2020 年 26 巻 6 号 p. 717-723
Thermal diffusivity is an important thermophysical property used in the modeling and computation of unsteady-state heat transfer in basic food processing. The objectives of this study were to propose a new determination method for thermal diffusivity and to examine the applicability of this method to materials of several shapes and sizes. The samples (burdock root, carrot, and radish) were heated in a water bath at 90±1 °C for durations of up to 10 minutes. Then, 2- and 3-dimensional unsteady-state heat conduction problems were solved numerically with a finite element method. The thermal diffusivity of each sample was determined by a sequential estimation method based on the temperature profiles of the sample. The thermal diffusivity values of the samples ranged from (1.1–1.5)×10−7 m2/s. The thermal diffusivity could be estimated regardless of size, shape, and 2 and 3 dimensions for carrot within the setting range of this experiment.
Thermal treatments such as pasteurization, concentration, drying, and cooling are frequently used in food processing, transportation, storage, and cooking. Knowledge of the thermophysical properties of foods is important not only for process design but also for the prediction and control of various changes that occur in food during thermal processing. Thermal diffusivity indicates how fast heat propagates through a material while heating or cooling under an unsteady-state condition (Rahman and Al-Saidi, 2009). Therefore, thermal diffusivity values are needed to predict temperature profiles for the analysis of unsteady-state heat transfer, i.e., the numerical simulation of temperature change, for example, the thermal pasteurization process, and to establish adequate heat treatments for the food industry. The thermal diffusivity values and measurement methods of foods have been reviewed by Singh (2006), Saravacos and Maroulis (2001), Rahman and Al-Saidi (2009), among others. However, there is a lack of information about the thermal diffusivity of food compared with thermal conductivity and specific heat.
The measurement methods of thermal diffusivity are classified into direct measurement, which the present work employs, and indirect measurement (Rahman and Al-Saidi, 2009). Thermal diffusivity can be obtained from the values of thermal conductivity, specific heat, and density in indirect measurements. The indirect method can easily determine the thermal diffusivity if these three physical properties can be measured under the same conditions. However, this measurement method requires considerable time and experimentation. Direct measurement methods include the pulse method (laser flash method), the probe method, and the inverse analysis based on the measurement result of the temperature profile in the material. The laser flash method can measure over a wide temperature range from room temperature to approximately 1 500 °C and has the advantages of high accuracy and high reproducibility (Araki, 2007; Akoshima, 2008). However, some direct measurement methods including the laser flash method require expensive and/or special devices. In addition, it is frequently necessary to perform complicated calculation procedures to determine thermal diffusivity under direct measurement methods.
Some analytical solutions of unsteady-state heat conduction equations have been used to estimate the thermal diffusivity of foods based on the measurement results of temperature changes of materials (Rahman and Al-Saidi, 2009). In these methods, the thermal diffusivity is usually determined from the analytical solution of a 1-dimensional unsteady-state heat conduction equation with appropriate boundary conditions for finite or infinite bodies. The disadvantages of using analytical solutions are that the material shape and size are limited, and the initial and boundary conditions must be strictly controlled. For example, when using an analytical solution in a 1-dimensional cylindrical coordinate system, the diameter and height of the material must be adjusted so that the 1-dimensional heat conduction phenomenon can be secured, or when using an analytical solution in a spherical coordinate system, the material must be shaped into a sphere. Therefore, it would be useful to easily determine thermal diffusivity with a simple and inexpensive device and without limitations on the material shape and size. However, there have been few reports that estimate the thermal diffusivity of foods while also changing the sample shape and size as well as the spatial dimensions in the analysis to verify the validity of the estimation method, as performed in this study.
The sequential procedure is developed using the matrix inversion lemma (Beck and Arnold, 2007) based on the Gaussian minimization method. Many researchers have used the ordinary least squares method for parameter estimation. For example, Mariani et al. (2008), Betta et al. (2009), Rinaldi et al. (2010), and Muramatsu et al. (2020) applied the least squares method for the estimation of thermal diffusivity of some foods. Sequential parameter estimation provides more insight into the estimation process. The advantage of sequential estimation over the ordinary least squares method is that it shows the updated parameters during the process. Improved estimation is achieved when estimating the parameters at each data addition (Mohamed, 2009). Under sequential estimation, if a parameter reaches a constant value after a certain reasonable time, then the experiment can be stopped, as further data will not improve the parameter estimate. Therefore, we expect the parameters to approach a constant as the number of observations increase. Mohamed (2009, 2010), Greiby et al. (2014), and Muramatsu et al. (2017) applied a sequential technique based on Gaussian minimization to estimate the thermal properties of food. Although the sequential estimation of parameters has some advantages over the ordinary least squares method, there have been few reports using the sequential estimation technique to predict the thermal diffusivity of food.
