Sedimentation equation, basic

Substituting the resistance force into equation 51 and expressing F and V in terms of d, the basic equation of sedimentation theory is obtained ... 

Basic Sedimentation Equilibrium Equation. Sedimentation equilibrium experiments are performed at constant temperature. The condition for sedimentation equilibrium is that the total molar potential, m, for all components i be constant everywhere in the solution column of the ultracentrifuge cell. Mathematically this can be expressed as... 

This then is experiments. our basic, working equation in sedimentation equilibrium To obtain the weight-average molecular weight, Mw, it is ... 

Estimation of the Ideal Values for d In c/d(r2) and dc/d . For the nonideal case we can use Equations 1-4 and 6 to obtain the basic sedimentation equilibrium equation for component i. In the Fujita notation (17) this equation is... 

There are three points to emphasize. First, the expressions for the concentration or concentration gradient distribution for non-sector-shaped centerpieces can be applied to other methods for obtaining MWD s, such as the Fourier convolution theorem method (JO, 15, 16), or to more recent methods developed by Gehatia and Wiff (38-40). The second point is that the method for the nonideal correction is general. Since these corrections are applied to the basic sedimentation equilibrium equation, the treatment is universal. The corrected sedimentation equilibrium equation (see Equation 78 or 83) forms the basis for any treatment of MWD s. Third, the Laplace transform method described here and elsewhere (11, 12) is not restricted to the three examples presented here. For those cases where the plots of F(n, u) vs. u will not fit the three cases described in Table I, it should still be possible to obtain an analytical expression for F(n, u) which is different from those in Table I. This expression for F (n, u) could then be used to obtain an equation in s using procedures described in the text (see Equations 39 and 44). Equation 39 would then be used to obtain the desired Laplace transform. 

Analysis of Mixed Associations from Conventional Sedimentation Equilibrium Experiments. In these experiments one measures a quantity Mieq (14, 28) instead of Mweq. The basic sedimentation equilibrium equation for each reactant is... 

Analysis by the Archibald Method or by Sedimentation Equilibrium Experiments at Different Speeds. Instead of using Mweq here, one uses Mwa, the apparent weight-average molecular weight. For the Archibald experiment one obtains Mwa,t at rm or r6 by the application of Equations 13-16. The extrapolation of Mwa>t to zero time gives Mwo. For sedimentation equilibrium experiments at different speeds, one can evaluate Mwa by two different methods here one uses either Equations 17 or 18. For a mixed association such as A + B AB, the basic sedimentation equilibrium equation can be written as... 

Most authors who have studied the consohdation process of soflds in compression use the basic model of a porous medium having point contacts which yield a general equation of the mass-and-momentum balances. This must be supplemented by a model describing filtration and deformation properties. Probably the best model to date (ca 1996) uses two parameters to define characteristic behavior of suspensions (9). This model can be potentially appHed to sedimentation, thickening, cake filtration, and expression. 

Increasing the radius of the suspended particles, Brownian motion becomes less important and sedimentation becomes more dominant. These larger particles therefore settle gradually under gravitational forces. The basic equation describing the sedimentation of spherical, monodisperse particles in a suspension is Stokes law. It states that the velocity of sedimentation, v, can be calculated as follows ... 

Stokes law is rigorously applicable only for the ideal situation in which uniform and perfectly spherical particles in a very dilute suspension settle without turbulence, interparticle collisions, and without che-mical/physical attraction or affinity for the dispersion medium [79]. Obviously, the equation does not apply precisely to common pharmaceutical suspensions in which the above-mentioned assumptions are most often not completely fulfilled. However, the basic concept of the equation does provide a valid indication of the many important factors controlling the rate of particle sedimentation and, therefore, a guideline for possible adjustments that can be made to a suspension formulation. 

The model is composed by different equations which in all cases can be used in unsubscribed format in a basic language program. An important point to highlight is that Qwasi takes into account both steady and unsteady state solutions for the equations for systems involving contamination of lakes (or rivers). The equations considered by Qwasi involve more than 15 physicochemical processes (such as partitioning, sediment transport, deposition, etc.) to estimate the fate of the studied system. These processes and the main involved variables and parameters are summarized in Fig. 2. 

Preferred fluid migration pathways are influenced by porosity and permeability, sedimentary sequences, facies architecture, and fractures. Porosity is a measure of pore space per unit volume of rock or sediment and can be divided into two types absolute porosity and effective porosity. Absolute porosity (n) is the total void space per unit volume and is defined as the percentage of the bulk volume that is not solid material. The equation for basic porosity is listed below ... 

