Compartment Mixing Model in a Stirred Tank Equipped Dual Rushton Turbine

Transcription

1 Iranian Journal of Chemial Engineering Vol. 9, No. (Summer), 01, IAChE Compartment Mixing Model in a Stirred Tank Equipped Dual Rushton Turbine F. Fakheri, J. Moghaddas Transport Phenomena Researh Center, Chemial Engineering Faulty, Sahand University of Tehnology, Tabriz, Iran Abstrat Knowledge of mixing time is of fundamental importane for investigation of mixing effiieny in agitation systems. The mixing time obtained by using the orrelation and formula in large sale mixing systems was inorret. Again, the number of available orrelations in this sale of mixing systems is limited. To predit the mixing time of stirred tanks with dual impellers ommonly used in industry, a thirdompartment mixing model was used. The time of homogenization of the harge (mixing time) was alulated from the time dependeny of the loal onentration of traer measured at various loations. Experimental data on mixing time were obtained with a ondutivity tehnique. In the present study distribution of traer in the bulk of the liquid was desribed by ompartment model (CM) as well as for stirred vessel with dual Rushton impellers. As for the model, a good agreement between the experimental data and the alulated values was apparent. Keywords: Mixing Time, Compartment Model, Rushton Turbine, Traer, Condutivity 1. Introdution Knowledge of mixing time is of fundamental importane for mixing investigation in agitation systems. In fat, this harateristi ontrols the performane of the mehanially agitated tanks and has a diret influene on the ost of the agitation proess [1]. Therefore, mixing time an be used as a omparative measure of the mixer's effiieny, with the same power input. Many attempts, however, have been made over the last 50 years to predit mixing times in stirred tanks []. There are two types of overall investigation of the mixing behavior in agitated system: mixing time from experimental measurements and the results from the simulation studies. For the mixing time measurements several models to approah the homogenizing proess have been proposed, e.g., the irulation model, the eddy diffusion model, the zone network model, the CFD model and the ompartment model. These models are developed based on the three mehanisms of the moleular diffusion, the eddy diffusion and bulk onvetion [ 5]. Corresponding author: Jafar.moghaddas@sut.a.ir 14

2 Fakheri, Moghaddas. Struture of the ompartment model Compared to single impeller arrangements, the flow within tall vessels stirred with multiple impellers is more omplex. Stirred vessel equipped with dual Rushton turbines generate three distintly different stable flow patterns depending on the offbottom learane of the lower impeller and the spaing between the two impellers [5, 6]. These flow patterns are defined as parallel, merging and diverging flow patterns depending upon the interation of the liquid irulation loops with eah other. When the impeller spaing is 1. times greater than the impeller diameter, the flow pattern generated is horizontal and parallel [7]. The flow struture reated by dual Rushton turbine in an aerated stirred tank onsists of irulation flow, exhange flow and indued flow [8]. The struture and parameter of the hydrodynami model is shown in Fig. 1. Aording to the ompartment struture in Fig. 1, ompartments for the impeller zones were used. To investigate the influene of radial distanes of the traer addition points and also to inrease the auray of the model, a large number of ompartments in the radial distane should be used.. Determination of model parameters Cirulation flow Cirulation inside the tank is promoted by the turbine pumping apaity Q P. It should be alulated by integrating the radialveloity profile and multiplying this by the orresponding area. Experimental measurements onfirm that Q P is proportional to ND in turbulent agitation [5, 9]. Q P P = K ND (1) Figure 1. Compartment model Sheme of a twostaged tank The flow number K P is the prinipal dimensionless group. The value of K P was found to depend on impeller geometry [5, 9]: W D K P = 6. ( )( ) D T T K P = 01. ( ) D () () Mahouast et al. found the irulation flow rate is usually onsidered to be twie the impeller pumping apaity at the front of the impeller [10]. Cui et al. therefore argued that the average irulation flow rate in the whole path should be used [5]. That rate was assumed to be 80% of the value given by Eq. 1 [, 5]. Q = K ND CF CF K = 0. 8 K = 1. K CF P P (4) 4. Exhange flow The exhange flow rate desribes the Interhange flow between adjaent stages in the axial diretion. For the stirred tank, in the fully turbulent regime and for the full range Iranian Journal of Chemial Engineering, Vol.9, No. 15

