Stagnant layer solution

The simple but representative chemical reaction is considered with one product, P, as 

The inclusion of more products only makes more details for us with no change in the basic content of the results. 

Let us examine the integration of the governing differential equations. From Equation (9.13), 

Since at y = 0, the mass flux pv is the mass loss rate of burning rate evolved from the condensed phase. From Equation (9.18) we also realize that we have a constant pressure process 

We have four equations but five unknowns. Although a constant, in this steady state case, nip is not known. We need to specify two boundary conditions for each variable. This is done by the conditions at the wall (y = 0) and in the free stream of the enviornment outside of the boundary layer (y = S). Usually the environment conditions are known. At y = 6, 

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Equation (9.41) constitutes a fundamental solution for purely convective mass burning flux in a stagnant layer. Sorting through the S-Z transformation will allow us to obtain specific stagnant layer solutions for T and Yr However, the introduction of a new variable - the mixture fraction - will allow us to express these profiles in mixture fraction space where they are universal. They only require a spatial and temporal determination of the mixture fraction/. The mixture fraction is defined as the mass fraction of original fuel atoms. It is as if the fuel atoms are all painted red in their evolved state, and as they are transported and chemically recombined, we track their mass relative to the gas phase mixture mass. Since these fuel atoms cannot be destroyed, the governing equation for their mass conservation must be... 

This crude approximation allows us to extend the stagnant layer solution to a host of convective heat transfer counterpart burning problems. Recall that for Equations (9.41) and (9.42), we can write... 

In other words, we replace Ahc by (1 — X, — XWjf)Ahc and L by Lrn everywhere in our stagnant layer solutions. The solution for flame temperature becomes, from Equation (9.56),... 

In the stagnant layer solution, an accounting for the water evaporated in the flame and the radiation loss would have modified the energy equation as (Equation (9.23)). 

The term in the bracket can be regarded as an equivalent heat of combustion for the more complete problem. If this effect is followed through in the stagnant layer solution of the ordinary differential equations with the more complete boundary condition given by... 

A situation which is frequently encountered in tire production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down tire cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at tire mouth of tire cavity, but since this is rare in vapour deposition processes, the assumption that the gas widrin dre cavity is stagnant is a good approximation. The appropriate solution of dre diffusion equation for the steady-state transport of material tlrrough the stagnant layer in dre cavity is... 

A typical example is the protonation of tetraphenylporphirin (TPP) at the dodecane-acid solution interface. The interfacial protonation rate was measured by a two-phase stop flow method [6] and a CLM method [9]. In the former method, the stagnant layer of 1.4 jxm still existed under the highly dispersed system. In the CLM method, the liquid membrane phase of 50-100 /am thickness behaved as a stagnant layer where the TPP molecule has to migrate according to its self-diffusion rate. 

Furthermore, we will take all other properties as constant and independent of temperature. Due to the high temperatures expected, these assumptions will not lead to accurate quantitative results unless we ultimately make some adjustments later. However, the solution to this stagnant layer with only pure conduction diffusion will display the correct features of a diffusion flame. Aspects of the solution can be taken as a guide and to give insight into the dynamics and interaction of fluid transport and combustion, even in complex turbulent unsteady flows. Incidentally, the conservation of momentum is implicitly used in the stagnant layer model since ... 

The introduction of Equation (9.71) for Equation (9.26e) makes this a new problem identical to what was done for the pure diffusion/convective modeling of the burning rate. Hence L is simply replaced by L,n to obtain the solution with radiative effects. Some rearranging of the stagnant layer case can be very illustrative. From Equations (9.61) and (9.42) we can write... 

This chapter began by discussing the steady burning of liquids and then extended that theory to more complex conditions. As an alternative approach to the stagnant layer model, we can consider the more complex case from the start. The physical and chemical phenomena are delineated in macroscopic terms, and represented in detailed, but relatively simple, mathematics - mathematics that can yield algebraic solutions for the more general problem. 

First, we must consider a gas-liquid system separated by an interface. When the thermodynamic equilibrium concentration is not reached for a transferable solute A in the gas phase, a concentration gradient is established between the two phases, and this will create a mass transfer flow of A from the gas phase to the liquid phase. This is described by the two-film model proposed by W. G. Whitman, where interphase mass transfer is ensured by diffusion of the solute through two stagnant layers of thickness <5G and <5L on both sides of the interface (Fig. 45.1) [1—4]. 

