Nan LI （李楠）， You-cai TUO （脫友才）， Yun DENG （鄧云）， Jia LI （李嘉）， Rui-feng LIANG （梁瑞峰），Rui-dong AN （安瑞冬）

State Key Laboratory of Hydraulics and Mountain River Engineering， Sichuan University， Chengdu 610065，China， E-mail: linanscu@163.com

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Heat transfer at ice-water interface under conditions of low flow velocities*

Nan LI （李楠）， You-cai TUO （脫友才）， Yun DENG （鄧云）， Jia LI （李嘉）， Rui-feng LIANG （梁瑞峰），Rui-dong AN （安瑞冬）

State Key Laboratory of Hydraulics and Mountain River Engineering， Sichuan University， Chengdu 610065，China， E-mail: linanscu@163.com

The heat transfer at the ice-water interface is closely related to the hydrodynamic and physical properties of the water body. It affects the ice cover thickness and the water temperature underlying the ice cover. This paper studies the heat transfer from the water to the ice cover. Based on the flume data， a linear relationship between the ice-water heat transfer coefficient and the flow velocity beneath the ice cover is established and the calculated dimensionless ice-water heat transfer coefficient is 1.1×10-3. This empirical relationship can be applied to estimate the ice-water heat transfer of reservoirs， lakes and other freshwater bodies when the flow velocity under the ice cover is in the range of 0.024 m/s-0.110 m/s.

ice cover， heat exchange， ice-water heat transfer coefficient， low flow velocity， laboratory experiment

Introduction

The formation of ice cover is an important phenomenon in cold regions［1，2］. Ice in surface water bodies changes the hydraulic and thermal conditions of rivers， lakes and reservoirs［3，4］. The ice cover influences the operation of water resource projects， leads to reductions in power generation［5，6］， causes ice disasters such as ice dam and ice flood［7，8］， and hinders energy and mass exchanges between air and water，resulting in adverse effects on biological environment［9］.

The thermal growth and decay of an ice cover is governed by heat-exchanges at the air-ice and icewater interfaces. The ice-water heat transfer coefficient， which reflects the rate of heat exchange between the ice cover and the underlying water， is an important parameter for quantifying the heat flux at the icewater interface， the thickness of ice cover， and the water temperature underneath the ice cover. During the past few decades， the heat-exchanges at the icewater interface in rivers， lakes and oceans were extensively studied. The turbulent heat transfer from the flowing river water to the ice cover is shown to have a significant effect on the thickness of the ice cover，especially during the decay period when the water temperature is above the freezing point［3］. The thermal growth and melting of the lake ice is primarily a vertical one-dimensional heat transfer process［10-12］. It is possible to estimate the heat exchange flux between ice and water by a bulk formula［13］

in which，ρwis the density of water，cwis the specific heat of water，Chis the dimensionless ice-water heat transfer coefficient，uwis the current speed，Twis the water temperature， and T0is the freezing point.

With considerations of the surface roughness， the ice thickness， the current and the temperature under ice， Hamblin and Carmack［14］estimated the dimensionless ice-water heat transfer coefficient Chto be（0.8±0.3）×10-3in lakes of Yukon River Basin， which is smaller than that was found in sea ice studies.Shirasawa et al.［15］obtained Chof a value 0.39×10-3from the HANKO and the BALTEX/BASIS experiments and used 2.0×10-3as the value ofChto calculate the ice-water heat flux in Saroma-ko Lagoon. Ji et al.［13］computedChto be（0.16-0.50）×10-3through field observations in Bohai Sea in different periods，and found that the coefficient had a positive relation with the thickness of the ice cover and the roughness of the bottom surface of the ice cover.

Fig.1 Schematic diagram of the experimental device

Fig.2 Detailed structures of the plexiglas flume （m）

However， the ice-water heat transfer coefficient is closely related to the hydrodynamic and physical properties of the water body. When the dimensionless ice-water heat transfer coefficientsChmentioned above are applied to particular situations， it is found that they are not satisfactory for the water body under low flow conditions such as a reservoir.

The field observation method to study ice-water heat exchange involves many difficulties and various uncontrollable factors. No detailed laboratory study has been made to determine the ice-water heat transfer coefficient. In this study， a laboratory experiment is conducted to investigate the heat exchange from water to ice cover to establish an empirical formula for the ice-water heat transfer coefficient under low flow conditions.

