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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:

[tex]

\dot{m}_fh=\dot{m}_aT_{t2}\left(\frac{T_{t4}}{T_{t2}}-\tauc{\tau_r}\right)

[/tex]

[tex]

\tau_c=1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}

[/tex]

[tex]

\tau_r=1+\frac{\gamma-1}{2}{M_0}^2

[/tex]

[tex]

T_{t2}=T_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)

[/tex]

[tex]

\frac{T_{t4}}{T_{t2}}=\left(\frac{compdesignline}{turbdesignline}\right)^2

[/tex]

[tex]

\dot{m}_a=\frac{P_{t2}}{P_{stp}}\frac{\dot{m}_{corr,2}}{\sqrt{\frac{T_{t2}}{T_{stp}}}}

[/tex]

[tex]

P_{t2}=P_0\left(1+\frac{\gamma-1}{2}{M_0}^2\right)

[/tex]

The Work:

Plug in the equations for [tex]\dot{m}_a, T_{t2}, \frac{T_{t4}}{T_{t2}}, \tau_c, \tau_r[/tex]

[tex]

\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)

[/tex]

Now I rewrite the radical as something I can deal with easier and substitute in for [tex]P_{t2}[/tex], and [tex]T_{t2}[/tex]. Shown in two steps

[tex]

\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)

[/tex]

[tex]

\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)

[/tex]

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

[tex]

A=\frac{P_0\dot{m}_{corr,2}\sqrt{T_{stp}}T_0}{P_{stp}\sqrt{T_0}}

[/tex]

[tex]

B=\left(\frac{compdesignline}{turbdesignline}\right)^2

[/tex]

[tex]

C=1 + \frac{\pi_c^{\frac{\gamma-1}{\gamma}}-1}{\eta_c}

[/tex]

[tex]

D=\dot{m}_fh

[/tex]

Then

[tex]

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)

[/tex]

And now letting

[tex]

E=1+\frac{\gamma-1}{2}{M_0}^2

[/tex]

[tex]

D=A\frac{E^2}{E^{\frac{1}{2}}}(B^2-CE^2)

[/tex]

Simplifying

[tex]

D=AE^\frac{3}{2}(B^2-CE^2)

[/tex]

Multiplying out

[tex]

D=AB^2E^\frac{3}{2}-ACE^{\frac{7}{2}}

[/tex]

To get rid of the 1/2 power I let [tex]F=E^\frac{1}{2}[/tex]

[tex]

D=AB^2F^3-ACF^7

[/tex]

Now I want to solve this for F, since F is a function of E and E is a function of [tex]M_0[/tex]...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.