Inlet and Boundary Conditions

The energy equation and the inlet and boundary conditions are given by... 

A comparison with Eq. 18.66 shows that, in writing, this equation, longitudinal heat conduction ( -direction), has been neglected in comparison to the lateral (y) and transverse (z) directions. The inlet and boundary conditions are ... 

The study of reaction kinetics in flow reactors to derive microkinetic expressions also rehes on an adequate description of the flow field and well-defined inlet and boundary conditions. The stagnation flow on a catalytic plate represents such a simple flow system, in which the catalytic surface is zero dimensional and the species and temperature profiles of the estabhshed boundary layer depend only on the distance from the catalytic plate. This configuration consequently allows the application of simple measurement and modehng approaches (Sidwell et al., 2002 Wamatz et al., 1994a). SFRs are also of significant technical importance because they have extensively been used for CVD to produce homogeneous deposits. In this deposition technique, the disk is often additionally forced to spin to achieve a thick and uniform deposition across the substrate (Houtman et al., 1986a Oh et al., 1991). 

With no density differences in the fluid, there will be no source term for the radial and tangential velocities. If the inlet and boundary conditions are axisymmetric, there will be no variation in the 0 direction. Removing the tangential variations will allow us to reduce the simulation fi om 3D to 2D. The symmetric boundary condition at the center of the tube can be formulated as... 

Boundary conditions may be written in terms of a known inlet pressure p 0) = po a vanishing velocity at the closed end U(L) = 0. Using these boundary conditions, formulate a discrete form of the governing equations. Be careful with the sense of the finite differences, considering the order of each derivative, upwind differencing, and boundary-condition information. 

The solution to this equation (unlike the partial differential equation 2.14) has been shown not to depend on the precise formulation of the inlet and outlet conditions, i.e. whether they are open or closed 051. In the following derivation, however, the reaction vessel is considered to be closed , i.e. it is connected at the inlet and outlet by piping in which plug flow occurs and, in general, there is a flow discontinuity at both inlet and outlet. The boundary conditions to be used will be those which properly apply to a closed vessel. (See Section 2.3.5 regarding the significance of the boundary conditions for open and closed systems.)... 

For single-phase turbulent reactor flows, the typical boundary conditions include impermeable solid walls, free surfaces, pressure boundaries, S3mimetry axis, inlet- and outlet conditions. 

The functions /,(, /) and fi(x,t) allow solutions to be derived for special cases of inlet and initial conditions. Note that 5 is a dummy variable of integration. In the case where the column has been washed with eluent and a sample has not been introduced, initial condition 3 from Table I applies, and f 1 (x,t) = 0. After a sample has been introduced and washed into the column by eluent, boundary condition 1 from Table I with c = 0 applies and f x,t) = 0. The eluent volume is Q — Avta. The two limiting cases mentioned above may be superimposed, offset from one another by A v a t = Q = VF where Q is the feed volume. This superimposition is applicable only in the case of linear equilibrium, which yields symmetric solutions. Figure 4 shows the results of these equations graphically for selected values. These values may be particularly applicable to proteins in size exclusion supports. 

The solution of the above equations requires both initial and boundary conditions. Usually the transient begins from the stationary state, so that at t=0 the profiles Ci., i0,z) and T O.z) are available. Inlet concentrations c, and temperature Tm coming from an upstream unit may be constant or fluctuate in time. 

We have previously pointed out that the use of the dispersion model changes the reactor analysis from an initial-value (PFR) to a boundary-value problem. As a result, we should worry about the form of the boundary conditions to use for equation (5-77). This is illustrated in Figure 5.16, where several possible configurations of inlet and outlet conditions are shown. Hopefully, this is not to make a... 

Consider a heterogeneous particle suspended in a straight microchannel (Fig. 3) while the inlet and the outlet are defined as open boundaries. The computational domain is fully covered by unstructured three-dimensional tetra meshes to increase the accuracy of the simulation. The motion of this heterogeneous particle in the microchannel can be simulated using the equations, initial conditions, and boundary conditions listed in above section. 

In the period of 1998-99, two sets of experiments focused on problems of rapid decrease of concentration of boric acid in reactor coolant at nuclear reactor core inlet were performed at the University of Maryland, US, under the auspices of OECD. The situation, when there is an inadvertent supply of boron-deficient water into the reactor vessel, could lead to a rapid (very probably local) increase of reactor core power in reactor, operated at nominal power, or to a start of fission reaction in shut-down reactor (secondary criticality). In the above mentioned experiments the transport of boron-deficient coolant through reactor downcomer and lower plenum was simulated by flow of cold water into a model of reactor vessel. These experiments were selected as the International Standard Problem ISP-43 and organisations, involved in thermal — hydraulic calculations of nuclear reactors, were invited to participate in their computer simulation. Altogether 10 groups took part in this problem with various CFD codes. The participants obtained only data on geometry of the experimental facility, and initial and boundary conditions. 

The inlet conditions also serve as the initial condition. A schematic representation of the cell assembly with the application of various heat source terms and boundary conditions is shown in Fig.4.2. 

The initial and boundary conditions for an initially sorbate free column subjected to a step change in sorbate concentration at the inlet at time zero are... 

Using the differential equation of energy change, derive the partial differential equation and boundary conditions needed for the case of laminar flow of a constant density fluid in a horizontal tube which is being heated. The fluid is flowing at a constant velocity u,. At the wall of the pipe where the radius. r = Tq, the heat flux is constant at. The process is at steady state and it is assumed at z = 0 at the inlet that the velocity profile is established. Constant physical properties will be assumed. 

Using L for the length of the vent line, and the subscripts o and L for the outlet and inlet, the boundary conditions are ... 

In the present paper, three kinds of inlet pressure boundary conditions for high speed thrust bearings operated in the turbulent flow regime are discussed in accordance with three types of lubrication conditions. The analytical expression of pressure distribution for each pressure boundary condition is derived and the static characteristics such as film pressure, load carrying capacity, lubricant flow-rate and the location of pressure center are calculated for a wide range of Reynolds numbers. 

Solutions to Equations 4.97 and 4.98 are dependent on the boundary conditions chosen, and these boundary conditions, in turn, are governed by the assumptions made concerning the reactor arrangement. In the case of chemical reactors, a so-called closed system is usually selected. By this we mean that no dispersion takes place either at the reactor inlet or outlet. Moreover, the dispersion inside the reactor is considered equal in each volume element of the system. For the closed system, the following initial and boundary conditions can be set ... 

A pressure difference and tangential velocity components or the total value of the velocity vector as the boundary conditions at the inlet and outlet is set for motion equations. We use a no-shp condition on the solid wall and boundary conditions of the second kind for the concentration equation. Some initial distribution for the concentration is also given [18]. 

The initial and boundary conditions for these equations would typically specify the reactant gas concentration at the inlet, the initial gaseous reactant profile within the bed, and the initial conversion of the solid reactant. Thus we have... 

The initial and boundary conditions for this system must specify that the initial temperatures and concentrations correspond to ambient conditions, and that ambient conditions be maintained at the inlet. 

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