Heat is conducted through a composite wall with heat generation in either wall A or wall B. Differential equations determine the temperature profile of each wall. When heat is generated in wall A:
$$
\frac{d^2T}{dz^2} = -\frac{Q}{k_A},
$$
$$
\frac{dT}{dz}|_{z=0} = 0,
$$
$$
T|_{z=L_A} = T_{s,5},
$$
where \(T\) is temperature (°C), \(z\) is wall thickness (m), \(Q\) is the volumentric heat generation (W/m\(^3\)), \(k_A\) is the thermal conductivity in wall A (W/[m K]), \(L_A\) is the thickness of wall A (m), and \(T_{s,5}\) is the surface temperature of the right side of wall A (°C), and \(T_{s,1}\) is the right side of wall C.
A thermal circuit is considered starting from the right side of the wall with heat transfer to the air by forced convection. If wall A generates heat, then the heat flux is:
$$
q = \frac{T_{s,1} - T_∞}{1/h} = \frac{T_{s,2} - T_{s,1}}{L_C/k_C} = \frac{T_{s,3} - T_{s,2}}{R_{tc}} = \frac{T_{s,4} - T_{s,3}}{L_B/k_B} = \frac{T_{s,5} - T_{s,4}}{R_{tc}}.
$$
The heat flux when wall A generates heat is:
$$
q = QL_A,
$$
where \(q\) is heat flux (W/m\(^2\)).
For generation in either wall, the surface temperatures are:
$$
T_{s,1} = T_∞ + \frac{q}{h},
$$
$$
T_{s,2} = T_{s,1} + q\frac{L_C}{k_C},
$$
$$
T_{s,3} = T_{s,2} + qR_{tc},
$$
where \(h\) is the heat transfer coefficient (W/[m\(^2\) K]), and \(R_{tc}\) is the contact resistance ([m\(^2\) K]/W).
For heat generation in wall A, the rest of the surface temperatures are:
$$
T_{s,4} = T_{s,3} + q\frac{L_B}{k_B},
$$
$$
T_{s,5} = T_{s,4} + qR_{tc},
$$
\(\quad T_{s,6} = T(z)\) the solution to the differential equations.
The heat flux when wall B generates heat is:
$$
q = QL_B
$$
and
$$
\frac{d^2T}{dz^2} = -\frac{Q}{k_B},
$$
$$
\frac{dT}{dz}|_{z=L_A} = 0,
$$
$$
T|_{z=L_A+L_B} = T_{s,3}.
$$
For heat generation in wall B, the remaining surface temperatures are:
\(\quad T_{s,4} = T(z)\) the solution to the differential equations,
$$
T_{s,6} = T_{s,5} = T_{s,4}.
$$
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