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Plane flow model of non-Newtonian turbulent stratified flow in wells and pipes
Journal of Petroleum Science and Engineering 44 (2004) 223 – 229 www.elsevier.com/locate/petrol
Plane flow model of non-Newtonian turbulent stratified flow in wells and pipes Li Hong-bo, Wu Chao *, Zheng Yong-gang State Key Laboratory of Hydraulics on High Speed flows, Sichuan University, Chendu 610065, China Received 9 July 2003; accepted 18 March 2004
Abstract Economical and efficient long-distance oil and natural gas pipe transportation technologies are greatly demanded. Nowadays, oil and gas mixed transportation is gradually put into practice and researchers pay much attention on the theories of two-phase flow in pipe. Methods that can efficiently solve problems of this field are being put forward including hydraulic approach. A new approach, the Rotatable Coordinate Axis, is proposed in this paper. It is used to study the hydraulic characters of two-phase stratified flow in pipe. Based on the new method, the plane flow model for stratified turbulent flow in pipe is built. Then, we can obtain the analytic formulas of velocity, discharge etc. in wells and pipes by this model. To prove the theory of the plane flow model, experiments of aeration in stratified pipe is conducted with the aeration experiment device. The experiment data showed that aeration could effectively achieve the resistance reduction in pipeline, which can offer great theory support to the development of oil and natural gas mixed transportation technology. D 2004 Elsevier B.V. All rights reserved. Keywords: Pipe; Turbulent flow; Rotatable Coordinate Axis; Aeration; Stratified flow
1. Introduction Along with the exploitation of the oil and natural gas in oceans and deserts, nowadays, the improvement of two-phase transportation in pipe is demanded and the theoretical study of two-phase flow in pipe has become more and more popular. The hydraulic characters of two-phase flow in pipe are complex, for they are related to physical characters, discharge, drag, flow regime (Wang et al., 1994), and many other factors. The stratified flow, the most typical flow regime, is very common in declining pipe and is the * Corresponding author. Fax: +86-28-85405148. E-mail address: [email protected] (W. Chao). 0920-4105/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2004.03.005
root of other flow regimes as well. Researchers had done a great deal of work on it (Prandtl et al., 1934; Govier and Aziz, 1972; Dou, 1988). There are also a few reports or researches on two-phase stratified pipe flow (Bishop and Deshpande, 1986; Zheng and Xie, 2000; Li and Wu, 2004). To ascertain the shearing stress, the application of empirical or semiempirical equation on the coefficient of frictional resistance made the researches less accurate. In this paper, the Rotatable Coordinate Axis is used in the research on cross-section, and stratified flow in pipe is approximately replaced by plane flow between two boundless flats. The cross-section of the pipe is divided into two parts of stratified flow section and a single-phase flow section. Under the
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condition of uniform stratified flow, viz. regardless interfacial liquid level and waves causing vortex motion and energy dissipation, a new flow model for the velocity distributing discipline of non-Newtonian stratified flow in pipe can be proposed. Then, by this model, analytic formulas of velocity, discharge and resistance reduction can be deduced without using average velocity, empirical or semiempirical equation on the coefficient of frictional resistance in wells and pipes as before. Thus, we can get more accurate quantity of velocity, discharge etc. in wells and pipes than before.
2. The plane flow model of two-phase flow in pipe and the velocity distributing discipline At the very beginning, a series of preconditions should be given: 1. As shown in Fig. 1, the flow in pipe is supposed to be full flow. 2. The two-phase flow is made up of two kinds of non-Newtonian fluid. 3. And the flow patterns concerned in this paper are turbulent flow. 4. The flow in pipe is uniform, which interfacial liquid level and waves causing vortex motion and energy dissipation are not concerned.
5. The flow in pipe is steady flow, which means parameters like velocity, pressure, density, temperatures, and the stress gradient are not time variation. As shown in Fig. 1, fluid 1 is top fluid, which is natural gas usually; and fluid 2, subjacent fluid, which should be oil. To describe this two-phase flow in pipe, here a rotatable coordinate system should be applied. It could be described as follows. First, random pipe cross-section was considered in Fig. 1. AOB was taken as rotatable coordinate axis. It can revolve around the point O, and there will be corresponding intersection angle (U) with the cross axis. In addition, it intersects the circle of pipe crosssection at points A and B. Then, two tangent lines to the circle can be obtained with the two points. Second, if the whole pipe was taken into account, the two tangent lines could be thought was two boundless parallel tangent planes with the interval of 2R to the pipe (R is the radius of the pipe). Accordingly, as shown in Fig. 1, the problem of pipe flow was transformed into plane flow approximately to study the stratified flow between two parallel planes. And the planes also revolved around the circle following the revolving coordinate axis AOB. Then, according to the angle U, the whole crosssection can be divided into two parts. Within the range of U0 V U V p-U0, there are two layers between the
Fig. 1. Stratified flow model in pipe.
