Thermophysical properties prediction of brown seaweed (Saccharina latissima) using artificial neural networks (ANNs) and empirical models

Figures & data

Table 1. Thermophysical properties calculated using Choi and Okos' (1986) model based on the proximate content of food

Thermophysical Parameter	Equation
Thermal Conductivity (W m^−1°C ^−1)	k = (0.57109 + 0.001762T – 6.7036 x 10^−6T^2) Xw + (0.18071–0.0027604T – 1.7749 x 10^−7T^2) Xf + (0.17881 + 0.001958T – 2.7178 x 10^−6T^2) Xp + (0.20141 + 0.0013874T – 4.3312 x 10^−6T^2) Xc  + (0.32961 + 0.001401T – 2.9069 x 10^−6T^2) Xa
Specific Heat Capacity (J kg^−1°C^−1)	C = (4176.2–0.0909T + 0.0054T^2) Xw + (1984.2 + 1.4373T – 0.0048T^2) Xf + (2008.2 + 1.2089T – 0.0013T^2) Xp + (1548.8 + 1.9625T – 0.00594T^2) Xc + (1092.6 + 1.8896T – 0.00368T^2) Xa
Density (kg m^−3)	ρ = (997.18 + 0.00314T – 0.00375T^2) Xw + (925.59–0.41757T) Xf + (1329.9–0.5184T) Xp + (1599.1–0.36589T) Xc + (2423.8–0.28063T) Xa

where X_i are the respective volume fractions of water, fats, protein, carbohydrate, and ash present in each sample. T (°C) is the temperature of the sample. The thermal diffusivity of the samples is calculated using formula (5).

Table 2. Thermal conductivity of sugar kelp measured using KD2 Pro and Choi and Okos' model

		Moisture Content (g water (100 g sample)^−1)
Analysis	Sample Temperature	6	10	30	50	70	90
Experimental Thermal Conductivity (W m^−1 K^−1)	30°C	0.147 ± 0.001 aAx	0.175 ± 0.004 aAx	0.226 ± 0.009 aBx	0.342 ± 0.008 aCx	0.421 ± 0.021 aDx	0.473 ± 0.011 aDx
	40°C	0.178 ± 0.005 bAx	0.205 ± 0.011 bAx	0.255 ± 0.014 bBx	0.374 ± 0.025 bCx	0.447 ± 0.025 bDx	0.497 ± 0.005 bDx
	50°C	0.200 ± 0.006 cAx	0.219 ± 0.004 cAx	0.260 ± 0.011 cBx	0.402 ± 0.057 cCx	0.508 ± 0.014 cDx	0.519 ± 0.025 cDx
	60°C	0.239 ± 0.016 dAx	0.252 ± 0.015 dAx	0.289 ± 0.019 dBx	0.444 ± 0.046 dCx	0.558 ± 0.048 dDx	0.543 ± 0.017 dDx
	70°C	0.349 ± 0.017 eAx	0.284 ± 0.021 eAx	0.329 ± 0.026 eBx	0.459 ± 0.030 eCx	0.626 ± 0.072 eDx	0.580 ± 0.067 eDx
Choi and Okos' Model Thermal Conductivity (W m^−1 K^−1)	30°C	0.158 ± 0.000 aAy	0.165 ± 0.000 aBy	0.234 ± 0.000 aCy	0.356 ± 0.000 aDy	0.498 ± 0.000 aEy	0.562 ± 0.000 aFy
	40°C	0.164 ± 0.000 bAy	0.171 ± 0.000 bBy	0.241 ± 0.000 bCy	0.366 ± 0.000 bDy	0.510 ± 0.000 bEy	0.575 ± 0.000 bFy
	50°C	0.171 ± 0.000 cAy	0.177 ± 0.000 cBy	0.248 ± 0.000 cCy	0.375 ± 0.000 cDy	0.520 ± 0.001 cEy	0.589 ± 0.000 cFy
	60°C	0.176 ± 0.000 dAy	0.182 ± 0.000 dBy	0.254 ± 0.000 dCy	0.384 ± 0.000 dDy	0.533 ± 0.000 dEy	0.602 ± 0.000 dFy
	70°C	0.181 ± 0.000 eAy	0.187 ± 0.000 eBy	0.260 ± 0.000 eCy	0.391 ± 0.001 eDy	0.541 ± 0.001 eEy	0.611 ± 0.000 eFy

Results are mean ± standard deviation (n = 3).

