Applicability of the generalised D Arcy and Watt model to description of water sorption on pineapple and other foodstuffs

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1 Journal of Food Engineering 79 (27) Applicability of the generalised D Arcy and Watt model to description of water sorption on pineapple and other foodstuffs Sylwester Furmaniak, Artur P. Terzyk *, Piotr A. Gauden, Gerhard Rychlicki Department of Chemistry, Physicochemistry of Carbon Materials Research Group, N. Copernicus University, Gagarin St. 7, 87-1 Toruń, Poland Received 6 December 25; received in revised form 2 February 26; accepted 26 February 26 Available online 5 April 26 Abstract In this paper we adapt (developed by us recently) the Generalised D Arcy and Watt (GDW) model to the description of water vapour sorption on foodstuffs. We show that this simple formula can successfully describe the set of sorption isotherms measured on pineapple (in the range of temperatures 2 5 C) and reported in this journal by Hossain et al. [Hossain, M. D., Bala, B. K., Hossain, M. A., & Mondol, M. R. A. (21). Sorption isotherms and heat of sorption of pineapple. Journal of Food Engineering, 48, 13 17]. Presented model also describes the isosteric enthalpy of sorption reported by those authors. We also show the applicability of GDW model to description of water sorption data on other foodstuffs (macaroni, sardine and pistachio nut paste) and we prove the inconsistency of the modified BET equation. Having strong kinetic basis, our model makes it possible to suggest the mechanism of water adsorption on foodstuffs, and can be successfully applied by researchers who are looking for mathematical equations to describe the sorption isotherms of food materials and/or calculate some parameters characterising water sorption mechanism. Ó 26 Elsevier Ltd. All rights reserved. Keywords: Enthalpy of sorption; Equilibrium moisture content; Isotherm model; Pineapple; Water sorption mechanism 1. Introduction and model description It is well known that sorption isotherms of foodstuffs are very important for design, modelling and optimization of many processes. Hossain, Bala, Hossain, and Mondol (21) pointed out the importance of those data in drying, aeration, predicting of stability and quality during packaging and storage of food. The same authors presented the literature review and concluding that the modified BET adsorption isotherm equation is the most popular to description of isotherms of foodstuffs. This BET equation, modified by Dincer and Esin (1996), can be written as: a ¼ ð1þ 1 bh r * Corresponding author. Tel.: ; fax: address: aterzyk@chem.uni.torun.pl (A.P. Terzyk). URL: (A.P. Terzyk). where is the equilibrium moisture content, a and b are the parameters, and h r is the relative humidity. Although Eq. (1) was shown to be applicable for the description of many experimental data it is incorrect from the theoretical point of view. This is caused by the absence of Henry s law limit; in other words, as h r tends to zero tends to finite value (a) and should tend to zero (Roquerol, Roquerol, & Sing, 1999; Rudzinski & Everett, 1992). Moreover the original BET model has limitations and is thermodynamically unrealistic (Hill, 1996; Roquerol et al., 1999; Seri-Levy & Avnir, 1993). The results of computer simulations of water adsorption on different materials show that the BET concept is in this case far from reality, and that the mechanism of water adsorption is rather of the clustering type (Brennan, Bandosz, Thomson, & Gubbins, 21; Mowla, Do, & Kaneko, 23). The problem of water adsorption has been the area of interest in our research group (Furmaniak et al., 25; Furmaniak, Gauden, Terzyk, Wesołowski, & Rychlicki, 25; Furmaniak, Terzyk, Gauden, & Rychlicki, 25; Gauden, /$ - see front matter Ó 26 Elsevier Ltd. All rights reserved. doi:1.116/j.jfoodeng

