Zero-order reaction effectiveness factor

The Effectiveness Factor for a Straight Cylindrical Pore Second- and Zero-Order Reactions. This section indicates the predictions of the straight cylindrical pore model for isothermal reactions that are zero- and second-... 

Figure 12.7 adapted from Satterfield (40) contains a plot of the effectiveness factor for a zero-order reaction versus the Thiele modulus... 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first order, whereas a second-order reaction has an apparent order of 3/2 when strongly influenced by diffusional effects. Because k and n are modified in the diffusion controlled region then, if the rate of the overall process is estimated by multiplying the chemical reaction rate by the effectiveness factor (as in equation 3.8), it is imperative to know the true rate of chemical reaction uninfluenced by diffusion effects. 

In assessing whether a reactor is influenced by intraparticle mass transfer effects WeiSZ and Prater 24 developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been showneffectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when the generalised Thiele modulus falls below a value of one. Since tj is about unity when 0 < ll for zero-order reactions, a quite general criterion for diffusion control of simple isothermal reactions not affected by product inhibition is < 1. Since the Thiele modulus (see equation 3.19) contains the specific rate constant for chemical reaction, which is often unknown, a more useful criterion is obtained by substituting l v/CAm (for a first-order reaction) for k to give ... 

An enzyme which hydrolyzes the cellobiose to glucose, /3-glucosidase is immobilized in a sodium alginate gel sphere (2.5 mm in diameter). Assume that the zero-order reaction occurs at every point within the sphere with k0 = 0.0795 mol/sm3, and cellobiose moves through the sphere by molecular diffusion with Ds = 0.6 x 10 5 cm2 /s (cellobiose in gel). Calculate the effectiveness factor of the immobilized enzyme when the cellobiose concentration in bulk solution is 10 mol/m3. 

To obtain a more realistic estimation of the behavior of an autocatalytic reaction under adiabatic conditions, it is possible to identify the kinetic parameters of the Benito-Perez model from a set of isothermal DSC measurements. In the example shown in Figure 12.11, the effect of neglecting the induction time assumes a zero-order reaction leading to a factor of over 15 during the time to explosion. Since this factor strongly depends on the initial conversion or concentration of catalyst initially present in the reaction mass, this method must be applied with extreme care. The sample must be truly representative of the substance used at industrial scale. For this reason, the method should be only be applied by specialists. 

The value of the overall effectiveness factor for a zero order reaction can be calculated with the following relation [4] ... 

In Fig. 7 the effectiveness factor is shown as a function of the generalized Thiele modulus pn for different reaction orders (flat plate). From this figure, it is obvious that, except for the case of a zero order reaction, the curves agree quite well over the entire range of interest. The asymptotic solution t = l/ pn is valid for any reaction order and for values of the modulus pn > 3. 

For this case two extremes are possible (i) a - 0, which reduces to the case of 1st order reaction (case 1) and (ii) a e - 1.0, for this case the reaction reduces to zero order reaction. Figure 5.54 shows the effect of on the effectivene.ss factor (Elnashaie and Mahfouz,... 

The effectiveness factor curve for a zero-order reaction is included in Figure (4.8). The curve lies about 40% above that for a first-order reaction and has a slope of —1 at high values of Note that for high values of reaction rate varies with C, and the reaction appears to be 0.5 order. 

Effectiveness Factors for a Straight Cylindricai Pore Second- and Zero-Order Reactions... 

Figure 4.5.21 shows the effectiveness factor as a function of tpn for a slab (fiat plate) and different reaction orders n. It is again obvious that except for a zero-order reaction (n = 0), all curves are similar, and again the asymptotic solution >jpore = IM is reached for (p > 2. 

For zero order reactions one should take into account that r=k only if c> 0 otherwise r=0. In a situation where the concentration reaches a zero value inside the particle at a point x=x we should replace the boundary condition (Ic) by f=0 and df/dx=0 at x=x, The effectiveness factor is now simply the ratio of the "utilized" particle volume and the total particle volume,i.e., r]=l-x. In the kinetic controlled regime ri-1 and in the pure diffusional regime n-yT/((), Again for any shape we get ... 

