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2 There s a momentum balance for each phase, and it accounts for pressure difference, buoyancy, drag, and stress (in terms of fluid viscosity or particle collisions). More about the momentum balance. To get a better understanding of the momentum balance, consider the example of a cannonball shot from a canon. The pressure buildup in the cannon pushes the cannonball out at high velocity. Like any object, the cannonball wants to travel from high pressure to low pressure, which is why we need to correct for pressure difference when calculating the momentum balance. Particle buoyancy is due to particle density differences in the presence of gravity. Because the cannonball is more dense than the air it s traveling through, the ball wants to fall to Earth. But if the ball were less dense than the air, it would want to travel into space. The drag forces are two equal but opposite forces: the friction the air imposes on the cannonball and the friction the ball imposes on the air. This two-way coupling of the drag forces can best be demonstrated by driving with your hand out the car window: You can feel the airflow s force on your hand, but your hand is also disturbing the airflow. Particle collisions are just that: the particles colliding with each other and the walls. As an example, if you re walking at a constant rate and bump into someone, you ll bounce in one direction and the other person will bounce in another. Your momentum transfers to the other person, and vice versa. If the person weighs more than you or is moving faster, the distance you bounce will be greater. Using the balance equations in CFD modeling. For many bulk solids operations, including fluidized beds, cyclones, pneumatic conveying lines, and hoppers containing small particles, we need to model multiphase flow: the flow of particles (the particle phase or solid phase), and the flow of fluid (typically a gas) that surrounds them (the gas phase or fluid phase). Two common ways of modeling this type of multiphase flow are the Eularian-Eularian and Eularian-Lagrangian methods. For these CFD models, the material and momentum balance equations are treated separately for the particle and gas phases because the phases are connected only by drag forces (as in our handout-the-car-window analogy, the drag forces are equal but opposite between the two phases). In the energy balance equation, both phases are typically treated as one mixture. Eularian-Eularian CFD models This method s name Eularian-Eularian indicates that it treats both the particle and gas phases using the Eularian framework developed by 18th-century mathematician Leonhard Euler. The Eularian framework, as shown in Figure 1a, models gas flow at specific, discrete locations through which the gas flows as time passes. The framework divides a large domain into an array of smaller domains (called cells), which are represented as three-dimensional cubes. If the cells are small enough, the overall gas flow can be described by vectors in X and Y coordinates. Although the overall (or macroscopic) flow may not be represented in an X and Y vector, the sum of the smaller cells X and Y vectors can reproduce the overall gas flow. Thus, for each cell in Figure 1a, the material, momentum, and energy balance equations are applied to each cell face (that is, each face of the cube) to keep track of what comes in and out of that cell. The equations we ll use in this method are Navier-Stokes equations. The equations with their mathematical terms are provided here just for completeness, but what s really important is what the equations stand for (explained in italics below each equation): Material balance: ( f ) ( f u) R rxn Mass accumulation + mass in and out reaction where t is time, is the particles void fraction, f is gas density, u is gas velocity, and R rxn is reaction rate. Momentum balance: ( f u) f u u P f (u v) f g Momentum change + momentum in and out = viscous losses + drag + gravity - pressure drop where P is pressure, f is viscosity effect (often described using Newton s law of gravity), is drag, and g is gravity acceleration constant. Energy balance: ( f C p ) ( f C p u) k 2 T R rxn H rxn Energy change + heat convection in and out = thermal conduction + reaction heat where C p is the particles heat capacity, k is thermal conductivity, T is temperature, and H rxn is reaction heat. For the material balance, mass accumulation is the result of the mass (particles) in and out of each cell and the reaction (such as attrition) in the cell. For the momentum balance, the momentum change results from the particle momentum in and out of each cell with respect to the viscosity losses, drag, and gravity minus the pressure drop. For the energy balance, the energy change in each cell results from the heat convection in and out, thermal conduction of the gas, and the reaction heat. The magnitude of the values we calculate for the material, momentum, and energy balances determines the direction of the particle flow, the particle momentum, and the temperature gradient in the process we re modeling.

