Any physical basis for this pressure vs flow model across flow restriction?

Recently read a journal paper on biomedical engineering where the authors used the following expression to model the expected volumetric gas flow, $Q$ from a pressure potential $\Delta P$ :

$$\Delta P=GQ^{\gamma}$$

here $\gamma$ is a constant that depends on the geometry of the flow restriction. For a perfect orifice you get close to the energy (Bernoulli) equation ($\gamma = 2$), but for more tortuous paths where viscosity becomes significant, $\gamma$ can be different from $2$ , at least when empirically fit to flow pressure data.

$G$ is also a constant.

So my question is does this expression have any physical basis? Can it perhaps be derived from first principles?

• Shouldn't $\gamma$ be 2 for Bernoulli, rather than 1/2? Also, for compressible flow like a gas, Q is not constant; so where is it measured? For incompressible flow through a tube, Hagen Poiseuille applies if viscous effects dominate, and $\gamma=1$ (unless the flow is turbulent, in which case $\gamma$ is closer to 2. – Chet Miller Jul 11 '18 at 3:14
• The particular expression was perhaps used purely for curve-fitting purpose. I have seen my colleagues in combustion research use such power laws simply for curve-fitting purpose without any physical justification whatsoever. I doubt it will one day turn out to have some physical basis. – Deep Jul 11 '18 at 3:32
• Reference/link for the journal paper? – sammy gerbil Jul 11 '18 at 12:18
• @ChesterMiller yes indeed. I'll correct it. An interesting attribute for this model is that it has the same structure if inverted (change the P and Q). Different constants, same structure. – docscience Jul 11 '18 at 14:50
• @Deep thanks for the deep thoughts - yes that's what I'm suspecting. – docscience Jul 11 '18 at 14:51

Any physical basis for this pressure vs flow model across flow restriction?

Recently read a journal paper on biomedical engineering where the authors used the following expression to model the expected volumetric gas flow, $Q$ from a pressure potential $ΔP$: $ΔP=GQ^γ$ - here $γ$ is a constant that depends on the geometry of the flow restriction. $G$ is also a constant.

Searching for "ΔP=GQγ" returns 42 results, and "ΔP=GQ" too many to guess which paper you refer to. A link to a specific paper or another article with the same equation is helpful.

$G$ and $γ$ might be within the range 0.95 to 1.05, they serve as a correction factor.

With such a simple equation it is only suitable for a specific set of conditions, variance in length, diameter, wall composition, etc. would require recalculation of the correction factors. You mention "biomedical engineering", with calculations involving Hemodynamics or transitional (Tracheobronchial or Mixed) flow a one size fits all equation isn't available.

Such a formula is more suitable for making calculations involving a Pitot tube.

In the paper "Causes of Extremely Negative Pulmonary Interstitial Fluid Pressure (Poutside) in Pulmonary Edema" by William C. Wilson, Jonathan L. Benumof, in Benumof and Hagberg's Airway Management, 2013, they wrote:

2 Airway Resistance

For air to flow into the lungs, a pressure gradient ($ΔP$) must also be developed to overcome the nonelastic airway resistance ($R_{AW}$) of the lungs to airflow. The $R_{AW}$ describes the relationship between ΔP and the rate of airflow (). $R(cm \; \text{H}_2\text{O}/L/sec)=\frac{ΔP(cm \; \text{H}_2\text{O})}{Δ \dot V(L/sec)} \qquad (4)$

The ΔP along the airway depends on the caliber of the airway and the rate and pattern of airflow.

There are three main patterns of airflow.

Laminar flow occurs when the gas passes down parallel-sided tubes at less than a certain critical velocity. With laminar flow, the pressure drop down the tube is proportional to the flow rate and may be calculated from the equation derived by Poiseuille: $ΔP=\dot V×8L×μ/πr^4 \qquad (5)$, where

$ΔP$ is the pressure drop (in cm H$_2$O), is the volume flow rate (in mL/sec), $µ$ is viscosity (in poises), $L$ is the length of the tube (in cm), and $r$ is the radius of the tube (in cm).

When flow exceeds the critical velocity, it becomes turbulent. The significant feature of turbulent flow is that the pressure drop along the airway is no longer directly proportional to the flow rate but is proportional to the square of the flow rate according to equation (6) for turbulent flow: $ΔP=\dot V^2pfL/4π^2r^5 \qquad (6)$, where

$ΔP$ is the pressure drop (in cm H$_2$O), is the volume flow rate (in mL/sec), $ρ$ is the density of the gas (or liquid), $f$ is a friction factor that depends on the roughness of the tube wall, and $r$ is the radius of the tube (in cm). With increases in turbulent flow (or orifice flow, as described in the next paragraph), $ΔP$ increases much more than and therefore $R_{AW}$ also increases more, as predicted by equation (4).

Orifice flow occurs at severe constrictions such as a nearly closed larynx or a kinked ETT. In these situations, the pressure drop is also proportional to the square of the flow rate, but density replaces viscosity as the important factor in the numerator.

The closest you are going to get to a remotely applicable equation is the ideal gas law which is: $PV = nRT$.

So my question is does this expression have any physical basis? Can it perhaps be derived from first principles?

Wikipedia: First principle

In physics and other sciences, theoretical work is said to be from first principles, or ab initio, if it starts directly at the level of established science and does not make assumptions such as empirical model and parameter fitting.

The only inputs into an ab initio calculation are physical constants.

Wikipedia: Physical Constant

A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.