mhd free convection within trapezoidal cavity with non-uniformly heated bottom wall
TRANSCRIPT
International Journal of Heat and Mass Transfer 69 (2014) 327–336
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International Journal of Heat and Mass Transfer
journal homepage: www.elsevier .com/locate / i jhmt
MHD free convection within trapezoidal cavity with non-uniformlyheated bottom wall
0017-9310/$ - see front matter Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.10.035
⇑ Corresponding author. Tel.: +880 1718210887.E-mail addresses: [email protected] (M.S. Hossain), [email protected]
(M.A. Alim).
Muhammad Sajjad Hossain a,⇑, Mohammad Abdul Alim b
a Department of Arts and Sciences, Ahsanullah University of Science & Technology (AUST), Dhaka - 1208, Bangladeshb Department of Mathematics, Bangladesh University of Engineering & Technology (BUET), Dhaka - 1000, Bangladesh
a r t i c l e i n f o
Article history:Received 6 April 2013Accepted 16 October 2013Available online 8 November 2013
Keywords:Free convectionFinite element methodTrapezoidal cavityNon-uniform heating
a b s t r a c t
The paper is connected to investigate numerical simulation of two-dimensional laminar steady-state onMHD free convection within trapezoidal cavity with non-uniformly heated bottom wall. The cavity con-sists of non-uniformly heated bottom wall, insulated top wall and isothermal side walls with inclinationangles (/). Heat flow patterns in the presence of free convection within trapezoidal enclosures have beenanalyzed with heatlines concept. The fluid is concerned for the wide range of Rayleigh number (Ra) from103 to 107 and Prandtl number (Pr) from 0.026, 0.7, 1000 with various tilt angles U = 45�, 30� and 0�(square). The physical problems are represented by non-dimensional governing equations along withthe corresponding boundary conditions and discretized by using Galerkin weighted residual method offinite element formulation. Results are presented in terms of streamlines, isotherms, local Nusselt num-ber along distance and average Nusselt number along Rayleigh number, Ra for non-uniform heating of thebottom wall, governing parameters namely, Prandtl number Pr, and at the three values of Rayleigh num-ber Ra, varying from 103 to 107, covering free convection dominated regimes. It is shown that the averageand local Nusselt number at the non-uniform heating of bottom wall of the cavity is depending on thedimensionless parameters and also tilts angles.
Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.
1. Introduction
In nature, there exist flows which are caused not only by thetemperature differences but also by concentration differences.These mass transfer differences do effect the rate of heat transfer.In industries, many transport processes exist in which heat andmass transfer takes place simultaneously as a result of combinedbuoyancy affect of thermal diffusion and diffusion through chemi-cal species. The phenomenon of heat and mass transfer frequentlyexist in chemically processed industries such as food processingand polymer production. Free convection flow involving coupledheat and mass transfer occurs frequently in nature. For this flow,the driving forces arise due to the temperature and concentrationvariations in the fluid. For example, in atmospheric flows, thermalconvection resulting from heating of the earth by sunlight is af-fected by differences in water vapour concentration. Magnetohy-drodynamics has attracted the attention of a large number ofscholars due to its diversified applications. In astrophysics and geo-physics, it is applied to study the stellar and solar structures, inter-stellar matter, radio propagation through the ionosphere etc. In
engineering it finds its application in MHD pumps, MHD bearingetc. The study of effects of magnetic field on free convection flowis important in liquid–metals, electrolytes and ionized gases. Thethermal physics of hydromagnetic problems with mass transferis of interest in power engineering and metallurgy. Moreover, thereare several engineering situations wherein combined heat andmass transport arise viz. humidifiers, dehumidifiers, desert coolers,chemical reactors etc. The usual way to study these phenomena isto consider a characteristic moving continuous surface. Free con-vection flow and heat transfer in a trapezoidal cavity have beenthe topic of many research in engineering studies. These studiesconsist of various technological applications such as in electroniccooling, ventilation of building and design of solar collectors etc.Most of the cavities commonly used in industries are cylindrical,rectangular, trapezoidal and triangular etc. Trapezoidal cavitieshave received a considerable attention for its application in variousfields. A brief review of the relevant literature is presented in thefollowing section.