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
225
two planes. The thickness of top fluid 1 is h1, governed by formula h1 = R h0/sinU, and the thickness of subjacent fluid 2 is h2, h2 = R + h0/sinU. In the left part: p U0 V U V 2p + U0, the flow between the parallel planes is single-phase flow of fluid 2. In this paper, only two-phase stratified flow section in pipe was studied. Supposing that x is the axial direction of the pipe and dp/dx = constant, the stratified flow model in pipe is:
as an example, viscous shearing stress expression, viz. constitutive equation of the top fluid should be
ds=dy ¼ ðdp=dxÞ=b
K1 ðdu=dyÞn1 ¼ sw1 ½1 ðy=y0 Þ
in which s is shearing stress, p is compressive stress and b is curvature parameter of pipe surface. When b = 1, it is a plane flow model; b = 2, a uniform pipe flow model. Let s = 0, and the coordinate of the point with the highest velocity can be obtained, that is y0 ¼ ðsw1 bÞ=ðdp=dxÞ Integrating it, under the boundary condition: when y = 0, the shearing stress s = sw1, we can obtain
s1 ¼ K1 ðdu=dyÞn1 where K1 is consistence coefficient of the top fluid, and n1 is flow index. And s = sw1[1-( y/y0)], so
Arrange the equation above in order and integral it on boundary condition: u = 0 ( y = 0), we can obtain n1 sw1 1=n1 u¼ n1 þ 1 K1 y0 h i ðn þ1Þ=n1 ðy0 yÞðn1 þ1Þ=n1 y0 1 When y = b1, the velocity is
s ¼ sw1 ½1 ðy=y0 Þ and
ub1
s ¼ ðdp=dxÞy=b þ sw1 The total shearing stress is s = sl + stSsl is the viscous shearing and st is the turbulent shearing stress. They are quite different according to the fluid position. In viscous sublayer, the turbulent shearing stress is much smaller than the viscous shearing stress, and could be ignored, while in the core section of turbulent flow, the viscous shearing stress can be ignored.
n1 sw1 1=n1 ¼ n1 þ 1 K1 y0 h i ðn þ1Þ=n1 ðy0 b1 Þðn1 þ1Þ=n1 y0 1
2.1.2. Core section of turbulent flow (U0VUVpU0, b1VyVh1) In this layer, viscous shearing stress could be ignored, and from the mixing length theory of Prandtl (Prandtl et al., 1934), the turbulent shearing stress is s1 ¼ q1 l 2 ðdu=dyÞ2
2.1. Top fluid 2.1.1. Viscous sub layer (U0VUVpU0, 0VyVb1) In this layer, turbulent shearing stress could be ignored. b1 is the thickness of viscous sublayer, and could be governed by the following formulas b1 ¼ a1 ðV1 =V*1 Þ
1=2
V*1 ¼ ðsw1 =p1 Þ
in which, a1 is proportional constant (top fluid), V1 is velocity. Take the power law fluid of non-Newtonian
in which q1 is the density of the top fluid and length l = ky, k: Karman constant, k = 0.4, no dimension. On boundary condition: u = ub1 ( y = b1) integral, we have u ¼ ub1 þ ð2V*1 =kÞ n o ½1 ðy=y0 Þ1=2 ½1 ðb1 =y0 Þ1=2 þ ðV*1 =kÞ ( ) 1 ½1 ðy=y0 Þ1=2 1 ½1 ðb1 =y0 Þ1=2 ln ln 1 þ ½1 ðy=y0 Þ1=2 1 þ ½1 ðb1 =y0 Þ1=2
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From y = h1, h1 = R-h0/sinU, the velocity of fluid interface could be obtained u1 ¼ u jy¼h1 ¼ ub1 þ ð2V*1 =kÞ n ½1 ððR h0 =sinUÞ=y0 Þ1=2 o ½1 ðb1 =y0 Þ1=2 þ V* =k ( 1 ½1 ððR h0 =sinUÞ=y0 Þ1=2 ln 1 þ ½1 ððR h0 =sinUÞ=y0 Þ1=2 ) 1 ½1 ðb1 =y0 Þ1=2 ln 1 þ ½1 ðb1 =y0 Þ1=2
distribution formula, the discharge can be obtained by integration in the cross-section of pipe. If all variables of the single-phase part in one pipe are marked with suffix 0, and top fluid 1, subjacent layer 2, the expressions of Q1 and Q2 are as follows Z pþU0 Z Z Z R urdrdU ¼ 2 dU udr Q1 ¼ S
