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Drying stresses and strains in a spherical food model
Computers and Chemical Engineering 33 (2009) 1805–1813
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Drying stresses and strains in a spherical food model L.S. Arrieche b,∗ , R.G. Corrêa a , D.J.M. Sartori a a b
Chemical Engineering Department, Federal University of São Carlos, P.O. Box 676, São Carlos, 13565-905, Brazil DECOM/CEUNES, Federal University of Espírito Santo, Humberto de Almeida Franklin, 257, São Mateus, ES, 29.933-480, Brazil
a r t i c l e
i n f o
Article history: Received 6 August 2008 Received in revised form 14 April 2009 Accepted 20 May 2009 Available online 27 May 2009 Keywords: Breakage Deformations Dehydration Modelling Rupture stress Modulus of Young
a b s t r a c t An approach to model the drying mechanical effects occurring in a single sphere, representing a food model, is proposed. The mechanical properties were evaluated through compression tests and drying kinetics data were obtained under a laminar fluid flow. The Fickian model was used to represent the moisture content profiles and the mechanical model was formulated from elasticity theory. Deformations and cracks predicted from numerical solution were in agreement with experimental observations. This work helped to understand the relationship between mass transfer, shrinkage, stresses, strains and physical degradation. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction When concerning quality of dehydrated foodstuffs, one of the main reasons for quality loss during drying is due to structural changes caused by shrinkage of the product. The shrinkage promotes loss in the permeability characteristics to the solvent and turns the material unstable and brittle. Different attributes of quality of the dehydrated products, such as density, crust formation, cracks and others are all related with the contraction process (Eichler, Ramon, Ladyzhinski, Cohen, & Mizrahi, 1997). With a polymeric structure similar to food and other types of biological materials, but a simple composition, the gels have been broadly used in researches as simulators (Achanta & Okos, 1996; Arrieche & Sartori, 2008; Gogus & Lamb, 1998; Mrani, Bénet, Fras, & Zrikem, 1997; Shah & Porter, 1973; Thakur, Vial, & Djelveh, 2008). Several gels are used as starch gel, agar gel and others, for the study of chemical reactions, shrinkage mechanisms and the great majority in drying studies, involving mathematical modelling. In many drying studies, the mathematical modelling is treated considering only the heat and mass transfers (Katekawa & Silva, 2006). These mathematical models may be reasonable to describe drying rates but are unable to predict shape evolutions and physical deteriorations that occur during convective drying of materials undergoing shrinkage. Under the structural point of view, few works consider the modelling of internal stresses, which appear due
to mass transfer and shrinkage (Arrieche & Sartori, 2008; Banaszak & Kowalski, 2005; Mezhericher, Levy, & Borde, 2008; Mrani et al., 1997; Ponsart, Vasseur, Frias, Duquenoy, & Méot, 2003; Shah & Porter, 1973). The non-uniform shrinkage owed to mass transfer will bring consequent shape evolution and physical deterioration, due to the appearance of stresses inside the structure. A consideration of these effects, when using spherical samples, contributes to supply the lack of mathematical modelling studies in literature, which include the prediction of shape evolutions during drying. In this sense, the main objective of our study is to analyze the drying and shape evolution of a spherical simulating material and to evaluate a simplified mathematical representation for this process, including the stress and strain fields inside the structure and its effects on quality loss by physical degradation. 2. Model presentation 2.1. Drying model Considering the radial diffusion component, the mass transfer from the inside of the solid to the drying air, can be represented by: ∂ = ∂
Rs2 def () df R2 ()
1 ∂( 2 (∂ /∂)) . ∂ 2
(1)
Eq. (1) is subject to the following initial and boundary conditions, as well as, to the symmetry condition of the radial mass flux: ∗ Corresponding author. Tel.: +55 27 3763 8680. E-mail address: [email protected] (L.S. Arrieche). 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.05.013
= 1 for = 0, ∀
(2)
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Nomenclature Bim c def df dwa d∝ D Deq E hl H H1 H2 k m r R Rs Re Sc Sh t T u Va Vl x1 X X¯ X0 Xe
rup s sa shr sim sp t T w wa 0 ∞
mass Biot number [≡ hl a x1 k/s def ] (–) dimensionless constant in Eqs. (24)–(26) (–) effective diffusion coefficient of water in solid (m2 /s) final diffusion coefficient at equilibrium (m2 /s) diffusion coefficient of water in air (m2 /s) momentum diffusion coefficient diameter (m) sphere equivalent diameter (m) modulus of Young (MPa) local convective mass transfer coefficient (m/s) height (m) dimensionless variable in Eq. (16) (–) dimensionless variable in Eq. (17) (–) coefficient of medium partition between sample and air moisture (–) mass (kg) radius (m) external radius (m) external radius (dried material) (m) Reynolds number [≡ Deq Va a /a ] (–) Schmidt number [≡ d∞ /dwa ] (–) Sherwood number [≡hl Deq /dwa ] (–) time (s) temperature (◦ C) radial displacement (m) air velocity (m/s) volume (m3 ) characteristic length of sphere (volume divided by area) [≡Deq /6] (m) local moisture content (kg water/kg dried solid) average moisture content (kg water/kg dried solid) initial moisture content (kg water/kg dried solid) equilibrium moisture content (kg water/kg dried solid)
Greek letters ˛ parameter in Eq. (20) (–) ˇ parameter in Eq. (20) (–) ε strain (–) azimuth angle (◦ ) dimensionless zenith angle [≡ / ] (–) viscosity (kg/m/s)
zenith angle (◦ )
sp separation zenith angle (◦ ) specific mass (kg/m3 ) stress (MPa) dimensionless time [≡ (df /Rs2 )(t)] (–)
Poisson coefficient (–) dimensionless radius [≡r/R] (–) dimensionless moisture [≡(X − Xe )/(X0 − Xe )] (–)
rupture dry solid sample related to shrinkage simulated separation tangential true water water in air initial free air stream
Fig. 1. Projected spherical volume element.
∂ ∂
= −3Bim () |=1 ,
∂ = 0 for ≥ 0, = 0 ∂
content
≥ 0, = 1
(3)
=1
(4)
In Eqs. (1)–(4), , and are dimensionless time, moisture content and radial position, respectively, R is the external radius, Rs is the external radius of the dried material, def is the effective diffusion coefficient of water in solid and df is the final diffusion coefficient at equilibrium. The mass Biot number (Bim ) is calculated in agreement with Cranck (1975). 2.2. Mechanical model
Subscripts a air e equilibrium E engineering ef effective eq equivalent exp experimental f final H Hencky r radial
Considering the radial moisture content distribution, there are radial and tangential stress components, according to Fig. 1. These stress components should satisfy the mechanical equilibrium condition of a volume element, in the radial direction: d r 2 + ( r − t ) = 0, r dr
(5)
where r is the radial stress, t is the tangential stress and r is the radial coordinate. Considering the gel as a predominantly elastic material, the law of Hooke is applied for the two total strains, taking
L.S. Arrieche et al. / Computers and Chemical Engineering 33 (2009) 1805–1813
into account the effect of mass transfer: 1 εr = ( r − 2 t ) + εshr , E εt =
(6)
1 ( t − ( r + t )) + εshr , E
du , dr u εt = , r εr =
(8) (9)
where u is the radial displacement. The analytical solution of Eqs. (5)–(9) is given by Timoshenko and Goodier (1970), for the case of constant mechanical properties. However, such simplifications can be inadequate in drying of food where the properties can vary with position, when moisture content presents a distribution during drying. For such cases the stress fields should be solved numerically. Eqs. (5)–(9) could be rearranged in the following form. From Eqs. (6) and (8): r − 2 t du = + εshr . E dr From Eq. (7): t =
u
1 E 1−
r
− εshr
(10)
+ r .
