International Journal of Numerical Methods for Heat & Fluid FlowA finite element analysis on combined convection and conduction in a channel with athick walled cavityM.M. Rahman Hakan Oztop S. Mekhilef R. Saidur A. Chamkha A. Ahsan Khaled S. Al-Salem

Article information:To cite this document:M.M. Rahman Hakan Oztop S. Mekhilef R. Saidur A. Chamkha A. Ahsan Khaled S. Al-Salem , (2014),"Afinite element analysis on combined convection and conduction in a channel with a thick walled cavity",International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24 Iss 8 pp. 1888 - 1905Permanent link to this document:http://dx.doi.org/10.1108/HFF-07-2013-0239

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Users who downloaded this article also downloaded:M. Sheikholeslami, R. Ellahi, Mohsan Hassan, Soheil Soleimani, (2014),"A study of natural convectionheat transfer in a nanofluid filled enclosure with elliptic inner cylinder", International Journal ofNumerical Methods for Heat & Fluid Flow, Vol. 24 Iss 8 pp. 1906-1927 http://dx.doi.org/10.1108/HFF-07-2013-0225M.A. Mansour, M.A. Bakeir, A. Chamkha, (2014),"Natural convection inside a C-shaped nanofluid-filledenclosure with localized heat sources", International Journal of Numerical Methods for Heat & FluidFlow, Vol. 24 Iss 8 pp. 1954-1978 http://dx.doi.org/10.1108/HFF-06-2013-0198Amgad Salama, Mohamed El Amin, Shuyu Sun, (2014),"Numerical investigation of natural convection in twoenclosures separated by anisotropic solid wall", International Journal of Numerical Methods for Heat &Fluid Flow, Vol. 24 Iss 8 pp. 1928-1953 http://dx.doi.org/10.1108/HFF-09-2013-0268

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A finite element analysis oncombined convection and

conduction in a channel witha thick walled cavity

M.M. RahmanDepartment of Mathematics, BUET, Dhaka,

Bangladesh and Tower Faculty of Engineering University of Malaya,Kuala Lumpur, Malaysia

Hakan OztopDepartment of Mechanical Engineering, Firat University, Elazig, Turkey

S. MekhilefDepartment of Electrical Engineering, University Malaya,

Kuala Lumpur, Malaysia

R. SaidurManufacturing Engineering Department,

The Public Authority for Applied Education and Training, Shuweikh, Kuwait

A. ChamkhaManufacturing Engineering Department,

The Public Authority for Applied Education and Training, Safat, Kuwait

A. AhsanDepartment of Civil Engineering, University Putra Malaysia,

Selangor, Malaysia, and

Khaled S. Al-SalemDepartment of Mechanical Engineering, King Saud University,

Riyadh, Saudi Arabia

Abstract

Purpose – The purpose of this paper is to examine the effects of thick wall parameters of a cavity oncombined convection in a channel. In other words, conjugate heat transfer is solved.Design/methodology/approach – Galerkin weighted residual finite element method is used to solvethe governing equations of mixed convection.Findings – The streamlines, isotherms, local and average Nusselt numbers are obtained andpresented for different parameters. It is found heat transfer is an increasing function of dimensionlessthermal conductivity ratio.Originality/value – The literature does not have mixed convection and conjugate heat transferproblem in a channel with thick walled cavity.

Keywords Combined convection, Conjugate heat transfer, Nonlinear, Open channel

Paper type Research paper

The current issue and full text archive of this journal is available atwww.emeraldinsight.com/0961-5539.htm

Received 29 July 2013Revised 22 February 2014Accepted 17 April 2014

International Journal of NumericalMethods for Heat & Fluid FlowVol. 24 No. 8, 2014pp. 1888-1905r Emerald Group Publishing Limited0961-5539DOI 10.1108/HFF-07-2013-0239

NomenclatureD dimensional thickness of solid wall (m)D dimensionless thickness of the solid wall

g gravitational acceleration (ms�2)H height of the enclosure (m)

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1. IntroductionAnalysis of both conduction and convection is an important problem in engineeringapplications such as district heating installation, solar photovoltaic/thermal, refrigerator,radiator, electronic cooling, channels and heat exchangers (Teo et al.,2012, Hasanuzzamanet al., 2009). In most of these applications, forced and natural convection can be occurred inthe same engineering systems which are called as mixed or combined convection. However,conduction heat transfer is taken into account in thick walls and the problem is solved forboth convection and conduction and is called as conjugate heat transfer problem.

