Analytical determination of the soil temperature distribution and freezing front position for linear arrangement of freezing pipes using the undetermined coefficient method

doi:10.1016/j.coldregions.2021.103253

Cold Regions Science and Technology

Volume 185, May 2021, 103253

https://doi.org/10.1016/j.coldregions.2021.103253 Get rights and content

Highlights

•
A novel analytical method—the undetermined coefficient method (UCM)—is proposed.
•
A model test about different seepage condition was carried out.
•
The UCM can be applied to real large-scale artificial ground freezing operations.
•
The optimal measurement point positions in UCM were determined.

Abstract

When on-site tests are performed in artificial ground freezing projects, the conventional analytical method for determining the soil temperature field often presents significant error under certain conditions, particularly when the temperature-measuring pipe is far removed from the freezing front. This paper proposes a new method—called the undetermined coefficient method (UCM)—that more accurately predicts the actual soil temperature field and position of the freezing front. The UCM comprises a new mathematical formula developed based on a series of model tests performed under three different conditions—no seepage, constant seepage, and sudden seepage—and a multiple data regression analysis-based method for determining the coefficients in the developed formula. The proposed UCM was compared to a conventional analytical method based on the results from model tests and from the construction of the cross-channel of Guangzhou Metro Line 11. The results indicated that the UCM presented higher stability and practical value than conventional alternatives and could accurately predict the actual soil temperature field and position of the freezing front. Therefore, the UCM is expected to be of high practical value in artificial ground-freezing operations.

Introduction

Artificial ground freezing (AGF) is an efficient ground stabilization method that is widely used in mines, tunnels, and other structures (Alzoubi et al., 2019; Schmall and Braun, 2006; Gallardo and Marui, 2016; Hu et al., 2018a, Hu et al., 2018b, Hu et al., 2018c; Russo et al., 2015). In AGF projects, the thickness of the frozen curtain determines the bearing capacity of the frozen soil and the average temperature determines its strength, thus, they are the two most important parameters. As these two parameters are obtained from the temperature field, calculation of the soil temperature field is the basis for AGF projects (Hu et al., 2019; Xiangdong, 2010).

The evolution of the soil temperature field is a transient process influenced by the coolant temperature, flow rate, soil properties, and size of the freezing pipe. However, at a given time, the calculation of the temperature field can be regarded as a steady-state problem and can be described by an analytical solution that does not contain a time-dependent term. Since the middle of the 20th century, many scholars have conducted research on the determination of the soil temperature field. In Russia, Trupak (1954) proposed an analytical solution for the steady-state temperature field of a single freezing pipe that ignored the influence of nearby freezing pipes. Bakholdin (1963) proposed a new analytical solution by assuming a linear freezing front. Sanger and Sayles (1979) divided the freezing process into three stages and proposed the form of the temperature distribution in each stage. In the last two decades, Hu et al., 2008, Hu et al., 2017, Hu et al., 2018a, Hu et al., 2018b, Hu et al., 2018c optimized a series of analytical solutions based on Bakholdin's analytical solution by using the thermal potential superposition method. Currently, the Bakholdin and Hu analytical solution is widely used in China.

However, in many projects, the result predicted by the above methods often differs from the actual conditions, especially in some conventional AGF projects. In this study, a new method was developed and model tests performed to validate its analytical temperature field solution under different working conditions, including non-seepage and seepage conditions. The developed method was used to calculate the freezing front in a cross-channel of Guangzhou Metro, and the optimal design scheme for the temperature measurement point was obtained based on numerical simulation results.

Section snippets

Theory

At present, the most commonly used analytical temperature field solutions are based on Bakholdin's and Hu's formula (Hu et al., 2017, Hu et al., 2019). Both Bakholdin's solution and Hu's solution calculate the thickness of the frozen curtain using the following formulaes: $T (x, y) = \frac{T_{CT}}{ln \frac{l}{2 π r_{0}} + \frac{πξ}{l}} \{\frac{πξ}{l} - m_{1} (x, y)\}$ $m_{1} (x, y) = \frac{1}{2} ln [2 (ch \frac{2 π}{l} y - cos \frac{2 π}{l} x)]$ $ξ = \frac{l}{π} \frac{T (x, y) ln \frac{l}{2 π r_{0}} + T_{CT} m_{1} (x, y)}{T_{CT} - T (x, y)}$ where T_CTis the temperature of the freezing pipe surface, ξ is the thickness of the frozen curtain, r₀ is the radius of the freezing pipe, x and y

