Determination and Modeling of Thermal Properties of Tofu

Abstract

Thermal properties of tofu were determined in the temperature range of 5–80°C (10–105°C for specific heat) and the moisture content range of 0.3–0.7 (wb). Thermal conductivity (k: 0.24–0.43 W m⁻¹ K⁻¹) and thermal diffusivity (α: 7.6 × 10⁻⁸–1.15 × 10⁻⁷ m² s⁻¹) were measured simultaneously using dual probe method, and specific heat (C _p : 2.52–3.69 kJ kg⁻¹ K⁻¹) with modulated differential scanning calorimetry. Good agreement was observed between perpendicular model and the thermal conductivity data, but not with parallel model. Simple empirical models were also developed as a function of the moisture content and temperature of the tofu.

Keywords:

• Specific heat
• Thermal conductivity
• Thermal diffusivity
• Perpendicular model
• Parallel model
• DSC

Introduction

Thermal properties of a product depend mostly on its composition and temperature. For porous products, such as bakery products, heat transfer by evaporation and condensation of water often affects the thermal properties as well.[1] Knowledge of thermal properties is important for mathematical modeling and simulation of heat and moisture transport. With the advent of high‐speed computers and inexpensive memory, thermo‐physical properties of target products can be treated as composition/temperature dependent variables instead of the average values over the whole process.

Tofu (soybean curd) is a versatile protein‐rich food and has traditionally been used as a source of protein or meat substitute in Eastern diets. The nutritional value (65% of net protein utilization) and functional properties (anti‐cancer, brain activity elevation) of tofu attract people across the world.[2] Tofu industry is continuously growing from domestic scale to mass scale, and various thermal processing, such as, drying, baking, frying is being applied for better organoleptic properties and microbial safety. During the cooking process, due to heat and mass transfer, temperature and moisture content change. Thus, in order to control the process and design new equipment, the thermal properties of tofu should be known for various ranges of moisture content and temperature. Little work has been published on the thermal properties of the tofu.

Kong et al.[3] determined thermal diffusivity of the tofu (fresh and frozen) at different moisture contents (mass fraction of water 1–0.65, initial moisture content: 0.8 wb) and at −11 to 3°C using 1‐dimensional unsteady heat transfer through the product. They confirmed that the tofu could be considered homogenous in terms of heat transfer, and effective thermal diffusivity was a function of composition and temperature.

The objectives of this study were to estimate the thermal properties of the tofu, and to develop prediction models of the thermal properties for possible use in the simulation of transport phenomenon during thermal processing of the tofu.

Materials and Methods

Tofu

Commercial tofu (extra firm variety, 350 g packets, Toronto based manufacturer) were obtained. For uniform quality, outer layer of tofu (∼0.4 cm) was removed using a sharp knife due to its possible unique hardness caused from mechanical pressure during its processing. Cylindrical samples (D: 5.7 cm, H: 3.5 cm) were cut using a sharp cylindrical cutter for thermal conductivity and diffusivity measurement. Initial chemical composition of the samples were 70.3% water, 15.6% protein, 10.7% fat, 2.0% carbohydrates, 1.1% fiber, and 0.29% ash. Moisture content was determined by drying method.[4] Other compositions were from chemical analysis provided by the manufacturer.

Dehydration

For five levels of moisture content in the range of 0.3–0.7 (wb), fresh tofu was dried in a convection oven (Fisher Isotemp oven® deluxe model/forced draft, Fisher Scientific Inc., Nepean, Ontario, Canada) at 70°C (5% relative humidity) for 0, 2, 8.3, 25, and 46.6 h. During drying, samples were placed on a metal screen support for the exposure of bottom surface to hot flowing air. For faster drying, 90°C air temperature was used in preliminary tests, however, it caused internal shrinkage which lead to inner structure breakage. Thermal deterioration effect if any affected the thermal properties of tofu, which is needed as the tofu is inherently a cooked product through its preparation, and the objective of this study was to evaluate the thermal properties of tofu for various thermal processes. After the completion of drying, samples were kept in a refrigerator (∼3°C) for one week for their internal moisture equilibrium.

