Comments Transient Material Balances
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1 Comments Transient aterial Balances Description of cell mass growth Qualitative ubstrates Cells extracelluar Products more Cells Quantitative X P nx i i toichiometry (example, aerobic) CHmO n a O b NH c CH O N d H O e CO subtrate biomass 2 1
2 aterial Balance Batch Reactor Cell Balances: VR VRgXVRk VRnetX 1 net X ubstrate Consumption & Product Growth: q q P 1 d X 1 dp X 3 aterial Balance Batch Reactor If net is constant then get exponential growth phase netx net X X ln net t X X0 exp net t X0 Followed by deceleration growth (unbalanced growth) & stationary (growth equal to death) phases 4 2
3 aterial Balance Batch Reactor Death phase is 1 st order in cell concentration & gives exponential decay kx d X ln k X X0 exp nett X0 5 ome Growth odels ubstrate-limited Growth (oser equation, onod for n=1) n m g K n ubstrate-limited Growth (Contois equation) m g K X X Noncompetitive ubstrate Inhibition m g K 1 1 K1 Competitive ubstrate Inhibition m g K 1 K1 Noncompetitive Product Inhibition m g K P 1 1 K p Competitive Product Inhibition m g P K 1 K p 6 3
4 onod Growth odel ubstrate-limited Growth / m g K g K 1 / K Also: m Limits: Constant growth rate at large substrate concentrations Proportional to substrate concentration at low concentrations m gk g K m g 7 aterial Balances Ideal Chemostat (ection 6.3.2) 8 4
5 aterial Balances Ideal Chemostat (CTR) Cell balance: VR FX0 FXVRgXVRk DX0 g kd DX where: D F/V R Usually feed is cell mass & product free g d net k D X D X 9 aterial Balances Ideal Chemostat (CTR) At steady state & negligible death rate 0 g D X g D Growth rate can be controlled by changing the dilution rate! However, if the dilution rate is too large then the cell mass is washed out the culture cannot reproduce fast enough to grow before it is removed 10 5
6 aterial Balances Ideal Chemostat (CTR) ubstrate balance d gx qx P VR F0 FVR mx s YX/ Y P/ At steady state g q D 0 P g qp 0 D0 m s X m s YX/ YP/ X YX/ YP/ Linear equation of substrate consumption Grow cell mass Create product Provide energy to the cell mass 11 aterial Balances Ideal Chemostat (CTR) If negligible product formation & maintenance, then: D 0 g D X YX/ 0 X YX/ g Y ubstrate (for onod eqn): m K g g K m g X/ 0 K g KD X YX/ 0 Y X/ 0 m g m D 12 6
7 aterial Balances Ideal Chemostat (CTR) Product formation steady state with introduction of cell mass (but no net growth): From cell balance: From substrate balance: 0 DX0 net D X X X0 From product yield definition: g qp 1 qpx 0 D0 ms X 0 YX/ YP/ DYP/ PP Y 0 P/ 0 13 Other Configurations Chemostat with Recycle 14 7
8 Other Configurations ulti-tage Chemostat 15 Other Configurations Fed Batch 16 8
9 Other Configurations Perfusion 17 Use of Batch Data in Flow Reactors For a batch reactor net X For a CTR it makes sense that the outlet concentration is related to the batch reactor s results such that: X X net 0 batch ttextent where t extent is some characteristic batch time that represents the extent of reaction 18 9
10 Use of Batch Data in Flow Reactors For a chemostat the dilution factor D controls the growth factor net You can relate the two systems & show performance by Plot / vs X for the batch data Plot a straight line through X 0 on the horizontal axis with a slope of D The intersection of the batch results curve & the chemostat performance line will give the value of X within the chemostat. The original batch X vs. t data will then give the corresponding t extent. Product composition can be determined either by: Find the corresponding P at t extent, or Do a similar DP/ vs. P analysis 19 Use of Batch Data in Flow Reactors Using data from Example 6.2, ethanol from glucose using. cerevisiae Time derivatives estimated from central differences 20 10
11 Use of Batch Data in Flow Reactors For a chemostat, D=0.05 h Use of Batch Data in Flow Reactors For a batch reactor productivity is the time-derivative increase in concentration vs. time. For a CTR the analogous term is the dilution factor times the concentration, e.g., D P 22 11
12 Details for Other Bioreactor Configurations 23 Other Configurations Chemostat with Recycle 24 12
13 aterial Balances Chemostat with Recycle Cell balance: 1 1 DX0 CX1 1 DX1 netx1 Ratio recycle flowrate to fresh feed rate 1 V FX0 F CX1 FX1VnetX1 C Concentration factor in Cell eparation At steady state with X 0 =0 1 1 C 0 D CX 1 DX X net D 1 1 net 1 25 aterial Balances Chemostat with Recycle Cell balance around Cell steady state: FX1F CX1 FX2 X2 CX1 ince C > 1 then X 2 < X
14 aterial Balances Chemostat with Recycle ubstrate balance d gx1 qx P 1 VR F0 F1FVR msx 1 YX/ YP/ d gx1 qx P 1 D0 m sx1 YX/ YP/ At steady state & growth limited gx1 qx P 1 0 D0 msx1 YX/ YP/ 1 X/ 0 1 g D YX/ 0 X Y X 1 1C 27 Other Configurations ulti-tage Chemostat 28 14
15 aterial Balances ulti-tage Chemostat Cell balance 2 reactors in series 1 V1 FX0 FX1 net,1xv V FX FX FFX net X V , st reactor looks like a single reactor. Focus on the downstream reactor(s) At steady state with X 0 =0 Now growth rate dependent on cell mass compositions F F F X F X 0 FX 1 F F X2 net,2x2v2 net,2 D2 V V X V X aterial Balances ulti-tage Chemostat ubstrate balance focus on 2 nd reactor d2 V 2 F1F0 FF 2 g,2 qp V2 ms X 2 YX/ YP/ At steady state with only cell mass growth: g,2x2 0 F 1 F 0 F F 2 V2 Y 2 X/ F 1 F 0 g X V2 F1 F 0 g X FF Y FF FF D Y,2 2,2 2 X/ 2 X/ 30 15
16 aterial Balances ulti-tage Chemostat ust simultaneously solve the 3 equations for cell mass & substrate concentrations as well as growth rate For onod eqn: F X1 g,2 D2 V2 X2 F 1 F 0 g X 2 F F DY 2 / m2 g,2 K 2,2 2 X 31 aterial Balances ulti-tage Chemostat Care must be taken to specify an iteration technique to solve this set of non-linear equations implest technique would be direct substitution, but it is doubtful that this would be a robust way to solve 32 16
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