Magnetohydrodynamics and Aspect Ratio Effects on Double Diffusive Mixed Convection and Their Prediction: Linear Regression Model

Magnetohydrodynamics and Aspect Ratio Effects on Double Diffusive Mixed Convection and Their Prediction:
Linear Regression Model

Shivananda Moolya ^a,b, Satheesh Anbalagan [*]^,a , Neelamegam Rajan Devi^c,
 Rekha Moolya^d

^aSchool of Mechanical Engineering, Vellore Institute of Technology, Vellore, Tamilnadu, INDIA-632014.

^bUniversity of Technology and Applied Sciences, PO Box 74, Al-Khuwair Postal code 133, Sultanate of Oman.

^cDepartment of Physics, Auxilium College (Autonomous), Tamilnadu, INDIA-632006.

^dGopalan College of Engineering and Management, Bangalore, Karnataka, India, 560048.

1. Introduction

Double diffusive mixed convection is widely used in many industrial and medical applications like solar ponds, material processing, Drug delivery systems, cancer treatments, solar air conditioning, packaging of electronic items, food processing, etc. Considering the wide application of double-diffusive mixed convection in the industrial and medical areas, further study is required to optimize the parameter influencing heat and mass transfer for maximum performance. Several numerical studies have been performed for the optimization of heat and mass transfer of the cavity flow. Rodrigues et al. [1] performed optimization of the geometry of the lid-driven cavity with two fins inside the cavity for mixed-type convection. They found that the asymmetric geometry of the fin plays a major role in controlling the temperature. Baag et al. [2] numerically studied the MHD effect using micropolar-type fluid flow on a vertical surface. They found that backflow is controlled by opposing buoyancy parameters. Khader and Megahed [3] used the Finite Difference Method to analyze the heat transfer on a liquid thin film over a stretching sheet using a numerical method. Li et al. [4] performed optimization of nanoparticle size and other parameters on natural convection. They found that aspect ratio and Rayleigh number are the most significant parameters to control the thermal performance of the cavity.  Iyi and Hasan [5] numerically performed the analysis on the effect of moistness using the buoyancy-driven flow of turbulence nature in a cavity. Li, et al. [6] have numerically optimized heat transfer in a microchannel using the field synergy principle. They found that internal cavity configuration plays a major role in controlling heat transfer. Mohammadi et al. [7] have optimized the heat transfer of flow of nanofluid inside a coil using Taguchi analysis. They found an optimum combination of a selected parameter for maximum performance. Srinivas, et al. [8] analyzed chemical reactions and thermal diffusion effect using MHD flow in a porous channel. The effect of different dimensional numbers has been considered in this analysis.  Nath and Krishnan [9] performed an optimization analysis using Taguchi and the utility method of a double-diffusive type of flow using a channel filled with nanofluid. They compared the results of those methods and found that the utility method predicts marginally less Nu and Sh. Raei [10] used a Taguchi method for optimizing the heat exchanger operating with aluminum oxide nanofluid.Kishore, et al. [11] have done an optimization analysis of thermal electric generators using the Analysis of Variance (ANOVA) and the Taguchi method. They compared Taguchi and full factorial ANOVA results and found an 11.8% deviation. Iftikha et al. [12] performed a study on natural convective heat and mass transfer with inclined MHD effect using fractional operators. Hatami et al. [13] performed an optimization analysis of heat transfer using natural convection in a circular cavity. The results of their findings are, the Lewis number plays a major role in controlling the Nusselt number. Optimization of the cooled condenser for forced convection is performed by Kumara et al. [14]. The results of the Taguchi method show that improved performance is observed for 2-4 tubes kept at a 3-5 mm gap. Mamourian et al. [15] have optimized heat transfer by mixed convection and generation of entropy in a lid-driven cavity using the Taguchi