On the conditions for fluidisation in cometary mantles

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Advances in Space Research 48 (2011) 862–873 www.elsevier.com/locate/asr

On the conditions for fluidisation in cometary mantles Mark S. Bentley ⇑, Norbert I. Ko¨mle, Gu¨nter Kargl, Erika Kaufmann, Erika Hu¨tter Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria Received 30 November 2010; received in revised form 22 March 2011; accepted 29 April 2011 Available online 7 May 2011

Abstract In our current understanding, active cometary nuclei comprise a volatile-depleted outer crust covering a mixture of dust and ices. During each perihelion passage the thermal wave penetrates the crust and sublimates a portion of these ices, which then escape the nucleus, dragging with them dust particles that replenish the coma and dust tail. The flux of released gases is likely to vary as a complex function of solar distance, nucleus structure, spin rate, etc. It has been previously hypothesised that at some point a fluidised state could occur, in which the gas drag is approximately equal to the weight of overlying dust and ice grains. This state is well understood and used in industrial processes where extensive mixing of the gas and solid components is desired. The literature on fluidisation under reduced gravity and pressure conditions is here reviewed and published relations used to predict the conditions under which fluidisation could occur in the near-surface of a cometary nucleus. The general trend is that the minimum fluidisation velocity decreases with reducing gravity but increases with reducing pressure. However, true fluidisation is no longer possible in free molecular flow, when the gas viscosity is meaningless. As a result, fluidisation is unlikely to be driven by H2O sublimation below a dust mantle, where pressures are too low to allow such flow. Requisite pressures could, however, be achieved at the crystalline/amorphous phase change boundary, which could support fluidisation if the overlying material is granular. Ó 2011 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Comet; Nucleus; Fluidisation; Dust mantle

1. Introduction Our current understanding of active comets suggests that with each perihelion passage, thermally evolved gases sublimated from within the icy nucleus are released and expand through the porous ice and dust layers, entraining dust grains, some of which escape the comet. In some scenarios, these upper layers are depleted of volatiles, leaving behind a dust layer or crust. In any case, the result is a differentiated nucleus with sublimation fronts for different ices found at different depths. Clearly many factors control the gas production rate and its subsequent velocity – the thickness of the overlying dust and ice layers, thermal properties of the mantle, orbital dynamics and temperature ⇑ Corresponding author. Tel.: +43 699 104 29 047; fax: +43 316 4120 490. E-mail address: [email protected] (M.S. Bentley).

profile of the nucleus, etc. For low production rates, the gas will simply diffuse through the mantle, whereas high rates can result in the impulsive ejection of large amounts of material. It is the intermediate range that is of interest here. Many industrial processes where two phases (e.g. a gas and a solid) are mixed use a so-called “fluidised bed” in which the solid phase exhibits fluid-like behaviour. In this regime the gas drag imparted on the solid particles by the gas flow is approximately equal to the weight of the particle bed. This results in efficient mixing of the phases, with a high degree of contact and thermal transfer between the two. It also allows typical fluid processes, such as buoyancy and convection, to apply also to the solid phase (Gidaspow, 1994). If this process was to occur on a comet, it could result in unexpected thermal regimes and morphological features. In this paper the literature on low gravity and pressure fluidisation is reviewed, and published relations are used to

0273-1177/$36.00 Ó 2011 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2011.04.039

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predict the conditions under which cometary fluidisation could occur. Fluidisation was first mentioned in the context of cometary science by Shul’man (1972) and expanded upon in later works, notably Shul’man (1987). Independently, Brin (1980) developed a similar theoretical basis and studied how a fluidised bed could result in enhanced downward heat transfer by convection of large grains. These authors considered the impact of a fluidised state on a more general model of cometary evolution, but did not deal with the detailed physics of the fluidisation process. Their interest was in studying how such a fluidised state could influence the development of a comet and its dust mantle. Fanale and Salvail (1984) also studied what they called a “disruptive mode” in which the pressure difference across the mantle disrupts the static nature of the dust layer, greatly increasing the gas flux. More recently, fluidisation has been proposed to explain several aspects of comet Tempel 1, namely the emplacement of smooth outflows and the nature of repetitive outbursts (Belton and Melosh, 2009). Cometary fluidisation is, of course, a specific case of the more general problem of understanding gas and particle interaction in cometary nucleii. Laboratory experiments can verify many aspects of cometary models (Ibadinov et al., 1991; Ko¨mle, 2005; Gru¨n et al., 1991) but the behaviour of dust and gas at both low gravity and low pressure are not readily accessible. The process of fluidisation is most easily described by a fluidisation curve, in which the gas velocity through a packed bed is steadily increased and the pressure drop across the bed recorded. A theoretical example (calculated from a two-fluid model under terrestrial conditions) is shown in Fig. 1. To begin, imagine a densely packed bed with particles having intimate contact, as in Fig. 2. Such a bed is characterised by a void fraction e, which is the vol-

Fig. 2. A schematic of fluidisation in a cometary context (left), and of the simplified scheme used for calculations (right).

