Chemical Engineering Science 58 (2003) 1777 – 1792
www.elsevier.com/locate/ces
Solution of hyperbolic PDEs using a stable adaptive
multiresolution method
P. Cruza , M. A. Alvesb , F. D. Magalhãesa , A. Mendesa;∗
a LEPAE-Departamento
de Engenharia Qumica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias,
Porto 4200-465, Portugal
b Departamento de Engenharia Qumica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias,
Porto 4200-465, Portugal
Received 22 February 2002; received in revised form 23 September 2002; accepted 4 November 2002
Abstract
An e2cient adaptive multiresolution numerical method is described for solving systems of partial di3erential equations. The grid is
dynamically adapted during the integration procedure so that only the relevant information is stored. The convection terms are discretised
with high-resolution methods, thus ensuring boundedness. The proposed method is general, but is particularly useful for highly convective
problems involving sharp moving fronts, a situation that frequently occurs in many chemical engineering problems, and where standard
procedures may lead to unphysical oscillations in the computed solution.
Numerical results for 7ve test problems are presented to illustrate the e2ciency and robustness of the method. The adaptive strategy is
found to signi7cantly reduce the computation time and memory requirements, as compared to the 7xed grid approach.
? 2003 Elsevier Science Ltd. All rights reserved.
Keywords: Simulation of hyperbolic PDEs; SMART high-resolution scheme; Adaptive grid; Numerical analysis; Dynamic simulation; Modelling
1. Introduction
The application of transient mass, energy and momentum balances to speci7c chemical engineering systems (e.g.
adsorption column, chemical reactor, :uid :ow) in which
the intrinsic properties (temperature, composition, pressure)
change in one or multiple dimensions, conduces to a system
of partial di3erential equations (SPDE). The utilisation of a
robust and e2cient numerical method is important for obtaining the solution, since an analytical solution is usually
impossible to derive.
Two numerical strategies can be applied in the solution
of such problems. The 7rst one consists in the simultaneous space and time discretisation of each partial di3erential equation (PDE) and posterior resolution of the resulting
non-linear algebraic system of equations. The other strategy
consists in the spatial discretisation of each PDE and subsequent integration of the resulting initial value system of
ordinary di3erential equations (ODE) with an appropriate
∗ Corresponding author. Tel.: +351-22508-1695;
fax: +351-22508-1449.
E-mail address: [email protected] (A. Mendes).
integrator (Finlayson, 1992). This second approach is used
in this work in the explicit formulation. The applications
presented in this work involve moving fronts, therefore
implying the use of small time steps in the time-integration
procedure. Under these circumstances, explicit schemes,
in addition to being easier to implement, are faster
than the corresponding implicit ones, for the same time
step.
When the solution of the PDE(s) presents a steep moving front (or fronts), which is a situation quite common
in the chemical engineering 7eld, two aspects are critical
for the proper problem resolution: the discretisation of the
convective terms and the grid re7nement in the vicinity of
the moving front. This work presents an integrated strategy
for dealing with these issues, as will be brie:y described
below.
When a PDE has a “smooth” solution, any conventional
higher-order discretisation scheme, applied to the convection terms (7rst-order space derivatives), conduces to stable
solutions, using a moderate number of grid points. However,
these schemes become inadequate in the presence of steep
moving fronts, leading to the appearance of non-physical
oscillations in the computed solution, or even to the
0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0009-2509(03)00015-0
1778
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
divergence of the numerical method. One exception is the
upwind di3erencing scheme (UDS) proposed by Courant,
Isaacson, and Rees (1952) that is unconditionally stable,
but due to its 7rst-order accuracy is not recommended
nowadays (Freitas, 1993). This work uses high-resolution
schemes (HRS), formulated in the context of the normalised variable and space formulation (NVSF) of Darwish
and Moukalled (1994). These are, by de7nition, bounded
higher-order schemes and will be brie:y described below.
The other essential issue is the ability to locally re7ne
the grid, in those regions where the solution exhibits sharp
features. This implies a strategy for dynamically identifying those critical regions and allocating extra grid points
accordingly. This is done in this work by taking advantage
of multiresolution data representation, introduced by Mallat
(1989) in the context of wavelet theory and later generalised by Harten (1996). This adaptive strategy allows for
a nearly constant discretisation error throughout the computational domain. In addition, because “smooth” regions are
represented by only the essential amount of data, memory
and CPU requirements are minimised.
The remaining of the paper is organised as follows: 7rst
the high-resolution schemes are described, followed by the
multiresolution grid adaptation strategy. Five representative
test cases are then presented in order to illustrate the e2ciency of the method. The paper ends with a summary of
the main conclusions.
2. Theory
This work considers generic time-dependent systems of
non-linear advection–di3usion–reaction equations of the
form
@u
@u
@F(x; t; u)
@
G(u)
+ S(x; t; u)
(1)
=−
+
@t
@x
@x
@x
with initial values u(t =0; x)=uo (x) and Dirichlet, Neumann
or Cauchy boundary conditions. The term appearing on the
left-hand side of Eq. (1) is called the inertia term, while on
the right-hand side one can identify a convective (advective), a di3usive and a source term, respectively. When Eq.
(1) is dominated by advection ( → 0), it is called a hyperbolic PDE, and special methods are needed for the treatment
of the convective term, otherwise strong unphysical oscillations may appear in the computed solution (Finlayson,
1992).
The discretisation of Eq. (1) is done in two stages. Firstly,
the space derivatives appearing in the right-hand side are
computed with appropriate schemes. Then, the resulting initial value system of ODEs is integrated explicitly to obtain
the grid point values at the next time step. This time integration is done with the package LSODA (Petzold, 1983),
and is brie:y described below.
2.1. High-resolution schemes
The high-resolution schemes implemented in this work
for the discretisation of the problematic convection term
