Three-dimensional analysis model of electric heating fabrics considering the skin metabolism

Abstract

In low temperature environment, electric heating clothing can provide extra heat for human body through built-in heat source, so it has better thermal insulation effect. The thermal analysis is the initial step for electric heating clothing design. The current thermal analysis of electric heating textiles focuses on the fabric itself instead of the effect of skin tissue metabolism and heat production. In order to improve the accuracy of skin surface temperature prediction, the biological heat transfer need be modeled to analyze the internal temperature distribution of the heating suit system. In this paper, a three-dimensional (3D) thermal analysis model of electric heating clothing combined with human skin tissue is established. Firstly, the coupling analysis of Fourier heat conduction and Pennes biological heat transfer equation is carried out. Then the reliability of the 3D thermal analysis model is verified by finite element analysis (FEA). The results show that the fitting error between the three-dimensional model analysis data and FEA simulation data is 5°C, which proves that the model can accurately predict the system temperature. Finally, we make further research about the effects of ambient temperature, clothing layer thickness, and input power on the maximum skin surface temperature. This study provides theoretical foundation for the design of wearable thermal management fabric.

Keywords

Electric heating clothing human skin FEA Pennes bio-heat transfer

Introduction

Electric heating clothing is an important barrier for people to resist the cold in life. It provides extra heat for the human body through the built-in heat source to maintain body temperature balance. In order to ensure the thermal comfort and safety of electric heating clothing, the research on its heat transfer mechanism becomes particularly important.

In the design process, not only clothing fibers themselves that keep warm in a passive way, but the design of the heat source is also particularly important.^1,2 Nowadays, the commonly used heating wire materials include metal wire, graphene, fiber, etc.³ Carbon fiber is the most widely used because of good flexibility, ductility, and flexible structure.⁴ The electric heating clothing made of carbon fiber keeps conformal contact with skin under pressure and tension,⁵ which has great application prospects in the field of biomedicine.

The electric heating clothing is close to the skin, so it is very important for the thermal analysis of the internal heat transfer to the skin surface, because high temperature will also cause skin thermal damage and adverse reactions. The heat transfer of electric heating clothing is complex, and it is not the sum of a single heat transfer mechanism, but includes heat conduction, convection, radiation, and combined heat transfer with the surrounding environment.

Several scholars use different methods to study the heat transfer mechanism of wearable flexible electronic devices or clothing to improve the user experience and security. Sun et al.⁶ used the influence of temperature and heat flux on the cross-section of fabric system by FEA, and found that the transient heat flux loss of fabric in contact with skin is the largest, which can be used as the basis for the prediction of contact comfort. Santo et al.^7,8 used computational fluid dynamics to study the microclimate conditions of cylindrical clothing. The results show that the natural convection increases with the thickness of the microclimate ratio, and the best gap between clothing and skin is 0.1 mm. Gong et al.⁹ established a one-dimensional heat transfer model of the fabric, and studied the thermal protection performance of the fabric using ANSYS software. Siddiqui and Sun^10
–12 used FEA method to predict the effective thermal conductivity and thermal resistance of fabrics according to the actual parameters of fabrics. Fan et al.¹³ proposed a three-layer branched chain structure bionic woven fabric, and analyzed the steady-state heat transfer performance of the fabric by using FEA method. The predicted effective thermal conductivity is in good agreement with the experimental results. Su et al.¹⁴ used organic disguised materials to change the thermal properties of fabrics, designed coated fabrics with bidirectional thermal regulation function, and evaluated their thermal protection and heat dissipation function. The results showed that although the disguised materials had good thermal protection, they improved the thermal hazard. Chen^15
–17 analyzed the heat transfer of protective clothing through FPGA simulation software and temperature sensor, and verified the transfer model by comparing with the experimental data. In their research, the thermal comfort performance of protective clothing was optimized, which could protect the skin from the thermal damage caused by low-intensity thermal radiation. However, they ignored that the skin burn can also come from fabric combustion.

The above researches were limited to the fabric itself or only involved the heat transfer performance of the garment in combination with the surrounding environment while ignoring the influence of biological metabolism heat generation on the thermal comfort of the skin. Pennes biothermal equation^18
–20 has been widely used to simulate the biological heat transfer of skin tissue and study the thermal damage process of skin. Cui et al.^21
–23 had developed the flexible electronic heating equipment combined with skin metabolic heat production, which was used for one-dimensional (1D) and 3D heat conduction analysis to explore the influence of the size, load, and material of the flexible electronic heating equipment on its heat transfer. Udayraj and Wang^24,25 established a 3D heat transfer model to analyze the heat transfer of thermal protective clothing under radiation heat exposure. A computational fluid dynamics model was used to analyze the heat transfer of the air gap between clothing and skin. However, the model ignored the influence of convection and other assumptions on the air gap of cold protective clothing, the deviation of heat transfer analysis and secondary combustion prediction was quite large. Due to the possible delamination between components and skin in practical application, Li et al.^26,27 established an axisymmetric model to analyze the influence of interface thermal resistance on the integrated system of electronic devices and skin. The results of FEA and experimental measurement are in good agreement. However, in all the above researches, only the heating element itself or the coupling analysis of the heating electronic equipment and the skin tissue had been taken into account. These researchers ignored the actual environment when the electric heating fabric was worn inside.

When electric heating clothing is worn on the human body under working conditions, part of the heat generated by the source is transmitted to the environment through the fibers of the garment. The other part is transferred to the skin surface to achieve a dynamic equilibrium with the surface temperature of the body. Compared with only analyzing the heat transfer of heating clothing, it will be more complicated to consider the influence of human biological metabolic heat production on the thermal comfort of heating clothing. In the whole process of thermal analysis, heat transfer is mainly through heat conduction, heat convection, and heat radiation, accompanied by different degrees of heat loss. Given the above two situations, it is of great significance to explore the whole process of thermal analysis by establishing mathematical models.

