Spectral Solutions of a Combined Multifluid–population Balance Model
Describing Bubbly Flow
A Numerical Study of weighted Residual Methods
Jannike Solsvik and Hugo A. Jakobsen
Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), Trondheim, Norway
Keywords:
Population Balance Equation, Multifluid Model, Weighted Residual Methods, Tau, Orthogonal Collocation,
Least-squares.
Abstract:
Fluid particle breakage and coalescence phenomena are important for optimal operation for industrial process
units like the bobble column reactors. The population balance equation (PBE) can be applied to describe the
evolution of populations of countable entities such as the bubbles in the bubble column. In recent literature,
the least-squares methods has been adopted for the approximate solution of population balance (PB) models.
Adopting a weighted residual method such as the least-squares method, the distribution function resolved
instead of obtaining only a few moments of the distribution function. The performance of the least-squares
method for PB problems should be compared to other techniques in the family of weighted residual methods.
The aim of the present study is to evaluate the orthogonal collocation, tau and last-squares methods for the
solution of a combined multifluid-PB model describing bubbly flows.
1 INTRODUCTION
Despite the simple construction of the bubble
columns and their widespread use for chemical re-
actions and separation in the process industries, our
understanding of the complex flows in these vessels
is still very limited. The complexity of the hydro-
dynamics in the bubble columns relates to the evo-
lution of phenomena such as breakage, coalescence,
growth and convective transport of the bubbles. In the
mathematical modeling framework, the PBE (Ramkr-
ishna, 2000; Jakobsen, 2008; Randolph and Larson,
1988; Sporleder et al., 2012) is considered a concept
for describing the evolution of populations of count-
able entities such as the bubbles in the bubble col-
umn. Adopting the PB modeling technique to bubbly
flows, the dispersed gas phase is treated as a popula-
tion of bubbles distributed not only in physical space
but also in an abstract property space. The quantity
of basic interest in PB modeling is the statistical den-
sity function representing the behavior of the popula-
tion of bubbles. The evolution of the statistical den-
sity function must take into account the different pro-
cesses that control the population of the bubbles in
the vessel, such as breakage, coalescence, growth and
convective transport. Thus the PBE provides a sta-
tistical description of the dispersed phase where the
Figure 1: Sketch of a bubble column reactor. The parameter
ξ is a characteristic property of the bubble, for example the
bubble diameter. Because of the different interaction pro-
cesses, the distribution of the bubbles evolves.
102
Jakobsen H. and Solsvik J..
Spectral Solutions of a Combined Multifluid–population Balance ModelDescribing Bubbly Flow - A Numerical Study of weighted Residual Methods.
DOI: 10.5220/0004477401020107
In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2013),
pages 102-107
ISBN: 978-989-8565-69-3
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
density function may be denoted by f(r,ξ,t) where r
is the spatial position vector, ξ is the property of inter-
est of the dispersed phase, and t is the time. Moreover,
f(r,ξ,t)dξ represents the average number of particles
per unit volume around the point r (r, r + dr) in the
time t, with the property between ξ and ξ + dξ. The
resulting PBE is characterized mathematically as a
non-linear integro-partial differential equation which
must be solved by a suitable numerical method.
In recent publications, the least-squares method
has been adopted for the solution of PB problems,
e.g. (Dorao and Jakobsen, 2006b; Dorao and Jakob-
sen, 2007b; Dorao and Jakobsen, 2008; Dorao and
Jakobsen, 2009; Dorao and Jakobsen, 2007a; Dorao
and Jakobsen, 2006a; Nayak et al., 2011; Sporleder
et al., 2011; Zhu et al., 2009a; Zhu et al., 2009b; Pa-
truno et al., 2009; Patruno, 2010; Borka and Jakobsen,
2012a). The least-squares method has also been ap-
plied to other chemical reactor problems such as the
pellet equations and fixed packed bed reactors (Rout
et al., 2011; Rout and Jakobsen, 2012; Solsvik and
Jakobsen, 2012; Sporleder et al., 2011). However,
the least-squares technique is a method in the family
of weighted residual. Thus, it is of interest to reveal
the properties of the least-squares method compared
to other methods in the weighted residual framework
like the orthogonal collocation and tau methods.
