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Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Determination of the kinetic constants of a chemical reaction in heterogeneous phase using parameterized metaheuristics

Jos´ e Mat´ ıas Cutillas Lozano and Domingo Gim´ enez

Departamento de Inform´ atica y Sistemas, University of Murcia

International Conference on Computational Science - Barcelona, June 2013

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Contents

1

Motivation

2

Determination of kinetic constants

3

Unified shared-memory scheme

4

Results

5

Conclusions

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Kinetic constants of a chemical reaction

Kinetic parameters of a chemical reaction are determined with metaheuristic methods. The processes occurring in the human stomach when neutralizing the acid with an antacid tablet are simulated. It is a reaction combined with mass transfer of carbonate ions present in the solid phase upon contact with an acid solution. Solving the problem requires the calculation of the whole chemical system using the Euler numerical method.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Kinetic constants of a chemical reaction

Kinetic parameters of a chemical reaction are determined with metaheuristic methods. The processes occurring in the human stomach when neutralizing the acid with an antacid tablet are simulated. It is a reaction combined with mass transfer of carbonate ions present in the solid phase upon contact with an acid solution. Solving the problem requires the calculation of the whole chemical system using the Euler numerical method.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Kinetic constants of a chemical reaction

Kinetic parameters of a chemical reaction are determined with metaheuristic methods. The processes occurring in the human stomach when neutralizing the acid with an antacid tablet are simulated. It is a reaction combined with mass transfer of carbonate ions present in the solid phase upon contact with an acid solution. Solving the problem requires the calculation of the whole chemical system using the Euler numerical method.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Kinetic constants of a chemical reaction

Kinetic parameters of a chemical reaction are determined with metaheuristic methods. The processes occurring in the human stomach when neutralizing the acid with an antacid tablet are simulated. It is a reaction combined with mass transfer of carbonate ions present in the solid phase upon contact with an acid solution. Solving the problem requires the calculation of the whole chemical system using the Euler numerical method.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Parallel-parameterized metaheuristics

Metaheuristic calculations are carried out with various parameters and functions. Many experiments are required to select a good metaheuristic and to tune it to the problem. A large number of optimization problems will be solved. We use a unified parallel-parameterized scheme of metaheuristics: Different metaheuristics obtained from the parameterized scheme are parallelized with the parallel parameters for

• ptimizing time execution.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Parallel-parameterized metaheuristics

Metaheuristic calculations are carried out with various parameters and functions. Many experiments are required to select a good metaheuristic and to tune it to the problem. A large number of optimization problems will be solved. We use a unified parallel-parameterized scheme of metaheuristics: Different metaheuristics obtained from the parameterized scheme are parallelized with the parallel parameters for

• ptimizing time execution.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Parallel-parameterized metaheuristics

Metaheuristic calculations are carried out with various parameters and functions. Many experiments are required to select a good metaheuristic and to tune it to the problem. A large number of optimization problems will be solved. We use a unified parallel-parameterized scheme of metaheuristics: Different metaheuristics obtained from the parameterized scheme are parallelized with the parallel parameters for

• ptimizing time execution.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Parallel-parameterized metaheuristics

Metaheuristic calculations are carried out with various parameters and functions. Many experiments are required to select a good metaheuristic and to tune it to the problem. A large number of optimization problems will be solved. We use a unified parallel-parameterized scheme of metaheuristics: Different metaheuristics obtained from the parameterized scheme are parallelized with the parallel parameters for

• ptimizing time execution.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Our search of the kinetic parameters of a chemical reaction that occurs in heterogeneous phase involves the simulation of the processes occurring on the human stomach. Depending on the value of the pH, there are three main ways in which the dissolution of calcium carbonate occurs:

By reaction with acetic acid. CaCO3 + H3O+ ↔ Ca2+ + HCO−

3 + H2O

(1) By reaction with carbonic acid. CaCO3 + H2CO3 ↔ Ca2+ + 2 · HCO−

3

(2) And by the hydrolysis reaction. CaCO3 + H2O ↔ Ca2+ + HCO−

3 + OH−

(3)

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Our search of the kinetic parameters of a chemical reaction that occurs in heterogeneous phase involves the simulation of the processes occurring on the human stomach. Depending on the value of the pH, there are three main ways in which the dissolution of calcium carbonate occurs:

By reaction with acetic acid. CaCO3 + H3O+ ↔ Ca2+ + HCO−

3 + H2O

(1) By reaction with carbonic acid. CaCO3 + H2CO3 ↔ Ca2+ + 2 · HCO−

3

(2) And by the hydrolysis reaction. CaCO3 + H2O ↔ Ca2+ + HCO−

3 + OH−

(3)

