1 Indian Journal of Chemical Technology Vol. 5, July 1998, pp Modelling of air-lift reactors based on bubble dynamics Saikat Chakraborty+, Kajari Kargupta & Avijit Bhowal* Department of Chemical Engineering, Jadavpur University, Calcutta , India Received 13 January 1998; accepted 8 June 1998 Predictions of hydrodynamic performance of an air-lift reactor (ALR) have traditionally considered bulk gas and liquid phase characteristics. This study aims to present an alternative approach of modelling an ALR by estimating the energy input to the reactor and gas hold-up in the riser based on dynamics of individual bubbles. The bubbles have been assumed to rise in discrete layers in the riser having a diameter in equilibrium with the surrounding pressure. The model predictions of gas hold-up and liquid velocity in the downcomer for ALRs have been compared with the reported experimental data. Airlift reactors are one of the most promising devices for gas-liquid mass transfer and are being increasingly used in biotechnological processes. In general, these reactors comprise a pool of liquid, divided into two vertical zones--a central zone called riser, where both the gas and the liquid flow upward and the liquid is sparged by a gas at the bottom, and an annular space called downcomer, where the liquid flows down from the top to the bottom of the reactor. Density difference of the fluids between the riser and the downcomer induces a circulation in the reactor-up flow in the riser and downflow in the downcomer. The primary hydrodynamic characteristics influencing the operation of an airlift reactor are gas hold-up and liquid circulation velocity. These affect the heat and mass transfer coefficients by controlling the extent of mixing in the reactor and the mean residence time of gas and hence process yield. Liquid circulation velocity in airlift reactors can be estimated by equating the energy input to the energy dissipated occurring as a result of liquid circulation 1.3. All the mechanical energy consuii1ed in an airlift reactor is introduced by the gas flow. The gas bubbles expanding through the riser provide the mechanical energy. This energy (E) counterbalances the friction losses between the fluids and devices (F) and the friction losses at the gas-liquid interface (S). The overall energy balance as given by Calvo3 is 'Forcorrespondence +Present address: Department of Chemical Engineering, Indian Institute of Science, Bangalore , India.. (1) To estimate the liquid velocity in the downcomer from the above equation, Calvo3 has equated the energy input, E to reversible isothermal work of expansion of the bulk gas i.e.,.. (2) and has calculated the average gas hold-up in the riser by the equation,.. (3) Besides this, other empirical correlations are also available for estimation of gas hold-up2.4. Thus, previous predictive studies concerning estimation of the hydrodynamic characteristics of ALR have not taken into account the disperse state of the gas in the riser. In order to present a more realistic approach towards modelling of an ALR, this study attempts to correlate the liquid circulation velocity and gas hold-up with the rise velocity, time of formation and volume of individual bubbles. Theory In this context, a literature review has been done to study the dynamics of bubble. Merchuk5
2 188 INDIAN J. diem. TECHNOL., JULY 1998 observed experimentally that bubbles in an ALR' ascend in straight lines, except at very high gas velocities and two bubbles may ascend side-byside a long distance without touching each other, i.e. coalescence is negligible. It was observed6 experimentally that the bubble rise velocity remains nearly constant throughout the column. This is due to the fact that over a short column height, the change in bubble diameter is not appreciable to cause significant change in bubble rise velocity. This velocity has been calculated from the expression given by Mendleson7,... (4) Since Vb is nearly constant, it may be calculated based on the bubble detachment diameter at the disperser nozzle, do. This diameter could be obtained by the correlation developed by Gaddis and Vogelpohl8 (given in Appendix). Eqs (AI) (A4) show that the bubble detachment diameter under constant flow conditions in a single nozzle depends only on the disperser hole diameter, apart from the physical properties of the gas and the liquid. The present model assumes that the formation of a bubble at the disperser nozzle is not influenced by the presence of adjacent bubbles. The model further assumes that bubbles formed at the disperser retain their spherical shape throughout their residence in the riser. The formation time of bubbles