flow field prediction and bubble trajectory model in gas ... - CiteSeerX

Proceedings of ETCE ‘99: Energy Sources Technology Conference & Exhibition Houston, Texas, February 1-2,1999

FLOW FIELD PREDICTION AND BUBBLE TRAJECTORY MODEL IN GAS-LIQUID CYLINDRICAL CYCLONE (GLCC) SEPARATORS Ivan Mantilla, Siamack A. Shirazi ([email protected]) and Ovadia Shoham ([email protected]) The University of Tulsa Tulsa, OK 74104

ABSTRACT Several mechanistic models have been already developed for predicting the onset of liquid carry-over in Gas-Liquid Cylindrical Cyclone (GLCC) Separators. However, currently no model is available to predict gas carry-under. A bubble trajectory model has been developed that can be used to determine the initiation of gas carry-under in the GLCC and to design GLCC for field applications. The bubble trajectory model uses a predicted flow field in GLCC that is based on swirl intensity. This paper describes the development of a general correlation to predict the decay of the swirl intensity. The correlation accounts for the effects of fluid properties (Reynolds number) as well as inlet geometry. Available experimental data as well as computational fluid dynamics (CFD) simulations were used to validate the correlation. The swirl intensity is used to calculate the local axial and tangential velocities. The flow model and improved bubble trajectory results agree with experimental observation and CFD results. Examples are provided to show how the bubble trajectory model can be used to design GLCC. INTRODUCTION The tendency of multiphase separation technology has always been to develop more efficient and less bulky equipment to decrease operation and maintenance costs. Conventional gravity based separators have been utilized for many years by different industries for multiphase flow separation. Despite the fact that they possess fairly good efficiency, conventional separators are massive and costly to operate and maintain. Therefore, alternative separators that are smaller and less expensive to operate have always been sought. One such an alternative is the cyclone-type separator that uses centrifugal forces to accelerate the separation of multiphase

mixtures. Among the small-size, high-efficiency cyclone separators, the Gas-Liquid Cylindrical Cyclone (GLCC) separator is an attractive alternative to the conventional separator due to its low cost, simple design and easy installation and operation. A schematic of the GLCC separator is shown in Fig. 1. The GLCC is a vertically installed pipe, mounted with a downward inclined tangential inlet. The gas and liquid outlets are provided at the top and the bottom of the pipe. Due to the tangential inlet, the flow forms a swirling fluid motion producing a centrifugal force. The gas and liquid phases of the incoming mixture are separated due to centrifugal and gravity forces. The rotation produces a gas-liquid interface with a parabolic shape. The liquid is forced radially towards the wall of the cylinder and is collected from the bottom, while the gas moves to the center of the cyclone and is taken out from the top. For some operational conditions, some liquid can flow up with the gas and be carried through the gas outlet at the top. This phenomenon is referred to as liquid carry-over. On the other hand, some gas can be carried down with the liquid and flow through the liquid outlet at the bottom. This is called gas carry-under. Several mechanistic models have been already developed for predicting the onset of liquid carry-over in GLCC (Arpandi, 1995). However, currently no model is available to predict gas carry-under. In order to predict gas carry-under, one needs to understand the complex flow inside the GLCC, including the distribution of the gas bubbles in the liquid phase below the inlet. Therefore, to design GLCC and estimate the required length of the GLCC below the inlet, a bubble trajectory model was developed by previous investigators (Marti et al., 1996).

