- #1
roldy
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I've posted on here in relation to an aerospace analysis project I'm doing. I'm stuck on one part of the project where I need to develop the performance envelope of the turbojet engine. 3 of the 9 plots that are required are Thrust vs. Mach number. On each of those plots I'll about 7 different lines that correspond to different fuel flow rates. My professor suggested to me that I find what the resulting mach number would be with a prescribed fuel flow rate and the other parameters of P_0, T_o, \pi_c, \eta_c, \dot{m}_{corr,2} that is stated in the problem.
What's important here is the manipulation of equations to arrive at something I can use. I've tried solving this by hand and using MATLAB various times but no luck. I'm hoping that if I post my work here someone can see if I made an error in my understanding of this.
Work:
<br /> \dot{m}_fh=\dot{m}_aT_{t2}\left(\frac{T_{t4}}{T_{t2}}-\tauc{\tau_r}\right)<br />
<br /> \tau_c=1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}<br />
<br /> \tau_r=1+\frac{\gamma-1}{2}{M_0}^2<br />
<br /> T_{t2}=T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)<br />
<br /> \frac{T_{t4}}{T_{t2}}=\left(\frac{compdesignline}{turbdesignline}\right)^2<br />
<br /> \dot{m}_a=\frac{P_{t2}}{P_{stp}}\frac{\dot{m}_{corr,2}}{\sqrt{\frac{T_{t2}}{T_{stp}}}}<br />
<br /> P_{t2}=P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)<br />
The Work:
Plug in the equations for \dot{m}_a, T_{t2}, \frac{T_{t4}}{T_{t2}}, \tau_c, \tau_r
<br /> \dot{m}_fh=\frac{P_{t2}}{P_{stp}}\frac{\dot{m}_{corr,2}}{\sqrt{\frac{T_{t2}}{T_{stp}}}}T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(\left(\frac{compdesignline}{turbdesignline}\right)^2-\left(1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}\right)\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
Now I rewrite the radical as something I can deal with easier and substitute in for P_{t2}, and T_{t2}. Shown in two steps
<br /> \dot{m}_fh=\frac{P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}{P_{stp}}\frac{\dot{m}_{corr,2}\sqrt{T_{stp}}}{\sqrt{T_{t2}}}}}T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(\left(\frac{compdesignline}{turbdesignline}\right)^2-\left(1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}\right)\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
<br /> \dot{m}_fh=\frac{P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}{P_{stp}}\frac{\dot{m}_{corr,2}\sqrt{T_{stp}}}{\sqrt{T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}}}}T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(\left(\frac{compdesignline}{turbdesignline}\right)^2-\left(1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}\right)\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
Now I collect the constants and call them some variable and name stuff in parenthesis a variable (the ones that don't have Mach number).
Let
<br /> A=\frac{P_0\dot{m}_{corr,2}\sqrt{T_{stp}}T_0}{P_{stp}\sqrt{T_0}}<br />
<br /> B=\left(\frac{compdesignline}{turbdesignline}\right)^2<br />
<br /> C=1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}<br />
<br /> D=\dot{m}_fh<br />
Then
<br /> D=A\frac{\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}{\sqrt{\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}}\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(B^2-C\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
And now letting
<br /> E=1+\frac{\gamma-1}{2}{M_0}^2<br />
<br /> D=A\frac{E^2}{E^{\frac{1}{2}}}(B^2-CE^2)<br />
Simplifying
<br /> D=AE^\frac{3}{2}(B^2-CE^2)<br />
Multiplying out
<br /> D=AB^2E^\frac{3}{2}-ACE^{\frac{7}{2}}<br />
To get rid of the 1/2 power I let F=E^\frac{1}{2}
<br /> D=AB^2F^3-ACF^7<br />
Now I want to solve this for F, since F is a function of E and E is a function of M_0...which is what I need.
Here's the MATLAB code I used to try and solve this along with the result
EDU>> syms A B C D F
EDU>>
EDU>> D=A*B^2*F^3-A*C*F^7
D =
A*B^2*F^3-A*C*F^7
EDU>> solve(D,F)
ans =
0
0
0
(B/C^(1/2))^(1/2)
-(B/C^(1/2))^(1/2)
i*(B/C^(1/2))^(1/2)
-i*(B/C^(1/2))^(1/2)
EDU>>
This is not what I need. I'm looking for a formula that as all the variables in it.
