A Stochastic Analysis of Liquid Mixing in Bubble Column

Transcription

1 Amerian Journal of Fluid Dynamis 013, 3(3): DOI: 193/j.ajfd A Stohasti Analysis of Liquid Mixing in Bubble Column Rajeev Parmar, Subrata Kumar Majumder * Department of Chemial Engineering, Indian Institute of Tehnology Guwahati, Guwahati, Assam, , India Abstrat In the present study, the mixing of liquid phase in bubble olumn is analysed by an entropy theory. The Kolmogorov entropy fator depending on the operating parameter is estimated to evaluate the onentration profile of the solute dispersed in the dynami two-phase system. The intensity of mixing is enuniated by the Kolmogorov entropy parameter and orrelated with the degree of mixing in the bubble olumn. Correlations have also been developed to predit the parameters by dimensional analysis. The analysis may be helpful for further understanding of the two-phase mixing in the multiphase system. Keywords Mixing, Bubble Column, Entropy, Two-Phase Flow 1. Introdution Bubble olumns are extensively used as gas-liqu id ontators in the hemial, petrohemial and biohemial industries. Effetive phase mixing leading to improve heat and mass transfer harateristis and other various advantages of bubble olumn reators render it an attrative hoie of reators for different hemial and bio-hemial proesses. The design and sale-up of these reators has been primarily based on empiriism. This an be attributed to the omplexity and sale dependeny of the hydrodynamis and transport parameters in these systems. Considerable effort has been direted in reent years to obtain fundamental models for the desription of hydrodynamis in bubble olumns. The bubble olumn has been modified in different way aording to the industrial need and appliations[1]. Bubble olumn with various type of gas-liquid distributer utilises the kineti energy of the high veloity liquid for the entrainment and dispersion of gas phas e. They provide several advantages during operation and maintenane suh as high heat and mass transfer rates, ompatness and low operating and maintenane osts[1]. The advantages laimed for bubble olumns ompared to other multiphase ontators are: i) absene of moving parts and hene low maintenane ost, ii) higher effetive interfaial area and volumetri mass transfer oeffiients, (ii) less floor spae requirement and iv) large liquid residene times espeially suited for slow reations, (v) high heat and mass transfer rate. A distint advantage of bubble olumns is that the mixing is effeted by the ation of rising bubbles, a proess signifiantly more * Corresponding author: skmaju@iitg.ernet.in (Subrata Kumar Majumder) Published online at Copyright 013 Sientifi & Aademi Publishing. All Rights Reserved energy-effiient than mehanial stirring. But the operation and performane of bubble olumn reator signifiantly depend on the fluid dynami harateristi of bubble olumn. Ejetor indued downflow bubble olumn has ertain advantages suh as uniform bubble size, finer bubbles, less power onsumption, high gas utilization, easy maintenane[-4]. For the analysis of performane of bubble olumn based on mixing harateristis, the axial dispersion model (ADM) is ommonly used. The liquid is assumed to be in plug flow with axial dispersion superimposed on it. This model lumps the different mehanisms of liquid mixing into a single axial dispersion oeffiient. The ADM laks a proper physial basis for bubble olumns but its popularity is reahed due to a single unknown parameter. There are many orrelations available in literature developed by different authors[5-7] to predit the axial dispersion oeffiient. As per Dekwer (199)[1], there is a wide satter in the data and orrelation preditions of the liquid mixing that are represented by simplified way. Other different phenomenologial based models developed by Kastane et al.(1973)[8], Nauman and Buffham (1983)[9], Dekwer and Shumpe (1993)[10], Krishna et al. (000) [11] and Majumder (008)[1] are also available for desribing liquid bakmixing in a way that an assoiate it with the fundamental mehanisms. Suh models have a strong physial basis for understanding the liquid mixing behavior without relying on the omplete solution of the fundamental hydrodynami equations[13]. From the literature it is seen that mixing harateristis have been extensively studied in bubble olumn reator and reported in the sientifi literature[1, 4, 5, 14 1]. Many of the studies overed in the literature use onventional liquid -phase mixing models to interpret the obtained experimental data and to quantify the effet of the different operating and/or design parameters. The axial dispersion

