natural convection heat transfer in inclined open …€¦ · natural convection heat transfer in...

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International journal of thermal engineering (IJTE)

Volume 5, Issue 2, July–Dec 2017, pp. 46-61, Article ID: IJTE_05_02_005

Available online at

http://www.iaeme.com/IJTE/issues.asp?JType=IJTE&VType=5&IType=2

© IAEME Publication

NATURAL CONVECTION HEAT TRANSFER IN INCLINED

OPEN ANNULUS PASSEGE HEATED FROM TWO SIDES

Mustafa Z. Ghani1, Yasin K. Salman2

1,2 Department of Energy Engineering, University of Baghdad

ABSTRACT

Natural convection is investigated experimentally in an open cylindrical annulus heated with

both annulus inner and outer sides under same constant heat flux condition to study the effect of

angle of inclination and heat flux on heat transfer. Heat transfer results are given for inclination

angles of 0o (horizontal), 30o, 60o and 90o (vertical) using annulus diameter ratio of 1.8, inner and

outer tube length 50 cm and heat flux from 70 W/m2 to 600W/m2. The results show that the local and

average Nusselt number increase as the heat flux increase and when angle of inclination changed

from 0o (horizontal) to 90o (vertical).An empirical correlations of average Nusselt number as a

function of Rayleigh number were deduced.

Keywords: Heat Transfer, Natural Convection, Inclined Annulus, Empirical Correlations

NOMENCLATURE

AS: Tube surface area (m2)

D1, D2=Inner and outer annulus diameters (m)

R1, R2:Inner and outer annulus radius (m)

Dh: Hydrolic diameter, 2(R2-R1)

F1-2: view factor between the inner and outer tube

Grm: Mean Grash of number,g 𝛽Dh

3 (ts−tb )

𝑣2

g: Gravitational acceleration (m/s2)

hX:Local heat transfer coefficient (W/m2.K)

h:Average heat transfer coefficient (W/m2.K)

K: Thermal conductivity (W/m.K)

L: Axial length of annulus (m)

X*: Dimensionless axial distance, X/Dh

NuX: Local Nusselt number, hX.Dh/K

Num: Mean Nusselt number

Pr: Prandtl number,µ . Cp/k

V: Heater voltage, volt.

I: Heater current. Amp.

QC: Heat transfer by convection (W)

QCd: Heat transfer by conduction (W)

Natural Convection Heat Transfer In Inclined Open Annulus Passege Heated From Two Sides

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Qt: Total heat input (W)

Qcr:Heat transfer by convetion and radiation (W)

qc: Convetion heat flux (W/m2)

qr: Radiation heat flux (W/m2)

qcr: Convetion –radiation heat flux (W/m2)

Ram: Mean Rayleigh number, Gm.Pr

Cp : Specific heat at constant pressure, (kJ/kg.Co)

( Tb )x: Local bulk air temperature

𝑏=Average bulk air temperature

(Ts1)x , (Ts2)x:Inner and outer annulus local surface temperatures (Co)

𝑇𝑆1 ,𝑇𝑆2

: Average inner and outer tube surface temperature (Co)

Greek symbols

𝛽=Coefficient for volumetric thermal expansion (K-1)

𝜀=Emissivity; inner surface and outer surface

µ=Fluid viscosity (kg/m.s)

𝑣=Kinematics viscosity (m2/s)

𝜌=Fluid density (kg/m3)

𝜎=Stefan-Boltzman constant (W/m2.K4)

𝜃=Inclination angle

1. INTRODUCTION

Natural convection heat transfer in the annulus between two concentric cylinders is an

important research topic due to its wide application in engineering problems. Applications are found

in energy conversion, storage and transmission systems. Examples of using annulus geometry

include solar collectors, phase change of material around pipes in thermal storage systems and

nuclear reactor design. Many experimental and theoretical investigations have been conducted in

recent years due to the wide range of applications as mentioned above. Kuehn and Goldstein [1]

presented numerical and experimental results for natural convection in horizontal annulus over a

wide range of Ram, Pr, and D2/D1. They obtained correlation equations for heat transfer by natural

convection using a conduction boundary layer model. Their results showed that the heat transfer

correlation is similar to that of heat transfer from a single horizontal cylinder as the outer cylinder

