JP2013210918A - Particle behavior simulation device, particle behavior simulation method, control program and recording medium - Google Patents

Abstract

PROBLEM TO BE SOLVED: To realize particle behavior simulation capable of shortening a calculation time under the consideration of the influence of an adhesive force.SOLUTION: A particle behavior simulation device 10 comprises: a contact force specification unit 13 for determining a corrected contact force using a corrected spring constant kd smaller than a spring constant kr when a contact force is expressed as a non-linear spring model; and a behavior calculation unit 12 for calculating the behaviors of particles using the corrected contact force as a contact force and a corrected adhesive force Fad which is set to a value smaller than an adhesive force Far according to the ratio between the spring constant kr and the spring constant kd as an adhesive force.

Description

  The present invention relates to a particle behavior simulation for predicting the behavior of a large number of particles.

  In a chemical synthesis process, a fluidized bed apparatus is widely used as a catalytic reaction apparatus for obtaining a reaction product from a gas. The fluidized bed apparatus is an apparatus that fluidizes powder by putting powder or the like in a processing container and blowing gas into the processing container from the lower part of the powder. For example, by putting catalyst particles in a processing container of a fluidized bed apparatus, supplying a raw material gas from the lower part of the processing container, and efficiently bringing the fluidized catalyst particles into contact with the raw material gas, a gaseous reaction A product can be obtained.

  It is important to simulate the behavior of the particles in the fluidized bed in order to design a fluidized bed apparatus that efficiently reacts or to find better conditions such as the amount of catalyst particles and gas flow rate.

As the catalyst particles used for such fluidized bed catalytic reaction, particles having an average particle diameter of 30 to 150 μm and a density of about 700 to 2500 kg / m ³ , particles classified as A particles in the so-called Geldart diagram are widely used. It is used. The classification of the particles in the Geldart diagram is an index for predicting the fluidized state of the particles in the fluidized bed. If particles classified as A particles in the Geldart diagram are used as the catalyst particles, the fluidized bed forms small bubbles, so that the reaction efficiency can be increased.

  One method for simulating the behavior of such particles in a fluidized bed is the discrete element method (DEM: Discrete Element Method). In the numerical analysis using the discrete element method, for each divided time step, the equation of motion is calculated for each particle based on the force acting on each particle, and the particle at the next time is advanced by the time step. Is obtained by numerical calculation. Then, the calculation for each time step is repeated to determine the behavior of the particles. The force acting on a certain particle includes contact force indicating repulsion due to collision with another particle or the inner wall of the processing container, fluid force received from a gas (fluid) around the particle, other particle or inner wall of the processing container There are adhesion force working between, gravity and so on.

  In the discrete element method, if the time interval (time step) of the simulation is small, an accurate simulation result can be obtained. However, if the time step is reduced, the amount of calculation for obtaining the behavior of the particles for a predetermined time (for example, 1 second) increases. For this reason, the calculation time required for the simulation becomes long. If the time step is increased, the calculation time can be shortened. On the other hand, there is an upper limit to the time step for performing the calculation stably (obtaining a reasonable simulation result).

  In Non-Patent Document 1, when a linear spring model representing the contact force of particle collision is assumed, the empirical upper limit of the time step for obtaining a reasonable simulation result is about 1 of the natural vibration period T of the spring. / 10. In this case, since the half cycle T / 2 is the contact time between the particles, about 1/5 of the contact time between the particles is an empirical upper limit of the time step. Since the period T of the natural vibration depends on the spring constant k, it is necessary to reduce the time step for particles having a large spring constant, that is, hard particles. Therefore, the simulation time of hard particles is longer than that of soft particles having a small spring constant.

Japan Powder Industrial Technology Association, “Fluidized Bed Handbook”, Baifukan Co., Ltd., March 25, 1999, p.125-144 Toshiaki Tanaka, Toshiya Ishida, and Yutaka Ishida, “Direct Numerical Simulation of Granular Plug Flow in a Horizontal Pipe (Without Adhesive Force)”, Transactions of the Japan Society of Mechanical Engineers (Part B), 57, 534, 1991 456-463

  Similarly, in the case of performing the simulation by applying the Hertz contact model instead of the linear spring, in the case of hard particles having a large spring constant, the contact time between the particles is shortened, and therefore the time step needs to be reduced.

  However, when the particle size is small, such as particles classified as A particles in the Geldart diagram, the adhesion force between particles due to van der Waals force and liquid cross-linking force increases, and the influence of adhesion force is ignored. become unable. The adhesion force is an attractive force that is applied when a particle comes close to another particle or the inner wall of the processing container. In systems where the adhesion force is large and the influence of the adhesion force cannot be ignored, if the spring constant is set smaller than the original spring constant and the time step is lengthened, the accuracy of the simulation deteriorates and a reasonable result cannot be obtained. There is. For this reason, in the system in which the influence of adhesion force cannot be ignored, it takes a long time to calculate the simulation.

  The present invention has been made in view of the above problems, and its purpose is to reduce the time for calculating the behavior of a plurality of particles using the discrete element method while considering the influence of adhesion force. It is to realize a particle behavior simulation that can be performed.

  A particle behavior simulation apparatus according to the present invention is a particle behavior simulation apparatus that predicts the behavior of a plurality of particles using a discrete element method, and when the particles are in contact with each other in order to solve the above problem When the spring force when the contact force acting on the particles is expressed by a nonlinear spring model is kr and the adhesion force between the particles is Far, the nonlinear spring model is used by using a modified spring constant kd smaller than the spring constant kr. A contact force specifying unit for obtaining a corrected contact force based on the above, and using the corrected contact force as the contact force, and the adhesion force depending on the ratio of the spring constant kr and the corrected spring constant kd as the adhesion force It is characterized by comprising a behavior calculation unit that calculates the behavior of particles by a discrete element method using a modified adhesion force Fad having a value smaller than Far.

