Enhancement of Natural Convection Heat Dissipation Using Longitudinal Elliptic Perforations Fins

Heat removal process from the components is major challenge, since it leads to risk of damaging. If the heat removal rate is slow or inadequate, overheating is another issue which directly influence the behavior of the working component. Hence the popular field called thermal engineering was getting popularity and leads to control and improvements in thermal system/components for better heat circulation. The popular key parameters used control the heat movement include heat transfer coefficient (surface to surrounding) and projected area to perform the heat interaction. Fins are been consider as the best candidate for the faster heat transfer since the surface area is large. Thus, this a good choice to directly attached such features to hot surface where cooling is needed almost [1, 2].

Fins can be considered as sheet like components, whose thickens and materials are varied in different dimensions based on the applications. Some of the applications are found in heat exchangers, boilers, industries burners, nuclear reactors, electric transformers, power plants and so on [3]. The fins shapes and designs have been studied for managing thermal performance, to meet heat transfer removal requirements, in the past decades. Commonly, Researchers uses a plate, triangular, and annular fin [4], but many other types of fins have been investigated such as tree-shaped fins (Y- shaped fins and T-shaped fins) [5, 6], uneven tree-like fins [7], gradient fins [8] with different fin number [9], fin layout [10], and fin materials [11].

This cooling technology has been developed due the huge demands to make products lightweight, compact, and economical [12]. Overall machining cost to get the marketable product for the fin based on the efficiency plays vital role, which need to be considered before the actual production or the design of the fin [13, 14]. Previous research work has given snipped outline on various methods adopted in fin design by making changes in the fin structure. Some of the design mainly focused on the getting cavities are periphery, generating holes, small slots in the junctions, providing groves and channels. Thus, the intension was to get more and more surface area [15-17].

Perforated plates also know was fins are the best way to representing surface interruption. One way is block reduction, which can enhance the performance of heat dissipation from most thermal applications such as novel heat exchanger for better colling rates, film cooling to enhance heat transfer, energy storage application, and solar collectors for under heat exchange [18]. Chin et al. [19] utilized perforated pin fin on investigating the heat dissipation of heat sink. They studied the effect of number and size of perforated region in each pin on forced convective heat transfer numerically and experimentally. Their results performed high enhancement Nusselt number reached to 48% comparing with solid fins when number of perforating increases. In addition, the pressure drop became lower. Li et al. [20] worked on the triangular perforated fin by focusing on the synergy with respect to fluid flow. Intension was to consider convective heat transfer. They concluded the perforated fins caused additional flows and vortices. Further, this phenomenon can be enhancing the heat transfer performance. Maji et al. [21] found improvement on heat transfer when they studied the performance of different configurations of perforated fin heat sink. Sundar et al. [22] investigated numerically the heat-dissipation performance of heat sinks using a staggered array of perforated fins. They studied the effect of fins porosity and angle of orientation. Their results showed 7-12% enhancement on heat resistance comparing with non-perforating fins. Maiti and Prasad [23] designed and analyzed a novel perforated pin fins numerically. In addition, performance was checked for forced type of heat transfer from the fin and heat sink. A detailed comparison was made at Reynolds number varying 2000 -11000, between perforated fins and solid pin fins. Outcome the research work led to the conclusion that perforated fins have performed way better than conventional way of design in terms of both heat transfer coefficient and pressure drop.

A turbulency flow happened due perforated fins is more usefulness than surface area in enhancing heat transfer coefficient. This phenomenon may increase the heat transfer coefficient immensely comparing with flat fins [24, 25]. On other side, perforated fins make an interruption of developing thermal boundary layers after each removing areas which also increasing heat transfer coefficient. Therefore, the shape and number of perforated fins is very important factors can control the heat dissipated rate [26].

Ibrahim et al. [27] investigated the effect of different shapes of perforated fins (rectangular, circular and triangular) on performance of heat exchanger. They compared the results with non-perforated fins. There results show significant enhancement on forced convection heat transfer of perforated fins over flat fins. The maximum enhancement was for the circular perforation which it was 51.29%. They pointed also; wider perforation area could develop the intensity of turbulence. Sudheer et al. [28] studied a performance of circular perforated fins numerically. They showed a significant decrease in temperature comparing with non-perforated fins. They studied the role of perforation array and diameter on the temperature profile of fines. They revealed that increasing perforated diameter caused higher reduction of fins temperature.

