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International Journal of Thermal Sciences 85 (2014) 40e46

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Statistical models for predicting the thermal resistanceof polyester/cotton blended interlock knitted fabrics

Ali Afzal a, *, Tanveer Hussain b, Muhammad Mohsin c, Abher Rasheed d, Sheraz Ahmad a

a Department of Materials and Testing, National Textile University, Sheikhupura Road, Faisalabad 37610, Pakistanb Department of Textile Processing, National Textile University, Faisalabad, Pakistanc Department of Textile Engineering, University of Engineering and Technology, Faisalabad, Pakistand Department of Garment Manufacturing, National Textile University, Faisalabad, Pakistan

a r t i c l e i n f o

Article history:Received 14 January 2014Received in revised form12 June 2014Accepted 12 June 2014Available online

Keywords:Statistical modelKnitted fabricPolyesterCottonThermal resistanceComfort

* Corresponding author. Tel.: þ92 322 407 5177; faE-mail address: aliafzalch89@gmail.com (A. Afzal)

http://dx.doi.org/10.1016/j.ijthermalsci.2014.06.0161290-0729/© 2014 Elsevier Masson SAS. All rights re

a b s t r a c t

Thermal resistance is one of the key aspects of thermo-physiological comfort of knitted clothes. The aimof this study was to develop statistical models for predicting the thermal resistance of polyester/cottonblended interlock knitted fabrics. Fabric samples were knitted at three different tightness levels, withyarns of three different linear densities each with three different polyester/cotton blend ratios. Based onthermal resistance results of the knitted samples, three different statistical models were developedcomprising different input variables as predictors. The validation results showed that the thermalresistance of polyester/cotton interlock fabrics could be predicted using the developed statistical modelswith an average percentage error of 4e8%. It could be concluded from the sensitivity analysis that themost influential factor affecting the thermal resistance was fabric stitch length, followed by yarn specificheat and fabric areal density.

© 2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

The primary needs of human being include textiles as a majorcommodity. It comprises all the items used to protect the body fromexternal environment. Textiles are used to cover as well as protectthe body.When the external climatic conditions outmatch the bodyrequirements, some specific fabrics are used to provide the opti-mum body demands for better comfort feelings. Comfort can bedefined as “a pleasant state of psychological, physiological andphysical harmony between a human being and the environment”[1]. Clothing plays a vital role in thermoregulatory process as italters heat loss from the skin and also changes the moisture lossfrom skin [2].

Different researchers investigated the effect of fibre, yarn andfabric properties on the thermal comfort performance of differentfabrics [3e8]. Cimilli et al. [3] investigated the effect of materialtype used for thermal comfort properties of plain jersey socks bymodal, micromodal, bamboo, soybean, chitosan, viscose and cottonfibres. The results obtained suggested that there was statisticalsignificant difference between the fibre type and the thermal

x: þ92 41 923 0098..

served.

resistance of fabrics. Schneider et al. [4] investigated the thermalconductivity of different fibres under moist conditions. Wan et al.[5], Schacher et al. [6] and Ramakrishnan et al. [7] explained theeffect of fibre fineness on thermal resistance of fabrics. According tothem, the micro-denier fibre gives low thermal conductivity andhigher thermal resistance. Oglakcioglu et al. [9] studied the thermalcomfort properties of 1�1 rib knit fabrics with different fibre blendratios of cotton and angora fibre.

Pac et al. [10] studied the effect of fibre morphology, yarn andfabric structure on thermal comfort properties of fabric. Ozdil et al.[11] investigated the effect of different yarn parameters on thermalcomfort of 1 � 1 rib knitted fabric. They explained that bydecreasing count and yarn twist, the thermal resistance increaseswhile moisture vapour permeability decreases. Majumdar et al.[12] found that by the use of finer yarn for knitted fabric formationof plain, rib and interlock structures by blend of bamboo and cottonfibres, the thermal conductivity of fabric reduces.

