ResearchArticle BioheatTransferEquationwithProtectiveLayer · 2021. 1. 25. · ResearchArticle...

Research ArticleBioheat Transfer Equation with Protective Layer

Kabita Luitel ,1 Dil Bahadur Gurung,2 Harihar Khanal,3 and Kedar Nath Uprety4

1Department of Mathematics, Bhaktapur Multiple Campus, Bhaktapur, Nepal2Department of Mathematics, Kathmandu University, Kavre, Dhulikhel, Nepal3Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL, USA4Central Department of Mathematics, Tribhuvan University, Kritipur, Nepal

Correspondence should be addressed to Kabita Luitel; [email protected]

Received 16 October 2020; Revised 25 December 2020; Accepted 13 January 2021; Published 25 January 2021

Academic Editor: Abdullahi Yusuf

Copyright © 2021 Kabita Luitel et al. +is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

+e human thermal comfort is the state of mind, which is affected not only by the physical and body’s internal physiologicalphenomena but also by the clothing properties such as thermal resistance of clothing, clothing insulation, clothing area factor, airinsulation, and relative humidity. In this work, we extend the one-dimensional Pennes’ bioheat transfer equation by adding theprotective clothing layer. +e transient temperature profile with the clothing layer at the different time steps has been carried outusing a fully implicit Finite Difference (FD) Scheme with interface condition between body and clothes. Numerically computedresults are bound to agree that the clothing insulation and air insulation provide better comfort and keep the body at the thermalequilibrium position. +e graphical representation of the results also verifies the effectiveness and utility of the proposed model.

1. Introduction

+e study of thermal comfort in the complex vascular ge-ometry of the human body with a protective clothing systemis not only the subjective domain but also the physiologicalfactors inside the body. According to the American Societyof Heating, Refrigerating, and Air Conditioning Engineers(ASHRAE), thermal comfort is the state of the mind whenone can feel and express the satisfaction with the thermalenvironment [1]. +ermoregulation is the process of con-trolling the internal body temperature through the hypo-thalamus heat production and heat loss center.+e body alsouses other processes of thermodynamical systems thatconstantly produce energy by metabolic activity togetherwith dilating or constricting blood vessels, shivering, andsweating [2, 3].

+e presence of clothes, on the other hand, plays a vitalrole for maintaining the thermal comfort at the equilibriumcondition of heat production and heat loss by the body.Clothing, the interface between the skin surface and theenvironment, works as an insulator and also transports theheat from the body to the outer environment around us. +ekey factors affecting thermal comfort are categorized as

(1) Personal factors(2) Environmental factors

1.1. *e Personal Factors. Metabolism, determined by age,sex, health, etc., is one of the human personal factors whichmakes a difference in thermal comfort. Besides, anothermajor factor which affects thermal comfort is clothing. Asignificant difference in thermal comfort can be caused bysmall changes in clothing layers. In the winter season,wearing a sweater and socks makes better comfort, whereasin summer, wearing light clothes makes better comfort in theworkplace.

1.2. Environmental Factors. Air temperature, moving air,radiant temperature, and relative humidity are the envi-ronmental factors that help to maintain thermal comfort bykeeping the room comfortable. +e radiation heat flux, onthe other hand, penetrates a certain depth passing entirelythrough the fabric, depending on the fabric structure andradiative wavelength. In contrast, the convection portion ofthe heat source could reach the fabric surface only [4].

HindawiMathematical Problems in EngineeringVolume 2021, Article ID 6639550, 12 pageshttps://doi.org/10.1155/2021/6639550

mailto:[email protected]://orcid.org/0000-0003-1573-1917https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2021/6639550

Usually, heat transfer in the body through the garmentoccurs from heat conduction and heat radiation; then, thetemperature rises and transfers into the air gap. Suitableclothing along with these environmental factors, therefore,maintains better comfort at home and workplace (business,office, study room, etc.).