Interestingly, at very low concentrations of micellised Qi(DS)2, the rate of the reaction of 5.1a with 5.2 was observed to be zero-order in 5.1 a and only depending on the concentration of Cu(DS)2 and 5.2. This is akin to the turn-over and saturation kinetics exhibited by enzymes. The acceleration relative to the reaction in organic media in the absence of catalyst, also approaches enzyme-like magnitudes compared to the process in acetonitrile (Chapter 2), Cu(DS)2 micelles accelerate the Diels-Alder reaction between 5.1a and 5.2 by a factor of 1.8710 . This extremely high catalytic efficiency shows how a combination of a beneficial aqueous solvent effect, Lewis-acid catalysis and micellar catalysis can lead to tremendous accelerations. 

Effectiveness Factors for Hougen-Watson Rate Expressions. The discussion thus far and the vast majority of the literature dealing with effectiveness factors for porous catalysts are based on the assumption of an integer-power reaction rate expression (i.e., zero-, first-, or second-order kinetics). In Chapter 6, however, we stressed the fact that heterogeneous catalytic reactions are more often characterized by more complex rate expressions of the Hougen-Watson type. Over a narrow range of... 

Method 1 By observation, [B] and [C] do not change [A] increases by a factor of 3. However, the reaction rate does not change. Therefore, changing [A] does not affect reaction rate and the reaction is zero order with respect to A. In all subsequent determinations, the effect of A can be ignored. 

To assess whether a reaction is influenced by intraparticle diffusion effects, Weisz and Prater [11] developed a criterion for isothermal reactions based upon the observation that the effectiveness factor approaches unity when the generalised Thiele modulus is of the order of unity. It has been shown that the effectiveness factor for all catalyst geometries and reaction orders (except zero order) tends to unity when... 

First-order reactions without internal mass transfer limitations A number of reactions carried out at high temperatures are potentially mass-transfer limited. The surface reaction is so fast that the global rate is limited by the transfer of the reactants from the bulk to the exterior surface of the catalyst. Moreover, the reactants do not have the chance to travel within catalyst particles due to the use of nonporous catalysts or veiy fast reaction on the exterior surface of catalyst pellets. Consider a first-order reaction A - B or a general reaction of the form a A - bB - products, which is of first order with respect to A. For the following analysis, a zero expansion factor and an effectiveness factor equal to 1 are considered. 

The actual reaction rate according to the distribution model with zero order is (4/3) r (R3-Rq)1c0. The rate without the diffusion limitation is (4/3)tt R3/c0. Therefore, the effectiveness factor, the ratio of the actual reaction rate to the rate if not slowed down by diffusion, is... 

The relationship between the concentration of reactants and the reaction rate is described by a factor known as the reaction order. In the previous example, the relationship between the reactants and the reaction rate was directly proportional, meaning that an increase in the concentration of one reactant caused proportionally the same increase in the rate. Doubling the concentration of a reactant doubled the rate of the reaction. This directly proportional relationship is known as a first-order relationship. If changing the concentration of a reactant had no effect on the reaction rate, the relationship would be described as a zero-order relationship. A second-order relationship is exponential in other words, doubling the concentration of a reactant will increase the rate by 4. The reaction order for a particular reactant is written as an exponent next to the concentration of that reactant. For instance, because the previous reaction was first order for substance A, we could represent this symbolically as [A] (the exponent 1 is understood). If A had a zero-order or second-order relationship, the symbols would be written [A]0 and [A]2, respectively. 

Transfer processes can be caused by monomer, counterion, and other components of the reaction mixture (additives, solvents, impurities). The latter reactions are sometimes called spontaneous because they are zero order in monomer. However, the spontaneous elimination of /3-protons is very unlikely, and proton elimination must be assisted by some basic reagent. The ratio of the rate constants of /8-proton elimination to that of electrophilic addition depends on several factors. The relative rate of transfer decreases with temperature, and therefore polymers with higher molecular weights are formed at sufficiently low temperatures. The effect of solvent and counterion is not yet sufficiently understood. 

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