Khoddami et al. [13] explained that by the use of hollow fibre,the fabric thickness increases which increases the thermal resis-tance of the fabric. Greyson [14], Havenith [15] and [16] presentedtheir findings that heat resistance increases by increasing the airentrapped in the fabric as well as fabric thickness. Ucar and Yilmaz[17] have worked on thermal insulation properties of different ribstructures made from cotton. Oglakcioglu and Marmarali [18] have

Nomenclature

Tt Yarn linear density, texC Specific heat of yarn, J g�1 K�1

l Stitch length, mmt Fabric thickness, mmm Fabric areal density, g m�2

Rct Fabric thermal resistance, m2 K mW�1

(m2 K W�1 � 103)

A. Afzal et al. / International Journal of Thermal Sciences 85 (2014) 40e46 41

studied the thermal comfort properties of different knitting struc-tures. The structures under considerations were single jersey,interlock and 1 � 1 rib constructions with polyester and cotton fi-bres. They explained that interlock structure in both fibre typesprovide the higher thermal resistance due to more thickness offabric.

It is observed that no study has been carried out on the influenceof fibre, yarn and fabric properties on thermal resistance of theinterlock knitted fabrics. The objective of this study is to determinethe effect of fibre type, composition, yarn and fabric properties onthermal resistance of the interlock knitted fabrics and statisticallymodel them for prediction purposes in future.

2. Experimental

2.1. Yarn preparation

Carded ring spun yarns of varying cotton/polyester blend ratiosin range from 60/40 to 35/65 were used with linear densities 36.9tex (Ne 16/1), 29.5 tex (Ne 20/1) and 24.6 tex (Ne 24/1) for knittinginterlock knitted fabric samples. The twist multipliers for each yarnlinear density of different blend ratios were kept the same, i.e., 3.43,3.45 and 3.47 for 36.9 tex, 29.5 tex and 24.6 tex respectively. Theproperties of the yarns used in this study are given in Table 1.

The raw materials and spinning machine settings were kept thesame during the production of all the yarns. No extra treatment wasapplied to these yarns before knitting them into fabric samples.

2.2. Fabric preparation

The as-spunyarns were used to knit the fabric samples in doublejersey interlock knit fabric structurewith varying tightness levels toachieve different fabric areal densities. The three tightness levelswere selected (slack, medium and tight) constituting stitch lengthsof 3.2 ± 0.1 mm, 3.5 ± 0.1 mm and 4.1 ± 0.1 mm. The change infabric structures with different tightness factors is evident from

Table 1Properties of the yarns used in this study.

Parameters Mean values

36.9 tex 29.5 tex

PES:CO (%) 40:60 52:48 65:35 40:60Tta (tex) 36.45 36.41 36.57 29.23U % 9.53 9.11 8.74 10.18Ntn (�50%) 0 0 0 0Ntk (þ50%) 35 34 20 65Nn (þ200%) 57 45 38 92H 8.10 7.76 7.65 7.81Ta (cN tex�1) 18.72 21.49 26.34 18.043b (%) 7.28 8.94 10.09 6.42Tm 544 544 543 611

PES e polyester, CO e cotton, Tta e actual yarn count, U e evenness, Ntn e number of thin1000 m, H e hairiness, Ta e tenacity, 3b e breaking elongation, Tm e number of twists

Fig. 1. The samples were fabricated on an 18 gauge Jacquard circularinterlock machine with positive yarn feeding system, 1728 totalneedle count and 30 inches diameter. The tight fabric samples for36.9 tex yarn could not be knitted on the selected machine due toknitting machine limits and therefore excluded from the currentstudy. A total number of 24 different fabric samples were made forthis study. The fabric constructions are given in Table 2.

2.3. Fabric processing

The knitted fabric samples were semi bleached at 110 �C usinghydrogen peroxide 50% (0.5759 g/L), stabilizing agents (Terminoxultra 50-L, 0.0384 g/L), caustic soda (5 g/L) and wetting agent(Viscavin FTC, 0.1920 g/L) for 10 min, followed by rinsing.

