Chapter 8, Problem 8.24P

Water at 20°C and a flow rate of $\text{0}\text{.1 kg/s}$ enters a heated, thin-walled tube with a diameter of 15 mm and length of 2 m. The wall heat flux provided by the heating elements depends on the wall temperature according to the relation $q_s^{"}\left(x\right)=q_{s,o}^{"}\left[1+\alpha \left(T_s-T_{\text{ref}}\right)\right]$ where $q_s^{"}={10}^{4}W/m^2,\alpha =0.2K^{-1},T_{ref}=20°C$ , and $T_s$ , is the wall temperature in °C. Assume fully developed flow and thermal conditions with a convection coefficient of $3000W/m^2\cdot K$ .

(a) Beginning with a properly defined differential control volume in the tube, derive expressions for the variation of the water, $T_m\left(x\right)$ , and the wall, $T_s\left(x\right)$ ,temperatures as a function of distance from the tube inlet.
(b) Using a numerical integration scheme, calculate and plot the temperature distributions, $T_m\left(x\right)$ and $T_s\left(x\right)$ on the same graph. Identify and comment on the main features of the distributions. Hint: The IHT integral function $DER\left(T_m,x\right)$ can be used to perform the integration along the length of the tube.
(c) Calculate the total rate of heat transfer to the water.