Convective heat transfer from an inclined isothermal fin array: A computational study

Journal Pre-proofs Convective heat transfer from an inclined isothermal fin array: A computational study Krishna Roy, Biplab Das PII: DOI: Reference:

S2451-9049(20)30009-3 https://doi.org/10.1016/j.tsep.2020.100487 TSEP 100487

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Thermal Science and Engineering Progress

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15 July 2019 15 January 2020 23 January 2020

Please cite this article as: K. Roy, B. Das, Convective heat transfer from an inclined isothermal fin array: A computational study, Thermal Science and Engineering Progress (2020), doi: https://doi.org/10.1016/j.tsep. 2020.100487

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Convective heat transfer from an inclined isothermal fin array: A computational study Krishna Roy1, Biplab Das*1,2 1

Department of Mechanical Engineering, National Institute of Technology, Silchar, Assam, 788010, India. 2

Centre for Sustainable Technologies, Ulster University, Northern Ireland, UK.

*Corresponding address: [email protected]/ [email protected]; Ph (m):+44-7563225168

Abstract Miniaturization of electronic devices and continuous evolution in new compact devices always seeks for the higher heat-releasing system. The use of rectangular fin is gaining popularity due to its ease of production and seeks an effort to analyze different configurations. In the present study, a computational analysis of shrouded rectangular fin array in an inclined channel is performed. The conservation equation of mass, momentum, and energy are solved by using finite volume method. For a range of inclination of 30°, 45° and 60° with non-dimensional spacing (S*=0.2, 0.3 and 0.4) and clearance (C*=0.1, 0.25 and 0.4) results are obtained. Also, the Reynolds number (Re) is varied from 640 to 1707 along with the Grashof number (Gr= 1.4×10 5 to 3.32×105), the effect on axial pressure defect and the non-dimensional bulk temperature is presented at each inclination. The increase of C* from 0.1 to 0.25 results in a maximum of 6.58% increase in the coefficient of heat transfer (h) at an inclination of 60°. Increase in Re enhances the heat transfer by about 12% and increase the pressure drop by about 30%, at a higher inclination. In general, higher inclination provides better heat transfer. Keywords: Inclined channel; Grashof number; Reynolds number; clearance; fin spacing 1. Introduction In many applications of thermo-electric components, an array of heat removal components are mounted on vertical, horizontal and inclined plated channel position and ambient at opposite ends. The cooling of such elements provides more reliability, performance, durability and cost

1

management. To achieve a maximum heat transfer rate is always been a prime objective. And in this paper of mixed convective heat transfer, particularly from an inclined channeled rectangular isothermal fin array is studied. Many researchers investigated both natural and mixed convection heat transfer for different orientations, physical geometry, and other parameters. An initial study in this field started for natural convection at three different orientations (i.e., 45° inclined, vertical and horizontal) with different fin height, spacing and shroud clearance, respectively [13]. Interestingly for mixed convection, Sparrow et al. [4] performed a study to show the effect of tip clearance for horizontal orientation and presented the significance of clearance, due to which the heat loss always increases throughout the fin from the base to the tip. Dogan and Sivrioglu [5] experimentally reported the average coefficient of convective heat transfer which increases first with fin spacing and reaches its maximum value thereafter it starts to reduce with the increase in fin spacing. For forced convection, Yang et al. [6] conducted a numerical investigation in a horizontal parallel plate channel with inline transverse fins on channel walls. Again, Giri and Das [7] investigated for mixed convection heat transfer from a vertical arrangement. They observed no variation for the fully-developed local Nu with Re. Another interesting result shows that it has an optimum clearance spacing, which further shows maximum fully-developed local Nu for the greater fin spacing. Furthermore, they have performed investigation for Non-Boussinésq laminar fluid flow and presented the effect of vortex generator for mixed convection heat transfer, respectively [8-9]. Liu et al. [10] investigated turbulent fluid flow and heat transfer from a cylindrical grooved rectangular channel. They found that the Grooves favorably reduce the pressure drop by weakening direct impingement of the mainstream on the strengthened configuration. Mohapatra and Mishra [11] studied the effect on heat transfer 2

