07 Oct 2011Prof. R. Shanthini1 Cellular kinetics and associated reactor design: Reactor Design for Cell Growth CP504 – Lecture 7

07 Oct 2011Prof. R. Shanthini2 Cell Growth Kinetics rXrX = μ C X (41) where μ : specific growth rate (per time) C X : cell concentration (dry cell weight per unit volume) Using the population growth model, we could write the cell growth rate (r X ) as

07 Oct 2011Prof. R. Shanthini3 V for volume of the reacting mixture at time t C X for concentration of the cells in V at time t (r X ) for cell growth rate in V at time t Mass balance for the cell: 0 + (r X ) V = 0 + d(VC X ) / dt which for a batch reactor with constant volume reacting mixture gives (42) Batch Fermenter dC X / dt = r X

07 Oct 2011Prof. R. Shanthini4 Combining (41) and (42), we get dC X = μ C X dt (43) Batch Fermenter If μ is a constant then integrating (43) gives, C X = C X0 exp[μ(t-t 0 )] (44) where C X = C X0 when t = t 0.

07 Oct 2011Prof. R. Shanthini5 Cell Growth Kinetics where μ max and K S are known as the Monod kinetic parameters. Mostly, however, μ is not a constant with time. It depends on C S, the substrate concentration. The most commonly used model for μ is given by the Monod model: Monod Model is an over simplification of the complicated mechanism of cell growth. However, it adequately describes the kinetics when the concentrations of inhibitors to cell growth are low. μ = K S + C S μ m C S (45)

07 Oct 2011Prof. R. Shanthini6 μ m C S = K S + C S (46) CXCX dC X dt Substituting μ in (43) by the Monod Model given by (45), we get Equation (46) could be integrated only if we know how C S changes with either C X or t. How to do that? Batch Fermenter

07 Oct 2011Prof. R. Shanthini7 Stoichiometry could have helped. But we don’t have such a relationship in the case of cellular kinetics. Therefore, we introduce a yield factor (Y X/S ) as the ratio between cell growth rate (r X ) and substrate consumption rate (-r S ) as follows: (47)Y X/S = r X / (-r S ) It is done as follows: Batch Fermenter We know (r X ) from (41) and/or (42). But we don’t know (-r S ). Therefore obtain an expression for (-r S ) as shown in the next slide.

07 Oct 2011Prof. R. Shanthini8 V for volume of the reacting mixture at time t C S for concentration of the Cells in V at time t (r S ) for substrate utilization rate in V at time t Mass balance for substrate: 0 = 0 + (-r S ) V + d(VC S ) / dt which for a batch reactor with constant volume reacting mixture gives (48)dC S / dt = -(-r S ) Batch Fermenter

07 Oct 2011Prof. R. Shanthini9 (48)dC S / dt = -(-r S ) Y X/S = - r S rXrX (47) (42)dC X / dt = r X Combining the above equations, we get dC X / dC S = -Y X/S which upon integration gives (C X – C X0 ) = Y X/S (C S0 – C S )(49) Batch Fermenter

07 Oct 2011Prof. R. Shanthini10 Substituting C S from (49) in (47) and integrating, we get μ m (t - t 0 ) = K S Y X/S C X0 + C S0 Y X/S + 1 ln C X0 CXCX + K S Y X/S C X0 + C S0 Y X/S ln CSCS C S0 (50) where (C X – C X0 ) = Y X/S (C S0 – C S )(49) Batch Fermenter () () )) ( (

07 Oct 2011Prof. R. Shanthini11 Exercise 1: The growth rate of E. coli be expressed by Monod kinetics with μ m = hr -1 and K S = 0.71 g/L. Assume that Y X/S is 0.6 g dry cells per g substrate. C X0 is 1 g/L and C S0 = 10 g/L when the cells start to grow exponentially (i.e., at t = 0). show how C X, C S, and dC X /dt change with respect to time. Batch Fermenter

07 Oct 2011Prof. R. Shanthini12 C S is varied from 10 g/L to 0. () () t = 0.71 x x ln 1 CXCX x x 0.6 () ln CSCS 10 () C X is calculated using (49) as Exercise 1 worked out using the calculator/spread sheet: C X = (10 – C S ) t is calculated using (50) as follows: C X is calculated using (46).

07 Oct 2011Prof. R. Shanthini13 specify C S Calculate C X using (49) Calculate t using (50) Calculate dC X /dt using (46) Continue until C S becomes 0 Exercise 1 worked out using the calculator/spread sheet:

07 Oct 2011Prof. R. Shanthini14 CSCS CXCX Exercise 1 worked out using the calculator/spread sheet:

07 Oct 2011Prof. R. Shanthini15 CSCS CXCX Exercise 1 worked out using the calculator/spread sheet:

07 Oct 2011Prof. R. Shanthini16 Programme written in MATLAB Exercise 1 worked out using an ODE solver: function dydt =CP504Lecture_07(t,y) %data given mumax = 0.935; % per hr Ks = 0.71; % g/L YXS = 0.6; %Monod model mu = mumax*y(2)/(Ks+y(2)); %rate equations rX = mu*y(1); rS = -rX/YXS; dydt=[rX; rS] [t,y] = 10]);

07 Oct 2011Prof. R. Shanthini17 Exercise 1 worked out using an ODE solver: plot(t,y(:,1),'b',t,y(:,2),'r') legend('Cell','Substrate') ylabel('Concentration (g/L)') xlabel('Time (h)')

07 Oct 2011Prof. R. Shanthini18 Exercise 1 worked out using an ODE solver: plot(t,y(:,1),'b',t,y(:,2),'r') legend('Cell','Substrate') ylabel('Concentration (g/L)') xlabel('Time (h)') mumax = 0.935; Ks = 0.71; mu= mumax*y(:,2)./(Ks+y(:,2)); rX = mu.*y(:,1); plot(t,rX,'g')

07 Oct 2011Prof. R. Shanthini19 F F C Xi, C Si C X, C S θ = V/F μ m θ = K S Y X/S C Xi + C Si Y X/S ( + 1 ) ln C Xi CXCX () + K S Y X/S C Xi + C Si Y X/S () ln CSCS C Si () (51) where (C X – C Xi ) = Y X/S (C Si – C S )(52) Plug-flow Fermenter at steady-state

07 Oct 2011Prof. R. Shanthini20 F C Xi, C Si F C X, C S V Continuous Stirred Tank Fermenter (CSTF) at steady-state Mass balance for cells over V: FC Xi + r X V = FC X (53)

07 Oct 2011Prof. R. Shanthini21 Equation (53) gives V F = C X - C Xi rXrX (54) Continuous Stirred Tank Fermenter (CSTF) at steady-state Introducing Dilution Rate D as = (55) F V D = 1 θ in (54), we get 1 D = C X - C Xi rXrX (56)

07 Oct 2011Prof. R. Shanthini22 Since r X = μ C X, (56) becomes 1 D = C X - C Xi μ C X (57) Continuous Stirred Tank Fermenter (CSTF) at steady-state If the feed is sterile (i.e., C Xi = 0), (57) gives C X (D – μ) = 0 (58) which means either C X = 0 or D = μ

07 Oct 2011Prof. R. Shanthini23 CSCS = (60) μ m - D K S D (59) can be rearranged to give C S as D = μ(59) μ m C S K S + C S = If D = μ, then To determine C X, we need to write the mass balance for substrate over the CSTF Continuous Stirred Tank Fermenter (CSTF) at steady-state

07 Oct 2011Prof. R. Shanthini24 F C Xi, C Si F C X, C S V Mass balance for substrate over V: FC Si = FC S + (-r S ) V Continuous Stirred Tank Fermenter (CSTF) at steady-state

07 Oct 2011Prof. R. Shanthini25 which is rearranged to give (-r S ) = D (C Si - C S )(61) Continuous Stirred Tank Fermenter (CSTF) at steady-state r X = D (C X - C Xi ) (56) gives Using the above equations in the definition of yield factor, we get (C X – C Xi ) = Y X/S (C Si – C S )(62)

07 Oct 2011Prof. R. Shanthini26 Since the feed is sterile, (62) gives C X = Y X/S (C Si – C S )(63) (60) is Therefore, we have CXCX = (64) Y X/S ( C Si - ) Continuous Stirred Tank Fermenter (CSTF) at steady-state CSCS = (60) μ m - D K S D μ m - D K S D

07 Oct 2011Prof. R. Shanthini27 which is valid only when D < μ m which is valid only when D < C Si μ m / (K S + C Si ) CSCS = (60) CXCX = (64) Y X/S ( C Si - ) Continuous Stirred Tank Fermenter (CSTF) at steady-state μ m - D K S D μ m - D K S D C Si > K S D / (μ m - D)

07 Oct 2011Prof. R. Shanthini28 Since D < C Si μ m / (K S + C Si ) < μ m D C = C Si μ m / (K S + C Si ) critical value of the Dilution Rate is as follows: Continuous Stirred Tank Fermenter (CSTF) at steady-state (65)

07 Oct 2011Prof. R. Shanthini29 If μ m equals or less than D C, then C X is negative. That is impossible. We need to take the solution C X = 0 of (58), not D = μ So, when μ m equals or less than D C, Substituting C X = 0 in C X = Y X/S (C Si – C S ) gives C S = C Si Continuous Stirred Tank Fermenter (CSTF) at steady-state

07 Oct 2011Prof. R. Shanthini30 C X = 0 means no cell in the reactor. Since the CSTF has a sterile feed (C Xi = 0), no reaction takes place unless we inoculate with the cells once again. So, CSTF gets into a WASHED OUT situation. To avoid CSTF getting into WASHED OUT situation, we need to maintain D = F / V < D C C S = C Si means substrate is not utilised. Continuous Stirred Tank Fermenter (CSTF) at steady-state

07 Oct 2011Prof. R. Shanthini31 Exercise 2 The growth rate of E. coli be expressed by Monod kinetics with μ m = hr -1 and K S = 0.71 g/L. Assume that Y X/S is 0.6 g dry cells per g substrate. The feed is sterile (C Xi = 0) and C Si = 10 g/L. show C X and C S changes with dilution rate. Continuous Stirred Tank Fermenter (CSTF) at steady-state

07 Oct 2011Prof. R. Shanthini32 Exercise 2 worked out using the calculator/spread sheet: D C = C Si μ m / (K S + C Si ) = 10 x / ( ) = per h CSCS = From (60): CXCX =0.6 ( 10 - ) D 0.71 D D 0.71 D Plot the following using excel / MATLAB From (64): g/L From (65):

07 Oct 2011Prof. R. Shanthini33 D C = Exercise 2 worked out using the calculator/spread sheet: