TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 108

4.4 CONVECTIVE HEAT AND MASS TRANSFER

In the case of heat transfer between a fluid and a solid wall, the heat flux component
normal to the wall may be related to the difference between the surface temperature of
the solid wall T w and the “ bulk ” fluid temperature T b :

j q , n = h T w − ð T b Þ ð Eq : 4 : 25 Þ

This relation is referred to asNewton s lawof cooling and is a pragmatic lawrather than a

fundamental law. The heat flow (per unit area of the wall) is assumed to be proportional
to the temperature difference. The proportionality constant h is called the heat transfer
coefficient and is a phenomenological quantity. It depends on the properties of the flow
near the wall and on the physical properties of the fluid. When temperature gradients are
present, an appropriate mean value between values at different temperatures is to be
used. For accurate results, the 1/3 rule should be used, stating that temperature-
dependent fluidproperties are evaluated at aneffective temperature andcomposition that
are weightedmeans of the values at the surface and in the surroundings (with a weight of
1/3 for the surroundings value) (see Hubbard et al., 1975). For two or more heat transfer
processes acting in parallel, heat transfer coefficients simply add. For two or more heat
transfer processes connected in series, heat transfer coefficients add inversely.

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Example 4.1 Heat transfer processes in series

Consider heat transferred from a fluid at bulk temperature T b2 to a fluid at bulk tem-
perature T b1 through a flat wall with thickness D and thermal conductivity λ w . This

can be considered as three heat transfer processes in series: convective heat transfer
from bulk fluid 2 to the wall, conductive heat transfer through the wall, and convec-
tive heat transfer from the other side of thewall to bulk fluid 1. For each step, the heat
flux can be expressed in terms of a temperature difference, giving three expressions

= λ w for the same heat flux: j q , n = h 1 T w1 − .

The first and last expressions represent the convective heat transfer at the two
sides of the wall. The middle expression represents the heat transfer by conduction.
By eliminating the wall temperatures from these equations, one obtains

ð T b1 Þ

D T w2 −

ð T w1 Þ = h 2 T b2 − ð T w2 Þ

j q , n = h combined ð T b2 − T b1 Þ

with the heat transfer coefficient for the combined process given by

h combined = 1

1

h 1 + D λ w + 1

h 2

Generally valid expressions for the heat transfer coefficient can be found by for-
mulating the dependence on flow conditions and other factors using dimensionless
numbers (see Table 4.1) and empirical constants. The higher the heat

transfer