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Temperature field analysis of a cold-region railway tunnel considering mechanical and train-induced ventilation effects
Accepted Manuscript Title: Temperature field analysis of a cold-region railway tunnel considering mechanical and train-induced ventilation effects Author: Xiaohan Zhou, Yanhua Zeng, Lei Fan PII: DOI: Reference:
S1359-4311(16)30020-5 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.01.070 ATE 7630
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
22-11-2015 24-1-2016
Please cite this article as: Xiaohan Zhou, Yanhua Zeng, Lei Fan, Temperature field analysis of a cold-region railway tunnel considering mechanical and train-induced ventilation effects, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.01.070. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
Temperature field analysis of a cold-region railway tunnel
2
considering mechanical and train-induced ventilation effects
3
Xiaohan Zhoua, Yanhua Zeng a*, and Lei Fanb
4
a
5
Jiaotong University, Chengdu, China
6
b
7
*Corresponding authors:
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Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest Jiaotong University,
9
Chengdu 610031, P.R. China
Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest
China Railway Eeyuan Engineering Group Co. Ltd, Chengdu, China
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Tel: +86-15895873903
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E-mail: [email protected]
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Highlights
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An unsteady-state finite-difference model of cold-region tunnel temperature fields is presented.
14
The coupled convection–conduction problem is considered.
15
Effects of mechanical ventilation and train-induced winds on the tunnel temperature distribution are
16 17
studied.
In situ observed frozen lengths in a cold-region railway tunnel are compared with the calculated results.
18 19
Abstract
20
In accordance with the unsteady-state finite-difference equations for heat transfer and heat convection in a
21
cold-region railway tunnel, an unsteady-state finite-difference computer model was developed to study the heat
22
convection between the air and the tunnel wall as well as the heat transfer in the surrounding rock and at interfaces
23
between different materials in the structure of the tunnel. The wind speed and wind direction of actual mechanical
24
ventilation wind produced during operation as well as the train-induced wind and natural wind in the tunnel were
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considered in the analysis of the temperature field distribution of a cold-region railway tunnel under actual
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periodic variations in entrance wind temperature. Good agreement was observed between the calculated frozen
27
length and the frozen length observed in situ at the entrance of the railway tunnel. The results show that
28
mechanical ventilation winds during operation and train-induced winds significantly influence the temperature
29
field distribution in the tunnel and that the variables associated with mechanical ventilation winds and
30
train-induced winds should be considered in the engineering design of cold-region tunnels. In this way, the
31
temperature field of a railway tunnel can be correctly predicted via finite differencing.
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Keywords: Cold-region railway tunnel, Mechanical ventilation wind, Train-induced wind, Temperature field,
33
Finite difference, Frozen length
34
1. Introduction
35
Research on cold-region tunnel design is becoming urgent because of the vast number of infrastructure
36
construction projects being undertaken for railways and highways in cold regions. The prediction of temperature
37
fields, which are used as references for relevant cold-proofing measures, is an important factor in cold-region
38
tunnel engineering design. Numerous studies have been conducted on temperature fields in cold-region tunnels.
39
Bonacina et al. proposed a finite-difference method for addressing melting and freezing with corresponding phase
40
changes; moreover, the problem formulation could be straightforwardly extended to multidimensional cases [1].
41
Comini et al. performed a finite-element analysis of the transient heat conduction problem with non-linear physical
42
properties and boundary conditions, in which latent heat effects were treated as heat capacity variations within a
43
narrow temperature range [2]. Lai et al. performed nonlinear analyses of related problems concerning temperature,
44
seepage and stress fields in cold-region tunnels based on the theory of heat transfer using finite-element formulas
45
[3], and related studies have demonstrated that good insulation should be installed to avoid substantial
46
frost-heaving damage to the tunnel lining [4]. He et al. predicted the freezing–thawing conditions in the rock
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surrounding tunnels in cold regions using a combined convection–conduction model based on factors including the
48
in situ air temperature, atmospheric pressure, and wind force conditions [5,6]. Lai et al. proposed an approximate
49
analytical solution for temperature fields in cold-region circular tunnels based on a dimensionless perturbative
50
method that can be used when the initial temperature is nearly 0°C [7]. Zhang et al. analyzed three-dimensional
51
temperature characteristics in cold-region tunnels using the finite-element method [8]. Zhang et al. performed a
52
forecast analysis of the refreezing of the Kunlunshan permafrost tunnel of the Qing-Tibet Railway in China [9].
53
Lai et al. [10] proved that the freezing-thawing behaviors of the rock surrounding a tunnel can be correctly
54
predicted even if the air temperature along the tunnel is unknown. Feng et al. adopted the Stehfest numerical
55
inversion and Laplace transform methods to address this problem [11].
56
To date, proper cold-proofing measures have been applied in China, Norway, Russia, Canada and Japan to
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decrease the frost damage experienced by tunnels in cold regions. In Japan, Okada et al. first introduced surface
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adiabatic treatments and adiabatic double lining to prevent water leakage and the formation of icicles and
59
described a cross-sectional model considering heat insulation of the lining surface and cyclically varying
60
temperatures in thin earth covers. In addition, these authors discussed the different temperature modes
61
corresponding to different heat insulator depths and different earth covers [12-14]. In Norway, heated cables, some
62
of which are fitted with double thermal insulation doors, whereas others have insulation installed, are used to heat
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certain drains [10]. Many useful methods, such as slurry injection and the use of insulation materials, thermal
64
insulation doors, snow sheds and drainage holes, have been proposed for cold-region tunnels in China [15]. A
65
three-dimensional nonlinear analysis of the coupled problem of heat transfer and seepage both with and without an
66
insulation layer was conducted for the Kunlun mountain tunnel by Zhang et al. [16]. An analysis of the refreezing
67
of the Feng Huoshan tunnel in the presence of thermal insulation material was conducted by Zhang et al. to study
68
the behavior of the thermal insulation material in a permafrost tunnel during the refreezing process [9]. To
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determine the appropriate scope of insulation materials or slurry injections for cold-regions tunnels, the predicted
70
scale of the tunnel structure at freezing temperatures must be confirmed via predictive calculations of the
71
temperature fields before construction.
