Effectiveness factor, particle

Intraparticle Mass Transfer. One way biofilm growth alters bioreactor performance is by changing the effectiveness factor, defined as the actual substrate conversion divided by the maximum possible conversion in the volume occupied by the particle without mass transfer limitation. An optimal biofilm thickness exists for a given particle, above or below which the particle effectiveness factor and reactor productivity decrease. As the particle size increases, the maximum effectiveness factor possible decreases (Andrews and Przezdziecki, 1986). If sufficient kinetic and physical data are available, the optimal biofilm thickness for optimal effectiveness can be determined through various models for a given particle size (Andrews, 1988 Ruggeri et al., 1994), and biofilm erosion can be controlled to maintain this thickness. The determination of the effectiveness factor for various sized particles with changing biofilm thickness is well-described in the literature (Fan, 1989 Andrews, 1988)... 

For this purpose, we introduce the particle effectiveness factor tj, the ratio of the observed rate of reaction for the particle as a whole to the intrinsic rate at the surface conditions, cAj and Ts. In terms of a reactant A,... 

We consider the effects of cA and Tseparately, deferring the latter to Section 8.5.5. In focusing on the particle effectiveness factor, we also ignore the effect of any difference in concentration between bulk gas and exterior surface (cAg and cAs.) in Section 8.5.6, we introduce the overall effectiveness factor to take this into account. 

For a flat-plate porous particle of diffusion-path length L (and infinite extent in other directions), and with only one face permeable to diffusing reactant gas A, obtain an expression for tj, the particle effectiveness factor defined by equation 8.5-5, based on the following... 

The definition of the particle effectiveness factor 77 involves the intrinsic rate of reaction, ( rA)int> for reaction A - products, at the exterior surface conditions of gas-phase concentration (cAs) and temperature (Ts). Thus, from equation 8.55,... 

The particle effectiveness factor 17 defined by equation 8.5-5 takes into account concentration and temperature gradients within the particle, but neglects any gradients from bulk fluid to the exterior surface of the particle. The overall effectiveness factor y)0 takes both into account, and is defined by reference to bulk gas conditions (cA, Tg) rather than conditions at the exterior of the particle (cAj, Ts) ... 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

Al-Dahhan and Dudukovic, 1996 Dudukovic et al., 1999). This way, more solid-liquid contact points over which the liquid flows are created and the bed porosity is reduced, especially near the reactor wall. Following a proper procedure for packing a trickle bed with catalyst particles and fines decouples the apparent kinetics from hydrodynamics, which is highly desirable. The addition of lines is not the same as reducing the particle size of the catalyst, as in the latter case the particle effectiveness factor is smaller. 

The apparent reaction rate ra at the level of one pore results from the exchange of mass between the liquid flow and the porous structure of the catalyst particle as depicted in the close-up of Figure 3. In the absence of external mass transfer limitations, ra equals the product of the intrinsic reaction rate r0 and the particle effectiveness factor rip, the variables being expressed... 

The approximative methods to calculate a catalyst particle effectiveness factor are always based on the assumption that the reactant concentration at the particle surface has one and the same value over the complete surface (see Fig. 17). Therefore, the use of these methods for predicting the BSR performance is—strictly spoken—allowed only if there is no external mass transfer limitation. The BSR can then be described with only one ODE ... 

The model that is formed by Eqs. (24) and (25), with the extra assumption that wall effects can be neglected (i.e., that the reactor is built up solely of central subchannel elementary cells such as the one depicted in Fig. 10), is called the catalyst bead (CB) model. Because of the assumption in the CB model that the value of the mass transfer coefficient at the flat ends of the particle equals the value at the cylindrical surface, it can be expected that this model overestimates the conversion obtained in a BSR, especially when the particle length-over-diameter ratio is much smaller than 0.5 and when the gap between consecutive particles on a string is small compared to the particle diameter. However, whether the CB model over- or underestimates the BSR performance also depends on the error made in the calculation of the particle effectiveness factor. 

In the present study the particle effectiveness factor was estimated with the Aris method presented by Ref. 18. In the estimation, the particles were modeled as normal cylindrical catalyst particles, i.e., finite cylinders with no inert kernel. This was necessary because approximative methods to estimate an effectiveness factor cannot account for an inert kernel in a catalyst particle such as the metal wire in the axial hole of the BSR beads. The cylinder diameter used in the calculations was defined in Fig. 18 (the... 

The influence of the pore diffusion on the effective rate of coke combustion is described by the particle effectiveness factor t and the Thiele-module O (see nomenclature), respectively ... 

Using the first-order kinetic equation, it is possible to apply well-known expression for the calculation of particle effectiveness factor q [20, 23]... 

The variables under < > are bed scale averaged. Pp r is the intrinsic reaction rate including the particle effectiveness factor, a and 6 are empirical factors. Henry et al. and Hears suggested a value of about 1/3 for 6. 

The effectiveness factor r] of the catalyst pellets, expressing the fraction of the intrinsic rate of reaction being exploited in the fixed-bed configuration, is extremely small (rj = 10 -10 ). This is due to the large catalyst pellet sizes used to avoid excessive pressure drop along the length of the reactor (18-20m). This severe limitation can be broken by using a fluidized-bed reactor with fine catalyst particles (effectiveness factor = 1.0) thus the full intrinsic activity of the catalyst is utilized. 

Catalyst particle effectiveness factor E as a function of the Thiele modulus (p for a flat plate and isothermal first-order reaction. 

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