at each nozzle is therefore the same and is given by, number of bubble-layers, given time is therefore, and the residence time of the bubbles in the column, Tn is N", in the column at any... (6)... (7) The average hold-up in the column, Bav, is then obtained as the total volume of the bubbles, Vr, entrapped in the column at any instant divided by the volume of the riser, Vr Bav = ~H where At any position in the column the equilibrium bubble radius, ri> commensurate to the surrounding liquid phase pressure in the column, is obtained by solving the Young-Laplace equation, ~ =p 00 _ j... (8).., (9)... (10) T=--=---- ;rrd~ Nb;rrd~ 6q' 6Qo... (5) Assuming isothermal expansion of the bubble Following the studies of Gaddis and Vogelpohl8 and model assumption made here, it is evident that Nb number of bubbles are detached simultaneously from the disperser at the bottom, after every interval of time T. Further since these bubbles rise in straight lines with a constant velocity, they will rise in discrete layers, separated by a constant vertical distance. This distance,, is the product of the time required for bubble formation, T, and the apparent velocity of the bubble, (vb+ Vir)' The The pressure at any height hi of the column, Eq. (10) is given by... (11) Poo, in... (12) neglecting liquid acceleration effects and frictional loss. 'I I'll I
3 CHAKRABORTY et at.: MODELLING OF AIR-Ll.FT REACTORS 189 Calculation of energy input Reversible isothermal work performed by a single bubble during expansion from a volume, Vb;' at the entrance of the riser to a volume, Vbb at height hi is given by Vbl Vb1 V. ~ = fl1dvb= f Pbo~dVb' Vb; Vbi Vb.. (13) Q 't) III :3 I 0 06 " Expt data +, Drc..ft-tube,,' diameter (m),./ / /' ,/ madel prectietiansrif " Calva /' P...-.t_ /',,,,,,,,, / /, where Vbl and Vb; are obtained by solving Eqs (10) (12). The total work of expansion, done by the l-th layer of bubbles during its ascend to a height, hb is then 0 00 o 123 Gas flow rate x 10~ mj5-' 4.. (14) According to the present model, the e.!: entire ~ ~ ~ height of the column is occupied by M layers of bubble which are separated by a distance ~. After every time interval, r, which is the time of formation of the bubbles, a new layer enters the column while all preceding layers move up by one layer's distance, ~, i.e., they occupy the positions of the previous layers and the topmost layer leaves the column. The work of expansion done by a single layer of bubbles during its rise from the bottom of the column to the top is Wo, where Wo is obtained from Eqs (13) and (14) for hl=h. On focussing attention on the layer which has just entered the column it is observed that during the residence time, Tr, of this layer within the riser, M-I, preceding layers leave the column while rising from their respective positions (i.e. the positions they occupied when the aforesaid layer enters the riser) and succeeding M-l layers enter the riser after r intervals of time. By the time this layer leaves the riser, each of the succeeding M-I layers attains different heights, hb from the bottom, where J Fig. I-Effect of gas flow rate on gas hold-up in the riser for different draft tube diameters madel + -- Draft d_(m) predictians tube af Pr..m.~,-,," c: '5 :J 'Wl 't) 0- ",." ti 0 Expt. ~u 0.00 '-~-,,,,.,,,, Gas flow rate xlo~ mj 5' Fig. 2-Effect of gas flow rate on liquid circulation velocity in the downcomer for different draft tube diameters. from different heights hi' Each of these preceding layers therefore performs a work of expansion which is Wo-~' The total work done during the residence time, Tn of a single layer is therefore the summation of the work done by the layer itself, layers preceding and layers succeeding it, 1= 1,2,...,(NI -1).. (15) The work of expansion done by each of these succeeding layers while attaining this height hi is ~. During the residence of the above mentioned layer in the reactor, N,-I layers that preceded the layer of interest, has left the riser while moving... (16)
