Non dimensional analysis of cassava transient drying in packing beds
H. Santamaria, N. Durango, A. Bula^{†} and M. Sanjuan
Mechanical Engineering Department, Universidad del Norte, Barranquuilla, Colombia.
^{†} abula@uninorte.edu.co
Abstract - Transient mass transfer process is analyzed for cassava drying (Manihot Esculenta Crantz) in a pack bed. Experiments were performed in a thermally insulated radial dryer, considering cylindrical pieces of non peeled cassava with three different thicknesses: 4, 6, and 8 mm. The void fractions considered were 0.22, 0.49, 0.64 and 0.66, while the temperature values were 50ºC and 70ºC. The humidity removed from the cassava was measured from 10 pieces randomly selected at the beginning of the process. These pieces were weighed every 15 minutes during a three hours period. From the data gathered a non linear regression model was attained as a function of non dimensional numbers, which is valid for the following ranges: 700≤Re≤1900, 10000≤Sc≤31000, 0<Fo≤4. The fitted regression coefficient is R^{2}_{adj} = 0.87, and the average error when comparing with experimental data is 24%.
Keywords - Drying; Mass Transfer; Transient; Cassava; Food.
]]> I. INTRODUCTION
Experiments conducted for food dehydration are considered important since they allow an adequate process design that ensures the appropriate drying of the products (Shukla, 2001). Cassava has been considered for the experiments because FAO (2004) (Food and Agriculture Organization of the United Nations) has concluded that this food stock is a vital element to fight famine in emerging and developing economies.
For the drying process, researchers have developed mathematical models for different drying materials, either organic or inorganic, under different conditions. Furthermore, particular solutions have been presented to these models considering numerical methods. These numerical solutions allow solving for different parameters such as mass diffusivity (Bon et al., 1998); heat and mass transfer inside the materials to be dried (Wu et al., 2003; Chourasia and Goswami, 2007). Additionally, the simulated drying process has also been used for materials such as sugar (Rastikian and Capart, 1998), bagasse (Vijayaraj and Saravanan, 2008), fruits (Lagunez-Rivera et al., 2007), and cassava (Durango et al., 2001a).
Some authors have worked in the drying process considering an experimental approach, fitting curves for different organic materials such as mate leaves (Zanoelo et al., 2006), penicillium (Friese et al., 2004), cassava (Durango et al., 2005), using natural variables such as temperature, air velocity and size. The regression models attained have good agreement with the data gathered presenting correlation coefficients around 90%. Moreover, parametric analysis for transient drying of crops has been assessed considering the non dimensional numbers governing the phenomena (Chen, 2005; Rahman et al., 2007).
The physical changes that the drying material undergoes during the process has been under study, considering specifically how the shrinkage affects the variables involved in the drying process, such as mass diffusivity (Hernández et al., 2000; 2004). The construction of the drying equipment and the influence in the materials has also been studied (Lee et al., 1992; Shukla, 2001; Durango et al., 2001b; Freire et al., 2005).
This paper studies the transient mass transfer process presented during the drying of cassava using non dimensional parameters, which will allow a better design of industrial dryers.
II. MATHEMATICAL MODEL
The drying process is generated by the non-equilibrium condition between the water vapor pressures at the drying media and at the surface of the material to be dried. Many mechanisms describing the process have been proposed in order to describe the water migration inside biological products during the drying process, such as: diffusion, capillary flow, interior pressure variation and combinations of these. The preponderance of each one of these mechanisms depends on the material, moisture content, link between the water and the solid constituent of the material, the temperature and the size of the porous. Most of these variables are complex to measure and some of them change as the drying process advances (Treybal, 1987; Johnson, 1999). Industrial drying processes are carried out most of the time in packing beds. These packing beds are made out of pieces of the material to be dried, and they are distributed in a random compact way. This configuration is used in order to enhance heat and mass transfer at the surface exposed (Incropera and De Witt, 1996). This formation allows a large area to volume ratio, leaving spaces for the fluid to go through the packing bed. The drying process in a packing bed is considered a boundary phenomena described by the following equation in cylindrical coordinates:
(1) |
To study the drying process in packing beds, it is necessary to define the void fraction:
(2) |
The area - volume ratio for the cassava pieces inside the packing bed can be written as follows,
(3) |
(4) |
For packing beds, the mass transfer expression is presented according to Eq. 5,
(5) |
The concentration difference can also be expressed as a function of mass fraction, and Eq. 5 can also be presented as,
(6) |
Notice that sub index surf indicates moisture fraction at the surface while sub index ∞ represents the moisture fraction in equilibrium with the surrounding media. Furthermore, the mass transfer coefficient can be defined in different ways, K_{G} and K"_{G}. An approximate conversion equation for air - water vapor systems is given by,
(7) |
The diffusive phenomena inside the root can be considered one dimensional when the cylindrical surface is sealed and the diffusion takes place in the axis direction. This was the case considered as the root dried, and the process can be described by the following approximation (Incropera and De Witt, 1996),
(8) |
where
(9) |
The characteristic length for the cassava pieces is obtained from the following equation,
(10) |
The void fraction can be presented as a function of the geometric characteristics of the material to be dried by modifying Eq. 2,
(11) |
III. EXPERIMENTAL SET UP
Twenty four (24) experiments were carried out, introducing 1 kg of cassava each time. Ten (10) samples were randomly selected at the beginning of the process in order to track the moisture content of the cassava in the packing bed. These samples, showed in Fig. 1, were randomly placed in the dryer.
]]>
Figure 1. Cassava pieces located in the radial dryer.
The time-span for each one of the experiments was three hours and the temperature values considered were 50ºC and 70ºC. The temperature selection was based on the maximum value allowed, which would not destroy the nutritional content of the food stock (Durango et al., 2005). Three characteristic lengths for the experiments were considered, 0.002 m, 0.004 m, and 0.006 m. The void fraction values under experimentation were 0.22, 0.49, 0.64 and 0.66.
IV. EXPERIMENTAL RESULTS
A. Statistical Analysis
The experimental procedure requires checking if the variables selected are really significant for the process since non-dimensional numbers will be built using the information obtained from these variables. The procedure consists in averaging the amount of water removed from the ten cassava samples placed at the dryer and these values are presented in Tab. 1. A covariance analysis for a general factorial experimental design was performed taking into consideration the levels used for each one of the significant variables. According to the ANOVA table, the variables are significant if the P-value is lower than 5x10^{-2}. Table 2 and 3 present the ANOVA results for the experiment. It is noticed that the significant variables to the transient mass transfer process are Temperature and Characteristic Length.
Table 1. Moisture removed in the experiments
Table 2. ANOVA for the main variables.
Table 3. ANOVA for the second and third order interactions.
From Table 3 it is noticed that third order interaction among Temperature - Void Fraction - Characteristic Length is also important for the process. The pattern of the residuals obtained from the experiments presented a normal distribution, confirming the normality assumption for the data. Homocedasticity was also checked using Barlett's test, confirming the assumption of equal variance.
A. Data Processing
To obtain the non dimensional numbers, the properties for the air were obtained. Kinematic viscosity and density were evaluated at the air supplied temperature, obtaining the following results:
(12) |
As presented in Johnson (1999).
]]> To calculate the velocity of the air at the entrance of the radial dryer, the following equation was used,(13) |
The velocity at the packing bed is calculated according to Eq. 14.
(14) |
The Reynolds number for the packing bed was calculated using Eq. 15, and the values obtained during the experiments are presented in Tab. 4.
(15) |
Table 4. Data for calculating the Packing Bed Reynolds number.
Table 5 and 6 present the mass variation and the wet based moisture content percentage variation with time for the different cassava pieces monitored. The values presented are for the 70°C experiment.
Table 5. Cassava mass variation with time (70ºC).
Table 6. Cassava wet based moisture variation with time (70ºC).
The mass transfer coefficient at the packing bed was calculated using the data gathered for the mass variation with time, and they are presented in Tab. 5 for 70°C. These values were used to calculate the water mass flow rate for the cassava pieces placed at the packing bed through Eq. 16,
(16) |
To calculate the mass transfer area, the initial weight, the initial density and the characteristic length are used for the experiments. These values are related by Eq. 17,
(17) |
The moisture content of the cassava at the mass transfer boundary was assumed to be saturated condition at the experiment temperature and it is represented by X_{surf}. The humidity of the drying air is calculated using the laboratory humidity at the drying air temperature and it is represented by X_{∞}. From a psychrometric chart at atmospheric pressure, the values for the humidity are obtained and they are presented in Tab. 7.
Table 7. Cassava and Drying air's moisture content at the surface.
The mass transfer coefficient is calculated using the values from Tab. 7 and 5 and it is presented in Tab. 8 for , and Tab. 9 for K_{G}.
]]> Table 8. Mass transfer coefficient, (10^{3} kg/m^{2} s)for the 70ºC experiments. L_{C} = 0.002m.
Table 9. Mass transfer coefficient, K_{G} (10^{3} m/s) for the 70ºC experiments. L_{C} = 0.002m
The mass diffusivity is required to have a complete set of non dimensional numbers and in combination with Eq. 8 can be rearranged to have a final expression such as Eq. 18,
(18) |
The mass diffusivity is obtained by combining the data from Tab. 5 and Eq. 18. The values attained are presented in Tab. 10. The average mass diffusivity shown in the final column presents a quasi constant value. For this reason, it is taken as an average for all the different measurements obtained during the experimental procedure. The value used in the non dimensional numbers is 5.49x10^{-10} m^{2} / s . The value attained in these experiments is close to the value reported by Hernández et al. (2000), 5.24x10^{-10} m^{2} / s, for the shrinking cassava at 61°C. The error between these values is 4.77%. Although in this paper the shrinkage was not considered, the values attained in the experiment are similar to those reported where shrinkage is taken into consideration.
Table 10. Mass diffusivity, (10^{10} m^{2} / s) for the 70ºC experiments, L_{C} = 0.002m ]]>
Using the information obtained, the non dimensional numbers are calculated according to the following equations,
(19) | |
(20) | |
(21) |
V. MODEL PROPOSED FOR THE NONDIMENSIONAL NONLINEAR REGRESSION
In order to establish the linear regression suitable for the process, different plots were developed considering how the Sherwood number responded as the non dimensional numbers were modified. Figure 2 presents how the Sherwood number changes as the Reynolds number increases, while the Schmidt and the Fourier numbers remain constant. It is noticed that the Sherwood number increases as the Reynolds number increases, presenting an almost linear behavior.
Figure 2. Sherwood number variation with respect to the Reynolds number. (Fo = 2.0, Sc = 22550)
Figure 3 shows that the Sherwood number exponentially decays as the Fourier number increases, while the Schmidt and Reynolds numbers remain constant. This behavior is typical for organic materials during a drying process.
]]> Figure 3. Sherwood number variations with respect to the Fourier number. (Re = 1300, Sc = 22550)
The Sherwood number variation with respect to the Schmidt number while Fourier and Reynolds numbers remain constant is presented in Fig. 4. It is noticed that the Sherwood number linearly increases as Schmidt number increases.
Figure 4. Sherwood number variation with respect to the Schmidt number. (Fo = 2.0, Re = 1300)
Figures 2, 3, and 4 help to identify the trend followed by the Sherwood number according to the Reynolds, Fourier and Schmidt numbers respectively. However, plotting the Sherwood number against two variables gives a better idea about the relation among them.
Figure 5 presents the Sherwood number variation with respect to the Reynolds and Fourier numbers while the Schmidt number remains constant. At the beginning of the transient process (Fo = 0), the main factor which establishes the maximum value for the Sherwood number is the Reynolds number. In contrast, as the transient process progresses, the Sherwood number tends to a minimum, with almost no significant contribution from the Reynolds number. It is noticed that the turning point where the Fourier number takes control of the process is around 2.0. After that point, it is fair to say that the Reynolds number does not have any influence in the process.
