05 book 2007/5/15 page 77 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.5. Staggering schemes 77 0 5e+07 1e+08 1.5e+08 2e+08 2.5e+08 3e+08 3.5e+08 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MAXIMUM FORCE (N) TIME NORMAL FORCE TANGENTIAL FORCE 0 5e+07 1e+08 1.5e+08 2e+08 2.5e+08 3e+08 3.5e+08 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MAXIMUM FORCE (N) TIME NORMAL FORCE TANGENTIAL FORCE Figure 7.14. The maximum force (and corresponding friction force) versus time imparted on the immovable obstacle surface, max( I). The top graph is with charging and the bottom is without charging. Notice that the maximum “signature” force is less with charging. 05 book 2007/5/15 page 78 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 78 Chapter 7. Advanced particulate flow models -5e+08 -4e+08 -3e+08 -2e+08 -1e+08 0 1e+08 2e+08 3e+08 4e+08 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TOTAL FORCE (N) TIME TOTAL X NORMAL FORCE TOTAL Y NORMAL FORCE TOTAL Z NORMAL FORCE TOTAL X TANGENTIAL FORCE TOTAL Y TANGENTIAL FORCE TOTAL Z TANGENTIAL FORCE -6e+08 -4e+08 -2e+08 0 2e+08 4e+08 6e+08 8e+08 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TOTAL FORCE (N) TIME TOTAL X NORMAL FORCE TOTAL Y NORMAL FORCE TOTAL Z NORMAL FORCE TOTAL X TANGENTIAL FORCE TOTAL Y TANGENTIAL FORCE TOTAL Z TANGENTIAL FORCE Figure 7.15. The total force (and corresponding friction force) versus time im- parted on the immovable obstacle surface, max( I). The top graph is with charging and the bottom is without charging. Notice that the total “signature” force is less with charging. 05 book 2007/5/15 page 79 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 7.5. Staggering schemes 79 X Y Z X Y Z X Y Z X Y Z Figure 7.16. Top to bottom and left to right, slow impact of charged clouds. The clouds combine into a larger cloud. 05 book 2007/5/15 page 80 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 80 Chapter 7. Advanced particulate flow models X Y Z X Y Z X Y Z X Y Z Figure 7.17. Top to bottom and left to right, fast impact of charged clouds. The clouds disperse. 05 book 2007/5/15 page 81 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ Chapter 8 Coupled particle/fluid interaction Until this point, we have ignored the presence of a fluid medium surrounding the particles. We now focus on the modeling and simulation of the dynamics of particles, coupled with a surrounding fluid, while bringing in several of the effects discussed earlier in the form of a model problem. Obviously, the number of research areas involving particles in a fluid un- dergoing various multifield processesis immense, andit would be futileto attempt to catalog all ofthe various applications. However, a common characteristic of such systems is that the various physical fields (thermal, mechanical,chemical, electrical, etc.) are strongly coupled. This chapter develops a flexible and robust solution strategy to resolve coupled sys- tems comprising large groups of flowing particles embedded within a fluid. A problem modeling groups of particles, which may undergo inelastic collisions in the presence of near-field forces, is considered. The particles are surrounded by a continuous interstitial fluid that is assumed to obey the compressible Navier–Stokes equations. Thermal effects are also considered. Such particle/fluid systems are strongly coupled due to the mechanical forces and heat transfer induced by the fluid on the particles and vice versa. Because the coupling of the various particle and fluid fields can dramatically change over the course of a flow process, a primary focus of this work is the development of a recursive “staggering” solution scheme, whereby the time steps are adaptively adjusted to control the error asso- ciated with the incomplete resolution of the coupled interaction between the various solid particulate and continuum fluid fields. A central feature of the approach is the ability to account for the presence of particles within the fluid in a straightforward manner that can be easily incorporated into any standard computational fluid mechanics code based on finite difference, finite element, or finite volume discretization. A three-dimensional example is provided to illustrate the overall approach. 