8 < y* ¼ 2R y
The total discharge of pipe ( Q) comprises the discharge of the single-phase part ( Q1) and that of the stratified part ( Q2). According to the velocity
pU0
1=n2
0
8 > < > :
R
b0
ub0 þ
2V*0 y 1=2 b0 1=2 1 1 R k R
y 1=2 1 1 V 06 R þ * 4ln y 1=2 k 1þ 1 R 39 b0 1=2 > = 1 1 7 R ln 1=2 5>ðR yÞdy ; 1 þ 1 bR0
With the similar methods for research on top fluid, we can establish coordinate y* in viscous sublayer section and core section of turbulent flow as shown in Fig. 1. The coordinate relationship of y and y*:
2.3. Discharge of subjacent fluid
2
2.2. Subjacent fluids
Similarly, the formulas about underlayer can be obtained by replacing y with y*. Furthermore, it is single-phase flow in the section p U0 V U V 2p + U0 and analytic distribution formulas of velocity and formulas of velocity in the center of the pipe can be confirmed. According to the junction condition of velocity on interphase, velocity of flow on the point of y = h1 of the top fluid is equal to that of underlayer. And according to the junction condition of the center of pipe, velocity of flow in the two-phase stratified flow section is equal to that in the center of pipe. Thus, two equations can be established in order to confirm b( y0) and sw1, in which b, sw1( y0,y* 0 ) are unknown functions of U.
b0
n2 sw0 ¼ 4U0 n þ 1 K 2 2 y0 0
Z n þ1 2 n2 þ1 n2 n2 ðy0 yÞ y0 ðR yÞdy þ 4U0
Thus, velocity on interphase is the function of U, under the initial conditions (U = U0, h1 = 0).
: * y0 ¼ 2R y0
Z
Then the discharge of the stratified part is Z pU0 Z Z Z RsinU0 =sinU Q2 ¼ urdrdU ¼ dU urdr S
U0
þ
Z
0
Z
2pU0
dU pþU0
R
urdr 0
in which, u¼
1 n2 þ1 n2 þ1 n2 sw0 n2 n ðy0 2R þ yÞ n2 y0 2 n2 þ 1 K 2 y0 " 2V*2 2R y 1=2 þ ub1 þ 1 2R y0 k 1=2 # V 2 b2 1 þ * 2R Y0 k ( 1 ½1 ð2R yÞ=ð2R y0 Þ1=2 ln 1 þ ½1 ð2R yÞ=ð2R y0 Þ1=2 ) 1 ½1 b2 =ð2R y0 Þ1=2 ln 1 þ ½1 b2 =ð2R y0 Þ1=2
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
And we can compared the total discharge of subjacent layer ( Q = Q1 + Q2), with that of a singlephase flow in one pipe, shown as Z Z Z R Q0 ¼ urdr urdrdU ¼ 2p S
8 > < > :
b0
0
n2 sw0 1=n2 h ðR yÞðn2 þ1Þ=n2 n þ 1 K y 2 2 2 0 Z R i ðn2 þ1Þ=n2 ðR yÞdy þ 2p R
¼ 2p
Z
b0
ub0 þ
2V*0 y 1=2 b0 1=2 1 1 R k R
2
y 1=2 1 1 V 06 R þ * 4ln y 1=2 k 1þ 1 R 39 b0 1=2 > = 1 1 7 R ln 5 ðR yÞdy 1=2 b0 > ; 1þ 1 R Whether drag reduction of stratified turbulent flow in one pipe is achieved or not can be ascertained with the following expression Dr ¼
Q Q0 Q0
227
driven air-entraining machine. Compressed air output by air entraining machine is first decompressed in decompression chamber, and then poured into testing pipe by a number of thin pipes, stratified aeration is provided by air entrance, 3– 16 holes with internal diameter of 3 mm on the upper are of the pipe. The main testing device of aeration experiments in this paper is a two-dimensional Laser Doppler Velocity produced by the company of DANTEC in Denmark. On the velocity-measuring cross-section, there are 15 – 20 spots which are used to describe distribution of velocity. We have applied two air entrances, upstream and downstream, to accomplish the aeration. 3.2. Basic data In this paper, fluid 1 (upper) is air, fluid 2 (lower fluid) is pure water, and so we have l1 ¼ 17:9 106 Pa s; l2 ¼ 1010 106 Pa s; q1 ¼ 1:21 kg=m3 ; q2 ¼ 1000 kg=m3 : From analysis of theory and experiments, we can get a1 = 0.116, a0 = 11.6, a2 = 11.6, B = 1.1. For stratified flow in pipe, the content of air is governed by U0. As U0 increases, content of air decrease; as U0 decreases, content of air increases. Relation between content of air on section of pipe e and U0 is (Table 1)
