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Mechanistic multidimensional analysis of horizontal two-phase flows
Nuclear Engineering and Design 240 (2010) 405–415
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Mechanistic multidimensional analysis of horizontal two-phase flows Elena A. Tselishcheva ∗ , Steven P. Antal, Michael Z. Podowski Rensselaer Polytechnic Institute, Troy, NY, USA
a r t i c l e
i n f o
Article history: Received 25 October 2008 Received in revised form 23 August 2009 Accepted 6 September 2009
a b s t r a c t The purpose of this paper is to discuss the results of analysis of two-phase flow in horizontal tubes. Two flow situations have been considered: gas/liquid flow in a long straight pipe, and similar flow conditions in a pipe with 90◦ elbow. The theoretical approach utilizes a multifield modeling concept. A complete three-dimensional two-phase flow model has been implemented in a state-of-the-art computational multiphase fluid dynamics (CMFD) computer code, NPHASE. The overall model has been tested parametrically. Also, the results of NPHASE simulations have been compared against experimental data for a pipe with 90◦ elbow. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The development of modeling and simulation capabilities of two-phase flow and heat transfer is very important for the design, operation and safety of nuclear reactors. Whereas significant progress in this field has been made over the recent years (Podowski, 2005), further advancements are clearly needed for new concepts of Advanced Boiling Water Reactors and future Generation-IV reactors (e.g., in the safety analysis of the proposed Sodium Fast Reactor). In the past, the complexity of various possible gas/liquid flow patterns has limited the scope and range of situations that could be analyzed using full three-dimensional models. In fact, most computer simulations of multidimensional gas/liquid two-phase flows performed to date have been concerned with vertical tubes or conduits. The reason for that was that in vertical flows gravity mainly affects the axial gas/to-liquid relative velocity, but does not induce any lateral asymmetry in either velocity or phase distribution. On the other hand, in the case of horizontal (or inclined) flows, the acceleration of gravity not only causes a significant flow asymmetry, but also imposes an extra vertical force across the main flow direction, which is typically much stronger that other interfacial forces such as the lift force, for instance. Needless to say, the level of difficulty increases further for two-phase flows in conduits of complex geometries and spatial orientations, where a variety of flow patterns have been observed, each characterized by different interfacial phenomena of mass, momentum and energy transfer. The purpose of this paper is to discuss the results of analysis of two-phase flow in horizontal tubes. Two pipe geometries have
∗ Corresponding author. E-mail addresses: [email protected] (E.A. Tselishcheva), [email protected] (S.P. Antal), [email protected] (M.Z. Podowski). 0029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2009.09.015
been considered: first, a long straight pipe, and then, a similar pipe but with 90◦ elbow in the mid-section along the flow. In the latter case, the geometry and flow conditions were similar to those used in the air/water experiments performed by Kim et al. (2005). The combined theoretical and computational approach is based on a complete three-dimensional multifield model which has been implemented in a state-of-the-art computational multiphase fluid dynamics (CMFD) computer code, NPHASE (Antal et al., 2000; Tiwari et al., 2006). NPHASE is a multicomponent/multiphase CFD code, which solves the individual transport equations for momentum, energy, and turbulence for each field. The key features of PHASE include the built-in mechanistic models of interfacial phenomena, robustness and numerical accuracy. Past experience with NPHASE clearly shows that this code employs several features uniquely qualifying it for use in the analysis of multiphase flows in complex geometries. The proposed model has been tested parametrically for both physical and numerical consistency. Then, the same model has been used to simulate selected experimental runs reported by Kim et al. (2005). The results of comparisons between the predictions and experimental data are shown in Section 5. 2. Overview of experimental facility and data A schematic diagram of the test facility documented in Kim et al. (2005) is shown in Fig. 1. The experimental test section was made of Pyrex glass tubes 50.3 mm in inner diameter. The measurement ports were made of clear acrylic. In the test section, a 90◦ elbow was installed at L/D = 197. The elbow had a radius of curvature of 76.2 mm with the length corresponding to L/D ratio of approximately 6. For the purpose of measurements, four local ports were installed along the test section, denoted as P1 through P4 in Fig. 1. Yet another port (P0) was located right after the two-phase mixing chamber,
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Fig. 1. A schematic diagram (Kim et al., 2005) of the 50.3 mm ID horizontal two-phase flow test facility with a 90◦ elbow junction (top view, not to scale).
but it was used only for the local pressure measurements. The next port, P1, was located at L/D = 197 (9.91 m from P0), just before the start of the elbow, whereas the ports, P2, P3 and P4, were located at the nondimensional distances, L/D = 225, 250 and 329 from the mixing chamber, or 1.39 m, 2.69 m and 6.65 m downstream of the start of the elbow, respectively. The measurements of void fraction and velocity were taken at P1, P3 and P4. Local two-phase flow parameters were measured using doublesensor and four-sensor conductivity probes. In order to capture the asymmetry of distribution of major flow parameters across the horizontal two-phase flow, the measurements were taken along the entire vertical diameter of the tube. In total, 15 different jf and jg combinations were investigated, all corresponding to slug flow conditions. Sample plots are presented in Fig. 2. As can be seen, there is almost no gas in the lower half of the pipe. It is interesting to notice that the void fraction in the upper half first increases dramatically, then stays almost constant and finally increases further near the top of the pipe wall. The results acquired from the experiments were the local flow rates, local pressure, and two-phase parameters, such as void fraction and bubble velocity. The area-averaged superficial velocity, evaluated as the sum of the products of local void fraction and gas velocity, was compared against the local superficial velocity measured by a flow meter. As mentioned before, the present measurements were at Port 1, Port 3 and Port 4, and no data was taken at Port 2. The evolution of void fraction along the experimental section is presented in Fig. 3. The experimental data shown in this figure include those at Port 1 (before the elbow), Port 3 and Port 4 (downstream after elbow). They correspond to Run 1, and to the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. It can be seen that all three curves follow a similar trend and cover the same range of values, although some difference in the void fraction profiles can also be observed. In particular, the highest average void fraction is acquired at Port 3, which is located 2.69 m after the start of the elbow; the minimum is at Port 1.