Specific Heat

Sample specific heats were determined using a modulated differential scanning calorimeter, (MDSC 2920, TA Instruments Inc., New Castle, DE) with a nitrogen cooling system. Temperature modulation (sinusoidal oscillation) of MDSC separates total heat flow [Eq. 1] into its reversing [specific heat related, MC _p (dT/dt)] and non‐reversing [kinetic, f (t,T )] components. The f (t,T ) denotes specific response that is kinetically hindered and takes a measurable amount of time to come to equilibrium when the temperature is changed, such as in glass transition, crystallisation, melting, chemical reactions, etc. For its calculation, the amplitudes of modulations in heat flow and heating rate, phase angle are required and they are measurable by the MDSC.

where, dQ/dt is heat flow rate. Thus, specific heat values obtained by MDSC are more precise than those obtained with a conventional DSC.[5] Into an aluminum pan, 10–15 mg of the sample was placed and the pan was hermetically sealed. A sealed empty pan was used as reference. The sample was scanned at a heating rate of 3°C/min over the temperature range (10–105°C). An 80 s modulation period (single cycle) and ± 1°C amplitude were used. Nitrogen was used as a purging gas (150 mL/min). The operating conditions were based on the report[5] for optimum accuracy. Distilled water was used for DSC calibration by comparing the measured C _p of pure water with published data.

Specific heat was also estimated from measured thermal conductivity, thermal diffusivity, and bulk density by applying the following relationship.

The apparent density of tofu sample was obtained as sample mass divided by its volume. The samples were cut with a sharp knife into regular shapes. The sample was then weighed, and each dimension was measured to calculate volume.

Thermal Conductivity and Thermal Diffusivity

A commercial thermal property‐meter (Thermolink TL‐1, Devices Inc., Pullman, WA) was used to measure thermal conductivity and thermal diffusivity of the tofu. The instrument was designed based on line heat source probe and additional thermocouple probe method. The merit of this method is the simultaneous measurement of both properties. The dual probes (1.0 mm o.d. and 30 mm long) consisted of a constantan heater wire and a chromel–constantan E‐type thermocouple (TLS‐1, Decagon Devices Inc., Pullman, WA). The probes were spaced 6 mm apart. Their location was in the following ranges suggested by Nix et al.[6]

where, r _d is the distance of temperature sensor from the probe heater, and t is operating time. A heating wire was inside one probe, and a type E (chromel/constantan) thermocouple was placed in the other. The two probes were placed in the sample, and 200 mA of current was passed through the heater wire, causing the heater probe to emit an 8 s‐long heat pulse. The probes have previously equilibrated with the sample temperature, so the temperature rise was monitored by the thermocouple for 60 s. The total power dissipated by the heater, Q was the product of the heater resistance (31.2 Ω) and the square of the current going through it. This was divided by the length of the heater and multiplied by the heat pulse length (8 s) to get Q (333 J m⁻¹ = 31.2 × 0.2² × 8/0.03).

The rate of temperature increase in the cylindrical heater is directly related to the diffusivity as shown in the following equations:

where, r is the radial distance. A simplified analytical solution[7] [Eq. 5] was used with a mathematical inverse method to find k and α from a set of measurements of ΔT vs. t for the tofu.

where, ΔT is the temperature rise measured by a thermocouple. The calculation was achieved by a microprocessor of the instrument with non‐linear least square fitting and the thermal property values were taken when R ² of the best‐fit curve was higher than 0.99. Before starting this study, k and α of agar gels (1, 1.5, 2 g agar/100 g water) were measured at 20°C using our instrument. The observed relationships were:

The intercept values were close enough to literature values[8] (0.597 W m⁻¹ K⁻¹ and 1.43 × 10⁻⁷ m² s⁻¹) of thermal properties of water with relative error of 2.4 and 2.1%, respectively.