The dyeing of polyester fibre content was followed throughdisperse dye, Navy blue Eco 300% (by Techron) (0.0762 g/L), Terasilyellow W-6GS (by Terasil) (0.0549 g/L) and Yellow Brown XF (byDianix) (0.1755 g/L), at 130 �C for 35min in the presence of levellingagent (Levenol V-505N, 0.2688 g/L). The fabric samples werereduction cleared followed by polyester dyeing at 95 �C for 25 min,in the presence of a wetting agent, sodium hydrosulphite andcaustic soda. The cotton fibre content was dyed with reactive dyesSynozol Navy blue KBF (0.3017 g/L), Drimagen E3R (0.384 g/L),Synozol Red K 3BS (0.0987 g/L) and Everzol Yellow 3RS H/C(0.5485 g/L) at 60 �C for 30min in the presence of soda ash (7.679 g/L)), anti-crease (Rucoline Jes, 0.192 g/L) and sequestrant (AlkaquestAM-700, 0.1536 g/L), followed by washing-off at 95 �C for10 min with detergent (Cotoblanc STG-L, 0.384 g/L), followed byrinsing and neutralization at 50 �C for 10 min with acetic acid(0.95 g/L). After dyeing, the knitted fabric samples were dried andstabilized in a compactor at speed of 22 m/min and 110 �Ctemperature.

2.4. Fabric testing

The knitted fabric samples were pre-conditioned at a temper-ature of 47 �C and relative humidity of 20% for four hours in hot-airoven followed by conditioning in standard atmosphere accordingto ASTM D 1776 [19] at temperature of 20 ± 2 �C and relative hu-midity of 65 ± 2%. The stitch length of the fabric was calculatedfrom loop length of 150 stitches. The thermal resistance testing wasperformed on an SDL Atlas M259B sweating guarded hotplate [20]according to ISO 11092:1993 [21]. This instrument is also known asskin model used to simulate the mass and heat transfer processeswhich occur next to the skin surface. The samples were placed onthe thermal plate enclosed in a controlled environment. The sam-ples were tested in standard conditions for the thermal resistancewhich were 20 ± 0.1 �C air temperature, 65 ± 3% R.H, 35 ± 0.1 �C

24.6 tex

52:48 65:35 40:60 52:48 65:3529.19 29.25 24.40 24.42 24.399.65 9.39 10.87 10.24 9.880 0 0 0 047 42 108 92 8876 72 181 161 1477.26 7.02 7.19 7.01 6.7820.60 25.71 17.27 20.35 25.687.57 8.94 5.81 7.41 8.66611 610 672 672 672

places on 1000 m, Ntk e number of thick places on 1000 m, Nn e number of neps onper 1 m.

Fig. 1. Fabric structures with different tightness factor (K) (a) K ¼ 15.78 (b) K ¼ 14.47 (c) K ¼ 12.41.

Table 3Correlation between fabric thermal resistance and different yarn and fabricparameters.

Parameter PCC P-value

PES% �0.621 0.000*Tt (tex) �0.192 0.035*C (J g�1 K�1) 0.621 0.000*l (mm) 0.591 0.000*t (mm) 0.133 0.146m (g m�2) �0.625 0.000*

PCC e Pearson correlation coefficient.*statistically significant.

A. Afzal et al. / International Journal of Thermal Sciences 85 (2014) 40e4642

thermal guard temperature, 1.00 ± 0.05 m/s air speed and35 ± 0.1 �Cmeasuring unit temperature. Total five readings for eachfabric sample were obtained and thermal resistance was recordedin the units of m2 K/mW.

3. Development of the prediction model

The total number of datasets consists of 120 input/output pat-terns comprising 24 individual samples with 5 replicates each. Thedata was subdivided into two groups, one for the development ofthe model while the other for the validation of the developedmodel. The datasets for the validation were randomly selected andwere separated for validation of the developed models. Minitabstatistical software was applied for data analysis and developmentof regression models.

The Pearson correlation coefficients between fabric thermalresistance and different yarn and fabric parameters are given inTable 3.

