Boundary condition step-change

Pick s second law of difflision enables predictions of concentration changes of electroactive material close to the electrode surface and solutions, with initial and boundary conditions appropriate to a particular experiment, provide the basis of the theory of instrumental methods such as, for example, potential-step and cyclic voltanunetry. 

The appropriate boundary conditions are the closed variety discussed in Section 9.3.1. The initial condition is a negative step change at the inlet. A full analytical solution is available but complex. For Pe = uL/D > 16, the following result is an excellent approximation ... 

For the step change condition, shown in Fig. 2.2, the initial conditions are given by y = 0, when t = 0. The solution to the differential equation, with the above boundary conditions, is given by... 

The boundary conditions are satisfied by Cq = 1 for a step change in feed concentration at the inlet, and by the condition that at the outlet C n+i = C n, which sets the concentration gradient to zero. The reactor is divided into 8 equal-sized segments. 

If we now consider a step change in tracer concentration in the feed to an open tube that can be regarded as extending to infinity in both directions from the injection point, the appropriate initial and boundary conditions on... 

Equation 11.1.33 is strictly applicable only to an ideal step change stimulus and the boundary conditions cited. The plant or laboratory situation may not correspond to these assumptions. 

For these conditions, Armaou and Christofides [4] determine the thickness profile, in Fig. 10.4-3, for the amorphous silicon film after 60 s, when the average thickness reaches 500 A. When characterizing the non-uniformity of the film, the sharp increase in thickness calculated near the outer edge of the wafer is assumed to be due to the boundary conditions, which assume step changes to zero concentrations at the edge. Brass and Lee (2003) disregard the profile from r = 3.6 to 4 cm, and compute the non-uniformity as ... 

Once the boundary conditions have been implemented, the calculation of solution molecular dynamics proceeds in essentially the same manner as do vacuum calculations. While the total energy and volume in a microcanonical ensemble calculation remain constant, the temperature and pressure need not remain fixed. A variant of the periodic boundary condition calculation method keeps the system pressure constant by adjusting the box length of the primary box at each step by the amount necessary to keep the pressure calculated from the system second virial at a fixed value (46). Such a procedure may be necessary in simulations of processes which involve large volume changes or fluctuations. Techniques are also available, by coupling the system to a Brownian heat bath, for performing simulations directly in the canonical, or constant T,N, and V, ensemble (2,46). 

The case of a fluid sphere moving at constant velocity and suddenly exposed to a step change in the composition of the continuous phase has been treated by solving Eq. (3-56), with Eqs. (3-40), (3-41), (3-42), and (3-57) as boundary conditions for potential flow (R14). The transient external resistance is given within 3% by... 

Sometimes the boundary conditions can be approximated as a step in concentration instead of a step in mass released. This subtle difference in boundary conditions changes the solution from one that is related to pulse boundaries (known mass release) to one resulting from a concentration front with a known concentration at one boundary. 

Consider a ternary diffusion couple in which each component has an initial step-function profile and boundary conditions similar to those given by Eq. 4.44. Integrating and changing variables, as in the development leading up to Eqs. 4.48 and 4.49 for the binary case,... 

Enter the loop, each time increasing the potential by Sp, and computing the new concentrations. When these have been calculated, apply the boundary conditions for the current potential, to get the boundary values at X = 0 write out the potential and current into the file (perhaps only if there has been a change in current greater than some set value, to reduce the volume of output). If half the total number of time steps tit has been done, flip the sign of Sp, so that the next half will go in the reverse direction, for CV. 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

There are many elaborations that have been developed over the years. One of the most important is the k + 1 rule. If a vertex has remained part of the simplex for k + 1 steps, perform the experiment again. The reason for this is that response surfaces may be noisy, so an unduly optimistic response could have been obtained because of experimental error. This is especially important when the response surface is flat near the optimum. Another important issue relates to boundary conditions. Sometimes there are physical reasons why a condition cannot cross a boundary, an obvious case being a negative concentration. It is not always easy to deal with such situations, but it is possible to use step 5 rather than step 4 above under such circumstances. If the simplex constantly tries to cross a boundary either the constraints are slightly unrealistic and so should be changed, or the behaviour near tire boundary needs further investigation. Starting a new simplex near tire boundary with a small step size may solve tire problem. 

Step-change in wall boundary condition It was anticipated that the discontinuity in wall temperature at x = 0, and the resulting steep local gradients, would lead to a locally poor approximation which might have adverse effects further downstream. It was soon found that mesh refinement in the axial direction improved the results considerably over the use of an equally-spaced mesh, whereas mesh refinement in the radial direction had little effect, and a fairly coarse uniform radial mesh was always found to be adequate. 

Consider now the effect of uncompensated iR on the shape of the potentiostatic transients. This was shown in Fig. 6D. The point to remember is that although the potentiostat may put out an excellent step function - one with a rise time that is very short compared to the time of the transient measured - the actual potential applied to the interphase changes during the whole transient, as the current changes with time (cf. Section 10.2). This effect is not taken into account in the boundary conditions used to solve the diffusion equation, and the solution obtained is, therefore, not valid. The resulting error depends on the value of R, and it is very important to minimize this resis-tance, by proper cell design and by electronic iR compensation. 

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