Calculating Pipe Leak Flow Rate

Question

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:

$$ Q = CA_0\sqrt{2g_c\frac{P_1-P_2}{\rho}}$$

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$\left(\frac{kg}{m^3}\right)$ $\left(\frac{2.2 lb}{kg}\right)$ $\left(\frac{1m}{3.28ft^3}\right)^3 $ $\left(\frac{1ft}{12in}\right)^3$ = $\frac{0.031lb_m}{in^3}$

$g_c$ = 32.2$\frac{lb_m*ft}{lb_f*s^2}$ = 386.4$\frac{lb_m*in}{lb_f*s^2}$

So I get the following:

$$Q = C3.14in^2\sqrt{\left(772.8\frac{lb_m*in}{lb_f*s^2}\right)\left(\frac{\frac{985.3lb_f}{in^2}}{\frac{0.031lb_m}{in^3}}\right)}$$

Is this the correct approach?

Answer 1

This is a correct method, given you have a good estimate of the discharge coefficient (C). There are a lot of resources out there on estimates for C for an orifice in a cross flow configuration such as this. "Handbook of Hydraulic Resistance" is a good resource, although, I am not sure what you would have available to you during the PE exam. A value such as 0.7 may be a good initial guess for C.