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Simultaneous estimation of the thermophysical properties of three kinds of fruit juices based on the measured result by a transient heat flow probe method
Journal of Food Engineering 96 (2010) 607–613
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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Simultaneous estimation of the thermophysical properties of three kinds of fruit juices based on the measured result by a transient heat flow probe method Yoshiki Muramatsu a,*, Eiichiro Sakaguchi b, Takahiro Orikasa c, Akio Tagawa d a
Department of Food Science and Technology, Faculty of Bioindustry, Tokyo University of Agriculture, 196 Yasaka, Abashiri-shi, Hokkaido 099-2493, Japan Department of Bioproduction and Environment Engineering, Faculty of Regional Environment Science, Tokyo University of Agriculture, 1-1-1 Sakuraoka, Setagaya-ku, Tokyo 156-8502, Japan c Department of Environmental Sciences, School of Food, Agricultural and Environmental Sciences, Miyagi University, 2-2-1 Hatadate, Taihaku-ku, Sendai-shi, Miyagi 982-0215, Japan d Graduate School of Horticulture, Chiba University, 648 Matsudo, Matsudo-shi, Chiba-ken 271-8510, Japan b
a r t i c l e
i n f o
Article history: Received 8 July 2009 Received in revised form 21 August 2009 Accepted 4 September 2009 Available online 10 September 2009 Keywords: Thermal conductivity Thermal diffusivity Specific heat Probe method Fruit juice
a b s t r a c t The thermophysical properties of three kinds of fruit juices (grape juice, orange juice, and pineapple juice) were measured at various temperatures (10–50 °C) and concentrations (10–50%). A new method for the simultaneous determination of thermophysical properties using a modified version of current probe theory method was proposed. The temperature changes of the probe upon insertion in the sample were fitted to an approximate solution of the heat conduction equation, and the values of two parameters in that solution were determined. Using the values of these parameters, the thermal conductivity and thermal diffusivity of each sample were determined. The specific heat of each sample was estimated from the definition of thermal diffusivity. These thermophysical properties were expressed as a function of concentration and temperature. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction 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 thus important not only for process design but also for the prediction and control of various changes that occur in food during thermal processing. The thermophysical properties of foods, such as thermal conductivity, thermal diffusivity, and specific heat, have been reviewed by Singh (1992), Sweat (1994), Saravacos and Maroulis (2001) and others. Thermal conductivities of some foods, such as dairy products and margarines (Sweat and Parmelee, 1979), milk (Reddy and Datta, 1994), and apple juice (Constenla et al., 1989), have been measured by a transient line source technique or probe method. In those reports, only the thermal conductivity of the material was determined from the temperature change in the line heat source or probe. We measured the thermal conductivity of apple juice (Muramatsu et al., 2000) by a twin probe method. The twin probe method is a relative probe measurement. In the twin probe method, the temperature changes of the probes inserted in the reference material (for example, water) and sample are simultaneously measured, and * Corresponding author. Tel.: +81 (0)152 48 3852; fax: +81 (0)152 48 2940. E-mail address: [email protected] (Y. Muramatsu). 0260-8774/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2009.09.008
the thermal conductivity of the sample is calculated by multiplying the thermal conductivity of the reference material by the ratio of the temperature rises of the two probes. The twin probe method can measure thermal conductivity with a high accuracy (Kasubuchi, 1977; Muramatsu et al., 2000). However, this method can only measure the thermal conductivity, and it has some disadvantages; for example, two probes and two amplifiers are needed, which increases the cost of the apparatus. Some researchers have reported the simultaneous determination of thermophysical properties such as thermal conductivity and thermal diffusivity using the probe method. Choi and Okos (1983) measured the thermal conductivity and thermal diffusivity of tomato juice concentrates by Nix’s method. The effective thermal conductivities, effective thermal diffusivities, and specific heats of beans (Tagawa and Murata, 1995) were predicted simultaneously from probe temperature changes by using Blackwell’s equation. These estimation methods are useful because some thermophysical properties can be simultaneously determined from the temperature change data. However, one disadvantage of these methods is that they require complex calculations. Some researchers reported the thermal conductivity of fruit juice (Constenla et al., 1989; Telis-Romero et al., 1998; Zainal et al., 2000; Muramatsu et al., 2000; Magerramov et al., 2006). Constenla et al. (1989) and Muramatsu et al. (2000) measured the specific heat of apple juice using a calorimeter. Thermal conductivities and the specific heats of guava juice were reported
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Nomenclature b
c Dh
j k 0 k kw
q qw cp C E I Ic Pc q 0
q
constant, b = exp (c) (–) Euler’s constant, 0.5772 (–) probe temperature difference (°C) thermal diffusivity of the sample (m2 s1) thermal conductivity of the sample (W m1 °C1) 1 reduced thermal conductivity, k0 ¼ kqw k1 (–) w q thermal conductivity of water (W m1 °C1) density of the sample (kg m3) density of water (kg m3) specific heat of the sample (kJ kg1 °C1) total solid content (%) voltage (V) current, 0.130 (A) correction coefficient, 0.3414 (°C) correction coefficient, 8.167 (m1) heat quantity per unit length of the probe and unit time (W m1) heat quantity of the probe per unit time (W), q0 ¼ E I ¼ I2 R
by Zainal et al. (2001) and Shamsudin et al. (2005). Gratao et al. (2005) and Tansakul and Chaisawang (2006) measured the thermal conductivity and the specific heat of passion fruit juice and coconut milk, respectively. The thermal diffusivity of fruit juice was reported by Telis-Romero et al. (1998) and Muramatsu et al. (2000). There has been no report that the thermal conductivity, thermal diffusivity, and specific heat of fruit juice were simultaneously measured under the same condition. In this study, the thermophysical properties (thermal conductivity, thermal diffusivity, and specific heat) of three kinds of fruit juices were measured under same conditions (total solid content levels and temperatures). Here, we proposed a new method for the simultaneous determination of thermophysical properties using a modified version of current probe theory method, and it easily determined the relevant thermophysical properties with sufficient accuracy. The objectives of this study were: (1) to develop a new method for the simultaneous determination of the thermophysical properties using the modified probe method; (2) to confirm the validity of this estimation method, and to carry out the simultaneous estimation of three kinds of thermophysical properties; (3) to represent these properties of the samples as a function of temperature and concentration. 2. Materials and methods 2.1. Samples Three kinds of fruit juices adjusted to various total solid content levels were used in this study, i.e., grape juice (total solid content levels of 10, 20, 30, 40, and 50%), orange juice (10, 20, 30, and 40%), and pineapple juice (10, 20, 30, 40, and 50%). Different solid content levels were obtained by diluting commercially available concentrated juices with distilled water. The total solid content of the sample solution was measured with a method that uses film with diatomite in which 5 g of the sample was dried under vacuum oven (Yamato Scientific Co., Ltd., DP-33, Tokyo, Japan) at 70 °C and 3.5 k Pa for 5 h (Tutumi, 1984). In this study, natural convection was avoided by using 1% agar to gel the sample solutions. The compositions of these sample solutions were measured according to the references (Tutumi, 1984; Resources Council, Science and Technology Agency, Japan, 2000), and given in Table 1.
