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A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors
Applied Mathematical Modelling xxx (2015) xxx–xxx
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A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors Zhi Shang ⇑, Jing Lou, Hongying Li Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
a r t i c l e
i n f o
Article history: Received 9 July 2013 Received in revised form 1 September 2014 Accepted 17 April 2015 Available online xxxx Keywords: Multiphase flow Lagrangian algebraic slip mixture model Eulerian model Lagrangian model Bubble column
a b s t r a c t A multi-dimensional Lagrangian algebraic slip mixture model has been developed to study multiphase flows. Through the single bubble Lagrangian movement equation, this model considers the accelerations of various forces on the bubble. Accordingly the slip velocities between bubbles and liquid were derived in Lagrangian manner to connect the effects of the various forces on bubbles with the diffusion flux velocities which appear in the Euler governing equations. Therefore this model realizes the connection between Eulerian model and Lagrangian model. Because the single bubble Lagrangian movement was extended to multi-dimension, the model is able to describe multi-dimensional movement of multiphase flows. Through the numerical simulations comparing with the experiments and the simulations of other models on bubble columns, the model is validated. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Bubble column reactors are widely used as gas–liquid and gas–liquid–solid contactors in many chemical, petrochemical and biochemical industries, such as absorption, oxidation, hydrogenation, catalytic slurry reaction, coal liquefaction, aerobic fermentation. The operation of these reactors is preferred because of the simple construction, ease of maintenance and low operating costs. When the gas is injected at the bottom of the column, it causes a turbulent stream to enable an optimum gas exchange. It is built in numerous forms of construction. The mixing is done by the gas sparging and it requires less energy than mechanical stirring. The liquid can be in parallel flow or counter-current. Usually the bubble column is particularly useful in reactions where the gas–liquid reaction is slow in relation to the absorption rate. A good understanding of the liquid dynamics of a bubble column will help the engineers to design a high efficient reactor under optimized operating parameters. Due to the complex two-phase or multi-phase flow and turbulence, normally the flow in bubble columns is under transient regime. The time average values of the parameters, such as gas holdup distributions, liquid phase back mixing, gas–liquid interface disturbing, mass and heat transfer between gas and liquid phases, bubble size distributions, bubble rise velocities etc., have to be considering the influence of turbulence. Although the operation of bubble columns is simple, the actual physical flow phenomena are still lack of complete understanding of the fluid dynamics [1]. Many experimental facilities and methods were introduced to study the multiphase flows in bubble columns. Krishna and Baten [2] used a modified Pitot tube, called as Pavlov tube, method to measure the liquid velocities and radial profiles. Sanyal et al. [1] used computer automated radioactive particle tracking (CARPT) method to directly measure the time-average ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Shang). http://dx.doi.org/10.1016/j.apm.2015.04.050 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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velocity and turbulence parameters of the liquid phase in the bubble column. Zhou et al. [3] used charge-coupled device (CCD) camera equipped with a variable electronic shutter to record the image of the flow field. The experimental methods can provide very useful information about the bubble columns at certain measurement points, but they are difficult to show the details of the flow fields and parameters inside the bubble columns. Following the development of computer technology, it is already allowed to use the numerical method to do the researches in the recent decades [4]. Therefore, many researchers employ the numerical method, called as computational fluid dynamics (CFD), to study the details of the flows inside bubble columns. Sanyal et al. [1] employed the algebraic slip mixture model (ASMM) and Euler–Euler two-fluid model provided by FLUENT commercial software to simulate a cylindrical bubble column from bubbly flow regime to churn flow regime respectively and compared with experiments. Zhou et al. [3] developed a second-order moment turbulence model based on Euler–Euler two-fluid model. Through comparisons with experiments, this model shows a good agreement on turbulent kinetic energy of fluid flows. However it has to deal with complex turbulence modeling. Zhang et al. [5] employed Euler–Euler two-fluid model provided by CFX commercial software to simulate a rectangle bubble column. They found the interfacial force coefficients had evident influences on the simulations. Through the former studies of CFD method, it can be seen a good mathematical model will not only help to obtain the agreeable simulation results, but also to be simple, efficient and accurate. A multi-dimensional Lagrangian algebraic slip mixture model was developed in this paper. This model was based on the idea of Shang [6,7]. It employed a mixture model to describe the multi-phase flows based on Eulerian model. The slip velocity, which can be developed from the dynamic equation of the dispersed phase based on Lagrangian model, was introduced to present the difference between dispersed and continuous phases. Owing to the Lagrangian model, the interfacial forces, such as buoyancy, drag force, lift force and virtual mass force etc., are enable to be involved in the multi-dimensional Lagrangian algebraic slip mixture model. It is therefore different from the traditional algebraic slip mixture model, which only considers the effects of gravity, centrifugal force and drag force, provided by FLUENT, CFX and other commercial software [1]. Through comparisons with experiments and other models on cylindrical and rectangle bubble columns, this model was validated. 2. Mathematical modeling Considering a problem of turbulent multi-component multi-phase flow with one continuous phase and several dispersed phases, the time average conservation equations of mass, momentum and energy for the multi-dimensional Lagrangian algebraic slip mixture model as well as the turbulent kinetic energy equation and the turbulent kinetic energy transport equation can be written as the following [1,6–9].
@ qm =@t þ r ðqm Um Þ ¼ 0;
ð1Þ
h i X @ðqm Um Þ=@t þ r ðqm Um Um Þ ¼ rp þ qm g þ r ðlm þ lt Þ rUm þ rUTm r ai qi Uim Uim
ð2Þ
@ðqm hm Þ=@t þ r ðqm Um hm Þ ¼ q þ r
@ðqm kÞ=@t þ r ðqm Um kÞ ¼ r
@ðqm eÞ=@t þ r ðqm Um eÞ ¼ r
lm Pr
þ
lt Prt
rhm r
X
ak qk hk Ukm ;
lm þ
lt rk þ G qm e; rk
lm þ
lt e re þ ðC 1 G C 2 qm eÞ; re k
ð3Þ
ð4Þ
ð5Þ
in which
qm ¼
lm ¼
X X
qm Um ¼
ai qi ;
ð6Þ
ai li ;
ð7Þ
X
ai qi Ui ;
Uim ¼ Ui Um ; G¼
i2 1 h l rUm þ ðrUm ÞT ; 2 t
lt ¼ C l qm
k
ð8Þ ð9Þ ð10Þ
2
e
;
ð11Þ
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 1 Constants of standard k–e turbulence model. Variable
Cl
rk
re
C1
C2
Constant
0.09
1.0
1.3
1.44
1.92
where, subscript m is mixture, i is the ith phase, q is the density, U are the velocity vectors, a is the volumetric fraction, p is pressure, g is the gravitational acceleration vector, Uim is the diffusion velocity vector of k dispersed phase relative to the averaged mixture flow, h is enthalpy, q is heat input, l is viscosity, lt is turbulent viscosity, Pr is molecular Prandtl number, Prt is turbulent Prandtl number, G is stress production. Cl, rk, re, C1, C2 are constants for standard k–e turbulence model [10], shown in Table 1. The subscript m stands for the averaged mixture flow, and k stands for k dispersed phase. Additional to the above equations, the following conservation equation for each dispersed phase is also necessary [1,6–9].