3 The cell size in the Eularian framework affects the model s accuracy, so the cells must be much smaller than the Figure 1 Modeling gas and gas-particle hydrodynamics using Eularian, Eularian-Eularian, and Eularian-Lagrangian frameworks a. Eularian framework for gas phase only Cell macroscopic flow we want to model. The problem is, the smaller these cells, the more equations we need to solve and the longer it takes our simulation to run. The power of Eularian framework modeling is most apparent in designing aircraft: Fifty years ago, wind tunnels were used to design planes. Today, the wind tunnels are virtually gone, and planes are designed with computers using Eularian framework CFD software. The Eularian-Eularian framework, as shown in Figure 1b, models the particle phase using equations similar to those for the gas phase: Y coordinates Gas Material balance: ((1 ) p ) ((1 ) p v) R rxn Mass accumulation + mass in and out = reaction rate X coordinates b. Eularian-Eularian framework for gas and particle phases Particles as fluid c. Eularian-Lagrangian framework for gas and particle phases Momentum balance: ((1 ) p v) (1 ) p v v P s (u v) (1 ) p g Momentum change + momentum in and out = particle pressure + particle viscosity + drag + gravity where p is particle density, v is particle velocity, and s is the particles viscosity effect. With these equations, the particle phase is actually modeled as a thick fluid. However, the particle pressure and particle viscosity need closure that is, parameters (terms in the equations) that have no physical or measurable value so you need another set of equations to fill the gap. In reality, the particle pressure and particle viscosity are the particles direct impact (called the normal stress) or glancing impact (shear stress) on other particles. Because we can t really measure pressure or viscosity for particles as easily as we can for a gas, we need to calculate these values. Particle velocity a The most common of many theories on how to calculate the particle pressure and particle viscosity in bulk solids flow is the kinetic theory of granular fluids. This theory views particles similarly to atoms so we can determine the particle pressure and viscosity. In short, here s how it works: An atom s vibrational mode is related to temperature, and a particle s velocity fluctuation during flow is related to the granular temperature. So, rather than being a form of heat, the granular temperature is a form of work: It s proportional to a particle s velocity fluctuation squared. We can use the granular temperature to calculate the particle pressure and particle viscosity for our model. a Note: The arrow direction represents the particle direction; arrow length is proportional to the particle velocity. One drawback: In most CFD software, these material and momentum balance equations are based on one represen-

4 tative particle size. But can we fully capture particle behavior during a process based on one particle size? If so, what should that size be? Most Eularian-Eularian modeling uses only the Sauter mean particle size (d p50 ), simply because it s the most readily available. However, new methods are being developed to trick the Eularian-Eularian framework to handle more particle sizes. As we ve discussed, the drag forces of the particle on the gas and of the gas on the particle are equal and opposite. This two-way coupling is how the two phases feel each other s momentum. You can use any of various equations to calculate these drag forces, but many are empirical, based on single-particle flow or packed-bed flow. Consider, however, whether the material flow in your unit operation typically fits either of these extremes: Chances are, it doesn t. For best results, work with your modeler to choose a drag equation that fits your application. In most granular-fluid flow problems, this equation has the greatest impact on the model. Eularian-Eularian modeling, also commonly called twofluid modeling, is the most common form of modeling for granular-fluid flow because a computer can use it to simulate a large-scale unit operation in a few weeks. However, to use this method, you re assuming that your particle phase behaves like a fluid. The small cells in your Eularian-Eularian framework (Figure 1b) must also be small enough to accurately depict your process s important hydrodynamics. You can test whether the cells are small enough by running two simulations with different cell sizes. If the results are the same, you re good. If not, run a simulation with a smaller cell size and test it again. But remember: Each simulation takes weeks! Eularian-Lagrangian CFD models What if we don t want to model our material flow based on assuming that our particles behave like a fluid? What if we want to model our entire particle size distribution, including the fines? Here s where we can consider a model based on the Eularian-Lagrangian framework, developed by 19th-century mathematician Joseph Luis Lagrange. In this framework, as shown in Figure 1c, the gas phase is modeled as in the previous frameworks (Figures 1a and b). However, the Eularian-Lagrangian framework treats the particle phase much differently than the other frameworks. It looks at the particle phase by: 1) tracking particles and collisions independently, and 2) tracking a component of each particle s drag force on the gas that s equal to and opposite from the gas s drag force on the particle. The Newtonian equation used to calculate momentum balance for the Eularian-Lagrangian framework is: F v P p (u v) g Particle momentum = pressure drag + gravity where F is the momentum transfer function. This equation applies to every particle in motion until the exact moment it collides with another particle, in which one particle s momentum translates into another particle s momentum. However, the problem with the Eularian-Lagrangian framework is that each particle and each collision must be modeled. One way around this is to use a hybrid method that models only the particle trajectories, not the collisions. This requires modifying the momentum balance equation this way: F v P p (u v) g s (1 ) p Particle momentum = pressure drag + gravity collisional stresses (These collisional stresses can be calculated by an empirical equation or some form of kinetic theory of granular flow equation; work with your modeler to choose a suitable equation.) By using this hybrid equation, we no longer need to track collisions because we re simply calculating how collisions affect the particle momentum. By eliminating the need to track collisions, the hybrid equation allows us to use a much larger time step (interval for calculations in time) in the Eularian-Lagrangian framework. Yet let s consider just how many equations we d need to use with the Eularian-Lagrangian framework to model bulk solids flow behavior in a 25-foot-diameter, 10-foot-tall fluidized bed containing 40 billion 100-micron particles. We d need three equations to simulate the gas flow in each cell in the framework; for this fluidized bed, we d most likely need about 500,000 cells, which leads to 1.5 million equations just for the gas phase. This isn t a problem for today s desktop computers. However, we need 40 billion equations for each particle we re tracking. That s way too many calculations to get simulation results in a reasonable amount of time. Anyone even considering such an attempt will probably end up making an involuntary career move! The Eularian-Lagrangian framework remains a useful tool, however, because it models the entire particle size distribution by assuming that particles of similar size and density in each cell can be modeled once. So if we group 10,000 particles of similar size and density in a cell (these groupings are called clouds or parcels), the number of equations we need is 1.5 million for the gas phase but only another 4 million for the particles (one equation per cloud). This gives us fewer than 2 million calculations total, so we can actually do them on a desktop computer. However, we have to carefully choose the number of cells and number of particles per cloud we use to avoid modeling errors. Using a trialand-error approach, in which we vary the number of cells and particles in a cloud each time, will help us converge on an accurate model.