Conjugate mixed convection heat transfer in a lid-driven enclosure with a thick bottomwall is analyzed by Oztop et al. (2008) at different Richardson numbers and thermalconductivity ratios. In another work (Oztop et al., 2009), lid-driven cavity was divided intotwo chests by using a solid conductive divider. This problem is highly interesting thatboth mixed convection and natural convection are formed in one system. Visualization ofheat flow using Bejan’s heatlines due to natural convection of water near 4oC in a thickwalled porous cavity is studied by Varol et al. (2010). They indicated that thedimensionless thickness of the wall is an effective parameter on heatlines.

Kanna and Das (2006) studied the conjugate heat transfer characteristics forbackward-facing step flow problem by using alternating direction implicit method.They compared the obtained results with the non-conjugate heat transfer case. Chiuet al. (2001) made both experimental and numerical study on conjugate heat transfer ina horizontal channel heated from below. In their work, they observed that the conjugateheat transfer revealed that the addition of conjugate heat transfer significantly affectsthe temperature and heat transfer rates at the surface of the heated region. Yang andTsai (2007) presented a numerical study of transient conjugate heat transfer in a highturbulence air jet impinging over a flat circular disk. Conjugate heat transfer in arectangular channel with lower and upper wall-mounted obstacles is investigatedusing the Lattice Boltzmann Method by Pirouz et al. (2011). Their results showed thatreducing the distance between obstacles made the flow deviate and accelerate in thevicinity of faces, and caused an increase in the rate of convective heat transfer fromobstacles. Other related works with conjugate heat transfer for natural or mixedconvection can be found in literature as Mobedi and Oztop (2008) and Varol et al. (2008).

kf fluid conductivity (Wm�1K�1)ks solid conductivity (Wm�1K�1)K thermal conductivity ratio (ks/kf)L length of cavity (m)Nu average Nusselt number (-)n any directionP dimensional pressure (Nm�2)P non-dimensional pressure (-)Pr Prandtl number (-)Re Reynolds number (-)Ri Richardson number (-)T temperature (K)Th hot wall temperature (K)(u, v) velocity components (ms�1)(U, V) dimensionless velocity componentw height of the channel (m)

(x, y) dimensional coordinates (m)(X, Y) dimensionless coordinates

Greek symbols

a thermal diffusivity (m2s�1)b coefficient of thermal expansion (K�1)m dynamic viscosity (Pa.s)n kinematic viscosity (m2s�1)y non-dimensional temperaturer density (kgm�3)c streamfunction

Subscripts

i inlet states Solidav average

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The detailed hydrodynamic and thermal problem for channel flow with cavity isstudied by Rahman et al. (2011a, b). Ozalp et al. (2010) made an experimental studyto compare the cavity shapes inside the open channel by using PIV technique. Theyselected the circular, rectangular and triangular cavities and found that rectangularand triangular cavities had the largest amplitudes while semi-circular cavity had thesmallest. Natural convection in trapezoidal and triangular cavities has been studiesfor heat transfer investigation (Hasanuzzaman et al. (2012a, b). Then, the flow past atriangular cavity in a channel by including magnetic force is numerically investigatedby Rahman et al. (2011a, b). An experimental work was performed by Migeon et al.(2000) to study the flow establishment phase inside square, rectangular and semi-circular closed cavities submitted to impulsive translation, from rest, of one of theirwalls at a Reynolds number of 1000. Flow induced vibrations and oscillations wereinvestigated by Lin and Rockwell (2001). Some other works related with channel withcavity flow can be found in Rahman et al. (2012a, b).

The main aim of this work is to examine the conduction-mixed convection heattransfer of laminar flow in a channel with thick walled cavity. Based on the authors’knowledge of the above literature survey and cited references, most of the workon channel flow with cavity is done for thin walled boundaries. Thus, solution ofconjugate problem of conduction, forced and natural convection in a single system isobtained in this work. This problem is important from the application of cooling ofelectronic equipments and space heating systems in engineering.