To compare the numerical solutions obtained using the NPHASE code with the experimental data, several inlet conditions have been chosen. These conditions are summarized in Table 1. In Table 1, jg,atm is the gas superficial velocity equivalent to the atmospheric pressure condition, jf is the average of the liquid superficial velocities acquired at four measurement ports for each run, jloc port-0 is the gas superficial velocity acquired by the flow meter at Port 0, Gg,loc is the mass flux calculated at Port 0 as a product of jloc port-0 and the gas density, and ˇ is the volumetric flow fraction at Port 0. The experimental data used as a reference in the NPHASE-based numerical solutions have been taken at Port 3 and at Port 4. Since the distance between the elbow and Port 4 was sufficiently long, the conditions there corresponded to a nearly fully developed flow, except for a small change in the fluid properties with decreasing pressure. The individual terms for the runs listed in Table 1 are given in detail in Table 2. They are defined as follows: jg,loc is the gas superficial velocity measured by flow meter, jg is the calculated area-averaged superficial velocity found using the average values of two-phase parameters, Gg,loc is the mass flux calculated as a product of jg,loc and the gas density, Gg is the mass flux calculated as a product of jg and the gas density, ˇloc is volumetric flow fraction corresponding to the local measurement, and ˇ is the volumetric flow fraction corresponding to area-averaged values. In view of measurement errors, the superficial gas velocity at given measurement port obtained by the flow meter was compared with that obtained by the area-averaged two-phase flow parame-
Table 1 Summary of test conditions used in the NPHASE simulations. Test number
jf
jg,atm
jloc port-0
Gg,loc
ˇ
1 2 3 4
0.56 1.65 0.56 1.65
0.287 0.293 0.587 1.218
0.2639 0.256 0.5384 1.0336
0.3439 0.3566 0.7058 1.4822
0.3203 0.1343 0.4902 0.3852
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Fig. 2. Typical profiles of local void fraction acquired along the vertical diameter of the tube with 90◦ elbow (Kim et al., 2005), corresponding to the inlet conditions: (a) jg,atm = 0.287 m/s, jf = 0.559 m/s; (b) jg,atm = 0.293 m/s, jf = 1.648 m/s; (c) jg,atm = 0.587 m/s, jf = 0.560 m/s; (d) jg,atm = 1.218 m/s, jf = 1.652 m/s.
ters at the same port. It can be seen from Table 2 that there are significant differences between the flow conditions obtained by using each method. For example, the maximum error between the local measured and calculated superficial velocities was nearly 20%. Another important observation is concerned with some discrepancies in the measurements of the volumetric flow rates at Port 3 and Port 4. Whereas, for ideal adiabatic flows (at constant pressure) the
volumetric flow rates of both phases should be constant along the flow, the actual air expansion due to the frictional pressure drop caused a slight increase in the gas volumetric flow rate (and superficial velocity). As can be seen in Table 2, the measurements for Cases 1, 3 and 4 show different trends. Needless to say, such experimental errors will have to be accounted for when using the present data for model validation.
Table 2 Summary of the data for air–water slug flow through a horizontal tube of 50.3 mm inner diameter with a 90◦ elbow. Test number
Gas superficial velocity
Local pressure, p (psig)
Jg,loc
jg
Port 3 1 2 3 4
0.2701 0.2689 0.5509 1.0946
0.2596 0.3077 0.4545 1.0921
1.0 1.0 1.0 1.4
Prot 4 1 2 3 4
0.2697 0.2704 0.5513 1.0991
0.2276 0.3321 0.4182 1.0844
1.0 1.3 1.0 1.5
Local air density, loc
Gas mass flux
Void fraction
Volumetric flow fraction
Gg,loc
Gg
˛
ˇloc
ˇ
1.286 1.286 1.286 1.319
0.3473 0.3458 0.7085 1.4438
0.3338 0.3957 0.5845 1.4405
0.2530 0.1150 0.4430 0.3380
0.3246 0.1403 0.4959 0.3988
0.3160 0.1573 0.4480 0.3983
1.286 1.311 1.286 1.327
0.3468 0.3545 0.7090 1.4585
0.2927 0.4354 0.5378 1.4390
0.2440 0.1230 0.4220 0.3470
0.3243 0.1410 0.4961 0.3998
0.2882 0.1677 0.4275 0.3966
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of the model’s state variables. For gas/liquid flows in vertical channels, the total interfacial forces are typically partitioned into several components, as shown below Mil−v = −Miv−l = FD + FVM + FL + FTD + FW FD
FVM
(6) FL
where is the drag force, is the virtual mass force, is the lift force, FTD is the turbulent dispersion force, and FW is the wall force. Whereas the expressions for the drag, virtual mass and lift forces have been extensively documented and used before (Ishii and Zuber, 1979; Anglart et al., 1997; Drew and Passman, 1998), the component forces which are particularly important for the present analysis are the turbulent dispersion force and the wall force. A consistent derivation of the mechanistic model of turbulent dispersion force (Podowski, 2009), yields the following expression FTD = CTD l ˛∇ [(1 − ˛)k] Fig. 3. Typical lateral profiles of local void fraction acquired at different measurement ports of the tube with 90◦ elbow (Kim et al., 2005), corresponding to the following inlet conditions: jg,atm = 0.287 m/s, jf = 0.559 m/s.