72
Tan et al. introduced a calculation model that includes control equations for the surrounding rock and air
73
temperatures to analyze the temperature field of the surrounding rock of the Galongla tunnel in Tibet and the
74
thermal insulation measures applied under ventilation conditions; the calculations were performed using the
75
commercial finite-element software package COMSOL Mutiphysics (version 4.2) [15]. It is known in the field of
76
railway tunnel ventilation that, in addition to the natural winds from the entrance or exit of a tunnel, mechanical
77
ventilation and train-induced winds are present inside the tunnel and cannot be ignored when the tunnel is in use.
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Thus, this paper primarily examines the influence of natural, mechanical ventilation and train-induced winds on
79
the temperature fields in a cold-region tunnel based on both theoretical finite-difference calculations and the results
80
of a field investigation. In addition, the effects of mechanical ventilation and train-induced winds on the freezing
81
scale in a cold-region tunnel are analyzed.
82
2. Tunnel temperature field model based on the coupled problem of
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the convective exchange between the air and tunnel lining and the
84
heat transfer in the lining structures and surrounding rock
85
2.1. Fundamental assumptions
86
The primary purpose of the current study was to determine how the mechanical ventilation wind and train-induced
87
wind in a tunnel influence the tunnel’s temperature field distribution. Thus, for simplicity, the following basic
88
assumptions are made regarding the transient heat transfer:
89
(1) The calculated cross section along the longitudinal direction of the tunnel is circular [6,7,17], and the
90
equivalent hydraulic radius is obtained in the computation.
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91 92
(2) At the beginning of the calculation, the temperature of the tunnel structure is equal to the initial temperature
of the surrounding rock, which is constant in the radial direction.
93
(3) The air is laminar and incompressible, and the air pressure does not vary with temperature [10,15].
94
2.2. Governing equations
95
The process of heat transfer in the surrounding rock-lining-wind system primarily involves the heat transfer
96
between the surrounding rock and the tunnel lining, the convective heat transfer between the tunnel wall and the
97
air, and any thermal radiation that occurs. The thermal radiation is ignored in this analysis because of its relatively
98
small magnitude.
99
The transmission of heat is described by the Fourier law [18] as follows:
100
(1) is the temperature gradient in the direction normal to the area A, λ is the heat transfer rate, and φ is the
101
where
102
heat that passes through a given area in unit time.
103
The temperature field of a tunnel consists of a set of temperatures at various points at every moment. The
104
temperature function in a three-dimensional rectangular coordinate system can be expressed as
105
(2)
106
According to the fundamental assumptions stated in Section 2.1, the governing equation of the longitudinal
107
transient temperature field of the surrounding rock and lining in cylindrical coordinates [10,17] is
108 109
(3)
The boundary conditions are
110
(4)
111
(5)
112
where T is the temperature of the tunnel lining and surrounding rock,
is the initial temperature of the
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surrounding rock, Tf is the wind temperature, r is the radial distance, a (
114
coefficient, t is time, ρ is the density of the tunnel lining and surrounding rock, and h is the convective heat transfer
115
rate.
116
The wind in the tunnel is regarded as a non-viscous, incompressible fluid exhibiting a laminar flow, non-viscous,
117
incompressible fluid, and the heating effect from the equipment in the tunnel is considered. The control equation
118
for the transient temperature of the axial wind in the tunnel [17] is
119 120
) is the thermal diffusion
(6)
The boundary conditions are
121
(7)
122
where
123
effect in the tunnel, and
124
At the frost front position s(t), the continuity condition and the conservation-of-energy requirement should be
125
satisfied, i.e.,
is the temperature of the tunnel wall, U is the perimeter of the cross section, qs is the equipment heating is the constant-pressure specific heat capacity of the surrounding rock.
126
(8)
127
Suppose that the phase change occurs in the temperature region denoted by Tm+
128
frozen regions ( ), the heat capacity of unfrozen regions ( ), the heat transfer rate in frozen regions (λf) and the
129
heat transfer rate in unfrozen regions (λu) do not depend on the temperature (T). Then, the following definitions
130
may be assumed [8]:
131
132
and that the heat capacity of
(9)
and
133
(10)
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134
2.3. Finite-difference discretization
135
Numerically calculating partial differential equations requires two steps. First, the continuous calculation range is
136
divided into finite and discrete point sets, and the partial differential equations and definite conditions are
137
transformed into algebraic equations at the corresponding points. Then, the computed results for the partial
138
differential equations at these points are obtained. Because there must be a smooth change between the computed
139
results for adjacent discrete points, approximate solutions over the entire calculation range can be obtained via
140
interpolation. Clearly, the accuracy of the computed results strongly depends on the number of discrete points
141
considered. Finally, the time domain should also be discretized for transient heat conduction problems.
142
Based on the actual structure of the tunnel, discrete processes in the spatial domain were adopted when
143
implementing the temperature field calculations in this study. The tunnel structures in a transverse section were
144
assumed to be circular with hydraulic radii[6,7,17], and the gridding was assumed to be centrosymmetric. A model
145
with equidistant gridding in the tunnel's axial direction was adopted. The mesh generated for the model used to
146
calculate the temperature field in the tunnel is detailed in Fig. 1 and Fig. 2.