Lowercase letters (a–e) denote row-wise significant difference between sample temperature.

Uppercase letters (A–F) denote significant difference between moisture content.

x/y denotes comparison between the experimental and model data.

Table 3. Thermophysical properties (k, D, and C) of terrestrially grown foods

Thermophysical Properties	Food	Temperature (°C)	Moisture (% Wet Basis)	Value	Reference
k (W m^−1 K^−1)	Potato	20	63-83	0.541	[48,49]
	Strawberry	28	87	0.462	[50]
	Spinach	21	93	0.347	[50]
	Rice flour	30	18	0.130	[51]
	Whole milk powder	30	3.8	0.091	[51]
	Veal meat	30	70	0507	[52]
D (mm^2 s^−1)	Apple	0-30	85	0.14	[53]
	Potato	0–70	63–83	0.13	[54]
	Strawberry	5	92	0.13	[55]
	Beef chuck	40–65	66	0.12	[56]
C (J kg^−1°C^−1)	Gooseberry	NA	88	3950	[57]
	Lemon	NA	87	3950	[57]
	Strawberry	NA	88	3940	[57]
	Mango	NA	82	3740	[57]
	Veal meat	NA	76	3650	[57]

NA: not available.

Table 4. Specific heat capacity of sugar kelp measured using KD2 Pro and Choi and Okos' model

		Moisture Content (g water (100 g sample)^−1)
Analysis	Sample Temperature (°C)	6	10	30	50	70	90
Experimental Specific Heat Capacity (J kg^−1°C^−1)	30	749.0 ± 5.1 aAx	1535.2 ± 180.3 aBx	2152.6 ± 156.0 aCx	2574.1 ± 149.4 aDx	2864.0 ± 166.5 aEx	3204.6 ± 89.7 aEx
	40	1218.8 ± 23.6 bAx	1720.6 ± 93.1 bBx	2343.3 ± 231.1 bCx	2722.6 ± 102.4 bDx	2787.4 ± 136.5 bEx	3287.4 ± 94.4 bEx
	50	1455.7 ± 112.7 bAx	1644.0 ± 109.8 bBx	2254.3 ± 146.8 bCx	2907.4 ± 310.4 bDx	3024.3 ± 301.5 bEx	3136.4 ± 401.4 bEx
	60	1507.9 ± 159.3 bAx	1783.1 ± 104.5 bBx	2253.2 ± 247.3 bCx	2902.3 ± 284.2 bDx	3116.3 ± 198.2 bEx	2963.4 ± 199.1 bEx
	70	2060.3 ± 260.6 cAx	1808.9 ± 99.0 cBx	2278.4 ± 278.7 cCx	2946.9 ± 443.3 cDx	3270.6 ± 219.4 cEx	3213.6 ± 418.2 cEx
Choi and Okos' Model Specific Heat Capacity (J kg^−1°C^−1)	30	1416.77 ± 0.04 Ay	1479.1 ± 0.4 By	1908.2 ± 0.3 Cy	2589.5 ± 0.2 Dy	3395.8 ± 0.3 Ey	3827.3 ± 0.6 Fy
	40	1423.8 ± 0.1 Ay	1486.0 ± 0.2 By	1915.7 ± 0.1 Cy	2599.0 ± 0.3 Dy	3407.2 ± 0.3 Ey	3858.5 ± 0.4 Fy
	50	1432.2 ± 0.4 Ay	1493.2 ± 0.2 By	1923.5 ± 0.3 Cy	2609.9 ± 0.4 Dy	3419.3 ± 1.4 Ey	3854.2 ± 1.1 Fy
	60	1438.7 ± 1.0 Ay	1500.0 ± 0.2 By	1932.8 ± 0.4 Cy	2622.5 ± 0.3 Dy	3437.8 ± 0.6 Ey	3874.2 ± 0.3 Fy
	70	1466.1 ± 0.8 Ay	1506.4 ± 0.2 By	1941.7 ± 0.7 Cy	2634.1 ± 2.1 Dy	3453.1 ± 2.9 Ey	3893.0 ± 0.4 Fy

Results are mean ± standard deviation (n = 3).

