Types of concentration gradient

Transport properties are often given a short treatment or a treatment too theoretical to be very relevant. The notion that molecules move when driven by some type of concentration gradient is a practical and easily grasped approach. The mathematics can be minimized. Perhaps the most important feature of the kinetic theory of gases is the recognition that macroscopic properties such as pressure and temperature can be derived by suitable averages of the properties of individual molecules. This concept is an important precursor to statistical thermodynamics. Moreover, the notion of a distribution function as a general... 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

Restrictions which may exist for the choice of a commercial reactor need not be imposed at the development stage. In some cases, a reactor of one type may be best for acquiring data in model characterisation, whereas a reactor of another type might be more suitable for full-scale production. (The cautions expressed in Sect. 4 must be taken into account.) Continuous flow back-mixed reactors can be very useful for kinetic studies because the absence of concentration gradients can reduce uncertainties in concentration measurements. When these reactors have attained a steady state, many of the problems associated with stiffness (see above) can be avoided. 

The emulsion liquid membrane for cephalosporins relies essentially on facilitated transport. There are basically, however, two types of facilitated transport in emulsion liquid membrane system, i. e.. Type I and Type II facilitation. In the first type, the concentration gradient of the membrane soluble solute/permeate... 

The observations discussed above may have implications for any system where water, dissolved salts and hydrophobic entities are present and there must be many. Currently the separation of hydrophobic proteins can be achieved using a hydrophobic chromatography column, by elution with salt solutions. There is no adequate theory for this process and present understanding is purely empirical. Suppose then, that while all salts reduce electrostatic forces, only those salts that reduce bubble coalescence also reduce the hydrophobic attraction. Further, these salts have a significant effect on the hydrophobic attraction only above their transition concentration. With this notion in mind the experimental results are explained. This then enables separations to be simplified, as the salt type and concentration gradient required are easily determined. 

This type of nonequilibrium results from the existence of a heterogeneous flow domain. Spatial (or temporal) heterogeneities in such properties as hydraulic conductivity or sorption capacity can result in nonuniform velocity fields. Conditions for diffusive mass transfer of solute may develop because of concentration gradients created by the nonuniform velocities. If this diffusive mass transfer is rate limited, nonequilibrium behavior (e.g., asymmetrical BTC) results. The influence of macroscopic heterogeneities (e.g., aggregates, macropores) on solute transport in soil has been well-documented [see Brusseau and Rao, (1990) for a recent review]. It should be noted that transport-related nonequilibrium (TNE) affects both sorbing and nonsorbing solute. 

Formation of concentration gradient should depend on (a) dissolution rate of PVC in PMA solution, (b) diffusion rate of PVC in PMA solution, and (c) interruption time of the diffusion due to completion of solvent evaporation. Factors for controlling the above phenomena are (1) type of solvent, (2) casting temperature, (3) molecular weight of PVC, and (4) amount of PMMA solution. 

The above equations are based on the assumption that the diffusion coefficient D is independent of the gas concentration. However, that may not be necessarily true. Also, the diffusion coefficient is usually dependent on temperature following an Arrhenius type relationship. For a sphere, the Equation 2 can be represented in terms of concentration gradient in radial direction (3c/3r) as follows ... 

A uniform distribution of sulfuric acid electrolyte between the plates of the battery is essential if the battery is to provide an optimum service. Two types of departure from this ideal are encountered in practice (i) a heterogeneous distribution of the liquid phase when saturation is incomplete (ii) and the development of concentration gradients within the liquid phase. 

As discussed previously, matter transport is due to the flux of atoms or vacancies driven by gradients in the concentration, which can be described by using the Pick s first law. This special case of mass transport is not applicable to those with other types of driving forces, such as gradients in pressure, electric potential, and so on. To address this issue, it is necessary to use chemical potential, instead of concentration gradients, as driving force of the diffusions. Definition and description of chemical potential can be found in various textbooks. 

In this chapter, different types of microstructures and their applications are presented. The results presented herein clearly indicate that MSR and structured catalysts offer superior mass transfer performance at lower energy dissipation in comparison to conventional equipment Therefore, the influence of concentration gradients between fluid and solid and within the porous catalyst can be effectively diminished or even suppressed, leading to high product selectivity and yield. 

A ternary gradient with increasing concentration of methanol and simultaneously decreasing concentration of acetonitrile may improve the resolution of the sample.Two specific types of ternary gradients are probably most useful in practice ... 

The diffusion gradients established on the interdiffusion of two solids may take a variety of forms. When two slabs of metal, e.g. Cu and Ni, are heated together, the simplest type of diffusion gradient which may be established is that illustrated in Pig. 85(40). The figure shows that the presence of a third substance may alter the shape of the concentration gradient. The form of the concentration gradient on either side of an interface is not always of this smooth form. Bramley and his co-workers (23) heated steel bars in various atmospheres from... 

Equation (3.4.8) relates the overall mass-transfer coefficient Kxg to the individual mass-transfer coefficients kxg and kxi- Relations between the overaU transfer coefficient and the individual phase transfer coefficients for other types of concentration driving gradients are also of considerable use. Consider... 

Though a porous medium may be described adequately under non-reactive conditions by a smooth field type of diffusion model, such as one of the Feng and Stewart models, it does not necessarily follow that this will still be the case when a chemical reaction is catalysed at the solid surface. In these circumstances the smooth field assumption may not lead to appropriate expressions for concentration gradients, particularly in the smaller pores. Though the reason for this is quite simple, it appears to have been largely overlooked,... 

The concentric cylinder viscometer described in Sec. 2.3, as well as numerous other possible instruments, can also be used to measure solution viscosity. The apparatus shown in Fig. 9.6 and its variations are the most widely used for this purpose, however. One limitation of this method is the fact that the velocity gradient is not constant, but varies with r in this type of instrument, as noted in connection with Eq. (9.26). Since we are not considering shear-dependent viscosity in this chapter, we shall ignore this limitation. 

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