Study of the drying kinetics and calculation of mass transfer properties in hot air drying of Cynara cardunculus

2.1 Samples

In this study, samples of thistle flower of the species C. cardunculus L. harvested in Viseu, situated in the centre region of Portugal, at the time of flowering, i.e., in June, were used. The samples were collected and immediately transported to the laboratory to proceed with the experiments, without any storage. The petals of the thistle flower were separated prior to drying. The petals are oblong, with about 10 mm long and 1 mm diameter.

2.2 Drying experiments

The thistle flowers were dried at different temperatures (35, 45, 55 and 65°C) in a forced convection chamber set to maintain a constant desired temperature. The choice of this temperature range was based on one hand on the factor that temperatures lower than 35–40°C are very long drying processes, and on the other hand that temperatures too high, above 65–70°C, may have an adverse effect on the phenolic compounds present in the thistle flower and also on their aspartic proteases which have the milk clogging capacity. The choice of these particular temperatures allowed covering the desired range of possible drying temperatures with a reduced number of experimental drying assays (four). The chamber used was a forced convection oven (model WTB Binder; BINDER GmbH company, Germany), with an air flow of 300 m³/h, corresponding to an air flow rate of 0.5 m/s. The initial moisture content of the thistle flower was 56.91% (wet basis) and was determined by the method of weight loss until the weight is constant on an oven at 105°C. On a periodical basis, the samples were evaluated for moisture content. For this, the whole tray was weighed, thus giving the weight of the batch, i.e., the whole set of thistle flowers in the drier. This allowed calculating the weight loss of the batch along time, and therefore the moisture content both on wet basis and dry basis. The measurements were made at intervals of 20 min for the fastest processes (temperatures of 55 and 65°C) and 30 min for the slowest dryings (temperatures of 35 and 45°C). Each drying process was completed when the sample reached a final moisture content of or lower than 5%. The drying time for the different assays was 15, 7, 5 and 4 h, respectively, for 35, 45, 55 and 65°C, and four dried samples resulted from these processes, designated as D35, D45, D55 and D65.

3.1 Thin layer model for transfer of moisture

The so-called thin-layer models consist in mathematical equations that correlate the changes in moisture during drying with certain parameters, such as the drying constant, k (1/s), or the lag factor, k₀ (dimensionless), which account for the combined effects of numerous transport phenomena taking place during drying. The Henderson and Pabis model is such an example of a thin-layer model and relates the moisture ratio (MR) to time (t) according to the following equation (Kouchakzadeh 2013; Avhad and Marchetti 2016; Vijayan et al. 2016):

where k₀ is the lag factor and k is the drying constant, as defined earlier, and the MR = (W − W_e)/(W₀ − W_e), with W₀, W_e and W, the moisture contents at the beginning, at equilibrium and at any time t, respectively. This equation can be expressed in the logarithmic form, corresponding to a linear function of the type:

in which the slope can be used to calculate the drying constant k.

3.2 Correlations for mass transfer

The Biot number for mass transfer, Bi_m, is a dimensionless number that correlates the convective mass transfer coefficient, h_m (m/s), with the diffusivity coefficient, D_e, (m/s²) (Dincer and Hussain 2002a):

where z is the characteristic dimension of the system (m). Equation (3) is valid for Bi > 0.1 (Sahin et al. 2002) and allows the estimation of h_m, if the value of Bi_m is known.

Dincer and Hussain (2002a) proposed an equation to associate the Biot number, Bi_m, with the dimensionless Dincer number, Di:

where the Dincer number is defined as:

where u is the flow velocity of drying air (m/s), k is the drying constant and z is the characteristic dimension. Furthermore, Dincer and Dost (1996) proposed an equation to relate the effective diffusivity, D_e, to the thin layer drying constant, k:

where μ₁ is a function of the Biot number, given by (Haghi and Amanifard 2008):

3.3 Diffusion model for mass transfer

Fick’s second law of diffusion has been widely recommended to predict with reasonable accuracy the moisture distribution inside many food materials during drying, even though significant changes occur in their physical–chemical properties (Danish et al. 2016; Nicolin et al. 2016; Silva et al. 2016). In this way, the non-steady-state diffusion for unidirectional mass transfer is expressed by (Haghi and Amanifard 2008):

where W(z,t) is the dry basis moisture content (kg water/kg dry solids), t is the time (s), D_e is the effective diffusivity (m²/s) and z is the characteristic dimension of the system (m). However, the partial differential equation of equation (8) can be simplified, if the following hypotheses are considered: (a) initial moisture content is uniform throughout the solid, (b) shape of the solid remains constant and shrinkage is negligible, (c) the effect of heat transfer on mass transfer is insignificant, (d) mass transfer occurs exclusively by diffusion and (e) moisture diffusion occurs in the z direction only. In this case, equation (8) is simplified into (Haghi and Amanifard 2008):

