Stagnant layer

In the study of steady evaporation of liquid, we derived in Equation (6.34), and given as Equation (9.6), 

We shall consider at first only steady burning. Moreover, we shall restrict our heating to only that of convection received from the gas phase. This is flame heating. Therefore, purely convective heating is given as 

By a process of applying the conservation of mass, species and energy to the control volume (Ay x Ax x unit length), and expressing all variables at y + Ay as 

Only laminar transport processes are considered in the transverse y direction. 

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  

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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... 

Ammonia gas is diffusing at a constant rate through a layer of stagnant air 1 mm thick. Conditions are such that the gas contains 50 per cent by volume ammonia at one boundary of the stagnant layer. The ammonia diffusing to the other boundary is quickly absorbed and the concentration is negligible at that plane. The temperature is 295 K and the pressure atmospheric, and under these conditions the diffusivity of ammonia in air is 1.8 x 10 5 m2/s. Estimate the rate of diffusion of ammonia through the layer. 

If the subscripts I and 2 refer to the two sides of the stagnant layer and the subscripts A and B refer to ammonia and air respectively, then the rate of diffusion through a stagnant layer is given by ... 

An interfacial reaction is accompanied by a diffusion process which provides reactants from the bulk phase to the interface. When the reaction rate is faster than the diffusion rate, the overall reaction has to be governed by the diffusion rate of the reactant. Diffusion, in general, takes place in the stagnant layers (the region on both sides of the interface) and is not disturbed even under the stirring of a bulk region. 

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. 

The dependence of the limiting current density on the rate of stirring was first established in 1904 by Nernst (N2) and Brunner (Blla). They interpreted this dependence using the stagnant layer concept first proposed by Noyes and Whitney. The thickness of this layer ( Nernst diffusion layer thickness ) was correlated simply with the speed of the stirring impeller or rotated electrode tip. 

The velocity of liquid flow around suspended solid particles is reduced by frictional resistance and results in a region characterized by a velocity gradient between the surface of the solid particle and the bulk fluid. This region is termed the hydrodynamic boundary layer and the stagnant layer within it that is diffusion-controlled is often known as the effective diffusion boundary layer. The thickness of this stagnant layer has been suggested to be about 10 times smaller than the thickness of the hydrodynamic boundary layer [13]. 

Figure 9.7 Stagnant layer model - pure convective burning...

Since diffusional effects are most important, we wish to emphasize these processes in the gas phase. For the control volume selected in Figure 9.7, the bold assumption is made that transport processes across the lateral faces in the x direction do not change - or change very slowly. Thus we only consider changes in the y direction. This approximation is known as the stagnant layer model since the direct effect of the main flow velocity (it) is not expressed. A differential control volume Ay x Ax x unity is selected. 

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... 

The stagnant layer analysis offers a pedagogical framework for presenting the essence of diffusive burning. For the most part the one-dimensional stagnant layer approximated a two-dimensional boundary layer in which 6 = <5(x), with x the flow direction. For a convective boundary layer, the heat transfer coefficient, hc, is defined as... 

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... 

The inclusion of radiative heat transfer effects can be accommodated by the stagnant layer model. However, this can only be done if a priori we can prescribe or calculate these effects. The complications of radiative heat transfer in flames is illustrated in Figure 9.12. This illustration is only schematic and does not represent the spectral and continuum effects fully. A more complete overview on radiative heat transfer in flame can be found in Tien, Lee and Stretton [12]. In Figure 9.12, the heat fluxes are presented as incident (to a sensor at T,, ) and absorbed (at TV) at the surface. Any attempt to discriminate further for the radiant heating would prove tedious and pedantic. It should be clear from heat transfer principles that we have effects of surface and gas phase radiative emittance, reflectance, absorptance and transmittance. These are complicated by the spectral character of the radiation, the soot and combustion product temperature and concentration distributions, and the decomposition of the surface. Reasonable approximations that serve to simplify are ... 

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... 

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),... 

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. 

This exactly reduces to the stagnant layer one-dimensional pure diffusion problem (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... 

The symbols are consistent with Section 9.2 of the text. We are to use the stagnant layer theory of huming with suitable approximations to analyze the burning of a droplet in natural convection. The droplet, suspended and burning in air, is assumed to remain spherical with diameter, D. 

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]. 

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