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A 3-D Analysis of Gas-Liquid Mixing, Mass Transfer and Bioreaction in a Stirred Bio-Reactor
0960–3085/00/$10.00+0.00 # Institution of Chemical Engineers Trans IChemE, Vol 79, Part C, December 2001
A 3-D ANALYSIS OF GAS-LIQUID MIXING, MASS TRANSFER AND BIOREACTION IN A STIRRED BIO-REACTOR H. HRISTOV1 , R. MANN 1 , V. LOSSEV2 , S. D. VLAEV3 and P. SEICHTER4 1
Department of Chemical Engineering, UMIST, Manchester, UK 2 Research Institute for Antibiotics, Razgrad, Bulgaria Institute of Chemical Engineering, Bulgarian Academy of Sciences, So a, Bulgaria 4 Techmix, Brno, Czech Republic
3
G
as-liquid stirred vessel two-phase mixing accompanied by bioreaction has been analysed using a 3-D networks-of-zones, in which non-axisymmetric phenomena can be included. The effect of the feed of liquid nutrient from a single dip-pipe can be incorporated so that previous 2-D limitations of axisymmetry are avoided1. The turbulent swirl ow created by the impeller uses clock-wise and anti-clock-wise swirl coef cients, which can be estimated using image-reconstruction 3-D visual modelling8. The simulations can provide detailed predictions of the local gas hold-up distribution, the local mass transfer area, the partial segregation of both the dissolved oxygen and the nutrient and the extent of oxygen depletion of bubbles. The overall gas hold-up and mass transfer area are obviously summations of the local values and the local and overall reaction rates can be predicted as well as the local and overall oxygen absorption uxes. Simulations are presented for a 3 ´ [2 ´ (10 ´ 10)) ´ 60] con guration of networks-of-zones for a 3 m3 triple-impeller industrial pilot-plant bioreactor. The theoretical predictions are demonstrated using colour-augmented 3D contour maps and solid-body isosurface images created by AVS graphics. Severe non-uniformity of gas hold-up distribution and consequently spatially uneven oxygen mass transfer create signi cant partial segregation of both oxygen and nutrient. The simulated bioreactor is predicted to be far from perfectly mixed, so that Tylosin producing microorganisms will experience large variations in dissolved oxygen and nutrient concentrations as they circulate around the stirred fermenter. Keywords: gas-liquid mixing; stirred bioreactor; oxygen mass transfer; networks-of-zones; partial segregation.
Recent analyses1–2, based upon a 2-D axisymmetric networks-of-zones model with O(103) zones, have con rmed the probable existence of very complex elds of partial segregation and associated bioreaction rates (based on illustrative second order elementary kinetics). This complexity suggest that simpler models with only a low degree of spatial sub-division3–5 will not be able to capture the complex interactions of gas-liquid mixing (in bubbly ow), oxygen masstransfer and bioreaction that together determine the time-wise evolution of the bio-product. Again, this may be especially important if the wide variation in internal composition can provoke undesired changes in metabolic pathways with possible consequent acceleration of by-product reactions. This paper presents a new development where the gasliquid two-phase mixing has been analysed by a fully 3-D networks-of-zones model. In this way it is possible to incorporate the feed of nutrient stream(s) by a single dippipe, which is expected to introduce asymmetry into any
INTRODUCTION Ef cient bioreactors are crucial to the successful manufacture of many high-value products, especially in the pharmaceutical industries. Stirred vessels are widely used, even though the internal complexities of their behaviour remain poorly de ned and inadequately understood. For large vessels, the simplifying assumptions of perfect mixing or plug ow will certainly not apply when the bioreaction rates are fast relative to the internal mixing rates generated by rotating impellers. Such macro-mixing effects will tend to cause a partial segregation of batch-fed components in semibatch operation. For typical fermenters both oxygen in the gas phase as well as liquid phase nutrient streams will be fed separately and (semi-) continuously as the manufacturing batch proceeds. Thus, gradients in dissolved oxygen in the liquid phase, as well as spatial variations in nutrient concentration, can be expected in practice. 232
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partially segregated concentration and bioreaction elds. Usually the oxygen is fed through an air sparge ring near the stirred vessel base, but in this case the aqueous nutrient stream is added nearer the upper surface. This again tends to intensify the partially segregated elds, with an oxygen-rich environment close to the base sparger and a nutrient-rich one around the upper dip-pipe. The use of a 3-D networksof-zones model of O(105) zones provides a much simpler and more tractable computational framework than can presently be achieved by CFD6–7. The new key element of the third-dimension swirling ow caused by impeller rotation is incorporated in clock-wise and anti-clock-wise swirl coef cients, which can be estimated using image reconstruction 3-D visual modelling8. The simulations of internal behaviour can predict, at any instant during the batch, the local gas hold-up distribution, the local mass-transfer area, the extent of oxygen absorbed by the bubble phase (throughout the liquid) as well as the 3D elds of oxygen and nutrient concentrations. Results are visualized for a 3m3 triple-impeller pilot-plant bioreactor in the form of colour-augmented 3-D see-through solid-body isosurfaces imaged by AVS graphics.
Figure 1. Pilot-plant bioreactor mapped onto networks-of-zones.
THEORY The complex gas-liquid two-phase ow can be analysed using a relatively advanced modelling technique. A 2-D networks-of-zones model has previously been used for numerical gas-liquid ow simulations to predict spatial detail of the local gas hold-up, dissolved oxygen and nutrient components and gas phase distribution. Nevertheless, this model imposed limitations because of the simplifying assumption of axi-symmetrical behaviour. In the earlier model, the individual zones are assembled into networks, which simultaneously combine the effect of convective ow and turbulent mixing. The 2-D networksof-zones model can be extended to 3-D by connecting axial=radial zone slices tangentially through the thirddimension swirling ow.
QL = KND3
(1)
There are n nested ow loops in the axial=radial networks and the overall ow is allocated uniformly amongst the m layers (number of zones along the vessel perimeter) in a k plane (see Figure 1). Hence each liquid loop ow rate is given by: qL = KND3 =(n ´ m)
(2)
where K is the impeller overall circulation constant, N is the impeller speed and D is impeller diameter. The basic scheme of the extension from 2-D to 3-D networks-of-zones is indicated in Figure 2.
Networks-of-Zones Assembly In line with the number of impellers (Ni) in the vessel, the uid phase is separated into Ni networks. Each network consists of an upper and lower half-network. The halfnetworks are divided into m ´ n ´ n number of zones, where m is the number of tangential zones along the vessel perimeter (see Figure 1) and n ´ n is a ‘square’ matrix of well mixed zones containing the axial=radial ows1. The zones are assumed to have equal volumes for each half-network and to be locally perfectly back-mixed. The impeller creates the axial=radial liquid circulation in the tank and also generates some tangential ow. The axial=radial ow is assembled into nested closed ow loops each with ow qL so as to follow the radial ow pattern typical of Rushton turbine impellers. In the tangential direction, the impeller creates a swirling liquid ow. The liquid rotates around the shaft of the tank due to this swirl ow. Both of the liquid ow rates (radial=axial and tangential) are linked to the overall impeller circulation ow QL9 given by: Trans IChemE, Vol 79, Part C, December 2001
Figure 2. Tangential assembly of a 3-D networks-of-zones con guration.
