Analyzing Gas Tank Filling Rates: Solving a Differential Equation

[nick5]

Hi all,

I hope this is right place to post this question (this is not a homework question).

I wanted to find out in what time a tank under low pressure (gas/air) is filled by the atmospheric pressure through a restriction.
(Tankpressure in relation of time or Flowrate through the restriction in relation of time.)

The flowrate is: V'(t)=delta_p(t)*C1 (Assuming a round restriction (pipe) and laminar flow C1=pi*r^{^4}/(8*viscosity*l))

delta_p(t)= po-ptank+C2/V(t)
(po=atmospheric_pressure, ptank=Intial_tank_pressure, V(t)=is the Volume of air which entered the tank through the restriction C2=n*R*T; p*V=n*R*T (actually the temperature T would also be a variable but for now I would assume it to be constant))

Therefore: V'(t)=C1*(po-ptank+C2/V(t))
(since po and ptank are constants they can be written as C3=po-ptank).

And I end up with this differential equation:

V'(t)-C1*C2/V(t)-C1*C3=0

Can anybody think of an analytical solution for this equation (I haven't dealt with differential equations for 15 years) or think of an alternative way to solve this problem?
If I were to plot the flowrate through the restriction, I would expect a curve that would look something like this: V'(t) = V'o*e^-t*C (Which is why I believe there should be an analytical solution.)

 

[tiny-tim]

Welcome to PF!

Hi nick5! Welcome to PF!

(have a delta: ∆ and a pi: π )

That's V' = AV + B,

so write it dV/(AV + B) = dt, and integrate.

 

[nick5]

Hi Tim,

thank you for your quick response and help.

The equation would actually be:

V'=A/V+B
dV/(A/V+B)=dt

and If integrated, I end up with:

t=(B*V-A*ln(A+B*V))/B^2

and if I try to solve this equation by V I don't get any further then this:
V=(e^{(B*V-t*B^2)/A}-A)/B

Is this even solvable by V?

 

[tiny-tim]

Hi nick5!

dV/(A/V+B)=dt

and If integrated, I end up with:

t=(B*V-A*ln(A+B*V))/B^2

hmm … I get the same …

I don't see how you can invert that, to a V = f(t) equation, without some numerical approximation method.

 

[nick5]

ok, that's what I was afraid of.

Well, maybe the initial equation I came up with is erroneous.

I'm certain that this part is correct:
V'(t)=∆_p(t)*C1

But maybe there is a better way to describe: ∆_p(t)

 

[Dr.D]

The V(t) that you have been discussing is pretty ill defined because volume is pressure and temperature dependent.

I'm not much of a fluids person, so I'm not the best person to advise on this. But, that said, what you need to write is a mass flow rate equation, and describe the quantity of mass in the tank. Then as a first approximation (not too bad really) you can use the ideal gas law to write
P*V = n*R*T
to relate the quantity of gas in the tank to the pressure, volume (which is fixed typically), and temperature, which usually goes up when the gas is compressed.

Try thinking in this direction and see if it helps.