Document Type : Full Lenght Research Article
Authors
^{1} Department of Mechanical Engineering,Shahid Chamran University of Ahvaz,Ahvaz, Iran
^{2} Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran
Abstract
Keywords
Main Subjects
Corresponding Author: R. Bahoosh , Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran Email : bahoosh@scu.ac.ir

Effective [.1] [.2] cooling and thermal management are essential and inevitable requirements in highenergy applications, such as those associated with batteries, motors, microelectronics, and micro/mini reactors. In these devices, singlephase micro/minichannel liquid cooling is suitable for high heat flux dissipation because they have a high areatovolume ratio, a large heat transfer coefficient, and a minimal requirement for coolant fluid. Conventional micro/minichannel heat sinks are characterized by poor flow mixing, thus rendering such equipment inadequate for effective heat transfer[.3] . An innovative solution to deficient heat transfer is the generation of secondary flow [1]. During the past two decades, many researchers have investigated several techniques for enhancing heat transfer in micro/minichannel heat sink, both theoretically and experimentally[.4] . Tukerman and Pease [2], for instance, studied microchannel cooling technology for heat dissipation in electronic cooling. The authors showed that a heat flux of 790 W/cm^{2} can be eliminated through the cooling of water flow under a maximum temperature difference of 71°C between silicon substrate and inlet flow. Qu and Mudawer [3] investigated the heat transfer and pressure drop across a rectangular microchannel heat sink. The results revealed that the outlet temperature of cooling fluid decreases with increasing Reynolds number and increases along the flow direction for each Reynolds number. The authors also found that at a fixed Reynolds number, pressure drop increases with increasing Reynolds number and effective heat flux. Xie et al. [4] numerically studied the effects of geometrical parameters and fluid flow velocity on heat transfer and pressure drop in a minichannel heat sink. The results indicated that pressure drop strongly depends on minichannel geometry. The authors further reported that a narrow channel enables more efficient heat transfer than does a wide channel. Laminar water flow in wavy microchannels with rectangular crosssections was numerically investigated by Sui et al. [5], who found that secondary flow is generated by water flow in wavy microchannels. The authors concluded that in a fixed crosssection of the microchannels, heat transfer exhibits an improvement over the levels observed in straight microchannels. Mohammed et al. [6] numerically analyzed the heat transfer and pressure drop in a microchannel heat sink with differently shaped channels. They found that in a fixed crosssection of a microchannel, zigzagshaped microchannels present the highest convective heat transfer coefficient and a maximum pressure drop. In another study [7], the authors numerically inquired into heat transfer and fluid flow in a wavy microchannel heat sink. They discovered that a decrease in the wave amplitude of the channel increases temperature and that the pressure drop in wavy microchannels is greater than that in straight microchannels. Lei et al. [8] also conducted a numerical investigation, with particular focus on heat transfer and fluid flow in a planar heat sink with sectional oblique fins. The results indicated that in designed oblique fins, the generation of secondary flow and the reduction of boundary layer thickness enhance heat transfer. Another numerical research was that conducted by Chai et al. [9], who probed into the heat transfer characteristics of fanshaped ribs on the side wall of microchannel heat sinks under laminar flow. They found that the ribs impede increases in wall temperature and that the local heat transfer coefficient increases with decreasing fanshaped spacing and increasing height. Xie et al.’s [10] study was devoted to heat transfer and fluid flow characteristics in microchannel heat sinks with rectangular flow obstructions. The researchers determined that heat transfer in the aforementioned heat sinks is greater than that in microchannels without flow obstructions. The authors added that the length of flow obstructions significantly affects the thermal performance of microchannel heat sinks. In numerically analyzing the heat transfer and fluid flow of Al_{2}O_{3}ethylen glycolwater nanofluid in wavy microchannels, Zhu et al. [11] found that such channels feature efficient heat transfer performance but that the pressure drop is higher than that occurring in plain microchannels. They also discovered that at low volume concentrations of nanofluid, thermal performance is superior to that achieved at high volume concentrations. Bovand et al. [.5] [12] numerically explored the effects of several parameters on heat transfer and flow structure in a porous solar heater. The authors reported that the Nusselt number increases with the incorporation of porous substrate into the solar heater.
Numerical analysis can also be directed toward entropy generation as a powerful approach to investigating system performance. In this regard, Bashi et al. [13] employed a numerical approach to explore entropy generation in a porous triangular duct. The researchers found that total entropy generation increases with the incorporation of porous materials into the duct and that frictional entropy generation dominates in the porous channel. Rashidi et al. [14] numerically examined the thermofluidic performance and entropy generation of new eccentric helical screw tape inserts in circular tubes under a turbulent regime. They reported that the performance factor improves with increasing eccentricity values at all ranges of the Reynolds number. They likewise declared that total thermal and frictional entropy generation increases with rising eccentricity values and Reynolds numbers. Rashidi et al. [15] performed a numerical study of corrugated channels at different Reynolds numbers, wave amplitudes, and wavelengths under a turbulent regime. The specific object of the study was thermohydraulic and entropy generation in the channels. They found that overall performance improves because of the corrugated channel and that total entropy generation is at its minimum at all wave amplitude and wavelength ranges and at a Reynolds number of 20000.
A number of researchers have experimentally studied heat transfer and fluid flow in planar micro/minichannel heat sinks under laminar fluid flow. Examples are Ho et al. [16], Lee et al. [17], Rimbault et al. [18], Zirakzadeh et al. [19], Lee et al. [20], Peyghambarzadeh et al. [21], and KhoshvaghtAliabadi and Nozan [22]. Most focused on planar micro/minichannel heat sinks, and fewer efforts have been devoted to the cooling of cylindrical heat sources, such as batteries, motors, and micro/mini reactors. Fan et al. [23, 24] experimentally and numerically delved into the heat transfer and fluid flow of water in a novel cylindrical minichannel heat sink. The minichannel heat sink, which also has cylindrical oblique fins, increases the average Nusselt number by 75.6% and decreases thermal resistance by 59.1%. [.6] Azizi et al. [25, 26] experimentally investigated the heat transfer and fluid flow of Cuwater nanofluid in a cylindrical microchannel heat sink. The maximum enhancement in the average Nusselt number is 23% at a 0.3% mass fraction of nanoparticles. The authors also reported that the entrance Nusselt number improves by 43% but that the friction factor increases by 45.5%.[.7]
The main aim of the present work was to numerically investigate heat transfer and fluid flow characteristics in a novel cylindrical heat sink with helical minichannels for laminar flow—issues that have not been explored in previous studies. The effects of helix angle, channel aspect ratio, and Reynolds number were investigated, and the heat transfer performance of the heat sink was examined on the basis of the performance factor and the augmentation entropy generation number.
2.1 Description of cylindrical heat sink with helical minichannels
Fig. 1 illustrates the physical model of the cylindrical heat sink and the computational domain pursued in this work. The cooling fluid and the material constituting the heat sink are water and copper, respectively. Each heat sink[.8] contains 36 minichannels with rectangular crosssections. Four helix angles of the rectangular minichannels, namely, 45°, 60°, 75°, and 90° (straight minichannel), were considered. For each helix angle, six aspect ratios (AR) (i.e., 1, 1.2, 3.6, 7.2, 10.8, and 16.2) were chosen for the same crosssectional area. All the parameters are presented in Table 1.
Table 1. Geometric parameters of minichannels at different aspect ratios (mm) = .
Case Number 
= 
H 
W 

