Numerical simulations of slip correction

Stoke's law

F=6πηRυ

is valid only in liquids. In order to apply this law to particles suspended in air, the slip correction is needed, especially for particles with diameters less than 1 μm.

The slip correction is included in Stoke's law by

with and the Knudsen number (ratio of the particle radius to the mean free path of the gas molecules), η the viscosity of the gas, R the particle radius and and υ the velocity of the particle. For large Knudsen numbers, this equation reduces to

In the present work a simple Monte-Carlo simulation model is used to determine slip corrections in the free molecule regime (Knudsen number Kn 1: The velocity of the air molecules are assumed to follow a Maxwellian distribution. The particle moves steadily in the gas, and the molecules impinge on its surface. The impaction points are distributed uniformly over the particle's surface. A simple criterium shows whether a molecule can in fact hit the surface at the selected point. If so, the transferred momentum is calculated. After sufficient iterations the slip correction is obtained by comparing the total transferred momentum with the expression for the Stokes drag force. Since only the free molecule regime is considered, the slipcorrection equals + β.     

Correlation between equation of state and temperature and pressure dependence of self-diffusion coefficient of polymers and simple liquids

Correlation between the equation of state and the temperature dependence of the self-diffusion coefficient D for polymers such as polystyrene (PS) and polydimethyl siloxane (PDMS) and simple liquids such as argon, methane and benzene and the pressure dependence of D for oligomers such as dimethyl siloxane (DMS) and simple liquids such as cyclohexane and methanol has been examined based on the equation of state derived previously. The experimental data used were published by Antonietti et al. and McCall et al. for polymers, by McCall for linear dimethylsiloxanes and by Jonas et al. and Woolf et al. for simple liquids. The expression for D in this work is given by

where A₁(M) is a function of molecular weight M_w, C₁(T) and P₁(T) are functions of temperature and B₁, n₁ and m₁ are constants determined experimentally. For simple liquids, the values of n₁ obtained range from 0.3 to 1.2, with an average , and m₁ is in the range 0.5–1.2, with . For polymers, values of n₁ are in the range 2.5–7.0 for PS and 0.5–1.3 for PDMS and m₁ for DMS is in the range 0.8–1.0. The relation Dη/T = f(M) is found to be useful for simple liquids over a wide range of temperature including the critical region and for pressures up to ≈5 kbar

1 kbar = 100 MPa There is a close correlation between ln(D/T) and _p and β_T through ln(D/T)ln D_c−⁻¹_p−β⁻¹_T, where D_c is D at the critical temperature and _p and β_T are the thermal expansion coefficient and compressibility, respectively. The molecular weight dependence of D for polymers and simple liquids is discussed based on the experimental data and recent theory of Doi and Edwards. A new model for the mechanism of self-diffusion in the liquid state is proposed.     

Reaction of NO with urea supported on activated carbons

Reaction of NO in air with urea supported on activated carbons (AC) was examined in the range of 100–1000 ppm NO at room temperature to establish a basic scheme for its reduction in open atmosphere. NO in atmosphere containing O₂ was found to be selectively reduced with urea supported on AC at the W/F of 2.5×10⁻³ to 1.5×10⁻³ ACF g min/ml and its reduction continued until the complete consumption of urea. The reduction of NO with urea supported on AC appears to proceed through the following steps.

NO₂+NO+(NH₂)₂CO→2N₂+CO₂+H₂O

Combining (1) and (2) steps results in the followed reaction equation.

Since (1) is rate determining, high NO oxidation activity is essential for AC to be active for the reaction. At the same time, activation of urea by AC is also necessary to reduce NO₂. The present scheme for NO reduction with urea on the AC is very effective to remove NO_x in the open atmosphere under ambient conditions.     

MAGNETOTHERMAL CONVECTION OF POLYATOMIC GASES IN HOMOGENEOUS MAGNETIC FIELDS Part I: Theory

The theory of magnetothermal convection of gases in homogeneous magnetic fields is developed from magnetogas-dynamic theory. Application is made to nonionized paramagnetic and diamagnetic gases between parallel vertical plates. An additive contribution to the natural thermal convection heat transfer results from the temperature dependence of the magnetic permeability of the gas. The result may be expressed parametrically in a dimensionless magnctoconvection number, which characterizes the magnetothermal contribution to the total heat transfer just as the Grashof number characterizes the natural thermal contribution. For oxygen gas at low pressure the magnetothermal contribution is shown to be proportional to p³H²(ΔT)² /ηT⁵ where p is the gas pressure, H is the magnetic field strength, ΔT is the temperature difference between plates, η is the viscosity coefficient and T is the absolute temperature.     