ume fraction not occupied by the solid phase. Consider the case in which gas flows vertically upwards through this bed with superficial velocity u (this is the velocity ignoring the presence of the particles), whilst gravity acts in the opposite direction. At zero gas velocity there is no pressure drop and the bed height (h) is at its initial value. As the flow velocity through the bed is increased, a pressure drop (DP = P1  P2) develops, increasing linearly with gas velocity. At a critical gas velocity, the frictional forces (i.e. the gas drag) acting upwards equal the gravitational forces (i.e. the bed weight) acting downwards. This is the onset of fluidisation. A simple force balance can be written to describe this scenario; the downward acting weight of the bed equals the buoyancy and drag forces acting upwards. This is given in Eq. (1), where DP is the pressure drop across the bed, A is the cross-sectional area, h is the bed height, qs and qg are the solid and gas densities, respectively and e is the bed voidage. Voidage is a common term in the fluidisation literature and refers to the volume fraction occupied by voids compared to the overall volume (of voids and granular solids). As such it is similar to porosity, but does not usually include contributions from, for example, closed pores in a solid particle. The pressure drop across the bed at the onset of fluidisation can then be expressed as DP/ h = (1  emf)(qs  qg)g, where emf is the voidage at the incipient fluidisation. The challenge is usually then to formulate an expression for this pressure drop as a function of measurable or useful parameters, such as the minimum superficial velocity to achieve fluidisation (umf), an important parameter in describing fluidisation, can be calculated (hereafter the minimum fluidisation velocity) DPA ¼ hAð1  eÞðqs  qg Þg

Fig. 1. Example of fluidisation curve calculated under terrestrial standard conditions using a two-fluid model. Particles of 2000 kg m3 are fluidised by air in a cylindrical column of radius 7 cm and a height of 1 m. The particles initially form a packed bed of height 0.5 m. Each point represents the pressure drop across the bed after 20 s of simulation time. The minimum fluidisation velocity is found from the intersection of the static bed pressure drop gradient and the constant post-fluidisation line.

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ð1Þ

From this point, increasing the gas velocity does not result in a further increase in the pressure drop (as seen in Fig. 1), but instead results in the expansion of the bed. The particles are no longer in intimate contact, but are separated and can move freely throughout the bed. The height of the bed is thus increased.

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It can immediately be seen that if this behaviour was to occur on a cometary nucleus, the heat transfer mechanism between dust grains could change from conduction via grain-to-grain contact, to radiative coupling (however, heat exchange between the vapour and solid phases would probably also be increased). In addition, the ability for particles to circulate through the bed could result in near-surface grains being heated by incident solar radiation and physically transporting this heat downwards to the colder ice layer. If the gas velocity is now reduced, the bed settles back into a static state, but typically with a higher void fraction; that is, the particles are in a less compact state than before fluidisation. Such curves can be measured experimentally, usually with decreasing gas inlet velocity to avoid these compaction effects. The minimum fluidisation velocity is then determined by the intersection of the pressure drop versus superficial velocity curve, and the pressure drop equals bed weight line (Gidaspow, 1994), as seen in Fig. 1. Clearly it is also of interest to study even higher gas velocities in order to know when particles will be ejected from the nucleus. A variety of fluidisation types can occur, depending on the flow regime, gas flux, particle size and density. These include homogeneous (“simple”) fluidisation and bubbling (heterogeneous) fluidisation in which bubbles of gas expand through the bed. Geldart (1973) developed four categories to characterise fluidised bed behaviour for different particle sizes and densities. Group A particles are small and/or low density, and can be uniformly fluidised up to a critical superficial gas velocity beyond which bubbling occurs. Larger (group B) particles exhibit bubbling from the onset of fluidisation. Still larger yet are group D particles, which are not commonly encountered in industrial systems. At the other end of the scale are very fine (group C) particles that exhibit high cohesion which can inhibit or even prevent fluidisation. This system provides a way to describe qualitatively the fluidisation regime and quality; the uniform fluidisation of group A particles, for example, provides the best conditions for heat and mass transfer between the two phases and is hence preferred for many industrial applications. It is derived, however, from experiments primarily carried out under terrestrial conditions and care must be taken when applying these classifications to powders in situations with non-standard temperature, gravity, pressure, etc. (Rietema, 1984). If fluidisation were to occur on a cometary nucleus, there are several effects that could be envisaged, including that 1. bed expansion separates particles and leads to a change in the thermal transfer mechanism, 2. bed expansion removes particles at low velocities from the subsurface, 3. particles exposed to solar heating could be transported to depth,

4. the fluidised state could allow sorting of dust particles according to size and/or density, 5. some types of fluidisation could result in impulsive particle ejection, 6. particles ejected onto the surface could themselves be fluidised in a “pyroclastic-like” process, resulting in a low viscosity flow. The details of these processes vary considerably, however all start from the requirement that fluidisation can occur. In the following sections, the literature on low pressure and low gravity fluidisation is reviewed, and predictions made of the minimum fluidisation velocity under such conditions. The calculated range of fluidisation velocities and fluxes are then compared with the results of a variety of cometary models in order to establish whether or not the fluidised state is likely to occur, or indeed possible, on a typical cometary nucleus. 2. Fluidisation in a cometary environment 2.1. From a furnace to a comet The conditions experienced in a terrestrial laboratory or industrial plant are very different from those we expect in the near-surfaces layers of a cometary nucleus. In particular, we expect a comet to have: 1. low overall pressure and high pressure gradients (vacuum above surface, gas build-up below bed), 2. low gravity (approximately 104 g for a typical comet nucleus), and 3. irregular particles (e.g. fluffy IDPs, rather than industrial spheres) having an 4. unknown size distribution. The first two of these are integral to the problem of studying fluidisation on a comet and will be discussed in detail. The following sections review the literature that exists on fluidisation under variable gravity and pressure, and employ published relations to predict the minimum fluidisation velocity for a range of possible cometary conditions. 2.1.1. The influence of gravity on fluidisation The effects of low gravity are amongst the most difficult encountered in trying to simulate cometary conditions in the laboratory due to the highly coupled nature of gravity to other parameters. Low-gravity studies can be performed in, for example, drop tower, parabolic flight, sounding rocket or orbital microgravity experiments. However, drop towers do not offer enough time for microgravity fluidisation to develop, whilst orbital experiments are typically cost and time prohibitive. Some parabolic flight experiments have been performed, and these results will be discussed later. However, the effects of variable gravity are