@ ðai qi Þ=@t þ r ðai qi Um Þ ¼ Ci r ðai qi Uim Þ;
ð12Þ
where Ui is the generation rate of the ith phase. In order to closure the governing Eqs. (1)-(12), it is necessary to determine the diffusion velocities Uim. The following equation is employed to covert the diffusion velocities to slip velocities that can be presented as Uic = Ui Uc where Uc is the velocity of continuous phase.
Uim ¼ Uic
X ai q
i
qm
Uic :
ð13Þ
Actually the above equation can be developed from the definition of the mixture density Eq. (6), the definition of mixture mass flux Eq. (8), the diffusion velocity Eq. (9) and the slip velocity Uic. Once the slip velocities are obtained, the whole governing equations will be closured. Because the slip velocities present the difference of the movement between the dispersed phase for instance gas and the continuous phase for instance liquid. The dispersed phase can be presented by its own law of motion. For example, in gas and liquid two phase flow system, the following equation can be use to describe the single bubble movement inside a liquid, which is normally called as Lagrangian equation of motion [11].
Fg ¼ Fbouyancy þ Fdrag þ Fvirtual þ Flift þ ;
ð14Þ
where Fg is the inertia force acting on the bubble due to its acceleration, Fbouyancy is the force due to gravity and buoyancy, Fdrag is force due to drag by the continuous liquid, Fvirtual is the force due to virtual mass effect, Flift is the force due to transverse lift, and so on the other forces can be added into Eq. (14). In this paper, only the forces of buoyancy, drag, virtual mass and transverse lift were considered. The expanded description about these forces can be presented as the following equation [3].
qi
dUi 3q C d dUc dUi þ C l qc ðUc Ui Þ r Uc ; ¼ ðqi qc Þg þ c ðUc Ui ÞjUc Ui j þ C v m qc dt 4di dt dt
ð15Þ
where g is gravity, Cd is drag force coefficient, Cvm is virtual force coefficient and Cl is lift force coefficient. These coefficients can be determined by empirical formulas or constants. If we define the dispersed phase as gas and the continuous phase as liquid, the slip velocity of Uic in Eq. (13) will be the slip velocity between gas and liquid as Ugl. The specific slip velocity between gas and liquid can be developed from Eq. (15) as the following.
g0 ¼ g
qg dUl dUg dUg þ C l Ul Ug r Ul þ Cvm ; ðql qg Þ dt dt ql qg dt
ql
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðql qg Þdg jgj g0 Ugl ¼ : 3ql C d jg0 j
ð16Þ
ð17Þ
The form is similar with the definition of the slip velocity between gas and the solid particle in Pericleous [12] and Shang [6]. In the slip velocity of Pericleous [12] and Shang [6], the slip velocity was for the single solid particle, hence the applications could be only for the dilute gas solid-particle two-phase flow. Later, Shang et al. [7] confirmed the applications. However for gas liquid bubbly flow, the differences from gas solid-particle two-phase flow are that the flow direction is revised by the revised gravity g0 and the density of the continuous phase is greater than the dispersed phase. Since the slip velocity is determined, the whole equations are closured to be solved. Considering the bubble size could be changed due to breakup and coalescence, same as Chen [13] and Chen et al. [14], the following experimental formula can be employed to estimate the distributions of the bubble size [15].
dg ¼ 3jgj0:44 r0:34 l0:22 q0:45 q0:11 jug j0:02 ; l l g
ð18Þ
where dg is bubble size, g is gravity, r is surface tension, ll is liquid viscosity, ql is liquid density, qg is gas density and ug is superficial gas velocity. From Eq. (18) it can be seen that the calculation of the bubble size dg considers the effect of the gas Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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holdup (i.e. gas volume fraction, ag) through the superficial gas velocity (ug = agUg). This consideration can potentially extend the slip velocity for swarm of bubbles. 3. Interfacial forces Owing to the multi-dimensional Lagrangian algebraic slip mixture model considering the interfacial forces between dispersed and continuous phases, the slip velocity of this model is able to deal with the non-uniform movement of bubbles. The interfacial forces, such as drag force, lift force and virtual mass force etc., have to be modeled before the simulations. The models about these forces are presented in Eq. (15). From the equation, it can be seen that once the coefficients of the forces are decided, the numerical simulation will be able to be performed. All the coefficients, such as drag force coefficient, virtual mass force coefficient and lift force coefficient, normally can be formulized as the functions of some non-dimensional numbers, for example Reynolds number and Eötvos number. The drag force coefficient can be estimated from drag law of a single bubble. The approximated formula developed by Schiller and Naumann [16] is used in this paper, shown in Eq. (19). Actually this formula had been used by many other researches for their simulations not only for solid particles but also for gas bubbles [1,6,14]. From their reports, it can be seen that this formula is able to produce very good results comparing with experiments.
( Cd ¼
24 Reb
1 þ 0:15Re0:687 Reb 6 1000 b
0:44
ð19Þ
Reb > 1000
where Reb is bubble Reynolds number that can be defined as the following formula.
Reb ¼
ql jUg Ul jdg : ll
ð20Þ
The lift force coefficient can be modeled with or without considering the bubble deformation, i.e. considering the bubble as sphere [17] or non-sphere [18,19]. Usually, when the bubble size is small (dg < 4.4 mm), the bubble can be treated as a sphere moves inside the liquid [19]. However, when bubble’s diameter is great, it will become flat along the motion direction [13,19]. Ideally it can be treated as an oval shape [13,19]. In this paper, the lift force formula based on the bubble deformation is adopted. After combining the shear lift force [20–22] and effects from slanted wake, the lift force model, which is Eötvos number dependent, had been put forward by Tomiyama [18] and Tomiyama et al. [19].
