pages116_143

Mass Transfer

5.1 Mass Transfer in Biological Reactors

Multiphase reaction systems usually involve the transport of material betweentwo or more phases. Usually one of the reactants is transferred from one phaseinto a second phase, in which the reaction takes place. The following cases areexamples of biological systems.

5.1.1 Gas Absorption with Bioreaction in the LiquidPhase

The gas phase is dispersed as gas bubbles within the liquid phase. Mass transferoccurs across the gas-liquid interface, out of the gas into the liquid, where thereaction occurs. The typical example is aeration of the bioreactor broth andthe supply of oxygen to the cells as shown in Fig. 5.1.

Figure 5.1. Absorption of oxygen from an air bubble to the liquid medium.

Biological Reaction Engineering, Second Edition, I. J. Dunn, E. Heinzle, J. Ingham, J. E. PfenosilCopyright 2003 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-30759-1

118 5 Mass Transfer

5.1.2 Liquid-Liquid Extraction with Bioreaction inOne Phase

An immiscible liquid phase is dispersed in a continuous liquid phase. Masstransfer of a reactant takes place across the liquid-liquid interface, shown here(Fig. 5.2) from the continuous phase into the dispersed phase, where reactionoccurs. An example might be the transfer of a substrate in an oil phase to anenzyme in the droplet aqueous phase.

Figure 5.2. Liquid-liquid extraction plus reaction.

5.1.3 Surface Biocatalysis

In this case, a liquid phase is in contact with solid biocatalyst. Substrates A andB diffuse from the liquid to the reaction sites on the surface of the solid, wherereaction occurs. The product C must similarly be transferred away from thesolid reaction surface, as shown in Fig. 5.3. Examples are found withimmobilized enzyme and cell systems. In Sec. 6.1 the modelling aspects of thistype of system are considered in detail.

5.2 Interface Gas-Liquid Mass Transfer 119

Figure 5.3. Reaction of two substrates on a solid biocatalyst surface.

5.1.4 Diffusion and Reaction in Porous Biocatalyst

Here a porous biocatalyst sphere is suspended in a liquid medium. Substratesdiffuse into the porous internal structure of the biocatalyst support and react.Similarly, the products must diffuse away from the reaction sites within thesolid to the outer surface, where they are then transported into the liquid.Detailed modelling of this process is treated in Ch. 6.

Figure 5.4. Reaction within a solid biocatalyst.

5.2 Interphase Gas-Liquid Mass Transfer

Concentration gradients are the driving forces for mass transfer. Actualconcentration gradients (Fig. 5.5) in the very near vicinity of the gas-liquid

120 5 Mass Transfer

interface, under mass transfer conditions, are very complex. They result froman interaction between the mass transfer process and the local fluidhydrodynamics, which change gradually from stagnant flow, close to theinterface, to perhaps fully-developed turbulence within each of the bulk phases.

According to the Two-Film Theory, the actual concentration profiles, asrepresented in Fig. 5.5 can be approximated by linear gradients, as shown inFig. 5.6.

A thin film of fluid is assumed to exist at either side of the interface. Awayfrom these films, each fluid is assumed to be in fully developed turbulent flow.There is therefore no resistance to mass transfer within the bulk phases, and theconcentrations, CG and CL, are uniform throughout each relevant phase. At thephase interface itself, it is assumed there is no resistance to mass transfer, andthe interfacial concentrations, CGI and CLI, are therefore in local equilibriumwith each other. All the resistance to mass transfer must, therefore, occur withinthe films. In each film, the flow of fluid is assumed to be stagnant, and masstransfer is assumed to occur only by molecular diffusion and therefore to be

Interface

Gas

Figure 5.5. Concentration gradients at a gas-liquid interface.

described by Pick's law, which says that the flux JA (mol/s m2) for the moleculardiffusion of some component A is given by,

dZ

5.2 Interface Gas-Liquid Mass Transfer 121

GasInterface

Liquid

Figure 5.6. Concentration gradients according to the Two-Film Theory.

where D is the molecular diffusion coefficient (m2/s) and dC/dZ is the steadystate concentration gradient (mol/m3). Thus applying the same concept to masstransfer across the two films,

JA = DG ZG ZLwhere DG and DL are the effective diffusivities of each film, and ZG and ZL arethe respective thicknesses of the two films.

