Solute diffusivities

In a few it has been demonstraled drat there is a solute molecular weight cutoff or size (MWC) above vdiid) a solme will not diffuse into a solid. Therefore, botdenecks in cell walls may control the rate of diffiiskm in solids. Solute molecules thm cannot pass through these bottlenecks cannot enter or leave the solid (mteepting diose portions in Mikh the cell walls are ruptured). This suggests that if MWC is known, f , ftrr a solute t can be estimated firmt die D, for a reference solute r and corresponding known values of l . and MW that is. 

Furthermore, if pairs of D P(., and MW values ate known, MWC can be estimated and used to estimate other D, values. In cases where P, fin- a iow-molecular-wdght solute must be estimated and (PA and MWC ate known but bodi Dt, values ate not known, (MW /MW,) can be substituted for in 

While ISq. (10.9-1) provides a reasonable basis for estititming D, in some cases, it should not be applicable when (1) sohite cause changes in the pore structure, (2) nongeometric pore-solute interactions affect the mobility of the srdute or its concemtation in the pores, (3) particular solines are sinbed selectively by the mate, or (4) fiicilitated or activated diffusion occurs. Selective sorption is particularly likely to occur 

As temperature, viscosity, and vary, the change in D, for a given solute will parallel the change in 1, in the occluded solvent that is, D, will te direcdy proportional to the absolute value of T and inversely proportional to n. 

for soyb oil during the initial period of its extraction from st beans with hexane is around 1.0 X 10 m /s but drops off to 0.12-0.2 times that value at the end of extraction. Both initial and final D, values for cottonse and flaxseed are roughly only 0.1 times as large as the corresponding initial and final values for soybeans and the extraction times required whoi these oils are extracted commercially is considerably longer than that for soybeans. 

In solids that contain vascular systems. D, parallel to the vascular bundle may be twice as large as it is at right angles to the bundle. Furtheimore. in such solids, D, as determined from betch leaching tests increases as the panicle size increases (below a threshold size of 3-5 mm). In dense cellular materials, on the other hand, raptured surface oells make up a progressively smaller fraction of the particle volume as size increases D, consequently decreases slightly as particle size increases. 

The corresponding NaCI sucrose Dt ratio is 3-4. Therefore, the cell walls in the solid hinder the difluskm of sucrose more than that of salts which have a lower molecular weight and voluttte. D, values for sucrose in materials with hard cell walls are between 0.1 and 0.2 tinres as large as Dt for sucrose with soft cell walls the corresponding D, Dt ratio is 0.3-0.9. 

Coefficient Infinite Slab Infinite Cylinder Sphere 

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One contribution to band broadening in which solutes diffuse from areas of high concentration to areas of low concentration. 

In general, the foUowing steps can occur in an overall Hquid—soHd extraction process solvent transfer from the bulk of the solution to the surface of the soHd penetration or diffusion of the solvent into the pores of the soHd dissolution of the solvent into the solute solute diffusion to the surface of the particle and solute transfer to the bulk of the solution. The various fundamental mechanisms and processes involved in these steps make it impracticable or impossible to describe leaching by any rigorous theory. 

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Solution—Diffusion Model. In the solution—diffusion model, it is assumed that (/) the RO membrane has a homogeneous, nonporous surface layer (2) both the solute and solvent dissolve in this layer and then each diffuses across it (J) solute and solvent diffusion is uncoupled and each is the result of the particular material s chemical potential gradient across the membrane and (4) the gradients are the result of concentration and pressure differences across the membrane (26,30). The driving force for water transport is primarily a result of the net transmembrane pressure difference and can be represented by equation 5 ... 

Equation 7 shows that as AP — oo, P — 1. The principal advantage of the solution—diffusion (SD) model is that only two parameters are needed to characterize the membrane system. As a result, this model has been widely appHed to both inorganic salt and organic solute systems. However, it has been indicated (26) that the SD model is limited to membranes having low water content. Also, for many RO membranes and solutes, particularly organics, the SD model does not adequately describe water or solute flux (27). Possible causes for these deviations include imperfections in the membrane barrier layer, pore flow (convection effects), and solute—solvent—membrane interactions. 

When it was recognized (31) that the SD model does not explain the negative solute rejections found for some organics, the extended solution—diffusion model was formulated. The SD model does not take into account possible pressure dependence of the solute chemical potential which, although negligible for inorganic salt solutions, can be important for organic solutes (28,29). 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, usiag software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Ke. To faciUtate these calculations, the foUowiag data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sb, is defined as Sh = 0.04 S c , where Sc is the Schmidt... 

Ultrafiltration separations range from ca 1 to 100 nm. Above ca 50 nm, the process is often known as microfiltration. Transport through ultrafiltration and microfiltration membranes is described by pore-flow models. Below ca 2 nm, interactions between the membrane material and the solute and solvent become significant. That process, called reverse osmosis or hyperfiltration, is best described by solution—diffusion mechanisms. 

