We may expand upon our earlier strategies for solving linear systems of temperatures by finding temperature profiles through these systems when no heat generation is occurring.
| Quantity/Equation | Cartesian | Cylindrical | Spherical |
|---|---|---|---|
| Conduction Equation | $\frac{d^2T}{dx^2}=0$ | $\frac 1r\frac d{dr}(r\frac{dT}{dr})=0$ | $\frac 1{r^2}\frac d{dr}(r^2\frac{dT}{dr})=0$ |
| Heat Flux | $k\frac{T_1-T_0}{x_1-x_0}$ | $\frac{k(T_1-T_0)}{r\ln(r_1/r_0)}$ | $\frac{4\pi k(T_1-T_0)}{r^2((1/r_0)-(1/r_1))}$ |
| Heat Transfer Rate | $kA\frac{T_1-T_0}{x_1-x_0}$ | $\frac{2\pi Lk(T_1-T_0)}{\ln(r_1/r_0)}$ | $\frac{4\pi k(T_1-T_0)}{(1/r_0)-(1/r_1)}$ |
| Conduction Resistances | $\frac{x_1-x_0}{kA}$ | $\frac{\ln(r_1/r_0)}{2\pi Lk}$ | $\frac{(1/r_0)-(1/r_1)}{4\pi k}$ |
| Temperature Profiles | $C_1x+C_0\Rarr\\T_0-(T_0-T_1)\frac{x-x_0}{x_1-x_0}$ | $C_1\ln(r)+C_0\Rarr\\T_1+(T_0-T_1)\frac{\ln(r/r_1)}{\ln(r_0/r_1)}$ | $C_1\frac 1r+C_0\Rarr\\T_0-(T_0-T_1)\frac{1-r_0/r}{1-r_0/r_1}$ |
In the case where heat generation does occur and is constant across the domain, the following general solutions are obtained
| Quantity/Equations | Cartesian | Cylindrical | Spherical |
|---|---|---|---|
| Conduction Equation | $k\frac{d^2T}{dx^2}+\dot g=0$ | $\frac kr\frac d{dr}(r\frac{dT}{dr})+\dot g=0$ | $\frac k{r^2}\frac d{dr}(r^2\frac{dT}{dr})+\dot g=0$ |
| Temperature Profile | $-\frac{\dot g}{2k}x^2+C_1x+C_2$ | $-\frac{\dot g}{4k}r^2+C_1\ln(r)+C_2$ | $-\frac{\dot g}{6k}r^2+C_1\frac 1r+C_2$ |