8 0:288 tanhð0:121Reb Þ ; f ðEog Þ ; Eog < 4 > < min C l ¼ f Eog ; 4 Eog 10 ; > : 0:29; Eog > 10
ð21Þ
where the function f(Eog) can be calculated by the following experimental formula [19].
f ðEog Þ ¼ 0:00105Eo3g 0:0159Eo2g 0:0204Eog þ 0:474;
ð22Þ
in which Eötvos number (Eog) is defined as: 2
Eog ¼
jgjðql qg Þdh
r
;
ð23Þ
in which r is surface tension, dh is the maximum horizontal bubble dimension when moving bubble has the deformation. Through dh, we can see the deformation effects on bubble are considered in the lift force. The maximum horizontal bubble dimension dh can be calculated through the formula of Wellek et al. [23].
1=3 dh ¼ dg 1 þ 0:163Eo0:757 ;
ð24Þ
where Eo is the Eötvos number depending on the sphere bubble diameter. The formula of Eo, shown in Eq. (25), is similar with Eq. (23). 2
Eo ¼
jgjðql qg Þdg
r
:
ð25Þ
The virtual mass force normally generated due to the motion of bubbles. When a bubble moves in the liquid, it disturbs the liquid around the bubble. Some of liquid near the bubble will be accelerated, accordingly the kinetic energy of the liquid surrounding the sphere bubble changes. Therefore, the movement of the bubble will be affected by the accelerated liquid. Accordingly Drew and Lahey [22] studied this phenomenon and developed a formula to describe the virtual mass force, shown in Eq. (15). The virtual mass force coefficient is generally bubble-shape and gas hold up dependent [13]. It seems case Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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Table 2 Relaxation factors for computed variables. Variable
Um
p
k
e
Factor
0.7
0.3
0.8
0.8
dependent according to the former studies. Mudde and Simonin [24] found the virtual mass force played important role on predicting the bubble plume oscillations. However, some of researches [2,25] found the virtual mass force can be negligible in their simulations. According to the former studies, in this paper the virtual mass force coefficient is treated as a small constant number. It means the virtual mass force is not negligible totally, but it has not a strong influence on the movement of bubble. 4. Numerical procedures During the simulations, all differential equations of the multi-dimensional Lagrangian algebraic slip mixture model are transformed into difference equations by the finite-volume integral method [26,27]. During the discretization of the equations, the derivatives of the all variables with respect to time are discretized using Euler implicit time scheme. The central-difference scheme and the QUICK scheme are used to discretize the diffusion terms and the convective transport terms of the equations of the multi-dimensional Lagrangian algebraic slip mixture model respectively [26]. The difference equations are solved numerically with an iterative solution procedure through the SIMPLE method [26,27]. During the numerical calculations, an under-relaxation method is utilized to accelerate the convergence. The relaxation factors used in the calculations are given in Table 2. The turbulent kinetic energy and turbulent kinetic energy transport of the mixture at inlet will be calculated by the method recommended by Shang [6]. The specific equations are shown as follows:
kin ¼ 0:005V 2m;in ;
ð26Þ
3=2
ein ¼
kin ; 0:025 ð2:5Dh Þ
ð27Þ
in which Dh is the hydraulic diameter of the inlet and Vm,in is the magnitude of the inlet velocity of the mixture which is normal to the inlet interface. The standard wall function was employed to predict the variables near wall [10]. Initially only the liquid fully fills the column. There is no gas, i.e. ag = 0 and al = 1, inside the column at the beginning. The gas is injected into the column at the bottom with ag = 1 and al = 0. The injected gas velocity therefore equals to the gas superficial velocity and the liquid velocity equals 0 m/s as the inlet condition. The gas is discharged at the top of column with ag = 1 and al = 0 as the outlet condition. It means that only gas flows out the column and the liquid flows back into the column at the outlet. 5. Numerical simulations Two typical kinds of bubble columns, which are cylindrical bubble column and rectangle bubble column, were chosen as the targets in this paper. The cylindrical bubble column is basically a cylindrical vessel with a gas distributor at the bottom. The gas is sparged in the form of bubbles into either a liquid phase or a liquid–solid suspension [28]. Similar with the cylindrical bubble column, the rectangle bubble column is basically a rectangle vessel with a gas distributor at the bottom. 5.1. Cylindrical bubble column reactor Sanyal et al. [1] did the experiment and numerical simulation on a cylindrical bubble column reactor. Hereafter, the experiment data has been cited by other researchers [13,14,29,30]. The column contained a batch liquid with unexpanded height of 104.5 and 95.0 cm, respectively. The distributor used in the experiments was a perforated plate, with 0.33 mm diameter holes in a square pitch, with an open area of 0.1% [1]. The gas superficial velocities were operated from 2.0 to 12.0 cm/s. The corresponding flow characteristic is from bubbly flow to churn turbulent flow. Owing to the wide range of the flow regimes, these experimental and numerical data were chosen as the comparisons in this paper. During the numerical simulations, all the parameters were used same as Sanyal et al. [1]. The air and water bubble columns were under atmospheric pressure operating at these superficial velocities with a perforated plate sparger. The grid size was employed same as Sanyal et al. [1], which was 300 (axial) 19 (radial) rectangular grid. The mesh cell size was 0.66 cm (axial) 0.5 cm (radial) uniformly. The finer meshes were employed to test the mesh sensitivity before the formal simulations. There were no evident differences of the simulation results between the current mesh and finer meshes. The mesh cell size used in this paper enables the simulation results on mesh independent. The formal simulations were solved Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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in a transient fashion with a time step of 0.01 s and carried out long enough to statistic the stable time-averaged variables same as Sanyal et al. [1]. On performing the numerical simulations, Sanyal et al. [1], Chen [13] and Chen et al. [14], Wang and Wang [29] made the similar analysis and obtained the same conclusion that the lift force and virtual mass force can be ignored for the cylindrical bubble column reactor, and only the buoyancy and drag force were considered. In this paper, same as the former researchers, the lift force is excluded during the simulations. Although the virtual mass force is not ignored, the virtual mass force coefficient was chosen as a small number, for example Cvm = 0.02, to weaken the effects of the virtual mass force. The buoyancy and drag force were kept during the simulations. Fig. 1 shows the comparisons of time average gas holdup (i.e. gas volume fraction, ag) and axial water velocity under the flow regime of gas superficial velocity Ug = 2 cm/s at the height z = 530 mm from the bottom of the bubble column reactor. From the comparisons in Fig. 1(a), it can be seen that the current multi-dimensional Lagrangian algebraic slip mixture model can have the predicted gas holdup values similar with the algebraic slip mixture model and two-fluid model. However, in Fig. 1(b), the predicted profile of the axial water velocity by the current multi-dimensional Lagrangian algebraic slip mixture model is more approaching to experiments than the algebraic slip mixture model and two-fluid model. The zero point of axial water velocity is at r/R = 0.7, same as the experiment and the predictions of the algebraic slip mixture model and two-fluid model. Fig. 2 shows the comparisons of time average gas holdup and axial water velocity under the flow regime of gas superficial velocity Ug = 12 cm/s at z = 530 mm. From the comparisons in Fig. 2(a), it can be seen that the current multi-dimensional Lagrangian algebraic slip mixture model predicted gas holdup values near experiments between the predictions of the algebraic slip mixture model and two-fluid model. The maximum value of the gas holdup approached to 0.38 of the experimental measure. It illustrated that the multi-dimensional Lagrangian algebraic slip mixture model can be used for both of dilute and dense dispersions in gas liquid two-phase flow, although the slip velocity model was based on the single bubble movement.
(a) distribution of gas holdup
(b) distribution of axial water velocity
Fig. 1. Comparisons under gas superficial velocity Ug = 2 cm/s.
(a) distribution of gas holdup
(b) distribution of axial water velocity
Fig. 2. Comparisons under gas superficial velocity Ug = 12 cm/s.
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Fig. 2(b) shows the predicted axial water velocities by the current multi-dimensional Lagrangian algebraic slip mixture model. It can be seen the predicted results are more accurate to experiments than algebraic slip mixture model and two-fluid model. The zero point of axial water velocity is at r/R = 0.69 near the experiment point. 5.2. Rectangle bubble column reactor Zhang et al. [5] did the simulations for a rectangle bubble column. The width, depth and height of the bubble column are, respectively, set to W = 0.15 m, D = 0.15 m, and H = 0.45 m or H = 0.90 m. Therefore there were two cases set up as H/D = 3 and H/D = 6. The gas distributor is situated in the bottom plane of the column at a distance of 0.06 m from each of the surrounding walls of the column. The entire column is initially filled with water, which acts as the continuous liquid phase. Air is employed as the dispersed gas phase and sparged at the bottom plane with the inlet superficial gas velocity of 4.9 mm/s. The gas–liquid flow regime is bubbly flow. The mesh size was employed same as Zhang et al. [5], which was Dx = Dy = Dz = 0.01 m along the three coordinates. Same as the cylindrical bubble column, the mesh sensitivity was tested before the formal simulations, and current the mesh size can guarantee the results on mesh independent. The formal simulations were solved transiently with a time step of 0.01 s and carried out long enough to statistic the stable time-averaged variables same as Zhang et al. [5]. Due to the non-symmetrical flow along angle direction, the flow situation inside the rectangle bubble column is different from the total symmetrical flow of the cylindrical bubble column. The transverse force acted on the bubbles cannot be ignored. The lift force therefore has to be considered [5]. Zhang et al. [5] found the bubble deformation occurred in the rectangle bubble column. Hence the lift force coefficient, which considers the influence of bubble deformation shown in Eq. (21), is employed in this paper. Same as the cylindrical bubble column, a small number of the virtual mass force coefficient was used to weaken the influence of the virtual mass force. The buoyancy and drag force were considered as the basic force on the bubble during the simulations. Fig. 3 shows the distributions of the timely averaged axial liquid velocity vector at x–z cross section. From Fig. 3, it can be seen that the liquid flows up in the middle, turns the flow direction at the top and flows down along the sides of the column towards the bottom. A big circular flow is generated inside the bubble column. Fig. 4 shows the comparisons of time-average gas and water axial velocity of the bubble column H/D = 3. Fig. 4(a)–(c) show the distributions of the time average axial gas velocity at different heights, and Fig. 4(d)–(f) show the distributions
(a) H/D = 3.0
(b) H/D = 6.0
Fig. 3. Time averaged axial liquid velocity vector at x–z cross section.
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(a) z/H=0.57
(d) z/H=0.57
(b) z/H=0.63
(e) z/H=0.63
(c) z/H=0.72
(f) z/H=0.72
Fig. 4. Comparisons of gas and liquid axial velocity under H/D = 3.0.
of the time-average axial liquid velocity at different heights. From the comparisons it can be seen that the current Lagrangian algebraic slip mixture model can predict the profiles of the velocity more approaching experiments, especially for gas velocities. Fig. 5 shows the comparisons of time-average gas and water axial velocity of the bubble column H/D = 6. Fig. 5(a) and (b) show the distributions of the time average axial gas velocity at different heights, and Fig. 5(c) and (d) show the distributions of the time-average axial liquid velocity at different heights. Same as the comparisons shown in Fig. 4, the current Lagrangian algebraic slip mixture model can predict the profiles of the velocity more approaching experiments.
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx
(a) z/H=0.57
(c) z/H=0.57
(b) z/H=0.73
(d) z/H=0.73
9
Fig. 5. Comparisons of gas and liquid axial velocity under H/D = 6.0.
6. Conclusion The multi-dimensional Lagrangian algebraic slip mixture model is developed based on the mixture multiphase flow model. The diffusion velocity between the dispersed phase and the mixture is closured through the slip velocity between the dispersed phase and the continuous phase. Accordingly the novel multi-dimensional Lagrangian algebraic slip mixture model is different from the general homogenous model that treats the dispersed phase and the continuous phase flowing under same velocity on multiphase flow. The slip velocity is modeled by the single bubble Lagrangian motion equation. The forces of buoyancy, drag, virtual mass, transverse lift and so on are therefore able to be considered as the usual two-fluid model. It makes an innovation on the traditional algebraic slip mixture model and combines both of the merits of the simplicity of the mixture model and accuracy of the two-fluid model. Through the comparisons of the numerical simulations between the multi-dimensional Lagrangian algebraic slip mixture model and experiments, the model is validated. Through comparisons of the numerical simulations between the multi-dimensional Lagrangian algebraic slip mixture model and the numerical simulations of traditional algebraic slip mixture model and two-fluid model, the efficiency and accuracy of the model is confirmed. Acknowledgements The authors would like to thank the support by Multiphase Flow for Deep-Sea Oil & Gas Down-hole Applications - SERC TSRP Programme of Agency for Science, Technology and Research (A⁄STAR) in Singapore (Ref #: 102 164 0075). References [1] J. Sanyal, S. Vasquez, S. Roy, M.P. Dudukovic, Numerical simulation of gas–liquid dynamics in cylindrical bubble column reactors, Chem. Eng. Sci. 54 (1999) 5071–5083. [2] R. Krishna, J.M. Baten, Scaling up bubble column reactors with the aid of CFD, Chem. Eng. Res. Des. 79 (3) (2001) 283–309. [3] L.X. Zhou, M. Yang, C.Y. Lian, L.S. Fan, D.J. Lee, On the second-order moment turbulence model for simulating a bubble column, Chem. Eng. Sci. 57 (2002) 3269–3281.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors Zhi Shang ⇑, Jing Lou, Hongying Li Institute of High Performance Computing, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore
a r t i c l e
i n f o
Article history: Received 9 July 2013 Received in revised form 1 September 2014 Accepted 17 April 2015 Available online xxxx Keywords: Multiphase flow Lagrangian algebraic slip mixture model Eulerian model Lagrangian model Bubble column
a b s t r a c t A multi-dimensional Lagrangian algebraic slip mixture model has been developed to study multiphase flows. Through the single bubble Lagrangian movement equation, this model considers the accelerations of various forces on the bubble. Accordingly the slip velocities between bubbles and liquid were derived in Lagrangian manner to connect the effects of the various forces on bubbles with the diffusion flux velocities which appear in the Euler governing equations. Therefore this model realizes the connection between Eulerian model and Lagrangian model. Because the single bubble Lagrangian movement was extended to multi-dimension, the model is able to describe multi-dimensional movement of multiphase flows. Through the numerical simulations comparing with the experiments and the simulations of other models on bubble columns, the model is validated. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Bubble column reactors are widely used as gas–liquid and gas–liquid–solid contactors in many chemical, petrochemical and biochemical industries, such as absorption, oxidation, hydrogenation, catalytic slurry reaction, coal liquefaction, aerobic fermentation. The operation of these reactors is preferred because of the simple construction, ease of maintenance and low operating costs. When the gas is injected at the bottom of the column, it causes a turbulent stream to enable an optimum gas exchange. It is built in numerous forms of construction. The mixing is done by the gas sparging and it requires less energy than mechanical stirring. The liquid can be in parallel flow or counter-current. Usually the bubble column is particularly useful in reactions where the gas–liquid reaction is slow in relation to the absorption rate. A good understanding of the liquid dynamics of a bubble column will help the engineers to design a high efficient reactor under optimized operating parameters. Due to the complex two-phase or multi-phase flow and turbulence, normally the flow in bubble columns is under transient regime. The time average values of the parameters, such as gas holdup distributions, liquid phase back mixing, gas–liquid interface disturbing, mass and heat transfer between gas and liquid phases, bubble size distributions, bubble rise velocities etc., have to be considering the influence of turbulence. Although the operation of bubble columns is simple, the actual physical flow phenomena are still lack of complete understanding of the fluid dynamics [1]. Many experimental facilities and methods were introduced to study the multiphase flows in bubble columns. Krishna and Baten [2] used a modified Pitot tube, called as Pavlov tube, method to measure the liquid velocities and radial profiles. Sanyal et al. [1] used computer automated radioactive particle tracking (CARPT) method to directly measure the time-average ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Shang). http://dx.doi.org/10.1016/j.apm.2015.04.050 0307-904X/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx
velocity and turbulence parameters of the liquid phase in the bubble column. Zhou et al. [3] used charge-coupled device (CCD) camera equipped with a variable electronic shutter to record the image of the flow field. The experimental methods can provide very useful information about the bubble columns at certain measurement points, but they are difficult to show the details of the flow fields and parameters inside the bubble columns. Following the development of computer technology, it is already allowed to use the numerical method to do the researches in the recent decades [4]. Therefore, many researchers employ the numerical method, called as computational fluid dynamics (CFD), to study the details of the flows inside bubble columns. Sanyal et al. [1] employed the algebraic slip mixture model (ASMM) and Euler–Euler two-fluid model provided by FLUENT commercial software to simulate a cylindrical bubble column from bubbly flow regime to churn flow regime respectively and compared with experiments. Zhou et al. [3] developed a second-order moment turbulence model based on Euler–Euler two-fluid model. Through comparisons with experiments, this model shows a good agreement on turbulent kinetic energy of fluid flows. However it has to deal with complex turbulence modeling. Zhang et al. [5] employed Euler–Euler two-fluid model provided by CFX commercial software to simulate a rectangle bubble column. They found the interfacial force coefficients had evident influences on the simulations. Through the former studies of CFD method, it can be seen a good mathematical model will not only help to obtain the agreeable simulation results, but also to be simple, efficient and accurate. A multi-dimensional Lagrangian algebraic slip mixture model was developed in this paper. This model was based on the idea of Shang [6,7]. It employed a mixture model to describe the multi-phase flows based on Eulerian model. The slip velocity, which can be developed from the dynamic equation of the dispersed phase based on Lagrangian model, was introduced to present the difference between dispersed and continuous phases. Owing to the Lagrangian model, the interfacial forces, such as buoyancy, drag force, lift force and virtual mass force etc., are enable to be involved in the multi-dimensional Lagrangian algebraic slip mixture model. It is therefore different from the traditional algebraic slip mixture model, which only considers the effects of gravity, centrifugal force and drag force, provided by FLUENT, CFX and other commercial software [1]. Through comparisons with experiments and other models on cylindrical and rectangle bubble columns, this model was validated. 2. Mathematical modeling Considering a problem of turbulent multi-component multi-phase flow with one continuous phase and several dispersed phases, the time average conservation equations of mass, momentum and energy for the multi-dimensional Lagrangian algebraic slip mixture model as well as the turbulent kinetic energy equation and the turbulent kinetic energy transport equation can be written as the following [1,6–9].
@ qm =@t þ r ðqm Um Þ ¼ 0;
ð1Þ
h i X @ðqm Um Þ=@t þ r ðqm Um Um Þ ¼ rp þ qm g þ r ðlm þ lt Þ rUm þ rUTm r ai qi Uim Uim
ð2Þ
@ðqm hm Þ=@t þ r ðqm Um hm Þ ¼ q þ r
@ðqm kÞ=@t þ r ðqm Um kÞ ¼ r
@ðqm eÞ=@t þ r ðqm Um eÞ ¼ r
lm Pr
þ
lt Prt
rhm r
X
ak qk hk Ukm ;
lm þ
lt rk þ G qm e; rk
lm þ
lt e re þ ðC 1 G C 2 qm eÞ; re k
ð3Þ
ð4Þ
ð5Þ
in which
qm ¼
lm ¼
X X
qm Um ¼
ai qi ;
ð6Þ
ai li ;
ð7Þ
X
ai qi Ui ;
Uim ¼ Ui Um ; G¼
i2 1 h l rUm þ ðrUm ÞT ; 2 t
lt ¼ C l qm
k
ð8Þ ð9Þ ð10Þ
2
e
;
ð11Þ
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 1 Constants of standard k–e turbulence model. Variable
Cl
rk
re
C1
C2
Constant
0.09
1.0
1.3
1.44
1.92
where, subscript m is mixture, i is the ith phase, q is the density, U are the velocity vectors, a is the volumetric fraction, p is pressure, g is the gravitational acceleration vector, Uim is the diffusion velocity vector of k dispersed phase relative to the averaged mixture flow, h is enthalpy, q is heat input, l is viscosity, lt is turbulent viscosity, Pr is molecular Prandtl number, Prt is turbulent Prandtl number, G is stress production. Cl, rk, re, C1, C2 are constants for standard k–e turbulence model [10], shown in Table 1. The subscript m stands for the averaged mixture flow, and k stands for k dispersed phase. Additional to the above equations, the following conservation equation for each dispersed phase is also necessary [1,6–9].
@ ðai qi Þ=@t þ r ðai qi Um Þ ¼ Ci r ðai qi Uim Þ;
ð12Þ
where Ui is the generation rate of the ith phase. In order to closure the governing Eqs. (1)-(12), it is necessary to determine the diffusion velocities Uim. The following equation is employed to covert the diffusion velocities to slip velocities that can be presented as Uic = Ui Uc where Uc is the velocity of continuous phase.
Uim ¼ Uic
X ai q
i
qm
Uic :
ð13Þ
Actually the above equation can be developed from the definition of the mixture density Eq. (6), the definition of mixture mass flux Eq. (8), the diffusion velocity Eq. (9) and the slip velocity Uic. Once the slip velocities are obtained, the whole governing equations will be closured. Because the slip velocities present the difference of the movement between the dispersed phase for instance gas and the continuous phase for instance liquid. The dispersed phase can be presented by its own law of motion. For example, in gas and liquid two phase flow system, the following equation can be use to describe the single bubble movement inside a liquid, which is normally called as Lagrangian equation of motion [11].
Fg ¼ Fbouyancy þ Fdrag þ Fvirtual þ Flift þ ;
ð14Þ
where Fg is the inertia force acting on the bubble due to its acceleration, Fbouyancy is the force due to gravity and buoyancy, Fdrag is force due to drag by the continuous liquid, Fvirtual is the force due to virtual mass effect, Flift is the force due to transverse lift, and so on the other forces can be added into Eq. (14). In this paper, only the forces of buoyancy, drag, virtual mass and transverse lift were considered. The expanded description about these forces can be presented as the following equation [3].
qi
dUi 3q C d dUc dUi þ C l qc ðUc Ui Þ r Uc ; ¼ ðqi qc Þg þ c ðUc Ui ÞjUc Ui j þ C v m qc dt 4di dt dt
ð15Þ
where g is gravity, Cd is drag force coefficient, Cvm is virtual force coefficient and Cl is lift force coefficient. These coefficients can be determined by empirical formulas or constants. If we define the dispersed phase as gas and the continuous phase as liquid, the slip velocity of Uic in Eq. (13) will be the slip velocity between gas and liquid as Ugl. The specific slip velocity between gas and liquid can be developed from Eq. (15) as the following.
g0 ¼ g
qg dUl dUg dUg þ C l Ul Ug r Ul þ Cvm ; ðql qg Þ dt dt ql qg dt
ql
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ðql qg Þdg jgj g0 Ugl ¼ : 3ql C d jg0 j
ð16Þ
ð17Þ
The form is similar with the definition of the slip velocity between gas and the solid particle in Pericleous [12] and Shang [6]. In the slip velocity of Pericleous [12] and Shang [6], the slip velocity was for the single solid particle, hence the applications could be only for the dilute gas solid-particle two-phase flow. Later, Shang et al. [7] confirmed the applications. However for gas liquid bubbly flow, the differences from gas solid-particle two-phase flow are that the flow direction is revised by the revised gravity g0 and the density of the continuous phase is greater than the dispersed phase. Since the slip velocity is determined, the whole equations are closured to be solved. Considering the bubble size could be changed due to breakup and coalescence, same as Chen [13] and Chen et al. [14], the following experimental formula can be employed to estimate the distributions of the bubble size [15].
dg ¼ 3jgj0:44 r0:34 l0:22 q0:45 q0:11 jug j0:02 ; l l g
ð18Þ
where dg is bubble size, g is gravity, r is surface tension, ll is liquid viscosity, ql is liquid density, qg is gas density and ug is superficial gas velocity. From Eq. (18) it can be seen that the calculation of the bubble size dg considers the effect of the gas Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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holdup (i.e. gas volume fraction, ag) through the superficial gas velocity (ug = agUg). This consideration can potentially extend the slip velocity for swarm of bubbles. 3. Interfacial forces Owing to the multi-dimensional Lagrangian algebraic slip mixture model considering the interfacial forces between dispersed and continuous phases, the slip velocity of this model is able to deal with the non-uniform movement of bubbles. The interfacial forces, such as drag force, lift force and virtual mass force etc., have to be modeled before the simulations. The models about these forces are presented in Eq. (15). From the equation, it can be seen that once the coefficients of the forces are decided, the numerical simulation will be able to be performed. All the coefficients, such as drag force coefficient, virtual mass force coefficient and lift force coefficient, normally can be formulized as the functions of some non-dimensional numbers, for example Reynolds number and Eötvos number. The drag force coefficient can be estimated from drag law of a single bubble. The approximated formula developed by Schiller and Naumann [16] is used in this paper, shown in Eq. (19). Actually this formula had been used by many other researches for their simulations not only for solid particles but also for gas bubbles [1,6,14]. From their reports, it can be seen that this formula is able to produce very good results comparing with experiments.
( Cd ¼
24 Reb
1 þ 0:15Re0:687 Reb 6 1000 b
0:44
ð19Þ
Reb > 1000
where Reb is bubble Reynolds number that can be defined as the following formula.
Reb ¼
ql jUg Ul jdg : ll
ð20Þ
The lift force coefficient can be modeled with or without considering the bubble deformation, i.e. considering the bubble as sphere [17] or non-sphere [18,19]. Usually, when the bubble size is small (dg < 4.4 mm), the bubble can be treated as a sphere moves inside the liquid [19]. However, when bubble’s diameter is great, it will become flat along the motion direction [13,19]. Ideally it can be treated as an oval shape [13,19]. In this paper, the lift force formula based on the bubble deformation is adopted. After combining the shear lift force [20–22] and effects from slanted wake, the lift force model, which is Eötvos number dependent, had been put forward by Tomiyama [18] and Tomiyama et al. [19].
8 0:288 tanhð0:121Reb Þ ; f ðEog Þ ; Eog < 4 > < min C l ¼ f Eog ; 4 Eog 10 ; > : 0:29; Eog > 10
ð21Þ
where the function f(Eog) can be calculated by the following experimental formula [19].
f ðEog Þ ¼ 0:00105Eo3g 0:0159Eo2g 0:0204Eog þ 0:474;
ð22Þ
in which Eötvos number (Eog) is defined as: 2
Eog ¼
jgjðql qg Þdh
r
;
ð23Þ
in which r is surface tension, dh is the maximum horizontal bubble dimension when moving bubble has the deformation. Through dh, we can see the deformation effects on bubble are considered in the lift force. The maximum horizontal bubble dimension dh can be calculated through the formula of Wellek et al. [23].
