MODEL-BASED RUN-TO-RUN OPTIMIZATION FOR PROCESS DEVELOPMENT

Luna, Martin F.; Martínez, Ernesto C.

doi:10.1590/0104-6632.20180353s20170212

Martin F. Luna

INGAR (CONICET-UTN), Avellaneda 3657, Santa Fe S3002 GJC, Argentina

Ernesto C. Martínez ^**Corresponding author. E-mail address: ecmarti@santafe-conicet.gov.ar

INGAR (CONICET-UTN), Avellaneda 3657, Santa Fe S3002 GJC, Argentina

*Corresponding author. E-mail address: ecmarti@santafe-conicet.gov.ar

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Figure 1
(a) Simulations for the PTM defined in eq (6) using different parameterizations of θ (in light blue) and its mean (in black). (b) The corresponding normalized pdf at two values of u.

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Figure 2
Block diagram for the optimization scheme proposed in this paper.

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Figure 3
An algorithm to shrink the operating region.

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Figure 4
Contour levels for the real performance (black lines), the prediction of the tendency models (doted lines) and the operating conditions (a) for the first run of an iteration and (b) for the second run, after the region is shrunk.

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Figure 5
(a) Actual region for process improvement; (b) model predictions for performance improvement; (c) Region where the model doesn't have a good tendency; (d) Zoom of the region around operating conditions used for model fitting.

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Figure 6
Reaction scheme for the case study.

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Figure 7
Reactor setup for the case study.

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Figure 8
Contour lines of the performance index and its restrictions for the example presented in this section.

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Figure 9
Reaction scheme used to develop the tendency model.

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Figure 10
Experimental data points (squares) and PTM predictions for the concentrations of (a) P and (b) I. The pdf of the concentration of species I predicted by the model is depicted in (c).

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Figure 11
(a) Best run from each iteration for the first implementation of the methodology. (b) The last run from iteration #1 and all runs from iteration #2.

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Figure 12
Learning curve for the first implementation of the methodology.

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Table 1
All runs from the first implementation of the methodology.

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Table 2
Summary of the implementations starting from operating point

u_{0}^{a}

.

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Table 3
Summary of the implementations for the starting points b, c and d.

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Table 4
Summary of the implementations for α₁=90% and without using PTMs.

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Table 5
Comparison of three cases presented.

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Table A.1.
Parameters used to simulate the experimental data

(1)

(2)

(3)

(4)

(5)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17.a)

(17.b)

(18)

(19)

(20)

(A.1)

(A.2)

ITER #	RUN #	$u_{1}^{*}$ [L/min]	$u_{2}^{*}$ [min]	J_R(u)*
0	1	1.20E-02	45.00	0.6942
1	2	8.73E-03	116.01	1.4055
2	3	6.98E-03	159.46	1.3781
	4	7.99E-03	110.05	1.3090
	5	8.40E-03	128.96	1.4184
3	6	7.12E-03	165.78	1.3826
	7	8.01E-03	132.29	1.4066
	8	8.35E-03	117.35	1.3870
	9	8.45E-03	128.46	1.4205
4	10	7.06E-03	166.88	1.3809
	11	8.57E-03	110.34	1.3691
	12	7.95E-03	133.42	1.4062
	13	8.44E-03	130.28	1.4220
5	14	7.10E-03	166.12	1.3829
	15	7.99E-03	133.05	1.4067
	16	8.34E-03	117.87	1.3876
	17	8.43E-03	130.55	1.4190
	18	8.45E-03	129.71	1.4175

#	Number of Runs	$u_{1}^{*}$ [L/min]	$u_{2}^{*}$ [min]	φ	J_R(u)*
1	18	8.44E-03	130.28	1.2207	1.4220
2	12	7.40E-03	154.22	9.8826	1.3972
3	17	8.28E-03	142.50	12.6217	1.4179
4	25	8.40E-03	134.25	15.8048	1.4227
5	12	8.35E-03	140.41	0.9878	1.4168