2. Role of Clothing

2.1.*ermal Resistance and Clothing Insulation. +e ratio oftemperature difference between two faces of material to therate of heat flow per unit area is defined as thermal resis-tance. In the study of thermal insulation of clothing, thermalresistance is a very important parameter defined as thefunction of thickness and thermal conductivity of clothes.+ermal Resistance Rth m2 · (°C/W) is given by [5]

Rth �ΔTq

�L

k, (1)

where L is the thickness of cloth (m), ∆T is the temperaturedifference (°C), and q is the heat flow rate (W/m2). +oughthe heat transfer in the clothed body consists of conductionand radiation, the primary determinant of the insulation isthe thickness of clothes on which the insulation is very muchdependent. +e limbs of human body with insulation pa-rameters can be seen in Figure 1.

+ermal insulation, together with the air gap, does notonly provides comfort at the skin surface and body from acold environment but also protects from burn injuries.While sitting by the fire, the air gap provides thermalinsulation that limits heat transfer to skin and protects theskin from excessive heat.

2.1.1. Clo Unit. Clo unit is the measure of thermal resistanceand includes the insulation provided by any layer of trappedair between skin and the insulation value of clothing itself. 1Clo is defined as the insulation of the clothing system thatrequires maintaining a sitting-resting average male com-fortable in a normally ventilated room. Clo is the thermalinsulation of overall clothing worn by a person. It has 0.1m/sair velocity at air temperature 21°C and relative humidity lessthan 50%. Among the total heat produced by the metabolicreaction, 24% heat is lost through evaporation and respi-ration. As 1 met� 50 kcal/m2h, the evaporative and respi-ratory heat loss� 1 met× 24%� 50× 0.24�12 kcal/m2h.Remaining 38 kcal/m2h is transmitted through the clothingsystem by conduction, convection, and radiation. +ecomfortable skin temperature is 33°C, so the total insulationof clothing and air layer (IT � Icl + Ia) is given by [8]

I �33 − 2138

� 0.32,

Ia � 0.14m2

· °C (h/kcal),

Icl � 0.32 − 0.14( m2

· °C (h/kcal) � 0.18m2 · °C (h/kcal),(2)

where Ia is the insulation of air and Icl is the insulation cloth.

Since 1 kcal/h� 1.163 watt (W), so 1 Clo unit is defined as

0.18m2 · °C �0.181.163≡ 0.155m2·(°C /W). (3)

2.2. Convective and Radiative Heat Transfer. Clothing worksas the mediator of heat exchange through convection as wellas radiation. +e standard measurement condition in ISO9920, four different manikins in three different laboratories,were used and determined the male and female clothingthermal insulation values of 52 nonwestern clothing con-figuration [9]. If hc(m2 · (°C/W)) and hr(m2 · (°C/W)) arethe heat exchange due to convection radiation, respectively,then the convective and radiative heat exchange is defined in[7, 9] by

hc + hr �ΔTIcl

. (4)

2.3. Clothing Area Factor. +e dimensionless parameter,clothing area factor fcl, is the ratio between the surface areaof clothed human body Ab (m2) and the surface area of nudehuman body Ab (m2) which is given by

fcl �Acl

Ab. (5)

+e prediction equation for clothing area factor fcl, basedon western clothing and listed in ISO Standard 9920-2009(ISO 2009), can also be found in various publications. +eequation for fcl is given [1, 9, 10] as

InClo, fcl � 1 + 0.31 Icl,

In m2 · (°C/W) , fcl � 1 + 1.97 Icl.(6)

2.4. Clothing Efficiency Factor. +e clothing efficiency fac-tor(dimensionless) depends upon the air insulation Icl(m2 ·(°C/W)) and the thermal insulation of overall clothing, notonly the particular garment but also the entire garment,including tops, bottoms, innerwear, and everything

Body

Enclosed airlayer

Surface airlayer

Clothing

IT

Ia

Icl

Figure 1: A cylindrical model with clothing and air insulation[6, 7].

