CFD study of wet agglomerate growth and breakage in a fluidized bed

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Kinetics, Catalysis, and Reaction Engineering

CFD study of wet agglomerate growth and breakage in a fluidized bed containing hot silica sand particles with evaporative liquid injection Amir Hossein Ahmadi Motlagh, Konstantin Pougatch, Anish Maturi, Martha Salcudean, John R. Grace, Dana Grecov, and Jennifer McMillan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05174 • Publication Date (Web): 25 Feb 2019 Downloaded from http://pubs.acs.org on March 2, 2019

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Industrial & Engineering Chemistry Research

CFD study of wet agglomerate growth and breakage in a fluidized bed containing hot silica sand particles with evaporative liquid injection A. H. Ahmadi Motlagh1,2*, Konstantin Pougatch 3, Anish Maturi 2, Martha Salcudean 2, John R. Grace1, Dana Grecov2, Jennifer McMillan4 1

Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada, V6T 1Z3

2

Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, Canada, V6T 1Z4 3

4

Coanda Research & Development, Vancouver, BC, Canada, BC V3N 4A3

Edmonton Research Centre, Syncrude Canada Ltd., Edmonton, AB, Canada T6N 1H4

1. ABSTRACT A computational model is developed to simulate evaporative spray injection into a hot gas– solid fluidized bed. Eulerian treatment of three phases – gas, droplets, and particles – is used, with a population balance model to account for variation in agglomerate sizes. Sub-models account for the interaction between droplets and hot particles, including momentum exchange, heat and mass transfer and vaporization of injected liquid as droplets or liquid film on the surface of agglomerates. An energy approach is utilized to assess the outcome of collisions between particles, ranging from rebound to agglomeration and breakage. Model predictions agree well with experimental data in terms of the amount of liquid vaporized. The effects of superficial gas velocity and bed temperature are compared to experimental data, leading to reasonable agreement for the mass distribution of agglomerates, but some over-prediction for small agglomerates. The new

*

Corresponding author E-mail: [email protected]

1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

model can be utilized for in-depth investigation of liquid injection into hot gas-solid fluidized beds with various operating conditions.

2. INTRODUCTION Liquid injection into gas-solid fluidized beds has a wide range of applications in the chemical, pharmaceutical and oil industries, e.g., in Fluid Coking, Fluid Catalytic Cracking (FCC), gas-phase polymerization and fluidized bed spray granulation. Despite many industrial applications and laboratory studies, the underlying physics of interactions between liquid, gas and solid phases are not well understood and in need of ongoing research. Liquid injection at the bottom of riser reactors becomes the most important step in determining unit performance, even dominating product distribution and quality (Mauleon and Courelle1; Mirgain et al.2). The effects of liquid vaporization and nozzle configuration have been evaluated in catalytic cracking reactors and pilot units, as well as in Fluid Coking, a process that utilizes a fluidized bed of hot coke particles to thermally crack and upgrade bituminous feeds. There have also been many attempts to simulate FCC units, almost all based on the assumption of instantaneous vaporization of feed at the riser entry (Gupta3; Gupta et al.4). Extensive research has also been conducted on hydrodynamics of Fluid Cokers to study local voidage profiles, particle velocities and mixture behavior in these reactors (Li et al.5; Song and co-workers6,7). In recent years, efforts have been made to study the mechanism of feed vaporization at the inlet of riser reactors. Research reports have been published on non-evaporative (Leach et al.8; Mohagheghi et al.9; Portoghese, et al.,

10),

and evaporative (Du et al.11; Gehrke and Wirth12) liquid (water and

liquid nitrogen) injection experiments. Both water (Bruhns and Werther13) and ethanol (Bruhns14; McMillan et al.15) have been used to study the contact between liquid droplets and fluidized particles. Bruhns and Werther13 injected water and ethanol into fluidized beds of quartz and FCC 2 ACS Paragon Plus Environment

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catalyst particles in a pilot scale unit. Based on experimental results, they introduced a new model for the mechanism of liquid injection which assumes instantaneous wetting of the fluidized particles at the nozzle exit and considers the transport of wetted particles by gross solids mixing of the fluidized bed. However, they did not include a detailed comparison of the new model with experimental results. Rapid vaporization of liquid droplets in FCC reactors leads to substantial increases in volumetric gas flow. This can significantly affect gas-solid mixing, temperature distribution, flow behaviour, process efficiency and product quality (Fan et al.16). Liu17 measured the temperature distribution of a three-phase mixture near the spray region, which is helpful for testing the fundamentals of spray jets in gas-solid flows. The results were then utilized to develop a parametric model which could simulate the phase interactions. Liu’s numerical results were reported to be in good agreement with experimental results. Li et. al18 simulated the vaporization of liquid feed in gas-solid fluidized beds. They considered three limiting cases, corresponding to different locations inside a fluidized bed at which the liquid vaporizes, to simplify the process. The vaporization rate was shown to have a significant impact on the hydrodynamics. Agglomeration and associated phenomena were not considered directly in their work. Pougatch and co-workers19, 20developed a model which can simulate the flow through the nozzle, fluidization and droplet-particle interactions. The primary atomization was treated as phase inversion to obtain initial droplet sizes. Their modelling predictions showed good agreement with experimental data. When liquid droplets collide with dry particles, some of the liquid can stick to the particles, making them wet. Boyce21 conducted a recent review of the prior works in microscale studies, as well as experimental and modeling techniques in wet fluidization. As wetted particles collide with each other or with dry particles, agglomeration may occur due to formation of a pendular liquid



bridge, whose formation is governed by the kinetic energy of collision, surface tension (or capillary effects) and viscous forces. These forces depend on the physical properties of the liquid and the shape of the bridge. Darabi22 developed a model which predicted the liquid redistribution between particles during collisions. His model focused on the viscous and capillary forces between particles. It was not limited to calculating liquid redistribution between wet particles, but also calculated redistribution between wet and dry particles. The effects of agglomerates and their properties on mass and heat transfer limitations in Fluid Cokers have been modelled by several researchers. Gray et al.23 proposed a model which predicted that thicker liquid films of heavy oil results in less yield due to more liquid being trapped inside the agglomerates. House et al.24 utilized a simplified model to show that liquid distribution has a major impact on the yield of valuable products in Fluid Cokers. The relatively long reaction time in Fluid Cokers would result in liquid staying more sticky and promote the formation of agglomerates. Morales and co-workers25, 26 conducted liquid injection experiments into a fluidized bed of sand particles and measured the mass distribution of agglomerates in the bed. The Plexiglas/acetone/pentane solution used by Morales25 was compared to the results obtained with bitumen and similar behavior was reported, making the mixture a relatively good surrogate for bitumen in experiments. There have been a number of CFD studies on three-phase flow in fluidized beds which investigated the effect of liquid injection. Sutkar et al.27 performed CFD-DEM simulations in a gas-solid spouted bed with liquid injection, resolving both particles and droplets as discrete phase. They included the coupling effects of mass and heat transfer and reported a fair agreement in terms of particulate flow pattern, velocities and pressure drop27. Pietsch et al.28studied liquid coating on particles in a 3D prismatic spouted bed both experimentally and numerically using CFD-DEM


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simulations under different operating conditions. Numerical studies on evaporative liquid sprays (Li et al.29), and utilization of a combination of DEM-PBM (Fries et al.30; Girardi et al.31; Sen et al.32) and Eulerian frameworks (Sivaguru et al.33) were performed to examine the effect of agglomeration in the presence of liquid injection in fluidized beds. However, none of the earlier studies included the effect of liquid vaporization, as well as agglomerate growth and breakage in a three-phase fluidized bed. The objective of this study is to seek better understanding of the hydrodynamics of the complex phenomena, including the aforementioned effects, in a three-phase fluidized bed the development of a CFD model in an Eulerian framework, accounting for interactions between liquid, hot particles and gas in a gas-solid fluidized bed with liquid injection.