resistance, 14.8 (X) outer diameter of the probe, 0.0005 (m) elapsed time (s) temperature (°C) constant (°C) constant (°C) constant (°C) constant (°C) constant (W m1 °C1 %1) constant (W m1 °C2) constant (W m1 °C1) constant (%1) constant (–) constant (m2 s1 %1) constant (m2 s1) constant (kJ kg1 °C1 %1) constant (kJ kg1 °C2) constant (kJ kg1 °C1)
R r t T A B F G a1 b1 d1 a2 b2 a3 b3 a4 b4 d4
2.2. Measurement apparatus Fig. 1 shows a schematic of the heat probe (Tokyo Riko Co., Ltd., Tokyo, Japan) used in this study. The probe consisted of a constantan heater wire (0.1 mm in diameter, resistance of 14.8 X) with a small temperature coefficient of electrical resistance, and a T-type thermocouple (0.1 mm in diameter) in a stainless steel tube (1 mm in diameter and 100 mm in length). To avoid contact between the heater wire and thermocouple, the space inside the stainless steel tube was filled with silicone oil. Fig. 2 presents a diagram of the thermophysical property measurement apparatus. The apparatus consisted of three units: heating, temperature control, and recording. The heating unit consisted of the probe, a DC power source (Metronix Co., Ltd., model 526, Tokyo, Japan), a DC ammeter (Yokogawa Electric Co., Ltd., model 2012, Tokyo, Japan), and a switch. A circulating water bath (Tokyo Riko Co., Ltd., TC-100, Tokyo, Japan) with a temperature control accuracy of 0.004 °C was used to hold the temperature constant throughout the tests. The temperature change in the heat source (probe), i.e., the change in the thermo-electromotive force, was amplified with an OP-amplifier (Tokyo Riko Co., Ltd., CA-25F, Tokyo, Japan) and recorded with a data logger (Advantest Co., Ltd.,
Table 1 Compositions (mass fraction) of each component for each sample. Sample
Composition (%) Water
Protein
Fat
Carbohydrate
Fiber
Ash
Grape juice
90.0 80.0 70.0 60.0 50.0
0.2 0.4 0.6 0.8 1.0
0.1 0.1 0.2 0.2 0.3
9.6 19.1 28.7 38.3 47.9
0.0 0.1 0.1 0.1 0.2
0.7 0.3 0.4 0.5 0.6
Pineapple juice
90.0 80.0 70.0 60.0 50.0
0.2 0.5 0.7 0.9 1.2
0.0 0.1 0.1 0.1 0.2
9.4 18.8 28.1 37.5 46.9
0.1 0.2 0.3 0.4 0.5
0.3 0.5 0.8 1.0 1.3
Orange juice
90.0 80.0 70.0 60.0
0.4 0.8 1.2 1.5
0.1 0.1 0.2 0.3
9.2 18.4 27.6 36.7
0.1 0.2 0.3 0.4
0.3 0.5 0.8 1.1
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containers were 30 mm in diameter and 150 mm in height. The reference point container was filled with 1% agar gel. 2.3. Measurement of thermophysical properties
Fig. 1. Schematic of the heat probe used to measure thermophysical properties.
The thermophysical properties of the samples were measured with the modified transient heat flow probe method at five different temperatures (10, 20, 30, 40, and 50 °C) for each concentration of each sample. After the water temperature in the water bath reached a preset temperature, the probe was inserted into the center of the sample. The sample container was submerged in the water bath; thus, the temperatures of the sample and reference point thermocouple (cold junction) were precisely controlled. When the temperature difference between the preset temperature and the sample temperature was less than 0.01 °C, which correspond to the thermo-electromotive force of the T-type thermocouple in the sample of 0.39–0.43 lV at measurement conditions, it was assumed that the temperatures had reached the preset temperature, i.e., they were in thermal equilibrium. Approximately 90 min were necessary to achieve this state of equilibrium. After the thermal equilibrium state had been established, the probe heater wire was energized, and the temperatures of the probe increased with time. The probe temperature changes were amplified and recorded with the data logger. In this study, the input and output range of the amplifier were adjusted to 100 lV and 10 mV, respectively, and the current applied to the heater wire in the probe was adjusted to 130 mA to raise the temperature of the probe to about 2 °C. Current was supplied to the heater wire of the probe for 180 s. In this study, the measurements of thermophysical properties were replicated five times for each experimental condition. 2.4. Estimation method of thermophysical properties When the sample is held at constant temperature, constant heat is continuously supplied to the heater wire in the probe inserted in the sample, and the elapsed heating time is long enough, the temperature change of the probe is approximately expressed as follows (Carslaw and Jaeger, 1988):
Dh ¼ A ln t þ B q A¼ 4pk q 4jb B¼ ln 2 : 4pk r
Fig. 2. Schematic of the experimental apparatus used to measure thermophysical properties.
R7326B, Tokyo, Japan). Because of the high thermal conductivity requirement, the cylindrical sample container and reference point container were made of copper plate (0.3 mm in thickness). These
ð1Þ ð2Þ ð3Þ
Eq. (1) shows that when the elapsed heating time is long enough, the relationship between the probe temperature difference and the logarithm of elapsed time is approximately linear, and the slope and intercept of this straight line are A and B, respectively. Many researchers have measured the thermal conductivities of various foods by utilizing values of A. If the relationship between the probe temperature difference and the time is expressed by Eq. (1) with a high accuracy, then thermal conductivity and thermal diffusivity can be calculated from Eqs. (2) and (3). The thermal conductivity and thermal diffusivity of the 1% agar gel were considered to be equal those of water, as the gel contained 99% water, and these parameters were determined by using Eqs. (2) and (3) at temperatures of 10–50 °C, and the estimated results were compared with values for water in the literature (Singh, 1992). In this case, the measured probe temperature changes in 30–180 s were fitted by a linear least squares method to Eq. (1), and the values of the parameters A and B were determined. The values of the root mean squared error (RMSE) and the coefficient of determination (R2) were about 1.0 103 °C1 and 0.9999 (–), respectively. The measured results agreed well with the calculated values. The ther-
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mal conductivity and thermal diffusivity of water were estimated from Eqs. (2) and (3) using the values of parameters A and B. Because the length of the probe was 100 mm, the heat quantity per unit length of the probe and unit time in Eq. (2) (q) was 0 q ¼ q0 10 ¼ E I 10 ¼ I2 R 10 (W/m). Where, q is the heat 0 quantity of the probe per unit time (W), and q ¼ E I ¼ I2 R. As a result, the thermal conductivity of water estimated from Eq. (2) was 10–15% higher than the literature value. Under some conditions, the thermal diffusivity of water calculated from Eq. (3) was a negative value, or 50% smaller than the literature value. Therefore, even if the measured results agreed well with the values calculated from Eq. (1), it is not suitable for determining the thermal conductivity and diffusivity from Eqs. (2) and (3). Although the length of the probe (length of the stainless pipe) was 100 mm, the heater wire was folded inside the stainless pipe, so the length of the heater wire was unknown. In addition, the values of A and B vary depending on the heat quantity of the probe, the heating time (supply time for the heat quantity), the range of the probe temperature change when fitted to Eq. (1) by the linear least squares method, and so on. This study introduced two new correction coefficients (Pc and Ic) to easily and accurately determine the thermophysical properties. Pc is the correction coefficient for the length of the heater wire in the probe, i.e., heat quantity per unit length of the probe, and q is rearranged: q ¼ q0 Pc ¼ I2 R Pc. Ic is the correction coefficient for the method of supplying the heat quantity to the heater wire in the probe (current, voltage, supply time of heat quantity), the data range of the probe temperature change when fitted to Eq. (1), and the error of measured temperature. In this study, these coefficients were introduced, Eqs. (1)–(3) were modified to following equations:
Dh ¼ F ln t þ G q0 Pc F¼A¼ 4pk q 4jb q0 Pc 4jb G ¼ B Ic ¼ ðln 2 Þ Ic ¼ ðln 2 Þ Ic: 4pk r 4pk r
ð4Þ ð5Þ ð6Þ
The correction coefficients (Pc and Ic) were determined using the measured results for the 1% agar gel at temperatures of 10– 50 °C. The method of supplying the heat quantity to the heater wire in the probe and the range of probe temperature changes when fitted to Eq. (4) were the same as those used for measuring the thermophysical properties of the sample. The values of the parameters F and G were determined by a linear least squares method. The known values (the heat quantity of the probe per unit time q0 ¼ I2 R, the outer diameter of the probe r, and the constant b) and the literature values of the thermal conductivity and thermal diffusivity of water (Singh, 1992) were substituted into Eqs. (5) and (6), and the values of the correction coefficients Pc and Ic were determined at each temperature. The values of the correction coefficients at temperatures from 10 °C to 50 °C ranged from 7.8 to 8.2 for Pc and from 0.39 to 0.30 for Ic. These correction coefficients had almost the same value regardless of temperature, remaining nearly constant from 10 °C to 50 °C. The values of Pc and Ic for the measurement conditions and apparatuses in this study were taken as the mean values of Pc and Ic in the temperature range of 10 °C to 50 °C, i.e., Pc = 8.167, Ic = 0.3414. To predict the thermophysical properties of the samples, the measured probe temperature changes in 30–180 s for samples were fitted by a linear least squares method to Eq. (4), and the values of the parameters F and G were determined. Using the values of F, G, Pc, and Ic, the thermal conductivity and thermal diffusivity of the sample were calculated from Eqs. (5) and (6), respectively. The specific heat of the sample was estimated by substituting the thermal conductivity and thermal diffusivity calculated from Eqs. (5)
and (6) and the density of the sample under the same condition into the following equation for thermal diffusivity (Singh, 1992):
j¼
k : cp q
ð7Þ
The values of the sample densities measured with a pycnometer (Figura and Teixeira, 2007) were used in Eq. (7). The method for estimating thermophysical properties used in this study was calibrated at 20 °C using 20%, 40%, and 60% sucrose solutions and 14%, and 23% sodium chloride solutions. As a result, the relative errors between the measured values and the literature values (Hayashi, 1989; The Japan Society of Mechanical Engineers, 1991; Singh, 1992) were 0–3% for thermal conductivity, 0–6% for thermal diffusivity, and 0–4% for specific heat. The measured values agreed well with the literature values. Therefore, Eqs. (4)–(7) and the values of the two correction coefficients (Pc and Ic) were used to determine the thermophysical properties of the samples. 3. Results and discussion 3.1. Temperature change of the probe The measured data for the probe temperature changes over 30– 180 s at each measurement condition were fitted by a linear least squares method to Eq. (4). A comparison between the measured results and the results calculated from Eq. (4) at a concentration of 10% and a temperature of 50 °C for orange juice is shown in Fig. 3. The solid line and symbols in Fig. 3 show the calculated values and the measured results, respectively. In this case, the values of parameters F and G, the RMSE and R2 in Eq. (4) were F = 0.2639, G = 0.4151, RMSE = 1.449 103 °C, and R2 = 0.9999 (–). As shown in Fig. 3, the measured data for the temperature changes of the probe over 30–180 s agreed well with the calculated values. Similar results were obtained for all other measurement conditions. Therefore, the values of parameters F and G determined from the probe temperature changes over 30–180 s were used to predict the thermophysical properties of the samples. 3.2. Thermal conductivities of the samples The values of the thermal conductivity calculated from Eq. (5) were 0.42–0.62 W m1 °C1 for grape juice and pineapple juice, and 0.46–0.61 W m1 °C1 for orange juice. The measured results of these samples were almost the same under the same conditions (temperature and concentration) because the major compositions
Fig. 3. Comparison of the observed temperature changes of the probe with the results calculated from Eq. (4) for orange juice at a concentration of 10% and a temperature of 50 °C.
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of all sample solutions were water and carbohydrate, and the compositions of these samples were similar (Table 1). The thermal conductivities of liquid foods, such as milk and fruit juice, can be expressed as functions of temperature and concentration (Constenla et al., 1989; Reddy and Datta, 1994; Telis-Romero et al., 1998; Zainal et al., 2000; Tansakul and Chaisawang, 2006). Riedel (1949) presented the following model to predict the thermal conductivities of fruit juice, sugar solutions, and milk (Singh, 1992):
tivity of the grape juice increased linearly with increasing temperature. Results similar to Fig. 4 were obtained for all samples. At the same temperature, the thermal conductivities of all samples decreased as the concentration increased, and a linear relationship existed between thermal conductivity and concentration. Therefore, we derived the following empirical equation to represent the relationship between thermal conductivity, concentration, and temperature:
k ¼ ð326:58 þ 1:0412T 0:00337T 2 Þ
The measured data for each sample were fitted by a linear least squares method to Eq. (9). The solid lines in Fig. 4 show the calculated results. For all samples, the calculated values agreed well with the measured data. The parameters, RMSE, and R2 of Eq. (9) for each sample are shown in Table 2. Eq. (9) accurately represented the relationships between thermal conductivity and both concentration and temperature for samples. Constenla et al. (1989) derived the following equation from the parallel heat transfer model based on the assumption that apple juice was consisted of two compositions; water and solid:
4
ð17:30 10
6
9:342 10 CÞ:
ð8Þ
The thermal conductivities calculated from Eq. (8) were compared with the measured thermal conductivities of the samples. As a result, the relative errors between the measured values and the calculated values were determined to be 0–3%. The fact that there was no significant difference between the measured values and the calculated values demonstrates that the method adopted in this study for estimating thermophysical properties was appropriate. Fig. 4 shows the relationships between thermal conductivity and temperature for grape juice, showing that the thermal conduc-
k ¼ a1 C þ b1 T þ d1 :
ð9Þ
k0 ¼ a2 C þ b2 ;
ð10Þ
0
0
where k represents a reduced thermal conductivity (k ¼ kqw k1 w q1 ). The values of the thermal conductivity of the sample k, thermal conductivity of water kw , density of the sample q, and density of water qw were known. Eq. (10) represents that the linear relationship exists between k0 and concentration C. The values of k0 for each sample were fitted by a linear least squares method to Eq. (10). The parameters, RMSE, and R2 of Eq. (10) for each sample are shown in Table 2. From Table 2, it is confirmed that the comparison of the measured data with the calculated value from Eq. (10) showed good agreement. 3.3. Thermal diffusivities of the samples
Fig. 4. Comparison of the measured thermal conductivity and the result calculated from Eq. (9) for grape juice.
The values of thermal diffusivity of the samples obtained from Eq. (6) were 1.2 107–1.5 107 m2 s1. The thermal diffusivities of orange juice as measured by the Dickerson method (Telis-Romero et al., 1998) were 1.0 107–1.4 107 m2 s1 at temperatures from 0.5 to 62 °C and water content levels from 30 to 75% (total solid contents from 25 to 70%), and were written as a function of temperature and water content. Comparing the measured data to the literature values for water (Singh, 1992) and
Table 2 The parameters, coefficients of determination (R2), and root mean squared errors (RMSE) of Eqs. (9)–(12) for each sample. Eq.