Dilute solution properties of carboxymethylchitins in high ionic-strength solvent

The hydrodynamic characteristics in aqueous solution at ionic strength I=0.2  of carboxymethylchitins of different degrees of chemical substitution have been determined. Experimental values varied over the following ranges: the translational diffusion coefficient (at 25.0°C), 1.1<10⁷×D<2.9 cm² s⁻¹; the sedimentation coefficient, 2.4<s<5.0 S; the Gralen coefficient (sedimentation concentration-dependence parameter), 130<k_s<680 mL g⁻¹; the intrinsic viscosity, 130<[η]<550 mL g⁻¹. Combination of s with D using the Svedberg equation yielded ‘sedimentation–diffusion' molecular weights in the range 40 000<M<240 000 g mol⁻¹. The corresponding Mark–Houwink–Kuhn–Sakurada (MHKS) relationships between the molecular weight and s, D and [η] were: [η]=5.58×10⁻³ M^0.94; D=1.87×10⁻⁴ M^−0.60; s=4.10×10⁻¹⁵ M^0.39. The equilibrium rigidity and hydrodynamic diameter of the carboxymethylchitin polymer chain is also investigated on the basis of wormlike coil theory without excluded volume effects. The significance of the Gralen k_s values for these substances is discussed.     

Deformation kinetics of ageing materials

A constitutive equation of time-dependent, chemically stable materials, which stems from the basic ideas of the irreversible thermodynamics of an internal variable and Eyring's absolute reaction rate theory, has been extended to chemically unstable materials. This formulation is quite general and, in principle, can be applied to many types of materials. In this paper, the ageing behaviour of time-dependent network polymers undergoing chain scission is considered. In the network scission process, we postulate that the energy barrier is affected by a changing of the chemical crosslink density. An explicit equation to account for the energy barrier change, which influences the relaxation process, is formulated. For the purpose of illustration, the effect of different chemical crosslink density, ν, on the relaxation rate has been considered, from which the following theoretical expression of relaxation modulus ΔE(t) is obtained:

ΔE = ΔE[t exp (γv/kT)]

It can be seen that a change in v leads to an effective change in the time scale, usually denoted by a_x. Here the analytical expression a_x = exp(γν/kT) correlates quite well with the experimental data.     

Two experiments for the introductory chemical reaction engineering course

This paper describes a pair of chemical reaction experiments developed for Rowan University's introductory course in chemical reaction engineering: an esterification reaction carried out in a packed bed, and a competitive reaction in which the kinetics were influenced by micromixing.

The first experiment is the esterification of ethanol and acetic acid to form ethyl acetate. Students first examine this reaction in their organic chemistry class. The experiment developed in this project re-examines this reaction from a chemical engineering perspective. For example, the reaction is reversible and equilibrium-limited, but in the organic chemistry lab, there is no examination of the kinetics. The complementary chemical engineering experiment examines the relationship between residence time and conversion.

The second experiment is a competitive system involving two reactions:

H₂BO₃⁻ + H⁺ ↔ H₃BO₃

5I⁻ + IO₃⁻ + 6H⁺ → 3I₂ + 3H₂O

The first reaction is essentially instantaneous. Thus, when H⁺ is added as the limiting reagent, a perfectly mixed system would produce essentially no I₂. Production of a significant quantity of I₂ is attributed to a local excess of H⁺; a condition in which all H₂BO₃⁻ in a region is consumed and H⁺ remains to react with I⁻ and IO₃⁻.

In the spring of 2005, for the first time, both experiments were integrated into the undergraduate chemical reaction engineering course. This paper describes the use of the experiments in the classroom and compares the performance of the 2005 students to the 2004 cohort, for whom the course included no wet labs at all.     

Mass transfer at cylindrical gas evolving electrodes

Mass transfer coefficients at cylindrical, H₂ evolving electrodes, were measured by determining the reduction rate of K₃Fe(CN)₆. The variables studied were: gas discharge rate V, diameter of the cylinder D, height and position of the cylinder. The diameters ranged from 0·2–2·5 cm, the cd from 25-380 mA/cm². For horizontal cylinders, the following correlation was found: log K = a + 2·17 log(V^0·11/D^0·08). The application of gas evolving cylindrical electrodes in industrial electrolysis is discussed in comparison with rotating electrodes.     

