Pores concentration dependence

The diffusion coefficient in these phases D,j is usually considerably smaller than that in fluid-filled pores however, the adsorbate concentration is often much larger. Thus, the diffusion rate can be smaller or larger than can be expected for pore diffusion, depending on the magnitude of the flmd/solid partition coefficient. 

Effectiveness As a reac tant diffuses into a pore, it undergoes a falling concentration gradient and a falling rate of reaction. The concentration depends on the radial position in the pores of a spherical pellet according to... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

This trend can be explained with the following mechanism. In the presence of NAPL, the extracted vapor concentration depends mainly on the vapor pressure of the contaminant. After the disappearance of free NAPL, the extracted vapor concentration becomes dependent on the partitioning of contaminants among the three other phases (see Table 14.3). As the air passes through the pores, the dissolved contaminants volatilize from the soil moisture to the gas phase, causing the desorption of contaminants from the surface of soil particles into the aqueous phase. As a result, the concentration in all three phases decreases, with a consequent decrease in removal rates. 

If the two competing reactions have the same concentration dependence, then the catalyst pore structure does not influence the selectivity because at each point within the pore structure the two reactions will proceed at the same relative rate, independent of the reactant concentration. However, if the two competing reactions differ in the concentration dependence of their rate expressions, the pore structure may have a significant effect on the product distribution. For example, if V is formed by a first-order reaction and IF by a second-order reaction, the observed yield of V will increase as the catalyst effectiveness factor decreases. At low effectiveness factors there will be a significant gradient in the reactant concentration as one moves radially inward. The lower reactant concentration within the pore structure would then... 

The concentration dependence of z/l vs. c/c0 is plotted in Figure 11.14a. It can be seen that from a Thiele modulus cp > 3 the educt does not reach the internal part of the pore. The inner part of the pore system is useless for catalysis. This is especially relevant if expensive metals serve as active components on a porous carrier, which are then wasted. There are chances to master this diffusion limitation, which will be discussed later in detail. Another important variable is the efficiency factor tj. The efficiency factor r is defined as the quotient of the speed of reaction rs to the maximal possible speed of reaction rsmax. r is related to q> as the quotient of the hyperbolic tangent of the Thiele modulus qy. 

As the concentration of MeOH increases, the divergent diffusion behavior between the two membrane types is a reflection of fhe difference in MeOH solubility and its concentration dependence within each membrane. This was verified by solvenf upfake measurements. Upon increasing MeOH concentration, Nafion 117 showed a steady increase in mass, while a sharp drop in total solution uptake was observed for BPSH 40. The lower viscosity of MeOH also affecfs fhe fluidity of the solution within the pores. The constant solvent uptake and the increased fluidity of the more concentrated MeOH solutions accounted for fhe slight increase in diffusion coefficienf of Nafion 117. For BPSH 40, increasing the MeOH concentration resulted in a decrease in MeOH diffusion. The solvent uptake measurements showed very similar behavior, indicating that the membrane excludes the solvent upon exposure to higher MeOH concentrations. 

First, the sample was examined by GPC, for which four columns of styragel of 106,10s, 104 and 103 A nominal pore size were used. The total number of theoretical plates as determined by acetone at a flow rate of 1 ml/min was ca. 26,000. The eluent was tetrahydrofuran. The chromatogram is shown in Figure 9, which indicates two peaks at ca. 21 and 24 counts. The former may be assigned to the tetra-chain, star-draped component, and the latter to the precursor. However, no complete separation of the two peaks was observed. For another comparison, velocity ultra-centrifugation was performed for the sample at 59,780 rpm using a 6-solvent for polystyrene, cyclohexane. The operation temperature was established at 35 °C, the 6-temperature, to minimize the concentration dependence of sedimentation velocity and other effects. A sedimentation pattern taken by UV-absorption is shown in Figure 10. It is seen that the separation of S-A sample into the two components was quite difficult even at a very low polymer concentration, 0.077 g/dl. 

Illustrative Example 9.2 Evaluating the Concentration Dependence of Sorption of Phenanthrene to Soil and Sediment POM Illustrative Example 9.3 Estimating Pore Water Concentrations in a Polluted Sediment... 

