determination for thickness of the multilayer thermal...

Research ArticleDetermination for Thickness of the Multilayer ThermalInsulation Clothing Based on the Inverse Problems

Bo Yu Qiao Chang Tingting Zhao and LinlinWang

College of Mathematical Sciences Dezhou University Dezhou 253023 China

Correspondence should be addressed to Bo Yu yubo51583817163com

Received 13 January 2019 Revised 30 March 2019 Accepted 28 April 2019 Published 9 June 2019

Academic Editor Babak Shotorban

Copyright copy 2019 Bo Yu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the optimal thickness of multilayer special clothing material under the high temperature operation based on theinverse problems is studied We analyze the parameters of the thickness of thermal insulation clothing material and the surfacetemperature of the dummy Using the least-squares fitting the function with the distribution of the dummy surface temperature isestablished Then the heat transfer model and optimization model to obtain the optimal thickness of different clothing layers arebuilt respectively Taking the true data as an example we give the application of thermal insulation clothing in fire area and theresults show that the models are feasible

1 Introduction

In fire metal smelting and other high temperature workstaffs are in the high temperature and high radiation envi-ronment This dangerous environment is generally dividedinto common dangerous and emergency [1] and differentenvironment will cause varying degrees of damage to humanbody So itwill need to provide themwith special work clothesto provide security for the high temperature operators

The material design of the thermal insulation clothinganalyzes the thermodynamic properties of the fabric froma scientific perspective and then determines the physicalparameters of the clothing thickness porosity and materialbased on the design objective It must prevent the heat sourceto injure the human body and provide the security safeguardfor the staffs under the high temperature condition Humanskin is very sensitive to temperature When the heat fluxdensity of body skin is 268 Jcm2 (the skin temperature is44∘C) people will have a burning sensation when the heatflux density of body skin is 502 Jcm2 (the skin temperatureis 72∘C) the human skin will get a second degree burn[2] When designing the thermal insulation clothing it isnecessary to deeply analyze the heat transfer regular insidethe thermal insulation clothing so as to provide theoreticalreference

In recent years many scholars have studied the heattransfer model of thermal insulation clothing Howeveraccording to whether the thermal insulation clothing adoptssingle-layer or multilayer material the existing models aredivided into single-layer model and multilayer model [3]In the single-layer model the thermal insulation clothingonly has a shell Scholars mainly study the radiation heatof its external flame the physical properties of the fabricand the influence of the thickness of the air layer betweenthe fabric and the skin on the performance of the thermalinsulation clothing [4] Gibson [5] first proposed the heatand mass transfer model of single-layer porousmedia at hightemperature Torvi [6] improved the model and establishedthe heat transfer model of thermal insulation clothingrsquos shellmaterial under the condition of long-term exposure to strongradiation and low radiation Chitrphiromsri [7] developedthe heat and moisture transport model inside the protectiveclothing of porous media based on the Gibson and Torvimodels Ghazy [8] made a further study on all aspects ofthe air layer skin system of thermal protective clothing andestablished the heat transfer model of single-layer fabricin motion PanB [4] introduced the heat transfer modelof single-layer thermal protective clothing and proposedrelevant inverse problems Some scholars also have studiedthe heat conduction multilayer model of thermal insulation

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 5193729 7 pageshttpsdoiorg10115520195193729

2 Mathematical Problems in Engineering

clothing based on the model of single-layer Mell [9] putforward a heat transfer model between layers of multilayerfabrics Lawson [10] established the heat and moisture trans-fer model of multilayer fabrics based on the influence ofmoisture on the protection effect Lu [3] built the heat transferequation for each layer of the three-layer thermal protectiveclothing and analyzed the influence of air layer and fabricthickness on the protective performance of the protectiveclothing

Previous studies show the heat transfer of the single-layer and multilayer thermal insulation clothing Howeverfew researchers study the thickness of each layer material ofthermal insulation clothing This paper studies three-layerthermal insulation clothing (shell waterproof and thermalinsulation) and the air layer based on the inverse problemsmethod and established the heat transfer model of each layerand optimization model Then the fabric thickness of eachlayer is optimized under different temperature and varyingworking hours on the premise of meet the security Thenthe high temperature insulation material performance underworking is concluded

