Analyte, solution combinations

In some specific cases, analytical solutions for the population balances can also be derived. For instance, Soares and Hamielec [73, 74] obtained analytical dynamic solutions to describe how the CLD of polyolefins varied as a function of time in stopped-flow reactors commonly used for mechanistic studies on olefin polymerization kinetics and mechanism [75-78]. These analytical solutions combine the power of full population balance numerical solutions with the ease and convenience of using closed form equations they are, unfortunately, difficult to attain for more complex cases. 

An analytical solution to this has already been attempted [25]. According to this model, the minimum concentration of fines would be that quantity required to coat each coarse particle with a monolayer of fines. Treating the particles as perfect spheres, the fractional change in combined particle volume due to additional film of fines is then ... 

Chakraborty S (2006) Analytical solutions of Nusselt number for thermally fully developed flow in microtubes under a combined action of electroosmotic forces and imposed gradients. Int J Heat Mass Transfer 49 810-813... 

Analytical Solution A stoichiometric relationship can be used to eliminate A>b-Combine the two ODEs to obtain... 

MEKC is a CE mode based on the partitioning of compounds between an aqueous and a micellar phase. This analytical technique combines CE as well as LC features and enables the separation of neutral compounds. The buffer solution consists of an aqueous solution containing micelles as a pseudo-stationary phase. The composition and nature of the pseudo-stationary phase can be adjusted but sodium dodecyl sulfate (SDS) remains the most widely used surfactant. 

Carfentrazone-ethyl, C-Cl-PAc, C-PAc, DM-C-Cl-PAc and HM-C-Cl-PAc stock solutions of 1000 xg mL were prepared by dissolving the appropriate amounts of the analytical standards in acetonitrile. Working solutions were prepared in volumetric flasks by appropriate dilutions of the stock solutions for each analyte or combination of analytes. Working solutions containing the parent were prepared only in acetonitrile and working solutions containing acid metabolites were prepared in acetonitrile (underivatized) or hexane (derivatized). Underivatized solutions (containing the parent and/or metabolites in acetonitrile) were used for fortification. Solutions of derivatized esters were prepared simultaneously with the samples. Standard solutions... 

Crosslinking of many polymers occurs through a complex combination of consecutive and parallel reactions. For those cases in which the chemistry is well understood it is possible to define the general reaction scheme and thus derive the appropriate differential equations describing the cure kinetics. Analytical solutions have been found for some of these systems of differential equations permitting accurate experimental determination of the individual rate constants. 

The goal of approximate and numerical methods is to provide convenient techniques for obtaining useful information from mathematical formulations of physical problems. Often this mathematical statement is not solvable by analytical means. Or perhaps analytic solutions are available but in a form that is inconvenient for direct interpretation. In the first case it is necessary either to attempt to approximate the problem satisfactorily by one which will be amenable to analysis, to obtain an approximate solution to the original problem by numerical means, or to use the two techniques in combination. 

In tests using the moving ID Hamiltonian harmonic oscillator, (5.25), a velocity Verlet integrator [24] combined with ttapezoidal integration of W (/.) performed well when compared to the analytic solution. An interesting analysis of how... 

Rate equations, such as equation 17.85, make no attempt to distinguish mechanisms of transfer within a pellet. All such mechanisms are taken into account within the rate constant k. A more fundamental approach is to select the important factors and combine them to form a rate equation, with no regard to the mathematical complexity of the equation. In most cases this approach will lead to the necessity for numerical solutions although for some limiting conditions, useful analytical solutions are possible, particularly that presented by Rosen(41). ... 

One choice of basis function, based on a quadrilateral patch, is illustrated in Figure 15.2c. In the figure the element in the fth row andyth column of the mesh is assumed to have a magnitude that varies within the patch the derivative properties may be important as well. The choice of fifix, y) is not arbitrary it is made to reflect certain mathematical qualities derived, perhaps, from prior knowledge of the general behavior of similar systems as well as properties that simplify the solution process to follow. One immediately practical constraint is that the fifix, y) must satisfy the boundary conditions. Another property is that the patches meet smoothly at the intersections this is usually obtained by continuity of fifix, y) to first and second order in the derivatives. It is also convenient in many applications to choose combinations of products of functions separately dependent on x and y, reminiscent of the analytic solution, Eq. (15.2). 

Central differences are applied to diffusion problems, and upwind differences are applied to convective problems, but most cases have both diffusion and convection. This conundrum led Spaulding (1972) to develop exponential differences, which combines both central and upwind differences in an analytical solution of steady, one-dimensional convection and diffusion. Consider a control volume of length Ax, in a flow fleld of velocity U, and transporting a compound, C, at steady state with a diffusion coefficient, D. Then, the governing equation inside of the control volume is a simphflcation of Equation (2.14) ... 

Abstract To design an adsorption cartridge, it is necessary to be able to predict the service life as a function of several parameters. This prediction needs a model of the breakthrough curve of the toxic from the activated carbon bed. The most popular equation is the Wheeler-Jonas equation. We study the properties of this equation and show that it satisfies the constant pattern behaviour of travelling adsorption fronts. We compare this equation with other models of chemical engineering, mainly the linear driving force (LDF) approximation. It is shown that the different models lead to a different service life. And thus it is very important to choose the proper model. The LDF model has more physical significance and is recommended in combination with Dubinin-Radushkevitch (DR) isotherm even if no analytical solution exists. A numerical solution of the system equation must be used. 

In Figure 4.27, some examples of theoretical breakthrough curves calculated from the analytical solutions for the Freundlich isotherm (Fr = 0.5) are presented. As is clear, the curve corresponds to the case of equal and combined solid and liquid-film diffusion resistances ([ = 1) which is between the two extremes, i.e. solid diffusion control (l = 10,000) and liquid-film diffusion control ( = 0.0001). 

Poiseuille flow also occurs in a simple shearing situation, but it presumes that there is a pressure gradient that drives the flow and that the solid boundaries are fixed. Flow in a pipe or tube is an example of Poiseuille flow. It is a straightforward matter to combine these Couette and Poiseuille effects, and still find an exact analytic solution. 

Algorithmic and computational solutions for model (or design) equations, combined with chemical/biological modeling, are the main subjects of this book. We shall learn that the complexities for generally nonlinear chemical/biological systems force us to use mainly numerical techniques, rather than being able to find analytical solutions. 

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