2. Model descriptionThe physical configuration for the problem with boundary conditions is depictedin Figure 1(a). The flow enters to the channel with a constant temperature and velocity.It is assumed that the height of the channel w¼ 0.25L. The channel is heated from thebottom wall of cavity with a heater at a constant temperature. The gravity also acts inthe y-direction. All of the boundaries are taken as adiabatic. Cyclic boundary conditionis applied to the exit of the channel. The thickness of the solid wall is depicted by d andthe dimensionless thickness of the solid wall is taken as D¼ d/(dþH).

3. Governing equationsThe fundamental laws used to solve the fluid flow and heat transfer problems are theconservation of mass (continuity equations), conservation of momentums (momentumequations), and conservation of energy (energy equations), which constitute a set ofcoupled, nonlinear, partial differential equations. For laminar incompressible thermal flow,the buoyancy force is included here as a body force in the v-momentum equation. Thegoverning equations for the two-dimensional steady flow after invoking the Boussinesqapproximation and neglecting radiation and viscous dissipation can be expressed as:

qU

qXþ qV

qY¼ 0 ð1Þ

UqU

qXþ V

qU

qY¼ � qP

qXþ 1

Rer2U ð2Þ

UqV

qXþ V

qV

qY¼ � qP

qYþ 1

Rer2V þ Ri y ð3Þ

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(b)

(c)3.18

3.14

3.10

3.06

3.02

2.985252 9330 14434 16962 20978

Number of Elements

Nu

(a)

x

y

g

adiabatic

ui, Ti

Th

H

outlet

2L

d

L

Figure 1.(a) Physical configuration

of the considered problem;(b) grid distribution;

(c) optimum grid

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UqyqXþ V

qyqY¼ 1

RePr

q2yqX2þ q2yqY 2

!ð4Þ

For the solid wall the energy equation is:

q2ys

qX2þ q2ys

qY 2¼ 0 ð5Þ

where Re ¼ uiL=u, Pr¼ v/a and Ri¼ gbDTL/ui2 are Reynolds number, Prandtl number

and Richardson number, respectively.The above equations were non-dimensionalized by using the following

dimensionless quantities:

X ¼ x

L; Y ¼ y

L; U ¼ u

ui

; V ¼ v

ui

; P ¼ pþ rgyð ÞL2

r u2i

; D ¼ d

L; y ¼ T � Tið Þ

Th � Tið Þ ; ys

¼ Ts � Tið ÞTh � Tið Þ

where X and Y are the coordinates varying along horizontal and vertical directions,respectively, U and V are the velocity components in the X and Y directions,respectively, y is the dimensionless temperature and P is the dimensionless pressure.

The boundary conditions for the present problem are specified as follows:

At the inlet: U¼ 1, V¼ 0, y¼ 0

At the outlet: qU/qX¼ 0, V¼ 0, qy/qX¼ 0

At the bottom wall of the cavity: U ¼ 0;V ¼ 0; qyqn

� �fluid¼ K qys

qn

� �solid

; at y ¼ d=H ; 0oxo1

U ¼ 0;V ¼ 0; y ¼ 1; at y ¼ 0; 0oxo1

�

At the channel as well as cavity walls (left and right): U¼V¼ qy/qn¼ 0

where N is the non-dimensional distances either X or Y direction acting normal to thesurface of the solid block and K is the dimensionless ratio of the thermal conductivity(K¼ ks/kf). Small values of K (Ko1) are only valid mathematically.

The local heat transfer rates on the surface of heat source is defined asNux ¼ �qy=qY jY¼0.

The average heat transfer rate on the surface of heat sources can be evaluated bythe average Nusselt, which is defined as Nuav ¼ �

R 1

0 qy=qY dX .

