Converging Diverging Nozzle problem

[nakas12]

Homework Statement

Air flows through a symmetric converging-diverging nozzle where the cross-sectional area of the nozzle (in meters^2) varies accoding to the relationship:

A(x) = 1.0 - 0.8X +.80X^(2) ; where X is in meters and the nozzle is 1 meter long.

P2/P1 = .90 (there is subsonic inflow) A normal shock also occurs near the exit.

A) Determine the inflow and outflow mach numbers
B) The stagnation pressure ratio across the normal shock
C)The X location of the shock (in meter) from the inflow to the normal shock.

Homework Equations

I have been using the isentropic and normal shock tables to solve this problem which involve ratios of pressures, areas, and temperature.

The Attempt at a Solution

I assume Mach number at the throat will be 1
Mth (throat) = 1
L =1
Area of the throat: X=0.5
A(x) = 1.0-0.8(.5) + 0.8(.5^(2)) = 0.8
Throat area: 0.8 = A*

Area of the inlet = 1 = A1

A1/A* = 1.25
From this I find the Mach number at the inlet equal to .5533
M1 = .5533



I honestly need help with only finding the Mach numbers at the inlet and exit. I'm a little confused due to the equation for finding the area given. I could really really use some help. Thanks!

 

[KataKoniK]

Thank you for your post. I am happy to help you with your problem.

To find the Mach numbers at the inlet and exit, we can use the isentropic relations for compressible flow. The area ratio A1/A* that you have calculated is correct. From this, we can find the Mach number at the inlet using the following equation:

M1 = (A1/A*)^((gamma-1)/2gamma)

Where gamma is the ratio of specific heats, which for air is approximately 1.4.

Plugging in the values, we get:

M1 = (1.25)^((1.4-1)/(2*1.4)) = 0.553

To find the Mach number at the exit, we can use the same equation, but with the ratio of pressures P2/P1 instead of the area ratio A1/A*. Since we know that P2/P1 = 0.9, we get:

M2 = (0.9)^((1.4-1)/(2*1.4)) = 0.539

Now, to find the stagnation pressure ratio across the normal shock, we can use the normal shock relations. The stagnation pressure ratio across a normal shock is given by:

(P02/P01) = ((gamma+1)*M1^2) / ((2+gamma)*M1^2 - (gamma-1))

Where P02/P01 is the ratio of stagnation pressures before and after the shock. Plugging in the values, we get:

(P02/P01) = ((1.4+1)*(0.553^2)) / ((2+1.4)*(0.553^2) - (1.4-1)) = 1.542

Finally, to find the location of the shock, we can use the relation:

X = L * (1 - (M1^2/((gamma+1)*M1^2 + (gamma-1)/2)))

Where X is the location of the shock, L is the length of the nozzle (in this case, 1 meter), and M1 is the Mach number at the inlet. Plugging in the values, we get:

X = 1 * (1 - (0.553^2/((1.4+1)*0.553^2 + (1.4-1)/2))) = 0.