5 The Eularian-Lagrangian framework model is growing more popular because it can handle the entire particle size distribution and capture the effects of attrition, agglomeration, and fines. (Although the mathematics are more difficult to implement for this model, that s the CFD software developer s problem!) Whether you use Eularian-Eularian or Eularian-Lagrangian modeling, the equations in today s CFD software allow you to simulate what happens in production-scale bulk solids equipment. The process is still slow, however, with 1 minute of simulation time taking anywhere from 1 week to about 1 month. The question we need to ask is: Can a simulation lasting a couple seconds or even minutes effectively describe the steady-state behavior of particles and gas in a unit operation? In many cases, production equipment achieves steady-state operation only several minutes or, in some cases, several hours after startup. Defining physical properties, initial conditions, and boundary conditions for modeling To produce good modeling results, regardless of the model we use, we also need to define the physical properties, initial conditions, and boundary conditions of the process we re simulating: These are the inputs for the model. Physical properties. Physical properties of both the particles and gas are straightforward: For the particle phase, we need the particle size distribution (or Sauter mean particle size [d p50 ]), particle density, and particle shape (for some drag equations). For the gas phase, we need the gas density and gas viscosity. If the process we re modeling doesn t have a constant temperature, we also need the heat capacity and thermal conductivity for both the particles and gas. Initial conditions. The initial conditions are the values we use to define a Eularian framework cell s properties at time equal zero that is, before the unit operation we re modeling begins. For most bulk solids simulations, we d need to define these initial conditions: gas velocity, particle velocity, particle concentration (as volume fraction), and temperature. So, for instance, let s look at the initial conditions for producing a three-dimensional Eularian-Lagrangian model of a pilot-scale (3-foot-diameter, 20-foot-tall) fluidized bed, as shown in Figure 2, where the fluidizing gas is air and the particles have a 70-micron median size with a 90-lb/ft 3 density. (The results in Figure 2 are shown as a slice view of a three-dimensional simulation after just 30 seconds and were produced using CPFD-Software s Barracuda Virtual Reactor program.) For this simulation, the air flowrate through the sparger (airflow grid) below the bed was set to obtain a 2-ft/s superficial gas velocity in the bed. The particle flowrate in and out of the vessel was set at 2.6 lb/s, and the pressure at the outlet was set at atmospheric. For this simulation, the initial conditions were set as following: The gas velocity was set at the operating superficial gas velocity (the air s volumetric flowrate divided by the cross-sectional flow area), the particle velocity was set at zero, the particle concentration was set at 0.45 (the value we expect during the fluidized bed s steady-state operation), and the temperature was set at room temperature. In this case, we re using the initial conditions to predefine the particle bed at fluidization that is, when it first becomes fluidized so the simulation can run faster. Boundary conditions. The boundary conditions are the conditions at the process inlets and outlets. While it s possible to get good simulation results with poorly defined initial conditions, we can t get good results with poorly defined boundary conditions. In the pilot-plant fluidized bed example in Figure 2, we had to define these inlet boundary conditions (shown at the figure s left): the fluidization air s entry through the sparger at the vessel s bottom and for the particle flow entering the inlet pipe halfway up the vessel. We also had to define the outlet boundary conditions for the particles exiting the vessel s bottom and top and the air exiting the vessel s top. For the fluidized-bed model, the vessel wall itself is another boundary condition, so we need to describe how the air and particles interact with the wall. In this case, the air velocity at the wall, flowing along the wall, was zero; particles colliding directly with the wall lost 99 percent of their momentum; and particles grazing the wall lost only 60 percent of their momentum. A close look at Figure 2 also reveals a boundary condition precaution we need to keep in mind for incoming particle flow. For this fluidized bed, we didn t want the model to simulate the vessel s inlet pipe right at the pipe s opening in the vessel wall; instead, we made the pipe s configuration, extending 1 foot out of the vessel wall, part of the model. Why? Because we don t know what the particle flow profile really looks like at the inlet pipe s opening in the vessel wall. We do know that the flow at this wall opening isn t uniform, as it would be along the vessel wall at other locations. So in Figure 2, we see that the boundary condition has been moved out 1 foot, along the inlet pipe, so that the particles in the model would have time to move to the pipe s bottom and then drop quickly into the bed, as they do in the pilot-plant vessel. If the boundary condition hadn t been moved out at this point, the model would have simulated a uniform boundary condition at the wall for the inlet pipe, which would show the particles falling farther out into the bed. This wouldn t represent what was really happening in the pilot-plant vessel. In fact, there are very few instances in bulk solids processing where a uniform boundary condition exists for incoming particle flow. So if you want to use a uniform boundary

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