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Optimization of a Complex Industrial Process – Role of Process Modelling



6th World Congresses of Structural and Multidisciplinary Optimization
Rio de Janeiro, 30 May - 03 June 2005, Brazil

Optimization of a Complex Industrial Process – Role of Proc

ess Modelling
Ryszard Pohoreckia, Jerzy Ba?dygaa, W?adys?aw Moniuka, Piotr T. Wierzchowskib
a

Faculty of Chemical and Process Engineering, Warsaw University of Technology, Warsaw, Poland, pohorecki@ichip.pw.edu.pl b Institute of Organic Chemistry, Polish Academy of Science, Warsaw, Poland

1. Abstract Oxidation of cyclohexane in the liquid phase is an important step in production of caprolactam, subsequently converted into polyamide plastics (Nylon 6, Nylon 66). One of the processes commercially employed to oxidize cyclohexane is the Polish CYCLOPOL process. In order to keep pace with other leading companies, the owners of the CYCLOPOL process continue to carry on research and development work, aimed at increasing productivity and selectivity of the process. A model of the process of the catalytic cyclohexane oxidation in the liquid phase, including both reaction kinetics and mass transfer, is presented. The reaction rate constant as well as the activation energies were determined on the basis of the experimental results obtained in a laboratory reactor. The model of the hydrodynamics of gas bubbling through a liquid layer is also presented. This model has been verified experimentally for different organic liquids and columns of different scale. The model has been subsequently used to establish a theoretically based correlation for the bubble diameter by means of a numerical experiment using “virtual liquids”. The two models (of chemical kinetics on the system hydrodynamics) have been integrated to give a comprehensive model of the whole process of cyclohexane oxidation in industrial reactors. Numerical simulation of industrial rectors gave good agreement with the operating data. 2. Keywords: process modelling, cyclohexane oxidation, chemical kinetics, hydrodynamics of gas bubbling 3. Introduction The process of cyclohexane oxidation is widespread in chemical industries all over the world, mostly to get cyclohexanone and cyclohexanol, being in turn processed into caprolactam, adipic acid, and – subsequently – polyamide fibers and plastics (as nylon 6, nylon 66). The amount of cyclohexane oxidized this way exceeds 4 millions t/year. The process is relatively a difficult one, since the desired products (i.e., cyclohexanone and cyclohexanol) are intermediates in a sequence of reactions, and the overoxidation results easily in a number of useless (or hardly recuperable) byproducts. Cyclohexane oxidation is a two – phase process, carried out in a gas – liquid system. The course of the process may be affected both by chemical kinetics and by hydrodynamic factors.

Fig. 1. Structure of the investigations program

1

One of the processes commercially employed to oxidize cyclohexane is the Polish CYCLOPOL process. In order to keep pace with other leading companies, the owners of the CYCLOPOL process continue to carry on research and development work, aimed at increasing productivity and selectivity of the process. In the present work an exhaustive research program, aimed at the optimization of the process is presented. The program was successfully completed in 2003 [1], and implemented on the industrial scale in 2004 [2]. The program included the following parts: investigation of the chemical kinetics (microkinetics) of the process; search for most suitable catalysts; development of a mathematical model of a complex reaction scheme in a multicomponent system (over 166 chemical species involved); exhaustive investigation of the process hydrodynamics, including the hydrodynamics of gas bubbling under elevated temperature and pressure, investigation of liquid mixing in a complicated geometry, and investigation of the mass transport in gas – liquid system; development of the mathematical model of the process in the reactor, comprising the hydrodynamics, mass transfer, and chemical reactions involved; verification of the model both in the laboratory and on the industrial scale; determination of optimal process conditions. The structure of the investigation program is shown in Fig. 1. 4. Chemical kinetics The mechanism of cyclohexane oxidation is a complicated, multistage, free – radical chain reaction with degenerated chain, comprising different chain inhibition, chain propagation and chain termination steps. A reaction scheme comprising up to 154 reactions has been developed by Tolman [3]. A number of simplified kinetic models of this reaction have been described in the literature [4 – 7]. All these simplified models proved to be inadequate. In order to obtain a more exact, but still workable model, we developed a new model of cyclohexanone oxidation. The model has been developed according to the following guidelines: It is impractical to use too many reactions in the model employed for practical calculations, as too complicated schemes require too many kinetic constants, which cannot be simultaneously determined with sufficient accuracy. It is therefore advisable to use lumped kinetic models, requiring a reasonable number of kinetic constants (of the order of 10). A more universal scheme should be modified/simplified, depending on the kind of catalyst used. The concentrations of the free radicals, which cannot be directly measured, are to be eliminated using the quasi – steady – state hypothesis. The following reaction are considered:
k RH + O 2 ??→ ROOH
O