The above equations can be expressed in terms of mass transfer coefficientskc and kL (m2/s) for the gas and liquid films,

JA = kG(CG-CGi) = kL(CLi-CL)

The total rate of mass transfer, Q (mol/s), is given by,Q = J A A = jA(aV)

where "A" is the total interfacial area available for mass transfer, and "a" isdefined as the specific area for mass transfer or interfacial area per unit liquidvolume (m2/m3). Thus for the total rate of mass transfer:

In terms of the total interfacial area A,

Q = kGA(CG-CGi) = kLA(CLi-CL)

In terms of a and VL,

Q = ko a (Co - CGi) VL = kLa(CLi-CL)VL

122 5 Mass Transfer

Since the mass transfer coefficient, k, and the specific interfacial area, a, dependon the same hydrodynamic conditions and system physical properties, they arefrequently combined and referred to as a "ka value" or more properly a masstransfer capacity coefficient.

In the above theory, the interfacial concentrations CGI and CLI cannot bemeasured, and are therefore of relatively little use, even if the values of the filmcoefficients are known. For this reason, by analogy to the film equations,overall mass transfer rate equations are defined, based on overall coefficients ofmass transfer, KG and KL, and overall concentration driving force terms, where:

Q = KGA(CG-CG*) = KLA(CL*-CL)

Here, CG* and CL* are the respective equilibrium concentrations, correspondingto the bulk phase concentrations, CL and CG, respectively, as shown in Fig. 5.6.

Equilibrium relationships for gas-liquid systems, at low concentrations ofcomponent A usually obey Henry's law, which is a linear relation between gaspartial pressure, PA, and equilibrium liquid phase concentration, CLA*:

PA=

where HA (bar m3/kg) is the Henry's law constant for component A in themedium. Henry's law is generally accurate for gases with low solubility, such asthe solubility of oxygen in water or in fermentation media. Thus from thisrelation, as shown in Fig. 5.7, the corresponding equilibrium concentrations canbe easily established.

C L C *

Figure 5.7. Equilibrium concentrations based on Henry's law.

For gases of low solubility, e.g., oxygen and carbon dioxide in water, theconcentration gradient through the gas film is very small, as compared to thatwithin the liquid film, as illustrated in Fig. 5.6. This results from the relatively

5.3 General Oxygen Balances for Gas-Liquid Transfer 123

low resistance to mass transfer in the gas film, as compared to the much greaterresistance to mass transfer in the liquid film. The main resistance to masstransfer is predominantly within the liquid film. This causes a large change inconcentration (Cy - CL), since the resistance is almost entirely on the liquidside of the interface.

At the interface, the liquid concentration, Cy, is in equilibrium with that ofthe gas, CGI, and since CGI is very close in magnitude to the bulk gasconcentration, CLI must then be very nearly in equilibrium with the bulk gasphase concentration, CG- This is known as liquid film control and correspondsto the situation where the overall resistance to mass transfer resides almostentirely within the liquid phase. The overall mass transfer capacity coefficientis KLa. Hence the overall mass transfer rate equation used for slightly solublegases in terms of the specific area is

Q = KLa (CL*-CL)VL

where CL* is in equilibrium with CG, as given by Henry's law,

CG= HCL*,

Mass transfer coefficients in fermentation are therefore generally spoken of asKL values or K^a values for the case of mass transfer capacity coefficients.

5.3 General Oxygen Balances for Gas-LiquidTransfer

In order to characterize aeration efficiency, to predict dissolved oxygenconcentration, or to follow the biological activity it is necessary to developmodels, which include expressions for the rate of oxygen transfer and the rateof oxygen uptake by the cells. Well-mixed phase regions, in which the oxygenconcentration can be assumed uniform, can be described by simple balancingmethods. Situations in which spatial variations occur require more complexmodels, as described in Sec. 5.4. The following generalized oxygen balanceequations are derived for well-mixed phases, using the well-mixed tankconcept. In the situation in Fig. 5.8, both the liquid and gas phases are definedby distinct well-mixed regions and by the total volumes of each phase, VL andVG.

