LAG TIME AND STEADY-STATE PERMEABILITY FOR A DIFFUSION SYSTEM WITH SERIAL AND/OR PARALLEL PATHS

A diffusion system is considered in which transport occurs along two parallel paths which, in turn, are in series with another path. Relations for the steady-state permeability and lag time are obtained in terms of the cross-sectional areas and lengths of diffusion, diffusivities and interfacial partition coefficients. The relations are derived from the slope and intercept (on the time axis) of the linear asymptote for the total amount of diffusant released into the receiver. The asymptote is calculated by repeated integration rather than a complete solution to the coupled diffusion equations. Equivalence relations are given for the transformation of the parallel paths into a single path, thus allowing for a treatment of the whole system in the form of two homogeneous paths in series. These equivalence relations include an effective partition coefficient as an average of its component's partition coefficient weighted by the corresponding fraction of cross-sectional area, and an effective diffusivity as an average of individual diffusivities weighted by the corresponding fraction of the product of cross-sectional area and partition coefficient.     

A FIRST-ORDER TIME CONSTANT ESTIMATION FOR NONLINEAR DIFFUSION PROBLEMS

A Laplace transform–based procedure was proposed to calculate the effective time constant for a class of nonlinear diffusion problems. The governing mathematical representation was first estimated with a linear model by omitting the nonlinear term. The solution to this problem was later introduced into the original equation, which was solved with Laplace transforms, resulting in a first-order approximation of the real system's behavior. A time constant was calculated using frequency-domain expressions. Two case studies were considered to illustrate the methodology. As the rate of heat supplied to a rod is raised, the speed at which the temperature reached an equilibrium value decreased. Increasing the maximum velocity in reaction-diffusion transport by a factor of three lowered the time constant by only 1.7%. The applications of this method range from biosensor dynamics to process control.     

RECIPROCITY OF LAG TIME FOR REACTIVE DIFFUSION THROUGH HETEROGENEOUS MEMBRANES: A GREEN'S FUNCTION APPROACH

Forward and reverse diffusion through heterogeneous membranes with position-dependent diffusivity, partition coefficient and reaction rate constant are considered. The reciprocity of the lag time is proved, utilizing the symmetry of the corresponding Green' function with respect to the exchange of the coordinate arguments. Simple relations between the permeabilities, reactive diffusion fluxes and total released amounts for forward and reverse diffusion direction are derived.     

MIXING WITH DIFFUSION AND FAST CHEMICAL REACTION IN SIMPLE SHEAR FLOW: A COMPARISON OF STRETCH MODEL TO TWO-DIMENSIONAL DIFFUSION WITH PERIODIC BOUNDARY CONDITIONS

The lamellar stretch model for describing mixing with simultaneous diffusion and fast chemical reaction is evaluated for simple shear flow between two parallel plates by comparing the predicted concentration profiles to those computed from the governing partial differential equation. A direct algorithm is devised to solve the sparse matrix originating from the discretization of the PDE. Under conditions where initial striation thickness equals the characteristic dimension of the flow geometry, the lamellar stretch model turns out to be quite accurate for k ≥ 10⁶ or k ≤ 1, where k is the product of the Reynolds number and the Schmidt number. In the intermediate range of k values, where mechanical and diffusional mixing are of comparable importance, the model is less accurate     

THE TWO-THIRDS POWER MODEL FOR PREDICTING DIFFUSION COEFFICIENTS IN LIQUIDS

Most of the models found in the literature for predicting diffusion coefficients in liquids take the viscosity of the solution as inversely proportional to the diffusivity. A model (previously derived for sucrose-water system)which considers the viscosity raised to the two-third power, is evaluated here using published data for the Benzene—Cylohexane, Ethanol-Water and Acetone-Chloroform systems. These results, and results for the sucrose-water system are compared with the experimental values, and with those obtained using the Hartley—Crank equation. Better agreement with the experimental values at low and in the middle of the concentration range was found when diffusivities were calculated using the model proposed than when using the Hartley-Crank equation.     

