Journal of Engineering Science and Technology

Vol. 15, No. 6 (2020) 4075 - 4090

© School of Engineering, Taylor’s University

EXPERIMENTAL STUDY FOR OPTIMUM FIN

SPACING OF RECTANGULAR FIN ARRANGEMENTS
UNDER THE INFLUENCES OF FREE CONVECTION

ABBAS J. JUBEAR AL-JASSANI

Department of Mechanical Engineering,

College of Engineering, Wasit University, Iraq
E-mail: abbasaljassani@uowasit.edu.iq

Abstract
Heat transfer enhancement systems are widely used in various industries, such
as thermal power plants, heating and air conditioning systems, and electronic
equipment. In this research, the thermal performance of a vertical heat sink with
rectangular fins geometry was investigated experimentally. The present study
mainly aims to find the optimum fin spacing that provides the highest heat
transfer rate and reduces the heat sink base temperature. An Aluminum heat sink
of six 300 mm length fins was considered in the test. The fin thickness and height
were kept constant at 4 mm and 45 mm, respectively. In the analysis, a range of
fins spacing (i.e., 10, 11, 12, 13, and 15 mm) was used in the experiments, and a
variety of Rayleigh number (i.e., from 7.6*106 to 1.48*108) was considered. The
difference of heat sink and surrounding temperatures were obtained at various
heat input and all the different parameters above. These temperature differences
are then used for the estimation of the heat transfer rate of the fin array. The
results showed that the optimum fin spacing was S = 12 mm, which gives the
highest heat transfer rate. Besides, a reduction of 18 % of the heat sink weight is
achieved due to the use of the optimum fin spacing compared to the weight of
using 15 mm as a fin spacing. The reduction of heat sink weight and size can no
doubt lead to a decrease in manufacturing cost, which consequently lead to the
spread of heat sink applications. Furthermore, the decline of the heat sink base
temperature will improve the working life span of the electronic devices that use
the heat sink. The results of the present study also show the development of a
new empirical equation predicting the Nusselt number in terms of Rayleigh
number and the fin spacing to fin height ratio. The developed equation shows the
high accuracy of prediction (e.g. about 90%).
Keywords: Fins, Fin spacing, Heat sink, Heat transfer rate, Natural convection.

4075
4076 A. J. J. Al-Jassani

1. Introduction
Natural convection phenomena in enclosures are necessary for an excellent
performance of high-power density electrics. Buoyancy drove flows have many
applications in widely preferred phenomena, such as air-cooled car engines, cooling
of generators, motors, refrigerators, transformers, and cooling of computer
processors. Natural convection is a method of heat dissipation that is relatively
inexpensive, discreet, and most dependable. The heat generated by electronic devices
can be controlled using fins. In the design of an efficient cooling system, generally,
55% of failure mechanisms in electronic devices are related to thermal effects Pascoe
[1]. However, many parameters, such as fins height and fins spacing, affect the rate
of convection heat dissipation. These parameters should be considered when the fins
best design or the enhancement of heat dissipation rate is targeted.
The subject of investigation of the thermal performance of finned systems still
attracts the interest of many researchers. A considerable number of researches have
been published. Part of these publications focuses on the development of
mathematical formulas that considered the fin arrangement specification. Some
other parts of the papers focus on the investigation on fins system arrangements
such as fins number, shapes, dimension, and fin spacing. For examples of the
former part of studies above (i.e., studies that develop mathematical correlation
formula), Elenbaas [2] studied the effect of small gap width between parallel plates
on heat dissipation under natural convection condition experimentally.
The results developed a proportional relation between the Nusselt number and
the Rayleigh number. Yazicioğlu and Yüncü [3] investigated the thermal
performance of the vertical heat sink of rectangular fins under different parameters.
The fin base–to-surrounding temperature difference ranges from 30 to 150 K. The
fin length is taken as 250 mm and 340mm. The space between the fins and the
height of the fin was from 4.5 to 85.5 mm and 5 to 25 mm, respectively. The authors
reported that the optimum value of fin spacing varies with the height of the fin,
which was from 6.1 to 11.9 mm. Besides, the rate of heat transfer enhanced
depends on the average base temperature, fin length, and fin spacing.
The influence of fin height, fin spacing, and fin orientation on the natural heat
transfer coefficient were studied experimentally by Walunj, et al. [4]. Bar-Cohen
and Rohsenow [5] conducted an analytical investigation for the natural convective
heat transfer from two parallel plates. Their work includes the development of a
correlation formula describing Nusselt number in terms of Rayleigh number for
isothermal and isoflux plates. The optimum fin spacing was also reported in the
same study.
Bodoia and Osterle [6] developed a numerical technique for the investigation
of flow in channels and heat transfer between symmetrically heated, isothermal
plate to predict the channel length required to achieve fully developed flow as a
function of the channel width and wall temperature. The heat transfer in inclined
interrupted fin channels was investigated in Fujii [7]. A mathematical correlation
was developed for fitting the experimental results. As was mentioned previously,
some publications investigated the influence of some fin geometrical properties
such as fins number, fins shapes, fins dimension, and fin spacing. Experimental
studies on vertical rectangular fin arrays ware carried out by Güvenç and Yüncü
[8] and Leung and Probert [9].

