To determine The maximum mass flow rate through the duct. The static temperature at inlet and exit of duct. The static pressure at inlet and exit of duct. The stagnation temperature at inlet and exit of duct. The stagnation pressure at inlet and exit of duct. The velocity at inlet and exit of duct. The sketch of temperature-entropy diagram for the flow. Explanation Write the equation for the continuity equation at entry and exit of the duct. ρ 1 A 1 V 1 = ρ 2 A 2 V 2 (I) Here, cross-sectional area of duct at entry is A 1 , the density of air at entry of duct is ρ 1 , velocity of air at entry of duct is V 1 , density of air at exit of duct is ρ 2 , cross-sectional area of duct at exit is A 2 , and velocity of air at exit of duct is V 2 . The flow is adiabatic. Therefore, stagnation temperature will be constant throughout. T 01 = T 02 Here, duct stagnation temperature at inlet is T 01 and duct stagnation temperature at outlet is T 02 . Write the expression for chocked flow. f ( l * − l 1 ) D = f ( l 2 − l 1 ) D (II) Here, friction factor is f and critical length of duct is l * and length of duct is ( l 2 − l 1 ) and diameter of duct D . Write the expression for critical temperature. T * = ( 2 k + 1 ) T 0 (III) Here, adiabatic index for air k and standard atmospheric temperature T 0 . Write the expression for critical velocity. V * = R T * k (IV) Here, gas constant is R . Write the expression for the temperature at inlet and critical condition. T 1 T * = 1.13 (V) Here, temperature at inlet of the duct is T 1 and critical temperature is T * . Write the expression for the volume at inlet and critical condition. V 1 V * = 0.6 (VI) Here, volume at inlet of the duct is V 1 and critical volume is V * . Write the expression for the pressure at inlet and critical condition. p 1 p * = 1.86 (VII) Here, pressure at inlet of the duct is p 1 and critical pressure is p * . Write the expression for the stagnation pressure at inlet and critical condition p 01 p 0 * = 1.22 (VIII) Here, stagnation pressure at inlet of the duct is P 1 and critical stagnation pressure is p 0 * . Write the expression for the mass flow rate m = P 1 R T 1 × π D 2 4 × V 1 (IX) Write the expression for stagnation pressure at inlet pressure. p 1 p 01 = [ 1 1 + [ ( k − 1 ) 2 M a 1 2 ] ] k k − 1 (X) Here, Mach number at entry is M a 1 . Write the expression for duct exit pressure. p 2 = ( p * p 1 ) × ( p 1 p 01 ) × p 01 (XI) The stagnation pressure at the entrance of the duct is same as the value of stagnation pressure through the isentropic nozzle. p 01 = 14.7 psia For the duct exit stagnation pressure. p 02 = p 0 * p 02 = p 0 * p 01 ( p 01 ) (XII) Write the expression for specific heat at constant pressure. C p = R k k − 1 (XIII) Write the expression for the change in entropy. s 2 − s 1 = c p ln ( T 2 T 1 ) − R ln ( p 2 p 1 ) (XIV) Conclusion: Substitute, 0.03 for f , 10 ft for ( l 2 − l 1 ) and 0.5 ft for D in Equation (II). f ( l * − l 1 ) D = ( 0.03 ) ( 10 ft ) 0.5 ft = 0.6 Substitute, ( 59 ° F ) for T 0 and 1.4 for k in Equation (III). T * = ( 2 1.4 + 1 ) ( 59 ° F ) = ( 2 1.4 + 1 ) ( 59 ° F+ 459.67 ) ° R = ( 2 1.4 + 1 ) ( 518.67 ° R ) = 432.5 ° R Substitute, ( 432.5 ° R ) for T * and 1.4 for k and 1716 ft .lb / slug . ° R in Equation (IV). V * = 1716 ft .lb / slug . ° R × ( 432.5 ° R ) × ( 1.4 ) = ( 1716 ft .lb slug ° R × ( 1 slug . ft s 2 1 lb ) ) × ( 432.5 ° R ) × ( 1.4 ) = 1020 ft / s Substitute ( 432 ° R ) for T * in Equation (V). T 1 ( 432 ° R ) = 1.13 T 1 = 488 ° R Substitute 1020 ft / s for V * in Equation (VI)

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

8th Edition

ISBN: 9781119080701

Author: Philip M. Gerhart, Andrew L. Gerhart, John I. Hochstein

Publisher: WILEY

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Question

Chapter 11.8, Problem 56P

To determine

The maximum mass flow rate through the duct.