The above literature mainly concerned about the effect of perforated fins on forced convection heat transfer where there is a lack of the research introduce a relation of perforated fins with natural convection heat transfer. Consequently, the study of surface coefficients is needed. The surface coefficients of perforated fins can be estimated by studying the shape geometry of the perforated fins (augmentation ratio and open area ratio). The main aim of this research is to investigate the effects of the geometry parameters of vertical longitudinal elliptic perforation on enhancing natural heat transfer rate of the fins. In addition, focus was given on the geometric values impact on the heat transfer rate. The study made a successful attempt to come up with optimum elliptic perforation size for enhanced heat transfer in a horizontal flat plate. The fin with optimum elliptic perforation introduces a good solution in cooling the electric devices with light weight and little cost of fins.

With acceptant of considered assumptions energy equation can be implied for fin as stated below [30]:

$k \frac{d^2 T}{\mathrm{dx}^2}=0$    (3)

For this general expression the boundary conditions imposed as below.

1. For the fin at base location (x=0):

$T=T_b$  (4)

2. In case of perforated surfaces:

$k \frac{\mathrm{dT}}{\mathrm{dx}} l_x+h_{\mathrm{ps}}\left(T-T_{\infty}\right)+h_{\mathrm{pc}}\left(T-T_{\infty}\right)=0$        (5)

According to the fin with longitudinal elliptic perforations that shown in Figure 1, the surface area of this fin including the tip can be expressed as:

$\begin{aligned} & A_{\mathrm{pf}}=\left(2 \mathrm{~W} * L-2 \mathrm{~N}_c * A_c\right)+(W * t)+\left(N_c * A_{\mathrm{pc}}\right) \\ & =A_f+N_c\left(A_{\mathrm{pc}}-2 \mathrm{~A}_c\right) \\ & \left.=A_f+N_x * N_y * \pi \mathrm{D}(t-\mathrm{D} / 2)\right)\end{aligned}$    (6)

A concept of weight reduction ratio RWF was used to show the relation between perforated vs non perforated fin. The relation is given below.

$\begin{aligned} & \mathrm{RWF}=W_{\mathrm{pf}} / \mathrm{W}_{\mathrm{sf}} \\ & =\left(L * W * t-N_x * N_y * A_c * t\right) /(L * W * t) \\ & =1-\left(N_x * N_y * A_c * t\right) /(L * W * t) \\ & =1-\pi \mathrm{D}^2 /\left(4\left(2 \mathrm{~S}_x+D\right)\left(2 \mathrm{~S}_y+D\right)\right.\end{aligned}$    (7)

The number of longitudinal elliptic holes N_x and N_y, must be integers, hence following expressions are used to get the final values used based on perforated fin for a considered length and width.

$N_x=\operatorname{Int}\left(\mathrm{L} /\left(2 \mathrm{~S}_x+D\right)\right)$   (8)

$N_y=\operatorname{Int}\left(\mathrm{W} /\left(2 \mathrm{~S}_y+D\right)\right.$       (9)

In the above relation the word “Int” implies to integer number function.

There are many popular techniques to get solution for the energy Eq. (1), based on the present boundary condition it was solved numerically by enabling finite-element technique to get best results. The computer program for one-dimensional problems, the subroutines called heat 1, adjust. decomp, and solve are used for the solution of this problem [30]. The next set of variational statement are as shown below [30]:

$\begin{aligned} & I_n=\frac{1}{2} \iiint_V k\left(\frac{\mathrm{dT}}{\mathrm{dx}}\right)^2 \mathrm{dV}+\frac{1}{2} \iint_{A_{\mathrm{ps}}} h_{\mathrm{ps}}\left(T-T_{\infty}\right)^2 \mathrm{dA}_{\mathrm{ps}}+ \\ & \frac{1}{2} \iint_{A_{\mathrm{pc}}} h_{\mathrm{pc}}\left(T-T_{\infty}\right)^2 \mathrm{dA}_{\mathrm{pc}}+\iint_{A_t} h_t\left(T_t-T_{\infty}\right) T \mathrm{dA}_t\end{aligned}$     (10)

The basis for computing the considered heat transfer situation with the boundary conditions can be referred from Figures 2 to 4.