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relatively easy to reproduce through modelling and are for the most part straightforward. A lower gravity results in a reduced bed weight, and hence the drag force needed to balance this weight is also reduced. Thus the minimum fluidisation velocity in a low-gravity environment should be considerably smaller than under terrestrial conditions. However, a secondary consideration is the force balance at the microscopic level. Under Earth gravity, fine particles are hard to fluidise. This is explained by the high inter-particle cohesive forces found in a fine-grained powder, as quantified by the ratio of the cohesive force to the particle weight. Under low gravity conditions inter-particle forces become much more important, and even for large particles fluidisation might be inhibited. Bakhtiyarov and Overfelt (1998) investigated the apparent viscosity of a fluidized bed under reduced gravity during a parabolic flight. They used a viscometer embedded in the flow to measure the viscosity and measured the pressure drop across the bed and the bed height whilst controlling the gas inlet rate. The advantage of such experiments is that a parameter that is usually strongly coupled to many others in the system (gravity) can be independently varied. The basic results of these experiments showed that decreasing gravity allowed bed expansion and the onset of incipient fluidisation at smaller gas velocities. It was also shown that the experimentally derived, semi-empirical Wen and Yu relation (see Section 2.2) often used to predict the minimum fluidisation velocity (Wen and Yu et al., 1966a) matches these experimental results very well and that bed expansion and voidage also show strong dependence on gravity – bed expansion in increased gravity is reduced; the opposite can be expected in reduced gravity. Thus one should expect that moderate gas fluxes can begin fluidisation and allow the bed to “swell”. Depending on the geometry of the situation, this can result in particles being lost from the bed before a high enough velocity to escape the comet is reached. This is one explanation that has been given for the smooth terrains seen on comet Tempel-1 (Belton and Melosh, 2009). Whilst reducing gravity is not easily achieved, the body force acting on a particle bed can be artificially increased by use of a centrifuge. Qian et al. (2001) used such a technique to study fluidisation quality as a function of “gravity”. They first showed theoretically that as gravity increases, the boundary between group A (easily fluidisable) and group C (cohesive) particles shifts to smaller particle sizes. Thus particles that behave cohesively under 1 g should be readily fluidised without aggregation at higher gravities since the particle and bed weight play a greater role than inter-particle forces in limiting fluidisation; the authors then demonstrated this experimentally. Wang and Rhodes (2004) used discrete element simulation to perform a similar study, where gravitation acceleration was increased to 100 g. They introduced a cohesive force that scaled in multiples of the single particle buoyant weight and investigated fluidisation behaviour for a variety

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of gravities and cohesive strengths. These simulations also confirmed that increasing gravity can convert a cohesive bed into a bubbling bed. More recently, fluidisation under reduced gravity conditions has been experimentally performed (Williams et al., 2008). These authors extrapolated from the model of Qian et al. (2001) to predict that in reduced gravity, particles that fluidise well under terrestrial conditions might exhibit cohesive behaviour under lunar or martian gravity. They verified this general trend on a parabolic flight, although the relatively short durations of microgravity compared with the time constant of the fluidised bed prevented a steady state being reached. In conclusion, the literature on reduced gravity fluidisation is scarce, but those studies that have been performed predict that the bed will fluidise at lower gas velocities, that the bed expansion and voidage scale inversely with gravity, and that the boundary between cohesive and easily fluidisable particles is shifted to larger particle sizes as gravity is reduced. 2.1.2. The influence of pressure on fluidisation One of the difficulties in building a complete model of gas and dust flow on a cometary nucleus is the wide range of expected pressures. Close to the nucleus, the gas density in the coma is high enough that it can be described as a collisional fluid, i.e. by continuum flow. Further away the density drops and it must be considered a free molecular flow. For gas passing through the porous nucleus itself, however, the size of the pores must be taken into account in deciding the appropriate flow regime. The Knudsen number (Kn), which is the ratio of the gas mean free path to the size of the pore space, is used to determine this. Depending on the gas production mechanism and the diffusivity of the overlying nucleus material, significant gas pressures can be obtained, in which the Knudsen number is small and continuum techniques can be applied (Kn  1). However, once the gas is free to expand beyond the mantle, it quickly becomes supersonic and the density falls such that it must be treated with molecular methods (Kn  1). It is the interim region, where the gases expand through the porous mantle, that is the most interesting to study and yet complex to model (Kn  1). Which techniques are valid depends critically on the gas flux, temperature distribution in the dust layer and the physical properties of the dust grains therein. The influence of low pressure, and in fact vacuum, conditions on fluidisation have been studied previously due to some interesting commercial applications, for example to reduce the risk of explosion when fluidising and drying some types of fine powder. Care has to be taken because the flow regime in such experiments can vary from the typically laminar condition, through slip flow, in which fluidisation is still possible, to molecular flow where the mean free path is too low to support fluidisation (Llop et al., 1996). These authors derived an expression to calculate the minimum fluidisation velocity using the force balance