2 S. Furmaniak et al. / Journal of Food Engineering 79 (27) Notation a adsorption value, mol g 1, the parameter of the mbet model, % (dry basis) a ml surface concentration of Langmuir-type primary adsorption sites, mol g 1 a prim the value of adsorption on primary sites, mol g 1 b the parameter of the mbet model, % 1 c kinetic constant related to the adsorption on secondary sites of Dubinin Serpinsky type DC the global determination coefficient value DC T the value of DC calculated for the data measured at the considered temperature h relative pressure of water h r relative humidity, % K, k kinetic constants, % 1 K, k entropic factors of the constants K and k, % 1 K L Langmuir constant, Pa 1 L the enthalpy of condensation of water, kj mol 1 equilibrium moisture content, % (dry basis) M o e;i observed moisture content for ith experimental point, % (dry basis) M t e;i M o e m N Q,q q st p s R T w theoretical value of the moisture content, % (dry basis) the average value of the observed moisture content, % (dry basis) maximum sorption on primary sites, % (dry basis) the number of temperatures for which the experimental data were measured the enthalpy values related to the kinetic constants K and k, kj mol 1 isosteric enthalpy of adsorption, kj mol 1 saturation water pressure at the studied temperature, Pa 1 the universal gas constant,.8314 kj mol 1 K 1 temperature, K the parameter determining what part of water molecules adsorbed on primary sites convert into the secondary adsorption sites 25; Gauden et al., 25; Terzyk, Gauden, & Rychlicki, 1999; Terzyk, Rychlicki, Ćwiertnia, Gauden, & Kowalczyk, 25; Terzyk, Rychlicki, & Gauden, 21; Wesołowski, Gauden, Terzyk, & Furmaniak, 24). Here of the simplest approach to analytical description of water adsorption is developed recently by us the Generalised D Arcy and Watt (GDW) model (Furmaniak, Terzyk et al., 25). In this approach all unrealistic assumptions of the BET concept are omitted. Therefore, the GDW assumes the existence of the primary (Langmuir-type) adsorption sites on solid surface. Each centre can adsorb only one water molecule. Those water molecules can be the secondary adsorption sites for the next molecules, and here the adsorption scenario is according to those proposed by Dubinin and Serpinsky (1981), Dubinin, Zaverina, and Serpinsky (1955) and D Arcy and Watt (197). The major difference here is the assumption that not all water molecules adsorbed on primary sites can be the secondary centres. Moreover, this model takes into account the possibility that one water molecule attached to primary site can create more than one secondary adsorption site. Simple kinetic considerations led to the following GDW adsorption isotherm equation: a ¼ a prim þ cwa primh 1 ch ¼ a 1 cð1 wþh prim 1 ch ¼ a mlk L h 1 cð1 wþh ð2þ 1 þ K p L h 1 ch s where a is water adsorption, h denotes the relative pressure (p s is water saturation pressure), a ml is the surface concentration of Langmuir-type sites, K L is the Langmuir constant, c is the kinetic constant associated with the secondary adsorption sites (of Dubinin Serpinsky type). The coefficient w determines the ratio of water molecules adsorbed on the primary sites which is converted into the secondary adsorption sites, and a prim is the value of adsorption on primary sites, given by: a prim ¼ a mlk L h ð3þ 1 þ K p L h s Converting Eq. (2) into the language of food engineering we obtain: ¼ mkh r 1 kð1 wþh r ð4þ 1 þ Kh r 1 kh r where is the equilibrium moisture content [% (dry basis)], h r is the relative humidity [%], and m, K and k are equivalent to a ml, p s K L and c in Eq. (2). In the case of description of adsorption data measured at different temperatures it is necessary to postulate the temperature dependence of the sorption isotherm equation parameters. Therefore from the basic assumptions of the GDW model we have: m; w 6¼ f ðt Þ ð5þ Q K ¼ K exp ð6þ k ¼ k exp RT q RT ð7þ

3 72 S. Furmaniak et al. / Journal of Food Engineering 79 (27) where Q and q are energies of sorption (diminished by the value of the enthalpy of water condensation) associated with the kinetic constants K and k, R is the gas constant and T the absolute temperature. K and k are almost independent of temperature pre-exponential entropic factors. 