Mass balance bubble phase

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

However with stirring and coalescence and breakup, both effects tend to mix the contents of the bubbles or drops, and this situation should be handled using the CSTR mass balance equation. As you might expect, for a real drop or bubble reactor the residence time distribution might not be given accurately by either of these limits, and it might be necessary to measure the RDT to correctly describe the flow pattern in the discontinuous phase. 

Reaction only takes place in the dense phase since that is where the catalyst particles are. Since the exchange is with a uniform environment where the concentration is cp, we can see that by the time the bubble has reached the top of the bed, the concentration of reactant in it is cp + (co - cp).exp -Tr, where c0 is the entering concentration, H the height of the bed and Tr = QH/UaV is a dimensionless transfer number. By doing a mass balance on the dense phase as a whole14 we obtain a linear equation for cp in terms of the... 

There is no need for an oxygen mass balance in the bubble phase because oxygen is assumed available in excess and the bubble is assumed to be free of solids. 

The reactor and regenerator mass and heat-balance equations for the dense phase and the bubble phase are given by equations (7.29) to (7.45). The catalyst activities in the reactor and regenerator are defined by the following two relations... 

The plots in Figures 7.8 and 7.9 make both Qer and Qeg infinite and therefore the dense phase and bubble phase conditions are identical and are equal to the output conditions of the reactor and the regenerator in this example. In case of finite exchange rates between the bubble and dense phases in reactor and regenerator, the output conditions from the reactor and the regenerator can be obtained by mass and heat balances for the concentration and the temperature of both phases and these expressions use the same symbols as before, but without the subscript D (used to signify the dense phase before). 

In this section we develop a dynamic model from the same basis and assumptions as the steady-state model developed earlier. The model will include the necessarily unsteady-state dynamic terms, giving a set of initial value differential equations that describe the dynamic behavior of the system. Both the heat and coke capacitances are taken into consideration, while the vapor phase capacitances in both the dense and bubble phase are assumed negligible and therefore the corresponding mass-balance equations are assumed to be at pseudosteady state. This last assumption will be relaxed in the next subsection where the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases of both the reactor and regenerator are assumed to be in a pseudosteady state. Based on these assumptions, the dynamics of the system are controlled by the thermal and coke dynamics in the dense phases of the reactor and of the regenerator. 

The mass balance on the inactive bubble phase for the monomer is given by... 

In this, the most common method, air is bubbled at a known rate into a volume of water containing the solute, so that the exit air achieves equilibrium with the water. By measuring the decrease in water concentration, KAW can be deduced from a mass balance. No air phase concentrations are measured. This method is ideal for fairly volatile chemicals, i.e., when Kaw exceeds 1(F3 but can be applied down to about 1(H. Yin and Hassett (1986) modified the method for less volatile chemicals the air phase solute concentration is measured by trapping solute from the exit air stream. Hovorka and Dohnal (1997) have refined the test conditions to achieve greater accuracy. 

This mass balance concerns the liquid phase, since oxygen must be dissolved in order to be used by the cells. Due to the difficulty in measuring the interfacial area (a), especially when oxygenation is carried out by bubble aeration, it is common to use the product of kL times a (kLa), known as the volumetric oxygen transfer coefficient, as the relevant parameter. 

Other work has been mainly concerned with the scale-up to pilot plant or full-scale installations. For example, Beltran et al. [225] studied the scale-up of the ozonation of industrial wastewaters from alcohol distilleries and tomato-processing plants. They used kinetic data obtained in small laboratory bubble columns to predict the COD reduction that could be reached during ozonation in a geometrically similar pilot bubble column. In the kinetic model, assumptions were made about the flow characteristics of the gas phase through the column. From the solution of mass balance equations of the main species in the process (ozone in gas and water and pollution characterized by COD) calculated results of COD and ozone concentrations were determined and compared to the corresponding experimental values. 

Given any complex system of heterogeneous catalytic first order reactions the mass balance on a differential volume element of the reactor at the height h yields the following system of differential equations for the j-th reaction component i) for the bubble phase... 

Similar to the situation in bubbling fluidized beds the two phases exchange gas with each other and are modeled by separate equations which are obtained from mass balances for each component in each phase. 

The dew point and bubble point calculations do not present peculiar problems, but the flash calculation does. Let X fx) be the mole fraction distribution in the feed to a flash, and let a be the vapor phase fraction in the flashed system. The mass balance is ... 

