## Applied Thermal Engineering

A simple moisture transfer model for drying of sliced foods

Bülent Yesilata *, Mehmet Azmi Aktacir

Harran University, Mechanical Engineering Department, Osmanbey Campus, 63300 Sanliurfa, Turkey

article info

Article history:

Received 1 January 2008

Accepted 30 March 2008

Available online 6 April 2008

Keywords:

Drying kinetics

Mathematical model

Moisture transfer

Slab

Effective diffusion coefficient

### Abstract

The mathematical formulation of mass transfer in drying processes is often based on the nonlinear

unsteady diffusion equation. In general, numerical simulations are required to solve these equations.

Very often, however, indirect and simplified methods neglecting fundamentals of the processes are used.

In this work, a new mathematical model approach for the mass transfer occurring during drying of sliced

foods is proposed. The model considers fundamentals of the drying process and takes internal resistance

to moisture transfer into account. The parameters in the formulation have physical meaning and permit

giving clear view of the moisture depletion process occurring during drying. The proposed model has an

analytical solution and allows finding effective diffusion coefficient accurately. The verification of the

model is made with basic drying experiments performed for chili red peppers sliced in slab form. The

results reveal that there is nearly perfect match between the drying curves obtained by the model and

the experiments.

1. Introduction

Drying of industrial solid matters is of practical importance in

engineering. New drying technologies along with better control

and operational strategies have contributed to better quality dried

products. Industrial drying processes have been fulfilled under the

condition of controlling drying air in order to reduce crop losses,

improve hygiene and quality of the dried products. This has caused

a strong transition from traditional sun or shade drying to indus-

trial driers despite of the fact that investment and operational costs

of the latter are significantly higher. The other major issue to be

solved for the drying industry is that there is yet no universally

or even widely applicable drying theory. Most mathematical mod-

els of drying still remain specific to product-equipment for a vari-

ety of reasons [1]. There is therefore a great need for drying models

to describe the drying process, help in its optimization, and assist

in the effective design of dryers or improve existing drying systems

[2,3].

In the literature, over 200 drying models (see [4,5], for excellent

discussions on the subject) have been offered for various foods,

which are formally characterized by two different, physical and

empirical, approaches. Food science is experiencing a transition

from empirically based approach for quantifying the dehydration

of dry food particulates to physically based models that are mainly

derived by assuming the food as a porous media [6–9]. For the time

being it is not yet possible to judge the practical usefulness and

reliability of these models that were mostly developed via com-

puter simulations. In general, there has been a need for more

experimental validation and for establishing the usefulness of the

approximate methods for fully modeling industrial drying pro-

cesses [5,10,11]. The complete description and accurate prediction

of the drying kinetics have not still been possible and different as-

pects of existing methods still need certain level of tuning due to

the complex transport phenomena involved in food drying [12,13].

In this study, a new mathematical model for the mass transfer

occurring during drying of sliced foods is proposed. The proposed

model is validated with drying experiments performed for chili

red pepper sliced in slab form. The drying kinetics of red peppers

has been studied in the literature up to some extent. Most of pre-

vious works have examined mass transfer rate of red pepper slices

using various empirical equations that are based on Fick’s second

diffusion law. These empirical equations were compared with their

correlations obtained from basic curve-fitting approaches of exper-

imental data. Either the Page model [14] or the modified Page

model [2,15] produced the best fit for basic drying curves of vari-

ous red peppers. These empirical equations on the other hand have

not provided any physical information about internal moisture

transfer mechanism.

2. Description of the proposed drying model

The schematic of the problem is given in Fig. 1. The solid body of

the food that is sliced in strip-form is hypothetically divided into

three different regions to explain logic of the model developed

here. The initial (t = 0) moisture content of the solid matter is

shown with wA0. The stagnant hot air contacts with the surface

at z(t = 0) = 0 when drying starts in the oven. A certain amount of

water is depleted from the region very next to the hot air (region I,

the outer depletion region) instantaneously leading to much lower

equilibrium concentration of wAS. The original solid–gas interface,

z(t = 0) = 0, is stationary because depletion of water content in

the solid food should not cause complete layer movement. The

interface between regions I and II moves with a velocity of v(t),

and its location is of interest to obtain a solution. Appropriate mod-

eling of the problem thus requires a reference frame that moves

with the hypothetical interface (i.e. z(t > 0) = 0 always represents

that interface). The amount of moisture transferred to the hot air

can then be calculated since these parameters are directly related

with the rate of moisture transfer from the solid matter.

