Rehydration Kinetics of Dried Spaghetti with Different Diameters

Abstract

The changes in moisture content over time were measured for commercially available dried spaghetti with five different diameters (1.17 mm to 1.90 mm) during the rehydration process at 30 °C to 80 °C. The change in moisture content over time at any temperature was modeled using a hyperbolic equation for all spaghetti diameters. The activation energy for the initial rehydration process did not depend on the diameter. The temperature dependence of the equilibrium moisture content differed at temperatures higher and lower than approximately 55 °C, which is close to the starch gelatinization temperature. The rehydration process of spaghetti with diameters of 1.17 mm and 1.42 mm differed from that of thicker spaghetti, with thinner spaghetti rehydrating faster at any temperature. A stochastic model was proposed to explain this phenomenon.

Durum semolina, a coarsely ground durum wheat flour (Triticum durum), contains the highest gluten content among wheat flours and is the primary ingredient of pasta. It consists of approximately 74 % carbohydrates, 11-13 % protein, 2 % fat, and 12 % moisture based on a wet basis [1]. Pasta is made by extruding a dough, prepared by kneading a mixture of durum semolina and water, through an extruder die. Among various pasta types based on shape and size, long cylindrical pasta is called spaghetti. Spaghetti is often distributed in a dried state for storage stability and transportation efficiency. Dried pasta is popular among consumers because of its affordability and extended shelf life [2]. Dried spaghetti is rehydrated or cooked before consumption to improve its cooking properties and digestibility through gelatinization of starch. Horigane et al. [3] expressed the rehydration process of dried pasta based on Fick's second law of diffusion and the dependence of the diffusion coefficient of water on moisture content. Cunningham et al. [4] considered two types of effective diffusion coefficients and expressed the rehydration process using the law. Phenomenologically, the rehydration process can be expressed based on the law, but the phenomena occurring during the rehydration process of spaghetti are more complicated [5, 6].

In this study, we measured the rehydration processes of spaghetti with different diameters at various temperatures to deepen the understanding of the effect of spaghetti diameter on rehydration behavior. The moisture content at a given rehydration time was higher for thinner spaghetti. We propose a stochastic model to explain this phenomenon.

The rehydration processes of spaghetti with five different diameters were measured at 30, 40, 50, 55, 60, 70, and 80 °C. As examples, the changes in moisture content X, based on the dry weight, over rehydration time, t, at 40 °C and 70 °C, which were below and above the gelatinization temperature of durum wheat starch (approximately 55-60 °C) [7], respectively, are shown in Figs. 1(A) and (B). According to our previous work [8], the rehydration process of spaghetti was described by a hyperbolic equation (Eq. (1)) in terms of t.

  

$$X=\mathit{at}/(b+t)+X_0\text{′}$$（1）

where a and b are constants. X₀′ is the moisture content immediately after rehydration, which was approximated by fitting the moisture content within the first 5 min of rehydration, measured at 1-min intervals, to a quadratic function of time t and estimated as the X value at t = 0 min. The difference between X₀′ and the initial moisture content X₀ of dried spaghetti is due to water seeping into minute cracks on the surface of dried spaghetti immediately after rehydration begins [9]. The constants a and b were determined using the Solver function of Microsoft Excel^® to minimize the sum of the residual squares of the measured values of moisture content X and the calculated values using Eq. (1). The curves in Figs. 1(A) and (B) were drawn using the a and b values obtained in this manner.

Fig. 1. Changes over time t in the moisture content X of spaghetti with different diameters at (A) 40 °C and (B) 70 °C. In (C) and (D), the moisture contents are plotted versus rehydration time t divided by the square of the initial diameter d, t/d². The initial diameters of dried spaghetti were (□) 1.17 mm, (△) 1.42 mm, (○) 1.60 mm, (▽) 1.78 mm, and (◇) 1.90 mm.

We previously reported the rehydration process of spaghetti with diameters of 1.4, 1.6, and 1.8 mm, and found that moisture content, X, could be represented by a single curve when potted against the rehydration time, t, divided by the square of the spaghetti diameter, d (i.e., t/d²), irrespective of the initial diameter of the spaghetti at a certain temperature [8]. Therefore, the moisture contents shown in Figs. 1(A) and (B) are plotted against t/d² (Figs. 1 (C) and (D)). For spaghetti with diameters of 1.60, 1.78, and 1.90 mm, all plots were represented by a single curve. However, the plot for spaghetti with a diameter of 1.42 mm was located slightly above the curve, and the plot for spaghetti with a diameter of 1.17 mm was located significantly above the curve. This suggests that the rehydration of spaghetti thinner than a certain diameter occurred more rapidly, suggesting that factors other than water diffusion were involved. This phenomenon is discussed later based on a stochastic model for the location of starch granules.