According to the correlation analysis, the relation between thepolyester content, yarn linear density and fabric areal density wereinversely proportional while specific heat of yarn, fabric stitchlength and fabric thickness were found in direct proportion withthe fabric thermal resistance. The p-value obtained for all the pa-rameters except fabric thickness were less than a value (0.05)

Table 2Fabric constructions used for modelling the thermal resistance of knitted interlockfabrics.

No. PES:CO (%) Tt (tex) C (J g�1 K�1) l (mm) K (√tex cm�1) t (mm) m (g m�2)

1 40:60 36.5 3.038 3.6 16.90 1.29 442.62 40:60 36.5 3.038 4.2 14.37 1.34 360.93 40:60 29.2 3.038 3.2 16.75 1.17 378.34 40:60 29.2 3.038 3.6 15.20 1.20 332.95 40:60 29.2 3.038 4.1 13.34 1.28 295.26 40:60 24.4 3.038 3.2 15.42 1.10 313.07 40:60 24.4 3.038 3.5 14.23 1.15 284.08 40:60 24.4 3.038 4.0 12.22 1.18 241.79 52:48 36.4 2.782 3.6 16.96 1.30 455.210 52:48 36.4 2.782 4.1 14.68 1.31 363.911 52:48 29.2 2.782 3.2 16.96 1.15 394.312 52:48 29.2 2.782 3.5 15.22 1.16 347.213 52:48 29.2 2.782 4.0 13.40 1.23 297.314 52:48 24.4 2.782 3.2 15.65 1.06 313.515 52:48 24.4 2.782 3.4 14.45 1.10 290.416 52:48 24.4 2.782 4.0 12.35 1.10 239.117 65:35 36.6 2.506 3.5 17.16 1.30 484.618 65:35 36.6 2.506 4.1 14.78 1.33 405.319 65:35 29.3 2.506 3.2 16.97 1.14 405.320 65:35 29.3 2.506 3.4 15.73 1.16 365.621 65:35 29.3 2.506 4.0 13.50 1.17 302.322 65:35 24.4 2.506 3.1 15.78 1.05 339.523 65:35 24.4 2.506 3.4 14.47 1.06 303.324 65:35 24.4 2.506 4.0 12.41 1.06 255.3

BR% - blend ratio, PES e polyester, CO e cotton, Tt e yarn count, C e specific heat ofyarn, l e stitch length, K e tightness factor, t e fabric thickness,m e areal density offabric.

expressing statistically significant effect on fabric thermal resis-tance of interlock knitted fabrics.

The measured thermal resistances of the interlock knitted fab-rics were within the range of 3.8e20.7 m2 K mW�1. It was observedfrom correlation analysis (Table 3) that fabric areal density andspecific heat of yarn were the dominating parameters which highlyinfluenced the thermal resistance of interlock knitted fabrics fol-lowed by fabric stitch length, yarn count and fabric thickness. It wasnoted that by increase in fabric areal density and yarn count, thethermal resistance of interlock knitted fabrics decreases propor-tionally and vice versa. This effect is also shown in Fig. 2. Thedecrease in thermal resistance with increase in fabric areal densitymay be explained by the decrease in amount of air trapped withinthe fabric structure at higher areal density. Since air is a badconductor of heat, its presence in the fabric structure results inbetter thermal resistance.

The effect of specific heat of yarn, stitch length and fabricthickness was in direct proportion with thermal resistance of theinterlock knitted fabrics. The relation between thermal resistanceof the interlock knitted fabric verses stitch length and specific heatof yarn is shown in Fig. 3.

500450400350300250200

22.5

20.0

17.5

15.0

12.5

10.0

7.5

5.0

Areal density (g/m2)

Ther

mal

resi

stan

ce(m

2K

/mW

)

24.4029.2236.48

(tex)Count

Yarn

Fig. 2. Scatter plot of thermal resistance verses areal density and yarn count.

4.24.03.83.63.43.23.0

22.5

20.0

17.5

15.0

12.5

10.0

7.5

5.0

Stitch length (mm)

Ther

mal

resi

stan

ce(m

2K

/mW

) 2.5062.7823.038

(g K))yarn (J/heat of

Specific

Fig. 3. Scatter plot of thermal resistance verses stitch length and specific heat of yarn.

