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8/7/2019 pp 533-541 HEAT AND MASS TRANSFER IN A SHRINKING CYLINDER DRYING
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Drying 2004 – Proceedings of the 14th International Drying Symposium (IDS 2004)
São Paulo, Brazil, 22-25 August 2004, vol. A, pp. 533-541
533
MATHEMATICAL MODELLING AND SIMULATION OF HEAT AND MASS
TRANSFER PHENOMENA IN A SHRINKING CYLINDER DURING DRYING
Bilel Hadrich and Nabil Kechaou
Unité de Recherche en Mécanique des Fluides Appliquée et Modélisation
Ecole Nationale d’Ingénieurs de Sfax, BP ‘W’ 3038, Sfax, Tunisie,e-mails : bilelhadrich@yahoo.fr & nabil.kechaou@enis.rnu.tn
Keywords: convective drying, mass and heat transfer, mathematical modeling, numerical simulation,
diffusion coefficient, banana
ABSTRACT
This work represents a contribution to the understanding of the mechanisms of heat and
mass transfer in a cylindrical shrinking material within the drying process. With
reference to a set of simplifying hypotheses (uncoupling of heat and material transfer
phenomena and unidirectional transfer), a simplified local model was adopted to study
the drying process of a cylindrical sample of banana. Based on certain experimentalresults of drying, this model was proved valid and the effective diffusion coefficient of
water in banana was identified. This mathematical model also provides the possibility of
establishing the distributions of water in the sample during the drying process.
INTRODUCTION
Modeling is essentially a mathematical way of representing processes or phenomena to explain the
observed data and to predict behaviour under different conditions.
A considerable amount of works have been carried on dying of cylindrical shrinking material. Drying of
biological materials, for example, is a complicated process involving simultaneous coupled heat and
mass transfer phenomena and the change of the volume and surface area arise simultaneously with the
loss of moisture, those occur inside the material being dried. Mathematical models are very useful in thedesign and analysis of simultaneous heat and moisture transfer processes. Among the mathematical
models the diffusionnal ones seem to be most popular, as they are easy to formulate and normally
provide reasonable results. The validating of drying models through experimentation is also an important
step.
In the present work, a drying model that takes into account uncoupling of heat and mass transfer
phenomena and the unidirectional shrinkage were used to predict the temperature and moisture profiles
inside the banana and the moisture diffusion coefficient of water in banana was identified. This
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mathematical model also provides the possibility of establishing the distribution of the amount of water
in the sample during the drying process. The effect of temperature on the moisture diffusion coefficient
in banana drying was investigated. The predicted temperature and mean moisture content of material
versus times were confirmed by experimental results.
MATHEMATICAL MODELLING
In a macroscopic description, the medium is considered to be the superposition of two continuousinteractions media: a solid matrix and a liquid "diffusing" within this matrix.
The following assumptions are made in this work:
o the initial distribution of moisture content and temperature are uniform,
o the water vaporization takes place only at the external surface level,
o the shrinkage is not negligible,
o uncoupling of heat and mass transfer phenomena,
o physical and thermal proprieties are assumed to be functions of local moisture content and
temperature,
o unidirectional transfer.
Mass Balance Equations
Conservation of the entire mass of the medium during a transformation is shown in the following
mass balance equations:
− Liquid phase : ( ) 0vdivt
eee =⋅ρ+
∂
ρ∂
(1)
− Solid phase : ( ) 0vdivt
sss =⋅ρ+
∂
ρ∂
(2)
In which eρ and sρ are respectively the apparent bulk mass densities of the liquid component and the
solid component. The density of material is considered as es ρ+ρ=ρ .
ev
is the velocity of the diffusion of the liquid and sv
is the velocity of the shrinkage of the solid.
