Chapter 2 Literature review
2.4 Numerical techniques for the solution of population balance equations
2.4.2 Numerical nonlinear model reduction approaches
2
2 0 2 1
var 2
0 0
0
( ) ( , )
( , )
m n
n
L L f L t dL
f L t dL
. (2.13)
The coefficient of variation, which quantifies the width of the distribution function relative to its mean:
var 0 2
2 1
. . 1
m
c v L . (2.14)
Number mean size providing information about the mean particle size (more sensitive for small particles sizes):
1 10
0
d . (2.15)
Weight mean crystal size, which provides information about the mean particle size (with more sensitivity towards larger particles):
43 4
3
d . (2.16)
Sauter mean diameter, which gives information related to the ratio between the volume of the dispersed phase to its surface area:
32 3
2
d . (2.17)
2.4.2 Numerical nonlinear model reduction approaches
The approaches considered in this category are commonly called “global methods” and are also considered as nonlinear model reduction techniques for the PBE. This category includes different variants of the “method of weighted residuals”, which have been frequently used in
the crystallisation-control community for general solution of the PBE. The algorithms are based on the idea of approximating the population density function (pdf) as a linear combination of chosen basis functions. The coefficients of the linear combination are obtained by minimizing the residuals between the approximation and the distribution such that for a chosen set of weighting functions, the residual will be orthogonal and hence will represent the pdf with a small number of terms. There are various possibilities to select the basic functions; e.g. Laguerre polynomials have been used with very good computational performance in Non-Linear Model Predictive Control (NMPC) schemes for crystallisation processes (Rawlings et al., 1993). The difficulty associated with these techniques is related to the choice of the proper basis functions. Note that the SMOM can be also considered as a model reduction technique, being the particular case when the weighting functions are considered as polynomials. Generally, the accuracy of the weighted residual approaches depends on the set of basis functions used for approximation. However it has been shown that if the only mechanism that governs the crystallisation is growth, good accuracy can be expected. This has been demonstrated for example for cases with growth rate proportional to the inverse of the characteristic crystal size by Kalani and Christofides, (2002).
Quadrature method of moments (QMOM)
The first approach discussed in this category is the quadrature method of moment (QMOM).
The method was proposed by McGraw (1997) and has been used for a wide variety of applications of population balance models. The method is a particular case of the generic weighted residual approach, which uses a particular form for the basis function (quadrature approximation given by the quadrature theory) that allows an explicit calculation of the weights from the moments, hence providing a solution to the moment closure problem from the SMOM. The QMOM proposed by McGraw (1997) is based on the product difference (PD) algorithm (Gordon, 1968). The PD algorithm is employed to calculate the weights (wi) and the abscissas (Li) from the moments following the quadrature approximation (McGraw, 1997),
0 1
( ) , where 0,1,2,..., .
Nq
k k
k n i i
i
f L L dL w L k (2.18)
where Nq is the number of quadrature points, wi are the corresponding quadrature weights and Li are the abscissas, which can be determined through the product-difference (PD) algorithms or via direct solution of a differential-algebraic equation (DAE) system (Gimbun et al., 2009), based on the idea of minimizing the error committed by replacing the integral from the moment definition with its quadrature approximation.
After applying the quadrature rule, equation (2.7) can be written as (Alopaeus et al., 2006):
1
1 1
nucleation
growth birth due to breakage
3 3 3
1 1
birth due to Aggregation
(0, ) ( )+ ( ) ( , )
1 ( ) ( , )
2
N N
k k
i i i i i i
i i
N N k
i j i j i j
i j
d k B k w L G L w g L b k L dt
w w L L F L L
1
death due to breakage
1 1
death due to aggregation
( )
( , )
N k
i i i
i
N N
k
i i j i j
i j
w g L L
w L w F L L
, (2.19)
Now the closure problem has been eliminated and hence the PBE given by (2.7) is solvable by means of the quadrature method of moments by following the evolution of wi and Li. The QMOM gives a coarse approximation of the change in the shape of the CSD, however the weights and abscissas do not directly provide the shape of the distribution. The application of the QMOM has been extended to aggregation, coagulation and breakage mechanisms (Fan et al., 2004; Marchisio et al., 2003b; Rosner et al., 2003; Rosner and Pyykonen, 2002; Wright et al., 2001).
The PD algorithm however is not always the best approach for computing the Gauss quadrature approximation, since the computation of the weights and abscissas is sensitive to small errors in the moments. Thus, the applicability of QMOM is limited to no more than six quadrature points (Gordon, 1968) and often even fewer for some cases, such as diffusion-controlled growth with secondary nucleation. Apart from the product difference algorithm based QMOM, there are several other variations of the approach, such as fixed quadrature method of moments (FQMOM) (Alopaeus et al., 2006), Jacobian matrix transformation (JMT) (McGraw and Wright, 2003) and direct quadrature method of moments (DQMOM) (Fan et al., 2004); however these techniques are not widely used.
