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Predicting the static liquid holdup for cylindrical packings of spheres in terms of the local structure of the packed bed
Chemical Engineering Science 65 (2010) 6181–6189
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Predicting the static liquid holdup for cylindrical packings of spheres in terms of the local structure of the packed bed S. Schwidder , K. Schnitzlein Brandenburg University of Technology, Department of Chemical Reaction Engineering, Burger Chaussee 2, 03044 Cottbus, Germany
a r t i c l e in fo
abstract
Article history: Received 20 May 2010 Received in revised form 25 August 2010 Accepted 2 September 2010 Available online 8 September 2010
A discrete 3D model has been developed for the prediction of static liquid holdup in cylindrical packed beds of spheres incorporating the local structural properties of the packing. The model predictions have been compared to experimental data covering a wide range of relevant physical properties of liquid– solid systems for aspect ratios dt =dP between 3.5 and 25. A notably good agreement between prediction and experiments is observed. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Static liquid holdup Multiscale analysis Multiphase reactor Multiphase flow Packed bed Hydrodynamics
1. Introduction The static liquid holdup in a packed bed is defined as the volume fraction of liquid referred to the total bed volume that remains in the bed after complete draining (Saez et al., 1991). In the characterization of the hydrodynamics of gas–liquid flow through packed beds the static liquid holdup is commonly used as a parameter. Moreover, the static liquid holdup is relevant in the modelling of mixing and transport processes in gas–liquid flow through packed beds, in which an amount of liquid equal to the static liquid holdup is commonly considered as a stagnant region under gas–liquid flow conditions. It is well established that the static liquid holdup depends on the liquid’s physical properties (e.g. density), the geometrical properties of the solid (e.g. particle size and shape), the gas–liquid interfacial properties (e.g. surface tension) and, in addition, on the local geometrical properties of the packing (Saez et al., 1991). Due to the high complexity of the local geometry finding, a relation which takes into account the local structural properties of the packing seemed to be a formidable problem. Therefore, in the past empirical relations have been proposed either neglecting the impact of local geometry (e.g. Charpentier and Favier, 1975) or using some averaged characteristic property of the packed bed (e.g. mean voidage (Kawabata et al., 2005; Kramer, 1998)) to
represent the impact of the local geometry on the static liquid holdup. Recent studies (Lappalainen et al., 2009; Ortiz-Arroyo et al., 2003; van der Merwe et al., 2004) on static liquid holdup assume that the liquid is present in pendular structures located at the contact points between particles. Thus, the static liquid holdup is obtained as a function of the volume of a single bond and the average number of contact points. Again, only averaged properties are used to represent the impact of geometrical properties of the packed bed. The most promising approaches for predicting static liquid holdup were compared to experimental data as obtained in our lab. The experiments were carried out by varying the liquid’s physical properties, particle shape, size, wettability and the properties of the packing as well. The static liquid holdup was measured for cylindrical packings of spheres by use of the drainage method. Details of the experimental procedures are given below. The results are shown in Fig. 1. The poor agreement between calculated and experimental data may be regarded as a hint that a significant effect on the static liquid holdup is not adequately accounted for in these correlations. As will be shown in this contribution, this effect is identified as the local structural properties of the packing.
2. Model development Corresponding author.
E-mail addresses: [email protected] (S. Schwidder), [email protected] (K. Schnitzlein). 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.09.004
Following the approach of Lappalainen et al. (2009) the static liquid holdup can be calculated by summing up the volumes of all
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pendular liquid bonds between two neighbouring spheres and between the spheres near the wall and the wall itself. It is important to note that if the particles are sufficiently small, the liquid retained in the bed is no longer the product of pendular liquid bonds but a consequence of the formation of liquid blobs that engulf several particles. Our analysis is restricted to particles large enough, so that the latter mechanism is of less importance.
Moreover, the analysis is confined to packings of spheres in a cylindrical container. 2.1. Volume of a single bond In order to represent the geometry of a single liquid bond, the model of Pietsch and Rumpf (1967) is used. Simplifying the real geometry of the liquid bond they represented the curvature of the liquid meniscus by means of a circular arc. As can be seen from Fig. 2 the predicted geometrical shape closely resembles real menisci for sphere–sphere contact points as well as for spherewall contact points. In addition, the geometrical details for a liquid bond connecting two spheres or a sphere and the cylinder wall are sketched below. Thus, the volume of a liquid bond between two spherical particles located at a given distance a can be calculated as n VS ¼ 2p ½R21,S þ ðR1,S þ R2,S Þ2 R1,S cosðbS þ yS Þ 1 ½R31,S cos3 ðbS þ yS Þ 3 h ðR1,S þ R2,S Þ R21,S cosðbS þ yS ÞsinðbS þ yS Þ p i bS yS þR21,S 2 1 3 dP ð2 þcosbS Þð1cosbS Þ2 24
ð1Þ
with the curvature radii R1,S ¼ R1 ðyS , bS Þ
Fig. 1. Comparison between measured and predicted static liquid holdup (20% variance).
and
R2,S ¼ R2 ðyS , bS Þ
ð2Þ
defined in terms of the contact angle yS and the half-filling angle between two spheres bS . A similar equation is set up for the volume of the liquid bond for sphere-wall contactpoints in terms of the contact angle Yw
Fig. 2. Sketch of a single liquid bond between two spheres (left, bottom) and between a sphere and the cylinder wall (right, bottom) in comparison with observed geometry (photography (top)).