Anandalakshmi and Basak [1] carried out for the energy distri-bution and thermal mixing in steady laminar natural convectiveflow through the rhombic enclosures with various inclination an-gles, / for various industrial applications. Basak et al. [2] also pre-sented natural convection flows in porous trapezoidal enclosureswith various inclination angles. Natural convection in trapezoidal
Nomenclature
B0 magnetic inductionCp specific heat at constant pressure (J/kg K)G gravitational acceleration (m/s2)Gr Grashof numberH convective heat transfer coefficient (W/m2 K)Ha Hartmann numberK thermal conductivity of fluid(W/m K)L height or base of trapezoidal cavity (m)k thermal conductivity ratio fluidN total number of nodesNuav average Nusselt numberNulocal local Nusselt numberP non-dimensional pressurep PressurePr Prandtl numberRa Rayleigh numberT temperature (k)Th temperature of non-uniformly heated bottom wall (k)Tc temperature of cold vertical wall (k)U x component of dimensionless velocityu x component of velocity (m/s)V y component of dimensionless velocityv y component of velocity (m/s)
V0 lid velocityx, y Cartesian coordinatesX, Y dimensionless Cartesian coordinates
Greek symbolsa thermal diffusivity (m2/s)b coefficient of thermal expansion (K�1)q density of the fluid (kg/m3)Dh temperature differenceH fluid temperaturel dynamic viscosity of the fluid (Pa s)P heatfunctionm kinematic viscosity of the fluid (m2/s)r fluid electrical conductivity(O�1 m�1)
Subscriptsb bottom walll left wallr right walls side wall
328 M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336
enclosures for uniformly heated bottom wall, linearly heated verti-cal wall(s) in presence of insulated top wall have been investigatednumerically with penalty finite element method by Basak et al. [3].Basak et al. [4] also performed the phenomena of natural convec-tion within a trapezoidal enclosure filled with porous matrix forlinearly heated vertical wall(s) with various inclination angles /.Basak et al. [5] studied a comprehensive heatline based approachfor natural convection flows in trapezoidal enclosures with the ef-fect of various walls heating. Basak et al. [6] also studied the phe-nomena of natural convection in a trapezoidal enclosure filled withporous matrix numerically. Also an analysis on entropy generationduring natural convection in a trapezoidal cavity with variousinclination angles (U = 45�, 60� and 90�) carried out for an efficientthermal processing of various fluids of industrial importance(Pr = 0.015, 0.7 and 1000) in the range of Rayleigh number(103 � 105) by Basak et al. [7]. Peric [8] studied natural convectionin trapezoidal cavities with a series of symmetrically refined grids10 � 10 – 160 � 160 control volume and observed the convergenceof results for grid independent solutions. A penalty finite elementanalysis with bi-quadratic elements is performed to investigatethe influence of uniform and non-uniform heating of bottom wallon natural convection flows in a trapezoidal cavity by Natarajanet al. [9]. Natarajan et al. [10] presented. a numerical study of com-bined natural convection and surface radiation heat transfer in asolar trapezoidal cavity absorber for Compact Linear FresnelReflector (CLFR). The numerical simulation results are presentedin terms of Nusselt number correlation to show the effect of theseparameters on combined natural convection and surface radiationheat loss. Saleh et al. [11] also studied the effect of a magnetic fieldon steady convection in a trapezoidal enclosure filled with a fluid-saturated porous medium by the finite difference method. Besides,a study of the natural heat and mass transfer in a trapezoidal cavityheated from the bottom and cooled from the inclined upper wall isundertaken by Boussaid et al. [12]. He obtained results show thatthe flow configuration depends on the h angle inclination of theupper wall. Hung et al. [13] made an attempt to analyze the non-linear instability of a magnetohydrodynamics film flow with phasechange at the interface. They pointed that increasing the stabilityof film flow by controlling magnetic field; a film flow with
optimum conditions could be obtained. Shanker and Kishan [14]studied the effects of mass transfer on the MHD flow past animpulsively started infinite vertical plate with variable tempera-ture or constant heat flux. Sattar and Hossain [15] also studied un-steady hydromagnetic free convection flow with Hall current andmass transfer along an accelerated porous plate with time depen-dent temperature and concentration. Besides Gnaneswara Reddyand Bhaskar Reddy [16] presented finite element analysis of Soretand Dufour effects on unsteady MHD free convection flow past animpulsively started vertical porous plate with viscous dissipation.Suresh Babu et al. [17] also presented finite element analysis offree convection flow with MHD micropolar and viscous fluids ina vertical channel with dissipative effects. Sparrow and Cess [18]attempted the effect of a magnetic field on free convection heattransfer. Basak et al. [19] also investigated heat flow patterns inthe presence of natural convection within trapezoidal enclosureswith heatlines concept. In this study, natural convection within atrapezoidal enclosure for uniformly and non-uniformly heated bot-tom wall, insulated top wall and isothermal side walls with incli-nation angle have been investigated.
