Some simple heat transfer formula derivations and questions

[mech-eng]

Hi. I would like to ask a simple question. Here is the link of the file I study on. Immediately before the formula 4.9 for Biot number. Lc=V/As but I cannot understand it and I think it is not clear enough. How it appears, for what the word "characteristic" stands for, for example a pipe? For what V and As stands for?
Is that characteristic length for only "spherical objects" or it is valid for cylindrical objects?
File: http://kntu.ac.ir/DorsaPax/userfiles/file/Mechanical/OstadFile/Sayyalat/Bazargan/cen58933_ch04.pdf

Thanks.

 

[SteamKing]

Hi. I would like to ask a simple question. Here is the link of the file I study on. Immediately before the formula 4.9 for Biot number. Lc=V/As but I cannot understand it and I think it is not clear enough. How it appears, for what the word "characteristic" stands for, for example a pipe? For what V and As stands for?
Is that characteristic length for only "spherical objects" or it is valid for cylindrical objects?
File: http://kntu.ac.ir/DorsaPax/userfiles/file/Mechanical/OstadFile/Sayyalat/Bazargan/cen58933_ch04.pdf

Thanks.

If you read the text preceding the formula for the Biot number, it becomes apparent that V is the volume of the body and As is the surface area of same. (See Section 4-1). By dividing volume by surface area, you are left with a "length" Lc, dimensionally speaking, which is taken as the characteristic length of this body. If the Biot No. is low, then the heat transfer characteristics can be treated using the lumped system analysis.

As far as what Lc is for a cylindrical object, I leave that calculation to you, since the quantities of interest have been explained.

The word "characteristic" in this context just means how you decide to treat the subsequent analysis, i.e., whether the lumped system is accurate or not.

For example, in calculating the Reynolds No. for internal pipe flow, the characteristic length to use is the inside diameter for circular pipes. If you are calculating the Reynolds No. for flow over an airfoil, you would use the chord length of the airfoil.

https://en.wikipedia.org/wiki/Reynolds_number

 

[BvU]

Clearly, V is volume and A_s is surface area. Figure 4-6 indeed shows a sphere yielding D/6, but the expression can be used for a cylinder too: it gives D/4 if you can ignore the ends. In my imagination L_c is something like the average distance to the surface.
For a pipe shape you'd get thickness/2 (*) -- but I have a hard time thinking of a practical application: a reaction in the space between two concentrical pipes or something.

(*)
$\ \pi r_o^2\ - \ \pi r_i^2\ \over 2\pi r_o + 2 \pi r_i$ with $\ r_o = r_i + d = r + d \$ this becomes $\ {2\pi r d \over 4 \pi r} = d/2$ (ignoring terms d²/r and higher)

Your question is good: wiki makes a mess of it and I hope someone can point us to a more extensive treatment. I looked in Carslaw and Jaeger ('standard reference') and didn't even find Biot mentioned.