3 Compartment Mixing Model in a Stirred Tank Equipped Dual Rushton Turbine of Newtonian working fluids the harateristi turbulent veloity and length sales are [, 5, 11] u = C V = C ND L u TIP u = C D L (5) The onstants C u and C L are a funtion of the impeller and tank geometry seleted. The dissipation, ε, is the rate of dissipation of turbulent kineti energy. The turbulent kineti energy must sale with u. The rate of dissipation of energy is taken to sale with u L, the harateristi time sale of the flow. This gives u ε (6) L For isotropi turbulent flow, the root mean square flutuation veloity of ontaining eddies an be alulated by Eq. 6 [, 5]. The exhange flow rate an be alulated 1 P from Eq.9 with K = π K N 4 [, 5] Q Z 4 1 i EF = u Zπ ) = K Z ND [( ) ( ) ] = Z ( T T H K D D ND EF (10) K EF is a fration oeffiient of the flow in axial diretion indued by an impeller and is proportional to the pumping apaity and the dimensionless geometry fator of a stirred tank. The dimensionless interstage exhange flow number for similar stirred tank is onsistent with the value 0.59 reported by Alves and Vasonelos and mathes the predition of for tanks of m diameter, aording to Vasonelos et al. [9]. From the measure of the axial exhange flow rate reported in the literature for Rushton turbines, the following relationship was found between K EF and T/D. K EF T = b (11) D u = ( L ε ) 1 (7) A seond estimate of turbulene harateristis whih avoids the need for C u is the power per unit mass of fluid in the tank [5]. Tank 5 P 4N P ρn D ε = ρv = (8) ρπt H i The ombination of Eq.7 with Eq.5 and 8 gives Eq. 9 [, 5]. 1 1 Z P i u = K N ND (T H ) (9) where b is in the small tank and 0.6 in the large tank (95% onfidene interval) [9]. 5. Indued flow Besides the axial turbulent liquid exhange and irulation flow, an extra flow pattern indued by air bubbles was found as a onsequene of the density differene in an aerated agitated tank. This flow is dependent on the gas flow rate and pumping apaity [, 1], thus in the liquid mixing system Indued flow is negligible. 6. Experimental The experimental setup is shown 16 Iranian Journal of Chemial Engineering, Vol. 9, No.

4 Fakheri, Moghaddas shematially in Fig.. All experiments were arried out in a transparent vessel with a flat bottom, having an inner diameter, T, of 0.m. The ylindrial tank body was made of glass and working heights liquid was 0.54 m. The stirred tank was fitted with four wall mounted baffles having a wih of 1/10th and thikness 1/100th that of the tank diameter (fully baffled ondition). Two six bladed Rushton turbine impellers with a diameter of D=T/ were plaed in the tank. The impeller wih, L, and the impeller blade height, W, were equal to D/4 and D/5. The off bottom learane is C 1 =0.55T, and the upper impeller is plaed ΔC=0.7T above the lower one. The stirred tank is filled with tap water as the main ontinuous phase fluid, the surfae of whih is C =0.55T above the upper impeller. A pump drives the turbines and the stirring speed is measured using a alibrated digital osillosope. A tahometer was used to measure the impeller speed for ertitude. For measuring the mixing time a small quantity of traer (60 ml of NaCl solution (100 g/l)) is added on the top of the dispersion surfae and 11 ports were used for traer detetion. The oordinates of these positions are listed in Table 1.The mixing time was estimated for eah of the probes as the time required to attain the final onentration within ±5% of the average onentration. 7. Results and disussion In the present paper, a CM was adopted in whih the vessel is theoretially divided into three ompartments per stage in the other two mixers orresponding to the number of impellers, in whih zones and 4 have the same ondition of flow harateristi affeted by veloity vetor. Sine the traer with a ertain amount of volume and onentration was injeted into the vessel, bulk onvetion mehanism is used in the homogenizing proess. Figure. Shemati of the experimental set and details of the tank Table 1. Positions of detetors and injetion point Position Injetion points R (r/t) Probe loations Z (m) (z/h) To verify the model, the theoretially predited onentration urves whih are Iranian Journal of Chemial Engineering, Vol.9, No. 17