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and the half-filling angle bW . 1 1 VW ¼ VS ðbW , yS Þ þ p ½R31,W cos3 ðbW þ yW Þ 2 3 þ ½R21,W þ ðR1,W þR2,W Þ2 R1,W cosðyW þ bW Þ ðR1,W þ R2,W Þ R21,W cosðbW þ yW ÞsinðbW þ yW Þ p io bW yW þ R21,W 2
ð3Þ
with the curvature radii R1,W ¼ R1 ðyW , bW Þ
and
R2,W ¼ R2 ðyW , bW Þ
ð4Þ
which can be calculated as R1 ðy, bÞ ¼
dP ð1cosbÞ þa 2cosðb þ yÞ
ð5Þ
R2 ðy, bÞ ¼
dP sinb þ R1 ½sinðb þ yÞ1 2
ð6Þ
2.2. Half-filling angles As can be seen from Fig. 2 the volume of a pendular liquid bond at a wall contact point depends on both the radius of the sphere and the radius of the cylindrical container. In the equations presented above, the impact of cylinder diameter is only implicitly accounted for by means of the half-filling angle bW at the wall. Consequentially, bW should be a function of cylinder diameter dt. In addition, both half-filling angles are considered to depend on the diameter of the spherical particles dP,
bS ¼ bS ðdP Þ and bW ¼ bW ðdP ,dt Þ These dependencies are exemplarily shown in Fig. 3 for the system water/glass as obtained experimentally. Pietsch and Rumpf (1967) proposed equations to calculate the maximal and minimal possible half-filling angle for a given liquid–solid system. While the minimal values are about zero
Fig. 3. Half-filling angle vs. particle diameter for water–glass system.
Fig. 4. Comparison between measured and calculated (Pietsch and Rumpf, 1967) half-filling angle (no separation distance, 20% variance).
degree, the predicted maximal values significantly differ from any observed value as shown in Fig. 4. Therefore, since to our best knowledge no better predictive correlations are available in the literature, the half-filling angles have to be obtained experimentally for any liquid–solid system under consideration.
2.3. Critical separation distance In order to establish a liquid bond, the two spheres need not to be in close contact to each other. As long as the separation distance, i.e. the distance between the surface of two neighbouring spheres, does not exceed a critical threshold a liquid bond can form. Actually, the term contact point is misleading in this sense. Therefore, some authors are using the term near points (Pietsch and Rumpf, 1967) or near contacts (Bernal and Mason, 1960). Nevertheless, in this contribution we will refer to these points as contact points. Generally, the critical separation distance is experimentally obtained by pulling two particles being in contact apart until the liquid bond ruptures (e.g. Lian et al., 1993). Several predictive correlations can be found in the literature which are either based on experimental data (Herminghaus, 2005; Lian et al., 1993) or on theoretical considerations (Megias-Alguacil and Gauckler, 2009; Pietsch and Rumpf, 1967) (see Table 1). While the first are restricted to the liquid–solid system used, i.e. di-n-butyl phthalate/liquid paraffin mixture suspended in 0.2% v/v Teepol L solution or water and polythene spheres (Lian et al., 1993) or water–glass (Herminghaus, 2005), the latter should be applicable for any liquid–solid system. But, as our experimental results indicate after drainage, no liquid bond can develop for a distance as obtained by using these correlations. In Fig. 5, the predictions are compared to experimental data obtained for a variety of liquid–solid systems. As can be seen, the predicted critical separation distance is generally overestimated. Therefore, the critical separation distance were obtained experimentally for any liquid–solid system under consideration.
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Table 1 Correlations from literature for calculating the critical separation distance. Author Lian et al. (1993) Herminghaus (2005)
Pietsch and Rumpf (1967) Megias-Alguacil and Gauckler (2009)
Correlation y 1=3 acrit ¼ 1 þ Vi 2 y 1=3 2=3 acrit ¼ 1 þ ðV~ i þ 0:1 V~ i Þ with 2 Vi V~ i ¼ ðdP =2Þ3 a Balance of forces and o 0,2 dP 1=3 dP y 4 p Vrel 1þ with acrit ¼ 3 2 2 3 Vi Vrel ¼ 4 p ðdP =2Þ3
Fig. 6. Critical separation distance vs. inclination angle a for water–glass system and dP ¼0.01 m.
Fig. 5. Comparison between measured critical separation distance (this work) and correlations from literature (inclination angle 03 , 20% variance).
2.4. Inclination angle Based on a simple balance of forces acting on a single liquid bond it can be shown that the inclination angle of the separation line connecting the centers of two contacting spheres should have an impact on the critical separation distance, too. But, most of the correlations available in the literature do not take this effect into account. According to our experiments the inclination angle largely influences the critical separation distance. As shown in Fig. 6 for the water–glass system the critical separation distance doubles when the two spheres are arranged vertically instead of horizontally. The given equation is obtained by fitting the experimental data. 2.5. Local structural properties of the packing Following the approach as outlined above the volume of every single bond existing at a contact point can be calculated provided the location of every spherical particle inside the cylindrical tube is exactly known. Unfortunately, these geometrical data can only be obtained with large experimental effort. As was shown by Schnitzlein (2001) computer generated sphere packings closely resemble real packings, i.e. predicted
Fig. 7. Computer generated sphere packing dt =dp ¼ 10.
radial voidage distributions are in good agreement with reported literature data. Therefore, as a remedy all the necessary informations about the local structural properties of the packed bed are retrieved from such simulated sphere packings. A small section of a simulated packed bed is shown in Fig. 7. It is well known that the geometrical structure of a spherical packing is considerably disturbed by the cylinder wall causing a pronounced radial voidage distribution. This is exemplarily shown in Fig. 8 for two packings with different aspect ratios dt =dP . Due to the larger voidage in the vicinity of the cylinder wall the number of contact points per sphere exhibits a radial variation (see Fig. 9). The average numbers of contact points per spherical particle as obtained by simulation are compared to predictions based on correlations from the literature applicable in the range 0:2595o e o 0:4764 (see Table 2). The observable difference between the predicted and the calculated average number of contact points may be explained by the fact that only the number of close contact points can be obtained via these correlations. Additionally considering the near contact points, as was done in this study, the determined average
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Fig. 8. Respective radial voidage distribution (left—dt =dP ¼ 5, right—dt =dP ¼ 10).
Fig. 9. Radial distribution of relevant sphere–sphere contact and near points for water–glass system (left—dt =dP ¼ 5, right—dt =dP ¼ 10).
