Two-phase discharge from a stratified region through two horizontal branches with centerlines falling on an inclined plane

International Journal of Multiphase Flow 97 (2017) 134–146

Contents lists available at ScienceDirect

International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow

Two-phase discharge from a stratified region through two horizontal branches with centerlines falling on an inclined plane M.K. Guyot, H.M. Soliman∗, S.J. Ormiston Department of Mechanical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada

a r t i c l e

i n f o

Article history: Received 12 May 2017 Revised 6 August 2017 Accepted 8 August 2017 Available online 12 August 2017 Keywords: Dual discharge Two-phase flow Horizontal branches Centerlines on inclined plane Experimental

a b s t r a c t The main objective of this study was to obtain new experimental data for conditions not previously tested for discharging two-phase flow through two 6.35 mm diameter branches with centerlines falling in an inclined plane. The present results are relevant to many industrial applications including headers and manifolds, multichannel heat exchangers and small breaks in horizontal pipes. In the experimental investigation, the critical heights for the onsets of liquid and gas entrainment (OLE and OGE, respectively) were obtained, analyzed and correlated for two different branch spacings and two different angles between the branches. For each combination of branch spacing and angle between the branches, a wide range of Froude numbers was used. Two-phase mass flow rate and quality results were also obtained and analyzed for a range of interface heights for 16 different combinations of branch spacing, inclination angle, test section pressure and pressure drop across each branch. New empirical correlations were developed to predict the dimensionless mass flow rate and quality. The new correlations show good agreement with the present data and with previous correlations. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Several engineering applications involve two-phase discharge from large pipes or headers through single or multiple branches (or breaks). Pertinent examples include the flow through small breaks in nuclear-reactors’ piping during loss-of-coolant accidents, the flow distribution among a number of streams discharging from a header, or multi-passage heat exchangers. Knowledge of flow phenomena involved in these applications is obviously essential for the design of such systems. Previous investigations on this topic involved a single branch in the horizontal, vertical up, vertical down, or inclined orientation (e.g., Smoglie and Reimann, 1986; Yonomoto and Tasaka, 1991; Hassan et al., 1998; Saleh et al., 2009; Bartley et al., 2010; Castiglia and Giardina, 2010). These studies succeeded in generating experimental data, models, and empirical correlations for the critical heights at the onset of gas entrainment (hOGE ) and the onset of liquid entrainment (hOLE ), as well as the discharging two-phase mass flow rate (m˙ TP ) and quality (x). Studies were also reported for the case of two horizontal branches mounted on a vertical plane wall with centerlines falling in a horizontal plane (Hassan et al., 1996a) and two horizontal branches with centerlines falling in a vertical plane (Hassan et al., 1996b). These studies generated data and

∗

Corresponding author. E-mail address: [email protected] (H.M. Soliman).

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2017.08.005 0301-9322/© 2017 Elsevier Ltd. All rights reserved.

empirical correlations for the critical heights for OLE and OGE, as well as the two-phase mass flow rate and quality at both branches. The geometry of a header (circular wall) discharging through two or three branches was also considered (Hassan et al., 1997; Bowden and Hassan, 2008,2011a,b; Saleh et al., 2010,2011). However, most of these studies focused only on the OLE and OGE phenomena. Due to industrial needs, research was directed to examine the two-phase flow phenomena in the geometry of a full header with 30 outlet branches (Teclemariam et al., 2003) and a similar geometry but with flow out of six downward branches (Shaban and Tavoularis, 2015). For the geometry of two horizontal outlet branches mounted on a vertical wall with centerlines falling in an inclined plane, experimental data and theoretical models for hOGE and hOLE were reported by Maier et al., (2001a,b). Only the entrainment at the branch closest to the interface was considered in that study (OGE at the top branch and OLE at the bottom branch). The objective of the present investigation is to extend the work of Maier et al. by generating data for hOGE and hOLE at both branches over wide ranges of Froude number. As well, data will be generated for m˙ TP and x in the two discharging branches over wide ranges of flow conditions. The test matrix was designed to allow for the study of the effect of each independent parameter separately.

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

135

Fig. 1. Geometrical and flow parameters.

2. Experimental parameters The geometrical and flow parameters that are relevant to this experimental investigation are shown in Fig. 1. Two circular branches of equal diameter d are located on the side of a large reservoir (test section) containing stratified layers of water and air. The upper branch (Branch A) and the lower branch (Branch B) direct the flow to separators A and B, respectively, where the air and water are separated before their flow rates are measured. The front view in Fig. 1 shows that the branch centerlines are separated by a distance L, and an angle θ measured relative to the horizontal plane. The interface heights, hA and hB , are measured upwards from their respective branch centerlines such that hA is positive if the interface is located above the centerline of Branch A and negative if it is located beneath the centerline of Branch A, and similarly for Branch B. For each experiment, the air pressure inside the test section is maintained constant at P0 and the pressures in separators A and B are maintained constant and equal to one another at PS,A = PS,B = PS . The pressure difference from the test section to each of the separators is therefore constant and equal to P = P0 − PS . The lengths of the two lines connecting the test section to the separators were adjusted such that when the interface is low and single-phase air flows through both branches, we get m˙ G,A = m˙ G,B , and when the interface is high and single-phase water flows through each branch, we get m˙ L,A = m˙ L,B . These line lengths were kept constant throughout all the experiments for this study. Under the above conditions, it is expected that the mass flow rates, m˙ TP , and qualities, x, of the two-phase flow through each of the branches are dependent on L, θ , P0 , P and h. For a fixed set of conditions (L, θ , P0 , and P), Fig. 2 shows schematically how the mass flow rate through the branches varies with the interface height. For this graph, hA was used for the xaxis for both Branches A and B so the critical onset heights for Branch B, hOLE, B and hOGE, B , are shown relative to Branch A. When the interface is located high above Branch A, single-phase water flows through both branches and m˙ TP,A = m˙ TP,B . As the interface is lowered, a critical height is reached where the onset of gas entrainment occurs at Branch A. At this instant, hA = hOGE,A and m˙ TP,A = m˙ L,OGE,A . Lowering the interface further results in twophase flow through Branch A, while the flow through Branch B is