#	Number of Runs	$u_{1}^{*}$ [L/min]	$u_{2}^{*}$ [min]	φ	J_R(u)*
$u_{0}^{b}$ = [ 1,8E-02 L/min; 30 min ]
1	26	8.04E-03	142.64	12.7983	1.4152
2	15	1.29E-02	66.64	24.9766	1.3086
3	14	8.21E-03	149.94	15.3505	1.4084
4	21	7.45E-03	152.47	18.2435	1.4000
5	26	8.10E-03	143.29	21.1343	1.4166
$u_{0}^{c}$ = [ 3E-03 L/min; 180 min ]
1	14	8.00E-03	141.40	1.6593	1.4157
2	18	8.03E-03	149.61	4.7823	1.4109
3	11	8.06E-03	146.66	1.2879	1.4150
4	16	7.50E-03	158.91	1.6744	1.3989
5	15	7.09E-03	165.13	2.5106	1.3832
$u_{0}^{d}$ = [ 7.5E-03 L/min; 70 min ]
1	24	8.21E-03	139.96	4.2448	1.4188
2	10	8.01E-03	147.11	1.1983	1.4107
3	7	8.32E-03	137.67	0.9730	1.4197
4	9	8.23E-03	145.83	3.9470	1.4140
5	10	7.92E-03	144.53	4.3775	1.4136

#	Number of Runs	$u_{1}^{*}$ [L/min]	$u_{2}^{*}$ [min]	φ	J_R(u)*
α1=90%
1	5	9.48E-03	104.16	1.3542	1.4011
2	5	8.29E-03	135.84	0.9026	1.4208
3	5	7.33E-03	142.42	1.6989	1.3873
4	5	8.30E-03	137.74	1.2697	1.4187
5	7	8.41E-03	136.64	6.4228	1.4217
With a non-probabilistic tendency model
1	17	8.11E-03	150.04	6.8170	1.4107
2	13	8.10E-03	134.37	6.6267	1.4157
3	15	7.89E-03	148.85	6.9433	1.4118
4	12	8.44E-03	138.43	6.6060	1.4207
5	14	1.36E-02	63.12	32.6206	1.3006

Case	Number of Failures	Mean φ	Mean J_R(u)*
α1 = 50%	12	8.1035	1.4153
α1 = 90%	2	2.3296	1.4099
No PTMs	19	11.9227	1.3919

\begin{matrix} {\bar{J}}_{(\bar{θ}, u)} \\ {pc}_{{\bar{J}}_{(\bar{θ}, u)}} = \Pr (C_{{\bar{J}}_{(\bar{θ}, u)}} \leq 0) \\ {pd}_{{\bar{J}}_{(\bar{θ}, u)}} = \Pr (D_{{\bar{J}}_{(\bar{θ}, u)}} = 0) \end{matrix}

\begin{matrix} \min_{U} φ_{(U)} \\ s . t . \forall i, i = 1,... n : \\ U_{i} = argmax J_{R (U_{i})} \\ C_{R}^{(Ui)} \leq 0 \\ D_{R}^{(U_{i})} = 0 \\ LB \leq U_{i} \leq UB \end{matrix}

\begin{matrix} \frac{dA}{dt} = - r_{1} - r_{3} - \frac{F}{V} A \\ \frac{dB}{dt} = - r_{1} - r_{2} - 2 r_{3} + \frac{F}{V} (B_{i} - B) \\ \frac{dP}{dt} = r_{1} - r_{2} - \frac{F}{V} P \\ \frac{dI}{dt} = {vr}_{3} - \frac{F}{V} I \\ \frac{dV}{dt} = F \end{matrix}

Parameter	Value
k ₁	1.225 E-01 [L.mol^-1.min^-1]
k ₂	1.870 E-02 [L.mol^-1.min^-1]
k ₃	7.0 E-03 [L.mol^-1.min^-1]
k ₄	3.0 E-02 [min^-1]
k ₅	3.0 E-01 [L.mol^-1.min^-1]

\begin{matrix} \frac{dA}{dt} = - r_{1} - r_{3} + r_{4} - \frac{F}{V} A \\ \frac{dB}{dt} = - r_{1} - r_{2} - r_{3} + r_{4} - r_{5} + \frac{F}{V} (B_{i} - B) \\ \frac{dP}{dt} = r_{1} - r_{2} - \frac{F}{V} P \\ \frac{dM}{dt} = r_{3} - r_{4} - r_{5} - \frac{F}{V} M \\ \frac{dI}{dt} = r_{5} - \frac{F}{V} S \\ \frac{dV}{dt} = F \end{matrix}

[1] *Corresponding author. E-mail address: ecmarti@santafe-conicet.gov.ar

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MODEL-BASED RUN-TO-RUN OPTIMIZATION FOR PROCESS DEVELOPMENT

Abstract