# Bioheat Transfer Equation with Protective Layer

Authors: Kabita Luitel; Dil Bahadur Gurung; Harihar Khanal; Kedar Nath Uprety
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

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


---

## Body

## 1. Introduction

The study of thermal comfort in the complex vascular geometry of the human body with a protective clothing system is not only the subjective domain but also the physiological factors inside the body. According to the American Society of Heating, Refrigerating, and Air Conditioning Engineers (ASHRAE), thermal comfort is the state of the mind when one can feel and express the satisfaction with the thermal environment [1]. Thermoregulation is the process of controlling the internal body temperature through the hypothalamus heat production and heat loss center. The body also uses other processes of thermodynamical systems that constantly produce energy by metabolic activity together with dilating or constricting blood vessels, shivering, and sweating [2, 3].The presence of clothes, on the other hand, plays a vital role for maintaining the thermal comfort at the equilibrium condition of heat production and heat loss by the body. Clothing, the interface between the skin surface and the environment, works as an insulator and also transports the heat from the body to the outer environment around us. The key factors affecting thermal comfort are categorized as(1)
Personal factors(2)
Environmental factors


### 1.1. The Personal Factors

Metabolism, determined by age, sex, health, etc., is one of the human personal factors which makes a difference in thermal comfort. Besides, another major factor which affects thermal comfort is clothing. A significant difference in thermal comfort can be caused by small changes in clothing layers. In the winter season, wearing a sweater and socks makes better comfort, whereas in summer, wearing light clothes makes better comfort in the workplace.


### 1.2. Environmental Factors

Air temperature, moving air, radiant temperature, and relative humidity are the environmental factors that help to maintain thermal comfort by keeping the room comfortable. The radiation heat flux, on the other hand, penetrates a certain depth passing entirely through the fabric, depending on the fabric structure and radiative wavelength. In contrast, the convection portion of the heat source could reach the fabric surface only [4]. Usually, heat transfer in the body through the garment occurs from heat conduction and heat radiation; then, the temperature rises and transfers into the air gap. Suitable clothing along with these environmental factors, therefore, maintains better comfort at home and workplace (business, office, study room, etc.).


## 1.1. The Personal Factors

Metabolism, determined by age, sex, health, etc., is one of the human personal factors which makes a difference in thermal comfort. Besides, another major factor which affects thermal comfort is clothing. A significant difference in thermal comfort can be caused by small changes in clothing layers. In the winter season, wearing a sweater and socks makes better comfort, whereas in summer, wearing light clothes makes better comfort in the workplace.


## 1.2. Environmental Factors

Air temperature, moving air, radiant temperature, and relative humidity are the environmental factors that help to maintain thermal comfort by keeping the room comfortable. The radiation heat flux, on the other hand, penetrates a certain depth passing entirely through the fabric, depending on the fabric structure and radiative wavelength. In contrast, the convection portion of the heat source could reach the fabric surface only [4]. Usually, heat transfer in the body through the garment occurs from heat conduction and heat radiation; then, the temperature rises and transfers into the air gap. Suitable clothing along with these environmental factors, therefore, maintains better comfort at home and workplace (business, office, study room, etc.).


## 2. Role of Clothing

### 2.1. Thermal Resistance and Clothing Insulation

The ratio of temperature difference between two faces of material to the rate of heat flow per unit area is defined as thermal resistance. In the study of thermal insulation of clothing, thermal resistance is a very important parameter defined as the function of thickness and thermal conductivity of clothes. Thermal ResistanceRth m2⋅°C/W is given by [5](1)Rth=ΔTq=Lk,where L is the thickness of cloth (m), ∆T is the temperature difference (°C), and q is the heat flow rate W/m2. Though the heat transfer in the clothed body consists of conduction and radiation, the primary determinant of the insulation is the thickness of clothes on which the insulation is very much dependent. The limbs of human body with insulation parameters can be seen in Figure 1.Figure 1
A cylindrical model with clothing and air insulation [6, 7].Thermal insulation, together with the air gap, does not only provides comfort at the skin surface and body from a cold environment but also protects from burn injuries. While sitting by the fire, the air gap provides thermal insulation that limits heat transfer to skin and protects the skin from excessive heat.


#### 2.1.1. Clo Unit

Clo unit is the measure of thermal resistance and includes the insulation provided by any layer of trapped air between skin and the insulation value of clothing itself. 1 Clo is defined as the insulation of the clothing system that requires maintaining a sitting-resting average male comfortable in a normally ventilated room. Clo is the thermal insulation of overall clothing worn by a person. It has 0.1 m/s air velocity at air temperature 21°C and relative humidity less than 50%. Among the total heat produced by the metabolic reaction, 24% heat is lost through evaporation and respiration. As 1 met = 50 kcal/m2h, the evaporative and respiratory heat loss = 1 met × 24% = 50 × 0.24 = 12 kcal/m2h. Remaining 38 kcal/m2h is transmitted through the clothing system by conduction, convection, and radiation. The comfortable skin temperature is 33°C, so the total insulation of clothing and air layer IT=Icl+Ia is given by [8](2)I=33−2138=0.32,Ia=0.14m2⋅C°h/kcal,Icl=0.32−0.14m2⋅C°h/kcal=0.18m2⋅C°h/kcal,where Ia is the insulation of air and Icl is the insulation cloth.Since 1kcal/h = 1.163 watt (W), so 1 Clo unit is defined as(3)0.18m2⋅C°=0.181.163≡0.155m2⋅C°/W.


### 2.2. Convective and Radiative Heat Transfer

Clothing works as the mediator of heat exchange through convection as well as radiation. The standard measurement condition in ISO 9920, four different manikins in three different laboratories, were used and determined the male and female clothing thermal insulation values of 52 nonwestern clothing configuration [9]. If hcm2⋅C°/W and hrm2⋅C°/W are the heat exchange due to convection radiation, respectively, then the convective and radiative heat exchange is defined in [7, 9] by(4)hc+hr=ΔTIcl.


### 2.3. Clothing Area Factor

The dimensionless parameter, clothing area factorfcl, is the ratio between the surface area of clothed human body Abm2 and the surface area of nude human body Abm2 which is given by(5)fcl=AclAb.The prediction equation for clothing area factorfcl, based on western clothing and listed in ISO Standard 9920-2009 (ISO 2009), can also be found in various publications. The equation for fcl is given [1, 9, 10] as(6)In Clo,fcl=1+0.31Icl,Inm2⋅C°/W,fcl=1+1.97Icl.


