Whitby and McMurry 1997). Several works exist in the literature where modal methods are embedded in CFD codes for the simulation of multidimensional ...
Aerosol Science and Technology, accepted, 2007
SECTIONAL MODELING OF AEROSOL DYNAMICS IN MULTIDIMENSIONAL FLOWS
Shortened running title: Sectional Modeling in Multi-Dimensional Flows
By
D. MITRAKOS1,2, E. HINIS2, C. HOUSIADAS1
1
“Demokritos” National Centre for Scientific Research, Institute of Nuclear Technology and Radiation Protection, Athens, 15310, Greece
2
National Technical University of Athens, Faculty of Mechanical Engineering Athens, 15780, Greece
1
Abstract The integration of computational fluid dynamics (CFD) with computer modeling of aerosol dynamics is needed in several practical applications. The use of a sectional size distribution is desirable because it offers generality and flexibility in describing the evolution of the aerosol. However, in the presence of condensational growth the sectional method is computationally expensive in multidimensional flows, because a large number of size sections is required to cope with numerical diffusion and achieve accuracy in the delicate coupling between the competing processes of nucleation and condensation. The present work proposes a methodology that enables the implementation of the sectional method in Eulerian multidimensional CFD calculations. For the solution of condensational growth a number conservative numerical scheme is proposed. The scheme is based on a combination of moving and fixed particle size grids and a remapping process for the cumulative size distribution, carried out with cubic spline interpolation. The coupling of the aerosol dynamics with the multidimensional CFD calculations is performed with an operator splitting technique, permitting to deal efficiently with the largely different time scales of the aerosol dynamics and transport processes. The developed methodology is validated against available analytical solutions of the general dynamic equation. The appropriateness of the methodology is evaluated by reproducing the numerically demanding case of nucleation-condensation in an experimental aerosol reactor. The method is found free of numerical diffusion and robust. Good accuracy is obtained with a modest number of size sections, whereas the computational time on a common personal computer remained always reasonable.
2
1. Introduction Computer modeling of aerosol dynamics is of importance in a wide spectrum of current applications, ranging form atmospheric chemistry and climate change, to a variety of technological fields, like nuclear reactor safety or production of nano-sized materials. In many circumstances the solution of the General Dynamics Equation (GDE) involves multiple spatial dimensions and complex flows and so Computational Fluid Dynamics (CFD) need to be combined with aerosol dynamics to accurately predict the behaviour of the aerosol flow. CFD-based aerosol simulations have gained much attention in experimental (Pyykönen and Jokiniemi 2000; Wilck and Stratmann 1997) or industrial (Mühlenweg et al. 2002; Schild et al. 1999) aerosol reactors, to design and control the system and improve the quality of the manufactured materials, usually nanoparticles. However, despite the increase of the computational capabilities, computational fluidaerosol dynamics still remains a challenging task, especially when simultaneous nucleation, condensational growth and coagulation take place within the flow. In such cases, very dense spatial and/or temporal resolutions are required to describe appropriately the coupling between processes characterized by largely different time scales and the strong nonlinearities that are introduced (Pyykönen et al. 2002; Pyykönen and Jokiniemi 2000). In such calculations the use of an efficient and accurate numerical representation of the particle size distribution is a key issue. Methods based on moments are widely used in aerosol dynamic simulations. The main advantage of these methods is their low computational cost, because a small number of additional equations, namely for the moments of the size distribution, need to be solved. The basic problem of the moment methods is that they require some kind of
3
closure. Usually an assumption is made on the functional form of the particle size distribution in order to achieve the closure of the transport equations (Modal methods, Whitby and McMurry 1997). Several works exist in the literature where modal methods are embedded in CFD codes for the simulation of multidimensional aerosol flows (Brown et al. 2006; Schwade and Roth 2003; Stratmann and Whitby 1989; Wilck and Stratmann 1997). Obviously, modal methods do not allow for arbitrary evolution of the size distribution because the functional form of the size distribution is specified beforehand and remains fixed throughout the whole simulation. Moreover, the use of a constant standard deviation may introduce inaccuracies in the calculation of coagulation and condensational growth (Zhang et al. 1999). McGraw (1997) overcame the problem of closure by proposing the quadrature-method of moments (QMOM). Because of its potential the QMOM has been employed in several works, demonstrating its applicability to more complex cases (e.g. Alopaeous et al. 2006, McGraw and Wright 2003). Also, QMOM has been used in combination with CFD (Marchisio et al. 2003, Rosner and Pyykönen 2002). The main advantage of the method is that no assumptions for the shape of the distribution, or the form of the growth law, are required to satisfy the closure of the moment equations. However, QMOM becomes quite challenging numerically in multivariate cases (Rosner and Pyykönen 2002). A general drawback of the QMOM is the non-uniqueness problem that arises in the reconstruction of the size distribution from its moments. The sectional approach (Gelbard et al. 1980) is conceptually straightforward and offers more degrees of freedom, ensuring therefore greater generality. On the other hand, the drawback is that the sectional method may be computationally very demanding
