The image presents a detailed explanation of a thermal model for the human head, simplified as a hemisphere with a radius of \( r_0 = 90 \, \text{mm} \). The head is assumed to have a single layer of brain tissue characterized by a thermal conductivity \( k_t = 0.5 \, \text{W/m°C} \). The hemisphere's bottom is insulated, while the surface is exposed to a convective environment with \( h = 8 \, \text{W/m}^2\text{K} \) and ambient temperature \( T_a = 25^\circ \text{C} \). ### Situations Considered:1. **Without Blood Perfusion**: - The volumetric metabolic heat generation \( q_{\text{m}} = 10000 \, \text{W/m}^3 \). - Governing equation: \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dT_f}{dr} \right) + \frac{\omega \rho_p c_b (T_a - T_f) + q_{\text{m}}}{k} = 0 \] - Boundary conditions are: - At \( r = 0 \): \(\frac{dT_f}{dr} = 0\) - At \( r = r_0 \): \(-k \frac{dT_f}{dr} = h(T_f - T_a)\)2. **With Uniform Blood Perfusion (\(\omega = 0.01 \, \text{s}^{-1}\))**: - Using the W-J approach, modify: - \( k_t \to k_{\text{eff}} = 3 \, k_t \)3. **Considering Local Blood Perfusion using the Pennes Model**: - Perfusion parameters: - Density \( \rho_p = 1050 \, \text{kg/m}^3 \), - Specific heat \( c_b = 3800 \, \text{J/kg°C} \), - Blood perfusion rate \( \omega = 0.01 \, \text{s}^{-1} \), - Arterial temperature \( T_a = 37^\circ \text{C} \). - The temperature distribution

Answered: Image /qna-images/answer/bae020b5-92a2-4399-97ba-b2796db8d211.jpg

Assume that the head can be modeled as a hemisphere (radius ro-90 mm) with a single layer of brain tissue (k-0.5 W/m °C). The bottom of the hemisphere is insulated. The head surface is assumed exposed to a convection environment (h-8 W/m²K, T-25°C). Based on the boundary conditions, the brain tissue temperature can be treated as 1-D, i.e., T.(r), where r is the radial variable in the spherical coordinates. The volumetric metabolic heat generation rate in the brain is uniformly distributed and is equal to q-10000 W/m³. Consider two situations, a) no blood perfusion is considered; b) blood perfusion rate is uniform in the brain and is equal to -0.01 s¹. The governing equation (steady state, 1-D heat transfer in the spherical coordinate system, with or without the Pennes perfusion term) and boundary conditions for both situations can be written as: 1 d +@pc,(T-T)+q=0 k₁ dr dr 0 dT, r=r₁ k₁. h(T, -T.) dr (a) For the first situation without considering blood perfusion via letting -0 in the above equation, derive the temperature field T.(r). Use the values given to plot the radial temperature distribution T.(r) vs. r from Excel. Calculate the head surface temperature (r=ro) and the maximal temperature (r=0) in the brain tissue. (b) For the second situation, we consider thermal effect of blood perfusion using the W-J approach. Let @-0 in the above equation, however, change k, by keff when keff=3 kr (the W- r=0 dT, dr J approach). The derived expression of T.(r) in (a) can be easily updated via replacing k, by Keff Again plot the radial temperature distribution T,(r) vs. r from Excel. Calculate the head surface temperature (r=ro) and the maximal temperature (r=0) in the brain tissue. This is the simulation prediction using the WJ equation. (c) For the third situation, we will simulate temperature fields using the Pennes model, where the local blood perfusion rate is considered (p=1050 kg/m³, c-3800 J/kg °C, a 0.01 s¹, and Ta-37°C), the temperature distribution is derived and expressed as: T,(r)={ T. + + p.c, k, + 9₁ wp.c sinhl/m,ca/k, r) kr+sinh(√mp,c./k, r.). √ep,c./k,cosh(√mp,c./k, r.) if r*0 T ifr=0 @p,c Use Excel to plot the temperature distribution in the same plot as in (a), and compare the three curves. What is the role of the blood perfusion rate playing on the temperature profile? Do you think that the W-J model is good to predict the temperature profile? Note that in the W-J model, the surface temperature does not change from (a), explain briefly why.