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The thermal resistance analogy simplifies composite wall problems. For slabs in series (heat flows sequentially through each): $R_{\text{total}} = \sum R_i = \sum L_i/(k_i A)$. Heat flow: $dQ/dt = \Delta T_{\text{total}}/R_{\text{total}}$. Interface temperatures: $T_n = T_{\text{hot}} - (dQ/dt) \sum_{i=1}^{n} R_i$.

For parallel paths (heat flows simultaneously through all): $1/R_{\text{total}} = \sum 1/R_i$. Total heat flow = sum of individual flows.

Effective conductivity: Series (equal thickness): $k_{\text{eff}} = 2k_1k_2/(k_1+k_2)$ (harmonic mean, always less than arithmetic mean). Parallel (equal area): $k_{\text{eff}} = (k_1+k_2)/2$ (arithmetic mean).

Y-junction problems (multiple rods meeting at a point): in steady state, net heat flow into the junction is zero. This is Kirchhoff's thermal current law: $\sum k_i A_i(T_i - T_{\text{junction}})/L_i = 0$.

For cylindrical and spherical geometries, thermal resistance formulas differ: $R_{\text{cyl}} = \ln(r_2/r_1)/(2\pi kL)$, $R_{\text{sphere}} = (1/r_1 - 1/r_2)/(4\pi k)$.$