Constant boundary conditions

The general case of non-uniform initial conditions and time-dependent boundary conditions can also be expressed analytically, and leads to convolution integrals in space and time. Nevertheless, for most practical applications, the assumptions of uniform initial conditions and constant boundary conditions are adequate. 

A3.2.1. Infinite medium, D is constant, boundary conditions at x=-[Pg.571]

Figure 22.3 One-dimensional concentration profiles at steady-state calculated from the diffusion/advec-tion/reaction equation (Eq. 22-7) for different parameter values D (diffii-sivity), x (advection velocity), and kr (first-order reaction rate constant). Boundary conditions at x = 0 and x - L are C0 and CL, respectively. Pe = 7. vx ID is the Peclet Number, Da = Dk/v] is the Damkohler Number. See text for further explanations.

For reactions of the type A + B = AB (or a+P = y), the situation is different. If one has a linear reaction geometry and the y product forms at different times and locations on the A/B interface, the patches of y eventually merge by fast lateral (interface) transport. Eventually, a full y layer is formed between a and / . At first, this layer has a non-uniform thickness (Fig. 6-4). In Chapter 11 we will show, however, that the uneven a/y and y/p interfaces are morphologically stable and become smooth during further growth. This leads to constant boundary conditions for the y formation after some time of reaction and eventually results in a parabolic rate law, as will be discussed later. 

In the case of internal flows extensive experimental data are available for turbulent pipe flow. The study of turbulent-friction coefficients in pipe flow has brought forth a number of effects displayed by flowing polymer solutions. Furthermore, many hydro-dynamic investigations in pipe flow have been made to elucidate the flow behavior (laminar and turbulent) of Newtonian fluids. Thus, the pipe is one of the most investigated and traditional pieces of test apparatus and one can easily compare the flow behavior of Newtonian fluids and polymer solutions under constant boundary conditions. 

A particularly simple case occurs when the diffusion is in a steady state and the composition profile is therefore not a function of time. Steady-state conditions are often achieved for constant boundary conditions in finite samples at very long times.3 Then dc/dt = 0, all local accumulation (divergence) vanishes, and the diffusion equation reduces to the Laplace equation,... 

The boundary conditions of the coarse grid were estimated from the forecast of the previous day. This method is compared with constant boundary conditions using average summer values. Total ozone column data was obtained from ECMWE data. 

Figure 18.2 shows the predicted maximum concentrations for ozone region 1. The course of ozone concentration from 1 day (Tag 1) and 2-days (Tag 2) model forecasts as well as the results from a backup run which considers constant boundary conditions only are compared to measurements (at 43 air quality stations). 

Fig. 18.2 (a) Gray area range between highest and lowest maximum observations (hourly average) at stations within the region with maximum predicted on the same and previous days backup run (constant boundary conditions), (b) Scatter-diagram of daily ozone maxima predicted versus observed in ozone region 1 for 2006 (maximum predicted on previous and on the same day, as well as maximum predicted by backup run lines information and alert threshold Directive 2002/3/EC))... 

Exceedances of threshold of 180 fig/m ( 90 ppbv) occurred in 2006 between middle of June and end of July. After that period the predicted concentrations were higher than the measurements. The light-blue line shows that the model run considering constant boundary conditions performs slightly better than the dynamic approach during that period. 

Turbulence is intrinsically unsteady, even when constant boundary conditions are imposed. Velocity and all other flow properties fluctuate in a random and chaotic way. Turbulent fluctuations always have a three-dimensional spatial character. There have been many attempts to analyze and to construct a physical picture of turbulence, following several different approaches. These different approaches, broadly classified into three categories, are discussed in this section. 

Heat conduction with a constant boundary condition at x =0 was considered in example 4.1. The same technique can be applied for time dependent boundary conditions. Consider the transient heat conduction problem in a slab.[4] The governing equation is ... 

