Frictional pressure drop correlations

The homogeneous mixture model is the simplest method for ealculating the frictional two-phase pressure drop, and has been found by Ungar and Cornwell (1992) to agree reasonably well with their experimental data representing the flow of two-phase ammonia in channels with d = 1.46—3.15 mm. 

The two-phase frictional pressure gradient is obtained from  

In the homogeneous flow model, pa is the homogenous mixture density defined 

Several models have been proposed to evaluate the two-phase mixture viscosity, and the model selected may affect the predicted two-phase frictional pressure drop  

The Lockhart and Martinelli (1949) correlation also uses a two-phase friction multiplier, defined by Eq. (5.16). The friction multiplier has been correlated in terms of the Lockhart-Martinelli parameter, X, given by 

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Friedel L (1979) Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow. In 3rd International European Two-Phase Group Meeting, Ispra, Italy, 1979, vol 18, issue 7... 

Figure 3.42 Two-phase friction pressure drop correlation for G = 1 x 106 lb/hr ft2. (From Baroczy. 1966. Copyright 1966 by Rockwell International, Canoga Park, CA. Reprinted with permission.)...

L. Friedel, Improved Friction Pressure Drop Correlations for Horizontal and Vertical Two-Phase Pipe Flow, Paper no. E2, European Two-Phase Flow Group Meeting, Ispra, Italy, 1979. 

H. Muller-Steinhagen, K. Heck, A simple friction pressure drop correlation for two-phase flow in pipes, Chem. Eng. Process., 1986, 20, 297-308. 

Table 9.2 Two-phase laminar-laminar frictional pressure drop correlations.

Dukler, A. E. Moye Wicks III, and Cleveland, R. G. <4. I. Ch. E. Jl. 10 (1964) 38. Frictional pressure drop in two-phase flow. A comparison of existing correlations for pressure loss and holdup. 

In the usual case h and hf are falling in the direction of flow and Ah and Ahf are therefore negative. Values of frictional pressure drop, — APtpf may conveniently be correlated in terms of the pressure drop —APL for liquid flowing alone at the same volumetric rate. Experimental results obtained for plug flow in a 25 mm. diameter pipe are given as follows by Richardson and Higson(6) ... 

For a micro-channel connected to a 100 pm T-junction the Lockhart-Martinelli model correlated well with the data, however, different C-values were needed to correlate well with all the data for the conventional size channels. In contrast, when the 100 pm micro-channel was connected to a reducing inlet section, the data could be fit by a single value of C = 0.24, and no mass velocity effect could be observed. When the T-junction diameter was increased to 500 pm, the best-fit C-value for the 100 pm micro-channel again dropped to a value of 0.24. Thus, as in the void fraction data, the friction pressure drop data in micro-channels and conventional size channels are similar, but for micro-channels, significantly different data can be obtained depending on the inlet geometry. 

Bubbly flow. In bubbly flow, the holdup is generally known and/or near homogeneous flow condition exists, and the frictional pressure drop can be correlated through similarity analysis (Dukler et al., 1964). The development shows that the frictional loss is expressed by a Fanning-type equation,... 

Figure 3.48 Comparison of potassium and sodium two-phase friction pressure drop data with Lock-hart-Martinelli correlation, and with a simple correlation [1/(1 - a)]. (From Fauskeand Grolmes, 1970. Copyright 1970 by American Society of Mechanical Engineers. Reprinted with permission.)...

Use of the friction factor chart or a correlation such as equation 2.19 enables calculation of the frictional pressure drop for a specified flow rate from equation 2.13. 

Fig. 7. Pressure drop correlation of Lockhart and Martinelli for frictional pressure losses in horizontal cocurrent flow.

It should be remembered that these correlations as originally devised by Lockhart and Martinelli were based almost entirely on experimental data obtained for situations in which accelerative effects were minor quantities. The Lockhart-Martinelli correlation thus implies the assumption that the static pressure-drop is equal to the frictional pressure-drop, and that these are equal in each phase. The Martinelli-Nelson approach supposes that the sum of the frictional and accelerational pressure-drops equals the static pressure-drop (hydrostatic head being allowed for) and that the static pressure-drop is the same in both phases. When acceleration pressure losses become important (e.g., as critical flow is approached), they are likely to be significantly different in the gas and liquid phases, and hence the frictional pressure losses will not be the same in each phase. In these circumstances, the correlation must begin to show deviations from experiment. 

In summary, the calculation of pressure drops by the Lockhart-Marti-nelli method appears to be reasonably useful only for the turbulent-turbulent regions. Although it can be applied to all flow patterns, accuracy of prediction will be poor for other cases. Perhaps it is best considered as a partial correlation which requires modification in individual cases to achieve good accuracy. Certainly there seems to be no clear reason why there should be a simple general relationship between the two-phase frictional pressure-drop and fictitious single-phase drops. As already pointed out, at the same value of X in the same system, it is possible to have two different flow patterns with two-phase pressure-drops which differ by over 100%. The Loekhart-Martinelli correlation is a rather gross smoothing of the actual relationships. 

Winkelmann et al. (54) have studied air-water flows in a corrugated heat exchanger. Flow visualization and two-phase pressure drop measurements have been performed. The flow visualizations have shown that the flow pattern is complex and that a wavy or a film flow occurs in most cases (Figure 29). The two-phase pressure drop depends on the total flow rate and vapor quality, and Chisholm-type correlation is proposed. More work is required to characterize the flow structure in compact heat exchangers and to develop predictive methods for the frictional pressure drop and the mean void fraction. 

With knowledge of Cf from the above correlations and using Equation (21) we can calculate frictional pressure drops in straight pipes. However, we also need to be able to evaluate frictional losses in pipe fittings. 

Achievements. Calculation of pipe frictional pressure drop based upon correlated experimental model data. 

The last term is the rate of viscous energy dissipation to internal energy, E = jy dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function <1> for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such cases additional information, which may come from empirical correlations, is needed. 

The pressure-drop correlation is given in Figures 6.18, 6.19a, 6.19b, and 6.20 in terms of the modihed Chilton-Genereaux friction factor as a function of Reynolds number, where... 

The flow pattern depicts a distinct topology (regarding the spatial and temporal distributions of vapor and liquid phases) of two-phase flow and greatly influences the resulting phenomena of heat transfer and friction. An important feature of a particular flow pattern is the direct relationships of the heat transfer and pressure drop characteristics to the pattern type, leading to an easy identification of important macroscopic heat transfer modes. Consequently, an approach to the selection of appropriate heat transfer and/or pressure drop correlations has to be preceded by an identification of the involved flow patterns. 

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