4.5 -4

-3.5 -3

-2.5 -2

-1.5 -1

-0.5 0

0.5 1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Re

1Q

M = 3

M = 5

M = 7


Fig.5: The amplitude Q1 of the first order skin friction 1τ versus Re for P =0.7, M = 1, G=10, Gm = 10, U = 1, E = 0.05

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Re

Q2

M=3

M=5

M=7

Fig.6: The amplitude Q2 of the first order Nusselt number 1Nu versus Re for P =0.7, S = 0.60, G=10, Gm = 10, U = 1, E = 0.05

S = 0.22

S = 0.60 S = 0.78

S = 1

-9-8-7-6-5-4-3-2-100.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Re

Q1


Q3

M=0

M=1 M=3

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

M=7M=5

Re

Fig.7: The amplitude Q3 of the first order Sherwood number 1Sh versus Re for S=0.60, G = 10, Gm = 10, U = 1, E = 0.05


That is to say that low viscosity or magnetic field effect or an increase in mass diffusivity leads to a steady drop in the first order mass flux (for z=0) from plate to the fluid. The same figures further indicate that Q3 is not affected by the Prandtl number Pr .

7.1 Comparison of results To compare the results of our paper with those cited in the references, we choose the paper by Gupta and Johari (2001) Table 1: The variation of the numerical values of the first order skin friction F1 at the plate against the Hartmann number M and the velocity ratio α for the paper by Gupta and Johari (2001) for Re=1

α M =0 M =2 M =4 0.5 1.0 1.5

1.6999 3.6796 5.9675

1.6967 3.6655 5.9341

1.6939 3.6524 5.9032

From the above table it is observed that the first skin friction F1 at the plate decreases as the Hartmann number M increases for Re=1.In the present paper also it is seen that for Re=1, the magnitude of the first order skin friction at the plate decreases as M increases (fig.4).Thus we see that the results concerning first order skin friction under effect of the parameter M are in a good agreement for both the papers. It may be mentioned that in the velocity ratio does not appear in the present work. It is adjusted in the boundary conditions of the flow problem under consideration. 8. Conclusions

The results obtained from our investigation lead to the following conclusions:

i) The transverse magnetic field or low viscosity or high Schmidt number causes a reduction to the viscous drag on the plate.

Fig. 8: The amplitude Q3 of the first order Sherwood number versus Re for M =1, G = 10, Gm = 10, U = 1, E = 0.05

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Re

3Q

22.0=S

1=S

78.0=S

60.0=S

Q3


ii) The transition from laminar to turbulent flow may be prevented up to a certain extent due to the application of a transverse magnetic field.

iii) The application of the transverse magnetic field or a reduction in mass diffusivity for small R causes the magnitude of amplitude of the perturbed part of the skin friction at the plate to fall.

iv) The rate of first order heat transfer (for z=0) from the plate to the fluid drops due to application of the transverse magnetic field or due to small suction..

v) The low viscosity or magnetic field effect or an increase in mass diffusivity leads a steady drop in the first order mass flux(for z=0) from plate to the fluid.

Nomenclature

B [-] magnetic induction vector

0B [Tesla] intensity of the applied magnetic field

pC [ Jkg K ] specific heat at constant pressure

C [ kmol/m 3 ] species concentration

∞C [ kmol/m 3 ] concentration of the fluid at infinity

wC [ kmol/m 3 ] concentration of the fluid at the plate D [ m 2 s-1 ] coefficient of chemical molecular mass diffusivity E [ -] Eckert number

0E [ - ] electric field Gr [ - ] Grashof number for heat transfer Gm [ - ] Grashof number for mass transfer g [m s-2 ] acceleration due to gravity

ˆˆ ˆ, ,i j k [-] unit vectors along the co-ordinate axes

J [-] electric current density

BJ × [-] Lorentz force per unit volume

σ

2J

[W m-3 ] Ohmic dissipation per unit volume

k [ W/mK] thermal conductivity L [m] wave length of the periodic suction velocity M [ -] Hartmann number Pr [ -] Prandtl number p [ N m-2 ] pressure

p∞ [ N m-2 ] gravitational pressure

q [- ] velocity vector Re [ -] Reynolds number Sc [- ] Schmidt number

T [K] temperature

∞T [K] temperature of the fluid at infinity

wT [K] temperature of the fluid at the plate

U [m s-1 ] free stream velocity

( )wvu ,, [m s-1 ] components of q u [-] dimensionless velocity in x-direction v [-] dimensionless velocity in y-direction

wv [ m s-1 ] suction velocity


0V [ m s-1 ] mean suction velocity w = dimensionless velocity in z-direction

( )zyx ,, [m] Cartesian coordinates zy, [-] dimensionless co-ordinates perpendicular to the free stream velocity

Greek Symbols α [m2 K-1 ] thermal diffusivity β [ K-1 ] co-efficient of volume expansion for thermal expansion

β [m3 /k mol] the volumetric co-efficient of expansion with concentration ε [-] small reference parameter ( 1<<ε ) φ [-] dimensionless concentration ϕ [W m-3 ] viscous dissipation per unit volume θ [-] dimensionless temperature ρ [kg/ m3 ] density of the fluid

ρ∞ [kg/ m3 ] density of the fluid in the free stream

σ [Ω -1 m-1 ] electrical conductivity

υ [ m2 s-1 ] kinematic viscosity Acknowledgement: The author is thankful to UGC for supporting this work under MRP(F. No. 36-96/2008(SR)) References Ahmed, N. and Sarma, D.,1997. Three-dimensional free convective flow and heat transfer through a porous medium, Indian J.

Pure Appt. Math., Vol .28, No.10,pp.1345-1353. Ahmed, N., Sarma, D. and Barua, D.P., 2006. Three dimensional free convective flow and mass transfer along a porous vertical

plate; Bulletin of the Allahabad Mathematical Society, Vol.21, pp.125-141. Acharya, M. , Dash, G.C. and Singh, L.P.,2000. Magnetic field effects on the free convection and mass transfer flow through

porous medium with constant suction and constant heat flux; Indian J: Pure Appl. Math.,Vol. 31 ,No.1, pp.1-18, Babu, D.C. and Prasad Rao, D.R.V.,2006. Free convective flow of heat and mass transfer past a vertical porous plate, Acta

Cienica Indica, Vol.32M, No.2, pp.673-684. Bejan, A. and Khair,K.R. ,1985. Mass transfer to natural convection boundary layer flow driven by heat transfer, ASME, J. Heat

Transfer, Vol.107, pp.1979-1981. Cramer, K.P. and Pai, S.I., 1973. Magneto Fluid Dynamics for Engineers and Applied Physics, New York: McGraw-Hill Book Co. Ferraro, V.C.A and Plumpton, C., 1966. An introduction to Magneto Fluid Mechanics, Oxford, Clarendon Press. Gupta, G.D. and Johari, Rajesh.,2001. MHD three dimensional flow past a porous plate, Indian J. Pure Appl. Math., Vol.32,

No.3,pp. 371-386. Jain, N.C. and P. Gupta, P. , 2006. Three dimensional free convection Couette flow with transpiration cooling, Journal of Zhejiang

University Science A, Vol.7, No.3, pp. 340-346. Lachman, G.V. ,1961.Boundary Layers and flow control, its principles and application, Vol. I and II, Pergamon Press,. Nikodijevic, D., Boricic,Z., Milenkovic,D. and Stamenkovic, Z. ,2009. Generalized similarity method in unsteady two-

dimensional MHD boundary layer on the body which temperature varies with time, Int.J.Eng.Sci.Tech., Vol.1, No.1, pp.206-215.

Raptis, A and Kafousias, N.,1982 . Magneto hydrodynamic free convective flow and mass transfer through a porous medium bounded by an infinite vertical porous plate with constant heat flux, Can. J. Phys., Vol. 60, pp.1725-1729.

Sanyal, D.C and Bhattacharya, S. ,1992. Similarly solutions of an unsteady incompressible thermal MHD boundary layer flow by group theoretic approach, In. J. Eng. Sci. Vol. 30, pp. 561-569.

Schlichting, H.,1950. Boundary- Layer Theory, Mc Graw-Hill, Inc,New York Singh, N.P. and Singh, Atul Kr. , 2000. MHD effects on Heat and mass transfer in flow of a viscous fluid with induced magnetic

field, Indian. J. Pure Appl. Phys. ,Vol.38, pp.182-189. Singh, N. P., Singh Atul Kr. and Singh, Ajay Kr. , 2007.MHD free convection MHD mass transfer flow part of flat plate, The

Arabian Journal for Science and Engineering, Vol.32, No. 1A, pp.93-112. Singh, P., Sharma, V.P. and Misra, U.N, 1978. Three-dimensional free convection flow and heat transfer along a porous vertical

plate, Appl. Sci. Res., Vol. 34, pp. 105-115.


Singh, K.D. and Sharma, Rakesh, 2001. Three dimensional Couette flow through a porous medium with heat transfer, Indian J. Pure, Appl. Math., Vol.32, No.12, pp.1819-1829.

Singh, K.D. and Sharma, Rakesh, 2002.Three dimensional free convective flow and heat transfer through a porous medium with periodic permeability, Indian J. Pure. Appl. Math., Vol.33, No.6, 941-949.

Singh,K.D.,1991. Hydro magnetic free convective flow past a porous plate, Indian. J. Pure Appl.Math.,Vol. 22, No.7, pp.591-599. Biographical notes Dr. N. Ahmed is a Reader (Associate Professor) in the Department of Mathematics, Gauhati University, Guwahat-781014, INDIA. He has been doing his research work in the field of Fluid Dynamics and Magneto Hydrodynamics since 1985. More than 50 research papers have been published in internationally reputed journals to his credit. Three research scholars have obtained Ph. D degree under his supervision. He is the principal investigator of a UGC major research project.

Received, March 2010 Accepted, March 2010 Final acceptance in revised form, April 2010

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Five-phase induction motor drive for weak and remote grid system

1Shaikh Moinoddin, 2Atif Iqbal* and 3Elmahdi M. Elsherif

1,3 Department of Electronics & Computer Engineering, Sebha University, Sebha, LIBYA 2*Department of Electrical Engineering, Aligarh Muslim University, Aligarh, INDIA

*Corresponding Author (e-mail: [email protected], Atif Iqbal; Tel. +919411210372)

Abstract Multi-phase (more than three-phase) motor drive systems have attracted much attention in recent years due to some inherent advantages which they offer when compared to the three-phase counterpart. Presently the grid power available is only limited to three-phase so the supply to multi-phase motors is invariably given from power electronic converters. Thus the paper focuses on the inverter controlled five-phase induction motor drive system for variable speed applications. The paper presents inverter control method for five-phase variable speed induction motor drives. The proposed solution may be employed in the applications not requiring very precise position and speed control such as water pumping especially in weak grid system with poor power quality. The inverter is operated in different operating modes with conduction angle varying from 180° to 108° conduction modes and the performance is evaluated in terms of the harmonic contents in the output phase voltages . It is shown that optimum performance is achieved by operating inverter at 144° conduction mode. Experimental and analytical results are included in the paper.

Keywords: Voltage source inverter, ten-step operation, conduction modes, power grid, Induction motor drive

1. Introduction Variable speed electric drives predominately utilise three-phase machines. However, since the variable speed ac drives require a power electronic converter for their supply (in vast majority of cases an inverter with a dc link), the number of machine phases is essentially not limited. This has led to an increase in the interest in multi-phase ac drive applications, since multi-phase machines offer some inherent advantages over their three-phase counterpart. Interesting research results have been published over the years on multi-phase drives and detailed review is available in Singh (2002), Jones and Levi (2002), Bojoi et al. (2006), Levi et al. (2007), Levi (2008a) and Levi (2008b). Major advantages of using a multi-phase machine instead of a three-phase machine are higher torque density, greater efficiency, reduced torque pulsations, greater fault tolerance, and reduction in the required rating per inverter leg (and therefore simpler and more reliable power conditioning equipment). Noise characteristics of multi-phase drives are better when compared three-phase drive as demonstrated by Hodge et al. (2002) and Golubev and Ignatenko (2000) . Higher Phase number yield smoother torque due to the simultaneous increase of the frequency of the torque pulsation and reduction of the torque ripple magnitude, as presented byWilliamson and Smith (2006) and Apsley (2006). Higher torque density in a multi-phase machine is possible because fundamental spatial field harmonic and space harmonic fields can be used to enhance total torque as presented by Xu et al. (2001a) and Xu et al. (2001b), Shi et al. (2001), Lyra and Lipo (2002), Duran et al. (2008) and Arahal and Duran (2009). This advantage of enhanced torque production stems from the fact that vector control of the machine’s flux and torque, produced by the interaction of the fundamental field component and the fundamental stator current component, requires only two stator currents (d-q current components). In a multi-phase machine, with at least five phases or more, there are therefore additional degrees of freedom, which can be utilised to enhance the torque production through injection of higher order current harmonics. The stability analysis of five-phase drive system for harmonic injection scheme is carried out by Duran et al. (2008) for both concentrated winding and distributed winding machines. It was concluded that the 3rd harmonic injection not only enhances the torque production but also offers a more stable control structure. The studies on multi-phase drive system carried out so far is for high performance variable speed applications. Multi-phase drive is seen as a serious contender for niche applications such as ship propulsion, traction, electric vehicles and in safety critical

Moinoddin et al. / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 136-154 137

applications requiring high degree of redundancy. However, general purpose drive applications using multi-phase machines are not yet investigated in detail. This paper advocates the use of a five-phase drive system for general purpose applications such as water pumping in remote and weak grid locations where the power quality is not adequate for operating sophisticated microprocessor based controllers due to their stringent power quality requirements. Further, the costs of such high performance drive systems are too high to be borne by poor farmers in remote locations. The question then arises why five-phase drive is at all required not conventional three-phase drive. The answer lies in the fault tolerant characteristic, reliable and higher efficiency of five-phase drive compared to three-phase drive (detailed by Apsley et al. (2006), Arhal and Duran (2008)). The power electronic converters supplying multi-phase drives are controlled using advanced digital signal processors (DSP) and Field programmable Gate Arrays (FPGA). Many modulation techniques implemented using DSPs and FPGAs are proposed in the literature for controlling the multi-phase power electronic converters, Iqbal and Levi (2005). In contrast this paper proposes simple, reliable and cheap controller circuit using analogue components and square wave operation of a five-phase voltage source inverter (VSI). In environment of weak grid with poor power quality, stepped operation of voltage source inverter may be considered as more viable solution in comparison to PWM mode. This paper thus analyses the performance of a five-phase induction motor drive supplied by a five-phase VSI operating in square wave mode. Conduction angle is varied from 180° leading to ten-step operation to 108°. A detailed comparison of inverter performance on various conduction angles is elaborated. It will be shown that a trade off exist between the fundamental output voltage and their harmonic content. Analytical, simulation and experimental results are provided. The paper is organised in ten different sections; first section details the literature review and laid down the need of the proposed drive topology, modelling of a five-phase VSI is illustrated in the second section, third section elaborates the proposed five-phase induction motor drive structure. Fourth section describes the gate drive circuit for the five-phase VSI and fifth section discusses the experimental results obtained for testing of the fabricated inverter, the analysis of results, and compares the magnitude of torque pulsation in a three-phase and a five-phase drive system then sixth section concludes the finding followed by the references.

2. Modelling of a five-phase VSI-review

Power circuit topology of a five-phase VSI, is shown in Figure 1. Each switch in the circuit consists of two power semiconductor devices, connected in anti-parallel. One of these is a fully controllable semiconductor, such as a bipolar transistor or IGBT, while the second one is a diode. The input of the inverter is a dc voltage, which is regarded further on as being constant. The inverter outputs are denoted in Figure 1 with lower case letters (a,b,c,d,e), while the points of connection of the outputs to inverter legs have symbols in capital letters (A,B,C,D,E), The shift between each phase voltage is (360/5)=72.

Vdc

va vb ve

VA VB VE

ae be een

N

Figure 1. Five-phase voltage source inverter power circuit.

Phase-to-neutral voltages of the star connected load are most easily found by defining a voltage difference between the star point n of the load and the negative rail of the dc bus N. The following correlation then holds true:

nNeE

nNdD

nNcC

nNbB

nNaA

vvv

vvv

vvv

vvv

vvv

+=+=+=+=+=

(1)

Since the phase voltages in a star connected load sum to zero, summation of the equations (1) yields


( )( )EDCBAnN vvvvvv ++++= 51 (2)

Substitution of (2) into (1) yields phase-to-neutral voltages of the load in the following form:

( ) ( )( )EDCBAa vvvvvv +++−= 5154

( ) ( )( )EDCABb vvvvvv +++−= 5154

( ) ( )( )EDBACc vvvvvv +++−= 5154 (3)

( ) ( )( )ECBADd vvvvvv +++−= 5154

( ) ( )( )DCBAEe vvvvvv +++−= 5154

3. Five-phase drive structure A simple open-loop five-phase drive structure is elaborated in Figure 2. The dc link voltage is adjusted from the controlled rectifier by varying the conduction angles of the thyristors. The frequency of the fundamental output is controlled from the IGBT based voltage source inverter. The inverter is operating in the quasi square wave mode instead of more complex PWM mode. Thus the overall control scheme is similar to a three-phase drive system. Since the inverter is operating in square wave mode the analogue circuit based controller is much simpler and cheaper compared to more sophisticated digital signal processor based control schemes. This type of solution is very cheap and convenient for use in coarse applications such as water pumping. These types of applications do not require fast dynamic response of drive systems and thus the need of high performance control schemes do not arise. The power quality of the remote locations in developing countries such as Indian subcontinents are not adequate for reliable and durable operation of sensitive microprocessors/microcontrollers/digital signal processors based controllers. It is thus intended to develop cheap and robust controller based on simple, and reliable analogue circuit components for such locations. The subsequent section describes the implantation issues of control of a five-phase voltage source inverter. The motivation behind choosing this structure lies in the fault tolerant nature of a five-phase drive system (Apsley et al., 2006).

5-p h

ase

IMDC

L IN

K

dcV f

Lf

Cf

Controlled Rectifier Inverter

Figure 2. Five-phase induction Motor Drive structure.

