Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259877827
Author: CENGEL
Publisher: MCG
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Chapter 12, Problem 74P
To determine

Pressure, temperature, velocity, Mach number, and stagnation pressure downstream of the shock and comparison of Helium undergoing the same shock.

Expert Solution & Answer
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Answer to Problem 74P

Downstream pressure P2=447.75kPa

Downstream temperature T2=604.34K

Downstream velocity V2=248.31m/s

Downstream Mach number Ma2=0.5039

Downstream stagnation pressure P02=532.69kPa

Comparison with helium for the normal shock under same conditions we have

Downstream pressure P2=475.65kPa

Downstream temperature T2=799.55k

Downstream velocity V2=908.27m/s

Downstream Mach number Ma2=0.5455

Downstream stagnation pressure P02=602.57kPa

Explanation of Solution

Given:

The upstream the shock is given by, Pressure of air P1=58kPa

Temperature of air T1=270K

Mach number Ma1=2.6

Calculation:

Downstream Mach number is given by Table A-14,For Ma1= 2.6 using the below formula.

The properties of air are

  k=1.4

  Ma2= ( k1 )Ma1 2 +2 2kMa1 2 k+1Ma2= ( 1.41 ) 2.62 +2 2×1.4× 2.62 1.4+1Ma2=0.5039

Downstream pressure is given by the formula,

  P2P1=( 1+kMa1 2 1+kMa2 2 )P2=P1( 1+kMa1 2 1+kMa2 2 )P2=58kPa( 1+1.4× 2.62 1+1.4× 0.50392 )P2=447.75kPa

Downstream temperature is given by the formula,

  T2T1=( 2+Ma1 2 ( k-1 ) 2+Ma2 2 ( k-1 ))T2=T1( 2+Ma1 2 ( k-1 ) 2+Ma2 2 ( k-1 ))T2=270K( 2+2.62 ( 1.4-1 ) 2+0.50392 ( 1.4-1 ))T2=604.34K

Downstream velocity of speed is given by the formula,

  V2=Ma2c2......(1)where,c2=kRT2equation(1) becomes,V2=Ma2kRT2V2=0.50391.4×0.287×604.34× 10001V2=248.31m/s

Downstream stagnation pressure is given by formula,

  P 02P1=( 1+kMa2 1 1+kMa2 2 )(1+ ( k-1)M a 2 2 2)k/(k-1)P02=58×1000( 1+1.4×2. 6 2 1+1.4×0.503 9 2( 1+ ( 1.4-1 )0.503 9 2 2))1.4/(1.4-1)P02=532.69kPa

Analysis for helium is to be found out,so we need to determine the downstream Mach number by the below formula

The properties of helium are

k = 1.667

  Ma2=( ( k1 )M a 1 2 +2 2kM a 1 2 k+1 )Ma2=( ( 1.6671 ) 2.6 2 +2 2×1.667× 2.6 2 1.667+1 )Ma2=0.5455

Calculate for the downstream pressure for the helium gas given by the formula,

  P2P1=1+kMa211+kMa22P2P1=1+1.667×2.621+1.667×0.54552P2=P1(8.2009)P2=58×1000(8.2009)P2=475.65kPa

Calculate for the downstream temperature for the helium gas given by the formula,

  T2T1=2+Ma21(k-1)2+Ma22(k-1)T2T1=2+2.62(1.667-1)2+0.54552(1.667-1)T2=T1(2.9612)T2=270(2.9612)T2=799.55k

Downstream velocity of speed is given by the formula,

  V2=Ma2c2V2=Ma2kRT2V2=0.54551.667×2.08×799.55× 10001V2=908.27m/s

Downstream stagnation pressure is given by formula,

  P 02P1=( 1+kMa2 1 1+kMa2 2 )(1+ ( k-1)M a 2 2 2)k/(k-1)P02=58×1000( 1+1.667×2. 6 2 1+1.667×0.545 5 2( 1+ ( 1.667-1 )0.545 5 2 2))1.667/(1.667-1)P02=602.57kPa

Conclusion:

Therefore, we can concludeMach number and velocity are greater for helium than for air because of the different values of .

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Chapter 12 Solutions

Fluid Mechanics: Fundamentals and Applications

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