Continuous flow systems. Distribution of residence times
TL;DR: In this article, the authors proposed a method to predict the distribution of residence-times in large systems by using distribution-functions for residencetimes, which can be used to calculate the etficiencies of reactors and blenders.
About: This article is published in Chemical Engineering Science.The article was published on 1995-12-01. It has received 1416 citations till now. The article focuses on the topics: Residence time distribution & Perfect mixing.
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01 Jan 1961
TL;DR: In this paper, a more direct method is presented for solving the differential equation governing the process of dispersion in a semi-infinite medium having a plane source at z = 0.
Abstract: Published papers indicate that most investigators use the coordinate transformation (x ut) in order to solve the equation tor dispersion of a moving fluid in porous media. Further, the boundary conditions O=0 at x=«> and 0=00 at x= «> for Z>0 are used, which results in a symmetrical concentration distribution. This paper presents a solution of the differential equation that avoids this transformation, thus giving rise to an asymmetrical concentration distribution. It is then shown that this solution approaches that given by symmetrical boundary conditions, provided the dispersion coefficient D is small and the region near the source is not considered. INTRODUCTION In recent years considerable interest and attention have been directed to dispersion phenomena in flow through porous media. Scheidegger (1954), deJong (1958), and Day (1956) have presented statistical means to establish the concentration distribution and the dispersion coefficient. A more direct method is presented here for solving the differential equation governing the process of dispersion. It is assumed that the porous medium is homogeneous and isotropic and that no mass transfer occurs between the solid and liquid phases. It is assumed also that the solute transport, across any fixed plane, due to microscopic velocity variations in the flow tubes, may be quantitatively expressed as the product of a dispersion coefficient and the concentration gradient. The flow in the medium is assumed to be unidirectional and the average velocity is taken to be constant throughout the length of the flow field. BASIC EQUATION AND SOLUTION Because mass is conserved, tl^e governing differential equation is determined to be d<7 (1) v where D=dispersion coefficient C= concentration of solute in the fluid u= average velocity of fluid or superficial velocity/ porosity of medium x= coordinate parallel to flow y,z coordinates normal to flow 2=time. In the event that mass transfer takes place between the liquid and solid phases, the differential equation becomes _ 5(7 d(7 &F where F is the concentration of the solute in the solid phase. The specific problem considered is that of a semiinfinite medium having a plane source at z=0. Hence equation 1 becomes Initially, saturated flow of fluid of concentration, (7=0, takes place in the medium. At t Q, the concentration of the plane source is instantaneously changed to (7=(70 . Thus, the appropriate boundary conditions are <7(co,£)=0; *> The problem then is to characterize the concentration as a function of x and t. To reduce equation 1 to a more familiar form, let (4) A-l 586211 61 2 A-2 FLUID MOVEMENT IN EARTH MATERIALS Substituting equation 4 into equation 1 gives The boundary conditions transform to = Co exp ( It is thus required that equation 5 be solved for a timedependent influx of fluid at 2=0. The solution of equation 5 may be obtained readily by use of Duhamel's theorem (Carslaw and Jaeger, 1947, p. 19): If C=F(x,y,z,t) is the solution of the diffusion equation for semi-infinite media in which the initial concentration is zero and its surface is maintained at concentration unity, then the solution of the problem in which the surface is maintained at temperature (t) is
875 citations
TL;DR: In this article, the authors present an evaluation and review of the transit time literature in the context of catchments and water transit time estimation and provide a critical analysis of unresolved issues when applied at the catchment-scale.
766 citations
Cites methods from "Continuous flow systems. Distributi..."
...Most methods are based on early adaptations from the chemical engineering and groundwater fields (e.g., Danckwerts, 1953; Eriksson, 1958; Maloszewski and Zuber, 1982; Haas et al., 1997; Levenspiel, 1999) and may not apply in catchments where there are complex and important controlling processes…...
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TL;DR: In this paper, it is shown that there are three components in a mixing process: convection, diffusion and shear, and a statistically satisfactory expression evolved for the state of a mixture.
Abstract: It is shown that there are three components in a mixing process: convection, diffusion and shear. The concepts involved in analysing complete and partial mixtures are examined and a statistically satisfactory expression evolved for the state of a mixture. A number of theories of mixing rate are examined and compared with the few published experimental results, and a new theoretical treatment is offered based on diffusion theory. This is shown to be in at least as good agreement with fact as existing theories, and to provide a better basis for extension to more complex cases.
560 citations
TL;DR: In this paper, the authors examined the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques and derived an analytical estimate of the flux rate into and out of the vortex neighbourhood.
Abstract: We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.
482 citations
TL;DR: In this paper, the pore water velocity was estimated by measuring the solute displacement within each subplot and the entire field and was found to be logarithmically normally distributed and in agreement with volumetric measures of water infiltration rates.
Abstract: Solute distributions within a soil profile during the leaching of water-soluble salts applied to the soil surface were measured at six depths to 182.4 cm within 20 subplots of a 150-ha field. Estimates of the pore water velocity based upon measures of solute displacement within each subplot and the entire field were found to be logarithmically normally distributed and in agreement with volumetric measures of water infiltration rates. Such agreement was only possible because it was recognized that the observed values were not normally distributed, and their mean values were calculated accordingly. The number of observations required to yield an estimate of the mean pore water velocity within a prescribed accuracy is shown to depend upon the nature and extent of the spatial variability of the field soil. For the field examined, 100 observations would allow the mean pore water velocity to be estimated within ±50% of its true value. The functional relation between field-measured values of the apparent diffusion coefficient, also found to be logarithmically normally distributed, and pore water velocity is examined and interpreted in terms of solute distributions likely to be measured at specific sampling sites.
444 citations
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876 citations
TL;DR: In this paper, it is shown that the important features of such mixtures can be expressed by two statistically defined quantities, the scale and the intensity of segregation, and methods of measuring these are suggested.
Abstract: The systematic study of mixing processes requires a quantitative method of expressing “goodness of mixing”, based on conveniently-made measurements. In this paper, mixtures of mutually soluble liquids, fine powders, or gases are considered. It is shown that the important features of such mixtures can be expressed by two statistically-defined quantities, the scale and the intensity of segregation, and methods of measuring these are suggested. The discussion also throws light on some of the factors which affect the efficiency of mixing processes.
698 citations