diameter tends to infinity and is similar to heat transfer to the fluid within a horizontal cylinder as the

inner cylinder diameter approaches zero. Takata et al. [2] studied natural convection analytically

and experimentally in an inclined cylindrical annulus enclosed in heated inner and cooled outer

cylinders. The three-dimensional structure of the fluid flow, temperature distribution and Nufor

different angles of inclination was investigated. They showed that the Nu slightly increased as the

angle of inclination from the horizontal increases for the case of D2/D1=2.0. Rao et al. [3]

investigated experimentally and theoretically the natural convection flow and temperature

distribution in horizontal cylindrical annuli. They compared the predictions with experimental results

for temperature and stream function distributions and determined the dominant flow pattern at a

given Ra and D2/D1.Hamad [4] investigated the natural convection heat transfer in an inclined

annulus. An empirical correlation for Nu has been given for Pr = 0.7, D2/D1=1.636 and 4×104 ≤

Ram≤125×104. His results showed that the Ra and angle of inclination had a very small effect on the

heat transfer coefficient through the annulus. Vafai and Ettefagh [5] carried out natural convection

between two horizontal, concentric cylinders open at both ends the axial velocity was found to

decrease away from the open end and a core region was observed inside the annulus where the flow

field was almost two-dimensional. Buoyancy-driven flow and heat transfer in a horizontal annulus

bounded by two boundaries in the axial direction are numerically investigated in this work. The


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results show that the temperature distribution remains unchanged in the core region provided that the

annulus length to outer radius ratio is larger than a critical value. Flow and temperature fields are

found to be symmetrical with respect to the mid-axial plane of the annulus. Akeel A. Mohammed

[6] carried out experimental study to find the local and average heat transfer by natural convection

in a vertical concentric cylindrical annulus. The experimental setup consists of an annulus has a

radius ratio of 0.555 and inner cylinder with a heated length 1.2m subjected to the constant heat flux

while the outer cylinder is subjected to the ambient temperature. The investigation covers heat flux

range from 58.2 W/m2 to 274.31 W/m2. Results show an increase in the natural convection as heat

flux increases leads to an improve in the heat transfer process. Asif Hussain Malik1et al.

[7]Studied buoyancy driven flow within bottom-heated vertical concentric cylindrical enclosure.

Experimental and numerical study of the axial temperature gradient and the heat transfer mechanism

within the enclosure were performed. The numerical simulations were validated by comparing the

numerical results with experimentally measured axial temperature. The numerical results of the

streamlines within the enclosure depicted the real picture of the buoyancy effects. The literature

survey indicates that most researchers have studied natural convection heat transfer through open and

closed horizontal and vertical annuli, but there was little information about the inclined cases. The

present study covers this lack and gives a clear view to actual physical behavior in the heat transfer

process by natural convection.

2. EXPERIMENTAL APPARATUS

A schematic diagram and photograph of the experimental setup of the apparatus are shown in

Fig. (1) and Fig.(2)a respectively. It consists essentially of an outer aluminum tube and inner coaxial

aluminum tube. The outer tube internal diameter is 46 mm and the inner tube external diameter is 26

mm. All of the two tubes are 500 mm in length. The inner tube mounted in the entrance on a teflon

piece (A) which have the same outer diameter for the inner tube. A well designed teflon bell mouth

(B) was fitted at the entrance of the outer tube which have the same inside diameter of the outer tube,

both of the teflon pieces are equal to 12 cm in length, another two ylindrical teflon pieces(C and D)

with the same lengths and diameters of (A) and (B) are fixed on the exit section of both inner and

outer tubes. Teflon was chosen because of its low thermal conductivity in order to reduce the heat

loss from the tubes ends. The tubes components are held by cross plate (M) tied together with the

tubes components by rivet-net and are mounted on wooden board (W) with four long rivets fitted

with nets on the board ,the inner tube teflon piece fitted on an cross plate that connected to the four

rivets by nets. The board can rotated around a horizontal spindle. The inclination of the cylinders to

the horizontal can thus be adjusted as required. The outer tube surface is electrically heated by means

of neickel –chromium wire (main heater) of 0.3 mm in diameter and 5 Ω per meter resistance. The

wire is electrically insulated by means of ceramic beads and is wound uniformaly along the tube

length with an asbestos rope of 5mm thickness in order to give a uniform heat flux. As seen in Fig.

(3) The main heater is covered by 30 mm thick asbestos ropes on which three pairs of thermocoupes

(A1/A2, B1/B2, C1/C2) are fitted at an aluminum plates with 10 mm thickness asbestos rope

between it and 10 mm thikness asbestos rope was wounded on it where an electric (guard-heater) is

uniformly wounded. For a certain main heater input the guard-heater input could adjusted so that the

thermocouple forming in each pair registered the same temperature ensuring that all heat generated

by the main heater flows to the inner surface of the outer tube .An asbestos rope of 10 mm thickness

covered the guard- heater. A fiber-glass layer of 7 mm thickness serves as an outside cover for the

heating system. The inner cylinder is internally heated by an electrical current passing through an 85

Ω resistance of 500 mm length, fixed in the center of the inner tube. The space between the heater

and the inner cylinder is filled totally with Magnesium oxide (MgO) powder, to avoid convection

currents. The axial temperature distribution of the inner and outer annulus surfaces have been

measured by using 34 Type K (chromel – alumel) thermocouples of 0.276 mm in size. 17


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thermocouples are fixed on both the inner and outer surfaces at equal distances along the axis of the

annulus, all of the thermocouples are fixed with defcon adhesive. Three additional thermocouples

were fixed at the midpoint of the outer surface annulus, spaced 90o, to measure the temperature

distribution in the circumferential direction. The temperature difference was found negligible in the

circumferential direction, hence the inner and outer cylinder was assumed to be circumferentially

isothermal. To measre the axial lagging losses two thermocouples are fitted with uniform distance in

the inner cylinder Teflon piece. One thermocouple is fixed in the entarance of the annulas to measure

the inlet temperature and three thermocouples are fixed in the exit part to measure the outlet

temperature .All thermocouples were used with leads, the thermocouples with and without lead were

calibrated against the melting point of ice made from distilled water and the boiling points of several

pure chemical substances.Fig. (2) b shows the photographs of instruments used in the test.The power

consumed by the heater was measured by an ammeter and voltmeter. A three variac units was used

to control the power supplied to the heaters by controlling the voltage across the heaters, a data

logger pico- (Tc-008) was used to record the thermocouple outputs to accuracy within 0.03 mV.

Figure (1) Schematic diagram of experimental apparatus: (A) Inner tube lower Teflon piece ;

(B) Outer tube lower Teflon piece(Bell mouth); (C) Upper Teflon piece for inner tube; (D)Upper

Teflon piece for outer tube; (E)Thermo couples of the outlet hole; (F)Inner tube heater;(G)Guard

heater;(H)Fiber glass layer; (I) Outer tube heater; (M) Inner tube support plate (R)Asbestos layer;

(K)Wooden box; (W) Wooden board;(N) Thermocouple of the inlet hole .


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( a) (b )

Fig. (2): Photographic of: (a) Test apparatus,(b) instruments used in the test

Figure (3) Cross-section through apparatus. (1) thermo couples of the inner tube; (2)thermo

couples of the outer tube; (3) outer tube heater with 5 mm (thickness) asbestos rope ; (4) 30 mm

(thickness) asbestos rope; (5) 10 mm (thickness) asbestos rope; (6) 10 mm (thickness) asbestos rope;

(7) Guard Heater; (8) 10 mm (thickness) asbestos rope (9) 7 mm (thickness) fiber glass layer.

3. EXPREMENTAL PROCEDURE

To achieve the experiments with working conditions, the following procedures were followed:

A. The test apparatus prepared to insure the well performance of all components.

B. Adjusting the required inclined angle.

C. The supply power to the electric elements was switched on, and it was adjusted by variac to

obtain the same required constant heat flux, then it was left in operation action for a period until

the surface temperature of the cylinders reached to steady state after about (6 hours) .


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D. During each experiment, at all selected temperature recording position the temperature recorded

by data logger for each interval time about of (15 minutes), together with the input voltage and

current.

