multidimensional management: can you help me?

[87527]

Hello, everyone. I'm struggling with a problem and I can't find any material about it.
In practice I have a square plate of thickness x and sides 300*300 mm.
on one of the two faces of area 300*300, in the center, I have a circular area of diameter 55 mm where the temperature is maintained at a certain value t fixed (t>t environment). on all the rest of the face, on the side faces and on the lower face I have room temperature.
I need to calculate the heat flow from this heated surface area propagates to the rest of the material and then to the environment, depending on the x variable thickness and the thermal conductivity of the plate.
do you know if there is a formula that allows me to do this?
Please, if you know or have any idea how to do it and you can tell me I would be extremely grateful.
or I would just have a formula that led me to have an equivalent resistance and that made the problem one-dimensional.
Daniele.
I attach a drawing.

I don't know if I've posted in the right section, at most I ask to the moderators to move (if possible) the discussion.

 

[paulpaul]

Hi.
It's a complex problem to solve analytically.
side faces (of area 300*x) and upper and lower (of area 300*300) can all exchange heat with the environment? is the circular zone at temperature t limited to the face or penetrates inside the plate as a cylinder diameter 55 mm and height x? with the external environment ipotizzi only convection or even irradiation (depending on temperature t)?

 

[87527]

Hi.
It's a complex problem to solve analytically.
side faces (of area 300*x) and upper and lower (of area 300*300) can all exchange heat with the environment? is the circular zone at temperature t limited to the face or penetrates inside the plate as a cylinder diameter 55 mm and height x? with the external environment ipotizzi only convection or even irradiation (depending on temperature t)?

Thank you for being interested.
then all sides, lateral, upper, lower can exchange heat with the environment except for the circular portion of the upper face which is maintained at constant t. the constant t bond is limited to the face.
the ambient temperature is that canon of 20-25 degrees so I would say that irradiation is negligible. we are in calm air conditions.
in literature I did not find similar cases and I have no idea where to look for them so I thank you for being interested.

 

[paulpaul]

the problems therefore are two:

1. conduction 3d with convective thermal flow boundary conditions imposed (and not the simplest case of pre-set or isolated face fear);

2. natural convection with different exchange coefficients for vertical and horizontal faces.

As far as the first point is concerned, I do not know approximate nor analytical solutions to determine the heat exchanged. maybe in specialized literature you could find something (e.g. sunderland-johnson on form factors).
As for the second, you must use the empirical correlations for determining the thermal exchange coefficient in natural convection for a flat plate varyingly oriented in space (for natural convection orientation is important). or, more quickly, you hypothesize a realistic coefficient.

I think the fastest way is a numerical approach with a fem.

Hi.

 

[87527]

then I am again struggling with a variant of this problem. I wrote what I want to look for and the steps in two photos I attach to the document. Can you tell me if he can stay as an idea? How could I prove the correctness of my observation?

 

[paulpaul]

Hi.
I see that you have slightly simplified the problem by placing the lower face at constant temperature. It seems to me that you have not excluded the convection, wanting instead to understand when it is possible to neglect it and when not, correct?