Optimum design of a series of CSTRs performing reversible Michaelis-Menten kinetics: a rigorous mathematical study

The optimum design of a given number of CSTRs in series performing reversible Michaelis-Menten kinetics in the liquid phase assuming constant activity of the enzyme is studied. In this study, the presence of product in the feed stream to the first reactor, as well as the effect of the product intermediate concentrations in the downstream reactors on the reaction rate are investigated. For a given number of N CSTRs required to perform a certain degree of substrate conversion and under steady state operation and constant volumetric flow rate, the reactor optimization problem is posed as a constrained nonlinear programming problem (NLP). The reactor optimization is based on the minimum overall residence time (volume) of N reactors in series. When all the reactors in series operate isothermally, the constrained NLP is solved as an unconstrained NLP. And an analytical expression for the optimum overall residence time is obtained. Also, the necessary and sufficient conditions for the minimum overall residence time of N CSTRs are derived analytically. In the presence of product in the feed stream, the reversible Michaelis-Menten kinetics shows competitive product inhibition. And this is, because of the increase in the apparent rate constant K^' _m that results in a reduction of the overall reaction rate. The optimum total residence time is found to increase as the ratio (‚₀) of product to substrate concentrations in the feed stream increases. The isomerization of glucose to fructose, which follows a reversible Michaelis-Menten kinetics, is chosen as a model for the numerical examples.     

Periodic operation of well mixed enzyme reactors with steady state multiplicities using relay feedback

Periodic operation of a well mixed enzyme reactor with input and output multiplicities is theoretically analyzed. The system is an isothermal continuously stirred tank enzyme reactor with a non ideal mixing represented by a Cholettes’ model and a reaction kinetics given by k ₁ c/(k ₁ c+1)². The system exhibits input multiplicities and output multiplicities. Some of the input steady-states correspond to unstable steady-states. The periodic operation in the inlet feed concentration cannot be done by open loop oscillation method. Relay feedback method is essential for the periodic operation of such systems. The improvement in the average conversion is calculated for the two unstable input steady-states. Higher average conversion is obtained at the largest value of the feed concentration. The limitation of relay feedback method for the stable input steady-state is also brought out.     

Optimum temperature operation mode for glucose isomerase reactor operating at constant glucose conversion

The optimum temperature operation mode required to achieve constant outlet glucose conversion is determined for immobilized glucose isomerase continuous packed bed reactor. The reactor design equation assumes reversible Michaelis-Menten kinetics with both enzyme deactivation and substrate protection. An increasing temperature profiles are determined for different operating periods, residence times and glucose conversions. The temperature increase with time is very small at low degree of glucose conversion and at relatively long residence time. The temperature rise with time increases at high degree of conversion and at relatively short residence time.     

Analysis of two-stage continuous operation of Escherichia coli containing bacteriophage λ vector

Bacteriophage u containing a cloned-gene is stably maintained in Escherichia coli in the lysogenic state while it is replicated and it overproduces a recombinant protein product in the lytic state. The host cell is eventually lysed in the lytic state. The kinetics of cell lysis and production induction were studied and are reported in this article through model equations. In two-stage continuous operation, the first tank is maintained in the lysogenic state for cell growth and cloned-gene stability while the second tank is in the lytic state for the overproduction of cloned-gene product. Individual cells in the second tank have different extent of the induction for product formation, since each has a different residence time. The different residence time for individual cells was taken into account using a population model. The numerical results show good agreement with the experimental data for the prediction of dilution rate in the second tank which gives the maximum product concentration.     

Process development of continuous hydrogen production by Enterobacter aerogenes in a packed column reactor

Hydrogen bioproduction from agro-industrial residues by Enterobacter aerogenes in a continuous packed column has been investigated and a complete reactor characterization is presented. Experimental runs carried out at different residence time, liable of interest for industrial application, showed hydrogen yields ranging from 1.36 to 3.02 mmol_H2mmol^у_glucose or, in other words, from 37.5% to 75% of the theoretical hydrogen yield. A simple kinetic model of cell growth, validated by experimental results and allowing the prediction of biomass concentration profile along the reactor and the optimization of superficial velocity, is suggested. By applying the developed approach to the selected operative conditions, the identification of the optimum superficial velocity v_0,opt of about 2.2 cm h^у corresponding to the maximum hydrogen evolution rate H£_2g,max, was performed.     