of an internally finned tube and observed that the highest heat transfer is from T-shaped fins compared to that of other shapes. Ayli et al. [12] examined experimentally and numerically for turbulent fully developed flow in a fin array of rectangular shapes. Particularly analyzing the effect of inclination for mixed convection situation is performed by Wang et al. [13] and Boutina and Bessaïh [14], but both studied for the localized heat source. And for natural convection heat transfer, the inclination effects are presented [15-19] at different conditions. The variation of heat transfer at the respective inclination is shown. The variations with other parameters are also compared. The impact of Non-Boussineśq fluid on a vertical plated non-isothermal fin array with variable property is investigated by Roy and Das [20] for natural convection heat transfer. Again, Wong and Lee [21] numerically studied the performance of circular fins in a horizontal tube for two different materials (i.e., aluminum and stainless steel). In recent years, various aspects are studied and analyzed such as the arrangement of fins and comparison of laminar and turbulent flow numerically by Mokhtari et al. [22] for a horizontal orientation. Chen et al. [23] studied both numerically and experimentally the mixed convective heat transfer and found the secondary vortices at both corners of the wind tunnel. Further, they have shown that the pressure drop has a large deviation in the particular range of the fin spacing. Interestingly, the strength of the secondary flow reduces with a decrease in the fin spacing. Alfarawi et al. [24] analyzed experimentally the improvement of heat transfer from a rectangular duct roughened by hybrid ribs. Again for the rectangular inclined channel, Hasan et al. [25] computationally studied two-dimensional heat transfer and presented the results for flow modulation.

3

Fonseca and Altemani [26] analyzed comparatively the performance of plated fins and inline strip fins heat sinks and for same conditions of operation, results found to have better performance of strip fins. Pua et al. [27] performed mixed convective heat transfer for mini electronic systems and found the higher experimental convection coefficient value than those observed from the correlations. Sastre et al. [28] studied both experimentally and numerically the flow topology. Specific geometries of typical tip clearance are also suggested and accordingly, better results are obtained for heat transfer and pressure drop. Again, applications of fins in solar heaters are studied by Singh and Singh [29] and investigated comparatively the performance of transverse ribs of uniform cross-section with square ribs and circular ribs. Khattak and Ali [30] presented a complete overview of thermal performance improvements, prevailing techniques, parametric optimization, and the material characteristics for a heat sink. Homod et al. [31] experimentally studied the effect of both lateral and longitudinal inclination for a fin array of single fixed spacing and clearance. From all the above mentioned pertinent literature it is observed to have a scope of investigation in isothermal fin conditions. And analyze the effect of Reynolds number, fin spacing, clearance, and angle of inclination in a mixed convective heat transfer problem. As the study of isothermal fin gives the maximum heat transfer values, this is highly important for any kind of device designed for practical usage with the non-isothermal fin. Thus, the present paper provides a better understanding of all the aforementioned effects with proper description. 2. Physical model and mathematical formulation 2.1 Physical problem In the inclined base plate, identical rectangular isothermal fins are kept vertically as presented in Fig. 1a. The inclination angle is denoted with ‘α’. An equidistance fin spacing of ‘S’ is 4

maintained throughout. To pass the working fluid flow, an adiabatic shroud is kept at a clearance gap of ‘C' from the fin tip. Representation of every fin height, thickness and length are 'H’, ‘t’ and ‘L’ respectively. As the flow passes through the identical ducts, it creates a symmetric half section in the duct as presented in Fig. 1b. Therefore, the convective heat transfer analysis of the half cross-section solves the problem. Figures illustrate the inclination with respect to y- and zdirection (i.e., fin height and length); whereas the x-direction remains completely parallel to the base. Also, the direction of gravity is acting from a vertically upward direction directly perpendicular to the x-direction. The base is maintained at a uniform temperature of ‘Tw', which causes coupled motion of cross-stream induction along with the axial forced flow forms the current problem of mixed convection. Mathematical formulations are described in the next section. 2.2 Mathematical model In the present investigation, the working fluid is air. The governing equations applied are the equations of conservation of mass, momentum and energy equations, considering the inclination effect in the gravity force term. Due to the lower flow velocity, it is assumed to be an incompressible flow and the viscous heat dissipation is negligible. The density variation effect is incorporated in the momentum equations by keeping the Boussinésq approximation [6] i.e.,

   o 1   T  To 

(1)

