Energy and entropy analysis in a square cavity with protruding body: effects of protruding body aspect ratio

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2002; 26:851–866 (DOI: 10.1002/er.824)

Energy and entropy analysis in a square cavity withprotruding body: effects of protruding body aspect ratio

B.S. Yilbasny, S.Z. Shuja and M.O. Iqbal

Mechanical Engineering Department, King Fahd University of Petroleum and Minerals,

Dhahran}31261, Saudi Arabia

SUMMARY

In the present study, the simulation of mixed convection in a square cavity with protruding body havingdifferent aspect ratios is carried out. The governing flow and energy equations are solved numerically usinga control volume approach. Air is used as fluid in the cavity while steel is considered as protruding body.To investigate the heat transfer characteristics due to different aspect ratios of the protruding body,Stanton number (St) variation with the aspect ratio is considered. The entropy analysis is carried out todetermine the irreversibility generated in the cavity for different aspect ratios. The heat transfer toirreversibility ratio is determined for each aspect ratio. It is found that the aspect ratio influences the heattransfer characteristics and the irreversibility generated in the cavity, in which case, the heat transferincreases at high aspect ratio while the irreversibility reduces. Moreover, heat transfer to irreversibilityratio improves considerably at an aspect ratio of 3. Copyright # 2002 John Wiley & Sons, Ltd.

KEY WORDS: Energy; entropy analysis; cavity flow; aspect ratio

1. INTRODUCTION

Energy savings is one of the important issues in electronic industry. In this case, electronicdevices are cooled either by natural or force convections. Proper selection of a cavity designedfor electronic boards minimizes the energy consumption for cooling and improves theperformance of the electronic device. Heat transfer involving mixed convection in cavities withprotruding body finds wide interest in electronic industry (Nansteel and Gief, 1981). This is dueto the fact that excessive temperature rise in the electronic equipment, which in turn results inthe failure of the electronic parts or unreliable operation. Moreover, when the effect of naturalconvection on the rate of heat transfer is comparable with the effect of forced convection, theflow is considered as mixed convection. Gebbart (1971), Llyod and Sparrow (1970), and Siebers(1983) examined the mixed flow situations for different geometrical configurations. However,most of these studies were limited to external flows.

The mixed convection in a cavity received a considerable attention in recent years. Combinedforced and natural convection heat transfer in a deep lid-driven cavity flow was investigated by

Received 6 June 2001Copyright # 2002 John Wiley & Sons, Ltd. Accepted 18 September 2001

nCorrespondence to: B.S. Yilbas, Mechanical Engineering Department, King Fahd University of Petroleum andMinerals, Dhahran 31261, Saudi Arabia.

yE-mail: [email protected]

Prasad and Koseff (1996). They analysed the mean heat flux over the entire boundary anddeveloped Nusselt and Stanton numbers correlation. Heat transfer optimization within open-ended annular cavities was studied by Desai et al. (1996). They introduced a finite elementanalysis for the natural convection due to open-ended cavities. They showed that changing thegeometric configuration of the cavity from cylinder to conical shape improves the heat transferrate considerably. Darbhe and Muralidhar (1996) experimentally examined the naturalconvection heat transfer from a discrete protruding surface. They indicated that the averageNusselt number calculated for the protruding body was larger than that reported in theliterature. Conjugate natural convection from an array of protruding heat sources was studiednumerically by Heindel et al. (1996). They showed that with increasing modified Rayleighnumber (1045Ra5109), the cavity flow approached boundary layer behaviour and more fluidpenetrated the regions between protrudes. Natural convection in rectangular enclosures withmultiple protruding heaters mounted on one sidewall was investigated by Desai et al. (1995).They showed that at low Rayleigh numbers (Ra51.5� 107), the flow was stable andcharacterized by the presence of a primary flow cell and a counter rotating secondary cell at thetop of the enclosure. At high Rayleigh numbers (Ra>3� 108), however, the isothermal top wallcaused a periodic flow pattern to develop within the enclosure. Dehghan and Behnia (1996)studied numerically natural convection air cooling of two heat-generating devices in verticalslot. They indicated that the wall conduction was important and for substrates with a finitethermal conductivity it was essential that the conjugate analysis should be employed. Numericalsimulation of natural convection in an enclosure with discrete protruding heaters was carriedout by Ju and Chen (1996). They presented local Nusselt number versus modified Rayleighnumber. Moreover, they indicated that the predictions agreed well with the experimentalfindings. Mixed convection in a partially divided rectangular enclosure was investigated byHsu et al. (1997). They showed that the average Nusselt number and the dimensionlessurface temperature highly depended on the location and the height of the divider.