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Gas Phase Distribution The circulation ows in the vessel generated from the impeller are contained within the liquid phase. The gas phase is assumed to be composed of spherical bubbles, which do not signi cantly in uence the liquid ow patterns in each of the radial, axial and tangential directions. Bubbles are assumed to keep their shape and volume constant and to respond to the liquid ow patterns created by the impeller. Thus, the gas phase (bubble) convection and mixing behaviour is determined by three mechanisms:
· Firstly, bubbles are driven by the convection of the liquid in radial direction qL, and are also swirling with the liquid with a ow in the clock-wise direction bLqL, and in the anti-clock-wise direction with bRqL. The assumption here is that the impeller is rotating clock-wise (the subscript L (then R) refers to leftwards looking down from the front and above, R refers to rightwards). The clock-wise bL and the anti-clock-wise bR factors de ne the swirl (tangential) ows. Thus, the net swirl ow in the direction of the rotation is given by their difference bL 7 bR8. · Secondly, bubbles of spherical size with diameter db rise buoyantly from zone to zone with a bubble rise velocity ub (corresponding to their single bubble rise velocity, which assumes that bubbles act independently of one another). · Thirdly, bubbles exchange between laterally adjacent nested loop zones in each m layer by an equal and opposite exchange turbulent mixing mechanism, with an exchange ow bqL, which is a simple ratio of the local loop ow. In the cross section of the impeller region this exchange rate is higher due to the higher turbulence generated in this area by the impeller. This particular exchange ow is given by b I qL. The material balance on the gas bubbly ow is schematically shown in Figure 3. Thus, for the local gas hold-up eG (i; j; k) in a general zone i; j; k a gas balance on the bubbly ow is given by:
k= m j= 2* Ni* n i= n
eG =
eG (i; j; k ) i= 1 j= 1 k= 1 2* Ni* n* n* m
+ b[eG (i; j - 1; k) + eG (i; j + 1; k)]} - {ub A0 + [1 + 2b) + b R + bL ]}qL eG (i; j; k)
(5)
The liquid surface level changes due to the hold-up of bubbles, as the overall gas hold-up changes due to the liquid level changes. It is thus necessary to correct the overall gas hold-up for the liquid level changes. The overall gas hold-up can be calculated from the de nition: k= m j= n i= n i= 1 j= 1 k= 1
h eG (i; j; k) n1 +
+
eG =
ub A0 eG (i; j - 1; k) + qL {eG (i - 1; j; k)
+ qL [bL eG (i; j; k - 1) + bR eG (i; j; k + 1)] = 0
Figure 3. The general i; j; k zone showing 3-D ows and exchanges.
k= m j= 3n i= n i= 1 j= n k= 1 k= m j= 5n i= n i= 1 j= 3n k= 1
eG (i; j; k)
h2 - H1 2n
h eG (i; j; k) n1 +
k= m j= 6n i= n i= 1 j= 5n k= 1
eG (i; j; k)
h2 - H1 2n
nmH (6)
(3)
[bubbles in by buoyancy] + [bubbles in by radial convection and turbulence] - [bubbles out by convection;buoyancy and turbulence] + [bubbles in by swirling ( tangential) flow] = 0 (4) Equations (3) and (4) are modi ed for the zones along the ring sparger to include incoming gas, but the modi cation only applies for those zones where a hole occurs in the ring sparger (a 3-D effect). The values of the spatial=local threedimensional gas hold-up distribution can be obtained by solving the complete set of simultaneous linear algebraic equations for all the zones (2 ´ Ni ´ n ´ n ´ m). The overall fractional gas hold-up can then be calculated using by summation:
In this equation, h1, h2 and h3 de ne the impeller axial position and the value for the aerated liquid level H is found through an iteration procedure. The local speci c interfacial area for bubbles of size db is then calculated directly from the gas hold-up data on the assumption that all bubbles are approximately spherical: a(i; j; k ) =
6eG (i; j; k ) db
(7)
Balances on the Reagents Liquid phase oxygen balances To calculate the spatial 3-D distribution of reagents, additional material balances for both the bulk liquid and dispersed gas phases should be applied. The oxygen source gas bubbles, as well as the nutrient liquid solution, are fed continuously during the semi-batch operation. Using a pseudo-steady-state assumption the accumulation terms are Trans IChemE, Vol 79, Part C, December 2001
A 3-D ANALYSIS IN A STIRRED BIO-REACTOR negligible so that dt i » 0: The oxygen (A) is determined by three balanced mechanisms: mixing ow (convective and turbulent), mass transfer and bio-reaction. For each zone the oxygen balance is given by: dC
[input of oxygen by mass transfer] + [net input of oxygen by convective radial flow] + [net input of oxygen by turbulent flow] + [net input of oxygen by swirl (tangential) flow] - [rate of abstraction by bio-reaction] = 0
(8)
Equation (8) is modi ed for the gas feed zones (via holes of the ring sparger). Within each zone, the oxygen (A) mass transfer rate from the bubbles to the bulk liquid phase according to the lm theory is: NA = kL a(i; j; k)V (i; j; k)[CA* (i; j; k) - CAb (i; j; k)]
(9)
The interface concentration CA* of the absorbing substance A (oxygen in this case) is de ned by a Henry’s constant and varies with the local pressure, temperature and partial pressure (gas phase concentration) of the solute, thus: C (i; j; k )p( j) CA* (i; j; k ) = AG H*
(10)
In the bulk liquid phase oxygen (A) is consumed by the bio-reaction. For illustration the bio-reaction is considered as a second order elementary reaction dependent upon oxygen (A) and nutrient (B) concentrations with reaction rate given by: - rA (i; j; k) = k2 CAb (i; j; k)CB (i; j; k)V (i; j; k)
´ [1 - eG (i; j; k)]
(11)
where k2 is a second order bio-reaction rate constant. The local liquid volume of the zone V(i,j,k) is corrected for the local gas hold-up. By taking into account all mechanisms, the oxygen balance for each zone is: qL [CAb (i - 1; j; k) + bCAb (i; j - 1; k) + bCAb (i; j + 1; k) - (1 + 2b + bL + b R )CAb (i; j; k)] + kL a(i; j; k)V (i; j; k)
´ [CAG (i; j; k)p(j)=H * - CAb (i; j; k)]
+ qL [bL CAb (i; j; k - 1) + bR CAb (i; j; k + 1)] - rA (i; j; k) = 0
(12)
It should be noted that any form of bioreaction kinetics may be incorporated, so that this approach is in no way limited by illustration using elementary second order kinetics in oxygen and nutrient.
Ammonia Nutrient Balances In the equation above, as can be seen from equation (11), the reaction rate also depends on the local nutrient concentration. A proper balance for the nutrient should also be carried out. The balance on the nutrient (B) in each zone is similar to the balance for the oxygen except that there is no mass-transfer term. The nutrient is introduced directly to the liquid phase. The nutrient balance can be written as: Trans IChemE, Vol 79, Part C, December 2001
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[net input of nutrient by convective radial flow] + [net input of nutrient by turbulent flow] + [net input of nutrient by swirl (tangential) flow] - [rate of consumption of nutrient by bio-reaction] = 0
(13)
Equation (13) is modi ed for the particular feeding zone of the nutrient, which in the 3-D networks-of zones is a single zone. The feeding point was in the centre of the radius of the vessel of the top j layer (corresponding to the actual operational conditions). The reaction rate for consumption of the nutrient uses a stoichiometry factor ‘f ’, which determines how many mols of the nutrient component B react for consumption of 1 mol of the oxygen (A), so that for the nutrient: - rB (i; j; k) = f k2 CAb (i; j; k)CB (i; j; k)V (i; j; k)
´ [1 - eG (i; j; k)]
(14)
The nutrient (B) balance for each zone is then: qL [CB (i - 1; j; k) + bCB (i; j - 1; k) + bCB (i; j + 1; k) - (1 + 2b + bL + bR )CB (i; j; k)] + qL [bL CB (i; j; k - 1) + bR CB (i; j; k + 1)] - rB (i; j; k) = 0 (15) The pressure in the vessel is a very important operational factor, since industrial bioreactors usually work under pressure for reasonsof sterility. In theverticalaxialdirection,the pressure changes due to the liquid depth are taken into account using: p( j) = p(1) -
p(1) - p(2Nin) ( j - 1) 2Ni n
(16)
Gas-Phase Oxygen Balance The gas bubbles are presumed to keep their shape and volume constant during the mixing process and in each zone they are locally perfectly mixed. The material balance for the oxygen in the gas phase is based on the ve mechanisms governing transport of oxygen: (i) buoyant rise = ub A0 eG (i; j - 1; k)CAG (i; j - 1; k) (ii) radial=axial convection = qL eG (i; j; k)CAG (i; j; k) (iii) tangential (swirl) = bL qL eG (i; j; k)CAG (i; j; k) + bR qL eG (i; j; k)CAG (i; j; k) (iv) turbulent exchange = qL beG (i; j; k)CAG (i; j; k) (v) mass transfer = kL a(i;j;k)V (i;j;k)[CAG (i;j;k)p( j)=H * - CAb (i;j;k)] The oxygen spatial distribution obviously tends to follow that of the gas hold-up, since the voidage determines the bubble surface area. The balance for each zone can be written: qL {eG (i - 1; j; k)CAG (i - 1; j; k) + b[eG (i; j - 1; k) ´ CAG (i; j + 1; k) + eGk (i; j + 1; k)CAG (i; j + 1; k)] + bL CAG (i; j; k - 1) + bR CAG (i; j; k + 1) - (1 + 2b + bL + bR )eG (i; j; k)CAG (i; j; k)} - ub A0 eG (i; j; k)CAG (i; j; k) - kL a(i; j; k)V (i; j; k)
´ [CAG (i; j; k)p( j)=H * - CAb (i; j; k)]
+ ub A0 eG (i; j - 1; k)CAG (i; j - 1; k) = 0
(17)
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The whole set of balance equations amount to [3 ´ 36,000] coupled equations, which should be solved simultaneously to obtain the pseudo-stationary spatial 3-D reagent distributions. The number of simultaneous equations is given by 2 ´ Ni ´ n ´ n ´ m ´ nf, where nf is the number of sets of equations for each reagent for each phase. In this particular case, with nf = 3, there is one set of equations for the nutrient in the bulk liquid, one for the oxygen also in the bulk liquid and one for the oxygen in the gas phase contained in the dispersed gas bubbles.
