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The original correlation, shown in Figure 4.4, consistently over-estimates the value of the liquid holdup (α L) in horizontal flow of two-phase gas-Newtonian liquid mixtures. Although it is based on experimental data for the flow of air–water mixtures in small diameter tubes (∼25 mm) at near atmospheric pressure and temperature, this correlation has proved to be quite successful when applied to other fluids and for tubes of larger diameters. Where (−Δ p L/ L) and (−Δ p G/ L) are, respectively, the pressure gradients for the flow of liquid and gas alone at the same volumetric flowrates as in the two-phase flow. Similar idealised models are available for the annular flow of gas and power-law liquids in horizontal pipes but most of them assume the liquid to be in streamline flow and the gas turbulent. This methodology has subsequently been extended to the stratified flow of a gas and power-law liquids. For instance, Taitel and Dukler assumed the interface to be smooth and the interfacial friction factor to be the same as that for the gas, but this model tends to underestimate the two-phase pressure drop. Although such analyses attempt to give some physical insight into the flow mechanism, they inevitably entail gross simplifications and empiricism. Taitel and Dukler have developed a semi-theoretical expression for the average liquid holdup and the two-phase pressure gradient for the stratified flow of mixtures of air and Newtonian liquids. Methods available for the prediction of the average value of liquid holdup fall into two categories: those methods which are based on models which utilise information implicit in the flow pattern and those which are entirely empirical. Richardson, in Non-Newtonian Flow in the Process Industries, 1999 Predictive methods for horizontal flow They have found that the liquid holdup and the pressure gradient as being predicted by Lockhart and Martinelli (1949) variable φ L 2 depending on X (should be presented later) and the power index, n. Heywood and Charles (1979) stretched the method to the stratified flow of a gas and non-Newtonian liquid having nature of both shear thinning and shear thickening, having a limit of 0.1≤ n≤2. The method was evolved from the physical analysis of the flow mechanism however, they have made significant simplifications and use of measurement values into their proposed method. They followed a semiempirical route to determine the value of mean liquid holdup and pressure gradient of the stratified gas–Newtonian liquid mixture.
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The predictive methods to determine the mean liquid holdup for Newtonian fluids are divided between two kinds: one is based on methods that have been evolved from information inherent in the flow pattern (a simplified theoretical scheme for stratified flow can be seen as Appendix B) and the second method is fully empirical due to Taitel and Dukler (1976). Khairuddin Sanaullah, Afrasyab Khan, in Advances in Sustainable Polymer Composites, 2021 7.4.1.3 Calculational methods for liquid holdup in horizontal flows