JOURNAL OF COAL SCIENCE & ENGINEERING

(CHINA) DOI 10.1007/s12404-012-0104-1

pp 18–24 Vol.18 No.1 Mar. 2012

Analytic computation on the forcible thawing temperature field formed by a single heat transfer pipe with unsteady outer surface temperature

ZHANG Chi1, YANG Wei-hao1, QI Jia-gen2, ZHANG Tao3 1. State Key Laboratory for Geomechanics & Deep Underground Engineering, China University of Mining & Technology,

Xuzhou 221116, China

2. Zhongtianhechuang Energy Corporation Limited, Ordos 017000, China

3. School of Civil Engineering, Yancheng Institute of Technology, Yancheng 224051, China

© The Editorial Office of Journal of Coal Science and Engineering (China) and Springer-Verlag Berlin Heidelberg 2012

Abstract A comprehensive and systematic research on the forcible thawing temperature field formed by a single heat transfer

pipe with unsteady outer surface temperature was carried out by analytic computation according to the theory of similitude. The

distribution law of thawing temperature field, calculation formulas of thawing radius b, heat flux density q and average thawing

temperature T were obtained. It theoretically explains that the main influential factors of thawing radius b, heat flux density q

and thawing average temperature T are K, f, Lλ and ω(f), but Lc affects little. Finally, based on the forcible thawing project of

Hulusu air shaft lining, the field data indicate that the analytical formulas of this article are comparatively accurate.

Keywords analytic computation, forcible thawing, temperature field, theory of similitude, artificial ground freezing

Received: 26 September 2011 Supported by the Natural Science Foundation of China (40801032) Tel: 86-516-83883170, E-mail: pxchi@126.com

Introduction

Artificial freezing technology is a geotechnical re-inforcement technology originating from natural freezing phenomena, which has been more than 140 years of history so far. Freezing engineering univer-sally adopts low temperature salt solution as refriger-ant, burying freezing pipes in soil body for absorbing heat, freezing soil, stabilizing soil body and preventing groundwater fast. Artificial freezing technology is widely used in mines, tunnels, subway, foundation engineering and some other fields. Meanwhile, freez-ing technology has been the only construction method which can be used in many projects in some cases (Насонов И. Д and Шуплик М Н, 1981).

As the inverse process of freezing, forcible thawing is a technology adopting artificial heating method to thaw frozen soil after the completion of freezing con-structions. In mining and municipal engineering, the forcible thawing process is usually combined with grouting process. Put previous freezing pipes as heat transfer pipes, and then circulate high temperature salt

water in heat transfer pipes to achieve the purpose of quick thawing, ground reinforcing and water preven-tion (Huang et al., 2005; Yue et al., 2008).

With the wide application of forcible thawing tech-nology in the construction of mine engineering and municipal engineering, some extremely valuable re-search results like frozen-thaw soil thermal physical parameters, thawing process and thawing temperature field are obtained (Yang et al., 1998; Liang et al., 2006; Wang et al., 2010). These research results can be used to control the frozen soil thawing settlement and un-equal foundation settlement. But objectively, these present research results seldom involve analytic com-putation and mainly concentrate on numerical simula-tion and in-situ measurement.

The actual forcible thawing projects mostly rely on multiple heat transfer pipes to melt frozen soil. But presently, the understanding of the rules of multi-pipes thawing temperature field is not comprehensive and systemic because of the multiple unknown influential factors. Therefore, it is necessary to study on the forcible thawing temperature field formed by a single

ZHANG Chi, et al. 2012, 18(1): 18-24 19

heat transfer pipe first, and then research on the multi-pipe thawing temperature field gradually.

Based on the analytic computation with the theory of similitude, a comprehensive and systematic re-search on the forcible thawing temperature field formed by a single heat transfer pipe with unsteady outer surface temperature is carried out. The detailed derivations of formulas of thawing temperature field are presented. Research results of this article can be used to calculate thawing soil radius quickly for tech-nicians and provide ideas for further analytic compu-tation on thawing temperature field with complex boundary conditions.

1 Mathematical model

As the inverse process of freezing, in any forcible thawing projects, the axial size of heat transfer pipe is much larger than radial size, the heat conduction in axial direction is much weaker than radial direction. Therefore, the forcible thawing temperature field can be simplified as a plane heat conduction model (Cui et al., 1998).