124 5 Mass Transfer

Gas CQO GO

Figure 5.8. The balance regions for well-mixed gas and liquid phases in a continuous reactor.

For the gas phase the oxygen balance can be developed as follows:

Rate ofaccumulation

of oxygenin gas ;

Flow of >oxygen in

inlet gas streamy

f Flow of \_ oxygen out _

Vin exit stream/

( Rate of ^transfer

of oxygenv from gas ,

Thus, for the gas phase,

dCGi KLa(CLi*- CLi)VL

where VQ represents the volume of gas in the dispersed phase, or the gasholdup.

For the liquid phase,

Rate of ^accumulationof oxygen

^ in liquid

/Flow ofoxygenin inletliquid

V stream J' Rate ofconsumption

of oxygenin liquid

( Flow ofoxygenout inexit

Rate oftransfer

of oxygen^ from gas

Rate of oxygen consumption = -rO2 = -qo2


Thus for the liquid phase,

dCLi- LiCL i + KLa(CL1*-CLi)VL -

The above equations include accumulation, convective flow, interphase transferand biological oxygen uptake terms. Here CLI* is the equilibrium solubility ofoxygen corresponding to the gas phase concentration, CGI , and is calculatedby Henry's law, according to the relationship:

Typical units are as follows: CG and CL (kg/m3); G and L (m3/s); K^a (1/s);VG and VL (m3); qO2 (kg/kg s); X (kg/m3).In the next sections, the general equations, given above, will be applied toimportant special situations.

5.3.1 Application of Oxygen Balances

5.3.1.1 Case A. Steady-State Gas Balance to Determine theBiological Uptake Rate

The convective terms in the generalized liquid balance equation can usually beneglected, owing to the low solubilities of oxygen in water (about 8 g/m3). Thisgives the steady state liquid balance, dCL/dt=0, relation as:

KLa(CLi*- CLI) = qo2XiThus at steady-state, the oxygen transfer rate is effectively equal to the oxygenuptake rate. Even during batch fermentations this is approximately true.

Substituting this relationship into the steady state gas balance gives,

0 = GO CGO - GI CGI - qo2 X VLThe above equation can also be derived from a steady state balance around theentire two-phase system. It shows that the biological oxygen uptake rate can becalculated from knowledge of the gas flow rates and the gas concentrations.

126 5 Mass Transfer

This application is very important in fermentation technology, since it permitsthe on-line monitoring of the rate of fermentation, by gas balancing methods(Heinzle and Dunn, 1991, Ingham and Dunn, 1991).

5.3.1.2 Case B. Determination of Ki,a Using the SulfiteOxidation Reaction

If a chemical reaction, classically the oxidation of sodium sulfite, is used to takeup the oxygen from solution, then the term qo2 X VL in the liquid phasebalance may be replaced by the chemical reaction term, ro2 VL- At steady-state,

KLa(CLi*-CLi) = r02Usually ro2 is obtained by taking samples and titrating for the fractionalconversion of sulfite, which can be related by stoichiometry to the oxygenconsumption. Since the chemical reaction causes the liquid dissolvedconcentration CLI to fall essentially to zero and with CLI* calculated from theoxygen concentration in the exit gas, the value of the overall mass transfercapacity coefficient, K^a, can be estimated. An improved method uses the gasbalance instead of titration to obtain ro2 in the manner outlined above forqO2 X and also provides a check on the sulfite measurements. The sulfitemethod is useful for comparing aeration systems, but the values are difficult torelate to actual fermentation conditions owing to the very different physicalconditions (coalescence, aeration rates) (Ruchti et al., 1985).

5.3.1.3 Case C. Determination of Ki,a by a Dynamic Method

If water is initially deoxygenated and is then re-aerated, the concentration ofthe dissolved oxygen will increase with time, from zero to effectively 100% airsaturation at the end of the experiment. The exact form of the response curveobtained depends on the values of KLa, the driving force, (CLI* - CLI), and themeasurement dynamics of the dissolved oxygen electrode. The liquid balance,for the unsteady state batch aeration condition, gives:

=

KLa(CLi*-CL i)VL


The classical dynamic KLa method assumes that K^a and CLI* are constant.Under these conditions, the differential equation can be integrated analyticallyto give the relationship: ( C * ^r * Lr = KLatCL - CL\}Plotting the natural logarithmic concentration function on the left side of theequation versus time, should, in principle, give a straight-line relationship, with^a as the slope.