Lowercase letters (a–c) denote row-wise significant difference between sample temperature.

Uppercasel letters (A–F) denote significant difference between moisture content.

x/y denotes the comparison between the experimental and model data.

Table 5. Thermal diffusivity of sugar kelp measured using KD2 Pro and Choi and Okos' model

		Moisture Content (g water (100 g sample)^−1)
Analysis	Sample Temperature (°C)	6	10	30	50	70	90
Experimental Thermal Diffusivity (mm^2 s^−1)	30	0.244 ± 0.001 aAx	0.153 ± 0.018 aBx	0.135 ± 0.006 aCx	0.149 ± 0.009 aCx	0.144 ± 0.005 aBCx	0.152 ± 0.006 aBx
	40	0.181 ± 0.002 aAx	0.158 ± 0.007 aBx	0.141 ± 0.004 aCx	0.153 ± 0.006 aCx	0.157 ± 0.005 aBCx	0.156 ± 0.003 aBx
	50	0.171 ± 0.012 aAx	0.177 ± 0.013 aBx	0.149 ± 0.005 aCx	0.154 ± 0.006 aCx	0.166 ± 0.015 aBCx	0.173 ± 0.023 aBx
	60	0.198 ± 0.024 bAx	0.187 ± 0.015 bBx	0.166 ± 0.009 bCx	0.171 ± 0.016 bCx	0.175 ± 0.006 bBCx	0.190 ± 0.012 bBx
	70	0.213 ± 0.028 cAx	0.208 ± 0.009 cBx	0.189 ± 0.031 cCx	0.176 ± 0.022 cCx	0.188 ± 0.021 cBCx	0.188 ± 0.021 cBx
Choi and Okos' Model Thermal Diffusivity (mm^2 s^−1)	30	0.138 ± 0.000 aAy	0.147 ± 0.000 aBy	0.162 ± 0.000 aCy	0.153 ± 0.000 aDy	0.143 ± 0.000 aEy	0.151 ± 0.000 aFy
	40	0.143 ± 0.000 bAy	0.152 ± 0.000 bBy	0.167 ± 0.000 bCy	0.157 ± 0.000 bDy	0.146 ± 0.000 bEy	0.154 ± 0.000 bFy
	50	0.148 ± 0.000 cAy	0.156 ± 0.000 cBy	0.171 ± 0.000 cCy	0.160 ± 0.000 cDy	0.149 ± 0.000 cEy	0.157 ± 0.000 cFy
	60	0.151 ± 0.000 dAy	0.160 ± 0.000 dBy	0.175 ± 0.000 dCy	0.163 ± 0.000 dDy	0.152 ± 0.000 dEy	0.160 ± 0.000 dFy
	70	0.155 ± 0.000 eAy	0.164 ± 0.000 eBy	0.178 ± 0.000 eCy	0.165 ± 0.000 eDy	0.153 ± 0.000 eEy	0.162 ± 0.000 eFy

Results are mean ± standard deviation (n = 3).

Lowercasae letters (a–e) denote row-wise significant difference between sample temperature.

Uppercase letters (A–F) denote significant difference between moisture content.

x/y denotes the comparison between the experimental and model data.

Table 6. Measured Bulk density, calculated porosity and the Choi and Okos' model porosity of sugar kelp

Moisture Content (g water/100g sample)	Measured Bulk Density (kg/m^3)	Calculated Porosity	Choi and Okos' Model Porosity
6	806.0 ± 8.3 A	0.49 ± 0.01 A	0.51 A
10	756.2 ± 2.7 B	0.51 ± 0.00 A	0.52 A
30	753.8 ± 11.8 B	0.45 ± 0.01 B	0.46 B
50	896.2 ± 16.8 C	0.29 ± 0.01 C	0.29 C
70	1022.4 ± 38.7 D	0.11 ± 0.03 D	0.10 D
90	969.6 ± 13.6 E	0.09 ± 0.01 D	0.06 E

Results are mean ± standard deviation (n = 6).

Uppercase letters (A–E) denote significant difference between moisture content.