are based on the NVSF of Darwish and Moukalled (1994),
which is an extension to non-uniform grids of the normalised
variable formulation (NVF) of Leonard (1988).
Consider a general non-uniform grid, as illustrated in
Fig. 1. The labelling of the nodes depends on the local velocity, a, calculated at face f by linear interpolation from
the surrounding grid points:
af=i+1=2 = (ai+1 + ai )=2
(2)
with ai ≡ (dF=du)i . For a given face f, the U and D nodes
refer to the upstream and downstream points, relative to node
P, which is itself upstream to the face f under consideration,
as shown in Fig. 1.
The :ux derivative at the general point i is evaluated with
(Fi+1=2 −Fi )(xi −xi−1 )2 +(Fi −Fi−1=2 )(xi+1 −xi )2
@F ;
=
@x i
(xi −xi−1 )(xi+1 −xi )(xi+1 − xi−1 )=2
(3)
where the unknown face :uxes Fi+1=2 and Fi−1=2 are interpolated from the neighbour grid point values using an appropriate discretisation scheme.
Several methods have been proposed in the literature to
accomplish this, such as the 7rst-order UDS of Courant
et al. (1952), the second-order linear upwind scheme
(LUDS) of Shyy (1985) or the third-order QUICK scheme
of Leonard (1979), which are all upwind biased. Central
schemes are often used, such as the second-order central
(CDS2) or the fourth-order central di3erences (CDS4). The
use of cubic splines is also an alternative (Cruz, Mendes, &
Magalhães, 2001a). All these methods, with the exception
of the 7rst-order UDS, su3er from lack of boundedness and,
for highly convective :ows, the appearance of unphysical
oscillations is usual, as will be demonstrated with some
numerical examples. The use of non-linear high-resolution
schemes is therefore adopted in this work, since these are
higher-order accurate and intrinsically bounded. According
to the NVSF, the face :uxes are interpolated as (Darwish
& Moukalled, 1994)
˙
Ff = FU + F f (FD − FU );
(4)
˙
where the normalised face :ux, F f , is calculated using an
appropriate non-linear limiter.
The following limiters were selected for this work:
(i) MINMOD (Harten, 1983)
˙
˙
xf ˙ 1 − xf ˙
˙
˙
F f = max F P ; min ˙ F P ;
˙ FP
xP
1 − xP
˙
˙
xf − xP
;
(5)
+
˙
1 − xP
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
1779
af=i+1/2 > 0
U
i– 2
af=i-1/2 > 0
D
D
P
U
i+ 1
i+ 2
i- 1
i
P
D
D
P
U
i
i+ 1
U
i– 2
P
i- 1
f = i - 1/2
f = i + 1/2
af=i+1/2 < 0
af=i-1/2 < 0
i+ 2
Fig. 1. De7nition of local variables.
(ii) Super-B (Roe, 1985)
˙
Ff
˙
˙
= max F P ; min 2F P ; 1; max
˙
+
˙
xf − xP
˙
1 − xP
˙
xf
˙
xP
V4
˙
FP ;
˙
1 − xf ˙
˙ FP
1 − xP
V3
˙
FP =
˙
xP =
˙
xf =
j= 3
k= 0 k= 1 k= 2 k= 3 k= 4 k= 5 k= 6 k= 7 k= 8
j= 2
V2
k= 0
;
(6)
˙ ˙
variables F P , x P
˙
and x f are calculated
FP − FU
;
FD − F U
(8)
xP − xU
;
xD − x U
(9)
x f − xU
:
xD − x U
(10)
More details on this issue, and other high-resolution
schemes, can be found in the works of Darwish and
Moukalled (1994) and Alves, Pinho, and Oliveira (2001).
For uniform meshes the normalised space coordinates de˙
˙
7ned in Eqs. (9) and (10) are simply x P = 12 and x f =
3=4, and the limiter functions (5)–(7) are greatly simpli7ed. For illustration, the SMART scheme is expressed in the
NVF as
˙
˙
˙ 3 ˙
3
(11)
F f = max F P ; min 3F P ; F P + ; 1 :
4
8
k= 1
k= 2
k= 3
k= 4
V1
j= 1
k= 0
(iii) SMART (Gaskell & Lau, 1988)
˙
˙
˙
˙
˙
x f (1 − 3 x P + 2 x f ) ˙ x f (1 − x f )
˙
˙
F f = max F P ; min
FP ; ˙
˙
˙
˙
x P (1 − x P )
x P (1 − x P )
˙ ˙
˙
x f(x f − x P )
˙
×F P +
;1
;
(7)
˙
1 − xP
where the normalised
using
j= 4
k= 1
k= 2
Fig. 2. Example of points in a dyadic grid.
3. Multiresolution representation of data
Consider a set of dyadic grids on the form
V j = {xkj ∈ R: xkj = 2−j k; k ∈ Z};
j ∈ Z;
(12)
where j identi7es the resolution level and k the spatial location (see Fig. 2).
Assuming that the function values are known on the
grid V j , the corresponding extension to the 7ner grid V j+1
is accomplished using the multiresolution approach. The
even-numbered grid point function values in V j+1 are
already present in V j ,
j+1
u2k
= ukj ;
(13)
while the function values in the odd-numbered grid points
in V j+1 are computed using an adequate interpolation
scheme, based on the known even-numbered grid points.
In wavelet-based methods, the use of symmetric interpolating polynomials is common (HolmstrOom, 1999; Cruz
et al., 2001a). In this work, however, we propose the use of
the bounded high-resolution interpolating schemes, in accordance with the discretisation strategy for the convection
term.
The interpolative error coe2cient, djk , is de7ned as the
j+1
), and the
di3erence between the interpolated value, I j (u2k+1
1780
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
Fig. 3. Example of a typical grid obtained with the multi-resolution adaptation strategy, evidencing the tree structure composed by those grid points
necessary for computing a given interpolated function value.
u2jk++11
ukj−1
ukj
I j ( u2jk+1+1 )
ukj+1
ukj+ 2
Fig. 4. Calculation of the interpolated value.
j+1
real one, u2k+1
,
j+1
j+1
djk = |u2k+1
− I j (u2k+1
)|:
(14)
If the djk value is below a given (small) threshold , then
j+1
can be rejected without loss of signi7the grid point x2k+1
cant information, since it can be reconstructed from the information preserved in the coarser grid V j . A function that
varies abruptly only in localised regions of the domain, will
have a small number of non-vanishing djk coe2cients. This
way, the information describing this function can be compressed with great e2ciency, without loss of accuracy.
It should be noted that the multiresolution approach must
incorporate a mechanism that, for each preserved grid point,
also selects associated relevant grid points. As illustrated
j+1
in Fig. 3, it is necessary to guarantee that, if point x2k+1
is retained in the adaptation algorithm, then the function
j
j
j
values at the locations xk−1
, xkj , xk+1
and xk+2
must also
be preserved. This procedure allows for the calculation of
j+1
the interpolated value I j (u2k+1
) and the evaluation of the djk
coe2cient in the next adaptation step.
The user must specify a maximum level of resolution
in order to avoid grid coalescence in problematic regions
(typically, in this work Jmax = 12 is used). The user must
also set the minimum level of resolution. All the grid points
pertaining to this resolution level are conserved throughout
the computations (typically, in this work Jmin = 4).
The calculation of the interpolated values is summarised
in the following steps (see Fig. 4):
(i) Calculate the face velocity, af , to identify the local
convective :ux direction:
af = (ajk + ajk+1 )=2:
(15)
(ii) Calculate the normalised face value of the advected
˙
variable, u f , using a high-resolution scheme. For example,
using the SMART high-resolution scheme, the normalised
face value is given by