Fig. 2. Variation of m˙ TP,A and m˙ TP,B with hA .

still single-phase water (m˙ TP,B = m˙ L,OGE,B ). As the interface is lowered further, a second critical height is reached where the onset of gas entrainment occurs at Branch B (hB = hOGE,B ). With further lowering of the interface, two-phase flow is now present in both Branches A and B until a third critical height is reached (onset of liquid entrainment at Branch A) where liquid stops flowing into Branch A at which time hA = hOLE,A and m˙ TP,A = m˙ G,OLE,A , while two phases continue to flow through Branch B. Finally, with further lowering of the interface, a last critical height is reached where liquid stops flowing into Branch B (hB = hOLE,B and m˙ TP,B = m˙ G,OLE,B ). Beyond this point, only single-phase gas flows into the branches as the interface is lowered. 3. Experimental investigation 3.1. Flow loop A schematic diagram of the apparatus is shown in Fig. 3. The test section consisted of a large tee-shaped reservoir containing stratified layers of water and air. The majority of the test section was fabricated with type 304 stainless steel, with an acrylic tube located near the outlet flange for visual observation of the flow

136

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Fig. 3. Schematic of the experimental apparatus.

phenomena. Distilled water was supplied to the bottom of the test section using a submersible pump. Water entered the test section through twelve 12.7 mm diameter holes located in three rows of four around the circumference of a 25.4 mm diameter copper tube entering through the bottom flange of the test section and closed from the other end. This inlet tube was designed to help disperse the water entering the test section horizontally and thus, prevent the formation of waves on the interface. The flow rate of water to the test section was controlled by a valve connected to the by-pass line. A cooling coil in the water tank was used to maintain the water at constant temperature. Filtered air was supplied to the top of the test section and maintained at a steady pressure Po using a pressure controller. Air entered the test section through an air inlet flange located at one end of the test section, as shown in Fig. 3. In order to reduce the incoming air velocity and prevent waves or ripples from forming on the interface, the air entered through four large outlet holes located on two sides of a triangular shaped dispersion box located inside the test section. Two pressure taps (one near the top in the air region and the other near the bottom in the

water region) connected to a pressure transducer were installed on the outlet flange to measure the interface height in the test section. The flow leaving the test section was directed through Branches A and B to two separators which separated the air and water. The discharging branches were two 6.35 mm diameter holes drilled through a brass block. The brass block was machined to allow for branch spacings of L/d = 1.5 and 3. Each branch was 127 mm in length to ensure a straight length of 20 diameters before any bends or area changes occur. The brass piece was installed onto the outlet flange with an assembly that allowed rotation at various angles. A pressure transducer was used to ensure that the two separators were at equal pressure. The water leaving the bottom of each separator was directed to one of four water rotameters with overlapping ranges, before it was returned to the water tank. The air, leaving the top of each separator was directed into separate air headers and then passed through one of four air rotameters with overlapping ranges, where its flow rate was measured. The air from both sides was then com-

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

137

Fig. 4. Photographs of OLE at both branches for P0 = 316 kPa and P = 123 kPa.

bined into a common header before being released into the atmosphere. A muffler was used for the air exiting the outlet header to minimize the noise. The temperature of the air and water in the test section and the air temperature in each air header were measured using type T thermocouples connected to a digital thermocouple reader. Bourdon pressure gauges were used to measure the pressure in the test section and in each separator and a digital pressure gauge was used to measure the pressure of air leaving the air flow meters. All measuring devices (rotameters, transducers, pressure gauges, and thermocouples) were calibrated in-house using standard devices. More details about the different components in the flow loop can be found in Guyot (2016). 3.2. Experimental procedure Data were collected for the OLE and OGE at both branches and for the two-phase mass flow rate and quality at interface heights between the OLE and OGE for both branches. For the OLE, the desired settings of P0 , P, L/d, and θ were established first with the interface lowered to a height below the expected hOLE, B . Precaution was exercised to maintain the same pressure in both separation tanks. With single-phase air flowing in both branches, the mass flow rates (m˙ G,OLE,A and m˙ G,OLE,B ) were measured and values of Froude number (FrG, OLE, A and FrG, OLE, B ) were calculated from,



0.5

FrG,OLE = (4 m˙ G,OLE /π )/ g d5 ρG (ρL − ρG )

(1)

