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I am writing my first math paper, and I am using a formula for the mass flow rate of a liquid through an orifice. I found this formula mentioned in some online videos (like this one) but I could not find the source of the formula. The formula is $$m' = \frac{(\Delta p)^{\frac{1}{\alpha}}}{R}$$ where $m'$ is the mass flow rate, $\Delta p$ is the pressure difference between the two sides of the orifice, $\alpha$ is 1 for laminar flow and 2 for turbulent flow, and $R$ is the resistance of the orifice.

How is this formula justified? I'm looking for a derivation with accompanying citations: either a paper in which the formula is derived (it need not be a main result), or citations for the assumptions of the derivation.

Some searching hints at the formula being derived from Bernoulli's Equation, or from the Poiseuille Equation (maybe Hagen-Poiseuille?), but I cannot find a clear match. Also, all the equations seem to deal specifically with laminar or turbulent flow.

I also found this question which seems to have some connection to my question (it mentions that the degree-one term dominates laminar flow, while the degree-two term dominates turbulent flow) but I cannot get a concrete reference from it.

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  • $\begingroup$ I've edited; see if that's okay. $\endgroup$
    – David Z
    Commented Aug 17, 2016 at 18:22
  • $\begingroup$ Relevant wiki: en.wikipedia.org/wiki/Darcy_friction_factor_formulae $\endgroup$ Commented Aug 17, 2016 at 19:11
  • $\begingroup$ It all depends on what $R$ is and how it is defined. Can you supply more information on the resistance please (incl. units). $\endgroup$ Commented Aug 17, 2016 at 19:12

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The laminar case with $\alpha=1$ (that is, $\frac{dm}{dt} \propto \Delta p$) does indeed follow directly from Pouseille's equation.

The turbulent case, $\frac{dm}{dt} \propto \sqrt{\Delta p}$, seems to be an upper limit on flow rate for incompressible fluids, derived from the Bernoulli equation $p + \frac12 \rho v^2 = \text{constant}$.

Note that the constant of proportionality (your $R$) for a given geometry will generally not be the same for laminar and for turbulent flow; in the two cases the constants $R_\text{laminar}, R_\text{turbulent}$ must even have different units.

The boundary between laminar flow and turbulent flow is messy; a good starting place for analysis is to pretend that flow is either laminar or turbulent and prepare to be unsurprised by subtleties.

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  • $\begingroup$ What worked for me is to calculate the friction factors $f$ for the upper and lower limit of the transient region and interpolate using the actual logarithm of the Reynold's number: $$ f = f_1 +(f_2-f_1) \left( \frac{\ln({\rm Re})-\ln({\rm Re}_1)}{\ln({\rm Re}_2)-\ln({\rm Re}_1)} \right) $$ $\endgroup$ Commented Aug 17, 2016 at 19:27
  • $\begingroup$ In Poiseuille's Equation, there is a factor for length of the pipe, which is zero in the case where the orifice is simply a hole. How do I account for that? $\endgroup$
    – shardulc
    Commented Aug 17, 2016 at 19:27
  • $\begingroup$ It is never zero. An orifice in cross section is still a pipe with length. The drop in pressure has to happen over a distance. There is probably an effective length which is larger than the actual length. $\endgroup$ Commented Aug 17, 2016 at 19:30
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Looking at the flow in a narrow pipe (laminar flow in pipe) we have the following relations

$$ \begin{array}{ll} \text{Mass flow rate} & m' = \rho A v \\ \text{Pressure Resistance} & \frac{\Delta p}{\rho g} = f \frac{\ell}{d} \frac{v^2}{2 g} \\ \text{Friction Factor} & f = \frac{64}{\rm Re} \\ \text{Reynold's number} & {\rm Re} = \frac{\rho d v}{\mu} \end{array}$$

Combined together yield

$$ m' = \frac{\rho A d^2}{32 \mu \ell} \Delta p $$

which matches the given equation for laminar flow if $R = \frac{32 \mu \ell}{\rho A d^2}$.

For turbulent flow the friction factor is more complex, and so they must have done some curve fitting to get to the $\sqrt{\Delta p}$ factor (probably a velocity related term since $\Delta p \propto \frac{\ell}{d} v^2$).

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  • $\begingroup$ Can't accept both answers, so I'm accepting this one because the derivation part is slightly clearer and the length issue with Poiseuille's Equation is cleared up. Thanks to ja72 and @rob! $\endgroup$
    – shardulc
    Commented Aug 17, 2016 at 19:47

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