Thermal equilibrium formula

[Damascus Road]

Hello all.

Been a long time since I took a thermal course. I'm looking for a formula to work with the thermal stabilization of something. Specifically, I want to minimize the time required for thermal stabilization to occur and how to mathematically calculate what that set temperatures should be. Presumably, for example, if I wanted something to stabilize at +40C from a starting point of 0C, I would want to have a set temperature of over 40C for some amount of time.

Any help would be appreciated!Edit: the item being cooled or heated is a container with various solids inside and is being heated or cooled by air. Imagine that there are temp sensors located on some of the solids within the container.

 

[anorlunda]

Your question can not be answered without a lot more information.

Thermal equilibrium for an object means that heat flowing in matches heat flowing out. You need equations for the heat transfer from the source and heat losses to surroundings. That will depend on the size and shape and the surface properties.

To minimize the time, you need to maximize the surface area. For a metal, you would flatten it to a foil, or evaporate it into a gas.

The specific heat of the material (energy per degree per unit of mass) is a property you must look up. It helps tell you how fast it will respond.

Your description sounds like an oven. If you set the oven to a temperature higher than 40, the object will warm faster, but the object's temperature won't stabilize at 40, it will continue going up until it reaches oven temperature.

You should rethink what you are actually trying to accomplish and then rephrase your question.

 

[Damascus Road]

Your question can not be answered without a lot more information.

Thermal equilibrium for an object means that heat flowing in matches heat flowing out. You need equations for the heat transfer from the source and heat losses to surroundings. That will depend on the size and shape and the surface properties.

To minimize the time, you need to maximize the surface area. For a metal, you would flatten it to a foil, or evaporate it into a gas.

The specific heat of the material (energy per degree per unit of mass) is a property you must look up. It helps tell you how fast it will respond.

Your description sounds like an oven. If you set the oven to a temperature higher than 40, the object will warm faster, but the object's temperature won't stabilize at 40, it will continue going up until it reaches oven temperature.

You should rethink what you are actually trying to accomplish and then rephrase your question.

apologies. you are correct, rapid thermal stability is what I am actually after, and it is essentially being heated or cooled in an oven.

The object being heated or cooled is an electronic device comprised of many different materials; metals, glass, electronics, and more.
Is there anything I can use to approximate the time to stability as a proof of concept?

The device will stabilize at some temperature over 40, within some margin, when the oven is set to 40. I would like a formula to approximate the effect of setting the chamber to 50 (for example) for a limited amount of time and adjusting the temperature of the oven to 40 in the best way to minimize the time required for stability.

Does that help at all?

 

[anorlunda]

The key thing you need then is the specific heat of the object. Perhaps you could approximate by just considering the specific heat of the material that composes the most mass in the object and ignoring the other materials.

Can you estimate the surface area of the object?

You also need the heat transfer coefficient between the object;s surface and the air.

Finally, you need to specify to what accuracy do you mean "thermally stabilized".

At best, you can only make a very rough estimate via calculations. You might be better off with experiments rather than calculation.

It would be faster to dip it in a bath of warm liquid rather than an oven full of air.

 

[Damascus Road]

The key thing you need then is the specific heat of the object. Perhaps you could approximate by just considering the specific heat of the material that composes the most mass in the object and ignoring the other materials.

Can you estimate the surface area of the object?

You also need the heat transfer coefficient between the object;s surface and the air.

Finally, you need to specify to what accuracy do you mean "thermally stabilized".

At best, you can only make a very rough estimate via calculations. You might be better off with experiments rather than calculation.

It would be faster to dip it in a bath of warm liquid rather than an oven full of air.

I can estimate the surface area, yes, and liquid isn't an option.
Since this is an electronic device, there are several temperature sensors within the device that are read, so when they stop fluctuating and remain fairly constant, thermal stability is reached.
Also, the air of the oven blows at the temperature the oven is set at.

I'm having trouble finding a formula for rough calculations to accompany experiments... does such a thing exist?

Thanks again.

 

[anorlunda]

Here's a very simple formula.

$\frac{dT(t)}{dt}=\frac{k\cdot{a}}{h\cdot{m}}(T_{oven}(t)-T)$

Where:
T(t) is the temperature of the object a a function of time
h is the specific heat of the material
k is the heat transfer coefficient of the surface to the moving air (may be a function of air speed)
a is the surface area
m is the mass of the object
Toven(t) is the temperature of the oven as function of time.

You could use that formula in a simple time simulation for any arbitrary Toven(t). You'll also need the initial conditions.

If would get more complicated if k is different for the surface facing the moving air than the surface on the other side.

Implicit in this is the assumption that temperatures are uniform throughout the interior of the object.

Your difficulty is not the formula, but rather estimating all those constants.

Good luck

 

[Damascus Road]

Here's a very simple formula.

$\frac{dT(t)}{dt}=\frac{k\cdot{a}}{h\cdot{m}}(T_{oven}(t)-T)$

Where:
T(t) is the temperature of the object a a function of time
h is the specific heat of the material
k is the heat transfer coefficient of the surface to the moving air (may be a function of air speed)
a is the surface area
m is the mass of the object
Toven(t) is the temperature of the oven as function of time.

You could use that formula in a simple time simulation for any arbitrary Toven(t). You'll also need the initial conditions.

If would get more complicated if k is different for the surface facing the moving air than the surface on the other side.

Implicit in this is the assumption that temperatures are uniform throughout the interior of the object.

Your difficulty is not the formula, but rather estimating all those constants.

Good luck

Thanks!
it's been a while since I took a thermal course...

I believe I should be able to use this to add in more oven points as well without error, yeah? What I want is to have several temperature set points to minimize the time.. so I might start at 60 for 10 minutes, move to 50, move to 45, and then to 40 over the course of some time period.

 

[Damascus Road]

Really, I don't even need to work with real specific heats and masses... all I really care about is comparing the time of the device sitting at exactly 40, vs. the time when we start at temperatures of above 40, which would result (I think) in a fraction of other result.