Chapter 4 Fluid Flow, Heat Transfer, and Mass Transfer:

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Chapter 4 Fluid Flow, Heat Transfer, and Mass
Transfer Similarities and Coupling 4.1
Similarities among different types of
transport 4.1.1 Basic laws
The transfer of momentum, heat , and species
A occurs in the direction of decreasing vz, T,
and wA, as summarized in Fig. 4.1-1. according to
Eqs. 1.1-2, 2.1-1, and 3.1-1
4.1-1 also 1.1-2
Newtons law of viscosity
Fouriers law of conduction
4.1-2 also 2.1-2
Ficks law of diffution
4.1-3 also 3.1-1
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The three basic laws share the same form as
follows Or The three-dimensional forms of
these basic laws are summarized in Table 4.1-1.
4.1-4
4.1-5
For constant physical properties, Eqs.
4.1-1 through 4.1-3 can be written as
follows
4.1-6
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4.1-7 4.1-8
These equations share the same form listed as
follows
4.1-9
In other words, n, a, and DA are the
diffusivities of momentum, heat, and mass,
respectively, and rvZ, rCvT, and rA are the
concentration of z momentum, thermal energy, and
species mass, respectively.
4.1.2 Coefficients of Transfer
Fig. 4.1-2 shows the transfer of z momentum,
heat, and species A from an interface, where
they are more abundant, to an adjacent fluid, and
from an adjacent fluid, where they are more
abundant, to an interface. The coefficients of
transfer, according to Eqs. 1.1-35, 2.1-14,
and 3.1-21, are defined as follows
4
(momentum transfer coefficient)
4.1-10
5
4.1-11
4.1-12
As mentioned in Sec. 3.1.6, Eq. 4.1-12 is
for low solubility of species A in the fluid.
These coefficients share the same form listed as
follows
4.1-13
or
4.1-14
It is common to divide Cf by rv/2 to make
denominator appear in the form of the kinetic
energy rv28/2. As shown in Eq. 1.1-36, the
so-called friction coefficient is defined by
4.1-15
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4.1.3 The Chilton-Colburn Analogy
The analogous behavior of momentum, heat, and
mass transfer is apparent from Examples 1.4-6,
2.2-5, and 3.2-4, where laminar flow over a flat
plate was considered. From Eqs. 1.4-67,
2.2-71, and 3.2-56, at a distance z from the
leading edge of the plate,
4.1-16
4.1-17
4.1-18
where
(local Reynolds number)
4.1-19
(Prandtl number)
4.1-20
(Schmidt number)
4.1-21
and ?8 is the velocity of the fluid approaching
the flat plate.
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Equations 4.1-16 through 4.1-18 can be
rearranged as follows
4.1-22
4.1-23
4.1-24
Since these equations have the same RHS, we see
4.1-25
Substituting Eqs. 4.1-19 through 4.1-21 into
Eq. 4.1-25, we obtain
4.1-26
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This equation, known as the Chilton-Colburn
analogy,1 is ofen written as follows
4.1-27
where the j factor for heat transfer
4.1-28
And the j factor for mass transfer
4.1-29
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The Chilton Colburn analogy for momentum,
heat and mass transfer has been derived here on
the basis of laminar flow over a flat plate.
However, it has been observed to be a reasonable
approximation in laminar and turbulent flow in
systems of other geometries provided no form drag
is present . From drag, which has no counterpart
in heat and mass transfer, makes Cf/2 greater
than jH and jD, for example, in flow around
(normal to) cylinders. However, when form drag is
present, the Chilton Colburn analogy between
heat and mass transfer can still be valid, that
is,
4.1-30
or
4.1-31
These equations are considered valid for
liquid and gases within the ranges 0.6 lt Sc lt
2500 and 0.6 lt Pr lt 100 . They have been observed
to be a reasonable approximation for various
geometries, such as flow over flat plates, flow
around cylinders, and flow in pipes.
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The Chilton Colburn analogy is useful in
that it allows one unknown transfer coefficient
to be evaluated from another transfer coefficient
which is known or measured in the same geometry.
For example, by use Eq. 4.1-26 the mass
transfer coefficient km(for low solubility of
species A in the fluid) can be estimated from a
heat transfer coefficient h already measured for
the same geometry. It is worth mentioning
that for the limiting case of Pr1, we see that
from Eq.4.1-26
4.1-32
Which is known as the Reynolds analogy , in honor
of Reynolds first recognition of the analogous
behavior of momentum and heat transfer in 1874.
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4.1.4 Integral-Balance Equations
The integral-balance equations governing
momentum, heat , and species transfer, according
to Eq. 1.4-3, 2.2-6, and 3.2-4,
respectively, are as follows
4.1-33
(momentum transfer)
4.1-34
(heat transfer)
4.1-35
(species transfer)
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In Eq. 4.1-33 the pressure term has been
converted from a surface integral to a volume
integral using a Gauss divergence type theorem
(i.e., Eq. A.4-2). Furthermore, the body force
fb and pressure gradient can be considered
as the rate of momentum generation due to these
force. In Eq. 4.1-34 the kinetic and potential
energy, and the pressure, viscous, and shaft work
are not included since they are either negligible
or irrelevant in most materials processing
problems. In Eq. 4.1-35 ?wA ?A . These
integral balance equations share the same form as
follows
4.1-36
or
4.1-37
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These equations are summarized in Table
4.1-2. The following integral mass-balance
equation ,Eq.1.2-4, is also included in the
table
4.1-38
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4.1.5 Overall Balance Equations
The overall balance equations for momentum, heat,
and species transfer according to Eqs.1.4-9,
2.2-8, and 3.2-7, respectively, are as follows
4.1.-39
(momentum transfer)
(heat transfer)
4.1-40
(species transfer)
4.1-41
These overall balance equations share the same
form as follows
4.1-42
or
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4.1-43
Where the total momentum, thermal energy, or
species A in the control volume O is
4.1-44
In Eq.4.1-39 the viscous force Fv at the
wall can be considered as the rate of momentum
transfer through the wall by molecular diffusion.
The pressure force Fp and the body force Fb , on
the other hand, can be considered as the rate of
momentum generation due to the action of these
forces. In Eq.4.1-40 Q is by conduction, which
is similar to diffusion.
The above equations are summarized in Table
4.1-3. The following overall mass balance
equation (i.e. Eq 1.2-6), is also included in
the table
4.1-45
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4.1.6 Differential Balance Equations
The differential balance equations governing
momentum, heat, and species transfer, according
to Eqs. 1.5-6, 2.3-5 and 3.3-5,
respectively, are as follows
(momentum transfer)
4.1-46
(heat transfer)
4.1-47
(species transfer)
4.1-48
In Eq. 4.1-47 the viscous dissipation is
neglected and in Eq. 4.1-48 ?wA ?A
These differential balance equations share
the same form as follows
4.1-49
or
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4.1-50
These equations are summarized in Table 4.1-4.
The following equation of continuity, Eq.
1.3-4, is also included in the table
4.1-51
Table 4.1-5 summarizes these equations for
incompressible fluids.
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Example