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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 & 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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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