In the field of clothing design, some scholars have also combined bio-heat transfer to analyze the safety performance of clothing, but only in special scenarios (such as fabric burning, high-intensity thermal radiation, etc.). Their research theories mostly focus on the material properties of clothing and do not involve flexible wearable electronic heating devices. For the bio-integrated application of electronic heating equipment such as electric heating clothing, the classical Fourier heat conduction ignores the influence of skin tissue heat production and blood infusion, and cannot accurately describe the heat transfer within tissues. In order to accurately analyze the bio-integrated applications of electric heating suits, Fourier heat conduction is adopted for electric heating suits, while Pennes bio-heat equation is adopted for skin tissues. There are few studies about the thermal analysis of the bio-integrated applications of heating suits and biological tissues. To accurately simulate the heat transfer mechanism between electric heating clothing and skin tissue, this paper established a three-dimensional thermal analysis model of electric heating clothing and skin tissue and verified it through finite element analysis to study the thermal performance of the integration of heating clothing and human skin. This analysis model can be extended to wearable flexible electronic heating devices with different layouts on different biological tissues. The transient heat transfer process is extremely complicated in the electrically heated clothing textiles. Heat is generated by the heat source and is transmitted through the air gap between the fabric and the skin. On the one hand, the heat is transferred to the outer surface of the fabric through the fabric system, until it reaches a thermal equilibrium by exchange heat with surrounding air on the outside. On the other hand, the heat is transferred to skin to reach a dynamic equilibrium on the skin surface with biological metabolism heat. During this process, there is varying degrees of heat loss in conducting heat, convecting heat, and radiating. Consequently, it is great significance to establish mathematical modeling to understand the process.

Although some scholars have studied biological heat transfer in clothing heat conduction, they are limited to the special situation of clothing itself (fabric burning, high-intensity heat radiation, etc.). Their researches can’t be directly applied to the thermal analysis of electrical heating clothing integrated with biological tissue. Fourier heat conduction can’t describe the heat transfer mechanism caused by biological tissue metabolism. In order to accurately describe the heat transfer mechanism in the system, a 3D thermal analysis model of electrically heated fabric on skin tissue has been modeled and simulated. This study will give a guide for thermal management of wearable electric heating fabrics in actual environments.

A 3D analysis model for heat conduction between skin and fabrics

Model establishment

The electric heating fabric is installed on the human body to form a multilayer structure of skin tissue-electric heating fabric-air gap and outer clothing. In order to simplify the model, the skin tissue is set to a single-layer structure (actually a multi-layer complex structure); the electric heating fabric is composed of clothing fibers-fusible interlining-carbon fiber heating elements. The thermal conductivity of carbon fiber heating elements is 600 W/mK, which is much larger than clothing fiber can be modeled as a heat source that is close to a rectangle and has a heat generation power density Q (W/m²).^23,28 Although the air gap exists between the electrically heated fabric and the outer clothing, the thickness is thin and the flow is stable, so the air gap heat convection and heat radiation can be ignored.²⁹ In addition, several factors affect the heat transfer of the fabric, such as: fabric, porosity, number of layers, bulkiness, etc. In order to simplify the complexity, the heat transfer model with a multilayer structure and good windproof performance fabrics is placed in an indoor environment. Based on the above-mentioned analysis and assumptions, the air gap is modeled as a uniform solid rectangle with constant thermal conductivity, ignoring the mass transfer through the fabric. The electric heating fabric is composed of clothing fiber and carbon fiber heating elements. The outer garment is a composite material composed of clothing fiber and air. The effective thermal conductivity of this layer is related to the thermal conductivity of clothing fibers, air, and its volume fraction.

In conclusion, an idealized model is shown in Figure 1. The heating plate inside the heating fabric contains wire, carbon fiber, fusible interling, and fabric. The one-dimensional model is simplified to a five-layer structure, h₁, h₂, h₃, h₄, and h₅ respectively represents skin tissue, clothing fiber, fusible interlining, carbon fiber heating element, and outer layer composition. As is shown in Figure 2, only a quarter geometry of the analytical model for the heating fabric on the skin need be analyzed by taking the advantage of the symmetry. The bottom center point of the three-dimensional model is the origin of the coordinates, the positive direction of the z-axis is from the skin layer to clothes, and the positive x-axis and y-axis directions follow the length and width directions of the carbon fiber heating element respectively. In the heat transfer analysis, the boundary conditions contain natural convection and heat radiation.

Figure 1.

Conceptual model of 3D analysis model with multi-layered structure: (a) schematic diagram of the layout and structure of heating plate inside the heating fabric and (b) schematic illustration of one-dimensional geometry of the analytic modeled heating fabric-skin system.

Figure 2.

The three-dimensional analytical model showing a quarter geometry of the heating fabric device integrated with skin tissue.

Coupling analysis of Fourier heat transfer and Pennes biological heat transfer

In the established three-dimensional model, heat transfer occurs at the interface of each layer, so the temperature difference caused by the heat transfer to each interface is used as the evaluation criterion. In the Cartesian coordinate system, h_(i_= 1–5) (Figure 1) respectively represents skin tissue, clothing fiber, fusible interlining, carbon fiber heating element, and outer layer composition, and n_(i_= 1–5) (Figure 2) represents the interface of each layer. The temperature difference of the skin tissue layer: $Δ T = T (x, y, z) - T_{h}$ . Among them, T(x,y,z) is the temperature of the coordinate point (x, y, z), and T_h is the heart temperature. In the entire heat transfer process, the heat transfer from the heating source of the electric heating fabric to the outer garment satisfies the Fourier heat conduction equation:

\nabla^{2} (Δ T_{i}) = 0 (i = 2, 3, 4, 5)

(1)

According to physiological structure, blood is evenly perfused into the entire skin tissue, and the blood temperature is usually close to the heart temperature: T_b = T_c.^22,23 The heat generated by metabolism will maintain the body temperature balance, so healthy people’s temperature can be set as constant. Therefore, the temperature change of the skin tissue layer satisfies the Pennes bioheat equation:

k_{1} \nabla^{2} (Δ T_{1}) - ω_{b} ρ_{b} c_{b} Δ T_{1} + q_{m e t} = 0

(2)

Where $ω_{b}$ is the blood infusion rate, $ρ_{b}$ is the blood density, $c_{b}$ is the blood specific heat capacity, $q_{m e t}$ is the skin metabolism, and $k_{1}$ is the thermal conductivity of skin tissue.

In the skin tissue layer (z = 0), the skin epidermal temperature is similar to body temperature, while natural convection and thermal radiation occur in the outermost layer (z = n₅). Based on the theory of natural convection and thermal radiation, the boundary temperature can be obtained:

Δ T_{1} |_{z = 0} = 0

(3)

- k_{5} \frac{\partial Δ T_{5}}{\partial z} |_{z = n_{5}} = Q_{c} + Q_{r}

(4)

Q_{c} = h_{c} (T_{c l o t h} - T_{a}), Q_{r} = ε σ (T_{c l o t h}^{4} - T_{r}^{4})

(5)

Where $Q_{c}$ is the natural convection heat, $Q_{r}$ is the radiant heat, $T_{a}$ is the ambient temperature, $T_{c l o t h}$ is the cloth temperature, $T_{r}$ is the radiation temperature, $ε$ is the blackbody emissivity, and $σ$ is the Stefan constant.