Therefore, mathematically the non-dimensional governing equations may be written as: Continuity equation

U V W   0 X Y Z

(2)

x-momentum equation

5

U U U P  2U  2U  2U U V W     X Y Z X X 2 Y 2 Z 2

(3)

y-momentum equation

U

V V V P *   2V  2V  2V    Gr cos  V W     X Y Z Y  X 2 Y 2 Z 2 

(4)

z-momentum equation

U

W W W P *   2W  2W  2W    Gr sin  V W     X Y Z Z  X 2 Y 2 Z 2 

(5)

Energy equation

   1   2  2  2   U V W     X Y Z Pr  X 2 Y 2 Z 2  (6) The above-mentioned equations are expressed in non-dimensional form by applying the following variables: X=x/H, Y=y/H, Z=z/H, U=uH/υ, V=vH/υ, W=wH/υ, P=p′H2/ρoυ2, P*= p H2/ρoυ2, Pr=υ/γ, Re=winH/υ, Gr= g β(Tw-To)H3/υ2, θ= (T-To)/(Tw-To), S*=S/H, C*=C/H 2.3 Initial and boundary conditions Uniform W-velocity is considered at the inlet; whereas U- and V-velocities are the cross-stream and assumed to be zero. The base is maintained to be no-slip and impermeable. The shroud surface is the same but completely insulated. The ambient temperature is kept at the entrance. Up to the tip of the fin, zero velocities and normal gradients of temperature are considered at the symmetry plane passing parallel through the fin. Therefore, the required inlet and boundary conditions are: For 0.5tf ≤ x ≤ 0.5(S+tf), 0 ≤ y ≤ (H + C) and z = 0 u=0, v=0, w=win and T = Tin

(7)

On the symmetry plane passing through the fin i.e., x = 0, 0 ≤ y ≤ H and 0 ≤ z ≤ L u=0, v=0, w=0 and

T 0 x

(8)

6

On the surface of the fin i.e., 0 < y ≤ H, x=0.5tf and 0 < z ≤ L u=0, v=0, w=0 and T = Tw

(9)

On the symmetry plane at the mid-section i.e., 0 < y < H+C, x=0.5(S+tf) and 0 < z ≤ L

u

v w T   0 x x x

(10)

At the clearance region of the y-z plane above the fin i.e., H < y ≤ H+C, x=0 and 0< z ≤ L

u

v w T   0 x x x

(11)

At the base of the configuration, i.e., 0≤ x ≤ 0.5(S+tf), y=0, 0< z ≤ L u=0, v=0, w=0 and T=Tw

(12)

At the shroud of the configuration, i.e., 0 ≤ x ≤ 0.5(S+tf), y = H + C and 0 ≤ z ≤ L u= v= w=

T 0 y

(13)

Conditions at the fin boundary at y = 0 and 0 ≤ z ≤ L, Tf = Tw at y= H and 0 ≤ z ≤ L,

T f y

(14a)

0

(14b)

2.4 Non-dimensionless bulk temperature The non-dimensional bulk temperature is calculated at any fin length (i.e., in the z-direction), H C 0.5 ( S t f )

H  C 0.5t f

  C wT  T dxdy   p



H

0

0

0

 C wT  T dxdy p

0

0.5t f

H C 0.5 ( S t f )  H C 0.5t f    (Tw  T0 )   C p wdxdy    C wdxdy p   H 0  0 0.5t f  

7

(15)