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3. MODEL DEVELOPMENT 3.1 Problem description

Solid particles

Gas and liquid injection

Fluidization gas

Figure 1. Schematic of liquid injection into fluidized bed (freeboard section) 0.15 m x 1.2 m x 1.5 m As a basis for the model, consider liquid injection into the fluidized bed shown schematically in Figure 1. Note that Figure 1 is only for understanding the problem addressed by model and does not reflect actual dimensions. The latter are included in Table 1. The bed contains hot solid particles, and the injected liquid that is atomized by a gas is a mixture of two components: one evaporative and the other not evaporating. Upon injection, a liquid droplet can collide with a solid particle and partially stick to the particle surface. We assume that the temperature difference between the phases is small, and the Leidenfrost effect (repulsion due to sudden vaporization) is not considered. As particles become wet, they may form aggregates upon collisions, due to capillary and viscous forces; the aggregates in turn may break apart due to collisional interactions


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in the bed. Inside the bed, the liquid (either in droplets or in outer layers on particles) can heat up because of convection and interactions with particles. The evaporative component may vaporize by either evaporation or boiling mechanisms. 3.2 Model description The complexities of the problem considered are evident. Multiple physical processes are taken place at the same time interacting with one another. Our objective is to develop a model that can reflect the problem complexity and predict the process at various scales and operating conditions. In order to achieve that, we choose to rely on the first principles, established physical laws and well-known correlations. Only when absolutely necessary, we augment them by our own development that involves adjustable parameters which are established based on available data. While simpler models could likely provide better fitting to some of the data, their extrapolations to industrially relevant conditions would be questionable due to limitations of many assumptions used. We consider three phases. The gas, which is the continuous phase, contains two species: fluidizing gas and evolved vapour coming from the injected liquid. The liquid droplets are the discrete phase, and they consist of evaporative and non-evaporative liquid. Another discrete phase is the particulate phase that includes everything containing solids. The particulate phase ranges from single solid particles to wetted particles and agglomerates. From this definition, it is evident that the particulate phase consists of three components: solid, evaporative liquid, and nonevaporative liquid. Note that some of these components may not be present in each particle. The gas, liquid and particulate phases are treated in this work as inter-penetrating continua, using an Eulerian approach. It means that each phase is characterized by its average local parameters, such as velocity, temperature, volume fraction, etc., throughout the domain. The 7 ACS Paragon Plus Environment


motion of the individual particles is not considered directly. Rather, it is approximated through the constitutive equations. Heat and mass transfer are included in the model to account for complex phenomena inside the bed. In line with an Eulerian approach, heat transfer through individual particles and droplets is not considered as they assumed to have a uniform temperature. This is justified by a low value of the Nusselt number, which is the ratio of the external to internal heat transfer. In order to be consistent with the targeted application, the gas is denoted as nitrogen, the evaporative liquid species as acetone, and the non-evaporative liquid species as PMMA. As the main equations are well-known and available elsewhere (ANSYS Fluent34), they are not discussed here. Rather, we focus on the constitutive equations that close the transport equation by providing a mathematical description of complex mechanisms involving inter-particle interactions, vaporization, agglomeration, and breakage. The conservation equations for each phase plus the solid phase equations with kinetic theory of granular flow and liquid bond energy expressions in this work are tabulated in a separate supporting information document as Table S1, S2 and S3, respectively. Liquid vaporization due to the effect of convective mass transfer (evaporation) and due to heating of liquid (boiling) is considered in the model. In addition, the model estimates the extent of agglomeration and breakage which are affected by the liquid content in the reactor. In order to properly take into account liquid transfer between droplets and particle phases, two species (acetone and PMMA) are defined for the liquid droplet phase. The solid phase consists of three species: acetone, PMMA and the silica sand solid core. The liquid acetone and PMMA species are transferred from the droplet phase to the same species in the particle phase due to collisions. This way, the liquid content of the solid phase, which


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grows as a result of particle-droplet collisions, can be tracked in the model. The gaseous phase includes nitrogen and vaporized acetone, enabling the evaporation extent in the bed to be tracked. Four distinct mass transfer steps between the phases are considered in the model: -

Liquid phase acetone species to particle phase acetone species,

-

Liquid phase PMMA species to particle phase PMMA species,

-

Liquid phase acetone species to gaseous phase vapor (liquid droplet vaporization)

-

Particle phase acetone species to gaseous phase vapor species (liquid film vaporization)

The corresponding terms related to the above mass transfer steps are added to mass, momentum and energy conservation equations and are shown in Table S1 in the supporting information section. The spreading of liquid film due to wet and dry particle collisions is accounted for by means of a diffusion term in the acetone/PMMA species equations for the solid phase. Figure 2 indicates the connection between different solution sections to facilitate understanding.



Liquid content

Collision scenarios

calculation Liquid vaporization

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(Section 3.3)

(Section 3.2.3)

(Section 3.2.1) (

Polydisperse representation of the solid phase

Solution of mass, momentum

(Section 3.2.2)

and energy equations for gas, liquid and solid phases/species

Figure 2. Solution diagram showing the connection between different sections in this text

The Gidaspow et al.35 drag correlation, widely utilized in dense fluidized bed simulations, is used to account for momentum exchange between gas and particulate phases:

𝛽𝑔𝑠 =

𝐶𝐷 =

𝑅𝑒𝑠 =

{

3 4𝐶𝐷

(

𝜀𝑠 𝜀𝑔𝜌𝑔 |𝑣𝑔 ― 𝑣𝑠|

150

𝑑𝑠

(

𝜀𝑠(1 ― 𝜀𝑔)𝜇𝑔 𝜀𝑔 𝑑2𝑠

)𝜀

―2.65 𝑔

, 𝜀𝑔 > 0.8

) + 1.75 (

𝜌𝑔 𝜀𝑠 |𝑣𝑔 ― 𝑣𝑠| 𝑑𝑠

), 𝜀

𝑔

≤ 0.8

24 [ 1 + 0.15(𝜀𝑔𝑅𝑒𝑠)0.687] 𝜀𝑔𝑅𝑒𝑠 𝜌𝑔 𝑑𝑠 |𝑣𝑔 ― 𝑣𝑠| 𝜇𝑔

The Schiller-Naumann36 correlation is used for gas-droplet phase momentum exchange:


Eq. 1

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𝛽𝑙𝑔 =

3𝐶𝐷𝜌𝑙𝜀𝑔 4𝑑𝑙

|𝑣𝑔 ― 𝑣𝑙|

{

24(1 + 0.15𝑅𝑒0.687) 𝐶𝐷 = 𝑅𝑒 0.44,

𝑅𝑒 =

Eq. 2 𝑖𝑓

𝑅𝑒 ≤ 1000

𝑖𝑓

𝑅𝑒 > 1000

𝜌𝑔 𝑑𝑙 |𝑣𝑔 ― 𝑣𝑙| 𝜇𝑔

In the above equation, the Reynolds number is calculated in a manner similar to Eq.1, by replacing solid parameters with droplet parameters. The initial diameter of the droplets is assumed to be constant in the simulations. The angle of spray is assumed to be zero. Momentum is exchanged between solid and liquid phases due to collisions. This transfer is accounted for by the approach proposed by Pougatch37, where colliding droplets partially transfer their momentum to the solid phase by leaving some liquid on the particles as they rebound. This mechanism is implemented in Eqs. 5 and 6 through mass transfer terms, with 𝜔 representing the stickiness ratio that determines the fraction of liquid which remains with the particle phase after collision: 𝜔 = 1 ― 0.35𝐵 𝑖𝑓 𝐵 < 2.86,

Eq. 3

𝑒𝑙𝑠𝑒 𝜔 = 0

𝑅𝑒0.37 𝑑𝑙 𝐵= 𝐿𝑝0.1 𝑑𝑠 𝑅𝑒 = 𝐿𝑝 =

𝜌𝑙|𝑣𝑙 ― 𝑣𝑠|𝑑𝑙 𝜇𝑙 𝜎𝜌𝑙𝑑𝑙 𝜇2𝑙

The stickiness ratio is also utilized to calculate mass transfer from liquid to particulate phase due to collisions (Pougatch37), 11 ACS Paragon Plus Environment


3

1

(

𝑚𝑎𝑠 = 𝜔2𝜀𝑙𝑥𝑎_𝑑𝑟𝑜𝑝𝜀𝑠𝑑 3𝜌𝑙 𝑠

3

𝑑𝑠 + 𝑑𝑙 2

1

2

(

𝑚𝑝𝑚𝑠 = 𝜔2𝜀𝑙𝑥𝑝𝑚_𝑑𝑟𝑜𝑝𝜀𝑠𝑑 3𝜌𝑙 𝑠

) |𝑣𝑙 ― 𝑣𝑠|

Eq. 4

𝑑𝑠 + 𝑑𝑙 2 2

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) |𝑣𝑙 ― 𝑣𝑠|

Eq. 5

where 𝑥𝑎_𝑑𝑟𝑜𝑝 and 𝑥𝑝𝑚_𝑑𝑟𝑜𝑝 are the volume fractions of acetone and PMMA in the liquid phase, obtained by solving species conservation equations. 3.2.1

Heat transfer, evaporation and boiling

Liquid vaporization can be attributed to two distinct mechanisms: (a) liquid reaching the boiling temperature and then vaporizing (boiling), and (b) liquid vaporizing from the interface due to convective mass transfer (evaporation). Boiling and evaporation can take place for all phases that contain evaporative liquid: droplets and particles. These terms are included in mass conservation equations. Boiling takes place in the whole volume occupied by the liquid, limited only by the availability of heat. The mass transfer rate from liquid droplets to the gas phase due to boiling is calculated as

𝑚𝑑𝑟𝑜𝑝_𝑏𝑜𝑖𝑙 =

6𝜀𝑙ℎ𝑑𝑟𝑜𝑝(𝑇𝑔 ― 𝑇𝑠𝑎𝑡)

Eq. 6

𝐻𝑙𝑔𝑑𝑙

where ℎ𝑑𝑟𝑜𝑝 is the heat transfer coefficient from gas to a droplet, calculated from the well-known Ranz and Marshall38 correlation: 𝑘𝑔

ℎ𝑑𝑟𝑜𝑝 = 𝑑𝑙 (2 + 0.6𝑅𝑒0.5𝑃𝑟1/3)

Eq. 7

The mass transfer from the liquid film (acetone component of solid phase) to the gas phase due to boiling is calculated in a similar way:


