MODELLING AND SIMULATION OF ULTRAFINE PARTICLE COAGULATION IN THE RESS PROCESS

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1 ISTP-16, 005, PRAGUE 16 TH INTERNATIONAL SYMPOSIUM ON TRANSPORT PHENOMENA MODELLING AND SIMULATION OF ULTRAFINE PARTICLE COAGULATION IN THE RESS PROCESS A. Ben Moussa 1,,*, H. Ksibi 1, and M. Baccar 1 IPEIS, B.P 805, 3018, Sfax, Tunisia CFDTP, ENIS, B.P W, 3038, Sfax, Tunisia * Ali.benMoussa@ipeis.rnu.tn, phone : , fax : Keywords: RESS, transport, spherical particle, coagulation Abstract Ultrafine particles play an integral role in a wide variety of physical and chemical processes. The rapid expansion of supercritical solution (RESS) process is one of promising applications for production of small particles with narrow size distribution. The RESS process involves an expansion of supercritical solution through a small nozzle to generate a rapid nucleation and then the formation of ultrafine particles. The formed particles are transported by a powerful jet developed in the expansion chamber. The size distribution of fine powders, obtained during the abrupt expansion, depends on the hydrodynamic conditions in the pre- and post-expansion unit, the nature of solute-solvent system, as well as on the nozzle geometry and diameter. In this wor, we perform a numerical simulation of both hydrodynamic of the jet and the particle transport in the expansion chamber, taing into account the Brownian coagulation and the particle growth. The largest nozzle orifice diameters involve a production of small particles, whereas, the highest expansion pressure is favourable to product great particles. We showed that these particles tae refuge on the jet boundaries where the coagulation is more pronounced. These particles are deposited on the flat plate as a ring. The goal is thus to predict optimal thermodynamic and geometrical conditions to have the narrowest distribution function of the deposited particles on the flat plate. 1. Introduction Varieties of organic substances, used in pharmaceutical and agro-alimentary industries, are often required as fine powders with controlled sizes. The RESS process (rapid expansion of supercritical solution) allows the micronization of thermally labile materials and the formation of small particles with a similar morphology and narrow size distributions. The advantages of the supercritical state allow a wide application in food and pharmaceutical industries where high purity of the product is recommended. Indeed, the production of fine powders with controlled particle sizes and morphology is of increasing importance particularly for medical applications. There are several difficulties to obtain a narrow PSD (particle size distribution) by using traditional mechanical methods as grinding. Several techniques of precipitation have been proposed, as recently reviewed by Jung and Perrut [1]. The Rapid Expansion of Supercritical Solution (RESS) is a precipitation process of powder based on supercritical fluid technique. Their potentialities have been demonstrated through several papers as Matson et al [], Tom et al [3], Ksibi et al [4], and Tür et al., 1999 [5]. In the RESS process, a component is dissolved in the supercritical fluid (dissolution stage), which expands rapidly through a narrow nozzle. Due to this important pressure drop, the fluid becomes gaseous, and the material is not soluble in the low-pressure fluid. An extremely fast phase change from the supercritical to the gas-lie state taes place 1