2 Mathematical Problems in Engineering

(including even socks and gloves). +e total clothinginsulation (IT � Icl � iIcli) (where Icli is the insulation ofeach fabric item) for a person in summer and winter is givenin Table 1. So, the clothing efficiency factor Fcl (dimen-sionless) is provided by [4, 7, 8]

Fcl �Ia

Icl + Ia/fcl( . (7)

Pennes [11], a famous researcher, established a bioheattransfer model in 1948 based on experimental observationincorporating the blood perfusion term for heat flow withintissue. Various methods related to the biological model usingPennes’ equation is tackled by many researchers one afteranother. Zhao et al. [12] developed a two-level finite dif-ference scheme for one-dimensional Pennes’ bioheatequation and established the stability and convergencecondition taking only one initial condition. Gurung andSaxena [13], Luitel et al. [2, 14, 15], Agrawal et al. [3],Acharya et al. [16], and Parsons [17] used Pennes’ model forsteady-state and unsteady-state temperature distributionwithin the human dermal part using finite element and finitedifference methods.

So far as the study of clothing is concerned, a number ofresearch studies [1, 4, 9, 10, 18] have turned significantstones in regards of thermal comfort in the human bodyunder the protective clothing by taking the dry and evap-orative heat loss with the significant clothing area factorwithin the different human manikins only by the experi-mental method taking the equation only in the clothing part.Havenith et al. [9] published a database of static clothingthermal insulation and vapor permeability values’ non-western ensembles for ASHARE standard 55, ISO-7730(International Organization for Standardization), and ISO-9920.

Gurung and Saxena [13] studied the transient temper-ature distribution in the human dermal part with a pro-tective layer at low atmospheric temperature, consideringthe bioheat equation to observe only the effect of thickness

and mass of the protective layer. So, the mathematical studyof the bioheat equation along with the impact of clothinginsulation, air insulation, and clothing area factor is stilllacking and is one of our research interests.

On the contrary, in the cold environment, the cold re-ceptor becomes active and vasoconstriction of blood vesseltakes place. Shivering of muscle also appears in this case. Ifthe body core temperature goes down around 30°C, then onegoes in the unconscious condition, and, if the body tem-perature goes down up to 27°C, his/her heart beating stops.Similarly, in the hot climatic condition, the hot receptorbecomes active and vasodilation takes place and sweatingoccurs. +e increment in body temperature up to 42°Ccauses hyperthermia [14–16, 19]. To overcome all theseunusual conditions, the human body needs proper well-designed protective clothing because only the internalphysiological conditions cannot cope up with such extremecold and hot climatic condition without wearing sufficientclothing in winter and putting on only light clothing insummer. So, this study, heat transfer in the human body withprotective clothing, is indispensable for proper managementof clothing to keep the body highly comfortable condition.

+e heat transfer in the human body with a protectiveclothing system is a heterogeneous phenomenon. +e an-alytical solution of such material is not an easy task. So, thestudy of the numerical approach for getting an approxi-mation solution is essential for this purpose. +e recentpaper aims to extend one-dimensional Pennes’ bioheatequation with unconditionally stable state and convergencewhich incorporates various personal and environmentalfactors on the one hand and protective clothing on the otherhand. +e graphical representation of convergence for theFD scheme will be shown for the use of the model.

3. Mathematical Formulation of the Model

+e bioheat transfer equation with the protective clothingsystem is given by

ρczT

zt� k

z2T

zx2 +

z2T

zy2 +

z2T

zz2 + wbcb Ta − T( + qm + P Tsk − Tcl( , (8)

where ρ is tissue density (kg/m3), c is tissue specific heat (J/kg°C), k is thermal conductivity (W/m°C), Wb is bloodperfusion rate (kg/m2·s), cb is blood specific heat (J/kg°C), Ta isarterial blood temperature (°C), qm is metabolic heat gen-eration (W/m3), and the symbol P in the last term is given by

P �kclAcl

Wm3 · °C

, (9)

where Acl � Abfcl, kcl is thermal conductivity of clothes(W/m·°C), Tsk is the skin temperature (°C), and (Tcl) is thecloth temperature (°C).

