Thermodynamics question

[Jadzia]

I was thinking about a little puzzle today, considering the flow of air in a network of connected chambers.

Picture.

The basic idea is that the chambers are fairly big compared to the thin tubes connecting them.

Imagine some random setup like this, and then drop a heater into one of the chambers and you get a build up of heat --> pressure --> flow. The flow should then stabilize and be totally driven by the convection of heat, as depicted by my coloured arrows.

The tubes at the top connect to the "outside", allowing the hot air to escape and cold air which will enter to displace it through a different tube.

Superficially, it all looks simple thermodynamics: Volume, pressure, temperature, density in each chamber. Pressure differences drive flow through each of the tubes. Gravity gives a potential gradient to displace the different gas densities over. Where fluid enters a chamber, it mixes with the gas already there, gently blending the temperatures and densities.

I want to simulate these flows, but I can't seem to get my head around it.

Does anyone here have any ideas?

 

[SamuraiBlue]

The basic idea is that the chambers are fairly big compared to the thin tubes connecting them.

Imagine some random setup like this, and then drop a heater into one of the chambers and you get a build up of heat --> pressure --> flow. The flow should then stabilize and be totally driven by the convection of heat, as depicted by my coloured arrows.

The tubes at the top connect to the "outside", allowing the hot air to escape and cold air which will enter to displace it through a different tube.

Superficially, it all looks simple thermodynamics: Volume, pressure, temperature, density in each chamber. Pressure differences drive flow through each of the tubes. Gravity gives a potential gradient to displace the different gas densities over. Where fluid enters a chamber, it mixes with the gas already there, gently blending the temperatures and densities.

I want to simulate these flows, but I can't seem to get my head around it.

Does anyone here have any ideas?

First you'll need to develop some guide line for experiment;
1.Heat energy will not be lost from walls of chamber and halls.
2.Temperature of all chambers are the same when experiment starts and the temperature of outside.
3. You'll also need to appropriate the amount of energy generated from the heat source.
4.You'll also need to decide the size of each chamber and length of each hallway so you can start calculating how much energy is required to heat (or cool) each chamber and hall until it reaches equilibrium.

Well going over the illustration depending on temperature difference between source A and outside temperature, route (route going out of E,3, and 4) may go either ways.There are also various chock points(route 1&5) where air may not flow due to equilibrium.

 

[Jadzia]

okay, I've looked at this again this morning.

For each chamber I have two independent variables:
mass m, temperature T, and one dependent variable pressure p

Each tube has one variable: mass flux f (mass per second)

I'm assuming that the tubes have negligible volume, so they deliver gas at the temperature of the chamber they're draining. Each chamber also has a constant volume V.

The equations I have, considering a single chamber first:

(1) dm/dt = SUM{ f_i } ; where i ranges over the tubes connected to this chamber.

Where there is a heat source (power P), we have the enthalpy equation: ΔE = constant . m . ΔT
(2) dT/dt = constant . P / m

Where there is gas mixing in the chamber from what comes in from the tubes, look at the enthalpies and deduce;
(3) dT/dt = SUM{ (T_j - T) . f_j / m } ; where j ranges over the inflowing tubes connected to this chamber.

Combining the effects of (3) and (2)
(4) dT/dt = SUM{ (T_j - T) . f_j / m } + constant . P / m

Then we have the gas law to calculate the dependent variable of pressure in each chamber: p = nρRT, which we can translate as:
(5) p = constant . (m/V) . T

Where there is a pressure difference between two linked chambers, there will be a flow, something like:
(6) f_i = Δp

However, this last equation cannot be complete, because it doesn't take gravity into consideration. I think that's what's stumping me. I'm not sure how I should use gravity. It will no doubt have something to do with the relative height of the chambers, and the density of gas in the tubes (creating a pressure head), but I'm not sure what.

If I'm forced to consider the flow speed in the tubes, then I'll be forced to consider a pressure drop (bernoulli's principle), which risks making this puzzle a lot more difficult.

 

[Jadzia]

Well I've had some success with the equations above. I've written a simulator this evening, and it does much what I expected it to do.

With equation (6) as given, there was just expansion of the whole system, and it overheated. There was no circulation of air. So I added an extra term for the pressure head between connected chambers, and it seems to work. I'm not confident about the term I've added, as it's more of a guess than something worked out... but it does appear to flow properly now.


In this demo, everything starts off cold, and the heater is active for the first 60 seconds. The system tends to an equilibrium.

After one minute, the heater is switched off, and the system quickly cools down by circulating out the warm air.

[yt]http://www.youtube.com/watch?v=Qor0e5OnA18[/yt]

 

[Tobin_Dax]

Since you are talking about the flow of a fluid, you should be using Bernoulli's Principle. You're building up to it, anyhow, and it will give you the depth-dependent pressure that you're looking for. The equation is

P + (1/2)*rho*v^2 + rho*g*h = constant

Substitute the ideal gas law in for the pressure term since it's dependent on density and temperature. You can rewrite the second term in terms of mass and your mass flux. (Mass flow rate is rho*area*velocity; you'd have to deal with the cross-sectional area somehow.)

That's all I have time to add right now. Hope it helps.

 

[Jadzia]

Thank-you for your advice.

I thought of using Bernoulli's principle on Friday, but the the flow velocity through the chambers is small. So v^2 is a very small (almost zero) term.

Also if we used a Bernoulli equation, it would have to be the compressible Bernoulli equation, since this is a thermodynamics problem.

The pipes at the moment are offering a resistance to flow. See Pouiseuille's pipe flow equation, where mass flux is proportional to the pressure gradient = delta pressure / pipe length.

Since rho can be different in each chamber, (rho*g*h) is not a term I can add to my equation for total pressure.

One thought from yesterday was that maybe the differences in rho will create some kind of weight vs buoyancy of fluid between different chambers.