If we consider that mass transfer follows the Fick’s law, we can write:
( ) ω∇⋅ω−
⋅ρ−=−⋅ρ
1
Dvv see (3)
Taking in consideration the Lagrangian co-ordinates system (noted by iξ ) is attached to the solid matrix,
we can combine equations (1) and (2) to obtain the following balance equation:
( )XDdivdt
dXss ∇⋅⋅ρ=
⋅ρ
ξ
(4)
in which variable X is defined as seX ρρ= and dtd is the time derivative following the movement of
the solid*.
Taking in consideration that ( )1Xes +ρ=ρ−ρ=ρ , equation (4) becomes:
∇
+
⋅ρ=
⋅ρ
ξ
XX1
Ddiv
dt
dXs
(5)
To simplify the problem, we can suppose that the axial shrinkage is neglected. Therefore, changing co-
ordinates system(*)
imposes to know the deformation ratio0ss ρρ . For the ideal deformation case
where shrinkage is linear versus moisture content (Kechaou, Maalej, 1994), this ratio writes:
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( )X4855.110s
s ⋅+=ρ
ρ(6)
Finally, the mass conservation with Fickian diffusion in a banana becomes:
( )
tteff
XT,XD
r
1
dt
dX
ξ∂
∂⋅⋅ξ
∂
∂⋅
ξ= (7)
with ( )( )
( )
2
eff r
²X4855.11
T,XDT,XD
ξ⋅
⋅+= (8)
Heat Balance Equation
Following the hypothesis that the temperature in the material (banana) at any time during the drying
may be considered uniform. The heat balance can be represented by the following equation:
( ) surf ,mvac ALTThAdt
dTCpm φ⋅⋅−−⋅⋅=⋅⋅ (9)
within ( )vasurf ,va
emsurf ,m PP
TR
Mk −
⋅
⋅=φ (10)
Initial and Boundary Conditions
For time zero, both temperature and water content are uniform:
w0 TTT == , 0XX = ξ∀ (11)
All gradients are nil in the center:
,0t >∀ ,0=ξ 0X
0
=
ξ∂
∂−
=ξ
(12)
At the surface ( )dr=ξ , mass flux density is given by Fick’s law:
,0t >∀ ,rd=ξ ( )dr
eff 0ssurf ,m
X
rT,XD
=ξ
ξ∂∂⋅ξ⋅⋅ρ−=φ (13)
The Solution of the Equations
As drying is a simultaneous heat and mass transfer process, correlation of moisture and temperature
changes in the product involves solution of coupled differential equations. If the parameters involved in
equations (7), (8), (9) and (10), such as thermal and physical properties, are functions of moisture content
or temperature, the equation (7) are not linear and must be solved through numerical methods. An
implicit finite difference method was used to solve this type of problem.
If we consider that the thickness of sample was divided into N finite difference points or nodes, the
system of differential equations to solve is a system of the first order of (N+1) dimensions (N from the
mass transfer equation and one from the heat transfer equation) with given boundary conditions. Using
the numerical method of resolution of differential equations of Runge-Kutta, to the 4th
order, with
variable step.
The computer program used to solve the above differential was written in MATLAB 6.5. At each time
step, the value of X was calculated at each node and the global value of T was obtained.
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The inputs to the mathematical models are: physical properties of drying air, initial sample dimension
and material shrinkage, density, thermal conductivity and specific heat as functions of moisture content
and/or temperature.
MATERIALS AND EXPERIMENTAL METHODS
Fresh bananas (Cavendish variety) purchased from a local market were used in the experiments. The
fruits were selected according to the required perfect cylindrical form, degree of ripeness and peeledmanually. In deed the longest cylindrical part in the fruit would be selected for drying with hot air under
forced convection.
The approximate weight of the individual sample was 30g with an overall length of about 50 mm. The
average approximate initial value of the banana moisture content was 3 kg/kg d.b.
The experimental drying system, presented on Figure 1, consisted of a pilot air dryer, composed of a
system for the provision of hot air, a humidifier and a drying test chamber.