+e bioheat equation (8) is the extension of the equationsuggested by Pennes’ in 1948. +e left-hand side is the totalheat storage; the first, second, and third terms of right-handside are diffusion, perfusion, and metabolic heat generation.respectively. +e extra term P (Tsk − Tcl), with clothingparameters, is the heat transfer from the skin to clothinglayer.

One-dimensional unsteady-state temperature profiles inthe cylindrical shape of the human body with the clothingsystem and the bioheat equation (8) in radial direction areperformed and given below:

Mathematical Problems in Engineering 3

ρczT

zt� k

z2T

zr2 +

1r

zT

zr + wbcb Ta − T( + qm + P Tsk − Tcl( ,

(10)

where r is radial distance from the center of core towardsskin surface (m).

+e study of heat transfer in such a nonhomogeneousphenomenon, the human body, and the protective clothingsystem is really cumbrous. So, we decouple equation (10)first and then combine by using the interface conditionbetween the body and clothing part.

3.1. Heat Transfer Equation for Body. As P � 0 in the bodypart so the bioheat equation for the body part is given by

ρczTt

zt� kt

z2Tt

zr2 +

1r

zTt

zr + wbcb Ta − Tt( + qm. (11)

3.2. Heat Transfer Equation for Clothing. +e heat equationfor clothing with Wb � 0 and qm � 0 is therefore given by

ρczTclzt

� kclz2Tcl

zr2 +

1r

zTclzr

+ P Tsk − Tcl( . (12)

3.3. Boundary Conditions. +e inner boundary condition ofthe living tissue is considered uniform and taken as

at r � 0,zTt

zr� 0. (13)

+ere is continuous heat flux between clothing surfaceand atmospheric environment as the outer surface of clothesis exposed to external environment [7, 17]. In this case, heatloss from the body via clothes is caused by convection andradiation. +e Robin boundary condition due to convectioncondition is guided by Newton’s law of cooling, and the termdue to radiation is guided by the Stefan Boltzmann law and isgiven as

at r � R, − kclzTcl

zr� Fcl hc Tcl − T∞( + ∈σ T

4cl − T

4∞ ,

(14)

where Fcl is the effective clothing area factor given inequation (14), hc is the heat transfer coefficient due toconvection, ∈, the emissivity that lies between 0 to 1,σ � 5.67×10− 8 is Stefan Boltzmann constant, and T∞is theatmospheric temperature.

+e bioheat problem becomes nonlinear when thenonlinear radiation term in the boundary condition appears.In this case, it becomes difficult to formulate. To avoid suchcomplexity, we apply the simplified form of boundarycondition as

− kclzTcl

zr� Fcl hc Tcl − T∞( + ∈σ Tcl − T∞( Tcl + T∞( T

2cl − T

2∞ ,

� Fcl Tcl − T∞( hc + ∈σ Tcl + T∞( T2cl − T

2∞ ,

� Fcl hc + hr( Tcl − T∞( ,

� hA Tcl − T∞( .

(15)

3.4. Initial Condition. For the time dependent boundaryvalue problem, the initial condition is given by

T(r, 0) � T0(r), where, T � T Tt, Tcl( . (16)

4. Solution of the Model

For the solution of model (10), we perform the followingsteps:

Table 1: Total insulation of clothes in Clo unit 8, 10.

Cloth item (summer) (Icli) Clo value Cloth item (winter) (Icli) Clo value

Half shirt 0.19 Full shirt 0.28Underwear 0.04 Underwear 0.04Pants 0.11 Pants/trousers 0.24Socks 0.02 Socks 0.03Shoes 0.02 Shoes 0.04— — Suit jacket 0.48Total 0.38 Total 1.11


(1) Construction of the Finite Difference (FD) schemefor models (11) and (12)

(2) Getting solution of (11)(3) Using the solution of (11) to get the solution of (12)

and applying the interface and boundary conditions(4) Representing the combined results in graph by

computer algebraic software

4.1. Construction of FD Scheme. One-dimensional form ofcylindrical tissue for the body part is divided into N discretepoints uniquely specified by spatial indices, ri � i∆r, in theradial direction. +e discretization of circular cross-sectionof peripheral human limb, where the temperature flow inaxial direction is uniform, is shown in Figure 2.

