nonlinear partial differential equations, and these equations must be solved .... two-phase flows are: (1) each phase has a volume fraction (or mass fraction, ... To avoid divergence, the denominator of this equation can be timed by. 1 1. P.
Chapter 8
The Solution Procedure for Modeling Multiphase Turbulent Reacting Flows In the preceding chapters, the fundamentals, basic equations, closure models, and their validation are discussed. In general, the equations are simultaneous nonlinear partial differential equations, and these equations must be solved numerically, and discretized using finite element or finite difference methods. For computational fluid dynamics, the finite difference method is predominant. For single-phase flows the numerical method is described in many books, for example in Reference [1]. For simulating fluid flows, heat transfer, and combustion, the most widely used is the so-called SIMPLE (semiimplicit pressure-linked) algorithm. In this chapter, only a brief introduction is given to the specific problems in some numerical procedures for simulating multiphase turbulent reacting flows.
8.1 THE PSIC ALGORITHM FOR EULERIANLAGRANGIAN MODELS The numerical method for EulerianLagrangian simulation of dispersed multiphase flows is the PSIC (Particle Source in Cell) method, first proposed by Crowe et al. [2]. It is now widely used and included in commercial software, such as ANSYS-FLUENT, STAR-CD, etc. In the PSIC method, for the gas flow field, the usual SIMPLE algorithm (SIMPLE, SIMPLER, SIMPLEC) may be adopted, i.e., the p 2 v correction, line-by-line or planeby plane TDMA, and under-relaxation iterations. The particle source term in the computational cell for the gas continuity can be written as X X X ΔVUnk Δmk 2ΔVUS 5 2 ΔV Δmk nk m_ k 5 2 ΔV nk 52 Δt Δt k k k XX 5 Nk;j ðmk;i 2 mk;e Þ k;c
j;c
where the first term on the far right-hand side is the summation of the mass of all particle-size trajectories crossing this cell, and the second term is the Theory and Modeling of Dispersed Multiphase Turbulent Reacting Flows. DOI: https://doi.org/10.1016/B978-0-12-813465-8.00008-9 © 2018 Elsevier Inc. All rights reserved.
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Theory and Modeling of Dispersed Multiphase Turbulent Reacting Flows
summation of the mass of the same-size particles with different initial positions and directions of trajectories. Nk,j is the total number flux of k-group particles, which keeps constant along the j-th trajectory. Similarly, the particle source terms in the momentum and energy equations are XX ΔVUSpvi 5 Nk;j ðmk;e vik;e 2 mk;i vik;i Þ ΔVUSph 5
k;c
j;c
k;c
j;c
XX
Nk;j ½cp Tðmk;e 2 mk;i Þ 2 Qk
where the subscripts i and e express the values at the entrances and exits of trajectories in the gas computational cell, respectively, and Qk is the convective heat transfer between the gas and particles. For the particle boundary conditions, the inlet particle concentration distribution, particle size, and velocity distribution, particle initial position and particle-wall collision conditions must be given. The gas pressure correction equation must be based on the gas continuity equation with particle source terms. To predict the particle trajectories and particle history along the trajectories, it is necessary to select an appropriate integral method for determining the crossing positions of trajectories with the walls of gas cells, including determining which wall to cross with, when the particle trajectory leaves a gas cell, and promptly stopping the particle trajectory computation in that cell. The particle trajectories can be obtained by solving the particle momentum equation using the RungKutta method: ðt ðt ðt xk 5 uk dt; rk 5 vk dt; θk 5 ðwk =rk Þdt 0
0
0
The problem is how to choose Δt. At first we may take Δxi re 2 ri Δtð1Þ 5 min ; uki vki and use the RungKutta method to calculate the particle position xk and rk in this time interval. If the point (xk, rk) cannot fall into the anticipated region in Fig. 8.1, then we should take Δxi re 2 ri ð2Þ ; Δt 5 min ðuki 2 uke Þ=2 ðvki 2 vke Þ=2 and repeat the trajectory computation, until the point (xk, rk) falls into the anticipated region in Fig. 8.1. Crowe et al. assume a constant gas velocity and a constant particle relaxation time in the gas cell, namely, vi 5 const;
τ rk 5 const
hence they give an analytical solution as uke 5 u 2 ðu 2 uki Þexpð2 Δt=τ rk Þ vke 5 v 2 ðv 2 vki Þexpð2 Δt=τ rk Þ
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FIGURE 8.1 Method 1 for computing particle trajectories.
FIGURE 8.2 Method 2 for computing particle trajectories.
