Skip to main content

Advertisement

SpringerLink
Log in

Capillary Pressure in Unsaturated Food Systems: Its Importance and Accounting for It in Mathematical Models

  • Published:
Food Engineering Reviews Aims and scope Submit manuscript

Abstract

Capillary pressure plays a critical role in driving fluid flow in unsaturated porous (pores not saturated with liquids but also containing air/gas) structures. The role and importance of capillary pressure have been well documented in geological and soil sciences but remain largely unexplored in the food literature. Available mathematical models for unsaturated food systems have either ignored the capillary-driven flow or combined it with the diffusive flow. Such approaches are bound to impact the accuracy of models. The derivation of the microscale definition of capillary pressure is overviewed, and the limitations of using the microscale definition at the macroscale are discussed. Next, the factors affecting capillary pressure are briefly reviewed. The parametric expressions for capillary pressure as a function of saturation and temperature, developed originally for soils, are listed, and their application for food systems is encouraged. Capillary pressure estimation methods used for food systems are then discussed. Next, the different mathematical formulations for food systems are compared, and the limitations of each formulation are discussed. Additionally, examples of hybrid mixture theory–based multiscale models for frying involving capillary pressure are provided. Capillary-driven liquid flow plays an important role in the unsaturated transport during the processing of porous solid foods. However, measuring capillary pressure in food systems is challenging because of the soft nature of foods. As a result, there is a lack of available capillary pressure data for food systems which has hampered the development of mechanistic models. Nevertheless, providing a fundamental understanding of capillary pressure will aid food engineers in designing new experimental studies and developing mechanistic models for unsaturated processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

\(A\) :

Pre-exponential factor (Eq. (43)) \(\left(\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right)\)

\(a\) :

Constant in the linear relation between \({\gamma }^{wn}\) and \(T\) (Eq. (22)) \((\mathrm{N}/\mathrm{m})\)

\({a}_{s}\) :

Radius of sphere (Eq. (42)) (\(\mathrm{m}\))

\({a}_{w}\) :

Water activity (Eq. (39)) (dimensionless)

\({a}_{0}^{ws}\) :

Coefficient in the power series expression for the interfacial energy per unit area (Eq. (28)) \(\left(\frac{\mathrm{N}}{\mathrm{m}}\right)\)

\({a}_{0}^{ns}\) :

Coefficient in the power series expression for the interfacial energy per unit area (Eq. (29)) \(\left(\frac{\mathrm{N}}{\mathrm{m}}\right)\)

\({a}_{1}^{ws}\) :

Coefficient in the power series expression for the interfacial energy per unit area (Eq. (28)) \(\left(\frac{\mathrm{N}}{\mathrm{mK}}\right)\)

\({a}_{1}^{ns}\) :

Coefficient in the power series expression for the interfacial energy per unit area (Eq. (29)) \(\left(\frac{\mathrm{N}}{\mathrm{mK}}\right)\)

\(B\) :

Mixture viscosity of the biopolymeric matrix (Eq. (48)) (\(\mathrm{Pa}.\mathrm{s}\))

\(b\) :

Constant in the linear relation between \({\gamma }^{wn}\) and \(T\) (Eq. (22)) \(\left(\frac{\mathrm{N}}{\mathrm{mK}}\right)\)

\(C\) :

Liquid concentration (Eq. (40)) (amount of substance)

\({C}_{0}\) :

Constant surface concentration (Eq. (42)) (amount of substance)

\({C}_{1}\) :

Initial uniform concentration (Eq. (42)) (amount of substance)

\({c}^{l}\) :

Mass concentration of liquid (Eq. (45)) \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)\)

\({c}^{ws}\) :

Constant (Eq. (30)) \(\left(\frac{\mathrm{N}}{\mathrm{mK}}\right)\)

\({c}^{ns}\) :

Constant (Eq. (31)) \(\left(\frac{\mathrm{N}}{\mathrm{mK}}\right)\)

\({C}_{p}^{\alpha }\) :

Specific heat for phase \(\alpha\) (Eq. (52)) \(\left(\frac{\mathrm{J}}{\mathrm{kgK}}\right)\)

\({D}_{eff}\) :

Effective diffusivity (Eq. (43)) \(\left(\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right)\)

\({D}^{\alpha }\) :

Diffusivity of phase \(\alpha\) (Eq. (48)) \(\left(\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right)\)  

\({D}_{c}\) :

Concentration gradient dependent capillary diffusivity (Eq. (46))  \(\left(\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right)\)

\({D}_{w}\) :

Capillary diffusivity of water (Eq. (47))  \(\left(\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right)\)

\({D}_{T}\) :

Temperature gradient dependent capillary diffusivity (Eq. (46))  \(\left(\frac{{\mathrm{m}}^{2}}{\mathrm{s}}\right)\)

\({E}_{a}\) :

Activation energy (Eq. (43)) \(\left(\frac{\mathrm{J}}{\mathrm{mol}}\right)\)

