Pore population distribution

Figure IB. Corresponding computed pore population distribution (probability density), n(r, t) (16). Each curve is labeled by the corresponding value of the injected charge Q. For Q = 25 and 20 nC (cases for which REB occurs), N increases to about 108 in less than 0.5 pus and then decays exponentially with a time constant of 4.5 pus. For Q — 15 nC, N increases rapidly to about 105 and remains almost constant for about 4 pus before the exponential decrease. For Q — 10 nC, N increases to about 2 X 103 in about 5 pus and remains almost constant for about 30 pus before the decay phase. The membrane in this case ruptures. For Q = 5 nC, N increases to about 40 in 80 pus. N will return to its initial value as the membrane discharges with a time constant of about 2 s.

Porosity and pore-size distribution usually are measured by mercury porosimetry, which also can provide a good estimate of the surface area (17). In this technique, the sample is placed under vacuum and mercury is forced into the pore stmcture by the appHcation of external pressure. By recording the extent of mercury intmsion as a function of the pressure appHed, it is possible to calculate the total pore volume and obtain the population of the various pore sizes in the range 2 nm to 10 nm. 

Figure 6 shows calibration curves for three other two column combinations, each representing 30,000 to 40,000 plates per set. The 10 A° plus 10 A° curve can be interpreted to show a deficiency in relative pore population in the range equivalent to about 50,000 to 600,000 molecular weight. The other two, properly calibrated, can conceivably be used for determination of molecular weight distributions. However, utility for resolution of specific polymodal mixtures is too difficult to assess from calibration curve alone. How much curvature of a calibration curve translates into utility or non-utility Calibration curves indicating pore size populations all have the same shape for given column combinations whether the plate count level is 5000 plates or 20,000 plates or 80,000 plates. 

Mercury poroslmetry data of these packings are given In Table IV. It Is of Interest to note that the pore-size distribution of CPG Is significantly more narrow than that of Syn-Chropak, a surface-modified porous silica (LlChrospher). These different physical characteristics may help to explain the existence of micropores In SynChropak. Because of the wide pore-size distribution of this packing. It seems reasonable that this material also contains a population of micropores which are only accessible to D2O. In mercury poroslmetry measurements, the lower pore size limit Is about 30A. 

From these studies with SynChropak SEC packings and controlled porosity glass, it is concluded that the silica packing contains a population of micropores which are differentially accessible to low molecular weight probes of total permeation volume. It is not known, however, if the microporosity in the 100 and 300A SynChropak SEC packings is the result of the rather wide pore-size distribution and whether all silicas contain micropores. 

The SAXS/TGA approach has been demonstrated to be a useful technique for time-resolution of porosity development in carbons during activation processes. Qualitative interpretation of the data obtained thus far suggests that a population balance approach focusing on the rates of production and consumption of pores as a function of size may be a fruitful approach to the development of quantitative models of activation proces.ses. These then could become useful tools for the optimization of pore size distributions for particular applications by providing descriptions and predictions of how various activating agents and time-temperature histories affect resultant pore size distributions. 

A combination of physical forces and diffusion governs pore evolution. As a result, pores with a wide range of sizes appear in the membrane. This distribution of sizes is described by a pore population function that is, a probability density function, n(r, t) At any time t, there are n(r, t) Ar pores with radii between r and r +Ar. 

Schechter and Gidley [128] utilized an approximation to the classical Graetz problem for the wall reaction to obtain v A, Cj t)), and then solved the population balance Eq. (a) by numerical methods. An example of the evolution of the pore area distribution for typical conditions for well acidation is shown in Fig. 1. 

Regarding the shapes of the pore size distribution plots, figure 3 shows that in all samples, with the exception of the pure titania sample, exist a small population of pores of around 40 A in diameter and a main pore population of around 130 A in diameter. The pore size distribution curve of the pure titania sample showed in contrast, the first maximum at 60 A and the indication of larger pores with the maximum beyond 300 A. 

Reyes and Jensen [1987] modeled the sulfation of calcined limestone by SO2, accounting for the pore size distribution and the possibility of pore plugging by expansion of the solid volume fraction. The continuity equations were of the type (4.4-2) and (4.4-7). The evolution of the pore size distribution with time was obtained through the simultaneous solution of a population balance equation (see Froment and Bischoff [1990], Chapter 12, Example 12.7.2). The evolution of the... 

Assessment of the development of the pore size distribution during carbon activation a population balance approach... 

C pores to B pores, carbon deposits should be densified or adsoibed carbon should be coked. The densification may occur at a relatively slow rate because it requires the change in the stmcture of the carbon deposit. The size of the pore B is assumed to be close to those of CO2 and O2 molecules. Thus, the B pores are reduced to the A pores faster than that of the C pores. Suppose the populations of pores A, B, C and D are 1, 2, 3 and 2, respectively. And suppose the relative speeds of the pore size reduction is 1.0, 0.5 and 1.5 for B to A, C to B, and D to C, the change in pore size distribution in time becomes as shown in Fig. 4.41b. The average pore size distribution is given by... 

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

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