5 Compartment Mixing Model in a Stirred Tank Equipped Dual Rushton Turbine obtained by the traer onentration in several loations an be ompared with those obtained by experiment. Conentration transients were simultaneously solved by the numerial solution of the system of ordinary differential equations whih obtained the transient mass balane of the traer in eah ompartment [, 5, 8, 9]. dc1 V1 = QinCin + QIFC QIFC1 + QCFC QCFC1 dc V = QCFC + QCFC QCFC + QIFC + QIFC QIFC dc V = QIFC4 + QIFC QIF C+ QEFC4 Q C + Q C Q C 1 1 EF CF CF dc 4 4 = IF 5+ IF IF 4 + EF V Q C Q C Q C Q C Q C + Q C Q C EF 4 CF 5 CF 4 dc 5 5 = CF 4 + CF 6 CF 5+ IF 4 V Q C Q C Q C Q C + Q C Q C V IF 6 IF 5 dc6 6 = QIFC5 QIFC6 + QCFC5 QCFC6 (1) where V i is volume of eah ompartment and C i represents the traer onentration in the ith ompartment. This system of firstorder differential equations was solved by Runge Kutta fourth order method. The initial values are omputed with the knowledge of the volume and onentration of the traer injeted into the vessel. It was assumed that, after injetion at time t = 0, the onentration C 1 prevails; and no traer onentration would be present in the other ompartments (assumption of ideal mixing) [, 8, 9]. In order to obtain mixing time, it is neessary to understand the model parameters. Computed exhange flow rate (Q EF ) and irulation flow rate (Q CF ) between the ompartments are shown in Table. The parameters of the model were omputed by Moghaddas et al. from experimental data [1] and the Eq. 11 [9]. The results obtained from previous information onfirm the high auray of Eq. 11 and Moghaddas et al. data, rather than Eq. and 10. Table. Exhange and irulation flow rate between the ompartments Model Moghaddas (004) [1] parameter (m /s) 00 RPM 400 RPM Q CF by Eq. Q CF by Eq. 11 Q EF by Eq. 10 Q EF The effet of different situation measurements of the traer onentration on the mixing time by using CM is shown in Fig. (a). It is obvious that the situation measurements of the traer onentration, being far from the injetion point, dereased the intensity of the peak. Fig. (b) shows the omparison between the measured and alulated urves of the onentration profiles and indiates that the model aurately desribes the behavior of a real mixed system. 18 Iranian Journal of Chemial Engineering, Vol. 9, No.

6 Fakheri, Moghaddas C(t)/C( ) Cell 1 Cell Cell Cell 4 Cell 5 Cell Mixing time (s) Mixing time (s) Compartment model Near wall of the tank 9.5 m far from wall of the tank Close to stirrer shaft Probe loation (m) (a) (a) m far from wall of the tank Close to stirrer shaft Compartment model C(t)/C( ) Compartment model Experimental Result" Mixing time (s) Mixing time (s) (b) Figure. (a). Response urves alulated from CM (b). Comparison of experimental response urve and that alulated from CM In the region between the upper and lower impellers, the radial pumping flow rate inreased. In this region homogenization of the liquid was done by both upper and lower irulation flow rate. The results of irulation loops turn bak a higher volume of the liquid to the bak of the blade. This aused the mixing time resulted from position three and four to be lower than the other position. The result from CM shows the effet of different situation measurements of Probe loation (m) (b) Figure 4. Effet of different situation measurements of the traer onentration on the mixing time: (a) 400rpm, (b) 00 rpm. the traer onentration on the mixing time, at first dereased and then inreased. The maximum mixing time obtained was from region one and six. In the experimental work, three traer injetion points on the liquid surfae were supposed (neighbor to wall of the tank, 9.5 m far from wall of the tank and lose to stirrer shaft). Mixing times obtained from the ompartment model for 400 rpm have a good agreement with loation of the traer injetion at 9.5 m far from wall of the tank. Average deviations of experimental Iranian Journal of Chemial Engineering, Vol.9, No. 19