Sample Grape juice
Pineapple juice
Orange juice
(9)
a1 (W m1 °C1 %1) b1 (W m1 °C2) d1 (W m1 °C1) R2 (–) RMSE (W m1 °C1)
3.328 103 1.471 103 5.796 101 0.9915 0.0048
3.456 103 1.405 103 5.814 101 0.9950 0.0037
3.244 103 1.337 103 5.778 101 0.9837 0.0053
(10)
a2 (%1) b2 (–) R2 (–) RMSE (W m1 °C1)
7.952 103 9.970 101 0.9868 0.0059
8.124 103 9.962 101 0.9912 0.0050
8.007 103 9.926 101 0.9862 0.0048
(11)
a3 (m2 s1 %1) b3 (m2 s1) R2 (–) RMSE (m2 s1)
4.243 1010 1.475 107 0.9765 9.302 1010
3.659 1010 1.497 107 0.9733 8.566 1010
4.725 1010 1.493 107 0.9888 5.622 1010
(12)
a4 (kJ kg1 °C1 %1) b4 (kJ kg1 °C2) d4 (kJ kg1 °C1) R2 (–) RMSE (kJ kg1 °C1)
2.579 102 1.053 102 3.938 0.9944 0.0294
2.743 102 9.788 103 3.883 0.9934 0.0337
3.084 102 9.929 103 4.135 0.9885 0.0401
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orange juice (Telis-Romero et al., 1998), the measured values were almost the same as the reported values for these liquids. The thermal diffusivities of the samples decreased with increasing concentration. This tendency was also seen in a report by Telis-Romero et al. (1998). For each concentration, the thermal diffusivity of each sample was almost constant over temperature. The relationships between the thermal diffusivity and concentration of the samples are shown in Fig. 5. The symbols in Fig. 5 show the measured results, and the average values of the measured data at 10–50 °C at each concentration and for each sample are also plotted in Fig. 5. The thermal diffusivities of the samples decreased linearly with increasing concentration. The values of thermal diffusivity plotted in Fig. 5 for each sample were fitted by a linear least squares method to the following equation:
j ¼ a3 C þ b3 :
ð11Þ
The values of the parameters are given in Table 2 along with the RMSE and R2 of Eq. (11) for each sample. The RMSE and R2 values show that the measured data agreed well with the calculated values. Thus, the thermal diffusivity of each sample was represented by Eq. (11) as a function of concentration only within the measurement conditions in this study. 3.4. Specific heats of the samples The specific heat of each sample was determined from Eq. (7) by substituting the thermal conductivity and thermal diffusivity
Fig. 5. Empirical relationships between the thermal diffusivity and the concentration for each sample.
calculated from Eqs. (5) and (6) and the density of the sample under the same condition into Eq. (7). Fig. 6 shows the relationships between specific heat and concentration for pineapple juice. The symbols in Fig. 6 show the measured results. The values of the specific heat of pineapple juice were 2.6–4.1 kJ kg1 °C1 under the measured conditions, and decreased linearly as the concentration rose. Relationships similar to Fig. 6 were obtained for all samples. The specific heats of the samples increased with increasing temperature for each concentration of each sample. Constenla et al. (1989) measured the specific heat of clarified apple juice over the range from 20–90 °C and 12 to 70 °Brix by using a DSC, and these values were related to temperature and concentration (°Brix). The values of the specific heat of apple juice measured with a twin isoperibol calorimeter at concentrations of 10–45% and temperatures of 10–50 °C were 3.1–4.2 kJ kg1 °C1, and were represented as a function of temperature and concentration (Muramatsu et al., 2000). There were no significant errors between the measured specific heats of the samples and the literature values for other fruit juices. Tansakul and Chaisawang (2006) reported that the specific heat of coconut milk measured with a DSC was related to temperature and concentration by the following equation:
cp ¼ a4 C þ b4 T þ d4 :
ð12Þ
In this study, because both the temperature dependency and the concentration dependency were represented by a linear function, the measured specific heat data for each sample were fitted by a linear least squares method to Eq. (12). The solid lines in Fig. 6 show the calculated results. As shown in Fig. 6, the calculations agreed well with the measured data. The values of the parameters, RMSE, and R2 of Eq. (12) for each sample are shown in Table 2. From Table 2, it is apparent that the relationships between specific heat and both concentration and temperature for each sample were accurately represented by Eq. (12). In this study, two new correction coefficients were introduced, and the thermal conductivities, thermal diffusivities, and specific heats of samples were simultaneously determined using a modified transient heat flow probe method, and these values were written as a function of temperature and concentration. The estimation method adopted in this study, expressed as Eqs. (4)–(7), is suitable for predicting thermal conductivity, thermal diffusivity, and specific heat with accuracy sufficient for practical use. This estimation method can be easily determined the three kinds of thermophysical properties and applied to various foods. The empirical equations, i.e., Eqs. (9)–(12), will be useful in the design of equipment and in calculations for the thermal processing of fruit juice.
4. Conclusion
Fig. 6. Empirical relationships between the specific heat and the concentration at five different temperatures for pineapple juice.
A new method for the simultaneous determination of thermophysical properties using a modified version of current probe theory method was proposed. Two new correction coefficients were introduced, and using the values of correction coefficients and an approximate solution of the heat conduction equation, the thermal conductivity and thermal diffusivity of three kinds of fruit juices (grape juice, orange juice, and pineapple juice) were measured at various temperatures (10–50 °C) and concentrations (10–50%), respectively. The specific heat of each sample was estimated from the definition of thermal diffusivity. The thermal conductivity and specific heat for each sample were represented by the empirical equations shown as a linear function of both concentration and temperature, respectively. The thermal diffusivity of each sample was represented as a linear function of concentration only within the measurement conditions in this study.
Y. Muramatsu et al. / Journal of Food Engineering 96 (2010) 607–613
References Carslaw, H.S., Jaeger, J.C., 1988. The use of sources and sinks in cases of variable temperature. In: Conduction of Heat in Solids. Oxford Univ. Press, New York, pp. 255–281. Choi, Y., Okos, M.R., 1983. The thermal properties of tomato juice concentrates. Trans. ASAE 26 (1), 305–311. Constenla, D.T., Lozano, J.E., Crapiste, G.H., 1989. Thermophysical properties of clarified apple juice as a function of concentration and temperature. J. Food Sci. 54 (3), 663–668. Figura, L.O., Teixeira, A.A., 2007. Mass and density. Food Physics. Springer, Berlin. pp. 41–72. Gratao, A.C.A., Silveira Junior, V., Polizelli, M.A., Telis-Romero, J., 2005. Thermal properties of passion fruit juice as affected by temperature and water content. J. Food Process Eng. 27 (6), 413–431. Hayashi, H., 1989. Thermophysical properties of food. Food physics. Yokendo Co. Ltd., Tokyo. pp. 84–125. Kasubuchi, T., 1977. Twin transient-state cylindrical-probe method for the determination of the thermal conductivity of soil. Soil Sci. 124 (5), 255–258. Magerramov, M.A., Abdulagatov, A.I., Azizov, N.D., Abdulagatov, I.M., 2006. Thermal conductivity of pear, sweet-cherry, apricot, and cherry-plum juices as a function of temperature and concentration. J. Food Sci. 71 (5), 238–244. Muramatsu, Y., Tagawa, A., Kasai, T., Sakai, H., Fukushima, M., 2000. Thermophysical properties of apple juice. J. Japan. Soc. Food Sci. Technol. 47 (7), 548–550. Reddy, Ch.S., Datta, A.K., 1994. Thermophysical properties of concentrated reconstituted milk during processing. J. Food Eng. 21 (1), 31–40. Resources Council, Science and Technology Agency, Japan, 2000. General compositions, 5th ed.. Standard Tables of Food Composition in Japan Bureau of Ministry of Finance in Japan, Tokyo. pp. 11–13. Riedel, L., 1949. Measurements of the thermal conductivity of sugar solution, fruits juices and milk. Chem. Ing. Technol. 21, 340–341.