4
because its accuracy is directly related with the number of discretization sections (size bins) that are used. A large number of sections is needed to reduce the numerical diffusion inherent in the numerical solution of condensational growth, making the sectional approach expensive to implement in CFD calculations (Mühlenweg et al. 2002; Pyykönen and Jokiniemi 2000). A moving sectional method, where particle size bins are allowed to move according to the growth law (Kim and Seinfeld 1990) can combat numerical diffusion, but introduces serious complications in coupling with the other aerosol processes, especially with transport. This limits the applicability of the moving sectional approach, which is usually implemented in zero-dimensional aerosol calculations along Lagrangian trajectories (Spicer et al. 2002; Tsantilis et al. 2002). Fixed particle size grid is a more suitable approach for multidimensional, elliptic aerosol flows and has been combined with CFD in multidimensional simulations in a number of works. However, it has been used either under an approximate Lagrangian transport frame (Johannessen et al. 2000; Pyykönen and Jokiniemi 2000) or in situations where growth is only due to coagulation/agglomeration (Jeong and Choi 2003; Kommu et al. 2004I, II; Lu et al. 1999; Mühlenweg et al. 2002; Park et al. 1999; Schild et al. 1999). When growth is also due to vapor condensation then the coupling with nucleation makes the calculation very demanding in terms of computational cost. Attempts to implement the sectional method in a fully Eulerian, multidimensional computational framework on the basis of standard CFD techniques revealed serious deficiencies (Pyykönen and Jokiniemi 2000; Pyykönen et al. 2002). A methodology for implementing efficiently the sectional approach in CFD-based aerosol calculations is therefore exceedingly needed in cases
5
characterized by strong coupling between nucleation and condensational growth. The aim of our work is to propose such a methodology. The reduction of the numerical diffusion is the greatest concern when implementing a sectional method within a multidimensional CFD calculation. Several methods have been proposed in the literature to combat numerical diffusion, while keeping reduced the number of the sections needed. “Three point” discretization schemes (Hounslow et al. 1998; Park and Rogak 2004) reduce numerical diffusion but introduce oscillations and negative values of the size distribution that can lead to significant errors, especially in cases where strong nucleation takes place. Lurmann et al. (1997) proposes to solve the growth equation using a moving grid and then re-map the result on the initial fixed grid, where all the other processes are treated. Re-mapping is done by using a cubic spline interpolation scheme, which can introduce oscillations and negative values when the size distribution is steep. Forcing these negative values to zero can result to significant errors in the conservation of the particle number and volume. Jacobson (1997) proposed the so-called moving center method in which the sections remain fixed through the simulation but the characteristic particle size of each section is allowed to vary. The particles grow according to the growth law and if the new size is out of the section all particles are transferred into a next section. The size characterizing each section is updated by averaging the transferred and preexisting particle volumes. Because particles are not spitted in adjacent size bins numerical diffusion is combated efficiently. Zhang et al. (1999) inter-compared several methods and qualified the moving center method as the most appropriate for calculating the condensational growth of atmospheric aerosol in air quality models. However, removing all particles from one size bin can result to
6
unrealistic empty bins and extra dents in the size distribution, as observed by Korhonen at al. (2004) in nucleation simulations. Furthermore, because several particle sizes are mixed through the transport processes, an averaging procedure must be performed repeatedly to compute the characteristic particle size in every section, which introduces a systematic bias in the results (Zhang at al. 1999). Yamamoto (2004) derived the condensational growth equation in terms of the cumulative size distribution function, which was solved with the help of semi-Lagrangian schemes. He showed that the use of the cumulative size distribution function allows the conservation of the number of particles and prevents overshoots, even in cases with sharp changes in the size distributions. The objective of the present work is to develop a methodology permitting to incorporate efficiently the sectional approach into a multidimensional CFD calculation. First, an efficient numerical solution of condensational growth is implemented by combining the advantages of the schemes of Lurmann et al. (1997) and Yamamoto (2004): a moving particle size grid is used and the results obtained are re-mapped on the initial fixed grid. The interpolation during the re-mapping step is not performed directly on the size distribution function, but on the cumulative size distribution, in order to ensure the conservation of the particle number concentration. The developed aerosol model includes nucleation, condensational growth, coagulation and all the major external processes like transport, diffusion and external forces. The aerosol dynamics calculations are one-way coupled with the CFD calculations, namely the velocity and temperature fields are taken as input for the aerosol model while the output of the aerosol calculations do not exert any influence on the flow. To cope with the strong coupling between aerosol
