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Chapter 13: Problem 78

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.

Short Answer

Expert verified
Downstream pressures attained after a normal shock and isentropic deceleration can be calculated using the formulas provided. The difference between these two pressures \( \Delta P = (p_{2a} - p_{2b})kPa \) will reveal how significantly the processes differ.

Step by step solution

01

Downstream pressure calculation using normal shock relations

To find the downstream static pressure, use the normal shock relation formula. \(p_{2}=p_{1}*\left[{1+\left({\gamma -1}\right)/2 * M_{1}^{2}}\right]/\left[{(\gamma +1)/(\gamma -1)}\right]\). Given upstream pressure \(p_{1} = p_{0_{1}}=650 kPa\), \(M_{1}=2.5\), and considering a calorically perfect gas with constant properties for air (\(\gamma =1.4\)), you can substitute these values into the formula to find \(p_{2}\).
02

Downstream pressure calculation using isentropic relations

For the second part of the exercise, you're required to find the downstream pressure assuming isentropic flow. Use the isentropic flow relation \(p_{02}=p_{01}*\left[1 + \left({\gamma -1}\right)/2 * M_{1}^{2}\right]^{-(\gamma/(\gamma -1))}\). Here, \(p_{01}=p_{0_{1}}=650 kPa\), \(M_{1}=2.5\) and \(\gamma =1.4\) for air. Substitute these values into the formula to find \(p_{02}\). The Mach number downstream from the shock is subsonic and equal to \((\gamma+1)/\[(2*\gamma*M_{1}^{2} - (\gamma - 1)\]/(\gamma - 1)\)^(0.5) which will be used to find \(p_{2}\) isentropically.
03

Comparison of the pressures

To compare the pressures downstream after the shock wave and after isentropic deceleration, express the pressures in kPa. If \(p_{2a}\) and \(p_{2b}\) denote the pressures calculated in step 1 and step 2 respectively, the difference can be noted as \((p_{2a} - p_{2b})kPa\). The values obtained after completion of both steps will reveal which process - normal shock or isentropic deceleration - results in a higher downstream pressure.

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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 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.

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 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.

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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