with the following initial and boundary conditions:

where Z is the half thickness of slab and h_m is the convective mass transfer coefficient. For the diffusion in transient conditions, assuming a uniform initial distribution of the moisture content and also a uniform concentration at the surface for t > 0, the solution of Fick’s law can be approximated by an infinite series, of the form (Crank 1975):

where MR is the moisture ratio (dimensionless), and W, W_e and W₀ are the moisture content at time t, the equilibrium moisture content and the initial moisture content, respectively, all expressed in dry basis (g water/g dry solids). Furthermore, considering that when the Fourier numbers are higher than 0.2, the second and following terms of the series can be neglected, then the solution of Fick’s equation is given by (Crank 1975):

if the following condition is satisfied,

where Fo is the dimensionless Fourier number.

Equation (14) can be expressed in the form of logarithm, giving place to a straight line of the form:

from which the value of D_e can be calculated from the slope.

3.4 Determination of the mass transfer properties

To determine the mass transfer properties for the different drying temperatures, two methodologies were used, based on the diffusion model and the thin layer model (Guiné et al. 2017):

Method A: Thin layer based model

• Estimate MR from the experimental drying data for every time t

• From a plot ln(MR) = f(t) estimate k and k₀ through equation (2)

• Calculate Di from equation (5)

• Calculate Bi_m from equation (4)

• Calculate m1 from equation (7)

• Calculate D_e from equation (6)

• Calculate h_m from equation (3)

Method B: Fick’s law based model

• Estimate MR from the experimental drying data for every time t

• From a plot ln(MR) = f(t) estimate D_e from the slope through equation (16)

• Estimate k and k₀ by combining equations (1) and (14): k = D_ep²/(4z²), k₀ = 8/p²

• Calculate Di from equation (5)

• Calculate Bi_m from equation (4)

• Calculate h_m from equation (3).

3.5 Activation energy for moisture diffusion and for convective mass transfer

The variation of effective diffusivity with temperature is expected to follow an Arrhenius function, as expressed by the following equation (Guiné et al. 2013):

where De0 is the diffusivity for an infinite temperature, E_d is the activation energy for moisture diffusion, R is the constant of gases (R = 8.31451 J mol⁻¹ K⁻¹) and T is the drying temperature (expressed in °C). A plot of ln(D_e) as a function of (1/(T + 273.15)) will result in a straight line with slope equal to (−E_d/R) and intercept equal to lnDe0, from which the parameters E_d and De0 can be estimated (Guiné et al. 2013).

In parallel, the dependence of the convective mass transfer coefficient, h_m, on temperature can also be expressed by an Arrhenius function like:

where hm0 is the Arrhenius constant and E_c is the activation energy for convective mass transfer.

Equations (17) and (18) can be linearized thus obtaining the corresponding equations of a straight line:

4.1 Drying curves

Figure 1 shows the drying curves obtained for the convective drying of thistle flower at different temperatures (35, 45, 55 and 65°C). The results in Figure 1(a) indicate that the drying time was highly variable as the temperature changed, although the kinetic profiles for the wet basis moisture content, H, for all temperatures were relatively similar, i.e., corresponding to a decreasing function of time. The drying time increased from about 5 to 15 h when the temperature decreased from 65°C to 35°C. The curves in Figure 1(b) show a very similar trend for the dry basis moisture content, W, with a very fast decrease particularly for the highest temperatures: 55°C and 65°C. The MR, represented in Figure 1(c), is a dimensionless function expressing the variations of moisture during drying and, in the case of thistle flower drying, the values showed a trend practically equal to that of the dry basis moisture content. The form of the curves for W corresponded for all temperatures to decreasing exponential functions, which is similar to what has been reported for the drying of several agricultural products (Vega-Gálvez et al. 2010; Wang et al. 2015; Romdhane et al. 2016). The drying rate, calculated as the difference in W divided by the corresponding difference in time (Figure 1(d)), was very high in the initial moments of drying for the temperatures 55°C and 65°C, a little lower for 45°C and even lower for 35°C. Nevertheless, for all temperatures in the first 2 h the drying rate decreased to almost zero. This is very important considering that faster processes are preferable for allowing us to obtain the products in less time, and therefore with less costs, provided that quality is guaranteed, namely considering the effects of temperature on the properties of the dried product.

Figure 1

Drying curves for the thistle flower: (a) wet basis moisture content, (b) dry basis moisture content, (c) MR and (d) drying rate.

