# On deriving a fluid flow pressure drop model using either physical theory or empirical methods

I'm trying to derive a model from physical principles of fluid flow, a simple quadratic model that tends to well fit actual pressure-flow data.

For (real) subsonic air flow (ideally adiabatic) through a tube, orifice, or any device that that leads to a measurable pressure drop across the device, the pressure-flow data can generally fit a mathematical model of the form $$\Delta P = K_2Q^2sgn(Q)+K_1Q$$ with a fairly good degree of accuracy, and $K_1$ and $K_2$ are constants for the particular device geometry.

One might just dismiss this as a 'good' and perhaps 'lucky' fit of data to an arbitrary polynomial function, but I suspect there is a good theoretical foundation in Bernoulli's energy equation that involves velocity (and thus flow) squared, and for the linear term, perhaps viscosity effects as in the Hagen-Poiseuille model. It's just not clear to me if there is a way to combine these two theories to derive the relation above.

Is there a way? Can anyone derive this model from physics principles? Or empirical methods that would give $K_1$ and $K_2$ in terms of physical parameters such as density, viscosity, area, length, etc. would be acceptable as well.

• Compressible flow leads to the situation where as the pressure drops, the density drops, and the velocity increases. This means that there is a small acceleration, and the Bernoulli equation is much more difficult to apply to this situation. There are published empirical equations for your problem; would you consider a more empirical approach to be "desirable"? – David White Aug 12 '15 at 17:47
• @DavidWhite If the empirical approach (such as Buckingham's PI approach and dimensionless numbers) could lead to the same model above and expressions that relate known physical parameters like viscosity, density, temperature, etc. to $K_1$ and $K_2$ - certainly yes. – docscience Aug 12 '15 at 17:53
• @docscience you sound well informed about fluid flow, so you know the answer for flow through a cylindrical tube (Poiseuille's equation). For more general geometries you are having to solve Navier-Stokes for whatever boundary conditions you need, which is more or less the definition of a hard problem. – gleedadswell Aug 12 '15 at 23:10
• @gleedadswell Navier Stokes is intractable by analytical means and I don't want to consider that route. I'm rather interested in using Bernoulli, etc. and maybe approximations to get to the model I've posed. – docscience Aug 12 '15 at 23:19
• But Bernoulli neglects viscosity. Hagen-Poiseulle accounts for viscosity but only applies to a cylindrical pipe. So, if you are talking about systems that can be thought of as combinations of cylindrical pipes this will work. But for both Bernoulli and Poiseuille $\Delta P \sim Q$, so I don't see how your $Q^2$ term could ever come out of a theory built with those ingredients. I agree that Navier-Stokes is intractable. That's what I meant by "definition of a hard problem". – gleedadswell Aug 13 '15 at 0:54

As I'm dealing with the same issue I though maybe sharing my findings would help the next person get one step closer. so please consider this as a sharing knowledge rather than an answer.

The quadratic equation you have shared highly resembles Prony equation:

$$\Delta p= \frac{L_t}{D_t}\left(a\nu+b\nu^2\right)\tag{1}$$

the first ever empirical model for pressure loss, dated back to early eighteen hundreds. However, this has been mostly replaced by Darcy-Weisbach equation:

$$\Delta p=f\frac{L_t}{D_t}\frac{\rho \nu^2}{2} \tag{2}$$

where $f$ is the friction factor. For laminar flow, Reynolds number $Re=\frac{\rho\nu D_t}{\mu}<2320$, the Hagen–Poiseuille equation implies:

$$f=\frac{64}{Re}\tag{3}$$

For turbulent $4000<Re<80000$ in general Colebrook-White formula is the most popular approximation:

$$\frac{1}{\sqrt{f}}=-2\log\left( \frac{\epsilon}{3.72 D_t} +\frac{2.51}{Re \sqrt{f}} \right)\tag{4}$$

Where $\epsilon$ is the roughness of the tube. There are dozens of approximate solutions to this equation but the most popular to my knowledge is the Moody's:

$$\frac{1}{\sqrt{f}}=-1.8\log\left( \left( \frac{\epsilon}{3.72 D_t} \right)^{1.11}+\frac{6.9}{Re} \right)\tag{5}$$

If the tube is smooth then Blasius formula can be used $(4000<Re<10^5)$:

$$f=\frac{0.3164}{Re^{\frac{1}{4}}} \tag{6}$$

And for highly turbulent flows $10^5<Re$ the Nikuradse formula can be used:

$$f=0.0032+0.221 {Re}^{-0.237} \tag{7}$$

if the fluid is compressible and for Mach number $Ma<0.6$, Voronin equation imposes one extra correction factor:

$$f_{comp}=f\left( 1+\frac{\gamma-1}{2}{Ma}^2 \right)^{-0.47} \tag{8}$$

I hope it helps.