The interface was then raised very slowly (1 mm/min) by pumping water into the test section and the interface was visually monitored through the acrylic section of the test section until water suddenly started flowing into Branch B. At this instant, the

interface height (hOLE, B ) was recorded. Raising the interface continued until OLE at Branch A occurred and hOLE, A was recorded. Fig. 4 shows photographs of the OLE-phenomenon at both branches. Parts (a) and (b) show the appearance of the phenomenon at Branch B for L/d = 1.5 and 3, respectively. In both cases, a water spout can be seen extending from the interface to the branch. Parts (c) and (d) illustrate the OLE at Branch A. For L/d = 1.5, the liquid entrainment into Branch A appears to be originating from the spout entering Branch B, while for L/d = 3, the liquid entrainment into Branch A appears to be originating from the interface. This suggests that the branches are behaving as two independent branches for L/d = 3, while for L/d = 1.5, there is interaction between the two branches. For the OGE-data, the interface was first raised above the expected hOGE, A with single-phase liquid flowing through both branches. The mass flow rate through both branches,m˙ L,OGE,A and m˙ L,OGE,B , were recorded while maintaining the desired settings of P0 , P, L/d, and θ , and steady liquid level and equal pressure in the separators. Froude number at both branches was then calculated from



0.5

FrL,OGE = (4 m˙ L,OGE /π )/ g d5 ρL (ρL − ρG )

(2)

The interface was then lowered very slowly (1 mm/min) until a continuous gas cone was observed entering Branch A, and the interface height hOGE, A was immediately recorded. The interface was lowered further until the OGE occurred at Branch B and hOGE, B was recorded. After completing the onsets data, two-phase flow results were obtained next for approximately 25 equispaced interface heights between hOGE, A and hOLE, B . For each combination of P0 , P, L/d, and θ , a steady interface height h was established and an equal pressure was set in the two separation tanks. After reaching steady

138

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

state with all readings of rotameters, pressure gauges, thermocouples, and liquid levels in the test section and the separation tanks remaining constant, the corresponding readings of h, m˙ G , and m˙ L corresponding to each branch were recorded. 3.3. Experimental uncertainty The experimental uncertainty was calculated using the method of Moffat (1988). For all the dimensional and dimensionless parameters, the uncertainties were low at high values of the parameter and generally increased as the value of the parameter decreased. For example, the uncertainty in M was typically within ±5% for most of the data; however, the uncertainty increased to a maximum of ±24.4% at H = 0.1. Similarly for H, the uncertainty was within ±5% for most of the data, but increased to a maximum of ±11.2% at H = 0.1. The uncertainties in FrG, OLE and FrL, OGE were within ±5.6% and ±4.8%, respectively, for the whole range. The uncertainties in m˙ TP , h, and x were calculated to be within ±8.6%, ±0.22 mm, and ±9.8%, respectively. For h/d, the uncertainty was found to be within ±5.7% for the whole range. 4. Results and discussion 4.1. Onsets of liquid and gas entrainment Data were collected for the four critical heights hOGE, A , hOLE, A , hOGE, B , and hOLE, B over a wide range of conditions (316 ≤ P0 ≤ 585 kPa, 40 ≤ P ≤ 300 kPa, L/d = 1.5 and 3, and θ = 30° and 60°). These conditions resulted in Froude number at the onsets in the ranges 15 < FrG, OLE < 40 and 15 < FrL, OGE < 50. A total of 144 data points were collected. Before starting the experimental campaign described in the above paragraph, some preliminary tests were done on cases where the two branches were oriented side by side (θ = 0°) and one on top the other (θ = 90°). Results for |hOLE, A |/d and hOGE, B /d are shown in Figs. 5(a) and (b), respectively, in comparison with Hassan’s (1995) correlation to ensure accuracy of the apparatus and to validate the experimental procedure. Good agreement can be seen in the two parts of Fig. 5. 4.1.1. OLE results Results were obtained for |hOLE, B |/d and |hOLE, A |/d with θ = 30° and 60°, and L/d = 1.5 and 3. A sample of these results is presented in Figs. 6 and 7 compared with previous models and correlations. The experimental data show the correct trend of increasing critical height with Froude number. Fig. 6 shows good agreement between the data of |hOLE, B |/d and the model of Maier (2001) as well as the correlation of Hassan (1995). The experimental data, the correlation equation, and the model exhibit small effects of θ and L/d. On the other hand, Fig. 7 shows that the correlation of Hassan (1995) predicts a significant decrease in |hOLE, A |/d as θ increases from 0° to 90°. The experimental data for θ = 30° and 60° fall within Hassan’s predictions for θ = 0° and 90°; however, the data do not show a significant θ -effect in the tested range. A possible explanation is that when θ = 90°, the two branches would be competing for liquid from the same region (which is not the case for θ = 30° and 60°), and consequently the critical height for OLE at Branch A decreases significantly. The following empirical correlations were developed for the two onset heights using the nonlinear least-squares Marquardt– Levenberg algorithm:

|hOLE,A |/d = 0.625 FrAG1,OLE,A

(3)

and

|hOLE,B |/d = 0.625 (B1 FrG,OLE,B )0.4

(4)

Fig. 5. Comparison of preliminary data for θ = 00 and 90° with Hassan (1995).

where





A1 = 0.4 exp −0.201 exp −0.0838

 + 0.404 exp −0.13 and



B1 = 1 + exp −0.683

 L 1.413  d

 L 0.408 d

 L 1.637  d

2

cosh

θ



cosh θ

sin θ − 0.219

(5)

 L 1.091 d

(6)

