Chapter 8, Problem 8.23P

An experimental nuclear core simulation apparatus consists of a long thin-walled metallic tube of diameter D and length L, which is electrically heated to produce the sinusoidal heat flux distribution $q_s^{"}\left(x\right)=q_o^{"}\sin \left(\frac{\pi x}{L}\right)$ where x is the distance measured from the tube inlet. Fluid at an inlet temperature $T_{m,i}$ flows through the tube at a rate of $\stackrel{\dot{}}{m}$ . Assuming the flow is turbulent and fully developed over the entire length of the tube, develop expressions for:
(a) the total rate of heat transfer, q, from the tube to the fluid;
(b) the fluid outlet temperature, $T_{m,o}$ ;
(e) the axial distribution of the wall temperature, $T_s\left(x\right)$ : and
(d) the magnitude and position of the highest wall temperature.
(e) Consider a 40-mm-diameter tube of 4-m length with a sinusoidal heat flux distribution for which $q_o^{"}=10,000W/m^2$ . Fluid passing through the tube has a flow rate of $\text{0}\text{.025 kg/s}$ , a specific heat of $\text{4180 J/kg}\cdot \text{ K}$ , an entrance temperature of $\text{25}°\text{C}$ , and a convection coefficient of ${\text{1000 W/m}}^2\cdot \text{ K}$ . Plot the mean fluid and surface temperatures as a function of distance along the tube. Identify important features of the distributions. Explore the effect of $\pm 25\%$ changes in the convection coefficient and the heat flux on the distributions.