Fixed multiplicity model

When only a single value is allowed, i.e ko = k =k, we call the model the fixed multiplicity model. Thus, for k = 2, the fixed multiplicity model reduces to the pairwise association. The normalization relation (7.98) for the fixed multiplicity model of monodisperse polymers (/ and n definite) is given by... 

To study the relative positions of CPMC and SMG concentration, we consider a fixed multiplicity model in which the multiplicity is fixed at a single value ko = km = k. The function m(x) in this fixed multiplicify model lakes the form m(x) = 1 +x , and leads to a set of equations... 

To calculate the gel point concentration, we introduce the reduced polymer concentration c = 2X(r)0/n, the total number density of the associative groups. From (10.25) for z = z, the concentration c at the gel point for the fixed multiplicity model takes the form... 

Fixed multiplicity model. This model allows a single value k = s > 2). We have... 

Specifically for the fixed multiplicity model of monodisperse primary chains discussed above, the gelation condition is given by a concentration... 

Fig. 4 The (scaled) gelation concentration (solid lines) of telechelic polymers and cmc of surfactants (dotted lines) plotted against the (scaled) concentration of surfactants for fixed multiplicity model. The multiplicity is varied frpm curve to curve. For s larger than 4, the gel concentration shows a minimum...

Allowing for the existence of differences between studies included in the metaanalysis is an intuitively sensible feature of the random-effects model, since in most circumstances when comparing multiple studies conducted by multiple independent research teams, it would be very surprising if the studies incorporated did not vary from each other. While the precise methods of quantitatively estimating the differences between studies (i.e., the additional component of the determination of the weight assigned to each study s treatment effect point estimate in addition to its precision) need not be discussed here, it is important to be aware of the consequences of employing a fixed-effects model when a random-effects model is the... 

It is the model library for fixed-bed catalytic reaction, fluidised-bed and various polymer reactors and so on. Different tools are available in gPROMS software for simulation and modelling of various systems. Some of the following are (i) multi-scale modelling of complex processes and phenomena, (ii) State-of-the-art model validation tools allow estimation of multiple model parameters from steady-state and dynamic experimental data, and provide rigorous model-based data analysis, (iii) The maximum amount of parameter information from the minimum number of experiments, (iv) The gPROMS-CFD Hybrid Multitubular interface provides ultimate accuracy in the modelling of... 

Steady-state mathematical models of single- and multiple-effect evaporators involving material and energy balances can be found in McCabe et al. (1993), Yannio-tis and Pilavachi (1996), and Esplugas and Mata (1983). The classical simplified optimization problem for evaporators (Schweyer, 1955) is to determine the most suitable number of effects given (1) an analytical expression for the fixed costs in terms of the number of effects n, and (2) the steam (variable) costs also in terms of n. Analytic differentiation yields an analytical solution for the optimal n, as shown here. 

There are two common methods for obtaining estimates of the fixed effects (the mean) and the variability the two-stage approach and the nonlinear, mixed-effects modeling approach. The two-stage approach involves multiple measurements on each subject. The nonlinear, mixed-effects model can be used in situations where extensive measurements cannot or will not be made on all or any of the subjects. 

The models used can be either fixed or adaptive and parametric or non-parametric models. These methods have different performances depending on the kind of fault to be treated i.e., additive or multiplicative faults). Analytical model-based approaches require knowledge to be expressed in terms of input-output models or first principles quantitative models based on mass and energy balance equations. These methodologies give a consistent base to perform fault detection and isolation. The cost of these advantages relies on the modeling and computational efforts and on the restriction that one places on the class of acceptable models. 

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