In heating and cooling processes, the energy transport is related to the irreversibility orentropy generation. The source of entropy generation includes the heat transfer and the viscousdissipation in the flow systems. When both the temperature and velocity fields are known, theentropy generated in the system can be computed (Bejan, 1996a). A comprehensive study ofentropy generation in fundamental convective heat transfer was carried out by Bejan (1996b).The second law aspects of heat transfer by forced convection were illustrated in terms of fourfundamental flow configurations. He showed as to how the flow parameters could be selectedin order to minimize the irreversibility associated with a specific convective heat transferproblem. Optimal paths for minimizing entropy generation in a common class of finite-timeheating and cooling process were investigated by Andreson and Gordon (1992). They indicatedthat the previously suggested proposal for the constant entropy generation rate for the optimalpath was only applicable for linear heat transfer cases. Carrington and Sun (1992) studied thesecond-law analysis of combined heat and mass transfer in internal and external flows. Theyindicated that the coupling term between heat and mass transfer should be used properly anddiffusion flux should be employed instead of absolute maximum flux in the analysis. Numericalpredictions of entropy generation for mixed convective flows in a vertical channel withtransverse fin arrays were carried out by Cheng and Ma (1994). They examined the variousphysical and geometric parameters and proposed a geometric configuration of the finnedchannel with higher second-law efficiency. Second-law analysis of a swirling flow in a circularduct with restriction was studied by Yilbas et al. (1999). They developed the dimensionless

Copyright # 2002 John Wiley & Sons, Ltd. Int. J. Energy Res. 2002; 26:851–866

B.S. YILBAS, S.Z. SHUJA AND M.O. IQBAL852

quantities for the entropy generation, heat transfer and irreversibility. They indicated that theeffect of swirling and restriction on the dimensionless quantities is more pronounced at highPrandtl numbers. Ledezma and Bejan (1997) investigated optimal geometric arrangement ofstaggered vertical plates in natural convection. They correlated the results predicted for theoptimal horizontal spacing using formulas derived from the theory of the intersection ofasymptotes.

The mixed convection flow studies are presented for fixed body geometries in a cavity andentropy analysis for such cases are omitted in the literature. Consequently, the investigation intoheat transfer characteristics and entropy generation in a cavity due to the varying aspect ratio ofa protruding body becomes necessary. In the present study, mixed convection cooling ofprotruding body in a square cavity is examined. The aspect ratio of the protruding body isvaried provided that the area of the protruding body remains constant, for fixed boundaryconditions. To accommodate the mixed convection situation, the inlet Reynolds number islimited to Re=65. Since Re is kept low, the natural convection effect in the cooling process isalso taken into account. The entropy analysis is carried out and the variation of irreversibilitywith the aspect ratio is computed.

2. MATHEMATICAL FORMULATION

The flow around the protruding bluff body is assumed to be incompressible and laminar sincethe Reynolds number considered at present is 65 due to mixed convection situation.

2.1. The flow field

The governing equations are given as follows:The continuity equation is

@u@x

þ@v@y

¼ 0 ð1Þ

The momentum equation isx-momentum:

u@u@x

þ v@u@y

¼ �1

r@p@x

þ u@2u@x2

þ@2u@y2

� �ð2Þ

y-momentum:

u@v@x

þ v@v@y

¼ �1

r@p@y

þgbDTr

þ u@2v@x2

þ@2v@y2

� �ð3Þ

The energy equation is

rCp u@T@x

þ v@T@y

� �¼ k

@2T@x2

þ@2T@y2

� �þ mf ð4Þ


ENERGY AND ENTROPY ANALYSIS WITH PROTRUDING BODY 853

where

f ¼ 2@u@x

� �2

þ@v@y

� �2" #

þ@v@x

þ@u@y

� �2

Boundary conditionsFigure 1 shows the schematic view of the cavity and the protruding body for aspect ratio 2,

and the boundary conditions as an example.At solid wall:u=0 and v=0 (no slip condition)At inlet port of the cavity, a uniform fluid temperature and uniform flow are assumed.

@j@x

¼ 0

where j is any of the fluid property such as velocity and temperature.

