Numerical Investigation into the Thermal Performance

Heat Transfer Engineering

ISSN: 0145-7632 (Print) 1521-0537 (Online) Journal homepage: http://www.tandfonline.com/loi/uhte20

Numerical Investigation into the Thermal Performance of Single Microchannels with Varying Axial Length and Different Shapes of Micro Pin-Fin Inserts Olayinka O. Adewumi, Tunde Bello-Ochende & Josua P. Meyer To cite this article: Olayinka O. Adewumi, Tunde Bello-Ochende & Josua P. Meyer (2017) Numerical Investigation into the Thermal Performance of Single Microchannels with Varying Axial Length and Different Shapes of Micro Pin-Fin Inserts, Heat Transfer Engineering, 38:13, 1157-1170, DOI: 10.1080/01457632.2016.1239927 To link to this article: http://dx.doi.org/10.1080/01457632.2016.1239927

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Date: 23 March 2017, At: 07:34

HEAT TRANSFER ENGINEERING , VOL. , NO. , – http://dx.doi.org/./..

Numerical Investigation into the Thermal Performance of Single Microchannels with Varying Axial Length and Different Shapes of Micro Pin-Fin Inserts Olayinka O. Adewumia , Tunde Bello-Ochendeb , and Josua P. Meyer

a

a

Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, Hatfield, South Africa; b Department of Mechanical Engineering, University of Cape Town, Cape Town, Rondebosch, South Africa

ABSTRACT

This study investigates numerically the thermal performance of combined microchannel heat sink with micro pin-fins with different cross-sectional shapes. The objective of this study is to investigate the best geometric configuration that maximizes the heat transfer from the heated base when the combined heat sink is subjected to a steady, laminar, incompressible convective fluid flow and heat transfer. The axial length of the solid substrate and microchannel is varied from 1 to 10 mm with fixed total volume of 0.9 mm3 while the number of rows of the different shapes of micro pin-fins was varied between three and seven. It was observed that best performance is obtained with a sixth row of circular-shaped micro pin-fins for the optimized combination of the microchannel and micro pin-fin heat sink. Results of the optimal axial length for fixed pressure drop range are also presented.

Introduction Numerical, analytical, and experimental investigations into microchannel and micro pin-fin heat sinks used to cool microelectronic devices have been carried out extensively over the years. Some of these investigations carried out involved the geometric optimization of the heat sinks to obtain optimal dimensions that give the best thermal performance depending on the operating conditions considered. Examples of such investigations on microchannels are numerical studies carried out by Ryu et al. [1], Muller and Frechette [2], and Li and Peterson [3, 4], analytical studies carried out by Knight et al. [5], Murakami and Mikic [6], and Upadhye and Kandlikar [7], and experimental investigations conducted by Naphon and Khonseur [8] and Zhang et al. [9]. Some investigations carried out on micro pin-fins were by Kim et al. [10], Khan et al. [11, 12], Yang and Peng [13], Sahiti et al. [14] and Galvis et al. [15]. Recently, combining microchannels and circularshaped micro pin-fins into a single heat sink design have been found to improve thermal performance of the heat sink when compared with the single microchannel [16]. The results from this previous study showed that combining microchannels with micro pin-fins embedded in a solid substrate with uniform surface heat flux on the heated base further improves the thermal performance

of the heat sink when compared to single microchannels without the micro pin-fin inserts. Only one shape of micro pin-fin inserts (circular-shape) was considered in our previous study. The constructal law technique, which involves the search for an optimal configuration subject to global constraints [17, 18], will also be used in this study for the geometric optimization of the combined heat sink. This technique has been used by many researchers to determine best geometric configuration of heat sinks that maximize heat transfer from heated surfaces or volumes to the cooling fluid. Muzychka [19] carried out an analytical study using constructal design approach to determine the optimal passage size to length ratio in terms of Bejan number for different shapes of cooling channels that maximizes heat transfer rate per unit volume. A simple optimal duct shape was developed in his study. In another investigation carried out by Muzychka [20], a constructal multi-scale design approach was presented which allowed for maximum heat dissipation in systems using circular microtubes. Results obtained from his study showed that the use of multi-scale design techniques gives greater performance of heat sink core structures when compared with conventional design approaches. Bello-Ochende et al. [21] also used the constructal approach to determine numerically the best possible

