Physics of Two-Phase Granular Systems

Single and two-phase granular system are of widespread use in industrial processes, e.g. rotating kilns in the cement industry, fluidized beds in the petrochemical, pharmaceutical and agricultural industry or mixers food and pharmceutical industry. However owing to their complexity granular systems have attracted significant interest from the engineering and more recently from the physics community. As granular systems can have the behaviour of a solid, fluid or gas it has not been possible so far to describe it in a mathematical form, i.e. something comparable to the Navier-Stokes equations for liquids does not exisits for granular media. However, for different 'states' of a granular media equations have been proposed, e.g. the kinetic theory, which is based on the kinetic theory of gases.
An issue which further complicates the understanding of granular system is the fact that measurements in granular media are intrinsically difficult due to their opaque nature. Therefore a lot of experiments has been restricted to observations in 2-D systems, i.e. systems where the third dimension is significantly smaller than the other two. An example would be a 2-D fluidized bed, a rectangular units with two vertical transparent parallel walls close together (e.g. 5-10 mm), so that bubbles bridge the gap between them. However, such arrangements cannot be free from the influence of the walls.
Other earlier measurement techniques included single probes, e.g. optical or capacitance probes. These probes can measure the voidage or if two probes are use the rise velocity of bubbles or slugs. However a short-coming of these probes is that they are often intrusive and, thus, disturbe the flow of both the fluid and solids.
More recently non-intrusive measurement have been developed for granular systems. These include Positron Emission Particle Tracking (PEPT), X-ray imaging or Electrical Capacitance Tomography (ECT), Diffusive-Wave Spectroscopy (DWS) and Magnetic Resonance (MR). Of these, MR offers the unique capability to image the particle distribution, i.e. voidage and the velocity of the particles.
So far the Combustion and Granular Physics group has mainly used Magnetic Resonance (MR), Particle Image Velocimetry (PIV), acetone planar luminescence induced fluorescence (PLIF) and pressure sensor measurements to measure the dynamics of single and two-phase granular systems. This has been done in collaboration with the groups of Prof. L. Gladden and Dr. C. Kaminski. It should be mentioned that due to the unfavourable relaxation characteristics (compared to liquids) MR imaging of granular matter is rather challanging. Usually several optimisation approaches to yield good signal-to-noise ratio or fast acquisition times are necessary. A review of some of these recent improvements made by our group has been published recently (Müller et al., 2008) another one has been accepted (Holland et al., 2008 accepted). Our main research has been focused on packed and fluidized beds. Fluidized beds are typically comprised of a bed of granular solids supported on a porous plate in a vertical column, through which a gas flows upwards. The solids become fluidized when the flow rate of gas is large enough for the pressure drop across the bed to become equal to the weight of the bed per unit area of the column. The superficial velocity of the gas, defined as volumetric flow rate divided by the cross-sectional area of the column, required to just fluidize the bed is U_mf. Gas in excess of that required for this minimum fluidization generally forms bubbles.
In the following some of our (published) results are presented. We try to update this webpage, as soon as submitted publications have been accepted.

1. Ultra-fast 1D-FLASH measurements to measure the bubble rise velocity in 3-D gas-fluidised beds and the frequency of bubble eruption and formation.

Fig. 1: MR measurements of the top of a deep fluidised bed. (a) Single slug eruption event, (b) multiple slug eruption events, (c) schematic of the eruption of a slug in a deep fluidised, (d) magnification of the red circled area in Fig. 1(a).

Ultra-fast 1-D FLASH (Haase et al., 1986) provides one-dimensional measurements (here along the axis of the fluidized bed) with a time-resultion of 1.34-2.08 ms and a spatial resolution of 1.25-0.625 mm. The fluidized bed was contained in a column of 50 mm i.d. and 1.8 m length.
In a deep gas fluidized the entering air forms small bubbles at the distributors. Bubbles rise through the fluidized bed and coalesce. If the bed is deep enough the diameter of the bubble aproaches the diameter of the bed containing column. Such a bubble is called slug.
Figure 1 shows MR images of a sequence of slugs breaking through the surface at the top of the bed (Müller et al., 2007a, Müller et al. 2007b). Figure. 1(a) shows images of four slugs breaking through; Fig. 1(b) is an enlargement of Fig. 1(a) showing the breakthrough of a single slug. Figure 1(c) shows the expected sequence from the two-phase theory of fluidization. The rise velocity of the top surface of the fluidized bed is U - U_mf. This is the slope of the boundary line between black (no particles) and light grey (particulate phase) at times t₁, t₂ in Fig. 1(b). Between times t₃ and t₄ the rising slug is in the field of view. Its rise velocity, U_S, is given by the (steep) slope of the (light grey) - (dark grey) boundary, marked in Fig. 1(a).