The existing data indicate that fcja is proportional to the square root of the solute-diffusion coefficient, and since the interfacial area a does not depend on Dl, it follows that /cl is proportional to Dl. An analysis of the design variables involved indicates that /cl should be proportional to Nsc when the Reynolds number is held constant. 

Temperature The temperature of the extraction should be chosen for the best balance of solubility, solvent-vapor pressure, solute diffusivity, solvent selectivity, and sensitivity of product. In some cases, temperature sensitivity of materials of construction to corrosion or erosion attack may be significant. 

FIGt 22-48 Transport mechanisms for separation membranes a) Viscous flow, used in UF and MF. No separation achieved in RO, NF, ED, GAS, or PY (h) Knudsen flow used in some gas membranes. Pore diameter < mean free path, (c) Ultramicroporoiis membrane—precise pore diameter used in gas separation, (d) Solution-diffusion used in gas, RO, PY Molecule dissolves in the membrane and diffuses through. Not shown Electro-dialysis membranes and metallic membranes for hydrogen. 

Basic Principles of Operation RO and NF are pressure-driven processes where the solvent is forced through the membrane by pressure, and the undesired coproducts frequently pass through the membrane by diffusion. The major processes are rate processes, and the relative rates of solvent and sohite passage determine the quality of the product. The general consensus is that the solution-diffusion mechanism describes the fundamental mechanism of RO membranes, but a minority disagrees. Fortunately, the equations presented below describe the obseiwed phenomena and predict experimental outcomes regardless of mechanism. 

Greenwood (1956) described the behaviour of an assembly of n groups of particles undergoing Ostwald ripening by solution-diffusion controlled transfer between particles according to a general relationship... 

In a packed column, however, the situation is quite different and more complicated. Only point contact is made between particles and, consequently, the film of stationary phase is largely discontinuous. It follows that, as solute transfer between particles can only take place at the points of contact, diffusion will be severely impeded. In practice the throttling effect of the limited contact area between particles renders the dispersion due to diffusion in the stationary phase insignificant. This is true even in packed LC columns where the solute diffusivity in both phases are of the same order of magnitude. The negligible effect of dispersion due to diffusion in the stationary phase is also supported by experimental evidence which will be included later in the chapter. 

Thus, during solute transfer between the phases, (t) is now the average diffusion time (to) and (o) is the mean distance through which the solute diffuses, Le., the depth or thickness of the film of stationary phase (df). Thus,... 

The reduced velocity compares the mobile phase velocity with the velocity of the solute diffusion through the pores of the particle. In fact, the mobile phase velocity is measured in units of the intraparticle diffusion velocity. As the reduced velocity is a ratio of velocities then, like the reduced plate height, it also is dimensionless. Employing the reduced parameters, the equation of Knox takes the following form... 

Atwood and Goldstein [16] examined the effect of pressure on solute diffusivity and an example of some of their results is shown in Figure 7. It is seen that the diffusivity of the solutes appears to fall linearly with inlet pressure up to 40 MPa and the slopes of all the curves appear to be closely similar. This might mean that, in column design, diffusivities measured or calculated at atmospheric pressure might be used after they have been appropriately corrected for pressure using correction factors obtained from results such as those reported by Atwood and Goldstein [16]. It is also seen that the... 

Figure 7. Graph of Solute Diffusivity against Pressure...

Thus, for significant values of (k") (unity or greater) the optimum mobile phase velocity is controlled primarily by the ratio of the solute diffusivity to the column radius and, secondly, by the thermodynamic properties of the distribution system. However, the minimum value of (H) (and, thus, the maximum column efficiency) is determined primarily by the column radius, secondly by the thermodynamic properties of the distribution system and is independent of solute diffusivity. It follows that for all types of columns, increasing the temperature increases the diffusivity of the solute in both phases and, thus, increases the optimum flow rate and reduces the analysis time. Temperature, however, will only affect (Hmin) insomuch as it affects the magnitude of (k"). 

The solute diffusivity will also depend on the nature of the mobile phase beitmay a gas or liquid. Very little work has been carried out on the effect of different carrier gases on column efficiency. Scott and Hazeldene [9] measured some HETP curves... 

The effect is much less when the fluid is a gas, where the solute diffusivity is high and the solute tends to diffuse rapidly across the tube and partially compensate for the nonlinear velocity profile. However, when the fluid is a liquid, the diffusivity is five orders of magnitude less and the dispersion proportionally larger. 

C. In their first series of experiments, six data sets were obtained for (H) and (u), employing six solvent mixtures, each exhibiting different diffusivities for the two solutes. This served two purposes as not only were there six different data sets with which the dispersion equations could be tested, but the coefficients in those equations supported by the data sets could be subsequently correlated with solute diffusivity. The solvents employed were approximately 5%v/v ethyl acetate in n-pentane, n-hexane, n-heptane, -octane, -nonane and n-decane. The solutes used were benzyl acetate and hexamethylbenzene. The diffusivity of each solute in each solvent mixture was determined in the manner of Katz et al. [3] and the values obtained are included... 

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