Effective diffusivities calculated

The average effective diffusivity calculated for the 44 compositions is 1.7 x 10"13 cm2/sec and the loss of tin 2.2%. These are in qualitative agreement with the results of laboratory testing. Cardarelli has also reported that antifouling rubber retains a considerable amount of organotin additives even after complete fouling (11). 

FIGURE 6.6. Sorption of li l hydrocarbons in 5A molecular sieve pellets (Davison G52I) under conditions of macropore diffusion control. (uptake curves for CjH at 323 K (6) concentration dependence of effective diffusivities calculated from the experirhental uptake curves (see Eq. (6.16)] (c) pressure dependence of pore diffusivities calculated from effective diffusivities D, Z> /(1 + (1 - t ) dq /dc)/ ]. (From ref. 9 reprinted from Canadian Journal of Chemical Engineering.)... 

These expressions provide a simple and convenient model for the analysis of experimental uptake curves when the equilibrium isotherm is highly favorable and micropore diffusion is rapid. These conditions are amply fulfilled for the adsorption of water at or near ambient temperature in molecular sieve adsorbents. Experimental uptake curves for this system measured by Kyte are shown in Figure 6.9. The experimental conditions and the effective diffusivity calculated according to Eq. (6.27) are giveii in Tables 6.4 and 6.5. Under the experimental conditions the estimated value of the Knudsen diffusivity [from Eq. (5.17)] is much larger than the molecular diffusivity... 

At the limit of Knudsen diffusion control it is not reasonable to expect that any of the proposed approximation methods will perform well since, as we know, percentage variations in pressure are quite large. Nevertheless it is interesting to examine their results, which are shown in Figure 11 4 At this limit it is easy to check algebraically that equations (11.54) and (11.55) become the same, while (11.60) differs from the other two. Correspondingly the values of the effectiveness factor calculated using the approximation of Kehoe and Aris coincide with the results of Apecetche et al., and with the exact solution, ile Hite and Jackson s effectiveness factors differ substantially. 

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

Principles of Rigorous Absorber Design Danckwerts and Alper [Trans. Tn.st. Chem. Eng., 53, 34 (1975)] have shown that when adequate data are available for the Idnetic-reaciion-rate coefficients, the mass-transfer coefficients fcc and /c , the effective interfacial area per unit volume a, the physical solubility or Henry s-law constants, and the effective diffusivities of the various reactants, then the design of a packed tower can be calculated from first principles with considerable precision. 

Sorption curves obtained at activity and temperature conditions which have been experienced to be not able to alter the polymer morphology during the test, i.e. a = 0.60 and T = 75 °C, for as cast (A) and for samples previously equilibrated in more severe conditions, a = 0.99 and T = 75 °C (B), are shown in Fig. 13. According to the previous discussion, the diffusion coefficient, calculated by using the time at the intersection points between the initial linear behaviour and the equilibrium asymptote (a and b), for the damaged sample is lower than that of the undamaged one, since b > a. The morphological modification which increases the apparent solubility lowers, in fact, the effective diffusion coefficient. 

A method of calculating the effective diffusivity /> in terms of each of the binary diffu-... 

A hydrocarbon is cracked using a silica-alumina catalyst in the form of spherical pellets of mean diameter 2.0 mm. When the reactant concentration is 0.011 kmol/m3, the reaction rate is 8.2 x 10"2 kmol/(m3 catalyst) s. If the reaction is of first-order and the effective diffusivity De is 7.5 x 10 s m2/s, calculate the value of the effectiveness factor r). It may be assumed that the effect of mass transfer resistance in the. fluid external Lo the particles may be neglected. 

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). 

If pore diffusion is controlUng, we repeat the effectiveness factor calculations in Chapter 10. Equation (10.29) has the form of Equation (11.48), and it includes both film resistance and pore diffusion. 

FIG. 21 Effective diffusion coefficients from Refs. 337 and 193 showing comparison of volume average results (Ryan) with models of Maxwell, Weisberg, Wakao, and Smith for isotropic systems (a), and volume averaging calculations (solid lines) and comparison with data for anisotropic systems (b). (Reproduced with kind permission of Kluwer Academic Publishers from Ref. 193, Fig. 3 and 12, Copyright Kluwer Academic Publishers.)... 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

For the purposes of this illustrative example, we wish to calculate the combined and effective diffusivities of cumene in a mixture of benzene and cumene at 1 atm total pressure and 510 °C within the pores of a typical TCC (Thermofor Catalytic Cracking) catalyst bead. For our present purposes, the approximation to the combined diffusivity given by equation 12.2.8 will be sufficient because we will see that the Knudsen diffusion term is the dominant factor in determining the combined diffusivity. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

Transport Properties. Because the feed is primarily air and because substantial amounts of N2 and 02 are present in the effluent stream, we will assume that the fluid viscosity is that of air for purposes of pressure drop calculations. For the temperature range of interest, the fluid viscosity may be taken as equal to 320 micropoise. The pressure range of interest does not extend to levels where variations of viscosity with pressure need be considered. The effective diffusivities of naphthalene and phthalic anhydride in the catalyst pellet may be evaluated using the techniques developed in Section 12.2. 