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𝑚𝑎𝑔𝑔_𝑓𝑖𝑙𝑚_𝑏𝑜𝑖𝑙 =

6𝜀𝑠ℎ𝑎𝑔𝑔(𝑇𝑔 ― 𝑇𝑠𝑎𝑡)

Eq. 8

𝐻𝑙𝑔𝑑𝑠

where ℎ𝑎𝑔𝑔 is the heat transfer coefficient from solid mixture to the gas, calculated from the Gunn39 correlation:

[

𝑘𝑔

ℎ𝑎𝑔𝑔 = 𝑑𝑠 (7 ― 10η𝑠 +

)(1 +

5η𝑠2

1

) + (1.33 ― 2.4η𝑠 +

3 0.7𝑅𝑒0.2 𝑠 𝑃𝑟

]

1

3 1.2η𝑠2)𝑅𝑒0.7 𝑠 𝑃𝑟

Eq. 9

Evaporation occurs along the surface of contact between the phases. Hence, the area of contact must be defined. Droplets are assumed to retain a spherical shape. The mass transfer rate for evaporation from liquid droplets to gas phase due to convection is calculated from:

(

𝑚𝑑𝑟𝑜𝑝𝑒𝑣 = 6𝑘𝑚𝑎𝑠𝑠𝑑𝑟𝑜𝑝

𝑃∗ 𝑃

― 𝑦∗

𝜀𝑙𝑥𝑎𝑑𝑟𝑜𝑝

)

𝑑𝑙

Eq. 10

𝜌𝑔

Note that the partial pressure of acetone on the particle surface is corrected according to Rault’s law; y* is the mole fraction of acetone in the gaseous phase and 𝑘𝑚𝑎𝑠𝑠_𝑑𝑟𝑜𝑝 is the mass transfer coefficient between droplets and gas, calculated from the mass transfer analog of the Ranz and Marshall36 equation: 𝑑𝑙

𝑆ℎ𝑑𝑟𝑜𝑝 = 𝑘𝑚𝑎𝑠𝑠_𝑑𝑟𝑜𝑝𝐷𝑐 = 2 + 0.6𝑅𝑒0.5𝑆𝑐1/3

Eq. 11

The structure of each agglomerate is assumed to consist of a random arrangement of primary particles, held together by liquid bonds. Therefore, there can be pores in the agglomerate that are penetrable by gas. The pores can also be partially filled by liquid, due to capillary forces. It is assumed that the liquid in the pores would be locked in and not available for evaporation. However, boiling can still occur for liquid inside the pores. For primary particles, the entire liquid is available for evaporation from the surface, whereas for larger agglomerates, depending on how 13 ACS Paragon Plus Environment


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much liquid penetrates the pores, some of the liquid is available for evaporation. In this work, a simple approach to estimate the fraction of available liquid is utilized, considering the effect of the surface-area-to-volume ratio by introducing the term “ALC”, standing for Available-LiquidContent, representing the proportion of liquid in the solid phase that is available for evaporation. ALC for agglomerates is then formulated as: 𝑑𝑝

Eq. 12

𝐴𝐿𝐶 = 𝑑𝑎

Liquid film h0

Agglomerate

2a

Figure 3. Schematic of liquid film formation on agglomerate surface For the liquid film, a hemispherical cap shape is assumed as shown in Figure 3, and the Sherwood number is calculated as above, based on the equivalent spherical diameter of the liquid in the solid phase. To estimate the evaporation rate from the liquid film on the agglomerate, we utilize a convective mass transfer expression with ALC as defined above:

𝑚𝑠𝑔_𝑒𝑣𝑎𝑝 = (2𝑎)𝑘𝑚𝑎𝑠𝑠

(

𝑃∗ 𝑃

)

― 𝑦 ∗ 𝑑𝑙𝑓𝑖𝑙𝑚

6𝜀𝑠𝑥𝑎_𝑓𝑖𝑙𝑚𝐴𝐿𝐶 𝑑𝑠3

𝜌𝑔

Eq. 13

where y* is the mole fraction of acetone in the vapor phase. The term 𝑑𝑙_𝑓𝑖𝑙𝑚 represents the equivalent diameter of the film, obtained by dividing the volume of the film by its area. The area of the film is calculated from the radius, a, and height, h0, as follows: 𝑉𝑓𝑖𝑙𝑚 ∗ 𝐴𝐿𝐶

Eq. 14

𝑑𝑙_𝑓𝑖𝑙𝑚 = 𝜋(𝑎2 + ℎ2) 0


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where, 𝜋𝑑𝑠3 𝑉𝑓𝑖𝑙𝑚 = 𝑥 6 𝑎_𝑓𝑖𝑙𝑚

𝑎=

[

3𝑉𝑓𝑖𝑙𝑚

𝑠𝑖𝑛3𝜃

𝜋

2 ― 3𝑐𝑜𝑠𝜃 + 𝑐𝑜𝑠3𝜃

ℎ0 = 𝑎

]

1 3

1 ― 𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃

Here θ is the contact angle for droplets on a solid surface. The source terms on the right-hand side of the energy equations account for heat transfer between phases. 𝑄𝑔𝑙 and 𝑄𝑔𝑠 are defined as:

𝑄𝑔𝑙 = 𝑄𝑔𝑠 =

6𝜀𝑙ℎ𝑑𝑟𝑜𝑝(𝑇𝑙 ― 𝑇𝑔)

Eq. 15

𝑑𝑙 6𝜀𝑠ℎ𝑎𝑔𝑔(𝑇𝑠 ― 𝑇𝑔)

Eq. 16

𝑑𝑠

Hg, Hl and Hs are the energy transfers associated with mass transferred due to liquid deposition on the particulate phase, boiling and evaporation: 𝐻𝑔 = (𝑚𝑑𝑟𝑜𝑝_𝑏𝑜𝑖𝑙 + 𝑚𝑎𝑔𝑔_𝑓𝑖𝑙𝑚_𝑏𝑜𝑖𝑙 + 𝑚𝑑𝑟𝑜𝑝_𝑒𝑣 + 𝑚𝑎𝑔𝑔_𝑓𝑖𝑙𝑚_𝑒𝑣)ℎ𝑔

Eq. 17

𝐻𝑙 = ― (𝑚𝑑𝑟𝑜𝑝𝑏𝑜𝑖𝑙 + 𝑚𝑑𝑟𝑜𝑝𝑒𝑣)ℎ𝑙 ― (𝑚𝑎𝑠 + 𝑚𝑝𝑚𝑠)ℎ𝑙

Eq. 18

(

)

𝐻𝑠 = ― 𝑚𝑎𝑔𝑔𝑓𝑖𝑙𝑚𝑏𝑜𝑖𝑙 + 𝑚𝑎𝑔𝑔𝑓𝑖𝑙𝑚𝑒𝑣 ℎ𝑠 + (𝑚𝑎𝑠 + 𝑚𝑝𝑚𝑠)ℎ𝑙


Eq. 19


Although turbulence in the gaseous phase can be significant in the vicinity of the injection nozzle, the flow in fluidized beds is mostly influenced by interactions with the particulate phase, and turbulent effects are commonly ignored. Constitutive relations to describe normal and tangential stresses in the particulate phases arising due to particle-particle interactions can be derived based on the kinetic theory of granular flow, allowing particles to undergo inelastic collisions. Frictional viscosity is neglected in this work as the bed is constantly fluidized and particle fraction does not reach the packing limit. Lun et al.40 and Gidspow et al.41correlations are used for bulk viscosity and kinetic viscosity of the solid phase. For brevity, the equations of granular kinetic theory with commonly-used correlations are summarized in Table S2 in supporting information. More details on granular kinetic theory are available from Gidaspow35. The strength of the liquid bridges can be characterized based on the energy required to break them. Liquid bridges are held together by capillary forces of the liquid. The external force has to overcome capillary and viscous forces, which contribute to the liquid resistance to particle motion for successful breakage. The bond breakage energy, 𝐸𝑏, can be taken as the summation of the energies needed to overcome the individual forces. Pitois et al.42 and Darabi22 derived expressions for the viscous and capillary energies which are adopted in this work and provided in Table S3.

