1. The space between two coaxial cylinders is filled with an incompressible fluid at constant temperature. The radii of the inner and outer wetted surfaces are кR and R, respectively. The angular velocities of rotation of the inner and outer cylinders are Qi and 0. See the image to the right. Derive the velocity profile of the fluid [Ve(r)] assuming equation 3.7-28 applies and determine the torques (Tz,outer and Tz,inner) exerted by the fluid on the two cylinders needed to maintain the motion. Ω kRi R Problem 3 Equations Equation of Continuity: Spherical coordinates (r,0,0): др + at 1 д (pr²v₁) + 1 д 1 д (pv sin 0) + r² dr r sin o de (pv) = 0 (B.4-4) r sin 0 дф Equations of Motion: Spherical coordinates (r,0,0): до dv % до P + + - ot dr г де r sin 0 do a² 11+ ar2 1 д 2 sin 0 de av, sin 0. de ave + at ave dr r de ve ave r =- 1 др dr r² sin² do² V dve ve- cot e + +P&r (B.6-7) 1 др + + =- r sin 0 дф r r de дор 1 д 1 д +1 + (v sin 0) 2 dr dr 12 до sin 0 де (0)) + 1 J²ve 2 dv, + r² sin² 0 do² 12 до 2 cote dv 2 sin 0 do + P&e (B.6-8) dve до vedo P v + + + at 1 д +μ 2 dr дт V du v + cot дг г де r sin 0 до д до 1 д 0 де e) r == r sin 0 до +()+((sin 0) + 12 sin' a des² + på sin o dig sin a do "The quantity in the brackets in Eq. B.6-7 is not what one would expect from Eq. (M) for [V. Vv] in Table A.7-3, because we have added to Eq. (M) the expression for (2/r)(V-v), which is zero for fluids with constant p. This gives a much simpler equation. 1 др 1 2 до do 2 cot dve + + P80 (B.6-9)

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

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Transcribed Image Text:1. The space between two coaxial cylinders is filled with an incompressible fluid at constant temperature. The radii of the inner and outer wetted surfaces are кR and R, respectively. The angular velocities of rotation of the inner and outer cylinders are Qi and 0. See the image to the right. Derive the velocity profile of the fluid [Ve(r)] assuming equation 3.7-28 applies and determine the torques (Tz,outer and Tz,inner) exerted by the fluid on the two cylinders needed to maintain the motion. Ω kRi R