Converging-Diverging Nozzle

[gfd43tg]

Homework Statement

2. Air flows from a large supply tank in which the pressure is 147 psig and the temperature is 160 F through a converging - diverging nozzle. The velocity at the throat of the nozzle is sonic. A normal shock occurs at a point in the diverging section of the nozzle where the cross-sectional area is 1.44 times the cross-sectional area of the throat.

a) Compute the Mach number, pressure, and temperature just upstream of the shock.
b) Compute the Mach number, pressure, and temperature just downstream of the shock.

Homework Equations

The Attempt at a Solution

I thought the slide 9 would be the equation to use for this problem, but if S* is what they mean, that is just the cross sectional area of the throat, right? What is S supposed to be without the Asterisk?

I am not sure how I could determine the mach numbers up and downstream of the shock using any of the equations give in the lecture slides. is Po, To, etc just at the ''back pressure'' (back pressure is the tank pressure, right?)

 

[SteamKing]

We have no idea what 'slide 9' refers to.

 

[gfd43tg]

My apologies, I intended to post the slides

 

[gfd43tg]

I am just using the equation on the 9th slide. Honestly, I can't really say I understand what the equation means. I know S* is the cross sectional area where the velocity is sonic (the throat), but my best guess is that S is an arbitrary cross section anywhere along the pipe. However, I wonder if it is only for the divergent part of the nozzle?

slide 10 summarizes all the equations, and of course T, P, ρ, etc are probably corresponding with S? It explicitly states T0, P0, etc. are at the reservoir, but doesn't mention what T, P, etc are.

I am uncertain of what it meant by ''just upstream of the shock'' or ''just downsteam'' and how I am supposed to calculate these. Just upstream can be anything upsteam, what are they asking for more precisely?

This table seems to agree with my calculation of the mach number being 1.80
http://www.cchem.berkeley.edu/cbe150a/isentropic_flow.pdf

 

[rezeew]

I would first clarify the given information and assumptions to ensure a thorough understanding of the problem. From the information provided, it seems that the air is flowing from a high pressure and temperature supply tank through a converging-diverging nozzle. The nozzle has a throat where the velocity is sonic, and a normal shock occurs at a point in the diverging section where the cross-sectional area is 1.44 times larger than the throat.

To solve this problem, I would first use the isentropic flow equations to determine the properties of the air at the throat of the nozzle. This would include the Mach number, pressure, and temperature. Then, I would use the normal shock equations to calculate the properties just upstream and downstream of the shock.

To clarify, S* in the isentropic flow equations represents the cross-sectional area of the throat, while S in the normal shock equations represents the cross-sectional area at the point where the shock occurs.

I would also assume that the back pressure mentioned in the problem refers to the pressure at the outlet of the nozzle, which is the tank pressure in this case.

Once all the necessary properties are calculated, I would check for consistency and accuracy by ensuring that the Mach number is less than 1 before the shock and greater than 1 after the shock, and that the pressure and temperature values are physically reasonable.

Overall, the solution to this problem requires a good understanding of fluid mechanics and the use of appropriate equations. It may also be helpful to draw a diagram of the nozzle and shock location to visualize the problem better.