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 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4077

Numerical studies on vertical rectangular fin arrays were investigated by

Fitzroy [10], Saikhedkar and Sukhatme [11]. Yüncü and Anbar [12] performed
experiments on the free convection heat transfer of the horizontal rectangular fin.
Fin thickness and fin length were kept constant at 3 and 100 mm, respectively,
whereas the other variables such as fin spacing and fin height varied at from 6mm
to 26mm and 6.2 to 83 mm respectively. The results showed that the fin spacing to
fin height ratio was the strongest influence on the free convection heat transfer
coefficient. However, the temperature difference decreases with the increase of fin
height. The optimum fin spacing was 11.6 and 10.4 mm, when fin height was 16
and 26 mm, respectively.
Yazicioglu and Yuncu [13] developed a new expression to predict the optimum
fin spacing of vertical rectangular fins under natural convection condition. The
study tested different parameters of an extended area such as fin spacing and fin
height. The results indicated that the rate of heat transfer enhanced when the fin
length, fin height, and fin base-to-surrounding temperature difference increased.
Kumar, et al. [14] investigated the performance of a triangular fin model within a
vertical orientation and with several influencing parameters such as the fin spacing.
The results developed an empirical correlation connecting the Nusselt number to
these parameters. It was found that the fin spacing has a significant impact on all
geometric parameters of the heat sink. Also, the relation between fin height and fin
spacing was reversed.
Shehab [15] investigated experimentally the effect of the horizontal heat sink
that has various fin spacing and number of fins on the heat transfer coefficient and
Nusselt number under natural convection condition. The results showed that the
fin spacing has a significant effect on the average Nusselt number. Karami, et al.
[16] examined three types of finned tube exchangers with square fins and different
spacing (5, 9, and 14) mm experimentally. The Nusselt number and heat transfer
coefficient enhanced with the increase of the fin spacing. The performance of the
vertical heat sink under natural convection is examined by Al-Jewaree [17].
Different materials and fin spacing are used in this study. The results showed that
the heat transfer rate improved when the fin spacing decreased and, the maximum
improvement was 22%.
An experimental based study has been conducted by Al-Jessani and Al-
Bugharbee [18] to examine the effect of circular perforations on the weight of the
heat sink. The results found that the optimum number of perforations per fin was
10. Also, the heat sink weight was reduced by 7%. Leung, et al. [19-22], Leung
and Probert [23, 24]and Van de Pol and Tierney [25] are some examples, which
were mostly focused on the effects of varying fin geometric parameters, the array,
and base plate orientation.
The heat transfer under natural convection and radiation of twelve vertical fins
system are investigated in Chaddock [26]. The fin width plate kept constant during
the experiments while the fin spacing and fin height is varied. The latter showed a
significant influence on the radiation of heat transfer.
Recently, the natural convection in porous fins and enclosures with a porous
medium is investigated in Ahmed, et al. [27], Ahmed, et al. [28], Hoseinzadeh, et
al. [29] and Hoseinzadeh, et al. [30]. The present study focuses on investigating
the effect of space between the fins on the rate of heat transfer, heat transfer
coefficient and fin base-to-surrounding temperature difference. The present study
offers an investigation for finding the optimum fin spacing, which provides best

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4078 A. J. J. Al-Jassani

heat dissipation of the heat sink. Consequently, this optimum spacing will no doubt
reduce the weight and manufacturing cost of the heat sink. In addition, an empirical
equation was developed for describing the relation between Nusselt number,
Rayleigh number, and dimensionless parameter of fins spacing and fins height.

