American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems
  • NHM Home
  • This Issue
  • Next Article
    Spectral stiff problems in domains surrounded by thin stiff and heavy bands: Local effects for eigenfunctions
March  2011, 6(1): 37-60. doi: 10.3934/nhm.2011.6.37

A mathematical model for spaghetti cooking with free boundaries

Antonio Fasano 1, , Mario Primicerio 1, and  Andrea Tesi 2,

1. 

Università degli Studi di Firenze, Dipartimento di Matematica, "Ulisse Dini", Viale Morgagni 67/A, I-50134, Firenze, Italy, Italy
2. 

Università degli Studi di Firenze, Dipartimento di Fisica, Via Sansone 1, I-50019, Sesto Fiorentino (FI), Italy

Received  January 2010 Revised  November 2010 Published  March 2011

We propose a mathematical model for the process of dry pasta cooking with specific reference to spaghetti. Pasta cooking is a two-stage process: water penetration followed by starch gelatinization. Differently from the approach adopted so far in the technical literature, our model includes free boundaries: the water penetration front and the gelatinization onset front representing a fast stage of the corresponding process. Behind the respective fronts water sorption and gelatinization proceed according to some kinetics. The outer boundary is also moving and unknown as a consequence of swelling. Existence and uniqueness are proved and numerical simulations are presented.
Keywords: Pasta cooking, nonlinear integral equations., free boundaries.
Mathematics Subject Classification: 76S05, 35R35, 45G1.
Citation: Antonio Fasano, Mario Primicerio, Andrea Tesi. A mathematical model for spaghetti cooking with free boundaries. Networks and Heterogeneous Media, 2011, 6 (1) : 37-60. doi: 10.3934/nhm.2011.6.37
References:
[1]

S. Cafieri, S. Chillo, M. Mastromatteo, N. Suriano and M. A. Del Nobile, A mathematical model to predict the effect of shape on pasta hydration kinetic during cooking and overcooking, J. Cereal Science, (2008).

[2]

E. Cocci, G. Sacchetti, M. Vallicelli and M. Dalla Rosa, Spaghetti cooking b microwave oven: Cooking kinetics and product quality, J. Food Eng., 85 (2008), 537-546. doi: oi:10.1016/j.jfoodeng.2007.08.013.

[3]

S. E. Cunningham, W. A. M. Mcminn, T. R. A. Magee and P. S. Richardson, Modelling water absorption of pasta during soaking, J. Food. Eng., 82 (2007), 600-607. doi: 10.1016/j.jfoodeng.2007.03.018.

[4]

M. J. Davey, K. A. Landman, M. J. McGuinness and H. N. Jin, Mathematical modelling of rice cooking and dissolution in beer production, AIChE Journal, 48 (2002), 1811-1826. doi: 10.1002/aic.690480821.

[5]

R. A. Grzybowski and B. J. Donnelly, Starch gelatinization in cooked spaghetti, J. Food Science, 42 (1977), 1304-1315. doi: 10.1111/j.1365-2621.1977.tb14483.x.

[6]

M. J. McGuinness, C. P. Please, N. Fowkes, P. McGowan, L. Ryder and D. Forte, Modelling the wetting and cooking of a single cereal grain, IMA J. Math. Appl. Business and Industry, 11 (2000), 49-70.

[7]

A. G. F. Stapley, P. J. Fryer and L. F. Gladden, Diffusion and reaction in whole wheat grains during boiling, AIChE Journal, 44 (1998), 1777-1789. doi: 10.1002/aic.690440809.

[8]

A. K. Syarief, R. J. Gustafson and R. V. Morey, Moisture diffusion coefficients for yellow-dent corn components, Trans. ASAE, 30 (1987), 522-528.

[9]

Ch. Xue, N. Sakai and M. Fukuoka, Use of microwave heating to control the degree of starch geletinization in noodles, J. Food. Eng., 87 (2007), 357-362. doi: 10.1016/j.jfoodeng.2007.12.017.

[10]

Tain-Yi Zhang, A. S. Bakshi, R. J. Gustafson and D. B. Lund, Finite element analysis of nonlinear water diffusion during rice soaking, J. Food Science, 49 (1984), 246-277. doi: 10.1111/j.1365-2621.1984.tb13719.x.

show all references

References:
[1]

S. Cafieri, S. Chillo, M. Mastromatteo, N. Suriano and M. A. Del Nobile, A mathematical model to predict the effect of shape on pasta hydration kinetic during cooking and overcooking, J. Cereal Science, (2008).

