Answered: Suggest a criterion for judging whether…

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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Question

Suggest a criterion for judging whether entrance effects may be neglected. If Re = 1500, R = 2 cm and

L = 250 m, can entrance effects be safely neglected?

Transcribed Image Text:Consider a cylindrical pipe of length L and diameter D = 2R. The angle that the axis of the pipe forms with the vertical direction is a. Assume that when the fluid enters the pipe its velocity is uniform (i.e., it has the same value over the entire cross-section of the pipe) and equal to U in the axial direction. In the radial and angular directions, the velocity is zero. So, it is: z = 0 : v(r, 0, z) = U e, (1.1) Here v is the fluid velocity and e, is a vector of unit magnitude parallel to the coordinate axis z; furthermore, we have assumed that the pipe inlet is located at z = 0. Near the entrance of the pipe, the velocity profile varies in the axial direction. But after a certain entrance length, the profile becomes fully developed, no longer changing with z. The evolution of the velocity profile is sketched in Fig. 1, where, for clarity, the pipe inclination is not shown. The entrance length is denoted by Le. For z > Le, the fluid velocity is no longer a function of the axial coordinate. In engineering design, pressure drop calculations are based on relations that hold only for fully developed flows. However, entrance effects are always present. To judge whether these effects are negligible, one must estimate the value of the entrance length. This is our main goal. Let p and u denote the density and viscosity of the fluid, respectively (assumed to be constants); we will restrict the analysis to steady, laminar flows in which Re = pUD/µ > 1. Inlet Developing Velocity Profile Fully Developed Velocity Profile Velocity Profile U 8(z) Le Figure 1: Evolution of the velocity profile in the pipe entrance region. Le is the entrance length, while 8(z) is the thickness of the "wall layer," i.e., the region where ô,„v, 7 0.

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