Specified Temperature Boundary Condition

For the statement of boundary conditions we begin with regular boundaries and consider the finite-difference systems shown in Fig. 4.10(a) for a corner node and a side node. We consider three types of boundary conditions specified temperature, specified heat flux, and heat transfer to ambient. 

In order to complete the problem, the initial and boundary conditions must be given. The temperature and degree of cine or crystallinity must initially (at time zero) be specified at every point inside the composite and the mandrel. For the latter only the temperature is required. As boundary conditions, the temperatures or heat fluxes at the composite outside diameter and mandrel inner diameter must be specified. 

Meanwhile, the idea was formulated about resolving the full set of primitive hydro- and thermodynamic equations with all the boundary conditions specified successively correcting the current model fields of the temperature, salinity, and SLE by their observed values with the use of this or that kind of assimilation algorithm [35,36]. This approach is sometimes referred to as a four-dimensional analysis. Strictly speaking, it has little in common with the initial diagnostic methods. They are joined only by the common goal - the hydrodynamic calculations of the fields of currents from the data of observations of the temperature, salinity, and sea level. Therefore, in this section, we consider the results of application of all the above-mentioned approaches. 

The boundary conditions on temperature at the wall depend on the thermal conditions specified at the wall. If the distribution of the temperature of the wall is specified then, since the fluid in contact with the wall must be at the same temperature as the wall, the boundary condition on temperature is ... 

The energy integral equation is applied in basically the same way as the momentum integral equation. The form of the boundary layer temperature profile, i.e., of the variation of (T - T ) with y, is assumed. In the case of laminar flow, for example, a polynomial form is again often used. The unknown coefficients in this assumed temperature profile are then determined by applying known boundary conditions on temperature at the inner and outer edges of the boundary layer. For example, the variation of the wall temperature Tw with x may be specified. Therefore, because at the outer edge of the boundary layer the temperature must become equal to the freestream temperature T, two boundary conditions on the assumed temperature profile in this case are ... 

Using Fourier s law, thq boundary condition on temperature at the wall in the specified heat flux case is ... 

To proceed further with the solution, the wall boundary conditions on temperature must be specified. Consideration will first be given to the case where the heat flux at the wall, qw, is uniform and specified. Some discussion of the solution for the case where the wall temperature is kept uniform will be given later. 

The coefficients in this equation, i.e., a, b, and c, are determined by applying the boundary conditions on temperature at the inner and outer edges of the thermal boundary layer. Because the case where the wall temperature variation is specified is being considered, these boundary conditions are ... 

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. 

These boundary conditions specify the value of the independent variable at the left boundary as a function of time/(0 (this may be the condition inside a furnace that is maintained at a preprogrammed temperature profile) and at the right boundary as a constant 7, (e.g., the room temperature at the outside of the furnace) (Fig. 6.2a). 

The boundary conditions for engineering problems usually include some surfaces on which values of the problem unknowns are specified, for instance points of known temperature or initial species concentration. Some other surfaces may have constraints on the gradients of these variables, as on convective thermal boundaries where the rate of heat transport by convection away from the surface must match the rate of conductive transport to the surface from within the body. Such a temperature constraint might be written ... 

The other boundary conditions are relatively simple. The temperature and species composition far from the disk (the reactor inlet) are specified. The radial and circumferential velocities are zero far from the disk a boundary condition is not required for the axial velocity at large x. The radial velocity on the disk is zero, the circumferential velocity is determined from the spinning rate W = Q, and the disk temperature is specified. 

No slip Is used as the velocity boundary conditions at all walls. Actually there Is a finite normal velocity at the deposition surface, but It Is Insignificant In the case of dilute reactants. The Inlet flow Is assumed to be Polseullle flow while zero stresses are specified at the reactor exit. The boundary conditions for the temperature play a central role in CVD reactor behavior. Here we employ Idealized boundary conditions In the absence of detailed heat transfer modelling of an actual reactor. Two wall conditions will be considered (1) adiabatic side walls, l.e. dT/dn = 0, and (11) fixed side wall temperatures corresponding to cooled reactor walls. For the reactive species, no net normal flux Is specified on nonreacting surfaces. At substrate surface, the flux of the Tth species equals the rate of reaction of 1 In n surface reactions, l.e. 