2.5

Overall Nusselt Number and coefficient of heat transfer

To construct any thermo-electric device, it is needed to know the Overall heat transfer rate. An evaluation of the actual heat transfer rate is problematic for a changing bulk temperature. For the fin-base system, with all the boundary conditions the equations are formulated to determine the total heat transfer (Q), the overall heat transfer coefficient (h), average heat flux (q), and overall Nusselt number (Nu) is as follows: Qtotal  Q f  Qb

 L H T   k  x 0 0

  L 0.5( S t f ) T dydz       k   y x 0.5( S t f )   0 0

 dxdz   y 0 

(16a)

q

Q H  0.5S  t f L

(16b)

h

q Tw  To 

(16c)

 H T  0   0 x Nu  

0.5t f

L

dy  x 0.5t f

 0

T y

0.5( S t f )



dx  yH

LH  0.5S  t f

T

w

0.5t f

 T0 

T y

 dx dz  y 0 

(17)

The overall Nusselt number is derived from the temperature difference between the fin-base system and the ambient, also on the average heat transfer from it. A similar equation is also used by Das and Giri [8], Roy et al. [34] and Pattak et al. [37]. 3. Computational method The conventional SIMPLER algorithm of Patankar [32] is used to solve the current problem. The continuity equation is compulsory to evaluate the pressure field. On a staggered mesh using a 8

power-law scheme complete discretization of both coupled convective and diffusive terms are carried out for the cross-stream coordinates. Appling backward difference method the convective terms are determined in the stream-wise coordinate. At every axial sectional location, appropriate numbers of the outer iteration are considered as an iterative implicit method. Further, solutions of the equations are obtained by implementing the Tri-Diagonal Matrix Algorithm (TDMA) accordingly. To reduce the computational errors and to improve accuracy suitable underrelaxation factors are applied. For the present problem, an adequately modified computational code originally developed by Giri and Das [7-9] is applied. 3.1 Grid sensitivity study and Code validation Different combinations of grid sizes are tested for S*=0.2, 0.3, and 0.4 to show its sensitivity as mentioned below in Table 1. Further, to examine the independence of the grids at a 45° inclination angle and Re= 853, 116 grids are employed in the axial flow direction (i.e., zdirection) along the fixed fin length. All the employed grids are raised in geometric progression. For various grid combinations, the result varies well within 1%. Moreover to validate the numerical results, another set of computational runs is performed for forced convection heat transfer for the same configuration with isothermal fin situation. Thus, Table 2.1 shows the comparison of results observed along with Sparrow et al. [4], and Giri and Das [7] and it indicates complete compatibility. In Table 2.2 the experimental results of laminar mixed convection investigated by Maughan and Incropera [35] for an isothermal condition are validated and the comparison of the results finds favorable agreement. 4. Results and Discussion As mentioned above the configuration of the present problem of isothermal fin array are as follows: dimensionless fin spacing (S*) and clearance (C*) are kept in the range of 0.20 to 0.40, 9

and 0.10 to 0.40, respectively, for the fin height of 0.03m and 0.04m. Three different inlet velocities of 0.4, 0.6 and 0.8 m/s are considered and the corresponding Reynolds number (Re) found to vary from 640 to 1707. The Prandtl number (Pr) is taken as 0.7. The ambient temperature is maintained at 20°C and the wall temperature is at 80°C. Accordingly, the Grashof number (Gr) is evaluated and observed to vary from 1.4×105 to 3.32×105. Mathematically, the product of Gr and Pr is defined as the Rayleigh number (Ra). As the value of Pr is fixed, it completely depends on the value of Gr. Thus in the present investigation, the value of Gr is varied. Calculation of air properties are as follows (Das and Giri (2014, 2015)) and T in the following is in Kelvin scale: Thermal Conductivity of air:

 T  1.195 10 k 3

3

(18)

T  118

Viscosity of Air:

 T  1.488 10  3

6

(19)

T  118

Density of Air:



