Bubble material balance

Reservoir engineers describe the relationship between the volume of fluids produced, the compressibility of the fluids and the reservoir pressure using material balance techniques. This approach treats the reservoir system like a tank, filled with oil, water, gas, and reservoir rock in the appropriate volumes, but without regard to the distribution of the fluids (i.e. the detailed movement of fluids inside the system). Material balance uses the PVT properties of the fluids described in Section 5.2.6, and accounts for the variations of fluid properties with pressure. The technique is firstly useful in predicting how reservoir pressure will respond to production. Secondly, material balance can be used to reduce uncertainty in volumetries by measuring reservoir pressure and cumulative production during the producing phase of the field life. An example of the simplest material balance equation for an oil reservoir above the bubble point will be shown In the next section. 

Knowing the bubble rise velocity, the bed expansion can be predicted from a material balance on the bubble phase gas. Thus, total gas flow through the bubble phase equals absolute bubble velocity times the volume fraction E of bubbles in the bed. 

Figure 23.7 Schematic representation of control volume for material balance for bubbling-bed reactor model...

In a typical slurry bubble column operation, the liquid velocity is one order of magnitude lower than the one of gas, and in general, is very low. This mode of operation can be approximated by a semibatch operation. The semibatch operation is frequently used and is the case where the liquid and the catalyst comprise a stationary phase (sluny) in the reactor. In this case, the material balance, eq. (3.122) is used along with the overall rate based on the bulk gas-phase concentration (see Section 3.4.6). In the following, the semibatch operation is presented. 

If the process is continuous and in the complete mixed-flow mode, for both the gas and slurry phases, the equations derived for agitated sluny reactors are valid (see Section 3.5.1) (Ramachandran and Chaudhari, 1980) by simply applying the appropriate mass transfer coefficients. Note that in sluiiy-agitated reactors, the material balances are based on the volume of the bubble-free liquid. Furthermore, in reactions of the form aA(g) + B(l) — products, if gas phase concentration of A is constant, the same treatment holds for the plug flow of the gas phase. 

In this case, the model equations derived for the slurry bubble column reactor are applicable. Note that if the gas-phase concentration is constant, the gas-phase material balance is not needed (where the two reactors have different model equations). 

In this case, the material balance in the liquid phase (3.238) is not applicable as both reactants are gases. Furthermore, as in sluny bubble columns, if the liquid is batch, the overall rate based on the bulk gas-phase concentration is used and the overall mass-transfer coefficient K° is found in the solution of the model (Chapter 5). 

In general, the material balances and the corresponding solutions for trickle and bubble bed reactors are the same, under the assumption that the plug-flow condition holds for both phases. Of course, the appropriate correlations should be used for the estimation of mass transfer coefficients. However, in packed bubble bed reactors, the liquid-phase is frequently found in a complete mixed state, and thus some adjustments have to be made to the aforementioned models. Two special cases will be presented here. 

Applying the appropriate material balances for the solids and the gas, the fraction of the bed occupied by the bubbles and wakes can be estimated using the Kunii-Levenspiel model. The fraction of the bed occupied by that part of the bubbles which does not include the wake, is represented by the parameter d, whereas the volume of the wake per volume of the bubble is represented by a. Consequently, the bed fraction in the wakes is a and the bed fraction in the emulsion phase (which includes the clouds) is 1 — <5 — ot<5. Then (Fogler, 1999)... 

Material balances can be written over a differential section of the bed (dz) for a reactant, in each of the three-phases (bubble, cloud, and emulsion). Then, the equations of the model are as follows ... 

Note that in this case, as in the case of slurry reactors, the material balances are based on the unit volume of the fluid (bubble) phase. The relationship between the rate expressions is (see Section 3.1.1 for derivation)... 

Emulsion phase gas in plug flow Solutions for bubble phase free of solids In the following, a simplified solution is presented under the following assumptions first-order reactions, gas flow only through the bubble phase (fh = 1), and absence of solids in the bubble phase (yb = 0). Under these conditions, the material balances (3.519) and (3.520) become the following. 

Under this condition, the reactant A is unlikely to reach the emulsion phase. Integrating the material balance for the bubble phase, eq. (3.534) yields the desired performance expression in terms of conversion ... 

