Question
I have the feeling gas cannot have an equivalent of Ohm's law, tying pressure and throughput via some kind of fluid resistance constant depending on the geometry of the obstacle considered. Certainly because gas can be compressible.
However I need a very rough estimate (not a number from experience, a first order model/formula) of the air throughput out of an obstacle of arbitrary geometry of which I know the hollow cross sectional area.
I have done my research but all I can find is Poiseuille's law or pipe flow formulas which apply to very long cylinders (what about if I'm looking at the "resistance" of a complex obstacle?)… And the venturi equation:
$$p_i-p_o=1/2(\rho_o v_o²-\rho_i v_i²)$$
With (conservation of mass flow)
$$\dot{m}=\rho_i A_i v_i=\rho_o A_o v_o$$
Which gives
$$\dot{m}=\sqrt{\frac{2(p_i-p_o)}{\frac{1}{\rho_o A_o^2}-\frac{1}{\rho_i A_i^2}}}$$
Knowing that
$$\rho=\frac{m}{V}=\frac{\frac{PVM}{RT}}{V}=\frac{PM}{RT}$$
(M is the molar mass of the gas, R the perfect gas constant)
Is it correct? It's not linear like Ohm's law, but it is a relationship.
Application
I would like this question to be generic, but as an application/illustration, attached is a simplified 3D model of the orifice – the scale is 15mm. I know the area of the side triangles and the front rectangle out of the conduit (top), and I'm wondering what the mass flow is through it.
Best Answer
An equivalent Ohms law can be applied to gas flow and pressure drop, but only for particular mechanical flow restrictions and limited to a range of flow. But more generally for orifices and tubes the relationship between pressure and flow is quadratic, explained predominantly by the energy equation for flow, also known as Bernoulli's equation.
In the testing of respiratory equipment, companies like Hans Rudolph provide 'linear' flow resistors which approach the ideal linear resistor given by Ohms law. The restrictions in these resistors are accomplished with a screen like diffuser, and their linearity is specified over a restricted range.
So geometry does govern the relationship, but to determine what geometry is required takes CFD software or repeated experimentation.