# A Horizontal Distribution Model of Static Ice Cover Generated by Static and Dynamic Water Considering the Heat Transfer of Riverbanks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Horizontal Distribution Model of Static Ice Cover Considering Heat Transfer of River Banks

#### 2.1. Temperature Control Equation

#### 2.1.1. Water Flow Temperature Calculation Model

_{w}is the water temperature, °C; Q

_{vd}is viscous dissipation, W/m

^{3}; q is the heat source term generated by electric heating and other related heating methods, W/m

^{3};

**u**is the velocity field of water flow, m/s; μ is fluid dynamic viscosity, (m

^{2})/s; ρ

_{w}is the density of water flow (kg/m

^{3}); p is water flow pressure, N/m

^{2};

**I**is the unit matrix; and

**F**is an external force acting on the water and is calculated using the Boussinesq approximation. ρ

_{wi}, C

_{pwi}and k

_{wi}are the density of ice-water mixture flow (kg/m

^{3}), constant pressure heat capacity (J/(kg·K)) and thermal conductivity (W/(m·K)), respectively, and Nu is a dimensionless Nusselt number that represents the degree of convective heat transfer. It can be expressed as a function of the dimensionless Reynolds number Re and the dimensionless Prandtl number Pr related to the hydraulic characteristics according to the experimental data [22,23], as shown in Equation (4).

_{T}is defined by the flow interface, and Pr

_{T∞}= 0.85 depends on the heat transfer turbulence model.

#### 2.1.2. Water Flow Icing Model

_{wi}is the effective density of the ice-water mixture; C

_{pwi}is the apparent heat capacity of the ice-water mixture; α

_{m}is the mass fraction; L is the latent heat of ice water phase change, kJ/kg; and θ

_{w}and θ

_{i}are the volume fractions of water and ice, respectively. ρ

_{w}and ρ

_{i}are the densities of water and ice, respectively. C

_{pw}and C

_{pi}are the constant pressure heat capacities of water and ice, respectively.

#### 2.1.3. Coupled Moisture-Heat Transport Model of Frozen Soil on Banks

_{v}are the equivalent constant pressure heat capacity J/(kg·K) and the equivalent thermal conductivity W/(m·K) of the soil, respectively, which are expressed by the calculated average of each component, Equations (14) and (15); and the subscripts sp, sw, si and sa represent soil particles, pore water, pore ice and gas phase, respectively. C is the specific water capacity, and the van Genuchten model (VG model) was used to describe the relationship between the unfrozen water content and the matrix potential and the permeability coefficient [34], Equations (16)–(19). a and b are the experimental fitting parameters of the freezing characteristic curve; ρ

_{d}is the dry density of soil, kg/m

^{3}.

_{s}and θ

_{r}are the soil saturation and residual water content, respectively. S

_{e}is the equivalent saturation; K

_{s}is the permeability coefficient of saturated soil, m/s; i is the ice impedance factor; and G

_{wT}is the temperature correction factor of the soil water characteristic curve (SWCC). γ is the surface tension of pore water, J/m

^{3}. K

_{lh}is the hydraulic conductivity, m/s, and K

_{lT}is the permeability coefficient caused by the thermal gradient, m

^{2}/(K·s).

#### 2.2. Model Boundary Condition and Solving Process

**n**is the normal vector of the upper surface; T

_{amb}and T

_{in}are the external environment temperature and the upper surface temperature of the river bank and water flow, respectively, °C; and h is the convective heat transfer coefficient, W/(m

^{2}·°C). The convective heat transfer coefficient on the surface of the water flow comprehensively considers the evaporation and convection of the water surface. The evaporation heat flux is expressed as the wind speed and the saturated water vapor pressure-related term [35,36], Equation (21). The convective heat transfer coefficient of the riverbed is only related to the wind speed, 15 W/(m

^{2}·°C) [27].

^{2}·°C) [38]; T > 0 °C, 10 W/(m

^{2}·°C) [27].

