Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 12, Problem 75P
To determine

Entropy change of air and helium across the normal shock wave.

Expert Solution & Answer
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Answer to Problem 75P

  (s2-s1)air =0.223kJ/kg.K

  (s2-s1)helium =1.27kJ/kg.K

Explanation of Solution

Given:

The properties of air are:

   k = 1.4R = 0.287 kJ/kg·K,cp = 1.005 kJ/kg·K

The properties of helium are,

  k = 1.667R = 2.0769 kJ/kg·K.cp = 5.1926kJ/kg·K

The air properties upstream the shock is, Pressure P1=58kPa

Temperature T1=270K

Mach number Ma1=2.6

Calculation:

Assumptions

  1. Air and helium are ideal gases with same specific heats.
  2. Flow through the nozzle is constant, single dimensional and isentropic before the shock occurs.

Figure shows the air flow with the Mach number,

Fluid Mechanics: Fundamentals and Applications, Chapter 12, Problem 75P

Downstream Mach number is given by,

  Ma2=( k1)M a 1 2+22kM a 1 2k+1Ma2=( 1.41) 2.6 2+22×1.4× 2.6 21.4+1Ma2=0.5039

Downstream pressure ratio is given by,

  P2P1=1+kMa211+kMa22P2P1=1+1.4×2.621+1.4×0.50392P2P1=7.72

Downstream temperature ratio is given by,

  T2T1=1+Ma21(k1)1+Ma22(k1)T2T1=1+2.62(1.41)1+0.50392(1.41)T2T1=2.2383

Calculate the air, entropy change across the shock ,

  s2-s1 = cp InT2T1-RInP2P1s2-s1 =1.005In(2.2383)-0.287In(7.7200)s2-s1 =0.223kJ/kg.K

Downstream Mach number for helium is given by,

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

Downstream pressure ratio is given by,

  P2P1=1+kMa211+kMa22P2P1=1+1.667×2.621+1.667×0.54552P2P1=8.2009

Downstream temperature ratio is given by,

  T2T1=1+Ma21(k1)/21+Ma22(k1)/2T2T1=1+2.62(1.6671)/21+0.54552(1.6671)/2T2T1=2.9606

Calculate the helium, entropy change across the shock ,

  s2-s1 = cp InT2T1-RInP2P1s2-s1 =5.1926In(2.9606)-2.0769In(8.2009)s2-s1 =1.27kJ/kg.K

Conclusion:

Hence, the entropy change of air across the normal shock wave is:

  (s2-s1)air =0.223kJ/kg.K

The entropy change of helium across the normal shock wave is:

  (s2-s1)helium =1.27kJ/kg.K

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Chapter 12 Solutions

Fluid Mechanics: Fundamentals and Applications

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