Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration
Abstract
:1. Introduction
- mathematical description of changes of dried apples characteristics (mass gain, volume increase, dry matter loss, rehydration indices, and colour) during their rehydration;
- application of the empirical models used in the literature, ANNs, and models not yet used for the description of the rehydration process;
- development of equations to calculate the constants of applied rehydration models;
- investigation of the effect of drying and rehydration conditions on changes of dried apple characteristics.
2. Materials and Methods
- for slices, L = s;
- for cubes, L−2 = 3 s−2;
- for the mass gain and the volume increase
- for the dry matter loss.
- for the Peleg model
- for the Pilosof–Boquet–Batholomai model
- for the Singh and Kulshrestha model
+ a8Tr2 + a9Tdvd + a10TdL + a11TdTr + a12vdL + a13vdTr + a14LTr,
- Index RI1 (in kgreh/kgdry):
- Index RI2 (in kgreht/kgraw):
- water absorption capacity WAC:
3. Results and Discussion
- for lightness L*
- for yellowness b*
- for hue angle h*
- for chroma C*
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Model No. | Model Equation | Model Name | References |
---|---|---|---|
(1) | * | Peleg | [60] |
(2) | Pilosof–Boquet–Batholomai | [61] | |
(3) | Sing and Kulshrestha | [62] | |
(4) | Lewis (Newton) | [63] | |
(5) | Henderson–Pabis | [64] | |
(6) | Page | [65] | |
(7) | Modified Page | [66] |
Model No. | Model Name | Variable | SSE | R2 | Adjusted R2 | RMSE |
---|---|---|---|---|---|---|
(1) | Peleg | mass gain | 0.0488–0.6590 | 0.9597–0.9966 | 0.9578–0.9964 | 0.0570–0.1910 |
volume increase | 0.0203–0.4924 | 0.8622–0.9950 | 0.8631–0.9946 | 0.0430–0.1496 | ||
dry matter loss | 0.0018–0.0385 | 0.9329–0.9976 | 0.9299–0.9974 | 0.0116–0.0454 | ||
(2) | Pilosof–Boquet–Batholomai | mass gain | 0.0488–0.6590 | 0.9597–0.9966 | 0.9578–0.9963 | 0.0570–0.1910 |
volume increase | 0.0203–0.4924 | 0.8622–0.9950 | 0.8531–0.9946 | 0.0430–0.1496 | ||
dry matter loss | 0.0018–0.0385 | 0.9329–0.9976 | 0.9299–0.9974 | 0.0116–0.0454 | ||
(3) | Singh and Kulshrestha | mass gain | 0.0488–0.6592 | 0.9597–0.9966 | 0.9578–0.9963 | 0.0570–0.1910 |
volume increase | 0.0203–0.5129 | 0.8452–0.9950 | 0.8349–0.9946 | 0.0430–0.1527 | ||
dry matter loss | 0.0018–0.0385 | 0.9329–0.9973 | 0.9299–0.9966 | 0.0116–0.0452 | ||
(4) | Lewis (Newton) | mass gain | 0.0763–0.2210 | 0.8840–0.9481 | 0.8840–0.9481 | 0.0738–0.1064 |
volume increase | 0.0671–0.3131 | 0.8071–0.9492 | 0.8071–0.9492 | 0.0731–0.1326 | ||