Inserting Eq. (11) into Eq. (5):
2
d r =− r dr
r −
1
u
1−
E
r
− εshr
(11)
+ r
Sh = 1.693
sin2
(2 + cos3 − 3 cos )
(Re)1/2 (Sc)1/2 ,
1/2
(14)
where Sh is the local Sherwood number, Re is the Reynolds number and Sc is the Schmidt number. Eq. (14) is valid for Sc < 1 and laminar flow with Re > O(102 ). From the above relation, the overall Sherwood number can be obtained as Sh = (Re Sc)1/2 [0.564H1 ( s ) + 0.156H2 ( s )]
(15)
with H1 ( sp ) = (2 + cos3 sp − 3 cos sp )
1/2
,
(16)
,
(17)
and H2 ( sp ) = (2 − cos3 sp + 3 cos sp )
1/2
where H1 and H2 are dimensionless variables. The angle for flow separation in the boundary layer, sp , described by Clift, Grace, & Weber (1978) and Lee and Barrow (1965), was determined from the following equation:
sp = 78 + 275Re−0.37
(18)
valid for 400 < Re < 3 × 105 .
.
(12) 4. Drying tests
Eqs. (10) and (12) compose a system of ordinary differential equations with the same boundary conditions used in the analytical solution: u = 0 for r = 0 and r = 0 for r = R. The numerical solution was obtained for a given moisture content distribution, established at each time in the drying process, considering the modulus of Young as a moisture content function. Once established the models for mass transfer and stress fields, it was necessary to determine the transport coefficients and mechanical properties. 3. Transport coefficients The effective mass diffusivity was obtained from the first term of the analytical solution of the Fickian model for spheres, where for long drying times only the first term of the infinity series solution is significant. This first term establishes a relation between drying rate and free water concentration. Then, for each experimental condition, it was possible to calculate the effective mass diffusivity from the following equation:
d X0 − X¯ ef = exp t X0 − Xe R2
boundary layer, over a surface of stationary spherical particle. This solution was based upon an assumption of potential flow in the energy and mass conservation equations, with a boundary layer approximation:
(7)
where εr is the total radial strain, εt is the total tangential strain, εshr is the strain due to shrinkage caused by the water loss, is the coefficient of Poisson and E is the modulus of Young. The total strains are not independent but related by geometry. In this case:
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(13)
where X¯ is the average moisture content, X0 is the initial moisture content, Xe is the equilibrium moisture content and t is the drying time. The effective mass diffusivity calculated from this procedure includes several mechanisms of mass transfer, including the effect of heat transfer, which depends on the convection condition and thickness. The coefficient of mass transfer by convection, hl , was determined using the Sherwood number, in agreement with Kendoush (1995), who proposed an analytical solution to the forced convective heat and mass transfer across a laminar incompressible
Granular solid agar was used as gelling agent to prepare the samples, because this material shows a linear volumetric shrinkage as a function of water content. In order to avoid cracks prior to the drying tests, the appropriated working concentration of agar was found to be 3%. The gel samples were prepared according to the method described by Committee on Food Chemicals Codex (2003). The drying tests were accomplished in classical air stream forced convection equipment, consisted of a circuit with a blower type radial compressor, electric heater connected to a voltage regulator, double tube heat exchanger, compartment to homogenize the air flow, drying compartment and an air drying section with silica gel. The equipment guarantees approximately 98.7% of uniform distribution of velocity and temperature of air, from 0.5 to 2.0 m/s and from 30 to 70 ◦ C, respectively, in the central drying region, keeping the relative humidity and temperature approximately constants. A detailed description of this equipment can be found elsewhere (Arrieche & Sartori, 2004). Experiments were performed to obtain drying kinetic from mass measurements and shrinkage from acquisition of digital images. The initial sample diameters were 2.66 × 10−2 and 1.61 × 10−2 m and the drying air velocities varied from 1.0 to 2.0 m/s, with air temperature maintained at 50 ± 1 ◦ C. Preliminary experiments were replicated to evaluate the reproducibility (Arrieche & Sartori, 2004). For measurement of drying air temperature and velocity, a digital thermo-anemometer was used, with a precision of 0.1 m/s for velocity and 1 ◦ C for temperature, located near the forward surface of the sample. The mass was measured with a precision of 10−6 kg. For measurement of relative humidity, thermal-couples of wet bulb and dry bulb were used (copper–constantan) with a precision of 0.25 ◦ C for temperature and 4% for relative humidity. The calculation of sample moisture content was accomplished according to the method of Lees (1982).
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5. Mechanical properties and apparent specific mass
7. Results and discussions
The mechanical properties, as the modulus of Young, the rupture stress and the rupture strain were obtained from compression tests, using cylindrical agar gel samples of 4.56 cm of length and 2.16 cm of diameter (H/D ∼ = 2), in a concentration range of 32.33–4.04 (g water/g solid). These dimensions were chosen in order to avoid cracking during compression. Once elaborated, the samples were placed in the equipment of mechanical tests, at room temperature and compressed at a strain rate of 10 mm/min (Weiner & Nussinovitch, 1994) until reaching rupture. The equipment used for mechanical tests was an Instron 5500 Universal Testing Machine (Instron International, UK), equipped with a load cell of 500 kgf (4903.32 N) and connected to a microcomputer through an analogical–digital converter. Subsequently, the instantaneous values of sample height and compression load were converted into true stress as a function of the Hencky strain:
7.1. Moisture content profiles
T ≡ E (1 − εE ),
(19)
εH ≡ − ln(1 − εE ),
(20)
where T is the true stress, E is the engineering stress, εE is the engineering strain and εT is the true strain. The modulus of Young, E, was obtained from a linear fitting of true stress and Hencky strain data: εH =
1 T . E
(21)
It is still necessary to relate the loss of moisture with the deformation due to shrinkage, εshr . This relationship (Appendix A) was obtained starting from the simulated moisture profiles and the equation of the linear volumetric shrinkage, Vg =
ms s
1 + X¯
s w
,
(22)
where Vg is the volume of the sample, ms is the mass of dry solid, s is the specific mass of the dry solid and w is the specific mass of water. The droplet diameter was calculated from the simulated average moisture content and Eq. (22). Then Vg was equated to the volume of a sphere to obtain a sphere equivalent diameter in each time of the drying simulation. The coefficient of Poisson, , which establishes the proportionality between the strain and the perpendicular stress applied was obtained from literature, because its value does not vary significantly for several materials, and it is around 0.5 for agar gel, according to Mrani et al. (1997). Other sample propriety calculated was the apparent specific mass: sa =
The operational conditions of Va = 1.0 m/s, D0 = 2.66 cm and Ta = 50 ◦ C were used in all simulations of the drying process. The model for mass transfer was considered a good approach to predict moisture content during the constant drying rate period, since the deviation was not greater than 10%. For the falling rate period, experimental data values were lower than simulated ones and it corresponded to a deviation up to 10%. This deviation can be attributed to sample degradation, which happened in this period. Because of the fissure formation, drying air reached the inner layers of the sample and evaporated the water from inside of the structure. Then, drying rate was raised and experimental moisture content values became lower than simulated ones, since the model considers water evaporation only on the surface. In spite of the high deviation of the model over the falling rate period, the simulated moisture content profiles allowed the prediction of drying stresses and physical degradation which happened during the constant drying rate period. Fig. 2 shows a simulated moisture content distribution as function of time and radial position. The steeper moisture content gradients happened near the surface with flat parabolic profiles in the centre. As a consequence of the radial mass transfer, the centre of the sample became more affected by mechanical degradation. During the constant drying rate period it was observed the formation of a central fissure, which made the samples completely hollow in the end of the drying process. The fast dehydration that happened on the surface of the samples, while the centre stayed with high moisture content, was related to this breakage in the centre.