1 
1 
1.095 
1.095 
1.095 
2 
1.2 
1.2 
1 
1.091 
3 
3.6 
2.078 
0.577 
0.904 
4 
7.2 
2.939 
0.408 
0.717 
5 
10.8 
3.6 
0.333 
0.610 
6 
16.2 
4.409 
0.272 
0.513 
To analyze threedimensional conjugate heat transfer in the cylindrical heat sink, the following assumptions were adopted: Flow is of a laminar, incompressible, and steadystate nature; natural convection, radiation, and viscous dissipation are negligible [27]; the thermophysical properties of fluids vary in a piecewise linear manner with fluid temperature [28]; and the thermophysical properties of solid are constant. In accordance with these assumptions, the governing equations for fluids and solids are as follows:
a)
b)
c)
Fig. 1. Cylindrical heat sink with helical minichannels;
(a) full domain, (b) computational domain, and (c) schematic view of the threedimensional mesh.
Continuity equation:
(1) 
Momentum equation:
(2) 
Energy equation for liquids:
(3) 
Energy equation for solids:
(4) 
The hydraulic diameter can be expressed using the following equation [29]:
(5) 
where is the aspect ratio, and is the Reynolds number, expressed thus:
(6) 
in which denotes the average velocity of fluids over the crosssectional area of the minichannels, and represents the inlet mass flow rate. The Darcy friction factor and pumping power are defined by
(7) 
(8) 
where , , and are the pressure drop in the minichannels, the length of the minichannels, and the number of minichannels, respectively.
The average Nusselt number and heat transfer coefficient are defined as
Table 2. Boundary conditions assigned for simulations.
Conditions 
Boundaries 
Uniform mass flow rate with uniform temperature 
Minichannel inlet 
Pressure outlet [31] 
Minichannel outlet 
Bottom wall of substrate heat sink 