147
Fig. 1 and Fig. 2 depict the horizontal and vertical grids, respectively. Finite-difference equations were established
148
for the heat transmission between the layers of tunnel lining, between the tunnel lining and the surrounding rock,
149
and inside the surrounding rock. By contrast, partial differential equations were used to model the heat convection
150
between the air and the tunnel wall. In combination with the law of energy conservation, the entire asynchronous
151
differencing scheme for the transient heat transfer calculation was established as follows [17].
152
The formula for approximating the temperature of a grid node within the surrounding rock is
153
(11)
154
where
155
cross-sectional plane;
is the temperature of node i,j at time n;
is the radial distance corresponding to the step size in the
is the longitudinal step size; F0 is the Fourier value,
; and
is the time step.
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The formula for approximating the temperature of a node at a convection boundary is
157 158
(12)
159
where
160
The formula for approximating the temperature of a node at an intersection point between materials is
is the wind temperature of node i at time n.
161 162
(13)
The longitudinal approximation formulas for the wind temperature in the tunnel are
163
(14)
164
where
165
the tunnel.
166
The approximation formulas given above for the temperatures of the wind and the surrounding rock include both
167
heat conduction and heat convection, and both distance and time steps are considered. Furthermore, the wind
168
temperature and wind speed are taken into account in Tf and v.
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3. Finite-difference analysis of a cold-region railway tunnel
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The Nan Shan railway tunnel, located in northeastern China, is analyzed here as an example. The natural wind,
171
mechanical ventilation wind and train-induced wind in the tunnel are the major factors considered in calculating
172
the temperature field using the finite-difference method presented in this paper.
173
3.1. Introduction to the Nan Shan tunnel
174
The Nan Shan railway tunnel is a cold-region tunnel in Jilin Province, China. The coldest monthly average
175
temperature in the region is -12.5°C, and the extreme low monthly average temperature is -17°C; the altitude is
is the wind temperature at node i at time n+1, A is the cross-sectional area, and v is the wind speed in
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approximately 1900 m. The Nan Shan tunnel is located in an area with a humid temperate continental monsoon
177
climate with long cold snowy winters and cool wet summers.
178
3.2. Model, conditions and parameters
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The Nan Shan tunnel has a length of 7,566 m (Bai-He to Long-He), a downward slope of 0.95%, and an elevation
180
ranging from 749.84 m to 821.72 m. The calculated cross-sectional area is 32.43 m2, the wetted perimeter is 21.76
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m, and the equivalent diameter of the tunnel is 5.96 m, which includes a 0.45-m secondary lining and a 0.25-m
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inner lining, as shown in the cross section in Fig. 4.
183
A longitudinal section of the Nan Shan tunnel is shown in Fig. 3. From knowledge of the buried depth and local
184
climate data of the tunnel, the initial temperature of the surrounding rock can be obtained by taking the
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temperature gradient to be 3°C/100 m and the local average soil temperature to be equal to the subsurface
186
temperature [10].
187
The average measured natural wind speed in the tunnel is 2.5 m/s, with the wind blowing from Bai-He to Long-He.
188
Train-induced wind is also considered in this paper. This latter wind speed is calculated as follows:
mm
-
189 190 191 192
m
mm
where
n n n
(12)
is the train-induced wind;
;
-l m
;
d
n
d
;
m
;
is the train speed;
-
is the train length; N is the resistance coefficient, where
; λr is the frictional resistance coefficient of the tunnel wall; and
is the blockage ratio, where
f
, with
193
fT being the cross-sectional area of a train and A being the cross-sectional area of the tunnel.
194
Three pairs of freight trains and one pair of passenger trains, all of which are driven by diesel locomotives, travel
195
through the Nan Shan tunnel on a daily basis. Using Equation 12 and a train speed of 60 km/h, the train-induced
196
wind speeds are found to be 6.8 m/s and 6.1 m/s when the train is moving with and against the direction of the
197
natural wind, respectively, as shown in Fig. 6. 9
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As shown in Fig. 4 and Fig. 5, the Nan Shan tunnel ventilation system consists of 10 jet fans and 4 axial flow fans
199
located at the Long-He end. The axial flow fans are installed at equal distances in two air holes at DK76+424 and
200
DK76+354, and the jet fans are hung at the exit of the tunnel at DK76+454. According to the relevant design data
201
and the literature, the mechanical ventilation time required to exhaust dusty air is 33 minutes after a train passes
202
through the Nan Shan tunnel during operation. After a train passes through the tunnel during actual operation, the
203
corresponding fans are activated to create a reverse wind. Four pairs of trains pass through the tunnel on a daily
204
basis, and the measured mechanical ventilation wind speed is 1.5 m/s from the exit to the entrance of the tunnel.
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The average specific heat capacities, densities, average thermal conductivity coefficients and other parameters of
206
the secondary lining, inner lining and surrounding rock can be obtained from the relevant experimental data given
207
in Table 1. The convective heat transfer rate is found to be h=3.06v+9.55
208
between the wind velocity and the convective heat transfer rate discussed by Zhang et al. [19]. According to the
209
climate data recorded at the Nan Shan tunnel over the past 20 years, the periodic entering wind temperature is
210
/(m2·°C), according to the relationship
(°C), where i represents time (in days).
211
3.3. Results and discussion
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The train-induced wind and the mechanical ventilation wind are both considered in the calculation. The
213
train-induced wind and mechanical ventilation wind primarily influence the speed and direction of the overall wind
214
in the tunnel, which in turn will influence the convective heat transfer between the air and the tunnel wall.
