Explanation Determine the relation of ideal gas speed of sound at the inlet. c 1 = k R T 1 (I) Here, the specific heat ratio of air is k , the gas constant of the air is R , and the exit temperature of the air is T 1 . Determine the speed of sound at the inlet. Ma 1 = V 1 c 1 V 1 2 = Ma 1 × c 1 (II) The inlet velocity of the air flow in the device is V 1 and the inlet Mach number is Ma 1 . Determine the inlet stagnation temperature of air. T 01 = T 1 + V 1 2 2 c p (III) Here, the inlet static temperature of ideal gas is T 1 , the specific heat of pressure for ideal gas is c p , and the inlet velocity of the ideal gas flow is V 1 . Determine the exit stagnation temperature from energy equation. q = c p ( T 02 − T 01 ) T 02 = T 01 + q c p (IV) Here, the heat transfer to the duct is q . Determine the maximum value of stagnation temperature is T 0 ∗ . T 0 ∗ = T 01 T 01 / T 0 ∗ (V) Determine the stagnation temperature ratio at the exit. Ratio stagnation = T 02 T 0 ∗ (VI) Here, the maximum value of stagnation temperature is T 0 ∗ . Determine the static temperature in the duct. T 2 T 1 = T 2 / T ∗ T 1 / T ∗ (VII) Here, the ratio of Rayleigh flow for inlet temperature is T 1 / T ∗ and the ratio of Rayleigh flow for outlet temperature is T 2 / T ∗ . Conclusion: From the Table A-2, “Ideal-gas specific heats of various common gases” to obtain value of universal gas constant, specific heat of pressure, and the specific heat ratio of air at 300 K temperature as 0.287 kJ / kg ⋅ K , 1.005 kJ / kg ⋅ K and 1.400 . Substitute 1.400 for k , 0.287 kJ / kg ⋅ K for R , and 300 K for T 1 in Equation (I). c 1 = ( 1.400 ) ( 0.287 kJ / kg ⋅ K ) × ( 300 K ) = ( 120.54 kJ / kg ) × ( 1000 m 2 / s 2 1 kJ / kg ) = 347.188 ≅ 347.2 Substitute 2 for Ma 2 and 347.2 for c 1 in Equation (II). V 1 = 2 × ( 347.2 m / s ) = 694.4 m / s Substitute 300 K for T 1 , 694.4 m / s for V 1 , and 1.005 kJ / kg ⋅ K for c p in Equation (III). T 01 = ( 300 K ) + ( 694.4 m / s ) 2 2 × ( 1.005 kJ / kg ⋅ K ) = ( 300 K ) + ( 694.4 m / s ) 2 ( 2.01 kJ / kg ⋅ K ) = ( 300 K ) + ( 482191 m 2 / s 2 ) × ( 1 kJ / kg 1000 m 2 / s 2 ) ( 2

9th Edition

ISBN: 9781259822674

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

Publisher: McGraw-Hill Education

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Chapter 17.7, Problem 106P

To determine

The exit Mach number of the duct.

The exit temperature in the duct.

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Chapter 17.7, Problem 105P

Chapter 17.7, Problem 107P

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The exit Mach number of the duct. The exit temperature in the duct.

Solution Summary: The author explains the exit Mach number of the duct and the temperature in it, and determines the ideal gas speed of sound at the inlet.