Analytical solution and numerical simulation of the liquid nitrogen freezing-temperature field of a single pipe

Artificial liquid nitrogen freezing technology is widely used in urban underground engineering due to its technical advantages, such as simple freezing system, high freezing speed, low freezing temperature, high strength of frozen soil, and absence of pollution. However, technical difficulties such as undefined range of liquid nitrogen freezing and thickness of frozen wall gradually emerge during the application process. Thus, the analytical solution of the freezing-temperature field of a single pipe is established considering the freezing temperature of soil and the constant temperature of freezing pipe wall. This solution is then applied in a liquid nitrogen freezing project. Calculation results show that the radius of freezing front of liquid nitrogen is proportional to the square root of freezing time. The radius of the freezing front also decreases with decreased the freezing temperature, and the temperature gradient of soil decreases with increased distance from the freezing pipe. The radius of cooling zone in the unfrozen area is approximately four times the radius of the freezing front. Meanwhile, the numerical simulation of the liquid nitrogen freezing-temperature field of a single pipe is conducted using the Abaqus finite-element program. Results show that the numerical simulation of soil temperature distribution law well agrees with the analytical solution, further verifies the reliability of the established analytical solution of the liquid nitrogen freezing-temperature field of a single pipe.Artificial liquid nitrogen freezing technology is widely used in urban underground engineering due to its technical advantages, such as simple freezing system, high freezing speed, low freezing temperature, high strength of frozen soil, and absence of pollution. However, technical difficulties such as undefined range of liquid nitrogen freezing and thickness of frozen wall gradually emerge during the application process. Thus, the analytical solution of the freezing-temperature field of a single pipe is established considering the freezing temperature of soil and the constant temperature of freezing pipe wall. This solution is then applied in a liquid nitrogen freezing project. Calculation results show that the radius of freezing front of liquid nitrogen is proportional to the square root of freezing time. The radius of the freezing front also decreases with decreased the freezing temperature, and the temperature gradient of soil decreases with increased distance from the freezing pipe. The radius of cool...


I. INTRODUCTION
Artificial ground freezing is a special ground-reinforcement technology that utilizes artificial refrigeration technology to circulate cryogenic refrigerants in freezing pipes set in the ground to absorb formation heat, converting soil moisture into ice in the ground, turning the natural rock into frozen soil, and greatly enhancing the strength and stability of ground.At the same time, a continuous frozen wall is formed around the underground engineering to resist soil pressure and isolate the connection between groundwater and underground engineering and to carry out the construction of underground project under the protection of frozen wall.Artificial ground freezing has been widely used in urban underground engineering because of its dual effect of ground reinforcement and waterproof sealing (Li et al., 2006;Pimentel et al., 2012;Afshani & Akagi, 2015;Russo et al., 2015).
Ammonia-brine circulating refrigeration technology is widely used in artificial ground freezing, whereas the liquid nitrogen refrigeration technology has started relatively late.In the 1960s, countries, such as the United States, Britain, Japan, and the former Soviet Union, applied liquid nitrogen freezing technology to urban underground engineering (Veranneman & Rebhan, 1979;Gallavresi, 1981).
Compared with ammonia-brine freezing, liquid nitrogen freezing has advantages of simple freezing system (solely inserting the freezing pipe into the ground without installing a freezing station on the construction site), high freezing speed (nearly 10 times of conventional ammonia-brine freezing speed), low freezing wall temperature, high strength of frozen soil, and no pollution.However, in the process of liquid nitrogen ground freezing in ground, many technical difficulties have emerged gradually, such as the uneven distribution of the freezing-temperature field, undefined range of the liquid nitrogen freezing, and thickness of frozen wall (Stoss & Valk, 1979;Weng, 1994).
The artificial freezing-temperature field is a transient heat-conduction problem with phase transition, moving boundary, and internal heat source.Its accurate solution process is relatively complicated.The freezing-temperature field of a single pipe is the basis of study on the artificial freezing-temperature field.By simplifying the one freezing pipe as the line heat source, some scholars have discussed analytical solutions of freezing-temperature field of a single pipe.For example, Lunardini & Varotta (1981), Li & Xia (2004), Jiang et al. (2009), and Wang et al. (2014) proposed analytical solutions of freezing-temperature field of a single pipe under the condition of constant heat flow boundary.Jiang et al. (2010) suggested an analytical solution of freezing-temperature field of a single pipe considering the constant temperature of freezing pipe wall.Zhou & Zhou (2012) established an analytical solution of freezing-temperature field of a single pipe considering the existence of certain unfrozen water in frozen soil.However, the above-mentioned analytical solutions are only used to analyze the freezing-temperature field of soil under brine freezing, and most of the aforementioned analytical solutions consider the freezing temperature of soil to be 0 • C.However, in practice, the water in soil is influenced by the surface energy, the solute, and the ground pressure; thus, the freezing temperature is mostly below 0 • C. For fine grained soil (such as silt and clay), the freezing temperature can even reach −5 • C; existing studies show that the freezing temperature of soil exerts a certain influence on freezing-temperature field (Watanabe et al., 2002;Lackner et al., 2005;Hu et al., 2008).
Therefore, in this study, the analytical solution of liquid nitrogen freezing-temperature field of a single pipe is established considering the freezing temperature of soil and the constant temperature of freezing pipe wall, and the numerical simulation method is used to verify the analytical solution.