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

When reaction occurs in several parallel paths independent of each other, the overall rate is simply the sum of all individual rates. So, the kinetic of dissolution of calcium carbonate is a function of the concentration of carbonic acid in the solution, the pH and the mass transfer area: 1 V dNCa2+ dt = −k1an1 H3O+n2 − k2an3 [H2CO3]n4 − k3 (4)

k1, k2 and k3 are the combined reaction rate constants. n1, n2, n3 and n4 are the reaction orders. a is the area of the tablet.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

When reaction occurs in several parallel paths independent of each other, the overall rate is simply the sum of all individual rates. So, the kinetic of dissolution of calcium carbonate is a function of the concentration of carbonic acid in the solution, the pH and the mass transfer area: 1 V dNCa2+ dt = −k1an1 H3O+n2 − k2an3 [H2CO3]n4 − k3 (4)

k1, k2 and k3 are the combined reaction rate constants. n1, n2, n3 and n4 are the reaction orders. a is the area of the tablet.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Using the notation for evolutionary algorithms, an individual is represented by a real vector of size seven that is the set of kinetic constants. The ranges of values for the constants are set following empirical criteria. Every time we have to evaluate the fitness of an individual, we must solve the whole chemical system: for i = 0 → N do Calculate at instant i:

• Ca2+

, a, [H3O+] , [HCO−] , [H2CO3] , pHcal, ∆

• Ca2+

, [CH3COOH] , [CH3COO−] Fitness = Fitness + (pHexp,i − pHcal,i)2 end for

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Using the notation for evolutionary algorithms, an individual is represented by a real vector of size seven that is the set of kinetic constants. The ranges of values for the constants are set following empirical criteria. Every time we have to evaluate the fitness of an individual, we must solve the whole chemical system: for i = 0 → N do Calculate at instant i:

• Ca2+

, a, [H3O+] , [HCO−] , [H2CO3] , pHcal, ∆

• Ca2+

, [CH3COOH] , [CH3COO−] Fitness = Fitness + (pHexp,i − pHcal,i)2 end for

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Using the notation for evolutionary algorithms, an individual is represented by a real vector of size seven that is the set of kinetic constants. The ranges of values for the constants are set following empirical criteria. Every time we have to evaluate the fitness of an individual, we must solve the whole chemical system: for i = 0 → N do Calculate at instant i:

• Ca2+

, a, [H3O+] , [HCO−] , [H2CO3] , pHcal, ∆

• Ca2+

, [CH3COOH] , [CH3COO−] Fitness = Fitness + (pHexp,i − pHcal,i)2 end for

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Parallel-parametrized scheme

Initialize(S,ParamIni,ThreadsIni) while (not EndCondition(S,ParamEnd,ThreadsEnd)) SS = Select(S,ParamSel,ThreadsSel) SS1 = Combine(SS,ParamCom,ThreadsCom) SS2 = Improve(SS1,ParamImp,ThreadsImp) S = Include(SS2,ParamInc,ThreadsInc) Independent parallelization of the functions, with parallelism parameters (number of threads) for each function. The optimum value of the parallelism parameters depends on the values of the metaheuristic parameters (the metaheuristic or combination of metaheuristics).

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Metaheuristics

Four pure metaheuristics: GRASP, Genetic algorithm (GA), Scatter search (SS), Tabu Search (TS) Eleven combinations: GRASP+GA, GRASP+SS, GRASP+TS, GA+SS, GA+TS, SS+TS, GRASP+GA+SS, GRASP+GA+TS, GRASP+SS+TS, GA+SS+TS, GRASP+GA+SS+TS Total: 15 metaheuristics or combinations of metaheuristics

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Metaheuristic Parameters

Metaheuristic and metaheuristic combination parameters used in the experiments

GR TS SS GA GR+TS GR+SS GR+GA TS+SS TS+GA Ini INEIni 50 50 20 20 50 50 50 20 20 FNEIni 1 1 10 20 1 10 20 10 20 PEIIni 100 100 100 100 100 100 100 IIEIni 15 15 15 15 15 15 15 STMIni 2 2 2 Sel NBESel 1 5 20 5 20 5 20 NWESel 5 5 5 Com NBBCom 15 10 15 10 15 10 NBWCom 20 20 20 NWWCom 15 15 15 Imp PEIImp 100 100 100 100 IIEImp 5 5 5 5 SMIImp 2 2 PEMImp 20 20 20 IMEImp 5 5 2 SMMImp 10 Inc NBEInc 1 5 20 5 20 5 20 LTMInc 3 3 3

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Metaheuristic Parameters

SS+GA GR+TS+SS GR+TS+GA GR+SS+GA TS+SS+GA GR+TS+SS+GA INEIni 20 50 50 50 20 50 FNEIni 15 10 20 15 15 15 PEIIni 100 100 100 100 100 100 IIEIni 15 15 15 15 15 15 STMIni 2 2 2 2 NBESel 8 5 20 8 8 8 NWESel 7 5 7 7 7 NBBCom 15 15 10 15 15 15 NBWCom 20 20 20 20 20 NWWCom 15 15 15 15 15 PEIImp 100 100 100 100 100 IIEImp 5 5 5 5 5 SMIImp 2 2 2 PEMImp 20 20 20 20 20 IMEImp 5 5 5 5 5 SMMImp 2 2 2 NBEInc 8 5 20 8 8 8 LTMInc 3 3 3 3

