233 Walls

At the impermeable wall boundaries of the solution domain, normally a 'no slip' boundary condition is employed. This is achieved by setting the transverse fluid velocity equal to that of the surface and setting the normal velocity to zero. Since the normal velocity at the wall is known, the value of pressure at the wall boundary is not required to be known. For species concentrations or temperatures, any of the following conditions can be specified at the wall boundaries:

• Surface temperature (or concentration) is specified.

• Heat (or mass) flux at the surface is specified.

• External heat transfer (or mass transfer) coefficient and external temperature (or concentration) at the surface is specified.

The specified heat or mass flux at the wall is then used to calculate the gradients of temperature or species concentrations at the wall by using the following equations:

w w where qw and jkw are the specified heat and mass (of species k) flux at the wall. At the impermeable, inert and insulated wall, the gradients of temperature and species concentration are set to zero (zero flux condition). When external heat or mass transfer coefficients are specified, along with the external temperature or species concentration, the flux at the walls are equated to the flux in terms of these external transfer coefficients:

qw = hext {Text — Tw) jkw = kmext {mkext — mw) (2.25)

Schematic representation of these three possible boundary conditions at the wall is shown in Fig. 2.5 for the enthalpy/temperature equation. For systems with conjugate heat transfer, continuity of the temperature and the normal component of fluxes are specified at the walls. For systems with reactions occurring on solid surfaces, generally, accumulation of species at the solid surface is neglected and the diffusive flux at the wall is equated to the surface reaction rate.

2.3.4. Symmetry/Periodic/Cyclic

Recognizing the intrinsic symmetry of the flow process or the repetitious nature of the process equipment can minimize the size and extent of the solution domain. If such a possibility exists, fictitious boundaries may be used to define the solution domain with special boundary conditions imposed on these such fictitious surfaces. Some of the commonly encountered cases are discussed below.

If the flow process is symmetric or in other words contains a surface across which the flux of all quantities is zero, the extent of the solution domain can be reduced. Two examples of such symmetric flow processes are shown in Fig. 2.6. At a symmetry surface, the normal velocity is set to zero and the normal gradients of all variables, except normal velocity, are set to zero. This ensures that there is no convective or diffusive flux across the symmetry surface. It must be noted that symmetric construction of the process equipment under consideration does not guarantee that the underlying flow processes are also symmetric. Even if the solution domain is geometrically symmetric, the flow process of interest may not be so. One example of such a case is shown in Fig. 2.6. When the flow processes are not symmetric, it is necessary to

FIGURE 2.5 Wall boundary conditions for enthalpy/temperature equation.

Symmetry planes

Cold

Two symmetry planes: (model needs to include only a 90° sector)

Symmetry planes

Cold

Hot fluid f rises

Cold

Not a plane of symmetry

FIGURE 2.6 Examples of symmetric flows.

include the whole vessel as the solution domain. However, wherever possible, the existence of symmetry planes must be recognized to reduce the size of the solution domain.

When the physical geometry of the problem under consideration or the expected flow pattern has a cyclically repeating nature, cyclic or periodic boundary conditions can be used to reduce the size of the solution domain. Two types of cyclic boundary condition can be distinguished. The first is for rotationally periodic flow processes, where all the variables at corresponding periodic locations on the cyclic planes are the same. The second is for translationally periodic flow processes, where all the variables, except pressure, at corresponding periodic locations on the cyclic planes are the same. Examples of these two types are shown in Fig. 2.7. Such cyclic planes are in fact part of the solution domain (by the nature of their definitions) and no additional boundary conditions are required at these planes, except the one-to-one correspondence between the two cyclic planes.

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