Flow Rate Through Pipe With Hole

Just curious and working on some hypothetical problems on my own…I’m currently studying for my PE license. Also, trying my best here with the Latex stuff.

Let’s say there’s a crude oil pipeline and it somehow gets a hole in it and begins to leak. If you wanted to calculate the flow rate through the hole, my understanding is that you would approach the problem in the following manner:

$$ Q = CA_0sqrt{frac{2g_c(P_1-P_2)}rho}$$

Where:

$A_0$ = cross-sectional area of orifice (leak hole) = $frac{pi*d^2}{4}$ = $frac{pi(2)^2}{4}$ = $3.14 in^2$

$P_1$ = 1,000 psi (pressure in pipeline)

$P_2$ = 14.7 psi (atmospheric pressure on outside of pipe)

$rho$ = density of crude oil = 870$(frac{kg}{m^3})$ $(frac{2.2 lb}{kg})$ $(frac{1m}{3.28ft^3})^3 $ $(frac{1ft}{12in})^3$ = $frac{0.031lb_m}{in^3}$

$g_c$ = 386.4$frac{lb_m*in}{lb_f^2*s^2}$

For now I’m going to ignore "C" because I’m not sure what a good value for that is. Let’s come back to it at the end.

So I get the following:

$$Q = 3.14in^2sqrt{(772.8frac{lb_m*in}{lb_f*s^2})(frac{frac{985.3lb_f}{in^2}}{frac{0.031lbs}{in^3}}})$$

I screwed up the Latex formatting somewhere. In either case…is this the correct approach? Under the square root you’ll end up getting $frac{in^2}{s^2}$. When all is said and done, you should get $frac{in^3}{s}$ which you are then free to convert to whatever units you see fit. I like $frac{ft^3}{min}$