The calculations of dispersion coefficients inside two‐dimensional randomly packed beds of circular particles

The longitudinal (D_L) and transverse (D_T) dispersion coefficients in two‐dimensional (2‐D) randomly packed beds of circular particles in a laminar flow regime are derived. A 2‐D discrete system of particles is divided into cells using modified Voronoi diagrams. The relationship between the variation of the stream function and the averaged vorticity is obtained from computational fluid dynamics (CFD) simulations. The whole flow pattern is then obtained by using the principle of energy dissipation rate minimization. The obtained values of D_L agree well with 3‐D experimental data for all velocities investigated. At very high velocities, D_T in 2‐D appears to be higher than 3‐D experimental data. In addition, the effects of particle‐size distributions, packing structure, and porosity on the D_L and D_T were studied. One result was that an increase in the width of the particle‐size distribution resulted in higher values of D_L and D_T at high velocities. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1002–1011, 2013     

Mixing in bubble columns: a new approach for characterizing dispersion coefficients

Use of bubble columns as photobioreactors requires a quantitative knowledge of radial mixing in these columns. A complete model of liquid-phase dispersion was used to simultaneously characterize axial and radial mixing in a relatively large (0.06 m³, 2.3 m tall, 0.193 m in diameter) bubble column photobioreactor. Axial and radial dispersion coefficients and mixing times were determined in tap water and sea water for superficial aeration velocities of up to . The measured axial dispersion coefficients (D_z) were generally consistent with the predictions of the well established correlations, thus validating the complete dispersion model used in the analysis. The D_z values ranged from ∼150 to and were highly reproducible. There was evidence that the existing literature data on D_z in bubble columns are slightly underestimated, as consistent underestimation was found to be a characteristic of the widely used dispersion model that disregards radial dispersion. The value of the radial dispersion coefficient was typically about 1% of the D_z value under any given condition. Except at incipient aeration, the radial dispersion coefficient was not as sensitive to the magnitude of the aeration rate as was the axial dispersion coefficient. The mixing time data were generally consistent with the existing correlations.     

Solids mixing in a down-flow circulating fluidized bed of 0.418-m in diameter

The axial and lateral solids mixing in a down-flow circulating fluidized bed of 0.418-m diameter was investigated by a pneumatic injection phosphor tracer technique (PIPTT). The axial and lateral solids dispersion were determined by measuring the solids RTD at same axial but different lateral positions using point sources for tracer injection. A two-dimensional dispersion model described the measured RTD curves satisfactorily. The results were compared to those obtained in the small scale downers and the scale-up effect was investigated. The axial solids Peclet number Pe_a is around 110 and invariable with changing U_g, G_s and ?_s, while the lateral solids Peclet number Pe_r is linearly increasing with ?_s. And Pe_r is found to decrease with the square root of inner diameter (ID) in comparison with the results obtained in small ID downers. Correlation of Pe_r, Pe_r = (15 + 70.7 ?_s)D^− 0.5, is proposed.     

Diffusion coefficients of diesel fuel and surrogate compounds in supercritical carbon dioxide

Facilitating a new concept of clean diesel combustion using supercritical fluids requires a better understanding of thermophysical properties of the diesel fuel/diluent system. Mass diffusivity is one such property that is important to understand diesel fuel/diluent mixing and spray and combustion of supercritical fuel mixtures. In this work, diffusion coefficients of diesel fuel and surrogate compounds in supercritical carbon dioxide were experimentally determined by the Taylor dispersion method at temperatures from 313.15 to 373.15 K and pressures up to 30 MPa. Difficulties were encountered to measure diffusion coefficients using the Taylor dispersion method near the critical region of CO₂ which resulted in curve-fitting errors greater than 5%. Predictive correlations including Wilke-Chang, Scheibel, and He-Yu were examined. Diffusivity data were also fitted by D₁₂/T − η and correlations. Results showed that the He-Yu correlation has the best prediction performance while the D₁₂/T − η correlation best fits the data with AAD% < 8%.     