(1)

ROOH + O 2 ?k ROH + ?→
1

1 O2 2

(2) (3) (4) (5) (6)

ROOH + O 2 ?k RO ?→
2

RH + ROOH ??→ 2ROH k ROH + RO * ??→ RO + RO * 2
kX
11

RO + RO * ?k ?→ D + RO * 2 2
7

ROOH + D ??→ RO + D' (7) k 2RO * ??→ D' (8) 2 where D is the reactive by-products in the liquid phase, D’ the nonreactive by-products in the gas phase. Reaction rate constant k0 is equal to k 0 = k 01 [ROOH ] + k 02 (9) Application of the quasi-steady-state hypothesis for the free radicals R*, RO* and RO2* and analysis of the resulting expressions led to the following conclusions: 1. The concentration of the radical [R*] varies following the changes in the cyclohexane concentration [RH], [R*]~[RH]: ' ' ' ' ' [R *] = [RH] {k 1 [O 2 ] + k 3 [RO * 2 ] + k' 4 [RO' *] + k 8 [ROOH] + k 19 [OH *] } (10) k '2 [O 2 ] + (k 13 + k 14 )[ROOH] 2. The concentration of the free radical [RO*] varies with the hydroperoxide concentration [ROOH], [RO*]~ [ROOH]: ' ' ' ' ' ' (11) [RO *] = [ROOH ] {k 5 + 2k 6 [ROH' ] + k 7 [RO' ] + k 8 [RO] + k 8 [RH ] + k 13 [R *] } k 4 [RH ] + k 9 [ROOH ] where: ki’ – reaction rate constans for non catalytic oxidation [7] 3. In the case of the concentration of the free radical RO*2 , the situation is more complex. Initially, there are low concentrations of all free radicals, so the rate of consumption of O2 is very slow. This results in high concentration of O2 in the liquid phase. When the concentrations of the free radicals increase, the rate of O2 consumption increases, decreasing its concentration significantly.
k 16
18

2

Most of the potentially active oxygen is then conserved in the hydroperoxide [ROOH] and the concentration of [RO*2] becomes proportional to [ROOH] with the “constant” of proportionality dependent on the concentrations of other radicals, [RO*2]~[ROOH]. The above observation are the basis of a new comprehensive model of the catalytic oxidation of cyclohexane (Fig. 2.).

Fig. 2. Scheme of reactions Following the suggestions of Berezin, Denisov and Emmanuel [8] resulting from the analysis of the catalytic oxidation process, we assume the following forms of dependency of the reaction rate constants on the catalyst concentration

k 01 ~ [K ] ; k x ~ and

[K ]

(12)

k 1 ~ [K ] ; k 2 ~ [K ] (13) Kinetic equation for the above – mentioned reactions then read: d[RH] = ? k s [K ][ROOH] + k 02 [O 2 ][RH] ? k sx [K ][RH][ROOH] (14) 01 dt d[ROOH] s = k s [K ][ROOH] + k 02 [O 2 ][RH] ? k 1 [K ][ROOH] ? k s2 [K ][ROOH] ? k sx [K ][RH][ROOH] ? k 16 [D][ROOH] (15) 01 dt d[ROH] s = k 1 [K ][ROOH] + 2k sx [K ][RH][ROOH] ? k 11 [ROH][ROOH] (16) dt d[RO] = k s2 [K ][ROOH] ? k 7 [RO][ROOH] + k 11 [ROH][ROOH] + k 16 [ROOH][D] (17) dt d[D] = k 7 [RO][ROOH] ? k 16 [ROOH][D] (18) dt d[D'] = k 16 [ROOH][D] + k 18 [ROOH] 2 (19) dt The change of the oxygen concentration in the liquid phase can be described by d[O 2 ] s = ? k s [K ][ROOH] + k 02 [RH][O 2 ] + 0.5k 1 [K ][ROOH] ? 1.5k 7 [RO][ROOH] (20) 01 dt The third term (1.5 k7[RO][ROOH]) on the right-hand side of Eq. (20) reflects oxygen consumption during oxidation of RO to produce by-products in reaction (8); the stoichiometric coefficient, 1.5 results from long-term industrial experiments. Eq. (20) describes only the change of oxygen concentration in the liquid phase resulting from the chemical reactions. For the full balance of oxygen in the liquid, the oxygen transfer from the gas phase to liquid phase must be taken into account. As pointed out earlier [9], the overall reaction of cyclohexane oxidation may be considered as an irreversible reaction, pseudofirst order with respect to oxygen. For such a reaction, the molar flux of oxygen equals [10]: [O 2 ] ? ? ? ? ? Hy O P ? (21) N = kL tanh ? ? cosh ? ? ? ? where

