Breakage of viscous and non-Newtonian drops in stirred dispersions

A model of breakage of drops in a stirred vessel has been proposed to account for the effect of rheology of the dispersed phase. The deformation of the drop is represented by a Voigt element. A realistic description of the role of interfacial tension is incorporated by treating it as a restoring force which passes through a maximum as the drop deforms and eventually reaching a zero value at the break point. It is considered that the drop will break when the strain of the drop has reached a value equal to its diameter. An expression for maximum stable drop diameter, dmax, is derived from the model and found to be applicable over a wide range of variables, as well as to data already existing in literature. The model could be naturally extended to predict observed values of dmax when the dispersed phase is a power law fluid or a Bingham plastic.

Continuously equivalent networks in state space

Schoeffler has derived continuously equivalent networks in the nodal-admittance domain. The letter derives a corresponding result in state space that combines the usefulness of Schoeffler's result and the power of the state-variable approach.

An economical, adaptable and user-friendly drip-perfusion bioreactor system designed for in vitro three dimensional cell culturing

This research project investigated a bioreactor system capable of high density cell growth intended for use in regenerative medicine and protein production. The bioreactor was based on a drip-perfusion concept and constructed with minimal costs, readily available components, and straightforward processes for usage. This study involved the design, construction, and testing of the bioreactor where the results showed promising three dimensional cell growth within a polymer structure. The accessibility of this equipment and the capability of high density, three dimensional cell growth would be suitable for future research in pharmaceutical drug manufacturing, and human organ and tissue regeneration.

Ferrous iron oxidation by foam immobilized Acidithiobacillus ferrooxidans: Experiments and modeling

Ferrous iron bio-oxidation by Acidithiobacillus ferrooxidans immobilized on polyurethane foam was investigated. Cells were immobilized on foams by placing them in a growth environment and fully bacterially activated polyurethane foams (BAPUFs) were prepared by serial subculturing in batches with partially bacterially activated foam (pBAPUFs). The dependence of foam density on cell immobilization process, the effect of pH and BAPUF loading on ferrous oxidation were studied to choose operating parameters for continuous operations. With an objective to have high cell densities both in foam and the liquid phase, pretreated foams of density 50 kg/m3 as cell support and ferrous oxidation at pH 1.5 to moderate the ferric precipitation were preferred. A novel basket-type bioreactor for continuous ferrous iron oxidation, which features a multiple effect of stirred tank in combination with recirculation, was designed and operated. The results were compared with that of a free cell and a sheet-type foam immobilized reactors. A fivefold increase in ferric iron productivity at 33.02 g/h/L of free volume in foam was achieved using basket-type bioreactor when compared to a free cell continuous system. A mathematical model for ferrous iron oxidation by Acidithiobacillus ferrooxidans cells immobilized on polyurethane foam was developed with cell growth in foam accounted by an effectiveness factor. The basic parameters of simulation were estimated using the experimental data on free cell growth as well as from cell attachment to foam under nongrowing conditions. The model predicted the phase of both oxidation of ferrous in shake flasks by pBAPUFs as well as by fully activated BAPUFs for different cell loadings in foam. Model for stirred tank basket bioreactor predicted within 5% both transient and steady state of the experiments closely for the simulated dilution rates. Bio-oxidation at high Fe2+ concentrations were simulated with experiments when substrate and product inhibition coefficients were factored into cell growth kinetics.

Modeling of heat and mass transfer for solid state fermentation process in tray bioreactor

A mathematical model is developed to simulate oxygen consumption, heat generation and cell growth in solid state fermentation (SSF). The fungal growth on the solid substrate particles results in the increase of the cell film thickness around the particles. The model incorporates this increase in the biofilm size which leads to decrease in the porosity of the substrate bed and diffusivity of oxygen in the bed. The model also takes into account the effect of steric hindrance limitations in SSF. The growth of cells around single particle and resulting expansion of biofilm around the particle is analyzed for simplified zero and first order oxygen consumption kinetics. Under conditions of zero order kinetics, the model predicts upper limit on cell density. The model simulations for packed bed of solid particles in tray bioreactor show distinct limitations on growth due to simultaneous heat and mass transport phenomena accompanying solid state fermentation process. The extent of limitation due to heat and/or mass transport phenomena is analyzed during different stages of fermentation. It is expected that the model will lead to better understanding of the transport processes in SSF, and therefore, will assist in optimal design of bioreactors for SSF.