3. Experiments and tests 3.1. Experiment device and other conditions The length of testing pipe is 17 m, and the inner diameter is 0.1 m. The part of measuring velocity box is made of plexiglass, and other part of pipe is PVC. A high water tower and a flat-water trough on the upstream of pipe provide steady water head and the decrease and pressure gradients are constant. Flow in pipe and velocity of flow are controlled by water level of upstream and gate at the exit of downstream. The source of air of aeration in testing pipe is provided by a power-
e¼
p 2U0 sinð2U0 Þ 2p
3.3. Experiment results Figs. 2 and 3 are the data comparisons of velocity distribution before and after aeration along with vertical radial (U = 90j) and horizontal radial (U = 0j), respectively. Under the conditions of aeration and water, we’ve got two groups of experiment data, which show that fluid velocity increases on the former condition. Fig. 4 is a comparison between
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Table 1 Data of experiments No.
1 2 3 4 5 6 7 8 9 10
Before aeration
Aeration density
After aeration
Q (10-3 m3/s)
V (m/s)
U0 (rad)
e (%)
Q (10-3 m3/s)
V (m/s)
0.990 1.800 1.166 1.166 1.118 0.512 0.199 0.199 0.199 2.330
0.126 0.229 0.148 0.148 0.142 0.065 0.025 0.025 0.025 0.297
0.8552 0.9861 0.8552 0.9861 0.8552 0.8552 0.7854 0.8552 0.9861 0.9250
7.0 4.0 7.0 4.0 7.0 7.0 9.1 7.0 4.0 5.3
1.040 1.831 1.177 1.214 1.139 0.513 0.200 0.202 0.205 2.400
0.132 0.233 0.15 0.155 0.145 0.065 0.025 0.026 0.026 0.306
DQ (%)
DV (%)
DA (error, %)
5.05 1.72 0.94 4.12 1.88 0.20 0.50 1.51 3.02 3.00
4.76 1.75 1.35 4.73 2.11 0.00 0.00 4.00 4.00 3.03
0.28 0.02 0.40 0.59 0.23 0.20 0.50 2.40 0.95 0.03
Q = discharge. V = velocity.
1. A new approach that is quite different from and more precise than conventional method, the method of Rotatable Coordinate Axle that can be
used to study the hydraulic characters of two-phase stratified flow in pipe, is proposed. Based on the method, the new model for stratified turbulent flow in pipe is built and the analytic formulas of velocity field are obtained. 2. The model of stratified flow in pipe is studied as stratified flow area and single-phase flow area. From the analysis of stress distribution rule, we’ve got the rule of velocity distribution of stratified flow in pipe.
Fig. 2. Velocity distribution along vertical radial.
Fig. 3. Velocity distribution along horizontal radial.
calculated result and experiment result and it can be seen that the two results are coincident with each other.
4. Conclusions
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Acknowledgements This research is supported by the Key Project of Ministry of Education, China [Grant No. (2000) 00108].
References
Fig. 4. Velocity distribution along horizontal radial (aeration).
3. Considering that the main discharge of the two-phase flow in pipe is larger than that of a single-phase flow in the aeration experiments, resistance reduction of stratified turbulent flow in pipe is proved.