Finally, the particle trajectory is obtained by linear interpolation as xke 5 xki 1 0:5ðuki 1 uke ÞΔt rke 5 rki 1 0:5ðvki 1 vke ÞΔt In fact, the particle mass, size, drag, and relative velocity change along the trajectories, therefore the present author suggests using an analytical solution of trajectories of particles with mass change [3] Δx 5 x 2 xi 5 t1 ½uf1 ðdÞ 2 ðu 2 ui Þf2 ðdÞ Δr 5 r 2 ri 5 t1 ½vf1 ðdÞ 2 ðv 2 vi Þf2 ðdÞ where (xi,ri) is the initial position of trajectories entering the gas call (Fig. 8.2)
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Theory and Modeling of Dispersed Multiphase Turbulent Reacting Flows
f1 ðdÞ 5 f2 ðdÞ 5
ð1
d dd 0:5ðm11Þ d 11Π
ð1
ðm11Þ
d dd 0:5ðm11Þ d 11Π
t1 5 ρs dki2 =½4Dρlnð1 1 BÞ 5 ρs dki2 cp =½4λlnð1 1 BÞ d 5 dk =dki ;
m 5 3:24=B
Π 5 0:25ðReki Þ1=2 For the given particle inlet position, velocity and size in a certain gas cell ð1Þ
xi ; uki :vki ; dki , at first take the value d near unity, then calculate f1 ðdÞ; f2 ðdÞ, and compare the predicted Δx(1), Δr(1) with the sizes of the gas cell Δxg, Δrg. If ΔX(1) is smaller than Δxg and Δr(1) is smaller than Δrg, may take ð2Þ
ð1Þ
d to be smaller than d , and calculate ΔX(2), Δx(2), until ΔX(n) at first reaches ΔXg or Δx(n) at first reaches Δxg. Using such a method, the position of trajectories in the gas cell is determined. The flow chart of the PSIC method is shown in Fig. 8.3.
8.2 THE LEAGAP ALGORITHM FOR EEL MODELING In simulating multiphase turbulent reacting flows, for example, pulverizedcoal combustion, a two-fluid-trajectory (gas Eulerian and particle Start
Solve single-phase velocity, temperaturespecies concentration and turbulence properties without particles
Calculate particle trajectories and particle velocity, mass and temperature change along their trajectories
Calculate particle source terms in the gas-phase equations
Add the particle source terms to the gas-phase equations and solve them again
Repeat the third step
No
Converged ? Yes Stop
FIGURE 8.3 The flow chart of the PSIC method.
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Start Solve isothermal gas flow field Solve gas combustion field Solve Eulerian particle flow field Solve Lagrangian particle equations to obtain particle trajectories, particle mass and temperature change and particle source terms in gas-phase equations Gas-phase equations convergent?
No
Yes Solve Eulerian particle equations again with particle mass change No
Particle equations converged? Yes Two-phase equations converged ? Yes Stop
FIGURE 8.4 The LEAGAP flow chart.
EulerianLagrangian, EEL) model was proposed by the author [4]; its algorithm is called LEAGAP (LagrangianEulerianEulerian Algorithm of Gas-Particle Flows). In this algorithm, in addition to the iteration inside the gas phase, there are coupling and iterations among the gas Eulerian, particle Eulerian, and particle Lagrangian computations. The LEAGAP flow chart is shown in Fig. 8.4.
8.3 THE PERT ALGORITHM FOR EULERIANEULERIAN MODELING A PERT algorithm (Pure Eulerian Algorithm of Reacting Two-phase Flows) was proposed by the author [4] for pure two-fluid or EulerianEulerian modeling of multiphase turbulent reacting flows. In this algorithm, in addition to the iteration inside the gas phase, there are coupling and iterations between the Eulerian gas and Eulerian particle computations. The flow chart of the PERT algorithm is shown in Fig. 8.5.
8.4 THE GENMIX-2P AND IPSA ALGORITHMS FOR EULERIANEULERIAN MODELING Spalding extended the SIMPLE algorithm to two-fluid modeling of twophase flows [5]. At first the diffusion terms in each equation of each phase
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Theory and Modeling of Dispersed Multiphase Turbulent Reacting Flows
Start Solve isothermal gas flow field Solve gas combustion field Solve Eulerian particle velocity, temperature and concentration without mass change Solve particle mass change and obtain particle source terms in gas-phase equations and solve gas-phase equations again Gas-phase equations convergent?