7
processes an operator splitting technique is employed (Oran and Boris 2001). The time scales of the aerosol dynamics, and especially of nucleation and growth, highly differ from these of the transport processes like convection and diffusion. Operator splitting is generally more efficient in coupling multiple time scales than a global-implicit scheme, which is based on a complete discretization of the equations. While the global implicit approach is stable and more straightforward, since it treats all the processes together, it can be very costly in multidimensional cases, when stiff processes are involved (Oran and Boris 2000), as in the present case of simultaneous nucleation and condensation. Recently, Mitsakou et al. (2004) and Kommu et al. (2004I, II) also used operator splitting techniques to solve numerically the GDE with a sectional representation of the size distribution. The method we are proposing is validated extensively. Comparisons are made with existing analytical solutions to assess the numerical solution of the GDE alone. To assess the appropriateness of the overall method we reproduce numerically an aerosol reactor case where all three major processes are coupled, i.e. nucleation, growth, coagulation are present. As such, we selected the homogeneous nucleation experiments performed by Ngyuen et al. (1987). These experiments are documented in detailed and have been extensively analyzed in the literature. So, they are an appropriate basis for benchmarking purposes.
8
2. Model The temporal and spatial variation of the particulate phase is described by the general dynamic equation (Friedlander 2000), which can be written as following: ∂ρ nm ∂ρ nm + ∇ ⋅ ( ρ (u + cth )nm ) − ∇ ⋅ ( ρ D p ∇nm ) = ∂t ∂t
+ nucl
∂ρ nm ∂t
+ growth
∂ρ nm ∂t
(1) coag
In the above equation nm is the size distribution function, expressed per unit mass of gas,
ρ is the density of the gas and Dp the particle diffusion coefficient, given by the StokesEinstein equation. In cases where the particles are small, inertia can be neglected and the particle velocity u can be taken equal to the gas velocity. The terms on the right-hand side of Eq. (1) describe the variations due to homogeneous nucleation, condensational growth and coagulation, respectively. Velocity cth corresponds to the transport velocity of the particles due to external forces (thermophoresis is the only mechanism considered). The thermophoretic velocity cth is calculated with the expression of Talbot et al. (1980), using the local values (position dependent) of temperature and temperature gradient. Equation (1) is coupled with the condensable vapor equation, which, neglecting thermal diffusion, is given by: ∂ρ Cm ∂ρ Cm + ∇ ⋅ ( ρ uCm ) − ∇ ⋅ ( ρ Dv∇Cm ) = − ∂t ∂t
− nucl
∂ρ Cm ∂t
(2) growth
where Cm is the vapor mass fraction and Dv is the vapor binary diffusion coefficient. The first and second terms on the right-hand side of the equation above represent the depletion of the condensable vapor due to homogeneous nucleation and condensational growth, respectively.
9
In the present analysis two theories have been adopted for the prediction of particle formation by homogeneous nucleation: the classical nucleation theory (Frenkel 1955) and the modified nucleation theory derived by Girshick at al. (1990). According to the classical nucleation theory the nucleation rate is:
J classical =
πσ d crit 2 Cm2 ρ 2 2σ exp − ρ p mm π mm 3k BT
d crit =
(3)
4σ vm k BT ln S
(4)
where d crit is the embryos critical diameter, σ the surface tension, vm the molecular volume, mm the molecular mass, ρ p the density of the particles and k B the Boltzman constant. The symbols S and T denote the saturation ratio and the temperature, respectively. According to the modified nucleation theory of Girshick et al. (1990) the nucleation rate J classical is multiplied by a correction factor derived from a self-consistency equilibrium cluster distribution, as following: J Girshick
3 36π v 2 σ 1 m = exp S k BT
J classical
(5)
Particles that are formed by homogeneous nucleation grow by condensational deposition of the existing vapor on their surface. The rate of change of the particle diameter due to condensational growth is given by a modified Mason equation (Mason 1971), which accounts for both mass and heat transfer (see, for example, in Jokiniemi et al. 1994): dd p dt
=
S − Sa 4 mass heat ρ p d p f mass / FFS + f heat / FFS
(6)
where
10
f mass =
L L RvT , f heat = − 1 Dv psat (T ) RvT T κ g
(7)
In Eq. (6) FFSmass and FFSheat are the Fuchs-Sutugin correction factors for mass and heat transfer, respectively:
FFSmass =
1 + Kng 1 + Knv , FFSheat = 2 1 + 1.71Knv + 1.333Knv 1 + 1.71Kng + 1.333Kng2
(8)
where the Knudsen number Knv ( Kng ) is defined as the ratio of the mean free path of the vapor (gas) to the droplet radius. The term S a = exp ( 4σ / d p Rv ρ pT ) in Eq. (6) accounts for the Kelvin effect. In Eqs. (7), L is the latent heat of condensation of the vapor species, Rv is the gas constant, κ g is the thermal conductivity of the carrier gas, and psat (T ) is the equilibrium vapor pressure over a flat surface.