For test at T < 10°C, tofu samples were directly transferred from the refrigerator into a insulation box. The probe was embedded inside the box cover (Fig. 1) and then the probe was fully inserted into the sample placed in the box with the linear movement through nuts to avoid breakage of the sample. For maintaining the probe spacing, a probe guide was placed on the end of the probe needles and then the needles were gently inserted all the way into the tofu samples. After each test, the spacing was verified by cutting inner part of the tofu sample and measuring the distance between two needle marks. A few wrong insertions made spacing errors, and the measured data were discarded for those experiments. The material of bolts and nuts was steel. Thermal conductivities of the insulation material (extruded polystyrene form) and the gasket (formed rubber) were 0.026 and 0.034 W m⁻¹ K ⁻¹ at 300 K, respectively. The samples were thermally equilibrated for at least 40 min. For higher temperatures, samples in the hermetic plastic bag were heated in water bath for ∼40 min, then samples were immediately moved from the platic bag into the sample holder, which was water proof. The samples were thermally equilibrated for 1.5–2 h based on preliminary test results. After each test, moisture content of the portion between two probes was determined. This moisture content was used in the regression analysis. At the beginning of this study, microwave heating was tried for the sample heating. In spite of the application of low power/long time, considerable pores were formed in the sample during heating. Water bath heating did not cause this problem. A sample diameter of 57 mm was selected to avoid edge effect, based on Campbell et al.[7] as follows:

Figure 1. Schematic of the measurement system for thermal conductivity and thermal diffusivity.

Data Analysis and Modeling

All experiments for each treatment were conducted in triplicate. A factorial experimental design was used. For C _p test, each different sample was used for three determinations; for k and α test, one same sample for three determinations. Statistical analysis was conducted using PROC REG[9] to develop regression models for the thermal properties of the tofu. The step‐wise regression procedure was used for selecting variables to be included in the model. The variables considered in the procedure were m, T, m ² and T ² and all interaction terms. Density was not included in the model to avoid multicollinearity among the variables since density often can be related to moisture content. To select the best‐fitting model, two criteria were considered, the correlation coefficient R ² and the F value of the model.

Results and Discussions

Specific Heat

Specific heat of the tofu containing moisture content of 0.33–0.68 (wb) was 2.52–3.69 kJ kg⁻¹ K⁻¹ for temperature range of 10–105°C. Specific heat of tofu increased with moisture content and temperature (Table 1a). Calculated C _p values from mass fraction model (∑ X _i ^w C _pi ) were compared with corresponding measured values. First, six major tofu components (water, protein, carbohydrate, fat, fiber, and ash) were considered; and then only two components were taken (water and total dry matter including fat). The C _p of dry matter was determined by thermal scanning of a fully dried tofu using MDSC. For other six components, literature data[10] were used. Visual comparison showed a good agreement with experimental data (Fig. 2). The RMS (root mean square deviation; , where, N = number of C _p values) between predictive values (C _pcal ) from two models and experimental data (C _pexp ) were 0.0987 and 0.103, respectively (Table 2) indicating the change at one decimal place. An empirical model as a function of moisture content and temperature was developed using stepwise method of PROC REG.[9] A summary of the stepwise selection of variables is given in Table 3. The suitable model is:

where, m is wet basis moisture content, w/w. The residuals vs. predicted C _p and each term in the model were normally distributed. The RMS (0.0684) of this model was lower than those models based on mass fraction. Effects of moisture content and temperature (m, m ² , T ) on C _p were significant (p < 0.01). Interaction between two variables was not significant (p > 0.05). A 3‐dimensional plot (Fig. 3) of C _p as a function of m and T is based on Eq. 9. The moisture content had higher effect on C _p than temperature, i.e., 10% of moisture content and temperature increase had 4.5% and 0.17% C _p increase, respectively for tofu with 0.5 moisture content and 50°C temperature.