These correlations are valid for L/d = 1.5 and 3, θ = 30° and 60°, and FrG, OLE, A = FrG, OLE, B = 15 to 40 and show good agreement with the experimental data, as shown in Fig. 8. These correlations also have the correct limits: as L/d → ∞, the branches behave as single branches and the coefficients A1 → 0.4 and B1 → 1 such that Eqs. 3 and 4 approach Craya’s (1949) analytical solution for a single branch (|h/d|OLE = 0.625 Fr0G.4 ), and as L/d → 0, B1 → 2 and Eq. 4 approaches Craya’s single branch solution with twice the Froude number. The RMS deviation between the experimental data and the correlations is 4.2% for |hOLE, A |/d and 2.5% for |hOLE, B |/d. 4.1.2. OGE results Fig. 9 shows the experimental results for hOGE, A /d and hOGE, B /d for θ = 30°, 60° and 90°, and L/d = 1.5 along with Hassan’s (1995) correlation for θ = 0° Part (a) of Fig. 9 shows that there are small differences between the results of hOGE, A /d for θ = 60° and 90°, but

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

139

Fig. 7. Data of |hOLE, A |/d for (a) L/d = 1.5 and (b) L/d = 3.

Fig. 6. Data of |hOLE, B |/d compared with (a) Hassan (1995) and (b) Maier et al. (2001).

and the general trend is for the onset height to increase with decreasing θ from θ = 60° to 0° As θ decreases towards zero, the distance between the bottom branch and the interface drops and the bottom branch aids the top branch more with gas entrainment. This trend was also observed for L/d = 3 with a smaller difference between θ = 30° and 60°, but still a slight increase in hOGE, A /d when θ drops to zero. For the onset of gas entrainment at the bottom branch, the experimental results for hOGE, B /d at θ = 30°, 60° and 90° are plotted in Fig. 9(b) for L/d = 1.5 along with Hassan’s (1995) correlation for θ = 0°. The results show that hOGE, B /d decreases significantly with increasing θ . At θ = 0°, Branch A is located beside Branch B and is assisting Branch B with gas entrainment, while when θ = 90°, Branch A is located directly above Branch B and is competing with Branch B for gas flow. Between these two limits, the upper branch provides more assistance (or less competition) to the lower branch with gas entrainment as θ decreases. A similar trend was seen for L/d = 3 (Guyot, 2016). Empirical correlations for hOGE, A /d and hOGE, B /d were developed based on the present data for θ = 30° and 60°, and Hassan’s (1995) data for θ = 0° The following correlations are valid over the ranges L/d = 1.5 and 3, θ = 0° to 60° and FrL, OGE, A =FrL, OGE, B = 15 to 50.





4 hOGE,A /d = 0.626 A02.4 + A03.4 Fr0L,.OGE ,A

(7)

hOGE,B /d = 0.626 FrBL,2OGE,B where,

(8)



1.516−0.771 sin θ

A2 = 2 exp −(0.975 sin θ + 0.485) (L/d )



2



−0.895

A3 = 1 − exp −0.025(L/d ) and,

B2 = 0.4 exp 1.274(L/d )

(9)



(10) cos θ − 1.014 (L/d )

−0.747



(11)

Eqs. (7)–(11) converge to the correct limits for large and small L/d. As L/d → ∞, Eqs. (9) to (11) give A2 = 0, A3 = 1, and B2 = 0.4. Consequently, Eqs. (7) and (8) approach the single-branch correlations of Lubin and Springer (1967) and Micaelli and Mem.4 ponteil (1989) given by hOGE /d = 0.626 Fr0L,OGE . Also, as L/d → 0, Eq. (7) approaches the single branch correlations of Lubin and Springer and Micaelli and Memponteil with twice the Froude number. Fig. 10 shows the good agreement between the experimental data and the correlations with an RMS deviation of 4.8% for hOGE, A /d and 5.8% for hOGE, B /d. 4.2. Two-phase flow results Experimental data for m˙ TP,A , m˙ TP,B , xA , and xB were collected for the 16 combinations of P0 , P, L/d and θ shown in Table 1. For

140

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Fig. 9. OGE data for L/d = 1.5: (a) Branch A and (b) Branch B.

Fig. 8. Comparison between OLE data and correlations: (a) Branch A and (b) Branch B.

each of these combinations, flow rate, temperature, pressure and interface height measurements were taken at approximately 25 interface heights located between hOLE, B and hOGE, A . This resulted in approximately 400 data points, not including the additional cases obtained for comparison and verification purposes. A representative sample of m˙ TP and x at Branches A and B is shown in Fig. 11. The data points corresponding to the onsets of gas and liquid entrainment are indicated by filled symbols. As hA decreases below hOGE at either branch, m˙ TP decreases and x increases at that branch. However, m˙ TP is always higher and x is lower in Branch B than in Branch A. The data for all 16 combinations in Table 1 are similar in trend to those in Fig. 11. 4.2.1. Effect of P Fig. 12 shows a representative sample of the effect of P on m˙ TP and x at the same values of P0 , L/d, and θ using data sets 1 and 2 in Table 1. The first and last points in each curve correspond to the onsets of gas and liquid entrainment. It can be seen that the values of hOGE , |hOLE |, and m˙ TP increase as P increases. On the other hand, the curves of x versus h appear to be crossing in Fig. 12(b) with x decreasing as P increases at low values of h, and the opposite at high h. This trend is explained by the schematic diagram in Fig. 13. Knowing that x = 0 at hOGE and x = 1 at hOLE