In the light of the above literature review, it appears that, visual-ization of heat flows via heatlines for magneto-hydrodynamics withnon-uniformly heated bottom walls were not reported for trapezoi-dal enclosures. It is also essential to study the heat transfer charac-teristics in complex geometries in order to obtain the optimaldesign of the container for various industrial applications. The aimof the present work is to present the effects on heat flow via heatlinesfor MHD free convection within trapezoidal cavity with non-uniformlyheated bottom wall. Results will be presented for different physicalparameters in terms of streamlines, stream functions, total heat flux,isotherms, heat transfer rate as well as the average temperature ofthe fluid in the cavity. The heatlines and thermal mixing will beillustrated for commonly used fluid with Pr = 0.026 � 1000 andRa = 103 � 107 in various industrial applications.
2. Physical model
The geometry of the problem herein studied is depicted inFig. 1. We consider a trapezoidal cavity of height L with the left
M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336 329
wall inclined at an angle U = 45�, 30�, 0� with Y axis. The heattransfer and the fluid flow for non-uniform heating in a two-dimensional trapezoidal cavity with a fluid whose left wall andright wall (i.e. side walls) are subjected to cold Tc temperature, bot-tom wall is subjected to hot Th temperature while the top wall iskept insulated. The boundary conditions for velocity are consideredas no-slip on solid boundaries.
3. Governing equations
The functioning fluid is assumed to be Newtonian, steady andincompressible with the flow is set to operate in the laminar freeconvection regime. The dimensionless governing equationsdescribing the flow under Boussinesq approximation are presentedas follows:
@U@Xþ @V@Y¼ 0 ð1Þ
U@U@Xþ V
@U@Y¼ � @P
@Xþ Pr
@2U
@X2 þ@2U
@Y2
!ð2Þ
U@U@Xþ V
@U@Y¼ � @P
@Yþ Pr
@2V
@X2 þ@2V
@Y2
!þ RaPrh� Ha2PrV ð3Þ
U@h@Xþ V
@h@Y¼ @2h
@X2 þ@2h
@Y2
!ð4Þ
Using the following dimensionless quantities, the Eqs. (1)–(4) arenon-dimensionalized:
X ¼ xL; Y ¼ y
L; U ¼ uL
a; V ¼ mL
a; P ¼ pL2
qa2 ; h ¼ T � Tc
Th � Tc;
Pr ¼ ma; Gr ¼ gbL3ðTh � TcÞ
m2 ; Ra ¼ gbL3ðTh � TcÞPrm2 ;
Ha2 ¼ rB20L2
l; a ¼ k
qCp
Fig. 1. Schematic diagram of the physical syste
Here Ha, Ra, Pr, Gr are Hartmann number, Rayleigh number, Prandtlnumber and Grashof number respectively. Thermal diffusivity, vol-umetric thermal expansion coefficient, dynamic viscosity, kine-matic viscosity, electrical conductivity, density and dimensionaltemperature difference of the fluid are represented by the symbolsa, b, l, m, r, q, DT, respectively.
The boundary conditions (also shown in Fig. 1), for the presentproblem are specified as follows
At the bottom wall:
U ¼ 0; V ¼ 0; h ¼ sinðpXÞ 8Y ¼ 0; 0 6 X 6 1
At the left wall:
U ¼ 0; V ¼ 0; h ¼ 0; 8X cos /þ Y sin / ¼ 0; 0 6 Y 6 1
At the right wall:
U ¼ 0; V ¼ 0; h ¼ 0; 8X cos /� Y sin / ¼ cos /; 0 6 Y 6 1
At the top wall:
U ¼ 0; V ¼ 0;@h@Y¼ 0; 8Y ¼ 1; � tan / 6 X 6 ð1þ tan /Þ
where X and Y are dimensionless coordinates varying along hori-zontal and vertical directions, respectively; U and V are dimension-less velocity components in X and Y directions, respectively; h is thedimensionless temperature.
The local Nusselt number at the heated surface of the cavitywhich is defined by the following expression: Nul ¼ Nur ¼ Nub ¼Nus ¼ � @h
@n ; where n denotes the normal direction on a plane.According to Singh and Sharif [20], the average Nusselt number
at the heated bottom wall, cold left and right walls and insulatedtop walls of the cavity based on the non-dimensional variablesmay be expressed as, Nu ¼
R 10 Nul dX ¼
R 10 Nur dX ¼
R 10 Nus dX ¼R 1
0 Nub dX.
m for (a) a = 45�, (b) a = 30� and (c) a = 0�.
Table 1Grid sensitivity check at Pr = 0.7, Phi = 45�, Ha = 50 and Ra = 107.
Nodes (Elements) 5858 (864) 8098 (1200) 16605 (2505) 22944 (3456) 25053 (3767) 31479 (4301) 43505 (6609)
Nu 11.54703 11.949276 12.08048 12.8922 12.8917 12.8233 12.6724Time (s) 8.047 9.172 18.703 26.75 31.968 38.594 64.75
Table 2Code validation for non-uniform heating of bottom wall with Pr = 0.7.
