Conjugate free convection with surface radiation in open top cavity

International Journal of Heat and Mass Transfer 89 (2015) 444–453

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Conjugate free convection with surface radiation in open top cavity Dwesh K. Singh a, S.N. Singh b,⇑ a b

HT Lab, Mechanical Engineering, ISM Dhanbad, Jharkhand 826004, India Mechanical Engineering, ISM Dhanbad, Jharkhand 826004, India

a r t i c l e

i n f o

Article history: Received 31 January 2015 Received in revised form 8 April 2015 Accepted 8 May 2015 Available online 5 June 2015 Keywords: Open cavity Volumetric heat generation Surface radiation Conjugate Convection

a b s t r a c t This paper reports the numerical study of two-dimensional, steady, incompressible, conjugate, laminar, natural convection with surface radiation in open top cavity having air as the intervening medium for different locations of uniform volumetric heat generating source at the left wall. For surface radiation calculations, radiosity–irradiation formulation has been used, while the view factors required therein, are calculated using the Hottel’s Crossed-string method. The analysis has been done for a constant property of fluid, with the Boussinesq approximation assumed to be valid. The effects of the magnitude of heat source, location, the material and surface properties of the heat source on both heat transfer and fluid flow have been studied and discussed. An important contribution from the present work is to find the location of heat source for efficient cooling and it is found at the top of the left wall. Based on a large set of numerical data, correlations have been developed for maximum non-dimensional temperature of heat source for three different locations at the left wall. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Open cavities are encountered in various engineering systems, such as open cavity solar thermal receivers, uncovered flat plate solar collectors having rows of vertical strips, electronic chips, passive systems, etc. The thermal performance of electronic packages containing a number of discrete heat sources has been studied extensively in the literature. The design problem in electronic packages is to maintain cooling of chips in an effective way to prevent overheating and hot spots. This is achieved generally by effective cooling by natural convection, mixed convection, surface radiation and finally by better design. In the latter case, the objective is to maximize heat transfer density so that the maximum temperature specified for safe operation of a chip is not exceeded. Thus, optimum placement of discrete heat source may be required. To date, many literature shows that there are numerous studies on heat transfer by natural convection numerical as well as experimental and by conjugate heat transfer. An excellent review of laminar natural convection has been presented by Ostrach [1]. Benchmark numerical solutions for natural convection in a square enclosure with two isothermal and two adiabatic walls have been obtained by de Vahl Davis and Jones [2]. Ho and Chang [3] studied the effect of aspect ratio

⇑ Corresponding author. Tel.: +91 3262235491; fax: +91 3262296563. E-mail addresses: [email protected] (D.K. Singh), [email protected] (S.N. Singh). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.05.038 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

for natural-convection heat transfer in a vertical rectangular enclosure with 2-dimensional discrete heating. Tou and Zhang [4] performed three-dimensional numerical simulation of natural convection in an inclined liquid-filled enclosure with an array of discrete heaters. Similarly, experimental studies have been performed using open cavities with an aspect ratio of one [5–8]. In the above literatures, only heat transfer by natural convection is considered, i.e. conduction and radiation are neglected. Theoretical conjugate heat transfer by natural convection and radiation has been studied in various other configurations. Balaji and Venkateshan [9] studied interaction of radiation with free convection in an open cavity. Dehghan and Behnia [10] studied both numerically and experimentally conjugate heat transfers in an open-top upright cavity having discrete heater and with the bottom side insulated. Gururaja Rao et al. [11] investigated for conjugate mixed convection with surface radiation from a vertical electronic board equipped with a movable flush-mounted discrete heat source and determined the best position for the heat source. Singh and Venkateshan [12] reported the results of numerical study of natural convection with surface radiation in side vented open cavities using mixed boundary condition for the opening above the right wall. Kuznetsov and Sheremet [13] performed a numerical study of two-dimensional transient natural convection in a rectangular enclosure having finite thickness heat-conducting walls. Martyushev and Sheremet [14,15] numerically studied transient laminar natural convection with surface radiation in a square and cubical enclosure with heat-conducting

D.K. Singh, S.N. Singh / International Journal of Heat and Mass Transfer 89 (2015) 444–453

445

Nomenclature A A1 d Fi,j g G’ G H h J’ J k ks Nuc Nuc Nur Nur Nrf m,n

aspect ratio, H/d geometric ratio, d/t spacing m view factor between element i and j acceleration due to gravity m/s2 elemental irradiation W/m2 elemental dimensionless irradiation, G’/rTref4 height of the cavity, m height of the heat source m elemental radiosity W/m2 elemental dimensionless radiosity, J’/rTref4 conductivity of air W/m-K conductivity of left wall W/m-K  @h  local convection Nusselt number,  1h @X RY average convection Nusselt number, 0 h Nuc dY local radiation Nusselt number, 1h Nrf f RY average radiation Nusselt number, 0 h Nur dY Nut Nuc + Nur radiation flow parameter, rTref4d/[DTrefk]