1=3 dh ¼ dg 1 þ 0:163Eo0:757 ;
ð24Þ
where Eo is the Eötvos number depending on the sphere bubble diameter. The formula of Eo, shown in Eq. (25), is similar with Eq. (23). 2
Eo ¼
jgjðql qg Þdg
r
:
ð25Þ
The virtual mass force normally generated due to the motion of bubbles. When a bubble moves in the liquid, it disturbs the liquid around the bubble. Some of liquid near the bubble will be accelerated, accordingly the kinetic energy of the liquid surrounding the sphere bubble changes. Therefore, the movement of the bubble will be affected by the accelerated liquid. Accordingly Drew and Lahey [22] studied this phenomenon and developed a formula to describe the virtual mass force, shown in Eq. (15). The virtual mass force coefficient is generally bubble-shape and gas hold up dependent [13]. It seems case Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx
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Table 2 Relaxation factors for computed variables. Variable
Um
p
k
e
Factor
0.7
0.3
0.8
0.8
dependent according to the former studies. Mudde and Simonin [24] found the virtual mass force played important role on predicting the bubble plume oscillations. However, some of researches [2,25] found the virtual mass force can be negligible in their simulations. According to the former studies, in this paper the virtual mass force coefficient is treated as a small constant number. It means the virtual mass force is not negligible totally, but it has not a strong influence on the movement of bubble. 4. Numerical procedures During the simulations, all differential equations of the multi-dimensional Lagrangian algebraic slip mixture model are transformed into difference equations by the finite-volume integral method [26,27]. During the discretization of the equations, the derivatives of the all variables with respect to time are discretized using Euler implicit time scheme. The central-difference scheme and the QUICK scheme are used to discretize the diffusion terms and the convective transport terms of the equations of the multi-dimensional Lagrangian algebraic slip mixture model respectively [26]. The difference equations are solved numerically with an iterative solution procedure through the SIMPLE method [26,27]. During the numerical calculations, an under-relaxation method is utilized to accelerate the convergence. The relaxation factors used in the calculations are given in Table 2. The turbulent kinetic energy and turbulent kinetic energy transport of the mixture at inlet will be calculated by the method recommended by Shang [6]. The specific equations are shown as follows:
kin ¼ 0:005V 2m;in ;
ð26Þ
3=2
ein ¼
kin ; 0:025 ð2:5Dh Þ
ð27Þ
in which Dh is the hydraulic diameter of the inlet and Vm,in is the magnitude of the inlet velocity of the mixture which is normal to the inlet interface. The standard wall function was employed to predict the variables near wall [10]. Initially only the liquid fully fills the column. There is no gas, i.e. ag = 0 and al = 1, inside the column at the beginning. The gas is injected into the column at the bottom with ag = 1 and al = 0. The injected gas velocity therefore equals to the gas superficial velocity and the liquid velocity equals 0 m/s as the inlet condition. The gas is discharged at the top of column with ag = 1 and al = 0 as the outlet condition. It means that only gas flows out the column and the liquid flows back into the column at the outlet. 5. Numerical simulations Two typical kinds of bubble columns, which are cylindrical bubble column and rectangle bubble column, were chosen as the targets in this paper. The cylindrical bubble column is basically a cylindrical vessel with a gas distributor at the bottom. The gas is sparged in the form of bubbles into either a liquid phase or a liquid–solid suspension [28]. Similar with the cylindrical bubble column, the rectangle bubble column is basically a rectangle vessel with a gas distributor at the bottom. 5.1. Cylindrical bubble column reactor Sanyal et al. [1] did the experiment and numerical simulation on a cylindrical bubble column reactor. Hereafter, the experiment data has been cited by other researchers [13,14,29,30]. The column contained a batch liquid with unexpanded height of 104.5 and 95.0 cm, respectively. The distributor used in the experiments was a perforated plate, with 0.33 mm diameter holes in a square pitch, with an open area of 0.1% [1]. The gas superficial velocities were operated from 2.0 to 12.0 cm/s. The corresponding flow characteristic is from bubbly flow to churn turbulent flow. Owing to the wide range of the flow regimes, these experimental and numerical data were chosen as the comparisons in this paper. During the numerical simulations, all the parameters were used same as Sanyal et al. [1]. The air and water bubble columns were under atmospheric pressure operating at these superficial velocities with a perforated plate sparger. The grid size was employed same as Sanyal et al. [1], which was 300 (axial) 19 (radial) rectangular grid. The mesh cell size was 0.66 cm (axial) 0.5 cm (radial) uniformly. The finer meshes were employed to test the mesh sensitivity before the formal simulations. There were no evident differences of the simulation results between the current mesh and finer meshes. The mesh cell size used in this paper enables the simulation results on mesh independent. The formal simulations were solved Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx
in a transient fashion with a time step of 0.01 s and carried out long enough to statistic the stable time-averaged variables same as Sanyal et al. [1]. On performing the numerical simulations, Sanyal et al. [1], Chen [13] and Chen et al. [14], Wang and Wang [29] made the similar analysis and obtained the same conclusion that the lift force and virtual mass force can be ignored for the cylindrical bubble column reactor, and only the buoyancy and drag force were considered. In this paper, same as the former researchers, the lift force is excluded during the simulations. Although the virtual mass force is not ignored, the virtual mass force coefficient was chosen as a small number, for example Cvm = 0.02, to weaken the effects of the virtual mass force. The buoyancy and drag force were kept during the simulations. Fig. 1 shows the comparisons of time average gas holdup (i.e. gas volume fraction, ag) and axial water velocity under the flow regime of gas superficial velocity Ug = 2 cm/s at the height z = 530 mm from the bottom of the bubble column reactor. From the comparisons in Fig. 1(a), it can be seen that the current multi-dimensional Lagrangian algebraic slip mixture model can have the predicted gas holdup values similar with the algebraic slip mixture model and two-fluid model. However, in Fig. 1(b), the predicted profile of the axial water velocity by the current multi-dimensional Lagrangian algebraic slip mixture model is more approaching to experiments than the algebraic slip mixture model and two-fluid model. The zero point of axial water velocity is at r/R = 0.7, same as the experiment and the predictions of the algebraic slip mixture model and two-fluid model. Fig. 2 shows the comparisons of time average gas holdup and axial water velocity under the flow regime of gas superficial velocity Ug = 12 cm/s at z = 530 mm. From the comparisons in Fig. 2(a), it can be seen that the current multi-dimensional Lagrangian algebraic slip mixture model predicted gas holdup values near experiments between the predictions of the algebraic slip mixture model and two-fluid model. The maximum value of the gas holdup approached to 0.38 of the experimental measure. It illustrated that the multi-dimensional Lagrangian algebraic slip mixture model can be used for both of dilute and dense dispersions in gas liquid two-phase flow, although the slip velocity model was based on the single bubble movement.
