# Heat Pump Drying of Kelp (Laminaria japonica): Drying Kinetics and Thermodynamic Properties

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{eff}), activation energy (E

_{a}), enthalpy (ΔH), entropy (ΔS), and Gibbs free energy (ΔG) were investigated. The results show that the Page model was effective in describing the moisture content change of kelp during heat pump drying. The D

_{eff}varied from 1.00 × 10

^{−11}to 13.00 × 10

^{−11}m

^{2}/s and the temperature, air velocity, humidity, and thickness had significant effects on drying time and moisture uniformity. Higher temperature and air velocity with proper humidity shortened the drying time and lessened the influence of thickness on moisture uniformity. The E

_{a}(16.38–26.66 kJ/mol) and ΔH (13.69–24.22 kJ/mol) were significantly increased by thickness. When the temperature was 40 °C, air velocity 1.3 m/s, and air humidity 40%, the moisture content was reduced to 18% in 5 h, with a homogeneous moisture content. This study clarifies the regularity of moisture change inside kelp and provides a theoretical reference for the development of macroalgae drying technology.

## 1. Introduction

^{−10}and 1.64 × 10

^{−10}m

^{2}/s over a temperature range of 37–42 °C. Almeida et al. [21] calculated the effective moisture diffusivity, activation energy, and the thermodynamic properties of Achachairu (Garcinia humilis) peels under the drying process. Their results provided important information about the moisture migration and the required energy during the process. A heat pump-assisted drying process can reduce the energy demand by 84% compared to traditional drying using fossil fuels, but possibly results in up to a 69% longer drying time due to the higher humidity [9]. To produce consistent-quality seaweed and shorten the drying time, Sarbatly et al. [8] studied the heat pump drying kinetic and thermodynamic properties of Eucheuma spinosum, and found that the difference in algae enthalpy values used in each experiment was up to 1.74 times, but the reason was not explained. Silva et al. [22] recommended standardizing and specifying the leaf thickness measuring points in the thermodynamic properties and drying kinetics of Bauhinia forficata leaves. Nadi et al. [23] also reported that the thickness of an apple slice affected the values of effective moisture diffusivity, enthalpy, entropy, and Gibbs free energy. All the above studies imply that the thickness of kelp may be one of the most important factors affecting the drying kinetics and thermodynamic characteristics, and a major cause of non-uniform moisture content.

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Heat Pump Dryer

#### 2.3. Drying Procedure

#### 2.4. Moisture Content and Moisture Ratio

_{db}) was calculated as follows:

_{wb}indicates the wet basis moisture content at a particular drying time t, g H

_{2}O/g w.b.; M

_{db}indicates the dry basis moisture content at a particular drying time t, g H

_{2}O/g d.b.; m

_{t}indicates the sample weight at a particular drying time t, g; m

_{0}indicates the initial weight of the sample, g; M

_{0}indicates the initial moisture content, g H

_{2}O/g w.b.

_{e}indicates the equilibrium moisture content, g H

_{2}O/g d.b.

_{e}) of kelp at temperatures of 40–70 °C and 25–50% humidity, and the M

_{e}value varied from 0.0707 to 0.1270 kg H

_{2}O/kg dry solids, without any significant difference. Since the M

_{e}was far less than the M

_{0}and M

_{t}, Formula (2) could be simplified as [25,26,27,28]

#### 2.5. Mathematical Modeling of Drying Curves

#### 2.6. Effective Moisture Diffusivity

_{eff}is the effective moisture diffusivity of the material, m

^{2}/s; L is the material thickness of the kelp, m; n is the positive integer; t is the drying time, s. Page [31] approximated the $\frac{8}{{\pi}^{2}}$ ratio as being equal to unity, and n is taken as 1 for longer drying times [27,33]. Thus, the equation was further simplified to the first term of the series and expressed in logarithmic form as follows:

#### 2.7. Thermodynamic Property Parameters

_{a}is calculated according to the formula

_{0}indicates the diffusion base in the material, m

^{2}/s; E

_{a}indicates the drying activation energy of the material, J/mol; R is the molar constant of gas, whose value is 8.314 J/(mol·K); T is the drying temperature of kelp, °C.

^{−1}) can be expressed as [23]

^{−23}J/K), h

_{p}is the Planck’s constant (6.626 × 10

^{−34}J·s), R is the universal gas constant (8.314 J/mol·K), and T

_{abs}is the absolute temperature (K).

