118.2. Viscous heating in laminar tube flow (asymptotic solutions). (a) Show that for fully developed laminar Newtonian flow in a circular tube of radius R, the energy equation becomes pè pouma [-- (R)]2+7() -Na (3) (113.2-1) if the viscous dissipation terms are not neglected. Here v may is the maximum velocity in the tube. What restrictions have to be placed on any solutions of Eq. 113.2-12 (b) For the isothermal wall problem (T = T, at r = R for 2 >0 and at z = 0 for all r), find the as- ymptotic expression for T() at large z. Do this by recognizing that at /az will be zero at large z. Solve Eq. 113.2-1 and obtain HUB T-T, = [1-() (113.2-2) 4K (c) For the adiabatic wall problem (q, = 0 at r = R for all z > 0) an asymptotic expression for large z may be found as follows: Multiply Eq. 113.2-1 by rdr and then integrate from r = 0 to r = R. Then integrate the resulting equation over z to get T. - To = (4uV, max/pCR) (113.2-3) in which T, is the inlet temperature at z = 0. Postulate now that an asymptotic temperature profile at large z is of the form T-T, = (44v,max/pĈ,RO)2 + for) (113.2-4) Substitute this into Eq. 113.2-1 and integrate the resulting equation for fr) to obtain 4 доллах T-T; 2 + pĈ R² (113.2-5) after determining the integration constant by an energy balance over the tube from 0 to z. Keep in mind that Eqs. 113.2-2 and 5 are valid solutions only for large z. The complete solu- tions for small z are discussed in Problem 110.2.