When solving how problems numerically, a Neumann boundary is described as an insulated boundary (or impermeable boundary), which means that there is no fiux at the boundary, while a Dirichlet boundary indicates that the value of head (potential, concentration, etc.) is constant at the boundary. Constant boundary conditions are not able to describe the nature of the electrokinetic transport realistically due to the existence of fiux boundaries caused by the electrode reactions and advection of fluid. In Cao s model, the boundary conditions apphed at the inlet and outlet of the soil column maintained the equahty between the flux of solute at the inside of the column and the flux of solute immediately outside of the column. The following boundary condition was used at the inlet (Lafolie and Hayot, 1993) ... 

This discrete equation Eq. 12.120ft can be solved using the calculus of finite difference (Chapter 5), to give a general solution in terms of arbitrary constants. Boundary conditions are necessary to complete the problem, if we wish to develop an iterative solution. The remainder of the procedure depends on the form of the specified boundary conditions. To show this, we choose the following two boundary conditions... 

At steady state, integrating eq. (7.5-4b) with respect to z and using constant boundary conditions at two ends of the capillary, we get the following steady state equation for the viscous flux through a capillary ... 

Having the viscous flow equation with slip at the wall (eq. 7.6-9), we integrate that equation for the case of constant boundary conditions at two ends of the capillary to obtain the following steady state flux ... 

Steady state flow of molecule through a zeolite membrane is a constant. It can be obtained by integrating eq. (10.2-8) subject to constant boundary conditions at two ends of the membrane ... 

The parameter characterising the diffusion through the medium is the Knudsen diffusivity, which could be determined from the time lag given in eq. (12.2-24) or from the short time solution (eq. 12.2-18). The long time solution for time lag is preferrable if the experimental data exhibit a linear asymptote behaviour at long time and the constant boundary conditions (12.2-4) are maintained throughout the course of the experiment. If the medium is rather impermeable and the time lag is practically too long to measure, then the application of the short time solution is the only possible choice. 

With this new initial condition, the solution for the concentration distribution inside the capillary subject to two constant boundary conditions (12.2-4) is ... 

This steady state flux is a constant as a result of the constant boundary conditions. Although it takes time for the steady state condition to be reached, if we assume this steady state flux is instantaneously attained from t = 0 the amount collected per unit area of the capillary from t = 0 to t is given by... 

Take the case of initially free adsorbate in the medium and the constant boundary conditions (12.2-4) imposed on the system, the mass balance equations (12.5-2) and (12.5-8) can be solved by Laplace transform (Appendix 12.1) to give the following solution for the amount collected in the receiving reservoir ... 

Integrating the flux equation with respect to x subject to the following constant boundary conditions ... 

The time lag method is shown to be a useful tool for the characterisation of a porous medium. Conditions are usually chosen in such a way that the constant boundary conditions are satisfied (12.2-4). This is usually possible but there are situations where the receiving reservoir is small and its pressure can not be maintained to satisfy the zero boundary condition (12.2-4b). In such cases, the pressure will rise and the boundary condition at the exit of the medium is replaced by ... 

Although mathematical solutions are always possible for time varying boundary conditions, experimental preparation should be exercised such that the constant boundary conditions hold during the course of experiment. This would then simplify the analysis and hence the ease of obtaining the diffusion coefficient, which is after all the main purpose of the time lag method. 

Physical modeling of CVD processes means solving the Navier-Stokes equations, partial differential equations for mass and heat transport in fluids given the constant boundary conditions of the reactor. These processes affect the uniformity of the deposit in all parts of the reactor. The proper name for CVD reactor physics is chemical vapor technology (CVT) and it is a subject of some significance for industrial reactor design. (This branch of continuum physics is outside the scope of this book, which is concerned with materials rather than machinery.)... 

FIG. 1 Theoretical mass fraction sorbed versus the square root of time using a particle diameter of 40 pm and the integrated solution to the mass balance with a constant boundary condition. 

---