Vessel boundary conditions

We should note one restriction on the E curve—that the fluid only enters and only leaves the vessel one time. This means that there should be no flow or diffusion or upflow eddies at the entrance or at the vessel exit. We call this the closed vessel boundary condition. Where elements of fluid can cross the vessel boundary more than one time we call this the open vessel boundary condition. 

Figure 13.9 The open-open vessel boundary condition.

Bischoff and Levenspiel (1962) have shown that as long as the measurements are taken at least two or three particle diameters into the bed, then the open vessel boundary conditions hold closely. This is the case here because the measurements are made 15 cm into the bed. As a result this experiment corresponds to a one-shot input to an open vessel for which Eq. 12 holds. Thus... 

Boundary Conditions There are two cases that we need to consider boundary conditions for closed vessels and open vessels. In the case of closed-closed vessels we assume that there is no dispersion or radial variation in concentration either upstream (closed) or downstream (closed) of the reaction section, hence tliis is a closed-closed vessel. In an open vessel, dispersion occurs both upstream (open) and downstream (open) of the reaction section hence this is an open-open vessel. These two cases are shown in Figure 14-8, where fluctuations in concentration due to dispersion are superimposed on the plug-flow velocity profile. A closed-open vessel boundary condition is one in which there is no dispersion in the entrance section but there is dispersion in the reaction and exit sections. 

Closed-Closed Vessel Boundary Condition For a closed-closed vessel, we have plug flow (no dispersion) to the immediate left of the entrance line (z = 0 ) (closed) and to the immediate right of the exit z = L z = (closed). However, between z = and z = L v/q have dispersion and reaction. The corresponding entrance boundary condition is... 

Open—Open Vessel Boundary Conditions When a tracer is injected into a packed bed at a location more than two or three particle diameters downstream from the entrance and measured some distance upstream from the exit, the open-open vessel boundary conditions apply. For an open-open system an analytical solution to Equation (14-21) can be obtained for a pulse tracer input. For an open-open system the bormdary conditions at the entrance are... 

This example uses the open-vessel boundary conditions where an inlet (upstream) section and an outlet (downstream) section are added to a tubular reactor where dispersion occurs but no reaction. 

Figure 4.10.57 Experimental set-up for open vessel boundary condition (a) and F functions for a large extent of dispersion (high deviation from plug flow) for an open vessel (b) and closed vessel (c) for different values of 6o = uL/Dax-Adapted from Levenspiel (1999).

The experimental curve is matched to the theoretical curves as given in Figure 4.10.56 (if we have an open vessel boundary condition). 

Figure 4.10.59 Example of an RTD (residence time distribution) curve (open vessel boundary condition, see Figures 4.10.55 and 4.10.57).

Figure 10-9 The dimensionless exit-age distribution (r (t)) for the dispersed plug-flow model as a function of the dimensionless time, t/r, for various values of the dispersion number, D/uL. Open vessel boundary conditions.

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... 

Turbulence may arise by two mechanisms. First, it may result either from a violent release of fuel from under high pressure in a jet or from explosive dispersion from a ruptured vessel. The maximum overpressures observed experimentally in jet combustion and explosively dispersed clouds have been relatively low (lower than 1(X) mbar). Second, turbulence can be generated by the gas flow caused by the combustion process itself an interacting with the boundary conditions. 

Most high-pressure process vessels will be under internal pressure only, the atmospheric pressure outside a vessel will be negligible compared with the internal pressure. The boundary conditions for this loading condition will be ... 

For other boundary conditions or for imperfect pulse injections, modifications must be made in these expressions. For example, for a closed vessel, Levenspiel and Bischoff (9) indicate that... 

In the early days, see, e.g., Bakker and Van den Akker (1994a), a black box representing the impeller swept volume was often used in RANS simulations, with boundary conditions in the outflow of the impeller which were derived from experimental data. The idea behind this approach was that such nearimpeller data are hardly affected by the rest of the vessel and therefore could be used throughout. Generally, this is not the case of course. Furthermore, this approach necessitates the availability of accurate experimental data, not only... 