The outermost layer temperature in the established three-dimensional thermal analysis model is expressed as $T_{c l o t h} = T_{5} |_{z = n_{5}}$ . In order to facilitate the calculation, $Q_{c}$ and $Q_{r}$ have a unified formula, and $Q_{r}$ can be:

Q_{r} = h_{r} (T_{c l o t h} - T_{r})

(6)

where $h_{r}$ is the linear thermal radiation coefficient.

The linear emissivity coefficient is related to the coat temperature and the radiation temperature,^22,30 so the boundary temperature of the outermost layer can be expressed as:

- k_{5} \frac{\partial Δ T_{5}}{\partial z} |_{z = n_{5}} = [h_{0} (T_{5} + T_{c} - T_{0})] |_{z = n_{5}}

(7)

h_{o} = h_{r} + h_{c}, T_{0} = \frac{T_{r} h_{r} + T_{a} h_{c}}{h_{r} + h_{c}}

(8)

where $h_{0}$ is the total heat transfer coefficient, $h_{c}$ is the natural convection coefficient, and $T_{0}$ is the system temperature.

In formula (4), k₅ is the effective thermal conductivity of the outer garment (a mixture of clothing fiber and air), which is calculated according to the effective medium theory²⁴:

{\begin{array}{l} v_{a} + v_{f} = 1 \\ v_{a} \frac{k_{a} - k_{5}}{k_{a} + 2 k_{5}} + v_{f} \frac{k_{f} - k_{5}}{k_{f} + 2 k_{5}} = 0 \end{array}

(9)

k_{5} = \frac{1}{4} [\begin{array}{l} (3 v_{a} - 1) k_{a} + (3 v_{f} - 1) k_{f} + \\ \sqrt{{((3 v_{a} - 1) k_{a} + (3 v_{f} - 1) k_{f})}^{2} + 8 k_{a} k_{f}} \end{array}]

(10)

where $V_{a}$ is the air volume fraction, $V_{f}$ is the volume fraction of clothing fiber, $k_{a}$ is the thermal conductivity of air, and $k_{f}$ is the thermal conductivity of clothing fiber.

This study assumes that the surrounding is an ideal radiation environment, and the radiation temperature is close to the ambient temperature.^21,23 Since the temperature difference between the two is small, the ambient temperature is taken as the outermost temperature of the coat. Based on the above settings, we can get:

T_{r} = T_{a}, T_{0} = T_{a}

(11)

h_{r} = 4 ε σ T_{a}^{3}

(12)

The temperature transfer is continuous at the interface (z = n₂) of the clothing fiber/adhesive interlining, so the heat flow at the interface meets the surface heat source conditions of the carbon fiber heating element, and we can get

{\begin{array}{l} Δ T_{2} |_{z = n_{2}^{-}} = Δ T_{3} |_{z = n_{2}^{+}} \\ - k_{3} \frac{\partial Δ T_{3}}{\partial z} |_{z = n_{2}^{+}} + k_{2} \frac{\partial Δ T_{2}}{\partial z} |_{z = n_{2}^{-}} = {\begin{array}{l} \frac{P}{4 a b}, (x, y) \in Ω \\ 0, (x, y) \notin Ω \end{array} \end{array}

(13)

Among them, $Ω = {(x, y) | 0 \leq x \leq a, 0 \leq y \leq b}$ represents the area range of the carbon fiber heating element at the interface of the clothing fiber/adhesive interlining. P is the input power of the carbon fiber heating element. Using the same expression, the temperature difference expression of each interface can be obtained as:

{\begin{array}{l} Δ T_{i} |_{z = n_{i}^{-}} = Δ T_{i + 1} |_{z = n_{i}^{+}} \\ - k_{i} \frac{\partial Δ T_{i}}{\partial z} |_{z = n_{i}^{+}} + k_{i + 1} \frac{\partial Δ T_{i + 1}}{\partial z} |_{z = n_{i}^{+}} = 0 (i = 1, 3, 4) \end{array}

(14)

According to the above, the skin metabolism heat production and electric heating heat production are analyzed separately. The linear superposition method has been used in the biological heat transfer expression (2), and the boundary constraints and carbon fiber heating elements (7), (8), and (13). According to the actual situation, the temperature difference ∆T at each interface needs to be solved by the following two actual conditions. One is that the carbon fiber heating element is not working and meets the non-uniform control equation. In this case, the calculated temperature difference corresponds to the heat transfer caused by metabolic heat production, natural convection, and thermal radiation. The second is that the carbon fiber heating element is in normal operation and satisfies the uniform control equation. The temperature difference calculated in this case corresponds to the heat transfer caused by the input power of the carbon fiber heating element, which includes metabolic heat production, natural convection, and heat radiation. The two cases are analyzed separately.

When the carbon fiber heating element is not working

According to the above conditions, skin tissue metabolism, heat production, natural convection, and heat radiation are constant, so it can be used as a one-dimensional heat transfer model along the thickness direction. The governing equation ∆T can be expressed as:

{\begin{array}{l} k_{1} \frac{d^{2} Δ T_{1}}{d z^{2}} - ω_{b} ρ_{b} c_{b} Δ T_{1} + q_{m e t} = 0 \\ \frac{d^{2} Δ T_{i}}{d z^{2}} = 0 (i = 2, 3, 4, 5) \end{array}

(15)

At the n = n_i_{(i = 1,3,4)} interface, the temperature transfer continuity and boundary conditions remain unchanged, while in the n = n₂ layer, the temperature change can be expressed as:

{\begin{array}{l} Δ T_{2} |_{z = n_{2}^{-}} = Δ T_{3} |_{z = n_{2}^{+}} \\ - k_{3} \frac{\partial Δ T_{3}}{\partial z} |_{z = n_{2}^{+}} + k_{2} \frac{\partial Δ T_{2}}{\partial z} |_{z = n_{2}^{_}} = 0 \end{array}

(16)

The expression of the temperature difference at each interface in equation (15) is:

{\begin{array}{l} \begin{array}{l} Δ T_{1, 1} = A_{1, 1} \exp (n_{1} \sqrt{η / k_{1}}) + B_{1, 1} \exp (- n_{1} \sqrt{η / k_{1}}) \\ + q_{m e t} / η \end{array} \\ Δ T_{i, 1} = A_{i, 1} + B_{i, 1} (i = 2, 3, 4) \end{array}