\(E\) :

Modulus of elasticity of the biopolymeric matrix (Eq. (48)) \((\mathrm{Pa})\)

\({}^{\beta }{\widehat{e}}^{\alpha }\) :

Source-sink term (Eq. (49)  \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}\mathrm{s}}\right)\)

\(F\) :

Resultant of the elastic surface forces acting on the contact line (Eq. (5))  \((\mathrm{N}/\mathrm{m})\)

\(g\) :

Gravitational acceleration \((\mathrm{m}/{\mathrm{s}}^{2})\)

\(h\) :

Capillary pressure head \((\mathrm{m})\)

\({h}_{b}\) :

Bubbling pressure head (Eq. (1)\(9\)) \((\mathrm{m})\)

\({\Delta }_{sn}^{sw}{h}^{s}\) :

Enthalpy of immersion per unit area (Eq. (21)) \((\mathrm{J}/{\mathrm{m}}^{2})\)

\(I\) :

Identity tensor (Eq. (3)) (dimensionless)

\({I}^{\sigma }\) :

\(I-NN\), Projected surficial identity tensor (Eq. (2)) (dimensionless)

\({j}_{m}\) :

Mass flux of liquid phase (Eq. (44)) \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{2}\mathrm{s}}\right)\)

\({K}^{\alpha }\) :

Permeability of phase \(\alpha\) (Eq. (48)) \(({\mathrm{m}}^{2})\)

\({k}^{\alpha }\) :

Thermal conductivity of phase \(\alpha\) (Eq. (52)) \(\left(\frac{\mathrm{W}}{\mathrm{mK}}\right)\)

\(k\) :

Restriction on fitting parameters of van Genuchten’s function (Eq. (17)) (dimensionless)

\(M\) :

Moisture content on a dry basis (Eq. (47)) \(\left(\frac{\mathrm{g}}{\mathrm{g solids}}\right)\)

\({M}_{W}\) :

Molecular weight (Eq. (37)) \(\left(\frac{\mathrm{kg}}{\mathrm{mol}}\right)\)

\(m\) :

Fitting parameter for van Genuchten’s function (Eq. (15)) (dimensionless)

\(N\) :

Unit vector normal to the interface (towards \(\alpha\) phase) (Eq. (1)) (dimensionless)

\(n\) :

Fitting parameter for van Genuchten’s function (Eq. (15)) (dimensionless)

\({p}^{\alpha }\) :

Pressure of phase \(\alpha\) (Eq. (45)) \(\left(\frac{\mathrm{N}}{{\mathrm{m}}^{2}}\right)\)

\({p}^{c}\) :

Microscale capillary pressure \((\mathrm{Pa})\)

\({P}^{c}\) :

Macroscopic capillary pressure \((\mathrm{Pa})\)

\(P\) :

Vapor pressure over the meniscus (Eq. (37)) \((\mathrm{Pa})\)

\({P}_{0}\) :

Vapor pressure over the planar surface (Eq. (37)) \((\mathrm{Pa})\)

\({P}^{b}\) :

Bubbling pressure (Eq. (18)) \((\mathrm{Pa})\)

\(R\) :

Universal gas constant \(\left(\frac{\mathrm{J}}{\mathrm{mol K}}\right)\)

\({R}_{c}\) :

Radius of curvature of the liquid surface in the tube (Eq. (37)) \((\mathrm{m})\)

\(r\) :

Capillary radius \((\mathrm{m})\)

\(r\) :

Radial distance \((\mathrm{m})\)

\({S}_{w}\) :

Wetting phase saturation (dimensionless)

\({S}_{e}\) :

Effective saturation of the wetting phase (dimensionless)

\({S}_{s}\) :

Saturated wetting phase saturation (dimensionless)

\({S}_{r}\) :

Residual wetting phase saturation (dimensionless)

\({S}^{\alpha \beta }\) :

Interface stress tensor (Eq. (1)) \(\left(\frac{\mathrm{N}}{\mathrm{m}}\right)\)

\(s\) :

Distance (Eq. (44)) \(\left(\mathrm{m}\right)\)

\({T}_{a}\) :

Stress tensor for adjacent phases (Eq. (1)) \(\left(\frac{\mathrm{N}}{{\mathrm{m}}^{2}}\right)\)

\(T\) :

Temperature \((\mathrm{K})\)

\({T}_{r}\) :

Reference temperature \((\mathrm{K})\)

\({T}_{f}\) :

Target temperature \((\mathrm{K})\)

\(U\) :

Velocity of the interface (Eq. (1)) \(\left(\frac{\mathrm{m}}{\mathrm{s}}\right)\)

\({u}^{s,\alpha \beta }\) :

Interfacial energy per unit area for an interface formed by phase \(\alpha\) and \(\beta\) (Eq. (27)) \(\left(\frac{\mathrm{J}}{{\mathrm{m}}^{2}}\right)\)