Calculation of coagulation is done on the basis of a modified Smoluchowski equation, appropriate for a sectional representation of the particle size distribution, as proposed by Jacobson et al. (1994). According to this formulation the variation of the number concentration of i − th section can be approximated as following: NB i i −1 dN i v = ∑∑ f i , j ,k K ( d j , d k ) k N j N k − vi N i ∑ (1 − fi , j ,i ) K ( di , d j ) N j dt vi j =1 k =1 j =1
(9)
The first term on the right-hand-side of Eq. (9) accounts for appearance of particles in the i − th size section due to collisions of smaller particles and the second term accounts for depletion of particles in the i − th size section due to collisions with all other particles. The coefficients f i , j ,k arise from the sectional representation of the size distribution, and represent the fraction of the new particles formed from collisions of diameters d j and d k that is partitioned into size section i. These coefficients are (Jacobson et al. 1994): 11
fi , j ,k
vk +1 − vi − v j vk +1 − vk = 1 − f i , j , k −1 1 0
vk vi + v j
vk ≤ vi + v j < vk +1 ; k < N B vk −1 < vi + v j < vk ; k > 1 , vi + v j ≥ vk ; k = N B
(10)
all other cases
where v is the particle volume. The coagulation kernel K is calculated as: K = K B + K LS
(11)
considering the kernels associated with Brownian coagulation ( K B ) and laminar shear coagulation ( K LS ). The Brownian kernel is determined from the standard Fuchs interpolation formula (Fuchs 1964), which is valid from the continuum to the free molecular regime:
K B (d j , d k ) = 2π ( d j + d k )( D p , j + D p ,k ) d j + dk d + d + 2 g2 + g2 k j k j
(
)
1/ 2
+
(d
1/ 2 2 2 c j + ck
8( D p , j + D p ,k ) j
+ dk )
(
−1
(12)
)
The laminar shear kernel is related with the velocity gradient in the direction normal to the flow as following (see, for example, in Drossinos and Housiadas 2006): K LS (d j , d k ) =
1 ∂u ( d j + d k )3 6 ∂y
(13)
In Eq. (12) ci = (8k BT / π mi )1/ 2 is the mean particle velocity and gi the so-called Fuch's length gi =
1 3 di + li ) − di2 + li2 ( 3di li
(
)
3/ 2
−d , i
(14)
where li = 8D p ,i /(π ci ) the mean free path of the aerosol particle.
12
3. Numerics 3.1. Internal aerosol processes To combat numerical diffusion, while using a computationally economical particle size grid resolution, we implemented a simple, hybrid method for solving condensational growth. The method is based on a combination of a moving particle size grid and a fixed particle size grid. More specifically, the condensational growth equation is integrated, in each time step, allowing the particles to grow to their actual sizes using the moving grid approach (Gelbard 1990; Kim and Seinfeld 1990). The integration is performed using a fourthorder Rosenbrock method with monitoring of the local truncation error to adjust the time step (Press et al. 1994). Then the cumulative distribution function is calculated on the moving grid using the following simple form, valid for a sectional formulation: i
i
1
1
Ci = ∑ N i = ∑ ρ nm,i ∆di
(15)
where ∆di is the width of the i − th size section of the moving grid. The cumulative distribution is then reallocated to the fixed size grid using a cubic spline interpolation. Let Ci* be the cumulative size distribution as inferred from the interpolation step. Then, the new number concentration Ni* can be simply calculated as: N i* = C *i − C *i −1
(16)
The use of a third order polynomial for the interpolation leads to significant reduction of numerical diffusion. Number conservation and enhanced stability are achieved by taking advantages of the mathematical properties of the cumulative size distribution. By
13
definition, the cumulative size distribution must be monotonic and no negative. Therefore, the following corrections are made: C *1 = 0, C * K = N total , for i =nmax − 1, 2, C *i =C *i +1 , if C *i ≥ C *i +1 and
(17)
for i = 2, nmax − 1, C *i =C *i −1 , if C *i ≤ C *i −1 where N total is the total number concentration of the particles and nmax the number of the used particle size sections. The above procedure ensures conservation of the total particle number in the condensational growth calculations, conferring to the fixed grid approach the appealing characteristics of the moving grid approach. Subsequently, we use the abbreviation CICR (Cubic Interpolation Cumulative Re-mapping) to refer to the method described above for the solution of growth. The implementation of the two other internal processes, i.e. nucleation and coagulation, is straightforward. Nucleation is resolved simply by adding particles in the size section that contains the critical diameter. Coagulation is directly calculated on the basis of Eqs. (9) and (10), using the semi-implicit scheme of Jacobson et al. (1994), which does not require iterations and is unconditionally stable.