MULTIPLE BILINEAR TIME SERIES MODELS

Abstract. A definition of multiple bilinear time series models is given. Sufficient conditions are obtained for the existence of strictly stationary solutions conforming to the model, and a brief discussion of the first and second order structure is included.     

A NOTE ON NON-STATIONARITY AND CANONICAL ANALYSIS OF MULTIPLE TIME SERIES MODELS

Abstract. Box and Tiao (1977) established the correspondence between non-stationary roots and canonical correlations of an AR(1) process. In this paper, we give an alternative, more direct, proof of the correspondence and extend a special case of that result to AR( p ) processes. The usefulness of these results for multiple time series modelling is also briefly discussed.     

DIFFERENCING MULTIPLE TIME SERIES: ANOTHER LOOK AT CANADIAN MONEY AND INCOME DATA

Abstract. In multiple time series analysis it is sometimes suggested to remove non-stationarities of the univariate subseries by differencing prior to the multivariate analysis. It is pointed out that, in general, this is not adequate if AR models are built even if stationarity of the univariate subseries can be induced by differencing. Canadian money and income data that were previously analysed by Hsiao are used to illustrate the resulting problems.     

ROBUST REGRESSION AND INTERPOLATION FOR TIME SERIES

Abstract. In this paper we shall consider the interpolation problem under the condition that the spectral density of a stationary process concerned is vaguely known (i.e., Huber's ε -contaminated model). Then we can get a minimax robust interpolator for the class of spectral densities S ={ g:g(x)=(1-ε)f(x)+εh(x)ε Ar D_o, 0<ε<1}, where f(x) is a known spectral density and D ₀ is a certain class of spectral densities. Also we shall consider the time series regression problem under the condition that the residual spectral density is vaguely known. Then we can get a minimax robust regression coefficient estimate for the class of the residual spectral densities S .     

EMPIRICAL IDENTIFICATION OF MULTIPLE TIME SERIES

Abstract. In the univariate case the problem of empirical identification consists in determining the order parameters p , d and q of ARIMA ( p, d, q ) processes. In this paper we introduce some new techniques for handling the corresponding problem for a multiple time series X ( t ) with the main emphasis on AR and MA models. Types of joint nonstationarity (or rather almost nonstationarity) are defined and a method of analyzing such structures based on the ordered eigenvalues of the function C ( t ) = K ( t ) K ^-1(0) is discussed, where K ( t ) is the covariance function of X ( t ). It is proposed that the further identification procedure should be based on two X ² statistics and on the estimated trace and eigenvalues of C ( t ), the matrix correlation function p ( t ) and the matrix partial correlation function P ( t ). The suitability for identification purposes of each of these functions is examined in terms of such properties as scale-invariance, existence of normalized eigenvalues and standard errors. The methods introduced are illustrated on a 5-dimensional economic time series first studied by Quenouille and on a 4-dimensional smulated MA series.     

