pp 533-541 heat and mass transfer in a shrinking cylinder drying

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 : [email protected] & [email protected]

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



534

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:



535

( )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.



536

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.



537

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



538

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



539

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



540

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



541

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

pp 533-541 heat and mass transfer in a shrinking cylinder drying

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