4. Analogue circuit based five-phase voltage source inverter To test the stepped operation in various conduction modes, a five-phase IGBT based prototype inverter is built in the laboratory, the block diagram of the control circuit of the Inverter is presented in Figure 3.


PhaseShifting Circuit

230V50Hz

9-0-9V

To gates of N-bankMosfets

+

-

Leg-Voltages

Non-Inv.Sh. Tr.&Wave

Shaping Ckt-1

Isolation & DriverCircuitP-1st

Non-Inv.Sh. Tr.&Wave

Shaping Ckt-n

Isolation& DriverCircuitP-nth

To gates of P-bank

IGBTs

InvertingSh.Tr.&Wave

Shaping Ckt-1

Isolation& DriverCircuitN-1st

InvertingSh.Tr.&Wave

Shaping Ckt-n

Isolation& DriverCircuitN-nth

Figure 3. Block diagram of the complete control circuit.

Electric supply is taken from a single-phase grid and is converted to 9-0-9 V using a transformer, which is fed to the phase shifting circuit, to provide appropriate phase shift (i.e. 72 between each phase) for operation at various conduction angles. Here the phase shift is achieved at 50 Hz. However, for other than 50 Hz the required phase shift can be achieved using variable resistances in RC network of phase shifting circuit. The phase shifted signal is then fed to the inverting/non-inverting Schmitt trigger circuit and wave shaping circuit block which contains zero crossing detector. The processed signal is then fed to the isolation and driver circuit which is then finally given to the gate of IGBTs. There are two separate circuits for upper and lower legs of the inverter. NOT gates are not used to give complemented gate drives for lower leg devices, because complement of 144 is 216 i.e. lower devices shall be ‘ON’ for 216. 5. Results and discussion 5. 1. Five-phase VSI testing Experiment is conducted for stepped operation of five-phase voltage source inverter with 180° (classical) and 144° (proposed) conduction modes for star connected five-phase resistive load at first. A single-phase supply is given to the control circuit through the phase shifting network. The output of the phase shifting circuit provides the required five-phase output voltage by appropriately tuning it as shown in Figure 4. These five-phase signals are then further processed to generate the pulses to the gate drive circuit. It is important to emphasise here that the poor power quality of the supply can be seen from the distorted waveforms of Figure 4. This is the power quality available in the laboratory setup, and thus the importance of the proposed solution can be understood at the remote locations.


Figure 4. Five-phase output obtained from phase shifting network.

5. 1.1. 180° Conduction mode Each switch is assumed to conduct for 180°, leading to the operation in the ten-step mode. Phase delay between firing of two switches in any subsequent two phases is equal to 360°/5 = 72°. The corresponding phase voltages thus obtained are shown in Figure 5, keeping the dc link voltage at 60 V. The waveform is in ten step and is in full compliance with the finding of Ward and Harer (1969).

Figure 5. Output phase ‘a-d’ voltages for 180° conduction mode.

There are two systems of line voltages in a five-phase system namely adjacent and non-adjacent with phase shifts of 72° and 144°, respectively. Adjacent line voltages have lower magnitudes compared to the non-adjacent line voltages. Non-adjacent line voltage thus obtained is shown in Figure 6. All currents are measured using ac/dc current probe giving output of 100 mV/A.


Figure 6. Non-adjacent line voltage for 180° conduction mode with dc link voltage equal to 180 V.

5. 1.2. 144° conduction mode

The gate drive signal is such that each power switch remains on for 144° (or 80% duty cycle) and remains floating for 36° (or 20% duty cycle). This mode thus provides an inherent dead band in the switching of two power switches of the same leg. The output from the wave shaping circuit and the gate drive for two legs are shown in Figure 7 and Figure 8, respectively. The corresponding phase-to-neutral output voltage for phase ‘a’ is shown in Figure 9. Non-adjacent line voltage is presented in Figure 10.

Figure 7. Output of wave shaping circuit for 144° conduction mode for leg A-B.


Figure 8. Gate Drive signals for leg A-B for 144° conduction mode.

Figure 9. Output phase ‘a-d’ voltage for 144° conduction mode.

Figure 10. Non-adjacent line voltage for 144° conduction mode with dc link voltage equal to 180 V.

5. 2. Harmonic profile and torque pulsation reduction

5. 2.1. Inverter output waveform analysis

This section presents the comprehensive analysis of experimental results. The performance of two different conduction modes are elaborated in terms of the harmonic content in the phase voltages, line voltages and the distortion in the ac side line current.

The Fourier series of the phase-to-neutral voltage for 180° conduction mode is obtained as;


++++++= ......13sin13

111sin

11

19sin

9

17sin

7

13sin

3

1sin

2)( ttttttVtv DC ωωωωωω

π (4)

From (4) it follows that the fundamental component of the output phase-to-neutral voltage has an RMS value equal to

DCdc VVV 45.02

1 ==π

. (5)

The Fourier series of the phase-to-neutral voltage for 144° conduction mode is obtained as;

( )( ) ( )( )

∑∞

=

−

−

−=

..3,2,112

12sin10

12cos2

n

dc

n

tnnV

tv

ωπ

π (6)

From (6) it follows that the fundamental component of the output phase-to-neutral voltage has an RMS value equal to

DCdc VVV 428.010

cos2

1 =

= ππ

(7)

The loss in fundamental voltage in 144° conduction mode is of the order of 4.89% compared to 180° conduction mode. This loss will affect the loss of torque in the driven machine and subsequently the laod will be affected. However, the drop in the torque is not very significant compared to the benefits obtained due to better harmonic performance. The harmonic analysis of, line voltage and input ac side line current is carried out for different conduction modes and the resulting waveforms are shown in Figures 11-14. The input side AC current of converter contains odd and even harmonics (Figure 14) for 144˚ conduction mode while a typical spectrum is depicted for 180˚ conduction mode.

Figure 11. Spectrum of non-adjacent line voltage (Vac) for 180° conduction mode for dc link voltageof 180 V.

.

Figure 12. Spectrum of input side current for 180° conduction mode.


Figure 13. Spectrum of non-adjacent line ‘Vac’ voltage for 144° conduction mode for dc link voltage of 180 V.

Figure 14. Spectrum of input side current for 144° conduction mode.

Performance comparison in terms of harmonic content in output phase voltage, output non-adjacent line voltage and input ac side current for different conduction modes are presented in Figures 15-17. It is clearly seen that the harmonic content reduces significantly with reduction in conduction angle. The harmonic content is largest in 180 degree conduction mode and it is least in 144 degree conduction mode. However, the best utilisation of available dc link voltage is possible with conventional ten step mode (180 degree conduction mode). It can thus be concluded that a trade-off exist between the loss in fundamental and corresponding gain in terms of lower harmonic content in output waveform is obtained by using 144 degree conduction mode.


Phase Voltage Harmonic Comparison

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10 11 12 13

Order of Harmonics

Per

cent

of F

unda

men

tal

180144

Figure 15. Harmonic content in output phase voltage for different conduction mode.

LL-nonadj-Harmonics Comparison

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10 11 12 13

Order of Harmonics

perc

ent o

f Fun

dam

enta

l

180144

Figure 16. Harmonic content in output phase voltage for different conduction mode.

Input Current Harmonics comparison

0

10

20

30

40

50

60

70

80

90

1 2 3 4 5 6 7 8 9 10 11 12 13

Order of Harmonics

Per

cet o

f Fun

dam

enta

l

180144

Figure 17. Harmonic content in input side ac current for different conduction mode.


A comparison of total harmonic distortion in the output phase voltages of five-phase voltage source inverter for different conduction angle is presented in Figure 18. The conduction angles considered are 180°, 162°, 144°, 126°, and 108°. Thus two more conduction states are included when compared to Figure 16 and Figure 17, to further prove the superiority of control at 144° conduction mode. It is observed from Figure 18 that the lowest THD is obtained for 144° conduction mode.

Comparison of Harmonics and THD

0

5

10

15

20

25

30

35

40

45

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 THD

Harmonics + THD

Per

cen

t o

f F

un

dam

enta

l

180 deg162 deg144 deg126 deg108 deg

Figure 18. Comparison of Total Harmonic distortion at various conduction angles.

5. 2.2 Five-phase induction motor drive using the quasi square wave 5.2.2.1180 degree Conduction Mode A Five-phase induction motor is supplied using the custom built five-phase inverter operating at 180˚ conduction mode and the dc link voltage is set to 200 V. The motor is allowed to run at one speed corresponding to 50 Hz output. The resulting non adjacent line voltage and the line current is depicted in Figure 19. It is observed from the waveform that current Ib leads the voltage Vac by 54° (approximately), that means Ib is lagging Vb by 36° because phase difference between Vac and Vb is 90°. The spectrum of the line voltage and current are shown in Figure 20 and Figure 21, respectively.

Figure 19. Non-adjacent line voltage Vac and Ib (180° conduction mode).


Figure 20. Line Current (Ib) harmonics (180° conduction mode)

Figure 21. Non-adjacent Voltage (Vac) Harmonics (180° conduction mode) 5.2.2.2144 degree Conduction Mode The five-phase induction motor is further tested with the inverter operating at 144˚ conduction mode. The resulting line voltage and current waveforms are presented in Figure 22. The corresponding harmonic spectrum are shown in Figures 23 and 24.

Figure 22. Non-adjacent L-L (Vac) Voltage and Line Current (Ib) (144° conduction mode)


Here, current Ib leading the voltage Vac by about 36°, that means Ib is lagging Vb by 54°. Here, even harmonics are negligible and 5th harmonics is not zero. It is because of current flowing through the freewheeling diodes on inductive load. But harmonics are reduced in comparison to 180° conduction mode. These experimental results are in full agreement with that of simulated and analytical findings.

Figure 23. Harmonics Spectrum of Non-adjacent L-L (Vac) Voltage (144° conduction mode)

Figure 24. Harmonic Spectrum of Line Current Ib (144° conduction mode)

5. 2.3. Torque pulsation comparison Pulsating torques are produced in an induction motor drive system when harmonic current interact with the fundamental air gap flux and also when the harmonic air gap flux interact with the fundamental rotor current, Bose (2002) and Krishnan (2001). The pulsating torques causes undesirable effects in the drive system by producing losses, vibration and noise. A quantitative assessment of torque pulsation in a three-phase drive and five-phase drive for stepped operation is done and presented in this section. 5.2.3.1 Three-phase drive The predominant harmonics in a three-phase induction motor drive are 5th and 7th, with 5th being backward rotating and 7th being forward rotating both leading to 6th harmonic pulsating torques, Bose (2002) and Krishnan (2001). The expression for the sixth harmonic pulsating torque is given as;

( ) ( ) ( ) ( )6 1 7 5 1 7 53

sin 6 cos 62e m r r s r m m sT P I I t I tω ω= Ψ − + Ψ + Ψ (8)

An expression is derived for the sixth harmonic pulsating torque in terms of fundamental voltage and equivalent circuit parameter and is obtained as;

( )

+

+

−

=° )6cos(25

)5sin(

49

)7sin(

2

1)6sin(

25

)5sin(

49

)7sin(

2

31806 t

ttt

ttKTe s

sss

ss ωωωωωω (9)

Where

=

πωDC

eqs

V

X

PVK

21 , and mkψψψψ is the peak of kth harmonic mutual flux, V1 is the fundamental applied voltage, Xeq

is the equivalent leakage reactance and P is the number of poles of induction machine


5.2.3.2 Five-phase drive The predominant harmonics in a five-phase machine are 9th and 11th, with 9th being backward rotating and 11th being forward rotating both leading to 10th harmonic torques, Iqbal et al. (2008). The detailed derivation of the torque pulsation is presented in Appendix 1. The tenth harmonic pulsating torque for 180° conduction mode is obtained as;

( ) ( ) ( )[ ])10()10(2

59111911118010 tCosItSinIIPT mmrrrme ωψψωψ ++−=o

(10)

The tenth harmonic pulsating torque for 144° conduction mode is obtained as;

( ) ( ) ( )[ ])10()10(2

59111911114410 tCosItSinIIPT mmrrrme ωψψωψ ++−=o

(11)

An expression is derived for the tenth harmonic pulsating torque in terms of fundamental voltage and equivalent circuit parameter and is obtained as;

( ) [ ])10()(2

2

52

18010 αωωπω

+

= tSintSin

V

X

PZT Dc

eqse o

( ) [ ])10()(10

2

2

5 22

14410 αωωππω

+

−= tSintSinCosV

X

PZT Dc

eqse o

(12)

Where;

+=

−=

===+= −

)9()9(

1)11(

)11(

1

2

1

)9()9(

1)11(

)11(

1

,),(),(,

22

22

122

tSintSinB

tSintSinA

A

BTanZSinBZCosABAZ

ss

ss

ωω

ωω

ααα

Thus the ratio of pulsating torques for a typical motor in two conduction modes is obtained as;

( )( )

9045084972.010

2

18010

14410 =

= πCos

T

T

e

e

o

o

(13)

( )( ) 0.7

102

104.116

15

18010

1806 == −

−

°

°

x

x

Te

Te (14)

( )( ) 78.7

108.1

104.116

15

14410

1806 == −

−

°

°

x

x

Te

Te (15)

The relations (13-15) show, there is reduction in torque ripples in five phase motor at 144° conduction mode by 10% (approx) when compared with 180° conduction mode of five phase motor, 700% when compared with 180° conduction modes of five phase motors, and 778% when compared with 180° conduction mode of three phase motor and 144° conduction mode of five phase motor. 6. Conclusion The paper presents a solution of electric drive system supplied from weak grid with poor power quality. The proposed solution lies in the use of inverter fed five-phase induction motor drive system with stepped operation of the inverter. The inverter is proposed to operate in newly proposed 144˚ conduction mode. This offers better harmonic performance and thus lower losses, higher efficiency and better running cost. A comparison in torque pulsation is also presented which suggest significant reduction in torque pulsation by adopting the proposed 144˚ conduction mode. Analytical and experimental approach is used to validate the findings. Nomenclature

10eT Tenth Harmonic torque

6eT Sixth Harmonic torque

1V Fundamental Voltage

dcV DC Link Voltage

nNv Common mode voltage


av Phase voltage

Av Leg Voltage

ψ Flux Linkage

Appendix I: Torque Pulsation for 180° conduction mode in a five-phase induction motor drive

( ) ( ) ( )[ ]

−+

−=

++−=

)10(9112

)10(9112

5

)10()10(2

5

91119111

9111911118010

tCosVV

X

VtSin

X

V

X

VVP

tCosItSinIIPT

ssseq

seqeqs

smmrsrrme

ωωω

ωω

ωψψωψo

(AI.1)

The above equation is the defining equation of the tenth harmonic pulsating torque as given in equation (9). After substituting the equations for V11 (magnitude of the eleventh harmonic voltage component), the following equation results;

+

+

−

= )10(

)9()9(

2

)11()11(

2

2

1)10()9(

)9(

2)11(

)11(

2)(

2

2

5

2

2

221 tCos

tSinV

tSinV

tSintSinV

tSinV

tSinV

X

Ps

sDC

sDC

ssDC

sDC

sDC

eqsω

ωπ

ωπ

ωωπ

ωπ

ωπω

(AI.2) Where the fluxes and currents are further given as;

,11

,9

,,112

,92

.2

1111

99

11

1111

99

11

eqr

eqr

eqr

sm

sm

sm X

VI

X

VI

X

VI

VVV===

×=

×==

ωψ

ωψ

ωψ

(AI.3)

After substituting the flux and current relations from A3 to A2, the equation can be written as;

++

−

=

)10()9()9(

1)11(

)11(

1

2

1

)10()9()9(

1)11(

)11(

1

)(2

2

5

22

222

tCostSintSin

tSintSintSin

tSinV

X

P

sss

sss

sDC

eqs ωωω

ωωωω

πω

(AI.4)

)11(11

2V ),9(

9

2V ),(

2V 1191 tSin

VtSin

VtSin

Vwhere s

DCs

DCs

DC ωπ

ωπ

ωπ

=== (AI.5)

( ) ( )[ ]

A

BTanBAZZSinBZCosAwhere

tCosZSintSinZCostSinV

X

P

tSintSinBtSintSinAwhere

sssDC

eqs

ssss

122

2

2222

, ),( ),(

)10()()10()()(2

2

5

)9()9(

1)11(

)11(

1

2

1 ,)9(

)9(

1)11(

)11(

1

−=+===

+

=

+=

−=

ααα

ωαωαωπω

ωωωω

(AI.6)

Finally the equation for the tenth harmonic pulsating torque is obtained as;

( ) [ ] ;)10()(2

2

52

18010 αωωπω

+

= tSintSin

V

X

PZT ss

DC

eqse o (AI.7)

Similarly the tenth harmonic torque pulsation for 144° conduction mode in a five-phase induction motor drive is derives as;

( ) ( ) ( )[ ])10()10(2

59111911114410 tCosItSinIIPT smmrsrrme ωψψωψ ++−=o (AI.8)

The above equation is the defining equation for the pulsating torque. After substituting the fluxes the following equation results;


+

+

−

=

−+

−=

)10()9(10

9

)9(

2)11(

10

11

)11(

2

2

1

)10()9(10

9

)9(

2)11(

10

11

)11(

2

)(10

2

2

5

)10(9112

)10(9112

5

22

221

91119111

tCostSinCosV

tSinCosV

tSintSinCosV

tSinCosV

tSinCosV

X

P

tCosVV

X

VtSin

X

V

X

VVP

ssDC

sDC

ssDC

sDC

sDC

eqs

ssseq

seqeqs

ωωππ

ωππ

ωωππ

ωππ

ωππω

ωωω

ωω

( AI.9)

)11(10

11

11

2V ),9(

10

9

9

2V ),(

10

2V 1191 tSinCos

VtSinCos

VtSinCos

Vwhere s

DCs

DCs

DC ωππ

ωππ

ωππ

=

=

= (AI.10)

,11

,9

,,112

,92

.2

1111

99

11

1111

99

11

eqr

eqr

eqr

sm

sm

sm X

VI

X

VI

X

VI

VVV===

×=

×==

ωψ

ωψ

ωψ

(AI.11)

