#### DMCA

## Experimental Investigation of Multiple Solutions for Liquid Holdup in Upward Inclined Stratified Flow Introduction

### BibTeX

@MISC{Smith_experimentalinvestigation,

author = {Steven P Smith and Garry A Gregory and Harvey W Yarranton},

title = {Experimental Investigation of Multiple Solutions for Liquid Holdup in Upward Inclined Stratified Flow Introduction},

year = {}

}

### OpenURL

### Abstract

Since the mid seventies, mechanistic models have been demonstrated to represent the most promising approach for improving the accuracy of pressure drop and liquid holdup calculations in multiphase flow systems. To date, such models generally involve a combination of rigorous development, classical fluid dynamics, and purely empirical relationships. They can be quite complex but usually account for most important system variables, such as fluid properties, pipe inclination angle, diameter, and roughness and can be capable of very good accuracy. Taitel and Dukler ͓1͔ presented a mechanistic model to describe transitions between five basic flow regimes in horizontal and inclined two-phase flow. Central to the flow pattern model is a generalized model for stratified flow. This is a one-dimensional twofluid model, in which separate steady state momentum balance equations are written for the gas and liquid phases. It has become the foundation for numerous subsequent models. Some years ago, it was discovered ͑Baker et al. ͓2,3͔͒ that for uphill flow at relatively low liquid flow rates, the Taitel-Dukler ͓1͔ model exhibits a region in which three solutions exist for the height of the interface in the stratified/wave flow regime. Each solution represents a different in situ liquid volume fraction. Other two-fluid models that follow the ground work laid by Taitel and Dukler ͓1͔ exhibit similar behavior. One such model, that closely parallels the work of Taitel and Dukler ͓1͔, was the Oliemans ͓4͔ mechanistic model. Several papers have been published which address the multiple solution problem, and the consensus was that one should always choose the solution that yields the minimum in situ liquid fraction. As a result of such publications and discussions of the phenomenon at both the 1991 and 1993 International Conferences on Multiphase Flow, a commercial software implementation ͑PIPE-FLO from Neotechnology Consultants Limited͒ of the Oliemans ͓4͔ model was revised to always select the smallest root as the correct solution. Shortly after the release of the software, the vendor was contacted by a client who claimed that the earlier version of the software, that typically selected one of the larger roots, was in much better agreement with their measured field data. The newer version was predicting liquid holdups that were too low by a factor of twenty or more, which had serious implications with respect to sizing liquid handling facilities. Following many discussions with the client, the vendor concluded that, in this case the larger root gave a better answer. The vendor, because of this experience, raised the issue of multiple roots at the 1995 International Conference on Multiphase Flow. Most of those present felt that the issue had been dealt with. However, representatives of two major European oil companies agreed that there was still uncertainty, based on their own field data and data from similar mechanistic models. Further, they announced that they were initiating internal projects to review the entire situation. To date, however, the author is not aware of any published results from either of those proposed studies. Thus, it is currently unclear whether one should always pick any particular root, or whether circumstances should dictate the choice of one solution over another. Background Taitel and Dukler Stratified Flow Model. In their paper, Taitel and Dukler ͓1͔ proposed a mechanistic model to describe transitions between five basic flow patterns. The model takes into account many of the parameters that can influence the flow regime, including gas and liquid flow rates, fluid properties, pipe diameter, and inclination of the pipe. Their approach begins with the assumption of stratified flow and then proposes mechanisms by which the flow regime might make a transition away from stratified flow. Thus, the basis of their flow pattern model is a generalized model for stratified flow. It is this model that exhibits multiple roots for the liquid level. As noted earlier, the stratified flow model presented by Taitel and Dukler ͓1͔ is a one-dimensional, two-fluid model. It assumes steady, isothermal flow with no mass transfer between the gas and liquid phases. For stratified flow, momentum balances on the two phases ͑liq-uid and gas͒ are given by: Assuming that the pressure gradient is the same in each phase allows these two equations to be combined by equating the pressure drop in the two phases. This yields: The shear stresses are defined in terms of the friction factors. The liquid and gas phase friction factors are assumed to be given by the classic Blasius ͓5͔ relation using hydraulic diameters evaluated as recommended by Agrawal et al. ͓6͔. The specific coefficients used by Taitel and Dukler ͓1͔ are taken from Lockhart and Martinelli ͓7͔. Following Gazley ͓8͔, Taitel and Dukler ͓1͔ assume that f i Ϸf g for smooth stratified flow. They further assume that, at conditions of interest, the gas phase velocity is much greater than the interfacial velocity. Thus, it is assumed that the interfacial shear is identical to the gas wall shear. Taitel and Dukler ͓1͔ then presented the following equation: where dimensionless variables are indicated by a tilde ͑ϳ͒. Reference variables are D for length, D 2 for area, and the superficial phase velocities for the gas and liquid velocities. X is the Lockhart-Martinelli ͓7͔ parameter, given by: The inclination parameter introduced by Taitel and Dukler ͑Y͒ is: where ͉(dP/dx) S ͉ is the pressure drop due to one phase flowing alone in the pipe. Each of the dimensionless variables depends only on the dimensionless liquid height given by: Taitel and Dukler ͓1͔ follow this outline of their model with the statement, ''Thus, each X-Y pair corresponds to a unique value of h L /D for all conditions of pipe size, fluid properties, flow rate, and pipe inclinations for which stratified flow exists.'' 