msa , Vleq
7.2. Mechanical properties In order to solve the mechanical model, it was necessary the determination of mechanical properties as the modulus of Young and the rupture stress, from the stress curves as a function of strain, for different moisture contents. The modulus of Young could be obtained with Eq. (21), through linear regression of Hencky strain data as a function of true strain. Statistical parameters were obtained in order to evaluate reproducibility. These results are shown in Table 1. From Table 1, it is observed that the modulus of Young was determined from Eq. (23), for different concentrations, with a high t value
(23)
where sa is the apparent specific mass of the sample, msa is the mass of the sample and Vleq is the equivalent volume. 6. Solution of the equations Eq. (1) was approximated by the method of finite-difference, applied for each discrete point in a mesh structure of the system spatial domain, including the boundary conditions. The resulting set of ordinary differential equations was integrated in time by the method of Runge-Kutta, using Matlab® . For the solution of the mechanical problem, the method of Runge-Kutta was also used. Due to the fact that the boundary conditions are in opposite extremities of the spatial domain, an interactive solution procedure was necessary. The integration of the ordinary differential equations began in r = 0 with an arbitrary value for r. The value of r in r = R was obtained, from which a new initial value for r in r = 0 was supposed. Interactive procedures as Newton Raphson were used in this case.
Fig. 2. Moisture content profiles as a radius function for various times.
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Table 1 Statistical parameters of the fitting of Eq. (21) to the experimental data of Hencky strain and true stress (significance level: 5%). Moisture (kg water/kg dry solid) 32.33 28.29 24.25 20.21 16.17 12.12 8.08 4.04
Modulus of Young (MPa) 0.0030 0.0032 0.0039 0.0050 0.0068 0.0111 0.0161 0.0196
Standard error (MPa) −5
1.2469 × 10 1.4667 × 10−5 1.2933 × 10−5 2.0833 × 10−5 1.2933 × 10−5 4.7234 × 10−5 11.4371 × 10−5 11.8788 × 10−5
(error at least 141 times less than the value of the parameter) and an explained variance of at least 96.21%. The reasonable adjustment, for all concentrations also indicates the reproducibility of stress and strain data. Once obtained the values of the modulus of Young it was possible to fit these values to an equation which best represented the modulus of Young as a function of moisture content. The modulus of elasticity, or modulus of Young, is strongly dependent of composition and morphology of the system. Several equations were tested to predict the dependence of the modulus as a function of mixture composition. The best results were found with the following equations, where moisture content is the independent variable: E = E1 c (X)
(24)
E = E1 exp(−cX)
(25)
t-test (–)
Correlation coefficient
Explained variance (%)
p-value
240 218 301 240 531 235 141 165
0.9947 0.9934 0.9968 0.9946 0.9988 0.9932 0.9839 0.9808
98.94 98.69 99.36 98.92 99.75 98.65 96.80 96.21
<0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001
equation: rup = 1 c (X)
(26)
where 1 = 0.842 MPa (rupture stress for dry solid) and c = 0.911, valid for the same moisture content domain as the modulus of Young. In Fig. 5 are shown the local modulus of Young distribution according to Eq. (26), for the moisture content profiles in Fig. 2. It can be observed high values for the modulus of Young on the surface, where occurs a strong dehydration, and low values in the centre, where moisture content is kept relatively high. During 60 and 150 min of the drying process it was observed the highest difference between values of the modulus of Young for the centre and surface. This difference is strongly related with breakage in the
where E1 is the rupture stress for dry solid and c is a dimensionless constant. From Table 2, it is possible to observe that Eqs. (24) and (25) were properly fitted according to the explained variance of 98.36% and significant parameters at a level of 95% of trust. Eq. (26) had a high t test and it was the function selected to represent the modulus of Young accordingly. Fig. 3 shows Eq. (24) fitted to the values obtained for the modulus of Young and Fig. 4 presents the graph of normal probability for the residues of this fitting. The points in Fig. 4 follow a straight line and indicate that residues are normally distributed. In spite of Eq. (24) is valid for 4.04 ≤ X ≤ 32.33 (kg water/kg dry solid), it can be used for low moisture contents, because when moisture content approaches zero the parameter E1 is interpreted as being the modulus of Young of the dry solid. On the other hand, for a high water concentration, the modulus of Young also approaches zero, as expected. The stress for rupture showed a similar behaviour because it is related to the modulus of Young, and it was also fitted to a similar
Fig. 4. Graph of normal probability of the residues for the fitting of Eq. (24) to the data of the modulus of Young.
Fig. 3. Modulus of Young as a function of moisture content.
Fig. 5. Modulus of Young from centre to surface as a time function.
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Table 2 Statistical parameters of the fitting of Eqs. (24) and (25) to the determinations of the modulus of Young (significance level: 5%). Equation
Parameters (–)
(24)
E1 c
(25)
E1 c
Standard error
t-test
Correlation coefficient
Explained variancy (%)
p-value
0.0279 0.9240
0.0015 0.0046
19 201
0.9918
98.36
<0.0001 <0.0001
0.0279 0.0787
0.0015 0.0053
19 15
0.9918
98.36
<0.0001 <0.0001
centre, because low values of the modulus of Young indicate low mechanical resistance. In the end of the drying process, the modulus of Young values rise for all radial positions and approach to equilibrium. The material becomes more rigid. This uniformity of the modulus of Young values at the end indicates that drying stresses will extinguish. With the modulus of Young and the moisture distribution know, it is possible to solve the mathematical model of the mechanical problem, in order to simulate the drying stresses and strains inside the spherical food models. 7.3. Drying stresses
stress distributions. It occurred in the centre after approximately 60 min of drying. The same observation could be done with the tangential stresses, because at the centre radial and tangential stresses have both the same value. This breakage determination is in agreement with observations made by Arrieche and Sartori (2008), Gogus and Lamb (1998) and Nicoleti, Telis-Romero, and Telis (2001), during the drying of spherical gel systems and persimmons, where the formation of a central fissure was reported. Because of this fissure, the samples became completely hollow in the end of the drying process. This was indicated by the apparent specific mass values, which rose during the beginning of the drying process and decreased during the falling drying rate period.
Fig. 6 shows typical results of radial stress profiles as a function of time. It was verified that radial stresses were null at zero time, because the initial moisture content distribution of the sample was considered uniform. As expected, stresses surged together with moisture content profiles. Radial stresses rose noticeably towards the centre of the sphere. This steep increase in stress values was related to the rupture of the structure in the centre, position where there was high moisture content and low mechanical resistance. In the end of the drying process, moisture content gradients became gentle and radial stress values decreased. The simulated tangential stress values changed from negative (near the surface) to positive (near the centre), as shown in Fig. 7. Conventionally, positive and negative stress values designate tensile and compressive stresses, respectively. The reason why tangential stresses were compressive near the surface is mainly due to the fast dehydration that happens on the surface, while the centre maintains high moisture content. Values for tangential stresses were found to be much higher than radial stresses. This observation was also found by Mezhericher et al. (2008), in the modelling of particle breakage during drying. Fig. 8 shows typical results of rupture and radial stresses. The rupture of the sample could be determined by crossing these two
Fig. 7. Local tangential stress profiles as a function of time.