method. They found the value of the optimum percentage of nanoparticles, wavelength, and Richardson number for better thermal performance of the cavity. Shirvan et al. [16] have numerically investigated and optimized the parameters involved in mixed convection using nanofluid flow in a cavity. The PISO algorithm is used to optimize the position of the heat sink and source in a closed enclosure Soleimani et al., [17]. An optimal combination for better thermal performance of the cavity is presented in this work. Position of multiple heat source optimization in a cavity was performed by Madadi and Balaji [18], using ANN and genetic algorithm. The result shows that a combination of ANN and GA will reduce the computational period. Paulo et al. [19] analyzed the mixed convection laminar flow in a domain where two openings as an inlet and outlet in the vertical walls. Alinejad and Esfahani [20] used the Taguchi design approach to optimize the mixed convection in an enclosure. Taguchi's analysis shows that all the selected parameters show a significant effect on output. An optimal level of the selected parameters is also presented in the work.  Gorobets et al. [21] published a numerical study on heat transfer with hydrodynamics effect in small diameter tube bundles.  Hatami et al. [22] used a T-shaped cavity filled with a porous material and nanofluid to analyze and optimize heat transfer. ANN and GA are used by Mathew and Hotta [23] for optimizing IC chip arrangements on SMPS for mixed convection heat transfer. Results of the work show that the hybrid arrangement of chips is more effective than ordinary methods. Pichandi and Anbalagan [24] used a 2D enclosure to analyze the fluid flow and heat transfer analysis with the sinusoidal wave. LBM method is used for the analysis. Mirzakhanlari et al. [25] performed a sensitivity analysis to improve mixed convection and to reduce the coefficient of drag of fluid flow in a cavity with a rotating cylinder. The result of the analysis gives the effective values to increase the mean Nusselt number and reduce the drag coefficient.  Sensitivity analysis and optimization are performed by Pordanjani et al. [26] to find the magnetic field effect on convection and entropy generation of nanofluid flow in an inclined domain. They found that Bejan and generation of entropy enhance with an increase in the thermal source length. Behbahan et al. [27] investigated the effect of aspect ratio on the melting thermal characteristics of metal foam. Investigation shows that aspect ratio plays a major role in changing the phase of copper foam. Selimefendigil and Öztop [28] Performed MHD convection optimization in a trapezoidal domain filled with nanofluid. Sudhakar et al. [29] used the ANN method to optimize the heat source configuration in a duct under mixed convection. Results show that CNN gives more accurate and fast results than computational studies. Tassone et al. [30] conducted optimization and heat transfer analysis of the cooling system using MHD flow in a WCLL blanket. Yigit and Chakraborty [31] performed a numerical analysis to find the effect of boundary condition and aspect ratio using Bingham fluids. Their investigation concludes that buoyancy forces improve flow by increasing the Rayleigh number. Hamzah et al. [32] published a research work on the effect of MHD on natural convection considering nanoparticle fluid flow in a U-Shape cavity. Yang and Yeh [33] optimized arrays of fins in a channel for mixed-type convection. A correlation of the aspect ratios for the maximum performance for different inclination angles is proposed in this study. Numerical investigation of the mixed convection effect in a square lid-driven cavity under the combined effects of thermal and mass diffusion is numerically investigated [34]. Izadi et al. [35] performed a numerical analysis to find the effect of heat source and cavity aspect ratio using a C-shaped cavity on natural convection. They found a linear relationship between Nu and the aspect ratio of the enclosure. Béghein et al. [36] performed a numerical study on double-diffusive convection using a rectangular cavity. Alsobaai [37] used optimized petroleum residue thermal cracking using a three-level