The Effect of Voidage on the Drag Force on Particles, Droplets and Bubbles in Dispersed Two-Phase Flow H. Rusche and R.I. Issa Department of Mechanical Engineering Imperial College of Science, Technology and Medicine Exhibtion Road, London SW7 2BX email:
[email protected] ABSTRACT In dispersed two-phase flow, experimental data indicate that the effective drag force on individual dispersed elements∗ is strongly influenced when the dispersed phase is present at high volume fractions. This appears to be true whether the DPEs are solid or fluid (i.e. droplets or bubbles). The paper first reviews the literature relating to the modelling of the drag force in densely dispersed solid-liquid, gas-liquid and liquid-liquid systems. The various models are then compared with an extensive set of data obtained from the literature in order to assess their validity. The paper will show that the various models often give widely different predictions and can even give the wrong trend. In order to improve the prediction capabilities, new correlations to express the influence of the value of phase fraction on the drag force in such systems are formulated. Theses are then validated by comparing their predictions for the slip velocity against available experimental data.
1.
INTRODUCTION
Many if not most industrial two-phase flow applications involve high phase fractions of the order of 10% or more. Therefore, flow simulations, using either the Eulerian twofluid model or the Lagrangian particle tracking approach, should take this effect into account, if the flow overall characteristics are to be predicted with reasonable accuracy. The effect of phase fraction on drag which in most flows is the dominant force has already been the subject of investigation by several workers. The results of such studies have been presented in different types of correlations. Some are based on the drift-flux model, some introduce a coefficient multiplier (friction factor) for the drift velocity, while others correlate the ratio of drag coefficient at high phase fraction to that of a single particle. 2.
DEFINITIONS
The momentum transfer rate per unit volume from a cluster of dispersed elements to a surrounding continuous phase is given as: Md =
αFd V
(1)
where α is the volume fraction of the dispersed phase, V is the volume of a dispersed element and Fd is the drag force on an individual dispersed element, which is calcu∗ In this paper we use the term “dispersed element” for bubbles, droplets and particles and the term “fluid particle” for bubbles and droplets
lated from the definition of the drag coefficient as : 1 Fd = Cd ρc A|Ur |Ur (2) 2 where ρc is the density of the continuous phase, A is the projected cross-sectional area and Ur is the relative velocity. For practical reasons, the projected area A is usually 2 calculated from A = π d4 , neglecting the influence of deformation on the projected area and defining d as the effective diameter, i.e. diameter of a sphere having the same volume V as the dispersed element. Some authors also use a friction coefficient to model the interphase momentum transfer due to drag directly as : Md = Cf Ur
(3)
Note that Cf is not dimensionless. The analysis of the motion of single dispersed elements can be extended to a group of dispersed elements if the differences are properly recognised and accounted for. When a dispersed element moves in a two-phase medium the magnitude of the buoyancy force is lower because of the effect of the mixture density and the balance between gravity and drag now has to account for this pseudo-hydrostatic effect of the mixture (see [4, 6, 13]). Thus for a dispersed element moving in equilibrium state in uniform flow : 1 (4) g∆ρV (1 − α) = ρc ACd |Ur |Ur 2 This can be rewritten in terms of the Archimedes and Reynolds number as : 3 (5) Ar(1 − α) = Cd Re2 4 where the Archimedes number and Reynolds number are defined in the usual manner (see nomenclature).
Modelling Approach
Particles
Friction Factor
Drag coefficient mutiplier
Drift Flux
–
Wallis [1] Richardson and Zaki [2]
–
new correlation
Droplets
–
Wallis [1]
Kumar and Hartland [6]
Bubbles
Schwarz and Turner [10]
Wallis [1]
Wen and Yu [11]
Pairwise Interaction
Mixture Viscosity
Ergun Type Correlation
–
Ishii and Zuber [3]
Barnea and Mizrahi [4]
Andersson [5]
new correlation
–
Ishii and Zuber [3]
Barnea and Mizrahi [7] Kumar et al. [8]
Pilhofer [9]
new correlation
Johansen and Boysan [12]
Ishii and Zuber [3]
–
–
Table 1: Modelling approaches for drag at high phase fraction 3.