2 Theoretical Analysis

21 e Distribution of the Dummy Surface TemperaturePrevious researches show that there is a relationship betweenthe temperature of the thermal insulation material andtime under high temperature conditions So we can obtainthe relationship between the surface temperature of thedummy and time under high temperature conditions byplotting the curve According to this curve we can obtaina regulation that the surface temperature of the dummyincreases with time and then becomes stable under thefixed working time thickness of each layerrsquos material andambient temperature Related researches show that thechange of temperature corresponds to a nonlinear functionThen we can fit this function using the least-squares fittingmethod

The least-squares fitting is introduced as followsSuppose a set of measures of multivariate function y =119891(1199091 119909119899) are (1199091119894 119909119899119894 119910119894) (119894 = 1 119898) we need to

obtain nonlinear function 120593 = 120593(1199091 1199092 119909119899 1205780 1205781 120578119873)including the parameters 120578119895 (119895 = 0 1 119873) For a set of pos-itive numbers 1205961 1205962 120596119898 the value of objective function119878(1205780 1205781 120578119873) = sum119898119894=1119908119894(119910119894minus120593(1199091119894 1199092119894 119909119899119894 1205780 1205781 120578119873))2is to minimize It is nonlinear least-squares fitting Thisproblem belongs to the optimization problem without con-straints And the general solution method is very com-plex so quasi-newton method is adopted to solve itusually

We directly observe the degree of fitting between themodel and the original data A good fitting degree indicatesthat the model well represents the relationship of the surfacetemperature of the dummy with time We use the sampledetermination coefficient (R2 ) tomeasure the degree of fittingmore accurately When the value ofR2 is close to 1 it indicatesthe degree of fitting is highThismodel is considered to be the

x0=0 x1 x2 x3 x4 x (mm)

75∘C 37 ∘C

T (∘C)＞1 ＞2 ＞3 ＞4

I II III IV

Figure 1 External environment-special clothing-dummy surfacesystem

distribution of the dummy surface temperature The R2 RSSand ESS are described as follows

1198772 = 1 minus 119864119878119878119877119878119878 (1)

119877119878119878 = 119899sum119894=1

(T119894 minus 1119899119899sum119894=1

T1015840119894)2

(2)

119864119878119878 = 119899sum119894=1

(T1015840119894 minus 1119899119899sum119894=1

T1015840119894)2

(3)

where 119877119878119878 is the sum of squared deviations of theoreticalvalues 119864119878119878 is the sum of squared residuals 119879119894 is the predictedvalue of the dummy surface temperature1198791015840119894 is the actual valueof the dummy surface temperature

22 e Heat Transfer Model of Each Layer Material andInverse Problems The whole system is divided into six partsincluding external environment special clothing the airlayer and structure of human skin surface and it is shownin Figure 1 Special clothing includes three-layer materialswhich are remarked layers I II and III respectively The airlayer is remarked layer IV The external temperature reachesthe surface of skin through three layers special clothingmaterials and an air layer Each layer absorbs heat and reducestemperature Therefore in order to show the gradual declineof temperature a heat transfer model is established

It is assumed that the temperature of the skin surfaceis the same as that of each layer at the initial stage Inthis paper we establish the heat transfer model among theexternal environment three-layer special clothing air layerand dummy skin with the initial conditions which certainexternal environment temperature and the parameters ofeach layer material

The heat transfer model in layer 119894 is119877119894 120597T119894120597119905 = 120597120597119909 (120582119894 120597119879119894120597119909 )

(119909 119905) isin Ω119894 times (0 119905) (119894 = 1 2 3 4)(4)

Mathematical Problems in Engineering 3

where Ω119894 = (D119894minus1D119894) (119894 = 1 2 3 4) D119894 = sum119894119896=1 119889119896 (119896 =1 2 3 4)subject to the following initial and boundary in layer 119894

respectively

Layer I

1198791 (1198631 0) = 1198791199011205971198791120597119905 (0 119905) = 01198791 (0 119905) = 119879119897119890

(5)

