12 - Conduction - One Dimensional Heat Conduction Equation

Starting from general one dimensional heat conduction equation, obtain the following expressions, for steady state heat transfer through flat plate.

At \(x=0\), \(T=T_1\); and, at \(x=L\), \(T=T_2\). With \(T_1>T_2\). \[{\frac{T(x)-T_1}{T_2-T_1} = \frac{x}{L}}\]

Starting from general one dimensional heat conduction equation, obtain the following expressions, for steady state heat transfer through cylindrical shell.

At \(r=r_1\), \(T=T_1\); and, at \(r=r_2\), \(T=T_2\). With \(T_1>T_2\), and \(r_2>r_1\). \(Q=qA\).

• \(T(r)=\dfrac{q_1r_1}{k}\ln\left(\dfrac{r_2}{r} \right) + T_2\)

• \(T(r)=\dfrac{q_2r_2}{k}\ln\left(\dfrac{r_1}{r} \right) + T_1\)

• \( \displaystyle \frac{T(r)-T_1}{T_2-T_1}=\frac{\ln(r/r_1)}{\ln(r_2/r_1)}\)