Similarity of the temperature field

According to dimensional analysis, the governing equation of the temperature field in dimensionless form is (Liu et al., 2018; Cai et al., 2019): $F (\frac{at}{r^{2}}, \frac{L}{cT}, \frac{r_{0}}{r}, \frac{T_{d}}{T_{0}}, \frac{T_{c}}{T_{0}}) = 0$ where a is the thermal diffusivity of the soil, t denotes time, r is the distance to the center of the freezing pipe, L is the latent heat of the soil, c is the specific heat of the soil, T denotes temperature, r₀ is the outer radius of the freezing pipe, T_d is the freezing temperature of the soil, Tc is the temperature of the

Ground freezing without seepage (Test 1)

Fig. 4 shows the obtained temperature field at different times. Over time, the frozen curtain continued to expand in the y-direction; its expansion in the x-direction was smaller than that in the y-direction. In the whole process, the temperature field was symmetrically distributed along 1 Introduction, 3 Model test. Further, the isotherm between two Section 2 approximated an elliptic curve in each side of Section 3.

Fig. 5 shows the temperature distribution on the X = 45 mm section (freezing

Feasibility and accuracy of the UCM

To confirm the possibility of determining the soil temperature field using the UCM, ten random sets of temperature data were extracted from Tests 1, 2, and 3; half of them were used to obtain the formula coefficients and the remaining data were used to calculate the error. For the same time value, the calculation results of the UCM were compared with those of the Bakholdin solution, which is the most commonly used method for AGF projects in China (see Eq. (1)). The results are listed in Table 5

Conclusions

This paper analyzed the result errors of a conventional analytical solution of the soil temperature field in AGF and highlighted that, when the conventional analytical solution uses a single temperature value for the calculations, the result error is increased substantially owing to the inappropriate location of the measurement point. Based on this phenomenon, the idea of using the undetermined coefficient method to determine the temperature field was proposed herein. Model tests were used to

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Acknowledgments

Funding: This work was supported by the State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University [grant numbers ZZ2020-07]; the Special Funds for Scientific and Technological Innovation of Tiandi Science and Technology Co., Ltd. [grant numbers 2019-TD-QN009] and National Nature Science Foundation of China [grant numbers 51804157].

References (18)

M.A. Alzoubi et al.
On the concept of the freezing-on-demand (FoD) in artificial ground freezing for long-term applications
Int. J. Heat Mass Transf.
(2019)
H. Cai et al.
Model test and numerical simulation of frost heave during twin-tunnel construction using artificial ground-freezing technique
Comput. Geotech.
(2019)
A.H. Gallardo et al.
The aftermath of the Fukushima nuclear accident: measures to contain groundwater contamination
Sci. Total Environ.
(2016)
X.D. Hu et al.
Analytical solution to steady-state temperature field for straight-row-piped freezing based on superposition of thermal potential
Appl. Therm. Eng.
(2017)
X. Hu et al.
Analytical solution to steady-state temperature field with typical freezing tube layout employed in freeze-sealing pipe roof method
Tunn. Undergr. Space Technol.
(2018)
X.D. Hu et al.
Analytical solution to temperature distribution in frozen soil wall with wavy boundaries by single-row-and double-row-piped freezing
Cold Reg. Sci. Technol.
(2018)
X.D. Hu et al.
Mathematical solution of steady-state temperature field of circular frozen wall by single-circle-piped freezing
Cold Reg. Sci. Technol.
(2018)
X. Hu et al.
Calculation of average temperature of frozen soil wall by single row-piped freezing
J. China Coal Soc.
(2019)
S. Huang et al.
Study on the influence of water flow on temperature around freeze pipes and its distribution optimization during artificial ground freezing
Appl. Therm. Eng.
(2018)

There are more references available in the full text version of this article.

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View full text

Analytical determination of the soil temperature distribution and freezing front position for linear arrangement of freezing pipes using the undetermined coefficient method

Highlights

Abstract

Introduction

Section snippets

Theory

Similarity of the temperature field

Ground freezing without seepage (Test 1)

Feasibility and accuracy of the UCM

Conclusions

Declaration of Competing Interest

Acknowledgments

Int. J. Heat Mass Transf.

Comput. Geotech.

Sci. Total Environ.

Appl. Therm. Eng.

Tunn. Undergr. Space Technol.

Cold Reg. Sci. Technol.

Cold Reg. Sci. Technol.

J. China Coal Soc.

Appl. Therm. Eng.