In this study, the sequential estimation algorithm was used to estimate the thermal diffusivity of 3 vegetables (burdock root, carrot, and radish) based on the 2- and 3-dimensional finite element analysis. The objectives of this study were to propose a new determination method of thermal diffusivity, to measure the thermal diffusivity of 3 vegetables (burdock root, carrot, and radish) using the new method, and to examine the applicability of this method using samples with different shapes and sizes. The thermal diffusivity was assumed to be constant with temperature, and because the temperature range was only 60 °C (20 to 80 °C), it was expected that the change in thermal diffusivity would be small. The predicted results agreed well with the measured results within the temperature range, which demonstrated that the proposed method was simple yet still accurate under the conditions used in this study. For larger temperature increases, such as more than 100 °C, we recommend using temperature-dependent thermal properties in the model.
Sample Three vegetables were used in this study: burdock root, carrot, and radish, which exhibit variability in size and shape, are usually heated before eating, and are readily available in Japan throughout the year. These samples were purchased from a local market. The moisture content of these samples was measured using a moisture analyzer (MX-50, A&D Co., Ltd., Tokyo, Japan). The moisture contents of the samples were 83±1% (wet basis) for burdock root, 89±1% for carrot, and 95±1% for radish. The burdock roots and radishes were cut into a cylindrical shape (diameter (D) 20 and height (H) 100 mm). To investigate the effects of size and shape on the estimated results, four carrot shapes were used for the estimation of thermal diffusivity: a long cylinder (D=20, H=100 mm; Carrot-A), a short cylinder (D=20, H=20 mm; Carrot-B), a disk (D=40, H=10 mm; Carrot-C), and a rectangular parallelepiped (Width=27, Depth=27, H=30 mm: Carrot-D).
Measurement of the temperature profile A sheath-type thermocouple (K-type) with a sheath diameter and length of 1.6 and 50 mm, respectively, was fitted to the sample to measure the center temperature. Another K-type thermocouple (0.3 mm in diameter) was set on the side surface of the sample to measure the side surface temperature.
After the initial temperature of the sample was uniform and reached approximately 20 °C (20.2±0.6 °C), the sample was submerged into a water bath. The water temperature was controlled at 89.7±1.0 °C. A water bath (SH-12, TAITEC Co., Ltd., Saitama, Japan), electric immersion heater (SWA1510, HAKKO Electric Co., Ltd., Tokyo, Japan) and circulating pump (SH-12, ASONE Co., Ltd., Osaka, Japan) were used to maintain the water temperature at approximately 90 °C throughout the tests. The initial temperature of the sample was adjusted by immersing in a water bath controlled at a water temperature of approximately 20 °C. By using the water bath, circulating pump, and immersion cooler (150LF, TAITEC Co., Ltd., Saitama, Japan), the water temperature was kept at about 20 °C. The changes in temperature at the center and side surface of the samples were measured and recorded with a data logger (GL200, Graphtec Co., Ltd., Kanagawa, Japan) every 10 s for durations up to 10 minutes. In this study, all experiments were replicated 5 times.
Heat transfer modeling: rotational axisymmetric two-dimensional unsteady-state heat conduction problem for radial coordinates The cylindrical and disk-shaped samples used in this study had an axially symmetric geometry in the radial direction at r=0, where r is the radial coordinate of the sample (m). Therefore, the axisymmetric 2-dimensional unsteady-state heat conduction was numerically solved to reduce the computational cost. The anisotropy and spatial distribution of thermophysical properties of the sample were not considered in this analysis. The sample was regarded as an isotropic medium and having the same thermophysical properties in all locations. This heat transfer phenomenon is mathematically expressed as follows:
The heat conduction in 2-dimensional and cylindrical coordinates is governed by:
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where T is the temperature of the sample (°C), t is time (s), z is the axial coordinate of the sample (m), R is the radius of the sample (m), H is the height of the sample (m), and α is the thermal diffusivity of the sample (m2/s).