2 Ali Ben Moussa, Hatem Ksibi, Mounir Baccar during the expansion as a supersonic free jet. This leads to a high supersaturation and subsequently to the particle formation. Since the solvent is a dilute gas after the expansion, the RESS process generates a very pure final product. The sizes and the morphologies of the formed particles depend on the geometric shape of the nozzle and the initial conditions of temperature and pressure. Several numerical wors are attached to simulation of supercritical fluid expansion, but none among them evoed the transport phenomenon of particles by the jet in the expansion chamber. In recent studies, the hydrodynamic of supercritical expansion of carbon dioxide has been studied according to the nozzle geometry and the initial thermodynamics conditions [6-8]. In this wor, we suppose that the jet hydrodynamics of the pure carbon dioxide is not affected by the presence of these particles. Thus, we simulated the transport phenomenon of the spherical particles numerically by the jet while taing into account of the Brownian coagulation.. Numerical Method:.1 Hydrodynamic modelling: The modeling of the RESS process involves the studies of several phenomena. The more important of them are the flow in both the nozzle and the expansion chamber, the nucleation of the solute at the nozzle exit and the transport of particles by the jet. A numerical code is developed to study the dynamics of the supercritical carbon dioxide expansion [8]. This code solved the full time dependent the conservation laws of mass, inetic quantity and energy of the flow by using a specific equation of state. The Naviers Stoes equations governing the instationary, bidimensional, axisymmetric, viscous, and compressible flow may be written in conservative form as : U ( FU ( )) ( GU ( )) H(U) =0 t r z r (1) Here, z represents the longitudinal axis, r refers to the radial direction. U is the solution vector of the conservative terms and F and G are fluxes in both directions including the Euler part and the dissipating terms. ρ ρu U= ρw ρe ρu ρu +P+τrr F= ρuw+τ rz T ( ρe+pu+uτ ) rr +wτ rz +κ r () ρw ρuw+τrz g= ρw +P+τ zz T ( ρe+p) w+uτ rz +wτ zz +κ z ρu ρu H= ρuw T ( ρe+pu+uτ ) rr+wτ rz +κ r Where the dissipation tensor τ is expressed by v u τ rr = μ - 3 r z u v τ zz = μ - (3) 3 z r u v τ rz = μ + r z Here ρ is the density, u and w are the components of the velocity in the radial and axial directions, respectively. E is the total energy. The coefficients of the viscosity and the thermal conductivity are denoted µ and κ, respectively. The Altunin and Gadetsii equation of state is chosen to describe the thermodynamic

3 MODELLING AND SIMULATION OF ULTRAFINE PARTICLE COAGULATION IN THE RESS PROCESS properties of the pure carbon dioxide [9]. Both the liquid and the gas states are represented accurately with this correlation. Except the very close region to the critical point, the supercritical domain can also be included up to 1000 bars. This correlation is written as follows P 9 6 j i Z= =1+ ρr bij( τ -1) ( ρr -1) (4) ρrt i=0 j=0 Where ρ r =ρ/ρ c and τ=t c /T. Z is the compressibility factor and R is the perfect gas constant. The c letter indicates the critical point properties. The b ij coefficients are 70 constants, which are tabulated in [9]. This set of equations is solved by using the Total Variation Diminishing algorithm. Harten and al. [10] introduced this numerical technique, which is suitable for transonic and supersonic flows. Since we loo for a steady solution, we want to use time step as large as possible. In this way, the convergence is accelerated by implicit linearisation using the altering direction implicit (ADI) formulation. To capture shoc waves and discontinuities, we used an approximate Riemann solver developed by Montagné et al. who adapted the Roe averaging to a nonconvex Riemann problem for a real gas [11]. Numerical simulations of supercritical carbon dioxide flow in the capillary nozzle and the expansion chamber are developed. We use the hydrodynamic results for studying the particles transport.. Coagulation modeling: The transport of the ultra fine particles is governed by the aerosol general dynamic equation (GDE). The evolution of the particle field is obtained by utilizing a sectional model to approximate the transport equation. The transport equation is written in discrete form as a population balance on each cluster or particle size. N r +div(j )=σ (5) t Where N is the particle density. The particle flux is due to the transport by the fluid and to the Brownian diffusion. It is given by r r j =Nu-D uuuuur grad N ( ) BK (6) Where u r is the fluid velocity and DB is the K Brownian diffusivity is given by CCu D B =KT B (7) 3πµd P where K B is the Boltzmann constant, C Cu is the Cuningham correction factor and d P is the particle diameter. The Cunningham correction factor due to Philips (1975) is given by 3 5+4K n+6k n +18Kn C = (8) 5-K +8+π K ( ) Cun n n Where K n is the Knudsen number. Aerosol particles suspended in a fluid may come into contact because of their Brownian motion. The source term σ represents the effects of particle-particle interactions. It is given by m m m 1 σ = β χ NN N β N (9) i,j i,j, i j i, i i = 1 j = 1 i = 1 The complete form of β i,m due to Fuchs [1] is given in table 1: ( i j) ( ) d+d 8 D+D i j β ij =π( D+D i j)( d+d i j) + d+d +g c d+d i 8T i i j ij ij i j c= πm ( ) 1 ( ) 1 g = g +g ij i j c = c +c ij i j ( ) g= i ( d+l i i) - ( d i) +li -di 3dl i i Table 1: Fuchs form of the Brownian coagulation coefficient χ i,j,m is given by v -v+v +1 i j v -v +1 v+v -v i j if v v+v <v i j +1 χ = if v v+v <v i, j, v -v -1 i j -1 0 otherwise -1 (10) 3

