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Numerical simulations on the effect of sloshing on liquid flow maldistribution of randomly packed column
Applied Thermal Engineering 112 (2017) 585–594
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Research Paper
Numerical simulations on the effect of sloshing on liquid flow maldistribution of randomly packed column Mengxian Zhang a, Yuxing Li a,⇑, Yan Li b, Hui Han a, Lin Teng a a b
Provincial Key Laboratory of Oil and Gas Storage and Transportation Safety in Shandong Province, China University of Petroleum, Qingdao 266580, China CNOOC Research Center, Beijing 100149, China
h i g h l i g h t s A model for randomly packed column under sloshing conditions is developed. The model predictions are validated with the experimental data. Both static and sloshing conditions are considered. Effect of tilt angles and aspect ratios are discussed.
a r t i c l e
i n f o
Article history: Received 6 February 2016 Revised 1 August 2016 Accepted 9 October 2016 Available online 11 October 2016 Keywords: Randomly packed column Numerical simulation Sloshing Liquid distribution
a b s t r a c t Randomly packed columns are key equipment for natural gas pre-treatment process of the FLNG (Floating Liquefied Natural Gas). It is of great importance to study on the effect of sloshing on liquid flow maldistribution at the column tray under severe sea state. A three-dimensional CFD model for randomly packed column under sloshing conditions was developed and the inter-phase drag force, porous resistance force, liquid dispersion and radial variation of volume porosity were incorporated into the volume averaged equations. Then the simulation results were validated with the experimental data. Both static and sloshing simulations have been carried out with varying angles and aspect ratios to map the distortion of liquid distribution. Quantitative analysis of liquid volume fraction, liquid velocity and maldistribution factor affected by tilt angles and aspect ratios were provided. The duration time of the dry area occupies about 40% of the period in the wall region and the total dry area occupies about 10% of the cross-sectional area. The maximum tilt angle should be less than 3° to prevent severe liquid flow maldistribution of the column with the aspect ratio of 3 (L/D = 3). Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Growing global demand for natural gas is pushing the industry to consider the development of remote offshore fields. Floating Liquefied Natural Gas (FLNG) offers a cost effective alternative, and is expected to be the next technological breakthrough for monetizing remote, offshore natural gas resources [1]. Packed columns have been used in acid gas removal, NGL recovery, and natural gas fraction on the FLNG vessel. However, these highest and heaviest equipment in natural gas liquefaction process (especially the pressure of feed gas is high) are easily influenced by the movements of hull [2]. In typical sloshing conditions under the severe sea state, there are three angular motions (roll, pitch and yaw) and three linear ⇑ Corresponding author. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.10.049 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
motions (surge, sway and heave). Angular motions divert the liquid from the vertical and cause liquid maldistribution close to the column wall especially with periods longer than 12 s. Horizontal accelerations affect the liquid distribution necessary to obtain the maximum column efficiency due to inertia effects, whereas vertical accelerations have a greater influence on the maximum capacity of the packed column. Static tilt causes liquid maldistribution similar as angular motions due to weather conditions or unbalanced product storage or offload conditions [3]. However, Cullinane et al. [4] illustrates faster motion dampens liquid maldistribution until inertial effects are more important than gravity and the gravity offset by inertia effects help to get the minimum liquid maldistribution of packed column. Up to now, many researchers such as Yin et al. [5,6], Sun et al. [7] and Liu et al. [8] have done studies on liquid flow maldistribution of randomly packed columns by experimental and CFD simulation study, which can predict the liquid flow distribution,
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Nomenclature as B C1 C2 Cl D dp de Fgl Fls Fgs Fpd Ffactor Gf Gg Gl g h Kc k L Lf M mstatic msloshing P0 rp rpd rpw R Rz r
specific surface area (1/m) body force vector (N/m3) constants in Robbins’s correlation (dimensionless) constants in Robbins’s correlation (dimensionless) empirical constant in Eq. (17) (dimensionless) column diameter (m) nominal diameter of packing particles (m) equivalent diameter of the packing (m) inter-phase drag force (N/m3) porous resistance by the porous medium to the liquid phase (N/m3) porous resistance by the porous medium to the gas phase (N/m3) packing factor (1/m) gas kinetic factor (Pa0.5) gas loading factor (kg/(m2 s)) gas superficial flow rate per unit cross-sectional area (kg/(m2 s)) liquid superficial flow rate per unit cross-sectional area (kg/(m2 s)) gravitational vector (kg/(m2 s)) holdup (dimensionless) constant in Eq. (14) (m2 s) turbulence kinetic energy (m2/s2) packed bed height (m) liquid loading factor (kg/(m2 s)) maldistribution factor (dimensionless) mass flux under static conditions (kg/s) mass flux under sloshing conditions (kg/s) operating pressure (Pa) pressure drop (Pa/m) dry pressure drop (Pa/m) wet pressure drop (Pa/m) radius of the column (m) axial resistance component (N/m3) radial coordinate (m)
pressure drop, concentration profile and column efficiency. Duss et al. [3] summarized the factors that influence liquid maldistribution and mass transfer: column diameter, bed height, packing type, hydraulic loads, physical properties, liquid distribution design, static tilt, motion conditions and the location of the column on the vessel. Jafari et al. [9] carried out numerical study on flow behavior through random packing of non-overlapping spheres. Dimensionless pressure drop was studied at different Reynolds numbers through randomly packed bed based on pore permeability and interstitial fluid velocity. Liu et al. [10] predicted the concentration and liquid velocity distribution as well as the turbulent mass transfer diffusivity in an industrial scale randomly packed distillation column based on c2 ec numerical model. Fourati et al. [11] proposed a gas-liquid porous media CFD model incorporated with the porous resistance, inter-phase momentum transfer and liquid dispersion. Pham et al. [12] developed the gas-liquid porous media CFD model integrated with mass transfer and chemical reaction in an amine absorber with Mellapak 500.X. TOTAL, PROSERNAT and IEPEN at Heriot-Watt University [13] built an early model of an absorption column subjected to motion based on theoretical approach and presented a summary of experimental measurement of liquid distribution with 5° at different heights of packing in the 0.6 m column. Pham et al. [14] developed a gas-liquid Eulerian porous media CFD model for an absorber with structured packing to remove CO2 from natural gas by mono-
T T0 t U Uslip u ug uav
period (s) operating temperature (K) time (s) interstitial velocity vector (m/s) slip velocity vector (m/s) liquid local superficial velocity (m/s) gas superficial velocity (m/s) liquid average superficial velocity (m/s)
Greek symbols e turbulence dissipation rate (m2/s3) eb bulk porosity of the column (dimensionless) ep porosity (dimensionless) c volume fraction (dimensionless) b the variation of flux ratio (dimensionless) / volume averaged variable /q intrinsic phase averaged variable q density (kg/m3) x angular velocity (1/s) C dispersion coefficient for volume fraction (kg/(m s)) h deviation angle of the column axis from the vertical axis (°) hmax the amplitude of the angular motion (°) l molecular viscosity (kg/(m s)) le effective viscosity (kg/(m s)) lt turbulent viscosity (kg/(m s)) rt turbulent Prandtl number (dimensionless) Subscripts g gas phase l liquid phase q phase index s solid phase t turbulent flow
ethanol-amine (MEA). As tilt angle increased, the liquid holdup and effective interfacial area decreased and CO2 removal efficiency was lowered. The uniformity of liquid holdup deteriorated by 10% for a 3° static tilt, and a rolling motion with 4.5° amplitude and 12 s period, respectively. CFD modeling generally confirms that same angle static tilt is the most severe design condition with the highest maldistribution factors [1,14]. Li et al. [15] investigated the effect of sloshing orientations, angles, frequencies and displacements on sloshing resistance and gas-liquid distribution performance in plate-fin heat exchangers. A mathematical model of FLNG spiral wound heat exchangers under rolling conditions was developed and the effect of rolling amplitude on the heat transfer performance was analyzed [16]. Experimental and theoretical studies of single-phase natural circulation flow and heat transfer under a rolling motion condition were performed [17]. However, there is still limited information and industrial experience available related to hydrodynamics and maldistribution of columns subjected to sloshing. It is necessary to do further research on the effect of sloshing on liquid flow distribution in randomly packed column to obtain critical conditions for column design to guarantee column performance under severe sea state. The gas-liquid porous media CFD model was developed to study on the effect of sloshing on liquid maldistribution of the randomly packed column. Under static conditions, effect of tilt angles and aspect ratios on gas-liquid distribution performance were
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quantitatively discussed. Different packed column depths were further discussed to analyze cross-sectional liquid maldistribution. Under sloshing conditions, the fluctuation of flux ratio along the column diameter was obtained and the duration time and percentage of the dry area were supplied.