Periodic 
Side walls of substrate heat sink 
Adiabatic 
Both end walls of substrate heat sink and top wall of minichannels and fins 
and 
Inner walls/fluid–solid interface 
(9) 
(10) 
where is the heat transfer absorbed by water, is the heat transfer area of the minichannels, is the average temperature of the minichannel wall, and is the average temperature of a fluid.
Overall thermal resistance is an important parameter for the thermal performance evaluation of micro/minichannel heat sinks. Such parameter is obtained using [4, 30–32]
(11) 
where indicates the highest temperature of the minichannel wall.
The local wall temperature of the minichannels at the location coordinate along the helix minichannel is defined as
(12) 
The thermal and hydrodynamic boundary conditions assigned to all cases are shown in Table 2.
The thermal performance factor at a fixed aspect ratio for evaluating the performance of different minichannels is defined thus [33]:
(13) 
where and denote the Nusselt number and friction factor at a helix angle of 90° (straight minichannel), respectively. When the thermal performance factor is greater than unity, heat transfer enhancement dominates over pressure drop.
Several researchers, such as Yue et al. [34], Chai et al. [35], and Ebrahimi et al. [36], evaluated the performance of microchannels by minimizing entropy generation. The rate of entropy generation is defined as [37].
(14) 
(15) 
(16) 
where and are the entropy generation rates due to heat transfer and fluid friction, respectively. The definitions of the Bejan number and the augmentation entropy generation number are provided below:
(17) 

(18) 
where is the entropy generation rate at a helix angle of 90°.
A finite volume method was implemented to discretize the governing equations (Eqs. (1)–(4)), and the commercial software Fluent 15 was used to perform simulations. The semiimplicit method for pressure linked equations [.9] algorithm and a standard scheme were applied for pressure and velocity coupling and pressure discretization, respectively. The convective term in the momentum and energy equations were discretized via the secondorder upwind method. The solution was converged at the point where the residuals of the continuity, momentum, and energy equations are smaller than , , and , respectively.
To ensure the accuracy of the results and reduce simulation time, the grid independence test was performed for all models. Four sets of uniform grids with hexahedral elements were generated in the entire computational domain for all the models. For the heat sink with a helix angle = 45°, AR = 1.2, and Re = 400, for instance, the relative errors in the maximum wall temperature and pressure drop were evaluated (Table 3). Relative error is computed using the following equation [38]:
(18) 
where represents the maximum wall temperature and pressure drop. On the basis of the percentage of relative errors in Table 3, 1.890 million grids were chosen for the computations.
Table 3. Grid independence results
Number of grid 