(17)

Through formulas (3), (7), (8), (14), and (16), the coefficients A_i,₁ and B_i,₁ in the formula can be obtained:

{\begin{array}{l} C = h_{0} k_{2} [\begin{array}{l} \frac{(n_{5} - n_{4})}{k_{5}} + \frac{(n_{4} - n_{3})}{k_{4}} \\ + \frac{(n_{3} - n_{2})}{k_{3}} + \frac{(n_{2} - n_{1})}{k_{2}} \end{array}] + k_{2} \\ D = \sqrt{\frac{η}{k_{1}}} . \frac{k_{1}}{k_{2}} \\ E = C D \cos h (n_{1} \sqrt{η / k_{1}}) + h_{0} \sin h (n_{1} \sqrt{η / k_{1}}) \\ F = D h_{0} [\begin{array}{l} (\cos h (n_{1} \sqrt{η / k_{1}}) - 1) \frac{q_{m e t}}{η} \\ + \cos h (n_{1} \sqrt{η / k_{1}}) (T_{c} - T_{0}) \end{array}] \end{array}

(18)

{\begin{cases} A_{1, 1} = \frac{1}{2} \frac{(- C D + h_{0}) e^{- n_{1} \sqrt{η / k_{1}}} \frac{q_{m e t}}{η} - h_{o} \frac{q_{m e t}}{η} - h_{0} (T_{c} - T_{o})}{E} \\ B_{1, 1} = - \frac{1}{2} \frac{(C D + h_{o}) e^{n_{1} \sqrt{η / k_{1}}} \frac{q_{m e t}}{η} h_{0} \frac{q_{m e t}}{η} - h_{0} (T_{c} - T_{o})}{E} \\ A_{2, 1} = - \frac{F}{E} \\ B_{2, 1} = \frac{(C + h_{0} n_{1}) D (\cosh (n_{1} \sqrt{η / k_{1}}) - 1) q_{m e t} / η}{E} \\ + \frac{+ (h_{0} n_{1} D \cosh (n_{1} \sqrt{η / k_{1}}) - h_{0} \sinh (n_{1} \sqrt{η / k_{1}})) (T_{c} - T_{o})}{E} \end{cases}

(19)

Carbon fiber heating element under working conditions

The carbon fiber heating element in the electric heating fabric is embedded in the designated area of the clothing fiber/fusible interlining. The thermal analysis of the whole system belongs to the three-dimensional heat conduction problem. The governing equation ∆T is:

k_{1} \nabla^{2} (Δ T_{1}) - ω_{b} ρ_{b} c_{b} Δ T_{1} = 0

(20)

Except for the skin tissue/clothing fiber interface, the governing equations ∆T of the other interfaces are as follows: $\nabla^{2} (Δ T_{i}) = 0, (i = 2, 3, 4)$ , where the temperature change formulas of each interface are (3), (13), and (14). The outermost layer temperature of the coat remains uniform and the expression for:

- k_{5} \frac{\partial Δ T_{5}}{\partial z} |_{z = n_{5}} = h_{0} Δ T_{5} |_{z = n_{5}}

(21)

According to the Fourier cosine transform of the established spatial rectangular coordinate system along the x- and y-axis, we can get:

Δ \hat{T} (α, β, z) = \int_{0}^{+ \infty} \int_{0}^{+ \infty} Δ T (x, y, z) \cos (α x) \cos (β y) d x d y

(22)

According to the above formula, the skin tissue layer and the electric heating suit are considered separately and calculated:

{\begin{array}{l} \frac{d^{2} Δ {\hat{T}}_{1}}{d z^{2}} - ((α^{2} + β^{2}) + \frac{η}{k_{1}}) Δ {\hat{T}}_{1} = 0 \\ \frac{d^{2} Δ {\hat{T}}_{i}}{d z^{2}} - (α^{2} + β^{2}) Δ {\hat{T}}_{i} = 0, (i = 2, 3, 4) \end{array}

(23)

The boundary temperature of the outermost layer of the 3D model and the heat transfer formula of other interfaces are transformed into:

{\begin{array}{l} Δ {\hat{T}}_{1} |_{z = 0} = 0 \\ Δ {\hat{T}}_{2} |_{z = n_{2}^{-}} = Δ {\hat{T}}_{3} |_{z = n_{2}^{+}} \\ - k_{3} \frac{d Δ {\hat{T}}_{3}}{d z} |_{z = n_{2}^{+}} + k_{2} \frac{d Δ {\hat{T}}_{2}}{d z} |_{z = n_{2}^{-}} = \frac{P \sin (α a) \sin (β b)}{4 a b α β} \\ - k_{5} \frac{d Δ {\hat{T}}_{5}}{\partial z} |_{z = n_{5}} = h_{0} Δ {\hat{T}}_{5} |_{z = n_{5}} \\ Δ {\hat{T}}_{i} |_{z = n_{i}^{-}} = Δ {\hat{T}}_{i + 1} |_{z = n_{i}^{+}} \\ - k_{i + 1} \frac{d Δ {\hat{T}}_{i + 1}}{d z} |_{z = n_{i}^{+}} + k_{i} \frac{d Δ \hat{T}}{d z} = 0 (i = 1, 3, 4) \end{array}

(24)

The solution formula of the above-mentioned expression is:

{\begin{cases} Δ {\hat{T}}_{1, 2} (α, β, z) = A_{1, 2} (α, β) \exp (n_{1} \sqrt{(α^{2} + β^{2}) + η / k_{1}}) \\ + B_{1, 2} (α, β) \exp (- n_{1} \sqrt{(α^{2} + β^{2}) + η / k_{1}}) \\ Δ {\hat{T}}_{i, 2} (α, β, z) = \\ A_{i, 2} (α, β) \exp (z_{i} \sqrt{α^{2} + β^{2}}) \\ + B_{i, 2} (α, β) \exp (- z_{i} \sqrt{α^{2} + β^{2}}) \end{cases}

(25)

The above A_i,₂ and B_i,₂ can be determined by equation (24) obtained from the boundary conditions in the heat transfer process and the continuous conditions of heat transfer at each interface, and then:

{\begin{cases} μ_{1} = k_{1} / k_{2} \sqrt{(α^{2} + β^{2} + η / k_{1}) / (α^{2} + β^{2})} \\ λ_{1} = μ_{1} \sqrt{α^{2} + β^{2} + η / k_{1}} \\ λ_{i} = (n_{i} - n_{i - 1}) \sqrt{α^{2} + β^{2}}, (i = 2, 3, 4, 5) \end{cases}

(26)

{\begin{cases} G_{1} = k_{4} (k_{3} \cosh (λ_{3}) \cosh (λ_{4}) + k_{4} \sinh (λ_{3}) \sinh (λ_{4})) \\ G_{2} = k_{5} (k_{3} \sinh (λ_{3}) \cosh (λ_{4}) + k_{3} \cosh (λ_{3}) \sinh (λ_{4})) \\ H_{1} = k_{2} (n_{1} \cosh (λ_{1}) \cosh (λ_{2}) + \sinh (λ_{1}) \sinh (λ_{2})) \\ H_{2} = k_{3} (n_{1} \cosh (λ_{1}) \sinh (λ_{2}) + \sinh (λ_{1}) \cosh (λ_{2})) \\ I_{1} = k_{3} (H_{1} \cosh (λ_{3}) + H_{2} \sinh (λ_{3})) \\ I_{2} = k_{4} (H_{2} \cosh (λ_{3}) + H_{1} \sinh (λ_{3})) \\ J_{1} = k_{4} (I_{1} \cosh (λ_{4}) + I_{2} \sinh (λ_{4})) \\ J_{2} = k_{5} (I_{2} \cosh (λ_{4}) + I_{1} \sinh (λ_{4})) \end{cases}

(27)

{\begin{array}{l} \begin{array}{l} A_{1, 2} = \frac{P \sin (α a) \sin (β b)}{8 a b α β \sqrt{α^{2} + β^{2}}} \times \\ \frac{[G_{1} + G_{2} \frac{h_{0}}{\sqrt{α^{2} + β^{2}} k_{5}}] \cosh (λ_{5}) + [G_{2} + G_{1} \frac{h_{o}}{\sqrt{α^{2} + β^{2}} k_{5}}] \sinh (λ_{5})}{[J_{1} + J_{2} \frac{h_{0}}{\sqrt{α^{2} + β^{2}} k_{5}}] \cosh (λ_{5}) + [J_{2} + J_{1} \frac{h_{0}}{\sqrt{α^{2} + β^{2}} k_{5}}] \sinh (λ_{5})} \end{array} \\ B_{1, 2} = - A_{1, 2} \\ A_{2, 2} = \frac{A_{1, 2} (n_{1} \cosh (λ_{1}) + \sinh (λ_{1}))}{e^{n_{1} \sqrt{α^{2} + β^{2}}}} \\ B_{2, 2} = - \frac{A_{1, 2} (n_{1} \cosh (λ_{1}) - \sinh (λ_{1}))}{e^{- n_{1} \sqrt{α^{2} + β^{2}}}} \end{array}

(28)

The calculation formula for the temperature difference at each interface is obtained through the Fourier arc cosine pair equation (25) conversion:

\begin{array}{l} Δ T (x, y, z) \\ = \frac{4}{π^{2}} \int_{0}^{+ \infty} \int_{0}^{+ \infty} Δ \hat{T} (α, β, z) \cos (α x) \cos (β y) d α d β \end{array}

(29)

The expression formula for the temperature difference at z = n₂ is:

\begin{array}{l} Δ T_{2, 2} (x, y, z) = \frac{4}{π^{2}} \int_{0}^{+ \infty} \int_{0}^{+ \infty} [A_{2, 2} (α, β) \exp (n_{2} \sqrt{α^{2} + β^{2}})] \cos (α x) \cos (β y) d α d β \\ + \frac{4}{π^{2}} \int_{0}^{+ \infty} \int_{0}^{+ \infty} [B_{2, 2} (α, β) \exp (- n_{2} \sqrt{α^{2} + β^{2}})] \cos (α x) \cos (β y) d α d β \end{array}

(30)

In the thermal analysis, the temperature of the carbon fiber heating element and the maximum temperature of the human skin surface need to be considered. The skin layer (x, y, z) = (0, 0, n₁) is brought into the equation:

Δ T_{s k i n}^{\max} = \frac{8}{π^{2}} \int_{0}^{+ \infty} \int_{0}^{+ \infty} A_{1, 2} \sinh (λ_{1}) d α d β

(31)

The temperature outside the carbon fiber heating element/clothing fiber contact area (z = n₂) can be calculated by the formula (32):

\begin{array}{l} Δ T_{2, 2} (x, y, n_{2}) \\ = \frac{8}{π^{2}} \int_{0}^{+ \infty} \int_{0}^{+ \infty} A_{1, 2} \frac{H_{2}}{k_{3}} \cos (α x) \cos (β y) d α d β \end{array}

(32)

Carbon fiber heating elements have high thermal conductivity and thin adhesive lining. In the area {(x,y,z)|0 ⩽ x ⩽ a, 0 ⩽ y ⩽ b, z = n₂}, the temperature tends to be uniform and continuous, so heat conduction in this area can be expressed by the temperature increase at the interface of the carbon fiber heating element and clothing fiber :

Δ T_{c a r b o n} = \frac{16}{π^{2} [2 a b + (a + b) h_{c a r b o n}]} \int_{0}^{+ \infty} \int_{0}^{+ \infty} A_{1, 2} \frac{H_{2}}{k_{3}} \frac{\sin (α a) \sin (β b)}{α β} d α d β

(33)

Analyze the above two situations together

According to the mathematical model of the carbon fiber heating element embedded in the electric heating fabric under working and non-working conditions, the linear superposition method is used to calculate the temperature calculation formula of the carbon fiber heating element/clothing fiber interface and the electric heating fabric/skin tissue contact surface:

T (x, y, n_{2}) = T_{h} + A_{2, 1} n_{2} + B_{2, 1} + Δ T_{2, 2} (x, y, n_{2})

(34)

T_{c a r b o n} = T_{h} + A_{2, 1} n_{2} + B_{2, 1} + Δ T_{c a r b o n}

(35)

T_{s k i n}^{\max} = T_{h} + A_{2, 1} n_{1} + B_{2, 1} + Δ T_{s k i n}^{\max}

(36)

Results

Simulation analysis based on ANSYS

In order to verify the validity and feasibility of the established analysis model, steady-state thermal simulation verification is performed through ANSYS. Outerwear is a mixture of clothing fiber and air, in which the air volume fraction is 75%.^31,32 According to equations (9) and (10), the thermal conductivity of the outerwear (k₅) can be calculated. The thermal conductivity and thickness of other layers are shown in Table 1.^20,33,34

Table 1.

