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shear-thinning liquid flows through pipes: Modeling and experiments
International Journal of Multiphase Flow 73 (2015) 217–226
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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Gas/shear-thinning liquid flows through pipes: Modeling and experiments Davide Picchi a, Yuri Manerba a, Sebastiano Correra b, Michele Margarone c, Pietro Poesio a,⇑ a
Università degli Studi di Brescia, Dipartimento di Ingegneria Meccanica e Industriale, Via Branze 38, 25123 Brescia, Italy eni spa, ISDEW, via Emilia, 1, 20097 San Donato Milanese, Italy c eni spa, ITEM, via Emilia, 1, 20097 San Donato Milanese, Italy b
a r t i c l e
i n f o
Article history: Received 4 November 2014 Received in revised form 2 March 2015 Accepted 8 March 2015 Available online 21 March 2015 Keywords: Shear-thinning fluid Flow pattern map Pressure gradient Slug frequency Stratified flow Slug velocity
a b s t r a c t In chemical and oil industry gas/shear-thinning liquid two-phase flows are frequently encountered. In this work, we investigate experimentally the flow characteristics of air/shear-thinning liquid systems in horizontal and slightly inclined smooth pipes. The experiments are performed in a 9-m-long glass pipe using air and three different carboxymethyl cellulose (CMC) solutions as test fluids. Flow pattern maps are built by visual observation using a high-speed camera. The observed flow patterns are stratified, plug, and slug flow. The effects of the pipe inclination and the rheology of the shear-thinning fluid in terms of flow pattern maps are presented. The predicted existence region of the stratified flow regime is compared with the experimental observation showing a good agreement. A mechanistic model valid for air/powerlaw slug flow is proposed and model predictions are compared to the experimental data showing a good agreement. Slug flow characteristics are investigated by the analysis of the signals of a capacitance probe: slug velocity, slug frequency, and slug lengths are measured. A new correlation for the slug frequency is proposed and the results are promising. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction Two-phase gas/liquid flows, where the fluid exhibits a shearthinning behavior, are often encountered in oil and process industries. However, only few studies have focused on gas/shear thinning fluid systems. Eisemberg and Weinberger (1979), Chhabra and Richardson (1984,), Chhabra et al. (1983), and Dziubinski and Chhabra (1989) were among the first to study gas/shear-thinning fluid mixtures in horizontal pipes: flow pattern maps were provided proposing empirical correlations for flow pattern transition starting from the map by Mandhane et al. (1974) and pressure drop data were analysed following the Lockhart and Martinelli (1949) model. Afterwords, Dziubinski (1995) proposed an empirical and a more general correlation for pressure drop predictions in the intermittent flow regime of gas/shear-thinning fluid mixtures in pipe and, more recently, Ruiz-Viera et al. (2006,) studied the wall-slip effects and the drag reduction due to air injection in the flow of a non-Newtonian lubricating grease giving a modified version of ⇑ Corresponding author. Tel.: +39 030 3715646. E-mail addresses: [email protected] (D. Picchi), [email protected] (P. Poesio). http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.03.005 0301-9322/Ó 2015 Elsevier Ltd. All rights reserved.
the Lockhart and Martinelli (1949) correlation. Flow patterns for gas/shear-thinning fluid in a vertical pipe were experimentally investigated by Dziubinski et al. (2004). These empirical approaches for pressure drop predictions are the simplest to study a complex situation like intermittent flows where one phase shows a shear-thinning behavior, see Chhabra and Richardson (2008) for a complete classification of gas/nonNewtonian fluid flow patterns and these experimental models for hold-up and pressure drop predictions in horizontal and vertical pipe. More sophisticated models can be used if information about the flow pattern are available. Concerning the stratified flow regime, Heywood and Charles (1979) extended the two-fluid model by Taitel and Dukler (1976) to gas/power-law fluids and Bishop and Deshpande (1986) tested that model with experimental data using air and a water-soluble polymer (7H4 SCMC) as test fluids. Xu et al. (2007, 2009) studied air/shear-thinning fluid flows in inclined pipes (test fluids were air and CMC-water solutions) for different pipe diameters and inclination angles: flow pattern maps were obtained by visual observation while hold-up and pressure drop data were collected. More recently, Picchi et al. (2014) studied gas/power-law fluid stratified pipe flow. In addition to the twofluid model given by Xu et al. (2007), a pre-integrated model was provided to obtain hold-up and pressure gradient predictions and
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the flow pattern transition from the stratified flow regime developed by Brauner and Moalem (1992) was extended to gas/ power-law fluid systems. The flow pattern transition predictions were validated with the experimental data by Xu et al. (2007). Regarding the slug flow regimes, a mechanistic model for holdup and pressure drop predictions was presented by Xu et al. (2007) and three-dimensional CFD simulations of gas/shear-thinning liquid mixtures were carried out by Jia et al., 2011. In addition to these works, where gas/shear-thinning pipe flows were investigated in terms of flow patterns, hold-up, and pressure drop, the study of slug flow characteristics has a practical relevance. The main characteristics of slug flows, slug frequency, and slug velocity for gas/Newtonian systems were deeply investigated, see Fabe and Liné (1992) and Hanratty (2013) and references therein. Only Rosehart et al. (1975) and Otten and Fayed (1977) presented an experimental characterization of air/shear-thinning fluid and air/Herschel–Bulkley fluid systems in horizontal pipes, respectively, and Dziubinski and Fidos (2004) for a vertical pipe. The aim of the present work is to present a new experimental campaign on two-phase gas/shear-thinning fluid flow in horizontal and slightly inclined pipes. The shear-thinning liquids chosen are CMC-water solutions: the introduction of small quantities of long-chain polymers into water has a practical relevance also for drag-reduction in gas–liquid flows (some studies on the drag-reducing effects on annular, stratified, and slug flow are mentioned in Hanratty (2013), references are therein) but in this work the polymers are added to modify the rheology of suspensions. Flow pattern maps are built by visual observations and the influence of the inclination angle and fluid rheology are investigated. The transition from stratified flow regime is compared with theoretical predictions by Picchi et al. (2014) and pressure drop data are compared to two-fluid model predictions for stratified flow and to a new model predictions for the slug flow, respectively. Slug frequency and slug velocity are measured by post-processing the signals of a capacitance sensor: slug frequency is compared to the Gregory and Scott (1969) correlation and a new correlation is proposed to take into account the rheology of the shear-thinning fluid; slug velocity is investigated studying the influence of the inclination and the polymer concentration on the distribution parameter and the drift velocity. Liquid slug length and bubble length data are also presented.
Experimental setup Flow loop The experiments are performed in setup schematically represented in Fig. 1. The system is composed by a 9 m-long glass pipe,
with an inner diameter of 22.8 mm. The pipe is mounted on a rigid beam and the inclination can be changed around the central pin. The investigated inclination angles are 0°, horizontal flow, 5°, downward flow, and + 5°, upward flow. CMC-water solutions and air, used as test fluids, are injected by the injector; L-injector in Fig. 1, see Grassi et al. (2008). Air (viscosity lg ¼ 1:8 105 Pa s, density qg ¼ 1:2 kg/m3) is supplied directly from the University network; air mass flow rate is measured prior to entering the test section by a thermal mass flow meter (accuracy is 3% of the full scale value) and, at the same spot, air pressure is monitored. Liquid supply is provided by a centrifugal pump, which draws CMC-water solution stored in a 0.5 m3 tank, and the flow rate is controlled by a frequency inverter. Liquid flow rate is measured with a turbine mass flow meter with an accuracy of 1% of the full scale value. A differential pressure transducer is placed 6 m downstream the injection point and the pressure drop is measured across a 1.5 m length (sensor accuracy is 2.5% of the full scale value). Two capacitance sensors (positioned lp ¼ 0:2 m apart) are placed 7 m downstream the injection device to characterize slug flows. Those capacitance probes have been developed in house, see Demori et al. (2010) and Strazza et al. (2011). Usually, capacitance sensors are used to investigate flows with non-conductive fluids, but in Strazza et al. (2011) a concave electrode sensor system is designed to work with conductive water introducing guarding electrodes and a high working frequency. The pipe-end is at atmospheric pressure and the fluids are discharged into the receiver tank. A glass box is inserted to reduce optical distortions and to allow a correct observation of the flows. A high-speed video camera is used to record the flows in the pipe in order to analyse the flow patterns. All the videos and the pictures are collected through the observation window. CMC-water solutions superficial velocity ranges from 0.05 m/s to 1.4 m/s and gas superficial velocity ranges from 0.1 m/s to 2 m/s. During the measurements, fluid temperature is in the range 24.5–25.5 °C. The sampling frequency used to acquire the capacitance probe, flow-meters, and pressure drop probes is set at 1 kHz.
Shear-thinning fluid characteristics The shear-thinning liquid phase is a water-CMC solution with three different polymer concentrations. The polymer is a high viscosity SCMC (sodium carboxymethyl cellulose) by Sigma Aldrich. Tap water is used for the solutions. The preparation of the solution is critical because CMC has a tendency to agglomerate or lump when added to water. During the preparation, water is stirred while the polymer is being added in small quantities.
Fig. 1. Sketch of the experimental set-up. A-Liquid tank, B-Receiver tank, C-Observation window, D-Turbine flow meter, E-Thermal flow meter, F-Centrifugal pump, G-Valve, I-Differential pressure transducer, L-Injector, M-Air injection, N-Valve, and O-Capacitance probe.
D. Picchi et al. / International Journal of Multiphase Flow 73 (2015) 217–226
The density of each solution is measured using a hydrometer with ± 0.5 kg/m3 accuracy. The rheology of the CMC water solutions is measured by a LVDV-3T Brookfield rheometer. The coaxial cylinder geometry sets are chosen to characterize the solutions. The CMC solutions exhibit a shear-thinning fluid behavior, which is described with a twoparameter power-law model. Assuming a power-law fluid the n shear stress s is related to the shear rate c_ by s ¼ mðc_ Þ , where the two fitting parameters m and n are the fluid consistency index and the flow behavior index, respectively. The fitting parameters are estimated using a least square interpolation method and the confident bounds chosen is 95%. Surface tension is measured by a Du Nouy ring tensiometer. The accuracy of the measurement is ± 2.5% of the measured value. Table 1 shows fluid properties of the different solutions used in the experiments. The increase of CMC concentration causes an increasing of the flow consistency index m, while the flow behavior index n decreases. The rheological behavior in terms of c_ s plot is shown for the three CMC solutions in the Electronic Annex 1.