Usually deoxygenation is accomplished with nitrogen, so that initially the gasphase consists of nitrogen, which is gradually displaced and mixed with air.Under these conditions, CLI* is nt constant, and a gas balance must beemployed to calculate the variation in CGI versus t. Since the liquid phaseconcentration, CLI, is measured by means of a membrane covered oxygenelectrode, the dynamics of measurement method usually cannot be neglected.The dynamics of the measurement electrode can be described, approximately,by a first-order lag equation,

dCE

where TE represents the electrode time constant, and CE is the measurementsignal.

The fractional response of the electrode for a step change in CL wouldappear as shown in Fig. 5.9.

time

Figure 5.9. Response of electrode for a step change in CE from zero to 100 % saturationaccording to a first-order lag model.

Note that TE corresponds to the time for the electrode to reach 63 % of the finalresponse. The overall process dynamics involves thus the gas phase, the liquid

128 5 Mass Transfer

phase and the electrode response. The three responses might appear as shownbelow:

time

Figure 5.10. Response of the gas, the liquid and the electrode measurement during a dynamicKLa experiment.

The values of three individual time constants determine the process response.These are TG = VG/G, (representing the dynamics of the gas phase), 1/KLa(representing the dynamics of the liquid phase mass transfer process), and IE(representing the measurement dynamics). This is illustrated in the simulationexample KLADYN, Sec. 8.5.5.

5.3.1.4 Case D. Determination of Oxygen Uptake Rates by aDynamic Method

Low oxygen uptake rates, as exist in slow growing systems (plant and animalcell cultures, aerobic sewage treatment processes, etc.), cannot easily bemeasured by a gas balance method, since the measured difference between inletand outlet oxygen gas phase concentrations is usually very small. Due to thelow solubility of oxygen in the liquid media, quite small oxygen uptake rateswill cause measurably large changes in the dissolved oxygen concentration.Thus it is possible to measure qo2 X either by taking a sample and placing it ina small chamber or by turning off the reactor air supply, according to theliquid balance equation

dCLi


Dissolved oxygen concentration decreases linearly and is equal to qo2 X asshown in Fig. 5.11.

Figure 5.11. Oxygen uptake rate determined by a dynamic method.

When the time required for an appreciable decrease in dissolved oxygen islarge, as compared to the electrode time constant, the method is quite accurateand no correction for the electrode measurement dynamics is required (Monaet al., 1979). If the response is too fast the sample can be diluted. This methodis illustrated by the simulation example OXDYN, Sec. 8.5.4. A similarsimulation example, ANAMEAS, Sec. 8.8.7, illustrates dynamic measurementsin anaerobic systems.

5.3.1.5 Case E. Steady-State Liquid Balancing to DetermineOxygen Uptake Rate

If the biomass is immobilized or retained by membranes within the reactor,oxygen can be supplied to the cells by means of a circulating liquid supply,which is aerated in a separate unit, external to the reactor.

130 5 Mass Transfer

CL1

CLO

Figure 5.12 Oxygen uptake rate determined by a steady-state liquid balance.

It then becomes possible to determine the oxygen uptake rate, simply bymeasuring the liquid flow rate and the difference in dissolved oxygen in theliquid inlet and outlet flow streams, according to the following steady-stateliquid phase balance equation:

0 = L(CLo - CLI) - qo2 Xi VLThus the rate of oxygen supply via the liquid is equal to the rate of oxygenuptake by the cells. This method provides a very sensitive way of measuringlow oxygen uptake rates (Keller et al., 1992, Tanaka et al., 1982). The case-study H in Sec. 5.3.1.8 is an example of this use for an experimental reactor.The simulation example FBR, Sec. 8.4.9, also demonstrates this method.

