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Unformatted text preview: ME 309 Fall 2008 Section 2 Homework # 36 Due Wednesday 3 December 2008 Problem 361 a) Convert the energy equation for an adiabatic flow with no work done into an equation for the stagnationtostatic temperature ratio as an algebraic function of Mach number. Take Station 1 as a location where the temperature is T 1 , and the velocity is u 1 ; and Station 2 as a location where the flow has been stopped so that the velocity is u 2 = 0, and the temperature has been increased to T 2 . Show all pertinent algebra. b) Using the Gibbs relation: dp dh Tds 1 = along with an isentropic process between Stations 1 and 2 (defined as in Part a) to provide a definition of the stagnation pressure. Relate this stagnation pressure to the static pressure and the Mach number. Show all pertinent algebra;p c) Convert the continuity equation in terms of static variables, uA m = & , into an algebraic relation dependent upon the Mach number, stagnation pressure and stagnation temperature. As a part of this step, define the Mach number, function, D. Show all pertinent algebra. d) Using the equation from Part c (after comparing with the notes to ensure it is correct), develop a table giving the mass flow per unit area for a flow that is being drawn from a quiescent reservoir at Station 1 where the static pressure is 101325 Pa and the static temperature is 300 K and the velocity is zero, to Station 2 where the Mach number ranges from 0 to 2 in increments of 0.2 and from 2 to 20 in increments of 1. What do you observe about the maximum flow per unit area at Station 2 (which is called choking )? How much flow would you expect per unit area if the Mach number is increased without bound? Does this expectation make physical sense? Solution : Assumptions : perfect gas, =1.4, constant specific heats Find: Relation for stagnation temperature, relation for Mach number, mass flow per area vs Mach number Analysis : a) Given: T 1 , u 1 , u 2 = 0 Energy Equation; No work, no heat addition, constant mass flow: 1 2 h h = Station 2 is the stagnated condition (velocity = 0) for Station 1: 2 2 1 1 1 2 2 u h h h h + = = + = Take constant specific heats:  + = + = + = RT u T R u T c u T T p 2 2 2 2 1 1 2 1 2 Final Result: ...
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This note was uploaded on 03/09/2010 for the course ME 309 taught by Professor Merkle during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 MERKLE

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