Robin boundary condition

In the finite element solution of the energy equation it is sometimes necessary to impose heat transfer across a section of the domain wall as a boundary condition in the process model. This type of convection (Robins) boundary condition is given as... 

Fig. 9.13. Stationary-state loci for the non-isothermal catalyst pellet (a) three stationary-state branches (b) unique response (c) five stationary-state branches, appropriate to Robin boundary conditions with a < 1.

Stationary-state solutions Robin boundary conditions... 

A different set of boundary conditions is that for which the concentration and temperature excess at the edge of the slab are determined by the two fluxes from the pellet. Thus, these Robin conditions have the form... 

In the world of numerical analysis, one distinguishes formally between three kinds of boundary conditions [283,528] the Dirichlet, Neumann (derivative) and Robin (mixed) conditions they are also sometimes called [283,350] the first, second and third kind, respectively. In electrochemistry, we normally have to do with derivative boundary conditions, except in the case of the Cottrell experiment, that is, a jump to a potential where the concentration is forced to zero at the electrode (or, formally, to a constant value different from the initial bulk value). This is pure Dirichlet only for a single species simulation because if other species are involved, the flux condition must be applied, and it involves derivatives. Therefore, in what follows below, we briefly treat the single species case, which includes the Cottrell (Dirichlet) condition as well as derivative conditions, and then the two-species case, which always, at least in part, has derivative conditions. In a later section in this chapter, a mathematical formalism is described that includes all possible boundary conditions for a single species and can be useful in some more fundamental investigations. 

The constants g, r and d can take on various values to express any given boundary condition. Thus, if we set g = d = 0, we are left with Co = 0, the Dirichlet (Cottrell) condition if we set r 0 and d 1, we have the Neumann or controlled current condition and setting g = 0 gives us Robin conditions. The constant r expresses the heterogeneous rate constant (this formula only considers a single species, so an irreversible reaction is implied). 

Fig. 3 One-dimensional loading profiles of benzene across a NaX zeolitic, single crystal membrane. The loading is the spatial average over planes perpendicular to the main diffusion direction (three-dimensional simulations are conducted periodic boundary conditions are employed in the transverse direction and Robin at the membrane interfaces exposed to the high- and low-pressure sides). The inset shows a schematic of the membrane. (View this art in color at www. dekker.com.)...

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

At boundary nodes where the variable values are given by Dirichlet conditions, no model equations are solved. When the boundary condition involve derivatives as defined by Neumann conditions, the boundary condition must be discretized to provide the required equation. The governing equation is thus solved on internal points only, not on the boundaries. Mixed or Robin conditions can also be used. These conditions consist of linear combinations of the variable value and its gradient at the boundary. A common problem does arise when higher order approximations of the derivatives are used at... 

Flux defined in terms of a mass transfer coefficient, k, with an external, known concentration, cq, or a heat transfer coefficient, h, with an external, known temperature, To (called a Robin condition or boundary condition of the third kind) ... 

There are three kinds of boundary conditions for elliptic equations. If the values of the unknown function are prescribed on the boundary, then the problem is called the Dirichlet problem. If the derivatives of the unknown function are prescribed on the boundary, then it is called the Neumann problem. If a linear combination of the function values and the derivatives is specified, then it is called the Robin problem. 

When a = 0, the boundary condition is called the Dirichlet boundary condition when P = 0, it is called the Neumann boundary condition. Otherwise, it is known as the Robin boundary condition. When y 0, the boundary condition is homogeneous. Otherwise, it is inhomogeneous. When a = 0 and P 0, m on the boundary is known and is described by y. When a 0 and p = 0, the derivative is described on the boundary, and u on the boundary is the unknown that must be solved. When both a 0 and P 0, both the u and its derivative are unknown. 

Transient conduction internal and external to various bodies subjected to the boundary conditions of the (1) first kind (Dirichlet), (2) second kind (Neumann), and (3) third kind (Robin) are presented in this section. Analytical solutions are presented in the form of series or integrals. Since these analytical solutions can be computed quickly and accurately using computer algebra systems, the solutions are not presented in graphic form. 

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

Days between -180...0 d correspond to the isothermal saturation period between the construction and the starting of the heaters. Days between 0... 53 d correspond to the heater adjustment period, which is approximated by a linear increase of the heater temperature to the final value assessed by the measurements. The temperature boundary condition at the rock is of the Robin type < =H(T-T,) with a calibrated value of the heat transfer coefficient H [W/(m K)]. 

Knowing that no analytical solution exists to this problem in the case of more than two layers or a mixed Robin boundary condition as shown in equation [13.1], we can find a complete numerical implementation in [NGU 13] with a detailed physical formulation in [VIT 1 la, VIT 07b]. In order to be concise, only the material balance at equihbrium for an assembly composed of n components or layers is presented. This type of formttlation coupled with transport equations, initial and boimdary conditions, forms ad hoc the basis of the FMECA method presented in section 13.3.3. 

Find the critical patch size for the RD equation with Robin boundary conditions... 

Linear Boundary Value Problem—Robin Boundary Condition 301... 

LINEAR BOUNDARY VALUE PROBLEM-ROBIN BOUNDARY CONDITION... 

The relevance of interphase gradients distinguishes between two different classes of problems, and this is reflected on the type of boundary condition at the pellet s surface. It is known that specifying the value of the concentration (or temperature) at the surfece (Dirichlet boundary condition) may not be realistic, and thus finite external transfer effects have to be considered (in a Robin-type boundary condition) [72]. Apart from these, a large number of additional effects have also been considered. Some examples include the nonuniformity of the porous pellet structure (distribution of pore sizes [102], bidisperse particles [103], etc.), nonuniformity of catalytic activity [104], deactivation by poisoning [105], presence of multiple reactions [106], and incorporation of additional transport mechanisms such as Soret diffusion [107] or intraparticular convection [108]. 

Equation 3.43g compares the timescale for radial heat dispersion in the solid phase with the one for internal heat conduction. For catalysts with good heat conduction properties and low particle-to-bed diameter ratios, A l. In this case, the surface boimdary condition is homogeneous and of Robin type, as given by the first terms on each side of (3.42b). A similar dimensionless number related with dispersion in the axial direction also appears, but its magnitude is considered much smaller than that of the other parameters in Equation 3.43, due to the geometrical reasons explained earlier. Note that Equations 3.32 and 3.34 are obtained by integrating Equation 3.41 with respect to over the pellet domain and using Equation 3.42 as boundary conditions. 

Assuming constant Helmholtz capacitance, Ch, leads to the following Robin boundary condition for potential. 

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