21.955  28.93 1.8T  273.26  491.69

(20)

4.1 Variation of axial pressure drop In Fig. 2 the axial variation of non-dimensional pressure drop (i.e., along Z-direction) is shown for different parameters. The variation at each clearance is illustrated in Fig. 2a, at S*=0.2. At any location along the fin length with the increase in clearance, the axial dimensionless pressure drop tends to reduce; this is directly due to the decrease in resistance with the increase in the clearance. The magnitude of the axial variation of non-dimensional pressure drop tends to 10

decrease, as the fin spacing increases (Fig. 2b). This trend is may be due to the increase in the area of the flow duct causing the reduction of pressure at inlet and outlet (Roy et al. [34] and Karki and Patankar [36]). Fig. 2c shows the variation of axial pressure at Re=1280 and Gr=3.32×105. For an initial increase in inclination angle (i.e., 30°≤ α ≤45°), about 27% of the increase in axial pressure drop at the exit is observed; and thereafter (i.e., 45°≤ α ≤60°) about 3035% of the increase is found, which directly shows the significance of higher inclination angle. Fig. 2d presents a direct increase in the axial pressure drop with the increase in the Re. Das and Giri [8-9] presented similar trends for an increase in the Re, but for vertical shrouded fin array. In general, all the value (i.e., magnitude) of P* is found to have increased at higher Gr. 4.2 Variation of non-dimensional bulk temperature The effect of clearance on the non-dimensional bulk temperature for an inclined fin array is shown in Fig. 3a along the fin length (i.e., Z-direction). The non-dimensional bulk temperature found to be reduced gradually with higher clearance. This may be due to the increase in the air mass flow rate with the increment in the inlet area along with a reduction of flow resistance. Again from Fig. 3b it is clearly illustrated, that the non-dimensional bulk temperature decreases with the increase in S*. The effect of inclination is presented in Fig. 3c, where the results indicate that the initial increase of inclination angle (i.e., 30°≤ α ≤45°) causes a marginal increase in the non-dimensional bulk temperature compared to that of 45°≤ α ≤60°. Thus, it can be concluded that higher inclination causes higher heat transfer. Similar trends of results are obtained by Homod et al. [31] experimentally, but for only one fixed fin spacing and clearance. An increase in Re tends to reduce the bulk temperature due to the higher mass flow rate of air coupled with a lower retention time of the working fluid in the channel. In Fig. 3d at S*=0.3, C*=0.1, α=60°, and Gr=3.32×105 a maximum reduction of 60-70% in dimensionless bulk temperature at the exit is 11

observed for 853 ≤Re≤ 1280 compare to that only about 20-30% of reduction is found for 1280 ≤Re≤ 1707. It particularly indicates the significance of the initial increase in Re. 4.3 Development of W-velocity The variation of W-velocity at the inlet (i.e., Z=1.83), intermediate (i.e., Z=5.90) and exit (i.e., Z=11.80) sections are plotted in Fig. 4a-c for the respective inclination. The W-velocity starts increasing as it reaches a fully developed condition for all the considered parameters. Further, the maximum peak is observed for higher inclination (i.e., 60°). And a direct comparison between Fig. 4a and Fig. 4c, it is found that the peak of the W-velocity is observed to be near the start for a higher inclination; whereas the peek moves away near the end of the fin height for lower inclination. This may be due to the higher heat removal rate at a higher inclination angle because of the chimney effect. Also Fig. 4c-e presents the effect of clearance, which is increased from 0.10 to 0.40 at an inclination angle of 60° for S*=0.2, Re=853, and Gr=3.32×105. The gap among the plotted W-velocity profile along the Y-axis between the inlet and exit diminishes, as the clearance is increased. This may be due to the increase in clearance causing the fresh inlet fluid flows over the region of inter fin ultimately keeping the hot fluid in the region and thereafter heat transfer also reduces. And if the clearance is further increased beyond a certain value than the effect of the inlet to the exit of the W-velocity profile may become very much marginal or reduced. For a horizontal setup, Dogan and Sivrioglu [5] experimentally presented a similar kind of effect due to which heat transfer rate reduces. 4.4 Isotherm variation In Fig. 5a-c, the variation of isotherms contours over the section at C*= 0.10, S*= 0.2, Gr= 3.32×105, α= 60° and 30°, at (a) Z= 1.83, (b) Z= 5.90 (c) Z= 11.80 are presented. The blue dashed line represents α=30° and the red line is for α=60°. Fig. 5a illustrates the variation near the inlet and shows a marginal 12

