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Research Papers

# Conjugate Mixed Convection Heat Transfer From a Shrouded Vertical Nonisothermal Heat SinkOPEN ACCESS

[+] Author and Article Information
Biplab Das

Department of Mechanical Engineering,
National Institute of Technology Silchar,
Silchar, Assam 788010, India
e-mail: biplab.2kmech@gmail.com

Asis Giri

Department of Mechanical Engineering,
North Eastern Regional Institute of Science
and Technology,
e-mail: measisgiri@rediffmail.com

Suman Debnath

Department of Mechanical Engineering,
National Institute of Technology Silchar,
Silchar, Assam 788010, India
e-mail: debnath.s1990@gmail.com

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF THERMAL SCIENCE AND ENGINEERING APPLICATIONS. Manuscript received March 10, 2016; final manuscript received January 10, 2017; published online March 21, 2017. Assoc. Editor: Hongbin Ma.

J. Thermal Sci. Eng. Appl 9(4), 041001 (Mar 21, 2017) (14 pages) Paper No: TSEA-16-1061; doi: 10.1115/1.4035970 History: Received March 10, 2016; Revised January 10, 2017

## Abstract

A computational analysis of conjugate mixed convection heat transfer from shrouded vertical nonisothermal heat sink on a horizontal base is performed. The overall Nusselt number and the product of friction factor (f) and Reynolds number (Re) are found to vary significantly with the spacing of heat sink as well as with the clearance between shroud and heat sink. By increasing the fin conductance by 200%, an enhancement of Nusselt number is noted to be around 58%, while the same Nusselt number enhancement is 134% for isothermal fin, within the range of parametric studies. The fRe value for smaller fin spacing shows a maximum with clearances, while the same for higher fin spacing remains the same or increases with clearances. Finally, overall Nusselt number and friction factor are well correlated with the governing parameters of the problem.

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## Introduction

Limited sources of depleting fossil fuels demand the efficient collection and utilization of energy from other sources as well as from fossil fuels. For effective removal and efficient utilization of heat energy, extended surfaces are commonly used in multiple applications, for example, solar collector, gas-cooled nuclear reactor, air-cooled automobile engine, aircraft engine cooling, cooling of electrical and electronics appliances, cooling of pipe line carrying oil and heat exchangers, etc. Extended surfaces are popularly known as fins. Fin may release heat by all the three modes (i.e., natural convection, forced convection, and radiation). Since fins are generally made of low emissivity materials like aluminum, copper, or steel, radiative heat transfer is limited and may be neglected. Heat transfer by natural convection is also limited by imposed buoyancy force. Hence, forced convection coupled with natural convection heat transfer is one of the most preferred method of relieving heat to keep the device operational.

Natural convection is prevalent in all systems. It is being a noise-free mode of heat transfer frequently encountered in many heat transfer appliances. Transport of heat by free convection from vertical and horizontal rectangular plate-finned array of heat sink is a topic of interest by many researchers, for instance, Starner and McManus [1], Welling and Wooldridge [2], Harahap and McManus [3], and Jones and Smith [4]. Considerable experimental studies on natural convection from plate-finned heat sink can also be found in Refs. [5,6], where Fisher and Torrance [5] have shown the evidence of chimney effect and Yazicioğlu and Yüncü [6] found the optimum condition of vertical fin array under natural convection. All these earlier investigations [16] are restricted to experimental studies. Computational study of heat transfer is generally measured as a complementary means, which can be used as an effective tool for the parametric studies. In this context, a note may be taken from the investigation of Karki and Patankar [7], where thorough computational study of natural convection from shrouded vertical rectangular isothermal plate-finned heat sink is presented. Consideration of negligible radiative transport in the earlier studies [17] has not affected the results significantly.

Rao and Venkateshan [8] have made investigation on simultaneous natural convection and radiation heat transfer from a rectangular slot akin to vertical plate-finned heat sink on a horizontal base. Later radiative heat transport from a slot of two vertical fins heat sink as side walls on a horizontal base under combined radiation, convection, and conduction is performed by Balaji and Venkateshan [9] and Rao et al. [10]. Additional experimental finding by Yüncü and Anbar [11] from vertical rectangular extended surface on a horizontal base reported simultaneous free convection and radiation, and convective heat transport is estimated by subtracting the radiative component from total heat transfer. Baskaya et al. [12] has also studied the heat transport from the vertical plate-finned heat sink projecting upward on a horizontal base using the finite volume-based cfd code. A plate-finned heat sink projecting downward underneath a horizontal base is not a preferred configuration and is therefore ignored in the past. Development of modern telecommunication devices considered such configuration. A study on such configuration has been reported by Dayan et al. [13].

Sparrow et al. [14] computationally examine laminar forced convection from plate-finned heat sink, while Sparrow and Kadle [15] and Kadle and Sparrow [16] report turbulent forced convection heat transfer from the plate-finned both experimentally and computationally. All these aforesaid studies [1416] focus on the cases of hydraulically and thermally developed flow. Since, away from entrance, essential physics remains unaffected by considering hydraulically and thermally developed flow; thus, a comprehensive study is possible under such developed flow conditions. Studying under such conditions, an intimate relation is prescribed between the heat transfer coefficient ratio and fin tip clearance to fin height ratio, but hardly any dependence is noted on the air flow rate and fin height [1416]. In a recent advancement, turbulent forced convective heat transfer is experimentally conducted from plate-finned heat sink under very high Reynolds number by El-Sayed et al. [17], where overall Nusselt number is related with the governing parameters. Further experimental study on forced convection by Elshafei [18] shows the effect of flow bypass on the performance of shrouded plate-finned heat sink. All these studies [1418] are related to forced convection in which natural convection plays nominal role. Hence, vertical and horizontal configurations played hardly any role.

To accommodate the increasing demands of high performance engineering devices, there remains an ever growing need in packaging density of equipment, which brings out a significant increase in temperature in the system. Under such a situation, the effect of forced convection coupled with natural convection plays pivotal role on heat transfer. Therefore, mixed convection heat transfer is a central topic and also a topic of discussion in this paper.

There exist several literature on vertical fin attached with a vertical base as seen in Zhang and Patankar [19], Al-Sarkhi et al. [20], Giri and Das [21], and Das and Giri [22,23]. Acharya and Patankar [24] report hydraulically and thermally developed mixed convection heat transport from a shrouded isothermal vertical fin array in a horizontal channel for a limited parameter. Previously, Zhang and Patankar [19] and later Al-Sarkhi et al. [20] considered the case of hydraulically and thermally developed mixed convection over a shrouded vertical isothermal fin array on a vertical base, while Giri and Das [21], and Das and Giri [22,23] attempted the case of entry region mixed convective flow over the vertical fin protruded from a vertical base at uniform temperature. All these studies [1921] considered isothermal plate-finned heat sink as an extended surface. It is worth noting that since plate-finned heat sinks are rarely isothermal in practice, the study of nonisothermal heat sinks will be more meaningful.