  The particle behavior simulation method according to the present invention is a particle behavior simulation method for predicting the behavior of a plurality of particles using a discrete element method, and the contact force acting on the particles when the particles are in contact with each other is a non-linear spring. When the spring constant when expressed by the model is kr and the adhesion force between the particles is Far, the contact force specifying unit of the simulation apparatus uses the modified spring constant kd smaller than the spring constant kr, and the nonlinear spring model. The contact force specifying step for obtaining a corrected contact force based on the above and the behavior calculation unit of the simulation apparatus use the corrected contact force as the contact force and the spring constant kr and the corrected spring constant kd as the adhesive force. The behavior of the particles is calculated by the discrete element method using the modified adhesion force Fad that is smaller than the adhesion force Far according to the ratio of It is characterized by including the behavior calculation step.

According to the above configuration, the corrected contact force is specified using the corrected spring constant kd smaller than the spring constant kr, and the behavior of the particles is calculated by the discrete element method using the corrected contact force. Therefore, since the contact time between particles becomes long, the time step can be lengthened and the time required for behavior calculation can be shortened. In the behavior calculation, the corrected spring constant kd is used, and at the same time, the corrected adhesive force Fad having a value smaller than the adhesive force Far is used according to the ratio of the spring constant kr and the corrected spring constant kd. Therefore, it is possible to obtain a reasonable simulation result while reducing the time required for the behavior calculation for the particle system for which the adhesion force should be considered.

  The nonlinear spring model representing the contact force acting on the particles may be a Hertz contact model. Further, the behavior calculation unit may treat the adhesion force between particles as a force acting on the particles only when the particles are closer than a predetermined distance.

  The ratio between the corrected adhesion force Fad and the adhesion force Far may be approximately the same as the 2 / 5th power of the ratio between the corrected spring constant kd and the spring constant kr.

  According to the above configuration, even when the behavior is calculated using a small corrected spring constant kd, the velocity vc that is a threshold value for whether two particles remain attached or leave after the collision between the particles is obtained. It does not change. Therefore, even if the behavior is calculated using a small correction spring constant kd, the behavior of the particles does not change. Therefore, it is possible to obtain a reasonable simulation result while reducing the time required for the behavior calculation for the particle system for which the adhesion force should be considered.

  Moreover, you may make it the said correction | amendment adhesive force Fad satisfy | fill the below-mentioned Formula (23) as conditions.

  According to the above configuration, the value of the corrected adhesive force Fad is 0.7 times to 1.3 times the value obtained by dividing the corrected spring constant kd by the spring constant kr and multiplying the 2/5 power by the adhesive force Far. It becomes the range.

  As described above, the corrected adhesion force Fad may be slightly deviated from the value obtained by multiplying the 2/5 power of the corrected spring constant kd by the spring constant kr and the adhesion force Far.

  The ratio between the corrected adhesion force Fad and the adhesion force Far may be the same order as the 2 / 5th power of the ratio between the corrected spring constant kd and the spring constant kr.

  The corrected spring constant kd may be 0.1 times or less of the spring constant kr.

  According to said structure, a time step can be lengthened according to the correction spring constant kd, and the time required for behavior calculation can be shortened.

  Further, based on the adhesion force Far, the corrected spring constant kd, and the spring constant kr, the ratio of the corrected adhesion force Fad to the adhesion force Far is the ratio of the correction spring constant kd to the spring constant kr. A configuration including a corrected adhesive force specifying unit that specifies the corrected adhesive force Fad that is the same as the 2/5 power may be used.

  According to the above configuration, the corrected adhesive force specifying unit specifies the corrected adhesive force Fad that does not change the behavior of the particles even if the behavior is calculated using a small corrected spring constant kd. Therefore, it is possible to obtain a reasonable simulation result while reducing the time required for the behavior calculation for the particle system for which the adhesion force should be considered.

The particle behavior simulation apparatus may be partially realized by a computer. In this case, a control program that causes the computer to operate as each unit, and a computer-readable recording medium that records the control program, It falls within the scope of the present invention.

  The particle behavior simulation apparatus according to the present invention is a particle behavior simulation apparatus that predicts the behavior of a plurality of particles using a discrete element method, and applies a contact force acting on the particles when the particles are in contact with each other to a nonlinear spring. A contact for obtaining a corrected contact force based on the nonlinear spring model using a corrected spring constant kd smaller than the spring constant kr, when the spring constant in the model is kr and the adhesion force between particles is Far. A corrected adhesive force that uses the corrected contact force as the contact force and the force specifying unit, and has a value that is smaller than the adhesive force Far depending on the ratio of the spring constant kr and the corrected spring constant kd as the adhesive force. And a behavior calculation unit that calculates the behavior of particles by a discrete element method using Fad.

  Therefore, it is possible to obtain a reasonable simulation result while reducing the time required for the behavior calculation for the particle system for which the adhesion force should be considered.

It is sectional drawing which shows schematic structure of the fluidized bed apparatus made into the object of simulation. It is the figure which showed a mode that two particle | grains collided for every time. It is a figure which shows two states after particle | grains collide with a wall surface. It is a block diagram which shows the functional structure of the particle behavior simulation apparatus of one Embodiment of this invention. It is a flowchart which shows the process of the said particle behavior simulation apparatus. It is a block diagram which shows the functional structure of the particle behavior simulation apparatus of another embodiment of this invention. The motion of one particle is numerically calculated based on the equation of motion in consideration of viscous damping and adhesion force, and the spring constant k and adhesion force Fa when the initial relative velocity vc of the limit for maintaining the adhesion state becomes a certain value. It is the graph which plotted and. It is the graph which plotted the value of speed | rate vc used as a threshold value when only the spring constant is changed by making adhesive force constant as a reference example. It is the graph which plotted the value of speed | rate vc used as a threshold value when changing a spring constant and adhesive force.

  In the following embodiment, the case of simulating the behavior of particles in a fluidized bed apparatus will be mainly described, but the present invention is not limited to this.

[Embodiment 1]
Hereinafter, the present embodiment will be described in detail with reference to FIGS. 1 to 5 and FIGS. 7 to 9. In the present embodiment, the particles in the fluidized bed apparatus shown in FIG.

