Journal of Food Engineering 72 (2006) 364–371 www.elsevier.com/locate/jfoodeng
Analysis of mass transfer parameters (changes in mass flux, diffusion coefficient and mass transfer coefficient) during baking of cookies Eren Demirkol, Ferruh Erdog˘du *, T. Koray Palazog˘lu Department of Food Engineering (Gıda Mu¨h. Bo¨l.), University of Mersin, C¸iftlikko¨y-Mersin 33343 Turkey Received 25 June 2004; received in revised form 15 October 2004; accepted 21 December 2004 Available online 17 February 2005
Abstract Increasing trade of ready to eat foods such as cookies highlights an interest in quality defects (cracking, checking due to nonhomogenous moisture distribution) during baking. The most important parameter resulting in adverse quality changes is nonhomogenous moisture distribution due to diffusion and external convective mass transfer coefficient. To control and optimize a baking process for quality purposes, actual values of mass transfer parameters should be known. Infinite external convective mass transfer coefficient and constant diffusion coefficient approaches are numerously applied in the literature. However, in natural convection cases, mass transfer coefficient value may not be infinite. In addition, the diffusion coefficient may be time and/or temperature dependent. To compare the finite and infinite external mass transfer and diffusion coefficient approaches, in this study, baking cookies was taken as an experimental case. For this purpose, moisture changes for cookies were estimated based on weight changes and initial moisture contents during baking at 190, 200 and 210 C. Then, infinite external convective mass transfer coefficient and constant diffusion coefficient value approaches were applied and compared to finite-variable external convective mass transfer coefficient and variable diffusion coefficient value approaches. These values were found to vary during baking, and they were a function of internal cookie temperature changes. It was also concluded that the variation in these values should be determined for effective mass transfer and optimization models. 2005 Elsevier Ltd. All rights reserved. Keywords: Mass flux; Mass transfer coefficient; Diffusion coefficient; Baking
1. Introduction Baking is a complex process that results in a series of physical and chemical changes in the product (Sablani, Marcotte, Baik, & Castaigne, 1998). Physically, it is a process involving simultaneous heat and mass transfer phenomena, and baking time and temperature are industrially important process considerations affecting the quality (Fahloul, Trystram, Duquenoy, & Barbotteau, 1994; Sablani et al., 1998). Dimensions and moisture content changes of cookies are significant, and large variability in these variables may cause break*
Corresponding author. Tel.: +90 5338120686; fax: +90 3243610032. E-mail addresses: ferruherdogdu@mersin.edu.tr, ferruherdogdu@ yahoo.com (F. Erdog˘du). 0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2004.12.016
down (cracking, checking, etc) problems (Cronin & Preis, 2000). All these are affected by baking time and temperature. Depending on the cookie type, baking time and temperature are generally 6 to 10 min or more, generally at 180 to 220 C (Manley, 1998). Chemical, rheological and structural changes, depending on the moisture content changes as a function of time-temperature combination of the process, are also important on the final product quality (Thorvaldson & Janestad, 1999). Therefore, another important parameter, other than baking time and temperature, that affects the quality is moisture content of the cookies (Manley, 1998). Moisture removal and distribution in baked products may be assumed to occur with external convective mass transfer and internal diffusion which are strong functions of temperature. During baking, heat is transferred
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Nomenclature surface area of cookies (m2) intercept of the linear part of the concentration ratio versus time curve a slope of the concentration ratio versus time curve (1/s or 1/min) c L Bi Biot number, Bi ¼ kD C1 medium (air) moisture concentration (kg water/kg dry matter or kg mol/m3) Ci initial moisture concentration of cookies (kg water/kg dry matter or kg mol/m3) Cs surface moisture concentration of cookies (kg water/kg dry matter or kg mol/m3) C(x, t) moisture concentration at a given location at a given time (kg water/kg dry matter or kg mol/m3) A A1
