W-21
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,
two-dimensional 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