pages259_292.pdf

8.4 Continuous Reactors 257

Run 3: 2021 steps in 0.15 seconds

120

100

80

- 60

40

20

^

\». ll

'-, 1\ \

1 1I

\!

1 1III 1

! 1. i i

v i|\ i

i ' \ *\! \l

•\\ \

-0.008

-0.007

-0.006

-0.005

-0.004 O

-0.003

-0.002

-0.001

-0

20 40 60 80 100 120 140 160 180 200

TIME

Figure 3. Influence of initial KLa value from 100 to 160 h"^ on the S and O profiles.

8.4 Continuous Reactors

8.4.1 Steady-State Chemostat (CHEMOSTA)

System

The steady state operation of a continuous fermentation having constantvolume, constant flow rate and sterile feed is considered here (Fig. 1).

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

258 8 Simulation Examples of Biological Reaction Processes Using Berkeley Madonna

D,S

Figure 1. Chemostat fermenter with model variables.

Model

The dynamic balance equations may be modified to apply only to the steadystate by setting the time derivatives equal to zero. The corresponding equationsare then:

For biomass,

For substrate,

Growth kinetics,

0 = - D X + rx

0 = D (S0 - S) + rs

rx = ^X

Substituting into the biomass balance gives

\i = Dwhere S is determined by the kinetics

The Monod relation results in,\i = f(S)

S =

The substrate balance gives,X = Y(S0 -S)


The productivity of the reactor for biomass is X D.

The above equations represent the steady state model for a chemostat withMonod kinetics. Using them it is possible to calculate the values of S and X,which result from a particular value of D, and to investigate the influence of thekinetic parameters.

Program

In Madonna programs, time can be used as a variable which will increase fromthe starting time. Here it is renamed D. Thus equations will be solved forincreasing values of the dilution rate. Fortunately X and S can be explicitlysolved for in this problem. If not, the ROOT FINDER facility of Madonna canbe used. The program is found on the CD-ROM.

Nomenclature

The nomenclature is the same as the example CHEMO, Sec. 8.1.2.

Exercises


Results

The steady state curves of X, S, and XD versus D are given Fig. 2. The resultsin Fig. 3 were obtained by varying K$ in each run. An interesting effect can be

observed on the position of the washout point.

Run 1:113 steps in 0 seconds

10 -

9-

8-

7-

6-

v- 5"

4-

3-

2-

1 -

0-

*S**~\ I\ !^r Y.-I 1 iim ~s;i \!

f*r —mm .

S" 1

// /{

-4

•3.5

-3

-2.5

•2 S

•1.5

• 1

•0.5

•0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2. Steady state curves of X, S and XD versus D.



Figure 3. Runs obtained by varying KS from 0.2 to 1.0.

8.4.2 Continuous Culture with Inhibitory Substrate(CONINHIB)

System

Inhibitory substrates at high concentrations reduce the specific growth ratebelow that predicted by the Monod equation. The inhibition function may beexpressed empirically as

where KI is the inhibition constant (kg/m3).

If substrate concentrations are low, the term S2/Kj is lower in magnitude thanKS and S, and the inhibition function reduces to the Monod equation. In batchcultures the term S2/Kj may be significant during the early stages of growth,even for higher values of K[. The inhibition function passes through amaximum at Smax = (Kg Ki)°-5. A continuous inhibition culture will often lead


to two possible steady states, as defined by the steady state condition JLI = D andas in shown Fig. 1.

D=

Figure 1. Possible steady states for a chemostat with inhibition kinetics.

One of these steady states (A) can be shown to be stable and the other (B) to beunstable. Thus, only state A and the washout state (S = SQ) are possible.

Model

A model of a chemostat with its variables is represented schematically in Fig. 2.

F,S0

+- F,S,X

Figure 2. Model variables.


Cell material balance,

VdXjj- = ^ i V X - F X

or,

where D is the dilution rate = F/V.

Substrate material balance,

VdS ^ V X— = F (S0-S) -~^f—

or,dS M « XdT = D (S0 - S) - —

where Y is the yield factor.

Program

When the system equations are solved dynamically, one of two distinct steadystate solutions are obtained, the stable condition A and the washout condition.The initial substrate and organism concentrations in the reactor will determinethe result. This is best represented as a phase-plane plot X versus S. All resultsindicate washout of the culture when the initial cell concentration is too low;higher initial substrate concentrations increases the likelihood of washout.

