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Chapter 11: Problem 68

Air enters a frictionless, constant area duct with \(\mathrm{Ma}_{1}=2.0\) \(T_{0,1}=59^{\circ} \mathrm{F},\) and \(p_{0,1}=14.7\) psia. The air is decelerated by heating until a normal shock wave occurs where the local Mach number is 1.5. Downstream of the normal shock, the subscnic flow is accelerated with heating until it chokes at the duct exit. Determine the static temperature and pressure, the stagnation temperature and pressure, and the fluid velocity at the duct entrance, just upstream and downstream of the normal shock, and at the duct exit. Sketch the temperature-entropy diagram for this flow.

Short Answer

Expert verified
By performing the above steps, we have calculated the required parameters. The temperature-entropy diagram will show an increase in temperature during heat addition and a decrease during heat extraction with a vertical jump representing the shock wave.

Step by step solution

01

Convert to SI units

Convert the given Fahrenheit temperatures to Kelvin using the relation \(T(K) = (T(°F) − 32) × 5/9 + 273.15\) and pressure from psia to pascal using the relation 1 psia = 6894.76 Pa.
02

Find values at the entrance

Use the given input: gas constant \(R = 287 J/kg-K\), specific heats at constant pressure and volume \(c_p=1005 J/kg-K, c_v=718 J/kg-K\), and the initial values for Mach number, temperature, and pressure to compute the velocity at the entrance using \(V_{entrance} = Ma * sqrt(gamma*R*T)\). Using the formula for stagnation temperature \(T_{0,1}=T_{entrance}(1 + ((gamma-1)/2) * Ma^2)\), we can find the stagnation temperature. The stagnation pressure can be found using \(p_{0,1}=p_{entrance}*(T_{0,1}/T_{entrance})^(gamma/(gamma-1))\).
03

Compute values near shock

Utilizing the provided Mach number of 1.5 just prior to the shock wave, we can calculate the static temperature and pressure, velocity just before and just after the shock using \(T_{2} = T_{1} * (2 * gamma * Ma^2 - (gamma - 1))/(gamma + 1)\), \(p_{2}=p_{1}*(2 * gamma * Ma^2 - (gamma - 1))/(gamma - 1)\) and \(V_{2} = Ma_{2} * sqrt(gamma * R * T_{2})\). The stagnation temperature bafter the shock is same as that before the shock, \(T_{0,2} = T_{0,1}\) and the stagnation pressure after the shock can be calculated using \(p_{0,2}=p_{2}*(T_{0,2}/T_{2})^(gamma/(gamma-1))\).
04

Calculate values at the exit

The air reaccelerates to supersonic speed and reaches choke condition at the exit. Use the isentropic flow tables for choked conditions and get the static temperature and pressure values. The stagnation temperature and pressure stay the same post shock wave, which is \(T_{0,exit}=T_{0,2}\) and \(p_{0,exit}=p_{0,2}\), respectively. The velocity at the exit point will be the sonic speed, calculated by \(V_{exit}=sqrt(gamma*R*T_{exit})\).
05

Sketch Temperature-Entropy diagram

The temperature-entropy (T-s) diagram will show how the temperature changes with respect to entropy. The process of heat addition and heat extraction are depicted by the vertical lines of constant pressure, and the shock wave will be represented by a vertical jump from the upstream to downstream state.

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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.

A normal shock occurs in a stream of oxygen. The oxygen flows at \(\mathrm{Ma}=1.8\) and the upstream pressure and temperature are 15 psia and \(85^{\circ} \mathrm{F}\) (a) Calculate the following on the downstream side of the shock: static pressure, stagnation pressure, static temperature, stagnation temperature, static density, and velocity. (b) If the Mach number is doubled to \(3.6,\) what will be the resulting values of the parameters listed in part (b)?

The static pressure to stagnation pressure ratio at a point in a gas flow field is measured with a Pitot-static probe as being equal to \(0.6 .\) The stagnation temperature of the gas is \(20^{\circ} \mathrm{C}\). Determine the flow speed in \(\mathrm{m} / \mathrm{s}\) and the Mach number if the gas is air. What error would be associated with assuming that the flow is incompressible?

The stagnation pressure and temperature of air flowing past a probe are \(120 \mathrm{kPa}(\mathrm{abs})\) and \(100^{\circ} \mathrm{C},\) respectively. The air pressure is \(80 \mathrm{kPa}(\text { abs }) .\) Determine the airspeed and the Mach number considering the flow to be (a) incompressible, (b) compressible.

A jet engine is to be designed for an altitude of \(12,000 \mathrm{m}\) where the atmospheric pressure is \(19.3 \mathrm{kPa}\). The jet nozzle has a supersonic exit Mach number and is perfectly expanded. The stagnation pressure and temperature of the gas are 100 kpa and \(600^{\circ} \mathrm{C}\) The flow rate of gas is \(45 \mathrm{kg} / \mathrm{s}\). Calculate the throat area, exit area, and exit velocity. Use \(k=1.4\) and \(R=260 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) for the gas.
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