After the beginning of preliminary computational work by Acharya and Patankar [24] on mixed convection over isothermal vertical plate-finned heat sink on a horizontal base, Maughan and Incropera [25,26] visited mixed convection from the same configuration by both numerical and experimental investigations. Acharya and Patankar [24] reported the result of Nu and fRe in the form of ratio of mixed convection value with forced convection value and the parameter range is very limited in the study. Since, no data are provided on forced convection, the results [24] found nominal significance and moreover, fins are rarely isothermal because of finite conductivity of fin. Experimental study by Maughan and Incropera [26] is made for entry region flow, while the computation [25] is performed for developed flow with isothermal fin. A series of experimental studies supported by limited computational work on the entry region mixed convection heat transfer from vertical aluminum plate-finned heat sink array in a horizontal channel are reported by Dogan and Sivrioglu [2729] for higher Rayleigh numbers. Therefore, the aforesaid studies [2429] are having nominal applicability for the fins with larger length. In this sense, recent report by Das and Giri [30] highlighted a computational study of hydraulically and thermally developed mixed convection from a nonisothermal fin array facing downward underneath a horizontal base can be considered as a case study of larger fin length.

A perusal of pertinent literature unveils the fact that this problem is a subject of active research and a topic of interest by many researchers. The studies, delineating the heat transfer from rectangular plate-finned heat sink, mostly considered natural or forced convection. In spite of a series of literature [1930] on mixed convection from a shrouded plate-finned heat sink, there remain considerable gaps in the results, which is in demand for practitioners. Nusselt number and pressure drop correlations are useful tools for every practitioner to design heat exchangers. Pressure drop correlation is completely missing in literature, albeit some form of correlation on Nusselt number [2729] exists for entry region flow in the case of vertical fin on a horizontal base. To fill up the above noted discrepancies, an attempt is made to study mixed convection from a shrouded nonisothermal vertical plate-finned heat sink in a horizontal channel to find correlations of Nusselt number and pressure drop for larger fin length. Application of such model may be found in solar air dryer. Additionally, the effect of finite conductivity of fin material is also considered in the report. Description of the physical model is presented in Sec. 2.

## Physical Model

Attention is now focused to a region where flow is thermally and hydraulically fully developed, i.e., sufficiently away from the entrance region. Problem undertaken for the consideration of heat transfer analysis is shown in Figs. 1(a) and 1(b). There is a series of vertical rectangular heat sink, which are glued to a lower horizontal base at equal distance apart. An adiabatic shroud is placed parallel to base and above the fin to force the flow over the fin surface. A series of horizontal channels is thus formed, each of which is equal in dimension. All these channels are identical to each other. Due to geometrical similarity, computational analysis is confined to only half cross section of a repeating channel (Fig. 1(b)). As depicted in figure, origin of the “x” and “y” coordinates is considered at midpoint of the channel. Coordinate x extends toward the fin, coordinate y extends along the fin height, and coordinate “z” is normal to cross section. Main flow is directed normal to the cross section of each channel (i.e., along z-direction) and secondary flow induced by the temperature gradient produces clockwise motion across the domain of interest. Main flow direction and clockwise motion are shown by arrow and circle, respectively, in Fig. 1(b). Fins are considered of thickness “t” and height “H,” and are spaced “S” distance apart and maintain a clearance “C” with the adiabatic shroud. To limit the governing parameters, it is assumed that the fins are thin (t$≪$H and t$≪$S). A constant heat input per unit axial length is supplied at the bottom of the base, which maintains the base at a uniform temperature, Tw at any axial location. The temperature induced secondary flow along with normal forced flow makes the present problem a mixed convection problem. Mathematical formulation is presented in Sec. 3.

## Mathematical Formulation

The well-known Navier–Stokes equations and the energy equations are solved for flow and temperature distributions. Compressibility and viscous dissipation effects are considered negligible. Thermo-physical properties of the working fluid (air) and solid are assumed to be independent of temperature.

The Oberbeck–Boussinésq approximation is invoked, i.e., Display Formula

(1)$ρ=ρw(1−β(T−Tw))$

where ρw is the base wall density and Tw is the corresponding temperature. A modified pressure ($p′$) is expressed as Display Formula

(2)$p′=p+ρwgy$

where “p” is the static pressure. Thus, the pressure term in the y-momentum equation may be read as Display Formula

(3)$−∂p′∂y=−∂p∂y−ρwg$

For a hydraulically fully developed flow u, v, and w remain unchanged along the axial z-coordinate, and the axial pressure gradient ($dp/dz$) implies the mean pressure gradient ($dp¯/dz$). Conservation of mass, momentum, energy, and fin conduction equations governing physical model in dimensional form can be cast as:

Mass conservation Display Formula

(4)$∂u∂x+∂v∂y=0$

x-momentum equation Display Formula

(5)$ρ(u∂u∂x+v∂u∂y)=−∂p∂x+μ(∂2u∂x2+∂2u∂y2)$

y-momentum equation Display Formula

(6)$ρ(u∂v∂x+v∂v∂y)=−∂p∂y−ρg+μ(∂2v∂x2+∂2v∂y2)$

z-momentum equation Display Formula

(7)$ρ(u∂w∂x+v∂w∂y)=−∂p¯∂z+μ(∂2w∂x2+∂2w∂y2)$

Energy equation Display Formula

(8)$u∂T∂x+v∂T∂y+w∂T∂z=k(∂2T∂x2+∂2T∂y2)$

Fin conduction equation Display Formula

(9)$kfin(t/2)d2Tfdy2=k∂T∂x|X=0$
Following nondimensional scheme is identified to nondimensionalize the various quantities appeared in the governing equations Display Formula
(10)$X=xH,Y=yH,Z=(zH)/(wavHα),C*=CH,S*=SH,U=uHυ,V=vHυ,W=−wμH2(dp¯/dz),θ=Tw−TQ/k,Gr=gβH3(Q/k)/υ2,P=p′H2ρwυ2$

Thus, the mass conservation equation and three x-, y-, and z-momentum equations in nondimensional form can be cast as Display Formula

(11)$∂U∂X+∂V∂Y=0$
Display Formula
(12)$U∂U∂X+V∂U∂Y=−∂P∂X+∂2U∂X2+∂2U∂Y2$
Display Formula
(13)$U∂V∂X+V∂V∂Y=−∂P∂Y+∂2V∂X2+∂2V∂Y2−Grθ$
Display Formula
(14)$U∂W∂X+V∂W∂Y=1+∂2W∂X2+∂2W∂Y2$

Further, for a thermally developed flow, like in the present case, dimensionless temperature distribution does not vary from one to another cross section. Applying Newton's law of cooling, wall heat flux per unit axial length at any axial z-location can be expressed as Display Formula

(15)$h(H+0.5S)(Tw−Tb)=Q$

where “h” is the heat transfer coefficient and $Tb$ is the bulk temperature. Therefore, difference between the base wall temperature and the bulk fluid temperature can be rewritten as Display Formula

(16)$Tw−Tb=Q/(h(H+0.5S))$

Since by assumption $Q$ is constant and further for hydraulically and thermally fully developed flow $h$ is also constant. In addition, for a fixed fin height and fin spacing, right-hand side of Eq. (16) is constant. Taking derivative of the above Eq. (16), one can write Display Formula

(17)$d(Tw−Tb)dz=0⇒dTwdz=dTbdz$

Further for the change of bulk fluid temperature $(ΔTb)$ in an axial length of $Δz$, one can write the following from the energy balance: Display Formula

(18)$m˙cpΔTb=QΔz⇒limΔz→0ΔTbΔz=dTbdz=Qmcp=constant$

For a hydraulically and thermally fully developed flow, dimensionless temperature $(θ=(Tw−T)/(Q/k))$ is invariant with axial direction [19,20,24]. Mathematically, it can be written as Display Formula

(19)$dθdz=d((Tw−T)/(Q/k))dz=0⇒dTwdz=dTdz$

Combining Eqs. (17)(19), we obtain Display Formula

(20)$dTwdz=dTbdz=dTdz=Qmcp=Q0.5S(H+C)ρwavcp=Constant$

The temperature gradient in the axial direction may be calculated by considering the overall energy balance. In dimensionless form Display Formula

(21)$ddZ(TwQ/k)=ddZ(TbQ/k)=10.5S*(1+C*)$

The energy equation for the fluid, in dimensionless form, can be expressed as Display Formula

(22)$U∂θ∂X+V∂θ∂Y=1Pr(∂2θ∂X2+∂2θ∂Y2)+1Pr(W/Wav)0.5S*(1+C*)$

Fins are considered to be nonisothermal along the Y-direction, and the energy balance of fin must satisfy the fin conduction equation Display Formula

(23)$Ωd2θfdY2=∂θ∂X|X=0$

where, $Ω=(kf(t/2))/(kH)$.