4. Numerical solution4.1 Solution procedureIn this investigation, the Galerkin weighted residual method of finite-elementformulation is used as a numerical procedure. The finite-element method begins bythe partition of the continuum area of interest into a number of simply shaped regionscalled elements. These elements may be different shapes and sizes. Within each element,the dependent variables are approximated using interpolation functions. In the presentstudy, erratic grid size system is considered especially near the walls to capture therapid changes in the dependent variables. The coupled governing Equations (2)-(5)are transformed into sets of algebraic equations using the finite-element method toreduce the continuum domain into discrete triangular domains. The system of algebraic

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equations is solved by iteration technique. The solution process is iterated until thesubsequent convergence condition is satisfied: Gmþ1 � Gm

�� ��p10�6 where m is numberof iteration and G is the general dependent variable.

4.2 Grid refinement checkGrid distribution for the physical model is depicted in Figure 1(b). As seen from thefigure, grid connections are given very regularly on the boundary of solid and fluidpart. Grid refinement check is also made to show the optimum grid in this work.It is given in Figure 1(c) to get optimum grid for D¼ 0.1, K¼ 0.2 and Ri¼ 1 and 16,962elements are used for executions.

4.3 Code validationResults of studied code are validated with an experimental work by done Ozalp et al.(2010) which is performed for flow past different shaped cavities. This study isperformed in a water tunnel and measurements by done PIV devices and laser-camera.In their work, just flow motion was performed and heat transfer is not included towork. A rectangular cavity without thick wall is chosen to compare with availabledata. The results are compared and showed in Figure 2. The experimental dataare taken from Ozalp et al. (2010). The flow is visualized by using PIV technique. Thecomparison of the two figures indicates that the obtained results from the written codeare acceptable.

5. Results and discussionA computational analysis has been performed in this work to investigate the effects ofdimensionless value of thermal conductivity, Richardson number and dimensionlessthickness value of the solid wall in a channel with a solid walled cavity. In this work,Prandtl number is taken as 7.0 for whole work, respectively.

Figure 3(a) show streamlines (on the left) and isotherms (on the right) for D¼ 0.1and K¼ 0.2 at different values of Richardson numbers. It is noticed that non-dimensional parameter of K is defined as the ratio of thermal conductivity of solid (ks)to fluid (kf). As well known from the literature, Richardson number is a measurementfor the ratio of forced convection to natural convection. When forced convectionbecomes dominant to natural convection a huge circulation cell is formed in clockwisedirection with cmin¼�0.199. However, the flow inside the cavity behaves as lid-drivencavity due to low Richardson number. Other two cells are formed in right bottomcorner and near the left corner. Physically, the flow tries to go into cavity from left andgo out from right vertical walls. In this case, the main flow through top side of thechannel does not affected from the presence of cavity. For K¼ 0.2, the thick wall hashigh conductivity and isotherms are formed as parallel to horizontal walls. They aremostly cumulated inside the cavity. The flow starts to heating from left vertical wall ofthe cavity. This result is supported with many studies in literature as Hasanuzzamanet al. (2012a). Thus, the heat is mostly transferred into fluid due to high conductivity ofbottom thick bottom wall. The distribution of isotherm also behaves as lid-drivencavity heated from the bottom wall. The main cell becomes smaller with increasing ofRichardson number to Ri¼ 1 and the cell moves to right bottom corner. The valuesof minimum stream function becomes as cmin¼ -0.021. It means that the flow strengthdecreases with increasing of Richardson number. This is clear from Figure 3(b).The flow goes into channel from inlet and it circulates due to presence of buoyancy.

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In this case, temperature contours are more parallel to horizontal walls as seen fromisotherms. Temperature value becomes higher inside the thick wall due to dominationof mixed convection. The circulating cell is disappeared with the highest value ofRichardson number due to increasing of domination of natural convection. Conductionmode of heat transfer is increased inside the cavity. The shape of the flow fits with thegeometry of the channel. As a general observation, variation of Richardson numberbecomes more effective on flow field than that of temperature field.

