Improvement of Free Convection Heat Transfer Rate of Rectangular Heatsink on Vertical Base Plates

In this paper, the laminar heat transfer of natural convection on vertical surfaces is investigated. Most of the studies on natural convection have been considered constantly whereas velocity and temperature domain, do not change with time, transient one are used a lot. Governing equations are solved using a finite volume approach. The convective terms are discretized using the power-law scheme, whereas for diffusive terms the central difference is employed. Coupling between the velocity and pressure is made with SIMPLE algorithm. The resultant system of discretized linear algebraic equations is solved with an alternating direction implicit scheme. Then a configuration of rectangular fins is put in different ways on the surface and heat transfer of natural convection on these surfaces without sliding is studied and finally optimization is investigated.


Introduction
This document is a natural convection is observed as a result of the movement of fluid because of density changes which is caused by heating process.A radiator used for warming the house is an example of practical equipment for heat transfer of natural convection.The movement of fluid, whether gas or liquid, in natural convection is caused by buoyancy force due to density reduction beside to heat transfer's surfaces in heating process.When the domain of an exterior force such as gravity, has no effect on the fluid then will be no buoyancy force and heat transfer will be in the form of conduction.But gravity is not the only force causing natural convection.When a fluid is confined in the rotating machine, centrifugal force is exerted on it and if one surface or so, with more or less temperature than that of the fluid are in touch with the fluid, natural convection flows will be experienced.The fluid which is adjacent to the vertical surface with constant temperature, and the fluid temperature is less than the surface temperature, a velocity boundary layer forms the natural convection.The velocity profile in this boundary layer is completely different with the velocity profile in forced convection.The velocity is zero on the wall due to lack of sliding .Then the velocity goes up and reaches its maximum and finally is zero on the external border of velocity boundary layer.Since the factor that causes the natural convection, is temperature gradient, the heating boundary layer appears too.The temperature profile has also the same value as the temperature of wall due to the lack of particles sliding on the wall, and temperature of particles goes down as approaching to external border of temperature boundary layer and it would reach the temperature of far fluids.The initial boundary layer enlargement is laminar , but in the distance from the uplifting edge, depending on fluid properties and the difference of temperature of wall and the environment, eddies will be formed and movement to turbulent zone will be started.
Today being more aware about energy sources' limitations and also increase demand for energy consumption from one side and considerable waste of energy in heating systems from the other side , have caused the societies to investigate on energy systems and find solutions for reducing energy dissipation from these systems.There are many appliances in engineering in which two environments with different temperatures are separated by a wall or a thin surface.The wall on one side is getting warm and on the other side is making it is beside fluid warm.Heat transfer between the two environments and the temperature of the wall, depends on the shape of the boundary layers on the wall surfaces.The amount of temperature on the wall will reach a balance of the temperature of both environments.Since the boundary layer of one side makes the boundary layer of the other side they are called conjugate boundary layers.
Although a lot of investigations have been done on natural convection heat transfer, the data the effect of geometrical complexes on natural convection heat transfer has many deficiencies.
Jofre and Barron have gathered some data on natural convection heat transfer, which the fluid is air, on the vertical surface with some swells in the shape of triangle ribs [1].They concluded that at Ra L = 10 9 .The average Nusselt number is increased 200% compared to the average Nusselt of turbulent flow achieved by Ekertand Jackson.Of course if the fact is considered that the flow on the flat surface is not turbulent on the entire surface, and the data are compared with upstream flow that is assumed laminar flow, the results show 100% increase as compared to the recent one.We can say that this big difference is caused by the lack of attention to the correct way of radiation heat transfer [2].
Bhavnani and Burgles [3] after several experiments proved that making special changes on vertical surfaces (horizontal little fins) reduces heat transfer in the natural convection heat transfer process.This conclusion can cause changes in the ways of insulating heat repelling surfaces and in this respect is of great importance.On the other hand we face many practical technologies causing natural convection heat transfer on the fins vertical surfaces.This matter is more obvious in the electronic circuit in which segment duty cycle is highly effected by the duty cycle.Rebert, Green, Chapman and others have offered five important advantages of algorithmic solving method [4,5].Numerical solution of the governing equations of boundary zones for vertical surfaces has been done by Helus and Churchil and step changes of surface temperature has been achieved [6].Most of the works [7] in this field are true for low Prandtl and in high Prandtl, computing errors cannot be negligible.
Although there are many articles about forced convection gas flow on vertical surfaces, there are quantitative researches done about natural convection flow on surfaces with rectangular fins.In this paper, a complete numerical solution including developing and developed enlarged hydrodynamic and heating parts is presented to analyze natural convection flow on the extended vertical surface.