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method, but using a pressure drop relation valid for slip, viscous and turbulent flow. They confirmed the validity of these results by comparison to laboratory experiments with absolute pressures ranging from 0.01 atm up to atmospheric pressure. They also noted that the effect of pressure on umf becomes more prominent with smaller particle sizes. Kusakabe et al. (1989) developed a theory of reduced pressure fluidisation, in which Knudsen flow is considered when the pressure drop across the bed is of the same order as the gas pressure. They verified this experimentally and also demonstrated that whilst a shallow bed of fine particles is readily fluidised under these conditions, deeper beds have a quiescent (inactive) lower section and a fluidised upper component. Experiments with pressures above ambient can also be useful to verify the trends with pressure and for pressures between 1 bar and 16 bar, results confirm that the minimum fluidisation velocity decreases with increasing pressure (Olowson and Almstedt, 1991). In conclusion, there is sufficient evidence in the literature that fluidisation of a particle bed with a very low pressure above the particle layer is possible, at least in the regime of slip flow. However, no fluidisation is possible in true free molecular flow. The usual equations used to predict the minimum fluidisation velocity (for example the Ergun equation, discussed later) cannot be used here, but formulations including gas flow at low pressures give broad agreement with experiments. In general the minimum fluidisation velocity increases as pressure is decreased (a higher gas flow is required to obtain the same drag force). The behaviour seen in low pressure laboratory experiments is that a strong vertical velocity gradient develops in the bed, resulting in a quiescent bed at depth but a fluidised upper layer if the bed is deep (Kusakabe et al., 1989). As the superficial gas velocity is increased, progressively more of the bed is fluidised (the fluidisation front moves downwards) until the entire bed is fluidised. However, at the point when the whole bed is fluidised, the upper layers may now have a considerably higher velocity than expected due to this gradient (Wraith and Harris, 1992).

some insight can be gained by the more simplistic calculation of balancing the weight of the particle bed with the drag force, as shown in Eq. (1). In this way one can derive the minimum fluidisation velocity (umf), which is one of the most important factors in determining if fluidisation in a cometary environment is possible. It is calculable from a simple force balance model by assuming that fluidisation begins when the drag force is equal to the weight of the particle bed, as shown in Eq. (2). In the cometary case, a term should also be included to account for the centrifugal force due to the comet’s rotation. This force acts to oppose gravity and depends on the latitude, varying from a maximum at the equator, to zero at the rotation poles. Centrifugal force is given by Fcent = mprc[(2p/T)cos(h)]2, where mp is the mass of a dust particle, rc is the radius of the nucleus, T is the rotation period of the comet and h is the latitude. The total body force acting can be written as the sum of these components, such that Feff = mp[g  (4p2rc/T2)cos2(h)]. Now writing the gravitational acceleration of a dust particle due to a spherical nucleus of bulk density qc and radius rc as g = (4p/3)Gqcrc and re-grouping terms, the force can be re-written as in Eq. (2) (Prialnik et al., 2004). This allows g in Eq. (1) to be replaced with an effective gravity taking into account the centrifugal force   3pcos2 h F eff ¼ mp g 1  ð2Þ Gqc T 2

2.2. Prediction of the minimum fluidisation velocity

DP 150lumf ð1  emf Þ2 1:75qg u2mf ð1  emf Þ ¼ þ h dp e3mf e3mf d 2p

There are several numerical modelling techniques that can be applied to the subject of fluidisation (van der Hoef et al., 2006). The most commonly used is the so-called twofluid model – an Euler–Euler model which treats both the solid phase (cometary dust particles) and the fluid phase (released volatiles) as inter-penetrating media. However, such models typically require constitutive relations derived, mostly, from experiments to create a closed equation set. Whilst such models are well-validated for the terrestrial case, their applicability to the parameters we require here is unknown. An additional difficulty is that most models of this type do not include inter-particle forces, and under low gravity these become important to consider. Such a model is currently under development, but in the interim

The challenge in solving Eq. (1) is then to find an expression for the pressure across the bed. One commonly used formulation is the Ergun equation (Eq. (3), Ergun, 1952). This allows calculation of the pressure drop (DP) as a function of the bed height (h), gas properties (density qg and viscosity l), particle properties (diameter dp and density qs), void fraction at minimum fluidisation emf and superficial gas velocity at minimum fluidisation umf. The use of this relation to predict the minimum fluidisation velocity will first be shown, before a more complex relation produced by Llop et al. (1996) will be used for low pressure calculations ð3Þ

The Ergun equation has two separate terms – the first describes viscosity dominated flow at low Reynolds number (a dimensionless term giving the ratio of inertial to viscous forces in fluid flow), whilst the second is appropriate for inertia dominated flow at high Reynolds number (Re > 1000). The viscous term can be derived from the Hagen–Poiseuille equation for flow in a straight tube, generalised to describe flow in a porous medium by replacing the pipe diameter with a “hydraulic diameter” for spherical particles and by introducing the void fraction. Numerically this results in an expression with a constant of 72, instead of 150 in Eq. (3). However the larger value was found to fit better with experiments (Ergun, 1952).

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In describing flow through porous media, an analogy is often made with flow through a set of capillaries. The interstitial velocity must then be used to describe flow through the channels, which takes place at an angle w relative to the superficial velocity vector (uc = u/e cos w) (Arnaldos et al., 1985). The degree to which the length of these capillaries (l) varies from a straight line (i.e. from the bed height) is described by the tortuosity which is then simply s = l/ h = 1/cos w. For a bed of spherical particles, w = 45° is often used. A final modification is made to account for non-spherical particles, by introducing the shape factor, or sphericity (u), into the equivalent diameter. This is the ratio of the surface area of a sphere (with a volume identical to the particle in question) to the surface area of this particle, and is hence unity for spherical particles. If these factors are included in the derivation of the pressure drop relation, Eq. (4) is found. This can be equated to Eq. (1) at the point of incipient fluidisation to give Eq. (5). As can be seen this yields a quadratic expression in minimum fluidisation velocity umf DP 72 umf lð1  eÞ2 ð1  eÞ ¼ þ 1:75 3 q u2 2 2 2 3 h cos w e ud p g mf e dpu ð1  emf Þðqs  qg Þg ¼

72 umf lð1  eÞ cos2 w e3 d 2p u2 þ 1:75

ð4Þ

2

ð1  eÞ q u2 e3 ud p g mf

ð5Þ

To simplify the solution of this equation, it is standard to introduce some constants that relate only to the voidage at minimum fluidisation and the particle shape. These are given in expression (6) using the same notation as Llop et al. (1996) C1 ¼

1 e3mf u

;

