## Predictive methods for upward vertical flow

The previous discussion on holdup related only to horizontal flow of gas - non-Newtonian liquid mixtures. Very few experimental results are available for holdup in vertical upward flow with shear-thinning liquids [Khatib and Richardson, 1984]. These authors used a y-ray attenuation method

Lockhart - Martinelli parameter, X

Figure 4.8 Experimental and predicted (equation 4.6) values of liquid holdup under turbulent conditions for liquid

Lockhart - Martinelli parameter, X

Figure 4.8 Experimental and predicted (equation 4.6) values of liquid holdup under turbulent conditions for liquid to measure the average as well as instantaneous values of liquid holdup for shear-thinning suspensions of china clay and air flowing upwards in a 38 mm diameter pipe. The average values of liquid holdup in streamline flow are in line with the predictions from equation (4.9).

Thus, in summary, average liquid holdup can be estimated using equation (4.6) for Newtonian liquids under all flow conditions, and for non-Newtonian liquids in transitional and turbulent regimes (ReMR > 2000).

For the streamline flow of shear-thinning fluids (ReMR < 2000), it is necessary to use equation (4.9). A further correction must be introduced (equation 4.14) for visco-elastic liquids. Though most of the correlations are based on horizontal flow, preliminary results indicate that they can also be applied to the vertical upward flow of mixtures of gas and non-Newtonian liquids.

4.2.5 Frictions! pressure drop

Generally, methods for determining the frictional pressure drop begin by using a physical model of the two-phase system, and then applying an approach similar to that for single phase flow. Thus, in the so-called separated flow model, the two phases are first considered to be flowing separately and allowance is then made for the effect of interfacial interactions. Irrespective of the type of flow and the rheology of the liquid phase, the total pressure gradient (—A pTP/L) in horizontal flow consists of two components which represent the frictional and acceleration contribution respectively, i.e.

Both a momentum balance and an energy balance for two-phase flow through a horizontal pipe may be written as expanded forms of those for single phase flow. The difficulty of proceeding in this manner is that local values of important variables such as in-situ velocities and holdups of the individual phases are not known and cannot readily be predicted. Some simplification is possible if it is assumed that each phase flows separately in the channel and occupies a fixed fraction of the pipe, but there are additional complications stemming from the difficulty of specifying interfacial conditions and the effect of gas expansion along the pipe length. As in the case of single phase flow of a compressible medium, the shear stress is no longer simply linked to the pressure gradient because the expansion of the gas results in the acceleration of the liquid phase. However, as a first approximation, it may be assumed that the total pressure drop can be expressed simply as the sum of a frictional and acceleration components:

For upward flow of gas-liquid mixtures, an additional term (—A pg) attributable to the hydrostatic pressure, must be included on the right hand side of equation (4.16), and this depends on the liquid holdup which therefore must be estimated.

Thus, complete analytical solutions for the equations of motion are not possible (even for Newtonian liquids) because of the difficulty of defining the flow pattern and of quantifying the precise nature of the interactions between the phases. Furthermore, rapid fluctuations in flow frequently occur and these cannot be readily incorporated into analysis. Consequently, most developments in this field are based on dimensional considerations aided by data obtained from experimental measurements. Great care must be exercised, however, when using these methods outside the limits of the experimental work.

Good accounts of idealised theoretical developments in this field are available for mixtures of gas and Newtonian liquids [Govier and Aziz, 1982; Hetsroni, 1982; Chisholm, 1983] and the limited literature on mixtures of gas and non-Newtonian liquids has also been reviewed elsewhere [Mahalingam, 1980; Chhabra and Richardson, 1986; Bishop and Deshpande, 1986].

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