Fig. 6. Radial stress from center to the surface, as time function.
Fig. 8. Rupture and radial stresses as a function of time (eight positions).
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Fig. 9. Tangential and rupture stress as time functions (eight positions).
Fig. 9 shows tangential and rupture stress profiles, as time functions, for eight positions, from centre to surface. It is noticeable that tangential stress curves on the surface do not cross rupture stress curves, time respectively. So, breakage determination was restricted only to the centre of the structure, as observed experimentally. 7.4. Drying strains The strain profiles were obtained using Eqs. (6) and (7). Initially, the term of the deformation due to shrinkage (εshr ) was excluded, in order to represent just the elastic strains. Figs. 10 and 11 show typical results of the elastic radial and tangential strains, in nine radial positions, as functions of drying time. Both distributions are quite similar but the tangential deformation presents negative values. That is due to a compressive global effect of tangential stresses, which present high values near the surface, when comparing to the central region. The peaks near the surface, in the beginning of the drying process, indicate a high mechanical deformation, due to fast dehydration in this area. It is observed that the peaks of radial and tangential strains happen after 130 min. This time lies between 60 and 150 min. This is the same time interval in which the central breakage was determined.
Fig. 10. Elastic radial strain as a time function.
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Fig. 11. Elastic tangential strains as a time function.
Also, in this time interval occurs the highest difference between the central and superficial absolute values of moisture contents. It is possible to infer that the breakage is related with the moisture content distribution. These results indicate that the fast superficial dehydration can be the cause for the physical degradation of the spherical samples in the centre. Fig. 12 shows typical results obtained for the profiles of shrinkage deformation (εshr ), in nine radial positions, as a time function. The shrinkage deformation increases with time and radial position (except in the centre), due to total loss of moisture content. The deformation due to shrinkage is distinct from elastic deformation since it is irreversible; once happened during the drying of the material, an equilibrium value is reached for εshr , while the elastic deformation tends to disappear in the end of the drying process. The term of the shrinkage deformation, in Eq. (7), represents the radial deformation developed by the sample due to the mass transfer and in Eq. (10) represents the deformation of the distributed perimeter of the sphere. When this term is added to the elastic deformation, it is had the total radial and tangential deformations of the structure which are shown in Figs. 13 and 14. Comparing Figs. 12–14, it is possible to observe that the elastic radial deformation acts against the shrinkage while the tangential elastic deformation tends to speed it.
Fig. 12. Deformation due to shrinkage as a time function.
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Fig. 13. Total radial deformation as a time function. Fig. 16. Distributed perimeter with compressive effect for various fixed coordinate positions, as a time function.
Figs. 15 and 16 show results for the distributed radius and perimeter in various fixed coordinate positions, as time functions. We can observe the tensile and compressive effects of drying stresses and strains during drying. In the end of the drying process, drying stresses tend to disappear and the total strain of the structure follows the equation for the linear volumetric shrinkage (Eq. (22)). These compensatory mechanical effects are action and reaction effects, which appear in order to maintain the control volumes united and to guarantee the integrity of the structure, until the limit of the rupture stress is reached.
8. Conclusions
Fig. 14. Total tangential deformation as a time function.
The model proposed was useful to predict drying kinetics in the constant drying rate period and the moisture content profiles inside the sample. The stress simulations turned possible to predict the central fissure, which happened in the constant rate period and deeply affected the quality of the product. The breakage simulated by the mechanical model was in agreement with experimental observations. The simulation and analysis of the drying stresses and strains allowed a better understanding of the mechanical effects occurring in the food models during drying. The mass transfer process could be evaluated in a more realistic way. The degradation observed experimentally could be explained, what was not possible with a model considering only mass transfer. This work is intended to be a contribution to the scientific development of the drying process, especially for isolated particles. The modelling of the mechanical effects resulting from drying can be also formulated for other geometries rather than spheres.
Acknowledgements
Fig. 15. Distributed radius with tensile effect for various fixed coordinate positions, as a time function.
The authors wish to thank the Brazilian Research and Project Financier (PRONEX/CNPq), the National Council of Scientific and Technological Development (CNPq), the Organization to the Improvement of Higher Learning Personnel (CAPES) and the Foundation for Research Support of São Paulo State (FAPESP) for their financial aid.
L.S. Arrieche et al. / Computers and Chemical Engineering 33 (2009) 1805–1813
Appendix A. Deformation due to shrinkage The following set of equation explains the procedure used to calculate the deformation due to shrinkage, εshr . It was based on the simulated moisture profiles and the equation of linear volumetric shrinkage. In these equations (N, NT) and (M, MT) are discrete points and total number of points, in radial and time dimensions, respectively: For N = 1, 2, 3, . . ., NT; M = 1, 2, 3, . . ., MT rN,M =
3 m s 4
s
1+
s XN,M w
1/3
(equivalent radius to local moisture)
(A.1)
For N = 1; M = 1, 2, 3, . . ., MT drN,M = 0
(differential radius in the sphere centre)
(A.2)
For N = 2, 3, . . ., NT; M = 1, 2, 3, . . ., MT drN,M =
+ rN−1,M ) (rN,M
2(NT − 1) (differential radius for other radial positions)
(A.3)
For N = 1, 2, 3, . . ., NT; M = 1, 2, 3, . . ., MT
N=NT
rN,M =
drN,M
(sphere radius)
(A.4)
N=1
uN,M = rN,1 − rN,M εshr,N,M =
uN,M rN,1
(radial displacement)
(deformation due to shrinkage)
(A.5) (A.6)
References Achanta, S., & Okos, M. (1996). Predicting the quality of dehydrated foods and biopolymers: research needs and opportunities. Drying Technology, 14, 1329. Arrieche, L. S., & Sartori, D. J. M. (2004). Dependence analysis of the shrinkage and shape evolution of a gel system with the forced convection drying periods. In
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M. A. Silva, & S. C. S. Rocha (Eds.), Proceedings of the 14th international drying symposium (IDS 2004) São Paulo, Brazil, (pp. 152–160). Arrieche, L. S., & Sartori, D. J. M. (2008). Fluid flow effect and mechanical interactions during drying of a deformable food model. Drying Technology, 26, 54. Banaszak, J., & Kowalski, S. J. (2005). Theoretical and experimental analysis of stresses and fractures in clay like materials during drying. Chemical Engineering and Processing, 44, 497. Clift, R., Grace, J. R., & Weber, M. E. (1978). Bubbles, drops and particles. New York: Academic Press. Committee on Food Chemicals Codex. (2003). Food chemicals codex. Washington, DC: National Academy Press. Cranck, J. (1975). The mathematics of diffusion (2nd ed.). England: Oxford University Press. Eichler, S., Ramon, O., Ladyzhinski, I., Cohen, Y., & Mizrahi, S. (1997). Collapse processes in shrinkage of hydrophilic gels during dehydration. Food Research International, 30, 719. Gogus, F., & Lamb, J. (1998). Choice of model gel systems for the food dehydration studies. Drying Technology, 16, 297. Katekawa, M. E., & Silva, M. A. (2006). A review of drying models including shrinkage effects. Drying Technology, 24, 5. Kendoush, A. A. (1995). Low Prandtl number heat transfer to fluids flowing past an isothermal spherical particle. International Journal of Heat & Mass Transfer, 16, 291. Lee, K., & Barrow, H. (1965). Some observations on transport processes in the wake of a sphere in low speed flow. International Journal of Heat & Mass Transfer, 8, 403. Lees, R. (1982). Food analysis: analytic methods and of quality control. Spain: Ed. Acribia. Mezhericher, M., Levy, A., & Borde, I. (2008). Modelling of particle breakage during drying. Chemical Engineering and Processing, 47, 1404. Mrani, I., Bénet, J. C., Fras, G., & Zrikem, Z. (1997). Two dimensional simulation of dehydration of a highly deformable gel: moisture content, stress and strains fields. Drying Technology, 15, 2165. Nicoleti, J. F., Telis-Romero, J., & Telis, V. R. N. (2001). Air-drying of fresh and osmotically pre-treated pineapple slices: fixed air temperature versus fixed slice temperature drying kinetics. Drying Technology, 19, 2175. Ponsart, G., Vasseur, J., Frias, J. M., Duquenoy, A., & Méot, J. M. (2003). Modelling of stress due to shrinkage during drying of spaghetti. Journal of Food Engineering, 57, 277. Shah, Y. T., & Porter, J. H. (1973). A model for gel drying. Journal of Applied Polymer Science, 17, 605. Thakur, R. K., Vial, C., & Djelveh, G. (2008). Effect of composition and process parameters on elasticity and solidity of foamed food. Chemical Engineering and Processing, 47, 474. Timoshenko, S., & Goodier, J. N. (1970). Theory of elasticity (3rd ed.). New York: McGraw Hill Higher Education. Weiner, G., & Nussinovitch, A. (1994). Succulent, hydrocolloid-based, texturized grapefruit products. Lebensmittel-Wissenschaft und-Technologie, 27, 394.