mainly by convection from heating medium (baking air) and by radiation from oven walls to the product surface followed by conduction inside the product. At the same time, moisture starts diffusing outward to the surface (Ahmad, Morgan, & Okos, 2001) with a simultaneous heat and mass transfer mechanism (Baik, Grabowski, Triui, Marcotte, & Castaigne, 1999). Thus, moisture content of the sample and its variation may be controlled by knowing the heat (convective heat transfer coefficient) and mass (external convective mass transfer coefficient and diffusion coefficients) transfer parameters. In the literature, heat transfer during baking has been numerously studied, and a lumped system approach has been widely applied to determine the external convective heat transfer coefficient (Huang & Mittal, 1995; Nitin & Karwe, 2001). For the case of mass transfer coefficient, an approach is to assume an infinite value resulting in estimating a constant effective diffusion coefficient (D). However, it is obvious that the diffusion coefficient value might be changing when the baking process proceeds as a result of increase in internal temperature. In fact, the sensitivity of moisture diffusivity to temperature changes is widely accepted (Guillard, Broyart, Guilbert, Bonazzi, & Gontard, 2004). Karathanos, Villalobos, and Saravacos (1990) discussed the use of method of slopes to estimate the effective variable moisture diffusivity at various moisture contents of drying samples and concluded that the method of slopes could be used to determine the effective diffusivity. Ozilgen and Heil (1994) used an empirical expression for moisture diffusivity in their mathematical model for transient heat and mass transport in a baking biscuit. Baik and Marcotte (2002) applied an Arrhenius type of equation to model the effect of temperature on the moisture diffusivity of
volume averaged moisture concentration of cookies at a given time (kg water/kg dry matter or kg mol/m3) D diffusion coefficient (m2/s) Fo Fourier number, Fo ¼ D t L2 kc mass transfer coefficient (m/s) L thickness of cookies (m) MWwater molecular weight of water (18 kg/kg mol) m mass of cookie during baking (kg) ms Initial moisture content of the cookies at the beginning of a baking process in a given time interval (kg) NA mass flux (kg mol/m2 s) t baking time (min) CðtÞ
a baking cake. Simal, Femenia, Garau, and Rossello (2005) also applied an Arrhenius type of equation to determine the variation of diffusion coefficient of kiwi fruit as a function of temperature during drying. Diffusion coefficient values changed from 3 · 10 10 (at 30 C) to 17.21 · 10 10 m2/s (at 90 C). In addition to the variation in the diffusion coefficient, another important consideration might be a finite variable convective mass transfer coefficient. This would be especially true under natural convection baking conditions. Fahloul et al. (1994) used a mass transfer coefficient of 2 · 10 8 to 4 · 10 8 kg/m2 s Pa. Broyart and Trystram (2002) used a steady-state mathematical model to calculate heat and mass transfer parameters during baking of biscuit type products. Tiwari, Kumar, and Prakash (2004) evaluated a variable external convective mass transfer coefficient during drying of jaggery. They found out that the external convective mass transfer coefficient was a strong function of temperature and relative humidity. Markowski (1997) developed a methodology to determine the mean value of external convective mass transfer coefficient under a simultaneous heat and mass transfer process. A general approach to determine the external convective mass transfer coefficient has been to use a heat-mass convection analogies, e.g., Chilton–Colburn analogy (Cengel, 1998). However, there are limitations of the heat-mass convection analogies. An important limitation in case of baking would be that surface moisture concentration may not be the same over the entire surface due to drying. Heat transfer correlations are developed for smooth and constant surface temperature situations. In addition to those, heat-mass convection analogy is valid for low mass flux cases so that mass transfer does not affect the surface flow (Cengel, 1998).
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Of course, it would not be an easy task to draw a line between low and high mass fluxes. Therefore, it would be valuable to analyze and determine the mass transfer parameters experimentally during baking under natural and forced convection conditions. Then, the moisture distribution inside the product might be predicted, and these values might be used in advanced simultaneous heat-mass transfer optimization studies. Stigter, Scheerlinck, Nicolai, and Van Impe (2001) applied the classical optimal theory to a heat conduction model with convective boundary conditions to develop optimal heating strategies for a convection oven. It would be more valuable to include the mass transfer, e.g. moisture content distribution, as a part of the objective functions or constraints in these optimization models. For these cases, a mathematical model to describe and quantify the simultaneous heat and mass transfer during baking would be required, and it relies on the heat and mass transfer parameters. Therefore, the objective of this study was to discuss the mathematical approaches to experimentally determine the mass transfer parameters (diffusion and external convective mass transfer coefficient) during baking under natural and forced convection conditions using mass changes of the baked cookies.