dry matter loss | 0.0501–0.2417 | 0.8398–0.9672 | 0.8398–0.9672 | 0.0646–0.1183 | ||
(5) | Henderson–Pabis | mass gain | 0.0660–0.1952 | 0.8997–0.9549 | 0.8930–0.9519 | 0.0580–0.1033 |
volume increase | 0.0610–0.2440 | 0.8327–0.9538 | 0.8215–0.9513 | 0.0636–0.1276 | ||
dry matter loss | 0.0476–0.2741 | 0.8590–0.9688 | 0.8523–0.9660 | 0.0652–0.1142 | ||
(6) | Page | mass gain | 0.0037–0.0582 | 0.9541–0.9981 | 0.9520–0.9970 | 0.0143–0.0619 |
volume increase | 0.0022–0.2092 | 0.8686–0.9983 | 0.8598–0.9982 | 0.0142–0.1130 | ||
dry matter loss | 0.0050–0.2267 | 0.8834–0.9969 | 0.8778–0.9966 | 0.0195–0.1039 | ||
(7) | Modified Page | mass gain | 0.0027–0.0843 | 0.9541–0.9981 | 0.9520–0.9980 | 0.0143–0.0619 |
volume increase | 0.0022–0.2092 | 0.8686–0.9983 | 0.8598–0.9982 | 0.0142–0.1130 | ||
dry matter loss | 0.0050–0.2267 | 0.8834–0.9969 | 0.8778–0.9966 | 0.0195–0.1039 |
Model No. | Model Name | Parameter Equations |
---|---|---|
(1) | Peleg | A1 = 0.087419 − 2.2 × 10−5·Td2 + 0.02319·L2 − 1.8·10−5·Tr2 + 0.00422·Td·vd − 0.08602·vd·L A2 = 0.329564 + 2.97 × 10−5·Td2 − 0.00178·Td·vd + 0.00042·Td·L + 0.0339644·vd·L − 0.00125·L·Tr |
(2) | Pilosof–Boquet–Batholomai | A3 = 3.045997 − 0.00028·Tr2 + 0.014144·Td·vd − 0.00385·Td·L − 0.31729·vd·L + 0.011896·L·Tr A4 = 0.280897 + 0.04096·L2 − 5.5 × 10−5·Tr2 |
(3) | Singh and Kulshrestha | A5 = 2.252442 + 0.021388·Td − 0.00031·Tr2 − 0.00693·Td·L − 0.0265·vd·L + 0.013017·L·Tr A6 = 4.083154 + 0.001954·Tr2 − 0.04949·L·Tr |
(4) | Lewis (Newton) | k = 2.327949 + 0.001058·Tr2 − 0.02735·L·Tr |
(5) | Henderson–Pabis | k = 1.940489 + 0.000939·Tr2 − 0.02355·L·Tr a = 0.895178 + 7.94 × 10−6·Td·Tr |
(6) | Page | k = 0.72067 + 0.00015·Tr2 n = 0.612353 − 9.7 × 10−6·Tr2 |
(7) | Modified Page | k = 2.20413 + 0.001114·Tr2 − 0.02795·L·Tr n = 0.513102 + 0.000599·Td·L − 1.5·10−5·Td·Tr |
Model No. | Model Name | Parameter Equations |
---|---|---|
(1) | Peleg | A1 = 0.432052 + 0.0144·L2 − 5.7 × 10−5·Tr2 A2 = 0.064866 + 0.003357·vd2 + 0.042058·L2 − 3.6 × 10−5·Tr2 |
(2) | Pilosof–Boquet–Batholomai | A3 = 1.797968 − 0.00243·L·Tr A4 = 0.33569 + 0.043488·L2 − 6.2 × 10−5·Tr2 |
(3) | Singh and Kulshrestha | A5 = 1.351868 − 0.00028·Tr2 − 0.00101·Td·vd +0.000959·Td·Tr − 0.01163·L·Tr A6 = 3.563119 + 0.001701·Tr2 + 0.058523·Td·vd − 0.00208·Td·Tr − 1.22396·vd·L |
(4) | Lewis (Newton) | k = 1.968387 + 0.000829·Tr2 − 0.02088·L·Tr |
(5) | Henderson–Pabis | k = 1.622107 + 0.000699·Tr2 − 0.01724·L·Tr a = 0.914742 |
(6) | Page | k = 1.367971 + 0.0004·Tr2 − 0.01023·L·Tr n = 0.623547 − 7.1 × 10−6·Tr2 |
(7) | Modified Page | k = 1.797016 + 0.000814·Tr2 − 0.01995·L·Tr n = 0.623556 − 7.1 × 10−6·Tr2 |