215
Therefore, the effects of the wind speed and wind temperature on the temperature distributions in the lining and
216
surrounding rock are studied below.
217
3.3.1. Temperature distributions in the lining and surrounding rock in the presence of
218
natural, train-induced and mechanical ventilation winds
219
The tunnel section DK73+139 (4211 m from the tunnel entrance) was selected for the analysis of the
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cross-sectional temperature distribution, as shown in Fig. 7. The natural wind speed is 2.5 m/s from Bai-He to
221
Long-He; the mechanical ventilation speed is 1.5 m/s from Long-He to Bai-He; the train-induced wind speeds are
222
6.8 m/s and 6.1 m/s when moving with and against the direction of the natural wind, respectively; and the wind
223
temperature at the tunnel entrance is
224
distributions at various radial points in section DK73+139 follow the same sine distribution rule as the inlet wind
225
temperature when the natural wind, train-induced wind and mechanical ventilation wind are all considered. Lesser
226
changes in temperature amplitude and greater time lags compared with the sine-wave profile of the entering wind
227
temperature are predicted for regions farther from the secondary lining surface. The changes in temperature
228
amplitude observed during the first cycle are 11.02, 8.58, 5.87, 1.38 and 0.29°C, with cycles of approximately 400,
229
410, 430, 540 and 640 days at points at distances of 0.45, 0.7, 17.6, and 27.6 m from the surface of the secondary
230
lining.
231
The temperature distributions in regions at different radial depths along the entire tunnel, under the natural wind,
232
mechanical ventilation wind, train-induced wind and temperature conditions given in Section 3.1, are shown in Fig.
233
8. The figure shows that different radial nodes generally follow the same distributional trends in the longitudinal
234
direction. After 300 days of the simulation, from the surface of the secondary lining to a distance of 17.7 m from
235
its surface, the longitudinal temperature begins to approximate the initial temperature of the surrounding rock. The
236
temperature distribution at a distance of 17.7 m from the surface of the secondary lining after 300 days of the
237
simulation is approximately identical to the original temperature distribution along the tunnel in the longitudinal
238
direction, except in the entrance and exit regions.
239
3.3.2. Influence of mechanical ventilation wind on temperature distributions in the
240
lining and surrounding rock
241
The temperature distributions along the entire tunnel at the surface of the secondary lining and at a distance of 0.7
(°C). The figure shows that the temperature
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242
m from its surface after 300 days of simulation, with the mechanical ventilation wind both considered and not
243
considered as described in section 3.3.1, are shown in Fig. 9. The entrance region (Bai-He) and the exit region
244
(Long-He) are found to follow opposite temperature distribution rules with and without mechanical ventilation
245
wind. Within 600 m of the entrance region, the average longitudinal temperature on the surface of the secondary
246
lining in the presence of mechanical ventilation wind is 1.1054°C higher than the average longitudinal temperature
247
without mechanical ventilation wind. Within 600 m of the exit region, the average longitudinal temperature on the
248
surface of the secondary lining in the presence of mechanical ventilation wind is 1.7954°C lower than the average
249
longitudinal temperature without mechanical ventilation wind. In addition, these temperature differences become
250
0.7722 and 0.4702°C, respectively, for points at a distance of 0.7 m from the surface of the secondary lining. Thus,
251
because of the direction of the mechanical ventilation fans (from Long-He to Bai-He), the exit region of the Nan
252
Shan tunnel is more sensitive to mechanical ventilation effects.
253
3.3.3. Influence of train-induced wind on temperature distributions in the lining and
254
surrounding rock
255
The temperature distributions along the entire tunnel at the surface of the secondary lining and at a distance of 0.7
256
m from its surface after 300 days of simulation, with the train-induced wind both considered and not considered as
257
described in section 3.3.1, are shown in Fig. 10. The train-induced wind travels in the same direction as the train;
258
thus, this wind amplifies the heat convection between the air and the tunnel wall without affecting the temperature
259
distribution rule governing the tunnel as a whole. The temperature distribution in the exit region exhibits a
260
substantial difference depending on whether the train-induced wind is considered because the train-induced wind
261
from Long-He to Bai-He pulls cold air into the tunnel, strongly cooling the lining and surrounding rock.
262
3.3.4. Influence of wind speed on temperature distributions in the lining and
263
surrounding rock 12
Page 12 of 25
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To provide a quantitative analysis of the influence of different wind speeds on the temperature distributions, a case
265
study is performed in this section. The temperature distributions in the lining and surrounding rock in a section at a
266
distance of 1620 m from the tunnel entrance (Bai-He) after 10 days of simulation at different wind speeds,
267
calculated under the conditions presented in section 3.3.1, are shown in Fig. 11. The figure shows that when the
268
wind speed is less than 5 m/s, an increase in the wind speed has a large influence on the temperature distribution.
269
By contrast, when the wind speed is higher than 10 m/s, the effect is very small, especially in regions at a depth of
270
greater than 3 m. The main reason for this difference in behavior is that the thermal exchange between the air and
271
the lining is stable when the wind speed is higher than 10 m/s [15]. Furthermore, Fig. 11 shows that the wind speed
272
has a greater influence on the temperature distribution at shallower depths.
273
3.3.5. Influence of wind temperature on temperature distributions in the lining and
274
surrounding rock
275
For the calculations presented here, the wind speed at the tunnel entrance (Bai-He) was fixed at 2.5 m/s and the
276
other conditions presented in section 3.3.1 were used. The temperature distributions in the lining and surrounding
277
rock in a section at a distance of 1620 m from the tunnel entrance (Bai-He) after 30 days of simulation at different
278
wind speeds are shown in Fig. 12. The figure illustrates that the wind temperature strongly influences the
279
temperature distribution in the lining and surrounding rock, primarily because a large temperature difference
280
between the air and the tunnel wall will promote heat convection.