The initial condition was a uniform temperature throughout the sample:
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where Ti is the initial temperature.
The axisymmetric condition was as follows:
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The boundary conditions were measured values of surface temperature on all sample surfaces:
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where Tsurf is the surface temperature of the sample, which stays at almost 90 °C but slightly changes with time.
The thermal diffusivities of burdock roots, Carrot-A, and radishes were estimated under 2 cases: including the temperature measurement device and only a sample. A statistically significant difference test was performed (significance level of 5%) using a test for the difference between population means for these 2 cases. As a result, no significant difference was found between these values. These results suggest that the temperature measurement device used in this study had no effect on the thermal diffusivity estimation results and that the object of the analysis was solely the sample under the measurement conditions in this study. Therefore, in terms of the measurement conditions, equipment, and analysis conditions adopted in this study, it was judged that the effect of the temperature measurement device on the analysis could be ignored and only the sample was analyzed as the object to reduce the computational cost.
Heat transfer modeling: three-dimensional unsteady-state heat conduction problem for Cartesian coordinates To confirm the usefulness and expandability of the thermophysical property estimation method proposed in this study, the heat transfer phenomena inside all samples were also analyzed in 3 dimensions.
Governing the heat conduction equation for heat conduction in 3-dimensional and Cartesian coordinates:
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where x, y, and z represent the x-, y-, and z-coordinates (m), respectively, D is the diameter or depth of the sample (m), and W is the diameter or width of the sample (m).
Initial condition:
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Boundary conditions:
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In the axisymmetric 2-dimensional analysis, it was shown that the temperature measurement device could be ignored in the parameter estimation in this study. Therefore, only the sample was analyzed in the 3-dimensional analysis.
Parameter estimation techniques The heat transfer equations, i.e., Eqs. 1–4 and 5–7, were numerically solved by the finite element method using COMSOL Multiphysics® (5.3a, COMSOL Inc., Stockholm, Sweden). Thermal diffusivity in Eq. 1 or 5 was estimated by the sequential estimation procedure using MATLAB® (2017b, MathWorks Inc., Natick, MA, USA). Specifically, the numerical solution for the temperature distribution change inside the sample was calculated by COMSOL®, and this solution was then inputted into the sequential estimation algorithm created by MATLAB®.
The sequential method of estimation updates parameters as new observations are added. The nonlinear maximum a posteriori (MAP) sequential estimation procedure given by Beck and Arnold (2007) was used in the sequential estimation procedure. The sequential iterative scheme used in this study was the same as that used previously (Sulaiman et al., 2013; Greiby et al., 2014; Mishra et al., 2016; Muramatsu et al., 2017). Under sequential estimation, one expects the parameter to approach a constant as the number of observations is increased (Mohamed, 2009). The sequential estimation procedures are as follows (Mishra et al., 2016):
The minimization function in the Gaussian method can be expressed as follows:
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where Y is the experimental response variable, Ŷ is the predicted response, µ is the prior information of parameter vector β, W is the inverse of the covariance matrix of errors and U is the inverse of the covariance matrix of parameters. The extremum of the function given by Eq. 8 can be evaluated by differentiating it with respect to β. The expression can be given as follows:
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Eq. 9 can be set to zero, and β is solved implicitly. Standard statistical assumptions that allow the use of sequential estimation include additive errors, zero means, uncorrelated errors, normally distributed errors, covariance matrix of completely known errors, no errors in independent variables and subjective prior information of parameters are known. One method used to achieve this outcome utilizes an iterative scheme (Beck and Arnold, 2007):
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where b*i+1 is the updated parameter (p×1, p is the number of parameters) vector at time step i+1; b*i is the parameter vector at the previous time step i; b is the parameter vector at the previous iteration; P is the covariance vector matrix of parameters (p×p), X is the sensitivity coefficient matrix (n×p, n is number of data), φ is the variance of the errors in Y, and e is the error vector. The scheme is started by providing parameter estimates, computing X and the error vector e, assuming variance of errors in the dependent variable, and assuming a matrix P. Because we did not have accurate prior information on the covariance matrix, we set P as a diagonal matrix of 105. Matrices P, X, and e are functions of b and not of b*. The index k is the number of iterations in the outer loop for the nonlinear regression. The stopping criteria for b can be given as follows:
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where j is the index for the number of parameters. The magnitude of δ was on the order of 10−4. The value of δ1 was set to a small value, such as 10−8, to avoid the problem where bkj tends to zero.