#### 2.8. Statistical Analyses

^{2}) of the model was calculated via the following equations [19].

_{pre,i}and MR

_{exp,i}represent the predicted value and the experimental value, respectively; and n is the number of data.

## 3. Results

#### 3.1. Effect of Heat Pump Drying on Dehydration Characteristics

#### 3.1.1. Effect of Temperature on Drying Characteristics

#### 3.1.2. Effect of Air Velocity on Drying Characteristics

#### 3.1.3. Effect of Humidity on Drying Characteristics

#### 3.1.4. Effect of Kelp Thickness on Drying Time

#### 3.2. Determination of Drying Model

#### 3.3. Verification of Kinetic Model

#### 3.4. Effective Moisture Diffusivity

_{eff}) was calculated (Table 5) in order to explore the moisture transfer characteristics of kelp in heat pump drying. It showed obviously that the D

_{eff}of heat-pump-dried kelp increased from 4.4307 × 10

^{−11}to 12.9957 × 10

^{−11}m

^{2}/s when the temperature rose from 20 to 50 °C (3.5–4.2 mm). When the air velocity rose from 0.3 to 1.3 m/s, the D

_{eff}increased from 5.9216 × 10

^{−11}to 12.2649 × 10

^{−11}m

^{2}/s (3.5–4.2 mm). When the humidity was increased from 20% to 50%, the D

_{eff}reduced from 8.2601 × 10

^{−11}to 4.1134 × 10

^{−11}m

^{2}/s (3.5–4.2 mm). When the thickness was increased from 0.8–2 mm to 3.5–4.2 mm, the D

_{eff}was improved from 1.2314 × 10

^{−1}

^{1}to 7.1409 × 10

^{−1}

^{1}m

^{2}/s (40 °C, 30 %, 0.3 m/s). This indicates that the temperature, air velocity, and thickness exert positive effects on the D

_{eff}value; conversely, humidity exerts a negative effect on the D

_{eff}increase. This is consistent with the findings of Bauhinia forficata link leaves [22], apple slices [23], and pepper leaves [29]. By contrast, the D

_{eff}of kelp is lower than that of Achachairu peels [21] and banana slices [14], but higher than that of Bauhinia forficata link leaves [22] and Piper umbellatum L. leaves [29] at the same drying temperature. The D

_{eff}value was related to the physicochemical properties and moisture content, as well as temperature. On the one hand, the kelp initial moisture content was up to 92.4–95.7%, which contributed to the D

_{eff}value. However, on the other hand, the kelp surface was compact and leathery, retarding the free diffusion of moisture. In addition, kelp is rich in dietary fiber, which has a strong water-holding capacity and inhibits the migration of water molecules.

#### 3.5. Thermodynamic Parameters of Kelp with Different Thicknesses

_{a}) is an important index to reflect the moisture bonding ability of materials, indicating the energy required for water molecules to change from a normal to an active state that is prone to dehydration. The E

_{a}of kelp with a thickness of 0.8–2.0 mm, 2.0–3.5 mm, and 3.5–4.2 mm were calculated to be 16.38 kJ/mol, 18.21 kJ/mol, and 26.66 kJ/mol, respectively. The E

_{a}values of various food materials range from 12.7 to 110 kJ/mol [40]. The energy threshold of the top part was 62.76% higher than that of the tail part, indicating that tail parts can start the dehydration process with lower energy supplied in the initial heating-up period. The E

_{a}difference was in line with the effective moisture diffusivity of kelp with different thicknesses, and this is the intrinsic reason for the faster drying of thinner kelp leaves.

## 4. Conclusions

_{eff}was calculated as 1.00 × 10

^{−11}–13.00 × 10

^{−11}m

^{2}/s, increasing with drying temperature, air velocity, and thickness but decreasing with humidity.