Above governing Eqs. (11)(14) and Eqs. (22) and (23) are solved with the following boundary conditions.

At fin surface (X = 0.5S* for 0 ≤ Y ≤ 1) Display Formula

(24)$U=V=W=0, θ=θf$

In Eq. (24), $θf$ comes out from the solution of fin conduction equation (23).

On the symmetry plane:

• (i)at the line passing through the midpoint of base and directed parallel to fin height (X = 0, for 0 < Y < 1 + C*) and
• (ii)at the line above the fin tip and along the fin height (X = 0.5S* for 1 < Y < 1 + C*) Display Formula
(25)$U=∂V∂X=∂W∂X=0, ∂θ∂X=0$

On the adiabatic shroud (0 ≤ X ≤ 0.5S* for Y = 1 + C*) Display Formula

(26)$U=V=W=0, ∂θ∂Y|Y=1+C*=0$
On the base wall (0 ≤ X ≤ 0.5S* for Y = 0) Display Formula
(27)$U=V=W=0, θ=0$

Boundary condition to solve fin conduction Eq. (23)Display Formula

(28)$θf=0, and dθfdY=0$

Bulk temperature, local and overall Nusselt number are evaluated from temperature and velocity field obtained by solving Eqs. (11)(28).

###### Bulk Temperature.

The dimensionless bulk temperature ($θb$) of the fluid is defined as Display Formula

(29)$θb=∫00.5S*∫01+C*θWdXdY∫00.5S*∫01+C*WdXdY$

###### Local Nusselt Number.

Local convective Nusselt number (Nul,f) of fin is expressed as Display Formula

(30)$Nul,f=hl,fHk=ql,fH(Tf−Tb)k=1(θf−θb)∂θ∂X|X=0.5S*$

Local Nusselt number (Nul,w) of base is written as Display Formula

(31)$Nul,w=hl,wHk=ql,wH(Tw−Tb)k=1θb∂θ∂Y|Y=0$

###### Overall Nusselt Number.

Overall Nusselt number (Nu) is expressed as Display Formula

(32)$Nu=1θb1(1+0.5S*)[−∫01(∂θ∂X|X=0.5S*)dY+∫00.5S*∂θ∂Y|Y=0dX]$

The complete mathematical formulation is expressed in Eqs. (11)(32). The governing parameters for the present problem are: Prandtl number (Pr), Grashof number (Gr), fin conductance parameter (Ω), dimensionless interfin spacing (S*), and dimensionless clearance spacing (C*). The computational procedure for solving the present problem is presented in Sec. 4.

## Computational Procedure

Since there exists no closed-form solution for the above described problem, computational procedure is favored and the problem is numerically solved using standard finite volume technique as outlined in Patankar [31]. The SIMPLER algorithm, elucidated in Patankar [31], is employed to decouple the velocity and pressure coupling. Power law scheme, which is very close to exact solution [31], is implemented for the combined convective-diffusive term in each X- and Y-directions separately for momentum, and energy equations. U and V velocity field are computed at the staggered locations to avoid any unrealistic solution induced by pressure variation in the domain. On the other hand, W velocity, pressure, and temperature are evaluated at the center of the computational cell. Computational domain is subdivided into a number of rectangular cells. Grids having rectangular in shape are engendered in both X- and Y-coordinate directions. Finer grids are deployed close to the solid walls as well as close to the fin tip in Y-direction. Away from the solid walls as well as from the fin tip, grid size increases with geometric progression and is shown in Fig. 2. A series of numerical experiments is carried out to test the grid sensitivity of the results obtained for the present problem. In the cross stream plane (i.e., in the X–Y plane), grids of 26 × 40, 30 × 60, and 36 × 60 for S*= 0.1, and 30 × 40, 36 × 60, and 42 × 60 for S*= 0.3, and other grids of 30 × 40, 36 × 60, and 42 × 60 for S*= 0.5 have been used. Results for the different grids differ within 1.5%. Results of comparison of different grids with different parameters are presented in Table 1. In the present investigation, grids of 36 × 60 for S*= 0.1, 36 × 60 for S*= 0.3, and 36 × 60 for S*= 0.5 are used. To validate our present results, numerical simulation is made for the experimental results of Maughan and Incropera [26] and is compared in Table 2. Maximum percentage deviation remains within 10%. It may be noted at this point that maximum uncertainty in the experimental results of Maughan and Incropera [26] is 17%. Rayleigh number (Ra) in Table 2 is defined the same way as defined in Ref. [26].

## Results and Discussion

Geometrical particulars of the present problem are varied in a way that results in Grashof number variation from 105 to 107, dimensionless interfin spacing (S*) variation from 0.1 to 0.5, and dimensionless clearance spacing (C*) variation from 0 to 0.30 for all the fin spacing. Fin conductance parameter (Ω) is varied from 10 to 100. Values of working fluid (air) properties are taken at the base wall temperature. The value of Prandtl number is assumed as 0.7.

###### Stream Function.

Present problem involves secondary flow across the cross-stream plane over the main axial flow due to the buoyancy force along Y-direction. Detailed information of secondary flow can be best observed from the streamline maps. The stream function can be defined as follows: Display Formula

(33)$U=∂ψ∂Y⇒ψ=∫0YUdY$

It is assumed that $ψ=0$ at X = Y = 0. Streamlines are displayed in Fig. 3. As seen in Figs. 3(a) and 3(c), single secondary motion in clockwise direction is created due to buoyancy force in Y-direction for smaller fin spacing (S*= 0.1) and Gr = 106. This buoyancy force is induced due to imposed high temperature at the fin surface. Whenever fin tip clearance is present, this eddying motion is strong around the clearance space since viscous action is relatively less in the clearance space. Secondary motion gains its strength with the increase of Grashof number by tenfold, i.e., due to the increase in buoyancy in the cross-stream Y-direction. By increasing fin spacing from S*= 0.1 to S*= 0.3, stronger secondary motion is observed for all Grashof numbers and the maximum strength of eddy shifts toward the base of the interfin region possibly due to relatively lower viscous action as well as higher buoyancy in the said region (Figs. 3(e)3(h)). At the highest fin spacing S*= 0.5 and Gr = 106, secondary motion is similar to S*= 0.3, but with increased Grashof number an additional counter-clockwise vortex is created near the corner of base and symmetry plane in conjunction with main secondary clockwise motion. These secondary motions will definitely influence the W-velocity profile, which in turn will influence the heat transfer.