CONTACT Professor Tunde Bello-Ochende [email protected] Department of Mechanical Engineering, University of Cape Town, Cape Town Private Bag X, Rondebosch , South Africa. Color versions of one or more of the figures in this paper can be found online at www.tandfonline.com/uhte. ©  Taylor & Francis Group, LLC

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O. O. ADEWUMI ET AL.

geometry for a microchannel heat sink with fixed axial length and volume which minimizes peak temperature when the pressure difference across the volume is fixed. Their results showed that optimal microchannel shape and minimized peak temperature were functions of the applied pressure difference and solid volume fraction. Some other studies that were carried out to investigate fluid flow and heat transfer in microchannel heat sinks using constructal approach were by Bello-Ochende et al. [22–24], Adewumi et al. [16], Olakoyejo et al. [25], Salimpour et al. [26], and Xie et al. [27] to mention a few. In the search for the optimal configuration of the microchannel carried out in these studies outlined above, both the volume and axial length of the solid were both fixed except for the investigation carried out by BelloOchende et al. [22] and Adewumi et al. [16] where results of optimized microchannel geometry were presented for relaxed axial length. In their studies, the variation of the surface heat flux q on the heated wall with changes in axial length was not considered because a constant surface heat flux of 100 W/cm2 was assumed. Microelectronic chips release heat load Q on the surface of the solid substrate on which they are mounted and the surface area of this solid substrate is used to calculate the surface heat flux q . This means that if the axial length and width of the solid substrate is free to morph, the surface heat flux q will vary. This variation in surface heat flux is taken into consideration in this present study with the objective of searching for the optimal geometric configuration of both the solid substrate and microchannel heat sink that minimizes the peak temperature on the heated surface when the total volume is fixed but axial length is varied and thereafter insert different shapes of micro pin-fins into the optimized microchannel to investigate if the peak temperature is further minimized. This numerical investigation is carried out using a computational fluid dynamics code with a goal-driven optimization tool.

Figure . Physical model of a microchannel heat sink.

Computational model In real applications, many microchannels are arranged in a solid substrate for effective cooling of the substrate. However, after applying symmetry conditions on the physical model, an elemental volume is taken as the computational domain. Using an elemental volume as the computational model is the constructal approach because we are starting from a basic construction unit [21]. Figure 1 shows a physical model of the microchannel heat sink while Figure 2 shows the computational domain of the combined heat sink with a circular-shaped micro pin-fin. The length N, height M and width W of the solid was allowed to morph while the total volume of

Figure . Computational domain of the combined microchannel and circular-shaped micro pin-fins. (a) Two-dimensional and (b) Three-dimensional.

HEAT TRANSFER ENGINEERING

microchannel V was fixed. The geometric dimensions t1 , t2 , t3 , Hc and Wc were also allowed to vary but subject to manufacturing constraints. The solid volume fraction or porosity φ is defined as the ratio of the solid volume to the total volume of the heat sink, which is only dependent on the cross-sectional area of the heat sink as shown in Eq. (1). φ=

Asolid N Asolid MW − HcWc Vsolid = = = V A A MW

(1)

The fixed global volume is defined as

The aspect ratio AR of the solid substrate is The hydraulic diameter Dh is defined as 2Hc (ARc + 1)

the microchannel and the circular-shaped micro pinHc fins [22] are W ≤ 20, t2 ≥ 50µm, M − t3 ≥ 50µm, 0.5 ≤ c Hf Df

≤ 4.0 and S1−n ≥ 50µm. In this study, the manufacH

turing constraint ( D ff ) is going to be applied to the aspect H

H

ratio ( L ff ) for the square- and ( 2Lff ) for the hexagonalshaped micro pin-fins, where Lf is the length of the each side of the square and hexagonal surface. A total fin volume constraint is also applied to the different shapes of micro pin-fins as outlined in the equations below: V f (circular) = V f (square) = V f (hexagonal)