Layer II

11987911003816100381610038161003816119909=1198631 = 11987921003816100381610038161003816119909=11986311205971198792120597119905 (0 119905) = 01198792 (1198632 0) = 119879119901

(6)

Layer III

11987921003816100381610038161003816119909=1198632 = 11987931003816100381610038161003816119909=11986321205971198793120597119905 (0 119905) = 01198793 (1198633 0) = 119879119901

(7)

Layer IV

11987931003816100381610038161003816119909=1198633 = 11987941003816100381610038161003816119909=11986331198794 (1198634 0) = 1198791199011198794 (1198634 119905) = 119879

(8)

where 120588119894 is the material density of layer 119894 (1198961198921198983)119888119894 is the material specific heat of layer 119894 (119869(119896119892 sdot 119870))119877119894 is the sensible heat capacity of layer 119894 (119896119869(1198983 sdot 119870))119879119894 is the temperature exchange value of layer 119894 (∘C)120582119894 is the material thermal conductivity of layer 119894Ω119894 (i = 1 2 3 4) is the range of values of 119909119889119896 (119896 = 1 2 3 4) is the thickness in layer 119896119863119894 (119894 = 1 2 3 4) is the total thickness from the first layer tothe i-th layer

The basic problem of inverse problems is to study theinverse process of various physical phenomena The basicmethod is to summarize the physical phenomena into acertain mathematical model and then quantitatively analyzethe physical process itself and its carrier through simulation[11]

23 e Optimal ickness of Layer II Clothing Material

231 Analyzing Clothing ickness from the Direct ProblemsWhen the density specific heat and heat conductivity ofevery layer and the thickness of the fourth layer are fixed we

only research the relationship of thickness and temperatureover time According to the heat transfer model establishedamong the external environment three-layer special cloth-ing air layer and dummy skin we can obtain the innermosttemperature of the first layer clothing 11987910158401 =T0minusT1 where T0is the outer temperature of the first layer clothing T1 is thetemperature variation of the first layer clothing

Then according to the heat transfer equation the tem-perature distribution of the innermost layer clothing in thefirst layer is taken as the initial condition of the temperaturedistribution in the second layer In this way the heat istransferred And the numerical solution of the temperaturedistribution in the fourth layer is obtained [12]

232 Analyzing Clothingickness from the Inverse ProblemsThe external environment special clothing air layer andskin mainly transfer energy in turn and thermal insulationproperties of clothing improve with the increase of thicknesswhile the thickness of clothing affects the comfort andconvenience of human So the thinner clothing in the premiseof thermal insulation is better

The dummies are put into the high temperature of 119879119890 Wecan get the thickness of clothing in layer II when the surfacetemperature is between 119879119898119894119899 and 119879119898119886119909 for not more than 119905minutes So the optimization model can be established asfollows

The objective function is

min 1198892 (9)

subject to the following physical constraints

max 119879 le Tmax

Tmin lt 119879 le Tmax

0 lt 119905 lt 1199051198981198981198892min lt 1198892 lt 1198892119898119886119909

(10)

According to the above model different values of d2can be obtained However considering that the thicknessof clothing affects peoplersquos comfort and convenience thethinner the clothing is the more comfortable and convenientpeople will be Therefore the minimum of all values ofd2 satisfying mentioned above conditions is the optimalthickness of the second layer

24 e Optimal ickness of the Second and Fourth LayerClothing Material

241 Analyzing Clothing ickness from the Direct ProblemsAccording to the heat transfer equations of each layer cloth-ing the temperature distribution in the fourth layer and thethickness in the second and fourth layers with time can beobtained finally The process of heat transfer is the same asthe mentioned above in Section 231

242 Analyzing Clothingickness from the Inverse ProblemsThe dummies are put into the high temperature of 119879119890


We can get the thickness of clothing in layers II and IVwhen the surface temperature is between 119879119898119894119899 and 119879119898119886119909for not more than 119905 minutes So the optimization modelis established according to the distribution of the dummysurface temperature mentioned above

The objective function

min 1198892max 1198894 (11)


1198892min lt 1198892 lt 1198892max

1198894min lt 1198894 lt 1198894max

max 119879 le 119879max

119879min lt 119879 le 119879max

0 lt 119905 lt 119905119898119898

(12)

