GELATIN DRYING PROCESS

GELATIN DRYING PROCESS

E.A.Silva^1*, I.Neitzel² and L.H.M.Silva3

¹Department of Chemical Engineering, UNIOESTE, Rua da Faculdade 2550, 85903-000,

Phone (55) [phone], Toledo - PR, Brazil.

Email: [e-mail]

2Department of Chemical Engineering, UEM, Av. Colombo 5790, 87020-900,

Phone: (55) [phone], Maringá - PR , Brazil.

Email: [e-mail]

3Department of Mathematics, UEM, Av. Colombo 5790, 87020-900,

Phone (55) (44) 226-2727, Maringá - PR , Brazil.

(Received: April 5, 2001 ; Accepted: November 10, 2001)

Abstract - In one of the stages of the gelatin production process, a highly concentrated solution of gel is cooled and extruded to form gelatin noodles, which are then laid on a drying belt. Gelatin is a molecular colloid that is not porous under these drying conditions, and as a consequence, water migration occurs solely by diffusive processes. To achieve a commercial standard of dryness, the dependence of the diffusion coefficient as a function of temperature is used. This set of circumstances favors the appearance of sharp concentration gradients inside the gel. In a numerical simulation of the drying process these characteristics create difficult conditions for use of the traditional methods for solution of time-dependent partial differential equation models. This paper evaluates an implementation of the boundary element method to determine surface conditions of the gelatin particle.

Keywords: gelatin, drying belt, molecular colloid.

INTRODUCTION

Development of the gelatin production process began in 1920 and today it is, nowadays, a well-mastered technology. Of the various production process stages, one of the most critical is the drying of gelatin noodles; this is due to the high consumption of energy and to their peculiar drying characteristics. Of these characteristics a few are worth mentioning: a tendency for liquefaction to occur during the first stage of the drying process, the formation of a dry crust on the surface of the noodles, volume contraction during the drying process and the possible loss of some of the gelatin’s commercial properties when it is subjected to high temperatures.

The gelatin moisture is fixed not only by physical forces but also by chemical forces, such as hydration, osmotic links, structural links, etc, which can be grouped under the denomination of physicochemical forces (Ward and Curtis, 1977). Substances showing this kind of behavior are characterized by the absence of macropores during the drying process; therefore, the migration of moisture inside the gel occurs solely by liquid diffusion to the surface, where evaporation takes place (Krischer and Kast, 1980).

The gelatin drying process can be divided in to two stages. During the first stage, the evaporation of unbound moisture takes place. During this period, the sole mechanism influencing the drying process is the mass transfer from the water vapor formed on the surface of the solid to the bulk gas. Therefore, drying occurs as if a water film covered the material, and thus the moisture gradients inside the gel are null. The second stage begins when the gel moisture equals the critical moisture; during this period the removal of bound moisture takes place and two mechanisms influence the drying process: the diffusion of water in its liquid phase from the inside the gel to the surface and the diffusion of water vapor from the solid surface to the bulk phase.

In this work the gelatin drying in fixed bed is modeled. The model governing equation are obtained by mass, energy and. In the model the formation of concentration gradients inside the is taken into a account gelatin noodles.

DRYING EXPERIMENTS

The gelatin used in these experiments was supplied by Indústrias Leiner do Brasil and was collected it after exited the extrusion machine, where the gelatin was in the form of noodles. The gelatin noodles used in these drying tests had the following standard measurements: a diameter of 0.3 x 10^-2m and a length of 5.0 x 10^-2m. The drying tests were conducted in a laboratory dryer with a diameter of 0.215 m and a length of 0.215 m, as shown in Figure 1-a. The gel and the bed properties are shown in Table 1.

The drying tests were conducted as follows: the gelatin noodles with known temperature and moisture was placed inside the dryer. Air with known properties was supplied continuously and at predetermined time intervals and measurements were taken of the weight of the sample and the temperature measurements of the exiting air.

The temperature profile used for the supply air was aimed to simulate the drying compartments on the dryer belt used by Indústrias Leiner do Brasil, Figure 1-b. The air-input conditions are different for each drying compartment. Therefore, the properties of the air (temperature, concentration, molar flow rate and humidity) injected in to the lab dryer remained unaltered during a predetermined period of time. The aim of this procedure was to simulate the behavior of a specific compartment of the industrial dryer. The air supply conditions in the drying experiments are in Table 2. After a predetermined period of time corresponding to the gel’s exposure a specific industrial dryer compartment, the air conditions were changed, simulating the gel’s passage to a new drying compartment. The results of the drying tests were reported by Neitzel (1987a). The temperature and the initial moisture of the gel are shown in Table 3.