###### W-Velocity.

Fully developed velocity profile is depicted in Figs. 4(a)4(d) for S*= 0.1 for different clearance and Grashof numbers. As evident in the figure, away from the base and shroud W-velocity is almost constant along Y-direction over the fin surface for Gr = 106 in the case of zero clearance (Fig. 4(a)). The W-velocity profile in the X-direction is parabolic in shape. The W-velocity profile is uniform along the fin height possibly due to the absence of viscous resistance away from base and shroud and also because of secondary motion. On the other hand, dimension in X-direction is relatively much smaller than Y-direction and hence viscous effects are felt throughout the X-direction. W-velocity profiles for Gr = 106 with clearance C*= 0.30 are shown in Fig. 4(c), where velocity profile is almost constant along Y-direction over the fin surface similar to zero clearance spacing. But sizeable amount of flow bypass is noted through the clearance due to higher viscous resistance in the interfin region and simultaneously lower viscous action in the clearance. Increasing the Grashof number by 10 fold, nominal deviation in the velocity profile is observed for zero clearance (Fig. 4(b)). However, at higher Grashof number (i.e., Gr = 107) with C*= 0.30, stronger secondary motion in the cross-stream direction due to enhanced buoyancy redistributes W-velocity profile near the central region of clearance, which is apparent by comparing Fig. 4(c) with respective Fig. 4(d). In the absence of clearance, secondary motion becomes weak due to strong viscous action. Similar development of velocity profile for S*= 0.3 for various Grashof numbers and different clearance spacing are presented in Figs. 4(e)4(h). In the case of S*= 0.3, secondary flow is relatively strong even for the case of Gr = 106. In most cases, the maximum strength of the secondary flow is near the base of the interfin region; therefore, the position of the maximum W-velocity profile shifts toward the base of the interfin region (Figs. 4(e) and 4(g)). Stronger secondary motion brings the fluid toward the base from the clearance region, thereby increasing velocity near the base. Increasing the Grashof number by ten times, even much stronger secondary flow is observed that causes more uniform W-velocity along X-direction near the base surface of the interfin region (Figs. 4(f) and 4(h)). The above described developments on W-velocity profile will definitely have a bearing on heat transfer as well as in the pressure drop.

###### Fin Temperature.

Fin temperature distributions along the fin height are presented in Figs. 5(a) and 5(b) for S*= 0.1 and S*= 0.3, respectively. The dimensionless fin temperature, as defined in the nomenclature, assumes zero value at the base and increases with the decreasing dimensional fin temperature. Fin temperature drop across the fin height is very much related with W-velocity. In smaller fin spacing (S*= 0.1), flow bypass is significant through the clearance. Thus, the region near the fin tip is effectively cooled by the cold fluid. This, therefore, increases fin temperature drop across the fin height. The higher the flow by-pass, the higher is the fin temperature drop. Fin temperature drop in the case of higher fin spacing (S*= 0.3) is shown in Fig. 5(b), in which flow bypass through the clearance is not significant. Especially, for high Grashof number (Gr = 107), flow is more near the base due to strong secondary flow. Hence, temperature drop across the fin height decreases with higher Grashof number for S*= 0.3.

###### Temperature Profile.

Attention may now be turned on to the dimensionless temperature of the fluid. Fully developed temperature profile is presented in Figs. 6(a)6(d) for S*= 0.1. The way this dimensionless temperature is defined means that the lower the dimensionless temperature, the higher the dimensional temperature. Dimensionless temperature profile for both the Grashof numbers (i.e., 106 and 107) with zero clearance indicates relatively lower value at the shroud. This is arguably due to the secondary motion of fluid across the section. Because of secondary motion of fluid, hot fluid over the fin surface is directed on to the shroud surface. However, the situations with clearance C*= 0.30 are somewhat different (Figs. 6(c) and 6(d)). Fluid, in the clearance, remains largely unheated, i.e., higher dimensionless fluid temperature. Comparing Fig. 6(c) with Fig. 6(d), it is realized that increasing Grashof number from 106 to 107 lowers dimensionless fluid temperature in the clearance, indicating efficient heat transfer by the fin to the fluid and there from hot fluid moves to the clearance due to the strong secondary motion at high Grashof number. This may perhaps be related to strong secondary motion at higher Grashof numbers. Figures 6(e)6(h) show the dimensional fluid temperature for the case of higher fin spacing (S*= 0.3). Similar to smaller fin spacing, the result of zero clearance indicates lower dimensionless temperature at the shroud due to secondary motion, which directs the fluid to shroud from the fin surface. At higher Grashof number (107), much stronger motion near the base directs cold fluid from the midplane to the base, which enhances heat removal capacity from the base. As clearance spacing is increased for the higher fin spacing, fluid in the clearance remains relatively cooler (i.e., higher dimensionless temperature) especially for the Grashof number 106. But at higher Grashof numbers, stronger secondary flow homogenizes the temperature distribution across the section.

###### Local Fin Nusselt Number.

Variation of local fin Nusselt number with different clearance spacing is shown in Fig. 7(a) for S*= 0.1. As seen in the figure, all the local fin Nusselt number starting at zero value at the base increases along the fin height very near base and thereafter it maintains almost constant value till near the fin tip, except for the case of zero clearance. Very near the fin tip, it rises again to a high value. Reason for this behavior may be understood from the axial velocity profile (i.e., W-velocity) in which velocity profile is more or less constant over the fin surface except near the base and the fin tip. Near the fin tip, fluid velocity is significantly enhanced due to least resistive path. With the increase in clearance spacing, the magnitude of local Nusselt number decreases indicating increased flow bypass through the clearances. Flow bypass is very much prominent for S*= 0.1. Increasing Grashof number from 106 to 107, i.e., an increase of ten fold in Grashof number induces higher local Nusselt number. This may be attributed to a relatively higher secondary flow as evident in the streamline plot (Fig. 3). Attention may now be turned on to the higher fin spacing (S*= 0.3) in Fig. 7(b). Local fin Nusselt number, for Gr = 106, shows similar trends as the smaller fin spacing. But for higher Grashof number, local fin Nusselt number along the fin height increases near the base, reaches a maximum value, and thereafter decreases continuously till near the fin tip. However, very near the fin tip, it shows increasing trend. This apparent difference in the results at the intermediate section of the fin surface with higher Grashof number is possibly due to the presence of relatively strong secondary flow near the base around the cross-stream direction. Due to strong secondary flow near the base, relatively hot fluid from the base and early part of fin come in contact with the later part of fin. Increasing trend near the fin tip is arguably due to the presence of relatively unheated fluid at the clearance.

###### Local Base Nusselt Number.

Attention will now be turned to the local base Nusselt number. Figure 8(a) shows the trend of local base Nusselt number variation for smaller fin spacing (S*= 0.1). In general, local base Nusselt number variation is parabolic along the X-direction assuming a minimum value at the base-fin corner and a maximum value at the point of symmetry. This is presumably due to the presence of cold fluid near the symmetry line brought by secondary clockwise motion. Fluid passing over the base loses its capacity to receive more heat while traveling toward the fin and hence minimum Nusselt number is noted at the base-fin corer. Similar to local fin Nusselt number, local base Nusselt number decreases with the increasing clearances possibly due to more flow bypass. By changing the Grashof number from 106 to 107, higher Nusselt number obtained is possibly due to higher buoyancy force. For the case of higher fin spacing (S*= 0.3) with Gr = 106, base Nusselt number (Fig. 8(b)) shows similar parabolic trend as observed in the case of smaller fin spacing. Like the smaller fin spacing, with increasing clearances, base Nusselt number decreases. However, with S*= 0.3 and Gr = 107, strong eddying motion is created near the base of the interfin region for all clearances resulting in maximum velocity near the base. Hence, it causes much higher base Nusselt number. Further decrease of local base Nusselt number with the increasing clearances (i.e., C*= 0.3) at all Grashof number is possibly due to increased flow bypass through clearance.