### 2.4. Clothing Efficiency Factor

The clothing efficiency factor(dimensionless) depends upon the air insulationIclm2⋅C°/W and the thermal insulation of overall clothing, not only the particular garment but also the entire garment, including tops, bottoms, innerwear, and everything (including even socks and gloves). The total clothing insulation IT=Icl=∑iIcli (where Icli is the insulation of each fabric item) for a person in summer and winter is given in Table 1. So, the clothing efficiency factor Fcl (dimensionless) is provided by [4, 7, 8](7)Fcl=IaIcl+Ia/fcl.Table 1
Total insulation of clothes in Clo unit8, 10.
Cloth item (summer)Icli Clo valueCloth item (winter)Icli Clo valueHalf shirt0.19Full shirt0.28Underwear0.04Underwear0.04Pants0.11Pants/trousers0.24Socks0.02Socks0.03Shoes0.02Shoes0.04——Suit jacket0.48Total0.38Total1.11Pennes [11], a famous researcher, established a bioheat transfer model in 1948 based on experimental observation incorporating the blood perfusion term for heat flow within tissue. Various methods related to the biological model using Pennes’ equation is tackled by many researchers one after another. Zhao et al. [12] developed a two-level finite difference scheme for one-dimensional Pennes’ bioheat equation and established the stability and convergence condition taking only one initial condition. Gurung and Saxena [13], Luitel et al. [2, 14, 15], Agrawal et al. [3], Acharya et al. [16], and Parsons [17] used Pennes’ model for steady-state and unsteady-state temperature distribution within the human dermal part using finite element and finite difference methods.So far as the study of clothing is concerned, a number of research studies [1, 4, 9, 10, 18] have turned significant stones in regards of thermal comfort in the human body under the protective clothing by taking the dry and evaporative heat loss with the significant clothing area factor within the different human manikins only by the experimental method taking the equation only in the clothing part. Havenith et al. [9] published a database of static clothing thermal insulation and vapor permeability values’ nonwestern ensembles for ASHARE standard 55, ISO-7730 (International Organization for Standardization), and ISO-9920.Gurung and Saxena [13] studied the transient temperature distribution in the human dermal part with a protective layer at low atmospheric temperature, considering the bioheat equation to observe only the effect of thickness and mass of the protective layer. So, the mathematical study of the bioheat equation along with the impact of clothing insulation, air insulation, and clothing area factor is still lacking and is one of our research interests.On the contrary, in the cold environment, the cold receptor becomes active and vasoconstriction of blood vessel takes place. Shivering of muscle also appears in this case. If the body core temperature goes down around 30°C, then one goes in the unconscious condition, and, if the body temperature goes down up to 27°C, his/her heart beating stops. Similarly, in the hot climatic condition, the hot receptor becomes active and vasodilation takes place and sweating occurs. The increment in body temperature up to 42°C causes hyperthermia [14–16, 19]. To overcome all these unusual conditions, the human body needs proper well-designed protective clothing because only the internal physiological conditions cannot cope up with such extreme cold and hot climatic condition without wearing sufficient clothing in winter and putting on only light clothing in summer. So, this study, heat transfer in the human body with protective clothing, is indispensable for proper management of clothing to keep the body highly comfortable condition.The heat transfer in the human body with a protective clothing system is a heterogeneous phenomenon. The analytical solution of such material is not an easy task. So, the study of the numerical approach for getting an approximation solution is essential for this purpose. The recent paper aims to extend one-dimensional Pennes’ bioheat equation with unconditionally stable state and convergence which incorporates various personal and environmental factors on the one hand and protective clothing on the other hand. The graphical representation of convergence for the FD scheme will be shown for the use of the model.


## 2.1. Thermal Resistance and Clothing Insulation

The ratio of temperature difference between two faces of material to the rate of heat flow per unit area is defined as thermal resistance. In the study of thermal insulation of clothing, thermal resistance is a very important parameter defined as the function of thickness and thermal conductivity of clothes. Thermal ResistanceRth m2⋅°C/W is given by [5](1)Rth=ΔTq=Lk,where L is the thickness of cloth (m), ∆T is the temperature difference (°C), and q is the heat flow rate W/m2. Though the heat transfer in the clothed body consists of conduction and radiation, the primary determinant of the insulation is the thickness of clothes on which the insulation is very much dependent. The limbs of human body with insulation parameters can be seen in Figure 1.Figure 1
A cylindrical model with clothing and air insulation [6, 7].Thermal insulation, together with the air gap, does not only provides comfort at the skin surface and body from a cold environment but also protects from burn injuries. While sitting by the fire, the air gap provides thermal insulation that limits heat transfer to skin and protects the skin from excessive heat.


### 2.1.1. Clo Unit

Clo unit is the measure of thermal resistance and includes the insulation provided by any layer of trapped air between skin and the insulation value of clothing itself. 1 Clo is defined as the insulation of the clothing system that requires maintaining a sitting-resting average male comfortable in a normally ventilated room. Clo is the thermal insulation of overall clothing worn by a person. It has 0.1 m/s air velocity at air temperature 21°C and relative humidity less than 50%. Among the total heat produced by the metabolic reaction, 24% heat is lost through evaporation and respiration. As 1 met = 50 kcal/m2h, the evaporative and respiratory heat loss = 1 met × 24% = 50 × 0.24 = 12 kcal/m2h. Remaining 38 kcal/m2h is transmitted through the clothing system by conduction, convection, and radiation. The comfortable skin temperature is 33°C, so the total insulation of clothing and air layer IT=Icl+Ia is given by [8](2)I=33−2138=0.32,Ia=0.14m2⋅C°h/kcal,Icl=0.32−0.14m2⋅C°h/kcal=0.18m2⋅C°h/kcal,where Ia is the insulation of air and Icl is the insulation cloth.Since 1kcal/h = 1.163 watt (W), so 1 Clo unit is defined as(3)0.18m2⋅C°=0.181.163≡0.155m2⋅C°/W.


## 2.1.1. Clo Unit

Clo unit is the measure of thermal resistance and includes the insulation provided by any layer of trapped air between skin and the insulation value of clothing itself. 1 Clo is defined as the insulation of the clothing system that requires maintaining a sitting-resting average male comfortable in a normally ventilated room. Clo is the thermal insulation of overall clothing worn by a person. It has 0.1 m/s air velocity at air temperature 21°C and relative humidity less than 50%. Among the total heat produced by the metabolic reaction, 24% heat is lost through evaporation and respiration. As 1 met = 50 kcal/m2h, the evaporative and respiratory heat loss = 1 met × 24% = 50 × 0.24 = 12 kcal/m2h. Remaining 38 kcal/m2h is transmitted through the clothing system by conduction, convection, and radiation. The comfortable skin temperature is 33°C, so the total insulation of clothing and air layer IT=Icl+Ia is given by [8](2)I=33−2138=0.32,Ia=0.14m2⋅C°h/kcal,Icl=0.32−0.14m2⋅C°h/kcal=0.18m2⋅C°h/kcal,where Ia is the insulation of air and Icl is the insulation cloth.Since 1kcal/h = 1.163 watt (W), so 1 Clo unit is defined as(3)0.18m2⋅C°=0.181.163≡0.155m2⋅C°/W.


