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.

 

Table 1: SII values at 0,1m3 and 1,0m3, with Ur and Ua.
State Variables	SIIcs	SIIns with Ur	SIIns with Ua
x1	6,52	6,52	13,57
x2	1,61	1,61	2,83
x3	125,34	125,34	261,97
x4	72,47	72,47	151,65
x5	2,74	2,74	5,85
x6	1,08	1,08	1,13

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.

 

Table 2: OP comparison at 0,1m3 and 1,0m3 with Ur and Ua.
Variables	Values at cs	Values at ns with Ur	Values at ns with Ua	SI Units
x1	1,3×10-1	1,3×10-1	5,0×10-4	kmol/m3
x2	5,5	5,5	1,3	kmol/m3
x3	2,0×10-3	2,0×10-3	5,2×10-1	kmol/m3
x4	4,9×101	4,9×101	4,7×102	kg/m3
x5	335	335	430	K
x6	297	297	301	K
y	25000	25000	907	kg/kmol

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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[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.

[3] M. C. Ruzicka, “On Dimensionless Numbers,” Chemical Engineering Research and Design, vol. 86, no. 8, pp. 835–868, Aug. 2008.

[4] M. Rüdisüli, T. J. Schildhauer, S. M. A. Biollaz, and J. R. van Ommen, “Scale-Up of Bubbling Fluidized Bed Reactors - A Review,” Powder Technology, vol. 217, no. 0, pp. 21–38, Feb. 2012.

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[6] A. Bisio and R. L. Kabel, Scaleup of Chemical Processes Conversion from Laboratory Scale Tests to Successful Commercial Size Design. New York: John Wiley & Sons, Inc., 1985, p. 699.

[7] V. J. Inglezakis and S. G. Poulopoulos, “6 - Reactors Scale-up,” in Adsorption, Ion Exchange and Catalysis Design of Operations and Environmental Applications, Amsterdam: Elsevier, 2006, pp. 523–550.

[8] H. Yin, Z. Zhu, and F. Ding, “Model Order Determination Using the Hankel Matrix of Impulse Responses,” Applied Mathematics Letters, vol. 24, no. 5, pp. 797–802, May 2011.

[9] L. A. Alvarez and J. J. Espinosa, “Methodology Based on SVD for Control Structure Design (in Spanish),” XIII Congreso Latinoamericano de Control Automático, p. 6, 2008.