Table 4Best subset regression models.

S. No. V R2 R2 (adj.) Cp S Tt C l t m

1 1 39.1 38.6 323.1 3.02 X2 1 38.6 38.1 327.1 3.04 X3 2 72.6 72.1 83.6 2.04 X X4 2 68.1 67.6 116.1 2.20 X X5 3 83.8 83.4 4.6 1.57 X X X6 3 83.7 83.3 5.2 1.58 X X X7 4 84.0 83.5 5.3 1.57 X X X X8 4 83.9 83.4 5.8 1.57 X X X X9 5 84.2 83.5 6.0 1.57 X X X X X

V e variables, S e standard deviation, Cp e Mallows Cp.

A. Afzal et al. / International Journal of Thermal Sciences 85 (2014) 40e46 43

It was noticeable that by the increase in stitch length and spe-cific heat of the yarn, the thermal resistance of the interlock knittedfabric increases. The fabric thickness was also found in direct pro-portion with the thermal resistance of interlock knitted fabrics asshown in Fig. 4. Thicker fabrics offer more resistance to heat flowacross the fabric.When the specific heat of the yarns is higher, moreheat is absorbed by the yarns for per unit increase in their tem-perature and thus less heat is transferred across the fabric leadingto better thermal resistance results. Higher stitch lengths result inbulkier fabrics which entrap higher amount of air, leading to betterthermal resistance of the fabric.

Initial statistical analysis on the obtained data showed thatpolyester content percentage was highly correlated with the spe-cific heat of the yarn. The Pearson correlation coefficient values forboth the parameters were found the same in correlation analysishaving p-value less than 0.05 for both of them, which explains theirsignificant effect on the fabric thermal resistance of the interlockknitted fabrics. The reason lies in the fact that these parameterswere actually interdependent and are influenced by variation ineither variable. Therefore, the polyester content percentage wasseparated and was not included in the development of the thermalresistance model of interlock knitted fabric.

In order to make sure that the variables selected for develop-ment of model were significant, best subset regression analysis wasperformed on the data before development of the model. Bestsubset regression is a statistical tool to sort out the less significantvariables from the data and help to develop model with fewer

1.41.31.21.11.0

22.5

20.0

17.5

15.0

12.5

10.0

7.5

5.0

Fabric thickness (mm)

Ther

mal

resi

stan

ce(m

2K

/ mW

)

24.4029.2236.48

(tex)Count

Yarn

Fig. 4. Scatter plot of thermal resistance verses fabric thickness and yarn count.

variables. The results of the best subset regression analysis aregiven in Table 4.

The results of best subset regression analysis suggested thatyarn count, specific heat of yarn and fabric stitch length were themost suitable combination of input variables for the developmentof thermal resistance model for interlock knitted fabrics, because itshowed lowest Mallows Cp and standard deviation while highervalues of R2 and R2 adjusted values. The R2 value expresses theaccuracy of the data fitting while adjusted R2 is a modified R2 valuewhich assures the selection of appropriate variables combinationfor the model. A good subset model should have higher values of R2

and R2 adjusted while small standard deviation and Mallows Cpvalue should be close to the number of input variables added withthe constant contained in the model. In case of three input vari-ables, the Mallows Cp should be less than double of the number ofvariables in the model.

Response surface regression analysis was utilized for thedevelopment of the models for thermal resistance of the interlockknitted fabrics. Response surface regression is selected instead ofmultiple linear regression, because of the fact that former is a fullycapable tool to model linear as well as nonlinear relationships ofthe variables which is not possible with the latter. As suggested bybest subset analysis, the yarn count, specific heat of yarn and fabricstitch length were taken as input parameters to develop first modelfor the thermal resistance of the interlock knitted fabrics. Thequadratic model developed using these input parameters, can beconsidered as prediction model for the thermal resistance (Rct) ofthe interlock knitted fabrics, and is given in Eq. (1).

