Bodenstein Number Research Articles

Axial dispersion in single pellet-string columns with non-porous packing

Using the chromatographic technique (carrier-gas: nitrogen; tracer-gases: H 2, He, Ar, C 2H 6) the axial dispersion in single pellet-string columns (SPSC) packed with non-porous spherical and cylindrical particles was studied. Dispersion due to extra column effects was eliminated via convolution of column responses for two column lengths. For spherical pellets the pellet/column diameter ratios ( d p / d c ) were in the range 0.57–0.79; for cylindrical pellets d p / d c =0.52–0.69 with pellet height/diameter ratios H/ D=0.78–1.5. With different tracer-gases and the same packing and column sizes analogous dependencies of Bodenstein numbers, Bo (Peclet number, Pe, with pellet size, d p , as characteristic dimension), versus the product of Reynolds and Schmidt numbers, ReSc, were obtained. For spherical column packing the Bo– ReSc curves depend also on the d p / d c ratio; for cylindrical pellets the sphericity (which depends on H/ D ratio) plays an additional role. Empirical interpolation equations for both shapes of pellets were set up. The obtained information on axial dispersion significantly increases the confidence of pore-diffusion characteristic when the chromatographic method with SPSC packed with porous pellets is applied.

Modeling and simulation of enzymatic membrane reactor for kinetic resolution of ibuprofen ester

The enzymatic hydrolysis of racemic ibuprofen ester by lipase from Candida rugosa using hollow fiber membrane reactor was studied. In process modeling and simulation of the enzymatic reaction in a horizontal hollow fiber membrane reactor system, two distinctive regions were studied: (i) diffusion in the fiber lumen; and (ii) reaction in the membrane matrix support. The model equations for both regions were developed and solved numerically by different approaches. In the first part, the modeling and simulation work was based on the diffusion of ( S)-ibuprofen acid in which it leaves the reactor from the lumen side of the capillary fibers. The proposed mathematical model containing non-linear partial differential equations was solved using the orthogonal collocation technique. The second part involved an enzymatic reaction taking place with the immobilized enzyme in the porous matrix support of the membrane, at which the substrate flows in from the shell side. The driving force was predominantly induced by pressure difference across the membrane where radial convection was the only mass transport in the porous support. The model was solved using two numerical techniques, the first technique based on the orthogonal collocation of weighted residuals in solving the non-linear partial differential equations of product diffusion in the capillary fibers; and the second technique was a collocation technique for solving ordinary differential equations (ODEs) of the enzymatic reaction in the porous support structure. The simulation was studied over a wide range of process parameters to investigate their effects on separation efficiency, in terms of enantiomeric excess, ee S and enantiomeric ratio, E, respectively. The model parameters include Peclet number, weight function parameters α, β and collocation points ( n), Thiele Modulus, Φ 2, dimensionless Michaelis constant, Θ and Bodenstein numbers, B 0. The performance of hollow fiber membrane reactor was found to be sufficiently effective at Θ=0.55, B 0=0.174 and ee S=74% at Φ 2<1.

Improving the prediction of liquid back‐mixing in trickle‐bed reactors using a neural network approach

AbstractCurrent correlations aimed at estimating the extent of liquid back‐mixing, via an axial dispersion coefficient, in trickle‐bed reactors continue to draw doubts on their ability to conveniently represent this important macroscopic parameter. A comprehensive database containing 973 liquid axial dispersion coefficient measurements (DAX) for trickle‐bed operation reported in 22 publications between 1958 and 2001 was thus used to assess the convenience of the few available correlations. It was shown that none of the literature correlations was efficient at providing satisfactory predictions of the liquid axial dispersion coefficients. In response, artificial neural network modeling is proposed to improve the broadness and accuracy in predicting the DAX, whether the Piston–Dispersion (PD), Piston–Dispersion–Exchange (PDE) or PDE with intra‐particle diffusion model is employed to extract the DAX. A combination of six dimensionless groups and a discrimination code input representing the residence‐time distribution models are used to predict the Bodenstein number. The inputs are the liquid Reynolds, Galileo and Eötvos numbers, the gas Galileo number, a wall factor and a mixed Reynolds number involving the gas flow rate effect. The correlation yields an absolute average error (AARE) of 22% for the whole database with a standard deviation on the AARE of 24% and remains in accordance with parametric influences reported in the literature.© 2002 Society of Chemical Industry

Radioisotope tracer study in trickle bed reactors

AbstractThe residence time distribution (RTD) of liquid phase in trickle bed reactors has been measured for air‐water system using radioisotope tracer technique. Experiments were carried out in a glass column of internal diameter of 0.152 m packed with glass beads and actual catalyst particles of two different shapes. From the measured RTD curves, mean residence time of liquid was calculated and used to estimate liquid holdup. The axial dispersion model was used to simulate the experimental data and estimate mixing index, ie. Peclet number. The effect of liquid and gas flow rates on total liquid holdup and Peclet number has been investigated. Results of the study indicated that shape of the packing has significant effect on holdup and axial dispersion. Bodenstein number has been correlated to Reynolds number, Galileo number, shape and size of the packing.