Ra Average Nusselt number, (Nuav)
Present work Basak et al. (March 2009)a = 45� a = 45�
103 1.883518 1.42156104 3.155096 2.87144105 4.822457 4.70209
330 M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336
4. Numerical technique
4.1. Mathematical formulation
The numerical procedure used to solve the governing equationsfor the present work is based on the Galerkin weighted residualmethod of finite-element formulation [21,22]. The non-linear para-metric solution method is chosen to solve the governing equations.This approach will result in substantially fast convergence assur-ance. A non-uniform triangular mesh arrangement is implemented
(a)
0.48
4.52
1.90
7.034.
1.65
0.10
0.50
1.601.10
0.60
0.20
0.200.20 0
0.8
1.
7.82
0.002
0.004
3.53
7.57
9.4
0.01
0
0.002
Fig. 2. Stream function (W), temperature (h), heat function or total heat flux (G) for nonU = 0� (b) U = 30� (c) U = 45�.
in the present investigation especially near the walls to capture therapid changes in the dependent variables.
The velocity and thermal energy Eqs. (1)–(4) result in a set ofnon-linear coupled equations for which an iterative scheme isadopted. To ensure convergence of the numerical algorithm thefollowing criteria is applied to all dependent variables over thesolution domain,X
wnij � wn�1
ij
��� ��� 6 10�5
where W represents a dependent variable U, V, P, and T; the indexesi, j indicate a grid point; and the index n is the current iteration atthe grid level. The six node triangular element is used in this workfor the development of the finite element equations. All six nodesare associated with velocities as well as temperature; only the cor-ner nodes are associated with pressure. This means that a lower or-der polynomial is chosen for pressure and which is satisfied throughcontinuity equation. The velocity components, the temperature dis-tributions and linear interpolation for the pressure distributionaccording to their highest derivative orders in the differential Eqs.(1)–(4) are as follows:
(b) (c)
0.30
06
1.75
0.48
3.622.46
1.521.27
0.10.50
0
60
0.10 0.200.50
0.60
0.80
1.30
1.80
1
.001
7.401
0.01
9.11
5.58
3.63
-uniform bottom heating h(X, 0) = sin(px) with Pr = 0.026, Ha = 50 and Ra = 103 (a)
(a) (b) (c)
89.46
63.32
28.2750.86
16.69
95.20
61.7557.20
15.9030.33
4.6
10.85
106.06
19.8945.26
48.1055.23
107.14
1.30
0.40
0.60
0.60
0.80
1.50
1.70 0.700.50
1.20
0.90 1.00
1.20
1.30
1.40
0.50
0.70
1.80
0.80 0.60
0.80 1.10 1.00
1.20
0.70
0.50 1.301.60
65.2963.04
75.33
20.25
68.67
50.37 39.07
68.09 40.88
66.67
50.0838.05
4.08
75.74
Fig. 3. Stream function (W), temperature (h), heat function or total heat flux(G) for non-uniform bottom heating h(X,0) = sin(px) with Pr = 0.026, Ha = 50 and Ra = 107 (a)U = 0� (b) U = 30� (c) U = 45�.
M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336 331
UðX;YÞ ¼ NaUa; VðX;YÞ ¼ NaVa; hðX;YÞ ¼ Naha; PðX;YÞ ¼ HkPk
where a = 1, 2, ... , 6; k = 1–3; Na are the element interpolation func-tions for the velocity components and the temperature, and Hk arethe element interpolation functions for the pressure.
To derive the finite element equations, the method of weightedresiduals finite-element formulation is applied to the Eqs. (1)–(4)asZ
ANa
@U@Xþ @V@Y
� �dA ¼ 0 ð5Þ
ZA
Na U@U@Xþ V
@U@Y
� �dA ¼ �
ZA
Hk@P@X
� �dAþ Pr
ZA
Na@2U
@X2 þ@2U
@Y2
!dA ð6Þ
ZA
Na U@V@Xþ V
@V@Y
� �dA ¼ �
ZA
Hk@P@Y
� �dAþ Pr
ZA
Na@2V
@X2 þ@2V
@Y2
!dA
þ RaPrZ
ANahdA� Ha2Pr
ZA
NaVdA ð7ÞZ
ANa U
@h@Xþ V
@h@Y
� �dA ¼
ZA
Na@2h
@X2 þ@2h
@Y2
!dA ð8Þ
So, the coefficients in element matrices are in the form of the inte-grals over the element area as,
Kabx ¼Z
ANaNb;xdA; Kaby ¼
ZA
NaNb;ydA; Kabcx ¼Z
ANaNbNc;xdA
Kabcy ¼Z
ANaNbNc;ydA; Kab ¼
ZA
NaNbdA; Sabxx ¼Z
ANa;xNb;xdA
Sabyy ¼Z
ANa;yNb;ydA; Malx ¼
ZA
HaHl;xdA; Maly ¼Z
AHaHl;ydA
Qau ¼Z
S0
NaSxdS0; Qam ¼Z
S0
NaSydS0; Qah ¼Z
Sw
Naq1wdSw
Qah s ¼Z
Sw
Naq2wdSw
Using reduced integration technique of Reddy [21] and Newton–Raphson iteration technique by first writing the unbalanced valuesfrom the set of the finite element equations, the nonlinear algebraicEqs. (5)–(8) so obtained are modified by imposition of boundaryconditions. These modified nonlinear equations are transferred intolinear algebraic equations by using Newton’s method. Finally, theselinear equations are solved by using Triangular Factorizationmethod.