Rct ¼ �234:11þ 3:11 Tt þ 117:97 C þ 7:73 l� 0:04 Tt2

� 17:57 C2 � 0:37 Tt � C (1)

where: Rct is thermal resistance (m2 K mW�1); Tt is yarn lineardensity (tex); C is specific heat of yarn (J g�1 K�1); l is stitch length(mm).

The coefficient of determination (R2 value) for the Eq. (1) wasfound 89.72% which explains that approximately 90% variation inthe data can be explained by the model. The interactions of input

Table 5Estimated regression coefficients in coded units for predicting model type.

Term Coefficient SE coefficient T-value P-value

Constant 12.75 0.27 46.34 0.000*Tt �2.14 0.19 �11.44 0.000*C 2.50 0.16 15.80 0.000*l 4.33 0.22 19.81 0.000*Tt2 �1.47 0.27 �5.44 0.000*

C2 �1.24 0.27 �4.59 0.000*Tt � C �0.59 0.19 �3.04 0.003*

* statistically significant.

Table 6Estimated regression coefficients in coded units for testing model type.

Term Coefficient SE coefficient T-value P-value

Constant 11.16 0.29 38.78 0.000*C 1.54 0.19 7.96 0.000*t 2.93 0.35 8.5 0.000*m �5.79 0.44 �13.19 0.000*C2 �0.98 0.32 �3.06 0.003*m2 �3.64 0.80 4.57 0.000*C � t �1.60 0.40 �3.98 0.000*C � m 1.60 0.55 2.93 0.004*T � m �4.98 0.99 �4.99 0.000*

* statistically significant.

Table 8Range of input parameters used in the statistical modeldevelopment of interlock fabrics.

Parameter Range

PES% 40e65Tt (tex) 24.4e36.6C (J g�1 K�1) 2.506e3.038l (mm) 3.1e4.2t (mm) 1.05e1.34m (g m�2) 239.1e484.6

A. Afzal et al. / International Journal of Thermal Sciences 85 (2014) 40e4644

variables which were found non-significant having p-value morethan 0.05 were deleted from the final developed model. All theterms in the final developed model have p-value less than 0.05which confirmed their significant contribution in the model ofthermal resistance of interlock knitted fabrics. The regression co-efficients in coded units for Eq. (1) are given in Table 5.

The prediction model given in Eq. (1) can be used to predict thethermal resistance of the interlock fabric based on the plannedknitting parameters, viz., yarn specific heat, yarn linear density andthe knitting stitch length. Another model was developed takinginto account only the fabric parameters, i.e. fabric thickness andfabric areal density along with specific heat of the yarn. This secondmodel is considered as testing model. In this quadratic model, onlysignificant terms having p value less than 0.05 were included andthe rest were excluded. The testing model is given in Eq. (2).

Rct ¼ �271:42þ 110:64 C þ 211:33 t � 0:05m� 13:85 C2

þ 0:0002 m2 � 37:65 C � t þ 0:05 C �m� 0:24 t �m

(2)

Where: Rct is thermal resistance (m2 K mW�1); C is specific heat ofyarn (J g�1 K�1); t is fabric thickness (mm); m is the fabric arealdensity (g m�2).

The fitting of developed model was obtained quite satisfactorybut less than the previous model with coefficient of determination(R2 value) of 87.85%. The estimated regression coefficients in codedunits for this model are given in Table 6.

A thirdmodel was developed taking into account the yarn lineardensity, specific heat of yarn, fabric stitch length, fabric thicknessand fabric areal density. The developed model is given in Eq. (3).

Rct ¼�157:44þ2:61 Ttþ40:82 C�38:31 lþ 274:92 t�0:23m

�0:05 Tt2 þ16:23 C� l� 98:88 C� tþ0:08 C�m

(3)

Table 7Estimated regression coefficient in coded units for generic model type.