\({v}^{\alpha ,s}\) :

Velocity of \(\alpha\) phase with respect to the solid phase (Eq. (48)) \(\left(\frac{\mathrm{m}}{\mathrm{s}}\right)\)

\({v}_{\alpha \beta }\) :

Normal to the contact line (Eq. (5)) (dimensionless)

\({v}_{w}\) :

Molar volume of water (Eq. (39)) \(({\mathrm{m}}^{3}\mathrm{ mo}{\mathrm{l}}^{-1})\)

\(v\) :

Velocity of adjacent phases at the interface (Eq. (1)) \(\left(\frac{\mathrm{m}}{\mathrm{s}}\right)\)

\(\Gamma\) :

Excess mass of the interface (Eq. (1)) \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{2}}\right)\)

\(\rho\) :

Density \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)\)

\({\rho }_{l}\) :

Liquid density (Eq. (37)) \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)\)

\({\rho }^{\alpha }\) :

Density of phase \(\alpha\) (Eq. (49)) \(\left(\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right)\)

\(\gamma\) :

Interfacial tension \(\left(\frac{\mathrm{N}}{\mathrm{m}}\right)\)

\({\gamma }^{wn}\) :

Interfacial tension between \(w\) and \(n\) (Eq. (2)) \(\left(\frac{\mathrm{N}}{\mathrm{m}}\right)\)

\(\Lambda\) :

Unit vector tangent to the contact line (Eq. (5)) (dimensionless)

\({\tau }^{wn}\) :

Viscous stress tensor for \(wn\) interface (Eq. (2)) \(\left(\frac{\mathrm{N}}{\mathrm{m}}\right)\)

\({\tau }^{\alpha }\) :

Viscous stress tensor for phase \(\alpha\) (Eq. (3)) \(\left(\frac{\mathrm{N}}{{\mathrm{m}}^{2}}\right)\)

\({\gamma }^{wns}\) :

Contact curve compression (Eq. (5)) \(\left(\mathrm{N}\right)\)

\(\theta\) :

Contact angle \((^\circ )\)

\(\lambda\) :

Fitting parameter for Brooks and Corey’s expression (Eq. (18)) (dimensionless)

\({\lambda }_{v}\) :

Latent heat of vaporization (Eq. (52)) \(\left(\frac{\mathrm{J}}{\mathrm{kg}}\right)\)

\(\alpha\) :

Fitting parameter for van Genuchten’s function (Eq. (15)) \(\left(\frac{1}{\mathrm{m}}\right)\)

\({\beta }_{0}\) :

Parameter in \({p}^{c}\left(T\right)\) expression (Eq. (33))

\({\beta }_{1}\) :

Parameter in \({p}^{c}\left(T\right)\) expression (Eq. (33))

\({\beta }_{2}\) :

Parameter in \({p}^{c}\left(T\right)\) expression (Eq. (33))

\(\mu\) :

Dynamic viscosity \(\left(\mathrm{Pa}.\mathrm{s}\right)\)

\({\mu }^{\alpha }\) :

Dynamic viscosity of phase \(\alpha\) (Eq. (48)) \(\left(\mathrm{Pa}.\mathrm{s}\right)\)

\(\beta\) :

Angle formed by the capillary tube with the vertical direction (Eq. (35))

\({\varepsilon }^{\alpha }\) :

Volume fraction of phase \(\alpha\) (Eq. (48)) (dimensionless)

\(\dot{{\varepsilon }^{\alpha }}\) :

Material time derivative of volume fraction of phase \(\alpha\) with respect to the solid phase (Eq. (48)) \(\left({s}^{-1}\right)\)

\(\phi\) :

Porosity (Eq. (51)) (dimensionless)

\(\alpha\) :

General representation for a phase

\(\beta\) :

General representation for a phase

\(w\) :

Wetting phase

\(n\) :

Non-wetting phase

\(l\) :

Liquid

\(w\) :

Water phase

\(o\) :

Oil phase

\(g\) :

Gas phase

\(\frac{{D}^{\sigma }}{Dt}\) :

Surface material derivative (Eq. (1)) \((1/\mathrm{s})\)

\({\nabla }^{\sigma }\) :

\({\nabla }^{\sigma }=\nabla -NN.\nabla\), Surface gradient operator (Eq. (1)) (\(1/\mathrm{m}\))

\({\nabla }^{c}\) :

\({\nabla }^{\mathrm{c}}=\mathrm{\Lambda \Lambda }.\nabla\), Curvilineal del operator (Eq. (5)) (\(1/\mathrm{m}\))

\(\langle \rangle\) :

Averaging operator (Eq. (12)) (dimensionless)

\({D}^{s}/Dt\) :

Material time derivative with respect to the solid phase (Eq. (49)) (\(1/\mathrm{s}\))

\(\left[\left[K\right]\right]\) :

\({\left[\left[K\right]\right]=K}^{\alpha }-{K}^{\beta }\), Jump in the quantity \(K\)

