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1. Chemical Engineering Science 61 (2006) 5070–5081
www.elsevier.com/locate/ces
Application of the least-squares method for solving population
balance problems in Rd+1
C.A. Dorao, H.A. Jakobsen∗
Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Received 1 June 2005; received in revised form 9 February 2006; accepted 6 March 2006
Available online 20 March 2006
Abstract
In multiphase chemical reactor analysis the prediction of the dispersed phase distribution plays a major role in achieving reasonable results.
The combined CFD–PBE (population balance equations) are computationally intensive requiring efficient numerical methods for dealing with
them. This paper presents the formulation and validation of a spectral least squares method for solving the steady state population balance
equations in Rd+1, with d the physical spatial dimension and 1 the internal property dimension. The least-squares method consists in minimizing
the integral of the square of the residual over the computational domain. Spectral convergence of the L2-norm error of the solution and of the
moments of the solution are verified for the zero- and one-dimensional cases using model problems with analytical solutions.
䉷 2006 Elsevier Ltd. All rights reserved.
Keywords: Chemical reactors; Population balance; Multiphase reactors; Multiphase flow; Least-squares method
1. Introduction
Population balance modeling is an active field of research due
to its application to several engineering and scientific problems.
This method is commonly used to study precipitation, polymer-
ization, particle size distribution, dispersed phase distribution
in multiphase flow problems, and so on. In particular in multi-
phase flow problems, the dispersed phase distribution presents
a strong effect in the hydrodynamic properties and phase distri-
bution. For that reason, considerable efforts have been made in
order to develop polydispersed multi-fluid models with an in-
herent population balance module that will be able to consider
the effects of the variations in the size and shape distributions
of the dispersed phase. In particular, the present status on PBE
modeling of bubbly flows has been examined (Jakobsen et al.,
2005).
Using a population balance approach the dispersed phase is
commonly treated using a density function, DF, for instance
f (r, , t) where r is the spatial vector position, is the prop-
erty of interest of the dispersed phase, and t the time. Thus,
f (r, , t) d can represent for example the average number of
∗ Corresponding author. Tel.: +47 73 594132; fax: +47 73 594080.
E-mail address: hugo.jakobsen@chemeng.ntnu.no (H.A. Jakobsen).
0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2006.03.019
particles per unit volume around the point x in the time t, with
the property between and +d. The evolution of this density
function must take into account the different processes that con-
trol particle population such as breakage, coalescence, growth
and convective transport of the particles. The resulting equa-
tion is a nonlinear partial integro-differential equation which
requires to be solved by a suitable numerical method, although
analytical approximations can sometimes be derived for some
particular cases, see for example Patil and Andrews (1998).
The method of moments is an efficient method to solve the
PBE, but it is only applicable to a limited number of problems
and gives no information about the shape of the distribution.
For example, Frenklach (1985) applied the method of mo-
ments to a coagulation process where the coagulation rate
was constant. For an arbitrary coagulation rate function, this
formulation results in an excess of unknowns compared to the
number of equations whichh is denoted as a closure problem
(Hulburt and Katz, 2003). One way to avoid this problem is to
assume the shape of the density function, and so the parame-
ters of the density function are related, closing off the set of
moments equations (Williams, 1986). A different alternative is
to express the density function as a truncated series of some
orthogonal polynomials (Hulburt and Katz, 2003). McGraw
(1997) suggested a modification of the method of moment

2. C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081 5071
which consists in using a quadrature approximation in order to
avoid the closure problem. McGraw (1997) based his method
on the product difference (PD) algorithm suggested by Gordon
(1968), calling this approach quadrature method of moments
(QMOM). Marchisio et al. (2003) and Marchisio et al. (2004a)
used this approach to study particulate systems. Later, McGraw
and Wright (2002) proposed a new moment closure method,
the Jacobian matrix transformation (JMT) which avoids the use
of the PD algorithm. Similarly, Marchisio et al. (2004b) ex-
tended the QMOM method to multifluid applications calling
this approach the direct quadrature method of moments (DQ-
MOM). Bove et al. (2005) presented the parallel parent and
daughter classes (PPDC) which used the PD algorithm for re-
ducing the computational cost of the QMOM. It is important
to mention that the PD algorithm is a numerical ill-conditioned
method for computing the Gauss quadrature rule (e.g. Lambin
and Gaspard, 1982). In general, the computation of the quadra-
ture rule based on the power moments of the density function is
quite sensitive to small errors as the number of moments used
becomes large (e.g. Golub and Welsh, 1969; Gautschi, 1994).