_{a}was 16.38–26.66 kJ/mol. Meanwhile, ΔH was 13.69–24.22 kJ/mol, ΔS was −403.60–−349.92 J/(mol·K), and ΔG was 127.42–142.50 kJ/mol. These results provide a theoretical basis for the thermodynamic calculation of a heat pump drying system for kelp and for improvements in macroalgae drying technology.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- FAO. The State of World Fisheries and Aquaculture 2020; Sustainability in Action; FAO: Rome, Italy, 2020. [Google Scholar]
- Wells, M.L.; Potin, P.; Craigie, J.S.; Raven, J.A.; Merchant, S.S.; Helliwell, K.E.; Smith, A.G.; Camire, M.E.; Brawley, S.H. Algae as nutritional and functional food sources: Revisiting our understanding. J. Appl. Phycol.
**2017**, 29, 949–982. [Google Scholar] [CrossRef] - Christine, D.; Rainer, S.; Gerhard, J. Amino acids, fatty acids, and dietary fibre in edible seaweed products. Food Chem.
**2007**, 103, 891–899. [Google Scholar] - Déléris, P.; Nazih, H.; Bard, J.M. Seaweeds in human health. In Seaweed in Health and Disease Prevention; Academic Press: Cambridge, MA, USA, 2016; pp. 319–367. [Google Scholar]
- Ali, M.K.M.; Fudholi, A.; Sulaiman, J.; Ruslan, M.H.; Yasir, S.M. Sauna technique, drying kinetic modeling and effectiveness on solar drying compared with direct drying in drying process of Kappaphycus striatum in Selakan Island Malaysia. Energy Power Eng.
**2014**, 06, 303–315. [Google Scholar] [CrossRef][Green Version] - Foscarini, R.; Prakash, J. Handbook on Eucheuma Seaweed Cultivation; FAO: Rome, Italy, 1990. [Google Scholar]
- Ling, A.L.M.; Yasir, S.; Matanjun, P.; Abu Bakar, M.F. Effect of different drying techniques on the phytochemical content and antioxidant activity of Kappaphycus alvarezii. J. Appl. Phycol.
**2015**, 27, 1717–1723. [Google Scholar] [CrossRef] - Sarbatly, R.; Wong, T.; Bono, A.; Krishnaiah, D. Kinetic and thermodynamic characteristics of seaweed dried in the convective air drier. Int. J. Food Eng.
**2010**, 6, 1–16. [Google Scholar] [CrossRef] - Jokiel, M.; Bantle, M.; Kopp, C.; Verpe, E.H. Modelica-based modelling of heat pump-assisted apple drying for varied drying temperatures and bypass ratios. Therm. Sci. Eng. Prog.
**2020**, 19, 100575–100582. [Google Scholar] [CrossRef] - Andriy, R.; Oleksandr, R.; Ronald, D. Heat Pumps in the Drying Industry; Academic Press: Cambridge, MA, USA, 2020; pp. 135–175. [Google Scholar]
- Hu, Z.Q.H.; Bao, Q.; Ge, F.; Gao, X. Study on drying characteristics and mathematical model of S. japonica knot heat pump. J. Jiangsu Univ.
**2019**, 40, 649–654. [Google Scholar] - Zhu, Z.; Zhao, Y.; Zhang, Y.; Wu, X.; Liu, J.; Shi, Q.; Fang, Z. Effects of ultrasound pretreatment on the drying kinetics, water status and distribution in scallop adductors during heat pump drying. J. Sci. Food Agric.
**2021**, 101, 6239–6247. [Google Scholar] - Deng, Y.; Liu, Y.; Qian, B.; Su, S.; Wu, J.; Song, X.; Yang, H. Impact of far-infrared radiation-assisted heat pump drying on chemical compositions and physical properties of squid (Illex illecebrosus) fillets. Eur. Food Res. Technol.
**2011**, 232, 761–768. [Google Scholar] [CrossRef] - Tunckal, C.; Doymaz, İ. Performance analysis and mathematical modelling of banana slices in a heat pump drying system. Renew. Energy
**2020**, 150, 918–923. [Google Scholar] - Aktaş, M.; Taşeri, L.; Şevik, S.; Gülcü, M.; Uysal Seçkin, G.; Dolgun, E.C. Heat pump drying of grape pomace: Performance and product quality analysis. Dry. Technol.
**2019**, 37, 1766–1779. [Google Scholar] - Ming, L.; Shan, W.; Rongtang, L.; Yan, J. Energy, exergy and economic analyses on heat pump drying of lignite. Dry. Technol.
**2019**, 37, 1688–1703. [Google Scholar] - Stévant, P.; Indergård, E.; Ólafsdóttir, A.; Marfaing, H.; Larssen, W.E.; Fleurence, J.; Roleda, M.Y.; Rustad, T.; Slizyte, R.; Nordtvedt, T.S. Effects of drying on the nutrient content and physico-chemical and sensory characteristics of the edible kelp Saccharina latissima. J. Appl. Phycol.
**2018**, 30, 2587–2599. [Google Scholar] [CrossRef] - Chapman, A.S.; Stévant, P.; Larssen, W.E. Food or fad? Challenges and opportunities for including seaweeds in a Nordic diet. Bot. Mar.
**2015**, 58, 423–433. [Google Scholar] [CrossRef] - Chang, A.; Zheng, X.; Xiao, H.; Yao, X.; Liu, D.; Li, X.; Li, Y. Short- and medium-wave infrared drying of cantaloupe (Cucumis melon L.) slices: Drying kinetics and process parameter optimization. Processes
**2022**, 10, 114–132. [Google Scholar] - Pereira, R.; Yarish, C. Encyclopedia of Ecology; Elsevier: Amsterdam, The Netherlands, 2008. [Google Scholar]
- Almeida, R.L.J.; Santos, N.C.; Alves, I.L.; André, A.M.M. Evaluation of thermodynamic properties and antioxidant activities of Achachairu (Garcinia humilis) peels under drying process. Flavour Fragr. J.
