Cost minimization in the predesign of an enzymatic CSTR: an overall approach

The balance equations pertaining to the modelling of a CSTR performing an enzyme-catalyzed reaction in the presence of enzyme deactivation are developed. Combination of heuristic correlations for the size-dependent cost of equipment and the purification-dependent cost of recovery of product with the mass balances was used as a basis for the development of expressions relating a (suitably defined) dimensionless economic parameter with the optimal outlet substrate concentration under the assumption that overall production costs per unit mass of product were to be minimized. The situation of Michaelis-Menten kinetics for the substrate depletion and first order kinetics for the deactivation of enzyme (considering that the free enzyme and the enzyme in the enzyme/substrate complex deactivate at different rates) was explored, and plots for several values of the parameters germane to the analysis are included.List of Symbols C _E mol m^–3 concentration of active enzyme - C _E,0 mol m^–3 initial concentration of active enzyme - C _p mol m^–3 concentration of product of interest - C _s mol m^–3 concentration of substrate - C _s,0 mol m^–3 initial concentration of substrate - I $ capital cost of equipment - k _d s^–1 deactivation constant of free enzyme - k _d s^–1 deactivation constant of enzyme in enzyme/substrate complex - K _m mol m^–3 Michaelis-Menten constant - K _m dimensionless counterpart of K _m - k _r s^–1 rate constant associated with conversion of enzyme/substrate complex into product - M _w kg mol^–1 molecular weight of product of interest - P $ kg^–1 cost of recovery of product of interest in pure form - Q m³s^–1 volumetric flow rate - V m³ volume of reactor - X $ kg^–1 global manufacture cost of product of interest in pure form - X dimensionless counterpart of X Greek Symbols ₁ $ m^–1.8 constant - ₂ $ m^–3 constant - t s useful life of CSTR - ₀ ratio of initial concentrations of enzyme and substrate - ratio of deactivation constant of free enzyme to rate constant of depletion of substrate - ratio of deactivation constants - univariate function expressing the dependence of the rate of enzyme deactivation on C _S - univariate function expressing the dependence of the rate of substrate depletion on C _S - dimensionless economic parameter     

Optimal temperature and concentration profiles in a cascade of CSTR's performing Michaelis-Menten reactions with first order enzyme deactivation

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.List of Symbols 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.k_v,0.C_E,0)/(Q.C_S,0)) - Da _min Minimum value of the overall Damköhler number - Da _tot Overall Damköhler number - 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 (C_S,i/C_{S, 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 Greek Symbols Dimensionless preexponential factor associated with the Michaelis-Menten constant (K _m,0/C_s,0) - Dimensionless activation energy of the step of enzymatic transformation of substrate (E _v/R.T₀)) - Dimensionless standard enthalpy change of the step of binding of substrate to the enzyme (E _m/(R.T₀)) - Dimensionless activation energy of the step of deactivation of the enzyme (E _d/(R.T₀)) - Dimensionless deactivation preexponential factor ((k _d,0.C_S,0)/(k_v,0.C_E,0)     

New solution method for equations of reaction and mass transfer in biocatalyst particles

Summary For numerical solution of the reaction-mass transfer equations for immobilised biocatalysts it may be better to start integration at the particle surface and proceed inwards: calculations are targetted on the region to which practically interesting changes are often confined (because concentrations are effectively zero in the interior); and during iterative solution wrong initial estimates may be rejected after detecting anomalies early in the integration.Symbols C_b substrate concentration in bulk (mol m^–3) - c dimensionless substrate concentration (C/C_b) (-) - D_e effective diffusion coefficient (m²s^–1) - Da Damkohler number (V.r_o ²/D_e.K_s) (-) - K_s substrate concentration kinetic coefficient (mol m^–3) - k_e external mass transfer coefficient (ms^–1) - r_o bead radius (m) - Sh Sherwood number (k_e.r_o/D_e) (-) - V maximum rate per unit volume in beads (mol m^–3s^–1) - x dimensionless distance from bead centre (r/r_o) (-) - dimensionless kinetic coefficient (K_s/C_b) (-) - _o effectiveness factor (-)     

Intraparticle mass-transfer resistance and apparent time stability of immobilized yeast alcohol dehydrogenase

