External surface concentrations

Notice that in the region of fast chemical reaction, the effectiveness factor becomes inversely proportional to the modulus h2. Since h2 is proportional to the square root of the external surface concentration, these two fundamental relations require that for second-order kinetics, the fraction of the catalyst surface that is effective will increase as one moves downstream in an isothermal packed bed reactor. 

When a solid acts as a catalyst for a reaction, reactant molecules are converted into product molecules at the fluid-solid interface. To use the catalyst efficiently, we must ensure that fresh reactant molecules are supplied and product molecules removed continuously. Otherwise, chemical equilibrium would be established in the fluid adjacent to the surface, and the desired reaction would proceed no further. Ordinarily, supply and removal of the species in question depend on two physical rate processes in series. These processes involve mass transfer between the bulk fluid and the external surface of the catalyst and transport from the external surface to the internal surfaces of the solid. The concept of effectiveness factors developed in Section 12.3 permits one to average the reaction rate over the pore structure to obtain an expression for the rate in terms of the reactant concentrations and temperatures prevailing at the exterior surface of the catalyst. In some instances, the external surface concentrations do not differ appreciably from those prevailing in the bulk fluid. In other cases, a significant concentration difference arises as a consequence of physical limitations on the rate at which reactant molecules can be transported from the bulk fluid to the exterior surface of the catalyst particle. Here, we discuss... 

Let us now consider how the external surface concentrations can be eliminated when our reaction follows simple irreversible first-order kinetics. In this instance equation 12.4.20 becomes... 

In the limit where the external surface concentration becomes very small compared to the bulk fluid concentration, we obtain the maximum temperature difference. 

For isothermal systems, it is occasionally possible to eliminate the external surface concentrations between equations 12.6.1 and 12.6.2 and arrive at a global rate expression involving only bulk fluid compositions (e.g., equation 12.4.28 was derived in this manlier). In general, however, closed form solutions cannot be achieved and an iterative trial and error procedure must be employed to determine thq global rate. One possible approach is summarized below. 

This ratio is sufficiently large that we may call it unity for purposes of determining the required reactor size. The errors associated with assuming that for the cases of interest there will not be significant differences between bulk fluid and external surface concentrations will be small compared to other sources of error, particularly the errors inherent in the use of a one-dimensional model. 

The effectiveness factor is a function of a dimensionless group termed the Thiele modulus, which depends on the diffusivity in the pore, the rate constant for reaction, pore dimension, and external surface concentration Cs-... 

Cs = external surface concentration, in moles per cubic centimeter m = reaction order (equal to 1 for first-order)... 

The initial concentration of pollutant A The concentration of pollutant at the reactor s inlet The concentration of pollutant at the reactor s outlet The mean concentration at the external surface Concentration at the surface of the photocatalyst The diffusion coefficient of the pollutant in air True rate constant in LH expression (mol min )... 

Hence the effectiveness factor is the ratio of the actual rate to that if the reactions were to occur at the external surface concentration, i.e., in absence of intraparticle diffusion resistance ... 

Regime Code Concentration on the external surface Concentration in the zeolite pwre Mobility in the zeolite pore... 

The definition of equation (7-1) does not envision differences between bulk and external surface concentrations, a point that will be discussed later. We will first treat the problem of transport limitations within the porous matrix (intraphase), then the combination of boundary-layer (interphase) transport with the intraphase effects. ... 

Since the reactant concentrations along the pores and within the particles are lower than the external surface concentrations, the overall effect of internal mass transfer resistances is to reduce the actually observed global rate below that measured at exterior surface conditions. It can be stated for isothermal effectiveness factors that r]concentration profile showing the pore diffusion-affected surface reaction is labeled as II in Figure 2.3. 

From a qualitative analysis of the competition between reaction and diffusion in an isothermal pellet one easily recognizes that the parameter governing the steady state behavior of the pellet is the ratio between time constants for diffusion and reaction,i.e.,T(j/xr-If Tr the reaction rate is much slower than the diffusion rate the concentration profile inside the pellet is then almost flat and equal to the external surface concentration.The effectiveness factor is around unity. However,when the concentration inside the pe-... 

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