4. EXPERIMENTAL DATA REDUCTION

The experimental apparatus described in section two has been used to provide the

experimental data for heat transfer calculations through the annulus. The inner and the outer tubes

was subjected to the uniform heat flux separately and together (with the same heat flux), depending

on the case of the study (inner tube heated, outer tube heated and both tubes heated). The total power

supplied to the inner or the outer tube was calculated in the same way as follows:

Qt =V×I ….. (1)

The convection radiation heat transferd from the any of the tubes suface is:

Qcr = Qt-Qcond …... (2)

Where Qcond is the axial conduction heat loss which was found experimentally equal to 3% of the

input power.The convection and radiation heat flux can be represented by:

qcr= (Qcr )/As …… (3)

where (AS=2πR1L) for the inner tube and (AS=2πR2L) for the outer tube.

The convection heat flux which is used to calculate the local heat transfer coefficient is obtained

after deduce the radiation heat flux from qcr value. The local radiation heat flux can be calculated as

follows:

qr = F1-2𝜎𝜀((𝑇𝑠1+ 273)4 - (𝑇𝑠2+ 273)4 ) ……. (4)

where:

F1−2 ≈ F2−1 ≈ 1

Hence the convection heat flux at any position is:

qc=qcr-qr …… (5)

The local heat transfer coefficient can be obtained as:

(hX) =𝑞𝑐

(Ts)x− (Tb)x …… (6)

All the air properties were evaluated at the mean film air temperature:

(Tf) x =(Ts)x+(Tb)x

2 …….. (7)

where:

(Tf)x is the local mean film air temperature at(Ts)x .


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The local nusselt number for the inner cylinder (Nux) then can be determine as:

(Nux) = (hx)Dh

k ……..….. (8)

Ts =1

L∫ (Ts)x dx

x=L

x=0 ………...... (9)

Tb =

1

L∫ (Tb)x dx

x=L

x=0 ……….. (10)

Tf= Ts +Tb

2 …………….… (11)

The averge heat transfer coefficient and the vavrege Nusselt number (Num) based on the

calculation of the averge tube surface temperature and the average bulk air temperature were

calculated as follows:

h =1

L∫ h𝑥 dx

x=L

x=0 ……….…. (12)

Num =q Dh

k(Ts −Tb ) ………..……. (13)

Grm =g 𝛽Dh

3 (Ts −Tb )

v2 …..…. (14)

where β = 1

(273+ Tf )

Pr=µ Cp

k ……. (15)

Ram = Grm . Pr …… (16)

All the air physical properties ρ, µ, v and k were evaluated at the average mean film temperature

(Tf)Holman [8].

5. EXPERIMENTAL UNCERTAINTY

Generally the accuracy of experimental results depends upon the accuracy of the individual

measuring instruments and the manufacturing accuracy of the circular tube. The accuracy of an

instrument is also limited by its minimum division (its sensitivity). In the present work, the

uncertainties in heat transfer coefficient (Nusselt number) and Rayleigh number were estimated

following Kline and McClintock differential approximation method reported by Holman [9]. For a

typical experiment, the total uncertainty in measuring the heater input power, temperature difference

(Ts-Tb), the heat transfer rate and the circular tube surface area were 0.38%, 0.48%, 2.6, and 1.3%

respectively. These were combined to give a maximum error of 2.43% in heat transfer coefficient

(Nusselt number) and maximum error of 2.36% in Rayleigh number.


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6. RESULTS AND DISCUSSION

6.1 Temperature variation

The variation of tubes surface temperature for different heat flux and for angle of inclination

𝜃 = 0°(horizontal) , 30o,45o 60°, and 90°(vertical) are shown in Figs.(4)-(7) respectively . It is

obvious from these figures that the surface temperature increases as heat flux increases because of

faster increasing of the thermal boundary layer as heat flux increases. It can be seen from Fig.(4)that

at𝜃 = 0o, the inner and outer tube surface temperature have no obvious change with the axial

distance except at the end of the tubes due the conduction end losess. this behavior explained that

there is no flow in the axial direction so the bouncy effect is just in the radial direction .For 𝜃=

30o,45o, 60°, and 90°, the distribution of the surface temperature (Ts) with tubes axial distance for

different heat fluxes have the same general shape as shown in Figs.(5)-(7). The surface temperature

distribution exhibits the following trend: the surface temperature gradually increases with the axial

distance at the same rate of the increasing for the inner and the outer tube until a certain limit to

reach a maximum value at approximately(X*= 17) beyond which it begins to decrease.