The depleted moisture is assumed to be replaced with some

type of vacancy (i.e. voids, pores) allowing water pass from region

II. Further stage of drying continues with moisture transfer from

the region II (inner depletion region) due to concentration gradient

of (wA0 wAS) between its two interfaces. The initial moisture con-

tent in the very thin region III of the solid matter (z ? 1) thus re-

mains always the same. The moisture is the only component that

diffuses through the interface, the system can then be reduced

from multi-diffusion to a binary diffusion problem (A: moisture

and B: sliced food). The resulting diffusion equation for species

‘A’ is given as

In the case of one dimensional mass transfer and constant material

properties (q=qA + qB) along with the assumption of no-dependence

of diffusion coefficient on position and composition (DAB = DBA =

Deff), the mass flux in a binary system can be described as

In Eq. (2), DAB = Deff is the effective diffusion coefficient of species ‘A’

in ‘B’ (i.e. moisture in the sliced solid food). The second term repre-

sents diffusion of moisture due to the moving coordinate. Eq. (2)

can be rewritten in simpler form by considering the relation of

where v(t) denotes the velocity of the active coordinate system,

which is directly related to the moisture transfer from the solid

matter. A parabolic-rate constant in mass is defined below to find

appropriate relationship between the mass loss and the velocity as

where t, Dm(t), and A are, respectively, drying time, mass loss, and

area perpendicular to diffusion flux. Eq. (4), at a specified constant

temperature, refers to a constant value of km since Dm is considered

to be proportional to t

1/2, and the value of km in practice only varies

with temperature [16]. The instantaneous mass variation of the

solid matter, which is always measured in basic drying experi-

ments, is described as,

It is important to notice that there is a proportional relationship

between z(t) and v(t); such as zðtÞ / R

vðtÞdt. The introduction of a

dimensionless limitation factor (/lim) is necessary when one of

the species is depleted from a solid body since mass transfer path

has complex obstructions. The idea behind this definition is similar

to Pilling–Bedworth ratio used in metal depletion from alloys [17]
but the prediction of /lim is somewhat easier here. Considering

maximum allowable location of z(t) at the end of the drying process

is one of the best way to satisfy with physical mechanism since z(t)

must never exceed the sample thickness. The instantaneous

location of the interface can then accordingly be defined as a

function of parabolic-rate constant in thickness (kz),

Such a definition is very beneficial to find appropriate relationship

between parabolic-rate constants in mass and in thickness as given

below,

The following dimensionless parameters can be defined to solve Eq.

with the boundary conditions of w(Z = 0) = 1, w(Z?1) = 0. The solu-

tion given below is well known for long years, after Arnold [18],

where a dimensionless parameter of W,

is used. The mass concentration profile determined in Eq. (9) can be

used to calculate the rate of water depletion from the solid matter

as follows:

The derivative is taken at z = 0 because the reference frame is

attached to the moving interface. In order to determine the loca-

tion of the reference frame at a particular time, the velocity given

by Eq. (8) is used. The rate of moisture depletion can now be used

to predict mass loss of the specimen due to drying of the food. Due

to the nature of basic drying experiments, the only unknown

parameter in the Eq. (13) is the effective diffusion coefficient of

DAB. The main object of the approach presented here is to deter-

mine DAB by using experimental mass transfer curve. For this pur-

pose, the equation can be arranged to give the instantaneous mass loss,

The last equation is now in more suitable form to compare with the

experimental curve since the mass variation of the food is usually

measured in experiments. The model can allow estimation of DAB

by seeking the best match between the experimental curve and

the curves calculated by the following equation:

### 3. Experimental verification of the model

#### 3.1. Experimentation

The experimental set-up used here consists of a drying oven

with thermostat and digital indicator, and a highly accurate digital

scale with sensitivity of 0.01 g. The mass transfer area and thick-

ness of red pepper samples used in experiments are respectively

A=10 cm2 (10 ± 0.01 cm 1 ± 0.01 cm) and d = 0.21 ± 0.01 cm. The

samples were sliced in strip-form so that one dimension mass

transfer model can be used. The three samples with identical

dimensions were tested at the same conditions and the maximum

deviations in time dependent mass variations remained within 2%.

The arithmetic mean mass value of three samples at each specified

time is taken into consideration for applying the mathematical

model. The drying temperature of 70 ± 1 C was applied. The initial

moisture contents of the samples were simply determined by con-

sidering the difference in fresh (wet) weight and dry weight that

was determined after applying relatively high drying temperature

of T = 110 C. During measurements of dry weights, the samples

were kept in the oven until the difference between two successive

mass measurements is less than 0.02 g.