j
j
ukj −uk−1
ukj −uk−1

max[
;
min(3
;

j
j
j
j

uk+1 −uk−1
uk+1 −uk−1




j
j


3 uk −uk−1

+ 38 ; 1)]
if af ¿ 0;

j
j
4
uk+1 −uk−1
˙
uf =
(16)
j
j
j
j

−uk+2
uk+1
−uk+2

 max[ uk+1
;
min(3
;
j
j
j
j

uk −uk+2
uk −uk+2




j
j


3 uk+1 −uk+2

+ 38 ; 1)]
if af ¡ 0:
4 uj −uj
k
k+2
(iii) Calculate the interpolated value:
j+1
)
uf = I j (u2k+1
j
˙
j
j
− uk−1
)
uk−1 + u f (uk+1
=
˙
j
j
j
uk+2 + u f (uk − uk+2 )
if af ¿ 0;
if af ¡ 0:
(17)
4. Adaptation strategy
During the time integration of the PDE, the grid is continuously adapted, so that it can adjust to the evolving solution.
The grid adaptation strategy involves the following steps:
(i) Knowing the function values ukj , in the grid V j at time
t = t1 , compute the interpolative error coe2cients djk
for Jmin 6 j 6 Jmax − 1 (Eq. (14)).
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
(ii) Identify the djk coe2cients that are above the predej+1
,
7ned threshold . The corresponding grid points, x2k+1
are retained.
j+1
(iii) Add the points x2(k+i)+1
, i=−NL , NR . These grid points
j+1
, at the same
are located to the right and left of x2k+1
resolution level. They are included to account for possible movement of the sharp features of the solution in
the next time-integration steps, and therefore reduce the
frequency of mesh adaptations. A conservative value
for the maximum time allowed between consecutive
mesh adaptation, for purely convective problems involving steep moving fronts, is
NL Sxmin NR Sxmin
Stadapt = max −
;
amin
amax
;
(18)
which guarantees that, between consecutive adaptations, the front will not move beyond a distance
NL Sxmin or NR Sxmin , where Sxmin corresponds
to the grid spacing at the maximum resolution
level.
j+2
j+2
(iv) Add the points x4k+1
and x4k+3
present at the resolution level immediately above. These grid points are
included to account for the possibility of the solution
becoming “steeper” in this region.
(v) Add all the grid points necessary for the calculation
of the interpolative error coe2cients at the next mesh
adaptation. This step is dependent on the interpolaj+1
tive scheme used to evaluate I j (u2k+1
). In this work
the use of high-resolution schemes is proposed, for
which the calculation of the coe2cient djk implies the
j
j
, xkj , xk+1
presence of the grid points at locations xk−1
j
and xk+2 .
(vi) Keep all the points associated to the lower resolution
level, Jmin . These are the “basic” grid points, which are
retained throughout all the computation.
5. Calculation of space derivatives
The space derivatives are calculated directly in the
adapted non-uniform grid, as suggested by Jameson (1998)
and Cruz et al. (2001a). Another possibility is the interpolation of the solution to the maximum resolution level
and calculation of the space derivatives in the generated
uniform grid (HolmstrOom, 1999). The latter approach is not
recommended, since it involves too many unnecessary interpolations, thus leading to a slow down in the integration
process.
The calculation of the convective terms in a non-uniform
grid based on the NVSF presented above is simple and
accurate, and the discretisation of the di3usive terms in a
non-uniform mesh is also easy to implement (Cruz, Mendes,
& Magalhães, 2002).
1781
6. Temporal integration
The time integration of the resulting system of ordinary
di3erential equations (ODE’s initial value problem) was
done with the solver LSODA (Petzold, 1983). This routine
solves initial boundary problems for sti3 or non-sti3 systems of 7rst-order ODEs. For non-sti3 systems, it utilises the
Adams method with variable order (up to 12th order) and
step size, while for sti3 systems it uses the Gear (or BDF)
method with variable order (up to 5th order) and step size.
Since our goal is to develop an e2cient and robust adaptive
spatial discretisation scheme, the error in time integration
was always set small enough to assure that the numerical errors are mainly due to inaccuracies in spatial discretisation.
The frequency of grid adaptations was dynamically adjusted in order to optimise the algorithm. A criterion, based
on the amount of change on the computed solution and on
the grid between two consecutive adaptations, was implemented for adjusting the time interval along which the grid
stays unchanged, Stadapt . This :exibility is very important
when dealing with problems involving di3erent time scales.
7. Results
In this section, 7ve examples are presented to illustrate
the e2ciency and robustness of the proposed strategy in
the solution of typical problems that arise in the chemical
engineering 7eld. All the simulations were performed in a
1:5 GHz Intel Pentium IVJ personal computer with 256 MB
SDRAM.
7.1. Example 1. Fixed-bed single-component adsorption
The 7xed-bed single-component adsorption model used
here assumes constant velocity plug :ow with axial dispersion, isothermal conditions and instantaneous equilibrium
between the :uid and adsorbed phases, and is described by
the following equation (Ruthven, Farooq, & Knaebel, 1994):
@c∗
1 @2 c∗
@c∗
∗
+
;
=
[1
+
!q
(c
)]
Pe @x2
@#
@x
(19)
where c∗ is the dimensionless :uid phase concentration,
c∗ = c=cref , x is the dimensionless spatial coordinate, x = z=L,
# is the dimensionless time, #=tu=L, Pe is the Peclet number,
Pe = Lu=Dax , ! is the capacity factor, ! = (1 − b )=b qref =cref ,
q (c∗ ) is the equilibrium isotherm derivative, u is the constant interstitial velocity, z is the spatial coordinate, L is the
column length, Dax is the e3ective axial dispersion coe2cient, b is the bulk porosity and qref is the reference concentration in the adsorbed phase. The equilibrium isotherm
considered has the general form
q∗ (c∗ ) =
Kc∗
;
1 + (K − 1)c∗
(20)
1782
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
