Generalized analytical solution to steady-state temperature field of double-circle-piped freezing

https://doi.org/10.1016/j.coldregions.2020.103076Get rights and content

Highlights

  • Boundary separation method is used to solve steady-state temperature field.

  • The analytical solution to double-circle-piped freezing is derived.

  • The analytical solution shows the consistency with the computational results.

  • The solutions for single-row- and double-row-piped freezing are improved.

  • The applications of analytical solution satisfy the requirements of projects.

Abstract

Temperature field distribution is the basis theory of artificial ground freezing, which is essential for mastering the temperature development of frozen soil wall. In practice, the double-circle-piped arrangements of freezing pipes are mostly applied. Concerning to all analytical solutions to the steady-state temperature field, there is no result of the double-circle-piped freezing. This paper establishes a model for the steady-state temperature field of double-circle-piped freezing. Then, according to conformal mapping and the boundary separation method, this model of the double-row-piped freezing is transformed into two special single-row-piped freezing models. Thereby, the solution to double-circle-piped freezing model is obtained by solving the single-row-piped freezing problem. And the analytical results for some typical double-circle-piped problems show good consistent with the numerical thermal results. Furthermore, the applications for calculating the thickness and average temperature under double-circle-piped condition are derived.

Section snippets

Notation

kthermal conductivity of the ground
qxdensity of the heat flux in unit time in the direction of x
qydensity of the heat flux in unit time in the direction of y
Zobject plane
x, ycoordinate in Z-plane
rpolar radius
θpolar angle
n1, n2number of inner freezing pipes and outer freezing pipes
βrotation angle
r0radius of freezing pipe
R1polar radius of the inner boundary of frozen soil wall
R2polar radius of the inner freezing pipes circle
R3polar radius of the outer freezing pipes circle
R4polar

Steady-state conduction theory

Conduction, convection and radiation are three manifestations of heat transfer in nature. As long as there is no special case where the frozen soil is directly exposed to the air, the influence of the convection and radiation on temperature field is negligible compared to the conduction (Carslaw, 1921). Furthermore, in a real AGF project, the longitudinal length of a freezing pipe is by far greater than its diameter and the temperature gradient along the freezing pipe is gentle, so the original

Mathematical model

For double-circle-piped freezing, the boundary of the frozen soil is wavy-shaped after connection of two adjacent freezing pipes. As the thickness of the frozen soil increases, we assume the boundary of the frozen soil to be circular which approximates to the real case (Bakholdin, 1963). In addition, the freezing temperatures of inner and outer freezing pipes are generally different in engineering applications, which is essential to be considered. The model of double-circle-piped freezing

Validation of the analytical solution

The boundary temperature of the freezing pipes is adopted only in one point in the deduction. In other points of the pipe edge, the surface temperature may not be identical. Such simplification may cause some errors in area close to the freezing pipes. Other simplifications were also used in the above derivation process, so the accuracy of the analytical solution is essential to be verified. Finite element analysis has proved its reliability in the simulation of steady-state temperature fields.

Application of solutions

In practical engineering applications, in order to evaluate the water sealing and strength of frozen soil wall, some temperature parameters, especially the thickness and the average temperature of frozen soil wall are of great significance. For freezing design, these two parameters have always been used to evaluate the freezing effect.

From the previous obtained analytical solution, the thickness and the average temperature of frozen soil can be derived. And we have also proved the reliability

Discussion

The underground water and frozen soil are both very important issues when dealing with engineering problem. There are couples of methods applying to discuss the similar freezing problem. In this study, we present an analytical solution for a commonly seen geometry. Though its validity is proved, there remain some important issues which need to be further discussed in actual application.

Firstly, some assumptions are used to get the analytical solution, which is essential for the analysis

Conclusions

The analytical solution to steady-state temperature field by double-circle-piped freezing has been derived and the conclusions obtained in this paper are as follows:

  • (a)

    The mathematical model of double-circle-piped freezing with unfrozen core has been established. By introducing the conformal mapping, the double-circle-piped freezing problem is transformed into a double-row-piped freezing problem, and according to the boundary separation of the harmonic equation, the double-row-piped freezing

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (No. 51478340), the Natural Science Foundation of Zhejiang Province, China (No. LZ13E080002), and the China Ministry of Communications Construction Science & Technology Projects (No. 2013318 J11300).

References (23)

Cited by (7)

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