2. Experimental Test Rig

An experimental device was designed and made from several parts, which are
assembled to install the test section unit. The instrument has been used for
measuring the base-to-surrounding temperature difference from a vertical
rectangular fin configuration to calculate heat transfer parameters. The tested frame
was manufactured from square cast iron material of a height of 320 mm and a cross-
sectional area (10*10) mm2. The structure is designed with rectangular shapes and
fastened on four stands. The frame was covered with sheet aluminium of 0.9 mm
thickness, which was insulated with a stratum of wool thermal of 15 mm thickness.
The unique plate material is used as a reflector of heat with a thickness of 0.5 mm.
Three heaters of 600 W are used in the test section. The heaters were covered by
a glass tube with a diameter of 20 mm. The fins are installed on the test section by
a sliding channel and are controlled by locking screws.
The fins have been manufactured with aluminium metal due to its high thermal
conductivity and low emissivity at 20 °C. These fins were formed by making
several longitudinal channels along the top face of the rectangular bar. Fins length
L and the fin height H are kept as 300 mm and 45 mm respectively during all the
experiment work. Fins arrangement width W are varied as 95, 100, 105, 110, and
120 mm. Different fin spacing S is used, namely 10, 11, 12, 13, and 15 mm. Fins
were kept integral with 7 mm thickness of the base plate, the fin thickness is
remained fixed at t = 4 mm, and the number of fins is n = 6. The geometry of in-
line continuous rectangular fins shown in Fig. 1. The base plate is kept at constant
heat flux, and the upper surface of the fins is at the environment temperature.

Fig. 1. Heat sink dimension (left) Locations of thermocouples (right).

3. Methodology
The methodology contains constructing a test rig of the finned heat sink for
conducting several experiments to obtain an average base and ambient temperature

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 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4079

at different heat inputs. The heat input can be determined by knowing the supplied
electrical voltage and current provided to the electrical heater. Then, the convection
heat transfer rate can be determined by subtracting the radiation and loss heat rates
from the generated heat transfer (e.g., input heat). Different convection heat
transfer coefficients can then be obtained at different input voltage and current,
different heat sink fins spacing. Eventually, an optimum fin spacing can be found
as that fin spacing providing heights heat transfer coefficients. Besides, an
empirical equation predicting the relation of the Nusselt number in terms of
Rayleigh number and the fin spacing to fin height ratio. The reduction of heat sink
weight and size can no doubt lead to a decrease in manufacturing cost, which
consequently lead to the spread of heat sink applications.

4. Experimental Procedure
The experiments were conducted at various heat flux. This variation is made through
controlling the heater voltage by a voltage regulator. The provided voltage to the
heater is varied from 50 to 150 V by 25 V step. The surrounding temperature was
measured via a thermometer. The fin base plate temperature was acquired at eight
positions to calculate the average temperature along with the base plate. The
temperature was measured using a temperature scanner thermometer (8 channel
thermocouple) manufactured by Altop Industries Ltd, Gujarat, India (Si. 1005164.
Model ADT 5003) with a resolution of 0.1. The average of these eight positions
represents the fin base temperature. Figure 2 shows the experimental rig with the
measuring instrument actions. The predefined heater inputs were modified with the
assistance of dimmer stat. The temperature of an aggregated fin heat sink array of
various locations and surrounding temperatures was registered during time intervals
of 30 min. It requires close to four hours to reach a steady condition. Consequently,
when the temperature reading does not vary by 0.3°C, in this case, the system reaches
a steady-state condition. The experimental tested was carry out in the Fluid Dynamics
Laboratory of the mechanical engineering department/Wasit University.

Fig. 2. Photograph of the experimental rig with the measuring instruments.

5. Analytical Analysis
The heat input of the heating base was dissipated based on two approaches,
primarily through the heat sink and, slightly by the insulation fins and heat sink

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4080 A. J. J. Al-Jassani

structure. A voltage V is provided through the voltage regulator to the heater, which
is fixed on the heat sink base. The temperature from the eight positions (predefined
in the previous section) as well as the ambient temperature is acquired. The
provided voltage and the received temperature are used for the calculation of
Qconv, as in the following equations Jubear [31].
𝑄𝑄𝑔𝑔𝑔𝑔𝑔𝑔. = 𝑄𝑄𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. + 𝑄𝑄𝑟𝑟𝑟𝑟𝑟𝑟. + 𝑄𝑄𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 (1)
where
𝑄𝑄𝑔𝑔𝑔𝑔𝑔𝑔. = 𝑉𝑉 × 𝐼𝐼 (2)
And the heat transfer by radiation was:
4 4
𝑄𝑄𝑟𝑟𝑟𝑟𝑟𝑟. = σ × 𝜀𝜀 × 𝑆𝑆𝑠𝑠𝑠𝑠𝑠𝑠 × 𝐴𝐴𝑡𝑡 (𝑇𝑇av. − 𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎 ) (3)
The loss is estimated according to the temperature difference between the heat
sink base and ambient, as follow Feng, et al. [32]:
𝑄𝑄𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = −9 ∗ 10−5 ∗ ∆𝑇𝑇 2 + 0.0202 ∗ ∆𝑇𝑇 (4)
In the present study, the heat loss for all tested was less than 4% of the overall
power input. Consequently, for these ranges of heat input, the heat loss is ignored.
Therefore, the convection heat transfer was calculated as follows:
𝑄𝑄𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. = 𝑄𝑄𝑔𝑔𝑔𝑔𝑔𝑔. − 𝑄𝑄𝑟𝑟𝑟𝑟𝑟𝑟. − 𝑄𝑄𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 (5)
Hence, the heat transfer coefficient in free convection was calculated from the
equation which, is known as Newton’s cooling law, as follows:
𝑄𝑄𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.
ℎ= (6)
𝐴𝐴𝑡𝑡 (𝑇𝑇𝑎𝑎𝑎𝑎. − 𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎)