## 2.2. Convective and Radiative Heat Transfer

Clothing works as the mediator of heat exchange through convection as well as radiation. The standard measurement condition in ISO 9920, four different manikins in three different laboratories, were used and determined the male and female clothing thermal insulation values of 52 nonwestern clothing configuration [9]. If hcm2⋅C°/W and hrm2⋅C°/W are the heat exchange due to convection radiation, respectively, then the convective and radiative heat exchange is defined in [7, 9] by(4)hc+hr=ΔTIcl.


## 2.3. Clothing Area Factor

The dimensionless parameter, clothing area factorfcl, is the ratio between the surface area of clothed human body Abm2 and the surface area of nude human body Abm2 which is given by(5)fcl=AclAb.The prediction equation for clothing area factorfcl, based on western clothing and listed in ISO Standard 9920-2009 (ISO 2009), can also be found in various publications. The equation for fcl is given [1, 9, 10] as(6)In Clo,fcl=1+0.31Icl,Inm2⋅C°/W,fcl=1+1.97Icl.


## 2.4. Clothing Efficiency Factor

The clothing efficiency factor(dimensionless) depends upon the air insulationIclm2⋅C°/W and the thermal insulation of overall clothing, not only the particular garment but also the entire garment, including tops, bottoms, innerwear, and everything (including even socks and gloves). The total clothing insulation IT=Icl=∑iIcli (where Icli is the insulation of each fabric item) for a person in summer and winter is given in Table 1. So, the clothing efficiency factor Fcl (dimensionless) is provided by [4, 7, 8](7)Fcl=IaIcl+Ia/fcl.Table 1
Total insulation of clothes in Clo unit8, 10.
Cloth item (summer)Icli Clo valueCloth item (winter)Icli Clo valueHalf shirt0.19Full shirt0.28Underwear0.04Underwear0.04Pants0.11Pants/trousers0.24Socks0.02Socks0.03Shoes0.02Shoes0.04——Suit jacket0.48Total0.38Total1.11Pennes [11], a famous researcher, established a bioheat transfer model in 1948 based on experimental observation incorporating the blood perfusion term for heat flow within tissue. Various methods related to the biological model using Pennes’ equation is tackled by many researchers one after another. Zhao et al. [12] developed a two-level finite difference scheme for one-dimensional Pennes’ bioheat equation and established the stability and convergence condition taking only one initial condition. Gurung and Saxena [13], Luitel et al. [2, 14, 15], Agrawal et al. [3], Acharya et al. [16], and Parsons [17] used Pennes’ model for steady-state and unsteady-state temperature distribution within the human dermal part using finite element and finite difference methods.So far as the study of clothing is concerned, a number of research studies [1, 4, 9, 10, 18] have turned significant stones in regards of thermal comfort in the human body under the protective clothing by taking the dry and evaporative heat loss with the significant clothing area factor within the different human manikins only by the experimental method taking the equation only in the clothing part. Havenith et al. [9] published a database of static clothing thermal insulation and vapor permeability values’ nonwestern ensembles for ASHARE standard 55, ISO-7730 (International Organization for Standardization), and ISO-9920.Gurung and Saxena [13] studied the transient temperature distribution in the human dermal part with a protective layer at low atmospheric temperature, considering the bioheat equation to observe only the effect of thickness and mass of the protective layer. So, the mathematical study of the bioheat equation along with the impact of clothing insulation, air insulation, and clothing area factor is still lacking and is one of our research interests.On the contrary, in the cold environment, the cold receptor becomes active and vasoconstriction of blood vessel takes place. Shivering of muscle also appears in this case. If the body core temperature goes down around 30°C, then one goes in the unconscious condition, and, if the body temperature goes down up to 27°C, his/her heart beating stops. Similarly, in the hot climatic condition, the hot receptor becomes active and vasodilation takes place and sweating occurs. The increment in body temperature up to 42°C causes hyperthermia [14–16, 19]. To overcome all these unusual conditions, the human body needs proper well-designed protective clothing because only the internal physiological conditions cannot cope up with such extreme cold and hot climatic condition without wearing sufficient clothing in winter and putting on only light clothing in summer. So, this study, heat transfer in the human body with protective clothing, is indispensable for proper management of clothing to keep the body highly comfortable condition.The heat transfer in the human body with a protective clothing system is a heterogeneous phenomenon. The analytical solution of such material is not an easy task. So, the study of the numerical approach for getting an approximation solution is essential for this purpose. The recent paper aims to extend one-dimensional Pennes’ bioheat equation with unconditionally stable state and convergence which incorporates various personal and environmental factors on the one hand and protective clothing on the other hand. The graphical representation of convergence for the FD scheme will be shown for the use of the model.


## 3. Mathematical Formulation of the Model

The bioheat transfer equation with the protective clothing system is given by(8)ρc∂T∂t=k∂2T∂x2+∂2T∂y2+∂2T∂z2+wbcbTa−T+qm+PTsk−Tcl,where ρ is tissue density (kg/m3), c is tissue specific heat (J/kg°C), k is thermal conductivity W/m°C, Wb is blood perfusion rate (kg/m2·s), cb is blood specific heat (J/kg°C), Ta is arterial blood temperature (°C), qm is metabolic heat generation W/m3, and the symbol P in the last term is given by(9)P=kclAclWm3⋅C°,where Acl=Abfcl, kcl is thermal conductivity of clothes W/m⋅°C, Tsk is the skin temperature (°C), and Tcl is the cloth temperature (°C).The bioheat equation (8) is the extension of the equation suggested by Pennes’ in 1948. The left-hand side is the total heat storage; the first, second, and third terms of right-hand side are diffusion, perfusion, and metabolic heat generation. respectively. The extra term PTsk−Tcl, with clothing parameters, is the heat transfer from the skin to clothing layer.One-dimensional unsteady-state temperature profiles in the cylindrical shape of the human body with the clothing system and the bioheat equation (8) in radial direction are performed and given below:(10)ρc∂T∂t=k∂2T∂r2+1r∂T∂r+wbcbTa−T+qm+PTsk−Tcl,where r is radial distance from the center of core towards skin surface (m).The study of heat transfer in such a nonhomogeneous phenomenon, the human body, and the protective clothing system is really cumbrous. So, we decouple equation (10) first and then combine by using the interface condition between the body and clothing part.


### 3.1. Heat Transfer Equation for Body

AsP=0 in the body part so the bioheat equation for the body part is given by(11)ρc∂Tt∂t=kt∂2Tt∂r2+1r∂Tt∂r+wbcbTa−Tt+qm.


### 3.2. Heat Transfer Equation for Clothing

The heat equation for clothing withWb=0 and qm=0 is therefore given by(12)ρc∂Tcl∂t=kcl∂2Tcl∂r2+1r∂Tcl∂r+PTsk−Tcl.