(

)

(

)

(

)

2

kL The pseudo-first-order reaction rate constant, k is equal:

?=

DA ?k

(22)

k = (k s 01

[K ][ROOH] + k ) [RH] + 1.5k [RO][ROOH] [O ]
7 02 2

(23)

Notice, that the oxygen production term (0.5k1s[K] [ROOH]) has been neglected in Eq. (23) because it does not affect the gradient of oxygen concentration in the boundary layer in the case of slow and very slow reactions [9]. Then, the oxygen concentration in the liquid-phase can be described by Eq. (24):

3

[O 2 ] ? ? k s [K ][ROOH] + k [RH][O ] + 0.5k s [K ][ROOH] ? 1.5k [RO][ROOH] d[O 2 ] ? ? ? ? Hy O P ? (24) = k La 01 02 2 1 7 dt tanh ? ? cosh ? ? ? ? Our experimental investigations (supported by the subject literature and industrial experience) revealed significant deactivation of the catalyst. We assume first order kinetics of deactivation with the kinetic constant dependent on the free radical concentrations [RO*] and [RO*2]; in the model their effect is represented by [ROOH] in agreement with the analysis based on the quasi-steady-state assumptions: d[K ] = ?k x [K ][ROOH] (25) dt In order to determine the reaction rate constants in Eqs. (14 – 19,25), measurements of kinetics of the catalytic cyclohexane oxidation were performed. Experiments were carried out in a stainless steel PARR reactor (PARR Instrument Company, Moline, Illinois, USA) with a capacity of 1?10-3m3. The mass transfer characteristics of the PARR reactor (ε, a and kL) have been previously determined using other reactive systems [11,12]. Oxygen diffusivity in cyclohexane was calculated from King, Hsueh and Mao correlations [13]. The value of Henry's constant was taken from Suresh, Sridhar and Potter data [14]. A comparison of the results calculated using the model (c=f (t)) with the experimental data is shown in Figs. 3 and 4.
2

(

)

Fig. 3. Comparison of the calculation and experimental results (chromium napthenate as a catalyst).

Fig. 4. Comparison of the calculation and experimental results (cobalt napthenate as a catalyst). 5. Hydrodynamics The CYCLOPOL process is carried out in gas – liquid reactors, in which the oxidizing gases (air, or air enriched in oxygen) are bubbling through a layer of liquid. A typical reactor is a horizontal cylinder about 20 m long and 4 – 6 m in diameter, subdivided into 4 – 6 compartments and equipped with a system of gas distributors (Fig. 5). The operating temperature is in the range 150 – 180oC, the pressure range being about 1 MPa.

