Prescribed diffusion fluxes, boundary value

In Sections II and III, we considered boundary value problems modeling the diffusion process for some substance in a homogeneous material. We have studied problems in which the unknown function takes a prescribed value on the boundary (in Section II), or the diffusion flux of the unknown function is given on the boundary (in Section III). The right-hand sides of the equations were supposed to be smooth. Boundary layers appear in these problems for small values of the parameter. 

When j8(-l) = /3(1) = 1, we arrive at the problem with prescribed values of the solution on the material surface, which was considered in Section II. In the case of /3(-l) =/3(1) = 0, problem (4.10) is the problem with given diffusion flux on the boundary, which was studied in Section III. On the set... 

Every partial differential equation needs an initial value or guess for numerical solver to start computing the equations. On the other hand, boundary conditions are specific for each conservation equation, described in Section 6.2. The variable in the continuity equation and momentum equations is the velocity vector, the variable in the energy equation is the temperature vector, and the variable in the species equation is the concentration vector. Therefore, appropriate velocity, temperature, and concentration values, which represent real-world values, need to be prescribed on each computational boundary, such as inlet, outlet, or wall at time zero. The prescribed values on boundaries are called boundary conditions. Each boundary condition needs to be prescribed on a node or line for 2D system or on a plane for 3D system. In general, there are several types of boundary conditions where the Dirichlet and Neumann boundary conditions are the most widely used in CFD and multiphysics applications. The Dirichlet boundary condition specifies the value on a specific boundary, such as velocity, temperature, or concentration. On the contrary, the Neumann boundary condition specifies the derivative on a specific boundary, such as heat flux or diffusion flux. Once the appropriate boundary conditions are prescribed to all boundaries on the 2D or 3D model, the set of the conservation equations is closed and the computational model can be executed. 

The carbon mass balance (2.1) takes into account the mixing of solids and the combustion reaction and is of reaction diffusion type with Neumann boundary conditions, i,e. the value of the normal derivative dCc/dn of the carbon concentration on the boundary is prescribed. The first two terms in the enthalpy balance (2.3) express the enthalpy flux due to the mixing of solids in the bed, the others the flue gas enthalpy flux, the heat sink due to the heat exchanger tubes and the heat source caused by the combustion. The balance is of convection diffusion type with third type Dirichlet-Neumann boundary conditions, i.e. the temperature values on the boundary depend on the corresponding gradients. Finally, the oxygen balance (2.2) considers the oxygen flux in upward direction and the combustion reaction. This ODE is explicitly solvable in dependence of the carbon concentration Cc and the temperature T ... 

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