Contents lists available at ScienceDirect
Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng
Drying stresses and strains in a spherical food model L.S. Arrieche b,∗ , R.G. Corrêa a , D.J.M. Sartori a a b
Chemical Engineering Department, Federal University of São Carlos, P.O. Box 676, São Carlos, 13565-905, Brazil DECOM/CEUNES, Federal University of Espírito Santo, Humberto de Almeida Franklin, 257, São Mateus, ES, 29.933-480, Brazil
a r t i c l e
i n f o
Article history: Received 6 August 2008 Received in revised form 14 April 2009 Accepted 20 May 2009 Available online 27 May 2009 Keywords: Breakage Deformations Dehydration Modelling Rupture stress Modulus of Young
a b s t r a c t An approach to model the drying mechanical effects occurring in a single sphere, representing a food model, is proposed. The mechanical properties were evaluated through compression tests and drying kinetics data were obtained under a laminar fluid flow. The Fickian model was used to represent the moisture content profiles and the mechanical model was formulated from elasticity theory. Deformations and cracks predicted from numerical solution were in agreement with experimental observations. This work helped to understand the relationship between mass transfer, shrinkage, stresses, strains and physical degradation. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction When concerning quality of dehydrated foodstuffs, one of the main reasons for quality loss during drying is due to structural changes caused by shrinkage of the product. The shrinkage promotes loss in the permeability characteristics to the solvent and turns the material unstable and brittle. Different attributes of quality of the dehydrated products, such as density, crust formation, cracks and others are all related with the contraction process (Eichler, Ramon, Ladyzhinski, Cohen, & Mizrahi, 1997). With a polymeric structure similar to food and other types of biological materials, but a simple composition, the gels have been broadly used in researches as simulators (Achanta & Okos, 1996; Arrieche & Sartori, 2008; Gogus & Lamb, 1998; Mrani, Bénet, Fras, & Zrikem, 1997; Shah & Porter, 1973; Thakur, Vial, & Djelveh, 2008). Several gels are used as starch gel, agar gel and others, for the study of chemical reactions, shrinkage mechanisms and the great majority in drying studies, involving mathematical modelling. In many drying studies, the mathematical modelling is treated considering only the heat and mass transfers (Katekawa & Silva, 2006). These mathematical models may be reasonable to describe drying rates but are unable to predict shape evolutions and physical deteriorations that occur during convective drying of materials undergoing shrinkage. Under the structural point of view, few works consider the modelling of internal stresses, which appear due
to mass transfer and shrinkage (Arrieche & Sartori, 2008; Banaszak & Kowalski, 2005; Mezhericher, Levy, & Borde, 2008; Mrani et al., 1997; Ponsart, Vasseur, Frias, Duquenoy, & Méot, 2003; Shah & Porter, 1973). The non-uniform shrinkage owed to mass transfer will bring consequent shape evolution and physical deterioration, due to the appearance of stresses inside the structure. A consideration of these effects, when using spherical samples, contributes to supply the lack of mathematical modelling studies in literature, which include the prediction of shape evolutions during drying. In this sense, the main objective of our study is to analyze the drying and shape evolution of a spherical simulating material and to evaluate a simplified mathematical representation for this process, including the stress and strain fields inside the structure and its effects on quality loss by physical degradation. 2. Model presentation 2.1. Drying model Considering the radial diffusion component, the mass transfer from the inside of the solid to the drying air, can be represented by: ∂ = ∂
Rs2 def () df R2 ()
1 ∂( 2 (∂ /∂)) . ∂ 2
(1)
Eq. (1) is subject to the following initial and boundary conditions, as well as, to the symmetry condition of the radial mass flux: ∗ Corresponding author. Tel.: +55 27 3763 8680. E-mail address: [email protected] (L.S. Arrieche). 0098-1354/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2009.05.013
= 1 for = 0, ∀
(2)
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Nomenclature Bim c def df dwa d∝ D Deq E hl H H1 H2 k m r R Rs Re Sc Sh t T u Va Vl x1 X X¯ X0 Xe
rup s sa shr sim sp t T w wa 0 ∞
mass Biot number [≡ hl a x1 k/s def ] (–) dimensionless constant in Eqs. (24)–(26) (–) effective diffusion coefficient of water in solid (m2 /s) final diffusion coefficient at equilibrium (m2 /s) diffusion coefficient of water in air (m2 /s) momentum diffusion coefficient diameter (m) sphere equivalent diameter (m) modulus of Young (MPa) local convective mass transfer coefficient (m/s) height (m) dimensionless variable in Eq. (16) (–) dimensionless variable in Eq. (17) (–) coefficient of medium partition between sample and air moisture (–) mass (kg) radius (m) external radius (m) external radius (dried material) (m) Reynolds number [≡ Deq Va a /a ] (–) Schmidt number [≡ d∞ /dwa ] (–) Sherwood number [≡hl Deq /dwa ] (–) time (s) temperature (◦ C) radial displacement (m) air velocity (m/s) volume (m3 ) characteristic length of sphere (volume divided by area) [≡Deq /6] (m) local moisture content (kg water/kg dried solid) average moisture content (kg water/kg dried solid) initial moisture content (kg water/kg dried solid) equilibrium moisture content (kg water/kg dried solid)
Greek letters ˛ parameter in Eq. (20) (–) ˇ parameter in Eq. (20) (–) ε strain (–) azimuth angle (◦ ) dimensionless zenith angle [≡ / ] (–) viscosity (kg/m/s)
zenith angle (◦ )
sp separation zenith angle (◦ ) specific mass (kg/m3 ) stress (MPa) dimensionless time [≡ (df /Rs2 )(t)] (–)
Poisson coefficient (–) dimensionless radius [≡r/R] (–) dimensionless moisture [≡(X − Xe )/(X0 − Xe )] (–)
rupture dry solid sample related to shrinkage simulated separation tangential true water water in air initial free air stream
Fig. 1. Projected spherical volume element.