Three boundary lines meet in a single point (shown by a red dot), called a triple point. All three phases are stable simultaneously at this unique combination of temperature and pressure. Notice that, although two phases are stable under any of the conditions specified by the boundary lines, three phases can be simultaneously stable only at a triple point. 

The wall temperature maps shown in Fig. 28 are intended to show the qualitative trends and patterns of wall temperature when conduction is or is not included in the tube wall. The temperatures on the tube wall could be calculated using the wall functions, since the wall heat flux was specified as a boundary condition and the accuracy of the values obtained will depend on their validity, which is related to the y+ values for the various solid surfaces. For the range of conditions in these simulations, we get y+ x 13-14. This is somewhat low for the k- model. The values of Tw are in line with industrially observed temperatures, but should not be taken as precise. 

The simplest boundary conditions for the catalyst pellet are those for which the concentration and temperature at the edges of the slab are specified as being equal to the respective reservoir values. These Dirichlet boundary conditions then give... 

The boundary-layer equations represent a coupled, nonlinear system of parabolic partial-differential equations. Boundary conditions are required at the channel inlet and at the extremeties of the y domain. (The inlet boundary conditions mathematically play the role of initial conditions, since in these parabolic equations x plays the role of the time-like independent variable.) At the inlet, profiles of the dependent variables (w(y), T(y), and Tt(y)) must be specified. The v(y) profile must also be specified, but as discussed in Section 7.6.1, v(y) cannot be specified independently. When heterogeneous chemistry occurs on a wall the initial species profile Yk (y) must be specified such that the gas-phase composition at the wall is consistent with the surface composition and temperature and the heterogeneous reaction mechanism. The inlet pressure must also be specified. 

Burner-Stabilized Flame While the governing equations are the same for either the burner-stabilized or the freely propagating flames, the boundary conditions differ. For the typical burner-stabilized case, the mass flux m" is specified, as is the temperature at the burner face (z = 0). The species boundary condition is specified through the mass-... 

It is generally good practice to represent the boundary conditions in residual form, even though in many cases a simple Dirichlet boundary condition could be imposed directly and not included in the y vector. For example, take the burner-face temperature specified as T (z = 0) = Tb. The residual form yields... 

The continuity equation at the inlet boundary can be viewed as a constraint equation. Referring to the difference stencil (Fig. 17.14), it is seen that this first-order equation itself is evaluated at the boundary and no explicit boundary condition is needed. Moreover, since the inlet temperature, pressure, and composition are specified, the density is fixed and thus dp/dt = 0. Therefore, at the boundary, the continuity equation (Eq. 17.15) has no time derivative it is an algebraic constraint. There is no explicit boundary condition for A. At the inlet boundary, the value of A must be determined in such a way that all the other boundary conditions are satisfied. Being an eigenvalue, A s effect is felt through its influence on the V velocity in the radial momentum equation, and subsequently by V s influence on u through the continuity equation. 

The reaction path in the T, Y plane could be plotted by solving the above set of equations with the appropriate boundary conditions. A reaction path similar to the curve ABC in Fig. 3.20 would be obtained. The size of reactor necessary to achieve a specified conversion could be assessed by tabulating points at which the reaction path crosses the constant rate contours, hence giving values of kYT which could be used to integrate the mass balance equation 3.87. The reaction path would be suitable provided the maximum temperature attained was not deleterious to the catalyst activity. 

Typically, there are two types of boundaries in reacting flows. The first is a solid surface at which a reaction may be occurring, where the flow velocity is usually set to zero (the no-slip condition) and where either a temperature or a heat flux is specified or a balance between heat generated and lost is made. The second type of boundary is an inflow or outflow boundary. Generally, either the species concentration is specified or the Dankwerts boundary condition is used wherein a flux balance is made across the inflow boundary (64). The gas temperature and gas velocity profile are usually specified at an inflow boundary. At outflow boundaries, choices often become more difficult. If the outflow boundary is far away from the reaction zone, the species concentration gradient and temperature gradient in the direction of flow are often assumed to be zero. In addition, the outflow boundary condition on the momentum balance is usually that normal or shear stresses are also zero (64). 

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