Table 1 Experimental matrix for two-phase flow. Set #

θ

P0

P0 (kPa)

P(kPa)

1 2 3 4

30°

1.5

316

40 123 97 235

517

5 6 7 8 9 10 11 12 13 14 15 16

3

316 517

60°

1.5

316 517

3

316 517

40 123 97 235 40 123 97 235 40 123 97 235

for any data set, and that hOGE and |hOLE | increase as P increases, Fig. 13 demonstrates that the curves for x versus h at two different values of P must cross. The data for both branches in all 16 data sets in Table 1 have the same trends as those seen in Fig. 12. The following dimensionless parameters were introduced by Hassan (1995) and shown to be capable of absorbing the effect of P :

M = (m˙ TP − m˙ G,OLE )/(m˙ L,OGE − m˙ G,OLE )

(12a)

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

141

Fig. 11. Data for Set #1: (a) m˙ TP versus hA and (b) x versus hA . Fig. 10. Comparison between OGE data and correlations: (a) Branch A and (b) Branch B.

and

H = (h − hOLE )/(hOGE − hOLE )

(12b)

When data sets 1 and 2 were plotted using M and H (see Fig. 14), the data for both sets collapsed indicating that the dimensionless groups succeeded in absorbing the P-effect and that external parameters, such as the flow patterns in the discharging tubes, have no significant effect on the values of M and x when plotted against H. This same observation was found to be true for all other geometries and flow conditions.

4.2.2. Effect of P0 Data from sets 1 to 4 corresponding to the same geometry (θ = 30° and L/d = 1.5) but different combinations of P0 and P are plotted in Fig. 15. Because it was demonstrated that nondimensional data with the same P0 and different P do collapse, the same symbol was used for sets 1 and 2, and similarly for sets 3 and 4. Thus, Fig. 15 demonstrates the effect of P0 . Fig. 15(a) shows that the effect of Po on M is largely absorbed, while Fig. 15(b) shows that x increases consistently as Po increases. These same observations were also true for all other geometries and flow conditions.

4.2.3. Effect of L/d Previous studies with θ = 0° and 90° (Hassan, 1995) showed that for discharging two-phase flow through two branches, the branches became more independent and had less influence on one another as L/d increased. This trend was examined in this study for θ = 30° and 60° by comparing the results of m˙ TP versus h, as well as x versus h, at Branch A against those at Branch B. When the branches are behaving as two independent branches, the results for m˙ TP,A (and xA ) versus hA fall on top of the results for m˙ TP,B (and xB ) versus hB , whereas when the branches are influencing one another the results of m˙ TP (and x) versus h at Branch A will differ from those at Branch B. Examining the results for L/d = 1.5, it was found that (for the flow conditions tested) there was always interaction between the two branches and that this interaction increased as P and θ increased. This trend is demonstrated in Fig. 16 by the increased deviation between m˙ TP,A and m˙ TP,B as P and θ increased from part (a) of the figure to part (b). The effect of L/d on m˙ TP is illustrated by Figs. 16 and 17. Figs. 16(a) and 17(a) correspond to the same values of P0 , P, and θ , but two different values of L/d, and the same for Figs. 16(b) and 17(b). The general trend is that the deviation between m˙ TP,A and m˙ TP,B decreases as L/d increases. Fig. 17(a) for L/d = 3 shows that the two branches are essentially independent for the given con-

142

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Fig. 13. Schematic of x versus h at two different P’s.

results in Fig. 20 have the same limits for m˙ TP,A at the onsets of liquid and gas entrainment. In addition, the magnitudes of the OLE and OGE interface heights increase with decreasing θ . To satisfy both these conditions, the results must cross as shown in Fig. 20. In general, for L/d = 3, the results showed a similar trend for the θ -effect on m˙ TP,A and x as those seen in Fig. 19; however the magnitude of this effect was lower than those observed for L/d = 1.5 (Guyot, 2016).

Fig. 12. Effect of P on m˙ TP and x in Branch B.

ditions; however, for higher values of P and θ , Fig. 17(b) shows that branch interaction can still exist for the same L/d. 4.2.4. Effect of θ There are two main aspects for the effect of θ on the twophase flow results. First, there is the effect on the degree of interaction between the two branches. Fig. 18 shows the data of m˙ TP,A and m˙ TP,B for the same P0 , P, and L/d with θ = 30° in part (a) and θ = 60° in part (b). In both parts of Fig. 18, the results from Hassan (1995) for θ = 90° are plotted as a guide for this comparison. These results demonstrate that the deviation between m˙ TP,A and m˙ TP,B increases (i.e., the interaction between the branches increases) as θ increases, and the same was found to be true for xA and xB . Second there is the effect of θ on the two-phase flow at either one of the two branches. Fig. 19 shows results for m˙ TP,A versus hA (in part (a)) and xA versus hA (in part (b)), for data sets 4 and 12 with P0 = 517 kPa, P = 235 kPa, and L/d = 1.5. Hassan’s (1995) correlation for θ = 90° is also shown. The results in these figures show that at low values of hA , m˙ TP,A decreases and xA increases with increasing θ . This trend occurs due to |hOLE, A | increasing with decreasing θ and thus more liquid enters Branch A at lower θ . However, in both m˙ TP,A and xA plots, as hA increases, the curves for θ = 30° and 60° cross right before the onset of gas entrainment height. The crossing of curves for different θ can be explained using the schematic shown in Fig. 20. The three sets of

