CC BY
41
UDC: 530.182
V. A. Arkhipov, S. S. Bondarchuk, A. S. Zhukov
MATHEMATICAL MODELING OF THE ALUMINA POWDER SYNTHESIS BY PLASMA CHEMICAL PROCESSES
Aspects of integrated effects of plasmatron mode parameters and heat-transfer medium properties on formation and evolution of droplet medium of injected solution by means of diffusion, filtration and heat transfer are considered. Mathematical model as well as calculation results for dependence of particle size and its liquid core on heating rate and initial mass fraction of dissolved salt are presented.
Key words: ceramic powders, metal oxides, plasma technology, barrel-type reactor, mathematical model, thermochemical controlled synthesis, precursor, concentration of equilibrium saturation, boundary of evaporation front, drop suspension, intra-drop diffusion.
Development of efficient technologies of production of ceramic powders (especially metal oxides) is a topical problem associated with the constantly increasing number of applications of these materials over the last 10-15 years.
The use of nanopowders in production of ceramic composites makes it possible to create products with unique strength characteristics, metal-replacing construction materials, protective coatings, membranes and filters with specified pore size (including nanoscale). One of the most efficient technologies of controlled production of oxide nanopowders are plasma technologies based on high-frequency generation. In this method superfine powders are produced by thermochemical decomposition of liquid sprayed agents in high-temperature heat transfer medium. This process takes place in direct-flow vertical barrel-type reactor.
Development of high-efficiency industrial units for high-output production of wide range of nanocrystal materials with specified properties can be enhanced by using advanced mathematical modeling of processes in such reactor.
Development of such mathematical apparatus requires multi-factor analysis of calculation of parameters of multicomponent double-phase chemically reacting environment and in-depth study of heat and mass transfer phenomena inside the drop of a sprayed agent as well as its interaction with gas phase. It should be noted that physical and mathematical models used in recent works are seriously simplified [1-3], which significantly reduce predictability. The aim of this work is to develop physical and mathematical model of processes of thermochemical decomposition of liquid sprayed agents in high-temperature heat-transfer medium based on simultaneous consideration of double-phase turbulent flow inside the reactor and heat and mass transfer in a drop of precursor solution (water solution of aluminum nitrate salt (hydrate Al(NO3)3*9H2O); hydrate mass fraction (0.06^0.40).
The solution is injected to reactor chamber through a centrifugal atomizer which sprays the agent providing specified drop size. High-frequency plasmatron
with air or argon as a plasma-forming gas is used for generation of high-temperature heat transfer medium.
Considering the abovementioned points, in work [4] the authors offer the following mathematical model of processes of the facility for thermochemical synthesis of ceramic powder materials (including nanoscale) as well as approaches to its numerical implementation. In accordance with problem description low-temperature plasma (heat transfer medium) is transferred to the reactor chamber through the plasma-tron outlet, at the same time the sprayed solution of the salt is injected through the centrifugal atomizer. Inlet parameters are specified and considered constant during reactor operation.
Gas dynamics and thermolysis processes were modeled based on the following assumptions:
- The flow of gas and condensed particles was considered stationary and axially symmetric.
- Turbulent shear stresses were determined by k-s-model constraints [5], the impact of solution drops on turbulence structure was neglected.
- There is no heat transfer between the gas and channel borders.
Gas dynamic and thermodynamic characteristics of reacting gas suspension were determined considering two phases in flow field taking into account heat, mass and momentum exchange between them. The equations of motions for gas phase were represented in Euler’s coordinate form based on assumption that the impact of local discontinuities in the flow caused by the presence of condensed particles in gas is insignificant. The equations of motions for the drops of precursor solution with finite number of fractions are represented in the Lagrange form; equations for particle size change caused by evaporation are formulated as well as equations of diffusion and heat balance; In order to take into account interaction between the phases the right-hand parts of equations for gas phase included additional members associated with interaction of particles with gas medium. Double component mixture, consisting of heat transfer medium (argon, air) and solvent evaporation products (water vapour) were considered
as a gas phase. Initial conditions for the set of equations describing particle motion were determined using spraying characteristics of the centrifugal atomizer [6] for accepted differential law of mass distribution of particles by size g(D) in the form of Rosin-Rammler law.
The following physical pattern was considered for liquid particles. In the evolution process of the drop of aluminum nitrate solution such phenomena as evaporation, diffusion and heat transfer as well as chemical reactions of salt breakdown take place. As the temperature rises diffusion and evaporation affect salt concentration. After reaching a certain temperature the process of chemical breakdown of salt accompanied by heat generation takes place in various layers of the drop. As evaporation rate is higher than diffusion rate, the concentration at the surface level will be higher. If concentration becomes higher than the value of critical oversaturation Kn, the process of salt precipitation takes place. However, there may be two scenarios of salt precipitation followed by aluminum oxide formation, depending on the parameters at which the investigated process takes place. These scenarios are used for assessment of particle morphology [2].