The above optimization model is multiobjective opti-mization problem However the goals of multiobjective opti-mization problems have conflict and no common measure-ment standards so it is much more difficult to solve multi-objective optimization problems than simple-objective opti-mization problemsThemultiobjective optimization problemcan be transformed into a simple-objective numerical opti-mization problem by linear weighting method The core ideaof linear weighting method is to give each goal a nonnegativeweight system according to its importance in the minds ofdecision makers then the weighted goals are added togetherto construct the evaluation function With this evaluationfunction the original multiobjective optimization problemcan be transformed into a simple-objective numerical opti-mization problem the final solution can be obtained bysolving the simple-objective problem [13] Since the abovetwo objective functions are of equal importance we assumethat the weighted coefficients are 05 to solve the single-objective numerical optimization problem

According to all the optimization conditions differentvalues of d2 and d4 satisfying the conditions can be obtainedthrough MATLAB programming Generally speaking thedensity of layer II fabric material is much higher than that oflayer IVmaterial Considering the research and developmentcosts of clothing the relatively small value among all valuesof d2 satisfying the above conditions is the optimal thicknessof the second layer

The heat transfer in the air layer is dominated by heatconduction without considering convection and the heatinsulation performance improves with the increase of airthickness The thickness of air layer will greatly affect thecomfort of human clothing and the convenience of actionTherefore the relatively large value of d4 is the optimalthickness of layer IV under the condition that d2 is theoptimal value

0 1000 2000 3000 4000 5000 600036

38

40

42

44

46

48

50

The original temperature

T (∘

C)

t (s)

Figure 2 The curve of temperature changing with time

3 Application

Staffs are often in the high temperature environment in firearea Different fire conditions can damage firemen body Soit needs special work clothes to provide security for themBased on the current situation of the fire field in China [14]this paper explores the application of the heat transfer modeland optimization model established above in the thermalinsulation clothing We take the data from CUMCM-2018-Problem-A-Chinese in China as an example

31 e Distribution of the Dummy Surface Temperature inFire Area At first the curve of temperature changing withtime is drawn by obtained data and it is shown in Figure 2

According to the obtained curve we get that the changeof temperature corresponds to a logarithmic function But thefitting function containing only logarithm function is greatlydifferent from the original data and the fitting is poor Itmay be due to the lack of a correct understanding of theerror distribution of the dependent variable using only thelogarithm function And the estimated parameter is a biasedestimator and the fitted curve deviates from the discretepoints [15]

So we establish the model combining logarithm functionand quadratic function can be given as

119879 = 119886 ln (t) + 1198871199052 + 119888119905 + 119889 (13)

where 119879 is the surface temperature of the dummy 119905 is time119886 119887 119888 and 119889 are parametersThe fitting function parameters are obtained through

MATLAB and then the fitting function is obtained as

119879 = 35708 ln (t) + 16964 times 10minus71199052 minus 00023119905+ 250400 (14)

Then this fitting function is compared with the originaldata by drawing a scatter diagram and it is shown in Figure 3


0 1000 2000 3000 4000 5000 600025

30

35

40

45

50

The original temperatureThe fitting temperauure

t (s)

Figure 3 The curves between the fitting function and the originaldata

As can be seen from Figure 3 the function curve fittedby optimized model agrees well with the original data but itneeds to be further verified in theory

The theoretical value obtained by fitting is compared withthe known experimental value and the sample determinationcoefficient R2 obtained through MATLAB is 09263 which isclose to 1 Therefore the fitting effect is good and the fittingfunction can better reflect the regulation of temperaturechange with time We obtain the distribution function of thesurface temperature of the dummy as follows


32 e Optimal ickness of the Second Layer ClothingMaterial in Fire Area The dummies are put into the hightemperature of 65∘C We can get the thickness of clothing inlayer II when the surface temperature is between 44∘C and47∘C for not more than 5 minutes According to the heattransfer model of each layer material and the optimizationmodel the minimum value satisfying all conditions is theoptimal thickness of the second layer The optimal thicknessof the second layer is 70133mm calculated with our programcode inMATLAB software by using a finite differencemethod[16] Then we can obtain the temperature distribution withtime under the optimal thickness of the second layer It isshown in Figure 4

