Abstract
Osmotic dehydration (OD) is a method to partially reduce water of fruits, vegetables, meat or fish aiming to increase the shelf life or as a pre-treatment in the processing of dehydrated foods. The aim of this work is to present the models most used in mathematical modelling of experimental data obtained from the OD processes, correlating the water loss and solid gain of the food product with process variables, and to perform a comparative and critical analyses of the different models and ability to fit data. The osmotic solution concentration, the temperature, the level of agitation and the geometry of the product are some of the operating parameters that will be mainly focused. Azuara’s, Peleg’s, Page’s, the Penetration, Magee’s, Weibull’s, Toupin et al.’s, Marcotte et al.’s, the Hydrodynamic mechanism, Spiazzi and Mascheroni’s, Seguí et al.’s, Crank’s, Hough et al.’s were the models approached. These were classified into empirical and semi-empirical, phenomenological and mechanistic, and the advantages and disadvantages were presented. An extensive list of applications of the different models to the osmotic dehydration, in variable ranges of operating conditions, of fruits, vegetables, meat and fish is provided in this work. Furthermore, equivalences between parameters of different models were established, based on the affinity of the functions used in the equations of the models, these equivalences allowing a better understanding of the adequacy of the different models to fit the same experimental data. A decision tree is provided in order to allow the selection of the most adequate model(s) to fit and predict experimental data from OD processes. All this information could assist and be helpful to researchers in the choice of the most adequate model(s) to fit experimental data, as well as to predict the water loss and solid gain of food products during OD processes.
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Abbreviations
- a :
-
Cylinder radius (m)
- A i :
-
Area (m2)
- A w :
-
Dehydration constant
- B w :
-
Page’s parameter
- A, x, y :
-
Magee’s parameters
- A m :
-
Membrane surface area (m2)
- a t :
-
Thermodynamic activity
- a w :
-
Water activity
- \(\hat{a}_{\text{wv}}\) :
-
Water activity of vacuole
- \(\hat{a}_{\text{wi}}\) :
-
Water activity of the extracellular space
- C 0 :
-
Initial solute concentration (kg m−3)
- C :
-
Mass fraction
- d :
-
Diameter (m)
- D apj :
-
Apparent diffusion coefficient (m2 s−1)
- D e :
-
Effective diffusivity (m2 s−1)
- d PM :
-
Thickness of plasma membrane (m)
- \(\bar{D}_{\text{s}}\) :
-
Apparent diffusivity of sucrose (m2 s−1)
- Fo:
-
Fourier number, D e·t/l 2
- J Pw :
-
Transmembrane molar water flux of the protoplast (kg mol m−2 s−1)
- J jm :
-
Transmembrane molar flux of species j, water or solute (kg mol m−2 s−1)
- J jp :
-
Symplastic molar flux of species j, water or solute (kg mol m−2 s−1)
- k 1 :
-
Peleg’s parameter (s kg dry solids kg water−1)
- k 2 :
-
Peleg’s parameter (kg dry solids kg water−1)
- k w1 , k s1 :
-
Peleg’s parameter (s kg total weight kg water or dry solids−1)
- k w2 , k s2 :
-
Peleg’s parameter (kg total weight kg water or dry solids−1)
- k w, k s :
-
Mass transfer coefficient for WL or SG (kg water or dry solids kg total weight s−1/2)
- k :
-
Rate parameter for WL or SG (mol kg−1 s−1/2)
- kj p :
-
Plasmalemmatic transmembranary mass transfer coefficients (m s−1)
- kj s :
-
Symplastic transmembrane mass transfer coefficients (m s−1)
- l :
-
Semi-thickness of sample (m)
- L kjm :
-
Macroscopic phenomenological coefficient of plasmalemma (kg mol2 J−1 m−2 s−1)
- L kjp :
-
