_{1}

^{*}

Fins are the extended surfaces through which heat transfer takes place to keep the surface cool. Fins of various configurations are presently used ranging from automobile engine cooling to cooling of computer parts. Note that in a fin majority of the heat transfer to atmosphere is by convection and therefore in the present research , and importance is given to variation of temperature along the length of the fin which in turn gives rate of heat transfer. In the present research a solid rectangular aluminum fin and the same rectangular fin with different perforations (2, 4, 8 and 10) were compared analytically, experimentally and its validity through finite element analysis for its temperature distribution along the length. From the present research it is observed that the mathematical and FEA for a solid rectangular fin without perforations are converging within ±1°C and rectangular fin with 10 perforations are converging within ±2°C and hence the validity.

The basic law that governs the convective heat transfer is Newton’s law of cooling given by:

where: Ts: the surface temperature ˚C, T_{f}: the fluid temperature ˚C, h: the convective heat transfer coefficient w/m^{2} ˚C, A: surface area, m^{2 }

Note that in the above equation, film coefficient of heat transfer or the convective heat transfer coefficient “h” is very important which mainly depends on the type of the surface, size, shape, its temperature, surface finish etc. Most of the research, mathematical analysis and dimensional analysis are confined in finding “h”. In addition to the above, for fins the temperature distribution along its length also become important.

The expulsion of overabundance warmth from framework parts is fundamental to abstain from harming impacts of overheating. Thusly the improvement of warmth exchange is a vital subject of warm designing [

Balances are utilized to improve convective warmth move in an extensive variety of designing applications and offer a handy method for accomplishing a substantial aggregate warmth exchange surface range without the utilization of an over the top measure of essential surface zone [

Along these lines, fins must be intended to accomplish most extreme warmth expulsion with least material use considering the simplicity of the balance fabricating. The change in warmth exchange coefficient is ascribed to the restarting of the warm limit layer after every an interference [

In the examination of warmth trade, cutting edges are surfaces that stretch out from a thing to manufacture the rate of warmth trade to or from nature by growing convection. The measure of conduction, convection, or radiation of an article chooses the measure of warmth it trades [

Finally, it is well known that major heat transfer from the fin is by convection and performance of a fin is evaluated by its efficiency and effectiveness.

a. Straight Fins (

b. Annular (circumferential) fins (

c. Spine or pin fins (

There are various examination related to warmth trade and weight drop of channels with pin cutting edges, which are limited to stick parities with round or couple of different cross territories. The genuine warmth trade takes by two modes i.e. by conduction took after by convection. Heat trade through the solid

to the surface of the solid happens through conduction whereas from the surface to the surroundings happens by convection. Further warmth trade may be by normal convection or by obliged convection.

Bayram Sahin and Alparslan Demir [

Metzger et al. [

Fins are the extended surfaces through which heat transfer takes place. In the present investigation the surface area is increased by providing perforations in the fins. This in turn increases effectiveness and efficiency of the fin. So far no attempt has been made to provide perforations in the solid aluminum fin to increase the surface area and hence the present research was under taken to fill the void.

The most popular energy balance equation used to find the heat transfer through fins mathematically (for steady state condition) is given by:

where Q: rate of hear transfer, watts, h: convective heat transfer coefficient w/m^{2}

˚C, P: the perimeter of the fin, m, K: thermal conductivity, w/m ˚C, A: area, m^{2} and C_{1}, C_{2} are constants obtained by applying the limits.

Above equation is modified to find temperature distribution based on the tip condition.

Equation (3) below is used to find analytically the fin temperature over a distance with given boundary conditions [

where,

where Nr = numerator term and Dr = denominator term

Correlations given below are used to find the heat transfer coefficient using dimensional analysis for vertical plate [

where,

(where constants c = 0.59, n = 0.25)

For analysis and comparison purpose, rectangular aluminum fin with 2, 4, 8 and 10 perforations was considered but for discussion only fin with 10 perforations is presented in this present paper.

Mathematically, the heat transfer coefficient is calculated using the vertical plate correlation discussed in Section 4.2. using the following details.