42 Remark. Although some portions of the presentation in this chapter may appear to be redundant with earlier parts of the monograph, there are subtle differences, and thus it is felt that a self-contained chapter is pedagogically superior to continual referral to previous portions of the monograph, which may lead to possible ambiguities. 42 It is assumed that the particles are small enough that their rotation with respect to their mass centers is deemed insignificant. However, even in the event that the particles are not extremely small, we assume that any “spin” of the particles is small enough to neglect lift forces that may arise from the interaction with the surrounding fluid. 81 05 book 2007/5/15 page 82 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 82 Chapter 8. Coupled particle/fluid interaction COMBINED PROBLEM PARTICLE-ONLY PROBLEMPROBLEM FLUID-ONLY = + Figure 8.1. Decomposition of the fluid/particle interaction (Zohdi [224]). 8.1 A model problem We consider a sufficiently complex model problem comprising of agroup of nonintersecting spherical particles (N p in total), each being small enough that their rotation with respect to their mass centers is deemed insignificant. The equation of motion for the ith particle in the system (Figure 8.1) is m i ¨ r i =  tot i (r 1 , r 2 , ,r N p ), (8.1) where r i is the position vector of the ith particle and  tot i represents all forces acting on particle i. In particular,  tot i =  drag i +  nf i +  con i +  fric i represents the forces due to fluid drag, near-field interaction, interparticle contact forces, and frictional forces. Clearly, under certain conditions one force may dominate over the others. However, this is generally impossible to ascertain a priori, since the dynamics and coupling in the system may change dramatically during the course of the flow process. Remark. Throughout this chapter, boldface symbols indicate vectors or tensors. The inner product of two vectors u and v is denoted by u · v. At the risk of oversimplification, we ignore the distinction between second-order tensors and matrices. Furthermore, we exclusively employ a Cartesian basis. Hence, if we consider the second-order tensor A with its matrix representation [A], then the product of two second-order tensors A · B is defined by the matrix product [A][B], with components of A ij B jk = C ik . The second-order inner product of two tensors or matrices is A : B = A ij B ij = tr([A T ][B]). Finally, the divergence of a vector u is defined by ∇·u = u i,i , whereas for a second-order tensor A, ∇·A describes a contraction to a vector with the components A ij,j . 8.1.1 A simple characterization of particle/fluid interaction We first consider drag force interactions between the fluid and the particles. The drag force acting on an object in a fluid flow (occupying domain ω and outward surface normal n)is defined as  drag =  ∂ω σ f · n dA, (8.2) where σ f is the Cauchy stress. For a Newtonian fluid, σ f is given by σ f =−P f 1 + λ f tr D f 1 + 2µ f D f =−P f 1 + 3κ f tr D f 3 1 + 2µ f D  f , (8.3) where P f is the thermodynamic pressure, κ f = λ f + 2 3 µ f is the bulk viscosity, µ f is 05 book 2007/5/15 page 83 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 8.1. A model problem 83 the absolute viscosity, D f = 1 2 (∇ x v f + (∇ x v f ) T ) is the symmetric part of the velocity gradient, tr D f is the trace of D f , and D  f = D f − trD f 3 1 is the deviatoric part of D f . The stress is determined by solving the following coupled system of partial differential equations (compressible Navier–Stokes): Mass balance: ∂ρ f ∂t =−∇ x · (ρ f v f ), Energy balance: ρ f C f  ∂θ ∂t + (∇ x θ f ) · v f  = σ f :∇ x v f +∇ x · (K f ·∇θ f ) + ρ f z f , Momentum balance: ρ f  ∂v f ∂t + (∇ x v f ) · v f  =∇ x · σ f + ρ f b f , (8.4) where, at a point, ρ f is the fluid density; v f is the fluid velocity; θ f is the fluid temperature; C f is the fluid heat capacity; z f is the heat source per unit mass; K f is the thermal conduc- tivity tensor, assumed to be isotropic of the form K f = K f 1, K f being the scalar thermal conductivity; and b f represents body forces per unit mass. The thermodynamic pressure is given by an equation of state: Z(P f ,ρ f ,θ f ) = 0. (8.5) The specific equation of state will be discussed later in the presentation. The fluid domain will require spatial discretization with some type of mesh, for exam- ple, usinga finite difference, finite volume, or finite elementmethod. Usually, it is extremely difficult to resolve the flow in the immediate neighborhood of the particles, in particular if there are several particles. However, if the primary interest is in the dynamics of the particles, as it is in this work, an appropriate approach, which permits coarser discretization of the fluid continuum, is to employ effective drag coefficients, for example, defined via C D def = || drag i || 1 2 ρ f  ω i ||v f  ω i − v i || 2 A i , (8.6) where (·) ω i def = 1 |ω i |  ω i (·)dω i is the volumetric average of the argument over the domain occupied by the ith particle, v f  ω i is the volumetric average of the fluid velocity, v i is the velocity of the ith (solid) particle, and A i is the cross-sectional area of the ith (solid) particle. For example, one possible way to represent the drag coefficient is with a piecewise definition, as a function of the Reynolds number (Chow [44]): • For 0 <Re≤ 1, C D = 24 Re . • For 1 <Re≤ 400, C D = 24 Re 0.646 . • For 400 <Re≤ 3 × 10 5 , C D = 0.5. • For 3 × 10 5 <Re≤ 2 ×10 6 , C D = 0.000366Re 0.4275 . • For 2 × 10 6 <Re<∞, C D = 0.18. Here, thelocal Reynolds number fora particle is Re def = 2b i ρ f  ω i ||v f  ω i −v i || µ and b i is the radius of the ith particle. The use of this simple concept makes it relatively straightforward to 05 book 2007/5/15 page 84 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 84 Chapter 8. Coupled particle/fluid interaction account for the presenceofthesolid particles in the fluid by augmentingtheflowcalculations with drag forces (Figure 8.1). Algorithmically speaking, one must compute the fluid flow with reaction forces due to the presence of the particles. To this end, one can use the volu- metric forces (b f ) and heat sources (z f ) within the fluid domain for this purpose by writing ρ f  ∂v f ∂t + (∇ x v f ) · v f  =∇ x · σ f + ρ f b f , b f = b D =−  drag i m i =− C D 1 2 ρ f  ω i ||v f  ω i −v i || 2 A i m i d  d = v f  ω i −v i ||v f  ω i −v i ||  , ρ f C  ∂θ f ∂t + (∇ x θ f ) · v f  = σ f :∇ x v f +∇ x · (K f ·∇ x θ f ) + ρ f z f , z f = z D = c v |b D · (v f  ω i − v i )|, (8.7) where the drag force on the fluid, b D (per unit mass), is nonzero if its location coincides with the particle domain and is zero otherwise. Here, z D is the heat source due to the rate of work done by the drag force on the fluid. 43 Such source terms are easily projected onto a finite difference or finite element grid. 44 This drag-based approach is designed to account for particles in the fluid using a coarse mesh. In other words, the smallest (mesh) scale allowable is that associated with the dimensions of the particles. Accordingly, we shall not employ meshes smaller than the particle length scale when simulations are performed later. Remark. More detailed analyses of fluid-particle interaction can be achieved in two primary ways: (1) direct, brute-force, numerical schemes, treating the particles as part of the fluid continuum (as another fluid or solid phase), and thus meshing them in a detailed manner, or (2) with semi-analytical techniques, such