factorial design. Sathiyamoorthi and Anbalagan [38] presented their numerical study on entropy generation in double-diffusive natural convection in a square enclosure with the presence of a rectangular block at the center. They found that an increase in the Rayleigh number significantly increases entropy generation. An increase in Hartmann number decreases total entropy generation. Aljabair et al. [39] published a research article on the mixed convection of nanofluid flow in a lid-driven domain of sinusoidal nature. Teamah [40] performed numerical analysis on double diffusion using the rectangular domain. Recently, Moolya and Satheesh [41,42] performed a numerical study on the effect of an inclined magnetic field on double-diffusive mixed convection by considering different inclination angles. In the study, it is found that the magnetic field has a negative impact and the inclination angle has a positive impact on heat and mass transfer. Further, the work is extended for finding the optimum combination of the selected parameter for maximum performance. In the analysis, it is found that the Richardson number is the least significant parameter compared to all the selected parameters. Hasan et al. [43] used a mechanical chamber of triangular shape to analyze the periodic natural convection effect using CNT nanofluid. Raju et al. [44] performed a study on heat transmission and flow characteristics of nanofluid in a radiated flexible porous channel. They found that the distribution of temperature is higher in case-1, but the opposite finding is observed in a momentum boundary layer. Hossain et al [45] conducted a numerical study on the MHD convection effect using nanofluid with radiation. This analysis would help in designing thermal devices that work on nanofluids. Priyadharshini et al. [46] investigated the use of heat and mass transfer in MHD flow over a stretching sheet using a machine learning technique. They found that machine learning technology reduces the cost of simulation in the analysis of metal flow in metallurgy. Azad et al. [47] performed numerical analysis on heat and mass transfer properties in an enclosure using variable buoyancy ratio. Results show that an increase in the dimensionless time supports heat and mass transfer. Kavya et al. [48] used a stretching cylinder to study the magnetic-hybrid nanoparticle effect suspended in Cu nanoparticles. Their major finding is magnetic field reduces fluid velocity. Farahani [58] conducted an efficacy study on the magnetic field effect on heat transfer using nanofluid in a flattened tube with nanofluid.  Senejani and Baniamerian [59] performed an optimization study of shell and tube heat exchangers using genetic, particle swarm, and Jaya optimization algorithms. Based on the above literature study, it is concluded that optimization of the geometry of the cavity and other related parameters to enhance the thermal performance of the cavity is needed to study further for better accuracy in the relevant field. Above mentioned literature it is found that aspect ratio also plays a major role in the present work, the aspect ratio is also considered one of the parameters. Also, by considering the importance of the Richardson number, it cannot be neglected without further investigation which was found as an insignificant parameter in the preceding study [41]. Therefore, to find the importance of the natural and forced convection effect, Grashof and Reynolds numbers are introduced. Hence in the present study, a two-dimensional domain, double-diffusive mixed convective laminar incompressible flow with MHD effect is considered. The performance of the cavity is determined by comparing Nusselt and Sherwood numbers for a different combination of dimensionless numbers. FVM-based numerical codes are used for the analysis and optimization is carried out using the Taguchi approach. The best combination of different dimensional numbers and the aspect ratio for maximum performance is obtained from the analysis. Residual plots obtained from linear regression models are used to correlate the fitted values with the observed values to check the accuracy of the obtained results. Following are the novelties of this study