3. Model description Applying the multifield modeling concept to adiabatic twophase gas/liquid flows yields the following form of phasic momentum equations for the liquid and gas phase, respectively: ∂[(1 − ˛)l vl ] + ∇ · [(1 − ˛)l vl vl ] ∂t = −(1 − ˛)∇ pl − (pil − pl )∇ ˛ + (1 − ˛)(∇ · ) l
+ ( i − ) · ∇ ˛ + (1 − ˛)l g + Miv−l l
(1)
l
∂(˛v vv ) + ∇ · (˛v vv vv ) = −˛∇ pv + (piv − pv )∇ ˛ + ˛(∇ · ) v ∂t − ( i − ) · ∇ ˛ + ˛v g + Mil−v v
v
(2)
where M denotes the total interfacial force, the superscript ‘i’ denotes the interfacial properties, and the remaining notation is conventional. Whereas Eqs. (1) and (2) apply to multicomponent flows of interpenetrating media (e.g., the flow of immiscible liquids), they are not, in general, applicable to dispersed “particle” flows such as gas/liquid bubbly flows. This is because the dispersed field is not capable of transmitting either the pressure force or the shear force (since the individual particles interact with the continuous field only, not with each other). However, it has been shown that Eqs. (1) and (2) can still be used to model bubbly flows provided two conditions are satisfied simultaneously (Podowski, 2009), namely pv = piv = pil = pl
and
= i = i = = v
v
l
l
(3)
where p is the only measurable continuous field pressure and is the corresponding total shear stress combining the molecular and turbulent components. Using Eq. (3), Eqs. (1) and (2) become ∂[(1 − ˛)l vl ] + ∇ · [(1 − ˛)l vl vl ] ∂t = −(1 − ˛)∇ p + (1 − ˛)(∇ · ) + (1 − ˛)l g + Miv−l ∂(˛v vv ) + ∇ · (˛v vv vv ) = −˛∇ p + ˛(∇ · ) + ˛v g + Mil−v ∂t
(4)
(5)
To complete the model given by Eqs. (4) and (5), closure laws must be formulated for the interfacial forces, Mil−v and Miv−l , in terms
(7)
where CTD = 1/3 for isotropic turbulence. Whereas for flows in vertical conduits, Eq. (7) already accounts for the fact that turbulence-induced force pushes the bubbles away from the channel surface in the near-wall region, in the case of horizontal flows, the wall reaction force to counterbalance the effect of buoyancy becomes the major vertical component of the overall lateral interfacial force. This is illustrated in Fig. 4 for the case of horizontal slug flows, which correspond to intermediate relative gas superficial velocities (or void fractions). As can bee seen in Fig. 4, the reaction force is limited to the conduit section occupied by the bubbles, and becomes zero elsewhere. If the elongated gas bubbles are not in direct contact with the walls, which is the situation corresponding to good wall-wettability conditions, this force will be transmitted across a thin liquid film between the bubbles and the wall, but it is practically independent of the film thickness. Furthermore, recognizing that this is a buoyancy-induced force (it disappears for neutrally/buoyant flows), its value per unit volume can be expressed as FR = CR (l − v )˛(1 − ˛)
g |g|
(8)
where the coefficient, CR , is defined as
CR =
g(1 − ε) 0
if y > yc if y ≤ yc
(9)
In Eq. (9), the parameter, yc , depends on the size of elongated bubbles and corresponds to the vertical position of the lower boundary of the bubbles. The adjustable constant, ε, is used to assess the effect of modeling uncertainties on the accuracy of predictions, such as: the effect of surface tension on the shape of elongated bubbles, the impact of nonuniform distribution of void fraction in the upper region of the conduit due to the shape variations of the elongated bubbles, and the presence of small dispersed bubbles. All such issues clearly deserve separate thorough investigations in the future. In horizontal flows of dispersed bubbles, the major interfacial force along the flow direction is the drag force. Thus, when the equilibrium is reached between the phases in ideal fully developed flows, the local bubble velocity is approximately equal to the velocity of the surrounding liquid and the local drag force reduces to nearly zero. At the same time, the force balance across the flow comprises of the following forces: gravity, buoyancy, turbulent dispersion, lift and wall reaction forces. In the present case, the effect of lift is practically negligible since the local relative velocity is very small at near-fully developed flow conditions. As mentioned before, since the wall reaction force is transmitted across the near-wall bubbles, for large bubbles characteristic to slug flows the range of this force may extend over a considerable distance from the upper wall.
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Fig. 4. A simplified schematic of slug flow in a horizontal conduit.
Fig. 5. An illustration of the three-dimensional numerical grid used in the NPHASEbased simulations of gas/liquid flow in a horizontal tube with 90◦ elbow: (a) the axial mesh and (b) the cross-sectional mesh.
4. Computational approach The model discussed in Section 3 has been implemented in the NPHASE code as a complete three-dimensional problem corresponding to the pipe geometry used in the experiments summarized in Section 2. The current formulation is applicable to both a straight pipe and to a pipe with 90◦ elbow. The computational grid used in the latter case is shown in Fig. 5. Fig. 5(a) shows the axial nodalization of the pipe with elbow, whereas Fig. 5(b) presents the unstructured grid used across the pipe. The pseudo-structured (regular) portion of the grid near the pipe wall helps to properly capture the wall effect on local velocity and phase distributions there. In the case of numerical model testing for a straight pipe, the same lateral
Fig. 6. The NPHASE-predicted radial void fraction profiles at the exit of a straight horizontal tube and a similar tube but with a 90◦ elbow, compared against the experimental data (Kim et al., 2005).
grid was combined with a regular axial discretization scheme. The effect of grid size has been investigated separately, showing that the current grid the needed accuracy level for NPHASE simulations. According to the system of coordinates shown in Fig. 5, the main flow direction gradually changes from that along x at the inlet to that along z at the outlet, whereas y always defines the vertical coordinate across the flow. Consequently, the gravity vector becomes, g = {gx , gy , gz }, where gx = gz = 0, gy = −g.
Fig. 7. Radial profiles of: (a) void fraction and (b) liquid and gas axial velocities, in a horizontal tube with 90◦ elbow, at Port 4 in Run 1 of the experiments (Kim et al., 2005).
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Fig. 8. Pressure along the centerline of straight horizontal tube and along at the outer and inner edge of horizontal tube with 90◦ elbow in Run 1 (a) and Run 2 (b) of the experiments reported in Kim et al. (2005).
Fig. 9. The radial void fraction and liquid and gas velocity profiles in a horizontal tube with 90◦ elbow, corresponding to Port P3 (a and b) and Port 2 (c and d) in Run 1 of the experiments reported in Kim et al. (2005).
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Fig. 10. The void fraction and liquid and gas velocity profiles along horizontal diameter in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 1 of the experiments reported in Kim et al. (2005).
Fig. 11. The radial void fraction and liquid and gas velocity profiles at 0.5 radius from the symmetry line in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 1 of the experiments reported in Kim et al. (2005).
Fig. 12. The radial void fraction and liquid and gas velocity profiles at several sections along the 90◦ elbow of a horizontal tube for Run 1 of the experiments reported in Kim et al. (2005).
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Fig. 13. The radial void fraction and liquid and gas velocity profiles in a horizontal tube with 90◦ elbow, corresponding to Port P3 (a and b) and Port 2 (c and d) in Run 2 of the experiments reported in Kim et al. (2005).
Fig. 14. The radial void fraction and liquid and gas velocity profiles in a horizontal tube with 90◦ elbow, corresponding to Port P4 in Run 2 of the experiments reported in Kim et al. (2005).
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Fig. 15. The void fraction and liquid and gas velocity profiles along horizontal diameter in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 2 of the experiments reported in Kim et al. (2005).