<Simulation system>
FIG. 1 is a cross-sectional view showing a schematic configuration of the fluidized bed apparatus 1. The fluidized bed apparatus 1 includes a processing container 2, a dispersion plate 3, a gas supply port 4, and an exhaust port 5. The dispersion plate 3 is horizontally arranged at the lower part in the processing container 2 so as to divide the inside of the container, and has fine through holes. Catalyst particles 6 are deposited on the dispersion plate 3 in 2 for processing. The dispersion plate 3 allows gas to pass but does not allow the catalyst particles 6 to pass. The gas supply port 4 is provided below the dispersion plate 3 of the processing container 2. The raw material gas is supplied into the processing container 2 from the gas supply port 4 and spouts up through the gap between the deposited particles from the bottom of the dispersion plate 3. The raw material gas and the reaction product thereof are discharged from an exhaust port 5 provided in the upper part of the processing container 2.

<Simulation method>
In the discrete element method, the position and motion state of a particle at the next time are calculated from the force acting on the particle of interest. The force F acting on the target particle is given by the following equation.

Here, Fd represents fluid force, Fg represents gravity, Fc represents contact force, and Fa represents adhesion force.

  The fluid force Fd is mainly an upward force received from a gas (raw material gas) blown up by particles. Gravity Fg is a force that acts on particles by their own weight. The contact force Fc represents a repulsive force due to contact between particles or contact between particles and the inner wall (or dispersion plate) of the processing container. The adhesion force Fa represents an attractive force due to van der Waals force, liquid cross-linking force, and the like, which is received when the particle approaches or comes into contact with another particle or the inner wall of the processing container. The fluid force Fd and the gravity Fg are forces that each particle always receives. However, the fluid force Fd is not constant, but varies depending on the position and the distribution (spatial density) of surrounding particles. On the other hand, the contact force Fc and the adhesion force Fa work only when, for example, the particle comes into contact with another particle or the inner wall.

  The contact force Fc and adhesion force Fa when the particles come into contact with other particles will be described. Here, when a particle contacts, the contact force is expressed using a model in which the particle undergoes elastic deformation and receives a repulsive force. Regarding the contact force that the particles having a radius r collide with each other, the force in the normal direction of the contact surface is expressed by a spring representing an elastic repulsion force and a dashpot representing a viscous damping force. The viscous damping coefficient is determined from the particle restitution coefficient. The force in the tangential direction of the contact surface is expressed by a combination of a spring and a dashpot similar to the normal direction when the amount of deformation of the particles is small, and when the amount of deformation of the particles is large, it represents slippage. Expressed using a friction slider.

  The contact force Fc in the normal direction in a nonlinear spring model (Hertz contact model) using Hertz's contact theory is expressed by the following equation.

Where k is the spring constant of the Hertzian contact model in the normal direction (elastic spring) representing the elastic repulsive force of the particle, η is the viscous damping coefficient in the normal direction representing the damp-pot damping, and x is the reference at the time of contact The displacement of the center position of the particles, v is the relative velocity vector of the particles, and n is the unit vector in the normal direction.

  FIG. 2 is a diagram showing how two particles collide at each time. Here, consider a case where two particles collide head-on in the direction of relative velocity. Let vij be the relative velocity of the particle i with respect to the particle j. At time t0, the particle i and the particle j are in contact with each other at a relative speed vij. The distance between the center (mass center) of particle i and particle j at time t0 is twice the radius of the particle. Then, the particle i and the particle j are elastically deformed by the collision, and the center of the particle further approaches (time t1). Then, due to the repulsive force, the particles ij move away at a relative speed vij ′ (time t2). The contact force Fc acts on the particles from when the two particles ij come into contact with each other (time t0) until the two particles ij begin to leave (time t2).

In the present embodiment, the adhesion force Fa is a constant force while the particles are in contact with each other.
We adopt a model that assumes that j acts to attract.

  When the adhesion force is not taken into account, a reasonable simulation result can be obtained even if calculation is performed using a modified spring constant that is somewhat smaller than the original spring constant. It has been confirmed that the influence of the spring constant is not recognized on the calculation result of the fluidized bed (see Non-Patent Document 1).

  However, when the adhesion force is considered, if a calculation is simply performed using a corrected spring constant smaller than the original spring constant, a reasonable result cannot be obtained. This is because, as the contact time becomes longer, the impulse of adhesion becomes larger and the behavior of the particles changes. Below, the influence of adhesive force is demonstrated.

(Influence of adhesion)
FIG. 3 is a diagram illustrating two states after the particles collide with the wall surface. When the particle i on which the adhesion force and the contact force act collide with the wall surface, when the relative velocity vij at the time of collision is equal to or less than a certain velocity vc, the adhesion force is superior, and the state where the particle i is adhered to the wall surface is maintained after the collision. (State A). When the relative speed vij is greater than a certain speed vc, the repulsive force due to the contact force prevails, and after the collision, the particle i moves away from the wall surface at the speed Vij ′ (state B). The velocity vc is a velocity serving as a threshold for dividing the state (behavior) of the particles after the collision.

  Thus, contact force and adhesion force are important factors for the repulsion / adhesion behavior of particles. Therefore, focusing on the contact force and adhesion force acting between the contacting particle and the wall surface, the relationship between the momentum change and impulse when the particle collides with the wall surface is expressed by the following equation.

Here, m is the mass of the particle, vij is the relative velocity of the particle before the collision, vij 'is the relative velocity of the particle after the collision, Tc is the contact time, Fc is the contact force, and Fa is the adhesion force. The left side represents the change in the momentum of the particle in the collision, and the right side represents the impulse received by the particle in the collision. The directions in which the contact force and the adhesion force work are opposite to each other.

  Assuming that vc is a critical initial relative speed at which particles remain attached after the collision, vij ′ = 0 when vij = vc. At this time, the relationship between momentum and impulse is expressed by the following equation.

Furthermore, assuming that the adhesion force Fa acting while the particles and the wall surface are in contact is constant, the following equation is obtained.

On the other hand, the equation of motion of the particle i in the collision in consideration of the contact force Fc and the adhesion force Fa when applying the Hertz contact model is expressed as follows.

Where k is the spring constant of the Hertzian contact model (non-linear spring) representing the elastic repulsive force of the particle, η is the viscous damping coefficient in the normal direction representing damping by the dashpot, and x is the particle's relative to the time of contact. This is the displacement of the center position. Here, k> 0, η> 0, and Fa> 0. It is difficult to analytically determine, for example, the contact time Tc from the above equation that takes into account the adhesion force Fa.

  Therefore, first, let us consider the motion of particles when viscosity damping and adhesion are not considered. The equation of motion of the particle i in the collision when the viscous damping and the adhesion force are not taken into consideration is expressed by the following equation.

