Effectiveness factor derivation

The effectiveness factor can be written as a function of a dimensionless independent variable called the Thiele modulus, which for a first-order reaction is defined below together with the corresponding effectiveness factor derived by integration of the corresponding diffusion-reaction equation (7-97) tiveness factor is ti = - (7-104) 0... 

These equations were nondimensionalized in terms of physically relevant parameters like the Thiele modulus and the Biot number for mass transfer, and an expression for overall effectiveness factor derived. An apparent rate constant for the organic reaction was derived as a function of both internal and external mass-transfer resistances... 

The above procedure produces blast parameters applicable to a completely symmetrical blast wave, such as would result from the explosion of a hemispherical vessel placed directly on the ground. In practice, vessels are either spherical or cylindrical, and placed at some height above the ground. This influences blast parameters. To adjust for these geometry effects, and 7 are multiplied by some adjustment factors derived from experiments with high-explosive charges of various shapes. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

The subscript refers to a spherical particle. One should also remember that we limited ourselves to a first-order irreversible reaction. Other expressions can be derived but are beyond the scope of this book. Nevertheless, Eq. (35) has important practical implications, since it is possible to discuss the effectiveness of the system by a single dimensionless parameter, (fig. Figure 5.35 shows the effectiveness factor as a function of O,. 

The quantitative measurement of toxicity level is expressed by parameters like NOEL (no observed effect level), NOAEL (no observed adverse effect level), and ADI (acceptable daily intake). The NOEL values are divided by 100 to obtain ADI values. The 100 safety factor derives from 10 x 10, where the 10s represent the animal-to-human conversion rate and the human variability factor. Currently, the most useful index of safety is the ADI, expressed as milligrams of test substance per kilogram of body weight (ppm), with the recommendation not to eat more than the ADI per day. The FDA, EU, and WHO agree on the ADI principle. 

Given this relationship between CA(t) and Co(t), where retention is factored in, we can proceed to convert Eq. (7.3) into Eq. (7.5), where a is the same as before, and b now needs to be multiplied by the partition-related factor, 1 — R. The so-modified ordinary differential, Eq. (7.5), is solved by standard methods, using integration limits from xLAG to t (not 0 to t), and the desired effective permeability derived as... 

To evaluate the effectiveness factor we require only the derivative of the concentration at the pore mouth. This parameter may be obtained by multiplying both sides of equation 12.3.31 by d(C/C0)/d(x/L) ... 

The expression under the radical gives the multiplying factor for the noise standard deviation for the computed derivative (or smoothed spectrum, but that is not our topic here, we will address only the question of the effect on derivatives), and can be computed solely from the convolution coefficients themselves, independently of the effect of the convolution on the signal part of the S/N ratio. 

The data derived from modeling at different conversion degrees (X = 5, 40, and 80%) were also compared to the results obtained from the calculation of the classical Thiele modulus. The calculated (by the Thiele modulus) and modeled (by Presto Kinetics) effectiveness factors showed comparable values. Hence, the usage of simulation software is not required to get a first impression of the diffusion limitations in a Fischer-Tropsch catalyst pore. Nevertheless, modeling represents a valuable tool to better understand conditions within a catalyst pore. 

Derive an expression for the catalyst effectiveness factor (17) for a spherical catalyst particle of... 

The heritabilities found in this last study are smaller than in the previous studies. The reason for this is simple—unreliability of measurement. Table 4 shows the MZA correlations for six common higher-order factors that can be estimated in two of the three studies and comparable factors derived from two instruments. As expected the correlations increase with the psychometric quality of the instruments. We have demonstrated the same effect in the twin registry sample (Lykken, Bouchard, McGue, Tellegen, 1993). 

The analysis below is given for the ORR, since the agglomerate and embedded models mainly examine the cathode reaction at the anode can be derived in a similar manner. The analysis is basically the same as that of reaction and diffusion in a catalyst pellet. For the analysis, an effectiveness factor is used, which allows for the actual rate of reaction to be written as (see eq 55)... 

Derive an expression for the effectiveness factor of a porous catalyst slab of thickness 2L that has an effective pore diameter = 4e/pcS with a first-order reaction r" = VCa- Can you justity 4 = 4ep Sl... 

The Canadian provinces (with the exception of Quebec) have now pooled resources to create a common drug review (through the independent CEDAC operating under the not-for-profit CCOHTA). The review factors transparent (public) cost-effectiveness estimates, derived from systematic reviews, back to provincial cost-effectiveness... 

Extensive studies on hydrogen exchange in aqueous sulfuric acid on 1 -methylpyrazole,252 isoxazole,328 and isothiazole328 and their methylated derivatives252,328 have been reported, but calculated partial rate factors derived from treatment of the form of Eq. (12) do not tally with those observed. This emphasizes the contribution from interactions between donor and acceptor groups or solvent effects which also lead to breakdown in the interpretation of nitration results.83... 

The unexpected overlap of this problem is more surprising than significant, for the first derivative of the effectiveness factor with respect to the Thiele modulus (Figs. 8 and 9) does not stand out as of particular interest. [Note added in proof (Jan. 99). I could not have been more wrong. See p. 37, fn. 11 and reference then cited.]... 

Derive an expression for the effectiveness factor of a spherical catalyst pellet in which a first-order isothermal reaction occurs. 

For spherical particle geometry, as in the case of a microbial floe, a pellet of mould or a bead of gel-entrapped enzyme, the expression for the effectiveness factor can again be derived by a procedure similar to that used in Chapter 3 for a spherical pellet of conventional catalyst. A material balance for the substrate across an elementary shell of radius r and thickness dr within the pellet will yield ... 

Derive the material and energy-balance design equations, including the equations of the catalyst pellets to calculate the effectiveness factor rj. 

Equations (14-37) and (14-42) represent two different ways of obtaining an effective factor, and a value of Ae obtained by taking the reciprocal of S, from Eq. (14-42) will not check exactly with a value of A, derived by substituting At = 1/Si and A2 = 1/S2 into Eq. (14-37). Regardless of this fact, the equations generally give reasonable results for approximate design calculations. 

To derive an equation for determining the overall effectiveness factor, we first introduce a Thiele modulus which is related to the unknown surface temperature Ts ... 

This type of reaction has been investigated by Smith and Amundsen [95] and Carberry [14]. Without derivation it may be stated that under isothermal conditions the same solution for the effectiveness factor is obtained as in the case of an irreversible reaction if a modified Thiele modulus rev is introduced [91] ... 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

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