Antibiotics production in a fluidized bed reactor utilizing a transverse magnetic field

A mathematical model is developed to describe the performance of a three-phase fluidized bed reactor utilizing a transverse magnetic field. The model is based on the axially dispersed plug flow model for the bulk of liquid phase and on the Michaelis-Menten kinetics. The model equations are solved by the explicit finite difference method from transient to steady state conditions. The results of the numerical simulation indicate that the magnetic field increases the degree of bioconversion. The mathematical model is experimentally verified in a three-phase fluidized bed reactor with Penicillium chrysogenum immobilized on magnetic beads. The experimental results are well described by the developed model when the reactor operates in the stabilized regime. At low and relatively high magnetic field intensities certain discrepancy in the model solution is observed when the model over estimates the product concentration.     

Stochastic modelling of axial dispersion in external-loop airlift bioreactors

The paper presents a model of the motion of a particle subjected to several transport processes in connection with mixing in two phase flow. A residence time distribution technique coupled with a one-dimensional dispersion model was used to obtain the axial dispersion coefficient in the liquid phase, D_ax. The proposed model of D_ax for an external-loop airlift bioreactor is based on the stochastic analysis of the two-phase flow in a cocurrent bubble column and modified for the specific flow in the airlift reactor. The model takes into account the riser gas superficial velocity, the riser liquid superficial velocity, the Sauter bubble diameter, the riser gas hold-up, the downcomer-to-riser cross sectional area ratio. The proposed model can be applied with an average error of ᆨ.     

Experimental behavior of a whole cell immobilized dextransucrase biocatalyst in batch and packed bed reactors

Dextransucrases from Leuconostoc mesenteroides have been used to produce a diversity of controlled structure oligosaccharides with potential industrial applications. This is the case of !(1̄) branched glucooligosaccharides produced by L. mesenteroides NRRL B-1299 dextransucrase. In order to establish an industrial scale process with the immobilized enzyme, a biocatalyst was produced by whole cell entrapment in alginate beads. The main physical and physicochemical properties of the biocatalyst were determined and the hydrodynamic behavior in a packed bed reactor studied. It was possible to produce spherical beads of 0.2 cm diameter containing the insoluble part of L. mesenteroides culture (cells and insoluble polymer) with an activity of 4 IU/g. Immobilization yield reached 93% with an effectiveness factor of 0.995 for particles of d_p < 0.2 cm. Due to the complexity of dextransucrase mechanism and kinetics, data obtained from initial rate measurements failed to describe the results obtained from the batch and continuous reactors. Therefore, apparent K_M and V_max data were used for the reactor modeling. It was found that under the conditions studied, the reaction rate was controlled by external mass transfer limitations.     

How performance of integrated systems of reaction and separation relates to that of parallel and sequential configurations

Given the thermodynamic and kinetic limitations which often constrain the extent of chemical reactions and post-reactional separation processes, and therefore constrain the yield and the degree of purity of the resulting products, integration of reaction and separation in a single unit has been under the scope of several bioengineering researchers in recent years. It is the aim of this work to compare the performance of a cascade of N reactor/separator sets, either in series or in parallel, with that of an integrated reaction/separation unit. In order to do so, a Michaelis-Menten reaction in dilute substrate solutions (i.e. a pseudo first order reaction) was considered to take place in either configuration and, under the same reaction and separation conditions, comparison of the performance and efficiency of these configurations was made in terms of fractional recovery of pure product, total time required to achieve such recovery and rate of recovery. It was concluded that: (i) the series combination of reactor/separator sets yields better results, both in terms of fractional amount of product recovered and time required to do so, than the parallel combination; and (ii) the integrated approach is much more time- and cost-effective than plain cascading, thus making it very attractive from an economic point of view.     

Continuous ethanol production by Zymomonas mobilis in a fluidized bed reactor. Part II: Process development for the fermentation of hydrolysed B-starch without sterilization

A continuous fluidized bed reactor operation system has been developed for ethanol production by Zymomonas mobilis using hydrolysed B-starch without sterilization. The operation system consists of two phases. In the first phase macroporous glass carriers in a totally mixed fluidized bed reactor were filled up totally with a monoculture of Z. mobilis by fast computer-controlled colonization, so that in the subsequent production phase no contaminants, especially lactic-acid bacteria, could penetrate into the carrier beads. In the production phase the high concentration of immobilized Z. mobilis cells in the fluidized bed reactor permits unsterile fermentation of hydrolysed B-starch to ethanol at short residence times. This results in wash-out conditions for contaminants from the substrate. Long-term experimental studies (more than 120 days) of unsterile fermentation of hydrolysed B-starch in the laboratory fluidized bed reactor (2.2 l) demonstrated stable operation up to residence times of 5 h. A semi-technical fluidized bed reactor plant (cascade of two fluidized bed reactors, each 55 l) was operated stably at a mean residence time of 4.25 h. Glucose conversion of 99% of the unsterile hydrolysed B-starch was achieved at 120 g glucose/l^–1 in the substrate, resulting in an ethanol concentration of 50 g·l^–1 and an ethanol space-time yield of 13 g·l^–1·h^–1. This is a factor of three compared to ethanol fermentation of hydrolysed B-starch with Z. mobilis in a continuous stirred tank reactor, which can only be operated stably under sterile conditions. Correspondence to: D. Weuster-Botz     