3.png

Note: I, IV: - Straight (Uniform) Region II, III: - Tapered Regions

Figure 3. All four regions are indicated with symmetricity for perforated fin

4.png

Figure 4. Symmetrical part for the considered four regions

With the help of Figure 2 with hatching indicates the symmetry part, while Figures 3 and 4 try to give more information to the four regions. Four regions are numbered as 1, 2, 3 and 4. This was marinated across all the considered perforation. A careful view leads to conclusion that regions 1 and 4 are uniform in nature, while 2 and 3 are more or less tapered in shape in the forward and then in backward directions, respectively.

The regions 1 and 4 are split into N_f elements for each case. While the regions 2 and 3 are classified as N_t elements each. The values of N_f and N_tare based on the mesh distribution. The expression to related, total number of elements N_e with total number of nodes N_n is as shown below.

$N_e=N_x\left(2 \mathrm{~N}_f+2 \mathrm{~N}_t\right)$       (11)

$N_n=N_e+1$     (12)

The problem is solved based on the variational approach with the concept of symmetric banded matrix width. The algebraic system obtained is solved by using the Choleski decomposition method as described by Rao [30]. Most of the variable remains same in case of heat dissipation from a fin, solid or perforated. The critical parameters need to focus are fin surface area and heat transfer coefficient. In case of solid fin, both features are recognized. The average value of h_ss is that for a single horizontal plate under natural convection mode can be represented below [31]:

$h_{\mathrm{ss}}=\mathrm{Nu} . \mathrm{k}_{\mathrm{air}} / \mathrm{L}_c$ （13）

$L_c=\mathrm{L}^* \mathrm{~W} /(2 \mathrm{~L}+2 \mathrm{~W})$    (14)

Nusselt number average was noted as Nu [31]:

$\mathrm{Nu}=\left(\mathrm{Nu}_u+\mathrm{Nu}_l\right) / 2$      (15)

$\begin{gathered}\mathrm{Nu}_u=\left[\left(1.4 / \ln \left(1+1.4 /\left(0.43 \mathrm{Ra}^{0.25}\right)\right)\right)^{10}\right.  \left.+\left(0.14 \mathrm{Ra}^{0.333}\right)^{10}\right]^{0.1}\end{gathered}$    （16）

$\mathrm{Nu}_l=\frac{0.527 \mathrm{Ra}^{0.2}}{1+(1.9 / \mathrm{Pr})^{0.9} 2 / 9}$      (17)

In present work fin tip was considered as vertical surface for which the Nusselt number, Nut are calculated as is given below [31], this was applied for both perforated and non-perforated type of fin.

$\mathrm{Nu}_t=0.5\left(\left(\frac{2.8}{\ln \left(1+\frac{2.8}{0.515 * \mathrm{Ra}^{0.25}}\right)}\right)^6+\left(0.103 * \mathrm{Ra}^{0.333}\right)^6\right)^{(1 / 6)}$     (18)

$h_t=\mathrm{Nu}_t \cdot \mathrm{k}_{\mathrm{air}} / \mathrm{L}_c$     (19)

$L_c=L \cdot t /(2 L+2 t)$    (20)

As far as discussion are considered for fin with respect to surface was covered in previous section. Now important can be given for three distinct heat transfer coefficients. Firstly, heat transfer coefficient for the solid portion in case of perforated surfaces (h_ss) can be determined using the below equation [12]:

$h_{\mathrm{ps}}=(1+0.75 \mathrm{ROA}) \cdot \mathrm{h}_{\mathrm{ss}}$    (21)

Next, when it comes to with perforation, we donate it as h_pc. Thereby Nusselt number (Nu_c) can be represented using below equation [1]:

$\mathrm{Nu}_c=\left[\left(\mathrm{Ra}_c / 16\right)^{-1.03}+\left(0.62 \mathrm{Ra}_c^{0.25}\right)^{-1.03}\right]^{-(1 / 1.03)}$    (22)

Finally, the last expression was related to the heat transfer coefficient at fin tip, which was denoted by ht. For this case Nusselt number (Nut) was given by Eq. (19). Since variation in temperature using perforated fin was occurred the length i.e., x direction. Now, the value of heat dissipation rate from the perforated fin was denoted as Q_pf, and is determined using below expression.