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Problem variables

Values of the problem variables considered in the experiments:

Series S1 S2 S3 S4 S5 S6 N 70 50 34 77 81 69 MT 1.33 1.33 1.31 1.32 1.32 1.30 [HAc]0 · 102 3.98 3.31 2.39 2.18 3.01 1.99 V 250 250 250 100 150 200 N: number of experimental points of time and pH to be integrated. MT (g): total mass of the tablet that is dissolving in acid. [HAc]0 (mol/L): initial concentration of acetic acid in each series. V (mL): volume of the solution. Other variables common to all series of experiments: MA (g) = 0.68 is the active mass of calcium carbonate in the tablet. a0 (cm2) = 7.11 is the initial area of the tablet.

• H2CO∗

3

• (mol/L) = 10−5 is the total concentration of carbonate species in the

solution. Equilibria constants involved have values: Ka = 10−4.76, K

′

a = 10−6.35, KH = 10−1.5.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Fitness for different metaheuristics and problem sizes

In most cases the final value of the objective function is very similar. The lower this value, the better the result. Combinations of metaheuristics produce good fitness values, highlighting GA metaheuristic and its combinations.

1 2 3 4 5 6 7 8 S1 S2 S3 S4 S5 S6 average

Fitness

GR TS SS GA GR+TS GR+SS GR+GA TS+SS TS+GA SS+GA GR+TS+SS GR+TS+GA GR+SS+GA TS+SS+GA GR+TS+SS+GA

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Iterations to optimum fitness for S5 series

Metaheuristics including GA have lower fitness values and generally more iterations to the optimum. GRASP, TS, SS and their combinations, have higher fitness values.

3.8 4 4.2 4.4 4.6 4.8 5 5.2 10 20 30 40 50 60

Fitness Iterations

GR TS SS GA GR+TS GR+SS GR+GA TS+SS TS+GA SS+GA GR+TS+SS GR+TS+GA GR+SS+GA TS+SS+GA GR+TS+SS+GA

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

pH and hydronium concentration over time

The pH calculated in the simulation is not far from the experimental, and the prediction made by the kinetic model is acceptable. Stomach acid simulated by acetic acid is gradually reduced.

3 3.5 4 4.5 5 5.5 6 5 10 15 20 25 30 35 40 pH Time (min) pHexp pHcal 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 5 10 15 20 25 30 35 40 Concentration (mol/L) Time (min) [H3O+]exp [H3O+]cal

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Concentration of carbonate and acetic species and calcium over time

The concentration of HCO−

3

remains at all times below H2CO3, which was expected in the pH range considered. Calcium concentration increases as the tablet is dissolving. [CH3COOH] decreases at the same rate as

• CH3COO−

increases, which is expected if the model is correct.

2e-06 4e-06 6e-06 8e-06 1e-05 1.2e-05 1.4e-05 5 10 15 20 25 30 35 40 Concentration (mol/L) Time (min) [H2CO3] [HCO3

• ]

0.005 0.01 0.015 0.02 0.025 0.03 5 10 15 20 25 30 35 40 Concentration (mol/L) Time (min) [Ca2+] [CH3COOH] [CH3COO-]

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Optimum values of the kinetic constants

The final values are: k1 = 6.41 · 10−5 n1 = 0.749 k2 = 9.68 · 10−2 n2 = 1.59 · 10−2 k3 = 8.87 · 10−7 n3 = 3.25 n4 = 1.10 We can see the effect that an antacid tablet can have on the human stomach. The simulation times give us a rough idea of the speed with which carbonate acts in the reaction medium, helping to predict the effects against heartburn. This can be used to design more efficient pills and with relatively low experimental costs.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Conclusions

The kinetic constants of a chemical reaction have been determined satisfactorily using a unified parameterized scheme for metaheuristics. The dissolution process of an antacid tablet has been satisfactorily simulated. The best results in terms of fitness function are obtained by combining metaheuristics: specifically the quaternary combination GR+GA+SS+TS and, in general, metaheuristics including GA.

Motivation Determination of kinetic constants Unified shared-memory scheme Results Conclusions

Future research

Application of hyperheuristics to obtain the best combination

• f parameters for the kinetic problem so as to facilitate the

collection of the best metaheuristic or combination. Application of the same methodology to other optimization problems of chemical parameters and processes. For problems with a high computational cost it is convenient to develop unified parameterized schemes in parallel systems: shared memory, message passing and GPUs.