In the time discretization, ∆t is denoted by the discretetime step size, and the total time to evaluate the temperatureis tn � nΔt.

4.2. FD Scheme for the Nodes in Body Part. We use FDscheme by writing equation (11) using implicit finite dif-ference (central difference) scheme for right-hand terms andforward difference for left-hand term.

4.2.1. FD Scheme at Boundary r� 0 (Body Core). +e cy-lindrical thickness r is measured from body core, as shown inFigure 2. At the body core, both r and the heat flux (zTt/zr)are zero; then, (1/r)(zT/zr) approaches to indeterminateform (0/0) as r⟶ 0.

+e use of Hospital rule then gives

1r

zTt

zr|r�0 �

(z/zr) zTt/zr( (z/zr)(r)

|r�0 �z2Tt

zr2|r�0. (17)

Now, equation (11) becomes

ρczTt

zt� 2D

z2Tt

zr2 + M Ta − Tt( + S, (18)

E0Tn+10 − 4αT

n+11 − F � T

n0, (19)

where D � (kt/ρc), α � (DΔ t/Δr2) , M � (wbcb/ρc),S � (qm/ρc), and F � Δt(MTa + S).

As shown in Figure 2, we solve the problem in two phasebody part and clothing part. For the body part, we take theinterior nodes from i � 1, 2, . . . N − 1, and the FD scheme ofequation (11) is given by

DiTn+1i− 1 + EiT

n+1i + BiT

n+11 − F � T

ni , (20)

where Di � (− α + (α/2i)), Ei � (1 + 2α + MΔt), andBi � (− α − (α/2i)) .

+e construction in (20) implies that FD schemes have atruncation error in the order O(Δr2 + Δt) for each interiorpoint (tn, ri), n≥ 1, 0< i

qN �T

n+1N+1 − T

n+1N

Δr/kt( + Δrc/kcl( ( . (28)

4.3.2. Interface Conditions at i � N (Left Boundary for ClothPart). +e heat flux occurs at the skin surface, and the leftmost boundary of clothes at i � N is given by

− ktzT

zr� qN. (29)

FD formulation for (29):

Tn+1N+1–T

n+1N− 1

2 zr� − K

Tn+1N+1–T

n+1N

Δr/kt( + Δrc/kcl( ( ,

Tn+1N− 1 � T

n+1N+1 − 2RcqN.

(30)

Now, equation (26) with left boundary condition can bewritten as

DNTn+1N− 1 + 2α1 − P1( T

n+1N+1 − FN � T

nN, (31)

where FN � 2DNRcqN.FD scheme for each interior gride point

(tn, ri), n≥ 1, N< i

Table 2: +ermophysical parameters related to the body part 15, 21.

Parameters Symbols Values Units+ermal conductivities kt 0.48 W/m°CBlood specific heat cb 3850 J/kg°CBlood density ρb 1000 kg/m3

Blood perfusion rate Wb 3 kg/s·m3

Metabolism qm 1085 W/m3Arterial temperature Ta 37 °C+ickness of tissue (domain) N 0.03 MTemperature at right boundary Tb 24 °C

Table 3: Physical parameters related to clothing properties.

