Four parameter model for

FIGURE 11.11 Creep response for (a) Maxwell model, (b) Voight-Kelvin model, and (c) four-parameter model for constant stress applied at r, and removed at if... 

Pitzer s equation - a three to four parameter model for mean molal activity coefficients... 

By substituting Equations (6.16) and (6.17) back into Equations (6.14) and (6.15), the four-parameter model for describing the temperature dependence of the LC refractive indices is derived, as [24]... 

Another four-parameter model for steady-state viscous flow is... 

FIGURE 15.8 Creep response of a four-parameter model for an input shear stress. Note that the model also works for tensile stress, [Pg.285]

Table 2. Four parameter model for the hydrophobic hydrogels...

Four Parameter Models. Two- and three-parameter theories are only accurate for simple, normal, and some slightly polar fluids. In order to accurately predict polar fluid behavior a fourth parameter is needed (80). The Stiel polarity factor, is one such fourth parameter and follows from the... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

Most screening designs are based on saturated fractional factorial designs. The firactional factorial designs in Section 14.8 are said to be saturated by the first-order factor effects (parameters) in the four-parameter model (Equation 14.27). In other words, the efficiency E = p/f = 4/4 = 1.0. It would be nice if there were 100% efficient fractional factorial designs for any number of factors, but the algebra doesn t work out that way. 

Use matrix least squares (regression an ysis) to fit the linear model y, = bl + hjx, + b]pc2i + byX + rii to the data. How are the least squares parameter estimates in the eight-parameter model related to the parameter estimates in this four-parameter model Why Draw the sums of squares and degrees of freedom tree. How many degrees of freedom are there for 55 55, f, and 55p Why ... 

Fig. 11.5 Ca titration of porcine calmodulin and fractional species calculation [25]. (A) Ca-titration for 15 pM of porcine calmodulin in 50 mM HEPES (pH 7.4, T = 21.5 °C, 90% D2O). Error bars were based on the deviation from two sets of Q-TOF data. The solid curve was the best fit for the average data using the four-parameter model. (B) Fractional species as a function of [Ca " ] for CaM interacting with four Ca ". ...

Fig. n.6 Ca titration of 15 aM porcine CaM in three different media (99% D2O) [25]. Line(a) 50 mM HEPES/0.1 M KCI, apparent pH 7.4. Line (b) 50 mM HEPES, apparent pH 7.4. Line (c) 2 mM NH4OAC, apparent pH 7.0. Error bars were based on two sets of LCQ titration data. Solid curves were taken from the four-parameter model and are the best fit for the average data. 

One may question the need for a four parameter enthalpy equation, i.e., whether describing an acid or base by two parameters each is redundant. The following simple matrix algebra shows the conditions whereby a four parameter model reverts to a less redundant two parameter equation. Letting A be the transformation matrix, E and C represent the parameters for the four parameter model, and a represent the acid parameters for the two parameter model, the following equation results ... 

In this application the AOM is a four-parameter model, eCTx, eaY, enx and enY being AOM parameters in each case. It is to be noted that the CFT/LFT parameterization of these complexes would also most usually be at the same level. Indeed, recalling equation (10), it is easy to write the correspondence between the two parameter sets, say for MX5Y... 

Summarizing The basic idea, mentioned in chapter 6, that creep of solid polymers could be represented by a simple four-parameter model (the Burgers model), composed of a Maxwell and a Kelvin-Voigt model in series, appears to be inadequate for three reasons ... 

In all of the above there is assumed no product inhibition. Nonlinear regression of the kinetic data revealed the four parameter model, Eq. 1, to be marginally better than the two parameter model of Eq. 2 with both models the maximum error was less than 20%, as shown in Figures 1 and 2. Eq. 2 is quite adequate for the range of variables studied and affords a more direct comparison of deactivation effects, hence it is used in the following. Fresh catalyst parameters are given below. 

Two quantities—Pi, the concentration of the isotactic triad and ai, the probability that an isotactic triad be followed by another isotactic triad—are sufficient to describe the sequence length of the isotactic units. The concentration of the various triad states can be expressed in terms of the various transition probabilities of the four-parameter model discussed previously (1). It can be shown that in the expressions for the probabilities of the various triads, the four transition probability parameters occur in such a combination that the concentration of the various triad states like Pi, etc. depend on only two independent parameters (Appendix I). 