Figure 1 : Experimental drying apparatus
This experimental dryer was designed to
perform hot air drying under the following
conditions: air temperature ranging from 25°C
to 80°C, air velocity between 0.3 to 5m/s andmoisture between the ambient and 90% relative
humidity.
A centrifugal blower, powered by a variable
speed motor used to control air velocity, forced
the air through two electrical heating elements.
If necessary, air was then moistened by steam
injection and heated again before flowing in the
test chamber. The maximum heating capacity
was 3 kW. Air flow rate was measured using a
hot wire anemometer with a precision of ± 0.01
m/s.
The test chamber consists of a rectangular tunnel (0.254 x 0.254 x 1m ). Hot air entered this tunnel and
flowed over the sample to be dried. In the drying chamber, a single banana was laid on a perforated tray
placed over an analytical balance (METTLER) with a precision of ± 0.01g. Periodic on-line weighing
was performed throughout the drying experiments at constant time period (10 min). The temperature of
the drying air was controlled using a thermo-hygrometer connected to a personnel computer.
At the end of each drying experiment, dry matter of the dried banana was determined by leaving it for 24
hours in an oven at 105° ± 2°C. EXPERIMENTAL RESULTS
A total of 6 drying experiments were run at different air conditions. Table 1 shows the air and material
conditions used in this work.
Drying air temperature was varied from 50 to 70°C, relative humidity from 3.6 to 11.5% and air velocity
from 3 to 4.5 m/s.
The plots of Figure 3 show the progress usual of the drying without constant rate periods that is non-
existent in the case of the biologic products. This particularity, noticed by several authors (Bimbenet,
1984 and Kechaou, 2001), is usual for the agro-alimentary products, that especially dry on the falling
rate period.
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The results indicated that diffusion is the most likely physical mechanism governing moisture movement
in banana.
Table 1. Air Drying Conditions and Parameters of Banana Used in the Experimental Tests
Test N° 1 2 3 4 5 6
Ta (°C) 51 51 51 59 67 69
Tw (°C) 23 24 25 26 27 29
(%) 7.6 9.5 11.4 6.4 3.6 5.7
(g/kg dry air) 6.120 7.636 9.219 7.595 6.052 9.690
va (m/s) 3 4 4.5 3 3 3
X0(kg/kg d.b.) 3.10 3.26 3.25 3.22 3.04 3.06
Xeq(kg/kg d.b.) 0.196 0.199 0.210 0.178 0.138 0.151
Initial radius of
banana(mm) 13.75 14.75 15 15 14.7 14
Initial length of
banana (mm) 49 49 50 50 50 49
On Figure 2, we present the dimensionless moisture ratio eq0
eq
XX
XX
−
−
at each time and, on Figure 3, the
drying rate as function of moisture content, for all the tests.
Figure 2 - Drying curves of banana at different air conditions Figure 3 - Drying kinetics of a single banana
Plate Profile of Temperature
To measure the temperature profile in the sample, three copper-constantan thermocouples with
diameter of 2 mm were used and placed in the bottom, the upper and in the axis of the sample. Figure 4represents the temperature profiles in the sample at different positions (thermocouples located at the
surface, the axis and at the upper of material) at a drying air temperature of 60°C. It was found that there
were no significant differences between the temperatures at the three points. This may indicate that the
temperature in banana at any time during the drying is considered uniform.
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 0,5 1 1,5 2 2,5 3 3,5
X (kg/kg d.b.)
