## Liquid Velocity

The liquid velocity is one of the most important parameters in the design of ALRs. It affects the gas holdup in the riser and downcomer, the mixing time, the mean residence time of the gas phase, the interfacial area, and the mass and heat transfer coefficients.

Circulation in ALRs is induced by the difference in hydrostatic pressure between the riser and the downcomer as a consequence of a difference in gas holdup. Liquid velocity—like gas holdup—is not an independent variable, because (see Fig. 5) the gas flow rate is the only variable that can be manipulated. As shown in Figure 5, the geometric design of the reactor will also influence the liquid velocity, but this remains constant during operation. Experiments have been carried out in devices specially designed to artificially change the resistance to flow, with the aim of studying the effect of the velocity at a fixed rate of aeration (71,79). The information emerging from these experiments indicates that an increase in the liquid velocity leads to a decrease in the mean residence time of bubbles in the riser and hence of the gas holdup in the riser. In practice, when the gas flow rate is increased, the higher liquid velocity increases the carryover of bubbles from the gas separator into the downcomer; the carryover dampens the liquid flow by reducing the hydrostatic driving force. As a result, the overall change in liquid velocity is tempered.

Liquid Velocity Measurement. Several different methods can be used for measuring the liquid velocity. The most reliable ones are based on the use of tracers in the liquid. If a tracer is injected and two probes are installed in a section of the tube, the velocity of the liquid traveling the distance between probes can be taken directly from the recorded peaks, as the quotient of the distance between the two electrodes and the time required by the tracer to travel from the one to the other. The latter is obtained as the difference of between the first moments of the two peaks. A second method is to calculate the liquid velocity (UL) from the circulation time (ic) and holdup (u) as:

where A is cross-sectional area.

In this case, only one electrode is necessary, u is the holdup at the point at which the electrode is installed, and the circulation time is obtained from two successive peaks recorded by the electrode.

Modeling of Liquid Flow. A number of expressions are available for the estimation of the liquid velocity. Two main methods have been used for the modeling of two-phase flow in ALRs—energy balances and momentum balances. Chakravarty et al. (58) used the energy balance approach to obtain a relationship between superficial gas velocity, holdup, and liquid velocity. Lee et al. (99) calculated UL by a similar type of model for a series of published data for concentric and external-loop ALRs and from their own results for split vessels. In both the above-mentioned models, constants accounting for friction losses were obtained by adjusting the models to the experimental data. Jones (100), on the other hand, managed to express the results of his energy balance (based on previous work of Niklin c

[101] and Freedman and Davidson [73]) in a relationship free of empirical constants. His results, however, fit the experimental data only qualitatively, and the fit is satisfactory only for very small diameters. An improvement of this method was suggested by Clark and Jones (102), who took into account the radial distribution of the gas holdup through the drift flux model. However, the values of the distribution coefficient C0 needed for satisfactory fitting of the experimental data for lager diameters is far from the range usual in this type of flow.

Chisti and Moo-Young (103) extended a model originally proposed by Bello (85), based on an energy balance over the airlift loop. Their expression for the average superficial liquid velocity is:

where UL is the superficial liquid velocity, Ar is the riser cross-sectional area, Ad is the downcomer cross-sectional area, Hd is the downcomer height, Kb and Kt are the hydraulic pressure loss coefficients, yr is the riser gas holdup, and is the downcomer gas holdup.

By choosing suitable values for the friction coefficients in each case, the authors showed that much of the published data on liquid velocity for the different types of ALRs could be satisfactorily correlated by equation 22. Only one coefficient has to be adjusted, since the authors assume that Kt, the friction coefficient at the top of the loop, is negligible in concentric-tube type reactors and that in external-loop reactors Kt can be taken as equal to Kb, the friction coefficient for the bottom of the loop. Equation 22 has thus been adopted by many scientists. Wachi et al. (104) claimed that their derivation of the same equation gives a clearer physical meaning to the adjustable parameters. Equation 22 can also be also derived from a simple momentum balance (77).

Chisti et al. (51) presented an empirical correlation for Kb obtained by comparison of results obtained from several sources:

where Ab is the minimal cross section at the bottom of the airlift reactor and Ad is the downcomer cross-sectional area.