5. Grid independence test
In order to obtain grid independent solution, a grid refinementstudy is performed for a trapezoidal cavity with Pr = 0.7, phi = 45�and Ra = 107. It is observed that grid independence is achieved with3456 elements where there is insignificant change in Nu with fur-ther increase of mesh elements. Seven different non-uniform gridsizes of 5858, 8098, 16605, 22944, 25053, 31479, 43505 nodesand 864, 1200, 2505, 3456, 3767, 4301, 6609 elements were
(a) (b) (c)
0 0 1 0 2 0 3 0 4 0 5 0 6
406.03
4.167.19
137.21 111.84 123.96
380.31
16.48
2.64
301.96381.08
6.775.97
125.36105.85
332.99
0.5 0 0.90
0.90
0.80
0.70
0.60
1.001.300.30 0.50
0.60
0.800.90
0.700.60
0.80
0.30
0.500.70
0.80
0.800.80
0.80
0.500.50
0.50
0.400.60
206.83 220.72
125.22131.66
590.84
601.06
384.16
45.4059.57
668.46
296.81
141.65143.38
Fig. 4. Stream function (W), temperature (h), heat function or total heat flux(G) for non-uniform bottom heating h(X,0) = sin(px) with Pr = 0.7, Ha = 50 and Ra = 107 (a) U = 0�(b) U = 30� (c) U = 45�.
332 M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336
considered for the grid refinement tests. Hence considering thenon-uniform grid system of 3456 elements is preferred for thecomputation of all cases (see Table 1).
6. Code validation
The model validation is a necessary part of a mathematicalinvestigation. Hence, the outcome of the present numerical codeis benchmarked against the numerical result of Basak et al. [19]for free convection in a trapezoidal cavity. Average Nusselt numberis calculated for three different Rayleigh numbers (Ra = 103, 104
and 105) and also an angle U = 45� while the Prandtl number isfixed i.e. Pr = 0.7 for non-uniform heating of bottom wall respec-tively in Table 2. The present result shows an outstanding agree-ment with these of Basak et al. [19].
7. Results and discussion
In this endorsement numerical studies have been studied onMHD free convection within trapezoidal cavity with non-uniformlyheated bottom. Results are discussed for the case of non-uniformheating bottom wall and also heat transfer rates for local and aver-age Nusselt numbers. Besides, results are found for various Ray-leigh number, Ra = 103 � 107 and Prandtl number, Pr = 0.026, 0.7,1000 with various angles, U = 45�, 30�, 0� (square cavity).
7.1. Non-uniform heating of bottom wall
Fig. 2 illustrate that the magnitudes of streamfunction are con-siderably smaller and heat transfer are primarily due to conduc-tion. For Ra = 103, Pr = 0.026 and U = 0� (square cavity) isotherms(temperature) with h = 0.10 � 0.20 occur symmetrically along side(left or right) walls and with h P 0.30 are smooth curves symmet-ric with respect to vertical symmetrical line (Fig. 2a). For Ra = 103,Pr = 0.026 and U = 30� the temperature contours withh = 0.10 � 0.40 occur symmetrically near the side walls of theenclosure and with h P 0.50 are smooth curves symmetricwith respect to central symmetrical line (Fig. 2b). Again forRa = 103, Pr = 0.026 and U = 45� isotherms (temperature) withh = 0.10 � 0.50 occur symmetrically near the side walls of theenclosure and with h P 0.60 are smooth curves symmetric with re-spect to vertical symmetrical line (Fig. 2c).The heatlines or totalheat flux or heat function are shown in panels of Fig. 2(a–c). Theheatlines illustrate similar feature that were observed for uniformheating cases.