4 Ali Ben Moussa, Hatem Ksibi, Mounir Baccar The governing transport equations involving the hydrodynamic field are solved using the finite volume method. The resolution of the equation system is made line by line using the Gauss s method, nowing that the first and the last equation are deduced from the mathematical formulation of boundary conditions. The initial conditions are taen as given in table. Pressure (bar) Temperature(K) Nozzle Inlet conditions Nozzle Outlet conditions Table : Initial hydrodynamic conditions The recrystallization chamber diameter and length are cm and 4 cm, respectively. The nozzle has a convergent shape for which the exit radius varies between 100 µms and 160 µms. 3 Results Figure 1 : velocity field The particle transport code is executed at the same thermodynamic and geometrical conditions quoted previously. The particles distribution is chosen as lognormal function with a geometric radius equal to 10 nm. Diameters of sectional particles are varied between 13 nm and 40 nm. 0,01 ms 0,03 ms 0,06 ms 0, ms Figure : temporal evolution of field of diameter s particles 9.3 nm 4

5 MODELLING AND SIMULATION OF ULTRAFINE PARTICLE COAGULATION IN THE RESS PROCESS 3.1 Temporal evolution and structure of fields of particle distributions The evolutions of density fields of sectional particles in the expansion chamber are conditioned by the hydrodynamic of the jet. We represent in figure 1 the velocity field of the fluid obtained by using diameter exit of nozzle equal to 10 µms. the thermodynamics conditions in the expansion chamber are fixed to: P outlet =1 bar ; T outlet =313 K The temporal evolution of the density field N shows that particles follow practically the lines of velocity field (figure ). The distributions fields of the sectional particles reach their stationary states after an approximate period equal to 0, ms. The structure of the velocity field have a repercussion on the stationary distributions of the different classes of particles. In the zone of silence which characterizes the jet, the numerical density of particles is negligible. Indeed, the speed out flow reaches maximal value in this region, from where particles are transported quicly outside. Therefore, particles get round the zone of silence that ends by the Mach dis, to concentrate on boundary of the jet. The high concentration of particles and the relatively important residence time favor the coagulation phenomenon. In the jet boundary, the N (m -3 ) 8 x 1015 z=mm 7 z=5mm z=10mm 6 z=0mm z=40mm Radius (m) x 10-3 Figure 4: Radial distribution of diameter s particles nm at different altitude - a - 10 x 1017 z=5mm z=10mm 8 z=0mm z=40mm N (m -3 ) b Radius (m) x 10-3 Figure 3: Radial distribution of diameter s particles.49 nm at different altitude Figure 5: radial density of diameter s particles at the flat plate: a -.49 nm b nm coagulation favours formation of large particles. These inds of particles are practically absent upstream of the Mach dis. While coming closer of the flat plate the large particle densities 5

6 Ali Ben Moussa, Hatem Ksibi, Mounir Baccar increase whereas fine particle densities decrease because of the coagulation (figure 3-4). As moving away of the center of the flat plate the particle sizes become more important (figure 5). 3. Effect of the nozzle geometry The hydrodynamic study shows that the structure of the jet developing upon the expansion chamber depends on the nozzle geometry. The Mach dis diameter and the jet size are much large than the nozzle exit diameter increases. We implemented the particle transport code by using hydrodynamic fields obtained with the same initial conditions of temperature and pressure. The exit section radius of the nozzle varies between 100 µms and 150 µms. We notice that for relatively small nozzle exit radius, the particles get round the silence zone to concentrate on the jet boundaries - a - and so favouring the coagulation phenomenon. The large particles, that result this phenomenon, form on the flat plate as a ring with small radius. The fraction of particles getting round the zone of silence and concentrated on jet boundaries decreases when the nozzle exit diameter increases. The spread of particles in the jet disfavours their coagulation and therefore the N (m -3 ) x 1016 Rs=100µm Rs=110µm Rs=10µm 1.5 Rs=130µm Rs=150µm radius (m) x 10-3 Figure 7: Radial distribution of diameter s particles nm for different nozzle exit radius density of large particles decreases. We represent in figure 7 the radial variation of the density of diameter s particles 3.17 nm for different nozzle exit radius at the flat plate. The figure 8 shows a small regression of the medium volume of particles at the flat plate according to the nozzle exit radius. Therefore, a great nozzle exit diameter generates finer particles. 7,5 7,0 medium radius (nm) 7,15 7,10 7,05 7,00 - b - Figure 6: Radial distribution of diameter s particles 3.17 nm for two nozzle exit radius: a- 100 µm b- 150 µm nozzle exit radius (µm) Figure 8 : particle medium radius for different nozzle exit radius. 6