Nomenclature

Symbols

D Dilution rate 1/hKI Inhibition constant kg/m3

KS Saturation constant kg/m3


SSmaxXY

Substrate concentration kg/m3Maximum in S for inhibition function kg/m3Biomass concentration kg/m3

Yield coefficient kg/kgSpecific growth rate 1/h

Indices

0Im

Refers to inletRefers to initial valueRefers to maximum

Exercises


Results


1-1-2

Figure 3. Time course of X, S and U.

10 15 20 25 30 35 40TIME


5 i

4.5 -

4

3.5

3

e/> 2.5

2

1.5

1

0.5

0

0.5 2.5

Figure 4. Phase-plane plot of X versus with varying ST from 0 to 5 kg/m3 using Batch Runswith overlay.


Run 9: 2000 steps in 0 seconds

5 -

4.5-

4 -

3.5-

3 •

</) 2.5-

2 •

1.5-

1 •

0.5-

0 •

X,

^^^>*" t**1

• ^ *t %%

\ * * C^1 v"| "% % V

• \ ^ x• i ! l

3:2 (0.5)3:3 (0.5857)

_-- 3:4(0.6714)3:5(0.7571)

— —3:6(0.8429)3:7(0.9286)3:8(1.014)3:9(1.1)

/ / / ; ,--^,f / / / ^' — . *x/ / J j * S ̂ "̂ -s^^-

f f / t / ,- rf'^T— K"̂ "̂ >*i'̂ ' ** -^:' / / f * / Jf*'^ ^^ *̂̂ ^$^S? •**! * \ f 1 C & ( ^^*^^ **

0.5

Figure 5. Phase plane plot of influence of the initial biomass Xi from 0.5 to 1.1 forSteady states upper left and lower right.

= 0.0.

Run 20:2000 steps in 0.0167 seconds

Figure 6. Influence on the inhibition function made by varying KI between 1 and 3.

Reference

Edwards, V.H, Ko, R.C. and Balogh, S.A. (1972). Dynamics and Control ofContinuous Microbial Propagators Subject to Substrate Inhibition Biotechnol.and Bioeng. 14, 939-974.


8.4.3 Nitrification in Activated Sludge Process(ACTNITR)

System

Nitrification is the process of ammonia oxidation by specialized organisms,called nitrifiers. Their growth rate is much slower than that of the heterotrophicorganisms which oxidize organic carbon, and they can be washed out of thereactors by the sludge wastage stream (Fs). In an activated sludge system(Fig. 1) when the organic load (F So/V) is high, then the high biomass growthrates require high waste rates. Nitrification will not be possible under theseconditions because the concentration of nitrifiers (Ni) will become very low.

O,FO 2, F4

Reieto*

2, F3

Figure 1. Configuration and streams for the activated sludge system.

Model

The dynamic balance equations can be written for all components around thereactor and around the settler. The settler is simplified as a well-mixed systemwith the effluent streams reflecting the cell separation.Organic substrate balance for the reactor:

= F0So + F2S2 -


Ammonia substrate balance in the reactor:

R2V t= FQ AQ + F2 A2 - FI A

Reactor balance for the heterotrophic organisms:

pj7 = p2 O2 — FI QI + RI Vj_

Reactor balance for the nitrifying organisms:

ldi = F2N2 - FiNi + R2Vi

Organic substrate balance in the settler:

V2dS2—j^— = FiSi - F3S2 - F4S2

Ammonia substrate balance for the settler:

V2dA2

— 3t — = FIAI -

Balance for heterotrophic organisms in the settler:

- F4A2

V2d02

dt = Fl °l - F3 02

Balance for nitrifying organisms in the settler:

V2 dN2— 34— = FiNi - F3N2

The equations for the flow rates are given below.

Recycle flowrate:F2 = F0R

where R is the recycle factor.

Reactor outlet flow:FI = F2 + F0 = FOR + FO

Flow of settled sludge:


where C is the concentration factor for the settler.

Flow of exit substrate:F4 = FI - F3>

Flow of exit sludge wastage:F5 = F3 - ?2.

Note that C and R must be chosen so that F5 is positive.

Monod-type equations are used for the growth rates of the two organisms.

Rl =

l^2maxR2 = ^Ni =

Program

The program is given on the CD-ROM.