281
3.3.6. Discussion
282
During tunnel operation, the temperatures of the lining and surrounding rock vary over time because of the
283
temperature differences that form between the air and the tunnel wall under the influence of periodic variations in
284
wind temperature. The temperatures of the lining and surrounding rock vary with the same periodic behavior as the
285
wind temperature but at a relative time delay. The temperature distributions of nodes at different radii generally
13
Page 13 of 25
286
follow the same distributional trends in the longitudinal direction.
287
The entrance region (Bai-He) and the exit region (Long-He) are found to follow opposite temperature distribution
288
rules with and without mechanical ventilation wind because of the reverse wind direction of the fans in the exit
289
region compared with the natural wind direction. Train-induced wind travels in the same direction as the train; thus,
290
train-induced wind amplifies the heat convection between the air and the tunnel wall without affecting the
291
temperature distribution rule governing the tunnel as a whole. The temperature distribution in the exit region
292
differs substantially depending on whether train-induced wind is considered because the train-induced wind from
293
Long-He to Bai-He pulls cold air into the tunnel, strongly cooling the lining and surrounding rock. Every railway
294
tunnel has its own properties of mechanical ventilation and train scheduling; thus, the direction and speed of the
295
ventilation fans are not common to all tunnels. Therefore, both mechanical ventilation and train-induced winds
296
should be considered when calculating the temperature field of a railway tunnel.
297
To provide a quantitative analysis of the influence of different wind speeds and wind temperatures on the
298
temperature distribution in a tunnel, two case studies were performed. The results show that both wind speed and
299
wind temperature considerably influence the temperature distribution because of their influence on the heat
300
convection between the air and the tunnel wall. A high wind speed results in the introduction of more cold air into
301
the tunnel and leads to a large convection coefficient; at the same time, a large temperature difference between the
302
air and the tunnel wall will also promote heat convection between the air and the wall.
303
3.4. Validation of the calculated result against the field-measured frozen
304
length
305
The Nan Shan tunnel opened to traffic in 2006, and a field investigation of this tunnel during the winter of 2009
306
showed that the frozen length at the entrance (Bai-He) reached a maximum of 1,472 m. The freezing sites included
307
the side drainage ditches and tunnel linings (see Fig. 13).
14
Page 14 of 25
308
To avoid damage from cold temperatures, the predicted scale of a tunnel structure under freezing temperatures is
309
important and useful in determining the appropriate scope of antifreezing measures, such as insulation layers in the
310
tunnel. The temperature field in the Nan Shan tunnel was calculated using the finite-difference model presented in
311
Section 3.3.1 given the actual wind speed and wind temperature conditions. Fig. 14 shows a maximum calculated
312
longitudinal length at temperatures below 0°C in the area between the inner lining and the surrounding rock in the
313
entrance region of the Nan Shan tunnel within the first year of operation of approximately 1,440 m, increasing to
314
approximately 1,500 m during the second year and subsequently remaining at approximately the same value,
315
which is consistent with the results of the field investigation conducted in 2009. Thus, the proposed
316
finite-difference model could be successfully applied to accurately calculate the basic temperature field
317
distribution of the Nan Shan tunnel.
318
4. Conclusions
319
Unsteady-state finite-difference equations for heat transfer and heat convection were established to calculate the
320
temperature field of a cold-region railway tunnel considering natural wind, train-induced wind and mechanical
321
ventilation wind. The temperature distributions in the tunnel structures and surrounding rock under natural wind,
322
train-induced wind and mechanical ventilation wind as well as the influence of wind temperature, wind speed and
323
wind direction on the temperature distribution were studied. The following conclusions were drawn based on a
324
combination of finite-difference calculations and the results of an in situ investigation.
325
(1) The wind temperature and wind speed inside a tunnel significantly influence the tunnel’s temperature
326
distribution. The temperature distributions at different radial points follow the same distribution rule as that
327
governing the natural wind temperature at the tunnel entrance,
328
radial distances experience smaller changes in temperature amplitudes and greater time lags with respect to the
329
wind temperature.
(°C). Points at greater
15
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330
(2) Train-induced wind and mechanical ventilation wind generally amplify the heat convection between the air and
331
the tunnel structures. Train-induced wind travels in the same direction as the train; thus, it amplifies the heat
332
convection between the air and the tunnel wall without significantly affecting the temperature distribution rule
333
governing the tunnel as a whole. Because of the reversed direction of the fans in the exit region compared with that
334
of the natural wind in the Nan Shan tunnel, the entrance and exit regions follow opposite temperature distribution
335
rules when mechanical ventilation effects are considered. These ventilation effects cannot be ignored when
336
predicting tunnel temperature fields.
337
(3) The accuracy of the temperature field calculation method for cold-region railway tunnels based on the
338
ventilation effect was verified by comparing the calculated results with the data from an in situ investigation of the
339
Nan Shan tunnel. The finite-difference model proposed in this paper can be used to guide the design of insulation
340
layers for cold-region railway tunnels, thereby overcoming the difficulties encountered because of the complexity
341
of modeling and stringent hardware requirements of general finite-element software.