1 It is now known that this statement is not true for all conditions. There is a region for which multiple solutions for h L /D exist. The Taitel and Dukler ͓1͔ paper goes on to discuss mechanisms by which transitions away from stratified flow can occur. Multiple Solution Region of Stratified Flow Models. Baker et al. ͑Baker and Gravestock ͓2͔, Baker et al. ͓3͔͒ introduced new pressure loss and liquid holdup correlations developed specifically for gas/condensate systems. The liquid holdup correlation introduced closely parallels the development by Taitel and Dukler ͓1͔. As such, some ''Notes on the Taitel-Dukler Momentum Balance'' were included in the papers. Since their liquid holdup model demonstrated some erratic behavior, a closer examination of the Taitel-Dukler momentum balance equations was undertaken. It was found that the equations yielded multiple solutions for the dimensionless liquid level for values of the Taitel-Dukler inclination parameter ͑Y͒, defined by Eq. ͑6͒, less than Ϫ3.8. Further, it was found that, of the five multiphase pipelines for which they presented data, four operated, at least partially, within the multiple solution region. To avoid the problem of multiple solutions, the model was adjusted to limit the minimum value of Y to Ϫ3.8. The authors offer no justification for this approach beyond the fact that there was no obvious physical explanation for the multiple solutions. Landman ͓9,10͔ dealt specifically with the multiple solution region of the Taitel-Dukler ͓1͔ model. Landman ͓9,10͔ began with a review of the Taitel-Dukler ͓1͔ model and showed how that model can be applied to flow in a rectangular duct. To verify the method of Taitel and Dukler ͓1͔, he showed that, for laminar flow in a square duct, both the holdup curve (E L versus X͒ developed from a modified set of Taitel-Dukler equations and that developed from an exact solution exhibit the multiple solution region. Landman ͓9,10͔ proposed a physical interpretation of the multiple solutions in terms of the competition between the gravity force and the axial pressure gradient. Landman ͓9͔ also presented a technique to define the boundaries of the multiple solution region. The critical value of Y, presented by Landman ͓9͔, was Ϫ3.737 for turbulent gas/turbulent liquid flow. For turbulent gas/ laminar liquid, the critical value of Y was given as Ϫ3.697. Landman ͓9,10͔ considered both modified Kelvin-Helmholtz ͑K-H͒ and linear stability criteria. These stability calculations show that when three roots for the liquid level are predicted to occur, the root with the lowest holdup is the most likely to be stable, the middle root either stable or unstable, and the upper root unstable. When multiple solutions were found to be stable, transient simulations demonstrated that hysteresis can occur. However, Landman ͓9,10͔ stressed that it was important that work be done to show whether the predictions contained in his work can be verified in the field or in the laboratory. Nitheanandan and Soliman ͓11͔ presented a mechanistic model for determining the stratified flow boundary for condensing flows in horizontal and slightly inclined pipes. Two formulations, a complete and a simplified, are considered for the equilibrium liquid level. The modelling approach followed the work of Taitel and Dukler ͓1͔. The complete formulation of their model yields a single solution for the liquid level while the simplified version can yield three solutions, similar to the multiple solution region of the Taitel-Dukler model. Nitheanandan and Soliman ͓11͔ present no stability criterion, only the fact that the single solution of the complete formulation of their model is always closest to the lowest solution ͑i.e., the minimum liquid holdup͒ in the multiple solution region. Based on this and the earlier works of Landman ͓9͔ and Barnea and Taitel ͓12͔, they conclude that the lowest solution should be accepted. Transactions of the ASME Stability Analyses for the Multiple Solution Region. In their paper ''Structural and Interfacial Stability of Multiple Solutions for Stratified Flow,'' Barnea and Taitel ͓12͔ present an extremely thorough treatment of stability analyses as applied to the multiple solutions for liquid level in stratified two-phase flow. A second work, ''Interfacial and Structural Stability of Separated Flow,'' ͑Barnea and Taitel ͓13͔͒ published in a special supplement to the International Journal of Multiphase Flow, combined the work contained in the 1992 paper and others into a thorough treatment of stability for both stratified and annular flow. Barnea and Taitel ͓12,13͔ recommend that a distinction be made between the stability of the interface and the physical stability of the solution itself. They propose that one should first perform a ''structural