2. Fitting the model to experimental data In this study we applied the data of sorption isotherms of pineapple measured at temperatures 2, 3, 4 and 5 C in the relative humidity range of 11 97% and reported in this journal by Hossain et al. (21). The approximation of theoretical Eqs. (4), (6) and (7) to the experimental data was performed applying the genetic algorithm (differential evolution (DE)) constructed by Storn and Price (1996, 1997) and applied successfully by us (Furmaniak et al., 25; Furmaniak, Terzyk et al., 25; Gauden, 25; Terzyk et al., 25). DE is a very simple heuristic approach for minimizing non-linear and nondifferentiable continuous space functions. To optimize the objective function (ofunc: (1 DC), where DC (see Eq. (9) below) is the determination coefficient) with DE the following settings for the input file are taken into account: DE/best/2/bin method is chosen (this time, the new vector to be perturbed is the best performing vector of the current generation); the number of parents (i.e. number of population members), NP is 1 times greater than the number of parameters of the objective function, D; weighting factor, F =.8 and crossover probabililty constant, CR =.5; the value to reach, VTR = 1 25 (the procedure stops when ofunc < VTR, if either the maximum number of iterations (generations) itermax is reached, or the best parameter vector bestmem has found a value f(bestmem) < = VTR). We maximised the value of the determination coefficient (DC T ) for the data measured in the whole temperature range: P 2 M o e;i M t e;i DC T ¼ 1 i P i 2 ð8þ M o e;i M o e where M o e;i is the observed moisture content for ith experimental point, M t e;i is the theoretical value of the moisture content calculated from Eq. (4), and M o e is the average observed moisture content. The global fitness parameter is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P T DC ¼ DC2 T N where N is the number of temperatures (in the studied case N = 4) at which the measurements were performed. The best fit parameters are: m, Q, q, w, K (at T = 293 K) and k (at T = 293 K). For the last two parameters the values of K and k were calculated (Eqs. (6) and (7)) since they were necessary for the calculation of K and k at the remaining temperatures. It should be pointed out that K and k are almost temperature independent. The simultaneous fitting of all adsorption data was performed to obtain the maximum value of the DC (Eq. (9)). Fig. 1 shows the obtained results, and the best-fitted values of the mentioned above parameters are tabulated in Table 1. It can be seen that the proposed model describes the data of sorption of water on pineapple very well. From the obtained parameters the following mechanism of water sorption can be proposed. Around one third of water sorbed by pineapple is attached to the strong primary sorption sites (the energy of sorption on those sites is kj/mol i.e. it is three times larger than the enthalpy of water condensation, m Equilibrium moisture content, % (dry basis) T 293 K (2 o C) 33 K (3 o C) 313 K (4 o C) 323 K (5 o C) Relative humidity, % ð9þ 1 Fig. 1. The comparison of the experimental (points) and theoretical (lines) data of water sorption on pineapple. The latter were calculated from Eq. (4) with Eqs. (6) and (7). Table 1 The results of the fitting of GDW (Eq. (4)) model to experimental data of water sorption on pineapple T [K] M [% (dry basis)] K [% 1 ] k [% 1 ] Q [kj mol 1 ] Q [kj mol 1 ] w K [% 1 ] k [% 1 ] DC T DC 293 (2 C) (3 C) (4 C) (5 C)

4 S. Furmaniak et al. / Journal of Food Engineering 79 (27) Isosteric enthalpy of sorption, q st - L, kj mole Equilibrium moisture content, % (dry basis) Fig. 2. The plot of the isosteric enthalpy of water sorption on pineapple, generated from the data collected in Table 1 and from Eq. (11). is around 2 and maximum equilibrium moisture content is around 6% at the lowest temperature studied). From those strongly bound molecules around 6% become the secondary adsorption sites for the next water molecules. The energy of sorption here is close to the enthalpy of water condensation. Basing on our model we can also explain the observed decrease in equilibrium moisture content with the rise in temperature. This is mainly caused by strong temperature dependence of K (caused by large Q value), therefore with the rise in temperature, since the sorption on primary sites is strongly exothermic, we observe their saturation at larger values of the relative humidity. 3. Isosteric enthalpy of sorption Classical phenomenological thermodynamics defines the formula describing the isosteric enthalpy of sorption. This formula adapted to the function = f(h r ) has the following form: [% (dry basis)] 2 1 Macaroni [% (dry basis)] 12 8 Sardine h r [%] h r [%] 5 4 [% (dry basis)] Pistachio Nut Paste h r [%] Fig. 3. The comparison of the applicability of GDW (Eq. (4) solid lines) and mbet (Eq. (1) dashed lines) to description of water sorption data (points) on different foodstuffs. The parameters are tabulated in Table 2.