Figure 1. Schematic view of the cavity and the solid body with the grid generated and boundary conditionsfor the aspect ratio of 0.25, where L=0.05m.



2.2. The protruding solid body

The energy equation is

rCp@T@t

� �¼ k

@2T@x2

þ@2T@y2

� �þ ’qq000 ð5Þ

where q000 is the uniform heat flux over the body.Boundary conditions:At solid wall:

�k @T =@n ¼ hðTw � T0Þ; Ts ¼ Tf and q000 ¼ 1250Wm�3

The dimensionless numbers used in the analysis are Stanton, Nusselt, and Grashof numbers.The Stanton number is St ¼ qw=brCpU1ðTw � T1Þc: The normalized Stanton number is St/St0,where St0 is the maximum value along all the surfaces. The Nusselt number is Nu ¼ hD=k where

h ¼ð�kð@T=@nÞÞwall

ðTw � T1Þ

and n is the direction normal to the surface. The Grashof number is Gr ¼ gbDTl3=u2

The aspect ratio of the protruding body is determined in accordance with the cavity size,which is shown in Figure 1. The size of the protruding body can be formulated as:

b ¼ 0:2Lffiffiffia

p and c ¼ 0:2ffiffiffia

pL

where b and c are length and height of the protruding body. By knowing the aspect ratio ‘‘a’’,the size of the protruding body can be determined. It should be noted that the area of theprotruding body is kept constant for all aspect ratios, i.e.: Area=bc (=constant).

2.3. Entropy analysis

The volumetric entropy generation is given in the previous study (Bejan, 1996a) as

’SS000gen ¼

kT 2

@T@x

� �2

þ@T@y

� �2" #

þmT

@u@x

� �2

þ@v@y

� �2( )

þ@u@y

þ@v@x

� �2" #

ð6Þ

where the first term is volumetric entropy generation due to heat transfer and the second term isdue to fluid friction.

The irreversibility in the flow system can be written as:

I ¼Z

To ’SS000gendc ð7Þ

In order to compare the effect of aspect ratio on the heat transfer and irreversibility generatedthe following non-dimensional heat and irreversibility ratios are introduced

QQo

andII0

where Q0 and I0 are the maximum value of heat transfer and irreversibility generated in thecavity.



3. NUMERICAL SOLUTION

The flow domain is overlaid with a rectangular grid as shown in Figure 1. The grid used in thepresent study has 100� 100 node points. The control volume approach is employed in thenumerical scheme. All the variables are computed at each grid point except the velocities, whichare determined midway between the grid points. The grid independent tests are conducted and100� 100 grid points are selected on the basis of less computation time without compromisingthe grid independence.

A staggered grid arrangement is used in the present study, which provides the pressurelinkages through the continuity equation and is known as SIMPLE algorithm (Imura et al.,1978). This procedure is an iterative process for convergence. The pressure link betweencontinuity and momentum is established by transforming the continuity equation into a Poissonequation for pressure. The Poisson equation implements a pressure correction for a divergentvelocity field (Patankar, 1980).

4. RESULTS AND DISCUSSION

The results are obtained for mixed flow at different aspect ratios of protruding body in a squarecavity. The inlet and exit ports are located in the side walls of the square cavity and the Rayleighnumber is considered as 11 000 during the simulations provided that the Rayleigh numbercalculation is based on the cavity length. The selection of cavity height is due to the developmentof the mixed flow conditions. The velocity, temperature and entropy field generated in the cavitydue to protruding body and the heat transfer characteristics due to different aspect ratios of theprotruding body are presented. The properties of air fluid and solid object employed in thesimulations are given in Table I.

In a mixed flow, the ratio of Gr/Re2 is important to identify the relative strength of the forcedconvection over natural convection, since the strength of shear driven motion is identified byReynolds number, while the strength of buoyancy is measured by Grashof number. It wasshown from the manipulation of Navier–Stoke’s equation that the buoyancy effect becomesnoticeable when Gr/Re2 approaches unity (Gebhart, 1971), i.e. for Gr/Re251, the forcedconvection component controls the heat transfer while Gr/Re241 buoyancy effectspredominate. The mixed convection limits for the flat plate in a horizontal stream are definedas 0.75Gr/Re2510 (Chen et al., 1977). In order to observe the influence of natural convectionon the overall heat transfer due to different aspect ratios, Figure 2 is plotted, in which, Gr/Re2

with aspect ratio is shown. Gr/Re2 is greater than one for all the aspect ratios. As the aspectratio increases Gr/Re2 reduces. In this case, the contribution of natural convection on theoverall heat transfer increases.