From Eq. (1), the initial rehydration rate dX/dt | _t=0 and equilibrium moisture content X_∞ are given by Eqs. (2) and (3), respectively.

  

$$\mathrm{d}X/\mathrm{d}t{\mid }_{t=0}=a/b$$（2）

  

$$X_{\infty }=a+X_0\text{′}$$（3）

The plots of the initial rehydration rate against the reciprocal of the absolute temperature T gave a straight line for spaghetti of any diameter (Fig. S1; see J. Appl. Glycosci. Web site). This indicates that the temperature dependence of the initial rehydration rate is expressed by the Arrhenius equation (Eq. (4)).

  

$$a/b=A\exp [-(E/R)(1/T)]$$（4）

where A is the frequency factor, E is the activation energy, and R is the gas constant. The activation energies for the initial rehydration process of spaghetti with diameters of 1.17, 1.42, 1.60, 1.78, and 1.90 mm, determined from the slopes of the straight lines in Fig. S1 (see J. Appl. Glycosci. Web site), were 29.8, 33.0, 35.6, 33.7, and 32.9 kJ/mol, respectively, and their mean and standard deviation were 33.0 ± 2.1 kJ/mol. These values were almost the same as the activation energy value (30.5 kJ/mol) that we previously reported for the initial rehydration process of spaghetti with diameters of 1.4, 1.6, and 1.8 mm [8].

The temperature dependence of the equilibrium moisture content can be expressed by the van't Hoff equation (Eq. (5)) [8].

  

$$d\ln X_{\infty }/d(1/T)=-\mathrm{∆}H/R$$（5）

where ∆H is the enthalpy change for rehydration. The initial velocity of rehydration depended on the diameter of the spaghetti. On the other hand, since the composition of the spaghetti used in this study was the same for all the spaghetti, the equilibrium moisture content should not depend on the diameter of the spaghetti. Therefore, the average value of the equilibrium moisture content X_∞ for all spaghetti of different diameters was plotted against the reciprocal of absolute temperature (Fig. S1; see J. Appl. Glycosci. Web site). The plot showed a stepwise change between 50 and 60 °C, attributed to the gelatinization of spaghetti starch [7]. The values of ∆H in the high- and low-temperature regions were evaluated to be 14.2 and 10.6 kJ/mol, respectively. The value of ∆H in the high-temperature region was smaller than the value reported previously (25.1 kJ/mol) [8]. On the other hand, the value of ∆H in the low-temperature region was larger than the previous one (1.44 kJ/mol). The reason for this difference is not clear, but it may be due to differences in the composition of the durum semolina used as the raw material for spaghetti or differences in the manufacturing method. In addition, the low reliability of the X_∞ value, due to the small moisture content at temperatures lower than the gelatinization temperature, also seems to be a reason for the large value of ∆H.

Although not physically realistic, infinitely thin spaghetti should represent a rehydration process that is unaffected by the diffusion rate of water. When the movement of water in spaghetti simply follows Fick's second law of diffusion and the water diffusion coefficient D is independent of diameter, the change in X can be expressed as a function of Dt/d² alone. Therefore, we attempted to estimate the moisture content $X_{d^2\to 0}$  of infinitely thin spaghetti by plotting the moisture content X at any given rehydration time against d². As examples, plots of X against d ² at different rehydration times at 40 °C and 70 °C are shown in Figs. 2 (A) and (B). The plots of X against t/d² for spaghetti with diameters of 1.60, 1.78, and 1.90 mm were connected by a line to estimate the moisture content $X_{d^2\to 0}$ of infinitely thin spaghetti.

Fig. 2. Example for estimating the moisture content $X_{d^2\to 0}$ of infinitely thin spaghetti at (A) 40 °C and (B) 70 °C.

　X is the moisture content at any given time. The rehydration times were (□) 5 min, (△) 10 min, (○) 20 min, (▽) 30 min, and (◇) 60 min.

Figure 3 shows the changes in $X_{d^2\to 0}$ over time at various temperatures. By assuming that the rehydration rate of infinitely thin spaghetti is proportional to the difference between the moisture content $X_{d^2\to 0}$ at any time and the equilibrium moisture content $X_{d^2\to 0,\infty }$, the rehydration rate is expressed by Eq. (6) [10, 11].

