Assignment4 Solution [PDF]

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Section 4 Homework! • Develop an iterative solver for!

γ − 1) 2 % , A 1 )" 2 % " ( = 1+ M '. +$ * ' $ A M *# γ + 1& # 2 &-

γ +1 2 (γ −1)

• Use any programming language you prefer! i.e. Fortran, C++, MATLAB, … ! • Clearly comment and document your code!

MAE 5420 - Compressible Fluid Flow!

1!

Section 4 Homework (cont’d)! • Plot the Mach number, pressure, and temperature! distribution along the SSME Nozzle for each of the ! following conditions! Pe=16.8727 KPa, P0=20.4 Mpa, T0=3500 Gas Properties: MW=22!

K!

γ = 1.22

• Hint: make sure nozzle is choked first, ! then program all functions you might need!

• Solve for Thrust and Isp in Vacuum! MAE 5420 - Compressible Fluid Flow!

2!

Section 4 Homework Solution (cont’d)! • SSME Nozzle profile ! Z!

X!

MAE 5420 - Compressible Fluid Flow!

SSME Nozzle Profile! X, cm !+Z, cm 0.00 !17.50 4.00 !15.50 8.00 !13.00 12.00 !12.25 16.00 !13.00 20.00 !15.50 25.00 !18.50 30.00 !22.00 50.00 !33.00 70.00 !41.50 90.00 !50.50 100.00 !54.50 120.00 !61.00 140.00 !68.00 160.00 !74.50 180.00 !80.50 200.00 !86.00 220.00 !91.00 240.00 !97.00 260.00 !101.00 280.00 !105.00 300.00 !107.50 305.00 !107.85

!-Z, cm! !-17.50! !-15.50! !-13.00! !-12.25! !-13.00! !-15.50! !-18.50! !-22.00! !-33.00! !-41.50! !-50.50! !-54.50! !-61.00! !-68.00! !-74.50! !-80.50! !-86.00! !-91.00! !-97.00! !-101.00! !-105.00! !-107.50! !-107.85! 3!

Compute Exit Mach Based on Given DATA! Pe=16.8727 KPa, P0=20.4 Mpa, T0=3500

K!

Gas Properties:!

γ =! 1.22 ! MW=22! • Compute Mach Number at Exit! γ −1 ) , γ 2 +# P0 & .= Me = − 1 . γ − 1 +%$ Pe (' +* .-

$ '$ 2 % '" # 1.22 − 1 "

1.22 − 1 $ % % 0.5 ' $ 20.4⋅ 1000 % 1.22 (( − 1( ( =4.858225! '" # " 16.8727 ##

MAE 5420 - Compressible Fluid Flow!

4!

Compute Ratio of Geometric" Throat Area to A*! Me = 4.858108! • Compute ratio of geometric throat area to Sonic throat area! SSME Nozzle Geometry! Ae/At = (107.85/12.25)^2 = 77.51190! γ +1 ) , 2 (γ −1) ) , ! $ ! $ ! $ ! $ γ − 1 At At ! Ae $ At + 1 2 ( )M 2 .= = = 1 + + . # & e * * &% . A #" Ae &% " A % #" Ae &% + M e *#" γ + 1 &% #" 2 +* .-

# 1.22 + 1 $ $ # ! " 1 &# $ # # 2 $ # 1 + # 1.22 − 1 $ ( 4.858225 ) 2 $ $ 2 ( 1.22 − 1 ) ' ⁄/ 77.519 &! ' 77.5119! = " !! "! ! " "" 1.22 + 1 2 ! 4.858225 "

MAE 5420 - Compressible Fluid Flow!

• Nozzle is Choked!! At = A*!

1.000!

5!

Compute Exit Mach Based on Given DATA (2)! OR USE A/A* assuming choked, isentropic flow!

• Compute Mach Number at Exit from A/A* equation! 2

Aexit 1 ! 107.85 $ #" &% = 77.5 = * = 12.25 A M exit

)! 2 $ ! , γ − 1) ( 2$ 1+ M exit & . +# & # 2 %*" γ + 1% "

γ +1 2 (γ −1)

=

1.22 + 1 2 % % 2 ( 1.22 − 1 )

1 $ 2 $ 1 + 1.22 − 1 4.858225 " " ## 4.858225 1.22 + 1 2

γ = 1.22

Mexit = 4.8582!

MAE 5420 - Compressible Fluid Flow!

Check!! 6!

Diameter, cm!

A/A*!

Compute " A/A* along nozzle! throat!

MAE 5420 - Compressible Fluid Flow!

7!

Solve for Mach Number Distribution in Nozzle! • Use Iterative Solver at each Nozzle station!

MAE 5420 - Compressible Fluid Flow!

8!

Solve for Pressure Distribution in Nozzle! P(x) =

P0 # γ −1 & M (x)2 ( %$ 1 + ' 2

MAE 5420 - Compressible Fluid Flow!

9!

γ γ −1

Solve for Temperature Distribution in Nozzle! γ −1) 2 ( T0 =1+ M T 2

MAE 5420 - Compressible Fluid Flow!

10!

Basic Program Structure! /* do this for each geometric point in nozzle */! /* external declarations */ ! extern double FOFM_(); ! extern double DFDM_();! extern double pratio_isen_();! extern double tratio_isen_();! for(i__ = 0; i__