2. Materials and methods 2.1. Experimental method The cookies were prepared using the following formula according to AACC Method: 10–50D (112.5 g wheat flour, 65 g sugar, 1.1 g salt, 1.25 g baking powder, 16.5 g dextrose solution, 32 g shortening and 26 ml distilled water). A single stage mixing method was used to prepare the cookie dough. All ingredients were mixed together for 3 min to obtain a homogenous mixture, and then the dough was cut to form a short cylindrical shape (63 mm in diameter and 7.8 mm in thickness). The cookies were baked at 190, 200 and 210 C in an oven (Memmert UP 600, 256L) under natural (0% fanning) and forced (100% fanning) convection conditions for 20 min. Mass losses of the samples were recorded at 2 min intervals using a balance (Scaltec SPB54, 0.01d). Since it was not possible to accommodate the balance inside the oven, 10 cookie samples from the same dough batch were used at each baking temperature. Since the same dough batch was used, and each sample was placed in the oven after the temperature equilibration, there did not seem to be any problem in the use of this experimental design. In addition, as explained below, the fact that the moisture content ratios will be used in the governing equation also enabled the use of this type of design. This way, the first sample was left in the oven for 2 min, weighed, and the second sample was baked
for 4 min, weighed, and data at 2 min intervals were obtained. Initial moisture content of the samples was determined by vacuum oven drying method for 2 h at 98–100 C and 25 mm-Hg pressure (AACC Method 44–32). Three replicates for 10 cookies were conducted at each temperature. Then, mass flux values for each condition were determined from experimental data, and the required governing differential equation and boundary conditions, given below, were applied to the experimental data to determine the mass transfer parameters. 2.2. Infinite external convective mass transfer and constant diffusion coefficient approach To determine the mass transfer parameters, governing mass transfer differential equation (Eq. (1)) for infinite slab geometry was solved using the given initial and boundary conditions (Eq. (2)). o2 C 1 oC ; Cðx; tÞ; ¼ ox2 D ot
0<x<L
Cðx; tÞjt¼0 ¼ C i oCðx; tÞ ¼0 ox x¼0 oCðx; tÞ ¼ k c ðCðx; tÞjx¼L C 1 Þ D ox x¼L
ð1Þ
ð2Þ
Solving Eq. (1) with the above given conditions led to: Cðx; tÞ C 1 Ci C1 1 X 2 sinðkn Þ x cos kn ¼ kn þ sinðkn Þ cosðkn Þ L n¼1 D t exp k2n 2 L
ð3Þ
where k tanðkÞ ¼
kc L ¼ Bi D
ð4Þ
To determine the diffusion coefficient (D), infinite and finite mass transfer coefficient approaches may be applied. An infinite external convective mass transfer coefficient, therefore an infinite Biot number approach has been widely used in the literature (Karathanos et al., 1990; Turhan & Ozilgen, 1991; Baik & Marcotte, 2002; Guillard et al., 2004). However, especially in the natural convection conditions, this approach may not be true and may result in inaccurate results. These approaches, either use of infinite or finite mass transfer coefficient values are based on a constant diffusion coefficient. It may not be constant either in this case due to the heating of cookie samples since it is a strong function of temperature and moisture content.