Method of characteristics (MOCH)
Another efficient technique to solve population balance equations is the method of characteristics (MOCH) (LeVeque, 1992). The technique provides an elegant way to determine the evolution of the crystal size distribution for crystallisation processes. It has been used for the crystal size determination in case of size-independent growth and nucleation by many researchers (see e.g. Hounslow and Reynolds (2006)). The method of characteristics for first order PDEs determines lines, called characteristic lines (or characteristics), along which the PDE degenerates into a set of ODEs. The ODE can be solved and transformed into a solution for the original PDE. The equation for each characteristic line L t( ) is:
0
( ) ( ( ))
t
L t G S t dt, i.e. dL G t( )
dt . (2.20)
To illustrate the technique it’s application to the solution of the PBE is illustrated next for a crystallisation process with size-independent growth the only governing phenomenon.
The population balance equation for a crystallisation process with one dimensional growth and no nucleation can be obtained by rearranging equation (2.5),
( , ) ( , ) ( , )
n 0
n G S L f L t
f L t
t L , (2.21)
where fn is the pdf, which can be a function of the characteristic size (L) and time (t). If the growth rate G only depends on the supersaturation S t( ), equation (2.21) can be transformed into a homogeneous hyperbolic equation,
( , ) ( , )
( ) 0
n n
f L t f L t
t G S L . (2.22)
The aim of the method of characteristics is to solve the PDE by finding curves in the (L t) plane, which reduces the partial differential equation to a system of ODEs. The (L t) plane can be expressed in a parametric form by L L( ) and t t( ), where the parameter gives the measure of distance along the characteristic curve. Therefore,
f L tn( , ) f Ln( ( ), ( ))t , (2.23)
and applying the chain rule gives,
dL fn dt fn dfn
d L d t d . (2.24) Comparing equations (2.22) and (2.24) a set of ODEs can be derived:
dt 1
d , (2.25)
dL G S( )
d , (2.26)
n 0 df
d , (2.27)
with initial conditions (corresponding to 0) t 0, L L0 andf Ln( , 0) fn,0( )L0 . From equation (2.25)-(2.27) t and the actual characteristic equations can be written as:
dL ( )
dt G S , (2.28)
0 dfn
dt . (2.29)
To obtain the dynamic evolution of the crystal size distribution,f L tn( , ), equation (2.26) and (2.27), with prescribed growth expressions can be integrated repeatedly for different initial values [ ,L f0 n,0( )]L0 where fn,0( )L0 is the seed distribution. The initial conditions start from along the L axis of the L t plane, with values calculated by choosing a discretisation interval L0 and using t0 0 and L0 max(0,L0,max k L0), k 0,1, ,N , where N is the number of discretisation points. The discretisation interval L0 will determine the number of integrations (the number of characteristic lines) and hence the resolution of the dynamic evolution of the seed CSD. The growth rate is a function of supersaturation, which is changing with time, S t( ) which can be determined from the mass balance and the third moment of the distribution obtained from the SMOM, using equation (2.8). Growth rate is a function of the supersaturation, G k Sg g, with g and kg being the growth kinetic parameters. The supersaturation (S) is expressed as the difference between the solute concentration C t( ) and equilibrium concentration Csat( )T at time t, S t( ) C t( ) C Tsat( ). The equilibrium concentration (solubility) is a function of the temperature (T), which is
generally a function of time during the batch, T t( ), hence Csat Csat( )t . The solute concentration can be calculated from the material balance:
C t( ) C(0) kv c( ( )3t 3(0)), (2.30)
where c is the density of crystals and kv is the volumetric shape factor. The third moments are computed from the SMOM.
To illustrate the methodology a simulation was performed for a real crystallisation process (crystallisation of potash alum in water) using the experimental conditions described in Chapter 5, however considering the following fictitious size-independent growth parameters:
growth rate constant kg 12.5 m s/ and growth order g 1.2. A batch time of 5400 s was used in the simulations. The actual temperature trajectory from the experiment (shown in Figure 5.4(a)) was used for the simulation. Because of the particular experimental conditions the supersaturation was changing during the batch. The initial conditions for the simulations were given based on the initial experimental concentration and on the measured seed distribution used in the experiments (described in Chapter 5).
Figure 2.4 (a) illustrates the evolution in time of the characteristic lines for the crystallization process using the size independent growth mechanism. All characteristic lines have the same slope at a given time and the pdf values are constant during the whole batch according to the size-independent growth mechanism and no nucleation. The CSD can be represented at any time by plotting the pdf values versus the corresponding L values obtained from the characteristic lines. The dynamic evolution of the CSD during the batch is illustrated in Figure 2.4 (c).
0 1000 2000 3000 4000 5000 0
200 400 600
Time (s)
Characteristic Length (m)
0 1000 2000 3000 4000 5000
0 5000 10000 15000
Time (s)
Number Density Function (a)
(b)
0 1000
2000 3000
4000
5400 0 100 200 300 400 500 600
0 5000 10000 15000
Crystal size (m) Time (s)
Number PDF
(c)
Figure 2.4: Simulation results of a crystallisation process with size-independent growth, using the method of characteristics. a) Characteristic lines for size (the slopes of all characteristic lines are the same due to the size independent growth mechanism). b) Characteristic lines for the
( , )
f L tn (showing constant values due to size independent growth and no nucleation). c) Evolution of the CSD obtained from the characteristic lines at different time steps.