3.2.2

Polydisperse representation of solid phase

As mentioned above, the particulate phase contains a wide range of particles, i.e., the distribution is polydisperse. This range needs to be specified in order to consider agglomeration and breakage. We utilize the population balance approach to account for the fraction of particles


Page 16 of 52

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in bins of different sizes. The Population Balance Equation (PBE) considered in this work can be written (ANSYS Inc.34) as: ∂ ( ) ∂𝑡 𝜀𝑖𝜌𝑠

+ ∇.(𝜀𝑖𝜌𝑠𝑣𝑠) = 𝜌𝑠𝑉𝑖(𝐵𝑎𝑔𝑔,𝑖 ― 𝐷𝑎𝑔𝑔,𝑖 + 𝐵𝑏𝑟,𝑖 ― 𝐷𝑏𝑟,𝑖)

Eq. 20

where 𝐵 and 𝐷 are the agglomerate birth and death terms due to agglomeration and/or breakage mechanisms: 𝑁 ― 1𝑁 ― 1

𝐵𝑎𝑔𝑔,𝑖 =

∑ ∑𝑎

𝑘𝑗𝑛𝑘𝑛𝑗𝑥𝑘𝑗𝜁𝑘𝑗

𝑘=0𝑗=0 𝑁―1

𝐷𝑎𝑔𝑔,𝑖 =

∑𝑎 𝑛 𝑛

𝑖𝑗 𝑖 𝑗

𝑗=0 𝑁―1

∑ 𝑔(𝑉 )𝑛 𝛽(𝑉 |𝑉 )

𝐵𝑏𝑟,𝑖 =

𝑗

𝑗

𝑖

𝑗

𝑗=𝑖+1

𝐷𝑏𝑟,𝑖 = 𝑔(𝑉𝑖)𝑛𝑖

3.2.3

Liquid content calculation

Morales25 reported only very minor changes in initial liquid content for different agglomerate sizes. Therefore, in this study we assume that the liquid fraction is independent of the agglomerate size. The mass of liquid per agglomerate in the solid phase can then be calculated as: 𝜌𝑙

𝑚𝑙𝑖 = 6 (𝑥𝑎_𝑠 + 𝑥𝑝𝑚_𝑠)𝜋𝑑3𝑖

Eq. 21

Part of 𝑚𝑙𝑖 forms a film on the surface of agglomerates that contributes to further agglomeration with other particles and agglomerates, whereas the remaining portion remains within the voids inside the agglomerate. This is considered in the model by the previouslyintroduced ALC coefficient (see Eq. 12). Therefore, 𝑚𝑙𝑖𝐴𝐿𝐶 is used to account for liquid film mass 17 ACS Paragon Plus Environment


Page 18 of 52

on the surface of agglomerates, with 𝑚𝑙𝑖(1 ― 𝐴𝐿𝐶) being the mass of liquid inside the pores. When two agglomerates collide, they are assumed to supply a portion of the liquid on the surface to the bridge. This portion of liquid is estimated by the model of Shi and McCarthy43:

(

1

𝑉𝑏 = 2(𝑉𝑙𝑝,𝑖 + 𝑉𝑙𝑝,𝑗) 1 ―

)= (

3 2

1 𝑚𝑙𝑖 2 𝜌𝑠 𝐴𝐿𝐶𝑖

+

𝑚𝑙𝑖

𝜌𝑠 𝐴𝐿𝐶𝑗

)(1 ― ) 3 2

Eq. 22

A different approach is followed to calculate the liquid bridge volume. We assume that all particles inside the agglomerate have identical properties. The liquid bridge volume is then calculated based on the number of bonds, using the coordination number:

𝑉𝑏_𝑎 =

𝑚𝑙(1 ― 𝐴𝐿𝐶)

Eq. 23

𝑁𝑝𝑖𝑘 2

3.2.4

Collision frequency

The frequency of collisions between two agglomerates is calculated (Pougatch37) as 𝜋 𝑑𝑎1 + 𝑑𝑎2 2 𝑉𝑟𝑒𝑙 2

𝑛𝑐𝑜𝑙𝑙 = 𝑛𝑎𝑖𝑛𝑎𝑗𝑔04

(

)

Eq. 24

where ai and aj refer to colliding agglomerates. The relative velocity in this work is calculated according to the approach of Hounslow44: 𝑉𝑟𝑒𝑙 =

(

32 3𝜃𝑚 1 𝜌 𝑑3𝑎𝑖 𝜋

1

+ 𝑑3

𝑎𝑗

)

Eq. 25

where θ𝑚 is the mixture granular temperature approximated (Rajniak et al.45) as: 𝜋

θm = 𝑚𝑠𝜃𝑠 = 𝜌𝑠6𝑑3𝑠 𝜃𝑠

Eq. 26

where 𝜃𝑠 is the granular temperature of a mono-disperse system.


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3.2.5

Liquid bond and collision energies

As shown in Table S3, the energies required to break the liquid bond between two primary particles inside an agglomerate “i” (Eb_a_i) and between two colliding agglomerates (Eb) are calculated following the approach of Pitois et. al42 and Darabi22, which provides a summation of capillary and viscous energies determined based on the available liquid content. The kinetic collision energy between two colliding agglomerates is then calculated from: 𝐸𝑐𝑖𝑗 = 0.5𝑚𝑖𝑗𝑉2𝑟𝑒𝑙_𝑖𝑗

Eq. 27

where i and j are agglomerate indices, and mij stands for the equivalent mass of agglomerate, estimated from 𝑚𝑖𝑗 = 𝑚𝑖𝑚𝑗 (𝑚𝑖 + 𝑚𝑗). 3.2.6

Collision outcome probabilities

Because there are many uncertainties involved in how agglomerates grow and break, the collision outcome in this work utilizes a statistical approach and probability functions, with a probability assign for each collision case. Agglomerate breakage is considered to happen only due to collision between agglomerates. Collisions with the wall are not considered in this work. Note that any collision between agglomerates are treated as events in which the summation of probabilities for agglomeration and rebound must be equal to one.

3.3 Collision scenarios Agglomeration and breakage are collision-based phenomena. Both can take place only as an outcome of collisions. Thus, by analyzing each collision we can determine rates associated with breakage and agglomeration. We utilize a probabilistic approach in evaluating collision outcomes. Based on the local conditions, we calculate probabilities of certain outcomes and distribute heat,



mass, and momentum exchange terms accordingly.

Page 20 of 52

Three potential collision scenarios are

considered in this work: 1) Collision between two primary particles of the smallest bin size, which corresponds to the initial particle size; see Figure 4. 2) A particle in the smallest bin collides with another from a larger bin (Figure 5); 3) Collision between agglomerates of two different bins (larger than primary particles) (Figure 6). We consider two types of agglomerate breakage: (a) abrasion, which accounts for the dislodgement and separation of single or multiple particles from the agglomerate surface, and (b) fragmentation, where agglomerates break through a cross-sectional plane. In this work, it is assumed that whenever fragmentation occurs, an agglomerate breaks in half.

3.3.1

Scenario I. Collision between two primary particles of the smallest bin size:

i

i

j

j

i

a) Agglomeration

j

b) Rebound

Figure 4 –Scenario I. Schematic of two identical colliding agglomerates of the smallest bin size and possible collision outcome


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In the scenario shown in Figure 4, both colliding particles are identical and of the smallest size. As a result of the collision, the particles may aggregate or rebound so that the sum of rebound and aggregation probabilities, used below in the agglomeration kernel, is unity: Eq. 28

𝑃𝑎𝑔𝑔 + 𝑃𝑟𝑏 = 1

In this work, the agglomeration probability is considered to consist of two terms (Rajniak et al.45): Eq. 29

𝑃𝑎𝑔𝑔 = Ψ𝑔𝑒𝑜𝑚Ψ𝑝ℎ𝑦𝑠

where Ψ𝑔𝑒𝑜𝑚 and Ψ𝑝ℎ𝑦𝑠 are geometrical and physical collision success factors, respectively. Rather than using a step-function approach, the physical success factor in this work is calculated based on an exponential function (Palanisamy46) to define the probability of agglomeration as a function of agglomerate strength and shear due to impacts, using the bond energy and kinetic energy of impact:

Ψ𝑝ℎ𝑦𝑠

(―𝛾 ) = 𝑒 𝐸𝑐𝑖𝑗 𝐸𝑏

Eq. 30

The coefficient γ was set to 25.0 in this work to provide a smooth transition of probability to the limiting values of zero or one, when the collision energy becomes comparable to (within an order of magnitude of) the liquid bond energy. The geometrical success factor is related to the fact that the liquid film on the agglomerate does not cover the entire surface, but takes a hemispherical shape due to the contact angle effect. Therefore, for a successful collision it is necessary that one of both colliding particles touch each other with parts covered by liquid. The area covered by liquid film can be calculated from: 𝑆𝑖 ≈ 𝜋𝑎2𝑖

Eq. 31



Page 22 of 52

Note that the amount of liquid trapped inside the agglomerate is not available and is not considered in calculating the liquid coverage area. The geometrical probability of agglomeration is then defined as the probability that both/either of the two colliding agglomerates make contact on the 𝑎2𝑖

𝑎2𝑗

𝑖

𝑗

wet area for further agglomeration. Considering 𝑑2 and 𝑑2 to be the ratios representing the wetted area on each colliding agglomerate, the probability can then be calculated as:

(

𝑎2𝑖

)(

𝑎2𝑗

)

Eq. 32

Ψ𝑔𝑒𝑜𝑚 = 1 ― 1 ― 𝑑2 1 ― 𝑑2 𝑖

𝑗

It is apparent that primary particles can break no further, so neither abrasion nor fragmentation can occur in this scenario.