4.1. Model validation The model presented in Sections 3 and 4 has been implemented in the NPHASE code and numerically tested for conditions similar to those used in the experiments (Kim et al., 2005). The test calculations have been performed for two geometries: a straight cylindrical tube and a tube with 90◦ elbow. The results illustrating the effect of elbow on phase distribution at the end of the pipe are shown in Fig. 6. As can be seen, the void profiles for the cases without and with the elbow are almost identical. These results confirm that the exit conditions in the experiments were nearly fully developed, as expected. At the same time, they show the consistency and numerical accuracy of the NPHASE solver. The purpose of Fig. 7 is to show the predicted void fraction and velocity profiles at Port 4 of the experimental Run 1 (Kim et al., 2005), including a comparison against the measured void fraction distribution. The calculations have been performed for the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. As it was mentioned previously, the results at Port 4 correspond to nearly fully developed flow conditions. As can be seen, the predictions are in a reasonable agreement with the measurements, especially given the high uncertainties of
the latter (schematically indicated by using typical error bars in Fig. 7(a)). Still, the observed differences in the shape of the void fraction curve indicate that the current modeling assumptions and formulations should be revisited and, possibly, modified to improve the predictive capabilities of the overall CMFD model in the NPHASE code. Fig. 8 shows the pressure distribution along the pipe with 90◦ elbow for two experimental conditions corresponding to Run 1 and Run 2. The change in pressure has been considered separately along the inner and outer edges of the elbow. It is interesting to notice that the pressure at the outer edge of the elbow is higher than at the inner edge. This is due to the formation of secondary flows inside the elbow. In Fig. 8(a), the pressure along the centerline of a straight horizontal tube is also shown. The axial pressure gradient along the straight sections of the pipe with elbow is practically the same as that for a straight tube. The differences in the calculated values of pressure are due to the different length of each pipe, and the assumed same pressure of reference at the outlet. Fig. 9 shows the calculated void fraction and velocity profiles at Port 3 (2.69 m from the start of elbow) and Port 2 (1.39 m from the start of elbow) of Run 1, including a comparison against the experimental results. The velocity and void fraction distributions
Fig. 16. The radial void fraction and liquid and gas velocity profiles at 0.5 radius from the symmetry line in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 2 of the experiments reported in Kim et al. (2005).
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Fig. 17. The radial void fraction and liquid and gas velocity profiles at several sections along the 90◦ elbow of a horizontal tube for Run 2 of the experiments reported in Kim et al. (2005).
refer to the downstream section of the tube. The calculations have been performed for the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. It can be seen that gravity has a strong effect on the velocity distribution along the vertical diameter of the tube. In particular, the velocity profiles are bottom-peaked at both locations. Fig. 10 shows the void fraction and velocity profiles along the horizontal diameter perpendicular to the gravity vector in a horizontal tube with 90◦ elbow. The results refer to the nearly fully developed region, corresponding to Port 4 of the test facility. The results in Fig. 11 correspond to Port 4, and refer to the location at half-radius from the centerline. The plots are made along the vertical diameter. The calculations have been performed for the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. The void fraction and gas velocity at different cross-sections along the horizontal tube with a 90◦ elbow are presented in Fig. 12. The superficial liquid and gas velocities are 0.56 m/s and 0.27 m/s, respectively. As can be seen, the axial velocity profile undergoes a gradual transition from nearly fully developed to top-peaked at the 90◦ angle. Downstream of the elbow, the velocity profile gradually changes again, and eventually approaches the bottom-peaked shape shown in Fig. 7(b). Fig. 13 shows the calculated void fraction and velocity profiles at Port 3 and Port 2 of Run 2 including a comparison against the experimental results for Port 3. There are no experimental data acquired at Port 2. The velocity and void fraction distributions refer to the downstream section of the tube. The calculations have been performed for the superficial velocities of liquid and gas of 1.65 m/s and 0.27 m/s, respectively. Fig. 14 shows the void fraction and velocity profiles at Port 4 of Run 2, including a comparison against the experimental data. The velocity and void fraction distributions refer to the outlet section of the pipe, where the flow is nearly fully developed. The calculations have been performed for the superficial velocities of liquid and gas of 1.65 m/s and 0.27 m/s, respectively. The effect of gravity-driven void fraction distribution on the axial velocity profile can be clearly seen. Fig. 15 shows the void fraction and velocity profiles along the horizontal diameter in a horizontal tube with 90◦ elbow at a location corresponding to Port 4 in Run 2. The superficial velocities of liquid and gas are 1.65 m/s and 0.27 m/s, respectively. The results in Fig. 16 correspond to Port 4 and refer to the location at half-radius from the centerline along the vertical diameter of
the tube. The calculations have been performed for the superficial velocities of liquid and gas of 1.65 m/s and 0.27 m/s, respectively. Major flow field characteristics at different cross-sections in the horizontal tube with a 90◦ elbow are presented in Fig. 17. The superficial liquid and gas velocities are 1.65 m/s and 0.27 m/s, respectively. Comparing with the results in Fig. 12, some differences can be noticed, including a larger change in the velocity for flow along the elbow. It can be seen that for a smaller velocity (Fig. 12, jf = 0.56 m/s) at the location corresponding to exit from the elbow, the void fraction started to return to fully developed conditions, whereas in case of a larger velocity, jf = 1.65 m/s, the effect of the elbow can still be noticed. 5. Summary and conclusions The results of a multidimensional analysis of gas/liquid twophase flows in horizontal conduits of different geometries have been presented. The NPHASE CMFD code has been used to perform extensive numerical model testing and validations against experimental data. It has been demonstrated that the proposed model is consistent both physically and numerically. Also, the results of predictions compare well against the experimental data. However, it is clear that additional work is needed to improve our understanding of the effect of gravity on the interfacial phenomena governing gas/liquid two-phase flows in horizontal pipes and channels. Furthermore, the current modeling formulations needed to be revisited to improve the predictive capabilities of the overall CMFD model in the NPHASE code. Acknowledgements The authors would like to acknowledge the financial support provided by the US Nuclear Regulatory Commission and the technical assistance of Dr. Shawn Marshall. References Anglart, H., Nylund, O., Kurul, N., Podowski, M.Z., 1997. CFD simulation of flow and phase distribution in fuel assemblies with spacers. Nuclear Science and Engineering 177, 215–228. Antal, S.P., Kunz, R., Ettore, S., Podowski, M.Z., 2000. Development of a next generation computer code for the prediction of multicomponent multiphase flows. In: Proceedings of the International Meeting on Trends in Numerical and Physical Modeling for Industrial Multiphase Flows, Cargese, France. Drew, D.A., Passman, S.L., 1998. Theory of Multicomponent Fluids. Springer-Verlag, NY, USA.
E.A. Tselishcheva et al. / Nuclear Engineering and Design 240 (2010) 405–415 Ishii, M., Zuber, N., 1979. Relative motion and interfacial drag coefficient in dispersed two-phase flow of bubbles, drops and particles. AIChE Journal 25, 843–855. Kim, S., Talley, J., Callender, K., 2005. Data Report for Slug Flow in Air/Water Horizontal Two-Phase Flow with 90◦ Elbow. UMR/NE-TFTL-05-03. Podowski, M.Z., 2005. Understanding multiphase flow and heat transfer; perception, reality, future needs. Archives of Thermodynamics 26 (3), 1–18.
