The Temperature Profile for the Innovative Design of the Perforated Fin

The development of the perforated fin had proposed in many studies to enhance the heat transfer from electronic pieces. This paper presents a novel derivative method to find the temperature distribution of the new design (inclined perforated) of the pin fin. Perforated with rectangular section and different angles of inclination was considered. Signum Function is used for modeling the variable heat transfer area. Set of parameters to handle the conduction and convection area were calculated. Degenerate Hypergeometric Equation (DHE) was used for modeling the Complex energy differential equation and then solved by Kummer's series. In the validation process, Ansys 16.0-Steady State Thermal was used. Two geometric models were considered. The big reliability of the presented model comes from the high agreement of the validation results about (0.25%). The results show the increase of the inclination leads to the enhancement of the temperature difference and heat transfer ratio. Improved of Heat transfer ratio is ranging from 13% to 50%.Article History: Received June 12th 2016; Received in revised form August 6th 2016; Accepted August 24th 2016; Available onlineHow to Cite This Article: Jasim, H.H and Soylemez, M.S. (2016). The temperature profile for the innovative design of the perforated fin. Int. Journal of Renewable Energy Development, 5(3), 259-266http://dx.doi.org/10.14710/ijred.5.3.259-266


Introduction
Development of the modern devices (communications devices, Mechatronics application, control package of the solar cells etc.) depended on the heat sink performance. Also the heat sources can be found in different positions and variable environment, according to the design conditions. For this reason, the researcher tries to find more effective heat sink based on the different techniques like perforated fin. In this field, most studies had changed the operation conditions and geometric shapes for the straight perforation to improve the thermal behavior.
In experimental studies, models with specified geometry were adopted, (Bayram et al. 2008 a ) studied the variation of the Nusselt Number and the friction factor of the circular perforated based on the change of the parameters (geometric and fluid flow), the efficiency increased by1. 1 and 1.9 based on the inter-fin spacing ratio and clearance ratio. Saurabh et al. (2014) investigated the influence of the metal on the performance of the circular perforated fin, the experimental tests show the higher heat transfer coefficient at a large number of perforated and a copper material having higher thermal conductivity. (Bayram et al. 2008 b ) proved the efficiency can be increased by 1.4 to 2.6 for circular pin fin with a circular section of perforated by optimized the Nusselt number and friction factor separately and then together. (Elshafei 2010) presented the study about the hollow fin with the circular perforated, the results show the increase of the heat transfer depended on the orientation and diameter ratio.
Finite volume is used by the most investigators to solve the governing equations, (Swee et al. 2013) presented the pin fin with the different specification (number and diameter of perforated) to study the recirculation effects. They found the heat transfer increased by 45%, also, the pressure drop of perforated fin decreased by 18%. Mohamed et al. (2014) determined the optimum number of pin fins that associated with a minimum value of thermal resistance to improve the heat transfer by use the circular perforated. Monoj et al. (2011) presented the numerical study to determine the heat transfer and pressure drop of elliptical pin fin with straight perforated. Results show the enhancement of heat transfer by 5.6%, a pressure drop improvement by 12% and the performance of the fin increased by 23%. Ashok et al. (2014) obtained a high increase in convection coefficient about (30% to 40%) for a perforated circular section of pin fin by using the different number of a circular straight perforated. Mohammad et al. (2012) solved The Navier-Stokes and energy equations by the finite volume procedure using the SIMPLE algorithm for the perforation fins. Results show the thermal entrance length is inversely proportional to the number of perforations.
Other investigations found the temperature distribution of perforated fin by analytical study, Kirpikov et al. (1972) reached to the temperature distribution equation of perforated flat plate based on the mathematical model that solved by Fourier series and Flocke's theory. Zan WU et al. (2012) evaluated the performance of the isolated isothermal perforation plates with Staggered pattern. The results show the increase of the total heat transfer rate depended on the effects of the open area ratio and inclined angle.
All recent studies, used the straight perforation at different geometries and various operation conditions to improve the performance of the fin. New approach introduces through this article. Two points of novelty propose in this study, firstly at geometric design, the inclined perforated with rectangular section is considered. Secondly, at analytical process, the new differential technique is used to derive the general form of the temperature distribution and regardless of the perforated shape.

