Finite Element Modeling of Coupled Heat and Mass Transfer ... Finite Element Modeling of Coupled...

Finite Element Modeling of Coupled Heat and Finite Element Modeling of Coupled Heat and

Mass Transfer of aMass Transfer of a Single Maize Kernel Based Single Maize Kernel Based

on Water Potential Using COMSOLon Water Potential Using COMSOL

MultiphysicsMultiphysics SimulationSimulation

University of West HungaryFaculty of Agricultural, Food and Environmental Sciences

Institute of Biosystems EngineeringAddress: 2 Vár, Mosonmagyaróvár, H-9200 Hungary

Tel.: +36 96 566 635 Fax: +36 96 566 641 E-mail: kovacsaj@mtk.nyme.hu

Authors:

Attila J. Kovács, E. Lakatos,

G. Milics and M. Neményi

Presented at the COMSOL Conference 2010 Paris

IntroductionIntroductionMain research area of our Institute is Main research area of our Institute is grain drying since the 1980’sgrain drying since the 1980’s..

Experimental convective drying tunnelTesting of grain dryers

Thermographic image of the dryer

−+ +=1a bX cTD e

For example, the mass diffusion relationship was

determined for different maize hybrids:

Besides experimental research drying Besides experimental research drying modeling of a single kernel was initiated. modeling of a single kernel was initiated.

where: D = moisture diffusion coefficient;

X = kernel average moisture content (dry basis);

T = average temperature of kernels;

a, b, c = constants.

(These constants for the Pioneer 3780 hybrid were

found: a = -3.1411; b = 3.2159; c = -6696.6)

Based on the measurements mathematical modeling Based on the measurements mathematical modeling

studies of drying were carried out using Finite Element studies of drying were carried out using Finite Element

method:method:

2 2( )

X X XD

∂ ∂ ∂= +

∂ ∂ ∂

2 2( )

T T T Xc k Lt x y t

ρ ρ∂ ∂ ∂ ∂

⋅ = + + ⋅∂ ∂ ∂ ∂

for heat transfer for mass transfer

where:X is moisture content d.b. [kg/kg]; D is diffusion coefficient [m2/s];ρ is density [kg/m3]; c is specific heat [J/kgK];T is temperature [K]; t is time [s];K is thermal conductivity [W/mK]; x,y are directions;L is latent heat of vaporisationof water [J/kg].

FE modelling with COSMOS 1.71

Mass transfer

35 min

50 min

85 min of drying

COMSOL MULTIPHYSICS simulation(FEMLAB 2.2 package)

2 2( )

X X XD

∂ ∂ ∂= +

∂ ∂ ∂

2 2( )

T T T Xc k Lt x y t

ρ ρ∂ ∂ ∂ ∂

⋅ = + + ⋅∂ ∂ ∂ ∂

Mass Transfer:

Heat Transfer:

FE modelling withFEMLAB 2.2

Before

10 min

20 min

30 min of drying

Magnetic resonance imaging (MRI) studies were carried out to evaluate the

accuracy of the FE models.

NMR magnet at BRUKER GMBHMRI image of a cornkernel before drying

FEMLAB model: themoisture content changes inside a single kernel during drying.

- the model is based on the m.c. gradients

MRImeasurement:

the m.c. gradients

FEMLAB model: themoisture content changes inside a single kernelduring drying.

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

MRImeasurement:

the m.c. gradients

Moisture content of the particles vs. Moisture content of the particles vs. wwhole hole maize kernelsmaize kernels (Florencia)(Florencia)

]endosperm

scutellum

0.05 0.10 0.15 0.20 0.25 0.30 0.35

Average moisture content of the whole kernel, X [kg/kg]

40 °C

Considering capillary (water) potentialConsidering capillary (water) potential

can be rewritten:

⋅∂∂

=Ψ∂∂ X

The modified Luikov’s model of mass transport,

where: X is moisture content;τ is time;D is mass diffusion coefficient.

graddiv ⋅=∂

∂⋅

Ψ∂∂+

⋅Ψ∂∂

∂⋅

Ψ∂∂=

∂Ψ∂ zD

where: Ψ is capillary (water) potential;K is hydraulic conductivity (K = D·∂X/∂Y).

and thus the equation gives:

graddiv ⋅=∂

EndospermScutellum

Desorption curves of maize kernel’s particlesDesorption curves of maize kernel’s particles

0.0 0.2 0.4 0.6 0.8 1.0Relative hunidity, ϕϕϕϕ

40 °C

ROTRONIC water activity meter

[ ] ϕ⋅

=Ψ lnV

where:

φ is the relative humidity of

the air (decimal);

RU is the universal gas

constant;50

al, Ψ

Endosperm

Scutellum

Water potential curves of maize particles Water potential curves of maize particles as a function of moisture contentas a function of moisture content

constant;

mv is the molecular weight of

vapour;

T is the absolute temperature;

X is moisture content, d.b.

0 0.1 0.2 0.3 0.4 0.5 0.6

Moisture content, d.b., X [kg/kg]

40 °C

Boundary conditionsBoundary conditions

∞=+∇⋅ ququucn )(

Neumann type:

where:

u is stands for the water potential (Ψ);

n is the normal vector.

Modelled

ΨΨΨΨ changesX changes based on grad Ψ

al, Ψ

Endosperm

Scutellum

0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]

40 °C

Water potential, [kJ/kg] Moisture cotent, [kg/kg]

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

Modelled

al, Ψ

Endosperm

Scutellum

40 °C

CONCLUSIONSCONCLUSIONS

• In the governing equation using water potential asthe driving force has exact physical meaning,instead of the moisture gradient drying force.

• One multiphysical equation system gives more

• One multiphysical equation system gives moreaccurate solutions instead of iteration solutions.

• Further studies needs to be done to determinewater potential above hygroscopic moisturecontent.

THANK YOU FOR YOUR ATTENTION!THANK YOU FOR YOUR ATTENTION!

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