- #1
dushak
- 3
- 0
(see also attached .doc)
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...
So what is wrong? Or anybody have a ready-for-use formula?
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...
So what is wrong? Or anybody have a ready-for-use formula?