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Saravacos, G.D., Maroulis, Z.B., 2001. Thermal conductivity and diffusivity of foods. Transport Properties of Foods. Marcel Dekker Inc., New York. pp. 269–358. Shamsudin, R., Mohamed, I.O., Yaman, N.K.M., 2005. Thermophysical properties of Thai seedless guava juice as affected by temperature and concentration. J. Food Eng. 66 (3), 395–399. Singh, R.P., 1992. Heating and Cooling Processes for Foods. In: Heldman, D.R., Lund, D.B. (Eds.), Handbook of Food Engineering. Marcel Dekker Inc., New York, pp. 247–276. Sweat, V.E., 1994. Thermal Properties of Foods. In: Rao, M.A., Rizvi, S.S.H. (Eds.), Engineering Properties of Foods, 2nd ed. Marcel Dekker, Inc New York, pp. 99– 138. Sweat, V.E., Parmelee, C.E., 1979. Measurement of thermal conductivity of dairy products and margarines. J. Food Process Eng. 2 (3), 187–194. Tagawa, A., Murata, S., 1995. Thermophysical properties of beans (Part 1). Prediction from results measured by transient heat flow method using probe. J. Japan. Soc. Agric. Machinery 57 (4), 27–35. Tansakul, A., Chaisawang, P., 2006. Thermophysical properties of coconut milk. J. Food Eng. 73 (3), 276–280. Telis-Romero, J., Telis, V.R.N., Gabas, A.L., Yamashita, F., 1998. Thermophysical properties of brazilian orange juice as affected by temperature and water content. J. Food Eng. 38 (1), 27–40. The Japan Society of Mechanical Engineers, 1991. Physical property, 4th ed.. JSME Data Book: Heat transfer Maruzen Co. Ltd., Tokyo. pp. 307–365. Tutumi, C., 1984. Moisture Content of Foods-Vegetables and Fruits-. In: Japanese Society of Food Science and Technology (Ed.), Methods of Food Analysis, 2nd ed. Korin Co. Ltd., Tokyo, pp. 76–83. Zainal, B.S., Abdul Rahman, R., Ariff, A.B., Saari, B.N., Asbi, B.A., 2000. Effects of temperature on the physical properties of pink guava juice at two different concentrations. J. Food Eng. 43 (1), 55–59. Zainal, B.S., Abdul Rahman, R., Ariff, A.B., Saari, B.N., 2001. Thermophysical properties of pink guava juice at 3 and 11 BRIX. J. Food Process Eng. 24 (2), 87–100.
Contents lists available at ScienceDirect
Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Simultaneous estimation of the thermophysical properties of three kinds of fruit juices based on the measured result by a transient heat flow probe method Yoshiki Muramatsu a,*, Eiichiro Sakaguchi b, Takahiro Orikasa c, Akio Tagawa d a
Department of Food Science and Technology, Faculty of Bioindustry, Tokyo University of Agriculture, 196 Yasaka, Abashiri-shi, Hokkaido 099-2493, Japan Department of Bioproduction and Environment Engineering, Faculty of Regional Environment Science, Tokyo University of Agriculture, 1-1-1 Sakuraoka, Setagaya-ku, Tokyo 156-8502, Japan c Department of Environmental Sciences, School of Food, Agricultural and Environmental Sciences, Miyagi University, 2-2-1 Hatadate, Taihaku-ku, Sendai-shi, Miyagi 982-0215, Japan d Graduate School of Horticulture, Chiba University, 648 Matsudo, Matsudo-shi, Chiba-ken 271-8510, Japan b
a r t i c l e
i n f o
Article history: Received 8 July 2009 Received in revised form 21 August 2009 Accepted 4 September 2009 Available online 10 September 2009 Keywords: Thermal conductivity Thermal diffusivity Specific heat Probe method Fruit juice
a b s t r a c t The thermophysical properties of three kinds of fruit juices (grape juice, orange juice, and pineapple juice) were measured at various temperatures (10–50 °C) and concentrations (10–50%). A new method for the simultaneous determination of thermophysical properties using a modified version of current probe theory method was proposed. The temperature changes of the probe upon insertion in the sample were fitted to an approximate solution of the heat conduction equation, and the values of two parameters in that solution were determined. Using the values of these parameters, the thermal conductivity and thermal diffusivity of each sample were determined. The specific heat of each sample was estimated from the definition of thermal diffusivity. These thermophysical properties were expressed as a function of concentration and temperature. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction 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 thus important not only for process design but also for the prediction and control of various changes that occur in food during thermal processing. The thermophysical properties of foods, such as thermal conductivity, thermal diffusivity, and specific heat, have been reviewed by Singh (1992), Sweat (1994), Saravacos and Maroulis (2001) and others. Thermal conductivities of some foods, such as dairy products and margarines (Sweat and Parmelee, 1979), milk (Reddy and Datta, 1994), and apple juice (Constenla et al., 1989), have been measured by a transient line source technique or probe method. In those reports, only the thermal conductivity of the material was determined from the temperature change in the line heat source or probe. We measured the thermal conductivity of apple juice (Muramatsu et al., 2000) by a twin probe method. The twin probe method is a relative probe measurement. In the twin probe method, the temperature changes of the probes inserted in the reference material (for example, water) and sample are simultaneously measured, and * Corresponding author. Tel.: +81 (0)152 48 3852; fax: +81 (0)152 48 2940. E-mail address: [email protected] (Y. Muramatsu). 0260-8774/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2009.09.008
the thermal conductivity of the sample is calculated by multiplying the thermal conductivity of the reference material by the ratio of the temperature rises of the two probes. The twin probe method can measure thermal conductivity with a high accuracy (Kasubuchi, 1977; Muramatsu et al., 2000). However, this method can only measure the thermal conductivity, and it has some disadvantages; for example, two probes and two amplifiers are needed, which increases the cost of the apparatus. Some researchers have reported the simultaneous determination of thermophysical properties such as thermal conductivity and thermal diffusivity using the probe method. Choi and Okos (1983) measured the thermal conductivity and thermal diffusivity of tomato juice concentrates by Nix’s method. The effective thermal conductivities, effective thermal diffusivities, and specific heats of beans (Tagawa and Murata, 1995) were predicted simultaneously from probe temperature changes by using Blackwell’s equation. These estimation methods are useful because some thermophysical properties can be simultaneously determined from the temperature change data. However, one disadvantage of these methods is that they require complex calculations. Some researchers reported the thermal conductivity of fruit juice (Constenla et al., 1989; Telis-Romero et al., 1998; Zainal et al., 2000; Muramatsu et al., 2000; Magerramov et al., 2006). Constenla et al. (1989) and Muramatsu et al. (2000) measured the specific heat of apple juice using a calorimeter. Thermal conductivities and the specific heats of guava juice were reported
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Nomenclature b
c Dh
j k 0 k kw
q qw cp C E I Ic Pc q 0
q
constant, b = exp (c) (–) Euler’s constant, 0.5772 (–) probe temperature difference (°C) thermal diffusivity of the sample (m2 s1) thermal conductivity of the sample (W m1 °C1) 1 reduced thermal conductivity, k0 ¼ kqw k1 (–) w q thermal conductivity of water (W m1 °C1) density of the sample (kg m3) density of water (kg m3) specific heat of the sample (kJ kg1 °C1) total solid content (%) voltage (V) current, 0.130 (A) correction coefficient, 0.3414 (°C) correction coefficient, 8.167 (m1) heat quantity per unit length of the probe and unit time (W m1) heat quantity of the probe per unit time (W), q0 ¼ E I ¼ I2 R
by Zainal et al. (2001) and Shamsudin et al. (2005). Gratao et al. (2005) and Tansakul and Chaisawang (2006) measured the thermal conductivity and the specific heat of passion fruit juice and coconut milk, respectively. The thermal diffusivity of fruit juice was reported by Telis-Romero et al. (1998) and Muramatsu et al. (2000). There has been no report that the thermal conductivity, thermal diffusivity, and specific heat of fruit juice were simultaneously measured under the same condition. In this study, the thermophysical properties (thermal conductivity, thermal diffusivity, and specific heat) of three kinds of fruit juices were measured under same conditions (total solid content levels and temperatures). Here, we proposed a new method for the simultaneous determination of thermophysical properties using a modified version of current probe theory method, and it easily determined the relevant thermophysical properties with sufficient accuracy. The objectives of this study were: (1) to develop a new method for the simultaneous determination of the thermophysical properties using the modified probe method; (2) to confirm the validity of this estimation method, and to carry out the simultaneous estimation of three kinds of thermophysical properties; (3) to represent these properties of the samples as a function of temperature and concentration. 2. Materials and methods 2.1. Samples Three kinds of fruit juices adjusted to various total solid content levels were used in this study, i.e., grape juice (total solid content levels of 10, 20, 30, 40, and 50%), orange juice (10, 20, 30, and 40%), and pineapple juice (10, 20, 30, 40, and 50%). Different solid content levels were obtained by diluting commercially available concentrated juices with distilled water. The total solid content of the sample solution was measured with a method that uses film with diatomite in which 5 g of the sample was dried under vacuum oven (Yamato Scientific Co., Ltd., DP-33, Tokyo, Japan) at 70 °C and 3.5 k Pa for 5 h (Tutumi, 1984). In this study, natural convection was avoided by using 1% agar to gel the sample solutions. The compositions of these sample solutions were measured according to the references (Tutumi, 1984; Resources Council, Science and Technology Agency, Japan, 2000), and given in Table 1.
