Optimum Configuration of a Double Jet Mixer Using DOE

Transcription

1 Optimum Configuration of a Double Jet Mixer Using DOE Manjula. P, Kalaichevi. P, Annadurai. G and Dheenathayalan. K Abstract Mixing is used for blending of liquids, flocculation, homogenization of mixtures, to ensure proper heat and mass transfer in various operations, prevention of deposition of solid particles etc. Earlier research aspects were focused on experimental estimation, CFD simulation and proposing suitable correlations for the prediction of mixing time. However, the significant effects and interactions of each variable involved in the process on mixing time were not analyzed. This report describes the effect of jet2 angle, jet2 location and radial angle on mixing time using DOE based on experimental mixing time results. The Box-Behnken design was applied in a cubic polynomial regression model to test the effects and interactions of variables using RSM. The statistical analysis shows that the cubic model gave a good fit with an R 2 value of The significance of variable on mixing time was identified and the optimum conditions for the minimum mixing time were obtained. Keywords Box-Behnken method, Double jet mixer, Design of experiments, Response surface methodology. J I. INTRODUCTION ET mixers are one of the simplest devices to achieve mixing. Side entry mixers or jet mixers are commonly used to achieve mixing in storage tanks and reactors. Impellers are the conventional devices used for mixing purpose in industries. But they are very expensive for large storage tanks and underground tanks. Jet mixers have become alternate to impellers for over 50 years in the process industry. In jet mixing, a part of liquid from the tank is circulated into the tank at high velocities with the help of pump through nozzles. The resulting jet of fluid entrains some of the surrounding fluid and creates a circulatory pattern, which leads to mixing in the tank. Jet mixing leaves fewer dead spots in a shallow or rectangular tank than does agitators. If system needs shear besides mixing, then jet mixer is more efficient. Researchers used various methods and techniques to examine the performance of impinging jets in an attempt to achieve a fundamental understanding of mixing. Manjula. P was with the Department of chemical Engineering, National Institute of Technology, Tiruchirapalli , Tamil Nadu, India. Kalaichevi. P is with the Department of chemical Engineering, National Institute of Technology, Tiruchirapalli , Tamil Nadu, India (phone: ; fax: ; kalai@nitt.edu ). Annadurai. G is with the Manonmanian Sundaranar University, Alwarkurchi India.(phone: ; fax ; gannadurai@hotmail.com ). Dheenathayalan. K is with the Department of chemical Engineering, National Institute of Technology, Tiruchirapalli , Tamil Nadu, India. Conductivity technique is used to measure mixing time in a jet mixer [1]. Mixing time is dependent on the jet Reynolds number [2]. For side entry jet mixing, a longer jet length produces a more effective mixing jet and therefore reduces mixing time [3]. In the circulation flow regime of mixing and for Reynolds number greater than there exits an optimum nozzle depth for rapid mixing, that ranges from the liquid surface level to three-quarters of the liquid depth when the liquid depth is equal to the tank diameter, and is the mid depth of the liquid when the liquid depth is smaller than the tank diameter [4]. The use of reactors with jet reagent mixing, which provides intense mixing due to the formation of stable recycling flows [5]. The flow patterns and mixing in jet mixed tanks equipped with alternating jets using CFD simulations [6]. CFD code is used to find the optimum shape needed for reduction in mixing time [7]. Comparing the experimental measurement with the simulation results over a wide range of jet velocities, nozzle angles, and nozzle diameters. He reported that the computational fluid dynamics modeling does not completely predict the mixing time in the tank rather it gives a fair estimation of the mixing time [8]. The effects of various parameters such as nozzle diameter, angle of inclination and jet velocity on mixing time. Mixing time is dependent on the angle of inclination of jet, because angle at which the jet