Purely convective cooling - with equations

Filias Cupio · Post by **Filias Cupio** » Thu Aug 31, 2006 9:13 pm

A little while ago, I was installing a Ninja and forgot to plug in my system fan - that being the sole fan in my system. (See my sig for system details.) I then ran my computer (not very hard) for several hours before any problem developed. (Interestingly, it was the optical drive which gave up first.) This got me thinking - how hard can it be to run a completely fanless system?

The setup I imagine is the computer at the bottom of a chimney. (E.g. a tall vertical tube protruding from the top of the case.) The hot air rising through the chimney pulls in cold air to cool the computer, providing convective cooling.

Are there any thoughts on how well this might work in reality?

The rest of this post is physics. The conclusion is that the effectiveness of the chimney increases slowly with height, and a bit better with increasing cross-section.

Consider a chimney (area A, length L) with a heating element (computer hardware) at the bottom. Let the temperature (in Kelvin) of the surrounding air be Ts, and the air in the chimney Tc (Tc > Ts). Similarly the air densities are Ds and Dc.

By Charles' law density is inversely proportional to temperature (at constant pressure) so
Ds Ts = Dc Tc

Now the boyancy force of the air in the chimney is
F = A L (Ds - Dc) g
where (A L) is the volume of the air, and g is the acceleration due to gravity.

Let v be the velocity of the air up the chimney. In small time dt, the mass of warm air entering the chimney is
dm = A v dt Dc
This air is accelerated by the force:
F = dm a
and it must be accelerated to velocity v in time dt:
a = v/dt
hence putting it all together:
F = dm v/dt = A v^2 Dc = A L (Ds - Dc) g
hence
v^2 = L g (Ds - Dc)/Dc = L g (Ds/Dc-1)
And from the gas law, Ds/Dc = Tc/Ts:
v^2 = L g (Tc/Ts-1)

I define the relative excess temperature of air in the chimney compared to the surroundings as
E = (Tc/Ts-1)
so
v^2 = L g E.

Given that the heating element/computer is putting out heat at a constant rate, the temperature excess will be inversely proportional to the airflow (v A):
E = k/(v A)
(for some constant k, depending on the power output of the computer and the heat capacity of the air). It is the airflow we wish to optimize for, so these equations can be combined and rearranged to
v A = (k^2 A^2 L g)^(1/3).

So the airflow increases with L (but slowly) and with A (a bit faster.)

There are some unrealistic assumptions in the above: there is no resistance to the airflow, the heated air loses no heat while in the chimney, and external sources of air movement can be ignored.

jaganath · Post by **jaganath** » Thu Aug 31, 2006 11:43 pm

Interesting...a lot of people here have theorised about using the stack effect to harness natural convection before, but this is the first time (AFAIAA) that somebody has actually done the sums; IIRC it has been dismissed in the past because you would require an impractically large and tall chimney, but maybe with today's low-heat parts it's become doable? Or maybe the chimney needs hotter parts to function more effectively?

Bluefront · Post by **Bluefront** » Fri Sep 01, 2006 2:40 am

Convection cooling definately can be made to work, but becomes difficult when confined to a standard computer case, and when using moderately hot components. There are obviously different pieces in a computer with different cooling requirements......difficult to manage with purely convection cooling.

In my current case project I'm combining positive intake pressure with mostly passive convection exhaust. So far this setup shows a lot of promise.

nici · Post by **nici** » Fri Sep 01, 2006 3:06 am

This wouldn't be very energy efficient, but wouldn't it be possible to have a "chimney" as mentioned, and then have some high power resistors in there to heat it up to increase flow..? Just thinking

jaganath · Post by **jaganath** » Fri Sep 01, 2006 3:13 am

I was thinking, maybe some shaping of the chimney would accelerate the flow (can we use the venturi effect here? like a cylinder with a narrow bit in the middle?); also, in industrial settings, the higher the chimney the lower the atmospheric pressure at the top, so the more effective it is, but that's not relevant for our purposes.

croddie · Post by **croddie** » Fri Sep 01, 2006 5:33 am

Why do you want to optimize for airflow? I should have thought speed would be more important given the area is enough to fit everything. You want to optimimize cooling: couldn't that be approximated by a function of air speed? Not sure what function; but if this is the case it doesn't matter what the function is because you will just want to maximize airspeed.

Post by **Rusty075** » Fri Sep 01, 2006 7:26 am

In the realm of "there's nothing new under the SPCR sun": Joe's Tower of Cooling Power from a little over three years ago.

Although Filias' equations are much more advanced, I suspect that he will come to the same conclusion: stack effect generates very little airflow, but with some careful planning, it can work. Joe and I were using ASHRAE equations that related more to building-sized chimney effects; Filias' numbers, if you work out the unknowns, might be more precise.

jaganath · Post by **jaganath** » Fri Sep 01, 2006 8:42 am

Looks like I had the same idea as fmah:

fmah@I'm wondering if you will get accelerated flow with a converging duct (largest at bottom).

http://www.aoxj32.dsl.pipex.com/NewFile ... ysics.html

When air moves through a straight pipe or passage, the pressure, velocity and temperature will remain constant.

