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Natural convection in a differentially heated cavity plays a ...

Natural convection in a differentially heated cavity plays a
major role in the understanding of flow physics and heat
transfer aspects of various applications. Parameters such as
Rayleigh number, Prandtl number, aspect ratio, inclination
angle and surface emissivity are considered to have either
individual or grouped effect on natural convection in an
enclosed cavity. In spite of this, simultaneous study of these
parameters over a wide range is rare. Development of
correlation which helps to investigate the effect of the large
number and wide range of parameters is challenging. The
number of simulations required to generate correlations for
even a small number of parameters is extremely large. Till
date there is no streamlined procedure to optimize the number
of simulations required for correlation development.
Therefore, the present study aims to optimize the number of
simulations by using Taguchi technique and later generate
correlations by employing multiple variable regression
analysis. It is observed that for a wide range of parameters,
the proposed CFD-Taguchi-Regression approach drastically
reduces the total number of simulations for correlation
generation.

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  • 1. nd th Proceedings of the 22 National and 11 International ISHMT-ASME Heat and Mass Transfer Conference December 28-31, 2013, IIT Kharagpur, India HMTC1300663 TAGUCHI BASED METHODOLOGY TO GENERATE CORRELATION VIA NUMERICAL ANALYSIS FOR NATURAL CONVECTION IN A DIFFERENTIALLY HEATED CAVITY D. Jaya Krishna Assistant Professor Department of Mechanical Engineering BITS PILANI Hyderabad campus, Hyderabad (INDIA) djayakrishna.iitm@gmail.com ABSTRACT Natural convection in a differentially heated cavity plays a major role in the understanding of flow physics and heat transfer aspects of various applications. Parameters such as Rayleigh number, Prandtl number, aspect ratio, inclination angle and surface emissivity are considered to have either individual or grouped effect on natural convection in an enclosed cavity. In spite of this, simultaneous study of these parameters over a wide range is rare. Development of correlation which helps to investigate the effect of the large number and wide range of parameters is challenging. The number of simulations required to generate correlations for even a small number of parameters is extremely large. Till date there is no streamlined procedure to optimize the number of simulations required for correlation development. Therefore, the present study aims to optimize the number of simulations by using Taguchi technique and later generate correlations by employing multiple variable regression analysis. It is observed that for a wide range of parameters, the proposed CFD-Taguchi-Regression approach drastically reduces the total number of simulations for correlation generation. NOMENCLATURE AR Avg. ER g H L Nu OA P Pr Ra T u aspect ratio average error acceleration due to gravity, (m/s2) cavity height, (m) cavity length, (m) Nusselt number Orthogonal array pressure, (N/m2) Prandtl number Rayleigh number temperature, (K) component of velocity in x-direction Ruturaj R. Deshpande M.E Thermal Department of Mechanical Engg. BITS PILANI Hyderabad campus, Hyderabad (INDIA) ruturaj1988@gmail.com v component of velocity in y-direction X, Y coordinates axis Greek symbols Coefficient of thermal expansion, (1/K) Density, (kg/m3) Kinematic viscosity, (m2/s) Subscripts c cold cor correlation E equally spaced levels h hot R randomly spaced levels Ref reference value sim simulation Superscripts i, i-1 present and previous grid size 1 INTRODUCTION The phenomenon of Natural convection in enclosures filled with fluid is an important study. It finds wide applications in solar collectors, double walled insulations, air circulation in buildings, electronic cooling systems, geophysics and nuclear science applications [1]. The importance can be quickly realized from the fact that, the amount of research carried in the last 30-40 years related to this phenomenon is enormous. A brief review of these studies is documented by Ostrach [2]. Natural convection in enclosures where the side walls are maintained at different temperatures and remaining sides being insulated has attracted many researchers to carry out research using both experimental as well as numerical methods. The contribution of de Val Davis [3], and Hortmann et al. [4] using numerical methods for square cavity are worth mentioning. These results are being used by several researchers for benchmarking their numerical studies. Ramesh [5] has considered the effect of surface radiation in his experimental work related to natural convection and has shown that natural convection in enclosures is not limited to a