forms with the horizontal determines the dead zone formed in the tank [9]. The effect of angle and elevation of mixing in a fluid jet agitated tank using computational fluid dynamics. They showed that the angle of injection is significant in determining the time required for mixing [10]. Optimum sizing procedure of autonomous PV/wind hybrid energy system with battery storage and a break-even analysis of the system and extension of transmission line [11]. Design of Experiment (DOE) is a structured, organized method and it is widely used in research and development, where a large proportion of the resources go towards solving optimization problems. Box Behnken design was used to optimize the range of experimentation and the Box Behnken model was found to be in close agreement with their experimental values, as indicated by the correlation coefficient value of [12]. Design of Experiment for studying the factors influencing keratinase production. The effect of each variable and interaction between them were determined from the responses obtained from the experimental design. Experimental values were found to be in accordance with the predicted values, the correlation coefficient is being [13]. 18

2 There are a number of papers published to date which present the experimental, correlation and CFD simulation of liquid flow with single and multi-jets, whereas as the significant effects and interactions of each variable involved in the process on mixing time were not studied. This shows that there is a strong need to investigate the interaction effects of operating parameters (jet2 angle, radial angle, jet2 location) involved in the process on the output response (mixing time) in a double jet mixer using design of experiments (DOE). In the present work, experimental mixing time data collected for different jet2 configurations foe a fixed jet1 configuration (2/3rd position, nozzle angle 45 0 and nozzle diameter 10 mm) were used. Mixing times were estimated for different jet2 configuration of jet angle (30 0, 45 0 and 60 0 ), radial angles (60 0, 120 0, ), jet diameter 5mm and located at different tank heights (26.66cm, 19.99cm and 13.33cm from the bottom of the tank). II. EXPERIMENTAL MIXING TIMING PREDICTION Experimental set up consist of a cylindrical tank of 0.4m internal diameter and with a working fluid height of 0.4m and total height of the tank 0.5m. There was a gap above the liquid surface, which is open to atmosphere. A one inch pipe was used to draw the liquid from the bottom of the tank and returned to the tank through side entry jets. Water was used as the working fluid and sodium chloride as an electrolyte to find the mixing time. Tap water was allowed to flow through both nozzles and the flow through jet1 and jet2 were adjusted with the help of two different rotameters (range: 1-12 lpm.). Once the steady state was reached, a small amount of sodium chloride solution (0.8% wt, 250ml) was introduced as the tracer pulse at the centre of the vessel at the top liquid surface. The samples were collected at the outlet of the tank at different intervals of time and conductivity of the samples was found using conductivity meter (ECTestr11+ Multi Range, Eutech instruments). When the conductivity of the samples reached constant value, the time was taken as the mixing time. Nozzle configuration for jet1 was fixed (2/3rd position, nozzle angle 45 0 and nozzle diameter 10 mm). Mixing times estimations were repeated for different jet2 configuration of jet angle (30 0, 45 0 and 60 0 ), radial angles (60 0, 120 0, ) and jet diameter 5mm and located at different tank heights (26.66cm, 19.99cm and 13.33cm from the bottom of the tank). The mixing time data collected during the experiments were used to optimize the range of experimentation. Design expert software (Version 6.0) was used for optimizing the operating variables of a double jet mixer considered in the present research as described below. III. STATISTICAL ANALYSIS The statistical design of experiments were carried out using Design Expert Software (version 6.0) using the predicted experiments mixing times. The response surface methodology (RSM) was applied to optimize the three most important independent operating variables: jet2 angle, radial angle and jet2 