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Chapter 11: Problem 55

Air is supplied to a convergent-divergent nozzle from a reservoir where the pressure is \(100 \mathrm{kPa}\). The air is then discharged through a short pipe into another reservoir where the pressure can be varied. The cross- sectional area of the pipe is twice the area of the throat of the nozzle. Friction and heat transfer may be neglected throughout the flow. If the discharge pipe anstant cross-sectional area, determine the range of static pressure in the pipe for which a normal shock will stand in the divergent section of the nozzle. If the discharge pipe tapers so that its cross- sectional area is reduced by \(25 \%\), show that a normal shock cannot be drawn to the end of the divergent section of the nozzle. Find the maximum strength of shock (as expressed by the upstream Mach number) that can be formed.

Short Answer

Expert verified
First, identify the static pressure range in the divergent of nozzle where a normal shock would occur using the given area ratio. When the area decreases by 25%, the new area ratio changes the possible location of the normal shock in the divergent part. Lastly, the maximum shock strength can be found using the given downstream Mach number and the normal shock relations to determine the upstream Mach number. Note, in reality, one would use tabulated values or gas dynamics relations to ascertain these Mach numbers and pressures.

Step by step solution

01

Identify the Given Parameters and Initialize

From the problem given: Reservoir pressure, \(P_1 = 100\) \(kPa\), Discharge pipe's cross-sectional area compared to the throat area is twice. The area, Mach number, and pressure ratios are all paramount components in solving nozzle flow problems, so identifying them first sets the groundwork to proceed to the next steps.
02

Analyze the Requirements for a Normal Shock to Occur

A normal shock occurs when the flow transitions from supersonic to subsonic. Using the area ratio for the given conditions (which is 2, since the pipe's cross-sectional area is twice the throat area), it's possible to find the corresponding Mach number using standard area-Mach number relations. These derived relations give us the Mach number at which the flow transitions from supersonic to subsonic (i.e., where the normal shock occurs). One can obtain these ratios from standard gas dynamics tables or relations. This will give us the range of pressures for which a normal shock stands in the divergent section.
03

Analyzing the Effect of Tapering

In the scenario where the discharge pipe tapers and its cross-sectional area reduces by 25%, the new area ratio will be \(2-(2*0.25)=1.5\). Again, using the area-Mach number relations for this modified condition, evaluate the change in Mach number where a shock can form.
04

Find the Maximum Shock Strength Expressed by Upstream Mach Number

To find the upstream Mach number, use the relations for normal shock i.e. the upstream and downstream properties of the shock. Given the Mach number downstream of the shock, find the corresponding upstream Mach number, which reveals the maximum strength of the shock.

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