Pore area radius

Here >f(>r,) is the total wall area, and T(>r,) the total length, of all pores having radius greater than r. ... 

Each of the procedures described in Section 3.6 for the calculation of pore size distribution involves a value of the pore area y4f for each successive group of pores. In the Roberts procedure 6A, can be immediately obtained from the corresponding pore volume and pore radius as (for... 

In this exercise we shall estimate the influence of transport limitations when testing an ammonia catalyst such as that described in Exercise 5.1 by estimating the effectiveness factor e. We are aware that the radius of the catalyst particles is essential so the fused and reduced catalyst is crushed into small particles. A fraction with a narrow distribution of = 0.2 mm is used for the experiment. We shall assume that the particles are ideally spherical. The effective diffusion constant is not easily accessible but we assume that it is approximately a factor of 100 lower than the free diffusion, which is in the proximity of 0.4 cm s . A test is then made with a stoichiometric mixture of N2/H2 at 4 bar under the assumption that the process is far from equilibrium and first order in nitrogen. The reaction is planned to run at 600 K, and from fundamental studies on a single crystal the TOP is roughly 0.05 per iron atom in the surface. From Exercise 5.1 we utilize that 1 g of reduced catalyst has a volume of 0.2 cm g , that the pore volume constitutes 0.1 cm g and that the total surface area, which we will assume is the pore area, is 29 m g , and that of this is the 18 m g- is the pure iron Fe(lOO) surface. Note that there is some dispute as to which are the active sites on iron (a dispute that we disregard here). 

Thermally stable AI2O3 was synthesized as in ref. 5, by hydrolysis of A1 isopropoxide (99.99+% Aldrich Chemicals) dissolved in 2-methylpentane-2,4-diol. The resulting solid was filtered, washed in 2-propanol, and dried in air at 373 K. Then, it was calcined in flowing dry air, while the temperature was raised at 1 K/min to 733 K, when 2.4% HjO was introduced to the flowing air. Afterwards, the temperature ramp was continued to 973 K. The sample was kept at 973 K for 2 h in 7% water. The isoelectric point of the resulting y-Al Oj was pH 8. The BET surface areas were 205 to 235 mVg, and the average pore size radius was around 8.3 nm... 

Influence on Electrolyte Conductivity In porous separators the ionic current passes through the liquid electrolyte present in the separator pores. Therefore, the electrolyte s resistance in the pores has to be calculated for known values of porosity of the separator and of conductivity, o, of the free liquid electrolyte. Such a calculation is highly complex in the general case. Consider the very simple model where a separator of thickness d has cylindrical pores of radius r which are parallel and completely electrolyte-filled (Fig. 18.2). Let / be the pore length and N the number of pores (all calculations refer to the unit surface area of the separator). The ratio p = Ud (where P = cos a > 1) characterizes the tilt of the pores and is called the tortuosity factor of the pores. The total pore volume is given by NnrH, the porosity by... 

Since mercury has a contact angle with most solids of about 140°, it follows that its cosine is negative (i.e., it takes applied pressure to introduce mercury into a pore). In a mercury porosimeter, a solids sample is evacuated in a cell, mercury is then intruded, and the volume, V, is noted (it actually reads out), and the pressure, P, is then increased stepwise. In this fashion it is possible to deduce the pore volume of a particular radius [corresponding to P by Eq. (21)]. A pore size distribution will give the total internal pore area as well, which can be of importance in dissolution. 

Brunauer s modelless method uses pore volume and pore area not as functions of the Kelvin radius but rather as functions of hydraulic radii that he defines as ... 

In real porous solids, the pores are not straight, and the pore radius can vary. Two parameters are used to describe the diffusion path through real porous solids the void fraction, e, defined as fhe ratio of pore area to fofal cross-sectional area, and the tortuosity, r, which corrects for the fact that pores are not straight. The resulting effective diffusivity is then... 

Fig. 14. Coion partition coefficients SN for uni-univalent electrolyte, CF/w = 21.7 mmol/dm3, Kpn = w, calculated by the EVM (line A) or the ESM for slit-shaped pores (lines B-F) (i) Single pores of radius r = 6 (line B) or 40 (line F) nm. (ii) Assemblies of two pores of equal surface charge density of mean radius 6 nm and individual radii (in nm) 5.306,40 (line C) 5.454,80 (line D) 4.583,40 (line E) where 98 % (C), 99 / (D), or 96 % (E) of the total pore wall surface area is in narrow pores 121). Note that the power law of Eq. (46) is obeyed over a considerable range in cases C and E...

Below the adsorption isotherm data, the detailed pore size distribution data are listed in seven columns. These include the pore radii corresponding to the 64 data points, the volume of liquid nitrogen desorbed at each step, the mean pore radii corresponding to each of the desorbed decrements, the pore volume per unit change in radius (AV/Ar), the cumulative pore volumes at each pore radius, the calculated surface area in each of the pore radius steps, and the cumulative pore areas in pores larger than each of the listed radii. The print-out sheet is completed with the two sections discussed in connection with Figure 2. 