A. Analytical solution of freezing-temperature field of a single pipe
For the freezing-temperature field of a single pipe, as shown in Fig. 1, the temperature field is divided into frozen and unfrozen areas according to the freezing front.The temperature of the soil in FIG. 1. Schematic of single-pipe freezing.
AIP Advances 8, 055119 (2018) the frozen area is T f and the temperature of the soil in the unfrozen area is T u , both of which are a function of time t and radial coordinate r.The radius of the freezing front is R(t), and soil temperature at the freezing front is the freezing temperature of soil T d , the initial temperature of soil is T 0 , and the outer radius of freezing pipe is R 0 .The freezing pipe wall temperature is assumed to be constant, and the value is T c .For this 2D heat-conduction problem, partial differential equations of heat conduction of the frozen and unfrozen areas can be written as follows: where α f and α u the thermal diffusivity of frozen and unfrozen soil in m 2 /d.The following condition exists: where k f and k u the thermal conductivity of frozen soil and unfrozen soil in kcal/(m•d• • C), c f and c u the specific heat of frozen and unfrozen soil in kcal/(kg• • C), ρ f and ρ u are the saturation density of frozen and unfrozen soil in kg/m 3 .The initial condition of differential equations at t = 0 is The boundary conditions of differential equations at r = R 0 , r = R(t), and r = ∞ are The heat balance equation at the phase-change boundary is where L is the soil latent heat of per unit volume in kcal/m 3 and is given by where L w is the unit mass of water latent heat of phase change with a value of 79.6 kcal/kg, ρ d is the dry density of soil in kg/m 3 , w 0 is the initial water content of soil, and w u is the unfrozen water content in frozen soil.
To solve the differential equations of the frozen and unfrozen areas, variables x f and x u are introduced.Thus, Substituting Equation (10) into Equations ( 1) and (2) yields The solution of Equations ( 11) and ( 12) can be used to obtain where A, B, C, and D are the undetermined constants and E i (x) is the exponential integral function.
The following condition exists: Substituting Equation (13) into Equations ( 5) and ( 6) yields Substituting Equation ( 14) into Equations ( 6) and ( 7) yields Then, the distribution law of temperature field in the frozen and unfrozen areas can be obtained as Substituting Equations ( 20) and ( 21) into Equation ( 8) yields where β is the undetermined constant.The following condition exists: Equation ( 23) shows that the radius of the freezing front is proportional to the square root of the freezing time.
For a single-pipe freezing problem, when the thermal physical parameters of frozen and unfrozen soil are determined under the condition of constant temperature of freezing pipe wall, the radius of freezing front can be solved by Equations ( 22) and ( 23), and the distribution of freezing-temperature field can be calculated by Equations ( 20) and ( 21).