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Podowski, M.Z., 2009. On the consistency of mechanistic multidimensional modeling of gas/liquid two-phase flows. Nuclear Engineering and Design 239, 933–940. Tiwari, P., Antal, S.P., Podowski, M.Z., 2006. Three-dimensional fluid mechanics of particulate two-phase flows in U-bend and helical conduits. Physics of Fluids 18 (4), 1–18.
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Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Mechanistic multidimensional analysis of horizontal two-phase flows Elena A. Tselishcheva ∗ , Steven P. Antal, Michael Z. Podowski Rensselaer Polytechnic Institute, Troy, NY, USA
a r t i c l e
i n f o
Article history: Received 25 October 2008 Received in revised form 23 August 2009 Accepted 6 September 2009
a b s t r a c t The purpose of this paper is to discuss the results of analysis of two-phase flow in horizontal tubes. Two flow situations have been considered: gas/liquid flow in a long straight pipe, and similar flow conditions in a pipe with 90◦ elbow. The theoretical approach utilizes a multifield modeling concept. A complete three-dimensional two-phase flow model has been implemented in a state-of-the-art computational multiphase fluid dynamics (CMFD) computer code, NPHASE. The overall model has been tested parametrically. Also, the results of NPHASE simulations have been compared against experimental data for a pipe with 90◦ elbow. © 2009 Elsevier B.V. All rights reserved.
1. Introduction The development of modeling and simulation capabilities of two-phase flow and heat transfer is very important for the design, operation and safety of nuclear reactors. Whereas significant progress in this field has been made over the recent years (Podowski, 2005), further advancements are clearly needed for new concepts of Advanced Boiling Water Reactors and future Generation-IV reactors (e.g., in the safety analysis of the proposed Sodium Fast Reactor). In the past, the complexity of various possible gas/liquid flow patterns has limited the scope and range of situations that could be analyzed using full three-dimensional models. In fact, most computer simulations of multidimensional gas/liquid two-phase flows performed to date have been concerned with vertical tubes or conduits. The reason for that was that in vertical flows gravity mainly affects the axial gas/to-liquid relative velocity, but does not induce any lateral asymmetry in either velocity or phase distribution. On the other hand, in the case of horizontal (or inclined) flows, the acceleration of gravity not only causes a significant flow asymmetry, but also imposes an extra vertical force across the main flow direction, which is typically much stronger that other interfacial forces such as the lift force, for instance. Needless to say, the level of difficulty increases further for two-phase flows in conduits of complex geometries and spatial orientations, where a variety of flow patterns have been observed, each characterized by different interfacial phenomena of mass, momentum and energy transfer. The purpose of this paper is to discuss the results of analysis of two-phase flow in horizontal tubes. Two pipe geometries have
∗ Corresponding author. E-mail addresses: [email protected] (E.A. Tselishcheva), [email protected] (S.P. Antal), [email protected] (M.Z. Podowski). 0029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2009.09.015
been considered: first, a long straight pipe, and then, a similar pipe but with 90◦ elbow in the mid-section along the flow. In the latter case, the geometry and flow conditions were similar to those used in the air/water experiments performed by Kim et al. (2005). The combined theoretical and computational approach is based on a complete three-dimensional multifield model which has been implemented in a state-of-the-art computational multiphase fluid dynamics (CMFD) computer code, NPHASE (Antal et al., 2000; Tiwari et al., 2006). NPHASE is a multicomponent/multiphase CFD code, which solves the individual transport equations for momentum, energy, and turbulence for each field. The key features of PHASE include the built-in mechanistic models of interfacial phenomena, robustness and numerical accuracy. Past experience with NPHASE clearly shows that this code employs several features uniquely qualifying it for use in the analysis of multiphase flows in complex geometries. The proposed model has been tested parametrically for both physical and numerical consistency. Then, the same model has been used to simulate selected experimental runs reported by Kim et al. (2005). The results of comparisons between the predictions and experimental data are shown in Section 5. 2. Overview of experimental facility and data A schematic diagram of the test facility documented in Kim et al. (2005) is shown in Fig. 1. The experimental test section was made of Pyrex glass tubes 50.3 mm in inner diameter. The measurement ports were made of clear acrylic. In the test section, a 90◦ elbow was installed at L/D = 197. The elbow had a radius of curvature of 76.2 mm with the length corresponding to L/D ratio of approximately 6. For the purpose of measurements, four local ports were installed along the test section, denoted as P1 through P4 in Fig. 1. Yet another port (P0) was located right after the two-phase mixing chamber,
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Fig. 1. A schematic diagram (Kim et al., 2005) of the 50.3 mm ID horizontal two-phase flow test facility with a 90◦ elbow junction (top view, not to scale).
but it was used only for the local pressure measurements. The next port, P1, was located at L/D = 197 (9.91 m from P0), just before the start of the elbow, whereas the ports, P2, P3 and P4, were located at the nondimensional distances, L/D = 225, 250 and 329 from the mixing chamber, or 1.39 m, 2.69 m and 6.65 m downstream of the start of the elbow, respectively. The measurements of void fraction and velocity were taken at P1, P3 and P4. Local two-phase flow parameters were measured using doublesensor and four-sensor conductivity probes. In order to capture the asymmetry of distribution of major flow parameters across the horizontal two-phase flow, the measurements were taken along the entire vertical diameter of the tube. In total, 15 different jf and jg combinations were investigated, all corresponding to slug flow conditions. Sample plots are presented in Fig. 2. As can be seen, there is almost no gas in the lower half of the pipe. It is interesting to notice that the void fraction in the upper half first increases dramatically, then stays almost constant and finally increases further near the top of the pipe wall. The results acquired from the experiments were the local flow rates, local pressure, and two-phase parameters, such as void fraction and bubble velocity. The area-averaged superficial velocity, evaluated as the sum of the products of local void fraction and gas velocity, was compared against the local superficial velocity measured by a flow meter. As mentioned before, the present measurements were at Port 1, Port 3 and Port 4, and no data was taken at Port 2. The evolution of void fraction along the experimental section is presented in Fig. 3. The experimental data shown in this figure include those at Port 1 (before the elbow), Port 3 and Port 4 (downstream after elbow). They correspond to Run 1, and to the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. It can be seen that all three curves follow a similar trend and cover the same range of values, although some difference in the void fraction profiles can also be observed. In particular, the highest average void fraction is acquired at Port 3, which is located 2.69 m after the start of the elbow; the minimum is at Port 1.