Figure 4 illustrates the streamlines (on the left) and isotherms (on the right) forD¼ 0.1 and K¼ 1. In this case, fluid and solid have the same value of thermal

(b)

(a)

Source: Ozalp et al. (2010)

Figure 2.Comparison of resultswith earlier work(a) results from presentstudy; (b) results fromexperimental work

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conductivity. When this figure compares with Figure 3 for K¼ 0.1, flow field showsalmost same behavior due to lower effects of buoyancy forces. Flow strength isincreased with increasing of thermal conductivity ratio for Ri¼ 0.1. However, presenceof solid thick wall of the bottom wall of the cavity becomes insignificant from thetemperature point of view. Thus, flow is more heated according to case of K¼ 0.1. Thisdistribution proves the temperature distribution inside the cavity heated from bottomand cooled from top because of flowing fluid from open side of the cavity. Non-lineartemperature distribution is observed. Figure 5 presents the streamlines and isothermsfor K¼ 10. In this case, the cavity is heated from the top side of the thick wall due tohigher thermal conductivity value of solid material.

As seen from the Figure 6 which is plotted for the case of D¼ 0.25 and K¼ 0.2,both flow field and temperature distribution are completely affected. There is nocirculation cell inside the cavity due to domination of forced convection flow atRi¼ 0.1. In this case, solid wall is chosen as insulated material. Thus, the fluid insidethe cavity is not well heated and the top point of the cavity the fluid is equal to inletfluid temperature. When buoyancy induced flow becomes effective two weakcirculation cells are formed. The biggest one rotates in clockwise and the other onerotates counterclockwise rotating direction which sits at the bottom wall. Figures 7and 8 illustrate the streamlines and isotherms for D¼ 0.25 and K¼ 1 and 10,respectively. The thick wall becomes insignificant from the temperature distribution

(c)

(a)

(b)

–0.199

0.05

0.05

0.25

0.75

0.3

0.5

0.85

–0.021

0.05

0.3

0.55

0.8

Figure 3.Streamlines (on the left)

and isotherms (on theright) for D¼ 0.1 and

K¼ 0.2, (a) Ri¼ 0.1;(b) Ri¼ 1, (c) Ri¼ 10

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point of view at K¼ 1. A huge circulation cell is formed inside the cavity withcmin¼�0.144. The flow is pushed to corner and impinges to bottom wall andjumped. Thus, a mini cell is formed on the bottom wall with cmax¼ 0.006 as seenfrom Figure 7(c). The mini cell is dissappeared for Ri¼ 1 and circle shaped cell isformed at the right corner (Figure 7(b)). Variation of Richarson number becomes lesseffective on isotherms. The solid wall is well conducted for K¼ 10 and the fluid isheated from the top side of solid wall in Figure 8. The Richardson number affects thetemperature distribution inside the solid region. It is mostly effective at the rightside of the solid wall due to recirculating flow inside the cavity. When thermalconductivity ratio increases, temperature values are also increased. Thermalboundary layer is a function of thermal conductivity ratio. Figure 9 showsstreamlines (on the left) and isotherms (on the right) for D¼ 0.5 and Ri¼ 10 atdifferent values of thermal conductivity ratio. In this case, half of the cavity isoccupied by solid wall. Thus, most of heat is captured inside the solid wall and top sidetemperature of the solid wall becomes colder according to other case of D values.However, natural convection heat transfer becomes effective as seen from Figure 9(a).The fluid exhibits wavy variation and two circulation cells are formed, one on the middleand the other at the right corner of the cavity. These circulations affect the thermalboundary layer at that points. These cells are dissappeared with increasing of

(b)

(c)

(a)

–0.266

–0.021

0.55

0.9

0.65

0.95

0.05

0.4

0.05

0.35

0.05

0.35

0.65

0.95

Figure 4.Streamlines (on the left)and isotherms (on theright) for D¼ 0.1 andK¼ 1, (a) Ri¼ 0.1;(b) Ri¼ 1, (c) Ri¼ 10

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dimensionless value of thermal conductivity. Figure 9(c) is plotted to show streamlines(left) and isotherms (right) for D¼ 0.5 and K¼ 10. In this case, solid wall occupies half ofthe cavity. The flow inlets to channel from input side of the duct and wavy variation isobtained above the solid wall. Mini circulation cell in streamline is observed on the wallbut the flow strength becomes very weak. Boundary layer becomes very thin aroundthe inlet and exit part of main duct. Isotherms are also showed wavy variation inside thecavity. Temperatures become constant inside the solid wall due to higher thermalconductivity value. Higher values of Richardson number become insignificant due tohigher thickness of the wall.