5.3.1.6 Case F. Steady-State Deoxygenated Feed Method forKLa

Feeding a deoxygenated liquid continuously to an aerated tank (Fig. 5.11)allows the oxygen transfer rate to be determined by difference measurement.Thus the liquid phase balance becomes

0 = L (CLO - CLI) + KLa (CLi* - CLi)VLKnowing the flow rate L, the oxygen liquid concentrations CLO and CLI and theoutlet oxygen in the gas phase (to determine CLI*) permits the calculation of


KLa. Another variation of this would be to gas with oxygen-enriched air orwith nitrogen, which would avoid the difficulty of producing a continuoussource of deoxygenated liquid. A similar steady state method has beenemployed to obtain steady oxygen concentration profiles in column (Meister etal., 1980), and tubular bioreactors (Ziegler et al., 1977). A suitable steady statemodel for the tubular reactor then allows calculating the unknown Kâ byparameter estimation (Shioya et al., 1978).

Deoxygenatedliquid CLO

t/"VX "N^ -^^^

nM&^/m^M:WM.Xiiiiiiiiiiji

r-^ ^ -~\_/~\^^x

^^^SRSSSSO^^ iÔ^wiBii

WiiiiilmiiiiiiiAir

Figure 5.12. Steady-state dissolved oxygen difference measurement for Kjâ.

5.3.1.7 Case G. Biological Oxidation in an Aerated Tank

A batch reactor liquid is aerated with a continuous flow of air to support abiological reaction, as shown in Fig. 5.13.

air

air

Figure 5.13. A batch bioreactor with continuous aeration.

132 5 Mass Transfer

The biological reaction in the liquid phase is first-order in oxygenconcentration. Since oxygen is relatively insoluble (approximately 8 g/m3saturation for air-water) the transfer rate is important to maintain a highdissolved oxygen concentration CL. The batch oxygen balance for the liquidphase is then:

f Rate of \accumulation of

=

V 02 in liquid )

dCL

, ^ f f N(Transfer rate o f \

l02 mto the hqmdj/ Uptake rate of^ by the cells

= KLa(CL*-CL)VL - kCLVL

A steady-state can be reached for which the mass transfer rate is equal to theoxygen uptake rate by reaction:

giving for CL0 = KLa (CL* - CL) - k CL

KLaCL*CL = KLa

Using this equation, the reaction rate constant, k, can be determined if CL ismeasured and K^a is known or measured. The equilibrium value, CL*, can becalculated from the gas phase concentration, and if there is little oxygendepletion it can be calculated from the inlet gas conditions.

If the oxygen depletion in the gas phase is appreciable, then the molefraction of oxygen in the exit may not be the same as in the inlet, and a gasphase balance must be applied to determine CL*:

Figure 5.14. Inlet and outlet oxygen mole fractions and total gas molar flow rates.


From the ideal gas law as shown before, assuming a well-mixed gas phase insteady state, N = (p / RT) F, where NO is the molar flow rate of air and F is theair volumetric flow rate.

/ Rate of O2 \ / Rate of O2 \ / Transfer rate of \0 =

V in by flow ) ~ Vout by flow ) ~ \ C>2 to the liquid )

Using the nomenclature in Fig 5.13,

0 = yoN 0 -y iNi -K L a(CL*-CL)VLwhere

Assuming NO = NI, these equations can be solved to obtain yi and CL-Solving for CL gives,

CL = k~" CL*

or for the apparent reaction rate,k ,

re = - k~ CL

Thus it is possible to distinguish between two different regimes for this system,transfer control and reaction control:

1 ) Reaction rate control applies for low values of k/KLa, when re approachesk CL*, and CL approaches CL*

2) Diffusion control applies for high values of k/KLa, when re approachesKL& CL* and CL approaches 0.

3) If KLa = k, then rc = - (k/2) CL*, and CL approaches (1/2) CL*.

5.3.1.8 Case H. Modelling Nitrification in a Fluidized BedBiofilm Reactor

Nitrification is a two-step microbiological process, in which the ammonium ionis oxidized to nitrite ion and further to nitrate ion as shown:

NH4+ - NO2- - NO3-

134 5 Mass Transfer

This reaction is important in waste water treatment because of the toxicity ofammonia and its large oxygen demand. Several known organisms can gainenergy from either of the two oxidation steps, but most commonlyNitrosomonas and Nitrobacter are responsible for steps (1) and (2),respectively. These organisms grow very slowly, obtaining their carbon fromdissolved carbonate. Due to the very slow grow rates, it is of interest to retainthe biomass within the reactor. One possibility considered here is toimmobilize the biomass as a natural biofilm on a fluidized bed of sand (Tanakaand Dunn, 1982).

