Jökull - 01.12.1973, Page 24

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o II 1- T = T0 o II f-
1 !i n m!
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x = 0 x=L/2 x = L
Fig. 3. The strip model of a convection cell.
the plane x = L/2; for an internally heated
fluid, there is no condition at the median
plane. There is negative buoyancy in the region
Oíxí L/2; and positive buoyancy in the
region L/2 x 5= L.
Due to the insulation of the walls, the tem-
perature will be constant transverse to the flow.
Then, we need only solve a one-dimensional
heat transport equation,
to determine the temperature T(x) in the strip.
Let the velocity u be the average velocity for
steady laminar flow between rigid horizontal
planes in the presence of a driving pressure
gradient. For a Newtonian fluid, this velocity
is given by (Lamb, 1932, p. 582).
T2 dP
12tj dx
(8)
where dP/dx is the pressure gradient. We re-
place (—dP/dx) by H/L' where H is the total
head driving the flow, and L' is the fluid flow
path length. This head results from the thermal
expansion of the fluid and is given by
H = pag [{ T(x)dx - J T(x)dx]
L/2 0
(9)
where T(x) is the temperature in the fluid.
The length L in (9) must be identified with
the length scale for heat conduction, i. e.
L = 2h. The fluid flow length must be deter-
mined from Fig. 2, resulting in
ture T(x). In order to determine the total
buoyant force this temperature distribution is
then integrated over the length of the strip
according to (9). Furthermore, equation (9)
combined with the flow equation (8) yields an
equation for the velocity u. As will be shown
below, this equation has only the solution
u = 0, unless a dimensionless number, the Ray-
leigh number, exceeds a certain critical value.
Thus the condition for the onset of thermal
instability can be obtained on the basis of this
relatively simple model.
Despite the difference in approach, the simi-
larities between the strip model and the Ray-
leigh model should be emphasized. Both models
assume laminar flow and neglect viscous dis-
sipation and horizontal convection of heat. The
strip model contains the additional simplifica-
tions of (1) neglecting horizontal heat conduc-
tion, (2) using an average flow velocity, (3)
assuming the cell size on physical grounds.
The essence of the two models is the same in
that the flow is driven by buoyant forces aris-
ing from the non-uniform density distribution
and that convection occurs only if the Rayleigh
number exceeds a certain critical value.
Before discussing the problem of convection
in two-phase systems we will demonstrate the
application of the strip model in two cases
which possess well-known solutions. The first
case involves the homogeneous fluid layer heat-
ed from below (Jeffreys, 1928) and the second
a layer of an internally heated fluid (Roberts,
1967).
APPLICATION TO A FLUID LAYER
HEATED FROM BELOW
For constant thermal conductivity and in the
absence of heat sources, equation (7) reduces to
d2T dT
K--------ncu------= 0 (11)
dx2 P dx v '
The boundary conditions are
T(0) = T(L) = 0; T(L/2) = T0 (12)
L' = \ + 2h —3T (10)
The general method of solution with the
strip model is to first determine the tempera-
Due to the discontinuity of the heat flow at
x = L/2, equation (11) must be solved separ-
ately in the regions 0:?x:£L/2and L/2 5=
xíLwe obtain
22 JÖKULL 23. ÁR