Macroscopic phenomenological coefficient of plasmodesmata (kg mol2 J−1 m−2 s−1)
- L wm :
-
Phenomenological coefficient of water (kg mol2 Pa−1 s−1 m−5)
- M 0 :
-
Initial moisture content (kg water kg dry solids−1)
- M :
-
Moisture content (kg water kg dry solids−1)
- M dm :
-
Mass of dry matter in the cellular volume (kg)
- M w :
-
Molecular weight of water (kg kg mol−1)
- nj cp :
-
Plasmalemmatic transmembranary mass flux (kg s−1 m−2)
- \(nj_{\text{cs}}\) :
-
Symplastic transmembranary transfer (kg s−1 m−2)
- nj 0 :
-
Extracellular mass flux (kg s−1 m−2)
- N i1 :
-
Molar flux (kg mol m−2 s−1)
- OD:
-
Osmotic dehydration
- P :
-
Pressure (N m−2)
- P 0c :
-
Cellular hydrostatic pressure (reference state at the temperature considered and at atmospheric pressure (N m−2)
- P c :
-
Cellular hydrostatic pressure (N m−2)
- r c :
-
Cylinder radius (m)
- r :
-
Sphere radius (m)
- R g :
-
Universal gas constant (J kg mol−1 K−1)
- R i :
-
Radius of the extracellular space (m)
- R wm :
-
Transmembrane transport of water (kg m−2 s−1)
- SG:
-
Solid gain (kg dry solids kg total weight−1)
- SG∞ :
-
Solid gain at equilibrium (kg dry solids kg total weight−1)
- s 1 , s 2 :
-
Azuara’s parameters (s−1)
- T :
-
Temperature (°C, K)
- t :
-
Time (s)
- t v :
-
Thickness of the transfer pathway (m)
- u :
-
Average velocity (m s−1)
- u b :
-
Barycentric velocity (m s−1)
- V :
-
Volume (m3)
- V c :
-
Cellular volume (m3)
- V 1 :
-
Intercellular volume (m3)
- \(\bar{V}_{\text{w}}\) :
-
Partial molar volume of water (m3 kg mol water−1)
- X :
-
Volumetric fraction of the total volume occupied by the liquid
- X v :
-
Volumetric fraction of the pore occupied by the liquid
- x w :
-
Water molar fraction
- z L :
-
Length (m)
- w 0 :
-
Initial weight of the sample (kg)
- WL:
-
Water loss (kg water kg total weight−1)
- WL∞ :
-
Water loss at equilibrium (kg water kg total weight−1)
- w s∞ :
-
Dry solids content of the sample at equilibrium (kg dry solids kg total weight−1)
- w s :
-
Dry solids content of the sample after treatment (kg dry solids kg total weight−1)
- w s0 :
-
Initial dry solids content of the sample (kg dry solids kg total weight−1)
- w w∞ :
-
Water content of the sample at equilibrium (kg water kg total weight−1)
- w w0 :
-
Initial water content of the sample (kg water kg total weight−1)
- w w :
-
Water content of the sample after treatment (kg dry solids kg total weight−1)
- α w , α s :
-
Weibull’s scale parameter (s)
- α n :
-
Positive roots of the equation
- β w , β s :
-
Weibull’s shape parameter
- γ i :
-
Number from the comparison between Page’s and Crank’s models
- ε e :
-
Effective porosity
- μ :
-
Liquid viscosity (Pa s)
- μ km :
-
Plasmalemma chemical potential (J kg mol−1)
- μ kp :
-
Plasmodesmata chemical potential (J kg mol−1)
- v :
-
Partial molar volume (m3 kg mol−1)
- π :
-
Osmotic pressure
- π* :
-
Osmotic pressure at equilibrium
- π 0 :
-
Initial osmotic pressure
- ρ :
-
Mass concentration (kg m−3)
- ρ s :
-
Volume mass of sucrose (kg m−3)
- ρj 0 :
-
Cellular mass concentrations (kg m−3)
- ρj c :
-
Extracellular mass concentrations (kg m−3)
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Acknowledgments
This work was supported by National Funds from FCT through Project PEst-OE/EQB/LA0016/2013. The first author acknowledges the financial support of CAPES (1528/13-0).
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Assis, F.R., Morais, R.M.S.C. & Morais, A.M.M.B. Mass Transfer in Osmotic Dehydration of Food Products: Comparison Between Mathematical Models. Food Eng Rev 8, 116–133 (2016). https://doi.org/10.1007/s12393-015-9123-1
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DOI: https://doi.org/10.1007/s12393-015-9123-1