References

  1. Aguilera JM, Michel M, Mayor G (2004) Fat migration in chocolate: diffusion or capillary flow in a particulate solid?—a hypothesis paper. J Food Sci 69(7):167–174

    Article  Google Scholar 

  2. Lumen Learning and OpenStax (2021) Fundamentals of Heat, Light & Sound. https://pressbooks.nscc.ca/heatlightsound/

  3. Rahman MM, Gu Y, Karim MA (2018) Development of realistic food microstructure considering the structural heterogeneity of cells and intercellular space. Food Struct 15:9–16

    Article  Google Scholar 

  4. Dadmohammadi Y, Datta AK (2020) Food as porous media: a review of the dynamics of porous properties during processing. Food Rev Int 1–33.

  5. Datta AK (2007a) Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: problem formulations. J Food Eng 80(1):80–95

  6. Shah Y, Takhar PS (2022) Pressure development and volume changes during frying and post-frying of potatoes. LWT 172:114243

    Article  CAS  Google Scholar 

  7. Vitrac O, Trystram G, Raoult-Wack AL (2000) Deep-fat frying of food: heat and mass transfer, transformations and reactions inside the frying material. Eur J Lipid Sci Technol 102(8–9):529–538

    Article  CAS  Google Scholar 

  8. Weerts AH, Lian G, Martin DR (2003) Modeling the hydration of foodstuffs: temperature effects. AIChE J 49(5):1334–1339

    Article  CAS  Google Scholar 

  9. Bansal HS, Takhar PS, Maneerote J (2014) Modeling multiscale transport mechanisms, phase changes and thermomechanics during frying. Food Res Int 62:709–717

    Article  Google Scholar 

  10. Ditudompo S, Takhar PS (2015) Hybrid mixture theory based modeling of transport mechanisms and expansion-thermomechanics of starch during extrusion. AIChE J 61(12):4517–4532

    Article  CAS  Google Scholar 

  11. Zhao Y, Kumar PK, Sablani SS, Takhar PS (2022) Hybrid mixture theory-based modeling of transport of fluids, species, and heat in food biopolymers subjected to freeze–thaw cycles. J Food Sci 87(9):4082–4106

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Cazzaniga A, Brousse MM, Linares AR (2022) Kinetics of moisture loss applied to the baking of snacks with pregelatinized cassava starch. J Food Sci

  13. Demiray E, Tulek Y (2017) Effect of temperature on water diffusion during rehydration of sun-dried red pepper (Capsicum annuum L.). Heat Mass Transf 53(5):1829–1834

  14. Zhang W, Chen J, Yue Y, Zhu Z, Liao E, Xia W (2020) Modelling the mass transfer kinetics of battered and breaded fish nuggets during deep-fat frying at different frying temperatures. J Food Qual 2020

  15. Lee KT, Farid M, Nguang SK (2006) The mathematical modelling of the rehydration characteristics of fruits. J Food Eng 72(1):16–23

    Article  Google Scholar 

  16. Vavra CL, Kaldi JG, Sneider RM (1992) Geological applications of capillary pressure: a review. AAPG Bull 76(6):840–850

    CAS  Google Scholar 

  17. Shi S, Belhaj H, Bera A (2018) Capillary pressure and relative permeability correlations for transition zones of carbonate reservoirs. J Pet Explor Prod Technol 8(3):767–784

    Article  CAS  Google Scholar 

  18. Kaliakin VN (2017) Example problems involving in situ stresses under hydrostatic conditions. In Soil Mech (pp. 205–242). Elsevier. https://doi.org/10.1016/b978-0-12-804491-9.00005-7

  19. Pinder GF, Gray WG (2008) Essentials of multiphase flow and transport in porous media. John Wiley & Sons

    Book  Google Scholar 

  20. Gu Z, Qi Z, Burghate R, Yuan S, Jiao X, Xu J (2020) Irrigation scheduling approaches and applications: a review. J Irrig Drain Eng 146(6):04020007

    Article  Google Scholar 

  21. Yadvinder-Singh Kukal SS, Jat ML, Sidhu HS (2014) Improving water productivity of wheat-based cropping systems in South Asia for sustained productivity. In Advances in Agronomy (pp. 157–258). Elsevier. https://doi.org/10.1016/b978-0-12-800131-8.00004-2

  22. Venturas MD, Sperry JS, Hacke UG (2017) Plant xylem hydraulics: what we understand, current research, and future challenges. J Integr Plant Biol 59(6):356–389

    Article  PubMed  Google Scholar 

  23. Miller CT, Bruning K, Talbot CL, McClure JE, Gray WG (2019) Nonhysteretic capillary pressure in two-fluid porous medium systems: definition, evaluation, validation, and dynamics. Water Resour Res 55(8):6825–6849

    Article  Google Scholar 

  24. Hassanizadeh SM, Gray WG (1993) Thermodynamic basis of capillary pressure in porous media. Water Resour Res 29(10):3389–3405