It is interesting to note that at the bottom corner point U = 0�(square cavity) is larger than for U = 45� and 30�. It is evident thatheatlines near the bottom portion of side walls are more dense forU = 45� and less dense for U = 0� (square cavity). The dense heat-lines is also indicating enhanced rate of heat transfer from the bot-tom to the side walls. Therefore isotherms with h = 0.05 � 0.35 areshifted for U = 45� toward the side walls. It is also observed thatheat transfer at the top portion of the cavity for U = 45� and 30�
(a) (b) (c)
871.52
1195.57
257.67 230.41
332.94
372.04
1406.29
243.94234.07
2047.78
220.13
726.08
296.42
0.60 0.80
0.90
0.90
0.70
0.50
0.50 1.30
0.80
0.80 0.80
0.70
0.80
0.70
0.70
1.40
0.900.90
0.80
0.80
0.900.70
0.90
1234.14
1519.30
299.82354.24
962.53
1564.62
302.77 254.69
1122.01
28.82129.14
262.59
Fig. 5. Stream function (W), temperature (h), heat function or total heat flux(G) for non-uniform bottom heating h(X,0) = sin(px) with Pr = 1000, Ha = 50 and Ra = 107 (a)U = 0� (b) U = 30� (c) U = 45�.
M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336 333
is higher compressed to U = 0� (square cavity) based on value ofheatfunction (P). As the heat transfer is quite large at the cornersof bottom wall, the thermal boundary layer was found to developnear the bottom edges and thickness of boundary layer is largerat the top portion of the cold wall signifying less heat transfer tothe top portion.
Fig. 3 illustrate that the magnitudes of streamfunction at largerRa = 107 with Pr = 0.026. It is interesting to detect that the stratifi-cation zone of temperature at the center vertical line near the bot-tom wall for Ra = 107 is suppressed whereas stratification zone oftemperature is larger for Ra = 103 due to increased convection. Itis also noted that stratification zone of temperature at bottom wallis thicker for U = 0� (square cavity) due to less intense circulationnear the top portion of the cavity. Isotherms (temperature) withh = 0.10 � 0.1.20, h = 0.10 � 1.20, h = 0.10 � 1.10 occur symmetri-cally near the side walls of the enclosure and with h P 1.30,h P 1.30, h P 1.20 are smooth curves symmetric with respect tocentral symmetrical line for Ra = 107, Pr = 0.026 and U = 45�, 30�,0� (square cavity) respectively. Heatlines indicate the heat transferfrom the hot wall to cold side walls and uniform heating flow cir-culation due to fluid circulation cells. Due to non-uniform heatingof bottom wall the heating rate near to corners of the wall is gen-erally lower and that induces less buoyancy effect resulting in lessthermal gradient throughout the domain. Heatlines near the topportion of the side walls are oscillatory due to secondary circula-tion for U = 45� and 30�. It is also observed that heatlines are quitedense near the central regime for U = 45� and 30� and implies
enhance thermal mixing near the central to top portion of thecavity. This is due to the fact that strong primary circulations occurnear to the corners to the bottom wall .
Fig. 4 represent that the magnitudes of streamfunction are cir-cular or elliptical near the core but the streamlines near the wall isalmost parallel to wall exhibiting large intensity of flow for Pr = 0.7and Ra = 107. Also isotherms h = 0.10 � 0.90, h = 0.10 � 0.80,h = 0.10 � 0.80 for 0.7 and Ra = 107 occur symmetrically near theside walls of the enclosure and h P 1.00, h P 0.90, h P 0.90 aresmooth curves symmetric with respect to central symmetrical linefor Ra = 107, Pr = 0.7 and U = 45�, 30�, 0� (square cavity) respec-tively. It is also fascinating that multiple correlations are absentfor Pr = 0.7 and Ra = 107. Due to enhanced flow circulations the iso-therms are highly compressed near the side walls except near thebottom wall especially for U = 45� and 30�. As Ra increases forPr = 0.7 the circulations near the central regime are stronger andconsequently the bottom portion contours start getting stretchedtoward the central regime of the bottom wall. It is indicated thatthe strength of circulations is more for non-uniform heating case.
Fig. 5 shows streamline. Isotherms (temperature) and heatlinesfor Ra = 103 � 107, Pr = 1000 and a = 45�, 30�, 0� (square cavity)respectively. Also for Pr = 1000 and Ra = 107 isotherms withh = 0.10 � 0.80, h = 0.10 � 0.80, h = 0.10 � 0.90 occur symmetricallynear the side walls of the enclosure and h P 0.90, h P 0.90,h P 1.00 are smooth curves symmetric with respect to centralsymmetrical line for Ra = 103 � 107, Pr = 0.7 and U = 45�, 30�, 0�(square cavity) respectively. As Pr increases from Pr = 0.7 to 1000
334 M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336
the flow intensity is found to be enhanced. Here it is seen that themagnitudes of streamlines is maximum and its shape is also circu-lar or elliptical. It is also indicate that the values of stream func-tions of non-uniform heating case are almost similar to uniformheating cases due to the lower heat from bottom wall and the lessintense heating effects near the central regime are attributed byless-dense heatline due to non-uniform heating effects.