3.3.2

Scenario II. Collision of one particle in the smallest bin with another from a

larger bin:

j

i

22

ACS Paragon Plus Environment

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j

j i

i

a) Agglomeration

b) Rebound

j

j

i

i b1) Rebound - unchanged

b2) Rebound with single or multiple abrasion

Figure 5 –Scenario II. Schematic of one particle in the smallest bin colliding with another from a larger bin and possible collision outcomes As in the first scenario, particles can aggregate or rebound, so that 𝑃𝑎𝑔𝑔 + 𝑃𝑟𝑏 = 1 The agglomeration probability is calculated in the same way as for scenario I by Eq. 29. However, if rebound occurs, more options are available. During collisions that did not lead to agglomeration, some energy is lost due to dissipation by capillary and viscous forces. The remaining energy,



Page 24 of 52

though, can be sufficient to cause abrasion of one or several particles from the outer layer of the agglomerate. Thus, the rebound probability can in turn be described as Eq. 33

𝑃𝑟𝑏 = 𝑃𝑟𝑏_𝑢𝑛𝑐ℎ + 𝑃𝑟𝑏_𝑎𝑏𝑟

Utilizing the smoothing exponential function, the abrasion probability for each collision is estimated as: 𝑃𝑟𝑏_𝑎𝑏𝑟𝑖

( ―𝛾 =𝑒

𝐸𝑏_𝑎 𝐸𝑐𝑖𝑝 ― 𝐸𝑏

)𝑃

( )

2 𝑔𝑒𝑜𝑚_𝑖 𝐾𝑐

Eq. 34

where Ecip - Eb is the kinetic energy between agglomerate “i” and primary particles that is available after dissipation due to liquid bond energy between the agglomerates, and Eb_a is the liquid bond energy between primary particles inside the agglomerate.

2 𝐾𝑐 is the coordination probability, related

to the number of bonds required to release a single particle from the surface, and 𝑃𝑔𝑒𝑜𝑚_𝑖 is the geometric probability, written as the ratio of the area of the active collision region to the total cross-sectional area of the agglomerate “i”: 𝑃𝑔𝑒𝑜𝑚_𝑖 =

9 𝑑𝑝2

Eq. 35

𝑑𝑖2

3.3.3

Scenario III. Collision between two agglomerates (larger than primary

particles):

j i


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j

j

i

i

a) Agglomeration

b) Rebound

Figure 6 –Scenario III. Schematic of single agglomerate colliding with another agglomerate and possible collision outcomes This scenario, depicted in Figure 6, accounts for collision between two agglomerates, both of which are larger than the smallest bin size. As for the other scenarios, the agglomerates can either aggregate further or rebound, so that 𝑃𝑎𝑔𝑔 + 𝑃𝑟𝑏 = 1 It is assumed that the collisions between agglomerates in this scenario do not carry enough energy to result in fragmentation. It can also be assumed that no abrasion is possible because the active area of a collision is much larger than the area covered by a single particle. The aggregation probability is calculated based on Eq. 29. 3.3.4

Energy of collisions over a time-span

In scenarios II and III, some of the collisional energy not utilized for particle abrasion may result in weakening of liquid bonds within agglomerates, which may lead to fragmentation when multiple collisions are combined over a certain time scale. This forms a basis of the fragmentation model.



Page 26 of 52

When collision between agglomerates takes place, we postulate that part of the energy is spent on abrading particles from the surface of the larger agglomerate, while the remaining portion is expended on fragmentation. Exact determination of how much energy is spent on the fragmentation and rebound processes is complex as it depends on how the agglomerate deforms and the dissipation of energy within the agglomerate through liquid bonds, as well as other losses in the system, such as those due to vibration. In this work, it was assumed that 50% of the energy available after the abrasion process is spent on fragmentation, with the rest dissipated due to rebounding. Therefore, the energy that contributes to fragmentation can be calculated as

(𝐸𝑐𝑖𝑗 ― 𝐸𝑏𝑖 ― 𝑛𝑎_𝑖𝐸𝑏_𝑎_𝑖)𝑋 where 𝑛𝑎_𝑖 refers to the number of liquid bonds broken as particles detach from the surface of agglomerate “i” due to abrasion, and X=0.5 is a factor which accounts for energy dissipation. Note that for scenario III, the same approach applies, with 𝑛𝑎_𝑖 = 0. Substituting for 𝑛𝑎_𝑖 and multiplying by the number of collisions, the total available kinetic energy that contributes to fragmentation can then be estimated from 𝑛𝑐𝑜𝑙𝑙𝑖𝑗

( )𝑡 (𝐸

𝐸𝑐_𝑓𝑟𝑖𝑠𝑐𝑒𝑛𝑎𝑟𝑖𝑜_2 = 0.5

𝑛𝑖

𝑓𝑟_𝑖

𝑐𝑖𝑗

― 𝐸𝑏_𝑖 ―

(

)𝑃

𝐸𝑐𝑖𝑗 ― 𝐸𝑏_𝑖 𝐸𝑏_𝑎_𝑖

𝑟𝑏_𝑎𝑏𝑟𝐸𝑏_𝑎_𝑖

)

Eq. 36

where tfr_i is the time scale at which agglomerate “i” undergoes fragmentation,

𝑛𝑐𝑜𝑙𝑙

( )𝑡𝑓𝑟_𝑖 is the 𝑛𝑖

number of collisions, 𝑃𝑟𝑏_𝑎𝑏𝑟 is the abrasion probability, explained in detail in subsequent sections, and

(

𝐸𝑐𝑖𝑗 ― 𝐸𝑏 𝐸𝑏_𝑎_𝑖

)𝑃

𝑟𝑏_𝑎𝑏𝑟

represents the number of liquid bonds that are ruptured as a result of abrasion

from the surface. The energy calculated from Eq. 36 can then be used to estimate the fragmentation probability, as explained in subsequent sections. 26 ACS Paragon Plus Environment

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The fragmentation probability is calculated using the exponential function related to the effect of bond energy and the available energy for fragmentation:

𝑃𝑓𝑟_𝑖

( =𝑒

―𝛾

(1 𝛼)(

𝑛𝑏𝑖𝐸𝑏_𝑎_𝑖 𝐸𝑐_ 𝑓𝑟 𝑖

))

Eq. 37

The accumulated energy available for fragmentation is then calculated from: 𝑁―1

Eq. 38

𝐸𝑐_𝑓𝑟𝑖 = ∑𝑗 = 0 𝐸𝑐_𝑓𝑟𝑖𝑗

𝑛𝑏𝑖 in Eq. 37 is the number of bonds in the middle plane of agglomerate “i”.

(1 𝛼) is a factor which

accounts for non-uniformity of the collision distribution. That is, the momentum of all impacts is not balanced, but rather results in a shearing force. α is the magnitude of the strain rate of the particulate phase multiplied by the fragmentation time. The strain rate is represented by the square root of the second invariant of the strain tensor. 𝑡𝑓𝑟_𝑖 is the fragmentation time during which the agglomerate undergoes breakage. It is estimated by assuming a constant angular velocity, which is equal to the collision velocity, and assuming that the top-half of the agglomerate needs to rotate through 90 degrees to break (see Figure 7): 𝑡𝑓𝑟_𝑖 =

( )𝜋 2 2𝑅 𝑉𝑟𝑒𝑙

Eq. 39

𝑉𝑟𝑒𝑙 is the relative collision velocity, calculated utilizing Eq. 25 and replacing dj by the mean diameter of agglomerates (d32) to obtain an average value.



Page 28 of 52

Figure 7 –Schematic of agglomerate breaking in half by rotation due to collisions 3.4 Breakage kernel Both the abrasion and fragmentation processes result in agglomerate breakage. The breakage kernel in this work consists of summation of the abrasion and fragmentation frequencies multiplied by their corresponding probabilities:

[

𝑔(𝑉𝑗) = (1 𝑡𝑓𝑟_𝑗)𝑃𝑓𝑟_𝑗 +

𝑛𝑐𝑜𝑙𝑙𝑗𝑝

𝐸𝑐_𝑗𝑝 ― 𝐸𝑏_𝑗

𝑛𝑗

𝐸𝑏_𝑎_𝑗

( )(

)𝑃

𝑟𝑏_𝑎𝑏𝑟𝑗

]

Eq. 40

3.5 Probability distribution function The Probability Distribution Function (PDF) indicates the agglomerate breakage distribution function for each class of the population. In this work, it is assumed that when fragmentation occurs, an agglomerate breaks in half. In addition to fragmentation, abrasion also results in breakage of agglomerates into one or several primary particles and a smaller agglomerate. Therefore, a combination of both abrasion and fragmentation are considered when calculating a PDF. Utilizing the fragmentation and breakage frequencies, the probabilites of agglomerate breakage in half and abrasion into single particles can be determined as follows, ensuring that the summation of probabilities is one:


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𝛾(𝑉𝑗 2) =

𝛾(𝑉𝑝) =

(1 𝑡𝑓𝑟_𝑗)𝑃 (1 𝑡𝑓𝑟_𝑗)𝑃

𝑓𝑟_𝑗

𝑛𝑐𝑜𝑙𝑙 𝑖𝑝 𝑛𝑖

( )( (1 𝑡𝑓𝑟_𝑖)𝑃

𝑓𝑟_𝑖

+

+

𝑛𝑐𝑜𝑙𝑙 𝑗𝑝 𝑛𝑗

( )(

𝑓𝑟_𝑗

𝐸𝑐_𝑗𝑝 ― 𝐸𝑏_𝑗 𝐸𝑏_𝑎_𝑗

)𝑃

𝐸𝑐_𝑖𝑝 ― 𝐸𝑏_𝑖 𝐸𝑏_𝑎_𝑖 𝑛𝑐𝑜𝑙𝑙 𝑖𝑝 𝑛𝑖

( )(

)𝑃

Eq. 41 𝑟𝑏_𝑎𝑏𝑟𝑗

𝑟𝑏_𝑎𝑏𝑟𝑖

)

