Average temperature calculation for straight single-row-piped frozen ...

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Sciences in Cold and Arid Regions 2011, 3(2): 0124–0131 DOI: 10.3724/SP.J.1226.2011.00124

Average temperature calculation for straight single-row-piped frozen soil wall XiangDong Hu 1, 2*, SiYuan She 1, RuiZhi Yu 1 1. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China 2. Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China *Correspondence to: Dr. XiangDong Hu, Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China. Tel: +86-21-65988771; Email: [email protected] Received: 19 September 2010

Accepted: 24 November 2010

ABSTRACT The average temperature of frozen soil wall is an essential parameter in the process of design, construction, and safety management of artificial ground freezing engineering. It is the basis of calculating frozen soil’s mechanical parameters, further prediction of bearing capacity and, ultimately, safety evaluation of the frozen soil wall. Regarding the average temperature of single-row-piped frozen soil wall, this paper summarizes several current calculation methods and their shortcomings. Furthermore, on the basis of Bakholdin’s analytical solution for the temperature field under straight single-row-piped freezing, two new calculation models, namely, the equivalent trapezoid model and the equivalent triangle model, are proposed. These two approaches are used to calculate the average temperature of a certain cross section which indicates the condition of the whole frozen soil wall. Considering the possible parameter range according to the freezing pipe layout that might be applied in actual construction, this paper compares the average temperatures of frozen soil walls obtained by the equivalent trapezoid method and the equivalent triangle method with that obtained by numerical integration of Bakholdin’s analytical solution. The results show that the discrepancies are extremely small and these two new approaches are better than currently prevailing methods. However, the equivalent triangle method boasts higher accuracy and a simpler formula compared with the equivalent trapezoid method. Keywords: artificial ground freezing; single-row-piped freezing; frozen soil wall; average temperature; equivalent trapezoid method; equivalent triangle method; Bakholdin’s solution

1. Introduction Application of artificial ground freezing is becoming more common in mine construction, tunneling, and municipal engineering. In the design of freezing schemes, construction management of ground freezing, and safety evaluation, the average temperature of the frozen soil wall is a crucial fundamental parameter. It is the basis on which the mechanical parameters and bearing capacity of frozen soil can be determined and further evaluation of the safety of the frozen soil wall can thus be made. In artificial ground freezing projects, double-row-pipe and multi-row-pipe are often used to freeze the soil, but sin-

gle-row-pipe freezing is the most widely used method. The study of temperature field and average temperature under single-row-pipe freezing is an essential part of artificial ground freezing theory. Calculation of the average temperature of frozen soil wall under single-row-pipe freezing has been the subject of intense interest, and a series of achievements have been already made (Trupak, 1954; Are, 1959; Bakholdin, 1963; Stepanova, 1966; Kractoshevsky, 1967a,b; Serebrianik, 1968, 1969). Sanger and Sayles from the United States (Sanger, 1968; Sanger and Sayles, 1979) developed a computation of temperature field produced by single-row-pipe freezing. The Japan Construction Mechanization Association (1978) summarized the calculation

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method of average temperature of frozen soil wall under single-row-pipe freezing, and Japanese scholars Tobe and Akimoto (1979) examined the temperature field of multi-pipe straight frozen soil wall, based on which Kato et al. (2007) proposed an approach to calculate the average temperature of frozen soil wall. Chinese scholars Chen and Tang (1982) developed their method of calculating the average temperature, namely, the Chengbing formula. However, due to the complexity of the temperature field of frozen soil wall and the limitations of conditions these researchers have considered, the average temperatures under single-row-pipe freezing obtained by a variety of calculation methods are quite different and have diverse degrees of error (Hu et al., 2008a; He and Hu, 2009). To analyze the accuracy of common calculations used in obtaining the average frozen soil wall temperature under single-row-pipe freezing, a more accurate solution is proposed in this paper. 2. Prevailing solutions of average temperature under single-row-pipe freezing The interpretation of average temperature of frozen soil wall is not universal. In some situations this represents the

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average of temperature distributions of a certain cross section along the thickness of frozen soil wall, while in other cases it represents the average temperature of the whole frozen soil wall. The value of the former may vary with the relative position of the calculation section and the freezing pipes (e.g., the "main section" or the "intersection section"). In mechanical calculation where the frozen soil wall is regarded as homogeneous material, the latter definition is usually adopted, and the average temperature of a typical position on the frozen soil wall can be used to epitomize the whole condition within the scope of permissible error. 2.1. Trupak’s solution Trupak (1954) saw the frozen soil wall under single-row-pipe freezing as geometrically connected frozen columns generated by multiple freezing pipes, without considering the interaction between the freezing pipes. Therefore, his temperature field model bears considerable error (Bakholdin, 1963; Hu et al., 2008a). Trupak regarded the arithmetic average value of the average temperatures at the main section (Section II-II, Figure 1) and at the intersection section (Section III-III, Figure 1) as that of the whole frozen soil wall.

