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Convective drying analysis of three-dimensional porous solid by mass lumping finite element technique

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Abstract

A numerical analysis of convective drying of a 3D porous solid of brick material is carried out using the finite element method and mass lumping technique. The energy equation and moisture transport equations for the porous solid are derived based on continuum approach following Whitaker’s theory of drying. The governing equations are solved using the Galerkin’s weighted residual method, which convert the governing equations into discretized form of matrix equations. The resulting capacitance matrices are made diagonal matrices by following the classical row-sum mass lumping technique. Hence with the use of the Eulerian time marching scheme, the final equations are reduced to simple algebraic equations, which can be solved directly without using an equation solver. The proposed numerical scheme is initially validated with experimental results for 1D drying problem and then tested by application to convective drying of 3D porous solid of brick material for four different aspect ratios obtained by varying the cross section of the solid. The mass lumping technique could correctly predict the wet bulb temperature of the solid under evaporative drying conditions. A parametric study carried out for three different values of convective heat transfer coefficients, 15, 30 and 45 W/m2 K shows an increased drying rate with increase in area of cross section and convective heat transfer coefficient. The proposed numerical scheme could correctly predict the drying behavior shown in the form of temperature and moisture evolutions.

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Abbreviations

A :

side (m)

B :

width (m)

C :

concentration (kg/m3), capacitance matrix

C p :

specific heat capacity (J/kg K)

D :

moisture diffusion coefficient (m2/s)

h c :

convective heat transfer coefficient (W/m2 K)

h m :

convective mass transfer coefficient (m/s)

H :

enthalpy (J/kg)

J :

mass flux (kg/m2 s)

k :

thermal conductivity (W/m K)

K :

permeability (m2)

L :

length (m)

\({\dot{m}}\) :

rate of evaporation/condensation (kg/m2 s)

M :

mass of solid (kg)

N :

shape function

p :

pressure (N/m2)

t :

time (s)

T :

temperature (K)

V :

volume (m3)

w :

moisture content (kg/kg of dry solid)

v :

species velocity (m/s)

x, y, z :

Cartesian coordinates

μ:

dynamic viscosity (N s/m2)

ϕ:

relative humidity

ρ:

density (kg/m3)

ψ:

tortuosity factor

c:

capillary

dv:

vapor diffusion

ini:

initial

l:

liquid

v:

vapor

ml:

isothermal liquid moisture

mv:

isothermal vapor moisture

Tl:

non-isothermal liquid moisture

Tv:

non-isothermal vapor moisture

w :

moisture

0:

dry solid

∞:

ambient value

References

  1. Luikov AV (1966) Heat and mass transfer in capillary porous bodies. Pergamon, New York

    MATH  Google Scholar 

  2. Whitaker S (1977) Simultaneous heat, mass and momentum transfer—a theory of drying. Adv Heat Transf 13:119–203

    Google Scholar 

  3. Kallel F, Galanis N, Perrin B, Javelas R (1993) Effects of moisture on temperature during drying of consolidated porous materials. J Heat Transf ASME Trans 115:724–733

    Article  Google Scholar 

  4. Ben Nasrallah S, Perre P (1988) Detailed study of a model of heat and mass transfer during convective drying of porous media. Int J Heat Mass Transf 31:957–967

    Article  MATH  Google Scholar 

  5. Huang CLD (1979) Multi-phase moisture transfer in porous media subjected to temperature gradient. Int J Heat Mass Transf 22:1295–1307

    Article  MATH  Google Scholar 

  6. Hussain MM, Dincer I (2003) Numerical simulation of two-dimensional heat and moisture transfer during drying of a rectangular object. Num Heat Transf Part A 43:867–878

    Article  Google Scholar 

  7. Pel L, Landman KA, Kaasschieter EF (2002) Analytic solution for the non-linear drying problem. Int J Heat Mass Transf 45:3173–3180

    Article  MATH  Google Scholar 

  8. Murugesan K, Seetharamu KN, Aswatha Narayana PA (1996) A one-dimensional analysis of convective drying of porous materials. Heat Mass Transf 32:81–88