E. Demirkol et al. / Journal of Food Engineering 72 (2006) 364–371
When an infinite external convective mass transfer coefficient approach is adapted, roots of Eq. (4) are determined as kn ¼ ð2 n 1Þ p ðn ¼ 1; 2; . . .Þ. Eq. (3) gives 2 the moisture concentration at a given time and a given location in an infinite slab shaped cookie. Since the experimental results were obtained for total moisture loss through the whole cookie, Eq. (3) was integrated throughout the total volume to result in Eq. (5) to enable the use of experimental data. The Eq. (5) gives the averaged moisture loss in the whole cookie as a function of time. CðtÞ C 1 Ci C1 1 X 2 sin2 ðkn Þ 2 D t exp kn 2 ¼ kn ½kn þ sinðkn Þ cosðkn Þ L n¼1 ð5Þ By applying the infinite external convective mass transfer coefficient, Eq. (5) becomes: " # 1 CðtÞ C 1 8 X 1 2 D t ¼ 2 exp kn 2 p n¼1 ð2 n 1Þ2 Ci C1 L ð6Þ Using experimental data and Eq. (6), a constant diffusion coefficient value (D) for moisture removal was determined using the procedure described below. 1 After a certain time, the concentration ratio ðCðtÞ C Þ C i C 1 versus time values change linearly. When Eq. (6) was approximated for n = 1 (only the 1st term of the series in Eq. (6) is used) to represent the linear part of this curve, the slope (a) basically becomes a function of diffusion coefficient: a ¼ k21 D¼
D ) L2
a 2 a L ¼ 2 L2 2 p k1 2
ð7Þ
This approach, as explained, is based on a constant diffusion coefficient value and an infinite external convective mass transfer coefficient. Therefore, the estimated diffusion coefficient should be defined as an effective diffusion coefficient. If the numerical value for the mass transfer coefficient were known, it might be easily used to determine k1 through Eq. (4) and then to determine the diffusion coefficient. 2.3. Finite mass transfer coefficient approach Mass transfer coefficient was determined applying the mass balance for moisture throughout the cookie sample, as defined by Markowski (1997): Z Z oCðtÞ dV ¼ N A ðtÞ dA ð8Þ ot V A
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where C is in kg-mol/m3 and NA is in kg mol/m2 s. The mass transfer flux (NA) is defined as: NA ¼
m2 m1 1 ¼ k c ðC s C 1 Þ MWwater A t
ð9Þ
Applying the mean value theorem to Eq. (8) results in: NA ¼
V oCðtÞ ¼ k c ðC s C 1 Þ A ot
ð10Þ
Then, the mass transfer coefficient (kc in m/s) is determined as: kc ¼
V oCðtÞ 1 A ot C s C 1
ð11Þ
This equation may be effectively used to evaluate a mean mass transfer coefficient during baking. It is obvious that the equation C = M Æ qs describes the relation between moisture concentration (C; kg/m3) and the moisture content (M; kg water/kg dry matter) where qs is the density of dry matter (kg/m3). Applying this relation to Eq. (11) results in: kc ¼
V oMðtÞ 1 A ot Ms
ð12Þ
where the Ms value may be taken as the initial total moisture content (kg water/kg dry matter) at the beginning of the baking process. The VA value is the total thickness of the cookie sample. During baking, the thickness might increase due to the volume expansion. If the change in thickness is determined, the L ¼ VA value may also be used as a function of time (L(t)). Since the mass flux will be a function of time, it is inevitable that the external convective mass transfer coefficient will be variable too. This makes sense due to the effect of internal and surface temperature of the cookies and of their variation during baking. Then, a time-averaged mass transfer coefficient value may be determined by integrating the time variable mass transfer coefficient values through the whole process time: Rt k c ðtÞ dt ð13Þ k¼ 0 t This constant finite external convective mass transfer coefficient value may be used to determine a constant diffusion coefficient. For these calculations, the diffusion coefficient may be determined by combining Eqs. (4) and (7): k tanðkÞ ¼
kc L 2
a L k2
ð14Þ
Solving for k value using a numerical procedure (e.g., Bisection method may be quite easy to apply) then leads to the value of Biot number and therefore the D-value through Eq. (4). This way, the estimated D-value is again based on a constant value approach.
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2.4. Variable diffusion coefficient approach As explained by Erdogdu (2004) and Unal, Erdogdu, Ekiz, and Ozdemir (2004), since the concentration ratios at certain time intervals are determined experimentally with the known initial and baking air moisture content, the equation to solve the D-value at the given time interval, based on Eq. (5), may be written as: CðtÞ C 1 2 2 D t w¼ ¼ A1 exp k1 2 k1 Ci C1 L 2 D t þ A2 exp k22 2 þ . . . ð15Þ k2 L sin2 kn kn þ sin kn cos kn
for n ¼ 1; 2; . . .