When air moves through a pipe or passage with a decreasing area (a convergent duct) the velocity of the air will increase but the pressure and temperature of the air will decrease.

peteamer · Post by **peteamer** » Fri Sep 01, 2006 9:10 am

That may be true for forced/powered air... but is the same true for convection?
Or will it restrict air flow more than powered air although with some velocity increase?

EndoSteel · Post by **EndoSteel** » Sun Sep 03, 2006 1:05 am

IMO, larger heatsinks would do the job more reliably.

justblair · Post by **justblair** » Sun Sep 03, 2006 3:17 am

Passive is the nirvanna of silencing IMHO, but I have to say I am with Endo on this one. While the chimney idea is an interesting experiment, I think that there are more practical solutions than this.

Heatpipes to big heatsinks give you a lot more control over the shape of your case, and offer a very practical method of producing passive systems that dont take over the room size wise.

Hyphe has a very good example of this method nearly complete. I have a somewhat less eloquent passive system about 70% complete using lots of mesh and smaller heatsinks.

Thats not to say that you shouldn't try it, it would be fascinating to see how you get on. Make a worklog so that we can share the trials and tribulations with you.

Azazel · Post by **Azazel** » Sun Sep 03, 2006 3:02 pm

Three words: Nuclear cooling tower.

The principles of these can be applied to help airflow in this manner. From memory, someone on procooling or OCAU did a fair bit of work on this, though from either an evoporative or water cooling perspective instead.

Longwalker · Post by **Longwalker** » Sun Sep 03, 2006 5:04 pm

More commonly known as 'bong coolers' these days, but original work on evaporative PC cooling goes back to at least 2001. See
Nuclear cooling tower. The downsides are many: noise, the need for frequent refills, cooling loop contamination risks, increased room humidity, and finally, a non-zero risk of culturing the bacteria that causes Legionnaires' Disease.

Filias Cupio · Post by **Filias Cupio** » Sun Sep 03, 2006 6:11 pm

nici: Yes, you could silently improve the airflow by adding extra heat above the computer. This thought had occured to me. I expect it would be expensive, unless you needed the heat in the room anyhow.

jaganath, croddie: We don't (directly) care about the velocity of the air in the chimney. All we care about is how much suction it is providing to draw fresh air through the computer. In the computer itself, we will try to direct our limited airflow to where it does the most good. For a large cross-section chimney, this will be a smaller cross-sectional area, and hence higher air velocity, than in the chimney itself.

Rusty075: Thanks for the link. Interestingly, they get a different formula:
flow proportional to A (g L E)^1/2.
It agrees with my formula if I change my exponent from 1/3 to 1/2. It looks to come from an authoritative source, so I guess I made an error. (I'm not going to look for it now.) Incidentally, this is the first time I've ever seen degrees Rankine actually used, as opposed to mentioned as a curiosity.

Incidentally, I have no plans to actually try this. For me, this is just an academic exercise.

Brian · Post by **Brian** » Tue Oct 03, 2006 7:05 pm

Filias, I ran some sample numbers through your calculation.

Assuming ambient = 25oC, average air temp in tower = 50oC, and a .3m tower, we get an impressive .5m/s (1mph) velocity.

However, I don't think an average air temperature of 50oC is anywhere near feasible. For one, a hard drive can not be immersed in 50oC air. For another, if you wish to cool a CPU to T=x, you must use air that's significantly colder than x.

So, perhaps a 2 m stack with 25oC ambient and 35oC exhaust (v=.3m/s) would be feasible and adequate.

Come to think of it, such an exhaust stack could be fitted to any system, passive or not.

I just measured the exhaust temperature of my PC: 32oC, with 20oC ambient. Tomorrow, I will try to determine the velocity increase caused by addition of a 2.5m PVC stack, accounting for drag losses in the pipe.

Brian · Post by **Brian** » Wed Oct 04, 2006 5:36 pm

I'll spare you the details of the mathematics, but I set velocity obtained from the specific heat equation (dQ/dt=(dm/dt)*Cp*deltaT) equal to velocity from the OP's equation and got:
H=1714/deltaT^3, where H is in meters and deltaT is in Kelvins, for a 100W heat source at the bottom of a smooth vertical pipe of dimensions 120mm x H.

So, with a 100W system at the base of a 2m tall exhaust pipe, with a case temperature elevation of about 10oC, you'd get about 7 CFM.

Only got 50W to power convection? You'll only get 3.5CFM. You can live with Tcase=Tambient+20oC? You should get roughly twice the convection.

It looks like you can cool the air in your case with convection, no problem. But can you cool a 50W CPU with convection? Well, I don't know; I don't have a Ninja. However, I think it's a definite possibility, if you can live with a CPU at 70oC. Maybe I'll give it a try when I feel the need to migrate to AM2.