Dispersion equation derivation

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

Equations that quantitatively describe peak dispersion are derived from the rate theory. The equations relate the variance per unit length of the solute concentration... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The equation derived by Troelstra and Kruyt is only valid for coagulating dispersions of colloids smaller than a certain maximum diameter given by the Rayleigh condition, d 0.10 A0. Equation 4 applies in cases where particles are transported solely by Brownian motion. Furthermore, the kinetic model (Equations 2 and 3) has been derived under the assumption that the collision efficiency factor does not change with time. In the case of some partially destabilized dispersions one observes a decrease in the collision efficiency factor with time which presumably results from the increase of a certain energy barrier as the size of the agglomerates becomes larger. 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

Again it will be assumed here that the continuous coalescing and redispersion of the drops does not influence the mass transfer into the drops. The effect of segregation will be calculated for a CSTR while the reaction takes place in the dispersed phase. For this derivation, use will be made of the equations derived in Section II,B,1. All drops are assumed to have the same diameter. 

In summary, the argument for writing the equations describing the reactor without accounting for axial dispersion is that this effect usually has very little importance, while the extra effort required to account for it is large. The equations derived in this section will be based on the assumptions that the axial dispersion is negligible, and that the conditions within the bed are sufficiently smooth functions of position to be related by differential equations. These assumptions involve the reservation that the bed is not extremely short. 

As Ic = 2ji / A this is the first example of a dispersion equation, giving the wave length as a function of frequency. It is historically interesting that [3.6.63] was already derived by Lord Kelvin very long ago. Hence it is called the Kelvin equation (for damping). 

From the dispersion equations Lucassen and van den Tempel l derived the following relation between K° and the (distance) damping coefficient p for longitudinal waves... 

The above simple considerations explain the origin of the middle phase, the change In structure associated with Its occurrence, as well as the fluctuations of the Interface between the continuous and dispersed medium which arise in some single phase microenulsions. While It Is difficult to obtain detailed quantitative information on the above behaviour, the thermodynamic equations derived in the following section provide a framework for further theoretical development as well as some additional Insight concerning the micropressures and various physical quantities involved. 

Thermoporometry. This method is based on the observation that the equilibrium conditions of solid, liquid and gaseous phases of a highly dispersed pure substance are determined by the curvature of the interface (s) (10,17). In the case of a liquid (in this work, pure water) contained in a porous material (the membrane), the solid-liquid interface curvature depends closely on the size of the pores. The solidification temperature therefore is different in each pore of the material. The solidification thermogram can be translated into a pore size distribution of the membrane with the help of the equations derived by Brun (17). For cylindrical pores, with water inside the pores, it leads to the following equations ... 

In this section we derive the algebraic-slip mixture model equations for cold flow studies starting out from the multi-fluid model equations derived applying the time- after volume averaging operator without mass-weighting [204, 205]. The momentum equations for the dispersed phases are determined in terms... 

Also calculate the conversion from the equation derived in part (c), using the above data, and DJuL — 0.05. This value of DJuL corresponds to an intermediate amount of dispersion. 

The equations derived in this section apply to either external or internal diffusion, or to both together, but probably not to longitudinal dispersion or diffusion as the controlling factor (as may occur particularly in gas chromatography). However, the equations of the next section, for trace chromatograms, should apply equally well regardless of which mechanism controls the shape to the curve. 

C. Maldarelli and R. K. Jain, The linear, hydrodynamic stability, of an interfacially perturbed, transversely isotropic, thin, planar viscoelastic film. I. General formulation and a derivation of the dispersion equation, J. Colloid Interface Sci. 90, 233-62 (1982). 

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