V = MW N = const

(2) M W

(3)

Hc . Where ARc is the channel aspect ratio defined as W c Figure 3 shows a two-dimensional view and the geometric dimensions of the different shapes of the micro pin-fins inserted into the microchannel investigated in this study. The manufacturing constraints for

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V f (square ) = V f 1 + V f 2 + ......... + V f n = cons tan t

(4) (5) (6)

V f (hexagonal ) = V f 1 + V f 2 + ......... + V f n = cons tan t

(7)

V f (circular) = V f 1 + V f 2 + ......... + V f n = cons tan t

The total heat load on the heated bottom wall Q is 100 W where Q is defined as Q = c.q N.W

(8)

where N.W is the elemental surface area that is heated and the number of microchannels c is assumed to be 100.

Figure . Two-dimensional view of the microchannel with the different shapes of micro pin-fin inserts.

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where the peak wall temperature difference T is

Basic equations and boundary conditions The present study assumes that the flow is steady and laminar. The cooling fluid, which is water, is also assumed to be incompressible with homogeneous and temperaturedependent thermophysical properties and it is driven through the microchannel by a fixed pressure drop P. The thermophysical properties of water were defined using the piecewise-linear function of temperature as shown in Eqn. (9) below [28]. Radiation and natural convection are assumed to be negligible. η (T ) = ηa +

ηa+1 − ηa (T − Ta ) Ta+1 − Ta

(9)

where 1 ࣘ a ࣘ L, L is the number of segments and η is any fluid property (this includes the thermal conductivity kf , density ρ f , thermal capacity Cpf and dynamic viscosity μf ). Heat transfer within the geometry is a conjugate problem that combines conduction and convective heat transfer. The temperature distribution within the geometry used in this study was determined by solving the conservation of mass, momentum, and energy Eqns. (10–13) numerically. The governing equations solved after applying the above assumptions are, ∇.v = 0

(10)

ρ (v .∇v ) = −∇P + μ∇ v 2

ρ f Cp f (v .∇T ) = k f ∇ T 2

(11) (12)

The energy equation for the solid regions can be written as: ks ∇ 2 T = 0

(13)

The heat flux between the interface of the fluid and the solid walls is coupled and its continuity between the interface of the solid and the liquid is given as: ∂T ∂T ks = k (14) f ∂n wall ∂n wall The inlet boundary conditions are specified as P = Pin , v = w = 0, T = Tin . At the outlet, the boundary condition is specified as, Pout = Patm . Symmetry boundary conditions are specified at the left and right side of the domain, i.e. the normal gradient of all the variables is = 0) and a no-slip boundary condition is speczero ( ∂T ∂x ified at the walls. A uniform heat load is applied on the bottom wall defined as Q = N.W.ks ∂T . The measure of ∂y thermal performance is the thermal conductance which is expressed as C=

Q T

(15)

T = Tmax − Tin where Tmax is the maximum temperature on the heated base. Numerical and optimization procedure The continuity, momentum, and energy Eqns. (10–13) along with the specified boundary conditions were solved numerically using a three-dimensional computational fluid dynamic code [28], which employs a finite volume method. The detail of this method is explained by Pantakar [29]. The computational domain was meshed using hexagonal/wedge, tetrahedron, and pyramid elements. A second-order upwind scheme was used to discretize the combined convection and diffusion terms in the momentum and energy equations and then the SIMPLE algorithm was employed to solve the coupled pressure-velocity fields in the equations. The solution is assumed to converge when the normalized residuals of the continuity and momentum equation fall below 10−5 while that of the energy equation falls below 10−7 . Several grid refinements tests were carried out for the different heat sink designs to ensure accurate numerical results. The convergence criterion for the peak temperature as the quantity monitored is, (T )i − (T )i−1 ≤ 0.01 (16) γ = (T )i where i is the mesh iteration index. The mesh was more refined [30] as i increases and i−1 mesh was selected as the appropriate mesh when Eqn. (16) was satisfied. A grid refinement test was carried out for the combined microchannel with the different shapes of pin-fins. Tables 1 and 2 show the dimensions of the geometry used for the grid refinement test while Tables 3 and 4 show the grid refinement test results for the combined design of the microchannel with square- and hexagonal-shaped micro pin-fins, respectively. The validation of code used Table . Dimensions of the microchannel and square-shaped micro pin-fin heat sink for grid refinement test. Hc (mm)

Wc (mm)

t (mm)

M−t (mm)

Hf to Hf (mm)

Lf to Lf (mm)

N (mm)

.

