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There is an ideal gas in the centrifuge; no flows, everything is steady (not very steady in fact - process is adiabatic). If gas is incompressible, it is very easy:

P=0.5 rho (w*r)^2

But my gas is compressible, and it's a trouble.

I started from distribution law of Bolzman (with C = Bolzman constant):

n=n0 exp(mgx/CT)

,hence

d(ln n)=(M * w^2 * r) dr / RT

,where M is molar mass of gas

after integrating this equation (also used PV/T=const and PV^k=const) I got:

P/P0 = { 1 + (k-1)*Mv^2/(2*R*T0) } ^ [k/(k-1)]

,where

k=Cp/Cv

v=w*r

And now goes the trouble. At low velocities (and low compression ratios) the last equation must transform into the first: deltaP=0.5 rho v^2 / 2

But it doesn't! Using the rule (1+x)^a = 1+a*x ,when x<<1 we have:

P/P0 = 1 + k*Mv^2 / 2RT0

deltaP/P0 = k*Mv^2 / 2RT0

deltaP = k*rho*v^2 / 2

,which is k time larger... :surprised

So what is wrong? Or anybody have a ready-for-use formula?

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# Centrifugal pressure

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