as those based on approximation of the interaction between the particles and the fluid, employing an analysis of the (interstitial) fluid gaps using lubrication theory. For a concise review of recent developments in such semi-analytical techniques, in particular methods that go beyond local analyses of flow in a single fluid gap, using discrete network approximations, which account for multiple hydrodynamic interactions, see Berlyand and Panchenko [30] and Berlyand et al. [31]. Although not employed here, discrete network approximations appear to be quite attractive for possibly improving the description of the interaction between the particles and the fluid, beyond a simple drag-based method (as adopted in this work), without resorting to detailed numerical meshing. 8.1.2 Particle thermodynamics Throughout the thermal analysis of the particles, we shall use relatively simple models. Consistent with the particle-based philosophy, it is assumed that the temperature within each particle is uniform (a lumped mass approximation). We remark that the validity of assuming a uniform temperature within a particle is dictated by the Biot number. A small Biot number indicates that such an approximation is reasonable. The Biot number for a 43 If the constant c v is not selected as unity, this can indicate endothermic or exothermic particle/fluid chemical reactions. 44 If the particlesaresignificantly smaller than the meshspacing,thenthe drag forces associated withthe particles are computed from the nearest node/particle center pair. 05 book 2007/5/15 page 85 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 8.1. A model problem 85 sphere scales with the ratio of particle volume (V ) to particle surface area (a s ), V a s = b 3 , which indicates that auniform temperaturedistributionis appropriate, since theparticlesare, by definition, small. Since it is assumed that the temperature fields are uniform within the particles, the gradient of the temperature within the particle is zero, i.e., ∇θ = 0. Therefore, a Fourier-type law for the heat flux will register a zero value, q =−K ·∇θ = 0. Under these assumptions, we consider an energy balance, governing the interconver- sions of mechanical, thermal, and chemical energy in a system, dictated by the first law of thermodynamics. Accordingly, we require the time rate of change of the sum of the kinetic energy (K) and stored energy (S) to be equal to the sum of the work rate (power, P) and the net heat supplied (H): d dt (K + S) = P +H, (8.8) where we assume that the stored energy is composed solely of a thermal part, S = mCθ, C being the heat capacity per unit mass. Consistent with the assumption that the particles deform negligibly during impact, the amount of stored mechanical energy is deemed in- significant. The kinetic energy is K = 1 2 mv · v. The mechanical power term is due to the forces acting on a particle: P = dW dt =  tot · v. (8.9) For the particles, it is assumed that a process of convection, for example, governed by Newton’s law of cooling and thermal radiation according to a simple Stefan–Boltzmann law, occurs. Accordingly, the first law reads m ˙ v ·v + mC ˙ θ    d(K+S) dt =  tot · v    power=P −h c a s (θ − θ o )    convection +mc v |b D · (v f  ω − v)|    drag −Ba s (θ 4 − θ 4 s )    radiation    H , (8.10) where θ o is the temperature of the ambient fluid, h c is the convection coefficient (using Newton’s law of cooling), and θ s is the temperature of the far-field surface (for example, a container surrounding the flow) with which radiative exchange is made. The Stefan– Boltzmann constant is B = 5.67 ×10 −8 W m 2 −K ;0≤  ≤ 1 is the emissivity, which indicates how efficiently the surface radiates energy compared to a black-body (an ideal emitter); 0 ≤ h c is the convection coefficient (Newton’s law of cooling); and a s is the surface area of a particle. It is assumed that the radiation exchange between the particles is negligible. 