Figure 1 Calculation scheme of single-row-piped frozen soil wall

The average temperature of the whole frozen soil wall tCP is expressed as (assuming soil freezing temperature is 0 C):

tСР 

t СР.Г  t СР.З 2

(1)

where tCP.Г is the average temperature of the main section:

tСР.Г 

tСТ

  r0  ln

  r0 ln   r0 ln r0  r0  

(2)

r0

and tCP.З is the average temperature of the intersection section:

tСР.З  tСТ

ln

2 l

2 ln



(3)

r0

As shown in Figure 1, tCT, l,  , and r0 are the temperature on the surface of the freezing pipe, the spacing of pipes in the same row, the distance from the freezing pipe to the outer surface of frozen soil wall (i.e., the flange extension of the frozen soil wall), and the outer radius of the freezing pipe, respectively.

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Due to the error of Trupak’s temperature field (Bakholdin, 1963; Hu et al., 2008a), this calculation method did not yield favorable accuracy (shown in Figure 7 and Table 1).



2.2. Bakholdin’s solution Bakholdin (1963) proposed an analytical solution for the temperature field of single-row-piped and double-row-piped frozen soil wall after the closure of ice columns. The solution for single-row-pipe freezing (assuming soil freezing temperature is 0 C) is as follows:

t  x, y  

tСТ     m1 ( x, y )   l   l   ln l 2r0

(4)

tК  tСТ

l ln

 ln 2

l

2r0





(6)

l

The temperatures of the intersection section are highest within the whole frozen soil wall. Hence, the approximation for the average temperature, called Bakholdin equivalent triangle method, is too conservative (as shown in Figure 7 and Table 1). 2.3. Chengbing formula

where

m1 ( x , y ) 

tion, and tK is the temperature at the intersection of the intersection section and the axial section (Section I-I, Figure 1), i.e., t(l/2,0):

On the basis of measured data of a great number of freezing constructions of shafts, Chen and Tang (1982) summarized the regression analysis and obtained the following formula for the average temperature of the frozen soil wall:

1   2 2   ln  2  ch y  cos x  2   l l 

Bakholdin’s temperature field of single-row-piped frozen soil wall is depicted in Figure 2. Due to the complexity of the formula, Bakholdin did not give the analytical solution of average temperature of frozen soil wall. He suggested using the average temperature of the intersection section to represent the condition of the whole freezing soil wall, and the average temperature of the intersection section can be approximately obtained by the triangularly distributed temperature field thereof (as shown in Figure 5):

 0.875 l  tСР  tСТ 1.135  0.352 l  3  0.266   0.466 E  E  (7) where E and l are the thickness of the frozen soil wall and the spacing of pipes in the same row, respectively. The average temperature of the frozen soil wall obtained by the Chengbing formula has a systematic discrepancy compared to the actual situation. For example, the actual average temperature of frozen soil decreases with decreasing freezing pipe spacing, whereas the Chengbing formula generates the opposite result (He and Hu, 2009). Therefore, the error of the Chengbing formula is large (as shown in Figure 7 and Table 1), especially when the spacing of the freezing pipes is small. 3. The calculation of average temperature based on Bakholdin’s model for temperature field under single-row-pipe freezing

Figure 2 Bakholdin’s temperature field of single-row-piped frozen soil wall

tСР 

1 tK 2

(5)

where tCP is the average temperature of the intersection sec-

The Bakholdin model for temperature field under single-row-pipe freezing is currently a favorable analytical solution, which has been proven to have excellent accuracy by the Bakholdin Experiment (Bakholdin, 1963), finite element numerical analysis (Hu and Zhao, 2010), and in the artificial ground freezing practices of many projects (Qiu and Hu, 2006; Zhao et al., 2006; Xiao et al., 2007). This paper proposes an average temperature calculation of frozen wall based on Bakholdin’s solution. This section presents the equivalent triangle method and the equivalent trapezoid method to compute the average temperature of the whole frozen soil wall. Bakholdin’s solution is applicable to a freezing pipe arrangement with uni-