Numerical Modeling
For analyzing the above mentioned matter, first the differential equation of the movement of this layer should be achieved.So y axis is on the horizon and x axis is erected vertically on the surface As in Figure 1.2dimentional flow is considered with physical properties, fixed wall temperature, more than the entrance free flow temperature.So upward flow is produced by buoyancy force in top of the surfaces.Two ends of the surface are open towards the environment with free flow temperature.Compressibility effect in the free flow with low velocity taken here can be negligible, and just laminar flow regime can be considered [8].Density conversion caused by temperature changes are exerted in buoyancy force is done by boussinesq approximation.The equation of conservation of mass, momentum and the energy for stable and laminar flow is described follow [9,10].
Continuity Equation: X-Momentum Equation: Y-Momentum Equation: Energy Equation: With boundary conditions: 0 0 Assuming that there are not any parameter changes on x direction, (2) is omitted.Pressure variation on "y" direction is caused by changes in height of the fluid.Out of velocity boundary layer, u, v, and their derivative are zero. .
With putting ( 5) in (3), we have: Difference of density can be defined according to Thermal Expansion Coefficient. 1 .
Above mentioned equation defines the momentum in natural convection velocity boundary layer, this momentum is caused by temperature gradient.Since temperature depends on the velocity and vice versa, momentum Equation ( 9) and energy Equation (4) should be solved simultaneously.Now considering boundary conditions, temperature and velocity distributions are guessed and put in equations.Governing equations are solved using a finite volume approach.The convective terms are discretized using the power-law scheme, whereas for diffusive terms the central difference is employed.Coupling between the velocity and pressure is made with SIMPLE algorithm.The resultant system of discretized linear algebraic equations is solved with an alternating direction implicit scheme [11].

Result
In this paper, first, vertical surface is considered, and thenheat sinksare put on the surface.Dimensions, distances between fins, the type of fins and the location they are put in, are considered.The dimensions are considered as shown in Table 1.
The mentioned heat sinks have shown in Figure 2.  The studies on different surfaces show that in fins with low distances, air flow between the fins is fully developed due to boundary layers interference, t and therefore heat transfer rate is achieved by 1 .
If n = w/s, then, Unlike the previous way, if the fin distances are great, the boundary layer thickness is so much less than the distance between the fins and the space between the fins is like the flow in parallel surfaces in which boundary T layers have no interference.Natural convection heat transfer from two surfaces of a fin is as follow: in which heat transfer coefficient is defined as: If A = H  L, then: As it has shown in Figure 3, the investigations show that in the same fin length and height and constant difference of temperature, heat transfer rate first increases with increasing the fins distance, until reaches the maximum point and then decreases.The maximum point is called optimum distance between the fins, is the distance in which there is the maximum heat transfer.We can have an equation for the optimum distance, by interfering Equations ( 12) and ( 16).
For dimensionless presentation of the order of magnitude of optimum fin spacing, the Rayleigh number is employed according to its definition.
And according to our investigations on mentioned dimensions the optimum distance equation calculated as follow: As a point of departure for the presentation of heat transfer rates from fin-arrays are plotted as a function of temperature differences as in Figure 4.
From these Figures it can also be seen that, at a given fin height and temperature difference, the convection heat transfer rates increases with increasing fin spacing and reaches a maximum.With further increases of fin spacing, heat transfer rate starts to decrease.The occurrence of this maximum has significant practical applications for optimum performance of fin-arrays.It would be appropriate tomanufacture the fin array with aforementioned fin height and spacing.The values of heat transfer coefficient obtained for an ambient air temperature of 27 º C and plate surface temperatures of 77, 102, 127 and 157 º C appear in Figures 5-7, As may be seen, the natural   The temperature at the upper surface of the heat sink base, the surface exposed to the operating fluid can then be found from (20), knowing the temperature at the bottom of the base by thermocouple measurement and the heat flux through the base. .
The heat flux at the base in W/m 2 is found with (20), knowing that the exposed base area is 4768 mm 2 . .
Finally, the heat transfer coefficient is calculated using (22) and the thermal resistance using (23).
As in Figure 8, average Nusselt number is increased with increasing average Rayleigh number and at a certain average Rayleigh number, average Nusseltnumber is increased by increasing the distance between the fins.

Conclusions
From these Figs it can also be seen that, at a given fin height and temperature difference, the convection rates increases with increasing fin spacing and reaches a maximum.With further increases of fin spacing, rate starts to decrease.The occurrence of this maximum has significant practical applications for optimum performance of fin-arrays.It would be appropriate to manufacture the fin array with aforementioned fin height and spacing.The values of coefficient obtained for an ambient air temperature of 27 º C and plate surface temperatures of 77, 102, 127 and 157 º C appear in    convective heat transfer rates from the fin arrays increases with fin height, fin length and base-to-ambient temperature difference.The heat transfer rate increases monotonously with temperature difference between fin base and surroundings, Tw-Ta.If the distance between the fins is selected properly, there will be no interference be ween boundary layers of two adjacent fins and the t   surfaces.For stating the magnitude of fin spacing for having the best convection rates of fins, we choose thedistance between the fins big enough so that the thickness of boundary layer is smaller than the distance between the fins and increases without any interfering.

Figure 1 .
Figure 1.Geometry of coordinate system and velocity Boundary layer.

Figure 3 .
Figure 3. Variation of convective rate with base-to-ambient temperature difference at H = 29.2mm and L = 80 mm.

Figure 4 .
Figure 4. Variation of convective heat transfer rate with fin spacing (fin height 29.2 mm).

Figure 5 .
Figure 5. Natural convection heat transfer coefficients for different heat sinks.

Figure 5 - 7 ,
As may be seen, the natural convection coefficient increases substantially as the gap between fins increases from 2.1 to 18.6 mm, and then flattens out with further increases in gap.It may be noted that the values shown for a gap of 18.6 mm are within a few per cent of those obtained using a correlation for an individual vertical plate.Figures 9-11 show that, the convective heat transfer rate from fin arrays depends on fin height, fin length, fin spacing and base-to-ambient temperature difference.The

Figure 6 .
Figure 6.Variation of heat flux rate with Average Rayleigh number at H = 29.2mm and L = 80 mm.

Figure 8 .
Figure 8. Variation of average nusselt number with average Rayleigh number.