2 76 Rajeev Parmar et al.: A Stohasti Analysis of Liquid Mixing in Bubble Column model (ADM) has been traditionally used to desribe and quantify the extent of liquid bakmixing in different reator systems[]. In the present study the mixing harateristis in bubble olumn has been analysed by a different a unified approah. In this ase the time dependent traer distribution whih is an estimation of intensity of mixing in bubble olumn reator has been analysed by Kolmogorov information entropy theory.. Theoretial Bakground The flow behaviour through a system is onveniently ahieved by determining the age distribution of the elements of the fluid in the exit stream or the residene time distribution (RTD) within the system. The RTD for flowing fluid is normally obtained by the so alled Stimulus response tehnique. This tehnique involves the injetion of a solute as a traer at the inlet streams and the orresponding response of traer onentration within the system. The variation of solute onentration is haraterized by moleular diffusion and onvetive diffusion in a system. The variation of onentration with a time step of t results the information about the solute dispersion intensity and the loal onentration at a ertain time. In general for P dimension ase, the experimentally obtained time signal of onentration C(t), in a time step t, the (m+1) values of onentration results as C(i, t) for i = 0, 1,, 3, 4 m. Thus a vetor information distribution for the traer partile an be expressed as[3]: Z i = [ C( i t), C( i t + λ),.., C( i t + ( P 1) λ)] (1) where i = 0, 1, (m-{p-1}k), λ = k. t, k = 1,, 3.and P is the dimension of the vetor Z(t). Therefore moving with time t, a series of P-dimensional vetors representing the P-dimensional portrait of system an be obtained. The Kolmogorov entropy, whih is a measured of degree of the rate of generation of information of the system, an be used to the dispersion behavior of the phase in a multiphase system. The rate of generation of information about onentration of solute at a point (for a partiular operating ondition) in a time t an be expressed as: Gi = It = 0 + KIt for t () where K is alled the Kolmogorov entropy. I t=0 is the information of solute dispersed at time t=0 and I t=t (t) is an information of solute dispersed at time t. This results in the distribution of solute with respet to time. From the experimental data, it is found that the distribution of solute follows the Gaussian type distribution whih an be represented as ( t tm ) G ( ) i t It= 0 1 = It = e (3) K π / where I t=0 is the initial information of the system. The Kolmogorov entropy K and parameter are to be found from the experimental data by fitting with the above distribution equation. t m = mean residene time of solute in system obtained by fitting the experimental data. The variane of the distribution equation (eq. (3)) is σ whih is obtained as: σ = (4) 4 As ompared to the axial dispersion model the variane an be expressed by the intensities of dispersion by fitting the axial dispersion model (ADM)[]. The varianes of the two models (ADM and eq.(3)) an be related as: σ = Nd (1 + 4Nd ) = (5) 4 Or 1 4 = N + (6) d Nd where, N d is the dispersion number ( D /( U i H ) ). Higher the dispersion number more the mixing of phases in the system. The dispersion number an be expressed in terms of parameter as: N d = (7) 4( ) inreases mean the dispersion of solute inreases and hene mixing inreases. 3. Results and Disussion The degree of dispersion examined by the parameter whih is known as Kolmogorov entropy (K) and the information harateristi fator () of spreading of solute distribution whih quantifies the intensity of the mixing of phases in the system. As the parameter inreases the mixing intensity inreases. In the following setion, the Kolmogorov entropy (K) and the effetive harateristi fator () of spreading of solute distribution in bubble olumn reator are enuniated based on the published data[5, 15, 16]. In the following setion effet of operating variables on mixing harateristi fators are enuniated Rate of Generati on of Information Fi gure 1. The rate of generation of information with respet to time

3 Amerian Journal of Fluid Dynamis 013, 3(3): The rate of generation of information (onentration of solute) at a point (for a partiular operating ondition) in a time t for different parameter to desribe the degree of mixing in bubble olumn is shown in Figure 1. From the figure it is seen that the present model is fitted well with the experimental data extrated from the published work at the speified ondition. gas veloity inreases with onstant liquid veloity, the uniform distribution of the solute may be hindered by the internal irulation of the liquid due to pressure differene in the olumn. The overall effets of the entropy may lead to the inrease in mixing harateristis () to represent the onentration profile of the traer partile inside the olumn. 