Chemical Engineering Science 61 (2006) 124 – 134 www.elsevier.com/locate/ces
Dynamic evolution of PSD in continuous flow processes: A comparative study of fixed and moving grid numerical techniques A.I. Roussosa, b , A.H. Alexopoulosb , C. Kiparissidesa, b,∗ a Department of Chemical Engineering, Aristotle University of Thessaloniki, P.O. Box 472, 541 24 Thessaloniki, Greece b Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 472, 541 24 Thessaloniki, Greece
Received 1 June 2004; accepted 1 December 2004 Available online 24 June 2005
Abstract The present study provides a comprehensive investigation on the numerical solution of the dynamic population balance equation (PBE) in continuous flow processes. Specifically, continuous particulate processes undergoing particle aggregation and/or growth are examined. The dynamic PBE is numerically solved in both the continuous and its equivalent discrete form using the Galerkin on finite elements method (GFEM) and the moving grid technique (MGT) of Kumar and Ramkrishna [1997. Chemical Engineering Science 52, 4659–4679], respectively. Numerical simulations are carried out over a wide range of variation of particle aggregation and growth rates till the dynamic solution has reached its final steady-state value. The performance of the two numerical methods is assessed by a direct comparison of the calculated particle size distributions and/or their moments to available steady-state analytical solutions. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Dynamic population balances; Particulate processes; Galerkin on finite elements method; Moving grid technique; Continuous flow reactors; Particle growth; Particle aggregation
1. Introduction An important property of many particulate processes is the particle size distribution (PSD) that controls key aspects of the process and affects the end-use properties of the product. Particulate processes are generally characterized by particle size distributions that can largely vary in time. The quantitative calculation of the evolution of the PSD in a reactive particulate system presupposes good knowledge of the particle growth and aggregation mechanisms. Particle growth due to chemical reaction results in an increase of the mean particle size and can affect the form of the PSD, particularly in size-dependent particle growth processes. Particle aggregation can result in significant changes in the form of ∗ Corresponding author. Department of Chemical Engineering, Aristotle
University of Thessaloniki, P.O. Box 472, 541 24 Thessaloniki, Greece. Tel.: +30 2310 996 211; fax: +30 2310 996198. E-mail address:
[email protected] (C. Kiparissides). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.12.056
the PSD. Moreover, in continuous flow systems particles are continuously introduced to the reactor via the inflow stream and removed by the outflow stream. The time evolution of the PSD in a particulate process is commonly obtained via the solution of the population balance equation (PBE). The general PBE for a continuous well-mixed particulate system is given by (Hulburt and Katz, 1964; Ramkrishna, 1985)
jn(V , t) jG(V )n(V , t) + jt jV =
nin (V , t) − n(V , t) V /2 + (V − U, U )n(V − U )n(U ) dU Vmin
− n(V )
Vmax Vmin
(V , U )n(U ) dU .
(1)
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In the above equation, n(V , t) and nin (V , t) denote the number density function in the well-mixed reactor and in the inflow stream, respectively. G(V ) is the particle volume growth rate function and (V , U ) is the aggregation rate kernel between particles of volumes V and U . Vmin and Vmax denote the corresponding minimum and maximum size of particles present in the system. In the above formulation of the PBE, the total reaction volume is assumed to remain constant, which is true for very dilute dispersions (i.e., the inflow and outflow rates are equal). Under these assumptions, denotes the mean residence time of the reactive phase. The PBE can be solved either in the continuous or in its equivalent discrete form. Notice that both approaches have been extensively used in the past for solving the general PBE (see Eq. (1)). In the continuous approach, the unknown solution is approximated by a series of basis functions with unknown coefficients that are determined by minimization of the integral of the weighted residuals so that the PBE holds true in an approximate sense (Finlayson, 1980; Mahoney and Ramkrishna, 2002). Based on this approach, several numerical methods have been developed for solving the PBE, including the method of weighted residuals (Ramkrishna, 2000), the orthogonal collocation on finite elements method (OCFE) (Gelbard and Seinfeld, 1978; Alexopoulos et al., 2004) and the Galerkin on finite elements method (GFEM) (Nicmanis and Hounslow, 1998; Rigopoulos and Jones, 2003; Sandu and Borden, 2003). In general, the numerical solution of the PBE in the continuous form produces accurate results but requires large computational times. Specifically, both the orthogonal collocation and the GFEMs have been shown to be very effective for solving problems involving combined particle nucleation, growth, and aggregation (Alexopoulos et al., 2004; Alexopoulos and Kiparissides, 2005; Roussos et al., 2005). In the discretized PBE (DPBE) approach, a particle number distribution, Ni (t), is defined Ni (t) =