The stability and, consequently, the lifetime of immobilized enzymes (IME) are important factors in practical applications of IME, especially so far as design and operation of the enzyme reactors are concerned. In this paper a model is presented which describes the effect of intraparticle diffusion on time stability behaviour of IME, and which has been verified experimentally by the two-substrate enzymic reaction. As a model reaction the ethanol oxidation catalysed by immobilized yeast alcohol dehydrogenase was chosen. The reaction was performed in the batch-recycle reactor at 303 K and pH-value 8.9, under the conditions of high ethanol concentration and low coenzyme (NAD⁺) concentration, so that NAD⁺ was the limiting substrate. The values of the apparent and intrinsic deactivation constant as well as the apparent relative lifetime of the enzyme were calculated.The results show that the diffusional resistance influences the time stability of the IME catalyst and that IME appears to be more stabilized under the larger diffusion resistance.List of Symbols C _A, C_B, C_E mol · m^–3 concentration of coenzyme NAD⁺, ethanol and enzyme, respectively - C _p mol · m³ concentration of reaction product NADH - d _p mm particle diameter - D _eff m² · s^–1 effective volume diffusivity of NAD⁺ within porous matrix - k _d s^–1 intrinsic deactivation constant - K _A, K_A, K_B mol · m^–3 kinetic constant defined by Eq. (1) - K _A ^x mol · m^–3 kinetic constant defined by Eq. (5) - r _A mol · m^–3 · s^–1 intrinsic reaction rate - R m particle radius - R _v mol · m^–3 · s^–1 observed reaction rate per unit volume of immobilized enzyme - t _E s enzyme deactivation time - t _r s reaction time - V mol · m^–3 · s^–1 maximum reaction rate in Eq. (1) - V ^x mol · m^–3 · s^–1 parameter defined by Eq. (4) - V _f m³ total volume of fluid in reactor - w _s kg mass of immobilized enzyme bed - factor defined by Eqs. (19) and (20) - kg · m^–3 density of immobilized enzyme bed - unstableness factor - effectiveness factor - Thiele modulus - relative half-lifetime of immobilized enzyme Index o values obtained with fresh immobilized enzyme     

Verification studies of a simplified model for the removal of dichloromethane from waste gases using a biological trickling filter

The removal of dichloromethane from waste gases in a biological trickling filter was studied experimentally as well as theoretically within the concentration range of 0–10,000 ppm. A stable dichloromethane elimination performance was achieved during two years of operation, while the start-up of the system only amounted to several weeks at constant inlet concentrations. The trickling filter system was operated co-currently as well as counter-currently.However, experimental and theoretical results revealed that the relative flow direction of the mobile phases did not significantly affect the elimination performance. Moreover, it was found that the gas-liquid mass-transfer resistance in the trickling filter bed applied was negligible, which leaves the biological process inside the biofilm to be the rate limiting step.A simplified model was developed, the Uniform-Concentration-Model, which showed to predict the filter performance close to the numerical solutions of the model equations. This model gives an analytical expression for the degree of conversion and can thus be easily applied in practice.The dichloromethane eliminating performance of the trickling filter described in this paper, is reflected by a maximum dichloromethane elimination capacity EC _max=157 g/(m³ · h) and a critical liquid concentration C _lcr=45 g/m³ at a superficial liquid velocity of 3.6 m/h, inpendent of the gas velocity and temperature.List of Symbols a _s m²/m³ specific area - a _w m²/m³ specific wetted area - A m² cross-sectional area - C _g g/m³ gas phase concentration - C _go g/m³ inlet gas phase concentration - C _gocr g/m³ critical gas phase concentration - C _g ^* C_g/C_go dimensionless gas concentration - C _l g/m³ liquid concentration - C _lcr g/m³ critical liquid concentration - C _lcr ^* mC_lcr/C_go dimensionless critical concentration - c _li g/m³ substrate concentration at liquid-biofilm interface - C _l ^* mC_l/C_go dimensionless liquid concentration - C _o g/m³ oxygen concentration inside the biofilm - C _oi g/m³ oxygen concentration at liquid-biofilm interface - C_s g/m³ substrate concentration inside the biofilm - C _si g/m³ substrate concentration at liquid-biofilm interface - D _eff m²/h effective diffusion coefficient in the biofilm - D _o m²/h effective diffusion coefficient for oxygen in the biolayer - E mu_g/u_l extraction factor - E _act kJ/mol activation energy for the biological reaction - EC g/(m³· h) K _o a _w : elimination capacity, or the amount of substrate degraded per unit of reactor volume and time - EC _max g/(m³ · h) K _o a_w: maximum elimination capacity - f degree of conversion - h m coordinate in height - H m height of the packed bed - K ₀ g/(m³ · h) _maxX_b/Y zeroth order reaction defined per unit of biofilm volume - k _og m/h overall gas phase mass transfer coefficient - K ^* dimensionless constant given by Eq. (A.5) - K _l ^* dimensionless constant given by Eq. (A.6) - K ₂ ^* dimensionless constant given by Eq. (A.6) - m C _g /C_l gas liquid distribution coefficient - N g/(m² · h) liquid-biofilm interfacial flux of substrate - N _og k_oga_wH/u_g number of gas phase transfer units - N _r k_o a_w H/u_g C_go number of reaction units - OL g/(m³· h) u _g C _go /H organic load - r _s g/(m³ ·h) zeroth order substrate degradation rate given by Eq. (1) - R _s g/(g TSS ·h) specific activity - T K absolute temperature - u _g m/h superficial gas velocity - u _t m/h superficial liquid velocity - X _b g TSS/m₃ biomass concentration inside biofilm - X _s g TSS/m₃ liquid suspended biomass concentration - x m coordinate inside the biofilm - Y g TSS/(g_DCM) yield coefficient Greek Symbols dimensionless parameter given by Eq. (2) - m averaged biofilm thickness - biofilm effectiveness factor given by Eqs. (7a)–(7c) - m penetration depth of substrate into the biofilm - _max d^–1 microbiological maximum growth rate - v _o stoichiometric utilization coefficient for oxygen - v _s stoichiometric utilization coefficient for substrate - dimensionless height in the filter bed - h H/u _g superficial gas phase contact time - _o (K ₀ /DC _ii )^1/2 - _o C _o /C _oi dimensionless oxygen concentration inside the biofilm - _s C _s /C _si dimensionless substrate concentration inside the biofilm Experimental results, verifying the model presented will be discussed Part II (to be published in Vol. 6, No. 4)     