6.898 
639.0078 
0.0822 
310.2170 
0.602 
0.808 
680.8090 
0.0318 
310.3735 
1.188 
0.752 
681.1921 
0.0119 
310.4351 
1.890 
baseline 
686.3353 
baseline 
310.4721 
2.520 
(a)
(b)
(c)
Fig. 2. Comparison of numerical solution with the experimental data of Chai et al. [35]; (a) , (b) , and (c) percentage of error.
2.4 Validation of the numerical model
For validation, the average Nusselt number and fanning friction factor were compared with the experimental data of Chai et al. [39] on straight channels with the same boundary conditions and dimensions[.10] . Fig. 2a indicates good agreement between the results in terms of the Nusselt number. The maximum deviation of the average Nusselt number is below 8%. Fig. 2b also shows strong agreement between the findings with respect to the fanning friction factor. The maximum deviation of the fanning friction factor is below 4%. The percentages of errors in the average Nusselt number and fanning friction factor are presented in Fig. 2c.
Numerical simulations were performed to evaluate the effects of helix angle and aspect ratio on heat transfer and pressure drop in the cylindrical heat sink with helical minichannels at a Reynolds number of 100 to 900.
3.1 Effects of helix angle on heat transfer and pressure drop
Fig. 3 shows variations in the local wall temperature along the flow direction at different helix angles, , and AR = 1.2. The local wall temperature increases along the flow direction and exhibits a similar trend at all helix angles. Such temperature decreases with decreasing helix angle in all locations. The local wall temperature of the helical minichannels is lower than that of the straight minichannel (helix angle = 90°). Additionally, the difference between the maximum and minimum temperatures of the helical minichannels is lower than that between the maximum and minimum temperatures of the straight minichannel. A helix angle of 45° enables the most uniform temperature distribution. To illustrate, the maximum wall temperature of the straight minichannel is 47.1°C, whereas the maximum wall temperature at a helix angle of 45° is 35.4°C. As a result, the local wall temperature decreases by 11.7°C—a value that is equivalent to 24.8%.
Fig. 4 presents variations in the average heat transfer coefficient with Reynolds number at different helix angles and AR = 1.2. The average heat transfer coefficient increases with Reynolds number at all helix angles, especially at high Reynolds numbers. This finding is attributed to the fact that the thickness of the thermal boundary layer decreases with increasing Reynolds [.11] number. High flow velocity improves flow mixing. Moreover, the average heat transfer coefficient at 45°, 60°, and 75° is higher than that observed at the straight minichannel. The enhancement in heat transfer coefficient is due to the generation of secondary flow by the curvature of the minichannels and the increase in heat transfer area through the enlargement in minichannel length. These results confirm that helical minichannels are the best configurations for heat dissipation in cylindrical heat sinks. As a case in point, at Re = 500, the average heat transfer coefficients at 45°, 60°, and 75° are 80.4%, 56.9% and 16.1% higher than that at the straight minichannel, respectively.
Fig. 5 displays the axial velocity contours at the middle position of the minichannels (ξ/L at different helix angles and Re = 500. econdary flow occurs at 45°, 60°, and 75°. With decreasing helix angle, centrifugal force increases, and the maximum velocity approaches the outer wall (right side) and moves outward at the middle of the minichannels’ crosssections. This phenomenon stems from the enhancement in heat transfer and flow mixing caused by the increase in secondary flow. The figure also shows that secondary flow is eliminated in the straight minichannel.
Fig. 3. Effects of axial distance on local wall temperature at different helix angles, Re = 500, and AR = 1.2.
Fig. 4. Effects of Reynolds number on average heat transfer coefficient at different helix angles and AR = 1.2.
Helix angle=45 (deg)
Helix angle=60 (deg)
Helix angle=75 (deg)
Helix angle=90 (deg)
Fig. 5. Axial velocity contours at different helix angles, Re = 500, AR = 1.2, and .
Helix angle=60 (deg)
Helix angle=75 (deg)
Helix angle=90 (deg)
Helix angle=45 (deg)
Fig. 6. Temperature contours at different helix angles,
Re = 500, AR = 1.2, and .
Fig. 7. Effects of Reynolds number on pressure drop at different helix angles and AR = 1.2.