33 e Optimal ickness of the Second and Fourth LayerClothing Material in Fire Area The dummies are put intothe high temperature of 80∘C We can get the thickness ofclothing in layers II and IV when the surface temperatureis between 44∘C and 47∘C for not more than 5 minutesSo the optimization model is established according to the

0 500 1000 1500 2000 2500 3000 3500 400036

37

38

39

40

41

42

43

44

45

46

47

The surface temperature

T (∘

C)

t (s)

Figure 4The surface temperature distribution with time under theoptimal thickness of the second layer

distribution of the dummy surface temperature mentionedabove According to the heat transfer model of each layermaterial and the optimization model the relatively smallvalue among all values of d2 satisfying the above conditionsis the optimal thickness of the second layer and the relativelylarge value of d4 is the optimal thickness of layer IV underthe condition that d2 is the optimal value So the optimalthickness of layer II and layer IV is iterated to be 57744 mmand 75773 mm respectively

Then we can obtain the temperature distribution withtime under the optimal thickness of the second layer and theforth layer It is shown in Figure 5

4 Conclusion

In this paper we mainly use the temperature to investigatethe thickness of thermal insulation clothing material andcombine logarithmic function and quadratic function toproperly govern the relationship of the surface temperature ofthe dummywith time at a certain thicknessThese parameterscan be estimated by the method of least-squares fitting

And this paper studies two theoretical models of heattransfer model and optimization model to optimize thethickness of thermal insulation clothing based on inverseproblems The theoretical models can be used for the designof fire-fighter protective clothing It has logical and concisethinking The obtained result confirms the validity and thegood behavior of the theoretical models

The inverse problems are used for design of the ther-mal insulation clothing which can provide theoretical basisand scientific reference for the improvement of protectiveperformance Under the premise of ensuring safety thispaper converts the multiobjective optimization problem intoa simple-objective problem based on the heat transfer model


0 500 1000 1500 2000 2500 300036

37

38

39

40

41

42

43

44

45

46

47


T (∘

C)

t (s)

Figure 5 The surface temperature distribution with time under the optimal thickness of the second layer and the forth layer

and the minimum degree of human burns This methodimproves the efficiency of optimization work

This application can promote the improvement of thework ability in fire protection fighting and rescue andpromote the comprehensive development of material sciencein the field of fire protection This paper has a reference valuefor the design of thermal insulation clothing under differenttemperature requirements

Appendix

TheMatlab code in Section 31function f = fun2(xtdata)f=x(1)lowastlog(tdata)+x(2)lowast(tdata)and2+x(3)lowast(tdata)+x(4)endclearclca=xlsread(C UsersBo YUDesktopTemperature

xlsxB3B5403)cdata=an=size(cdata2)tdata=1nplot(tdatacdata)The figure of the original temperature

in Figure 2xlabel(t(s) )ylabel(T(∘C) )x0=[01010101]x=lsqcurvefit(fun2x0tdatacdata)f=fun2(xtdata)plot(tdatacdata rtdataf b) The curves between the

fitting function and the original data in Figure 3xlabel(t(s) )ylabel(T(∘C))The Matlab code in Section 32clear

clct=13600f=35708lowastlog(t)+(16964e-07)lowasttand2-00023lowastt+250400y=35617lowastlog(t)+(12364e-07)lowasttand2-00023lowastt+210404a=[3561735708 1236416964 2104042504]b=mean(a)if length(find(ygt44))lt=300ampmax(y)lt=47d2=60bendd2plot(ty)xlabel(t(s) )ylabel(T(∘C) )axis([0 4000 36 47])set(gca YTick[363738394041424344454647])The Matlab code in Section 33clearclct=13000m=009778lowast(35708lowastlog(t)+(16964e-7)lowasttand2-

00023lowastt+2504)+089510lowast(35600lowastlog(t)+(12364e-7)lowasttand2-00023lowastt+210404)

b=length(find(mgt=44))c=max(m)d2=009978lowast6+099510lowast7013d4=009778lowast5+09610lowast55plot(tm)xlabel(t(s) )ylabel(T(∘C) )axis([0 3000 36 47])set(gca YTick[363738394041424344454647])