Experimental procedure The following procedure is used during the experimental campaign: 1. a water-CMC solution is introduced at the lowest velocity U ls ; 2. air is introduced at the lowest value U gs ; 3. observations and data acquisitions at 1 kHz frequency for (U ls ; U gs ) pair are completed; 4. water-CMC solution flow-rate is increased and acquisitions for the new (U ls ; U gs ) pair are carried out; 5. step 4 is repeated from the lowest to the maximum liquid flowrate; 6. air flow-rate is increased and the procedure is started again from the lowest liquid flow-rate until the maximum is reached. Such a procedure is repeated for three different inclination angles: 0° and ± 5°.
From the analysis of the capacitance probe signals, we obtain the average length of the bubbles, of the slug and of the slug unit. We calculate the average time for the bubble, the slug, and the slug unit to transit (tf ; t s , and t) from the binarized signals. The lengths of bubble (‘f ), slug (‘s ), and slug unit (‘) are calculated multiplying U s , obtained as mentioned above and considering that the bubbles are translating with the slug velocity (U b ¼ U s ), and the average times t f ; t s , and t. Theoretical considerations Slug flow The mechanistic model proposed by Taitel and Barnea (1990) was reformulated by Orell (2005), but these models are valid for horizontal air-Newtonian fluid slug flows. We propose a model extending the one by Orell (2005) to inclined flows and considering the rheology of the power-law fluid. The model is based on a mass and momentum balance on the slug unit of length ‘ that consists in two sections (Fig. 2): a liquid slug zone of length ‘s and a film zone of length ‘f , which is composed by a liquid film with a constant height overlaid by a gas bubble (‘ ¼ ‘s þ ‘f ). The mass balances on the slug unit yield
‘f ‘s þ U f Hf ; ‘ ‘ ‘f ‘s U gs ¼ U s ð1 Hs Þ þ U g ð1 Hf Þ ; ‘ ‘
U ls ¼ U s Hs
ð1bÞ
U s ¼ U m ¼ U gs þ U ls :
ð2Þ
A liquid mass balance relative to a coordinate system that travels at the translational velocity of the slug unit, U t , of the slug unit is given by
ð3Þ
where the translational velocity is computed for inclined slug flows as
Slug frequency, slug velocity, slug length, and bubble length are obtained from the analysis of the capacitance probe signals. The slug frequency is considered as the number of slugs units (N s ) through a pipe cross-section during a given time period (Dt). The signals are processed by a threshold technique to obtain the equivalent rectangular wave from probe signals, see Angeli and Hanratty (2000) for details, and the frequency is estimated as f s ¼ N s =Dt. The slug velocity is obtained cross-correlating the two signals: the time interval between the signals (Dt) is given by the crosscorrelation and then the slug velocity is calculated dividing the distance between the probes by the time gap as U s ¼ lp =Dt.
U t ¼ C 0 U m þ ud ;
Conc. (%)
ql (kg/m3)
m (Pa sn )
n (–)
r (mN/m)
– 1 3 6
997.5 ± 0.5 998.0 ± 0.5 999.0 ± 0.5 1002.0 ± 0.5
0.001 ± 0.001 0.007 ± 0.001 0.061 ± 0.002 0.264 ± 0.010
1 0.942 ± 0.010 0.875 ± 0.011 0.757 ± 0.010
71.1 ± 1.8 72.1 ± 1.8 73.7 ± 1.8 76.1 ± 1.9
ð4Þ
where C 0 is the distribution parameter and ud is the drift velocity. The distribution parameter is usually set to 1.2 and 2, for turbulent and laminar velocity profile in the slug zone, respectively; the drift velocity is usually modeled as suggested by Bendiksen (1984). We used the value of C 0 and ud obtained from the experimental data, see Table 2.
Table 1 Physical properties of the test fluids at 25 °C and atmospheric pressure.
Water CMC-1 CMC-3 CMC-6
ð1aÞ
where Hs ; Hf ; U ls ; U gs , U s ; U f ; U g are the liquid holdup in the slug zone, the liquid holdup in the film zone, the liquid superficial velocity, the gas superficial velocity, the slug zone velocity, the liquid velocity in the film zone, and the gas velocity in the film zone, respectively. As proposed by Taitel and Barnea (1990), the slug zone velocity is supposed to be at the mixture velocity U m
ðU t U f ÞHf ¼ ðU t U s ÞHs ; Capacitance probe signals processing
219
Fig. 2. Slug flow in a pipe with a circular cross section.
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The momentum balance in the film zone, considering a uniform liquid thickness, can be written as for the stratified flow. Thus, eliminating the pressure gradient, the momentum balance yields
sl
Sl 1 1 Sg þ sg ðql qg Þg sin b ¼ 0; þ s i Si þ Ag Al Al Ag
ð5Þ
where Ag;l ; Sg;l ; sg;l ; U g;l and, qg;l are cross section, the wetted perimeter, the wall shear stress, the phase velocity, and the density of the gas and liquid, respectively. b is the inclination angle (positive for upward inclination) and si is the interfacial shear stress. The following closures are chosen for the shear stresses:
sl ¼ f l
ql U 2l 2
sg ¼ f g si ¼ f i
qg U 2g 2
ml
f l ¼ C l Rel
; ;
Rel ¼
;
g f g ¼ C g Rem ; g
qg ðU g U l ÞjU g U l j 2
Dnl U 2n q l ln ; m8n1 1þ3n 4n
Reg ¼
Dg U g qg
lg
;
ð6aÞ ð6bÞ ð6cÞ
;
where the constants are chosen as C l;g ¼ 16 and ml;g ¼ 1 for the laminar flow regime and as C l;g ¼ 0:079 and ml;g ¼ 0:25 for the turbulent flow regime. lg is the gas dynamic viscosity and the interfacial friction factor is chosen as f i ¼ 0:014, see Cohen and Hanratty (1968). Geometrical relations for the film zone, see Orell (2005) for details, and a model for the liquid hold-up in the slug zone are needed to close the model. We consider the correlation proposed by Andreussi et al. (1993)
Hs ¼ 1
pffiffiffiffiffiffi U m = gD F 0 pffiffiffiffiffiffi ; U m = gD þ 2400Bo3=4
D0 ; F 0 ¼ 2:6 1 2 D
ð7Þ
where Bo ¼ ðql qg ÞgD2 =r is the Bond number, D0 ¼ 0:025 m, and pffiffiffi F 0 ¼ 0 if D < 2D0 . For given fluid properties, geometry and fluid superficial velocities the model solves simultaneously Eqs. (1a), (1b), (3), and (5). Gas and liquid velocities in the film zone, U f and U g , the liquid hold-up in the film zone, Hf , and ‘s;f =‘ predictions are obtained. Then, the average pressure gradient of the slug unit is given by
‘f dP f q U 2 ‘s 4 ¼2 s s s þ ðsl Sl þ sg Sg Þ þ qsu g sin b; dx D ‘ pD2 ‘
ð8Þ
where qs ¼ ql Hs þ qg ð1 Hs Þ; f s is the slug friction factor, and qsu is the average density of the slug unit. The average density of the slug unit is evaluated as qsu ¼ ð1 Hsu Þqg þ Hsu ql , where Hsu ¼ Hs ‘s =‘ þ Hf ‘f =‘ is the average liquid hold-up of the slug unit. The liquid slug friction factor is modeled as the friction factor for a power-law fluid considering mixture conditions, yielding s f s ¼ C s Rem ; s
Res ¼
2n Dn U m qs ; n1 1þ3nn m8 4n
ð9Þ
Table 2 Values of C 0 and ud for the different air-CMC solution systems. Fluid
b (°)
C 0 (–)
ud (m/s)
R2 (–)
CMC-1 CMC-1 CMC-1
0 5 +5
1.08 1.23 1.18
0.42 0.26 0.18
0.924 0.643 0.950
CMC-3 CMC-3 CMC-3
0 5 +5
1.37 1.40 1.49
0.61 0.16 0.32
0.875 0.819 0.840
CMC-6 CMC-6 CMC-6
0 5 +5
1.52 1.49 1.53
0.32 0.25 0.36
0.971 0.942 0.967
where C l ¼ 16 and ml ¼ 1 for the laminar flow regime and as C l ¼ 0:079 and ml ¼ 0:25 for the turbulent flow regime. Results and discussion The experimental results are presented in this section. The considerations are organized considering separately flow pattern maps, pressure drop, and slug flow characteristics. Flow pattern maps In this section the experimental flow pattern maps obtained by visual observations are presented. We distinguish the following flow regimes: stratified, plug, and slug flow patterns. Churn and annular flow are not observed. We follow the flow pattern classification given by Chhabra and Richardson (2008); the distinction between plug and slug flow regime is due to the presence of entrained gas bubbles within the liquid slug. We also plotted the ‘‘zero characteristics’’ (ZRC) boundary, which gives the region of existence of the stratified flow regime computed from the twofluid model predictions and considering constant shape factors (cg ¼ cl ¼ 1), see Picchi et al. (2014) for details. The main difference between the slug flow pattern of a gas/ shear-thinning fluid system with respect to gas/water system concerns the structure of the liquid slug: there are fewer distinctly dispersed bubbles, in particular for higher concentration water-CMC solutions. The same behavior was observed by Xu et al. (2007) for air/CMC-solution systems and by Al-Sarkhi and Hanratty (2001) for air/water-polymer solution annular flows; Al-Sarkhi and Hanratty (2001) suggested that the disturbance waves at the interface are reduced by the polymer addition and the drop entrainment is reduced. In Fig. 3 two pictures taken with a high speed camera of typical plug and slug flows (CMC-6 solution) are shown: in plug flow there is no gas entrainment within the liquid, as opposite to slug flow where air bubbles are present within the liquid body. In Fig. 4 a slug of CMC-6 solution is shown in detail, showing the bubble head (Fig. 4(a)), middle part (Fig. 4(b)), tail (Fig. 4(c)), and the liquid slug with gas entrainment (Fig. 4(c)). Fig. 5 shows the experimental flow pattern maps obtained in the case of horizontal flow for the three different water-CMC solution systems. The different rheology of the three CMC solutions does not affect the transition from plug flow to slug flow. Only for the CMC-1 solution, one condition indicates the transition from plug to bubbly flow; in fact, increasing the CMC concentration the liquid superficial velocity given by the centrifugal pump decreases. The theoretical ZRC boundary, is plotted, but no stratified flow conditions are experimentally observed in the predicted existence region of the stratified flow regime. In Fig. 6 the results obtained for upward inclined flows (b ¼ þ5 ) are reported: plug and slug flows are observed. Only for the CMC-1 solution two points that give the transition from plug to bubbly flow are observed at low gas and high liquid superficial velocities. Increasing the CMC concentration and as a consequence, decreasing n and increasing m, plug flow regime is present at higher gas velocity for a given liquid superficial velocity. The ZRC boundary is not plotted because the calculation algorithm gives no solutions for these upward inclined flows. As discussed by Brauner and Moalem (1992) the shape of the ZRC boundary varies dramatically with inclination: in particular for upward inclination angles the well-posed region is reduced significantly and only limited areas of existence of the stratified flow regime are obtained. Fig. 7 shows the flow pattern maps for downward inclined flows (b ¼ 5 ), where, in addition to slug and plug, the stratified flow
D. Picchi et al. / International Journal of Multiphase Flow 73 (2015) 217–226
221
Fig. 3. Picture of the plug flow (CMC-6, horizontal flow, U gs ¼ 0:20 m/s and U ls ¼ 0:44 m/s) and picture of the slug flow (CMC-6, horizontal flow, U gs ¼ 1:20 m/s and U ls ¼ 0:46 m/s) taken with the high-speed camera.
