As introduced in Chapter 1, gas fluidization occurs when gas lifts-up a certain mass of powder and starts to behave fluid-like. The fluidized bed behavior is dependent on superficial gas velocity (u0) and powder/- particle properties. Most influential powder properties are particle density (Â½p ) and particle diameter (Dp ). The fluidization behavior can be generally classified by Geldartâs powder classification using both quantities. Figure 2.1 gives the classification diagram for ambient conditions . The particle diameter on the x-axis indicates Sauter mean diameter (D3,2) (dsv Ã D3,2) . Geldart A and B type powders are most widely used as industrial catalyst particles. In this work, base refers to such powders, fines refers to Geldart C or small group A powders (Dp Ã 45 &m). Type A shows homogeneous fluidization and expansion between minimum fluidization velocity (umf) and minimum bubbling velocity (umb) whereas type B starts bubbling right at the point of minimum fluidization. Both show excellent particle mixing and heat transfer while within the bubbling regime. Geldart C is associated with high cohesion resulting in channeling during fluidization and lacks good particle mixing. This type is hard to fluidize without assistance such as vibration, mechanical stirring or pulsating flow . As was introduced in Chapter 1 as well, fines are found to influence bubble size. It is therefore required to understand the behavior and role of bubbles in fluidized beds. This will be discussed in next section. As the main purpose of fluidized beds is to convert reactants into products, the widely used Kunii and Levenspiel
model for the conversion in bubbling fluidized beds is discussed afterwards. First, the criterion for minimum
fluidization is given.
2.1.1. MINIMUM FLUIDIZATION
A powder bed becomes fluidized when gas enters at a velocity above minimum fluidization velocity (umf). As gas travels through the void spaces in a static bed, a pressure drop (Â¢P) over the bed is created. It is said that when Â¢P equals the weight of the powder bed, the bed reaches the fluidized state. Among other existing correlations, the Ergun equation can be used to calculate umf: with gas viscosity (Â¹g ), void fraction (Â²), superficial gas velocity (u0) and powder bed height (H). The Sauter mean diameter (D3,2) should be used for the particle diameter (Dp ). Equation 2.1 can be solved for umf with Â¢P Ã Â½p (1Â¡Â²mf) gHmf. According to Equation 2.1, umf is independent on inner column diameter (D) and powder properties such as cohesion. Although this assumption is not completely true for small particles and narrow columns, Equation 2.1 can still be used with relatively high accuracy . A drawback is however that the void fraction at minimum fluidization (Â²mf) is often unknown beforehand. Therefore, the correlation of Baeyens is often preferred for particles smaller than 100 &m:
with gas density (Â½g ) and gravitational acceleration constant (g ).
Within the bubbling regime, bubbles motion dominates the fluidized bed behavior of Geldart A and B powders. Since fluidized beds do not consist of two distinct phases, but could be described as such (see Appendix J for a brief discussion on two-fluid simulations), the termbubbles does in fact indicate parts of the fluidized bed with local low solid holdup. As bubbles move upwards, particles flow around in both up- and downwards direction. This rather chaotic behavior allows for excellent heat and mass transfer in fluidized beds. As was previously indicated, Geldart A can be homogeneously fluidized, i.e. non-bubbling. The powder visually appears to be static and is in a solid- or fluid-like regime [23, 24]. This regime is further discussed in Section 2.5. Bubbles generally grow while rising through a fluidized bed of Geldart B powder. This behavior can be mainly attributed to bubble coalescence . A rising bubble leaves a lower pressure in its path, resulting in preferred bubble pathways, often referred to as bubble catch-up and coalescence. Bubble size reduction can thus be achieved by diminishing this mechanism. This is explanatory for the positive effect of multi-stage gas injection where gas is fed at different positions in the powder bed . Apart from bubble coalescence, bubble splitting does also occur. Geldart A powders have a maximum bubble size, with respect to bed height, due to a coalescence-splitting equilibrium. The splitting rate mainly depends on particle diameter and increases for smaller particles (Â» DÂ¡0.5 p ) . This may be attributed to better powder flowability or lower apparent
viscosity voeg (see Section 2.4) [5, 27].