BACK MIXING AND LIQUID HOLD-UP IN A COCURRENT UP-FLOW PACKED BED BIOREACTOR

The hydrodynamics of an up-flow three phase packed bed bioreactor were investigated experimentally using tracer techniques. The effects of the liquid and gas velocities, and the particle size on the Peclet number (Pe) and on the liquid hold-up (ϵL) were determined. While the range of gas and liquid flow rate were chosen according to biotic conditions. It was found that the Pe was not influenced from liquid flow rate, however decreased with increasing gas flow rate and particle/reactor diameter ratio (dp/Dc ). Also, it was found that liquid hold-up was not influenced from gas flow rate, however increased with increasing liquid flow rate and decreased with increasing dp/Dc ratio. The correlation which connected the Bodenstein number to gas phase particle Reynolds number and dp/Dc ratio and the correlation which connected the liquid hold-up, liquid phase particle Reynolds number and dp/Dc ratio were found using Marquardt-Levenberg approach.

Theories of chromatographic efficiency applied to expanded beds

Various quantities such as plate height (HETP), number of plates ( N), axial dispersion coefficient ( D ax) and Bodenstein number (Bo) are used to describe the efficiency of, and dispersion in chromatographic columns. Different quantities highlight different aspects of the performance. Due to the expansion of expanded-bed columns, the information contained in some of these quantities is not the same for expanded beds as for packed beds. In this article the mentioned quantities are described and discussed both theoretically and related to experimental data. It is concluded that they are often used in a confusing way. Quantities modified to be more informative when comparing beds of different expansions are developed ( N EB= N/expansion 2 and HETP EB=HETP·bed expansion) and recommendations of which quantity to use in what situation are given.

Microchannel reactors for fast periodic operation: the catalytic dehydration of isopropanol

A microchannel reactor specially designed for periodic operation and a process to deposit a catalyst, γ-alumina, inside the reactor channels were developed. The hydrodynamics of the reactor was characterised. Although the flow in the microchannels is laminar, a uniform radial concentration profile and consequently a narrow residence time distribution were obtained. Depending on the manufacturing method the Bodenstein number was found to be Bo= ud/ D m ≅70. The catalytic coating had no influence on this distribution indicating a uniform deposition of the catalyst within the microchannels. As the inlet function was only slightly modified in the reactors, frequencies of up to 1 Hz could be imposed for periodic operation. The dehydration of isopropanol to propene was chosen as model reaction to study the dynamic behaviour of a microreactor under periodic concentration variations. The kinetics was firstly studied in a conventional fixed-bed reactor with the catalyst developed for the microreactor. Experimental results confirmed the theoretically predicted increase of the reactor performance under periodic operation exceeding the maximum obtainable performance under steady-state conditions.

Estimation of axial dispersion in horizontal rotating tubular bioreactor by means of a structured model

A horizontal rotating tubular bioreactor (HRTB) is designed as the combination of a “thin layer bioreactor” and a “biodisc” reactor. The investigation of mixing in HRTB was done by the temperature step method in a wide range of process conditions [residence time (t z =360036000 s) and bioreactor rotation speed (n=0.0830.917 s−1)]. In all experiments heat losses were detected. A mathematical model based on “tank in series” concept was developed to describe the mixing in HRTB – a “spiral flow” model (SFM) which has incorporated heat losses. However, the simulations of SFM could be used for calculation of temperature response curves for the case when there is no heat losses. These corrected curves were used then to estimate Bodenstein number as a parameter of standard dispersion model (SDM). The obtained Bodenstein numbers were in the range 10–17. The simulations showed that SFM was more capable to describe the mixing in HRTB giving better fitting with experimental measurements than SDM, indicating that mixing pattern in HRTB is too complex to be described with this relatively simple, one-parameter model.