4.2.5. Experimental repeatability In addition to the comparisons that were made with previous experimental data and correlations, an effort was made during the generation of the data to further ensure the accuracy of the present data by conducting some repeatability experiments. These repeatability tests included experiments for the critical onset heights, as well as experiments for two-phase flow. The results from these repeatability tests (Guyot, 2016) produced a maximum deviation in the critical onset heights within ±1.82%. For the two-phase-repeatability tests in both branches, the percent deviation was within ±5.8% in m˙ TP and within ±5.3% in x. These small deviations further confirm the accuracy of the present results. 4.2.6. Empirical correlations Based on the present experimental data for sets 1 to 16 and the correlations developed by Hassan (1995) for θ = 0° and 90°, new correlations were developed for MA (HA , L/d, θ ), MB (HB , L/d, θ ), xA (HA , L/d, θ , ρ L /ρ G ), and xB (HB , L/d, θ , ρ L /ρ G ). The correlations are valid for air-water over the following ranges: • • • •

P0 = 316 to 517 kPa P = 40 to 235 kPa θ = 0° to 90° L/d = 1.5 to 8

For these correlations, the angle between the branches must be measured from the horizontal line through the center of the bottom branch towards the line connecting the centerlines of the two branches. (a) MA correlation:





MA = HA(2+A4 ) exp −1.84 HA2 1 − HA2

A 5

(13)

where,

A4 = exp [(1.123 − 0.722 L/d ) sinh θ ] (sin θ )

(14)

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Fig. 15. Effect of P0 on M and x.

Fig. 14. Data from sets 1 and 2 presented in dimensionless form.

and

A5 = 1.318 + (7.873 sin θ −5.973 ) (sin

θ ) (L/d )2.085 sin (1.635θ ) −3.197 . (15)

As L/d → ∞ or when θ = 0°, A4 → 0, A5 → 1.318, and Eq. (13) converges to Hassan et al. (1998) single-branch correlation for M. When θ = 90°, Eq. (13) converges to the correlation developed from Hassan’s data for θ = 90°. Fig. 21(a) shows the comparison between the experimental results for MA and the correlation; good agreement was found with an RMS deviation of 11.4% between the limits of 0.1 ≤ MA ≤ 1 and 19.0% between the limits 0.01 ≤ MA ≤ 1. (b) MB correlation:



2

 2 B3 +B4 HB

MB = HB2 exp −1.84 HB 1 − HB Where,

B3 = 1.318 + 5.78 (sin

θ ) exp



(16)

−0.854

θ )−2.101(L/d )

2

− 0.37 (L/d ) (sin



xA = 0.2 exp [6HA (1 − A6 HA )/(1 − HA )] + 0.8 1 − A7 HA2

1 . 3

(A8 E )HA (19)

where,



A6 = 11.67 − 9.67 sin θ exp −0.01(L/d )

3

+ (0.883(L/d ) − 1.468)θ − 0.0859(L/d ) A7 = exp



134.2(L/d )

−6.343

2.279



46.7−14.41(L/d )

cos θ (sin θ )



cos θ , (20)

,

(21)

(17) A8 = 1 + 0.92 exp [−0.75(L/d )

and

B4 = −39.94(L/d )

As L/d → ∞ or when θ = 0°, B3 → 1.318, B4 → 0 and Eq. (16) reduces to Hassan et al. (1998) single-branch correlation. As well, 2 B3 → 1.318 + 5.78 exp [−0.37(L/d ) ] (sin θ ) and B4 → 0 when θ = 90°, and consequently Eq. (16) reduces to the correlation for MB developed by Hassan et al. (1996b) for dual discharge with θ = 90°. Between these limits of θ = 0° and 90°, the correlation showed excellent agreement with the experimental data for sets 1 to 16 (from Table 1) with an RMS deviation of 8.1% between the limits of 0.1 ≤ MB ≤ 1 and 14.1% between the limits of 0.01 ≤ MB ≤ 1 (Guyot, 2016). (c) xA correlation



−0.911 − 4.551 (L/d ) (cos θ ) −0.803−1.134 sin θ

143

(cos θ ) (sin θ )

2.664

.

(18)



−0.953

+ 6.79(L/d )

  θ 1.546−0.534(L/d ) cos θ sin θ ,

(22)

144

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Fig. 17. Effect of P and θ on m˙ TP at Branches A and B for L/d = 3. Fig. 16. Effect of P and θ on m˙ TP at Branches A and B for L/d = 1.5.

and and



E = −0.0122 + 0.42/ 1 +



ρL / ρG .