Therefore, the applicability of QMOM is limited to no more
than 12 moments, although in certain applications it is claimed
that only a few moments are enough for obtaining reliable re-
sults (McGraw et al., 1997).
A different way of avoiding the closure problem is discussed
by Frenklach (2002) in the method of moments with interpola-
tive closure (MOMIC). In this case, the natural logarithmic of
the moments is expressed by a polynomial in the moment or-
der, and thus, the required moments are interpolated or extrap-
olated. Further discussion about the closure for the method of
moments can be found in Diemer and Olson (2002).
An alternative strategy is to employ projection methods, such
as finite element methods (FEM), in which the solution is ap-
proximated as a linear combination of the basis functions over
a finite number of sub-domains. Chen et al. (1996) developed
a wavelet-Galerkin method for solving PBEs for the treat-
ment of particle-size distribution in problems of a continuous,
mixed-suspension and mixed-product removal crystallizer with
effects of breakage. Niemanis and Hounslow (1998) applied
FEM to the steady-state PBE, finding more accurate solutions
than using the sectional methods and using less computational
power. Liu and Cameron (2001) proposed the use of a wavelet-
based method for the treatment of problems involving particle
nucleation, growth and agglomeration. Niemanis and Houn-
slow (2002) showed an a posteriori error estimate of the FEM
applied to PBE. A posteriori error estimate is an important
characteristic of the projection methods, which allows to quan-
titatively assess the quality of an obtained numerical solution.
This characteristic is not commonly presented in the previous
discussed methods. Due to the fact that for some applications
such as chemical reactor simulations the computational cost of
the solver of the PBE requires to be reduced, high-order poly-
nomial approximation methods could be an option, improving
the behavior of FEM solvers. The global approximation of the
solution, compared with the local one of FEM or sectional
methods, permits to reduce the final computational cost since
less points are required for the same accuracy. These methods,
including FEM, can be presented in the framework of the
methods of weighted residuals (MWR). Depending on the
election of the trial and test functions different methods can
be obtained (Canuto et al., 2000). Subramain and Ramkrishna
(1971) presented a Tau method for solving the distribution of
the population of microbial cells that present growth and break-
age processes. Mantzaris et al. (2001) discussed the Galerkin,
Tau and pseudo-spectral methods as a tool for solving multi-
variable cell population balance models that present growth
and breakage. Recently, Dorao and Jakobsen (2005a, 2006)
discussed the applicability of the least squares method (LSM)
(Jiang, 1998a; Bochev, 2001; Proot and Gerritsma, 2002;
Pontaza and Reddy, 2003) for solving the population balance
equation.
The LSM consists in finding the solution which minimizes
the L2 norm of the residual over the computational domain.
The LSM can also be considered a special case of the MWR
where the trial and test functions are equal to the residual
equation (Finlayson, 1972). The interest on the LSM has in-
creased quite a lot during the last decade as a consequence of its
properties:
• Independent of the underlining equation, least squares al-
ways leads to symmetric positive-definite systems of linear
algebraic equations, which can be efficiently solved.
• For first order problems, e.g. an advective transport equation,
the LSM does not require any special numerical treatment
like the up-wind discretization in the case of finite difference.
Thus, no numerical diffusion is introduced.
• The evaluation of the accuracy of the approximate solution
is many areas of engineering and applied science is rather
important. The LSM meets the need for a posteriori error
analysis by supplying an error indicator in the form of the
residuals that are minimized by the procedure. In particular,
this is a very reliable indicator which can be used for example
for grid refinement.
• Finally, the LSM is formulated in a very general setting.
Thus, the programming can be done in a very systematic
way and new applications requires a minimum work reduc-
ing drastically cost and programming errors in code devel-
opment.
Dorao and Jakobsen (2005a, 2006) applied the least-squares
spectral method to the population balance equation involving
breakage and coalescence processes using Legendre polynomi-
als for the particle space discretization and Crank–Nicolson for
the time discretization. Thus, the solver shown spectral con-
vergence in the property space while algebraic convergence
rate for time. Later, Dorao and Jakobsen (2005b) discussed the
space–time least-squares formulation for solving the PBE. In
this space–time formulation, time is treated as an additional di-
mension, which allows high-order accuracy both in space and
in time (e.g. De Maerschalck, 2003; Pontaza and Reddy, 2004).
