Boundary conditions dimensionless

Boundary condition Dimensionless form Boundary condition ... 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

Identical dimensionless sets of boundary conditions, including geometry, in the room and in the model ... 

The requirement of identical dimensionless boundary conditions is met when the model is geometrically similar to full scale in all details that are important for the volume flow, the energy flow and the contaminant flow see Fig. 12.24. 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

Reactor wall thermal boundary conditions can have a strong effect on the gas flow and thus the deposition. Here, for example, we indicate how cooling the reactor walls can enhance deposition uniformity. We consider the results of three simulations comparing the effects of two different wall boundary conditions. Figure 4 shows how the ratio of the computed susceptor heat flux to the onedimensional heat flux varies with the disk radius for the different conditions (the Nusselt number Nu is a dimensionless surface heat flux). In two cases the reactor walls are held at 300 K (0 = 0), and in one case the walls are insulated ( 0/ r —... 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

Solving the system of the equations tacking into account the boundary conditions according to [3] we receive the following expression for the dimensionless polarization characteristic of the elementary cell. 

From the continuity and momentum equations for the fluid and solid phases along with the boundary conditions, the following groups of independent dimensionless parameters are found to control the hydrodynamics, noting our assumption that the particle-particle forces are only dependent on hydrodynamic parameters,... 

From the boundary conditions, show the dimensionless parameters that the critical Damkohler number will depend on. 

The P-i concept can also be used to collapse the results of a number of dynamic response calculations for structural elements into compact dimensionless design curves. A number of illustrations are given in Ref. 15, with one for blast-loaded beams with various boundary conditions appearing in Fig. 20. These curves give predictions of maximum dynamic bending strains and displacements for beams with a variety of boundary conditions. Details appear in Ref. 15, so we do not try to define all parameters here. 

The dimensionless boundary conditions are satisfied by C0=l for a step change in feed concentration at the inlet and by the condition that at the outlet... 

Dimensionless reactant balances for Nelem sections, with "closed-end" boundary conditions d/dt(c[l])=(co-c[l])/deiz-(c[l]... 

The dimensionless model equations are used in the program. Since only two boundary conditions are known, i.e., S at X = l and dS /dX at X = 0, the problem is of a split-boundary type and therefore requires a trial and error method of solution. Since the gradients are symmetrical, as shown in Fig. 1, only one-half of the slab must be considered. Integration begins at the center, where X = 0 and dS /dX = 0, and proceeds to the outside, where X = l and S = 1. This value should be reached at the end of the integration by adjusting the value of Sguess at X=0 with a slider. 

An alternative to Lees-Edwards boundary conditions is the formalism put forth by Parrinello and Rahman for the simulation of solids under constant stress.52,53 They described the positions of particles by reduced, dimensionless coordinates ra, where the ra can take the value 0 < ra < 1 in the central image. Periodic images of a given particle are generated by adding or subtracting integers from the individual components of r. 

Linearizing the kinetic term as before, a set of three unknown linear equations is obtained, which is completed by the finite difference expression of the initial and boundary conditions. Inversion of the ensuing matrix allows the calculation of C at each node of the calculation grid and finally, of the current flowing through the electrode, or of the corresponding dimensionless function, by means of its finite difference expression. Calculation inside thin reaction layers may thus be more efficiently carried out than with explicit methods. The combination of the Crank-Nicholson... 

The pure kinetic conditions still apply if electron transfer is not unconditionally fast and Nemst s law has to be replaced by the law that governs the electron transfer kinetics as boundary condition, that is, in dimensionless terms,... 

The two successive electron transfer reactions are assumed to obey the Butler-Volmer law with the values of standard potentials, transfer coefficient, and standard rate constants indicated in Scheme 6.1. It is also assumed, matching the examples dealt with in Sections 2.5.2 and 2.6.1, that the reduction product, D, of the intermediate C, is converted rapidly into other products at such a rate that the reduction of B is irreversible. With the same dimensionless variables and parameters as in Section 6.2.4, the following system of partial derivative equations, and initial and boundary conditions, is obtained ... 

Thus, the dimensionless current-potential curves depend on the dimensionless parameters 1, A, A , oq, and a2. Simulating the dimensionless cyclic voltammograms then consists of finite difference resolutions of equations (6.57) and (6.58), taking into account all initial and boundary conditions. Examples of such responses are given in Section 2.5.2 (Figure 2.35). 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

In the gas/vapour phase the dimensionless distance tj ranges from 0 to 1, where tj — 1 corresponds to the position of the interface. In the liquid phase this parameter ranges from 0 to 1 for the mass transfer film and from 0 to Le for the heat transfer film. Hence, rj = 0 corresponds to the position of the interface and rj = I and t] = Le correspond, respectively, to the boundaries of the mass and heat transfer film. The mass and energy fluxes can now be calculated by solving the differential equations (4) and (8)-(12) subject to the boundary conditions (15). Due to the non-linearities a numerical solution procedure has been used which will be discussed subsequently. 

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