Two limiting mechanisms for solute retention can be imagined to occur in RPC binding to the stationary phase surface or partitioning into a liquid layer at the surface. In the previous treatment we assumed that retention is caused by eluite interaction with the hydrocarbonaceous surface, i.e., the first type of mechanism prevails. When the eluent is a mixed solvent, however, the less polar solvent component could accumulate near the apolar surface of the stationary phase. In the extreme case, an essentially stagnant layer of the mobile phase rich in the less polar solvent could exist at the surface. As a result eluites could partition between this layer and the bulk mobile phase without interacting directly with the stationary phase proper. 

When transport is not able to do its job adequately and there is a change in the interfacial concentrations of electron acceptors and donors from the bulk values, there is a variation of concentration with distance from the interface toward the bulk of the solution. What matters, however, as far as the charge-transfer reaction is concerned, is the gradient of concentration at the interface because it is this gradient that drives the diffusion flux Jjy Even when there is convection with a laminar flow of electrolyte, the transport in the (assumed) stagnant layer adjacent to the electrode is by diffusion... 

The removal of O at the electrode surface sets up a concentration gradient across the stagnant solution layer. Species O diffuses across this layer to the electrode surface where it is electrolyzed to R, which then diffuses back across the stagnant layer to the bulk solution. 

The thickness of the stagnant layer 5o is inversely proportional to the tangential velocity of the solution near the electrode. So, as it becomes thinner, the limiting current /l increases with the stirring rate. 

If the depletion layer is completely inside the stagnant layer, the current is not affected by the change of flow. From (7.18), we know that this happens when the electrode radius becomes small. For Clark-type electrodes, the flow insensitivity is obtained even for larger diameters of the electrode, because of the additional confining effect of the membrane which has lower oxygen transmissivity, DmSm, than that of the solution. 

In this solution, subscripts 1 and 2 refer to the liquid surface and vapour side of the stagnant layer respectively and subscripts B and T refer to benzene and toluene. 

Another approximate limiting solution for a vertical enclosure (i.e., < 90°) is obtained, as mentioned before, by assuming that the flow consists of boundary layers on the hot and cold walls with an effectively stagnant layer between them and that the presence of these end walls has a negligible effect on the boundary layer flows. The assumed flow is therefore as shown in Fig. 8.31. 

Interestingly, if the drug concentration profile inside the two stagnant layers and the membrane has always a linear trend, is time independent and the drug diffusion coefficient is concentration independent (these conditions are usually met for thin membranes and well stirred donor and receiver compartments), the proposed numerical model has the following analytical solution ... 

The maximum useful current density through the membrane is normally limited by a phenomenon known as polarization. Concentration polarization is caused due to the depletion of the transported ion at the membrane surface, because of its faster electrolytic transport through membrane phase and its comparatively slower rate of transport through the solution phase. This causes excessive resistance at the stagnant layer near the membrane-solution interface. It is therefore necessary to avoid stagnant layers at the membrane-solution interfaces by operating at high Reynolds number or with turbulence promoters. 

From an electrochemical point of view it is easily inferred that the solution in a cell near an electrode is separable into two parts a stagnant layer adjacent to the electrode in which no convective motions occur, and the remainder of the solution, which is homogeneous (bulk solution). Yet this is not a particularity of electrochemical methods since the same phenomena occur at any solid/liquid interface, as when metal particles (reductions by Zn or Na, for example) or any heterogeneous reagent is used in organic homogeneous chemistry, as well as in phase-transfer catalysis or related methods. 

On the basis of the dichotomous representation of the solution near the electrode surface, Eq. (133) simplifies in the two domains. Within the stagnant layer and in the presence of an excess of supporting electrolyte, the two first terms are negligible and one obtains Eq. (134). Conversely, when x > 5conv the solution is macroscopically homogeneous the diffusional contribution then vanishes, and the flux is given in Eq. (135). 

Figure 1. Schematic representation of the electrochemical cell. First insert concentration profile of the reactant in the stagnant layer in the vicinity of the electrode. Second insert mass transfer black box at the boundary between the stagnant layer and the bulk solution (x = (5).

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