**2020**, 36, 213–222. [Google Scholar] - Silva, F.P.D.; Siqueira, V.C.; Martins, E.A.; Miranda, F.; Melo, R.M. Thermodynamic properties and drying kinetics of Bauhinia forficata Link leaves. Rev. Bras. Eng. Agrícola Ambient.
**2017**, 21, 61–67. [Google Scholar] - Nadi, F.; Tzempelikos, D. Vacuum drying of apples (cv. Golden Delicious): Drying characteristics, thermodynamic properties, and mass transfer parameters. Heat Mass Transf.
**2018**, 54, 1853–1866. [Google Scholar] - Sappati, P.K.; Nayak, B.; Van Walsum, G.P. Effect of glass transition on the shrinkage of sugar kelp (Saccharina latissima) during hot air convective drying. J. Food Eng.
**2017**, 210, 50–61. [Google Scholar] [CrossRef] - Tham, T.C.; Ng, M.X.; Gan, S.H.; Chua, L.S.; Aziz, R.; Chuah, L.A.; Hii, C.K.; Ong, S.P. Effect of ambient conditions on drying of herbs in solar greenhouse dryer with integrated heat pump. Dry. Technol.
**2017**, 35, 1721–1732. [Google Scholar] [CrossRef] - Doymaz, I.; Pala, M. Hot-air drying characteristics of red pepper. J. Food Eng.
**2002**, 55, 331–335. [Google Scholar] [CrossRef] - Shi, Q.; Zheng, Y.; Zhao, Y. Mathematical modeling on thin-layer heat pump drying of yacon (Smallanthus sonchifolius) slices. Energy Convers. Manag.
**2013**, 71, 208–216. [Google Scholar] [CrossRef] - Dadalı, G.; Kılıç Apar, D.; Özbek, B. Microwave drying kinetics of Okra. Dry. Technol.
**2007**, 25, 917–924. [Google Scholar] [CrossRef] - Dorneles, L.D.N.S.; Goneli, A.L.D.; Cardoso, C.A.L.; da Silva, C.B.; Hauth, M.R.; Oba, G.C.; Schoeninger, V. Effect of air temperature and velocity on drying kinetics and essential oil composition of Piper umbellatum L. leaves. Ind. Crops Prod.
**2019**, 142, 111846–111853. [Google Scholar] [CrossRef] - Henderson, S.M.; Pabis, S. Grain drying theory II: Temperature effects on drying coefficients. J. Agric. Eng. Res.
**1961**, 6, 169–174. [Google Scholar] - Page, G.E. Factors Influencing the Maximum Rates of Air-Drying of Shelled Cron in Thin Layers. Master Thesis, Purdue University, West Lafayette, IN, USA, 1949. [Google Scholar]
- Lewis, W.K. The rate of drying of solid materials. J. Ind. Eng. Chem.
**1921**, 13, 427–432. [Google Scholar] [CrossRef] - Ojediran, J.O.; Okonkwo, C.E.; Adeyi, A.J.; Adeyi, O.; Olaniran, A.F.; George, N.E.; Olayanju, A.T. Drying characteristics of yam slices (Dioscorea rotundata) in a convective hot air dryer: Application of ANFIS in the prediction of drying kinetics. Heliyon
**2020**, 6, e03555. [Google Scholar] [CrossRef] - Chen, Z.; Zhou, Y.; Fan, Y.; Chen, Y.; Chen, Y. Study on drying technology of S. japonica extract. Food Sci. Technol.
**2011**, 36, 221–224. [Google Scholar] - Xiao, M.; Zhu, Y.; Liu, H. Moisture diffusion from thin layer media during drying process. J. Huaqiao Univ.
**2003**, 2, 184–188. [Google Scholar] - Xu, W.; Islam, M.N.; Cao, X.; Tian, J.; Zhu, G. Effect of relative humidity on drying characteristics of microwave assisted hot air drying and qualities of dried finger citron slices. LWT
**2021**, 137, 110413–110422. [Google Scholar] [CrossRef] - Namkanisorn, A.; Murathathunyaluk, S. Sustainable drying of galangal through combination of low relative humidity, temperature and air velocity. Energy Rep.
**2020**, 6, 748–753. [Google Scholar] [CrossRef] - Singhanat, P.; Supawinee, S. Effect of two stage, tray and heat pump assisted-dehumidified drying on drying characteristics and qualities of dried ginger. Food Bioprod. Processing
**2011**, 89, 429–437. [Google Scholar] - Hao, W.; Zhang, H.; Liu, S.; Mi, B.; Lai, Y. Mathematical modeling and performance analysis of direct expansion heat pump assisted solar drying system. Renew. Energy
**2021**, 165, 77–87. [Google Scholar] [CrossRef] - Maroulis, Z.B.; Saravacos, G.D.; Panagiotou, N.M.; Krokida, M.K. Moisture diffusivity data compilation for foodstuffs: Effect of material moisture content and temperature. Int. J. Food Prop.
**2001**, 4, 225–237. [Google Scholar] [CrossRef] - Costa, C.F.; Corrêa, P.C.; Vanegas, J.D.B.; Baptestini, F.M.; Campos, R.C.; Fernandes, L.S. Mathematical modeling and determination of thermodynamic properties of jabuticaba peel during the drying process. Rev. Bras. Eng. Agrícola Ambient.
**2016**, 20, 576–580. [Google Scholar] [CrossRef][Green Version] - Corrêa, P.C.; Oliveira, G.H.H.; Botelho, F.M.; Goneli, A.L.D.; Carvalho, F.M. Mathematical modeling and determination of thermodynamic properties of coffee (Coffea arabica L.) during the drying process. Rev. Ceres
**2010**, 57, 595–601. [Google Scholar] [CrossRef]

