Derivation of a Simplified D'Arcy's Law Equation

[SpaceDuck127]

Homework Statement

Done for a research essay on physics models for water filtration, and what I am focusing on is the change in speed after water passes through a porous material

Relevant Equations

q = -k*∆p/(µ*L) Darcy’s Law (flux rate)
∆p = f(L/D)(𝜌V^2/2) Darcy-Weisbach equation
Re = ρVD/µ Reynolds Number equation
f = 64/Re Friction factor equation

By substituting the darcy-weisbach equation into darcy’s law we get
q = -kf/µL * (L/D) * (𝜌V^2/2)
This can be further simplified by substituting the equation for friction factor for laminar flow, f = 64/Re , with the equation for reynolds number, Re = ρVD/µ substituted in such that:
q = (-k/µL)(64µ/ρVD)*(L/D)(𝜌V^2/2)
Which can be simplified from crossing out variables into:
q =-32kV/D^2

Based on physics and research i've done on filtration mechanics this makes kinda perfect sense, but I haven't found any evidence of this substitution online.

 

[Steve4Physics]

Relevant Equations: q = -k*∆p/(µ*L) Darcy’s Law (flux rate)
∆p = f(L/D)(𝜌V^2/2) Darcy-Weisbach equation
Re = ρVD/µ Reynolds Number equation
f = 64/Re Friction factor equation

By substituting the darcy-weisbach equation into darcy’s law we get
q = -kf/µL * (L/D) * (𝜌V^2/2)
This can be further simplified by substituting the equation for friction factor for laminar flow, f = 64/Re , with the equation for reynolds number, Re = ρVD/µ substituted in such that:
q = (-k/µL)(64µ/ρVD)*(L/D)(𝜌V^2/2)
Which can be simplified from crossing out variables into:
q =-32kV/D^2

Not my area but…

d’Arcy’s law is about a fluid passing through a porous material.

Reynold’s number is essentially about the transition between laminar and turbulent flow in a ‘free’ fluid.

Does it make sense to combine these when they apply to such different situations?

Beware of combining equations merely because they have some common parameters.

 

[erobz]

Homework Statement: Done for a research essay on physics models for water filtration, and what I am focusing on is the change in speed after water passes through a porous material
Relevant Equations: q = -k*∆p/(µ*L) Darcy’s Law (flux rate)
∆p = f(L/D)(𝜌V^2/2) Darcy-Weisbach equation
Re = ρVD/µ Reynolds Number equation
f = 64/Re Friction factor equation

By substituting the darcy-weisbach equation into darcy’s law we get
q = -kf/µL * (L/D) * (𝜌V^2/2)
This can be further simplified by substituting the equation for friction factor for laminar flow, f = 64/Re , with the equation for reynolds number, Re = ρVD/µ substituted in such that:
q = (-k/µL)(64µ/ρVD)*(L/D)(𝜌V^2/2)
Which can be simplified from crossing out variables into:
q =-32kV/D^2

Based on physics and research i've done on filtration mechanics this makes kinda perfect sense, but I haven't found any evidence of this substitution online.

What do you mean "change in speed after the water passes through a porous material"? Do you instead mean "as it is going through the porous material"?

 

[Chestermiller]

Your analysis would work if you had an array of parallel pores running through your medium. Then, for each pore, you would have the Poiseulle equation: $$-\frac{dP}{dL}=\frac{128Q\mu}{\pi D^4 }=\frac{32\mu v}{D^2}$$where v is the pore velocity. The pore velocity is related to the superficial velocity q by $$q=\epsilon v$$where ##\epsilon## is the porosity. So, we have Darcy's law for such a medium being: $$-\frac{dP}{dL}=\frac{32q\mu}{\epsilon D^2}$$or $$q=-\frac{dP}{dL}\frac{\epsilon D^2}{32 \mu}$$So, the permeability for such a medium is $$k=\frac{\epsilon D^2 }{32}$$