# Measuring temperature profiles

1. May 30, 2005

### Tsunami

Here's the thing. This year we had to work on a project, mine's about building a thermoacoustic refrigerator. I'll first give some background, for those who are not familiar with it, and believe they need to know these details to answer my question:

Essentially, it's made of the following component:
a resonator tube, closed at one end, and half a wavelength long;
a loudspeaker, set at the other end, in such a way that it more or less behaves like a closed end of the tube, and sending in an acoustic pressure wave operating at resonant frequency; (so, a standing wave is created)
last element is a stack, an element of small solid layers placed into the tube --the thermal contact between the stack layers and the fluid (which is air) evokes a temperature gradient over the two sides of the stack, and thus a heat pump is made (and of course, if you have a heat pump you can have refrigeration).

A part of this project involves simulating the velocity stream function and temperature profile of the waves in this resonator tube, and comparing them to the streaming and temperature profile of a resonator tube without stack, and closed at both ends. These simulations are done in 2D (there is axial symmetry) in Fluent.

When I look at the velocity stream function of the closed tube without stack, I get vortex streaming (eddies) as expected - there are 2 times 2 eddies, placed symmetrically about the longitudinal axis and also mirrored around the centre of the tube.

When I look at the temperature profile (plotted against the longitudinal distance), I get something that looks more or less like a cosine, with its minimum in the centre of the tube.
Now, I think that's a bit funny. When you have a standing pressure wave in your tube:

p = P1 cos(omega*t-k*x) +P0 , P0 being p_atmosphere

You should get a pressure profile that looks like (for t=0):

p= P1 | cos(k*x) | in absolute value . This is right, yes?

They told me it's like this because the pressure you measure is simply the difference with reference pressure, and it doesn't matter if it's a greater or a lower pressure.

Now, since temperature follows pressure, shouldn't I get a temperature profile that looks like:

T= T1 |cos(k*x)| ?

... I've been thinking a whole number of things:

1. for temperature it's different than pressure, you don't measure in relation to your reference, you simply measure its value;
2. the fact that you can't measure negative pressure is a measurement problem, but the relative pressure amplitude itself does become negative.
3. My simulation is simply wrong.

Maybe it's a good thing to mention that there is a good reason to doubt that my simulations are reliable. :shy:

2. May 31, 2005

### OlderDan

You know more about what you are talking about than I do, but let me throw out some observations. If they make any sense to you, great; if not you can ignore them
This does not look right to me. Assuming P1 is a constant, this would represent a traveling wave. What you should have for a standing wave is a spatial profile multiplid by a time dependent sinusiod. If x = 0 is the driving end, and the location of maximum pressure variation, the spatial profile would be something like cos(kx), with k chosen to give you a node at the closed end of the tube. That would be modulated by the time varying function. So I think you should have something like

p = P1 cos(k*x)*cos(omega*t) +P0 , P0 being p_atmosphere

At t=0, this becomes your next equation, but there is no need for the absolute value for a half-wavelength tube. You would want it for higher order harmonics.

Actually, I think maybe your tube is a quarter wavelength with the speaker driver at the open end being an antinode, as illustrated here,

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/clocol.html#c1

but it would not have to be. The speaker end could be near a node of a half wavelength tube that would have an antinode in the middle. That seems to fit better with your comparison tube that is closed at both ends. If that is the case, the spatial function would have to be in terms of x = 0 in the middle of the tube. Alternatively, the spatial function could be a sine function that has zeros at both ends of the tube.

As for the temperature following the pressure, I think you are on the right track. This article seems to be what you are talking about, though it apperas to me they are working with a quarter wavelength tube. It explains the heat pumping process you are dealing with

http://www.americanscientist.org/template/AssetDetail/assetid/21006/page/2#23065

Last edited: May 31, 2005