Measurements of buoyancy for some Antarctic notothenioid fishes from the South Shetland Islands

Buoyancy was measured for 258 specimens representing 13 species of adult and sub-adult nototheniids, bathydraconids, and channichthyids from the South Shetland Islands. Measurements were expressed as percentage buoyancy (%B)=W_water/W_air᎒². There were no neutrally buoyant species and mean values for %B were 3.07-6.11%, with channichthyids at the low end and benthic nototheniids and bathydraconids at the high end. All species showed an ontogenetic decrease in %B with increasing body weight. With the exception of Champsocephalus gunnari, there was no sexual dimorphism in %B within this sample. With some exceptions, values for %B were consistent with life-history information. Sub-adult Dissostichus mawsoni were not neutrally buoyant, as are large adults. Notothenia rossii had a significantly lower %B than closely related N. coriiceps. Benthic Gobionotothen gibberifrons had a lower %B than semipelagic Lepidonotothen larseni. Although they exhibit some diversification in life history, the four channichthyids in the sample were similar in %B. Neutral buoyancy is rare in notothenioids and may be confined to a single nototheniid clade.     

Scale down of recombinant protein production: a comparative study of scaling performance

A large bioreactor is heterogeneous with respect to concentration gradients of substrates fed to the reactor such as oxygen and growth limiting carbon source. Gradient formation will highly depend on the fluid dynamics and mass transfer capacity of the reactor, especially in the area in which the substrate is added. In this study, some production-scale (12 m³ bioreactor) conditions of a recombinant Escherichia coli process were imitated on a laboratory scale. From the large-scale cultivations, it was shown that locally high concentration of the limiting substrate fed to the process, in this case glucose, existed at the level of the feedpoint. The large-scale process was scaled down from: (i) mixing time experiments performed in the large-scale bioreactor in order to identify and describe the oscillating environment and (ii) identification of two distinct glucose concentration zones in the reactor. An important parameter obtained from mixing time experiments was the residence time in the feed zone of about 10 seconds. The size of the feed zone was estimated to 10%. Based on these observations the scale-down reactor with two compartments was designed. It was composed of one stirred tank reactor and an aerated plug flow reactor, in which the effect of oscillating glucose concentration on biomass yield and acetate formation was studied. Results from these experiments indicated that the lower biomass yield and higher acetate formation obtained on a large scale compared to homogeneous small-scale cultivations were not directly caused by the cell response to the glucose oscillation. This was concluded since no acetate was accumulated during scale-down experiments. An explanation for the differences in results between the two reactor scales may be a secondary effect of high glucose concentration resulting in an increased glucose metabolism causing an oxygen consumption rate locally exceeding the transfer rate. The results from pulse response experiments and glucose concentration measurements, at different locations in the reactor, showed a great consistency for the two feeding/pulse positions used in the large-scale bioreactor. Furthermore, measured periodicity from mixing data agrees well with expected circulation times for each impeller volume. Conclusions are drawn concerning the design of the scale-down reactor.     

Reactor kinetics for submerged aerobic biofilms

Sclerotium rolfsii ATCC 15205 was cultivated in continuous culture (48 l reactor volume) for scleroglucan production. The maximum volumetric productivity (Q_Pvmax) amounted to 7.2 g/ld and was more than twice as high as in comparable batchwise cultivations. In addition, the relation between (a) volumetric productivity (Q_P [g/ld]) and product yield (Y_{Glucan/Glucose} [1]), (b) volumetric productivity and product quality (M_W [g/mol]), and (c) product quality and impeller tip speed (Nd [m/s]) was studied in continuous culture. It was found, that an increase in volumetric productivity led to a decline in product yield and product quality. Furthermore, an impeller tip speed of >0.7 m/s decreased the attainable product quality considerably. Based on these results, the impact of the operational setpoint of the process in terms of oxygen supply and reactor scale on the economics of scleroglucan production was discussed. In contrast to batchwise cultivation, scleroglucan production in continuous culture proved to be not feasible under non-aseptic conditions.     

