Modeling of antibiotics production in magneto three-phase airlift fermenter

A mathematical model is developed to describe the performance of a three-phase airlift reactor utilizing a transverse magnetic field. The model is based on the complete mixing 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 airlift reactor with P. chrysogenum immobilized on magnetic beads. The experimental results are well described by the developed model when the reactor operates in the stabilized regime. At relatively high magnetic field intensities a certain discrepancy in the model solution was observed when the model over estimates the product concentration.     

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.     

A model of microbial growth in a plug flow reactor with wall attachment

A mathematical model of microbial growth for limiting nutrient in a plug flow reactor which accounts for the colonization of the reactor wall surface by the microbes is formulated and studied analytically and numerically. It can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each, and for the long term persistence of the bacteria in the reactor.     

Predicting the performance of immobilized enzyme reactors using reversible Michaelis–Menten kinetics

A general mathematical model is developed in the present work for predicting the steady state performance of immobilized enzyme reactor performing reversible Michaelis - Menten kinetics. The model takes into account the effect of external diffusional limitations, the axial dispersion and the equilibrium constant on reactor performance quantified as relative substrate conversion and yield. The performance of reactor is characterized using the dimensionless parameters of Damkohler number, Stanton number, Peclet number, the equilibrium constant and the dimensionless input substrate concentration. The reactor performance is described for the two extreme cases of plug flow reactor (PFR) and continuous stirred tank reactor (CSTR) in addition to the intermediate case of dispersed plug flow reactor (DPFR). The performance of reactor is compared for the two cases of zero order and reversible first order kinetics.     

Design and analysis of a solar reactor for anaerobic wastewater treatment

The aim of this research was to design a solar heated reactor system to enhance the anaerobic treatment of wastewater or biological sludge at temperatures higher than the ambient air temperature. For the proposed reactor system, the solar energy absorbed by flat plate collectors was transferred to a heat storage tank, which continuously supplied an anaerobic-filter reactor with water at a maximum temperature of 35 degrees C. The packed reactor was a metallic cylindrical tank with a peripheral twin-wall enclosure. Inside this enclosure was circulated warm water from the heat storage tank. Furthermore, a mathematical model was developed for the prediction of the temperature distribution within the reactor under steady state conditions. Preliminary results based on model simulations performed with meteorological data from various geographical regions of the world suggested that the proposed solar reactor system could be a promising and environmentally friendly approach for anaerobic treatment of wastewater and biological sludge.     

Semi-continuous xylose-to-xylitol bioconversion by Ca-alginate entrapped yeast cells in a stirred tank reactor

Candida guilliermondii FTI 20037 cells were entrapped in Ca-alginate beads and used for xylose-to-xylitol bioconversions during five successive batches in a stirred tank reactor. Supplemented sugarcane bagasse hemicellulosic hydrolysate was used as the fermentation medium. The average volume of the Ca-alginate beads was reduced by about 30% after the 600 h taken to perform the five bioconversion cycles, thus demonstrating physical instability under the conditions prevailing in the reactor vessel. In spite of this, almost steady bioconversion rates and yields were observed along the repeated batches. In average values, a production of 51.6 g l(-1), a productivity of 0.43 g l(-1 )h(-1) and a yield of 0.71 g g(-1) were attained in each batch, variation coefficients being smaller than 10%.     

Modeling of continuous Ph-stat stirred tank reactor with Lactococcus lactis ssp. lactis bv. diacetylactis immobilized in calcium alginate gel beads

A dynamic diffusion-reaction-growth model is proposed for the study of lactic fermentation, the bioconversion of citric acid, and cell release in an immobilized cell reactor [pH-stat continuous stirred tank-reactor (CSTR)]. The model correctly simulates the onset of fermentation and colonization of the gel, followed by the steady state. External diffusion is nonlimiting and internal diffusion is limited by high cell densities at the periphery of the gel beads. Lactose-citrate cometabolism in the gel is related to the distribution of active included biomass within the gel and to gradients of substrates (lactose, citrate) and products (lactate, pH) in the beads. The utilization of lactose is limited by reaction, whereas that of citrate is limited by diffusion. Cell release from gel to the liquid medium occurs in the external spherical cap of the beads. In this peripheral zone viability is maintained at around 90%. (c) 1995 John Wiley & Sons Inc.     