4.2 Estimation of mass transfer properties from the thin layer model

Table 1 summarizes the values obtained for the slope and intercept of the linearization of equation (2), which is the logarithmic form of the thin-layer model Henderson and Pabis, and which was used to calculate the parameters drying constant (k) and lag factor (k₀) by method A. The values of the regression coefficient (R²) were relatively close to 1, varying in the range 0.8687–0.9440, indicating that the fit obtained to the linear functions was good in all cases, with just exception for the data obtained at 55°C, in which case the value was lower than 0.9 (R² = 0.8687). This lower value found for the regression coefficient was because in this drying experiment the measured values of the moisture content were not fitted to the linear model with the same accuracy as that in the other cases, for which the model best fitted the experimental values.

Table 2 presents the results obtained when calculating the different mass transfer properties for the evaluated temperatures through the algorithm of method A. According to the values in Table 2, it is possible to verify that the drying constant, k, increased when the temperature was raised from 35°C to 65°C, which would be expected having in consideration that a higher temperature provides more heat that will in turn promote a higher rate of water removal. The values of the drying constant for thistle flower in the temperature range studied varied between 5.2791 × 10⁻⁵ s⁻¹ and 3.2989 × 10⁻⁴ s⁻¹. These values stand in the same range as those for the drying constant in the Henderson and Pabis thin layer model reported by Guiné et al. (2017) for the drying of kiwi and eggplant in a temperature range from 50°C to 80°C and at the same hot air flow rate used in this work (0.5 m/s) (values comprised between 1.1530 × 10⁻⁴ and 6.0176 × 10⁻⁴), as well as those reported by Darıcı and Şen (2015), for kiwi dried at 60°C and the same air velocity (2.696 × 10⁻² min⁻¹ corresponding to 4.493 × 10⁻⁴ s⁻¹). The values of the lag factor, k₀, found in the present study, are in the range from 0.3911 to 0.6089, which are lower than those reported for the convective drying of kiwi and eggplant in the work by Guiné et al. (2017) (1.3539–1.5046) and also lower than the values reported by Darıcı and Şen (2015) (0.9741).

The values of the Di number in Table 2 varied in the range from 5.0523 × 10⁵ to 3.1571 × 10⁶, and these values are somewhat similar to those reported by Guiné et al. (2013) for the drying of pears (3.7135 × 10⁵–2.3174 × 10⁶). However, Dincer and Hussain (2002a) found values of Di number considerably lower for the hot air drying of spherical potatoes (1.2356 × 10⁴), being these differences probably owing to the much different size and geometry of the samples, as well as the drying conditions or even considering the different nature of the product.

Table 2 also reveals the values found for the Biot number, which varied in the range 0.0908–0.1805, increasing with increasing temperature and which are very similar to those reported by Guiné et al. (2017) for the drying of two food products (0.1217–0.2261, respectively, for kiwi and for eggplant). As one can confirm, these values are all over 0.1, which makes it possible to use equation (3) for the estimation of the mass transfer coefficients. Guiné et al. (2013) reported values of Bi between 0.1020 and 0.2026 for the drying of pears under variable drying conditions, and Dincer and Hussain (2002a) obtained a value of Bi for the convective drying of spherical potatoes equal to 0.3119. These values are of the same magnitude of those found in this study for the drying of thistle. Nevertheless, Srikiatden and Roberts (2006) reported values for Bi for mass transfer in potato and carrot samples dried by convective hot air drying considerably higher than the ones found in the present study (1.44 × 10⁴–9.70 × 10⁴ for 1.5 m/s at 50–70°C and 0.97 × 10⁵–2.77 × 10⁵ for 3.0 m/s at 40–70°C). These differences are probably due to the air velocities used in their study, which were three and six times higher than that used in the present work for the drying of thistle.

Table 2 further presents the values of μ₁, which were found to vary from 0.4129 to 0.4601, which are very similar to the ones encountered by Guiné et al. (2017) for the drying of two food products (0.4294–0.4652 for kiwi and 0.4439–0.4832 for eggplant). In contrast, the value reported by Dincer and Hussain (2002a) for the air drying of spherical potatoes was lower (0.2781).