3.2. Coupling with CFD The computation of fluid and aerosol dynamics is carried out in two stages. First, CFD calculations are made to determine the gas velocity and temperature fields in the reactor. These fields are used as input to the aerosol dynamics code. The CFD step is performed using a general-purpose commercial CFD package (code ANSWER; ACRi 2001). The Nodal Point Integration (NPI) technique is used for the integration of the equations (Runchal 1987a). The continuity, momentum and energy equations are solved
14
in a two-dimensional cylindrical uniform grid. Since the effect of heat release from condensation on the gas temperature is neglected (one-way coupled system), only the convective, diffusive terms are taken into account in the energy equation. The flow is laminar. Standard numerical methods were selected in our CFD calculations. More specifically, the hybrid scheme (Runchal 1972) was used for the discretization of the convective terms, while the ADI method (Fletcher 1991) was used for the solution of the algebraic system of equations. The sensitivity of the results to the scheme used in the solution of the convection term is examined by using also the QUICK scheme (Leonard 1979). Practically, no difference was found in the overall calculations. The problem of fluid flow and heat transfer in laminar flow aerosol reactors has been addressed extensively in the literature. Parameters, concerning the fluid flow and heat transfer in the reactor, that could affect the overall aerosol dynamics calculations have been examined by employing both CFD-based and analytical methods (Pyykönen and Jokiniemi 2000; Wilck and Stratmann 1997; Housiadas et al. 2002; Housiadas et al. 2000). Therefore, no particular focus was given to the fluid flow and heat transfer problem in the reactor. The aerosol dynamic calculations are then performed according to the following methodology. The fully Eulerian, multidimensional problem described by Eqs. (1) and (2) is numerically advanced in time using an operator splitting technique. The source terms for each of the aerosol dynamics processes are explicitly calculated in each time step by using the methods described previously and then combined (added) to give the right-hand sides (overall source terms) for Eqs. (1) and (2). These source terms consist input for the integration of Eqs. (1) and (2), which is performed by using an implicit finite volume scheme (Patankar 1980). The convection terms are treated using the hybrid or the
15
power law scheme (Patankar 1980), while the ADI method is adopted for the solution of the algebraic equations. To accelerate the calculation an adaptive time step process is implemented as following. At every computational grid point j the local relative change of the vapor mass fraction, due to condensational growth, is forecasted using the time step of the previous iteration ∆t 0 . If this change is higher than a pre-specified value (usually, a relative change of 0.5% was specified in the runs of the present work) a local time step ∆t1j is calculated on the basis of a pre-specified relative change for the number
concentration due to nucleation (usually 30%). Otherwise, the local time step is taken as ∆t1j = 2∆t 0 . The global time step ∆t1 with which the integration with the finite volume method is performed is selected as the minimum of the local time steps ∆t1j . The doubling of the local time step in the locations where vapor depletion is small, and consequently there is no important effect on the calculation of the saturation ratio, was found to accelerate significantly the convergence of the solution. Steady state problems are solved using a pseudotransient approach. Starting from arbitrary initial conditions, the solution procedure marches along the time, exactly like in unsteady problems, until the converged, steady-state solution is reached. The spatial CFD and aerosol dynamics grids are independent. A spatially nonuniform grid, finer at the nucleation zone is usually needed in aerosol dynamics calculations. The temperature and the gas flow field at the aerosol dynamics grid nodes are calculated by multidimensional linear interpolation (Press et al. 1994) on the output data provided by the CFD calculations.