The effects of helix angle on the temperature field at the middle position of the minichannels (ξ/L are depicted in Fig. 6. The fluid temperature near the minichannel walls is at its maximum at all helix angles. At 45°, 60°, and 75°, the maximum fluid temperature decreases relative to that in the straight minichannel. At a helix angle of 45°, for example, the maximum fluid temperature is 11 K lower than that in the straight minichannel. The temperature field is nonuniform because of the formation of vortexes at 45°, 60°, and 75° and thereby improves convective heat transfer.
Fig. 7 illustrates variations in pressure drop versus Reynolds number at different helix angles and AR= 1.2. With increasing Reynolds number, pressure drop rises at 45°, 60°, and 75° compared with the level observed in the straight minichannel. With decreasing helix angle, pressure drop rises. Pressure drop increases primarily because the helix angles reduce secondary flow and increase channel length. Reaching a high Reynolds number therefore necessitates more energy. [.12]
Fig. 8 shows variations in overall thermal resistance versus Reynolds number at different helix angles and AR = 1.2. Overall thermal resistance decreases with increasing Reynolds number and decreasing helix angle. The maximum wall temperature decreases with increasing Reynolds number at each helix angle. As an illustration, at Re = 500, the overall thermal resistance levels at 45°, 60°, and 75° are 50.1%, 39.9%, and 18.5% lower than that in the straight minichannel, respectively.
Fig. 8. Effects of Reynolds number on overall thermal resistance at different helix angles and AR = 1.2.
Fig. 9. Effects of helix angle on average heat transfer coefficient at different channel aspect ratios and Re = 500.
Fig. 10. Effects of helix angle on pressure drop at different channel aspect ratios and Re = 500.
3.2 Effects of channel aspect ratio on heat transfer and pressure drop
Fig. 9 illustrates variations in the average heat transfer coefficient versus helix angle at different aspect ratios and Re = 500 in equal crosssectional areas. The average heat transfer coefficient increases with rising aspect ratio at all helix angles—a phenomenon related to the increase in the wetted perimeter. As mentioned previously, the average heat transfer coefficient decreases with rising helix angle because of the reduction in secondary flow and minichannel length. At aspect ratios below 7.2, the average heat transfer coefficient exhibits a sharp decrement with increasing helix angle. This behavior is especially pronounced at low aspect ratios. At high aspect ratios, however, the reduction in the average heat transfer coefficient with decreasing helix angle smoothens, indicating that the effects of helix angle on the average heat transfer coefficient decrease at high aspect ratios.
Fig. 10 shows variations in pressure drop versus helix angle at different aspect ratios and fixed crosssectional areas. Pressure drop increases with rising aspect ratio at all helix angles, similar to the trend reported by Wang et al. [32]. The increase in aspect ratio in unchanged crosssectional areas reduces the hydraulic diameter (Table. 1). Consequently, the water flow rate in the minichannels with small hydraulic diameters creates a high pressure drop.
Fig. 11 exhibits variations in overall thermal resistance versus helix angle at different aspect ratios and equal crosssectional areas. Given that the maximum wall temperature increases with increasing helix angle, overall thermal resistance grows with rising helix angle. At a fixed helix angle, overall thermal resistance decreases with increasing aspect ratio. On this basis, the effects of helix angle on overall thermal resistance lessen at high aspect ratios. For instance, the overall thermal resistance at a helix angle of 45° to 90° decreases by 50.1% and 17.7% at aspect ratios of 1.2 and 16.2, respectively.
3.3 Performance evaluation criteria
On the basis of the results, heat transfer enhancement is associated with pressure drop increment in helical minichannels. We therefore evaluated the overall performance of each helical minichannel heat sink. To this end, we used the thermal performance factor and the minimization of entropy generation.