To compare the numerical solutions obtained using the NPHASE code with the experimental data, several inlet conditions have been chosen. These conditions are summarized in Table 1. In Table 1, jg,atm is the gas superficial velocity equivalent to the atmospheric pressure condition, jf is the average of the liquid superficial velocities acquired at four measurement ports for each run, jloc port-0 is the gas superficial velocity acquired by the flow meter at Port 0, Gg,loc is the mass flux calculated at Port 0 as a product of jloc port-0 and the gas density, and ˇ is the volumetric flow fraction at Port 0. The experimental data used as a reference in the NPHASE-based numerical solutions have been taken at Port 3 and at Port 4. Since the distance between the elbow and Port 4 was sufficiently long, the conditions there corresponded to a nearly fully developed flow, except for a small change in the fluid properties with decreasing pressure. The individual terms for the runs listed in Table 1 are given in detail in Table 2. They are defined as follows: jg,loc is the gas superficial velocity measured by flow meter, jg is the calculated area-averaged superficial velocity found using the average values of two-phase parameters, Gg,loc is the mass flux calculated as a product of jg,loc and the gas density, Gg is the mass flux calculated as a product of jg and the gas density, ˇloc is volumetric flow fraction corresponding to the local measurement, and ˇ is the volumetric flow fraction corresponding to area-averaged values. In view of measurement errors, the superficial gas velocity at given measurement port obtained by the flow meter was compared with that obtained by the area-averaged two-phase flow parame-
Table 1 Summary of test conditions used in the NPHASE simulations. Test number
jf
jg,atm
jloc port-0
Gg,loc
ˇ
1 2 3 4
0.56 1.65 0.56 1.65
0.287 0.293 0.587 1.218
0.2639 0.256 0.5384 1.0336
0.3439 0.3566 0.7058 1.4822
0.3203 0.1343 0.4902 0.3852
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Fig. 2. Typical profiles of local void fraction acquired along the vertical diameter of the tube with 90◦ elbow (Kim et al., 2005), corresponding to the inlet conditions: (a) jg,atm = 0.287 m/s, jf = 0.559 m/s; (b) jg,atm = 0.293 m/s, jf = 1.648 m/s; (c) jg,atm = 0.587 m/s, jf = 0.560 m/s; (d) jg,atm = 1.218 m/s, jf = 1.652 m/s.
ters at the same port. It can be seen from Table 2 that there are significant differences between the flow conditions obtained by using each method. For example, the maximum error between the local measured and calculated superficial velocities was nearly 20%. Another important observation is concerned with some discrepancies in the measurements of the volumetric flow rates at Port 3 and Port 4. Whereas, for ideal adiabatic flows (at constant pressure) the
volumetric flow rates of both phases should be constant along the flow, the actual air expansion due to the frictional pressure drop caused a slight increase in the gas volumetric flow rate (and superficial velocity). As can be seen in Table 2, the measurements for Cases 1, 3 and 4 show different trends. Needless to say, such experimental errors will have to be accounted for when using the present data for model validation.
Table 2 Summary of the data for air–water slug flow through a horizontal tube of 50.3 mm inner diameter with a 90◦ elbow. Test number
Gas superficial velocity
Local pressure, p (psig)
Jg,loc
jg
Port 3 1 2 3 4
0.2701 0.2689 0.5509 1.0946
0.2596 0.3077 0.4545 1.0921
1.0 1.0 1.0 1.4
Prot 4 1 2 3 4
0.2697 0.2704 0.5513 1.0991
0.2276 0.3321 0.4182 1.0844
1.0 1.3 1.0 1.5
Local air density, loc
Gas mass flux
Void fraction
Volumetric flow fraction
Gg,loc
Gg
˛
ˇloc
ˇ
1.286 1.286 1.286 1.319
0.3473 0.3458 0.7085 1.4438
0.3338 0.3957 0.5845 1.4405
0.2530 0.1150 0.4430 0.3380
0.3246 0.1403 0.4959 0.3988
0.3160 0.1573 0.4480 0.3983
1.286 1.311 1.286 1.327
0.3468 0.3545 0.7090 1.4585
0.2927 0.4354 0.5378 1.4390
0.2440 0.1230 0.4220 0.3470
0.3243 0.1410 0.4961 0.3998
0.2882 0.1677 0.4275 0.3966
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of the model’s state variables. For gas/liquid flows in vertical channels, the total interfacial forces are typically partitioned into several components, as shown below Mil−v = −Miv−l = FD + FVM + FL + FTD + FW FD
FVM
(6) FL
where is the drag force, is the virtual mass force, is the lift force, FTD is the turbulent dispersion force, and FW is the wall force. Whereas the expressions for the drag, virtual mass and lift forces have been extensively documented and used before (Ishii and Zuber, 1979; Anglart et al., 1997; Drew and Passman, 1998), the component forces which are particularly important for the present analysis are the turbulent dispersion force and the wall force. A consistent derivation of the mechanistic model of turbulent dispersion force (Podowski, 2009), yields the following expression FTD = CTD l ˛∇ [(1 − ˛)k] Fig. 3. Typical lateral profiles of local void fraction acquired at different measurement ports of the tube with 90◦ elbow (Kim et al., 2005), corresponding to the following inlet conditions: jg,atm = 0.287 m/s, jf = 0.559 m/s.
3. Model description Applying the multifield modeling concept to adiabatic twophase gas/liquid flows yields the following form of phasic momentum equations for the liquid and gas phase, respectively: ∂[(1 − ˛)l vl ] + ∇ · [(1 − ˛)l vl vl ] ∂t = −(1 − ˛)∇ pl − (pil − pl )∇ ˛ + (1 − ˛)(∇ · ) l
+ ( i − ) · ∇ ˛ + (1 − ˛)l g + Miv−l l
(1)
l
∂(˛v vv ) + ∇ · (˛v vv vv ) = −˛∇ pv + (piv − pv )∇ ˛ + ˛(∇ · ) v ∂t − ( i − ) · ∇ ˛ + ˛v g + Mil−v v
v
(2)
where M denotes the total interfacial force, the superscript ‘i’ denotes the interfacial properties, and the remaining notation is conventional. Whereas Eqs. (1) and (2) apply to multicomponent flows of interpenetrating media (e.g., the flow of immiscible liquids), they are not, in general, applicable to dispersed “particle” flows such as gas/liquid bubbly flows. This is because the dispersed field is not capable of transmitting either the pressure force or the shear force (since the individual particles interact with the continuous field only, not with each other). However, it has been shown that Eqs. (1) and (2) can still be used to model bubbly flows provided two conditions are satisfied simultaneously (Podowski, 2009), namely pv = piv = pil = pl
and
= i = i = = v
v
l
l
(3)
where p is the only measurable continuous field pressure and is the corresponding total shear stress combining the molecular and turbulent components. Using Eq. (3), Eqs. (1) and (2) become ∂[(1 − ˛)l vl ] + ∇ · [(1 − ˛)l vl vl ] ∂t = −(1 − ˛)∇ p + (1 − ˛)(∇ · ) + (1 − ˛)l g + Miv−l ∂(˛v vv ) + ∇ · (˛v vv vv ) = −˛∇ p + ˛(∇ · ) + ˛v g + Mil−v ∂t
(4)
(5)
To complete the model given by Eqs. (4) and (5), closure laws must be formulated for the interfacial forces, Mil−v and Miv−l , in terms
(7)
where CTD = 1/3 for isotropic turbulence. Whereas for flows in vertical conduits, Eq. (7) already accounts for the fact that turbulence-induced force pushes the bubbles away from the channel surface in the near-wall region, in the case of horizontal flows, the wall reaction force to counterbalance the effect of buoyancy becomes the major vertical component of the overall lateral interfacial force. This is illustrated in Fig. 4 for the case of horizontal slug flows, which correspond to intermediate relative gas superficial velocities (or void fractions). As can bee seen in Fig. 4, the reaction force is limited to the conduit section occupied by the bubbles, and becomes zero elsewhere. If the elongated gas bubbles are not in direct contact with the walls, which is the situation corresponding to good wall-wettability conditions, this force will be transmitted across a thin liquid film between the bubbles and the wall, but it is practically independent of the film thickness. Furthermore, recognizing that this is a buoyancy-induced force (it disappears for neutrally/buoyant flows), its value per unit volume can be expressed as FR = CR (l − v )˛(1 − ˛)
g |g|
(8)
where the coefficient, CR , is defined as
CR =
g(1 − ε) 0
if y > yc if y ≤ yc
(9)
In Eq. (9), the parameter, yc , depends on the size of elongated bubbles and corresponds to the vertical position of the lower boundary of the bubbles. The adjustable constant, ε, is used to assess the effect of modeling uncertainties on the accuracy of predictions, such as: the effect of surface tension on the shape of elongated bubbles, the impact of nonuniform distribution of void fraction in the upper region of the conduit due to the shape variations of the elongated bubbles, and the presence of small dispersed bubbles. All such issues clearly deserve separate thorough investigations in the future. In horizontal flows of dispersed bubbles, the major interfacial force along the flow direction is the drag force. Thus, when the equilibrium is reached between the phases in ideal fully developed flows, the local bubble velocity is approximately equal to the velocity of the surrounding liquid and the local drag force reduces to nearly zero. At the same time, the force balance across the flow comprises of the following forces: gravity, buoyancy, turbulent dispersion, lift and wall reaction forces. In the present case, the effect of lift is practically negligible since the local relative velocity is very small at near-fully developed flow conditions. As mentioned before, since the wall reaction force is transmitted across the near-wall bubbles, for large bubbles characteristic to slug flows the range of this force may extend over a considerable distance from the upper wall.
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Fig. 4. A simplified schematic of slug flow in a horizontal conduit.
Fig. 5. An illustration of the three-dimensional numerical grid used in the NPHASEbased simulations of gas/liquid flow in a horizontal tube with 90◦ elbow: (a) the axial mesh and (b) the cross-sectional mesh.
4. Computational approach The model discussed in Section 3 has been implemented in the NPHASE code as a complete three-dimensional problem corresponding to the pipe geometry used in the experiments summarized in Section 2. The current formulation is applicable to both a straight pipe and to a pipe with 90◦ elbow. The computational grid used in the latter case is shown in Fig. 5. Fig. 5(a) shows the axial nodalization of the pipe with elbow, whereas Fig. 5(b) presents the unstructured grid used across the pipe. The pseudo-structured (regular) portion of the grid near the pipe wall helps to properly capture the wall effect on local velocity and phase distributions there. In the case of numerical model testing for a straight pipe, the same lateral
Fig. 6. The NPHASE-predicted radial void fraction profiles at the exit of a straight horizontal tube and a similar tube but with a 90◦ elbow, compared against the experimental data (Kim et al., 2005).
grid was combined with a regular axial discretization scheme. The effect of grid size has been investigated separately, showing that the current grid the needed accuracy level for NPHASE simulations. According to the system of coordinates shown in Fig. 5, the main flow direction gradually changes from that along x at the inlet to that along z at the outlet, whereas y always defines the vertical coordinate across the flow. Consequently, the gravity vector becomes, g = {gx , gy , gz }, where gx = gz = 0, gy = −g.
Fig. 7. Radial profiles of: (a) void fraction and (b) liquid and gas axial velocities, in a horizontal tube with 90◦ elbow, at Port 4 in Run 1 of the experiments (Kim et al., 2005).
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Fig. 8. Pressure along the centerline of straight horizontal tube and along at the outer and inner edge of horizontal tube with 90◦ elbow in Run 1 (a) and Run 2 (b) of the experiments reported in Kim et al. (2005).
Fig. 9. The radial void fraction and liquid and gas velocity profiles in a horizontal tube with 90◦ elbow, corresponding to Port P3 (a and b) and Port 2 (c and d) in Run 1 of the experiments reported in Kim et al. (2005).
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Fig. 10. The void fraction and liquid and gas velocity profiles along horizontal diameter in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 1 of the experiments reported in Kim et al. (2005).
Fig. 11. The radial void fraction and liquid and gas velocity profiles at 0.5 radius from the symmetry line in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 1 of the experiments reported in Kim et al. (2005).
Fig. 12. The radial void fraction and liquid and gas velocity profiles at several sections along the 90◦ elbow of a horizontal tube for Run 1 of the experiments reported in Kim et al. (2005).
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Fig. 13. The radial void fraction and liquid and gas velocity profiles in a horizontal tube with 90◦ elbow, corresponding to Port P3 (a and b) and Port 2 (c and d) in Run 2 of the experiments reported in Kim et al. (2005).
Fig. 14. The radial void fraction and liquid and gas velocity profiles in a horizontal tube with 90◦ elbow, corresponding to Port P4 in Run 2 of the experiments reported in Kim et al. (2005).
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Fig. 15. The void fraction and liquid and gas velocity profiles along horizontal diameter in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 2 of the experiments reported in Kim et al. (2005).