16
4. Validation tests The algorithm and methods employed to calculate aerosol dynamics are assessed by comparing with available analytical solutions of the general dynamic equation for a number of idealised cases. The first comparison with theory is based on the analysis of combined condensation and coagulation of Ramabhadran et al. (1976). In this analysis the coagulation kernel K is assumed constant, while the growth rate is taken as a linear function of the particle volume:
dv =σv dt
(18)
The initial aerosol distribution function has the form of a first order gamma function:
n0 (v) =
v N0v exp − 2 v0 v0
(19)
where N 0 is the initial number concentration and v0 the mean volume of the initial distribution. Fig. 1 shows the evolution of the normalized number and volume concentrations of the aerosol as a function of the dimensionless time. As can be seen, the particle number decreases as the particles coagulate, whereas the aerosol volume increases due to condensational growth. The change in the aerosol properties was numerically reproduced using a size grid of 6 particle size sections/decade and a dimensional time step equal to 0.1. The numerical results are in excellent agreement with the analytical solution both in terms of volume and number. Although the CICR method does not take into account the conservation of volume, the results in fig. 1 indicate that, in practice, the errors introduced in the calculation of the particle volume during the re-mapping step are very small. Fig. 2 shows the calculated dimensionless size distribution function of the
17
particles. From the comparison with the analytical solution it is concluded that numerical diffusion is satisfactorily combated even using a low resolution of 6 particle size sections/decade.
Figure 1. Comparison between calculated and analytical normalized number and volume concentrations, as a function of the dimensionless time τ = σ t , for a system with Λ = 0.3 . (characteristic dimensionless quantity Λ = σ / KN 0 ).
18
Figure 2. Calculated size distribution in comparison with the analytical solution at τ = 5 for the first analytical test. The initial size distribution of the aerosol is also shown ( τ = 0 ).
The second validation test is concerned with the analytical solution of an evolving size distribution under the influence of pure condensational growth (Seinfeld and Pandis 1998):
dd p dt
=
A dp
(20)
The aerosol has initially a lognormal size distribution:
n0 =
ln 2 (d p / d pI ) 1 exp − 2 ln 2 σ I d p 2π ln σ I N0
(21)
19
where A is a constant and N 0 is the initial total number concentration. A case with initial median diameter D pI = 0.1 µ m , initial geometric standard deviation σ I = 1.15 ,
A = 1.0 ×10−10 cm 2 s −1 and total simulation time T f = 1 s was chosen because it involves steep evolution of the size distribution. Note that the same test conditions have been also employed in previous assessment exercises (Test III in Yamamoto 2004). In the calculations a particle diameter range from 0.06 µ m to 0.3 µ m was used to cover all the sizes encountered during the simulation. Fig. 3 shows the volume distribution of particles, as calculated using the CICR method (fig. 3a) and the moving center method (fig. 3b), as a function of the number of size sections in which the size range is divided. As the number of size sections increases the accuracy of the CICR method improves steadily, converging to the analytical solution. Due to the spline interpolation used in the remapping step and the imposed corrections (Eq. (17)) some oscillations can be artificially introduced in the distribution. The results in Fig. 3a indicate that, in practice, such effects are insignificant. In all investigated cases artificial peaks and deeps have been obtained only at points where the distribution is vanishing (for example in the horizontal tail of the distribution of fig. 3a). On the contrary, the moving center method predicts an oscillating, unrealistic distribution. Moreover, it does not converge uniformly to the analytical solution, but instead overshoots. This behavior is due to the frequent occurrence of empty sections in the course of the calculation.
20
a).
b).
Figure 3. Volume distribution at T f = 1 s for various particle size grid resolutions, as calculated using a) the CICR method, b) the moving center method, and comparison with the analytical solution.
5. Results
To illustrate our model we reproduce a real nucleation-condensation case in an aerosol reactor. We used the data reported (meticulously) by Ngyuen et al. (1987), as obtained from homogeneous nucleation experiments. In these experiments dry air saturated with DBP is conducted through a hot tube, whose temperature is kept equal to that of the saturator, into a cooler tube, the “condenser”. A transition zone of 10 cm exists between the hot tube and the condenser. The condenser (length of 52 cm, inner diameter of 1 cm) is rapidly cooled, causing the gas to become highly supersaturated. Particles are formed by homogeneous nucleation and grow mainly by condensation. Nguyen et al. (1987) reported extensive measurements of both the number concentration and size distribution of the formed particles at the outlet of the condenser, for different conditions with respect to the flow rate and saturator temperature.