Figure 10 illustrates the local Nusselt number along the heated side for differentRichardson number and thermal conductivity ratio at D¼ 0.1. As seen from the figure,local Nusselt values are linearly increased along the heated side for all values ofRichardson numbers. As an expected results, values of local Nusselt number increaseswith increasing of thermal conductivity ratios. Also, higher values are obtained whennatural convection becomes dominant to forced convection, namely higher valuesof Richardson number. Values are closer to each other with increasing of thermalconductivity ratio due to increasing of heat transfer from bottom to top. Cold fluidpasses the cavity at lowest value of the Richardson number. At the left side of thecavity, values are almost same at Ri¼ 0.1 and 1 due to stagnant flow at that side.Similarly, Figure 11 gives an opportunity to compare results on local Nusselt numbers

(b)

(c)

(a)

–0.310

0.0030.003

0.05

0.5

0.65

0.95

0.05

0.45

0.75

0.95

–0.029

0.05

0.35

0.65

0.95

Figure 5.Streamlines (on the left)

and isotherms (on theright) for D¼ 0.1 and

K¼ 10, (a) Ri¼ 0.1;(b) Ri¼ 1, (c) Ri¼ 10

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between dimensionless solid wall thicknesses for different dimensionless value ofthermal conductivity. Results are decreased with increasing of wall thickness due tolow heat transfer from outside to inside of the cavity. Nevertheless, values are increasedalong the bottom wall. In case of dominant regime of mixed convection and forcedconvection values are very close to each other for all values of thermal conductivity dueto high flow velocity which passes from the cavity. On the contrary a huge differencesare obtained for Ri¼ 10 especially near the outlet port of the cavity due to increasingof flow velocity at that part. Figure 12 is plotted to show variation of local Nusseltnumber for D¼ 0.5. In this case, all values are close to each other which are oppositetrend of other thickness values as given in Figures 11 and 12.

Table I lists the variation of average Nusselt number for different parameters forstudied parameters such as dimensionless thickness of the solid wall, dimensionlessthermal conductivity and Richardson number. For higher value of dimensionlessthickness, Richardson number is not an effective parameter for the lower value of thethermal conductivity such as K¼ 0.01 and 0.2 due to low heat transfer. It must be notedthat studied value of K o 1 is only valid for mathematically. Also, average Nusseltnumber decreases with increasing of dimensionless thermal conductivity value for allvalues of Richardson numbers. However, average Nusselt number increases withincreasing of Richardson number except D¼ 0.5. Average Nusselt number decreaseswith increasing of dimensionless wall thickness.

(b)

(c)

(a)

0.008

–0.073

0.05

0.2

0.9

0.05

0.25

0.9

0.05

0.25

0.9

Figure 6.Streamlines (on the left)and isotherms (on theright) for D¼ 0.25 andK¼ 0.2, (a) Ri¼ 0.1;(b) Ri¼ 1, (c) Ri¼ 10

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(b)

(c)

(a)

–0.144

0.006

0.05

0.45

0.95

0.05

0.35

0.55

0.95

0.003

0.05

0.3

0.65

0.95

Figure 7.Streamlines (on the left)

and isotherms (on theright) for D¼ 0.25 and

K¼ 1, (a) Ri¼ 0.1,(b) Ri¼ 1, (c) Ri¼ 10

(b)

(c)

(a)

–0.203

0.006

0.05

0.5

0.65

0.95

0.05

0.45

0.75

0.95

–0.002

0.05

0.4

0.75

0.95

Figure 8.Streamlines (on the left)

and isotherms (on theright) for D¼ 0.25 and

K¼ 10, (a) Ri¼ 0.1,(b) Ri¼ 1, (c) Ri¼ 10

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6. ConclusionsA numerical analysis has been performed to investigate the conjugate mixedconvection problem in an open channel with thick walled cavity. Important findingscan be listed as follows:

. The presence of thick wall inside the cavity affects both flow distribution andtemperature distribution. The thickness of the cavity is directly an effectiveparameter with flowing fluid inside cavity. It is important for lower Richardsonnumber, namely, domination of forced convection.