7 Compartment Mixing Model in a Stirred Tank Equipped Dual Rushton Turbine data at the 00 rpm for the loation of the traer injetion at 9.5 m far from the wall of the tank and lose to stirrer shaft are +7.9% and.% respetively. Experimental results show that the effet of traer injetion point at the liquid surfae on the mixing time was negligible. Therefore, the mixing time omputed from CM in the radial distanes was from the stirrer shaft to 9.5 m far from the wall of the tank. 8. Conlusions In this work, the ompartment model was used for the alulation of mixing time (t 95%) in single phase stirred tanks by dual Rushton turbines. A good agreement between the time dependene of the onentration obtained experimentally and that alulated from theory was obtained. The result from CM shows the effet of different situation measurements of the traer onentration on the mixing time first dereased and then inreased. Nomenlature H Liquid height (m) T D W L ΔC C 1 N Q P Q CF tank diameter(m) impeller diameter(m) impeller blade height(m) impeller blade wih(m) impeller spaing(m) off bottom learane of the lower turbine(m) impeller rotational speed (rpm) pumping apaity (m /s) irulation flow rate (m /s) Q EF Q IF C i C(t) C( ) exhange flow rate (m /s) indued flow rate (m /s) traer onentration in the ith ompartment (gr/lit) traer onentration at time t (gr/lit) final traer onentration (gr/lit) N P power number () turbulent kineti energy ε u dissipation rate (m /s ) harateristi turbulent veloity sale (m/s) L harateristi length sale (m) volumetri traer flow rate Q in C in Hi V TIP V i (m /s) traer onentration (gr/lit) denotes ompartment height (m) Impeller tip veloity (m/s) denotes ompartment volume (m ) Referenes [1] Houine, I., Plasari, E., and David, R., "Effets of the stirred tank's design on power onsumption and mixing time in liquid phase", Chem. Eng. Tehnol.,, , (000). [] Guillard, F. and Tragardh, C., "Mixing in industrial Rushton turbineagitated reators under aerated onditions", Chem. Eng. and Pro., 4, 7 86, (00). [] Vrabel, P., Van Der Lans, R., Cui, Y.Q., Luyben, K. and Ch, A.M., "Compartment model approah: Mixing in large sale aerated reators with 0 Iranian Journal of Chemial Engineering, Vol. 9, No.

8 Fakheri, Moghaddas multiple impellers", Chem. Eng. Res. Des., 77, 91 0, (1999). [4] Hiraoka, S., Tada, Y., Kato, Y., Matsuura, A., Yamaguhi, T. and Lee, Y.S., "Model analysis of mixing time orrelation in an agitated vessel with paddle impeller", J. Chem. Eng. Jpn., 4, , (001). [5] Cui, Y.Q., Lans, R., Noorman, H.J. and Luyben, K., "Compartment mixing model for stirred reators with multiple impellers", Chem. Eng. Res. Des., 74, 6171, (1996). [6] Shewale, S.D. and Pandit, A.B., "Studies in multiple impeller agitated gasliquid ontators", Chem. Eng. Si., 61, , (006). [7] Khopkar, A.R. and Tanguy, P.A., "CFD simulation of gasliquid flows in stirred vessel equipped with dual Rushton turbines: Influene of parallel, merging and diverging flow onfigurations", Chem. Eng. Si., 6, 81080, (008). [8] Kasat, G.R. and Pandit, A.B., "Mixing time studies in multiple impeller agitated reators", Can. J. Chem. Eng., 8, 89904, (008). [9] Vasonelos, J.M.T., Alves, S.S. and Barata, J.M., "Mixing in gasliquid ontators agitated by multiple turbines", Chem. Eng. Si., 50, 4 54, (1995). [10] Mahouast, M., Fontaine, P. and Mallet, J., "A two ompartment LDV benh for automated determination of flow rate and energy transfer generated by agitators", Proeedings of the 7 th European Conf. on Mixing, BHRA, Brugge, pp , (1991). [11] Edward, P.L., AtiemoObeng, V.A. and Kresta, S.M., Handbook of industrial mixing: siene and pratie, Hoboken, NJ, John Wiley & Sons, (004). [1] Vasonelos, J.M.T., Alves, S.S., Nienow, A.W. and Bujalski, W., "Saleup of mixing in gassed multiturbine agitated vessels", Can. J. Chem. Eng., 76, 98404, (009). [1] Moghaddas, J. S., Tragardh, C., Ostergren, K. and Revste, J., "A omparison of the mixing harateristis in singleand two phase gridgenerated turbulent flow systems, Chemial Engineering & Tehnology, 7(6), 66670, (004). Iranian Journal of Chemial Engineering, Vol.9, No. 1