21
The model was applied to reproduce the fluid and particle dynamics in the condenser. The analysis is made on a two-dimensional grid in cylindrical coordinates, assuming symmetry with respect to the condenser axis. The flow is laminar and becomes fully developed before the entrance of the transition zone. Therefore, at the tube inlet the gas velocity is taken parabolic, and the temperature is taken uniform and equal to that of the saturator. Previous works have showed the importance of choosing correctly the boundary conditions in the analysis of nucleation experiments in laminar flow aerosol reactors (Housiadas et al. 2000; Wilck and Stratmann 1997). The temperature values on the wall nodes are set on the basis of a linear interpolation on the measured wall temperature data, as provided by Nguyen et al. (1987). The wall boundary values for the vapor mass fraction are those corresponding to equilibrium at the wall temperature. Also, a zero particle number concentration is imposed at the wall (for all sizes), assuming the tube wall to be a totally absorbing boundary. The physical properties of DBP are those given by Nguyen et al. (1987) and the properties of the carrier gas (air) are taken all as temperature dependent. One-dimensional (1-D) calculations were also performed, based on a previously developed one-dimensional, semi-Lagrangian model (Mitrakos et al. 2004). This model was further elaborated by implementing the operator splitting technique and the previously described CICR method for the solution of growth. This model is used to further test the capabilities of the 1-D description of aerosol flow reactors and also to intercompare easily the CICR method and the moving center method in a real nucleationcondensation case.
22
Fig. 4 shows the bulk average number concentration of the particles at the exit of the reactor, as calculated with our model. For comparison, the experimental data are also shown, as well as simulation results obtained with other models. Our CFD calculations were done with an equally spaced spatial grid consisted of 700 axial and 40 radial nodes. In the two-dimensional (2-D) calculations aerosol dynamics are solved over a grid of 600 axial and 20 radial nodes, non-uniform in the axial direction (grid sizes ranging from about 0.3 mm to 3 mm). As discussed before an adaptive time step was used, which, typically, ranged between ~2 ms and 20 ms. In the 1-D calculations an adaptive axial grid was used, similar to that used by Im et al. (1985) and Jokiniemi et al. (1994). The axial step is derived from the maximum allowable relative changes for the particle number concentration and the vapor mass fraction. The modified nucleation theory of Girshick et al. (1990) was used in the present calculations for the estimation of the nucleation rate. As commonly made in homogeneous nucleation analyses, the calculated nucleation rate needs to be multiplied by a correction factor to get agreement with the experimental data. In the present calculations the nucleation rate was multiplied by a factor C=5 ×10-4 for the 2-D runs and C=1.2×10-3 for the 1-D runs, which are both consistent with the factors used by Pyykönen and Jokiniemi (2000) (C=3.2×10-4) and Wilck and Stratmann (1997) (C=1×10-3). As the competition of nucleation and growth becomes stronger, or equivalently as the number concentration and the subsequent vapor depletion become more important the time step needs to decrease in order to accurately calculate the nucleation rate. Hence, the running time of a simulation depends on the intensity of coupling between these two processes. The running time is roughly proportional to the resolution used in the discretization of the particle size grid. For a resolution of 10 size
23
sections/decade the CPU time using a personal computer (Pentium IV 3 GHz, 1 GB) ranges from 25 minutes for the lower saturator temperature (left part of fig. 4) to 1 h and 40 minutes for the most demanding case, namely for the higher saturator temperature (right part of fig. 4). Coagulation has an effect less than 0.5% in the final number concentration of particles, but increases computational time roughly by a factor of three. Therefore, in the runs coagulation was usually turned-off. The CPU time for the 1-D calculation was trivial (less than a minute in all the cases).
