Boundary layers parabolic equations

Boundary layer similarity solution treatments have been used extensively to develop analytical models for CVD processes (2fl.). These have been useful In correlating experimental observations (e.g. fi.). However, because of the oversimplified fiow description they cannot be used to extrapolate to new process conditions or for reactor design. Moreover, they cannot predict transverse variations In film thickness which may occur even In the absence of secondary fiows because of the presence of side walls. Two-dimensional fully parabolized transport equations have been used to predict velocity, concentration and temperature profiles along the length of horizontal reactors for SI CVD (17,30- 32). Although these models are detailed, they can neither capture the effect of buoyancy driven secondary fiows or transverse thickness variations caused by the side walls. Thus, large scale simulation of 3D models are needed to obtain a realistic picture of horizontal reactor performance. 

When the boundary-layer approximations are applicable, the characteristics of the steady-state governing equations change from elliptic to parabolic. This is a huge simplification, leading to efficient computational algorithms. After finite-difference or finite-volume discretization, the resulting problem may be solved numerically by the method of lines as a differential-algebraic system. 

The boundary-layer equations represent a coupled, nonlinear system of parabolic partial-differential equations. Boundary conditions are required at the channel inlet and at the extremeties of the y domain. (The inlet boundary conditions mathematically play the role of initial conditions, since in these parabolic equations x plays the role of the time-like independent variable.) At the inlet, profiles of the dependent variables (w(y), T(y), and Tt(y)) must be specified. The v(y) profile must also be specified, but as discussed in Section 7.6.1, v(y) cannot be specified independently. When heterogeneous chemistry occurs on a wall the initial species profile Yk (y) must be specified such that the gas-phase composition at the wall is consistent with the surface composition and temperature and the heterogeneous reaction mechanism. The inlet pressure must also be specified. 

The method of lines is a computational technique that is particularly suited for solving coupled systems of parabolic partial-differential equations (PDE). The boundary-layer equations can be solved by the method of lines (MOL), although the task is facilitated considerably by casting the problem in a differential-algebraic setting [13]. As an introductory illustration, consider the heat equation... 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

Some consideration must be given to the conditions existing along the initial i = 1 line which were assumed to be known in the above discussion. The actual conditions will depend on the nature of the problem. For flow over a flat plate, because the boundary layer equations are parabolic in form, the use of these equations requires that the plate have no effect on the flow upstream of the plate. Hence, in this case, the variables will have their freestream values at all the nodal points (except at the point which lies on the surface) on the initial line which is coincident with the leading edge. At the nodal point on the surface, the known conditions at the surface must apply. This is illustrated in Fig. 3.24. 

Because of the parabolic nature of the assumed form of the governing equations, a forward-marching solution in the Z-directien can be used. The solution starts with known conditions on the first /Mine. Finite-difference forms of the governing equations are then used to obtain the solution on the next /Mine. Once this is obtained, the same procedure can be used to get the solution on the next /Mine and so on. This is the same procedure as was used in Chapter 3 in obtaining a solution to the boundary layer equations. 

In this equation, q",.dA is the net radiative heat flux at the moving material surface imposed by external sources such as radiant burners/heaters or electric resistance heaters. Both parabolic, boundary layer [80], and full, elliptic [61,81] problem solutions have been reported. Because of the nature of the problem, the heat transfer results can t be given in terms of correlations. The interested reader is referred to Refs. 62 and 79 for citation of relevant references. 

For the linear equation, the rate of oxidation is constant, or dy/dt = k and y = kt + const, where k is a constant. Hence, the thickness of scale, y, plotted with time, t, is linear (Fig. 11.3). This equation holds whenever the reaction rate is constant at an interface, as, for example, when the environment reaches the metal surface through cracks or pores in the reaction-product scale. Hence, for such metals, the ratio MdInmD is usually less than unity. In special cases, the linear equation may also hold even though the latter ratio is greater than unity, such as when the controlling reaction rate is constant at an inner or outer phase boundary of the reaction-product scale for example, tungsten first oxidizes at 700-1000 °C, in accord with the parabolic equation, forming an outer porous WO3 layer and an inner compact oxide scale [14]. When the rate of formation of the outer scale becomes equal to that of the inner scale, the linear equation is obeyed. 

