Question

Consider a pipe (diameter of 50 mm) with air flowing through it. The inlet conditions are: Mach 3; total pressure = 1000 kPa and 550 K. The friction coefficient (f) is 0.004. The exit Mach number decreases with the length of the pipe. Assume adiabatic, steady flow.

Write a short Matlab code and plot the following, while the exit Mach number changes from 2.5 to 0.99 with increments of ?M=0.01. Find the length of the pipe that is going to give the desired exit Mach number. Find the pipe exit pressure, pipe exit temperature, and pipe exit total pressure.

Answer 1

For given problem flow is adiabatic and steady..Air is assumed as working fluied with constant =1.4 .

f is constant..This is a fanno flow problem where given relation can be used -

the given relation can be found easily for fanno flow..

other relation that are used -

Using these a matlab code is here -

y=1.4; %flow coefficent
Po1=1000 ;%inlet toal pressure (in kPa)
T1=550 ;%inlet temperature in kelvin
f=0.004; %friction coefficient
M1=3; %inlet mach number
D=50*10^-3; %Diameter of the pipe

M2=2.5:-0.01:0.99; %exit mach no.

%Fanno flow relation
jj=(y+1)/(2*y);
L=-(D/f)*((jj*log((1+(y-1)/2*M2.^2)/(1+(y-1)/2*M1^2)))-(1/y*(1./M2.^2-1/M1^2)-jj*log(M2.^2/M1^2)));

P1=Po1/(1+(y-1)/2*M1^2)^(y/(y-1)); %inlet pressure
tt=((1+(y-1)/2*M1^2)./(1+(y-1)/2*M2.^2)); %T2/T1 (temperature ratio)
T2=tt*T1; %exit temperature
P2=P1*(M1./M2).*sqrt(tt); %exit pressure
Po2=P2.*(1+(y-1)/2*M2.^2).^(y/(y-1)); %exit total pressure

plot(M2,L);
xlabel('M2');
ylabel('L(in cm)')
title('Exit mach no. variation with length of pipe')
figure;
plot(M2,Po2);
xlabel('M2');
ylabel('Po2(in kPa)')
title('Exit mach no. variation with exit total pressure')
figure;
plot(M2,T2);
xlabel('M2');
ylabel('T2(in K)')
title('Exit mach no. variation with exit temperature')
figure;
plot(M2,P2);
xlabel('M2');
ylabel('P2(in K)')
title('Exit mach no. variation with exit pressure')