Data Availability

The data in this paper are based on China UndergraduateMathematical Contest in Modeling-2018 Problem-A which


are authentic and available Sharing data can increase theimpact and visibility of this manuscript and enhance theimage of our research group If you have any questions pleasecontact the author Bo Yu e-mail yubo51583817163com

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The paper is supported by the Ministry of EducationHumanities and Social Sciences Research Youth Founda-tion (Grant no 19YJC910011) and the Project of ShandongProvince Higher Educational Science and Technology Pro-gram (J18KB099)

References

[1] R L Barker A Review of Gaps and Limitations in Test Methodfor First Responder Protective Clothing and Equipment NationalPersonal Protection Technology Laboratory Pittsburgh Penn-sylvania 2005

[2] Y YuMathematical Modeling of Heat-Moisture Transfer withinTextiles and e Corresponding Inverse Problems Zhejiang Sci-Tech University Zhejiang China 2016

[3] L Lu Mathematical Model of Heat Transfer within Multilayerermal Protective Clothing and Corresponding Optimal Param-eter Determination Zhejiang Sci-Tech University ZhejiangChina 2018

[4] B PanMathematical Modeling of Heat Transfer within ermalProtective Clothing and Inverse Problems of Parameters Determi-nation Zhejiang Sci-Tech University Zhejiang China 2017

[5] P W Gibson ldquoMultiphase heat and mass transfer throughhygroscopic porous media with applications to clothing mate-rialsrdquo Fiber vol 53 no 5 pp 183ndash194 1996

[6] D A Torvi Heat Transfer in in Fibrous Materials underHigh Heat Flux Conditions University of Alberts EdmontonCanada 1997

[7] P Chitrphiromsri and A V Kuznetsov ldquoModeling heat andmoisture transport in firefighter protective clothing during flashfire exposurerdquo Heat and Mass Transfer vol 41 no 3 pp 206ndash215 2005

[8] A Ghazy ldquoNumerical study of the air gap between fire-protective clothing and the skinrdquo Journal of Industrial Textilesvol 44 no 2 pp 257ndash274 2014

[9] W E Mell and J R Lawson ldquoA heat transfer model forfirefightersrsquo protective clothingrdquo Fire Technology vol 36 no 1pp 39ndash68 2000

[10] J R Lawson W E Mell and K Prasad ldquoA heat transfer modelfor firefightersrsquo protective clothing continued developments inprotective clothingmodelingrdquo Fire Technology vol 46 no 4 pp833ndash841 2010

[11] P Qitian Inverse Problems of Parameters Determination forree-Layer Porous Textile Material Based on A Coupled Heatand Moisture Model Zhejiang Sci-Tech University ZhejiangChina 2017

[12] G Cao G Wang and X Ren ldquoFundamental solution to one-dimensional heat conduction equationrdquo Journal of ShandongPolytechnic University vol 4 pp 77ndash80 2005

[13] A Liu R Liu and H Liu ldquoThe study on making the multi-objective optimized parameters into simple-objective opti-mized parameterrdquo Journal of Henan Science vol 30 no 11 pp1605ndash1609 2012

[14] Y Li ldquoDiscussion on the application of data in the field of fireprotectionrdquo Fire Services (Electronic) vol 7 pp 111ndash113 2016

[15] ZWang and QWang ldquoNonlinear regressionmethod with zerosum of residual errors and its applicationrdquo Journal NortheastForestry University vol 39 no 2 pp 125ndash127 2011

[16] S Ce ldquoHeat conduction equation finite difference method toachieve the matlabrdquo Journal of XianyangNormal University vol4 pp 27ndash29 2009

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clothing based on the model of single-layer Mell [9] putforward a heat transfer model between layers of multilayerfabrics Lawson [10] established the heat and moisture trans-fer model of multilayer fabrics based on the influence ofmoisture on the protection effect Lu [3] built the heat transferequation for each layer of the three-layer thermal protectiveclothing and analyzed the influence of air layer and fabricthickness on the protective performance of the protectiveclothing