    Article  Google Scholar 

  25. Bear J, Bachmat Y (1991) Introduction to modeling of transport phenomena in porous media. Kluver Academic Publisher, Netherlands

    Google Scholar 

  26. Starnoni M, Pokrajac D (2020) On the concept of macroscopic capillary pressure in two-phase porous media flow. Adv Water Resour 135:103487

    Article  CAS  Google Scholar 

  27. Li Y, Liu C, Li H, Chen S, Lu K, Zhang Q, Luo H (2022) A review on measurement of the dynamic effect in capillary pressure. J Petrol Sci Eng 208:109672

    Article  CAS  Google Scholar 

  28. Drelich J, Fang C, White CL (2002) Measurement of interfacial tension in fluid-fluid systems. Encyclopedia of surface and colloid science 3:3158–3163

    Google Scholar 

  29. Hebbar RS, Isloor AM, Ismail AF (2017) Contact angle measurements. In Membrane characterization (pp. 219–255). Elsevier

  30. Alghunaim A, Kirdponpattara S, Newby BMZ (2016) Techniques for determining contact angle and wettability of powders. Powder Technol 287:201–215

    Article  CAS  Google Scholar 

  31. Dana D, Saguy IS (2006) Mechanism of oil uptake during deep-fat frying and the surfactant effect-theory and myth. Adv Coll Interface Sci 128:267–272

    Article  Google Scholar 

  32. Alam T, Takhar PS (2016) Microstructural characterization of fried potato disks using X-ray micro computed tomography. J Food Sci 81(3):E651–E664

    Article  CAS  PubMed  Google Scholar 

  33. Reeve RM, Neel EM (1960) Microscopic structure of potato chips. Am Potato J 37(2):45–52

    Article  Google Scholar 

  34. Spiruta SL, Mackey A (1961) French-fried potatoes: palatability as related to microscopic structure of frozen par-fries. J Food Sci 26(6):656–662

    Article  Google Scholar 

  35. Harimi B, Ghazanfari MH, Masihi M (2020) Modeling of capillary pressure in horizontal rough-walled fractures in the presence of liquid bridges. J Petrol Sci Eng 185:106642

    Article  CAS  Google Scholar 

  36. Maqsoud A, Bussière B, Aubertin M, Mbonimpa M (2012) Predicting hysteresis of the water retention curve from basic properties of granular soils. Geotech Geol Eng 30(5):1147–1159

    Article  Google Scholar 

  37. Likos WJ, Lu N, Godt JW (2014) Hysteresis and uncertainty in soil water-retention curve parameters. J Geotech Geoenviron Eng 140(4):04013050

    Article  Google Scholar 

  38. Abbasi F, Javaux M, Vanclooster M, Feyen J (2012) Estimating hysteresis in the soil water retention curve from monolith experiments. Geoderma 189:480–490

    Article  Google Scholar 

  39. Nuth M, Laloui L (2008) Advances in modelling hysteretic water retention curve in deformable soils. Comput Geotech 35(6):835–844

    Article  Google Scholar 

  40. Pedroso DM, Williams DJ (2010) A novel approach for modelling soil–water characteristic curves with hysteresis. Comput Geotech 37(3):374–380

    Article  Google Scholar 

  41. Šimůnek J, Kodešová R, Gribb MM, van Genuchten MT (1999) Estimating hysteresis in the soil water retention function from cone permeameter experiments. Water Resour Res 35(5):1329–1345

    Article  Google Scholar 

  42. Yakhshi-Tafti E, Kumar R, Cho HJ (2011) Measurement of surface interfacial tension as a function of temperature using pendant drop images. Int J Optomechatronics 5(4):393–403

    Article  Google Scholar 

  43. Grant SA, Salehzadeh A (1996) Calculation of temperature effects on wetting coefficients of porous solids and their capillary pressure functions. Water Resour Res 32(2):261–270

    Article  CAS  Google Scholar 

  44. Kang W, Chung WY (2009) Liquid water diffusivity of wood from the capillary pressure-moisture relation. J Wood Sci 55(2):91–99

    Article  CAS  Google Scholar 

  45. Spolek GA, Plumb OA (1981) Capillary pressure in softwoods. Wood Sci Technol 15(3):189–199

    Article  Google Scholar 

  46. Philip JR, De Vries DA (1957) Moisture movement in porous materials under temperature gradients. EOS Trans Am Geophys Union 38(2):222–232

    Article  Google Scholar 

  47. Nimmo JR, Miller EE (1986) The temperature dependence of isothermal moisture vs. potential characteristics of soils. Soil Sci Soc Am J 50(5):1105–1113

  48. Van Genuchten MT (1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci Soc Am J 44(5):892–898

    Article  Google Scholar 

  49. Chen J, Hopmans JW, Grismer ME (1999) Parameter estimation of two-fluid capillary pressure–saturation and permeability functions. Adv Water Resour 22(5):479–493