Vi+1 Vi
n(V , t) dV
(2)
that corresponds to the number of particles in the “i” volume element [Vi+1 , Vi ]. Accordingly, an equivalent to Eq. (1) discretized PBE is derived in terms of Ni (t). Among the first implementations of the DPBE were those of Batterham et al. (1981) and Hounslow et al. (1988). Litster et al. (1995) developed a widely-used DPBE formulation based on the geometric fractional discretization of the volume domain. Kumar and Ramkrishna (1996) proposed a more general fixed-pivot technique that can be applied to any type of discretization of the particle volume domain. While fixed-grid DPBE methods are capable of producing accurate numerical solutions for problems involving particle aggregation and/or breakage, they do not perform as well in growth-dominated processes (Alexopoulos et al., 2004). In the moving-grid implementation of the pivot technique developed by Kumar and Ramkrishna (1997), the volume grid
125
moves according to the particle growth rate. The moving grid technique (MGT) was shown to be very effective for particulate processes undergoing combined particle growth and aggregation in the absence of particle nucleation. Despite the wide application of the aforementioned techniques no specific guidelines are available regarding the selection and implementation of a numerical method for the accurate calculation of the time evolution of the PSD in a particulate process. The limitations of the various methods in terms of accuracy and stability of the numerical solution have not been thoroughly investigated. Finally, there is no comprehensive study on the applicability of the Galerkin finite element and MGT for solving the general PBE (1) over a wide range of variation of aggregation and growth rates, functional forms of growth rate and inflow distributions. In the present study, a brief overview of the two numerical methods is presented. Subsequently, simulation results on the application of both numerical methods to several continuous flow particulate processes, undergoing simultaneous particle growth and aggregation, are presented. Whenever possible, the calculated numerical results are compared with available analytical solutions on the steady-state PSD and/or its respective leading moments. Finally, several improvements regarding the performance (i.e., accuracy, stability, CPU time) of both numerical methods are discussed.
2. Numerical solution of the population balance equation The continuous form of the general PBE (see Eq. (1)) was solved using the GFEM. Accordingly, the volume domain is first discretized into “ne” elements, each element containing “np” nodal points. Subsequently, the number density function, n(V , t), is approximated over each element in terms of its respective values at the nodal points using standard Lagrange interpolation polynomials. n(V , t) =
np i=1
ei (V )nei (t),
(3)
where nei (t) denotes the value of the number density function at the “i” nodal point in element “e”. Then, a weighted residual equation is derived for each node by multiplying all the terms in Eq. (1) by an appropriate weighting function and integrating the resulting equation over the element volume. A brief description of the GFEM is presented in Appendix A. A more detailed presentation of the GFEM as well as its numerical implementation is given in Roussos et al. (2005). In the present study, the resulting system of ne · np ordinary differential equations was integrated in time using the IMSL solver DASPG. To solve the PBE using the MGT, the initial volume domain is first discretized into a number of moving bins. The velocity of the moving bins is set equal to the particle volume growth rate (Kumar and Ramkrishna, 1997). A brief de-
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scription of the MGT is presented in Appendix B. The resulting system of ODEs was integrated in time using the IMSL solver IVPAG. Notice that for problems involving particle nucleation or any other fixed-volume particle source and/or requiring the satisfaction of a boundary condition at Vmin , the moving grid must be expanded to accommodate the particles being generated or flowing into the system. Thus, after a specified number of time integration steps defined by the bin addition frequency parameter, f , new bins are added to the moving grid at V = Vmin . This results in a significant increase in the number of equations that is the main drawback of the MGT.
