Consider a C-D nozzle with a throat and exit diameter of 252 mm and 657 mm, resp

Question

Consider a C-D nozzle with a throat and exit diameter of 252 mm and 657 mm,
respectively. The reservoir pressure and temperature is 10 atm and 1000 K, respectively.

a) Calculate the back pressure (Pb) for each of these scenarios:
i) a choked nozzle with isentropic, subsonic flow throughout the nozzle
ii) normal shock at the exit of the nozzle
iii) isentropic, supersonic flow at the exit with no shocks and/or expansions

b) What is the range of Pb required for each of these scenarios:
i) isentropic, subsonic flow throughout the nozzle
ii) normal shock somewhere in the divergent portion of the nozzle
iii) overexpanded nozzle
iv) perfectly expanded nozzle
v) underexpanded nozzle

c) Massflow through the nozzle.
i) Using the exit plane properties, calculate the massflow through the nozzle for each
scenario in Part (a). What do you find?
ii) Use the Eq. in Prob. 10.5 to calculate massflow.
iii) What conclusions can you draw from this?
iv) If the flow through the C-D nozzle were entirely subsonic, how would the massflow
compare to that you found above? Equal, less, or more? Why? Justify?

Explanation / Answer

The area ratio of the given nozzle= Exit to throat = (657/252)2=6.797 ;

a)

i) Using subsonic tables, Mach number at exit =  0.08551514 ;

also, P/P0 = 0.99489781;

given P0= 1 MPa;

P (back pressure) = 0.99489781* 1 MPa = 0.99489781 MPa;

ii)

From normal shock relations,

P2/p1 at M=3.5 ( since M should be greater than one, supersonic case, to have a normal shock at the exit) = 14.125;

Hence, back pressure = P2 = 14.125*0.01308922 MPa = 0.185 MPa.

iii)

Using supersonic relationship , M exit = 3.5

also, P/P0 = 0.01308922;

given P0= 1 MPa;

P (back pressure) = 0.01308922* 1 MPa = 0.01308922 MPa;

b)

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