The aim of the present study is to compare the per-
formance of different solution techniques in the fam-
ily of weighted residual methods. The solution meth-
ods is evaluated for a combined multifluid-PB model
describing bubbly flow. The evaluation should con-
sider residual measures, computational time, imple-
mentation issues, and complexity of the algebraic the-
ory of the methods.
2 THE MODEL
The combined multifluid-PB model considered in the
present study is based upon the work of (Nayak et al.,
2011; Borka and Jakobsen, 2012a; Borka and Jakob-
sen, 2012b; Borka and Jakobsen, 2012c). The steady-
state model holds one dimension in physical space
and one dimension in the property space (bubble di-
ameter). The model equations is outlined in the se-
quel.
The main advantages of the novel model by
(Nayak et al., 2011) is that both the PBE and the
momentum equation of the dispersed gas phase is in
terms of the internal coordinate ξ. Hence, the inner
coordinate space physics can be resolved provided
that a sufficient numerical method is available. In par-
ticular, the least-squares, tau, Galerkin and orthogonal
collocation methods in the weighted residual frame-
work can be applied to solve the model proposed by
(Nayak et al., 2011).
In the following, the multifluid–PB model is out-
lined for bubbly flows in a cold-flow column.
2.1 The dispersed Phase
The two-dimensional (one dimensiona in the physical
space and one dimension in the property space) PBE
is presented by:
z
v
z
(ξ,z) f
d,m
(ξ,z)
+
∂ξ
v
ξ
(ξ,z) f
d,m
(ξ,z)
= b(ξ) f
d,m
(ξ,z)
+V(ξ)
Z
ξ
max
ξ
h(ξ,ζ)b(ζ)
f
d,m
(ζ,z)
V(ζ)
dζ
f
d,m
(ξ,z)
Z
(ξ
3
max
ξ
3
)
1/3
ξ
min
c(ξ,ζ)
f
d,m
(ζ,z)
ρ
d
(z)V(ζ)
dζ
+
ξ
2
V(ξ)
2
Z
(ξ
3
ξ
3
min
)
1/3
ξ
min
c([ξ
3
ζ
3
]
1/3
,ζ)
[ξ
3
ζ
3
]
2/3
×
f
d,m
(ζ,z)
ρ
d
(z)V(ζ)
f
d,m
([ξ
3
ζ
3
]
1/3
,z)
V(ξ) V(ζ)
dζ
(1)
in which f
d,m
is the mass density function [kg/m
3
/m].
The bubbles are transported in physical space accord-
ing to velocity v
z
and in the property space according
to velocity v
ξ
. Moreover, the internal coordinate is
the bubble diameter. The breakage rate and daughter
size redistribution function proposed by (Coulaloglou
and Tavlarides, 1977), and the coalescence model by
(Prince and Blanch, 1990) are adopted. The breakage
frequency yields:
b(z,ξ) =
k
1
ε
1/3
ξ
2/3
exp
σk
2
ρ
l
ε
2/3
ξ
5/3
(2)
where k
1
and k
2
are empirical parameters, which de-
pend on the system properties. The daughter size re-
distribution function is given as:
h(ξ,ζ) =2P(ξ,ζ) = 2
2.4
π
2
ξ
2
V(ζ)
× exp
4.5
[2V(ξ) V(ζ)]
2
V(ζ)
2
!