Totally exposed area

𝐴𝐴𝑡𝑡 = 𝑛𝑛 ∗ 𝐴𝐴𝑓𝑓 + 𝐿𝐿 ∗ 𝑠𝑠 ∗ (𝑛𝑛 − 1) (7)
The ratio of the convection heat flux to the conduction heat flux is known as
the Nusselt Number and written as follows:
ℎ 𝐿𝐿
𝑁𝑁𝑁𝑁 = (8)
𝐾𝐾𝑓𝑓

The Rayleigh Number was defined as the ratio between buoyancy and viscosity
and written as follows:
𝐿𝐿3 𝑔𝑔 𝛽𝛽 (𝑇𝑇𝑎𝑎𝑎𝑎. −𝑇𝑇𝑎𝑎𝑎𝑎𝑎𝑎)
𝑅𝑅𝑅𝑅 = 𝑃𝑃𝑃𝑃 (9)
𝜈𝜈2

6. Results and Discussion

The experimental study used the vertical rectangular fin, based on six fins model
with different configurations of fin spacing (S = 10, 12 and 15 mm), length L =
300mm, height H = 45 mm and thickness t = 4 mm, respectively, under a different
heat flux (78, 176, 317, 496 and 715 W). The experiments were performed using
the Rayleigh Number (7.6*106 – 1.48*108). The results show the effect of spacing
among the fins on the heat transfer coefficient, Nusselt number (Nu), and the
Rayleigh Number (Ra). This parameter can be discussed, given the following.

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 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4081

6.1. Model validation

The present experimental results have been compared with Yazicioğlu and Yüncü
[3], Tari and Mehrtash [33], Tari and Mehrtash [34] and Mehrtash and Tari [35].
The values of optimum fin spacing were 11.9 and 11.75 mm. It was noted there is
a good agreement between the present work and the other research, as shown in
Table 1, and the error is no more than 2%.

Table 1. Validation of experimental model.

Fin Base Fin Fin Fin Optimum
Reference
Length width height Thickness Spacing Fin
No.
(L) (W) (H) (t) (S) Spacing
5.8-
3 250,340 180 5,15,25 3 10.4-11.9
85.5
5-
33,34,35 250,340 180 5-25 3 11.75
85.5
Present 100- 10-
300 45 4 12
Work 150 15

6.2. Temperature distribution

To clarify the experimental results by a more reliable technique the Fluke Ti32
Thermal Imager (hereafter “the Imager”) was used to show the temperature
distribution on the fin surface from different sides (top and side view). The Thermal
Imager uses various colours or shades of grey to show the temperature distribution
on the fin surface over the Imager’s domain of view, as shown in Fig. 3.

Fig. 3. Snapshots of temperature distribution imager.

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4082 A. J. J. Al-Jassani

Figure 3 is presented for further visualisation purpose as several considerations

should be taken for achieving accurate readings. This is because both local
emissivity and the background temperature affect the IR system. The accuracy of
the IR reading is also influenced by several factors such as air temperature and
wind speed.

6.3. Temperature difference variation

The influence of the fin’s base-to-surrounding temperature difference on the
coefficient of heat transfer, the rate of heat transfer, and Nusselt number according
to different fins spacing of (S= 10, 11, 12, 13, and 15mm) are presented in Figs. 4
to 6. These parameters possibly increase, according to increasing of the fin base-
to-surrounding temperature difference for all fin spacing values. The behaviour of
the heat transfer coefficient with the fin base-to-surrounding temperature
difference is shown in Fig. 4. The coefficient of heat transfer improves with the
increase of the temperature difference between the average base temperature of the
heat sink and the surrounding temperature. Besides, the heat transfer coefficient
has a maximum value when the fin spacing is 12 mm; despite that, the temperature
difference has been reduced. The figure also shows that the benefits of the heat
transfer coefficient become closer when the temperature difference is slight. In
other words, the heat flux on the base of the heat sink is weak, while it diverged as
a result of the high range of the heat flux. Finally, the enhancement of the fin base-
to-surrounding temperature difference (reduction of the heat sink base) at 12 mm
fin spacing was 25% compared with 15 mm.