具有级间短路的多股进料和出料的系统的停留时间分布

本文工作是前文工作的一些补充。进一步讨论了具有多股进料和出料、多级、具有级间短路的系统的流动模型,给出其停留时间分布函数的表达式。     

AN EXPERIMENTAL STUDY OF POLYMER DIFFUSION IN CONCENTRATED SOLUTION: IMPLICATIONS FOR DIFFUSION IN POLYMERIZATION

The Trommsdorf or gel effect in free radical polymerization is due to the fact that the termination reaction becomes strongly diffusion controlled above a critical concentration associated with the onset of molecular entanglements. Therefore, an understanding of polymer self-diffusion in entangled systems becomes essential to understanding the Trommsdorf effect. Our group has previously proposed a molecular model for the gel effect which uses a specific theory for polymer diffusion (reptution). The present work represents an experimental attack on the same problem Experimental studies of polymer self-diffusion in entangled systems are scarce. Quasielastic light scattering from ternary systems composed of solvent(l)-polymer(2)-polymer(3), in which species (3) is isorefractive to the solvent (i.e. 0n/0c ₃ = 0), offers an attractive way to study the tracer diffusion coefficient of species (2) in a binary mixture of composition c ₃. In regimes of low momentum transfer (qR _G < 1,) where q is the scattered wave vector and R _G is the polymer radius of gyration, we have shown that the correlation function of the scattered electric field should decay with a single exponential decay time, given by (D₂₂ q ²)^?1 where D ₂₂(c ₂, c ₃) is the main ternary diffusion coefficient of component (2). Extrapolation to zero concentration of 2) at fixed concentration of (3) yields the tracer diffusion coefficient of (2) in the binary mixture of (1) and (3). The systems toluene(l)-polystyrene(2)-polymethylmethacrylate(3) (0n/0c ₃ ≈ 0.007 at 25°C) and toluene(l)-polystyrene(2)-polyvinylmethylether (3) (0n/0c ₃ ≈ 0.012 at 60°C) very nearly satisfy the above criteria. In both systems, we have found that the tracer diffusion coefficient of the polystyrene decreases with increasing concentration of the isorefractive polymer. Further studies have focused upon the dependence of the tracer diffusion coefficient upon polystyrene molecular weight, and upon the effect of incomplete index matching.     

TESTING FOR GAUSSIANITY AND LINEARITY OF A STATIONARY TIME SERIES

Abstract. Stable autoregressive (AR) and autoregressive moving average (ARMA) processes belong to the class of stationary linear time series. A linear time series { } is Gaussian if the distribution of the independent innovations {ε( t )} is normal. Assuming that E ε( t ) = 0, some of the third-order cumulants c_xxx= Ex ( t ) x ( t + m ) x ( t + n ) will be non-zero if the ε( t ) are not normal and E ε³( t )≠O. If the relationship between { x ( t )} and {ε( t )} is non-linear, then { x ( t )} is non-Gaussian even if the ε( t ) are normal. This paper presents a simple estimator of the bispectrum, the Fourier transform of { c _xxx( m, n )}. This sample bispectrum is used to construct a statistic to test whether the bispectrum of { x ( t )} is non-zero. A rejection of the null hypothesis implies a rejection of the hypothesis that { x ( t )} is Gaussian. Another test statistic is presented for testing the hypothesis that { x ( t )} is linear. The asymptotic properties of the sample bispectrum are incorporated in these test statistics. The tests are consistent as the sample size N →-∞     

DIFFUSION AND MASS TRANSFER IN MICELLAR SYSTEMS

Research on diffusion and mass transfer in micellar systems is reviewed. Models for interfacial transport at liquid/liquid interfaces, solubilization at solid/liquid and at liquid/liquid interfaces, and for detergency are discussed and similarities between these phenomena are elucidated. The literature on diffusion in micellar systems is discussed, especially in the region of the critical micelle concentration where previous investigators have presented results in disagreement with each other. Finally, the mechanism of micellar transport through membranes is also reviewed.     

NONPARAMETRIC ESTIMATORS FOR TIME SERIES

Abstract. Kernel multivariate probability density and regression estimators are applied to a univariate strictly stationary time series X _r We consider estimators of the joint probability density of X _t at different t -values, of conditional probability densities, and of the conditional expectation of functionals of X _v given past behaviour. The methods seem of particular relevance in light of recent interest in non-Gaussian time series models. Under a strong mixing condition multivariate central limit theorems for estimators at distinct points are established, the asymptotic distributions being of the same nature as those which would derive from independent multivariate observations.     