Figure 4. Bulk average number concentration at the outlet of the reactor as a function of the saturator temperature, as predicted from our models, in comparison with the measured data and previous modeling results (System B of Nguyen et al. (1987) with gas flow rate 0.5 lt/min and condenser temperature 21.4 oC). 24
As the results of fig. 4 indicate, our model predicts with very good accuracy most of the measured points and agrees closely with previous models. The previous numerical results have been obtained with different computational approaches. The calculations of Wilck and Stratmann (1997) are done with a fully 2-D Eulerian model, based on the modal method. The calculations of Pyykönen and Jokiniemi (2000) are made with a sectional, quasi-2-D model, based on a Lagrangian approach (called the “stream-tube” approach). Instead, previous attempts to reproduce these data with a sectional, 2-D Eulerian approach based on the full discretization of the governing equations failed (took CPU days to converge, and gave unacceptably overestimated predictions; see, discussions in Pyykönen et al. 2002; Pyykönen and Jokiniemi 2000; Stratmann and Witby 1989). Our model reproduced successfully these experimental data, demonstrating the appropriateness of the proposed methodology as to the way of using the sectional method in a multi-dimensional, Eulerian computational framework. As the results of Fig. 4 indicate, there are discrepancies between calculated and experimental data for saturator temperatures higher than about 100˚C, for all the models. Seemingly, all models underestimate the nucleation rate and overestimate the growth rate over this range of saturator temperatures. The reason of this behavior is not yet clear. Wilck and Stratmann (1997) tried to explain these discrepancies by artificially suppressing the growth rate, but they did not reach to a definite conclusion. Finally, it is interesting to notice that in this case the 1-D model gave satisfactory results for the number concentration, in comparison with the 2-D approach, using practically the same correction factor. This interesting behavior was reported in a previous communication
25
(Mitrakos et al. 2004). Similar trends in 1-D aerosol reactor simulations have been also observed by other investigators (Jeong and Choi 2003; Park et al. 1999). The validity limits of an 1-D approximation cannot be easily established. Moreover, as it will become apparent below, the 1-D solution fails in predicting the size distribution, and, hence, the use of the 1-D approximation is not generally recommended.
Figure 5. Bulk average number concentration at the outlet of the reactor as a function of the saturator temperature, as calculated with various particle size grid resolutions (same experimental case as in figure 4).
The characteristics of our model in terms of stability and accuracy are assessed by examining the sensitivity of the results to the number of sections that are used to divide
26
the particle size spectrum. The influence of the particle size grid resolution on the calculated particle number is illustrated in fig. 5, which shows the exit number concentration of particles as calculated for various numbers of size sections. As can be observed, the impact of the particle size grid resolution on the calculated number concentration increases towards the right (higher particle number), i.e. as the competition between nucleation and growth becomes stronger. The bias introduced by the CICR method in the calculation of the particle total volume during the re-mapping step and the unavoidable errors arising from the discretization of the particle size on the fixed grid are the reasons for this trend. By construction, the CICR method is number conservative, but not volume conservative. The small numerical errors on particle volume reflect on the vapor depletion calculations, and have therefore a positive feedback on the error in the nucleation rate calculation because of the extreme sensitivity of the nucleation rate on the saturation ratio. The moving center method is less sensitive to this feedback effect because is volume conservative. However, the moving center method was found to predict unrealistic particle size distributions, as previously discussed in the validation tests and as it will also become apparent latter. In fact, this was the motivating reason to seek for a new numerical method (CICR). The remedy to the previously described error propagation loop is the increase of the number of size sections. In this respect, the proposed method performed satisfactorily. As the results of Fig. 5 indicate, the accuracy of our calculations using 10 size sections/decade can be considered as adequate. Note that this particle size grid resolution is similar to that used in previous analogous analyses (Pyykönen and Jokiniemi 2000). Very good accuracy is achieved using 20 sections/decade. With this resolution, the computational time for the most demanding
27
run, namely, for the higher saturator temperature, reached 3h and 50 min on our computer, therefore remaining tolerable in all cases. Figure 6 shows the calculated size distribution at the exit of the reactor, for a test for which Nguyen et al. (1987) reported detailed size distribution measurement data. The calculations results of Nguyen et al. (1987) are also shown, which were obtained with a particular semi-analytical model, tailored to this application, on the basis of the theory of Green functions (Pesthy et al. 1983). Results for the calculated size distribution are not provided by Pyykönen and Jokiniemi (2000) and Wilck and Stratmann (1997), and so it is not possible to perform a direct comparison with other CFD-based computational schemes. Following Nguyen et al. (1987) in this case we use the classical nucleation theory. A nucleation rate correction factor C=2.1×104 was used in our calculations, which is very close to that used by Nguyen et al. (1987) (C=4×104). The calculated size distribution presents a similar dependence on the particle size grid resolution with that previously discussed. Adequate accuracy is achieved and numerical diffusion is efficiently combated even with 10 size sections/decade. The shape of the size distribution remains smooth and a fast convergence is achieved as the number of size sections increases. 20 sections/decade are adequate for an accurate representation of the size distribution. The agreement with the experimental data is satisfactory. Our CFD-based predictions are close to the theoretical, semi-analytical results of Nguyen et al. (1987). Note, however, that the numerical model performs better than the semi-analytical over the small size range (
