Chapter 8, Problem 8.14P

Consider a cylindrical nuclear fuel rod of length L and diameter D that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate $\stackrel{\dot{}}{m}$ , and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, $\stackrel{\dot{}}{q}\left(x\right)={\stackrel{\dot{}}{q}}_o\sin \left(\pi x/L\right),\text{where}{\stackrel{\dot{}}{q}}_o\left(W/m^3\right)$ is a constant. A uniform convection coefficient h may be assumed to exist between the surface of the rod and the water.

(a) Obtain expressions for the local heat flux $q\text{'}\text{'}\left(x\right)$ and the total heat transfer q from the fuel rod to the water.
(b) Obtain an expression for the variation of the mean temperature $T_m\left(x\right)$ of the water with distance xalong the tube.
(c) Obtain an expression for the variation of the rod surface temperature $T_s\left(x\right)$ with distance x along the tube. Develop an expression for the x-location at which this temperature is maximized.