(a) front of bubble
(b) middle of bubble
(c) back of bubble Fig. 4. Picture of the slug flow taken with the high-speed camera (CMC-6, horizontal flow, U gs ¼ 1:20 m/s and U ls ¼ 0:81 m/s).
regime is observed. The transition from plug to slug flow, keeping fixed the liquid superficial velocity, occurs at higher gas superficial velocities with respect to the horizontal case. The stratified flow is observed for all CMC solutions: the interface is wavier and more irregular for the CMC-1 solution and becomes noticeably smoother increasing the polymer concentration. In fact, adding polymer the interfacial activity is reduced, decreasing the interfacial roughness, see Soleimani et al. (2002). The existence region of the stratified flow regime is compared to the experimental data of Fig. 7: the predicted ZRC boundary agrees very well with the transition from the stratified flow in the experimental maps for CMC-1 and CMC-3 solution systems, and slightly over-predicts the transition for CMC-6 solution system. Increasing the CMC concentration, the existence region of the stratified flow moves to lower liquid velocities for a given gas superficial velocity and this behavior is well predicted by the theoretical criteria, see Fig. 7.
Pressure drop In this section the measured pressure drops are presented in terms of total pressure gradients. We have no holdup measurements, so we cannot estimate the frictional pressure gradient for inclined flows.
Stratified flow The experimental pressure drop data are compared to the predictions of the steady fully developed two-fluid model, see Picchi et al. (2014) for details. Usually, the pressure drop for the stratified flow are presented in in terms of dimensionless liquid pressure gradient U2l as a function of the Lockhart–Martinelli parameter X 2
U2l ¼
ðdp=dxÞtp ; ðdp=dxÞsl
X2 ¼
ðdp=dxÞsl 2f ls ql U 2ls =D ¼ ; ðdp=dxÞsg 2f gs qg U 2gs =D
ð10Þ
where ðdp=dxÞtp is the measured two-phase pressure gradient, f ls and f gs are superficial gas and liquid friction factors evaluated for the superficial conditions. In Fig. 8 the experimental data are compared to the theoretical predictions: it can be noted that the agreement is not so satisfactory. This behavior is underlined both in Xu et al. (2007) and Picchi et al. (2014): in Fig. 8 the data by Xu et al. (2007) are also reported to show that the model fails the predictions when the total pressure drop is close to zero for inclined flows. The not perfect agreement can be due to the error in the hold-up prediction in inclined conditions and to the fact that for these downward inclined flows the pressured drop are close to zero. In fact, the pressure drop can be divided into two contributes, the frictional and the gravitational one: for downward inclined flows they have an
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plug slug plug/bubbly ZRC line
Uls (m/s)
0
10
−1
10
−1
10
0
10
Ugs (m/s) (a) CMC-1
plug slug ZRC line
0
10
−1
10
0
Uls (m/s)
Uls (m/s)
1
10
compatible with the model and the solution algorithm is built to find the solution of Eq. (5) in the range of U f > 0. In Fig. 9 the experimental pressure drop for the three waterCMC solution systems are compared to the predictions by the model, showing a good agreement. The model predicts 78% of the experimental data within ± 30% error, giving an acceptable agreement between the predicted and the measured pressure drop. In the Electronic Annex 1 the experimental pressure drop for the three solution systems (horizontal flow) compared to the model predictions are plotted as function of mixture and gas superficial velocities: increasing gas velocity, and keeping fixed the mixture velocity, the pressure drop increases; on the other hand increasing the mixture velocity, keeping fixed the superficial gas velocity, the pressure drop increases. In Fig. 9 the experimental pressure drops are also compared to the predictions by Dziubinski (1995), where a modified Lockhart and Martinelli (1949) model was presented. The Dziubinski’s correlation is a simple way to obtain pressure drop
−2
10
−1
10
0
10
Ugs (m/s) (b) CMC-3
10
1
10
plug slug plug/bubbly
−1
10
−1
0
10 plug slug ZRC line
0
10
−1
0
10
10
Uls (m/s)
Uls (m/s)
10
Ugs (m/s) (a) CMC-1
−2
10
−1
10
0
10
Ugs (m/s) (c) CMC-6
1
−1
10
10
plug slug −1
10
Fig. 5. Flow pattern maps for horizontal flow for the three water–CMC solutions systems. The predicted well-posedness boundary (ZRC) is plotted.
0
Ugs (m/s) (b) CMC-3
10
opposite sign, so, in this case, the total pressure drop has a magnitude close to zero. 0
10
Uls (m/s)
Slug flow The experimental pressure drops are compared to the model presented in Section 3.1. The measured pressure drops are the time average pressure drop during the experimental acquisition. Considering the operative conditions of this work (working fluids, pipe diameter and fluid superficial velocities), the model gives pressure drop predictions for horizontal and downward flows, but gives no-solution for upward b ¼ þ5 inclination flows. The mass balance given in Eq. (5) has to be satisfied to obtain a valid solution for the the liquid hold-up in the film zone. The model gives a negative liquid velocity (U f ) in the film zone from Eq. (1a) for the operative condition of the upward flow. The result of a negative liquid velocity in the film zone is not consistent with the formulation of the model: back-flow in the film zone is not
−1
10
plug slug −1
10
0
10
Ugs (m/s) (c) CMC-6
Fig. 6. Flow pattern maps for the upward (b ¼ þ5 ) flow for the three water–CMC solutions systems.
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and Scott (1969) correlation is valid for air–water gas slug flows and it reads
plug slug stratified ZRC line
f s ¼ 0:0226
Uls (m/s)
0
10
−1
10 −1 10
0
1
10
10
Ugs (m/s) (a) CMC-1
plug slug stratified ZRC line
0
Uls (m/s)
er ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 f s;corr f s;exp f s;exp
1
N
ð12Þ
;
where f s;corr ; f s;exp , and N are the predicted slug frequency, the measured slug frequency, and the number of experiments, respectively. We extend the Gregory and Scott (1969) correlation to account for the rheology of the shear-thinning fluid: the structure of the correlation is respected adding two contributes. The extended correlation yields
−1
10
−1
10
0
1
10
10
Ugs (m/s) (b) CMC-3
f s ¼ 0:0448
0
Rels ¼
−1
10
−1
10
0
1
10
10
Ugs (m/s) (c) CMC-6
0:88 0:07 U ls 32:2014 Rels þ Um ðnÞ2:85 ; Um gD Rews
ð13Þ
where Rels is the Reynolds number for a power-law fluid considering superficial conditions and Rews is the water Reynolds number considering superficial conditions
plug slug stratified ZRC line
10
Uls (m/s)
ð11Þ
where f s is the slug frequency, U ls is the liquid velocity, U m is the mixture velocity, g is the gravitational acceleration and D is the pipe diameter. In Fig. 10 the comparison between the measured and the predicted frequency by the Gregory and Scott (1969) correlation is presented. The frequency for the CMC-1 solution is well predicted by the Gregory and Scott (1969) correlation: this result is reasonable due to the fact that the rheological behavior of the CMC-1 solution is close to a Newtonian fluid (n ¼ 0:9424 and m ¼ 0:0065 Pa sn ). The correlation does not hold acceptable for the other two solutions, CMC-3 and CMC-6. In Fig. 10 the average relative error er is shown; er is defined as
PN
10
1:2 U ls 19:75 þ Um ; Um gD
Dnl U 2n ls ql n ; m8n1 1þ3n 4n
Rews ¼
qw U ls D : lw
ð14Þ
The proposed correlation is written to keep the same structure of the Gregory and Scott (1969) correlation, considering two additional contributes: the fluid behavior index and the ratio between the Reynolds numbers. The correlation gives the same coefficients of the Gregory and Scott (1969) interpolating their gas–water data (the two new terms collapse to unity for air–water systems).
0.4
Fig. 7. Flow pattern maps for the downward (b ¼ 5 ) flow for the three water– CMC solutions systems. The predicted well-posedness boundary (ZRC) is plotted.
0.35
predictions, but, respect to the mechanistic model (Section 3.1), it has an emiprical nature: it was obtained from the fitting of experimental data of gas/shear-thinning slug flows from the literature. The experimental data error of this work is around 50%, see Fig. 9, but this can be due to the experimental nature of the correlation. Slug flow characteristics In this section, measured slug flow characteristics are presented: slug frequency, slug velocities, and slug length are measured from the analysis of the capacitance probe signals. Slug frequency The measured slug frequency is compared with the empirical correlation proposed by Gregory and Scott (1969). The Gregory
Predicted Φl2 (−)
0.3 0.25 0.2 0.15 0.1 0.05 CMC−1 CMC−3 CMC−6 Xu et. al (2007) data
0 −0.05 −0.1 −0.1
0
0.1
0.2
0.3
0.4
Experimental Φ2l (−) Fig. 8. Dimensionless experimental pressure gradient U2l of this work for the three CMC-solution systems and by Xu et al. (2007) compared to the predictions by the steady fully developed two-fluid model. Dotted lines represent 30%.
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5
4000
4.5
3500
4
3000
3.5
Ub (m/s)
4500
2500 2000
3 2.5 Exp. β=0° β=0°: U =1.08 U +0.42
1500
2
1000
1.5
Exp. β=−5° β=−5°: U =1.23U +0.26
1
Exp. β=+5° β=+5°: U =1.18U +0.18
Dziubinski (1995) correlation CMC−1 CMC−3 CMC−6
500 0 0
1000
2000
3000
b
m
b
m
b
0.5 0.5
4000
1
1.5
2
2.5
m
3
3.5
4
Um (m/s) Fig. 9. Experimental pressure drop of this work for the three water–CMC solution systems compared to the predictions by the proposed model and by Dziubinski (1995) correlation. Dotted lines represent 30%.