2.1.3. CONVERSION MODEL FOR A BUBBLING FLUIDIZED BED
The Kunii and Levenspielmodel is widely used to predict catalytic reaction conversion in a bubbling fluidized bed . Themodel divides the bed in three distinct phases: (1) bubble phase, (2) cloud and wake surrounding the bubbles and (3) dense emulsion phase. The model assumes that all gas travels vertically in the bubble phase and there is no convective flowthrough the dense phase. Catalytic reaction takes place in all phases and is dependent on local solids concentrations. Consequently, reaction is fastest in the dense phase and there exists a species gradient in the radial/horizontal direction of the bed. Without presenting the full derivation (for details see ), the reaction conversion (X) for a first order catalytic reaction can be given by: with bubble holdup (Â±), reaction rate constant (k) and effective reaction rate constant (Keff). Kbc and Kce indicate the transfer rate from the bubble to cloud and from the cloud to the emulsion phase respectively. Similarly subscripts b, c and e of solids fraction (Ã) indicate the corresponding phase, i.e. bubble, cloud or emulsion phase respectively. Equation 2.4 of Keff can be seen as a series transfer resistance over the three phases. Assuming that minimum fluidization conditions apply for the dense phase, Â± can be given by: Combination of Equations 2.5 and 2.7 indicate that smaller bubble diameter (Db) leads to lower ubr and thus increased Â±. According to Equation 2.4, increased Â± increases Keff and thus X. Furthermore, the individual transfer rates Kbc and Kce increase for smaller bubble size too, partially due to increased surface to volume ratio (for details see ). The positive effect of fines on conversion can be explained by a decrease in Db . Furthermore, the positive effect of catalytic fines could also be explained by an increase of the solids holdup in the bubble phase (Ãb) .
2.2. FORCES DURING FLUIDIZATION
A number of forces act on particles while fluidizing. These can be separated into three types: (1) forces due to particle-gas interaction, (2) interparticle (cohesive) forces and (3) collisional forces. These are sequentially discussed next.
2.2.1. PARTICLE-GAS INTERACTION
Drag forces and gravity are the dominant forces in a fluidized bed. One can easily set up a force balance for an isolated single particle in a gas stream: dvp
With particle velocity (vp ), interstitial gas velocity (vg ) and drag coefficient (CD) for Reynolds number (Re) Re Ã 1. Equation 2.8 does however not take the effects of nearby particles on the gas flowinto account. Therefore, it is better to model the powder bed as a whole as the particulate phase. Furthermore, as was indicated before, particles interact with each other during fluidization. Particle stresses are especially important during homogeneous expansion (see Section 2.5). This can be taken into account by setting up a force balance over a horizontal slice of a fluidized bed reactor [15, 16, 23, 29]. This yields a one-dimensional model for the particulate and gas phase respectively: With inner column diameter (D) and particle stresses (Â¾s ). The last termtakes wall friction into account and is often neglected. The Â§-sign indicates that wall friction can be in either upwards or downwards direction, dependent on the direction of bed expansion. A powder cohesion term should be included similarly. The Richardson-Zaki drag coefficient (Â¯) is given by [10, 15, 16, 23, 29]:
Where the Richardson-Zaki exponent (n) is an empirical parameter which is often n Â¼ 5. The terminal fall velocity (vt ) can be estimated for creeping flow by solving Equations 2.8 and 2.9 for dvp/dt Ã 0: Equation 2.11 and 2.13 can be combined to yield a relation similar to the Ergun Equation 2.1. Summarized, the force of gas exerted on a single particle is given as the first term on the right hand side of Equation 2.8. The force of the gas on an entire particle phase is given in Equation 2.11. The latter can be rewritten to the force per particle and compared to the result in Equation 2.8. The difference between both approaches is the estimation of the actual gas velocity past the particle and corresponding effect on drag force.
2.2.2. INTERPARTICLE FORCES
Powder or particle cohesion is a result of interparticle forces. Generally, three types of forces can be identified. These are from strong to relatively weak: (1) capillary forces, (2) Van der Waals forces (FVdW) and (3) electrostatic forces . The magnitude and importance of these forces do heavily depend on particle diameter (Dp ). As Figure 2.2 indicates, interparticle forces become more dominant for small particles when compared to particle weight. This can influence fluidization as agglomerates may be formed due to strong cohesive forces. Capillary forces aries when liquid bridges are formed due to the presence of small amounts of water in a fluidized bed. This can be either due to gas humidity or particle water content. Capillary forces are generally the stronger forces of all interparticle forces. The Van der Waals forces (FVdW) include dipolar and non-polar forces which exist between molecules. These forces do also arise between the surfaces of a particle and other objects (e.g. other particles or wall). FVdW can be estimated using: with surface separation (s) and Hamaker constant (AH) which is typically in the order of magnitude AH Â»10Â¡19 J. Equation 2.14 is only valid for small particle separation. According to Equation 2.14, the force rises to infinity when particles are attached to each other. In practice, there is a minimum separation due to surface asperities. The contact force as shown in Figure 2.2 is calculated with a minimum separation of 0.2 and 0.4 nm . FVdW is only important for small particles at contact. Since fines have Dp Ã 45 &m, FVdW could have a large effect according to Figure 2.2. Electrostatic charging can occur during fluidization due to friction between two particles or by friction with the vessel wall. Charging occurs more readily at high collision velocity. Particle charging is very dependent on material properties. It is known to occur due friction between different-sized particles . This phenomenon could play a role in the effect of fines on fluidization. Furthermore, the bed gains a net charge when fines with the opposite charge are elutriated fromthe powder bed. Figure 2.2 shows however that electrostatic forces are a few orders of magnitude lower than Van der Waals forces at particle contact. Therefore, these forces are often neglected . In addition, electrostatic charges can be minimized using conducting vessel material such as stainless steel . Humid air can also be used but has capillary forces as a consequence.