Estimation of oxygen penetration depth in immobilized cells

Summary A simple method is proposed for calculating oxygen pentration depth in immobilized cells by assuming zero order kinetics in the presence of several external oxygen transport resistances. Calculations indicate that typical penetration depths of oxygen for immobilized microbial cells are in the range of 50–200 and those for immobilized or encapsulated animal and plant tissue culture are about 500–1000 . Based on calculations, oxygen transport in microencapsulation and microcarriers for tissue cultures are not transport-limited, but a slight limitation is expected for those in a hollow fiber reactor.Nomenclature a_s specific area of a support (cm) - Bi Biot number - dimensionless - C_b oxygen concentration in the bulk liquid (mM) - C _b C _b ^* -C_cr (mM) - C _b ^* bulk oxygen concentration in equilibrium with air (mM) - C_cr critical oxygen concentration (mM) - C_s oxygen concentration in the solid phase (mM) - d_p diameter or thickness of a support (cm) - D_eff effective diffusivity of oxygen in the solid phase (cm²/s) - k_m membrane permeability of oxygen (cm/s) - k _m ^* D_eff/_m - k_La_L liquid phase mass transfer rate coefficient (1/s) - k_sa_s solid phase mass transfer rate coefficient (1/s) - (OUR)_v volumetric oxygen uptake rate (mmol O₂/l) - p geometry parameter, p=0 for slab, p=1 for cylinder, p=2 for sphere - P_d oxygen penetration depth (cm) - P _d oxygen penetration depth in the absence of external diffusion limitation (cm) - Q volumetric oxygen uptake rate, (mmol O₂/l·h) - specific oxygen uptake rate (mmol O₂gm biomass (dry)·h) - r length coordinate (cm) - r_c oxygen penetration depth for sphere (cm) - r _c r_c in the absence of external diffusion limitation (cm) - r _c ^* oxygen penetration depth for cylinder (cm) - r _c ^* r _c ^* in the absence of external diffusion limitation (cm) - r_com combined mass transfer rate resistance (s) - r_d location where C_s becomes zero or C_cr (cm) - r_i radius of cylinder or sphere, half thickness of slab (cm) - U_sg superficial gas velocity (cm/s) - X cell concentration (g/l) Greek letters Thiele modulus, dimensionless - _L, _s liquid and solid phase volume fraction, respectively, dimensionless - effectiveness factor On sabbatical leave from KAIST, Seoul, Korea     

Observer design for differential-algebraic model of an aerobic culture of a recombinant yeast

The mathematical model of an aerobic culture of recombinant yeast presented in work by Zhang et al. (1997) is given by a differential-algebraic system. The classical nonlinear observer algorithms are generally based on ordinary differential equations. In this paper, first we extend the nonlinear observer synthesis to differential-algebraic dynamical systems. Next, we apply this observer theory to the mathematical model proposed in Zhang et al. (1997). More precisely, based on the total cell concentration and the recombinant protein concentration, the observer gives the online estimation of the glucose, the ethanol, the plasmid-bearing cell concentration and a parameter that represents the probability of plasmid loss of plasmid-bearing cells. Numerical simulations are given to show the good performances of the designed observer.Symbols C ₁ activity of pacing enzyme pool for glucose fermentation (dimensionless) - C ₂ activity of pacing enzyme pool for glucose oxidation (dimensionless) - C ₃ activity of pacing enzyme pool for ethanol oxidation (dimensionless) - E ethanol concentration (g/l) - G glucose concentration (g/l) - k _a regulation constant for (g glucose/g cell h^–1) - k _b regulation constant for (dimensionless) - k _c regulation constant for (g glucose/g cell h^–1) - k _d regulation constant for (dimensionless) - K _m1 saturation constant for glucose fermentation (g/l) - K _m2 saturation constant for glucose oxidation (g/l) - K _m3 saturation constant for ethanol oxidation (g/l) - L ( t) time lag function (dimensionless) - p probability of plasmid loss of plasmid-bearing cells (dimensionless) - P recombinant protein concentration (mg/g cell) - q _G total glucose flux culture time (g glucose/g cell h) - t culture time (h) - t _lag lag time (h) - X total cell concentration (g/l) - X ⁺ plasmid-bearing cell concentration (g/l) - Y ^F _{X / G} cell yield for glucose fermentation pathway (g cell/g glucose) - Y ^O _{X / G} cell yield for glucose oxidation pathway (g cell/g glucose) - Y _{X / E} cell yield for ethanol oxidation pathway (g cell/g ethanol) - Y _{E / X} ethanol yield for fermentation pathway based on cell mass (g ethanol·g cell) - ₂ glucoamylase yield for glucose oxidation (units/g cell) - ₃ glucoamylase yield for ethanol oxidation (units/g cell) - µ₁ specific growth rate for glucose fermentation (h^–1) - µ₂ specific growth rate for glucose oxidation (h^–1) - µ₃ specific growth rate for ethanol oxidation (h^–1) - µ_1max maximum specific growth rate for glucose fermentation (h^–1) - µ_2max maximum specific growth rate for glucose oxidation (h^–1) - µ_3max maximum specific growth rate for ethanol oxidation (h^–1)     

