HW_22_Soln

HW_22_Soln - ME 309 Fall 2008 Section 2(Merkle Homework 22...

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ME 309 Fall 2008 Section 2 (Merkle) Homework # 22 Due Wed 22 Oct 2008 Problem 22-1 . The kinetic energy coefficient, α , provides a correction for the energy equation so we can use the average velocity in a pipe as the kinetic energy term. This quantity is defined as: 2 3 2 3 U m dA u udA U dA u A A A = = ρ where U is the average velocity in a pipe. For laminar flow, the velocity profile is given by: 2 max 1 - = R r u u (eq. 8.14) where u max = 2 U (eq. 8.13e). For turbulent flow, the velocity profile may be approximated by n R r u u 1 max 1 - = where m = 1/ n is used to simplify the integration. a. Show that ( )( ) 1 2 1 2 2 max + + = n n n u U Eq. 8.24. Note: Using m = 1/ n simplifies the integration. b. Show that =2 for laminar flow (pg 327) and is given by Eq. 8.27 for turbulent flow. Solution: Given: velocity profiles and definition of alpha; Assumptions: incompressible, no body forces, fully developed Find : kinetic energy coefficient, average velocity for turbulent flow Analysis: Laminar flow: First calculate average velocity in terms of max velocity: - = - = = = R R A R r d R r R r u R rdr R r u A dA u A Q U 0 2 max 2 0 2 max 1 2 2 1 π Replace r/R by y: { } { } 2
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HW_22_Soln - ME 309 Fall 2008 Section 2(Merkle Homework 22...

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