2. Randomly packed column The randomly packed column packed with 25.4 mm stainless steel Pall rings was used for the CFD simulation and geometric characteristics of the packing were presented in Table 1. The dimension of the randomly packed column was obtained from the literature [18] with 0.6 m in diameter and 1.8 m in height. As shown in Fig. 1, the liquid is injected at the top of the column through the whole part to ensure the initial uniform liquid distribution, and the gas enters from the bottom of the column in order to facilitate counter-current calculations. The center of rotation is situated at the center of the packing bottom and h is measured as a deviation of the column axis from the vertical axis. Fig. 1 also shows the mesh structure of the packed column. The hexahedral cells were built in the porous zone with refined meshes near the wall. The coarse, medium, and fine meshes (9000, 21,000, and 50,000 cells, respectively) were implemented to the porous zone in order to investigate the effect of the mesh number on the simulation results.
3. Mathematical modeling development The following assumptions are given below: (1) The fluid flow in column is assumed to be isothermal and incompressible, where both phases are Newtonian, with no mass transfer at the gas-liquid interface and no chemical reaction. (2) The porosity (ep) of packing is anisotropic. (3) The sloshing can be assumed to be regular harmonic wave motion, which means it is sinusoidal and centered at a vertical position.
Fig. 1. Mesh structure in CFD calculation domain.
!
(1) Three additional momentum terms for the porous zone (F gl , 3.1. Fluid dynamic equations
!
A gas-liquid Euler-Euler model was developed to solve the mass and momentum balances and the volume averaging procedure was applied to Navier-Stokes equations to predict the hydrodynamic characteristics in randomly packed column. However, there was a lot of information lost in the averaging process. Therefore, the constitutive parameters in the model should be clearly determined and the complicated two-phase flow behavior characteristics were modeled using several user-defined functions (UDFs) as follows:
Parameters
Value
Robbins coefficients
C1 = 4.002 102, C2 = 1.99 102 Fpd = 174 (for 25.4 mm Pall rings) as = 207 (for 25.4 mm Pall rings) eb = 0.94 (for 25.4 mm Pall rings) Kc = 2.9 103, r = 1.0 102
Specific surface area (m2/m3) Bulk porosity of the column (m3/m3) Modification factors of dispersion coefficient Densities of gas and liquid (kg/m3) Viscosities of gas and liquid [kg/(m s)]
The volume-averaged fluid dynamic equations for the gas and liquid phases are given as follows. Continuity Equation: ! @ðep cq qq Þ þ r fep ðcq qq Uq Cq rcq Þg ¼ 0 q ¼ 1; g @t
Table 1 Porous media model parameters.
Packing factor (m1)
!
F ls , F gs ) in Eqs. (7), (12) and (13) (2) Dispersion coefficient for volume fraction (C) in Eqs. (15) and (16) (3) Radial variation of volume porosity (ep) in Eq. (20) (4) Angular velocity of ship sloshing in Eq. (23)
qg = 1.2, ql = 1.0 103 lg = 1.79 105, ll = 1.0 103
ð1Þ
Momentum Equation: !
! ! ! ! T @ðep cq qq Uq Þ þ r fep cq ½qq Uq Uq leq ðrUq þ ðrUq Þ Þg @t !
!
¼ ep cq ðBq rpÞ þ F q
q ¼ 1; g:
ð2Þ
where ep is the porosity, c is the volume fraction, q is the fluid density, le is the effective viscosity of the liquid phase, C is the disper!
sion coefficient for volume fraction, U is the interstitial velocity
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M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594 !
!
vector, B is the body force vector, p is the pressure, F is the interphase drag force vector, and q is the phase index.
!
3.2. Inter-phase drag force F
F gs
For irrigated packing, the pressure drop Dp is greater than that of the dry bed Dpd and Dpw represents the increased pressure drop created by the interfacial drag force, which can be described by the Robbins correlation.
Dpd ¼ C 1 G2f 10C2 Lf
ð3Þ
Dpw ¼ 0:774
Lf 20; 000
0:1 ðC 1 G2f 10C2 Lf Þ
4
ð4Þ
where Gf is the gas loading factor, Lf is the liquid loading factor.
(
Gf ¼
Gg ð1:2=qg Þ0:5 ðF pd =65:62Þ0:5
P 6 1atm 0:0187qg
Gg ð1:2=qg Þ ðF pd =65:62Þ 10 0:5
( Lf ¼
0:5
Gl ð1000=ql ÞðF pd =65:62Þ0:5 l0:2 l Gl ð1000=ql Þð65:62=F pd Þ
0:5
l
0:1 l
P > 1atm
F pd P 15 F pd < 15
ð5Þ
ð6Þ
F gl ¼
DP w !
jUslip j
!
!
ð7Þ
Uslip
!
!
jUslip j ¼ jUl Ug j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU l U g Þ2 þ ðV 1 V g Þ2 þ ðW l W g Þ2
ð8Þ
!
where Uslip is the slip velocity. A REV (Representative Elementary Volume) is defined as a minimum volume within which measurable variables (velocity, density, etc.) become continuum quantities inside a porous media. The variable in the fluid can be averaged over the entire volume V of REV and the partial volume of each phase Vl or Vg. The two volume averaged quantities / and /q are related by the holdup of each phase, / = hq/q. The gas interstitial velocity and liquid interstitial velocity are defined as: !
jUg j ¼
Gg
g
P > 1atm
ð13Þ
3.4. Dispersion coefficient for volume fraction C According to Yin et al. [18], the dispersion coefficient for volume fraction is linearly proportional to the adverse gradient of the resistance along the direction of liquid main flow and can be expressed as:
C ¼ K c rRz
ð14Þ
where Kc is a proportionality constant that depends upon the structure and size of the packing. For the liquid phase, the inertial resistance is the major porous resistance to flow in the packed column under the usual operating conditions. Then C is determined by taking the inertial term of Eq. (12) to replace Rz :
C ¼ 1:75K c
!
!
qj U j U ð1 eÞqj U j ! re 3:5K c rU 2 de e ede
ð15Þ
where the first term presents the effect of the spatially porosity on liquid spreading, and the second term shows the effect of the nonuniform initial liquid distribution. Another important factor should be considered is the unstable turbulence flow under the usual operating conditions. The turbulent dispersion coefficient Ct is obtained using the eddy diffusivity hypothesis.
Ct ¼
lt I rt
lt ¼ qC l
ð16Þ 2
k
ð17Þ
e
where rt is the turbulent Prandtl number. The turbulent viscosity lt can be solved with the standard k-e model where Cl is an empirical constant, k is the turbulence kinetic energy and e is the dissipation rate. For the gas phase, the main turbulent dispersion is calculated in the same way as the liquid phase replacing the liquid turbulent viscosity with gas turbulent viscosity.
Gl
ð10Þ
ep ql cl
The standard k-e model consists of the following equations: !
!
Body force includes the gravitational acceleration and the porous resistances defined as follows: !
!
ð11Þ
For the liquid phase, the porous resistance was derived from a generalized Ergun correlation.
150ð1 ep Þ2 l
e
2 de 2p
!
U
1:75ð1 ep Þq ! ! jUjU de ep
ð12Þ
where the first term is the viscous resistance, the second term is the inertial resistance, de is the equivalent diameter of the random packing, de = 6(1 ep)/as, and as is the specific surface area of the packing.
lt rk re
! ! ! T ¼ hlt r U ðr U þðr UÞ Þ qhe !
!
Bq ¼ qq g þF qs
F ls ¼
r ðqh U kÞ r h l þ
3.3. Body force B
!