∂ ∂
= −3Bim () |=1 ,
∂ = 0 for ≥ 0, = 0 ∂
content
≥ 0, = 1
(3)
=1
(4)
In Eqs. (1)–(4), , and are dimensionless time, moisture content and radial position, respectively, R is the external radius, Rs is the external radius of the dried material, def is the effective diffusion coefficient of water in solid and df is the final diffusion coefficient at equilibrium. The mass Biot number (Bim ) is calculated in agreement with Cranck (1975). 2.2. Mechanical model
Subscripts a air e equilibrium E engineering ef effective eq equivalent exp experimental f final H Hencky r radial
Considering the radial moisture content distribution, there are radial and tangential stress components, according to Fig. 1. These stress components should satisfy the mechanical equilibrium condition of a volume element, in the radial direction: d r 2 + ( r − t ) = 0, r dr
(5)
where r is the radial stress, t is the tangential stress and r is the radial coordinate. Considering the gel as a predominantly elastic material, the law of Hooke is applied for the two total strains, taking
L.S. Arrieche et al. / Computers and Chemical Engineering 33 (2009) 1805–1813
into account the effect of mass transfer: 1 εr = ( r − 2 t ) + εshr , E εt =
(6)
1 ( t − ( r + t )) + εshr , E
du , dr u εt = , r εr =
(8) (9)
where u is the radial displacement. The analytical solution of Eqs. (5)–(9) is given by Timoshenko and Goodier (1970), for the case of constant mechanical properties. However, such simplifications can be inadequate in drying of food where the properties can vary with position, when moisture content presents a distribution during drying. For such cases the stress fields should be solved numerically. Eqs. (5)–(9) could be rearranged in the following form. From Eqs. (6) and (8): r − 2 t du = + εshr . E dr From Eq. (7): t =
u
1 E 1−
r
− εshr
(10)
+ r .
Inserting Eq. (11) into Eq. (5):
2
d r =− r dr
r −
1
u
1−
E
r
− εshr
(11)
+ r
Sh = 1.693
sin2
(2 + cos3 − 3 cos )
(Re)1/2 (Sc)1/2 ,
1/2
(14)
where Sh is the local Sherwood number, Re is the Reynolds number and Sc is the Schmidt number. Eq. (14) is valid for Sc < 1 and laminar flow with Re > O(102 ). From the above relation, the overall Sherwood number can be obtained as Sh = (Re Sc)1/2 [0.564H1 ( s ) + 0.156H2 ( s )]
(15)
with H1 ( sp ) = (2 + cos3 sp − 3 cos sp )
1/2
,
(16)
,
(17)
and H2 ( sp ) = (2 − cos3 sp + 3 cos sp )
1/2
where H1 and H2 are dimensionless variables. The angle for flow separation in the boundary layer, sp , described by Clift, Grace, & Weber (1978) and Lee and Barrow (1965), was determined from the following equation:
sp = 78 + 275Re−0.37
(18)
valid for 400 < Re < 3 × 105 .
.
(12) 4. Drying tests
Eqs. (10) and (12) compose a system of ordinary differential equations with the same boundary conditions used in the analytical solution: u = 0 for r = 0 and r = 0 for r = R. The numerical solution was obtained for a given moisture content distribution, established at each time in the drying process, considering the modulus of Young as a moisture content function. Once established the models for mass transfer and stress fields, it was necessary to determine the transport coefficients and mechanical properties. 3. Transport coefficients The effective mass diffusivity was obtained from the first term of the analytical solution of the Fickian model for spheres, where for long drying times only the first term of the infinity series solution is significant. This first term establishes a relation between drying rate and free water concentration. Then, for each experimental condition, it was possible to calculate the effective mass diffusivity from the following equation:
d X0 − X¯ ef = exp t X0 − Xe R2
boundary layer, over a surface of stationary spherical particle. This solution was based upon an assumption of potential flow in the energy and mass conservation equations, with a boundary layer approximation:
(7)
where εr is the total radial strain, εt is the total tangential strain, εshr is the strain due to shrinkage caused by the water loss, is the coefficient of Poisson and E is the modulus of Young. The total strains are not independent but related by geometry. In this case:
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(13)
where X¯ is the average moisture content, X0 is the initial moisture content, Xe is the equilibrium moisture content and t is the drying time. The effective mass diffusivity calculated from this procedure includes several mechanisms of mass transfer, including the effect of heat transfer, which depends on the convection condition and thickness. The coefficient of mass transfer by convection, hl , was determined using the Sherwood number, in agreement with Kendoush (1995), who proposed an analytical solution to the forced convective heat and mass transfer across a laminar incompressible
Granular solid agar was used as gelling agent to prepare the samples, because this material shows a linear volumetric shrinkage as a function of water content. In order to avoid cracks prior to the drying tests, the appropriated working concentration of agar was found to be 3%. The gel samples were prepared according to the method described by Committee on Food Chemicals Codex (2003). The drying tests were accomplished in classical air stream forced convection equipment, consisted of a circuit with a blower type radial compressor, electric heater connected to a voltage regulator, double tube heat exchanger, compartment to homogenize the air flow, drying compartment and an air drying section with silica gel. The equipment guarantees approximately 98.7% of uniform distribution of velocity and temperature of air, from 0.5 to 2.0 m/s and from 30 to 70 ◦ C, respectively, in the central drying region, keeping the relative humidity and temperature approximately constants. A detailed description of this equipment can be found elsewhere (Arrieche & Sartori, 2004). Experiments were performed to obtain drying kinetic from mass measurements and shrinkage from acquisition of digital images. The initial sample diameters were 2.66 × 10−2 and 1.61 × 10−2 m and the drying air velocities varied from 1.0 to 2.0 m/s, with air temperature maintained at 50 ± 1 ◦ C. Preliminary experiments were replicated to evaluate the reproducibility (Arrieche & Sartori, 2004). For measurement of drying air temperature and velocity, a digital thermo-anemometer was used, with a precision of 0.1 m/s for velocity and 1 ◦ C for temperature, located near the forward surface of the sample. The mass was measured with a precision of 10−6 kg. For measurement of relative humidity, thermal-couples of wet bulb and dry bulb were used (copper–constantan) with a precision of 0.25 ◦ C for temperature and 4% for relative humidity. The calculation of sample moisture content was accomplished according to the method of Lees (1982).