of kL = 2.67 10- 3 m s- 1 and k2 = 0.208 m3 kmol- 1 s- 1 provide the combination that exactly matches the experimental values of exit air oxygen concentration (hence the overall rate of oxygen mass transfer) and the local level of dissolved oxygen measured close to the lowest impeller. The overall kLa value of 0.036 s- 1 is typical of stirred fermenters14. NETWORKS-OF-ZONES PREDICTIONS Spatial Distribution of Gas Hold-Up
NETWORKS-OF-ZONES MODELLING Industrial Bioreactor Operational Details The pilot-plant 3m3 pressured baf ed bioreactor (with 2.5 bar pressure above the upper liquid level) was agitated by three 8-blade Rushton turbines attached to a central shaft. The impeller speed was 3 rps. Air was fed axisymmetrically at a typical rate for a fermentation of 1 vvm through a ring sparger mounted just below the lowest impeller. Ammonium nitrate solution (10% wt) was fed at 2.33 10- 4 dm3 s- 1 onto the free surface between the 60th and 144th hour of production as a source of nitrogen for the micro-organisms. Figure 1(a) shows the layout and the main dimensions of the fermenter, which were: initial liquid level H = 2.26 m, diameter T = 1.3 m, impeller diameter D = 0.52 m, baf es width 0.1 m, clearance between the lowest impeller and the base 0.47 m, between the lower impeller pair 0.57 m and between the top impellers 0.56 m. On-line measurements of a number of key parameters were taken to form the basis for a comprehensive interpretation of the mass transfer within the fermenter during one production cycle. The exit gas phase oxygen concentration was measured in the outgoing gas ow throughout a 7-day-batch, which gives the time varying overall rate of oxygen transfer. Dissolved oxygen was monitored continuously close to and level with the lowest impeller (see Figure 1). Changes in viscosity and biomass growth and decay were also recorded throughout the batch. In addition, at the end of the production cycle, the liquid level was observed to provide an estimate of the overall gas phase voidage. Networks-of-Zones Parameters All the predictions of the model correspond to a ‘snapshot’ of the conditions in the fermenter after exactly 60 hours on stream. The networks-of-zones model in general is based upon an overall circulation ow constant K for each impeller. Measurements of mixing times of viscous nonNewtonian uids at two semi-batch scales (0.2 m and 0.4 m) have established the validity of the Norwood and Metzner9, correlation indicating that K » 0.24 for the bio uid viscosity of 100cp after 60 hours on stream. This value has been used for the ows generated by each of the impellers. The gas phase bubbles were assumed to have typical diameter of db = 5 mm 10 with a rise velocity of ub = 0.185 m s- 1 11. The value de ning the turbulent exchange was b = 0.2, which has been estimated from visualizations of tracer mixing12, as well as the values of the swirling ow of bL = 1.0 and bR = 0.1 8. It is also necessary to identify the bioreaction stoichiometry factor f. For a representative value of f = 0.003 and for H* = 0.82 atm m3 kmol- 1 13, values
The predicted gas hold-up distribution in 3-D is presented in Figure 4 in the form of solid-body isosurfaces using AVS graphics (AVS V5.0). In particular, the complex space-wise geometry of gas voidage, which comprises 36,000 values of gas voidage in each model voxel, demands some visualization formatting to convey the intrinsic complexity of the bubbly ow gas-liquid mixing. This is achieved in Figure 4 by using solid-body isosurfaces for a sequence of ‘cut-off’ voidage values. However, Figure 4(a) actually shows the zone gas hold-up values for all the outer peripheral zones visible in a forward isometric view. The colour key shows blue to red from the minimum value close to zero to a maximum value of some 91%. In Figure 4(a) the peripheral hold-up values are relatively low, especially below the lowest of the three impellers and all the peripheral zones show blue and green, although higher at the upper uid surface. The overall predicted voidage eG is 17%, so that the rst dark green isosurface in Figure 4(b), which corresponds to 17% gas hold-up, encompasses approximately half of the bioreactor uid volume. Inside this double-hour-glass shaped contour, the voidage is higher than 17%. Outside it is less than this value. The solid-body contour in Figure 4(b) is perfectly axisymmetrical in consequence of gas injection through an axisymmetrical sparge ring. The 30% isosurface in Figure 4(c) now shows a single hour-glass shape with an annular ring detached from it around the topmost impeller. Figure 4(c) indicates that higher voidages are associated with the impeller regions where recirculating ows join together. As the cut-off voidage is increased, Figure 4(d) and (e) again show that the highest voidages lie around the impellers. In fact, these highest values arise just below the impeller discs, because they impede the upward ow of bubbles by buoyant rise. The highest values above 70% shown in Figure 4(f ) occur just above each of the sparge ring holes. This effect is then shown in greater detail by enlargement in Figure 5. The effect of localized gas ow through the 42 individual sparger holes can be clearly seen, with a maximum value close to 90%. Distribution and Mass-Transfer of Oxygen Oxygen is the gas phase reagent being fed to the fermenter in an air stream through the sparge ring. It undergoes mass transfer from the air bubbles, dissolving into the bioreactor uid where it is subsequently consumed by aerobic bioreaction. Therefore, both the bubble gas phase oxygen composition and the dissolved oxygen can be spatially distributed, as consequently can the associated local rates of oxygen mass transfer (in each voxel). These effects are captured in Figures 6, 7 and 8. Trans IChemE, Vol 79, Part C, December 2001
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Figure 4. Local gas hold-up distribution depicted as solid-body isosurfaces.
Figure 5. Ring-sparger 3D gas hold-up effect.
The format of the visualization follows that already used to illustrate gas voidage. Thus Figure 6 shows gas phase oxygen distribution. It is predicted to vary over the range from 11.37 to 20.89% (the gas feed value in air). Figure 6(a) shows the wall and upper surface values. These are higher adjacent to the lowest impeller (showing orange), somewhat lower below the lowest impeller and around the middle impeller (yellow) and green at the upper surface which corresponds to the measured exit gas composition of 17.35%. The sequence of Figures 6(b)–(f ) then details the internal shape of the bubble gas phase composition eld. The composition is close to the feed air in the region between the sparge ring and the lowest impeller (into which the feed air rises) as shown in Figure 6(b). At a cut-off of 19.25%, Figure 6(c) shows that bubbles spread outwards and upwards to the middle impeller. Figure 6 Trans IChemE, Vol 79, Part C, December 2001
indicates that some annular shaped pockets of gas corresponding to the exit gas composition are con ned to regions just above each of the impellers. Within these annular regions lie zones containing gas bubbles exhibiting low values from the contour of 15.35% down to the very lowest value of 11.37%. These annular pockets with the lowest composition bubbles lie, in each case, above the impeller discs. Values are lowest here as gas is recirculating downwards to the impellers before mixing and recirculating with the gas rising from beneath each impeller. The downward ow (against their buoyant rise) allows a longer time period for oxygen to be stripped from these bubbles, with a consequently locally lower residual level of oxygen. Figure 7 shows that the levels of dissolved oxygen cover a much wider variation than the gas bubble composition. Thus, the peripheral zones (with low voidage) show the lowest level
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Figure 6. Oxygen in the gas phase.