1. Experimental setup

1.1 Flume design

The experiment is conducted in a small plexiglas flume of 4.0 m （length）×3.0 m （width）×2.0 m （height）in a cold room （Fig.1）. The flume is wrapped with polyethylene plastic foam to prevent the heat exchange from sidewalls. A screen is installed in the entrance section， which makes the flow uniformly distributed. An ice cover is formed in the flume in each test. Figure 2 shows the detailed structures of the plexiglas flume.

1.2 Instrumentation

The measured data in the experiment include the flow velocity， the ice thickness variation， the ice temperature and the water temperature. Figure 3 is a photo of the experimental device and the measuring instruments.

Fig.3 Photo of the experimental device and measuring instruments

The flow measuring section is 0.50 m from the entrance （Section 1， Fig.2）. A Vectrino velocimeterwith a resolution of 0.001 m/s is used to measure the flow velocity. The velocities at three depths （0.07 m，0.14 m and 0.21 m） are measured before the experiment. The data indicate that the velocities distrubute evenly in depth. Therefore， the flow velocity at the mid-depth is measured in the experiment， which may be taken as the depth-averaged velocity.

The ice thickness variation measuring section is 1.50 m from the entrance （Section 2， Fig.2）. The bottom surface of the ice cover is flat， and the variation of the ice thickness is uniform along the ice cover. The vertical distance between the bottom surface of the ice cover and a fixed point at the beginning of the experiment is h1， and it becomes h220 min later. The measured ice thickness variation is then h2-h1. The vertical distance is measured by a micrometer， of accuracy of 0.0001 m （Fig.4）.

Fig.4 Schematic diagram of ice thickness variation measurement

The temperature measuring section is 1.60 m from the entrance （Section 3， Fig.2）. A LG93-22 temperature recorder is used to measure the water and ice temperatures， with an accuracy of 0.1oC. The ice temperature is measured by No.1 and No.2 temperature probes， and the water temperature is measured by No.3 to No.11 temperature probes. The layout of the temperature probes is shown in Fig.5.

Fig.5 Layout of temperature probes （m）

1.3 Experimental procedure

A flow regime test is made first， under 4 flow conditions with the depth-averaged flow velocities of 0.110 m/s， 0.084 m/s， 0.055 m/s and 0.024 m/s.

The flow regime test is made by recording the instantaneous velocities （ux，uyand uz） every 0.04 s. The average turbulence intensity under each condition is determined by

in which，Tuis the instantaneous turbulence intensity，u′x，u′y，u′zare the fluctuating velocities inx，y，z directions，ux，uy，uzare the instantaneous velocities inx，y，zdirections，Tuis the average turbulence intensity， andNis the number of instantaneous turbulence intensities in each test.

The inflow velocity is stable under all 4 conditions and the calculated average turbulence intensities are 14.7%， 14.4%， 13.7% and 9.8%， respectively. According to Wang et al.［16］， the boundary layer under a flat surface is of turbulence when the water turbulence intensity reaches 3.5%. Hence， the flows under these 4 conditions are of turbulence and are valid for the experiment.

Main steps of the experiment include: （1） Set the temperature in the cold room to -15oC， and freeze an ice cover （about 1.70 m long and 0.02 m thick） in the flume. （2） Stabilize the temperature in the cold room to 0oC， start the pump to make the water flow， and use the valve to control the flow velocity. （3） Start the LG93-22 temperature recorder and measure the vertical distance between the bottom surface of the ice cover and the fixed point， when the flow field in the flume is stable. （4） Repeat Steps （1） to （3） in each experiment.

1.4 Experimental results

A total of 22 experiment runs are made by combining these 4 velocity conditions with different inflow temperatures （Table 1）. The measured ice thickness reductions at the bottom of the ice cover and the inflow temperature over a 20 min period are given in Table 1. Figure 6 shows the vertical temperature profiles under various velocity conditions. Influenced by the air temperature in the cold room， the ice temperature reaches 0oC gradually. No. 3 to No. 11 temperatureprobes are in the flow where the water temperature is mixed evenly.

Table 1 Summary of test conditions

2. Results and analyses

2.1 Analyses of the heat flux process

A definition of the heat flux between the ice cover and the flowing water is presented in Fig.7 for analyzing the heat flux process. The air temperature in the cold room is 0oC， and the whole temperature of the ice cover reaches a stable 0oC. Accordingly， there is no conductive heat flux at the air-ice interface and the inner ice. The melting of the bottom surface of the ice cover is caused by the turbulent heat exchange between ice and water， and the heat balance equation at the bottom of the ice cover can be written as

in which，qwiis the turbulent heat exchange between ice and water，ρiis the ice density，Liis the latent heat of the ice melting， and dh/dtis the rate of the ice thickness variation.

