1. Prof. R. Shanthini 23 Nov 2012 1 Enzyme kinetics and associated reactor design: Enzyme Reactor Design CP504 – ppt_Set 05

2. Prof. R. Shanthini 23 Nov 2012 2 V for volume of the reacting mixture at time t C S for concentration of the substrate in V at time t (-r S ) for substrate utilization rate in V at time t Mass balance for the substrate: 0 = 0 + d(VC s )/dt + (-r S )V - dC s /dt = -r S For a batch reactor with constant volume reacting mixture, the above becomes (29) Batch Enzyme Reactor

3. Prof. R. Shanthini 23 Nov 2012 3 Substituting (-r S ) for the simple enzyme reaction in (29), we get (30) r max C S K M + C S = dC S dt - Rearranging (30), we get r max dt = K M + C S CSCS - dC S ( ∫ ) C S0 CSCS ∫ 0 t (31) Integrating (31) gives (32) r max t = K M ln CSCS (C S0 – C S ) ) C S0 ( + Batch Enzyme Reactor

4. Prof. R. Shanthini 23 Nov 2012 4 (33) t = - K M + (C S0 – C S ) ln(C S0 /C S ) r max (32) in linear form becomes t ln(C S0 /C S ) (C S0 – C S ) ln(C S0 /C S ) r max - K M Determination of M-M kinetic parameters Batch Enzyme Reactor

5. Prof. R. Shanthini 23 Nov 2012 5 Mass balance for the substrate over dV: FC S = F(C S + dC S ) + (-r S ) dV The above can be simplified to F F C S0 C Sf dV F C S +dC S F CSCS - FdC S / dV = -r S dV for small volume of the reacting mixture C S for concentration of the substrate (-r S ) for substrate utilization rate in dV F for the steady flow rate through the reactor Plug-flow Enzyme Reactor at steady-state

6. Prof. R. Shanthini 23 Nov 2012 6 F F C S0 C Sf dV F C S +dC S F CSCS dV for small volume of the reacting mixture C S for concentration of the substrate (-r S ) for substrate utilization rate in dV F for the steady flow rate through the reactor Plug-flow Enzyme Reactor at steady-state Introducing space-time θ ( = V/F), we get - dC S / dθ = -r S which is very similar to (29) (34)

7. Prof. R. Shanthini 23 Nov 2012 7 Plug-flow Enzyme Reactor at steady-state Substituting (-r S ) for the simple enzyme reaction in (34), we get (35) r max C S K M + C S = dC S dθdθ - Rearranging (33), we get r max dθ = K M + C S CSCS - dC S ( ∫ ) C S0 CSCS ∫ 0 θ (36) Integrating (34) gives (37) r max θ = K M ln CSCS (C S0 – C S ) ) C S0 ( +

8. Prof. R. Shanthini 23 Nov 2012 8 (38) θ = - K M + (C S0 – C S ) ln(C S0 /C S ) r max (37) in linear form becomes θ ln(C S0 /C S ) (C S0 – C S ) ln(C S0 /C S ) r max - K M Determination of M-M kinetic parameters Plug-flow Enzyme Reactor at steady-state

9. Prof. R. Shanthini 23 Nov 2012 9 V for the volume of the reacting mixture C S for concentration of the substrate in the reactor and at the exit (-r S ) for substrate utilization rate in V F for the steady flow rate through the reactor F C S0 F CSCS V CSCS Continuous Stirred Tank Enzyme Reactor at steady-state

10. Prof. R. Shanthini 23 Nov 2012 10 F C S0 F CSCS V CSCS Continuous Stirred Tank Enzyme Reactor at steady-state Mass balance for the substrate over V: FC S0 = FC S + (-r S ) V(39)

11. Prof. R. Shanthini 23 Nov 2012 11 Continuous Stirred Tank Enzyme Reactor at steady-state Introducing space-time θ ( = V/F) in (39), we get C S0 = C S + (-r S ) θ(40) Substituting (-r S ) for the simple enzyme reaction in (40), we get (41) r max C S K M + C S + C S0 = C S θ

12. Prof. R. Shanthini 23 Nov 2012 12 Continuous Stirred Tank Enzyme Reactor at steady-state (42) (41) in linear form becomes CSCS CS θCS θ = - K M + (C S0 – C S ) r max CS θCS θ (C S0 -C S ) CSCS r max - K M Determination of M-M kinetic parameters