Thermal conductivity and thickness of each material.

	Thermal conductivity (W/m/K)	Thickness (mm)
Skin tissue	0.470	4.0
Carbon fiber heating element	700	0.2
Fiber (95% cotton)	0.049	0.3
Fusible interlining	0.420	0.3
Air gap	0.023	0.1
Outer layer	0.0282	1.5

In order to ensure that the contact area of each layer is large enough to get a good convergence result, the length, width, and height are set as 14 mm × 14 mm × 6.2 mm. The size of the carbon fiber heating element embedded in the fabric is: 2a × 2b × H_carbon = 0.2 mm × 4 mm × 0.2 mm. The geometric model is divided into a mapped grid, and the grid cell size is 0.1 mm. Before the simulation, material parameter definition and boundary condition constraints are required. Under standard atmospheric pressure, the setting parameters and constant values^{21,33,35–38} are shown in Table 2. The above linear thermal radiation coefficient h_r can be calculated by formula (12),^34,39 the external environment temperature is set to T_a = 283 K, the physical properties of blood tissue under the skin tissue surface:

Table 2.

Parameters setup.

Variables	Symbols	Values
Natural convection coefficient	h _c	15 W/m²/K
Skin metabolism	q _met	368 W/m³
Ambient temperature	T _a	283 K
Blackbody emissivity	$ε$	0.95

\begin{array}{l} ρ_{b} = 1069 k g / m^{3}, c_{b} = 3659 J / (k g \cdot K),^{22, 28, 40, 41} \\ ω_{b} = 0.0005 m l / (m l \cdot s) . \end{array}

In order to understand the heat transfer state in the electric heating fabric-skin tissue, the heat source temperature inside the electric heating fabric is transferred to the skin surface and the external environment.^42,43 Taking the center point of the bottom layer as the origin of the coordinates, the horizontal and vertical coordinates are x and y. When the input power is 50 mW, the thermal map distribution along the y-axis is shown in Figure 3. The heat is transferred to the outermost boundary, causing the temperature of each part to be different, causing the surrounding cold and hot air to flow relatively, the hotter part of the airflow rises, the colder part of the airflow drops, circulating flow. The temperature is transferred down to the surface of the skin to reach thermal steady-state equilibrium.

Figure 3.

The dependence of the thermal map on the thickness and thermal conductivity of the substrate for the heating fabric-skin tissue size of 14 mm × 14 mm × 6.2 mm, and the heat generation power of 50 mW along the y-axis cross section.

Thermal analysis model and finite element data fitting

We used finite element calculation to verify the correctness of the established three-dimensional thermal analysis model, which is consistent within the allowable error range. When the input power is P = 50 mW, the temperature distribution of the finite element calculation and thermal analysis model at the interface (z = n₂) of the fusible interlining and the clothing fiber along the positive direction of the x- and y-axis varies with distance as shown in Figure 4. As shown in Figure 4(a) and (b), the analysis and prediction results are in good agreement with the finite element analysis results, and the error is about 0.5°C, thus verify the feasibility of the established three-dimensional thermal analysis model. The maximum temperature occurs inside the region of the carbon fiber heating plate and keeps as a uniform value at around 58.3°C. Outside the heating plate region, the temperature decays quickly as far from the boundary of the heating plate and then begins to flatten gradually.

Figure 4.

Temperature distribution along the positive x- and y-axis (z = n₂): (a) the temperature distribution along the x-axis and (b) the temperature distribution along the y-axis.

Discussion

In this paper, a three-dimensional thermal analysis model of metabolic heat of biological tissues and heating clothing is established based on the real environment. For the thermal analysis model of clothing and biological tissue, several scientists come from different fields have different research focuses and adopt different methods. Santos et al.⁷ established a two-dimensional model of the horizontal cylindrical structure of skin and clothing based on fluid dynamics. Shen et al.⁴⁴ simulated the thermal resistance of the fabric heat transfer and established a new heterogeneous model, the transverse and longitudinal heat transfer can be processed in parallel. Cui et al.²¹ analyzed blood infusion rate and adipose decompose biological heat by one-dimensional thermal analysis model. Compared with the above model, the three-dimensional FEA model can predict the skin surface temperature more accurately.

Moreover, the model verifies the heat transfer to the skin surface is affected by the thickness of the garment. For a planar heat source, the geometric size, input power, and ambient temperature are kept constant. The results predicted from the analytical model in formal³⁶ agree well with FEA (Figure 5), which validate the reliability of the model. The thickness of the clothing fiber layer increases from 0.1 to 0.5 mm, the maximum temperature change for heat transfer to the skin surface is from 38.4°C to 34.1°C. The thickness of fabric increases from 0.1 to 0.4 mm, the maximum skin surface temperature decreased by about 4.3°C. But the thickness exceeds 0.4 mm and the temperature is maintained at around 34°C, the skin surface temperature is no longer affected by the thickness of clothing. The maximum temperature at the heating fabric/skin interface varies with the thickness of the clothes fiber, which is very helpful to design the electric heating clothing integrated with human skin. Zhu and Lu⁴⁵ pointed out that the critical temperature of thermal pain which can be felt by human is 39°C, while Lovik et al.⁴⁶ considered 43°C as the threshold of tissue injury. By simulating a more realistic environment, the maximum temperature of the heating fabric/skin interface was respectively 38.6°C and 38.4°C in the analytical model and FEA. On the basis of previous research, more accurate and safe temperature predictions are demonstrated. It can be understood that if the clothing is thicker and the porosity is higher, the thermal performance of electric heating clothing will be reduced.

Figure 5.

Relation between the temperature and the width of fiber.

Although we verified the influence of a single variable on the distribution of skin surface temperature, we further explored the influence of different input power and ambient temperature on the distribution of skin surface temperature. As the ambient temperature changes, and the input power is 50 and 70 mW, the temperature in the skin tissue/clothing layer decrease with the x-axis as shown in Figure 6(a) and (b). When the ambient temperature is −5°C, 5°C, and 15°C,^47–49 FEA analysis is in good agreement with the analytical model, and the influence trend on the temperature is consistent. The lower the ambient temperature, the higher the input power can improve the insulation efficiency of the electrically heated fabric. Therefore, the insulation effect can be achieved in a low temperature environment by adjusting the heating power to increase the heat. For example, as the rated power is 70 mW and the ambient temperature drops from 15°C to −5°C, the maximum skin surface temperature drops by around 17.6℃ after the heat transfer reaches the steady-state equilibrium.