C2 ¼

1  emf ; e3mf u2

K1 ¼

72 cos2 w

ð6Þ

With these substitutions and some algebraic manipulation, we reach Eq. (7). The dimensionless Reynolds number (Remf = umfqgd/l) and Archimedes number (Ar = d3qg (qs  qg)g/l2) (which describes the ratio of buoyancy to inertial forces) are now introduced such that a simple quadratic in Remf is the end result, Eq. (8) u2mf q2g d 2 d 3 qg ðqs  qg Þg umf qg d þ 1:75C ¼ K C 1 2 1 l2 l l2

ð7Þ

1:75C 1 Re2mf þ K 1 C 2 Remf  Ar ¼ 0

ð8Þ

This equation can now be solved for a given set of particles to give the particle Reynolds number at the point of minimum fluidisation and hence the minimum fluidisation velocity (Eq. (9)) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 21 C 22 þ 7C 1 Ar  K 1 C 2 l umf ¼ ð9Þ qg d 3:5C 1

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In the derivation of these equations, some assumptions have to be made about the voidage at minimum fluidisation (emf), which appears in the force balance equation to derive the weight per unit length of the bed. The voidage and the bed height are of course intrinsically linked. The voidage in this case equates to a pressure independent value of 0.40, assuming spherical particles. In reality, there is some variation in the voidage at incipient fluidisation with pressure, trending towards an increasing voidage (and hence increased bed height) with higher pressures (Wank et al., 2001). In solving Eq. (9), strictly following the definitions of K1, C1 and C2 above yield the values shown in the first column of Table 1, assuming spherical particles and a voidage of 0.4 (typical for packed beds). In fact, these factors are often modified to better match experimental data, yielding semi-empirical forms. Table 1 also shows the coefficients used to give the experimentally derived Wen and Yu relation (Wen and Yu, 1966b). In their work on low pressure fluidisation, Llop et al. (1996) noted that for experiments at reduced pressure, the pressure-drop expression should be modified to properly account for free molecular and slip flow. They thus derive an expression valid for a wide range of pressures by combining the contributions due to free molecular, slip and viscous flow, presented in Eq. (10). This is essentially the Ergun equation with the linear addition of terms representing the pressure drop per unit length in free molecular and slip flow DP u qffiffiffiffiffiffi ¼ 2 2 ud e 16 2 h þ cos72 w cos2 w 1e pqP 45

e3 u2 d 2 lð1eÞ2

þ 1:75

ð1  eÞ 2 qu e3 ud ð10Þ

This can be re-formulated in the same way as the Ergun equation, with the use of additional constants K2 = 45p/ 32 cos2 w and C3 = 1/emf2u and Eq. (11) (in which Knp is the particle Knudsen number, k/d) to give another quadratic, which can be solved in the same way as before yielding the final expression Eq. (12) Z¼

1 Knp K 2 C3

þ

1 K 1C2

ð11Þ

In solving, the values for the particle parameters are for round particles, and are summarised in the “Llop et al.” Table 1 Parameters defining particle properties (shape, etc.). The theoretical column is calculated for spherical particles, assuming an interstitial angle of 45° and voidage at minimum fluidisation of 0.4.

K1 K2 C1 C2 C3

Theory

Wen and Yu

Llop et al.

102 – 15.6 9.4 –

150 – 14 11 –

150 9.2 16 11 5.5

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column of Table 1. The voidage at incipient fluidisation (emf) is taken here to be approximately 0.4 and the mean free path and viscosity of an ideal gas are used " #1=2 2 Z Ar Z Remf ¼ þ  ð12Þ 3:5C 1 1:75C 1 3:5C 1 Using this formulation the minimum fluidisation velocity umf can be calculated for a variety of pressures, particles sizes, temperatures and gravities, for comparison with the conditions expected to be found on a cometary nucleus. Fig. 3 shows such a plot of umf versus pressure for particles of a variety of diameters, each of density 2000 kg m3, fluidised by water vapour at 200 K, under a surface gravity of 104 g. As expected, as pressure is decreased, a higher velocity is required to maintain the drag force. In addition, the particle size plays a major role; as might be intuited, smaller particles are easier to fluidise. Because true fluidisation cannot occur when the fluidising gas is in free molecular flow, the curves are truncated at Kn = 10. For the smallest particles (and pore spacings) the gas is already in slip flow at atmospheric pressure. A similar plot can be produced to show the effects of gravity for a fixed particle size (Fig. 4). Plotted are curves for the 1 g case, for Mars (0.38 g), and for comet 67P/CG (1.8  104 g) with and without the effects of centrifugal acceleration (assuming a bulk density of 370 kg m3 and a rotation period of 12.4 h (Lamy et al., 2007)), which further decreases the surface gravity by 19%. It is clear that the reduced gravity produces a marked decrease in the minimum fluidisation velocity, which decreases linearly with gravity as the weight of the bed is decreased. In order to establish whether fluidisation is possible in the cometary context, these values must be compared with the expected gas velocities for evolved gases expanding

0

Fig. 3. A plot of the minimum superficial gas velocity needed to achieve fluidisation (umf) versus pressure for a variety of particle sizes. Calculations are performed for particles of density 2000 kg m3 fluidised by water vapour at 200 K under a surface gravity of 104 g. The void space is assumed to be the same as the particle diameter. Each curve is truncated at low pressures when the Knudsen number is greater than 10, and fluidisation can no longer occur.