The stoichiometric relations for the reaction steps (1) and (2) are:, 3 ,

NH4+ + j O2 -> NO2" + H2O + 2 H+

O2 -> NO3"

Summing the above steps (1) and (2) gives

NH4+ + 2O2 -

The reactor of volume, Vr, consisted of a conical sand bed column, which wasfluidized by the liquid recycle stream flowing up through the bed. The recyclestream was oxygenated in a separate, baffled, tank contactor of volume VT, withturbine impeller and air or oxygen sparging. The reactor and oxygenator werethus separate parts of a recycle loop configuration. This could be operatedbatchwise or with a continuous feed and effluent stream flow to and from thesystem. When operating at high recycle rates, the whole system actedeffectively as one well-mixed tank system.

The reactor-oxygenator recycle loop can be analyzed as a total system orbroken down into its individual components as shown in Fig. 5.14. Theseinclude liquid phase balance over the reactor and combined phase, liquid phaseand gas phase balances over the oxygenator and over the total system.


Figure 5.15. Mass balancing regions for the fluidized bed reactor nitrification system.

The mass balances to be considered are those for oxygen and the nitrogen-containing reactants and products. The oxygen balance taken over the totalsystem can be simplified by neglecting the accumulation terms and the liquidflow terms, that will be small compared to the gas rates and the consumption byreaction, owing to the relatively low solubility of oxygen in the liquid medium.Thus the oxygen balance becomes,

0 =

Here Vr is the volume of the reactor column.The nitrogen (N) components, NH4+, NCV, and NOs', in the liquid phase

can be balanced around the total system by considering the accumulation, flow,and reaction terms for each of the N-containing components. For the totalsystem each component equation has the form,

VdCpdt

= F(CNi-CN2)

When the reactor is operated as a batch system, F = 0, and when used as acontinuous steady state reactor, dCN2/dt = 0. This equation can be used incolumn systems for very low single-pass conversion, when the differences inlocal reaction rate at the reactor inlet and outlet are not large. Although thereactions actually occur in the solid phase, because of the high solid-liquidinterfacial area, the system is treated here as being quasi-homogeneous.

The gas-liquid interfacial mass transfer area will often be small enough to beimportant for the overall process, and it is therefore useful to consider the gas

136 5 Mass Transfer

and liquid phases as separate balance regions. The absorption tank can bedescribed by the oxygen balances for the liquid phase:

0 = FR(CL4-CL3) + KLa(CL*-CL3)VTand for the gas phase:

0 = G(CGi - CG2) - KLa (CL* - CL3) VTThe liquid phase oxygen balance for the total system is

0 = KL

where ro2 is the oxygen uptake rate by the reaction. These equations, whichassume ideally mixed phases, are useful in designing the gas absorberaccording to the required oxygen transfer coefficient.

Balancing the oxygen around the reactor gives

0 =

Since CL4 at the reactor outlet is usually very low, then,

FR CL3 = - ro2 Vr

which says that the oxygen uptake rate by reaction must be equal to the supplyrate from the oxygenation tank. This is the condition of reaction-ratelimitation by the oxygen transfer in the absorber.

From the stoichiometry, the relationships between the molar reaction rates(rNH4> rO2 rH* r2,NO2 an^ fNO3) can be found. Thus, for example, the firstnitrification step gives

2TNH4 = T r l ,O2 = ~ri ,NO2

and the total rate for 02 is given by the sum of the rates for steps (1) and (2).rO2 = rl,O2 + r2,O2

From the measured concentration dependency of these rates, the reactionkinetics of the individual steps can be determined. The dependency of theserates on the individual concentrations can then be used to establish the reactionkinetic model. This model is the basis of the simulation example NITRIF, Sec.8.5.3. A similar type of recycle, fluidized-bed reactor is the theme ofsimulation examples FBR, Sec. 8.4.9 and DCMDEG, Sec. 8.4.6.