Figure 6.

The influence of ambient temperature and input power on the temperature of skin tissue/clothing fiber (z = n₁): (a) the distribution of skin/cloth fiber interface temperature along the x-axis under −5°C, 5°C, and 15°C with the input power 50 mW and (b) the distribution of skin/cloth fiber interface temperature along the x-axis under 5°C, 5°C, and 15°C with the input power 70 mW.

Due to the simplification of heat radiation, heat mass transfer, and air gap heat convection, a three-dimensional thermal analysis model of electric heating fabric and biological tissue is established under the premise of natural convection air and airtight clothing. The accuracy of FEA calculation is affected by the simplification of physical model, material properties, and environmental boundary conditions. The thermal characteristics and blood infusion rate of biological tissues vary from person to person, and the convection coefficient is affected by the environment. To ensure the accuracy of the above thermal prediction, the uncertainty caused by the skin thermal characteristics and blood infusion rate should be considered. Several scholars have discussed the uncertainty problem.^50–52 Here we consider conditions with uncertain values (input power, ambient temperature, and clothing thickness). Figure 6(a) and (b) presents the largest differences in the temperatures for the three conditions are approximately 9.5°C, which means that these factors are sensitive to the maximum surface temperature of the skin. In the actual environment, more factors can affect the heat preservation effect, therefore a more comprehensive and widely applicable model needs to be developed for electric heating suits.

Conclusions

In this paper, an integrated three-dimensional thermal analysis model of electric heating fabric and human skin is established, the temperature distribution of the skin surface is predicted. Under the condition of constant power and environmental temperature remain unchanged, the biological heat transfer of the skin tissue and the Fourier heat conduction in the electric heating clothes are introduced for coupling analysis, and the calculated prediction results fit well with the finite element simulation data, thus verifying the feasibility of the three-dimensional thermal analysis model. The formula for calculating skin surface temperature is obtained from the above, and the influence of clothing thickness, ambient temperature, and input power of electric heating clothing on skin surface temperature is studied. The research results can be used to quantitatively evaluate the thermal comfort of human skin, which can avoid adverse reactions such as heat damage to the skin caused by excessively high temperature of the electric heating clothes. The research results provide new ideas and methods for the design of wearable electric heating devices to reduce adverse thermal effects and power consumption in bio-integrated applications. Theoretical results discussed above also provide new ideas for the thermal design of bio-electronic devices.

Footnotes

Author contributions

Xiao Li: Investigation, Methodology, Writing – original draft, Writing – review and editing, Project administration, Funding acquisition. Bo Kuai: Investigation, Methodology, Formal analysis, Visualization, Writing – review and editing. Xikai Tu: Resources, Formal analysis, Supervision, Validation. Jiahao Tan: Investigation, Conceptualization, Methodology, Writing – original draft. Xuan Zhou: Methodology, Writing – original draft.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support from National Natural Science Foundation of China (Grant No. 52005169), the Doctoral Scientific Research Foundation of Hubei University of Technology (Grant Nos. BSQD2019007), and Key Laboratory of Modern Manufacturing Quality Engineering of Hubei Province (Grant No. KFJJ-2020007).

ORCID iD

Xiao Li

References

Thilagavathi

Muthukumar

Kannaian

(2017) Development and characterization of electric heating fabric based on silver coated nylon yarn. J Text Eng Fashion Technol 1: 224–226.

Senis

Golosnoy

Dulieu-Barton

, et al. (2019) Enhancement of the electrical and thermal properties of unidirectional carbon fibre/epoxy laminates through the addition of graphene oxide. J Mater Sci 54: 8955–8970.

Tembei

SAN

Hessein

Fath El-Bab

AMR

, et al. (2020) A low voltage, flexible, graphene-based electro-thermal heater for wearable electronics and localized heating applications. Mater Today Proc 33: 1840–1844.

Papageorgiou

Terzopoulou

Fina

, et al. (2018) Enhanced thermal and fire retardancy properties of polypropylene reinforced with a hybrid graphene/glass-fibre filler. Compos Sci Technol 156: 95–102.

Wang

Zhou

(2020) Research on the electric heating performance of heating materials for electric heating sheet. In: 2020 13th International Symposium on Computational Intelligence and Design (ISCID), Hangzhou, China, 12–13 December 2020, pp.34–37. New York, NY: IEEE.

Sun

Chen

Cheng

, et al. (2010) Study of heat transfer through layers of textiles using finite element method. Clothing Sci Technol 22: 161–173.

Santos

Oliveira

Campos

, et al. (2018) Numerical analysis of the flow and heat transfer in cylindrical clothing microclimates – influence of the microclimate thickness ratio. Int J Heat Mass Transf 117: 71–79.

Cheng

Chen

Wang

(2020) Effect of underwear on microclimate heat transfer in clothing based on computational fluid dynamics simulation. Text Res J 90: 1262–1276.

Gong

Zhang

Mao

(2010) Numerical simulation on thermal protection properties of textile materials. Dong Hua Univ 36: 115–127.

10.

Siddiqui

MOR

Sun

(2013) Finite element analysis of thermal conductivity and thermal resistance behaviour of woven fabric. Comput Mater Sci 75: 45–51.

11.

Siddiqui

MOR

Sun

(2016) Automated model generation of knitted fabric for thermal conductivity prediction using finite element analysis and its applications in composites. J Ind Text 45: 1038–1061.

12.

Siddiqui

MOR

Sun

(2018) Thermal analysis of conventional and performance plain woven fabrics by finite element method. J Ind Text 48: 685–712.

13.

Fan

Wang

Cheng

, et al. (2015) Thermal analysis of bio-mimic woven fabric based on finite element method. J Therm Anal Calorim 121: 737–742.

14.

Zhu

Tian

, et al. (2020) Intelligent bidirectional thermal regulation of phase change material incorporated in thermal protective clothing. Appl Therm Eng 174: 1359–4311.

15.

Chen

(2021) Simulation of heat transfer process of thermal protective clothing based on FPGA and sensor processing system. Microprocess Microsyst 81: 141–149.

16.

Eryuruk

Gidik

Koncar

, et al. (2021) Heat and moisture transfer properties of a firefighter clothing with a new fire-resistant underwear. J Ind Text 105: 1326–1340.

17.

Łapka

Furmański

(2020) Modeling and analysis of the influence of the protective garment movement on the skin temperature and burn degree. Fire Saf J 111: 379–385.