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r z

θ

Tangential Velocity

GAS-LIQUID INTERFACE

Axial Velocity

Fig. 2 - Schematic of Typical Axial and Tangential Velocity Profiles and GLCC Coordinate System Fig. 1 - Schematic of the Gas-Liquid Cylindrical Cyclone Separator However, this bubble trajectory model used a simple velocity distribution inside the GLCC separators and a simplified force balance to determine the trajectory of individual bubbles. But, simple velocity distributions that were assumed may not be a good representation of swirling flow that occurs in GLCC below the gas-liquid interface. Many investigators have studied single-phase gas or liquid flow in pipes with tangential injection and indicated that the resulting swirling flow field is very complex. For example, Ito et al. (1979) indicated that the tangential velocity distribution has two flow regions: forced-vortex flow near the center of the tube and a free-vortex region near the wall. The axial velocity distribution shows a region of flow reversal near the center of the tube. Fig. 2 shows schematically typical axial and tangential velocity profiles that have been observed for high swirl intensities The objective of this investigation is to develop a method for predicting the flow field and to improve the previous bubble trajectory model for GLCC separators. The tools used for this investigation include experimental data provided in the literature and a commercial computational fluid dynamics (CFD) code called CFX (CFX-4.2, 1997). NOMENCLATURE A = cross sectional area of GLCC Ais = inlet slot area Cd = drag coefficient d = diameter of GLCC g = gravity acceleration I = inlet factor L = length

Mt MT m& n r rrev R Re u vsl w z µ ρ Ω

= tangential momentum flux = total momentum flux = mass flow rate = number of tangential inlets (for n>2) = radial coordinate = flow reversal radius = radius = Reynolds Number = axial velocity = superficial liquid velocity = tangential velocity = axial coordinate = viscosity = density = swirl intensity

Subscripts av = average b = bubble d = drag g = gas is = inlet slot L = liquid m = mixture r = radial s = slip t = tangential z = axial Abbreviations CFD = computational fluid dynamics CFX = CFD commercial version GLCC = gas-liquid cylindrical cyclone

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MMM = modified mechanistic model for bubble trajectories. PMM = previous mechanistic model for bubble trajectories. Flow Field Prediction The analysis of several sets of experimental data available in the literature (Chang and Dhir, 1994; Kitoh, 1991; Ito et al., 1979; Nissan and Bressan, 1961; Algifri et al., 1988) for swirling flows led to the development of a set of correlations to predict the flow field. The correlations developed do not only account for the key operating variables of the flow, but also incorporate the physical phenomena. For example, a parameter that is referred to as the swirl intensity is used to develop correlations for the velocity field in GLCC separators. Swirl Intensity Decay The swirling motion decays as a result of wall friction, and the swirl intensity is used to characterize this decay. For axisymmetric and single-phase flow, the swirl intensity, Ω , is defined as the ratio of the rate of tangential to total momentum flux at any axial location (Chang and Dhir, 1994).

that for axial distances smaller than two diameters from the inlet, the swirl appears to decay at a much slower rate, thus, the correlation is valid for z > 2 . d

The momentum flux ratio at the inlet, M t , for single-phase MT

flow, can be calculated as the ratio of the inlet to GLCC area: m& M t m&vtis vtis ?Ais A = = = = (3) & m M T m&uav uav Ais ?A vtis is the tangential velocity at the inlet, m& is the mass flow rate, Ais and A are the inlet slot area and cross sectional area of the pipe, respectively. For two-phase flow that enters the GLCC separator, the liquid velocity at the inlet is affected by the gas velocity. A method that was developed by Gomez (1998) can be used to compute the tangential velocity of the liquid at the inlet. Thus, the momentum flux ratio at the inlet for GLCC application can be calculated as: M t  vtis  =   MT  uav L

R

O=

2p? ∫uwrdr

(1)

0

2 p?R 2 uav

where uav is the average axial velocity, R is the tube radius and ρ is the fluid density. The numerator in Eq. (1) corresponds to the tangential momentum flux integrated over the cross section, while the denominator is the total momentum flux based on the average axial velocity. Several published experimental data sets indicate that, the swirl intensity decays exponentially with axial distance (Chang and Dhir, 1994; Kitoh, 1991; Algifri et al., 1988). However for small values of swirl intensities (< 0.5), the rate of decay increases and deviates from exponential behavior, as shown by Kitoh (1991). Chang and Dhir (1994), based on their experimental data, developed a correlation to predict the swirl intensity as a function of the inlet momentum ratio and the axial location. Based on analysis of the experimental data in the literature and computational fluid dynamics (CFD) simulations, during this investigation, a more general correlation was developed to account for fluid properties and inlet geometry effects. The new correlation for the swirl intensity is: 0.93