4

off-gas outlet air inlet

cyclohexane inlet

oxidate outlet

Fig. 5. Schematic diagram of the oxidation reactor The hydrodynamic parameters of such a reactor include gas hold – up, bubble diameter (or rather distribution of bubble diameters), interfacial area and liquid mixing. In spite of widespread industrial application of bubble systems, the hydrodynamics description of such systems is far from being complete. There exists a number of models, describing gas-liquid flows at different scales (interface tracking models for single bubble, Euler-Lagrange models for bubble swarms, Euler-Euler models for the whole apparatus). These models can be coupled to give a multi-level model [15,16]. In many cases, however, the bubble diameter has to be assumed a priori, and to this end empirical and semi-empirical correlations are normally used. Such correlations are usually based on a limited number of experiments, and often give highly divergent results when applied to a case at hand. To develop a mathematical model of the bubbling process, a simplified version of the Prince and Blanch [17] model was used. The population balance equation used was of the form suggested by Fleisher et al. [18]: ? ? ? ? ? ? (26) n (z, d b , t ) + [n (z, d b , t )u r (z, d b )] + n (z, d b , t ) d b (z, d b )? = G (z, d b , t ) ?t ?z ?d b ? ?t ? ? where the first term describes the change of bubble number concentration with time, the second is the convection term, the third describes bubble growth, and the right hand side is the generation function. For the equilibrium region considered in this work, we observed experimentally that the bubble size distribution does not change in time or along the column axis. Moreover, in the absence of mass transfer and with sufficiently small pressure change, one can assume that all the terms on the left hand side are equal to zero. Dividing the total bubble population into N classes one can write Eq. (26) as: Gi = 0 (27) where Gi is the generation function for bubbles of class “i”. The generation function is the difference between bubble birth and death functions. The bubble “births” in a given class result from breaking a bigger bubble, or from the coalescence of smaller bubbles. Assuming that a bubble can be broken into two smaller bubbles of equal volume (which is rather arbitrary assumption) or be formed by coalescence of two smaller ones, we can write: N 1 N N G i = ∑∑ C i ,kl ? ∑ C + 2B m ? Bi (28) 2 k =1 l=1 j=1 where υ m = 2υ i (29) and ?C kl if υ k + υ l = υi C i ,kl = ? (30) ?0 if υ k + υ l ≠ υ i The model assumes that: (1) the bubble coalescence rate is equal to the product of the bubble collision rate and the collision efficiency; (2) the bubble collisions may be caused by turbulence, buoyancy or laminar shear; the bubble break-up rate is equal to the product of collision rate of bubbles and turbulent eddies and collision efficiency; (3) (4) bubbles are broken by eddies of the same size as the bubble or smaller (but not smaller than 20% of bubble diameter); the bubble-eddy collision efficiency depends on eddy kinetic energy. (5) The details of the model used and the results obtained using this model have been described in our earlier papers [19,20]. In the conditions considered the bubble size distribution can be described by a log-normal distribution. The parameters of the distribution were selected so as to minimize the ∑ G i2 values (Eq. 28). Bubble concentration can be obtained from the gas hold-up
ij

i

and the average bubble diameter using the equation [17]:
ni = 3εD 2 H L xi 2d 3 32

(31)

To validate the model more completely, the bubble diameter distributions have been measured experimentally for a number of pure liquids, using two different columns. Two different columns were used:

5

A laboratory column 9 cm diameter and 200 cm high (125 cm clear liquid head), operated at atmospheric pressure and low temperature, with seven different liquids: acetaldehyde, acetone, cyclohexane, isopropanol, methanol, n-heptane, and toluene. A pilot plant column, 30.4 cm diameter and 3.99 m high, operated at elevated pressure (up to 1.1 MPa) and temperature (up to 160oC), with two different liquids: cyclohexane and water. The bubble diameters were measured by the photographic method. The gas hold-up was determined on the basis of froth height and clear liquid head measurements. The detailed descriptions of the experimental stands, as well as the details of the measuring techniques, have been described in our earlier papers [18,19]. The mixing of liquid in the reactor was investigated by three methods: classical stimulus-response technique using radioactive tracer with Br-82 isotope to measure the residence time distribution curve; a specially devised technique, consisting in the employment of a wooden sphere of density identical to that of the reaction mixture, and containing a radioactive isotope La-140. The trajectories of this sphere in the reactor were observed; a computational fluid dynamics (CFD) method, used to establish the velocity patterns inside the reactor (Fig. 6-7). In this Figures velocity vectors are expressed in m/s.

Fig. 6. Liquid velocity vectors in cross – section.

All the three methods gave a good picture of the flow pattern in the reactor. From the investigations of the liquid mixing, it was concluded that liquid circulation in the reactor could be described by the Tilton and Russell model [21].

6

Fig. 7. Liquid velocity vectors in longitudinal section.
6. Mathematical model of the reactor The model of the reaction kinetics described above, together with a suitable description of an industrial reactor hydrodynamics (gas holdup, bubble diameter, interfacial area, liquid mixing), have been used to develop a mathematical model of the industrial reactor in the CYCLOPOL process. The mathematical model of an industrial reactor enabled calculation of the following parameters: degree of conversion, selectivity towards cyclohexanol, selectivity towards cyclohexanone, selectivity towards cyclohexyl hydroperoxide, amount of the by-products (reactive and nonreactive), concentration profile of all the above species, concentration profile of dissolved oxygen, concentration of oxygen in the outlet gases.

