Outlet, boundary condition

To solve the model, inlet velocities and temperatures are imposed as boundary conditions. Outlet pressures are also imposed. External surfaces are assumed as adiabatic. No turbulence model has been considered. 

Typically the exit velocity in a flow domain is unknown and hence the prescription of Dirichlet-type boundary conditions at the outlet is not possible. However, at the outlet of sufficiently long domains fully developed flow conditions may be imposed. In the example considered here these can be written as... 

Two types of boundary conditions are considered, the closed vessel and the open vessel. The closed vessel (Figure 8-36) is one in which the inlet and outlet streams are completely mixed and dispersion occurs between the terminals. Piston flow prevails in both inlet and outlet piping. For this type of system, the analytic expression for the E-curve is not available. However, van der Laan [22] determined its mean and variance as... 

The criterion for the validity of Equation 8-141 is Npg 1.0. A rough rule-of-thumb is Npg > 10. If this condition is not satisfied, the correct equation depends on the boundary conditions at the inlet and outlet. A procedure for determining dispersion coefficient Dg [ is as follows ... 

Estimate the Peclet number from Equation 8-141. If the value of Npg is 10 or greater, accept it. Otherwise, use a refinement of the theory, which accounts for the boundary conditions at the outlet, or formulate another model. 

Parameters specified are mass flow or velocity. Usually at one outlet, pressure equal to a constant is specified in incompressible flow. If several outlets are present, this pressure boundary condition can only be applied to one outlet, as there are some (unknown) pressure differences between the different outlets. The flow conditions in the rooms are better represented by taking the outlet mass flows when they are known. 

The micro-channels utilized in engineering systems are frequently connected with inlet and outlet manifolds. In this case the thermal boundary condition at the inlet and outlet of the tube is not adiabatic. Heat transfer in a micro-tube under these conditions was studied by Hetsroni et al. (2004). They measured heat transfer to water flowing in a pipe of inner diameter 1.07 mm, outer diameter 1.5 mm, and 0.600 m in length, as shown in Fig. 4.2b. The pipe was divided into two sections. The development section of Lj = 0.245 m was used to obtain fully developed flow and thermal fields. The test section proper, of heating length Lh = 0.335 m, was used for collecting the experimental data. 

One particular characteristic of conduction heat transfer in micro-channel heat sinks is the strong three-dimensional character of the phenomenon. The smaller the hydraulic diameter, the more important the coupling between wall and bulk fluid temperatures, because the heat transfer coefficient becomes high. Even though the thermal wall boundary conditions at the inlet and outlet of the solid wall are adiabatic, for small Reynolds numbers the heat flux can become strongly non-uniform most of the flux is transferred to the fluid at the entrance of the micro-channel. Maranzana et al. (2004) analyzed this type of problem and proposed the model of channel flow heat transfer between parallel plates. The geometry shown in Fig. 4.15 corresponds to a flow between parallel plates, the uniform heat flux is imposed on the upper face of block 1 the lower face of block 0 and the side faces of both blocks... 

If the range of the channel height is limited to be above 10 pm, then the no-slip boundary condition can be adopted. Furthermore, with the assumptions of uniform inlet velocity, pressure, density, and specified pressure Pout at the outlet, the boundary conditions can be expressed as follows ... 

The boundary conditions (10.12-10.14) correspond to the flow in a micro-channel with a cooled inlet and adiabatic receiver (an adiabatic pipe or tank, which is established at the exit of the micro-channel). Note, that the boundary conditions of the problem can be formulated by another way, if the cooling system has another construction, for example, as follows x = 0, Tl = IL.o, x = L, Tg = Tg.oo, when the inlet and outlet are cooled x = 0, dT /dx = 0, x = L, Tg = Tg.oo in case of the adiabatic inlet and the cooled outlet, etc. 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

The constants Ci and C2 are evaluated using the boundary conditions. Equations (9.16) and (9.17). The outlet concentration is found by setting z = L. Algebra gives... 

Solution An open system extends from —oo to +oo as shown in Figure 9.9. The key to solving this problem is to note that the general solution. Equation (9.18), applies to each of the above regions inlet, reaction zone, and outlet. If k = Q then p=. Each of the equations contains two constants of integration. Thus, a total of six boundary conditions are required. They are... 

The outlet boundary condition for this unsteady but closed system is a generalization of Equation (9.17) ... 

At the solid walls, the boundary conditions state that the velocity is zero (i.e. no slip). Also at the walls, the temperature is either fixed or a zero-gradient condition is applied. At the surface of the spinning disk the gas moves with the disk velocity and it has the disk temperature, which is constant. The inlet fiow is considered a plug fiow of fixed temperature, and the outlet is modeled by a zero gradient condition on all dependent variables, except pressure, which is determined from the solution. 

An Eulerian-Eulerian (EE) approach was adopted to simulate the dispersed gas-liquid flow. The EE approach treats both the primary liquid phase and the dispersed gas phase as interpenetrating continua, and solves a set of Navier-Stokes equations for each phase. Velocity inlet and outlet boundary conditions were employed in the liquid phase, whilst the gas phase conditions consisted of a velocity inlet and pressure outlet. Turbulence within the system was account for with the Standard k-e model, implemented on a per-phase basis, similar to the recent work of Bertola et. al.[4]. A more detailed description of the computational setup of the EE method can be found in Pareek et. al.[5]. 

The boundary conditions determine the form of balance equation for the inlet and outlet sections. These require special consideration as to whether diffusion fluxes can cross the boundaries in any particular physical situation. The physical situation of closed ends is considered here. This would be the case if a smaller pipe were used to transport the fluid in and out of the reactor, as shown in Figs. 4.13 and 4.14. 

Referring to Fig. 4.15, it is seen that the concentration and the concentration gradient are unknown at Z=0. The above boundary condition relation indicates that if one is known, the other can be calculated. The condition of zero gradient at the outlet (Z=L) does not help to start the integration at Z=0, because, as Fig. 

The representation of the boundary conditions for both the top and bottom of the column are really more mathematical than practical in nature and fail to take into account the actual geometry and construction of the upper and lower parts of the column and the relative positioning of the inlet and outlet connections. They may therefore require special modelling appropriate to the particular form of construction of the column, as discussed previously in Sec. 3.3.1.10. 

The consideration of the boundary conditions again follows Franks (1967). The position at the tube inlet and shell outlet section, segment number 1 is shown in Fig. 4.28. 

The boundary conditions are satisfied by Cq = 1 for a step change in feed concentration at the inlet, and by the condition that at the outlet C n+i = C n, which sets the concentration gradient to zero. The reactor is divided into 8 equal-sized segments. 

Pressure differentials at the exit of the length steps are used to calculate cross-flows. Crossflows are then used to calculate pressure differentials at the outlet of the previous (upstream) length steps. These pressure differentials are saved for use during the next iteration. At the exit of the last length step, the boundary condition of zero pressure differential is imposed and crossflows at the exit are calculated on this basis. The iterative procedure forces agreement with the assumed boundary condition. 

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