Table 2 Correlations for calculating the number of contact points, and calculated average number of sphere contact points. Author
van der Merwe et al. (2004) Kramer (1998)
Correlation
Application dt =dp ¼ 5 e ¼ 0:46
dt =dp ¼ 10 e ¼ 0:42
NC ¼ 22 ð1e Þ2 3:1 NC ¼
6.4
7.4
6.7
7.4
3:42
6.3
7.0
6.7
7.4
7.4
8.2
e
Lappalainen et al. (2009) Ortiz-Arroyo et al. (2003) This work
NC ¼ NC ¼
e
1:18
3:1
e
Simulation
number of contact points is in good agreement with the experimental data (Bernal and Mason, 1960). Near the wall the two spheres comprising a contact point are found to be more distant than in the interior of the packing leading to a decreased number of contact points and an increased volume of the individual liquid bond. One is tempted to expect that these two effects may cancel each other out. According to Fig. 10 this is not the case. There the volume, as obtained by summing up the volumes of all liquid bonds with respect to bed volume, is plotted as a function of radial position for the
same packings as used in Figs. 8 and 9. A small but significant increase of liquid holdup with increasing radial position is found to occur. In the near vicinity of the wall the static liquid holdup is comprised solely by the liquid volume retained in the wall contact points. As can be seen from Fig. 10, the impact of wall contact points on static liquid holdup becomes more pronounced with decreasing curvature radius of the cylinder.
3. Model validation In order to validate the model predictions a large number of experiments have been carried out varying the physical properties of the liquid, the size and material of the spherical particles, the wettability and the properties of the packed bed as well. The respective experimental conditions are summarized in Table 3. The surface tension and the contact angle for each liquid–solid ¨ DAS 10 mk2 tensiometer. system were determined using a Kruss The half-filling angle were obtained by analysing and evaluating photographic images of pendular liquid bonds. In order to determine the critical separation distance a simple arrangement of two single spheres located at a certain distance was used. Thus, for a given distance and inclination angle the appearance of a liquid bond could be checked by draining the initially soaked spheres. The data were fitted yielding a simple
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Fig. 10. Static liquid holdup for water–glass system (left—dt =dP ¼ 5, right—dt =dP ¼ 10).
Table 3 Varied parameter. Parameter
Range of variation
Solid material Surface tension Liquid density Liquid viscosity Contact angle liquid/solid Particle diameter Cylinder diameter Packing height
glass, polyoxymethylene, vitrified clay 0.032–0.079 N/m 991–1181 kg/m3 1–2.5 and 100 mPa s 0–751 0.004–0.014 m 0.05 and 0.10 m 0.004–0.40 m
Table 4 Parameter for calculating the critical separation distance between two spheres with Eq. (7) for different liquids. Liquid Water Water Water Water Water Water
with with with with with
surfactant 5 wt% ethanol 20 wt% NaCl 15 wt% sugar 0.1 wt% xanthan
rL (kg/m3)
sL (N/m)
ZL (mPa s)
A (dimensionless)
B (dimensionless)
998 998 991 1181 1038 998
0.0725 0.032 0.0555 0.079 0.07325 0.0725
1.002 1.002 1.222 1.400 2.125 100
0.00095 0.0009 0.0009 0.0005 0.00045 0.002
0.00085 0.00045 0.00065 0.0008 0.001 0.0011
function representing the critical separation distance in terms of liquid properties and inclination angle.
4. Discussion 4.1. Aspect ratio
dP acrit ¼ ðAsina þ BÞ 0:01
ð7Þ
Respective values for parameters A and B are summarized in Table 4 for all liquid–solid systems under consideration. The static liquid holdup was determined by draining the initially soaked packed bed and measuring the amount of liquid retained in the packed bed gravimetrically as well as volumetrically. Fig. 11 shows a comparison of model predictions with experimental data of static liquid holdup. It is noted that these experimental data were already used in Fig. 1. By comparing the two figures the improvement as obtained by taking into account the local structural properties of the packed bed is obvious. Nevertheless, there is still a scattering of the data indicating that other effects may be important, too, which are not considered in this study.
Usually, the aspect ratio is used to represent the impact of cylinder diameter on static liquid holdup (e.g. Hochman and Effron, 1969). In order to assess this dependency the static liquid holdup was determined experimentally and by simulation for two different container diameters as a function of particle diameter. The results are shown in Fig. 12. It is obvious that the observed trends cannot be explained in terms of aspect ratio alone. Both, particle diameter and cylinder diameter need to be considered explicitly. The observed behaviour can be explained in terms of the two different contributions for static liquid holdup resulting from both the sphere contact points and the wall contact points. In addition, the simulated contributions are shown in Fig. 12 as dashed lines. As can be seen there the contribution to static liquid holdup resulting from the sphere contact points increases with aspect
S. Schwidder, K. Schnitzlein / Chemical Engineering Science 65 (2010) 6181–6189
ratio irrespective of the cylinder size. In contrast, the contribution resulting from the wall contact points becomes more pronounced for the small cylinder diameter, so that in this case the superposition of both contributions yields a maximum for the static liquid holdup for an aspect ratio of about 8. For the larger cylindrical diameter the impact of the wall contact points is almost neglectable.
that the static liquid holdup cannot be represented in terms of ¨ os ¨ number alone. Eotv As was shown in the previous section the sole use of the aspect ratio is insufficient for adequately predicting static liquid holdup even on a global level. Therefore, a new correlation was developed
estat ¼ 0:3804Eo€
0:4661
4.2. Global correlation
¨ os ¨ number as a function of a modified Eotv
For any practical application it is desirable to have at hand a simple correlation relating the static liquid holdup to ¨ os ¨ number representing both the liquid’s physical the Eotv properties and the wettability properties of the liquid–solid system. The comparison of predictions based on correlations available in the literature (Fukutake and Rajakumar, 1982; Kawabata et al., 2005; Lappalainen et al., 2009; Saez et al., 1991; van der Merwe et al., 2004; Weigert and Ripperger, 1999) and our experimental data as shown in Fig. 13 clearly indicate
Eo€ ¼
Fig. 11. Comparison between calculated and measured static liquid holdup for different liquid–solid systems and geometrical properties (20% variance).
6187
rL g d2p dt 1 1 sL dP 1 þ cosyS ðsin2bS Þ4
ð8Þ
ð9Þ
which incorporates the half-filling angle bS which is shown to depend on the sphere diameter (cf. Fig. 3). In order to apply Eqs. (8) and (9) for arbitrary liquid–solid systems the respective contact angle as well as the half-filling angle need to be experimentally determined, e.g. by means of tensiometer measurements and image analysis. As Fig. 14 shows the experimental data can be represented well by means of this global correlation.