When the force of the above equation works, according to Hertz's contact theory, the contact time Tc when two spheres (particles) of equal material and size collide with each other at a relative velocity v ₀ is expressed by the following equation.

Here, ρ is the particle density, ν is the Poisson's ratio of the particle, E is the Young's modulus of the particle, and R is the particle radius. Further, in the initial collision condition (t = 0), the position x of the particle i is 0, and the relative velocity of the particle ij is v ₀ . If the constant in the above equation is omitted, the contact time Tc can be expressed as the following equation.

  On the other hand, the spring constant k in the normal direction when the sphere of the same material and the wall surface are in contact is expressed by the following equation.

  For reference, the spring constant k in the normal direction when two spheres (particles) having the same material and size are in contact is expressed by the following equation.

  If the constant of the formula (10) or the formula (11) is omitted, the spring constant k based on the Hertz contact model can be expressed as the following formula in any case.

From the above equation and equation (9), the relationship between the contact time Tc and the spring constant k is expressed by the following equation.

That is, it can be seen that the contact time Tc is proportional to the -2 / 5th power of the spring constant k of the Hertz contact model.

On the other hand, according to Hertz's contact theory, the maximum overlap amount x _max when two spheres (particles) having the same material and size are in contact is expressed by the following equation.

The maximum overlap amount x _max represents the length in which the radii of the particles overlap when the particles are closest to each other. The maximum overlap amount x _max can be rephrased as the displacement of the particle i when the particles are closest to each other with x = 0 at the time of collision.

From equation (7), the contact force Fc _max when the particles are closest to each other can be expressed as follows using the maximum overlap amount x _max .

From the equations (14) and (15), the relationship between the (maximum) contact force Fc _max and the spring constant k when the particles are closest to each other is as follows.

  Further, from the equation (13), the relationship between the contact time Tc and the spring constant k is as follows.

From the equations (16) and (17), the maximum contact force Fc _max is proportional to the 2 / 5th power of the spring constant k, and the contact time Tc is proportional to the −2 / 5th power of the spring constant k. I understand. That is, the impulse of contact force (the integral term of Fc on the right side of Equation (5)) can be regarded as constant regardless of the spring constant k.

  Here, when a corrected spring constant smaller than the original spring constant k is used for shortening the calculation time of the simulation, in order to obtain an appropriate simulation result, the particles after the collision are used even if the corrected spring constant is used. It is important that the behavior does not change. If the behavior of particles in the collision (attachment or separation) does not change, a reasonable simulation result can be obtained even if a simulation is performed using a modified spring constant smaller than the original spring constant k. That is, it is possible to accurately simulate the state of the particle system. If a small correction spring constant is used, the contact time of the collision becomes long. Therefore, even if the time step (time step) is set large in the simulation calculation, a reasonable simulation result can be obtained. By setting a large time step, it is possible to reduce the calculation load of the simulation.

  The fact that the behavior of the particles after the collision does not change even if the spring constant is changed means that the change in the momentum in the collision represented by the left side of Equation (3) does not change. In other words, the relative velocity vc (the limit of the initial relative velocity vc that maintains the state in which particles are attached after the collision) may be constant even if the spring constant is changed. That is, assuming that the left side is constant in Equation (5), the impulse of the contact force on the right side (the integral term of Fc) is constant regardless of the spring constant, so the impulse of the right side adhesive force also changes in the spring constant. Regardless, it must be constant.

  That is, when a modified spring constant smaller than the original spring constant k is used, the following equation must be satisfied.

From the above equation and equation (13), the following equation is obtained.

Here, ρ, R, and vc are constants irrelevant to the spring constant, so that the following expression representing the relationship between the adhesion force Fa and the spring constant k is obtained.

  That is, when a simulation is performed using a modified spring constant smaller than the original spring constant k, the adhesion force Fa must also be reduced according to the above equation. Otherwise, the initial relative velocity vc serving as a threshold value changes, and the adhesion / repulsion behavior of particles changes.

  In addition, even when viscous damping and adhesion force are taken into account as in equation (6), the relational expression (formula (20)) between the spring constant and adhesion force that does not change the behavior of the particles holds in the same manner. This will be described below.

The forces acting on the particles when one particle collides with the wall surface of the same material as the equation (6), by performing numerical calculations for one particle by finely time step, particles which collide at a certain relative velocity v ₀ It can be calculated whether it adheres to or leaves from the wall after the collision. Therefore, it is possible to determine the limit initial relative velocity vc for maintaining the attached state.

FIG. 7 shows a numerical value of the motion of one particle based on the equation of motion (equation (6)) considering the viscous damping and the adhesion force, and the initial initial relative velocity vc for maintaining the adhesion state becomes a certain value. It is the graph which plotted the spring constant k and the adhesion force Fa. When the plotted points were fitted with a curve, a relationship of Fa = 4 × 10 ⁻¹¹ × k ^0.4 was obtained. That is, the relationship between the adhesion force Fa and the spring constant k so that the initial relative velocity vc that maintains the adhered state is constant is expressed by the following equation.

This agrees with the relational expression (formula (20)) when the viscous damping and the adhesion force are not considered. For reference, the following relationship was obtained for the initial relative velocity v ₀ and the contact time Tc ₂ when colliding with the wall surface.

This agrees with the relational expression (formula (13)) when the viscous damping and the adhesion force are not considered.

  In the present embodiment, a modified spring constant kd smaller than the original spring constant kr is used as the spring constant of the Hertz contact model, and at the same time, a modified adhesion force Fad smaller than the original adhesion force Far is used as the adhesion force. I do. The corrected adhesion force Fad is preferably selected so that the threshold velocity vc does not change so as not to change the behavior of the particles. That is, the corrected adhesive force Fad is set so that the ratio of the original spring constant kr and the corrected spring constant kd to the second power is approximately the same as the ratio of the original adhesive force Far and the corrected adhesive force Fad. To do. This relationship can be expressed as follows.

If the corrected spring constant kd and the corrected adhesion force Fad satisfying the relationship of the above equation are used, the threshold velocity vc does not change, that is, the behavior related to particle adhesion does not change, so that a reasonable simulation result can be obtained. it can. Therefore, the spring constant used for the calculation can be reduced and the time step Δt in the simulation can be increased, so that the calculation time can be shortened.