Fig. 2: Rise velocity of bubbles and slugs (MR measurements)

Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.S., Gladden, L.F. Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J.,The rise of bubbles and slugs in gas-fluidized beds using Ultra-fast Magnetic Resonance Imaging. Fifth World Congress on Particle Technology 2006 , Orlando, FL, USA. Paper Number 242a.

Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.S., Gladden, L.F. Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J., Real time measurement of bubbling phenomena in a 3-D gas-fluidized bed using ultra-fast Magnetic Resonance Imaging. Phyical Review Letters 96, 154504, 2006.

Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.S., Gladden, L.F. Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J., Rise velocities of bubbles and slugs in gas-fluidised beds: ultra-fast Magnetic Resonance Imaging, Chemical Engineering Science, 62, 82-93, 2007.

Frequency of bubble eruption and formation

Fig. 3: Ultra-fast MRmeasurements, showing the rise of bubbles in a shallow fluidise bed.

It is possible to extract the frequency of bubble passage from MR data shown in Fig. 1. If the top of the bed or the vicinity of the distributor are chosen, the frequency represents the frequency of bubble or slug eruption and bubble formation, respectively. Figure 3 gives a very graphic example on how the bubble frequency changes with increasing fluidization velocity. A rather shallow bed (H₀ = 26 mm) is imaged. The striations for U > U_mf correspond to bubbles rising up through the fluidized bed. The striations show a clear periodicity, so it can be concluded that bubbles form and later erupt at the top of the bed in a periodic way, rather than by a random process. This maybe, however, a feature of the rather small diameter column the bed is contained in.
For further result the interested reader is refered to:

Müller, C.R., Davidson, J.F., Dennis, J.S., Fennell, P.S., Gladden, L.F. Hayhurst, A.N., Mantle, M.D., Rees, A.C., Sederman, A.J., Oscillations in Gas-Fluidized Beds: Ultra-Fast Magnetic Resonance Imaging and Pressure Sensor Measurements, Powder Technology, 177, 87-98, 2007.

Velocity measurements of the particulate phase in a fluidised bed using MR

An advantage of MR over other experimental techniques is that, besides measurements of the voidage it can also measure the velocity of particles. An exampple is given in Fig. 4, which shows a typical image of time averaged particle velocities in a shallow (H₀= 30 mm) 3D fluidized bed. Particle velocities in a vertical slice, going through the centre of the bed, are plotted; the vectors indicate the direction of the flow whereas the colours indicate the magnitude of the velocity. The flow pattern shown in Fig. 4 is often described as `gulf-streaming'.

Fig. 4: Particle velocity profile

A.C. Rees, J.F. Davidson, J.S. Dennis, P.S. Fennell, L.F. Gladden, A.N. Hayhurst, M.D. Mantle, C.R. Müller and A.J. Sederman, The nature of the flow just above the perforated plate distributor of a gas-fluidised bed, as imaged using magnetic resonance. Chemical Engineering Science, 61, 6002-6015, 2006.

Müller, C.R., Holland, D.J., Sederman, A.J., Mantle, M.D., Gladden, L.F. and Davidson, J.F. Magnetic Resonance Imaging of Fluidized Beds. Powder Technology, 183, 53-62, 2008

Holland, D.J., Müller, C.R., Davidson, J.F., Dennis, J.S., Gladden, L.F., Hayhurst, A.N., Mantle M.D., and Sederman, A.J., Time-of-flight variant to image mixing of granular media in a 3D fluidized bed. Journal of Magnetic Resonance , 187, 199-204 (2007).