(a) (b)

Fig.(4) : Surface temperature variation with the axial distance for different heat fluxes,𝜃=0o (a) Inner

tube , (b) Outer tube.

(a) (b)


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Fig.(5) : Surface temperature variation with the axial distance for different heat fluxes,𝜃=30o (a)

Inner tube , (b) Outer tube.

(a) (b)

Fig.(6) : Surface temperature variation with the axial distance for different heat fluxes,𝜃=60o (a)


(a) (b)

Fig.(7) : Surface temperature variation with the axial distance for different heat fluxes,𝜃 =90o (a)


Figs.(8)-(10) show the effect of angle of inclination on the temperature distribution along the inner

and the outer tube surface .It is clear that the surface temperature increases as angle of inclination

moves from vertical to horizontal position . This behavior can be attributed to the fact that says as

the air is heated and dilates, the difference between air density near the wall and the annulus center

causes a circulation which displaces the wall air in a direction parallel to the gravity vector. When

the heat transfers through the wall of a horizontal tube, the warmer air moves upward along the side

walls, and by continuity the heavier air near the smallest temperature wall of the tube flows

downward. As a result, a two symmetrical spiral, like motion is formed along the annulus. This air

motion is slightly small due to very small tempareture different between the tubes surface in this case

because both tubes heated with the same heat flux. The circulation is driven by radial temperature

variation, and at the same time it reduces this temperature variation. These two spiral vortex weak as

the angle of inclination moves from horizontal to vertical position to be single vortex only and the

flow would be totally in the axial direction in the vertical position. Therefore; it is expected that the

convection heat transfer process in vertical position is better than that in other positions.


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(a) (b)

Fig.(8) : Surface temperature variation with the axial distance for different inclination angles , q=70

W/m2 (a) Inner tube , (b) Outer tube.

(a) (b)

Fig.(9) : Surface temperature variation with the axial distance for different inclination angles , q=300

W/m2 (a) Inner tube , (b) Outer tube.

(a) (b)

Fig.(10) : Surface temperature variation with the axial distance for different inclination angles ,

q=600 W/m2 (a) Inner tube , (b) Outer tube.

6.2 Variation of local Nusselt number


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The local Nusselt number variation along the tubes surfaces for different heat fluxs (70

W/m2 to 600 W/m2) and for angle of inclination𝜃 = 00 (horizontal) , 30° ,60°, and 90° (vertical);are

shown by plotting the local Nusselt number with the dimensionless axial distance in Figs.(11)-(14)

respectively. Generally, It is obvious from these figures that the local Nusselt number values increase

as the heat flux increases because of increasing natural convection currents which improves the heat

transfer process. At the higher heat flux, the results of Nux were higher than the results of lower heat

flux. This may be attributed to the secondary flow effect that increases as the heat flux increases

leading to higher heat transfer coefficient. Therefore, as the heat flux increases, the fluid near the

wall becomes hotter and lighter than the bulk fluid in the core. As a consequence, in the vertical

position two upward currents flow along the sides walls, where for the horizontal case the flow near

the tubes walls would be in the radial direction depending on the small temperature difference

between the walls caused by inner and outer tube surface shape. For inclined positions the flow will

be combined of the axial and radial direction and by continuity, the fluid near the tube center flows

downstream.

(a) (b)

Fig.(11) :Local Nusselt number variation with the axial dimensionless distance for different heat

fluxes, 𝜃=0o(a) Inner tube , (b) Outer tube.

(a) (b)

Fig.(12) Local Nusselt number variation with the axial dimensionless distance for different heat

fluxes , 𝜃 =30o (a) Inner tube , (b) Outer tube.


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(a) (b)


fluxes , 𝜃 =60o (a) Inner tube , (b) Outer tube.

(a) (b)


fluxes, 𝜃=90o(a) Inner tube , (b) Outer tube.

The effects of angle inclination on the local Nusselt number variation are shown in Figs(15)-

(17). For the horizontal position it can be seen that the values of Nux as they should , are constant

and independent of x.As be expected ,it is clear that , the local Nusselt number increases relatively as

angle of inclination moves from horizontall to vertical position for the same heat fluxs of the inner

and the outer tubes .