**Figure 5.**Drying time of kelp with different thicknesses under different drying conditions (T: temperature; V: air velocity; H: humidity; *: extremum value).

Specimen | Top Part | Main Part | Tail Part |
---|---|---|---|

Thickness (mm) | 3.5–4.2 | 2.0–3.5 | 0.8–2.0 |

Moisture content (%) | 94.5 | 95.7 | 92.4 |

Group | Temperature (°C) | Air Velocity (m/s) | Humidity (%) | Thickness (mm) |
---|---|---|---|---|

1 | 20 | 0.8 | 40 | 0.8–2.0(A) 2.0–3.5(B) 3.5–4.2(C) |

2 | 30 | 0.8 | 40 | |

3 | 40 | 0.8 | 40 | |

4 | 50 | 0.8 | 40 | |

5 | 40 | 0.3 | 40 | |

6 | 40 | 0.8 | 40 | |

7 | 40 | 1.3 | 40 | |

8 | 40 | 0.3 | 20 | |

9 | 40 | 0.3 | 30 | |

10 | 40 | 0.3 | 40 | |

11 | 40 | 0.3 | 50 |

Model Name | Model | After Linearization | Reference |
---|---|---|---|

Henderson–Pabis | $MR=A\mathrm{exp}\left(-Kt\right)$ | $-\mathrm{ln}MR=-\mathrm{ln}A+Kt$ | [30] |

Page model | $MR=\mathrm{exp}\left(-K{t}^{n}\right)$ | $\mathrm{ln}\left[-\mathrm{ln}\left(MR\right)\right]=\mathrm{ln}K+n\mathrm{ln}t$ | [31] |

