Temperature Profile for One Dimensional Heat Flow

Temperature profile for one dimensional steady state conduction, for systems with constant thermal conductivity are obtained as below: (here, \(T_1>T_2\)   and   \(b>a\)) \[\begin{aligned} \text{Flat plate:} & \qquad & T(x) &= (T_2-T_1)\frac{x}{L} + T_1 \\ \text{Cylinder:} & & \frac{T(r)-T_1}{T_2-T_1} &= \frac{\ln(r/a)}{\ln(b/a)} \\ \text{Sphere:} & & T(r) &= \frac{a}{r}\cdot\frac{b-r}{b-a}\cdot T_1 \, + \, \frac{b}{r}\cdot\frac{r-a}{b-a}\cdot T_2 \end{aligned}\]

  • For flow across flat surfaces: \(T\) vs. \(x\) is linear; \(\dfrac{dT}{dx} = \text{ constant}\).

  • For flow across cylindrical surfaces: \(T\) vs. \(\ln r\) is linear; \(\dfrac{dT}{dr} \propto \dfrac{1}{r}\)

  • For flow across spherical surfaces: \(T\) vs. \(1/r\) is linear; \(\dfrac{dT}{dr} \propto \dfrac{1}{r^2}\)