45 Because dK dt = m ˙ v ·v =  tot ·v = P, we obtain a simplified form of the first law, dS dt = H, and therefore Equation (8.10) becomes mC ˙ θ =−h c a s (θ − θ o ) + mc v |b D · (v f  ω − v)|−Ba s (θ 4 − θ 4 s ), (8.11) where θ o =θ f  ω is the local average of the surrounding fluid temperature. Remark. To account for the convective exchange between the fluid and the particles, we amend the source term in Equation (8.7) for the fluid to read z f = z D = c v |b f · (v f  ω − v)|+ h c a s (θ − θ o ) m . (8.12) 45 Various arguments for such an assumption can be found in the classical text of Bohren and Huffman [33]. 05 book 2007/5/15 page 86 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 86 Chapter 8. Coupled particle/fluid interaction If the fluid is “radiationally” thick, then we assume that no radiation enters the system from the far field, namely, that Ba s θ 4 s = 0 in Equation (8.11), and that any emission from the particle gets absorbed by the fluid. Thus, in that situation, we can again amend the source term to read z f = z D = c v |b f · (v f  ω − v)|+ h c a s (θ − θ o ) + Ba s θ 4 m . (8.13) Remark. We assume that various phenomena, such as near-field interaction, particle contact, interparticle friction, and particle thermal sensitivity, are similar for the wet and dry particulate flow problems, with the primary difference being that drag forces from the surrounding fluid need to be determined via numerical discretization of the Navier–Stokes equations, which is next. 46 8.2 Numerical discretization of the Navier–Stokes equations We now develop a fully implicit staggering scheme, in conjunction with a finite difference discretization, to solve the coupled system. Generally, such schemes proceed, within a discretized time step, by solving each field equation individually, allowing only the corre- sponding primary field variable (ρ f , v f ,orθ f ) to be active. This effectively (momentarily) decouples the system of differential equations. After the solution of each field equation, the primary field variable is updated, and the next field equation is solved in a similar man- ner, with only the corresponding primary variable being active. For accurate numerical solutions, the approach requires small time steps, primarily because the staggering error accumulates with each passing increment. Thus, such computations are usually computa- tionally intensive. First, let us consider a finite differencediscretizationof the derivatives in the governing equations where, for brevity, we write (L indicates the time step counter, v L f def = v f (t), v L+1 f def = v f (t +t), etc.) for each finite difference node (i,j,k) ρ i,j,k,L+1 f = ρ i,j,k,L f − t  ∇ x · (ρ f v f )  i,j,k,L+1 , Z(P i,j,k,L+1 f ,ρ i,j,k,L+1 f ,θ i,j,k,L+1 f ) = 0, θ i,j,k,L+1 f = θ i,j,k,L f − t (∇ x θ f · v f ) i,j,k,L+1 +  t ρ f C f (σ f :∇ x v f +∇ x · (K f ·∇ x θ f ) + ρ f z f )  i,j,k,L+1 , v i,j,k,L+1 f = v i,j,k,L f − t (∇ x v f · v f ) i,j,k,L+1 + t ρ f  ∇ x · σ f + ρ f b f  i,j,k,L+1 , (8.14) 46 Clearly, the wetting of the particle surfaces, breaking of hydrodymanic films, etc., are nontrivial, but are outside the scope of this introductory treatment. [...]... K ), where ˆ ˆ def zrK = and def zvK = ˆ ErK TOLr ˆ EvK TOLv and then a minimum scaling factor cles  φrK =  def and TOLr Er0 ErK Er0 and 1 pKd ˆ def φvK =  ˆ TOLv ˆ Ev0 ˆ EvK ˆ Ev0 1 pKd 1 pK (8.41) ˆ Eθ K , TOLθf (8.42) ˆ ˆ = min(φrK , φθ K , φvK , φθf K ), where, for the parti   1 , and for the fluid  def zθf K = ˆ Eθ K TOLθ def K 1 pK def zθ K =  , def φθ K =  TOLθ Eθ 0 Eθ K Eθ0 pKd 1... || f f (b) Etot,K = , def , Eθ K = def ˆ Eθf K = Np i=1 ||θiL+1,K − θiL+1,K−1 || Np i=1 ||θiL+1,K − θiL || L+1,K L+1,K−1 ||θf − θf || L+1,K L ||θf − θf || , ; ˆ ˆ w1 ErK + w2 Eθ K + w3 ErK + w4 Eθ K , w1 + w 2 + w 3 + w 4 w1 TOLr + w2 TOLθ + w3 TOLrf + w4 TOLθf ; w1 + w 2 + w 3 + w 4  1  TOLtot = (c) K  = def TOLtot Etot,0 Etot,K Etot,0 pKd 1 pK  ; (5) IF TOLERANCE MET (Etot,K ≤ 1) AND K < Kd... = TOL, where TOL is a tolerance and Kd is the desired number of iterations.49 If the error tolerance is not met in the desired number of iterations, the contraction constant η is too large Accordingly, we can solve for a new smaller step size, under the assumption that S is constant:50 2  ttol =  t TOL ||E L+1,0 || ||E L+1,K || ||E L+1,0 || 1 pKd 1 pK    (8.35) The assumption that S is constant... constant) and (2) one assumes that the error within an iteration behaves approximately according to (S( t)p )K ||E L+1,0 || = ||E L+1,K ||, K = 1, 2, , where ||E L+1,0 || is the initial norm of the iterative error and S is a function intrinsic to the system.48 Our goal is to meet an error tolerance in exactly a preset number of iterations To this end, we write this in the approximate form (S( ttol... Remark In order to determine the thermal state of the particles when impact-induced reactions are significant, we shall decompose the heat generation and heat transfer processes into two stages Stage I describes the extremely short time interval when impact occurs, δt t, and accounts for the effects of chemical reactions, which are relevant in certain applications, and energy release due to mechanical straining... TIME STEP: (c) SELECT MINIMUM t = min( t t= lim K t; , t) AND GO TO (0); (6) IF TOLERANCE NOT MET (Etot,K > TOL) AND K = Kd , THEN (a) CONSTRUCT NEW TIME STEP: t= K t; (b) RESTART AT TIME = t AND GO TO (0) Algorithm 8.1 ✐ ✐ ✐ ✐ ✐ ✐ ✐ 8.5 A numerical example 05 book 2007/5/15 page 95 ✐ 95 order, second order, etc.), divided by the sum of the number of equations using numerical time derivatives in the system... where N is the number of field equations where a numerical derivative was used and Oi is the order of the time differentiation (first order, second order, etc.) of the individual field equation i Remark An alternative and more severe way to measure the error is to define “violation ratios,” i.e., the measure of which field is relatively more in error, compared to def its corresponding tolerance, via ZK = max(zrK... will generate a significant amount Thus, when values of ξ are chosen such that ρCb 47 of heat Thereafter (Stage II, postimpact), it is assumed that a process of convection, for example, governed by Newton’s law of cooling and radiation according to a simple Stefan– L δH Boltzmann law, occurs Since δt t we assign θ = θ(t + δt) = θ(t) + mC and replace L θ with it in Equation (8.23) to obtain θ (t + t) =... and radiative effects, as discussed earlier As before, we consider an energy balance, governing the interconversions of mechanical, thermal, and chemical energy in a system, dictated by the first law of thermodynamics, d (K + S) = P + H, with the previous assumptions leading to dS = H For Stage I, the dt dt primary source of heat is the chemical reactions that occur upon impact due to the presence of. .. for each particle, is recast as θ L+1,K = mC tBas (θ L+1,K−1 )4 − θs4 θL − mC + hc as t mC + hc as t + mcv t|bD · ( v L+1,K f ω − v L+1,K )| mC + hc as t (8.23) hc as tθo + mC + hc as t and is added into the fixed-point scheme with the equations of momentum balance and the equations governing the fluid mechanics Concisely, the equations for the particle mechanics problem can be addressed in an abstract . t = min(t lim ,t )AND GO TO (0); (6) IF TOLERANCE NOT MET (E tot,K > TOL) AND K = K d , THEN (a) CONSTRUCT NEW TIME STEP: t =  K t; (b) RESTART AT TIME = t AND GO TO (0). Algorithm 8.1 05. Y Z Figure 7. 16. Top to bottom and left to right, slow impact of charged clouds. The clouds combine into a larger cloud. 05 book 2007/5/15 page 80 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 80 Chapter 7. Advanced particulate. Accordingly, we require the time rate of change of the sum of the kinetic energy (K) and stored energy (S) to be equal to the sum of the work rate (power, P) and the net heat supplied (H): d dt (K