MODELING

Gelatin Bed Drying

The used drying model is based on the published work of Medeiros and Massarani (1982), Neitzel (1987b) and Neitzel (1987c) modified to account for the concentration variation of particles.

The following assumptions were for the model:

i) The air behaves as an ideal gas;

ii) the properties of the air vary only in the direction of the gas flow;

iii) the process is adiabatic since the dryer used in these experiments is thermally insulated;

iv) the gel-specific heat is a function of only the moisture present in the solid;

v) the convective processes prevail over the diffusive ones in respect to the phenomena heat and mass transfer to the gas phase; and

vi) the bed volume is constant.

Based upon these assumptions we can obtain the following set of equations:

a) Momentum balance:

b) Gas mass balance:

c) Water mass balance in the gas phase:

d) Gas energy balance:

e) Solid mass balance:

f) Solid energy balance:

The initial conditions are

The drying rate for the first and second stages is defined by the following equations:

Constant drying rate stage:

Decreasing drying rate stage:

The B.E.T desorption isotherm (equation (15)) was used to calculate the moisture of equilibrium, whose parameters (equation (16)) were determined by Greg and Sing (1967) at 20ºC.

where j is of the air relative humidity.

To determine the drying rate during the second stage (equation (8)) it is necessary to know the gelatin noodle moisture, u º u (t,r), at the surface (r = R).

Noodle Drying

The mathematical modeling of the noodle drying process assumed the air to be circling transversally through the gelatin noodles and the following assumptions were made:

i) The water diffusion coefficient inside the gel is constant;

ii) moisture equilibrium is a function related solely to the average air humidity and to the gelatin;

iii) the driving force for the diffusion of water inside the gel comes exclusively from the liquid water concentration gradients;

iv) particle volume remains constant;

v) evaporation takes place only at the lateral surface because its area is 30 times larger than the transversal surface area;

vi) heat desorption from the gelatin-water system equals the water vaporization heat; and vii) the temperature gradients inside the gelatin particles are negligible.

After then, the following set of equations can be written:

a) Mass balance in the gelatin noodle:

The initial condition is

The boundary conditions are

The particle drying rate per evaporation area unit is expressed by the following equation:

NUMERICAL SIMULATION AND RESULTS

The equations for bed drying, (equations (1-16)), and particle drying, (equations (17-21)), were coupled and solved simultaneously.

Gelatin Bed Drying

The set of equations (1-6) referring the bed drying model were discretized at the z variable (bed length) using backwards finite differences, transforming the set of six partial differential equations into an ordinary differential equation system of (21 x n), where n is the number of discretized cells.

To solve the ordinary differential equations, we have used the Gear method with a variable step because this method is stable, even when the characteristic values associated with the differential operator have very different orders of magnitude.

Gel Noodle Drying Model

To solve the one dimensional diffusion equation cylindrical coordinates for the modeling of the noodles, we have used the boundary element technique. To construct the boundary element integral formulation, we have used the direct formulation, in which a reciprocity relation between the real state (unknown) and the auxiliary state is established. This was chosen to conduct the integral representation covering the quantities directly related to the problem under analysis (Cruze and Rizzo, 1968).

The fundamental solution of the diffusion operator G º G (x, r, t_F,t) was chosen as auxiliary state. The Green’s function is obtained solving the equation of diffusion, due to a source em x = x and t = t_F, for homogenous boundary conditions in an infinite medium (Chuang and Szekely, 1971). Replacing the fundamental solution in the initial relation the we obtain an integral formulation defined on the domain of the problem. Then, this representation is taken to the boundary, using Green's third identity and some properties of the fundamental solution, resulting in the following equation:

where:

when x Î G (boundary) in the case x =R ;

c(x) =1 , when x Î W (domain).

The fundamental solution for the associate operator of the one dimensional diffusion equation in cylindrical coordinates is (Chuang and Szekely, 1972):

Differentiating (23) with respect to the r variable, we obtain

where

I₀(z) is the Bessel modified function of the first kind of order zero.

I₁(z) is the Bessel modified function of the first kind of order one.

t = t_F - t

A more detailed description of the construction of the boundary element techniques integral formulation for the diffusion problem in one-dimensional cylindrical coordinates is given by Silva (1995).

In the numerical solution of equation (22), we have adopted the step-by-step time procedure due to its smaller capacity for storing the intermediate results, since it is necessary to store only the results of the dependent variables at an earlier time. In addition, as we have used a constant time step, the integrals are assessed only once.

In equation (22), the boundary variables were discretized using time linear interpolation functions, i.e.,

for t_F-1 < t < t_F ;

where

and h = t_F - t_F-1

The dependent variable, u(r,t), was interpolated in the r direction by quadratic functions. The number of elements (E) used is 5, as shown in Figure (2).