###### Overall Nusselt Number.

Overall Nusselt number variation against clearance spacing is depicted in Figs. 9(a)9(c) and 9(d)9(f) for $Ω$  = 10 and $Ω$  = 30, respectively. It may be noted from the figures that for each of S*= 0.1, 0.2, 0.3, and 0.5, overall Nusselt number maintains a fixed value up to certain clearance, but it decreases thereafter with the clearance. The value of overall Nusselt number is enhanced with the Grashof number, except S*= 0.1. Nusselt number for smaller fin spacing (i.e., S*= 0.1 and 0.2) is, in general, higher than larger fin spacing (S*= 0.3 and 0.5) up to definite value of clearance. As clearance increases, the overall Nusselt number for smaller fin spacing decreases rapidly and assumes a lower value than the higher fin spacing. Lower Grashof numbers (i.e., Gr = 105, and 106) with larger clearance, S*= 0.5 seem to be an optimum fin spacing, but at higher Grashof number (i.e., Gr = 107), S*= 0.3 shows maximum Nusselt number for larger clearance spacing. Further, it may be noted from Fig. 9 that higher conductance parameter causes higher overall Nusselt number, especially at lower fin spacing in which effect is more prominent. By increasing the fin conductance parameter from 10 to 30, i.e., an increase of 200%, Nusselt number is enhanced by around 58% at lower fin spacing (S*= 0.1), while the same increase of Nusselt number is around 134% for the enhancement of fin conductance parameter to 100 representing isothermal fin. In other words, isothermal fin predicts overall Nusselt number 2.34 times as high as that of finite conductive fin within the range of parametric studies. Thus, it may be concluded that the results obtained with the consideration of isothermal fin (e.g., Al-Sarki et al. [20], Acharya and Patankar [24]) overrate significantly from practice especially at lower fin spacing.

Presently computed overall Nusselt number is correlated reasonably well with the governing parameters of the problem as below (Fig. 10)

Display Formula

(34)$4Ω0.002Nu0.56=[18.6Grm−2.3+(0.06Grm)+0.31]0.75$

where $Grm=(0.235+C*)−1.386S*0.65Gr0.12$. The above correlation (Eq. (34)) is developed with 264 data points. Correlation coefficient is found to be 0.98. The validity of the above correlation is for the parameter range of 0.1 ≤ S*≤ 0.5, 0 ≤ C*≤ 0.3, 105 ≤ Gr ≤ 107, 10 ≤ Ω ≤ 100. Experimental results of Dogan and Sivrioglu [2729] are also plotted in Fig. 10 for the sake of comparison. Experimental results agreed reasonably well with the proposed correlation indicating usefulness of the present result.

###### Mass Bypass and Friction Factor.

Fraction of mass by-pass (dashed line) through clearance and friction factor (solid line) is plotted in Figs. 11(a)11(c) for various Grashof numbers. Fraction of mass bypass means: the ratio of mass flow through the clearance to the total mass flow through the section (Das and Giri [32,33]). It can be clearly seen that in the case of smallest fin spacing (S*= 0.1), mass flow fraction through clearances is significantly enhanced even for a clearance spacing of C*= 0.1. Other cases, in particular S*= 0.2, mass bypass is reasonable up to clearance spacing C*= 0.15. Thereafter, flow bypass is significantly enhanced as compared to clearance area ratio. It may be recalled that clearance space remains mostly unheated for smaller fin spacing. Hence, significant flow bypass cannot be encouraged. But in the case of higher fin spacing, higher clearances up to certain limit can be tolerated keeping in view that more homogeneous temperature distribution is observed. Clearance is an important parameter, since larger clearance means larger area of flow that causes more fluid flow through the channel. Hence, heat removal capacity of the base-fin system increases. Again, it may be pointed out that very large clearance decreases the thermal performance of the fin even for larger fin spacing.

Numerically obtained values of the product of friction factor (f) and Reynolds number (Re) are plotted against clearance spacing in Figs. 11(a)–11(c) for different Grashof numbers (solid line). Friction factor is calculated as follows: Display Formula

(35)$f=2(−dpdz)Deρwav2$

where the hydraulic diameter De for the case of a typical module is expressed as Display Formula

(36)$De=4(H+C)S2/(H+S)$

From the above two Eqs. (35) and (36), the following relation: Display Formula

(37)$fRe=(8/Wav)((1+C*)S*1+S*)2$

is obtained. Numerical results obtained thus are presented in the figure mentioned above. Result of smaller fin spacing shows that fRe increases with the clearance, reaches a maximum, and thereafter decreases or remains the same with the clearance spacing for the range of parameters considered presently. For higher fin spacing (S*= 0.3) with Gr = 105, fRe initially increases with the clearance spacing, reaches a maximum, and afterward maintains the same value with the increasing clearances. But in the case of higher fin spacing with high Grashof numbers fRe continuously increases with the Grashof numbers. Since it is a product of two parameters, namely, f and Re, its nature depends on the influence of these parameters with the clearances. When clearance increases, pressure drop decreases due to less resistance (for same inlet velocity), while on the other hand, Reynolds number increases due to the increase of hydraulic diameter. With the increase in Grashof numbers fRe increases for higher fin spacing is possibly due to relatively strong cross-stream secondary flows which causes larger pressure drop. To complete the presentation of fRe of the present result, a useful form correlation of fRe involving the governing parameter of the problem is obtained as below (Fig. 12)

Display Formula

(38)$Gr0.95(fRe−4)0.7=(0.2Prm−4.9+0.11Prm+0.15Prm0.445)0.5$

where $Prm=5((1+C*)−1.897S*0.67Ω0.06)$. The correlation is developed with 304 data points. The correlation coefficient of the above correlation is 0.96. The correlation is valid in the parameter range of 0.1 ≤ S *≤ 0.5, 0 ≤ C*≤ 0.3, 105 ≤ Gr ≤ 107, 10 ≤ Ω ≤ 100.

## Conclusions

A numerical study of conjugate mixed convection heat transfer from a shrouded nonisothermal vertical fin heat sink in horizontal channel has been made. Attention is paid to the region of flow, where flow is hydraulically and thermally developed. The involving parameters fin spacing, clearance, fin conductance, and Grashof numbers are varied from 0.1 to 0.5, 0 to 0.3, 10 to100, and 105 to 107, respectively. Prandtl number is kept fixed at 0.7.