Pr p P

number of grid points in horizontal and vertical direction Prandtl number, t/a pressure Pa non-dimensional pressure, (pp1)H2/qa2

qv

uniform volumetric heat generation rate, W/ m3

qc,qr ⁄

Ra t T Tref

convective and radiative heat flux W/m2 modified Rayleigh number, gbDTrefH3/(ta) thickness of left wall m temperature K reference temperature of left wall K

solid walls of finite thickness. Bejan and Sciubba [16] presented experimental work to optimize the spacing between two parallel plates using laminar forced convection. Hajmohammadi et al. [17] applied two reliable methods to cope with the rising temperature in an array of heated segments exposed to forced convective boundary layer flow. Hajmohammadi et al. [18] studied for controlling the heat flux distribution by changing the thickness of heated wall. Hajmohammadi et al. [19] investigated for improvement of forced convection cooling due to the attachment of heat sources to a conducting thick plate. da Silva et al. [20] used constructal theory for optimal distribution of discrete heat sources on a plate with laminar forced convection. Lindstedt and Karvinen [21] performed a simple analytical study for a flat plate cooled on one surface by forced or natural convection while the other side remains at uniform temperature. Hajmohammadi et al. [22] applied semi-analytical method for conjugate heat transfer. Hajmohammadi and Nourazar [23] reported the results of conjugate forced convection heat transfer from a heated flat plate of finite thickness and temperature-dependent thermal conductivity. A conjugate analysis with finite volume approach is performed by Hajmohammadi et al. [24] to study the effects of a thick plate on the excess temperature of an iso-heat flux heat source cooled by laminar forced convection flow. From a careful review of the literature, it is clear that a very few works have been done in the present geometry having displaced volumetric heat generating source at the left wall and dissipates heat due to adjacent fluid due to conjugate free convection with surface radiation So, the main objectives of the present study are to investigate the influence of local volumetric heat generating

T1 u,v U,V

DTref x,y X,Y Yh DY DYh

ambient temperature K vertical and horizontal velocity m/s dimensionless vertical and horizontal velocity, uH/a and vH/a modified reference temperature difference, qvHt/ks K horizontal and vertical coordinate m dimensionless horizontal and vertical coordinate, x/H and y/H dimensionless height of the heat source, h/H dimensionless height of the wall element dimensionless height of the heat source element

Greek symbols a fluid thermal diffusivity m2/s b isobaric coefficient of volumetric thermal expansion, 1/T 1/K e emissivity of the walls c thermal conductance parameter, kd/kst f

dimensionless radiative heat flux, qr/rT4ref

t

kinematic viscosity of the fluid m2/s dimensionless temperature, (TT1)/DTref

h hmax W’

W

r

dimensionless maximum temperature, (TT1)/DTref stream function m2/s dimensionless stream function, w’/a Stefan Boltzmann constant, 5.67  108 W/m2 K4

Subscripts 1 ambient i,j any two arbitrary area element rf radiation flow

discrete heat source on flow and heat transfer and subsequently fixing the best location of the same for cooling purposes. Based on a large set of numerical data, a correlation for maximum non-dimensional temperature has been also developed. 2. Mathematical formulation 2.1. Formulation for convection and conduction The two-dimensional, steady, incompressible, laminar natural convection heat transfer from left wall with very thin volumetric heat generating source in open top cavity with fixed height H, spacing d, is considered using the system of coordinates as shown in Fig. 1. There is a volumetric heat generating source of height h (25% of the left wall) and thermal conductivity, ks in which heat conduction is one-dimensional and thickness t provided in the left vertical wall of the cavity. Conductivity of heat source is independent of temperature level. It is possible to change the location of the heat source along the left wall of the cavity such that the heat source can take up three different position viz. (a) bottom (b) mid (c) top in the cavity. The governing equations for mass, momentum and energy for a constant property fluid under the Boussinesq approximation, in the non-dimensional form are:

@U @V þ ¼0 @X @Y

ð1Þ

" # @U @U @P @2U @2U U þV ¼ þ Pr þ @X @Y @X @X 2 @Y 2

ð2Þ

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element on a boundary of the cavity the non- dimensional radiosity is given by the equation:

Y,V

J i ¼ ei

Qcond, (Y+ΔY)

ΔY

qvΔyt

Qconv Qrad

g t

Qcond, Y

Adiabac walls, qc +qr=0

X,U

Fig. 1. Schematic of the Geometry.