Fig. 12 (a and b) illustrates variations in the Nusselt number ratio and friction coefficient ratio with Reynolds number at different helix angles. Here, and are the Nusselt number and friction factor of the straight minichannel, respectively. The helical minichannels improve heat transfer to levels higher than that observed in the straight minichannel (Fig. 12a). With an increment in Reynolds number, the Nusselt number ratio increases, especially at high Reynolds numbers. At a fixed Reynolds number (Re = 500), for example, the Nusselt number improves by 1.81, 1.57, and 1.16 times higher than that in the straight minichannel at 45°, 60°, and 75°, respectively. With decreasing helix angle, heat transfer enhancement is more prominent because the decrement in helix angle increases secondary flow, improves flow mixing, and enhances heat transfer. As indicated in Fig. 9b[.13] , an increasing Reynolds number elevates the friction coefficient ratio, and the rate of increase in this ratio at helix angles of 45° and 60° is higher than that at 75°. Because of the decrement in helix angle, the value of increases given the corresponding increase in secondary flow. The increase in at 45° and 60° is higher than that at 75°. The variations in the range at helix angles of 45°, 60°, and [.14] 75° are 1.11 to 1.6, 1.04 to 1.4 and 1 to 1.1, respectively.
Fig. 13 indicates variations in the thermal performance factor versus Reynolds number at different helix angles and AR = 1.2. The thermal performance factor is higher than unity at all helix angles. The thermal performance factor grows with increasing Reynolds number. At each Reynolds number, the thermal performance factor increases with decreasing helix angle. At a fixed Reynolds number (Re = 500), for instance, the thermal performance factors are 1.6, 1.4, and 1.3 at helix angles of 45°, 60°, and 75°, respectively. Additionally, an increase in Reynolds number from 100 to 900 enhances the thermal performance factor of the minichannels with helix angles of 45°, 60°, and 75° by 59.3%, 51.4%, and 21%, respectively.
Fig. 11. Effects of helix angle on overall thermal resistance at different channel aspect ratios and Re = 500.
(a)
(b)
Fig. 12. Effects of Reynolds number on (a) Nusselt number ratio and (b) friction coefficient ratio at different helix angles and AR = 1.2.
Fig. 13. Effects of Reynolds number on thermal performance factor at different helix angles and AR = 1.2.
Fig. 14 (a and b) shows variations in the entropy generation rate at different helix angles and AR = 1.2. With increasing Reynolds number, the thermal entropy generation rate decreases (Fig. 14a). With decreasing helix angle, such rate decreases at all Reynolds numbers, and the thermal entropy generation rate at all helix angles is lower than that observed in the straight minichannel. This finding is attributed to the increased heat transfer caused by the increment in Reynolds number and the decrement in helix angle (see Fig. 4). With increasing Reynolds number, the frictional entropy generation rate increases at all helix angles (Fig. 14b). A decrease in helix angle augments the frictional entropy generation rate; an increase in Reynolds number and a decrease in helix angle increases pressure drop. The frictional entropy generation rate is lower than the thermal entropy generation rate, with the latter dominating over the former at all helix angles and Reynolds numbers.
Fig. 14c shows the Bejan numbers at different Reynolds numbers, different helix angles, and an aspect ratio of 1.2, as a reflection of the investigation into the contribution of thermal entropy and frictional entropy generation to total entropy generation. The results showed that increasing Reynolds number and decreasing helix angle reduce the Bejan number. This finding implies that an increase in Reynolds number and a decrease in helix angle elevates the contribution of fluid friction to total entropy generation. The Bejan numbers at all helix angles are very close to unity, indicating that entropy generation is dominated by thermal entropy generation in the adopted range of Reynolds numbers and helix angles.
Fig. 15 presents variations in the augmentation entropy generation number at different helix angles and an aspect ratio of 1.2. The best performance is gained at low augmentation entropy generation numbers. Such number decreases with an increment in Reynolds number and a decrement in helix angle. The augmentation entropy generation number is also lower than unity, indicating that the heat sink with helical minichannels exhibits less irreversibility than does the heat sink with the straight minichannel.
(a)
(b)
(c)
Fig. 14. Effects of the Reynolds number entropy generation rate and Bejan number at different helix angles and AR = 1.2; (a) thermal entropy generation, (b) frictional entropy generation, and (c) Bejan number.
Fig. 15. Effects of helix angle on augmentation entropy generation at different helix angles and AR = 1.2.
(a)
(b)
Fig. 16. Effects of helix angle on (a) Nusselt number ratio and (b) friction coefficient ratio at different aspect ratios and Re = 500.
Fig. 17. Effects of helix angle on thermal performance factor at different aspect ratios and Re = 500.