### 3.3. Boundary Conditions

The inner boundary condition of the living tissue is considered uniform and taken as(13)atr=0,∂Tt∂r=0.There is continuous heat flux between clothing surface and atmospheric environment as the outer surface of clothes is exposed to external environment [7, 17]. In this case, heat loss from the body via clothes is caused by convection and radiation. The Robin boundary condition due to convection condition is guided by Newton’s law of cooling, and the term due to radiation is guided by the Stefan Boltzmann law and is given as(14)atr=R,−kcl∂Tcl∂r=FclhcTcl−T∞+∈σTcl4−T∞4,where Fcl is the effective clothing area factor given in equation (14), hc is the heat transfer coefficient due to convection, ∈, the emissivity that lies between 0 to 1, σ = 5.67 × 10−8 is Stefan Boltzmann constant, and T∞is the atmospheric temperature.The bioheat problem becomes nonlinear when the nonlinear radiation term in the boundary condition appears. In this case, it becomes difficult to formulate. To avoid such complexity, we apply the simplified form of boundary condition as(15)−kcl∂Tcl∂r=FclhcTcl−T∞+∈σTcl−T∞Tcl+T∞Tcl2−T∞2,=FclTcl−T∞hc+∈σTcl+T∞Tcl2−T∞2,=Fclhc+hrTcl−T∞,=hATcl−T∞.


### 3.4. Initial Condition

For the time dependent boundary value problem, the initial condition is given by(16)Tr,0=T0r,where,T=TTt,Tcl.


## 3.1. Heat Transfer Equation for Body

AsP=0 in the body part so the bioheat equation for the body part is given by(11)ρc∂Tt∂t=kt∂2Tt∂r2+1r∂Tt∂r+wbcbTa−Tt+qm.


## 3.2. Heat Transfer Equation for Clothing

The heat equation for clothing withWb=0 and qm=0 is therefore given by(12)ρc∂Tcl∂t=kcl∂2Tcl∂r2+1r∂Tcl∂r+PTsk−Tcl.


## 3.3. Boundary Conditions

The inner boundary condition of the living tissue is considered uniform and taken as(13)atr=0,∂Tt∂r=0.There is continuous heat flux between clothing surface and atmospheric environment as the outer surface of clothes is exposed to external environment [7, 17]. In this case, heat loss from the body via clothes is caused by convection and radiation. The Robin boundary condition due to convection condition is guided by Newton’s law of cooling, and the term due to radiation is guided by the Stefan Boltzmann law and is given as(14)atr=R,−kcl∂Tcl∂r=FclhcTcl−T∞+∈σTcl4−T∞4,where Fcl is the effective clothing area factor given in equation (14), hc is the heat transfer coefficient due to convection, ∈, the emissivity that lies between 0 to 1, σ = 5.67 × 10−8 is Stefan Boltzmann constant, and T∞is the atmospheric temperature.The bioheat problem becomes nonlinear when the nonlinear radiation term in the boundary condition appears. In this case, it becomes difficult to formulate. To avoid such complexity, we apply the simplified form of boundary condition as(15)−kcl∂Tcl∂r=FclhcTcl−T∞+∈σTcl−T∞Tcl+T∞Tcl2−T∞2,=FclTcl−T∞hc+∈σTcl+T∞Tcl2−T∞2,=Fclhc+hrTcl−T∞,=hATcl−T∞.


## 3.4. Initial Condition

For the time dependent boundary value problem, the initial condition is given by(16)Tr,0=T0r,where,T=TTt,Tcl.


## 4. Solution of the Model

For the solution of model (10), we perform the following steps:(1)
Construction of the Finite Difference (FD) scheme for 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 of cylindrical tissue for the body part is divided intoN discrete points uniquely specified by spatial indices, ri = i∆r, in the radial direction. The discretization of circular cross-section of peripheral human limb, where the temperature flow in axial direction is uniform, is shown in Figure 2.Figure 2
Discretization in radial direction with interface.In the time discretization, ∆t is denoted by the discrete time step size, and the total time to evaluate the temperature is tn=nΔt.


### 4.2. FD Scheme for the Nodes in Body Part

We use FD scheme by writing equation (11) using implicit finite difference (central difference) scheme for right-hand terms and forward difference for left-hand term.


#### 4.2.1. FD Scheme at Boundaryr = 0 (Body Core)

The cylindrical thicknessr is measured from body core, as shown in Figure 2. At the body core, both r and the heat flux ∂Tt/∂r are zero; then, 1/r∂T/∂r approaches to indeterminate form 0/0 as r⟶0.The use of Hospital rule then gives(17)1r∂Tt∂r|r=0=∂/∂r∂Tt/∂r∂/∂rr|r=0=∂2Tt∂r2|r=0.Now, equation (11) becomes(18)ρc∂Tt∂t=2D∂2Tt∂r2+MTa−Tt+S,(19)E0T0n+1−4αT1n+1−F=T0n,where D=kt/ρc, α=DΔt/Δr2, M=wbcb/ρc, S=qm/ρc, and F=ΔtMTa+S.As shown in Figure2, we solve the problem in two phase body part and clothing part. For the body part, we take the interior nodes from i=1,2,…N−1, and the FD scheme of equation (11) is given by(20)DiTi−1n+1+EiTin+1+BiT1n+1−F=Tin,where Di=−α+α/2i, Ei=1+2α+MΔt, and Bi=−α−α/2i.The construction in (20) implies that FD schemes have a truncation error in the order OΔr2+Δt for each interior point tn,ri,n≥1,0<i<N.Ati=N, the skin surface is the interface between body and clothes. So, the FD scheme at i=N is(21)DNTN−1n+1+ENTNn+1+BNTN+1n+1−F=TNn.


#### 4.2.2. Interface Condition ati=N (at Skin Surface)

The interface temperature between the skin surface and clothes ati=N. The right-most Dirichlet boundary at this point is(22)Tt|i=N=Tcl|i=N=Tint.FD scheme, ati=N, is(23)DNTN−1n+1+ENTNn+1+BNTintn+1−F=TNn.This construction implies that FD schemes have an error in the orderOΔr2+Δt for the gride point tn,rN, at i=N.The system of equations (19), (20), and (23) can be written in the matrix form as(24)A1Ttn+1=Ttn+B1,where Ttn=T0n,T1n,T3n,…,TNn′.A1 is the corresponding tridiagonal matrix of order N+1×N+1 and Ttn+1 and B1 are of column vectors of N+1×1. A1 matrix is diagonally dominant since the absolute value of each leading diagonal element satisfies the relation:(25)ajj≥∑i=0,i≠jNaij.


### 4.3. FD Scheme for Clothing Part

As we have mentioned above, the skin surface is the interface between the body and clothing part ati=N; the FD scheme for (12) yields(26)DNTN−1n+1+ENTNn+1+BNTN+1n+1=TNn,where EN=EN+1=EN+2=⋯=ER=1+2α+P1.