I. If concentration of solution K(r) by volume re [0, R^t)] of the drop is greater or equal to concentration of equilibrium saturation Kn at the temperature of the drop with diameter RK(t), then precipitation takes place over the whole volume of the drop (volume precipitation, the upper branch of the process in Fig. 1).
II. If K(r) differs from Kn, then precipitation takes place only in the part of the drop, where K(r) > Kn (layer-by-layer precipitation, the upper branch of the process in Fig. 1).
surface zones increases. After the temperature of the solution reaches boiling point, a corresponding moving boundary of evaporation front RK(t,r) is formed. At the same time if concentration in any part reaches Kn -the concentration of equilibrium saturation - this is considered to be the condition of salt precipitation. After reaching the effective reaction temperature the precipitate forms a porous oxide structure.
Equations are written for elementary volume dV, bounded by a moving (in general) surface dS, with speed - u and external normal direction determined by a unit vector n (Fig. 2). Current droplet sizes and boiling front location are determined by corresponding radii Rs and RK(t).
Fig. 2. Calculation area
In accordance with assumptions made, heat transfer equation (change of temperature TK) in solution is as follows:
dT
dn
dS +
m dmK
+cKTK--------= 0,
K dt
(1)
where t is time; q, c, X are density, heat capacity and heat conductivity factor, index “K” indicates solution.
Boundary condition for equation (1) prior to salt precipitation is as follows:
a(T - TK ) + gz(t4 - TK ) = -X K
dT^
Fig. 1. Particle formation scenarios
Equations describing thermal condition regime of the drop were formulated on the basis of the following assumptions:
- Equation of transient heat conduction is written for the case of spherical symmetry.
- Only solvent (water) evaporates.
- Heat conductivity, density and heat capacity in the drop are considered to be constant over the volume and time.
Physics of the process dynamics is regarded as follows. The solvent evaporates through the surface of the drop at zero time as a result of heating; the diameter of the drop decreases and salt concentration in sub-
dr
1 dm
4nRK dt
(2)
where T is local temperature of heat-transfer medium, a is Stefan-Boltzmann constant, s is spectral transparency coefficient of “gas - drop surface” system, qK,Cp are heat of vaporization and isobaric heat capacity of steam.
Due to forced convection, heat transfer coefficient a is determined using Nusselt number Nu calculated for flow parameters [4].
After salt precipitation, boundary condition binds heat fluxes of solution and precipitate as well as their temperatures TK, TS with equilibrium vaporization temperature TV:
1 dTs
~A c--------
S dr
= -1,
dTr
dr
qK dm
4nRK dt '
T = T
K \r=RK S \r=rk
= T
1V-
(3)
Mass rate of vaporization dm/dt as well as linear speeds of movement of solution boundary, u0 and
“grid” intra-drop, u, are determined from boundary
conditions (2) and (3)
1 dm r
U0 — 2 ? u — U0 .
4nRKqk dt RK
Equation for variation of salt precipitate temperature TS (parameters marked with “S ”) is as follows:
d_
dt
jQscsTsdV -J
X S
dn
dS +
+(i -9)Hs dm.=cktk
dmK
~dt
(4)
where HS is enthalpy of salt-to-oxide phase transition.
Balance energy relation is used to solve equation (4) on the surface of salt precipitate.
a (T - TK) + os(T4-TK) = -X;
T
dr
CP{T - Tr) dm
nD
dt
The change of oxide fraction 9 in salt precipitate is obviously determined by the following relation
S-]<?QsdV = ( -<?)dm*
dt •
dt
and mass rate of salt transformation to oxide dmS/dt is determined by reaching “transition” temperature Tox:
( - Tox )QscsdV = HSdmS•
Mass transfer in the drop for intra-drop diffusion process description in accordance with Fick’s law is as follows:
d_
dt
jKdv-j
V S
0 ^ r ^ RK (t)
dK
D---------+ (u, n )K
dn v ’
dS+—=0,
Qk dt
by the value of non-dimensional parameter u0 RK/D (Fig. 5). Mass variation of precipitate over the full period of particle exposure to heat is shown in Fig. 6.
where D is dissolved salt diffusion coefficient.
Mass rate of precipitation dmK/dt, determined by saturation concentration Kn, is calculated using the following equation:
(K - Kn)QKdV = dmK.