Due to their complexity, the model equations will not be derived or presented here. Details can be found elsewhere [Adris, 1994 Abdalla and Elnashaie, 1995]. Basically mass and heat balances arc performed for the dense and bubble phases. It is noted that associated reaction terms need to be included in those equations for the dense phase but not for the bubble phase. Hydrogen permeation, the rate of which follows Equation (10-51b) with n=0.5, is accounted for in the mass balance for the dense phase. Hydrodynamic parameters important to the fluidized bed reactor operation include minimum fluidization velocity, bed porosity at minimum fluidization, average bubble diameter, bubble rising velocity and volume fraction of bubbles in the fluidized bed. The equations used for estimating these and other hydrodynamic parameters are taken from various established sources in the fluidized bed literature and have been given by Abdalla and Elnashaie [1995]. 

The above discussions pertain to models assuming three regions the dense phase, bubble phase and separation side of the membrane. The membrane is assumed to be inert to the reactions. There are, however, cases where the membrane is also catalytic. In these situations, a fourth region, the membrane matrix, needs to be considered. The mass and heat balance equations for the catalytic membrane region will both contain reaction-related terms. 

Another useful relation results from the mass balance for the gas phase of the bubble column ... 

The equation of continuity for the liquid phase of the bubble column is obtained from a mass balance on tracer material for differential gas-liquid mixed phases. With the same procedure as for a homogeneous flow, the following equation is obtained when 4>(c) = 0 and the aspect ratio L/Dt 1 ... 

Heat and mass transfer constitute fundamentally important transport properties for design of a fluidized catalyst bed. Intense mixing of emulsion phase with a large heat capacity results in uniform temperature at a level determined by the balance between the rates of heat generation from reaction and heat removal through wall heat transfer, and by the heat capacity of feed gas. However, thermal stability of the dilute phase depends also on the heat-diffusive power of the phase (Section IX). The mechanism by which a reactant gas is transferred from the bubble phase to the emulsion phase is part of the basic information needed to formulate the design equation for the bed (Sections VII-IX). These properties are closely related to the flow behavior of the bed (Sections II-V) and to the bubble dynamics. 

In the Westinghouse electrochemical step, hydrogen is released at the cathode interface. In the numerical study, bubble generation is assumed to be localized in the first row of fluid cells neighbouring the electrode. In this special zone analogue to a boundary layer, the rate of gas production is assumed equal to the rate of the reduction process. It is modelled by the source term S2 of the dispersed phase mass-balance equation, assuming a 100% Faradic yield. 

In the case of a total condenser, the vapor-phase compositions used in the calculation of the equilibrium relations and the summation equations are those that would be in equilibrium with the liquid stream that actually exists. That is, for a total condenser, the vapor composition used in the equilibrium relations is the vapor composition determined during a bubble point calculation based on the actual pressure and liquid compositions found in the condenser. These compositions are not used in the component mass balances since there is no vapor stream from a total condenser. 

The corresponding mass (mole) balance for species A in the bubble phase... 

In chap 8 the basic bubble column constructions and the principles of operation of these reactors are described. The classical models for two- and three phase simple bubble column reactors are defined based on heat and species mass balances. The state of the art on fluid djmamic modeling of bubble column reactors is then summarized including a few simulations of reactive flows. 

Chapter 10 contains a literature survey of the basic fluidized bed reactor designs, principles of operation and modeling. The classical two- and three phase fluidized bed models for bubbling beds are defined based on heat and species mass balances. The fluid dynamic models are based on kinetic theory of granular flow. A reactive flow simulation of a particular sorption enhanced steam reforming process is assessed. 

Another interesting example is that of gas bubbles dispersed in a continuous liquid phase with which mass is exchanged. Also for this case the rate of change of the internal coordinates due to mass transfer is written starting from a simple mass balance for a single bubble. Following the standard notation for gas-liquid systems, the single-particle mass balance becomes... 

To model the measured transient foam displacements, equations 2 through 12 are rewritten in standard implicit-pressure, explicit-saturation (IMPES) finite difference form, with upstream weighting of the phase mobilities following standard reservoir simulation practice (10). Iteration of the nonlinear algebraic equations is by Newton s method. The three primitive unknowns are pressure, gas-phase saturation, and bubble density. Four boundary conditions are necessary because the differential mass balances are second order in pressure and first order in saturation and bubble concentration. The outlet pressure and the inlet superficial velocities of gas and liquid are fixed. No foam is injected, so Qh is set to zero in equation... 

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