  

$${\mathrm{d}X}_{d^2\to 0}/\mathrm{d}t=k(X_{d^2\to 0,\infty }-X_{d^2\to 0})$$（6）

where k is the rehydration rate constant. By solving Eq. (6) under the initial condition of $X_{d^2\to 0}=X_{d^2\to 0,0}$ at t = 0, the moisture content of infinitely thin spaghetti $X_{d^2\to 0}$ at any rehydration time t is expressed by Eq. (7).

  

$$X_{d^2\to 0}=X_{d^2\to 0,\infty }-(X_{d^2\to 0,\infty }-X_{d^2\to 0,0})e^{-\mathit{kt}}$$（7）

Fig. 3. Changes over time in the moisture content $X_{d^2\to 0}$ of infinitely thin spaghetti at (□) 30 °C, (■) 40 °C, (△) 50 °C, (▲) 55 °C, (○) 60 °C, (●) 70 °C, and (◇) 80 °C.

The parameters, k and $X_{d^2\to 0,\infty }$, were estimated using the Solver function in Microsoft Excel^® to minimize the sum of the residual squares of $X_{d^2\to 0}$  obtained by extrapolating X to d²→0 and the value calculated using Eq. (7). The temperature dependence of k and $X_{d^2\to 0,\infty }$ would be expressed by Eqs. (4) and (5), respectively.

The activation energy for the rehydration process of infinitely thin spaghetti was estimated to be 8.50 kJ/mol (Fig. S2; see J. Appl. Glycosci. Web site). This value was smaller than the activation energy for the rehydration process of dried spaghetti with various diameters. During the rehydration of dried spaghetti with a finite diameter, the relaxation of the gluten network is accompanied by several other phenomena, including water diffusion and moisture sorption by the starch granules. The observed activation energy is influenced by these phenomena. On the other hand, because the rate constants for the rehydration process in boiling water of infinitely thin dried spaghetti and dried spaghetti-like noodle made of gluten alone were the same [11], the activation energy obtained here is considered to reflect the relaxation of the gluten network.

The van't Hoff plot for the rehydration process of infinitely thin spaghetti showed a significant change around 55 °C, with ∆H values of 17.9 and 7.12 kJ/mol at the high- and low-temperature regions, respectively. Even though infinitely thin spaghetti contains starch, it is reasonable that the X_∞ changes significantly around its gelatinization temperature.

Because the apparent density of dried spaghetti was 1.44 g/cm³ irrespective of its diameter, the volume of 100 g of dried spaghetti equates to 69.4 cm³. It is assumed that the protein and carbohydrate in the spaghetti are gluten and starch, respectively. Given that the weight fraction of protein is 0.117 and the density of gluten is 1.31 g/cm³ [12], the volume fraction, f_vG, of gluten in dried spaghetti is estimated to be 0.089. Similarly, given that the weight fraction of carbohydrate is 0.750 and the density of wheat starch is 1.6 g/cm³ [13], the volume fraction of starch, f_vS, is estimated to be 0.675.

Consider a cross-section of cylindrical spaghetti. Starch exists as granules, with a diameter of several 10 µm [14]. Its volume fraction, f_vS, is, as mentioned above, relatively large at 0.675. Therefore, the fraction of starch presents in the circular cross-section, f_sS, closely approximates its volume fraction, f_vS, and it is reasonable to assume f_sS ≈ f_vS. On the other hand, gluten, a tangled string-like macromolecule with a diameter of approximately 5 µm [15], occupies a smaller fraction of the area in the circular cross-section, f_sG, than the volume fraction of gluten, f_vG, suggesting that f_sG ≤ f_vG.

The cross-sectional area of spaghetti of diameter d is given by πd²/4. The length L of a square with an equal area is defined as $L=\sqrt{{\mathit{\pi d}}^2/4}=\sqrt{\pi }d/2$   (Fig. S3; see J. Appl. Glycosci. Web site). A square of length L is further divided into a lattice of length h, where h corresponds to the size of a starch granule. It is assumed that the starch is uniformly distributed over the entire cross-section [16, 17], and the lattice in which starch granules exist is considered to be determined at random. The total number, N_t, of lattices with length h per side is given by N_t=πd²⁄(4h²). By assuming that the area occupied by starch in the cross-section is equal to the volume fraction of starch (f_sS ≈ f_vS), the number of lattices, N_S, in which starch exists is N_S=f_vSN_t=f_vSπd²⁄(4h²).

On the other hand, the lattice number N_C on the four sides of the square is determined by $N_{\mathrm{C}}=4\left[\left(\sqrt{\pi }d/2\right)/h-1\right]=2\sqrt{\pi }d/h-4$. The number of lattices on the four sides of the square where starch is present, N_CS, is obtained by multiplying the number of lattices on the surface, N_C, and the fraction of the lattice in which starch exists, f_vS, thus, $N_{\mathrm{CS}}=f_{\mathrm{vS}}{N}_{\mathrm{S}}=f_{\mathrm{vS}}(2\sqrt{\pi }d/h-4)$. Therefore, the fraction of starch granules exposed on the surface, F_S, is

  

$$F_{\mathrm{S}}=N_{\mathrm{CS}}/N_{\mathrm{S}}=f_{\mathrm{vS}}(2\sqrt{\pi }d/h-4)/(f_{\mathrm{vS}}{\mathit{\pi d}}^2/(4h^2))=(8h\sqrt{\pi }d-16h^2)/({\mathit{\pi d}}^2)$$（8）

and is independent of the volume fraction of starch, f_vS.

Based on Eq. (8), for spaghetti with a diameter in the range of d = 0.8 to 2.5 mm and a lattice length per side (corresponding to the size of the starch granules) h of 25 µm, 50 µm, 75 µm, and 100 µm, the relationship between the square of the diameter d² and the fraction of starch granules exposed on the surface, F_S, is shown in Fig. 4. The fraction F_S shows an almost linear relationship for spaghetti diameters d > 1.4 mm (d²≈2 mm²), but below 1.4 mm, and F_S increases rapidly as the spaghetti becomes thinner. During the rehydration process of dried spaghetti, water moves from the surface towards the interior. With smaller diameters of spaghetti, the fraction of starch granules exposed on the surface, which easily absorb water, becomes extremely high. This may explain why thinner spaghetti has a faster increase in moisture content than predicted by the diffusion phenomenon.

Fig. 4. Relationship between the initial diameter d of the dried spaghetti and the fraction of starch granules exposed on the surface, F_S. The length h of the lattice in which starch exists is a: 25 µm, b: 50 µm, c: 75 µm, and d: 100 µm.

EXPERIMENTAL

Materials. All five types of spaghetti with different diameters, d = 1.17 mm, 1.42 mm, 1.60 mm, 1.78 mm, and 1.90 mm, were purchased from Barilla Japan K.K. (Tokyo, Japan). The apparent density of the spaghetti was estimated to be 1.440 ± 0.042 g/cm³, independent of the diameter, by measuring the weight of each diameter at various lengths. The compositions of durum wheat semolina, indicated on the product, were 11.7 g of protein, 2.0 g of fat, 75.0 g of carbohydrates, and 0.01 g of salt equivalent per 100 g of spaghetti, regardless of the spaghetti diameter.

Measurement of the rehydration process of spaghetti. A cylindrical stainless-steel container filled with 1 L of distilled water (Milli-Q water; Merck Millipore Corporation, Burlington, MA, USA) was placed in a thermostatic water bath (EOS-200RD, As One Corp., Osaka, Japan) with a stirrer rotation function, and tap water was added to the bath. The water in the cylindrical container was gently stirred by rotating a stirrer (7.8 mm × 40.8 mm) at approximately 140 rpm. A digital temperature controller (AD-5605H, A & D Corp., Tokyo, Japan), connected to a K-type thermocouple fixed near the inner surface of the container, controlled the water temperature in the container to a constant value between 30 °C and 80 °C.

More than 84 pieces of spaghetti, cut into approximately 5 cm lengths, were placed in the container to rehydrate the spaghetti. Since the moisture content of spaghetti rapidly increases immediately after rehydration due to water seeping into minute cracks near the surface [9], six samples (spaghetti) were taken out at 1-min intervals until 5 min after the start of rehydration, and at 10, 15, 20, 30, 45, 60, 75, and 90 min thereafter. The water on the sample surface was lightly wiped off using Kimtowel and Kimwipes (Nippon Paper Crecia Co., Ltd., Tokyo, Japan). The samples were then placed on a pre-weighed heat-resistant sheet with an identification number and weighed. The weight of the tare was subtracted from the gross weight to obtain the wet weight of the sample, m_w. The samples were placed in a dryer (DO-300FA, As One Corp.) at 130 °C for approximately 2 h to obtain the dry weight, m_d. The moisture content, X, at the rehydration time, t, was calculated using Eq. (9).

  

$$X=(m_{\mathrm{w}}-m_{\mathrm{d}})/m_{\mathrm{d}}$$（9）

The moisture content at each rehydration time was measured in triplicate, and the mean value and standard deviation were calculated. Due to the complexity of the graphs, only the mean values are shown, except for the rehydration enthalpy of spaghetti with various diameters.

CONFLICTS OF INTERESTS

The authors have no conflicts of interest to declare.

ACKNLOWLEDGMENTS

The authors thank Mr. M. Kohsaka for his technical assistance.

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