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Chapter 11: Problem 15

An ideal gas flows with velocity \(V\), pressure \(p\), temperature \(T,\) and density \(\rho .\) Determine a set of equations for stagnation properties, including entropy, if the stagnation process is defined to be isothermal \((T=\text { constant ) rather than isentropic }(s=\text { constant })\).

Short Answer

Expert verified
The set of equations for stagnation properties under an isothermal process include: the pressure equation which is \(p_2 = p_1 \times e^{(-\Delta s/R)}\), the temperature equation which is \(T = T_0 - V^2/2C_p\), and the density equation is \(\rho=\rho_0\). The entropy remains unchanged in an isentropic process, and thus \(\Delta s = 0\).

Step by step solution

01

State the conservation laws and definitions

Stagnation properties are defined as the conditions obtained when a flowing fluid is brought to rest isentropically. For an isothermal process, the equation for entropy change is given by \(\Delta s = c_p \ln(T_2/T_1) - R \ln(p_2/p_1)\) whereas for an isentropic process, the change in entropy is zero. Here, \(c_p\) is the specific heat at constant pressure, \(R\) is the gas constant, \(T_1\) is the initial temperature, \(T_2\) is the final temperature, \(p_1\) is the initial pressure, and \(p_2\) is the final pressure.
02

Apply the appropriate laws and definitions

In case of an isothermal process, \(T_2 = T_1\), and thus the equation for entropy change simplifies to \(\Delta s = -R \ln(p_2/p_1)\). For this process, it's also true that \(p_2 = p_1 \times e^{(-\Delta s/R)}\). This is the equation for pressure. The Bernoulli equation (in the form \(H=T+V^2/2C_p\) where \(H\) is the total enthalpy) can be used to write the equation for temperature as \(T = T_0 - V^2/2C_p\). These are the main equations for the stagnation properties pressure and temperature.
03

Derive an equation for stagnation density

By definition, the density \(\rho\) of an ideal gas is given by \(p/R/T\). Substituting our derived isothermal pressure equation into this, we obtain \(\rho = \rho_0 \times (T_0/T)\). Since \(T\) is constant, we have \(\rho=\rho_0\). This represents the equation for stagnation density under an isothermal process. Since by definition \(\Delta s = 0\) for an isentropic process, the expression for entropy remains unchanged for an isothermal process.

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Most popular questions from this chapter

Air flows isentropically through a duct to a section where \(p_{1}=25 \mathrm{kPa}, T_{1}=300 \mathrm{K},\) and \(V_{1}=900 \mathrm{m} / \mathrm{s} .\) For these conditions: (a) Determine the stagnation conditions for the flow. (b) What is the Mach number at station \(1 ?\) Show a \(T-s\) diagram displaying stagnation and static conditions. (c) Is the flow choked? Is the throat behind or ahead of section \(1 ?\) Label this state on the \(T-s\) diagram.

An air heater in a large coal-fired steam generator heats fresh air entering the steam generator by cooling flue gas leaving the steam generator. One million lbm/hr of air at \(100^{\circ} \mathrm{F}\) and 1.1 million lbm/hr of flue gas at \(720^{\circ} \mathrm{F}\) enters the air heater. The flue gas leaves at \(310^{\circ} \mathrm{F}\). Flue gas has \(c_{p}=0.26 \mathrm{Btu} / \mathrm{lbm}^{\circ} \mathrm{F}\) and \(k=1.39 .\) Pressure changes are small and may be neglected. Calculate the temperature of the air leaving the air heater and the total entropy change for the process.

Prove that, in Rayleigh flow, the Mach number at the point of maximum temperature is \(1 / \sqrt{k}\).

An aircraft cruises at a Mach number of 2.0 at an altitude of \(15 \mathrm{km} .\) Inlet air is decelerated to a Mach number of 0.4 at the engine compressor inlet. A normal shock occurs in the inlet diffuser upstream of the compressor inlet at a section where the Mach number is \(1.2 .\) For isentropic diffusion, except across the shock, and for standard atmosphere, determine the stagnation temperature and pressure of the air entering the engine compressor.

An ideal gas is to flow isentropically from a large tank where the air is maintained at a temperature and pressure of \(59^{\circ} \mathrm{F}\) and 80 psia to standard atmospheric discharge conditions. Describe in general terms the kind of duct involved and determine the duct exit Mach number and velocity in \(\mathrm{ft} / \mathrm{s}\) if the gas is air.
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