Transversal moving-front patterns: Criteria and simulations for two-bed and cylindrical shell packed-bed reactors

We show that a moving-front solution in a cylindrical shell packed-bed catalyzing a first-order activated reaction may bifurcate into transversal patterns when Pe_C/Pe_T<ΔT_ad/ΔT_m, i.e. when the ratio of the mass to heat Pe numbers is smaller than the ratio of the adiabatic to maximal temperature rises. This coincides with the previous condition of transversal patterns to emerge in stationary fronts [Pe_C/Pe_T<1 [Viswanathan, G., Bindal, A., Khinast, J., Luss, D., 2005. Stationary transversal hot zones in adiabatic packed-bed reactors. A.I.Ch.E. Journal 51, 3028-3038]] and extends the bifurcations condition to the case of moving fronts. The novel condition cannot be satisfied in a downstream propagating front (ΔT_m/ΔT_ad>1), but for an upstream propagating front (toward the cold reactor inlet) ΔT_m/ΔT_ad<1 and the symmetry breaking can be obtained within a feasible domain of operating conditions (Pe_C/Pe_T>1). It was also assumed that the axial and the transversal Pe numbers vary consistently, i.e. κ_C=Pe_C⊥/Pe_C=κ_T=Pe_T⊥/Pe_T. A similar condition was also obtained using a simplified model composed of two 1-D beds with heat and mass exchange between them.Bifurcation diagram showing domains of transversal patterns is constructed using a learning two-bed model. These predictions are verified by direct numerical simulations of the continuous 2-D cylindrical shell model showing various types of moving transversal patterns within a feasible domain of the state parameters with Pe_C>Pe_T. In the case of varying ratio (κ_C≠κ_T) the pattern domain can be significantly extended toward larger Pe_C/Pe_T.     

Numerical derivation of dispersion coefficients for flow through three‐dimensional randomly packed beds of monodisperse spheres

The longitudinal (D_L) and transverse (D_T) dispersion coefficients for flow through randomly packed beds of discrete monosized spherical particles are studied. The three‐dimensional (3‐D) porous‐medium model consists of thousands of spherical particles that are divided into cells using Voronoi diagrams. The relationship between the variation of the dual stream function and the vorticity between neighboring particles is derived using Laurent series. The whole flow pattern at low particle Reynolds number is then obtained by minimization of the dissipation rate of energy with respect to the dual stream function. The D_L is obtained by fitting the resulting effluent curve to a 1‐D solution of a continuous model. The D_T is obtained by fitting the numerical concentration profile to an approximate 2‐D solution. The derived D_L and D_T values are in agreement with 3‐D experimental data from the literature enabling a study of the effects of pore structure and porosity on D_L and D_T. © 2013 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J 60: 749–761, 2014     

The oxidation of aqueous sodium sulphite solutions

In this paper new experimental work on the kinetics of the absorption of oxygen in an aqueous solution of sodium sulphite with cobaltous sulphate as a catalyst, has been presented and compared with already published data. It is shown that the reaction of oxygen with sodium sulphite is zero order in sulphite, first order in cobalt and second order in oxygen for 2 × 10⁴ N/m² <p_O2 < 10⁵ N/m², 150°C <T < 60°C, 3 × 10^?6 kmol/m³ <c_Co⁺⁺ < 3 × 10^?3 kmol/m 7.50 < pH < 8.50 and 0.4 kmol/m² <c_SO2-- < 0.8 kmol/m³. The reaction rate constant can be varied by more than two decades by changing the cobalt concentration, pH and temperature.The specific interfacial areas, a, and mass transfer coefficients, k_L, measured by DeWaal and Beek [14, 15] using an incorrect kinetic model of this reaction, are reinterpreted with the results of the present kinetic investigation. It is shown that their values of a and k_L are a factor 2.2 too low and too high, respectively, but that their values of k_La are correct.     

The influence of axial pressure on relaxor properties of BaBi2Nb2O9 ceramics

On the basis of our studies it results that dielectric properties of BaBi₂Nb₂O₉ ceramics are sensitive to axial pressure applied. The pressure causes an increase of dispersion in the real part of dielectric permittivity ?′(T,f) and a rise in the temperature T_m at which the maximum in ?′(T,f) dependence occurs. The applied pressure induces in the ?′(T) dependence an additional step-like anomaly, which appears at the temperature T_A < T_m. The applied pressure shifts both T_m and T_A at the same rate, i.e. dT_A/dX = dT_m/dX = +14 °C/kbar at high axial pressure range, above the threshold pressure X_thresh. The Vogel–Fulcher relationship is employed to determine the axial pressure influence on relaxor properties of BBN ceramics. The simulated order parameter q takes non-zero values below Burn‘s temperature T_B, where the polar clusters appear on cooling. For pressures higher than 0.8 kbar, the T_B changes at the rate dT_B/dX = −200 °C/kbar. The decrease in the difference between Burn's T_B and the freezing T_f temperatures induced by the applied axial pressure is observed. This could be ascribed to the narrowing of temperature range of relaxor behavior.     

