TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 109

TABLE 4.1 Selected dimensionless numbers relevant for heat and mass transfer
Number

Definition Interpretation

Bi = h 1 L
λ 2

Biot number (heat) Ratio of the boundary layer thermal resistance

to the internal thermal resistance of a solid;
subscripts 1 and 2 denote two different
media

Fo = α t
L 2

Fourier number (heat) Ratio of the heat conduction rate to the rate of

thermal energy storage in a solid.
Dimensionless time

L 3

Gr = g

β ð T w − T b Þ

Grashof number Ratio of buoyancy force to viscous force

3

ν

Le = D

Lewis number Ratio of the thermal and species diffusivities

Nu = h L
λ

Nusselt number Dimensionless temperature gradient at the

surface

Pr = ν
α

Prandtl number Ratio of the momentum and thermal

diffusivities

Re = u L
ν

Reynolds number Ratio of inertial and viscous forces

Sc = D

Schmidt number Ratio of momentum and species diffusivities

Sh = k L
D

Sherwood number Dimensionless concentration gradient at the

surface

coefficient, the thinner the boundary layer and the larger the temperature gradient
orthogonal to the wall. The temperature gradient normalized by the temperature dif-
ference is the Nusselt number, Nu, and its value is related to the value of other dimen-
sionless numbers characterizing the problem at hand. For example, a well-known
correlation is the Ranz

Marshall correlation for flow around a solid sphere. Its form
and range of validity are given by (see, e.g., Bird et al., 2007, p. 681)

-

6Re 2 Pr 3 ,0

RePr 3 <5×10 4 : ≤ ð Eq : 4 : 26 Þ

Nu = 2 + 0

where Re is the Reynolds number based on the particle diameter and on the difference
between the particle velocity and the velocity of the fluid ahead of the particle and Pr is
the Prandtl number of the fluid.
In a similar way, in the case of mass transfer between a solid wall and a surrounding
fluid, the mass flux component normal to the wall may be related to the difference
between the concentration of a species at the wall surface c i,w and the concentration
in the bulk fluid c i,b :

j i , n = k c i , w − c i , b ð Eq : 4 : 27 Þ

The mass flow (per unit area of wall) is assumed proportional to the concentration
difference. The proportionality constant k is called the mass transfer coefficient.