Diffusion dimensionless groups

The differential material balances contain a large number of physical parameters describing the structure of the porous medium, the physical properties of the gaseous mixture diffusing through it, the kinetics of the chemical reaction and the composition and pressure of the reactant mixture outside the pellet. In such circumstances it Is always valuable to assemble the physical parameters into a smaller number of Independent dimensionless groups, and this Is best done by writing the balance equations themselves in dimensionless form. The relevant equations are (11.20), (11.21), (11.22), (11.23), (11.16) and the expression (11.27) for the effectiveness factor. 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

Where surface-active agents are present, the notion of surface tension and the description of the phenomena become more complex. As fluid flows past a circulating drop (bubble), fresh surface is created continuously at the nose of the drop. This fresh surface can have a different concentration of agent, hence a different surface tension, from the surface further downstream that was created earlier. Neither of these values need equal the surface tension developed in a static, equiUbrium situation. A proper description of the flow under these circumstances involves additional dimensionless groups related to the concentrations and diffusivities of the surface-active agents. 

Sohd Catalysts Processes with solid catalysts are affected by diffusion of heat and mass (1) within the pores of the pellet, (2) between the fluid and the particle, and (3) axially and radially within the packed bed. Criteria in terms of various dimensionless groups have been developed to tell when these effects are appreciable. They are discussed by Mears (Ind. Eng. Chem. Proc. Des. Devel., 10, 541-547 [1971] Jnd. Eng. Chem. Fund., 15, 20-23 [1976]) and Satterfield (Heterogeneous Cataly.sls in Practice, McGraw-Hill, 1991, p. 491). 

For many cooling waters, including seawater and also drinking water, where corrosion rates are 70 to 100% of the limiting diffusion current, the use of dimensionless group analysis can then be applied. 

It may be noted that many of these dimensionless groups are ratios. For example, the Nusselt group h/(k/l) is the ratio of the actual heat transfer to that by conduction over a thickness l, whilst the Prandtl group, (p/p)/(k/Cpp) is the ratio of the kinematic viscosity to the thermal diffusivity. 

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

It is also assumed that axial diffusion of mass, heat and momentum are negligible. The velocity profiles are assumed to be locally fully developed in length. For this simplified case the polymer concentration and secondary flows are determined by three dimensionless groups ... 

The term in brackets is a dimensionless group that plays a key role in determining the limitations that intraparticle diffusion places on observed reaction rates and the effectiveness with which the catalyst surface area is utilized. We define the Thiele modulus hT as... 

The dimensionless group Pep is essentially the ratio of the rate of convective transport to the rate of diffusive transport. Similarly, Nr describes the relative importance of radioactive decay to convective flow as a method of removing radon from the soil pores. In the case of Pep >>1/ diffusion can be neglected and the first term in equation (1) drops out. If in addition Nr >>1, then radioactive decay can be neglected as a removal term. If Pep 1, then diffusive radon migration dominates, and the second term in equation (1) can be neglected. 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

An interesting problem arises when we consider solutions or colloidal sols where the diffusing component is much larger in size than the solute molecules. In dilute systems Equation (1.14) would give an adequate value of the Peclet number but not so when the system becomes concentrated, i.e. the system itself becomes a condensed phase. The interactions between the diffusing component slow the motion and, as we shall see in detail in Chapter 3, increase the viscosity. The appropriate dimensionless group should use the system viscosity and not that of the medium and now becomes... 

Drew number phys chem A dimensionless group used in the study of diffusion of a solid material A into a stream of vapor initially composed of substance B, equal to Za(Ma Mg) + Mg My... 

Consider the mass transfer across a flat plate. In this case, the important variables are (dimensions in parentheses) the mass transfer coefficient k (L/T), the bulk fluid velocity u (L/T), the kinematic viscosity of the fluid v (L2/T), the solute diffusion coefficient I) (L2/T), and the plate length / (L). The number of independent variables n = 5 and the number of the involved dimensions m = 2. Hence, die number of dimensionless groups Pi = n — m = 3. 

In the solution of mass transport problems, several dimensionless groups are used in order to reduce the number of variables. Diffusion layer thicknesses etc. are expressed in much of the literature in terms of these dimensionless variables. 

Examples of dimensionless groups that specify ratios of transport mechanisms are listed next in Table II and depend on the size and shape of the domain. The Peclet numbers for heat (Pet) and solute (Pes) and momentum (Re) transport are ratios of scales for convective to diffusive transport and depend on the magnitudes of the velocity field and the length scale for the diffusion gradient. Boundary layers form at large Peclet numbers (Pet or Pes) or Reynolds numbers (Re). The fonnation of a boundary layer at a large Re is particularly important in crystal growth from the melt, because the low... 

The Hatta number is a dimensionless group used to characterize the importance of the speed of reaction relative to the diffusion rate. 

The analysis is concerned with a one-dimensional model of electrodes in which reaction rates are distributed unevenly due to diffusion as well as a variation in electrode potential.4,5 The treatment of the problem of a simultaneous variation in electrolyte concentration and potential distribution in the electrode is treated in an analogous manner to that of non-isothermal chemical reactions in porous catalysts.16 The results show that several dimensionless groups or numbers control the electrode behavior. Figure 8 shows a back fed porous anode used in the model. 

Knowing the viscosity and density of the reaction mixture, the flow channel diameter, void fraction of the bed, and the superficial fluid velocity, it is possible to determine the Reynolds number, estimate the intensity of dispersion from the appropriate correlation, and use the resulting value to determine the effective dispersion coefficient Del or I). Figures 8-32 and 8-33 illustrate the correlations for flow of fluids in empty tubes and through pipes in the laminar flow region, respectively. The dimensionless group De l/udt = De l/2uR depends on the Reynolds number (NRe) and on the molecular diffusivity as measured by the Schmidt number (NSc). For laminar flow region, DeJ is expressed by ... 

The rate of deposition is calculated for a spherical, cylindrical, or rotating disc collector the dimensionless rate of deposition was found to be a function of a single dimensionless group B, involving the rate constant, characteristic velocity, diffusion coefficient, and (except in the case of a disc) the collector size. 

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