17-12-2012, 05:43 PM
Optimum Geometry of Plate Fins
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Abstract
A method and practical results are presented for finding the geometries of fixed volume
plate fins for maximizing dissipated heat flux. The heat transfer theory used in optimization
is based on approximate analytical solutions of conjugated heat transfer, which couple
conduction in the fin and convection from the fluid. Nondimensional variables have
been found that contain thermal and geometrical properties of the fins and the flow, and
these variables have a fixed value at the optimum point. The values are given for rectangular,
convex parabolic, triangular, and concave parabolic fin shapes for natural and
forced convection including laminar and turbulent boundary layers. An essential conclusion
is that it is not necessary to evaluate the convection heat transfer coefficients
because convection is already included in these variables when the flow type is specified.
Easy-to-use design rules are presented for finding the geometries of fixed volume fins that
give the maximum heat transfer. A comparison between the heat transfer capacities of
different fins is also discussed.
Introduction
Extended surfaces are used when convective heat transfer from
a surface must be improved. There is extensive literature concerning
fin heat transfer, and excellent books have been published on
the subject [1]. If the heat transfer coefficient is given, the performance
of a fin is easily predicted. However, a reliable prediction
of convection heat transfer from a fin is of vital importance,
and its evaluation is much more difficult than the solving of the
heat conduction problem with a known heat transfer coefficient.
In actual practice, conduction in a fin and convection from it are
coupled together, which means that the two problems should be
solved simultaneously because the heat transfer coefficient
depends on the fin temperature distribution. This is especially true
for laminar flow in cases of forced or natural convection.
In practice, it is important that the fins are designed in such a
way that the heat transfer rate is as high as possible with a given
amount of material. Thus, we should maximize the heat flux by
changing the fin geometry. The authors have already presented a
method for finding the optimum shape of a single rectangular
plate fin with a given amount of material that maximizes the heat
flux [2]. Some preliminary results for other cross-sections have
also been published [3]. The method used in previous studies will
now be extended, and the comparison of performance between
different fin cross-sections is shown.
Effect of Nonisothermal Fin Temperature on
Convection
In actual practice, the heat transfer coefficient is not a constant
but depends on the surface temperature distribution of the fin,
especially in the case of a laminar boundary layer. In the case of
forced convection, the effect of nonisothermal surface temperature
and the coupling between convection and conduction has
been studied [5], where convection heat transfer has been treated
using a step change solution of the surface temperature and a
superposition technique [14]. The efficiencies of plate fins with
different cross-sections can be represented by a single curve using
a nondimensional variable
Design Procedure and Discussion
The optimum geometry of a fin could also be obtained by performing
multivariable optimization numerically with L, t0, and l
as optimization variables for each combination of parameters
from Eqs. (1)–(3). This type of approach gives one solution. A
great benefit in our approach is that a single variable, X for forced
or x for natural convection, with a fixed value gives the optimum
dimensions for all possible thermal and flow values and makes the
fin design easy without the designer needing to go through an
optimization procedure. Again, it is worth mentioning that the
correct value of a heat transfer coefficient is already included in
these variables.
In order to become familiar with the design procedure of fins,
some examples are given below. Typical values of cooling fins in
power electronics are used. It must be noted that the arrays of fins
are used in many applications and if the channel width between
fins is small the results above are no longer valid.
Conclusions
A method is presented for finding the fin geometries that maximize
the dissipated heat flux of fixed volume plate fins. An essential point of the method is that the total heat transfer of a
nonisothermal fin can be expressed using the mean heat transfer
coefficient of an isothermal surface. This type of treatment, which
is very accurate when compared to the numerical solution of the
conjugated problem, is employed to combine both conduction and
convection in a fin. The optimization procedure produced very
simple design rules to find the optimal fin geometry. When one of
the fin dimensions, in addition to the volume, is fixed, a nondimensional
variable with a fixed value determines the best geometry.
The advantage of our results is that there is no need to
evaluate convective heat transfer coefficients, because they are
included in nondimensional variables, which determine the optimum
fin geometry for forced and natural convections with either a
laminar or a turbulent boundary layer.