Thermal Conductivity of an Infinite Thin Plate with a heater on it

[kuba]

Homework Statement

It's not a homework problem, but an engineering problem I try to solve before I delve into experimentation

I have a source of heat (SSR - a solid state relay) that's well thermally coupled to the inside of a steel box. The thermal resistance between the SSR and the box is to be ignored. The SSR has constant power. The steel box has simple rectangular sides and air is flowing in it. It is much larger than the SSR.

Known are:

$$q$$ - the power generated by the SSR,
$$k$$ - thermal conductivity of the material the box is made of,
$$\delta$$ - the thickness of the material.

I know that there's some air flow above the metal surface inside of the box, so that the heat transfer coefficients of the top/bottom of the plate are known. I ignore heat transfer between the SSR and the air, heat only goes SSR to the box, then BOX to air in it. Outside of the box is assumed to be insulated (negligible heat transfer).

I've decided to model this as an infinite thin plate with heated circular hole.

So, known are also:

$$h_1$$ - the heat transfer coeffcient of the top of the plate (where the SSR resides, and where there's airflow),
$$h_2$$ - the heat transfer coefficient of the bottom of the plate,
$$T_1$$ - temperature of the air on top of the plate,
$$T_2$$ - temperature of the air on the bottom of the plate.

I need to know the temperature $$T$$ of the SSR in steady state. I only need to know rough estimate for feasibility determination, it will be tested afterwards of course.

Homework Equations

From the horrible Remsburg's "Thermal Design of Electronic Equipment" , I've got the equations that follow.

$$\frac{k \delta \Delta_{T-T_\infty}}{q} = \frac{K_0(\frac{Br}{\delta})}{2\pi(\frac{Br_1}{\delta})K_1(\frac{Br_1}{\delta})}$$

where

$$r_1$$ is the radius of the hole (modelling the SSR in contact with the plate),
$$r$$ is the distance from the hole to the point of interest,
$$B=\sqrt{Bi_1+Bi_2}$$,
$$T_\infty=\frac{T_1+HT_2}{1+H}$$,
$$H=\frac{Bi_1}{Bi_2}$$,

where

$$Bi$$ is the Biot Number. In the case of a plate, for either top or bottom surface, it is $$Bi_i=h_i\delta/k$$

They don't explain two constants there -- $$K_0$$ and $$K_1$$ -- what are they?

The Attempt at a Solution

I already chose the model (infinite thin plate with a hole), I only need to know how to apply the equations given above. The book author (Remsburg) gives no hints as to what $$K_0$$ and $$K_1$$ are in the equation above.

Solving for SSR temperature rise, I presume I can set $$r=r_1$$, such that a lot of the equation vanishes:

$$\frac{k \delta \Delta_{T-T_\infty}}{q} = \frac{K_0}{2\pi K_1}$$

The $$K_0$$ and $$K_1$$ look pretty important here

Any hints? The Remsburg book is nearly useless, it presents bunches of equations with little explanation and no examples, it's a recipe for disaster more than anything. But that's what I've found in our library. Any recommendations for another book where conduction in thin plates with heat sources is addressed (in forms of ready equations)? Or better yet, a clue as to what those K's are?

Kuba

 

[AlephZero]

K_0 and K_1 are almost certainly some sort of Bessel or Hankel functions.

But I'm not going to guess exactly which sort, because IIRC not everybody uses the same notation for them.

Is there a table of the standard notations used in the book? A table of the function values would be even better of course.

Update: The Wikipedia definitions look like ones you want: http://en.wikipedia.org/wiki/Bessel_function

 

[piercas]

I would suggest looking for additional resources or consulting with a colleague who has experience in this area. The equations provided in the Remsburg book may not be sufficient for solving this problem, as they are not clearly explained and lack examples. It may also be helpful to reach out to experts in the field of thermal conductivity or heat transfer for guidance.

In terms of the K_0 and K_1 constants, it is possible that they are specific to the Remsburg book and may not have a standard meaning. It would be best to clarify this with the author or another expert in the field. Alternatively, you could try searching for similar equations in other resources or conducting your own experiments to determine the values of these constants.

Overall, it is important to thoroughly understand the equations and assumptions being made in order to accurately solve this problem. I would recommend seeking additional resources and guidance to ensure a successful solution.