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5. Mechanical properties and apparent specific mass
7. Results and discussions
The mechanical properties, as the modulus of Young, the rupture stress and the rupture strain were obtained from compression tests, using cylindrical agar gel samples of 4.56 cm of length and 2.16 cm of diameter (H/D ∼ = 2), in a concentration range of 32.33–4.04 (g water/g solid). These dimensions were chosen in order to avoid cracking during compression. Once elaborated, the samples were placed in the equipment of mechanical tests, at room temperature and compressed at a strain rate of 10 mm/min (Weiner & Nussinovitch, 1994) until reaching rupture. The equipment used for mechanical tests was an Instron 5500 Universal Testing Machine (Instron International, UK), equipped with a load cell of 500 kgf (4903.32 N) and connected to a microcomputer through an analogical–digital converter. Subsequently, the instantaneous values of sample height and compression load were converted into true stress as a function of the Hencky strain:
7.1. Moisture content profiles
T ≡ E (1 − εE ),
(19)
εH ≡ − ln(1 − εE ),
(20)
where T is the true stress, E is the engineering stress, εE is the engineering strain and εT is the true strain. The modulus of Young, E, was obtained from a linear fitting of true stress and Hencky strain data: εH =
1 T . E
(21)
It is still necessary to relate the loss of moisture with the deformation due to shrinkage, εshr . This relationship (Appendix A) was obtained starting from the simulated moisture profiles and the equation of the linear volumetric shrinkage, Vg =
ms s
1 + X¯
s w
,
(22)
where Vg is the volume of the sample, ms is the mass of dry solid, s is the specific mass of the dry solid and w is the specific mass of water. The droplet diameter was calculated from the simulated average moisture content and Eq. (22). Then Vg was equated to the volume of a sphere to obtain a sphere equivalent diameter in each time of the drying simulation. The coefficient of Poisson, , which establishes the proportionality between the strain and the perpendicular stress applied was obtained from literature, because its value does not vary significantly for several materials, and it is around 0.5 for agar gel, according to Mrani et al. (1997). Other sample propriety calculated was the apparent specific mass: sa =
The operational conditions of Va = 1.0 m/s, D0 = 2.66 cm and Ta = 50 ◦ C were used in all simulations of the drying process. The model for mass transfer was considered a good approach to predict moisture content during the constant drying rate period, since the deviation was not greater than 10%. For the falling rate period, experimental data values were lower than simulated ones and it corresponded to a deviation up to 10%. This deviation can be attributed to sample degradation, which happened in this period. Because of the fissure formation, drying air reached the inner layers of the sample and evaporated the water from inside of the structure. Then, drying rate was raised and experimental moisture content values became lower than simulated ones, since the model considers water evaporation only on the surface. In spite of the high deviation of the model over the falling rate period, the simulated moisture content profiles allowed the prediction of drying stresses and physical degradation which happened during the constant drying rate period. Fig. 2 shows a simulated moisture content distribution as function of time and radial position. The steeper moisture content gradients happened near the surface with flat parabolic profiles in the centre. As a consequence of the radial mass transfer, the centre of the sample became more affected by mechanical degradation. During the constant drying rate period it was observed the formation of a central fissure, which made the samples completely hollow in the end of the drying process. The fast dehydration that happened on the surface of the samples, while the centre stayed with high moisture content, was related to this breakage in the centre.
msa , Vleq
7.2. Mechanical properties In order to solve the mechanical model, it was necessary the determination of mechanical properties as the modulus of Young and the rupture stress, from the stress curves as a function of strain, for different moisture contents. The modulus of Young could be obtained with Eq. (21), through linear regression of Hencky strain data as a function of true strain. Statistical parameters were obtained in order to evaluate reproducibility. These results are shown in Table 1. From Table 1, it is observed that the modulus of Young was determined from Eq. (23), for different concentrations, with a high t value
(23)
where sa is the apparent specific mass of the sample, msa is the mass of the sample and Vleq is the equivalent volume. 6. Solution of the equations Eq. (1) was approximated by the method of finite-difference, applied for each discrete point in a mesh structure of the system spatial domain, including the boundary conditions. The resulting set of ordinary differential equations was integrated in time by the method of Runge-Kutta, using Matlab® . For the solution of the mechanical problem, the method of Runge-Kutta was also used. Due to the fact that the boundary conditions are in opposite extremities of the spatial domain, an interactive solution procedure was necessary. The integration of the ordinary differential equations began in r = 0 with an arbitrary value for r. The value of r in r = R was obtained, from which a new initial value for r in r = 0 was supposed. Interactive procedures as Newton Raphson were used in this case.
Fig. 2. Moisture content profiles as a radius function for various times.
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Table 1 Statistical parameters of the fitting of Eq. (21) to the experimental data of Hencky strain and true stress (significance level: 5%). Moisture (kg water/kg dry solid) 32.33 28.29 24.25 20.21 16.17 12.12 8.08 4.04
Modulus of Young (MPa) 0.0030 0.0032 0.0039 0.0050 0.0068 0.0111 0.0161 0.0196
Standard error (MPa) −5
1.2469 × 10 1.4667 × 10−5 1.2933 × 10−5 2.0833 × 10−5 1.2933 × 10−5 4.7234 × 10−5 11.4371 × 10−5 11.8788 × 10−5
(error at least 141 times less than the value of the parameter) and an explained variance of at least 96.21%. The reasonable adjustment, for all concentrations also indicates the reproducibility of stress and strain data. Once obtained the values of the modulus of Young it was possible to fit these values to an equation which best represented the modulus of Young as a function of moisture content. The modulus of elasticity, or modulus of Young, is strongly dependent of composition and morphology of the system. Several equations were tested to predict the dependence of the modulus as a function of mixture composition. The best results were found with the following equations, where moisture content is the independent variable: E = E1 c (X)
(24)
E = E1 exp(−cX)
(25)
t-test (–)
Correlation coefficient
Explained variance (%)
p-value
240 218 301 240 531 235 141 165
0.9947 0.9934 0.9968 0.9946 0.9988 0.9932 0.9839 0.9808
98.94 98.69 99.36 98.92 99.75 98.65 96.80 96.21
<0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001
equation: rup = 1 c (X)
(26)
where 1 = 0.842 MPa (rupture stress for dry solid) and c = 0.911, valid for the same moisture content domain as the modulus of Young. In Fig. 5 are shown the local modulus of Young distribution according to Eq. (26), for the moisture content profiles in Fig. 2. It can be observed high values for the modulus of Young on the surface, where occurs a strong dehydration, and low values in the centre, where moisture content is kept relatively high. During 60 and 150 min of the drying process it was observed the highest difference between values of the modulus of Young for the centre and surface. This difference is strongly related with breakage in the
where E1 is the rupture stress for dry solid and c is a dimensionless constant. From Table 2, it is possible to observe that Eqs. (24) and (25) were properly fitted according to the explained variance of 98.36% and significant parameters at a level of 95% of trust. Eq. (26) had a high t test and it was the function selected to represent the modulus of Young accordingly. Fig. 3 shows Eq. (24) fitted to the values obtained for the modulus of Young and Fig. 4 presents the graph of normal probability for the residues of this fitting. The points in Fig. 4 follow a straight line and indicate that residues are normally distributed. In spite of Eq. (24) is valid for 4.04 ≤ X ≤ 32.33 (kg water/kg dry solid), it can be used for low moisture contents, because when moisture content approaches zero the parameter E1 is interpreted as being the modulus of Young of the dry solid. On the other hand, for a high water concentration, the modulus of Young also approaches zero, as expected. The stress for rupture showed a similar behaviour because it is related to the modulus of Young, and it was also fitted to a similar
Fig. 4. Graph of normal probability of the residues for the fitting of Eq. (24) to the data of the modulus of Young.
Fig. 3. Modulus of Young as a function of moisture content.
Fig. 5. Modulus of Young from centre to surface as a time function.