3. Results and discussion The dynamic PBE (see Eq. (1)) was numerically solved by the two methods (i.e., GFEM and MGT) for several test problems described in Table 1. For cases (a) and (d), the inflow stream contained particles having either an exponential or a Gaussian number density function. On the other hand, for cases (b) and (c), it was assumed that particles were continuously formed as a result of the imposed boundary condition n(Vmin , t) = 1 at V = Vmin . This form of particle source acted as an effective nucleation term and resulted in PSDs characterized by very steep moving fronts. For cases (a) and (b), the GFEM and MGT were validated by a direct comparison of the calculated steady-state PSDs with available analytical solutions provided by Nicmanis and Hounslow (1998). On the other hand, for cases (c) and (d), the numerical results were compared with available analytical solutions for the leading moments of the distribution. To obtain a direct comparison of the two numerical techniques, the MGT results on the number distribution, Ni (t), were transformed into a corresponding average number density function for each volume element, using the following approximation: n¯ i ≈ Ni /(Vi+1 − Vi ). 3.1. Size-independent particle aggregation The general PBE was first solved for a constant aggregation kernel and an exponential particle inflow distribution (see case (a) in Table 1). In Fig. 1, the calculated steady-state PSDs by the GFEM and MGT are compared with the corresponding analytical solutions for two values of the mean residence time (i.e., = 102 and 104 ). It should be pointed out that for both methods a logarithmic discretization of the volume domain and the same number of nodes were employed. It is apparent that there is an excellent agreement between the two numerical solutions and the analytical one even for values of the number density function as small as 10−15 . Moreover, both methods provide very accurate numerical estimates of the leading moments of the steady-state distribution given by the analytical solution (i.e., with relative error less than 2%).
Table 1 Continuous flow test problems Case Aggregation
Growth
Particle source nin (V , t) = (N0 /V0 ) exp(−V /V0 ) n(Vmin , t) = 1 n(Vmin , t) = 1 √ nin (V , t) = (1/ 2) exp[(V − V0 )2 /22 ]
(a)
= 0
0
(b) (c) (d)
= 0 = 0 = 0
G = G0 G = G0 (V /V0 ) G = G0
N0 = 1 in all cases. V0 = 1 for cases (a) and (c) and V0 = 10 in case (d).
In Fig. 2, the evolution of the number density function calculated by the GFEM and the MGT is depicted at different times (t = 10, 102 , 104 and 106 ) for a mean residence time of =104 . Notice that the two methods produce slightly different results in the large volume range of the PSD, corresponding to values of the number density function less than 10−12 . The observed deviation of the MGT solution from the corresponding GFEM one, at small simulation times, is due to a diffusion-like “over prediction error” that is inherent to all sectional methods in particle aggregation problems (Kumar and Ramkrishna, 1996). However, at longer simulation times, this deviation decreases and practically disappears as the dynamic PSD approaches its final steadystate. 3.2. Combined constant particle aggregation and growth The time evolution of the PSD in a continuous flow process, undergoing combined constant particle aggregation and constant particle growth was also examined (see case (b) in Table 1). In Fig. 3, the steady-state PSDs calculated by the GFEM and the MGT are compared with the analytical solutions for a mean residence time of = 1, an aggregation rate constant of 0 = 1 and three different values of the growth rate constant (i.e., G0 = 1, 10 and 100). As can be seen, the steady-state PSDs calculated by the GFEM are in excellent agreement with the corresponding analytical solutions whereas the MGT results exhibit a slight deviation in the large-volume portion of the PSD. In Fig. 4, the calculated steady-state PSDs are compared with the analytical solutions for a mean residence time of = 1, a growth rate constant of G0 = 1 and three different values of the aggregation rate constant (i.e., 0 = 1, 10 and 100). It is evident that, the calculated steady-state PSDs are in excellent agreement with the corresponding analytical solutions. However, the diffusion error of the MGT in the large-volume portion of the PSD increases with the value of 0 . The observed agreement between the analytical and the GFEM numerical solutions (see Figs. 3 and 4) is remarkable, considering the fact that the distribution evolves from an initially very abrupt front. It should be noted that, the dynamic simulations leading to the steady-state PSDs in Figs. 3 and 4, were initiated from a zero initial condition for the PSD (i.e., n(V , 0) = 0). Other initial conditions were also tested including a
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127
Number Density Function
1.E+00
1.E-04
1.E-08
1.E-12 τ=102 1.E-16
1.E-20 1.E-03
Analytical GFEM MGT 1.E-01
τ=104 1.E+01 Volume
1.E+03
1.E+05
Fig. 1. Comparison of steady-state PSDs for constant particle aggregation (Case (a) in Table 1: 0 = 1). GFEM (ne = 20 and 30, np = 3, nt = 61 and 81); MGT (nt = 60 and 80).