¨ os ¨ number. Fig. 13. Static liquid holdup vs. Eotv
Fig. 12. Static liquid holdup vs. aspect ratio for the system water–glass and dP ¼0.01 m (left—dt ¼0.05 m, right—dt ¼ 0.1 m).
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holdup can be considerably improved by taking into account the local geometrical properties of the packing. Thus, the developed model is not only able to predict the static liquid holdup in a good agreement with experimental data but, in addition, provides detailed 3D information about the distribution of static liquid inside the packing. Knowing the location of every liquid bond inside the cylindrical container both the mixing and the transport processes occurring in trickling flow can be simulated on a particle scale (Schwidder and Schnitzlein, 2010).
Notation a acrit A,B
¨ os ¨ number. Fig. 14. Static liquid holdup vs. modified Eotv
dP dt Eo¨ * Fg Fh g M1, M2 NC Q0 r R R1,R2 Vbed,i Vi VS VW
separation distance, m critical separation distance, m parameters for calculating the critical separation, dimensionless distance in dependence of the liquid particle diameter, m cylinder diameter, m ¨ os ¨ number, dimensionless modified Eotv force of gravity, N adhesive force, N gravitational acceleration, m/s2 center of spheres, dimensionless averaged number of particle contact points per, dimensionless particle distribution of sphere–sphere contact and near points, dimensionless local radius, m cylinder radius, m radii of curvature, m volume of one zone of the particle bed, m3 volume of a liquid bond, m3 volume of a liquid bond between two spheres, m3 volume of a liquid bond between a sphere and the wall, m3
Greek letters
a bS bW
e erad estat ZL Fig. 15. Comparison between with Eq. (8) calculated and measured static liquid holdup (20% variance).
Despite the fact that the local structural properties are not accounted for by such a global correlation function the predicted static liquid holdups are in acceptable agreement with the experimental data (see Fig. 15). Only for large static liquid holdups considerable deviations are to be seen. These large holdups are usually encountered in packed beds with a small diameter ratio. As was shown above, in this case the influence of the cylinder wall becomes more and more significant. Since this influence is not accounted for in the global correlation the observed deviations are reasonable. 5. Conclusion As was shown by detailed experiments using random packings of spheres in cylindrical containers the predictions of static liquid
y yS yW
rL sL
inclination angle, deg half-filling angle between two spheres, deg half-filling angle between a sphere and the wall, deg void fraction of the packed bed, dimensionless radial voidage, dimensionless static liquid holdup, dimensionless dynamic viscosity, mPa s contact angle, deg contact angle on the sphere material, deg contact angle on the wall material, deg density of the liquid, kg/m3 surface tension of the liquid, N/m
Acknowledgement The German Research Foundation (DFG) is kindly acknowledged for their financial support. References Bernal, J., Mason, J., 1960. Packing of spheres: co-ordination of randomly packed spheres. Nature 188, 910–911. Charpentier, J., Favier, M., 1975. Some liquid holdup experimental data in tricklebed reactors for foaming and nonfoaming hydrocarbons. A.I.Ch.E. J. 21 (6), 1213–1218.
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Fukutake, T., Rajakumar, V., 1982. Liquid holdup and abnormal flow phenomena in packed beds under conditions simulating the flow in the dropping zone of a blast furnace. Trans. ISIJ 22, 355–364. Herminghaus, S., 2005. Dynamics of wet granular matter. Adv. Phys. 54, 221. Hochman, J., Effron, E., 1969. Two-phase cocurrent downflow in packed beds. Ind. Eng. Chem. Fundam. 8 (1), 63–71. Kawabata, H., Shinmyou, K., Harada, T., Usui, T., 2005. Influence of channeling factor on liquid hold-ups in an initially unsoaked bed. ISIJ Int. 45 (10), 1474–1481. Kramer, G., 1998. Static liquid hold-up and capillary rise in packed beds. Chem. Eng. Sci. 53 (16), 2985–2992. Lappalainen, K., Manninen, M., Alopaeus, V., Aittamaa, J., Dodds, J., 2009. An analytical model for capillary pressure-saturation relation for gas–liquid system in a packed bed of spherical particles. Transp. Porous Med. 77 (1), 17–40. Lian, G., Thornton, C., Adams, M., 1993. A theoretical study of the liquid bridge forces between two rigid spherical bodies. J. Colloid Interface Sci. 161, 138–147. Megias-Alguacil, D., Gauckler, L., 2009. Capillary forces between two solid spheres linked by a concave liquid bridge: regions of existence and forces mapping. A.I.Ch.E. J. 55 (5), 1103–1109.
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Ortiz-Arroyo, A., Larachi, F., Iliuta, I., 2003. Method for inferring contact angle and for correlating static liquid hold-up in packed beds. Chem. Eng. Sci. 58, 2835–2855. ¨ Pietsch, W., Rumpf, H., 1967. Haftkraft, Kapillardruck, Flussigkeitsvolumen und ¨ ¨ Grenzwinkel einer Flussigkeitsbr ucke zwischen zwei Kugeln. Chem. Ing. Tech. 39 (15), 885–893. Saez, A., Yepez, M., Cabrera, C., Soria, E., 1991. Static liquid holdup in packed beds of spherical particles. A.I.Ch.E. J. 37 (11), 1733–1736. Schnitzlein, K., 2001. Modelling radial dispersion in terms of the local structure of packed beds. Chem. Eng. Sci. 56, 579–585. Schwidder, S., Schnitzlein, K., 2010. Prediction of liquid flow distribution in trickle bed reactors in terms of liquid/solid properties. In: 19th International Congress of Chemical and Process Engineering CHISA 2010 and Seventh European Congress of Chemical Engineering - ECCE-7, Prague. van der Merwe, W., Maree, C., Nicol, W., 2004. Nature of residual liquid holdup in packed beds of spherical particles. Ind. Eng. Chem. Res. 43, 8363–8368. Weigert, T., Ripperger, S., 1999. Calculation of the liquid bridge volume and bulk saturation from the half-filling angle. Part. Part. Syst. Charact. 16, 238–242.