資源処理プロセスの伝達関数表示とその応用

Transfer function representations of various plants still have significant positions in design and analysis of control systems, because new frequency-domain control techniques, such as H∞ control theory, are now easily available. This paper describes applications of transfer function representations of resources processing plants. The authors at first have showed examples of processes when the idea proposed in a paper of Åström et al. (1992) is applied to resources processing plants.The authors have, subsequently, calculated 4 transfer functions of one-dimensional reactive-diffusive models, which are typically found in resources processing plants, under 4 types of boundary conditions. Then, behaviors under Bo → ∞, 0 have been established based on the transfer functions calculated above. The transfer function representation of a free-flow flotation process proposed by Niemi (1966) seems to be lack of generality, because of the boundary conditions adopted. The authors have showed a new transfer function representation of the flotation process, under appropriate boundary conditions. The new transfer function model has been investigated to validate the idea of Åström et al. (1992), which attempts to apply Ziegler Nichols's PID tuning method through analyzing dimensionless numbers. The main results are summarized as follows : · Achievable performance of the PID control system is not affected by normalized reaction constant K, but by Bodenstein number Bo. · Relation between normalized process gain κ and normalized deadtime θ can empirically be described by an equation κ = 1 + 0.46 / θ 1.67. Relation κ = 2 (11 θ + 13) / (37 θ - 4) for finite-dimensional systems also gives rather a good approximation. · Within recommended region 0.1< θ < 0.6 of Ziegler-Nichols's ultimate gain method, crude approximations κ λ ≈ 1.4 and τ ≈ 1 hold.The latter half of the second term and the whole of the third term mentioned above show the validity of the idea of Åström et al. (1992), while the reactive-diffusive models are infinite-dimensional. This suggests usefulness of dimensionless numbers.Relations between other tuning methods and dimensionless numbers, applications to back-mixing models which are also familiar models in resources processing plants are the remaining themes to be studied.

Superporous agarose monoliths as mini-reactors in flow injection systems. On-line monitoring of metabolites and intracellular enzymes in microbial cultivation processes.

A new type of agarose material, superporous agarose, was used as a support material in an analytical system designed for monitoring of bioprocesses with respect to metabolites and intracellular enzymes. The superporous agarose was used in the form of miniaturised gel plug columns (15 x 5.0 mM I.D. monolithic gel bed). The gel plugs were designed to have one set of very large pores (about 50 microm in diameter) through which cells, cell debris and other particulate contaminants from the bioreactor could easily pass. The material also had normal diffusion pores (300 A) characteristic of all agarose materials, providing ample surface for covalent attachment of antibodies and enzymes used in the analytical sequence. The superporous agarose gel plug columns were characterised with respect to flow properties and handling of heavy cell loads as well as dispersion of injected samples (a Bodenstein number of about 40 was observed with acetone tracer at a flow rate of 1 ml min(-1)). To evaluate the practical performance of the superporous gel plug columns, two applications were studied: (1) on-line determination of glucose in cultivation broth (gel plug with immobilized glucose oxidase) and (2) immunochemical quantification of intracellular beta-galactosidase in E. coli (gel plug with lysozyme to achieve cell lysis and gel plug with antibodies against beta-galactosidase).

Modelling mixing parameters in concentric-tube airlift bioreactors

Axial dispersion of the liquid phase was investigated in a concentric-tube airlift bioreactor (RIMP: VL=0.70 m3) as a whole and in the separate zones (riser, downcomer, gas-separator) using the axial dispersion model. The axial dispersion number Bo and the axial dispersion coefficient, Dax were determined from the output curves to an initial Dirac pulse, using the tracer response technique. They were analyzed in relation to process and geometrical parameters, such as: gas superficial velocity, νSGR; top clearance, hS; bottom clearance, hB, and resistances at downcomer entrance expressed as Ad/AR ratio. Correlations between Bodenstein numbers in the overall bioreactor and riser and downcomer sections (BoT,BoR,BoD) and the geometrical and process parameters were developed, which can allow to assess the complex influence of these parameters on liquid axial dispersion.