Fig. 3 and Fig. 4 show dependences of particle size and its liquid core on heating rate (Nu value) for a particle in the flow (Fig. 3) and initial mass fraction of dissolved salt K0 (Fig. 4). It should be noted, that until the moment, when concentration reaches the level of critical oversaturation Kn, the curvature of its radial distribution in the drop is predominantly determined
Rc.hm
200
150
100
50
particle s ze
Kc II 0 0 O)
. liquic core size
2 2.2 2.4 2.6 2.8 Nu
Fig. 3. Particle size dependence on Nusselt number
Rc.hm
350
250
150
50
pa •tide si; e
N1 =2.5
liq jid core size
0.05
0.15
0.25
0.35 K0 Fig. 4. Particle size dependence on initial mass fraction of dissolved salt
Fig. 5. Distribution of mass fraction of salt in solution at the beginning of precipitation process
Fig. 6. Mass variation of precipitate over the period of particle exposure to heat
Thus a mathematical model and calculation results ing into account the early stage of evaporation of salt
are presented. The model makes it possible to assess solution, it’s transition to “boiling” regime as well as
the integral impact of parameters of plasmatron’s reactions, which take place when precipitation zones
modes, heat transfer medium properties on formation are formed inside the drops and their transformation
and evolution of drop environment of injected solution into porous oxide structure.
by means of diffusion, filtration and heat transfer, tak-
References
1. Larin V. K., Kondakov V. M., Malyi E. N. et al. Plasmochemical method of production of ultra-fine (nano-) powders of metal oxides and their prospective applications // Izvestiya vuzov. Non-ferrous metallurgy. 2003. No. 5. P. 59-64.
2. Gary L. Messing, Shi-Chang Zhang, and Gopal V. Jayanthi. Ceramic powder synthesis by spray pyrolisis // J. American Ceram. Soc. 1993. Vol. 76, No.11. P. 2707-2724.
3. Suris A. L. Plasmochemical processes and apparatus. M.: Chemistry, 1989. 304 p.
4. Zhukov A. S., Bondarchuk S. S. Heat and mass transfer in the process of production of metal oxides by means of plasma-chemical method // Izvestiya vuzov. Physics. 2010. Vol. 53, No.12/2. P. 96-101.
5. Spalding D. B. Mathematical models of turbulent flames: A review // Combust. Sci. Technol. 1976. Vol. 13. P. 3-25.
6. Arkhipov V. A., Berezikov A. P., Bondarchuk S. S. et al. Laser diagnostics of spray cone structure of centrifugal atomizer // Izvestiya vuzov. Aviation technology. 2009. No. 1. P. 75-77.
Arkhipov V. A.
Tomsk State University.
Pr. Lenina, 36, Tomsk, Russia, 634050.
E-mail: [email protected]
Bondarchuk S. S.
Tomsk State Pedagogical University.
Ul. Kievskaya, 60, Tomsk, Russia, 634061.
E-mail: [email protected]
Zhukov A. S.
Tomsk State University.
Pr. Lenina, 36, Tomsk, Russia, 634050.
E-mail: [email protected]
Received 14.03.2011.
В. А Архипов, С. С. Бондарчук, А. С. Жуков МОДЕЛИРОВАНИЕ ПРОЦЕССА ПЛАЗМОхИМИЧЕСКОГО СИНТЕЗА ПОРОШКА ОКСИДА АЛЮМИНИЯ
Рассмотрены процессы комплексного воздействия параметров режимов работы плазмотрона, свойств теплоносителя на формирование и эволюцию капельной среды инжектируемого раствора механизмами диффузии, фильтрации и теплопереноса. Представлена математическая модель и результаты вычислений зависимости размера частицы и ее жидкого ядра от темпа нагрева и начальной массовой доли растворенной соли.
Ключевые слова: керамические порошки, оксиды металлов, плазменная технология, прямоточный реактор, математическая модель, термохимический управляемый синтез, прекурсор, концентрация равновесного насыщения, граница фронта испарения, осаждение, внутрикапельная диффузия.
Архипов В. А., доктор физико-математических наук, зав. отделом.
Томский государственный университет.
Пр. Ленина, 36, Томск, Россия, 634050.
E-mail: [email protected]
Бондарчук С. С., доктор физико-математических наук, профессор. Томский государственный педагогический университет.
Ул. Киевская, 60, Томск, Россия, 634061.
E-mail: [email protected]
Жуков А. С., докторант, кандидат физико-математических наук, научный сотрудник. Томский государственный университет.
Пр. Ленина, 36, Томск, Россия, 634050.
E-mail: [email protected]