.

.

.

.

.

.

Table . Dimensions of the microchannel and hexagonal-shaped micro pin-fin heat sink for grid refinement test. Hc (mm)

Wc (mm)

t (mm)

M-t (mm)

Hf to Hf (mm)

Lf to Lf (mm)

N (mm)

.

.

.

.

.

.

.


Table . Grid refinement test results for combined heat sink with square-shaped pin-fins (P =  kPa). Number of cells

T (K )

     

. . .

|

(T )i −(T )i−1 | (T )i

— . .

was done in our previous study by comparing numerical results with analytical results found in open literature [16]. The computational fluid dynamics code used in this study [28] has a goal-driven optimization (GDO) tool which is used to carry out the geometric optimization of the microchannel and combined heat sink investigated in this study. The main objective of the GDO is to identify the relationship between the performance of a product and the design variables. It uses the response surface methodology (RSM) for its optimization process. The RSM uses a group of mathematical and statistical techniques to develop an adequate functional relationship between a response of interest and a number of input variables, which has been thoroughly explained in literature [31, 32]. Once a model is created and parameters defined, a response surface is created. Based on the number of input parameters, a given number of solutions (design points) are required to build this response surface (an approximation of the response of the system). After inserting a response surface, the design space is defined by giving the minimum and maximum values to be considered for each input variable. The design of experiments (DOE) part of the response surface system creates the design space sampling and when updated, a response surface is created for each output parameter. The GDO is an optimization technique that finds design candidates from the response surfaces. The objective of the numerical study is set in the GDO and then the optimization problem is solved. The accuracy of the response surface for the design candidates is checked by converting it to a design point and thereafter a full simulation is carried out for that point to check the validity of the output parameters [28].

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Numerical simulations and optimization were carried out for a fixed total volume V of 0.9 mm3 with varied axial length N of 1 to10 mm. The temperature of water pumped across the micro-channel was 20°C at the inlet and total heat load applied to the bottom wall was 100 W. The design space for the response surface for the fixed total volume was defined as 50 ࣘ Wc ࣘ 80 µm, 50 ࣘ M − t3 ࣘ 60 µm, and 500 < Hc < 3000 µm. The optimized design point chosen was required to meet the manufacturing constraints for pressure drops of 10 to 60 kPa. In the GDO, the objective function was to minimize the peak temperature based on the design space. For the micro pin-fins, the spacing between the pins (S1 to Sn ) was fixed as 50 µm and they were positioned centrally in the microchannel.

Results and discussion In our previous study [16], the thermophysical properties of water, which is the cooling fluid, were assumed constant. In this present study, temperature-dependent properties of water were considered and results of minimized peak temperature (Tmax )min were compared to those of the previous study. Figure 4 shows that at the lowest pressure drop of 10 kPa used in this study, the result of minimized peak temperature (Tmax )min when constant fluid properties assumptions was made was 5% more than when temperature-dependent properties of fluid was used for the numerical simulation. This variation reduces as the pressure drop is increased and for P = 60 kPa, the difference between the two results is less than 1%. It can be concluded from the results shown

Table . Grid refinement test results for combined heat sink with hexagonal-shaped pin-fins (P =  kPa). Number of cells

T (K )

     

. . .

|

(T )i −(T )i−1 | (T )i

— . .

Figure . Effect of pressure drop on minimized peak temperature.