Figure 7. Dissolved oxygen in bio uid.
of oxygen saturation from 8%. The cut-off isosurfaces show a clear monotone geometry, with the highest values being con ned to the lowest impeller with a maximum of 48.2%. It is evident from this set of gures that the bioreaction speed required to balance the mass transfer, gives rise to a highest saturation which is even so less than 50%. In fact from Figure 7(d), some 75% of the uid has an oxygen level less than 33% of saturation. Figure 7 con rms that the bacteria operate throughout in an oxygen lean environment. The dual interaction of the gas phase and the liquid phase composition creates a relatively complex 3-D eld of local oxygen absorption uxes. This can be seen in Figure 8. Then Figure 8(c) shows three intense zones of oxygen mass transfer,
whereas Figure 8(d) shows two zones separated at the top and bottom of the reactor. However, as Figure 8(e) and (f) indicate, the most intensive oxygen uxes occur just above the sparge ring. Note also from Figure 8(d), the asymmetric feed of nutrient produces a small departure from axisymmetry in the absorption ux annulus just above the topmost impeller (just visible on the right hand side). Distribution of Ammonia Nutrient Concentration The corresponding ‘geometry’ of the ammonia nutrient concentration eld is detailed in Figures 9 and 10. The networks-of-zones model predicts that the ammonia nutrient Trans IChemE, Vol 79, Part C, December 2001
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Figure 8. Local absorption uxes.
eld will show a maximum value of 1.1 mmol dm- 3 and a minimum value of 0.403 mmol dm- 3. This is for a nutrient feed rate of 2.33 10- 4 dm3 s- 1 at a feed concentration of 1070 mmol dm- 3. The dip pipe feeds into zone (6,60,1). Figures 9(a)–(f ) indicate that the spatial concentration variation of the nutrient (for second order bioreaction kinetics) is much less pronounced than for dissolved oxygen. Thus, Figures 9(b)–(e) show that for around 90% of the bioreactor uid, the concentration of ammonia varies
only from 0.403 to 0.447 mmol dm- 3. However, there is still a distinctive meso-mixing plume which ows and swirls from the dip-pipe position, as can be seen in Figure 9(f ). This is shown enlarged in Figure 10. The dip-pipe close to the surface experiences an inward owing uid which draws nutrient towards the impeller shaft (Figure 10(c)). This plume simultaneously experiences the net clock-wise swirl created by the impeller rotation, so it spreads out tangentially. On reaching the impeller shaft (Figure 10(b)), nutrient
Figure 9. Ammonia nutrient concentration eld.
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Figure 10. Enlarged view of the nutrient plume near the topmost impeller.
is mixing and being drawn downwards towards the topmost impeller, whilst continuing to be spread tangentially by the swirling ow. The complex shape of the meso-mixing plume as shown very clearly in Figure 10 using these different isosurface cut-offs.
Local Bioreaction Rates in 3-D Figure 11 shows how the local bioreaction rates for assumed second order bioreaction with respect to dissolved oxygen and nutrient are con gured in space. Although the highest reaction rate is around the lower impeller (as Figures 11(e)–(f ) show), at the intermediate rates of 0.037 in Figure 11(d) there are localized maxima adjacent to the lowest and middle impellers. Figure 11(c) then shows that there is a slight but clearly visible ‘reaction rate’ plume which twists and swirls away from the nutrient feed point. This re ects the higher bioreaction rates associated with higher ammonia concentration. Figure 11(a) also shows that the low dissolved O2 values around the walls and upper uid surface are re ected in relatively low reaction rates, assuming there is negligible surface aeration. However, Figure 11 con rms
that there is a ´ 55-fold variation in potential second order reaction speeds, indicating that the micro-organisms will experience large variations in composition as they circulate round the vessel. Whether such segregated environments can have an adverse effect upon the aerobic vitality and metabolism of the Streptomyces fradiae needs further research. However, it is clear that a networks-of-zones model with O(105) zones in full 3-D offers a computationally tractable framework for quantifying these complex interacting partially segregated concentration elds. CONCLUSIONS
· a further development of networks-of-zones modelling into full 3-D has been successfully applied to a 3 m3 triple-impeller pilot-plant bioreactor, which uses Streptomyces fradiae to manufacture tylosin; · a 3 ´ (2 ´ (10 ´ 10)) ´ 60 con guration comprised of O(105) zones is able to incorporate important thirddimensional effects, including single point dip-pipe feeding of nutrient as well as the effect of hole-spacing on the air feed sparge ring;
Figure 11. Local bioreaction rates.
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· visualization derived values for the turbulent exchange and swirl- ow coef cients have been applied to exactly match the overall gas hold-up, the overall oxygen absorption rate and a local dissolved oxygen measurement (close to the lowest impeller) in the pilot bioreactor; · applying the key parameters of kL = 2.67 10- 3 m s- 1 and a second order bioreaction rate constant k2 = 0.208 dm3 mmol- 1 s- 1, it is possible to predict the concentration elds in all 36,000 voxels for bubble phase oxygen, dissolved oxygen and liquid nutrient concentrations. Results can be compactly presented by 3-D solidbody isosurfaces imaged by AVS graphics; · a high-degree of partial segregation of oxygen and nutrient is predicted, so that circulating micro-organisms will experience large variations in their uid environment. Dissolved oxygen varied spatially from 8% to 48% saturation, hence, the simulated batch operates with a fairly oxygen lean liquid phase. The nutrient stream forms a complex shaped swirling plume from the dip-pipe addition point, with ammonium ion concentration varying from 0.403 to 1.100 mmol dm- 3. The consequent complex reaction rate eld can also be readily visualized by AVS graphics. NOMENCLATURE A0 a CB CAG CAb CA* D db f H H* h1,2,3 i j K k kL k2 m N NA Ni n p QG QL qL - rA T ub V(i,j,k,)
zone area, m2 speci c interfacial area, m- 1 concentration of nutrient B in the bulk liquid phase, kmol m- 3 gas phase concentration of A, kmol m- 3 concentration of A in the bulk liquid phase, kmol m- 3 interface concentration of A, kmol m- 3 impeller diameter, m bubble diameter, m stoichiometric factor (mols of B consumed per mol of A respired) liquid level in the vessel, m Henry constant, atm m3 kmol- 1 impellers positions on the shaft, m radial zone position axial zone position overall impeller ow constant tangential zone position mass transfer coe cient, m s- 1 second order bioreaction rate constant, m3 kmol- 1 s- 1 number of zones along the vessel perimeter impeller speed, rps mass transfer of A, kmol h- 1 m- 3 number of impellers number of zones in axial=radial plane (half network) pressure, bar feed rate of the feed gas, m3 s- 1 overall liquid circulation by the impeller, m3 s- 1 liquid ow rate in nested (axial=radial) ow loops, m3 s- 1 bioreaction rate (based on oxygen), kmol s- 1 vessel diameter, m bubble rise velocity, m s- 1 zone volume, m3
Greek letters eG(i,j,k) fractional gas hold-up in zone i,j,k eG fractional overall gas hold-up b lateral turbulent exchange coef cient bI turbulent exchange coef cient in the impeller plane
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bL bR
clock-wise swirl coef cient anti-clock-wise swirl coef cient
Indexes A B b G
oxygen nutrient bubble gas
REFERENCES 1. Vlaev, D., Mann, R., Lossev, V., Vlaev, S. D., Zahradnik, J. and Seichter, P., 2000, Macro-mixing and Streptomyces fradiae: Modelling oxygen and nutrient segregation in an industrial bioreactor, Trans IChemE, Part A, Chem Eng Res Des, 78(A3): 354–362. 2. Zahradnik, J., Mann, R., Fialova, M., Vlaev, S. D., Lossev, V. and Seichter, P., 2001, A networks-of-zones analysis of mixing and mass transfer in three industrial bioreactors, Chem Eng Sci, 56: 485–492. 3. Cui, Y. Q., Van der Lans, R. G. J. M., Noorman, H. J. and Luyben, K. C. A. M., 1996, Compartment mixing model for stirred reactors with multiple impellers, Trans IChemE, Part A, Chem Eng Res Des, 74(A2): 261–271. 4. Alves, S. S., Vasconcelos, J. M. T. and Barata, J. M., 1997, Alternative compartment models of mixing in tall tanks agitated by multi-Rushton turbines, Trans IChemE, Part A, Chem Eng Res Des, 75(A2): 334–338. 5. Vasconcelos, J. M. T., Bujalski, W., Nienow, A. W. and Alves, S. S., 1997, Spatial characteristics of oxygen transfer in dual impeller agitated aerated tank, 4th Int Conf on Bioreactor and Bioprocess Fluid Dynamics (Ed. A. W. Nienow), BHR Conf Series 25: 361–377. 6. Bakker, A. and van den Akker, H. E. A., 1994, A computational model for the gas-liquid ow in stirred reactors, Trans IChemE, Part A, Chem Eng Res Des, 72(A5): 594–604. 7. Lane, G., Schwarz, M. P. and Evans, G. H., 2000, Modelling of the interactions between gas and liquid in stirred vessel, Proc 10th European Conf. On Mixing (Delft), Van den Akker H. E. A. and Derksen, J. J. (eds.) (Elsevier, Amsterdam). 8. Rahimi, M., Senior P. R. and Mann, R., 2000, Visual 3-D modelling of stirred vessel mixing for an inclined-blade impeller, Trans IChemE, Part A, Chem Eng Res Des, 78(A3): 348–353. 9. Metzner, A. B. and Norwood, K. W., 1960, Flow patterns and mixing rates in agitated vessels, AIChE J, 6: 432–437. 10. Barigou, M. and Greaves, M., 1992, Bubble size distribution in a mechanically agitated gas-liquid contactor, Chem Eng Sci, 47: 2009– 2025. 11. Chhabra, R. P., 1993, Bubbles Drops and Particles in Non-Newtonian Fluids (CRC Press, Boca Raton, USA). 12. Mann, R., Pillai, S. K., El-Hamouz, A. M., Ying, P., Togatorop, A. and Edwards, R. B., 1995, Computational uid mixing for stirred vessels: Progress from seeing to believing, Chem Eng J, 59: 39–50. 13. Atkinson, B. and Mavituna, F., 1983, Biochemical Engineering and Biotechnology Handbook (Macmillan, UK), pp 734. 14. Kawase, Y., Halard, B. and Moo-Young, M., 1992, Liquid-phase mass transfer coef cients in bioreactors, Biotech Bioeng, 39: 1133–1140.