difference when the angle of inclination is increased. As we move near the exit the development of isotherm contours is reported in Fig. 5b and c. Also, the difference between inclinations is quite significant. The chimney effect is more at 60° inclination as it is near the vertical position than that of 30°. As the Y-axis represents the dimensional fin height and X-axis is the base, the isotherms are near to the Yaxis (i.e., fin) at lower inclination (i.e., α=30°) and its deviation at a higher inclination (i.e., α=60°) can be observed clearly. Moreover, it shows more heat transfer at a higher inclination, which is further explained in section 4.5.

4.5 Variation of convective heat transfer coefficient Clearance plays a critical role in heat transfer improvement. The effect of clearance for the convective heat transfer coefficient (h) at each inclination angle is shown in Fig. 6a-b. The initial increase in clearance causes an immediate increase in the mass flow rate and the capacity of heat removal is increased. At an inclination angle of 60°, a maximum of 6.5-8% increase in convection coefficient is found for an increase in clearance from 0.1 to 0.25. The rate of enhancement is 2-3% with an increase in clearance from 0.25 to 0.4. The present results find its resemblance to the results of Boutina and Bessaïh [14]. Again for fin spacing, as shown in Fig. 6c and d the rate of increase in h are directly obtained to be more for 0.2 ≤ S*≤ 0.3, which increases further for 0.3≤ S* ≤ 0.4, but at a lower rate. For higher and lower Gr the results are plotted at respective inclinations (i.e., α=30°, 45°, and 60°). A maximum of 10% increase in h is obtained at 60° inclination and 4.5% increase at 30° inclination for S*=0.2 to 0.4. Heat transfer at different configurations has also been shown a similar trend by Souza et al. [33] Roy et al. [34] and Karki and Patankar [36] for natural convection. The effect of Re is plotted for the same parameters at each angle of inclination in Fig. 6e and f for higher and lower Gr, respectively. The magnitude of the h value increases directly with an

13

increase in Re. At higher Gr the increase in Re from 853 to 1280 causes a maximum of about 12% increase in h for 60° inclinations. This increase in h value continues to increase thereafter (i.e., 1280≤Re≤1770) but with a lower rate of increase comparatively (i.e., 5-6% only). The magnitude of h is gradually increased when the Gr is increased for all the cases. Finally for the present investigation, a correlation associating the Nu with the governing parameters by utilizing 180 computational data is presented in Fig. 7.





Nu.1.2 Re 0.12 Gr 0.16  3.81 L*0.52 S *0.45C *0.41 (sin  ) 0.183  0.4

(21)

The correlation coefficient is found to be 0.9887. The relation is appropriate for a range of 640 ≤ Re ≤ 1770, 1.4×105 ≤ Gr ≤ 3.32×105, 12.5 ≤ L* ≤ 16.67, 0.2 ≤ S*≤ 0.4, 0.10 ≤ C* ≤ 0.40 and 30° ≤ α ≤ 60°. 5. Conclusion The main scope of this study is to analyze the impact on an isothermal inclined fin array with different combinations. The specific objective includes evaluation of temperature variation, pressure drop and convective heat transfer coefficient with various angles of inclination underspecified set of fluid flow velocity. The results presented in this study can be concluded as follows: 

An increase in clearance increases the heat transfer by a maximum of 6.5-8% for clearance of 0.1 to 0.25 and 2-4% for clearance 0.25 to 0.40.