Number Density Function
1.E+00
1.E-04
1.E-08
steady-state
1.E-12
1.E-16
GFEM MGT
1.E-20 1.E-03
1.E-01
t=10
1.E+01
t=102
1.E+03
t=104
t=106
1.E+05
Volume Fig. 2. Comparison of dynamic PSDs for constant particle aggregation (Case (a) in Table 1: 0 = 1, = 104 ). GFEM (ne = 40, np = 3, nt = 81); MGT (nt = 80).
constant value, n(V , 0) = n0 , an exponential one, n(V , 0) = e−V , and a gamma function, n(V , 0) = V e−V . For all the aforementioned cases, the calculated steady-state solutions were identical. For both numerical methods, a single volume element was assigned to the volume subdomain Vmin =10−5 to V =10−1 , followed by a logarithmic discretization of the remaining volume domain. For the GFEM a total number of 81 nodal points were employed. The initial grid of the MGT included 30 nodes while the initial value of Vmax was chosen so that the final grid expanded over the same volume domain as in the GFEM. The effect of the bin addition frequency parameter f on the resolution of the PSD was investigated by altering its value from 5 to 50 time steps. In Fig. 5, the ef-
fect of the grid updating frequency on the calculated PSD is depicted for a growth rate constant G0 = 100, a mean residence time =1 and a total number of 1000 time integration steps. As can be seen, when the moving grid is updated at longer time intervals (i.e., large values of parameter f ), the updated grid lacks the required resolution to properly enforce the boundary condition at V = Vmin . As a result, the MGT fails to resolve the left part of the PSD. Moreover, the deviation of the calculated PSD from the corresponding analytical solution increases with the growth rate constant. From the results of Table 2, one can easily conclude that a more frequent updating of the moving grid (i.e., smaller values of parameter f ) largely improves the accuracy of the calculated moments of the steady-state PSD.
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Number Density Function
1.E+00
1.E-04
1.E-08
G0=1
1.E-12
1.E-16
G0=10
G0=102
Analytical GFEM MGT
1.E-20 1.E-01
1.E+01
1.E+03
1.E+05
Volume Fig. 3. Comparison of steady-state PSDs for combined constant particle aggregation and growth (Case (b) in Table 1: 0 = 1, = 1). GFEM (ne = 30, np = 3, nt = 81); MGT (nt 0 = 30, f = 20, nt = 80).
Number Density Function
1.E+00
1.E-04
1.E-08
1.E-12
1.E-16
β0=1
β0=102
Analytical GFEM MGT
β0=10
1.E-20 1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
Volume Fig. 4. Comparison of steady-state PSDs for combined constant particle aggregation and growth (Case (b) in Table 1: G0 = 1, = 1). GFEM (ne = 30, np = 3, nt = 81); MGT (nt 0 = 30, f = 20, nt = 80).
In Fig. 6, the evolution of the number density function is depicted at different times for 0 = 1, G0 = 1 and = 1. As can be seen, the initial distribution evolves through very steep fronts to the final steady-state distribution given by the analytical solution (Nicmanis and Hounslow, 1998). It should be noted that both numerical methods converge to the same final distribution. However, at early simulation times, a large number of nodes are required to resolve adequately the dynamic evolution of the distribution. Moreover, the use of special types of discretization and, in some cases, the introduction of a small amount of artificial diffusion can significantly improve the GFEM solution (e.g., the solution becomes smoother and more robust). In general, if only the final steady-state PSD is needed, the volume discretization requirements are significantly less than
those required for the calculation of the dynamic evolution of PSD. 3.3. Combined constant particle aggregation and linear particle growth The third test problem (see case (c) in Table 1) refers to a constant particle aggregation rate kernel and a linear particle growth rate model (i.e., G(V ) = G0 V /V0 ). In terms of computational efficiency, the MGT is especially suited for linear growth problems because the particle aggregation mapping between different bins does not change (Kumar and Ramkrishna, 1997). On the other hand, for a constant particle growth rate, the aggregation mapping has to be updated continuously. In Fig. 7, the
A.I. Roussos et al. / Chemical Engineering Science 61 (2006) 124 – 134
129
1.2
Number Density Function
1.0 0.8 0.6 0.4
f=5 f=10 f=20 Analytical
0.2 0.0 1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
Volume Fig. 5. Steady-state PSDs calculated with the MGT for three different values of the parameter f . (Case (b) in Table 1: G0 = 100, 0 = 1, = 1, nt 0 = 30).