location. It is an empirical model used to evaluate the relation between controlled variables and observed results. The dependent parameter (mixing time) was analyzed to obtain the optimum values of jet2 angle, radial angle and jet2 location. The following cubic model was used in response surface methdology (RSM) to explain the mathematical relationship between the independent variables (jet2 angle, radial angle and jet2 location) and the dependent responses (mixing time). Y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + β 11 x β 22 x β 33 x β 12 x 1 x 2 + β 13 x 1 x 3 + β 23 x 2 x 3 + β 111 x β 222 x β 333 x 3 + β 112 x 1 2 x 2 + β 113 x 1 2 x 3 + β 22 x 1 x β 133 x 1 x β 223 x 2 2 x 3 + β 233 x 2 x β 123 x 1 x 2 x 3 (1) Where β 0 is the constant, β 1,β 2, β 3,β 11, β 22,β 33, β 12, β 13, β 23, β 111,β 222, β 333, β 112, β 113, β 122, β 133, β 223, β 233, β 123 are regression coefficients and x 1,x 2,x 3 are the independent variables, y is the dependent variable which is to be optimized. This regression model contains 3 linear (x 1, x 2,x 3 ), 3 quadratic (x 1 2,x 2 2,x 3 2 ), 3 cubic (x 1 3,x 2 3,x 3 3 ) and 10 interaction (x 1 x 2,x 1 x 3,x 2 x 3,x 1 2 x 2,x 1 2 x 3,x 1 x 2 2,x 1 x 3 2,x 2 2 x 3,x 2 x 3 2,x 1 x 2 x 3 ) terms, plus 1 block term. All terms regardless of their significance are included in the above equation. In this case, dependent variable y represents mixing time and independent variables x 1, x 2 and x 3 represents jet2 angle, radial angle and jet2 location respectively. The interaction between the process variables (jet2 angle, radial angle, jet2 location) and the responses (mixing time) were examined using the analysis of variance (ANOVA) results. The quality of the fit polynomial model was expressed by the coefficient of determinationr 2. Model terms were evaluated by the P-value (probability) with 95% confidence level. Three-dimensional plots and their respective contour plots were obtained for jet2 angle, radial angle and jet2 location based on their effects on mixing time at three levels. IV. RESULTS AND DISCUSSIONS A. Statistical analysis The relationship between the three independent variables (jet2 angle, radial angle and jet2 location) and the dependent process response (mixing time) were analyzed using response surface methodology (RSM) (Annadurai et al) and its results are presented in table (1). The P values shown in the last column of the ANOVA table were less than 0.05 indicating the significant model terms. The R 2 value shown in the ANOVA table gives the proportion of the total variation in the response predicted by the model, indicating ratio of sum of squares due to regression (SSR) to total sum of squares (SST). A high R 2 value, close to 1, is desirable and also shows good agreement with adjusted R 2. In this study, the obtained coefficient of determination (R 2 ) value presented in table (1) was , which shows that the chosen model equation was very reliable for the system considered in the present study. The terms which were statistically insignificant (A 3, B 3, C 3, AC 2, B 2 C, BC 2, ABC) were eliminated from (1), based on the ANOVA results presented in table (1) to arrive at a good fit of cubic equation for the model considered. This reduced the (1), to the following form (in terms of coded factors) 19

3 Y = A B + 4 C A B C A B A C A2 B A2 C A B2 P (2) Where, A, B, C represents jet2 location, jet2 angle, radial angle and Y represents mixing time in terms of coded factors. The coefficient of variance (CV) is the ratio of the standard error of estimate to the mean value of the observed response defines reproducibility of the model. The value of CV was observed to be 0.3% from the ANOVA table and since it was less than 10%, it was considered to be reproducible. SOURCE TABLE I REGRESSION ANALYSIS FOR THE MIXING TIME FOR CUBIC RESPONSE SURFACE MODEL FITTING (ANOVA) Coefficient Value DF Mean Square F Value Prob>F E+007 <0.001* significant A E+007 <0.001* B E+007 <0.001* C E+007 <0.001* A E+007 <0.001* B E+007 <0.001* C E+007 <0.001* AB E+007 <0.001* AC E+007 <0.001* BC E+007 <0.001* A B C A 2 B E+007 <0.001* A 2 C E+007 <0.001* AB E+007 <0.001* AC B 2 C BC ABC A = jet2 location, B = jet2 angle, C = radial angle. B. Process Analysis and Optimization 