Modeling of Mixing in The Liquid Phase For Bubble Column

Hydrodynamic characteristics (mixing in the liquid phase) in a bubble column with a non-Newtonian liquid phase (aqueous solutions of carboxymethylcellulose, or CMC, at different concentrations) were m easured and correlated. Experi ments in a 0.2 -m diameter, 2.4-m-high bubble column were carried out to determine degree of mixing in the liquid phase at various gas and liquid flow rates. The axial dispersion model was used in the two operating modes, batch and continuous, and the t anks-in-series model was used just in the case of continuous mode. The axial dispersion model with closed-closed boundary conditions fit experimental data quite well and thus was used to estimate the axial dispersion coefficient. This parameter was higher in batch mode than in continuous mode, a nd it s trend was to increase as su perficial gas velocity increased.


Introduction
Many models can be used to describe bubble column reactors.Some aspects to take into account when it is proposed a model are the mathematical nature of the equations and the degree of complex of their solution (Deckwer, 1992).The rising bubbles cause turbulent stochastic diffusion processes and large-scale steady circulation flows (Riquarts, 1981).

Theory
Some of the mathematical models found in literature are presented in this section.Models as perfect mixing (CSTR), partial mixing (ADM) and tubular flow (PFR) may be found in gas and liquid phases operations (Deckwer, 1992  Where L is the exit point (i.e., the length of the reactor).
In case of a step injection, the exit concentration will be given by: …….. ( 6) Where is the maximum concentration that corresponds to feed concentration of tracer and H(t) is the Heaviside function that gives the step form of the obtained answer.

Tank-in-Series Model
This model is a modified CSTR model, where a mass balance of the tracer is made in a generic tank "n" of a series of identical tanks that constitute the system.When the resulting equation from the mass balance is manipulated and the initial and the boundary conditions are applied, the final expression is obtained for each form of tracer injection: For step injection tests (Levenspiel, 1999): Where N is the number of tanks in the system.

The dispersion model
The The boundary conditions used by authors are: While the initial condition is: Where p is a height filled with tracer.
The solution to the differential equation is: Where C E is related through the expression: ......

Results and discussion
This section.Additionally, the fit of experimental data by a proposed correlation through the minimization of the sum of squares of errors using a Mathcad @ subroutine is presented, together with the best estimate of the parameter and the moments calculated from the parameter.
The liquid phase mixing has an important effect on mass transfer capabilities of bubble column.Mixing in bubble columns is due to liquid circulation caused by the rise of the bubbles through liquid phase, reducing or eliminating the concentration gradient, in the system.Because of the high ratio of length to diameter, the radial gradients are often neglected compared to axial gradients (Walter and blanch, 1983).Several mathematical models have been proposed in the literature to describe mixing based on conservation laws or simply based on empirical relations.It is common to use an injection of a tracer at the feed and then measure tracer concentrations at the exit.These collected data are analyzed using, for example, the moment's theory or the transfer function of a mathematical model that could represent the behavior of these experimental data.The disadvantage of the moments method is that moments can be quite sensitive to measurement errors at the tail of the function E(t), (Ostergaard and Michelsen, 1969).In the case of bubble columns, mixing or back mixing of each phase (degree of turbulence) is due to flow or movement of the fluids through the column.
Plug-Flow Reactors (PFR)In an ideal, plug-flow reactor (considered tubular), the fluid is assumed to travel through the system at uniform velocity and in straight streamlines; therefore there are no radial concentration gradients.Under these conditions, the concentration in the reactor, c(t,z), is a function of time and axial position in the reactor.A tracer mass balance on a differential element of fluid inside the reactor, taking into account as initial condition, PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.27,No.10,2009 Modeling of Mixing in The Liquid Phase For Bubble Column 1995 c(0,z) = 0 in the case of an impulse injection, gives the final expression: …… (5)

Figure ( 4 )
Figure(4) shows the response curves for impulse injection of tracer in these boundary conditions.On the other hand, when there are at least two phases in the system, the equation (9) needs to be modified as follows: ...... (22) D Z is obtained from the fitting of Eq. (20) to experimental data, where the authors plotted c/ c E as a function of and took as the distance in the abscise marked when the value of c/ c E = 0.7 and c/ c E = 0.3 intercept the curve obtained from the model for a z/ L. The authors correlated the data and proposed two PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.27,No.are: a column made up by two cylindrical sections of Plexiglas of 0.20 m of inner diameter and an entrance cone, a self-metering pump, two plastic feed tanks, filter devices, a rotameter to measure the gas flow rate, and a pressure transducer connected to a data acquisition system.Two types of experiments were carried out: continuous (the gas and liquid phases are fed continuously in the column in the bottom of the column, flowing in this case in ascendant and cocurrent mode) and semicontinuos modes (the gas phase flow in ascendant mode while the liquid phase was charged to the column at the beginning of the operation).First the CMC solution was prepared in one of the feed tanks.Runs were carried out at various gas and liquid superficial through the axial dispersion model and tank-inseries model using the moment theory and direct fit of the experimental data to the model.As a result of this procedure, the number of tanks in series, Bodenstein number and axial dispersion coefficient in the liquid phase were found, taking into account the multiphase system used in this case PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal, Vol.27,No.10,2009 Modeling of Mixing in The Liquid Phase For Bubble Column 2000 (gas and liquid phases involved in the experiments).Axial dispersion coefficient data were correlated for each flow regime as a function of superficial velocities of gas and liquid phases, and rheological parameters of the power-law model.

Figure ( 6
Figures (8) and (9) for 0.20 and 0.40% CMC solutions.In continuous mode, Eq. (8) for the tanks-in-series model and Eq.(15) for the axial dispersion model with boundary conditions open-open were programmed in Mathcad®.

Figure ( 10 )
Figure (10) shows the experimental data and the models tested for tap water.It is observed that the axial dispersion model (ADM) with openopen boundary conditions does not fit the experimental data.Figures (11)
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