Problem 118 Room air is drawn into an insula... [FREE SOLUTION] (2024)

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Chapter 13: Problem 118

Room air is drawn into an insulated duct of constant area through a smoothlycontoured converging nozzle. Room conditions are \(T=80^{\circ} \mathrm{F}\) and\(p=14.7\) psia. The duct diameter is \(D=1\) in. The pressure at the duct inlet(nozzle outlet \()\) is \(p_{1}=13\) psia. Find \((a)\) the mass flow rate in theduct and (b) the range of exit pressures for which the duct exit flow ischoked.

Short Answer

Expert verified

The mass flow rate in the duct is approximately \( 1.08 \) lbm/sec. The duct exit flow is choked for any exit pressure less than \( 10.15 \) psia.

Step by step solution

Calculate absolute temperature

First, convert the given room temperature from Fahrenheit to Rankin because Rankin is the absolute scale required for these calculations. Use the formula \( T(R) = T(F) + 459.67 \) to find that the absolute room temperature is \( 539.67 R \).

Calculate the area of the duct inlet

Next, calculate the cross-sectional area of the duct at the inlet using \( A_{1} = \pi D^2/4 \), where D is the duct diameter. Given that \( D = 1 \) in, the area is \( A_{1} = 0.7854 \) sq.in. Converting to square feet (since velocity will be in ft/s) for the mass flow rate calculation, one gets \( A_{1} = 0.005454 \) sq. ft.

Find the mass flow rate

The formula for the mass flow rate in the duct, \( \dot{m} \), for ideal gases is \( \dot{m} = p_{1} A_{1} \sqrt{\gamma/R T_{1}} \), where \( p_{1} \) is the duct inlet pressure, \( A_{1} \) is the duct inlet area, \( \gamma \) is the heat capacity ratio, R is the specific gas constant, and \( T_{1} \) is the absolute room temperature. Given that \( \gamma \) for air is 1.4, \( R \) is 1716 ft lb/slug R and \( T_{1} = 539.67 R \), and remembering to convert the given pressures from psia to psfa and area from sq. in to sq. ft, we find that \( \dot{m} = 0.0335 \) slugs/sec. Converting to lbm/sec (since 1 slug = 32.2 lbm), we find the mass flow rate to be about \( 1.08 \) lbm/sec.

Establish the choked flow conditions

The flow is said to be 'choked' when the Mach number is 1. Thus, the exit pressure for choked flow, \( p^* \), can be found using the isentropic relation \( p/p^* = (1 + (\gamma - 1)/2)^{\gamma/(\gamma-1)} \). Solve for \( p^* \) to get \( p^* = p/(1 + (\gamma - 1)/2)^{\gamma/(\gamma-1)} \). Substituting the given duct inlet pressure and \( \gamma \) for air yields \( p^* \approx 10.15 \) psia. Therefore, the flow will be choked for any pressure lower than \( 10.15 \) psia.

Conclusion

In conclusion, the mass flow rate in the duct is approximately \( 1.08 \) lbm/sec and the range of exit pressures for which the duct exit flow is choked is less than \( 10.15 \) psia.

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Most popular questions from this chapter

Air flows into a converging duct, and a normal shock stands at the exit of theduct. Downstream of the shock, the Mach number is \(0.54 .\) If \(p_{2} /p_{1}=2,\) compute the Mach number at the entrance of the duct and the arearatio \(A_{1} / A_{2}\).An airplane is flying at Mach \(5,\) where \(T_{1}=216.67 \mathrm{K}\) Obliqueshocks form in the intake prior to entering the combustion chamber ?. Thenozzle expansion ratio is \(A_{s} / A_{4}=5 .\) The inlet and exit areas areequal, \(A_{1}=A_{5}=0.2 \mathrm{m}^{2}\) Assuming isentropic flow with\(M_{2}=4, \quad M_{3}=3.295,\) and \(M_{4}=1.26,\) calculate the exit Mach numberand the exhaust jet velocity ?. Hint: Calculate the temperature ratios in eachsection.An airplane is flying at Mach 5 at an altitude of \(16,764 \mathrm{m},\) where\(T_{1}=216.67 \mathrm{K}\) and \(p_{1}=9.122 \mathrm{kPa} .\) The airplane uses ascramjet engine. Two oblique shocks are formed in the intake ? prior toentering the combustion chamber ? at supersonic speed. The inlet and exitareas are equal, \(A_{1}=A_{5}=0.2 \mathrm{m}^{2} .\) Calculate the stagnationtemperature \(T_{2} / T_{1},\) and the Mach number in the intake ?.Consider frictionless flow of air in a duct with \(D=10 \mathrm{cm}\) At section? the temperature and pressure are \(0^{\circ} \mathrm{C}\) and \(70\mathrm{kPa}\) the mass flow rate is \(0.5 \mathrm{kg} / \mathrm{s}\). How muchheat may be added without choking the flow? Evaluate the resulting change instagnation pressure.Testing of a demolition explosion is to be evaluated. Sensors indicate thatthe shock wave generated at the instant of explosion is \(30 \mathrm{MPa}\)(abs). If the explosion occurs in air at \(20^{\circ} \mathrm{C}\) and \(101\mathrm{kPa}\), find the speed of the shock wave, and the temperature and speedof the air just after the shock passes. As an approximation assume \(k=1.4 .\)(Why is this an approximation?)

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Problem 118 Room air is drawn into an insula... [FREE SOLUTION] (2024)

Short Answer

Step by step solution

Calculate absolute temperature

Calculate the area of the duct inlet

Find the mass flow rate

Establish the choked flow conditions

Conclusion

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Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Solid State Physics

Modern Physics

Rotational Dynamics

Work Energy and Power

Torque and Rotational Motion

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Privacy Overview

References