Strategy for the simulation of batch reactors when the enzyme-catalyzed reaction is accompanied by enzyme deactivation

The balance equations pertaining to the modelling of batch reactors performing an enzyme-catalyzed reaction in the presence of enzyme deactivation are developed. The functional form of the solution for the general situation where both the rate of the enzyme-catalyzed reaction and the rate of enzyme deactivation are dependent on the substrate concentration is obtained, as well as the condition that applies if a maximum conversion of substrate is sought. Finally, two examples of practical interest are explored to emphasize the usefulness of the analysis presented.List of Symbols C _E mol/m³ concentration of active enzyme - C _E,O mol/m³ initial concentration of active enzyme - C _S mol/m³ concentration of substrate - C _S,O mol/m³ initial concentration of substrate - C _S,min mol/m³ minimum value for the concentration of substrate - k 1/s first order rate constant associated with conversion of enzyme/substrate complex into product - k ₁ 1/s first order deactivation constant of enzyme (or free enzyme) - k ₂ 1/s first order deactivation constant of enzyme in enzyme/substrate complex form - K _m mol/m³ Michaelis-Menten constant - p mol/(m³s) time derivative of C _S - q mol/m³ auxiliary variable - t s time elapsed after reactor startup Greek Symbols 1/s univariate function expressing the dependence of the rate of enzyme deactivation on C _S - mol/m³ dummy variable of integration - mol/m³ dummy variable of integration - 1/s univariate function expressing the dependence of the rate of substrate depletion on C _S - m³/(mol s) derivative of with respect to C _S     

Determination of the inherent kinetics of immobilized glucose oxidase using a diffusion cell

The enzyme glucose oxidase (GO) was covalently immobilized onto a poly(vinyl alcohol) hydrogel, cross-linked with glutardialdehyde and a polyazonium salt. To compare the kinetic parameters of immobilized GO with the known kinetic parameters of soluble GO, the diffusion cell method was used.Between two compartments, containing solutions with different glucose concentrations, a GO-containing hydrogel membrane was placed. Simultaneous diffusion through and enzymatic reaction in the membrane occurred. In this way diffusional effects of the membrane could be eliminated from the effective kinetic parameters to yield the inherent kinetic parameters.It appeared that the enzymatic reaction is independent of the oxygen concentration at oxygen concentrations 0.22 mol m^–3 (Michaelis constant for oxygen < 0.22 mol m^–3). Further, the Michaelis constant for glucose does not change dramatically after immobilizing the enzyme. The maximal reaction rate is depending on the enzyme concentration. As the enzyme concentration in the membrane is not exactly known (mainly due to leakage of enzyme out of the membrane during membrane preparation), only an estimation of the turnover number can be made.The diffusion cell method is easy to carry out. Still, some recommendations can be made on the performance.List of Symbols _g , _0x partition coefficient of glucose and oxygen, respectively - thickness of the wetted membrane (m) - A _m surface area of membrane (m^–2) - C constant (mol² m^–3) - c _g , c _0x concentration of glucose and oxygen, respectively (mol m^–3) - c _g,0 c _g, glucose concentration at the filter-paper/membrane interface next to compartment A and B, respectively (mol m^–3) - c _{g, A} c _{g, B} glucose concentration in compartment A and B, respectively (mol m^–3) - c _GO glucose oxidase concentration (mol m^–3) - D _eff effective diffusion coefficient (m² s^–1) - D _m , D _sl diffusion coefficient in, respectively, the membrane and the solution layer (m² s^–1) - d _dl , d _df , d _sl thickness of, respectively, the diffusion layer, the filter-paper and the solution layer (m) - h _B initial slope of concentration versus time curve of compartment B (mol m^–3 s^–1) - J flux (mol m^–2 s^–1) - J ₀ flux in the membrane at membrane/filter-paper interface next to compartment A and B, respectively (mol m^–2 s^–1) - J _A , J _B flux leaving compartment A and entering compartment B, respectively (mol m^–2 s^–1) - J _m flux through the membrane (mol m^–2 s^–1) - k total mass transfer coefficient (m s^–1) - k ₁ , k ₂ rate constant of a particular reaction step (m³ mol^–1 s^–1) - k_–1, k_–2 rate constant of a particular reaction step (s^–1) - k _cat (intrinsic) catalytic constant of turnover number (s^–1) - k _cat ^* inherent catalytic constant, determined by inserting D _m (s^–1) - k _cat ^** inherent catalytic constant, determined by inserting D _eff (s^–1) - k _m (g) (intrinsic) Michaelis constant for glucose (mol m^–3) - k _m (o) (intrinsic) Michaelis constant for oxygen (mol m^–3) - k _m ^* (g) inherent Michaelis constant for glucose (mol m^–3) - k _m ^* (o) inherent Michaelis constant for oxygen (mol m^–3) - m _GO number of moles of GO present (mol) - P _m permeability of glucose in the mebrane (m s^–1) - P _eff effective permeability (m s^–1) - V volume (m³) - v ₀ initial reaction velocity (mol m^–3 s^–1) - V _max ^** inherent maximal reaction velocity, determined by inserting D_eff (mol m^–3 s^–1) - x distance (m)     