#### 3.2. Results

The variation of instantaneous masses for the three samples and

the corresponding variations in mean moisture content with the

drying time are shown in Fig. 2. The drying curve by using arithme-

tic mean values is indicated with solid line in Fig. 2a. The drying

behaviors of all three samples are similar and the maximum devi-

ation from the mean never exceeds 2%. The dimensionless mois-

ture content of the red pepper (wA) given in Fig. 2b is based on

its dry weight and calculated with the following equation:

where mT and mD are respectively total (or wet) and dry weights of

the sample. The initial weight of the sample is denoted by mT0.

The application of the model for predicting effective mass diffu-

sion coefficient of DAB becomes now relatively easier since the

instantaneous values of moisture (mA) transferred to the hot air

are known. The experimental values of mA as a function of drying

time are shown in Fig. 3a with filled symbols. The solid lines are

obtained by using Eq. (14) along with Eq. (13). Eq. (14) enables

us to determine DAB by seeking the best match between the exper-

imental and theoretical curves. The best match between these two

curves is obtained for DAB = 0.0087 cm2

/min (1.45 108 m2/s). Eq.(14) can be considered to be very useful since the experimental

determination of DAB otherwise is quiet difficult. The other two

curves with mismatching values of DAB are also included in the fig-

ure to demonstrate the usefulness of the method and the signifi-

cant effect of DAB on the drying rate.

Fig. 2. (a) The variation of the sample mass with drying time (the symbols denote

experimental data for samples 1, 2, and 3, and the solid line represents their mean

values). The dashed line is for the mean dry mass, (b) the variation of the mean

relative humidity with drying time.

Fig. 3. (a) Variation of depleted moisture mass with drying time for various values

of DAB (the lines show calculations with the model and symbols show experimental data). The bold solid line represents the best matching value of DAB, (b) mass var-

iation of red pepper with drying time (the solid line show calculations with the best matching value of DAB and the symbols show experimental data).

Fig 4.(a) Variation of the interface velocity with drying time, and (b) variation of

interface location with drying time.

The comparison between the mass variations of the red pepper

obtained from experimental measurements and the model with

the predicted DAB value of 0.0087 cm2/min is shown in Fig. 3b.

The agreement of the two curves is nearly perfect with the excep-

tion of initial and final stages of drying. The deviations occurred in

these regions can be expected due to the using constant mass dif-

fusion coefficient during the whole process. It is well known that

the initial mass transfer rate is higher due to the perfect contact

of the outer surface with the hot air. The drying rate is conversely

lower when approaching to the final stage due to shrinking of the

mass transfer area. However, as can be seen from the figure, both

initial and final stages last relatively for very short time and the

deviations from experimental curve are also at acceptable level.

The analytical model presented here can thus be considered as

quiet beneficial for industrial drying of the red pepper.

The variation of instantaneous interface velocity calculated

with the model, v(t), is shown in Fig. 4a to demonstrate physical

nature of the problem. The initial velocities are significantly higher

and decrease then sharply. The model appears to overpredict the

initial velocities, which may be one of the causes for that higher

mass transfer rates than the real case are obtained in Fig. 3b. The

decreasing rate of velocity becomes then much smoother until

the end of drying process. These intermediate times confirm the

parabolic-rate approach that result in nearly perfect agreement

with experimental data during major part of drying.

The instantaneous locations of the hypothetical interface line

are illustrated in Fig. 4b. The initial location is started from the original food surface (z = 0) and it moves with parabolic-rate. The final

location is adjusted by /lim such that it does not pass through very

thin outer region of 0.1 mm in thickness (region III or the bright

shell of the red pepper).

Although the primary objective in this paper is to introduce ma-

jor logic and utilization procedure of the proposed model, rather

than a parametrical investigation, the model validation was veri-

fied at another air temperature (Tair = 60 C) as well. The physical

behaviors of the drying curves given in Fig. 3 remained nearly

the same at this temperature, as expected. Only the values were

shifted due to that predicted value of the effective diffusion coeffi-

cient was decreased to 0.0056 cm2/min (0.93 108 m2/s). The effect of air temperature on the drying kinetics is indeed a well established matter, regardless of the nature and complexity of

any reasonable model considered. It is well known that the air

temperature proportionally affects the diffusion coefficient [2,5].

