Diffusion Dimensional analysis

On the basis of scaling arguments, general functional dependencies can also be derived. For example, dimensional analysis shows that the center of mass diffusion coefficient DG for Zimm relaxation has the form... 

In order to simplify the situation, we assume that our porous sample under investigation covers the bottom of an open straight-walled can and fills it to a height d (Figure 1). Such a sample will exhibit the same areal exhalation rate as a free semi-infinite sample of thickness 2d, as long as the walls and the bottom of the can are impermeable and non-absorbant for radon. A one-dimensional analysis of the diffusion of radon from the sample is perfectly adequate under these conditions. To idealize the conditions a bit further we assume that diffusion is the only transport mechanism of radon out from the sample, and that this diffusive transport is governed by Fick s first law. Fick s law applied to a porous medium says that the areal exhalation rate is proportional to the (radon) concentration gradient in the pores at the sample-air interface... 

The point of view based on a physical model started with the 1935 paper of Higbie [30], While the main problem treated by Higbie was that of the mass transfer from a bubble to a liquid, it appears that he had recognized the utility of his representation for both packed beds and turbulent motion. The basic idea is that an element of liquid remains in contact with the other phase for a time A and during this time, absorption takes place in that element as in the unsteady diffusion in a semiinfinite solid. The mass transfer coefficient k should therefore depend on the diffusion coefficient D and on the time A. Dimensional analysis leads in this case to the expression... 

The product D0 (dCo/dx)x=0 t is the flux or the number of moles of O diffusing per unit time to unit area of the electrode in units of mol/(cm2 s). (The reader should perform a dimensional analysis on the equations to justify the units used.) Since (3Co/3x)x=01 is the slope of the concentration-distance profile for species O at the electrode surface at time t, the expected behavior of the current during the chronoamperometry experiment can be determined from the behavior of the slope of the profiles shown in Figure 3. IB. Examination of the profiles for O at x = 0 reveals a decrease in the slope with time, which means a decrease in current. In fact, the current decays smoothly from an expected value of oo at t = 0 and approaches zero with increasing time as described by the Cottrell equation for a planar electrode,... 

Chapter 7 covers the kinetic theory of gases. Diffusion and the one-dimensional velocity distribution were moved to Chapter 4 the ideal gas law is used throughout the book. This chapter covers more complex material. I have placed this material later in this edition, because any reasonable derivation of PV = nRT or the three-dimensional speed distribution really requires the students to understand a good deal of freshman physics. There is also significant coverage of dimensional analysis determining the correct functional form for the diffusion constant, for example. 

Often, the same expressions from this model are applied to the reductions of pellets, in which cases such structural factors as particle size distribution, porosity and pore shape, its size distribution, etc. should really affect the whole kinetics. Thus, the application of this model to such systems has been criticised as an oversimplification and a more realistic model has been proposed [6—12,136] in which the structure of pellets is explicitly considered to consist of pores and grains and the boundary is admitted to be diffusive due to some partly reduced grains, as shown in Fig. 4. Inevitably, the mathematics becomes very complicated and the matching with experimental results is not straightforward. To cope with this difficulty, Sohn and Szekely [11] employed dimensional analysis and introduced a dimensionless number, a, given by... 

There is no established relationship between variables affecting the above diffusion. However, if we assume that the slippage forces are negligible as compared to the pressures applied to the particulate mass, and if we confine the mass within an immovable boundary, by means of dimensional analysis we may arrive at a sort of possible diffusion relationship. Let the diffusion 3D be regarded as a function of some average particle diameter dy the apparent density of the mass pa and the pressure intensity py that is... 

Third, a serious need exists for a data base containing transport properties of complex fluids, analogous to thermodynamic data for nonideal molecular systems. Most measurements of viscosities, pressure drops, etc. have little value beyond the specific conditions of the experiment because of inadequate characterization at the microscopic level. In fact, for many polydisperse or multicomponent systems sufficient characterization is not presently possible. Hence, the effort probably should begin with model materials, akin to the measurement of viscometric functions [27] and diffusion coefficients [28] for polymers of precisely tailored molecular structure. Then correlations between the transport and thermodynamic properties and key microstructural parameters, e.g., size, shape, concentration, and characteristics of interactions, could be developed through enlightened dimensional analysis or asymptotic solutions. These data would facilitate systematic... 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

Engineers at Mobil Oil Corporation are satisfied that a one-dimensional analysis is suitable for treating reaction kinetics in these beds, simply using an appropriate Peclet number to represent the effective axial gas diffusivity (Avidan, 1982 Krambeck et al, 1987). Inputs for Mobil s analysis are two (1) the Peclet number expected for a commercial fluid bed in question—they estimate this to be 7 for beds they contemplate for carrying out Mobil s methanol-to-gasoline or methanol-to-olefin reactions—and (2) kinetic data from a pilot fluid bed, which can be expected to reflect, reasonably well, whatever top-to-bottom mixing of powder will occur in the commercial bed. 

Dimensional analysis of the coupled kinetic-transport equations shows that a Thiele modulus (4> ) and a Peclet number (Peo) completely characterize diffusion and convection effects, respectively, on reactive processes of a-olefins [Eqs. (8)-(14)]. The Thiele modulus [Eq. (15)] contains a term ( // ) that depends only on the properties of the diffusing molecule and a term ( -) that includes all relevant structural catalyst parameters. The first term introduces carbon number effects on selectivity, whereas the second introduces the effects of pellet size and pore structure and of metal dispersion and site density. The Peclet number accounts for the effects of bed residence time effects on secondary reactions of a-olefins and relates it to the corresponding contribution of pore residence time. 

The probability of readsorption increases as the intrinsic readsorption reactivity of a-olefins (k,) increases and as their effective residence time within catalyst pores and bed interstices increases. The Thiele modulus [Eq. (15)] contains a parameter that contains only structural properties of the support material ( <>, pellet radius Fp, pore radius 4>, porosity) and the density of Ru or Co sites (0m) on the support surface. A similar dimensional analysis of Eqs. (l9)-(24), which describe reactant transport during FT synthesis, shows that a similar structural parameter governs intrapellet concentration gradients of CO and H2 [Eq. (25)]. In this case, the first term in the Thiele modulus (i/>co) reflects the reactive and diffusive properties of CO and H2 and the second term ( ) accounts for the effect of catalyst structure on reactant transport limitations. Not surprisingly, this second term is... 

Figure 3.37. Computed velocity fields (m/s) for flows in adjacent interdigitated oxygen channels (with gas diffusion layer on the left side) of a PEM fuel cell (A inlet, C outlet, in x-y plane). In the middle (B), the flow from one gas channel through the gas diffusion layer to the adjacent gas channel is shown in the z-y plane, for the midpoint of the cell extension in the z-direction. The flows in A and C are for the midpoint value of z. (From S. Um and C. Wang (2004). Three-dimensional analysis of transport and electrochemical reactions in polymer electrolyte fuel cells. /. Power Sources 125, 40-51. Used with permission from Elsevier.)...

In all cases, the mass-transfer coefficient depends upon the diffusivity of the transferred material and the thickness of the effective film. The latter is largely determined by the Reynolds number of the moving fluid, that is, its average velocity, density, and viscosity, and some linear dimension of the system. Dimensional analysis gives the following relation ... 

The coefficients fco and kj, have been experimentally determined for many mass transfer systems and correlated with gas and liquid flow rates, liquid density and viscosity, the diffusivity of A in the gas and the liquid, and the physical dimensions of the systems. Dimensional analysis suggests that dimensionless quantities of the form... 

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