• Analysis of the effect of nonlinear MHD effect on double-diffusive mixed convection by considering different aspect ratios
• The impact of thermal and momentum diffusivity on the heat transfer property of the cavity is analyzed
• A most significant parameter which controls the heat and mass transfer rate is analyzed
• An optimum combination of the selected parameter to maximize the rate of heat and mass transfer is determined by optimization analysis.
• Prediction models are developed.template, created in Microsoft Word 2013. Margins, column widths, line spacing, and type styles are built-in throughout this document. Author can use this document as a template and simply type own text into it.

2. Problem description and governing equations

A horizontal two-dimensional cavity model selected in this study is shown in Figure 1. The aspect ratio (L/H) is varied from 0.5 to 2.0 for the analysis. A horizontal magnetic field (Bo) is induced to the cavity in a horizontal direction. Vertical sides of the cavity are kept with zero temperature and concentration gradients. Maximum temperature and concentration are kept to the top and bottom walls of the cavity. The upper wall is made to move towards positive x-direction to give a force convection effect and all other walls are kept stationary. The Newtonian and incompressible fluid is assumed for the analysis. All the thermal and physical properties of the fluid are assumed as constant except the density term. The variation of density term in the buoyancy force is calculated using a model developed by Boussinesq as shown in equation 1.

where β_C is the concentration volumetric coefficient of expansion and β_T is the thermal volumetric coefficient of expansion. The equations shown below are the governing equations used for flow, heat, and mass transfer respectively.

here, ν (kinematic viscosity), ρ (density), α (thermal diffusivity), Bo (magnetic induction), σ (electrical conductivity), and D (mass diffusivity) respectively. The dimensionless form of the equations mentioned above is written as follows:

Equations from (3) to (7) are written in the dimensionless form using the above dimensionless group.

Figure 1. Schematic demonstration of the present problem

Following are the boundary condition of the problem selected,

Upper wall: 

Lower wall:

Left and Right walls:  

3. Numerical Solution

The discretization of governing equations was done by FVM using uniform grid sizes. Pressure-velocity coupling equations were solved using the SIMPLE algorithm technique derived by (50). Convection-diffusion terms are solved by a central difference scheme. Equations obtained after discretization was solved iteratively using C++ code. In this selected criterion the convergence of the solution for the equations is 10-7. Nusselt number and Sherwood number are used to analyze the behavior of flow and the thermal characteristics of the fluid inside the cavity. Convective heat and species transfer inside the cavity are analyzed using flow behavior. Local and average Nu and Sh of the cavity hot wall are calculated as follows:

The following equations are used to calculate Local Sh and average Sh,

4.      Result and Discussion

4.1. Grid Independence Study and Validation of Codes

A rectangular domain is used for the grid study. Several computations for the average Nu and Sh were performed for the uniform structured grid size from 41×21 to 201×101. The aspect ratio (L/H) selected for the grid-independent analysis is 2.0, therefore the number of grid sizes in the x-axis is two times the y-axis. Results of the simulation show an error of 0.6385 between the grid size 41×21 and 81×41 in average Nu and 10.63 in average Sh, 0.7918 in average Nu, and 2.3226 in average Sh between grid size 81×41 and 121×61, 0.5901 in average Nu and 0.368 in average Sh between grid size 121×61 and 161×81, 0.3551 in average Nu and 0.0736 in average Sh between grid size 161×81 and 201×101 is observed. There is a marginal variation in error is observed for both average Nu and Sh when the grid size changes from 121×61 to 161×81. Therefore 121×61 grid size is used for further analysis. Temperature and streamline contours of the present study are compared with Ji, et al. [49] as shown in Figure 2, and found good agreement between both results. Table 1 presents the validation of codes for the average Nu and the literature results. Obtained results show an excellent match with the literature results.

Table 1. Comparison of avg. Nu of present work with the literature results for Grt=100, Pr=0.71 at different Re.

4.2. Effect of the Magnetic Field, Prandtl number, and Aspect ratio

The temperature, concentration, and flow fields are shown below for different Ha, Pr, and aspect ratios. Thermal and concentration lines are orthogonal to the left and right walls since they are adiabatic walls. This ensures no heat and mass transfer through the walls. In streamline contour, a vertex is seen on the right portion of the domain due to the movement of the upper edge in the positive x-direction and the diffusion due to the buoyancy force. Figures 3(a), 3(b), and 3(c) show the variation of the contours of temperature, concentration, and streamlines for different Ha. The pattern of the contour plots indicates, the magnetic field greatly affects the temperature and concentration distribution inside the cavity. The negative effect is observed inside the cavity for both Nu and Sh as the value of Ha increases. This negative effect is mainly due to a decrease in the flow velocity due to the increase in the magnetic effect.

