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HW_36_Soln

# HW_36_Soln - ME 309 Fall 2008 Section 2 Homework 36 Due...

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ME 309 Fall 2008 Section 2 Homework # 36 Due Wednesday 3 December 2008 Problem 36-1 a) Convert the energy equation for an adiabatic flow with no work done into an equation for the stagnation-to-static 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 ρ 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 ρ = & , 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, γ =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: 0 1 0 2 h h = Station 2 is the stagnated condition (velocity = 0) for Station 1: 2 0 2 1 1 0 1 2 0 2 u h h h h + = = + =

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HW_36_Soln - ME 309 Fall 2008 Section 2 Homework 36 Due...

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