Temperature rise due to conduction

aurthurg

O.k. so I know am going to kick myself when I get the answer but its a long time since I studied thermo and I need a little help

Here s the problem I have a heated surface conducting through an insulator to a cold one.

Heated surface starts at 20c heated to 450c in 45 mins

Insulator has a conductivity of .06W/m/K

Contact area = 100 sq cm

Cold surface starts at 20c

What i need to know is what will the temperature of my cold surface be at 1hr 2hr etc.

Any help would be appreciated

Cheeble

You need to know the specific heat capacities and conductivities of the various layers and the thermal resistance from the cold surface to the ultimate heat sink (usually ambient).

If the cold surface is an infinite, perfectly conducting layer then after 1hr it will still be at 20c.

If the cold surface is an infinitely thin sheet of perfectly insulated foil, it will heat up at the same rate as the hot surface.

Cheeble-eers

rubadub

As said above we would need to know a lot more info on this, is this just a theoretical setup or something you are really doing? In practise I actually rarely use formulas, those I do are ones I derive myself, some based on real thermodynamics but with empirical losses etc factored in.

e.g. does your 20C surface get to 450C in 45mins not matter what? i.e. to do that it really needs variable power, since if you have the cold thing heating up it will draw out power.

Does it get to 450C and remain there? e.g. a block of aluminum from an oven will stay hotter longer, and hold more energy than a thin sheet of aluminium from the same oven.

aurthurg

Cheers for the responses guys

Yep this is a real set up for my thesis.

I ve already manufactured the system but now need to show the math. Basically I have a hollow steel cylinder with lids at both ends. I heat the main body of the cylinder using some cartridge heaters to about 500c.

Now one of these lids i ve decided i need to keep cold so i ve made up a ceramic
plate that goes between it and the rest of the reactor.

Now am pretty sure i can figure out the theoretical energy flow to the lids with simple formula but i need to convert this to a theoritical temperature rise and then compare it to a thermocouple reading.

I do understand the complexities of getting an exact answer but a basic one would do

aurthurg

Sorry should probably add that the insulating barrier is 10mm thick

mawk

i know its not a solution but ill chime in that the hinge will need be made of an insulating material and you'll to look up the emissivity of the materials to work out the radiative heat losses as well as the conductive losses

aurthurg

About the hinge the lids are actually bolted on. I did realise the bolts would conduct heat to the lid but decided that the amount of heat they would transfer would be minimal. I also took steps to reduce this e.g. short bolts, larger holes through the lid so the only direct contact is where the head of the nut tightens on the lid... about 2cm sq approx

rubadub

aurthurg wrote: »

only direct contact is where the head of the nut tightens on the lid... about 2cm sq approx

I have done a bit of work reducing conduction in a few machines, you can use the bigger hole or put a sleeve on the bolt, also you can use washers so there is no direct contact of steel at all.

It is quite complex stuff, I find it much easier to do your tests first and get your empirical results, and then apply the theory to it, when it doesn't add up you can think more and discover what you left out. Any photos of the rig would probably help.

aurthurg

I ve have isolated the bolt with washers and a bigger hole through the lid. I m actually pretty confident in my design, I ve tested with this ceramic insulating board before and found it to work. I would just like to show in my thesis write up that I had a basic matheamatical reasoning to support making this component not just engineering gut instinct

I not sure how to post pics btw, do I have to host them on tinypic etc. first?

Cheeble

You still need more information.

The mathematical approach is to examine the rate of enery transfer into the cold material and then the rate of energy transfer away from it. Knowing the material properties you can then solve dynamically for the rate of temperature rise of the various components.

The first part comes from knowing the temperature difference, and the thermal path, from the hot surface to the cold surface.

The latter comes from knowing about the thermal path from the cold surface to the eventual heat sink (as I said before, normally ambient).

So, you need to evaluate the thermal path from the cold lid to ambient. If the whole experiment is in a vacuum, the cold surface will heat up quickly, if it's in free air, the rate will depend on thermal resistance from the lid to ambient (e.g. does it have cooling fins? are you blowing air over it? etc). Once you have a controlled set of conditions, this is easily determined experimentally. It's quite hard to do it theoretically as there are so many variables and unknowns. You might get an initial estimate from comparing to similar structures, otherwise you're looking for some very expensive thermal modelling software.

Cheeble-eers

Turbulent Bill

Cheeble wrote: »

You still need more information.

The mathematical approach is to examine the rate of enery transfer into the cold material and then the rate of energy transfer away from it. Knowing the material properties you can then solve dynamically for the rate of temperature rise of the various components.

The first part comes from knowing the temperature difference, and the thermal path, from the hot surface to the cold surface.

The latter comes from knowing about the thermal path from the cold surface to the eventual heat sink (as I said before, normally ambient).

So, you need to evaluate the thermal path from the cold lid to ambient. If the whole experiment is in a vacuum, the cold surface will heat up quickly, if it's in free air, the rate will depend on thermal resistance from the lid to ambient (e.g. does it have cooling fins? are you blowing air over it? etc). Once you have a controlled set of conditions, this is easily determined experimentally. It's quite hard to do it theoretically as there are so many variables and unknowns. You might get an initial estimate from comparing to similar structures, otherwise you're looking for some very expensive thermal modelling software.

Cheeble-eers

+1, try to isolate your rig as much as possible to get standard known boundary conditions, otherwise there'll be too many variables for a simple model to have validity. You could also increase the number of monitoring points and build a more complex model, but analysing this would be a nightmare if you don't know fundamentally how the system works.

I'd try a 1-D transient counductive heat simulation for the lid, compare the results to your measurements and work from there.

Gurgle

aurthurg wrote: »

Here s the problem I have a heated surface conducting through an insulator to a cold one.

Heated surface starts at 20c heated to 450c in 45 mins

Insulator has a conductivity of .06W/m/K

Contact area = 100 sq cm

Cold surface starts at 20c

What i need to know is what will the temperature of my cold surface be at 1hr 2hr etc.

Any help would be appreciated

Erm, whats the thickness of your insulator?

aurthurg

Cheers guys all advice was greatly appreciated

I might try a simple energy flow in energy flow out equations to put in my right up

I am however doing my first real test run tomorrow and will have 8 thermo couples reading on the reactor and the lid so will have experimental results to work with. I m fairly confident that my insulator will do the job.

Will post how the run goes