Superficial gas velocity (cm/s)

Figure 25. Bodenstein numbers of the first and second peaks of the gas residence time distribution in an 80-L internal-loop ALR, as a function of the gas superficial velocity. From Frehlich et al. (131).

Superficial gas velocity (cm/s)

Figure 25. Bodenstein numbers of the first and second peaks of the gas residence time distribution in an 80-L internal-loop ALR, as a function of the gas superficial velocity. From Frehlich et al. (131).

column performance and therefore have limited application for ALRs. Recently, a method facilitating the prediction of the distribution of the energy dissipated in an ALR, based on a simple thermodynamic approach, has been developed (77). Energy dissipation was considered to occur in the ALR by two main mechanisms, wall friction and bubble-associated dissipation (ideal gas behavior was assumed). The work done by the gas on the liquid (and vice versa) was expressed assuming isothermal expansion of the bubbles. The energy dissipation inside the gas phase was considered negligible. The general energy balance (135) was written as:

In this equation, the first term represents the flow work lost by the system under consideration, ED is the energy dissipated per unit of time, and WS is the shaft work done by the surroundings on the system under consideration. The schematic representation of the concentric-tube ALR in Figure 26 indicates the different points in the reactor considered in the mathematical expressions. The expressions found for the energy dissipated in each zone were the following:

Bottom

Results of this model of an ALR can also be used to estimate the global shear rate in each region of the reactor, according to the global approach presented Merchuk and Ben-Zvi (Yona) (96). The shear stress in the liquid of each region of the reactor can be defined as the energy dissipated divided by the mean path of circulation in the region and by the sum of the areas of all the bubbles. For the region i in the ALR

where ti is the residence time of the liquid, hi is the effective length, and ai is the specific interfacial area, in the region i.

A global shear rate yi can be calculated for each region

Riser

(Ed)R = Ql(P4 - P5) - QiPigh - QP ln(P5) (34) Gas separator

(Ed)s = Ql(P5 - P2) - QinP4 ln(P1 - QdP4 ln(P2) (35)

Figure 26. Schematic description of the variables in the ther-modynamic model for energy dissipation distribution in an ALR. From Merchuk and Berzin (77).

Figure 26. Schematic description of the variables in the ther-modynamic model for energy dissipation distribution in an ALR. From Merchuk and Berzin (77).

Si 1

where 1 is the effective viscosity of the fluid.

For liquids exhibiting different types of rheological behavior, the corresponding constitutive equation must be used. Such an approach has been used for the interpretation of shear effects on mammalian cells (136) and algal growth (43).

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