    Article  CAS  Google Scholar 

  50. Yang X, You X (2013) Estimating parameters of van Genuchten model for soil water retention curve by intelligent algorithms. Applied Mathematics & Information Sciences 7(5):1977

    Article  Google Scholar 

  51. Li B, Tchelepi HA, Benson SM (2013) Influence of capillary-pressure models on CO2 solubility trapping. Adv Water Resour 62:488–498

    Article  CAS  Google Scholar 

  52. Webb SW (2000) A simple extension of two-phase characteristic curves to include the dry region. Water Resour Res 36(6):1425–1430

    Article  Google Scholar 

  53. Troygot O, Saguy IS, Wallach R (2011) Modeling rehydration of porous food materials: I. Determination of characteristic curve from water sorption isotherms. J Food Eng 105(3):408–415

  54. Alonso EE, Pereira JM, Vaunat J, Olivella S (2010) A microstructurally based effective stress for unsaturated soils. Géotechnique 60(12):913–925

    Article  Google Scholar 

  55. Mualem Y (1976) A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour Res 12(3):513–522

    Article  Google Scholar 

  56. Burdine N (1953) Relative permeability calculations from pore size distribution data. J Petrol Technol 5(03):71–78

    Article  Google Scholar 

  57. Brooks RH, Corey AT (1964) Hydraulic properties of porous media and their relation to drainage design. Transactions of the ASAE 7(1):26–0028

    Article  Google Scholar 

  58. Ippisch O, Vogel HJ, Bastian P (2006) Validity limits for the van Genuchten-Mualem model and implications for parameter estimation and numerical simulation. Adv Water Resour 29(12):1780–1789

    Article  Google Scholar 

  59. Durner W (1994) Hydraulic conductivity estimation for soils with heterogeneous pore structure. Water Resour Res 30(2):211–223

    Article  Google Scholar 

  60. Rogers C, Stallybrass MP, Clements DL (1983) On two phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation. Nonlinear Anal Theory Methods Appl 7(7):785–799

    Article  Google Scholar 

  61. Russo D (1988) Determining soil hydraulic properties by parameter estimation: on the selection of a model for the hydraulic properties. Water Resour Res 24(3):453–459

    Article  Google Scholar 

  62. Silin D, Patzek T, Benson SM (2009) A model of buoyancy-driven two-phase countercurrent fluid flow. Transp Porous Media 76(3):449–469

    Article  CAS  Google Scholar 

  63. Grant SA, Bachmann J (2002) Effect of temperature on capillary pressure. Geophys Monogr-Am Geophys Union 129:199–212

    Google Scholar 

  64. Gaines GL Jr (1972) Surface and interfacial tension of polymer liquids–a review. Polym Eng Sci 12(1):1–11

    Article  CAS  Google Scholar 

  65. Jasper JJ (1972) The surface tension of pure liquid compounds. J Phys Chem Ref Data 1(4):841–1010

    Article  CAS  Google Scholar 

  66. Grant SA (2003) Extension of a temperature effects model for capillary pressure saturation relations. Water Resour Res 39(1), SBH-1

  67. She HY, Sleep BE (1998) The effect of temperature on capillary pressure-saturation relationships for air-water and perchloroethylene-water systems. Water Resour Res 34(10):2587–2597

    Article  CAS  Google Scholar 

  68. McPhee C, Reed J, Zubizarreta I (2015) Capillary pressure. In Developments in Petroleum Science (Vol. 64, pp. 449–517). Elsevier

  69. Halder A, Datta AK, Spanswick RM (2011) Water transport in cellular tissues during thermal processing. AIChE J 57(9):2574–2588

    Article  CAS  Google Scholar 

  70. Stevenson CD, Dykstra MJ, Lanier TC (2013) Capillary pressure as related to water holding in polyacrylamide and chicken protein gels. J Food Sci 78(2):C145–C151

    Article  CAS  PubMed  Google Scholar 

  71. Hamraoui A, Nylander T (2002) Analytical approach for the Lucas-Washburn equation. J Colloid Interface Sci 250(2):415–421

    Article  CAS  PubMed  Google Scholar 

  72. Cai J, Jin T, Kou J, Zou S, Xiao J, Meng Q (2021) Lucas-Washburn equation-based modeling of capillary-driven flow in porous systems. Langmuir 37(5):1623–1636

    Article  CAS  PubMed  Google Scholar 

  73. Carbonell S, Hey MJ, Mitchell JR, Roberts CJ, Hipkiss J, Vercauteren J (2004) Capillary flow and rheology measurements on chocolate crumb/sunflower oil mixtures. J Food Sci 69(9):E465–E470

    Article  CAS  Google Scholar 

  74. Saguy IS, Marabi A, Wallach R (2005) New approach to model rehydration of dry food particulates utilizing principles of liquid transport in porous media. Trends Food Sci Technol 16(11):495–506