[2]

E. Cocci, G. Sacchetti, M. Vallicelli and M. Dalla Rosa, Spaghetti cooking b microwave oven: Cooking kinetics and product quality, J. Food Eng., 85 (2008), 537-546. doi: oi:10.1016/j.jfoodeng.2007.08.013.

[3]

S. E. Cunningham, W. A. M. Mcminn, T. R. A. Magee and P. S. Richardson, Modelling water absorption of pasta during soaking, J. Food. Eng., 82 (2007), 600-607. doi: 10.1016/j.jfoodeng.2007.03.018.

[4]

M. J. Davey, K. A. Landman, M. J. McGuinness and H. N. Jin, Mathematical modelling of rice cooking and dissolution in beer production, AIChE Journal, 48 (2002), 1811-1826. doi: 10.1002/aic.690480821.

[5]

R. A. Grzybowski and B. J. Donnelly, Starch gelatinization in cooked spaghetti, J. Food Science, 42 (1977), 1304-1315. doi: 10.1111/j.1365-2621.1977.tb14483.x.

[6]

M. J. McGuinness, C. P. Please, N. Fowkes, P. McGowan, L. Ryder and D. Forte, Modelling the wetting and cooking of a single cereal grain, IMA J. Math. Appl. Business and Industry, 11 (2000), 49-70.

[7]

A. G. F. Stapley, P. J. Fryer and L. F. Gladden, Diffusion and reaction in whole wheat grains during boiling, AIChE Journal, 44 (1998), 1777-1789. doi: 10.1002/aic.690440809.

[8]

A. K. Syarief, R. J. Gustafson and R. V. Morey, Moisture diffusion coefficients for yellow-dent corn components, Trans. ASAE, 30 (1987), 522-528.

[9]

Ch. Xue, N. Sakai and M. Fukuoka, Use of microwave heating to control the degree of starch geletinization in noodles, J. Food. Eng., 87 (2007), 357-362. doi: 10.1016/j.jfoodeng.2007.12.017.

[10]

Tain-Yi Zhang, A. S. Bakshi, R. J. Gustafson and D. B. Lund, Finite element analysis of nonlinear water diffusion during rice soaking, J. Food Science, 49 (1984), 246-277. doi: 10.1111/j.1365-2621.1984.tb13719.x.

[1]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
[2]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[3]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[4]

David Jerison, Nikola Kamburov. Free boundaries subject to topological constraints. Discrete and Continuous Dynamical Systems, 2019, 39 (12) : 7213-7248. doi: 10.3934/dcds.2019301
[5]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure and Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
[6]

Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317
[7]

M. R. Arias, R. Benítez. Properties of solutions for nonlinear Volterra integral equations. Conference Publications, 2003, 2003 (Special) : 42-47. doi: 10.3934/proc.2003.2003.42
[8]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[9]

Chonghu Guan, Xun Li, Zuo Quan Xu, Fahuai Yi. A stochastic control problem and related free boundaries in finance. Mathematical Control and Related Fields, 2017, 7 (4) : 563-584. doi: 10.3934/mcrf.2017021
[10]

Lei Li, Jianping Wang, Mingxin Wang. The dynamics of nonlocal diffusion systems with different free boundaries. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3651-3672. doi: 10.3934/cpaa.2020161
[11]

Yihong Du, Mingxin Wang, Meng Zhao. Two species nonlocal diffusion systems with free boundaries. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1127-1162. doi: 10.3934/dcds.2021149
[12]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[13]

Hongxia Zhang, Ying Wang. Liouville results for fully nonlinear integral elliptic equations in exterior domains. Communications on Pure and Applied Analysis, 2018, 17 (1) : 85-112. doi: 10.3934/cpaa.2018006
[14]

Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012
[15]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[16]

Meng Zhao, Wan-Tong Li, Wenjie Ni. Spreading speed of a degenerate and cooperative epidemic model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 981-999. doi: 10.3934/dcdsb.2019199
[17]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256
[18]

Meng Zhao. The longtime behavior of the model with nonlocal diffusion and free boundaries in online social networks. Electronic Research Archive, 2020, 28 (3) : 1143-1160. doi: 10.3934/era.2020063
[19]

Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2201-2238. doi: 10.3934/dcdsb.2020360
[20]

Yuan Wu, Jin Liang. Free boundaries of credit rating migration in switching macro regions. Mathematical Control and Related Fields, 2020, 10 (2) : 257-274. doi: 10.3934/mcrf.2019038

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (195)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy