Abstract
A necessary condition is found for the intermediate temperatures and substrate concentrations in a series of CSTR's performing an enzyme-catalyzed reaction which leads to the minimum overall volume of the cascade for given initial and final temperatures and substrate concentrations. The reaction is assumed to occur in a single phase under steady state conditions. The common case of Michaelis-Menten kinetics coupled with first order deactivation of the enzyme is considered. This analysis shows that intermediate stream temperatures play as important a role as intermediate substrate concentrations when optimizing in the presence of nonisothermal conditions. The general procedure is applied to a practical example involving a series of two reactors with reasonable values for the relevant five operating parameters. These parameters are defined as dimensionless ratios involving activation energies (or enthalpy changes of reaction), preexponential factors, and initial temperature and substrate concentration. For negligible rate of deactivation, the qptimality condition corresponds to having the ratio of any two consecutive concentrations as a single-parameter increasing function of the previous ratio of consecutive concentrations.
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Abbreviations
- C E,0 mol.m−3 :
-
Initial concentration of active enzyme
- C E,i mol.m−3 :
-
Concentration of active enzyme at the outlet of the i-th reactor
- C S,0 mol.m−3 :
-
Initial concentration of substrate
- C S,i mol.m−3 :
-
Concentration of substrate at the outlet of the i-th reactor
- Da i :
-
Damköhler number associated with the i-th reactor ((V i.kv,0.CE,0)/(Q.CS,0))
- Da min :
-
Minimum value of the overall Damköhler number
- Da tot :
-
Overall Damköhler number \(\left( {\sum\limits_{j = 1}^n {Da_j } } \right)\)
- E d J.mol−1 :
-
Activation energy of the step of deactivation of the enzyme
- E m J.mol−1 :
-
Standard enthalpy change of the step of binding of substrate to the enzyme
- E v J.mol−1 :
-
Activation energy of the step of enzymatic transformation of substrate
- i :
-
Integer variable
- j :
-
Dummy integer variable
- k :
-
Dummy integer variable
- k d,i s−1 :
-
Kinetic constant associated with the deactivation of enzyme in the i-th reactor (k d,o·exp{−E d/(R.T i})
- k d,0 s−1 :
-
Preexponential factor of the kinetic constant associated with the deactivation of the enzyme
- K m,i mol.m−3 :
-
Equilibrium constant associated with the binding of substrate to the enzyme in the i-th reactor, (k m,o·exp{−E m}(R.T i})
- K m,0 mol.m−3 :
-
Preexponential factor of the Michaelis-Menten constant associated with the binding of substrate to the enzyme
- k v,i s−1 :
-
Kinetic constant associated with the transformation of the substrate by the enzyme in the i-th reactor (k v,o·exp{−E v/(R.T i}))
- k v,0 s−1 :
-
Preexponential factor of the kinetic constant associated with the transformation of the substrate by the enzyme
- N :
-
Number of reactors in the series
- Q m3.s−1 :
-
Volumetric flow rate of reacting liquid through the reactor network
- R J.K−1.mol−1 :
-
Ideal gas constant
- T i K:
-
Absolute temperature at the outlet of the i-th reactor
- T 0 K:
-
Initial absolute temperature
- V i m3 :
-
Volume of the i-th reactor
- v max mol.m−3.s−1 :
-
Maximum rate of reaction under saturation conditions of substrate
- x i :
-
Normalized concentration of substrate (CS,i/CS, 0)
- x i,opt :
-
Optimum value of the normalized concentration of substrate
- y i :
-
Dimensionless temperature (exp{−T 0/T i})
- y i,opt :
-
Optimum value of the dimensionless temperature
- α :
-
Dimensionless preexponential factor associated with the Michaelis-Menten constant (K m,0/Cs,0)
- β :
-
Dimensionless activation energy of the step of enzymatic transformation of substrate (E v/R.T0))
- γ :
-
Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E m/(R.T0))
- δ :
-
Dimensionless activation energy of the step of deactivation of the enzyme (E d/(R.T0))
- η :
-
Dimensionless deactivation preexponential factor ((k d,0.CS,0)/(kv,0.CE,0)
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Paiva, A.L., Malcata, F.X. Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation. Bioprocess Eng. 9, 77–82 (1993). https://doi.org/10.1007/BF00369034
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DOI: https://doi.org/10.1007/BF00369034