# Evaluation of the Mass Diffusion Coefficient and Mass Biot Number Using a Nondominated Sorting Genetic Algorithm

^{*}

## Abstract

**:**

^{2}+ 0.000024316T

^{2}− 0.12478256sT, D = 1.27547936∙10

^{−7}− 2.3808∙10

^{−5}s − 5.08365633∙10

^{−9}T + 0.0030005179s

^{2}+ 4.266495∙10

^{−11}T

^{2}+ 8.33633∙10

^{−7}sT or Bi = 0.764714 + 10.1689091s − 0.003400089T + 948.715738s

^{2}+ 0.000024316T

^{2}− 0.12478252sT, D = 1.27547948∙10

^{−7}− 2.3806∙10

^{−5}s − 5.08365753∙10

^{−9}T + 0.0030005175s

^{2}+ 4.266493∙10

^{−11}T

^{2}+ 8.336334∙10

^{−7}sT. The results of statistical analysis for the Biot number and moisture diffusion coefficient equations were as follows: R = 0.9905672, MAE = 0.0406375, RMSE = 0.050252 and R = 0.9905611, MAE = 0.0406403 and RMSE = 0.050273, respectively.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Moisture Transfer Analysis

- Initial condition: M at any point of the sample is the same at the beginning of drying:

- Boundary conditions of the third kind: The moisture flux from the surface of the sample is described in terms of moisture content difference between the surface and the equilibrium moisture content:

#### 2.3. Multi-Objective Optimization

_{1}dominates solution S

_{2}if S

_{1}has a lower cost than S

_{2}for at least one of the objective functions and is not worse with respect to the remaining objective functions.

#### 2.4. The Optimization Tasks

## 3. Results and Discussion

#### 3.1. Optimization of Bi and D

#### 3.2. Optimization of Parameters of the Functions for Determining Bi and D.

_{Bi}, b

_{Bi}, c

_{Bi}, d

_{Bi}, e

_{Bi}, f

_{Bi}) and water diffusion coefficient D (a

_{D}, b

_{D}, c

_{D}, d

_{D}, e

_{D}, f

_{D}). The results of the tests are shown in Table 1 and Table 2 and Figure 5.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a_{Bi}, b_{Bi}, c_{Bi}, d_{Bi}, e_{Bi}, f_{Bi} | constants in Equation (7) (-) |

a_{D}, b_{D}, c_{D}, d_{D}, e_{D}, f_{D} | constants in Equation (8) (-) |

Bi | mass Biot number |

D | mass diffusion coefficient (m^{2} s^{−1}) |

k | mass transfer coefficient (m s^{−1}) |

L | characteristic dimension (m) |

M, M_{0}, M_{e} | moisture content, initial moisture content, equilibrium moisture content (kg H_{2}O kg^{−1} d.m.) |

MR, MR_{exp}, MR_{pred} | moisture ratio, moisture ratio from experiment, predicted moisture ratio (-) |

