Chapter 17 Compressible Flow Study Guide in PowerPoint to accompany Thermodynamics An Engineering Approach 8th edition by Yunus A engel and Michael A Boles 1 Stagnation Properties V velocity y Consider a fluid flowing g into a diffuser at a temperature p T pressure p P and enthalpy h etc Here the ordinary properties T P h etc are called the static properties that is they are measured relative to the flow at the flow velocity The diffuser is sufficiently long and the exit area is sufficiently large that the fluid is brought to rest zero velocity at the diffuser exit while no work or heat transfer is done The resulting state is called the stagnation state We apply the first law per unit mass for one entrance one exit and neglect the potential t ti l energies i L Lett th the iinlet l t state t t b be unsubscripted b i t d and d th the exit it or stagnation t ti state have the subscript o 2 2 o net net o q V h w 2 V h 2 2 Since the exit velocity work and heat transfer are zero ho h 2 V 2 The term ho is called the stagnation enthalpy some authors call this the total enthalpy It is the enthalpy the fluid attains when brought to rest adiabatically while no work is done If in addition the process is also reversible the process is isentropic and the inlet and exit entropies are equal so s The stagnation enthalpy and entropy define the stagnation state and the isentropic stagnation pressure Po The actual stagnation pressure for p stagnation g irreversible flows will be somewhat less than the isentropic pressure as shown below 3 2 V 2 Example 17 1 Steam at 400oC 1 0 MPa and 300 m s flows through a pipe Find the properties of the steam at the stagnation state At T 400oC and P 1 0 MPa h 3264 5 kJ kg s 7 4670 kJ kg K 4 Then 2 V ho h 2 m 300 kJ s 3264 3264 5 5 kg 2 2 kJ kg m2 1000 2 s kJ 3309 5 kg and kkJ so s 7 4670 kg K ho h Po so 5 We can find Po by trial and error or try the EES solution for problem 3 27 in the text The resulting stagnation properties are Po 1 16 1 16 MPa To 422 2o C Ideal Gas Result 1 kg o 3 640 3 vo m Rewrite the equation defining the stagnation enthalpy as ho h 2 V 2 For ideal gases with constant specific heats heats the enthalpy difference becomes CP To T 2 V 2 6 where To is defined as the stagnation temperature To T 2 V 2C P For the isentropic process process the stagnation pressure can be determined from or Using variable specific heat data PR To Po Po Preff P P Pref PR T 7 Example 17 2 An aircraft flies in air at 5000 m with a velocityy of 250 m s At 5000 m air has a temperature of 255 7 K and a pressure of 54 05 kPa Find To and Po 2 kJ m 2 250 V kg s To T 255 7 K 2 kJ m 2CP 2 1 005 1000 2 kg K s 255 7 31 1 K 286 8 K 8 Conservation of Energy for Control Volumes Using Stagnation Properties The steady flow conservation of energy for the above figure is Since ho h 2 V 2 9 For no heat transfer one entrance one exit this reduces to If we neglect the change in potential energy this becomes For ideal gases with constant specific heats we write this as Conservation of Energy for a Nozzle We assume steady flow no heat transfer no work one entrance and one exit and neglect elevation changes then the conservation of energy becomes E in E out m 1ho1 m 2 ho 2 10 But m 1 m 2 Then ho1 ho 2 Thus the stagnation enthalpy remains constant throughout the nozzle nozzle At any cross section in the nozzle the stagnation enthalpy is the same as that at the entrance For ideal gases this last result becomes To1 To 2 Thus the Th h stagnation i temperature remains i constant through h h out the h nozzle l A At any cross section in the nozzle the stagnation temperature is the same as that at the entrance Assuming an isentropic process for flow through the nozzle we can write for the entrance and exit states 11 So we see that the stagnation pressure is also constant through out the nozzle for isentropic flow NOTE It iis iimportant NOTE t t tto understand d t d why h th the stagnation t ti enthalpy th l stagnation t ti temperature for an ideal gas and stagnation pressures are constant in adiabatic nozzle flow Velocity of Sound and Mach Number We want to show that the stagnation properties are related to the Mach number M of V the flow where V is the velocity M C and C is the speed of sound in the fluid But first we need to define the speed of sound in the fluid A pressure disturbance propagates through a compressible fluid with a velocity dependent upon the state of the fluid fluid The velocity with which this pressure wave moves through the fluid is called the velocity of sound or the sonic velocity Consider a small pressure wave caused by a small piston displacement in a tube filled with an ideal gas as shown below 12 It is easier to work with a control volume moving with the wave front as shown below 13 Apply the conservation of energy for steady flow with no heat transfer no work and neglect the potential energies 2 C 2 C2 h 2 h 2 C dV d h dh 2 2 2 C 2CdV dV h dh 2 Cancel terms and neglect dV 2 we have dh CdV 0 Now apply the conservation of mass or continuity equationm AV to the control volume volume AC d A C dV AC A C dV Cd d dV Cancel terms and neglect the higher order terms like d dV We have C d dV 0 14 Also we consider the property relation dh T ds v dP dh T ds 1 dP Let s assume the process to be isentropic then ds 0 and 1 dh dP Using the results of the first law dh From the continuity equation 1 dP C dV C d dV Now 15 Thus dP 2 C d Since the process is assumed to be isentropic the above becomes For a general thermodynamic substance the results of Chapter 12 may be used to show that the speed of sound is determined from where k is the ratio of specific heats k CP CV 16 Ideal Gas Result For ideal gases Notice that the temperature used for the speed of sound is the static normal temperature Example 17 3 Find the speed of …