Determination of optimized membrane area from long-term flux decline data

Due to membrane fouling, long-term flux decline follows a general trend: it starts with a steep decline followed by a gradual decrease and approaches its final value asymptotically. Coupled with the concentration effect, our data indicate that this flux behavior can be represented with the equation F = f(t) (k₁ 1n r + k₂) with f(t) = a₀ + exp [a₁ + a₂t + a₃t²] where t is time, a₀, a₁, a₂, a₃, k₁, and k₂ are constants and r is the concentration ratio.

To take advantage of early high flux, ultrafiltration is usually discontinued after a period of time t_u, to be replaced by membrane cleaning period of time t_c. Only then the ultrafiltration cycle is repeated. The total membrane area required for multistage ultrafiltration can be shown as

Full-size image

where Q₀ is the ultrafiltration capacity, n is the number of stages, and rr_i is the concentration ratio at stage i. An optimization method can be utilized to determine minimum value of A_T. Results will be presented for the case of ultrafiltration of palm oil mill effluent.     

Cathodic behaviour of samarium(III) in LiF–CaF2 media on molybdenum and nickel electrodes

The electrochemical behaviour of SmF₃ is examined in molten LiF–CaF₂ medium on molybdenum and nickel electrodes. A previous thermodynamic analysis suggests that the reduction of SmF₃ into Sm proceeds according to a two-step mechanism:

Sm^III + e⁻ = Sm^II

Sm^II + 2e⁻ = Sm

The second step occurs at a potential lower than the reduction potential of Li⁺ ions.

Cyclic voltammetry, chronopotentiometry and square-wave voltammetry were used to confirm this mechanism and the results show that it was not possible to produce samarium metal in molten fluorides on an inert cathode (molybdenum) without discharging the solvent. The electrochemical reduction of SmF₃ is limited by the diffusion of SmF₃ in the solution. The diffusion coefficient was calculated at different temperatures and the values obtained obey Arrhenius’ law.

For the extraction of the samarium from fluoride media, the use of a reactive cathode made of nickel leading to samarium–nickel alloys is shown to be a pertinent route. Cyclic voltammetry and open-circuit chronopotentiometry were used to identify and to characterise the formation of three alloys: liquid Sm₃Ni and a compact layer made of SmNi₃ and SmNi₂.     

SINTERING KINETICS AND TRANSPORT PROPERTY EVOLUTION OF LARGE MULTI-PARTICLE AGGREGATES

Ultrafine (“nano”-) particles produced from highly supersaturated vapors or liquids are usually aggregated, often containing thousands of small 'primary' particles bound together in tenuous structures characterized by mass fractal dimensions less than 3. Such aggregates have large initial surface area but are metastable with respect to more compact configurations. Available restructuring mechanisms include surface energy driven coalescence, which, in the case of viscous flow at high gas temperatures, is ultimately able to obliterate all evidence of the original (“primary”) particles. We here exploit the notion that, provided an aggregate is sufficiently large, it can be treated like a spatially non-uniform porous medium, undergoing finite-rate surface energy driven viscous flow sintering leading to final collapse to a single dense sphere. For this purpose, after a D_ƒ ≌s const stage of sintering [associated with a corresponding increase in mean apparent primary particle ('grain') size], we use an extension of the sintering rate models of Mackenzie and Shuttleworth (1949) and Scherer (1977), treating the material of the restructuring aggregate to be a Newtonian viscous fluid. We predict and report here the time-dependent increase in fractal dimension, D_ƒ, and associated decreases in: aggregate outer (maximum) radius, mobility radius, and changes in accessible surface area with dimension-less time [real time in multiples of the characteristic sintering time, μ (R₁)_t=0/σ cr, where u is the material's viscosity (R_l)_t=0 is the effective initial grain radius and a the material surface tension]. In these units, we find that the total required coalescence time does not increase with N as sensitively as N^1/3 an important observation for processes involving very large aggregates. With validation and the indicated extensions, our pseudo-continuum methods are efficient enough to be used for estimating the morphological- and transport property-evolution of entire populations of restructuring aggregates, perhaps characterized by some non-separable probability density function pdf(N, D_ƒ,R₁,) locally, in non-isothermal combustion-synthesis reactors.     