  FIG. 8 is a graph plotting the value of the speed vc serving as a threshold when only the spring constant is changed with the adhesive force being constant as a reference example. Here, the motion at the time of collision of one particle is numerically calculated based on the equation of motion (equation (6)) in consideration of viscous damping and adhesion force. Note that glass beads having a diameter of 60 μm are assumed as particles. As the adhesive force, the physical property values of the glass beads are used. According to FIG. 8, it can be seen that when the spring constant k is changed without changing the adhesive force (when the corrected spring constant is used), the threshold speed vc changes. That is, the result shown in FIG. 8 indicates that if the calculation is performed using the corrected spring constant with the adhesion force kept constant, the behavior of the particles changes, and an appropriate simulation result cannot be obtained.

  FIG. 9 is a graph plotting the value of the speed vc serving as a threshold when the spring constant and the adhesive force are changed in accordance with the relationship of Expression (23). Here, the motion at the time of collision of one particle is numerically calculated based on the equation of motion (equation (6)) in consideration of viscous damping and adhesion force. The physical property values and the like are the same as in FIG. According to FIG. 9, when the spring constant and the adhesive force are changed according to the relationship of Expression (23) (when the corrected spring constant and the corrected adhesive force are used), it can be seen that the threshold speed vc does not change. . If the threshold velocity vc does not change, the behavior of particles after impact (whether they adhere or leave) does not change. That is, when the result shown in FIG. 9 is calculated using the corrected spring constant and the corrected adhesion force according to the relationship of the equation (23), the behavior of the particles does not change from the original behavior, and an appropriate simulation result is obtained. It shows that you can.

  Note that the simulation method of the present embodiment for calculating the particle behavior using the corrected spring constant kd and the corrected adhesion force Fad is preferably used when the original adhesion force Far is 0.1 times or more than the gravity Fg. be able to.

(Time step)
The appropriate time steps for obtaining reasonable simulation results are described below.

When a linear spring model is applied as the contact force model, the particle contact time Tc ₃ is expressed by the following equation, where the linear spring constant is kl.

Here, m is the mass of the particles.

It is empirically known that in a discrete element method simulation of a fluidized bed using a linear spring model, stable and reasonable calculation results can be obtained if the simulation is performed in time increments in which the contact time Tc ₃ is divided into five or more. (See Non-Patent Document 1). Therefore, a reasonable range of time step Δt in the case of the linear spring model is expressed by the following equation.

As can be seen from the above equation, the upper limit of the time step Δt depends on the particle mass m and the linear spring constant kl, and when the particle mass m is small or the particle is hard (the linear spring constant kl is large), Smaller and longer calculation time.

  On the other hand, when the Hertz contact model is applied as the contact model, the contact time of the particles varies depending on the initial relative velocity at the time of contact and cannot be obtained analytically. Therefore, consider a one-degree-of-freedom vibration system expressed by the following equation.

In this non-linear vibration system, the frequency increases as the amplitude increases. For this reason, the stability of the simulation (numerical integration) with time depends on the initial conditions and the length of the time step. Therefore, assuming the initial expected relative velocity v ₀ at the maximum contact of the particle system to be actually simulated, simulation of this vibration system (Equation (26)) is performed, and the limit of the stable calculation is possible. By specifying the time step, an appropriate time step can be determined (reference: Non-Patent Document 2). It should be noted that the time step at which the simulation of the vibration system can be stably performed without determining the limit may be adopted as the time step used for the particle system that actually performs the simulation.

  Moreover, you may employ | adopt the time step which divided | segmented contact time Tc into 5 or more division | segmentation. The particle contact time Tc in the vibration system is expressed by the following equation.

Here, ρ is the particle density, ν is the Poisson's ratio of the particle, E is the Young's modulus of the particle, and R is the particle radius. The contact time Tc becomes longer due to the influence of adhesive force. Therefore, if simulation is performed at time steps obtained by dividing the contact time Tc into five or more, an appropriate simulation result can be obtained. Therefore, a reasonable range of time step Δt in the case of the Hertz contact model is expressed by the following equation.

In this nonlinear vibration system when the Hertz contact model is applied, the contact time Tc becomes shorter as the initial relative velocity v ₀ of the collision is larger. Therefore, assuming an initial relative velocity v ₀ at the maximum expected contact, a time step Δt may be obtained by dividing the predicted minimum contact time Tc obtained therefrom by five.

As mentioned above, in the discrete element method simulation of the fluidized bed of the linear spring model that does not consider the adhesion force, there is no problem even if the calculation is performed using a modified spring constant smaller than the spring constant that represents the repulsion of the actual particles. It has been known. For example, in the collision of hard particles, even if the calculation is performed by setting the correction spring constant of the linear spring model so that the elastic deformation amount of the particles is within a few percent (for example, within 1%) of the particle diameter, It is empirically known that can be obtained. Even in the case of the Hertz contact model, calculation may be performed by setting the modified spring constant of the Hertz contact model so that the amount of elastic deformation of the particle falls within several percent (for example, within 1%) of the particle diameter in the collision of hard particles. . The relative speed at the time of collision varies, but the above condition may be satisfied in the case of the average relative speed of the collision. For example, the above condition may be satisfied in 95% of the collisions. Also good. In addition, even when the gas is not blown up, the corrected spring constant of the Hertz contact model can be set so that the amount of elastic deformation of the lowermost particle is within a few percent of the particle diameter depending on the weight of the accumulated particles. Good. When the correction spring constant of the Hertzian contact model is set under such conditions, the correction spring constant kd can be set to a value not more than 0.1 times the original spring constant kr for hard particles. For example, if the corrected spring constant kd of the Hertz contact model is 0.1 times the original spring constant, the time step can be set to about 2.5 times (10 ^2/5 times).

<Configuration of particle behavior simulation device>
FIG. 4 is a block diagram showing a functional configuration of the particle behavior simulation apparatus 10 of the present embodiment. The particle behavior simulation apparatus 10 includes a condition input unit 11, a behavior calculation unit 12, and a contact force calculation unit (contact force specifying unit) 13.