Bishop, A.A., Deshpande, S.D., 1986. Non-Newtonian liquid – air stratified flow through horizontal tubes—II. International Journal of Multiphase Flow 12 (6), 977 – 996. Dou, G.-R., 1988. Hydrodynamics on Turbulent Flow. Higher Education Press, Beijing, China. Govier, G.W., Aziz, K., 1972. The Flow of Complex Mixtures in Pipes. Van Nostrand Reinhold, New York. Li, H.-B., Wu, C., 2004. Theoretical and experimental study of nonNewtonian turbulent stratified flow in pipe. Natural Gas Industry 24 (1), 89 – 91. Prandtl, L., Tietjens, O.G., Hartjog, J., 1934. Applied Hydro and Aeromechanics. McGraw-Hill, London, England. Wang, C.-B., Wei, Z.-S., Chen, J.-L., 1994. The flow regime distribution of gas and power-law liquid two-phase horizontal flow in pipe. Pipeline Technology and Facility 1994 (3), 5 – 7. Zheng, Y.-G., Xie, C.-L., 2000. Stratified flow and the rule of drag reduction of non-Newtonian flow in pipe. Journal of Hydrodynamics, Ser. A 15 (3), 314 – 319.
Plane flow model of non-Newtonian turbulent stratified flow in wells and pipes Li Hong-bo, Wu Chao *, Zheng Yong-gang State Key Laboratory of Hydraulics on High Speed flows, Sichuan University, Chendu 610065, China Received 9 July 2003; accepted 18 March 2004
Abstract Economical and efficient long-distance oil and natural gas pipe transportation technologies are greatly demanded. Nowadays, oil and gas mixed transportation is gradually put into practice and researchers pay much attention on the theories of two-phase flow in pipe. Methods that can efficiently solve problems of this field are being put forward including hydraulic approach. A new approach, the Rotatable Coordinate Axis, is proposed in this paper. It is used to study the hydraulic characters of two-phase stratified flow in pipe. Based on the new method, the plane flow model for stratified turbulent flow in pipe is built. Then, we can obtain the analytic formulas of velocity, discharge etc. in wells and pipes by this model. To prove the theory of the plane flow model, experiments of aeration in stratified pipe is conducted with the aeration experiment device. The experiment data showed that aeration could effectively achieve the resistance reduction in pipeline, which can offer great theory support to the development of oil and natural gas mixed transportation technology. D 2004 Elsevier B.V. All rights reserved. Keywords: Pipe; Turbulent flow; Rotatable Coordinate Axis; Aeration; Stratified flow
1. Introduction Along with the exploitation of the oil and natural gas in oceans and deserts, nowadays, the improvement of two-phase transportation in pipe is demanded and the theoretical study of two-phase flow in pipe has become more and more popular. The hydraulic characters of two-phase flow in pipe are complex, for they are related to physical characters, discharge, drag, flow regime (Wang et al., 1994), and many other factors. The stratified flow, the most typical flow regime, is very common in declining pipe and is the * Corresponding author. Fax: +86-28-85405148. E-mail address: [email protected] (W. Chao). 0920-4105/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2004.03.005
root of other flow regimes as well. Researchers had done a great deal of work on it (Prandtl et al., 1934; Govier and Aziz, 1972; Dou, 1988). There are also a few reports or researches on two-phase stratified pipe flow (Bishop and Deshpande, 1986; Zheng and Xie, 2000; Li and Wu, 2004). To ascertain the shearing stress, the application of empirical or semiempirical equation on the coefficient of frictional resistance made the researches less accurate. In this paper, the Rotatable Coordinate Axis is used in the research on cross-section, and stratified flow in pipe is approximately replaced by plane flow between two boundless flats. The cross-section of the pipe is divided into two parts of stratified flow section and a single-phase flow section. Under the
224
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
condition of uniform stratified flow, viz. regardless interfacial liquid level and waves causing vortex motion and energy dissipation, a new flow model for the velocity distributing discipline of non-Newtonian stratified flow in pipe can be proposed. Then, by this model, analytic formulas of velocity, discharge and resistance reduction can be deduced without using average velocity, empirical or semiempirical equation on the coefficient of frictional resistance in wells and pipes as before. Thus, we can get more accurate quantity of velocity, discharge etc. in wells and pipes than before.
2. The plane flow model of two-phase flow in pipe and the velocity distributing discipline At the very beginning, a series of preconditions should be given: 1. As shown in Fig. 1, the flow in pipe is supposed to be full flow. 2. The two-phase flow is made up of two kinds of non-Newtonian fluid. 3. And the flow patterns concerned in this paper are turbulent flow. 4. The flow in pipe is uniform, which interfacial liquid level and waves causing vortex motion and energy dissipation are not concerned.