Parameters Symbols Values Units References for the values+ermal conductivities kcl 0.305 W/m°C Abbas et al. [22]+ickness of clothes Lcl 0.0050 m Gurung and Saxena [13]Total thickness (tissue and cloth) R 0.035 m —Density of clothes ρcl 1550 kg/m3 Holmer et al. [4]Specific heat of clothes ccl 1340 J/kg°C Holmer et al. [4]Clothing insulation Icl 0.17 m2 · (°C/W) Holmer et al. [4], Havenith et al. [9]Air insulation Ia 0.0992 m2 · (°C/W) Havenith et al. [9]Area of nude body Ab 1.7 (m2) http://www.medicinenet.com

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

33.0

T (r

)

r0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

t = 0 sect = 60 sec

t = 120 sect = 180 sec

(a)

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

T (r

)

r0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

t = 0 sect = 60 sec

t = 120 sect = 180 sec

(b)

Figure 3: Continued.


http://www.medicinenet.com/

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

T (r

)

r0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

t = 0 sect = 60 sec

t = 120 sect = 180 sec

(c)

Figure 3: Temperature profile with various mesh sizes at time step Δt � 0.01 sec. (a) Mesh size Δr � 0.001m, Δt � 0.01 s. (b) Mesh sizeΔr � 0.0005m, Δt � 0.01 s. (c) Mesh size Δr � 0.00005m, Δt � 0.01 s.

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

33.0

T (r

)

r0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

t = 0 sect = 60 sec

t = 120 sect = 180 sec

(a)

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

T (r

)

r0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

t = 0 sect = 60 sec

t = 120 sect = 180 sec

(b)

Figure 4: Continued.


(24) and (35) have been performed and tabulated in (a–c) ofFigure 3 in time step Δt � 0.01 sec and (a–c) of Figure 4 intime step Δt � 0.4 sec with different mesh sizes 0.001m,0.0005m, and 0.00005m respectively.

In Figures 3(a) and 4(a), the curves are slightly deviatedwhen rN � NΔrN � 0.030m at the skin surface due to theinterface condition between two materials having nonho-mogeneous behavior.+e curves in Figures 3(b) and 4(b) areless deviated than Figures 3(a) and 4(a), while the graphs inFigure 3(c) and Figure 4(c) are smoother than in previousfour Figures. On the one hand, the comparison in graphsconcerns that the increment in numbers of grid pointsmakes the graphs smoother, more accurate, and reliable; onthe other hand, the graphs are independent of mesh sizes. Allthe graphs in Figures 3 and 4 indicate that the temperatureremains steady up to certain distance (0.02m) from the bodycore, then decreases towards the skin surface and further

then towards clothes. +e temperature profile obtained in 60seconds, in 120 seconds, and in 180 seconds are, respectively,the same no matter the mesh sizes are how small and large.+e interface temperature (skin surface temperature) ob-tained from results having different mesh sizes at time steps0.01 second and 0.4 second are, respectively, shown in Ta-bles 4 and 5 and graphically shown in Figures 5(a) and 5(b).Similarly, the temperature in Tables 6 and 7 and Figures 6(a)and 6(b) represent these results exactly same way as in theprevious case at the skin surface. As the graphs presented inFigure 7 and obtained temperature profile in Table 6 forbody part and in Table 7 for clothes coincide, respectively, itcan be concluded that the numerical solution of the model isstable and convergence with respect to the grid.

+e temperatures in Table 8 and graphs in Figure 7 attime steps Δt � 0.05 s, Δt � 0.1 s, Δt � 0.4 s, and Δt � 0.5 swith Δr � 0.0005m, respectively, coincide. So, all graphs are

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

T (r

)

r0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

t = 0 sect = 60 sec

t = 120 sect = 180 sec

(c)

Figure 4: Temperature profile with various mesh sizes at time step Δt � 0.4 sec. (a) Mesh size Δr � 0.001m, Δt � 0.4 s. (b) Mesh sizeΔr � 0.0005m, Δt � 0.4 s. (c) Mesh size Δr � 0.00005m, Δt � 0.4 s.

Table 4: Temperature profile at the interface (skin surface) when Δt � 0.01 s.

∆r (m) Temperature in 60 (s) Temperature in 120 (s) Temperature in 180 (s)0.001 36.76374598 36.30866076 35.857151290.0001 36.76037207 36.30920285 35.865976630.00005 36.7602856 36.309688591 35.86747444

Table 5: Temperature profile at the interface (skin surface) when Δt � 0.4 sec.



37.0

36.8

36.6

36.4

36.2

36.0Te

mpe

ratu

re (°

C)

0 25 50 75 100 125 150 175Time (S)

∆r = 0.001m∆r = 0.0001m∆r = 0.00005m

(a)

37.0

36.8

36.6

36.4

36.2

36.0

35.80 25 50 75 100 125 150 175

Tem

pera

ture

(°C)

Time (s)

∆r = 0.001m∆r = 0.0001m∆r = 0.00005m

(b)

Figure 5: (a) Temperature profile at the interface (skin surface) at Δt � 0.01 s. (b) Temperature profile at the interface (skin surface) atΔt � 0.4 s.

Table 6: Temperature profile when Δt � 0.01 s.

∆ r(m) Temperature in 60 (s) Temperature in 120 (s) Temperature in 180 (s)0.001 34.8052701 33.88093058 33.193366810.0001 34.80522479 33.9068697 33.250057730.00005 34.80551148 33.91018652 33.25706422

Table 7: Temperature profile when Δt � 0.4 sec.



independent of the time step sizes as well. +ese results helpto verify the stability and convergence of the FD scheme forthe model.

6. Conclusion

A one-dimensional time dependent bioheat transfer modelwith a protective clothing system has been established and

solved using the fully implicit, unconditionally stable finitedifference method. Because of the heterogeneous domainhaving two distinct physiological and physical behaviors ofbody and clothes, the differential equation models for twodistinct parts (body and clothes) are solved separately usingimplicit scheme and then combined by using interfacecondition. +e model is the extension of Pennes’ bioheatequation with nonlinear Robin’s boundary condition. +e

∆r = 0.05 sec∆r = 0.1 sec

∆t = 0.4 sec∆t = 0.5 sec

755025 100 125 1751500Time (s)

33.0

33.5

34.0

34.5

35.0

35.5

36.0

36.5

37.0

Tem

pera

ture

(°C)

Figure 7: Temperature profile when Δr � 0.0005m at different time steps.

Table 8: Temperature profile when Δr � 0.0005m at different time steps.

∆t (s) Temperature in 60 (s) Temperature in 120 (s) Temperature in 180 (s)0.05 34.80467674 33.8804583 33.192941270.1 34.803935 33.87986792 33.192409340.4 34.79948362 33.87632483 33.189217820.5 34.79799948 33.8751435 33.18815399

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

Tem

pera

ture

(°C)

0 25 50 75 100 125 150 175Time (s)

∆r = 0.001m∆r = 0.0001m∆r = 0.00005m

(a)

37.0

36.5

36.0

35.5

35.0

34.5

34.0

33.5

Tem

pera

ture

(°C)

0 25 50 75 100 125 150 175Time (s)

∆r = 0.001m∆r = 0.0001m∆r = 0.00005m

(b)

Figure 6: (a) Temperature profile at the interface (skin surface) at Δt � 0.01 s. (b) Temperature profile at the interface (skin surface) atΔt � 0.4 s.


developed model with the clothing phenomena is stable,consistent, and convergent on the basis of grid points as wellas time step sizes. +e numerical verification of the con-vergence and stability of the model has also been representedgraphically. +e result shows that suitable management ofclothing, in cold and hot climatic condition, keeps the bodyin a highly satisfactory and comfortable level. +e numericalcomputational results in this paper seem to agree with thesimilar values of clothing parameters which are experi-mentally verified in [1, 9].+e proposedmodel may be usefulfor the clothing and environmental designers as well asbiomedical researchers.

Future work should include extension of the model tohigher dimensions and use of fractional derivatives asstudied in [23–25].

Data Availability

+e data used for supporting the findings of this study areincluded within the article.

Conflicts of Interest

+e authors declare that there are no conflicts of interest.

Acknowledgments

+e first author would like to gratefully acknowledgeUniversity Grant Commission (UGC), Nepal, for providingwith PhD Fellowship award 2019, fellowship with award no.PhD-75/76-S &T-10.

References

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