Lewis model | $MR=\mathrm{exp}\left(-Kt\right)$ | $-\mathrm{ln}MR=Kt$ | [32] |

No. | $\mathbf{ln}\mathit{M}\mathit{R}\u2014\mathit{t}$ | $\mathbf{ln}\left[-\mathbf{ln}\left(\mathit{M}\mathit{R}\right)\right]\u2014\mathbf{ln}\mathit{t}$ | ||||
---|---|---|---|---|---|---|

lnA | K | R^{2} | lnK | n | R^{2} | |

1A | 0.4672 | 0.2096 | 0.9479 | −4.283 | 2.381 | 0.966 |

1B | 0.4662 | 0.1313 | 0.8826 | −6.137 | 2.618 | 0.945 |

1C | 0.4353 | 0.1061 | 0.8749 | −5.711 | 2.209 | 0.964 |

2A | 0.4124 | 0.2487 | 0.9837 | −2.591 | 1.646 | 0.977 |

2B | 0.5470 | 0.1867 | 0.8537 | −5.378 | 2.648 | 0.959 |

2C | 0.4214 | 0.1647 | 0.9128 | −4.433 | 1.983 | 0.994 |

3A | 0.3928 | 0.2635 | 0.9857 | −2.170 | 1.503 | 0.993 |

3B | 0.5583 | 0.2353 | 0.8132 | −4.797 | 2.790 | 0.958 |

3C | 0.4750 | 0.1934 | 0.8849 | −3.991 | 2.085 | 0.987 |

4A | 0.6750 | 0.4139 | 0.8849 | −2.941 | 2.484 | 0.995 |

4B | 0.8562 | 0.2614 | 0.7441 | −5.176 | 3.110 | 0.948 |

4C | 0.5998 | 0.3112 | 0.8842 | −3.704 | 2.476 | 0.978 |

5A | 0.4833 | 0.2252 | 0.9334 | −3.666 | 2.059 | 0.989 |

5B | 0.4079 | 0.1111 | 0.8973 | −5.658 | 2.295 | 0.956 |

5C | 0.4740 | 0.1418 | 0.9288 | -5.399 | 2.350 | 0.989 |

6A | 0.3928 | 0.2635 | 0.9857 | −2.170 | 1.503 | 0.993 |

6B | 0.5583 | 0.2353 | 0.8132 | −4.797 | 2.790 | 0.958 |

6C | 0.4750 | 0.1934 | 0.8849 | −3.991 | 2.085 | 0.987 |

7A | 0.6824 | 0.4135 | 0.9216 | −2.330 | 1.894 | 0.984 |

7B | 0.6204 | 0.3023 | 0.6865 | −5.104 | 3.380 | 0.921 |

7C | 0.6077 | 0.2937 | 0.8461 | −4.185 | 2.664 | 0.967 |

8A | 0.4315 | 0.2850 | 0.977 | −2.112 | 1.475 | 0.999 |

8B | 0.5719 | 0.1979 | 0.6182 | −5.849 | 2.849 | 0.885 |

8C | 0.5268 | 0.1978 | 0.8614 | −4.658 | 2.356 | 0.984 |

9A | 0.3339 | 0.2230 | 0.985 | −2.219 | 1.381 | 0.989 |

9B | 0.5192 | 0.1543 | 0.7807 | −5.993 | 2.653 | 0.955 |

9C | 0.4818 | 0.1710 | 0.8853 | −5.105 | 2.461 | 0.984 |

10A | 0.4833 | 0.2252 | 0.9334 | −3.666 | 2.059 | 0.989 |

10B | 0.4079 | 0.1111 | 0.8973 | −5.658 | 2.295 | 0.956 |

10C | 0.4740 | 0.1418 | 0.9288 | −5.399 | 2.350 | 0.989 |

11A | 0.4176 | 0.1808 | 0.9366 | −3.964 | 2.043 | 0.995 |

11B | 0.3440 | 0.0873 | 0.8544 | −5.787 | 2.187 | 0.986 |

11C | 0.3865 | 0.0985 | 0.7726 | −5.428 | 2.084 | 0.976 |

No. | Linear Regression Fitting Formula | D_{eff}(10 ^{−11} m^{2}·s^{−1}) | R^{2} | No. | Linear Regression Fitting Formula | D_{eff}(10 ^{−11} m^{2}·s^{−1}) | R^{2} |
---|---|---|---|---|---|---|---|