In this way, space–time can be solved at once, or per time-step
on a space–time slab in a kind of semi-discrete formulation.
The main goal of this paper is to extend the previous mathe-
matical framework for solving the PBE including the physical

3. 5072 C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081
spatial dimensions, i.e., in Rd+1
with d = 0, 1, 2, 3 the spatial
dimension and with 1 the property dimension. The focus in this
work is on a steady state population balance equation. In order
to emphasize the relevant numerical issues. Time-dependent
problems can easily be derived following the work of Dorao
and Jakobsen (2006) for the case of finite difference approxi-
mation for the time, or the work of Dorao and Jakobsen (2005b)
for the case of spectral approximation for the time.
In Section 2, the population balance equation is presented.
Section 3 discusses the LSM, presenting the general derivation.
The computational cost for the general problem is discussed
in Section 5. In Section 4 the main implementation issues are
discussed for the zero- and one-dimensional case. In Section 6,
numerical experiments are performed for studying the conver-
gence property of the method. Finally, Section 7 presents the
main conclusions of this work.
2. The population balance equation
The population balance equation models the evolution of a
density function, DF, representing the behavior of a population
of particles such as bubbles, droplets or solid particles. The
evolution of this density function must take into account the
different processes that control the population such as break-
age, coalescence, growth and convective transport of the parti-
cles. The PBE without considering advection or growth can be
written as
jf (r, , t)
jt
= Lf (r, , t) + g(r, , t) in (1)
with r is the spatial coordinate, the property coordinate and t
the time. The domain is give as =sp × ×[0, T ] where
sp ∈ Rd
with d = 0, 1, 2, 3 is the spatial domain, ∈ R1
is
the property domain and [0, T ] is the simulation time interval.
The first term on the left-hand side of Eq. (1) represents the
change with respect to time of the density function. The first
term on the right-hand side of Eq. (1) represents the change of
the density function due to different processes that affect the
population such as breakage and coalescence. The last term,
g(r, , t) is a sink or source of particles, representing some
mechanism that introduces or removes particles from the sys-
tem.
In this work, we focus on the steady-state case which can be
expressed as
Lf (r, ) + g(r, ) = 0 in (2)
with =sp ×. In this case, the population balance operator
is given like
Lf (r, ) = − b(r, )f (r, ) +
h(r, , s)b(r, s)f (r, s) ds
− f (r, )
a(r, , s)f (r, s) ds
+
1
2
a(r, − s, s)f (r − s, )f (r, s) ds, (3)
where the first term in the RHS represents the change in
the population due to loss of individuals in the population,
for example due to a breakage process, where b(r, ) is the
breakage rate of the particles of type . The second term
in the RHS gives us the change in the population due to
the arrival of new individuals with property . In the case
of a breakage process, the breakage of particles of type s
will produce particles of type according to the break-
age yield function, h(r, , s). For simplicity, we can write
k(r, , s) = h(r, , s)b(r, s). The third term in the RHS rep-
resents the change in the population due to the loss of indi-
viduals due to pair interactions such as a coalescence process;
a(r, , s) is the coalescence rate between particles of type
and s. Finally, the last term in the RHS represents the arrivals
of new individuals due to the pair interaction. One particle of
type 1 that coalesces with a particle of type 2 will produce
a particle of type = 1 + 2. Depending on the modeled
process, the definition of the operator (3) can present some
variations (e.g. Ramkrishna, 2000). Due to the fact that the
focus of this work is on mathematical issues more than in the
modeling ones, we consider the previous PBE like a general
case.
3. The least-squares spectral method
The LSM is a well-established numerical method for solving
a wide range of mathematical problems, (e.g. Jiang, 1998a;
Bochev, 2001; Proot and Gerritsma, 2002; Pontaza and Reddy,
2003). The basic idea in the LSM is to minimize the integral
of the square of the residual over the computational domain. In
the case when the exact solutions are sufficiently smooth the
convergence rate is exponential. For time-dependent problems,
the space–time formulation, i.e., time is treated as an additional
dimension, allows high-order accuracy both in space and in
time (e.g. De Maerschalck, 2003; Pontaza and Reddy, 2004).
In this way, space–time can be solved at once, or per time-step
on a space–time slab in a kind of semi-discrete formulation.
The application of LSM to PBE were previously discussed by
Dorao and Jakobsen (2005a,b, 2006).