Figure 2. Comparison between experimental and calculated specific heats.

Figure 3. Specific heat calculated from an empirical model [Eq. 9] as a function of moisture content and temperature of the tofu.

Table 1a. Specific heat (kJ kg⁻¹ K⁻¹) data of the tofu at different moisture contents and temperatures

	Temperature, °C
m (wb), %	10	30	50	70	90	105
0.677	3.450–3.583	3.479–3.610	3.519–3.649	3.541–3.669	3.544–3.679	3.556–3.689
0.604	3.160–3.311	3.283–3.333	3.310–3.363	3.321–3.378	3.325–3.384	3.329–3.391
0.510	2.980–3.198	2.987–3.188	3.026–3.197	3.115–3.216	3.160–3.268	3.197–3.282
0.428	2.661–2.998	2.701–2.907	2.732–2.924	2.770–2.931	2.779–2.951	2.851–2.884
0.333	2.530–2.649	2.626–2.666	2.648–2.690	2.657–2.722	2.660–2.707	2.663–2.713

Table 1b. Thermal conductivity and thermal diffusivity data of the tofu at different moisture contents and temperatures

m (wb), %	T, °C	ρ, kg m ^−3	k, W m^−1 K^−1	α × 10^7, m^2 s^−1
0.702	7.5–12.8	1,078	0.35–0.38	1.00–1.01
0.704	20.8–21.2	1,054	0.37–0.40	1.06–1.07
0.732	49.4–53.2	1,059	0.39–0.42	1.10–1.15
0.724	48.1–65.4	1,053	0.40–0.42	1.12–1.13
0.712	62.7–74.2	1,059	0.39–0.43	1.11–1.12
0.616	6.4–11.3	1,112	0.33–0.34	0.90–0.95
0.594	20.9–21.4	1,118	0.33–0.36	0.96–1.01
0.617	46.6–54.6	1,112	0.34–0.37	0.95–1.00
0.626	48.4–56.9	1,113	0.35–0.38	1.00–1.01
0.623	64.5–73.4	1,082	0.36–0.37	1.02–1.03
0.512	7.1–11.8	1,134	0.28–0.30	0.82–0.85
0.510	21.6–21.7	1,180	0.29–0.31	0.82–0.84
0.506	44.4–48.1	1,133	0.32	0.90–0.91
0.539	50.3–63.9	1,157	0.32–0.33	0.89–0.91
0.524	65.2–73.1	1,157	0.31–0.33	0.89–0.90
0.420	6.04–12.7	1,203	0.24–0.28	0.78–0.79
0.406	22.0–22.2	1,207	0.25–0.27	0.77–0.78
0.418	42.9–43.9	1,187	0.27–0.29	0.82
0.423	52.8–63.0	1,211	0.28–0.30	0.81–0.83
0.434	64.8–73.6	1,206	0.28–0.31	0.83–0.84
0.344	7.8–12.4	1,225	0.23–0.27	0.76–0.78
0.342	21.5–21.7	1,251	0.26–0.28	0.78–0.81
0.358	45.3–48.9	1,215	0.27–0.28	0.82
0.365	37.4–54.2	1,215	0.24–0.28	0.78–0.79
0.366	64.6–72.7	1,235	0.26–0.28	0.78–0.80

Table 2. Thermal property models and root mean square errors in comparison to experimental data

References	Equation	RMS
[15]		0.0987
[16]		0.1030
Present study		0.0684
[15]		0.0118
		0.0118
Present study		0.0113
[14]		3.01 × 10^−8
Present study		2.224 × 10^−9

Table 3. Summary of the stepwise selection of variables in the empirical model of specific heat of the tofu

Input variables	Variables selected	Estimate	SEE ^a	Pr > F
m, T, m × T, m ^2,	m ^2	1.596	0.3228	< 0.0001
T ^2, m × T ^2,	T	1.06 × 10^−3	1.38 × 10^−4	< 0.0001
m ^2 × T, m ^2 × T ^2	m	1.064	0.328	0.0013
	Intercept	2.055	0.08	< 0.0001

a: Standard error of estimate.