Fig. 16 (a and b) illustrates variations in the Nusselt number ratio and the friction coefficient ratio with helix angle at different aspect ratios and Re = 500. As can be seen in Fig. 16a, an increase in helix angle decreases the Nusselt number ratio in the entire range of aspect ratios; however, the rate of reduction at an aspect ratio below 7.2 is higher than that at high aspect ratios. This finding means that the decrement in Nusselt number ratio at an aspect ratio of 1.2 is higher than that at other aspect ratios in all helix angles. As presented in Fig. 16b, increasing helix angle elevates the friction coefficient ratio. With an increment in aspect ratio, the friction coefficient ratio decreases.
Fig. 17 depicts variations in the thermal performance factor versus helix angle at different aspect ratios and equal crosssectional areas. The thermal performance factor declines as helix angle increases. At aspect ratios below 7.2, the thermal performance factor sharply decreases with increasing helix angle, particularly at low aspect ratios. At high aspect ratios, however, the rate of reduction in the thermal performance factor with decreasing helix angle smoothens. For example, the thermal performance factors at helix angles of 45° to 75° decrease by 28.9% and 9.8% at aspect ratios of 1.2 and 16.2, respectively.
Fig. 18 (a and b) depicts variations in the entropy generation rate versus helix angle at different aspect ratios, equal crosssectional areas, and Re = 500. With increasing helix angle, the thermal entropy generation rate increases because when helix angle increases, heat transfer decreases (Fig. 18a). With a decrement in aspect ratio, the thermal entropy generation increases because when aspect ratio decreases, heat transfer decreases (Fig. 9).
(a)
(b)
(c)
Fig. 18. Effects of the helix angle on entropy generation rate and Bejan number at different aspect ratios and Re = 500; (a) thermal entropy generation, (b) frictional entropy generation, and (c) Bejan number.
Fig. 19. Effects of helix angle on augmentation entropy generation number at different aspect ratios and Re = 500.
The thermal entropy generation rate sharply increases at low aspect ratios . With an increment in helix angle, the frictional entropy generation rate decreases, and with increasing aspect ratio, the frictional entropy generation rate increases (Fig. 18b). Fig. 18c shows variations in the Bejan number versus helix angle at different aspect ratios, equal crosssectional areas, and Re = 500. The results showed that with increasing helix angle and decreasing aspect ratio, the Bejan number increases. At a fixed helix angle and decreasing aspect ratio, the contribution of frictional entropy generation to total entropy generation decreases. Such contribution is very close to unity at low aspect ratios .
Fig. 19 graphically represents variations in the augmentation entropy generation number versus helix angle at different aspect ratios, equal crosssectional areas, and Re = 500. The augmentation entropy generation number increases with an increment in helix angle. With increasing aspect ratio, the augmentation entropy generation number increases at all helix angles. The heat sink with an aspect ratio of 1.2 exhibits a low augmentation entropy generation number and the best thermodynamic performance.
Heat transfer and laminar fluid flow in a novel cylindrical heat sink with helical minichannels were investigated numerically by a finite volume method. The effects of helix angle, aspect ratio, and Reynolds number were also examined. On the basis of the numerical results, the following conclusions were derived:
Nomenclature


Crosssectional area of minichannel, 

Convection heat transfer area, 

Minichannel aspect ratio 

Bejan number 

Specific heat transfer, J/kg K 

Heat sink diameter, 





Hydraulic diameter, 

Heat source diameter, 

Friction factor 

Minichannel height, 

Heat transfer coefficient, W/ K 

Thermal conductivity, W/m K 

Minichannel length, 

Mass flow rate, kg/s 

Nusselt number 

Pressure, 

Pumping power, 

Heat gain by water, 

Total thermal resistance, K/W 

Reynolds number 

Total entropy generation rate, /K 

Frictional entropy generation rate, /K 

Heat transfer entropy generation rate, /K 

Reynolds number 

Heat sink length, 

Temperature, K 

Velocity, m/s 

Minichannel width, 

Greek symbols 

µ 
Dynamic viscosity, Pa.s 
Density, kg/ 

Thermal performance factor 

ζ 
Location coordinate along a helix minichannel , 
Augmentation entropy generation number 

Subscripts 

Average 

Fluid 

Inlet 

Maximum 

Wall 
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