Applying Immersed or Embedded Boundary Methods (Mittal and Iaccarino, 2005) circumvents the whole issue of the friction between the more or less steady overall flow in the bulk of the vessel and the strongly transient character of the flow in the zone of the impeller. These methods are introduced below. In the context of a LES, Derksen and Van den Akker (1999) introduced a forcing technique for both the stationary vessel wall and the revolving impeller. They imposed no-slip boundary conditions at the revolving impeller and at the stationary tank wall (including baffles). To this purpose, they developed a specific control algorithm. 

The results again demonstrate that, since there is significant backmixing within the vessel (Pe, relatively small), the different boundary conditions and method of solution lead to markedly different values of PeL. 

The boundary conditions for a closed-vessel reactor are analogous to those for a tracer in a closed vessel without reaction, equations 19.4-66 and -67, except that we are assuming steady-state operation here. These are called the Danckwerts boundary conditions (Danckwerts, 1953).1 With reference to Figure 19.18,... 

Open end vessel is one in which there are no discontinuities (abrupt changes) in concentration at the inlet and outlet where both bulk and dispersion flow occur. The boundary condition are C - C0 when z=0 and dC/dz - 0 when z=co. 

A second order reaction with kC0t = 5 is conducted in a vessel with Pe = 10, The feed is partially converted with flnlet = 0.8. Find the effluent yield, and- compare with the yield when finjet = i,o. Closed end boundary conditions. 

A reaction is conducted in an annular vessel with catalyst whose bulk density is pc. Porosity of the bed is e. The reaction is first order with specific rates kc per unit mass on the catalyst and kh per unit volume in the space not occupied by the granules. The two walls are maintained at temperatures Tx and T2 The reaction also proceeds on the walls with corresponding specific rates kwl and kw2 per unit area. Only radial diffusion occurs. Write equations of the material balances and the boundary conditions. 

Boundary conditions are part of the mathematical description of a process. For the energy balance, the condition at the vessel wall is that the rate of heat transfer by conduction equals the rate of transfer to the heat transfer medium. Similarly the rate of mass transfer at the wall equals the rate of reaction on the wall if that is catalytic, or equals zero when the wall is inert and impermeable. Clearly, the temperature, composition and pressure of the inlet to the reactor are part of the problem specification. 

One final reminder, the relationship between Cpui g and the E curves only holds exactly for vessels with closed boundary conditions. 

These relationships show how stimulus-response experiments, using either step or pulse inputs can conveniently give the RTD and mean flow rate of fluid in the vessel. We should remember that these relationships only hold for closed vessels. When this boundary condition is not met, then the and E curves differ. The Cp ise curves of the convection model (see Chap. 15) clearly show this. 

Fortunately, for small extents of dispersion numerous simplifications and approximations in the analysis of tracer curves are possible. First, the shape of the tracer curve is insensitive to the boundary condition imposed on the vessel, whether closed or open (see above Eq. 11.1). So for both closed and open vessels... 

Let us consider two types of boundary conditions either the flow is undisturbed as it passes the entrance and exit boundaries (we call this the open b.c.), or you have plug flow outside the vessel up to the boundaries (we call this the closed b.c.). This leads to four combinations of boundary conditions, closed-closed, open-open, and mixed. Figure 13.7 illustrates the closed and open extremes, whose RTD curves are designated as E c and E. ... 

Now only one boundary condition gives a tracer curve which is identical to the E function and which fits all the mathematics of Chapter 11, and that is the closed vessel. For all other boundary conditions you do not get a proper RTD. 

The shape of the F curve depends on D/wL and the boundary conditions for the vessel. Analytical expressions are not available for any of the F curves ... 

The simplest type of boundary conditions to use in solving Eq. (16) are the so-called infinite pipe conditions. With these conditions, the vessel is assumed to extend from—oo to Physically this means that... 

We have to make a distinction here between electric controllers (e.g. PID controllers) with a proportional valve as actuator and mechanical diaphragm controllers. In a regulation system w/ith electric controllers the coordination between controller and actuator (piezoelectric gas inlet valve, inlet valve A/ith motor drive, butterfly control valve, throttle valve) is difficult because of the very different boundary conditions (volume of the vessel, effective pumping speed at the vessel, pressure control range). Such control circuits tend to vibrate easily when process malfunctions occur. It is virtually impossible to specify generally valid standard values. 

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