" # @V @V @P @2V @2V þ Ra Prh U þV ¼ þ Pr þ @X @Y @Y @X 2 @Y 2

4

þ ð1  ei Þ

n X F ij J j

ð3Þ

@h @h @ 2 h @ 2 h þV ¼ þ @X @Y @X 2 @Y 2

ð4Þ

@w @w ; V ¼ @Y @X

2.2. Temperature distribution along heat source at the left wall



Energy balance of conduction (Qcond), convection (Qconv) and radiation (Qrad) at the differential element of heat source provided on the left wall, which is shown, enlarged in the inset of Fig. 1, yields the following equation for the temperature distribution along the heat source (except end points) at the left wall, in the normalized form:

Q cond;y þ qv Dyt ¼ Q cond;ðyþDyÞ þ Q conv þ Q rad The above equation after normalization and subsequent simplification leads to

     @h A1 e   þc þ cNrf @X X¼0 A 1e @Y 2

@h ¼ Nrf ðJ  GÞ @X

Open top:

U ¼ 0;

@2h

@h ¼ Nrf ðJ  GÞ @Y

Right wall:

U ¼ 0; V ¼ 0 and

The stream function is calculated from its definition as

U¼

ð8Þ

The velocity boundary conditions on the solid rigid walls are based on the assumption that the trapped air does not sleep on the walls. Temperature boundary condition at the top open section is based on flow condition. At the adiabatic bottom, right and rest parts of the left wall convection and radiation balance each other (see Fig. 1). Hence Bottom wall:

U ¼ 0; V ¼ 0 and U

i ¼ 1; 2ðm þ n  2Þ

j¼1

2.4. Boundary conditions

Differential element of heat source

d

Ti Th

The first term on the right-hand side represents the emission term and the second term is the reflected radiation. The summation is over all the irradiation terms for that particular element. After all the radiosities are known, the irradiations are evaluated and net radiant flux from each zone is calculated.

H

h



"

T T ref

4

# J ¼0

ð5Þ

For the elements at interface of the bottom and top ends of the heat source the equations for temperature variation has been separately derived by making energy balance as below:

"  #   4 @h DY @h A 1 DY  e  DY T þc þ  cNrf J ¼0 @Y 2 @X X¼0 A 2 T ref 1e 2 ð6Þ "  #   4 @h DY @h A 1 DY  e  DY T c  þ cNrf J ¼0 @Y 2 @X X¼0 A 2 T ref 1e 2 ð7Þ

@h @Y

@V ¼0 @Y

 ¼ 0;

hin ¼ 0

out

Left wall:

U ¼ 0 and V ¼ 0 @h ¼ Nrf ðJ  GÞ for the non-heat source parts of the left wall: @X 3. Solution procedure The governing Eqs. (1)–(4) are discretized by the finite volume method on a non-uniform staggered grid system using the SIMPLE algorithm of Patankar [25] with first order UPWIND scheme for convective - diffusive terms . The set of discretized equations were then solved by a line-by-line procedure of the tri-diagonal matrix algorithm (TDMA). In radiosity-irradiation equation view factors are evaluated using Hottel’s crossed string method [26]. A computer code is developed under FORTRAN platform. As convergence criteria, 103 is chosen for all dependent variables. Based on grid refinement test, numbers of grid points are taken as 51  41. Results of grid refinement test are presented in the ensuing section. A cosine and semi- cosine function have been chosen to generate the non-uniform grids respectively along X and Y directions. For derivative boundary conditions, three point formulae using second degree Lagrangian polynomial have been used. 4. Results and discussion

2.3. Formulation for radiation The radiosity-irradiation formulation is used to describe surface radiation. The walls are assumed diffuse and gray. For an area

Table 1 shows the range of parameters considered in the present study. Thermal conductivity varies from 0.25 to 0.5 W/m-k. One example is epoxy glass (coated with Mylar) having thermal

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4.2. Code validation

Table 1 Range of parameters. Parameters

Range

3

4

qv, W/m ks, W/m-K Ra⁄ Nrf

6

5  10 –1  10 0.25–0.50 1  105–1  107 17.11–58.758 1.359–2.719 0.05–0.85

c e

conductivity 0.26 W/mk. Calculations have been made keeping in view the objective of evolving useful correlations for hmax.

In order to validate the employed method and check the code, results for isothermal boundary condition at the left wall in an air-filled open top cavity of aspect ratio, A = 4, Rayleigh number Ra = 5  105, e = 0 and e = 1were obtained and compared with the Balaji and Venkateshan [9] in terms of local convection Nusselt number (Nuc). The convection Nusselt number distribution along the left wall for the two cases of emissivity is given in Fig 2. Since, the convection Nusselt number is locally distributed at the left wall. Therefore, for quantitative comparison, maximum and minimum convection Nusselt number at the leading and trailing edge of right vertical wall is considered for the two values of emissivity as shown in Table 3. The comparison between the local