2.2.3. COLLISION AND COEFFICIENT OF RESTITUTION
The first part of this section addressed the hydrodynamics of fluidization and paid only little attention to particle interaction. The most obvious type of particle interaction is particle collision. This is best understood when looking at two colliding objects rather than the fluidized bed as whole (see Appendix J for Discrete ElementMethod (DEM)). Collision can be described bymomentum and energy conservation. These balances are respectively: with particle velocity before collision (v), particle velocity after collision (u) and collisional kinetic energy dissipation (Edis). Indices address different particles. For ideal collisions (i.e. Edis Ã 0), the above equations are readily solved for ui and uj . Particle collisions are however seldom ideal. As was briefly addressed in Chapter 1, kinetic energy dissipation can be described by the coefficient of restitution (COR). The COR is defined as the relative particle velocity after collision divided by the relative velocity before: where ei j indicates COR which has a value of 0 for sticking collisions and 1 for ideal elastic collisions. Most collisions have 0 Ã ei j Ã 1. The three Equations 2.15 to 2.17 can be solved simultaneously to evaluate ui , uj and Edis: Obviously, Edis has the highest value for sticking collision (i.e. ei j Ã 0) and vanishes for ideal collision (i.e. ei j Ã 1). It becomes clear from above equations that when mi Â¿mj , the collision behavior is dominated by the heavy particle j . Since collisions are heavily influenced by the COR, it is important to use proper valueswhen modeling collisions in fluidized beds. Surface adhesion effects and sticking collision become more important for smaller particles and low impact velocity, especially for Dp Ã 100 &m . A number of models has been published to theoretically evaluate COR as a function of impact velocity . Here the Elastic-Adhesion (EA) model of [33, 35] is used. The model evaluates COR as function of the normal impact velocity (vn). As one can intuitively imagine, particles will stick more readily to a surface when the approach velocity is low. As its name implies, the EA model is based on the principle that energy dissipates due to elastic or Hertzian deformation and adhesion forces. Although the model was developed for particle impact on flat surfaces, it can be modified to particle-particle collision using the effective mass (Mi j ) for the overall particle mass and with surface roughness coefficient (CR) and circumferential tension ( f0). The combined stiffness (K) is a function of E of both particles. The adhesion work (wA) is dependent on the surface adhesion energies of both particles. The critical celocity (vc ) indicating the minimum velocity for non-sticking collision is simply evaluated by solving Equation 2.21 for ei j Ã 0. This yields: Further details of the EA model and corresponding MATLAB script are included in Appendix H. The combined stiffness (K) is a measure of the material elasticity. Low K indicates elastic material which can easily deform. Soft material generates a relatively large contact surface area and has low circumferential tension ( f0). Equation 2.22 shows that COR decreases for smaller particles. This becomes clear considering that Mi j is dependent on Ri j via Mi j Â» R3
i j . Similar, vc decreases for smaller particles. This agrees well with the fact that interparticle forces are more dominant for small particles. It should however be taken into account that smaller particles of the same material have higher E, thus higher K and higher COR [36, 37]. Furthermore, the porosity of the material should be taken into account since this is known to have a strong effect on E. E is typically ca. 80% lower for material with a porosity of 60% .
