/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 71 A convergent-divergent nozzle op... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 71

A convergent-divergent nozzle operates at NPR of \(14.86\) in an altitude where the ambient pressure is \(p_{0}=30\) \(\mathrm{kPa}\). Assuming the flow in the nozzle is isentropic with \(\gamma=\) 1.3, calculate (a) nozzle area ratio, \(A_{9} / A_{8}\), if the nozzle is perfectly expanded 5 (b) exit Mach number, \(M_{9}\)

Short Answer

Expert verified
To answer part (a), the calculated nozzle area ratio is obtained by substituting the given values into the isentropic relations formula. For part (b), exit Mach number is evaluated using the formula from the isentropic gas relations, validating its value makes sure our assumption of isentropic flow holds true.

Step by step solution

01

Calculate exit pressure

To start with, we will calculate the exit pressure \(p_9\) which is given by the formula, \(p_9 = p_0 / NPR\) for a perfectly expanded nozzle. Here, \(p_0 = 30 \, \text{kPa}\) is the ambient pressure and NPR is the nozzle pressure ratio, \(NPR = 14.86\). Thus, substituting the given values we get the exit pressure.
02

Calculate the nozzle area ratio

Next, we calculate the nozzle area ratio \(A_9/A_8\). This ratio can be obtained from the isentropic relations and is given by the formula \(A_9/A_8 = [(1/M_9) * (2/(饾浘+1) * (1+((饾浘-1)/2) * M_9^2))^((饾浘+1)/(2*(饾浘-1))]\). Here, the exit Mach number \(M_9 = 1\), since the flow is perfectly expanded and \(饾浘 = 1.3\) is the specific heat ratio. Substituting these values yields \(A_9/A_8\).
03

Calculating the exit Mach number

The exit Mach number \(M_9\) is obtained from the isentropic gas relations. The formula to find this is \(M_9 = sqrt [(2/(饾浘-1)) * ((p_0/p_9)^((饾浘-1)/饾浘) - 1)]\). Substituting the values of \(饾浘 = 1.3\), \(p_0 = 30\) kPa, and the calculated value of \(p_9\) allows us to find \(M_9\). Simultaneously, also validate if the obtained Mach number is greater than zero and less than infinity, which verifies our assumption of isentropic flow.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

convergent-divergent nozzle
A convergent-divergent nozzle is used to accelerate a gas to supersonic speeds. This type of nozzle features two sections: a converging section where the cross-sectional area decreases, and a diverging section where the area increases again. As gas moves through this nozzle, the converging part compresses the gas, increasing its velocity. When the flow reaches the throat of the nozzle (the narrowest part), it can reach sonic speed, or Mach 1. At the diverging part, the gas can further accelerate, often reaching supersonic speeds (greater than Mach 1). Convergent-divergent nozzles are crucial in applications requiring controlled supersonic flow, such as in rocket engines and jet turbines.
  • The design ensures efficient expansion and acceleration of gases.
  • When perfectly expanded, the exit pressure matches the external pressure, ensuring ideal efficiency.
nozzle pressure ratio
The nozzle pressure ratio (NPR) is a critical factor in assessing how a nozzle operates, particularly in isentropic flows. NPR is defined as the ratio of the upstream stagnation pressure to the downstream (or ambient) pressure. This ratio dictates how gases expand when passing through a nozzle. For instance, a higher NPR indicates a greater pressure difference, allowing more extensive expansion and acceleration of the flow to supersonic speeds in convergent-divergent nozzles.
  • An NPR greater than 1 indicates accelerating flow within the nozzle is possible.
  • At specific NPRs, nozzles can achieve maximum efficiency by perfectly expanding the flow.
The correct NPR ensures the effective transition of subsonic to supersonic flow without any shock waves that might reduce efficiency.
specific heat ratio
The specific heat ratio (\( \gamma \)) is the ratio of the heat capacity at constant pressure (\( C_p \)) to the heat capacity at constant volume (\( C_v \)) for a gas. This is a vital parameter in thermodynamics and aerodynamics, describing how a gas behaves under different thermodynamic processes.
  • For air, the typical value of \( \gamma \) is around 1.4.
  • In this exercise, \( \gamma \) = 1.3, tailoring the calculations to the particular gas used in the nozzle analysis.
In isentropic flow conditions鈥攚hich assume no heat exchange and no friction鈥攖he specific heat ratio influences how the gas's pressure, density, and temperature change. Specifically, it affects calculations for things like exit pressures and Mach numbers in nozzle flows. Systems with different values of \( \gamma \) need distinct considerations in the engineering process.
Mach number
The Mach number is a dimensionless unit representing the ratio of an object's speed to the speed of sound in the given medium. In aerodynamics and fluid dynamics, it provides key information about the flow characteristics. When the Mach number is below 1, the flow is subsonic; at exactly 1, it is sonic; and above 1, it becomes supersonic.
  • The exit Mach number, \( M_9 \), is crucial for understanding how exhaust gases behave as they leave the nozzle.
  • In our exercise, achieving an exit Mach number of 1 means the gas flow matches the ideal conditions for complete supersonic acceleration.
Determining the Mach number involves analyzing pressure changes and the specific heat ratio. Additionally, supersonic flows can introduce complications like shock waves, which is why understanding the Mach number is essential for designing efficient nozzles and engines.