Suppose the initial freezing area is a semi-infinite space and the uniform initial temperature is td (℃). Radius of the initial freezing area is much larger than the thaw phase interface radius rM(m). Analytical model is shown in Fig.1.

Fig.1 Analytical model

Formulas of heat conduction are 2

1 1 11 0 M2

1( 0, )

t t ta r r r

r r r

(1)

22 2 2

2 M2

1( 0, )

t t ta r r

r r r

(2)

where, t1 and t2 are temperatures at area 1 and area 2, ℃, area 1 is the thawing zone, area 2 is the initial freezing zone; τ is time, s; a1 and a2 are the thermal diffusivity, m2/s, a1=λ1/c1, a2=λ2/c2; r is radial coordi-nate for the origin to the middle of heat transfer pipe and r0 is the external radius of heat transfer pipe, m.

Uniform temperature of initial freezing area is td.

2 d( ,0)t r t (3)

Temperature far away from thaw phase interface is

td, ℃.

2 d( , )t t (4)

Thawing temperature on phase interface is th (℃).

1 M 2 M h( , ) ( , )t r t r t (5)

Outer surface temperature of heat transfer pipe can be expressed as tc (℃) during forcible thawing.

t dc 1 0 d

k

( , )t t

t t r t

(6)

here, τk is an empirical coefficient reflecting the speed of temperature variation. The smaller τk is, the faster the previous temperature rises. The later steady tem-perature out of the heat transfer pipe is tt (℃).

Thermal balance formula on both sides of the thaw phase interface is

M M

1 2 M1 2

d

dr r r r

t t rΨ

r r

(7)

where, ψ is the latent heat of phase change of unit volume soil, W/m; λ1 and λ2, W/m·℃, are the heat conductivity.

Absolute value of outer surface heat flux density of the heat transfer pipe is (W/m2).

0

11

r r

t

r

(8)

To make Formula (1) to (8) dimensionless, we as-suming that (Cui, 1990)

c h t h1 h 2 h1 2 c t

d h d h d h d h

2 2 k 1k λ2 2

0 0 0 2 d h 2

01 λ Mc a

2 c 0 1 d h

, , , ,

, , , , ,

, , , .( )

t t t tt t t tT T T T

t t t t t t t t

a a rf f s K L

r r r c t t

rc L rL L b q

c L r t t

where, T1 and T2 are the dimensionless temperature at area 1 and area 2; Tc is the dimensionless outer surface temperature of heat transfer pipe; Tt is the dimen-sionless later steady temperature out of the heat trans-fer pipe; f is dimensionless time; fk is the dimen-sionless empirical coefficient reflecting the speed of temperature variation; s is the dimensionless external radius of heat transfer pipe; K is the dimensionless latent heat of phase change; Lλ is the ratio of thawing soil heat conductivity to frozen soil; Lc is the ratio of thawing soil volumetric specific heat to frozen soil; La is the ratio of Lλ to Lc; b is dimensionless thawing ra-dius; q is dimensionless heat flux density; c1 and c2 are the volumetric specific heat, J/(m3·℃).

Substituting them into Formula (1) to (8), we have Formula (9) to (16),

21 1 1

a 2

1 ( 0,1 )

T T TL f s b

f s s s

(9)

Journal of Coal Science & Engineering (China) 20

22 2 2

2

1 ( 0, )

T T Tf b s

f s s s

(10)

2 ( ,0) 1T s (11)

2 ( , ) 1T f (12)

1 2( , ) ( , ) 0T b f T b f (13)

c 1 tk

(1, ) 1 1 ( )f

T T f T ff f

(14)

1 2λ

d

ds b s b

T T bL K

s s f

(15)

1

1s

Tq

s

(16)

For variable substitution, we assuming 1a

1

4L f ,

2

1

4 f , 2

1 1x s and 22 2x s . Therefore, For-

mula (9) and (10) can be transformed to Formula (17) and (18).

21 1

21 1 1

d d11 0

d d

T T

x x x

(17)

22 2

22 2 2

d d11 0

d d

T T

x x x

(18)

General solutions of Formula (9) and (10) can be obtained from Formula (17) and (18). Then, the steady-state solutions T1 and T2 can be found by mak-ing use of boundary conditions like Formula (11) to (14).