Fig. 4. The influence of temperature difference on heat transfer coefficient.

Figure 5 shows the effect of fin base-to-surrounding temperature difference on

convective heat transfer. The result illustrates that the rate of heat transfer of the
heat sink is improved when the temperature difference increased for all values of
fin spacing. Also, the space between the fins and the fin base-to-surrounding
temperature difference parameters have strongly influenced on the heat transfer
rate of the heat sink. The rate of heat transfer is enhanced dramatically, according
to the fin base-to-surrounding temperature difference. The behaviour of the Nusselt

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 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4083

number with the fin base-to-surrounding temperature difference is similar to the

action of heat transfer coefficient conduct, as shown in Fig. 6. A straight line cannot
represent the result, and as a result, an exponential equation has been used.

Fig. 5. The effect of temperature Fig. 6. The effect of temperature

difference on heat transfer rate. difference on Nusselt number.

6.4. Heat transfer coefficient variation

The relation between heat transfer coefficient and several variables such as rate of
convection heat transfer, and Nusselt number with different fin spacing (S = 10,
11, 12, 13, and 15 mm) can be seen in Figs. 7 and 8. The proportional relation
between the heat transfer coefficient and the heat transfer rate is shown in Fig. 7.
This figure exhibited, the convection heat transfer rate is decreased when space
between fins was increased, reached a minimum value at fin spacing 12 mm, and
then grew and reached a maximum value at fin spacing 15 mm. It can be observed
that the same behaviour was correct in each of the previous action for the Nusselt
number, as shown in Fig. 8.

Fig. 7. The effect of heat transfer Fig. 8. The effect of heat transfer
coefficient on heat transfer rate. coefficient on Nusselt number.

6.5. Fin spacing variation

The influence of fin spacing on the fin base-to-surrounding temperature difference,
free convection heat transfer coefficient and Nusselt number can be observed in

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4084 A. J. J. Al-Jassani

Figs. 9 to 11. The effect of heat input on the fin base-to- surrounding temperature
difference can be seen more clearly in Fig. 9. In this figure, the fin base-to-
surrounding temperature difference is plotted as a function of fin spacing. It can be
deduced that the fin base-to-surrounding temperature difference from a fin array
decreases with the space of fins, and reach to a minimum level when the fin spacing
was 12 mm, and then increasing. The corresponding fin spacing value of the
minimum base-to-ambient temperature difference is named the optimum fin
spacing. It was noted that the optimum fin spacing of this study was S = 12 mm.
This result reveals that the optimum value of fin spacing was hypersensitive to the
variations in heat generated and temperature difference parameters.

Fig. 9. The effect of fin spacing on base to surrounding

temperature difference with various heat input.

The values of the natural convection heat transfer coefficient obtained for
different heat inputs are shown in Fig. 10. It can be seen that when the space
between two fins (i.e., gap) increases from 10 to 12 mm, the free convection heat
transfer coefficient ultimately increases. Besides, a further increase in the fin
spacing leads to a decrease in this coefficient. If the gaps between the fins are
evenly matched or small, the free convection heat transfer coefficient is minimized
as mixing of the thermal boundary layer occurs. Figure 10 clearly shows that the
heat transfer coefficient decreases as the gap between fins decreases. However, if
the fins are closely spaced, there is also a more dissipating surface area (more fins
for a given volume). The additional surface area can counteract the reduced heat
transfer coefficient.
More precisely, a variation of the convective heat transfer coefficient with
different heat input is studied in this figure. Initially, the heat transfer coefficient
enhanced when the space among the fins increases. It reaches a maximum value at
fin spacing (S = 12 mm), and with further additions of fin spacing, it begins to
decrease. The space between the fins is called optimum value when the convection
heat transfer coefficient has maximized. Inspection of Fig. 10, reveals that the
optimum fin spacing value was (S = 12 mm) at the different heat input. One of the
essential parameters in designing a heat sink is the fin spacing, S. The enhancement

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 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4085

of the heat transfer coefficient due to optimum fin spacing was 27%. The same
behaviour is shown in Fig. 11.

Fig. 10. The effect of fin spacing on heat Fig. 11. The effect of fin spacing on Nusselt
transfer coefficient of various heat input. number of various heat input.