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Figure . Effect of varying axial length on surface heat flux.

in Figure 4 that it is more accurate to take into consideration temperature-dependent fluid properties for water at low pressure drops. Effect of varying axial length on the thermal performance of the single microchannel Figure 5 shows how varying the axial length of the solid substrate affects the surface heat flux q on the heated wall for the constant heat load Q of 100 W and fixed total volume V of 0.9 mm3 . It was observed that as the length is increased from 1 to 10 mm, q reduces. The results also shows decreasing surface heat flux as the solid substrate aspect ratio AR was decreased for all the axial lengths investigated. A small aspect ratio results in a large width and short height while the reverse is the case for a large aspect ratio. The effect of varying the axial length of the solid substrate on the minimized peak temperature (Tmax )min is shown in Figure 6. The results shows that at fixed AR of 3, the optimal axial length Nopt was 3 mm when P = 10 kPa, but as P increased, Nopt was discovered to be 5 mm. Increasing the axial length above 5 mm did not further improve the thermal performance of the microchannel rather the minimized peak temperature increased. Figure 7 shows results of the effect of varying AR on the minimized peak temperature at a fixed pressure drop P = 20 kPa. It was observed that the optimal AR was 3 for axial length ࣘ 5 mm but became greater than 3 as N increased. Observations made from Figures 6 and 7 showed that global ARopt that minimizes the peak temperature was 3. Varying the axial length of the solid substrate and microchannel heat sink affects the temperature

Figure . Effect of varying axial length on minimized peak temperature for AR = .

distribution within the solid substrate as observed in Figure 8. When the axial length N was 1 mm with solid substrate aspect ratio AR of 3 and P of 20 kPa, there was poor heat transfer from the heated wall to the working fluid. The peak temperature for this configuration was 31.28°C as shown in Figure 8a. As the axial length was further increased to 3 mm while keeping AR fixed as three, there was an improvement in heat transfer from the heated wall to the working fluid giving 10% reduction in peak temperature as observed from the temperature contours shown in Figure 8b. Further increase in the axial length to 5 mm as shown in Figure 8c makes the

Figure . Effect of varying solid substrate aspect ratio AR on minimized peak temperature at fixed pressure drop.


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Figure . Effect of varying axial length on the temperature distribution within the solid substrate with AR =  (a)  mm (b)  mm (c)  mm (d)  mm.

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Figure . Effect of varying axial length on (a) volume flow rate and minimized peak temperature (b) volume flow rate and channel aspect ratio (AR = , P =  kPa).

geometric configuration more slender and does not improve the thermal conductance significantly. The peak temperature was 28.12°C which was just 0.2% reduction in peak temperature when compared to the 3 mm axial length geometry. As the geometry becomes more slender (with increased axial length and reduced solid substrate width), there is an increase in the peak temperature. This is observed in Figure 8d for the axial length of 7 mm with peak temperature of 29.53°C which is an increase of about 5% when compared to the axial length of 5 mm. These changes in peak temperature as the length varies are as a result of changes in the volume flow rates and channel aspect ratios as shown in Figure 9. Figure 9 shows how varying the axial length of the solid substrate affects the volume flow rate of the fluid (uin .Ac ), peak temperature and the channel aspect ratio for