Figure 19.17 Spherical macroparticle with radius ra consisting of an aggregate of microparticles separated by micropores filled with water. A chemical with constant concentration C° diffuses into the pore volume of the macroparticle. The local dissolved pore concentration Cw is at instantaneous equilibrium with the local sorbed phase C ( K d is microscopic equilibrium coefficient). Note that the macroscopic distribution coefficient Kd is time dependent (see Eq. 19-78.)...

Other results also confirm the important role of internal diffusion. Experimental activation energies (67—75 kJ mol"1) of the sucrose inversion catalysed by ion exchangers [506—509] were considerably lower than those of a homogeneously catalysed reaction (105—121 kJ mol"1) [505, 506,508] and were close to the arithmetic average of the activation energy for the chemical reaction and for the diffusion in pores. The dependence of the rate coefficient on the concentration in the resin of functional groups in the H+-form was found to be of an order lower than unity. A theoretical analysis based on the Wheeler—Thiele model for a reaction coupled with intraparticle diffusion in a spherical bead revealed [510,511] that the dependence of the experimental rate coefficient on acid group concentration should be close to those found experimentally (orders, 0.65 and 0.53 for neutralisation with Na+ and K+ ions respectively [511] or 0.5 with Na+ ions [510]). 

Sample molecules that are too large to enter the pores of the support material, which is commercially available in various pore dimensions, are not retained and leave the column first. The required elution volume Ve is correspondingly small. Small molecules are retained most strongly because they can enter all the pores of the support material. Sample molecules of medium size can partly penetrate into the stationary phase and elute according to their depth of penetration into the pores (Fig. 7.3). No specific interactions should take place between the molecules of the dendrimer sample and the stationary phase in GPC since this can impair the efficiency of separation by the exclusion principal. After separation the eluate flows through a concentration-dependent detector (e.g. a UV/VIS detector) interfaced with a computer. One obtains a chromatogram which, to a first approximation, reflects the relative contents of molecules of molar mass M. If macromolecules of suitable molar mass and narrow molar mass distribution are available for calibration of the column, the relative GPC molar mass of the investigated dendrimer can be determined via the calibration function log(M) =f( Vc). 

A monolithic hydrophobic polymer formed by photoinitiated polymerization for on-chip solid-phase extraction is shown in Figure 5.6. The polymer mixture includes butyl methacrylate (BMA) and ethylene dimethacrylate (EDMA), with the pore size controlled by the composition of the hexane/methanol porogenic mixture. The degree of pre-concentration depends on the flow rate, as shown in the pre-concentration of GFP at three flow rates (see Figure 5.7). The factors of pre-concentration were 355, 756, and 1002 for the flow rates of 3, 1.03, and 0.53 rE/min, respectively [342]. 

More recent investigations suggest that many different mechanisms (for example path windiness, pore geometry, or concentration dependant diffusion) may be important (2A). By casting the diffusion equation in the appropriate geometry, the physical mechanisms of release can be elucidated without resort to any simple empirical factors. 

Where CA is the concentration of gas A within the pores. Equation 3.25 is known as Fick s law of diffusion. In general, the effective diffusivity is concentration dependent. The total pressure gradient may also have an influence on the values of DcA. 

Among the other mechanism of action tested it was recently found that palytoxin caused efflux in an hybrid between the Na-K-ATPAse and the H-K-ATPase, converting this enzyme into and open channel (Farley et al. 2001). Furthermore, palytoxin stimulated a cation-dependent current in rat distal and proximal colon in a concentration-dependent manner when applied to the mucosal surface of the tissue whose apical membrane does not express Na-K-ATPase. The observation that the palytoxin-induced current was blocked by vanadate but was resistant to ouabain supported the suggestion that the toxin was able to convert a vanadate-sensitive H-K-ATPase into an electrogenic cation transporter and, consequently, that the pore-forming action of palytoxin was not restricted to Na-K-ATPase since it was also observed with the colonic H-K-ATPase (Scheiner-Bobis et al. 2002). The toxin was also found to interfere with the sarcolemmal Ca pump in cardiac myocytes (Kock-skamper et al. 2004). 

In this paper, we will present a procedure to separate pore diffusion and surface diffusion in GAC adsorption by use of a stepwise linearization. Furdiermore, a simplified method to estimate concentration dependency of apparent diffiisivities from breakthrough curves will be proposed. 

---