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Luikov’s Analytical Solution with Complex Eigenvalues in Intensive Drying

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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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Notes

  1. For more details on the TM model and its solution, the reader is encouraged to see the references Mikhailov and Özisik (1984) and Pandey et al. (1999).

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

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Acknowledgements

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

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Correspondence to Tahar Loulou.

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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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