3.. Effet of Phase Veloity on Mixing Charateristi Fators () From the experimental data on mixing in bubble olumn published by Dekwer et al. (1974)[5], Zaharandnik and Fialova (1996)[15], Kelkar et al. (1983)[16]. The variation of mixing harateristi fator with gas phase veloities is shown in Figure. It is observed that the mixing harateristi fator () inreases with inrease in gas veloity. At onstant liquid veloity inreasing gas veloity indiates a higher dispersion or higher mixing in the olumn. K (-) Zaharandnik and Fialova (1996) [15] Deekwer et al (1974) [5] Kelkar et al (1983) [16] = m/s Majumder et al. (005) [4] (s) Zaharandnik and Fialova (1996) [15] Dekwer et al (1974) [5] Kelkar et al (1983)[16] = m/s Majumder et al. (005) [4] Fi gure. Variation of mixing harateristi fator with phase veloities As the gas veloity inreases the momentum exhange in the olumn inreases whih inrease the internal irulation of the gas phase in the olumn whih may lead to the more enhanement of spreading of traer moleule and onsequent the more mixing in the olumn as shown in Figure. At low gas veloity, bubbles are relatively s maller and rise uniformly without muh interation with the liquid at higher gas veloity, large fast-rising bubbles appear whih disrupt the system ontents, results inrease in the liquid dispersion and results the inrease in mixing harateristi fator with inrease in gas veloity 3.3. Effet of Phase Veloity on Entropy (K) From the data analysis by the entropy theory it is seen that the Kolmogorov entropy (K) dereases with inreases in gas veloity as shown in Figure 3. The entropy inreases with inrease in liquid veloity but dereases with inrease in gas veloity. The inrease in entropy with liquid veloity indiates more dispersion of solute in the olumn but as the Fi gure 3. Variation of entropy with phase veloities 3.4. Effet of Phase Veloity on Mean Residene Time (MRT) The mean residene time of the traer partiles inside the olumn depends on the liquid and the gas veloities. Geometri onfigurations have also effets on the mean residene time of the partile. As the liquid or gas veloity inreases the mean residene time of the traer partiles dereases. The mean residene time an be alulated from the relation, α g H/. As the gas veloity inreases the gas holdup inreases in the olumn whih results in overall derease of mean residene time of the liquid in the olumn. The same effet of liquid veloity on the gas holdup may lead to the derease of mean residene time of the liquid in the olumn. The parameter t represents the mean residene time in the present analysis. The variation of the mean residene time with gas veloity at different liquid veloity is shown in Figure 4. Some typial data s of different mixing harateristis vary with different operating variables are obtained by present analysis fro m d ifferent authors are shown in Table 1. From the table it is seen that parameters are affeted not only by dynami variables but also with the geometri variables like height and diameters of the bubble olumn Correlations for Model Parameters The present model is fitted with the experimental data of different authors, to find the different parameters. It is seen that the parameters are affeted by the different operating variables as shown in Table 1. Based on the different operating variables, orrelations are developed to predit the parameters by multiple regression analysis with the help of Mirosoft exel data analysis tool.

4 78 Rajeev Parmar et al.: A Stohasti Analysis of Liquid Mixing in Bubble Column t m (s) Zaharandnik and Fialova (1996) [15] Dekwer et al (1974) [5] Majumder et al. (005) [4] Correlation for : usl = d H Correlation for t m : sg ρusld u sg ρusld 1 tmusl = (10) d H usl µ The orrelations predit the parameters satisfatorily. The orrelation oeffiients and the standard errors of the orrelation equations (8), (9) and (10) are found to be (0.97, 0.05), (0.98, 0.11) and (0.98, 0.14) respetively. Parities of experimental values versus predited values are shown in Figure 5. sl µ (9) Fi gure 4. Variation of mean residene time of traer with phase veloity Table 1. Different paramet ers obtained by fitting present model with experimental published data by different authors % error (average) Author d (m) H (m) t m (s) (s) K (-) Predited data % error (average) Zahradnik and Fialová [15] K /d t m /d Experimental data Fi gure 5. Experimental versus predited values of different parameters Dekwer et al.[5] Kelkar et al.,[16] The orresponding orrelations are as follows: Correlation for K: sg ρusld K = (8) H u sl µ 4. Conlusions The mixing harateristis in a bubble olumn have been analysed in the present study by an entropy theory. The entropy theory desribes the intensity of dispersion in the bubble olumn under different operating variables. Different authors desribed the dispersion in multiphase system with axial dispersion model. But in this present study, an alternative way has been enuniated to desribe the mixing harateristis in the bubble olumn reator. The degree of dispersion and the information harateristi fator of spreading of solute distribution quantify the intensity of the mixing of phases in the system. The overall affets of the entropy plays an important role in mixing harateristis to represent the onentration profile of the traer partile. The flow behavior through a system an be analysed with the developed orrelations whih are ahieved by determining the age distribution of the elements of the fluid in the exit stream. The orrelations developed to predit the parameters are found to be well fitted with the experimental data. The analysis may be useful for further understanding to analyse the mixing harateristis in multiphase system.