To what extent is a constant volume design worse than a minimum volume design for a series of CSTR's?

The balance equations for substrate in a cascade of CSTR's undergoing an enzyme-catalyzed reaction following Michaelis-Menten kinetics are developed in dimensionless form. Analytical expressions relating the intermediate concentrations are independently obtained for the cases of minimum overall volume and constant volume. The fractional deviations between the overall volumes following these two design criteria are calculated and presented for several values of the relevant parameters. For situations of practical interest, the fractional deviation is below 10%. Increasing values of the Michaelis-Menten parameter, K _m(or decreasing values of the number of reactors in the cascade, N) lead to lower values of the maximum deviation; this maximum deviation is attained at lower conversions of substrate when K _mis increased or N decreased.List of Symbols C _{S, i}mol.m^–3 concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i} normalized concentration of substrate at the outlet of the i-th reactor - C _* ^{S, i, eq} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of constant volume - C _* ^{S, i, opt} normalized concentration of substrate at the outlet of the i-th reactor using the design criterion of minimum overall volume - C _{S, 0} mol.m^–3 concentration of substrate at the inlet to the first reactor - Da _i Damköhler number for the i-th reactor - Da _eq constant Damköhler number for each reactor of the cascade - Da _{tot, eq} overall Damköhler number for the cascade assuming equal-sized reactors - Da _{tot, min} minimum overall Damköhler number for the cascade - Er fractional deviation between the overall volumes using the two different design criteria - K _mmol. m^–3 Michaelis-Menten constant - K _* ^M dimensionless Michaelis-Menten constant - N number of reactors of the cascade - Q m³. s^–1 volumetric flow rate - V _im³ volume of the i-th reactor - v _max mol. m^–3. s^–1 reaction rate under saturation conditions of the enzyme with substrate - V _{tot, opt} m³ minimum overall volume of the cascade - V _{tot, eq} m³ overall volume of the cascade assuming equal-sized reactors     

The effect of temperature on the growth of A. niger in solid state fermentation

A kinetic model of solid state fermentation with temperature deactivation of microorganisms is presented. The experimental results of cultivation of Aspergillus niger on a mixture of wheat bran and beet pulp in temperature range from 26 °C to 40 °C were used to estimate the parameters of the model. The activation energies of growth, thermal deactivation and maintenance have been calculated.List of Symbols C _CX mol/g proportionality coefficient - E _d J/mol energy of activation for thermal deactivation - E _g J/mol energy of activation for growth - J _CO2 mol/gh carbon dioxide evolution rate - k _d h^–1 thermal deactivation constant - k _g h^–1 growth kinetic constant - k _x h^–1 net growth constant - m h^–1 maintenance coefficient - N _CO2 mol amount of carbon dioxide - N _{m, CO2} mol maximum amount of carbon dioxide generated by growth - t h time - X g dry biomass weight - X _m g maximum biomass weight - X dimensionless biomass weight - X _0,r g real mass of inoculum - X _0,a g apparent mass of inoculum - X ₀ dimensionless apparent mass of inoculum - dimensionless maintenance coefficient This work was supported by the Committee of Scientific Research under grant No 3 3401 91 02.     