Substituting equations A10 and A11 into equation A9 yield;

++

−

=

)10()9()9(

1)11(

)11(

1

2

1

)10()9()9(

1)11(

)11(

1

)(10

9

10

2

2

5

22

222

tCostSintSin

tSintSintSin

tSinCosCosV

X

P

sss

sss

sDC

eqsωωω

ωωω

ωπππω

(AI.12)

( ) ( )[ ]

( ) ( )[ ])10()()10()()(10

2

2

5

)9()9(

1)11(

)11(

1

2

1 ,)9(

)9(

1)11(

)11(

1

)10())10()(10

9

10

2

2

5

10

11

10

9

22

2222

2

tCosZSintSinZCostSinCosV

X

P

tSintSinBtSintSinAwhere

tCosBtSinAtSinCosCosV

X

P

CosCosSince

sssDC

eqs

ssss

sssDC

eqs

ωαωαωππω

ωωωω

ωωωπππω

ππ

+

−=

+=

−=

+

=

=

(AI.13)

−=

=+=== −1010

9 , , ),( ),( 122 ππααα CosCos

A

BTanBAZZSinBZCosAwhere

(AI.14)

Finally the torque equation for tenth harmonic pulsation for 144° conduction is obtained as

( ) [ ] ;)10()(10

2

2

5 22

14410 αωωππω

+

−= tSintSinCos

V

X

PZT ss

DC

eqse o (AI.15)


Appendix II: Switching diagram for the two conduction mode:

05

3π5

π5

2π5

4π π5

6π5

7π5

8π5

9π π2

s1

s2

s3

s4

s5

s6

s7

s8

s9

s10

Switching Diagram For 180° Conduction Mode

05

3π5

π5

2π5

4π π5

6π5

7π5

8π5

9π π2

s1

s2

s3

s4

s5

s6

s7

s8

s9

s10

Switching Diagram For 144° Conduction Mode


Appendix III Photograph of the entire set-up:

IGBTs

Isolating TransformersPhase shifterSnubber Circuit

Driver Circuit

Appendix IV: Table of components and their cost (For Driver Circuit Only)

Name of components Specifications Quantity Cost

(in Indian Rupees)

Step down Transformer 230 V to 12-0-12, 100mA 8 240/- Operational Amplifier uA-741 10 50/- Voltage Regulator IC 7812 7 50/-

Opto-couplers 4N35 10 50/- Presets 47K 10 20/-

Resistances 1K, 10K, etc (1/4 W) 100 20/- Capacitors 0.1 microF, 1000 microF, etc 20 100/-

Printed Circuit Boards general 6 60/- Hardware Screws, nuts & bolts, etc - 20/-

Wooden board - - 30/- Terminals 5 amp 20 100/-

Total Cost (in Indian Rupees) 740/- Total Cost (in US$) 16.44

Acknowledgements Authors greatfully acknowledge the support provided by CSIR, New Delhi, India project no. 22(0420)/EMR-II/07 for the work. References Apsley, J.M., Williamson, S., Smith, A.C., and Barnes, M., 2006. Induction motor performance as a function of phase number,

IEE Proc. Electric Power Applications, Vol. 153, No. 6, pp. 898-904, 2006. Arahal, M.R. and Duran, M.J., 2009. PI tuning of Five-phase drives with third harmonic injection, Control Engg. Practice, Vol.

17, pp. 787-797, Feb.


Bojoi, R., Farina, F., Profumo, F., and Tenconi, 2006. Dual three induction machine drives control - A survey, IEEE Tran. On Ind. Appl.,Vol. 126, No. 4, pp. 420-429.

Bose, B.K., 2002. Power Electronics and AC Drives, Prentice Hall, New Jersey. Duran, M.J., Salas, F., Arahal, M.R., 2008. Bifurcation analysis of five-phase induction motor drives with third harmonic injection,

IEEE Trans. on Ind. Elect. Vol. 55, No. 5, pp. 2006-2014, May. Golubev, A.N., and Ignatenko, S.V., 2000. Influence of number of stator-winding phases on the noise characteristics of an

asynchronous motor, Russian Electrical Engineering, Vol. 71, No. 6, pp. 41-46, 2000. Hodge, C., Williamson, S., and Smith, S., 2002. Direct drive propulsion motors, Proc. Int. Conf. on Electrical Machines ICEM,

Bruges, Belgium, Paper no. 087. Iqbal, A., and Levi, E., 2005. Space vector modulation scheme for a five-phase voltage source inverter, Proc. European Power

Electronics and Appl. Conf., EPE, Dresden, Germany, CD-ROM paper 0006. Iqbal, A., Moinuddin, S., and Moin, A, Sk. 2008. Quantitative assessment of torque pulsation in inverter fed five-phase induction

motor drive system”, Proceedings of IEEE Region 10 Conference (TENCON 2008), University of Hyderabad, Hyderabad, Andhra Pradesh, India. Paper No. 1569126802.

Jones, M., and Levi, E., 2002. A literature survey of state-of-the-art in multiphase ac drives, Proc. 37th Int. Universities Power Eng. Conf. UPEC, Stafford, UK, pp. 505-510.

Krishnan, R., 2001. Electric Motor Drives-Modelling, Analysis, and Control, Prentice Hall, New Jersey. Levi, E., Bojoi, R., Profumo, F., Toliyat, H.A and Williamson, S., 2007. Multi-phase induction motor drives-A technology status

review, IET Elect. Power Appl. Vol. 1, No. 4, pp. 489-516. Levi, E., 2008a. Guest editorial, IEEE Trans. Ind. Electronics, Vol. 55, No. 5, May, pp. 1891-1892. Levi, E., 2008b. Multi-phase machines for variable speed applications, IEEE Trans. Ind. Elect., Vol. 55, No. 5, May, pp. 1893-

1909. Lyra, R.,O., C., and Lipo, T., A. 2002. Torque density improvement in a six-phase induction motor with third harmonic current

injection”, IEEE Trans. Ind. Appl., Vol. 38, No. 5, pp. 1351-1360, Sept./Oct. Singh, G.K., 2002. Multi-phase induction machine drive research – a survey, Electric Power System Research, Vol. 61, pp. 139-

147. Shi, R., Toliyat, H.A., and El-Antably, A., 2001. Field oriented control of five-phase synchronous reluctance motor drive with

flexible 3rd harmonic current injection for high specific torque, Conf. Rec. IEEE IAS Annual Meeting, Chicago, IL, pp. 2097-2103.

Ward, E.E., and H¨arer, H., 1969. Preliminary investigation of an inverter-fed 5-phase induction motor, Proc. IEE Vol. 116, No. 6, pp. 980–984.

Xu, H., Toliyat, H.A, and Petersen, L. J., 2001a, Five-phase induction motor drives with DSP-based control system,” Proc. IEEE Int. Elec. Mach. And Drives Conf. IEMDC2001, Cambrdige, MA, pp. 304-309.

Xu, H., Toliyat, H.A, and Petersen, L. J., 2001b. Rotor field oriented control of a five-phase induction motor with the combined fundamental and third harmonic injection, Proc. IEEE Applied Power Elec. Conf. APEC, Anaheim, CA, pp. 608-614, 2001.

Williamson, S., and Smith, S., 2003. Pulsating torque and losses in multiphase induction machines, IEEE Trans. On Ind. Appl. Vol. 39, No. 4, pp. 986-993.

Biographical notes

Shaikh Moinuddin received his B.E. and M.Tech (Electrical), PhD on multi-phase inverter modeling and control, degrees in 1996, 1999 and 2009 respectively, from the Aligarh Muslim University, Aligarh, India. He is recipient of University Gold Medals for standing first in Electrical branch and in all branches of Engineering in 1996 B.E. exams. He has served Indian Air Force from 1971 to 1987. He is employed in the University Polytechnic, Aligarh Muslim University since 1987 where he is currently working as a Assistant Professor. His principal area of research interest is Power Electronics and Electric Drives. Atif Iqbal received his B.Sc. and M.Sc. Engineering (Electrical) degrees in 1991 and 1996, respectively, from the Aligarh Muslim University, Aligarh, India and PhD in 2006 from Liverpool John Moores University, UK. He has been employed as Lecturer in the Department of Electrical Engineering, Aligarh Muslim University, Aligarh since 1991 and is working as Associate Professor in the same university. He is presently with Texas A&M University at Qatar on research assignment. He is recipient of Maulana Tufail Ahmad Gold Medal for standing first at B.Sc. Engg. Exams in 1991 at AMU, and EPSRC, Govt. Of UK, fellowship from 2002-2005 for pursuing PhD studies. His principal research interest is Modelling and Control of Power Electronics Converters & Drives. Elmahdi M. Elsherif received his B. Sc. on Electrical and Electronics Engineering degree in 1987 from the Garyounes University, Benghazi, Libya, M.Sc. on Electrical Power Engineering and Management and PhD on load frequency control, in 1994 and 2002 respectively, from Dundee University Scotland. He is working as Head of Department of Electronics and Computer Engineering, Sebha University, Libya. His principal research interest is load flow analysis, control of automatic generation & drives.

Received, January 2010 Accepted, February 2010 Final acceptance in revised form, April 2010

MultiCraft


Vol. 2, No. 2, 2010, pp. 155-164




Influencing parameters on performance of a mantle heat exchanger for a

solar water heater - a simulation study

G. Naga Malleshwara Rao1*, K. Hema Chandra Reddy2, M. Sreenivasa Reddy3

1* Intell Engineering College, Ananthapur, INDIA, 2 Academics and Planning, JNTUA, Ananthapur,INDIA

3 JNTUCollege of Engineering, Ananthapur, INDIA *Corresponding Author, Email: [email protected], Mobile: +91-9441222930

Abstract: Thermal performance of a solar water heater mainly depends on the thermal stratification. Thermal stratification in solar water tanks is essential for a better performance of energy storage systems where these tanks are integrated. In this research work, the performance of a solar water heater with a mantle heat exchanger is investigated experimentally. The heat exchanger is assessed for a range of operating conditions to quantify both the mantle side and the tank side heat transfer coefficients and the effect of thermal stratification in the tank. The experiments arte simulated and validated by using CFD tool ANSYS-CFX and a good agreement is obtained between experiments and simulations. The objective of this paper is to investigate the influence of location of hot fluid inlet, mass flow rate of mantle fluid and type of hot fluid on the performance of the mantle heat exchanger. Keywords: Performance of mantle heat exchanger, mantle side heat transfer coefficient, tank side heat transfer coefficient, operating conditions, ANSYS-CFX, hot fluid inlet, mass flow rate, type of hot fluid 1. Introduction

There are many good methods and sources used to store warm thermal energy. These include solar heaters, solar ponds, geothermal storage methods, and many others. The advantage of warm thermal energy storage is that usually, the warm thermal energy storage is obtained from an abundant and ecologically friendly source, such as the sun. As a result, heat storage is usually very cost friendly and good for the environment. The main aim of this paper to investigate the influence of certain operating parameters which can affect the performance of the mantle heat exchangers. Location of hot fluid inlet into the mantle heat exchanger, mass flow rate of mantle fluid, heat transfer fluid are the three parameters that influence the performance of a mantle heat exchanger. An effort is made through this work to study the effect of the above parameters on the performance of mantle heat exchanger. Further details on the parameters considered are discussed in section 3. Figure 1 shows a solar domestic hot water system with a vertical mantle heat exchanger. The mantle fluid is slowly pumped down through the mantle in order to exchange heat between the mantle fluid and the water in the tank. 2. Solar water heater with mantle heat exchanger A mantle tank is a cylindrical storage tank surrounded by an annulus through which hot liquid from the collector flows thereby transferring energy to the tank contents. The separating wall is the heat exchange surface. Wrap-around coil and tank-in-tank systems are similar in this respect. Wrap-around coil designs usually have a smaller fluid inventory than mantle tanks and operate with higher flow rates to produce turbulent conditions within the coil. Tank-in- tank systems allow the top and bottom of the water storage tank, in addition to the sides, to serve as heat exchanger surface. Mantle heat exchangers are an interesting alternative to external heat exchangers because they reduce the complexity of the system by combining the heat exchanger and the storage unit in one element. In a mantle heat exchange system, fluid flow from the heat source does not pass through the tank. A possible

Rao et al. / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 155-164

156

advantage of this flow configuration is reduced internal tank mixing and, as a consequence, improved temperature stratification resulting in higher solar collector efficiency (Furbo and Knudsen, 2005; Knudsen and Furbo, 2004).

Figure 1: Solar water heater with mantle heat exchanger

2.1. Salient Features of the Mantle Heat Exchanger System: A). Thermal Stratification: The mantle tank system is one of the simplest ways of producing high heat exchanger effectiveness

while promoting thermal stratification. The mantle configuration provides a large heat transfer area and effective distribution of the collector loop flow over the wall of the tank. Most of the incoming mantle fluid seeks the thermal equilibrium level in the mantle, and thermal stratification in the mantle and the inner tank is not disturbed (Shah and Furbo, 1998). B). Cost: The cost of tank is less compared to the increased performance initiated by the fine thermal stratification (Shah, 2000).

C). Size limitation: The mantle tank design is not suitable for large low flow SDHW systems, as the heat transfer area gets too small for tanks with volumes over 800-1000 lts. (Shah et al., 1999). 3. Experimental equipment, procedures and experimental parameters 3.1. Description of experimental equipment In this paper, the convective heat transfer at the mantle wall and at the tank wall is investigated under operation conditions. In this study, the convective heat transfer in the mantle and in the inner tank is analysed by means of dimensionless heat transfer theory in order to obtain heat flux and heat transfer coefficients for vertical mantle heat exchangers. It is not enough only to be able to predict the heat transfer at the mantle wall and at the tank wall to model the thermal stratification in the inner tank, it is also necessary to know the heat transfer at different levels due to the heat flows inside the inner tank. An effort is also made to model heat transfer in energy storage tank with mantle heat exchanger. A schematic diagram of a solar domestic water heater with a mantle heat exchanger as shown in Figure 1. An experimental arrangement of a vertical mantle heat exchanger for the solar water heater as illustrated in Figure 2. The purpose of this experimentation is to evaluate the overall heat transfer characteristics of the mantle heat exchanger over a range of mantle flow rates and thermal boundary conditions. These experiments are carried out at heat transfer laboratory of JNTU College of Engineering, Ananthapur, Andhra Pradesh, India. The mantle is constructed with an annular spacing of 30 mm wrapped around the bottom half of a stainless steel (SS 304 grade) tank. The tank is insulated with glass wool. In the experiments hot water is supplied to the hot side of the heat exchanger from a conventional hot water tank as the heat source and the cold water is supplied from the bottom side of the storage tank. Eight thermocouples (copper-constantan) are fitted at different levels inside the core of the tank and four thermocouples (copper-constantan) are mounted on the mantle tank wall side of the heat exchange surface, to measure tank temperatures at different locations as shown in Figure 3 to know the stratification. The inlet and outlet temperatures and flow rate of the hot water (mantle fluid) through the heat exchanger are also measured. A 3 1/2 digit digital display unit is used to display the temperatures. All the instruments used in these experiments are well calibrated with standard instruments.


157

Figure 2: Photographic view of Experimental set-up. Figure 3: Temperature measuring points for Solar water tank with mantle heat exchanger. The details of mantle type heat exchanger fabricated for the purpose of experimentation are given in Table 1.

Table 1. Details of Mantle heat exchanger Data Of Mantle Tank Stainless Steel Properties: Grade: SS304

Volume in inner tank [m3] 0.5585 Specific heat [ J/Kg.K ] 460 Volume in inner tank above mantle [m3] 0.246 Density[Kg/m3] 7820

Tank height [m] 0.900 Thermal Conductivity[W/m2.K] 15

Inner diameter of tank [m] 0.1976 Thickness of tank wall [m] 0.0012 INSULATION:

Mantle height [m] 0.400 Material Glass Wool Mantle tank diameter[m] 0.260 Insulation top, [m] 0.006 Mantle top (distance from bottom of tank) [m] 0.500 Insulation side above/below

mantle, [m] 0.006

Mantle bottom(distance from bottom of tank) [m] 0.100 Mantle gap width [m] 0.030

Insulation side mantle,[m] 0.006

Mantle inlet position from top of mantle[m] 0.05 / 0.10/ 0.15

Mantle inlet size [ inner diameter ] [m] 0.012 Hot water tank volume[m3] 0.073

Insulation bottom, [m] 0.006

Inner tank inlet size[m] 0.0254 Mantle Fluid Water 3.2. Experimental procedures In this study, with a mantle inlet size of 12 mm, tests with two different locations of the mantle inlet port are carried out, that is top inlet position and lower inlet position. The top (first) inlet is positioned 50 mm below from the top of mantle tank and that of lower (second) inlet is positioned at 100 mm below the top of the mantle tank (one fourth of the total mantle height). All the experiments are performed with two mass flow rates of mantle fluid at 0.5 lit/min (0.00833 kg/sec) and 0.75 lit/min (0.0125 kg/sec). Two heat transfer fluids viz., water and 30%propylene glycol and 70% water mixture are used to study the effect of the


158

heat transfer fluid on the performance of mantle heat exchanger. All these experiments are performed at two inlet temperatures of mantle fluid i.e., 50°C and 70°C respectively.

Table 2. Description of experiments conducted in Laboratory Experiment Mantle

inlet position

Mantle flow rate(Kg/sec)

Mantle fluid Mantle fluid inlet temp oC

Fi-M0.5-T50 Fi 0.00833 Water 50 Fi-M0.5-T70 Fi 0.00833 Water 70 Si-M0.5-T50 Si 0.00833 Water 50 Si-M0.5-T70 Si 0.00833 Water 70 Fi-M0.75-T50 Fi 0.0125 Water 50 Fi-M0.75-T70 Fi 0.0125 Water 70 Fi-M0.5-T50-G Fi 0.00833 Glycol/Water Mix. 50 Fi-M0.5-T70-G Fi 0.00833 Glycol/Water Mix. 70

Note: The abbreviations EXP and CFD used in these experiments represent experiment and simulation respectively.