1A | $\mathrm{ln}MR=-5.822\times 10{-}^{5}\mathrm{t}+0.4672$ | 1.1574 | 0.9479 | 7A | $\mathrm{ln}MR=-1.148\times 10{-}^{4}\mathrm{t}+0.6824$ | 2.2833 | 0.9216 |

1B | $\mathrm{ln}MR=-3.647\times 10{-}^{5}\mathrm{t}+0.4662$ | 2.7975 | 0.8826 | 7B | $\mathrm{ln}MR=-8.397\times 10{-}^{5}\mathrm{t}+0.6204$ | 6.4408 | 0.6865 |

1C | $\mathrm{ln}MR=-2.947\times 10{-}^{5}\mathrm{t}+0.4353$ | 4.4307 | 0.8749 | 7C | $\mathrm{ln}MR=-8.158\times 10{-}^{5}\mathrm{t}+0.6077$ | 12.2649 | 0.8461 |

2A | $\mathrm{ln}MR=-6.908\times 10{-}^{5}\mathrm{t}+0.4124$ | 1.3733 | 0.9837 | 8A | $\mathrm{ln}MR=-7.916\times 10{-}^{5}\mathrm{t}+0.4315$ | 1.5738 | 0.977 |

2B | $\mathrm{ln}MR=-5.186\times 10{-}^{5}\mathrm{t}+0.547$ | 3.9778 | 0.8537 | 8B | $\mathrm{ln}MR=-5.497\times 10{-}^{5}\mathrm{t}+0.5719$ | 4.2165 | 0.6182 |

2C | $\mathrm{ln}MR=-4.575\times 10{-}^{5}\mathrm{t}+0.4214$ | 6.8779 | 0.9128 | 8C | $\mathrm{ln}MR=-5.494\times 10{-}^{5}\mathrm{t}+0.5268$ | 8.2601 | 0.8614 |

3A | $\mathrm{ln}MR=-7.319\times 10{-}^{5}\mathrm{t}+0.3928$ | 1.4550 | 0.9857 | 9A | $\mathrm{ln}MR=-6.194\times 10{-}^{5}\mathrm{t}+0.3339$ | 1.2314 | 0.985 |

3B | $\mathrm{ln}MR=-6.536\times 10{-}^{5}\mathrm{t}+0.5583$ | 5.0133 | 0.8132 | 9B | $\mathrm{ln}MR=-4.286\times 10{-}^{5}\mathrm{t}+0.5192$ | 3.2875 | 0.7807 |

3C | $\mathrm{ln}MR=-5.972\times 10{-}^{5}\mathrm{t}+0.4750$ | 8.0764 | 0.8849 | 9C | $\mathrm{ln}MR=-4.75\times 10{-}^{5}\mathrm{t}+0.4818$ | 7.1409 | 0.8853 |

4A | $\mathrm{ln}MR=-1.149\times 10{-}^{4}\mathrm{t}+0.675$ | 2.2855 | 0.8449 | 10A | $\mathrm{ln}MR=-6.255\times 10{-}^{5}\mathrm{t}+0.4833$ | 1.2435 | 0.9334 |

4B | $\mathrm{ln}MR=-7.261\times 10{-}^{5}\mathrm{t}+0.5862$ | 5.5694 | 0.7441 | 10B | $\mathrm{ln}MR=-3.086\times 10{-}^{5}\mathrm{t}+0.4079$ | 2.3671 | 0.8973 |

4C | $\mathrm{ln}MR=-8.644\times 10{-}^{5}\mathrm{t}+0.5998$ | 12.9957 | 0.8842 | 10C | $\mathrm{ln}MR=-3.938\times 10{-}^{5}\mathrm{t}+0.4740$ | 5.9216 | 0.9288 |

5A | $\mathrm{ln}MR=-6.255\times 10{-}^{5}\mathrm{t}+0.4833$ | 1.2435 | 0.9334 | 11A | $\mathrm{ln}MR=-5.022\times 10{-}^{5}\mathrm{t}+0.4176$ | 0.9984 | 0.9366 |

5B | $\mathrm{ln}MR=-3.086\times 10{-}^{5}\mathrm{t}+0.4079$ | 2.3671 | 0.8973 | 11B | $\mathrm{ln}MR=-2.425\times 10{-}^{5}\mathrm{t}+0.3440$ | 1.8600 | 0.8544 |

