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Zap-o
09-22-2011, 11:25 PM
I have a though experiment wand want to bring it to life.

An aluminum block that is rectangular 3" long x 1" x 1". I apply a heating element to one of the long ends and measure the temperature on that same end (I can measure up to 100th of a degree). I want to know how to calculate the temperature at the opposing end.

Shouldn't the temperature be different. I think so but by how much? I know its going to be a small factor but whats the math behind this?

Pliers
09-23-2011, 12:21 AM
The factors would be how much energy you are applying and how much energy are you losing from convection.
Even if you did this in a vacuum you would still be losing energy in the form of infrared radiation.

There are many articles on the web about this.

Example - Heat Transfer by Conduction
A plane wall constructed of solid iron with thermal conductivity 70 W/moC, thickness 50 mm and with surface area 1 m by 1 m, temperature 150 oC on one side and 80oC on the other.

Conductive heat transfer can be calculated as:

q = (70 W/moC) (1 m) (1 m) ((150 oC) - (80 oC)) / (0.05 m)

= 98,000 W

= 98 kW

Zap-o
09-23-2011, 12:53 AM
Will the shape of the block have an affect on the temperatures?

Searider
09-23-2011, 01:29 AM
Will the shape of the block have an affect on the temperatures?

I think the surface area is what counts. More Surface Area = more area to radiate and conduct heat out of the block. Remember that a rough surface will have more surface area than a polished surface. The tempurature of the surounding medium (Air?) will effect the rate of conduction but I don't think it will effect the rate of radiation.

ElectricAye
09-23-2011, 01:49 AM
...

Shouldn't the temperature be different.....

Yes, the temperature will be different. But calculating it to any degree of accuracy can be a very involved process, best done with some sort of heat transfer modeling software.

The comments made about convection are right on. In fact, convection can play such an important role, it matters how your bar is oriented in space - with its length parallel to horizontal or parallel to vertical, etc. Surface finish will affect radiative heat transfer. How well your heat source is thermally coupled to the bar matters, too, otherwise the heat source might lose most of its heat via its own skin and transfer only a small percentage into the bar.

Because cross section of the bar affects conductive heat transfer, the shape of the bar matters, too.

Probably the cheapest and most accurate thing to do is just build a test rig and make the measurements. But then you have to make sure your temperature sensors are thermally coupled well enough to the bar.

Best thing at this point might be to get a book on heat transfer and read some of the basics. Many people get confused about heat transfer because temperature isn't the same thing as heat, and the human intuitive sense of temperature can sometimes trick you into doing dumb things.

Tracy Allen
09-23-2011, 04:23 AM
It's complicated math. Fourier series came about to solve the heat equation (http://en.wikipedia.org/wiki/Heat_equation). Go about a quarter of the way down the wiki page; there is a picture like your bar, only circular cylinder instead of square.

Fourier's biography (http://en.wikipedia.org/wiki/Joseph_Fourier) is interesting to place the Théorie analytique de la chaleur in context.