The details of the experiments that are performed in the laboratory are given in the Table 2. All the experiments are specified with a specific notation as shown in the table 2. The first digit in the specification ‘Fi’ or ‘Si’ represents the first inlet or the second inlet, the second digit ‘M0.5’ or ‘M0.75’ represents mass flow rate of mantle fluid (0.5 lit/min or 0.75 lit/min). The letter ‘T50’ or ‘T70’ gives the mantle fluid inlet temperature i.e., 50°C or 70°C. The letter ‘G’ stands for 30% glycol/water mixture.

3.3. Measurement of experimental parameters In the heat storage test facility it is possible to control the flow rate in the mantle and mantle inlet temperature. During the experiments the following parameters are measured.

a) The mantle flow rate b) The mantle inlet temperature c) The mantle outlet temperature d) The temperature of the domestic water in the inner tank at eight points inside the tank e) The temperature at four points on the outside of the mantle wall f) The ambient temperature.

3.4. Specifications of measuring devices: The specifications of the equipment used I these experiments are as follows. Tank with mantle heat exchanger: Stainless Steel of grade SS 304. Thermocouples: copper –constantan with the specifications of P-200°C (PT100) Digital temperature indicator panel: 216Y0811DTI Measuring jar: 1000 ml made up of glass to measure mantel flow rate. 4. CFD Modeling

The CFD code used in this work uses CFX-5 11.0 (Ansys, 2003) and is based on a finite volume method which uses an unstructured mesh containing tetrahedral and prism elements. This has the advantage that local numerical diffusion is reduced and is therefore suitable for complex flows with e.g. flow reversal. To reduce the number of iterations required for convergence, a false-time stepping method is imposed which guides the approximate solutions in a physically based manner to a steady-state solution. Buoyancy is modeled using the Boissinesq approximation in which the forces are modeled as source terms in the momentum equations (Cruickshank and Harrison, 2006). Various models exist in CFX-5 code for modeling turbulent flow. Two-equation models based on the eddy-viscosity concept include the κ-ε (Launder and Spalding, 1974), κ - ω (Wilcox, 1998) and Shear Stress Transport (SST) κ - ω based models (Menter, 1994). Compared to the commonly used κ-ε turbulence model, the κ - ω model implies a new formulation for the near wall treatment which provides an automatic switch from a wall-function to a low-Reynolds number formulation based on the near-wall grid spacing. This makes it more accurate and more robust. The turbulence viscosity is assumed to be linked to the turbulence kinetic energy (κ-equation) and turbulent frequency (ω-equation) instead of the turbulence dissipation rate (ε-equation in the κ -ε model). To overcome the sensitivity of the k-ω model to free stream conditions, the SST model is developed. It blends the k-ω model near the surface with the k-ε model in the outer region. In contrast, Reynolds Stress Turbulence models such as the standard Launder-Reece-Rodi Isotropic Production (LRR-IP) model (Launder et al., 1975) and Second Moment Closure-ω (SMC-ω) model (Wilcox, 1998), do not use the eddy-viscosity hypothesis, but solve transport equations for all components of the Reynolds


159

stresses. This makes Reynolds Stress models more suited to complex flows. However, practice shows that they are often not superior to two-equation models because convergence difficulties often occur. The LRR-IP model is based on the k-ε model, whereas the SMC- ω is based on the k-ω model with the advantages already explained. The influence of mesh resolution on temperatures and velocities in the wall boundary layer and on heat transfer at the heated plate are investigated by varying three key parameters for prism layer sizing: first prism size (use of 0.1mm, 0.2mm, 0.4mm, 1.0mm and 2.0mm), core prism size (use of 1.33mm, 2mm, 3mm and 4mm) and prism inflation (use of 1.5, 2.0 and 2.5). For investigations of constant core prism scales (by setting the prism inflation factor to unity) and prism inflation, a first prism size of 0.1mm is used (refer Figure 5). In the sensitivity study of first prism sizes, a constant core prism scale of 2.0mm is chosen. The first prism size determines the prism edge length in conversed direction of the surface normal for the first prism layer adjacent to the wall. The core prism size determines the prism edge length for the other prism elements towards the edge of the wall boundary layer and the outer region. A high mesh quality with finite element scales similar to the neighbouring elements reduces the risk of convergence problems. Therefore, with prism inflation, the prism layer size can be increased gradually towards the core such that the last prism layer size has a similar scale to the edges of the neighbouring tetrahedral elements. The experiments are first simulated by taking into account the steady state natural convective flow on the tank side. A 3-D CFD model of a mantle heat exchanger coupled with a storage tank is developed with the same dimensions as the prototype unit. The CFD modeling of this type of heat exchanger would lead to very time consuming CFD simulations. In order to make the computational solution viable, the computational domain is simplified by modeling the mantle tank system into three regions which makes the problem simpler. Although, the model appears symmetric, grid restrictions require a full three dimensional analysis of the model. The purpose of the configuration is to investigate the influence of the flow on the stratification in the tank, the heat transfer inside the tank and the natural convection in the tank loop. The buoyancy for an incompressible fluid with constant properties except density and viscosity is modeled by using the Boussinesq approximation in ANSYS CFX 11.0. No slip and adiabatic boundary conditions are applied to the walls of the tank. The Boussinesq approximation uses the thermal expansion coefficient (β) to capture natural convection. Thus, if β is presumed to be constant, a linear dependency of the density of water on temperature is used. The computational domain is simplified by modeling the top and bottom of the tank as flat walls, and the mantle-tank volume as the same as that of the prototype unit. High concentration mesh is used in the high temperature gradient regions near the heat transfer wall between the mantle and the storage tank. A total of 537077 grid points are used in the computational domain within the mantle gap and the inner tank. The number of grid points is given in Table 3. For each simulation, the inlet temperature, flow rate of mantle fluid and flow rate of cold fluid from the experiments are specified as inputs to the CFD model. The typical running time for simulation is approximately 3 days. On the tank side, the temperature profile along the tank height is initialized based on the measured data. Heat loss from the tank is modeled using a constant convective heat transfer coefficient on the outer surfaces of the tank. All physical properties of water and glycol/water mixture are assumed to be constant except the density and viscosity in the buoyancy term in order to obtain faster numerical convergence. In order to have sufficient number of grid points for solving the flow and heat transfer in mantle (steady state model) particularly near the inlet jet impingement region and the high temperature gradient near the heat transfer wall, a mesh sensitivity check is undertaken by comparing the numerical results for different mesh sizes. Figures 4 and 5 show the geometry and mesh models of the work respectively.

Figure 4: 3-D modeling of solar water tank with mantle Figure 5: 3-D mesh model of solar water tank heat exchanger with mantle heat exchanger.


160

Table 3. Mesh information Domain Nodes Elements Region1 47456 256660 Region2 486540 1352471 Region3 3081 11108

4.1. Validation of models In order to examine the validity of the CFD simulation models used for heat transfer studies in mantle heat exchangers, simulated results are compared to experimental data. Figures 6 & 7 show a comparison of tank temperature stratification between experiments and simulations of the cold start test with inlet temperatures of 50oC & 70oC with a flow rate of 0.5lit/min (0.00833 kg/sec). As observed, the predicted tank temperature stratification is in reasonable agreement with the measured temperature profiles.

00.10.20.30.40.50.60.70.80.9

24 25 26 27 28 29 30 31 32

Hei

ght

of T

ank[

M]

Temperature profile CFD-Fi-M0

EXP-Fi-M0

0

0.10.20.30.40.50.60.70.80.9

25 27 29 31 33 35

Hei

ght

of T

ank[

M]

Temperature profile CFD-Fi-M0

EXP-Fi-M0

Figure 6 Figure 7

Figures 6 and 7: Simulated and measured core tank temperature distribution along the height of the storage tank for mantle inlet temperatures of 50oC and 70oC and at the flow rate of 0.5 lit/min at top inlet. 5. Heat transfer coefficients Heat Transfer between the hot fluid in the mantle heat exchanger and cold fluid in the storage tank is governed by convective heat transfer on the hot side of the heat exchanger, conduction through the tank wall and natural convection circulation inside the tank (Baur et al., 1993). The mean convective heat transfer coefficients (h) for the two convection processes in the mantle heat exchanger are determined using equations 1 and 2 based on the measured mantle side wall temperatures and the core temperatures (Soo Too et al., 2004).

Heat transfer coefficient on the mantle side,

,( )m w m

QhmA T T

=−

(1)

Heat transfer coefficient on the tank side,

,( )w j t

QhtA T T

=−

(2)

where A is the available heat transfer area of the mantle heat exchanger(m2), Tw,t is the measured inner wall temperature on the tank side (°C),

, ,w

w m w tw

QtT Tk A

= + is the wall temperature on the mantle side of the wall (°C),

tw, kw are the thickness(m) and thermal conductivity(W/mk) of the tank wall (Mercer et al., 1967; Morrison et al., 1999).


161

6. Results and discussions Investigating the process of heat transfer inside the storage tank for various governing parameters is the main objective of the present study. In this paper, the effect of following three parameters on heat transfer in energy storage systems are considered in order to study and quantify the thermal stratification in energy storage tanks. 1. Location of inlet position of mantle heat exchanger. 2. Mass flow rate of mantle fluid. 3. Type of heat transfer fluid. 6.1. Effect of Location of Mantle Inlet Position on Heat Transfer Coefficients

60

80

100

120

140

160

180

Hea

t Tr

ansf

er C

oeff

(w/m

2 K)

Mantle side heat transfer coeff.hm

EXP-Fi-M

EXP-Si-M

50

70

90

110

130

150

Hea

t Tr

ansf

er C

oeff.

(w/m

2 k)

Tank side heat transfer coeff. ht

EXP-Fi-M

EXP-Si-M

Figure 8 Figure 9 Figures 8 & 9: Mantle and Tank side heat transfer coefficients, hm & ht for 50°C mantle fluid inlet temperature at both top and lower inlets. By observing the mantle side heat transfer coefficient (hm) graph (Figure 8), it can be noticed that, hm value is less in the beginning for the top inlet and increases continuously, where as hm value (Figure 8) is almost constant for the second inlet. From the tank side heat transfer coefficient ht graph (Figure 9), it is observed that, ht values are greater for the second inlet flow than the first (top) inlet as there is a more heat transfer at the bottom of the mantle tank. It can be observed that, higher mantle fluid inlet temperature transfers more heat to the tank contents.

6.2. Effect of mantle flow rate on heat transfer coefficients Figures 10 and 11 show the heat transfer coefficients on the mantle side and tank side of the heat exchanger for cold start test with mantle flow rates of 0.5 and 0.75 lit/min for inlet fluid temperatures of 50°C. The mantle side heat transfer coefficient ‘hm’ is found to be 165.4 w/m2k (Figure 10) while the tank side heat transfer coefficient ‘ht’ varies from 72.17 w/m2k to 96.47 w/m2k due to change of viscosity with temperature. The results obtained from cold start test with a mantle flow rate of 0.5 lit/min show that heat transfer coefficient on the mantle side hm is a factor which is 1.5 times higher than that on the tank side. When the mantle flow rate is increased to 0.75 lit/min, the hm increases due to the impingement effect near the inlet and better flow distribution across the mantle at high flow rate. Even though higher mantle side heat transfer coefficients remain unchanged (Figure 10) the limiting factor on the performance of the mantle heat exchanger is primarily the natural convection in the storage tank. As the inlet flow rate is increased, the mixing due to impingement could extend over the mantle, hence increasing the heat transfer coefficients.


162

90

110

130

150

170

190

210

230

Hea

t Tr

ansf

er C

oeff.

(w/m

2k)

Mantle side heat transfer coeff. hm

EXP-Fi-M

EXP-Fi-M

406080

100120140160180200

0 30 60 90 120

Hea

t Tr

ansf

er C

oeff.

(w

/m2

k)

Tank side heat transfer coeff. ht

EXP-Fi-M0

EXP-Fi-M0

Figure 10 Figure 11 Figures 10 & 11:Mantle and Tank side heat transfer coefficient, hm & ht for 50°C mantle inlet temperature for 0.5 lit/min and 0.75 lit/min at top inlet.

6.3. Effect of type of heat transfer fluid heat transfer coefficients

0

50

100

150

200

250

eat

Tran

sfer

Coe

ff. (

w/m

2k)

Mantle side heat transfer coeff. hm

EXP-Fi-M0.

EXP-Fi-M0.

40

60

80

100

120

eat

Tran

sfer

Coe

ff. (

w/m

2k)

Tank side heattransfer coeff. ht

EXP-Fi-M0.5

EXP-Fi-M0.5

Figure 12 Figure 13 Figures 12 and 13: Mantle and Tank side heat transfer coefficient, hm & ht for 50°c mantle inlet temperature for water and glycol-water mixture.

In practice, a 30% propylene glycol-water mixture with high Prandtl number (21.77 at 45°c) and viscosity is typically used in the heat exchanger in order to provide freeze protection for solar water heaters. By utilising propylene glycol-water mixture as the mantle fluid, the lower range of Reynold’s numbers can be extended. From the Figures 12 and 13, it can be seen that, the higher viscosity of glycol mixture decreases the Reynolds’s number when compared to water. Higher specific heat and thermal conductivity of glycol-water mixture results in higher values of mantle side and tank side heat transfer coefficients. The viscosity of glycol is more in the beginning of the experiment which means that the flow will tend to become laminar and heat transfer is less. As the experiment is continued, the viscosity of glycol is reduced by absorbing heat and then the flow becomes turbulent and thus heat transfer is increased. It is observed that the heat transfer rate is more for the glycol-water mixture irrespective of mantle fluid inlet temperature.


163

7. Conclusions The heat transfer characteristics of a mantle heat exchanger with a single pass flow arrangement are investigated under controlled indoor conditions. Measurements showed that the tank is well mixed above the mantle level of the heat exchanger. The influence of mantle inlet location, the mantle fluid flow rate and the type of mantle fluid on the flow and heat transfer coefficients in a mantle is investigated using experimental and CFD simulated temperatures. It can be noticed that, mantle side heat transfer coefficient (hm) value is low (110 w/m2K) for the first 30 minutes from the beginning of the experiment for the top inlet (50 mm below the top of the mantle) and is found to be increasing continuously up to 160 w/m2K between 30 minutes and 150 minutes, where as, the hm value is observed to be almost constant 140 w/m2K for the second inlet i.e., 100 mm below the top of the mantle tank throughout the test duration. In both the cases it is observed that heat exchange is taking place effectively from 0.1 m to 0.5 m height only. The heat transfer coefficients on either side of mantle heat exchanger are noticed to be higher for 70°C mantle fluid inlet temperature. The difference in heat transfer coefficients for 70°C and 50°C inlet temperatures is found to be marginal on mantle side whereas it is considerable on tank side. The tank side heat transfer coefficient, ht values are greater for the second inlet than the first (top) inlet as there is a more heat transfer at the bottom of the mantle tank. It can also be observed that, at higher mantle fluid inlet temperature (70°C), more heat is transferred to the tank contents. The increased mass flow rate from 0.5 lit/min to 0.75 lit /min is resulting in increased mantle side heat transfer coefficients. The experimental results showed that hm value is 190 w/m2K for 0.75 lit/min flow rate and 120 w/m2K for 0.5 lit/min 90 minutes after the beginning of the experiment. This can be for the reason that the inlet jet impingement on the back wall adjacent to the inlet induces a region of localized turbulent flow. With propylene glycol-water mixture as the mantle fluid, the Reynold’s number is found to be low due to the higher viscosity of glycol mixture when compared to water. However higher specific heat and thermal conductivity of glycol-water mixture results in higher values of mantle side and tank side heat transfer coefficients 200 w/m2K and 130 w/m2K respectively. The disadvantage with the glycol-water mixture is that, glycol should be removed as hazardous waste at regular intervals. The second inlet position (100 mm below the top of the mantle tank) for the mantle heat exchanger at high inlet temperature of 70°C, high mass flow rates of mantle fluid and the mantle fluid with high thermal properties can improve the performance of the mantle heat exchangers. It is also observed that both experimental and simulated temperatures are in good agreement with each other. Nomenclature SDHWS Solar domestic hot water system Ti heat exchanger inlet temperature (0K) To Heat exchanger outlet temperature (0K) Tw Tank wall temperature (0K) Tt Temperature of fluid in tank (0K) A Available heat exchange area (m2) hm Mantle side convective heat transfer coefficient (w/m2K) ht Tank side convective heat transfer coefficient (w/m2K) m Mantle fluid mass flow rate (kg/sec) Q Heat supplied to the system (W) ∆Tlm log-mean temperature difference (0K) T Time (sec) tw tank material wall thickness (m) kw thermal conductivity of tank material(w/m.k) References Ansys CFX, 2003: CFX-5 Solver Models and Theory User Manuals, Version 5.6. Baur, J.M., Klein, S., Beckman, W.A., 1993. Simulation of water tanks with mantle heat exchangers. Proceedings, ASES Solar

Vol. 93, pp. 286– 291. Cruickshank, C.A. and Harrison, S.J. 2006. Simulation and testing of stratified multi-tank, thermal storages for solar heating

systems. Proceedings of the Euro Sun 2006 Conference, Glasgow, Scotland. Furbo, S., Knudsen, S., 2005. Heat transfer correlations in mantle–tanks, ISES Solar World Congress, Orlando, USA. Knudsen, S., Furbo, S., 2004. Thermal stratification in vertical mantle heat exchangers with application to solar domestic hot water

systems. Applied Energy, Vol.78, pp. 257–272. Launder B.E., Spalding D.B., 1974. The numerical computation of turbulent flows, Comp. Meth. in Applied Mech. Eng., Vol 3, pp

269-289.


164

Launder B., Reece G. and Rodi W., 1975. Progress in the development of a Reynolds-stress turbulence closure, J. Fluid Mech., Vol. 68, No. 3, pp 537-566.

Mercer, W.E., Pearce, W.M., Hitchcock, J.E., 1967. Laminar forced convection in the entrance region between parallel flat plates. ASME Journal of Heat Transfer, Vol. 89, 251–257.

Menter, F.R., 1994. Two-equation eddy viscosity turbulence models for engineering applications, AIAA Journal, Vol. 32, No. 8, pp. 1598-1605.