  • 2. particular set of parameters but may contain extra influential parameters. Successful attempts were made by Berkovsky and Poleviko [6], and Balaji and Venkateshan [7] to develop correlations considering a fairly wide range of parameters to have a better understanding of the phenomenon. To develop a correlation, large number of experimentssimulations is to be carried out and each experimentsimulation follows a plan. Each plan has predetermined values of parameters, and a method in which all possible experiments are considered is known as full factorial design. For a full factorial design, the number of experimentalsimulation runs to be carried out is extremely large. There is an additional toll on the resources to carry out experimentssimulations and to handle large quantity of data a highly evolved statistical tools are required. With the above mentioned disadvantages conventional full factorial design in any type of engineering analysis is almost impossible [8]. Various attempts were made to come up with a fractional factorial design to overcome the disadvantages of full factorial design. The most successful and widely used methods are Taguchi method, Central Composite Design (CCD), Optimal Space Filling Design (OSFD), and Box-Behnken Design (BBD). These methods were developed for a specific problem. Over a period of time these are evolved into more robust, userfriendly and accurate methods to study multiple parameters affecting the dependent parameter [9]. Taguchi method for design of experiments was developed after the World War II in Electrical Communication Laboratories (ECL-JAPAN) by Dr. Taguchi. The motivation behind the development of Taguchi method was to optimize the process of engineering experimentation. Originally the Taguchi method was used for quality control. This method was later used by companies like Ford Motor for quality control and product improvement [8]. Taguchi method along with Analysis of Variance (ANOVA) is successfully used by researchers to cut down on the number of experimental runs with the help of orthogonal arrays [10-12]. Till date a full theory that includes all the possible number of plans is not yet developed [13]. A set of orthogonal arrays which serve as a plan for experiments to be carried out can be prepared by the use of computer codes. However, one can also use orthogonal arrays available at various resource libraries [14]. Even though selected and less number of experiments are carried out, conclusions drawn from Taguchi-ANOVA analysis is valid over an entire experimental region spanned by the independent factors. The first attempt to apply the Taguchi method to investigate the statistical performance of CFD solver and its interaction was done by Stephen [15]. Taguchi method was used by Follett et al. [16] to optimize different nozzle configurations for bell annular tri-propellant engine. This approach was adopted to overcome difficulties such as little or no previous experience, time constraints, and nearly infinite number of parameters. Juan and Piniella [17] used CFDTaguchi method to study flow over four digit NACA airfoil using signal to noise ratio (S/N) concept. S/N concept was used by modifying the Reynolds number reported by NACA. Mousavi et al. [18] used Taguchi method to effectively optimize biomass production. Zeta Dynamics Ltd. used Taguchi method in conjunction with the CFD package PHOENICS-VR 3.4 to find the optimum design of a piece of hardware known as the Zeta Linear Coanda Unit (ZLCU) [19]. Jafari et al. [20] used CFD-Taguchi method for evaluation of the controllable parameters effect on thermomagnetic convection in ferro-fluids. Recently, Ravinder [21] successfully optimized cold spraying of Hydroxyapatite using CFD-Taguchi-ANOVA approach where Taguchi orthogonal array helped to plan the simulations. Once all the experimentalsimulation outcomes are noted, a mathematical tool is required for the analysis and development of correlation. Regression analysis serves as a tool to find out the relationship between dependent and independent parameters. It also helps to predict and explain the behavior of the dependent variable. However, the computation required for multivariable regression analysis is highly complicated, and is performed using computer codes. Various models are used for regression analysis such as linear, exponential, weighted averages, parabolic, hyperbolic etc. An incorrectly specified model may cause biased estimates of parameters. An initial model may be based on a combination of educated guesses, availability of data, and observation from the scatter plot. The model selected for regression analysis may not be optimal for several reasons some of them are: (i) Model may contain too many variables and is said to be over specified (ii) Model may not contain right variables (iii) The relation between parameters differ from the one used in the model. If the model is inadequate for either or both of the last two reasons then the model exhibits specification error i.e. the inputs for regression analysis are not correct. It is difficult to judge the error due to omission of variables and can hardly be avoided. Parity plots are useful to find out errors due to mathematical relationship. A general thumb rule is that the spectrum of Parity plot must be as small as possible [22]. Normally 4(independent parameters) are the number of data points required to come up with a fairly accurate correlation [23]. This shows that even for a small number of parameters the number of required data points is huge. Traditionally Taguchi method is used for the optimization of parameters by performing very few experimental runs. Application of Taguchi method to understand the phenomenon itself is rarely explored. Use of CFD-Taguchi-Regression analysis will not only help to optimize the whole phenomenon but also to understand and predict the behavior of dependent parameters. Perhaps for the first time, an attempt is being made, to use CFD-Taguchi-Regression analysis to come up with a systematic approach for correlation development. The success of this approach lies in the realization of a fairly accurate correlation with less number of simulations than the number which is conventionally suggested. In this study the numerical analysis for a differentially heated cavity has been carried out using a commercial software Ansys-Fluent. The independent parameters considered for the study are Rayleigh number (103≤Ra≤106), Prandtl number (0.71≤Pr≤10), and aspect ratio (1≤AR≤10). The dependent parameter under study is Nusselt number. 2 MATHEMATICAL MODEL A schematic representation of a differential heated cavity is shown in Fig. 1, where L and H are the length and height of the cavity in X and Y directions respectively .The standard gravitation field is acting in the negative Y-direction. 2.1 Governing Equations The flow and heat transfer solutions are obtained by solving a simultaneous system of Eqs. (1- 4) describing the conservation of mass, momentum and energy. Continuity equation (1)
  • 3. than 0.5%. Table 1 provides details pertaining to mesh sizes employed in the present study. X-Momentum equation (2) Y-Momentum equation Energy equation ( ) (3) (4) 2.2 Boundary Conditions The left and right walls of the cavity shown in Fig. 1 are maintained at Tc and Th respectively. Lower and upper walls of the cavity are insulated. No-slip condition is applied at the walls. Velocity Boundary conditions. (5) (7) To check the consistency of the present numerical methodology, comparison of the simulation results is carried out with the results available in the literature [3, 24-26]. Table 2 shows an extensive comparison of the present results with benchmark data for an average Nusselt number. From Table 2 it may be noted that the present numerical methodology is rigorously validated and the results agree very well with the results available in the literature. Figure 2 provides isotherms (top) and stream function (bottom) for Prandtl number 1(left) and 10 (right) for a fixed Rayleigh number (106) and aspect ratio (5). It may be noted that with the increase in Prandtl number a marginal increase in vortex strength may be observed which in turn lead to the marginal increase of Nusselt number. To study the influence of aspect ratio on thermal hydraulics Fig. 3 is plotted for AR=1 and 10 for a fixed Rayleigh number (106) and Prandtl number (5.35). It may be noted that with the increase in aspect ratio in spite of increase in vortex strength the Nusselt number is observed to reduce due to the increase of the boundary layer. Table 1. MESH SIZES CONSIDERED TO GENERATE RESULTS Temperature Boundary conditions (6) AR Mesh 1 90×90 2 90×180 4 8 90×280 90×320 10 90×320 For lower and upper walls Table 2. COMPARISON OF PRESENT RESULTS WITH LITERATURE S. No 1 103 0.71 1 1.114 1.116 [3] 2 Figure 1. SCHEMATIC REPRESENTATION OF DIFFERENTIAL HEATED CAVITY Ra 10 4 0.71 1 2.198 2.234 [3] 10 5 0.71 1 4.494 4.510 [3] 10 6 0.71 1 8.805 8.798 [3] 5 10 3 0.71 4 1.110 1.090 [24] 6 105 0.71 8 3.373 3.390 [24] 7 10 3 10 1 1.113 1.141 [25] 8 10 6 10 1 9.236 9.232 [25] 9 104 7 8 1.843 1.844 [25] 0.71 10 1.68 1.706 [26] 3 4 The governing equations are solved by using finite volume method. SIMPLE algorithm is employed for pressure-velocity coupling. The convective terms of the momentum and energy equations are discretized using the second order upwinding scheme. Convergence is assumed to be achieved when all the residuals are reduced to 10-6. 2.3 Grid Independence and Validation Careful study is carried out to establish grid independence using Eq. (7), where represents the average Nusselt number, i and i-1 represents present and previous grid size respectively. A non-uniform grid with suitable growth factor is considered. Grid independence is assumed to be achieved when the percentage deviation for the average Nusselt number is less 10 10 4 Pr AR Average Nusselt number Present Literature Study Reference