resistance, 14.8 (X) outer diameter of the probe, 0.0005 (m) elapsed time (s) temperature (°C) constant (°C) constant (°C) constant (°C) constant (°C) constant (W m1 °C1 %1) constant (W m1 °C2) constant (W m1 °C1) constant (%1) constant (–) constant (m2 s1 %1) constant (m2 s1) constant (kJ kg1 °C1 %1) constant (kJ kg1 °C2) constant (kJ kg1 °C1)
R r t T A B F G a1 b1 d1 a2 b2 a3 b3 a4 b4 d4
2.2. Measurement apparatus Fig. 1 shows a schematic of the heat probe (Tokyo Riko Co., Ltd., Tokyo, Japan) used in this study. The probe consisted of a constantan heater wire (0.1 mm in diameter, resistance of 14.8 X) with a small temperature coefficient of electrical resistance, and a T-type thermocouple (0.1 mm in diameter) in a stainless steel tube (1 mm in diameter and 100 mm in length). To avoid contact between the heater wire and thermocouple, the space inside the stainless steel tube was filled with silicone oil. Fig. 2 presents a diagram of the thermophysical property measurement apparatus. The apparatus consisted of three units: heating, temperature control, and recording. The heating unit consisted of the probe, a DC power source (Metronix Co., Ltd., model 526, Tokyo, Japan), a DC ammeter (Yokogawa Electric Co., Ltd., model 2012, Tokyo, Japan), and a switch. A circulating water bath (Tokyo Riko Co., Ltd., TC-100, Tokyo, Japan) with a temperature control accuracy of 0.004 °C was used to hold the temperature constant throughout the tests. The temperature change in the heat source (probe), i.e., the change in the thermo-electromotive force, was amplified with an OP-amplifier (Tokyo Riko Co., Ltd., CA-25F, Tokyo, Japan) and recorded with a data logger (Advantest Co., Ltd.,
Table 1 Compositions (mass fraction) of each component for each sample. Sample
Composition (%) Water
Protein
Fat
Carbohydrate
Fiber
Ash
Grape juice
90.0 80.0 70.0 60.0 50.0
0.2 0.4 0.6 0.8 1.0
0.1 0.1 0.2 0.2 0.3
9.6 19.1 28.7 38.3 47.9
0.0 0.1 0.1 0.1 0.2
0.7 0.3 0.4 0.5 0.6
Pineapple juice
90.0 80.0 70.0 60.0 50.0
0.2 0.5 0.7 0.9 1.2
0.0 0.1 0.1 0.1 0.2
9.4 18.8 28.1 37.5 46.9
0.1 0.2 0.3 0.4 0.5
0.3 0.5 0.8 1.0 1.3
Orange juice
90.0 80.0 70.0 60.0
0.4 0.8 1.2 1.5
0.1 0.1 0.2 0.3
9.2 18.4 27.6 36.7
0.1 0.2 0.3 0.4
0.3 0.5 0.8 1.1
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containers were 30 mm in diameter and 150 mm in height. The reference point container was filled with 1% agar gel. 2.3. Measurement of thermophysical properties
Fig. 1. Schematic of the heat probe used to measure thermophysical properties.
The thermophysical properties of the samples were measured with the modified transient heat flow probe method at five different temperatures (10, 20, 30, 40, and 50 °C) for each concentration of each sample. After the water temperature in the water bath reached a preset temperature, the probe was inserted into the center of the sample. The sample container was submerged in the water bath; thus, the temperatures of the sample and reference point thermocouple (cold junction) were precisely controlled. When the temperature difference between the preset temperature and the sample temperature was less than 0.01 °C, which correspond to the thermo-electromotive force of the T-type thermocouple in the sample of 0.39–0.43 lV at measurement conditions, it was assumed that the temperatures had reached the preset temperature, i.e., they were in thermal equilibrium. Approximately 90 min were necessary to achieve this state of equilibrium. After the thermal equilibrium state had been established, the probe heater wire was energized, and the temperatures of the probe increased with time. The probe temperature changes were amplified and recorded with the data logger. In this study, the input and output range of the amplifier were adjusted to 100 lV and 10 mV, respectively, and the current applied to the heater wire in the probe was adjusted to 130 mA to raise the temperature of the probe to about 2 °C. Current was supplied to the heater wire of the probe for 180 s. In this study, the measurements of thermophysical properties were replicated five times for each experimental condition. 2.4. Estimation method of thermophysical properties When the sample is held at constant temperature, constant heat is continuously supplied to the heater wire in the probe inserted in the sample, and the elapsed heating time is long enough, the temperature change of the probe is approximately expressed as follows (Carslaw and Jaeger, 1988):
Dh ¼ A ln t þ B q A¼ 4pk q 4jb B¼ ln 2 : 4pk r
Fig. 2. Schematic of the experimental apparatus used to measure thermophysical properties.