## 3. Results

#### 3.1. Analysis of Ice Thickness Growth Generated by Static and Dynamic Water

#### 3.2. Coupling Analysis of Ice Thickness Growth and Riverbank Freezing Process

## 4. Discussion

#### 4.1. The Influence of Temperature and Water Velocity on the Formation of Static Ice Cover

#### 4.1.1. Analysis of Model Coupling Parameters

#### 4.1.2. Analysis of the Horizontal Difference of Static Ice Cover

#### 4.1.3. The Influence of Riverbed Heat Transfer on Water and Heat Exchange

#### 4.2. Horizontal Distribution Regression Model of Ice Cover

_{ice}is the thickness of ice cover, cm; H

_{w}is the water depth of the section, cm; x′ is the distance from the shore, m; B is the width of the water surface, m; and

**a**and

**c**are the parameters of the empirical formula to describe the transverse distribution of the ice cover. The fitting parameters and their correlation coefficients are shown in Table 3.

**a**and

**c**were related to the cumulative negative temperature of the initial ice. As shown in Table 3, parameter

**a**was positively correlated with the ratio of cumulative negative temperature to initial ice cumulative negative temperature, and parameter

**c**was negatively correlated with the ratio of initial ice cumulative negative temperature to cumulative negative temperature. As freezing progresses, the horizontal distribution shape parameters of the ice cover increased linearly with the multiple cumulative negative temperatures of the initial ice. The regression model based on numerical model generalization is a simple calculation, with clear physical meaning of fitting parameters and high accuracy, which is convenient for the direct application of practical engineering and design.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Saadé, R.G.; Sarraf, S. Simulation of Ice Cover Melting in Turbulent Flow. Int. J. Numer. Methods Heat Fluid Flow
**1995**, 5, 647–663. [Google Scholar] [CrossRef] - Rokaya, P.; Budhathoki, S.; Lindenschmidt, K.-E. Trends in the Timing and Magnitude of Ice-Jam Floods in Canada. Sci. Rep.
**2018**, 8, 5834. [Google Scholar] [CrossRef] - Adalaiti, H.J.; Yu, S.S. Study and Prospect of Ice Damage Prevention and Control in Xinjiang’s Water Conveyance Projects. J. Water Resour. Archit. Eng.
**2010**, 8, 46–49. (In Chinese) [Google Scholar] - Yang, K. Advances of ice hydraulics, ice regime observation and forecasting in rivers. J. Hydraul. Eng.
**2018**, 49, 81–91. (In Chinese) [Google Scholar] [CrossRef] - Guo, X.; Wang, T.; Fu, H.; Pan, J.; Lu, J.; Guo, Y.; Li, J. Progress and Trend in the Study of River Ice Hydraulics. Theor. Appl. Mech.
**2021**, 53, 655–671. (In Chinese) [Google Scholar] - Shen, H.T. River Ice Processes. In Advances in Water Resources Management; Wang, L.K., Yang, C.T., Wang, M.-H.S., Eds.; Springer International Publishing: Cham, Switzerland, 2016; pp. 483–530. ISBN 978-3-319-22923-2. [Google Scholar]
- Peters, M.; Dow, K.; Clark, S.P.; Malenchak, J.; Danielson, D. Experimental Investigation of the Flow Characteristics beneath Partial Ice Covers. Cold Reg. Sci. Technol.