A multi-stage model for drop breakage in stirred vessels

Existing models for dmax predict that, in the limit of μd → ∞, dmax increases with 3/4 power of μd. Further, at low values of interfacial tension, dmax becomes independent of σ even at moderate values of μd. However, experiments contradict both the predictions show that dmax dependence on μd is much weaker, and that, even at very low values of σ,dmax does not become independent of it. A model is proposed to explain these results. The model assumes that a drop circulates in a stirred vessel along with the bulk fluid and repeatedly passes through a deformation zone followed by a relaxation zone. In the deformation zone, the turbulent inertial stress tends to deform the drop, while the viscous stress generated in the drop and the interfacial stress resist deformation. The relaxation zone is characterized by absence of turbulent stress and hence the drop tends to relax back to undeformed state. It is shown that a circulating drop, starting with some initial deformation, either reaches a steady state or breaks in one or several cycles. dmax is defined as the maximum size of a drop which, starting with an undeformed initial state for the first cycle, passes through deformation zone infinite number of times without breaking. The model predictions reduce to that of Lagisetty. (1986) for moderate values of μd and σ. The model successfully predicts the reduced dependence of dmax on μd at high values of μd as well as the dependence of dmax on σ at low values of σ. The data available in literature on dmax could be predicted to a greater accuracy by the model in comparison with existing models and correlations.

A multi-stage model for drop breakage in stirred vessels

Existing models for dmax predict that, in the limit of μd → ∞, dmax increases with 3/4 power of μd. Further, at low values of interfacial tension, dmax becomes independent of σ even at moderate values of μd. However, experiments contradict both the predictions show that dmax dependence on μd is much weaker, and that, even at very low values of σ,dmax does not become independent of it. A model is proposed to explain these results. The model assumes that a drop circulates in a stirred vessel along with the bulk fluid and repeatedly passes through a deformation zone followed by a relaxation zone. In the deformation zone, the turbulent inertial stress tends to deform the drop, while the viscous stress generated in the drop and the interfacial stress resist deformation. The relaxation zone is characterized by absence of turbulent stress and hence the drop tends to relax back to undeformed state. It is shown that a circulating drop, starting with some initial deformation, either reaches a steady state or breaks in one or several cycles. dmax is defined as the maximum size of a drop which, starting with an undeformed initial state for the first cycle, passes through deformation zone infinite number of times without breaking. The model predictions reduce to that of Lagisetty. (1986) for moderate values of μd and σ. The model successfully predicts the reduced dependence of dmax on μd at high values of μd as well as the dependence of dmax on σ at low values of σ. The data available in literature on dmax could be predicted to a greater accuracy by the model in comparison with existing models and correlations.

Breakage and coalescence of drops in turbulent stirred dispersions

The various existing models for predicting the maximum stable drop diameterd max in turbulent stirred dispersions have been reviewed. Variations in the basic framework dictated by additional complexities such as the presence of drag reducing agents in the continuous phase, or viscoelasticity of the dispersed phase have been outlined. Drop breakage in the presence of surfactants in the continuous phase has also been analysed. Finally, the various approaches to obtaining expressions for the breakage and coalescence frequencies, needed to solve the population balance equation for the number density function of the dispersed phase droplets, have been discussed.

A new model for the breakage frequency of drops in turbulent stirred dispersions

A model of drop breakage in turbulent stirred dispersions based on interaction of a drop with eddies of a length scale smaller than the drop diameter has been developed. It predicts that, unlike the equal breakage assumed by earlier models, a large drop reduces in size due to stripping of smaller segments off it through unequal breakage. It is only when the drop nears the value of the maximum stable drop diameter that it breaks into equal parts. This new model of drop breakage, coupled with the pattern of interaction of drops with eddies of different sizes existing in the vessel, has been used to evaluate not only the breakage frequency, but also the size distribution of the daughter droplets(which was hitherto assumed). The model has been incorporated in the population balance equation and the resulting cumulative size distributions compared with those availble in the literature.