R7326B, Tokyo, Japan). Because of the high thermal conductivity requirement, the cylindrical sample container and reference point container were made of copper plate (0.3 mm in thickness). These
ð1Þ ð2Þ ð3Þ
Eq. (1) shows that when the elapsed heating time is long enough, the relationship between the probe temperature difference and the logarithm of elapsed time is approximately linear, and the slope and intercept of this straight line are A and B, respectively. Many researchers have measured the thermal conductivities of various foods by utilizing values of A. If the relationship between the probe temperature difference and the time is expressed by Eq. (1) with a high accuracy, then thermal conductivity and thermal diffusivity can be calculated from Eqs. (2) and (3). The thermal conductivity and thermal diffusivity of the 1% agar gel were considered to be equal those of water, as the gel contained 99% water, and these parameters were determined by using Eqs. (2) and (3) at temperatures of 10–50 °C, and the estimated results were compared with values for water in the literature (Singh, 1992). In this case, the measured probe temperature changes in 30–180 s were fitted by a linear least squares method to Eq. (1), and the values of the parameters A and B were determined. The values of the root mean squared error (RMSE) and the coefficient of determination (R2) were about 1.0 103 °C1 and 0.9999 (–), respectively. The measured results agreed well with the calculated values. The ther-
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mal conductivity and thermal diffusivity of water were estimated from Eqs. (2) and (3) using the values of parameters A and B. Because the length of the probe was 100 mm, the heat quantity per unit length of the probe and unit time in Eq. (2) (q) was 0 q ¼ q0 10 ¼ E I 10 ¼ I2 R 10 (W/m). Where, q is the heat 0 quantity of the probe per unit time (W), and q ¼ E I ¼ I2 R. As a result, the thermal conductivity of water estimated from Eq. (2) was 10–15% higher than the literature value. Under some conditions, the thermal diffusivity of water calculated from Eq. (3) was a negative value, or 50% smaller than the literature value. Therefore, even if the measured results agreed well with the values calculated from Eq. (1), it is not suitable for determining the thermal conductivity and diffusivity from Eqs. (2) and (3). Although the length of the probe (length of the stainless pipe) was 100 mm, the heater wire was folded inside the stainless pipe, so the length of the heater wire was unknown. In addition, the values of A and B vary depending on the heat quantity of the probe, the heating time (supply time for the heat quantity), the range of the probe temperature change when fitted to Eq. (1) by the linear least squares method, and so on. This study introduced two new correction coefficients (Pc and Ic) to easily and accurately determine the thermophysical properties. Pc is the correction coefficient for the length of the heater wire in the probe, i.e., heat quantity per unit length of the probe, and q is rearranged: q ¼ q0 Pc ¼ I2 R Pc. Ic is the correction coefficient for the method of supplying the heat quantity to the heater wire in the probe (current, voltage, supply time of heat quantity), the data range of the probe temperature change when fitted to Eq. (1), and the error of measured temperature. In this study, these coefficients were introduced, Eqs. (1)–(3) were modified to following equations:
Dh ¼ F ln t þ G q0 Pc F¼A¼ 4pk q 4jb q0 Pc 4jb G ¼ B Ic ¼ ðln 2 Þ Ic ¼ ðln 2 Þ Ic: 4pk r 4pk r
ð4Þ ð5Þ ð6Þ
The correction coefficients (Pc and Ic) were determined using the measured results for the 1% agar gel at temperatures of 10– 50 °C. The method of supplying the heat quantity to the heater wire in the probe and the range of probe temperature changes when fitted to Eq. (4) were the same as those used for measuring the thermophysical properties of the sample. The values of the parameters F and G were determined by a linear least squares method. The known values (the heat quantity of the probe per unit time q0 ¼ I2 R, the outer diameter of the probe r, and the constant b) and the literature values of the thermal conductivity and thermal diffusivity of water (Singh, 1992) were substituted into Eqs. (5) and (6), and the values of the correction coefficients Pc and Ic were determined at each temperature. The values of the correction coefficients at temperatures from 10 °C to 50 °C ranged from 7.8 to 8.2 for Pc and from 0.39 to 0.30 for Ic. These correction coefficients had almost the same value regardless of temperature, remaining nearly constant from 10 °C to 50 °C. The values of Pc and Ic for the measurement conditions and apparatuses in this study were taken as the mean values of Pc and Ic in the temperature range of 10 °C to 50 °C, i.e., Pc = 8.167, Ic = 0.3414. To predict the thermophysical properties of the samples, the measured probe temperature changes in 30–180 s for samples were fitted by a linear least squares method to Eq. (4), and the values of the parameters F and G were determined. Using the values of F, G, Pc, and Ic, the thermal conductivity and thermal diffusivity of the sample were calculated from Eqs. (5) and (6), respectively. The specific heat of the sample was estimated by substituting the thermal conductivity and thermal diffusivity calculated from Eqs. (5)
and (6) and the density of the sample under the same condition into the following equation for thermal diffusivity (Singh, 1992):
j¼
k : cp q
ð7Þ
The values of the sample densities measured with a pycnometer (Figura and Teixeira, 2007) were used in Eq. (7). The method for estimating thermophysical properties used in this study was calibrated at 20 °C using 20%, 40%, and 60% sucrose solutions and 14%, and 23% sodium chloride solutions. As a result, the relative errors between the measured values and the literature values (Hayashi, 1989; The Japan Society of Mechanical Engineers, 1991; Singh, 1992) were 0–3% for thermal conductivity, 0–6% for thermal diffusivity, and 0–4% for specific heat. The measured values agreed well with the literature values. Therefore, Eqs. (4)–(7) and the values of the two correction coefficients (Pc and Ic) were used to determine the thermophysical properties of the samples. 3. Results and discussion 3.1. Temperature change of the probe The measured data for the probe temperature changes over 30– 180 s at each measurement condition were fitted by a linear least squares method to Eq. (4). A comparison between the measured results and the results calculated from Eq. (4) at a concentration of 10% and a temperature of 50 °C for orange juice is shown in Fig. 3. The solid line and symbols in Fig. 3 show the calculated values and the measured results, respectively. In this case, the values of parameters F and G, the RMSE and R2 in Eq. (4) were F = 0.2639, G = 0.4151, RMSE = 1.449 103 °C, and R2 = 0.9999 (–). As shown in Fig. 3, the measured data for the temperature changes of the probe over 30–180 s agreed well with the calculated values. Similar results were obtained for all other measurement conditions. Therefore, the values of parameters F and G determined from the probe temperature changes over 30–180 s were used to predict the thermophysical properties of the samples. 3.2. Thermal conductivities of the samples The values of the thermal conductivity calculated from Eq. (5) were 0.42–0.62 W m1 °C1 for grape juice and pineapple juice, and 0.46–0.61 W m1 °C1 for orange juice. The measured results of these samples were almost the same under the same conditions (temperature and concentration) because the major compositions
Fig. 3. Comparison of the observed temperature changes of the probe with the results calculated from Eq. (4) for orange juice at a concentration of 10% and a temperature of 50 °C.
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of all sample solutions were water and carbohydrate, and the compositions of these samples were similar (Table 1). The thermal conductivities of liquid foods, such as milk and fruit juice, can be expressed as functions of temperature and concentration (Constenla et al., 1989; Reddy and Datta, 1994; Telis-Romero et al., 1998; Zainal et al., 2000; Tansakul and Chaisawang, 2006). Riedel (1949) presented the following model to predict the thermal conductivities of fruit juice, sugar solutions, and milk (Singh, 1992):
tivity of the grape juice increased linearly with increasing temperature. Results similar to Fig. 4 were obtained for all samples. At the same temperature, the thermal conductivities of all samples decreased as the concentration increased, and a linear relationship existed between thermal conductivity and concentration. Therefore, we derived the following empirical equation to represent the relationship between thermal conductivity, concentration, and temperature:
k ¼ ð326:58 þ 1:0412T 0:00337T 2 Þ
The measured data for each sample were fitted by a linear least squares method to Eq. (9). The solid lines in Fig. 4 show the calculated results. For all samples, the calculated values agreed well with the measured data. The parameters, RMSE, and R2 of Eq. (9) for each sample are shown in Table 2. Eq. (9) accurately represented the relationships between thermal conductivity and both concentration and temperature for samples. Constenla et al. (1989) derived the following equation from the parallel heat transfer model based on the assumption that apple juice was consisted of two compositions; water and solid:
4
ð17:30 10
6
9:342 10 CÞ:
ð8Þ
The thermal conductivities calculated from Eq. (8) were compared with the measured thermal conductivities of the samples. As a result, the relative errors between the measured values and the calculated values were determined to be 0–3%. The fact that there was no significant difference between the measured values and the calculated values demonstrates that the method adopted in this study for estimating thermophysical properties was appropriate. Fig. 4 shows the relationships between thermal conductivity and temperature for grape juice, showing that the thermal conduc-
k ¼ a1 C þ b1 T þ d1 :
ð9Þ
k0 ¼ a2 C þ b2 ;
ð10Þ
0
0
where k represents a reduced thermal conductivity (k ¼ kqw k1 w q1 ). The values of the thermal conductivity of the sample k, thermal conductivity of water kw , density of the sample q, and density of water qw were known. Eq. (10) represents that the linear relationship exists between k0 and concentration C. The values of k0 for each sample were fitted by a linear least squares method to Eq. (10). The parameters, RMSE, and R2 of Eq. (10) for each sample are shown in Table 2. From Table 2, it is confirmed that the comparison of the measured data with the calculated value from Eq. (10) showed good agreement. 3.3. Thermal diffusivities of the samples
Fig. 4. Comparison of the measured thermal conductivity and the result calculated from Eq. (9) for grape juice.