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versal spacing, so the temperature distribution is periodic along the x-axis. As a result, the average temperature of the whole frozen soil wall equals that of the soil strip ranging in the interval of 0 ≤ x < l/2 in the coordinate system as shown in Figure 1. Therefore, the average temperature, tCP, can be expressed by the average value of t(x, y) in the interval of 0 ≤ x < l/2, 0 ≤ y < ξ as follows:

tСР

2 2l     t ( x, y )dydx l 0 0

(8)

The expression of t(x, y) itself (Equation (4)) is so complicated that the direct solving of the average temperature calculation formula (Equation (8)) could be extremely difficult. A simple method is proposed below. In all the cross sections, the average temperature of the main section is the lowest and that of the intersection section is the highest. Thus, the average temperature of the whole frozen soil wall should be between the two. It is feasible that within the interval of 0 ≤ x < l/2 we can find a cross section in which the average temperature is equal to that of the whole frozen soil wall. The temperature curve in the cross sections resembles a triangle or trapezoid shape; therefore, for the convenience of calculation and application, the equivalent trapezoid or triangle is adopted to calculate the average temperature of the equivalent cross section. 3.1. Equivalent trapezoid method

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curves t(x0, y) on feature sections of x0＝ 0, l/4, l/5, and l/2 show that we can choose curves t(l/4, y) and t(l/5, y) for trial calculation (Figure 3). The area under curve t(l/4, y) or t(l/5, y) is counted as that of the equivalent trapezoid for each (the equivalent trapezoid method). For the dimensions of the equivalent trapezoid, its upper base is 2r0 and its lower base is the total thickness of frozen soil wall (i.e., 2ξ), while the height is the y-axis intercept (tK) of each curve (Figure 3). Therefore, the average temperature of Section x0 is:

tСР 

r0   tК 2

(9)

For section x0 = l/4, tK = t(l/4, 0), the formula is:

 tК  tСT

l ln

 l

2r0

ln 2 2 



(10)

l

For section x0 = l/5, tK = t(l/5, 0), the formula is:

tK 

  1   2 tСТ  ln 21  cos  l   l 2   5  ln 2 r0 l

Investigation of the shape of the frozen soil temperature

Figure 3 Calculation scheme for average temperature by the equivalent trapezoid method

      (11)

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In order to check the accuracy of the above approach, we solved Equations (8) and (9) employing the numerical method. Examining the above formulae, we found that the average temperature is greatly influenced by the relative magnitude of borehole spacing (l) and the flange extension of the frozen soil wall (). Therefore, numerical solutions were checked in the parameter scope of :l = 0.5–1.5 m and l = 0.4–1.4 m, which is reasonable and common for freezing pipe layout in engineering practice. Layout parameters beyond that range could be regarded as irrational or very infrequent in ground freezing practice. The results of tentative calculation trials in section x0 = l/4

and x0 = l/5 are shown in Figure 4, and they describe two extreme cases of parameter scope, namely, :l = 0.5 and :l = 1.5; common situations are between the two. As shown in Figure 4, the average temperature derived by integral calculation following Bakholdin’s formula agrees well with that by the equivalent trapezoid method on Sections x0 = l/4 and x0 = l/5. Compared with the result of Section x0 = l/4, the result of Section x0 = l/5 agrees more favorably with that of the integral calculation. Therefore, section x0 = l/5 is recommended to be the calculative section for the equivalent trapezoid method (refer to Equations (9) and (11)).

Figure 4 Comparison of calculated average temperatures by the equivalent trapezoid method

3.2. Equivalent triangle method Because r0 is smaller than ξ, the equivalent trapezoid closely resembles a triangle. Furthermore, the position of Section x0 = l/5 is close to the freezing pipe where the temperature curve closely resembles a triangle according to the characteristic of the temperature field. Therefore, it is more appropriate to use the equivalent triangle to calculate the average temperature of the equivalent section. As for the dimensions of the equivalent triangle, its lower base is the total thickness of frozen soil wall (i.e., 2ξ), while its height is the y-axis intercept (tK) of each curve (Figure 5). Therefore, the average temperature of Section x0 is:

1 tСР  tK 2

(12)

where tK is the temperature of point y = 0 on Section x0 and can be computed as follows. For Section x0 = l/5, tK = t(l/5, 0), the formula is Equation (11), while for Section x0 = l/6, tK = t(l/6, 0), the formula is:

 tK 

ln

l

l

2r0



 l

tСТ

(13)

Curves t(l/5, y) and t(l/6, y) were chosen for the tentative calculation trial and the result of the calculation is shown in Figure 6. As shown in Figure 6, the average temperature derived by integral calculation following Bakholdin’s formula agrees well with that by the equivalent triangle method on Sections x0 = l/5 and x0 = l/6. Compared with the result of Section x0 = l/5, the result of Section x0 = l/6 agrees more favorably with that of the integral calculation. Therefore, Section x0 = l/6 is recommended to be the calculation section for the equivalent triangle method (refer to Equations (12) and (13)). 4. Discussion 4.1. The accuracy of the recommended methods In Figure 7 the results of currently prevailing methods and our recommended methods are compared. The resulting discrepancies between the new approaches and Bakholdin’s analytical solution are listed in Table 1. As shown in Table 1, the discrepancies of computations developed by Trupak, Bakholdin, and the Chengbing formula used widely in China are much larger than those of our two recommended methods.