ACKNOWLEDGEMENT The authors gratefully acknowledge nancial support from the Commission of the European Communities under INCO-COPERNICUS contract No. ICIS-CT98-0502.
ADDRESS Correspondence concerning this paper should be addressed to Professor R. Mann, Department of Chemical Engineering, UMIST, PO Box 88, Manchester, M60 1QD, UK. E-mail: [email protected] The manuscript was received 10 October 2000 and accepted for publication after revision 28 August 2001.
A 3-D ANALYSIS OF GAS-LIQUID MIXING, MASS TRANSFER AND BIOREACTION IN A STIRRED BIO-REACTOR H. HRISTOV1 , R. MANN 1 , V. LOSSEV2 , S. D. VLAEV3 and P. SEICHTER4 1
Department of Chemical Engineering, UMIST, Manchester, UK 2 Research Institute for Antibiotics, Razgrad, Bulgaria Institute of Chemical Engineering, Bulgarian Academy of Sciences, So a, Bulgaria 4 Techmix, Brno, Czech Republic
3
G
as-liquid stirred vessel two-phase mixing accompanied by bioreaction has been analysed using a 3-D networks-of-zones, in which non-axisymmetric phenomena can be included. The effect of the feed of liquid nutrient from a single dip-pipe can be incorporated so that previous 2-D limitations of axisymmetry are avoided1. The turbulent swirl ow created by the impeller uses clock-wise and anti-clock-wise swirl coef cients, which can be estimated using image-reconstruction 3-D visual modelling8. The simulations can provide detailed predictions of the local gas hold-up distribution, the local mass transfer area, the partial segregation of both the dissolved oxygen and the nutrient and the extent of oxygen depletion of bubbles. The overall gas hold-up and mass transfer area are obviously summations of the local values and the local and overall reaction rates can be predicted as well as the local and overall oxygen absorption uxes. Simulations are presented for a 3 ´ [2 ´ (10 ´ 10)) ´ 60] con guration of networks-of-zones for a 3 m3 triple-impeller industrial pilot-plant bioreactor. The theoretical predictions are demonstrated using colour-augmented 3D contour maps and solid-body isosurface images created by AVS graphics. Severe non-uniformity of gas hold-up distribution and consequently spatially uneven oxygen mass transfer create signi cant partial segregation of both oxygen and nutrient. The simulated bioreactor is predicted to be far from perfectly mixed, so that Tylosin producing microorganisms will experience large variations in dissolved oxygen and nutrient concentrations as they circulate around the stirred fermenter. Keywords: gas-liquid mixing; stirred bioreactor; oxygen mass transfer; networks-of-zones; partial segregation.
Recent analyses1–2, based upon a 2-D axisymmetric networks-of-zones model with O(103) zones, have con rmed the probable existence of very complex elds of partial segregation and associated bioreaction rates (based on illustrative second order elementary kinetics). This complexity suggest that simpler models with only a low degree of spatial sub-division3–5 will not be able to capture the complex interactions of gas-liquid mixing (in bubbly ow), oxygen masstransfer and bioreaction that together determine the time-wise evolution of the bio-product. Again, this may be especially important if the wide variation in internal composition can provoke undesired changes in metabolic pathways with possible consequent acceleration of by-product reactions. This paper presents a new development where the gasliquid two-phase mixing has been analysed by a fully 3-D networks-of-zones model. In this way it is possible to incorporate the feed of nutrient stream(s) by a single dippipe, which is expected to introduce asymmetry into any
INTRODUCTION Ef cient bioreactors are crucial to the successful manufacture of many high-value products, especially in the pharmaceutical industries. Stirred vessels are widely used, even though the internal complexities of their behaviour remain poorly de ned and inadequately understood. For large vessels, the simplifying assumptions of perfect mixing or plug ow will certainly not apply when the bioreaction rates are fast relative to the internal mixing rates generated by rotating impellers. Such macro-mixing effects will tend to cause a partial segregation of batch-fed components in semibatch operation. For typical fermenters both oxygen in the gas phase as well as liquid phase nutrient streams will be fed separately and (semi-) continuously as the manufacturing batch proceeds. Thus, gradients in dissolved oxygen in the liquid phase, as well as spatial variations in nutrient concentration, can be expected in practice. 232
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partially segregated concentration and bioreaction elds. Usually the oxygen is fed through an air sparge ring near the stirred vessel base, but in this case the aqueous nutrient stream is added nearer the upper surface. This again tends to intensify the partially segregated elds, with an oxygen-rich environment close to the base sparger and a nutrient-rich one around the upper dip-pipe. The use of a 3-D networksof-zones model of O(105) zones provides a much simpler and more tractable computational framework than can presently be achieved by CFD6–7. The new key element of the third-dimension swirling ow caused by impeller rotation is incorporated in clock-wise and anti-clock-wise swirl coef cients, which can be estimated using image reconstruction 3-D visual modelling8. The simulations of internal behaviour can predict, at any instant during the batch, the local gas hold-up distribution, the local mass-transfer area, the extent of oxygen absorbed by the bubble phase (throughout the liquid) as well as the 3D elds of oxygen and nutrient concentrations. Results are visualized for a 3m3 triple-impeller pilot-plant bioreactor in the form of colour-augmented 3-D see-through solid-body isosurfaces imaged by AVS graphics.
Figure 1. Pilot-plant bioreactor mapped onto networks-of-zones.
THEORY The complex gas-liquid two-phase ow can be analysed using a relatively advanced modelling technique. A 2-D networks-of-zones model has previously been used for numerical gas-liquid ow simulations to predict spatial detail of the local gas hold-up, dissolved oxygen and nutrient components and gas phase distribution. Nevertheless, this model imposed limitations because of the simplifying assumption of axi-symmetrical behaviour. In the earlier model, the individual zones are assembled into networks, which simultaneously combine the effect of convective ow and turbulent mixing. The 2-D networksof-zones model can be extended to 3-D by connecting axial=radial zone slices tangentially through the thirddimension swirling ow.
QL = KND3
(1)
There are n nested ow loops in the axial=radial networks and the overall ow is allocated uniformly amongst the m layers (number of zones along the vessel perimeter) in a k plane (see Figure 1). Hence each liquid loop ow rate is given by: qL = KND3 =(n ´ m)
(2)
where K is the impeller overall circulation constant, N is the impeller speed and D is impeller diameter. The basic scheme of the extension from 2-D to 3-D networks-of-zones is indicated in Figure 2.
Networks-of-Zones Assembly In line with the number of impellers (Ni) in the vessel, the uid phase is separated into Ni networks. Each network consists of an upper and lower half-network. The halfnetworks are divided into m ´ n ´ n number of zones, where m is the number of tangential zones along the vessel perimeter (see Figure 1) and n ´ n is a ‘square’ matrix of well mixed zones containing the axial=radial ows1. The zones are assumed to have equal volumes for each half-network and to be locally perfectly back-mixed. The impeller creates the axial=radial liquid circulation in the tank and also generates some tangential ow. The axial=radial ow is assembled into nested closed ow loops each with ow qL so as to follow the radial ow pattern typical of Rushton turbine impellers. In the tangential direction, the impeller creates a swirling liquid ow. The liquid rotates around the shaft of the tank due to this swirl ow. Both of the liquid ow rates (radial=axial and tangential) are linked to the overall impeller circulation ow QL9 given by: Trans IChemE, Vol 79, Part C, December 2001
Figure 2. Tangential assembly of a 3-D networks-of-zones con guration.