AR = 3 and P = 20 kPa. In Figure 9a, the volume flow rate decreases with increase in the axial length and as the volume flow rate decreases, the peak temperature decreases but begins to increase when the axial length is greater than 5 mm. Figure 9b shows that the decreasing volume flow rate as the axial length increases is due to decreasing channel aspect ratio. The geometric configuration with axial length of 10 mm has the highest peak temperature, lowest volume flow rate and lowest channel aspect ratio. The results in Figures 8 and 9 shows that as length is increased there is improvement in heat transfer until a point where an optimum is reached and thereafter there is poor heat transfer from the heated base to the working fluid due to further decrease in flow rates and channel aspect ratio. This confirms that geometry configuration is very critical in achieving maximum thermal performance in heat sinks. When the aspect ratio of the channel is very large or very small, heat transfer from the heated base to the fluid is very poor. The results of optimal channel aspect ratio, solid volume fraction, channel hydraulic diameter, solid substrate aspect ratio and axial length for different pressure drops considered are shown in Figure 10. For the lowest P of 10 kPa, (ARc )opt ≈ 11, φopt ≈ 0.8 and Nopt = 3mm but when P was increased from 20 to 60 kPa, the optimal channel aspect ratio and solid volume fraction were approximately 8 and 0.7, respectively, while Nopt = 5mm. The optimal channel hydraulic diameter (Dh )opt ≈ 140µm and the solid substrate aspect ratio (AR)opt = 3 for all pressure drops considered in this study. The results presented in Figure 10 shows that at higher pressure drops (above 40 kPa), the optimal microchannel and solid substrate dimensions do not vary with changes in the pressure drop. The results of maximized thermal conductance and corresponding surface heat flux for the optimal single microchannel and solid substrate dimensions are shown in Figure 11. As the pressure drop was increased, the maximized thermal conductance Cmax increased from 10.31 to 16.95 W/°C, while the surface heat flux q was discovered to decrease from 105.41 W/cm2 when P = 10 kPa to a constant value of 81.65 W/cm2 as the P was increased from 20 to 60 kPa. The change in the surface heat flux is as a result of the surface area of the heated wall for the optimal configuration as shown in Figure 10a. The thermal performance of the combined heat sink made up of single microchannel with the different shapes of micro pin-fin inserts are discussed below.

Combined design (CD) with circular-, square-, and hexagonal-shaped micro pin fins Three to seven rows of circular-shaped micro pin-fins were inserted into the optimized single microchannel to


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Figure . Optimal microchannel and solid substrate dimensions.

investigate whether the thermal performance of the heat sink will be further improved. The results of minimized peak temperature for fixed total volume V of 0.9 mm3 H were reported. The aspect ratios ( D ff ) of the circularshaped micro pin-fins which met the manufacturing constraints were 0.53, 0.61, 0.69, 0.75, and 0.81 for three to seven rows of micro pin-fins, respectively. Results showed

that combining the single microchannel with circularshaped micro pin-fins reduced the minimized peak temperature further by 2 to 3% for pressure drops of between 10 to 60 kPa as shown in Figure 12a. This reduction in minimized peak temperature gave a corresponding increase in the maximized thermal conductance of about 8 to 10%. The optimal number of rows of micro pin-fins

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Figure . Effect of pressure drop on (a) maximized thermal conductance and (b) surface heat flux.

for the combined microchannel heat sink with circularshaped pin-fin inserts that minimized the peak temperature was discovered to be six. Figure 12b shows the results of maximized thermal conductance for the combined microchannel and three to seven rows of micro pinfins when pressure drop is 60 kPa. The combined heat sink with six rows of micro pin-fins had the best thermal performance. A similar trend was also observed when square- and hexagonal-shaped micro pin-fins were inserted in the optimized single microchannel. For the single microchannel with three to six rows of square-shaped micro pin-fin

Figure . Effect of number of rows on the minimized peak temperature for single microchannels with micro pin-fin inserts (a) square-shaped and (b) hexagonal-shaped. H

inserts, the aspect ratios ( L ff ) of the square-shaped micro pin-fins which met the manufacturing constraints were 0.60, 0.69, 0.77, and 0.86 for three to six rows of micro pin-fins, respectively. The thermal performance of the combined design with square-shaped micro pin-fins was also better than the single microchannel with a reduction of 2 to 3% in minimized peak temperature which gave a corresponding 7 to 10% increase in maximized thermal conductance for all the pressure drops considered in this study. The optimal number of rows for the square-shaped micro pin-fins was discovered to be four as shown in Figure 13a. Four to six rows of micro pin-fins were used for the combined single microchannel and hexagonal-shaped


Figure . Comparison between minimized peak temperature for single microchannel and combined heat sink at different pressure drops.

