Calculating mass flow - given pressure differential, density, etc. 1

[JustinMEEE]

Suppose there are two closed volumes of gas connected by a small pipe line of area A and length L. The pipe has an L/d (length/diameter) ratio greater than 1 and less than 10. The pipe line has a valve that initially closed. Suppose also that all thermodynamic properties (pressure, temperature, density, etc.) are known. The flow is subsonic. The total mass of the system is constant.

Does anybody know an equation that describes the mass flow rate with respect to time, once the valve is opened?
My intuition tells me that the mass flow rate is exponentially decaying.

Another way for me to solve my problem is to obtain an equation that describes the pressure differential across both volumes with respect to time, once the valve is opened. I believe that this will also be an exponentially decaying function.

Thanks,
Justin

 

[vpl]

Is this a homework question?

Patricia Lougheed

******

Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of the Eng-Tips Forums.

 

[JustinMEEE]

This is not a homework problem; the way I've set this up is intended for clear communication. Part of the actual problem would also disclose proprietary information.
(I graduated in December 2007.)

 

[stanier]

I suggest you download a copy of AFTs Arrow (demo) and model it.

http://www.aft.com

You will find the help manual of use also.

 

[IRstuff]

While the solution to such differential problems are often exponential, there are some problems that result in an error function solution:

http://en.wikipedia.org/wiki/Fick's_law

However the solution looks, it's a time-decaying function with asymptotic behavior.

TTFN

FAQ731-376

 

[SomptingGuy]

The solution is likely to be non-trivial and will involve pressure waves.

- Steve

 

[rb1957]

based on your fairly massive assumptions, the system is going to equalise the pressure in both tanks. flow rate is determined by pressure differential, the smaller the differential, the lower the flow rate. and flow rate determines the pressure decay/rise ... i'm sure the flow rate isn't constant, it sounds like a "simple" differential equation to solve, but i'm sure there a plenty of canned solutions as well.

 

[SomptingGuy]

The evolution of pressures in the volumes and the connecting pipe strongly depends on the sizes of the volumes. If they are large compared with the pipe, you can probably ignore momentum effects and treat it as fully developed nozzle flow - you'll get a smooth transition as the pressures even out. If the volumes are small (e.g. spherical and 10x the pipe diameter), expect some very interesting dynamics.

- Steve

 

[israelkk]

To my best knowledge there is no closed solution for such a case for real gases. You need to write your own equation and solve numerically. Depends on the case you may need to include heat transfer too. The temperature of the gas indide the supplying volume decreases, while the temperature of the gas in the receiving volume increases during this process.

Many years ago I used similar analysis to push a large mass by discharging a high pressure stored gas inside a vessel into an inflating flexibile volume.

 

[JustinMEEE]

Thanks iraelkk.
Assuming constant pressure differential across some arbitrary pipe, is there an equation for the mass flow rate? (Just trying to get a starting point.)

 

[israelkk]

You may start her

http://www.air-dispersion.com/feature2.html

but you may need to study incompressible flow from an appropriate books. You may also start a NASA

http://www.grc.nasa.gov/

http://WWW/K-12/airplane/shortc.html.

 

[ione]

Isralkk,

I was just thinking of the approach used in accidental gas release from a pressure vessel (accidental release source-terms) for a non-chocked application. There are correlations out there which work for this matter.

http://en.wikipedia.org/wiki/Accidental_release_source_terms

 

[sheiko]

JustinMEEE,

Why do you point out that "The pipe has an L/d (length/diameter) ratio greater than 1 and less than 10"?
I ask this because i don't believe it is of any relevancy to solve this problem.

I also fail to understand why you want to know the flowrate profile?

Anyway, below is how i would obtain the mass flowrate

If,
W = mass flowrate of gas through the valve (g/s)
M = average molecular weight of gas(g/mol)
n = number of moles of gas (mol)
P = pressure (Pa)
V = volume (m3)
R = 8,314 Pa.m3/mol/K
T = temperature (K) - assumed constant
Z = compressibility coefficient (-)
Qcv = control valve flowrate at a certain % opening (m3/s)
e stands for "at equilibrium"

Material balance: accumulation = in - out
So, upstream the valve: dn/dt = 0 - W/M

If you assume PV = ZnRT, then:
dn/dt = (1/ZRT)*(V*dP/dt + P*dV/dt)

The change in volume can be expressed by:
dV/dt = - Qcv

At equilibrium, dP/dt = 0 (no more accumulation in pressure), and then:

W = (Pe*M/ZRT)*Qcv

As you can see, your flowrate profile will depend on the characteristic of your control valve: linear?, equal-percentage? quick-opening?

"We don't believe things because they are true, things are true because we believe them."

 

[sheiko]

If you know your inital pressure Pi and the volumes Va and Vb respectively upstream ad dowmnstrean the control valve, the the equilibrulum pressure will equal:

Pe = Pi*Va/(Va+Vb), given that Vb is initially empty.


"We don't believe things because they are true, things are true because we believe them."

 

[zekeman]

"Assuming constant pressure differential across some arbitrary pipe, is there an equation for the mass flow rate? (Just trying to get a starting point.)"

Not for the transient solution you want, since along the pipe, you have each of the parameters functions of time and position
v=v(x,t)
V=V(x,t)
T=T(x,t)
and you don't have continuity of mass, so you can't use
Rho*A*V= constant.= as you would for steady state .
If you add in gas laws, you have to write 3 equations at any position of the pipe, namely momentum conservation, energy conservation and conservation of mass at each point. This will lead to nonlinear partial differential equations which can only be solved numerically.