Previous studies show the heat transfer of the single-layer and multilayer thermal insulation clothing Howeverfew researchers study the thickness of each layer material ofthermal insulation clothing This paper studies three-layerthermal insulation clothing (shell waterproof and thermalinsulation) and the air layer based on the inverse problemsmethod and established the heat transfer model of each layerand optimization model Then the fabric thickness of eachlayer is optimized under different temperature and varyingworking hours on the premise of meet the security Thenthe high temperature insulation material performance underworking is concluded

2 Theoretical Analysis

21 e Distribution of the Dummy Surface TemperaturePrevious researches show that there is a relationship betweenthe temperature of the thermal insulation material andtime under high temperature conditions So we can obtainthe relationship between the surface temperature of thedummy and time under high temperature conditions byplotting the curve According to this curve we can obtaina regulation that the surface temperature of the dummyincreases with time and then becomes stable under thefixed working time thickness of each layerrsquos material andambient temperature Related researches show that thechange of temperature corresponds to a nonlinear functionThen we can fit this function using the least-squares fittingmethod

The least-squares fitting is introduced as followsSuppose a set of measures of multivariate function y =119891(1199091 119909119899) are (1199091119894 119909119899119894 119910119894) (119894 = 1 119898) we need to

obtain nonlinear function 120593 = 120593(1199091 1199092 119909119899 1205780 1205781 120578119873)including the parameters 120578119895 (119895 = 0 1 119873) For a set of pos-itive numbers 1205961 1205962 120596119898 the value of objective function119878(1205780 1205781 120578119873) = sum119898119894=1119908119894(119910119894minus120593(1199091119894 1199092119894 119909119899119894 1205780 1205781 120578119873))2is to minimize It is nonlinear least-squares fitting Thisproblem belongs to the optimization problem without con-straints And the general solution method is very com-plex so quasi-newton method is adopted to solve itusually

We directly observe the degree of fitting between themodel and the original data A good fitting degree indicatesthat the model well represents the relationship of the surfacetemperature of the dummy with time We use the sampledetermination coefficient (R2 ) tomeasure the degree of fittingmore accurately When the value ofR2 is close to 1 it indicatesthe degree of fitting is highThismodel is considered to be the

x0=0 x1 x2 x3 x4 x (mm)

75∘C 37 ∘C

T (∘C)＞1 ＞2 ＞3 ＞4

I II III IV

Figure 1 External environment-special clothing-dummy surfacesystem

distribution of the dummy surface temperature The R2 RSSand ESS are described as follows

1198772 = 1 minus 119864119878119878119877119878119878 (1)

119877119878119878 = 119899sum119894=1

(T119894 minus 1119899119899sum119894=1

T1015840119894)2

(2)

119864119878119878 = 119899sum119894=1

(T1015840119894 minus 1119899119899sum119894=1

T1015840119894)2

(3)

where 119877119878119878 is the sum of squared deviations of theoreticalvalues 119864119878119878 is the sum of squared residuals 119879119894 is the predictedvalue of the dummy surface temperature1198791015840119894 is the actual valueof the dummy surface temperature

22 e Heat Transfer Model of Each Layer Material andInverse Problems The whole system is divided into six partsincluding external environment special clothing the airlayer and structure of human skin surface and it is shownin Figure 1 Special clothing includes three-layer materialswhich are remarked layers I II and III respectively The airlayer is remarked layer IV The external temperature reachesthe surface of skin through three layers special clothingmaterials and an air layer Each layer absorbs heat and reducestemperature Therefore in order to show the gradual declineof temperature a heat transfer model is established

It is assumed that the temperature of the skin surfaceis the same as that of each layer at the initial stage Inthis paper we establish the heat transfer model among theexternal environment three-layer special clothing air layerand dummy skin with the initial conditions which certainexternal environment temperature and the parameters ofeach layer material

The heat transfer model in layer 119894 is119877119894 120597T119894120597119905 = 120597120597119909 (120582119894 120597119879119894120597119909 )

(119909 119905) isin Ω119894 times (0 119905) (119894 = 1 2 3 4)(4)



respectively

Layer I

1198791 (1198631 0) = 1198791199011205971198791120597119905 (0 119905) = 01198791 (0 119905) = 119879119897119890