Enhancement of product selectivity via enzyme immobilization in sequential degradation reactions of polymeric substrates

The balance equations pertaining to the modelling of a slap-shaped bead containing immobilized enzyme uniformly distributed which catalyzes the sequential reactions of degradation of a polymeric substrate were written and analytically solved in dimensionless form. The effect of the Thiele modulus on the selectivity of consumption of each multimeric product was studied for a simple case. Whereas plain diffusional regime leads to lower selectivities than plain kinetic regime, improvements in selectivity of species A _i relative to species A_i+1 may be obtained at the expense of higher Thiele moduli within a limited range when the diffusivity of A _i is larger than that of A _i +1, or when the pseudo first order kinetic constant describing the rate of consumption of A _i is lower than that of A_i+1.List of Symbols A _i polymeric substrate containing i monomeric subunits - C _i mol·m^–3 normalized counterpart of C _i - C _i mol·m^–3 concentration of substrate A _i - C _i,0 mol·m^–3 initial concentration of substrate A _i - C _i,0 normalized counterpart of C _i,0 - D _ap,i m²·s^–1 apparent diffusivity of substrate A _i - k _i s^–1 pseudo-first order rate constant - K _m,i mol·m^–3 Michaelis-Menten constant associated with substrate A _i - L m half-thickness of the catalyst slab - N number of monomeric subunits of the largest substrate molecule - Th Thiele modulus - V _i mol·m^–3·s^–1 rate of rection of substrate A _i - V_max,i mol·m^–3·s^–1 maximum rate of reaction under saturating conditions of substrate A _i - x m longitudinal coordinate - S _i,i+1 selectivity of enzyme with respect to substrates with consecutive numbers of monomeric subunits Greek Symbols _i ratio of maximum rates of reaction - _i ratio of apparent diffusivities     

Process engineering aspects of the bioleaching of copper ores

The bioleaching of minerals is a complex process that is affected by a number of biological, mineralogical, electrochemical and engineering factors. This work presents and discusses the most significant process engineering aspects involved in the bacterial leaching of copper ores, i.e. bacterial population, type of mineral and particle size, nutrients and inhibitors, oxygen and carbon dioxide, temperature and pH, leaching kinetics and operation mode.It is concluded that more work is needed in this area in order to gain a deeper insight in the many factors that govern this process. This would allow to significantly improve its overall productivity.List of Symbols C _L kg/m³ dissolved oxygen concentration - C ^* kg/m³ equilibrium oxygen concentration - d, e, f, g % percentage of C, H, O and N in the cell - D m impeller diameter - K consistency index - K _S, K₁, K_c constants - k _La h^–1 volumetric oxygen transfer coefficient - M _b mol/kg biomass apparent molecular weight - N s^–1 rotation frequency - n behavior index - P kg/m³ ungassed agitation power, product concentration - P _g kW/m³ gassed agitation power - p % pulp density - Q m³/h air flow rate - S kg/m³ limiting substrate concentration - W kg/(m³ · h) mass transfer rate per unit volume - X cells/cm³ biomass concentration - Y _o g cells/g Fe oxygen cell yield - Y _x g cells/g Fe substrate cell yield - h^–1 specific growth rate - _m h^–1 maximum specific growth rate     

Simulation of a multicolumn recirculated packed bed reactor (MRPBR) for penicillin acylase

Enzyme reactors for the industrial hydrolysis of penicillin are analyzed in terms of biocatalyst stability to pH. A multicolumn system with packed beds placed in parallel and operating under recirculating conditions is proposed as an adequate reactor for this process. The system is studied both experimentally and with the aid of a simulation program.List of Symbols A transversal area (cm²) - C _A ammonia concentration in the reaction mixture (M) - C ₁ concentration of KH₂PO₄ in buffer (M) - C ₂ concentration of K₂HPO₄ in buffer (M) - d _p biocatalyst diameter (cm) - E enzyme or biocatalyst concentration (gcat l^–1) - K _APA APA non competitive inhibition constant (M) - K _IS excess substrate inhibition constant (M) - Km constant Michaelis-Menten (M) - K _PAA PAA competitive inhibition constant (M) - Q recirculation flow rate (cm³ min^–1) - Q _T recirculation flow rate per column (cm³ min^–1) - Re Reynolds number - S _E substrate concentration entering the neutralization tank (M) - S ₀ initial substrate concentration (M) - S _T substrate concentration in neutralization tank (M) - t time (min) - v _i initial reactor rate (mol min^–1 gcat^–1) - V _s superficial velocity (cm seg^–1) - V _T volume of neutralization tank (cm³) - X _E substrate conversion entering tank - X _T substrate conversion in neutralization tank - X conversion - Z reactor length (cm) - z axial position in reactor (cm) - z ^* non-dimensional axial position in reactor - biocatalyst's density (gcat cm^–3) - p pressure drop in the packed-bed reactor     

