Problem 9:
The thin steel cylinder wall of a small industrial+ reciprocating internal combustion engine is initially at
the known temperature $T_o$. Suddenly the engine is started and run at constant speed, exposing the inside
surface of the cylinder wall to a gas whose temperature varies with time according to the relation:
$T_G = T_M cos(wt)$
Where $T_M$ is the average gas temperature, $h$ and $w$ (angular frequency) are known constants. There is a
constant surface coefficient of heat transfer, $h$ between the gas and the inside cylinder wall and the very
low (compared with $h$) surface heat transfer coefficient on the outside of the cylinder wall effectively
insulates the outside surface. The conductivity of the cylinder is sufficiently high so that temperature
gradients within the cylinder can be neglected. Predict the cylinder wall temperature as a function of time.
The mass, $M$ and the heat capacity, $C_p$, and the inside heat transfer area, $A$ of the cylinder wall are known.
Note: $\int e^{ax}cos(bx)dx = e^{ax}\frac{acos(bx)+bsin(bx)}{a^2+b^2}$