The least-squares formulation is based on the minimization of
a norm-equivalent functional. This consists in finding the min-
imizer of the residual in a certain norm. The norm-equivalent
functional for Eq. (2) is given by
J(f ; g) ≡ 1
2 Lf + g2
Y() (4)
with the norm • 2
Y() defined like
• 2
Y() = •, •Y() =
• • d. (5)
Based on variational analysis, the minimization statement is
equivalent to:
find f ∈ X() such that
lim
→0
dJ(f + v; g, f0)
d
= 0 ∀v ∈ X(), (6)

4. C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081 5073
where X() is the space of the admissible functions. Conse-
quently, the necessary condition can be written as:
find f ∈ X() such that
A(f, v) = F(v) ∀v ∈ X() (7)
with
A(f, v) = Lf, LvY(), (8)
F(v) = g, LvY(), (9)
where A : X × X → R is a symmetric, continuous bilinear
form, and F : X → R a continuous linear form.
The discretization statement consists in searching the solu-
tion in a reduced subspace, i.e., fN (r, ) ∈ XN () ⊂ X().
Therefore, fN can be expressed like
fN (r, ) =
N
l=0
fll(r, ), (10)
where l(r, ) are multidimensional basis functions. Inserting
approximation (10) into Eq. (7), and choosing systematically
v = 0, . . . , N, we get the final algebraic system
Af = F, (11)
where the matrix A ∈ R(N+1)×(N+1)
, and vectors F, f ∈
R(N+1)×1
are defined as
[A]ij = A(j , i) = Lj , LiY(), (12)
[F]i = F(i) = g, LiY(), (13)
[f]i = fi. (14)
The final system of equations is symmetric, positive definite,
so the solution of such a system can be obtained in an efficient
way using standard matrix solvers. The algorithm presented in
this section can be applied for solving the population balance
equation in d spatial dimensions. It is important to mention that
the mathematical framework presented in this section is quite
general, being suitable for solving a large range of complex
problems. Further details about the mathematical derivation,
properties and application of the LSM can be found in Jiang
(1998a), Bochev (2001), Proot and Gerritsma (2002), Pontaza
and Reddy (2003, 2004).
In the next section a zero- and one-dimensional cases will
be discussed in detail. In particular, the main practical issues in
the discretization step will be highlighted, i.e., we will focus on
how to obtain the final matrix A and the vector F in Eq. (11).
4. Application examples
4.1. Zero-dimensional population balance equation
The zero-dimensional PBE considers only the property de-
pendency of the density function, f (), i.e., the physical spatial
effects are ignored. Thus, a zero-dimensional PBE considering
only breakage can be written like
Lf () + g() = 0 in = [0, L] (15)
with g() the source term, and Lf () the breakage operator
defined as
Lf () = −b()f () +
L
0
k(, s)f (s) ds. (16)
4.1.1. The least-squares spectral approximation
The least-squares functional is defined like
J(f ; g) ≡ 1
2 Lf + g2
Y() (17)
with the norm
• 2
Y() = •, •Y() =
• • d =
L
0
• • d. (18)
The discretization statement consists in searching the solu-
tion in a reduced subspace, i.e., fN () ∈ XN () ⊂ X().
Therefore, if we use high-order polynomial for expanding the
subspace XN (), i.e.,
XN () = span{0(), . . . , N ()}, (19)
the unknown function fN () ∈ XN () can be written like
fN () =
N
j=0
fj j () with fj = fN (j ), (20)
where j (x) are the nodal basis functions used to expand the
approximation. For this particular problem, the nodal basis
functions are chosen to be the Lagrange interpolant polynomi-
als of order N based on the Gauss–Legendre (GL) quadrature
points, Fig. 1. A different option can be for example to use
the Gauss–Lobatto-Legendre (GLL) quadrature points, Fig. 2,
if the boundary points are required.
Repeating the steps from the previous section, we get the
following algebraic system
Af = F, (21)
where the matrix A ∈ R(N+1)×(N+1)
and vectors F, f ∈
R(N+1)×1
are defined as
[A]ij = A(j , i) = Lj , LiY(), (22)
[F]i = F(i) = g, LiY(), (23)
[f]i = fi = f (xi). (24)
4.1.2. Numerical issues
In order to obtain an efficient algorithm, it is necessary to
reduce the number of operations involved in the computation
of the elements of the matrix A and vector F. For that reason,
the integral expressions can be approximated in an efficient
way by using numerical quadrature (Deville et al., 2002). For
example, the integral of a function b() in the reference domain

5. 5074 C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
0
()
2
()
3
()
1
()
Fig. 1. Lagrange interpolant of third order using the GL quadrature points.