Effect of time delay in specific growth rate on the periodic operation of bioreactors with input multiplicities

The effect of time delay in specific growth rate () on the periodic operation of bioreactors with input multiplicities is theoretically analyzed for productivity improvement. A periodic rectangular pulse is applied either in feed substrate concentration (S_f) or in dilution rate (D). Periodic operation under feed substrate concentration cycling gives improvement in productivity at lower value of ¯S_f of the two steady-state multiplicities of S_f only when the time delay in is larger. Whereas the larger value of ¯S_f gives improvement in average productivity for all values of time delay. Dilution rate (D) cycling gives an improvement in average productivity particularly for larger time delay in . This improvement in average productivity is obtained only at smaller value of dilution rate out of the two steady-state input multiplicities of D.List of Symbols D 1/h dilution rate - F memory function - g dummy variable - K_i g/l substrate inhibition constant - K_m g/l substrate saturation constant - P g/l product concentration - P_m g/l product saturation constant - Q g/(hl) product cell produced per unit time - S g/l substrate concentration - S_f g/l feed substrate concentration - S_f,p g/l feed substrate concentration during fraction of a period - X g/l biomass concentration - Y_X/S g/g cell mass yield - w variable either S or Z - Z g/l weighted average of substrate concentration Greek Letters 1/h time delay parameter - ₁ , ₂ product yield parameters, g/g and 1/h - pulse width expressed as a fraction of a period - 1/h specific growth rate - _m 1/h maximum specific growth rate - h period of oscillation - – average value     

Nonlinear feedforward control of bioreactors with input multiplicities

A steady-state nonlinear feedforward controller (FFC) for measurable disturbances is designed for a continuous bioreactor, which is represented by Hammerstein type nonlinear model wherein the nonlinearity is a polynomial with input multiplicities. The manipulated variable is the feed substrate concentration (S_f) and the disturbance variable is the dilution rate (D). The productivity (Q=DP) is considered as the controlled variable. The desired value of Q=3.73 gives two values of feed substrate concentration. The nonlinearity in the gain is considered for relating output to the manipulated variable and separately for the relation between output to disturbance variable. The FFC is also designed for the overall linearized system. The performance of the FFC is evaluated on the nonlinear differential equation model. The FFC is also designed for the model based on a single nonlinear steady-state equation containing both D and S_f. This nonlinear FFC gives the best performance. The nonlinear FFC is also designed by using only linear gain for the disturbance and nonlinear gain for the manipulated variable. Similarly, nonlinear FFC is also designed by using linear gain for the manipulated variable and the nonlinear gain for the disturbance variable. The performances of these FFC schemes are compared.     

Mass-transfer limitations for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor

A mathematical model has been developed for immobilized enzyme-catalyzed kinetic resolution of racemate in a fixed-bed reactor in which the enzyme-catalyzed reaction (the irreversible uni-uni competitive Michaelis-Menten kinetics is chosen as an example) was coupled with intraparticle diffusion, external mass transfer, and axial dispersion. The effects of mass-transfer limitations, competitive inhibition of substrates, deactivation on the enzyme effective enantioselectivity, and the optical purity and yield of the desired product are examined quantitatively over a wide range of parameters using the orthogonal collocation method. For a first-order reaction, an analytical solution is derived from the mathematical model for slab-, cylindrical-, and spherical-enzyme supports. Based on the analytical solution for the steady-state resolution process, a new concise formulation is presented to predict quantitatively the mass-transfer limitations on enzyme effective enantioselectivity and optical purity and yield of the desired product for a continuous steady-state kinetic resolution process in a fixed-bed reactor.     

Evaluation of effectiveness factor of immobilized enzymes using Runge-Kutta-Gill method: how to solve mathematical undetermination at particle center point?

The internal effectiveness factor of immobilized enzymes was analysed assuming the Michaelis-Menten kinetics to be valid. This factor can be analytically evaluated for cases where C_sdK_M or C_s€K_M. When this kinetic equation is used in its original form, the effectiveness factor can be evaluated only using a numerical technique, such as the Runge-Kutta-Gill method. This method presents a 0/0 type undetermination when applied to the center of a cylindrical or spherical catalyst particle. The undetermination was eliminated here by using L'Hôpital's rule and an expression valid only for the center of the particle was developed.     

