For many real-world applications such as parallelly stacked printed circuit boards,
radiative air heaters, etc, the flow of air
between the components can be simplified to that of a natural convection between parallel plates.
The flow and heat transfer characteristics can vary depending on the orientation of these plates
i.e., whether they are aligned vertically or inclined, and on the thermal conditions of
each of the plates. In this lesson, we will take a deeper look at each of these situations.
Based on their orientation, a parallel plate configuration can either be categorized as
vertical or inclined. In a vertical configuration, the direction of buoyancy force is vertical,
and a fluid flow is established in this vertical or streamwise direction. In the
inclined configuration, buoyancy force has two components i.e., streamwise and normal
to the surface and as a result leads to strong secondary flows. If we consider the ratio L/S
where L is the length of the plates and S is the separation between the plates,
for small ratios i.e., for short channels or for channels with really large plate separation,
the boundary layers develop independently over each plate. In such a situation,
the plates can be analyzed individually, and heat transfer rates can be estimated based
on external flow relationships developed for an isolated plate in quiescent fluid. However,
for longer channels or channels with narrow spacing, the boundary layers on the plates
interact with each other eventually attaining a fully developed profile.
In such a situation, new correlations are needed to predict the heat transfer rates.
These correlations strongly depend on the surface conditions of the plates
i.e., symmetric or asymmetric, isothermal or isoflux conditions.
Let us now look at these situations in a bit more detail.
For a vertical channel at symmetric isothermal conditions i.e., Ts1=Ts2, Elenbaas
proposed the following correlation. This correlation is applicable only for air
and for Rayleigh numbers in the range shown here. The Rayleigh number used in this correlation can
be calculated using the following relation, where the characteristic length S is the separation
between the parallel plates. The average Nusselt number is given by the equation shown here.
For channels where S/L is small, a fully developed limit is reached.
At this limit, the correlation for calculating the Nusselt number reduces to the form shown here.
The subscript fd refers to the fully developed condition.
For a situation where one of the plates is heated isothermally and the other adiabatically, the
average Nusselt number for the isothermal plate can be obtained using the following correlation.
Bar-Cohen and Rohsenow obtained the following correlation which is valid for any value of S/L
by combining the correlations for both the isolated plate and the fully developed limit.
The values of the constants depend on the surface condition and are shown in this table.
It can be inferred that the heat transfer from each plate reduces with decreasing plate
separation distance. From this correlation, it is possible to obtain optimal plate spacing, Sopt,
which maximizes the heat transfer for an array of isothermal vertical plates. Let us assume that the
fins of a heat sink can be represented by an array of vertical parallel plates over a fixed with W.
The fluid flow driven by buoyancy forces must overcome the viscous forces due to the fins.
The mass flow rate between two adjacent fins is in fact a balance between the buoyancy and the
viscous forces. As the spacing between the plates is decreased, the viscous forces increase leading
to reduced mass flow rates and hence lower heat transfer rates. However, reducing the
plate separation distance also increases the total number of plates in the given volume.
As a result, the total plate surface area of heat transfer, As, increases.
At a specific value of S referred to as Sopt, the product of hbar and As is maximum and
consequently, is the condition where maximum heat transfer is obtained from a parallel plate array.
For values of S less than Sopt, the reduction in heat transfer due to viscous effects overpowers
the increase of heat transfer surface area and on the other hand, for S greater than Sopt,
the decrease in the number of plates per specific volume reduces the heat transfer
surface area overpowering the increase in heat transfer levels due to reduced viscous effects.
The following table shows the empirical relations that can used to calculate the optimum plate
separation distance. To maximize the heat transfer from each plate, the distance between
the plates should be large enough to prevent the merging of boundary layers from adjacent plates.
The value of S at this condition is generally referred to as Smax.
Till now, we analyzed the vertical plates at isothermal condition.
Let us now look at the constant heat flux condition.
For convenience, we define a local Nusselt and Rayleigh numbers as shown here.
Ts,L refers to the plate temperature at x=L where the temperature is maximum. For symmetric
conditions i.e., qs1=qs2, the fully developed limit is given by the following correlation.
A similar limit is obtained for an asymmetric heating condition with one surface at constant
heat flux condition q’’s,1 and the other at an adiabatic condition i.e., q’’s,2=0. Bar-Cohen
and Rohsenow also proposed a correlation for local Nusselt number at the constant heat flux condition
which is valid for all values of S/L. For an array of vertical plates with constant heat
flux condition, the total volumetric heating rate increases with decreasing plate spacing.
The primary objective of the heat transfer analysis with constant wall flux condition
is to maintain the surface temperature value below a certain prescribed limit. Therefore,
the optimal plate spacing Sopt is defined based on maximum volumetric heat dissipation
per unit temperature difference Ts(L)-Tinfty. For a prescribed heat flux value, the maximum spacing
Smax is defined as the spacing between the plates that leads to the lowest possible
surface temperature without accounting for any volumetric considerations.
The values of C1, C2, and the relations used to obtain Sopt for the constant heat flux conditions,
both symmetric and asymmetric are shown in the following table.
Let us switch gears and quickly analyze the inclined parallel plates configuration.
Sparrow and Azevedo obtained the heat transfer relations for inclined channels by conducting
experiments with water as the working fluid. For small angles of inclination, they obtained
the following correlation for both symmetric isothermal plates and isothermal-adiabatic plate
configurations. The experimental data was observed to depart from the given correlation at large
inclination angles with bottom surface heating, which was attributed to enhancement in heat
transfer because of the formation of 3D secondary flows. This brings us to the end of the lesson.