Model No. | Model Name | Parameter Equations |
---|---|---|
(1) | Peleg | A1 = 1.199471 + 0.03494·L2 − 0.01674·Td·vd + 0.000595·Td·Tr + 0.345962·vd·L − 0.0122·L·Tr A2 = −0.09904 + 0.105514·L2 + 0.018807·Td·vd − 8.7 × 10−5·Td·Tr − 0.38681·vd·L |
(2) | Pilosof–Boquet–Batholomai | A3 = 0.65515 − 0.00546·L2 − 2.9·10−5·Tr2 + 0.003547·Td·vd − 0.07355·vd·L + 0.00107·L·Tr A4 = −0.02683 + 0.063203·L2 + 0.018608·Td·vd − 5.4 × 10−5·Td·Tr − 0.38314·vd·L |
(3) | Singh and Kulshrestha | A5 = 0.658516 + 0.003843·Td·vd − 0.07913·vd·L A6 = 3.234311 − 0.00966·Td·L + 0.000608·Td·Tr |
(4) | Lewis (Newton) | k = 2.388268 − 0.06332·L2 |
(5) | Henderson–Pabis | k = 1.048803 + 0.000156·Tr2 a = 0.825358 − 4.6·10−5·Tr2 8.45 × 10−5·Td·Tr |
(6) | Page | k = 1.353281 − 0.02825·L2 + 6.64 × 10−5·Tr2 n = 0.510797 + 0.005854·L2 |
(7) | Modified Page | k = 1.098296 + 0.00019·Tr2 n = 0.510797 + 0.005854·L2 |
Rehydration Index | Weights and Biases between Input and Hidden Layer | Weights and Biases between Hidden and Output Layer | ||||||
---|---|---|---|---|---|---|---|---|
No. | Weights | Bias | Weights | Bias | ||||
RI1 | i | D1i | D2i | D3i | D4i | Di | Wi | Bi |
1 | 0.76302 | −4.30465 | 6.1141 | 4.1011 | 3.2308 | 3.8703 | −1.0468 | |
2 | −0.18017 | 5.2373 | −2.251 | −9.0609 | −0.41154 | 0.46843 | ||
3 | −5.4656 | 2.1371 | −5.5557 | 4.6212 | −1.931 | 4.4227 | ||
4 | −8.7999 | 2.4317 | −0.35888 | 3.979 | 0.66115 | −6.7914 | ||
RI2 | 1 | −1.8885 | 2.0685 | 1.1286 | −0.37259 | −2.3498 | 3.0742 | 0.79132 |
2 | 0.50806 | 0.55895 | −7.7709 | −0.84271 | 0.62434 | −2.1788 | ||
3 | −3.4158 | −0.11008 | −10.5825 | −1.202 | 0.85564 | 8.3004 | ||
4 | 0.32057 | −4.9892 | 2.4163 | −5.1056 | 1.52 | 3.3142 | ||
WAC | 1 | −0.32603 | −0.53705 | −0.88337 | 2.8526 | −2.0969 | −1.0055 | 0.42304 |
2 | 9.2834 | −46.5133 | −60.2563 | 122.4035 | 0.81527 | 7.7133 | ||
3 | −22.1376 | −52.8683 | 0.59559 | 19.7618 | 0.54159 | −62.5777 |
Rehydration Index | R | RMSE | χ2 |
---|---|---|---|
RI1 | 0.9424 | 0.0756 | 0.0065 |
RI2 | 0.9376 | 0.0787 | 0.0070 |
WAC | 0.9270 | 0.0802 | 0.0070 |
Rehydration Index | Omitted Parameter | R | RMSE | χ2 |
---|---|---|---|---|
RI1 | Td | 0.76 (2) * | 0.34 (1) | 0.130 (1) |
vd | 0.84 (4) | 0.14 (4) | 0.021 (4) | |
Tr | 0.78 (3) | 0.15 (3) | 0.026 (3) | |
L | 0.21 (1) | 0.29 (2) | 0.096 (2) | |
RI2 | Td | 0.79 (3) | 0.18 (2) | 0.035 (2) |
vd | 0.84 (4) | 0.13 (4) | 0.020 (4) | |
Tr | 0.78 (2) | 0.15 (3) | 0.026 (3) | |
L | 0.45 (1) | 0.23 (1) | 0.062 (1) | |
WAC | Td | 0.76 (2) | 0.14 (2) | 0.022 (2) |
vd | 0.85 (4) | 0.12 (4) | 0.016 (4) | |