Local convective fin Nusselt number shows remarkable variation along the fin height and indicates hardly any general trend. However, almost all cases highlight large convective heat transfer from the fin tip except the cases of zero fin tip clearance. Local convective base Nusselt number variation is parabolic in nature with a minimum value at the base-fin corners and a maximum value at the symmetry plane. Overall Nusselt number shows noticeable variation with the fin spacing as well as with the clearance. By increasing the fin conductance parameter from 10 to 30, i.e., enhancing the conductance by 200%, an enhancement of Nusselt number is noted to be around 58%, while the same Nusselt number enhancement is 134% for the fin conductance parameter 100 representing isothermal fin. This means that isothermal fin predicts overall Nusselt number as high as 2.34 times that of nonisothermal fin with the range of parameters. Whenever clearance is small, smaller fin spacing seems to be more efficient for the purpose of heat removable. However, larger fin spacing is more effective when the clearance is relatively large. Another important outcome of the present investigation is the product of friction factor and Reynolds number. Results of fRe for smaller fin-spacing indicate that it increases initially, reaches a maximum, and then decreases with the clearance spacing for all Grashof numbers. With the increase in clearance, fRe for higher fin spacing increases initially and thereafter remains same or increases possibly due to secondary motion. Finally, total overall Nusselt number and fRe are correlated with the governing parameters of the problem.

## Nomenclature

• Ac =

cross-sectional area of fin geometry, 0.5S (H + C) (m2)

• C =

fin tip to shroud clearance (m)

• C* =

dimensionless tip clearance, C/H

• g =

gravitational acceleration (m/s2)

• Gr =

thermal Grashof number, gβQH3/(ν2k)

• h =

heat transfer coefficient (W/m2 K)

• H =

fin height (m)

• k =

thermal conductivity of fluid (W/m K)

• kf =

thermal conductivity of fin (W/m K)

• Nu =

Nusselt number

• p =

pressure (Pa)

• $p¯$ =

average pressure over the section (Pa)

• Pr =

Prandtl number, ν/α

• Q =

heat input per unit axial length (W/m)

• Re =

Reynolds number, wavH/ν

• S =

fin spacing (m)

• S* =

dimensionless fin spacing, S/H

• t =

fin thickness (m)

• T =

temperature (K)

• u, v =

velocity component in x, y direction, respectively (m/s)

• U, V =

nondimensional velocity in x, y direction, respectively (uH/ν, vH/ν)

• w =

velocity component in z-direction (m/s)

• W =

dimensionless velocities in Z-directions, −/(H2(d$p¯$/dz))

• x,y,z =

cross stream and axial coordinates (m)

• X,Y,Z =

dimensionless cross stream and axial coordinates, x/H, y/H and (z/H)/(wavH/α)

Greek Symbols
• $α$ =

thermal diffusivity (m2/s)

• $β$ =

thermal volumetric expansion coefficient, $−(1/ρw)(∂ρ/∂T)=1/Tw$ (1/K)

• $ΔT$ =

scaling temperature difference, Q/k (K)

• $θ$ =

dimensionless temperature, (Tw − T)/(Q/k)

• ν =

momentum diffusivity (m2/s)

• $ρ$ =

density (kg/m3)

• Ω =

thermal conductance parameter, kf(t/2)/(kH)

Subscripts
• av =

average quantity

• b =

bulk

• c =

convection

• f =

fin

• l =

local quantity

• s =

shroud

• w =

base

• 0 =

ambient/reference

## References

Starner, K. E. , and McManus, H. N. , 1963, “ An Experimental Investigation of Free Convection Heat Transfer From Rectangular Fin-Arrays,” ASME J. Heat Transfer, 85(3), pp. 273–278.
Welling, J. R. , and Wooldridge, C. V. , 1965, “ Free Convection Heat Transfer Coefficients From Rectangular Vertical Fins,” ASME J. Heat Transfer, 87(4), pp. 439–444.
Harahap, F. , and McManus, H. N. , 1967, “ Natural Convection Heat Transfer From Horizontal Rectangular Fin Arrays,” ASME J. Heat Transfer, 89(1), pp. 32–38.
Jones, C. D. , and Smith, L. F. , 1970, “ Optimum Arrangement of Rectangular Fins on Horizontal Surfaces for Free Convection Heat Transfer,” ASME J. Heat Transfer, 92(1), pp. 6–10.
Fisher, T. S. , and Torrance, K. E. , 1999, “ Experiments on Chimney-Enhanced Free Convection,” ASME J. Heat Transfer, 121(3), pp. 603–609.
Yazicioğlu, B. , and Yüncü, H. , 2007, “ Optimum Fin Spacing of Rectangular Fins on a Vertical Base in Free Convection Heat Transfer,” Heat Mass Transfer, 44(1), pp. 11–21.
Karki, K. C. , and Patankar, S. V. , 1987, “ Cooling of a Vertical Shrouded Fin Array by Natural Convection: A Numerical Study,” ASME J. Heat Transfer, 109(3), pp. 671–676.
Rao, V. R. , and Venkateshan, S. P. , 1996, “ Experimental Study of Free Convection and Radiation in Horizontal Fin Arrays,” Int. J. Heat Mass Transfer, 39(4), pp. 779–789.