2.3. KINETIC THEORY OF GRANULAR FLOW
The most widely used method to describe granular flow, via a two-fluid model, is the Kinetic Theory of Granular Flow (KTGF). Thismethod describes the powder-gas emulsion in a fluidized bed as a continuum. All flow equations for the KTGF are based on the Navier-Stokes equation. The theory is based on the kinetic theory of gases but takes non-ideal particle interaction and gas-solid drag into account (see Section 2.2). It includes the effects of local particle concentrations and fluctuating particle velocities due to particle collisions . It does however not take cohesive effects into account. Although no simulations are performed in this work, the KTGF remains a useful tool to gain understanding of particle flow and fluidization behavior. Appendix J
discusses simulation methods for gas-fluidized beds in further detail. Without performing simulations, the KTGF can be used to gain insight into the influence of fines on fluidized bed behavior. Some of these aspects are:
1. Apparent viscosity (Â¹app)
2. Collision frequency (F)
3. Mean free path (Â¸)
4. Mean particle impact velocity (hCi j i)
The equations of the KTGF which should be used to evaluate these quantities are given in Appendix G. A MATLAB script where the equations are implemented is also given. The equations are adopted from , similar to the work of . As particles in a fluidized bed move in a rather random manner, particles will collide with each other. During collision, particle interaction such as cohesive forces and friction play a role. Furthermore, particle collisions are one way, next to particle-gas interaction, how kinetic energy is transported within a fluidized bed. It is therefore interesting to investigate the effect of fines on collision frequency since it may provide further insight into fluidized bed behavior. It is necessary to introduce the granular temperature of the particle mixture (Âµs ), which describes the mean square fluctuation velocity (hC2 i
i) of particles. This is analogue to the kinetic gas theory were gas molecules have an average velocity (e.g. due to convective flow) but strongly fluctuate due to collisions. The same applies to the description of fluidized granular flow in the KTGF. Generally, different sized particles (described as different phases in KTGF) have different granular temperature of particle phase (Âµi ). Âµi and Âµs are defined as : with particle mass (mi ), number of particle phases (NP) and number of particles per unit volume (ni ). Note that Âµs has units in kg m2 sÂ¡2. In other work, mi is often omitted in the definition in Equation 2.25, but included elsewhere in the derivation. Note that Âµs can differ depending on the location within the fluidized bed. When no full simulation is performed, which is the case for this work, Âµs should be treated as input variable. The mean free path (Â¸) and mean particle impact velocity (hCi j i) are a direct result of Âµs . The equations are further given in Appendix G. These can be used to quantify the apparent viscosity (Â¹app) of a fluidized bed. The physical meaning of Â¹app is described in Section 2.4.
2.4. APPARENT VISCOSITY
Gas-fluidized beds behave very similar to fluids and are sometimes visually described as boiling fluids. Therefore, some fluid properties (e.g. bed density and viscosity) can be assigned to a fluidized bed. This can become particularly useful to gain understanding of bed behavior even though the fluid-like properties are very dependent on applied gas velocity. This section introduces the concept of apparent viscosity (Â¹app): its physical meaning, its possible relation to the effect of fines on fluidized beds and how the quantity can be experimentally estimated. Similar to the viscosity ofNewtonian fluids, Â¹app describes the powder flowability in a fluidized bed. It can also be described as the inverse of particle mobility, i.e. the velocity change due to a small applied force,where high viscosity indicates low mobility [27, 39]. High viscosity means that the powder has higher resistance to flow and that kinetic energy dissipates faster through the bed. Â¹app is thus also descriptive for energy dissipation and frictional forces in a fluidized bed. As opposed to Newtonian fluids (i.e. gases and most liquids) which exhibit a constant viscosity, fluidized beds generally show non-Newtonian behavior [14, 40â” 42]. Generally, powders start to move easier when their flow rate is increased. This behavior is similar to shear thinning when explained in terms of fluid properties. This, in combination with the effect on bed porosity makes Â¹app for fluidization highly dependent on applied gas velocity. Several researches have put effort into the experimental determination of Â¹app. Generally, three experimental methods have been used: (1) falling/rising sphere [28, 43], (2) Couette or stirrer type powder rheometer [14, 40â”42, 44, 45] and (3) spherical cap bubble inclined angle [46, 47]. The falling or rising sphere method is based on the principle of buoyant and gravitational forces on a sphere submerged in a fluidized bed. The fall or rise velocity can be used to evaluate Â¹app based on Stokesâ law. Viscosity measurements with a Couette type rheometer is done by measuring the torque required while rotating an inner cylinder inside a fluidized bed. Other stirrer type rheometers use a rotating blade, plate or geometry instead of an inner cylinder. The FT4 Powder Rheometer by Freeman Technology is based on the blade type stirring method and may be used for evaluating a measure for Â¹app. The spherical cap bubble inclined angle uses visual observations of the bubble shape to investigate Â¹app. The last method can be used to qualitatively