5. The flow in pipe is steady flow, which means parameters like velocity, pressure, density, temperatures, and the stress gradient are not time variation. As shown in Fig. 1, fluid 1 is top fluid, which is natural gas usually; and fluid 2, subjacent fluid, which should be oil. To describe this two-phase flow in pipe, here a rotatable coordinate system should be applied. It could be described as follows. First, random pipe cross-section was considered in Fig. 1. AOB was taken as rotatable coordinate axis. It can revolve around the point O, and there will be corresponding intersection angle (U) with the cross axis. In addition, it intersects the circle of pipe crosssection at points A and B. Then, two tangent lines to the circle can be obtained with the two points. Second, if the whole pipe was taken into account, the two tangent lines could be thought was two boundless parallel tangent planes with the interval of 2R to the pipe (R is the radius of the pipe). Accordingly, as shown in Fig. 1, the problem of pipe flow was transformed into plane flow approximately to study the stratified flow between two parallel planes. And the planes also revolved around the circle following the revolving coordinate axis AOB. Then, according to the angle U, the whole crosssection can be divided into two parts. Within the range of U0 V U V p-U0, there are two layers between the
Fig. 1. Stratified flow model in pipe.
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
225
two planes. The thickness of top fluid 1 is h1, governed by formula h1 = R h0/sinU, and the thickness of subjacent fluid 2 is h2, h2 = R + h0/sinU. In the left part: p U0 V U V 2p + U0, the flow between the parallel planes is single-phase flow of fluid 2. In this paper, only two-phase stratified flow section in pipe was studied. Supposing that x is the axial direction of the pipe and dp/dx = constant, the stratified flow model in pipe is:
as an example, viscous shearing stress expression, viz. constitutive equation of the top fluid should be
ds=dy ¼ ðdp=dxÞ=b
K1 ðdu=dyÞn1 ¼ sw1 ½1 ðy=y0 Þ
in which s is shearing stress, p is compressive stress and b is curvature parameter of pipe surface. When b = 1, it is a plane flow model; b = 2, a uniform pipe flow model. Let s = 0, and the coordinate of the point with the highest velocity can be obtained, that is y0 ¼ ðsw1 bÞ=ðdp=dxÞ Integrating it, under the boundary condition: when y = 0, the shearing stress s = sw1, we can obtain
s1 ¼ K1 ðdu=dyÞn1 where K1 is consistence coefficient of the top fluid, and n1 is flow index. And s = sw1[1-( y/y0)], so
Arrange the equation above in order and integral it on boundary condition: u = 0 ( y = 0), we can obtain n1 sw1 1=n1 u¼ n1 þ 1 K1 y0 h i ðn þ1Þ=n1 ðy0 yÞðn1 þ1Þ=n1 y0 1 When y = b1, the velocity is
s ¼ sw1 ½1 ðy=y0 Þ and
ub1
s ¼ ðdp=dxÞy=b þ sw1 The total shearing stress is s = sl + stSsl is the viscous shearing and st is the turbulent shearing stress. They are quite different according to the fluid position. In viscous sublayer, the turbulent shearing stress is much smaller than the viscous shearing stress, and could be ignored, while in the core section of turbulent flow, the viscous shearing stress can be ignored.
n1 sw1 1=n1 ¼ n1 þ 1 K1 y0 h i ðn þ1Þ=n1 ðy0 b1 Þðn1 þ1Þ=n1 y0 1
2.1.2. Core section of turbulent flow (U0VUVpU0, b1VyVh1) In this layer, viscous shearing stress could be ignored, and from the mixing length theory of Prandtl (Prandtl et al., 1934), the turbulent shearing stress is s1 ¼ q1 l 2 ðdu=dyÞ2
2.1. Top fluid 2.1.1. Viscous sub layer (U0VUVpU0, 0VyVb1) In this layer, turbulent shearing stress could be ignored. b1 is the thickness of viscous sublayer, and could be governed by the following formulas b1 ¼ a1 ðV1 =V*1 Þ
1=2
V*1 ¼ ðsw1 =p1 Þ
in which, a1 is proportional constant (top fluid), V1 is velocity. Take the power law fluid of non-Newtonian