4.1. Model validation The model presented in Sections 3 and 4 has been implemented in the NPHASE code and numerically tested for conditions similar to those used in the experiments (Kim et al., 2005). The test calculations have been performed for two geometries: a straight cylindrical tube and a tube with 90◦ elbow. The results illustrating the effect of elbow on phase distribution at the end of the pipe are shown in Fig. 6. As can be seen, the void profiles for the cases without and with the elbow are almost identical. These results confirm that the exit conditions in the experiments were nearly fully developed, as expected. At the same time, they show the consistency and numerical accuracy of the NPHASE solver. The purpose of Fig. 7 is to show the predicted void fraction and velocity profiles at Port 4 of the experimental Run 1 (Kim et al., 2005), including a comparison against the measured void fraction distribution. The calculations have been performed for the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. As it was mentioned previously, the results at Port 4 correspond to nearly fully developed flow conditions. As can be seen, the predictions are in a reasonable agreement with the measurements, especially given the high uncertainties of
the latter (schematically indicated by using typical error bars in Fig. 7(a)). Still, the observed differences in the shape of the void fraction curve indicate that the current modeling assumptions and formulations should be revisited and, possibly, modified to improve the predictive capabilities of the overall CMFD model in the NPHASE code. Fig. 8 shows the pressure distribution along the pipe with 90◦ elbow for two experimental conditions corresponding to Run 1 and Run 2. The change in pressure has been considered separately along the inner and outer edges of the elbow. It is interesting to notice that the pressure at the outer edge of the elbow is higher than at the inner edge. This is due to the formation of secondary flows inside the elbow. In Fig. 8(a), the pressure along the centerline of a straight horizontal tube is also shown. The axial pressure gradient along the straight sections of the pipe with elbow is practically the same as that for a straight tube. The differences in the calculated values of pressure are due to the different length of each pipe, and the assumed same pressure of reference at the outlet. Fig. 9 shows the calculated void fraction and velocity profiles at Port 3 (2.69 m from the start of elbow) and Port 2 (1.39 m from the start of elbow) of Run 1, including a comparison against the experimental results. The velocity and void fraction distributions
Fig. 16. The radial void fraction and liquid and gas velocity profiles at 0.5 radius from the symmetry line in a horizontal tube with 90◦ elbow, corresponding to Port 4 in Run 2 of the experiments reported in Kim et al. (2005).
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Fig. 17. The radial void fraction and liquid and gas velocity profiles at several sections along the 90◦ elbow of a horizontal tube for Run 2 of the experiments reported in Kim et al. (2005).
refer to the downstream section of the tube. The calculations have been performed for the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. It can be seen that gravity has a strong effect on the velocity distribution along the vertical diameter of the tube. In particular, the velocity profiles are bottom-peaked at both locations. Fig. 10 shows the void fraction and velocity profiles along the horizontal diameter perpendicular to the gravity vector in a horizontal tube with 90◦ elbow. The results refer to the nearly fully developed region, corresponding to Port 4 of the test facility. The results in Fig. 11 correspond to Port 4, and refer to the location at half-radius from the centerline. The plots are made along the vertical diameter. The calculations have been performed for the superficial velocities of liquid and gas of 0.56 m/s and 0.27 m/s, respectively. The void fraction and gas velocity at different cross-sections along the horizontal tube with a 90◦ elbow are presented in Fig. 12. The superficial liquid and gas velocities are 0.56 m/s and 0.27 m/s, respectively. As can be seen, the axial velocity profile undergoes a gradual transition from nearly fully developed to top-peaked at the 90◦ angle. Downstream of the elbow, the velocity profile gradually changes again, and eventually approaches the bottom-peaked shape shown in Fig. 7(b). Fig. 13 shows the calculated void fraction and velocity profiles at Port 3 and Port 2 of Run 2 including a comparison against the experimental results for Port 3. There are no experimental data acquired at Port 2. The velocity and void fraction distributions refer to the downstream section of the tube. The calculations have been performed for the superficial velocities of liquid and gas of 1.65 m/s and 0.27 m/s, respectively. Fig. 14 shows the void fraction and velocity profiles at Port 4 of Run 2, including a comparison against the experimental data. The velocity and void fraction distributions refer to the outlet section of the pipe, where the flow is nearly fully developed. The calculations have been performed for the superficial velocities of liquid and gas of 1.65 m/s and 0.27 m/s, respectively. The effect of gravity-driven void fraction distribution on the axial velocity profile can be clearly seen. Fig. 15 shows the void fraction and velocity profiles along the horizontal diameter in a horizontal tube with 90◦ elbow at a location corresponding to Port 4 in Run 2. The superficial velocities of liquid and gas are 1.65 m/s and 0.27 m/s, respectively. The results in Fig. 16 correspond to Port 4 and refer to the location at half-radius from the centerline along the vertical diameter of
the tube. The calculations have been performed for the superficial velocities of liquid and gas of 1.65 m/s and 0.27 m/s, respectively. Major flow field characteristics at different cross-sections in the horizontal tube with a 90◦ elbow are presented in Fig. 17. The superficial liquid and gas velocities are 1.65 m/s and 0.27 m/s, respectively. Comparing with the results in Fig. 12, some differences can be noticed, including a larger change in the velocity for flow along the elbow. It can be seen that for a smaller velocity (Fig. 12, jf = 0.56 m/s) at the location corresponding to exit from the elbow, the void fraction started to return to fully developed conditions, whereas in case of a larger velocity, jf = 1.65 m/s, the effect of the elbow can still be noticed. 5. Summary and conclusions The results of a multidimensional analysis of gas/liquid twophase flows in horizontal conduits of different geometries have been presented. The NPHASE CMFD code has been used to perform extensive numerical model testing and validations against experimental data. It has been demonstrated that the proposed model is consistent both physically and numerically. Also, the results of predictions compare well against the experimental data. However, it is clear that additional work is needed to improve our understanding of the effect of gravity on the interfacial phenomena governing gas/liquid two-phase flows in horizontal pipes and channels. Furthermore, the current modeling formulations needed to be revisited to improve the predictive capabilities of the overall CMFD model in the NPHASE code. Acknowledgements The authors would like to acknowledge the financial support provided by the US Nuclear Regulatory Commission and the technical assistance of Dr. Shawn Marshall. References Anglart, H., Nylund, O., Kurul, N., Podowski, M.Z., 1997. CFD simulation of flow and phase distribution in fuel assemblies with spacers. Nuclear Science and Engineering 177, 215–228. Antal, S.P., Kunz, R., Ettore, S., Podowski, M.Z., 2000. Development of a next generation computer code for the prediction of multicomponent multiphase flows. In: Proceedings of the International Meeting on Trends in Numerical and Physical Modeling for Industrial Multiphase Flows, Cargese, France. Drew, D.A., Passman, S.L., 1998. Theory of Multicomponent Fluids. Springer-Verlag, NY, USA.
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