Abstract
Luikov’s equations of heat and mass transfer with pressure gradient have significant applications especially in the case of intensive drying in porous media. The established analytical solution of Luikov’s equations including pressure gradient effect in a complete form is presented in this work. The existence of complex roots in the analytical solution describing intensive drying with pressure gradient was overlooked in literature. A Matlab function capable of searching and sorting out the complex eigenvalues is also showcased. Three test cases are analyzed and compared with numerical solutions: one theoretical case to emphasize the importance of complex roots in analytical solution while two cases of drying process in ceramic and barley kernel to ensure practical applicability. Excellent matching between analytical and numerical results is noticed when complex eigenvalues are included.
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Abbreviations
- a :
-
Diffusivity (m\(^2\)/s)
- \(A_{nj}\) :
-
Coefficient of analytical solution
- \(B_{nj}\) :
-
Coefficient of analytical solution
- \(b_{nj}\) :
-
Coefficient of analytical solution
- Bi:
-
\(\dfrac{h_{m,q}L}{k_{m,q}}\) Biot number
- Bu:
-
\(\delta _p\dfrac{\Delta P}{\Delta t}\) Bulygin number
- C :
-
Specific heat capacity (J/kgK)
- \(C_m\) :
-
Specific mass transfer capacity (kg/kg M)
- \(C_p\) :
-
Capillarity capacity (kgm\(^2\)/kgN)
- \(C_{nj}\) :
-
Coefficient of analytical solution
- \(D_{nj}\) :
-
Coefficient of analytical solution for average variable
- h :
-
Convective heat transfer coefficient (W/mK)
- \(h_m\) :
-
Convective mass transfer coefficient (kg/m\(^2\)s M)
- k :
-
Thermal conductivity (W/mK)
- \(k_m\) :
-
Moisture conductivity (kg/ms M)
- \(k_p\) :
-
Capillarity effect (kgm/sN)
- Ko:
-
\(\dfrac{\lambda }{c}\dfrac{X_r}{T_r}\) Kossovitch number
- L :
-
Overall length (m)
- Lu:
-
\(\dfrac{a_m}{a_q}\) Luikov number
- Lu\(_p\) :
-
\(\dfrac{a_p}{a_q}\) Luikov pressure number
- \(n_t\) :
-
Number of roots
- P :
-
Pressure (Pa)
- \(P_{nj}\) :
-
Coefficient of transcendental equation
- Pn:
-
\(\delta \dfrac{T_r}{X_r}\) Possnov number
- \(Q_{nj}\) :
-
Coefficient of transcendental equation
- r :
-
Root mean square
- T :
-
Temperature (K)
- t :
-
Time (s)
- X :
-
Moisture content
- FDM:
-
Finite difference method
- FEM:
-
Finite element method
- GITT:
-
Generalized integral transform technique
- TM model:
-
Model concerning temperature and moisture content
- TMP model:
-
Model concerning temperature, moisture content and pressure
- \(\alpha \) :
-
Parameter
- \(\beta \) :
-
Parameter
- \(\gamma \) :
-
Parameter
- \(\Delta \) :
-
Criteria for nature of cubic equation
- \(\delta \) :
-
Thermogradient
- \(\epsilon \) :
-
Phase conversion factor
- \(\lambda \) :
-
Latent heat of vaporization or roots of cubic polynomial (with subscript index j)
- \(\mu _n\) :
-
Coefficient of analytical solutions—eigenvalues of transcendental equation
- \(\nu _j\) :
-
Coefficient of analytical solution—cubic roots
- \(\Omega \) :
-
Criteria for nature of transcendental equation
- \(\omega \) :
-
Elements of \(\Omega \)
- \(\chi _{nj}\) :
-
Coefficient of transcendental equation
- \(\pi _{1,2}\) :
-
Coefficients of polynomial
- \(\psi _n\) :
-
Coefficient of analytical solution
- \(\rho \) :
-
Density (kg/m\(^3\))
- \(\sigma \) :
-
Coefficient of analytical solution
- \(\overline{\theta }\) :
-
Average variable
- a:
-
Analytical
- i:
-
Initial
- n:
-
Numerical
- *:
-
Equilibrium
- \(+\) :
-
Non-dimensional
- m:
-
Mass
- o:
-
Initial
- p:
-
Pressure
- q:
-
Thermal
- r:
-
Reference
References
Abahri, K., Trabelsi, A., Belarbi, R.: Contribution to analytical and numerical study of combined heat and moisture transfer in porous building materials. Build. Environ. 46(7), 1354–1360 (2011)
Chang, W.-J., Weng, C.-I.: An analytical solution to coupled and moisture diffusion transfer in porous materials. Int. J. Heat Mass Transf. 43(19), 3621–3632 (2000)
Comsol Mutiphysics ® v. 5.4. Comsol AB, Stockholm, Sweden. www.comsol.com (2018)
Conceição, R.S.G., Macêdo, E.N., Pereira, L.B.D., Quaresma, J.N.N.: Hybrid integral transform solution for the analysis of drying in spherical capillary-porous solids based on luikov equations with pressure gradient. Int. J. Therm. Sci. 71, 216–236 (2013)
Cotta, R.M.: Integral Transforms in Computational Heat and Fluid Flow. CRC Press, Boca Raton (1993)
Cotta, R.M., Mikhailov, M.D.: Heat Conduction: Lumped Analysis, Integral Transforms. Symbolyc Computation. Wiley, New York (1997)
Dantas, L.B., Orlande, H.R.B., Cotta, R.M.: Estimation of dimensionless parameters of luikov’s system for heat and mass transfer in capillary porous media. Int. J. Therm. Sci. 41, 217–227 (2002)
Dantas, L.B., Orlande, H.R.B., Cotta, R.M.: An inverse problem of parameter estimation for heat and mass transfer in capillary porous media. Int. J. Heat Mass Transf. 46, 1587–1598 (2003)
Dantas, L.B., Orlande, H.R.B., Cotta, R.M.: Improved lumped-differential formulation and hybrid solution methods for drying in porous media. Int. J. Therm. Sci. 46, 878–889 (2007)
Lobo, P.D., Mikhailov, M.D., Özisik, M.N.: On the complex eigenvalues of luikov system of equations. Drying Technol. 5(2), 273–286 (1987)
Luikov, A.V.: Heat and Mass Transfer in Capillary Porous Bodies. Pergamon Press, Oxford (1966)
Luikov, A.V.: Analytical Heat Diffusion Transfer. Academic Press, New York (1968)
Luikov, A.V.: Systems of differential equations of heat and mass transfer in capillary porous bodies (review). Int. J. Heat Mass Transf. 18, 1–14 (1975)
Luikov, A.V., Mikhailov, Y.A.: Theory of Energy and Mass Transfer. Prentice-Hall, Upper Saddle River (1961)
Luikov, A.V., Mikhailov, Y.A.: Theory of Heat and Mass Transfer. Israel Program for Scientific Translations, Jerusalem (1965)
Mikhailov, Y.A.: Higly intensive heat and mass transfer in dispersed media. Int. J. Heat Mass Transf. 1, 37–45 (1960)
Mikhailov, M.D.: General solutions of the coupled diffusion equations. Int. J. Eng. Sci. 11, 235–241 (1973)
Mikhailov, M.D.: General solutions of the diffusion equations coupled at boundary conditions. Int. J. Heat Mass Transf. 16, 2155–2164 (1973)
Mikhailov, M.D., Özisik, M.N.: Unified Analysis and Solution of Heat and Mass Diffusion. Wiley, New York (1984)
Mikhailov, M.D., Shishedjiev, B.K.: Temperature and moisture distributions during contact drying of a moist porous sheet. Int. J. Heat Mass Transf. 18, 15–24 (1975)
Naveira-Cotta, C.P., Cotta, R.M., Orlande, H.R.B., Fudyme, O.: Eigenfunction expansions for transient diffusion in heterogeneous media. Int. J. Heat Mass Transf. 52, 5029–5039 (2009)
Özisik, M.N.: Finite Difference Methods in Heat Transfer. CRC Press, Boca Raton (1994)
Pandey, R.N., Pandey, S.K.: Complete and satisfactory solutions of luikov equations of heat and mass transport in spherical capillary porous body. Int. Commun. Heat Mass Transf. 27(7), 975–984 (2000)
Pandey, R.N., Srivastava, S.K., Mikhailov, M.D.: Solution of luikov equations of heat and mass transfer in capillary porous bodies through matrix calculus: a new approach. Int. J. Heat Mass Transf. 42, 2649–2660 (1999)
Rossen, J.L., Hayakawa, K.: Simultaneous heat and moisture transfer in dehydrated food: a review of theoretical models. Symp. Ser. AIChE 163, 71–81 (1977)
Saker, L.F., Orlande, H.R.B.: Simultaneous estimation of spatially-dependent mass and heat transfer coefficients of drying bodies. Inverse Probl. Sci. Eng. 12, 549–561 (2004)
Smirnov, M.S.: On a system of differential equations for higly intensive heat and mass transfer. Int. J. Heat Mass Transf. 5, 521–524 (1962)
Smith, G.D.: Numerical Solution of Partial Differential Equations. Finite Difference Methods. Clarendon Press, Oxford (1985)
Tripathi, G., Shukla, K.N., Pandey, R.N.: Intensive drying of an infinite plate. Int. J. Heat Mass Transf. 20, 451–458 (1977)
Yun, W.: Effect of Pressure on Heat and Mass Transfer in starch based food systems. Ph.D. thesis, University of Saskatchewan (1997)
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The authors are grateful for the research funding provided by the Brittany Region (Région de Bretagne).
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A Matlab Function for Searching Roots
A Matlab Function for Searching Roots
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Jayapragasam, P., Le Bideau, P. & Loulou, T. Luikov’s Analytical Solution with Complex Eigenvalues in Intensive Drying. Transp Porous Med 130, 923–946 (2019). https://doi.org/10.1007/s11242-019-01348-1
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DOI: https://doi.org/10.1007/s11242-019-01348-1