I am trying to follow the calculations made in this blog post for a $50$ lbf. IPA/Nitrous Oxide engine, but cannot seem to get the values the author made for their propellant flow rates. Since the mixture ratio isn't mentioned, I figured I'd derive the net flow rate through the injector and check if the fuel and oxidizer flow rates sum to the net flow rate.
Since the flow rates for fuel and oxidizer are given in volume per unit time, I converted them to mass per second using the densities as
\begin{align*} \rho_{\text{IPA}} & =\frac {786\text{ kg}}{\text{m}^3}=\frac {6.56\text{ lbm}}{\text{gal}}\\ \rho_{\text{N}_2\text{O}} & =\frac {1220\text{ kg}}{\text{m}^3}=\frac {10.181\text{ lbm}}{\text {gal}} \end{align*}
My first attempt was using specific impulse. Since $I_{sp}=2224.3$ sec, then
$$\dot{w}=0.02248\text{ lbf/s}\qquad\implies\qquad\dot{m}=6.98\times10^{-4}\text{ lbm/s}\tag{1}$$
Where I have divided by $g=32.2\text{ ft}^2\text{/s}$ to convert from units of force to units of mass. This value is supposedly for net flow rate into the engine and the value is so small. For our propellants, using the values mentioned in the blog post, then
\begin{align*} \dot{m}_{\text{IPA}} & =\frac {12.24\text{ gal}}{\text{hr}}\frac {\text{6.56}\text{ lbm}}{\text{gal}}=0.0221\text{ lbm/s}\tag2\\ \dot{m}_{\text{N}_2\text{O}} & =\frac {75.74\text{ gal}}{\text{hr}}\frac {10.81\text{ lbm}}{\text{gal}}=0.2142\text{ lbm/s}\tag3 \end{align*}
By the way, this would imply that the mixture ratio is
$$r=\frac {0.2142}{0.0221}=9.69$$
Which is oxidizer-rich than the supposed stoichiometric mixture ratio of $6.591$ (I obtained this value by balancing the chemical reaction). Does this sound right?
Adding the two m-dots obviously does not give the m-dot found in $(1)$. Am I using the specific impulse wrong? I double checked all my calculations and am sure that I did not make a mistake.
My second attempt was to use the $\dot{m}$ formula found in Sutton.
$$\dot{m}=A_t(p_c)_{\text{ns}}\sqrt{\frac {\gamma}{R(T_c)_{\text{ns}}}\left(\frac 2{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}$$
I correctly calculated the area of the throat as $A_t=0.124\text{ in}^2$ and referencing the CEA output in the link above: $(T_c)_{\text{ns}}=6035.958^{\circ}\text{R}$, $\gamma=1.2592$, $(p_c)_{\text{ns}}=294\text{ psia}$, and
$$R=\frac {R_u}{\mathfrak{M}}=53.326\text{ lbf ft/lbm}^{\circ}\text{R}$$
Then
$$\dot{m}=0.0424\text{ lbm/s}$$
This value is also wrong again because the m-dot for $(3)$ is greater than the supposed total m-dot from the equation. I've spent a couple of hours on this and have no idea what I'm doing wrong, can anyone help?