6.6. Nusselt number variation

Figures 12 and 13 show the variations of Rayleigh number and fin heat transfer
rate with Nusselt number for all the fin spacing. Figure 12 shows the continuous
increase in Nusselt number with a rate of heat transfer as reported for all the fin
spacing. A straight line cannot represent the relation between the Nusselt number
and the heat transfer rate. As a result, an exponential equation has been used. The
variations of the Nusselt number with fin spacing have been illustrated in Fig. 13,
at a particular fin height value. The Nusselt number increases continuously with an
increase in fin spacing for all the Rayleigh number values. It reaches a maximum
value at the fin spacing 12 mm. An increase in the Nusselt number immediately
indicates an enhanced convection heat transfer rate.

Fig. 12. The relation between heat Fig. 13. The relation between
transfer rate and Nusselt number. Nusselt number and Rayleigh number.

It is essential to mention that 18% of heat sink weight is achieved due to the
use of the optimum fin spacing compared to use the 15 mm. The reduction
percentage can be calculated either by 1) weighing the heat sink at using fin spacing

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4086 A. J. J. Al-Jassani

of 15mm and optimum fin spacing (e.g.12 mm) or 2) calculating the heat sink
volume and multiplying by the material density to fins the mass.

6.7. Correlation equation

In the current study, different correlation formulas were investigated to find the
best representation of the experimental readings. The developed empirical equation
was in the form of a nonlinear equation, which is a general form as in Eq.10.
𝑆𝑆 𝐶𝐶3
𝑁𝑁𝑢𝑢 = 𝐶𝐶1 + 𝐶𝐶2 ∗ � � + 𝐶𝐶4 ∗ 𝑅𝑅𝑅𝑅𝐶𝐶5 (10)
𝐻𝐻

where C1 = -1095, C2 = -389, C3 = 222.8, C4 = 0.8 𝑎𝑎𝑎𝑎𝑎𝑎 C5 = 0.1

Figure 14 compares the predicted and experimental Nusselt number readings.
In this figure, the x-axis represents the data points of the experimental test; the y-
axis represents the Nusselt number. The data points refer to the number of recorded
points. It can be seen from this figure that both predicted and experimental results
have the same trend and fluctuation.

Fig. 14. The relation between experimental and predicted data.

The accuracy of the empirical equation can be statistically assessed using the
following measures Zubaidi, et al. [36]:
𝑦𝑦𝑜𝑜 −𝑦𝑦p 2
%𝑅𝑅𝑅𝑅 = ∑𝑁𝑁
𝑚𝑚=1 � � ∗ 100 (11)
𝑦𝑦𝑜𝑜

2
∑𝑁𝑁
𝑚𝑚=1�𝑦𝑦𝑜𝑜 −𝑦𝑦p �
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 = 1 − � (12)
𝑁𝑁

∑𝑁𝑁
𝑚𝑚=1(𝑦𝑦𝑜𝑜 −𝑦𝑦
����) �����
𝑜𝑜 �𝑦𝑦𝑝𝑝 −𝑦𝑦𝑝𝑝
𝑅𝑅 = 2
(13)
�∑𝑁𝑁
𝑚𝑚=1(𝑦𝑦𝑜𝑜 −𝑦𝑦
2 𝑁𝑁
𝑜𝑜 ∑𝑚𝑚=1�𝑦𝑦𝑝𝑝 −𝑦𝑦
����) �����
𝑝𝑝

where:
RE Relative error.
RMSE Root mean square error
NRMSE normalized root means square error

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 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4087

R Pearson’s product moment correlation

yo Experimental reading
yp Predicted reading.
Figure 15 shows the residuals of the predicted and experimental Nusselt
number. It is clear from the figure that the residuals fluctuating around zero and
has randomly distributed.

Fig. 15. The residuals of the experimental and predicted data.

The statistical measures of the predicted and experimental results show that
%RE = 10, NRMSE= 0.43 and R= 0.84

For the present work, the uncertainty of the experimental results is obtained
using mathematical formulas in Moffat [37]. The uncertainty parameter for the
measured base temperature was ∓1.2℃. The correlation coefficient of the
innovative and predicted Nusselt values are computed and found 0.82.

7. Conclusions
In this study, the effect of different fin spacing, heat flux, and Rayleigh number
(Ra) on the performance of vertical rectangular fins under free convection heat
transfer was investigated experimentally. The heat transfer coefficient and base
temperature are obtained analytically. From the different results, it was found that
the optimum fin spacing is S=12 mm. This optimum S was selected in terms of
providing maximum heat transfer coefficient, maximum Nusselt number, and heat
transfer rate. Additionally, the fin base-surrounding temperature increases by 25%
at this fin spacing. The selection of optimum S also leads to a18% of fin weight
reduction. In the present study, a precise empirical correlation equation was also
developed for describing the Nusselt in terms of Rayleigh number and the fin
spacing to fin height (S/H). The maximum error of the equation prediction was 10
%. The studies on the selection of fin spacing are still of great of importance as it
provides practical guidelines for manufacturing heat sinks with lighter weights,
less expensive cost and smaller size.

