Solve the following problem:
Q: 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(x) = q"o sin(px/L) where x is the distance measured from the tube inlet. Fluid at an inlet temperature Tm.i flows through the tube at a rate of In. 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, Tm,o;
(c) The axial distribution of the wall temperature, Ts(x); 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 = 10,000 W/m2. Fluid passing through the tube has a flow rate of 0.025 kg/s, a specific heat of 4180 J/kg · K, an entrance temperature of 25°C, and a convection coefficient of 1000 W/m2 · 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 ±25% changes in the convection coefficient and the heat flux on the distributions.