Although this introduction outlines the basic principles of centrifugation, it does not take into account other factors that influence the rate of particle sedimentation. Centrifuged particles migrate at a rate that depends on the mass, shape, and density of the particle and the density of the medium. The centrifugal force felt by the particle is defined by Equation 7.1. The term m is the effective mass of the particle, that is, the actual mass,... 

In this chapter, we introduced the reader to some basic principles of solution chemistry with emphasis on the C02-carbonate acid system. An array of equations necessary for making calculations in this system was developed, which emphasized the relationships between concentrations and activity and the bridging concept of activity coefficients. Because most carbonate sediments and rocks are initially deposited in the marine environment and are bathed by seawater or modified seawater solutions for some or much of their history, the carbonic acid system in seawater was discussed in more detail. An example calculation for seawater saturation state was provided to illustrate how such calculations are made, and to prepare the reader, in particular, for material in Chapter 4. We now investigate the relationships between solutions and sedimentary carbonate minerals in Chapters 2 and 3. 

The use of stable isotopic tracers is based on following a labeled compound from one pool (the source pool) to another (the target pool) (Sheppard, 1962). The principles described below are the same regardless of whether one is measuring rates in the water column or sediment. What changes are the protocols used to isolate the different fractions prior to isotopic analysis. The basic equation to calculate the flux rate from one pool to another is ... 

Before beginning a size determination, it is customary to look at the material, preferably under a microscope. This examination reveals the approx size range and distribution of the particles, and especially the shapes of the particles and the degree of aggregation. If microscopic examinatiori reveals that the ratios between max and min diameters of individual particles do not exceed 4, and indirect technique for particle size distribution based on sedimentation or elutria-tion may be used. Sedimentation techniques for particle size determination were first used by Hall (Ref 2) in 1904. He showed that the rate of fall of individual particles in a fluid was directly related to the particle size by the hydrodynamic law derived by Stokes from Newton s law of fluids in 1849 (Ref 1). This basic equation of the motion of a particle suspended in a fluid assumes that when subjected to constant driving force the particle acceleration is opposed by the... 

The diffusion coefficient, D, is involved in many calculations because it must be overcome in order to completely sediment small particles and is necessary to determine molecular weights. The ultracentrifuge method is to form a sharp boundary in the cell using a high speed, then to lower the speed and allow the boundary to spread, but not sediment it. The most accurate method is to use the interference fringes as they change over time. The basic equation is 39-13, and the correction equation is 39-14. 

Additional practical methods of aerosol removal include filtration and centrifugation. Centrifugation, of course, is basically the same as sedimentation except that the force of gravity is replaced by artificial forces of greater strength. Equation (13.6) continues to apply in that case, although the value of g must be multiphed by the appropriate factor. 

In centrifugal sedimentation the basic equation for the conversion of settling velocity to particle size is analogous to equation 2.44 ... 

Recall equation (10) from our previous discussion of basic principles of sedimentation. 

This chapter presents our recent efforts to develop and calibrate a sand transport model that is suited for practical applications but contains the basic mechanics of sand suspension and bedload movement on beaches. The hydrodynamic input required for the sand transport model is limited to the variables of irregular waves and currents which can be predicted efficiently and fairly accurately using a combined wave and current model based on time-averaged continuity, momentum, and energy equations. More advanced but computationally-demanding wave and current models may not improve the accuracy of the sand transport model with errors of a factor of about 2. Moreover, practical coastal sediment problems require the prediction of sediment transport rates for a duration of days to years. The computational efficiency is hence essential for practical applications. 

The two equations governing planform evolution are the conservation equation (also called the continuity equation ) and the sediment transport equation. The conservation equation simply performs a book-keeping role to ensure that no sediment is lost or gained. This basic equation is written as... 

Concentrations of minor elements in lake water are usually determined by adsorption onto settling particles, desorption from settling particles and interaction of sediments and bottom water. Thus, one dimensional vertical model assuming laterally homogeneous concentration is useful for the analysis of distribution of minor element concentration. Imboden and Schwarzenbach (1985) calculated tetrachloroethylene concentration in Zurich lake, Switzerland based on this model. Basic equations used by them are mass balance equation concerning solutes and particles, mass balance equation concerning minor elements on particles in sedimentary column, and diffusion equation at the boundary between sediment surface and bottom water. Results of calculation based on these equations and vertical concentration profile are shown in Fig. 6.19. 

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