Table 2 Calculated moments of the steady-state distribution for constant particle aggregation and growth
0 = 1, G0 = 1
MGT (f = 20) MGT (f = 10) GFEM Analytical
0 = 100, G0 = 1
0 = 1, G0 = 100
m0
m1
m2
m0
m1
m2
m0
m1
m2
0.8368 0.7832 0.7325 0.7320
0.8105 0.7690 0.7326 0.7320
2.14 2.05 2.003 1.9998
0.2636 0.1904 0.1365 0.1319
0.1453 0.1380 0.1371 0.1318
2.027 2.25 2.156 2.0007
26.36 19.03 13.20 13.18
1485.7 1413.8 1325.0 1317.7
2346610 2217420 2029000 1999874
Number Density Function
GFEM (ne = 30, np = 3, nt = 81); MGT (nt 0 = 30, f = 20, nt = 80).
1.E-02
steady–state t=0.01
1.E-06
t=0.1
t=1
t=10
GFEM MGT 1.E-10 1.E-05
1.E-03
1.E-01
1.E+01
1.E+03
Volume Fig. 6. Comparison of dynamic PSDs for constant particle aggregation and growth: (Case (b) in Table 1: 0 = 1, G0 = 1, = 1). GFEM (ne = 40, np = 3, nt = 81), MGT (nt 0 = 30, f = 20, nt = 80).
calculated steady-state PSDs by the two numerical methods are compared for different values of the growth rate constant, G0 . Although both methods result in nearly identical
steady-state PSDs, the GFEM is more accurate in terms of the calculated moments of the steady-state PSD (see Table 3).
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Number Density Function
1.E+00
1.E-04 G0=0.5 1.E-08
1.E-12
1.E-16
G0=0.05
G0=0.1 GFEM MGT
1.E-20 1.E-03
1.E-02
1.E-01
1.E+00
Volume Fig. 7. Comparison of steady-state PSDs for combined constant particle aggregation and linear growth: (Case (c) in Table 1: 0 = 1, = 1). GFEM (ne = 30, np = 3, nt = 61); MGT (nt 0 = 10, f = 20, nt = 60).
Table 3 Calculated moments of the steady-state distribution for constant particle aggregation and linear particle growth G0 = 0.05
MGT (f = 20) GFEM Analytical
G0 = 0.1
G0 = 0.5
m0
m1
m2
m0
m1
m2
m0
m1
m2
7.82 × 10−5 6.00 × 10−5 5.00 × 10−5
8.26 × 10−8 6.36 × 10−8 5.26 × 10−8
8 × 10−9 6.5 × 10−9 5 × 10−9
1.37 × 10−4 1.09 × 10−4 1 × 10−4
1.52 × 10−7 1.21 × 10−7 1.11 × 10−7
2.1 × 10−8 1.4 × 10−8 2 × 10−8
6.15 × 10−4 5.14 × 10−4 5.00 × 10−4
1.23 × 10−6 1.02 × 10−6 1 × 10−6
5.8 × 10−7 7.1 × 10−7 7.33 × 10−7
GFEM (ne = 30, np = 3, nt = 61), MGT (nt 0 = 10, f = 20, nt = 60).
In this case, the moments mainly depend on the values of the number density function near V = Vmin . In fact, in this part of the distribution, the particle growth rate becomes much smaller than the particle aggregation rate. As a result, the distribution is particularly sensitive to the initial discontinuity at t = 0. Consequently, neither method can correctly predict the actual numerical values of the moments and have difficulties resolving the aggregation-dominant behavior of the distribution at small times. This is a particularly difficult problem to solve, especially, for small values of the growth rate constant G0 (i.e., particle aggregation rate is dominant, see Table 3). Again, the accuracy of the MGT is directly linked with the value of the bin addition frequency parameter f . For example, for G0 = 0.05, the numerical values of the leading moments of the distribution, calculated by the MGT with f = 5, were: m0 = 5.62 × 10−5 and m1 = 5.92 × 10−8 . These values are significantly closer to those given by the analytical solution (see Table 3). On the other hand, the numerical values of the leading moments calculated by the GFEM, using a total number of 101 nodal points, were: m0 =5.14×10−5 and m1 =5.41×10−8 . Clearly, an increase in the number of nodal points improves the accuracy of the GFEM. However, it should be noted that an increase in the number of points in the GFEM is much less computationally demanding than a decrease in the parameter f in the MGT.