Therefore, we can write the following equation:

where

Substituting equations (25)-(27) in to (22), we obtain

Defining:

and substituting equations (29)-(33) in to (28), we obtain:

The coefficient values in equation (34) depend on the application value point (x). To solve the integrals defined by equations (29-33), the modified Bessel functions of the first kind (I₀ and I₁) had their value by series calculated, expansion according Abramowitz and Stegun (1965).

The integral defined on the domain represented by equation (33) depends only on the value of at dependent variable at time t_{F - 1}, and the integrals defined at the boundary represented by equations (29-32) do not present singularities where x ¹ R values were calculated by the Gaussian quadrature rule with 10 points.

When x = R , the integral defined at the boundary shows a singularity (Wrobel and Brebbia, 1981) and its values analytically calculated. When the argument of modified Bessel functions of the first kind (I₀ and I₁) is greater to 3.75, we can use the following expression to calculate these functions:

where

z = x / 3.75 .

Substituting equation (35) to (29) and (30) and analytically integrating

On the other hand, substituting equation (35) and (36) in to (31) and (32), respectively, and analytically integrating, we obtain

Assuming that gelatin temperature and air humidity remain constant, during the time step used in the boundary element technique, we can then rewrite equation (20) as

where

Substituting equation (41) in to (34) and rearranging it, we obtain

When x =R and c(x)=1/2, equation (43) becomes:

We, obtain a single linear equation, whose the unknown value is the moisture value on the gelatin surface, u º u (R, t). Following solution, of this equation, we can calculate the moisture gradient on the gelatin surface from equation (41).

Knowing the moisture values inside the particles is necessary to calculate the integral of the B domain, used in the next time step. When x Î W (domain), then c(x)=1, which was substituted into equation (34), giving us:

When we utilize the boundary element technique the time advance step value must be selected properly. As Dt approaches 0, the integrand in the domain integral (the fundamental solution) becomes less smooth, its limit being a Dirac delta function. The difficulty of numerically integrating a function with this type behavior may introduce numerical problems into the solution (Brebbia et al., 1984). We've adopted h = 0,02 hours as the time advance step.

Coupling of Drying Models

During the first stage the concentration gradients inside the gel are null; thus only the bed drying model is solved as described by the system of partial differential equations (1-6) with respective initial conditions and together with equation (10) which mathematically represents the drying behavior during this stage.

The second drying stage, represented by equation (14), has as one of its unknown values the moisture value at the noodle surface so for models the bed and its particles should be solved simultaneously. In the boundary element technique we have assumed that gel temperature and air humidity remain constant, during the time-step advance (h). However, this assumption is not appropriate, and therefore we have adopted the following strategy: first we solve the particle drying model at time (t_F-1 ^{+ h}) and we calculate the moisture inside the particle at time (t _F-1 + h/100) using the time linear interpolation functions.

The model parameters used in the simulation are shown in Table 4. Three of the, model parameters, the heat transfer coefficient, the water diffusion coefficient in gel and the mass transfer coefficient were determined by minimizing the objective function represented by equation (46), using the simplex method (Nelder and Mead, 1965).

where and are the experimental mass bed values and the calculated values using the model respectively and ne is the number of experimental points. The mass of the bed determined by the model was calculated by integration of moisture into the interval [0,L]. The heat capacity calorific of dried gel was determined by Skuratov and Shkitov (1946).

The experimental and simulated results of the drying curves are shown in Figure 3. It can be observed in this figure that the model appropriately represented the drying of the gelatin. In Figure 4 the concentration profile for the water inside the gel is shown.

CONCLUSIONS

In this paper we have discussed a process of coupled drying of a gelatin bed and its noodles. To solve the differential equation that describes the drying of gelatin noodle the we have used the boundary element technique. To solve the differential equations which describe the drying of the gelatin bed, we have used Gear's method.

The boundary element technique proved to be efficient for the solution of the equation that describes the behavior of the gelatin noodle drying process. Depending on the drying conditions, high gradients of water concentration can appear inside the gel, as observed in the figures showing the simulation results in Figure 3. In this case, the use of traditional solution methods for the partial differential equations require a more refined treatment, implying an increase size of the equation set. When used to solve this type of problem the boundary element technique, captures these gradients more adequately, without requiring any additional effort. In addition, the boundary element technique reduces the dimension of the problem under analysis because, when we take the variables at the boundary, the dimension of the problem which is (m) becomes (m – 1). For the problem under consideration, after assumptions we made, the dimension of the mathematical model was m=2. Using the boundary integral formulation, m became 1 (one) and taking advantage of the symmetry, the dimension became zero, i.e., the boundary was reduced to a single point. Therefore, due to this fact, there is a reduction in computational time.