4.1. Grid refinement test

25.0

ε =0 ε =1

A=4

22.5 20.0 17.5 15.0

Nu c

To study the effect of grid size on the solution, a case with qv = 5  105 W/m3, Ra* = 2.307  106, A = 2, e = 0.8 and ks = 0.5 W/m-K is considered, and the results are shown in Table 2. The present problem involves an interaction between free convection and surface radiation. Since the grid size affects these two to different extents. It is necessary to look at the effect of grid size on both the average convection and radiation Nusselt number in order to decide the grid size. The grid sensitivity analysis is done in two parts – (a) ‘M’ fixed with varying ‘N’ and (b) ‘N’ fixed with varying ‘M’ and optimum grid size is arrived at. Table 2(a) shows when M is fixed, the differences in Nuc between the grids size of 31  21 and 31  31 is 0.178% which is the lowest. However, the differences in Nur , Nut and hmax are the biggest. Also, 31  21 is a relatively coarse grid. The difference in Nur between the grids size of 31  51 and 31  61 is 0.240% which is the lowest. However, the differences in Nuc , Nut and hmax are comparatively higher than for other grids size. The difference in Nut between the grids size of 31  41 and 31  51 is 0.025% which is the lowest and its Nuc , Nur and hmax are comparatively smaller than others. Based on these observations N has been fixed as 41. Similarly, when N is fixed Table 2(b) shows that the difference in Nut between the grids size of 51  41 and 61  41 is 0.575% which is the lowest and Nuc ; Nur and hmax have a comparatively smaller value than others. Based on these observations the M is taken as 51. Thus, the grid pattern used in the present work is taken as 51  41.

12.5 10.0 7.5 5.0 2.5

0

1

2

3

4

Y Fig. 2. Comparison of present work for e = 0; e = 1. Rayleigh number, Ra = 5  105.

Table 3 Comparison of present work with Balaji and Venkateshan [9].

Balaji and Venkateshan [9] Present results Deviation (%)

Nuc (Leading edge)

Nuc (Trailing edge)

e=0

e=1

e=0

e=1

23.75 22.60 5.08

16.40 16.15 1.54

6.95 7.22 3.88

7.05 7.11 0.85

Table 2a Grid refinement test M = 31, N varied. MN

Nuc

Nur

Nut

hmax

%change in Nuc

%change in Nur

%change in Nut

%change in hmax

31  21 31  31 31  41 31  51 31  61 31  71

17.365 17.396 17.347 17.305 17.261 17.202

6.364 6.082 6.179 6.227 6.242 6.261

23.729 23.479 23.527 23.533 23.503 23.463

0.864 0.855 0.861 0.859 0.856 0.852

– 0.178 0.281 0.242 0.254 0.342

– 4.431 1.569 0.771 0.240 0.303

1.054 0.204 0.025 0.127 0.170

1.042 0.696 0.232 0.349 0.467

Table 2b Grid refinement test N = 41, M varied. MN

Nuc

Nur

Nut

hmax

%change in Nuc

%change in Nur

%change in Nut

%change in hmax

21  41 31  41 41  41 51  41 61  41 71  41

18.233 17.347 17.245 17.221 17.200 17.161

6.204 6.179 6.107 6.294 6.447 6.714

24.438 23.527 23.353 23.511 23.647 23.875

0.838 0.861 0.866 0.856 0.859 0.851

– 4.859 0.588 0.139 0.122 0.227

– 0.403 1.165 2.971 2.373 3.976

3.872 0.739 0.672 0.575 0.955

2.671 0.577 1.155 0.349 0.931

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Nusselt numbers at the leading and trailing edge show the percentage difference of 5.08% and 3.88% for e = 0 and 1.54% and 0.88% for e = 1. Thus, comparison results clearly show the good agreement between the Balaji and Venkateshan [9] results and those of the present study. 4.3. Typical results for open top cavity Having validated the present code with previous results of Balaji and Venkateshan [9], a detailed parametric study has been undertaken. Typical results from this study are presented here for open top cavity having discrete volumetric heat generation source at different locations viz. (a) bottom position (b) mid position and (c) top position at the left wall of the cavity. 4.3.1. Flow patterns and isotherms Different locations of heat source on the left wall plays a decisive role with regard to the flow pattern and heat transfer that resides in the cavity. Fig. 3 shows the streamlines and isotherms for the two values of emissivity, e = 0.05 and e = 0.85 with ks = 0.25 W/m-K, qv = 1  105 W/m3 and Ra⁄ = 9.231  105. Considering the weak radiation case i.e. e = 0.05 when heat source is placed at the bottom of the left wall as shown in Fig. 3(a), ambient air enters into the cavity up to the bottom wall. After heating of air by the heat source placed at the bottom of the left wall, the same start moving upward only along the left wall forming a single loop circulation as it is confirmed by the corresponding isotherms which show a thin boundary layer on the left wall. On placing the heat source at the bottom of the left wall for strong radiation i.e. e = 0.85 as shown in Fig. 3(b), the adiabatic walls of the cavity get heated due to intense surface radiation heat exchange as evidenced by the thermal boundary layer development. Thus, the air adjacent to the walls of the cavity is heated due to convection heat transfer. As a result both the side walls are induced by up-flow forming two loop circulations as shown in Fig. 3(b). When heat source is placed at the mid of the left wall as shown in Fig. 3(a) for weak radiation i.e. e = 0.05, two zones of fluid circulation are observed in the contour of streamlines. In the first zone

