Isentropic Flow with Area Change -1 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Isentropic Flow with Area Change Examine mass and momentum equations for reversible and adiabatic conditions dx p+dp  +d  v+dv A+dA pvApvA xx Mass (VI.9) Momentum (VI.10) 0 (no viscous stress/friction, reversible) * Combine

Isentropic Flow with Area Change -2 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Mach Number Relation But isentropic, so p+dp  +d  v+dv A+dA pvApvA (VI.16) From * Derived using only mass/momentum conservation, and speed of sound Valid for all simple comp. substances

Isentropic Flow with Area Change -3 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Isen. Flow - Mach Number Dependence For M<1:dA, dp same signA  p  dA, dv opposite signA  v  For M>1:dA, dp opposite signA  p  dA, dv same signA  v  How does area change effect flow properties? dp, dv always opposite signs (v   T   p  ) Expansion dp<0 Compression dp>0 M<1 M>1 Nozzle dv>0 (dM>0) Diffuser dv<0 (dM<0)

Isentropic Flow with Area Change -4 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Sonic Throat Requirement Need to go through M=1 For M=1:dA/dv = 0 but dv  0 (flow accel. or decel.) dA=0: maximum or minimum in area So need a throat to transition, and M=1 at throat (sonic condition) How to transition from subsonic to supersonic (or vica versa)? M<1  M  M>1  M   M<1M>1M=1 M>1M<1M=1

Isentropic Flow with Area Change -5 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Area Ratio For isentropic flow, look at effect of area change on M by comparing A at any point to area at sonic point (A*) –* refers to properties of a flow if isen. accel./decel. to M=1 sonic conditions (e.g.,  *, T*,…) –alternative to stagnation as ref. state Use mass conservation from VI.8 from VI.6 (VI.17) TPG, CPG A* A

Isentropic Flow with Area Change -6 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Area Ratio Results Two (isentropic) solutions for a given A/A* –one subsonic –one supersonic A  A* always Accel. to high M requires large A/A* Supersonic Subsonic  =1.4 – maximum mass flux at M=1 (throat)

Isentropic Flow with Area Change -7 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Mass Flux and Stagnation Properties Examine mass flux in terms of stagnation conditions from VI.6,7 CPG from VI.2, TPG (VI.18) For given isen. flow, all stagnation (and sonic) properties constant, including mass flow rate

Isentropic Flow with Area Change -8 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Choked Flow For fixed stagnation properties and flow area SupersonicSub-  =1.4  Choked Flow –if throat is sonic, can’t get more by changing downstream conditions (e.g., back pressure) A front p back p o T o reservoir AtAt For nozzle with fixed stagnation properties

Isentropic Flow with Area Change -9 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Choked Flow (con’t) For nozzle with fixed stagnation properties and initially sonic throat SupersonicSub- –if reduce throat area A t, flow at throat stays sonic (A front /A t  ) and  (since A , & same) –if increase A t,  and eventually throat not sonic (A front /A*  ) A front p back p o T o reservoir AtAt

Isentropic Flow with Area Change -10 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Choked Mass Flowrate Maximum flow rate when choked (M=1 at throat) Choked mass flowrate from f( ,1) (VI.19) rule of thumb for choked gas flows To increase mass flowrate –increase A * (throat size) –increase p o, decrease T o (increase stagnation density) f( ,1) typically near 0.7

Isentropic Flow with Area Change -11 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example 1 For a supersonic wind tunnel with an incoming flow with M 1. What area throat required to produce a test section Mach number of M=3 in test section with 0.2 m 2 cross-section? –Assume isentropic flow, calor./thermally perfect gas,  =1.4. Answer

Isentropic Flow with Area Change -12 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example 2 Converging-diverging supersonic nozzle with M 1 =2.0 and M 2 =3.0 What is A 2 /A 1 ? –Assume isentropic flow, calor./thermally perfect gas,  =1.4. Answer: A* same for 1 and 2 (isen. in between)

Isentropic Flow with Area Change -13 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example 3 For nozzle shown, inlet area A 1 =0.50 m 2, M 1 =0.20 and p o, T o fixed How far can A 2 be reduced from A 1 without changing mass flowrate in nozzle? –Assume isentropic flow, calor./thermally perfect gas,  =1.4. Answer: When flow becomes choked (M=1 throat).

Isentropic Flow with Area Change -14 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example 4 For nozzle shown (A 1 =0.060m 2, A 2 =0.020m 2 ) and p o, T o fixed For isentropic flow in the nozzle, what are the limits on the allowed inlet Mach numbers (M 1 )? –Assume tpg/cpg,  =1.4. Answer: When throat sonic. Flow not choked if A 2 >A*(A 1 /A*>A 1 /A 2 ) 1 2 1

Isentropic Flow with Area Change -15 School of Aerospace Engineering Copyright © 2001 by Jerry M. Seitzman. All rights reserved. AE3450 Example 5 Same nozzle as Example 4, air and p o =500kPa, T o = 300K What is max. possible mass flowrate through the nozzle? –Assume tpg/cpg,  =1.4. Answer: Choked mass flowrate.