    Article  Google Scholar 

  75. Kalogianni EP, Papastergiadis E (2014) Crust pore characteristics and their development during frying of French-fries. J Food Eng 120:175–182

    Article  Google Scholar 

  76. Saguy IS, Marabi A, Wallach R (2005) Liquid imbibition during rehydration of dry porous foods. Innov Food Sci Emerg Technol 6(1):37–43

    Article  Google Scholar 

  77. Lisgarten ND, Sambles JR, Skinner LM (1971) Vapour pressure over curved surfaces-the Kelvin equation. Contemp Phys 12(6):575–593

    Article  CAS  Google Scholar 

  78. Dhall A, Datta AK (2011) Transport in deformable food materials: a poromechanics approach. Chem Eng Sci 66(24):6482–6497

    Article  CAS  Google Scholar 

  79. Billings SW, Bronlund JE, Paterson AHJ (2006) Effects of capillary condensation on the caking of bulk sucrose. J Food Eng 77(4):887–895

    Article  CAS  Google Scholar 

  80. Rosa GS, Moraes MA, Pinto LA (2010) Moisture sorption properties of chitosan. LWT-Food Sci Technol 43(3):415–420

    Article  CAS  Google Scholar 

  81. Labuza TP, Rutman M (1968) The effect of surface active agents on sorption isotherms of a model food system. Can J Chem Eng 46(5):364–368

    Article  CAS  Google Scholar 

  82. Al-Rub FAA, Datta R (1998) Theoretical study of vapor pressure of pure liquids in porous media. Fluid Phase Equilib 147(1–2):65–83

    Article  Google Scholar 

  83. Fisher LR, Israelachvili JN (1979) Direct experimental verification of the Kelvin equation for capillary condensation. Nature 277(5697):548–549

    Article  CAS  Google Scholar 

  84. Haynes JM (1973) Pore size analysis according to the Kelvin equation. Matériaux et Construction 6(3):209–213

    Article  Google Scholar 

  85. Svintradze DV (2020) Generalization of the Kelvin equation for arbitrarily curved surfaces. Phys Lett A 384(20):126412

    Article  CAS  Google Scholar 

  86. Wang Y, Shardt N, Lu C, Li H, Elliott JA, Jin Z (2020) Validity of the Kelvin equation and the equation-of-state-with-capillary-pressure model for the phase behavior of a pure component under nanoconfinement. Chem Eng Sci 226:115839

    Article  CAS  Google Scholar 

  87. Shereshefsky JL (1928) A study of vapor pressures in small capillaries. Part I. Water vapor.(A). Soft glass capillaries1, 2. J Am Chem Soc 50(11):2966–2980

  88. Shereshefsky JL, Carter CP (1950) Liquid—vapor equilibrium in microscopic capillaries. 1 I. Aqueous system. J Am Chem Soc 72(8):3682–3686

  89. Folman M, Shereshefsky JL (1955) Liquid-vapor equilibrium in microscopic capillaries. I. Non-aqueous Systems. J Phys Chem 59(7):607–610

  90. Coleburn NL, Shereshefsky JL (1972) Liquid-vapor equilibrium in microscopic capillaries. III. Toluene. J Colloid Interface Sci 38(1):84–90

  91. McGavack J Jr, Patrick WA (1920) The adsorption of sulfur dioxide by the gel of silicic acid. J Am Chem Soc 42(5):946–978

    Article  CAS  Google Scholar 

  92. Everett DH, Whitton WI (1955) A thermodynamic study of the adsorption of benzene vapour by active charcoals. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 230(1180):91–110

  93. Pierce C, Wiley JW, Smith RN (1949) Capillarity and surface area of charcoal. J Phys Chem 53(5):669–683

    Article  CAS  Google Scholar 

  94. Jaros M, Cenkowski S, Jayas DS, Pabis S (1992) A method of determination of the diffusion coefficient based on kernel moisture content and its temperature. Drying Technol 10(1):213–222

    Article  CAS  Google Scholar 

  95. Kang S, Delwiche SR (1999) Moisture diffusion modeling of wheat kernels during soaking. Transactions of the ASAE 42(5):1359

    Article  Google Scholar 

  96. Shivhare US, Raghavan GSV, Bosisio RG (1994) Modelling the drying kinetics of maize in a microwave environment. J Agric Eng Res 57(3):199–205

    Article  Google Scholar 

  97. Suarez C, Viollaz P, Chirife J (1980) Diffusional analysis of air drying of grain sorghum. Int J Food Sci Technol 15(5):523–531

    Article  Google Scholar 

  98. Crank J (1975) Diffusion in a sphere. The mathematics of diffusion

  99. Datta AK (2007b) Porous media approaches to studying simultaneous heat and mass transfer in food processes. II: property data and representative results. J Food Eng 80(1):96–110

  100. Halder A, Dhall A, Datta AK (2007a) An improved, easily implementable, porous media based model for deep-fat frying: part I: model development and input parameters. Food Bioprod Process 85(3):209–219