closer to the left wall the incoming ambient air comes down and turn around towards the open top of the cavity after taking heat from the source before reaching the bottom wall which is not heated due to weak radiation as it is confirmed by the corresponding isotherms. The trapped air is recirculated in the non-heating zone near the bottom and right wall of the cavity as clearly shown in Figure. It is to be noted that as the emissivity increases from e = 0.05 to e = 0.85 as shown in Fig. 3(b) by placing the heat source at the mid of the left wall, flow patterns are almost similar except the increment of the strength of second circulation due to convection heat transfer from the adiabatic walls to the air. When heat source is placed at the top of the left wall under weak radiation case i.e. e = 0.05 as shown in Fig. 3(a), air inside the cavity forms a single loop flow pattern adjacent to the left wall due to the absence of heat source at the bottom wall and weak radiation. Keeping the position of the heat source as earlier case for strong radiation (e = 0.85) as shown in Fig. 3(b), it results in the radiative heating of adiabatic walls due to which both the side walls are induced by up-flow forming two loops circulation as shown in Fig. 3(b). Transition from single loop to double loop circulation takes place at different values of emissivity and volumetric heat generation. Effect of conductivity is studied by varying the conductivity of heat source as shown in Fig. 4 for different locations. Results are obtained for qv = 1  105, e = 0.5, Ra⁄ = 9.231  105 and 4.615  105. For ks = 0.50 W/m-K as shown in Fig. 4(b), stream lines and isotherms exhibit similar trends of Fig. 4(a) for ks = 0.25 W/m-K except for the case when heat source is at top of left wall. In the case when heat source is at top of the left wall a secondary vortex is developed at the bottom right corner. Conductive heat transfer increases on increasing the conductivity of heat source from ks = 0.25 W/m-K to ks = 0.50 W/m-K which leads to decrement in temperature of heat source. Due to this decrement in temperature radiative heat flux from the heat source decreases and results in the elimination of secondary vortex developed due to radiative heating of adiabatic walls as shown in Fig. 4(b). Increment in conductivity does not affect the intensity of stream function for the case when heat source is at the bottom

Fig. 3. Streamline and isotherm for different location of the heat source (bottom, mid and top) arranged left to right at (a) e = 0.05 (b) e = 0.85.

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D.K. Singh, S.N. Singh / International Journal of Heat and Mass Transfer 89 (2015) 444–453

Fig. 4. Streamline and isotherm for different location of the heat source (bottom, mid and top) arranged left to right at (a) ks = 0.25 W/m-K (b) ks = 0.50 W/m-K.

0.50

qv=1x105, Ra*=9.231x105

0.40

qv=1x106, Ra*=9.231x106

0.35 0.30

0.4

0.45

qv=1x105,Ra*=9.231x105 qv=1x106,Ra*=9.231x106

0.40 0.35 0.30 0.25

0.25 0.20

0.0

0.1

0.2

0.3

0.4

0.5

0.7

0.8

Yh

(a)

0.9

1.0

Y

h

(b)

1.1

1.2

Dimensionless temperature ( θ)

0.45

Dimensionless temperature ( θ)

Dimensionless temperature (θ)

0.50

0.3

0.2

qv=1x105,Ra*=9.231x105 qv=1x106,Ra*=9.231x106

0.1

0.0 1.4

1.5

1.6

1.7

1.8

1.9

2.0

Y

h

(c)

Fig. 5. Variation of dimensionless temperature (h) along the heat source (a) Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

of the left wall, whereas, the increment in conductivity from ks = 0.25 to ks = 0.50 W/m-K suppresses the intensity of stream function for the case when heat source is at mid and top of the left wall. Changes in conductivity of heat source alters the temperature distribution inside the cavity for all the different locations of heat source as shown in Figs. 4(a) and (b).

4.3.2. Variation of non-dimensional temperature with parameters (qv,

e and c) Fig. 5 shows the variation of non-dimensional temperature along the volumetric heat generating source for different locations viz. (a) When heat source is placed at the bottom of the left wall (b) heat source is placed at the mid of the left wall and (c) heat source is placed at the top of the left wall. Results are presented for two different values of qv and Ra⁄ as shown in figure with e = 0.5 and ks = 0.25 W/m-K. Fig. 5 shows that for all the three different locations non-dimensional temperature increases along the heat source and reaches a maximum value at the end of the heat source.

