Explanation Write the formula stagnation temperature of helium at the inlet. T 01 = T 1 + V 1 2 2 c p (I) Here, actual (static) temperature of helium at the inlet is T 1 , specific heat at constant pressure is c p and stagnation temperature of helium at inlet is T 01 . Since, the flow is isentropic. Write the formula for stagnation pressure of helium at inlet (isentropic condition). P 01 = P 1 ( T 01 T 1 ) k / ( k − 1 ) (II) Here, the static pressure of helium at inlet is P 1 , the stagnation pressure of helium at inlet is P 01 , and the specific heat ratio is k . It is given that the temperature and pressure of helium to be found at the location where the velocity of helium equals to the velocity of the sound. This statement indicates that the exit Mach number is 1. Since the flow is isentropic, the inlet and exit stagnation properties are equal. T 01 = T 02 P 01 = P 02 Write the formula for the ratio of critical (throat) temperature to the stagnation temperature. T ∗ T 0 = 2 k + 1 T ∗ = T 0 ( 2 k + 1 ) (III) Write the formula for the ratio of critical (throat) pressure to the stagnation pressure. P ∗ P 0 = ( 2 k + 1 ) k / ( k − 1 ) P ∗ = P 0 ( 2 k + 1 ) k / ( k − 1 ) (IV) Here, the superscript * indicates the critical state (throat) and the subscript 0 indicates stagnation state. Write the formula for velocity of sound at the inlet conditions. c 1 = k R T 1 (V) Here, speed of sound at the inlet condition is c 1 , gas constant of helium is R , actual temperature of helium at the inlet is T 1 and ratio of specific heats is k . Write formula for the upstream Mach number ( Ma 1 ) . Ma 1 = V 1 c 1 (VI) Here, the velocity of helium is V 1 and the velocity of sound is c 1 . Write the formula for ratio of the inlet area to the critical (throat) area. A 1 A ∗ = 1 Ma 1 [ ( 2 k + 1 ) ( 1 + k − 1 2 Ma 1 2 ) ] ( k + 1 ) / [ 2 ( k − 1 ) ] (VII) Refer Table A-1, “Molar mass, gas constant, and critical-point properties”. The gas constant ( R ) of helium is 2.0769 kJ/kg ⋅ K or 2.0769 kPa ⋅ m 3 /kg ⋅ K . Refer Table A-2, “Ideal-gas specific heats of various common gases”. The specific heat at constant pressure ( c p ) of helium is 5.1926 kJ/kg ⋅ K . The specific heat ratio ( k ) of helium is 1.667 . Conclusion: Substitute 600 K for T 1 , 120 m/s for V 1 , and 5.1926 kJ/kg ⋅ K for c p in Equation (I). T 01 = 600 K + ( 120 m/s ) 2 2 ( 5.1926 kJ/kg ⋅ K ) = 600 K + 14400 m 2 /s 2 × 1 kJ/kg 1000 m 2 /s 2 10.3852 kJ/kg ⋅ K = 600 K + 1.3866 K ≃ 601.4 K Substitute 0

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ISBN: 9781259822674

Author: Yunus A. Cengel Dr., Michael A. Boles

Publisher: McGraw-Hill Education

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Chapter 17.7, Problem 123RP

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The pressure and temperature of the helium at a location where the velocity equals the speed of sound and what is ratio of the area at this location to the entrance area.

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Chapter 17.7, Problem 122RP

Chapter 17.7, Problem 124RP

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Chapter 17 Solutions

Thermodynamics: An Engineering Approach

Ch. 17.7 - A high-speed aircraft is cruising in still air....Ch. 17.7 - What is dynamic temperature?Ch. 17.7 - Prob. 3P Ch. 17.7 - Prob. 4P Ch. 17.7 - Prob. 5P Ch. 17.7 - Prob. 6P Ch. 17.7 - Calculate the stagnation temperature and pressure...Ch. 17.7 - Prob. 8P Ch. 17.7 - Prob. 9P Ch. 17.7 - Prob. 10P

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The pressure and temperature of the helium at a location where the velocity equals the speed of sound and what is ratio of the area at this location to the entrance area.

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The pressure and temperature of the helium at a location where the velocity equals the speed of sound and what is ratio of the area at this location to the entrance area.

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