ON THE EXISTENCE OF A GENERAL MULTIPLE BILINEAR TIME SERIES

Abstract. A sufficient condition is derived for the existence of a strictly stationary solution of the general multiple bilinear time series equations (without assuming subdiagonality). The condition is shown to reduce to the condition of Stensholt and Tjostheim in the special case which they consider. Under this condition a solution is constructed which is shown to be casual in the sense we define, strictly stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, i.e. the multiple autoregressive moving-average (ARMA) model. the condition given here reduces to the well-known sufficient condition for the existence of a casual stationary solution.     

INTRAPARTICLE CONVECTION,DIFFUSION AND ZERO ORDER REACTION IN POROUS CATALYSTS

An assessment is made of the influence of convective effects in the behaviour of calalyst pellets with large pores

Concentration profiles and effectiveness factors are calculated for zero order reactions in isothermal regimes using a model which involves convective transport as well as diffusion and reaction processes in the pellet

Errors made in current reactor design are discussed. A predictive method for calculating the intraparticle fluid velocity in industrial cases is reported. Finally, the analysis is applied to the design of biological reactors.     

MASS TRANSFER AND DIFFUSION COEFFICIENT DETERMINATION IN THE WET AND DRY SALTING OF MEAT

Dehydrated salted meat is widely used in Brazil as a very important source of animal protein. The main objective of this kind of processing is water removal. initially by osmotic pressure changes and then by drying, resulting in a product with intermediate moisture levels.

In this work, mass transfer and salt diffusion in pieces of meat submitted to wet and dry salting were studied. Slabs of beef m. trapezius with an infinite plate geometry were salted in a NaCl saturated solution or in a dry salt bed, at two temperatures (10 and 20°C) and different time exposures (120 min and 96 hours). Equilibration studies were extended up to six days.

It was observed that water loss increased with salt uptake, for increasing periods of times. At 20°C the moisture loss was higher than it was at 10°C in both salting processes. On the other hand, the kinetics of salt uptake and moisture loss were of greater importance in the process of dry salting than in that of wet salting.

The salt diffusion coefficient for wet salting was 0.26 × 10^-10m²/s at20°C and 0.25 × 10^-10 m²/s at 10°C and for the dry salting the values were 19.37 × 10^-10 m²/s at 20°C and 17.21 × 10^-10 m²/s at 10°C.     

MASS TRANSFER AND DIFFUSION COEFFICIENT DETERMINATION IN THE WET AND DRY SALTING OF MEAT

ABSTRACT

Dehydrated salted meat is widely used in Brazil as a very important source of animal protein. The main objective of this kind of processing is water removal. initially by osmotic pressure changes and then by drying, resulting in a product with intermediate moisture levels.

In this work, mass transfer and salt diffusion in pieces of meat submitted to wet and dry salting were studied. Slabs of beef m. trapezius with an infinite plate geometry were salted in a NaCl saturated solution or in a dry salt bed, at two temperatures (10 and 20°C) and different time exposures (120 min and 96 hours). Equilibration studies were extended up to six days.

It was observed that water loss increased with salt uptake, for increasing periods of times. At 20°C the moisture loss was higher than it was at 10°C in both salting processes. On the other hand, the kinetics of salt uptake and moisture loss were of greater importance in the process of dry salting than in that of wet salting.

The salt diffusion coefficient for wet salting was 0.26 × 10^?10m²/s at20°C and 0.25 × 10^?10 m²/s at 10°C and for the dry salting the values were 19.37 × 10^?10 m²/s at 20°C and 17.21 × 10^?10 m²/s at 10°C.     

INTRAPARTICLE CONVECTION, DIFFUSION AND ZERO ORDER REACTION IN POROUS CATALYSTS

An assessment is made of the influence of convective effects in the behaviour of calalyst pellets with large pores

Concentration profiles and effectiveness factors are calculated for zero order reactions in isothermal regimes using a model which involves convective transport as well as diffusion and reaction processes in the pellet

Errors made in current reactor design are discussed. A predictive method for calculating the intraparticle fluid velocity in industrial cases is reported. Finally, the analysis is applied to the design of biological reactors.