Chapter 3, Problem 3.1P

Consider a flat plate or a plane wall with a thickness L and a long cylinder of radius $r_0$. Both of these are made of materials such that they can be treated as lumped capacitances $\left(\text{Bi }<0.1\right)$. Show that in each case, the characteristic length ${\mathcal{l}}_c$, defined ${\mathcal{l}}_c=(V/A_s)$, can be approximated as $(L/2)$ and $(r_o/2)$, respectively.

To determine

The characteristic length for a long cylinder and flat plate

Explanation of Solution

Characteristic length for long cylinder:

For long cylinders heat transfer through ends can be neglected, as the surface area at ends is negligible in comparisons with the total surface area of cylinder.

$\text{Characteristic length}=\frac{\text{Volume}}{\text{Surface Area}}=\frac{{\text{πr}}^2\text{L}}{2\text{πrL}}=\frac{\text{r}}{2}$

“r” is radius of the cylinder.

“L” is length of the cylinder.

Characteristic length for plane wall or flat plate:

The plate should be very thin so that internal resistance of the plate is negligible and lumped parameter analysis can be used.

For thin plates heat transfer through ends can be neglected as, heat transfer area at ends is negligible in compression with total surface area of the plate.

$\text{Characteristic length}=\frac{\text{Volume}}{\text{Surface Area}}=\frac{\text{L}\times \text{b}\times \text{h}}{2\text{bh}}=\frac{\text{L}}{2}$

L = thickness of the plate.

“b” and “h” are width and height of the plate.