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:

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


$A_0$ = 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?

  • $\begingroup$ I would use SI units in your calculation $\endgroup$ Commented Feb 12, 2021 at 14:28
  • $\begingroup$ Yeah, that certainly seems to make it look cleaner. I'm mainly just going through the unit conversion olympics here to challenge myself. $\endgroup$
    – Ryan_C
    Commented Feb 12, 2021 at 15:47

1 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.

  • $\begingroup$ Okay great, thanks. I was mainly looking for confirmation that it is okay to calculate it this way. As far as PE exam references go, the environmental PE exam is now computer based so NCEES provides the examinee with an electronic copy of a reference manual. $\endgroup$
    – Ryan_C
    Commented Feb 12, 2021 at 19:53

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