Fractionation models dimensionless parameter

Haynes apd Leung (1983) formulated a similar configurational diffusion model combining the effects of active site poisoning as well as pore plugging on the HDM reaction. In this case the reaction form in the conservation equation is multiplied by a deactivation function which accounts for the loss of intrinsic activity, (1 - ) is frequently chosen, where x is the fractional coverage of the sites. Other forms of the site deactivation function have been discussed by Froment and Bischoff (1979). The deactivation was found to depend on a dimensionless parameter given by... 

The former is valid for Gaussian chains (Chapter 2) and the latter for straight rods. Hence the wormlike chain takes on a variety of conformations intermediate between Gaussian coils and rods depending on the value of a dimensionless parameter L/q. It is due to this property that the wormlike chain is used to model polymer molecules with stiffness. However, what can be obtained with wormlike chains is only a fraction of the infinitely numerous conformations realized by actual polymer molecules. 

Figure 12.10 DLVO term for steric repulsion. The first term (the one squared in parenthesis) is simply the volume fraction of the polymer in the layer. Different expressions are obtained for the three different geometries Indicated here as the overlap volume (vq) dp = sphere diameter (m), H = distance between two objects (m), X = thermodynamic interaction parameter (estimation requires thermodynamic model) (dimensionless), M2 = molecular weight of adsorbed polymer (kg mol ), p = molar volume (m moC ), C2 = surface layer concentration of adsorbed polymer (mol m ), 8 = adsorbed layer thickness (m)...

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

In the structure with all the surfactant molecules located at monolayers, the volume fraction of surfactant should be proportional to the average surface area times the width of the monolayer divided by the volume, i.e., Ps (X Sa/V. The proportionality constant is called the surfactant parameter [34]. This is true for a single surface with no intersections. In our mesoscopic description the volume is measured in units of the volume occupied by the surfactant molecule, and the area is measured in units of the area occupied by the amphiphile. In other words, in our model the area of the monolayer is the dimensionless quantity equal to the number of amphiphiles residing on the monolayer. Hence, it should be identified with the area rescaled by the surfactant parameter of the corresponding structure. 

Zhou et al. [55], The most effective method to assess the capacity is the flow simulation which includes volumetric formulas and more reservoir parameters rather than other methods [56], Mass balance and constitutive relations are accounted in mathematical models to capacity assessment and dimensional analysis consists of fractional flow formulation with dimensionless assessment and analytical approaches [33], From the formulations demonstrated by Okwen and Stewart for analytical investigation, it can be deduced that the C02 buoyancy and injection rate have affected the storage capacity [57], Zheng et al. have indicated the equations employed in Japanese and Chinese methodology and have noted that some parameters in Japanese relation can be compared to the CSLF and DOE techniques [58]. 

A mathematical model has been proposed to account for the mutual synergistic action of either particle component on the other in increasing the value of the dimensionless time 0 as shown in Fig. 53, in terms of the mass fraction x2 of fines, and two empirical parameters n, and n2 ... 

The fraction of cells in solution, 7, as calculated using the present model, is presented in Fig. 2 as a function of the dimensionless time 8, a and fi being parameters. The curve for a = 0 is the exponential decay predicted by the previous model (and by the present model when z1 = t2), and is independent of the value of / . 

The equation addresses the molar volume of a segment of the polymer v, the subscript m indicating a segmental molar quantity. The parameters of the equation are segmental parameters thus is the excluded volume of a segment q = bJ(Av is the density of the fluid expressed in a reduced dimensionless form, also called the packing fraction and is the attractive-pressure parameter of the segment. The eccentricity parameter of the rotators is e, equal to 1.078 for the model rotator. Results of data reduction indicate the attractive parameter to vary weakly and linearly with temperature. 

Monte Carlo simulation data and the two-timescale model, eq. (6) in the time domain, for which a closed-form expression of the fractional coverage may be obtained (8]. In this figure, r is the dimensionless time tD/L, where L = 1 l.u. It can be shown from dimensional ancdysis and from simulation results that the two parameters entering into... 

Introducing the dimensionless variables and the model parameter, we calculate starting from [10.21] the fraction of grains having nucleated at time ... 

In these cases, the models (see Appendices A.4 and A.6) reveal the parameter defined by relation [10.31]. But we cannot directly compare an experimental curve of the rate versus time with those resulting from the models because these are built in dimensionless rate and time. To reach this point, we calculate, for different value of the fractional extent, a, a theoretical reduced rate defined by ... 

These tables give, for various values of the model parameter, the remarkable properties of the kinetic curves fractional extent and dimensionless time at the point of inflection (if there exists) and dimensionless time at a fractional extent of 0.5 in two-process models with anisotropic growths. 

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