${\overline{MR}}_{exp}$ | average moisture ratio from experiment |

N | number of data (-) |

s | half of plane (slice) thickness (m) |

t | time (s) |

T | temperature (°C) |

## References

- Lewicki, P.P. Water as the determinant of food engineering properties. A review. J. Food Eng.
**2004**, 61, 483–495. [Google Scholar] [CrossRef] - Doulia, D.; Tzia, K.; Gekas, V. A knowledge base for the apparent mass diffusion coefficient of foods. Int. J. Food Prop.
**2000**, 3, 1–14. [Google Scholar] [CrossRef] - Gros, J.B.; Ruegg, M. Physical Properties of Foods-2. In COST 90bis Final Seminar Proceedings; Jowitt, R., Escher, F., Kent, M., McKenna, B., Roques, M., Eds.; Elsevier Applied Science: London, UK, 1987; p. 71. [Google Scholar]
- Warin, F.; Gekas, V.; Voirin, A.; Dejmek, P. Sugar Diffusivity in Agar Gel/Milk Bilayer Systems. J. Food Sci.
**1997**, 62, 454–456. [Google Scholar] [CrossRef] - Rattanakijsuntorn, K.; Penkova, A.; Sadha, S.S. Mass diffusion coefficient measurement for vitreous humor using FEM and MRI. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 297, 012024. [Google Scholar] - Efremov, G.; Markowski, M.; Białobrzewski, I.; Zielinska, M. Approach to calculation time-dependent moisture diffusivity for thin layered biological materials. Int. Commun. Heat Mass
**2008**, 35, 1069–1072. [Google Scholar] [CrossRef] - Zamel, N.; Astrath, N.G.C.; Li, X.; Shen, J.; Zhou, J.; Astrath, F.B.G.; Wang, H.; Liu, Z.-S. Experimental measurements of effective diffusion coefficient of oxygen–nitrogen mixture in PEM fuel cell diffusion media. Chem. Eng. Sci.
**2010**, 65, 931–937. [Google Scholar] [CrossRef] - Chan, C.; Zamel, N.; Li, X.; Shen, J. Experimental measurement of effective diffusion coefficient of gas diffusion layer/microporous layer in PEM fuel cells. Electrochim. Acta
**2012**, 65, 13–21. [Google Scholar] [CrossRef] [Green Version] - García-Salaberri, P.A.; Hwang, G.; Vera, M.; Weber, A.Z.; Gostick, J.T. Effective diffusivity in partially-saturated carbon-fiber gas diffusion layers: Effect of through-plane saturation distribution. Int. J. Heat Mass Transf.
**2015**, 86, 319–333. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Combe, C.; Clark, M.C. The effects of pH and calcium on the diffusion coefficient of humic acid. J. Membr. Sci.
**2001**, 183, 49–60. [Google Scholar] [CrossRef] - Stewart, P.S. A Review of Experimental Measurements of Effective Diffusive Permeabilities and Effective Diffusion Coefficients in Biofilms. Biotechnol. Bioeng.
**1998**, 59, 261–272. [Google Scholar] [CrossRef] - Perré, P.; Pierre, F.; Casalinho, J.; Ayouz, M. Determination of the Mass Diffusion Coefficient Based on the Relative Humidity Measured at the Back Face of the Sample During Unsteady Regimes. Dry Technol.
**2015**, 33, 1068–1075. [Google Scholar] [CrossRef] - Piot, A.; Woloszyn, M.; Brau, J.; Abele, C. Experimental wooden frame house for the validation of whole building heat and moisture transfer numerical models. Energ. Build.
**2011**, 43, 1322–1328. [Google Scholar] [CrossRef] - Liu, J.Y.; Simpson, W.T.; Verrill, S.P. An inverse moisture diffusion algorithm for the determination of diffusion coefficient. Dry Technol.
**2001**, 19, 1555–1568. [Google Scholar] [CrossRef] [Green Version] - Bruin, S.; Luyben, K.C.A.M. Recent developments in dehydration of food materials. In Food Process Engineering; Food processing Systems; Applied Science Publishers: London, UK, 1980; Volume 1. [Google Scholar]
- Sitompul, J.P.; Istadi; Widiasa, I.N. Modeling and simulation of deep-bed grain dryers. Dry Technol.
**2001**, 19, 269–280. [Google Scholar] [CrossRef] [Green Version] - Rovedo, C.O.; Suarez, C.; Viollaz, P. Analysis of moisture profiles, mass Biot number and driving forces during drying of potato slabs. J. Food Eng.
**1998**, 36, 211–231. [Google Scholar] [CrossRef] - Shi, Q.; Zheng, Y.; Zhao, Y. Mathematical modeling on thin-layer heat pump drying of yacon (Smallanthus sonchifolius) slices. Energy Convers. Manag.
**2013**, 71, 208–216. [Google Scholar] [CrossRef] - Zielinska, M.; Markowski, M. Drying behavior of carrots dried in a spout–fluidized bed dryer. Dry Technol.
**2007**, 25, 261–270. [Google Scholar] [CrossRef] - Giner, S.A.; Irigoyen, R.M.T.; Cicuttín, S.; Fiorentini, C. The variable nature of Biot numbers in food drying. J. Food Eng.
**2010**, 101, 214–222. [Google Scholar] [CrossRef] - Ruiz-López, I.I.; Ruiz-Espinosa, H.; Arellanes-Lozada, P.; Bárcenas-Pozos, M.E.; García-Alvarado, M.A. Analytical model for variable moisture diffusivity estimation and drying simulation of shrinkable food products. J. Food Eng.
**2012**, 108, 427–435. [Google Scholar] [CrossRef] - Gigler, J.; van Loon, W.K.P.; van der Berg, J.V. Natural wind drying of willow stems. Biomass Bioenerg.
**2000**, 19, 153–163. [Google Scholar] [CrossRef] - Arévalo-Pinedo, A.; Murr, F.E.X. Kinetics of vacuum drying of pumpkin (Cucurbita maxima): Modeling with shrinkage. J. Food Eng.
**2006**, 76, 562–567. [Google Scholar] [CrossRef] - Falade, K.O.; Abbo, E.S. Air-drying and rehydration characteristics of date palm (Phoenix dactylifera L.) fruits. J. Food Eng.
**2007**, 79, 724–730. [Google Scholar] [CrossRef] - Reyes, A.; Mahn, A.; Vásquez, F. Mushrooms dehydration in a hybrid-solar dryer, using a phase change material. Energy Convers. Manag.
**2014**, 83, 241–248. [Google Scholar] [CrossRef] - Kaleta, A.; Górnicki, K.; Winiczenko, R.; Chojnacka, A. Evaluation of drying models of apple (var. Ligol) dried in a fluidized bed dryer. Energy Convers. Manag.
**2013**, 67, 179–185. [Google Scholar] [CrossRef] - Saguy, I.S.; Marabi, A.; Wallach, R. New approach to model rehydration of dry food particulates utilizing principles of liquid transport in porous media. Trends Food Sci. Tech.
**2005**, 16, 495–506. [Google Scholar] [CrossRef] - Sanjuán, N.; Simal, S.; Bon, J.; Mulet, A. Modelling of broccoli stems rehydration process. J. Food Eng.
**1999**, 42, 27–31. [Google Scholar] [CrossRef] - García-Pascual, P.; Sanjuán, N.; Bon, J.; Carreres, J.E.; Mulet, A. Rehydration process of Boletus edulis mushroom: Characteristics and modelling. J. Sci. Food Agric.
**2005**, 85, 1397–1404. [Google Scholar] [CrossRef] - García-Pascual, P.; Sanjuán, N.; Melis, R.; Mulet, A. Morchella esculenta (morel) rehydration process modelling. J. Food Eng.
**2006**, 72, 346–353. [Google Scholar] [CrossRef] - Resio, A.N.C.; Aguerre, R.J.; Suárez, C. Study of some factors affecting water absorption by amaranth grain during soaking. J. Food Eng.
**2003**, 60, 391–396. [Google Scholar] [CrossRef] - Cunningham, S.E.; Mcminn, W.A.M.; Magee, T.R.A.; Richardson, P.S. Effect of processing conditions on the water absorption and texture kinetics of potato. J. Food Eng.
**2008**, 84, 214–223. [Google Scholar] [CrossRef] - Maldonado, S.; Arnau, E.; Bertuzzi, M.A. Effect of temperature and pretreatment on water diffusion during rehydration of dehydrated mangoes. J. Food Eng.
**2010**, 96, 333–341. [Google Scholar] [CrossRef] - Ramallo, L.A.; Albani, O.A. Water diffusion coefficient and modeling of water uptake in packaged yerba mate. J. Food Process. Preserv.
**2007**, 31, 406–419. [Google Scholar] [CrossRef] - Markowski, M. Air drying of vegetables: Evaluation of mass transfer coefficient. J. Food Eng.
**1997**, 34, 55–62. [Google Scholar] [CrossRef] - Dincer, I. Moisture transfer analysis during drying of slab woods. Int. J. Heat Mass Transf.
**1998**, 34, 317–320. [Google Scholar] [CrossRef] - Miketinac, M.J.; Sokhansanj, S.; Tutek, Z. Determination of Heat and Mass Transfer Coefficients in Thin Layer Drying of Grain. Trans. ASAE
**1992**, 35, 1853–1858. [Google Scholar] [CrossRef] - Wang, N.; Brennan, J.G. A mathematical model of simultaneous heat and moisture transfer during drying of potato. J. Food Eng.
**1995**, 24, 47–60. [Google Scholar] [CrossRef] - Białobrzewski, I. Determination of the mass transfer coefficient during hot-air-drying of celery root. J. Food Eng.
**2007**, 78, 1388–1396. [Google Scholar] [CrossRef] - Górnicki, K.; Kaleta, A. Drying curve modelling of blanched carrot cubes under natural convection condition. J. Food Eng.
**2007**, 82, 160–170. [Google Scholar] [CrossRef] - Huang, C.-H.; Yeh, C.-Y. An inverse problem in simultaneous estimating the Biot numbers of heat and moisture transfer for a porous material. Int. J. Heat Mass Transf.
**2002**, 45, 4643–4653. [Google Scholar] [CrossRef] - Chen, X.D.; Peng, X. Modified Biot number in the context of air drying of small moist porous objects. Dry Technol.
**2005**, 23, 83–103. [Google Scholar] [CrossRef] - Li, Z.X.; Renault, F.L.; Gómez, A.O.C.; Sarafraz, M.M.; Khan, H.; Safaei, M.R.; Filho, E.P.B. Nanofluids as secondary fluid in the refrigeration system: Experimental data, regression, ANFIS, and NN modeling. Int. J. Heat Mass Transf.
**2019**, 144, 118635. [Google Scholar] [CrossRef] - Sarafraz, M.M.; Tlili, I.; Tian, Z.; Bakouri, M.; Safaei, M.R. Smart optimization of a thermosyphon heat pipe for an evacuated tube solar collector using response surface methodology (RSM). Phys. Stat. Mech. Its Appl.
**2019**, 534, 122146. [Google Scholar] [CrossRef] - Sarafraz, M.M.; Safaei, M.R.; Goodarzi, M.; Arjomandi, M. Experimental investigation and performance optimisation of a catalytic reforming micro-reactor using response surface methodology. Energy Convers. Manag.
**2019**, 199, 111983. [Google Scholar] [CrossRef] - Benyounis, K.Y.; Olabi, A.G. Optimization of different welding processes using statistical and numerical approaches—A reference guide. Adv. Eng. Softw.
**2008**, 39, 483–496. [Google Scholar] [CrossRef] [Green Version] - Kopsidas, G. Multiobjective optimization of table olive preparation systems. Eur. J. Oper. Res.
**1995**, 85, 383. [Google Scholar] [CrossRef] - Kiranoudis, C.; Markatos, N. Pareto design of conveyor-belt dryers. J. Food. Eng.
**2000**, 46, 145–155. [Google Scholar] [CrossRef] - Therdthai, N.; Zhou, W.; Adamczak, T. Optimization of the temperature profile in bread baking. J. Food. Eng.
**2002**, 55, 41–48. [Google Scholar] [CrossRef] - Erdogdu, F.; Balaban, M. Complex method for nonlinear constrained optimization of thermal processing multi-criteria. J. Food Process. Eng.
**2003**, 26, 357–375. [Google Scholar] [CrossRef] - Gergely, S.; Bekassy-Molnar, E.; Vatai, G. The use of multiobjective optimization to improve wine filtration. J. Food Eng.
**2003**, 58, 311–316. [Google Scholar] [CrossRef] - Hadiyanto, M.; Boom, R.; Straten, G.; Boxtel, A.; Esveld, D. Multi-objective optimization to improve the product range of baking systems. J. Food Process. Eng.
**2009**, 32, 709–729. [Google Scholar] [CrossRef] - Zhang, Y.; Hidajat, K.; Ray, A.K. Optimal design and operation of SMB bioreactor: Production of high fructose syrup by isomerization of glucose. Biochem. Eng. J.
**2004**, 21, 111. [Google Scholar] [CrossRef] - Kawajiri, Y.; Biegler, L.T. Nonlinear programming superstructure for optimal dynamic operations of simulated moving bed processes. Ind. Eng. Chem. Res.
**2006**, 45, 8503. [Google Scholar] [CrossRef] - Hakanen, J.; Kawajiri, Y.; Miettinen, K.; Biegler, L.T. Interactive multi-objective optimization for simulated moving bed processes. Control Cybern.
**2007**, 36, 50. [Google Scholar] - Abakarov, A.; Sushkov, Y.; Almonacid, S.; Simpson, R. Multiobjective Optimization Approach: Thermal Food Processing. J. Food Sci.
**2009**, 74, E477–E487. [Google Scholar] [CrossRef] [PubMed] - Winiczenko, R.; Górnicki, K.; Kaleta, A.; Martynenko, A.; Janaszek-Mańkowska, M.; Trajer, J. Multi-objective optimization of convective drying of apple cubes. Comput. Electron. Agric.
**2018**, 145, 341–348. [Google Scholar] [CrossRef] - Winiczenko, R.; Górnicki, K.; Kaleta, A.; Janaszek-Mankowska, M.; Trajer, J. Multi-objective optimization of the apple drying and rehydration processes parameters. EJFA
**2018**, 30, 1–9. [Google Scholar] - Seng, C.; Rangaiah, S. Multi-objective optimization in food engineering. In Optimization in Food Engineering; Erdogdu, F., Ed.; Taylor & Francis Book: Abingdon, UK, 2008; Chapter 4; p. 800. [Google Scholar]
- Górnicki, K.; Winiczenko, R.; Kaleta, A. Estimation of the Biot Number Using Genetic Algorithms: Application for the Drying Process. Energies
**2019**, 12, 2822. [Google Scholar] [CrossRef] [Green Version] - Górnicki, K.; Kaleta, A. Modelling convection drying of blanched parsley root slices. Biosyst. Eng.
**2007**, 97, 51–59. [Google Scholar] [CrossRef] - Luikov, A.V. Analytical Heat Diffusion Theory; Academic Press Inc.: New York, NY, USA, 1970. [Google Scholar]
- Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, UK, 1975; ISBN 978-0-19-853344-3. [Google Scholar]
- Pabis, S.; Jayas, D.S.; Cenkowski, S. Grain Drying: Theory and Practice; John Wiley: New York, NY, USA, 1998; ISBN 978-0-471-57387-6. [Google Scholar]
- Kaleta, A.; Górnicki, K. Some remarks on evaluation of drying models of red beet particles. Energy Convers. Manag.
**2010**, 51, 2967–2978. [Google Scholar] [CrossRef] - Kaleta, A.; Górnicki, K. Evaluation of drying models of apple (var. McIntosh) dried in a convective dryer. Int. J. Food Sci. Technol.
**2010**, 45, 891–898. [Google Scholar] [CrossRef] - Górnicki, K.; Kaleta, A.; Choińska, A. Suitable model for thin-layer drying of root vegetables and onion. Int. Agrophysics
**2020**, 1, 79–86. [Google Scholar] [CrossRef] - Foneseca, C.M.; Flemming, P. Genetic algorithms for multi-objective optimization: Formulation, discussion, and generalization. In Proceedings of the 5th International Conference on Genetic Algorithms, Urbana-Champaign, July 17–21; Morgan Kaufmann: San Francisco, CA, USA, 1993; pp. 416–423. [Google Scholar]
- Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - MATLAB 7.6 R2008a, Documentation R; MathWorks, Inc.: Natick, MA, USA, 2008.

**Figure 2.**Statistics for Pareto optimal sets for Bi, L = 6 mm; thin line—MAE = f (R), dashed line—RMSE = f (R), thick line—RMSE = f (MAE).

**Figure 3.**Statistics for Pareto optimal sets for Bi, T = 70 °C; thin line—MAE = f (R), dashed line—RMSE = f (R), thick line—RMSE = f (MAE).

**Figure 4.**Pareto optimal sets for (

**a**) mass Biot number (Bi) and (

**b**) mass diffusion coefficient (D/s

^{2}); L – characteristic dimension, s – half of plane (slice) thickness, T – temperature.

**Figure 5.**Pareto optimal sets for constants in Equations (6) and (7): (

**a**) MAE = f(R), (

**b**) RMSE = f(R), (

**c**) RMSE = f(MSE).

**Table 1.**Pareto optimal set for constants (a

_{Bi}, b

_{Bi}, c

_{Bi}, d

_{Bi}, e

_{Bi}, f

_{Bi}) in Equation (6) and considered statistics; R - coefficient of correlation, MAE - mean absolute error, RMSE - root mean square error.

ID | a_{Bi} | b_{Bi} | c_{Bi} | d_{Bi} | e_{Bi} | f_{Bi} | R | RMSE | MAE |
---|---|---|---|---|---|---|---|---|---|

S_1 | 0.7647141 | 10.1689977 | −0.003400086 | 948.715758 | 0.000024316 | −0.12478256 | 0.9905672 | 0.050289 | 0.0406375 |

S_2 | 0.7647140 | 10.1689891 | −0.003400089 | 948.715738 | 0.000024316 | −0.12478252 | 0.9905665 | 0.050287 | 0.0406376 |

S_3 | 0.7647142 | 10.1689057 | −0.003400089 | 948.715760 | 0.000024319 | −0.12478129 | 0.9905653 | 0.050284 | 0.0406381 |

S_4 | 0.7647142 | 10.1689752 | −0.003400082 | 948.715784 | 0.000024316 | −0.12478196 | 0.9905647 | 0.050282 | 0.0406383 |

S_5 | 0.7647143 | 10.1689954 | −0.003400083 | 948.715759 | 0.000024317 | −0.12478102 | 0.9905638 | 0.050280 | 0.0406389 |

S_6 | 0.7646614 | 10.1689890 | −0.003400072 | 948.715763 | 0.000024330 | −0.12478277 | 0.9905631 | 0.050278 | 0.0406391 |

S_7 | 0.7647142 | 10.1689346 | −0.003400066 | 948.715773 | 0.000024316 | −0.12477951 | 0.9905627 | 0.050277 | 0.0406395 |

S_8 | 0.7646615 | 10.1689980 | −0.003400071 | 948.715758 | 0.000024330 | −0.12478214 | 0.9905619 | 0.050275 | 0.0406397 |

S_9 | 0.7647164 | 10.1689977 | −0.003400082 | 948.715752 | 0.000024316 | −0.12478000 | 0.9905611 | 0.050273 | 0.0406403 |

S_10 | 0.7646749 | 10.1689859 | −0.003400075 | 948.715762 | 0.000024323 | −0.12478259 | 0.9905604 | 0.050271 | 0.0406404 |

S_11 | 0.7647171 | 10.1689683 | −0.003400077 | 948.715759 | 0.000024316 | −0.12478368 | 0.9905593 | 0.050268 | 0.0406411 |

S_12 | 0.7646708 | 10.1689920 | −0.003400071 | 948.715757 | 0.000024324 | −0.12478274 | 0.9905588 | 0.050267 | 0.0406412 |

S_13 | 0.7646981 | 10.1689694 | −0.003399883 | 948.715758 | 0.000024320 | −0.12478335 | 0.9905584 | 0.050267 | 0.0406418 |

S_14 | 0.7647143 | 10.1689797 | −0.003400077 | 948.715764 | 0.000024316 | −0.12478110 | 0.9905574 | 0.050264 | 0.0406421 |

S_15 | 0.7646723 | 10.1689885 | −0.003400092 | 948.715759 | 0.000024324 | −0.12478362 | 0.9905565 | 0.050261 | 0.0406425 |

S_16 | 0.7647146 | 10.1689610 | −0.003400070 | 948.715796 | 0.000024316 | −0.12478125 | 0.9905553 | 0.050259 | 0.0406432 |

S_17 | 0.7647112 | 10.1689946 | −0.003400011 | 948.715761 | 0.000024316 | −0.12478272 | 0.9905544 | 0.050257 | 0.0406438 |

S_18 | 0.7646966 | 10.1689920 | −0.003400075 | 948.715743 | 0.000024323 | −0.12478189 | 0.9905537 | 0.050256 | 0.0406442 |

S_19 | 0.7647146 | 10.1689886 | −0.003400073 | 948.715721 | 0.000024317 | −0.12478261 | 0.9905528 | 0.050254 | 0.0406447 |

S_20 | 0.7647145 | 10.1689228 | −0.003400067 | 948.715776 | 0.000024316 | −0.12478094 | 0.9905517 | 0.050252 | 0.0406452 |

**Table 2.**Pareto optimal set for constants (a

_{D}, b

_{D}, c

_{D}, d

_{D}, e

_{D}, f

_{D}) in Equation (7) and considered statistics.

ID | a_{D} | b_{D} | c_{D} | d_{D} | e_{D} | f_{D} | R | RMSE | MAE |
---|---|---|---|---|---|---|---|---|---|

S_1 | 0.000000127547936 | −0.000023808 | −0.00000000508365633 | 0.0030005179 | 0.00000000004266495 | 0.0000008336330 | 0.9905672 | 0.050289 | 0.0406375 |

S_2 | 0.000000127547948 | −0.000023806 | −0.00000000508365753 | 0.0030005175 | 0.00000000004266493 | 0.0000008336334 | 0.9905665 | 0.050287 | 0.0406376 |

S_3 | 0.000000127547939 | −0.000023803 | −0.00000000508365666 | 0.0030005216 | 0.00000000004266494 | 0.0000008336333 | 0.9905653 | 0.050284 | 0.0406381 |

S_4 | 0.000000127547938 | −0.000023801 | −0.00000000508365663 | 0.0030004841 | 0.00000000004266493 | 0.0000008336339 | 0.9905647 | 0.050282 | 0.0406383 |

