finite element modeling of coupled heat and mass transfer ......finite element modeling of coupled...
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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: [email protected]
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..
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: [email protected]
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.
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: [email protected]
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
2 2( )
X X XD
t x y
∂ ∂ ∂= +
∂ ∂ ∂
2 2
2 2( )
T T T Xc k Lt x y t
ρ ρ∂ ∂ ∂ ∂
⋅ = + + ⋅∂ ∂ ∂ ∂
for heat transfer for mass transfer
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: [email protected]
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
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: [email protected]
5 min
35 min
50 min
85 min of drying
COMSOL MULTIPHYSICS simulation(FEMLAB 2.2 package)
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: [email protected]
2 2
2 2( )
X X XD
t x y
∂ ∂ ∂= +
∂ ∂ ∂
2 2
2 2( )
T T T Xc k Lt x y t
ρ ρ∂ ∂ ∂ ∂
⋅ = + + ⋅∂ ∂ ∂ ∂
Mass Transfer:
Heat Transfer:
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: [email protected]
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.
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: [email protected]
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:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
FEMLAB model: themoisture content changes inside a single kernelduring drying.
- the model is based on the m.c. gradients
MRImeasurement:
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: [email protected]
the m.c. gradients
Moisture content of the particles vs. Moisture content of the particles vs. wwhole hole maize kernelsmaize kernels (Florencia)(Florencia)
0.2
0.3
0.4
0.5
0.6
Mo
istu
re c
on
ten
t o
f th
e c
on
stitu
en
ts,
X
[kg
/kg
]endosperm
scutellum
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: [email protected]
0.0
0.1
0.05 0.10 0.15 0.20 0.25 0.30 0.35
Average moisture content of the whole kernel, X [kg/kg]
Mo
istu
re c
on
ten
t o
f th
e c
on
stitu
en
ts,
40 °C
Considering capillary (water) potentialConsidering capillary (water) potential
can be rewritten:
Ψ∂
⋅∂∂
+
Ψ∂
⋅∂∂
+
Ψ∂
⋅∂∂
=Ψ∂∂ X
DX
DX
DX
The modified Luikov’s model of mass transport,
where: X is moisture content;τ is time;D is mass diffusion coefficient.
XDX
graddiv ⋅=∂
∂
τ
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: [email protected]
∂⋅
Ψ∂∂+
∂
⋅Ψ∂∂
+
∂⋅
Ψ∂∂=
∂Ψ∂ zD
zyD
yxD
xτ
where: Ψ is capillary (water) potential;K is hydraulic conductivity (K = D·∂X/∂Y).
and thus the equation gives:
ψτ
graddiv ⋅=∂
Ψ∂
Ψ∂
∂K
X
0.2
0.3
0.4
0.5
0.6
0.7
Eq
uil
ibri
um
mo
istu
re c
on
ten
t,
d.b
., X
[kg
/kg
]
EndospermScutellum
Desorption curves of maize kernel’s particlesDesorption curves of maize kernel’s particles
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: [email protected]
0.0
0.1
0.2
0.0 0.2 0.4 0.6 0.8 1.0Relative hunidity, ϕϕϕϕ
Eq
uil
ibri
um
mo
istu
re c
on
ten
t,
40 °C
ROTRONIC water activity meter
[ ] ϕ⋅
=Ψ lnV
U
Xm
TR
where:
φ is the relative humidity of
the air (decimal);
RU is the universal gas
constant;50
100
150
200
Wat
er p
ote
nti
al, Ψ
[kJ/
kg]
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
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: [email protected]
constant;
mv is the molecular weight of
vapour;
T is the absolute temperature;
X is moisture content, d.b.
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6
Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Boundary conditionsBoundary conditions
∞=+∇⋅ ququucn )(
Neumann type:
where:
u is stands for the water potential (Ψ);
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: [email protected]
ψ∂
∂=
Xda
u is stands for the water potential (Ψ);
n is the normal vector.
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
Modelled
ΨΨΨΨ changesX changes based on grad Ψ
100
150
200
Wat
er p
ote
nti
al, Ψ
[k
J/k
g]
Endosperm
Scutellum
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: [email protected]
0
50
0 0.1 0.2 0.3 0.4 0.5 0.6Moisture content, d.b., X [kg/kg]
Wat
er p
ote
nti
al,
40 °C
Water potential, [kJ/kg] Moisture cotent, [kg/kg]
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
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: [email protected]
• 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!
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: [email protected]