The drying model presented in this paper has been shown in be efficient to describing the phenomenon of the gelatin noodle drying on a fixed bed, as can be observed in Figure 4 by the good agreement of the bed moisture results obtained by solution of the model and the results of the drying experiments.

NOMENCLATURE

a transfer area by bed unit volume (m²/m³) C gas concentration (kgmol air/m³) Cp specific heat at constant pressure [molar] (J/kgmol K) specific heat at constant pressure [mass] (J/Kg K) C_V specific heat at constant volume (J/Kgmol K) d_s dryer diameter D diffusion coefficient for water in gel(m²/h) F molar rate (kgmol air/m²h) F_obj objective function G Green’s function in cylindrical coordinate system h

time-step advance

h_c convective heat transfer coefficient (J/m²K h) I₀ Bessel modified function of the first kind of order zero I_I Bessel modified function of the first kind of order one k_y mass transfer coefficient (kg water/m² h) L bed length (m) M molecular weight (kg/kgmol) m ^exp mass of experimental bed m ^mod mass of the bed model n discretization number for bed length ne number of experimental points

r

cylindrical coordinate system axis (m)

R

particle radius (m) R_G ideal gases constant (J/kgmol K) t Time (h) t_F time of instantaneous source T Temperature (K)

u

particle moisture (water kg/dry gel kg) average particle moisture (water kg/dry gel kg) X particle moisture (water kg/dry gel kg) X_c critical moisture (water kg/dry gel kg) X_e moisture equilibrium (water kg/dry gel kg) Y air humidity (water kgmol/moisturized air kgmol) air humidity (water kg/dry air kg)

z

ordinate axis parallel to the air flow (m)

Z_s

dryer length (m) e bed porosity

r_a

apparent density (kg/m³) DH_s gelatin desorption heat (J/dry gel Kg) f₁ time linear interpolation function

f₂

time linear interpolation function x position of instantaneous source j relative humidity of the air (water in the air kgmol/saturated water kgmol) y drying rate per bed volume unit (water kg/m³ h) y_p particle drying rate (water kg/m²h)

Index

air air g gelatin s dry gelatin w water sat saturated 0 initial

• Abramowitz, M. and Stegun I. A., Handbook of Mathematical Functions. Dover, New York (1965).
• Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C., Boundary Element Techniques, Springer Verlag, Berlin (1984).
• Chuang, Y. K. and Szekely, J., On the Use of Greens Functions for Solving Melting of Solidification Problems. Int. J. Heat Mass Transfer, 14, 1285 (1971).
• Chuang, Y. K. and Szekely, J., The Use of Greens Functions for Solving Melting of Solidification Problems in the Cylindrical Coordinate System. Int. J. Heat Mass Transfer, 15, 1171 (1972).
• Cruze, T. A. and Rizzo, F. J., A Direct Formulation and Numerical Solution of the General Transient Elastodynamic Problem. Int. J. Math. Anal. Appl., 22, 244 (1968).
• Greg, S. J. and Sing, K. S. W., Adsorption, Surface Area and Porosity, Academic Press, London, (1967).
• Krischer, O. and Kast, W., Die Wissenschaftlichen Grundlagen der Trocknungstechnik, Band I Dritte Auflage, Springer Verlag (1978).
• Medeiros, J. L. and Massarani, G., Secagem de Bagaço de Cana - III Săo Carlos, Anais do X ENEMP (1982).
• Neitzel, I., Secagem de Gelatina, Relatório Interno (DEQ/Univ. Est. de Maringá.), (1987a).
• Neitzel^b, I., Secagem de Gelatina, Relatório Interno, (DEQ/Univ. Est. de Maringá), 1, (1987b).
• Neitzel^c, I., Secagem de gelatina, Relatório Interno, (DEQ/Univ. Est. de Maringá), 2, (1987c).
• Nelder J. A. and Mead R. A Simplex Method for Function Minimization. The Computer Journal, 7, 308 (1965).
• Silva, E. A., Análise do Perfil de Concentraçăo e Temperatura no Interior do Gel, Masters thesis, Univ. Est. de Maringá (1995).
• Skuratov, S. M. and Shkitov, M. S., Specific Heat of Water Bound by High Polymeric Substances, Physical Chemistry, 53, No (7), 627 (1946).
• Ward, A. G. and Courts, A., The Science and Technology of Gelatin, Academic Press, London (1977).
• Wrobel, L. C. and Brebbia, C. A., A Formulation of the Boundary Element Method for Axisymmetric Transient Heat Conduction, Int. J. Heat Mass Transfer, 24, 843 (1981).

*

To whom correspondence should be addressed

Publication Dates

• Publication in this collection
  17 Apr 2002
• Date of issue
  Dec 2001

History

• Received
  05 Apr 2001
• Accepted
  10 Nov 2001