Effect of electro-chemical properties of sodium salts of various anions on their diffusional parameters in symmetrical cellulose acetate membranes

A relation was obtained between electro-chemical properties of sodium salts (NaCl, NaBr, and Na₂SO₄), and the thermodynamic property of permeability in symmetrical cellulose acetate membranes, the distribution coefficient K and the kinetic property, the overall diffusion coefficients D. K and D were obtained by the method we proposed using measured unsteady- and steady-state dialysis data. The K values increase with the increase of water content and are in the range of 10⁻² for sodium halides and 10⁻³ for Na₂SO₄. D is found to increase with the increase of the solute concentration, and the extrapolated values of D to zero concentration D⁽⁰⁾ are obtained as 0.015–0.03 μm²/s and increase with the increase of water content in the membrane. D can be divided into the concentration independent diffusion coefficients in the dense part of the membrane D_d and in the porous D_p, applying a two-part (perfect or dense and imperfect or porous) model of the membrane. Contrary to D_d, D_p increases with the increase of W_w and can be correlated as D_p,c = _d exp (γ × W_w). It is shown that the averaged D_d, _D increases with the increase of the quantity of the ionic mobility u of the solutes at infinite dilution divided by valence, and that the parameter γ increases with the increase of the ionic mobility u. The value of K increases slightly with the increase of water content and decreases with the increase of the Flory—Huggins parameter χ. The Flory—Huggins parameter χ is calculated from the measured values of distribution coefficients and data obtained from the literature. And it was found that the gradient of linear decrease of χ (λ_cation) depends on equivalent ionic conductivity of anion of salt, λ_an.     

Attrition resistance of cobalt F–T catalysts for slurry bubble column reactor use

There exists much current interest in the use of supported Co catalysts and slurry bubble column reactors (SBCR) for the conversion of natural gas to higher hydrocarbons via the Fischer–Tropsch (F–T) synthesis. Catalyst attrition resistance is extremely important in the operation of slurry-phase reactor systems because of potential problems with plugging of system filters and/or contamination of the liquid products. This paper addresses the effects of different supports, promoters, and preparation methods on the attrition resistance of Co F–T catalysts for SBCR use.

The calcined supports had attrition resistances (inversely related to % fines <11 μm generated during attrition testing) as follows:

−Al₂O₃>TiO₂(rutile)SiO₂

Loading of Co onto the supports improved the attrition resistances of both alumina and silica significantly. It has essentially no effect on titania. The resulting catalysts had attrition resistances in the order

Co/Al₂O₃>Co/SiO₂>Co/TiO₂(rutile)>Co/TiO₂(anatase)

The addition of small amounts of metal (Ru, Cu) and oxide (La, Zr, K, Cr) promoters had mainly small effects on the attrition resistance of the supported Co catalysts. However, it would appear that the addition of Zr to Co/alumina had a negative impact on its attrition resistance. The different preparation methods used in this study (aqueous impregnation, non-aqueous impregnation, and kneading) did not appear to have a significant effect on catalyst attrition resistance.     

The kinetics of photocatalytic degradation of trichloroethylene in gas phase over TiO2 supported on glass bead

In this investigation, a packed bed filled with coated titanium dioxide glass beads to study the kinetics of photocatalytic degradation of trichloroethylene under irradiation of 365 nm UV light. In the range of 100–500 ml/min of flowrate, the reaction rate for 6 μM TCE increased with an increasing flowrate upto 300 ml/min, while was not affected by the flowrate at the values higher than 300 ml/min. For moisture in the range of 9.4–1222.2 μM, the reaction rate of TCE was decreased with an increasing humidity. The adsorbed water on the catalyst surface could compete with the adsorption of TCE on the sites. The reaction rate of 6 μM TCE increased as the light intensity increased, and was proportional to the 0.61 order of the light intensity. Among the three L–H bimolecular form models, the experimental data had the best fit for one of models:

     

A general strength distribution function for brittle materials

A new strength distribution function for brittle materials is developed, which applies to materials with an inhomogeneous distribution of flaws.

The probability of failure is

F=1−exp[−N_c,s]

where N_c,s is the mean number of critical defects in the specimen of size S. The well-known Weibull statistics are a special case of the new statistics for a special flaw size distribution.

Several aspects of the relationships between the Weibull statistics and material structure are analysed in the light of the new formalism. Examples are materials with several different flaw distributions or rising crack resistance. The conditions necessary to get a Weibull distribution as well as the reasons why Weibull distributions are observed so often in the daily material testing practice are discussed. Finally, the minimum number of test specimens necessary to guarantee a reliable prediction of the component's reliability using Weibull's theory is given. This number depends on the necessary reliability as well as on the loaded (effective) volumes of the test specimens and components, respectively.     