342 343 344 345 346 347 348 349 350 351
Nomenclature
352 353 354 355 356 357 358 359 360 361
A
cross-sectional area [m2] temperature gradient in the direction normal to the area A
λ
heat transfer rate [W/m·K]
φ
heat passing through a given area in unit time [W/m2·s]
T
temperature of the tunnel lining and surrounding rock [°C] initial temperature of the surrounding rock [°C]
Tf
wind temperature [℃]
r
radial distance [m]
a
thermal diffusion coefficient (
t
time [s]
ρ
density [kg/m3]
h
convective heat transfer rate [W/m2·K]
U
cross-sectional perimeter [m]
qs
equipment heating effect in the tunnel [W]
)
tunnel wall temperature [℃]
constant-pressure specific heat capacity [J/m3·K] temperature of node i,j at time n [°C] radial distance corresponding to the step size in the cross-sectional plane [m] longitudinal step size [m] 16
Page 16 of 25
362
F0
363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
Fourier value, time step [s] wind temperature at node i at time n [°C] wind temperature at node i at time n+1 [°C]
v
wind speed in the tunnel [m/s]
i
time [d] train-induced wind [m/s] train speed [m/s] train length [m]
N r
resistance coefficient of a train [1/m] frictional resistant coefficient of the tunnel wall blockage ratio
fT
cross-sectional area of a train [m2] heat capacity in frozen regions [J/m3·K] heat capacity in unfrozen regions [J/m3·K]
λf
heat transfer rate in frozen regions [W/m·K]
λu
heat transfer rate in unfrozen regions [W/m·K]
References 379
[1]
380 381
Heat Mass Transf. 16 (1973) 1825-1832.
[2]
382 383
[3]
Y.M. Lai, Z.W. Wu, Y.L. Zhu, L.N. Zhu, Nonlinear analysis for the coupled problem of temperature,
seepage and stress fields in cold-region tunnels, Tunnelling Undergr. Space Technol. 13 (1998) 435–440.
[4]
386 387
G. Comini, S. Del Guidice, R.W. Lewis, O.C. Zienkiewicz, Finite element solution of non-linear heat
conduction problems with special reference to phase change, Int. J. Numer. Methods Eng. 8 (1974) 613-624.
384 385
C. Bonacina, G. Comini, A. Fasano, M. Primicerio, Numerical solution of phase-change problems, Int. J.
Y. Lai, Z. Wu, Y. Zhu, L. Zhu, Nonlinear analysis for the coupled problem of temperature and seepage fields
in cold-region tunnels, Cold Regions Sci. Technol. 29 (1999) 89–96.
[5]
C.X. He, Z.W. Wu, Preliminary prediction for the freezing –thawing situation in rock surrounding
388
DabanShan tunnel, in: Proceedings of the Fifth National Conference on Glaciology and Geocryology,
389
Culture Press of Gan Su, Lan Zhou, 1996, pp. 419–425.
17
Page 17 of 25
390
[6]
391 392
in the surrounding rock wall of a tunnel in permafrost regions, Sci. China Ser. D 29 (1999) 1–7.
[7]
393 394
[8]
X.F. Zhang, Y.M. Lai, W.B. Yu, S.J. Zhang, Nonlinear analysis for the three-dimensional temperature fields
in cold region tunnels, Cold Regions Sci. Technol. 35 (2002) 207-219.
[9]
397 398
Y.M. Lai, S.Y. Liu, Z.W. Wu, W.B. Yu, Approximate analytical solution for temperature fields in cold
regions circular tunnels, Cold Regions Sci. Technol. 13 (2002) 43–49.
395 396
C.X. He, Z.W. Wu, L.N. Zhu, A convection – conduction model for analysis of the freeze – thaw conditions
X.F. Zhang, Y.M. Lai, W.B. Yu, S.J. Zhang, J.Z. Xiao, Forecast analysis for the re-frozen of Kunlunshan
permafrost tunnel on Qing-Tibet railway in China, Cold Regions Sci. Technol. 38 (2004) 12–22.
[10]
Y.M. Lai, X.F. Zhang, W.B. Yu, S.J. Zhang, Z.Q. Liu, J.Z. Xiao, Three-dimensional nonlinear analysis for
399
the coupled problem of the heat transfer of the surrounding rock and the heat convection between the air and
400
the surrounding rock in cold-region tunnel, Tunnelling Undergr. Space Technol. 20 (2005) 323-332.
401
[11]
402
Q. Feng, B.S. Jiang, Analytical calculation on temperature field of tunnels in cold region by LaPlace integral
transform, J. Min. Saf. Eng. 29 (2012) 391-395.
403
[12]
K. Okada, Lcile prevention by adiatic treatment of tunnel lining, Jpn. Railw. Eng. 26 (1985) 75-80.
404
[13]
K. Okada, F. Toshishige, S. Tomoyasu, K. Katsuhisa, I. Shigehiro, Adaptability of icicle prevention work by
405
adiabatic treatment in thin-earth-covering tunnel: Part. 1 (Tunnel cross section model and cyclic changes in
406
temperature), Bull. Sci. Eng. Res. Inst. 17 (2005) 1-8.
407
[14]
K. Okada, F. Toshishige, S. Tomoyasu, K. Katsuhisa, I. Shigehiro, Adaptability of icicle prevention work by
408
adiabatic treatment in thin-earth-covering tunnel: Part. 2 (Temperature responses in tunnel cross section and
409
adaptability of icicle prevention work), Bull. Sci. Eng. Res. Inst. 18 (2006) 10-23.
18
Page 18 of 25
410
[15]
X. Tan, W. Chen, D. Yang, Y. Dai, G. Wu, J. Yang, H. Yu, H. Tian, W. Zhao, Study on the influence of
411
airflow on the temperature of the surrounding rock in a cold region tunnel and its application to insulation
412
layer design, Appl. Therm. Eng. 67 (2014) 320-334.
413
[16]
414 415
X.F. Zhang, J.Z. Xiao, Y.N. Zhang, S.X. Xiao, Study of the function of the insulation layer for treating water
leakage in permafrost tunnels, Appl. Therm. Eng. 27 (2007) 637-645.
