Abstract
Many processes involve simultaneous transport phenomena in a variety of environments. The intertwined effects of gradients of temperature, velocity, and species concentration in the considered media are impossible to study by analytical methods and prohibitive to attack with experimental procedures. That is why numerical computations of such complex processes, or virtualization, come into play. Being the CFD our tool of choice to integrate the ensemble of governing PDEs, supplemented with proper PCB notations, one can enforce the virtualization of transfer phenomena to extract all of the information needed without recurring to systematic experimental assessment, provided that modeling has been properly validated with the corresponding data. In this last chapter, our attention is devoted to actual modeling: that is, the choice of proper governing Equations (with related initial and boundary conditions and ancillary statements, where applicable) that will be integrated and that will yield for all sort of graphical restitution of results.
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Change history
31 March 2019
In the original version of the book, the belated corrections from author.
Notes
- 1.
As performed by Dirita, C., De Bonis, M.V., Ruocco, G.: Analysis of food cooling by jet impingement, including inherent conduction (2007) https://doi.org/10.1016/j.jfoodeng.2006.10.002..
- 2.
As performed by De Bonis, M.V., Caccavale, P., Ruocco, G.: Convective control to microwave exposure of moist substrates. Part I: Model methodology. International Journal of Heat and Mass Transfer (2015) https://doi.org/10.1016/j.ijheatmasstransfer.2015.03.037.
- 3.
As attributed to the work by the German mathematician J.H. Lambert at the end of the eighteenth century.
- 4.
\(f(z')\) depends on media’s dielectric properties, that dictate the transmission and reflection coefficients, and the attenuation and phase factors. A complete account on \(f(z')\) for a variety of moist substrate can be found in the accompanying paper by Marra, F., De Bonis, M.V., Ruocco, G.: Combined microwaves and convection heating: A conjugate approach. Journal of Food Engineering (2010) https://doi.org/10.1016/j.jfoodeng.2009.09.012.
- 5.
After Swedish physical chemist S.A. Arrhenius, early twentieth century:
$$\begin{aligned} K=A\exp \left( E_\mathrm {a}/RT_\mathrm {s}\right) \end{aligned}$$with the pre-exponential factor \(A=925\)Â (1/s), RÂ (J/kmolK) the universal gas constant, and \(E_\mathrm {a}\)Â (J/mol) a dehydration activation energy, whose value depends on the heating physics employed.
As an example, \(E_\mathrm {a}=4\times 10^4\) for air convection dehydration, while for a combined air convection/microwave dehydration \(E_\mathrm {a}=3.1\times 10^4\) (being the later mechanism less demanding in terms of energy budget for drying). This approach has been first exploited, in a conjugate heat and mass transfer framework, by De Bonis, M.V., Ruocco, G.: A generalized conjugate model for forced convection drying based on an evaporative kinetics. Journal of Food Engineering (2008) https://doi.org/10.1016/j.jfoodeng.2008.05.008.
- 6.
Dry/basis humidity XÂ (kg/kg d.b.) and wet/basis humidity UÂ (kg/kg w.b.) values and concepts are often interchanged, with their implied meaning included in the dimensions. X and U are related by
$$\begin{aligned} U=\frac{X}{X+1} \end{aligned}$$while the following conversion formulas with the molar concentrations hold:
$$\begin{aligned} c_\mathrm {l}=\frac{1000U\rho _\mathrm {s}}{M}\qquad c_\mathrm {v}=\frac{1000\omega _\mathrm {a}\rho _\mathrm {a}}{\left( \omega _\mathrm {a}+1\right) M} \end{aligned}$$where the T-dependent density of humid air \(\rho _\mathrm {a}\) and the specific humidity \(\omega _\mathrm {a}\)Â (g/kg dry air) can be deduced by using a common psychrometric chart.
The specific humidity \(\omega _\mathrm {a}\) in the humid air has a parallel in the moist substrate with its water activity \(a_\mathrm {w}\).
- 7.
A complete account on the experimental validation can be found in the accompanying paper.
- 8.
As reviewed for example by Carrieri, G., De Bonis, M.V., Pacella, C., Pucciarelli, A., Ruocco, G. Marra, F., De Bonis, M.V., Ruocco, G.: Modeling and validation of local acrylamide formation in a model food during frying. Journal of Food Engineering (2009) https://doi.org/10.1016/j.jfoodeng.2009.04.017 and Carrieri, Anese, M., G., Quarta, B., De Bonis, M.V., Ruocco, G.: Evaluation of acrylamide formation in potatoes during deep-frying: The effect of operation and configuration. Journal of Food Engineering (2010) https://doi.org/10.1016/j.jfoodeng.2009.12.011.
- 9.
Verseux, C. et al.: Sustainable Life Support on Mars - The Potential Roles of Cyanobacteria. International Journal of Astrobiology (2016) https://doi.org/10.1017/S147355041500021X.
- 10.
De Bonis, M.V., Ruocco, G.: A Heat and Mass Transfer Perspective of Microbial Behavior Modeling in a Structured Vegetable Food. Journal of Food Engineering (2016) https://doi.org/10.1016/j.jfoodeng.2016.06.015.
- 11.
Baranyi, J. et al.: Brochothrix Thermosphacta at Changing Temperature. International Journal of Food Microbiology (1995) https://doi.org/10.1016/0168-1605(94)00154-X.
- 12.
Tang, L. et al.: Computational Modeling of 3-D Tumor Growth and Angiogenesis for Chemotherapy Evaluation. PloS one (2014) https://doi.org/10.1371/journal.pone.0083962.
Reference
Gerald, C.F., Wheatley, P.O.: Applied Numerical Analysis. Addison-Wesley, Reading (1984)
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Ruocco, G. (2018). Modeling Examples of PCB Processes. In: Introduction to Transport Phenomena Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-66822-2_6
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DOI: https://doi.org/10.1007/978-3-319-66822-2_6
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