Morrison, G.L., Rosengarten, G., Behnia, M., 1999. Mantle heat exchangers for horizontal tank thermosyphon solar water heaters, Solar Energy, 67, 53–64.

Shah, L.J., Furbo, S., 1998. Correlation of experimental and theoretical heat transfer in mantle–tanks used in low flow SDWH systems. Solar Energy, Vol. 64, 245–256.

Shah, L.J., 2000. Heat transfer correlation for vertical mantle heat exchangers. Solar Energy, Vol. 69, 157–171. Shah, L.J., Morrison, G.L., Behnia, M., 1999. Characteristics of vertical mantle heat exchangers for solar water heaters. Solar

Energy, Vol. 67, pp. 79–91. Soo Too Y.C., Morrison GL and Behnia M, 2004. Vertical mantle heat exchangers for solar water heaters, Proceedings of ANZSES

Annual Conference, December, Perth, Australia. Wilcox D.C., 1998: Turbulence modeling for CFD, 2nd edn., DCW Industries Inc., Canada. Biographical notes G. Naga Malleshwara Rao is an Associate Professor in the Department of Mechanical Engineering at Intel Engineering College, Ananthapur of Andhra Pradesh State, India. His current area of research includes Heat transfer and Renewable energy. He has presented many articles in national and international conferences. He also worked as Training and Placement Officer at Intel Engineering College between 2006 and 2009. He is a life member of IE (India). He is having a total experience of 17 years of which 6 years is Industrial and 11 years academic. Dr K. Hema Chandra Reddy received MTech and PhD. from Jawaharlal Nehru Technological University (JNTU), Hyderabad, India. He is a Professor in the Department of Mechanical Engineering, JNTU College of Engineering Ananthapur. He also worked as Training and Placement Officer at JNTU College of Engineering Ananthapur and during this tenure many of their students are globally placed. Currently he is the Director of Academics and Planning at Jawaharlal Nehru Technological University, Ananthapur. He is a Fellow of IE (India), and life member of various professional bodies. Many research scholars are pursuing their PhDs under his esteemed guidance and some of them have been awarded. M. Sreenivasa Reddy is working as a Technical Assistant at JNTU College of Engineering, Ananthapur. He received his MTech from JNTU, Hyderabad. He submitted his PhD in JNTU, Ananthapur and waiting for the final viva-voce. His area of research includes Internal Combustion Engines.

Received, March 2010 Accepted, April 2010 Final acceptance in revised form, April 2010

MultiCraft


Vol. 2, No. 2, 2010, pp. 165-173




Evaluation of thermal characteristics of oscillating combustion

J. Govardhan1*, G.V.S. Rao2

1* PRRM Engineering College, Shabad-509 217, JNTUH, Hyderabad, A.P.,INDIA 2 PIRM Engineering College, Chevella, JNTUH, Hyderabad, A.P. 509 217, INDIA *Corresponding Author: e-mail: [email protected] Tel +91-95735 94984

Abstract In view of the economy and environmental impacts of the energy utilization, most of the heat transfer industries such as steel mills, glass plants and forging shops, foundry process and furnaces are focusing on energy efficient strategies and implementing new technologies. Gas Technology Institute (GTI) and Air Liquide Chicago Research Centre (ALCRC) have applied Oscillating Combustion Technology (OCT) on high temperature forged furnaces and reheat furnace for melting steel. The oscillating combustion requires a new hardware to incorporate on the fuel flow ahead of the burner. Solid State Proportionate (SSP) valves were used to create oscillations in the fuel flow. Natural gas was used as fuel and the technology was applied with air-gas, oxygen-gas, and excess level of air during the oscillating combustion. The present work deals with the implementation of OCT on liquid fuels at ambient conditions for melting aluminum metal in a fuel-fired crucible furnace which is of importance to foundry. Also, carrying out a study over the enhanced performance characteristics of oscillating combustion and comparing its thermal effects with those of the conventional combustion mode. The oscillating device, developed by the author, unlike other oscillating valves used earlier is a cam operated electro mechanical valve cause oscillations on the fuel flow. Experiments were conducted at varying air-fuel ratio, aluminum stocks, frequency and amplitude of the oscillating valve. The results when compared to the conventional combustion led to low fuel and specific energy consumption, enhanced heat transfer rate, increased furnace efficiency with visibly low volumes of flue gases with reduced emissions. The increased heat transfer rate and furnace efficiency was found to be in agreement with the results of GTI and ALCRC experiments. The reasons for such improvements in performance characteristics were verified by conducting experiments in the furnace by measuring the temperature distribution at designated point and calculating the heat transfer rate both for conventional and oscillating combustion mode. The analyses presented in this paper are for two levels of air-fuel ratios above and below the stoichiometric ratio, three different loads at 100 & 200amplitude and 5 & 10Hz frequency of oscillating valve. Key words: Furnace efficiency, fuel fired furnace, heat transfer, oscillating combustion, specific energy consumption

1. Introduction Energy is one of the most critical input resources in the heat transfer industries requires new combustion concepts to increase the thermal efficiency with reduced emissions and alternate sources of energy (Im et al., 1996; Kim et al., 2002). The increased demand, depleted energy resources, rising cost of fossil fuels and considerable environmental impacts are to be viewed and attempts must be made for higher energy utilization of fuel and thermal energy (Fadare et al., 2009). The furnaces which operate at high temperature produce large quantities of emissions are sometimes less productive and less efficient. The aim of steady state combustion technology is to enhance the energy efficiency and consequential fuel cost savings as well to reduce emission levels (www.energy.com). Industrial manufactures are to implement new technologies to increase the thermal efficiency and stringent pollution norms require development of new combustion concepts (Gupta, 1977). Oscillating combustion system is a low-cost, low NOx, high efficient technology and can be integrated in any combustion system, whose principle is based on a cyclical perturbation of the gas line (Delabroy et al., 2001). Oscillating combustion sometimes referred as the derivative of pulse combustion. Pulse combustion occurs when fuel and oxidizer react chemically in the presence of a standing acoustic wave. Although fuel and oxygen generally admitted to the combustion can be applied to gaseous, liquid and solid fuels (Research in pulse combustion). Oscillating combustion is a retrofit technology that involves forced oscillations of the fuel flow rate to a furnace. The oscillations of the fuel

Govardhan and Rao / International Journal of Engineering, Science and Technology, Vol. 2, No. 2, 2010, pp. 165-173 166

create successive, fuel-rich and fuel-lean zones within the furnace. The fuel rich zone flames are more luminous and longer causes more heat transfer from the flame to the load. This act of increased heat transfer to the load is the result of break up of the thermal boundary layer (Energy matters, 2002). Oscillating flames results in lower peak temperature of the furnace since the fuel- rich and fuel-lean zones mix only when the thermal energy of the fuel rich zone flame transferred the heat to the load thereby causing low peak furnace temperature and reduction in additional NOx formation (John C and Wagner, 2004). Modeling at Air Liquide by Wagner and John C showed that oscillation flames have lower peak temperature and longer length than non-oscillating flames, which corroborates the burner test results. OCT has led to increased heat transfer and the increased heat transfer results in improved furnace productivity and efficiency (Wagner, C John, 2002). In order to optimize the furnace few parameters are mentioned as amplitude, frequency, duty cycle and phasing (gti.osc.combus.com). Oscillating combustion can be applied to many types of furnaces used in steel and high-temperature process industries such as glass, petrochemical, aluminum, cement and metal heating. OCT has been extensively tested and evaluated by Institute of Gas Technology, Air-Liquide, Ceram Physics.Inc. Gas Research Institute, GT Development Corporation etc.(Energy matters, 2002). The tests have shown enhanced furnace efficiency in terms of low fuel consumption with visibly low volumes of flue gases. The concept of heat transfer or so called film co-efficient was introduced by Newton which depends on the properties of flowing fluid, thermal conductivity, viscosity, density and specific heat are termed as transport properties which play crucial role for better heat transfer (Arora.). When oscillations occur, the pressure amplitude is sufficient enough to produce significant variations in axial velocity within the nozzle annulus. These axial velocities can vary during the oscillating combustion. The swirl vanes on the surface of the fuel gun of the burner would provide a variation in tangential velocity. Due to this the flow around the nozzle’s annulus is having high and low regions of tangential velocity convected along the main axial flow of the fuel. The magnitude of heat transfer in any furnace from hot gases to load depends upon the temperature distribution inside the furnace. Generally, the temperature distribution through out a body varies with location and time. Temperature distribution in the crucible of a furnace is an important operation variable that is a function of the materials used in construction, temperature in the metal-refractory interface etc.(Luis Felip Verdeja, Roberto Gonzales et al, 2002). Also, the amount of heat release depends upon the variations in the axial velocity of the fuel due to the oscillations introduced by the oscillating valve and the variations in the tangential velocity of combustion air. Oscillating combustion is a retrofit technology that involves forced oscillations of the fuel flow rate to a furnace. When temperature gradient exist within, it is experienced that heat is transferred from high temperature to the low temperature region and the heat transfer is proportional to the temperature gradient and the area normal to the direction of flow of heat. The heat flux to the load from the temperature distribution inside the furnace is analyzed and calculated for non-oscillating and oscillating mode of combustion. During the oscillating combustion mode the turbulent flame travels rapidly upward in radial direction (in ‘y’ direction) greater than the axial direction (‘x’ direction towards the load) to the location of designated temperature points. When a load is heated up, generally its corners are heated faster than any other region. Heat penetrates into the load in all the directions or x, y, z directions. However, highest temperature spots are located at the corners of the load than in the inner area of the load surfaces. Therefore, the temperature gradient also exists in the surfaces of the load at different zones is due to the fast heating of the corners at the surface. In this mode of combustion, the radiation heat flux or the temperature of hot gases at the stack zone found to be less in comparison to steady state. It is because the load at its position has received large portion of radiation flux or heat from the gases and radiation from the furnace wall. Thereby the temperature found to be low at higher position. However the gases escape with high temperature into the stack. The cumulative heat transfer from hot gases to the load directly or indirectly via refractory to loads is a function of time. Higher velocity shortens the time for heat transfer to be accomplished within a given flow path length or furnace size. The radiation emitted by hot gases impinges on the furnace refractory brick wall as well as stock or load. Temperature gradient is the main cause for driving radiation flux from the furnace wall as well from the convective and radiative heat transfer from the hot gases to the load or stock. When large radiation flux is prevailing between hot gases and load it is understood that large heat flux is formed around the crucible or load. Since the thermal gradient between the load and hot gases is bigger than that of between furnace wall and hot gases, most of the radiation flux from hot gases flows into the load, thereby load heats up faster. So, the thermal gradient is found between the furnace wall and hot gases and hot gases and load. The oscillating combustion mode results in successive fuel-rich zone which enables to break up the thermal boundary layer developed around the furnace load due to thermal gradient between hot gases and load. The main aim of the research is to implement the concept of “Oscillating Combustion Technology” on liquid fuels with ambient conditions on a crucible furnace installed with an “Oscillating Valve”. The oscillating valve should be simple, low cost and highly reliable during the operation. It should be a retrofit into the furnace and should necessitate no major modifications. With the oscillating valve installed in the experimental setup exclusive analyses was carried out on temperature distribution and heat transfer for optimization of thermal efficiency with less emission.


2. Materials and methods 2.1. Description of the Experimental Equipment The experimental setup shown in Figure1 (A) has a small furnace with different crucibles, a blower with motor and an air-box. Air-box is connected to ‘U’ tube manometer to calculate the amount of air flow in to the furnace. A gun type burner was used has an adjustment to vary the amount of fuel and air passing through the nozzle is shown as schematic Figure 1 (B). Test equipment is equipped with digital temperature indicators and thermocouples placed at different positions in the furnace to record the temperatures and to analyze the heat transfer rate from the experimental data by using the empirical equations. An oil drum located at approximately 2.5 m above the level of burner has a 3-way cock and piezometer tubes to measure the fuel consumption from time to time during the operation for consistent values. An oscillating valve installed on the fuel line ahead of the burner to create oscillations in the fuel is shown as Figure 1(C).

A B C

Figure 1. Experimental setup (A), Schematic of gun type burner (B), Oscillating valve (C)

2.1.1 Specifications of Instrumentation Electrical ratings and settings: Manufacturer: Lucky Industries, Model: 3 PH Induction Motor; Speed: 2880 rpm, Power: 2.2 kW, 3 HP, 4.5 A, 415 V. Burner data: Manufacturer: Bajaj Engineering; Model: Gun Type Blower data: Manufacturer: Bajaj Engineering; Nominal Motor Power: 2.2 kW Furnace dimensions: i) Inner diameter of the furnace = 0.47m : ii) Height of the furnace (L) = 0.45m ;

iii) Total inner volume = 0.0829 m3

Fuel drum: Inner diameter = 0.457m : Area of cross section of drum = 0.164 m2. Chromel-Alumel (K-type) thermocouples used - Temperature range of -180C to 13720C; Accuracy = ± 0.5 % ; Sensitivity = 40 µV/0C. These thermocouples have long life and low thermal conductivity and suitable for aluminum alloys. Digital temperature indicators are connected with the thermocouples measure and display the designated furnace temperatures All the thermocouples and sensing probe used in this instrumentation system are properly calibrated in the range of investigation. 2.1.2 Brief description of oscillating valve The proposed oscillating valve which is simple, reliable, low cost, easily installed as retrofit has a swivel disc incorporated on the fuel flow pipe and the whole oscillating valve unit was installed in the fuel flow ahead of the burner is shown Figure 2 with different schematic views.

A B C

Figure 2. Schematic of oscillating valve in different views. The axis of the swivel disc is perpendicular to the axis of fuel flow through the pipe. When the swivel disc is actuated, it rotates either side of its axis and controls the volume of the pipe. When the swivel disc is oscillated from its position, due to the cam and follower action it causes reduction in the volume. Thereby, restricts the fuel flow through the valve to the burner. When the cam resumes its normal position, the spring attached to the follower brings back the swivel disc to its original position or to its starting


point. The swivel valve could open and close in 1/10th of a second, and can vary according to the control by potentiometer. The oscillations of the swivel disc are adjusted electromechanically and the amplitude of the swivel disc is adjusted according to the size of the cam or cam profile. The motor consumes 4mA of current during its operation. 2.1.3 Description of experimental variables 1) Frequency (Hz): Number of oscillation cycles per unit time. The oscillating valve was operated at 5 and 10Hz of frequency. 2) Amplitude (Degree): The relative change in mass of fuel flow rate above or below the average flow rate. It is the magnitude of closing of the swivel disc from its initial position. Valve was operated at 100 and 200 of amplitude. 3) Load: Aluminum used (10, 15 and 20 kg.). 4) Air-fuel ratio: Above and below the stoichiometric ratio. Two levels of air- fuel ratios above and below the stoichiometric ratio.( !3:1,14:1. 15:1, 16:1 and 17:1) 2.3. Experimental procedure The experimental work on oscillating combustion was carried out on liquid fuel fired crucible furnace. Prior to the testing of oscillating combustion, few modifications were carried out on the furnace and incorporated the oscillating valve in the experimental set up. Adjustment for variation of fuel flow into the burner is provided to set up required air- fuel ratios during the experiments. A custom valve control was used to oscillate the air-fuel ratio around the stochiometric ratio creating alternate fuel- rich and fuel-lean zones in the furnace during the oscillating combustion for any assumed air- fuel ratio. Initially, experiments were carried out on conventional mode of combustion. Tests include different air-fuel ratios, mass of stocks or loads. Oscillating combustion experiments were carried out with the same parameters with the oscillating valve incorporated as retrofit system. The data gathered during the experiments were used to find out the fuel consumption, specific energy consumption, melting time, temperature distribution, heat transfer rate and thermal efficiency for 13:1 and 15:1 air fuel ratios. Since the thermal characteristics were found to be good during the oscillating combustion at 13:1 air fuel ratio, 200amplitude, 5 Hz frequency and 20 kg load, the results were compared with those of stochiometric air fuel ratio i.e. 15:1(approx.). Analysis is made with the available experimental data for such improvements in the performance characteristics. 2.4. Measurement of experimental parameters. The consumption of air supplied during the combustion was measured with an air box measurement apparatus. For consistent measurement of fuel consumption during the operation a three way cock burette was used. Also a piezometer tube fixed to the fuel drum provides the fuel consumption. Aluminum melting time was recorded by logging it with starting to melting time at equal intervals of time. Thermocouples, sensing probe with digital temperature indicators were used to measure the temperature at designated points in the furnace. A weighing machine was used to weigh the aluminum load for the experiments. 3. Results and discussions

Experiments were performed for conventional and oscillating combustion mode for the evaluation of thermal characteristics. The oscillating valve was operated at different air-fuel ratios, at 13:1, 14:1, 15:1, 16:1 and 17:1. Optimum results were found at 13:1 and 15:1 air-fuel ratio for both modes of combustion. This can be explained as the higher heat release rate around the stoichiometric ratio resulting higher thermal energy available in the furnace. The temperature variation at the aluminum load at 13:1 and for stoichiometric ratio (15:1 appx.) for conventional and oscillating combustion mode was observed and presented. The effect of varying temperatures on the fuel consumption, specific energy consumption, thermal efficiency of furnace and reasons for such improvements due to the enhanced heat transfer rate for both modes of combustion was analyzed. The results and discussions are broadly based on 13:1 and 15:1 air-fuel ratio at different parameters. They are discussed here and are shown in as graphical representations. 3.1 Temperature Profile at Load in the Furnace Here the manifestation of temperature distribution for the stock of 20 kg load inside the furnace at stoichiometric ratio and at a fuel rich condition of air-fuel ratio 13:1 is represented in the Table 1 and shown as Figure 2 for 200 amplitude and 5 Hz frequency. From the Figure 3 for 13:1 air-fuel ratio the point T1, the oscillating combustion mode shows increase in the temperature magnitude than the non-oscillating mode and the higher temperature is observed at the end of 40 to 50 minutes of operation. The temperature difference between oscillating and non-oscillating mode found to be around 1300 C at the end of 20 minutes of operation and 760 C after 40 minutes of operation. It shows that the maximum difference prevails till the point the furnace attains steady state and high radiant condition. The difference is decreased at the end due to the minimum thermal gradient at the position irrespective of the combustion mode. However, the temperature obtained during the oscillating mode must have ensured the process operation of melting the aluminum due to increased heat transfer and thermal conductivity in the load. T1= Temperature measured at the load during melting operation


Table 1.Temperature profile at the Load (T1)

0

100

200

300

400

500

600

700

800

10 20 30 40 50

Time interval (minutes)Te

mpe

ratu

re (

0 C)

13:1 T1 Without Oscillations15:1 T1 Without Oscillations13:1 T1 With Oscillations15:1 T1 With Oscillations

Figure 3. Graphic display of factors effects on temperature distribution The temperature distribution and heat transfer in the furnace can be seen from the Table I and Figure 3 for 15:1 air-fuel ratio that the temperature point T1 is in increasing trend during the oscillating combustion mode. There is a spurt in the temperature for both modes of operation during the time interval of 20 to 30 minutes. The magnitude of temperature at the load found to be greater in the oscillation mode. The temperature difference was around 140 0C after 20 minutes of time interval and gradually reduced after 40 minutes of time interval. The maximum difference in T1 prevails till the point in the furnace attains radiant condition. The high degree of temperature during the oscillating combustion mode is an indication of high thermal gradient between the gases and load resulting greater absorption of heat by the load especially due to the fuel-rich flame. As the fuel-rich zone flame is more luminous and longer in length enhances heat transfer to the load. It can be seen that for any assumed interval of time the magnitude of temperature was found to be higher during the oscillating mode than the steady state mode for both air-fuel ratios. This can be attributed that enhanced heat transfer rate depends upon the temperature gradients inside the furnace which are generally found that the top surface temperature always stays higher than the bottom surface temperature. Oscillating combustion is a novel retrofit and efficient method makes use of oscillating flow field with different zones of flames thereby enhanced heat transfer to the load. This causes decrease in energy consumption and can directly reduce the emission levels.

