Transient Conduction into a Semi-infinite Solid: Example Problems
Try to solve these problems before watching the solutions in the screencasts.
Example Problem 1
Consider a part made of steel. It is very thick. It has an initial temperature of \(T_i\) = 300°C. The temperature of its surface is somehow maintained at a constant value of \(T_S\) = 25°C. The steel has a density of \(\rho\) = 7800 kg/m3, a heat capacity of \(c\) = 480 J/kg-K, and a thermal conductivity of \(k\) = 50 W/m-K.
a. Calculate the temperature of the steel, \(T\), at a depth of \(x\) = 50 mm below the surface after \(t\) = 500 seconds has elapsed.
b. At what depth, \(x\), does the temperature \(T\) = 90°C after a time of \(t\) = 500 seconds has elapsed?
Example Problem 2
Consider a thick slab made of some material. It has an initial temperature of \(T_i\) = 40°C. Air at a temperature of \(T_∞\) = 750°C flows over the surface, resulting in a heat transfer coefficient of \(h\) = 25 W/m2-K. The material has a density of \(\rho\) = 600 kg/m3, a heat capacity of \(c\) = 2400 J/kg-K, and a thermal conductivity of \(k\) = 0.2 W/m-K. How long does it take the surface of the material to warm up to a temperature of \(T_S\) = 325°C?