Robin-type boundary conditions

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... 

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. 

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. 

We prescribe fluxes at the catalyst layer-GDL boundary. At the channel-GDL boundary, we prescribe Robin-type conditions to account for boundary layers introduced in the gas channel flow by the fluxes from the GDL. 

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. 

In the limit of no external mass transfer resistance (Bi, - - ) the Robin type boundary condition (12) becomes a simpler Dirichlet condition of u = 1. This problem was solved also. 

At the exposed surfaces, the current in the normal direction is no longer a fixed value but depends on the local potential value. This type of boundary condition is the mixed or Robin type. Symbolically, the mixed boundary condition is displayed below ... 

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