Boundary conditions, inlet

The main purpose with flow visualization is to make the airflow field or the emission and transport of air contaminants visible and thereby possible to study. In technical terms, flow visualization gives possibilities to study airflow field and contaminant dispersion and changes in it depending on general changes in geometry, boundary conditions, inlet and exhaust airflow, etc. It is... 

The theoretical and numerical basis of computational flow modeling (CFM) is described in detail in Part II. The three major tasks involved in CFD, namely, mathematical modeling of fluid flows, numerical solution of model equations and computer implementation of numerical techniques are discussed. The discussion on mathematical modeling of fluid flows has been divided into four chapters (2 to 5). Basic governing equations (of mass, momentum and energy), ways of analysis and possible simplifications of these equations are discussed in Chapter 2. Formulation of different boundary conditions (inlet, outlet, walls, periodic/cyclic and so on) is also discussed. Most of the discussion is restricted to the modeling of Newtonian fluids (fluids exhibiting the linear dependence between strain rate and stress). In most cases, industrial... 

The treatment of alternative boundary conditions (inlet premixing zone,... 

Identical initial DCPIPox concentrations within the reactor and transient asymmetric boundary conditions (inlet DCPIPox concentration or incident light intensities). ... 

BOUNDARY CONDITIONS IN POLYMER PROCESSING MODELS 95 Inlet conditions... 

Typically velocity components along the inlet are given as essential (also called Dirichlet)-type boundary conditions. For example, for a flow entering the domain shown in Figure 3.3 they can be given as... 

For axial dispersion in a semi-infinite bed with a linear isotherm, the complete solution has been obtained for a constant flux inlet boundary condition [Lapidiis and Amundson,y. Phy.s. Chem., 56, 984 (1952) Brenner, Chem. Eng. Set., 17, 229 (1962) Coates and Smith, Soc. Petrol. Engrs. J., 4, 73 (1964)]. For large N, the leading term is... 

Boundary Conditions In normal operation with closed ends, reactant is brought in by bulk flow and carried away by both bulk and dispersion flow. At the inlet where L = 0 or r = 0,... 

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 ... 

Subject to the boundary conditions that must be imposed at the axis, at the inlet, and on the wall boundaries of the cyclone, Bloor and Ingham found that the solution for 4 may be approximated by the expression... 

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. 

Integrating Equation (3.21) and applying the inlet boundary condition gives... 

The inlet and centerline boundary conditions associated with Equation (8.52) are similar to those used for mass transfer ... 

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... 

These boundary conditions are really quite marvelous. Equation (9.16) predicts a discontinuity in concentration at the inlet to the reactor so that ain a Q+) if D >0. This may seem counterintuitive until the behavior of a CSTR is recalled. At the inlet to a CSTR, the concentration goes immediately from to The axial dispersion model behaves as a CSTR in the limit as T) — 00. It behaves as a piston flow reactor, which has no inlet discontinuity, when D = 0. For intermediate values of D, an inlet discontinuity in concentrations exists but is intermediate in size. The concentration n(O-l-) results from backmixing between entering material and material downstream in the reactor. For a reactant, a(O-l-) [Pg.332]

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... 

Thus, the value for the next, j+, point requires knowledge of two previous points,/ and j— 1. To calculate U2, we need to know both and uq. The boundary conditions. Equations (9.16) and (9.17), give neither of these directly. In finite difference form, the inlet boundary condition is... 

The propagation rate is assumed to be second order with respect to the end-group concentration,. p = ka. The boundary conditions are a specified inlet concentration, zero flux at the wall, and symmetry at the centerline. 

The appropriate boundary conditions are the closed variety discussed in Section 9.3.1. The initial condition is a negative step change at the inlet. A full analytical solution is available but complex. For Pe = uL/D > 16, the following result is an excellent approximation ... 

This is a second-order ODE with independent variable z and dependent variable k C t,z), which is a function of z and of the transform parameter k. The term C(t, 0) is the initial condition and is zero for an initially relaxed system. There are two spatial boundary conditions. These are the Danckwerts conditions of Section 9.3.1. The form appropriate to the inlet of an unsteady system is a generalization of Equation (9.16) to include time dependency ... 

To use these boundary conditions with Equation (15.36), they must be transformed. The result for the inlet is... 

Turbulent inlet conditions for LES are difficult to obtain since a time-resolved flow description is required. The best solution is to use periodic boundary conditions when it is possible. For the remaining cases, there are algorithms for simulation of turbulent eddies that fit the theoretical turbulent energy distribution. These simulated eddies are not a solution of the Navier-Stokes equations, and the inlet boundary must be located outside the region of interest to allow the flow to adjust to the correct physical properties. 

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. 

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