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Gas Phase Distribution The circulation ows in the vessel generated from the impeller are contained within the liquid phase. The gas phase is assumed to be composed of spherical bubbles, which do not signi cantly in uence the liquid ow patterns in each of the radial, axial and tangential directions. Bubbles are assumed to keep their shape and volume constant and to respond to the liquid ow patterns created by the impeller. Thus, the gas phase (bubble) convection and mixing behaviour is determined by three mechanisms:
· Firstly, bubbles are driven by the convection of the liquid in radial direction qL, and are also swirling with the liquid with a ow in the clock-wise direction bLqL, and in the anti-clock-wise direction with bRqL. The assumption here is that the impeller is rotating clock-wise (the subscript L (then R) refers to leftwards looking down from the front and above, R refers to rightwards). The clock-wise bL and the anti-clock-wise bR factors de ne the swirl (tangential) ows. Thus, the net swirl ow in the direction of the rotation is given by their difference bL 7 bR8. · Secondly, bubbles of spherical size with diameter db rise buoyantly from zone to zone with a bubble rise velocity ub (corresponding to their single bubble rise velocity, which assumes that bubbles act independently of one another). · Thirdly, bubbles exchange between laterally adjacent nested loop zones in each m layer by an equal and opposite exchange turbulent mixing mechanism, with an exchange ow bqL, which is a simple ratio of the local loop ow. In the cross section of the impeller region this exchange rate is higher due to the higher turbulence generated in this area by the impeller. This particular exchange ow is given by b I qL. The material balance on the gas bubbly ow is schematically shown in Figure 3. Thus, for the local gas hold-up eG (i; j; k) in a general zone i; j; k a gas balance on the bubbly ow is given by:
k= m j= 2* Ni* n i= n
eG =
eG (i; j; k ) i= 1 j= 1 k= 1 2* Ni* n* n* m
+ b[eG (i; j - 1; k) + eG (i; j + 1; k)]} - {ub A0 + [1 + 2b) + b R + bL ]}qL eG (i; j; k)
(5)
The liquid surface level changes due to the hold-up of bubbles, as the overall gas hold-up changes due to the liquid level changes. It is thus necessary to correct the overall gas hold-up for the liquid level changes. The overall gas hold-up can be calculated from the de nition: k= m j= n i= n i= 1 j= 1 k= 1
h eG (i; j; k) n1 +
+
eG =
ub A0 eG (i; j - 1; k) + qL {eG (i - 1; j; k)
+ qL [bL eG (i; j; k - 1) + bR eG (i; j; k + 1)] = 0
Figure 3. The general i; j; k zone showing 3-D ows and exchanges.
k= m j= 3n i= n i= 1 j= n k= 1 k= m j= 5n i= n i= 1 j= 3n k= 1
eG (i; j; k)
h2 - H1 2n
h eG (i; j; k) n1 +
k= m j= 6n i= n i= 1 j= 5n k= 1
eG (i; j; k)
h2 - H1 2n
nmH (6)
(3)
[bubbles in by buoyancy] + [bubbles in by radial convection and turbulence] - [bubbles out by convection;buoyancy and turbulence] + [bubbles in by swirling ( tangential) flow] = 0 (4) Equations (3) and (4) are modi ed for the zones along the ring sparger to include incoming gas, but the modi cation only applies for those zones where a hole occurs in the ring sparger (a 3-D effect). The values of the spatial=local threedimensional gas hold-up distribution can be obtained by solving the complete set of simultaneous linear algebraic equations for all the zones (2 ´ Ni ´ n ´ n ´ m). The overall fractional gas hold-up can then be calculated using by summation:
In this equation, h1, h2 and h3 de ne the impeller axial position and the value for the aerated liquid level H is found through an iteration procedure. The local speci c interfacial area for bubbles of size db is then calculated directly from the gas hold-up data on the assumption that all bubbles are approximately spherical: a(i; j; k ) =
6eG (i; j; k ) db
(7)
Balances on the Reagents Liquid phase oxygen balances To calculate the spatial 3-D distribution of reagents, additional material balances for both the bulk liquid and dispersed gas phases should be applied. The oxygen source gas bubbles, as well as the nutrient liquid solution, are fed continuously during the semi-batch operation. Using a pseudo-steady-state assumption the accumulation terms are Trans IChemE, Vol 79, Part C, December 2001
A 3-D ANALYSIS IN A STIRRED BIO-REACTOR negligible so that dt i » 0: The oxygen (A) is determined by three balanced mechanisms: mixing ow (convective and turbulent), mass transfer and bio-reaction. For each zone the oxygen balance is given by: dC
[input of oxygen by mass transfer] + [net input of oxygen by convective radial flow] + [net input of oxygen by turbulent flow] + [net input of oxygen by swirl (tangential) flow] - [rate of abstraction by bio-reaction] = 0
(8)
Equation (8) is modi ed for the gas feed zones (via holes of the ring sparger). Within each zone, the oxygen (A) mass transfer rate from the bubbles to the bulk liquid phase according to the lm theory is: NA = kL a(i; j; k)V (i; j; k)[CA* (i; j; k) - CAb (i; j; k)]
(9)
The interface concentration CA* of the absorbing substance A (oxygen in this case) is de ned by a Henry’s constant and varies with the local pressure, temperature and partial pressure (gas phase concentration) of the solute, thus: C (i; j; k )p( j) CA* (i; j; k ) = AG H*
(10)
In the bulk liquid phase oxygen (A) is consumed by the bio-reaction. For illustration the bio-reaction is considered as a second order elementary reaction dependent upon oxygen (A) and nutrient (B) concentrations with reaction rate given by: - rA (i; j; k) = k2 CAb (i; j; k)CB (i; j; k)V (i; j; k)
´ [1 - eG (i; j; k)]
(11)
where k2 is a second order bio-reaction rate constant. The local liquid volume of the zone V(i,j,k) is corrected for the local gas hold-up. By taking into account all mechanisms, the oxygen balance for each zone is: qL [CAb (i - 1; j; k) + bCAb (i; j - 1; k) + bCAb (i; j + 1; k) - (1 + 2b + bL + b R )CAb (i; j; k)] + kL a(i; j; k)V (i; j; k)
´ [CAG (i; j; k)p(j)=H * - CAb (i; j; k)]
+ qL [bL CAb (i; j; k - 1) + bR CAb (i; j; k + 1)] - rA (i; j; k) = 0
(12)
It should be noted that any form of bioreaction kinetics may be incorporated, so that this approach is in no way limited by illustration using elementary second order kinetics in oxygen and nutrient.
