Application Bioheat Equation For Heat Transfer Model Of Firefighter’s Burn Injury

Mathematical Model

Pennes (1948) introduced a modification of heat transfer in living tissue phenomena by adding metabolic heat generation and the heat exchange of thermal energy of flowing blood and the surrounding tissue. Both factors can be identified as the heat sources of heat in the heat transfer equation as the following:

$\frac{d^{\mathrm{2}}T}{{dx}^{\mathrm{2}}}+\frac{\stackrel{\dot{}}{q_m}+\stackrel{\dot{}}{q_p}}{k}=\mathrm{0}$ (1)

Where $\stackrel{\dot{}}{q_m}$ and $\stackrel{\dot{}}{q_p}$ is the metabolic and profusion heat rates. While the thermal conductivity k is assumes to be constant.

$\stackrel{\dot{}}{q_p}=w{\rho }_b{c}_b(T_a-T)$ (2)

Equation 2 represents the heat transfer rate of the blood flowing in the small capillaries. The inlet (arterial temperature) and exit temperatures of the blood are denoted as Ta and T, respectively. The rate at which the skin tissue layer gains the heat is the rate at which the blood loses the heat. The blood profusion rate is denoted as w (m³/s of the volumetric blood flow) while ${\rho }_b$ and $c_b$ are the blood density and specific heat respectively.Consider a human body that has a muscle thickness, Lm=34.2mm and a skin fat thickness, Lsf of 12.08mm (Ogaswara, 2012) as illustrated in the Figure 1 . The surface area of the skin is estimated 1.8m² (Onofrei et al., 2015). The core body temperature Tc and arterial temperature Ta are assumed to be 37ºC respectively. The metabolic heat generation rate of the person in a sedentary situation at the upper arm $\stackrel{\dot{}}{q_m}$=684W/m3(Fiala, Havenith, Bröde, Kampmann, & Jendritzky, 2012). The radiation coefficient hr is 5.49 W/m2.K. The surrounding temp $T_{\mathrm{\infty }}$ is assumed 30°C. Table 1 shows the material properties of the human’s muscle, skin and blood.

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The rate of heat transfer between the skin and the adjacent air can be described as:

$\stackrel{\dot{}}{q}=\mathrm{}\frac{T_i-T_{\mathrm{\infty }}}{R_{tot}}\mathrm{}$(3)

Where the total resistance, R _tot is

$$R_{tot}=\frac{L_{sf}}{k_{sf}A}\mathrm{}{\left(\frac{\mathrm{1}}{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$h_r$}\right.A}\right)}^{-\mathrm{1}}$$

$$=\frac{\mathrm{1}}{A}\left(\frac{L_{sf}}{k_{sf}}+\frac{\mathrm{1}}{\mathrm{}{h}_r}\right)$$

The radiation, h _r heat transfer coefficients of the air gap are 5.49 W/m².K.

The excess temperature at the boundary given by (Bergman, Incropera, DeWitt, & Lavine, 2011)are;

$\theta \left(\mathrm{0}\right)=\mathrm{}{T}_c-T_a-\frac{\stackrel{\dot{}}{q_m}}{w\mathrm{}{\rho }_b{c}_b}=\mathrm{}{\theta }_c$(4)

$\theta \left(L_m\right)=T_i-T_a-\frac{\stackrel{\dot{}}{q_m}}{w\mathrm{}{\rho }_b{c}_b}=\mathrm{}{\theta }_i$ (5)

The prescribed temperature involving two boundary conditions given by (Bergman et al., 2011)is;

$\frac{\theta }{{\theta }_c}=\mathrm{}\frac{\left({\theta }_i/{\theta }_c\right)\sinh \tilde{m}x+\sinh \tilde{m}\mathrm{}(L_m-x)}{\sinh \tilde{m}\mathrm{}{L}_m}\mathrm{}$ (6)

The heat leaving the muscle equal heat transfer through the skin/fat. The heat transfer rate at L _m is

$q_{|x=Lm}=\mathrm{}-k_{m}A\frac{dT}{dx}{|}_{x=Lm}=\mathrm{}-k_{m}A\frac{d\theta }{dx}{|}_{x=Lm}=\mathrm{}-k_{m}A\tilde{m}{\theta }_c\frac{\left(\raisebox{1ex}{${\theta }_i$}\!\left/ \!\raisebox{-1ex}{${\theta }_c$}\right.\right)\cosh \tilde{m}{L}_m-\mathrm{1}}{\sinh \tilde{m}{L}_m}$ (7)

Combining the equation 3 and 7 will solve the surface temperature at muscle T _i

$T_i=\frac{T_{\mathrm{\infty }}sinh\tilde{m}{L}_m+\mathrm{}{k}_{m}A\tilde{m}{R}_{tot}\left[{\theta }_{c\mathrm{}}+\left(T_a+\frac{\stackrel{\dot{}}{q_m}}{w{\rho }_b{c}_b}\right)\cosh \tilde{m}{L}_m\right]}{\sinh \tilde{m}{L}_m+\mathrm{}{k}_{m}A\tilde{m}{R}_{tot}cosh\tilde{m}{L}_m}$(8)

Where $\tilde{m}=\sqrt{\frac{w{\rho }_b{c}_b}{k_m}}$

The skin/fat temperature T _sf is obtained by equating the heat transfer by conduction at skin/fat layer to heat by radiation $q_{rad}$at the air layer. Which the heat transfer through muscle layer, $q_{|x=Lm}$ is equal to heat transfer through skin/fat, $q_{cond|x=Lsf}$. Thus;

$$q_{cond|x=Lsf}=q_{rad}$$

$$q_{cond|x=Lsf}=\mathrm{}{h}_r(T_{sf}-T_{\mathrm{\infty }})$$

Therefore the skin temperature is

$T_{sf}=\frac{q_{cond|x=Lsf}+h_r{T}_{\mathrm{\infty }}}{h_r}$ (9)

Finite Element Model

A simplified two-dimensional model of multi-layer clothing materials was developed using ANSYS transient thermal finite element software as shown in Figure 2 . The model also includes a human skin and air gap layers. The clothing materials consist of three layers, namely, outer shell, thermal liner and moisture barrier. The air gap was placed between the clothing and skin layers. The effects of the air gap thickness on the skin temperature, T_i was evaluated based on six different thicknesses, specifically 1 mm, 2 mm, 3 mm, 4 mm,5 mm and 6 mm. The analysis excluded the moisture effect in the clothing materials.

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Figure 3 shows the boundary condition prescribed on the model. Bioheat equation was used to obtain the initial value of the inner surface skin temperature precisely, at the E-layer. The initial inner surface temperature was fixed at 38.8ºC. Heat flux was prescribed at the outermost layer for describing the real flame, specifically at the D-layer. The effects of heat flux on the skin temperature were evaluated based on five different intensities, they were 1200 W/m²,1000 W/m², 800 W/m², 600 W/m² and 400 W/m². The analysis considered the effect of radiation heat transfer by specifying emissivity values on each A-layer, B-layer and C-layer, respectively as shown in Figure 4 . The type of material was prescribed as aramid since it has a high insulator thermal protective performance fabric commonly used during fire suppression. The outer shell is made of polyurethane coated with 100% aramid, while the thermal barrier and moisture barrier are made of 100% aramid.. The simulation was performed under a transient condition in a duration of 1000 seconds. The initial ambient temperature was specified at 22°C as prescribed by the (ISO, 2002)

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