2 21 1

1 21 1

( ) ( ) ( )

( ) ( )

f G s G bT

G G b

(19)

22

2 22

( )1

( )

G sT

G b

(20)

Obviously, if only the changing regularity of di-mensionless thawing radius b is obtained, then the dimensionless thawing temperature field T1 and T2 can be found. So, substitute Formula (19) and (20) into Formula (15) and assuming z=b2, we have

12λ

2 1 1

4 ( )e4e d

( ) ( ) ( ) d

zz L f zK

G z G G z f

(21)

There, e

( ) du

yG y u

u

, which is called well

function (Zhu and Liang, 1994; Jiang et al., 2009). Noticing that the initial condition of Formula (21) is

z(0)=1, so z can be solved by numerical method, and then b is obtained easily.

2 Analytic computation and discussion

2.1 Approximation of well function The series expansion of well function G(y) can be

expressed as (Zhu and Liang, 1994)

2

( )( ) ln( )

!

k

k

yG y y y

kk

(22)

where, γ is Euler’s constant and approximately equal to 0.577 216; e is the base of natural logarithm and approximately equal to 2.718 28.

It is noted that 2

( )

!

k

k

y

kk

is an alternating series in

Formula (22). When 0<y≤1, the calculation error may be less than 0.02% according to approximate Formula (23). When 0<y≤0.05, the calculation error may be less than 2% according to approximate Formula (24); when 0<y≤0.03, the calculation error is still less than 2% according to approximate Formula (25).

5

2

( )( ) ln( )

!

k

k

yG y y y

kk

(23)

( ) ln( )G y y y (24)

( ) ln( )G y y (25)

2.2 Approximate analytic solutions of forcible

thawing temperature field For forcible thawing temperature field, when di-

mensionless time f is big enough to ensure that 21s ,

21b and 1 are very small (such as less than 0.03),

we can approximatively replace Formula (22) with (25). Then, we have,

21 a( ) 2ln 2ln 2 ln lnG s s L f (26)

21 a( ) 2 ln 2ln 2 ln lnG b b L f (27)

1 a( ) 2 ln 2 ln lnG L f (28)

Thus, Formula (19) of forcible thawing temperature field T1 can be approximately simplified as

1

ln1 ( ) ( 0, 1 )

ln

sT f f s b

b

(29)

We can get the distribution of approximate tem-perature field in thawing zone by substituting b into Formula (29). From Formula (29), we can find that the temperature distribution law along the radius direction of heat transfer pipe is logarithmic curve in thawing zone.

On the contrary, if testing the temperature T1 at some points in thawing zone, we can also calculate the dimensionless thawing radius b quickly.

1

( )

( )

f

f Tb s

(30)

We can compute dimensionless heat flux density q outside the heat transfer pipe further.

1

1

( )

lns

T fq

s b

(31)

Accordingly, the dimensionless average temperature

ZHANG Chi, et al. 2012, 18(1): 18-24 21

T in thawing zone is

11

1 1 1d ( )

1 ln 1

bT T s f

b b b (32)

Similarly, if only dimensionless time f is big enough, calculated point is not far away from thaw phase in-terface in the non-thawing zone (chiefly the tempera-ture rising zone), 2

2s and 22b are very small

(such as less than 0.03), then we have 2

2( ) 2ln 2ln 2 lnG s s f (33)

22( ) 2ln 2ln 2 lnG b b f (34)

So, in the non-thawing zone (chiefly the tempera-ture rising zone), Formula (20) can be approximately simplified as

2

2ln 2ln ( 0, )

2ln 2ln 2 ln

b sT f b s

b f

(35)

It is clear that the temperature distribution law along the radius direction of heat transfer pipe is also loga-rithmic curve in non-thawing zone (chiefly the tem-perature rising zone).

Similarly, we can calculate the dimensionless radius of the non-thawing zone (chiefly the temperature ris-ing zone) according to Formula (36).