  The condition input unit 11 includes a correction spring constant acquisition unit 14, a correction adhesion acquisition unit 15, and a time step acquisition unit 16. The condition input unit 11 reads a setting file in which various conditions necessary for the simulation are described, and receives an input of simulation conditions. The simulation conditions include the shape of the container, the number of particles, the particle diameter, the particle mass, the corrected spring constant in the collision between particles, the corrected adhesion force in the collision between particles, the restitution coefficient in the collision between particles, the lower side of the dispersion plate There are the flow velocity (flow rate) of the gas flowing in from the gas, the density of the gas, the viscosity coefficient of the gas, and the time step. There are other simulation conditions, but they are omitted. The condition input unit 11 may include an input device such as a keyboard, and may receive an input of simulation conditions directly from the user.

  The corrected spring constant acquisition unit 14 acquires the corrected spring constant kd and the viscous damping coefficient η among the input simulation conditions, and outputs the corrected spring constant kd and the viscous damping coefficient η to the contact force calculation unit 13. The corrected spring constant kd is a value smaller than the original spring constant kr.

  The corrected adhesion force acquisition unit 15 acquires the corrected adhesion force Fad among the input simulation conditions, and outputs the corrected adhesion force Fad to the behavior calculation unit 12. The corrected adhesion force Fad is a value smaller than the original adhesion force Far according to the ratio (2/5) of the original spring constant kr and the correction spring constant kd.

  The time step acquisition unit 16 acquires the time step td from the input simulation conditions, and outputs the time step td to the behavior calculation unit 12. The time step td is a time step whose value is increased in accordance with the reduced correction spring constant kd, and can be called a correction time step. The correction time step td can be set to a large value according to the square root of the ratio between the original spring constant kr and the correction spring constant kd. In order to obtain a stable and appropriate result, for example, it is only necessary to satisfy the condition that the calculation is performed in a time step of 1/5 or less of the contact time Tc of the particle in the collision. When a small correction spring constant kd is used, the contact time Tc becomes long.

  The condition input unit 11 outputs other conditions for the acquired simulation to the behavior calculation unit 12.

  The behavior calculation unit 12 calculates the behavior (position and motion) of particles at each time for each correction time step td using each simulation condition. The behavior calculation unit 12 obtains a force acting on each particle at a certain time t. Here, the force F acting on one particle of interest is expressed by the following equation.

When the particle of interest is in contact with another particle (or wall) (the distance between the particle of interest and the other particle is within a predetermined distance), the behavior calculation unit 12 has contact force acting on the particle of interest. Is obtained, the distance between the particle of interest and other particles (or walls) and the relative velocity are output to the contact force calculator 13.

  The contact force calculation unit 13 uses the distance and relative speed between the target particle and other particles, the corrected spring constant kd, and the viscous damping coefficient η, and the contact force (corrected contact force) Fcd that acts on the target particle. Ask for. The corrected contact force Fcd in the normal direction is expressed by the following equation.

The contact force calculation unit 13 outputs the calculated corrected contact force Fcd acting on the target particle to the behavior calculation unit 12.

  The behavior calculation unit 12 uses the corrected contact force Fcd applied to the target particle received from the contact force calculation unit 13, the corrected adhesion force Fad, the fluid force Fd applied to the target particle, and the gravity Fg of the target particle. The position and speed after the correction time step td are calculated. Similarly, the behavior calculation unit 12 calculates the position and velocity after the correction time step td for all particles. Note that the behavior calculation unit 12 may further obtain the rotation angular velocity of the particles as the motion state of the particles.

  In this way, the behavior calculation unit 12 obtains the position and motion of each particle at each time for each correction time step td. The behavior calculation unit 12 outputs the position and motion information of each particle at each time to a storage device (not shown). For example, the particle behavior simulation apparatus 10 may display the position and movement of each particle in a display device (not shown) sequentially with a dot or a circle for each time and display the particle behavior to the user. .

<Flow of particle behavior simulation>
Next, a parameter determination method for particle behavior simulation and a processing flow in the particle behavior simulation apparatus 10 will be described.

(Specification of spring constant kr)
In order to perform the particle behavior simulation of the present embodiment, the corrected spring constant kd and the corrected adhesion force Fad are required. Therefore, the original spring constant kr and the original adhesion force Far are required.

  The spring constant kr when the contact force acting on the particles when the particles are in contact with each other is represented by an elastic repulsion model using a non-linear spring is determined by the following procedure.

  According to Hertz's contact theory, the spring constant in the normal direction when a sphere (particle) collides with a wall of the same material is expressed by equation (10). The spring constant in the normal direction when two spheres (particles) having the same size as the material collide with each other is expressed by Expression (11). That is, the original spring constant kr of the Hertzian contact model can be determined from the Young's modulus E of the particle, the Poisson's ratio ν of the particle, and the radius R of the particle.

  The viscosity damping coefficient η is specified from the measured value of the coefficient of restitution.

(Specification of adhesive force Far)
The adhesion force Far can be specified by actual measurement. For example, the adhesion force can be a force required to peel two particles against each other and then peel them off. Note that the adhesion force Far may be specified (assumed) based on Hamaker theory or the like.

(Specification of corrected spring constant kd)
Next, a corrected spring constant kd smaller than the spring constant kr is specified (set). As described above, for example, in the collision of hard particles, calculation may be performed by setting a correction spring constant so that the elastic deformation amount of the particles is within several percent (for example, within 3% or within 1%) of the particle diameter. It is empirically known that reasonable simulation results can be obtained. In the present embodiment, the modified spring constant kd is set so that the amount of elastic deformation of the particle due to the main collision (distance x (particle overlap distance) at time t1 in FIG. 2) is within 1% of the particle diameter. To do.

(Specification of corrected adhesive force Fad)
Next, a corrected adhesion force Fad having a value smaller than the adhesion force Far is specified in accordance with the ratio (2/5) of the spring constant kr and the correction spring constant kd. The corrected adhesion force Fad can be obtained by Expression (23).

(Specification of correction time step td)
Assuming a maximum initial relative velocity v ₀ at the time of a collision, a correction time step td used for calculation is obtained. In the present embodiment, 1/5 or less of the particle contact time Tc in the collision is set as the correction time step td. Therefore, the correction time step td is determined by the equation (28).