1. Diagnostic plots Diagnostic plot of predicted mixing time values versus actual mixing time values is shown in Fig.1. Adequate precision (AP) compares the range of the predicted values at the design points to the average prediction error. The AP values greater than 4 indicate adequate model discrimination. The AP value of 1 could be observed from Fig.1 indicating an adequate agreement between the model predicted data and the experimental data. C. Response surface plots Response surface plots are used to evaluate a relation between set of independent variables (jet2 angle, radial angle, jet2 location) and dependent variable (mixing time). The 3D surfaces plots corresponding to the present model are shown in Fig.2 to Fig.4 These Figures give the mixing time values predicted by the statistical model using (2), [12]. Fig. 1 Actual mixing time vs predicted mixing time. 1. Effect of jet2 location on mixing time In order to study the effect of Jet2 location on mixing time, the mixing time values obtained from the model were plotted against with jet2 location for different jet2 angles, radial angles and it is shown in the form of 3D response surface plots in Fig.2 (a,b,c).the mixing time values (13.3 sec to 28.6sec) obtained from the model for varies combinations of jet2 locations (13.33cm to 26.66cm) and jet2 angles (30 0 to 60 0 ) for radial angle of 60 0,120 0,180 0 are in Figures 2(a),2(b) and 2(c) respectively. The response surface plots shown in Fig.2 (a,b,c), are approximately symmetrical in shape with circular contours. The response plots show clear lower peaks, implying that the optimum conditions for minimum values of the mixing time are attributed to jet2 angle and jet2 location in the design space. Two-dimensional representation of the mixing time on the contour plot shows concentrically closed curves whose centers represent the optimum conditions. It was observed from Fig.2 (a) that minimum mixing time value of 14sec could be obtained when jet2 located at a height of 26cm from the bottom of the tank at 60 0 radial positions. Fig. 2(b) shows the optimum mixing time of 16.3sec, when jet2 located at 24.5cm from the bottom of the tank and at radial angle. The response surface plot in Fig. 2(c) showed an optimum mixing time of 17.8sec for a radial angle of 180, for jet2 location of 23cm. Also the response surfaces plots, Fig. 2(a), 2(b), 2(c) show an increasing mixing time efficiency as the height of the jet2 increases from the bottom of the tank. 2. Effect of jet2 angle on mixing time The effect of jet2 angle on mixing time was studied by calculating the mixing time from the model eq.(2) for different combinations of radial angles (60 to 180 ) and jet2 angles (30 0 to 60 0 ) for fixed jet2 location (13.33cm, 19.99cm, 26.66cm). The mixing time values (13.3 sec to 28.6sec) found from the model equation for jet2 angles of 30 0 to 60 0, radial angle of 60 0 to 180 0, for fixed jet2 locations are shown in form of 3D response surface plots in Figures 3(a) to 3(c). From the surface plot shown in Fig. 3(a), it was observed that minimum mixing time value of 20sec for jet2 angle of 44 0, located at a height of 13.33cm from the bottom of the tank. The 3D response surface plot (Fig. 3(b)) shows the optimum mixing time of 16.6sec for jet2 inclination of 40 0, for the jet2 located at 19.99cm from the bottom of the tank. Fig. 3(c) gives the optimum mixing time of 14.1sec for jet2 angle of 42 0 and 20

4 jet2 located at a height of 26.66cm from the bottom of the tank. Fig. 3(a) Analysis of mixing time from 3D surface graph for optimization: radial angle (degrees) vs jet2 angle (degrees) at a jet2 location of 13.33cm. Also the response surface plots shown in Fig.3 (a) to Fig.3(c), it was observed that mixing time values was decreased as the jet2 angle increases from 30 0 to 45 0 and starts increases when jet2 angle increases from 45 0 to Fig. 2(a) Analysis of mixing time from 3D surface graph for optimization: jet2 location (cm) vs. jet2 angle (degrees) at 60 degree Fig. 3(b) Analysis of mixing time from 3D surface graph for optimization: radial angle (degrees) vs jet2 angle (Degrees) at a jet2 location of 19.99cm. Fig. 2(b) Analysis of mixing time from 3D surface graph for optimization: jet2 location (cm) vs jet2 