3.4. Constant particle aggregation and growth with a Gaussian particle inflow stream Subsequently, the time-evolution of the distribution in a continuous flow reactor, undergoing combined constant particle aggregation and constant particle growth at small residence times was examined. It was assumed that the particle inflow stream followed a normal Gaussian distribution (see case (d) in Table 1). In Fig. 8, the steady-state PSDs calculated by the GFEM and the MGT for an aggregationdominated case (i.e., 0 = 100, G0 = 1) are plotted. It can be seen that the predicted distributions by the two numerical methods are almost identical. It should be noted that an increase in the aggregation rate constant results in significantly smaller values of the number density function but the resulting distribution qualitatively has a similar shape. In the inset of Fig. 8, the effect of the number of points, nt 0 , of the initial grid on the resolution of the steady-state secondary peaks is depicted. It is evident that as nt 0 increases the resolution of the secondary peaks improve dramatically. In order to minimize the domain error in the calculation of the leading moments, an appropriately large volume domain had to be selected. Thus, for a value of the aggregation rate constant 0 = 1, 10 and 100, the corresponding values of Vmax in the GFEM and the final volume in the MGT
A.I. Roussos et al. / Chemical Engineering Science 61 (2006) 124 – 134
0.15
Volume Density Function
GFEM MGT
0.06
131
GFEM MGT (nt0 = 80) MGT (nt0 = 40) MGT (nt0 = 20)
0.05 0.04 0.03
0.1
0.02 0.01 0 15
0.05
20
25
30
35
40
45
0 0
10
20
30 Volume
40
60
50
Fig. 8. Comparison of steady-state PSDs for combined constant particle aggregation and growth and a Gaussian particle inflow stream. (Case (d) in Table 1: 0 = 100, G0 = 1, = 1, = 2). GFEM (nt = 121); MGT (nt 0 = 80, f = 40, nt = 105).
Table 4 Numerical parameters for the growth dominant case: ( = 1, 0 = 1, nt 0 = 30) GFEM
G0 = 10 G0 = 20 G0 = 30
MGT
Vmax
nt
CPU time (s)
f
nt
CPU time (s)
200 300 400
93 97 101
108 224 348
18 9 6
86 141 196
298 1196 3454
0.2 Dynamic Steady-State Volume Density Function
0.15
0.1 t=0.1 t=0.2
t=10
0.05
0 0
10
20
30 Volume
40
50
60
Fig. 9. Dynamic PSDs calculated by the GFEM for combined constant particle aggregation and growth and a Gaussian inflow stream. (Case (d) in Table 1: 0 = 100, G0 = 1, = 1, = 2).
were equal to 102 , 103 , and 104 (Table 4). In Fig. 9, the dynamic PSDs calculated by the GFEM for an aggregationdominated case are depicted. The distributions display four distinct peaks at discrete multiples of V0 = 10.
In Fig. 10, the steady-state distributions calculated by the GFEM and the MGT for a growth-dominated case (i.e., 0 = 1, G0 = 30) are plotted. It can be seen that the predicted distributions by the two numerical methods are almost
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0.03
Number Density Function
0.023 0.02 GFEM MGT (f = 6) MGT (f =15) MGT (f =20)
0.018
0.01
0.013 7
13
19
25 GFEM MGT
0 0
20
40
60
80
100
Volume Fig. 10. Comparison of steady-state PSDs for combined constant particle aggregation and growth and a Gaussian inflow stream. (Case (d) in Table 1: 0 = 1, G0 = 30, = 1, = 2). GFEM (nt = 101), MGT (nt 0 = 30, f = 6, nt = 196).