Contents lists available at ScienceDirect
Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Predicting the static liquid holdup for cylindrical packings of spheres in terms of the local structure of the packed bed S. Schwidder , K. Schnitzlein Brandenburg University of Technology, Department of Chemical Reaction Engineering, Burger Chaussee 2, 03044 Cottbus, Germany
a r t i c l e in fo
abstract
Article history: Received 20 May 2010 Received in revised form 25 August 2010 Accepted 2 September 2010 Available online 8 September 2010
A discrete 3D model has been developed for the prediction of static liquid holdup in cylindrical packed beds of spheres incorporating the local structural properties of the packing. The model predictions have been compared to experimental data covering a wide range of relevant physical properties of liquid– solid systems for aspect ratios dt =dP between 3.5 and 25. A notably good agreement between prediction and experiments is observed. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Static liquid holdup Multiscale analysis Multiphase reactor Multiphase flow Packed bed Hydrodynamics
1. Introduction The static liquid holdup in a packed bed is defined as the volume fraction of liquid referred to the total bed volume that remains in the bed after complete draining (Saez et al., 1991). In the characterization of the hydrodynamics of gas–liquid flow through packed beds the static liquid holdup is commonly used as a parameter. Moreover, the static liquid holdup is relevant in the modelling of mixing and transport processes in gas–liquid flow through packed beds, in which an amount of liquid equal to the static liquid holdup is commonly considered as a stagnant region under gas–liquid flow conditions. It is well established that the static liquid holdup depends on the liquid’s physical properties (e.g. density), the geometrical properties of the solid (e.g. particle size and shape), the gas–liquid interfacial properties (e.g. surface tension) and, in addition, on the local geometrical properties of the packing (Saez et al., 1991). Due to the high complexity of the local geometry finding, a relation which takes into account the local structural properties of the packing seemed to be a formidable problem. Therefore, in the past empirical relations have been proposed either neglecting the impact of local geometry (e.g. Charpentier and Favier, 1975) or using some averaged characteristic property of the packed bed (e.g. mean voidage (Kawabata et al., 2005; Kramer, 1998)) to
represent the impact of the local geometry on the static liquid holdup. Recent studies (Lappalainen et al., 2009; Ortiz-Arroyo et al., 2003; van der Merwe et al., 2004) on static liquid holdup assume that the liquid is present in pendular structures located at the contact points between particles. Thus, the static liquid holdup is obtained as a function of the volume of a single bond and the average number of contact points. Again, only averaged properties are used to represent the impact of geometrical properties of the packed bed. The most promising approaches for predicting static liquid holdup were compared to experimental data as obtained in our lab. The experiments were carried out by varying the liquid’s physical properties, particle shape, size, wettability and the properties of the packing as well. The static liquid holdup was measured for cylindrical packings of spheres by use of the drainage method. Details of the experimental procedures are given below. The results are shown in Fig. 1. The poor agreement between calculated and experimental data may be regarded as a hint that a significant effect on the static liquid holdup is not adequately accounted for in these correlations. As will be shown in this contribution, this effect is identified as the local structural properties of the packing.
2. Model development Corresponding author.
E-mail addresses: [email protected] (S. Schwidder), [email protected] (K. Schnitzlein). 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.09.004
Following the approach of Lappalainen et al. (2009) the static liquid holdup can be calculated by summing up the volumes of all
6182
S. Schwidder, K. Schnitzlein / Chemical Engineering Science 65 (2010) 6181–6189
pendular liquid bonds between two neighbouring spheres and between the spheres near the wall and the wall itself. It is important to note that if the particles are sufficiently small, the liquid retained in the bed is no longer the product of pendular liquid bonds but a consequence of the formation of liquid blobs that engulf several particles. Our analysis is restricted to particles large enough, so that the latter mechanism is of less importance.
Moreover, the analysis is confined to packings of spheres in a cylindrical container. 2.1. Volume of a single bond In order to represent the geometry of a single liquid bond, the model of Pietsch and Rumpf (1967) is used. Simplifying the real geometry of the liquid bond they represented the curvature of the liquid meniscus by means of a circular arc. As can be seen from Fig. 2 the predicted geometrical shape closely resembles real menisci for sphere–sphere contact points as well as for spherewall contact points. In addition, the geometrical details for a liquid bond connecting two spheres or a sphere and the cylinder wall are sketched below. Thus, the volume of a liquid bond between two spherical particles located at a given distance a can be calculated as n VS ¼ 2p ½R21,S þ ðR1,S þ R2,S Þ2 R1,S cosðbS þ yS Þ 1 ½R31,S cos3 ðbS þ yS Þ 3 h ðR1,S þ R2,S Þ R21,S cosðbS þ yS ÞsinðbS þ yS Þ p i bS yS þR21,S 2 1 3 dP ð2 þcosbS Þð1cosbS Þ2 24
ð1Þ
with the curvature radii R1,S ¼ R1 ðyS , bS Þ
Fig. 1. Comparison between measured and predicted static liquid holdup (20% variance).
and
R2,S ¼ R2 ðyS , bS Þ
ð2Þ
defined in terms of the contact angle yS and the half-filling angle between two spheres bS . A similar equation is set up for the volume of the liquid bond for sphere-wall contactpoints in terms of the contact angle Yw
Fig. 2. Sketch of a single liquid bond between two spheres (left, bottom) and between a sphere and the cylinder wall (right, bottom) in comparison with observed geometry (photography (top)).