7 MODELLING AND SIMULATION OF ULTRAFINE PARTICLE COAGULATION IN THE RESS PROCESS 3. Effect of the pressure We study the particle transport phenomenon by the jet and then we control the variations of particle distributions according to the recrystallization pressure. Here, the recrystallization pressure varies between 1 and 40 bars. Relating to the expansion hydrodynamics, the jet becomes very thin and its velocity decreases when the recrystallization pressure increases (figure 9). This effect favours the coagulation phenomenon and therefore the formation of larger particles. There is no Mach dis when the recrystallization pressure overcomes 10 bars. Therefore, at high pressures of recrystallization, large particles are uniformly distributed in all the jet. The variation of the medium radius of particles on the flat plate (figure 10) according to the recrystallization pressure shows a logarithmic tendency. P inlet = bar P inlet = 5 bar P inlet = 10 bar Figure 9: celerity field in the expansion chamber and the radial distribution of diameter s particles on the flat plate 7

8 Ali Ben Moussa, Hatem Ksibi, Mounir Baccar medium radius (nm) 8,4 8,0 7,6 7, Conclusion crystallisation pressure (bar) Figure 10 : particle medium radius for different recrystallisation pressure. In this wor, we show by numerical simulations that the particle coagulation during their transport by the jet depends on the initial thermodynamic parameters and the nozzle exit diameter. We notice that the deposition of fine particles on the flat plate is favoured with a low expansion pressure and a bigger nozzle exit diameter. The obtained results verified the experimental investigations given in several wors. Whereas, experimental results showed a more pronounced effects of both the nozzle diameter and the recrystallisation pressure. Indeed, improving numerical model, should tae into account the particle growth and the turbulence phenomena. supercritical fluid, European Journal of Mechanics /B Fluids 15 (4) (1996) [7] Ben Moussa A, Ksibi H and Baccar M. Capillary nozzles in th RESS process: Hydrodynamic modeling, Proc. 6 th international symposium on supercritical fluids. Versailles, France, Vol. 3, PP 1719 (003). [8] Ben Moussa A, Ksibi H, Tenaud C and Baccar M. Paramètres géométriques de contrôle de la détente d un fluide supercritique. International journal of thermal science 005, in press. [9] Altunin V V and Gadetsii O G. Method of formulating the fundamental equations of state for pure substances from varied experimental data, Teplofizia Vysoih Temperatur 9 (3) (1971) [10] Harten A and Yee H C. Implicit TVD schemes for hyperbolic conservation lows in curvilinear coordinates, AIAA Journal 5 (1986) [11] Montagné J L, Yee H C and Vinour M. Comparative study of high-resolution shoccapturing schemes for a real gas, 7th GAMM Conference on Numerical Methods in Fluids Mechanics, Louvain-la-neuve, Belgium, septembre [1] Fuchs N A. Mechanics of aerosols, Pergamon, New Yor (1964) References [1] Jung J and Perrut M. Particle design using supercritical fluids: literature and patent survey. The Journal of supercritical fluids, 0 (001) 179. [] Matson D W, Fulton J L. Peterson R C. and Smith R D. Ind. Eng. Chem. Res., 6 (1987) 98. [3] Tom J and Debenedetti P G. Particle Formation with Supercritical Fluids - A Review. J. Aerosol Sci., (1991) 555. [4] Ksibi H. Subra P. and Garrabos Y. Formation of fine powders of caffeine by RESS. Advanced Powder Technology J., 6-1 (1995) 5. [5] Tür M. Formation of small organic particles by RESS: experimental and theoretical investigations. The Journal of supercritical fluids, 15-1 (1999) 79. [6] Ksibi H, Tenaud C, Subra P and Garrabos Y. Numerical simulation of rapid expansion of 8