18.

Seffen

(2008) Biothermomechanics of skin tissues. J Mech Phys Solids 56: 1852–1884.

19.

Luo

(2019) The Dynamics and Mechanism of Human Thermal Adaptation in Building Environment: A Glimpse to Adaptive Thermal Comfort in Buildings[M]. Springer Nature.

20.

Ezzat

AlSowayan

Al-Muhiameed

ZIA

, et al. (2014) Fractional modelling of Pennes’ bio-heat transfer equation. Heat Mass Transf 50: 907–914.

21.

Cui

Xing

, et al. (2016) One-Dimensional thermal analysis of the flexible electronic devices integrated with human skin. Micromachines 7: 210–265.

22.

Cui

Xing

, et al. (2017) Thermal design of rectangular microscale inorganic light-emitting diodes. Appl Therm Eng 122: 653–660.

23.

Cui

Xing

, et al. (2018) Three-dimensional thermal analysis of rectangular micro-scale inorganic light-emitting diodes integrated with human skin. Int J Therm Sci 127: 321–328.

24.

Udayraj Wang

(2018) A three-dimensional conjugate heat transfer model for thermal protective clothing. Int J Therm Sci 130: 28–46.

25.

Gao

Zhang

Malyarchuk

, et al. (2014) Epidermal photonic devices for quantitative imaging of temperature and thermal transport characteristics of the skin. Nat Commun 5: 4938–4949.

26.

Zhang

Xing

, et al. (2018) Thermal analysis of epidermal electronic devices integrated with human skin considering the effects of interfacial thermal resistance. AIP Adv 8: 055102.

27.

Chen

Xing

, et al. (2017) Thermal management of micro-scale inorganic light-emittng diodes on an orthotropic substrate for biointegrated applications. Sci Rep 7: 38–57.

28.

Kong

Bodelot

Lebental

, et al. (2018) Novel three-dimensional carbon nanotube networks as high performance thermal interface materials. Carbon N Y 132: 359–369.

29.

Yuan

(2005) Mathematical simulation of human heat regulation system, vol. 113. Beijing, China: Beihang University Press, pp.1123–1135.

30.

Zhang

Xie

Hong

, et al. (2021) Impact of metabolism and the clothing thermal resistance on inpatient thermal comfort. Energy Built Environ 2: 223–232.

31.

Zupin Ž, Hladnik

Dimitrovski

(2012) Prediction of one-layer woven fabrics air permeability using porosity parameters. Text Res J 82: 117–128.

32.

Lian

Wang

, et al. (2020) A multifunctional wearable E-textile via integrated nanowire-coated fabrics. J Mater Chem C 8: 8399–8409.

33.

Plourde

Vallez

Nelson-Cheeseman

, et al. (2017) Transcutaneous recharge: a comparison of numerical simulation to in vivo experiments. Neuromodulation 20: 613–621.

34.

Špelić

Petrak

Mahnić Naglić

(2021) The correction of clothing insulation and ergonomic design through 3D CAD reverse engineering. Int J Occup Saf Ergon 27: 737–753.

35.

Kim

Lee

(2020) Thermal insulation property of graphene/polymer coated textile based multi-layer fabric heating element with aramid fabric. Sci Rep 10: 17586–18168.

36.

Kim

Lee

(2021) Effect of relative humidity condition on electrical heating textile coated with graphene-based on cotton fabric. Fibers Polym 22: 276–284.

37.

Carson

Lovatt

Tanner

, et al. (2005) Thermal conductivity bounds for isotropic porous materials. Int J Heat Mass Transf 48: 2150–2158.

38.

Yen

Chen

Huang

, et al. (2013) Numerical study of anisotropic thermal conductivity fabrics with heating elements. Int J Numer Methods Heat Fluid Flow 23: 750–771.

39.

Yang

Wang

Song

, et al. (2020) Effects of clothing size and air ventilation rate on cooling performance of air ventilation clothing in a warm condition. Int J Occup Saf Ergon 114: 1–10.

40.

Webb

Bonifas

Behnaz

, et al. (2013) Ultrathin conformal devices for precise and continuous thermal characterization of human skin. Nat Mater 12: 938–944.

41.

Kadoya

Matsunaga

Nagashima

(1985) Viscosity and thermal conductivity of dry air in the gaseous phase. J Phys Chem Ref Data 14: 947–970.

42.

Silva

Freitas

Andrade

, et al. (2020) Computational modelling of the bio-heat transfer process in human skin subjected to direct heating and/or cooling sources: a systematic review. Ann Biomed Eng 48: 1616–1639.

43.

Lou

Shou

Park

, et al. (2020) Thermoelectric air conditioning undergarment for personal thermal management and HVAC energy saving. Energy Build 226: 378–389.

44.

Shen

Xie

Shi

, et al. (2019) Analysis of heat transfer characteristics in textiles and factors affecting thermal properties by modeling. Text Res J 89: 4681–4690.

45.

Zhu

(2010) A multi-scale view of skin thermal pain: from nociception to pain sensation. Philos Trans A Math Phys Eng Sci 368: 521–559.

46.

Lovik

Abraham

Sparrow

(2011) Surrogate human tissue temperatures resulting from misalignment of antenna and implant during recharging of a neuromodulation device. Neuromodulation 14: 501–511.

47.

Yutong

(2021) Influence of local electric heating on human thermal response in cold environment. Donghua Univ 198: 98–112.

48.

Park

Hwang

Lee

, et al. (2016) Impact of electrical heating on effective thermal insulation of a multi-layered winter clothing system for optimal heating efficiency. Int J Clothing Sci Technol 28: 254–264.

49.

Wang

Gao

Holmér

(2010) Effects of air velocity and clothing combination on heating efficiency of an electrically heated vest (EHV): a pilot study. J Occup Environ Hyg 7: 501–505.

50.

Stark

Romero

Gorman

, et al. (2016) Modulated-Power implantable neuromodulation devices and their impact on surrounding tissue temperatures. J Biomed Sci Eng 9: 545–562.

51.

Gasperin

Juricić

(2009) The uncertainty in burn prediction as a result of variable skin parameters: an experimental evaluation of burn-protective outfits. Burns 35: 970–982.

52.

Wang

Liu

Yang

, et al. (2017) A novel method of non-probabilistic reliability-based topology optimization corresponding to continuum structures with unknown but bounded uncertainties. Comput Methods Appl Mech Eng 326: 573–595.