micro pin-fins because the manufacturing constraint for H ( 2Lff ) was not met when the number of rows was less H

than four. The aspect ratios ( 2Lff ) of the hexagonal-shaped micro pin-fins which met the manufacturing constraints were 0.558, 0.624, and 0.683 for four to six rows of micro pin-fins, respectively. The thermal performance of the combined design with hexagonal-shaped micro pin-fins was also better than the single microchannel with further decrease in minimized peak temperature of 2 to 3% which also resulted in an 8 to 9% reduction in the minimized thermal resistance. The optimal number of rows of hexagonal micro pin-fins was also four for all pressure

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Figure . Comparison between the thermal performance of the combined microchannel and different shapes of micro pin-fins (a) minimized peak temperature and (b) maximized thermal conductance.

drops considered in this study as shown in Figure 13b. The optimized micro pin-fin size for all the different shapes gave a total fin volume Vf of 0.00023 mm3 . We can conclude that an addition of 0.025% of micro pin-fin volume gave an increase in thermal performance of almost 10%.

Comparison between the combined heat sink with circular-, square-, and hexagonal-shaped pin-fins Figure 14 shows results of the comparison between the thermal performances of the optimized combined heat

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Figure . Effect of the number of rows of micro pin-fins on the thermal performance of the combined heat sink.

sink design with the different shapes of micro pin-fins. When the optimal designs were compared, the combined heat sink with six rows of circular-shaped micro pinfins had the best thermal performance but with an average decrease of 0.05% in minimized peak temperature and average increase of 0.2% in maximized thermal conductance when compared with the combined heat sink with square- and hexagonal-shaped micro pin-fins. From these results, we can conclude that considering the optimal number of rows, the shape of the micro pin-fin does not have a significant effect in improving the thermal performance of the combined heat sink design. Figure 15 shows the effect of the number of rows of micro pin-fins on the thermal performance of the combined microchannel and different shapes of micro pin-fin inserts for a fixed pressure drop. It was observed from the graph that the effect of the different shapes of micro pinfins inserted into the microchannel had different trends. For the circular-shaped micro pin-fins, as the number of rows increased, the thermal conductance increased but beyond six rows the thermal conductance began to reduce. The square-shaped pin-fins showed an increase in thermal conductance up till when the number of rows was four and thereafter the thermal conductance began to decrease. Increasing the number of rows of the hexagonalshaped micro pin-fins above four worsened the thermal conductance of the combined heat sink. The global optimal nopt was clearly shown to be six rows of circularshaped micro pin-fins.

Conclusions Numerical investigations were carried out on forced convection heat transfer and fluid flow in a single

microchannel heat sink inserted with circular-, square and hexagonal-shaped micro pin-fins. The solid substrate with a constant heat load of 100 W on its heated surface and the single microchannel with varying axial length of 1 to 10 mm were first optimized and thereafter inserted with different shapes of micro pin-fins. The thermal performance of the combined heat sink design were compared and the observations made from the numerical results obtained are as follows. The single microchannel with axial length of 3 mm had the best thermal performance for pressure drop of 10 kPa but when the pressure drop was increased to 20 to 60 kPa, the single microchannel with axial length of 5 mm had the best performance. It was also observed that varying the axial length of the solid substrate and the single microchannel had an effect on the surface heat flux on the heated wall. The combined heat sink design had better thermal performance than the optimized single microchannel without the pin-fins for all the different shapes of pin-fin inserts considered in this study but it was also observed that the optimized single microchannel with six rows of circular-shaped micro pin-fin inserts had the best thermal performance.

Nomenclature A Ac AR ARc c C Cp CD Dh Df GDO Hc Hf ks kf Lf M n N P Q q S1−n T t1 t2

Area (m2 ) Cross-sectional area of microchannel (m2 ) Aspect ratio of solid substrate Aspect ratio of microchannel Number of microchannels Thermal Conductance (W/°C) Heat capacity (J/K) Combined design Channel hydraulic diameter (m) Circular pin-fin diameter (m) Goal Driven Optimization Channel height (m) Pin-fin height (m) Thermal conductivity of solid wall (W/mK) Thermal conductivity of fluid (W/mK) Length of sides ( square and hexagonal pin-fins) (m) Height of computational volume (m) Number of rows of micro pin-fins Length (m) Pressure (Pa) Heat load (W) Heat flux (W/m2 ) Micro pin-fin spacing (m) Temperature (°C) Half-width thickness of vertical solid (m) Channel base thickness (m)


t3 u,v, w V Vf W Wc Wt x,y,z

Channel base to height distance (m) Velocities in the x-, y- and z-directions (m/s) Computational domain volume (m3 ) Volume of micro pin-fin (m3 ) Computational domain width (m) Channel width (m) Total width of solid substrate (m) Cartesian coordinates