Fig. 12. Mixture velocity versus slug velocity for air/CMC-1 flow.
5
6 CMC−1 CMC−3 CMC−6
5
4.5
3.5
4
Ub (m/s)
Predicted fs (1/s)
4
3
2
3 2.5 Exp. β=0° β=0°: U =1.37U +0.61
2 1.5
er = 34.4 %
1
b
b
m
Exp. β=+5° β=+5°: U =1.49U +0.32
1 0
m
Exp. β=−5° β=−5°: U =1.40U +0.16
b
0
1
2
3
4
5
6
0.5 0.5
Experimental fs (1/s)
2
2.5
3
3.5
4
Fig. 13. Mixture velocity versus slug velocity for air/CMC-3 flow.
5
CMC−1 CMC−3 CMC−6
5
1.5
Um (m/s)
Fig. 10. Experimental slug frequency for the three water–CMC solution systems compared to the predictions by the Gregory and Scott (1969) correlation. Dotted lines represent 20%.
6
1
m
4.5
3.5
Ub (m/s)
Predicted fs (1/s)
4 4
3
2
3 2.5
Exp. β=0° β=0°: U =1.48U +0.38
2
er = 22.5 %
1
0
b
1.5
Exp. β=+5° β=+5°: U =1.53U +0.35
1 0
1
2
3
4
5
b
6
Experimental fs (1/s) Fig. 11. Experimental slug frequency for the three water–CMC solution systems compared to the predictions by the proposed correlation. Dotted lines represent 20%.
0.5 0.5
m
Exp. β=−5° β=−5°: Ub=1.49+0.25
1
1.5
2
2.5
3
m
3.5
Um (m/s) Fig. 14. Mixture velocity versus slug velocity for air/CMC-6 flow.
4
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Despite the inclusion in Eq. (13) of the term n is quite unorthodox, it allows to improve the agreement with experimental data and it accounts for different shear-thinning fluid behavior. In Fig. 11 the comparison between the predicted frequency by Eq. (13) and the measured one is presented for the three waterCMC solution systems. The aim is to propose a correlation valid for all the CMC concentration and the agreement is reasonably good: the relative average error is er ¼ 22:5%, about 35% less that using the Gregory and Scott (1969) correlation. This first attempt to propose a correlation for the slug frequency valid for gas/ shear-thinning systems is encouraging. In fact, in Rosehart et al. (1975) and Otten and Fayed (1977), the frequencies are correlated using the Gregory and Scott (1969) relation, which uses different values of the coefficients for each shear-thinning fluid series data. In the Electronic Annex 1 the measured slug frequencies are reported. Slug velocity The slug velocity is obtained by post-processing capacitance probe signals, see Section 2.4 for details. The data are presented 450 o
400
f /D
(−)
350
β=0
o
β=−5
o
β=+5
300 250 200 150 100 50 0 0.1
450
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o
β=+5
(−) f /D
300 250 200 150 100 50
0.5
0.6
0.7
0.8
0.9
1
Ugs /Um (−) (b) CMC-3 o
β=0 400
o
β=−5
o
β=+5
f /D
(−)
350 300 250 200 150 100 50 0 0.4
0.5
where C 0 is the distribution parameter and ud is the drift velocity. The values of the estimated distribution parameter and the drift velocity are shown in Table 2. The values of C 0 for the CMC-1 solution systems are close to the ones for air–water systems: C 0 1:1 for the horizontal case, as observed by Woods and Hanratty (1996) for air/water systems at low gas velocities, and C 0 1:2 for inclined systems. The distribution parameter C 0 increases while the CMC concentration increases. Otten and Fayed (1977) observed the same trend in C 0 due to the addition of the polymer, assuming the velocity equal to zero (ud ¼ 0) since all the experiments are performed for horizontal flows. The drift velocity is ud – 0 for all the experimental data of this work even in horizontal conditions. Slug length The bubble length ‘f is an important feature to describe slug flows and in Fig. 15 it is plotted as a function of the ratio U gs =U m : increasing the ratio U gs =U m the bubble length increases. The liquid slug length ‘s is always considered as a constant parameter, see Nicholson et al. (1978), and equal to 30 pipe diameter. In the Electronic Annex 1 the liquid slug length is presented as a function of the mixture velocity for all the experimental data of this work. The average length for air/CMC-1, CMC-3, and CMC-6 systems are ‘s;CMC1 =D ¼ 34, ‘s;CMC3 =D ¼ 42, and ‘s;CMC6 =D ¼ 26, respectively.
In this work an experimental investigation on gas/shearthinning fluid pipe flow was presented. The experiments were performed in a 9-m-long glass pipe studying three different air/CMC solution systems and the different inclination angles. Flow pattern maps were presented and the influence of the rheology of the water–CMC solutions and of the inclination angle were investigated. The existence region of the stratified flow regime was compared to the ZRC boundary, predicted by the model given by Picchi et al. (2014), showing a good agreement. Increasing polymer concentration (decreasing the flow behavior index n and increasing the flow consistency index m) the existence region of the stratified flow moves to lower liquid velocities for a given gas superficial velocity and this behavior is well predicted by the criteria. Furthermore, as mentioned by Soleimani et al. (2002), we observed that the addition of polymers reduces the interfacial activities of the stratified flow and we investigated how the transition from plug to slug flow was influenced by the inclination angle and the rheology of the shear thinning fluid. The pressure drop data of the stratified flow regime were compared to the steady fully developed two-fluid model, see Picchi et al. (2014) for details, showing that the model does not predict them with a high accuracy. A model for gas/power-law fluid slug flow was presented and the predictions were compared to the experimental data of this work. The model predicts the experimental pressure drops with a satisfactory agreement: the 78% of the experimental data were predicted within ± 30% error. This comparison demonstrates the validity of the present model. Slug characteristics were studied from the analysis of a capacitance probe signals: slug velocity, slug frequency and slug lengths were investigated. Slug frequency was at fist compared to the
350
450
ð15Þ
(a) CMC-1 o
0.4
U b ¼ C 0 U m þ ud ;
Conclusion
β=−5o
0
in Figs. 12–14 distinguishing among three polymer concentrations and the inclination angles. A linear least-square fitting on experimental data is done to estimate the coefficients of the Nicklin et al. (1962) relations for bubble velocity, yielding
Ugs /Um (−)
β=0 400
225
0.6
0.7
0.8
0.9
1
Ugs /Um (−) (c) CMC-6 Fig. 15. Bubble length ‘f for the three different air/CMC systems as a function of the ratio U gs =U m .
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Gregory and Scott (1969) correlation and then a new correlation for the slug frequency was proposed to take into account the rheology of the shear-thinning fluid. The comparison between the experimental data and the new correlation prediction was promising. Slug velocity is investigated in terms of Nicklin et al. (1962) model: the distribution parameter increases increasing the polymer concentration and the drift velocity is different from 0 for all the case investigated. Liquid slug length and bubble length data are presented. Appendix A. Supplementary material Supplementary material associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j. ijmultiphaseflow.2015.03.005. References Al-Sarkhi, A., Hanratty, T.J., 2001. Effect of drag-reducing polymers on annular gas– liquid flow in horizontal pipe. Int. J. Multiphase Flow 27, 1151–1162. Andreussi, P., Minervini, A., Paglianti, A., 1993. Mechanistics model of slug flow in near-horizontal pipes. AIChE J. 39, 1281–1291. Angeli, P., Hanratty, T.J., 2000. Flow structure in horizontal oilwater ow. Int. J. Multiphase Flow 26, 1117–1140. Bendiksen, K., 1984. An experimental investigation of the motion of long bubbles in inclined tubes. Int. J. Multiphase Flow 10, 467–483. Bishop, A.A., Deshpande, S.D., 1986. Non-Newtonian liquid–air stratified flow trough horizontal tubes II. Int. J. Multiphase Flow 12, 977–996. Brauner, N., Moalem Maron, D., 1992. Stability analysis of stratified liquid–liquid flow. Int. J. Multiphase Flow 18, 103–121. Brauner, N., Moalem Maron, D., 1992. Analysis of stratified/nonstratified transitional boundaries in inclined gas–liquid flows. Int. J. Multiphase Flow 18, 541–557. Chhabra, R.P., Richardson, J.F., 1984. Prediction of flow patterns for concurrent flow of gas and non-Newtonian liquid in horizontal pipes. Can. J. Chem. Eng. 62, 449– 454. Chhabra, R.P., Richardson, J.F., 2008. Non-Newtonian Flow and Applied Rheology: Engineering Applications. Butterworth-Heinemann, Oxford. Chhabra, R.P., Farooqi, S.I., Richardson, J.F., 1983. Co-current flow of air and shear thinning suspensions in pipes of large diameter. Chem. Eng. Res. Des. 61, 56–61. Chhabra, R.P., Farooqi, S.I., Richardson, J.F., 1984. Isothermal two-phase flow of air and aqueous polymer solutions in a smooth horizontal pipe. Chem. Eng. Res. Des. 62, 22–32. Cohen, L.S., Hanratty, T.J., 1968. Effect of waves at gas–liquid interface on a turbulent air flow. J. Fluid Mech. 31, 467–479. Demori, M., Strazza, D., Ferrari, V., Poesio, P., 2010. A capacitive sensor system for the analysis of two-phase flows of oil and conductive water. Sens. Actuat. A 163, 172–179. Dziubinski, M., 1995. General correlation for two-phase pressure drop in intermittent flow of gas and non-Newtonian liquid mixtures in a pipe. Chem. Eng. Res. Des. 73, 528–534. Dziubinski, M., Chhabra, R.P., 1989. Predicting 2-phase pressure drop for the flow of gas non-Newtonian liquid mixtures in horizontal pipes. Int. J. Eng. Fluid Mech. 2, 63–78. Dziubinski, M., Fidos, M., 2004. A simple model for slug flow of a gas and nonNewtonian liquid mixtures in a vertical pipe. In: 3rd International Symposium on Two-Phase Flow Modelling and Experimentation, 22–24 September. Pisa, Italy.