The variation in fin spacing from 0.2 to 0.3 causes more heat transfer than that of 0.3 to 0.4.



An increase in Re from 853 to 1280 causes a maximum of about 12% increase in h for 60° inclinations.

14



The maximum increase in mixed convective heat transfer for the present problem is found to be highest at an inclination of 60° for S*=0.2 to 0.3 and C*=0.1 to 0.25. Acknowledgment

The fund received from DBT, Govt of India, under, ’Overseas Associate’ to pursue research at Ulster University, UK, is highly acknowledged. Nomenclature C C

*

Fin tip to shroud clearance (m)

S*

Non-dimensional fin spacing, S/H

Dimensionless fin tip to shroud clearance, C/H

T

Temperature (K)

t

Fin thickness (m)

2

g

Gravitational acceleration, (m/s )

Gr

Grashof number, g β(Tw-To)H /υ

H

Fin height (m)

h

Coefficient of convective heat transfer, (W/m2K)

U,V,W Dimensionless velocity components in x,y and z directions, uH/υ, vH/υ and wH/υ

k

Thermal conductivity, (W/mK)

x,y,z

L

Fin length (m)

L*

Dimensionless fin length, L/H

X,Y,Z Dimensionless cross-stream and axial coordinates, x/H, y/H, and z/H

Nu

Overall Nusselt number

Greeks

p

Total pressure defect, po-ps (Pa)

α

Angle of inclination

p′

Cross-stream pressure, (Pa)

β

p

Average pressure defect over the cross-section (Pa)

Coefficient of thermal volumetric expansion, 1/To

γ

thermal diffusivity (m2/s)

ΔT

Scaling temperature difference, (TwTo)

ρ

Density (kg/m3)

µ

Viscosity (Ns/m2)

3

u,v,w Velocity components in x,y and z directions (m/s)

2

P*

Dimensionless axial pressure defect, p H2/ρoυ 2

P

Dimensionless cross-stream pressure defect, p′H2/ρoυ 2

Cross stream and axial coordinates (m)

Pr

Prandtl number, υ/γ

υ

Momentum diffusivity (m2/s)

Q

Overall heat transfer

θ

Ra

Rayleigh number, (Gr×Pr), g β(TwTo)H3/υ γ

Non-dimensional temperature, (TTo)/(Tw-To)

Subscripts

Re

Reynolds Number, winH/υ

f

Fin

S

Inter-fin spacing (m)

l

Local

15

in

Inlet

o

Ambient

m

Mean

w

Wall

Reference [1]

K.E. Starner, H.N. McManus, An experimental investigation of free convection heat transfer from rectangular fin-arrays, ASME J. Heat Transf. 85(1963) 273-278.

[2]

J.R. Welling, C.V. Wooldridge, Free convection heat transfer coefficients from rectangular vertical fins, ASME J. Heat Transf. 87 (1965) 439-444.

[3]

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[10] J. Liu, G. Xie, T. W. Simon, Turbulent flow and heat transfer enhancement in rectangular channels with novel cylindrical grooves, Int. J. of Heat and Mass Transfer 81 (2015) 563– 577. [11] K. Mohapatra, D.P. Mishra, Effect of fin and tube configuration on heat transfer of an internally finned tube, Int. J. of Numerical Methods for Heat & Fluid Flow 25 8 (2015) 1978-1999. [12] E Ayli, O. Bayer, S. Aradag, Experimental investigation and CFD analysis of rectangular profile fins in a square channel for forced convection regimes, Int. J. of Therm. Sci. 109 (2016) 279-290. [13] X. Wang, L. Robillard, Mixed convection in an inclined channel with localized heat sources, Numerical Heat Transf. Part A 28 (1995) 355-373. [14] L. Boutina, R. Bessaïh, Numerical simulation of mixed convection air-cooling of electronic components mounted in an inclined channel, Appl. Therm. Eng. 31 (2011) 2052-2062. [15] G. Mittelman, A. Dayan, K. D. Turjeman, A. Ullmann, Laminar free convection underneath a downward facing inclined hot fin array, Int. J. Heat Mass Transf. 50 13 (2007) 2582–2589. [16] M. Mehrtash, I. Tari, A correlation for natural convection heat transfer from inclined platefinned heat sinks, Appl. Therm. Eng. 51 (2013) 1067-1075. [17] I. Tari, M. Mehrtash, Natural convection heat transfer from inclined plate-fin heat sinks, Int. J. Heat Mass Transf. 56 1 (2013) 574–593. [18] Q. Shen, D. Sun, Y. Xu, T. Jin, X. Zhao, Orientation effects on natural convection heat dissipation of rectangular fin heat sinks mounted on LEDs, Int. J. of Heat and Mass Transf. 75 (2014) 462–469.