(a) distribution of gas holdup
(b) distribution of axial water velocity
Fig. 1. Comparisons under gas superficial velocity Ug = 2 cm/s.
(a) distribution of gas holdup
(b) distribution of axial water velocity
Fig. 2. Comparisons under gas superficial velocity Ug = 12 cm/s.
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Fig. 2(b) shows the predicted axial water velocities by the current multi-dimensional Lagrangian algebraic slip mixture model. It can be seen the predicted results are more accurate to experiments than algebraic slip mixture model and two-fluid model. The zero point of axial water velocity is at r/R = 0.69 near the experiment point. 5.2. Rectangle bubble column reactor Zhang et al. [5] did the simulations for a rectangle bubble column. The width, depth and height of the bubble column are, respectively, set to W = 0.15 m, D = 0.15 m, and H = 0.45 m or H = 0.90 m. Therefore there were two cases set up as H/D = 3 and H/D = 6. The gas distributor is situated in the bottom plane of the column at a distance of 0.06 m from each of the surrounding walls of the column. The entire column is initially filled with water, which acts as the continuous liquid phase. Air is employed as the dispersed gas phase and sparged at the bottom plane with the inlet superficial gas velocity of 4.9 mm/s. The gas–liquid flow regime is bubbly flow. The mesh size was employed same as Zhang et al. [5], which was Dx = Dy = Dz = 0.01 m along the three coordinates. Same as the cylindrical bubble column, the mesh sensitivity was tested before the formal simulations, and current the mesh size can guarantee the results on mesh independent. The formal simulations were solved transiently with a time step of 0.01 s and carried out long enough to statistic the stable time-averaged variables same as Zhang et al. [5]. Due to the non-symmetrical flow along angle direction, the flow situation inside the rectangle bubble column is different from the total symmetrical flow of the cylindrical bubble column. The transverse force acted on the bubbles cannot be ignored. The lift force therefore has to be considered [5]. Zhang et al. [5] found the bubble deformation occurred in the rectangle bubble column. Hence the lift force coefficient, which considers the influence of bubble deformation shown in Eq. (21), is employed in this paper. Same as the cylindrical bubble column, a small number of the virtual mass force coefficient was used to weaken the influence of the virtual mass force. The buoyancy and drag force were considered as the basic force on the bubble during the simulations. Fig. 3 shows the distributions of the timely averaged axial liquid velocity vector at x–z cross section. From Fig. 3, it can be seen that the liquid flows up in the middle, turns the flow direction at the top and flows down along the sides of the column towards the bottom. A big circular flow is generated inside the bubble column. Fig. 4 shows the comparisons of time-average gas and water axial velocity of the bubble column H/D = 3. Fig. 4(a)–(c) show the distributions of the time average axial gas velocity at different heights, and Fig. 4(d)–(f) show the distributions
(a) H/D = 3.0
(b) H/D = 6.0
Fig. 3. Time averaged axial liquid velocity vector at x–z cross section.
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
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(a) z/H=0.57
(d) z/H=0.57
(b) z/H=0.63
(e) z/H=0.63
(c) z/H=0.72
(f) z/H=0.72
Fig. 4. Comparisons of gas and liquid axial velocity under H/D = 3.0.
of the time-average axial liquid velocity at different heights. From the comparisons it can be seen that the current Lagrangian algebraic slip mixture model can predict the profiles of the velocity more approaching experiments, especially for gas velocities. Fig. 5 shows the comparisons of time-average gas and water axial velocity of the bubble column H/D = 6. Fig. 5(a) and (b) show the distributions of the time average axial gas velocity at different heights, and Fig. 5(c) and (d) show the distributions of the time-average axial liquid velocity at different heights. Same as the comparisons shown in Fig. 4, the current Lagrangian algebraic slip mixture model can predict the profiles of the velocity more approaching experiments.
Please cite this article in press as: Z. Shang et al., A multi-dimensional Lagrangian algebraic slip mixture model for bubble column reactors, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.04.050
Z. Shang et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx
(a) z/H=0.57
(c) z/H=0.57
(b) z/H=0.73
(d) z/H=0.73
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Fig. 5. Comparisons of gas and liquid axial velocity under H/D = 6.0.
6. Conclusion The multi-dimensional Lagrangian algebraic slip mixture model is developed based on the mixture multiphase flow model. The diffusion velocity between the dispersed phase and the mixture is closured through the slip velocity between the dispersed phase and the continuous phase. Accordingly the novel multi-dimensional Lagrangian algebraic slip mixture model is different from the general homogenous model that treats the dispersed phase and the continuous phase flowing under same velocity on multiphase flow. The slip velocity is modeled by the single bubble Lagrangian motion equation. The forces of buoyancy, drag, virtual mass, transverse lift and so on are therefore able to be considered as the usual two-fluid model. It makes an innovation on the traditional algebraic slip mixture model and combines both of the merits of the simplicity of the mixture model and accuracy of the two-fluid model. Through the comparisons of the numerical simulations between the multi-dimensional Lagrangian algebraic slip mixture model and experiments, the model is validated. Through comparisons of the numerical simulations between the multi-dimensional Lagrangian algebraic slip mixture model and the numerical simulations of traditional algebraic slip mixture model and two-fluid model, the efficiency and accuracy of the model is confirmed. Acknowledgements The authors would like to thank the support by Multiphase Flow for Deep-Sea Oil & Gas Down-hole Applications - SERC TSRP Programme of Agency for Science, Technology and Research (A⁄STAR) in Singapore (Ref #: 102 164 0075). References [1] J. Sanyal, S. Vasquez, S. Roy, M.P. Dudukovic, Numerical simulation of gas–liquid dynamics in cylindrical bubble column reactors, Chem. Eng. Sci. 54 (1999) 5071–5083. [2] R. Krishna, J.M. Baten, Scaling up bubble column reactors with the aid of CFD, Chem. Eng. Res. Des. 79 (3) (2001) 283–309. [3] L.X. Zhou, M. Yang, C.Y. Lian, L.S. Fan, D.J. Lee, On the second-order moment turbulence model for simulating a bubble column, Chem. Eng. Sci. 57 (2002) 3269–3281.
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