**2017**, 142, 69–78. [Google Scholar] [CrossRef] - Pan, J.; Shen, H.T. Modeling Ice Cover Effect on River Channel Bank Stability. Environ. Fluid Mech.
**2022**, 22, 1121–1133. [Google Scholar] [CrossRef] - Duan, W.; Huang, G.; Yang, J.; Liu, M. Ice regime analysis and safe dispatch research on long distance water diversion project in winter. South-North Water Transf. Water Sci. Technol.
**2016**, 14, 96–104. (In Chinese) [Google Scholar] [CrossRef] - Shen, H.T. Mathematical Modeling of River Ice Processes. Cold Reg. Sci. Technol.
**2010**, 62, 3–13. [Google Scholar] [CrossRef] - Svensson, U.; Billfalk, L.; Hammar, L. A Mathematical Model of Border-Ice Formation in Rivers. Cold Reg. Sci. Technol.
**1989**, 16, 179–189. [Google Scholar] [CrossRef] - Huang, F.; Shen, H.T.; Knack, I. Modeling Border Ice Formation and Cover Progression in Rivers. In Proceedings of the IAHR International Symposium on Ice, Dalian, China, 11–15 June 2012. [Google Scholar]
- Mao, Z.; Dong, Z.; Chen, C. Review on mathematical simulation of river ice. Water Power
**1996**, 12, 58–61. (In Chinese) [Google Scholar] - Mao, Z.; Chen, C. Simulation of Heat Transfer between Streambed and River Flow. Water Resour. Hydropower Eng.
**1999**, 5, 11–13. (In Chinese) [Google Scholar] [CrossRef] - Bernard, M. Comparison of Field Data with Theories on Ice Cover Progression in Large Rivers. Can. J. Civ. Eng.
**2011**, 11, 798–814. [Google Scholar] [CrossRef] - Shen, H.T.; Yapa, P.D. A Unified Degree-Day Method for River Ice Cover Thickness Simulation. Can. J. Civ. Eng.
**1985**, 12, 54–62. [Google Scholar] [CrossRef] - Yang, K.; Guo, X.; Wang, T.; Fu, H.; Pan, J. Effects of solar radiation and ground temperature on water temperature under ice cover. J. Hydraul. Eng.
**2022**, 53, 530–538+548. (In Chinese) [Google Scholar] [CrossRef] - Lian, J.; Zhao, X. Radiation degree-day method for predicting the development of ice cover thickness under the hydrostatic and non-hydrostatic conditions. J. Hydraul. Eng.
**2011**, 42, 1261–1267. (In Chinese) [Google Scholar] [CrossRef] - Gordon, M. Greene Simulation of Ice-Cover Growth and Decay in One Dimension on the Upper St. Lawrence River. Available online: https://repository.library.noaa.gov/view/noaa/10432 (accessed on 30 June 2023).
- Khan, Z.H.; Ahmad, R.; Sun, L. Effect of Instantaneous Change of Surface Temperature and Density on an Unsteady Liquid–Vapour Front in a Porous Medium. Exp. Comput. Multiph. Flow
**2020**, 2, 115–121. [Google Scholar] [CrossRef] - Peters, G.W.M.; Baaijens, F.P.T. Modelling of Non-Isothermal Viscoelastic Flows. J. Non-Newton. Fluid Mech.
**1997**, 68, 205–224. [Google Scholar] [CrossRef] - Xiao, H.; Dong, Z.; Long, R.; Yang, K.; Yuan, F. A Study on the Mechanism of Convective Heat Transfer Enhancement Based on Heat Convection Velocity Analysis. Energies
**2019**, 12, 4175. [Google Scholar] [CrossRef] - Lienhard, J.H. A Heat Transfer Textbook; Phlogiston Press: Cambridge, MA, USA, 2011; Available online: https://ahtt.mit.edu (accessed on 3 August 2023).
- Kays, W.M. Turbulent Prandtl Number—Where Are We? Asme Trans. J. Heat Transf.
