12 - Conduction - One Dimensional Heat Conduction Equation

Rate of heat conduction through any sort of surface, from surface 1 to surface 2 can be given by \[Q = \frac{T_1-T_2}{R}\] with \[R = \frac{\Delta x}{k\, A_m}\] where, \(\Delta x\) = thickness of surface through which heat is getting transferred; and, \(A_m\) = mean heat transfer area. \[A_m = \left\{\begin{array}{ll} A_{am} & \text{= arithmetic mean, for rectangular coordinates} \\ A_{lm} & \text{= logarithmic mean, for cylindrical coordinates} \\ A_{gm} & \text{= geometric mean, for spherical coordinates} \end{array} \right.\]

\[\begin{aligned} {For \ flat \ surface,} A_m = A_{am} &= \frac{A_1+A_2}{2} = \frac{A+A}{2} = A \\ \ \ {For \ cylindrical \ surface,} A_m = A_{lm} &= \frac{A_1-A_2}{\ln\dfrac{A_1}{A_2}} = \frac{2\pi aH - 2\pi bH}{\ln \dfrac{2\pi aH}{2\pi bH}} = \frac{2\pi H(b-a)}{\ln(b/a)} = \frac{2\pi H\Delta x}{\ln(b/a)} \\ \ \ {For \ spherical \ surface,} A_m = A_{gm} &= \sqrt{A_1A_2} = \sqrt{(4\pi a^2)(4\pi b^2)} = 4\pi ab \end{aligned}\]

During steady one-dimensional heat conduction in a spherical (or cylindrical) container, the rate of heat transfer (\(Q\)) remains constant, but the heat flux (\(q\)) decreases with increasing radius.