Jet aeration as alternative to overcome mass transfer limitation of stirred bioreactors

In industrial biotechnology increasing reactor volumes have the potential to reduce production costs. Whenever the achievable space time yield is determined by the mass transfer performance of the reactor, energy efficiency plays an important role to meet the requirements regarding low investment and operating costs. Based on theoretical calculations, compared to bubble column, airlift reactor, and aerated stirred tank, the jet loop reactor shows the potential for an enhanced energetic efficiency at high mass transfer rates. Interestingly, its technical application in standard biotechnological production processes has not yet been realized. Compared to a stirred tank reactor powered by Rushton turbines, maximum oxygen transfer rates about 200% higher were achieved in a jet loop reactor at identical power input in a fed batch fermentation process. Moreover, a model‐based analysis of yield coefficients and growth kinetics showed that E. coli can be cultivated in jet loop reactors without significant differences in biomass growth. Based on an aerobic fermentation process, the assessment of energetic oxygen transfer efficiency [kgO₂ kW^?1 h^?1] for a jet loop reactor yielded an improvement of almost 100%. The jet loop reactor could be operated at mass transfer rates 67% higher compared to a stirred tank. Thus, an increase of 40% in maximum space time yield [kg m^?3 h^?1] could be observed.     

Attainable region analysis for continuous production of second generation bioethanol

Background

Despite its semi-commercial status, ethanol production from lignocellulosics presents many complexities not yet fully solved. Since the pretreatment stage has been recognized as a complex and yield-determining step, it has been extensively studied. However, economic success of the production process also requires optimization of the biochemical conversion stage. This work addresses the search of bioreactor configurations with improved residence times for continuous enzymatic saccharification and fermentation operations. Instead of analyzing each possible configuration through simulation, we apply graphical methods to optimize the residence time of reactor networks composed of steady-state reactors. Although this can be easily made for processes described by a single kinetic expression, reactions under analysis do not exhibit this feature. Hence, the attainable region method, able to handle multiple species and its reactions, was applied for continuous reactors. Additionally, the effects of the sugars contained in the pretreatment liquor over the enzymatic hydrolysis and simultaneous saccharification and fermentation (SSF) were assessed.

Results

We obtained candidate attainable regions for separate enzymatic hydrolysis and fermentation (SHF) and SSF operations, both fed with pretreated corn stover. Results show that, despite the complexity of the reaction networks and underlying kinetics, the reactor networks that minimize the residence time can be constructed by using plug flow reactors and continuous stirred tank reactors. Regarding the effect of soluble solids in the feed stream to the reactor network, for SHF higher glucose concentration and yield are achieved for enzymatic hydrolysis with washed solids. Similarly, for SSF, higher yields and bioethanol titers are obtained using this substrate.

Conclusions

In this work, we demonstrated the capabilities of the attainable region analysis as a tool to assess the optimal reactor network with minimum residence time applied to the SHF and SSF operations for lignocellulosic ethanol production. The methodology can be readily modified to evaluate other kinetic models of different substrates, enzymes and microorganisms when available. From the obtained results, the most suitable reactor configuration considering residence time and rheological aspects is a continuous stirred tank reactor followed by a plug flow reactor (both in SSF mode) using washed solids as substrate.

     

Optimal temperature policy for immobilized enzyme packed bed reactor performing reversible Michaelis-Menten kinetics using the disjoint policy.

The optimal temperature policy that maximizes the time-averaged productivity of a continuous immobilized enzyme packed bed reactor is determined. This optimization study takes into consideration the enzyme thermal deactivation with substrate protection during the reactor operation. The general case of reversible Michaelis-Menten kinetics under constant reactor feed flow rate is assumed. The corresponding nonlinear optimization problem is solved using the calculus of variations by applying the disjoint policy. This policy reduces the optimization problem into a differential-algebraic system, DAE. This DAE system defines completely the optimal temperature-time profiles. These profiles depend on the kinetic parameters, feed substrate concentration, operating period, and the residence time and are characterized by increasing form with time. Also, general analytical expressions for the slopes of the temperature and residual enzyme activity profiles are derived. An efficient solution algorithm is developed to solve the DAE system, which results into a one-dimensional optimization problem with simple bounds on the initial feed temperature. The enzymatic isomerization of glucose into fructose is selected as a case study. The computed productivities are very close to that obtained by numerical nonlinear optimization with simpler problem to solve. Moreover, the computed conversion profiles are almost constant over 90% of the operating periods, thus producing a homogeneous product.