What's important here is the manipulation of equations to arrive at something I can use. I've tried solving this by hand and using MATLAB various times but no luck. I'm hoping that if I post my work here someone can see if I made an error in my understanding of this.
Work:
<br /> \dot{m}_fh=\dot{m}_aT_{t2}\left(\frac{T_{t4}}{T_{t2}}-\tauc{\tau_r}\right)<br />
<br /> \tau_c=1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}<br />
<br /> \tau_r=1+\frac{\gamma-1}{2}{M_0}^2<br />
<br /> T_{t2}=T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)<br />
<br /> \frac{T_{t4}}{T_{t2}}=\left(\frac{compdesignline}{turbdesignline}\right)^2<br />
<br /> \dot{m}_a=\frac{P_{t2}}{P_{stp}}\frac{\dot{m}_{corr,2}}{\sqrt{\frac{T_{t2}}{T_{stp}}}}<br />
<br /> P_{t2}=P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)<br />
The Work:
Plug in the equations for \dot{m}_a, T_{t2}, \frac{T_{t4}}{T_{t2}}, \tau_c, \tau_r
<br /> \dot{m}_fh=\frac{P_{t2}}{P_{stp}}\frac{\dot{m}_{corr,2}}{\sqrt{\frac{T_{t2}}{T_{stp}}}}T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(\left(\frac{compdesignline}{turbdesignline}\right)^2-\left(1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}\right)\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
Now I rewrite the radical as something I can deal with easier and substitute in for P_{t2}, and T_{t2}. Shown in two steps
<br /> \dot{m}_fh=\frac{P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}{P_{stp}}\frac{\dot{m}_{corr,2}\sqrt{T_{stp}}}{\sqrt{T_{t2}}}}}T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(\left(\frac{compdesignline}{turbdesignline}\right)^2-\left(1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}\right)\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
<br /> \dot{m}_fh=\frac{P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}{P_{stp}}\frac{\dot{m}_{corr,2}\sqrt{T_{stp}}}{\sqrt{T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}}}}T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(\left(\frac{compdesignline}{turbdesignline}\right)^2-\left(1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}\right)\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
Now I collect the constants and call them some variable and name stuff in parenthesis a variable (the ones that don't have Mach number).
Let
<br /> A=\frac{P_0\dot{m}_{corr,2}\sqrt{T_{stp}}T_0}{P_{stp}\sqrt{T_0}}<br />
<br /> B=\left(\frac{compdesignline}{turbdesignline}\right)^2<br />
<br /> C=1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}<br />
<br /> D=\dot{m}_fh<br />
Then
<br /> D=A\frac{\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}{\sqrt{\left(1+\frac{\gamma-1}{2}{M_0}^2\right)}}\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\left(B^2-C\left(1+\frac{\gamma-1}{2}{M_0}^2\right)\right)<br />
And now letting
<br /> E=1+\frac{\gamma-1}{2}{M_0}^2<br />
<br /> D=A\frac{E^2}{E^{\frac{1}{2}}}(B^2-CE^2)<br />
Simplifying
<br /> D=AE^\frac{3}{2}(B^2-CE^2)<br />
Multiplying out
<br /> D=AB^2E^\frac{3}{2}-ACE^{\frac{7}{2}}<br />
To get rid of the 1/2 power I let F=E^\frac{1}{2}
<br /> D=AB^2F^3-ACF^7<br />
Now I want to solve this for F, since F is a function of E and E is a function of M_0...which is what I need.
Here's the MATLAB code I used to try and solve this along with the result
EDU>> syms A B C D F
EDU>>
EDU>> D=A*B^2*F^3-A*C*F^7
D =
A*B^2*F^3-A*C*F^7
EDU>> solve(D,F)
ans =
0
0
0
(B/C^(1/2))^(1/2)
-(B/C^(1/2))^(1/2)
i*(B/C^(1/2))^(1/2)
-i*(B/C^(1/2))^(1/2)
EDU>>
This is not what I need. I'm looking for a formula that as all the variables in it.