Studies of mixing in a concentric tube airlift bioreactor with different spargers

Gas hold-up, ε, liquid velocity, J Lr , axial dispersion coefficient, E z , and mixing time, t m , have been measured in a concentric tube airlift bioreactor (12 × 10 −3 m 3 in volume) using sea water, as a function of the superficial gas velocity, J Gr , (up to 0.21 m/s). Seven different spargers were tested. Four of them were cylindrical (pore size from 60 μm to 1 × 10 −3 m) and three were porous plates (pore size from 30 to 120 μm). Three different flow regimes are observed in hold-up and liquid velocity over the range of J Gr covered: uniform bubbly flow at low gas velocities, heterogeneous flow at high gas velocities and a transition flow between bubbly and heterogeneous. The spargers with smaller pore sizes produce more hold-up and slow down velocity, because more gas recirculates into the downcomer. The change from uniform bubbly flow to transition flow appears because of the start of coalescence. In heterogeneous flow, where the bubble size is set by the degree of coalescence, no influence of the difference between the spargers used could be detected. Gas hold-up in the riser, ε r , could be represented, over the whole range of J Gr used, by E r=α J Gr J Lr β When the Bodenstein number for the gas-liquid mixture, Bo LG , and the Froude number, Fr, are used to represent axial dispersion, three flow regimes appear which correspond to those observed when hold-up and liquid velocity are plotted vs J Gr . The axial dispersion coefficient, E z , could be represented by E z=K 5d e J Gr E n 4 where d e is the equivalent diameter of the reactor. This equation suggests that E z is dominated by bubble slip, and fits 78% of the measured data with less than 20% error. It has also been applied satisfactorily to the results obtained by other authors who used different systems and liquid phases. Mixing time depends on sparger geometry and pore size only at low gas velocities. At high gas velocities, mixing time is practically independent of sparger and gas velocity.

Characterisation of liquid and gas flow in jet loop reactors

The fluid dynamics in jet loop fermenters has been subject to experimental and theoretical investigation. It is demonstrated that by determination of Euler number, Bodenstein number, and residence time distribution for the gas phase it is possible to perform a reliable characterization of the fermenters. It can be shown that the investigated jet loop fermenters with internal loop closely resemble ideally mixed tanks.

The influence of particle size distribution and operating conditions on the adsorption performance in fluidized beds

The influence of matrix properties and operating conditions on the performance in fluidized-bed adsorption has been studied using Streamline diethyl-aminoethyl (DEAE), an ion exchange matrix based on quartz-weighted agarose, and bovine serum albumin (BSA) as a model protein. Three different particle size fractions (120-160 microm, 120-300 microm, and 250-300 microm) were investigated. Dispersion in the liquid phase was reduced when particles with a wide size distribution were fluidized compared to narrow particle size distributions. When the mean particle diameter was reduced, the breakthrough capacities during frontal adsorption were enlarged due to a shorter diffusion path length within the matrix. At small particle diameters the effect of film mass transfer became more relevant to the adsorption performance in comparison to larger particles. Therefore matrices designed for fluidized-bed adsorption should have small particle diameter and increased mean particle density to ensure small diffusion path length in the particle and a high interstitial velocity to improve film mass transfer. Studies on the influence of sedimented matrix height on axial mixing showed an increased Bodenstein number with increasing bed length. Higher breakthrough capacities were also found for longer adsorbent beds due to reduced dispersion and improved fluid and particle side mass transfer. With increasing bed height the influence of flow rate on breakthrough capacity was reduced. For a settled bed height of 50 cm breakthrough capacities of 80% of the equilibrium capacity for flow rates varying from 3 to 9 cm/min could be achieved.

The influence of particle size distribution and operating conditions on the adsorption performance in fluidized beds

The influence of matrix properties and operating conditions on the performance in fluidized-bed adsorption has been studied using Streamline diethyl-aminoethyl (DEAE), an ion exchange matrix based on quartz-weighted agarose, and bovine serum albumin (BSA) as a model protein. Three different particle size fractions (120–160 μm, 120–300 μm, and 250–300 μm) were investigated. Dispersion in the liquid phase was reduced when particles with a wide size distribution were fluidized compared to narrow particle size distributions. When the mean particle diameter was reduced, the breakthrough capacities during frontal adsorption were enlarged due to a shorter diffusion path length within the matrix. At small particle diameters the effect of film mass transfer became more relevant to the adsorption performance in comparison to larger particles. Therefore matrices designed for fluidized-bed adsorption should have small particle diameter and increased mean particle density to ensure small diffusion path length in the particle and a high interstitial velocity to improve film mass transfer. Studies on the influence of sedimented matrix height on axial mixing showed an increased Bodenstein number with increasing bed length. Higher breakthrough capacities were also found for longer adsorbent beds due to reduced dispersion and improved fluid and particle side mass transfer. With increasing bed height the influence of flow rate on breakthrough capacity was reduced. For a settled bed height of 50 cm breakthrough capacities of 80% of the equilibrium capacity for flow rates varying from 3 to 9 cm/min could be achieved. © 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 54–64, 1997.