Fig. 4. Minimum fluidisation velocity (umf) versus pressure for several surface gravities, assuming 100 lm particles of density 2000 kg m3 fluidised by water vapour at 200 K. Plotted are curves for the Earth and Martian gravities and for comet 67P/C-G (1.8  104 g) with and without the effects of centrifugal acceleration.

through a porous bed and into vacuum. Of course in this case we expect a range of velocities, as also seen in laboratory vacuum fluidisation experiments, as the gas accelerates. 2.3. Limitations of this approach The above calculations give good results according to laboratory experiments on vacuum fluidisation and provide an insight into the conditions necessary for fluidisation on a cometary nucleus. However, they are subject to some limitations, which will be discussed next. Firstly, the preceding analysis of the minimum fluidisation velocity does not explicitly include inter-particle forces. These can include van der Waals, electrostatic and capillary forces due to the presence of liquids (Krupp, 1967). In a cometary environment this last force can probably be ignored, but the first two may play an important role. Electrostatic forces arise from charging due to, for example, sunlight or charged particles impinging on the surface. Van der Waals forces, however, can be expected to occur between all particles. It is also well-known from terrestrial fluidisation that very fine particles exhibit different fluidisation behaviour due to their cohesive nature; whereas larger particles fluidise smoothly, fine grains can be so cohesive that a high pressure builds up below the bed, sometimes lifting the entire bed itself, resulting in a more or less impulsive release of pressure via channels forced through the bed. On the other hand, cohesive forces can result in clumping of smaller grains to form aggregates that must then be considered the fluidised solids, with their own weight, cross-sectional diameter and hence drag force (Wank et al., 2001).

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In order to obtain a first-order idea of the relative magnitude of such forces, the weight of a single particle and the magnitude of the van der Waals force between two identical particles can be compared as a function of gravity. Fig. 5 shows the weight of individual particles of density 2000 kg m3 and the van der Waals force between two identical particles. For this latter calculation, a separation ˚ is used, and a Hamaker constant of of 1.65 A 20 6.5  10 J (the value for fused silica, used here as an analogue for a refractory grain (Israelachvili, 1992)); the Hamaker constant is related to the geometry and the materials involved and describes the strength of the attractive force. It is immediately clear that in a cometary environment, cohesive forces potentially play a rather dominant role, that has not been considered in the analysis so far. It should again be noted that geometric factors also play a role in determining the Van der Waals forces, and for non-spherical and porous particles, as might be expected at a comet, surface properties likely dominate over bulk properties (Seville et al., 2000). In calculating the Van der Waals force in particles with severe asperities, for example, the radius of these features can be much smaller than the bulk particle radius, and hence the cohesive force can be reduced (Visser, 1989). For example, contaminant sub-particles on supposedly spherical grains can result in a reduction in the cohesive force of orders of magnitude (Massimilla and Donsı`, 1976). In other words the shape and surface roughness of the particles should be considered. Direct evidence as to the microscopic nature of cometary dust is to date rather scarce. Porous anhydrous interplanetary dust particles (IDPs) are believed to originate in comets, and are aggregates of many sub-micron grains, showing a very high porosity, up to 90% (Flynn, 2005). The Stardust mission also returned definitively cometary particles captured in aerogel (Tsou et al., 2004), although fragile aggregates were found to break up during deceleration (Flynn, 2008) and it is very difficult to remove all traces of aerogel to study the particle surfaces in detail. Looking to the future, the MIDAS atomic force microscope will make measurements

Fig. 5. A comparison of the attractive van der Waals force versus weight for a range of particle sizes.

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of ejected dust particles as part of the Rosetta mission, providing sub-nm resolution images of particles collected at low relative velocity (Riedler et al., 2007). Since the van der Waals forces depend critically on interparticle separation, they are dominant when the bed is in a packed state; following the onset of fluidisation the bed expands and the particle spacing is increased such that they drop off rapidly. Thus, once a cohesive bed is successfully fluidised, it is relatively easy to keep it in this state. In industrial systems, a variety of techniques such as vibrating the bed, are used to enable fluidisation. It could be that on a comet an impact may provide the initial impulse to overcome cohesive forces and allow a granular layer to be fluidised. Although at the point of impact, melt production (and the subsequent creation of agglutinates) could locally inhibit fluidisation, the propagation of vibration waves could influence the regolith to some distance from the impact. It is believed that even without a gas or fluid, granular materials can undergo vibration induced convectionlike processes, often resulting in the sorting of material by grain size, for example as seen on asteroid Itokawa (Miyamoto et al., 2007). As indicated by Visser (1989), the cohesive forces depend on the particle density, porosity and surface roughness and the minimum fluidisation velocity is governed by this force when it is of the order of, or larger than, the gravitational and fluid forces in the bed; as was seen in Fig. 5 this is likely the case for a comet. The cohesive force can also be included in the force-balance described earlier to derive the minimum fluidisation velocity. Wank et al. (2001) incorporated an expression for the Van der Waals force per unit area into their derivation. This was a function of the number of particles per unit volume, and the number of contact points per particle (the coordination number), where this latter parameter was expressed as a function of the voidage at the minimum fluidisation velocity. In this analysis it is assumed that the cohesive bed behaves as an elastic medium where the bed can expand prior to the onset of fluidisation. In this regime a moderate gas flux causes some of the contact points between particles to be broken, and allows bed expansion (and subsequently an increase in voidage) without full fluidisation (Jaraiz et al., 1992). This reduces the cohesive force and eventually allows fluidisation, at a higher gas velocity than otherwise would be needed. Such an expression is somewhat harder to apply here since the voidage at minimum fluidisation (emf) is not known a priori. In addition, this parameter is hard to define for a deep low pressure fluidised bed which includes both quiescent and fluidised regions. If umf and emf are measured experimentally (usually by decreasing the gas velocity) at the point where the bottom of the bed becomes quiescent, emf decreases with lower pressures (Wank et al., 2001) although the range of voidages is still quite small (0.40–0.55) and comparable to the constant value of 0.4 implied in the previous expression for umf. Thus the inclusion of the cohesive force in the study of cometary fluidisation is necessary, but hard to quantify