MODELLING APPROACHES
Over the years several models have been proposed to calculate drag on dispersed elements at elevated phase fractions. They can be grouped into modelling approaches which are summarised in table 1. A short description of the main features of each category is given below : Friction Factor: The drag is given in the form of a friction factor and Eq. (3) is used in the determination of the momentum transfer due to drag. Drift flux: Drift flux correlations are derived by plotting the drift flux j21 ≡ α(1 − α)Ur as a function of the phase fraction. The data is then fitted to a function of the phase fraction and sometimes the Reynolds number [2]. Drag coefficient multiplier: In this approach, the ratio of the drag coefficient to its single dispersed element value is fitted to a function of the phase fraction, i.e. Cd Cd0 = f (α) where Cd0 is the drag coefficient for a single dispersed element. Pairwise Interaction: The effect of the phase fraction on the drag is derived from considering the pairwise interactions between two particles. Mixture Viscosity: Here the presence of other dispersed elements is taken into account by considering their effect on a mixture viscosity. This mixture viscosity is then used to define a Reynolds number which can in turn be used with a suitable drag correlation. Ergun type correlation: The correlation by Ergun [14] provides the pressure drop across a closely packed bed of particles as a function of the Reynolds number. It has been extended to cover fluidised beds of particles and swarms of particles and droplets by introducing additional, empirical parameters. 4.
NEW CORRELATIONS
Most of the correlations discussed in the previous sections suffer from the limitation that they do not revert to a single dispersed element drag correlation when the phase fraction
approaches zero. Even if they incorporate a single dispersed element model, the model often does not represent the current state of the art. This is especially the case for correlation which cover systems of droplets. It therefore seems desirable to formulate the effects of the presence of other dispersed elements independently from the model used in the low phase fraction limit. In this respect the new approach is similar to the drift flux approach. In the new correlation, the drag coefficient is expressed as a correction to the drag coefficient on a single dispersed element : Cd = Cd0 f (α)
(6)
where f (α) is a function which takes into account the effects arising from the presence of other dispersed elements. An obvious constraint on f (α) is that it must approach unity when the phase fraction vanishes. f (α) is probably not only a function of the phase fraction, but also a function of the Archimedes or Reynolds numbers as well as turbulence. As a first step, f (α) is modelled here as a function of the phase fraction only. Hereby, a suitable functional form for f (α) is chosen in the light of the experimental data described in more detail in section 5. and the model constants are then evaluated using a non-linear fitting procedure. Experimental values for the function f (α) are derived from the data and are presented in Figure 1 for particles, droplets and bubbles. In general, f (α) increases with increasing phase fraction. It is also evident, that f (α) tends to unity for very low phase fractions, although for bubbles there is a large amount of scatter indicating that not all important effects are taken fully into account. This will be discussed further in section 5.3.. For particles, a weak but discernible and systematic dependence on the Archimedes number is noticeable from figure 1a. This gives rise to the conclusion that f (α) should also be a function of the Archimedes or Reynolds number. Unfortunately, for fluid particles, a systematic dependency cannot be found (figure 1b and c) where other effects seem to have a stronger influence. With reference to figure 1a and b it is assumed that f (α) increases exponentially for high phase fractions, whereas for lower phase fractions up to α = 0.3, f (α) seems to initially rise very rapidly and then tail off. Therefore,
a combination of a power law and an exponential function is chosen as a fitting function for f (α) :
a)
8
6
f [−]
f (α) = exp(K1 α) + αK2
new Correlation (P) 2 Ar = 8.3 10 5 Ar = 2.1 10 5 Ar = 1.2 10 6 Ar = 1.9 10 6 Ar = 1.8 10
(7)
where model coefficients K1 and K2 are determined by using a non-linear fitting procedure. Their values are given in table 2. 5.
4
APPLICATIONS AND RESULTS
In this section the various models are compared to each other in order to assess their validity and to recommend models suitable for flow simulations.
2
5.1. Particles 0
b)
0
0.2
0.4 α [−]
0.6
8 new Correlation (D) 30 < Ar < 91 10 < Ar < 82 2 2 1.1 10 < Ar < 4.2 10 5 6 3.5 10 < Ar < 1.4 10 5 6 5.6 10 < Ar < 1.3 10 4 Ar = 8.6 10 5 Ar = 1.2 10 5 Ar = 1.2 10 5 Ar = 1.1 10
f [−]
6
4
2
0
c)
0
0.2
0.4 α [−]
0.6
0.8
10
8
f [−]
6
4 new Correlation (B) 4 5 5.4 10 < Ar < 1.5 10 6 6 1.5 10 < Ar < 2.5 10 6 7 2.5 10 < Ar < 5.0 10 6 7 5.0 10 < Ar < 1.0 10
2
0
0
0.2
0.4 α [−]
0.6
0.8
Figure 1: Experimental values for the function f (α) and best fits to Eq. (7). a) Particles, b) Droplets, c) Bubbles.