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and the half-filling angle bW . 1 1 VW ¼ VS ðbW , yS Þ þ p ½R31,W cos3 ðbW þ yW Þ 2 3 þ ½R21,W þ ðR1,W þR2,W Þ2 R1,W cosðyW þ bW Þ ðR1,W þ R2,W Þ R21,W cosðbW þ yW ÞsinðbW þ yW Þ p io bW yW þ R21,W 2
ð3Þ
with the curvature radii R1,W ¼ R1 ðyW , bW Þ
and
R2,W ¼ R2 ðyW , bW Þ
ð4Þ
which can be calculated as R1 ðy, bÞ ¼
dP ð1cosbÞ þa 2cosðb þ yÞ
ð5Þ
R2 ðy, bÞ ¼
dP sinb þ R1 ½sinðb þ yÞ1 2
ð6Þ
2.2. Half-filling angles As can be seen from Fig. 2 the volume of a pendular liquid bond at a wall contact point depends on both the radius of the sphere and the radius of the cylindrical container. In the equations presented above, the impact of cylinder diameter is only implicitly accounted for by means of the half-filling angle bW at the wall. Consequentially, bW should be a function of cylinder diameter dt. In addition, both half-filling angles are considered to depend on the diameter of the spherical particles dP,
bS ¼ bS ðdP Þ and bW ¼ bW ðdP ,dt Þ These dependencies are exemplarily shown in Fig. 3 for the system water/glass as obtained experimentally. Pietsch and Rumpf (1967) proposed equations to calculate the maximal and minimal possible half-filling angle for a given liquid–solid system. While the minimal values are about zero
Fig. 3. Half-filling angle vs. particle diameter for water–glass system.
Fig. 4. Comparison between measured and calculated (Pietsch and Rumpf, 1967) half-filling angle (no separation distance, 20% variance).
degree, the predicted maximal values significantly differ from any observed value as shown in Fig. 4. Therefore, since to our best knowledge no better predictive correlations are available in the literature, the half-filling angles have to be obtained experimentally for any liquid–solid system under consideration.
2.3. Critical separation distance In order to establish a liquid bond, the two spheres need not to be in close contact to each other. As long as the separation distance, i.e. the distance between the surface of two neighbouring spheres, does not exceed a critical threshold a liquid bond can form. Actually, the term contact point is misleading in this sense. Therefore, some authors are using the term near points (Pietsch and Rumpf, 1967) or near contacts (Bernal and Mason, 1960). Nevertheless, in this contribution we will refer to these points as contact points. Generally, the critical separation distance is experimentally obtained by pulling two particles being in contact apart until the liquid bond ruptures (e.g. Lian et al., 1993). Several predictive correlations can be found in the literature which are either based on experimental data (Herminghaus, 2005; Lian et al., 1993) or on theoretical considerations (Megias-Alguacil and Gauckler, 2009; Pietsch and Rumpf, 1967) (see Table 1). While the first are restricted to the liquid–solid system used, i.e. di-n-butyl phthalate/liquid paraffin mixture suspended in 0.2% v/v Teepol L solution or water and polythene spheres (Lian et al., 1993) or water–glass (Herminghaus, 2005), the latter should be applicable for any liquid–solid system. But, as our experimental results indicate after drainage, no liquid bond can develop for a distance as obtained by using these correlations. In Fig. 5, the predictions are compared to experimental data obtained for a variety of liquid–solid systems. As can be seen, the predicted critical separation distance is generally overestimated. Therefore, the critical separation distance were obtained experimentally for any liquid–solid system under consideration.
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Table 1 Correlations from literature for calculating the critical separation distance. Author Lian et al. (1993) Herminghaus (2005)
Pietsch and Rumpf (1967) Megias-Alguacil and Gauckler (2009)
Correlation y 1=3 acrit ¼ 1 þ Vi 2 y 1=3 2=3 acrit ¼ 1 þ ðV~ i þ 0:1 V~ i Þ with 2 Vi V~ i ¼ ðdP =2Þ3 a Balance of forces and o 0,2 dP 1=3 dP y 4 p Vrel 1þ with acrit ¼ 3 2 2 3 Vi Vrel ¼ 4 p ðdP =2Þ3
Fig. 6. Critical separation distance vs. inclination angle a for water–glass system and dP ¼0.01 m.
Fig. 5. Comparison between measured critical separation distance (this work) and correlations from literature (inclination angle 03 , 20% variance).
2.4. Inclination angle Based on a simple balance of forces acting on a single liquid bond it can be shown that the inclination angle of the separation line connecting the centers of two contacting spheres should have an impact on the critical separation distance, too. But, most of the correlations available in the literature do not take this effect into account. According to our experiments the inclination angle largely influences the critical separation distance. As shown in Fig. 6 for the water–glass system the critical separation distance doubles when the two spheres are arranged vertically instead of horizontally. The given equation is obtained by fitting the experimental data. 2.5. Local structural properties of the packing Following the approach as outlined above the volume of every single bond existing at a contact point can be calculated provided the location of every spherical particle inside the cylindrical tube is exactly known. Unfortunately, these geometrical data can only be obtained with large experimental effort. As was shown by Schnitzlein (2001) computer generated sphere packings closely resemble real packings, i.e. predicted
Fig. 7. Computer generated sphere packing dt =dp ¼ 10.
radial voidage distributions are in good agreement with reported literature data. Therefore, as a remedy all the necessary informations about the local structural properties of the packed bed are retrieved from such simulated sphere packings. A small section of a simulated packed bed is shown in Fig. 7. It is well known that the geometrical structure of a spherical packing is considerably disturbed by the cylinder wall causing a pronounced radial voidage distribution. This is exemplarily shown in Fig. 8 for two packings with different aspect ratios dt =dP . Due to the larger voidage in the vicinity of the cylinder wall the number of contact points per sphere exhibits a radial variation (see Fig. 9). The average numbers of contact points per spherical particle as obtained by simulation are compared to predictions based on correlations from the literature applicable in the range 0:2595o e o 0:4764 (see Table 2). The observable difference between the predicted and the calculated average number of contact points may be explained by the fact that only the number of close contact points can be obtained via these correlations. Additionally considering the near contact points, as was done in this study, the determined average
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Fig. 8. Respective radial voidage distribution (left—dt =dP ¼ 5, right—dt =dP ¼ 10).
Fig. 9. Radial distribution of relevant sphere–sphere contact and near points for water–glass system (left—dt =dP ¼ 5, right—dt =dP ¼ 10).