Equation 22 has the particularly that the gas flow rate, which is the main, and often the only, manipulable variable in the operation, is not present directly, but exerts its influence through the gas holdup. Therefore, either experimental data or a valid mathematical expression for the gas holdup in both the riser and the downcomer are required.

Chisti and Moo-Young (103) extended this model further in order to facilitate the prediction of liquid circulation in ALRs operating with pseudoplastic fluids, such as mold suspensions. This improvement is very important, since many commercial fermentation processes involve such non-Newtonian liquids. Kemblowski et al. (86) presented a method for the prediction of gas holdup and liquid circulation in external-loop ALRs. In their experiments there was almost no gas recirculation, because of the large size of the gas separators used.

Garcia Calvo (105) presented an ingenious model based on energy balances and on an idea originally proposed by Richardson and Higson (106), and Garcia Calvo and Leton (107) extended the model to bubble columns. The model is based on the assumption that the superficial gas velocity (JG) in any region can be considered to be the sum of two streams (J and J) as follows. The J stream has a velocity equal to that of the liquid and can therefore be treated by the laws of homogeneous two-phase flow (no slip between the bubbles and the liquid). The second stream (J) is considered to be responsible for all the energy loses at the gasliquid interface. The concept in itself is simple and elegant, and it is possible to envisage its application even to the flow in the downcomer, where UG < UL. In such a case, we would divide the gas flow rate into two parts as follows: One part would be larger than the actual flow rate, i.e., it would have the same velocity as the liquid. In order to arrive at the actual gas flow rate, the second flow rate must have the reverse direction. This type of gas flow can actually be seen under certain conditions, such as when there is coalescence of bubbles and larger bubbles ascend along the walls of the downcomer.

Another technique used by several researchers to predict liquid velocity is the momentum balance of the ALR. This method has been used by Blenke (108) in jet-loop reactors and by Hsu and Dudukovic (109), Kubota et al. (36), Bello (85), Koide et al. (47), and Merchuk and Stein (71). The latter authors presented a simple model for the prediction of the liquid velocity as a function of the gas input in an ALR. They assumed that the pressure drop between the bottom and the top of their external-loop reactor could be expressed as a continuation of the downcomer, using an equivalent length LE. This length was set as an adjustable parameter describing the pressure loss in the loop. Kubota et al. (36) used a similar approach for the analysis of Imperial Chemical Industries' deep-shaft reactor. They were able to simulate the operation of the reactor and to predict the minimum air supply required to prevent flow reversal.

Verlaan et al. (76) used a similar model, in combination with the expression of Zuber and Findlay (74), to calculate the friction coefficients from experimental data reported by several authors for a wide range of reactor volumes. Koide et al. (47) presented an analysis of the liquid flow in a concentric-tube ARL that was also based on a momentum balance. The main difference between this model and that used by Merchuk and Stein (71) was that Koide et al. used a convergence-divergence flow model for the bottom and the top of the loop. At the bottom, the effect of flow reversal on the pressure drop was included in the effective width of the gas-liquid flow path under the lower end of the draft tube hl, which was smaller than the actual gap. Miyahara et al. (57), who studied both the bubble size distribution in an internal-loop ALR and the pressure drop at the top and the bottom of the draft tube, also presented a model facilitating the prediction of the liquid velocity.

Other models use the drift-flux model (59) presented in equations 5-7, as:

UGJ can be taken from equation 10. The range of variation of C0 is rather narrow, as shown in the previous section, and therefore it is not difficult to make a judicious guess as to the value of C0 in an unknown system. The drift flux model has also been used together with energy balances (110) or with the momentum balance (111). Some studies on liquid measurement present the results in the form of empirical correlations (42,52,112). The usefulness of these correlations depends on the amount of data and the number of parameters taken into account.

Most of those correlations are shown on Table 3, and some of them are presented in Figure 18. In general, the superficial liquid velocity increases with an increase in the

Table 3. Liquid Circulation Velocity in ALRs

Formula

Ref.

l ELR ILR

2 Bubbly JLr — 0.024JGr322(A

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