Comparing among the three locations of heat source, sharp increase of h is not found in the case when heat source is at bottom position. From physical standpoint of view this is justified, because bottom wall gets heated due to radiative heat flux emitted (e = 0.5) by the heat source and transfer heat convectively to the air. Also, from the Fig. 5(a)–(c), one can find that difference in minimum and maximum temperature within the heat source is greater when heat source is at the top position and it is smaller for bottom position. So, it is observed that degree of increase in non-dimensional temperature within the heat source is smaller for bottom position and larger for top position of left wall. Since reference temperature Tref and Ra⁄ increase with the increase in qv, which lead to increase in buoyancy force caused by temperature difference resulting in the decrement of non-dimensional temperature as it is depicted in Fig. 5(a)–(c). The maximum non-dimensional temperature is obtained for the case when heat source is at the mid position and it is smaller when heat source is at the top position of the left wall where it is more exposed to ambient condition.

D.K. Singh, S.N. Singh / International Journal of Heat and Mass Transfer 89 (2015) 444–453

0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8

at the bottom of the cavity at the mid of the cavity at the top of the cavity

0.4

0.3

0.2

0.1

0.0

1.0

0.0

Width of the cavity

0.2

0.4

0.6

0.8

at the bottom of the cavity at the mid of the cavity at the top of the cavity

0.4

Dimensionless temperature (θ)

at the bottom of the cavity at the mid of the cavity at the top of the cavity

0.4

Dimensionless temperature (θ)

Dimensionless temperature (θ)

450

0.3

0.2

0.1

0.0

1.0

0.0

Width of the cavity

(a)

0.2

0.4

0.6

0.8

1.0

Width of the cavity

(b)

(c)

Fig. 6. Variation of dimensionless temperature (h) along the width of the cavity (a) Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

Fig. 6 shows non-dimensional local temperature distribution across the width of the cavity for different locations of heat source at three different section of the cavity. The parameters set is taken as Ra⁄ = 9.231  105, qv = 1  105 W/m3, e = 0.85 and ks = 0.25 W/m-K. It can be seen that bottom wall is substantially heated because of radiation which is more pronounced in the case when heat source is at the bottom position as discussed earlier for Fig. 5. Curves also show that for all the three different locations of heat source at the left wall of cavity, maximum temperature is generated at the left wall within the heat source and substantially decreases up to one-half of width of the cavity. It then again start increasing up to the right wall due to the radiation exchange. Further when heat source is at bottom position, it is found that temperature at the top of the left wall is lower than the centre of the left wall of the cavity which clearly shows that heat is carried out by air from bottom section of the cavity but the same trend is not shown in case when heat source is placed at the mid and top of the left wall. Because air after getting heat from the source starts moving upward and escapes out of the cavity. Fig. 7 shows the variation of hmax inside the cavity with emissivity (e) for different values of heat generation (qv). As shown in Figs 7(a)–(c), ten different values of e and four different values of qv are considered with conductivity ks = 0.25 W/m-K and Ra⁄ = 1  105 for different locations viz. (a) bottom position (b) mid position and (c) top position. Figures show that for a particular

qv=5 X104 qv=1 X105 qv=5 X105

0.60

0.60

qv=1 X106

0.55

0.50

θ max

θ max

0.55

0.45 0.40

0.50

0.55 0.50 0.45

0.45

0.40 0.35

0.35

0.30

0.30

0.30

0.0

0.2

0.4

0.6

Emissivity (ε)

(a)

0.8

1.0

0.25

qv=5 X104 qv=1 X105 qv=5 X105 qv=1 X106

0.60

0.40

0.35

0.25

qv=5 X104 qv=1 X105 qv=5 X105 qv=1 X106

0.65

θ max

0.65

value of qv maximum non-dimensional temperature decreases substantially for all the three different locations of heat source at the left wall of the cavity. For different values of qv and locations of heat source, hmax undergoes a marked drop in its value as e increases from 0.05 to 0.85. When heat source is at bottom and top position degree of decrement in hmax is 21% as e changes from 0.05 to 0.85 for qv = 5  104, 1  105 and 1  106 W/m3 whereas it is 20% for qv = 5  105 W/m3. For the mid position, hmax drops down by 25% for qv = 5  104 and 1  105 W/m3 which shows that effect of emissivity between the same limit is somewhat higher for these two values of qv at this position. Whereas, for the same position the decrement in hmax for qv/ = 5  105 W/m3 is 20% and for qv = 1  106 W/m3 degree of decrement in hmax is 22% are observed. The effect of conductance (c) on hmax is plotted in Fig. 8 for six different values of c at four different values of volumetric heat generation (qv) with e = 0.85 as shown in figures. From Fig. 8(a)–(c) for different locations of heat source at the left wall, it is investigated that with increase in c, maximum non-dimensional temperature hmax substantially decreases. It is to be noted, the case considered here when heat source is at bottom and mid position of the left wall for e = 0.85, hmax decreases by 50% when c changes from 1.359 to 2.719 for all the four different values of qv. In the same range of c, however, in Fig 8(c) when heat source is located at the top of the left wall reveals the decrement in hmax is about 45% for qv = 5  104 and 1  105 W/m3 and the same decreases