A new equation for correlating a pipe flow reactor with a cascade of mixed reactors

For many reactions, a pipe flow reactor and a cascade of ideally mixed reactors yield similar results, if we use a proper correlation between the Bodenstein number of the pipe flow reactor and the number of stages for the cascade. However, existing equations for this correlation, which are derived from residence time considerations, apply only for Bodenstein numbers greater than 7 or stage numbers greater than 6. In this paper a new equation is developed by comparing the governing equations for the reaction in both systems. It is shown that this correlation yields good agreement of the output concentrations for all stage numbers down to a single stage, respectively, zero Bodenstein number, even for a successive reaction.

Theoretical studies on biospecific adsorption for large-scale affinity separations

A dynamic model has been developed in order to estimate values of the parameters characterizing the interaction of transport phenomena with biospecific adsorption in fixed-bed affinity separations. The derived dimensionless model equations comprise the hydrodynamic behaviour on the basis of the Bodenstein number, influences of mass transfer by the Stanton number and biochemical driving potentials of the adsorption by the Gamma number. Model parameters were estimated on the basis of experimental results obtained from fixed-bed experiments. Results obtained from fixed-bed experiments and model predictions show that transport resistances have an important influence on the efficiency of biospecific separations and have to be considered in the design of large-scale systems as well as in the development of functionalized supports.

Does the fluid elasticity influence the dispersion in packed beds?

AbstractReasons are given why the axial dispersion in a gas flowing through a packed bed may be influenced by the elasticity ‐ or compressibility ‐ of the fluid. To support this hypothesis, experiments have been done in a packed column at pressures from 0.13 to 2.0 MPa. The elasticity E of a gas is proportional to the pressure P and the compressibility to 1/P. The axial dispersion coefficients as determined were found to be a function of the pressure in the packed bed in the turbulent flow region of 3 &lt; Rep &lt; 150 if the Bodenstein number is plotted as a function of the particle Reynolds number. This is shown to be an artifact. The pressure influence is eliminated, if Bom, ax is plotted versus the ratio of the kinetic forces over the elastic forces ϱu2/E. Regrettably, Bom, ax seems to be independent of ϱu2/E. For the moment we only can conclude that Bom, ax in the turbulent region is a unique function of the velocity of the gas which flows through the packed bed. Although the fact that a constant Bo value is obtained when plotted against ϱu2/E, the experimental results are so intriguing we wanted to make them public already now. The experimental work proceeds.

Axial dispersion in gases flowing through a packed bed at elevated pressures

Axial dispersion in upward gas flow is investigated by pulse and displacement experiments in a vertical, packed column with different concentrations of the tracer and at pressures up to 1.5 MPa. The responses to the introduced pulse and step changes are measured at two locations and the extent of axial dispersion, respresented by the Bodenstein number, is determined by curve fitting in the time domain. The performed experiments demonstrate that the residence time distribution is considerably affected by density differences between the tracer and carrier gas, particularly at elevated pressures. Obtained Bodenstein numbers for step changes from nitrogen to a helium/nitrogen mixture and vice versa differ by as much as a factor ten, depending on the helium concentration and column pressure. The difference in axial dispersion may be ascribed to gravitation-driven instabilities as due to vertical density gradients in the case of a heavy gas displaced by a light gas; density gradients in the step changes from a light to heavy gas evidently inhibit axial dispersion. The presented observations are of major importance for the description of flow behaviour of gases in packed bed reactors where density gradients exist due to temperature and concentration gradients, particularly because many processes operate at elevated pressures.

Axial dispersion in packed fiber beds

The application of fibrous sorbent materials in gas and liquid separations as an alternative to conventional granular sorbents has attracted increased attention over the last decade. Though axial dispersion of flow in granular beds has been investigated extensively, published experimental data concerning the dispersion phenomena in porous fiber beds are sparse. The study presented involves the experimental determination of axial dispersion for water flow in packed beds of randomly aligned fibers, using a tracer pulse response technique. The influence of the bed properties (fiber diameter, bed void fraction) and the flow velocity on the dispersion coefficient was determined. The measured Bodenstein numbers vary between 0.001 and 0.1. the experimental results are compared with predicted values obtained from a theoretical model, based on isotropic fiber structures, showing substantial disagreement between the experimental and predicted values attributed to inhomogeneities of the fiber structure. The effect of inhomogeneities is similar for various types of random fiber structure. An empirical correlation for the axial dispersion coefficient for water flow in random fiber structures is presented.