Figures 4(a), 4(b), and 4(c) show the effect of Pr on the thermal performance of the cavity. More concentration and temperature distribution are observed as the value of Pr increases. An increase in the avg. Nu and Sh at the hot wall indicate improvement in the heat and mass transfer rate.

Compare to the thermal distribution, the magnitude of the concentration distribution is observed more because Le is maintained as 5. Not much change is observed in streamline contours because Ha is maintained as high and at the high magnetic field, flow velocity retards.

The effect of an aspect ratio of the cavity on heat and the species transfer rate is shown in Figure 5. Three different aspect ratios are taken for the analysis. Flow is observed as very weak for an aspect ratio less than one but increases for an aspect ratio greater than one. This is mainly due to an increase in the supply of heat at the top side of the cavity.

CFD analysis for the optimization of the performance of lid-driven cavity for heat and mass transfer has been performed by selecting the aspect ratio, thermal Grashof number, Reynolds number, Hartmann number and Prandtl number as the control parameters and average Nu and Sh are the responses.

Taguchi method of experimental design is used in this analysis for optimizing the performance of lid-driven cavities. In the following session, the method used for the present numerical analysis has been discussed.

4.3. Taguchi Method

Taguchi method is used to find the optimal combination of selected parameters for maximum performance of the domain in terms of average Nusselt and Sherwood numbers.

The parameters considered for the analysis are the cavity aspect ratio (Ar), Thermal Grashoff number (Gr), Reynold’s number (Re), Hartmann number (Ha), and Prandtl number (Pr), and their level are shown in Table II.

Normally in the Taguchi method, L8, L16, L18, and L27 orthogonal arrays are used for parameter assignment. In this analysis, the L27 orthogonal array is considered which consists of 27 samples data shown in Table III.

The CFD codes were run as per Taguchi L27 combinations. The software used for the Taguchi analysis is Minitab 15.0.

The results of the Taguchi analysis and CFD simulation are shown in Table IV. Taguchi method reduces the number of experiments a researcher is required to perform which intern reduces the cost involved. The signal to Noise Ratio (S/N ratio) plot is used to discover the dominant level of the selected parameters.

The rank for each factor is assigned depending on the delta value obtained from the S/N ratio response table. The minimum rank in the table shows the maximum influence of factors on the output response. Five active parameters at three different levels were analyzed.

An optimal combination of levels of these factors for extreme performance of the domain is found. Graphs and contour plots are also used to investigate the heat and species transfer pattern in the cavity.

The present analysis uses L27 orthogonal with DOF 26. Five parameters and each parameter at 3 levels require 35=243 experiments in a full factorial design, however, the Taguchi design requires only 27 runs which will reduce the time and cost.

Table 2. Levels of independent parameters

The S/N ratio is used to measure the variation in the Taguchi analysis method. The choice of the S/N ratio definition is based on the property response parameter or the characteristic value. Three different types of characteristic values are NB (Normal is the Best), SB (Smaller is the Best), and LB (Larger is the Better).

The main aim of the present analysis is to maximize the heat and mass transfer rate; therefore, LB is selected for the analysis. The S/N ratio calculated for the selected characteristic values is shown in Table IV and is calculated using the following expression, Eq. (15).

Table 3. L27 Taguchi orthogonal array table

Where y_j is the value of jth response data and r is the iteration number. The optimization of the selected parameters has been obtained by linking the Taguchi and ANOVA method. Figures 6 and 7 show the effect of independent variables on the average Nusselt and Sherwood numbers based on the S/N ratio. Steep variations of S/N ratios for Pr and Re indicate that these factors have the maximum influence to control the heat and mass transfer rates. As the aspect ratio increases the heat transfer rate also increases, this is mainly due to an increase in the length of the edge compared to height. The maximum value of 11.164 is found at level 3 of the aspect ratio and then the sharp reduction in the S/N ratio is found in other levels of Ar. The effect of the thermal Grashof number is the least preference to control the value of Nu and Sh is observed in the same graph. No difference in the value of S/N ratios of Nu and Sh is observed by varying the value of Ha from 30 to 60. As Reynold’s number increases the heat and mass transfer rate also increases this is mainly because of the increase in the mean velocity of the fluid. The effect of Ha and Grt in controlling the heat and mass transfer is comparatively less. Tables V and VI describe the rank of each parameter in controlling the thermal efficiency of the cavity. The order of significance is Pr, Re, Ar, Ha, and Grt in controlling both heat and mass transfer rates. The same pattern is observed in the S/N graph. Level 3 of Pr, Re, and Ar, and level 1 of Grt and Ha are the more significant levels. The corresponding values of the S/N ratio of average Nu are 12.040, 11.103, 11.164, 10.333, and 7.820 and similarly, the same order and level of significance are observed in controlling the mass transfer rate. The corresponding values of the S/N ratio of average Sh are 18.522, 14.080, 14.532, 12.229, and 13.527 respectively.

Figure 6. Graph of Independent variables effect on
 avg. Nu mean S/N ratios

Figure 7. Graph of Independent variables effect on
avg. Sh mean S/N ratios

4.4. Analysis of variance

The significance of different parameters to control heat and mass transfer in a lid-driven cavity is performed by adapting the method of analysis of variance (ANOVA). Five parameters at 3 levels have been considered for the analysis. The main purpose of this study is to find the finest arrangement of the selected factors to maximize the heat and species transfer rate. The significance of factors to control the thermal properties of the cavity is analyzed by taking a 95 % confidence level. The temperature and species transfer rates are analyzed using average Nu and Sh values. Tables VII and VIII elucidate the significance and sum of squares of average Nu and Sh respectively. The contribution of each parameter to the total sum of squares (SS) is also seen in the same tables. This contribution percentage gives a clear idea about the importance of each factor in deciding the response. Using all those tables, a combination of parameters for optimal performance of the cavity can be easily predicted.

Table 4. S/N ratio table for average Nu and Sh

Table 5. Response table of S/N ratio on avg. Nu

Table 6. Response table of S/N ratio on avg. Sh

Table 7. ANOVA table for avg. Nu

Table 8 shows the analysis of variance for average Nu. From the P-value, it is clear that Pr, Ar, and Re are the significant factors to control the heat and mass transfer rate. The percentage contribution of Pr is more in both heat and mass transfer and which are 29.98% and 53% respectively. The contribution of Grt is comparatively very less in both cases; therefore, this parameter can be neglected for further analysis. The most optimal combination is Ar = 2.0, Grt =103, Re = 1000, Ha = 0 and Pr = 13.0. The residual plots were obtained by feeding the data obtained from the numerical studies into analytical software as shown in Figure 8 for average Nu and Figure 9 for average Sh. A normal distribution is observed in the residual plots indicating good agreement about the validity of the results (Joardar, Das, and Sutradhar, [57]). The normal probability plot shows the linear relationship which indicates error terms are distributed normally. The histogram of residuals indicates that residuals are distributed normally. From the above plots, it is clear that there is a good correlation between fitted values and the observed values.

The regression model equation for average Nu and Sh for the selected parameters is shown in equations 16 and 17. The standard error (S) of the average Nu regression model is found as 1.57and for the average Sh regression model is obtained as 2.32. The smaller values of S indicate that the model offers better fits due to minor residuals.

Avg Nu = -2.12+2.07Ar+0.000002Grt+0.00326Re-0.0343Ha+0.310Pr                                                          (16)

Avg Sh = -3.56+2.98Ar+0.000002 Grt+0.00333Re-0.0284Ha+0.626Pr                                                           (17)

From these equations, it is clear that Ar, Grt, Re, and Pr have a positive impact on average Nu and average Sh, whereas Ha has a negative impact.