Mt 2  O = 1.48   M I   T 

0.35 0.7   1M − 0.16  z   t 4  exp−  I Re     2 M d     T  

(2)

(4)

where the velocities correspond to the liquid phase. The Reynolds number in Eq. (2) is defined as for pipe flow, based on the diameter of the GLCC. The inlet factor, I, is assumed to be function of the number of tangential inlets, n.  n I = 1 − exp−   2

(5)

This correction factor is based on analysis of experimental data for 2 and more tangential inlets. However, no data are available for one inlet. During this investigation, with the aid of CFD simulations, it was observed that the initial swirl decay for one tangential inlet is similar to two inlets. Thus, a value of n = 2 can be approximately used for one tangential inlet geometry until more information is available. Tangential Velocity The swirl intensity is related, by definition, to the local axial and tangential velocities (Eq. (1)). Therefore, it is assumed that, for a specific axial location, the swirl intensity prediction can be used to calculate the velocity profiles. The tangential velocity distribution, except in the vicinity of the wall, can be approximated by a Rankine Vortex. Algifri et al. (1988) proposed the following equation for the tangential velocity profile:

where M t is the ratio of the tangential momentum flux to the

  r 2  T  w   = m 1 − exp − B    u av  r   R          R 

MT

total momentum flux at the inlet, I is the inlet geometry factor, Re is the Reynolds number, z is the axial distance and d is the diameter of the GLCC. It was reported (Chang and Dhir, 1994)

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(6)

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where w is the local tangential velocity, r is the radial location, Tm is related to the maximum moment of the tangential velocity and B is related to the radial location of the maximum velocity. Algifri et al. (1988) calculated Tm and B from their own data. However, to obtain a general correlation, in this investigation expressions for Tm and B were developed by curve-fitting several sets of available experimental data. Tm = 0 .9 O − 0.05

(7)

 O  B = 3.6 + 20 exp −   0.6 

(8)

Axial Velocity A radial as well as an axial pressure gradient develops as a result of the swirling motion and the tangential velocity in the GLCC. These pressure gradients, in turn, influence the flow field and lead to a complex flow phenomenon. If the swirling motion is strong enough, a positive pressure gradient in the axial direction could result in the development of flow reversal in the main flow (see Fig. 2). In the limit, when the swirl intensity decays to nearly zero, the flow becomes purely axial (pipe flow). Based on the distribution of the axial velocity (Fig. 2), a third-order polynomial was selected to approximately represent axial velocity profile, u (r )= a1r 3 + a2 r 2 + a3 r + a4

(9)

Four conditions were applied to determine the coefficients of the polynomial. It was assumed that the boundary layer is very thin, thus, the velocity has a maximum at the wall (in the same direction as the axial flow). Also, the velocity is zero at the location of flow reversal (rrev) and minimum at the GLCC axis. The conservation of mass is included in the set of conditions. Substituting the conditions in Eq.(9) yields: 3

2

u 2  r  3  r  0.7 =   -   + + 1.0 u av C  R  C  R  C

 r 2  r C =  rev   3 - 2 rev  R    R   r rev = 0.174 O 0.63 R

    - 0.7   

(10)

(11)

Radial Velocity The magnitude of the radial velocity, according to experimental data and CFD simulations, is two orders of magnitude smaller than the corresponding tangential or axial velocities. Therefore, no attempt was made to predict the radial velocity profile, since it would have a very small impact on the bubble trajectories. Bubble Trajectory Model The starting point of the model is a force balance on the bubble with the assumption of local momentum equilibrium, similar to the study by Marti et al. (1996). Details of the bubble trajectory equations are given by Mantilla (1998); only a brief description is given here. In the radial direction, the drag force is balanced with the centripetal force yielding a expression for the radial slip bubble velocity, vsr,: vsr (r ) =

w (r )2 d b 1   r C v (r )  d sb

(13)

where w(r) is given by Eq. (6) and vsb is the resultant slip velocity of the bubble. Similarly, in the axial direction the drag force is equal the buoyancy force and the slip bubble velocity in the axial direction is determined: 4  ρm − ρ g  d b 1  vsz (r )=  g 3  ρL  Cd vsb (r )