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Table 2 Statistical parameters of the fitting of Eqs. (24) and (25) to the determinations of the modulus of Young (significance level: 5%). Equation
Parameters (–)
(24)
E1 c
(25)
E1 c
Standard error
t-test
Correlation coefficient
Explained variancy (%)
p-value
0.0279 0.9240
0.0015 0.0046
19 201
0.9918
98.36
<0.0001 <0.0001
0.0279 0.0787
0.0015 0.0053
19 15
0.9918
98.36
<0.0001 <0.0001
centre, because low values of the modulus of Young indicate low mechanical resistance. In the end of the drying process, the modulus of Young values rise for all radial positions and approach to equilibrium. The material becomes more rigid. This uniformity of the modulus of Young values at the end indicates that drying stresses will extinguish. With the modulus of Young and the moisture distribution know, it is possible to solve the mathematical model of the mechanical problem, in order to simulate the drying stresses and strains inside the spherical food models. 7.3. Drying stresses
stress distributions. It occurred in the centre after approximately 60 min of drying. The same observation could be done with the tangential stresses, because at the centre radial and tangential stresses have both the same value. This breakage determination is in agreement with observations made by Arrieche and Sartori (2008), Gogus and Lamb (1998) and Nicoleti, Telis-Romero, and Telis (2001), during the drying of spherical gel systems and persimmons, where the formation of a central fissure was reported. Because of this fissure, the samples became completely hollow in the end of the drying process. This was indicated by the apparent specific mass values, which rose during the beginning of the drying process and decreased during the falling drying rate period.
Fig. 6 shows typical results of radial stress profiles as a function of time. It was verified that radial stresses were null at zero time, because the initial moisture content distribution of the sample was considered uniform. As expected, stresses surged together with moisture content profiles. Radial stresses rose noticeably towards the centre of the sphere. This steep increase in stress values was related to the rupture of the structure in the centre, position where there was high moisture content and low mechanical resistance. In the end of the drying process, moisture content gradients became gentle and radial stress values decreased. The simulated tangential stress values changed from negative (near the surface) to positive (near the centre), as shown in Fig. 7. Conventionally, positive and negative stress values designate tensile and compressive stresses, respectively. The reason why tangential stresses were compressive near the surface is mainly due to the fast dehydration that happens on the surface, while the centre maintains high moisture content. Values for tangential stresses were found to be much higher than radial stresses. This observation was also found by Mezhericher et al. (2008), in the modelling of particle breakage during drying. Fig. 8 shows typical results of rupture and radial stresses. The rupture of the sample could be determined by crossing these two
Fig. 7. Local tangential stress profiles as a function of time.
Fig. 6. Radial stress from center to the surface, as time function.
Fig. 8. Rupture and radial stresses as a function of time (eight positions).
L.S. Arrieche et al. / Computers and Chemical Engineering 33 (2009) 1805–1813
Fig. 9. Tangential and rupture stress as time functions (eight positions).
Fig. 9 shows tangential and rupture stress profiles, as time functions, for eight positions, from centre to surface. It is noticeable that tangential stress curves on the surface do not cross rupture stress curves, time respectively. So, breakage determination was restricted only to the centre of the structure, as observed experimentally. 7.4. Drying strains The strain profiles were obtained using Eqs. (6) and (7). Initially, the term of the deformation due to shrinkage (εshr ) was excluded, in order to represent just the elastic strains. Figs. 10 and 11 show typical results of the elastic radial and tangential strains, in nine radial positions, as functions of drying time. Both distributions are quite similar but the tangential deformation presents negative values. That is due to a compressive global effect of tangential stresses, which present high values near the surface, when comparing to the central region. The peaks near the surface, in the beginning of the drying process, indicate a high mechanical deformation, due to fast dehydration in this area. It is observed that the peaks of radial and tangential strains happen after 130 min. This time lies between 60 and 150 min. This is the same time interval in which the central breakage was determined.
Fig. 10. Elastic radial strain as a time function.
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Fig. 11. Elastic tangential strains as a time function.
Also, in this time interval occurs the highest difference between the central and superficial absolute values of moisture contents. It is possible to infer that the breakage is related with the moisture content distribution. These results indicate that the fast superficial dehydration can be the cause for the physical degradation of the spherical samples in the centre. Fig. 12 shows typical results obtained for the profiles of shrinkage deformation (εshr ), in nine radial positions, as a time function. The shrinkage deformation increases with time and radial position (except in the centre), due to total loss of moisture content. The deformation due to shrinkage is distinct from elastic deformation since it is irreversible; once happened during the drying of the material, an equilibrium value is reached for εshr , while the elastic deformation tends to disappear in the end of the drying process. The term of the shrinkage deformation, in Eq. (7), represents the radial deformation developed by the sample due to the mass transfer and in Eq. (10) represents the deformation of the distributed perimeter of the sphere. When this term is added to the elastic deformation, it is had the total radial and tangential deformations of the structure which are shown in Figs. 13 and 14. Comparing Figs. 12–14, it is possible to observe that the elastic radial deformation acts against the shrinkage while the tangential elastic deformation tends to speed it.
Fig. 12. Deformation due to shrinkage as a time function.
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Fig. 13. Total radial deformation as a time function. Fig. 16. Distributed perimeter with compressive effect for various fixed coordinate positions, as a time function.
Figs. 15 and 16 show results for the distributed radius and perimeter in various fixed coordinate positions, as time functions. We can observe the tensile and compressive effects of drying stresses and strains during drying. In the end of the drying process, drying stresses tend to disappear and the total strain of the structure follows the equation for the linear volumetric shrinkage (Eq. (22)). These compensatory mechanical effects are action and reaction effects, which appear in order to maintain the control volumes united and to guarantee the integrity of the structure, until the limit of the rupture stress is reached.
8. Conclusions
Fig. 14. Total tangential deformation as a time function.
The model proposed was useful to predict drying kinetics in the constant drying rate period and the moisture content profiles inside the sample. The stress simulations turned possible to predict the central fissure, which happened in the constant rate period and deeply affected the quality of the product. The breakage simulated by the mechanical model was in agreement with experimental observations. The simulation and analysis of the drying stresses and strains allowed a better understanding of the mechanical effects occurring in the food models during drying. The mass transfer process could be evaluated in a more realistic way. The degradation observed experimentally could be explained, what was not possible with a model considering only mass transfer. This work is intended to be a contribution to the scientific development of the drying process, especially for isolated particles. The modelling of the mechanical effects resulting from drying can be also formulated for other geometries rather than spheres.
Acknowledgements
Fig. 15. Distributed radius with tensile effect for various fixed coordinate positions, as a time function.
The authors wish to thank the Brazilian Research and Project Financier (PRONEX/CNPq), the National Council of Scientific and Technological Development (CNPq), the Organization to the Improvement of Higher Learning Personnel (CAPES) and the Foundation for Research Support of São Paulo State (FAPESP) for their financial aid.
L.S. Arrieche et al. / Computers and Chemical Engineering 33 (2009) 1805–1813
Appendix A. Deformation due to shrinkage The following set of equation explains the procedure used to calculate the deformation due to shrinkage, εshr . It was based on the simulated moisture profiles and the equation of linear volumetric shrinkage. In these equations (N, NT) and (M, MT) are discrete points and total number of points, in radial and time dimensions, respectively: For N = 1, 2, 3, . . ., NT; M = 1, 2, 3, . . ., MT rN,M =
3 m s 4
s
1+
s XN,M w
1/3
(equivalent radius to local moisture)
(A.1)
For N = 1; M = 1, 2, 3, . . ., MT drN,M = 0
(differential radius in the sphere centre)
(A.2)
For N = 2, 3, . . ., NT; M = 1, 2, 3, . . ., MT drN,M =
+ rN−1,M ) (rN,M
2(NT − 1) (differential radius for other radial positions)
(A.3)
For N = 1, 2, 3, . . ., NT; M = 1, 2, 3, . . ., MT
N=NT
rN,M =
drN,M
(sphere radius)
(A.4)
N=1
uN,M = rN,1 − rN,M εshr,N,M =
uN,M rN,1
(radial displacement)
(deformation due to shrinkage)
(A.5) (A.6)
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