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
1.5
Nodal Points
0
()
2
()
3
()
1
()
Fig. 2. Lagrange interpolant of third order using the GLL quadrature points.
ˆ
=[−1, 1] can be evaluated numerically using GL integration
as
1
−1
b() d ≈
P
q=0
b(q)wq, (25)
where q are the quadrature points and q the quadrature
weights. Generally, the same GL-roots are used for the evalu-
ation of the integrals as for the approximation of the solution,
i.e., P = N, due to simplicity and implementation aspects. The
Gaussian quadrature based on the GL-roots is even exact when
the integrand b() is a polynomial of degree 2P − 1 or lower.

6. C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081 5075
Thus, it is possible to determine a priori the optimal order of P
for a given problem.
The coefficients of the matrix A can be computed in the
following way
[A]ij = Li, Lj Y() (26)
=
Li()Lj () d (27)
≈
N
q=0
Li(q)Lj (q) q (28)
≈
N
q=0
[L]iq[]qq[L̂]jq (29)
≈ [L]T
i [L]j , (30)
where the set {q, wq}N
q=0 is the quadrature rule, chosen in this
case to coincide with the nodal points of the expansion, i.e.,
j (q) = jq. Thus, the matrix A and the vector F are given
by
A = LT
L, (31)
F = LT
g, (32)
with the matrices L, ∈ R(N+1)×(N+1)
and vector g ∈ R(N+1)
defined like
[L]ij = Lj (i), (33)
[]ij = j ij , (34)
[g]i = g(i). (35)
In particular, the computation of the coefficients of the matrix
L is the most expensive part of the setting process. This matrix
is formed by the contribution from all the processes involved.
For example, considering only breakage the element of the
matrix L can be obtained like
[L]ij = Lj (i) (36)
= − b(i)j (i) +
L
0
k(i, s)j (s) ds, (37)
where the integral term can be again approximated by numerical
quadrature like
L
0
k(i, s)j (s) ds ≈
P
q=0
k(i, sq)j (sq)q. (38)
4.2. One-dimensional population balance equation
In the one-dimensional PBE the density function, f (x, ),
presents a dependence on the internal coordinate or property
and on the spatial coordinate x. The model problem is given like
Lf (x, ) + g(x, ) = 0 in = [0, Lx] × [0, L] (39)
with g(x, ) the source term, and
Lf (x, )=−b(x, )f (x, )+
L
0
k(x, , s)f (x, s) ds (40)
the breakage operator.
4.2.1. The least square spectral approximation
The least-square functional for this problem is defined as
J(f ; g) ≡ 1
2 Lf + g2
Y() (41)
with the norm given as
• 2
Y() = •, •Y() =
• • d
=
Lx
0
L
0
• • d dx. (42)
In this case, the discretization statement consists in approx-
imating the unknown function f (x, ) by
fN (x, ) =
N
i=0
N
j=0
fij ij (x, ) (43)
=
N
i=0
N
j=0
fij i(x)j (), (44)
where the two-dimensional basis functions ij (x, ) are built
like the product of two one-dimensional basis functions, i.e.,
ij (x, ) = i(x)j (x). The one-dimensional basis functions
i(x) and j () are Lagrange interpolant polynomials of order
N. The nodal values are defined like fij such that fij =f (xi, j ).
For instance, we can consider the internal coordinate ex-
panded based on the GL points Lagrange interpolant, while
the spatial dimension based on the GLL points Lagrange in-
terpolant. In order to simplify the numerical manipulation, the
approximation can be expressed based on a global numbering
procedure. In this way, defining the index l =i +j(N +1) with
0i, j N, and N = (N + 1)2
− 1, we can write (43) in the
following way
fN (x) =
N
l=0
fll(x), (45)
where x = (x, ) and l = ij , which inherits the property that
l(xm) = lm, 0l, mN. (46)
In Fig. 3, the nodal points are shown in the reference domain
ˆ
=[−1, 1]2 with the corresponding global numbering. An ex-
ample of a two-dimensional function is shown with the corre-
sponding global numbering in Fig. 4.
