Biot number

The Biot Number (Bi) is the ratio of heat transfer resistance due to internal conduction vs the surface convection. At the interface of the solid, heat transfer is taking place with the surrounding via convection. This could be forced by external convection or natural convection. The heat transfer taking place inside the solid is through conduction.

For highly conductive and relatively small objects, Biot number usually turns out to be really small.

$\mathrm{B}\mathrm{i}=\frac{h{\ell }_c}{k}$

(1)

where $h$ is the convective heat transfer coefficient, ${\ell }_c$ is a characteristic length, and $k$ is the thermal conductivity of the solid.

Introduction

Energy transfer between a body and a fluid experiences resistance from the heat transfer boundary and from the solid itself. If Bi << .1, convective resistance dominates in the system, and thus a uniform temperature distribution throughout the solid can be assumed. A small ${\ell }_c$ can also cause this assumption to be reasonable, due to the lack of distance for temperature gradients to form in. This assumption allows for the lumped capacitance method to be used, which states that

$T_{\mathrm{\infty }}-T=(T_{\mathrm{\infty }}-T_0)(e^{\frac{-h{A}_s}{\rho c_{p}V}t})$

(2)

where $\rho$ is the density of the solid, $c_p$ is the heat capacity of the solid, $V$ is the volume of the solid, $T_{\mathrm{\infty }}$ is the temperature of the bulk fluid, $T_0$ is the initial temperature of the solid, $A_s$ is the surface area of the solid, and $t$ is time. Before completing an unsteady-state heat transfer problem, the Biot Number should be calculated to determine whether the lumped sum model or a Heisler chart would be a more appropriate model of deriving heat transfer.

The lumped capacitance method can also be written using the Bi and the Fourier Number (Fo) as,

$\frac{T-T_{\mathrm{\infty }}}{T_0-T_{\mathrm{\infty }}}=(e^{-BiFo})$

(3)

Characteristic Length

The characteristic length is dependent on the geometry of the solid. Unless otherwise said, it is typically derived by dividing the volume (V) by the surface area (A).

• For a sphere,

$x=\frac{V}{A}=\frac{\frac{4}{3}\pi r^3}{4\pi r^2}=\frac{r}{3}$

(4)

• For a long cylinder,

$x=\frac{V}{A}=\frac{\pi D^2\frac{L}{4}}{\pi DL}=\frac{D}{4}=\frac{r}{2}$

(5)

• For a long square rod,

$x=\frac{V}{A}=\frac{4x^{2}L}{8xL}=\frac{x}{2}$

(6)

Mass Transfer

There is a mass transfer equivalent for Bi generally notated as $B{i}_m$, which is used during mass diffusion processes.

$B{i}_m=\frac{h_m{\ell }_c}{D_{AB}}$

(7)

where

• $h_m$ is the film mass transfer coefficient
• ${\ell }_c$ is the characteristic length discussed above
• $D_{AB}$ is the diffusion coefficient