4 190 INDIAN 1. CHEM. TECHNOL., JULY 1998 Therefore total work transferred by the bubbles to the liquid per unit time per unit area of the riser is given by,... (17) Eqs (1) and (4)-(17) are solved simultaneously using successive substitution method to obtain the liquid velocity in the downcomer and the gas holdup in the riser. Initial guess values of V1d and Cay are chosen. The liquid circulation velocity in the downcomer, Vir is related to Vld by the continuity equation,... (18) predicts the experimental data with greater accuracy over a large range of draft tube diameters and gas flow rates. A maximum error of ± 15% is obtained for gas hold-up while the corresponding value for liquid velocity is ± 20%. Conclusion The present study attempts to rationalise the method for estimation of energy input and gas hold-up in an internal loop airlift reactor. In contrast to previous models which assume gas phase to be continuous, the present work models this gas-dispersed liquid continuous system based on bubble dynamics. This model satisfactorily predicts liquid circulation and gas hold-up in these devices over a broad range of draft tube diameters and scale of operation. Nomenclature Based on this value Vir, the number of layers in the A =cross sectional area, m2 reactor and the residence time of the bubbles in the d =draft tube diameter, m reactor are estimated from Eqs (6) and (7) d. =equivalent bubble diameter, m db =disperser nozzle diameter, m respectively, alongwith Eqs (4) and (5). The radius do =bubble diameter at the disperser nozzle, m of the bubble in different layers, rl is calculated E =rate of energy input, Js 1 solving Eqs (10)-(12) and (15) simultaneously. g =acceleration due to gravity, ms-2 hi Following this, new value of =height in the column from the bottom of the draft tube, Cay is obtained from m Eqs (8) and (9). The estimate of E from Eq. (17), H =column height, m obtained after solution of Eq. (16), is substituted in K f =total friction coefficient Eq. (1) to calculate a new value of Vld' Iterations NI =number of layers in the column =number of bubbles in a single layer--same as the are continued till both the values of V1d and Cay Nb number of nozzles in the disperser converge. =pressure at the top of the column, Nm-2 =pressure inside the bubble, Nm-2 Results and Discussions =pressure inside the bubble at the time of formation, Nm-2 The model is used to compute the hydrodynamic characteristics of an internal airlift reactor for the experimental conditions of Jones I. The reactor has qo a volume of 0.06m', the draft tubes were 1.22m in q' length, located 0.1 m above the base. The value of rl the slip velocity, Vs used in Eq_ (1) was 0.25ms"\ as ; proposed by different workers The draft tube V ḃ diameters were 0.07m, 0.121m and 0.146m for Vbf which the corresponding values of total friction Vbl coefficient, Kf, as reported by Calv03 are 144.0, vbo 30.0 and 22.0 respectively. vt Figs 1 and 2 compare the results of the present model with the experimental data of Jones' alongwith the model predictions of Calv03, for gas hold-up in the riser and the liquid circulation velocity in the downcomer respectively. The model p 00 =pressure at any height hi of the column, Nm-2 =volumetric flow rate of air, m-3s-1 =volumetric air flow rate through each nozzle, m- 's-i =radius of the bubble at any height hi, m =bubble radius at the disperser nozzle, m =residence time of a bubble in the column, s =bubble volume, m3 =bubble volume at the top of the tower, m3 =bubble volume in the l-th layer at height hi' m3 =bubble volume at the disperser nozzle, m3 =total volume entrapped in the column at any instant, m3 Vb =velocity of bubble, ms-i vg =superficial velocity of gas in the riser, ms-i V1d =liquid circulation velocity in the downcomer, ms-i Vir =liquid circulation velocity in the riser, ms-i 'I' "" ""',' I "'!I''''!'I'"I "I'" """II'!I'I"'" 1"1'11'" I""r ""'I ''', "'I! 'I' "II,,," 'I "'1'1'11'1 I' "'!'
5 CHAKRABORTY et al.: MODELLING OF AIR-LIFT REACTORS 191 v, =slip velocity, ms-i Wb =work done by a single bubble during its residence time in the reactor, J WI =work done by a single layer of bubbles during its rise frpm bottom of the column to its present position, J Wn =work done by a single layer of bubbles during its residence time in the reactor, J Greek syrnbols Cav =average gas voidage.,., =dynamic viscosity of the continuous phase, Pa s P =density, kg m-j (j =interfacial tension, Nm-I r =formation time of a single bubble, s ~ =vertical distance between successive layer of bubble, m Subscripts c =continuous phase d =downcomer g I r =gas phase =Iayers =riser References I Jones A G, Chern Eng SCi, 40 (1985) Hsu Y C & Dudukovic M P, Chern Eng Sci, 35 (1980) Calvo E G, Chern Eng Sci, 44 (1989) Chisti M Y, Halard B, & Moo-Young N, Chern Eng Sci, 43 (1988) Merchuk J C, Chern Eng Sci, 41 (1986) II. 6 Jones J C, M Sc Thesis, UMIST, Manchester, Mendleson H D, AIChEJ, 13 (1967) Gaddis E S & Vogelpohl A, Chern Eng Sci, 41 (1986) Calvo E G & Leton P, Chern Eng Sci, 46 (1991) Hsu Y C, D Sc Thesis, Washington University, St. Louis, Mo Appendix The bubble detachment diameter, do, has been calculated using the following equation proposed by Gaddis and Vogelpohl8, d3 = s+ L T o -+do d; S, Land T are constants defined as L = Tr(Pc 811]Q' - Pg)g... (AI)... (A2)... (A3)... (A4)....(A5)