5 722 S. Furmaniak et al. / Journal of Food Engineering 79 (27) Table 2 The parameters obtained during fitting of GDW and mbet models to experimental data determined for different foodstuffs Foodstuffs References T ( C) GDW (Eq. (4)) mbet (Eq. (1)) m [% (db)] K [% 1 ] k [% 1 ] w DC a [% (db)] b [% 1 ] DC Macaroni Arslan and Togrul (25) Sardine Bellagha et al. (25) Pistachio nut paste Maskan and Gogus (1997) q st L ¼ RT 2 o ln h r ð1þ ot where L is the enthalpy of water condensation. The equation defining the enthalpy of sorption related to Eq. (2) was developed by us previously (Furmaniak, Terzyk et al., 25). The application of this equation to Eq. (4) leads to the following equation: K 1 þ wkhr q st ð1þkh L ¼ rþ 2 1 kh r Q þ Khr wk 1þKh r q ð1 kh rþ 2 ð11þ 1 þ wkhr 1 kh r þ Khr wk 1þKh r K ð1þkh rþ 2 ð1 kh rþ 2 Fig. 2 shows the plot of the isosteric enthalpy of sorption generated from Eq. (11) and from the parameters collected in Table 1. Obviously, since we fitted the family of isotherms, the obtained theoretical enthalpy in the range of moisture content 2 4%, is exactly the same as calculated from isosteres by Hossain et al. (21). This is independent confirmation of the validity of the GDW model. 4. The comparison of GDW with the modified BET model sorption on other foodstuffs As we mentioned above the modified BET (mbet) model, although inconsistent, is successfully applied to description of data on different foodstuffs. In this paragraph we test and compare the applicability of GDW (Eq. (2)) and mbet (Eq. (1)) models to description of water sorption isotherm of macaroni (Arslan & Togrul, 25), isotherm of sardine (Bellagha, Sahli, Glenza, & Kechaou, 25) and sorption isotherm of pistachio nut paste (Maskan & Gogus, 1997). We applied the same numerical procedure and algorithm as described above. The results are compared in Fig. 3 and the parameters are tabulated in Table 2. Obtained results show that the GDW model describes the data with higher accuracy than the mbet model. Moreover, the data from Fig. 3, especially those measured for macaroni, show the inapplicability of mbet equation. It can be observed that due to, mentioned above, inconsistency of this equation (i.e. the absence of Henry s law limit) mbet model fails at low h r limits. Contrary, GDW model describe the data well in the whole measured h r range. 5. Conclusions In this paper we show that the GDW model can be successfully applied to description of water vapour sorption on foodstuffs (in the range of relative humidity 1% and above). This simple formula can successfully describe the set of sorption isotherms measured on pineapple reported in this journal by Hossain et al. (21). It also describes the isosteric enthalpy of sorption reported by those authors. We also show that this model is more consistent than the mbet model, describing water sorption data measured on other foodstuffs better. Strong kinetic basis of our model gives insights into the mechanism of water sorption on foodstuffs, and can be successfully applied by researchers who are looking for mathematical equations to describe the sorption isotherms of food materials. References Arslan, N., & Togrul, H. (25). Modelling of water sorption isotherms of macaroni stored in a chamber under controlled humidity and thermodynamic approach. Journal of Food Engineering, 69, Bellagha, S., Sahli, A., Glenza, A., & Kechaou, N. (25). Isohalic sorption isotherm of sardine (Sardinella aurita): experimental determination and modeling. Journal of Food Engineering, 68, Brennan, J. K., Bandosz, T. J., Thomson, K. T., & Gubbins, K. E. (21). Water in porous carbon. Colloids and Surfaces A, , D Arcy, R. L., & Watt, I. C. (197). Analysis of sorption isotherms of nonhomogeneous sorbents. 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