K1 K2
Particles 2.68 0.430
Droplets 2.10 0.249
Bubbles 3.64 0.864
Table 2: Model coefficients for in new correlations
The dynamics of falling particles has been researched quite extensively and a large body of literature is available. The existing experimental data have been reviewed by [4, 22]. In this study, 4 data sets from 3 sources have been chosen for comparison so that a wide range of the Archimedes number is covered. They are tabulated in table 3. Figure 2 shows the relative velocity normalised by single particle values for data sets P1 and the relative velocity for data set P4. The quantities are plotted as functions of the phase fraction. The model by Andersson [5] is not valid for α < 0.05 and we use dashed lines to indicate this. Dataset P1 contains data for glass and steel spheres falling in glycerine-water solution for very low Reynolds numbers as evidenced in table 3 and the flow is therefore in the creeping flow regime. For this data set it is apparent from figure 2a that all models predict the same trend, namely that the relative velocity decreases with increasing phase fraction. All models give very similar results, although some differences are noticeable for the model by Barnea and Mizrahi [4] and the new correlation for α < 0.3. The results for much higher Archimedes and Reynolds numbers are plotted in figure 2b (P4). It is evident that the Ishii and Zuber [3] model substantially underpredicts the drag for low phase fractions and overpredicts it for large ones. There is noticeable difference between the formulations for the undistorted and Newton regime, largely due to the different formulations in the single particle limit. Furthermore, the Ur − α relationship is a concave function of α which is not in agreement with the experimental findings or with the predictions of the other models. This is a result of the expression for the mixture velocity, which seems to underpredict the mixture viscosity at low phase fractions and to overpredict it at high ones. Barnea and Mizrahi [4], whose model also uses a mixture viscosity approach, seem to compensate for this deficiency by modifying the definition for Cd by a factor of (1 − α)1/3 . This leads to a sharp increase in the drag coefficient for small α which gives better agreement with the experimental data for high Archimedes numbers, but introduces a problem at low Archimedes numbers where the relative velocity for α < 0.2 is underpredicted. The model by Wen and Yu [11] performs well for data set P3 (not shown), but largely overestimates the relative velocity for α < 0.4. This seems to be due to the underlying single particle model (Schiller and Naumann [23]), which is not valid for Re0 > 1000.
Set
System steel spheres, glass spheres in glycerinewater solution
Armin
Armax
0.05 α < 0.05
1.5 Ur / U0 [m/s]
Hartland [6]). Plots showing the results for kerosene droplets rising in water are given in figure 3b (D4). The model predictions by Pilhofer [9], Kumar et al. [8], Barnea and Mizrahi [7] and the new correlation are in good agreement with the experimental results. From figure 3a and b it is noticeable that Pilhofer’s [9] correlation exhibits a small jump at α = 0.55 where the model formulation is not continuous. This is unphysical and could lead to serious numerical problems when used in a CFD code. In essence, it is evident from the previous results that the dynamics of fluid particle systems is much more complex than that of solid particle systems : firstly the predictions of the various models deviate more strongly; and secondly we find larger differences between the model predictions and the experimental results. But overall, the models given by Pilhofer [9] and Kumar and Hartland [6] as well as the new correlation seem to give the best predictions for the terminal velocity of droplets in liquids. As mentioned earlier, Pilhofer’s [9] correlation suffers from the deficiency that it cannot be applied to systems at low phase fraction, where α < 0.06. On the other hand, Kumar and Hartland’s [6] correlation does not use a state of the art single droplet drag law which limits its applicability. Hence both correlations lack the generality required for wide use and we recommend to use our new correlation.