Note: Model R ² = 0.959, degree of freedom of error = 296.

Specific heat was also calculated from measured thermal conductivity, diffusivity, and density. The apparent density of tofu was 1053–1251 kg m⁻³ for the moisture content range of 0.34–0.73 (wb). Linear relationship was observed between density and moisture content:

where, SEE is standard error of estimate.

The residuals vs. predicted ρ and the moisture content term in the model were normally distributed. For shrinkage modeling of a solid material, the following assumption is often applied. Thermal expansion of a sample is negligible compared to shrinkage due to moisture migration for a non‐porus solid material. Good agreements were reported between the calculated data from this assumption and experimental data.[11],[12] Therefore, temperature effect was neglected in the correlation. The calculated specific heat was slightly lower (∼1.7%) than those measured with MDSC, however, the difference was not significant (p = 0.17) when using t‐test (DF = 132). Direct measurement using MDSC should be more accurate as calculated C _p already has inherent deviation from experimental errors of three other thermophysical properties.

Thermal Conductivity

Thermal conductivity of the tofu was 0.24–0.42 W m⁻¹ K⁻¹ for moisture content range of 0.34–0.72 and temperature range of 6–74.2°C (Table 1b). Yano et al.[13] reported thermal conductivity of frozen and fresh tofu (physical characteristics, such as variety, moisture content, temperature, and density of tofu were not specified) as 0.49 and 0.30 W m⁻¹ K ⁻¹, respectively. Thermal conductivity of the tofu increased with increase in temperature and moisture content. Predicted values from the following structural models were compared with experimental data (Fig. 4).

Where, X _i ^v = (X _i ^w /ρ _i )/(∑ X _i ^w /ρ _i ), k _i and ρ _i were taken from literature.[10] In parallel distribution, multiple phases are considered in parallel with respect to the direction of heat flow. In the case of perpendicular distribution, a structural arrangement of the phases is in perpendicular to heat flow.[14] Parallel and perpendicular models were selected for comparison due to their practical applications. Application of other structural models requires k values of continuous and discontinuous phases in the sample, which is generally not available.

Figure 4. Comparison between experimental and calculated thermal conductivity from structural models and empirical model.

Predicted values from the parallel model did not match well with experimental values, and indicating a RMS of 0.119. The perpendicular model provided good predictions with a RMS of 0.0118. This was consistent with the observation of Yano et al.[13] Statistical analysis showed that the effects of moisture content (m ²) and interaction between moisture content and temperature (m × T ) were significant ( p < 0.0001) (Table 4). The model is:

A 3‐dimensional surface mesh of thermal conductivity was plotted based on Eq. 13 (Fig. 5). The effect of moisture content was predominant as the case of specific heat. At 0.5 moisture content (wb) and 50°C temperature of the tofu, a 10% increase in moisture content and temperature produced 5.9% and 0.72% of k increase, respectively.

Figure 5. Thermal conductivity calculated from an empirical model [ Eq.13] as a function of moisture content and temperature of the tofu.

Table 4. Summary of the stepwise selection of variables in the empirical model of thermal conductivity of the tofu

Input variables	Variables selected	Estimate	SEE	Pr > F
m, T, m × T, m ^2,	m ^2	0.3077	0.0119	< 0.0001
T ^2, m × T ^2,	m × T	8.943 × 10^−4	1.207 × 10^−4	< 0.0001
m ^2 × T, m ^2 × T ^2	Intercept	0.2112	0.00357	< 0.0001

Note: Model R ² = 0.944, degree of freedom of error = 69.

