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.
 A: 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.