𝐸𝑐_𝑖𝑝 ― 𝐸𝑏_𝑖 𝐸𝑏_𝑎_𝑖

Eq. 42 𝑃𝑟𝑏_𝑎𝑏𝑟𝑖

One can see that the contribution of abrasion and fragmentation mechanisms are included in derivation of Eqs. 41 and 42, and the summation of the fragmentation and abrasion probabilities is equal to one. In order to make the calculated probabilities usable in the population balance equation, there are two physical conditions that should be met for each class (ANSYS Inc.34): The normalized number of breaking particles/agglomerates must sum to unity, and the masses of the fragments must sum to the original agglomerate mass. Applying these physical constraints, considering the fact that two agglomerates are produced as a result of fragmentation into two equal parts and

(


)𝑃


primary particles are produced as a result of abrasion, the following

PDFs are calculated: 𝛽(𝑉𝑝|𝑉𝑗) =

[(


)𝑃

]( )𝛾(𝑉 ) 𝑉𝑝


𝑉𝑗

Eq. 43

𝑝

𝛽(𝑉𝑗 2|𝑉𝑗) = 𝛾(𝑉𝑗 2)

(( (

𝛽 𝑉𝑗 ―


)𝑃

𝑟𝑏_𝑎𝑏𝑟𝑗𝑉

Eq. 44

)|𝑉 ) = [1 ― ( )(

𝑝

𝑗

(( (

where 𝛽(𝑉𝑝|𝑉𝑗), 𝛽(𝑉𝑗 2|𝑉𝑗) and 𝛽 𝑉𝑗 ―

𝑉𝑝

𝐸𝑐_𝑗𝑝 ― 𝐸𝑏_𝑗

𝑉𝑗

𝐸𝑏_𝑎_𝑗


)𝑃

)𝑃



]𝛾(𝑉 ) 𝑝

Eq. 45

)|𝑉 ) are, respectively, the PDFs for

𝑝

𝑗

the primary particles, agglomerates which break in half and new agglomerates produced as a result of abrasion. While the PDF for abrasion of agglomerate into single/multiple primary particles, 𝛽

(𝑉𝑝|𝑉𝑗), should be applied in total to the primary particle bin, the breakage in half results in two 29 ACS Paragon Plus Environment


Page 30 of 52

fragments, each with half the volume of the original agglomerate, which may not exist in the population bin distribution. Therefore, the corresponding PDF should be distributed based on the fractions of neighboring bins:

𝑖𝑓(𝑉𝑙 + 1 < 𝑉𝑗 2 < 𝑉𝑙)→ 𝑉𝑗 2 = 𝑥𝑙𝑉𝑙 + (1 ― 𝑥𝑙)𝑉𝑙 + 1 𝑥𝑙 =

𝑉𝑗 2 ― 𝑉𝑙 + 1 𝑉𝑙 ― 𝑉𝑙 + 1

𝛽(𝑉𝑙|𝑉𝑗) = 𝑥𝑙𝛽(𝑉𝑗 2|𝑉𝑗)

𝛽(𝑉𝑙 + 1|𝑉𝑗) = (1 ― 𝑥𝑙)𝛽(𝑉𝑗 2|𝑉𝑗) The same approach should be followed for new agglomerates produced as a result of abrasion, with volume 𝑉𝑗 ―

(


)𝑃


, and the corresponding PDF should be

𝑝

distributed as:

(

[

𝑖𝑓 𝑉𝑚 + 1 < 𝑉𝑗 ― (


)𝑃


)

]

𝑉𝑝 < 𝑉𝑚 → 𝑉𝑗 ― (


(1 ― 𝑥𝑚)𝑉𝑚 + 1 𝑉𝑗 ―

𝑥𝑚 =

(


)

𝑃𝑟𝑏_𝑎𝑏𝑟𝑗𝑉𝑝 ― 𝑉𝑚 + 1

𝑉𝑚 ― 𝑉𝑚 + 1

(( (

𝛽(𝑉𝑚|𝑉𝑖) = 𝑥𝑚𝛽 𝑉𝑗 ―


)𝑃


(( (

𝛽(𝑉𝑚 + 1|𝑉𝑖) = (1 ― 𝑥𝑚)𝛽 𝑉𝑗 ―

)|𝑉 ), 𝑗

𝑝


)

))

𝑃𝑟𝑏_𝑎𝑏𝑟𝑗𝑉 |𝑉𝑗 𝑝


)𝑃


𝑝

= 𝑥𝑚𝑉 𝑚 +

Page 31 of 52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3.6 Simulation setup The experimental data reported by Morales and co-workers25,

26

was selected for

comparison with the model. In the experiments, a volatile liquid was injected into a fluidized bed of hot sand particles. Morales et al.26 reported agglomeration data, as well as the extent of liquid vaporization, both of which can be compared with model predictions. The experimental equipment dimensions and physical properties of the materials utilized in the experiments are provided in Table 1. As the heat capacity of the solid inventory is quite large and the amount of liquid vaporization is high, there is little change in temperature throughout the bed. Therefore, a constant value of the viscosity was assumed in this study. A schematic of the experimental apparatus is depicted in Figure 8. It should be noted that, as schematically depicted in Figure 1, the expander section was not considered in this work and Figure 8 is only for representation purposes. The liquid injection nozzle penetrated 50 mm into the bed and was located 0.38 m above the bottom of the column. In the simulations, the superficial velocity of the fluidizing gas was 0.3 m/s. Atmospheric pressure was assumed at the top, and no slip at the walls. The particle-particle restitution coefficient was set at 0.9. ANSYS Fluent34 was used for the simulations, supplemented by customdesigned sub-routines. A three-dimensional domain was considered. The laminar model was solved for the gas phase, using the Phase-Coupled SIMPLE algorithm for pressure-velocity coupling. Second-order upwind discretization schemes were applied for the convection terms. A varying time-step between 10-4 and 2.5×10-4 s was chosen to reach convergence. In their experiments, Morales et al.26 used a mixture of acetone (80 wt%), pentane (10 wt %) and polymer resin, commonly known as Acrylic or Plexiglas - PMMA (10 wt %), as liquid solution. For the sake of simplicity and based on the high weight percentage of acetone, a mixture of 90% acetone and 10% PMMA was assumed in our simulations. 9 population bins of height 31 ACS Paragon Plus Environment


0.0101, 0.0063, 0.0038, 0.0023, 0.0015, 0.0009, 0.00055, 0.0003 and 0.0002 m were selected for the population balance simulations to correspond to the agglomerate sizes measured in the experiments.

Figure 8- Schematic of experimental fluidized bed, Reprinted (Adapted or Reprinted in part) with permission from Morales, C. B., Jamaleddine, T. J., Berruti, F., McMillan, J., and Briens, C. (2016). Low-Temperature Experimental Model of Liquid Injection and Reaction in a Fluidized Bed. Can. J. Chem. Eng., 94(5), 886–895. Copyright 2016 John Wiley and sons Table 1 – Physical and geometrical experimental properties in experiments and simulations Reprinted (Adapted or Reprinted in part) with permission from Morales, C. B., Jamaleddine, T. J., Berruti, F., McMillan, J., and Briens, C. (2016). Low-Temperature Experimental Model of Liquid Injection and Reaction in a Fluidized Bed. Can. J. Chem. Eng., 94(5), 886–895. 32 ACS Paragon Plus Environment

Page 32 of 52

Page 33 of 52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Copyright 2016 John Wiley and sons Property

Value

Particle type

Silica sand

Particle diameter

210 µm

Particle asperity diameter

1% of particle diameter

Particle density

2650 kg/m3

Superficial velocity

0.3 m/s

Fluidizing gas density

1.2 kg/m3

Acetone density

791 kg/m3

PMMA density

1180 kg/m3

Acetone surface tension

0.0226 N.m

Contact angle

32o

Gas-to-liquid ratio (GLR)

3.6 % (wt)

Liquid viscosity

0.0022 Pa.s

Column width

0.15 m

Column length

1.2 m

Column height

1.5 m

Bed weight

150 kg

Particle-particle restitution

0.90

coefficient

4. RESULTS AND DISCUSSION 33 ACS Paragon Plus Environment


A mesh independence study was first conducted on grids with three different numbers of cells (38K, 64K and 117K), and the converged solutions were compared to find a suitable mesh. The 38K, 64K and 117K cases correspond to 15x24x105, 25x40x64 and 35x53x64 cells in each direction, respectively. In fluidized beds that involve the kinetic theory of granular flow, a definitive mesh-independence solution cannot be achieved due to small-scale closure issues. While this is quite an active topic of discussion in the research community, it lies outside of the scope of this manuscript. On the other hand, the inter-connections between different sub-models in this study make it even more difficult to perform a mesh study. Therefore, the choice was to turn off the agglomeration and breakage model and only test mesh dependency by comparing values for common fluidization features. To compare and evaluate the solutions, the time-averaged pressure profiles along the centerline of the bed were plotted. In addition, we looked at overall representations of the bubbling fluidized bed and the expanded bed height to ensure correct predictions of the fluidization. As shown in Figure 9, the curves for 64K and 117K cells are close to one another, whereas the simulation with 38K cells shows significant deviations near the point where the change in slope of the curves appears, related to the height at which the column expands. Qualitative observation also confirmed that bubble formation was not captured properly in the 38K cell case compared to the two finer meshes. Therefore, on a balance between accuracy and practicality, the computational grid with 64K cells was selected for all the simulations presented in this study.