S_5 | 0.000000127547934 | −0.000023799 | −0.00000000508365680 | 0.0030005241 | 0.00000000004266493 | 0.0000008336334 | 0.9905638 | 0.050280 | 0.0406389 |

S_6 | 0.000000127547927 | −0.000023797 | −0.00000000508365755 | 0.0030005275 | 0.00000000004266495 | 0.0000008336336 | 0.9905631 | 0.050278 | 0.0406391 |

S_7 | 0.000000127547932 | −0.000023796 | −0.00000000508365677 | 0.0030005201 | 0.00000000004266492 | 0.0000008336339 | 0.9905627 | 0.050277 | 0.0406395 |

S_8 | 0.000000127547934 | −0.000023794 | −0.00000000508365671 | 0.0030005301 | 0.00000000004266494 | 0.0000008336337 | 0.9905619 | 0.050275 | 0.0406397 |

S_9 | 0.000000127547936 | −0.000023792 | −0.00000000508365669 | 0.0030005201 | 0.00000000004266494 | 0.0000008336338 | 0.9905611 | 0.050273 | 0.0406403 |

S_10 | 0.000000127547930 | −0.000023790 | −0.00000000508365669 | 0.0030005199 | 0.00000000004266495 | 0.0000008336335 | 0.9905604 | 0.050271 | 0.0406404 |

S_11 | 0.000000127547935 | −0.000023787 | −0.00000000508365634 | 0.0030004820 | 0.00000000004266493 | 0.0000008336332 | 0.9905593 | 0.050268 | 0.0406411 |

S_12 | 0.000000127547929 | −0.000023786 | −0.00000000508365667 | 0.0030005267 | 0.00000000004266494 | 0.0000008336337 | 0.9905588 | 0.050267 | 0.0406412 |

S_13 | 0.000000127547934 | −0.000023785 | −0.00000000508365706 | 0.0030005213 | 0.00000000004266494 | 0.0000008336336 | 0.9905584 | 0.050267 | 0.0406418 |

S_14 | 0.000000127547935 | −0.000023782 | −0.00000000508365694 | 0.0030004793 | 0.00000000004266493 | 0.0000008336337 | 0.9905574 | 0.050264 | 0.0406421 |

S_15 | 0.000000127547933 | −0.000023780 | −0.00000000508365703 | 0.0030005265 | 0.00000000004266494 | 0.0000008336336 | 0.9905565 | 0.050261 | 0.0406425 |

S_16 | 0.000000127547937 | −0.000023777 | −0.00000000508365643 | 0.0030004974 | 0.00000000004266494 | 0.0000008336336 | 0.9905553 | 0.050259 | 0.0406432 |

S_17 | 0.000000127547934 | −0.000023775 | −0.00000000508365667 | 0.0030005259 | 0.00000000004266494 | 0.0000008336337 | 0.9905544 | 0.050257 | 0.0406438 |

S_18 | 0.000000127547929 | −0.000023773 | −0.00000000508365675 | 0.0030005227 | 0.00000000004266494 | 0.0000008336335 | 0.9905537 | 0.050256 | 0.0406442 |

S_19 | 0.000000127547931 | −0.000023771 | −0.00000000508365740 | 0.0030005313 | 0.00000000004266494 | 0.0000008336340 | 0.9905528 | 0.050254 | 0.0406447 |

S_20 | 0.000000127547938 | −0.000023768 | −0.00000000508365654 | 0.0030005173 | 0.00000000004266493 | 0.0000008336333 | 0.9905517 | 0.050252 | 0.0406452 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Winiczenko, R.; Górnicki, K.; Kaleta, A.
Evaluation of the Mass Diffusion Coefficient and Mass Biot Number Using a Nondominated Sorting Genetic Algorithm. *Symmetry* **2020**, *12*, 260.
https://doi.org/10.3390/sym12020260

**AMA Style**

Winiczenko R, Górnicki K, Kaleta A.
Evaluation of the Mass Diffusion Coefficient and Mass Biot Number Using a Nondominated Sorting Genetic Algorithm. *Symmetry*. 2020; 12(2):260.
https://doi.org/10.3390/sym12020260

**Chicago/Turabian Style**

Winiczenko, Radosław, Krzysztof Górnicki, and Agnieszka Kaleta.
2020. "Evaluation of the Mass Diffusion Coefficient and Mass Biot Number Using a Nondominated Sorting Genetic Algorithm" *Symmetry* 12, no. 2: 260.
https://doi.org/10.3390/sym12020260