. The number of circulation cells inside the cavity is directly related with theRichardson number for all values of the thermal conductivity and the solid wallthickness. The cell number is decreased with decreasing of the Richardsonnumber and cells are disappeared in the case of forced convection dominatedregime.

. The local Nusselt numbers are very close to each other at the left side of thecavity almost for all studied parameters. The values increase to right side ofthe cavity.

. Heat transfer is an increasing function of the thermal conductivity ratio for allcases. It is calculated from the top side of solid wall. Thus, it is increased withincreasing of Richardson number.

(b)

(c)

(a)

0.005

0.05

0.4

0.8

0.95

0.05

0.3

0.7

0.95

0.05

0.35

0.7

0.95

Figure 9.Streamlines (on the left)and isotherms (on theright) for D¼ 0.5 andRi¼ 10, (a) K¼ 0.2;(b) K¼ 1, (c) K¼ 10

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1.3Ri = 0.1

Ri = 1

Ri = 10

Ri = 0.1Ri = 1Ri = 10

Ri = 0.1Ri = 1Ri = 10

1.1

0.9

Nu l

ocal

Nu l

ocal

Nu l

ocal

0.7

0.5

2.4

1.9

1.4

0.9

0.4

3

2.5

2

1.5

1

0.5

0

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

X

X

X

(a)

(b)

(c)

Figure 10.Local Nusselt number

for different Richardsonnumber at D¼ 0.1,

(a) K¼ 0.2; (b) K¼ 1and (c) K¼ 10

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(a)

(b)

(c)

0.7

0.6

0.5

0.40 0.2 0.4 0.6 0.8 1

Ri = 0.1Ri = 1Ri = 10

Ri = 0.1

Ri = 1

Ri = 10

Ri = 0.1

Ri = 1

Ri = 10

X

0 0.2 0.4 0.6 0.8 1

X

0 0.2 0.4 0.6 0.8 1

X

Nu l

ocal

Nu l

ocal

Nu l

ocal

1.7

1.4

1.1

0.8

0.5

2.5

2.0

1.5

1.0

0.5

0.0Figure 11.Local Nusselt numberfor different Richardsonnumber at D¼ 0.25,(a) K¼ 0.2; (b) K¼ 1 and(c) K¼ 10

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00.28

1.1

0.9

0.7

0.5

2.3

2.0

1.7

1.4

1.1

0.8

0.5

1.3

0.30

0.32Nu l

ocal

Nu l

ocal

Nu l

ocal

0.34

0.36

0.38

Ri = 0.1

Ri = 1

Ri = 10

Ri = 0.1

Ri = 1

Ri = 10

Ri = 0.1

Ri = 1

Ri = 10

0.2

X

0.4 0.6 0.8 1

0 0.2

X

0.4 0.6 0.8 1

0 0.2

X

0.4 0.6 0.8 1

(a)

(b)

(c)

Figure 12.Local Nusselt number

for different Richardsonnumber at D¼ 0.5,

(a) K¼ 0.2; (b) K¼ 1and (c) K¼ 10

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The results of this theoretical work can be used for applications of cooling of electronicequipments in the range of Richardson numbers. The study can be extended toturbulent flow and different boundary conditions as a future work.

References

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KD Ri 0.01 0.2 1 10

0.1 0.1 1.505136 2.464541 5.000679 9.438668

1 1.882581 2.992809 5.539639 9.45668

10 2.859337 4.184681 6.620178 9.562642

0.25 0.1 1.503638 2.127217 3.111915 3.93377

1 1.735948 2.363287 3.202081 3.934454

10 2.730468 3.133054 3.551391 3.941136

0.5 0.1 1.842747 1.872564 1.891989 1.999554

1 1.921745 1.968558 1.991879 2.02955410 2.055648 2.251022 2.318075 2.469555

Table I.Variation of averageNusselt number fordifferent parametersas D, K and Ri

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Corresponding authorDr Hakan Oztop can be contacted at: hfoztop1@yahoo.com

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