angle (degrees) at 120 degree Fig. 3(c) Analysis of mixing time from 3D surface graph for optimization: radial angle (degrees) vs jet2 angle (degrees) at a jet2 location of 26.66cm. Fig. 2(c) Analysis of mixing time from 3D surface graph for optimization: jet2 location (cm) vs. jet2 angle (degrees) at 180 degree 3. Effect of radial angle on mixing time To analyze the influence of radial angle on mixing time, the mixing time values found from the model equation (2) were plotted against with jet2 angle and radial angle for fixed jet2 locations and are shown in the response surface plots from Fig. 4(a) to 4(c). The mixing time values (13.3 sec, 28.6sec) obtained from the model for varies combinations of radial angle 60 0 to 180 0, jet2 locations 13.33cm to 26.66cm for fixed jet2 angles of 30 0,45 0 and 60 0 were shown in form of response surface plots. The response surface plot Fig.4 (a) gave the minimum mixing time value of 17.1sec when jet2 was located at a radial angle of 63 0 for the fixed jet2 angle of Fig. 4(b) showed the optimum mixing time of 14.9sec, for the radial angle of 61 0 when jet2 inclination was It was observed from Fig.4(c) that for a jet2 angle of 60, the optimum mixing time was found to be 18.2sec for the jet2 located at a radial angle of

5 The 3D surface graphs 4(a), 4(b) and 4(c) also shows a decreasing in the mixing time efficiency as the radial angle of jet2 with respect to jet1 increases. Fig. 4(a) Analysis of mixing time from 3D surface graph for angle of 30 degree. Fig. 4(b) Analysis of mixing time from 3D surface graph for angle of 45 degree. inclination between 40 0 and 45 0 give the minimum mixing time. This shows that mixing time decreases as the length of the jet2 increases. Increase in radial angle of jet2 with respect to jet1 increases the mixing time. The minimum mixing time of 14 to 15 sec could be achieved using jet2 angle of 40 0 to 45 0, radial angle of 60 0 to 62 0 and jet2 location of 24cm to 26cm, which concludes the optimum jet2 configuration for the double jet considered in the present analysis. REFERENCES [1] H. Fossett and L. E. Prosser, The application of free jets to the mixing of fluids in bulk, Journal of Institute of Mechanical Engineers, 160, pp , [2] E. A Fox and V. Gex, E. Single Phase Blending Of Liquids, A.I.Ch.E Journal, 2, pp , [3] P. W. Coldrey, Jet mixing, Industrial chemistry engineering course, paper, University of Bradford, England, [4] T. Maruyama, Y. Ban and T. Mizushina, Jet mixing of fluids in tanks, Journal of Chemical Engineering, Japan, 15, pp , [5] V. V. Kafarov, Effective use of Reactors with Jet Mixing of Reagents, Foundation of Chemical Engineering, 22, pp , [6] V. V. Ranade, Towards better mixing protocols by designing spatially periodic flows: The case of a jet mixer, Chemical Engineering Science, 51, pp , [7] S. Jayanthi, Hydrodynamics of jet mixing in vessels, Chemical engineering science, 56, pp , [8] A.W. Patwardhan, CFD Modeling of jet mixed tanks, Chemical engineering science, 57, pp , [9] A. W.Patwardhan and S. G. Galkwad, Mixing in Tanks Agitated by Jets, Chemical engineering science, 57, pp , [10] D. Zughbi and IqtedarAhmad Mixing in Liquid Jet Tanks: Effect of Jet Asymmetric, Industrial and Engineering Chemistry Research, American Chemical Society, 2005, 44, pp [11] Orhanekren, Y. Banu, Ekren and Barisozerdem Break-even analysis and size optimization of a PV/wind hybrid energy conversion system with battery storage A case study, Applied Energy, 86, pp , [12] Yi-ling Lai, GurusamyAnnadurai, Fu- chuang Huang and Jiunn-fwu Lee Biosorption of Zn(II) on the different Ca-alginate beads from aqueous solution, Bioresource Technology, 99, pp , [13] P.Anbu, S.C.B. Gopinath, A. Hilda, T. Lakshmipriya and G. Annadurai, Optimization of extracellular keratinase production by poultry farm isolate Scopulariopsisbrevicaulis, Bioresource Technology,98,pp ,2007 Fig. 4(c) Analysis of mixing time from 3D surface graph for angle of 60 degree V. CONCLUSION Based on the analysis of mixing times obtained from the model using design of experiments for a double jet mixer considered in the present study the following points are concluded. Mixing time efficiency increases as the height of the jet2 increases from the bottom of the tank. A jet2 22