Figure 5. Sherwood number variations with respect to the Reynolds and Fourier numbers. (Sc = 22550)
Figure 6. Sherwood number variations with respect to the Schmidt and Fourier numbers. (Re = 1300)
It is noticed that at the beginning of the transient process (Fo = 0), the maximum value attained by the Sherwood is dependent of the Schmidt number, but as the transient process continues, the Sherwood number attains a minimum value, with almost no significant contribution from the Schmidt number. Similar to the previous case, the turning point where the Fourier number takes control of the process is around 2.0. This trend is quite similar to the one presented in Fig. 5 for the Sherwood number depending on Reynolds and Fourier numbers.
Figure 7 presents the Sherwood number variation with respect to the Schmidt and Reynolds numbers as the Fourier number remains constant. The maximum value for the Sherwood number is attained when the Reynolds and Schmidt number present their maximum. It is also noticed that a minimum value is attained by the Sherwood when the Reynolds and Schmidt numbers have their minimum value.
Figure 7. Sherwood number variations with respect to the Schmidt and Reynolds numbers. (Fo = 2.0)
From Fig. 2 to 7, it is concluded that the Schmidt number and the Reynolds number must follow a power law trend, while the Fourier number must have a negative exponential factor which allows Sherwood to decrease as the transient process progresses. Due to this, the following model is proposed:
]]> Sh = d · Re^{a} · Sc^{b} · e^{-c.Fo} (22)A nonlinear regression was used to obtain the constants a, b, c, and d, and the following values were obtained,
Sh = 0.001048 Re ^{0.504} Sc ^{1.3817} e^{-1.5049 Fo} | (23) |
700 ≤ Re ≤ 1900 | |
10000 ≤ Sc ≤ 31000 | |
0 < Fo ≤ 4 |
The fitted regression coefficient is = 87% and the average error introduced when comparing with experimental data is 24%.
VI. SUMMARY
A non linear regression was used to obtain an expression which introduces non dimensional parameters in order to have a suitable expression for designing purposes. The design of experiment results showed that the temperature, characteristic length, and the temperature - characteristic length - void fraction combination, are relevant to the drying process. The model used to obtain the transient mass diffusivity showed to be adequate for the definition of the non dimensional numbers. Finally, the non dimensional model proposed, which is a power law - exponential combination, proved to be valid to correlate the moisture present in the transient process along with geometric, thermal and kinetics factors involved in the packing beds drying process.
SYMBOLS
]]>A ]]>D ]]>g ]]>K_{G} ]]>L_{c} ]]>m ]]>PM ]]>R ]]>Re ]]>Sh ]]>T ]]>Vol ]]> ]]> ]]> ]]> ]]> ]]> ]]>Mass transfer area, (m^{2}) | |
C | Mass concentration, (kg/m^{3}) |
Mass diffusivity, (m^{2}/s) | |
Fo | Fourier number, |
Gravity, (m/s^{2}) | |
H | Height, (m) |
Mass transfer coefficient, (m/s) | |
K"_{G} | Mass transfer coefficient, (kg/m^{2}s) |
Characteristic length, (m) | |
Mass flow, (kg/s) | |
Mass, (kg) | |
P | Absolute Pressure, (kPa) |
Molecular weight, (kg/mol) | |
r | Radius, (m) |
Gas universal constant, (kPa m^{3}/mol K) | |
Fitted regression coefficient | |
Reynolds number, Re=(L_{c} V_{pb}/v) | |
Sc | Schmidt number, |
Sherwood number, | |
t | Time elapsed in the drying process, (s) |
Temperature (KºC) | |
V | Velocity, (m/s) |
Volume, (m^{3}) | |
X | Moisture content, |
Greek Symbols | |
ε | Void fraction ]]> |
ρ | Density, (kg/m^{3}) |
υ | Kinematics viscosity, (m^{2}/s) ]]> |
Sub index | |
0 | Initial condition |
∞ | At the infinite or inlet |
cas | Cassava |
equil | At the equilibrium or final condition |
ext | External |
H_{2}O | Humidity |
int | Internal |
t | At any time of the transient process |
pb | At the packing bed |
surf | At the surface |
stag | Stagnation condition |
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Accepted: March 18, 2010.
Recommended by Subject Editor Walter Amborsini. ]]>