5.4 Models for Oxygen Transfer in Large Scale Bioreactors 137

5.4 Models for Oxygen Transfer in Large ScaleBioreactors

Large-scale industrial fermenters can generally be expected to exhibitdeviations from the two idealized flow conditions of perfect mixing or perfectplug flow. Thus the assumption of completely mixed gas or liquid phases maynot be valid. Little experimental information is available on concentrationinhomogeneities or concentration gradients within large bioreactors. Residencetime distribution information, from which a physical and mathematical modelcould be established, is also generally not available.

Convection currents within the liquid phase of a bioreactor are usually causedby the mechanical energy inputs of agitation and aeration. It is oftenreasonable to assume that slowly changing quantities, such as biomassconcentration, substrate concentration, pH and temperature are uniform withinthe whole mass of bioreactor liquid. Oxygen must be considered, however, as arapidly changing substrate, owing to its low solubility in fermentation media. Itis therefore necessary to consider that differences in oxygen transfer anduptake rates will create oxygen concentration gradients throughout the reactor.

Buoyancy forces carry the gas from the lower gas inlet point up to the topliquid surface. In the absence of mechanical agitation, the gas phase mightmove from the bottom to the top of the reactor in an approximate plug flowmanner, with very little backmixing. If the stirring power supplied to thefermenter, however, is sufficient to create liquid velocities, that are greater thanthe free rise velocity of a bubble (about 26 cm/sec) then the bubbles willcirculate around the fermenter, before eventually escaping. Very high powerinputs can cause the smaller bubbles to circulate many times within the vesseland spend an appreciable time before reaching the surface. Under suchconditions, if no bubble coalescence occurs, the gas phase would contain afraction of small bubbles, depleted of oxygen but with a large surface area.Obviously any well-mixed phase assumption becomes difficult to justify.

The gas phase flow conditions in large scale industrial fermenters usually liesomewhere between the extreme cases of idealized plug flow and perfectmixing. Experimental residence-time distribution information, obtained byhelium tracer techniques under actual operating conditions, are then necessaryto characterize the gas phase flow. Unfortunately very little experimentation onindustrial scale equipment has been reported.

Hydrostatic pressure gradients in tall fermenters will cause large differencesin the oxygen solubility, CL*, with regard to the depth position in the tank. In a10m tall reactor, the oxygen solubility for a given gas composition will be twicethat at the bottom of the tank as compared to the top surface, since the totalpressure is effectively doubled. This is seen by Henry's law which can be writtenas:

138 5 Mass Transfer

d* = y2pH

where yo2 is the mole fraction of oxygen in the air and p is the total pressure atsome point in the tank.

The possibility that oxygen gas compositions, dissolved oxygenconcentrations, oxygen solubilities, gas holdup volumes, bubble sizes and othertransfer parameters can vary with depth in a tall fermenter introduces a muchgreater degree of complexity to the problem of modelling the reactor. Thismakes it difficult to obtain data on oxygen mass transfer coefficients.Although it is impossible to give specific recommendations that apply to anyparticular situation, a further discussion of possible models and theirunderlying assumptions may help to define the problem. Incorporated into themore complex models, discussed below, are such factors as gas and liquid phaseflow pattern, gas composition gradients and the effects of hydrostatic pressure.Great caution and wisdom must be exercised to avoid creating a model that istoo complex to verify by experimentation. Experienced engineers will say"Keep it simple!" and "Avoid too much model!".

All large scale reactors, whether multi-impeller tanks or column fermenters,will display some axial dissolved oxygen concentration gradients. The mostgeneral method for modelling is to represent the reactor using balances in aseries of sections or stages. Mass balances in multi-stage process are easy toformulate, since both the liquid and gas phases may be assumed to be well-mixed, for any given stage of the cascade.

Figure 5.16. A single gas-liquid stage with backmixing of the liquid phase.

The formulation of the mass balances for a single stage, as shown in Fig.5.16, follows closely that described previously, except that now the reactor ismade up of many stages which are interconnected by the flows of gas andliquid between stages and by diffusive mass transfer mechanisms.