0.0

0.2

0.4

0.6

Emissivity (ε)

(b)

0.8

1.0

0.25

0.0

0.2

0.4

0.6

Emissivity (ε)

0.8

1.0

(c)

Fig. 7. Variation of hmax with emissivity for different heat source position (a) Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

451

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qv=5 X104 qv=1 X105 qv=5 X105 qv=1 X106

0.8

θmax

0.7 0.6

0.9 0.8

θmax

0.9

qv=5 X104 qv=1 X105 qv=5 X105 qv=1 X106

1.0

0.5

0.7 0.6

0.7 0.6

0.5

0.4

0.5 0.4

0.4

0.3

qv=5 X104 qv=1 X105 qv=5 X105 qv=1 X106

0.8

θ max

1.0

0.3

0.3

0.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.2 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Conductance (γ)

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Conductance (γ)

Conductance (γ)

(a)

(b)

(c)

ε=0.1, qv=5x104 ε=0.8, qv=5x104 ε=0.1, qv=1x106 ε=0.8, qv=1x106

30 25 20 15 10

0.0

0.1

0.2

0.3

0.4

34 32 30 28 26 24 22 20 18 16 14 12 10 0.7

ε=0.1, qv=5x104 ε=0.8, qv=5x104 ε=0.1, qv=1x106 ε=0.8, qv=1x106

0.8

0.9

Yh

1.0

1.1

Local Convective Nusselt number

35

Local Convective Nusselt number

Local Convective Nusselt number

Fig. 8. Variation of hmax with conductance for different heat generating position (a)Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

45

ε=0.1, qv=5x104 ε=0.8, qv=5x104 ε=0.1, qv=1x106 ε=0.8, qv=1x106

40 35 30 25 20 15 10 1.5

1.2

1.6

(a)

1.7

1.8

1.9

Yh

Yh

(b)

(c)

Fig. 9. Variation of local convective Nusselt number along the heat source (a) Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

7 6 5

ε=0.1, qv=5x104 ε=0.8, qv=5x104 ε=0.1, qv=1x106 ε=0.8, qv=1x106

4 3 2 1

8 7 6 5

ε=0.1, qv=5x104 ε=0.8, qv=5x104 ε=0.1, qv=1x106 ε=0.8, qv=1x106

4 3 2 1

0.0

0.1

0.2

0.3

0.4

0.7

0.8

0.9

1.0

Yh

Yh

(a)

(b)

1.1

Local Radiative Nusselt number

Local Radiative Nusselt number

8

Local Radiative Nusselt number

9

9

10 8 6

ε=0.1, qv=5x104 ε=0.8, qv=5x104 ε=0.1, qv=1x106 ε=0.8, qv=1x106

4 2 0

1.5

1.6

1.7

1.8

1.9

Yh

(c)

Fig. 10. Variation of local radiative Nusselt number along the heat source (a) Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

by 47% for qv = 5  105 and 1  106 W/m3. With this quantitative study it is investigated that the effect of c is more pronounced for the case when heat source is located at the bottom and mid position of the left wall inside the cavity. 4.3.3. Variation of local Nusselt number Variation of local convective and radiative Nusselt number Nuc and Nur along the heat source is shown in Figs. 9 and 10 for three

different position of heat source with various combinations of governing parameters. Value of emissivity e and volumetric heat generation qv (W/m3) are shown in figures and the corresponding modified Rayleigh number is taken as Ra⁄ = 4.615  105 and 9.231  106 and Ks = 0.25 W/m-K. For all the different position of heat source, Fig. 9 shows that local convective Nusselt number decreases along the heat source with the exception when heat source is at bottom position. At this position there is a considerable

D.K. Singh, S.N. Singh / International Journal of Heat and Mass Transfer 89 (2015) 444–453

qv=1x105 Ra*=1x105 qv=1x106 Ra*=1x106

, ,

600 400 200 0 -200 -400 -600 -800

0.0

0.2

0.4

0.6

Width of the cavity (X)

0.8

1.0

400

qv=1x105 Ra*=1x105 qv=1x106 Ra*=1x106

, ,

800

Dimensionless vertival velocity

Dimensionless vertival velocity

800

600

Dimensionless vertival velocity

452

400 200 0 -200 -400 -600

qv=1x105 Ra*=1x105 qv=1x106 Ra*=1x106

, ,

200

0

-200

-400

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

Width of the cavity (X)