ð16Þ
In Eq. (15), the only unknown is the diffusion coefficient. When the Biot number, and therefore the k values through Eq. (4), is known, the diffusion coefficient may be easily obtained applying Newton–Raphson or any other numerical technique to Eq. (15). When Newton–Raphson method is applied, iterative solution of Eqs. (17)–(19) give the result for diffusion coefficient at any given time with a known concentration ratio (Erdogdu, 2004). 1 X 2 2 D t f ðDÞ ¼ An exp kn 2 ð17Þ w¼0 kn L n¼1 "
1 X 2 kn t 2 D t A exp k f ðDÞ ¼ n n L2 L2 n¼1 0
Dnþ1 ¼ Dn
f ðDn Þ f 0 ðDn Þ
3.1. Infinite external convective mass transfer and constant diffusion coefficient approach Using the concentration ratio changes and Eq. (5), constant D-values were determined, h Fig.i1 and Fig. 2
where A values are given by: An ¼
concentrations of the cookies at each baking time were determined, and these values were used to determine the concentration ratios (Eqs. (5) or (6)) and then the diffusion coefficient changes. The mass fluxes were directly determined from the mass changes of the cookies (Eq. (9)), and the mass transfer coefficients were also evaluated from these data (Eq. (12)). All experiments were done three times, and results were reported as the averages of these replicates.
# ð18Þ
ð19Þ
The constant diffusion coefficient assumption may also be checked with this procedure using the experimental data obtained at different times. With a finite-variable mass transfer and variable diffusion coefficient approach, a complex iteration procedure combined with an optimization methodology should be applied through the Eqs. (4), (5) and (17)–(19) since, in this case, the unknowns in Eq. (4) would be both the D and k values.
3. Results and discussion As explained in Section 2, mass changes of the cookies were recorded during baking. Average initial moisture content of the cookie dough prepared for different baking treatments was 21% (d.b., kg water/kg dry matter). The baking medium moisture concentration was assumed to be 0 due to a very high temperature and a large oven volume (0.256 m3). Based on the initial moisture content and weight changes, the moisture
show the concentration ratio
ln
CðtÞ C 1 C i C 1
changes at
each baking temperature for natural (0% fanning) and forced (100% fanning) convection conditions. The diffusion coefficient values determined based on the slope of the linear part of these curves were shown in Table 1. This method is widely used in the literature, and it might be valid for a constant diffusion coefficient when the external convective mass transfer coefficient is known. However, there is an interesting fact of this method and the analytical solution used to represent whether it may be used to determine the diffusion coefficient or to evaluate whether the diffusion coefficient is constant or not. In the analytical solution when using the 1st termi h of the series, natural logarithm of A1 ¼
2 sin2 ðk1 Þ k1 ½k1 þsinðk1 Þ cosðk1 Þ
gives the intercept of the linear part of the concentration ratio curve. The value of the intercept must change between certain values (0 and 0.209) whether the Biot number is 0 or infinite, respectively. For these cases, k value, according to Eq. (4), becomes 0 or p2 resulting in ln(A1) to be 0 or 0.209: lim A1 ¼ 1 ) lnðA1 Þ ¼ 0 k!0
limp A1 ¼ k!2
8 p2
ð20Þ
) lnðA1 Þ ¼ 0:209
Experimental results showed that the intercept of the linear part of the concentration ratio versus time curve Concentration Ratio
368
0 -0.5 190 °C -1
200 °C 210 °C
-1.5 -2 0
4
8
12
16
20
Baking Time(min) h i 1 Fig. 1. The concentration ratio ln CðtÞ C changes for different C i C 1
baking temperatures under natural convection conditions (0% Fanning).
E. Demirkol et al. / Journal of Food Engineering 72 (2006) 364–371
-1 190 °C
-1.5
200 °C
-2
210 °C
-2.5 -3 -3.5
0
4
8
12
16
20
Baking Time (min) h i 1 changes for different Fig. 2. The concentration ratio ln CðtÞ C C i C 1 baking temperatures under forced convection conditions (100% Fanning).