Term Co SE Co T-value P-value

Constant 9.97 2.57 3.87 0.000*Tt 2.85 1.16 2.47 0.015*C 2.80 0.26 10.71 0.000*l 3.74 0.95 3.94 0.000*t 0.13 1.08 0.12 0.904**m �1.67 1.79 �0.94 0.352**Tt2 �0.75 0.13 �5.99 0.000*C � l 2.42 0.72 3.37 0.001*C � t �4.21 0.96 �4.38 0.000*C � m 2.69 0.99 2.72 0.008*

* statistically significant, ** statistically non significant.

where: Rct is thermal resistance (m2 K mW�1); Tt is yarn lineardensity (tex); C is specific heat of yarn (J g�1 K�1); l is stitch length(mm); t is fabric thickness (mm); m is the fabric areal density(g m�2).

The R square value for this model obtained was 89.83% whichshowed satisfactory fitting of data in the model. The estimatedregression coefficient in coded units for this generic model is givenin Table 7.

4. Results and discussion

4.1. Validation of the models

The validation datasets which were separated from the maindata before the development of the models were used to evaluatethe performance of the developed models. The range of the inputparameters used for the development and validation of the statis-tical models of interlock fabrics are given in Table 8.

It was observed that predicted values were in close approxi-mationwith actual values of thermal resistance of interlock knittedfabrics depicting good prediction level of the developed models.The percentage error was calculated to determine the deviationbetween predicted and actual values using the following formula.

Percentage error ¼ Actual value� predicted valueActual value

� 100

The results of the thermal resistance validation along with thepercentage error are given in Table 9. The average percentage errorobtained for predicting, testing and generic models were 4.68%,7.04% and 6.88% respectively. These results confirmed that pre-dictability of the models on unseen data is quite satisfactory.Furthermore, a fitted line plot of all models is shown in Fig. 5, whichshows good agreement between predicted and actual values offabric thermal resistance.

4.2. Sensitivity analysis

Sensitivity analysis was performed to identify the relativeimportance of different factors on the thermal resistance of inter-lock knitted fabrics by using response surface regression basedmodel [22]. The influence of single variable at its minimum andmaximum values on thermal resistance was analysed keeping allthe other variables at their mean values. The range of the thermalresistance obtained by changing single input variable at itmaximum and minimum levels generates an idea howmuch it willinfluence the response as a percentage of total range of thermalresistance of interlock fabrics. The developed models had variousinput parameters, therefore to analyse the effect of all variablesModel 3was used for sensitivity analysis. The maximum, minimumandmid level values of input variables used for analysis are given inTable 10. The extreme levels and range of response variable aregiven in Table 11.

Table 9Predicted and actual values of thermal resistance of interlock fabrics.

S. No. Tt (tex) C (J g�1 K�1) l (mm) t (mm) m (g m�2) Rct A.V M1 M2 M3

P.V %E P.V %E P.V %E

1 36.48 2.78 3.53 1.30 455.7 8.0 8.1 1.7 8.6 7.1 7.1 10.92 36.48 2.78 3.59 1.32 450.8 7.7 8.6 11.6 8.6 12.3 7.6 1.33 36.48 2.78 3.55 1.30 457.7 8.4 8.3 1.3 8.5 1.3 7.2 13.84 36.48 2.78 3.54 1.30 461.4 7.9 8.2 3.9 8.4 6.4 7.1 9.85 36.48 2.78 3.57 1.30 450.5 8.2 8.4 2.9 8.7 6.3 7.5 8.96 29.22 2.51 3.20 1.14 408.3 5.5 5.6 2.4 5.5 0.4 5.6 0.97 29.22 2.51 3.17 1.12 407.9 5.2 5.4 3.9 5.1 1.3 5.0 4.88 29.22 2.51 3.17 1.14 404.3 5.8 5.4 6.9 5.6 3.9 5.6 3.29 29.22 2.51 3.17 1.14 406.2 5.3 5.4 1.9 5.5 4.3 5.6 4.810 29.22 2.51 3.22 1.14 399.8 5.7 5.8 1.5 5.7 0.1 5.9 3.311 24.40 2.51 4.00 1.10 249.9 10.8 11.6 7.8 12.6 16.3 11.3 4.712 24.40 2.51 3.97 1.06 252.2 10.9 11.4 4.7 10.1 7.0 10.1 7.613 24.40 2.51 4.01 1.06 254.3 10.6 11.7 10.6 10.0 5.8 10.1 4.714 24.40 2.51 3.95 1.06 255.9 10.7 11.3 5.2 9.9 7.8 9.9 7.515 24.40 2.51 3.97 1.04 264.3 11.0 11.4 3.8 8.2 25.2 9.1 17.1A.%E 4.68 7.04 6.88

A.V e actual value, M1 e 1st model, M2 e 2nd model, M3 e 3rd model, P.V e predicted value, E � error, A.%E � average percentage error.