5 Amerian Journal of Fluid Dynamis 013, 3(3): NOTATIONS C Conentration (kg/m 3 ) d Column diamet er D Dispersion oeffiient (m /s) G Rate of generat ion (1/s) H Height of the olumn (m) i Index (-) I Rate of generat ion of Initial information (1/s) K Kolmogorov entropy (-) N d Dispersion no. P dimension t Time (s) Superfiial gas veloity Superfiial liquid veloity Charateristi fator (s) Z Information distribution funtion σ Variane ρ Density of liquid (kg/m 3 ) µ Visosity of liquid (kg/m.s) REFERENCES [1]. -D. Dekwer, Bubble olumn reators, iley, New York, 199. [] H. F. Svendsen, H. A. Jakobsen and R. Torvik, Loal flow strutures in internal loop and bubble olumn reators, Chemial Engineering Siene, vol. 47, pp , 199. [3] A. Sokoliehin and G. Eigenberger, Gas-liquid flow in bubble olumns and loop reators: Part I. Detailed modeling and numerial simulation, Chemial Engineering Siene, vol 49, pp , [4] Majumder S. K., Kundu G. and Mukherjee D., Mixing Mehanism in Ejetor Indued Downflow Bubble Column, Chemial Engineering Journal, vol 11, no. 1-3, pp , 005. [5]. -D., Dekwer, R. Burkhart and G. Zoll, Mixing and mass transfer in tall bubble olumns, Chemial Engineering Siene, vol. 9, pp , [6] J. B. Joshi and M. M. Sharma, Liquid phase bak mixing in sparged ontators, Canadian Journal of Chemial Engineering, vol. 56, pp , [7] M.V. Kantak, S. A. Shetty and B. G. Kelkar, Liquid phase bakmixing in bubble olumn reators -A new orrelation, Chemial Engineering Communiation, vol. 17, pp. 3 34, [8] F. Kastane, J. Zahradnik, Gas liquid reators I. The residene time distribution in multistage bubble reators, Colletion Czehoslovia Chemial Communiation, vol. 38, pp , [9] E. B. Nauman, B. A. Buffham, Mixing in Continuous Flow Systems, iley, New York, [10]. -D. Dekwer and A. Shumpe, Improved tools for bubble olumn reator design and sale-up, Chemial Engineering Siene, vol. 48, pp , [11] R. Krishna, M. I. Urseanu, J. M. van Baten, and J. Ellenberger, Liquid phase dispersion in bubble olumns operating in the hurn-turbulent flow regime, Chemial Engineering Journal, vol. 78, pp , 000. [1] S. K. Majumder, G. Kundu and D. Mukherjee, Predition of intensity of liquid axial dispersion in a modified downflow bubble olumn, Institute of Engineers (India)-CH, vol. 87, pp. 8-33, 007. [13] S. Degaleesan, M. P. Duduković, Liquid bakmixing in bubble olumns and the axial dispersion oeffiient, Amerian Institute of Chemial Engineer Journal, vol. 44, no. 11, pp , [14] S. Moustiri, G. Hebrard, S. S. Thakre and M. A. Roustan, Unified Correlation for Prediting Liquid Axial Dispersion Coeffiient in Bubble Columns, Chemial Engineering Siene, vol. 56, pp , 001. [15] J. Zahradnik and M. Fialová, The effet of bubbling regime on gas and liquid phase mixing in bubble olumn reators, Chemial Engineering Siene, vol. 51, no. 10, pp , [16] B. G. Kelkar, S. R. Phulgaonkar and Y. T. Shah, The effet of eletrolyte solutions on hydrodynami and bakmixing harateristis in bubble olumns, The Chemial Engineering Journal, vol. 7, no. 3, pp , [17] M. P. Dudukovi, J. C. Chen, L. S. Fan, S. Degaleesan, P. Gupta, M. H. Al-Dahhan, A. B. Toseland, Fluid dynami parameters in bubble olumns with internals, Chemial Engineering Siene, vol. 54, pp , [18] S. N. Palaskar, J. K. De, A. B. Pandit, Liquid phase RTD studies in setionalized bubble olumn, Chemial Engineering Tehnology, vol. 3, no. 1, pp.61 69, 000. [19] M. Vinaya,, Y. B. G. Varma, Some aspets of hydrodynamis in multistage bubble olumns, Bioproess Engineering, vol., pp , [0] E. Blass,. Cornelius, The residene time distribution of solid and liquid in multistage bubble olumns in the o-urrent flow of gas, liquid, and suspended solids, International journal of Multiphase Flow, vol. 3, pp , [1] S. K. Majumder, Effiieny of non-reative isothermal bubble olumn based on mass transfer, Asia-Paifi Journal of Chemial Engineering, vol. 3 no. 4, pp , 008. [] O. Levenspiel, Chemial Reation Engineering, iley, New York, 196 [3] Y. J. Cho, P. S. Song, S. H. Kim, Y. Kang and S. D. Kim, Stohasti analysis of gas-liquid-solid flow in three-phase irulating fluidized bed, Journal of Chemial Engineering of Japan, vol. 34, no., pp , 001