The values of thermal diffusivity of the samples obtained from Eq. (6) were 1.2 107–1.5 107 m2 s1. The thermal diffusivities of orange juice as measured by the Dickerson method (Telis-Romero et al., 1998) were 1.0 107–1.4 107 m2 s1 at temperatures from 0.5 to 62 °C and water content levels from 30 to 75% (total solid contents from 25 to 70%), and were written as a function of temperature and water content. Comparing the measured data to the literature values for water (Singh, 1992) and
Table 2 The parameters, coefficients of determination (R2), and root mean squared errors (RMSE) of Eqs. (9)–(12) for each sample. Eq.
Sample Grape juice
Pineapple juice
Orange juice
(9)
a1 (W m1 °C1 %1) b1 (W m1 °C2) d1 (W m1 °C1) R2 (–) RMSE (W m1 °C1)
3.328 103 1.471 103 5.796 101 0.9915 0.0048
3.456 103 1.405 103 5.814 101 0.9950 0.0037
3.244 103 1.337 103 5.778 101 0.9837 0.0053
(10)
a2 (%1) b2 (–) R2 (–) RMSE (W m1 °C1)
7.952 103 9.970 101 0.9868 0.0059
8.124 103 9.962 101 0.9912 0.0050
8.007 103 9.926 101 0.9862 0.0048
(11)
a3 (m2 s1 %1) b3 (m2 s1) R2 (–) RMSE (m2 s1)
4.243 1010 1.475 107 0.9765 9.302 1010
3.659 1010 1.497 107 0.9733 8.566 1010
4.725 1010 1.493 107 0.9888 5.622 1010
(12)
a4 (kJ kg1 °C1 %1) b4 (kJ kg1 °C2) d4 (kJ kg1 °C1) R2 (–) RMSE (kJ kg1 °C1)
2.579 102 1.053 102 3.938 0.9944 0.0294
2.743 102 9.788 103 3.883 0.9934 0.0337
3.084 102 9.929 103 4.135 0.9885 0.0401
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orange juice (Telis-Romero et al., 1998), the measured values were almost the same as the reported values for these liquids. The thermal diffusivities of the samples decreased with increasing concentration. This tendency was also seen in a report by Telis-Romero et al. (1998). For each concentration, the thermal diffusivity of each sample was almost constant over temperature. The relationships between the thermal diffusivity and concentration of the samples are shown in Fig. 5. The symbols in Fig. 5 show the measured results, and the average values of the measured data at 10–50 °C at each concentration and for each sample are also plotted in Fig. 5. The thermal diffusivities of the samples decreased linearly with increasing concentration. The values of thermal diffusivity plotted in Fig. 5 for each sample were fitted by a linear least squares method to the following equation:
j ¼ a3 C þ b3 :
ð11Þ
The values of the parameters are given in Table 2 along with the RMSE and R2 of Eq. (11) for each sample. The RMSE and R2 values show that the measured data agreed well with the calculated values. Thus, the thermal diffusivity of each sample was represented by Eq. (11) as a function of concentration only within the measurement conditions in this study. 3.4. Specific heats of the samples The specific heat of each sample was determined from Eq. (7) by substituting the thermal conductivity and thermal diffusivity
Fig. 5. Empirical relationships between the thermal diffusivity and the concentration for each sample.
calculated from Eqs. (5) and (6) and the density of the sample under the same condition into Eq. (7). Fig. 6 shows the relationships between specific heat and concentration for pineapple juice. The symbols in Fig. 6 show the measured results. The values of the specific heat of pineapple juice were 2.6–4.1 kJ kg1 °C1 under the measured conditions, and decreased linearly as the concentration rose. Relationships similar to Fig. 6 were obtained for all samples. The specific heats of the samples increased with increasing temperature for each concentration of each sample. Constenla et al. (1989) measured the specific heat of clarified apple juice over the range from 20–90 °C and 12 to 70 °Brix by using a DSC, and these values were related to temperature and concentration (°Brix). The values of the specific heat of apple juice measured with a twin isoperibol calorimeter at concentrations of 10–45% and temperatures of 10–50 °C were 3.1–4.2 kJ kg1 °C1, and were represented as a function of temperature and concentration (Muramatsu et al., 2000). There were no significant errors between the measured specific heats of the samples and the literature values for other fruit juices. Tansakul and Chaisawang (2006) reported that the specific heat of coconut milk measured with a DSC was related to temperature and concentration by the following equation:
cp ¼ a4 C þ b4 T þ d4 :
ð12Þ
In this study, because both the temperature dependency and the concentration dependency were represented by a linear function, the measured specific heat data for each sample were fitted by a linear least squares method to Eq. (12). The solid lines in Fig. 6 show the calculated results. As shown in Fig. 6, the calculations agreed well with the measured data. The values of the parameters, RMSE, and R2 of Eq. (12) for each sample are shown in Table 2. From Table 2, it is apparent that the relationships between specific heat and both concentration and temperature for each sample were accurately represented by Eq. (12). In this study, two new correction coefficients were introduced, and the thermal conductivities, thermal diffusivities, and specific heats of samples were simultaneously determined using a modified transient heat flow probe method, and these values were written as a function of temperature and concentration. The estimation method adopted in this study, expressed as Eqs. (4)–(7), is suitable for predicting thermal conductivity, thermal diffusivity, and specific heat with accuracy sufficient for practical use. This estimation method can be easily determined the three kinds of thermophysical properties and applied to various foods. The empirical equations, i.e., Eqs. (9)–(12), will be useful in the design of equipment and in calculations for the thermal processing of fruit juice.
4. Conclusion
Fig. 6. Empirical relationships between the specific heat and the concentration at five different temperatures for pineapple juice.
A new method for the simultaneous determination of thermophysical properties using a modified version of current probe theory method was proposed. Two new correction coefficients were introduced, and using the values of correction coefficients and an approximate solution of the heat conduction equation, the thermal conductivity and thermal diffusivity of three kinds of fruit juices (grape juice, orange juice, and pineapple juice) were measured at various temperatures (10–50 °C) and concentrations (10–50%), respectively. The specific heat of each sample was estimated from the definition of thermal diffusivity. The thermal conductivity and specific heat for each sample were represented by the empirical equations shown as a linear function of both concentration and temperature, respectively. The thermal diffusivity of each sample was represented as a linear function of concentration only within the measurement conditions in this study.
Y. Muramatsu et al. / Journal of Food Engineering 96 (2010) 607–613
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