The characteristic drying curves of the same material at various

drying temperatures collapse into a single curve by using an appro-

priate temperature shift factor. This factor is usually described by

the Arrhenius law, where the logarithm of the diffusivity perfectly

exhibits a linear behavior against the reciprocal of the absolute

temperature [4]. The decrease in the effective diffusion coefficient

with decreasing air temperature (from 70 C to 60 C) obtained

here is thus in accord with the Arrhenius law.

#### 4. Conclusion

A simple unsteady and one dimensional diffusion model based

on moisture depletion from solid body in slab form is introduced.

The model is validated with isothermal drying experiments per-

formed for sliced chili red pepper. It is shown that by using this

model, the effective diffusion coefficient of the depleted moisture

in the solid food can accurately be predicted. This accomplishment

allows calculating the instantaneous mass variation of the solid

food during the drying process.

In contrast to most semi-theoretical models offered in the liter-

ature, which simplify general series solution of the Fick’s second

law, the proposed model considers the fundamentals of drying pro-

cess for slab-shaped products. The parameters in the formulation

have physical meaning and allow giving clear view of the moisture

depletion process occurring during drying. The observations made

so far indicate that the model could be applicable for wide ranges

of products and external air conditions providing that significant

shrinking do not occur.

References

[1] A. Mujumdar, An overview of innovation in industrial drying: current status

and R&D needs, in: Drying of Porous Materials by Stefan Kowalski, Springer,

Netherlands, 2007, pp. 3–18.

[2] A. Vega, P. Fito, A. Andres, R. Lemus, Mathematical modeling of hot-air drying

kinetics of red bell pepper (var. Lamuyo), Journal of Food Engineering 79

(2007) 1460–1466.

[3] S. Kooli, A. Fadhel, A. Farhat, A. Belghith, Drying of red pepper in open sun and

greenhouse conditions: mathematical modeling and experimental validation,

Journal of Food Engineering 79 (2007) 1094–1103.

[4] J. Crank, The Mathematics of Diffusion, second ed., Oxford University Press,

New York, NY, 1975.

[5] W.J. Coumans, Models for drying kinetics based on drying curves of slabs,

Chemical Engineering and Processing 39 (2000) 53–68.

[6] J.R. Philip, D.A. de Vries, Moisture transport in porous materials under

temperature gradients, Transactions-American Geophysical Union 38 (1957)

222–232.

[7] K.M. Waananen, J.B. Litchfield, M.R. Okos, Classification of drying models for

porous solids, Drying Technology 11 (1993) 1–40.

[8] K.A. Landman, L. Pel, E.F. Kaasschieter, Analytic modelling of drying of

porous materials, Mathematical Engineering in Industry 8 (2001) 89–

122.

[9] I.S. Saguy, A. Marabi, R. Wallach, New approach to model rehydration of dry

food particulates utilizing principles of liquid transport in porous media,

Trends in Food Science and Technology 16 (2005) 495–506.

[10] I. Ceylan, M. Aktas, H. Dog ̆an, Mathematical modeling of drying characteristics

of tropical fruits, Applied Thermal Engineering 25 (2007) 1931–1936.

[11] A. Can, An analytical method for determining the temperature dependent

moisture diffusivities of pumpkin seeds during drying process, Applied

Thermal Engineering 27 (2007) 682–687.

[12] M. Aversa, S. Curcio, V. Calabro, G. Iorio, An analysis of the transport

phenomena occurring during food drying process, Journal of Food

Engineering 78 (2007) 922–932.

[13] M. Batista Lucia, A. da Rosa Cezar, A.A. Pinto Luiz, Diffusive model with

variable effective diffusivity considering shrinkage in thin layer drying of

chitosan, Journal of Food Engineering 81 (2007) 127–132.

[14] I. Doymaz, M. Pala, Hot-air drying characteristics of red pepper, Journal of Food

Engineering 55 (2002) 331–335.

[15] E.K. Akpinar, Y. Bicer, C. Yildiz, Thin layer drying of red pepper, Journal of Food

Engineering 59 (2003) 99–104.

[16] B. Yesilata, Analytical modeling of unsteady aluminum depletion in thermal

barrier coatings, Turkish Journal of Engineering and Environmental Science 25

(2001) 675–680.

[17] N. Birks, G.H. Meier, Introduction to High Temperature Oxidation of Metals,

Edward Arnold (Publishers) Ltd., London, 1983.

[18] J.H. Arnold, Studies in diffusion: III unsteady-state vaporization and

absorbtion, Transactions A.I.Ch.E. 40 (1944) 361–378.