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Chapter 13: Problem 57

Air flows isentropically through a converging-diverging nozzle attached to a large tank, in which the pressure is 251 psia and the temperature is \(500^{\circ} \mathrm{R}\). The nozzle is operating at design conditions for which the nozzle exit pressure, \(p_{e},\) is equal to the surrounding atmospheric pressure, \(p_{a}\). The exit area of the nozzle is \(A_{e}=1.575\) in. \(^{2}\). Calculate the flow rate through the nozzle. Plot the mass flow rate as the temperature of the tank is progressively increased to \(2000^{\circ} \mathrm{R}\) (all pressures remaining the same). Explain this result (e.g., compare the mass flow rates at \(500^{\circ} \mathrm{R}\) and \(2000^{\circ} \mathrm{R}\) ).

Short Answer

Expert verified
The flow rate of air through the nozzle increases as the temperature in the tank is increased from \(500^{\circ} \mathrm{R}\) to \(2000^{\circ} \mathrm{R}\). This is because the increase in temperature leads to a higher velocity of the air through the nozzle, resulting in a higher mass flow rate.

Step by step solution

01

Understand the problem and identify the given data

The problem is to calculate the flow rate of air through a rozzle attached to a large tank. The data provided include: the pressure in the tank (\(p\)) is 251 psia, the exit area of the nozzle (\(A_e\)) is 1.575 square inches, the nozzle exit pressure (\(p_e\)) is equal to the surrounding atmospheric pressure (\(p_a\)), and the initial temperature is \(500^{\circ} \mathrm{R}\)
02

Convert pressure to absolute units

The pressure is given in psia which is already an absolute unit. Therefore, no conversions are necessary in this step
03

Apply the formula for flow rate

The flow rate (\(q\)) can be calculated using the formula \(q = A_v \cdot v\) where \(A_v\) is the nozzle exit area and \(v\) is the velocity of the air. The velocity can be calculated from the isentropic flow relations for a perfect gas. The equation becomes \(q = A_e \cdot \sqrt{\gamma R T} \cdot M \cdot \left( \frac{2}{\gamma+1} \right)^{(\gamma+1)/(2(\gamma-1))}\) where \(R\) is the specific gas constant, \(T\) is the absolute temperature, and \(M\) is the Mach number
04

Calculate the flow rate at the initial temperature

Substitute the given values into the flow rate equation to calculate the initial flow rate. Keep in mind that the Mach number at the exit of the nozzle is 1 when the nozzle is operating at design condition
05

Increase the temperature and recalculate the flow rate

Progressively increase the temperature to \(2000^{\circ} \mathrm{R}\) while keeping the pressures constant, and calculate the new flow rates. This will provide the variation of the mass flow rate with the temperature
06

Explain the result

Compare the mass flow rates at \(500^{\circ} \mathrm{R}\) and \(2000^{\circ} \mathrm{R}\). The increase in temperature causes the air to expand, leading to a higher velocity through the nozzle and hence a higher mass flow rate

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Most popular questions from this chapter

A normal shock occurs in the diverging section of a converging-diverging nozzle where \(A=25 \mathrm{cm}^{2}\) and \(M=\) \(2.75 .\) Upstream, \(T_{0}=550 \mathrm{K}\) and \(p_{0}=700 \mathrm{kPa}(\mathrm{abs}) .\) The nozzle exit area is \(40 \mathrm{cm}^{2}\). Assume the flow is isentropic except across the shock. Determine the nozzle exit pressure, throat area, and mass flow rate.

A normal shock stands in a constant-area duct. Air approaches the shock with \(T_{0_{1}}=550 \mathrm{K}, p_{0_{1}}=650 \mathrm{kPa}(\mathrm{abs})\) and \(M_{1}=2.5 .\) Determine the static pressure downstream from the shock. Compare the downstream pressure with that reached by decelerating isentropically to the same subsonic Mach number.

A supersonic aircraft cruises at \(M=2.7\) at \(60,000 \mathrm{ft}\) altitude. A normal shock stands in front of a pitot tube on the aircraft; the tube senses a stagnation pressure of 10.4 psia. Calculate the static pressure and temperature behind the shock. Evaluate the loss in stagnation pressure through the shock. Determine the change in specific entropy across the shock. Show static and stagnation states and the process path on a \(T s\) diagram.

Air is flowing steadily through a series of three tanks. The first very large tank contains air at \(650 \mathrm{kPa}\) and \(35^{\circ} \mathrm{C}\). Air flows from it to a second tank through a converging nozzle with exit area \(1 \mathrm{cm}^{2}\). Finally the air flows from the second tank to a third very large tank through an identical nozzle. The flow rate through the two nozzles is the same, and the flow in them is isentropic. The pressure in the third tank is 65 kPa. Find the mass flow rate, and the pressure in the second tank.

Air flows adiabatically through a duct. At the entrance, the static temperature and pressure are \(310 \mathrm{K}\) and \(200 \mathrm{kPa}\) respectively. At the exit, the static and stagnation temperatures are \(294 \mathrm{K}\) and \(316 \mathrm{K},\) respectively, and the static pressure is \(125 \mathrm{kPa}\). Find (a) the Mach numbers of the flow at the entrance and exit and (b) the area ratio \(A_{2} / A_{1}\).
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