[17]
416
X.H. Zhou, Study on the influence of ventilation on frost resistance and reasonable range of resistance for
tunnel in cold region, Master dissertation, Southwest Jiaotong University, 2012, pp. 29-31 (in Chinese).
417
[18]
Z.R. Zhang, Heat Transfer, High Educational Press, Beijing, 1989.
418
[19]
J.R. Zhang, Z.Q. Liu, A study on the convective heat transfer coefficient of concrete in wind tunnel
419
experiment, China Civil Eng. J. 9 (2006) 39-42 (in Chinese).
Table and Figure Captions 420
Fig. 1. The partition schematic of the calculated tunnel cross section, indicating the numbering of the
421
computational nodes
422 423
Fig. 2. The distribution of the internal nodes in the tunnel along the axial direction
424
19
Page 19 of 25
425 426
Fig. 3. A longitudinal section of the Nan Shan tunnel with mileage and elevations, including the 600-m entrance
427
region from DK 68+928 to DK69+048 and the 600-m exit region from DK75+894 to DK76+574
428 429 430
Fig. 4. A cross section of the tunnel
431 432 20
Page 20 of 25
433
Fig. 5. The mechanical ventilation system of the Nan Shan tunnel, including 10 jet fans and 2 axial flow fans
434 435 436
Fig. 6. The changes in wind speed and direction in the tunnel throughout the day caused by train-induced wind
437 438 439
Fig. 7. The temperatures at various radial nodes in section DK73+139 as a function of time (considering natural
440
wind, train-induced wind and mechanical ventilation wind)
Surface of secondary lining 0.45 m (radial direction) 0.6 m (radial direction) 17.6 m (radial direction) 27.6 m (radial direction)
12 10
Temperature /℃
8 6 4 2 0 -2
0
100 200 300 400 500 600 700 800 900 100011001200130014001500160017001800
Time/day
441 21
Page 21 of 25
442
Fig. 8. The calculated temperatures at various axial nodes after 300 days (considering natural wind, train-induced
443
wind and mechanical ventilation wind)
Temperature /℃
10 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -14
surface of the secondary lining 0.45 m away from the secondary lining 0.7 m away from the secondary lining 17.7 m away from the secondary lining original temperature 0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Longitudinal distance along the tunnel /m
444 445 446
Fig. 9. The calculated temperatures at various axial nodes after 300 days (with and without mechanical ventilation
447
wind)
10 8 6 4
Temperature/℃
2 0 -2 -4 -6 -8
0.7m away from the surface of the secondary lining (under mechanical ventilation wind) 0.7m away from the surface of the secondary lining (no mechanical ventilation wind) the surface of the secondary lining (with mechanical ventilation wind) the surface of the secondary lining (no mechanical ventilation wind)
-10 -12 -14
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Longitudinal distance along the tuunel/m
448 449 450
Fig. 10. The calculated temperatures at various axial nodes after 300 days (with and without train-induced wind)
22
Page 22 of 25
6 4 2
temperature/℃
0 -2 -4 -6 -8 0.7m away from the surface of the secondary lining (under train-induced wind) 0.7m away from the surface of the secondary lining (no train-induced wind) the surface of the secondary lining (with train-induced wind) the surface of the secondary lining (no train-induced wind)
-10 -12 -14
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Longitudinal distance along the tunnel/m
451 452
Fig. 11. Temperature distributions in the lining and surrounding rock in a section at a distance of 1620 m from the
453
tunnel entrance for various wind speeds
8
temperature/℃
7
1 m/s 5 m/s 10 m/s 15 m/s 20 m/s
6
5
4
3
0
1
2
3
4
5
6
7
8
depth/m
454 455 456
Fig. 12. Temperature distributions in the lining and surrounding rock in a section at a distance of 1620 m from the
457
tunnel entrance for various wind temperatures
23
Page 23 of 25
20 15
temperature/℃
10 5 0 -5
-10 -15 0
1
2
3
4
5
6
-30℃ -20℃ -10℃ 0℃ 10℃ 20℃ 30℃ 7
8
depth/m
458 459 460
Fig. 13. Illustration of freezing sites in the Nan Shan tunnel
461 462 463
Fig. 14. The maximum calculated frozen lengths in the entrance region of the Nan Shan tunnel under actual
464
operating conditions (between the inner lining and the surrounding rock, below 0°C)
24
Page 24 of 25
1520
Frozen length/m
1500
1480
1460
1440
1420
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time/year
465 466 467
Table 1. Experimental thermal and physical parameters of dielectric materials at approximately 5°C Materials
Thickness (m)
Density
Specific heat capacity at constant
Thermal conductivity
pressure (J/(kg·℃))
( /(m·℃))
3
(kg/m )
surrounding rock
—
2400
850
2.5
initial liner
0.25
2500
1046
1.74
second lining
0.45
2500
1046
1.74
air flow
—
1.2
1005
—
468 469
25
Page 25 of 25
S1359-4311(16)30020-5 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.01.070 ATE 7630
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
22-11-2015 24-1-2016
Please cite this article as: Xiaohan Zhou, Yanhua Zeng, Lei Fan, Temperature field analysis of a cold-region railway tunnel considering mechanical and train-induced ventilation effects, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.01.070. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
Temperature field analysis of a cold-region railway tunnel
2
considering mechanical and train-induced ventilation effects
3
Xiaohan Zhoua, Yanhua Zeng a*, and Lei Fanb
4
a
5
Jiaotong University, Chengdu, China
6
b
7
*Corresponding authors:
8
Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest Jiaotong University,
9
Chengdu 610031, P.R. China
Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, Southwest
China Railway Eeyuan Engineering Group Co. Ltd, Chengdu, China
10
Tel: +86-15895873903
11
E-mail: [email protected]
12
Highlights
13
An unsteady-state finite-difference model of cold-region tunnel temperature fields is presented.