Table 1. Results of the calculations and experiments Degree of conversion (%) Cyclohexanol (w%) A Calc 4.11 2.56 Exp 4.09-4.30 2.32-2.88 A Calc 3.95 2.42 Exp 3.81-4.36 2.08-3.06 A Calc 4.01 2.56 Exp 4.09-4.30 2.32-2.88 B Calc 3.77 2.60 Exp 3.88 2.58-2.96 B Calc 3.66 2.44 Exp 3.68 2.33-2.54 B Calc 3.81 2.50 Exp 3.71 2.06-2.57 A reactor volume = 254 m3, B reactor volume = 102 m3

Cyclohexanone (w %) 1.16 1.11-1.37 1.12 1.17-1.35 1.16 1.14-1.49 1.27 1.15-1.130 1.25 1.24-1.40 1.31 1.29-1.53

The results of simulations performed using this model have been compared with the data from different industrial reactors. A few examples of this comparison are shown in Table 1. As can be seen, very good agreement of the simulation results and the industrial data has been obtained. On this basic, the model has

7

been applied to optimize the industrial process. The optimize process has been successfully implemented on the industrial scale [1,2].
7. Conclusions A new, comprehensive model of the process of the catalytic cyclohexane oxidation in liquid phase, including both reaction kinetics and mass transfer has been developed. The model can be modified/simplified depending on the kind of catalyst used. A coalescence – redispersion model, based on that originally proposed by Prince and Blanch [17], was used to determine the Sauter average bubble diameter values for eight different liquids and two bubble columns of different size. The values obtained from the model are in very good agreement with experimental data. The agreement is better than between the experimental values and those obtained from the existing correlations. The two models (of chemical kinetics and the system hydrodynamics) were integrated, to give a comprehensive model of the whole process in industrial reactors. Numerical simulation of industrial reactor gave good agreement with the operating data, and have been used for optimization of the industrial process. 8. Nomenclature a interfacial area per unit volume of liquid, m2/m3 B break-up rate, 1/m3s C coalescence rate, 1/m3s c concentration, kmol/m3 D column diameter, m D reactive by-products in the liquid phase D’ non-reactive by-products in the gas phase DA oxygen diffusivity in the liquid, m2/s bubble diameter, m db d32 Sauter bubble diameter, m G generation function, 1/m3s H Henry's constant, kmol/m3 MPa clear liquid height, m HL K catalyst k pseudo-first reaction rate constant, 1/s k0 reaction rate constant defined by Eq. (11), m3/kmol s k02 reaction rate constant, m3/kmol s s proportionality constant, m7.5/kmol2.5 ski k 01 k’i reaction rate constants for non catalytic oxidation, m3/kmol s, 1/s reaction rate constants, m3/kmol s ki ksi proportionality constants, m3/kmol s catalyst deactivation rate constants, m3/kmol s kK kL physical mass transfer coefficient in the liquid phase, m/s proportionality constant, m4.5/kmol1.5 s ksx N molar flux, kmol/m2s n bubble concentration per unit volume, 1/m3 P pressure, MPa RH cyclohexane RO cyclohexanone ROH cyclohexanol ROOH cyclohexyl hydroperoxide t time, s ur bubble rise velocity, m/s v liquid volume, m3 Vm gas molar volume, m3 x bubble fraction y mole fraction in the gas phase z axial co-ordinate [] concentration, kmol/m3 Greek letters