Thermal Diffusivity

For moisture content range of 0.34–0.72 (wb) and temperature range of 6–74.2°C, the thermal diffusivity of the tofu were 7.6 × 10⁻⁸–1.15 × 10⁻⁷ m² s⁻¹. These values are slightly lower than those (1.1 × 10⁻⁷–1.3 × 10⁻⁷ m² s⁻¹) of the tofu with moisture content range of 0.52–0.8 (wb) and temperature range of 0–3°C reported by Kong et al.[3]

Martens'regression model[14] [ Eq.14] based on 246 published values of thermal diffusivity of many food products were compared with experimental data.

Figure 6 shows that Martens' model consistently predicted higher values than experimental data indicating a RMS of 3.01 × 10⁻⁸. This suggests that the thermal diffusivity of the tofu is lower than that of average values of various foods. From statistical analysis, moisture content (m, m ²) and interaction between moisture content and temperature (m ² × T, m ² × T  ²) had significant effect ( p < 0.05) on the thermal diffusivity of the tofu. Developed empirical model is:

Details on the stepwise selection of variables are listed in Table 5. The RMS between this model and experimental data was 2.224 × 10⁻⁹. A 3‐dimensional surface mesh of the predicted thermal diffusivity from Eq. 15 was plotted (Fig. 7). Similar to specific heat and thermal conductivity, sensitivity of moisture term was higher than that of temperature. A 10% increase in moisture content and temperature caused 5.0% and 0.21% increase in α, respectively for tofu with 0.5 moisture content (wb) and 50°C temperature.

Figure 6. Comparison between experimental and calculated thermal diffusivity from Martens' and empirical model.

Figure 7. Thermal diffusivity calculated from an empirical model [Eq. 15] as a function of moisture content and temperature of the tofu.

Table 5. Summary of the stepwise selection of variables in the empirical model of thermal diffusivity of the tofu

Input variables	Variables selected	Estimate	SEE	Pr > F
m, T, m × T, m ^2,	m ^2	1.164 × 10^−7	2.056 × 10^−8	< 0.0001
T ^2, m × T ^2,	m ^2 × T	6.866 × 10^−10	1.87 × 10^−10	0.0005
m ^2 × T, m ^2 × T ^2	m	−5.682 × 10^−8	2.157 × 10^−8	0.0105
	m ^2 × T ^2	−5.17 × 10^−12	2.35 × 10^−12	0.0315
	Intercept	8.16 × 10^−8	5.51 × 10^−9	< 0.0001

Note: Model R ² = 0.965, degree of freedom of error = 67.

Conclusions

Thermal properties of tofu were measured for various ranges of moisture content and temperature. Predicted properties from mass fraction model and perpendicular model showed good agreement with experimental data of specific heat and thermal conductivity, respectively. Empirical regression models were developed as a function of moisture content and temperature.

Acknowledgments

The first author thanks Fond FCAR (Le Fonds pour la Formation de Chercheurs et l'Aide à la Recherche) for providing post‐doctoral scholarship for this research. The authors gratefully acknowledge NSERC (Natural Science and Engineering Research Council of Canada) for research funding.

Nomenclature
a	=	Agar content in weight percentage (g agar/100 g water)
C p	=	Specific heat, kJ kg−1 K−1
k	=	Thermal conductivity, W m−1 K−1
M	=	Mass, kg
m	=	Wet basis moisture content in mass ratio
Q	=	Power dissipated by probe heater, J m−1 or heat provided to the sample, J
r d	=	Distance in radial direction, m
r s	=	Sample radius, m
T	=	Temperature, °C
t	=	Time, s
X i w , X i v	=	Mass fraction, volume fraction
α	=	Thermal diffusivity, m2 s −1
ρ	=	Apparent density, kg m−3

Subscript
cal	=	calculated
exp	=	experimental
i	=	unit component