Bulk flow reactant concentration

We now want to apply the component continuity equation for reactant A to a small differential slice of width dz, as shown in Fig. 2.4. The inflow terms can be split into two types bulk flow and diffusion. Diffusion can occur because of the concentration gradient in the axial direction. It is usually much less important than bulk flow in most practical systems, but we include it here to see what it contributes to the model. We will say that the diffusive flux of A, (moles of A per unit time per unit area), is given by a Pick s law type of relationship... 

Modeling considerations for monoliths are similar to those of gauze catalysts however, since the flow and temperature in each channel may be assumed to be identical to those in the next channel the solution for a single channel may reflect the performance of the reactor. For an application in which the reaction rate is mass-transfer-limited, the reactant concentration at the wall of the catalyst is much lower than in the bulk and may be neglected. In such a case, the fractional conversion % is... 

In laminar gas flow, the reactant concentrations and gas velocity are zero at the substrate surface and increase to the bulk concentration and bulk stream velocity at some distance, 6, from the substrate surface and is given by, ... 

Here t is time, x the coordinate directed along the catalytic element axis, T, To the temperature of the catalyst and the reaction medium, C, Cq the reactant concentrations at the catalyst surface and in the bulk-flow, m, k the heat capacity per unit volume and thermal conductivity of the catalyst, D the reactant diffusion coefficient for the gaseous phase, k = koexp — EIRT) the Arrhenius rate constant for a first-order reaction, q the thermal effect of the reaction, 3 the characteristic size of the effective film, " a and b numerical coefficients of the order of unity related to the element geometry (for simplicity, let a = f> = 1). 

The limiting current density is an important parameter for the analysis of mass transfer controlled electrochemical processes and represents the maximum possible reaction rate for a given bulk reactant concentration and fluid flow pattern. During anodic metal dissolution, a mass transfer limiting current does not exist because the surface concentration of the dissolving ion (e.g., Cu + when the anode is composed of copper metal) increases with increasing current density, eventually leading to salt precipitation that blocks the electrode surface. 

Identifying the limiting current density (ij) during electrodeposition is important since it represents the maximum rate of metal plating for a given bulk reactant concentration and hydrodynamic flow pattern. Also, the stracture of the electrodeposit varies with current density see Figure 26.23 [92]. 

In the previous section the following assumptions have been made (1) Perfect mixing of the catalyst and adsorbent particles, (2) The adsorption isotherm is of the Langmuir type, (3) Mass transfer resistance is restricted to the external fllm, (4) Reactions are first order in reactants concentrations, (5) The intermediate product adsorbs exclusively, (6) Isothermal operation. Supplementary to the choices described above, the following assumptions are made (7) Ideal gas law is obeyed, (8) Axially dispersed plug flow in the bed, (9) Negligible pressure drop over the bed, (10) Bed of constant voidage, bulk density... 

As illustrated in Sections 30-1 and 30-2, all intrapellet resistances can be expressed in terms of f-A, surface a, intrapeiiet and Ea mtrapeiiet approaches zero near the central core of the catalyst when the intrapellet Damkohler number is very large. For small values of the intrapellet Damkohler number, effectiveness factor calculations within an isolated pellet allow one to predict Ca, intrapeUet in terms of CA,sur ce via the dimensionless molar density profile. All external transport resistances can be expressed in terms of Ca, buit gas — Ca, surface, and integration of the plug-flow mass balance allows one to calculate the bulk gas-phase concentration of reactant A. The critical step involves determination of Ca, surface via effectiveness factor formalism. Finally, a complete reactor design strategy is... 

In a continuous flow system, reactions are performed at steady state, which makes it possible to achieve better control and reproducibility. Furthermore, the ability to manipulate reactant concentrations in both space and time also provides a high level of reaction control than that of bulk stirred reactors. The spatial and temporal controls of chemical reactions in microfluidic devices are useful to control and alter chemical reactivity according to the prefiminary design. And usually multistep synthesis can produce particles with fairly complex shapes and functionalities. However, the coalescence between droplets, the stability of flows after several times of mixing, and the controllability of the fluid by multistep stiU remain to be improved. 