Two-dimensional analysis Fig. 1 shows the sequential estimation process for the thermal diffusivity of Carrot-A with time. A comparison between the measured center temperature and the predicted temperature for Carrot-A is shown in Fig. 1(a). The solid line and symbols in Fig. 1(a) show the predicted values and the measured results, respectively. In this case, the values of estimated thermal diffusivity (α) and root mean square error (RMSE) between the measured data and the results calculated from Eq. 1 were α = 1.423×10−7 m2/s and RMSE = 0.343 °C, respectively. In the case that the thermal property of the material has a very high temperature dependency, the residual between the observed and predicted temperatures will be large in the low-temperature area or the high-temperature area, and the value of RMSE will be large. The predicted and observed temperatures matched well in the entire temperature range, as shown in Fig. 1(a). This result demonstrates that the temperature dependency of the thermal diffusivity is weak and negligible in this study. As shown in Fig. 1(b), the estimated thermal diffusivity attained a steady value at 200 s after the start. This result suggests that the experiment for this sample could be stopped at approximately 200 s because further data will not improve the parameter value. In Fig. 1, the value of the sequential estimate for the thermal diffusivity of Carrot-A was 1.409×10−7 m2/s at 200 s and 1.423×10−7 m2/s at the final time. Comparing these values, the error in the parameter at 200 s is approximated as less than 1%. In this way, the sequential estimation can reduce the computational cost, although the heating time in which the parameter approaches the convergence value will vary for each experimental condition.
Estimation results of thermal diffusivity for Carrot-A based on the 2-dimensional analysis. (a) temperature profile; (b) sequentially estimated parameter α versus time.
Fig. 2 shows the comparison of observed center temperature changes with the results calculated from Eq. 1 for Carrot-B and Carrot-C. The predicted and observed temperatures matched well for these samples. In this case, the RMSE values were 0.207 °C for Carrot-B and 0.278 °C for Carrot-C. For other samples not shown in Figs. 1 and 2, the predicted temperature matched well with the observed value.
Comparison of the observed temperature changes with the results calculated from Eq. 1 for (a): Carrot-B and (b): Carrot-C (2-dimensional analysis).
The mean of the final sequential estimated thermal diffusivities and RMSE for all samples by the 2-dimensional analysis are presented in Table 1. The literature values of thermal diffusivity are 1.24×10−7 m2/s for burdock root (Japan Society of Thermophysical Properties, 2008), (1.39–1.55)×10−7 m2/s for carrot (Rahman and Al-Saidi, 2009; Japan Society of Thermophysical Properties, 2008), and 1.35×10−7 m2/s for radish (calculated from the literature values of thermal conductivity (Souma et al., 2004) and specific heat (Japan Society of Thermophysical Properties, 2008), and the measured value of density). There were no significant differences between the measured values and the literature values. This finding indicates that the estimation method and procedure of the thermal diffusivity proposed in this study are appropriate.
| Sample | α (×10−7 m2 s−1) | RMSEa (°C) |
|---|---|---|
| Burdock | 1.372±0.093 | 0.769±0.475 |
| Carrot-Ab | 1.402±0.109 | 0.496±0.182 |
| Carrot-Bc | 1.347±0.064 | 0.587±0.258 |
| Carrot-Cd | 1.337±0.088 | 0.352±0.138 |
| Radish | 1.147±0.024 | 0.754±0.160 |
| (mean ± standard deviation) | ||
Multiple comparison tests were performed on these values using a one-way analysis of variance (5% significance level) for the values of thermal diffusivity estimated by 2-dimensional analysis for Carrot-A, -B, and -C. As a result of the test, there were no significant differences in these values. Thus, it is shown that this estimation method can appropriately determine the thermal diffusivity of cylindrical samples that have a diameter of 20–40 mm and a height of 10–100 mm.
Three-dimensional analysis A 3-dimensional unsteady-state heat transfer analysis was also performed to estimate the thermal diffusivity of 3 cylindrical and disk-shaped vegetables. A comparison of the observed center temperature changes with the results calculated from Eq. 5 for Carrot-A is shown in Fig. 3(a). In this case, the values of α and RMSE between the measured data and the results calculated from Eq. 5 were 1.426×10−7 m2/s and 0.379 °C, respectively. The results estimated by the 3-dimensional analysis were almost the same as those estimated by the 2-dimensional analysis.
Comparison of the observed temperature changes with the results calculated from Eq. 5 for (a): Carrot-A and (b): Carrot-D (3-dimensional analysis).
The mean of the final sequential estimated thermal diffusivities and RMSE for 3 cylindrical and disk-shaped vegetables by the 3-dimensional analysis are presented in Table 2.
| Sample | α (×10−7 m2 s−1) | RMSEa (°C) |
|---|---|---|
| Burdock | 1.372±0.088 | 0.806±0.431 |
| Carrot-Ab | 1.402±0.111 | 0.491±0.184 |
| Carrot-Bc | 1.345±0.056 | 0.630±0.195 |
| Carrot-Cd | 1.338±0.079 | 0.405±0.145 |
| Carrot-De | 1.467±0.052 | 0.790±0.368 |
| Radish | 1.151±0.025 | 0.715±0.157 |
| (mean ± standard deviation) | ||
The values of thermal diffusivity estimated by the 2-dimensional and 3-dimensional analyses for each dimension of each sample were tested for differences between population means, and there was no significant difference (5% significance level) between these values. This result suggests that this estimation method could be applied to thermal diffusivity estimation for materials having no axial symmetry.
To confirm this result, we measured the temperature change of the rectangular parallelepiped carrot (Carrot-D) and estimated the thermal diffusivity. The means of the final sequential estimated thermal diffusivity and RMSE for Carrot-D are presented in Table 2. Although RMSE values were slightly larger than those of the other geometries, the predicted temperature from Eq. 5 agreed with the measured data, as shown in Fig. 3(b). A multiple comparison test using a one-way analysis of variance (significance level of 5%) showed no significant differences between the values of Carrot-A, -B, -C, and -D. This result suggests that the proposed estimation method for the thermal diffusivity could be applied to the estimation of thermal diffusivity for samples with no asymmetric geometry.
In this study, we proposed a new estimation method of thermophysical properties, using the measurement results of the temperature change of the material and the numerical solution of the heat transfer phenomenon, and estimated the thermal diffusivity of 3 vegetables by using the sequential estimation technique. The value of thermal diffusivity estimated by the proposed method is useful basic data for heat transfer simulations, thermal calculations for thermal unit operations and so on. The results obtained in this study indicate that the proposed estimation method could potentially be applied to materials that have an arbitrary shape and size. The numerical assumptions and restrictions used in previous papers are eliminated in this proposed estimation method. There is a possibility that this method could be used to estimate the thermophysical properties of intricately shaped material and realistically shaped and sized material based on adequately measured temperature changes at a certain location. In addition, this method has applicability or extensibility to simultaneously estimate the thermophysical properties at several locations in the material or investigate the anisotropy of the thermophysical properties by using the temperature changes at several locations in the material. The precise prediction of the temperature change of vegetables during food processing, such as cooking, by using thermal diffusivity can be useful for the determination of cooking time or seasoning time, since the softening of vegetables and the diffusion of components like NaCl are highly affected by the temperature. Other foods should adhere to the same principles from the viewpoint of the temperature dependence of various kinds of reaction rates.
This study provided a new sequential determination method for the thermal diffusivity of food by using numerical simulation software. The advantage of this method is that the device and the estimation method are simple and require no special equipment. This method has the potential to be applied to materials that have an arbitrary shape and size. The thermal diffusivity values of the samples ranged from (1.1–1.5)×10−7 m2/s. No significant differences were observed among the rotational axisymmetric 2-dimensional and 3-dimensional analyses for all samples. There were also no significant differences in the thermal diffusivity values of carrots for all sizes and shapes.