Table I. Properties of air and steel used in the simulations.

Air Steel

Density (kgm�3) 1.89 7836Specific heat (J kg�1K�1) 1005 969Thermal conductivity (Wm�1K�1) 0.02565 28.2Viscosity (m2 s�1) 1.544� 10�5 }



Figure 3 shows the velocity contours that resulted in the cavity due to different aspect ratiosof a protruding body. In general, three circulation cells, two in front and one behind theprotruding body, are developed in the cavity for aspect ratios equal and greater than one.Moreover, one extra circulation cell below the protruding body is developed for aspect ratiosless than one. The protruding body extends vertically in the cavity for the aspect ratio greaterthan one. In this case, the fluid next to the protruding body sidewalls accelerates due to thebuoyancy effect. This generates a low pressure gradient in the region close to the left bottomcorner of the body. Consequently, the flow entering into cavity from the inlet port passes aroundthe protruding body. This generates a fluid circulation in front of the solid body. The fluidcaptured between this circulation cell and the cavity wall generates a secondary clockwisecirculation cell due to the flow developed between the inlet and exit ports of the cavity. Thelocation of the circulation cells in front of the body does not change significantly withthe change in aspect ratio. However, the cell with the counter rotation inclines slightly towardsthe cavity wall because of the slight change of velocity gradient close to the protruding bodywall. The circulation cell developed behind the protruding body is considerably large due to alow velocity gradient attainment in this region, i.e. the heated fluid is captured in the cavitybehind the protruding body because of the location of the exit port, which is located in the lefttop corner of the cavity. As the aspect ratio reduces, the circulation cell with counter clockwiserotation is developed below the solid body. In this case, heated fluid close to the bottom surfaceof the protruding body generates almost a stagnation region, which in turn almost blocks theflow entering the cavity from the inlet port reaching this region. However, a relatively lowpressure gradient developed behind the protruding body enhances the motion of the fluid toflow into this region. Consequently, a circulation cell is generated in the cavity below theprotruding body. As the aspect ratio reduces, the circulation cell becomes more apparent. Thesize of the circulation cell in front of the protruding body becomes smaller as the aspect ratioreduces.

Figure 4 shows the temperature contours in the cavity at different aspect ratios. Thetemperature contours close to the left bottom corner of the protruding body is highly

Figure 2. Variation of Gr/Re2 with aspect ratio.



concentrated and the thermal boundary layer becomes thin in this region. This is because thevelocity close to the front surface of the body attains a high gradient due to buoyancy effect. Theflow entering into the cavity from the inlet port spills and accelerates towards the high velocitygradient region. The thermal boundary layer increases in the region behind the body as theaspect ratio increases. This is because the circulation cell formed behind the body enhances thethickness of the thermal boundary layer. As the aspect ratio reduces, thermal boundary layerbehind the body becomes relatively thinner. This is due to the change of orientation ofcirculation cell behind the body. Moreover, the temperature contours below the body are highlyconcentrated for all aspect ratios. In this case, the heated fluid expands and develops a radialvelocity gradient in the region close to the protruding wall. The extension of temperatureprofiles towards the exit port of the cavity is due to convective current generated between theinlet and exit ports.

Figure 3. Velocity contours at different aspect ratios.



In order to assess the variation of Stanton number (St) across the surfaces of the protrudingbody with aspect ratio, St is normalized with its maximum value (St0) over the surfaces. Ingeneral, St/St0 decreases with increasing aspect ratio for almost all the surfaces (Figure 5). Theenhancement in St/St0 for a low aspect ratio is due to the convective current generated over thesurfaces, in which case, it enhances the convective heat transfer from the protruding body. Asthe aspect ratio increases, the orientation of circulation cell changes which in turn results inrelatively large thermal boundary layers. Consequently, convective cooling of the surfaces issuppressed. St/St0 corresponding to the top surface of the body attains relatively higher valuesof St/St0 as compared to its counter parts corresponding to other surfaces. Moreover, thisreverses for the bottom surface. This is because of the contribution of the natural convection forcooling the body. In this case, the buoyancy effect generates relatively high velocity gradients inthe region close to the top surface of the body. This enhances the convective current close to the

Figure 4. Temperature contours at different aspect ratios.