#### 4.3.1. Interface Condition ati=N (between Skin and Cloth)

Before taking the interface temperature between skin surface and clothes, it is necessary to evaluate the interface thermal conductivityK. The nonhomogeneous material such as body and clothes which consists different physiological properties, has nonuniform thermal conductivity. Thus, the proper formulation for nonuniform K is highly desirable. The interface conductivity K is to assume a linear variation of K between two points N and N+1 is given by [20](27)K=fNkt+1−fNkcl,where fN is the interpolation factor defined by fN=Δrc/Δr and ∆rc is the mesh size in protective layer.For the interface grid pointN, we consider the control volume surrounding N is filled with the uniform conductivity kt of body tissue, one around N+1 with a conductivity kcl of clothes. The good representation for a heat flux over the composite domain between N and N+1 leads to(28)qN=TN+1n+1−TNn+1Δr/kt+Δrc/kcl.


#### 4.3.2. Interface Conditions ati=N (Left Boundary for Cloth Part)

The heat flux occurs at the skin surface, and the left most boundary of clothes ati=N is given by(29)−kt∂T∂r=qN.FD formulation for (29):(30)TN+1n+1–TN−1n+12∂r=−KTN+1n+1–TNn+1Δr/kt+Δrc/kcl,TN−1n+1=TN+1n+1−2RcqN.Now, equation (26) with left boundary condition can be written as(31)DNTN−1n+1+2α1−P1TN+1n+1−FN=TNn,where FN=2DNRcqN.FD scheme for each interior gride pointtn,ri,n≥1,N<i<R in the clothing part with a truncation error of the order OΔr2+Δt is given by(32)DiTi−1n+1+EiTin+1+BiT1n+1−F=Tin,where Di=−α1+α1/2i, Ei=1+2α1+MΔt, Bi=−α1−α1/2i, and α1=DclΔt/Δrc2.


#### 4.3.3. Boundary Conditions ati=R (at the Surface of the Cloth)

The heat flux occurs at the outer surface of the right-most boundary of clothes ati=R which is given by(33)−kcl∂Tcl∂rc=hATcl−T∞,TR+1n+1=TR−1n+1−2hARcTcl−T∞.FD scheme ati=R is now given by(34)−2α1−P1TR−1n+1+ER−2BRhARcTRn+1−FR=TRn.The system of equations (31), (32), and (34) can be written in the matrix form as(35)A2Tcln+1=Tcln+B2,where Tcln=TNn,TN+1n,TN+2n,…,TR−1n,TRn′.A2 is the corresponding tridiagonal matrix of order R−N×R−N, and Tcln+1 and B2 are of column vectors of orderR−N×1.A2 is diagonally dominant matrix since the absolute value of each leading diagonal element of A2 satisfies the relation:(36)ajj≥∑i=N,i≠jRaij.Since each matricesA1 and A2 are diagonally dominant, so A1−1 and A2−1 exist and systems (24) and (35) are separately solvable.The constructed FD scheme (20) and (31) in our model has truncation error τΔr,Δt=OΔr2+Δt. So, as Δr,Δt⟶0 as τΔr,Δt⟶0 separately for body part 0≤i≤N and clothes part N≤i≤R, hence, the model is consistent.The notion of the second matrix norms of invertible matricesA12,A22 and their inverses A1−12,A2−12 together with Gregorian Theorem [12] imply the relation(37)En+12≤E02where En+1=Ttn+1−Tt∗n+1,Tcln+1−Tcl∗n+1.Tt∗n+1 and Tcl∗n+1 are small perturb in Ttn+1 and Tcln+1, respectively.Then, by Lax–Richtmyer theorem, one can claim that the model is unconditionally stable with respect to initial data. Finally, the stability and consistent imply the convergence. Hence, the model is unconditionally convergent.


## 4.1. Construction of FD Scheme

One-dimensional form of cylindrical tissue for the body part is divided intoN discrete points uniquely specified by spatial indices, ri = i∆r, in the radial direction. The discretization of circular cross-section of peripheral human limb, where the temperature flow in axial direction is uniform, is shown in Figure 2.Figure 2
Discretization in radial direction with interface.In the time discretization, ∆t is denoted by the discrete time step size, and the total time to evaluate the temperature is tn=nΔt.


## 4.2. FD Scheme for the Nodes in Body Part

We use FD scheme by writing equation (11) using implicit finite difference (central difference) scheme for right-hand terms and forward difference for left-hand term.


### 4.2.1. FD Scheme at Boundaryr = 0 (Body Core)

The cylindrical thicknessr is measured from body core, as shown in Figure 2. At the body core, both r and the heat flux ∂Tt/∂r are zero; then, 1/r∂T/∂r approaches to indeterminate form 0/0 as r⟶0.The use of Hospital rule then gives(17)1r∂Tt∂r|r=0=∂/∂r∂Tt/∂r∂/∂rr|r=0=∂2Tt∂r2|r=0.Now, equation (11) becomes(18)ρc∂Tt∂t=2D∂2Tt∂r2+MTa−Tt+S,(19)E0T0n+1−4αT1n+1−F=T0n,where D=kt/ρc, α=DΔt/Δr2, M=wbcb/ρc, S=qm/ρc, and F=ΔtMTa+S.As shown in Figure2, we solve the problem in two phase body part and clothing part. For the body part, we take the interior nodes from i=1,2,…N−1, and the FD scheme of equation (11) is given by(20)DiTi−1n+1+EiTin+1+BiT1n+1−F=Tin,where Di=−α+α/2i, Ei=1+2α+MΔt, and Bi=−α−α/2i.The construction in (20) implies that FD schemes have a truncation error in the order OΔr2+Δt for each interior point tn,ri,n≥1,0<i<N.Ati=N, the skin surface is the interface between body and clothes. So, the FD scheme at i=N is(21)DNTN−1n+1+ENTNn+1+BNTN+1n+1−F=TNn.


### 4.2.2. Interface Condition ati=N (at Skin Surface)

The interface temperature between the skin surface and clothes ati=N. The right-most Dirichlet boundary at this point is(22)Tt|i=N=Tcl|i=N=Tint.FD scheme, ati=N, is(23)DNTN−1n+1+ENTNn+1+BNTintn+1−F=TNn.This construction implies that FD schemes have an error in the orderOΔr2+Δt for the gride point tn,rN, at i=N.The system of equations (19), (20), and (23) can be written in the matrix form as(24)A1Ttn+1=Ttn+B1,where Ttn=T0n,T1n,T3n,…,TNn′.A1 is the corresponding tridiagonal matrix of order N+1×N+1 and Ttn+1 and B1 are of column vectors of N+1×1. A1 matrix is diagonally dominant since the absolute value of each leading diagonal element satisfies the relation:(25)ajj≥∑i=0,i≠jNaij.


