What is the equation to calculate the Thrust coefficient within a set of parameters?

In the book, Design of Liquid-Propellant Rocket Engines, there is a practice problem written as such.

"Determine the design values of $(C_*), (C_f), and (I_s)_t$ for the engine thrust chambers of the stages of the hypothetical Alpha vehicle, with the following assumed design parameters."

I am given the fuel, the fuel mixture ratio, the nozzle stagnation pressure $(P_c)_n$, the specific heat ratio, and the nozzle expansion ratio.

I can calculate $(C_*)$ fairly easy, the problem is calculating $(C_f)$. The book says that you can derive a theoretical vacuum $(C_f)$ using a figure. However when I look at the figure its listing, It does not give me a clear path to derive the equation.

What are the equations I need to calculate $(C_f)$ or the theoretical vacuum $(C_f)$ ( I have the equation used to convert the theoretical vacuum into the actual thrust coefficient. )

Here are the pages I'm talking about.

Fig 1.17 is the referenced figure, however it does not show the equation it uses to graph its chart. What would be the equation needed to graph the chart used in fig 1.17

So I figured it out, I first took this equation.

$(C_F) = (CT_F) - (E)(P_a)/(P_C)_N$

Then I isolated $(CT_f)$ and substituted $(C_F)$ equation resulting in

$C = \sqrt{\frac{2y^2}{y-1)}*\frac{2}{y+1}^{y+1/y-1}*(1-\frac{P_e}{P_C}^{\frac{y-1}{y}})+e(\frac{P_e - P_a}{P_C})} + e\frac{P_e}{P_C}$

Then I solved for P_e using this equation

$E = \frac{(\frac{2}{y+1})^\frac{1}{y-1}*(\frac{(P_C)_n}{P_e})^\frac{1}{y}}{\sqrt{\frac{y+1}{y-1}*(1-\frac{P_e}{(P_C)_N})^(\frac{y-1}{y})}}$

The value $(P_a)$ is ambient pressure, so I will plugin in 14.6959488 in the place of $(P_a)$. 14.6959488 is the pressure in psi at ground level ( I hate imperial but the book's units are imperial ).

This will give me a value for $(CT_F)$ which I can then plug back into the original equation

$(C_F) = (CT_F) - (E)(P_a)/(P_C)_N$

Which will give me the value of $(C_f)$

only took me two weeks.

• As a side note, I figured out that there was a error in the book and that the figure I should be referencing is figure 14. Jul 12 '19 at 12:43