Dziubinski, M., Fidos, M., Sosno, M., 2004. The flow pattern map of a two-phase nonNewtonian liquid–gas flow in the vertical pipe. Int. J. Multiphase Flow 30, 551– 563. Eisemberg, F.G., Weinberger, C.B., 1979. Annular two-phase flow of gases and nonNewtonian liquids. AIChE J. 25, 240–245. Fabe, J., Liné, A., 1992. Modelling of two-phase slug flow. Ann. Rev. Fluid Mech. 24, 21–46. Grassi, B., Strazza, D., Poesio, P., 2008. Experimental validation of theoretical models in two-phase high-viscosity ratio liquid liquid flows in horizontal and slightly inclined pipes. Int. J. Multiphase Flow 34, 950–965. Gregory, G.A., Scott, D.S., 1969. Correlation of liquid slug velocity and frequency in horizontal cocurrent gas–liquid slug flow. AIChE J. 26, 933–935. Hanratty, T., 2013. Physics of Gas–Liquid Flows. Cambridge University Press, Cambridge. Heywood, N.I., Charles, M.E., 1979. The stratified flow of a gas and non-Newtonian liquid in horizontal pipes. Int. J. Multiphase Flow 5, 341–352. Jia, N., Gourma, M., Thompson, C.P., 2011. Non-Newtonian multi-phase flows: on drag reduction, pressure drop and liquid wall friction factor. Chem. Eng. Sci. 66, 4742–4756. Lockhart, R.W., Martinelli, R.C., 1949. Proposed correlation of data for isothermal two-phase, two-components flow in pipes. Chem. Eng. Progress 45, 39–48. Mandhane, J.M., Gregory, G.A., Aziz, K., 1974. A flow pattern map for gas–liquid flow in horizontal pipes. Int. J. Multiphase Flow 1, 537–553. Nicholson, M.K., Aziz, K., Gregory, G.A., 1978. Intermittent two phase flow in horizontal pipes: predictive models. Can. J. Chem. Eng. 56, 653–663. Nicklin, D.J., Wilkes, J.O., Davidson, J.F., 1962. Two-phase flow in vertical tubes. Trans. Inst. Chem. Engrs. 40, 61–68. Orell, A., 2005. Experimental validation of a simple model for gas and liquid slug flow in horizontal pipes. Chem. Eng. Sci. 60, 1371–1381. Otten, L., Fayed, S., 1977. Slug velocity and slug frequency measurements in concurrent air/non-Newtonian slug flow. Trans. Inst. Chem. Engrs. 55, 64–67. Picchi, D., Correra, S., Poesio, P., 2014. Flow pattern transition, pressure gradient, hold-up predictions in gas/non-Newtonian power-law fluid stratified flow. Int. J. Multiphase Flow 63, 105–115. Rosehart, R.G., Rhodes, E., Scott, D.S., 1975. Studies of gas–liquid (non-Newtonian) slug flow: void fraction meter, void fraction and slug characteristics. Chem. Eng. J. 10, 57–64. Ruiz-Viera, M.J., Delgado, M.A., Franco, J.M., Sanchez, M.C., Gallegos, C., 2006. On the drag reduction for the two-phase horizontal pipe flow of highly viscous nonNewtonian liquid/air mixtures: case of lubricating grease. Int. J. Multiphase Flow 32, 232–247. Ruiz-Viera, M.J., Delgado, M.A., Franco, J.M., Gallegos, C., 2006. Evaluation of wall slip effects in the lubricating grease/air two-phase flow along pipelines. J. NonNewton. Fluid Mech. 139, 190–196. Soleimani, A., Al-Sarkhi, A., Hanratty, T.J., 2002. Effect of drag-reducing polymers on pseudo-slugs interfacial drag and transition to slug flow. Int. J. Multiphase Flow 28, 1911–1927. Strazza, D., Demori, M., Ferrari, V., Poesio, P., 2011. Capacitance sensor for hold-up measurement in high-viscous-oil/conductive-water core annular flows. Flow Measur. Instrum. 22, 360–369. Taitel, Y., Barnea, D., 1990. A consistent approach for calculating pressure drop in inclined slug flow. Chem. Eng. Sci. 35, 779–793. Taitel, Y., Dukler, A.E., 1976. A model for predicting flow regime transitions in horizontal and near horizontal gas liquid flow. AIChE J. 22, 47–55. Woods, B.D., Hanratty, T.J., 1996. Relation of slug stability to shedding rate. Int. J. Multiphase Flow 22, 809–828. Xu, J., Wu, Y., Shi, Z., Lao, L., Li, D., 2007. Studies on two-phase co-current air/nonNewtonian shear-thinning fluid flows in inclined smooth pipes. Int. J. Multiphase Flow 33, 948–969. Xu, J., Wu, Y., Li, H., Guo, J., Chang, Y., 2009. Studies of drag reduction by gas injection for power-law fluid flow in horizontal stratified and slug flow regimes. Chem. Eng. J. 147, 235–244.
Contents lists available at ScienceDirect
International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w
Gas/shear-thinning liquid flows through pipes: Modeling and experiments Davide Picchi a, Yuri Manerba a, Sebastiano Correra b, Michele Margarone c, Pietro Poesio a,⇑ a
Università degli Studi di Brescia, Dipartimento di Ingegneria Meccanica e Industriale, Via Branze 38, 25123 Brescia, Italy eni spa, ISDEW, via Emilia, 1, 20097 San Donato Milanese, Italy c eni spa, ITEM, via Emilia, 1, 20097 San Donato Milanese, Italy b
a r t i c l e
i n f o
Article history: Received 4 November 2014 Received in revised form 2 March 2015 Accepted 8 March 2015 Available online 21 March 2015 Keywords: Shear-thinning fluid Flow pattern map Pressure gradient Slug frequency Stratified flow Slug velocity
a b s t r a c t In chemical and oil industry gas/shear-thinning liquid two-phase flows are frequently encountered. In this work, we investigate experimentally the flow characteristics of air/shear-thinning liquid systems in horizontal and slightly inclined smooth pipes. The experiments are performed in a 9-m-long glass pipe using air and three different carboxymethyl cellulose (CMC) solutions as test fluids. Flow pattern maps are built by visual observation using a high-speed camera. The observed flow patterns are stratified, plug, and slug flow. The effects of the pipe inclination and the rheology of the shear-thinning fluid in terms of flow pattern maps are presented. The predicted existence region of the stratified flow regime is compared with the experimental observation showing a good agreement. A mechanistic model valid for air/powerlaw slug flow is proposed and model predictions are compared to the experimental data showing a good agreement. Slug flow characteristics are investigated by the analysis of the signals of a capacitance probe: slug velocity, slug frequency, and slug lengths are measured. A new correlation for the slug frequency is proposed and the results are promising. Ó 2015 Elsevier Ltd. All rights reserved.
Introduction Two-phase gas/liquid flows, where the fluid exhibits a shearthinning behavior, are often encountered in oil and process industries. However, only few studies have focused on gas/shear thinning fluid systems. Eisemberg and Weinberger (1979), Chhabra and Richardson (1984,), Chhabra et al. (1983), and Dziubinski and Chhabra (1989) were among the first to study gas/shear-thinning fluid mixtures in horizontal pipes: flow pattern maps were provided proposing empirical correlations for flow pattern transition starting from the map by Mandhane et al. (1974) and pressure drop data were analysed following the Lockhart and Martinelli (1949) model. Afterwords, Dziubinski (1995) proposed an empirical and a more general correlation for pressure drop predictions in the intermittent flow regime of gas/shear-thinning fluid mixtures in pipe and, more recently, Ruiz-Viera et al. (2006,) studied the wall-slip effects and the drag reduction due to air injection in the flow of a non-Newtonian lubricating grease giving a modified version of ⇑ Corresponding author. Tel.: +39 030 3715646. E-mail addresses: [email protected] (D. Picchi), [email protected] (P. Poesio). http://dx.doi.org/10.1016/j.ijmultiphaseflow.2015.03.005 0301-9322/Ó 2015 Elsevier Ltd. All rights reserved.
the Lockhart and Martinelli (1949) correlation. Flow patterns for gas/shear-thinning fluid in a vertical pipe were experimentally investigated by Dziubinski et al. (2004). These empirical approaches for pressure drop predictions are the simplest to study a complex situation like intermittent flows where one phase shows a shear-thinning behavior, see Chhabra and Richardson (2008) for a complete classification of gas/nonNewtonian fluid flow patterns and these experimental models for hold-up and pressure drop predictions in horizontal and vertical pipe. More sophisticated models can be used if information about the flow pattern are available. Concerning the stratified flow regime, Heywood and Charles (1979) extended the two-fluid model by Taitel and Dukler (1976) to gas/power-law fluids and Bishop and Deshpande (1986) tested that model with experimental data using air and a water-soluble polymer (7H4 SCMC) as test fluids. Xu et al. (2007, 2009) studied air/shear-thinning fluid flows in inclined pipes (test fluids were air and CMC-water solutions) for different pipe diameters and inclination angles: flow pattern maps were obtained by visual observation while hold-up and pressure drop data were collected. More recently, Picchi et al. (2014) studied gas/power-law fluid stratified pipe flow. In addition to the twofluid model given by Xu et al. (2007), a pre-integrated model was provided to obtain hold-up and pressure gradient predictions and
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the flow pattern transition from the stratified flow regime developed by Brauner and Moalem (1992) was extended to gas/ power-law fluid systems. The flow pattern transition predictions were validated with the experimental data by Xu et al. (2007). Regarding the slug flow regimes, a mechanistic model for holdup and pressure drop predictions was presented by Xu et al. (2007) and three-dimensional CFD simulations of gas/shear-thinning liquid mixtures were carried out by Jia et al., 2011. In addition to these works, where gas/shear-thinning pipe flows were investigated in terms of flow patterns, hold-up, and pressure drop, the study of slug flow characteristics has a practical relevance. The main characteristics of slug flows, slug frequency, and slug velocity for gas/Newtonian systems were deeply investigated, see Fabe and Liné (1992) and Hanratty (2013) and references therein. Only Rosehart et al. (1975) and Otten and Fayed (1977) presented an experimental characterization of air/shear-thinning fluid and air/Herschel–Bulkley fluid systems in horizontal pipes, respectively, and Dziubinski and Fidos (2004) for a vertical pipe. The aim of the present work is to present a new experimental campaign on two-phase gas/shear-thinning fluid flow in horizontal and slightly inclined pipes. The shear-thinning liquids chosen are CMC-water solutions: the introduction of small quantities of long-chain polymers into water has a practical relevance also for drag-reduction in gas–liquid flows (some studies on the drag-reducing effects on annular, stratified, and slug flow are mentioned in Hanratty (2013), references are therein) but in this work the polymers are added to modify the rheology of suspensions. Flow pattern maps are built by visual observations and the influence of the inclination angle and fluid rheology are investigated. The transition from stratified flow regime is compared with theoretical predictions by Picchi et al. (2014) and pressure drop data are compared to two-fluid model predictions for stratified flow and to a new model predictions for the slug flow, respectively. Slug frequency and slug velocity are measured by post-processing the signals of a capacitance sensor: slug frequency is compared to the Gregory and Scott (1969) correlation and a new correlation is proposed to take into account the rheology of the shear-thinning fluid; slug velocity is investigated studying the influence of the inclination and the polymer concentration on the distribution parameter and the drift velocity. Liquid slug length and bubble length data are also presented.