Width of the cavity (X)

(b)

(c)

(a)

1.0

Fig. 11. Variation of dimensionless vertical velocity at the top of the cavity (a) Heat source at bottom position (b) Heat source at mid position (c) Heat source at top position.

difference among Nusselt number distribution at the bottom of the heat source. This is because of adiabatic condition of bottom wall as already discussed in Section 4.3.2. It can be seen from figures that increase in qv leads to the increment in local convective Nusselt number distribution. Further, the higher value of surface emissivity causes a reduction of convective heat transfer from the heat source, but at the same time increases radiative heat transfer as shown in Fig. 10, because of intense surface radiation interchange among the walls. From Fig 10 for local radiative Nusselt number distribution it is observed that radiative Nusselt number increases along the heat source. This increment can be explained by the increment in non-dimensional temperature along the heat source which results in more radiative heat flux along the heat source. Effect of qv on radiative Nusselt number is studied for two different values. Since, radiation flow parameter (Nrf) is the increasing function of qv, therefore heat transfer by radiation increases with the increase in qv. 4.3.4. Variation of vertical velocity Variation of non-dimensional vertical velocity, V at the top section across the width of the cavity is presented in Fig. 11. Results are depicted for two different values of qv and Ra⁄ as shown in figure with ks = 0.25 W/m-K, e = 0.5. In the vertical velocity profiles shown in Fig. 11(a)–(c), for different locations of heat source, two peaks are observed, longer one near the heated left wall and a second smaller one close to the right adiabatic wall which confirm two loop circulations and boundary layer development at the right adiabatic vertical wall in presence of surface radiation as described in the earlier Section 4.3.1. Radiation heat transfer from the left wall heats both the right and bottom walls and under equilibrium these two walls lose heat convectively to the air. Thus, there are two plumes leaving the cavity. In addition to the above, it is also found that maximum non dimensional velocity attained by the air is approximately same for the cases when heat source is at bottom and mid position. While for the case when heat source is at top position, the maximum non dimensional vertical velocity is almost half for both the values of Ra⁄ and qv in comparison to the bottom and mid position of heat source. The possible reason for this cause is that air after heating at the top position instantly escapes out from the cavity and fresh ambient air re-enters into the cavity which disturb the flow coming out from the cavity. 5. Correlations The range of parameters for which calculation have been done are shown in Table 1. All the walls were assumed to be of the same

emissivity. Based on a large set of data, non- dimensional maximum temperature, hmax for different locations are correlated as: When heat source at the bottom of the left wall: hmax ¼ 21:186ð1 þ Ra Þ

0:208

ðNrf =ð1 þ Nrf ÞÞ0:303 ð1 þ eÞ0:428 ð1 þ cÞ0:794

When heat source at the mid of the left wall: hmax ¼ 23:183ð1 þ Ra Þ

0:211

0:078

ðNrf =ð1 þ Nrf ÞÞ

0:460

ð1 þ eÞ

0:798

ð1 þ cÞ

When heat source at the top of the left wall: 0:161

hmax ¼ 9:116ð1 þ Ra Þ

ðN rf =ð1 þ N rf ÞÞ

1:011

ð1 þ eÞ

0:367

0:710

ð1 þ cÞ

As the modified Rayleigh number Ra⁄ enhances heat transfer by convection which decreases the maximum temperature hmax inside the cavity, as it is confirmed by the negative exponent. The exponent of emissivity and conductance are negative which signify that hmax decreases with the increase in said parameters. In evolving the above correlation, (1 + parameter) term has been used as the most appropriate form , because in multi-mode heat transfer , even, when one mode is not considered or parameter set to zero, hmax would be non-zero, as it still contains heat transfer by other modes. Minimum correlation coefficient of 0.996 and maximum standard error of 2.19% indicates the goodness of fit. 6. Conclusions Conjugate laminar natural convection with surface radiation in a two-dimensional open top cavity due to a volumetric heat generating source for three different locations on the left vertical sidewall is numerically investigated. Main efforts are focused on the location of heat source at the left wall and the effect of independent variables on the fluid flow and heat transfer characteristics, and the following conclusions are obtained. From design point of view for cooling of such type of system one should not ignore the location of heat source which plays a significant role in natural convection heat transfer. The best possible position of heat source is found at the top of the left wall inside the cavity. Surface radiation changes the basic flow physics and enhances the radiative heat transfer as a result of which heat transfer by convection decreases. So, ignoring of surface radiation is not feasible for this kind of problem. Thermal conductance and volumetric heat generation decrease the non-dimensional maximum temperature. Volumetric heat generating rate enhances the convective and radiative heat transfer. Further, a useful correlation has been developed for maximum non-dimensional temperature (hmax).

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