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without experimental data or detailed models. An expression for the minimum fluidisation velocity including particle cohesion is readily formulated, but includes the voidage at minimum fluidisation which is unknown. Future efforts should take this into account. For now the calculated values of minimum fluidisation velocity should be considered as lower limits since the effects of cohesion act against fluidisation by raising the minimum velocity. In the preceding calculations, another problem is that a constant temperature profile has been assumed for the bed. In actuality a strong temperature gradient can exist through the mantle, and the temperature and hence velocity of the gases flowing through it can be influenced. Finally, the above analysis assumes that the entire bed comprises particles of a single size, but what little we currently know of the size distribution of dust particles in a cometary mantle points towards a broad size distribution. Unfortunately there is no simple analytical solution that can deal with this complexity. Multiphase fluid or discrete element models can support multiple particle sizes, but their complexity is beyond this simple first-order approach. However, the effect of the particle size distribution (PSD) on fluidisation parameters has been studied experimentally for industrial applications, and some general results can be summarised. Whilst the PSD has very little effect on bubble size or frequency, the minimum fluidisation velocity is found in practice to depend upon the size distribution (Gauthier et al., 1999). For a gaussian distribution, the fluidisation curve looks much the same as for a narrow PSD and has a well defined value of umf. For bi-modal and flat distributions, however, the situation is different. In both of these cases segregation occurs, with smaller particles rising to the top of the bed and larger particles falling to the bottom. It is interesting to note that this is the opposite effect to that seen with vibration fluidisation (the so-called “Brazil but effect”) where larger particles rise (Daleffe et al., 2008). The fluidisation curve in these cases still has a linear packed bed section, and a flat region when the entire bed is fluidised, but now also shows a transition region between the point of incipient fluidisation for the most easily fluidised particles and the entire bed. Whilst it is difficult to extrapolate from these studies to the cometary scenario, any future attempts including more detailed modelling should take a polydisperse approach to modelling the dust particles. 3. Application to comets Comets are now known to be rather low bulk density bodies, which, given the densities of the ice and dust grains likely to comprise them, implies a high porosity in the range of 0.6–0.8 (Basilevsky and Keller, 2006). How this porosity translates to the macroscopic or microscopic structure of the interior is not well-known (Prialnik et al., 2004). Porosity is defined as the ratio of the volume of free space to volume containing material. This should be con-

trasted with the term “voidage” used in the fluidisation literature, which refers to the ratio of void volume to the total volume of voids and particles. Voidage thus defined does not include the porosity of the particles themselves (e.g. closed voids within a particle). For a set of monodisperse spheres, voidage is determined by the packing structure and varies between 0.26 for hexagonal packing and 0.48 for regular cubic packing (Woodcock and Mason, 1988). A key problem in cometary science is that, apart from a few recent spacecraft encounters, all data are derived either from telescopic observations of inactive, distant and low albedo nuclei or from measurements in which only the coma is visible and source regions and activity have to be inferred from models. Therefore there are, as yet, no direct measurements of the near-surface gas pressure or velocity, but only calculated results. In addition, a clear requirement for fluidisation is the presence of granular material on the comet, either in the form of a refractory dust mantle, a layer of icy grains, or both. The limited number of in situ encounters with cometary nuclei to date has shown a wide variety of surface features and activity levels. As such it is very difficult to generalise and each comet must be more or less treated separately. For example, thermal modelling of the nucleus of the Rosetta target comet 67P/Churyumov-Gerasimenko (hereafter 67P/C-G), in which an initially homogeneous comet is modelled over several perihelion passages, has shown that the formation and retention of a substantial dust crust) is difficult (Sanctis et al., 2005). In this model, a dust crust can be produced by a “trapping mechanism” in which large grains are collected in the upper layers of the comet and prevent smaller grains from being ejected (Shul’man, 1972). Such a dust layer is likely to be millimetres to centimetres thick and is already enough to dramatically reduce the amount of heat transported to the interior due its low thermal conductivity (Ko¨mle and Steiner, 1992). A shallow layer like this has been found to fluidise uniformly even under low-pressure conditions. As a first step, the Knudsen number for various hypothetical cometary scenarios can be used to determine which flow regime is applicable, and hence if fluidisation is possible. As previously described, the Knudsen number is the ratio of the gas mean free path length to a representative physical scale length, in this case the size of the pores through which the gases flow. Thus for very small numbers, intermolecular collisions are dominant and Poiseuille flow is assumed. Very high Knudsen numbers imply that gas collisions with the grains in the dust bed are dominant, and that the flow is indeed free molecular. In between (for Kn  1) there exists a transition regime known as slip flow. As a first approximation the void space between particles is taken to be the same size as the dust grains themselves. Thus the particle Knudsen number can be calculated for a range of pressures and grain sizes. In Fig. 6 the Knudsen number is plotted against pressure for a variety of particle sizes, assuming that the gas is water vapour

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at 200 K. This shows the limits at which fluidisation can occur (taken here as at Kn < 10). Naturally for smaller grain sizes, and hence smaller voids, a higher gas pressure is required (with a correspondingly smaller mean free path) to ensure that gas molecules interact with each other more than with the dust grains, and hence that viscosity is meaningful and fluidisation can occur. Regardless of the exact form of the upper layers (ice-free dust, icy grains, or a mixture) the pressure at the surface and at the source of the vapour production must be known to estimate the gas fluxes and velocities that could drive fluidisation. There are two distinct mechanisms thought to be responsible for gas release in cometary nuclei and currently included in most models. The first is direct sublimation of ices (at the surface or below it) in response to solar irradiation and the second is the release of volatile gases trapped in amorphous ice. Such ice forms at low temperatures (<120 K) and remains stable until it experiences a temperature of 150 K or higher, when it undergoes an exothermic transformation to a crystalline state, releasing trapped gases (Prialnik, 2002). These two mechanisms can result in rather different pressures. For a direct sublimation mechanism, the gas produced is usually considered to be at its equilibrium vapour pressure, which itself is a strong function of temperature. Supersaturation is possible, but modelling shows that in practice it is rare and that deviations only occur very close to the sublimation surface (Ko¨mle et al., 1992). Taking water vapour at 200 K, for example, and using the familiar Clausius– Clapeyron relation and appropriate constants (e.g. Fanale and Salvail (1984)) the pressure at the water-ice front is found to be 0.164 Pa. This results in a mean free path of approximately 5 cm, which is clearly much larger than most conceivable pore sizes, and hence the water vapour is in the Knudsen, or free molecular, regime, with intermolecular collisions being far less frequent than those with

