The heat flux \(q \) from the wall with uniform generation is to the right and is:
$$
q = QL_i
$$
where \(q \) is in \(W/m^2 \), \(Q \) is the volumetric heat generation rate in \( W/m^3 \), and \(L_i \) is the thickness of the wall with uniform generation (either A or B) in \(m \).
A thermal circuit is modeled from the right side of the wall with generation to the air with forced convection. If wall A generates heat, then the heat flux is:
$$
q = \frac{T_{s,1}\ – \ T_∞}{1/h} = \frac{T_{s,2} \ – \ T_{s,3}}{L_C/k_C} = \frac{T_{s,3} \ – \ T_{s,2}}{R_{tc}} = \frac{T_{s,4} \ – \ T_{s,3}}{L_B/k_B} = \frac{T_{s,5} \ – \ T_{s,4}}{R_{tc}}
$$
where \(R_{tc} \) is the contact resistance (\([m^2 \ K]/W \)), \(T_{s,1} \) is the temperature on the right surface of wall C and \(T_{s,5} \) is the temperature on right side of wall A (°C). However, if wall B generates heat, then the thermal circuit can only be used to solve up to \(T_{s,3} \), the temperature on the right side of wall B.
The temperature profile for heat generation in either wall A or B is calculating using:
$$
\frac{d^2T}{dx^2} = -\frac{Q}{k_i}
$$
where \(k_i \) is the thermal conductivity of the wall generating heat.
The boundary conditions are:
$$
\frac{dT}{dx} = 0
$$
$$
T(L_i) = T_{s, \ right \ side \ of \ wall}
$$
The left side \((L_i = 0 \)) is well insulated so the flux is zero. The temperature of the right side of the wall with generation is determined from the thermal circuit.
References:
-
T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Introduction to Heat Transfer, 6th ed., Hoboken: John Wiley and Sons, 2011.
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