where K is a parameter which can be selected in order to
change the isotherm behaviour.
The problem was solved numerically in the interval
0 6 x 6 1, subjected to the following initial and boundary
conditions:
c∗ (# = 0; x) = 0;
1 @c∗
Pe @x
@c∗
@x
∗
= c∗ − cin
(#);
= 0;
x = 0;
x = 1:
∀#
This example is divided into three distinct cases. Firstly,
the performance of the di3erent discretisation schemes is
analysed in order to demonstrate the advantage of bounded
schemes in solving problems with sharp moving fronts. This
is done in cases 1.1 and 1.2, using a 7xed grid with uniform
mesh spacing. After this preliminary discussion, dynamic
grid adaptation is included and discussed in the second half
of cases 1.2 and 1.3.
Case 1.1: In this 7rst test case, an inlet concentration pulse
of the following form is assumed:
1=T if 0 6 t 6 T;
∗
cin (t) = 'T (t) where 'T (t) =
0
if t ¿ T
with T = 10−3 . The other parameters used in the 7xed-bed
single-component adsorption model are !=1, K =1 and Pe=
104 . In all cases, the di3usive term was discretised with cubic
splines, as in a previous work (Cruz et al., 2002), but other
approaches could be used, such as second- or fourth-order
central di3erences, without any signi7cant di3erences in the
computed results. The numerical results obtained at # = 1
for two 7xed meshes, with 28 + 1 and 210 + 1 grid points,
are presented in Fig. 5 for the various schemes used in the
treatment of the convection term.
Fig. 5 shows that all the di3erencing schemes used for
the treatment of the convection term provide satisfactory
results when the 7ner grid is used. The only exception is
the upwind scheme which, due to its 7rst-order accuracy,
is excessively di3usive and therefore is not recommended
(Freitas, 1993). The unbounded schemes CDS2, CDS4 and
QUICK show signi7cant unphysical oscillations when the
coarse grid is used (negative concentrations), which are signi7cantly reduced with mesh re7nement. The cubic splines
method applied to the discretisation of the convection term
is also unbounded, but to a signi7cantly lesser extent, and
the observed oscillations in the computed solution are almost negligible, especially with the 7ner grid.
The high-resolution MINMOD and SMART schemes
are intrinsically bounded, and the computed solution shows
good agreement with the theoretical result without any type
of oscillations, even for the coarse mesh. The computed
pro7le obtained with the Super-B scheme in the 7ner mesh
presents an exaggerated peak value, which is due to an arti7cial reduction of the physical di3usion of the model. This
anomalous high compressibility of the Super-B scheme is
common, and therefore it is usually not recommended in
convection/di3usion problems (Leonard, 1991). The accuracy of the SMART scheme is found to be signi7cantly
superior to that of the MINMOD and Super-B schemes, as
seen by the di3erences in the peak values shown in Figs.
5(f) – (h). The error obtained in the 7nest mesh for the
peak values are 16.0%, 8.7% and 1.6% for the MINMOD,
Super-B and SMART schemes, respectively. This clearly
illustrates the superiority of the SMART high-resolution
scheme. Based on the results obtained for this simple
convection–di3usion example, one concludes that the best
alternatives for the treatment of the convection term are
the cubic splines (although slightly unbounded) and the
SMART high-resolution scheme. However, in the next test
case it will be shown that HR schemes are actually superior when solving truly hyperbolic PDEs, i.e. when no
stabilising di3usive terms are present.
Liu, Cameron, and Wang (2000) also studied this problem, using a 7xed-grid wavelet-collocation method, for
the case ! = 0. The numerical solutions obtained for this
“smooth” problem exhibited some oscillations, which disappear with mesh re7nement. Cruz et al. (2001a) pointed
out that this approach is actually equivalent to CDS4, and
proposed the use of an adaptive grid wavelet-collocation
method. The results obtained evidenced the main advantages
of using an adaptive grid strategy, namely the reduction in
memory and CPU requirements.
Case 1.2: This case considers the same problem as before, but with the :ow being driven only by advection, i.e.
Pe → ∞, and there is no adsorption (! = 0). This is an academic case, but, due to the di2cult numerical solution, it is
an excellent test to the discretisation schemes used for treating the convection term. In this problem, the injected pulse
travels along the column with constant velocity and without
changing its original shape.
Fig. 6 shows the numerical results obtained for an injected
pulse with T = 10−1 . Once again, the upwind scheme leads
to very di3usive results. The unbounded CDS2, QUICK
and cubic splines discretisation schemes fail in the vicinity
of the discontinuity, leading to strong oscillations in the
computed solution, without any physical meaning. These
oscillations are seen to decrease in magnitude and increase
in frequency with mesh re7nement, but are always present,
even for very re7ned meshes. This is an important result
that clearly illustrates the superiority of the high-resolution
schemes for convection-dominated problems, as illustrated
in Fig. 6(e) – (f).
The proposed adaptive approach is general and can be
coupled with any of the high-resolution schemes described
(or others that seem to be more suitable for a speci7c problem) for the treatment of the convective term. However,
since the SMART scheme was found to be the most robust and accurate in the preceding test cases, it is therefore
adopted in the treatment of the convection term in the following examples dealing with dynamical grid adaptation.
This 7xed-bed single-component adsorption problem