Experimental setup Flow loop The experiments are performed in setup schematically represented in Fig. 1. The system is composed by a 9 m-long glass pipe,
with an inner diameter of 22.8 mm. The pipe is mounted on a rigid beam and the inclination can be changed around the central pin. The investigated inclination angles are 0°, horizontal flow, 5°, downward flow, and + 5°, upward flow. CMC-water solutions and air, used as test fluids, are injected by the injector; L-injector in Fig. 1, see Grassi et al. (2008). Air (viscosity lg ¼ 1:8 105 Pa s, density qg ¼ 1:2 kg/m3) is supplied directly from the University network; air mass flow rate is measured prior to entering the test section by a thermal mass flow meter (accuracy is 3% of the full scale value) and, at the same spot, air pressure is monitored. Liquid supply is provided by a centrifugal pump, which draws CMC-water solution stored in a 0.5 m3 tank, and the flow rate is controlled by a frequency inverter. Liquid flow rate is measured with a turbine mass flow meter with an accuracy of 1% of the full scale value. A differential pressure transducer is placed 6 m downstream the injection point and the pressure drop is measured across a 1.5 m length (sensor accuracy is 2.5% of the full scale value). Two capacitance sensors (positioned lp ¼ 0:2 m apart) are placed 7 m downstream the injection device to characterize slug flows. Those capacitance probes have been developed in house, see Demori et al. (2010) and Strazza et al. (2011). Usually, capacitance sensors are used to investigate flows with non-conductive fluids, but in Strazza et al. (2011) a concave electrode sensor system is designed to work with conductive water introducing guarding electrodes and a high working frequency. The pipe-end is at atmospheric pressure and the fluids are discharged into the receiver tank. A glass box is inserted to reduce optical distortions and to allow a correct observation of the flows. A high-speed video camera is used to record the flows in the pipe in order to analyse the flow patterns. All the videos and the pictures are collected through the observation window. CMC-water solutions superficial velocity ranges from 0.05 m/s to 1.4 m/s and gas superficial velocity ranges from 0.1 m/s to 2 m/s. During the measurements, fluid temperature is in the range 24.5–25.5 °C. The sampling frequency used to acquire the capacitance probe, flow-meters, and pressure drop probes is set at 1 kHz.
Shear-thinning fluid characteristics The shear-thinning liquid phase is a water-CMC solution with three different polymer concentrations. The polymer is a high viscosity SCMC (sodium carboxymethyl cellulose) by Sigma Aldrich. Tap water is used for the solutions. The preparation of the solution is critical because CMC has a tendency to agglomerate or lump when added to water. During the preparation, water is stirred while the polymer is being added in small quantities.
Fig. 1. Sketch of the experimental set-up. A-Liquid tank, B-Receiver tank, C-Observation window, D-Turbine flow meter, E-Thermal flow meter, F-Centrifugal pump, G-Valve, I-Differential pressure transducer, L-Injector, M-Air injection, N-Valve, and O-Capacitance probe.
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The density of each solution is measured using a hydrometer with ± 0.5 kg/m3 accuracy. The rheology of the CMC water solutions is measured by a LVDV-3T Brookfield rheometer. The coaxial cylinder geometry sets are chosen to characterize the solutions. The CMC solutions exhibit a shear-thinning fluid behavior, which is described with a twoparameter power-law model. Assuming a power-law fluid the n shear stress s is related to the shear rate c_ by s ¼ mðc_ Þ , where the two fitting parameters m and n are the fluid consistency index and the flow behavior index, respectively. The fitting parameters are estimated using a least square interpolation method and the confident bounds chosen is 95%. Surface tension is measured by a Du Nouy ring tensiometer. The accuracy of the measurement is ± 2.5% of the measured value. Table 1 shows fluid properties of the different solutions used in the experiments. The increase of CMC concentration causes an increasing of the flow consistency index m, while the flow behavior index n decreases. The rheological behavior in terms of c_ s plot is shown for the three CMC solutions in the Electronic Annex 1.
Experimental procedure The following procedure is used during the experimental campaign: 1. a water-CMC solution is introduced at the lowest velocity U ls ; 2. air is introduced at the lowest value U gs ; 3. observations and data acquisitions at 1 kHz frequency for (U ls ; U gs ) pair are completed; 4. water-CMC solution flow-rate is increased and acquisitions for the new (U ls ; U gs ) pair are carried out; 5. step 4 is repeated from the lowest to the maximum liquid flowrate; 6. air flow-rate is increased and the procedure is started again from the lowest liquid flow-rate until the maximum is reached. Such a procedure is repeated for three different inclination angles: 0° and ± 5°.
From the analysis of the capacitance probe signals, we obtain the average length of the bubbles, of the slug and of the slug unit. We calculate the average time for the bubble, the slug, and the slug unit to transit (tf ; t s , and t) from the binarized signals. The lengths of bubble (‘f ), slug (‘s ), and slug unit (‘) are calculated multiplying U s , obtained as mentioned above and considering that the bubbles are translating with the slug velocity (U b ¼ U s ), and the average times t f ; t s , and t. Theoretical considerations Slug flow The mechanistic model proposed by Taitel and Barnea (1990) was reformulated by Orell (2005), but these models are valid for horizontal air-Newtonian fluid slug flows. We propose a model extending the one by Orell (2005) to inclined flows and considering the rheology of the power-law fluid. The model is based on a mass and momentum balance on the slug unit of length ‘ that consists in two sections (Fig. 2): a liquid slug zone of length ‘s and a film zone of length ‘f , which is composed by a liquid film with a constant height overlaid by a gas bubble (‘ ¼ ‘s þ ‘f ). The mass balances on the slug unit yield
‘f ‘s þ U f Hf ; ‘ ‘ ‘f ‘s U gs ¼ U s ð1 Hs Þ þ U g ð1 Hf Þ ; ‘ ‘
U ls ¼ U s Hs
ð1bÞ
U s ¼ U m ¼ U gs þ U ls :
ð2Þ
A liquid mass balance relative to a coordinate system that travels at the translational velocity of the slug unit, U t , of the slug unit is given by
ð3Þ
where the translational velocity is computed for inclined slug flows as
Slug frequency, slug velocity, slug length, and bubble length are obtained from the analysis of the capacitance probe signals. The slug frequency is considered as the number of slugs units (N s ) through a pipe cross-section during a given time period (Dt). The signals are processed by a threshold technique to obtain the equivalent rectangular wave from probe signals, see Angeli and Hanratty (2000) for details, and the frequency is estimated as f s ¼ N s =Dt. The slug velocity is obtained cross-correlating the two signals: the time interval between the signals (Dt) is given by the crosscorrelation and then the slug velocity is calculated dividing the distance between the probes by the time gap as U s ¼ lp =Dt.
U t ¼ C 0 U m þ ud ;
Conc. (%)
ql (kg/m3)
m (Pa sn )
n (–)
r (mN/m)
– 1 3 6
997.5 ± 0.5 998.0 ± 0.5 999.0 ± 0.5 1002.0 ± 0.5
0.001 ± 0.001 0.007 ± 0.001 0.061 ± 0.002 0.264 ± 0.010
1 0.942 ± 0.010 0.875 ± 0.011 0.757 ± 0.010
71.1 ± 1.8 72.1 ± 1.8 73.7 ± 1.8 76.1 ± 1.9
ð4Þ
where C 0 is the distribution parameter and ud is the drift velocity. The distribution parameter is usually set to 1.2 and 2, for turbulent and laminar velocity profile in the slug zone, respectively; the drift velocity is usually modeled as suggested by Bendiksen (1984). We used the value of C 0 and ud obtained from the experimental data, see Table 2.
Table 1 Physical properties of the test fluids at 25 °C and atmospheric pressure.
Water CMC-1 CMC-3 CMC-6
ð1aÞ
where Hs ; Hf ; U ls ; U gs , U s ; U f ; U g are the liquid holdup in the slug zone, the liquid holdup in the film zone, the liquid superficial velocity, the gas superficial velocity, the slug zone velocity, the liquid velocity in the film zone, and the gas velocity in the film zone, respectively. As proposed by Taitel and Barnea (1990), the slug zone velocity is supposed to be at the mixture velocity U m
ðU t U f ÞHf ¼ ðU t U s ÞHs ; Capacitance probe signals processing
219
Fig. 2. Slug flow in a pipe with a circular cross section.