2

2

2 ln [ ln(4 )]exp

2 2

s T fb

T

(36)

2.3 Main influencing factors of forcible thawing

temperature field (1) Dimensionless criterion formulas of b, q and T According to the analysis as above, when dimen-

sionless time f is big enough to ensure that 1z , 2 z and 1 are very small (such as less than 0.03), in this instance, dimensionless thermal balance Formula (21) can be simplified through Formula (25).

12λ4 ( )e4e d

2ln 2 ln ln ln d

zz L f zK

f z z f

(37)

Meanwhile, when dimensionless time f is big enough, 1e z and 2e z will approach to 1. So di-mensionless thermal balance Formula (21) can be fur-ther approximated as

λ4 ( )4 d

2ln 2 ln ln ln d

L f zK

f z z f

(38)

According to Formula (38), the dimensionless

thawing radius b z is mainly affected by K, f, Lλ and ω(f). Thus,

b λ[ , , , ( )]b z F K f L f (39)

Similarly, the simplified Formula (31) of dimen-sionless heat flux density q outside the heat transfer pipe and Formula (32) of dimensionless average tem-perature T in thawing zone are only related to ω(f)

and b. So the dimensionless criterion formulas of q and T can be expressed as

q λ[ , , , ( )]q F K f L f (40)

λ[ , , , ( )]TT F K f L f (41)

The significance of Formula (39) to (41) is that theo-retically proving the dimensionless thawing radius b, the dimensionless heat flux density q and the dimen-sionless thawing average temperature T are mainly affected by K, f, Lλ and ω(f), but Lc affects little.

(2) Relationship between b and the influence factors According to the thermal physical properties of

thawing soil and frozen soil (Yang, 2003; Zhang, 2006), and also refer to the Monograph of Frozen Soil Physics (Xu et al., 2001), the typical value ranges of K, f, Lλ and ω(f) are shown in Table 1.

Table 1 Value table of influence factors of thawing soil

radius

Influence factor Value range Step size Level

K 0.1~5.3 0.4 14

f 23~323(n3) Δn=3 11

Lλ 0.5~1.0 0.05 11

Lc 1.0~2.0 0.2 5

ω( f ) -0.2~-7.0 -0.4 18

For a comprehensive study on relationship between

b and the influence factors, an all-combination calcu-lation is taken by substituting the typical values of K, f, Lλ and ω(f) into Formula (21).

In Figs.2 to 6, mark (a) stands for the factor combi- nation mode when thawing radius b is the minimum, mark (b) represents the factor combination mode when

Fig.2 Relation curves of b vs. K1/2

Journal of Coal Science & Engineering (China) 22

Fig.3 Relation curves of b vs. f 1/3

Fig.4 Relation curves of b vs. Lλ

thawing radius b is the maximum. According to the curves shown in Figs.2 to 6, we have: ① The dimen-sionless thawing radius b is mainly affected by K, f, Lλ and ω(f), but Lc affects little (Fig.5). ② The dimen-sionless thawing radius b decreases linearly with K1/2 and increases linearly with Lλ (Fig.2, Fig.4). b is the quadratic function of f1/3 and decreases nolinearly with f1/3 (Fig.3). b is the quadratic function of ω(f) and de-creases nolinearly with ω(f) (Fig.6).

3 Engineering example

The air shaft lining of Hulusu coal mine is located in Hujierte coal area in Chinese Erdos. Artificial

ground freezing technology is adopted for the con-struction of shaft lining and the frozen depth is 670 m. Within the limit of shaft lining vertical depth, the en-gineering geologic condition is surface soil stratum, Cretaceous stratum and Jurassic stratum from top to bottom.

Fig.5 Relation curves of b vs. Lc

Fig.6 Relation curves of b vs. ω(f)

According to the construction documentations, it is about 70 days from air shaft lining stopping freezing to May 2011. The frozen wall has extended about 4.0 m to the center of shaft lining and extended about 5.5 m outside, so the total effective thickness of frozen wall has reached to about 9.5 m (Fig.7). In order to cut off the hydraulic connection in the annular space of

ZHANG Chi, et al. 2012, 18(1): 18-24 23

freezing holes and create a better construction envi-ronment for shaft inset tunneling, 21 freezing pipes were used for perforation grouting in the depth of -380 m to -382 m.