Analysis and interpretation of residence time distribution experimental curves in FM01-LC reactor using axial dispersion and plug dispersion exchange models with closed-closed boundary conditions

The liquid phase mixing flow pattern at low (20 < Re < 120) and intermediate liquid flow rate (120 < Re < 400) was studied by means of residence time distribution (RTD) experimental curve in an up-flow Filter Press electrochemical reactor (FM01-LC) bench scale. For this purpose, a plastic turbulence promoter was used with stainless-steel and platinised titanium structural meshes as electrodes in channel configuration. To visualize and determine the mixing flow pattern in the liquid phase, the stimulus-response technique was employed using dextran blue (D_M = 1.058 × 10⁻¹¹ m² s⁻¹, 25 °C, in water) as model tracer. A theoretical analysis and approximation RTD experimental curves with axial dispersion model (ADM) and plug dispersion exchange model (PDE), with “closed-closed vessel” boundary conditions were used in order to establish a better approximation of the axial dispersion, stagnant zones, channelling and by-pass (preference flow) effects present at low and intermediate Re. RTD curves show that the liquid flow pattern in the FM01-LC deviates considerably from axial dispersion model at low Re, where the FM01-LC exhibits large channelling, stagnant zones, and dead zone. The PDE model represents fairly this deviation from ideal flow (less dead zone).     

Axial dispersion in gases flowing through small bore helical column

The results of an experimental study on the axial dispersion of gases flowing in helical columns under laminar flow conditions are reported. The ranges of variables covered are 26.6 < λ < 98; 10 < N_Re < 100; 0.176 < N_Sc < 1.359. The measured dispersion coefficients are correlated with Reynolds, Schmidt and Dean groups. A single dimensionless parameter, N_Dc (N_Se^0,5, was found to correlate the data well. Up to N_De (N_Sc)^0,5 =10, dispersions in straight and coiled tubes exhibit very similar axial dispersion behavior.     

Lamella reorientation in thin films of a symmetric poly(l-lactic acid)-block-polystyrene upon crystallization at different temperatures

The thin films of a symmetric crystalline-coil diblock copolymer of poly(l-lactic acid) and polystyrene (PLLA-b-PS) formed lamellae parallel to the substrate surface in melt. When annealed at temperatures well above the glass transition temperature of PLLA block (T_g^PLLA), the PLLA chains started to crystallize, leading to reorientation of lamellae. Such reorientation behavior exhibited dependence on the correlation between the crystallization temperature (T_c), the glass transition temperature of PS (T_g^PS), the peak melting point of PLLA crystals (T_m^PLLA), and the end melting point of PLLA crystals (T_m,end^PLLA). When annealed at (T_c=) 80 °C (T_c < T_g^PS < T_ODT, order-disorder transition temperature), 123 °C (T_g^PS < T_c < T_m^PLLA < T_ODT), 165 °C (T_g^PS < T_m^PLLA < T_c < T_m,end^PLLA < T_ODT), the parallel lamellae became perpendicular to the substrate surface, exclusively starting at the edge of surface relief patterns. Meanwhile, the corresponding lamellar spacing was significantly enhanced. The PLLA crystallization between PS layers was hypothesized to account for the lamella reorientation during annealing. The crystallization, chain conformation, and possible chain folding mechanisms were discussed, based on detailed analysis of the lamellar structure before and after crystallization.     

Sorption of dissolved organics from aqueous solution by polystyrene resins—II. External and internal mass transfer

External and internal mass transfer were characterized for the sorption of 2,4-dichlorophenol and p-nitrophenol on polystyrene resins. An evaluation method is presented which estimated the external mass-transfer coefficient, κ_L, from the concentration uptake in an integral fixed-bed reactor. The experimental κ_L values are compared with values predicted using four different empirical correlations. The effect of axial dispersion on the values of κ_L was shown to be small under the given conditions.To quantify the intraparticle mass transfer, an effective diffusion coefficient, D_eff, was estimated using the pore diffusion model interpretation of data from a differential column reactor within a recirculated batch system. The D_eff values depend on both resin structure and solute properties. Values of D_eff between 0.3 × 10⁻⁹ and 1.8 × 10⁻⁹ m²/s were observed for macroporous resins, which exceed values that are characteristic of pore diffusion, and a value of 0.6 × 10⁻¹¹ m²/s was estimated for a microporous resin.     