Greek symbols ρ μ φ γ

Density (kg/m3 ) Dynamic viscosity (kg/ms) Volume fraction Convergence criterion

Subscripts a atm f in max min opt out 1−n

Piecewise points Atmosphere Fluid inlet Maximum Minimum Optimum Outlet Number of rows of pin-fins

Acknowledgments The funding obtained from DST/the National Research Foundation (NRF), the Tertiary Education Support Programme (TESP), University of Pretoria, the South African National Energy Research Institute (SANERI)/South African National Energy Development Institute (SANEDI), the Council of Scientific and Industrial Research (CSIR), the Energy-efficiency and Demand-side Management (EEDSM) Hub and NAC is acknowledged and duly appreciated.

Notes on contributors Olayinka O. Adewumi is a PhD student at the University of Pretoria, Department of Mechanical and Aeronautical Engineering, South Africa. She obtained her B.Eng. degree in Mechanical Engineering at the Ekiti State University, Ado-Ekiti, Nigeria, in 1999 and her M.Sc. degree in Mechanical Engineering at the University of Lagos, Akoka, Nigeria, in 2002. Her current research work is based on constructal design and numerical optimization of combined singleand multi-layered microchannels with micro pin-fin inserts

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under the supervision of Prof. Tunde Bello-Ochende and Prof. Josua P. Meyer. Tunde Bello-Ochende is currently a Professor at the Department of Mechanical Engineering, University of Cape Town. His research focuses on the theoretical, experimental, and numerical optimization of heat transfer devices from macro to micro scales. These include compact heat exchanger, microchannel heat sink, and heat transfer augmentation using micro-fins. His recent work are in the areas of sustainable/renewable energy (solar, geothermal, and fuel cells). He is the author of over 110 papers in referred journals and proceedings. He is a member of America Society of Mechanical Engineer (MASME). He received his Bachelor’s, Master’s degree from the University of Ilorin (1995, 1999) and doctorate from Duke University (December 2004), all in Mechanical Engineering. Josua P. Meyer obtained his BEng (cum laude) in 1984, MEng (cum laude) in 1986, and his PhD in 1988, all in mechanical engineering from the University of Pretoria and is registered as a professional engineer. After his military service (1988–1989), he accepted a position as Associate Professor in the Department of Mechanical Engineering at the Potchefstroom University in 1990. He was Acting Head and Professor in Mechanical Engineering before accepting a position as Professor in the Department of Mechanical and Manufacturing Engineering at the Rand Afrikaans University in 1994. He was Chairman of Mechanical Engineering from 1999 until the end of June 2002, after which he was appointed Professor and Head of the Department of Mechanical and Aeronautical Engineering at the University of Pretoria from 1 July 2002. At present, he is the Chair of the School of Engineering. He specializes in heat transfer, fluid mechanics and thermodynamic aspects of heating, ventilation, and air-conditioning. He is the author and co-author of more than 450 articles, conference papers and patents and has received various prestigious awards for his research. He is also a fellow or member of various professional institutes and societies such as the South African Institute for Mechanical Engineers, South African Institute for Refrigeration and Air-Conditioning, American Society for Mechanical Engineers, American Society for Air-Conditioning, Refrigeration and Air-Conditioning, and is regularly invited as a keynote speaker at local and international conferences. He has also received various teaching and exceptional achiever awards. He is an associate editor of Heat Transfer Engineering and Editor of the Journal of Porous Media.

ORCID Josua P. Meyer

http://orcid.org/0000-0002-3675-5494

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