(23)

At the limit of L/d → ∞ or when θ = 0°, A6 → 11.67, A7 → 1, A8 → 1, and Eq. (19) reduces to Hassan et al. (1998) single-branch correlation for x. Also, A7 → 1, A8 → 1 + 0.92 exp[−0.75 (L/d )], 3 and A6 → 11.67 − 9.67 exp[−0.01(L/d ) ] when θ = 90° , and consequently, Eq. (19) reduces to the correlation for xA developed by Hassan et al. (1996b) for dual discharge with θ = 90°. For sets 1 to 16, this correlation predicts the experimental data, as shown in Fig. 21(b), with an RMS deviation of 19.1% between the limits of 0.01 ≤ xA ≤ 1 and 24.0% between the limits of 0.001 ≤ xA ≤ 1. (d) xB correlation:



xB = 0.2 exp [6HB (1 − B5 HB )/(1 − HB )] + 0.8 1 − HB2

1 . 3

(B6 E )HB (24)

where,



3

B5 = 11.67 − 3.03 exp −0.016(L/d )

+ 1.108( (L/d ) − 1.448 ) cos θ ] sin θ +89.415 (L/d )

−1.657



B6 = 1 − 0.71 exp −0.03(L/d )

(cos θ )2.25 sin θ

(25)



+ 0.793(L/d )

1.614

4

  θ − 1.332(L/d )−0.331 cos θ sin θ .

(26)

Eq. (24) also converges to the single-branch correlation for x (Hassan et al., 1998) for L/d → ∞ and for θ = 0°. For L/d → ∞ and θ = 0°, B5 → 11.67, B6 → 1 and Eq. (24) reduces to the same single-branch correlation for x. For θ = 90°, B5 = 11.67 − 3 4 3.03 exp[−0.016 (L/d ) ], B6 = 1 − 0.71 exp[−0.03 (L/d ) ], and the correlation reduces to the correlation for xB developed by Hassan et al., (1996b) for dual discharge with θ = 90°. Comparison between the correlation and the experimental data of Table 1 resulted in RMS deviation of 17.2% within the limits of 0.01 ≤ xB ≤ 1 and 18.8% within the limits of 0.001 ≤ xB ≤ 1 (Guyot, 2016). 5. Summary and conclusions Throughout the experimental investigation, data were collected for the OLE and OGE interface heights over wide ranges of Froude numbers, as well as m˙ TP and x for discharging two-phase air-water flow from a stratified region through two horizontal branches. For the OLE experiments, results were obtained for L/d = 1.5 and 3, θ = 30° and 60°, and FrG,OLE,A = FrG,OLE,B = 15 to 40. The results showed that for both L/d = 1.5 and 3, there is very little effect of θ on the OLE at Branch B. However, for the OLE at Branch A, as θ

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Fig. 18. Effect of θ on the difference between m˙ TP,A and m˙ TP,B at the same P0 , P, and L/d.

increased, Branch B competed more with Branch A for liquid and |hOLE, A |/d decreased significantly. This effect of θ on |hOLE, A |/d was found to decrease as L/d increased. Empirical correlations were developed based on the present data and Hassan’s (1995) data to predict |hOLE, A |/d and |hOLE, B |/d. The developed correlations showed excellent agreement with the data with RMS deviations of 4.2% and 2.5% for |hOLE, A |/d and |hOLE, B |/d, respectively, for L/d = 1.5 and 3, θ = 30° to 60° and FrG,OLE,A = FrG,OLE,B = 15 to 40. For the OGE, data were obtained for L/d = 1.5 and 3, θ = 30°, 60°, and 90°, and FrL,OGE,A = FrL,OGE,B = 15 to 50. For the OGE at Branch A, as θ decreased, Branch B provided more assistance to Branch A with gas entrainment and hOGE, A /d increased. A larger θ effect was observed for the OGE at Branch B. As θ increased, Branch A competed more with Branch B for gas entrainment and hOGE, A /d decreased. These trends were observed for both values of L/d; however, the magnitude of the effect decreased as L/d increased. From the present data and Hassan’s (1995) data, empirical correlations were developed to predict hOGE, A /d and hOGE, B /d with an RMS deviation of 4.8% and 5.8%, respectively, for the data over the range of L/d = 1.5 to 8, θ = 30° to 90° and FrL,OGE,A = FrL,OGE,B = 15 to 50. For interface heights between hOLE, B and hOGE, A , results were obtained for 16 data sets covering the following ranges of parameters: P0 = 316 and 517 kPa; P = 40 to 235 kPa; L/d = 1.5 and 3; and

145

Fig. 19. Effect of θ on the two-phase flow in Branch A for L/d = 1.5.

Fig. 20. Schematic of m˙ TP versus h at Branch A for θ = 30°, 60°, and 90°.

θ = 30° and 60°. For this range of parameters, the effects of P, P0 , L/d, and θ on m˙ TP and x were analyzed and the following observations were made:

146

M.K. Guyot et al. / International Journal of Multiphase Flow 97 (2017) 134–146

Acknowledgment The financial assistance provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the University of Manitoba is gratefully acknowledged. The authors also acknowledge the technical support by Mr. Zeev Kapitanker. References

Fig. 21. Comparison between data and correlation (a) for MA and (b) for xA .

•

•

•

•

•

•

•

•

For both Branches A and B, as P increases, m˙ TP increases and x decreases at low values of h and increase at high values of h. When the results were plotted in terms of the dimensionless parameters M versus H or x versus H, the effects of P were absorbed. The effect of L/d on the two-phase flow distribution in the branches is dependent on both P and θ but, in general, as L/d increased, the branches had less influence on one another. The branches had less influence on one another as θ or P decreased. For both Branches A and B, at high values of h, m˙ TP decreased and x increased as θ increased. For Branch A, at low values of hA , m˙ TP,A increased and xA decreased with increasing θ . For Branch B, at low values of hB , θ had very little effect on m˙ TP,B and xB because hOLE, B is nearly independent of θ . Empirical correlations were developed to predict MA , MB , xA , and xB . The correlations show good agreement with the experimental data for P0 = 316 to 517 kPa, P = 40 to 235 kPa, θ = 0 to 90° and L/d = 1.5 and 3.