stability analysis'' in which it is determined which solution is physically stable; and that only the physically stable solutions be subjected to the K-H stability analysis. The K-H stability criteria deal with the stability of the interface. The structural stability approach, previously applied to annular flow by Barnea ͓14͔, can be applied to evaluate the physical stability of the flow. A non-linear structural stability analysis may show that the upper root is unstable to large disturbances. This type of instability has been shown ͑Barnea and Taitel ͓15͔͒ to lead to a transition to slug flow. It is shown that non-linear structural stability occurs over only a very narrow range of conditions. Of the three solutions, the lowest is always structurally stable, the middle is always unstable by linear structural stability, and the upper is usually unstable by non-linear structural stability. It is also noted that the upper solution is usually unstable to K-H stability. Application of Theory Much of the theory from the works outlined above was incorporated into a computer program. At the heart of this computer program is a routine that solves the Taitel-Dukler ͓1͔ equations for the liquid level and, where there are multiple solutions, evaluates the stability of each solution. The program allows for detailed comparison of the measured data and the theoretical predictions. While the details of the proprietary computer code must remain confidential, Experimental Program Materials. The liquid used was a low viscosity mineral oil. Air was supplied by the building compressors. The air stream was passed through a pair of filters to remove any entrained liquids. From the filter, the air stream passed through a pressure regulator which restricted the pressure to 351.6 kPa. Apparatus University of Calgary Multiphase Flow Loop. The pipeline test section was mounted on a twenty-four meter long triangular truss that could be inclined to a range of angles ͑Ϯ5°͒. The pipeline test section was made up of 1.22 meter flanged sections of transparent acrylic pipe. The inside diameter of the pipe was 50.8 mm. The oil storage system was equipped with a secondary loop that allowed the oil from the storage tanks to be pumped through a bed of calcium chloride pellets and returned to the storage tanks. This ensured that the oil contained no water, which is critical to ensure the accuracy of the volume fraction measurements. The oil storage system was also equipped with a shell and tube heat exchanger. The flow of water ͑used as the coolant͒ through the heat exchanger was controlled by a temperature sensor on the liquid line near the test section inlet manifold. Liquid pumping was provided by one of two available pumps. A bypass line with a pressure regulator allowed excess oil, not required for the specific flow rate being investigated, to be returned to the storage tanks. The liquid flow could be controlled by either an orifice or one of four rotameters. Both control options were tied to the data acquisition system. As noted earlier, air was supplied to the system from the building compressors. Similar to the liquid flow, the gas flow could be controlled by either an orifice or one of two rotameters. Three capacitance type volume fraction sensors were built at the University of Calgary and mounted along the length of the test section. These were mounted in such a way that they did not result in any disturbance of the flow. A complete description of the sensors can be found in Gregory and Mattar ͓16͔. Computerized Data Acquisition System. The data acquisition system associated with the University of Calgary multiphase flow laboratory was developed using the LabVIEW™ software ͑National Instruments ͓17͔͒. The system guides the user through the startup procedure for the multiphase flow loop, incorporates all controls and data output into a single screen displayed while Results The following sections present comparisons of the measured data and the corresponding calculated values. The results of the Landman ͓9͔ parameterization and the search for solutions to the Taitel-Dukler equation are compared. Within the multiple solution region, the results of the various stability criteria are reviewed. Where statistical analyses are presented, the error is defined as: Errorϭ100 E L͑Calc͒ ϪE L͑ M eas͒ E L͑ M eas͒ LaminarÕTurbulent Transition Criteria. The Blasius-type friction factors incorporated into the Taitel-Dukler equation mean that there are actually two forms of the equation depending on whether the liquid phase is characterized as being in the laminar or turbulent flow regime. Background. As noted earlier, the Blasius-type friction factor equations used by Taitel and Dukler ͓1͔ had previously been presented by Lockhart and Martinelli ͓7͔. The coefficients to be used in these equations vary depending on whether the flow is laminar or turbulent. Lockhart and Martinelli ͓7͔ propose that a transition Reynolds number, based on the superficial velocity of the phase, of 1000 should be reasonable. Govier and Aziz ͓18͔ state that deviations from true laminar flow begin to occur at Reynolds numbers of approximately 1225. Fully turbulent flow evolves over the range of Reynolds numbers between 1225 and 3000. Govier and Omer ͓19͔ studied the flow of gas-liquid mixtures in horizontal pipes. Among their observations is that, as the gasliquid ratio increases, transition Reynolds number moves from a single phase value of approximately 2000 to a value of approximately 500 for an input gas-liquid volume ratio of 200.