## THERMAL SCIENCE

International Scientific Journal

### MATHEMATICAL MODELING OF FISH BURGER BAKING USING FRACTIONAL CALCULUS

**ABSTRACT**

Tilapia (Oreochromis sp.) is the most important and abundant fish species in Brazil due to its adaptability to different environments. The development of tilapia-based products could be an alternative in order to aggregate value and increase fish meat consumption. However, there is little information available on fishburger freezing and cooking in the literature. In this work, the mathematical modeling of the fish burger baking was studied. Previously to the baking process, the fishburgers were assembled in cylindrical shape of height equal to 8mm and diameter 100mm and then baked in an electrical oven with forced heat convection at 150ºC. A T-type thermocouple was inserted in the burger to obtain its temperature profile at the central position. In order to describe the temperature of the burger during the baking process, lumped-parameter models of integer and fractional order and also a nonlinear model due to heat capacity temperature dependence were considered. The burger physical properties were obtained from the literature. After proper parameter estimation tasks and statistical validation, the fractional order model could better describe the experimental temperature behavior, a value of 0.91±0.02 was obtained for the fractional order of the system with correlation coefficient of 0.99. Therefore, with the better temperature prediction, process control and economic optimization studies of the baking process can be conducted.

**KEYWORDS**

PAPER SUBMITTED: 2016-04-22

PAPER REVISED: 2016-05-24

PAPER ACCEPTED: 2016-06-15

PUBLISHED ONLINE: 2016-10-01

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**Issue 1**, PAGES [41 - 50]

- Bainy, E.M., et al., Effect of Grilling and Baking on Physicochemical and Textural Properties of Tilapia (Oreochromis niloticus) Fish Burger, Journal of Food Science and Technology, 52 (2015), 8, pp. 5111-5119
- Bainy, E.M, et al., MK Measurement of Freezing Point of Tilapia Fish Burger Using Differential Scanning Calorimetry (DSC) and cooling curve method. Journal of Food Engineering, 161 (2015) pp. 82-86
- Mitterer-Daltoe, M.L., et al., Are Fish Products Healthy? Eye Tracking as a New Food Technology Tool for a Better Understanding of Consumer Perception, LWT-Food Science and Technology, 55 (2014), 2, 459-465
- Juarez, M., et al., O. Buffalo Meat Composition as Affected by Different Cooking Methods. Food and Bioproducts Processing, 88 (2010), C2-3, pp. 145-148
- Yang, X.-J., et al., Jordan Nonlinear dynamics for local fractional Burgers' equation arising in fractal flow, Nonlinear Dynamics, 84 (2016), 1(SI), pp. 3-7
- Datta, A.K., Toward Computer-Aided Food Engineering: Mechanistic Frameworks for Evolution of Product, Quality and Safety During Processing, Journal of Food Engineering, 176 (2016), pp. 9-27
- Feyissa, A.H., et al., 3D Modelling of Coupled Mass and Heat Transfer of a Convection-Oven Roasting Process, Meat Science, 93 (2013), 4, pp. 810-820
- van der Sman, R.G.M. Modeling Cooking of Chicken Meat in Industrial Tunnel Ovens with the Flory-Rehner Theory, Meat Science, 95 (2013), 4, 940-957
- Kondjoyan, A., et al., JB Combined Heat Transfer and Kinetic Models to Predict Cooking Loss During Heat Treatment of Beef Meat. Meat Science, 95 (2013), 2, pp. 336-344
- Liu, S.X., et al., Modeling of Fish Boiling Under Microwave Irradiation. Journal of Food Engineering, 140 (2014), pp. 9-18
- Isleroglu, H., Kayrnak-Ertekin, F., Modelling of Heat and Mass Transfer During Cooking in Steam-Assisted Hybrid Oven, Journal of Food Engineering, 181 (2016), pp. 50-58
- Ramallo-González, A.P., et al., Lumped Parameter Models for Building Thermal Modelling: An Analytic Approach to Simplifying Complex Multi-Layered Constructions, Energy and Buildings, 60 (2013), pp. 174-184.
- Hristov, J., Transient Heat Diffusion with a Non-Singular Fading Memory: From the Cattaneo Constitutive Equation with Jeffrey's Kernel to the Caputo-Fabrizio Time-Fractional Derivative. Thermal Science, 20 (2016), 2, pp. 757-762
- Isfer, L.A.D., et al., Generalization of Internal Model Control Loops Using Fractional Calculus, Latin American Applied Research, 42 (2012), 2, pp. 149-154.
- Lenzi, E.K., et al., Fractional Diffusion Equation and Green Function Approach: Exact Solutions, Physica A-Statistical Mechanics and its Applications, 360 (2006), 2, pp. 215-226
- Friesen, V.C., et al., Modeling Heavy Metal Sorption Kinetics Using Fractional Calculus Mathematical Problems in Engineering, 2015, Manuscript Number: 549562
- Hristov, J., A Unified Nonlinear Fractional Equation of the Diffusion-Controlled Surfactant Adsorption: Reappraisal and New Solution of the Ward-Tordai Problem, Journal of King Saud University-Science, 28 (2016), 1, 7-13.
- Simpson, R., et al., Fractional Calculus as a Mathematical Tool to Improve the Modeling of Mass Transfer Phenomena in Food Processing, Food Engineering Reviews, 5 (2013), 1, pp. 45-55.
- Ma, L., BarbosaCanovas, G.V., Simulating Viscoelastic Properties of Selected Food Gums and Gum Mixtures Using a Fractional Derivative Model, Journal Of Texture Studies, 27 (1996), 3, pp. 307-325
- Caputo, M., Linear Models Of Dissipation Whose Q Is Almost Frequency Independent-2, Geophysical Journal of The Royal Astronomical Society 13 (1967), 5, pp. 529-&
- Isfer, L.A.D., et al., Fractional Control of an Industrial Furnace, Acta Scientiarum-Technology, 32 (2010), 3, pp. 279-285
- Lenzi, E.K., et al., Specific Heat in the Nonextensive Statistics: Effective Temperature and Lagrange Parameter Beta, Physics Letters A, 292 (2002), 6, pp. 315-319
- Gomes, E.M., et al., Parametric Analysis of a Heavy Metal Sorption Isotherm Based on Fractional Calculus, Mathematical Problems In Engineering, 2013, Manuscript number: 642101
- Isfer, L.A.D., et al., Identification of Biochemical Reactors Using Fractional Differential Equations, Latin American Applied Research, 40 (2010), 2, pp. 193-198
- Lenzi, E.K., et al., Non-Markovian Diffusion Equation and Diffusion in a Porous Catalyst, Chemical Engineering Journal, 172 (2011), 2-3, pp. 1083-1087
- Corless, R.M., et al., On the Lambert W Function, Advances in Computational Mathematics, 5 (1996), pp. 329-359.
- Giona, M.; Roman, H.E., A Theory of Transport Phenomena in Disordered-Systems. Chemical Engineering Journal and The Biochemical Engineering Journal, 49 (1992), 1, pp. 1-10