Nomenclature

Symbols

A Ammonia substrate concentration kg/m3

C Concentrating factor for settler -F Flow rate m3/hFo-5 Flow rates, referring to the figure m3/hKI Saturation constant of heterotrophs kg/m3

K2 Saturation constant of nitrifying organisms kg/m3

N Concentration of nitrifiers kg/m3

O Concentration of heterotrophs kg/m3

R Recycle factor -


RlR2sVY

Hi

Growth rate of heterotrophicsGrowth rate of nitrifying organismsOrganic substrate concentrationVolumesYield coefficientsSpecific growth rate of heterotrophs

kg/m3hkg/m3hkg/m3

m3

kg/kg1/h

Specific growth rate of nitrifying organisms 1/h

Indices

Flow and concentration indices referring to Fig. 1 are as follows:0 Refers to feed and initial values1 Refers to reactor and organic oxidation2 Refers to settler and ammonia oxidation3 Refers to recycle4 Refers to settler effluent5 Refers to sludge wastagem Refers to maximum

Exercises


Results

The results in Fig. 2 demonstrate the influence of flow rate on the effluentorganics 82- The ammonia in the effluent A2 is seen, in Fig. 3, to respondsimilarly to FQ, but for a very high value of FQ = 1000 m3/h the nitrificationceases, and A2 becomes the same as the inlet value AQ. This corresponds towashout of the nitrifiers, which would be seen by plotting NI versus time.


0.9 •,

0.8 -I ,/"

°M ifIf JII /It

0.3- rr• J 82:1(20)

02. II 82:2(180)" 82:3(340)

0-1 JM | 82:4(500)

°" .6 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

TIME

Figure 2. Transient of S2 at various flow rates F0 (20 to 500m3/h, bottom to top).


0.1 -I

0.09-

0.08-

3 0.06 -|-_- A2:1(20)— — A2:2(180)

005 J* -_A2:3(340)1 S. I A2:4(5QO)

0.04-

0.03-

0.02J

0 2 4 6 8 10 12 14 16 18 20TIME

Figure 3. Ammonia in the effluent (A2) at various flow rates F0 (5 to lOOOm^/h, bottom totop).


8.4.4 Tubular Enzyme Reactor (ENZTUBE)

System

A tubular, packed-bed, immobilized-enzyme reactor is to be investigated bysimulation. The flow is assumed to be ideal plug flow. The distribution of theenzyme is not uniform and varies linearly from the inlet to higher values at theoutlet, as shown in Fig. 1.

Enzymeconcentration

Enzyme distribution

Distance along reactor, Z

Figure 1. Distribution of enzyme along the tubular reactor.

Model

The equations for steady state operation are given below.

Substrate balance,dS 1dZ = ~ v

Kinetics,

The linear flow velocity is increased by the presence of the solid enzyme carrierparticles according to


FV7 = ~L Ae

The reaction velocity depends on the enzyme concentration,

vm = KE

and the linear distribution of enzyme distribution given by,

E = E0 + mZ

Program

The model is solved by renaming the independent variable, TIME, to be thereactor length coordinate Z. The program is given on the CD-ROM.

Nomenclature

Symbols

A Reactor tube cross section m2

F Flow rate m3/hK Rate constant 1/hKM Michaelis-Menten constant kg/m3

m Enzyme distribution constant kg/m3 mr Reaction rate kg/m3 hS Substrate concentration kg/m3

vm Maximum reaction velocity kg/m3hvz Linear flow velocity m/hZ Reactor length me Void volume fraction of packing

E Enzyme concentration kg/m3


Indices

0S

Refers to inletRefers to substrate

Exercises

Results

Flow rate is the primary operating variable, along with enzyme loading andinlet concentration. In Fig. 2 the influence of F is seen in the steady-state, axial,substrate profile.



7

10 12 14 16 18 20

Figure 2. Substrate profile under the influence of F (1 to 10 m^/h, bottom to top).

8.4.5 Dual Substrate Limitation (DUAL)

System

In defined-nutrient growth media, one substrate can usually be made to belimiting by adjusting its concentration relative to those of the other mediumcomponents. In general, however, more than one substrate may limit the cellgrowth rate. In this case the yield coefficients for the various components,Yxsi> may vary depending upon the growth regime. This situation wasdiscussed by Egli et al. (1989), who examined results at steady state with dualnutrient limitation. The present mathematical model simulates the transientbehaviour of such a dual (Si -carbon, 82 -nitrogen) nutrient-limited systemwhen carried out in a chemostat. The model assumes that the yield coefficientsare each a function of the ratio 81/82, i.e. the ratio of the carbon-nitrogensubstrate concentrations in the vessel. The original paper took the carbon-nitrogen ratio in the feed stream as the controlling parameter. Here theconcentrations in the reactor are assumed to be controlling.

Model

Assuming a perfectly mixed, constant volume continuous-flow stirred-tankreactor, the mass balance equations for the cells and for the two limitingsubstrates are as follows:


1= D (SlFeed - Si) -

D /O O \ I »^__i^^_ I i i "V(o2pee(j — o2) — lvYQO J ]LlA

where D = F/V.

The specific growth rate is modelled as

Si V S2

The yield coefficients are assumed to vary with the carbon-nitrogen ratio in thereactor.

SiRATIO = ^

The yield coefficients are varied according to RATIO using the following logic:

and YXS2 = Y2min if RATIO <BiYXSl=Yim i n and YXS2 = Y2max if RATIO > B2

where,_ Y2min Y2maxBi = \r - and Bo = v~, — T"1 Imax -1 1mm

The boundaries of the three growth regimes in Fig. 1 are defined by thequantities BI and B2.


XSi

C limitation Double limitation

rXS1

N limitation

10

0.8

B2S2

Figure 1. Limitation regions for carbon and nitrogen showing influence on yield.

The yield coefficients for biomass on nitrogen and carbon take maximum orminimum values when only one substrate is limiting and vary linearly withopposing tendencies in the double-limitation region.

Program

Note that the programing of this example is rather more complicated than usualowing to the need to allow for the logical conditions of carbon limitation,nitrogen limitation or both substrates together causing limitation. A partiallisting is seen below and the full program is on the CD-ROM.

(CALCULATION OF YIELD VALUES)YXSl=if (RATIO < Bl) then YlMAX else ( if (RATIO >B2) then Y1MIN else (Y1MAX+(RATIO-B1)/(B2-B1)*(Y1MIN-Y1MAX)) )YXS2 = if (RATIO < Bl) then Y2MIN else ( if (RATIO >B2) then Y2MAX else (Y2MIN+(RATIO-B1)/(B2-

Bl)*(Y2MAX-Y2MIN)) )


Nomenclature

Symbols

BiB2

CcCnDFKSR

SiS2

XYH

Indices

12

Ratio of Y2min/Yimax

Ratio of Y2max/Yimin

Carbon source concentrationNitrogen source concentrationDilution rateVolumetric feed rateAffinity constantReaction ratesCarbon source concentrationNitrogen source concentrationBiomass concentrationYield coefficientSpecific growth rate

Refers to carbon sourceRefers to nitrogen source

_-

kg/m3

kg/m3

1/hm3/hkg/m3

kg/m3 hkg/m3

kg/m3

kg/m3

kg/kg1/h

Exercises


Results

The startup of a continuous culture is shown in Fig. 2. Note that the nitrogenlevel 82 in the reactor drops to a low level after 15 h and causes a change in theyield coefficients. The influence of dilution rate on the system was investigatedby varying D from 0 to 1.5 as shown in Fig. 3.

3-c


1

Figure 2. Startup of a continuous culture.

Run 4: 305 steps in 0 seconds

X 1.5

Figure 3. Variation of D from 0.1 to 1.5 (top to bottom).


Reference

Egli, Th., Schmidt, Ch. R. (1989). "On Dual-Nutrient-Limited Growth ofMicrobes, with Special Reference to Carbon and Nitrogen Substrates", inProceed. Microb. Phys. Working Party of Eur. Fed Biotech. Eds. Th. Egli, G.Hamer and M. Snozzi, Hartung-Goree, Konstanz, 45-53.

This example was developed by S. Mason, ETH-Zurich.

8.4.6 Dichloromethane in a Biofilm Fluidized SandBed (DCMDEG)

System

The process involves the removal of dichloromethane (DCM) from a gas streamand the subsequent degradation by microbial action. The reactor consists ofbiofilm sand bed column with circulation to an aeration tank, into which thesubstrate and oxygen enters in the gas phase, or the substrate can be fed in aliquid stream, as shown in Fig. 1. The column is approximated by a series of sixstirred tanks. The reaction is treated with homogeneous, double saturationkinetics with dichloromethane (DCM) inhibition. The oxidation of one mole ofDCM produces 2 moles of HC1, making a hydrogen ion balance for pH im-portant. The yield with respect to oxygen is 4.3 mg DCM/mg 62. In practice,care must be taken to prevent stripping of DCM to the air stream.


CSR6>

SRin» C jn , pH jn

Figure 1. Schematic of fluidized bed column with external aeration vessel.

Model

The model does not include a gas balance on the aeration tank, since it isassumed that the gas phase dynamics are comparatively fast and hence anequilibrium with the inlet concentration of oxygen and DCM may be assumed.The biomass is assumed to grow slowly, and growth rates are therefore also notmodelled. The model for pH changes does not include buffering effects.

For the inlet section 1 at the bottom of the column the balances are as follows:

O2 balance,dCQ1 ^

dt


DCM balance,

H+ ion balance,

dCSrl _ CSrin~CSrldt

dCHi

t

i -CHI 2rS i84900

Here T is the residence time of the liquid in one section of the column. Theconstant 84,900 converts grams to moles and includes the stoichiometry.

pHi = -0.434 log |Cm|

Evaluation of rates for the inlet section 1:

VmaxCSrl

KI )

-01

For the aeration tank the 62 and DCM balances are:

KLa02(Co2eq-Coin)

dC • R— -~ = — (CSr6 - CSrin )at V DCM (Cs2eq - Tr- (CSFO ~ CSrin

Program

The program constants describe DCM entering the reactor in the gas stream.The DCM concentration in the liquid feed is set to zero. The program is on theCD-ROM.


Nomenclature

oin

srin

CSFOCSGFKIKLa

KS

pHn

R

VR

VTVmaxYSO

H+ ion concentration in section n kg mol/m3

Inlet dissolved oxygen concentration g/m3

Oxygen saturation constant g/m3

DCM saturation constant g/m3

DCM inlet concentration g/m3

DCM concentration in section n g/m3

Oxygen concentration in section n liquid g/m3

DCM concentration in feed g/m3

DCM gas concentration g/m3

Feed rate m3/hInhibition constant g/m3

Transfer coefficients for DCM and ©2 1/hSaturation constants g/m3

pH in n section n pH unitsRecirculation rate m3/hOxygen uptake rate in section n g/m3 hSubstrate uptake rate in section n g/m3 hReactor volume m3

Volume of aeration tank m3

Maximum degradation rate g/m3 hYield coefficient for DCM/oxygenLiquid residence time in one section h

Exercises

•11


Results

The concentrations in the stream leaving the top of the column (CSr6) duringstartup of the fluidized bed are shown in Fig. 2 for four values of F (0.5 to 10)The change of the pH for one flow rate (F = 0.5) is shown in Fig. 3.



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5TIME

Figure 2. Fluidized bed startup for four values of F (0.5 to 10, bottom to top).


3.5

3

2.5

: 2

1.5

1

0.5

0

— CSR6:1... PH6:1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

TIME

Figure 3. Change of carbon substrate and pH in the top section 6 during startup.

Reference

D. Niemann Ph.D. Dissertation 10025, ETH, 1993.


8.4.7 Two-Stage Chemostat with Additional Stream(TWOSTAGE)

System

Two chemostats are arranged in series (Fig. 1) with the intention that the firstoperates at a relatively high rate of cell growth, while the second operates at lowgrowth rate, but high cell density, for secondary metabolite production.Additional substrate may be fed to the second stage.

X1.S!

Figure 1. Two-stage chemostat with two feed streams.

Model

HiUS

, 810

The balance equations are written for each component in each reactor.

Stage 1 with sterile feed,

= F[SO-S!] -


Stage 2 with additional substrate feed and an input of cells and substrate fromStage 1,

V2 - [F + Fi]X2

V2 ^2.= F [Si - S2] + F! [Sio - S2] - vdt Y

Productivity for biomass:

First stage,

Both stages,

KS + S2

Prodi =

Prod2 =

V,

Program

The program is on the CD-ROM.

Nomenclature

Symbols

FKSProdSVXY

Volumetric feed rateSaturation constantProductivity for biomassSubstrate concentrationReactor volumeBiomass concentrationYield coefficientSpecific growth rate

m3/hkg/m3

kg/m3 hkg/m3

3

kg/m3

kg/kg1/h


Indices

01210m

Refers to tank 1 inletRefers to tank 1 and inlet of tank 2Refers to tank 2 and outlet of systemRefers to separate feed for tank 2Refers to maximum

Exercises

Results

The results in Fig. 2 give biomass concentrations and productivities for bothtanks during a startup with a constant feed stream to the first tank (F = 0.5). InFig. 3 the influence on X2 of feed to the second tank Fl (0 to 1.0) withconstant F is shown.



•T5

35 40

Figure 2. Biomass (Xj X2) and productivities for both tanks (F = 0.5).


5

4.5

4

3.5

3

32.52

1.5

1

0.5

010 15 20 25

TIME30 35 40

Figure 3. Influence on X2 of feed to the second tank (Ft = 0 to 1.0, curves right to left).

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