P 6 1atm
3.5. Turbulence model
!
jUl j ¼
ð9Þ
ep qg cg
8 F pd > < C 1 e2p c2g q2g 1:2 j U! j U! qg 65:62 ¼ F pd > : C 1 e2p c2g q2g 1:2 100:0374qg j U! j U! q 65:62
!
where Gg is the gas superficial flow rate per unit cross-sectional area, Gl is the liquid superficial flow rate per unit cross-sectional area. The inter-phase drag force can be determined by: !
For the gas phase, the porous resistance can be modeled from the dry pressure drop part of the Robbins equation.
r ðqh U eÞ r h l þ
lt re re
ð18Þ
! ! ! T e e2 ¼ c1 hlt r U ðr U þðr UÞ Þ c2 qh k k
ð19Þ
The constants in the above equations are set to be: Cl = 0.09,
rk = 1.0, re = 1.0, c1 = 1.44 and c2 = 1.92. 3.6. Geometric relations
Referring to Sun et al. [7], the porosity of randomly packed column with Pall rings is determined by:
M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594
ep ¼ 1 ð1 eb Þf1 expð2ððR rÞ=dp Þ2 Þg
ð20Þ
where eb is the bulk porosity of the column, R is the radius of the column, r is the position in radial direction, dp is the nominal diameter of packing particles. Fig. 2 shows the radial variation of volume porosity in the packed column. From the figure, it can be seen that the radial variation of volume porosity occurs within one packing element diameter from the wall and in the bulk region the porosity is constant. 3.7. The sloshing conditions The sloshing conditions cause a serious liquid maldistribution and a decreased column performance in the offshore packed column. The moving mesh method is used to perform the sloshing of the packed column. There are two kinds of velocity vector. !
!
One is the fluid velocity such as Ug and Ul in the stationary refer!
ence frame. The other is v mesh , the mesh velocity arising from the ship motion defined as the outer product of the angular velocity !
!
(x) and the position vector ( r ). !
!
!
v mesh ¼ x r
h ¼ hmax cos
x ¼ hmax
ð21Þ 2p p tþ T 2
p 2p 180
T
sin
2p p tþ T 2
ð22Þ
ð23Þ
where j and k are the direction vectors of y and z axes and z is the vertical axis of the randomly packed column, hmax is the amplitude of the angular motion, T is the period.
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4. Model parameters and boundary conditions Table 1 shows the porous media model parameters. The Robbins coefficients C1 and C2 were reported by Robbins [19]. The packing factor, specific surface area and bulk porosity of the column were the geometric characteristics of 25.4 mm Pall rings. The modification factors of dispersion coefficient of the water-air system were recommended by Sun et al. [7]. The values of Kc and rt were determined in order to minimize the deviations between the predicted and measured liquid velocity profiles at different bed heights. The viscosities and densities were given for waterair system. In the present study, the experimental data was obtained from Yin et al. [18]. in order to validate the simulation model. Table 2 presents the boundary conditions and simulation parameters. The multiphase Eulerian model was calculated by the steady-state, pressure-based, and implicit solver. The SIMPLE algorithm was chosen for pressure-velocity coupling and the second-order upwind spatial discretization scheme was used for all differential equations. For counter-current multiphase flow, it is possible to have flows that enter and leave the same flow boundary simultaneously. Therefore, the liquid inlet boundary condition is used for the gas outlet boundary condition and the gas inlet boundary condition is used for the liquid outlet boundary condition. The top-side boundary condition (BC) was set to the velocity-inlet and the bottom-side boundary condition (BC) was set to the mass-flow-inlet for gas and liquid. The domain was limited by the asymmetry boundary condition and the wall boundary condition. The negative velocity represents the opposite direction. The packed column was operated at P0 = 1 atm and T0 = 288.16 K. 5. Results and discussions The Eulerian porous media CFD model was first performed on three different mesh number of the porous zone. Then the simulation results were validated with experimental data such as the liquid relative velocity, liquid volume fraction and pressure drop. Finally, both static and sloshing simulations were carried out with varying angles and aspect ratios. 5.1. Mesh independent test The coarse, medium and fine meshes (9000, 21,000, and 50,000 cells, respectively) were built in the porous zone. The effect of the mesh number for the porous zone was performed on the liquid relative velocity in the radial direction, as shown in Fig. 3. Liquid relative velocity is defined as u/uav, where u is the liquid local superficial velocity and uav is the liquid average superficial velocity over the column cross section.
Table 2 Boundary conditions (BC) and simulation parameters. Parameters
Value
Mesh member Eulerian multiphase model Top-side BC
About 9000, 21,000, and 50,000 cells Steady state, pressure-based, and implicit solver
Bottom-side BC
Fig. 2. Radial variation of volume porosity.
Wall BC Axis BC
Gas outlet (gas velocity = 0.7 m/s), liquid inlet (liquid velocity = 0.082 m/s), liquid volume fraction = 6.2 102, 1 atm and 288.16 K Gas inlet (gas mass flux = 0.75 kg/(m2s), liquid outlet (liquid mass flux = 4.78 kg/(m2 s), liquid volume fraction = 6.2 102, 1 atm and 288.16 K Non-slip and no penetration Asymmetry
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As shown in Fig. 5, the liquid volume fraction remains constant in the bulk region of the column, however, the liquid volume fraction increases significantly near the wall and the peak value reaches about 9.3%. Fig. 6 shows the radial distribution of the liquid velocity. A higher liquid velocity can be seen in the wall region due to the higher liquid volume fraction, however, the adverse velocity gradient will prevent further spreading of liquid towards the wall. Fig. 7 shows the pressure drop per unit length versus the Ffactor pffiffiffiffiffiffi (gas kinetic factor) defined as ug qg , where ug is the gas superficial velocity. The simulation results are in good agreement with the experimental data with the maximum relative error of 8.3%. However, the relative error gets larger with the Ffactor increases due to the gas-liquid flow situation close to the flooding point under the operating condition.
5.3. Effect of tilt angles Fig. 3. Mesh independent test on coarse, medium and fine meshes.
From the figure, the numerical results were not sensitive to the mesh number due to the simple structure and the porous media in randomly packed column. However, the liquid relative velocity above 0.1 m in the radial direction on the coarse mesh shows relatively lower than those of the medium and fine meshes. Finally, the medium mesh was selected from the mesh independence test as a compromise between the accuracy and computational efficiency.
In this study, four tilt angles (1°, 3°, 6°, 9°) were analyzed to investigate the effect of tilt angles on gas-liquid distribution performance in randomly packed columns. Different packed column depths (0.1 m, 0.4 m, 0.8 m, 1.2 m, 1.6 m) were further discussed to analyze cross-sectional liquid maldistribution.
5.2. Validity of simulation model Fig. 4 shows the comparison of the predicted liquid relative velocity with experimental data (water-air system) for the uniform liquid inlet distribution in the packed column. The prediction results agree well with the experiment with the maximum relative error of 7.8%. Due to the constant porosity in the center region of the column, only a little variation in the liquid velocity profile can be seen. However, there is a slight drop and then a significant increase by about 9% near the wall. The liquid spreading towards the non-irrigated zone can easily cause the accumulation of liquid near the wall due to the non-uniform packed-bed structure and the radial variation of volume porosity as well as the no-slip boundary at the column wall surface, known as the wall effect [10].
Fig. 4. Verification of liquid relative velocity distribution.
Fig. 5. Radial distribution of liquid volume fraction.
Fig. 6. Radial liquid velocity contours.
M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594
Fig. 7. Comparison of the predicted pressure drop with experimental data.
Fig. 8 shows the liquid volume fraction distribution along the packed column height under different static tilt conditions. In order to show the effect of tilt angles on cross-sectional liquid maldistribution in the packed column clearly, the maximum liquid volume fraction in the legend was set to 0.05. From the figure, the liquid maldistribution becomes more severe as tilt angle increases and deeper bed experiences more severe maldistribution especially at the base of the column. In the tilted column, the crosssectional area can be divided into two parts with liquid overloading on the tilted side and dry area of the opposite side. Fig. 9 quantifies the effect of tilt angles on the liquid volume fraction distribution along the column height (0.1 m, 0.4 m, 0.8 m, 1.2 m, 1.6 m). The liquid maldistribution becomes more severe as tilt angle increases, for example, the maximum liquid vol-
591
Fig. 9. Maximum liquid volume fraction along the column height (+1°, 3°, 6°, 9° tilt angle).
ume fraction along the column height is less than 6% for a tilt angle of 1° and the value is more than 5% for a tilt angle of 9°. The maximum liquid volume fraction decreases as packed column depths increases, therefore, the bottom of the packed column is considered to be the area with the most severe liquid maldistribution. The peak value of the maximum liquid volume fraction reaches 56.1% at the bottom of the column for a tilt angle of 9°, which is 50 times more than the averaged value of the vertical column. Fig. 10 shows the effect of tilt angles on liquid velocity distribution of the bottom section. As tilt angle increases, the dry area in which the void is occupied by vapor increases. For a static tilt of 9°, the area with the liquid velocity around 0.01 m/s occupies
Fig. 8. Contours of liquid volume fraction along the column height (+1°, 3°, 6°, 9° tilt angle).
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M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594
Fig. 10. Effect of tilt angles on contours of liquid velocity.
about 13% of the total cross-sectional area and the insufficient gasliquid interfacial area will contribute to a drop in mass transfer performance. Another way to represent flow non-uniformity is the maldistribution factor defined as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u1 Xn u uav 2 t M¼ n¼1 n uav
ð24Þ
Fig. 11 provides the effect of tilt angles on the maldistribution factor of liquid velocity at the base of the column. As tilt angle increases, the maldistribution factor of liquid velocity significantly increases and the value increases by almost 40% for a static tilt of 9°. The tendency of the value corresponds to the polynomial function to extrapolate to various tilt angles in typical sloshing conditions under the severe sea state. 5.4. Effect of aspect ratios In this study, five aspect ratios (L/D = 1.5, 2, 3, 4, 5) were analyzed to investigate the effect of aspect ratios on gas-liquid distribution performance in randomly packed columns taking a 6° tilt angle as an example. The aspect ratio of the packed column (L/D) is likely to have a greater influence on liquid flow maldistribution. Column and bed size can vary widely depending on the process and design philosophy. In general, industrial sized columns for conventional oil and gas production are 1–7 m in diameter and 2.5–8 m in heights [4]. At a static tilt angle and column diameter, higher packed bed height will experience more severe maldistribution. As shown in Fig. 12, the maldistribution factor of liquid velocity increases slightly as the aspect ratio (L/D) increases and the ten-
Fig. 11. Effect of tilt angles on maldistribution factor of liquid velocity (L/D = 3).
dency corresponds to the polynomial function to extrapolate results to larger industrial columns. A rule that the aspect ratio of the packed column should be less than 2 (L/D 6 2) provides guidance for liquid distribution quality under static conditions [4]. At the aspect ratio of 2 (L/D = 2) of the packed column, the maldistribution factor of liquid velocity is 3.6 for a 6° tilt angle and based on this value, the maximum tilt angle should be less than 3° to prevent the severe liquid flow maldistribution of the column with the aspect ratio of 3 (L/D = 3).
M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594
Fig. 12. Effect of packed bed heights on maldistribution factor of liquid velocity (+6° tilt angle).
5.5. Effect of sloshing conditions Fig. 13 shows the row mapping for liquid maldistribution tests. Dimensionless b shows the variation of flux ratio in different rows under static and sloshing conditions at the bottom of the column. b should be -100% if there is no flux under sloshing conditions and should be 0 if the flux ratio under sloshing conditions is equal to the flux ratio under static conditions. The area is supposed to be not sensitive to the sloshing conditions if b is close to 0.
b¼
msloshing 1 100% mstatic
Fig. 14. Variation of flux ratio along the column diameter (T = 8 s, hmax = 6°). Row 10 r = 0.03 m; Row 6 r = 0.15 m; Row 16 r = 0.18 m; Row 19 r = 0.27 m.
of flux ratio below 25% can be seen in the center of the packed column. However, the flux ratio close to the column’s edge can be 3 times more than that under static condition, which is more sensitive than the center of the column. The affected time and area were provided in different rows along the column diameter. The flux ratios in row 16 and row 19 both reach -100%, which means the area is occupied by vapor. The duration time of the dry area occupies 24.8% in Row 16 and 40% in Row 19 of the period and the total dry area occupies about 10% of the cross-sectional area. The affected area is defined as jbj 6 50%, which occupies more than 40% of the cross-sectional area in the packed column.
ð25Þ
where msloshing is the mass flux under sloshing condition and mstatic is the mass flux under static condition. Fig. 14 shows the variation of flux ratio along the column diameter under sloshing condition (T = 8 s, hmax = 6°). Small fluctuation
Fig. 13. Row mapping for liquid maldistribution tests.
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6. Conclusion A three-dimensional CFD model for randomly packed column with Pall rings under sloshing conditions was developed and mesh independent test was performed. Both static and sloshing conditions were carried out with varying angles and aspect ratios to map the distortion of liquid distribution and the following conclusions can be made. (1) The gas-liquid porous media CFD model was developed to study the effect of sloshing on liquid maldistribution, liquid volume fraction and fractional dry area in randomly packed column. In order to accurately predict the hydrodynamic features, the inter-phase drag force, porous resistance force, liquid dispersion and radial variation of volume porosity were included. (2) Effect of tilt angles (+1°, 3°, 6°, 9°) on liquid volume fraction distribution along the column height was analyzed quantitatively. The liquid maldistribution becomes more severe as tilt angle increases and the maximum liquid volume fraction decreases as packed column depths increases. The peak value of the maximum liquid volume fraction reaches 56.1% at the bottom of the column for a tilt angle of 9°, which is 50 times more than the averaged value of the vertical column. (3) Effect of tilt angles and packed bed heights on maldistribution factor of liquid velocity at the bottom of the column were analyzed and the polynomial functions were given to extrapolate results in typical sloshing conditions. The maximum tilt angle should be less than 3° to prevent the severe liquid flow maldistribution of the column with the aspect ratio of 3 (L/D = 3).
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(4) The center region is less sensitive than the wall region affected by sloshing conditions and the affected time and area were provided. The duration time of the dry area occupies 24.8% in Row16 and 40% in Row19 of the period and the total dry area occupies about 10% of the cross-sectional area. The affected area is defined as jbj 6 50%, which occupies more than 40% of the cross-sectional area in the packed column.
Acknowledgements The paper is supported by Important National Science and Technology Major Project of China (2011ZX05026-006-07). References [1] Z. Kayat, M. Schott, S. Doong, R. Subris, D. Farr, Pretreatment of Acid Gas in feed for Petronas floating LNG facility (2012) 1–11. [2] W. Wang, Y. Li, L. Zang, B. Xie, X. Yu, Offshore Applicability Analysis of the LNGFPSO Production Process, in: The Twentieth International Offshore and Polar Engineering Conference, International Society of Offshore and Polar Engineers, 2010, pp. 466–469. [3] M. Duss, M. Roza, Columns subject to motion: Maldistribution sensitivity and susceptibility, in: 10th International Conference on Distillation & Absorption, Friedrichshafen, Germany, 2014, pp. 14–17. [4] J.T. Cullinane, N. Yeh, E. Grave, Effects of tower motion on packing efficiency, in: Brasil Offshore Conference and Exhibition. Macaé, Brazilpp, SPE: 143766, pp. 1–14. [5] F.H. Yin, A. Afacan, K. Nandakumar, K.T. Chuang, CFD simulation and experimental study of liquid dispersion in randomly packed metal pall rings, Chem. Eng. Res. Des. 80 (2) (2002) 135–144.
[6] F. Yin, Z. Wang, A. Afacan, K. Nandakumar, K.T. Chuang, Experimental studies of liquid flow maldistribution in a random packed column, Can. J. Chem. Eng. 78 (3) (2000) 449–457. [7] C.G. Sun, F.H. Yin, A. Afacan, K. Nandakumar, K.T. Chuang, Modelling and simulation of flow maldistribution in random packed columns with gas-liquid countercurrent flow, Chem. Eng. Res. Des. 78 (3) (2000) 378–388. [8] S. Liu, J. Long, Gas-liquid countercurrent flows through packed towers, J. Porous Media 3 (2) (2000) 99–113. [9] A. Jafari, P. Zamankhan, S.M. Mousavi, K. Pietarinen, Modeling and CFD simulation of flow behavior and dispersivity through randomly packed bed reactors, Chem. Eng. J. 144 (3) (2008) 476–482. [10] G.B. Liu, K.T. Yu, X.G. Yuan, C.J. Liu, A numerical method for predicting the performance of a randomly packed distillation column, Int. J. Heat Mass Transf. 52 (23–24) (2009) 5330–5338. [11] M. Fourati, V. Roig, L. Raynal, Liquid dispersion in packed columns: experiments and numerical modeling, Chem. Eng. Sci. 100 (2013) 266–278. [12] D.A. Pham, Y.I. Lim, H. Jee, E. Ahn, Y. Jung, Porous media Eulerian computational fluid dynamics (CFD) model of amine absorber with structured-packing for CO2 removal, Chem. Eng. Sci. 132 (2015) 259–270. [13] G. White, C. Weiss, E. Huguet, J. Magné-Drisch, P. Alix, G. Perdu, C. Salais, How Waves Can Significantly Impact Performance of Amine Unit Installed on a FLNG?, Offshore Technology Conference 599–601 (1) (2014) 1871–1874 [14] D.A. Pham, Y.I. Lim, H. Jee, E. Ahn, Y. Jung, Effect of ship tilting and motion on amine absorber with structured-packing for CO2 removal from natural gas, AIChE J. 61 (12) (2015) 4412–4425. [15] Y. Li, Y. Li, Q. Hu, W. Wang, B. Xie, X. Yu, Sloshing resistance and gas–liquid distribution performance in the entrance of LNG plate-fin heat exchangers, Appl. Therm. Eng. 82 (2015) 182–193. [16] T. Wang, G. Ding, T. Ren, J. Chen, H. Pu, A mathematical model of floating LNG spiral-wound heat exchangers under rolling conditions, Appl. Therm. Eng. 99 (2016) 959–969. [17] S. Tan, G. Su, P. Gao, Experimental and theoretical study on single-phase natural circulation flow and heat transfer under rolling motion condition, Appl. Therm. Eng. 29 (14–15) (2009) 3160–3168. [18] F. Yin, Liquid Maldistribution and Mass Transfer Efficiency in Randomly Packed Distillation Columns (Ph.D. thesis), University of Alberta, 2000. [19] L.A. Robbins, Improve pressure-drop prediction with a new correlation, Chem. Eng. Prog. 87 (1991) 87–91.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical simulations on the effect of sloshing on liquid flow maldistribution of randomly packed column Mengxian Zhang a, Yuxing Li a,⇑, Yan Li b, Hui Han a, Lin Teng a a b
Provincial Key Laboratory of Oil and Gas Storage and Transportation Safety in Shandong Province, China University of Petroleum, Qingdao 266580, China CNOOC Research Center, Beijing 100149, China
h i g h l i g h t s A model for randomly packed column under sloshing conditions is developed. The model predictions are validated with the experimental data. Both static and sloshing conditions are considered. Effect of tilt angles and aspect ratios are discussed.
a r t i c l e
i n f o
Article history: Received 6 February 2016 Revised 1 August 2016 Accepted 9 October 2016 Available online 11 October 2016 Keywords: Randomly packed column Numerical simulation Sloshing Liquid distribution
a b s t r a c t Randomly packed columns are key equipment for natural gas pre-treatment process of the FLNG (Floating Liquefied Natural Gas). It is of great importance to study on the effect of sloshing on liquid flow maldistribution at the column tray under severe sea state. A three-dimensional CFD model for randomly packed column under sloshing conditions was developed and the inter-phase drag force, porous resistance force, liquid dispersion and radial variation of volume porosity were incorporated into the volume averaged equations. Then the simulation results were validated with the experimental data. Both static and sloshing simulations have been carried out with varying angles and aspect ratios to map the distortion of liquid distribution. Quantitative analysis of liquid volume fraction, liquid velocity and maldistribution factor affected by tilt angles and aspect ratios were provided. The duration time of the dry area occupies about 40% of the period in the wall region and the total dry area occupies about 10% of the cross-sectional area. The maximum tilt angle should be less than 3° to prevent severe liquid flow maldistribution of the column with the aspect ratio of 3 (L/D = 3). Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Growing global demand for natural gas is pushing the industry to consider the development of remote offshore fields. Floating Liquefied Natural Gas (FLNG) offers a cost effective alternative, and is expected to be the next technological breakthrough for monetizing remote, offshore natural gas resources [1]. Packed columns have been used in acid gas removal, NGL recovery, and natural gas fraction on the FLNG vessel. However, these highest and heaviest equipment in natural gas liquefaction process (especially the pressure of feed gas is high) are easily influenced by the movements of hull [2]. In typical sloshing conditions under the severe sea state, there are three angular motions (roll, pitch and yaw) and three linear ⇑ Corresponding author. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.10.049 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
motions (surge, sway and heave). Angular motions divert the liquid from the vertical and cause liquid maldistribution close to the column wall especially with periods longer than 12 s. Horizontal accelerations affect the liquid distribution necessary to obtain the maximum column efficiency due to inertia effects, whereas vertical accelerations have a greater influence on the maximum capacity of the packed column. Static tilt causes liquid maldistribution similar as angular motions due to weather conditions or unbalanced product storage or offload conditions [3]. However, Cullinane et al. [4] illustrates faster motion dampens liquid maldistribution until inertial effects are more important than gravity and the gravity offset by inertia effects help to get the minimum liquid maldistribution of packed column. Up to now, many researchers such as Yin et al. [5,6], Sun et al. [7] and Liu et al. [8] have done studies on liquid flow maldistribution of randomly packed columns by experimental and CFD simulation study, which can predict the liquid flow distribution,
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Nomenclature as B C1 C2 Cl D dp de Fgl Fls Fgs Fpd Ffactor Gf Gg Gl g h Kc k L Lf M mstatic msloshing P0 rp rpd rpw R Rz r
specific surface area (1/m) body force vector (N/m3) constants in Robbins’s correlation (dimensionless) constants in Robbins’s correlation (dimensionless) empirical constant in Eq. (17) (dimensionless) column diameter (m) nominal diameter of packing particles (m) equivalent diameter of the packing (m) inter-phase drag force (N/m3) porous resistance by the porous medium to the liquid phase (N/m3) porous resistance by the porous medium to the gas phase (N/m3) packing factor (1/m) gas kinetic factor (Pa0.5) gas loading factor (kg/(m2 s)) gas superficial flow rate per unit cross-sectional area (kg/(m2 s)) liquid superficial flow rate per unit cross-sectional area (kg/(m2 s)) gravitational vector (kg/(m2 s)) holdup (dimensionless) constant in Eq. (14) (m2 s) turbulence kinetic energy (m2/s2) packed bed height (m) liquid loading factor (kg/(m2 s)) maldistribution factor (dimensionless) mass flux under static conditions (kg/s) mass flux under sloshing conditions (kg/s) operating pressure (Pa) pressure drop (Pa/m) dry pressure drop (Pa/m) wet pressure drop (Pa/m) radius of the column (m) axial resistance component (N/m3) radial coordinate (m)
pressure drop, concentration profile and column efficiency. Duss et al. [3] summarized the factors that influence liquid maldistribution and mass transfer: column diameter, bed height, packing type, hydraulic loads, physical properties, liquid distribution design, static tilt, motion conditions and the location of the column on the vessel. Jafari et al. [9] carried out numerical study on flow behavior through random packing of non-overlapping spheres. Dimensionless pressure drop was studied at different Reynolds numbers through randomly packed bed based on pore permeability and interstitial fluid velocity. Liu et al. [10] predicted the concentration and liquid velocity distribution as well as the turbulent mass transfer diffusivity in an industrial scale randomly packed distillation column based on c2 ec numerical model. Fourati et al. [11] proposed a gas-liquid porous media CFD model incorporated with the porous resistance, inter-phase momentum transfer and liquid dispersion. Pham et al. [12] developed the gas-liquid porous media CFD model integrated with mass transfer and chemical reaction in an amine absorber with Mellapak 500.X. TOTAL, PROSERNAT and IEPEN at Heriot-Watt University [13] built an early model of an absorption column subjected to motion based on theoretical approach and presented a summary of experimental measurement of liquid distribution with 5° at different heights of packing in the 0.6 m column. Pham et al. [14] developed a gas-liquid Eulerian porous media CFD model for an absorber with structured packing to remove CO2 from natural gas by mono-
T T0 t U Uslip u ug uav
period (s) operating temperature (K) time (s) interstitial velocity vector (m/s) slip velocity vector (m/s) liquid local superficial velocity (m/s) gas superficial velocity (m/s) liquid average superficial velocity (m/s)
Greek symbols e turbulence dissipation rate (m2/s3) eb bulk porosity of the column (dimensionless) ep porosity (dimensionless) c volume fraction (dimensionless) b the variation of flux ratio (dimensionless) / volume averaged variable /q intrinsic phase averaged variable q density (kg/m3) x angular velocity (1/s) C dispersion coefficient for volume fraction (kg/(m s)) h deviation angle of the column axis from the vertical axis (°) hmax the amplitude of the angular motion (°) l molecular viscosity (kg/(m s)) le effective viscosity (kg/(m s)) lt turbulent viscosity (kg/(m s)) rt turbulent Prandtl number (dimensionless) Subscripts g gas phase l liquid phase q phase index s solid phase t turbulent flow
ethanol-amine (MEA). As tilt angle increased, the liquid holdup and effective interfacial area decreased and CO2 removal efficiency was lowered. The uniformity of liquid holdup deteriorated by 10% for a 3° static tilt, and a rolling motion with 4.5° amplitude and 12 s period, respectively. CFD modeling generally confirms that same angle static tilt is the most severe design condition with the highest maldistribution factors [1,14]. Li et al. [15] investigated the effect of sloshing orientations, angles, frequencies and displacements on sloshing resistance and gas-liquid distribution performance in plate-fin heat exchangers. A mathematical model of FLNG spiral wound heat exchangers under rolling conditions was developed and the effect of rolling amplitude on the heat transfer performance was analyzed [16]. Experimental and theoretical studies of single-phase natural circulation flow and heat transfer under a rolling motion condition were performed [17]. However, there is still limited information and industrial experience available related to hydrodynamics and maldistribution of columns subjected to sloshing. It is necessary to do further research on the effect of sloshing on liquid flow distribution in randomly packed column to obtain critical conditions for column design to guarantee column performance under severe sea state. The gas-liquid porous media CFD model was developed to study on the effect of sloshing on liquid maldistribution of the randomly packed column. Under static conditions, effect of tilt angles and aspect ratios on gas-liquid distribution performance were
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quantitatively discussed. Different packed column depths were further discussed to analyze cross-sectional liquid maldistribution. Under sloshing conditions, the fluctuation of flux ratio along the column diameter was obtained and the duration time and percentage of the dry area were supplied.
2. Randomly packed column The randomly packed column packed with 25.4 mm stainless steel Pall rings was used for the CFD simulation and geometric characteristics of the packing were presented in Table 1. The dimension of the randomly packed column was obtained from the literature [18] with 0.6 m in diameter and 1.8 m in height. As shown in Fig. 1, the liquid is injected at the top of the column through the whole part to ensure the initial uniform liquid distribution, and the gas enters from the bottom of the column in order to facilitate counter-current calculations. The center of rotation is situated at the center of the packing bottom and h is measured as a deviation of the column axis from the vertical axis. Fig. 1 also shows the mesh structure of the packed column. The hexahedral cells were built in the porous zone with refined meshes near the wall. The coarse, medium, and fine meshes (9000, 21,000, and 50,000 cells, respectively) were implemented to the porous zone in order to investigate the effect of the mesh number on the simulation results.
3. Mathematical modeling development The following assumptions are given below: (1) The fluid flow in column is assumed to be isothermal and incompressible, where both phases are Newtonian, with no mass transfer at the gas-liquid interface and no chemical reaction. (2) The porosity (ep) of packing is anisotropic. (3) The sloshing can be assumed to be regular harmonic wave motion, which means it is sinusoidal and centered at a vertical position.
Fig. 1. Mesh structure in CFD calculation domain.
!
(1) Three additional momentum terms for the porous zone (F gl , 3.1. Fluid dynamic equations
!
A gas-liquid Euler-Euler model was developed to solve the mass and momentum balances and the volume averaging procedure was applied to Navier-Stokes equations to predict the hydrodynamic characteristics in randomly packed column. However, there was a lot of information lost in the averaging process. Therefore, the constitutive parameters in the model should be clearly determined and the complicated two-phase flow behavior characteristics were modeled using several user-defined functions (UDFs) as follows:
Parameters
Value
Robbins coefficients
C1 = 4.002 102, C2 = 1.99 102 Fpd = 174 (for 25.4 mm Pall rings) as = 207 (for 25.4 mm Pall rings) eb = 0.94 (for 25.4 mm Pall rings) Kc = 2.9 103, r = 1.0 102
Specific surface area (m2/m3) Bulk porosity of the column (m3/m3) Modification factors of dispersion coefficient Densities of gas and liquid (kg/m3) Viscosities of gas and liquid [kg/(m s)]
The volume-averaged fluid dynamic equations for the gas and liquid phases are given as follows. Continuity Equation: ! @ðep cq qq Þ þ r fep ðcq qq Uq Cq rcq Þg ¼ 0 q ¼ 1; g @t
Table 1 Porous media model parameters.
Packing factor (m1)
!
F ls , F gs ) in Eqs. (7), (12) and (13) (2) Dispersion coefficient for volume fraction (C) in Eqs. (15) and (16) (3) Radial variation of volume porosity (ep) in Eq. (20) (4) Angular velocity of ship sloshing in Eq. (23)
qg = 1.2, ql = 1.0 103 lg = 1.79 105, ll = 1.0 103
ð1Þ
Momentum Equation: !
! ! ! ! T @ðep cq qq Uq Þ þ r fep cq ½qq Uq Uq leq ðrUq þ ðrUq Þ Þg @t !
!
¼ ep cq ðBq rpÞ þ F q
q ¼ 1; g:
ð2Þ
where ep is the porosity, c is the volume fraction, q is the fluid density, le is the effective viscosity of the liquid phase, C is the disper!
sion coefficient for volume fraction, U is the interstitial velocity
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M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594 !
!
vector, B is the body force vector, p is the pressure, F is the interphase drag force vector, and q is the phase index.
!
3.2. Inter-phase drag force F
F gs
For irrigated packing, the pressure drop Dp is greater than that of the dry bed Dpd and Dpw represents the increased pressure drop created by the interfacial drag force, which can be described by the Robbins correlation.
Dpd ¼ C 1 G2f 10C2 Lf
ð3Þ
Dpw ¼ 0:774
Lf 20; 000
0:1 ðC 1 G2f 10C2 Lf Þ
4
ð4Þ
where Gf is the gas loading factor, Lf is the liquid loading factor.
(
Gf ¼
Gg ð1:2=qg Þ0:5 ðF pd =65:62Þ0:5
P 6 1atm 0:0187qg
Gg ð1:2=qg Þ ðF pd =65:62Þ 10 0:5
( Lf ¼
0:5
Gl ð1000=ql ÞðF pd =65:62Þ0:5 l0:2 l Gl ð1000=ql Þð65:62=F pd Þ
0:5
l
0:1 l
P > 1atm
F pd P 15 F pd < 15
ð5Þ
ð6Þ
F gl ¼
DP w !
jUslip j
!
!
ð7Þ
Uslip
!
!
jUslip j ¼ jUl Ug j ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU l U g Þ2 þ ðV 1 V g Þ2 þ ðW l W g Þ2
ð8Þ
!
where Uslip is the slip velocity. A REV (Representative Elementary Volume) is defined as a minimum volume within which measurable variables (velocity, density, etc.) become continuum quantities inside a porous media. The variable in the fluid can be averaged over the entire volume V of REV and the partial volume of each phase Vl or Vg. The two volume averaged quantities / and /q are related by the holdup of each phase, / = hq/q. The gas interstitial velocity and liquid interstitial velocity are defined as: !
jUg j ¼
Gg
g
P > 1atm
ð13Þ
3.4. Dispersion coefficient for volume fraction C According to Yin et al. [18], the dispersion coefficient for volume fraction is linearly proportional to the adverse gradient of the resistance along the direction of liquid main flow and can be expressed as:
C ¼ K c rRz
ð14Þ
where Kc is a proportionality constant that depends upon the structure and size of the packing. For the liquid phase, the inertial resistance is the major porous resistance to flow in the packed column under the usual operating conditions. Then C is determined by taking the inertial term of Eq. (12) to replace Rz :
C ¼ 1:75K c
!
!
qj U j U ð1 eÞqj U j ! re 3:5K c rU 2 de e ede
ð15Þ
where the first term presents the effect of the spatially porosity on liquid spreading, and the second term shows the effect of the nonuniform initial liquid distribution. Another important factor should be considered is the unstable turbulence flow under the usual operating conditions. The turbulent dispersion coefficient Ct is obtained using the eddy diffusivity hypothesis.
Ct ¼
lt I rt
lt ¼ qC l
ð16Þ 2
k
ð17Þ
e
where rt is the turbulent Prandtl number. The turbulent viscosity lt can be solved with the standard k-e model where Cl is an empirical constant, k is the turbulence kinetic energy and e is the dissipation rate. For the gas phase, the main turbulent dispersion is calculated in the same way as the liquid phase replacing the liquid turbulent viscosity with gas turbulent viscosity.
Gl
ð10Þ
ep ql cl
The standard k-e model consists of the following equations: !
!
Body force includes the gravitational acceleration and the porous resistances defined as follows: !
!
ð11Þ
For the liquid phase, the porous resistance was derived from a generalized Ergun correlation.
150ð1 ep Þ2 l
e
2 de 2p
!
U
1:75ð1 ep Þq ! ! jUjU de ep
ð12Þ
where the first term is the viscous resistance, the second term is the inertial resistance, de is the equivalent diameter of the random packing, de = 6(1 ep)/as, and as is the specific surface area of the packing.
lt rk re
! ! ! T ¼ hlt r U ðr U þðr UÞ Þ qhe !
!
Bq ¼ qq g þF qs
F ls ¼
r ðqh U kÞ r h l þ
3.3. Body force B
!
P 6 1atm
3.5. Turbulence model
!
jUl j ¼
ð9Þ
ep qg cg
8 F pd > < C 1 e2p c2g q2g 1:2 j U! j U! qg 65:62 ¼ F pd > : C 1 e2p c2g q2g 1:2 100:0374qg j U! j U! q 65:62
!
where Gg is the gas superficial flow rate per unit cross-sectional area, Gl is the liquid superficial flow rate per unit cross-sectional area. The inter-phase drag force can be determined by: !
For the gas phase, the porous resistance can be modeled from the dry pressure drop part of the Robbins equation.
r ðqh U eÞ r h l þ
lt re re
ð18Þ
! ! ! T e e2 ¼ c1 hlt r U ðr U þðr UÞ Þ c2 qh k k
ð19Þ
The constants in the above equations are set to be: Cl = 0.09,
rk = 1.0, re = 1.0, c1 = 1.44 and c2 = 1.92. 3.6. Geometric relations
Referring to Sun et al. [7], the porosity of randomly packed column with Pall rings is determined by:
M. Zhang et al. / Applied Thermal Engineering 112 (2017) 585–594
ep ¼ 1 ð1 eb Þf1 expð2ððR rÞ=dp Þ2 Þg
ð20Þ
where eb is the bulk porosity of the column, R is the radius of the column, r is the position in radial direction, dp is the nominal diameter of packing particles. Fig. 2 shows the radial variation of volume porosity in the packed column. From the figure, it can be seen that the radial variation of volume porosity occurs within one packing element diameter from the wall and in the bulk region the porosity is constant. 3.7. The sloshing conditions The sloshing conditions cause a serious liquid maldistribution and a decreased column performance in the offshore packed column. The moving mesh method is used to perform the sloshing of the packed column. There are two kinds of velocity vector. !
!
One is the fluid velocity such as Ug and Ul in the stationary refer!
ence frame. The other is v mesh , the mesh velocity arising from the ship motion defined as the outer product of the angular velocity !
!
(x) and the position vector ( r ). !
!
!
v mesh ¼ x r
h ¼ hmax cos
x ¼ hmax
ð21Þ 2p p tþ T 2
p 2p 180
T
sin
2p p tþ T 2
ð22Þ
ð23Þ
where j and k are the direction vectors of y and z axes and z is the vertical axis of the randomly packed column, hmax is the amplitude of the angular motion, T is the period.
589
4. Model parameters and boundary conditions Table 1 shows the porous media model parameters. The Robbins coefficients C1 and C2 were reported by Robbins [19]. The packing factor, specific surface area and bulk porosity of the column were the geometric characteristics of 25.4 mm Pall rings. The modification factors of dispersion coefficient of the water-air system were recommended by Sun et al. [7]. The values of Kc and rt were determined in order to minimize the deviations between the predicted and measured liquid velocity profiles at different bed heights. The viscosities and densities were given for waterair system. In the present study, the experimental data was obtained from Yin et al. [18]. in order to validate the simulation model. Table 2 presents the boundary conditions and simulation parameters. The multiphase Eulerian model was calculated by the steady-state, pressure-based, and implicit solver. The SIMPLE algorithm was chosen for pressure-velocity coupling and the second-order upwind spatial discretization scheme was used for all differential equations. For counter-current multiphase flow, it is possible to have flows that enter and leave the same flow boundary simultaneously. Therefore, the liquid inlet boundary condition is used for the gas outlet boundary condition and the gas inlet boundary condition is used for the liquid outlet boundary condition. The top-side boundary condition (BC) was set to the velocity-inlet and the bottom-side boundary condition (BC) was set to the mass-flow-inlet for gas and liquid. The domain was limited by the asymmetry boundary condition and the wall boundary condition. The negative velocity represents the opposite direction. The packed column was operated at P0 = 1 atm and T0 = 288.16 K. 5. Results and discussions The Eulerian porous media CFD model was first performed on three different mesh number of the porous zone. Then the simulation results were validated with experimental data such as the liquid relative velocity, liquid volume fraction and pressure drop. Finally, both static and sloshing simulations were carried out with varying angles and aspect ratios. 5.1. Mesh independent test The coarse, medium and fine meshes (9000, 21,000, and 50,000 cells, respectively) were built in the porous zone. The effect of the mesh number for the porous zone was performed on the liquid relative velocity in the radial direction, as shown in Fig. 3. Liquid relative velocity is defined as u/uav, where u is the liquid local superficial velocity and uav is the liquid average superficial velocity over the column cross section.
Table 2 Boundary conditions (BC) and simulation parameters. Parameters
Value
Mesh member Eulerian multiphase model Top-side BC
About 9000, 21,000, and 50,000 cells Steady state, pressure-based, and implicit solver
Bottom-side BC
Fig. 2. Radial variation of volume porosity.
Wall BC Axis BC
Gas outlet (gas velocity = 0.7 m/s), liquid inlet (liquid velocity = 0.082 m/s), liquid volume fraction = 6.2 102, 1 atm and 288.16 K Gas inlet (gas mass flux = 0.75 kg/(m2s), liquid outlet (liquid mass flux = 4.78 kg/(m2 s), liquid volume fraction = 6.2 102, 1 atm and 288.16 K Non-slip and no penetration Asymmetry
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As shown in Fig. 5, the liquid volume fraction remains constant in the bulk region of the column, however, the liquid volume fraction increases significantly near the wall and the peak value reaches about 9.3%. Fig. 6 shows the radial distribution of the liquid velocity. A higher liquid velocity can be seen in the wall region due to the higher liquid volume fraction, however, the adverse velocity gradient will prevent further spreading of liquid towards the wall. Fig. 7 shows the pressure drop per unit length versus the Ffactor pffiffiffiffiffiffi (gas kinetic factor) defined as ug qg , where ug is the gas superficial velocity. The simulation results are in good agreement with the experimental data with the maximum relative error of 8.3%. However, the relative error gets larger with the Ffactor increases due to the gas-liquid flow situation close to the flooding point under the operating condition.
5.3. Effect of tilt angles Fig. 3. Mesh independent test on coarse, medium and fine meshes.
From the figure, the numerical results were not sensitive to the mesh number due to the simple structure and the porous media in randomly packed column. However, the liquid relative velocity above 0.1 m in the radial direction on the coarse mesh shows relatively lower than those of the medium and fine meshes. Finally, the medium mesh was selected from the mesh independence test as a compromise between the accuracy and computational efficiency.