3.2 Effect of Mass on Fuel Consumption

0

1

2

3

4

5

6

10 15 20

Mass of stock (kg)

Fuel con

sum

ption (k

g)

13:1 n.o

13:1 max

13:1 min

14:1 n.o

14:1 max

14:1 min

15:1 n.o

15:1 max

15:1 min

16:1 n.o

16:1 max

16:1 min

17:1 n.o

17:1 max

17:1 min

0

1

2

3

4

5

6

10 15 20

Mass of stock (kg)

Fuel

con

sum

ptio

n (k

g)

13:1 n.o

13:1 max

13:1 min

14:1 n.o

14:1 max

14:1 min

15:1 n.o

15:1 max

15:1 min

16:1 n.o

16:1 max

16:1 min

17:1 n.o

17:1 max

17:1 min A B

Figure 4. Graphic display of factor effects on fuel consumption at 100amplitude (A) and 200amplitude (B)

From Figure 4(A), it can be seen that conventional mode of combustion the fuel consumption tends to increase marginally at higher loads. Optimum fuel consumption was observed at 15:1 air-fuel ratio for 10 kg, 15 kg, and 20 kg of aluminum load stocks. The same trend was continued for other air-fuel ratios but the melting time was found to be longer. In the oscillating combustion

Time interval (minutes) Air-

fuel ratio

Condition- (Temp.0C) After

10 min.

After 20

min.

After 30

min.

After 40

min.

After 50

min.

13:1 Without Oscillation 205 375 553 625 705

15:1 Without Oscillation 197 342 480 578 650

13:1 With Oscillation 302 505 648 701 --

15:1 With Oscillation 288 474 612 680 723


mode the fuel consumption came down drastically for 100 of amplitude operation at 10 Hz & 5 Hz frequency. Low fuel consumption was observed among the oscillating combustion for 13:1 air-fuel ratio for a 20 kg of load. However, there is only marginal increase in the fuel consumption for other air-fuel ratios for any given load. This can be considered to be a minor factor. The fuel consumption during the conventional mode of combustion for the same 13:1 air-fuel ratio and 20 kg was found to be 3.98 kg when compared to the oscillating mode fuel consumption which was 2.92 kg. A variation of almost 1.0 kg of fuel was less consumed. This can be discussed as heat is always lost from flame to the furnace walls and the propagation of the flame becomes slower as the flame gets closer to the quenching position and the flame velocity is reduced with the reduced flame temperature. Gradually heat is lost to the furnace walls during the steady state combustion mode. In the oscillating combustion mode, within a reasonable short time the furnace wall temperature becomes more or less uniform because time scale of the flame propagating is less and its velocity is faster due to more luminous flame from the fuel-rich zone of the flame. Fuel consumption tends to become low due to less time taken for the load to melt. Figure 4(B) shows the fuel consumption during the oscillating combustion mode for 200amplitude for all air-fuel ratios and especially at 5Hz frequency operation was found to be at its lower value. The restriction of fuel passage by the swivel valve of the oscillating combustion valve at 200amplitude plays an important role in creating greater amplitude with lower frequency. The lower frequency gives enough time to match the higher amplitude in perturbing the fuel flow to introduce oscillations. This causes the air-fuel ratio to oscillate above and below the stoichiometric ratio, thereby producing alternating fuel-rich and fuel-lean zones in the flame. This increases the heat transfer rate from flame to the load, hence significantly less fuel consumption 3.3. Effect of Mass on Specific Energy Consumption (SEC) Specific energy consumption (SEC) is the ratio of quantity of fuel or energy consumed to the quantity of metal processed. Furnace utilization factor and standby losses plays important role in achieving the low SEC. Utilization has a critical effect on SEC and is a factor that is often neglected. If the furnace is at a temperature then standby losses of a furnace occur whether or not a product is in the furnace. Energy is also lost from the charge or its enclosure in the way of heat.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

10 15 20

Mass of stock (kg)

Spec

ific

ener

gy c

onsu

mpt

ion

(1x4

1,80

0kJ/

kg)

13:1 n.o

13:1 10Hz

13:1 5Hz

14:1 n.o

14:1 10Hz

14:1 5Hz

15:1 n.o

15:1 10Hz

15:1 5Hz

16:1 n.o

16:1 10Hz

16:1 5Hz

17:1 n.o

17:1 10Hz

17:1 5Hz

I

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

10 15 20

Mass of stock (kg)

Spec

ific

ener

gy c

onsu

mpt

ion

(1x4

1,80

0kJ/

kg)

13:1 n.o

13:1 10Hz

13:1 5Hz

14:1 n.o

14:1 10Hz

14:1 5Hz

15:1 n.o

15:1 10Hz

15:1 5Hz

16:1 n.o

16:1 10Hz

16:1 5Hz

17:1 n.o

17:1 10Hz

17:1 5Hz

I

A B

Figure 5. Graphic display of factor effects on specific energy consumption at 100amplitude (A) and 200amplitude (B) The effect of oscillations on SEC for different air-fuel ratios and loads has been shown in Figure 5(A). For the steady state combustion mode without oscillations the SEC was found to be more for the rich mixtures and in decreasing trend for lean mixtures. But when compared for the same air-fuel ratio for different loads the SEC was lower for higher loads and lean air-fuel ratios. However the time taken for the melting operation for these low SEC loads was marginally higher during the steady state mode operation. But when compared the same with oscillating mode combustion, the SEC has dramatically found to be at its lower side and especially for 20kg of load i.e. 0.146 kJ/kg at 5 Hz frequency at 13:1 air-fuel ratio and 0.163 kJ/kg at 5 Hz frequency at 14:1 air-fuel ratio. The significant reduction in oscillating combustion can be stated as the oscillated fuel, creating successive fuel-rich and fuel-lean zones during which the heat transfer rate from the flame to the load is maximum for a higher load and due to maximum utility of the furnace. Here, the effect of oscillations especially during the oscillating combustion mode for all air-fuel ratios and loads operating at 200 amplitude and 5 Hz frequency is seen in Figure 5(B). SEC has been low among the oscillating combustion for all air-fuel ratios. This was due to the enhanced rate of heat transfer to the loads due to radiation and convection of hot gases to load and internal conduction in the load during oscillating combustion. Noticeable difference was observed in SEC at 15:1 air-fuel ratio. This was found to be low among the oscillating conditions i.e.0.156 kJ/kg. Significant reduction in SEC was observed from steady state to oscillating combustion from 0.400 kJ/kg to 0.156 kJ/kg. However for all other air-fuel ratios the SEC was found to be less at 200 amplitude operation. At 200 amplitude the restriction of the fuel quantity was more and due to smaller frequency of the swivel disc operation causing more time for the restricted fuel to flow with oscillations to the burner. This resulted in optimum release of


energy during the fuel-rich zone and higher heat transfer. There was a slight variation among the oscillating mode of operation but the difference was significant when compared to the steady state combustion. 3.4 Effect of Air-fuel ratio on Efficiency

0

2

4

6

8

10

12

14

16

18

10 15 20Mass of stock (kg)

Effic

ienc

y (%

)

13:1 n.o

13:1 10

13:1 20

14:1 n.o

14:1 10

14:1 20

15:1 n.o

15:1 10

15:1 20

16:1 n.o

16:1 10

16:1 20

17:1 n.o

17:1 10

17:1 20

I

0

5

10

15

20

25

10 15 20Mass of stock (kg)

Effic

ienc

y (%

)

13:1 n.o

13:1 10

13:1 20

14:1 n.o

14:1 10

14:1 20

15:1 n.o

15:1 10

15:1 20

16:1 n.o

16:1 10

16:1 20

17:1 n.o

17:1 10

17:1 20

I

A B

Figure 6. Graphic display of factor effects on efficiency 10Hz (A) and 20Hz (B) The relation between air-fuel ratio and efficiency for different mass of stocks at different amplitudes and frequencies with and without oscillations has been shown in Figures 6(A) & (B). For oscillating modes the efficiency was found to be increasing from 13:1 to 15:1 and there is marginally decreasing for lean air-fuel ratios. This is due to the release of optimum energy at nearly stoichiometric ratio and further the mixture was lean which tend to decrease the efficiency at 16:1 and 17:1 air-fuel ratios. But the efficiency was on the increasing trend for the higher loads due to the maximum utility of the furnace capacity and absorption of heat energy. In the same oscillating modes of operations, again it was demonstrated that the efficiency was in increasing trend up to 16:1 air fuel ratio and higher loads. The efficiency was found to be remarkable at air-fuel ratio 13:1 and 20° amplitude, 5 Hz frequency and 20 kg of load during oscillating combustion. 3.5 Effect of Mass on Melting Time

A B

Figure 7. Graphic display of factor effects on melting time 100amplitude (A) and 200amplitude (B) The effect of different loads of mass of stock on melting time is shown in Bar Chart as Figure 7(A) for 10° amplitude at different air-fuel ratios and frequencies with and without oscillations. Melting time for all the masses was found to be less for lower frequencies. It took less time for 10 kg at 13:1 air-fuel ratio but marginally took more time for 15 and 20 kg for the same 13:1 air fuel ratio. Though, the time taken for melting 10 kg of mass was less than 15 and 20 kg but the maximum heat energy available in the furnace due to oscillating combustion could not be used. However, the melting time for other masses was marginally high but processed larger amounts of mass The relation between mass of the stock and melting time at 20°amplitude is shown Bar Chart as Figure.7 (B) at different air-fuel ratios and frequencies with and without oscillations. Different air-fuel ratios gave different melting times. It is seen that the time taken for melting different masses was reduced further due to oscillations of lower frequency at 20°amplitude. This was again due to the radiant heat which was existing already and the optimum use of available heat energy in the furnace. In oscillating modes of operation the oscillating valve is able to open and close steadily at higher amplitude and lower frequency facilitating break up of thermal boundary layer which shortens heat up time.


4. Conclusions

Experimental investigation was performed to investigate the temperature distribution and radiative heat transfer characteristics of aluminum loads in a fuel fired crucible furnace with conventional and oscillating combustion modes of operations at different parameters. The temperature measurements inside the furnace at load were recorded using thermocouples with digital indicators and the data has been used to analyze the heat transfer rate inside the furnace. Thermal boundary layer was a major concern during the normal combustion mode in a furnace. It has ill effect on the heat transfer rate to the load. The break up of thermal boundary layer is the result of increased heat transfer rate in the oscillating combustion from hot gases to load. Oscillating combustion results were compared with conventional combustion results. The results showed that due to oscillations in the fuel flow by the oscillating valve there are improvements in the performance characteristics in the oscillating combustion. Also, the oscillating combustion results were compared with the work carried out by the GTI and ALRC. The results were found to be in good agreement in terms of furnace efficiency, heat transfer rate and fuel costs. The results obtained during the oscillating combustion are promising.

• 2% to 6% increase in efficiency, • 7% to 27% of fuel savings • reduction in specific energy consumption (SEC) from 16.5% to 32% • reduced melting time and increased productivity rate with reduced flue gas volumes

It can be concluded that successful testing of oscillating combustion which recorded improvements in the performance characteristics would directly facilitate the deployment of this technology in the process industries, heat transfer industries. Acknowledgment The author is grateful to the management of P.R.R.M. Engineering College, Shabad, R. R. Dist., Andhra Pradesh, India, for providing the facilities for the execution of this experimental analysis in the Production Technology Laboratory of the Department of Mechanical Engineering

References Arora C.P, Text book, Heat and Mass Transfer. Delabroy O., Louedin O., Tsiava R., Gouefflec G. Le, and Bruchet P., 2001. Oxycombustion for reheating furnaces: major benefits

based on ALROLLTM, a mature technology, AFRC/JFRC/IEA 2001 Joint Int. Combust. Symp., Hawaii, Sep. 9-12, Vol. 5, pp. 217-240.

Energy Matters, 2002. Spring – Office of Industrial Technologies. Energy Matters. 2002. National Renewable Energy Laboratory; DOE/GO-102002-1573 Fadare D.A, Bamiro, O.A., and Oni, A.O, 2009. Energy analysis for production of powered and palletized organic fertilizer in

Nigeria. ARPN Journal of Engineering and Applied Science, Vol.4, No.4, pp. 75-82. Gupta A.K, 1997. Gas turbine combustion prospects and challenges, Energy Converts, Mgmt,Vol.38. No.10-13, pp 1311-1318. gti-osc. combus.com. Im H.G., Bechtold J.K. and Law C.K., 1996. Response of counter flow pre-mixed flames to oscillating strain rates. Combustion

and Flame, Vol. 105, No. 3, pp. 358-372. John ,C and Wagner, 2004, NOx emission reduction by oscillating combustion, GTI. Kim J., Won S.H., Shin M.K. and Chung S.H., 2002. Numerical simulation of oscillating lifted flames in co-flow jets with highly

diluted propane. Proceedings of Combustion Institute, Vol. 29, No. 2, pp. 1589-1595. Ruiz, Roberto. et al.Oscillating combustion: An innovative NOx emission control approach. International Gas Research

Conference 1995: V2: 2242-2249. Verdeja L.F, Gonzales R. and Ordonez A.. 2000. Using FEM to determine temperature distribution in a blast furnace crucible.

JOM Journal of the Minerals, Metals and Materials Society, Vol. 52, No. 2, pp. 74-77. Wagner, C. John. 2002. Demonstration of oscillating combustion on a reheat furnace. Final Report, Reports, Publications and

Software Vol. 56: GRI- 02/0144. www.chaos.engr.utk.edu/index.html- Research in pulse combustion (Accessed 27 April, 2010) www. clean energy.com (Accessed 27 April, 2010). Zheng M., Asad U., Reader G.T., Tan Y., Wang M. 2009. Energy efficiency improvement strategies for a diesel engine in low-

temperature combustion. International Journal of Energy Research, Vol. 33, No. 1, pp. 8-28. Biographical notes Dr.G.V.S.Rao graduated in Science and Engineering from Andhra University in 1966 and 1969 respectively. He received his Master’s degree (M.E) with Thermal Science as specialization from Regional Engineering College, Warangal in 1972 under Osmania University. He pursued in Doctorial Work and received Ph.D from Indian Institute of Technology, Madras in 1978. He served in Research and Development of Bharath Heavy Electricals Limited (BHEL) and designed, developed combustion system for M.H.D Power generation of a pilot plant, and National Project with participation of BHEL and BARC of DAE. For 23 years in BHEL served in different Engineering areas and R&D. His main areas of interest are combustion, gasification and new energy developments.


J.Govardhan is a Ph.D candidate at the Department of Mechanical Engineering, Osmania University. He got his Bachelor’s degree in Mechanical Engineering from Institution of Engineers (India) and Master’s degree in Thermal Engineering from Delhi University. His area of research is Oscillating Combustion. He developed an oscillating valve and incorporated it in the test equipment and studied the effects of oscillations during the combustion and its performance characteristics such as processing time, specific energy consumption, thermal efficiency and emissions. Received November 2009 Accepted March 2010 Final acceptance in revised form April 2010

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Persistence and stability of a two prey one predator system

T. K. Kara* and Ashim Batabyalb

a Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, INDIA

b Department of Mathematics, Bally Nischinda Chittaranjan Vidyalaya, Bally Ghoshpara, Howrah, INDIA E-mails:([email protected], T. K. Kar), *Corresponding author); [email protected] (Ashim Batabyal)

Abstract The purpose of this work is to offer some mathematical analysis of the dynamics of a two prey one predator system in the presence of a time delay due to gestation. We derived criteria which guarantee the persistence of the three species and the global dynamics of the model system. Our study also shows that, the time delay may play a significant role on the stability of the system. Lastly, some numerical simulations are given to illustrate analytical results. Keywords: Prey predator, persistence, extinction, global stability, Hopf bifurcation. AMS Subject classifications: 34C07, 34D23, 92D25.