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Figure 5 Calculation scheme for average temperature by the equivalent triangle method

Figure 6 Comparison of calculated average temperatures by the equivalent triangle method

Figure 7 Comparison of calculated average temperatures by different methods

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Table 1 Error comparison of the different methods

:l

0.5

1.5

l (m) 0.4 0.6 0.8 1.0 1.2 1.4

Equivalent trapezoid (x0 = l/5) 14.57% 3.75% −1.18% −4.04% −5.82% −7.06%

Equivalent triangle (x0 = l/6) 0.57% −1.99% −2.93% −3.45% −3.68% −3.81%

Bakholdin’s triangle (x0 = l/2) −43.81% −45.24% −45.77% −46.05% −46.18% −46.26%

0.4 0.6 0.8 1.0 1.2 1.4

5.35% 2.14% 0.59% −0.32% −0.93% −1.35%

0.09% −0.22% −0.32% −0.36% −0.39% −0.41%

−14.63% −14.90% −14.98% −15.02% −15.04% −15.06%

Bakholdin only dealt with the case of t0 = 0 °C. Hu et al. (2008b) proposed a modification of the Bakholdin solution, considering the fact that the freezing temperature is usually below 0 °C in actual situations and the modified average temperature field of straight single-row-piped frozen soil wall is:

tСТ  t0      m1 ( x, y )   t0 l   l  ln  l 2r0

(14)

r0    t0 2

(15)

For Section x0 = l/5, tK = t(l/5, 0), the formula is:

tK 

tСТ  t0 l  ln  l 2 r0

  1   2    ln 21  cos    t0  2   5    l

(16) The average temperature (tCP) of the equivalent triangle method can be expressed as:

tСР 

1  t K  t 0   t0 2

(17)

For Section x0 = l/6, tK = t(l/6, 0), the formula is:

 tK 

ln

l

l

2r0





 tСТ  t0   t0

−56.24% −51.42% −48.48% −46.50% −45.02% −43.88%

−98.59% −71.56% −52.24% −38.22% −27.87% −20.27%

−43.60% −45.54% −46.69% −47.49% −48.10% −48.59%

−46.53% −30.17% −20.88% −15.38% −12.19% −10.54%



tCP 

tСТ  t0  t0 l  2 ln  2r0 l l

(19)

Since the case of t0 = 0 °C has been discussed, the accuracy of our conclusions in the case of t0 ≤ 0 °C is apparent. 5. Conclusions

where t0 is the freezing temperature of ground soil, t0 ≤ 0 °C. Hence, the average temperature (tCP) of the equivalent trapezoid method can be expressed as:

tСР  t К  t0 

Chengbing formula

perature of straight single-row-piped frozen soil wall, and it can be expressed as:

4.2. Consideration of freezing temperature depression

t  x, y  

Trupak’s solution

(18)

l

The equivalent triangle formula is finally recommended as the preferred calculation method for the average tem-

The Bakholdin model has favorable agreement with the actual situation. However, accurate solutions based on it can only rely on tedious numerical methods. In this paper, a more convenient and accurate formula is proposed. These calculations can be expressed as follows: (1) Based on Bakholdin’s analytical solution for temperature field under single-row-pipe freezing, the equivalent trapezoid method was proposed to calculate the average temperature of frozen soil wall. Section x0 = l/5 is recommended to be the calculation section for this method, and the average temperature thereof was reckoned as that of the whole frozen soil wall. The bases of the equivalent trapezoid are the diameter of the freezing pipes and the thickness of the frozen soil wall, and the height is the temperature at the intersection of the axis section and the calculation section. (2) Based on Bakholdin’s analytical solution for temperature field under single-row-pipe freezing, the equivalent triangle method was also proposed to calculate the average temperature of frozen soil wall. Section x0 = l/6 is recommended to be the calculation section for this method, and the average temperature thereof was reckoned as that of the whole frozen soil wall. The basis of the equivalent triangle method is the thickness of frozen soil wall, and the height is the temperature at the intersection of the axis section and the calculation section. (3) Taking the range of the freezing pipe layout parame-