In this study, four tilt angles (1°, 3°, 6°, 9°) were analyzed to investigate the effect of tilt angles on gas-liquid distribution performance in randomly packed columns. Different packed column depths (0.1 m, 0.4 m, 0.8 m, 1.2 m, 1.6 m) were further discussed to analyze cross-sectional liquid maldistribution.
5.2. Validity of simulation model Fig. 4 shows the comparison of the predicted liquid relative velocity with experimental data (water-air system) for the uniform liquid inlet distribution in the packed column. The prediction results agree well with the experiment with the maximum relative error of 7.8%. Due to the constant porosity in the center region of the column, only a little variation in the liquid velocity profile can be seen. However, there is a slight drop and then a significant increase by about 9% near the wall. The liquid spreading towards the non-irrigated zone can easily cause the accumulation of liquid near the wall due to the non-uniform packed-bed structure and the radial variation of volume porosity as well as the no-slip boundary at the column wall surface, known as the wall effect [10].
Fig. 4. Verification of liquid relative velocity distribution.
Fig. 5. Radial distribution of liquid volume fraction.
Fig. 6. Radial liquid velocity contours.
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Fig. 7. Comparison of the predicted pressure drop with experimental data.
Fig. 8 shows the liquid volume fraction distribution along the packed column height under different static tilt conditions. In order to show the effect of tilt angles on cross-sectional liquid maldistribution in the packed column clearly, the maximum liquid volume fraction in the legend was set to 0.05. From the figure, the liquid maldistribution becomes more severe as tilt angle increases and deeper bed experiences more severe maldistribution especially at the base of the column. In the tilted column, the crosssectional area can be divided into two parts with liquid overloading on the tilted side and dry area of the opposite side. Fig. 9 quantifies the effect of tilt angles on the liquid volume fraction distribution along the column height (0.1 m, 0.4 m, 0.8 m, 1.2 m, 1.6 m). The liquid maldistribution becomes more severe as tilt angle increases, for example, the maximum liquid vol-
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Fig. 9. Maximum liquid volume fraction along the column height (+1°, 3°, 6°, 9° tilt angle).
ume fraction along the column height is less than 6% for a tilt angle of 1° and the value is more than 5% for a tilt angle of 9°. The maximum liquid volume fraction decreases as packed column depths increases, therefore, the bottom of the packed column is considered to be the area with the most severe liquid maldistribution. The peak value of the maximum liquid volume fraction reaches 56.1% at the bottom of the column for a tilt angle of 9°, which is 50 times more than the averaged value of the vertical column. Fig. 10 shows the effect of tilt angles on liquid velocity distribution of the bottom section. As tilt angle increases, the dry area in which the void is occupied by vapor increases. For a static tilt of 9°, the area with the liquid velocity around 0.01 m/s occupies
Fig. 8. Contours of liquid volume fraction along the column height (+1°, 3°, 6°, 9° tilt angle).
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Fig. 10. Effect of tilt angles on contours of liquid velocity.
about 13% of the total cross-sectional area and the insufficient gasliquid interfacial area will contribute to a drop in mass transfer performance. Another way to represent flow non-uniformity is the maldistribution factor defined as:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u1 Xn u uav 2 t M¼ n¼1 n uav
ð24Þ
Fig. 11 provides the effect of tilt angles on the maldistribution factor of liquid velocity at the base of the column. As tilt angle increases, the maldistribution factor of liquid velocity significantly increases and the value increases by almost 40% for a static tilt of 9°. The tendency of the value corresponds to the polynomial function to extrapolate to various tilt angles in typical sloshing conditions under the severe sea state. 5.4. Effect of aspect ratios In this study, five aspect ratios (L/D = 1.5, 2, 3, 4, 5) were analyzed to investigate the effect of aspect ratios on gas-liquid distribution performance in randomly packed columns taking a 6° tilt angle as an example. The aspect ratio of the packed column (L/D) is likely to have a greater influence on liquid flow maldistribution. Column and bed size can vary widely depending on the process and design philosophy. In general, industrial sized columns for conventional oil and gas production are 1–7 m in diameter and 2.5–8 m in heights [4]. At a static tilt angle and column diameter, higher packed bed height will experience more severe maldistribution. As shown in Fig. 12, the maldistribution factor of liquid velocity increases slightly as the aspect ratio (L/D) increases and the ten-
Fig. 11. Effect of tilt angles on maldistribution factor of liquid velocity (L/D = 3).
dency corresponds to the polynomial function to extrapolate results to larger industrial columns. A rule that the aspect ratio of the packed column should be less than 2 (L/D 6 2) provides guidance for liquid distribution quality under static conditions [4]. At the aspect ratio of 2 (L/D = 2) of the packed column, the maldistribution factor of liquid velocity is 3.6 for a 6° tilt angle and based on this value, the maximum tilt angle should be less than 3° to prevent the severe liquid flow maldistribution of the column with the aspect ratio of 3 (L/D = 3).
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Fig. 12. Effect of packed bed heights on maldistribution factor of liquid velocity (+6° tilt angle).
5.5. Effect of sloshing conditions Fig. 13 shows the row mapping for liquid maldistribution tests. Dimensionless b shows the variation of flux ratio in different rows under static and sloshing conditions at the bottom of the column. b should be -100% if there is no flux under sloshing conditions and should be 0 if the flux ratio under sloshing conditions is equal to the flux ratio under static conditions. The area is supposed to be not sensitive to the sloshing conditions if b is close to 0.
b¼
msloshing 1 100% mstatic
Fig. 14. Variation of flux ratio along the column diameter (T = 8 s, hmax = 6°). Row 10 r = 0.03 m; Row 6 r = 0.15 m; Row 16 r = 0.18 m; Row 19 r = 0.27 m.
of flux ratio below 25% can be seen in the center of the packed column. However, the flux ratio close to the column’s edge can be 3 times more than that under static condition, which is more sensitive than the center of the column. The affected time and area were provided in different rows along the column diameter. The flux ratios in row 16 and row 19 both reach -100%, which means the area is occupied by vapor. The duration time of the dry area occupies 24.8% in Row 16 and 40% in Row 19 of the period and the total dry area occupies about 10% of the cross-sectional area. The affected area is defined as jbj 6 50%, which occupies more than 40% of the cross-sectional area in the packed column.
ð25Þ
where msloshing is the mass flux under sloshing condition and mstatic is the mass flux under static condition. Fig. 14 shows the variation of flux ratio along the column diameter under sloshing condition (T = 8 s, hmax = 6°). Small fluctuation
Fig. 13. Row mapping for liquid maldistribution tests.
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6. Conclusion A three-dimensional CFD model for randomly packed column with Pall rings under sloshing conditions was developed and mesh independent test was performed. Both static and sloshing conditions were carried out with varying angles and aspect ratios to map the distortion of liquid distribution and the following conclusions can be made. (1) The gas-liquid porous media CFD model was developed to study the effect of sloshing on liquid maldistribution, liquid volume fraction and fractional dry area in randomly packed column. In order to accurately predict the hydrodynamic features, the inter-phase drag force, porous resistance force, liquid dispersion and radial variation of volume porosity were included. (2) Effect of tilt angles (+1°, 3°, 6°, 9°) on liquid volume fraction distribution along the column height was analyzed quantitatively. The liquid maldistribution becomes more severe as tilt angle increases and the maximum liquid volume fraction decreases as packed column depths increases. The peak value of the maximum liquid volume fraction reaches 56.1% at the bottom of the column for a tilt angle of 9°, which is 50 times more than the averaged value of the vertical column. (3) Effect of tilt angles and packed bed heights on maldistribution factor of liquid velocity at the bottom of the column were analyzed and the polynomial functions were given to extrapolate results in typical sloshing conditions. The maximum tilt angle should be less than 3° to prevent the severe liquid flow maldistribution of the column with the aspect ratio of 3 (L/D = 3).
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(4) The center region is less sensitive than the wall region affected by sloshing conditions and the affected time and area were provided. The duration time of the dry area occupies 24.8% in Row16 and 40% in Row19 of the period and the total dry area occupies about 10% of the cross-sectional area. The affected area is defined as jbj 6 50%, which occupies more than 40% of the cross-sectional area in the packed column.
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