An integral dynamic model for the UASB reactor

In this article a dynamic model of a continuous working UASB reactor is described. It results from the integration of the fluid flow pattern in the reactor, the kinetic behavior of the bacteria (where inhibition and limitation were taken into account), and the mass transport phenomena between different compartments and different phases. The mathematical equations underlying the model and describing the important mechanisms were programmed and prepared for computations and simulations by computer. The settler efficiency has to be over 99% to prevent the reactor from wash-out. When the settler efficiency is over 99%, the total sludge content of the reactor increases steadily, so the reactor is hardly ever in a steady state. This implies dynamic modeling. The model is able to predict the various observable and nonobservable or difficult to observe state variables, e.g., the sludge bed height, the sludge blanket concentration, the short-circuiting flows over bed and blanket, and the effluent COD concentration as a function of the hydrodynamic load, COD load, pH, and settler efficiency. The optimal pH value is between 6.0 and 8.0; fatty acid shock loadings are difficult to handle outside this optimal pH range.     

Model-based analysis on growth of activated sludge in a sequencing batch reactor

A mathematical model is developed to describe the growth of multiple microbial species such as heterotrophs and autotrophs in activated sludge system. Performance of a lab-scale sequencing batch reactor involving storage process is used to evaluate the model. Results show that the model is appropriate for predicting the fate of major model components, i.e., chemical oxygen demand, storage polymers (X _STO), volatile suspended solid (VSS), ammonia, and oxygen uptake rate (OUR). The influence of sludge retention time (SRT) on reactor performance is analyzed by model simulation. The biomass components require different time periods from one to four times of SRT to reach steady state. At an SRT of 20 days, the active bacteria (autotrophs and heterotrophs) constitute about 57% of the VSS; the remaining biomass is not active. The model established demonstrates its capacity of simulating the reactor performance and getting insight in autotrophic and heterotrophic growth in complex activated sludge systems.     

Modeling and experimental validation of carbon dioxide evolution in alkalophilic cultures

The chemical reactions involving carbon dioxide in mineral culture media are considered. A mathematic model is set up, based on published data, which is valid at pH values below 9, and in which the nonideality of the solution is taken into account. The crucial parameter is the constant expressing the equilibrium between carbon dioxide and bicarbonate, K(1).The reactions were studied in three different aqueous solutions: water, mineral salt medium, and a suspension with nongrowing bacterial cells. For each situation, three methods were compared for the determination of the bicarbonate concentration in the solution: equilibrium state total carbon analysis, dynamic monitoring of the rate of acid or alkali addition, and dynamic measurement of the carbon dioxide gas phase mole fraction.In a batch-stirred tank reactor, the equilibrium constant K(1) agreed with the published value, and the three bicarbonate analysis methods give the same results. If the nonideality is not taken into account, the result significantly differed from the published value and is likely to be incorrect.A real alkalophilic process, using Acinetobacter calcoaceticus in a continuous stirred tank reactor at steady state, also gave results that are in accord with the literature. However, the results do not allow validation of the equation expressing the nonideality.The steady state in the batch system and in continuous culture can be well described with the mathematical model. However, in the transient state there are some unexplained differences between simulation and measurement.     

Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems

When searching for hosts, parasitoids are observed to aggregate in response to chemical signalling cues emitted by plants during host feeding. In this paper we model aggregative parasitoid behaviour in a multi-species host-parasitoid community using a system of reaction-diffusion-chemotaxis equations. The stability properties of the steady-states of the model system are studied using linear stability analysis which highlights the possibility of interesting dynamical behaviour when the chemotactic response is above a certain threshold. We observe quasi-chaotic dynamic heterogeneous spatio-temporal patterns, quasi-stationary heterogeneous patterns and a destabilisation of the steady-states of the system. The generation of heterogeneous spatio-temporal patterns and destabilisation of the steady state are due to parasitoid chemotactic response to hosts. The dynamical behaviour of our system has both mathematical and ecological implications and the concepts of chemotaxis-driven instability and coexistence and ecological change are discussed. I. G. Pearce gratefully acknowledges the financial support of the NERC.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.     