(Calculation of particle behavior)
FIG. 5 is a flowchart showing processing of the particle behavior simulation apparatus 10.

  The condition input unit 11 of the particle behavior simulation apparatus 10 acquires simulation conditions including the corrected spring constant kd, the corrected adhesion force Fad, and the correction time step td specified as described above (S1).

  The correction spring constant acquisition unit 14 acquires the correction spring constant kd and the viscous damping coefficient η and outputs them to the contact force calculation unit 13.

  The corrected adhesion force acquisition unit 15 acquires the corrected adhesion force Fad and outputs it to the behavior calculation unit 12.

  The time step acquisition unit 16 acquires the time step td and outputs it to the behavior calculation unit 12.

  When the target particle is in contact with another particle (Yes in S2), the behavior calculation unit 12 outputs the distance and relative velocity between the target particle and the other particle to the contact force calculation unit 13 (S3). .

  The contact force calculation unit 13 uses the distance and relative velocity between the target particle and other particles, the modified spring constant kd, and the viscous damping coefficient η, and the contact force acting on the target particle from Equation (12). (Corrected contact force) Fcd is obtained (S4). The contact force calculation unit 13 outputs the corrected contact force Fcd acting on the target particle to the behavior calculation unit 12.

  When the target particle is not in contact with other particles (No in S2), the contact force and the adhesion force do not work (Fcd = Fad = 0).

  After No in S2 or after S4, the behavior calculation unit 12 uses the corrected contact force Fcd, the corrected adhesion force Fad acting on the target particle, the fluid force Fd, and the gravity Fg, and the target particle after the correction time step td. The position and speed of are determined (S5).

  If there are still particles for which the position and velocity after the correction time step td have not been obtained (No in S6), the processing of S2 to S5 is repeated focusing on the remaining particles.

  When the positions and velocities after the correction time step td are obtained for all particles at a certain time (Yes in S6), it is determined whether the calculation has been performed up to a predetermined end time (time increment) (S7). Here, when the behavior of the particle behavior for 1 second is performed, the end time is one second after the first time.

  When the calculation has not been performed until the predetermined end time (No in S7), the time is advanced by the correction time step td, and the calculation (S2 to S6) at the next time is repeated.

  When the calculation is completed up to a predetermined end time (Yes in S7), the process ends.

(Summary of Embodiment 1)
According to the present embodiment, the particle behavior is calculated by the discrete element method using the corrected spring constant kd smaller than the original spring constant kr and the corrected adhesion force Fad smaller than the original adhesion force Far accordingly. To do. Therefore, the time step can be increased according to the small correction spring constant kd. Therefore, even when the adhesive force is taken into consideration, the simulation calculation time can be shortened and an appropriate simulation result can be obtained.

Further, the corrected adhesion force Fad only needs to be reduced in accordance with the corrected spring constant kd, and a reasonable result can be obtained as long as it is within a certain range even if the relationship of the expression (23) is not strictly satisfied. Can do. For example, the corrected adhesion force Fad may be set within the range of the following equation.

Alternatively, the corrected adhesion force Fad may be set so that the ratio between the corrected adhesion force Fad and the adhesion force Far is in the same order as the 2 / 5th power of the ratio between the correction spring constant kd and the spring constant kr.

  In this embodiment, a Hertzian contact model is assumed as a model of the corrected contact force. Further, as the contact force, a non-linear spring obtained by combining a plurality of Hertz contact model springs may be considered.

  Moreover, although the above demonstrated the collision of particle | grains, about a collision with particle | grains and a wall, you may obtain | require a correction spring constant and correction adhesion force separately, and may use it for calculation.

[Embodiment 2]
Hereinafter, the present embodiment will be described in detail with reference to FIG. For convenience of explanation, members / configurations having the same functions as those in the drawings described in the first embodiment are given the same reference numerals, and detailed descriptions thereof are omitted. In the present embodiment, the particles in the fluidized bed apparatus shown in FIG.

<Configuration of particle behavior simulation device>
FIG. 6 is a block diagram showing a functional configuration of the particle behavior simulation apparatus 20 of the present embodiment. The particle behavior simulation apparatus 20 includes a condition input unit 21, a behavior calculation unit 12, a contact force calculation unit (contact force specifying unit) 13, and a modified adhesion force specifying unit 22.

  The condition input unit 21 includes a spring constant acquisition unit 23, an adhesion force acquisition unit 24, a modified spring constant acquisition unit 25, and a time step acquisition unit 16. The condition input unit 21 reads a setting file in which various conditions necessary for the simulation are described, and receives an input of simulation conditions.

  The spring constant acquisition unit 23 acquires the original spring constant kr among the input simulation conditions, and outputs the spring constant kr to the corrected adhesion force specifying unit 22.

  The adhesive force acquiring unit 24 acquires the original adhesive force Far among the input simulation conditions, and outputs the adhesive force Far to the corrected adhesive force specifying unit 22.

  The correction spring constant acquisition unit 25 acquires the correction spring constant kd and the viscous damping coefficient η among the input simulation conditions, and outputs the correction spring constant kd and the viscous damping coefficient η to the contact force calculation unit 13. Further, the corrected spring constant acquisition unit 25 outputs the corrected spring constant kd to the corrected adhesive force specifying unit 22.

  The corrected adhesive force specifying unit 22 specifies the corrected adhesive force Fad such that the ratio of the corrected adhesive force Fad to the adhesive force Far is the same as the 2/5 power of the ratio of the corrected spring constant kd to the spring constant kr. To do. That is, the corrected adhesive force specifying unit 22 specifies the corrected adhesive force Fad according to the equation (23). The corrected adhesive force specifying unit 22 outputs the specified corrected adhesive force Fad to the behavior calculating unit 12.

  The condition input unit 21 also outputs other conditions for the acquired simulation to the behavior calculation unit 12.

  Since the configurations of the behavior calculation unit 12 and the contact force calculation unit 13 are the same as those in the first embodiment, the description thereof is omitted.

  In the present embodiment, the corrected adhesive force specifying unit 22 obtains an appropriate corrected adhesive force Fad based on the original spring constant kr, the original adhesive force Far, and the corrected spring constant kd. Therefore, when considering the adhesive force, a reasonable simulation result can be obtained in a short calculation time.