in which q1 is the density of the top fluid and length l = ky, k: Karman constant, k = 0.4, no dimension. On boundary condition: u = ub1 ( y = b1) integral, we have u ¼ ub1 þ ð2V*1 =kÞ n o ½1 ðy=y0 Þ1=2 ½1 ðb1 =y0 Þ1=2 þ ðV*1 =kÞ ( ) 1 ½1 ðy=y0 Þ1=2 1 ½1 ðb1 =y0 Þ1=2 ln ln 1 þ ½1 ðy=y0 Þ1=2 1 þ ½1 ðb1 =y0 Þ1=2
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From y = h1, h1 = R-h0/sinU, the velocity of fluid interface could be obtained u1 ¼ u jy¼h1 ¼ ub1 þ ð2V*1 =kÞ n ½1 ððR h0 =sinUÞ=y0 Þ1=2 o ½1 ðb1 =y0 Þ1=2 þ V* =k ( 1 ½1 ððR h0 =sinUÞ=y0 Þ1=2 ln 1 þ ½1 ððR h0 =sinUÞ=y0 Þ1=2 ) 1 ½1 ðb1 =y0 Þ1=2 ln 1 þ ½1 ðb1 =y0 Þ1=2
distribution formula, the discharge can be obtained by integration in the cross-section of pipe. If all variables of the single-phase part in one pipe are marked with suffix 0, and top fluid 1, subjacent layer 2, the expressions of Q1 and Q2 are as follows Z pþU0 Z Z Z R urdrdU ¼ 2 dU udr Q1 ¼ S
8 < y* ¼ 2R y
The total discharge of pipe ( Q) comprises the discharge of the single-phase part ( Q1) and that of the stratified part ( Q2). According to the velocity
pU0
1=n2
0
8 > < > :
R
b0
ub0 þ
2V*0 y 1=2 b0 1=2 1 1 R k R
y 1=2 1 1 V 06 R þ * 4ln y 1=2 k 1þ 1 R 39 b0 1=2 > = 1 1 7 R ln 1=2 5>ðR yÞdy ; 1 þ 1 bR0
With the similar methods for research on top fluid, we can establish coordinate y* in viscous sublayer section and core section of turbulent flow as shown in Fig. 1. The coordinate relationship of y and y*:
2.3. Discharge of subjacent fluid
2
2.2. Subjacent fluids
Similarly, the formulas about underlayer can be obtained by replacing y with y*. Furthermore, it is single-phase flow in the section p U0 V U V 2p + U0 and analytic distribution formulas of velocity and formulas of velocity in the center of the pipe can be confirmed. According to the junction condition of velocity on interphase, velocity of flow on the point of y = h1 of the top fluid is equal to that of underlayer. And according to the junction condition of the center of pipe, velocity of flow in the two-phase stratified flow section is equal to that in the center of pipe. Thus, two equations can be established in order to confirm b( y0) and sw1, in which b, sw1( y0,y* 0 ) are unknown functions of U.
b0
n2 sw0 ¼ 4U0 n þ 1 K 2 2 y0 0
Z n þ1 2 n2 þ1 n2 n2 ðy0 yÞ y0 ðR yÞdy þ 4U0
Thus, velocity on interphase is the function of U, under the initial conditions (U = U0, h1 = 0).
: * y0 ¼ 2R y0
Z
Then the discharge of the stratified part is Z pU0 Z Z Z RsinU0 =sinU Q2 ¼ urdrdU ¼ dU urdr S
U0
þ
Z
0
Z
2pU0
dU pþU0
R
urdr 0
in which, u¼
1 n2 þ1 n2 þ1 n2 sw0 n2 n ðy0 2R þ yÞ n2 y0 2 n2 þ 1 K 2 y0 " 2V*2 2R y 1=2 þ ub1 þ 1 2R y0 k 1=2 # V 2 b2 1 þ * 2R Y0 k ( 1 ½1 ð2R yÞ=ð2R y0 Þ1=2 ln 1 þ ½1 ð2R yÞ=ð2R y0 Þ1=2 ) 1 ½1 b2 =ð2R y0 Þ1=2 ln 1 þ ½1 b2 =ð2R y0 Þ1=2
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
And we can compared the total discharge of subjacent layer ( Q = Q1 + Q2), with that of a singlephase flow in one pipe, shown as Z Z Z R Q0 ¼ urdr urdrdU ¼ 2p S
8 > < > :
b0
0
n2 sw0 1=n2 h ðR yÞðn2 þ1Þ=n2 n þ 1 K y 2 2 2 0 Z R i ðn2 þ1Þ=n2 ðR yÞdy þ 2p R
¼ 2p
Z
b0
ub0 þ
2V*0 y 1=2 b0 1=2 1 1 R k R
2
y 1=2 1 1 V 06 R þ * 4ln y 1=2 k 1þ 1 R 39 b0 1=2 > = 1 1 7 R ln 5 ðR yÞdy 1=2 b0 > ; 1þ 1 R Whether drag reduction of stratified turbulent flow in one pipe is achieved or not can be ascertained with the following expression Dr ¼
Q Q0 Q0
227
