(218b) Scale-Up of Continuous Reactors Using Phenomenological-Based Models

SCALE-UP OF CONTINUOUS
REACTORS USING PHENOMENOLOGICAL-BASED MODELS

 

For years scaling up has been a
sort of art in which the expertise, rules of thumb, trial and error, particular
solutions and subjective decisions have been used to obtain a proper result at
a new scale [1].
Traditionally, chemical processes scale-up is performed based on dimensional
analysis perspective, similarity principle criteria, rules of thumb, empirical
relations constructed from a set of process data, and in recent years using
phenomenological-based models supported by empirical relationships [2].
This shows that it does not exist an infallible method for scaling up a given
process, and that a huge effort and proficiency is required to achieve an
acceptable outcome of the process at industrial scale [1].

Within the drawbacks of traditional
scale-up methods, the main problem is to make a complete list of the relevant
independent variables. Here the science meets the art: the choice of the
variables is highly subjective and without any rigor [3],
[4]. In addition, during scale-up mass transfer, energy transfer,
momentum transfer and kinetic reaction could change significantly between
scales, and traditional scale-up methods do not take into account how avoiding
these changes [5].
This forces the designer to decide by expertise which mechanism is the most important
(dominant) of the process. Here, the real governing regime (Dominant Operating
Regime, DOR) is unknown and there is no direct method for finding it [1],
[2]. This fact illustrates that traditional scale-up methods do not
always leads into the best commercial unit design.

It is well-known that a model can
represent precisely a given system. However, the main inconvenient to use a
model for scaling up a process is its validation at several scales; especially because
the model parameters vary with the change of scale [6],
[7]. As a way to overcome this inconvenient and taking into account
that a phenomenological-based model is a fundamental tool to comprehend the
behavior of a given system, this work presents a new methodology for scaling up
continuous processes using Hankel matrix for capturing process dynamic behavior
and scaling up a given process including its dynamic behavior. The methodology
herein proposed is based in a previous one developed by Ruiz and Alvarez [2],
but overcoming its limitations.

The proposed procedure uses an
extension of the discrete form of a control tool called Hankel matrix which has
been widely used in model reduction, systems identification, digital filter
design [8],
and recently in controllers design for establishing inputs and outputs pairings
[9].
This methodology uses a Phenomenological-Based Semiphysical Model (PBSM)
to represent the process system and Hankel matrix for: (i) representing
the dynamic behavior of the process (design variables - state variables terms),
(ii) determining the effect of the design variables as a whole over each
state variable and (iii) scaling up the process maintaining its dynamics
hierarchy, establishing the real scale factors.

For determining the dynamics
hierarchy the State Impactability Index (SII) of each state variable is
computed. This index represents the impactability of process designs variables
as a whole over a k-th given state [9].
According to this, the main dynamic (slowest dynamic) is the state variable
with the highest SII in the process, i.e., the governing mechanism is
intimately related with the main dynamic allowing finding the Dominant
Operating Regime (DOR). SII is computed by means of Singular
Value Decomposition (SVD) of the matrix obtained by multiplying the discrete
observability (O_b) and controllability (C_o)
matrices; such product is known as the Hankel matrix.

The methodology consists of the
following steps:

1. Define
capacity variable at the current and new scales.

2. Obtain
a PBSM of the process under investigation.

3. Define
the state variables, design variables, synthesis parameters and
design-variables-dependent parameters from the model obtained in second step.

4. Fix
the Operating Point (OP), considering that it must be stable.

5. Explicit
each design-variables-dependent parameter as a function of the design
variables.

6. Linearize
the obtained model around the OP, considering the design variables as
manipulated inputs.

7. Numerically
condition B and C matrices obtained from linearization to make
inputs (design variables) and outputs (state variables) dimensionless and
normalized.

8. Discretize
the linear model obtained.

9. Compute
the Observability, Controllability and Hankel matrices from the discrete model.

10. Compute
singular values from Hankel matrix.

11. Compute
SII of each state variable.

12. Repeat
ninth to twelfth steps until the capacity variable's value is equal to its
value at the new scale.

13. Compare
SII values at each scale. If SII values at the current scale (cs)
are approximately equal to SII values at the new scale (ns),
continue with the next step. On the opposite case, a successful scale-up is not
possible from the process synthesis established.

14. Compute
design-variables-dependent parameters as a function of the design variables at
the ns (scale factors).

15. Simulate
the process with the synthesis parameters, design variables and
design-variables-dependent parameters values at the ns in order to prove
if scale-up task is successfully developed.

The proposed methodology is
applied to a non-isothermal polymerization reactor for scaling up this process
from 0,1m³ to 1,0m³. To do this, SII
index is computed considering two cases: (i) the overall heat transfer
coefficient required (U_r) and (ii) the overall heat
transfer coefficient available (U_a) (maintaining the
geometrical similarity). In Table 1 the SII values of the state variables
is shown at the cs and ns. Here, the dynamics hierarchy in both
cases is the same, i.e., SII values of x₃>x₄>x₁>x₅>x₂>x₆
respectively.

 

In both cases, x₁
(which represents total inactive polymer concentration) is the main (slowest)
dynamic. In addition, in Table 1 is shown that the SII values remain
constant with U_r and increase with U_a. If
it is considered that changes in SII values show the deterioration of the
OP with changes in the operating scale, in Table 2 state variables
values at OP and the polymer average molecular weight (y) are compared
for both cases at each scale demonstrating this deterioration.

 

This shows that preserving
geometrical similarity do not guarantee the same yield of the process at the
new scale, and that in order to satisfy the energy requirements of the process,
the reactor's jacket must be redesigned by adding baffles. It is also showed that
by maintaining the dynamics hierarchy it is possible to obtain the same process
yield accomplished at cs.

The main contribution of this
work is the integration of an index (SII) to the scale-up task which
allows the establishment of the real scale factors of a given process, and the DOR
(governing mechanism) which was quite difficult to establish with traditional
methods. In addition, the proposed procedure allows scaling up the process maintaining
the same dynamics hierarchy through changes of scale.

REFERENCES

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Investigate the Performance of Bioreactors,? Enzyme and Microbial Technology,
vol. 9, no. 7, pp. 386?398, Jul. 1987.

[2] A. A. Ruiz and H. Alvarez,
?Scale-up of Chemical and Biochemical Processes Using a Phenomenological Model
(in Spanish),? Información Tecnológica, vol. 22, no. 6, pp. 33?52, 2011.

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[6] A. Bisio and R. L. Kabel, Scaleup
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[7] V. J. Inglezakis and S. G.
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[8] H. Yin, Z. Zhu, and F.
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[9] L. A. Alvarez and J. J.
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