Mass transfer characteristics of low H/D bubble columns

The mass transfer characteristics of 0.2, 0.6 and 1.0 m diameter bubble columns having a low height to diameter ratio (0.6 < H/D < 4) and operated at low superficial gas velocities (0.01 < V_G < 0.08 m/s) were investigated. Different types of spargers were used to study their effect on the column performance. The values of effective interfacial area, a , and volumetric mass transfer coefficient, k_L a , were measured by using chemical methods. The values of a and k_L a were found to vary from 40 to 420 m²/m³ of clear liquid volume and from 0.01 to 0.16 s^?1, respectively, in the range of V_G, and V_L covered in this investigation. The value of the liquid-side mass transfer coefficient, k_L, was found to vary from 3 × 10^?4 to 7 × 10⁴ m/s. The effect of the physical properties of the system on the values of a was also investigated. The height to diameter ratio and the column diameter did not have significant effect on the values of gas holdup, a and k_L a . It was found that the sparger design is not of critical importance, provided multipoint/multiorifice gas spargers are used. The comparative performance of bubble columns having low H/D with horizontal sparged contactors and tall bubble columns has been considered.     

Application of √t-type, logarithmic and half-time methods in desorptive measurements of diffusivity in narrow humidity ranges

This paper concerns the use of non-stationary desorptive measurement techniques for defining the mass diffusivity of cement based materials. Three different procedures are presented: √t-type calculation; logarithmic; and half-time procedures. Cement mortars of different water to cement ratios (w / c), equal to 0.50, 0.65 and 0.80 were selected as the model environment for testing the usability of the above-mentioned desorptive techniques. The study was carried out at the temperature (T) of 20 °C within narrow relative humidity (φ) ranges: from φ₁ = 30% to φ₂ = 12%, and 50% → 30%, 75% → 50%, 85% → 75%, 97% → 85%. The results obtained are used to evaluate the conformity of these methods. The conformity is analyzed with regard to each mortar in all the above humidity ranges Δφ. The values of diffusivity D_m, defined by means of the √t-type calculation and the logarithmic procedure, demonstrated rather high conformity, all relative differences between D_m(√t) and D_m(ln) did not exceeded 20%. However, the half-time procedure can be applied for rough estimation of the diffusivity only. That is because deviations between D_m(t_1 / 2) and the values found by means of the two other methods were too large.     

Study of geometry and operational conditions on mixing time, gas hold up, mass transfer, flow regime and biomass production from natural gas in a horizontal tubular loop bioreactor

A horizontal tubular loop bioreactor (HTLB) was used for production of biomass from natural gas. Hydrodynamic characterizations (mixing time and gas hold up) and mass transfer coefficients were considered in the HTLB (L=2.2 m, H=0.4 m and D=0.03 m) as functions of design parameters, i.e., horizontal length to diameter ratio (L/D) and volume of gas-liquid separator (S) as well as operational parameters, i.e., superficial gas and liquid velocities (U_sG, U_sL). In addition, flow regime in different gas and liquid flow rates was investigated. It was observed from experimental results that U_sL has remarkable effects on gas hold up and k_La due to its influence on mixing time. The volumetric mass transfer coefficients for oxygen (k_La_O2) and methane (k_La_CH4) were determined at different geometrical and operational factors. In average, the amount of oxygen consumption for metabolism is approximately 1.4 times higher than that of methane. In bubble flow regime, the HTLB was used for biomass production, too. A gas mixture of 50% methane and 50% oxygen (based on results of dry cell weight, optical density and doubling time) was the best gas mixture inlet for biomass production. The empirical correlations for mixing time, gas hold up and k_La in terms of U_sG, U_sL, L/D and volume of gas-liquid separator were obtained and expressed separately.     

Solids mixing behavior in a nano-agglomerate fluidized bed

The nano-particles mixing behavior in a nano-agglomerate fluidized bed (NAFB) using R972, a kind of nano-SiO₂ powder, was investigated by the nano-particle coated phosphors tracer method. The axial and radial solids dispersion coefficients in this system were two orders of magnitude lower than those in fluid catalytic cracking (FCC) catalyst systems. The axial solids dispersion coefficient increased with increasing superficial gas velocities, and ranged between 9.1 × 10^− 4 and 2.6 × 10^− 3 m²/s. There was a step increase in the axial solids dispersion coefficient between the particulate fluidization regime and bubbling and turbulent fluidization regimes. As the superficial gas velocity increased, the radial solids dispersion coefficient increased gradually, from 1.2 × 10^− 4 to 4.5 × 10^− 4 m²/s. The much smaller D_a and D_r, compared to regular fluidized systems, is mainly due to the reduced density difference between the fluidized particles and fluidizing medium. To validate this, the solids dispersion coefficients in the NABF were compared with literature values for liquid-solid particulate systems in the particulate fluidization regime and FCC systems in the bubbling and turbulent fluidization regimes. The density difference between the fluidized particles and fluidizing medium and kinetic viscosity of the fluidizing medium, and other hydrodynamic factors like the superficial velocity of the fluidizing medium and the average diameters of the fluidized particles, were the key factors in the solids mixing in the fluidized beds. Empirical correlations are given to describe the results.     

Residence time distribution and liquid holdup in Kenics KMX static mixer with hydrogenated nitrile butadiene rubber solution and hydrogen gas system

A Kenics^® KMX static mixer that has curved-open blade internal structure was investigated to study its hydrodynamic performance related to residence time distribution and liquid holdup in a gas/liquid system. The static mixer reactor had 24 mixing elements arranged in line along the length of the reactor such that the angle between two neighboring elements is 90°. The length of the reactor was 0.98 m with an internal diameter of 3.8 cm and was operated cocurrently with vertical upflow. The fluids used were hydrogen (gas phase), monochlorobenzene (liquid phase) and hydrogenated nitrile butadiene rubber solution (liquid phase). In all the experiments, the polymer solution was maintained as a continuous phase while hydrogen gas was in the dispersed phase. All experiments were conducted in the laminar flow regime with the liquid side hydraulic Reynolds number in the range of 0.04-0.36 and the gas side hydraulic Reynolds number in the range of 3-18. Different polymer concentrations and different operating conditions with respect to gas/liquid flow rates were used to study the corresponding effects on the hydrodynamic parameters such as Peclet number (Pe) and the liquid holdup (ε_L). Empirical correlations were obtained for the axial dispersion coefficient (D_a) and liquid holdup in liquid system alone and for the gas/liquid system separately. It was observed that the Peclet number decreased with the introduction of gas in to the reactor while in the liquid system alone, an increase in viscosity decreased the Peclet number. The liquid holdup was empirically correlated as a function of the physical properties of the fluids used in addition to the operating flow rates.     

Effects of enzyme microcapsule shape on the performance of a nonisothermal packed-bed tubular reactor

The effects of enzyme microcapsule shape (spherical, cylindrical and flat plate) on the performance of a nonisothermal, packed-bed reactor have been modeled as a function of Biot number and Peclet number for mass and heat transfer (Bi_m, Bi_h, Pe_m and Pe_h), and dimensionless heat of reaction α. Under the given simulation conditions, only higher values of Bi_m and Bi_h (>2·5) confirm the influence of microcapsule shape on the reactor performance such that the axial and overall conversion and bulk temperature decrease as follows: spherical > cylindrical > flat plate. In terms of the shape-independent modified Biot number, Bi* = Bi/{(n + 1)/3}, this order is retained for 2 < Bi* < 8. The influence of increasing Pe_m, Pe_h, and α on conversion and bulk temperature also follows the above order. For the flat plate, the exit conversion and temperature are not influenced by Pe_m and Pe_h, that is, mass transfer and thermal backmixing effects, respectively. On the other hand, for the spherical and cylindrical microcapsules, overall backmixing effects are negligible only beyond a critical value of Pe_m (∼7) and Pe_h (∼1·75). The conversion and bulk temperature increase with the increase in α, independent of the microcapsule shape. The spherical and cylindrical microcapsules, unlike the flat plate, cannot be considered isothermal.     

Effect of time on the hydration and temperature-induced phase separation in aqueous polymer solutions. H NMR study

Dehydration during temperature-induced phase separation in D₂O solutions of poly(vinyl methyl ether) (PVME), poly(N-isopropylmethacrylamide) (PIPMAm) and poly(N-isopropylacrylamide) (PIPAAm) was followed from time dependences of NMR spin-spin relaxation times T₂ of HDO. Both the time characterizing the exclusion of the water from mesoglobules (manifested by the increase in T₂ values) and the induction period which precedes the increase in T₂ values, increased in the order PVME < PIPMAm < PIPAAm. For D₂O solutions of PIPMAm/PVME (or PIPMAm/PIPAAm) mixtures a direct connection between the state of the mesoglobules (hydrated or dehydrated) formed by the component with lower LCST (PVME, PIPAAm) and the temperatures of the phase transition of the PIPMAm component was established by NMR spectroscopy.