Basic Thermodynamics: Calculating Temperatures Over Time

[JeffD]

Given a system with known materials of known geometries, and an initial temperature profile, it seems relatively straight forward to calculate a final temperature profile. I assume there are computers that can do this for all kinds of arbitrary systems.

What about determining the temperatures over time. The decay of the "hot spots" and the rise of the cooler areas. Are there predetermined time constants for different materials? What assumptions about the materials are required (isomorphic in all directions?). Are these time constants empirical, or derived. How do they compare with "real world" experiences?

My background is in electric power. Thermo is not one of my strong subjects, so please be nice. :)

 

[DEvens]

The usual thing to do is not based on "time constants" but thermal conductivity and heat capacity. Those parameters are typically determined from measurements.

Basically, you start with heat capacity as a function of temperature, thermal conductivity also as a function of temperature, the material at each location, and some sources and sinks of heat. Then you put in some boundary conditions. Then you solve the time dependent heat diffusion equation, usually using some numerical method.

If the problem can be treated as 1-D, and if it is at equilibrium, then you have the simplest situation you are likely to get. It becomes a 1-D differential equation with boundary conditions. So, for example, a simple spherical object, generating heat through radioactive decay for example, in a cooling bath, could be treated as a radial problem with symmetry. The equilibrium situation would have a temperature profile that did not change over time. As simple as you are likely to get. If you have an analytical function for the thermal conductivity, you might even be able to get an analytical solution for temperature. And because it is time independent, you do not even need the heat capacity.

The general case can be very complicated indeed. You could have a material that changes composition and density over time. For example a chemical reaction. It could generate different amounts of heat depending on the conditions. And it could have different capability of getting rid of heat, again depending on the conditions. Or even the history. For example, a nuclear reactor fuel assembly could start producing one amount of heat. Then if there is a power excursion this could rise. Then this could cause the coolant to boil off, changing both the material present and the ability to get rid of heat. And both thermal conductivity of the materials, and the heat capacity, are non-trivial functions of temperature. And it might require a full 3-D plus time solution to get an accurate picture. A computer code that could accurately do that would be a substantial job.

 

[Chestermiller]

Hi DEvens,

I think when you used the term equilibrium in your post, you meant to say steady state.

Chet

 

[JeffD]

DEvens, thank you for your response. One key thing I am missing, I think, in your explanation. Probably something obvious that thermo-folks take for granted that everyone understands.

I get that the heat capacity and thermal conductivity of materials are functions of the material. How does the "time dependence" get into the heat diffusion equation? There may be a time function for the heat sources or the heat sinks, but how to account for the heat transfer time across a material?

I would assume this is all a solved problem. I was thinking of temperature controllers, thermostats if you want, and whether a high tech thermostat in an industry application would be more than what I have at home, which is: turn on full when temperature is above X, and turn off when not. You know, some sophisticated PID control algorithm that take into account how far from X the actual temperature is and the heating and cooling rates through materials.

This is the kind of stuff nerds think about when its -2 F outside, and its seems to take forever to heat up the office in the morning.

 

[Chestermiller]

To get the temperature variation over time, you need to use the transient heat balance equation. This breaks the body down into tiny parcels, and does the following balance on each of them: rate of heat input - rate of heat output = rate of heat accumulation. The rate of accumulation involves the partial derivative of temperature with respect to time. You end up with a partial differential equation, with the temperature varying with both time and spatial position.

Chet

 

[anorlunda]

If you combine what Chet said with "finite element method" (look it up), then you can solve for the transient temperatures for nearly any aribtrsry object and initial conditions.

 

[JeffD]

Gotcha. So Chet, where does one find the rate of heat input, and output, for each little incremental parcel. The rate is a property of the material, empirically derived I assume, that I can look up somewhere? Say for dense wood, or for aluminum, or air or water.

 

[Chestermiller]

Gotcha. So Chet, where does one find the rate of heat input, and output, for each little incremental parcel. The rate is a property of the material, empirically derived I assume, that I can look up somewhere? Say for dense wood, or for aluminum, or air or water.

If you're talking about heat conduction, then you probably already know that the local heat flux is equal to the thermal conductivity of the material times the derivative of temperature with respect to distance. If you don't remember this, you need to go back and review intro physics. If you want all the gory details of transient heat conduction behavior in bodies, see Transport Phenomena by Bird, Stewart, and Lightfoot.

Chet

 

[JeffD]

No, thermo is not my strong suit. That said, what you said makes sense. I think, then, that its the thermal conductivity of a material that I am after.

Thanks all.