determine Â¹app without disturbing the fluidized bed but can only be used when bubbles are clearly visible, which can be achieved in a pseudo-2D column. The first two methods alter the bed hydrodynamics around and above the falling/rising sphere or stirring blade, but can be used to give quantitative estimations. It can however be experimentally difficult to track the falling sphere accuratelywithin a powder bed. The same drawback applies strongly to the spherical cap method. From these methods, a Couette type viscometer seems best to determine Â¹app since it only slightly disturbs the bed hydrodynamics, while the torque is readily measured. The rotational velocity should however be low enough . It has been proposed that the addition of fines reduces the granular viscosity of a fluidized bed [1, 6, 14]. Several causes could explain this reduction. Most of the energy dissipation occurs due to sliding and frictional dissipation (ca. 80% [19, 48]). As fines may position themselves on the surface of base particles, they could act as a lubricant similar to ball bearings . On the other hand, fines tend to increase bed expansion [1, 6, 12]. This would also lead to lower Â¹app. Lower viscosity leads to better powder flowability and faster bubble splitting which could be explanatory for the observed bubble size reduction due to addition of fines . This theory does however not cover all observed effects. Furthermore, experimental results on the effects of fines on Â¹app are contradictory [14, 44]. This couldwell be due to differences in applied stirring speed for the torque measurements. As discussed in Section 2.1, Geldart A powder fluidizes homogeneously at velocities above umf but below umb. This fluidization regime can be described as the particulate regime were interparticle forces prevail above gas-solid interactions. The fluidized bed is said to be in a solid- or fluid-like regime [24, 50â”52]. When homogeneously fluidizing, the powder bed expands and gas travels homogeneously through the small void spaces in-between particles. It remains uncertain how particles behave during homogeneous fluidization and whether they float and collide with each other or whether they are subject to endured particle-particle contact. The behavior of stable bed expansion suggests that particle stresses play a significant role and that stability cannot be fully attributed to fluid hydrodynamics [10, 15, 23, 24, 51, 52]. Furthermore, the role of particle stresses has been investigated to gain more insight into the origin of bubble formation . Several researchers provided evidence for particle stresses during (homogeneous) fluidization. One of the experiments is the tilting bed experiment. Here, a powder bed is homogeneously fluidized and slowly tilted until the bed shears off. It was shown that stable bed expansion can be sustained until a critical angle . This clearly shows that interparticle forces play a role and suggest that particles are in sustained contact. This theory was strengthened by measuring a surprisingly good electrical conductivity through a homogeneously fluidized bed, which can only be explained if particles were in continuous contact . It was proposed that a homogeneous expanded bed consists out of a particle microstructure as schematically shown in Figure 2.3
. In the terminology of the KTGF, it was measured that the granular temperature of the particle mixture (Âµs ) was essentially zero during homogeneous expansion which indicates nearly stagnant particles [16, 54, 55]. Further proof was given by investigation of the pressure drop (Â¢P) over a fluidized bed as function of applied gas velocity. As indicated in Section 2.1, fluidization occurs when Â¢P over the bed equals the bed weight. When slowly increasing the gas velocity to umf and beyond, it was found that the pressure drop does actually increase to values above the bed weight. When decreasing the gas velocity to zero flow, Â¢P gradually decreases to zero without generating an overshoot. This hysteresis is again proof of an important role of particle-particle and particle-wall stresses close to minimumfluidization [10, 15, 18, 23, 51, 53]. Experiments have shown that the pressure overshoot can be mainly attributed to particle-wall friction [16, 23, 29, 56]. In addition, DEM simulations have shown that interparticle forces such as Van der Waals forces play a significant role [17, 57, 58]. The pressure overshoot can be regarded as a threshold force which must be applied to overcome resistance and initiate fluidization. Apart from the overshoot, it was also found that Â¢P over a fluidized bed is generally just below the bed weight. This is said to be due to bed stability: the powder being partly supported by particles on the bottom and partly by friction with the wall [23, 58]. These observations could bemodeled with the linear momentum balance introduced in Section 2.2. The idea of amicrostructure (Figure 2.3) for homogeneous expansion allows for assigning solid-like properties to the powder bed. Interparticle forces give the fluidized bed a certain bed elasticity [51â”53]. It is shown that the bed elasticity increases when the spread in Particle Size Distribution (PSD) also increases. This result is mainly due to the increased number of particle contact points . In addition, as fines have stronger relative cohesive forces, stability is expected to further increase. Bed stability and a particle microstructure could thus partially explain the effect of fines on fluidization . As bubble formation could be seen as the result of a perturbation, this has lead to a criterion forminimumbubbling conditions . If the bed is more stable,
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