was also used to assess the performance of the grid
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
20
15
15
10
10
c*
c*
20
5
5
0
0
-5
-5
0.4
0.45
0.5
0.55
0.4
0.6
0.45
20
20
15
15
10
10
0.55
0.6
0.55
0.6
0.55
0.6
0.55
0.6
c*
c*
0.5
x
(b)
x
(a)
5
5
0
0
-5
-5
0.4
0.45
0.5
0.55
0.6
0.4
0.45
(d)
x
(c)
20
15
15
10
10
c*
c*
0.5
x
20
5
5
0
0
-5
-5
0.4
0.45
0.5
0.55
0.6
x
(e)
0.4
0.45
0.5
x
(f)
20
25
20
15
15
10
c*
c*
1783
10
5
5
0
0
-5
-5
0.4
(g)
0.45
0.5
x
0.55
0.4
0.6
(h)
0.45
0.5
x
Fig. 5. E3ect of mesh re7nement for the case T = 10−3 , ! = 1, K = 1 and Pe = 104 . Results for # = 1 obtained in a 7xed mesh with 2J + 1 grid
) J = 8, (
) J = 10, —– Theoretical). Discretisation of the convective term with (a) UDS, (b) CDS2, (c) CDS4, (d) Cubic splines,
points (
(e) QUICK, (f) MINMOD, (g) Super-B and (h) SMART scheme.
1784
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
12
10
10
8
8
6
6
c*
c*
12
4
4
2
2
0
0
-2
-2
0.4
0.45
0.5
0.55
0.6
x
(a)
0.4
0.45
0.5
0.55
0.6
0.55
0.6
0.55
0.6
x
(b)
12
10
10
8
8
6
6
c*
c*
12
4
4
2
2
0
0
-2
-2
0.4
0.45
0.5
0.55
x
(c)
0.4
0.6
10
10
8
8
6
6
c*
12
c*
0.5
x
12
4
4
2
2
0
0
-2
-2
0.4
(e)
0.45
(d)
0.45
0.5
0.55
0.4
0.6
x
(f)
0.45
0.5
x
Fig. 6. E3ect of mesh re7nement for the linear advection test case: T = 10−1 , ! = 0 and Pe → ∞. Results for # = 0:55 obtained in a 7xed mesh
J = 8,
J = 10, Theoretical). Discretisation of the convective term with (a) UDS, (b) CDS2, (c) QUICK, (d) Cubic
with 2J + 1 grid points (
splines, (e) MINMOD and (f) SMART scheme.
adaptation strategy proposed here. The SMART highresolution scheme is used to discretise the convective term
directly in the adapted non-uniform grid. An alternative
approach would be the complete reconstruction of the mesh
from the stored relevant information, followed by computation of the convective term in the reconstructed uniform
mesh. This would be, however, very time consuming and
is therefore not recommended.
The results obtained at # = 0:55 with the adaptive algorithm are presented in Fig. 7(a), for the model parameters
T = 10−1 , ! = 0 and Pe → ∞. As can be seen, the adaptation algorithm works well and the pulse is predicted with
excellent accuracy. The spatial location of the adapted grid
points and the corresponding resolution levels are presented
in Fig. 7(b). For this simple constant velocity problem, the
location of the higher grid density regions could be easily
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
10
1785
12
11
8
10
9
j
c*
6
4
8
7
6
2
5
4
0
0.0
0.2
0.4
0.6
0.8
x
(a)
0.0
1.0
(b)
0.2
0.4
0.6
0.8
1.0
x
Fig. 7. Solution of the 7xed bed single-component adsorption problem with T = 10−1 , ! = 0 and Pe → ∞: (a) dimensionless :uid phase concentration
pro7le at # = 0:55; (b) corresponding distribution of the grid points in terms of spatial location and resolution level. The parameters used in the adaptive
algorithm are Jmin = 4, Jmax = 12, = 10−3 and NR = NL = 2.
190
ε = 10 -5
nº of grid points
160
ε = 10 -3
130
100
70
40
10
0.0
0.4
θ
0.8
1.2
Fig. 8. Number of grid points used by the adaptation algorithm for
= 10−3 and 10−5 .
predicted, but the proposed method does not need such information and works in a self-su2cient dynamic way.
The number of grid points used by the adaptation algorithm for two values of the threshold parameter ( = 10−3
and 10−5 ) is presented in Fig. 8. The threshold parameter is
a direct measure of the error involved in the approximation
of the solution by a reduced set of grid points. Higher values imply fewer equations to integrate but lower accuracy.
The number of grid points used in the adaptive process approximately doubles when # = 0:1, which corresponds to the
instant when the pulse completely enters the computational
domain (with the corresponding appearance of the second
discontinuity in the solution). This e2cient variation of the
number of grid points along the integration process is of extreme importance. Because it uses only the grid points that
are necessary to attain a given precision, the present method
is more e2cient and versatile than methods that use a constant number of grid points, even in the case of moving mesh
methods.
To illustrate the performance of the adaptive strategy
proposed in this work, as compared to the uniform mesh
approach, the CPU times required to solve the problem in
a uniform mesh with 2Jmax + 1 grid points and the corresponding CPU times for the adaptive strategy are presented
in Table 1. The speedup factors presented in Table 1 are de7ned as the ratio of the CPU times required by the uniform
mesh and the adaptive algorithm. The accuracy of the solution is similar for both approaches, for each Jmax , but the
computed speedup factors are high, and increase with mesh
re7nement (almost at the optimum rate of 2), thus proving
the e2ciency of the proposed adaptive strategy.
Case 1.3: This example illustrates the inclusion of a
non-linear term in Eq. (19), through the adsorption equilibrium isotherm Eq. (20). K = 1:5 is used, together with
T = 10−1 , ! = 1 and Pe → ∞. For this particular isotherm
shape, higher concentrations tend to move faster than lower
concentrations. This causes the formation of a shock wave
at the pulse front and to a progressively deforming di3usive
wave at the back of the pulse (Cruz, Mendes, & Magalhães,
2001b), as shown in Fig. 9(a). The adaptation algorithm
places a higher density of grid points simultaneously in the
pulse front and in the tail of the di3usive wave, where the
changes in concentration gradient are higher, as illustrated
in Fig. 9(b).
The number of grid points used throughout the integration
is plotted in Fig. 10. After an initial increase in the total
number of grid points, due to the entrance of the pulse in