5.4.1. Case Studies

5.4.1.1 Case A. Model for Oxygen Gradients in a BubbleColumn Bioreactor

The application of the stagewise modelling approach is shown below, where abubble column reactor is modelled as a five-stage reactor system. The reactorwill be assumed to operate cocurrently, as would be also the case for the riser ofan airlift bioreactor.

Exit Gas

Gas Feed

Exit Liquid

Liquid Feed

A CQB A CLS

Gas ill

GO L ' LO

Figure 5.17. Stagewise model of a bubble column bioreactor.

140 5 Mass Transfer

The oxygen balance equations for the gas and liquid phases of each stage areas follows:

if = FG(CGn-l-CGn) - KLa(CLn*-CLn)VLf\(~^

VL-dT = FL(CLn-i-CLn) + KLa(CLn*-CLn)VL - rn VLwhere,

r * - 1 r r c * -

P02 for each stageas functions of time and also yields the resulting final steady state values.

Note that the biomass concentrations Xn are assumed constant, otherwisebiomass balance and growth kinetics equations would have to be added to themodel. Using simulation methods, other effects, such as the effect ofhydrostatic pressure on CG or on bubble size could be included. Thesimulation example DCMDEG, Sec. 8.4.6, demonstrates some aspects of thestagewise modelling approach.

5.4.1.2 Case B. Model for a Multiple Impeller Fermenter

Mixing in a tank reactor is complex, and it would be necessary to considerliquid flow in both directions. It is generally assumed, however, that theintensity of mixing is such that no radial variations occur. Fig. 5.16 representsa multiple impeller reactor with well-mixed liquid zones in the region of eachimpeller. The reactor can be described approximately by means of a three-stage model. Mixing of the liquid in a direction which is directly opposite tothat of the main flow liquid (here upwards) can be incorporated into the model,by the assumption of a backmixing stream, with flow rate FB- This backmixingstream accounts for a flow interaction between the mixing zones and fordeviations from ideal stage mixing. To determine FB, a tracer experiment


would need to be performed to obtain the necessary information regarding thedegree of backmixing actually existing in the reactor.

Exit Gas Exit Liquid c G3 A A C

L3Fd \_^Gas

G2 L3

Gas

G1

L2

Gas

'LIFL+FB

Inlet Gas Liquid FeedI F G- C GO \\>cu>

Figure 5.18. Stagewise approximation for stirrer regions in multi-stirrer tank.

To model this system, the liquid-phase impeller zones are assumed to be well-mixed, and the plug-flow gas is described by a series of well-mixed phases,together with an arithmetic-mean, concentration-driving-force approximation.Here the flow rates and mass transfer coefficients are assumed constant.

Stage 1:

VG

dCLiVL -ar = FLCLO + FBCL2 - (FL + FB) CLI ++ KLa(CLi*-CLi)VL + riVL

dCGl """' xx-< f\ \ -rr- //~1 ^ /"I \A 7-CGI) -KLa(CLi -CLI)VL

where the plug flow nature of the gas is partially accounted for by

142 5 Mass Transfer

and

= - Qo2rCLI

Stage 2:

dCL2VL -gf = (FL+ FB) CLI + FBCL3 - (FL+ FB) CL2 - FB CL2

KLa(CL2*-CL2)VL

where=

FG (CGI - CG2) - KLa (CL2* - CL2)VL

CL2. .

and n CL2r2 = -Qo2mK0 + CL2

Stage 3:

dCL3

KLa(CL3*-CL3)VL+ r3VL

where

VG - CG3) - KLa (CL3*- CL3) VL

CL3. .

and = -QO2nCL3

K0 + CL3

The above equations describe the dynamic oxygen concentrations in the multi-impeller continuous bioreactor. Note that the liquid phase balances for the twoend stages 1 and 3 differ from that of the intermediate stage 2, owing to the


absence of any backmixing flow contribution exterior to the column. A batchreactor would be described by setting the liquid flow, FL, equal to zero. Sincethe biomass balance and growth kinetics are not included here, the solutionwould be valid at only one time during the fermentation, corresponding to theassumed value of Qo2m> which is proportional to the value of X existing at thattime. Variations in X are, however, easily incorporated into the model byadding cell and substrate balance equations.

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