The effect of micro- and macromixing on enzyme reaction in a real CSTR

The effect of micromixing and macromixing on enzyme reaction of Michaelis-Menten type in a real continuously stirred tank reactor (CSTR) is considered. The effect of bypassing of a fraction of feed stream, dead space, initial enzyme concentration and Michaelis-Menten constant on substrate conversion is evaluated. Bypass reduces the substrate conversion significantly compared with other parameters in the case of micro and macromixing. Micromixing predicts higher substrate conversions compared with macromixing. The effect of micro and macromixing on substrate conversion is negligible at low and high conversions.List of Symbols C kmol/m³ concentration of reactant - ¯C kmol/m³ average concentration of reactant - C_A kmol/m³ exit concentration of reactant A - C_Aa kmol/m³ exit concentration of reactant A from active zone - C_AO kmol/m³ initial concentration of reactant A - C_EO kmol/m³ initial enzyme concentration - C_O kmol/m³ initial concentration of reactant - E(t) 1/s exit age distribution function - k 1/s reaction rate constant - M kmol/m³ Michaelis-Menten constant - r kmol/(m³s) rate of reaction - –r_A kmol/(m³s) rate of reaction with respect to A - t s time - v m³/s volumetric feed rate - v_a m³/s volumetric feed rate entering the active zone - v_b m³/s volumetric feed rate entering the bypass stream - V m³ total volume of the vessel - V_a m³ active volume of the vessel - V_d m³ volume of dead space - X_A conversion of A Greek Letters fraction of feed stream bypassing the vessel (v_b/v) - fraction of the total volume as dead space (V_d/V) - (t) 1/s Dirac delta function, an ideal pulse occurring at time t = 0 - s life expectancy of a molecule - 1/s intensity function or escape probability function - s space time or mean residence time     

Emission of microorganisms from biofilters

Experiments are reported on the discharge of microbial germs by biofilter systems used for the treatment of waste gases containing volatile organic compounds. The systems investigated concern six full-scale filter installations located in the Netherlands in several branches of industry, as well as a laboratory-scale installation used for modelling the discharge process. It is concluded that the number of microbial germs (mainly bacteria and to a much smaller extent moulds) in the outlet gas of the different full scale biofilters varies between 10³ and 10⁴ m^–3, a number which is only slightly higher than the number encountered in open air and of the same order of magnitude encountered in indoor air. It is furthermore concluded that the concentration of microorganisms of a highly contaminated inlet gas is considerably reduced by the filtration process. On the basis of the experiments performed in the laboratory-scale filter bed, it is shown that the effect of the gas velocity on the discharge process results from two distinctive mechanisms: capture and emission. A theoretical model is presented describing the rate processes of both mechanisms. The model presented and the experimentally determined data agree rather well.List of Symbols a _s m^–1 specific area of the packing material - C m^–3 microbial gas phase concentration - C _e , C _i m^–3 microbial concentration in the exit and inlet gas resp. - CFU colony-forming-units - d _c , d _m m diameter of collecting and captured particle resp. - D m diameter of the filter bed - E single particle target efficiency - H m bed height - k _c s^–1 first order capture rate constant per unit of bedvolume - k _e m^–3 emission rate constant per unit of bedvolume - n number of observations - r _c , r _e m^–3 s^–1 capture and emission rate per unit of bed-volume - Re = Reynolds number - S _t = Stokes number - u m s^–1 superficial gas velocity - u _m m s^–1 superficial gas velocity at which C _e = C _i Greek Symbols void fraction of the filter bed - kg m^–3 density of the gas phase - _m kg m^–3 density of captured particle - Pa s dynamic gas phase viscosity - = filter bed efficiency     

Reactor design for the enzymatic isomerization of glucose to fructose

A comprehensive methodology is presented for the design of reactors using immobilized enzymes as catalysts. The design is based on material balances and rate equations for enzyme action and decay and considers the effect of mass transfer limitations on the expression of enzyme activity. The enzymatic isomerization of glucose into fructose with a commercial immobilized glucose isomerase was selected as a case study. Results obtained are consistent with data obtained from existing high-fructose syrup plants. The methodology may be extended to other cases, provided sound expressions for enzyme action and decay are available and a simple flow pattern within the reactor might be assumed.List of Symbols C kat/kg specific activity of the catalyst - D m²/s substrate diffusivity within the catalyst particle - Dr m reactor diameter - d d operating time of each reactor - E kat initial enzyme activity - E _i kat initial enzyme activity in each reactor - F m³/s process flowrate - F _i m³/s reactor feed flowrate at a given time - F ₀ m³/s initial feed flowrate to each reactor - H number of enzyme half-lives used in the reactors - K mole/m³ equilibrium constant - K _S mole/m³ Michaelis constant for substrate - K _P mole/m³ Michaelis constant for product - K _m mole/m³ apparent Michaelis constant f(K, K _s, K_p, s₀) - k mole/s · kat reaction rate constant - k _d d^–1 first-order thermal inactivation rate constant - L m reactor height - L _r m height of catalyst bed - N _R number of reactors - P _i kg catalyst weight in each reactor - p mole/m³ product concentration - R m particle radius - R _P ratio of minimum to maximum process flowrate - r m distance to the center of the spherical particle - s mole/m³ substrate concentration - s _0i mole/m³ substrate concentration at reactor inlet - s ₀ mole/m³ bulk substrate concentration - s mole/m³ apparent substrate concentration - T K temperature - t d time - t _i d operating time for reactor i - t _s d time elapsed between two successive charges of each reactor - V m³ reactor volumen - V _m mole/m³ s maximum apparent reaction rate - V _p mole/m³ s maximum reaction rate for product - V _R m³ actual volume of catalyst bed - V _r m³ calculated volume of catalyst bed - V _S mol/m³ s maximum reaction rate for substrate - v mol/m³ s initial reaction rate - v _i m/s linear velocity - v _m mol/m³ s apparent initial reaction rate f(K_m, s,V_m) - X substrate conversion - X _eq substrate conversion at equilibrium - =s/K dimensionless substrate concentration - ₀=s₀/K bulk dimensionless substrate concentration - _eq=s_eq/K dimensionless substrate concentration at equilibrium - local effectiveness factor - mean integrated effectiveness factor - Thiéle modulus - =r/R dimensionless radius - _s kg/m³ hydrated support density - substrate protection factor - s residence time     

