Model to predict time needed to cool down steel billets.

[capterdi]

Hi,

I´m looking for a model or formula to calculate the time that
is needed to cool under natural conditions (not forced air flow, water, etc.) a pile of steel billets produced at a melt shop. The process is as follows: steel billets with square
cross section of 140 mm x 140 mm and 12.7 m. long are produced via the casting of liquid steel. These billets are placed separated one from the other on a cooling bed, until they reach aprox. 450 °C. Then they are stored in a yard, where they are
placed side by side. Here they sit tightly close one to another; each billet touches the next one along the face that has a horizontal measure of 12.7 m. When the yard has 20 contiguous billets (we can call this a bed of 20 billets), then the next
billets are piled on top of this bed, until another bed of 20 billets is formed. The process continues in such way, until 25 beds are completed. So, we have a pile 2.8 m wide, 3.5 m. high
& 12.7 m. depth.

I´m faced with this problem: these steel billets will be shipped later to another industrial plant, that is far away. So flat bed trailers will be used to transport the billets. The platforms of the trailers are made from steel and wood; so the billets will
sit on top of wood, and we have to be careful in allowing the billets to cool down to a temperature of 150 °C or less, in order to rest assured that the wood won´t catch fire.

I have observed that it´s not the same to ship billets from the top beds; these ones have cooled to a lower temperature. Suppose that we want to ship the hole pile, truck after truck, with no loading delays. We will see that after finishing with the uppermost beds, the next ones will be hotter, due to the temperature gradient that exists from the center of the pile to each of the sides.

In short, about the adding and removing of billets: we can assume at this point that once they are piled, they will remain were they are, until the time for shipment comes. Then, let´s assume that once we begin removing billets from top, and continuing toward the bottom of the pile, the time for this operations is equal to zero, so that there is no additional complication for the theoretical model. In other words, what I would like is this:

Input data: initial temperature (eg. 450 °C)
Size of solid body: as described above

Based on this, there is an amount of energy stored in the form of heat, and a temperature gradient begins to develop. I would like the model to predict which is the innermost temperature of the body (eg. center) after 24, 36, ...etc. hours.

So, I´m looking for a model or formula to predict the temperature of the center of a billet depending on the bed (lowest one, second lowest, etc.), the time that has
passed since the pile was formed and the initial temperature.

Thank you.

 

[russ_watters]

The equation is from Newton's law of cooling: http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html

Unfortunately, the reality is that it is extrordinarily difficult to find an analytical solution to a heat transfer problem like this. Experimentation followed by curve fitting is the best way.

 

[capterdi]

Russ,

Thank you...let me check that link...

 

[capterdi]

Ok..I´ve gone thru that material, and the example that´s included. This is fine. Thank you.

Now, a question: really this law was found by Newton? Then, could it be the 4th law? Where can I find more material about this law, like its origin, examples, etc?

 

[FredGarvin]

Any heat transfer text will have a section on natural convection.

Here is an on-line heat transfer book:
http://web.mit.edu/lienhard/www/ahtt.html

 

[capterdi]

Fred,

Thank you...!

 

[AlephZero]

Unfortunately, the reality is that it is extrordinarily difficult to find an analytical solution to a heat transfer problem like this. Experimentation followed by curve fitting is the best way.

Certainly, experimenation is the quickest way to get a reliable heat transfer coefficient for your problem. You may have done the experiment already if you know the rate of cooling for the complete stack.

Given the HTC, an analytical solution for a rectangular block is not too difficult. In fact Fourier invented Fourier series to solve the differential equations for similar problems.

The standard reference for analytical solutions is "Conduction of heat in solids", Carslaw & Jaeger, 1959.

You may be able to do some useful scaling of cooling times without getting a full solution. As a first approximation the heat capacity of the blocks will be proportional to the volume, and the rate of heat loss will be proportional to the surface area.