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 4µo max 1 a r dr () (11B.2-1) az R2 if the viscous dissipation terms are not neglected. Here v,max is the maximum velocity in the tube. What restrictions have to be placed on any solutions of Eq. 11B.2-1? (b) For the isothermal wall problem (T = T, at r= R for z> 0 and at z = 0 for all r), find the asymptotic expression for T(r) at large z. Do this by recognizing that aT/dz will be zero at large z. Solve Eq. 11B.2-1 and obtain T(r) – T, = 4k (11B.2-2) (c) For the adiabatic wall problem (q, = 0 atr R for all z> 0) an asymptotic expression for large z may be found as follows: Multiply Eq. 11B.2-1 by rdr and then integrate from r = 0 to r = R. Then integrate the resulting equation over z to get T, - T = (4µ0 max/pČ,R²)z (11B.2-3) in which T, is the inlet temperature at z = 0. Postulate now that the asymptotic temperature profile at large z is of the form T(r,z) – T, = (4µ0zmax/pC,R?)z +f(r) (11B.2-4) Substitute this into Eq. 11B.2-1 and integrate the resulting equation for f(r) to obtain z,max T(r,z) – T, = - z,max (11B.2-5) k after determining the integration constant by an energy balance of the tube from 0 to z. Keep in mind that the solutions in Eqs. 11B.2-2 and 11B.2-5 are valid solutions only for large z.

Introduction to Chemical Engineering Thermodynamics

8th Edition

ISBN:9781259696527

Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart

Chapter1: Introduction

Section: Chapter Questions

Problem 1.1P

See similar textbooks

Related questions

Question

Transcribed Image Text: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 4µo max 1 a r dr () (11B.2-1) az R2 if the viscous dissipation terms are not neglected. Here v,max is the maximum velocity in the tube. What restrictions have to be placed on any solutions of Eq. 11B.2-1? (b) For the isothermal wall problem (T = T, at r= R for z> 0 and at z = 0 for all r), find the asymptotic expression for T(r) at large z. Do this by recognizing that aT/dz will be zero at large z. Solve Eq. 11B.2-1 and obtain T(r) – T, = 4k (11B.2-2) (c) For the adiabatic wall problem (q, = 0 atr R for all z> 0) an asymptotic expression for large z may be found as follows: Multiply Eq. 11B.2-1 by rdr and then integrate from r = 0 to r = R. Then integrate the resulting equation over z to get T, - T = (4µ0 max/pČ,R²)z (11B.2-3) in which T, is the inlet temperature at z = 0. Postulate now that the asymptotic temperature profile at large z is of the form T(r,z) – T, = (4µ0zmax/pC,R?)z +f(r) (11B.2-4) Substitute this into Eq. 11B.2-1 and integrate the resulting equation for f(r) to obtain z,max T(r,z) – T, = - z,max (11B.2-5) k after determining the integration constant by an energy balance of the tube from 0 to z. Keep in mind that the solutions in Eqs. 11B.2-2 and 11B.2-5 are valid solutions only for large z.

Expert Solution