    Article  CAS  Google Scholar 

  101. Ni H, Datta AK (1999) Moisture, oil and energy transport during deep-fat frying of food materials. Food Bioprod Process 77(3):194–204

    Article  Google Scholar 

  102. Ni H, Datta AK, Torrance KE (1999) Moisture transport in intensive microwave heating of biomaterials: a multiphase porous media model. Int J Heat Mass Transf 42(8):1501–1512

    Article  CAS  Google Scholar 

  103. Halder A, Dhall A, Datta AK (2007b) An improved, easily implementable, porous media based model for deep-fat frying: part II: results, validation and sensitivity analysis. Food Bioprod Process 85(3):220–230

    Article  CAS  Google Scholar 

  104. Gulati T, Zhu H, Datta AK (2016) Coupled electromagnetics, multiphase transport and large deformation model for microwave drying. Chem Eng Sci 156:206–228

    Article  CAS  Google Scholar 

  105. Warning A, Dhall A, Mitrea D, Datta AK (2012) Porous media based model for deep-fat vacuum frying potato chips. J Food Eng 110(3):428–440

    Article  CAS  Google Scholar 

  106. Rakesh V, Datta AK, Walton JH, McCarthy KL, McCarthy MJ (2012) Microwave combination heating: coupled electromagnetics-multiphase porous media modeling and MRI experimentation. AIChE J 58(4):1262–1278

    Article  CAS  Google Scholar 

  107. Ni H (1997) Multiphase moisture transport in porous media under intensive microwave heating. Cornell University

  108. Takhar PS (2011) Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: coupled fluid transport and stress equations. J Food Eng 105(4):663–670

    Article  Google Scholar 

  109. Takhar PS, Maier DE, Campanella OH, Chen G (2011) Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: validation and simulation results. J Food Eng 106(4):275–282

    Article  Google Scholar 

  110. Takhar PS (2014) Unsaturated fluid transport in swelling poroviscoelastic biopolymers. Chem Eng Sci 109:98–110

    Article  CAS  Google Scholar 

  111. Bansal HS, Takhar PS, Alvarado CZ, Thompson LD (2015) Transport mechanisms and quality changes during frying of chicken nuggets—hybrid mixture theory based modeling and experimental verification. J Food Sci 80(12):E2759–E2773

    Article  CAS  PubMed  Google Scholar 

  112. Sandhu JS, Takhar PS (2018) Verification of hybrid mixture theory based two-scale unsaturated transport processes using controlled frying experiments. Food Bioprod Process 110:26–39

    Article  Google Scholar 

  113. Ozturk OK, Takhar PS (2020) Hybrid mixture theory-based modeling of moisture transport coupled with quality changes in strawberries and carrots. Drying Technol 39(7):932–949

    Article  Google Scholar 

  114. Collman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction. Arch Ration Mech Anal 13:167

    Article  Google Scholar 

  115. Sansano M, De los Reyes, R., Andrés, A., & Heredia, A. (2018) Effect of microwave frying on acrylamide generation, mass transfer, color, and texture in french fries. Food Bioprocess Technol 11(10):1934–1939

    Article  CAS  Google Scholar 

  116. Sansano M, Juan-Borrás M, Escriche I, Andrés A, Heredia A (2015) Effect of pretreatments and air-frying, a novel technology, on acrylamide generation in fried potatoes. J Food Sci 80(5):T1120–T1128

    Article  CAS  PubMed  Google Scholar 

  117. Sandhu J, Bansal H, Takhar PS (2013) Experimental measurement of physical pressure in foods during frying. J Food Eng 115(2):272–277

    Article  Google Scholar 

  118. Parikh A, Takhar PS (2016) Comparison of microwave and conventional frying on quality attributes and fat content of potatoes. J Food Sci 81(11):E2743–E2755

    Article  CAS  PubMed  Google Scholar 

Download references

Funding

The USDA-NIFA provided financial support under award numbers 20–67017-31194, ILLU-698–308, and ILLU-698–926.

Author information

Authors and Affiliations

  1. Department of Food Science and Human Nutrition, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Yash Shah & Pawan Singh Takhar

Authors
  1. Yash Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pawan Singh Takhar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Y. S. wrote most of the manuscript text, reviewed the literature, gathered the figures, organized the contents, and acquired figure permissions. P. S. T. provided overall guidance, perceived the main idea, provided examples of applications to be incorporated, edited the text, and wrote parts of the manuscript.

Corresponding author

Correspondence to Pawan Singh Takhar.

Ethics declarations

Competing Interests

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Corresponding author has been previously published as Pawan P. Singh.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shah, Y., Takhar, P.S. Capillary Pressure in Unsaturated Food Systems: Its Importance and Accounting for It in Mathematical Models. Food Eng Rev 15, 393–419 (2023). https://doi.org/10.1007/s12393-023-09341-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12393-023-09341-7

Keywords

Search

Navigation