Chapter 8, Problem 8.10P

When viscous dissipation is included. Equation 8.48 (multiplied by $\rho c_p$ ) becomes

   $\rho c_{p}u\frac{\partial T}{\partial x}=\frac{k}{r}\frac{\partial }{\partial r}\left(r\frac{\partial T}{\partial r}\right)+\mu {\left(\frac{du}{dr}\right)}^2$

This problem explores the importance of viscous dissipation. The conditions under consideration are laminar, fully developed flow in a circular pipe, with u given by Equation 8.15.

(a) By integrating the left-hand side over a section of a pipe of length L and radius $r_o$ , show that this term yields the right-hand side of Equation 8.34.

(b) Integrate the viscous dissipation term over the same volume.

(c) Find the temperature rise caused by viscous dissipation by equaling the two terms calculated above. Use the same conditions as in Problem 8.9.