Actual open area = 10 * (pi/4 * 12^{2}) = 1130.9 mm^{2 }

Maximum possible perforation open area = 15 * (pi/4 * 12^{2}) = 1696.45 mm^{2 }

Due to perforations, boundary layer detachment and re-attachment phenomenon occurs. This is accommodated by a multiplying factor in the formula given below where h_{ps} is the convective heat transfer coefficient.

Element types used for the analysis are: SOLID70 element and SURF152 element. The steps involved in the present FEA are a) 3D rectangular fin modelling and meshing b) Creation of surf elements for the modeling c) Applying the boundary conditions and source temperature d) Applying the material property (aluminum) e) Obtaining the steady state thermal contours.

Finally the mathematical, experimental and ANSYS FE analysis are compared for the validity. Initially this method was used for solid rectangular fin under steady state condition and later to rectangular fin with perforations.

Note that for a solid fin, only mathematical and FE analysis was performed whereas for a rectangular fin with perforations, mathematical analysis and experimental (temperature measurement) results are compared as a check and finally the validity by FEA.

It is observed from

Analytic calculation to find temperature at distance x | ||||||
---|---|---|---|---|---|---|

L, Length of the fin | 0.15 m | |||||

w, width of the fin | 0.1 m | |||||

t, thickness of the fin | 0.015 m | |||||

h, heat transfer coeff. | 5.489 w/m2-k (for Al) | |||||

p, perimeter | 0.23 m | |||||

Ac, cross section Area | 0.0015 m2 | |||||

k, thermal conductivity | 236 w/m-k (for Al) | |||||

m | 1.888 | |||||

Tbase | 200˚C | |||||

Tinfinity | 20˚C | |||||

L(inm) | x(in m) | Numerator | Denominator | Theta(a)/theta(b) | T(x), ˚C | |

0.15 | 0 | 1.0439 | 1.0439 | 1.0000 | 200.0 | |

0.015 | 1.0358 | 1.0439 | 0.9923 | 198.6 | ||

0.03 | 1.0286 | 1.0439 | 0.9823 | 197.4 | ||

0.045 | 1.0222 | 1.0439 | 0.9792 | 196.3 | ||

0.06 | 1.0166 | 1.0439 | 0.9738 | 195.3 | ||

0.075 | 1.0118 | 1.0439 | 0.9692 | 194.5 | ||

0.09 | 1.0078 | 1.0439 | 0.9654 | 193.8 | ||

0.105 | 1.0047 | 1.0439 | 0.9624 | 193.2 | ||

0.12 | 1.0023 | 1.0439 | 0.9601 | 192.8 | ||

0.135 | 1.0008 | 1.0439 | 0.9586 | 192.6 | ||

0.15 | 1.0000 | 1.0439 | 0.9579 | 192.4 | ||

along the length of the fin with a power supply of 40 watts for a rectangular fin with different perforations. Temperature was measured using thermocouples along the length of the fin.

Figures 10-12 below shows the FE models of thermal contour (steady state), thermal gradient and thermal heat flux for a rectangular fin containing 10 perforations.

It is observed from

increases with perforations having full connectivity between the base and the fin. It is observed from the present research that keeping the base (root) temperature at 200˚C with power supply of 40 watts, rectangular fin with 10 perforations reaches 184.2˚C at the tip. This shows that rectangular fin with perforations removes more heat compared to a solid fin.

From

From

Extended surfaces i.e., fins are used to remove heat from the component to keep it cool. It is observed from the present research that keeping the base (root) temperature at 200˚C with power supply of 40 watts, a solid rectangular fin reaches a temperature of 192˚C at the tip whereas the same fin with 10 perforations reaches 184˚C at the tip. This shows that rectangular fin with perforations removes more heat compared to that of a solid fin. It is also observed from the research that this temperature fall gradually from fin with 2, 4, 8 and 10 perforations thus the heat removal gradually decreases with increase in perforations. Heat flux also follows the same pattern along the length of the fin as that of the temperature.

In the present research it is observed from the mathematical and FEA that for a solid rectangular fin without perforations are converging within ±1˚C and rectangular fin with 10 perforations are converging within ±2˚C and hence the validity.

Hemanth, J. (2017) Experimental, Mathematical and Finite Element Analysis (FEA) of Temperature Distribution through a Rectangular Fin with Circular Perforations. Modeling and Numerical Simulation of Material Science, 7, 19-32. https://doi.org/10.4236/mnsms.2017.72002