(3)
The coalescence model by (Prince and Blanch, 1990)
is defined as the product of a collision volume rate
h
c
(ξ,ζ) and the coalescence probability λ
c
(ξ,ζ):
c(ξ,ζ) = h
c
(ξ,ζ)λ
c
(ξ,ζ) (4)
SpectralSolutionsofaCombinedMultifluid--populationBalanceModel DescribingBubblyFlow-ANumericalStudyof
weightedResidualMethods
103
The expression for the coalescence probability is
given as:
λ
c
(ξ,ζ) = exp
r
3
c
(ξ,ζ)ρ
l
16σ
1/2
ε
1/3
ln
h
0
h
f
r
c
(ξ,ζ)
2/3
(5)
where the equivalent radius is defined as:
r
c
(ξ,ζ) =
1
4
1
ξ
+
1
ζ
1
(6)
The collision volume rate is given as:
h
c
(ξ,ζ) =
π
4
[ξ+ ζ]
2
β[εξ ]
2/3
+ β[εζ]
2/3
1/2
(7)
The bubble growth velocity is estimated as:
v
ξ
(z,ξ) =
ξv
z
3ρ
d
(z)
ρ
d
(z)
z
(8)
The momentum equation of the dispersed phase is
given as:
z
f
d,m
(ξ,z)v
z
(ξ,z)v
z
(ξ,z)
=
∂ξ
f
d,m
(ξ,z)v
ξ
(ξ,z)
f
d,m
(ξ,z)
ρ
d
(z)
ρ
d
(z)
z
+ f
d,m
(ξ,z)g+ f
drag
(ξ,z)
(9)
The dispersed phase fraction:
α
d
(z) =
Z
0
f
d,m
(ξ,z)
ρ
d
(z)
dξ (10)
The gas-phasepressure is assumed equal to the liquid-
phase pressure: P = P
d
= P
l
. The ideal gas law
gives the relationship between the pressure and the
dispersed phase density:
p(z)
p
0
ρ
0
d
ρ
d
(z) = 0 (11)
The drag force is defined as:
f
drag
=
3
4
ρ
l
C
D
ξ
f
d,m
(ξ,z)
ρ
d
(z)
|v
r
(ξ,z) v
l
(z)|
×
v
r
(ξ,z) v
l
(z)
(12)
where
C
D
=
max
min
16
Rep
(1+ 0.15Re
0.687
p
),
48
Re
p
,
8
3
Eo
Eo+ 4
(13)
Re
p
=
ρ
l
|v
z
(z,ξ) v
l
(z)|ξ
µ
l
(14)
Eo =
g(ρ
l
ρ
g
)ξ
2
σ
(15)
2.2 The Continuous Phase
The liquid continuity equation:
d
dz
α
l
(z)ρ
l
v
l
(z)
= 0
(16)
or integrated:
α
l
(z)ρ
l
v
l
(z) = α
l
(z = 0)ρ
l
v
l
(z = 0) (17)
The momentum equation:
α
l
(z)ρ
l
v
l
(z)
d
dz
v
l
(z) =
α
l
(z)
dp(z)
dz
1
2
ρ
l
f
w
(z)
D
v
l
(z)
2
Z
0
f
drag
(ξ,z)dξ + α
l
(z)ρ
l
g
(18)
The summation of the phase area fraction must be
unity:
α
d
+ α
l
= 1 (19)
The wall friction factor is calculated as:
f
w
(z) =
0.79ln
Re
l
(z)
1.64
2
(20)
3 NUMERICAL METHODS
Basically, spectral methods are based upon using a
representation of the solution function over the entire
computational domain via a truncated series expan-
sion (Finlayson, 1972):
f
ex
(ξ) f
P
(ξ) =
P
j=0
α
j
ϕ
P
j
(ξ) (21)
in which ϕ
P
j
(ξ) denotes the basis function and α
j
de-
notes the basis coefficients. A nodal basis is obtained
if the truncated series expansion (21) is given in terms
of Lagrangian basis polynomials
P
j
(ξ):
f(ξ) f
P
(ξ) =
P
j=0
f
P
j
P
j
(ξ) (22)
in which the basis coefficients are meaningful in the
sense that they correspond to the solution function
values at the collocation points.