Kbe can also be accounted for mechanically from its relationship with and Kce given in Eq. (12.77). Following the analysis of Davidson and Harrison (1963) for Kbc and that of Kunii and Levenspiel (1968) for Kce for bubbling fluidized beds based on the bubble configuration described in 9.4.2.1, a material balance on a single rising bubble gives... 

For a given pressure, zl ranges from the liquid composition at the bubble point to the vapor composition at the dew point. Material balance ... 

The theta method. This method has been primarily applied to the Thiele-Geddes equations but a form of the theta method equation has also been applied to the equations of the Lewis-Matheson method. The main independent variable of the method is a convergence promoter, theta (or 6). The convergence promoter 0 is used to force an overall component and total material balance and to adjust the compositions on each stage. These new compositions are then used to calculate new stage temperatures by an approximation of the dew- or bubble-point equation called the Kb method. The power of the Kb method is that it directly calculates a new temperature without the sort of failures that occur when iteratively solving the bubble- or dew-point equations. 

In the Newton-Raphson methods, the bubble point equations and energy balances are solved simultaneously for the stage temperatures and vapor flow rates the liquid flow rates follow from the total material balances [Tierney and coworkers, AlChE J., 13, 556 (1967) 15, 897 (1969) Billingsley and Boynton, AIChE /., 17, 65 (1971)]. Similar methods are described by Holland (op. cit.) and by Tomich [AIChE J., 16,229 (1970)]. 

L. Number of Bubbles in a Compartment, N. With compartment height based on the diameter of the cloud, the number of bubbles can be computed from material balance considerations as well as some assumptions concerning the average solids volume fraction in the bed (14)... 

Because bubble diameter is a function of the height from the distributor, and the height from the distributor is taken to the center of the bubble in question, an iterative procedure is used to determine D]. The initial guess is taken to be the bubble diameter computed for the previous compartment. For each compartment there are three material balance equations with three unknowns, the concentrations in each phase (bubble, cloud and emulsion). The total number of equations then is three times the total number of compartments. These may be solved by a matrix reduction scheme or a trial and error procedure. The average superficial gas velocities in each phase are first determined from Eqns. (4) - (6). The computational sequence for the remaining parameters in Eqn. (1) is given in Table 1. 

Generalized Material Balance Equation for Reservoirs with an Initial Gas Cap and Water Encroachment. Constant volume reservoirs, such as the one considered in the previous section, are seldom met in actual practice. Usually the volume of the reservoir decreases as production progresses because formation water encroaches the reservoir. Furthermore, actual reservoirs often exist with the initial pressure below the bubble point so that a gas cap is present. Consequently, the material balance equation must be extended to include the initial gas cap and the effect of water encroachment. 

Material Balance Equation for a Reservoir Producing Above the Saturation Pressure. When a reservoir exists at a pressure above its bubble point oil can be produced by expansion of the reservoir fluid as the pressure is reduced to the saturation pressure. This process is shown for a constant volume reservoir in Figure 95. The pressure declines from the original reservoir pressure to a pressure which is equal to or greater than the saturation pressure. 

As an illustration, a simplified model will be considered here to show the importance of the nonisothermal effect in the dilute phase. We assume that

bubble phase is negligible, and temperature in the dense phase is uniform. Then the material balance equation for the dense phase is ... 

Bubble Separation Process Descriptions and Definitions Based on the Techniques Used for Bubble Generation Bubble Separation Process Descriptions and Definitions According to the Techniques Used for Solids Separation Bubble Separation Process Descriptions and Definitions According to the Operational Modes Surface Adsorption Bubble Phenomena Multiphase Flow Material Balances... 

In the batch adsorptive bubble separation processes, a feed solution was introduced to a bubble separation column (or chamber) containing an aqueous solution of surface-active materials. Surface-active solutes or complexes that are hydrophobic and readily attachable to the air bubbles are carried up to the surface of the water by the bubbles. The enriched material at the top (whether collapsed foam from a foam separation column or overflow liquid from a nonfoaming bubble separation column) and the clarified drain solution at the bottom are withdrawn from the system. The overall material balance for the process is as follows ... 

In case of continuous adsorptive bubble separation processes, the following set of material balance equations at the steady state will be obtained. 

If represents the molar flow rate of component i in the vapor phase, L = L r, the total vapor flow rate, a the interfacial area per unit volume of froth. Ay the froth height, and Afj the active bubbling area, then the component material balance for the vapor phase may be written as... 

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