5C | $\mathrm{ln}MR=-3.938\times 10{-}^{5}\mathrm{t}+0.4740$ | 5.9216 | 0.9288 | 11C | $\mathrm{ln}MR=-2.736\times 10{-}^{5}\mathrm{t}+0.3865$ | 4.1134 | 0.7726 |

6A | $\mathrm{ln}MR=-7.319\times 10{-}^{5}\mathrm{t}+0.3928$ | 1.4550 | 0.9857 | ||||

6B | $\mathrm{ln}MR=-6.536\times 10{-}^{5}\mathrm{t}+0.5583$ | 5.0133 | 0.8132 | ||||

6C | $\mathrm{ln}MR=-5.372\times 10{-}^{5}\mathrm{t}+0.4750$ | 8.0764 | 0.8849 |

Thickness | Drying Conditions | E_{a} (kJ/mol) | ΔH (kJ/mol) | ΔS (J/mol·K) | ΔG (kJ/mol) |
---|---|---|---|---|---|

0.8–2.0 mm | 20 °C | 16.38 | 13.94 | −398.26 | 130.69 |

30 °C | 13.86 | −398.96 | 134.8 | ||

40 °C | 13.78 | −400.47 | 139.18 | ||

50 °C | 13.69 | −398.6 | 142.5 | ||

0.3 m/s | 13.78 | −401.78 | 139.59 | ||

0.8 m/s | 13.78 | −400.47 | 139.18 | ||

1.3 m/s | 13.78 | −396.72 | 138.01 | ||

20% | 13.78 | −399.82 | 138.98 | ||

30% | 13.78 | −401.86 | 139.62 | ||

40% | 13.78 | −401.78 | 139.59 | ||

50% | 13.78 | −403.6 | 140.16 | ||

2.0–3.5 mm | 20 °C | 18.21 | 15.77 | −384.68 | 128.54 |

30 °C | 15.69 | −384.08 | 132.12 | ||

40 °C | 15.61 | −384.34 | 135.96 | ||

50 °C | 15.52 | −385.53 | 140.11 | ||

0.3 m/s | 15.61 | −390.58 | 137.92 | ||

0.8 m/s | 15.61 | −384.34 | 135.96 | ||

1.3 m/s | 15.61 | −382.26 | 135.31 | ||

20% | 15.61 | −385.78 | 136.41 | ||

30% | 15.61 | −387.85 | 137.06 | ||

40% | 15.61 | −390.58 | 137.92 | ||

50% | 15.61 | −392.58 | 138.54 | ||

3.5–4.2 mm | 20 °C | 26.66 | 24.22 | −352.03 | 127.42 |

30 °C | 24.14 | −351.65 | 130.74 | ||

40 °C | 24.06 | −353.39 | 134.72 | ||

50 °C | 23.97 | −352.33 | 137.83 | ||

0.3 m/s | 24.06 | −355.97 | 135.53 | ||

0.8 m/s | 24.06 | −353.39 | 134.72 | ||

1.3 m/s | 24.06 | −349.92 | 133.63 | ||

20% | 24.06 | −353.21 | 134.66 | ||

30% | 24.06 | −354.42 | 135.04 | ||

40% | 24.06 | −355.97 | 135.53 | ||

50% | 24.06 | −359 | 136.48 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Q.; Li, S.; Zhang, M.; Mu, G.; Li, X.; Zhang, G.; Xiong, S.
Heat Pump Drying of Kelp (*Laminaria japonica*): Drying Kinetics and Thermodynamic Properties. *Processes* **2022**, *10*, 514.
https://doi.org/10.3390/pr10030514

**AMA Style**

Zhang Q, Li S, Zhang M, Mu G, Li X, Zhang G, Xiong S.
Heat Pump Drying of Kelp (*Laminaria japonica*): Drying Kinetics and Thermodynamic Properties. *Processes*. 2022; 10(3):514.
https://doi.org/10.3390/pr10030514

**Chicago/Turabian Style**

Zhang, Qian, Shiyu Li, Minqi Zhang, Gang Mu, Xiuchen Li, Guochen Zhang, and Shanbai Xiong.
2022. "Heat Pump Drying of Kelp (*Laminaria japonica*): Drying Kinetics and Thermodynamic Properties" *Processes* 10, no. 3: 514.
https://doi.org/10.3390/pr10030514