Repeating the procedure from Section 3, we get the following
algebraic system:
Af = F, (47)
where
[A]ij = A(j , i) = Lj , LiY(), (48)

7. 5076 C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x
x0 x1 x2 x3
x4 x5 x6 x7
x8 x9 x10 x11
x12 x13 x14 x15
Fig. 3. The distribution of the points GLL–GL points, xi, in ˆ
.
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
1.5
x
Φ
9
(x)
=
23
(x,ξ)
Fig. 4. One of the 16 basis functions l(x) = ij (x, ).
[F]i = F(i) = g, LiY(), (49)
[f]i = fi = f (xi). (50)
Fig. 5 shows an example of the computation of the one-
dimensional PBE. The dots are the nodal points where the
solution is directly available.
4.2.2. Numerical issues
The coefficients of the matrix A can be computed like
[A]ij = Li, Lj Y() (51)
=
Li(x)Lj (x) dx (52)
≈
N
q=0
Li(xq)Lj (xq)q (53)
≈
N
q=0
[L]iq[]qq[L]jq (54)
≈ [L]T
i [L]j , (55)
0
0.2
0.4
0.6
0.8
1 0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2
2.5
x
f
(x,)
Fig. 5. Example of the solution of the one-dimensional population balance
equation. The dots are the nodal points where the solution is directly available.
where dx ≡ d dx, and the quadrature rules used {i, i}N
i=0,
correspond to the two-dimensional quadrature rules.
Thus, the matrix A ∈ R(N+1)×(N+1)
and vector F ∈
R(N+1)×1
are given by
A = LT
L, (56)
F = LT
g, (57)
where the matrices L, and g are defined like
[L]ij = Lj (xi), (58)
[]ij = j ij , (59)
[g]i = g(xi). (60)
5. Computational complexity
The computational complexity for a Rd+1
population bal-
ance problem is discussed. For simplicity we assume that all
the dimensions are approximated with the same polynomial de-
gree N. The computational cost for assembling the final alge-
braic system can be decomposed as the computational cost for
computing the matrix L plus the cost of getting the matrix A
and the vector F. The matrix L requires
WL
≈ 3(P + 1)(N + 1)2(d+1)
(61)
operations. For simplicity we consider only the leading term
in (38). The factor 3 in expression (61) is related to the two
product operations plus the addition operation for computing
each term in the sum of expression (38). It is important to
note that if P = N, we have j (q) = jq, so the number of
operations is reduced to
WL
≈ (N + 1)2(d+1)
. (62)
The matrix A and the vector F can be computed as
WA
≈ 3(N + 1)2(d+1)+1
, (63)

8. C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081 5077
100 101 102
10-15
10-10
10-5
100
LSQ - 0D PBE
Machine Precision 10-16
10-7
Case 1
Case 2
||
||
L
2
N
∋
Fig. 6. Convergence rate of the L2 error vs. the approximation order N.
WF
≈ 3(N + 1)(d+1)+1
, (64)
respectively. So, the operation count gives
W = WA
+ WL
+ WF
≈ O((N + 1)(d+1)+1
), (65)
where only the leading term was considered. The memory re-
quirement scales as O((N + 1)(d+1)
), given by the memory
required for storing A. Actually, the matrix A is symmetric
so it requires to store only a half of the matrix. It is noted that
the operation count and memory requirement for a low-order
method will be equivalent if we assume that N is the num-
ber of discrete points in the low-order approximation. The rea-
son why high-order methods are competitive compared with
low-order methods is the fact that for sufficiently smooth so-
lutions and stringent error requirements, high order methods
need less number of unknowns than a low-order method for
satisfying the requirements. Finally, it is important to note
that the final algebraic system is symmetric, positive definite
100
101
102
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
LSQ - 0D PBE - Case 1
0.001 % error
0th
Moment
1st
Moment
2nd
Moment
3rd
Moment
4th
Moment
0th
Moment
1st
Moment
2nd
Moment
3rd
Moment
4th
Moment
100
||
k
||L
1
[%]
N
100
101
102
N
LSQ - 0D PBE - Case 2
0.001 % error
∋
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
100
||
k
||L
1
[%]
∋
Fig. 7. Convergence rate of the percentual error in the moment computation vs. the expansion order N.
and therefore can be solved in an optimal way with standard
solvers such as the conjugate gradient. On the other hand, most
of the low-order methods produce a final algebraic systems with
no particular matrix property that could improve the efficiency
of the solution procedure. This fact can be considered an impor-
tant drawback of most of the low-order methods for handling
the population balance problem in several dimensions.