Tr | 0.82 (3) | 0.14 (3) | 0.021 (3) | |
L | 0.30 (1) | 0.21 (1) | 0.048 (1) |
Variable | SSE | R2 | Adjusted R2 | RMSE |
---|---|---|---|---|
L* | 0.0077–0.2359 | 0.6306–0.9889 | 0.6306–0.9889 | 0.0394–0.2068 |
b* | 0.0009–0.5514 | 0.3516–0.9986 | 0.3516–0.9986 | 0.0177–0.3031 |
h* | 0.0105–0.6388 | 0.2601–0.9865 | 0.2601–0.9865 | 0.0419–0.3574 |
C* | 0.0012–0.6660 | 0.3611–0.9980 | 0.3611–0.9980 | 0.0241–0.3085 |
No. | Weights and Biases between Input and Hidden Layer | Weights and Biases between Hidden and Output Layer | ||||||
---|---|---|---|---|---|---|---|---|
Weights | Bias | Weights | Bias | |||||
i | D1i | D2i | D3i | D4i | D5i | Di | Wi | Bi |
1 | −0.3936 | 0.068912 | 7.7881 | −0.37068 | −0.29036 | −4.083 | 9.4829 | 4.3014 |
2 | 5.5244 | 3.3242 | 0.99513 | −1.4537 | −2.1599 | 1.3043 | −1.7557 | |
3 | 5.143 | −0.15515 | 5.6857 | −2.9232 | −7.4673 | 0.73944 | −3.9588 | |
4 | −0.67089 | −5.7064 | 2.1326 | 0.36501 | 2.9711 | 0.87002 | −3.3684 | |
5 | −4.0698 | 3.8803 | 2.1008 | 1.5752 | 3.0415 | 0.7498 | 1.5977 | |
6 | 2.7802 | 5.5689 | 3.5537 | −2.3719 | −3.6688 | −1.6085 | −2.9859 | |
7 | 0.13428 | −0.70078 | 0.3338 | −7.8718 | 1.8079 | −0.83412 | 4.0532 | |
8 | 7.6823 | 0.47277 | −0.4144 | 8.8348 | 6.2643 | −0.78616 | 5.2191 |
Parameter | All Parameters Considered | Omitted L | Omitted Td | Omitted vd | Omitted Tr | Omitted t | |
---|---|---|---|---|---|---|---|
Statistical Test Method | |||||||
R | 0.9048 | 0.1395 | 0.4424 | 0.4482 | 0.4859 | 0.6721 | |
RMSE | 0.0567 | 0.2654 | 0.2953 | 0.1980 | 0.1246 | 0.1022 | |
R2 | 0.8187 | 0.0195 | 0.1957 | 0.2009 | 0.2361 | 0.4517 | |
Adjusted R2 | 0.8177 | 0.0143 | 0.1915 | 0.1967 | 0.2321 | 0.4488 | |
SSE | 0.0573 | 0.1332 | 0.1207 | 0.1203 | 0.1176 | 0.0996 |
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Górnicki, K.; Kaleta, A.; Kosiorek, K. Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration. Appl. Sci. 2022, 12, 5495. https://doi.org/10.3390/app12115495
Górnicki K, Kaleta A, Kosiorek K. Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration. Applied Sciences. 2022; 12(11):5495. https://doi.org/10.3390/app12115495
Chicago/Turabian StyleGórnicki, Krzysztof, Agnieszka Kaleta, and Krzysztof Kosiorek. 2022. "Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration" Applied Sciences 12, no. 11: 5495. https://doi.org/10.3390/app12115495
APA StyleGórnicki, K., Kaleta, A., & Kosiorek, K. (2022). Mathematical Description of Changes of Dried Apple Characteristics during Their Rehydration. Applied Sciences, 12(11), 5495. https://doi.org/10.3390/app12115495