7.2. Heat transfer rates: local Nusselt number vs distance
Fig. 6(a) show the effects of various inclinations of angles i.e. (i)U = 0�, (ii) U = 30�, (iii) U = 45�, in presence of non-uniform heat-ing of bottom wall with Pr = 0.026 for local Nusselt number vs dis-tance. Here the heat transfer rates are discussed for Ra = 103. Sincethe bottom wall is non-uniformly heated and side walls are coldand top wall is thermal insulated so that, for adjacent wall it is ob-served that heat transfer rate is maximum near edge of the leftwall and the rate is step down from left side and it is straightlymoving and then it is seen that heat transfer rate is minimum nearthe bottom edge of the wall and then also it goes up to right side.The heat transfer rate is changing for the effect of Grashof numberbecause the Prandtl number is fixed. Here the heat transfer ratesare almost same for U = 30�, 45� except U = 0� [square cavity].
(a) (b)
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60 phi = 0, Ra = 1e7phi = 30, Ra = 1e7phi = 45, Ra = 1e7
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.50102030405060708090
Phi = 0, Ra = 1e7Phi = 30, Ra = 1e7Phi = 45, Ra = 1e7
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60phi = 0, Ra = 1e5phi = 30, Ra = 1e5phi = 45, Ra = 1e5
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60Phi = 0, Ra = 1e5Phi = 30, Ra = 1e5Phi = 45, Ra = 1e5
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60phi = 0, Ra = 1e4phi = 30, Ra = 1e4phi = 45, Ra = 1e4
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60
Phi = 0, Ra = 1e4Phi = 30, Ra = 1e4Phi = 45, Ra = 1e4
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60 Phi = 0, Ra = 1e3phi = 30, Ra = 1e3phi = 45, Ra = 1e3
Distance, X
LocalNusseltNumber,Nub
0.1 0.2 0.3 0.4 0.5010
2030
40
50
60 Phi = 0, Ra = 1e3Phi = 30, Ra = 1e3Phi = 45, Ra = 1e3
Fig. 6. Variations of local Nusselt numbers (Nub) with distance for (a) Pr = 0.026, (b)Pr = 0.7 and Ra = 103, 104,105, 107 and for various inclination of angles U = 0�, 30�,45� in presence of non-uniform heating of bottom walls.
Fig. 6(a) also present similar effects of local Nusselt number withdistance for various inclination tilt angles for Pr = 0.026 in case ofnon-uniform heating. But here the values of heat transfer rate in-crease a little and the heat transfer rates are shown for Ra = 103,104. Fig. 6(a) detects the variation of local Nusselt number withdistance for various inclination tilt angles i.e. (i) U = 0�, (ii)U = 30�, (iii) U = 45�, with Pr = 0.026 for non-uniform heating ofbottom wall. Here heat transfer rates are discussed for Ra = 107.It is pragmatic that at left corner heat transfer rate is very highand reduces towards the middle position of the bottom wall asthe comparison of temperature contours is minimum at the centerof wall irrespective of /s with Pr = 0.026 and it is also seen thatheat transfer rates arises to right side.
Fig. 6(b) explain the effects for various inclination angles whenRa = 103 and Pr = 0.7 in presence of non-uniform heating of bottomwalls. Here it is seen that the value of heat transfer rates increasesas Pr increases. It is also interesting to observe that the heat trans-fer rates (Nub) for U = 30�, and U = 45� are almost alike exceptU = 0�. It is also observed that thermal gradient is minimum atthe center of bottom wall as seen from dispersed isotherm con-tours at the center of the wall for irrespective of as. Fig. 6(b) epit-omize that similar effects of various inclination angles U = 45�, 30�,0� for Ra = 104, 105, 107 with Pr = 0.7 in presence of non-uniformbottom heating. Here we also see that, the heat transfer rate of leftwall is very high at the top edge of left wall is very high and heattransfer rate is almost not uniform near the bottom edge of hotvertical wall. As Ra increases then the magnitudes of heat transferrates increases. It is seen that the thermal gradient is minimum atthe center of bottom wall for dispersed isothermal contour for irre-spective of a. The larger heat transfer rate for U = 0� occurs due tohighly compressed isotherms as seen Fig. 6(b). Here it is also seenthat the heat transfer rates for U = 30� and 45� are almost same ex-cept U = 0�. But at larger Ra = 107 it is experiential that the oscilla-tion of local heat transfer rates occur due to presence of secondarycirculations which result of isotherm contours at various places ofbottom wall of non-uniform heating.