No
Yes Solve Eulerian particle equations again with particle mass change No
Particle equations converged? Yes Stop
FIGURE 8.5 The flow chart of the PERT algorithm.
are modeled in the form of volume fraction gradient. So, the conservation equations of all the dependent variables of each phase are expressed in the following form: @ @ @ @ϕk ðαk ρk ϕk Þ 1 ðαk ρk vkj ϕk Þ 5 Γ ϕk 1 αk Sϕk @t @xj @xj @xj where αk is the volume fraction of phase k, φ is a property of phase k, Γ φk is the transport coefficient inside phase k, and Sφk is the source term, including the source term inside phase k and the source term of interaction between phase k and other phases. Hence the original numerical methods, like GENMIX and SIMPLE algorithms, can be extended to two-phase flows, called GENMIX-2P and IPSA algorithms. The specific features of solving two-phase flows are: (1) each phase has a volume fraction (or mass fraction, apparent density); (2) need to give the turbulence properties of each phase and the interactions between two phases, like drag force and heat transfer between two phases; (3) the partial pressures of two phases and their pressure correction. The basic steps of the GENMIX-2P algorithm for twodimensional parabolic flows (jets, boundary layers, pipe flows) are: 1. Introduce a stream function ψ, and transform the xy coordinate system into an xω coordinate system, where ω is a relative stream function; 2. Presume or guess the two-phase volume fractions; 3. Solve the gas momentum equation in the x direction and energy equation using the TDMA method;
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259
4. Find the position in the y-coordinate of the boundary grid nodes; 5. Solve gas continuity equation to find the velocity v; 6. Solve the gas momentum equation in the y direction to find the pressure distribution in the y direction; 7. Solve particle-phase equations to find the particle velocity, temperature, and volume fraction; 8. Compare the predicted particle volume fraction with the initially presumed value and make corrections, and repeat the computation until the correction is smaller than a certain value. For the IPSA algorithm in modeling of two-dimensional elliptic twophase flows, at first the two-phase differential equations are discretized into finite difference equations X X ϕ ϕ ϕk 5 A i ϕi 1 b = Ai Let g 5 ρva 5 density 3 velocity 3 cross-sectional surface be the mass flux across the wall of a cell, and the subscripts and e denote the inlet and outlet values, respectively, then the continuity equation of the first phase is h X i. X ϕ1 g1 1 ΔVS g1;e ϕ1 5 i
To avoid divergence, the denominator of this equation can be timed by P P ðϕ1 g1 Þi 1 ΔVS ðϕ2 g2 Þi 1 ΔVS P P 2 ϕ1 1 2 ϕ2 11 g1;e g2;e In general this term is unity, hence we have P P g2;e ½ ðϕ1 g1 Þi 1 ΔVS P P P P P ϕ1 5 P g1;e U ðϕ2 g2 Þi 1 ð g2;e 2 g1;e ÞΔVS g2;e U ðϕ1 g1 Þi 1 P where expresses the summation over the walls of the cell. This expression is called the equation of volume fraction. The pressure correction accounts for two phases. If @u1 @u2 @v1 @v2 ; ; ; @p @p @p @p is known, the error of u1 and u2 is e1, and the error of v1 and v2 is e2, then @e1 @e2 ; , and the pressure correction is we can find @p @p . @e2 . e1 e2 0 @e1 p 1 ρ1 1 ρ 52 @p @p 2 ρ1 ρ2 where ρ1 and ρ2 are the material density of two phases, respectively. The flow chart of the IPSA algorithm is shown in Fig. 8.6.
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Theory and Modeling of Dispersed Multiphase Turbulent Reacting Flows
Presume initial two-phase velocity field
Solve two-phase energy equations to find the source terms in the volume fraction equation Solve the volume fraction equation to find ϕ1 and ϕ2 = 1– ϕ2 Solve the two-phase mixture momentum equation to find the pressure p Solve momentum equations for each phase
Establish pressure correction equations to find the corrected pressure, velocity and volume fraction
Solve other equations No
Converged?
Yes
Stop FIGURE 8.6 The flow chart of the IPSA algorithm.
The IPSA algorithm has been applied to simulate two-phase flows with large variation of two-phase volume fractions. However, due to lack of the dispersed-phase turbulence models it is not widely used in practical engineering applications.
REFERENCES [1] S.V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere, New York, 1980. [2] C.T. Crowe, M.P. Sharma, D.E. Stock, The particle-source-in-cell (PSIC) method for gas-droplet flows., J. Fluid Eng. 99 (1977) 325332. [3] L.X. Zhou, Combustion Theory and Reacting Fluid Dynamics (in Chinese)., Science Press, Beijing, 1986. [4] L.X. Zhou, Dynamics of Multiphase Turbulent Reacting Fluid Flows (in Chinese)., Defense Industry Press, Beijing, 2002. [5] D.B. Spalding, Numerical computation of multiphase fluid flow and heat transfer., in: C. Taylor (Ed.), Recent Advances in Numerical Mechanics, Pinerage Press, New Jersey, 1980.