Bartley, J.T., Soliman, H.M., Sims, G.E., 2010. Experimental investigation of two-phase discharge from a stratified region through a small branch mounted on an inclined wall. Int. J. Multiphase Flow 36, 588–597. Bowden, R.C., Hassan, I.G., 2008. Incipience of liquid entrainment from a stratified gas-liquid region in multiple discharging branches. J. Fluids Eng. 130, 011301 Article No. Bowden, R.C., Hassan, I.G., 2011a. The onset of gas entrainment from a flowing stratified gas-liquid regime in dual discharging branches – part I: flow visualization and related phenomena. Int. J. Multiphase Flow 37, 1358–1370. Bowden, R.C., Hassan, I.G., 2011b. The onset of gas entrainment from a flowing stratified gas-liquid regime in dual discharging branches – part II: critical conditions at low to moderate branch Froude numbers. Int. J. Multiphase Flow 37, 1371–1380. Castiglia, F., Giardina, M., 2010. A semi-empirical approach for predicting two-phase flow discharge through branches of various orientations connected to a horizontal main pipe. Nucl. Eng. Des. 240, 2779–2788. Craya, A., 1949. Theoretical research on the flow of non-homogeneous fluids. La Houille Blanche 4, 44–55. Guyot, M.K., 2016. Discharging two-phase flow through single and multiple branches: experiments and CFD modelling Ph.D. Thesis. University of Manitoba, Canada. Hassan, I., 1995. Single, dual and triple discharge from a large, stratified, two-phase region through small branches Ph.D. Thesis. University of Manitoba, Canada. Hassan, I.G., Soliman, H.M., Sims, G.E., Kowalski, J.E., 1996a. Experimental investigation of the two-phase discharge from a stratified region through two side branches oriented horizontally. Exp. Therm. Fluid Sci. 13, 117–128. Hassan, I.G., Soliman, H.M., Sims, G.E., Kowalski, J.E., 1996b. Discharge from a smooth stratified two-phase region through two horizontal side branches located in the same vertical plane. Int. J. Multiphase Flow 22, 1123–1142. Hassan, I.G., Soliman, H.M., Sims, G.E., Kowalski, J.E., 1997. Single and multiple discharge from a stratified two-phase region through small branches. Nuc. Eng. Des. 176, 233–245. Hassan, I.G., Soliman, H.M., Sims, G.E., Kowalski, J.E., 1998. Two-phase flow from a stratified region through a small side branch. ASME J. Fluids Eng. 120, 605–612. Lubin, B.T., Springer, G.S., 1967. The formation of a dip on the surface of a liquid draining from a tank. J. Fluid Mech. 29, 385–390. Maier, M.R., Soliman, H.M., Sims, G.E., Armstrong, K.F., 2001a. Onsets of entrainment during dual discharge from a stratified two-phase region through horizontal branches with centerlines falling in an inclined plane: part 1 – analysis of liquid entrainment. Int. J. Multiphase Flow 27, 1011–1028. Maier, M.R., Soliman, H.M., Sims, G.E., 2001b. Onsets of entrainment during dual discharge from a stratified two-phase region through horizontal branches with centerlines falling in an inclined plane: part 2 – experiments on gas and liquid entrainment”. Int. J. Multiphase Flow 27, 1029–1049. Micaelli, J.C., Memponteil, A., 1989. Two-phase flow behavior in a tee-junction – the CATHARE model. In: Proceedings of the 4th International Topical Meeting on Nuclear Reactor Thermal-Hydraulics, 2. Karlsruhe, F.R.G., pp. 1024–1030. Moffat, R.J., 1988. Describing uncertainties in experimental results. Exp. Therm. Fluid Sci. 1, 3–17. Saleh, W., Bowden, R.C., Hassan, I.G., Kadem, L., 2009. A hybrid model to predict the onset of gas entrainment with surface tension effects. J. Fluids Eng. 131, 011305 Article No. Saleh, W., Bowden, R.C., Hassan, I.G., Kadem, L., 2010. Two-phase flow structure in dual discharges – stereo PIV measurements. Exp. Therm. Fluid Sci. 34, 1016–1028. Saleh, W., Bowden, R.C., Hassan, I.G., Kadem, L., 2011. Theoretical analysis of the onset of gas entrainment from a stratified region through multiple branches. Int. J. Multiphase Flow 37, 1348–1357. Shaban, H., Tavoularis, S., 2015. Distribution of downward air-water flow in vertical tubes connected to a horizontal cylindrical header. Nuc. Eng. Des. 291, 90–100. Smoglie, C., Reimann, J., 1986. Two-phase flow through small branches in a horizontal pipe with stratified flow. Int. J. Multiphase Flow 12, 609–625. Yonomoto, T., Tasaka, K., 1991. Liquid and gas entrainment to a small break hole from a stratified two-phase region. Int. J. Multiphase Flow 17, 745–765. Teclemariam, Z., Soliman, H.M., Sims, G.E., Kowalski, J.E., 2003. Experimental investigation of the two-phase flow distribution in the outlets of a horizontal multi-branch header. Nuc. Eng. Des. 222, 29–39.