Table 2 Correlations for calculating the number of contact points, and calculated average number of sphere contact points. Author
van der Merwe et al. (2004) Kramer (1998)
Correlation
Application dt =dp ¼ 5 e ¼ 0:46
dt =dp ¼ 10 e ¼ 0:42
NC ¼ 22 ð1e Þ2 3:1 NC ¼
6.4
7.4
6.7
7.4
3:42
6.3
7.0
6.7
7.4
7.4
8.2
e
Lappalainen et al. (2009) Ortiz-Arroyo et al. (2003) This work
NC ¼ NC ¼
e
1:18
3:1
e
Simulation
number of contact points is in good agreement with the experimental data (Bernal and Mason, 1960). Near the wall the two spheres comprising a contact point are found to be more distant than in the interior of the packing leading to a decreased number of contact points and an increased volume of the individual liquid bond. One is tempted to expect that these two effects may cancel each other out. According to Fig. 10 this is not the case. There the volume, as obtained by summing up the volumes of all liquid bonds with respect to bed volume, is plotted as a function of radial position for the
same packings as used in Figs. 8 and 9. A small but significant increase of liquid holdup with increasing radial position is found to occur. In the near vicinity of the wall the static liquid holdup is comprised solely by the liquid volume retained in the wall contact points. As can be seen from Fig. 10, the impact of wall contact points on static liquid holdup becomes more pronounced with decreasing curvature radius of the cylinder.
3. Model validation In order to validate the model predictions a large number of experiments have been carried out varying the physical properties of the liquid, the size and material of the spherical particles, the wettability and the properties of the packed bed as well. The respective experimental conditions are summarized in Table 3. The surface tension and the contact angle for each liquid–solid ¨ DAS 10 mk2 tensiometer. system were determined using a Kruss The half-filling angle were obtained by analysing and evaluating photographic images of pendular liquid bonds. In order to determine the critical separation distance a simple arrangement of two single spheres located at a certain distance was used. Thus, for a given distance and inclination angle the appearance of a liquid bond could be checked by draining the initially soaked spheres. The data were fitted yielding a simple
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Fig. 10. Static liquid holdup for water–glass system (left—dt =dP ¼ 5, right—dt =dP ¼ 10).
Table 3 Varied parameter. Parameter
Range of variation
Solid material Surface tension Liquid density Liquid viscosity Contact angle liquid/solid Particle diameter Cylinder diameter Packing height
glass, polyoxymethylene, vitrified clay 0.032–0.079 N/m 991–1181 kg/m3 1–2.5 and 100 mPa s 0–751 0.004–0.014 m 0.05 and 0.10 m 0.004–0.40 m
Table 4 Parameter for calculating the critical separation distance between two spheres with Eq. (7) for different liquids. Liquid Water Water Water Water Water Water
with with with with with
surfactant 5 wt% ethanol 20 wt% NaCl 15 wt% sugar 0.1 wt% xanthan
rL (kg/m3)
sL (N/m)
ZL (mPa s)
A (dimensionless)
B (dimensionless)
998 998 991 1181 1038 998
0.0725 0.032 0.0555 0.079 0.07325 0.0725
1.002 1.002 1.222 1.400 2.125 100
0.00095 0.0009 0.0009 0.0005 0.00045 0.002
0.00085 0.00045 0.00065 0.0008 0.001 0.0011
function representing the critical separation distance in terms of liquid properties and inclination angle.
4. Discussion 4.1. Aspect ratio
dP acrit ¼ ðAsina þ BÞ 0:01
ð7Þ
Respective values for parameters A and B are summarized in Table 4 for all liquid–solid systems under consideration. The static liquid holdup was determined by draining the initially soaked packed bed and measuring the amount of liquid retained in the packed bed gravimetrically as well as volumetrically. Fig. 11 shows a comparison of model predictions with experimental data of static liquid holdup. It is noted that these experimental data were already used in Fig. 1. By comparing the two figures the improvement as obtained by taking into account the local structural properties of the packed bed is obvious. Nevertheless, there is still a scattering of the data indicating that other effects may be important, too, which are not considered in this study.
Usually, the aspect ratio is used to represent the impact of cylinder diameter on static liquid holdup (e.g. Hochman and Effron, 1969). In order to assess this dependency the static liquid holdup was determined experimentally and by simulation for two different container diameters as a function of particle diameter. The results are shown in Fig. 12. It is obvious that the observed trends cannot be explained in terms of aspect ratio alone. Both, particle diameter and cylinder diameter need to be considered explicitly. The observed behaviour can be explained in terms of the two different contributions for static liquid holdup resulting from both the sphere contact points and the wall contact points. In addition, the simulated contributions are shown in Fig. 12 as dashed lines. As can be seen there the contribution to static liquid holdup resulting from the sphere contact points increases with aspect
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ratio irrespective of the cylinder size. In contrast, the contribution resulting from the wall contact points becomes more pronounced for the small cylinder diameter, so that in this case the superposition of both contributions yields a maximum for the static liquid holdup for an aspect ratio of about 8. For the larger cylindrical diameter the impact of the wall contact points is almost neglectable.
that the static liquid holdup cannot be represented in terms of ¨ os ¨ number alone. Eotv As was shown in the previous section the sole use of the aspect ratio is insufficient for adequately predicting static liquid holdup even on a global level. Therefore, a new correlation was developed
estat ¼ 0:3804Eo€
0:4661
4.2. Global correlation
¨ os ¨ number as a function of a modified Eotv
For any practical application it is desirable to have at hand a simple correlation relating the static liquid holdup to ¨ os ¨ number representing both the liquid’s physical the Eotv properties and the wettability properties of the liquid–solid system. The comparison of predictions based on correlations available in the literature (Fukutake and Rajakumar, 1982; Kawabata et al., 2005; Lappalainen et al., 2009; Saez et al., 1991; van der Merwe et al., 2004; Weigert and Ripperger, 1999) and our experimental data as shown in Fig. 13 clearly indicate
Eo€ ¼
Fig. 11. Comparison between calculated and measured static liquid holdup for different liquid–solid systems and geometrical properties (20% variance).
6187
rL g d2p dt 1 1 sL dP 1 þ cosyS ðsin2bS Þ4
ð8Þ
ð9Þ
which incorporates the half-filling angle bS which is shown to depend on the sphere diameter (cf. Fig. 3). In order to apply Eqs. (8) and (9) for arbitrary liquid–solid systems the respective contact angle as well as the half-filling angle need to be experimentally determined, e.g. by means of tensiometer measurements and image analysis. As Fig. 14 shows the experimental data can be represented well by means of this global correlation.
