Fundamental equations concentration distribution

The meanings of Gn and Ggi are the number of the ion-pairs per 100 eV of radiation absorbed which never recombine together and the number of ion-pairs which recombine in the absence of a scavenger (or electron or cation) (see Sect. 3.3 and Fig. 32, p. 185). An expression of the form of eqn. (174) can be used to estimate the lifetime distribution of ion-pairs [365] or, rather more fundamentally, the initial distribution of ion-pair distances (see Sect. 3.3). Clearly, for small scavenger concentrations, G(P) is represented equally well by the two equations above. Comparing either with the probability of scavenging [eqn. (172)], it follows that... 

Abstract The fluxes of water and solutes across membranes are expressed as functions of differences of the hydraulic and osmotic pressures at both sides. Such difference equations are deduced from more fundamental differential equations. The distributions of concentration and pressure in a series array of membranes are derived. The order in which the individual membranes are placed exerts a strong influence upon the effects of the applied differences of hydraulic and osmotic pressures. The effect of the interchange of two membranes in a series array of an arbitrary number of membranes can be summarized in four simple rules. The special case of reversal of the flow is also discussed. 

Poisson-Boltzmann Equation A fundamental equation describing the distribution of electric potential around a charged species or surface. The local variation in electric-field strength at any distance from the surface is given by the Poisson equation, and the local concentration of ions corresponding to the electric-field strength at each position in an electric double layer is given by the Boltzmann equation. The Poisson-Boltzmann equation can be combined with Debye-Hiickel theory to yield a simplified, and much used, relation between potential and distance into the diffuse double layer. 

This is the name given to the study of the kinetics of drug absorption, distribution, metabolism, and excretion, all of which are rate-controlled. The earliest studies, which were concerned with inhaled anaesthetics (Widmark, 1920 Dominguez, 1933), were not suited for general application. The fundamental equations were introduced by T. Teorell (1937) in his studies of insulin action. He provided simple kinetic formulae to monitor the concentration of... 

Fluid flow pattern may be described in two ways a fluid mechanisms approach and a global phenomenological approach based on the Residence Time Distribution (R. T. D) concept. The fluid mechanics approach tries to determine the velocity, concentration and temperature profiles within the reactor on the basis of fundamental equations of fluid flow hydrodynamics. This approach, when successful, leads to complex mass and heat balance equations requiring cumbersome numerical computations and yielding too detailed informations when a macroscopic description of the process is required by the chemical engineer. 

In fundamental SEC studies retention is often described in terms of a distribution coefficient. The theoretical distribution coefficient Kg is defined as the ratio of solute concentration inside and outside of the packing pores under size exclusion conditions. The experimental distribution coefficient as defined in Equation 1, is a measurable quantity that can be used to check the theory. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

Clearance (Cl) and volumes of distribution (VD) are fundamental concepts in pharmacokinetics. Clearance is defined as the volume of plasma or blood cleared of the drug per unit time, and has the dimensions of volume per unit time (e.g. mL-min-1 or L-h-1). An alternative, and theoretically more useful, definition is the rate of drug elimination per unit drug concentration, and equals the product of the elimination constant and the volume of the compartment. The clearance from the central compartment is thus VVklO. Since e0=l, at t=0 equation 1 reduces to C(0)=A+B+C, which is the initial concentration in VI. Hence, Vl=Dose/(A+B-i-C). The clearance between compartments in one direction must equal the clearance in the reverse direction, i.e. Vl.K12=V2-k21 and VVkl3=V3-k31. This enables us to calculate V2 and V3. 

Maintenance of unequal concentrations of ions across membranes is a fundamental property of living cells. In most cells, the concentration of K+ inside the cells is about 30 times that in the extracellular fluids, while sodium ions are present in much higher concentration outside the cells than inside. These concentration gradients are maintained by the Na+-K+-ATPase by means of the expenditure of cellular energy. Since the plasma membrane is more permeable to K+ than to other ions, a K+ diffusion potential maintains membrane potentials which are usually in the range of -30 to -90 mV. H+ ions do not behave in a manner different from that of other ions. If passively distributed across the plasma membrane, then the equilibrium intracellular H+ concentration can be calculated from the Nernst equation via... 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

An aerosol distribution can be described by the number concentrations of particles of various sizes as a function of time. Let us define Nk(t) as the number concentration (cm-3) of particles containing k monomers, where a monomer can be considered as a single molecule of the species representing the particle. Physically, the discrete distribution is appealing since it is based on the fundamental nature of the particles. However, a particle of size 1 pm contains on the order of 1010 monomers, and description of the submicrometer aerosol distribution requires a vector (N2, N-j,..., N10io) containing 1010 numbers. This makes the use of the discrete distribution impractical for most atmospheric aerosol applications. We will use it in the subsequent sections for instructional purposes and as an intermediate step toward development of the continuous general dynamic equation. 

Equations 10.14, 10.15 and 10.17 clearly suggest that the computation of the loading dose necessary to attain the desired plasma concentration of a drug instantaneously requires the knowledge of two fundamental pharmacokinetic parameters of a drug the apparent volume of distribution and/or the elimination half life. 

The fundamental approaches to definition of turbulent flows macro-kinetics and macro-mixing processes are considered in [136-139]. Special attention was focused on micro-mixing models in the context of method based on equation for density of random variables probabilities distribution. Advantage of this method is that we can calculate average rate of chemical reaction if know the corresponding density of concentration and temperature possibility distribution. 

Cij and Di represent the area-averaged concentration, the intersticial velocity and the dispersion coefficient, in channel i. It woiild be interesting to derive a similar equation for bed scale averaged variables. Unfortunately, it is impossible to derive such equation in an exact manner because diffusion and percolation processes are ruled by fundamentally different elementary mechanisms, (see e.g. Broadbent et al., 1957). Actually, the stochastic model defined by Eq.5 > describes the liquid velocity distribution and could also be used to characterize nximerically the distribution of residence times i.e., the dispersion process. Achwal et al. (1979) drew attention to a procedure using a Markov chain model which led to similar results for the velocity distribution. This model remained essentially numerical and rather cumbersome. Even if Eq.5 has a simple analytical form, its mamerical application to estimate the dispersion process is also too complex for practical purposes. 

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