The values of the effective diffusion coefficient, D_e, shown in Table 2, varied between 2.7866 × 10⁻⁹ and 1.4027 × 10⁻⁸ m²/s, rising with increasing temperature, as expected. These values are similar to those found by Guiné et al. (2017) for the drying of kiwi and eggplant (5.6281 × 10⁻⁹–2.3152 × 10⁻⁸ m²/s) and to those reported by Tripathy and Kumar (2009) for potato (2.43 × 10⁻⁸–6.09 × 10⁻⁸ m²/s). Nevertheless, many authors have reported values of D_e much lower for the drying of variable food products: 1.997 × 10⁻¹¹ for rough rice (Silva et al. 2010), 3.320 × 10⁻¹⁰ to 9.000 × 10⁻⁹ m²/s for berberis fruits (Aghbashlo et al. 2008), 6.7904 × 10⁻¹⁰ to 4.7722 × 10⁻⁹ m²/s for pears (Guiné et al. 2013), 8.94 × 10⁻¹¹ to 9.63 × 10⁻¹¹ m²/s for cocoa beans (Dina et al. 2015), 2.7906 × 10⁻¹¹ to 1.8489 × 10⁻¹⁰ m²/s for mango (Dissa et al. 2011), 1.20 × 10⁻¹¹ to 1.33 × 10⁻¹¹ m²/s for mint (Sallam et al. 2015) and 5.00 × 10⁻¹⁰ to 1.95 × 10⁻⁹ m²/s for mushroom (Reyes et al. 2014). In fact, although there are many studies evaluating the diffusion coefficients of variable food products, it is not possible to accurately compare the data obtained in the present study with other findings, due to differences in the drying conditions, the nature of the products or even the shape and size of the elements dried.

Finally, Table 2 also presents the values of the convective mass transfer coefficient, h_m, which were found to vary from to 8.4335 × 10⁻⁸ to 8.4400 × 10⁻⁷ m/s. Other values reported in the literature for other diverse food products include 3.798 × 10⁻⁷ for rough rice (Silva et al. 2010), 2.2831 × 10⁻⁷–1.7882 × 10⁻⁶ m/s for kiwi and eggplant (Guiné et al. 2017), 1.3520 × 10⁻⁸–1.3976 × 10⁻⁹ m/s for pears (Guiné et al. 2013) and 3.2665 × 10⁻⁵ m/s for potatoes (Dincer and Hussain 2002b). Again, like it was observed for the effective diffusion coefficient, accurate comparisons of the mass transfer coefficient are difficult across variable products and drying conditions.

4.3 Estimation of mass transfer properties from the diffusional model

Table 3 shows the values obtained in this work for the different mass transfer properties evaluated with the algorithm described in method B. The values of the diffusion coefficient, D_e, in this case varied from 1.9256 × 10⁻¹⁰ to 1.2033 × 10⁻⁹ m²/s, which are lower when compared to those found using the algorithm in method A, which proves that the algorithm used for the calculation of diffusivity is conditioned according to the fundamentals of mass transfer considered: use of Fick’s law of diffusion or use of thin layer empirical models. Fick’s equation is of a fundamental origin, but has some assumptions that might eventually lead to a discrepancy between the theoretical model and the reality of the moisture diffusion inside the specific matrix of the food considered. On the other hand, the thin layer models rely on empirical equations, which, even though resulting from an attempt to fit experimental data with accuracy, also have some limitations as to their applicability. The values of the mass transfer coefficient range from 5.8277 × 10⁻⁹ to 7.2398 × 10⁻⁸ m/s, which are lower than those in Table 2.

4.4 Calculation of activation energy

Table 4 presents the results obtained through the linearization (by linear regression) of the functions ln(D_e) or ln(h_m) versus 1(T + 273.15), which express the variations in diffusivity and mass transfer coefficient with temperature by means of Arrhenius-type equations (equations (19) and (20)). The values of the regression coefficient are high in both cases, with R² close to 1, which would be the perfect fit.

Table 5 shows the calculated values of the constants in equations (19) and (20), including the activation energies for moisture diffusion and for convective mass transfer. The results obtained for the constants De0 and hm0 were 0.1146 and 6766.2 m/s, respectively, while the activation energy for moisture diffusion, E_d, was 51.5 kJ/mol, and the activation energy for convective mass transfer, E_c, was 70.9 kJ/mol. It is once again difficult to compare the values with those from the literature, because there are no studies about the drying of the thistle flower stems, and therefore any comparisons include necessarily other types of food products and eventually other drying conditions. Reported values for De0 include 4.2458 × 10⁻⁴ for kiwi and 2.3640 × 10⁻⁴ for eggplant (Guiné et al. 2017). The value obtained for hm0 by Thripathy and Kumar (2009) for the solar drying of potato slices was 0.285 m/s and those obtained by Guiné et al. (2017) for the drying of kiwi and eggplant slices were 3.0738 and 1.5752 m/s, respectively. Values reported for activation energy for moisture diffusion, E_d, in other works include 18.6 kJ/mol for tomato (Fiorentini et al. 2015) or the range 110.837–124.356 for berberis fruits (Aghbashlo et al. 2008). Other authors have reported values of the activation energy for convective mass transfer, E_c, for different foods, for example, for pumpkin 86.25 kJ/mol (Guiné et al. 2012) or for potato slices 4.42 kJ/mol (Tripathy and Kumar 2009).

The knowledge of the mass transfer properties and activation energy has practical implications in the optimization of the drying processes and, namely, in the energy input for the process.