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ter that may appear in engineering practice as the calculation scope, numerically computed results of average temperature following the equivalent trapezoid method, the equivalent triangle method, and Bakholdin’s formula were examined. The discrepancies proved to be small enough to meet the accuracy requirements for engineering application, qualifying both the equivalent trapezoid method and the equivalent triangle method as feasible approaches in calculating average temperature for single-row-pipe freezing. (4) However, the equivalent triangle method boasts higher accuracy and a simpler formula compared with the equivalent trapezoid method. (5) The discrepancies among the approaches developed by Trupak, Bakholdin, and Chen/Tang are much larger than that of the equivalent triangle method. Hence, the equivalent triangle method is recommended to calculate the average temperature of straight single-row-piped frozen soil wall. Acknowledgments: The work was supported by the National Natural Science Foundation of China (No. 50578120), and the National High Technology Research and Development Program of China (863 Program) (No. 2006AA11Z118). REFERENCES Are FE, 1959. Determining the average temperature of frozen soil wall. Shaft Construction, (3): 22–23. Bakholdin BV, 1963. Selection of Optimal Ground Freezing Mode for Construction Purposes. Gosstroiizdat, Moscow. Chen WB, Tang ZB, 1982. The average temperature in ice wall and the diameter of frozen circle in Panji Coal Field. Journal of China Coal Society, (1): 46–52. He TX, Hu XD, 2009. Study on adaptability of "Chengbing" formula for average temperature of frozen soil wall. Low Temperature Architecture Technology, 31(5): 77–81. Hu XD, Bai N, Yu F, 2008a. Analysis of Trupak and Bakholdin formulas for temperature field of single-row-pipe frozen soil wall. Journal of Tongji

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University (Natural Science), 36(7): 906–910. Hu XD, Huang F, Bai N, 2008b. Models of artificial frozen temperature field considering soil freezing point. Journal of China University of Mining and Technology, 37(4): 550–555. Hu XD, Zhao JJ, 2010. Research on precision of Bakholdin model for temperature field of artificial ground freezing. Chinese Journal of Underground Space and Engineering, 6(1): 96–101. Japan Construction Mechanization Association, 1978. Ground Freezing: Planning, Designing and Construction. Gihodo Shuppan Co., Ltd., Tokyo. Kato T, Izuta H, Kushida Y, 2007. Bending strength of frozen soil beam with temperature distribution and consideration of average temperature evaluation method. Doboku Gakkai Ronbunshuu F, 63(1): 97–106. Kractoshevsky GM, 1967a. Determining the average temperature of frozen cylinders. Designing and Construction of Coal Mines, (7): 53–55. Kractoshevsky GM, 1967b. Temperature field of cooled zone in ground freezing. Design and Construction of Coal Mines, (12): 59–63. Qiu F, Hu XD, 2006. 3D visualization analysis of the frozen soil curtain for the cross-passage construction in tunnels. In: The 2nd Conference of Geo-Engineering, China. Science Press, Beijing. 79–83. Sanger FJ, 1968. Ground Freezing in Construction. Journal of the Soil Mechanics and Foundations Division, Proceedings of ASCE, 94(SM1): 131–158. Sanger FJ, Sayles FH, 1979. Thermal and rheological computations for artificially frozen ground construction. Engineering Geology, 13(1–4): 311–337. Serebrianik OV, 1968. Determining the average temperature of frozen soil retaining wall. Shaft Construction, (8): 13–16. Serebrianik OV, 1969. Determining the average temperature for designing frozen soil retaining wall. Design and Construction of Coal Mines, (3): 26–28. Stepanova EM, 1966. Calculation of the Average Strength of Frozen Ground Depending on the Scheme and the Temperature of Freezing by Means of Electronic Digital Computers. Skochinsky Institute of Mining, Моscow. Tobe N, Akimoto O, 1979. Temperature distribution formula in frozen soil and its application. Refrigeration, 54(622): 3–11. Trupak NG, 1954. Ground Freezing in Shaft Sinking. Ugletehizdat, Moscow. Xiao ZY, Hu XD, Zhang QH, 2007. Characteristics of temperature field in frozen soil wall with multirow freeze-tubes and limited depth freezing. Chinese Journal of Rock Mechanics and Engineering, 26(S1): 2694–2700. Zhao YH, Hu XD, Zhao GQ, 2006. Experiment on artificial frozen soil boundary ground penetrating radar detection in tunnels. Journal of Tongji University (Natural Science), 34(4): 508–513.