υ ε φ Subscripts i,j,k,l,m L

bubble volume, m3 gas holdup parameter defined by Eq. (22)

bubbles belonging to class i, j, k, l, m liquid

8

9. References 1. R. Pohorecki, S. Oczkowicz, S. Rygiel, J. Wais, A. Kozio?, M. Gruszka, A. Kondrat, P.T. Wierzchowski, W. Moniuk, A. Krzysztoforski, W. Mija?, W. Wójcik, M. ?yliński and J. Szymczak. Polish Patent Application, P 358357 ,2003 2. R. Pohorecki, S. Oczkowicz, S. Rygiel, J. Wais, A. Kozio?, M. Gruszka, A. Kondrat, P.T. Wierzchowski, W. Moniuk, A. Krzysztoforski, W. Mija?, W. Wójcik, M. ?yliński and J. Szymczak. Polish Patent Application, P 365299 ,2004 3. C.A. Tolman. Resume. Research accomplishments and interests. http://copland.udel.edu/~tolman/resume.html, 1997 4. M. Spielman. Selectivity in hydrocarbon oxidation. AIChE Journal, 1964, 10: 496 – 501 5. J. Alagy, F. Defoor and S. Franckowiak. Progrès récents realizes dans l'oxydation du cyclohexane en cyclohexanone. Revue de l'Institut Fran?ais du Pétrole, 1964, 19: 1380 – 1390 6. J. Alagy, P. Trambouze and H. van Landeghem. Designing a cyclohexane oxidation reactor. Ind. Eng. Chem., Proc. Des. and Dev., 1974, 13: 317 – 323 7. T.V. Kharkova, I.L. Arest-Yakubovich, and V.V. Lipes. Kinetic model of the liquid-phase oxidation of cyclohexane. I. Homogeneous proceeding of the process. Kinetika i Kataliz (Kinetics and Catalysis), 1989, 30: 954 – 958 (in Russian) 8. I.V. Berezin, E.T. and N.M. Emmanuel. The oxidation of cyclohexane. New York, Pergamon Press, 1966 9. K. Krzysztoforski, Z. Wójcik, R. Pohorecki and J. Ba?dyga. Industrial contribution to the reaction engineering of cyclohexane oxidation. Ind. Eng. Chem., Proc. Des. and Dev., 1986, 25: 894 – 898. 10. P.V. Danckwerts. Gas – liquid reactions. New York, McGraw-Hill Book Co., 1970 11. W. Moniuk, R. Pohorecki and A. Zdrójkowski. Measurements of interfacial area in stirred bubbling reactor. Reports of the Faculty of Chemical and Process Engineering at the Warsaw University of Technology, 1997a, 24: 51 – 70 (in Polish) 12. W. Moniuk, R. Pohorecki and A. Zdrójkowski. Measurements of mass transfer coefficients in liquid phase in stirred reactor. Reports of the Faculty of Chemical and Process Engineering at the Warsaw University of Technology, 1997b, 24: 91 – 111 (in Polish) 13. W. Moniuk, R. Pohorecki. Comparison of the calculation methods of the nitrogen diffusion coefficients in cyclohexane. Chemical and Process Engineering, 1998, 19: 551 – 555 (in Polish) 14. A.K. Suresh, T. Sridhar and O.E. Potter. Mass transfer and solubility in autocatalytic oxidation of cyclohexane. AIChE Journal,1988, 34: 55 – 68 15. N.G. Deen, M. van Sint Annaland, and J.A.M. Kuipers. Multi-scale modeling of dispersed gas-liquid two-phase flow, Chem Eng Sci, 2004, 59(8-9): 1853-1861 16. M. van Sint Annaland, N.G. Deen and J.A.M. Kuipers. Multi-level modelling of dispersed gas-liquid two-phase flows, in Series: Heat and Mass Transfer, M. Sommerfeld and D. Mewes (eds), Springer, Berlin, 2003 17. M.J. Prince and H.W. Blanch. Bubble coalescence and break-up in air-sparged bubble columns. AIChE Journal, 1990, 36(10): 1485-1499 18. G. Fleisher, S. Becker and G. Eigenberger. Detailed modeling of the chemisorption of CO2 into NaOH in bubble column. Chem. Eng. Sci., 1996, 51 (10): 1715 – 1724 19. R. Pohorecki, W. Moniuk, P. Bielski and A. Zdrójkowski. Modelling of the coalescence/redispersion processes in bubble columns. Chem. Eng. Sci., 2001, 56 (21-22): 6157 – 6164 20. R. Pohorecki, W. Moniuk, A. Zdrójkowski and P. Bielski. Hydrodynamics of a pilot plant bubble column under elevated temperature and pressure. Chem. Eng. Sci., 2001, 56 (3): 1167 – 1174 21. J.N. Tilton and T.W.F. Russell. Designing gas-sparged vessel for mass transfer. Chemical Engineering, 1982, 29: 61 – 68

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