1
0.5
0
0
0.1
0.2 0.3 Turbulence intensity [−]
0.4
Figure 5: Plot of the normalised bubble velocity against turbulence intensity plotted against the turbulence intensity in figure 5. Data points pertaining to phase fraction less than 5% are plotted as red circles. The strong effect of the turbulence intensity in reducing the rise velocity is clearly noticeable. Although it has to be noted that bubble size is not constant and the bubble diameter varies in the range between 1.8 and 2.8 mm. Plots showing the results for bubbles with a diameter of between 3.7 and 4.7 mm are given in figure 4b (B4). As mentioned earlier, the relative velocity is a concave function of the phase fraction for the model by Ishii and Zuber [3]. Consequently, the model substantially underpredicts the drag for α = 0.3 and overestimates the relative velocity by up to 50%. Not surprisingly, large differences between the distorted and the undistorted formulations of the model are noticeable. However, it is interesting to note that the distorted formulation predicts lower relative velocities and therefore agrees better with the experimental data than the undistorted formulation. This is not surprising, since the E¨ otv¨ os numbers in this regime are high enough to justify the use of Harmathy’s [25] simple drag law. Furthermore, large discrepancies between the different model predictions are noticeable from figure 4 and it is surprising that the models do not even give a consistent trend for the influence of the phase fraction. Two models do not predict decrease of the relative velocity with increasing phase fraction, contrary to experimental evidence; they are the models of Johansen and Boysan [12] and Schwarz and Turner [10]. The latter assumes a constant slip velocity and a reduction of the drag coefficient is a consequence of this modelling assumption. The model by Johansen and Boysan [12] is derived by considering the special case of two particles behind each other for which experimental results are available. The result is then cast into a relationship for the drag coefficient. With reference to the experimental data it is obvious that these modelling assumptions do not hold. Unfortunately, the uncertainties in the experimental data, especially related to turbulence effects, make a quantitative assessment very difficult. Nevertheless, it is evident from the previous results that overall, the new correlation seems to give the best predictions for the terminal
velocity of bubbles in liquids at high phase fraction. 6.
CONCLUSIONS
This paper reviewed and formulated models to determine the drag force in dispersed two phase systems at high concentrations of the dispersed phase. It presented a comprehensive literature review of the state-of-the art in drag modelling for particles, droplets and bubbles at low and elevated phase fractions. New correlations for modelling drag at high phase fractions are put forward. The models are then compared against experimental data from the literature as well as recently obtained data. In the case of drag models for solid particles at high phase fractions it can be concluded that a number of models give reliable results. Although, it should be noted that the new correlation proposed requires some refinement at low Archimedes/Reynolds numbers. Overall, the model by Garside and Al-Diboui [22] meets the requirements in terms of accuracy and generality best. When comparing models for liquid-liquid systems, it becomes evident that the dynamics of droplets is not as well understood as that of solid particle systems. This is apparent from the deviation between the various models as well as the large differences between the model predictions and the experimental results. It should also be noted that currently no accurate and reliable model for the drag on single droplets exists. This issue needs to be addressed by future research. Nevertheless, some of the models for elevated phase fractions were able to give reasonable predictions, although some of them lacked generality. A large number of models for the drag on single bubbles in an infinite domain have been published. However, the effect of turbulence, especially on smaller bubbles is not well understood. This uncertainty limits the quality of the predictions at higher values of the phase fraction and should be addressed in future research effort. Overall, it is evident from the results that the new correlation gives the best predictions for the rise of swarms of bubbles. 7.
ACKNOWLEDGEMENTS
Financial support for this work was provided by the Commission of the European Communities in the framework of the BRITE-EURAM III programme.
NOMENCLATURE Roman Symbols Sym. A
Description Area
Ar
Archimedes number ≡
Cd Cf d Eo F g j21 K M Re U V
d3 gρc |∆ρ| µ2c Fd Drag coefficient ≡ 1 ρ A|U r |Ur 2 c Friction coefficient diameter g|∆ρ|d2 E¨ otv¨ os number ≡ σ Force (eg. for drag force) Acceleration due to gravity Drift flux Dimensionless coefficient Interfacial momentum transfer rate ρc |Ur |d Reynolds number ≡ µc Velocity Volume
Units m2 − − kg/(m3 s) m − N m/s2 m/s − kg/(m2 s2 ) − m/s m3
Greek Symbols Sym. α µ ρ σ
Description Volume fraction Dynamic viscosity Density Surface tension
Units − N s/m2 kg/m3 N/m
Subscripts Sym. Qc Qd Qr Q0
Description Value of Q in the continous phase Value of Q in the dispersed phase Relative Value of Q between two phases Value of Q for a single DPE
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