Table 1 Constant diffusion coefficients (m2/s) determined based on an infinite mass transfer coefficient for natural and forced convection conditions Baking temperature ( C)
190 200 210
D · 108 (m2/s) Natural convection condition
Forced convection condition
1.99 ± 0.08 2.49 ± 0.08 3.11 ± 0.08
3.62 ± 0.03 4.02 ± 0.08 6.62 ± 0.10
y = -0.0548x + 0.2654
0
R2= 0.9923
10
-0.5 -1 -1.5
190 °C 0% Fanning
-2
210 °C 100% Fanning
y = -0.2345x + 1.4081 2 R = 0.964
-2.5
8 6 4
5
Concentration Ratio
mass transfer coefficients during baking. Fig. 4 shows the change of mass flux (NA) versus baking time at 190 to 210 C under natural (0% fanning) and forced (100% fanning) convection conditions. The NA values increased steadily under natural convection conditions. However, under forced convection conditions, they increased until a certain baking time ( 12–14 min) and then decreased presumably due to the reduction in the moisture content of the baked cookies. In parallel to these results, the external convective mass transfer coefficient values also showed a similar trend and increased steadily. Under both natural and forced convection conditions, this increase continued until the end of baking (Fig. 5). In order to determine an average mass transfer coefficient value, time variable values were integrated through the whole baking time (Eq. (13)). The time averaged mass transfer coefficient values were 4.05 · 10 6, 4.84 · 10 6 and 5.68 · 10 6 m/s under natural (0% fanning) and 6.19 · 10 6, 6.54 · 10 6 and 7.10 · 10 6 m/s under forced (100% fanning) convection conditions at 190, 200 and 210 C, respectively. In the literature, a common way to use Eq. (10) to determine the variable mass transfer coefficients from the mass flux data is as follows:
NAx10 (kg-mol/m² -s)
Concentration Ratio
0 -0.5
369
-3 -3.5 0
4
8
12
16
20
2 0
Baking Time (min)
0
4
8
12
16
20
Baking Time (min)
Fig. 3. Intercepts of linear parts of the concentration ratio versus time curves for natural (0% fanning) convection at 190 C and forced (100% fanning) convection at 210 C cases.
3.2. Finite external convective mass transfer coefficient approach Eqs. (9) and (12) were applied to the experimental data to determine the mass flux changes and then the
200˚C 0% Fanning
210˚C 0% Fanning
190˚C 100% Fanning
200˚C 100% Fanning
210˚C 100% Fanning
Fig. 4. Change in mass flux (NA) values for natural (0% fanning) and forced convection (100% fanning) conditions.
Mass Transfer Coefficient x106(m/s)
always passed through a positive value, proving that the diffusion coefficient was not constant, or the use of the slope method was not suitable. Fig. 3 shows this for natural (190 C, 0% fanning) and forced (210 C, 100% fanning) convection conditions. As seen, the slope was steeper at 210 C–100% fanning because of faster heating, faster drying and greater internal temperature changes. Due to the fact that the diffusion coefficient is a function of temperature, this result might be expected, and therefore the experimental results should be carefully investigated before applying the slope method.
190˚C 0% Fanning
10 8 6 4 2 0 0
4
8
12
16
20
Baking Time (min) 190˚C 0% Fanning
200˚C 0% Fanning
210˚C 0% Fanning
190˚C 100% Fanning
200˚C 100% Fanning
210˚C 100% Fanning
Fig. 5. Change in mass transfer coefficient (kc) values for natural (0% fanning) and forced (100% fanning) convection conditions.