B. Case calculation
Referring to the construction of cross passages for shield tunnels by means of liquid nitrogen freezing method (Fang et al., 2010), the diameter of a single freezing pipe is 108 mm and the outer radius of freezing pipe R 0 = 54 mm.Assuming a constant wall temperature of freezing pipe, T c = −80 • C, and the initial temperature of soil T 0 = 20 • C. The values of thermal physical parameters of soil are shown in Tables I-II.
In accordance with Equation ( 9), the latent heat of phase change for a unit volume of soil is calculated as follows: The freezing temperatures of soil T d are 0 • C, −1 • C, and −2 • C, and the thermal physical parameters of soil are substituted into Equations ( 21) and ( 22).Maple math software is used to compile the calculation program.The variation in the radius of freezing front with the freezing time is shown in Fig. 2.
As shown in Fig. 2, the freezing front exhibits a fast expansion in the initial stage of liquid nitrogen freezing.The freezing front expansion rate slows down with the prolongation of freezing time, and the radius of the freezing front is proportional to the square root of the freezing time.
At the same time, the radius of the freezing front decreases with decreased freezing temperature.For example, after freezing for 2 days, the radius of the freezing front is 300.1 mm when the freezing temperature is 0 • C. When the freezing temperature is −2 • C, the radius of the freezing front is 289.5 mm.The difference between the two radii is 10.6 mm.With prolonged freezing time, the FIG. 2. Variation in the radius of freezing front with the freezing time.
AIP Advances 8, 055119 ( 2018) difference in the radius of the freezing front caused by different freezing temperatures increases.For example, after 10 days, the freezing front radii of 0 • C and −2 • C are 552.5 and 528.3 mm, respectively.The difference between the two radii is 24.2 mm.For a single-pipe liquid nitrogen freezing problem, the freezing temperature is −1 • C when the temperature of the pipe wall is constant at −80 • C.After freezing for 2 days, the radius of the freezing front is 294.8 mm because the outer radius of the freezing pipe is 54 mm.The thickness of frozen soil formed is 240.8 mm, and the growth velocity of frozen soil is 120.4 mm/day.After freezing for 6 days, the radius of the freezing front is 444.3 mm, the thickness of frozen soil is 390.3 mm, and the growth velocity of frozen soil is 65.05 mm/day.After freezing for 10 days, the radius of the freezing front is 540.3 mm, the thickness of frozen soil is 486.3 mm, and the growth velocity of frozen soil is 48.63 mm/day.
When the freezing temperature of soil is −1 • C, the freezing time is taken by t = 2, 4, 6, 8, 10 days.The thermal physical parameters of soil and the calculated radius of the freezing front are substituted into Equations ( 19) and ( 20), and the distribution law of freezing-temperature field of soil under the condition of the freezing temperature of −1 • C as shown in Fig. 3.
As shown in Fig. 3, the soil temperature in frozen and unfrozen areas decreases gradually with prolonged freezing time.At different freezing times, the distribution law of freezing-temperature field of soil is similar, the temperature gradient near freezing pipe is large, and the temperature gradient away from freezing pipe is small.In the unfrozen area, the temperature gradient gradually decreases and reaches zero with increased distance from the freezing pipe, that is, the soil temperature returns to the initial temperature of 20 • C.
The radius of the cooling zone in the unfrozen area is around four times the radius of the freeing front.For example, after freezing for 6 days, the radius of the freezing front is 444.3 mm and the radius of cooling area is approximately 1.8 m.That is, the soil temperature of the radial coordinate r of 1.8 m is 20 • C, which is not affected by the freezing of liquid nitrogen.