(14)

The resultant slip velocity of the bubble, vsb, is obtained by 2 2 combining Eqs. (13) and (14), note that v sb . = v sz2 + v rb 4  ?m − ?g  d b   2   vsb = C g + 3 ?  L  d 

 w (r )2     r   

2

1

   

4

(15)

The drag coefficient is calculated using the correlation developed by Mei et al. (1994) and reported by Magnaudet (1997): −1    16   8 1 3 . 315   + 1 + Cd = 1 +  (16)    Reb (r )  Re ( r ) 2 ( ) Re r b  b      In the expression for the bubble Reynolds number, the mixture density and viscosity are used, (Devulapalli and Rajamani (1996)). Reb (r )=

(12)

Equation (12) was originally developed by Chang and Dhir (1994) and is used to predict the location of the flow reversal.

4  ρm − ρ g  3  ρL

?m vsb (r )d b µm

(17)

The trajectory of the bubble (change of position in the z direction, ∆z, for a small displacement in the radial direction ∆r) is computed using the following numerical integration:

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r

z (r )= ∑ R

u (r )− vsz (r )   v (r )  ? r   r

10

(18)

where u is the liquid axial velocity and vr is equal to vsr, since the radial velocity of the liquid is neglected. CFD Simulations A commercially available CFD code called CFX (CFX4.2, 1997) was used to aid in the development of the correlation for the swirl intensity in the GLCC. The influence of the Reynolds number was investigated by simulating different flow conditions and fluids. Also, the effects of one and two tangential inlets were investigated with the CFD code. To verify the present bubble-trajectory model, the CFD code was also used to track the trajectories of bubbles of different sizes. Details of the CFD simulations are presented by Mantilla (1988); a bubble trajectory comparison with the CFD code is included in this paper. Model Verification The swirl intensity predictions achieved with Eq.(2) are compared with results of Eq. (1) using different sets of experimental data for u and w. Fig. 3 summarizes the results. Ito et al. (1979) and Nissan and Bressan (1961) used water in their experiments with two tangential inlets. Chang and Dhir (1994) used air with 4 and 6 tangential inlets. Kitoh (1991) used air or water and guide vanes to generate the swirling motion. For this case, the swirl intensity is not very strong. Overall, good agreement in magnitude and trends is observed between the correlation and the data. Also, it is shown that the decay of swirl intensity is faster for higher inlet momentum ratios. Representative velocity profile predictions are presented in Fig. 4-6. Fig. 4 shows the comparison of the tangential velocity predictions with Ito et al. (1979) data for three different axial locations. The agreement is good in magnitude and decay rate. Axial and tangential velocity predictions for Kitoh’s data are shown in Fig. 5. The axial location is z/d = 6. The tangential velocity prediction is good as well as the axial velocity prediction except for the near wall region since in the correlations the boundary layer effects were not considered. The predictions for Chang and Dhir’s data are depicted in Fig. 6. The axial location is z/d = 7.06. The predicted axial and tangential velocities are in agreement with the data. Very few experimental data are available in the literature for bubble trajectories in swirling flows. In the work by Guo and Dhir (1989), trajectories for 2.2 mm and 4.0 mm bubble were reported for a 3.5” cylinder. A four-inlet tangential geometry created the swirling flow with water as the working fluid. The bubbles were released five diameters from the inlet, at the wall (i.e., r = R) and the axial displacement, δ , was

1

Ito, Re = 6600, Mt/MT = 50, n = 2 Prediction, (Ito) Chang, Re = 12500, Mt/MT = 7.8, n = 4 Prediction, (Chang), Mt/MT = 7.8 Chang, Re = 12500, Mt/MT = 2.67, n = 6 Prediction, (Chang), Mt/MT = 2.67 Nissan, Re = 10000, Mt/MT = 8, n = 2 Prediction, (Nissan) Kitoh, Re = 50000, Mt/MT = 1 Prediction, (Kitoh)