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The momentum balance in the film zone, considering a uniform liquid thickness, can be written as for the stratified flow. Thus, eliminating the pressure gradient, the momentum balance yields
sl
Sl 1 1 Sg þ sg ðql qg Þg sin b ¼ 0; þ s i Si þ Ag Al Al Ag
ð5Þ
where Ag;l ; Sg;l ; sg;l ; U g;l and, qg;l are cross section, the wetted perimeter, the wall shear stress, the phase velocity, and the density of the gas and liquid, respectively. b is the inclination angle (positive for upward inclination) and si is the interfacial shear stress. The following closures are chosen for the shear stresses:
sl ¼ f l
ql U 2l 2
sg ¼ f g si ¼ f i
qg U 2g 2
ml
f l ¼ C l Rel
; ;
Rel ¼
;
g f g ¼ C g Rem ; g
qg ðU g U l ÞjU g U l j 2
Dnl U 2n q l ln ; m8n1 1þ3n 4n
Reg ¼
Dg U g qg
lg
;
ð6aÞ ð6bÞ ð6cÞ
;
where the constants are chosen as C l;g ¼ 16 and ml;g ¼ 1 for the laminar flow regime and as C l;g ¼ 0:079 and ml;g ¼ 0:25 for the turbulent flow regime. lg is the gas dynamic viscosity and the interfacial friction factor is chosen as f i ¼ 0:014, see Cohen and Hanratty (1968). Geometrical relations for the film zone, see Orell (2005) for details, and a model for the liquid hold-up in the slug zone are needed to close the model. We consider the correlation proposed by Andreussi et al. (1993)
Hs ¼ 1
pffiffiffiffiffiffi U m = gD F 0 pffiffiffiffiffiffi ; U m = gD þ 2400Bo3=4
D0 ; F 0 ¼ 2:6 1 2 D
ð7Þ
where Bo ¼ ðql qg ÞgD2 =r is the Bond number, D0 ¼ 0:025 m, and pffiffiffi F 0 ¼ 0 if D < 2D0 . For given fluid properties, geometry and fluid superficial velocities the model solves simultaneously Eqs. (1a), (1b), (3), and (5). Gas and liquid velocities in the film zone, U f and U g , the liquid hold-up in the film zone, Hf , and ‘s;f =‘ predictions are obtained. Then, the average pressure gradient of the slug unit is given by
‘f dP f q U 2 ‘s 4 ¼2 s s s þ ðsl Sl þ sg Sg Þ þ qsu g sin b; dx D ‘ pD2 ‘
ð8Þ
where qs ¼ ql Hs þ qg ð1 Hs Þ; f s is the slug friction factor, and qsu is the average density of the slug unit. The average density of the slug unit is evaluated as qsu ¼ ð1 Hsu Þqg þ Hsu ql , where Hsu ¼ Hs ‘s =‘ þ Hf ‘f =‘ is the average liquid hold-up of the slug unit. The liquid slug friction factor is modeled as the friction factor for a power-law fluid considering mixture conditions, yielding s f s ¼ C s Rem ; s
Res ¼
2n Dn U m qs ; n1 1þ3nn m8 4n
ð9Þ
Table 2 Values of C 0 and ud for the different air-CMC solution systems. Fluid
b (°)
C 0 (–)
ud (m/s)
R2 (–)
CMC-1 CMC-1 CMC-1
0 5 +5
1.08 1.23 1.18
0.42 0.26 0.18
0.924 0.643 0.950
CMC-3 CMC-3 CMC-3
0 5 +5
1.37 1.40 1.49
0.61 0.16 0.32
0.875 0.819 0.840
CMC-6 CMC-6 CMC-6
0 5 +5
1.52 1.49 1.53
0.32 0.25 0.36
0.971 0.942 0.967
where C l ¼ 16 and ml ¼ 1 for the laminar flow regime and as C l ¼ 0:079 and ml ¼ 0:25 for the turbulent flow regime. Results and discussion The experimental results are presented in this section. The considerations are organized considering separately flow pattern maps, pressure drop, and slug flow characteristics. Flow pattern maps In this section the experimental flow pattern maps obtained by visual observations are presented. We distinguish the following flow regimes: stratified, plug, and slug flow patterns. Churn and annular flow are not observed. We follow the flow pattern classification given by Chhabra and Richardson (2008); the distinction between plug and slug flow regime is due to the presence of entrained gas bubbles within the liquid slug. We also plotted the ‘‘zero characteristics’’ (ZRC) boundary, which gives the region of existence of the stratified flow regime computed from the twofluid model predictions and considering constant shape factors (cg ¼ cl ¼ 1), see Picchi et al. (2014) for details. The main difference between the slug flow pattern of a gas/ shear-thinning fluid system with respect to gas/water system concerns the structure of the liquid slug: there are fewer distinctly dispersed bubbles, in particular for higher concentration water-CMC solutions. The same behavior was observed by Xu et al. (2007) for air/CMC-solution systems and by Al-Sarkhi and Hanratty (2001) for air/water-polymer solution annular flows; Al-Sarkhi and Hanratty (2001) suggested that the disturbance waves at the interface are reduced by the polymer addition and the drop entrainment is reduced. In Fig. 3 two pictures taken with a high speed camera of typical plug and slug flows (CMC-6 solution) are shown: in plug flow there is no gas entrainment within the liquid, as opposite to slug flow where air bubbles are present within the liquid body. In Fig. 4 a slug of CMC-6 solution is shown in detail, showing the bubble head (Fig. 4(a)), middle part (Fig. 4(b)), tail (Fig. 4(c)), and the liquid slug with gas entrainment (Fig. 4(c)). Fig. 5 shows the experimental flow pattern maps obtained in the case of horizontal flow for the three different water-CMC solution systems. The different rheology of the three CMC solutions does not affect the transition from plug flow to slug flow. Only for the CMC-1 solution, one condition indicates the transition from plug to bubbly flow; in fact, increasing the CMC concentration the liquid superficial velocity given by the centrifugal pump decreases. The theoretical ZRC boundary, is plotted, but no stratified flow conditions are experimentally observed in the predicted existence region of the stratified flow regime. In Fig. 6 the results obtained for upward inclined flows (b ¼ þ5 ) are reported: plug and slug flows are observed. Only for the CMC-1 solution two points that give the transition from plug to bubbly flow are observed at low gas and high liquid superficial velocities. Increasing the CMC concentration and as a consequence, decreasing n and increasing m, plug flow regime is present at higher gas velocity for a given liquid superficial velocity. The ZRC boundary is not plotted because the calculation algorithm gives no solutions for these upward inclined flows. As discussed by Brauner and Moalem (1992) the shape of the ZRC boundary varies dramatically with inclination: in particular for upward inclination angles the well-posed region is reduced significantly and only limited areas of existence of the stratified flow regime are obtained. Fig. 7 shows the flow pattern maps for downward inclined flows (b ¼ 5 ), where, in addition to slug and plug, the stratified flow
D. Picchi et al. / International Journal of Multiphase Flow 73 (2015) 217–226
221
Fig. 3. Picture of the plug flow (CMC-6, horizontal flow, U gs ¼ 0:20 m/s and U ls ¼ 0:44 m/s) and picture of the slug flow (CMC-6, horizontal flow, U gs ¼ 1:20 m/s and U ls ¼ 0:46 m/s) taken with the high-speed camera.
(a) front of bubble
(b) middle of bubble
(c) back of bubble Fig. 4. Picture of the slug flow taken with the high-speed camera (CMC-6, horizontal flow, U gs ¼ 1:20 m/s and U ls ¼ 0:81 m/s).
regime is observed. The transition from plug to slug flow, keeping fixed the liquid superficial velocity, occurs at higher gas superficial velocities with respect to the horizontal case. The stratified flow is observed for all CMC solutions: the interface is wavier and more irregular for the CMC-1 solution and becomes noticeably smoother increasing the polymer concentration. In fact, adding polymer the interfacial activity is reduced, decreasing the interfacial roughness, see Soleimani et al. (2002). The existence region of the stratified flow regime is compared to the experimental data of Fig. 7: the predicted ZRC boundary agrees very well with the transition from the stratified flow in the experimental maps for CMC-1 and CMC-3 solution systems, and slightly over-predicts the transition for CMC-6 solution system. Increasing the CMC concentration, the existence region of the stratified flow moves to lower liquid velocities for a given gas superficial velocity and this behavior is well predicted by the theoretical criteria, see Fig. 7.
Pressure drop In this section the measured pressure drops are presented in terms of total pressure gradients. We have no holdup measurements, so we cannot estimate the frictional pressure gradient for inclined flows.
Stratified flow The experimental pressure drop data are compared to the predictions of the steady fully developed two-fluid model, see Picchi et al. (2014) for details. Usually, the pressure drop for the stratified flow are presented in in terms of dimensionless liquid pressure gradient U2l as a function of the Lockhart–Martinelli parameter X 2
U2l ¼
ðdp=dxÞtp ; ðdp=dxÞsl
X2 ¼
ðdp=dxÞsl 2f ls ql U 2ls =D ¼ ; ðdp=dxÞsg 2f gs qg U 2gs =D
ð10Þ
where ðdp=dxÞtp is the measured two-phase pressure gradient, f ls and f gs are superficial gas and liquid friction factors evaluated for the superficial conditions. In Fig. 8 the experimental data are compared to the theoretical predictions: it can be noted that the agreement is not so satisfactory. This behavior is underlined both in Xu et al. (2007) and Picchi et al. (2014): in Fig. 8 the data by Xu et al. (2007) are also reported to show that the model fails the predictions when the total pressure drop is close to zero for inclined flows. The not perfect agreement can be due to the error in the hold-up prediction in inclined conditions and to the fact that for these downward inclined flows the pressured drop are close to zero. In fact, the pressure drop can be divided into two contributes, the frictional and the gravitational one: for downward inclined flows they have an
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plug slug plug/bubbly ZRC line
Uls (m/s)
0
10
−1
10
−1
10
0
10
Ugs (m/s) (a) CMC-1
plug slug ZRC line
0
10
−1
10
0
Uls (m/s)
Uls (m/s)
1
10
compatible with the model and the solution algorithm is built to find the solution of Eq. (5) in the range of U f > 0. In Fig. 9 the experimental pressure drop for the three waterCMC solution systems are compared to the predictions by the model, showing a good agreement. The model predicts 78% of the experimental data within ± 30% error, giving an acceptable agreement between the predicted and the measured pressure drop. In the Electronic Annex 1 the experimental pressure drop for the three solution systems (horizontal flow) compared to the model predictions are plotted as function of mixture and gas superficial velocities: increasing gas velocity, and keeping fixed the mixture velocity, the pressure drop increases; on the other hand increasing the mixture velocity, keeping fixed the superficial gas velocity, the pressure drop increases. In Fig. 9 the experimental pressure drops are also compared to the predictions by Dziubinski (1995), where a modified Lockhart and Martinelli (1949) model was presented. The Dziubinski’s correlation is a simple way to obtain pressure drop
−2
10
−1
10
0
10
Ugs (m/s) (b) CMC-3
10
1
10
plug slug plug/bubbly
−1
10
−1
0
10 plug slug ZRC line
0
10
−1
0
10
10
Uls (m/s)
Uls (m/s)
10
Ugs (m/s) (a) CMC-1
−2
10
−1
10
0
10
Ugs (m/s) (c) CMC-6
1
−1
10
10
plug slug −1
10
Fig. 5. Flow pattern maps for horizontal flow for the three water–CMC solutions systems. The predicted well-posedness boundary (ZRC) is plotted.
0
Ugs (m/s) (b) CMC-3
10
opposite sign, so, in this case, the total pressure drop has a magnitude close to zero. 0
10
Uls (m/s)
Slug flow The experimental pressure drops are compared to the model presented in Section 3.1. The measured pressure drops are the time average pressure drop during the experimental acquisition. Considering the operative conditions of this work (working fluids, pipe diameter and fluid superficial velocities), the model gives pressure drop predictions for horizontal and downward flows, but gives no-solution for upward b ¼ þ5 inclination flows. The mass balance given in Eq. (5) has to be satisfied to obtain a valid solution for the the liquid hold-up in the film zone. The model gives a negative liquid velocity (U f ) in the film zone from Eq. (1a) for the operative condition of the upward flow. The result of a negative liquid velocity in the film zone is not consistent with the formulation of the model: back-flow in the film zone is not
−1
10
plug slug −1
10
0
10
Ugs (m/s) (c) CMC-6
Fig. 6. Flow pattern maps for the upward (b ¼ þ5 ) flow for the three water–CMC solutions systems.