Table 8. ANOVA table for avg. Sh

4.5. Confirmation Test

A confirmation test was performed for the optimum combination. The simulations are performed using the optimal combination of Ar, Grt, Re, Ha, and Pr. The obtained results from this combination are compared with the results of all the simulations used in the analysis.

It is observed that the highest value of Nu and Sh is obtained in optimum combination. This confirms that the optimum combination is the best combination for maximizing heat and mass transfer.

The thermal, concentration, and flow fields for the optimal combination obtained from Taguchi and ANOVA results are offered in the form of isotherms, isoconcentration, and streamlines in Figure10.

The nature of the contours is perfectly validated with the obtained results. Considering the wide range of applications mentioned in the introduction section about the MHD flows and diffusion effect in controlling the thermal performance in medical and biomedical, power production, furnace design, and production of glass sheets, the present numerical results help to select the best combinations to improve the performance.

The value of average Nu and Sh obtained from the confirmation test for the optimal combination is 18.176 and 34.668 which clearly shows the highest values compare to all other combinations used in the analysis.

Figure 8. Average Nu residual plot

Figure 9. Average Sh residual plot

Figure 10. Temperature, concentration and streamline contours at an optimized combination

Conclusion

In this work effect of different dimensionless numbers and their optimization for the supreme performance of the selected domain is numerically performed using a lid-driven cavity. The performance of the cavity is arbitrated using avg Nu and Sh values. The results obtained from the analysis are as follows.

• With an increase in Pr value to the third level, the heat transfer rate increases by 79.98%, and mass transfer by 74.92% which is the most significant parameter.
• Similarly, an increase in Re improves the heat transfer rate by 68.76% and the mass transfer rate by 44.140%. Re is the second significant parameter for the selected range of values.
• An increase in the aspect ratio improves the heat transfer rate by 61.98% and the mass transfer rate by 38.79%.
• The optimum level for the maximum performance of the cavity is obtained as Ar = 2.0, Grt = 103, Re = 1000, Ha = 0.0 and Pr = 13.0.
• The average Nu and Sh are dependent maximum on Pr, Re, and Ar and the effect of Grt and Ha are comparatively negligible.
• The value of average Nu and Sh obtained from the confirmation test for the optimal combination is 18.176 and 34.668 which clearly shows the highest values compare to all other combinations used in the analysis.
• The regression model shows a positive impact of all selected parameters on average Nu and Sh and a negative impact with Ha.

Nomenclature

Ar       Aspect ratio

Bo       Magnetic induction (tesla)

c         Concentration

C         Dimensionless concentration

D        Mass diffusivity(m2s-1)

g         Gravitational acceleration(ms-2)

GrC     Grashof number concentration

GrT     Grashof number temperature

h         Heat transfer coefficient

hs       Solutal transfer coefficient

H        Enclosure height(m)

Ha      Hartmann number

L         Enclosure length(m)

Le       Lewis number

N        Buoyancy ratio

Nu      Nusselt number

p         Pressure

P         Non-dimensional pressure

Pr       Prandtl number

Re       Reynolds number

Ri        Richardson number

Sc       Schmidt number

Sh       Sherwood number

T         Dimensional temperature(K)

u,v      Dimensional velocity component (ms-1)

U,V     Dimensionless velocity components

x,y      Dimensional co-ordinates

X,Y     Dimensionless Cartesian coordinates

r         Iteration number

Greek symbol

α         Thermal diffusivity

β         Fluid thermal expansion coefficient

θ         Dimensionless temperature

ν         Effective kinematic viscosity

ρ         Density

σ         Fluid electrical conductivity  

Subscripts

avg     Average

C         Cold, Concentration

h         Hot

T         Thermal

max    Maximum

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[*]Corresponding Author: Satheesh Anbalagan

   Email: satheesh.a@vit.ac.in;egsatheesh@gmail.com