(-dX/dt) (kg/kg d.b. h)
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
0
0,2
0,4
0,6
0,8
1
1,2
0 100 200 300 400 500 600
t (min)
Moisture ratio (-)
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
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Figure 4 - Experimental temperature profiles in the sample at different positions at dying air temperature of 60°C
IDENTIFICATION OF MOISTURE EFFECTIVE DIFFUSION COEFFICIENT
The moisture effective diffusion coefficients were determined by comparing the calculated drying
curve with the observed one, using a trial and error method. The banana shrinkage was considered when
calculating the drying curve. First we start with an initial D0 founded from literature (Kechaou, 2001):
( )( )
⋅
⋅γ −⋅β+α−⋅
⋅+
⋅+−⋅=
TR
Xexpexp
Xdc
XbaexpDT,XD 000
(14)
within ,s/ ²m1095.6D6
00−⋅= ,1cba === ,5.12d = ,mol/ J1022 3⋅=α mol/ J1017 3⋅=β and .2=γ
Using the theoretical model developed previously to calculate the changes of average moisture content,
we can compare the calculated drying curve with the experimental one. If the sum of squared derivation
between the experimental and theoretical drying curve was not minimum, another D0 with new
parameters was used to repeat the procedures, until the minimum derivation was obtained. Different air
temperature, absolute humilities and velocities, were used in the experiments to determine the diffusion
coefficient of cylinder of banana. The moisture diffusion coefficient parameters identified through themodel are shown in equation (15). The computation was performed with experimental data of average
moisture content.
( )( )
⋅
⋅−⋅⋅+⋅−⋅
⋅+
⋅+−⋅⋅= −
TR
X5.1exp105.171022exp
X141
X221exp1095.6T,XD
336 (15)
The influence of the temperature of material on the diffusion coefficient was assumed to be Arrenius -
type relationship.
Equation (15) has been used in the moisture diffusion model to calculate the moisture and the
temperature profiles within the banana sample.
VALIDATION OF THE MODEL
The model validation was carried out through a comparison of predicted variable values with those
obtained experimentally. In fact Figure 5 represents the comparison between the experiment results, the
simulated values using the moisture diffusion coefficient extracted to the literature and the simulated
values for test 1.
Ta = 60°C, RH = 22%, va = 1.6 m/s
0
10
20
30
40
50
60
70
0 100 200 300 400 500 600 700 800t (min)
Tempe
rature (°C)
Taxis
Tair
Tsurf Tbottom
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Figure 5 - Predicted and experimental means moisture content and temperature during the drying of banana for test 1
The slight misfit observed can be attributed to a large extent to the sum of errors in the experiment
conditions and to the simplify hypothesis. Nevertheless, we can consider that our mathematical modeling
was largely satisfactory for an overall experimental measurement; especially if we compare the results
obtained with this predicted coefficient to results obtained with others, we can notice that this new
coefficient were the more meadows of experimental results. The computing is sufficiently accurate toprovide a perspective tool to analyze the different physical phenomena governing the drying process.
Field of Moisture Content
Figure 6 shows the moisture distribution predicted, by using the equations, in the sample as function
of effective radius and drying time during drying in air at 51°C. The shrinkage effect was taken into
account during the calculation of moisture contents in the sample. A uniform moisture profile was
assumed at the beginning of drying, which were characteristic of the molecular diffusion (Fick’s law).
Moisture content evolution shows a falling parabolic period during the drying and after a certain time it
was ending at the equilibrium moisture content within the slab.
Figure 6 - The predicted moisture distribution in the sample as
functions of both position and drying time at 51°C with taken
into account the shrinkage and the evolution the radius’s sample.
Every curve of Figure 6 refers to the moisturedistribution from the bottom to the surface of
the sample after the time drying indicated on
the curve. It can be seen that the surface
moisture content during the drying decreases
sharply with the increase in the drying time.
The moisture content gradients from surface to
bottom of the sample decrease with increase in
the drying time while the moisture content
near the bottom of the sample decreases
steadily with drying time.
Experimental measurement of moisture
profiles in banana is difficult and has limitedthe ability to validate our dying model.
0 100 200 300 400 500 6000.5
1
1.5
2
2.5
3
3.5
t (min)
M
oisture co
ntent (kg water/kg d.b.) Experimental
Simulated with D0
Simulated with the new D
0 100 200 300 400 500 60020
25
30
35
40
45
50
t (min)
Tem
perature (°C)
Experimental
Simulated with D0
Simulated with the new D
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CONCLUSIONS
A simplified mathematical model presented in cylindrical coordinates allows describing the heat and
mass transports during drying of deformable media considered as an immiscible two-phase system. A
finite difference method was used to solve the equations, applied to the banana drying process.
The results of this model were compared to experimental ones during the drying of a sample cylindrical
banana.
This model showed that there is not a concordance between the simulated and the experimental resultswith an effective diffusion coefficient of water in the banana pulled from previous works.
To validate the model better, another effective diffusion coefficient has been identified while adjusting
the theoretical results with the experimental results while using a trial and error method.
With this new effective diffusion coefficient, the mathematical model enables the prediction of
temperature and mean moisture content along the drying process, when the shrinkage of material is
considered.
The moisture distribution inside the cylindrical banana sample can be predicted at different drying times.
NOTATION
A area surface of material m²
absolute humidity kg/kg dry airCp specific heat of material J/kg K
D diffusion coefficient m²/sd.b. dry basis -hc convective heat transfer coefficient W/m² K
k m mass transfer coefficient m/sLv specific latent heat of vaporisation J/kg
m mass of humid sample kgMe molar mass of water vapour g/molP pressure Pa
r radial distance m
R universal gas constant (8,31451) J/ mol K relative humidity %
t drying time sT temperature of sample K
v velocity m/s
X water content of sample kg water/kg d.b.
Greek Symbols
φm mass-transfer flux mol/m² s
ρ density kg/m3
ω mass fraction of water %
ξ solid coordinate in Lagrangian space -
∇
grad -
Subscripts Superscripts
0 initial 0 initial state
a air
d dry materiale liquid phase
eff effective
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eq equilibriums solid phase
surf at surface
v vapour phasew wet-bulb
LITERATURE
Anderson, R. B. (1946), Modification of the Brunauer Emmett and Teller equation, Journal of American
Chemical Society, 68, pp. 686-691
Aregba, A. W. (1989), Séchage d’un gel et d’une pâte-processus internes et procédé adaptés, Thèse de
doctorat de l’Université de Bordeaux I, France
Bimbenet, J. J. (1984), Le séchage dans les industries agricoles et alimentaires, Cahier du Génie
Industriel Alimentaire (GIA), SEIPAC, PARIS
Bizot, H. (1983), Using the GAB model to construct sorption isotherms, Physical Properties of foods,
R. Jowitt et al., pp. 43-53
Guggenheim, E. A. (1966), Applications of Statistical Mechanics, Clarendon Press, 2nd
Ed., Oxford,
U.K., pp. 186-206
Gunes, M. (2000), Correlations for some thermo-physical properties of air. Proceedings of the 12th
International Drying Symposium IDS’2000, paper N°215, The Netherlands
Jomaa, W. (1991), Séchage de matériaux fortement déformables : prise en compte de la vitesse de retrait,
Thèse de doctorat de l’Université de Bordeaux I, France
Kechaou, N. (2001), Etude théorique et expérimentale du processus de séchage des produits
agroalimentaires, Thèse de doctorat de la Faculté des Sciences de Tunis, Tunisie
Kechaou, N.; Maâlej, M. (1994), Evaluation of diffusion coefficient in the case of banana drying, Drying
Technology vol. A, Mujumdar A. S. (Ed.), pp. 841-848
Kechaou, N., Maâlej, M. (1999), Desorption isotherms of imported Banana-Application of the GAB
theory. Drying Technology, Vol. 13, no. 6, pp. 1201-1213
Langmuir, J. (1918), Journal of American Chemistry Society, 46, pp. 1361
Nadeau, J. P.; Puiggali, J. R. (1995), Séchage des processus physiques aux procédés industriels,
Technique et documentation, Lavoisier
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