¨ os ¨ number. Fig. 13. Static liquid holdup vs. Eotv
Fig. 12. Static liquid holdup vs. aspect ratio for the system water–glass and dP ¼0.01 m (left—dt ¼0.05 m, right—dt ¼ 0.1 m).
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holdup can be considerably improved by taking into account the local geometrical properties of the packing. Thus, the developed model is not only able to predict the static liquid holdup in a good agreement with experimental data but, in addition, provides detailed 3D information about the distribution of static liquid inside the packing. Knowing the location of every liquid bond inside the cylindrical container both the mixing and the transport processes occurring in trickling flow can be simulated on a particle scale (Schwidder and Schnitzlein, 2010).
Notation a acrit A,B
¨ os ¨ number. Fig. 14. Static liquid holdup vs. modified Eotv
dP dt Eo¨ * Fg Fh g M1, M2 NC Q0 r R R1,R2 Vbed,i Vi VS VW
separation distance, m critical separation distance, m parameters for calculating the critical separation, dimensionless distance in dependence of the liquid particle diameter, m cylinder diameter, m ¨ os ¨ number, dimensionless modified Eotv force of gravity, N adhesive force, N gravitational acceleration, m/s2 center of spheres, dimensionless averaged number of particle contact points per, dimensionless particle distribution of sphere–sphere contact and near points, dimensionless local radius, m cylinder radius, m radii of curvature, m volume of one zone of the particle bed, m3 volume of a liquid bond, m3 volume of a liquid bond between two spheres, m3 volume of a liquid bond between a sphere and the wall, m3
Greek letters
a bS bW
e erad estat ZL Fig. 15. Comparison between with Eq. (8) calculated and measured static liquid holdup (20% variance).
Despite the fact that the local structural properties are not accounted for by such a global correlation function the predicted static liquid holdups are in acceptable agreement with the experimental data (see Fig. 15). Only for large static liquid holdups considerable deviations are to be seen. These large holdups are usually encountered in packed beds with a small diameter ratio. As was shown above, in this case the influence of the cylinder wall becomes more and more significant. Since this influence is not accounted for in the global correlation the observed deviations are reasonable. 5. Conclusion As was shown by detailed experiments using random packings of spheres in cylindrical containers the predictions of static liquid
y yS yW
rL sL
inclination angle, deg half-filling angle between two spheres, deg half-filling angle between a sphere and the wall, deg void fraction of the packed bed, dimensionless radial voidage, dimensionless static liquid holdup, dimensionless dynamic viscosity, mPa s contact angle, deg contact angle on the sphere material, deg contact angle on the wall material, deg density of the liquid, kg/m3 surface tension of the liquid, N/m
Acknowledgement The German Research Foundation (DFG) is kindly acknowledged for their financial support. References Bernal, J., Mason, J., 1960. Packing of spheres: co-ordination of randomly packed spheres. Nature 188, 910–911. Charpentier, J., Favier, M., 1975. Some liquid holdup experimental data in tricklebed reactors for foaming and nonfoaming hydrocarbons. A.I.Ch.E. J. 21 (6), 1213–1218.
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Fukutake, T., Rajakumar, V., 1982. Liquid holdup and abnormal flow phenomena in packed beds under conditions simulating the flow in the dropping zone of a blast furnace. Trans. ISIJ 22, 355–364. Herminghaus, S., 2005. Dynamics of wet granular matter. Adv. Phys. 54, 221. Hochman, J., Effron, E., 1969. Two-phase cocurrent downflow in packed beds. Ind. Eng. Chem. Fundam. 8 (1), 63–71. Kawabata, H., Shinmyou, K., Harada, T., Usui, T., 2005. Influence of channeling factor on liquid hold-ups in an initially unsoaked bed. ISIJ Int. 45 (10), 1474–1481. Kramer, G., 1998. Static liquid hold-up and capillary rise in packed beds. Chem. Eng. Sci. 53 (16), 2985–2992. Lappalainen, K., Manninen, M., Alopaeus, V., Aittamaa, J., Dodds, J., 2009. An analytical model for capillary pressure-saturation relation for gas–liquid system in a packed bed of spherical particles. Transp. Porous Med. 77 (1), 17–40. Lian, G., Thornton, C., Adams, M., 1993. A theoretical study of the liquid bridge forces between two rigid spherical bodies. J. Colloid Interface Sci. 161, 138–147. Megias-Alguacil, D., Gauckler, L., 2009. Capillary forces between two solid spheres linked by a concave liquid bridge: regions of existence and forces mapping. A.I.Ch.E. J. 55 (5), 1103–1109.
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Ortiz-Arroyo, A., Larachi, F., Iliuta, I., 2003. Method for inferring contact angle and for correlating static liquid hold-up in packed beds. Chem. Eng. Sci. 58, 2835–2855. ¨ Pietsch, W., Rumpf, H., 1967. Haftkraft, Kapillardruck, Flussigkeitsvolumen und ¨ ¨ Grenzwinkel einer Flussigkeitsbr ucke zwischen zwei Kugeln. Chem. Ing. Tech. 39 (15), 885–893. Saez, A., Yepez, M., Cabrera, C., Soria, E., 1991. Static liquid holdup in packed beds of spherical particles. A.I.Ch.E. J. 37 (11), 1733–1736. Schnitzlein, K., 2001. Modelling radial dispersion in terms of the local structure of packed beds. Chem. Eng. Sci. 56, 579–585. Schwidder, S., Schnitzlein, K., 2010. Prediction of liquid flow distribution in trickle bed reactors in terms of liquid/solid properties. In: 19th International Congress of Chemical and Process Engineering CHISA 2010 and Seventh European Congress of Chemical Engineering - ECCE-7, Prague. van der Merwe, W., Maree, C., Nicol, W., 2004. Nature of residual liquid holdup in packed beds of spherical particles. Ind. Eng. Chem. Res. 43, 8363–8368. Weigert, T., Ripperger, S., 1999. Calculation of the liquid bridge volume and bulk saturation from the half-filling angle. Part. Part. Syst. Charact. 16, 238–242.