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and Scott (1969) correlation is valid for air–water gas slug flows and it reads
plug slug stratified ZRC line
f s ¼ 0:0226
Uls (m/s)
0
10
−1
10 −1 10
0
1
10
10
Ugs (m/s) (a) CMC-1
plug slug stratified ZRC line
0
Uls (m/s)
er ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 f s;corr f s;exp f s;exp
1
N
ð12Þ
;
where f s;corr ; f s;exp , and N are the predicted slug frequency, the measured slug frequency, and the number of experiments, respectively. We extend the Gregory and Scott (1969) correlation to account for the rheology of the shear-thinning fluid: the structure of the correlation is respected adding two contributes. The extended correlation yields
−1
10
−1
10
0
1
10
10
Ugs (m/s) (b) CMC-3
f s ¼ 0:0448
0
Rels ¼
−1
10
−1
10
0
1
10
10
Ugs (m/s) (c) CMC-6
0:88 0:07 U ls 32:2014 Rels þ Um ðnÞ2:85 ; Um gD Rews
ð13Þ
where Rels is the Reynolds number for a power-law fluid considering superficial conditions and Rews is the water Reynolds number considering superficial conditions
plug slug stratified ZRC line
10
Uls (m/s)
ð11Þ
where f s is the slug frequency, U ls is the liquid velocity, U m is the mixture velocity, g is the gravitational acceleration and D is the pipe diameter. In Fig. 10 the comparison between the measured and the predicted frequency by the Gregory and Scott (1969) correlation is presented. The frequency for the CMC-1 solution is well predicted by the Gregory and Scott (1969) correlation: this result is reasonable due to the fact that the rheological behavior of the CMC-1 solution is close to a Newtonian fluid (n ¼ 0:9424 and m ¼ 0:0065 Pa sn ). The correlation does not hold acceptable for the other two solutions, CMC-3 and CMC-6. In Fig. 10 the average relative error er is shown; er is defined as
PN
10
1:2 U ls 19:75 þ Um ; Um gD
Dnl U 2n ls ql n ; m8n1 1þ3n 4n
Rews ¼
qw U ls D : lw
ð14Þ
The proposed correlation is written to keep the same structure of the Gregory and Scott (1969) correlation, considering two additional contributes: the fluid behavior index and the ratio between the Reynolds numbers. The correlation gives the same coefficients of the Gregory and Scott (1969) interpolating their gas–water data (the two new terms collapse to unity for air–water systems).
0.4
Fig. 7. Flow pattern maps for the downward (b ¼ 5 ) flow for the three water– CMC solutions systems. The predicted well-posedness boundary (ZRC) is plotted.
0.35
predictions, but, respect to the mechanistic model (Section 3.1), it has an emiprical nature: it was obtained from the fitting of experimental data of gas/shear-thinning slug flows from the literature. The experimental data error of this work is around 50%, see Fig. 9, but this can be due to the experimental nature of the correlation. Slug flow characteristics In this section, measured slug flow characteristics are presented: slug frequency, slug velocities, and slug length are measured from the analysis of the capacitance probe signals. Slug frequency The measured slug frequency is compared with the empirical correlation proposed by Gregory and Scott (1969). The Gregory
Predicted Φl2 (−)
0.3 0.25 0.2 0.15 0.1 0.05 CMC−1 CMC−3 CMC−6 Xu et. al (2007) data
0 −0.05 −0.1 −0.1
0
0.1
0.2
0.3
0.4
Experimental Φ2l (−) Fig. 8. Dimensionless experimental pressure gradient U2l of this work for the three CMC-solution systems and by Xu et al. (2007) compared to the predictions by the steady fully developed two-fluid model. Dotted lines represent 30%.
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5
4000
4.5
3500
4
3000
3.5
Ub (m/s)
4500
2500 2000
3 2.5 Exp. β=0° β=0°: U =1.08 U +0.42
1500
2
1000
1.5
Exp. β=−5° β=−5°: U =1.23U +0.26
1
Exp. β=+5° β=+5°: U =1.18U +0.18
Dziubinski (1995) correlation CMC−1 CMC−3 CMC−6
500 0 0
1000
2000
3000
b
m
b
m
b
0.5 0.5
4000
1
1.5
2
2.5
m
3
3.5
4
Um (m/s) Fig. 9. Experimental pressure drop of this work for the three water–CMC solution systems compared to the predictions by the proposed model and by Dziubinski (1995) correlation. Dotted lines represent 30%.
Fig. 12. Mixture velocity versus slug velocity for air/CMC-1 flow.
5
6 CMC−1 CMC−3 CMC−6
5
4.5
3.5
4
Ub (m/s)
Predicted fs (1/s)
4
3
2
3 2.5 Exp. β=0° β=0°: U =1.37U +0.61
2 1.5
er = 34.4 %
1
b
b
m
Exp. β=+5° β=+5°: U =1.49U +0.32
1 0
m
Exp. β=−5° β=−5°: U =1.40U +0.16
b
0
1
2
3
4
5
6
0.5 0.5
Experimental fs (1/s)
2
2.5
3
3.5
4
Fig. 13. Mixture velocity versus slug velocity for air/CMC-3 flow.
5
CMC−1 CMC−3 CMC−6
5
1.5
Um (m/s)
Fig. 10. Experimental slug frequency for the three water–CMC solution systems compared to the predictions by the Gregory and Scott (1969) correlation. Dotted lines represent 20%.
6
1
m
4.5
3.5
Ub (m/s)
Predicted fs (1/s)
4 4
3
2
3 2.5
Exp. β=0° β=0°: U =1.48U +0.38
2
er = 22.5 %
1
0
b
1.5
Exp. β=+5° β=+5°: U =1.53U +0.35
1 0
1
2
3
4
5
b
6
Experimental fs (1/s) Fig. 11. Experimental slug frequency for the three water–CMC solution systems compared to the predictions by the proposed correlation. Dotted lines represent 20%.
0.5 0.5
m
Exp. β=−5° β=−5°: Ub=1.49+0.25
1
1.5
2
2.5
3
m
3.5
Um (m/s) Fig. 14. Mixture velocity versus slug velocity for air/CMC-6 flow.
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Despite the inclusion in Eq. (13) of the term n is quite unorthodox, it allows to improve the agreement with experimental data and it accounts for different shear-thinning fluid behavior. In Fig. 11 the comparison between the predicted frequency by Eq. (13) and the measured one is presented for the three waterCMC solution systems. The aim is to propose a correlation valid for all the CMC concentration and the agreement is reasonably good: the relative average error is er ¼ 22:5%, about 35% less that using the Gregory and Scott (1969) correlation. This first attempt to propose a correlation for the slug frequency valid for gas/ shear-thinning systems is encouraging. In fact, in Rosehart et al. (1975) and Otten and Fayed (1977), the frequencies are correlated using the Gregory and Scott (1969) relation, which uses different values of the coefficients for each shear-thinning fluid series data. In the Electronic Annex 1 the measured slug frequencies are reported. Slug velocity The slug velocity is obtained by post-processing capacitance probe signals, see Section 2.4 for details. The data are presented 450 o
400
f /D
(−)
350
β=0
o
β=−5
o
β=+5
300 250 200 150 100 50 0 0.1
450
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
o
β=+5
(−) f /D
300 250 200 150 100 50
0.5
0.6
0.7
0.8
0.9
1
Ugs /Um (−) (b) CMC-3 o
β=0 400
o
β=−5
o
β=+5
f /D
(−)
350 300 250 200 150 100 50 0 0.4
0.5
where C 0 is the distribution parameter and ud is the drift velocity. The values of the estimated distribution parameter and the drift velocity are shown in Table 2. The values of C 0 for the CMC-1 solution systems are close to the ones for air–water systems: C 0 1:1 for the horizontal case, as observed by Woods and Hanratty (1996) for air/water systems at low gas velocities, and C 0 1:2 for inclined systems. The distribution parameter C 0 increases while the CMC concentration increases. Otten and Fayed (1977) observed the same trend in C 0 due to the addition of the polymer, assuming the velocity equal to zero (ud ¼ 0) since all the experiments are performed for horizontal flows. The drift velocity is ud – 0 for all the experimental data of this work even in horizontal conditions. Slug length The bubble length ‘f is an important feature to describe slug flows and in Fig. 15 it is plotted as a function of the ratio U gs =U m : increasing the ratio U gs =U m the bubble length increases. The liquid slug length ‘s is always considered as a constant parameter, see Nicholson et al. (1978), and equal to 30 pipe diameter. In the Electronic Annex 1 the liquid slug length is presented as a function of the mixture velocity for all the experimental data of this work. The average length for air/CMC-1, CMC-3, and CMC-6 systems are ‘s;CMC1 =D ¼ 34, ‘s;CMC3 =D ¼ 42, and ‘s;CMC6 =D ¼ 26, respectively.
In this work an experimental investigation on gas/shearthinning fluid pipe flow was presented. The experiments were performed in a 9-m-long glass pipe studying three different air/CMC solution systems and the different inclination angles. Flow pattern maps were presented and the influence of the rheology of the water–CMC solutions and of the inclination angle were investigated. The existence region of the stratified flow regime was compared to the ZRC boundary, predicted by the model given by Picchi et al. (2014), showing a good agreement. Increasing polymer concentration (decreasing the flow behavior index n and increasing the flow consistency index m) the existence region of the stratified flow moves to lower liquid velocities for a given gas superficial velocity and this behavior is well predicted by the criteria. Furthermore, as mentioned by Soleimani et al. (2002), we observed that the addition of polymers reduces the interfacial activities of the stratified flow and we investigated how the transition from plug to slug flow was influenced by the inclination angle and the rheology of the shear thinning fluid. The pressure drop data of the stratified flow regime were compared to the steady fully developed two-fluid model, see Picchi et al. (2014) for details, showing that the model does not predict them with a high accuracy. A model for gas/power-law fluid slug flow was presented and the predictions were compared to the experimental data of this work. The model predicts the experimental pressure drops with a satisfactory agreement: the 78% of the experimental data were predicted within ± 30% error. This comparison demonstrates the validity of the present model. Slug characteristics were studied from the analysis of a capacitance probe signals: slug velocity, slug frequency and slug lengths were investigated. Slug frequency was at fist compared to the
350
450
ð15Þ
(a) CMC-1 o
0.4
U b ¼ C 0 U m þ ud ;
Conclusion
β=−5o
0
in Figs. 12–14 distinguishing among three polymer concentrations and the inclination angles. A linear least-square fitting on experimental data is done to estimate the coefficients of the Nicklin et al. (1962) relations for bubble velocity, yielding
Ugs /Um (−)
β=0 400
225
0.6
0.7
0.8
0.9
1
Ugs /Um (−) (c) CMC-6 Fig. 15. Bubble length ‘f for the three different air/CMC systems as a function of the ratio U gs =U m .
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