The melting rate of the bottom surface of the ice cover is related to the water temperature gradient in the thermal boundary layer［17］. The plots of the temperature in Fig.6 show a thermal boundary layer with a sharp temperature gradient close to the ice cover. The heat flux from the water to the ice cover in the boundary layer contributes to the heat flux for the melting of the bottom surface of the ice cover. The heat flux from the water is determined by the temperature gradient at the ice-water interface， which is expressed by the Fourier's law［18，19］

in which，kwis the thermal conductivity of water，?T/?zis the temperature gradient.

Fig.6 Vertical temperature profiles under various velocity conditions

Many processes influence the millimeter thick thermal boundary， and it is difficult to accurately measure the temperature gradient at the ice-water interface.

The turbulent heat exchange between ice and water can also be expressed by the Newton's law of cooling［20］

Fig.7 Definition of the heat flux

Solving Eq.（4） and Eq.（6） for hwi， we have

Solving Eq.（4） and Eq.（5） for ?z， we have

in which，?his the ice thickness variation，?tis the duration of each experiment， and

The calculated ice-water heat transfer coefficients （Table 2） show that the relative error between the coefficients under various conditions and their average value is in the range of -14%-16%， and the standard deviation of the coefficients under each condition is 51.2， 17.7， 21.7 and 10.2， respectively. The calculation error mainly results from the flow water temperature and the instability of the flow field. Besides， this error can be caused by the ice thickness variation measurement. The average values of the velocity， the ice thickness variation and the vertical temperature in experiment runs are used herein for error reduction.

The calculated thickness of the thermal boundary layer ?z（Table 2） shows that the thickest thermal boundary layer is less than 0.006 m， and it has an inverse relationship with the velocity.

Table 2 Comparison of calculated results

Fig.8 Linear fitting for the average ice-water heat transfer coefficient and the flow velocity

2.2 Correlation analysis

The influence factors of the heat transfer coefficient include the flow velocity， the salinity， the specific heat capacity， and the density［21］. In this study， the inflow temperature and the flow velocity are variables，and the velocity is the governing factor. Figure 8 shows a linear relationship between the average value of the ice-water heat transfer coefficient under each condition and its flow velocity.

The result shows a positive linear correlation between the average ice-water heat transfer coefficient and the flow velocity beneath the ice cover. The regression coefficient is 0.9982 and the regression equation is

in which，hwiis the ice-water heat transfer coefficient，uwis the depth-averaged velocity.

2.3 Comparison of the dimensionless ice-water heat transfer coefficients

The bulk formula mentioned above provides a method to determine the dimensionless ice-water heat transfer coefficient Ch. The bulk formula and the empirical formula Eq.（9） describe the same heat transfer process at the ice-water interface. From Eq.（1），Eq.（6） and Eq.（9）， the dimensionless ice-water heat transfer coefficient is obtained as

The comparisons of Chbetween this study and previous researches［13］are shown in Table 3. The value ofChin this paper is 1.1×10-3， which is between the maximum value （3.8×10-3） and the minimum value（0.16×10-3）， and similar to theChcalculated by Hamblin and Carmack［14］. The variation of Chin Table 3 may be the results of the hydrodynamics conditions， the properties of water body and the roughness of the ice cover.

Table 3 Comparison of Chbetween this study and pre-

3. Conclusion

In this study， the flume experiment is carried out to determine the ice-water heat transfer coefficient under low flow velocity conditions. Based on the flume data and data analyses， a positive linear correlation between the ice-water heat transfer coefficient and the flow velocity beneath the ice cover is established and an empirical formulais obtained. This empirical formula provides a convenient way to estimate the ice-water heat transfer of reservoirs， lakes and other freshwater bodies when the flow velocity under the ice cover is in the range of 0.024 m/s-0.11 m/s. However， there are still some important issues that should be further studied， such as the icewater heat transfer coefficient under extremely low flow conditions and the verification of this empirical formula in field work.

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10.1016/S1001-6058（16）60664-9

October 16， 2014， Revised April 4， 2015）

* Project supported by the National Natural Science Foundation of China （Grant Nos. 51309169， 51179112）.

Biography: Nan LI （1987-）， Male， Ph. D. Candidate

You-cai TUO，

E-mail: tuoyoucai@scu.edu.cn

2016，28（4）:603-609