Single porosity

Discussing pore size selection of columns leads directly to the issue of using single porosity columns or so-called linear or mixed-bed columns, which contain mixtures of different pore sizes in a single column (3,19). Both types of columns have advantages and disadvantages, as shown in Table 9.6. 

TABLE 9.6 Comparison of Single Porosity Type and Linear Mixed-Bed SEC Columns... 

Single porosity Efficient Optimized Flexible Low cost Good for QC Viscous fingering... 

Since a permeability coefficient of 2 x 10 cm/sec was used in our theoretical calculation for case B, this is equivalent to a single porosity value for all calculations with Equation 6. For any given set of ethanol concentration at the boundary, the appropriate ethanol concentration distance profile was calculated from the experimentally determined profile for the pure ethanol/water system and taken as that of the system. For any given set of ethanol concentrations at the boundaries the appropriate portion of the experimentally determined ethanol concentration distance profile, was calculated from the experimentally determined profile for the pure ethanol/water system. For any given ethanol concentration distance profile, the corresponding solubility profile of 3-estradiol was obtained by interpolation of experimental solubility data (Figure 2). 

Disequilibrium single porosity models. In addition to complete equilibrium transport, several other variations to the basic model have been proposed. The first relaxes the assumption that moving melt remains in chemical equilibrium with the solid at all times (Spiegelman and Elliott, 1993), although instantaneous melts are assumed to be in chemical equilibrium with the mantle that produced them. For stable elements, this disequilibrium transport produces a residue that reflects perfect fractional melting and the melts have compositions identical to accumulated fractional melts. These models are similar to the dynamic melting models in the limit that but... 

It is also straightforward to extend the equations to allow for only partial equihbration during transport. Iwamori (1993a) presents a one-dimensional steady-state single-porosity model for stable elements that includes diffusive re-equili-bration between melt and solid. He does not extend it to radioactive nuclides in this paper but includes this effect in his two porosity model (Iwamori, 1994) (see Section 3.14.4.3.4). The expected effects of chemical disequilibrium should be similar to those in the Qin (1992) dynamic melting model, namely he effective bulk partition coefficients of all elements will be driven towards unity. 

The melt velocity estimated from transport models is a bit slower but still comparable to estimates from the dynamic melting models. For example, if we assume that Ra-excesses are produced at the bottom of a column 90 km deep and need to move to the surface in —3 half-lives, then wq —20myr. It should be stressed that this is a constraint on the average melt velocity across the entire melting column rather than a constraint on the maximum velocity near the surface. Moreover, the constraint from Equation (9) assumes that there is only a single porosity near the surface. Two-porosity models (next section) relax this constraint somewhat. 

Another problem with the transport models is that they assume complete equilibrium between melt and solid throughout the melting regime. This implies that the solid residues at the top of the column (e.g., near the Moho), should be in chemical equilibrium with MORE. However, another key observation of MORBs is that they are out of equilibrium with abyssal peridotities near the Moho for both major and trace elements. A quick fix for the transport models is to assume that melts remain in chemical equilibrium up to some depth and then melt fractionally for the remaining distance (e.g., see Kelemen et al., 1997). However, different attempts to explain both U-series and stable elements in melts and residues in the single-porosity transport models has motivated much of the development of the two-porosity models in next section. 

Both models assume a single porosity at any height in the melting region and produce well mixed melts at the top. Thus, these models cannot fractionate elemental uranium from thorium and cannot produce... 

Terzaghi for saturated elastic medium with a single porosity. The Duhamel-Neumann extension to Hooke s law gives the thermoelastic constitutive relationship as,... 

CONFIDENCE LIMITS OF PREDICTED PERFORMANCE DATA USING SINGLE-POROSITY ESTIMATES AT 0.20% MEASUREMENT ERROR... 

Figure 3 shows the confidence limits of the predicted bottom-hole flowing pressures using single-porosity estimate at 0.20 percent measurement error. The confidence interval is about 186 psi which is practically acceptable. The true pressures are all contained within the confidence interval also. As can be seen in Figure 2, the confidence regions for joint estimation of porosity and permeability at 0.20 percent measurement error indicate that even at the lowest confidence level of 95 percent, the confidence interval for porosity is very wide. The orientation and shape of the ellipses show that porosity is much less well determined than permeability. It seems, therefore, that porosity estimation is very sensitive to measurement error. Also, porosity estimates are not reliable when joint estimation of porosity and other parameter(s) is made or when there is a significant error in the matched performance data. 

A more familiar looking version of the film transport equation is obtained for a single-porosity model. This result can be formally obtained by setting p = 0 and e = Eg in Eq. fl8-4RbT... 

Note that Eq. (18z56a) is very similar to Eq. (18=48b) except that c replaces the concentration of the pore fluid Cp j-e and the lunped parameter mass transfer coefficient l, replaces the film coefficient kf. As expected, the lunped parameter expressions using the single-porosity model are sirtpler. 

In a classic paper Lapidus and Amundson (1952) studied liquid chromatography for isothermal operation with linear, independent isotherms when mass transfer is very rapid, but axial dispersion is inportant. Although the two-porosity model can be used (Wankat, 1990), the solution was originally obtained for the single-porosity model. Starting with Eq. [18-551. we substitute in the equilibrium expression Eq. [18-6al to remove the variable q (solid and fluid are assumed to be in local equilibrium). Since the fluid density is essentially constant in liquid systems, the interstitial fluid velocity Vj ter can be assumed to be constant. The resulting equation for each solute is... 

C5. Derive the equation for the solute velocity for linear isotherms for systems using a single-porosity... 

Table 5. Comparison of single porosity-type and linear mixed-bed CPC columns.

A very simple approach can be applied to increase the resolution and/or the separation range. Instead of just using one column, multiple columns are combined to a column combination or a column bank. Two to four columns (plus a pre- or guard column) are typical in SEC. A column combination or column bank provides more available pore volume for more efficient separations. If two columns with the same pore sizes (single porosity or linear/mixed bed/ multipore) are combined, the calibration curve becomes flatter and the resolution increases by a factor of 1.4 whereas the... 

Disadvantages of column banks are that price, pressure, analysis time, and eluent consumption inaease. An inCTeased pressure might result in the need to reduce the flow rate and/ or to increase the temperature to have better chromatographic conditions, especially for high molar mass macromolecules. In addition, there is the potential danger of porosity mismatch for all column types, linear/mixed bed or single porosity alike. 

The end modeling results are coupled immiscible flow equation systems, containing twice as many input parameters as the more rational single-porosity model would have two sets of relative permeability curves, two sets of capillaiy pressure curves, and so on. Consequently, such hopelessly ill-defined approaches, given the dearth of real-world data, not to mention errors likely to be found in laboratory measurement, may never see complete validation. Simpler flow models for periodic shales and fractures, such as those introduced in Chapter 5, shed greater physical insight. 

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