identical. However, in these growth-dominated cases, the MGT has an inherent difficulty in properly resolving the first peak of the distribution. In the inset of Fig. 10 the effect of the grid updating frequency on the resolution of the main peak of the steady-state PSD is depicted. It is evident that as the update frequency increases (i.e., value of the parameter f decreases) the resolution of the main peak of the PSD improves significantly. Therefore, a more frequent addition of new grid points is required leading to a large number of nodal points. On the other hand, for a growth-dominated process, the initial grid of the MGT does not need to be as fine as the one required for an aggregation-dominated process (Fig. 8). In general, for a volume-dependent growth rate function, G(V ), the volume increase, V , of a particle of initial size Vmin over a time period of t will be given by
Vmin +V Vmin
dV = t. G(V )
(4)
Thus, if the moving grid is not properly updated, a fraction of the particles in the inflow stream corresponding to the volume section [Vmin , Vmin + V ] will be lost during the time period, t. In general, the fraction of particles lost from the inflow stream, Eloss , will be equal to: Eloss =
Vmin +V
Vmin
nin (V ) dV
Vmax
Vmin
nin (V ) dV .
(5)
Eq. (5) is used to control the rate of grid updating. For a constant growth rate, the volume increase of a particle size of Vmin in two successive updates of the grid will be equal to:
V = G0 tend f/nsteps,
(6)
where tend is the total simulation time and “nsteps” is the total number of time integration steps from t=0 to tend . It was found that a simulation time of approximately tend =10 was required to reach the final steady-state. For a normal inflow distribution, nin (V ) Eq. (5) can be simplified by assuming that the grid is updated when V . Thus, the following expression for the grid update parameter, f , is obtained: f = · nsteps/10G0 .
(7)
According to Eq. (7), for = 1, G0 = 30, nsteps = 1000, and = 2 the optimal value of the grid update parameter is f = 6 as depicted in Fig. 10. Overall, by proper selection of the initial grid and the grid update parameter, the main and secondary PSD peaks can be properly resolved, in which case the two methods (i.e., the MGT and GFEM) predict identical PSD moments. The effect of the aggregation rate constant on the CPUtime of the MGT and GFEM was examined for an aggregation dominated case. The numerical parameters of both techniques were changed in order to correspond to similar grids and steady-state PSDs of similar accuracy for the aggregation-dominated case. Specifically, the volume domain of the GFEM was increased at a constant node density (on a logarithmic volume scale) and the initial grid of the MGT was also increased. The required CPU-time of the MGT was found to be approximately seven times the CPUtime required by the GFEM for all values of the aggregation rate constant. In Table 4, numerical details regarding the implementation of the two methods are reported, for a growth dominated case and the effect of the growth rate constant on the CPU-time is shown. Note that, in contrast to the case with 0 =1 and G0 = 1 (for which Vmax = 100) a slightly larger volume domain was required (due to the larger growth rate) to minimize the domain error. Thus, for a value of G0 equal to 10, 20 and
A.I. Roussos et al. / Chemical Engineering Science 61 (2006) 124 – 134
30, the corresponding values of Vmax in the GFEM and the final volume in the MGT were 200, 300 and 400. In general, larger growth rates required a more frequent addition of new grid points in the MGT to obtain an adequate resolution of the main distribution peak. Finally, as can be seen in Table 4, the required CPU-time for the GFEM increases linearly with the growth rate constant. On the other hand for the MGT, it follows a geometric dependence with respect to the value of G0 .
4. Conclusions In the present study, two numerical methods (i.e., the GFEM and the MGT) were employed to follow the dynamic evolution of the PSD in continuous flow reactors. It was found that the MGT was in general more stable but not as accurate as the GFEM. For aggregation-dominant problems, the MGT displays the typical diffusion-like error of the discretized population balance methods. The MGT is perfectly suited for combined aggregation and growth problems in the absence of a fixed-volume particle source. However, in the presence of such sources, the MGT requires frequent updating of the discretization grid, leading to a large number of points. For problems with a fixed-volume particle source, the GFEM with proper discretization of the volume domain was more accurate and efficient than the MGT. It should be emphasized that, the MGT requires a frequent addition of new grid points to resolve the first peak of the distribution, especially for growth dominated problems. Moreover, a sufficiently fine initial grid is required to resolve the secondary peaks of the PSD, especially for aggregationdominated problems. In order to render the MGT applicable to a wide-range of particulate processes under the combined action of particle aggregation, growth, nucleation and a particle inflow stream the updating of the moving-grid must become more sophisticated. For example, the performance of the MGT can be improved by allowing for the removal of unnecessary grid points or/and the variation of the grid-point addition frequency with time.