Holland, D.J., Fennell, P.S., Müller, C.R., Dennis, J.S., Gladden, L.F. and Sederman, A.J. In situ measurement of Dynamic Mixing in Gas-Solid Fluidized Beds using Magnetic Resonance, Fluidization XII, Vancouver, Canada 2007

Jets in packed and fluidised beds

Design of the distributor is important to ensure the desired overall performance of the fluidized bed reactor. For example, maldistribution of the fluidizing gas can lead to a partially defluidized and partially spouting bed. Furthermore, poor mixing of the solids near the distributor in a fluidized coal combustor can lead to low rates of heat transfer within the particulate phase at that point giving, in turn, defluidizsation and consequently sintering of the bed material. A variety of experimental techniques has been applied by previous workers to study both the nature and geometry of the voids formed by gas discharging at a distributor. There are also conflicting conclusions about whether the discharge of gas at an orifice forms a permanent cavity, i.e. a jet, or instead a stream of bubbles. We used MR, which provides both ultra-fast as well as time-averaged measurements of the distribution of gas and solids.
Figure 5 shows 3 air jets in a bed of seeds of diameter d_p = 0.5 mm. The distributor for the 50 mm diameter bed was a plate containing three holes of diameter d_o = 1 mm.

Fig. 5: Gas jets in fluidised beds

Fig. 6: Velocity distribution around a jet in a packed bed.

A.C. Rees, J.F. Davidson, J.S. Dennis, P.S. Fennell, L.F. Gladden, A.N. Hayhurst, M.D. Mantle, C.R. Müller and A.J. Sederman, The nature of the flow just above the perforated plate distributor of a gas-fluidised bed, as imaged using magnetic resonance. Chemical Engineering Science, 61, 6002-6015, 2006.

Müller, C.R., Holland, D.J., Sederman, A.J., Mantle, M.D., Gladden, L.F. and Davidson, J.F. Magnetic Resonance Imaging of Fluidized Beds. Powder Technology, 183, 53-62, 2008

Ultra-fast 2D imaging of bubbles in gas-fluidised beds

Ultrafast magnetic resonance has been applied to measure the geometry of bubbles and slugs in a 3D gas-fluidized bed. A tailored FLASH (Haase et al., 1986) sequence was applied, reducing achieving an acquisitiontime of down to 25 ms and a spatial resolution down to 1.7 mm. Both jets, bubbles and slugs were imaged in horizontal and vertical planes. Due to the ultrafast character of these measurements, it is not only possible to evaluate published correlations, such as those for the bubble diameter, but also study complex hydrodynamic phenomena, such as the splitting and coalescence of bubbles or the detachment of jets.

Fig. 7: Ultra-fast MR imaging of the bubble detachment at an orfice in a bed of particles.

For more information on the technique the interested reader is refered to:

Müller, C.R., Holland, D.J., Davidson, J.F., Dennis, J.S., Gladden, L.F., Hayhurst, A.N., Mantle M.D., and Sederman, A.J., Rapid Two-Dimensional Imaging of Bubbles and Slugs in a Three-Dimensional, Gas-Solid, Two-Phase Flow System using Ultra-fast Magnetic Resonance, Physical Review E, 75, 020302 2007.

Particle Image Velocimetry (PIV) measurements of the motion and eruption of bubbles in a 2D fluidised bed.

Fig. 8: Velocity and Vorticity distribution around a rising bubble in a 2D fluidized bed.

A very interesting area of fluidized bed research is the behavior at the top of the bed. The phenomena occuring include bubble eruption and collapse and the associated elutriation and entrainment of particles. An understanding of the underlying physics of these processes is important for a better understanding of two-phase granular physics, but is also of industrial relevance. Here, measurements were made on a 2D fluidized bed of height, width, and horizontal thickness 500 mm, 194 mm, and 10 mm, respectively. The bed was fluidized by compressed air. Starting from the calculated particle velocities, it is possible to calculate the vorticity. Information on the vorticity in a fluidized bed is of great importance, since various correlations are based on potential flow theory, which assumes zero vorticity.Figure 8 shows the flow field around a bubble approaching the top of a bed. In Fig. 8(a), the velocity, that is, the magnitude and direction, is given by the vectors, whereas the color indicates the magnitude of the velocity. In Fig. 8(b), the same velocity vectors are shown, but the color gives the magnitude of the vorticity, calculated using the measured velocity field as a function of x and y. The area of high particle velocity and high vorticity is marked by an arrow in Fig. 8. It is very interesting that, except for the wake region, the fluidized bed has a vorticity close to zero. The blue and red regions, Fig. 8(b), indicate high vorticities with rotation in opposite directions.

For more information on the technique the interested reader is refered to:

Müller, Davidson, J.F., Dennis, Hayhurst, A.N., A study of the motion and eruption of a bubble at the surface of a two-dimensional fluidized bed using of particle image velocimetry (PIV), Industrial & Engineering Chemistry Research, 46, 1642-1652, 2007.