Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
8th Edition
ISBN: 9780073398174
Author: Yunus A. Cengel Dr., Michael A. Boles
Publisher: McGraw-Hill Education
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Chapter 17.7, Problem 105P

Air is heated as it flows subsonically through a duct. When the amount of heat transfer reaches 67 kJ/kg, the flow is observed to be choked, and the velocity and the static pressure are measured to be 680 m/s and 270 kPa. Disregarding frictional losses, determine the velocity, static temperature, and static pressure at the duct inlet.

Expert Solution & Answer
Check Mark
To determine

The static temperature in the duct.

The static pressure in the duct.

The velocity in the duct.

Answer to Problem 105P

The static temperature in the duct is 1172K_.

The static pressure in the duct is 351.3kPa_.

The velocity in the duct is 533.9m/s_.

Explanation of Solution

Determine the speed of sound at the exit.

Ma2=V2c2c2=V2Ma2 (I)

The exit velocity of the air flow in the device is V2 and the exit Mach number is Ma2.

Determine the relation of ideal gas speed of sound at the exit.

c2=kRT2 (II)

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 T2.

Determine the exit stagnation temperature of air.

T02=T2+V222cp (III)

Here, the exit static temperature of ideal gas is T2, the specific heat of pressure for ideal gas is cp, and the exit velocity of the ideal gas flow is V2.

Determine the inlet stagnation temperature from energy equation.

q=cp(T02T01)T01=T02qcp (IV)

Here, the heat transfer to the duct is q.

Determine the stagnation temperature ratio at the inlet.

Ratiostagnation=T01T0 (V)

Here, the maximum value of stagnation temperature is T0.

Determine the static temperature in the duct.

T2T1=T2/TT1/T (VI)

Here, the ratio of Rayleigh flow for inlet temperature is T1/T and the ratio of Rayleigh flow for outlet temperature is T2/T.

Determine the static pressure in the duct.

P2P1=P2/PP1/P (VII)

Here, the ratio of Rayleigh flow for inlet pressure is P1/P and the ratio of Rayleigh flow for outlet pressure is P2/P.

Determine the velocity in the duct.

V2V1=V2/VV1/V (VIII)

Here, the ratio of Rayleigh flow for inlet velocity is V1/V and the ratio of Rayleigh flow for outlet velocity is V2/V.

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 300K temperature as 0.287kJ/kgK, 1.005kJ/kgK and 1.400.

Substitute 1 for Ma2 and 680m/s for V2 in Equation (I).

c2=(680m/s)1=680m/s

Substitute 1.400 for k, 0.287kJ/kgK for R, and 680m/s for c2 in Equation (II).

(680m/s)=(1.400)(0.287kJ/kgK)×T2(680m/s)=(1.400)(0.287kJ/kgK)×(1000m2/s21kJ/kg)×T2T2=1151K

Substitute 1151K for T2, 680m/s for V, and 1.005kJ/kgK for cp in Equation (III).

T0=(1151K)+(680m/s)22×(1.005kJ/kgK)=(1151K)+(680m/s)2(2.01kJ/kgK)=(1151K)+(462400m2/s2)×(1kJ/kg1000m2/s2)(2.01kJ/kgK)=1381K

Substitute 1381K for T02, 67kJ/kg for q and 1.005kJ/kgK for cp in Equation (V).

T01=1381K67kJ/kg1.005kJ/kgK=1381K1314K=67K

Substitute 1314K for T01 and 1381K for T0 in Equation (VI).

Ratiostagnation=1314K1381K=0.951480.9515

Refer to Table A-34, “Rayleigh flow function for an ideal gas with k=1.4”, to obtain the value inlet Mach number at 0.9515 ratio of stagnation temperature using interpolation method of two variables.

Write the formula of interpolation method of two variables.

y2=(x2x1)(y3y1)(x3x1)+y1 (IX)

Here, the variables denote by x and y is ratio of stagnation temperature and Mach number.

Show the ratio of stagnation temperature at 0.7 and 0.8 as in Table (1).

S. No

ratio of stagnation temperature

(x)

Mach number

(y)

10.90850.7
20.9516y2=?
30.96390.8

Calculate inlet Mach number at 0.9515 ratio of stagnation temperature using interpolation method.

Substitute 0.9085 for x1, 0.9516 for x2, 0.9639 for x3, 0.7 for y1, and 0.8 for y3 in Equation (IX).

y2=(0.95160.9085)(0.80.7)(0.96390.9085)+0.7=0.7777980.7778

From above calculation the inlet Mach number at 0.9515 ratio of stagnation temperature is 0.7778.

Repeat the Equation (IX), to obtain the value of inlet ratio of temperature, pressure, and velocity at 0.7778 inlet Mach number as:

T1/T=1.018P1/P=1.301V1/V=0.7852

From the Table A-34, “Rayleigh flow function for an ideal gas with k=1.4”, to obtain the value of the outlet ratio of temperature, pressure, and velocity at 1 outlet Mach number as:

T2/T=1P2/P=1V2/V=1

Substitute 1151 K for T2, 1 for T2/T, and 1.018 for T1/T in Equation (VI).

(1151K)T1=11.018T1=(1.018)×(1151K)T1=1171.7KT11172K

Thus, the static temperature in the duct is 1172K_.

Substitute 270 kPa for P2, 1 for P2/P, and 1.301 for P1/P in Equation (VII).

(270kPa)P1=11.301P1=(1.301)×(270kPa)P1=351.27kPaP1351.3kPa

Thus, the static pressure in the duct is 351.3kPa_.

Substitute 680m/s for V2, 1 for V2/V, and 0.7852 for P1/P in Equation (VIII).

(680m/s)V1=10.7852V1=(0.7852)×(680m/s)V1=533.9m/s

Thus, the velocity in the duct is 533.9m/s_.

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Chapter 17 Solutions

Thermodynamics: An Engineering Approach

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