A. Finite-element calculation model
Considering the influence range of liquid nitrogen freezing, the finite-element calculation model of single-pipe freezing is established as shown in Fig. 4 by using ABAQUS finite-element program.The radius of the calculation model is 4 m, the model is divided into 1000 units, and the unit type selects the 4-node heat transfer physical unit DC2D4.
In this numerical model, the thermal physical parameters of soil such as the density, thermal conductivity, specific heat, and latent heat are listed in Tables I-II.The phase-change temperature of soil is taken as [−1 • C, 0 • C], where −1 • C is the freezing temperature of soil T d and 0 • C is the melting temperature of soil T r , which are defined as Solidus Temp and Liquidus Temp in the ABAQUS program (Song et al., 2016).The initial temperature of soil T 0 is 20 • C, and the temperature of freezing pipe wall is −80 • C.

B. Calculation results and analysis
Fig. 5 provides a numerical simulation of the freezing frontal radius over time and is compared with the analytic solution.The distribution law of temperature field of soil after freezing for 2 and 10 days by numerical simulation is shown in Fig. 6.
Fig. 5 shows that, at the same freezing time, the radius of the freezing front obtained by numerical simulation is slightly larger than that of analytical solution.For the single-pipe liquid nitrogen freezing problem, when the pipe wall temperature is constant at −80 • C, the freezing temperature is taken as −1 • C.After freezing for 2 days, the radius of the freezing front is obtained by numerical simulation calculation as 311 mm, which is slightly larger than the analytical solution (294.8 mm) by 16.2 mm.After freezing for 10 days, the radius of the freezing front is 566.1 mm, which is slightly larger than the analytic solution (540.3 mm) by 25.8 mm.
This result is due to the fact that in the numerical simulation, the element shape and size, the phase-change temperature range, and other factors affect the calculation results to some extent.However, the variation in freezing frontal radius with time calculated by numerical simulation is grossly consistent with that by analytical solution.
After freezing for 2 and 10 days, the distribution law of soil temperature calculated by numerical simulation and analytical solution are shown in Figs.7 and 8, respectively.Figs.7 and 8 show that the calculated results of numerical simulation and analytical solution are in good agreement with each other, and the reliability of the analytical solution is verified.

IV. CONCLUSIONS
(1) The analytical solution of freezing-temperature field of a single pipe is established on the basis of the condition of constant temperature of freezing pipe wall and considering the freezing temperature of soil, and the formulas of the radius of freezing front and temperature field distribution of frozen and unfrozen areas are obtained.The analytical solution is applied in liquid nitrogen freezing project.The calculation results show that the radius of liquid nitrogen freezing front is proportional to the square root of the freezing time.At the same time, the radius of the freezing front decreases with decreased freezing temperature.For the liquid nitrogen freezing problem of a single pipe, when the pipe wall temperature is constant at −80 • C, the freezing temperature is taken as −1 • C, the average growth velocity of frozen soil after freezing for 2 days is 120.4 mm/day, and the average growth velocity of frozen soil after freezing for 10 days is 48.63 mm/day.The temperature gradient of soil decreases with increased distance from the freezing pipe.The radius of the cooling zone in the unfrozen area is approximately four times the radius of the freezing front.
(2) The finite-element calculation model of single-pipe freezing is established using ABAQUS finite-element program in consideration of the influence range of liquid nitrogen freezing.The numerical simulation of freezing-temperature field distribution of single-pipe liquid nitrogen under the condition of the pipe wall temperature of −80 • C and the freezing temperature of −1 • C is studied.The results show that, at the same freezing time, the radius of the freezing front obtained by the numerical simulation is slightly larger than that of the analytical solution.The numerical simulation of soil temperature distribution is in good agreement with the analytical solution.This agreement further verifies the reliability of the established analytical solution of liquid nitrogen freezing-temperature field of a single pipe.

TABLE I .
Physical parameters of soil.