If the mass transfer is accompanied by a chemical reaction at the catalyst surface on the reactor wall, the mass transfer depends on the reaction kinetics [55]. For a zero-order reaction, the rate is independent of the concentration and the mass flow from the bulk to the wall is constant, whereas the reactant concentration at the catalytic wall varies along the reactor length. For this situation the asymptotic Sh in circular tube reactors becomes Sh. = 4.36 [55]. The same value is obtained when reaction rates are low compared to the rate of mass transfer. If the reaction rate is high (very fast reactions), the concentration at the reactor wall can be approximated to zero within the whole reactor and the asymptotic value for Sh is = 3.66. As a consequence, the Sh in the reacting system depends on the ratio of the reaction rate to the rate of mass transfer characterized by the second Damkohler number defined in Equation 6.11. 

E.g. in the so-called "pseudo-equilibrium model, developed by Sylvester [53-56], the same design procedure is used as in a single phase catalytic gas phase reaction, where the mass transfer resistance is replaced by a suitable overall term. Bulk flow and dispersion of the liquid phase are neglected and the whole transport mechanisms are lumped into the equilibrium of the reactant concentrations between gas-, liquid- and particle phase. It is an application of the same principle used successfully in fluid/fluid reactions [57], But the necessary precondition is that the rate of reaction is slow compared to the transfer rate across the phase boundaries, so that equilibrium can really by assured. This might be justified in some of the hydrotreating processes, but certainly not in case of an aqueous liquid phase, existing in waste water treating. Earlier models used in petroleum industry have taken in-... 

Note that the term nFCi/si is common to all flow geometries. That is, the limiting current is proportional to bulk reactant concentration. The limiting current is increased upon increases in diffusivity, flow velocity, and rotational speed of rotating electrodes. Likewise, it is decreased upon... 

From Equation 6.131a for the limiting current density with mass transfer limited by diffusion resistance, the reactant concentration in the bulk gas flow in the channels is written as... 

Equation 6.141 represents the activation overpotential on the basis of the reactant gas concentration in bulk gas flow. The mass transfer loss can be estimated on the basis of the changes in activation overpotential owing to the variation in reactant gas concentration from the bulk flow to the reaction surface as follows ... 

The performance of enzyme electrodes— particularly those with three-dimensional porous architecture—is affected by both enzyme kinetics and mass transfer to, and through, the electrode architecture. Understanding these mechanisms requires an approach that models the reactant concentration at the enzyme surface based on bulk concentrations, which is measured in situ by online spectrophotometric detection or ex situ by liquid chromatography. In this approach, the mass transfer characteristics of the electrodes are estimated by comparison of experimental data with model predictions of system performance in the presence of continuous flow. Thereafter, a combined mass transfer parameter can be calculated and enzyme kinetics can be evaluated. The primary advantages of this approach are to rapidly screen immobilization protocols and to provide some insight into mass transport of reactants, metabolic by-products, and electrons. Specifically, it is possible to estimate the actual Umax of tho electrode, and hence quantify the amount of active enzyme available for bioelectrocatalysis. 

The reader will recall from Eqs. (2.2.1) and (2.2.2) that the mass flux from a fluid to a solid surface (or vice versa) is given by the product of the mass transfer coefficient and the concentration driving force. For systems involving single particles this driving force is well defined as the difference between the concentration in the bulk of the gas and at the surface of the solid (or some suitable modification thereof in which account is taken of bulk flow effects). At any point within a packed bed system the correct value for the concentration driving force is the difference between the reactant concentration in the void space at that particular point in the bed and the concentration at the surface of the solid particle adjacent to this fluid element. As noted earlier, this reactant concentration in the fluid is usually not known a priori but has to be calculated with the aid of subsidiary equations. 

Provided that the bulk concentrations remain constant, the flux J of Eq. (9-3) is also constant. Consider a spherical distribution of potential reactants B around a particular molecule of A. The surface area of a sphere at a distance r from A is 4irr2. Thus, the expression for the flow of B toward A is... 

Gas-liquid reactors. Gas-liquid reactors are quite common. Gas-phase components will normally have a small molar mass. Consider the interface between a gas and a liquid that is assumed to have a flow pattern giving a stagnant film in the liquid and the gas on each side of the interface, as illustrated in Figure 7.2. The bulk of the gas and the liquid are assumed to have a uniform concentration. It will be assumed here that Reactant A must transfer from the gas to the liquid for the reaction to occur. There is diffusional resistance in the gas film and the liquid film. 

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