Balaji, C. , and Venkateshan, S. P. , 1995, “ Combined Conduction, Convection and Radiation in a Slot,” Int. J. Heat Fluid Flow, 16(2), pp. 139–144.
Rao, V. D. , Naidu, S. V. , Rao, B. G. , and Sharma, K. V. , 2006, “ Heat Transfer From a Horizontal Fin Array by Natural Convection and Radiation–A Conjugate Analysis,” Int. J. Heat Mass Transfer, 49(19–20), pp. 3379–3391.
Yüncü, H. , and Anbar, G. , 1998, “ An Experimental Investigation on Performance of Rectangular Fins on Horizontal Base in Free Convection Heat Transfer,” Heat Mass Transfer, 33(5), pp. 507–514.
Baskaya, S. , Sivrioglu, M. , and Ozek, M. , 2000, “ Parametric Study of Natural Convection Heat Transfer From Horizontal Rectangular Fin Array,” Int. J. Therm. Sci., 39(8), pp. 797–805.
Dayan, A. , Kushnir, R. , Mittelman, G. , and Ullmann, A. , 2004, “ Laminar Free Convection Underneath a Downward Facing Hot Fin Array,” Int. J. Heat Mass Transfer, 47(12–13), pp. 2849–2860.
Sparrow, E. M. , Baliga, B. R. , and Patankar, S. V. , 1978, “ Forced Convection Heat Transfer From a Shrouded Fin Array With and Without Tip Clearance,” ASME J. Heat Transfer, 100(4), pp. 572–579.
Sparrow, E. M. , and Kadle, D. S. , 1986, “ Effect of Tip to Shroud Clearance on Turbulent Heat Transfer From a Shrouded, Longitudinal Fin Array,” ASME J. Heat Transfer, 108(3), pp. 519–524.
Kadle, D. S. , and Sparrow, E. M. , 1986, “ Numerical and Experimental Study of Turbulent Heat Transfer and Fluid Flow in Longitudinal Fin Array,” ASME J. Heat Transfer, 108(1), pp. 16–23.
El-Sayed, S. A. , Mohamad, S. M. , Abdel-Latif, A. M. , and Abouda, A. E. , 2002, “ Investigation of Turbulent Heat Transfer and Fluid Flow in Longitudinal Rectangular-Fin Arrays of Different Geometries and Shrouded Fin Array,” Exp. Therm. Fluid Sci., 26(8), pp. 879–900.
Elshafei, E. A. M. , 2007, “ Effect of Flow Bypass on the Performance of a Shrouded Longitudinal Fin Array,” Appl. Therm. Eng., 27(13), pp. 2233–2242.
Zhang, Z. , and Patankar, S. V. , 1984, “ Influence of Buoyancy on the Vertical Flow and Heat Transfer in a Shrouded Fin Array,” Int. J. Heat Mass Transfer, 27(1), pp. 137–140.
Al-Sarkhi, A. , Abu-Nada, E. , Akash, B. A. , and Jaber, J. O. , 2003, “ Numerical Investigation of Shrouded Fin Array Under Combined Free and Forced Convection,” Int. Commun. Heat Mass Transfer, 30(3), pp. 435–444.
Giri, A. , and Das, B. , 2012, “ A Numerical Study of Entry Region Laminar Mixed Convection Over Shrouded Vertical Fin Arrays,” Int. J. Therm. Sci., 60, pp. 212–224.
Das, B. , and Giri, A. , 2014, “ Non-Boussinesq Laminar Mixed Convection in a Non-Isothermal Fin Array,” Appl. Therm. Eng., 63(1), pp. 447–458.
Das, B. , and Giri, A. , 2015, “ Mixed Convective Heat Transfer From Vertical Fin Array in the Presence of Vortex Generator,” Int. J. Heat Mass Transfer, 82, pp. 26–41.
Acharya, S. , and Patankar, S. V. , 1981, “ Laminar Mixed Convection in Shrouded Fin Array,” ASME J. Heat Transfer, 103(3), pp. 559–565.
Maughan, J. R. , and Incropera, F. P. , 1990, “ Mixed Convection Heat Transfer With Longitudinal Fins in a Horizontal Parallel Plate Channel—Part I: Numerical Results,” ASME J. Heat Transfer, 112(3), pp. 612–618.
Maughan, J. R. , and Incropera, F. P. , 1990, “ Mixed Convection Heat Transfer With Longitudinal Fins in a Horizontal Parallel Plate Channel—Part II: Experimental Results,” ASME J. Heat Transfer, 112(3), pp. 619–624.
Dogan, M. , and Sivrioglu, M. , 2009, “ Experimental Investigation of Mixed Convection Heat Transfer From Longitudinal Fins in Horizontal Rectangular Channel: In Natural Convection Dominated Flow Regimes,” Energy Convers. Manage., 50(10), pp. 2513–2521.
Dogan, M. , and Sivrioglu, M. , 2010, “ Experimental Investigation of Mixed Convection Heat Transfer From Longitudinal Fins in Horizontal Rectangular Channel,” Int. J. Heat Mass Transfer, 53(9–10), pp. 2149–2158.
Dogan, M. , and Sivrioglu, M. , 2012, “ Experimental and Numerical Investigation of Clearance Gap Effects on Laminar Mixed Convection Heat Transfer From Fin Array in a Horizontal Channel–A Conjugate Analysis,” Appl. Therm. Eng., 40, pp. 102–113.
Das, B. , and Giri, A. , 2014, “ Conjugate Conduction and Convection Underneath a Downward Facing Non-Isothermal Extended Surface,” Energy Convers. Manage., 88, pp. 15–26.
Patankar, S. V. , 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere/McGraw-Hill, New York.
Das, B. , and Giri, A. , 2014, “ Second Law Analysis of an Array of Vertical Plate-Finned Heat Sink Undergoing Mixed Convection,” Int. Commun. Heat Mass Transfer, 56, pp. 42–49.
Das, B. , and Giri, A. , 2016, “ Combined Energy and Exergy Analysis of a Non-Isothermal Fin Array With Non-Boussinesq Variable Property Fluid,” ASME J. Therm. Sci. Eng. Appl., 8(3), p. 031010.
View article in PDF format.