A. Afzal et al. / International Journal of Thermal Sciences 85 (2014) 40e46 45

The contribution ratios (C.R) of the input variables were calcu-lated by dividing the range of single input variable by the overallrange of the thermal resistance data as mentioned in the followingEq. (4).

C:R ¼ Predicted range of the Rct by the variableOverall range of Rct

; (4)

The relative contribution percentage (C.R (%)) of single variablewas calculated considering the contribution ratios (C.R) of eachvariable with respect to other variables according to the followingformula.

C:R ð%Þ ¼ C:R of single variablesum of all variable0s C:R

� 100; (5)

The obtained relative contribution percentage from Eq. (5) wasplotted in form of pie chart to illustrate the relative influence of thevariables in percentage on thermal resistance of interlock fabricstructures as shown in Fig. 6.

It was observed from the results of sensitivity analysis that thepercentage relative influence of yarn count, specific heat of yarn,stitch length, fabric thickness and areal density were 15%, 29%, 38%,1% and 17% respectively. It is evident that stitch length significantlyaffects the thermal resistance of the interlock knitted fabrics fol-lowed by specific heat of the yarn. The concluded results confirmthat by increase in stitch length, the fabric's ability to entrap free air

R² = 0.9847R² = 0.8703

R² = 0.9443

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

4.5 6.5 8.5 10.5

Pred

icte

d th

erm

al r

esis

tanc

e

Actual thermal resistance

M1

M2

M3

Power (M1)

Power (M2)

Power (M3)

Fig. 5. Fitted line plot between predicted and actual values of thermal resistance ofinterlock knitted fabrics.

increases due to change in pore size of the fabric which resultantlyincreases the overall specific heat of the fabric structure and henceincreases the thermal resistance of the knitted fabric. It is alsoevident that influence of fabric thickness was not as significant asthat of stitch length and specific heat of yarn which is due to thereason that specific heat of the structure is more influential forthermal resistance of interlock knitted fabric than orientation ofmaterial.

5. Conclusions

Three different statistical models were developed and comparedfor the prediction of thermal resistance of polyester/cotton inter-lock knitted fabrics. The first was based on yarn specific heat, yarnlinear density and knitting stitch length. The second model wasbased on specific heat of the yarn, fabric areal density and fabricthickness. The third model was based on yarn specific heat, yarnlinear density, knitting stitch length, fabric areal density and fabricthickness. The R-sq values of the three models were 89.72%, 87.85%and 89.83% with corresponding percentage prediction error of4.68%, 7.04% and 6.88% respectively. It was found from the results ofsensitivity analysis that the percentage relative influence of yarncount, specific heat of yarn, stitch length, fabric thickness and fabricareal density on the fabric thermal resistance were 15%, 29%, 38%,1% and 17% respectively. The results of this study may be helpful forthe fabric engineer to optimize knitting fabric parameters formaximizing or minimizing the fabric thermal resistance as desired.

Table 10Extreme level and mid level values of input variables for sensitivity analysis.

V Max Le Min Le Mid Le

Tt 36.48 24.4 30.44C 3.038 2.506 2.772l 4.23 3.11 3.67t 1.36 1.04 1.2m 492.7 236.6 364.65

Max Le e maximum level, Min Le e minimum level, Mid Le e mid level.

Table 11Extreme levels and range of response variable for sensitivity analysis.

V Max Le Min Le R

Rct 20.7 3.8 16.9

R e range.

Fig. 6. Relative contribution percentages of input variables on thermal resistance asdetermined by sensitivity analysis.

A. Afzal et al. / International Journal of Thermal Sciences 85 (2014) 40e4646

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