Fig.7 Heat transfer pipes of Hulusu air shaft lining

Perforation grouting process can be divided into 3 steps, they are forcible thawing, jet perforating and pressure slip casting (Huang et al., 2005). Here we only make an analysis of forcible thawing progress. Forcible thawing in air shaft lining makes the present 21 main freezing pipes as heat transfer pipes and re-verse circulating hot water of 45 ℃ in them (Fig.7). The diameter of freezing pipe is 133 mm and the planned thawing soil diameter is 400 mm. So the di-mensionless thawing radius b=400/133≈3.

According to the thermal physical properties of soil and the construction documentations (Ma, 2008), dif-ferent dimensionless factors of thawing temperature field are as follows: K=4.62, Lλ=0.77, La=0.55. Ap-proximately adopting average temperature of initial freezing zone to replace td, so td=-12 . Because r℃ e-verse circulating hot water process generally adopts the technical measures of first heating and post circu-lating, it can be approximately thought that the tem-perature outside heat transfer pipes will quickly stabi-lize at tc=45 ℃. Therefore, dimensionless temperature outside the heat transfer pipes is ω(f)=45/-12= -3.75. The time for hot water circulating in heat trans-fer pipes is one day (τ=1), corresponding to dimen-sionless time f=5.23.

Based on the thermal physical parameters above, it is easy to get the relation curves between dimen-sionless thawing radius b and dimensionless time f according to Formula (21) (Fig.8).

According to Fig.8, it is clear that the relationship between b and f is nonlinear. And if the dimensionless thawing radius b=3, f should approximately equal to 13.5 and the corresponding thawing days τ should equal to 2.58. The theoretical thawing days of 2.58 is consistent with the practical thawing 3 days in engi-neering applications. So we can conclude that the ana-lytical solutions of this article are available for refer-ence for forcible thawing engineering.

Fig.8 Relation curves of b vs. f

Substituting dimensionless thawing radius b into Formulas (19) and (20), we can get the relation curves between dimensionless temperature T and dimen-sionless radial coordinate s. According to Fig.9, we have the followings.

Fig.9 Relation curves of T vs. s

(1) The temperature distribution law along the ra-dius direction of heat transfer pipe is logarithmic curve in both thawing zone and non-thawing zone (chiefly the temperature rising zone).

(2) Under the condition of dimensionless time f=13.5 in non-thawing zone (chiefly the temperature rising zone), dimensionless temperature T2 tends to be a constant value td=1 in the place of s=10. Meanwhile, radius of initial freezing area of single pipe is ap-proximatively 0.5 times the distance of the adjacent freezing pipes, so the radius of initial freezing area is about 0.5×262 3=1 311.5 mm, corresponding to the dimensionless value 19.7>s=10. This means that when the planned thaw phase interface radius rM=200 mm (corresponding to b=3, f=13.5), even if 21 freezing pipes shown in Fig.7 starting thawing together, the thawing zone and non-thawing zone (chiefly the tem-perature rising zone) of adjacent freezing pipes will not overlap. Therefore, forcible thawing temperature field in Hulusu air shaft lining can be approximatively recognized as semi-infinite space.

4 Conclusions

(1) The temperature distribution law along the ra-dius direction of heat transfer pipe is logarithmic curve

Journal of Coal Science & Engineering (China) 24

in both thawing zone and non-thawing zone (chiefly the temperature rising zone).

(2) When dimensionless time f is big enough, we can approximately use formula 1( )/[ ( ) ]f f Tb s to calculate dimensionless thawing radius b.

(3) When dimensionless time f is big enough, the dimensionless thawing radius b, the dimensionless heat flux density q and the dimensionless thawing av-erage temperature T are mainly affected by K, f, Lλ and ω(f), but Lc affects little.

(4) The dimensionless thawing radius b decreases linearly with K1/2 and increases linearly with Lλ. b is the quadratic function of f1/3 and decreases nolinearly with f1/3. b is the quadratic function of ω(f) and de-creases nolinearly with ω(f).

(5) Based on the forcible thawing project of Hulusu air shaft lining, the relation curve between dimen-sionless thawing radius b and dimensionless time f is obtained. The theoretical thawing days of 2.58 is con-sistent with the 3 practical thawing days in engineer-ing application. So we can conclude that analytical solutions of this article are available for reference for forcible thawing engineering.

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