Ammonia Nutrient Balances In the equation above, as can be seen from equation (11), the reaction rate also depends on the local nutrient concentration. A proper balance for the nutrient should also be carried out. The balance on the nutrient (B) in each zone is similar to the balance for the oxygen except that there is no mass-transfer term. The nutrient is introduced directly to the liquid phase. The nutrient balance can be written as: Trans IChemE, Vol 79, Part C, December 2001
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[net input of nutrient by convective radial flow] + [net input of nutrient by turbulent flow] + [net input of nutrient by swirl (tangential) flow] - [rate of consumption of nutrient by bio-reaction] = 0
(13)
Equation (13) is modi ed for the particular feeding zone of the nutrient, which in the 3-D networks-of zones is a single zone. The feeding point was in the centre of the radius of the vessel of the top j layer (corresponding to the actual operational conditions). The reaction rate for consumption of the nutrient uses a stoichiometry factor ‘f ’, which determines how many mols of the nutrient component B react for consumption of 1 mol of the oxygen (A), so that for the nutrient: - rB (i; j; k) = f k2 CAb (i; j; k)CB (i; j; k)V (i; j; k)
´ [1 - eG (i; j; k)]
(14)
The nutrient (B) balance for each zone is then: qL [CB (i - 1; j; k) + bCB (i; j - 1; k) + bCB (i; j + 1; k) - (1 + 2b + bL + bR )CB (i; j; k)] + qL [bL CB (i; j; k - 1) + bR CB (i; j; k + 1)] - rB (i; j; k) = 0 (15) The pressure in the vessel is a very important operational factor, since industrial bioreactors usually work under pressure for reasonsof sterility. In theverticalaxialdirection,the pressure changes due to the liquid depth are taken into account using: p( j) = p(1) -
p(1) - p(2Nin) ( j - 1) 2Ni n
(16)
Gas-Phase Oxygen Balance The gas bubbles are presumed to keep their shape and volume constant during the mixing process and in each zone they are locally perfectly mixed. The material balance for the oxygen in the gas phase is based on the ve mechanisms governing transport of oxygen: (i) buoyant rise = ub A0 eG (i; j - 1; k)CAG (i; j - 1; k) (ii) radial=axial convection = qL eG (i; j; k)CAG (i; j; k) (iii) tangential (swirl) = bL qL eG (i; j; k)CAG (i; j; k) + bR qL eG (i; j; k)CAG (i; j; k) (iv) turbulent exchange = qL beG (i; j; k)CAG (i; j; k) (v) mass transfer = kL a(i;j;k)V (i;j;k)[CAG (i;j;k)p( j)=H * - CAb (i;j;k)] The oxygen spatial distribution obviously tends to follow that of the gas hold-up, since the voidage determines the bubble surface area. The balance for each zone can be written: qL {eG (i - 1; j; k)CAG (i - 1; j; k) + b[eG (i; j - 1; k) ´ CAG (i; j + 1; k) + eGk (i; j + 1; k)CAG (i; j + 1; k)] + bL CAG (i; j; k - 1) + bR CAG (i; j; k + 1) - (1 + 2b + bL + bR )eG (i; j; k)CAG (i; j; k)} - ub A0 eG (i; j; k)CAG (i; j; k) - kL a(i; j; k)V (i; j; k)
´ [CAG (i; j; k)p( j)=H * - CAb (i; j; k)]
+ ub A0 eG (i; j - 1; k)CAG (i; j - 1; k) = 0
(17)
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The whole set of balance equations amount to [3 ´ 36,000] coupled equations, which should be solved simultaneously to obtain the pseudo-stationary spatial 3-D reagent distributions. The number of simultaneous equations is given by 2 ´ Ni ´ n ´ n ´ m ´ nf, where nf is the number of sets of equations for each reagent for each phase. In this particular case, with nf = 3, there is one set of equations for the nutrient in the bulk liquid, one for the oxygen also in the bulk liquid and one for the oxygen in the gas phase contained in the dispersed gas bubbles.
of kL = 2.67 10- 3 m s- 1 and k2 = 0.208 m3 kmol- 1 s- 1 provide the combination that exactly matches the experimental values of exit air oxygen concentration (hence the overall rate of oxygen mass transfer) and the local level of dissolved oxygen measured close to the lowest impeller. The overall kLa value of 0.036 s- 1 is typical of stirred fermenters14. NETWORKS-OF-ZONES PREDICTIONS Spatial Distribution of Gas Hold-Up
NETWORKS-OF-ZONES MODELLING Industrial Bioreactor Operational Details The pilot-plant 3m3 pressured baf ed bioreactor (with 2.5 bar pressure above the upper liquid level) was agitated by three 8-blade Rushton turbines attached to a central shaft. The impeller speed was 3 rps. Air was fed axisymmetrically at a typical rate for a fermentation of 1 vvm through a ring sparger mounted just below the lowest impeller. Ammonium nitrate solution (10% wt) was fed at 2.33 10- 4 dm3 s- 1 onto the free surface between the 60th and 144th hour of production as a source of nitrogen for the micro-organisms. Figure 1(a) shows the layout and the main dimensions of the fermenter, which were: initial liquid level H = 2.26 m, diameter T = 1.3 m, impeller diameter D = 0.52 m, baf es width 0.1 m, clearance between the lowest impeller and the base 0.47 m, between the lower impeller pair 0.57 m and between the top impellers 0.56 m. On-line measurements of a number of key parameters were taken to form the basis for a comprehensive interpretation of the mass transfer within the fermenter during one production cycle. The exit gas phase oxygen concentration was measured in the outgoing gas ow throughout a 7-day-batch, which gives the time varying overall rate of oxygen transfer. Dissolved oxygen was monitored continuously close to and level with the lowest impeller (see Figure 1). Changes in viscosity and biomass growth and decay were also recorded throughout the batch. In addition, at the end of the production cycle, the liquid level was observed to provide an estimate of the overall gas phase voidage. Networks-of-Zones Parameters All the predictions of the model correspond to a ‘snapshot’ of the conditions in the fermenter after exactly 60 hours on stream. The networks-of-zones model in general is based upon an overall circulation ow constant K for each impeller. Measurements of mixing times of viscous nonNewtonian uids at two semi-batch scales (0.2 m and 0.4 m) have established the validity of the Norwood and Metzner9, correlation indicating that K » 0.24 for the bio uid viscosity of 100cp after 60 hours on stream. This value has been used for the ows generated by each of the impellers. The gas phase bubbles were assumed to have typical diameter of db = 5 mm 10 with a rise velocity of ub = 0.185 m s- 1 11. The value de ning the turbulent exchange was b = 0.2, which has been estimated from visualizations of tracer mixing12, as well as the values of the swirling ow of bL = 1.0 and bR = 0.1 8. It is also necessary to identify the bioreaction stoichiometry factor f. For a representative value of f = 0.003 and for H* = 0.82 atm m3 kmol- 1 13, values
The predicted gas hold-up distribution in 3-D is presented in Figure 4 in the form of solid-body isosurfaces using AVS graphics (AVS V5.0). In particular, the complex space-wise geometry of gas voidage, which comprises 36,000 values of gas voidage in each model voxel, demands some visualization formatting to convey the intrinsic complexity of the bubbly ow gas-liquid mixing. This is achieved in Figure 4 by using solid-body isosurfaces for a sequence of ‘cut-off’ voidage values. However, Figure 4(a) actually shows the zone gas hold-up values for all the outer peripheral zones visible in a forward isometric view. The colour key shows blue to red from the minimum value close to zero to a maximum value of some 91%. In Figure 4(a) the peripheral hold-up values are relatively low, especially below the lowest of the three impellers and all the peripheral zones show blue and green, although higher at the upper uid surface. The overall predicted voidage eG is 17%, so that the rst dark green isosurface in Figure 4(b), which corresponds to 17% gas hold-up, encompasses approximately half of the bioreactor uid volume. Inside this double-hour-glass shaped contour, the voidage is higher than 17%. Outside it is less than this value. The solid-body contour in Figure 4(b) is perfectly axisymmetrical in consequence of gas injection through an axisymmetrical sparge ring. The 30% isosurface in Figure 4(c) now shows a single hour-glass shape with an annular ring detached from it around the topmost impeller. Figure 4(c) indicates that higher voidages are associated with the impeller regions where recirculating ows join together. As the cut-off voidage is increased, Figure 4(d) and (e) again show that the highest voidages lie around the impellers. In fact, these highest values arise just below the impeller discs, because they impede the upward ow of bubbles by buoyant rise. The highest values above 70% shown in Figure 4(f ) occur just above each of the sparge ring holes. This effect is then shown in greater detail by enlargement in Figure 5. The effect of localized gas ow through the 42 individual sparger holes can be clearly seen, with a maximum value close to 90%. Distribution and Mass-Transfer of Oxygen Oxygen is the gas phase reagent being fed to the fermenter in an air stream through the sparge ring. It undergoes mass transfer from the air bubbles, dissolving into the bioreactor uid where it is subsequently consumed by aerobic bioreaction. Therefore, both the bubble gas phase oxygen composition and the dissolved oxygen can be spatially distributed, as consequently can the associated local rates of oxygen mass transfer (in each voxel). These effects are captured in Figures 6, 7 and 8. Trans IChemE, Vol 79, Part C, December 2001
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Figure 4. Local gas hold-up distribution depicted as solid-body isosurfaces.
Figure 5. Ring-sparger 3D gas hold-up effect.