Modeling of an aerobic biofilm reactor with double-limiting substrate kinetics: bifurcational and dynamical analysis

A mathematical model of an aerobic biofilm reactor is presented to investigate the bifurcational patterns and the dynamical behavior of the reactor as a function of different key operating parameters. Suspended cells and biofilm are assumed to grow according to double limiting kinetics with phenol inhibition (carbon source) and oxygen limitation. The model presented by Russo et al. is extended to embody key features of the phenomenology of the granular‐supported biofilm: biofilm growth and detachment, gas–liquid oxygen transport, phenol, and oxygen uptake by both suspended and immobilized cells, and substrate diffusion into the biofilm. Steady‐state conditions and stability, and local dynamic behavior have been characterized. The multiplicity of steady states and their stability depend on key operating parameter values (dilution rate, gas–liquid mass transfer coefficient, biofilm detachment rate, and inlet substrate concentration). Small changes in the operating conditions may be coupled with a drastic change of the steady‐state scenario with transcritical and saddle‐node bifurcations. The relevance of concentration profiles establishing within the biofilm is also addressed. When the oxygen level in the liquid phase is <10% of the saturation level, the biofilm undergoes oxygen starvation and the active biofilm fraction becomes independent of the dilution rate. © 2011 American Institute of Chemical Engineers Biotechnol. Prog., 2011     

Effect of wall growth on scale-up problems and dynamic operating characteristics of the biological reactor

If inhibitory substrates are being utilized in a well-stirred biological reactor, microbiological growth on the walls of the reactor can create a scale-up problem. A simple model is proposed which shows that without such growth, of the three existing steady states only one is stable and nontrivial, but with wall growth the trivial, stable, steady state (washout) is impossible. In addition, wall growth reduces the region over which three steady states are feasible and reduces the minimum residence time for which there is only one steady state that corresponds to a high conversion. Thus, a laboratory process with a high surface area to volume ratio can give an over optimistic prediction of both necessary residence; time and stability of the full scale process unless wall growth is accounted for.     

Regression and regrowth of tumour cords following single-dose anticancer treatment

In this paper, the evolution of a tumour cord after treatment is investigated by extensive numerical simulations on the basis of a mathematical model developed by Bertuzzi et al. (submitted). The model is formulated in cylindrical symmetry adopting the continuum approach, and takes into account the influence of oxygen level on the proliferation and death rate of cells, the volume reduction due to disgregation of dead cells, and the cell killing effects of radiation and drugs. Some extensions of the model are proposed to represent more accurately the radioresistance of hypoxic cells and the cytotoxic action of anticancer drugs. The steady state of the cord, and the cord evolution from the steady state after the delivery of a single dose of an anticancer agent, are computed for various combinations of model parameters and for different choices of the functions describing the effects of treatments. The results of the numerical computations show that, in spite of its many simplifications, the model behaviour appears to be reasonable in view of the available experimental observations. The model allows having a better insight into some complex treatment-related events, such as cell reoxygenation and repopulation.     

Three steady state situation in an open chemical reaction system. I

In an isothermal continuously stirred tank reactor (open chemical reaction system) fed by sulphuric acid solutions of bromate, bromide and cerium)(III) bistability (three steady state situation) is experimentally observed. This remarkable behavior, based on the instability of one steady state, has important consequences for the understanding of excitability and biochemical control mechanisms. The mass-balance equations for the reactor and the chemical mechanism of the reaction are combined into a simple mathematical model. The behavior of the resulting nonlinear differential equations is examined analytically and by a graphical integration procedure (method of isoclines). Using realistic kinetic data, the model shows the same behavior as observed in the experiment.     

Mathematical analysis of a metapopulation model with space-limited recruitment

We investigate a mathematical aspect of a multi-species' sessile metapopulation model with space-limited recruitment proposed by Iwasa et al. in 1986. We define some basic reproduction numbers to show the threshold condition for the stability of trivial steady state and the existence of coexistent steady state. We show the existence of steady state where all species exist when some reproduction numbers are greater than one by the fixed point theorem. And we construct the Lyapunov function to show the global stability of trivial steady state when some basic reproduction numbers are not greater than one.