top surface. However, the secondary convective current generated behind the body carries aheated fluid towards the top surface. This increases the temperature of the fluid in the regionabove the top surface. Consequently, heat transfer rate from the top surface reduces leading to arelatively low St/Sto corresponding to the top surface. In the case of bottom surface, the verticalvelocity gradient is zero and the radial velocity gradient is non-zero, i.e. the buoyancy effectenhancing the convective cooling is minimal. Therefore, the Nusselt number reduces so that thenormalized Stanton number St/St0 reduces to a minimum at an aspect ratio of 3 for the rearsurface, which in turn indicates reduced heat transfer. In this case, the effect of circulation celland their orientation have an adverse effect on the heat transfer characteristics.

Figure 6 shows the overall normalized Stanton number (St/St0) with the aspect ratio. Thenormalization is carried out with the maximum Stanton number corresponding to all the aspectratios. The overall St/St0 initially reduces to a minimum at the aspect ratio of 0.333, then itincreases with increasing aspect ratio and reaches its maximum at the aspect ratio of 4. As theaspect ratio increases, the natural convection contribution to heat transfer increases, sinceGr/Re2 reduces with aspect ratio as shown in Figure 2. This indicates that the convectivecurrent generated due to buoyancy effect leads to the enhancement of heat transfer from theprotruding body. In this case, the convective current due to flow circulates without substantialspilling around the body. This may be more pronounced at low aspect ratios. Consequently,the effect of forced convection on the heat transfer at present simulation conditions may not besubstantial as compared to natural convection contribution.

Figure 7 shows the non-dimensional Nusselt number (Nu/Gr/Re2) variation with the inverseof aspect ratio (1/a). Nu/Gr/Re2 reduces as 1/a increases to 1. Moreover, as 1/a increasesfurther, Nu/Gr/Re2 increases. This indicates that a nonlinear relation between Nu/Gr/Re2 and1/a exists. After using a curve fit technique and introducing a polynomial equation, the relationyields

y ¼ 0:2555x4 � 2:3801x3 þ 8:1538x2 � 10:366xþ 14:429

The polynomial relation gives excellent agreement with the predictions (as seen from the figure)and the mean standard error is in the order of less than 1% (R=0.993). Consequently, the

Figure 5. Variation of mean St/St0 along each face with aspect ratio.



Nusselt number variation with the aspect ratio is nonlinear and the influence of aspect ratio onthe Nusselt number is considerable.

Figure 8 shows the normalized irreversibility variation with aspect ratio. The normalization iscarried out by the maximum value of irreversibility. The normalized irreversibility reduces as theaspect ratio increases provided that a slight reduction in normalized irreversibility is observedfor the aspect ratio of 0.25. The high magnitude of normalized irreversibility at low aspect ratios(a>1) is the indication of a relatively high overall entropy generation. This is because the shapeof the protruding body at low aspect ratio generates large entropy production in the vicinity ofthe protruding body. In this case, the influence of thermal boundary layer thickness in thevicinity of the surfaces on the temperature gradient is significant. Consequently, the current

Figure 6. Variation of overall St/St0 with aspect ratio.

Figure 7. Variation of Nu/Gr/Re2 with the inverse of aspect ratio.



driven by the buoyancy effect and forced convection at a low aspect ratio influence the thermalboundary layer around the protruding body such that the high temperature gradient isgenerated towards the end of the thermal boundary layer, which slightly reduces the heattransfer from the protruding body as is evident from Figure 9. The normalized heat transferwith aspect ratio is shown in Figure 9. The heat transfer from the protruding body attainsslightly higher values at high aspect ratios (a>2), as compared to its counterpart at a low aspectratio (a52), except the aspect ratio of 0.25. The normalization process is carried out to its

Figure 8. Variation of I/I0 with aspect ratio.

Figure 9. Variation of Q/Q0 with aspect ratio.



maximum value for all the aspect ratios. The increase in heat transfer indicates thin thermalboundary layer with a high-temperature gradient across that layer. The normalized heat transferattains its maximum value at an aspect ratio of 3. Therefore, the convective and conductioncooling of the body improves considerably at this aspect ratio for a cavity configurationintroduced in the present study.