7.3. Heat transfer rates: average Nusselt number vs Rayleigh number
The effects upon the heat transfer rates are presented inFig. 7(a–c), where distributions of average Nusselt number of bot-tom wall respectively are plotted vs the logarithmic Rayleigh num-ber. Here Fig. 7(a–c) illustrate non-uniform heating of bottom wallrespectively. It may be noted that average Nusselt number is ob-tained by considering temperature gradient. It also be noted thatas Ra increases then the average Nusselt number increases. It isseen in Fig. 7(a) that as Ra increases from 103 � 106 then averageNusselt number is straightly moving but as Ra increases more, thenaverage Nusselt number is increasing for Pr = 0.026. As Pr increases(Fig. 7(b)) then conduction dominant heat transfer is narroweddown. It also seen from Fig. 7(c) that, as Pr increases more thanfrom non-uniform heating case it is analyzed that average Nusseltnumber for bottom wall is also little increasing during the entireRayleigh number regime. This illustrates that the conduction dom-inant heat transfer for different Prandtl number regime irrespec-tive of Ra. As Pr increases then for conduction dominant heattransfer, the average Nusselt number is generally constant irre-spective of Ra. It is observed that Nub at the middle portion of bot-tom wall for U = 0� is larges for non-uniform heating case whereasfor U = 30� and 45� heat transfer rates are identical. It is also ob-served that when Pr = 0.026 then heat transfer rates for U = 45�is swiftly increasing except U = 30� and 0�. At larges Pr(Pr = 1000) of non-uniform bottom heating it is seen that as Ra in-crease from 103 to 107, then average heat transfer rates (Nub) alsoincreases.
(c)
RaNu a
v
10 3 10 4 10 5 10 6 10 70
5
10
15 Phi = 0Phi = 30Phi = 45
Ra
Nu a
v
10 3 10 4 10 5 10 6 10 70
5
10
15Phi = 0Phi = 30Phi = 45
Ra
Nu a
v
10 3 10 4 10 5 10 6 10 70
5
10
15 Phi = 0Phi = 30Phi = 45
(a) (b)
Fig. 7. Variations of Average Nusselt Number vs Rayleigh number for (a) Pr = 0.026, (b) Pr = 0.7, (c) Pr = 1000 and for various inclination of angles U = 0�, 30�, 45� in presenceof non-uniform heating of bottom wall.
M.S. Hossain, M.A. Alim / International Journal of Heat and Mass Transfer 69 (2014) 327–336 335
8. Conclusions
A computational study is performed to consider two-dimen-sional laminar steady state MHD free or natural convection withintrapezoidal cavity for non-uniformly heated of bottom with heat-lines concept, whereas, a finite element method is used. The de-rived finite element conservation of mass, momentum, andenergy equations are nonlinear requiring an iterative techniquesolver. To solve these nonlinear equations for solutions of the nodalvelocity component, temperature, and pressure by consideringPrandtl numbers of 0.026, 0.7, 1000, Hartman numbers of 50 andalso Rayleigh numbers of 103–107 are applied Galerkin weightedresidual method. The stream line, total heat flux and isothermsas well as characteristics of heat transfer process particularly itsexpansion has been evaluated for non-uniform heating. The fol-lowing conclusions may also be drawn from the current analysis:
� The heat transfer rate is maximum near the edge of the wall andthe rate is minimum near the center of the wall irrespective ofall angles (/) for non-uniform heating of the bottom wall forRayleigh number 103–107 gradually.� The average Nusselt number (Nu) at the non-uniform heating of
bottom wall is the highest for the angle 0� when Rayleigh num-ber 107, whereas the lowest heat transfer rate for the angle 45�when Rayleigh number 103.� Heat transfer depends on Prandtl number and heat transfer rate
is maximum near the edge of the wall and the rate is minimumnear the center of the wall irrespective of all angles (/) for non-uniform heating of the bottom wall for different Prandtlnumber.� Thermal boundary layer thickness is thinner for increasing of
Rayleigh number due to upsurge of convective heat transfer rate.� Local Nusselt number for non-uniform bottom heating is largest
at the bottom edge of the side wall, and thereafter thatdecreases sharply upto a point which is very near to the bottomedge.� The heat transfer rate average Nusselt Number, Nuav increases
with the increase of Rayleigh number, Ra, for non-uniform heat-ing of bottom wall.
� Various vortices entering into the flow field and secondary vor-tex at the vicinity boundary wall and bottom wall of the cavityis seen in the streamlines.
Acknowledgements
We would like to express our gratitude to the Department ofMathematics, Bangladesh University of Engineering and Technol-ogy (BUET) and Department of Arts and Sciences, Ahsanullah Uni-versity of Science and Technology (AUST), Dhaka, Bangladesh, forproviding computing facility during this work.
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