Generalized analytical solution to steady-state temperature field of double-circle-piped freezing
Section snippets
Notation
k thermal conductivity of the ground qx density of the heat flux in unit time in the direction of x qy density of the heat flux in unit time in the direction of y Z object plane x, y coordinate in Z-plane r polar radius θ polar angle n1, n2 number of inner freezing pipes and outer freezing pipes β rotation angle r0 radius of freezing pipe R1 polar radius of the inner boundary of frozen soil wall R2 polar radius of the inner freezing pipes circle R3 polar radius of the outer freezing pipes circle R4 polar
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).
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