Figure 10 shows the heat transfer to irreversibility ratio (Q/I) (variation with aspect ratio).The value of Q/I is very high, since the irreversibility generated in the cavity is very low ascompared to the heat transfer. Q/I improves as aspect ratio increases beyond two. In this case,irreversibility reduces while heat transfer increases, which is as evident from Figures 9 and 10.Q/I attains the maximum value at an aspect ratio of 3. Consequently, the optimum coolingcondition for the protruding body with minimum irreversibility generation is achieved at anaspect ratio of 3.

Figure 11 shows the ratio of heat transfer difference corresponding to forced and naturalconvection (Q�Q0) to heat transfer taking place in forced convection (Q). (Q�Q0)/Q reduces asthe aspect ratio increases to 3, then it increases with aspect ratio. The range of aspect ratio,where low (Q�Q0)/Q occurs, indicates that the contribution of natural convection to mixedconvection is significant. This behaviour is due to the current generated by the buoyancy effectand the orientation of the circulation cells in the cavity.

5. CONCLUSIONS

The simulation of mixed convection due to a protruding body with different aspect ratios in thecavity is carried out numerically. In order to include the effect of buoyancy-driven convectionon the heat transfer characteristics, the Reynolds number at the inlet port of the cavity is takenas 65, which results in Gr/Re2 in the order of 6. In general, the aspect ratio influences the heattransfer characteristics and irreversibility generated in the cavity. The heat transfer toirreversibility ratio reaches its maximum at an aspect ratio of 3. Moreover, the heat transfer

Figure 10. Variation of Q/I with aspect ratio.



from the protruding body reduces while irreversibility increases at low aspect ratios. The specificconclusions derived from the present study can be listed as follows:

1. The flow developed in front and back of the protruding body generates circulation cells.The circulation cells are due to the convective current developed due to buoyancy andforced convection effects. In addition, two circulation cells with counter rotation areobserved in front of the body due to secondary flow. As the aspect ratio reduces, thecirculation is developed below the protruding body. This occurs because of the mixing ofconvection current entering the cavity and the buoyancy-driven current in the radialdirection (along the y-axis).

2. The temperature profiles close to the top surface of the protruding body extends towardsthe exit part of the cavity. The buoyancy effect on the temperature contours in the sidesurface of the cavity is visible, in this case, thermal boundary layer thickness extends intothe fluid, which is more pronounced at high aspect ratios. The pattern of temperaturecontours changes as the aspect ratio reduces.

3. Entropy generated due to heat transfer dominates over the fluid friction contribution. Thevolumetric entropy generation amplifies in this region close to the left corner of theprotruding body. In this case, the cold fluid entering the cavity from the inlet port mixeswith the heated fluid in this region. The entropy generation close to the left and bottomsurfaces of the body substantiate at high aspect ratios. This changes as the aspect ratioreduces, in which case, the entropy generation close to the rear surface of the bodysubstantiates.

4. The irreversibility reduces with increasing aspect ratios, in which case, the heat transfercontribution of entropy generation increases, since viscous dissipation has little effect onthe irreversibility generation. Consequently, the temperature gradient in the fluid close tothe solid body attains high values at high aspect ratios.

5. The heat transfer to irreversibility ratio increases as the aspect ratio increases. It attains themaximum at an aspect ratio of 3. In this case, the optimum cooling condition forthe protruding body with minimum irreversibility generation is achieved at an aspect ratioof 3.6.

Figure 11. Variation of (Q�Q0) with aspect ratio.



The overall St/St0 increases as the aspect ratio increases, in which case Gr/Re2 reduces. Thisindicates that the effect of buoyancy-driven current on the heat transfer from the protrudingbody dominates over the forced convection contribution.

ACKNOWLEDGEMENTS

Acknowledgements are due to King Fahd University of Petroleum and Minerals.

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NOMENCLATURE

A aspect ratioB length of the protruding body (m)C height of the protruding body (m)D size of the solid body (m)H heat transfer coefficient (Wm�2K�1)I irreversibility (Wm�3)K thermal conductivity (Wm�1K�1)Nu Nusselt numberP pressure (Pa)Ra Rayleigh numberS- volumetric entropy generation (Wm�3K�1)T temperature (K)U velocity in x-axis (m)V velocity in y-axis (m)X distance in x-axis (m)Y distance in y-axis (m)

Greek letters

a thermal diffusivity (m2 s�1)b expansion coefficient (K�1)j any flow variablem dynamic viscosity (Nsm�3)u kinematic viscosity (m2 s�1)r density (kg/m3)



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Energy and entropy analysis in a square cavity with protruding body: effects of protruding body aspect ratio