1. Introduction The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Over the past decades, mathematics has made a considerable impact as a tool to model and understand biological phenomena. In return, biologists have confronted the mathematicians with a variety of challenging problems, which have stimulated developments in the theory of nonlinear differential equations. Such differential equations have long played important role in the field of theoretical population dynamics, and they will, no doubt, continue to serve as indispensable tools in future investigations. Differential equation models for interactions between species are one of the classical applications of mathematics to biology. The development and use of analytical techniques and the growth of computer power have progressively improved our understanding of these types of models. Although the predator-prey theory has seen much progress, many long standing mathematical and ecological problems remain open. Theoretical ecology remained silent about the astonishing array of dynamical behaviors of three species models for a long time. Of course, the increasing number of differential equations and the increasing dimensionality raise considerable additional problems both for the experimentalist and theoretician. Freedman and Waltman (1984) considered three level food webs – two competing predators feeding on a single prey and a single predator feeding on two competing prey species. They obtain criteria for the system to be persistent. Kar and Chaudhuri (2004) considered a two-prey one-predator harvesting model with interference. The model is based on Lotka-Volterra dynamics with two competing species which are affected not only by harvesting but also by the presence of a predator, the third species. Optimal harvesting policy and the possibility of existence of a bioeconomic equilibrium is discussed. Dubey and Upadhyay (2004) proposed a two predator one prey system with ratio dependent predator growth rate. Criteria for local stability, instability and global stability of the non-negative equlibria are obtained. They also discussed about the permanent co-existence of the three species. Braza (2008) considered a two predator; one prey model in which one predator interferes significantly with the other predator is analyzed. The analysis centers on bifurcation diagrams for various levels of interference in which the harvesting is the primary bifurcation parameter. Zhang et al. (2006) studied the stability of three species population model consisting of an endemic prey (bird), an alien prey (rabbit) and an alien predator (cat). It may be pointed out here that all the above studies are based on the traditional prey dependent models. Recently, it has been observed that in some situations, especially when a predator have to search for food and have two different choice of food, a more suitable predator-prey theory should be based on the so-called ratio-dependent theory, in which the per-capita growth rate should be function of the ratio of prey to predator abundance, and should be the so-called predator functional response (Abrams and Ginzburg, 2000; Akcakaya et al., 1995; Arditi et al., 1991; Arditi and Saiah, 1992).

Kar and Batabyal / International Journal of Engineering, Science and Technology, Vol. 2, No.2, 2010, pp. 174-190

175

Recently, Kesh et al. (2000) proposed and analyzed a mathematical model of two competing prey and one predator species where the prey species follow Lotka-volterra dynamics and predator uptake functions are ratio dependent. They derived conditions for the existence of different boundary equilibria and discussed their global stability. They also obtain sufficient conditions for the permanence of the system. Hsu et al. (2001) studied the qualitative properties of a ratio dependent predator-prey model. They showed that the dynamics outcome of interactions depend upon parameter values and initial data. Three general forms of functional response are commonly used in ecological models: linear, hyperbolic, and sigmoidal. How predators respond to changes in prey availability (functional response) is an issue of particular importance. There is evidence from several models that the type of functional response specified can greatly affect model predictions (Gao et al., 2000; Kar and Chaudhury, 2004). To have a perfect model we would need to consider so many factors, namely, growth rate, death rate, carrying capacity, stage structure, predation rate, stochasticity etc. However, it is obvious that a perfect model can not be achieved because even if we could put all these factors in a model, the model could never predict ecological catastrophes of Mother Nature caprice. Therefore, the best we can do is to look for analyzable model that describes as well as possible the reality. In this paper, we shall study the dynamical behaviors of a two prey one predator system. Before we introduce the model and its analysis we would like to present a brief sketch of the construction of the model which may indicate the biological relevance of it. (i) There are three populations namely, two prey whose population densities are x and y, the predator whose population density

is denoted by z. (ii) In the absence of the predator the prey population density grows according to logistic law of growth. (iii) Two prey species are competitive in nature. (iv) One prey is much higher in abundance and more vulnerable compare to other. (v) Handling time for one prey is negligible, where as the predator needs sufficient handling time for other prey. These are

incorporated using Holling type I and II functional response. (vi) There is a reaction time for predator. On the basis of the above assumptions, our proposed model is as follows:

.)(

)()(

,)(

1

,1

2211

22

11

cztym

ztybztxbdtdz

ymyzxya

Lysy

dtdy

xzxyaKxrx

dtdx

−−+

−+−=

+−−⎟

⎠⎞

⎜⎝⎛ −=

−−⎟⎠⎞

⎜⎝⎛ −=

ττωτω

ω

ω

(1)

In model (1), r and s are the intrinsic growth rate of two prey species, K and L are their carrying capacities, c is the mortality rate coefficient of the predator, a1, a2 are inter-species interference coefficient of two prey species, ω1 is the first prey specie’s searching efficiency and ω2 is the second type prey specie’s searching efficiency of the predator, b1 and b2 are the conversion factors denoting the number of newly born predators for each captured of first and second prey respectively, and m is the half saturation co-efficient. A discrete time delay ( 0≥τ ) is introduced to the functional response term involved with the growth equation of predator to allow for a reaction time (Liu, 1994). Section 2 deals with the determination of plausible steady states and their existence conditions. Dynamical behavior of these steady states is discussed in section 3. Global stability and persistence of the system is studied in section 4. Section 5 deals with simulation and discussion of the problem. 2. Equilibrium analysis It can be checked that the system (1) has seven non-negative equilibria and three of them namely Eo (0, 0, 0), Ex (K, 0, 0), Ey (0, L, 0) always exist. We show the existence of other equilibria as follows: Existence of Exy (x4

*, y4*, 0)

Here x4*, y4

* are the positive solution of the following algebraic equations.

01,01 21 =−⎟⎠⎞

⎜⎝⎛ −=−⎟

⎠⎞

⎜⎝⎛ − xa

Lysya

Kxr (2)

Solving (2), we get


176

KLaarsKasrLy

KLaarsLarsKx

21

2*4

21

1*4

)(,)(−−

=−

−= (3)

Thus, the equilibrium Exy (x4*, y4

*, 0) exists if ( )Lar 1− and ( )Kas 2− are of same sign, that is either

Lar 1> and Kas 2> (4a)

Or, Lar 1< and Kas 2< (4b) Existence of Eyz (0, y5

*, z5*)

Here y5*, z5

* are the positive solution of the following algebraic equations.

.0,01 222 =−+

=+

−⎟⎠⎞

⎜⎝⎛ − c

ymyb

ymz

Lys ωω

(5)

Solving (5), we get

( )*5

*5

2

*5

22

*5 1and ym

Lysz

cbcmy +⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

−=

ωω (6)

It can be seen that Eyz (0, y5*, z5

*) exists if ( ) .22 cmcbL >−ω (7)

Existence of Exz (x6*, 0, z6

*) Here x6

*, z6* are the positive solution of

.0,01 111 =−=−⎟⎠⎞

⎜⎝⎛ − cxbz

Kxr ωω (8)

Solving (8), we get

⎟⎟⎠

⎞⎜⎜⎝

⎛−==

111

*6

11

*6 1,

ωωω Kbcrz

bcx (9)

It can be seen that Exz (x6

*, 0, z6*) exists if

.11 cKb >ω (10) Existence of Exyz (x7

*, y7*, z7

*) Here ( )*

7*

7*

7 ,, zyx is the positive solution of the system of algebraic equations given below:

,01 11 =−−⎟⎠⎞

⎜⎝⎛ − zya

Kxr ω (11a)

,01 22 =

+−−⎟

⎠⎞

⎜⎝⎛ −

ymzxa

Lys ω

(11b)

.02211 =−+

+ cym

ybxb ωω (11c)

Solving (11a) & (11b), we get ( ) ,0, =zyf (12)

where ).()())(()(),( 212122 ymKLzaymKLyaaLzrymyLrsymLrKazyf +−+−++−−+= ωω

Also solving (11a) & (11c), we obtain ( ) ,0, =zyg (13)

where ( ) .)()()()(, 1

211112211 zymbKymybKaybymcrymKrbzyg +−+−−+−+= ωωωω

From (12) we note the following. When z→ 0, then y→ ya, where


177

.)(

21

2

KLaarsKasrLya −

−= (14)

We note that ya > 0, if the inequality (4a) holds.

Also from the equation (12), we have1

1

BA

dzdy

= , where

,)( 2211 LrymKLaA ωω −+=

and zaKLaaKLrsymKL

KsarKLB 212121 )2()( ω−⎟

⎠⎞

⎜⎝⎛ −++−= .

It is clear that 0>dzdy

, if either

holdBandAii

orBandAi00)

,00)

11

11

<<>>

(15)

Again from (13), we note that when z→ 0, then y → yb, where

,2

4

2

222

22

ACABB

yb

−+−=

KbaA 1112 ω−= ,

)()()( 22111221112 cbrKmarbrcrbKmarbB −+−=−+−= ωωωω ,

( )cKbmrC −= 112 ω . Clearly A2 < 0 and C2 > 0 if the inequality (10) is satisfied.

We also have

ygzg

dzdy

∂∂∂∂

−= .

We note that 0<dzdy

, if either

.00)(

,00)(

holdzgand

ygii

zgand

ygi

<∂∂

<∂∂

>∂∂

>∂∂

(16)

From the above analysis we note that two isoclines (12) and (13) intersect at a unique (y7*, z7

*) if in addition to conditions (4a), (10), (15) and (16), the inequality ya < yb (17) holds. Knowing the value of y7

* and z7*, the value of x7

* can be calculated from

.)(

)()(

)(

11*

7

*722

11*

7

*722

*7*

7 ωω

ωω

bymycbcm

bymybymcx

+−−

=+

−+= (18)

It may be noted that for x7* to be positive, we must have

.)( *722 ycbcm −> ω (19)

This completes the existence of ),,( *7

*7

*7 zyxExyz .

3. Linear stability analysis


178

The characteristic equation for the model (1) is given by 0=−+ − IeBA λλτ ,

where, 33)( ×= ijaA , zyaKxra 1111

21 ω−−⎟⎠⎞

⎜⎝⎛ −= , ,, 113112 xaxaa ω−=−=

,)(

21, 222

222221 ymyz

ymzxa

Lysayaa

++

+−−⎟

⎠⎞

⎜⎝⎛ −=−=

ωω

ymya+

−= 223

ω

0333231 === aaa

1 3 3 11 12 13 21 22 23( ) , 0, 0,jB b b b b b b b×= = = = = = =

zzebb λω −=1131

.c)ym(

ybxbb,e

)ym(

zmbb, −

++=

+= − 22

113322

32ω

ωω λτ

We first consider the case where τ =0: At ( )0,0,00E the characteristic equation becomes (λ –r) (λ –s) (λ +c) =0. So, we arrive at a conclusion: Theorem 3.1. Eo is always a saddle node and there can not be total extinction of the system (1) for positive initial conditions For )0,0,(KEx the characteristic equation becomes ( )( )( ) ,0112 =−−−−−− λωλλ cKbKasr

and for )0,0( LEy , the characteristic equation becomes 0))(( 221 =⎟

⎠⎞

⎜⎝⎛ −−

+−−−− λ

ωλλ c

LmLbsLar

Observing the sign of eigenvalues we can state the following theorem: Theorem 3.2 a) Ex is a saddle point with locally stable manifold in x directions and with locally unstable manifold in y – z plane if s – a2K > 0 and Kb1ω1 < c hold, but if s – a2K < 0 and Kb1ω1 < c, then Exz does not exist and in that case Ex is locally asymptotically stable in x – y – z space.

b) If inequalities r – a1L > 0 and ( ) .22 cmcbL >−ω hold then Ey is a saddle point with locally stable manifold in y –

direction and with locally unstable manifold in x – z plane. But if r – a1L < 0 and 022 <−+

cLmLb ω

, then Eyz does not exist and in

that case Ey is locally asymptotically stable in x – y – z space

For, ,0,)(,)(

21

2

21

1⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−

KLaarsKasrL

KLaarsLarsKExy the characteristic equation becomes

( ) ( )

( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

0

00221

211

21

111

221

21

21

2

21

22

21

11

21

11

21

1

=

−−−+−

−+

−−

−+−−−

−−

−−

−−−

−−−

−−−

−−

λωω

ωλ

ωλ

cKasrLKLaarsm

KasrLbKLaars

LarsKbKasrLKLaarsm

KassKKLaarssKars

KLaarsKasrLa

KLaarsLarsK

KLaarsLarsKa

KLaars)rLa(rs

One of the eigenvalue of the system is cKasrLKLaarsm

KasrLbKLaars

LarsKb−

−+−−

+−

−)()(

)()(

221

222

21

111 ωω

and sum and product of other two eigenvalues are ⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+

−−

KLaarssKars

KLaarsrLars

21

2

21

1 )()( and

)())((

21

21

KLaarsKasLarrs

−−−

respectively. Clearly, when the inequality (4a) holds, the sum of other two eignevalues is negative and

product is positive. But when the inequality (4b) holds then the product of other two eignevalues is negative. Hence we state out the following theorem: Theorem 3.3. If the inequality (4a) holds then Exy exists and is asymptotically stable in x – y plane but if the inequality (4b) holds then Exy exists and in that case it will be unstable in x-y plane. Moreover, it will be stable in x – y – z space if


179

cKasrLKLaarsm

KasrLbKLaars

LarsKb<

−+−−

+−

−)()(

)()(

221

222

21

111 ωω (20)

For, ,))(/1(,where),,,0(2

*5

*5*

522

*5

*5

*5 ωω

ymLyszcb

cmyzyEyz+−

=−

=

we state out the following theorem: Theorem 3.4 Eyz exists and is asymptotically stable in y – z plane if the inequality 2222 )( ωω mbmccbLcm +<−< (21)

holds. Also if the inequality 0zyar *51

*51 <ω−− holds, then it will be asymptotically stable in x – y – z space

For ),,0,( *6

*6 zxExz where ,1,

111

*6

11

*6 ⎟⎟

⎠

⎞⎜⎜⎝

⎛−==

ωωω Kbcrz

bcx , the characteristic equation becomes

0

11

010

111

22

111

111

2

11

2

111

1

11

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

−−−−

λωω

ωω

λωω

ωω

ωλ

ω

Kbc

mb

Kbcrb

Kbc

mr

bcas

bc

bca

Kbrc

So as before we state out the following theorem: Theorem 3.5 If Exz exists, then it is asymptotically stable in x – y – z space if the following inequality holds.

sKb

cmr

bca

>⎟⎟⎠

⎞⎜⎜⎝

⎛−+

111

2

11

2 1ωω

ωω

(22)

Now, to investigate the local stability of interior equilibrium, we first linearize the system (1) using the following transformations

Zzz,Yyy,Xxx *7

*7

*7 +=+=+= , (23a)

where X, Y, Z is small perturbation about ( )*7

*7

*7 ,, zyxExyz , and then the linear form of (1) is given by

.)(

)(

,)(

,)()(

2*7

*7

*722

*7

*722

11*

7

*7*

7

22*

7

*72

*7*

7*

72

1*

71*

7*

7

Yym

zybymzbXbzZ

Zyymym

YzyYyLsXyaY

ZxYaxXxK

rX

⎥⎦

⎤⎢⎣

⎡

+−

++=

+−

++−−=

−+−+−

=

•

•

•

ωωω

ωω

ω

(23b)

We consider the following positive definite function

2*

7

22*

7

12*

7 2221 Z

zd

Yyd

Xx

U ++= , where d1, d2 are positive constants to be chosen later.

Differentiating U with respect to time t along the solution of linear model 23(b) it can be seen that •

U is negative definite if we

choose ( ) .0)(

4and)(

,12*

7

*7212

121*71

21

12 ≤⎟⎟

⎠

⎞⎜⎜⎝

⎛

+−−+

+==

ymz

Ls

Kdrdaa

ymbmbd

bd ω

23(c)

Hence we arrive at a conclusion:


180

Theorem 3.6 If ( ) ,0)(

42*

7

*7212

121 ≤⎟⎟⎠

⎞⎜⎜⎝

⎛

+−−+

ymz

Ls

Kdrdaa ω

then )z,y,x(E *7

*7

*7xyz is locally

asymptotically stable 4. Global stability and persistence of the system Theorem 4.1 a) If the equilibrium Exy exists and is locally asymptotically stable in the interior of positive quadrant of x – y plane then it will be globally asymptotically stable there. b) If Eyz exists and is locally asymptotically stable in y-z plane then it will be globally asymptotically stable in the region 2

+R of y-

z plane, where .0)(,0,0:),( 222

⎭⎬⎫

⎩⎨⎧ >−+>>≡+ z

sLymzyzyR ω

c) If Exz exists and is locally stable then it will be globally asymptotically stable in the positive quadrant of x – z plane

Proof. For Exy, let xyaKxrxyxh

xyyxH 11 1),(,1),( −⎟

⎠⎞

⎜⎝⎛ −== and

xyaLysyyxh 22 1),( −⎟⎠⎞

⎜⎝⎛ −= .

Clearly H(x,y) > 0 in the interior of positive quadrant of x – y plane. Then we have

.0)()(),( 21 <−−=∂∂

+∂∂

=ΔxLs

yKrHh

yHh

xyx

Clearly Δ (x, y) does not change sign and is not identically zero in the positive quadrant of x – y plane. Therefore, by Bendixson-Dulac criterion there exists no limit cycle in the positive quadrant of x-y plane. So, if Exy is locally asymptotically stable then it will be globally asymptotically stable in the interior of positive quadrant of x – y plane (Hale, 1969).

For Eyz, let ym

yzLysyzyh

yzzyH

+−−== 2'

1 )1(),(,1),( ω and

.),( 22'2 cz

ymyzbzyh −

+=

ω

Clearly H(y, z) > 0 in the interior of the positive quadrant of y-z plane. Then we have

0)ym(Lz

s)Hh(z

)Hh(y

)z,y( 22'

2'

1 <+

ω+−=

∂∂

+∂∂

=Δ when .0)( 22 >−+ zs

Lym ω

Hence Eyz is globally asymptotically stable in the region 2+R of y-z plane, where

⎭⎬⎫

⎩⎨⎧ >−+>>≡+ 0)(,0,0:),( 2

22 zsLymzyzyR ω .

For Exz, let xzKxrxzxh

xzH 1

"1 1),(,1 ω−⎟

⎠⎞

⎜⎝⎛ −== and zcxzb)z,x(h 111

"1 −ω= .