## 4.2.1. FD Scheme at Boundaryr = 0 (Body Core)

The cylindrical thicknessr is measured from body core, as shown in Figure 2. At the body core, both r and the heat flux ∂Tt/∂r are zero; then, 1/r∂T/∂r approaches to indeterminate form 0/0 as r⟶0.The use of Hospital rule then gives(17)1r∂Tt∂r|r=0=∂/∂r∂Tt/∂r∂/∂rr|r=0=∂2Tt∂r2|r=0.Now, equation (11) becomes(18)ρc∂Tt∂t=2D∂2Tt∂r2+MTa−Tt+S,(19)E0T0n+1−4αT1n+1−F=T0n,where D=kt/ρc, α=DΔt/Δr2, M=wbcb/ρc, S=qm/ρc, and F=ΔtMTa+S.As shown in Figure2, we solve the problem in two phase body part and clothing part. For the body part, we take the interior nodes from i=1,2,…N−1, and the FD scheme of equation (11) is given by(20)DiTi−1n+1+EiTin+1+BiT1n+1−F=Tin,where Di=−α+α/2i, Ei=1+2α+MΔt, and Bi=−α−α/2i.The construction in (20) implies that FD schemes have a truncation error in the order OΔr2+Δt for each interior point tn,ri,n≥1,0<i<N.Ati=N, the skin surface is the interface between body and clothes. So, the FD scheme at i=N is(21)DNTN−1n+1+ENTNn+1+BNTN+1n+1−F=TNn.


## 4.2.2. Interface Condition ati=N (at Skin Surface)

The interface temperature between the skin surface and clothes ati=N. The right-most Dirichlet boundary at this point is(22)Tt|i=N=Tcl|i=N=Tint.FD scheme, ati=N, is(23)DNTN−1n+1+ENTNn+1+BNTintn+1−F=TNn.This construction implies that FD schemes have an error in the orderOΔr2+Δt for the gride point tn,rN, at i=N.The system of equations (19), (20), and (23) can be written in the matrix form as(24)A1Ttn+1=Ttn+B1,where Ttn=T0n,T1n,T3n,…,TNn′.A1 is the corresponding tridiagonal matrix of order N+1×N+1 and Ttn+1 and B1 are of column vectors of N+1×1. A1 matrix is diagonally dominant since the absolute value of each leading diagonal element satisfies the relation:(25)ajj≥∑i=0,i≠jNaij.


## 4.3. FD Scheme for Clothing Part

As we have mentioned above, the skin surface is the interface between the body and clothing part ati=N; the FD scheme for (12) yields(26)DNTN−1n+1+ENTNn+1+BNTN+1n+1=TNn,where EN=EN+1=EN+2=⋯=ER=1+2α+P1.


### 4.3.1. Interface Condition ati=N (between Skin and Cloth)

Before taking the interface temperature between skin surface and clothes, it is necessary to evaluate the interface thermal conductivityK. The nonhomogeneous material such as body and clothes which consists different physiological properties, has nonuniform thermal conductivity. Thus, the proper formulation for nonuniform K is highly desirable. The interface conductivity K is to assume a linear variation of K between two points N and N+1 is given by [20](27)K=fNkt+1−fNkcl,where fN is the interpolation factor defined by fN=Δrc/Δr and ∆rc is the mesh size in protective layer.For the interface grid pointN, we consider the control volume surrounding N is filled with the uniform conductivity kt of body tissue, one around N+1 with a conductivity kcl of clothes. The good representation for a heat flux over the composite domain between N and N+1 leads to(28)qN=TN+1n+1−TNn+1Δr/kt+Δrc/kcl.


### 4.3.2. Interface Conditions ati=N (Left Boundary for Cloth Part)

The heat flux occurs at the skin surface, and the left most boundary of clothes ati=N is given by(29)−kt∂T∂r=qN.FD formulation for (29):(30)TN+1n+1–TN−1n+12∂r=−KTN+1n+1–TNn+1Δr/kt+Δrc/kcl,TN−1n+1=TN+1n+1−2RcqN.Now, equation (26) with left boundary condition can be written as(31)DNTN−1n+1+2α1−P1TN+1n+1−FN=TNn,where FN=2DNRcqN.FD scheme for each interior gride pointtn,ri,n≥1,N<i<R in the clothing part with a truncation error of the order OΔr2+Δt is given by(32)DiTi−1n+1+EiTin+1+BiT1n+1−F=Tin,where Di=−α1+α1/2i, Ei=1+2α1+MΔt, Bi=−α1−α1/2i, and α1=DclΔt/Δrc2.


### 4.3.3. Boundary Conditions ati=R (at the Surface of the Cloth)

The heat flux occurs at the outer surface of the right-most boundary of clothes ati=R which is given by(33)−kcl∂Tcl∂rc=hATcl−T∞,TR+1n+1=TR−1n+1−2hARcTcl−T∞.FD scheme ati=R is now given by(34)−2α1−P1TR−1n+1+ER−2BRhARcTRn+1−FR=TRn.The system of equations (31), (32), and (34) can be written in the matrix form as(35)A2Tcln+1=Tcln+B2,where Tcln=TNn,TN+1n,TN+2n,…,TR−1n,TRn′.A2 is the corresponding tridiagonal matrix of order R−N×R−N, and Tcln+1 and B2 are of column vectors of orderR−N×1.A2 is diagonally dominant matrix since the absolute value of each leading diagonal element of A2 satisfies the relation:(36)ajj≥∑i=N,i≠jRaij.Since each matricesA1 and A2 are diagonally dominant, so A1−1 and A2−1 exist and systems (24) and (35) are separately solvable.The constructed FD scheme (20) and (31) in our model has truncation error τΔr,Δt=OΔr2+Δt. So, as Δr,Δt⟶0 as τΔr,Δt⟶0 separately for body part 0≤i≤N and clothes part N≤i≤R, hence, the model is consistent.The notion of the second matrix norms of invertible matricesA12,A22 and their inverses A1−12,A2−12 together with Gregorian Theorem [12] imply the relation(37)En+12≤E02where En+1=Ttn+1−Tt∗n+1,Tcln+1−Tcl∗n+1.Tt∗n+1 and Tcl∗n+1 are small perturb in Ttn+1 and Tcln+1, respectively.Then, by Lax–Richtmyer theorem, one can claim that the model is unconditionally stable with respect to initial data. Finally, the stability and consistent imply the convergence. Hence, the model is unconditionally convergent.


## 4.3.1. Interface Condition ati=N (between Skin and Cloth)

Before taking the interface temperature between skin surface and clothes, it is necessary to evaluate the interface thermal conductivityK. The nonhomogeneous material such as body and clothes which consists different physiological properties, has nonuniform thermal conductivity. Thus, the proper formulation for nonuniform K is highly desirable. The interface conductivity K is to assume a linear variation of K between two points N and N+1 is given by [20](27)K=fNkt+1−fNkcl,where fN is the interpolation factor defined by fN=Δrc/Δr and ∆rc is the mesh size in protective layer.For the interface grid pointN, we consider the control volume surrounding N is filled with the uniform conductivity kt of body tissue, one around N+1 with a conductivity kcl of clothes. The good representation for a heat flux over the composite domain between N and N+1 leads to(28)qN=TN+1n+1−TNn+1Δr/kt+Δrc/kcl.