Notation f G0 G G mi nej n
grid updating parameter growth rate constant, m3-a /s growth rate function, m3 /s derivative of the growth rate function “ith” dimensionless moment of the number density function number density function at node j in element e, m−6 number density function, m−6
nin
133
number density function in the inflow stream, m−6 average number density function in element i, m−6 total number of elements total number of nodes in a finite element total number of time integration steps number distribution in the“i” element characteristic particle number, m−3 weighted residual of PBE at node i of element e time, s total simulation time, s volume, m3 characteristic particle volume, m3 minimum particle volume, m3 maximum particle volume, m3
n¯ i ne np nsteps Ni N0 Rie t tend V, U V0 Vmin Vmax
Greek letters
0 j
constant aggregation rate kernel, m3 /s standard deviation of inflow distribution, m3 residence time, s Lagrange basis functions
Appendix A. Solution of the PBE with the GFEM In order to form a weighted residual statement the PPE is multiplied with an appropriate weighting function and integrated over the volume domain. The weighted residuals at the i point of element e have the form: Rie
e Vnp
jn(V , t) jG(V )n(V , t) + jt jV nin (V , t) − n(V , t) + − B(V ) + D(V ) dV = 0, (A.1)
=
V1e
ei (V )
where B(V ) and D(V ) are the aggregation birth and death terms and ei (V ) are the polynomial Lagrange weighting function of order np − 1. The unknown number density function n(V , t), is approximated over each element in terms of its nodal values according to Eq. (3). The solution is forced to be continuous at element boundaries by considering the global element contributions. After applying continuity, Eq. (A.1) is transformed in a system of ne(np−1)+1 ODEs which can be written in matrix notation as [A]ij · +
jnv (V , t) e 1 e + · [A] + [E] + [C] ij ij ij · nj jt j
1 · [F ]i − [B]i + [D]i = 0,
(A.2)
134
A.I. Roussos et al. / Chemical Engineering Science 61 (2006) 124 – 134
where the matrices [A]ij , [E]ij , [C]ij , [F ]i , [B]i and [D]i have the form: Vnp e [A]ij = ei (V )ej (V ) dV , [E]ij = [C]ij =
V1e Vnp e
V1e Vnp e V1e e Vnp
[F ]i =
V1e
[D]i =
dV
e Vnp
e Vnp
V1e
×
ei (V )nin (V , t) dV , ei (V )
Vmax Vmin
n(V , t)
(V , U )n(U, t) dU dV .
(A.3)
The calculation of the matrices is done numerically, using an appropriate Simpson’s or Gauss quadrature rule (Roussos et al., 2005). Appendix B. Solution of the PBE with the MGT In the MGT the population of particles is assumed to exist only at characteristic sizes Vi , which are called pivots (Kumar and Ramkrishna, 1996). If a particle is born due to aggregation in an intermediate size U , it is automatically assigned to the two adjoining pivots in a way that any two desired moments of the distribution are preserved. For the preservation of total particle number and volume, the equation that describes the variation of the particle number at the characteristic size Vi has the form: dNi = dt
j k j,k Vi−1 (t) (Vj (t)+Vk (t)) Vi+1 (t)
agg.
× i,k,j j,k Nj Nk − Ni
1 1 − j,k 2
ne k=1
i,k Nk ,
(B.1)
where agg.
i,k,j
(B.3)
References
G (V )ei (V )ej (V ) dV ,
ei (V )
dVi = G(Vi ). dt
dV ,
V −Vmin 1 (V − U, U ) e 2 V1 Vmin × n(V − U, t)n(U, t) dU dV ,
[B]i =
G(V )ei (V )
dej (V )
In order to account for the contribution of particle growth the pivots Vi move to larger sizes with a velocity equal to the particle growth rate, thus
V (t) − U i+1 , Vi (t) U Vi+1 (t), Vi+1 (t) − Vi (t) = U − Vi−1 (t) , Vi−1 (t) U Vi (t). Vi−1 (t) − Vi−1 (t)
(B.2)
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