Fluorination of this catalyst altered the pore structure in the following way. The sample pore volume increased from 0.78 to 1.42 cc. per gram, and the median pore radius increased to 154 A. However, the BET surface area of the sample remained close to 228 sq. meters per gram. The total calculated pore area changed only from 226 to 216 sq. meters per gram. The surface area had probably de-... 

For a simplified model of the membrane having cylindrical pores of radius r and length / with Ap pores per unit surface area, Poiseuille s law expresses the volume flow Q for a pressure difference AP as follows ... 

Values of t are obtained from adsorption data of the same adsorptive on a non-porous surface of the same nature, as in fig. 1.28. Substituting bulk values for y and, the pore size distribution a(p) or d(p) is obtainable. The occurrence of hysteresis implies that this gives different results for the two branches (ascending and descending) of the curve. In fact, the difference between the two metastable states is a characteristic of the type of pores. For non-connected pores, usually the downward curve is analyzed, because then the menisci have already been formed but for connected networks the ascending one may be more appropriate. After each stepwise change of p, the radius is calculated and from that the exposed pore volume and pore area. This yields a cumulative distribution which, if so desired, can be differentiated. 

The measured surface area consists of both external and internal area where internal surface area includes all cracks or connected pores that are deeper than they are wide, varying from subatomic defects to pores of extreme size (Gregg and Sing, 1982). For example, micropores are dehned as pores with radius <2 nm, mesopores as pores with radius from 2 nm to 50 nm, and macropores as those with pores of diameter >50 nm. The main distinction between internal and external surface is that advection can control transport to and away from external surface while diffusion must control transport for internal pore space (Hochella and Banheld, 1995). Porosity may be related to crystallization or replacement processes (Putnis, 2002). 

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores ). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slitshaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution. 

The equilibrium partition coefficient, Keq, depends on the pore size in the porous particle as well as the size and conformation of the solute molecule. Giddings et al. [60] have shown that the characteristic size parameter for all shapes of molecules is the mean length of the molecular projection. For a rigid spherical solute this is simply the radius a. By considering the ratio of the area available for the solute molecule inside the cylindrical pore to the actual pore area, and neglecting any interaction between the solute and pore wall, we get... 

From a practical point of view, this is a very common and a very important problem. Holes may originate from defects in membrane backings, Inhomogeneltles in the casting solutions, airborne dust particles, etc. For simplicity, let us consider a membrane,area that contains N pores of radius r3 and N holes of radius r3. Water flux, J3 through this membrane is given by ... 

Based on the data of Table 8.4, what proportion of the cross-sectional area of a cylindrical soil pore of radius 15 fxm is influenced by the electric double layer if monovalent ions predominate at a salt concentration of 10 1 M ... 

Equations 7.6 and 7.7 can be used to describe diffusion down a straight cylindrical pore. A porous solid does not consist of straight cylindrical pores, each having the same length and radius. Models for pore structure have been proposed that describe the pore size distribution and orientation as a function of location within the pellet [2]. These microscopic descriptions can be used to predict the porosity, pore size distribution, pore volume and pore area, all of which can be measured experimentally. 

More general models for the porous structure have also been developed by Johnson and Stewart [60] and by Feng and Stewart [43], called the parallel cross-linked pore model. Here, Eqs. 3.5.b-4 to 6 or Eq. 3.5.b-7 are considered to apply to a single pore of radius r in the solid, and the diffusivities interpreted as fte actual values rather than effective diffusivities corrected for porosity and tortuosity. A pore size and orientation distribution function /(r, Q), similar to Eq. 3.4-2, is defined. Then /(r, Q)dri is the fraction open area of pores with radius r and a direction that forms an angle Q with the pdlet axis. The total porosity is then... 

The same data can be used for an evaluation of the specific surface area. On noting that the internal surface area of cylindrical pores with radius in the range / ,... 

The original semiempirical parallel-pore model represents a monodisperse pore-size distribution and makes use of the measurable physical properties, Sg, Vg, ps, and Gp. The complex particle with porosity Gp is replaced by an array of straight and parallel cylindrical pores of radius a, much like a honeycomb structure. The mean pore radius a is simply calculated by assuming that the sum of the inner surface areas of all the n pores in an array nlnaL) is equal to the total surface area Sg and the sum of all the pore volumes nna L) is equal to the experimental pore volume V [5] ... 

Effect of time and ageing temperature on the whole surface area (m /g), and the surface area (m /g) corresponding to pores with radius less than 8nm. 

Tp = the pore radius or half the distance across the pore = the radius of the immobile surface atom or ion = fraction of the excluded area compared to the hard-sphere ratio 5bet = the slope of the transformed BET plot... 

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