## References

Starner, K. E. , and McManus, H. N. , 1963, “ An Experimental Investigation of Free Convection Heat Transfer From Rectangular Fin-Arrays,” ASME J. Heat Transfer, 85(3), pp. 273–278.
Welling, J. R. , and Wooldridge, C. V. , 1965, “ Free Convection Heat Transfer Coefficients From Rectangular Vertical Fins,” ASME J. Heat Transfer, 87(4), pp. 439–444.
Harahap, F. , and McManus, H. N. , 1967, “ Natural Convection Heat Transfer From Horizontal Rectangular Fin Arrays,” ASME J. Heat Transfer, 89(1), pp. 32–38.
Jones, C. D. , and Smith, L. F. , 1970, “ Optimum Arrangement of Rectangular Fins on Horizontal Surfaces for Free Convection Heat Transfer,” ASME J. Heat Transfer, 92(1), pp. 6–10.
Fisher, T. S. , and Torrance, K. E. , 1999, “ Experiments on Chimney-Enhanced Free Convection,” ASME J. Heat Transfer, 121(3), pp. 603–609.
Yazicioğlu, B. , and Yüncü, H. , 2007, “ Optimum Fin Spacing of Rectangular Fins on a Vertical Base in Free Convection Heat Transfer,” Heat Mass Transfer, 44(1), pp. 11–21.
Karki, K. C. , and Patankar, S. V. , 1987, “ Cooling of a Vertical Shrouded Fin Array by Natural Convection: A Numerical Study,” ASME J. Heat Transfer, 109(3), pp. 671–676.
Rao, V. R. , and Venkateshan, S. P. , 1996, “ Experimental Study of Free Convection and Radiation in Horizontal Fin Arrays,” Int. J. Heat Mass Transfer, 39(4), pp. 779–789.
Balaji, C. , and Venkateshan, S. P. , 1995, “ Combined Conduction, Convection and Radiation in a Slot,” Int. J. Heat Fluid Flow, 16(2), pp. 139–144.
Rao, V. D. , Naidu, S. V. , Rao, B. G. , and Sharma, K. V. , 2006, “ Heat Transfer From a Horizontal Fin Array by Natural Convection and Radiation–A Conjugate Analysis,” Int. J. Heat Mass Transfer, 49(19–20), pp. 3379–3391.
Yüncü, H. , and Anbar, G. , 1998, “ An Experimental Investigation on Performance of Rectangular Fins on Horizontal Base in Free Convection Heat Transfer,” Heat Mass Transfer, 33(5), pp. 507–514.
Baskaya, S. , Sivrioglu, M. , and Ozek, M. , 2000, “ Parametric Study of Natural Convection Heat Transfer From Horizontal Rectangular Fin Array,” Int. J. Therm. Sci., 39(8), pp. 797–805.
Dayan, A. , Kushnir, R. , Mittelman, G. , and Ullmann, A. , 2004, “ Laminar Free Convection Underneath a Downward Facing Hot Fin Array,” Int. J. Heat Mass Transfer, 47(12–13), pp. 2849–2860.
Sparrow, E. M. , Baliga, B. R. , and Patankar, S. V. , 1978, “ Forced Convection Heat Transfer From a Shrouded Fin Array With and Without Tip Clearance,” ASME J. Heat Transfer, 100(4), pp. 572–579.
Sparrow, E. M. , and Kadle, D. S. , 1986, “ Effect of Tip to Shroud Clearance on Turbulent Heat Transfer From a Shrouded, Longitudinal Fin Array,” ASME J. Heat Transfer, 108(3), pp. 519–524.
Kadle, D. S. , and Sparrow, E. M. , 1986, “ Numerical and Experimental Study of Turbulent Heat Transfer and Fluid Flow in Longitudinal Fin Array,” ASME J. Heat Transfer, 108(1), pp. 16–23.
El-Sayed, S. A. , Mohamad, S. M. , Abdel-Latif, A. M. , and Abouda, A. E. , 2002, “ Investigation of Turbulent Heat Transfer and Fluid Flow in Longitudinal Rectangular-Fin Arrays of Different Geometries and Shrouded Fin Array,” Exp. Therm. Fluid Sci., 26(8), pp. 879–900.
Elshafei, E. A. M. , 2007, “ Effect of Flow Bypass on the Performance of a Shrouded Longitudinal Fin Array,” Appl. Therm. Eng., 27(13), pp. 2233–2242.
Zhang, Z. , and Patankar, S. V. , 1984, “ Influence of Buoyancy on the Vertical Flow and Heat Transfer in a Shrouded Fin Array,” Int. J. Heat Mass Transfer, 27(1), pp. 137–140.
Al-Sarkhi, A. , Abu-Nada, E. , Akash, B. A. , and Jaber, J. O. , 2003, “ Numerical Investigation of Shrouded Fin Array Under Combined Free and Forced Convection,” Int. Commun. Heat Mass Transfer, 30(3), pp. 435–444.
Giri, A. , and Das, B. , 2012, “ A Numerical Study of Entry Region Laminar Mixed Convection Over Shrouded Vertical Fin Arrays,” Int. J. Therm. Sci., 60, pp. 212–224.
Das, B. , and Giri, A. , 2014, “ Non-Boussinesq Laminar Mixed Convection in a Non-Isothermal Fin Array,” Appl. Therm. Eng., 63(1), pp. 447–458.
Das, B. , and Giri, A. , 2015, “ Mixed Convective Heat Transfer From Vertical Fin Array in the Presence of Vortex Generator,” Int. J. Heat Mass Transfer, 82, pp. 26–41.
Acharya, S. , and Patankar, S. V. , 1981, “ Laminar Mixed Convection in Shrouded Fin Array,” ASME J. Heat Transfer, 103(3), pp. 559–565.
Maughan, J. R. , and Incropera, F. P. , 1990, “ Mixed Convection Heat Transfer With Longitudinal Fins in a Horizontal Parallel Plate Channel—Part I: Numerical Results,” ASME J. Heat Transfer, 112(3), pp. 612–618.
Maughan, J. R. , and Incropera, F. P. , 1990, “ Mixed Convection Heat Transfer With Longitudinal Fins in a Horizontal Parallel Plate Channel—Part II: Experimental Results,” ASME J. Heat Transfer, 112(3), pp. 619–624.
Dogan, M. , and Sivrioglu, M. , 2009, “ Experimental Investigation of Mixed Convection Heat Transfer From Longitudinal Fins in Horizontal Rectangular Channel: In Natural Convection Dominated Flow Regimes,” Energy Convers. Manage., 50(10), pp. 2513–2521.
Dogan, M. , and Sivrioglu, M. , 2010, “ Experimental Investigation of Mixed Convection Heat Transfer From Longitudinal Fins in Horizontal Rectangular Channel,” Int. J. Heat Mass Transfer, 53(9–10), pp. 2149–2158.
Dogan, M. , and Sivrioglu, M. , 2012, “ Experimental and Numerical Investigation of Clearance Gap Effects on Laminar Mixed Convection Heat Transfer From Fin Array in a Horizontal Channel–A Conjugate Analysis,” Appl. Therm. Eng., 40, pp. 102–113.
Das, B. , and Giri, A. , 2014, “ Conjugate Conduction and Convection Underneath a Downward Facing Non-Isothermal Extended Surface,” Energy Convers. Manage., 88, pp. 15–26.
Patankar, S. V. , 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere/McGraw-Hill, New York.
Das, B. , and Giri, A. , 2014, “ Second Law Analysis of an Array of Vertical Plate-Finned Heat Sink Undergoing Mixed Convection,” Int. Commun. Heat Mass Transfer, 56, pp. 42–49.
Das, B. , and Giri, A. , 2016, “ Combined Energy and Exergy Analysis of a Non-Isothermal Fin Array With Non-Boussinesq Variable Property Fluid,” ASME J. Therm. Sci. Eng. Appl., 8(3), p. 031010.

## Figures

Fig. 1

(a) Schematic diagram of shrouded plate finned heat sink on a horizontal base and (b) computational domain

Fig. 2

Allocation of computational grid arrangement

Fig. 3

Streamline contours of the flow domain for Ω = 30: (a) S*= 0.1, C*= 0, Gr = 106, ψmax = 0.0215, (b) S*= 0.1, C*= 0, Gr = 107, ψmax = 0.215, (c) S*= 0.1, C*= 0.30, Gr = 106, ψmax = 0.127, (d) S*= 0.1, C*= 0.30, Gr = 107, ψmax = 3.3, (e) S*= 0.3, C*= 0, Gr = 106, ψmax = 1.876, (f) S*= 0.3, C*= 0, Gr = 107, ψmax = 9.54, (g) S*= 0.3, C*= 0.30, Gr = 106, ψmax = 3.0, and (h) S*= 0.3, C*= 0.30, Gr = 107, ψmax = 10.35

Fig. 4

W-velocity profile for Ω = 30: (a) S* = 0.1, C*= 0, Gr = 106, (b) S* = 0.1, C*= 0, Gr = 107, (c) S* = 0.1, C*= 0.30, Gr = 106, (d) S* = 0.1, C*= 0.30, Gr = 107, (e) S*= 0.3, C*= 0, Gr = 106, (f)S*= 0.3, C*= 0, Gr = 107, (g) S*= 0.3, C*= 0.30, Gr = 106, and (h) S*= 0.3, C*= 0.30, Gr = 107

Fig. 5

Variation of fin temperature along the fin height for Ω = 30: (a) S*= 0.1 and (b) S*= 0.3

Fig. 6

Temperature profile for Ω = 30: (a) S*= 0.1, C*= 0, Gr = 106, (b) S*= 0.1, C*= 0, Gr = 107, (c) S*= 0.1, C*= 0.30, Gr = 106, (d) S*= 0.1, C*= 0.30, Gr = 107, (e) S*= 0.3, C*= 0, Gr = 106, (f) S*= 0.3, C*= 0, Gr = 107, (g) S*= 0.3, C*= 0.30, Gr = 106, and (h) S*= 0.3, C*= 0.30, Gr = 107

Fig. 7

Local fin Nusselt number variation along the fin height for Ω = 30: (a) S*= 0.1 and (b) S*= 0.3

Fig. 8

Local base Nusselt number variation along the base for Ω = 30: (a) S*= 0.1 and (b) S*= 0.3

Fig. 9

Overall Nusselt number variation with clearance for (a) Ω = 10, Gr = 105, (b) Ω = 10, Gr = 106, (c) Ω = 10, Gr = 107, (d) Ω = 30, Gr = 105, (e) Ω = 30, Gr = 106, and (f) Ω = 30, Gr = 107

Fig. 10

Computed and correlated overall Nusselt number

Fig. 11

Mass bypass and fRe for Ω = 30: (a) Gr = 105, (b) Gr = 106, and (c) Gr = 107

Fig. 12

Computed and correlated fRe

## Tables

Table 1 Comparison of results of various grids for nonisothermal fin with C*= 0.25, Ω = 30, Ra = 10,000
Table 2 Comparison of present results with the experimental results of Maughan and Incropera [26]

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