    Article  Google Scholar 

  9. Lewis RW, Malan AG (2005) Continuum thermodynamic modeling of drying capillary particulate materials via an edge-based algorithm. Comput Methods Appl Mech Eng 194:2043–2057

    Article  MATH  Google Scholar 

  10. Comini G, Lewis RW (1976) A numerical solution of two-dimensional problems involving heat and mass transfer. Int J Heat Mass Transf 19:1387–1392

    Article  MATH  Google Scholar 

  11. Murugesan K, Suresh HN, Seetharamu KN, Aswatha Narayana PA, Sundararajan TS (2001) A theoretical model of brick drying as a conjugate problem. Int J Heat Mass Transf 44:4075–4086

    Article  MATH  Google Scholar 

  12. Murugesan K, Seetharamu KN, Aswatha Narayana PA, Thomas HR, Ferguson WJ (2000) Study of shrinkage stresses for drying of brick as a conjugate problem. Int J Numer Methods Eng 48:37–53

    Article  MATH  Google Scholar 

  13. Le Bray Y, Prat M (1999) Three-dimensional pore network simulation of drying in capillary porous media. Int J Heat Mass Transf 42:4207–4224

    Article  MATH  Google Scholar 

  14. Perre P, Turner IW (1999) A 3-D version of trans pore: a comprehensive heat and mass transfer computational model for simulating the drying of porous media. Int J Heat Mass Transf 42:4501–4521

    Article  MATH  Google Scholar 

  15. Narasimhan TR (1978) A perspective on numerical analysis of the diffusion equation. Adv Water Res 1:147–155

    Article  Google Scholar 

  16. Chin-Joe-Kong MJS, Mulder WA, Van Veldhuizen M (1999) Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. J Eng Maths 35:405–426

    Article  MATH  MathSciNet  Google Scholar 

  17. Cohen G, Joly P, Roberts JE, Tordjman N (2001) Higher order triangular finite elements with mass lumping for the wave equation. SIAM J Numer Anal 38:2047–2078

    Article  MATH  MathSciNet  Google Scholar 

  18. Philip J, DeVries DA (1957) Moisture movement in porous materials under temperature gradients. Trans Am Geophys Union 18:222–232

    Google Scholar 

  19. Dinulescu HA, Eckert ERG (1980) Analysis of the one dimensional moisture migration caused by temperature gradients in porous medium. Int J Heat Mass Transf 23:1069–1077

    Article  MATH  Google Scholar 

  20. De Vries DA (1958) Simultaneous transfer of heat and moisture in porous materials. Trans Am Geophys Union 39:909–916

    Google Scholar 

  21. Murugesan K (1998) Finite element analysis of convective drying of porous materials. PhD thesis, Department of Mechanical Engg., Indian Institute of Technology Madras, Chennai, India

  22. Malan AG, Lewis RW (2003) Modelling coupled heat and mass transfer in drying non-hygroscopic capillary particulate materials. Commun Numer Methods Eng 19:669–677

    Article  MATH  Google Scholar 

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Acknowledgments

The National Science Council of Taiwan is gratefully acknowledged for providing financial support under the grant no. NSC 94-2211-E-002-071 to carry out the present research and greatly appreciated.

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, IIT Roorkee, Roorkee, India

    K. Murugesan

  2. Department of Civil Engineering and Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan

    D. C. Lo, D. L. Young, C. W. Chen & C. M. Fan

Authors
  1. K. Murugesan
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  2. D. C. Lo
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  3. D. L. Young
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  4. C. W. Chen
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  5. C. M. Fan
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Correspondence to D. L. Young.

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Murugesan, K., Lo, D.C., Young, D.L. et al. Convective drying analysis of three-dimensional porous solid by mass lumping finite element technique. Heat Mass Transfer 44, 401–412 (2008). https://doi.org/10.1007/s00231-007-0260-9

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  • DOI: https://doi.org/10.1007/s00231-007-0260-9

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