Activation of methane with organopalladium complexes

Selective, direct oxidation of methane to methanol is a process of scientific interest and industrial importance. Reports have appeared in the literature describing the use of organometallic complexes to effect this transformation [1–5]. Investigation of one of these reaction schemes in our laboratory has produced interesting results. Our research effort was an extension of work reported by Sen et al. [3]. The reported reaction occurs between methane (at 800 psig 5.52 MPa) and palladium(II) acetate in trifluoroacetic acid at 80°C (Eq. (1)). The product, methyl trifluoroacetate, is readily hydrolyzed to produce methanol and trifluoroacetic acid. It is reported that methyl trifluoroacetate is produced with reported conversions, calculated on palladium metal recovery, of 60 percent.

     

Surface dynamics during latex film formation

Surface dynamics during latex film formation has been investigated theoretically and experimentally by atomic force microscopy. The peak-to-valley distance, y(t), of the latex particles in the surface plane of the latex film decayed exponentially with time during film formation. A theoretical relationship between y(t) and time, t, is given by y(t)=y(0) exp[−t/τ], where y(0) is the value of y(t) when t is zero. τ is a characteristic constant related to the nature of polymer, the particle radius, the surface diffusion coefficient and the temperature. The relationship between the surface diffusion coefficient, D_s, y(0), the radius of the latex particles, R, temperature, T, and τ is given approximately by D_s=1.2×10⁻²⁰y(0)²[2R−y(0)]²T/τ (cm²/s), where the units are manometers for y(0) and R, kelvin for temperature, and seconds for τ. By measuring the decay of y(t) with time, the surface diffusion coefficient can be obtained. The surface diffusion coefficient for a poly(methyl methacrylate-co-butylacrylate) (50:50) copolymer latex film was found to be A×10⁻¹³ cm²/s, A is temperature-dependent.     

Mass transfer resistances in the palladium-catalysed hydrogenation of 1-methoxy-2,4-(nitrophenyl)-ethane

Hydrogenation of 1-methoxy-2,4-(nitrophenyl)-ethane (MNPE) to 1-methoxy-2,4-(aminophenyl)-ethane (MAPE) over Pd/C catalyst was studied in a three-phase stirred slurry reactor. The reaction was performed in a kinetic region (no internal and external mass transfer resistances) at temperatures ranging from 293 to 320 K, hydrogen pressure 200 kPa using powdered pellets of 1–4% Pd on carbon support. The rates have been correlated with power law and Hougen–Watson kinetic models. The rate expression which best fit the data is

which postulates a reaction between adsorbed MNPE (marked by B) and hydrogen (marked by A).     

Generating hydrogen-rich fuel-cell feeds from dimethyl ether (DME) using physical mixtures of a commercial Cu/Zn/Al2O3 catalyst and several solid–acid catalysts

Homogeneous physical mixtures containing a commercial Cu/ZnO/Al₂O₃ catalyst and a solid–acid catalyst were used to examine the acidity effects on dimethyl ether hydrolysis and their subsequent effects on dimethyl ether steam reforming (DME-SR). The acid catalysts used were zeolites Y [Si/Al = 2.5 and 15: denoted Y(Si/Al)], ZSM-5 [Si/Al = 15, 25, 40, and 140: denoted Z(Si/Al)] and other conventional catalyst supports (ZrO₂, and γ-Al₂O₃). The homogeneous physical mixtures contained equal amounts, by volume, of the solid–acid catalyst and the commercial Cu/ZnO/Al₂O₃ catalyst (BASF K3-110, denoted as K3). The steam reforming of dimethyl ether was carried out in an isothermal packed-bed reactor at ambient pressure.

The most promising physical mixtures for the low-temperature production of hydrogen from DME contained ZSM-5 as the solid–acid catalyst, with hydrogen yields exceeding 90% (T = 275 °C, S/C = 1.5, τ = 1.0 s and P = 0.78 atm) and hydrogen selectivities exceeding 94%, comparable to those observed for methanol steam reforming (MeOH-SR) over BASF K3-110, with values equaling 95% and 99%, respectively (T = 225 °C, S/C = 1.0, τ = 1.0 s and P = 0.78 atm). Large production rates of hydrogen were directly related to the type of acid catalyst used. The hydrogen production activity trend as a function of physical mixture was