The methods of weighted residual is presented by
the following generalized inner product:
Z
R (ξ; f
P
0
, f
P
1
,..., f
P
P
)w
i
(ξ)d= 0
for i = 0, 1, ...,P
(23)
where w
i
are weighting functions and R is the resid-
ual. The particular choice of the weighting function
defines the particular solution approximation tech-
nique in the family of weighted residual methods:
SIMULTECH2013-3rdInternationalConferenceonSimulationandModelingMethodologies,Technologiesand
Applications
104
the least-squares method
w
i
=
f
P
i
R (ξ, f
P
0
, f
P
1
,..., f
P
P
) (24)
the tau method
w
i
=
P
i
(ξ) (25)
the orthogonal collocation method
w
i
= δ(ξ ξ
i
) (26)
The tau (Lanczos, 1938) and least-squares (Jiang,
1998; Bochev and Gunzburger, 1998; Pontaza, 2003)
methods have not obtained the same popularity as the
orthogonal collocation method (Villadsen and Stew-
ard, 1967; Villadsen, 1970; Villadsen and Michelsen,
1978; Michelsen and Villadsen, 1981) in the chem-
ical reactor engineering society. However, the tau
method is similar to the Galerkin method (Galerkin,
1915). The essential difference between the Galerkin
and the tau methods is the treatment of the boundary
conditions. Theory of the least-squares method from
a chemical engineering point of view is given by (Do-
rao, 2006; Zhu, 2009; Sporleder, 2011). Furthermore,
the algebra and implementation issues using the least-
squares method for the solution of a PB problem is
provided by (Solsvik and Jakobsen, 2013).
4 RESULTS
The solution of the PBE (1) is presented in figure 2.
The PB model has been solved with the orthogonal
collocation, tau and least-squares methods.
0
0.01
0.02
0
1
2
3
0
5
10
15
z [m]
ξ [m]
f [kg/m
3
m]
Figure 2: Solution of a PB problem. The bubble size distri-
bution along the bubble column height.
Two residual measure definitions have been em-
ployed to evaluate the accuracy of the solution meth-
ods. The residual measure definitions are based on
(i) the problem operator form and (ii) the algebraic
system form. The former residual definition measure
the residual of the governing equation, while the lat-
ter definition gives the residual of the algebraic equa-
tion system that is actually solved. Noticeable, for the
orthogonal collocation method the problem operator
form and the algebraic system form coincide. Fig-
ure 3 shows the residual versus the number of itera-
tions adopting the residual measure definition based
on the algebraic system form. Based on this partic-
ular residual measure, the tau method is significantly
favorable above the orthogonal collocation and least-
squares methods. The orthogonal collocation and
least-squares methods obtain the same accuracy, but
with differences in the number of iteration to reach
convergence, in which the least-squares method re-
quires less iteration to reach its minimum residual
value. In figure 4 the residual measure definition
based on the problem operator form is adopted. For
this particular residual measure definition, the tau and
orthogonal collocation show the same performance.
On the other hand, the least-squares method obtains a
significant larger value of the residual measure. Thus,
the least-squares method does not fulfill the govern-
ing equation as well as the orthogonal collocation and
tau methods. The least-squares method requires fur-
ther attention to reveal the significantly larger resid-
0 10 20 30 40
10
−20
10
−10
10
0
Residual
Iterations
Collocation
LSQ
Tau
Machine accuracy
Figure 3: Residual (algebraic system form) as a function of
the number of iterations.
0 10 20 30 40
10
−20
10
−10
10
0
Residual
Iterations
Collocation
LSQ
Tau
Machine accuracy
Figure 4: Residual (problem operator form) as a function of
the number of iterations.
SpectralSolutionsofaCombinedMultifluid--populationBalanceModel DescribingBubblyFlow-ANumericalStudyof
weightedResidualMethods
105
ual measure on the problem operator form compared
to the orthogonal collocation and tau methods.
The orthogonal collocation method is favorable
considering the simplicity of implementation com-
pared to the tau and leas-squares methods. Moreover,
the orthogonal collocation method uses less computa-
tional costs per iteration than the relatively more com-
putational demanding tau and least-squares methods.
5 CONCLUSIONS
The accuracy of the orthogonal collocation, tau and
least-squares method can be evaluated on different
residual measure definitions. Dependent on the resid-
ual measure definition adopted, the relatively perfor-
mance of the numerical methods may change signifi-
cantly. However, the simulation results of the present
PB problem indicate that the orthogonal collocation
and tau method are favorable above the least-squares
method considering accuracy. Nevertheless, the or-
thogonal collocation method uses less computational
costs per iteration than the tau and least-squares meth-
ods. Furthermore, the orthogonal collocation method
holds the simplest algebraic theory, and is thus asso-
ciated with the simplest implementation issues.
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SpectralSolutionsofaCombinedMultifluid--populationBalanceModel DescribingBubblyFlow-ANumericalStudyof
weightedResidualMethods
107