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

Construction of an Asymptotic Expansion for the Parabolic Problem Other Problems with Corner Boundary Layers Nonisothermal Fast Chemical Reactions Contrast Structures in Partial Differential Equations A. Step-Type Solutions in the Noncritical Case Step-Type Solutions in the Critical Case Spike-Type Solutions Applications... 

The method of Vishik-Lyusternik is well developed not only for elliptic but also for parabolic and hyperbolic partial differential equations. Some nonlinear problems can be solved using this method as well. The main advantage of this method is its simplicity. Usually the boundary layer functions are defined as solutions of ordinary differential equations in which the independent variable is the stretched distance along the normal to the boundary (the variable p in our case). However, there are some problems when the boundary layer functions are described by more complicated equations, for example, by parabolic equations. A survey of results obtained by the Vishik-Lyusternik method with many references can be found in [28]. 

To construct special schemes, it is possible to use well-developed methods either fitted methods or methods with special condensed grids. Fitted methods are attractive because they allow one to use grids with an arbitrary distribution of nodes, in particular, uniform grids (see, e.g., [13, 14, 30-34]). However, even for the simplest singularly perturbed non-stationary diffusion equation, fitted methods are found to be inapplicable for the construction of e-uniformly convergent schemes. Fitted methods are inapplicable for more general elliptic and parabolic equations in the case when parabolic boundary layers, that is, layers described by parabolic equations, appear [4, 23, 29]. Therefore, the use of methods with special condensed grids is necessary for the construction of special schemes. 

Hence, from equation (4.83), it may be noted that the diffusion layer thickness is of the order of one tenth that of the viscous layer for Sc = 10. Equation 4.83 also shows that the diffusion layer growth is parabolic in the streamwise distance, x, which is similar to that of the viscous boundary layer. It may be noted that similar expressions are derived in case of thermal boundary layer thickness that is. 

Assuming that the contribution made by the free oxygen was controlled by boundary layer diffusion [35] whereas those made by carbon dioxide and water vapour were controlled by the rates of surface reactions [57], the authors derived separate equations to calculate the components on the right-hand side of Equation (8.8). Based on laboratory examination results, the authors believed that when the steel was oxidized in dilute O2-N2 atmospheres, the oxidation rate followed a linear kinetics law until the scale thickness was 400-500 microns. Thereafter, the oxidation kinetics gradually changed from linear to parabolic. 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

Assuming a constant surface area, dissolution at a solution-solid interface (Case I) results in linear kinetics in which the rate of mass transfer is constant with time (equation 1). Analytical solutions to the diffusion equation result in parabolic rates of mass transfer (, 16) (equation 2). This result is obtained whether the boundary conditions are defined so diffusion occurs across a progressively thickening, leached layer within the silicate phase (Case II), or across a growing precipitate layer forming on the silicate surface (Case III). Another case of linear kinetics (equation 1) may occur when the rate of formation of a metastable product or leached layer at the fresh silicate surface becomes equal to the rate at which this layer is destroyed at the aqueous... 

Furthermore, the growth of the layer may involve two boundaries and, consequently, two diffusion coefficients. Similarly, the diffusion in the adjoining layers is dependent of their thickness, which, in turn, may partly depend on diffusion in yet other layers. Solutions of the diffusion equations, for given boundary conditions, have been evaluated [66]. Van Horn [96] and others have obtained equations, together with nomographic solutions, relating the motion of a phase boundary to the diffusion rates in a simple two-phase binary system using the parabolic law as a basis. A relationship between the diffusivities in each phase was found ... 

This is known as lander equation, which has two oversimplifications that limit its applicability and the capability to predict the rates of most chemical reactions. Firstly, the parabolic growth law for the thickness of the product layer is only valid for one-dimensional reaction across a planar boundary, but not suflhcient for the reactions involving spherical particles. In other words, this assumption is only valid for the initial stage of the reaction when y r. Second, the changes in molar volumes of the reactants and the products are not taken into account. To address this problem, a more comprehensive equation should be used, which is [51, 52] ... 

Equation (2.16), referred to as the Jander equation, suffers from two oversimplifications that limit its applicability and the range over which it adequately predicts reaction rates. First, the parabolic growth law assumed for the thickness of the reaction layer is valid for one-dimension reaction across a planar boundary and not for a system witli spherical geometry. At best, it is expected to be valid only for the initial stages of the powder reaction when y r. Second, any change in molar volume between the reactant and the product is neglected. These two oversimplifications have been taken into account by Carter (37), who derived the following equation ... 

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