W21
6S.1
Derivation of the Convection Transfer
Equations
In Chapter 2 we considered a stationary substance in which heat is transferred by
conduction and developed means for determining the temperature distribution
within the substance. We did so by applying
conservation of energy
to a differential
control volume (Figure 2.11) and deriving a differential equation that was termed
the
heat equation
. For a prescribed geometry and boundary conditions, the equation
may be solved to determine the corresponding temperature distribution.
If the substance is not stationary, conditions become more complex. For exam
ple, if conservation of energy is applied to a differential control volume in a moving
±uid, the effects of ±uid motion (
advection
) on energy transfer across the surfaces
of the control volume must be considered, along with those of conduction. The
resulting differential equation, which provides the basis for predicting the tempera
ture distribution, now requires knowledge of the velocity ²eld. This ²eld must, in
turn, be determined by solving additional differential equations derived by applying
conservation of mass
and
Newton’s second law of motion
to a differential control
volume.
In this supplemental material we consider conditions involving ±ow of a
vis
cous Fuid
in which there is concurrent
heat
and
mass transfer
. Our objective is to
develop differential equations that may be used to predict velocity, temperature, and
species concentration ²elds within the ±uid, and we do so by applying Newton’s
second law of motion and conservation of mass, energy, and species to a differential
control volume. To simplify this development, we restrict our attention to
steady,
twodimensional Fow
in the
x
and
y
directions of a Cartesian coordinate system. A
unit depth may therefore be assigned to the
z
direction, thereby providing a differ
ential control volume of extent (
).
6S.1.1
Conservation of Mass
One conservation law that is pertinent to the ±ow of a viscous ±uid is that matter
may neither be created nor destroyed. Stated in the context of the differential con
trol volume of Figure 6S.1, this law requires that, for steady ±ow,
the net rate at
which mass enters the control volume
(in±ow
2
out±ow)
must equal zero
. Mass
enters and leaves the control volume exclusively through gross ±uid motion. Trans
port due to such motion is often referred to as
advection
. If one corner of the control
volume is located at (
x, y
), the rate at which mass enters the control volume through
the surface perpendicular to
x
may be expressed as
, where
is the total mass
density (
) and
u
is the
x
component of the
mass average velocity
. The
control volume is of unit depth in the
z
direction. Since
and
u
may vary with
x
, the
r
r
5
r
A
1
r
B
r
(
r
u
)
dy
dx
z
dy
z
1
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6S.1
j
Derivation of the Convection Transfer Equations
rate at which mass leaves the surface at
x
+
dx
may be expressed by a Taylor series
expansion of the form
Using a similar result for the
y
direction, the conservation of mass requirement
becomes
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 Spring '11
 Anthony
 Thermodynamics, Energy, Heat, Couette flow, convection transfer equations

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