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Chapter 13: Problem 14

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 \(A_{2} / A_{1}\).

Short Answer

Expert verified
The Mach numbers at the entrance and exit are calculated by solving the isentropic flow relations in step 1 for \(M_{1}\) and \(M_{2}\) respectively. The area ratio \(A_{2} / A_{1}\) is calculated using the equation found in step 3.

Step by step solution

01

Finding the Mach Numbers at the Entrance and Exit

Using the isentropic flow relation, we have \(\frac{T_{1} / T_{0 1}}{(1 / \gamma)}=1+[(\gamma-1) / 2] \cdot M_{1}^{2}\), where \(T_{1}\) is the static temperature at the entrance, \(T_{01}\) is the stagnation temperature at the entrance, \(\gamma\) is the ratio of specific heats (which is approximately 1.4 for air), and \(M_{1}\) is the Mach number at the entrance. Similarly, we have: \(\frac{T_{2} / T_{02}}{(1 / \gamma)}=1+[(\gamma-1) / 2] \cdot M_{2}^{2}\), where \(T_{2}\) is the static temperature at the exit, \(T_{02}\) is the stagnation temperature at the exit, and \(M_{2}\) is the Mach number at the exit. These equations can be solved simultaneously for \(M_{1}\) and \(M_{2}\) respectively.
02

Finding the Pressure Ratio

We use the isentropic flow relation to find the pressure ratio: \(\frac{P_{1}}{P_{2}}=\left(\frac{T_{1} / T_{0 1}}{T_{2} / T_{0 2}}\right)^{\gamma /(\gamma-1)}\), where \(P_{1}\) is the static pressure at the entrance and \(P_{2}\) is the static pressure at the exit.
03

Finding the Area Ratio \(A_{2} / A_{1}\)

We then find the area ratio using the equation: \(A_{2} / A_{1}=M_{1} /[(M_{2} / M_{1})^{2}\left(2+(\gamma-1) M_{1}^{2}\right) /(\gamma+1)\right]^{1 /(\gamma-1)}\), where \(A_{2}\) and \(A_{1}\) are the areas at the exit and the entrance respectively.

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Most popular questions from this chapter

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 flows into a converging duct, and a normal shock stands at the exit of the duct. 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 area ratio \(A_{1} / A_{2}\).

A converging-diverging nozzle is attached to a large \(\operatorname{tank}\) of air, in which \(T_{0_{1}}=300 \mathrm{K}\) and \(p_{0_{1}}=250 \mathrm{kPa}(\mathrm{abs}) . \mathrm{At}\) the nozzle throat the pressure is \(132 \mathrm{kPa}(\text { abs }) .\) In the diverging section, the pressure falls to \(68.1 \mathrm{kPa}\) before rising suddenly across a normal shock. At the nozzle exit the pressure is 180 kPa. Find the Mach number immediately behind the shock. Determine the pressure immediately downstream from the shock. Calculate the entropy change across the shock. Sketch the \(T s\) diagram for this flow, indicating static and stagnation state points for conditions at the nozzle throat, both sides of the shock, and the exit plane.

At a section in a passage, the pressure is \(150 \mathrm{kPa}(\mathrm{abs})\) the temperature is \(10^{\circ} \mathrm{C},\) and the speed is \(120 \mathrm{m} / \mathrm{s}\). For isentropic flow of air, determine the Mach number at the point where the pressure is \(50 \mathrm{kPa}\) (abs). Sketch the passage shape.

A converging-diverging nozzle, with a throat area of 2 in. \(^{2}\) is connected to a large tank in which air is kept at a pressure of 80 psia and a temperature of \(60^{\circ} \mathrm{F}\). If the nozzle is to operate at design conditions (flow is isentropic) and the ambient pressure outside the nozzle is 12.9 psia, calculate the exit area of the nozzle and the mass flow rate.
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