the domain, a signi7cant decrease is observed due to the
formation of the di3usive wave. Once again, the adaptation
strategy performs as expected, dynamically adjusting the
grid to the computed solution in an e2cient way.
7.2. Example 2. Inviscid Burgers’ equation
Burgers’ equation results from the application of the
Navier–Stokes equation to unidirectional :ow without
a pressure gradient. This is a problem that also arises
from Fokker–Planck equation and equations governing the
1786
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
Table 1
Speedup factors in CPU time consumption for the 7xed-bed single-component adsorption model with T = 10−1 , ! = 0 and Pe → ∞ (# = 0:55)
Uniform mesh
Adaptive
Jmax
NGPa
CPU (s)
NGPa
CPU (s)
Speedup
7
8
9
10
11
12
129
257
513
1025
2049
4097
1.59
5.51
21.1
78.3
303.9
1225
51
68
86
107
129
147
0.99
1.97
4.76
10.1
20.5
42.2
1.61
2.80
4.43
7.75
14.8
29.0
a Number
of grid points. For the adaptive strategy corresponds to # = 0:55.
4.0
12
11
3.0
10
2.0
j
c*
9
8
7
6
1.0
5
4
0.0
0.0
0.2
0.4
0.6
0.8
x
(a)
0.0
1.0
(b)
0.2
0.4
0.6
0.8
1.0
x
Fig. 9. Solution of the 7xed-bed single-component adsorption model with K = 1:5, T = 10−1 , ! = 1 and Pe → ∞. (a) Dimensionless :uid phase
concentration pro7le at # = 1; (b) corresponding distribution of the grid points in terms of spatial location and resolution level for Jmin = 4, Jmax = 12,
= 10−3 and NR = NL = 2.
220
nº of grid points
190
160
130
100
70
40
0
0.2
0.4
θ
0.6
0.8
1
Fig. 10. Number of grid points used by the adaptation algorithm. The
parameters used are the same as in Fig. 9.
spreading of liquids on solids. In the inviscid limit, Burgers’
equation is expressed in conservation form as (Fletcher,
1991):
@ ∗2
@u∗
+
(u =2) = 0;
@#
@x
(21)
where u∗ is the dimensionless velocity u∗ = u=uref , x is the
dimensionless space coordinate x = z=L, # is the dimensionless time variable # = turef =L, uref is the reference velocity,
z is the space coordinate, L is the characteristic length and
t is the time variable. This is a classical test for a numerical
method, since the problem is non-linear and, consequently,
discontinuities can develop from smooth solutions (Carlson
& Miller, 1998).
The problem was solved in the interval 0 6 x 6 2 subjected to the following initial and boundary conditions as
suggested by Finlayson (1992):
1; 0:5 ¡ x 6 1:5;
u(0; x) =
0 elsewhere;
u(#; 0) = 0:
For this type of initial discontinuous pro7le there is a
moving front with velocity equal to the average of the velocity just before and just after the shock, a = (a+ + a− )=2 =
(1 + 0)=2 = 0:5. The trailing edge does not move at all, but
the solution with u∗ = 1 moves with constant velocity a = 1.
Fig. 11(a) shows the numerical results, at # = 0:5, computed with the adaptive strategy using Jmax = 12 and the
SMART scheme to calculate the convective term in the
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
1
1787
12
11
0.8
10
9
j
u*
0.6
0.4
8
7
6
0.2
5
0
0.0
0.4
(a)
0.8
1.2
1.6
4
0.0
2.0
x
0.4
0.8
1.2
1.6
2.0
x
(b)
Fig. 11. Solution of Burgers’ equation at # = 0:5. (a) Dimensionless velocity pro7le; (b) corresponding distribution of the grid points in terms of spatial
location and resolution level. The parameters used in the adaptation algorithm were Jmin = 4, Jmax = 12, = 10−3 and NR = NL = 2.
adaptive grid. In Fig. 11(b) the grid points are shown in
terms of spatial location and resolution level. The number
of grid points used by the adaptive procedure is practically
constant along the integration with approximately 125 grid
points (not shown).
The speedup ratios obtained for this problem are presented
in Table 2. Once again the e2ciency of the proposed adaptive method is remarkable. The solution of this hyperbolic
problem with the unbounded cubic splines, CDS2 and CDS4
schemes leads, once again, to severe unphysical oscillations.
7.3. Example 3. Buckley–Leverett problem
The Buckley–Leverett problem results from the application of the material balance equations to two immiscible
:uids, one being displaced by the other in a porous media.
Neither :uid completely 7lls the space before nor after the
injection front, so the problem is solved in terms of a dimensionless saturation variable, s∗ (fraction of the space 7lled
with one phase). The equation that describes such problem
is (Kurganov & Tadmor, 2000):
@s∗
@s∗
@F(s∗ )
@
G(s∗ )
;
(22)
+
=
@#
@x
@x
@x
where # and x are the dimensionless time and space coordinate variables, respectively. The function F(s∗ ) is the ratio between the mobility of the two phases. Kurganov and
Tadmor (2000) also analysed this problem with
F(s∗ ) =
s∗2
;
∗2
s + (1 − s∗ )2
G(s∗ ) = 4s∗ (1 − s∗ )
(23)
(24)
and = 0:001, in the interval 0 6 x 6 1, subjected to the
following initial and boundary conditions:
s∗ (0; x) = 0;
s∗ (#; 0) = 1:
This highly non-linear problem has a moving front with
variable velocity. The simulation results obtained in an
adapted grid, with Jmax = 12, = 10−3 , NR = NL = 2
and SMART high-resolution scheme, are presented in
Fig. 12(a). The adaptation strategy proves to be capable
of dealing with highly non-linear equations that include
di3usive terms, as seen by the correct allocation of the grid
points shown in Fig. 12(b).
Observation of the time evolution of the number of grid
points shows that, after a shortly initial stage, it stays practically unchanged around 70, which corresponds to a data
compression of about 98%, relatively to the 7xed mesh approach.
Now consider the same problem without the numerically
stabilising di3usive term (=0), and including gravitational
e3ects in f(s∗ ) (Kurganov & Tadmor, 2000)
F(s∗ ) =
s∗2
[1 − 5(1 − s∗ )2 ]
s∗2 + (1 − s∗ )2
(25)
for the interval 0 6 x 6 1, and subjected to the following
initial and boundary conditions:


 0; 0 6 x ¡ 1 − √1 ;
∗
2
s (0; x) =

 1 elsewhere;
s∗ (#; 0) = 0:
This problem has initially one steep front, which propagates and creates a second one. The moving front shown on
the left part of Fig. 13(a) is self-sharpening, thus conducing to fewer points in the front vicinity as illustrated in Fig.
13(b). In this case, the number of grid points used by the
adaptive strategy increases initially, due to the creation of
the second step, and then stays practically unchanged around
85, corresponding to a data compression of about 98%.
1788
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
Table 2
Speedup factors in CPU time consumption for Burgers’ equation (# = 0:5)
Uniform mesh
Adaptive
Jmax
NGPa
CPU (s)
NGPa
CPU (s)
Speedup
8
9
10
11
12
257
513
1025
2049
4097
1.16
4.62
18.8
75.1
340.4
75
85
102
114
120
0.39
1.15
2.58
5.87
13.7
2.97
4.02
7.29
12.8
24.8
a Number
of grid points. For the adaptive strategy corresponds to # = 0:5.
1.0
12
11
0.8
10
9
8
j
s*
0.6
0.4
7
6
0.2
5
0.0
0.0
4
0.2
0.4
0.6
0.8
x
(a)
0.0
1.0
0.2
0.4
(b)
0.6
0.8
1.0
x
Fig. 12. Solution of Buckley–Leverett problem: (a) dimensionless saturation pro7le at # = 0:2; (b) grid points distribution in terms of spatial location
and resolution level. The parameters used in the adaptation algorithm are Jmin = 4, Jmax = 12, = 10−3 and NR = NL = 2.
12
1.0
11
0.8
10
9
j
s*
0.6
0.4
8
7
6
0.2
5
4
0.0
0.0
(a)
0.2
0.4
0.6
0.8
x
1.0
0.0
(b)
0.2
0.4
0.6
0.8
1.0
x
Fig. 13. Solution of the Buckley–Leverett problem including gravitational e3ects: (a) dimensionless saturation pro7le at # = 0:2; (b) distribution of the grid
points in terms of spatial location and resolution level. The parameters used in the adaptation algorithm are Jmin = 4, Jmax = 12, = 10−3 and NR = NL = 2.
7.4. Example 4. Counter-current heat exchanger
This example illustrates the application of the adaptive
strategy to a system of partial equations that models the
dynamic start-up of a counter-current heat exchanger. The
model assumes plug-:ow, negligible heat losses to the exterior, constant heat capacity and density of both :uids and
negligible thickness and thermal inertia of the wall that
separates both :uids. For the inner :uid the thermal balance
leads to (Rice & Do, 1995):
h i Pi
@Ti
@Ti
+i Cpi
+ +i Cpi ui
+
(Ti − To ) = 0;
(26)
@t
@z
Ai
while for the outer :uid
@To
@To
h o Po
+o Cpo
(Ti − To ) = 0;
− +o Cpo uo
−
@t
@z
Ao
(27)
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
where T is the temperature, t is the time, u is the velocity, +
and Cp are the density and the heat capacity of the :uids, A
is the transverse area, h is the overall heat transfer coe2cient
and P is the heat exchange perimeter. The subscripts i and
o refer to the inner and outer :uid, respectively.
It is a better alternative to rewrite Eqs. (26) and (27) in
dimensionless form
@Ti∗
@T ∗
= − i − Hi (Ti∗ − To∗ );
@#
@x
(28)
@T ∗
@To∗
= o + Ho (Ti∗ − To∗ )
@#
@x
(29)
r
with Ti∗ = (Ti − Ti; inlet )=Ti; inlet , To∗ = (To − Ti; inlet )=Ti; inlet , x =
z=L, # = t(ui =L), r = ui =uo , Hi = hi Pi L=(+i Cpi Ai ui ) and Ho =
ho Po L=(+o Cpo Ao uo ), where Ti; inlet is the inlet temperature
in the inner tube.
The problem was solved in the interval 0 6 x 6 1 subjected to the following initial and boundary conditions:
Ti∗ (0; x) = 0;
To∗ (0; x) = 0;
The numerical results for Hi =1, Ho =1, r=1 and To;∗ inlet =1
are presented in Fig. 14(a) and, once again, the data compression is high as seen in Fig. 14(b).
7.5. Example 5. Non-isothermal catalytic reactor model
This example describes the startup of a tubular catalytic
reactor undergoing a pseudo-homogeneous 7rst-order irreversible exothermic reaction. The main assumptions of the
model are: plug-:ow, negligible radial gradients, constant
interstitial velocity, kinetic constant dependence on temperature according to Van’t Ho3 equation, constant heat capacities and densities, thermal equilibrium between the stationary and mobile phases, negligible heat accumulation in the
column wall, and negligible pressure drop. The conservation
laws for mass and energy are expressed in dimensionless
form as (Cruz et al., 2002):
@c∗
1
@c∗
∗
E
;
(30)
=−
− Da c exp R 1 − ∗
@#
@x
T
@T ∗
@T ∗
=−
− NW (T ∗ − 1)
@#
@x
1
∗
E
+ 2c exp R 1 − ∗
T
(31)
with
Da =
kref 3b
;
b
T ∗ = T=Tref ;
RE = −
RH =
NW =
2h3b
Rb b + Cp
and
b Cp + + (1 − b ) Cps +s
;
b Cp +
where c is the :uid phase concentration, T is the absolute
temperature, t is the time variable, L is the column length, u
is the velocity, k is the velocity reaction constant, SH is the
heat of reaction, h is the overall heat transfer coe2cient, Rb
is the inner column radius, Cps is the solid heat capacity, +s
is the solid density, z is the axial space coordinate, Cp is the
gas heat capacity, + is the gas density, b is the bed porosity,
E is the activation energy and R is the ideal gas constant.
The “∗ ” superscript designates dimensionless variables and
the subscript “ref ” means reference.
The initial and boundary conditions are
c∗ (0; x) = 0;
T ∗ (0; x) = 1;
T ∗ (#; 0) = 1:
To∗ (#; 1) = To;∗ inlet :
c∗ = c=cref ;
(−SH )kref 3b cref
;
b Tref + Cp
c∗ (#; 0) = 1;
Ti∗ (#; 0) = 0;
RH
2=
1789
# = t=3b ;
E
;
RTref
3b = L=u;
A similar problem, but incorporating a stabilising di3usive term (axial dispersion plug :ow), was solved by Cruz
et al. (2002) using a wavelet-based adaptive grid procedure,
and also by Coimbra, Sereno, and Rodrigues (2001) using
a moving 7nite-element method. In the present work, on the
other hand, the limiting case of plug :ow (Pe → ∞) is
studied. The mentioned methods would fail, in this situation,
leading to the appearance of oscillations in the computed solution when strong gradients develop, due to the inexistence
of stabilising di3usive terms in Eqs. (30) and (31).
The results obtained with Da=1, 2=1, RE =20, RH =5000
and NW = 30 are plotted in Fig. 15(a) for a short dimensionless time, # = 0:5. The reactant’s concentration front is moving through the reactor, but the temperature still remains essentially unchanged. The grid points are placed correctly, at
the locations of the strong concentration gradients, as shown
in Fig. 15(b).
Fig. 16 shows the time evolution of the dimensionless
temperature pro7les for # = 300, 500, 700 and 900 and # →
∞ (steady state solution). As time increases, the reactant is