driven air-entraining machine. Compressed air output by air entraining machine is first decompressed in decompression chamber, and then poured into testing pipe by a number of thin pipes, stratified aeration is provided by air entrance, 3– 16 holes with internal diameter of 3 mm on the upper are of the pipe. The main testing device of aeration experiments in this paper is a two-dimensional Laser Doppler Velocity produced by the company of DANTEC in Denmark. On the velocity-measuring cross-section, there are 15 – 20 spots which are used to describe distribution of velocity. We have applied two air entrances, upstream and downstream, to accomplish the aeration. 3.2. Basic data In this paper, fluid 1 (upper) is air, fluid 2 (lower fluid) is pure water, and so we have l1 ¼ 17:9 106 Pa s; l2 ¼ 1010 106 Pa s; q1 ¼ 1:21 kg=m3 ; q2 ¼ 1000 kg=m3 : From analysis of theory and experiments, we can get a1 = 0.116, a0 = 11.6, a2 = 11.6, B = 1.1. For stratified flow in pipe, the content of air is governed by U0. As U0 increases, content of air decrease; as U0 decreases, content of air increases. Relation between content of air on section of pipe e and U0 is (Table 1)
3. Experiments and tests 3.1. Experiment device and other conditions The length of testing pipe is 17 m, and the inner diameter is 0.1 m. The part of measuring velocity box is made of plexiglass, and other part of pipe is PVC. A high water tower and a flat-water trough on the upstream of pipe provide steady water head and the decrease and pressure gradients are constant. Flow in pipe and velocity of flow are controlled by water level of upstream and gate at the exit of downstream. The source of air of aeration in testing pipe is provided by a power-
e¼
p 2U0 sinð2U0 Þ 2p
3.3. Experiment results Figs. 2 and 3 are the data comparisons of velocity distribution before and after aeration along with vertical radial (U = 90j) and horizontal radial (U = 0j), respectively. Under the conditions of aeration and water, we’ve got two groups of experiment data, which show that fluid velocity increases on the former condition. Fig. 4 is a comparison between
228
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
Table 1 Data of experiments No.
1 2 3 4 5 6 7 8 9 10
Before aeration
Aeration density
After aeration
Q (10-3 m3/s)
V (m/s)
U0 (rad)
e (%)
Q (10-3 m3/s)
V (m/s)
0.990 1.800 1.166 1.166 1.118 0.512 0.199 0.199 0.199 2.330
0.126 0.229 0.148 0.148 0.142 0.065 0.025 0.025 0.025 0.297
0.8552 0.9861 0.8552 0.9861 0.8552 0.8552 0.7854 0.8552 0.9861 0.9250
7.0 4.0 7.0 4.0 7.0 7.0 9.1 7.0 4.0 5.3
1.040 1.831 1.177 1.214 1.139 0.513 0.200 0.202 0.205 2.400
0.132 0.233 0.15 0.155 0.145 0.065 0.025 0.026 0.026 0.306
DQ (%)
DV (%)
DA (error, %)
5.05 1.72 0.94 4.12 1.88 0.20 0.50 1.51 3.02 3.00
4.76 1.75 1.35 4.73 2.11 0.00 0.00 4.00 4.00 3.03
0.28 0.02 0.40 0.59 0.23 0.20 0.50 2.40 0.95 0.03
Q = discharge. V = velocity.
1. A new approach that is quite different from and more precise than conventional method, the method of Rotatable Coordinate Axle that can be
used to study the hydraulic characters of two-phase stratified flow in pipe, is proposed. Based on the method, the new model for stratified turbulent flow in pipe is built and the analytic formulas of velocity field are obtained. 2. The model of stratified flow in pipe is studied as stratified flow area and single-phase flow area. From the analysis of stress distribution rule, we’ve got the rule of velocity distribution of stratified flow in pipe.
Fig. 2. Velocity distribution along vertical radial.
Fig. 3. Velocity distribution along horizontal radial.
calculated result and experiment result and it can be seen that the two results are coincident with each other.
4. Conclusions
L. Hong-bo et al. / Journal of Petroleum Science and Engineering 44 (2004) 223–229
229
Acknowledgements This research is supported by the Key Project of Ministry of Education, China [Grant No. (2000) 00108].
References
Fig. 4. Velocity distribution along horizontal radial (aeration).
3. Considering that the main discharge of the two-phase flow in pipe is larger than that of a single-phase flow in the aeration experiments, resistance reduction of stratified turbulent flow in pipe is proved.
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