Model two-parameter

Since the accuracy of experimental data is frequently not high, and since experimental data are hardly ever plentiful, it is important to reduce the available data with care using a suitable statistical method and using a model for the excess Gibbs energy which contains only a minimum of binary parameters. Rarely are experimental data of sufficient quality and quantity to justify more than three binary parameters and, all too often, the data justify no more than two such parameters. When data sources (5) or (6) or (7) are used alone, it is not possible to use a three- (or more)-parameter model without making additional arbitrary assumptions. For typical engineering calculations, therefore, it is desirable to use a two-parameter model such as UNIQUAC. 

A proportionality between the theoretical spall strength and the bulk modulus is obtained when a two-parameter model is chosen to represent the intermolecular potential. Other two-parameter representations of the intermolecular potential, such as the Lennard-Jones 6-12 potential, will yield a similar proportionality although the numerical coefficients will differ slightly. 

Although two-parameter models are rather restrictive, three-parameter models of the intermolecular potential have been developed which provide reasonable descriptions of the thermodynamic behavior of solids. Examples include the Morse potential, the exponential-six potential, and, more recently, a form proposed by Rose et al. (1984) for metals. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

A useful two-parameter model is the power-law model, or Ostwald-de Waele law to identify its first proponents. The relation between shear stress and shear rare is given by ... 

Temperature (K) Experimental Pressure (MPa) Calculated Pressure with three-parameter model (MPa) Calculated Pressure with two-parameter model (MPa)... 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

As discussed in Chapter 3, the Carreau viscosity model is one of the most general and useful and reduces to many of the common two-parameter models (power law, Ellis, Sisko, Bingham, etc.) as special cases. This model can be written as... 

We obtain an r.m.s. deviation of 0.84 kcal/mol with an optimal a of 0.181. One can also note the similarity between the a value of this model and that of the two-parameter model with a free a and /3. This suggests that the model is robust in the sense that the actual polar and non-polar free energy contributions are more or less invariant, as long as deviations from linear response are taken into account in a proper way. The FEP-derived model could be considered preferable to the two-parameter model since it contains only one free parameter, viz. oc. The results of adding a constant yto the new model was also investigated. Remarkably, the optimal value for such a y was found to be -0.02 kcal/mol, i.e. virtually zero. 

Figure4.9 shows the best Freundlich quenching model plots. While more complex models could fit the data, this simple two-parameter model fits all the data within experimental error. Since the decays are highly nonexponential, the parameters represent lumped parameters that are some average over the available sites, lifetimes, and A sv s. A detailed discussion of the different models tried is given elsewhere.

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 ... 

However, there is no reason to use more complicated isotherm models if two-parameter models, such as Langmuir and Freundlich, can fit the data well. It should be clarified that these models are only mathematical functions and that they hardly represent the adsorption mechanisms. 

Diffusion-type models are two-parameter models, involving kt or Ds and La, while BDST models are one-parameter models, involving only 0, as gmax is an experimentally derived parameter. The determination of La requires the whole experimental equilibrium curve, and in case of sigmoidal or other non-Langmiur or Freundlich-type isotherms, these models are unusable. From this point of view, BDST models are more easily applied in adsorption operations, at least as a first approximation. 

Historical data were obtained on U.S. refinery operations and the production of sulfur from U.S. refineries from 1971-1977. A two-parameter model was developed correlating past levels of sulfur recovery with the quality and rate of crude processed and the qualities and amounts of end products. An overall correlation coefficient of 0.91 was obtained. 

In the cases above, a two-parameter model well represents the data. A model with more parameters would be more flexible, but by using a partition constant, K, or a desorption rate constant ka and k, , for the mass-transfer coefficients, the data are well described (see Figs. 3.4-15 and 3.4-13). While K would be a value experimentally determined, kp can be estimated from eqn. (3.4-97) with the external mass-transfer coefficient, km, estimated from the correlation of Stiiber et al. [25] or from that of Tan et al. [27], and the effective diffusivity from the Wakao Smith model [36], Typical values of kp obtained by fitting the data of Tan and Liou are shown in Fig. 3.4-16. As expected, they are below the usual mass-transfer correlations, because internal resistance diminishes the global mass transfer coefficient. These data correspond to the regeneration of spent activated carbon loaded with ethyl acetate, using high-pressure carbon dioxide, published by Tan and Liou [45]. 

To summarise, the fracture data and type of crack growth over a wide range of rates and temperatures may all be rationalised by the concept of a critical stress acting over a critical distance and this two-parameter model provides a unique failure criterion for thermosetting polymers. 

Luedeking and Piret<56) proposed that the formation of microbial products could be described by a two parameter model. This was based on their studies of lactic acid fermentation using Lactobacillus delbruekii in which the accumulation of the acid is not directly linked to the growth rate. The function used was ... 

Let us carry out a so-called screening experiment in which we attempt to discover if the fermentation temperature (factor x ) has a significant effect on the response (% alcohol content). We will choose two levels of temperature, 23°C and 27°C. This is the minimal number of factor levels required to fit the two-parameter model... 

These results are familiar from the trivial invariance of the two-parameter model (cf. Eq. (8.4)). As a consequence, the terms ne 2 taken along in the two-parameter model drop out from Eqs. (8.27), (8.28), leaving the n-independent sublcadmg terms to determine A (A) and Bi(A). A little calculation yields... 

A more convenient extrapolation technique is to approximate the experimental data with a viscosity model. The Power Law, shown in Eq. 6, is the most commonly used two-parameter model. The Bingham model, shown in Eq. 7, postulates a linear relationship between x and y but can lead to overprediction of the yield stress. Extrapolation of the nonlinear Casson model (1954), shown in Eq. 8, is straightforward from a linear plot of x°5 vs y05. Application of the Herschel-Bulkley model (1926), shown in Eq. 9, is more tedious and less certain although systematic procedures for determining the yield value and the other model parameters are available (11) ... 

Three empirical models were utilized to fit the rheologic characteristics of the wet grain slurries power law, Herschel-Bulkley, and Casson. The power law and Casson models are two-parameter models and are ideal for... 

The nucleotide sequences of gastrin precursors in Phylip format are given below. Preform phylogenetic analysis by neighbor joining method using the Kimura two-parameter model. 

Although both the BDST and two-parameter models fit the experimental data points so well, care should be taken when they are being used in fixed-bed design. Consequently, Chu [106] in contradicting an early work of Ko et al. [38], suggests that these models should not be used to predict breakthrough a priori. The modeling approach presented here is thus for the purpose of illustration only. 

Fig. 10. Effect of bed height on fluoride removal from water. Two-parameter model simulation is represented by continuous line.

Since the Margules expansions represent a convergent power series in the mole fractions,8 they can be summed selectively to yield closed-form model equations for the adsorbate species activity coefficients. A variety of two-parameter models can be constructed in this way by imposing a constraint on the empirical coefficients in addition to the Gibbs-Duhem equation. For example, a simple interpolation equation that connects the two limiting values of f (f°° at infinite dilution and f = 1.0 in the Reference State) can be derived after imposing the scaling constraint... 

Another two-parameter model of widespread application results from imposing the constraint that the activity of an adsorbate species can be a power-law expression in its exchanger charge fraction (cf. Eq. 5.10) ... 

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