Kinetic behavior of penicillin acylase immobilized on acrylic carrier

The usefulness of Lilly's kinetic equation to describe penicillin G hydrolysis performed by immobilized penicillin acylase onto the acrylic carrier has been shown. Based on the experimental results characteristic kinetic constants have been estimated. The effect of noncompetitive inhibition of 6-amino penicillanic acid has not been found. Five components of reaction resistance have been defined. These components were also estimated for the reaction of the native enzyme as well as the Boehringer preparation.List of Symbols C _E g/m³ enzyme concentration - C _P,C _Q mol/m³ product concentrations - C _S mol/m³ substrate concentration - C _SO mol/m³ initial substrate concentration - K _A mol/m³ constant which defines the affinity of a substrate to the enzyme - K _iS mol/m³ substrate inhibitory constant - K _iP mol/m³ PhAA inhibitory constant - K _iQ mol/m³ 6-APA inhibitory constant - k ₃ mol/g/min constant rate of dissociation of the active complex - R(1) concentrational component of reaction resistance - R(2) resistance component derived from substrate affinity - R(3) resistance component due to the inhibition of the enzyme by substrate - R(4) resistance component due to the inhibition of the enzyme by PhAA - R(5) resistance component due to inhibition of the enzyme by 6-APA - r = dC_s/dt mol/m³ min rate of reaction - t min reaction time - (i) relative resistance of reaction     

A fedbatch strategy for optimal red pigment production by monascus ruber

According to the measurement of the pigment production and some material balance treatments, the monosodium glutamate (MSG), which represents a main substrate involved in the production of red pigment by Monascus ruber, is recovered on-line. Fedbatch operation then represents an alternative for increasing the production of pigment. A nonlinear quotient control scheme is expressed to regulate the monosodium glutamate substrate at an optimal value determined from batch studies.List of Symbols A absorbancy units (AU) - L lightness parameter - Pig red pigment quantity (Gmol) - vol fermentor active volume (1) - empirical coefficient - MW _pig red pigment molecular weight (g) - E elementary matrix - E ^# pseudo-inverse elementary matrix - r conversion rates vector (g· l^–1·h^–1) - E _m measured part of the elementary matrix - r _m measured conversion rates vector - E _c non-measured part of the elementary matrix - r _c non-measured conversion rates vector - v fictitious controller input - h fictitious controller output - u controller input - y controller output - y controller setpoint - k ₁ controller parameter - k ₂ controller parameter - k time iteration - MSG monosodium glutamate concentration (g·l^–1) - MSG _init monosodium glutamate initial quantity (g) - MSG _added total quantity of monosodium glutamate added (g) - MSG _cons total quantity of monosodium glutamate consumed (g) - MW _msg monosodium glutamate molecular weight (g)     

A heuristic approach to fed-batch optimisation of streptokinase fermentation

Previous studies have shown that the rate of formation of streptokinase, a secondary metabolite, in batch fermentation is proportional to the specific growth rate of the biomass, which in turn is inhibited by its substrate and the primary product (lactic acid). These kinetics suggest the suitability of fed-batch operation to increase the yield of streptokinase. A near-optimal feed policy has been calculated by the chemotaxis algorithm, and it shows a substrate feed rate decreasing nonlinearly and vanishing after 11 hours. This is followed by batch fermentation for a further 8 hours, at the end of which 12% more streptokinase is generated than by purely batch fermentation. Further improvements in productivity are possible.List of Symbols k _dh^–1 decay constant for active cells - k _ph^–1 decay constant for streptokinase - K _Igl^–1 inhibition constant for lactic acid - K_S gl^–1 inhibition constant for substrate - M gl^–1 lactic acid concentration - P gl^–1 streptokinase concentration - Q 1h^–1 substrate feed rate - S gl^–1 substrate concentration - S _ingl^–1 inlet concentration of substrate - t h time - t _bh end-point of batch fermentation - t _fh end-point of fed-batch fermentation - V l volume of broth in fermenter - V ₀ l initial value of V (at t=0) - V _ml maximum value of V - X gl^–1 total biomass concentration - X _agl^–1 concentration of active biomass - Y _MX yield coefficient for lactic acid from biomass - Y _PX yield coefficient for streptokinase from biomass - Y _XS yield coefficient for biomass from substrate Greek Letters h^–1 specific growth rate of biomass - _mh^–1 maximum specific growth rate