  • 4. Figure 3. COMPARISON OF TEMPERATURE CONTOURS (LEFT) AND STREAM FUNCTION (RIGHT) FOR (a) AR=1; (b) 6 AR=10 WITH FIXED Ra(10 ) AND Pr (5.35). 3 RESULTS AND DISCUSSION 3.1 Taguchi Method Figure 2. COMPARISON OF (a) TEMPERATURE CONTOURS ; (b) STREAM FUNCTION CONTOURS FOR 6 Pr=1(LEFT) AND Pr=10 (RIGHT) WITH FIXED Ra(10 ) AND AR (5). The smallest Orthogonal Array (OA) suggested by the Taguchi method for 3 independent parameters is L-4 OA. The number of data points available from the L-4 OA is 4 but considering the complex nature of the problem these are very few in number. Formulation of correlation based on the L-4 OA array is mathematically impossible for 3 independent parameters and one dependent parameter which are related in a non-linear fashion. The next smallest OA suggested by Taguchi is L-9 OA. The L-9 OA has 9 data points i.e. it suggests 9 simulations to be carried out which are less than the number of simulations suggested by conventional analysis procedures (4(independent parameters) = 64) [23]. In the present study selection of values for different levels for each independent parameter is carried out by following two rules: (i) Simulation plan must have combinations of extreme levels of considered independent factors. If not, extra simulations are to be planned with all levels set to extreme values (ii) Selection of levels other than extreme levels should be unbiased. This is achieved by choosing equally spaced intervals or randomly generated levels. Table 3 gives different levels for independent parameters using equal interval and also by using random level. Since the original L-9 OA does not suggest a simulation when all the levels are set to 3 an additional simulation i.e. number 10 is carried out. Table 3. EQUAL AND RANDOM INTERVAL LEVELS Ra Pr 3 3.16×10 10 6 6×10 10 6 0.71 5 Random 2 4 Equidistant Random 103 Random Equidistant 103 Equidistant Levels 1 AR 0.71 1 1 5.35 7 5.5 5 10 10 10 10
  • 5. 3.2 Regression Analysis As mentioned earlier multivariable regression analysis is considered to be more complicated and is performed using computer codes. Hence, Regression analysis is carried out using commercial code-LAB Fit. The input given to Lab Fit are (i) Tabular data containing the values of dependent and independent variables and (ii) The form of the equation to be fitted to this data [27]. The general selection of the particular form of an equation is based on either pure guess or literature or on the trend of dependent variables. It may be noted that the present study focuses to provide a streamlined methodology to optimize the number of simulations/experiments to generate correction. Therefore to avoid the influence due to the form/type, the following equations i.e. Eqs. (8): power law, (9): Berkovsky and Poleviko [6] and (10): modification of Eq. (9) have been employed for the data provided in Table 4. Even though three different forms of equations are used to carry out regression analysis, Fig. 5 shows the inability of correlation suggested by equation 10R to accurately predict the Nusselt number over the whole range of parameters. This helps to eliminate the form of equation suggested by Eq. (10). The correlation developed from Eqs. (8 and 9) seem to be valid for the entire range of independent parameters, the analysis of parity plot shows Eq. (9) is the best equation available after regression analysis. It is to be noted that the correlation suggested as Eqs. (8E, 9E, 8R, and 9R) are valid for the entire range of the independent parameter. In order to further test the validity of obtaining correlations (Eqs. 8 E, 9E, 8R, and 9R) 10 more simulations have been considered with randomly generated parameters. (8) (9) (10) Where a, b, c, d and e are constants. The constants for Eqs. (8 –10) are obtained by employing equidistant values (represented by suffix E) and also by randomly spaced values (represented by suffix R) for independent variables. The values pertaining to the generation of correlation may be referred from Table 3. The constants for Eqs. (8-10) are obtained through regression analysis by carrying out simulations based on L-9 OA (Table 4). It may be noted that the curve fitting using multiple linear regression analysis would approximately require 4 n solutions, where ‘n’ is the number of independent parameters [23]. As the present study has