(5)

Layer II

11987911003816100381610038161003816119909=1198631 = 11987921003816100381610038161003816119909=11986311205971198792120597119905 (0 119905) = 01198792 (1198632 0) = 119879119901

(6)

Layer III

11987921003816100381610038161003816119909=1198632 = 11987931003816100381610038161003816119909=11986321205971198793120597119905 (0 119905) = 01198793 (1198633 0) = 119879119901

(7)

Layer IV

11987931003816100381610038161003816119909=1198633 = 11987941003816100381610038161003816119909=11986331198794 (1198634 0) = 1198791199011198794 (1198634 119905) = 119879

(8)










min 1198892 (9)


max 119879 le Tmax


0 lt 119905 lt 1199051198981198981198892min lt 1198892 lt 1198892119898119886119909

(10)








min 1198892max 1198894 (11)


1198892min lt 1198892 lt 1198892max

1198894min lt 1198894 lt 1198894max

max 119879 le 119879max

119879min lt 119879 le 119879max

0 lt 119905 lt 119905119898119898

(12)




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T (∘

C)

t (s)


3 Application





119879 = 119886 ln (t) + 1198871199052 + 119888119905 + 119889 (13)






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T (∘

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t (s)




4 Conclusion





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Data Availability






Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018










respectively

Layer I

1198791 (1198631 0) = 1198791199011205971198791120597119905 (0 119905) = 01198791 (0 119905) = 119879119897119890

(5)

Layer II

11987911003816100381610038161003816119909=1198631 = 11987921003816100381610038161003816119909=11986311205971198792120597119905 (0 119905) = 01198792 (1198632 0) = 119879119901

(6)

Layer III

11987921003816100381610038161003816119909=1198632 = 11987931003816100381610038161003816119909=11986321205971198793120597119905 (0 119905) = 01198793 (1198633 0) = 119879119901

(7)

Layer IV

11987931003816100381610038161003816119909=1198633 = 11987941003816100381610038161003816119909=11986331198794 (1198634 0) = 1198791199011198794 (1198634 119905) = 119879

(8)










min 1198892 (9)


max 119879 le Tmax


0 lt 119905 lt 1199051198981198981198892min lt 1198892 lt 1198892119898119886119909

(10)








min 1198892max 1198894 (11)


1198892min lt 1198892 lt 1198892max

1198894min lt 1198894 lt 1198894max

max 119879 le 119879max

119879min lt 119879 le 119879max

0 lt 119905 lt 119905119898119898

(12)




0 1000 2000 3000 4000 5000 600036

38

40

42

44

46

48

50


T (∘

C)

t (s)


3 Application





119879 = 119886 ln (t) + 1198871199052 + 119888119905 + 119889 (13)






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t (s)







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t (s)




4 Conclusion





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T (∘

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Appendix








Data Availability






Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018











min 1198892max 1198894 (11)


1198892min lt 1198892 lt 1198892max

1198894min lt 1198894 lt 1198894max

max 119879 le 119879max

119879min lt 119879 le 119879max

0 lt 119905 lt 119905119898119898

(12)




0 1000 2000 3000 4000 5000 600036

38

40

42

44

46

48

50


T (∘

C)

t (s)


3 Application





119879 = 119886 ln (t) + 1198871199052 + 119888119905 + 119889 (13)






0 1000 2000 3000 4000 5000 600025

30

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45

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t (s)







0 500 1000 1500 2000 2500 3000 3500 400036

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T (∘

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t (s)




4 Conclusion





0 500 1000 1500 2000 2500 300036

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T (∘

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Appendix








Data Availability






Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018









0 1000 2000 3000 4000 5000 600025

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t (s)







0 500 1000 1500 2000 2500 3000 3500 400036

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T (∘

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t (s)




4 Conclusion





0 500 1000 1500 2000 2500 300036

37

38

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T (∘

C)

t (s)




Appendix








Data Availability






Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018









0 500 1000 1500 2000 2500 300036

37

38

39

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T (∘

C)

t (s)




Appendix








Data Availability






Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018












Acknowledgments


References
























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








determination for thickness of the multilayer thermal...

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