Figure 3. Experimental and theoretical plots of the crystalline mass fraction vs. temperature for a stereoregular polypropylene oxide fraction X—four-parameter model [H-four-parameter model with "car parking correction.

E3 = — + and E4 = - 7 where the + and - define the relative configuration of two adjacent monomer units in the polymer chain. In accordance with the nomenclature of Bovey (7), we call the sequence (+ +) isotactic, (+ -), or (- +) heterotactic, and (- -) syndiotactic. The transition probabilities between various states were defined for the four-parameter model (1). 

The critical chain length for the four-parameter model can be found in a similar manner to that used above. In this case, as discussed earlier (Equation 1), the probability P+( ) that a given sequence of + s is exactly of length is given by ... 

There are four dimensionless parameters in eqns. (4.3.7 to 4.3.9) o, r,y, xji, and kjj. This version of the WS mixing rule can be used as a four-parameter model to correlate the behavior of complex mixtures or in several ways that have fewer adjustable parameters. For example, one can solve the two relations obtained from eqn. (4.3.8) in the infinite dilution limit bi... 

Theory for Nafion. To apply the general theory quantitatively it is necessary to express explicitly the relationships between rh and nh, rc and nc, and the force constant Ke. A four-parameter model (fCe,pi,p2,p3) was formulated to accomplish this goal. First, we define three packing factors pi, p2, and p3 as pi = packing factor for water molecules in a purely aqueous hydration shell p2 = packing factor for water molecules in hydration shells containing a cation, for example, ion-dipole hydration shells and p3 = packing factor of hydration shells in a cluster. 

There are five adjustable parameters per molecule X, the dispersion parameter q, the induction parameter x, the polarity parameter a, the hydrogen-bond acidity parameter and p, the hydrogen-bond basicity parameter. The induction parameter q often is set to a value of 1.0, yielding a four-parameter model. The terms fj and are asymmetry factors calculated from the other parameters. A database of parameter values for 150 compounds, determined by regression of phase equilibrium data, is given by Lazzaroni et al. [Ind. Eng. Chem. Res., 44(11), pp. 4075-4083 (2005)]. An application of MOSCED in the study of liquid-liquid extraction is described by Escudero, Cabezas, and Coca [Chem. Eng. Comm., 173, pp. 135—146 (1999)]. Also see Frank et al., Ind. Eng. Chem. Res., 46, pp. 4621-4625 (2007). 

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

The VSM-W isotherm equation is a four parameters model, A i, K and C s). The pairwise interaction constants Aj and A j have been found to be highly correlated. To avoid this problem, Cochran et al. (1985) used the Flory-Huggin equation for the activity coefficient instead of the Wilson equation ... 

By eliminating various elements in the four-parameter model the response of a Maxwell fluid, Kelvin solid, three-parameter solid (a Kelvin and a spring in series) can be obtained and the model can be used to represent thermoplastic and/or thermoset response as illustrated in Fig. 3.13. For example the creep response of a three-parameter solid is obtained by eliminating the free damper in Eq. 3.44 and gives the creep and creep recovery response shown in Fig. 3.13 for a crosslinked polymer. 

An example of creep deflection in a tensile bar for an epoxy at different temperatures is shown in Fig 5.12. It will be noticed that the creep response for a temperature of 155° C still has a positive slope after seven hours. Without knowing the type of material, one might expect the response to be that of a viscoelastic fluid. The creep response for 165° C and 170° C clearly have reached a limit and has the character of a thermoset. Because of the nature of the response, the epoxy could be best characterized by a viscoelastic fluid model such as the four-parameter fluid for both the 155° C and 160° C data. On the other hand, the epoxy could best be characterized by a viscoelastic solid model such as the three-parameter solid for temperatures above 160° C. To characterize the material over all time and temperature ranges would require a generalized model with a large number of elements. Methods to accomplish this will be discussed in subsequent sections. 

When the risk is not monotonous with the traffic variable, a supplementary term (here parabolic) is introduced. However, due to an insufficient number of accidents, it was not possible to estimate four parameters models are then estimated on datasets excluding rain, where the rain occurrence is replaced by the square of the traffic variable in that case, the name of the model (POWER or EXPO) is indexed by NoRain and by the abbreviation Para for parabolic y is the coefficient of the square of the indicator the direction of variation of the risk depends on whether the traffic variable F, is below/above the value -p/(2.y). 

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