three independent parameters based on multiple linear regression analysis we would require a minimum of 64 solutions. But by making use of Taguchi’s L9OA we could make use of only 10 solutions to generate correlations. The correlations thus obtained are provided in Table 5. Figs. 4 and 5 show the parity plot for Eqs. (8 – 10) developed based on equidistant levels (8E, 9E and 10 E) and random levels (8R, 9R and 10 R). Figure 4. PARITY PLOT FOR EQUATIONS DEVELOPED FROM EQUIDISTANT LEVELS Table 4. L-9 OA FOR EQUALLY SPACED AND RANDOMLY GENERATED LEVELS Simulation plan no Ra Pr AR 1 1 1 1 2 1 2 2 3 1 3 3 4 2 1 2 5 2 2 3 Eq. no 6 2 3 1 8E 7 3 1 3 9E 8 3 2 1 9 3 3 2 10 3 3 3 Figure 5. PARITY PLOT FOR EQUATIONS DEVELOPED FROM RANDOM LEVELS Table 5. DIFFERENT FORMS OF CORRELATIONS OBTAINED THROUGH REGRESSION ANALYSIS 10E Equation/correlation ( ( ) )
  • 6. 8R ( 9R 10R ( ) traditional multiple linear regression analysis. The independent parameters have been considered based on both equidistant and random values for Taguchi’s orthogonal array. Three different forms of equations (Eqs. (8 – 10)) have been considered. Based on the study the following conclusions may be drawn. )  The values for average Nusselt number for 10 randomly considered parameters based on simulations and Eqs. (8 E, 9E, 8R, and 9R) are given in Table 6. The percentage error between the simulated result and respective correlation is given by Eq. (11). From Table 6 it may be observed that the maximum percentage error based on Eq. (9E) is -5.41%. Based on the study it may be noted that the present methodology can be an attractive option in the generation of correlation with few number of simulations. (11) Conclusions:      A systematic approach for generating correlation with reduced number of simulations is developed based on Taguchi’s orthogonal arrays. Regression analysis has to be considered for multiple proposed forms of equations. An optimum form of the equation is to be found by comparing the correlation generated from equally spaced and randomly generated levels. The final form of the equation is to be obtained by comparing the correlations generated from equally spaced and randomly generated levels. Parity plots can be used to select the best form of equation. The correlation thus developed by making use of 10 solutions is observed to have a maximum percentage error of -5.41% for average Nusselt number. In the current work a problem of natural convection in a differentially heated cavity has been considered for generating correlation with fewer number of solutions. The parameters considered for the study are 103≤ Ra≤106, 0.71≤ Pr≤10 and 1≤ AR≤10. The numerical methodology has been rigorously validated with literature. The study deals with the application of Taguchi method for correlation development instead of Table 6. AVERAGE NUSSELT NUMBER FOR RANDOMLY CONSIDERED PARAMETERS BASED ON SIMULATIONS AND EQS. 8E, 8R, 9E AND 9R Average Nusselt number Ra Pr AR Simulation 104 104 105 106 103 105 103 106 104 104 7 7 0.71 0.71 0.71 0.71 10 10 7 0.71 4 2 1 1 4 8 1 1 8 10 Equation 8E % Error Equation 8R % Error Equation 9E % Error Equation 9R % Error 2.179 2.39 4.494 8.805 1.11 3.373 1.113 9.236 1.843 1.68 2.04 2.308 4.787 8.692 1.115 3.221 1.466 8.77 1.788 1.7 6.38 3.43 -6.52 1.28 -0.45 4.51 -31.72 5.05 2.98 -1.19 1.945 2.223 4.657 8.833 0.991 3.118 1.347 9.193 1.702 1.575 10.74 6.99 -3.63 -0.32 10.72 7.56 -21.02 0.47 7.65 6.25 2.121 2.415 4.737 8.591 1.111 3.208 1.111 9.063 1.863 1.697 2.66 -1.05 -5.41 2.43 -0.09 4.89 0.18 1.87 -1.09 -1.01 1.953 2.232 4.608 8.742 0.98 3.089 1.347 9.197 1.71 1.56 10.37 6.61 -2.54 0.72 11.71 8.42 -21.02 0.42 7.22 7.14 REFERENCES [1] Kakac, S., Aung, W., and Viskanta, R., Natural convection: fundamentals and applications, eds., Hemisphere, Washington, DC, 1985. [2] Ostrach, S., Natural convection in enclosures, Journal of Heat Transfer, vol. 110, pp. 1175–1190, 1988. [3] de Vahl Davis, G., Natural convection of air in a square cavity: a bench-mark numerical solution, International Journal of Numerical Methods in Fluids, vol. 3, pp. 249–264, 1983. [4] Hortmann, M., Peric, M., and Schenere, G., Finite volume multigrid prediction of laminar natural convection : benchmark solutions, International Journal of Numerical Methods in Fluids, vol. 11, pp. 189-207, 1990. [5] Ramesh, N., Experimental study of interaction of natural convection with surface radiation Ph. D. Thesis. Dept. of Mechanical Engineering, Indian Institute of Technology, Madras, India 1999. [6] Berkovsky, B. M., and Polevikov, V. K., Numerical study of problems on high-intensive free convection. D.B. Spalding, H. Afgan (Eds.), Heat Transfer and Turbulent Buoyant Convection, Hemisphere, Washington, DC, pp. 443–455, 1977.
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