Clearly H(x, z) > 0 in the interior of the positive quadrant of x-z plane. Then we have

0zKr)Hh(

z)Hh(

x)z,x( "

2"

1 <−=∂∂

+∂∂

=Δ in the positive quadrant of x - z plane. Hence Exz is globally asymptotically

stable in the positive quadrant of x - z plane Theorem 4.2 Let the following hold

,))((

4)( *7

*72112

211 ⎟⎟⎠

⎞⎜⎜⎝

⎛

++−≤+

ymymzd

Lsd

Krada ω

(24)


181

where d1, d2 are same as defined in 23(c), then the positive equilibrium Exyz is globally asymptotically stable with respect to all solutions initiating in the interior of 3

+R

Proof. Consider the following positive definite function about ).,,( *7

*7

*7 zyxExyz

.lnlnln *7

*7

*72*

7

*7

*71*

7

*7

*7 ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

zzzzzd

yyyyyd

xxxxxV

Differentiating V with respect to t along the solution of model (1), we get

zzzzd

yyyyd

xxxxV

••••

−+−+−= )()()( *72

*71

*7 .

Using the system of equation (1), we get

))()(()()( *7

*7211

2*71

2*7 yyxxadayy

Lsdxx

KrV −−+−−−−−=

•

.)())((

))(())((

))(())((

)())()((

2*7*

7

*721*

7*

7*7

*721

*7

*7*

7

2212*7

*72111

yyymym

zdzzyyymym

yd

zzyyymymbddmzzxxdb

−++

+−−++

−

−−++

−−−−−−

ωω

ωωω

The above equation can further be written as

[ ]

,))(())((

)())(()(*

7*

723*

7*

713

2*722

*7

*712

2*711

zzyybzzxxb

yybyyxxbxxbV

−−+−−+

−+−−+−−=•

where, )1(),(, 211132111211 dbbadabKrb −−=+== ω

*7

*7211222

23*7

*7211

22 ))(()(,

))(( ymymyddbdmb

ymymzd

Lsdb

++−−

=++

−=ωωω

Let us choose1

21b

d = , and)( *

71

21 ymb

mbd+

= , then the sufficient condition for .

V to be negative definite is 011 >b , and

04 22112

12 ≤− bbb . (25) We note that b11 > 0 always. Also (24) ⇒ (25), Hence V is a Lipunov function with respect to Exyz. To examine the permanence of the system (1) we shall use the method of “average lyapunov function” ( Gard and Hallam, 1997; Hafbauer, 1981). This method was first applied by Hutson and Vickers (1983) to ecological problems.

Let the average Liapunov function for the system (1) be σ(X) = 21 ppp zyx , where p, p1and p2 are positive constants. Clearly in the interior of R+

3 we have

⎥⎦

⎤⎢⎣

⎡+

−−⎟⎠⎞

⎜⎝⎛ −+⎥

⎦

⎤⎢⎣

⎡−−⎟

⎠⎞

⎜⎝⎛ −=

++==Ψ

••••

ymzxa

Lyspzya

Kxrp

zzp

yyp

xxp

XXX

22111

21

11

)()()(

ωω

σσ

.22112 ⎥

⎦

⎤⎢⎣

⎡−

+++ c

ymybxbp ωω

Let us assume that inequalities (4a), (7) and (10) hold. Also the hypothesis of theorem 4.1 holds.


182

Then Exy, Eyz and Ezx exist, further there are no periodic orbits in the interior of x – y plane, x – z plane and in the region R+2 of y

– z plane. Thus to prove the uniform persistence of the system, it is enough to show that ψ(X) > 0 in the domain D of R+

3 where

⎭⎬⎫

⎩⎨⎧ >−+>>>≡ 0)(,0,0,0:),,( 22 z

sLymzyxzyxD ω

for a suitable choice of p, p1 and p2 > 0. That is one that has to satisfy the following conditions.

0)( 21 >−+=Ψ cpspprEo , 26(a)

0)()()( 112212112211 >−+−=−+−=Ψ cKbpKaspcpKbpKapspEx ωω , 26(b)

0)()( 2221

2221 >⎟

⎠⎞

⎜⎝⎛ −

++−=⎟

⎠⎞

⎜⎝⎛ −

++−=Ψ c

LmLbpLarpc

LmLbpLpaprEy

ωω, 26(c)

0)()(

)()()(221

222

21

1112 >⎥

⎦

⎤⎢⎣

⎡−

−−−−

+−

−=Ψ c

KasrLKLaarsmKasrLb

KLaarsLarsKbpExy

ωω, 26(d)

0)(

1)(22222

1

22

1 >⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−

−=Ψcb

cmmcbL

cmscb

cmarpEyz ωωωω

ω, 26(e)

01)(111

2

11

21 >⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=Ψ

ωωω

ω Kbc

mr

bcaspExz . 26(f)

We note that by increasing p to sufficiently large value, ψ(Eo) can be made positive. Thus inequality 26(a) holds. Inequalities (4a) & (10) imply that 26(b) holds. So we state out the following theorem. Theorem 4. 3 In addition to inequalities (4a) and (10) let the hypotheses of theorem 4.1 hold, and then the system (1) is uniformly persistent if the following inequalities hold

cKasrLKLaarsm

KasrLbKLaars

LarsKb>

−−−−

+−

−)()(

)()(

221

222

21

111 ωω , 27(a)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+−

>cb

cmmcbL

cmscb

cmar22222

1

22

1

)(1

ωωωω

ω, 27(b)

01111

2

11

2 >⎟⎟⎠

⎞⎜⎜⎝

⎛−−−

ωωω

ω Kbcr

mbcas 27(c)

Theorem 4. 4. If 23(c) holds, then the equilibrium Exyz is asymptotically stable for τ < τo and unstable for τ > τo. Further, as τ increases through τo, , Exyz (x7

*, y7*, z7

*) bifurcates into small amplitude periodic solutions, where τo = τon as n = 0 . Proof. The characteristic equation for the case where τ ≠ 0 is given by λ3 + λ2 (A + B) + λ (AB -C) + e-λτ λ (D+E) + AD + BE – F = 0 28(a) where A= ( r/K )x *

7 , B= ,,)/()()/( *7

*721

2*7

*7

*72

*7 yxaaCymzyyLs =+− ω

( ) ( ) .)/()()(,,/ 2*7

*7112211

*7

*7

*721

*7

*7

211

3*7

*7

*7

222 ymybababamzyxFzxbEymzymbD +++==+= ωωωω

Now λ = iω (ω> 0) in 28(a) gives -iω3 - ω2 (A + B) + iω (AB -C) + (cos ωτ - i sin ωτ)iω(D+E)+ AD + BE – F = 0. Comparing real and imaginary parts we get, − ω3 +ω (AB -C) = (AD + BE – F) sin ωτ - (D+E)ω cos ωτ, 28(b) − ω2 (A + B) = −(D+E)ω sin ωτ − (AD + BE – F) cos ωτ. 28(c) Squaring and adding 28(b) and 28(c) we get, ω6 + Q1ω4 + Q2ω2 + Q3 = 0, 28(d) where, .)(,)()(),2( 2

322

222

1 FBEADQEDCABQCBAQ −+−=+−−=++=


183

So equation 28(d) has unique positive solution ωo2 irrespective of the sign of 2Q , as 1Q > 0

and 3Q < 0. Now, from 28 (b) and 28 (c) we get,

222

24

)()())(())(()(cos

FBEADEDEDCABFBEADBAED

−++++−−−++++

=ω

ωωωτ . 28(e)

So, corresponding to λ= iω0, there exists τ0n such that,

0222

24

00

2)()(

))(())(()(cos1ωπ

ωωω

ωτ n

FBEADEDEDCABFBEADBAEDarcn +⎥

⎦

⎤⎢⎣

⎡−+++

+−−−++++= . 28(f)

n= 0, 1, 2, - - - - -

Now differentiation of 28(a) with respect to τ gives,

λτ

λλλλλλ

τλ

λτ −−+++

++

−+++−+++

=⎟⎠⎞

⎜⎝⎛

−

−

)()()()(23 21

FBEADEDED

FBEADEDeCABBA

dd

λτ

λλλλλλλ

λτ −−+++

++

−+++−+++

= − )()()()(23

2

23

FBEADEDED

FBEADEDeCABBA

=λτ

λλλλλλλ

λτ

λτ

−−+++

++

−+++−+++−++

−

−

)()()()(2

2

23

FBEADEDED

FBEADEDeFBEADEDeBA

=λτ

λλλλλλλλλ

−−−+++

++

−+++−++

2232

23 1)()()(

)(2FBEADED

EDCABBA

BA

=λτ

λλλλλλλ

−−+++

−+−

−+++−++

)()()()(2

222

2

FBEADEDFBEAD

CABBABA

00200

200

01

)()()()()(2

0ωτ

ωωωωωω

τλ

ωλ iEDiFBEADFBEAD

CABBAiBAi

dd

i

−++−+

−++

−+++−++−

=⎟⎠⎞

⎜⎝⎛∴

−

=

=0

20

2220

022

022

00

0200

)()()())((

)()()()(2

ωτ

ωωω

ωωωωωω i

EDFBEADEDiFBEADFBEAD

BACABBAiCABBAi

+++−+

+−−+−++

++−−+−−−++−

.

⎥⎦

⎤⎢⎣

⎡++−+

−++⎥

⎦

⎤⎢⎣

⎡++−−

+++=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛

−

= )()()(

)()(2)2(Re 2

0222

0

2

220

220

20

221

0ωωωω

ωτλ

ωτ EDFBEADFBEAD

BACABCBA

dd

i

> 0. Therefore the

transversability condition holds and hence Hopf-bifurcation occurs at τ = τo . This completes the proof. 5. Simulation and discussion In this paper we studied the dynamical behaviors of a two prey one predator system. . Holling type I response function is taken to represent the interaction between one of the prey and predator. The interaction between the other prey and the predator is assumed to be governed by a Holling type II response function. Such different choices of functional responses may be particularly useful when handling time for one prey is negligible, whereas the predator needs sufficient handling time for other prey. A good example of a two prey one predator system is minke whale (predator) and two of its main prey juvenile herring and capelin. To illustrate the results numerically, choose r=3.5, K=150, a1=0.001, w1=0.24, s=4.5, L=150, a2=0.1, w2=0.21, m=15, b1=0.5, b2=0.6, c=3.9 in appropriate units. With the above parameter values, system (1) has a positive equilibrium (31.72, 42.89, 11.32), which is globally asymptotically stable (see Figs. 1, 2).


184

0 2 4 6 8 10 12 14 16 18 200

50

100

150

time

popu

latio

ns

x

y

z

Fig.1. Time evolution of all the population for the model system (1).

Population converges to the positive equilibrium (31.72, 42.89, 11.32).

0 20 40 60 80 100 120

0

100200

0

20

40

60

80

100

120

yx

z

Fig.2. Phase portrait of the system (1), corresponding to different initial levels. The figure clearly indicates that the interior

equilibrium point (31.72, 42.89, 11.32) is globally asymptotically stable. Often we come across several biological systems in nature exhibiting cycle behavior. Due to this cyclic nature some population exhibit periodic fluctuation in abundance, with periodic crashes. One could avoid such crashes and stabilize the population by controlling one of the interacting species. (Hudson et al., 1998). Thus it is relevant to find conditions under which a multispecies system exhibits cyclic behavior and it is equally important to find conditions under which cycles can be prevented in a multispecies system. By using Liu’s criterion (see appendix) It is interesting to observe that, when the inter-species interference co-efficient a1 of two prey species is increased, the positive equilibrium losses its stability and a Hopf- bifurcation occurs when a1 passes a critical value. With parameter values r=3.5, K=150, w1=0.24, s=4.5, L=150, a2=0.2731614, w2=0.21, m=15, b1=0.5, b2=0.6, c=1.7 in appropriate units, a super critical Hopf bifurcation occurs when a1*=0.01981331. Now, if we gradually increase the value of a1, keeping other parameters fixed, then Exyz losses its stability as a1 crosses its critical value a1

*=0.01981331 (see Figs 2-4).


185

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

time

popu

latio

ns

x

y

z

Fig. 3. When a1 =0.01<a1*, clearly the populations approach their equilibrium values in finite time. Here parameter values are

r=3.5, K=150, w1=0.24, s=4.5, L=150, a2=0.273164, w2=0.21, m=15, b1=0.5, b2=0.6, c=1.7 in appropriate units.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40

45

time

popu

latio

ns

Fig.4. Unstable solution of system (1). Here all the parameters are same as in Fig.3. except a1 = 0.025>a1*.


186

0 10 20 30 40 50 60020

40

0

5

10

15

20

25

30

35

xy

z

Fig.5. For a1 = a1*, there is a limit cycle near Exyz.

Also as before we can consider c (the mortality rate co-efficient of the predator) as the bifurcation parameter. With parameter values r=3.5, K=150, w1=0.24, s=4.5,L=150, a1=0.015, a2=0.27, w2=0.21, m=9.0, b1=0.5, b2=0.6 in appropriate units, a supercritical Hopf bifurcation occurs when c* = 1.674233.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40

time

popu

latio

ns

x

y

z

Fig.6. When c= 1.7 >c*, clearly the populations approach their equilibrium values in finite time .Here

parameters values are r=3.5, K=150, w1=0.24, s=4.5, L=150, a1=0.015, a2=0.27, w2=0.21, m=9.0, b1=0.5, b2=0.6 in appropriate units.


187

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

40

45

time

popu

latio

ns

Fig.7. Unstable solutions of system (10). Here all the parameters are same as Fig.6. except c = 1.5<c*.

0 10 20 30 40 50 60

020

4060

0

5

10

15

20

25

30

35

yx

z

Fig.8. For c = c* there is a limit cycle near Exyz.

The numerical study presented here shows that, using parameter a1 or c as control, it is possible to break unstable behaviour of the system (1) and drive it to a stable state. In the similar fashion we can consider a2 as a control parameter. Also it is possible to keep the population levels at a required state using the above control. So, we see that in our model dynamics competition plays an important role. Our results established criteria which guarantee the persistence of the three species and the global dynamics of the model system. It has long been recognized that most of the studies of continuous time deterministic models reveal two basic patterns: approach to an equilibrium or to a limit cycle. The basic rationale behind such type of analysis was the implicit assumption that most food chains we observe in nature correspond to stable equilibria of the model. From this viewpoint, we presented the stability and bifurcation of the most important equilibrium point Exyz. We see that Exyz (x7

*, y7*, z7

*) is locally asymptotically stable in the absence of delay. Now for the same values of parameters as for the first figure, it is seen from the Theorem 4.4, that there exists a critical value of τ = τ0=0.0726532 and Exyz losses its stability as τ crosses the critical value τ0.


188

We have also given some graphical representation in favors of our numerical results.

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

x

y

z

τ =0.06

time

popu

latio

ns

Fig.9. When τ =0.06 <τ0, clearly the populations approach their equilibrium values in finite time.

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

90τ =0.08

time

popu

latio

ns

Fig.10. Unstable solution of system (1). Here all the parameters are same as in figure 1, except τ =0.08 >τ0.


189

020406080100120

0

50

100

0

10

20

30

40

50

60

x

Tau=0.0726532

y

z

Fig.11. For τ = τ 0 = 0.0726532, there is a limit cycle near Exyz.

In most of the ecosystems, population of one species does not respond instantaneously to the interactions with other species. To incorporate this idea in modeling approach, the time delay models have been developed. Our result indicates the fact that a sufficient large time delay has ability to destabilize the model system. Considering gestation delay as a bifurcation parameter we have shown that the system undergoes Hopf-bifurcation as ‘τ’ passes through its critical value τ 0 from lower to higher and individual population exhibit small amplitude oscillations around their steady-state value. Appendix. Liu (1994), derived a criteria of Hopf bifurcation without using the eigenvalues of the variational matrix of the interior equilibrium point. We specify below the Liu’s criterion. Liu’s criterion: If the characteristic equation of the interior equilibrium point is given by, ,0)()()( 32

21

3 =+++ θλθλθλ aaa where )(),()()()(),( 33211 θθθθθθ aaaaa −=Δ are smooth function of θ in an

open interval about R∈*θ such that (I) 0)(,0)(,0)( *

3**

1 >=Δ> θθθ aa

(II) ,0*

≠Δ

=θθθdd

then a simple Hopf bifurcation occurs at θ= θ*.

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Math. Appl. Sci., Vol.23, pp. 347-356. Liu W. M., 1994. Criterion of Hopf bifurcation without using eigenvalues. J. Math. Anal. Appl., Vol.182, pp.250-256. Martin A. and Ruan S., 2001. Predator-prey models with delay and prey harvesting. J. Math. Biol., Vol.43 pp. 247-267. Kar T. K. and Chaudhury K. S., 2004. Harvesting in a two-prey one-predator fishery. ANZIAM J., Vol.45, pp. 443-456. Steele J. H. and Henderson E. W., 1992. The role of predation in plankton models Journal of Plankton Research. Vol. 14, pp.157-

172. Zhang J. Fan M. and Kuang Y., 2006. Rabbits Killing birds revisited. Mathematical Biosciences, Vol.203, pp.100 – 123. Acknowledgement Research of T. K. Kar is supported by the Council of Scientific and Industrial Research (C S I R), India (Grant no. 25(0160)/ 08 / EMR-II dated 17.01.08) Biographical notes Dr. T. K. Kar is an Associate Professor at the Department of Mathematics, Bengal Engineering and Science University, Shibpur, in India. His research interests include Dynamical systems, stability and bifurcation theory, population dynamics, mathematical modeling in ecology and epidemiology, management and conservation of fisheries, bioeconomic modeling of renewable resources. He wrote around 60 academic papers on those topics. He also supervised several students of master and doctor degree. Ashim Batabyal is an assistant teacher of Mathematics, Bally Nischinda Chittaranjan vidyalaya, Howrah, in India. He is currently doing his Ph.D. under the guidance of Dr. T. K. Kar in the Department of Mathematics, Bengal Engineering and Science University, Shibpur, India. His research topic is “Mathematical modelling on the dynamics of ecological systems with special emphasis on epidemiological problems”. He has obtained his post graduate degree in Mathematics from the University of Burdwan in 1995. Received January 2010 Accepted March 2010 Final acceptance in revised form April 2010

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