In combination with DFT calculations, the time- and depth-dependent phonon frequency allows to estimate the effective diffusion rate of 2.3 cm2 s 1 and the electron-hole thermalization time of 260 fs for highly excited carriers. A recent experiment by the same group looked at the (101) and (112) diffractions in search of the coherent Eg phonons. They observed a periodic modulation at 1.3 THz, which was much slower than that expected for the Eg mode, and attributed the oscillation to the squeezed phonon states [9]. 

For reasons of simplicity, the Thiele modulus will be defined and calculated for a catalyst plate with pore access at both ends of the plate and not at the bottom or top. Note that for most cases in real-life applications the assumptions have to be modified using polar coordinates for the calculations. The Thiele modulus q> is therefore defined as the product of the length of the catalyst pore, /, and the square root of the quotient of the constant of the speed of the reaction, k. divided by the effective diffusion coefficient DeS ... 

The activity in terms of 1st order rate constant khcalc was calculated in Table 2 from (8) and (9) with effective diffusivity Dejf=5.3-10 6 m2/s and intrinsic rate constant =33000 Nm3/h/m3 = 23 s"1 fitted to the measurements. This simple and useful method models the measured influence of particle size satisfactorily for a first optimization of particle size and shape. The 35% higher activity measured for the 9-mm Daisy compared to the 12-mm Daisy, however, exceeds the 25% expected from (8), and this illustrates the importance of measuring the activity of the actual shape. 

Miller has shown that TBTO will prevent fouling attachment at leaching rates as low as 1.25 yg/cm2/day (18). It is thus reasonable to assume that fouling commences when the rate of release falls below 0.5 yg Sn/cm2/day. Based on this, the effective dif-fusivities are calculated, using Crank s rate equation. The calculated effective diffusivities are then substituted in the integral form of Crank s equation to estimate the amount of Sn lost. 

Carlos and Latif both fluidised glass particles in dimethyl phthalate. Data on the movement of the tracer particle, in the form of spatial co-ordinates as a function of time, were used as direct input to a computer programmed to calculate vertical, radial, tangential and radial velocities of the particle as a function of location. When plotted as a histogram, the total velocity distribution was found to be of the same form as that predicted by the kinetic theory for the molecules in a gas. A typical result is shown in Figure 6.11(41 Effective diffusion or mixing coefficients for the particles were then calculated from the product of the mean velocity and mean free path of the particles, using the simple kinetic theory. 

To calculate the release through diffusion of an entrapped residue, Barraclough et al. (2005) considered the size of organic matter particles (effective radius 10" to 10 cm) and the effective diffusion coefficient of small organic molecules in a sorbing medium (D 10 cm s )- The time for 50% of the material in a sphere to diffuse out is given by... 

The solute concentration Wg from each run has been expressed as a step function of the distance z, from which (-Wg/Wo) is calculated. The values of all the experimental parameters, D/uL, Dg/uL, Lu, /LD, Cpyti/k and (h f g/yM. ) ( AT) etc. are calculated from the values of the related physical properties in the literature (19). In Figure 3, g is correlated with z/L for all the thirteen runs, for which the values of effective diffusivity Dg in the melted zone have been predicted from the Kraussold correlation (7) using the experimental values of PrGr number and the values of the parameter Pe calculated show that little improvement has been made by using Pe instead of P in the correlation. 

Efremov, G. and Kudra, T., Calculation of the effective diffusion coefficients by applying a quasi-stationary equation for drying kinetics. Drying Tech., 22 (2004) 2273-2279. 

The critical input parameters are then (1) the grain size, which should be known for each case, (2) the Aci temperature which is calculated from thermodynamics, (3) the effective diffusion activation energy, Qea, and (4) the empirical constants aj for each element. Qea and aj were determined by empirically fitting curves derived using Eq. (11.12) to experimentally observed TTT curves, and the final formula for calculating r was given as... 

These calculations were done using the typical thickness of a washcoated SCT substrate ( 7 micron) and the effective diffusivity based on the diffusion coefficient (4.148 x 10 m s ) calculated for a real gas matrix and taking representative values of the tortuosity (3.0), porosity (0.4) and constriction (0.8) factors. 

---