Fig. 6. The Knudsen number plotted as a function of pressure for a variety of particle sizes. The shaded region indicates where the Knudsen number is greater than 10, and fluidisation is no longer possible.

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the pore walls. True fluidisation is not possible in this regime. This, however, does not discount fluidisation due to gases released during the amorphous ice phase transition, such as the supervolatile CO which is typically seen in the coma when a comet is far from the Sun (too far for heat to have penetrated deep enough to sublimate fresh CO in a comet that has undergone at least a few apparitions). Modelling has shown that such transitions, when triggered, can be self-sustaining for some time due to their exothermic nature, and so CO produced in this way is released in “spurts” (Prialnik and Bar-Nun, 1987, 1988). The amount of CO trapped in amorphous ice is found to be a complex function of the initial ice composition (Capria, 2000) but is often estimated as a few to ten mass percent of the ice (e.g. Prialnik, 2002; Capria et al., 2002). Some models of CO production have produced rather large pressures at the phase change boundary in this way, for example that of Prialnik and Bar-Nun (1990) which results in a pressure of several atmospheres at the boundary, decreasing gradually towards the surface. Such a situation would support fluidisation, but is only possible because of the fixed solid matrix restricting the gas flow, which negates a fundamental assumption in fluidisation, namely that the particles are free to move (once cohesive forces have been overcome). In their study of possible cometary fluidisation, Belton and Melosh (2009) examined the role of CO released at the amorphous/crystalline boundary as a fluidising agent in the subsurface of comet Tempel 1. They used the model predictions of Tancredi et al. (1994) to provide the CO partial pressure as a function of depth and demonstrated that this was substantially larger than both the overburden (weight) and the upper bounds of the expected tensile strength of the nucleus. They also discuss the implications of the low yield strength derived from the Deep Impact results, which support the concept of a granular surface layer of ice/dust aggregates. Studies of the plume excavated by the Deep Impact spacecraft suggest yield strengths of between one and a few tens of kPa, but there is some debate over whether this plume was entirely a result of impact cratering, or other interior processes (A’Hearn, 2008). The Tancredi model generates CO with a pressure of 104 Pa and a Knudsen number of 0.2 near the phase change layer during active crystallisation, and thus from pressure considerations fluidisation could indeed occur. However, both the gas pressure and the superficial gas velocity must be sufficient. In order to consider if such a scenario could create a fluidised bed, one must know the porosity of the overlying material, which determines the pressure built up, and the depth of the crystallisation layer below the surface. The Tancredi model found this depth to be 138 m. Model calculations usually put the boundary between crystalline and amorphous ice at a few tens of metres (Huebner et al., 2006), up to a few hundred metres, although results from the Deep Impact mission point towards the former

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(A’Hearn, 2008). A recent model of 67P/C-G which starts from a homogeneous amorphous ice composition show that the depth of the crystallisation front varies over the surface, according to the solar incidence angle. After the five revolutions for which the model was run, a maximum value of about 5 m was predicted (Rosenberg and Prialnik, 2009). Thus to fully quantify the possibility of fluidisation in a cometary mantle, the outputs of various models in terms of pressure and gas velocity as a function of depth should be examined and compared with the limits of fluidisation. Only then can a definitive prediction be made as to the existence and type of fluidisation that might occur. 4. Conclusions and further work Fluidisation has been suggested as a possible mechanism for several observed cometary phenomena, including the creation of smooth plains by bed expansion and overflowing, and periodic outbursts by “slugging” type behaviour. It also has the possibility to create a situation in which the thermal properties of the upper layers of the mantle could be dramatically changed. However, the details of fluidisation in low-gravity and low-pressure environments have not yet been explored in sufficient detail. The literature shows that true fluidisation is possible at low pressures, at least in the slip-flow regime, but not in free-molecular flow. As pressure is reduced, the minimum fluidisation velocity increases, first gradually and then more quickly. As gravity is reduced, the bed weight decreases and the minimum fluidisation velocity decreases linearly with it. However, a more realistic treatment needs to take into account the forces acting at the microscopic level, in particular van der Waals and electrostatic forces. These will act to make the bed more cohesive and increase the minimum fluidisation velocity needed to begin fluidisation. In a cometary context, it is clear that sublimated water vapour cannot generate a high enough pressure to drive fluidisation, but that volatiles released during the amorphous/crystalline water phase transition could well be able to do this. These conditions combine to make true fluidisation, in which particles are free to move around the bed, unlikely at a comet. Some kind of partial fluidisation probably still occurs, in which particles can be partially separated and removed at low speeds from the subsurface, but without the high degree of mixing that typifies a completely fluidised bed state. Calculation of the minimum fluidisation velocity, as performed here, only shows the gas flux at which fluid-like effects become apparent in a granular bed. A more detailed model is required to describe the particle and gas flow beyond this point. A two-fluid model, used to generate Fig. 1, is currently being evaluated for simulating fluidisation in low pressure and gravity by including partial-slip boundary conditions. However the pressures mostly found at a cometary nucleus require a more rigorous approach.

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