Mach Number as a Function of Area Ratio

[minger]

Hi there, I have an assignment in my Gas Dynamics class where were supposed to compute the flow inside a nozzle given stagnation pressure, temperature and area ratio (throat to exit). I am 80% sure this is given a choked condition.

Anyways, we have charts to find the Mach number as a function of area ratio, but the assignment is to create a program in either Excel or MathCad. The only thing I can find is the Area Ratio (area to reference area) as a function of Mach number. Since this equation has two solutions (the sub and supersonic solution) I was wondering if anyone could find out the supersonic solution to the equation (i.e. the top part of the graph).

I probably could get an estimate typing in pages and pages of charts, and then drawing a line of best fit, but this hardly seems worthwhile, and accurate.

If anyone could give a hand, I'd be very grateful.

 

[FredGarvin]

Are you referring to a plot that looks almost parabolic, with the lowest point near Ma=1? If so, I have it as being defined for air as:

$$\frac{A}{A*} = \left\{\frac{1}{Ma}\right\} \left\{\frac{1+[(k-1)/2]Ma^2}{1+[(k-1)/2]}\right\}}^{(k+1)/[2(k-1)]}$$

Where k is the ratio of specific heats.

 

[hakkad]

Are you referring to a plot that looks almost parabolic, with the lowest point near Ma=1? If so, I have it as being defined for air as:

$$\frac{A}{A*} = \left\{\frac{1}{Ma}\right\} \left\{\frac{1+[(k-1)/2]Ma^2}{1+[(k-1)/2]}\right\}}^{(k+1)/[2(k-1)]}$$

Where k is the ratio of specific heats.

Since area ration is a function of Mach number, which is squared in the equation, there will be 2 values for mach numbers, one for each side of the critical area of the converging-diverging nozzle, one subsonic and one supersonic, as there are 2 roots of the squared Mach number in the equation.

For a given area ratio, in what basis should I decide which Mach I should consider, if the problem does not include specific information about whether the flow is in the convergent or in the divergent section?

 

[CFDFEAGURU]

Man that's a boring assignment. You should jazz it up with some shock wave capturing. If you want to know how let me know. I would help you out.

Thanks
Matt

 

[hakkad]

Hi there, I have an assignment in my Gas Dynamics class where were supposed to compute the flow inside a nozzle given stagnation pressure, temperature and area ratio (throat to exit). I am 80% sure this is given a choked condition.

Anyways, we have charts to find the Mach number as a function of area ratio, but the assignment is to create a program in either Excel or MathCad. The only thing I can find is the Area Ratio (area to reference area) as a function of Mach number. Since this equation has two solutions (the sub and supersonic solution) I was wondering if anyone could find out the supersonic solution to the equation (i.e. the top part of the graph).

I probably could get an estimate typing in pages and pages of charts, and then drawing a line of best fit, but this hardly seems worthwhile, and accurate.

If anyone could give a hand, I'd be very grateful.

A. Well, at the throat area, the critical one, the ratio of the pressure to the stagnation pressure should be less than or equal 0.5283, M = 1, in order to establish a sonic flow in the throat, whereas the flow is likely to be choked and the mass flow rate is maximum.

B. From this point on, the flow should undergo an isentropic expansion if the pressure ratio lowered to a pressure ratio 0.02722, M = 3, since the length is so short that a tiny and negligible amount of heat transfer is likely to happen. The flow in the divergent section will be supersonic. The static pressure in the pressure ratio is the design pressure, the exit pressure.

C. Between A and B there will be a region where, shock waves are likely to happen. Shock wave is not an isentropc phenomenon.

 

[FredGarvin]

Man that's a boring assignment. You should jazz it up with some shock wave capturing. If you want to know how let me know. I would help you out.

Thanks
Matt

Well, considering Minger was doing that problem back in 2006, I think he's moved on.

 

[CFDFEAGURU]

LOL, I never looked at the original post date.

Thanks
Matt

 

[minger]

We will not bring up Minger's pre-employment posts where he was (more) young and dumb. :rofl:

 

[CFDFEAGURU]

lol, yeah we all have some embarrasing posts here from the past. My problem is that I leave one everyday. LOL

Matt

 

[minger]

I was just trying to find a post I did a few weeks ago, and ended up going through a couple years of posts. How much I've learned...how much I've forgotten.

The important thing is that now I sound more like I know what I'm talking about. Very key.