$\begin{gathered}Q_{\mathrm{pf}}=2 \mathrm{~N}_y \cdot \sum_{I=1}^{N_e}\left(\frac{T_I+T_{I+1}}{2}-\right.  \left.T_{\infty}\right)\left(h_{\mathrm{ps}}\left(\frac{P(I)+P(I+1)}{2}\right) \operatorname{Le}(I)+h_{\mathrm{pc}} \cdot \mathrm{L}_{\mathrm{pc}}(I) \cdot \mathrm{t}\right)+Q_t\end{gathered}$        (23)

where, Q_t is the heat dissipation from the tip of the perforated fin, which can be reported with the below expression.

$Q_t=A_t * h_t *\left(T_t-T_{\infty}\right)$    （24）

In case to compare performance of the perforated fin with that of the solid (non-perforated) one of the same dimensions, the following equations are used [32].

$\frac{T_x-\mathrm{T}_{\infty}}{T_b-T_{\infty}}=\frac{\operatorname{Cosh}(\mathrm{m}(L-x))+\left(\frac{h_t}{m * \mathrm{k}}\right) \operatorname{Sinh}(\mathrm{m}(\mathrm{L}-\mathrm{x}))}{\left.\operatorname{Cosh}(m * \mathrm{~L})+\left(h_t /(m * k)\right) \operatorname{Sinh}(m * \mathrm{~L})\right)}$   (25)

$T_{\mathrm{sf}}=T_{\infty}+\frac{T_b-T_{\infty}}{\operatorname{Cosh}(\mathrm{m} * L)+\left(\frac{h_t}{m * k}\right) \operatorname{Sinh}(m * L)}$        (26)

$Q_{\mathrm{sf}}=k * A * m\left(T_b-T_{\infty}\right) \frac{\operatorname{Sinh}(m * L)+\left(\frac{h_t}{m * k}\right) \operatorname{Cosh}(m * L)}{\operatorname{Cosh}(m * L)+\left(\frac{h_t}{m * k}\right) \operatorname{Sinh}(m * L)}$     (27)

$\begin{gathered}m=\sqrt{\frac{h * P_{\mathrm{sf}}}{k * A}}, P_{\mathrm{sf}}=2 . \mathrm{W}, A=W * \mathrm{t}, A_{\mathrm{sf}}  =L * P_{\mathrm{sf}}+W * t\end{gathered}$      (28)

The relation between heat dissipation with respect to the perforated fin was given with RQF, as shown below:

$\mathrm{RQF}=Q_{\mathrm{pf}} / \mathrm{Q}_{\mathrm{sf}}$        (29)

Present works reports the behavior of longitudinal Elliptic Perforations fin and its surface parameters on natural heat convection to dissipate the excess heat from light electrical devices. Intensive applications of perforation fins are found in heat exchangers, boilers, industries burners, nuclear reactors, electric transformers and power plants. The methodology used was numerically approach with finite element method for one- dimensional model. At specific values of the perforated geometry, the temperature gradient between the plate and solid fins was less than the temperature drops along the perforated fin. The rate at which heat was dissipated using perforated fins depends on the thickness of the fins, the thermophysical characteristics of the fin materials, and the surface characteristics of the perforated area. The main surface parameters could impact mainly on the heat dissipation rate in elliptical perforated areas are perforation diameter (D_o) and optimum spacing (Syo). In contrast, the highest heat transfer rate achieved by using these parameters ideal values. The perforation of fins increases heat dissipation rates while at the same time reducing the material consumption of the fins. This work can further be elaborated by considering 3-D approach and other perforated geometries.