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4088 A. J. J. Al-Jassani

Nomenclatures

Af Exposed fin area, m

At Total exposed area, m
g Gravitational acceleration, m/s2
h Convection heat transfer coefficient, W/m2 °C
I Electrical current, Amper
Kf Thermal conductivity, W/m °C
L Fin length, m
n Number of fins
Nu Nusselt number
Pr Prandtl number
Qconv. Convection heat transfer, W
Qgen. Heat generated, W
Qrad. Radiation heat transfer, W
Ra Rayleigh number
s Fin spacing, m
Ssur The shape factor
Tair Air temperature, °C
Tav. Average temperature, °C
V Electrical voltage, volt

Greek Symbols
σ Steven-Boltzmann, W/m2 K4
β The volume coefficient of expansion, K-1
ε Emissivity
ν Kinematic viscosity, m2/s
References
1. Pascoe, N. (2011) Reliability Technology: Principles and Practice of Failure
Prevention in Electronic Systems: John Wiley & Sons.
2. Elenbaas, W. (1942). Heat dissipation of parallel plates by free convection,
Physica, 9(1), 1-28.
3. Yazicioğlu, B.; and Yüncü, H. (2007). Optimum fin spacing of rectangular
fins on a vertical base in free convection heat transfer, Heat and Mass
Transfer, 44(1), 11-21.
4. Walunj, A.; Daund, V.; and Palande, D. (2013). Parametric analysis of plate-
fin heat sink over heat transfer, International Journal of Research, 1(4).
5. Bar-Cohen, A.; and Rohsenow, W. (1984). Thermally optimum spacing of
vertical, natural convection cooled, parallel plates.
6. Bodoia, J.; and Osterle, J. (1962). The development of free convection
between heated vertical plates.
7. Fujii, M. (2007). Enhancement of natural convection heat transfer from a
vertical heated plate using inclined fins, Heat Transfer—Asian Research: Co‐
sponsored by the Society of Chemical Engineers of Japan and the Heat
Transfer Division of ASME, 36(6), 334-344.

Journal of Engineering Science and Technology December 2020, Vol. 15(6)

 Experimental Study for Optimum Fin Spacing of Rectangular Fin . . . . 4089

8. Güvenç, A.; and Yüncü, H. (2001). An experimental investigation on

performance of fins on a horizontal base in free convection heat transfer, Heat
and mass transfer, 37(4-5), 409-416.
9. Leung, C.; and Probert, S. (1997). Heat-exchanger performance: Influence of
gap width between consecutive vertical rectangular fin-arrays, Applied energy,
56(1), 1-8.
10. Fitzroy, N.D. (1971). Optimum spacing of fins cooled by free convection,
Journal of Heat Transfer, 93(4), 462-463.
11. Saikhedkar, N.; and Sukhatme, S. (Year). Heat transfer from rectangular cross-
sectioned vertical fin arrays. in Proceedings of the sixth national heat and
mass transfer conference, HMT, 1981, pp. 9-81.
12. Yüncü, H.; and Anbar, G. (1998). An experimental investigation on
performance of rectangular fins on a horizontal base in free convection heat
transfer, Heat and Mass Transfer, 33(5-6), 507-514.
13. Yazicioglu, B.; and Yuncu, H. (2009). A correlation for optimum fin spacing
of vertically-based rectangular fin arrays subjected to natural convection heat
transfer, Journal of thermal science and technology, 29(11), 99-105.
14. Kumar, G.; Sharma, K.R.; DWIVEDI, A.; Yadav, A.S.; and Patel, H. (2014).
Experimental Investigation of Natural Convection from Heated Triangular Fin
Array within a Rectangular Enclosure, International Review of Applied
Engineering Research. ISSN, 2248-9967.
15. Shehab, S.N. (2017). Experimental Study of Free-Convection from
Rectangular Fins Array on a Heated Horizontal Plate with Notch Effects, Al-
Nahrain Journal for Engineering Sciences, 20(1), 140-148.
16. Karami, M.; Yaghoobi, M.; and Keyhani, A. (2018). Experimental study of
natural convection from an array of square fins, Experimental Thermal and
Fluid Science, 93(409-418.
17. Al-Jewaree, H.M. (2018). An experimentally investigate the fin thermal
performance to the different fin spaces by natural convections, Al-Kitab
Journal for Pure Sciences, 2(1).
18. Al-Jessani, A.J.; and Al-Bugharbee, H.R. (Year). An experimental
investigation of free convection heat transfer rate enhancement of rectangular
fins with circular perforations. in 2018 International Conference on Advance
of Sustainable Engineering and its Application (ICASEA), 2018, pp. 233-237.
19. Leung, C.;Probert, S.; and Shilston, M. (1985). Heat exchanger: optimal
separation for vertical rectangular fins protruding from a vertical rectangular
base, Applied Energy, 19(2), 77-85.
20. Leung, C.;Probert, S.; and Shilston, M. (1985). Heat exchanger design:
optimal uniform separation between rectangular fins protruding from a vertical
rectangular base, Applied energy, 19(4), 287-299.
21. Leung, C.;Probert, S.; and Shilston, M. (1985). Heat exchanger design:
thermal performances of rectangular fins protruding from vertical or
horizontal rectangular bases, Applied energy, 20(2), 123-140.
22. Leung, C.;Probert, S.; and Shilston, M. (1986). Heat transfer performances of
vertical rectangular fins protruding from rectangular bases: effect of fin length,
Applied Energy, 22(4), 313-318.

