TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 110

As for the heat transfer coefficient, generally valid expressions can be formulated
using dimensionless numbers and empirical constants. The larger the concentration
gradient orthogonal to the surface, the higher the rate of mass transfer. This is char-
acterized by the Sherwood number, which is the normalized concentration gradient at
the surface. The Ranz Marshall correlation for mass transfer between a sphere and a

surrounding fluid is given by

-

6Re 2 Sc 3 ,0 ReSc 3 <5×10 4

Sh = 2 + 0 : ≤ ð Eq : 4 : 28 Þ

where Sc is the Schmidt number of the fluid.
Chilton and Colburn have introduced two additional dimensionless numbers, one
for heat transfer j H and one for mass transfer j D (Bird et al., 2007):

Nu
Re Pr 0 : 33 ,

Sh
Re Sc 0 : 33

j H =

j D = ð Eq : 4 : 29 Þ

At large Reynolds number (turbulent flow), these two numbers turn out to be equal,
expressing the analogy between heat transfer and mass transfer. This gives a relation
for h / k :

0 : 33 0 : 67

h

k = D

Pr
Sc

Sc
Pr

c p Le 0 : 67

= ρ c p = ρ ð Eq : 4 : 30 Þ

The transport properties in this expression are those of the flowing medium. The rela-
tion Equation (4.30) is of great practical use because in general it is easier to measure
the heat transfer coefficient than the mass transfer coefficient.

4.5 TRANSFER OF HEAT AND MASS WITH PHASE CHANGE

In the case of phase changes, as is the case in melting, evaporation, pyrolysis, and
combustion, the transport equations have to be set up for the separate phases (gaseous,
liquid, and solid), and in addition to heat transfer between the phases, also the mass
transfer has to be described. When setting up the energy balance, the latent heat
involved in the phase change has to be taken into account. In the following, we discuss
two cases: the evaporation of a biofuel droplet and the conversion to char of a biomass
particle.

4.5.1 Evaporation of a Single-Component Fuel Droplet

Let us consider the evaporation of a spherical fuel droplet. The rate of change of the
droplet diameter depends on properties of the droplet and on properties of the sur-
roundings. The temperature and composition of the droplet and surrounding gas,
and also the relative velocity, have a strong influence on the evaporation rate. The
simplest situation arises when a spherical single-component droplet of diameter d d
is put in, e.g., hot air at constant temperature and composition.