6. Numerical experiments
The capability of the method for solving PBEs is studied with
test problems with known analytical solution, see Appendix A.
In order to show how good the numerical solutions of the above
problems are in comparison with the exact ones, the L2 error
is defined like
L2 =
N
q=0
[f (xq) − fN(xq)]2 (66)
with xq the nodal points of the approximation and with N =
(N + 1)d+1
where d is the physical spatial dimension. In this
case fN represents the numerical solution while f is the ana-
lytical solution.
Besides, for showing the error in the prediction of the mo-
ments of the density function, the weighted 1-norm k(t)1 is
defined by
k1 =
k − k,N
k
, (67)
where k(t) and k,N (t) are the exact and numerical moments
respectively, which are defined like
k =
f (x)xk
d
=
L
0
f ()k
d, x ∈ R0+1
(68)
=
Lx
0
L
0
f (x, )xk
k
d dx, x ∈ R1+1
. (69)

9. 5078 C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081
100 101 102
10-15
10-10
10-5
100
LSQ - 1D PBE
Machine Precision 10-16
10-7
Case 1
Case 2
||
||L
2
N
∋
Fig. 8. Convergence rate of the L2 error vs. the approximation order N.
6.1. A zero-dimensional problem
The convergence properties of the least squares solution of
a zero-dimensional pure breakage steady-state equation is an-
alyzed. The model problem is given as
−b()f () +
1
0
k(, s)f (s) ds = 0 in = [0, 1]. (70)
The kernel functions are given in Appendix A. The test cases
are defined to study the impacts of complex kernel functions
in the convergence rate. In Fig. 6 the dependence of the er-
ror with the expansion order N is shown, where the expected
spectral convergence rate is observed. The convergence rate is
strongly affected by the capability of the solver for solving the
integral term (15) correctly. In this concern, high-order meth-
ods present an important advantage compared with low-order
methods such as the discrete method (e.g. Ramkrishna, 2000)
LSQ - 1D PBE - Case 1
0.001 % error
LSQ - 1D PBE - Case 2
0.001 % error
100
101
102
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
100
||
k
||L
1
[%]
N
∋
100
101
102
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
100
||
k
||L
1
[%]
N
∋
0th
Moment
1st
Moment
2nd
Moment
3rd
Moment
4th
Moment
0th
Moment
1st
Moment
2nd
Moment
3rd
Moment
4th
Moment
Fig. 9. Convergence rate of the percentual error in the moment computation vs. the expansion order N.
or the multigroup method (e.g. Carrica et al., 1999). While low-
order methods have an algebraic convergence rate for solving
the mentioned integral term, high-order methods can have a
spectral convergence rate if the integrand is analytical.
The convergence rate is also affected by the characteristics
of the solution. The more complex the solution, a higher value
of N is required. The dependence on the solution characteristics
is not presented in this work.
Case 1 shows a slow convergence rate compared to case 2,
due to the nature of the kernel function. In particular, case 1
requires a higher order representation which affects the conver-
gence rate.
In some engineering applications the first moments are the
desired result of the computation. Fig. 7 shows the percentual
error in the first five moments of the distribution. It is important
to remark that the moments are computed correctly even using
a low value of N. For this particular model problem, using
only 11 basis functions, N = 10, the percentual error in the
computation of the moments is approximately 0.001% in case
1, while for case 2 with only five basis function, N=4, the same
order in the percentual error is achieved. Therefore, problems
containing kernels with a low-order polynomial representation
can be expected to require a low value of N.
An important characteristic of this methods is that the er-
ror in the properties of the distribution depends on the order
of the expansion. Therefore, it is possible to reduce these er-
rors by increasing the order of the expansion, thus avoiding
the introduction of heuristic rules to satisfy a certain accuracy
in the values of some moments of the distribution (e.g. mass
conservation).
6.2. A one-dimensional problem
A one-dimensional pure breakage steady-state equation is
analyzed. The model problem is given as
− b(x, )f (x, ) +
1
0
k(x, , s)f (x, s) ds = g(x, )
in = [0, 1]2
. (71)

10. C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081 5079
The kernel functions are given in Appendix A. The main pur-
pose of these test cases is to study the effects of a complex
kernel function in the convergence rate. The dependency of
the error with the one-dimensional expansion order N is shown
in Fig. 8. In the figure, spectral behavior is observed for both
cases, although case 1 shows a slow convergence rate compared
to case 2 due to the nature of the kernel function.