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Most popular questions from this chapter

The bypass ratio in a turbofan engine is \(\alpha=3.5\). We intend to mix the cold fan and the hot core flows in a mixer to enhance the engine gross thrust. Assuming the hot core temperature of the gas is \(T_{\mathrm{h}}=3 T_{\mathrm{c}}\), calculate (a) the mixed-out temperature of the gas \(T_{\mathrm{m}} / T_{\mathrm{c}}\) (b) the percentage increase in ideal gross thrust as a result of mixing the cold and hot streams in the mixer You may assume a reversible adiabatic mixer flow with \(\gamma_{\mathrm{h}}=\gamma_{\mathrm{c}}=1.4\).

A fixed-geometry, convergent-divergent, internalcompression inlet is designed for \(M_{\mathrm{D}}=2.0\) and a self-starting capability. Calculate (a) \(A_{1} / A_{\text {th }}\) (b) \(M_{\mathrm{th}}\) (c) inlet total pressure recovery for the "best" backpressure

A subsonic aircraft cruises at \(M_{0}=0.85\) and its inlet operates with a capture ratio of \(A_{0} / A_{1}=0.7\). First, calculatethe lip Mach number \(M_{1}\). Second, assuming an engine becomes inoperative and the inlet lip Mach number drops to \(0.3\) (the so-called engine wind-milling condition), calculate the additive drag \(D_{\text {add }}\) for an inlet area of \(A_{1}=4 \mathrm{~m}^{2}\) and the ambient static pressure of \(p_{0}=16.6 \mathrm{kPa}\).

A supersonic nozzle operates in an underexpanded mode with an area ratio \(A_{9} / A_{\text {th }}=2\). We know that the exhaust plume turns outward by \(15^{\circ}\), as shown. Assuming the flow inside the nozzle is isentropic and \(\gamma=1.4\), calculate (a) exit Mach number \(M_{9}\) and (b) the nozzle pressure ratio \(p_{\mathrm{L}, \mathrm{moz}} / p_{0}\) (note that \(p_{10}=\) \(p_{0}\) ) (c) the Mach number after the expansion waves \(M_{10}\) (d) the NPR if the nozzle was perfectly expanded (e) the nozzle pressure ratio if a normal shock appears at the exit

A normal-shock inlet is operating in a supercritical mode, with the shock inside the inlet. If the flight Mach number is \(M_{0}=1.6\) and the shock is located at \(A_{s} / A_{t}=1.2\), calculate (a) Mach number ahead of the shock wave, \(M_{\mathrm{x}}\) (b) percentage total pressure gain if the inlet were to operate in the critical mode
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