rapidly consumed in the initial portion of the reactor, originating a strong temperature “hot spot”. The steady-state
solution (system of 2 ODEs) was also computed with the
initial-value problem ODE integrator LSODA. The admissible error in the ODE integration was varied in the range
10−9 –10−10 and the computed results did not di3er significantly, thus the steady-state solution presented in Fig. 16
may be regarded as “exact”. For large time values, the computed solution with the proposed adaptive strategy is clearly
approaching the “exact” solution for steady-state conditions,
thus proving the good accuracy and stability of the proposed
strategy for PDE solution.
1790
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
1.0
11
To*
10
0.8
9
0.6
j
T*
8
7
0.4
6
0.2
5
Ti *
0.0
0.0
4
0.2
0.4
0.6
0.8
1.0
x
(a)
(b)
0.0
0.2
0.4
x
0.6
0.8
1.0
Fig. 14. Solution of the counter-current heat exchanger model with Hi = 1, Ho = 1, r = 1 and To;∗inlet = 1: (a) dimensionless temperature pro7les for
# = 0:5; (b) distribution of the grid points in terms of spatial location and resolution level. The parameters used in the adaptation algorithm are = 10−3 ,
Jmin = 4, Jmax = 12 and NR = NL = 2.
11
1.0
10
0.8
0.6
8
c *,
j
T
*
9
7
0.4
6
0.2
5
0.0
4
0.0
0.2
0.4
0.6
0.8
x
(a)
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
(b)
Fig. 15. Solution of the non-isothermal catalytic reactor problem. (a) Dimensionless :uid phase concentration and temperature pro7les at # = 0:5; (b)
corresponding distribution of the grid points in terms of spatial location and resolution level. The parameters used in the adaptation algorithm are
= 10−4 , NR = NL = 2, Jmin = 4 and Jmax = 11.
220
1.7
1.6
180
∞
nº of grid points
θ
1.5
T*
1.4
1.3
θ = 700
1.2
θ = 900
θ = 500
1.1
1.0
0.0
140
100
60
θ = 300
0.1
0.2
0.3
x
20
0.01
0.1
1
10
100
1000
10000
θ
Fig. 16. Non-isothermal catalytic reactor. Time evolution of the dimensionless temperature pro7le. The parameters used are the same as Fig. 15.
Fig. 17. Number of grid points used by the adaptation algorithm for the
solution of the non-isothermal catalytic reactor model. The parameters
used in the adaptive algorithm are the same as Fig. 15.
The number of grid points used by the adaptation algorithm is shown in Fig. 17. At # = 1, a signi7cant decrease in
the number of grid points occurs, due to the exiting of the
concentration front. As time increases, the progressive onset
of the temperature “hot spot” leads to a signi7cant increase
in the total number of grid points.
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
As illustrated by Fig. 17, the time scale of the concentration front is of order O(1) while the time scale of the
temperature is of order O(103 ). This problem illustrates the
need that the frequency of the grid adaptations must be estimated based on the time evolution of the computed solution, and conveniently adjusted during the time integration
process in order to maximise the e2ciency of the method
in terms of computation speed.
In all the examples presented the time-integration error
was set small enough to ensure that the main errors are due
to the spatial discretisation.
8. Conclusions
An adaptive numerical method for solution of PDEs is
described and applied to several problems involving sharp
moving fronts. The method is based on the multiresolution
representation of data, and utilises a high-resolution scheme
to discretise the convective terms in the PDEs, thus ensuring boundedness and good accuracy of the computed solution. The dynamic adaptive strategy was found to be robust
and e2cient, leading to signi7cant reductions in the required
CPU times, as compared to the equivalent 7xed mesh approach. The solution of numerically di2cult systems, such
as hyperbolic PDEs, is illustrated. The method proved to be
very e2cient in terms of speed and memory savings, with
data compressions up to 98%.
Extension of the adaptive strategy to multiple dimensions
is possible, and is currently under investigation.
Pe
q
r
R
Rb
RE
RH
s∗
S
t
T
T
u
V
x
x
z
1791
Peclet number, Lu=Dax
adsorbed phase concentration, mol=m3
velocity ratio, r = ui =uo
universal gas constant, J=(mol K)
inner column radius, m
dimensionless parameter, −E=RTref
dimensionless parameter, b Cp + + (1 − b ) Cps +s =
b Cp +
dimensionless :uid saturation
source term
time variable, s
dimensionless pulse concentration time
temperature, K
:ow velocity, m/s
dyadic grid space
spatial location
dimensionless spatial coordinate
spatial coordinate, m
Greek letters
2
SH
b
#
5
+
+s
3b
dimensionless parameter, (−SH )kref 3b cref =b Tref + Cp
heat of reaction, J/mol
threshold parameter
bed porosity
dimensionless time variable
capacity factor, (1 − b )=b qref =cref
:uid density, kg=m3
solid density, kg=m3
bed residence time, s, L=u
Superscripts
Notation
a
c
Cp
Cps
d
Da
Dax
dtout
E
F
G
h
Hi
Ho
Jmin
Jmax
k
K
L
NL
NR
NW
P
local velocity
:uid phase concentration, mol/m3
:uid phase heat capacity, J=(K kg)
solid heat capacity, J=(K kg)
interpolative error
DamkOohler number, kref 3b =b
e3ective axial dispersion coe2cient, m2 =s
time interval along which the grid stays unchanged
activation energy, J/mol
convective :ux
di3usive function
overall heat transfer coe2cient, W=(m2 K)
dimensionless parameter, Hi = hi Pi L=+i Cpi Ai ui
dimensionless parameter, Ho = ho Po L=+o Cpo Ao uo
minimum resolution level
maximum resolution level
reaction rate constant, m3 =(mol s)
isotherm parameter
column length, m
number of collocation points added to the left
number of collocation points added to the right
dimensionless parameter, 2h3b =Rb b + Cp
heat exchange perimeter, m
j
∗
in
resolution level
dimensionless variable
inlet
Subscripts
i
o
D
f
k
P
ref
U
inner :uid
outer :uid
downstream node to face f
face
spatial index
upstream node to face f
reference
upstream node to P
Acknowledgements
The work of P. Cruz was supported by FCT (grant
BD/21483/99) and by funds of Sapiens Project 38067/EQU/
2001. M.A. Alves wishes to thank Universidade do Porto
and his colleagues at Departamento de Engenharia QuXYmica,
FEUP, for a temporary leave of absence and acknowledges the 7nancial support provided by Fundagão Calouste
Gulbenkian.
1792
P. Cruz et al. / Chemical Engineering Science 58 (2003) 1777 – 1792
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