Jökull - 01.12.1973, Page 23

T= 0
-----X/2
T = T0 or dT/dz = 0
Fig- 1. Two dimensional convection cell model.
where c is the specific heat at constant pressure.
The heat transport equation (2) has been simpli-
fied so that it involves only the vertical com-
ponent of the convective heat transport uzy3-
The horizontal convective heat transport terms
contained in (5) have been neglected.
Tlie resulting linearized Rayleigh model is
physically quite simple. Laminar viscous fluid
wrculation is driven by a pressure gradient re-
sulting from the thermal expansion of the fluid.
Heat is transported vertically by conduction
and convection, but thermal conduction is the
°nly mode of horizontal heat transfer.
In this light, we consider the model of a
two-dimensional convection cell of wavelength
X depicted in Fig. i. The homogeneous fluid
ls confined between rigid horizontal planes
separated by a distance h with tlie upper sur-
face maintained at T = 0. The walls of the
cell are assumed rigid and thermally insulated.
It is further assumed that horizontal heat con-
duction can be neglected and that the flow
takes place around rigid, thermally insulating
cores. This assumption simplifies the calcula-
tions considerably and does not seriously alter
the convection process. Two different condi-
tions will be used at the lower boundary of the
cell. If the fluid is heated from below, the
condition at the lower surface is T = 'I’o- If
the fluid is internally heated, the condition of
zero heat flux, dT/dz = 0, is employed.
The model in Fig. 1 is shown again in Fig. 2.
The flow is assumed to circulate around the
core through a channel of width t- Clearly, the
stability problem presented here is of the same
nature as the Rayleigh problem. Convection will
not occur unless a certain critical temperature
difference between the upflowing and the
downflowing sections can be maintained in
order to provide the required buoyancy forces.
The essential difference between a two-dimen-
sional Rayleigh model of fixed wavelength and
the present model lies in the neglecting of the
horizontal heat conduction. Mathematically, the
term 32T' / 3x2 has been dropped from the
perturbation equation (2).
To arrive at the strip model we now make
the further simplification that the convective
flow is uniform about the core and the flow
velocity is constant over the width t- The work-
ing model for calculating the temperature then
becomes that shown in Fig. 3. The cell has
been cut along the dotted line (Fig. 2), and
the flow channel has been stretched out as a
strip. The bottom of the channel has been
folded into the plane x = L/2. Fleat losses
through the upper surface of the cell now place
through the ends of the strip. The problem
then reduces to that of determining the tem-
perature distribution in a strip of length L and
width t- The ends of the strip are held at
T = 0; there are no heat losses through the
rigid horizontal surfaces and the flow is uni-
form in the x direction. For a fluid heated
from below, the condition T = To exists at
T = 0
X/2
T = T0 or dT/dz = 0
Fig. 2. Equivalent convection model.
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