## 4.3.2. Interface Conditions ati=N (Left Boundary for Cloth Part)

The heat flux occurs at the skin surface, and the left most boundary of clothes ati=N is given by(29)−kt∂T∂r=qN.FD formulation for (29):(30)TN+1n+1–TN−1n+12∂r=−KTN+1n+1–TNn+1Δr/kt+Δrc/kcl,TN−1n+1=TN+1n+1−2RcqN.Now, equation (26) with left boundary condition can be written as(31)DNTN−1n+1+2α1−P1TN+1n+1−FN=TNn,where FN=2DNRcqN.FD scheme for each interior gride pointtn,ri,n≥1,N<i<R in the clothing part with a truncation error of the order OΔr2+Δt is given by(32)DiTi−1n+1+EiTin+1+BiT1n+1−F=Tin,where Di=−α1+α1/2i, Ei=1+2α1+MΔt, Bi=−α1−α1/2i, and α1=DclΔt/Δrc2.


## 4.3.3. Boundary Conditions ati=R (at the Surface of the Cloth)

The heat flux occurs at the outer surface of the right-most boundary of clothes ati=R which is given by(33)−kcl∂Tcl∂rc=hATcl−T∞,TR+1n+1=TR−1n+1−2hARcTcl−T∞.FD scheme ati=R is now given by(34)−2α1−P1TR−1n+1+ER−2BRhARcTRn+1−FR=TRn.The system of equations (31), (32), and (34) can be written in the matrix form as(35)A2Tcln+1=Tcln+B2,where Tcln=TNn,TN+1n,TN+2n,…,TR−1n,TRn′.A2 is the corresponding tridiagonal matrix of order R−N×R−N, and Tcln+1 and B2 are of column vectors of orderR−N×1.A2 is diagonally dominant matrix since the absolute value of each leading diagonal element of A2 satisfies the relation:(36)ajj≥∑i=N,i≠jRaij.Since each matricesA1 and A2 are diagonally dominant, so A1−1 and A2−1 exist and systems (24) and (35) are separately solvable.The constructed FD scheme (20) and (31) in our model has truncation error τΔr,Δt=OΔr2+Δt. So, as Δr,Δt⟶0 as τΔr,Δt⟶0 separately for body part 0≤i≤N and clothes part N≤i≤R, hence, the model is consistent.The notion of the second matrix norms of invertible matricesA12,A22 and their inverses A1−12,A2−12 together with Gregorian Theorem [12] imply the relation(37)En+12≤E02where En+1=Ttn+1−Tt∗n+1,Tcln+1−Tcl∗n+1.Tt∗n+1 and Tcl∗n+1 are small perturb in Ttn+1 and Tcln+1, respectively.Then, by Lax–Richtmyer theorem, one can claim that the model is unconditionally stable with respect to initial data. Finally, the stability and consistent imply the convergence. Hence, the model is unconditionally convergent.


## 5. Numerical Results and Discussion

The numerical solution of heat transfer in human body with a protective clothing layer is obtained from the bioheat equation (10) by decoupling it into equations (11) and (12) and applying finite difference scheme separately. The temperature obtained from the system of (11) (skin surface) is used in equation (12) to calculate the temperature for clothes layer on the body. For the numerical experiment, the following parametric values in Tables 2 and 3 are chosen.Table 2
Thermophysical parameters related to the body part15, 21.
ParametersSymbolsValuesUnitsThermal conductivitieskt0.48W/m°CBlood specific heatcb3850J/kg°CBlood densityρb1000kg/m3Blood perfusion rateWb3kg/s·m3Metabolismqm1085W/m3Arterial temperatureTa37°CThickness of tissue (domain)N0.03MTemperature at right boundaryTb24°CTable 3
Physical parameters related to clothing properties.
ParametersSymbolsValuesUnitsReferences for the valuesThermal conductivitieskcl0.305W/mC°Abbas et al. [22]Thickness of clothesLcl0.0050mGurung and Saxena [13]Total thickness (tissue and cloth)R0.035m—Density of clothesρcl1550kg/m3Holmer et al. [4]Specific heat of clothesccl1340J/kg°CHolmer et al. [4]Clothing insulationIcl0.17m2⋅C°/WHolmer et al. [4], Havenith et al. [9]Air insulationIa0.0992m2⋅C°/WHavenith et al. [9]Area of nude bodyAb1.7(m2)http://www.medicinenet.com


### 5.1. Graphical Representation

The effects of different parameters mentioned in Tables1–3 have been investigated for heat transfer in a cylindrical-shaped clothed human body. Different mesh sizes are taken to demonstrate the validity and applicability of the developed numerical FD schemes (20) and (32). The tests of the combined solution of systems (24) and (35) have been performed and tabulated in (a–c) of Figure 3 in time step Δt=0.01sec and (a–c) of Figure 4 in time step Δt=0.4sec with different mesh sizes 0.001 m, 0.0005 m, and 0.00005 m respectively.Figure 3
Temperature profile with various mesh sizes at time stepΔt=0.01sec. (a) Mesh size Δr=0.001m, Δt=0.01s. (b) Mesh size Δr=0.0005m, Δt=0.01s. (c) Mesh size Δr=0.00005m, Δt=0.01s.
(a)(b)(c)Figure 4
Temperature profile with various mesh sizes at time stepΔt=0.4sec. (a) Mesh size Δr=0.001m, Δt=0.4s. (b) Mesh size Δr=0.0005m, Δt=0.4s. (c) Mesh size Δr=0.00005m, Δt=0.4s.
(a)(b)(c)In Figures3(a) and 4(a), the curves are slightly deviated when rN=NΔrN=0.030m at the skin surface due to the interface condition between two materials having nonhomogeneous behavior. The curves in Figures 3(b) and 4(b) are less deviated than Figures 3(a) and 4(a), while the graphs in Figure 3(c) and Figure 4(c) are smoother than in previous four Figures. On the one hand, the comparison in graphs concerns that the increment in numbers of grid points makes the graphs smoother, more accurate, and reliable; on the other hand, the graphs are independent of mesh sizes. All the graphs in Figures 3 and 4 indicate that the temperature remains steady up to certain distance (0.02 m) from the body core, then decreases towards the skin surface and further then towards clothes. The temperature profile obtained in 60 seconds, in 120 seconds, and in 180 seconds are, respectively, the same no matter the mesh sizes are how small and large. The interface temperature (skin surface temperature) obtained from results having different mesh sizes at time steps 0.01 second and 0.4 second are, respectively, shown in Tables 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 the previous case at the skin surface. As the graphs presented in Figure 7 and obtained temperature profile in Table 6 for body part and in Table 7 for clothes coincide, respectively, it can be concluded that the numerical solution of the model is stable and convergence with respect to the grid.Table 4
Temperature profile at the interface (skin surface) whenΔt=0.01s.
∆r (m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00136.7637459836.3086607635.857151290.000136.7603720736.3092028535.865976630.0000536.760285636.30968859135.86747444Table 5
Temperature profile at the interface (skin surface) whenΔt=0.4sec.
∆r (m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00136.7582983836.3075262435.855708260.000136.7603720736.3092028535.865976630.0000536.758280936.3080991635.866122888Figure 5
(a) Temperature profile at the interface (skin surface) atΔt=0.01s. (b) Temperature profile at the interface (skin surface) at Δt=0.4s.
(a)(b)Table 6
Temperature profile whenΔt=0.01s.
∆ r(m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00134.805270133.8809305833.193366810.000134.8052247933.906869733.250057730.0000534.8055114833.9101865233.25706422Table 7
Temperature profile whenΔt=0.4sec.
∆r (m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00134.7994836233.8763248333.189217820.000134.8052247933.906869733.250057730.0000534.7999476733.9058485933.25312605Figure 6
(a) Temperature profile at the interface (skin surface) atΔt=0.01s. (b) Temperature profile at the interface (skin surface) at Δt=0.4s.
(a)(b)Figure 7
Temperature profile whenΔr=0.0005m at different time steps.The temperatures in Table8 and graphs in Figure 7 at time steps Δt=0.05s, Δt=0.1s, Δt=0.4s, and Δt=0.5s with Δr=0.0005m, respectively, coincide. So, all graphs are independent of the time step sizes as well. These results help to verify the stability and convergence of the FD scheme for the model.Table 8
Temperature profile whenΔr=0.0005m at different time steps.
∆t (s)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.0534.8046767433.880458333.192941270.134.80393533.8798679233.192409340.434.7994836233.8763248333.189217820.534.7979994833.875143533.18815399