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Figure 9 – Time-averaged pressure profile along centerline for mesh study. 4.1 Comparison between modeling predictions and experimental results The predicted acetone vaporization rate, based on the combined effects of boiling and evaporation, is compared with experimental results in Figure 10. Liquid injection was started after 3 s of fluidization in the simulations to avoid numerical complications. The liquid started to vaporize at a fast rate, reaching a steady rate after about 10 s of injection. The total mass of acetone remaining in the bed after 45 s of injection was found to be slightly more than 1% of the total injected acetone, compared to 5% remaining acetone reported by Morales et al.26 based on their experiments. The possible reason for this deviation could be the effect of ALC (available liquid content), i.e. liquid available for evaporation. In reality, the amount of evaporated liquid is a complex function of the local temperature drop and the number of voids within the agglomerate. 35 ACS Paragon Plus Environment


Figure 10 – Simulated acetone vaporization evolution with time for U=0.3 m/s and bed temperature of 68˚C.

Further simulations were performed to compare modelling predictions with experimental results. The effect of superficial gas velocity and bed temperature were studied and compared with experimental data. Because cumulative weight percentages for agglomerates were reported by Morales et al.26, the same quantity was selected as a basis of comparison in this paper. The experimental data points are reported for macro-agglomerates, defined as agglomerates of diameter > 600 µm (Morales et al.26). Due to a slight difference in bin size definition between


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experiments and simulations, the simulation results shown in subsequent figures correspond to agglomerates > 550 µm.

4.2 Effect of superficial gas velocity Simulations were performed for U = 0.2, 0.3 and 0.4 m/s at a constant bed temperature of 68˚C. The predictions are compared with experimental data in Figures 11-13. One can see that when the superficial gas velocity is increased, fewer large agglomerates are predicted, in agreement with the trend shown by the experimental results. This is expected as the higher superficial gas velocity results in a more agitated bed that leads to higher breakage rates and less agglomeration in the bed. The results demonstrate reasonable agreement between simulations and experimental results. The under-prediction of cumulative weight percentage for agglomerates larger than 2000 µm may be due to the effect of the simplifying assumption that agglomerates only break in half when fragmentation takes place, whereas, in reality, the agglomerates may break into smaller pieces. This would increase the fragmentation rate and, at the same time, reduce the probability of abrasion in the PDF according to Eqs. 64 and 65. The effect would be more significant at higher superficial gas velocity and could improve the agreement with experimental data. It is interesting to note the qualitative agreement between the simulations and the experiments, showing a sharp increase in weight percentage of agglomerates smaller than the 2000-3000 µm range. However, the increase in cumulative weight percentage of smaller agglomerates is more abrupt in the simulation results, and the model tends to over-predict the number of smaller agglomerates (< 2000 µm) compared to the experiments. While the exact reasons cannot be determined, we suspect that the reason may be due to the effect of the higher 37 ACS Paragon Plus Environment


abrasion rate for smaller agglomerates as the geometric probability (Eq. 58) increases, resulting in a larger number of small agglomerates.

Figure 11 – Comparison of measured and simulated agglomerate diameter distribution after 45 s of injection (U=0.2 m/s, T = 68˚C).


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Figure 12 – Comparison of measured and simulated agglomerate diameter distribution after 45 s of injection (U=0.3 m/s, T = 68˚C).



Figure 13 – Comparison of measured and simulated agglomerate diameter distribution after 45 s of injection (U=0.4 m/s, T = 68˚C). 4.3 Effect of bed temperature The effect of bed temperature on the weight distribution of agglomerates is indicated in Figure 14. It is predicted that an increase in temperature from 68 to 88˚C results in less agglomeration in the bed, in agreement with the experimental results. There is reasonable quantitative agreement with experimental data for larger agglomerates, similar to what was observed in Figures 11-13. Faster acetone vaporization leads to a lower agglomeration rate for higher bed temperature due to less liquid being available for bridge formation.


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Figure 14 – Comparison of measured and simulated agglomerate diameter distributions after 45 s of injection (U=0.4 m/s) Figures 15 and 16 show the predicted contours of liquid film (acetone specie) mass fraction and vaporized acetone mass fraction after 45 s of liquid injection in the middle cross-section plane. As indicated in Figure 16, most of the liquid vaporization occurs in the region where the jet cavity ends, as most interactions between droplets and hot bed particles take place in this region. This is also consistent with the contour shown in Figure 15 where more liquid film is forming beyond the jet cavity, providing a source for liquid vaporization in that region. Some of the injected liquid travels back to the side wall where the injection occurs due to the circulatory motion of the bed induced by the injection as depicted in Figure 15. Figure 17 shows the solid phase distribution in



the bed and exhibits penetration of liquid phase inside the bed in the region where the liquid jet reaches its maximum length.

Liquid injection

Figure 15 – Ratio of liquid-film acetone to solid mass fraction contour after 45 s of liquid injection at U=0.3 m/s and bed temperature of 68˚C.


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Liquid injection

Figure 16 – Vaporized acetone mass fraction contour after 45 s of liquid injection at U=0.3 m/s and bed temperature of 68˚C



Page 44 of 52

Figure 17 – Volume fraction of solid phase contour after 45 s of liquid injection at U=0.3 m/s and bed temperature of 68˚C

5. CONCLUSIONS A comprehensive Eulerian CFD model is developed to predict particle agglomeration and agglomerate breakage in hot gas-fluidized beds with liquid injection. The model considers gas, droplets, and particles that can exchange mass, heat and momentum due to collisions and vaporization. The model utilizes the population balance approach to account for interactions between agglomerates of different sizes. Two distinct breakage mechanisms are included: abrasion and fragmentation. Reasonably good agreement is obtained between modelling predictions and experimental measurements of Morales and co-workers25,

26

for experiments

involving acetone, pentane and PMMA. At the same time, some over-prediction of the number of 44 ACS Paragon Plus Environment

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small agglomerates is evident for all cases considered. This is likely an outcome of limiting agglomerate breakage to division into two equal parts. There was good agreement also between the simulated and measured amounts of liquid vaporized. Qualitative analysis of results revealed that most liquid vaporization takes place in the vicinity of the jet cavity, where more liquid film forms around the agglomerates. The model qualitatively captures the vaporization, and the effects of superficial gas velocity and temperature in the bed. The problem addressed in this paper is certainly a complex one, for which simple models are incapable of capturing with acceptable accuracy. Also, there is little experimental data to guide the modeling, adding to the complexity. The problem and system addressed are important from an industrial standpoint, and there is a need to begin the process of modeling by including the important physics. The promising agreement in predicting the trends and also the good quantitative agreement for large agglomerates suggest that a suitable framework has been built which can be extended as more relevant data become available, or possibly with the aid of microscale modeling approaches.

6. ACKNOWLEDGMENTS The authors gratefully acknowledge funding by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Syncrude Canada Limited.

SUPPORTING INFORMATION Tables of conservation equations, kinetic theory of granular flow and liquid bond energy expressions.