Geometric Model
The inclined perforated of the pin fin is shown in Figure 1. In the general form, an undefined section of perforated for the pin fin was considered. Inclination started from ( 0   ) at straight perforated. The fin has a rectangular cross section area. Also, the base fin located at the x-y plane and the fin length at y-axis. According to above geometry, the heat transfer area is changing with y-direction and inclination angle

Energy analysis and assumption
Applied Energy balance to the element shown in Figure 2, to obtain the energy equation of the perforated region.
. Heat transfer analysis for inclined perforated region relies on a set of assumptions: 1-Impairment value of the Biot number at z-axis and x-axis leads to the one-dimensional heat transfer. 2-Steady heat conduction with no heat generation. 3-Constant conductivity. 4-Constant base temperature. 5-Insulation tip fin.
6-Radiation effects are neglected. Uniform ambient temperature and convection heat  transfer coefficient assumed to be uniform over all surfaces of the fin. 7-Convection coefficient is divided into three types (external non-perforated (h1), external perforated (h2) and internal perforated (h3)). Based on these assumptions, the energy equation can be written as shown in equation (1

General solution
Expanded conduction heat transfer term of Equation (1) Equation (2) According to kummer's series (Hazewinkel 1995), the general solution of (DHE) is: The boundary conditions at original form can be written as below: (5) solved based on the boundary conditions. The general solution can be obtained after rewriting the parameters by re-compensation to all substitutions in original form.

Inclined perforated with rectangular section
Fin with inclined perforated is classified into three regions (solid, inclined perforated and then solid) as illustrated in Figure 3. Conduction area ( ) and convection perimeter ( The solid region was solved based on the (Incropera et al. 2007), while the perforated region was solved based on the equation (6). Matlab (R2014a) language was used to find the results according to heat transfer by natural convection and Rayleigh number (Ra=10 6 ). In this study, two models (I and II) were adopted as described below:

Coefficient of the convection heat transfer
Various convection heat transfer coefficients were appeared due to inclined perforated. Moreover, Convection coefficients depended on the properties of cooling fluid, specifications of the perforated fin and the open perforated ratio (ROP). ROP represents the ratio of actual perforated area to maximum perforation effects.
Empirical correlations from (Incropera et al. 2007;Zan et al. 2012) were used to find the convection coefficients h1 and h2 respectively. While the correlation from (Raithby et al. 1998) was modified to become useful for calculated the h3 that is used for the inclined perforated.

Numerical solution and simulation
Thermal analysis have been simulated using Ansys 16.0-Steady State Thermal for the models I and II. For each case, several grids are studied to ensure that the solutions are independent grids and the results are shown in table (2). The minimum temperature was compared until independent grids were succeeded. Some of grids configurations are illustrated in fig. (4).

Validation of analytical model
In the validation process, three inclination angles were considered. The distribution of the fin temperature (θ/θb) was calculated along the fin length (y/LT) to show the convergence between results. Figures  (5&6) are illustrating the temperature distribution of the Model I and Model II, respectively. All figures contain the temperature contours that are obtained from the simulation work. Increase of the inclination angle leads to growth the convection area and decrease in the conduction area. Also the increased of the perforated size can be lead to the same improvement in the convection area. Increase of the convection effects leads to big drop of the fin temperature through the perforated region. Agreements between results are starting from complete resemblance to a maximum difference of (0.25%). When using the inclined perforated, Multi advantages can be happened:  Increase of the inner convection area  Distribute the inner convection area on the longer length in the y-axis.  Increase of the external perforated area.  Development of the recirculations in y-z as well as the x-z planes).

Effects of inclination (heat ratio)
Heat transfer is possible to increase, according to the improvement of the temperature distributions and convection heat transfer coefficients. The heat transfer ratio ( s p q q / ) is described as the ratio between the heat transfer of a perforated fin to heat transfer of the solid fin at the same properties.

Conclusion
In addition to the new geometry (inclined perforated), this paper presented a novel technique at Citation: Jasim, H.H and Soylemez, M.S. (2016). The temperature profile for the innovative design of the perforated fin. Int. Journal of Renewable Energy Development, 5(3), 259-266, doi: 10.14710/ijred.5.3.259-266 P a g e | 266 © IJRED -ISSN: 2252-4940, October 15 th 2016, All rights reserved two major points, modeling of the variable heat transfer area based on Signum function and solves energy differential equation for complex geometry by (DHE). The high agreement of the validation results leads to being sure; the present mathematic model has big reliability. In this study, the use of the common perforated shapes (rectangular) showed the accuracy level and flexibility of the mathematic model. Consequently, general solution for present study taken as the basis to resolve any inclined perforation shape based on modeling of the area. Fin performance can be improved based on the advantages of inclined perforated. Increasing of the inclination angle leads to enhance the temperature distribution and heat transfer, on the other hand, all previous advantages could be better if the perforated size was increased.