The format of the visualization follows that already used to illustrate gas voidage. Thus Figure 6 shows gas phase oxygen distribution. It is predicted to vary over the range from 11.37 to 20.89% (the gas feed value in air). Figure 6(a) shows the wall and upper surface values. These are higher adjacent to the lowest impeller (showing orange), somewhat lower below the lowest impeller and around the middle impeller (yellow) and green at the upper surface which corresponds to the measured exit gas composition of 17.35%. The sequence of Figures 6(b)–(f ) then details the internal shape of the bubble gas phase composition eld. The composition is close to the feed air in the region between the sparge ring and the lowest impeller (into which the feed air rises) as shown in Figure 6(b). At a cut-off of 19.25%, Figure 6(c) shows that bubbles spread outwards and upwards to the middle impeller. Figure 6 Trans IChemE, Vol 79, Part C, December 2001
indicates that some annular shaped pockets of gas corresponding to the exit gas composition are con ned to regions just above each of the impellers. Within these annular regions lie zones containing gas bubbles exhibiting low values from the contour of 15.35% down to the very lowest value of 11.37%. These annular pockets with the lowest composition bubbles lie, in each case, above the impeller discs. Values are lowest here as gas is recirculating downwards to the impellers before mixing and recirculating with the gas rising from beneath each impeller. The downward ow (against their buoyant rise) allows a longer time period for oxygen to be stripped from these bubbles, with a consequently locally lower residual level of oxygen. Figure 7 shows that the levels of dissolved oxygen cover a much wider variation than the gas bubble composition. Thus, the peripheral zones (with low voidage) show the lowest level
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Figure 6. Oxygen in the gas phase.
Figure 7. Dissolved oxygen in bio uid.
of oxygen saturation from 8%. The cut-off isosurfaces show a clear monotone geometry, with the highest values being con ned to the lowest impeller with a maximum of 48.2%. It is evident from this set of gures that the bioreaction speed required to balance the mass transfer, gives rise to a highest saturation which is even so less than 50%. In fact from Figure 7(d), some 75% of the uid has an oxygen level less than 33% of saturation. Figure 7 con rms that the bacteria operate throughout in an oxygen lean environment. The dual interaction of the gas phase and the liquid phase composition creates a relatively complex 3-D eld of local oxygen absorption uxes. This can be seen in Figure 8. Then Figure 8(c) shows three intense zones of oxygen mass transfer,
whereas Figure 8(d) shows two zones separated at the top and bottom of the reactor. However, as Figure 8(e) and (f) indicate, the most intensive oxygen uxes occur just above the sparge ring. Note also from Figure 8(d), the asymmetric feed of nutrient produces a small departure from axisymmetry in the absorption ux annulus just above the topmost impeller (just visible on the right hand side). Distribution of Ammonia Nutrient Concentration The corresponding ‘geometry’ of the ammonia nutrient concentration eld is detailed in Figures 9 and 10. The networks-of-zones model predicts that the ammonia nutrient Trans IChemE, Vol 79, Part C, December 2001
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Figure 8. Local absorption uxes.
eld will show a maximum value of 1.1 mmol dm- 3 and a minimum value of 0.403 mmol dm- 3. This is for a nutrient feed rate of 2.33 10- 4 dm3 s- 1 at a feed concentration of 1070 mmol dm- 3. The dip pipe feeds into zone (6,60,1). Figures 9(a)–(f ) indicate that the spatial concentration variation of the nutrient (for second order bioreaction kinetics) is much less pronounced than for dissolved oxygen. Thus, Figures 9(b)–(e) show that for around 90% of the bioreactor uid, the concentration of ammonia varies
only from 0.403 to 0.447 mmol dm- 3. However, there is still a distinctive meso-mixing plume which ows and swirls from the dip-pipe position, as can be seen in Figure 9(f ). This is shown enlarged in Figure 10. The dip-pipe close to the surface experiences an inward owing uid which draws nutrient towards the impeller shaft (Figure 10(c)). This plume simultaneously experiences the net clock-wise swirl created by the impeller rotation, so it spreads out tangentially. On reaching the impeller shaft (Figure 10(b)), nutrient
Figure 9. Ammonia nutrient concentration eld.
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Figure 10. Enlarged view of the nutrient plume near the topmost impeller.
is mixing and being drawn downwards towards the topmost impeller, whilst continuing to be spread tangentially by the swirling ow. The complex shape of the meso-mixing plume as shown very clearly in Figure 10 using these different isosurface cut-offs.
Local Bioreaction Rates in 3-D Figure 11 shows how the local bioreaction rates for assumed second order bioreaction with respect to dissolved oxygen and nutrient are con gured in space. Although the highest reaction rate is around the lower impeller (as Figures 11(e)–(f ) show), at the intermediate rates of 0.037 in Figure 11(d) there are localized maxima adjacent to the lowest and middle impellers. Figure 11(c) then shows that there is a slight but clearly visible ‘reaction rate’ plume which twists and swirls away from the nutrient feed point. This re ects the higher bioreaction rates associated with higher ammonia concentration. Figure 11(a) also shows that the low dissolved O2 values around the walls and upper uid surface are re ected in relatively low reaction rates, assuming there is negligible surface aeration. However, Figure 11 con rms
that there is a ´ 55-fold variation in potential second order reaction speeds, indicating that the micro-organisms will experience large variations in composition as they circulate round the vessel. Whether such segregated environments can have an adverse effect upon the aerobic vitality and metabolism of the Streptomyces fradiae needs further research. However, it is clear that a networks-of-zones model with O(105) zones in full 3-D offers a computationally tractable framework for quantifying these complex interacting partially segregated concentration elds. CONCLUSIONS
· a further development of networks-of-zones modelling into full 3-D has been successfully applied to a 3 m3 triple-impeller pilot-plant bioreactor, which uses Streptomyces fradiae to manufacture tylosin; · a 3 ´ (2 ´ (10 ´ 10)) ´ 60 con guration comprised of O(105) zones is able to incorporate important thirddimensional effects, including single point dip-pipe feeding of nutrient as well as the effect of hole-spacing on the air feed sparge ring;
Figure 11. Local bioreaction rates.
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· visualization derived values for the turbulent exchange and swirl- ow coef cients have been applied to exactly match the overall gas hold-up, the overall oxygen absorption rate and a local dissolved oxygen measurement (close to the lowest impeller) in the pilot bioreactor; · applying the key parameters of kL = 2.67 10- 3 m s- 1 and a second order bioreaction rate constant k2 = 0.208 dm3 mmol- 1 s- 1, it is possible to predict the concentration elds in all 36,000 voxels for bubble phase oxygen, dissolved oxygen and liquid nutrient concentrations. Results can be compactly presented by 3-D solidbody isosurfaces imaged by AVS graphics; · a high-degree of partial segregation of oxygen and nutrient is predicted, so that circulating micro-organisms will experience large variations in their uid environment. Dissolved oxygen varied spatially from 8% to 48% saturation, hence, the simulated batch operates with a fairly oxygen lean liquid phase. The nutrient stream forms a complex shaped swirling plume from the dip-pipe addition point, with ammonium ion concentration varying from 0.403 to 1.100 mmol dm- 3. The consequent complex reaction rate eld can also be readily visualized by AVS graphics. NOMENCLATURE A0 a CB CAG CAb CA* D db f H H* h1,2,3 i j K k kL k2 m N NA Ni n p QG QL qL - rA T ub V(i,j,k,)
zone area, m2 speci c interfacial area, m- 1 concentration of nutrient B in the bulk liquid phase, kmol m- 3 gas phase concentration of A, kmol m- 3 concentration of A in the bulk liquid phase, kmol m- 3 interface concentration of A, kmol m- 3 impeller diameter, m bubble diameter, m stoichiometric factor (mols of B consumed per mol of A respired) liquid level in the vessel, m Henry constant, atm m3 kmol- 1 impellers positions on the shaft, m radial zone position axial zone position overall impeller ow constant tangential zone position mass transfer coe cient, m s- 1 second order bioreaction rate constant, m3 kmol- 1 s- 1 number of zones along the vessel perimeter impeller speed, rps mass transfer of A, kmol h- 1 m- 3 number of impellers number of zones in axial=radial plane (half network) pressure, bar feed rate of the feed gas, m3 s- 1 overall liquid circulation by the impeller, m3 s- 1 liquid ow rate in nested (axial=radial) ow loops, m3 s- 1 bioreaction rate (based on oxygen), kmol s- 1 vessel diameter, m bubble rise velocity, m s- 1 zone volume, m3
Greek letters eG(i,j,k) fractional gas hold-up in zone i,j,k eG fractional overall gas hold-up b lateral turbulent exchange coef cient bI turbulent exchange coef cient in the impeller plane
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bL bR
clock-wise swirl coef cient anti-clock-wise swirl coef cient
Indexes A B b G
oxygen nutrient bubble gas
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ACKNOWLEDGEMENT The authors gratefully acknowledge nancial support from the Commission of the European Communities under INCO-COPERNICUS contract No. ICIS-CT98-0502.
ADDRESS Correspondence concerning this paper should be addressed to Professor R. Mann, Department of Chemical Engineering, UMIST, PO Box 88, Manchester, M60 1QD, UK. E-mail: [email protected] The manuscript was received 10 October 2000 and accepted for publication after revision 28 August 2001.