In Fig. 9, the dependency of the first five moments as a func-
tion of the expansion order N is presented. For this problem, the
percentual error obtained is relative low for a low value of N. In
order to obtain a percentual error around 0.001% in the value
of the moments, it is neccessary to have N =10 for case 1, and
N=6 for case 2. The slightly worse convergence rate in this case
as compared with the zero-dimensional case can be attributed
to the spatial dependence of the yield breakage function (see
Appendix A), which must also be solved. For this type of kernel
functions, a non-equal order expansion could be beneficial.
7. Conclusions
In this work, a spectral LSM is applied to solve the popula-
tion balance equation. The main concern was to derive a gen-
eral mathematical framework for multidimensional problems
and, at the same time, to reduce the computational cost of the
setting process. For that reason, the discretization procedure
was based on Lagrange interpolant functions, which reduce the
computational cost involved in the numerical evaluation of the
integral expressions.
The zero- and one-dimensional model problems were tested
numerically using analytical test cases. In this way, an accurate
error analysis is made possible. It was observed that for the
problems tested the method presented good numerical proper-
ties, in particular the expected spectral convergence rate.
An important characteristic of the LSM applied to PBEs is
that the error in the properties of the distribution, such as the
first moments of the distribution, depend on the order of the
expansion. It is possible to reduce these errors by increasing
0
0.2
0.4
0.6 0.8 1 0
0.2
0.4
0.6
0.8
1
0
2
4
6
8
10
12
h
(,
)
h
(,
)
0
0.2
0.4 0.6 0.8
1 0 0.2 0.4 0.6 0.8
1
0.4
0.5
0.6
0.7
0.8
0.9
1
~
~
~
~
Fig. A1. Breakage yield function. Case 1 on the left and Case 2 on the right.
the order of the expansion, thus avoiding the introduction of
heuristic rules to satisfy a certain accuracy in the values of some
moments of the distribution (e.g. mass conservation).
In future works, the method will be applied for simulating
some physical systems. Besides, further work is required for
coupling the present solver with available multifluid solvers for
simulating multiphase flows, which constitute the final goal in
the ongoing research.
Acknowledgements
The Ph.D. fellowship (Dorao, C.A.) financed by the Research
Council of Norway through a Strategic University Program
(CARPET) is gratefully appreciated.
Appendix A. Test cases definition
A.1. The zero-dimensional case
The model problem for the zero-dimensional PBE is defined
like
b() = 1,
h(, ˜
) =
p2
h
((p − )2
+ p2
h) ((ps − ˜
)2
+ p2
h)
,
g() = 1 − ph
arctan( ps
ph
) + arctan(1−ps
ph
)
p2
h + (p − )2
, (A.1)
with the analytical solution
f () = 1 (A.2)
with p =0.3, ps =0.3, and ph =0.3 for the Case 1, or ph =1.0
for the Case 2. Cases 1 and 2 differ in the characteristics of the
yield function h(, s). Case 1 presents an stronger peak than
Case 2 to resolve that demands a higher order approximation,
see Fig. A1.

11. 5080 C.A. Dorao, H.A. Jakobsen / Chemical Engineering Science 61 (2006) 5070–5081
0 0.2
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
1
2
3
4
5
6
7
8
h(x=0.5,
=0.5,
x,
)
0 0.2
0.4 0.6 0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
~
~
h(x=0.5,
=0.5,
x,
)
~
~
0.4
x
~
~
x
~
~
Fig. A2. Breakage yield function. Case 1 on the left and Case 2 on the right.
A.2. The one-dimensional case
The model problem for the one-dimensional PBE is defined
like
b(x, ) = 1,
h(x, , x̃, ˜
) =
p2
he− x̃
((p − )2
+ p2
h) ((ps − ˜
)2
+ p2
h)
,
g(x, ) = 1 − ph
1
−
e−
×
arctan( ps
ph
) + arctan(1−ps
ph
)
p2
h + (p − )2
, (A.3)
with the analytical solution
f (x, ) = 1 (A.4)
with p = 0.3, ps = 0.3, = 10, and ph = 0.3 for the Case 1,
or ph = 1.0 for the Case 2. Cases 1 and 2 differ in the char-
acteristics of the yield function h(x, , x̃, ˜
). Case 1 presents
an stronger peak than Case 2 to resolve that demands a higher
order approximation similar to the zero-dimensional problem.
The factor determines the spatial characteristics of the yield
function, see Fig. A2.
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