**1994**, 116, 284–295. [Google Scholar] [CrossRef] - Thonon, M.; Fraisse, G.; Zalewski, L.; Pailha, M. Towards a Better Analytical Modelling of the Thermodynamic Behaviour of Phase Change Materials. J. Energy Storage
**2020**, 32, 101826. [Google Scholar] [CrossRef] - Moench, S.; Dittrich, R. Influence of Natural Convection and Volume Change on Numerical Simulation of Phase Change Materials for Latent Heat Storage. Energies
**2022**, 15, 2746. [Google Scholar] [CrossRef] - Jiang, H.; Liu, Q.; Wang, Z.; Gong, J.; Li, L. Frost Heave Modelling of the Sunny-Shady Slope Effect with Moisture-Heat-Mechanical Coupling Considering Solar Radiation. Sol. Energy
**2022**, 233, 292–308. [Google Scholar] [CrossRef] - Liu, Q.; Wang, Z.; Li, Z.; Wang, Y. Transversely Isotropic Frost Heave Modeling with Heat–Moisture–Deformation Coupling. Acta Geotech.
**2020**, 15, 1273–1287. [Google Scholar] [CrossRef] - Harlan, R.L. Analysis of Coupled Heat-Fluid Transport in Partially Frozen Soil. Water Resour. Res.
**1973**, 9, 1314–1323. [Google Scholar] [CrossRef] - Hansson, K.; Imnek, J.; Mizoguchi, M.; Lundin, L.C.; Van Genuchten, M.T. Water Flow and Heat Transport in Frozen Soil: Numerical Solution and Freeze–Thaw Applications. Vadose Zone J.
**2004**, 3, 693–704. [Google Scholar] [CrossRef] - Liu, X.; Liu, J.; Tian, Y.; Shen, Y.; Liu, J. A Frost Heaving Mitigation Method with the Rubber-Asphalt-Fiber Mixture Cylinder. Cold Reg. Sci. Technol.
**2020**, 169, 102912. [Google Scholar] [CrossRef] - Li, S.; Zhang, M.; Tian, Y.; Pei, W.; Zhong, H. Experimental and Numerical Investigations on Frost Damage Mechanism of a Canal in Cold Regions. Cold Reg. Sci. Technol.
**2015**, 116, 1–11. [Google Scholar] [CrossRef] - Xu, X. Physics of Frozen Soil; Physics of Frozen Soil: Beijing, China, 2010; ISBN 978-7-03-028867-7. [Google Scholar]
- Lu, N.; Likos, W.J. Unsaturated Soil Mechanics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2004; Available online: https://webapps.unitn.it/Biblioteca/it/Web/EngibankFile/5802626.pdf (accessed on 15 June 2023).
- Nan, L.I.; Tuo, Y.C.; Deng, Y.; Jia, L.I.; Liang, R.F.; Rui-Dong, A.N. Heat Transfer at Ice-Water Interface under Conditions of Low Flow Velocities. J. Hydrodyn.
**2016**, 28, 603–609. [Google Scholar] [CrossRef] - Sarraf, S.; Zhang, X.T. Modeling Ice-Cover Melting Using a Variable Heat Transfer Coefficient. J. Eng. Mech.
**1996**, 122, 930–938. [Google Scholar] [CrossRef] - Dong, S.; Cui, H. Analysis of Calculating Formula and Improvement of Empirical Formula for Saturation Vapour Pressure. Q. J. Appl. Meteorol.
**1992**, 3, 501–508. (In Chinese) [Google Scholar] - Yang, K. Heat exchange model between river-lake and atmosphere during ice age. J. Hydraul. Eng.
**2021**, 52, 556–564+577. (In Chinese) [Google Scholar] [CrossRef] - Hanley, T.O.; Michel, B. Laboratory Formation of Border Ice and Frazil Slush. Can. J. Civ. Eng.
**1977**, 4, 153–160. [Google Scholar] [CrossRef] - Michel, B.; Ramseier, R.O. Classification of River and Lake Ice. Can. Geotech. J.
**1971**, 8, 36–45. [Google Scholar] [CrossRef] - Turcotte, B.; Morse, B. A Global River Ice Classification Model. J. Hydrol.
**2013**, 507, 134–148. [Google Scholar] [CrossRef] - Blokhina, N.S.; Ordanovich, A.E. The Influence of Ice Cover on a Reservoir on the Development of a Spring Thermal Bar. Mosc. Univ. Phys.
**2012**, 67, 109–115. [Google Scholar] [CrossRef] - Marsh, P.; Prowse, T.D. Water Temperature and Heat Flux at the Base of River Ice Covers. Cold Reg. Sci. Technol.
**1987**, 14, 33–50. [Google Scholar] [CrossRef] - Yu, S.; Li, Q.; Song, L. Forecast of Ice State in Open Canal with Water Delivered in Ice Periods. China Rural Water Hydropower
**2008**, 9, 108–109+113. (In Chinese) [Google Scholar]

