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.