Page 204 - Modelling in Transport Phenomena A Conceptual Approach

184 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

Transport Gradient of

Transfer Difference in )]
= [ ( coefficient ) ( Quantity/Volume fluid (7.1-10)
The gradient of driving force is expressed in the form
Difference in driving force
Gradient of driving force = (7.1-11)
Characteristic length
On the other hand, “Difference in Quantity/Volume” can be expressed as
Difference in ) = (’Thnsport property
Quant ity/Volume Diffusivity driving force ) (7.1-12)
Substitution of Eqs. (7.1-11) and (7.1-12) to the left- and right-hand sides of Eq.
(7.1-10), respectively, gives

(7.1-13)
Diffusivity

in which Bi designates the Biot number defined by
(Difference in driving force)sotid
Bi = (7.1-14)
(Difference in driving
Therefore, the Biot numbers for heat and mass transfer are defined as

(7.1-15)

(7.1-16)

It is important to distinguish the difference between the Biot and the Nusselt
(or, the Sherwood) numbers. The transport properties in the Biot numbers, Eqs.
(7.1-15) and (7.1-16), are referred to the solid, whereas the transport properties in
the Nusselt and the Sherwood numbers, Eqs. (3.411) and (3.412), are referred to
the fluid.
When the Biot number is small, one can conclude from €2q. (7.1-14) that
( driving force )solid ( Difference in (7.1-17)
Difference in

Therefore, dependent variables may be considered uniform within the solid phase
only if Bi << 1.