Hyperbolic streams

Suppose an aerosol is flowing from an opening and impacting against an infinite wall with the air moving in a so-called hyperbolic stream. The velocity components of the air will have the form... 

The first mode may occur when a droplet is subjected to aerodynamic pressures or viscous stresses in a parallel or rotating flow. A droplet may experience the second type of breakup when exposed to a plane hyperbolic or Couette flow. The third type of breakup may occur when a droplet is in irregular flow patterns. In addition, the actual breakup modes also depend on whether a droplet is subjected to steady acceleration, or suddenly exposed to a high-velocity gas stream.[2701[2751... 

The initial conditions for the velocity components are set up so that there is a tubular shear layer aligned along the 2 -direction at time t = 0. The tv-velocity has a top-hat profile with a tan-hyperbolic shear layer. Stream wise and azimuthal perturbations are introduced to expedite roll-up and the development of the Widnall instability. The details can be found in [7]. The initial velocity field is made divergence-free using the Helmholtz decomposition. The size of the computational domain (one periodic cubical box) is 4do on each side. 

Our intent here is not to suggest a solution method but rather to use the stream-function-vorticity formulation to comment further on the mathematical characteristics of the Navier-Stokes equations. In this form the hyperbolic behavior of the pressure has been lost from the system. For low-speed flow the pressure gradients are so small that they do not measurably affect the net pressure from a thermodynamic point of view. Therefore the pressure of the system can simply be provided as a fixed parameter that enters the equation of state. Thus pressure influences density, still accommodating variations in temperature and composition. Since the pressure or the pressure gradients simply do not appear anywhere else in the system, pressure-wave behavior has been effectively filtered out of the system. Consequently acoustic behavior or high-speed flow cannot be modeled using this approach. 

One troublesome aspect of solving low-speed flow problems numerically is dealing with the hyperbolic characteristics of pressure waves. Since the pressure waves usually have no importance in these problems, they are mainly a mathematical and computational nuisance. Therefore techniques to filter pressure waves are often desirable. The stream-function-vorticity approach accomplishes this filtering, but it is not now used very much... 

This set of hyperbolic partial differential equations for the gasifier dynamic model represents an open or split boundary-value problem. Starting with the initial conditions within the reactor, we can use some type of marching procedure to solve the equations directly and to move the solution forward in time based on the specified boundary conditions for the inlet gas and inlet solids streams. 

Typical ratios of gas velocity to solids velocity are about 400, 4200, 1200, at the top of the reactor, in the burning zone, and at the bottom of the reactor, respectively. The solids and gas velocities represent the two characteristic directions for our hyperbolic system. If we plot these velocity curves on a reactor length versus time graph, the characteristic curves for the gas will be essentially horizontal in comparison to the solids stream characteristic because of the large gas to solids velocity ratios. 

The continuity equations for mass and energy were used to derive an adiabatic dynamic plug flow simulation model for a moving bed coal gasifier. The resulting set of hyperbolic partial differential equations represented a split boundary-value problem. The inherent numerical stiffness of the coupled gas-solids equations was handled by removing the time derivative from the gas stream equations. This converted the dynamic model to a set of partial differential equations for the solids stream coupled to a set of ordinary differential equations for the gas stream. 

The fundamental postulate of the time on stream theory is that the activity of the catalyst in a given reaction is purely a function of time. There are different types of catalyst decay according to this theory. The decay may be linear, exponential, hyperbolic and power function as given in Table 7.2. 

Attempts were made to fit the time-on-stream data to simple mathematical forms for comparison among the different catalysts. The exponential form, dA7dt = exp(-tt), where X is the conversion and is a deactivation constant, fit the data rather poorly. The power law of the form -dZ/dt = kX in general fit all the data better. However, it was not possible to obtain a satisfactory fit for all catalysts over the entire 18 h of reaction, using one single value of n. The most useful method was the hyperbolic form ... 

Figure 4.35 shows the time-on-stream behavior for a single CSTR at three levels of holding time according to the exponential decay function. It can be seen that the reactor performance is very sensitive to the relative magnitudes of the decay constant a and the residence time i. Use of the empirical time-on-stream correlation of equation (4-157) amounts to disregarding the kinetics expressed in equation (4-156). Just as a reminder we will repeat that the nature of the deactivation function—exponential, linear, hyperbolic, etc.—will not change the qualitative nature of the results... 

Figure 7.20B2 depicts a typical flow pattern. On each side, two fluids of different refractive indices merge to form co-injected laminar flows and establish a CaCl2 concentration distribution. The convergence of co-injected streams from both sides eventually leads to a complete hyperbolic secantlike refractive index distribution in the main channel. The refractive index profile within the main channel can be adjusted readily by changing the flow rates from different inlets [20]. 

Dimensional gain of the composition process can always be found by writing a material balance across it. If composition of an effluent stream is controlled by manipulating an influent stream, as in this example, the process is linear. But if effluent composition is controlled by manipulating the effluent flow, the process is hyperbolic ... 

Another substantial body of work dealing with cracking reactions has been reported over the past several years by Wojciechow-ski and coworkers. Results with cumene cracking over La-Y zeolite catalyst are summarized in entry 9 of Table 3. Catalyst deactivation in these studies is also treated as separable, but an hyperbolic function of time on stream is used for correlation of activity. The use of this type of correlation for a number of applications was summarized in a 1974 review (70). Since that time, in addition to recent work on cumene cracking (69) the model has been applied to an extensive series of studies on the cracking of gas oil distillates (71,72,73,74), as well as being employed in the correlation of Jacobs, et al. (77). 

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