Exit conditions

BOUNDARY CONDITIONS IN POLYMER PROCESSING MODELS 97 Exit conditions 

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 

To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developed a method called traction boundary conditions . In this method starting from an initial guess, outflow condition is updated in an iterative procedure which ensures its consistency with the flow regime immediately upstream. This method is successfully applied to solve a number of turbulent flow problems. 

Papanastasiou et al. (1992) suggested that in order to generate realistic solutions for Navier-Stokes equations the exit conditions should be kept free (i.e. no outflow conditions should be imposed). In this approach application of Green s theorem to the equations corresponding to the exit boundary nodes is avoided. This is eqvrivalent to imposing no exit conditions if elements with 

FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES 

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These equations are consistent with the isentropic relations for a perfect gas p/po = (p/po), T/To = p/poY. Equation (6-116) is valid for adiabatic flows with or without friction it does not require isentropic flow However, Eqs. (6-115) and (6-117) do require isentropic flow The exit Mach number Mi may not exceed unity. At Mi = 1, the flow is said to be choked, sonic, or critical. When the flow is choked, the pressure at the exit is greater than the pressure of the surroundings into which the gas flow discharges. The pressure drops from the exit pressure to the pressure of the surroundings in a series of shocks which are highly nonisentropic. Sonic flow conditions are denoted by sonic exit conditions are found by substituting Mi = Mf = 1 into Eqs. (6-115) to (6-118). 

The turbine exit condition (for reversible cooled cycles)... 

Shift Rotation and Night Work 3.5.3.2 Level of Description 3.5.3.3 Specification of Entry/Exit Conditions 3.5.3.4 Quality of Checks and Warnings 3.5.3.5 Degree of Fault Diagnostic Support... 

When you can determine that the overall coefficient U or fluid properties vary markedly from the inlet to the exit conditions of the unit, the arithmetic mean is no longer satisfactory for fluid property evaluation. For this case, the proper temperature of each stream is termed the caloric temperature for each fluid. The F fraction is the smallest of the values calculated and applies to both streams. Although the caloric temperature... 

Obtain from Figure 10-116, for Equation 10-181 for both the average conditions and the exit conditions... 

Fauske (1962) developed a phases in equilibrium but separated flow model for a long pipe, which could be used even at exit conditions, by assuming ... 

The header illustrated is 0.5 in. in diameter and feeds three branch lines, each 0.25 in. in diameter. The five nodes are labeled in the diagram. The fluid exits each of the branches at atmospheric pressure and the same elevation, so each of the branch exit points is labeled 5 because the exit conditions are the same for all three branches. The distance between the branches on the header is 60 ft, and each branch is 200 ft long. Water enters the header (node 1) at a pressure of 100 psi and exits the branches (nodes 5) at atmospheric pressure. The entire network is assumed to be horizontal. Each branch contains two globe valves in addition to the 200 ft of pipe and the entrance fitting from the header to the branch. We must determine the... 

Timely rendering of medical assistance and evacuation of people can depend on carrying capacity of escalators, exits, condition of roads and other ways of evacuation, availability of proper evacuation means, and evacuation periods. 

Solution procedure. Because the differential equations must be solved numerically, a two-stage flow of information is needed in the computer program used to solve the problem. Examine Figure E 14.2c. The code GRG2 (refer to Chapter 8) was coupled with the differential equation solver LSODE, resulting in the following exit conditions ... 

If you were called to review such a refinement, you should check that (1) the claimed invariant is bound to be satisfied on first entry to the loop (it is satisfied here, because there s nothing beyond top) (2) if the invariant is true before any iteration of the body, it is bound to be true after (3) the invariant and the exit condition together guarantee the effect claimed for the whole thing (true in this case because when top gets to 0, the whole queue must be sorted). 

Another practical consequence of the large number of cells is that to sample the exit condition of all cells is extremely time consuming. As a result, in the case of exit brine strength for example, only a representative sample of cells was actually measured. The size of this sample was based on the variation in the individual measurements and statistical indicators that show how accurately the global average can be predicted from a small sample. 

In equations 17.103 and 17.104, cps and cpg are the mean specific heats over the ranges of temperature and concentrations encountered. Properties with subscripts 1 and 3 are known from inlet and exit conditions respectively. If the plateau values represented by subscript 2 are in equilibrium, then the values C2, Cv2 and 73 may be found from the equations for any known form of the adsorption isotherm CS2 = f(C2). 

The reactor was optimized using (27) with the direct enforcement error criterion and the reduced SQP algorithm. Here the approximation error tolerance, e, was set to 10, and the dummy elements were added only at elements with active error constraints. In addition, four different choices of initial number of elements (NE = 2,3,4, and 5) were considered in initializing the element partition. The initial and final element partitions are shown in Table IV. The number of SQP iterations and the error norms, for each of these four cases, are also presented there. Initial and final optimal values for the state variables, measured at exit conditions, and the objective function are given in Table V. In addition, the calculated values of exit ammonia... 

Initial and Optimal Exit Conditions with Variable U ... 

If Vj. is to be found and the exit conditions are known, then the procedure is direct. Pick a number of values and integrate graphically. 

Since complete mixing is assumed, Hs and Hg are exit conditions of the fluidized bed. 

That notorious pair, the Danckwerts boundary conditions for the tubular reactor, provides a good illustration of boundary conditions arising from nature. Much ink has been spilt over these, particularly the exit condition that Danckwerts based on his (perfectly correct, but intuitive) engineering insight. If we take the steady-state case of the simplest distributed example given previously but make the flux depend on dispersion as well as on convection, then, because there is only one-space dimension,/= vAc — DA dddz), where D is a dispersion coefficient. Then, as the assumption of steady state eliminates... 

Air saturated 65°F (65°F wet-bulb) has an enthalpy of Hls = 30.4 Btu/lb air at the outlet temperature of 80°F (value interpolated from Table 5.4). This is the exit conditions of the tower and thus represents one endpoint on the operating line (point C). The liquid and air loadings determine the slope of the operating line starting at point C ... 

Note that this approximates the exit condition of Example 17. 

For high-permeability barriers, predictions at the field scale are less sensitive to the form of the specified BCs than low-permeability systems. For most applications, a specified-concentration entrance condition should provide an adequate representation of the source. For the exit condition, the nature of the transition from sorbing to nonsorbing material suggests the use of a zero-derivative condition, although sensitivity analysis indicates that minimal error is introduced by the use of the common semi-infinite condition, which is more amenable to analytical solution. 

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