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(19) Pougatch, K., Salcudean, M., Chan, E., and Knapper, B., A Two-Fluid Model of GasAssisted Atomization Including Flow through the Nozzle, Phase Inversion, and Spray Dispersion.Int. J. Multiphase Flow, 2009, 35(7), 661–675. (20) Pougatch, K., Salcudean, M., and McMillan, J., Three-Dimensional Numerical Modelling of Interactions between a Gas–Liquid Jet and a Fluidized Bed. Chem. Eng. Sci., 2012, 68(1), 258–277. (21) Boyce, C. M., Gas-Solid Fluidization with Liquid Bridging: A Review from a Modeling Perspective. Powder Technol., 2018, 336, 12-29. (22) Darabi, P., Mathematical Modeling of Interaction of Wet Particles and Application to Fluidized Beds. PhD Thesis, University of British Columbia, Vancouver, Canada, 2011. (23) Gray, M. R., Le, T., McCaffre, W. C., Berruti, F., Soundararajan, S., Chan, E., Thorne, C., Coupling of Mass Transfer and Reaction in Coking of Thin Films of an Athabasca Vacuum Residue. Ind. Eng. Chem. Res., 2001, 40(15), 3317–3324. (24) House, P. K., Saberian, M., Briens, C. L., Berruti, F., & Chan, E., Injection of a Liquid Spray into a Fluidized Bed: Particle-Liquid Mixing and Impact on Fluid Coker Yields. Ind. Eng. Chem. Res., 2004, 43(18), 5663–5669. (25) Morales, C. B., Development and Application of an Experimental Model for the Fluid Coking Process. M.Sc Thesis, Western University, London, Canada, 2013. (26) Morales, C. B., Jamaleddine, T. J., Berruti, F., McMillan, J., and Briens, C., LowTemperature Experimental Model of Liquid Injection and Reaction in a Fluidized Bed. Can. J. Chem. Eng., 2016, 94(5), 886–895. (27) Sutkar, S. V., Deen, N. G., Patil, A. V., Salikov, V., Antonyuk, S., Heinrich, S., Kuipers, J.A.M., CFD–DEM Model for Coupled Heat and Mass Transfer in a Spout Fluidized Bed with Liquid Injection. Chem. Eng. J., 2016, 288, 185-197. (28) Pietsch, A., Kieckhefena, P., Heinrich, S., Müller, M., Schönherr, M., Jäger, F. K., CFDDEM Modelling of Circulation Frequencies and Residence Times in a Prismatic Spouted Bed. Chem. Eng. Res. Des., 2018, 1105-1116. (29) Li, T., Pougatch, K., Salcudean, M., and Grecov, D., Numerical Modeling of an Evaporative Spray in a Riser. Powder Technol., 2010, 201(3), 213–229. (30) Fries, L., Dosta, M., Antonyuk, S., Heinrich, S., and Palzer, S., Moisture Distribution in Fluidized Beds with Liquid Injection. Chem. Eng. Technol., 2011, 34(7), 1076–1084. (31) Girardi, M., Radl, S., and Sundaresan, S., Simulating Wet Gas–Solid Fluidized Beds using Coarse-Grid CFD-DEM, Chem. Eng. Sci., 2016, 144, 224–238. (32) Sen, M., Barrasso, D., Singh, R., and Ramachandran, R., A Multi-Scale Hybrid CFD-DEMPBM Description of a Fluid-Bed Granulation Process. Processes, 2014, 2(1), 89–111. (33) Sivaguru, K., Begum, K. M. M. S., and Anantharaman, N., Hydrodynamic Studies on Three-Phase Fluidized Bed using CFD Analysis. Chem. Eng. J., 2009, 155(1–2), 207–214. (34) ANSYS Inc., FLUENT 18.2 Student Version (non-public) Theory Documentation, Canonsburg, PA, 2017. (35) Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. New York: Academic Press, 1994. (36) Schiller, L. and Naumann, A., A Drag Coefficient Correlation. Zeitschrift des Vereins Deutscher Ingenieure, 77, 318-320, 1935. (37) Pougatch, K., Mathematical Modelling of Gas and Gas-Liquid Jets Injected into a Fluidized Bed. PhD Thesis, University of British Columbia, Vancouver, Canada, 2011. (38) Ranz, W. E., and Marshall, W. R. Evaporation from Drops - Part 1. Chem. Eng. Prog., 47 ACS Paragon Plus Environment


48(3), 1952, 141–148. (39) Gunn, D. J., Transfer of Heat or Mass to Particles in Fixed and Fluidised Beds. Int. J. Heat Mass Transfer, 1978, 21(4), 467–476. (40) Lun, C., Savage, S., Jeffrey, D. and Chepurniy, N., Kinetic Theories for Granular Flow: Inelastic Particles in Couette Flow and Slightly Inelastic Particles in a General Flow Field. J. Fluid Mech., 1984, 140, 223-256. (41) Gidaspow, D., Bezburuah, R. and Ding, J., Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In Fluidization VII, Proceedings of the 7th Engineering Foundation Conference on Fluidization, pages 75-82, 1992. (42) Pitois, O., Moucheront, P., and Chateau, X., Rupture Energy of a Pendular Liquid Bridge. The European Physical Journal, 2001, 86, 79–86. (43) Shi, D., and McCarthy, J. J., Numerical Simulation of Liquid Transfer between Particles. Powder Technol., 2008, 184(1), 64–75. (44) Hounslow, M. The Population Balance as a Tool for Understanding Particle Rate Processes. KONA Powder Part. J., 1998, 16, 179-193. (45) Rajniak, P., Stepanek, F., Dhanasekharan, K., Fan, R., Mancinelli, C., and Chern, R. T., A Combined Experimental and Computational Study of Wet Granulation in a Wurster Fluid Bed Granulator. Powder Technol., 2009, 189(2), 190–201. (46) Palanisamy, K., Modeling of Agglomerate Abrasion in Fluidized Beds. PhD Thesis, University of British Columbia, Vancouver, Canada, 2016.


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NOMENCLATURE A

surface area, [m2]

𝐴𝑖

surface area of interface, [m2]

ALC

Available Liquid Content, [-]

a

radius of spherical cap, [m]

B

dimensionless number, [-]

Bi

birth (agglomeration) rate, [s/m3]

CD

drag coefficient, [-]

d

diameter, [m]

Dc

mass diffusivity, [m2/s]

Dl

mass diffusivity in droplet phase, [m2/s]

Dm

mass diffusivity in solid phase, [m2/s]

Dg

mass diffusivity in gas phase, [m2/s]

𝐷𝑠𝑟𝑢𝑝𝑡

liquid bond rupture distance, [m]

De

death rate, [s-1 m-3]

e

coefficient of restitution, [-]

E

energy, [J]

𝐸𝑎𝑏𝑟

rate of energy dissipation due to abrasion, [J/s]

𝐸𝑏_𝑎

liquid bond energy inside agglomerates, [J]

𝑓𝑖

volume fraction of bin, [-]

𝑓(𝐷𝑚)

term used in viscous energy, [N]

g

gravitational acceleration, [m/s2]

g0

radial distribution function, [-]

h

heat transfer coefficient, [W/m2K]

he

enthalpy, [J/kg]

h0

height of film, [m]

ℎ𝑎

height of asperities, [m]

Hlg

latent heat of vaporization, [J/kg]

I

square-root of second invariant of strain rate tensor, [1/s]

k

conductive heat transfer coefficient, [W/m.K]

kmass

diffusive mass transfer coefficient, [m/s] 49 ACS Paragon Plus Environment


𝑘𝜃

diffusion coefficient for granular energy, [kg/m.s]

𝑚

mass transfer rate, [kg/s]

n

number density, [m-3]

𝑛𝑎𝑏𝑟

abrasion rate, [m-3s-1]

𝑛𝑐𝑜𝑙𝑙

collision frequency, [m-3s-1]

P

pressure, [Pa]

𝑃𝑎𝑔𝑔

probability of agglomeration, [-]

P*

saturation vapor pressure, [Pa]

Pr

Prandtl number, [-]

Pbond,abr

liquid bond breakage probability in abrasion model, [-]

Pbond,fr

liquid bond breakage probability in fragmentation model, [-]

Pgeom

geometric breakage probability in abrasion model, [-]

Q

heat transferred between phases, [W]

r

radius, [m]

R

agglomerate radius, [m]

Re

Reynolds number, [-]

S

surface area, [m2]

t

time, [s]

T

temperature, [K]

Tsat

saturation temperature, [K]

U

superficial gas velocity, [m/s]

V

volume, [m3]

Vrel

relative velocity, [m/s]

𝑉𝑏

volume of bridge, [m3]

v

velocity, [m/s]

x

volume fraction used in species conservation equations, [-]

y*

mole fraction of acetone in vapor phase, [-]

Greek letters



drag coefficient, [kg/m.s]

𝛾𝜃

collision dissipation of energy, [kg/m.s3] 50 ACS Paragon Plus Environment

Page 50 of 52

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𝛾

strain rate, [1/s]



volume fraction, [-]

Ψ𝑔𝑒𝑜𝑚

geometrical probability of agglomeration, [-]

Ψ𝑝ℎ𝑦𝑠

physical probability of agglomeration, [-]

𝜃

contact angle, [radians]

𝜃𝑔𝑟𝑎𝑛

granular temperature, [m2/s2]

𝜃𝑚𝑖𝑥

granular temperature of mixture, [m2/s2]

𝜆𝑠

solid bulk viscosity, [kg/m.s]

η

voidage fraction, [-]

µ

shear viscosity, [kg/m.s]



density, [kg/m3]

σ

liquid surface tension, [N/m]

τ

stress tensor, [Pa]

𝜔

stickiness ratio, [-]

𝜑𝑔𝑠

transfer rate of kinetic energy, [kg/m.s3]

Subscripts a

agglomerate

ag

acetone vapor in gaseous phase

a_drop

acetone species in droplet phase

abr

abrasion

agg_film

film species in solid phase

as

acetone in solid phase

b

liquid bond

boil

boiling

c

collision

as

asperity

cap

capillary

drop

droplet phase

ev

evaporation

fr

fragmentation 51 ACS Paragon Plus Environment


g

gas

ib

index for bin ‘i’

ij

averaged value for bins ‘i’ and ‘j’

j

index for fragmentation plane

jb

index for bin ‘j’

l

liquid

lb

largest bin in particle size distribution

n

fragmentation plane number

N2

nitrogen

p

particle

pm_drop

PMMA species in droplet phase

pms

PMMA species in solid phase

r

relative

s

solid

sb

smallest bin in particle size distribution

sc

solid-core

visc

viscous

x,y, z

x-, y- and z- coordinates


Page 52 of 52

CFD study of wet agglomerate growth and breakage in a fluidized bed

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