**Figure 1.**Horizontal distribution of static and dynamic water ice cover: (

**a**) reservoir, Karabelli Water Conservancy Project, which is located in Wuqia County, Kezhou, China; (

**b**) canal, diversion canal of power station, which is located in Xie Tongmen County, Tibet, China.

**Figure 5.**Horizontal ice thickness distribution. (

**a**) Development of ice thickness. (

**b**) Development of width of border ice.

**Figure 9.**The variation of the Nusselt number of static and dynamic water. (

**a**) Static water; (

**b**) dynamic water.

**Figure 11.**Maximum heat flux ratio between riverbed and water surface. (

**a**) The variation of maximum heat flux ratio with environmental temperature; (

**b**) The variation of maximum heat flux ratio with flow velocity; (

**c**) The variation of maximum heat flux ratio with slope coefficient; (

**d**) The variation of maximum heat flux ratio with water depth. The red line in (

**c**) represents the variation of maximum heat flux ratio of low slope ratio.

**Figure 12.**Empirical formula for ice thickness. (

**a**) Vertical thickening formula; (

**b**) horizontal distribution formula. The red dash line in (

**b**) represents the connecting part of the channel slope and the trough.

**Table 1.**Model parameter values used in the simulation [27].

Variable | Value | Variable | Value |
---|---|---|---|

ΔT (K) | 0.1 | λ_{sa} (W/(m·K)) | 0.024 |

ρ_{sp} (kg/m^{3}) | 2700 | L (kJ/kg) | 334 |

ρ_{sw} (kg/m^{3}) | 1000 | θ_{s} (1) | 0.43 |

ρ_{si} (kg/m^{3}) | 931 | θ_{r} (1) | 0.03 |

C_{sp} (kJ/(m^{3}·K)) | 2000 | α (1/m) | 0.38 |

C_{sw} (kJ/(m^{3}·K)) | 4220 | m (1) | 0.36 |

C_{si} (kJ/(m^{3}·K)) | 1935 | K_{s} (m/s) | 2 × 10^{−7} |

λ_{sp} (W/(m·K)) | 1.5 | a (1) | 6.89 |

λ_{sw} (W/(m·K)) | 0.55 | b (1) | 0.57 |

λ_{si} (W/(m·K)) | 2.22 | ρ_{d} (kg/m^{3}) | 0.7 |

Case | T /°C | V /(m·s ^{−1}) | H_{w}/m | Slope Coefficient m | Initial Nu | Initial Ice Time /h | Freeze-Up Period /h | Ice Cover Thickness /cm | Thickness Difference /cm |
---|---|---|---|---|---|---|---|---|---|

1 | −5 | 0.00 | 0.2 | 1.0 | 20 | 3.5 | 6.0 | 4.16 | 0.21 |

1 | −10 | 0.00 | 0.2 | 1.0 | 23 | 2.3 | 4.3 | 5.42 | 0.22 |

1 | −20 | 0.00 | 0.2 | 1.0 | 68 | 1.9 | 2.8 | 12.06 | 0.42 |

2 | −20 | 0.00 | 0.3 | 1.0 | 68 | 1.8 | 3.4 | 11.47 | 1.75 |

3 | −20 | 0.00 | 0.2 | 1.1 | 68 | 1.8 | 2.8 | 12.08 | 0.52 |

4 | −20 | 0.00 | 0.2 | 1.5 | 68 | 1.5 | 2.9 | 12.17 | 0.54 |

5 | −20 | 0.12 | 0.2 | 1.0 | 80 | 1.9 | 3.0 | 9.73 | 1.45 |

6 | −20 | 0.20 | 0.2 | 1.0 | 95 | 3.4 | 4.7 | 9.78 | 1.23 |

Freezing Time (h) | a | c | R^{2} |
---|---|---|---|

2.7 | 0.00221 | −1.3839 | 0.99632 |

3.0 | 0.00366 | −1.33003 | 0.99769 |

24 | 0.06690 | −1.08838 | 0.99426 |

48 | 0.12089 | −1.08153 | 0.99391 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Xue, B.; Wang, Z.; Liu, Q.; Li, H.
A Horizontal Distribution Model of Static Ice Cover Generated by Static and Dynamic Water Considering the Heat Transfer of Riverbanks. *Water* **2023**, *15*, 3893.
https://doi.org/10.3390/w15223893

**AMA Style**

Xue B, Wang Z, Liu Q, Li H.
A Horizontal Distribution Model of Static Ice Cover Generated by Static and Dynamic Water Considering the Heat Transfer of Riverbanks. *Water*. 2023; 15(22):3893.
https://doi.org/10.3390/w15223893

**Chicago/Turabian Style**

Xue, Boxiang, Zhengzhong Wang, Quanhong Liu, and Hanxiang Li.
2023. "A Horizontal Distribution Model of Static Ice Cover Generated by Static and Dynamic Water Considering the Heat Transfer of Riverbanks" *Water* 15, no. 22: 3893.
https://doi.org/10.3390/w15223893