Journal of Engineering Science and Technology December 2020, Vol. 15(6)

4090 A. J. J. Al-Jassani

23. Leung, C.; and Probert, S. (1987). Natural-convective heat exchanger with
vertical rectangular fins and base: design criteria, Proceedings of the
Institution of Mechanical Engineers, Part C: Journal of Mechanical
Engineering Science, 201(5), 365-372.
24. Leung, C.; and Probert, S. (1989). Thermal effectiveness of short-protrusion
rectangular, heat-exchanger fins, Applied energy, 34(1), 1-8.
25. Van de Pol, D.; and Tierney, J. (1974). Free convection heat transfer from
vertical fin-arrays, IEEE Transactions on Parts, Hybrids, and Packaging,
10(4), 267-271.
26. Chaddock, J. (1970). Free convection heat transfer from vertical rectangular
fin arrays, ASHRAE journal, 12(8), 53-&.
27. Ahmed, S.E.; Hussein, A.K.; Mohammed, H.; Adegun, I.; Zhang, X.; Kolsi,
L. (2014). Viscous dissipation and radiation effects on MHD natural
convection in a square enclosure filled with a porous medium, Nuclear
Engineering and Design, 266(34-42.
28. Ahmed, S.E.; Hussein, A.K.; El-Aziz, M.A.; and Sivasankaran, S. (2016).
Conjugate natural convection in an inclined square porous enclosure with
finite wall thickness and partially heated from its left sidewall, Heat Transfer
Research, 47(4).
29. Hoseinzadeh, S.; Heyns, P.; Chamkha, A.; and Shirkhani, A. (2019). Thermal
analysis of porous fins enclosure with the comparison of analytical and
numerical methods, Journal of Thermal Analysis and Calorimetry, 1-9.
30. Hoseinzadeh, S.; Moafi, A.; Shirkhani, A.; and Chamkha, A.J. (2019).
Numerical validation heat transfer of rectangular cross-section porous fins,
Journal of Thermophysics and Heat Transfer, 1-7.
31. Jubear, A.J. (2017). An experimental investigation of heat transfer
enhancement for vertical interrupted fin in free convection, International
Journal of Recent Scientific Research, 8(6), 17297-17284.
32. Feng, S.; Li, F.; Zhang, F.; and Lu, T.J. (2018). Natural convection in metal
foam heat sinks with open slots, Experimental Thermal and Fluid Science,
91(354-362.
33. Tari, I.; and Mehrtash, M. (2013). Natural convection heat transfer from
inclined plate-fin heat sinks, International Journal of Heat and Mass Transfer,
56(1), 574-593.
34. Tari, I.; and Mehrtash, M. (2013). Natural convection heat transfer from
horizontal and slightly inclined plate-fin heat sinks, Applied Thermal
Engineering, 61(2), 728-736.
35. Mehrtash, M.; and Tari, I. (2013). A correlation for natural convection heat
transfer from inclined plate-finned heat sinks, Applied Thermal Engineering,
51(1-2), 1067-1075.
36. Zubaidi, S.L.; Dooley, J.; Alkhaddar, R.M.; Abdellatif, M.; Al-Bugharbee, H.;
and Ortega-Martorell, S. (2018). A Novel approach for predicting monthly
water demand by combining singular spectrum analysis with neural networks,
Journal of hydrology, 561(136-145.
37. Moffat, R.J. (1988). Describing the uncertainties in experimental results,
Experimental thermal and fluid science, 1(1), 3-17.

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