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Transport in Porous Media

, Volume 130, Issue 3, pp 923–946 | Cite as

Luikov’s Analytical Solution with Complex Eigenvalues in Intensive Drying

  • Puvikkarasan Jayapragasam
  • Pasacal Le Bideau
  • Tahar LoulouEmail author
Article
  • 34 Downloads

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.

Keywords

Drying Luikov’s equation Analytical solution Porous material 

List of symbols

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

Abbreviations

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

Greek Letters

\(\alpha \)

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

Superscripts

a

Analytical

i

Initial

n

Numerical

*

Equilibrium

\(+\)

Non-dimensional

Subscripts

m

Mass

o

Initial

p

Pressure

q

Thermal

r

Reference

Notes

Acknowledgements

The authors are grateful for the research funding provided by the Brittany Region (Région de Bretagne).

Supplementary material

References

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Centre de Recherche, IRDLUMR CNRS 6027, Université de Bretange SudLorientFrance

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