Table 7. Prediction errors in the thermophysical properties with different ANN configurations and Choi and Okos' Model

Thermophysical Properties	Model	Neurons	Performance Measures
			MAE	MRE	SE	R^2
Thermal Conductivity, k	ANN	2	0.024	0.065	0.002	0.945
		4	0.020	0.055	0.002	0.957
		6	0.019	0.049	0.002	0.960
		8	0.019	0.049	0.002	0.961
		10	0.019	0.051	0.002	0.960
		12	0.020	0.054	0.005	0.956
		14	0.023	0.060	0.002	0.935
		16	0.020	0.054	0.003	0.957
	Choi and Okos	NA	0.051	0.143	0.005	0.853
Thermal Diffusivity, D	ANN	2	0.012	0.070	0.001	0.640
		4	0.011	0.063	0.001	0.705
		6	0.013	0.077	0.001	0.576
		8	0.012	0.069	0.001	0.599
		10	0.012	0.067	0.001	0.619
		12	0.010	0.060	0.001	0.720
		14	0.010	0.054	0.001	0.750
		16	0.010	0.056	0.001	0.749
	Choi and Okos	NA	0.024	0.128	0.002	0.014
Specific Heat Capacity, C	ANN	2	194.232	0.092	18.550	0.884
		4	184.356	0.082	17.726	0.893
		6	156.648	0.066	15.880	0.915
		8	177.051	0.080	16.854	0.905
		10	152.255	0.065	15.291	0.920
		12	163.899	0.070	16.611	0.907
		14	176.824	0.087	16.877	0.903
		16	164.575	0.077	15.890	0.914
	Choi and Okos	NA	380.167	0.172	35.235	0.758

One hidden layer was considered and the neurons in that layer are listed in the table.

MAE, MRE, SE, and R² are the mean absolute error, the mean relative error, the standard error and the coefficient of determination of thermophysical properties (thermal conductivity, thermal diffusivity, specific heat capacity) of sugar kelp, respectively.

Table 8. Regression parameters for predicting the thermophysical properties of sugar kelp with the best ANN configuration

IW		LW														B1	B2
Thermal Conductivity, k for 8 neurons
1.60	-5.18	1.58	-0.19	-0.05	-0.44	-0.22	-0.09	-0.15	-0.23							-7.76	1.18
-4.60	-8.92															7.30
6.93	-6.40															-3.22
-0.98	-2.61															0.07
-2.52	6.85															-4.88
-2.45	-2.55															-2.41
5.88	-3.41															5.31
-1.39	-3.60															-5.01
Thermal Diffusivity, D for 14 neurons
-4.12	-4.43	-0.63	-0.29	-0.56	-0.34	-0.04	-0.27	-0.19	-0.15	0.15	-0.12	-0.29	-0.23	0.72	-2.93	4.92	2.30
-4.39	2.36															4.83
4.59	2.30															-3.98
-3.52	2.68															2.59
-1.39	-5.06															1.88
-3.14	-4.04															1.27
3.91	3.42															-0.47
-0.65	-5.30															-0.21
4.75	-2.16															1.50
-3.03	4.14															-1.95
4.75	-2.37															2.66
-3.40	3.81															-3.86
3.88	2.76															5.23
1.90	4.91															6.74
Specific Heat Capacity, C for 10 neurons
-5.08	8.04	0.35	-0.74	0.79	-0.22	2.04	-0.17	-1.19	0.24				0.24	-0.24		9.81	-0.96
4.16	1.05															-3.64
8.48	2.12															-7.66
5.95	-3.42															-2.83
0.57	3.33															4.48
-4.93	0.26															0.80
-1.53	1.58															3.12
-1.09	-3.13															3.93
0.91	5.72															0.68
0.21	-3.98															4.00

where Y is the output parameter, x is the input matrix, B1 and B2 are bias value for layers 1 and 2, respectively, IW is the input weight matrix, and LW is layer weight matrix.

Figure 1. Correlation of experimental versus neutral network values of thermophysical properties of sugar kelp with training data set (a) thermal conductivity, (b) specific heat capacity, (c) thermal diffusivity. The best ANN configuration included 8, 10, and 14 neurons in each layer for (a) thermal conductivity, (b) specific heat capacity, (c) thermal diffusivity, respectively

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