## 5.1. Graphical Representation

The effects of different parameters mentioned in Tables1–3 have been investigated for heat transfer in a cylindrical-shaped clothed human body. Different mesh sizes are taken to demonstrate the validity and applicability of the developed numerical FD schemes (20) and (32). The tests of the combined solution of systems (24) and (35) have been performed and tabulated in (a–c) of Figure 3 in time step Δt=0.01sec and (a–c) of Figure 4 in time step Δt=0.4sec with different mesh sizes 0.001 m, 0.0005 m, and 0.00005 m respectively.Figure 3
Temperature profile with various mesh sizes at time stepΔt=0.01sec. (a) Mesh size Δr=0.001m, Δt=0.01s. (b) Mesh size Δr=0.0005m, Δt=0.01s. (c) Mesh size Δr=0.00005m, Δt=0.01s.
(a)(b)(c)Figure 4
Temperature profile with various mesh sizes at time stepΔt=0.4sec. (a) Mesh size Δr=0.001m, Δt=0.4s. (b) Mesh size Δr=0.0005m, Δt=0.4s. (c) Mesh size Δr=0.00005m, Δt=0.4s.
(a)(b)(c)In Figures3(a) and 4(a), the curves are slightly deviated when rN=NΔrN=0.030m at the skin surface due to the interface condition between two materials having nonhomogeneous behavior. The curves in Figures 3(b) and 4(b) are less deviated than Figures 3(a) and 4(a), while the graphs in Figure 3(c) and Figure 4(c) are smoother than in previous four Figures. On the one hand, the comparison in graphs concerns that the increment in numbers of grid points makes the graphs smoother, more accurate, and reliable; on the other hand, the graphs are independent of mesh sizes. All the graphs in Figures 3 and 4 indicate that the temperature remains steady up to certain distance (0.02 m) from the body core, then decreases towards the skin surface and further then towards clothes. The temperature profile obtained in 60 seconds, in 120 seconds, and in 180 seconds are, respectively, the same no matter the mesh sizes are how small and large. The interface temperature (skin surface temperature) obtained from results having different mesh sizes at time steps 0.01 second and 0.4 second are, respectively, shown in Tables 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 the previous case at the skin surface. As the graphs presented in Figure 7 and obtained temperature profile in Table 6 for body part and in Table 7 for clothes coincide, respectively, it can be concluded that the numerical solution of the model is stable and convergence with respect to the grid.Table 4
Temperature profile at the interface (skin surface) whenΔt=0.01s.
∆r (m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00136.7637459836.3086607635.857151290.000136.7603720736.3092028535.865976630.0000536.760285636.30968859135.86747444Table 5
Temperature profile at the interface (skin surface) whenΔt=0.4sec.
∆r (m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00136.7582983836.3075262435.855708260.000136.7603720736.3092028535.865976630.0000536.758280936.3080991635.866122888Figure 5
(a) Temperature profile at the interface (skin surface) atΔt=0.01s. (b) Temperature profile at the interface (skin surface) at Δt=0.4s.
(a)(b)Table 6
Temperature profile whenΔt=0.01s.
∆ r(m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00134.805270133.8809305833.193366810.000134.8052247933.906869733.250057730.0000534.8055114833.9101865233.25706422Table 7
Temperature profile whenΔt=0.4sec.
∆r (m)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.00134.7994836233.8763248333.189217820.000134.8052247933.906869733.250057730.0000534.7999476733.9058485933.25312605Figure 6
(a) Temperature profile at the interface (skin surface) atΔt=0.01s. (b) Temperature profile at the interface (skin surface) at Δt=0.4s.
(a)(b)Figure 7
Temperature profile whenΔr=0.0005m at different time steps.The temperatures in Table8 and graphs in Figure 7 at time steps Δt=0.05s, Δt=0.1s, Δt=0.4s, and Δt=0.5s with Δr=0.0005m, respectively, coincide. So, all graphs are independent of the time step sizes as well. These results help to verify the stability and convergence of the FD scheme for the model.Table 8
Temperature profile whenΔr=0.0005m at different time steps.
∆t (s)Temperature in 60 (s)Temperature in 120 (s)Temperature in 180 (s)0.0534.8046767433.880458333.192941270.134.80393533.8798679233.192409340.434.7994836233.8763248333.189217820.534.7979994833.875143533.18815399


## 6. Conclusion

A one-dimensional time dependent bioheat transfer model with a protective clothing system has been established and solved using the fully implicit, unconditionally stable finite difference method. Because of the heterogeneous domain having two distinct physiological and physical behaviors of body and clothes, the differential equation models for two distinct parts (body and clothes) are solved separately using implicit scheme and then combined by using interface condition. The model is the extension of Pennes’ bioheat equation with nonlinear Robin’s boundary condition. The developed model with the clothing phenomena is stable, consistent, and convergent on the basis of grid points as well as time step sizes. The numerical verification of the convergence and stability of the model has also been represented graphically. The result shows that suitable management of clothing, in cold and hot climatic condition, keeps the body in a highly satisfactory and comfortable level. The numerical computational results in this paper seem to agree with the similar values of clothing parameters which are experimentally verified in [1, 9]. The proposed model may be useful for the clothing and environmental designers as well as biomedical researchers.Future work should include extension of the model to higher dimensions and use of fractional derivatives as studied in [23–25].



---

*Source: 6639550-2021-01-25.xml*