0.1 0

10

20

30

40

z/d

Fig. 3 - Swirl Intensity Predictions vs. Experimental Data recorded until the bubble reached near the center of the cylinder (i.e., r = 0.1R). The experimental data and the predictions of the bubble trajectory model are compared in Fig. 7. It can be observed that the predictions agree well with the measured axial displacement of the bubbles. No experimental data are yet available for very small bubbles in GLCC separators. Therefore, the mechanistic model bubble trajectory results are compared with the trajectories using the CFD code. Fig. 8 shows a bubble trajectory comparison (in a r-z plane), for superficial liquid velocity of 1.0 ft/s and superficial gas velocity of 10 ft/s in a 3” GLCC, using air and water as the working fluids. The trajectories are calculated with the previous mechanistic model for bubble trajectories (PMM) of Marti et al. (1996), the modified mechanistic model (MMM) and the CFD code (denoted by CFX). The present model (MMM) incorporates a more realistic velocity field and an improved bubble trajectory model than the previous model (PMM). Thus, the trajectories calculated with the present model (MMM) agree better with the CFD results (CFX) than the previous model (PMM).

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12

8

z/d = 9 Prediction z/d = 17 Prediction z/d = 31 Prediction

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Tangential Velocity Data Tangential Velocity Pred. Axial Velocity Data Axial Velocity Pred.

6

8

4

6

2

4

0 2

-2 0

0 0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1

r/R

1

r/R

Fig. 6 - Velocity Predictions vs. Chang and Dhir’s Data, Mt/MT = 7.84, Re = 12500, d = 3.5”, n = 4, z/d = 7.06

Fig. 4 - Tangential Velocity Predictions vs. Ito et al. Data, Mt/MT = 50, Re = 6600, d = 6.3”, n = 2

4 Data, 2.2 mm bubble Prediction, 2.2 mm bubble Data, 4 mm bubble Prediction, 4 mm bubble

2 3

0

2

Tangential Velocity Data Tangential Velocity Pred. Axial Velocity Data AxialVelocity Pred.

1

-2 0

0.2

0.4

0.6

0.8

1 0

r/R

0.05

Fig. 5 - Velocity Predictions vs. Kitoh’s Data, Mt/MT =1, Re = 50000, d = 5.9”, n = ∞ , z/d = 6

0.1

0.15

0.2

0.25

uav (m/s)

Fig. 7 - Comparison of Mechanistic Model Predictions and Experimental Data, Ghuo and Dhir (1989), d = 3.5”

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50 µm bubble

10

Bubble size, µm

Inlet

0

Viscosity: 1cp

50 100

5 4

0

8

0

0.5

1

vsl (ft/s) 12

10 Bubble size, µm Viscosity: 10 cp

Outlet

16 0

0.5 r/R

5

1

PMM MMM CFX

0 0

Fig. 8 - Bubble Trajectory Comparison, PMM, MMM and CFX, vsl = 1.0 ft/s, v sg = 10 ft/s, 3” GLCC

200 300 500

0.5 vsl (ft/s)

1

Fig. 9 - Bubble Axial Displacement, v sg = 10 ft/s, 3” GLCC (Release point: 2d, below the inlet)

Application The present model allows the study of the effect of different operational parameters on the trajectories of bubbles, and their influence on the design of the GLCC length. Fig. 9, for example, shows the length (below the inlet and measured from the release point) required to separate different sizes of bubbles, δ (z/d), as a function of superficial liquid velocity and viscosity. The bubbles are released 2d below the inlet. It is observed from the figure that for low superficial liquid velocities, or high viscosities, the bubbles travel longer distances. Thus, requiring longer GLCCs to separate them. Also, can be observed that for a viscosity of 10 cp, the length traveled by a 200-µm bubble is large for superficial velocities below 0.5 ft/s. This indicates that smaller bubbles may not be separated for the same operational conditions. Fig. 10 depicts the relationship between d100 (the smallest bubble that is separated 100%) and the Reynolds number for different inlet momentum ratios. It can be observed that for high Reynolds numbers (high liquid flow in GLCC), the inlet momentum ratios do not significantly change the predicted value of d100. Also, it can be observed that for a constant Reynolds number, d100 is smaller for higher inlet momentum ratio, as expected.

300

250 vtis/uav

200