  Finally, each block of the particle behavior simulation devices 10 and 20, particularly the condition input unit 11, the behavior calculation unit 12, the contact force calculation unit 13, the modified spring constant acquisition unit 14, the corrected adhesion force acquisition unit 15, and the time step acquisition unit 16. The condition input unit 21, the corrected adhesive force specifying unit 22, the spring constant acquiring unit 23, the adhesive force acquiring unit 24, and the corrected spring constant acquiring unit 25 may be configured by hardware logic, or as described below. It may be realized by software using a (central processing unit).

  That is, the particle behavior simulation apparatus 10/20 includes a CPU that executes instructions of a control program that realizes each function, a ROM (read only memory) that stores the program, a RAM (random access memory) that expands the program, A storage device (recording medium) such as a memory for storing programs and various data is provided. The object of the present invention is to record the program code (execution format program, intermediate code program, source program) of the control program of the particle behavior simulation apparatus 10/20, which is software that realizes the above-described functions, in a computer-readable manner. This can also be achieved by supplying the recording medium to the particle behavior simulation apparatuses 10 and 20, and reading and executing the program code recorded on the recording medium by the computer (or CPU or MPU (microprocessor unit)).

  Examples of the recording medium include a tape system such as a magnetic tape and a cassette tape, a magnetic disk such as a floppy (registered trademark) disk / hard disk, a CD-ROM (compact disc read-only memory) / MO (magneto-optical) / Disk systems including optical disks such as MD (Mini Disc) / DVD (digital versatile disk) / CD-R (CD Recordable), card systems such as IC cards (including memory cards) / optical cards, or mask ROM / EPROM ( An erasable programmable read-only memory) / EEPROM (electrically erasable and programmable read-only memory) / semiconductor memory system such as a flash ROM can be used.

Further, the particle behavior simulation apparatuses 10 and 20 may be configured to be connectable to a communication network, and the program code may be supplied via the communication network. The communication network is not particularly limited. For example, the Internet, intranet, extranet, LAN (local area network), ISDN (integrated services digital network), VAN (value-added network), CATV (community antenna television) communication. A network, a virtual private network, a telephone line network, a mobile communication network, a satellite communication network, etc. can be used. In addition, the transmission medium constituting the communication network is not particularly limited. For example, IEEE (institute of electrical and electronic engineers) 1394, USB, power line carrier, cable TV line, telephone line, ADSL (asynchronous digital subscriber loop) line Infrared data association such as IrDA (infrared data association) and remote control, Bluetooth (registered trademark), 802.11 wireless, HDR
(High data rate), mobile phone network, satellite line, terrestrial digital network, etc. can also be used.

  The present invention is not limited to the above-described embodiments, and various modifications are possible within the scope shown in the claims, and embodiments obtained by appropriately combining technical means disclosed in different embodiments. Is also included in the technical scope of the present invention.

  The present invention can be used for particle behavior simulation for predicting particle behavior.

DESCRIPTION OF SYMBOLS 1 Fluidized bed apparatus 2 Processing container 3 Dispersing plate 4 Gas supply port 5 Exhaust port 6 Catalyst particle | grains 10 and 20 Particle behavior simulation apparatus 11 and 21 Condition input part 12 Behavior calculation part 13 Contact force calculation part (contact force specific part)
14, 25 Modified spring constant obtaining unit 15 Modified adhesion obtaining unit 16 Time step obtaining unit 22 Modified adhesion identifying unit 23 Spring constant obtaining unit 24 Adhesive force obtaining unit

Claims (10)

A particle behavior simulation apparatus for predicting the behavior of a plurality of particles using a discrete element method,
When the contact force acting on the particles when the particles are in contact with each other is represented by a nonlinear spring model, the spring constant is kr, and the adhesion force between the particles is Far.
A contact force specifying unit for obtaining a corrected contact force based on the nonlinear spring model using a corrected spring constant kd smaller than the spring constant kr;
Using the corrected contact force as the contact force, and using the corrected adhesion force Fad having a value smaller than the adhesion force Far according to the ratio of the spring constant kr and the correction spring constant kd as the adhesion force, A particle behavior simulation apparatus comprising: a behavior calculation unit that calculates particle behavior by a discrete element method.   The particle behavior simulation apparatus according to claim 1, wherein the nonlinear spring model is a Hertzian contact model. The corrected adhesive force Fad is as a condition:

The particle behavior simulation apparatus according to claim 1, wherein:   The ratio between the corrected adhesion force Fad and the adhesion force Far is the same order as the 2 / 5th power of the ratio between the corrected spring constant kd and the spring constant kr. Particle behavior simulation equipment.   The particle behavior simulation apparatus according to any one of claims 1 to 4, wherein the corrected spring constant kd is 0.1 times or less of the spring constant kr.   The ratio between the corrected adhesion force Fad and the adhesion force Far is substantially the same as the 2 / 5th power of the ratio between the corrected spring constant kd and the spring constant kr. Particle behavior simulation equipment.   Based on the adhesive force Far, the corrected spring constant kd, and the spring constant kr, the ratio of the corrected adhesive force Fad to the adhesive force Far is 2 / of the ratio of the corrected spring constant kd to the spring constant kr. The particle behavior simulation apparatus according to claim 1, further comprising a modified adhesion force identifying unit that identifies the modified adhesion force Fad that is the same as the fifth power. A particle behavior simulation method for predicting the behavior of a plurality of particles using a discrete element method,
When the contact force acting on the particles when the particles are in contact with each other is represented by a nonlinear spring model, the spring constant is kr, and the adhesion force between the particles is Far.
A contact force specifying step in which the contact force specifying unit of the simulation device obtains a corrected contact force based on the nonlinear spring model using a corrected spring constant kd smaller than the spring constant kr;
The behavior calculation unit of the simulation apparatus uses the corrected contact force as the contact force, and the adhesion force is set to a value smaller than the adhesion force Far according to the ratio of the spring constant kr and the corrected spring constant kd. And a behavior calculation step of calculating the behavior of the particles by the discrete element method using the corrected adhesion force Fad.   A control program for causing a computer to function as the particle behavior simulation apparatus according to any one of claims 1 to 7, wherein the control program causes the computer to function as each unit described above.   A computer-readable recording medium on which the control program according to claim 9 is recorded.

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