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N A ¼ k c ðC s C 1 Þ ¼ k c
Ps P1 R Ts R T1
ð21Þ
where NA is in kg mol/m2 s, kc is in m/s, C’s are in kg mol/m3, Ps is surface vapor pressure (atm), Ts is surface temperature (K), R is universal gas constant (atm L/kg mol K), and P1 (atm) and T1 (K) are vapor pressure and temperature of baking air. To apply this equation, surface temperature should be measured experimentally first. With a known surface temperature, surface vapor pressure may be determined using the temperature-saturated vapor pressure tables or empirical equations (e.g., Antoine equation). However, the surface should be saturated to use this methodology, and in baking cookies, the surface is not expected to be saturated especially when the initial moisture content is already low (e.g., 21% d.b.). Therefore, the applied methodology developed by Markowski (1997) was proven to be useful to be applied in these situations. As explained in the Material and Methods, Eq. (14) may be easily used to determine the diffusion coefficient value when a finite mass transfer coefficient was known or when it was assumed to be infinite. However, the application of this equation is also based on a constant diffusion coefficient. Above in Section 2, it was shown that the diffusion coefficient might be changing with temperature (internal temperature increases). Therefore, a variable diffusion coefficient approach was applied to see the variation in the diffusion coefficient value. 3.3. Variable diffusion coefficient approach The experimental data and its mathematical analysis showed that the slope method was not suitable to apply because of the effects of internal temperature changes on the diffusion coefficient. Using Eqs. (15)–(19), variation in D-values was determined applying an infinite mass transfer coefficient approach. The time averaged mass transfer coefficient values might also be used for a finite mass transfer case. However, in this case, there would be
2-unknown in Eq. (4) (diffusion coefficient and k) requiring the application of iteration procedures combined with optimization methodologies. Table 2 showed the changes in diffusion coefficient values for an infinite mass transfer coefficient approach. Since the internal temparature of the cookies were low at the beginning of the baking process, the diffusion coefficient values were also low. As baking proceeds and the internal temperature increases started, the values of diffusion coefficient values also increased. All these showed that a constant diffusion coefficient approach was not applicable during baking of cookies. With a further effort, an averaged diffusion coefficient value may be easily obtained through the use of some numerical optimization techniques (i.e. least of some of squares) through the given analytical solutions and experimental data (Erdogdu, 2004). 3.4. Conclusion and suggestions In baking, control of moisture content and its distribution is significantly important since it may cause quality problems in the later stages after production. Moisture distribution is a function of mass transfer, which is affected by baking time and temperature. To optimize the process for a uniform moisture distribution, mass transfer parameters (diffusion coefficient and external convective mass transfer coefficient) and their changes with baking time and temperature should be known. In this study, methodologies to determine the diffusion and external convective mass transfer coefficients were investigated, and useful mathematical relationships were utilized to determine the time-temperature dependent mass transfer coefficients. Since baking is a simultaneous heat and mass transfer phenomena, temperature changes inside the cookies should also be determined, and combination of heat and mass transfer models might then be used to optimize the moisture distribution to result in products of higher quality.
Table 2 Variable diffusion coefficients (D · 1010 m2/s) as a function of baking time under natural (0% fanning) and forced (100% fanning) convection conditions Baking time (min)
Natural convection condition
Forced convection condition
190 C
200 C
210 C
190 C
200 C
210 C
2 4 6 8 10 12 14 16 18 20
0.32 ± 0.11 0.68 ± 0.04 2.57 ± 0.34 10.2 ± 1.23 22.2 ± 5.39 36.1 ± 2.50 51.5 ± 8.41 72.7 ± 2.76 77.0 ± 8.42 91.0 ± 5.46
0.92 ± 0.04 1.42 ± 0.33 6.19 ± 0.39 25.9 ± 1.06 34.5 ± 5.66 68.4 ± 10.6 93.2 ± 8.06 103.0 ± 0.71 121.0 ± 12.7 142.0 ± 19.1
0.54 ± 0.23 3.05 ± 1.07 11.2 ± 2.89 30.4 ± 4.50 71.1 ± 13.4 98.2 ± 8.56 125.0 ± 13.4 145.0 ± 5.77 177.0 ± 8.89 182.0 ± 4.58
1.57 ± 0.03 21.8 ± 6.65 51.0 ± 0.64 93.9 ± 4.60 133.0 ± 3.54 163.0 ± 4.95 201.0 ± 24.0 230.0 ± 18.8 247.0 ± 4.95 236.0 ± 12.0
2.09 ± 0.38 18.4 ± 2.89 59.0 ± 6.93 108.8 ± 1.53 161.0 ± 13.8 197.0 ± 4.16 217.0 ± 7.37 246.0 ± 8.72 270.0 ± 7.21 297.0 ± 4.95
7.53 ± 0.37 30.8 ± 6.50 110.0 ± 9.54 155.0 ± 24.0 203.0 ± 7.09 259.0 ± 1.41 302.0 ± 3.54 373.0 ± 21.2 423.0 ± 16.1 495.0 ± 12.7
E. Demirkol et al. / Journal of Food Engineering 72 (2006) 364–371
Acknowledgement This research was supported by the Scientific and _ ¨ BITAK), Technical Research Council of Turkey (TU Project No: TOGTAG 3201 (Agriculture, Forestry and Food Technologies Research Group).
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