1. Dynamic Optimization of Batch HPLC Separation Processes
Anders Holmqvista,, Fredrik Magnussonb, Bernt Nilssona
aDepartment of Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden
bDepartment of Automatic Control, Lund University, P.O. Box 118, SE-221 00 Lund, Sweden
Abstract
This contribution describes the realization of a new
off-line dynamic optimization framework for batch
HPLC separation processes. A large-scale dynamic
optimization problem (DOP) constrained by PDEs
is formulated for simultaneous optimization of gen-eral
elution trajectories, parameterized with piece-wise
constant control signals, and target component
pooling decisions (two fractionation or cut-times)
with respect to the recovery yield. The advantages
of this methodology are illustrated through the solu-tion
of a specific challenging ternary complex mixture
separation problem, with the intermediately elut-ing
component as the target, by hydrophobic inter-action
chromatography (HIC). The realistic multi-component
system dynamics required for analysis
were generated by numerical solution of the reactive-dispersive
model with experimentally validated kinet-ics.
Dynamic Optimization
The recovery yield of the th component collected in
the pooling interval [0,, f,] is defined as:
load,
dY
dt
= c(t, zf)(t, 0,, f,), (1)
where (t, 0,, f,) 2 [0, 1] is a continuously differ-entiable
rectangular function and ,load is the loaded
sample amount. A quadratic cost on the differences
of the piecewise constant control flows ui is added
to the cost function:
L(tf ) = −
tf
Z
t0
dY
dt
dt +
Ni
Xi=1
(ui)TU(ui), (2)
in order to influence the smoothness u(t). L(tf) was
optimized while fulfilling the requirement placed on
purity of the target component fractionation:
X(tf ) =
tf
Z
t0
load,
dY
dt
dt

X8
tf
Z
t0
load,
dY
dt
−1

. (3)
X(tf ) was incorporated in the DOP as a terminal
inequality constraint:
0 Xmin,(tf ) − X(tf ), (4)
where Xmin,(tf ) is the assigned purity requirement.
Limit Cycle Criteria
The limit cycle criteria ensure that the state at the
initial time is retained at the end of the batch cy-cle,
i.e. the modifier, S, and all components, 2
{A,B,C}, loaded have to be completely eluted:
0 =
tf
Z
t0
c(t, zf)dt − load,, (5a)
0 = cS(t0, z) − cS(tf , z), (5b)
0 = u(t0) − u(tf ), (5c)
and can be classified into terminal equality con-straints.
Optimization Environment
The DOP was transcribed into a nonlinear program
(NLP) using the direct collocation method in the
JModelica.org framework (www.JModelica.org).
◮ The control and state variables were fully dis-cretized
in the temporal domain [t0, tf ] using
Radau collocation on finite elements.
◮ The PDE system was approximated using an adap-tive,
high order finite volume weighted essentially
non-oscillatory (WENO) scheme.
◮ The NLP was solved using the primal-dual inte-rior
point method IPOPT and the Jacobian and
Hessian matrices were computed using automatic
differentiation techniques.
Benchmarking
In order to assess the performance of the optimized
elution trajectories, these are benchmarked with op-timized
nonlinear gradients:
ug(t) = ug(0) + [ug(f ) − ug(0)]
t − 0
f
S
, (6)

where S is the gradient shape factor.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.5
1.35
1.2
1.05
0.9
0.75
0.6
0.45
Xmin,B(tf ) = 0.99
YB(tf ) = 0.74
cA
cB
cC
ug
cS
0.2 0.3
0.1
0.15
0 0
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25
t (CV)
¯c(t, zf ) (−)
¯cS(t, zf ), ¯ug(t) (−)
Figure 1: Elution profile generated with optimal
nonlinear gradients for Xmin,B(tf ) = 0.99.
General Elution Trajectories
a) 1
1.5
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.35
1.2
1.05
0.9
0.75
0.6
0.45
Xmin,B(tf ) = 0.90
YB(tf ) = 0.99
0.2 0.3
0.1
0.15
0 0
¯c(t, zf ) (−)
¯cS(t, zf ), ¯u(t) (−)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.5
1.35
1.2
1.05
0.9
0.75
0.6
0.45
Xmin,B(tf ) = 0.975
YB(tf ) = 0.95
0.2 0.3
0.1
0.15
0 0
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25
c)
cA
cB
cC
u
cS
t (CV)
¯c(t, zf ) (−)
¯cS(t, zf ), ¯u(t) (−)
b) 1
1.5
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.35
1.2
1.05
0.9
0.75
0.6
0.45
Xmin,B(tf ) = 0.95
YB(tf ) = 0.98
0.2 0.3
0.1
0.15
0 0
t (CV)
¯c(t, zf ) (−)
¯cS(t, zf ), ¯u(t) (−)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
1.5
1.35
1.2
1.05
0.9
0.75
0.6
0.45
Xmin,B(tf ) = 0.99
0.2 0.3
0.1
0.15
0 0
0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25
d)
YB(tf ) = 0.85
t (CV)
¯c(t, zf ) (−)
¯cS(t, zf ), ¯u(t) (−)
Figure 2: Elution profile generated with optimal trajectories for Xmin,B(tf ) 2 [0.90, 0.95, 0.975, 0.99] and
Ni = 50 piecewise constant control signals. Markers indicate the solution at the Radau collocation points.
The colored area represents the pooling interval endpoints [0,B, f,B].
Concluding Remarks
By comparison of the elution profiles depicted in Figure 1 and 2d, it is apparent that the general elution
trajectories are superior to the nonlinear gradients in terms of recovery yield. In the case study considered, a
13% increase in recovery yield was obtained. Additionally, the optimal terminal cut-time is significantly lower
for the elution profile generated with the general elution trajectory. In the context of optimizing productivity,
being the mass of target component per unit mass of packing and per unit time, the magnitude of terminal
cut-time is essential since the productivity metric is inversely proportional to this parameter.
The authors acknowledge the support of the strategic innovation program Process Industrial IT and Automation
(PiiA) through the Process Industrial Center at Lund University (PIC–LU).
,
Corresponding author. Tel.: +46 46 222 4925 www.chemeng.lth.se
E-mail address: anders.holmqvist@chemeng.lth.se www.pic.lu.se