A new model for coalescence efficiency of drops in stirred dispersions

A model for coalescence efficiency of two drops embedded in an eddy has been developed. Unlike the other models which consider only head-on collisions, the model considers the droplets to approach at an arbitrary angle. The drop pair is permitted to undergo rotation while they approach each other. For coalescence to occur, the drops are assumed to approach each other under a squeezing force acting over the life time of eddy but which can vary with time depending upon the angle of approach. The model accounts for the deformation of tip regions of the approaching drops and, describes the rupture of the intervening film, based on stability considerations while film drainage is continuing under the combined influence of the hydrodynamic and van der Waals forces. The coalescence efficiency is defined as the ratio of the range of angles resulting in coalescence to the total range of all possible approach angles. The model not only reconciles the contradictory predictions made by the earlier models based on similar framework but also brings out the important role of dispersed-phase viscosity. It further predicts that the dispersions involving pure phases can be stabilized at high rps values. Apart from explaining the hitherto unexplained experimental data of Konno et al. qualitatively, the model also offers an alternate explanation for the interesting observations of Shinnar.

Adaptive optimization of continuous bioreactor using neural network model

An adaptive optimization algorithm using backpropogation neural network model for dynamic identification is developed. The algorithm is applied to maximize the cellular productivity of a continuous culture of baker's yeast. The robustness of the algorithm is demonstrated in determining and maintaining the optimal dilution rate of the continuous bioreactor in presence of disturbances in environmental conditions and microbial culture characteristics. The simulation results show that a significant reduction in time required to reach optimal operating levels can be achieved using neural network model compared with the traditional dynamic linear input-output model. The extension of the algorithm for multivariable adaptive optimization of continuous bioreactor is briefly discussed.

Transient analysis of mammalian cell growth in hollow fibre bioreactor

A mathematical model describing the dynamics of mammalian cell growth in hollow fibre bioreactor operated in closed shell mode is developed. Mammalian cells are assumed to grow as an expanding biofilm in the extra-capillary space surrounding the fibre. Diffusion is assumed to be the dominant process in the radial direction while axial convection dominates in the lumen of the bioreactor. The transient simulation results show that steep gradients in the cell number are possible under the condition of substrate limitation. The precise conditions which result in nonuniform growth of cells along the length of the bioreactor are delineated. The effect of various operating conditions, such as substrate feed rate, length of the bioreactor and diffusivity of substrate in different regions of the bioreactor, on the bioreactor performance are evaluated in terms of time required to attain the steady-state. The rime of growth is introduced as a measure of effectiveness factor for the bioreactor and is found to be dependent on two parameters, a modified Peclet number and a Thiele modulus. Diffusion, reaction and/or convection control regimes are identified based on these two parameters. The model is further extended to include dual substrate growth limitations, and the relative growth limiting characteristics of two substrates are evaluated. (C) 1997 Elsevier Science Ltd.

Prediction of reaeration rates in square, stirred tanks

Experiments were conducted on the oxygen transfer coefficient, k(L)a(20), through surface aeration in geometrically similar square tanks, with a rotor of diameter D fitted with six flat blades. An optimal geometric similarity of various linear dimensions, which produced maximum k(L)a(20) for any rotational speed of rotor N by an earlier study, was maintained. A simulation equation uniquely correlating k = k(L)a(20)(nu/g(2))(1/3) (nu and g are kinematic viscosity of water and gravitational constant, respectively), and a parameter governing the theoretical power per unit volume, X = (ND2)-D-3/(g(4/3)nu(1/3)), is developed. Such a simulation equation can be used to predict maximum k for any N in any size of such geometrically similar square tanks. An example illustrating the application of results is presented. Also, it has been established that neither the Reynolds criterion nor the Froude criterion is singularly valid to simulate either k or K = k(L)a(20)/N, simultaneously in all the sizes of tanks, even through they are geometrically similar. Occurrence of "scale effects" due to the Reynolds and the Froude laws of similitude on both k and K are also evaluated.