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Table 1: Grid independence examination C*=0.25, Re= 853, Gr=3.32×105. S*

Grid Size

Nu

(X ×Y × Z)

0.2

0.3

0.4

18 × 42 ×116

7.8901

20 × 44 ×116

7.8867

22 × 46 ×116

7.8820

22 × 42 ×116

8.6002

24 × 44 ×116

8.5952

26 × 46 ×116

8.5909

26 × 42 ×116

8.9898

28 × 44 ×116

8.9821

30 × 46 ×116

8.9799

21

Table 2.1: Test for validation. S*

C*

Nu [4]

Nu [7]

Nu [present]

0

34.02

34.27

34.861

0.25

0.605

0.616

0.636

0

5.806

5.920

6.012

0.25

4.448

4.385

4.567

0.1

0.5

Table 2.2: Validation with previous researcher's experimental results. S* H= 0.02m, C*= 1.0, L= 0.914m, Re= 1000

Ra

Nu [35]

Nu [present]

13500

6.85

7.07

29000

7.6

7.91

13500

5.1

5.39

29000

5.9

6.32

2.0

4.0

22

Fig. 1. (a) Inclined shrouded rectangular isothermal fin array placed on an inclined base plate. (b) Cross-section of one isothermal half duct. (c) Boundary conditions for symmetric half flow duct. 23

Fig. 2. Variation of dimensionless axial pressure drop at different (a) clearance (b) spacing (c) angle of inclination and (d) Reynolds number.

24

Fig. 3. Variation of non-dimensional bulk temperature at different (a) clearance (b) spacing (c) angle of inclination and (d) Reynolds number.

25

Fig. 4. Effect of W-velocity variation for S*=0.2, C*=0.10, Re=853, Gr=3.32×105 at an inclination angle of (a) 30°, (b) 45°, (c) 60°, and at different clearance (d) C*=0.25 and (e) C*=0.40.

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Fig. 5. Isotherms contours across the section at S* = 0.2, C* = 0.10, Gr=3.32×105, α=60° and 30° (a) Z= 1.83, (b) Z= 5.90 (c) Z= 11.80. 27

Fig. 6. Variation of coefficient of convective heat transfer (h) (W/m2K) at each inclination at different (a) clearance with lower Re and Gr and (b) with higher Re and Gr, (c) fin spacing with lower Re and Gr and (d) with higher Re and Gr and (e) Re with lower Gr and (f) with higher Gr.

28

Fig. 7. Correlation developed for overall Nusselt number. Highlights 

A performance study of heat sinks in a inclined channel is reported numerically.



Presence of fin tip to shroud clearance enhances the heat transfer by about 8%.



Increase in Re enhances the heat transfer and pressure drop 12% and 30%, respectively.



Higher inclination always increase the heat transfer.

Title: Convective heat transfer from an inclined fin array: A computational study

This is to certify that there is no conflict of interest with the present study.

Mr. Krishna Roy Dr, Biplab Das 29

Title: Convective heat transfer from an inclined isothermal fin array: A computational study This is to certify that   

The corresponding author is responsible for ensuring that the descriptions are accurate and agreed by all authors. The role(s) of all authors are listed, using the relevant above categories. Authors have contributed in multiple roles.

30