Residence time distribution boundary conditions

The overall set of partial differential equations that can be considered as a mathematical characterization of the processing system of gas-liquid dispersions should include such environmental parameters as composition, temperature, and velocity, in addition to the equations of bubble-size and residence-time distributions that describe the dependence of bubble nucleation and growth on the bubble environmental factors. A simultaneous solution of this set of differential equations with the appropriate initial and boundary conditions is needed to evaluate the behavior of the system. Subject to the Curie principle, this set of equations should include the possibilities of coupling effects among the various fluxes involved. In dispersions, the possibilities of couplings between fluxes that differ from each other by an odd tensorial rank exist. (An example is the coupling effect between diffusion of surfactants and the hydrodynamics of bubble velocity as treated in Section III.) As yet no analytical solution of the complete set of equations has been found because of the mathematical difficulties involved. To simplify matters, the pertinent transfer equation is usually solved independently, with some simplifying assumptions. 

As the Figure 8.12 reveals, the flow pattern deviates from plug flow. The residence time distribution function E(l) is calculated from the experimentally recorded responses, after which the F(t) function was obtained from integration of E(t). The experimental functions are compared to the theoretical ones. The expressions of E(t) and F(t) obtained from the analytical solution of the dynamic, non-reactive axial dispersion model with closed Danckwerts boundary conditions were used in comparison. A comparison of the results shown in Figure 8.12 suggests that a reasonable value for the Peclet number is Pe=3. 

In this equation, t = t/il with T being the mean residence time. The Bodenstein number is therefore the parameter of the dispersion model used in quantifying the residence time distribution, and it may be obtained from the experimentally measured curves using Equ. 3.6a with particular boundary conditions (see Levenspiel and Smith, 1957) ... 

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).

The axial dispersion model has been discussed exhaustively in the literature. The reader is referred to Levenspiel (57), Nauman and Buffham (4), Wen and Fan (58), and Levenspiel and Bischoff (91) for numerous available references. The appropriateness of various boundary conditions has been debated for decades (92-95) and arguments about their effect on reactor performance continue to the present day (96). We now know that the Danckwerts boundary conditions make the model closed so that a proper residence time distribution can be obtained from the model equations given below (when the reaction rate term is set to zero) ... 

Correct modeling of the flow near the front of a stream requires a rigorous solution of the hydrodynamic problem with rather complicated boundary conditions at the free surface. In computer modeling of the flow, the method of markers or cells can be used 124 however this method leads to considerable complication the model and a great expenditure of computer time. The model corresponds to the experimental data with acceptable accuracy if the front of the streamis assumed to be flat and the velocity distribution corresponds to fountain flow.125,126 The fountain effect greatly influences the distribution of residence times in a channel and consequently the properties of the reactive medium entering the mold. 

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