(a) (b)


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Fig.(15) :Local Nusselt number variation with the axial dimensionless distance for different angles

of inclination , q= 70 W/m2 (a) Inner tube , (b) Outer tube

(a) (b)


of inclination , q= 300 W/m2 (a) Inner tube , (b) Outer tube.

(a) (b)


of inclination , q= 600 W/m2 (a) Inner tube , (b) Outer tube.

6.3 Average Nusselt number

Figs.(18)-(21) show the logarithmic of mean Nusselt number versus logarithmic Rayleigh

number for q=70 W/m2 to 600 W/m2 ,at𝜃 = 0° (horizontal) , 30° , 60° , and 90° (vertical) ;

respectively. An empirical equations have been deduced from these figures as follows:-

For inner tube:

Num= 0.00325 Ra𝑚0.63272𝜃 =0o

Num= 0.00138 Ra𝑚0.73465𝜃 =30o

Num= 0.00946 Ra𝑚0.56092𝜃 =60o

Num= 0.00178Ra𝑚0.72184𝜃 =90o


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For outer tube:

Num= 0.04102 Ra𝑚0.4067𝜃 =0o

Num= 0.23739 Ra𝑚0.2666𝜃 =30o

Num= 0.20731 Ra𝑚0.27696𝜃 =60o

Num= 0.00794 Ra𝑚0.58395𝜃 =90o

(a) (b)

Fig.(18) : Logarithm Average Nusselt Number Versus log(Ram) ,𝜃=0o(a) Inner tube,(b) Outer tube.

(a) (b)



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(a) (b)


(a) (b)


7. CONCLUSIONS

1. The extent of the local mixing increases as the heat flux increases.

2. The heat transfer process improves as heat flux increases and as angle of inclination moves

from horizontal to vertical to horizontal.

3. The effect of buoyancy is small at the cylinder entrance and increases downstream.

REFERENCES

[1] Kuehm, T. H. and Goldstein, R. J., “An experimental and theoretical study of natural

convection in an annulus between horizontal concentric cylinders”. J. Fluid Mech., Vol. 74.

(1976).

[2] Takata, Y., Iwashige, K., Fukuda, K. and Hasegawa, S., “Three-dimensional natural

convection in an inclined cylindrical annulus”. Inti J. Heat Mass Transfer vol. 27, pp. 747-

754 (1984).

[3] Rao, Y. F., Miki, Y., Fukuda, K., Takata, Y. and Hasegawa, S., “Flow patterns of natural

convection in horizontal cylindrical annuli”. Int. J. Heat Mass Transfer, Vol.28 (1985).

[4] Hamad, F. A., Experimental study of natural convection heat transfer in inclined cylindrical

annulus. Solar and Wind Technology Vol.6 (1989).

[5] Vafai, K. and Ettefagh, J. “An investigation of transient three dimensional buoyancy driven

flow and heat transfer in a closed horizontal annulus”. Inti J. Heat Mass Transfer vol. 34,pp.

2555-2570 (1991).


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[6] Akeel Abdullah Mohammed “ Natural convection heat transfer in a vertical concentric

annulus” J. of Engineering , a scientific Refereed Journal Published by college of

Engineering University of Baghdad, Vol. 13 pp1-14 ( 2007).

[7] Asif Hussain Malik, Shahab Khushnood and Ajmal Shah “Experimental and numerical

study of buoyancy driven flow within a bottom heated vertical concentric cylindrical

enclosure”, Natural Science vol.5, No.7, pp.771-782 (2013).

[8] Jack P. Holman,”Heat transfer”,10th edition, McGraw-Hill Series in Mechanical

Engineering (2010).

[9] Jack P. Holman.“Experimental methods for engineers”, 8th ed. McGraw-Hill Series in

Mechanical Engineering (2011).

[10] D. Subramanyam, M. Chandrasekhar and R. Lokanadham, “Experimental Analysis of

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ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.

[11] Yogesh Dhote and S.B. Thombre, “A Review on Natural Convection Heat Transfer

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Engineering & Technology (IJARET), Volume 4, Issue 7, 2013, pp. 170 - 175, ISSN Print:

0976-6480, ISSN Online: 0976-6499.

[12] Omar Mohammed Ali and Ghalib Younis Kahwaji, “Numerical Investigation of Natural

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