The static temperature at inlet and exit of duct.

The static pressure at inlet and exit of duct.

The stagnation temperature at inlet and exit of duct.

The stagnation pressure at inlet and exit of duct.

The velocity at inlet and exit of duct.

The sketch of temperature-entropy diagram for the flow.

Expert Solution & Answer

Chapter 11 Solutions

Munson, Young and Okiishi's Fundamentals of Fluid Mechanics, Binder Ready Version

Ch. 11.1 - Prob. 1P Ch. 11.1 - Air flows steadily between two sections in a duct....Ch. 11.1 - Consider the flow process in Fig. P11.3. Does the...Ch. 11.1 - Prob. 4P Ch. 11.1 - Prob. 5P Ch. 11.1 - Prob. 6P Ch. 11.1 - Prob. 7P Ch. 11.1 - Prob. 8P Ch. 11.1 - Air flows in a 15-cm-diameter horizontal pipe. At...Ch. 11.1 - An air heater in a large coal-fired steam...

Ch. 11.2 - Determine the static pressure to stagnation...Ch. 11.2 - Prob. 12P Ch. 11.2 - Prob. 13P Ch. 11.2 - Prob. 14P Ch. 11.2 - Prob. 15P Ch. 11.3 - Prob. 16P Ch. 11.3 - Prob. 17P Ch. 11.3 - Prob. 18P Ch. 11.3 - Prob. 19P Ch. 11.3 - Prob. 20P Ch. 11.3 - Prob. 21P Ch. 11.3 - Prob. 22P Ch. 11.4 - Prob. 23P Ch. 11.4 - Prob. 24P Ch. 11.4 - Prob. 25P Ch. 11.4 - Prob. 26P Ch. 11.4 - Prob. 27P Ch. 11.4 - Air flows in a constant-area, insulated duct. The...Ch. 11.5 - Prob. 29P Ch. 11.5 - The Pitot tube on a supersonic aircraft (see Video...Ch. 11.5 - An aircraft cruises at a Mach number of 2.0 at an...Ch. 11.5 - Prob. 32P Ch. 11.5 - Prob. 33P Ch. 11.5 - Prob. 34P Ch. 11.5 - Prob. 35P Ch. 11.6 - The stagnation pressure and temperature of air...Ch. 11.6 - Prob. 37P Ch. 11.6 - Prob. 38P Ch. 11.6 - Prob. 39P Ch. 11.6 - Prob. 40P Ch. 11.6 - Prob. 41P Ch. 11.6 - The static pressure to stagnation pressure ratio...Ch. 11.7 - Air flows steadily and isentropically from...Ch. 11.7 - Prob. 44P Ch. 11.7 - Prob. 45P Ch. 11.7 - Prob. 46P Ch. 11.7 - At a certain point in a pipe, air flows steadily...Ch. 11.7 - Prob. 48P Ch. 11.7 - Prob. 49P Ch. 11.7 - Prob. 50P Ch. 11.7 - A jet engine is to be designed for an altitude of...Ch. 11.7 - Prob. 52P Ch. 11.7 - A convergent–divergent nozzle has an exit throat...Ch. 11.7 - An ideal gas flows isentropically through a...Ch. 11.8 - Standard atmospheric air (T0 = 59 F, p0 = 14.7...Ch. 11.8 - Prob. 57P Ch. 11.8 - Prob. 58P Ch. 11.8 - Prob. 59P Ch. 11.8 - Prob. 64P Ch. 11.9 - Prob. 66P Ch. 11.9 - Prob. 67P Ch. 11.9 - Prob. 68P Ch. 11.9 - Air enters a frictionless, constant area duct with...Ch. 11.9 - Prob. 70P Ch. 11.9 - Prob. 71P Ch. 11.9 - Prob. 72P Ch. 11.9 - Prob. 73P

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