14
The coupled convection–conduction problem is considered.
15
Effects of mechanical ventilation and train-induced winds on the tunnel temperature distribution are
16 17
studied.
In situ observed frozen lengths in a cold-region railway tunnel are compared with the calculated results.
18 19
Abstract
20
In accordance with the unsteady-state finite-difference equations for heat transfer and heat convection in a
21
cold-region railway tunnel, an unsteady-state finite-difference computer model was developed to study the heat
22
convection between the air and the tunnel wall as well as the heat transfer in the surrounding rock and at interfaces
23
between different materials in the structure of the tunnel. The wind speed and wind direction of actual mechanical
24
ventilation wind produced during operation as well as the train-induced wind and natural wind in the tunnel were
1
Page 1 of 25
25
considered in the analysis of the temperature field distribution of a cold-region railway tunnel under actual
26
periodic variations in entrance wind temperature. Good agreement was observed between the calculated frozen
27
length and the frozen length observed in situ at the entrance of the railway tunnel. The results show that
28
mechanical ventilation winds during operation and train-induced winds significantly influence the temperature
29
field distribution in the tunnel and that the variables associated with mechanical ventilation winds and
30
train-induced winds should be considered in the engineering design of cold-region tunnels. In this way, the
31
temperature field of a railway tunnel can be correctly predicted via finite differencing.
32
Keywords: Cold-region railway tunnel, Mechanical ventilation wind, Train-induced wind, Temperature field,
33
Finite difference, Frozen length
34
1. Introduction
35
Research on cold-region tunnel design is becoming urgent because of the vast number of infrastructure
36
construction projects being undertaken for railways and highways in cold regions. The prediction of temperature
37
fields, which are used as references for relevant cold-proofing measures, is an important factor in cold-region
38
tunnel engineering design. Numerous studies have been conducted on temperature fields in cold-region tunnels.
39
Bonacina et al. proposed a finite-difference method for addressing melting and freezing with corresponding phase
40
changes; moreover, the problem formulation could be straightforwardly extended to multidimensional cases [1].
41
Comini et al. performed a finite-element analysis of the transient heat conduction problem with non-linear physical
42
properties and boundary conditions, in which latent heat effects were treated as heat capacity variations within a
43
narrow temperature range [2]. Lai et al. performed nonlinear analyses of related problems concerning temperature,
44
seepage and stress fields in cold-region tunnels based on the theory of heat transfer using finite-element formulas
45
[3], and related studies have demonstrated that good insulation should be installed to avoid substantial
46
frost-heaving damage to the tunnel lining [4]. He et al. predicted the freezing–thawing conditions in the rock
2
Page 2 of 25
47
surrounding tunnels in cold regions using a combined convection–conduction model based on factors including the
48
in situ air temperature, atmospheric pressure, and wind force conditions [5,6]. Lai et al. proposed an approximate
49
analytical solution for temperature fields in cold-region circular tunnels based on a dimensionless perturbative
50
method that can be used when the initial temperature is nearly 0°C [7]. Zhang et al. analyzed three-dimensional
51
temperature characteristics in cold-region tunnels using the finite-element method [8]. Zhang et al. performed a
52
forecast analysis of the refreezing of the Kunlunshan permafrost tunnel of the Qing-Tibet Railway in China [9].
53
Lai et al. [10] proved that the freezing-thawing behaviors of the rock surrounding a tunnel can be correctly
54
predicted even if the air temperature along the tunnel is unknown. Feng et al. adopted the Stehfest numerical
55
inversion and Laplace transform methods to address this problem [11].
56
To date, proper cold-proofing measures have been applied in China, Norway, Russia, Canada and Japan to
57
decrease the frost damage experienced by tunnels in cold regions. In Japan, Okada et al. first introduced surface
58
adiabatic treatments and adiabatic double lining to prevent water leakage and the formation of icicles and
59
described a cross-sectional model considering heat insulation of the lining surface and cyclically varying
60
temperatures in thin earth covers. In addition, these authors discussed the different temperature modes
61
corresponding to different heat insulator depths and different earth covers [12-14]. In Norway, heated cables, some
62
of which are fitted with double thermal insulation doors, whereas others have insulation installed, are used to heat
63
certain drains [10]. Many useful methods, such as slurry injection and the use of insulation materials, thermal
64
insulation doors, snow sheds and drainage holes, have been proposed for cold-region tunnels in China [15]. A
65
three-dimensional nonlinear analysis of the coupled problem of heat transfer and seepage both with and without an
66
insulation layer was conducted for the Kunlun mountain tunnel by Zhang et al. [16]. An analysis of the refreezing
67
of the Feng Huoshan tunnel in the presence of thermal insulation material was conducted by Zhang et al. to study
68
the behavior of the thermal insulation material in a permafrost tunnel during the refreezing process [9]. To
3
Page 3 of 25
69
determine the appropriate scope of insulation materials or slurry injections for cold-regions tunnels, the predicted
70
scale of the tunnel structure at freezing temperatures must be confirmed via predictive calculations of the
71
temperature fields before construction.
72
Tan et al. introduced a calculation model that includes control equations for the surrounding rock and air
73
temperatures to analyze the temperature field of the surrounding rock of the Galongla tunnel in Tibet and the
74
thermal insulation measures applied under ventilation conditions; the calculations were performed using the
75
commercial finite-element software package COMSOL Mutiphysics (version 4.2) [15]. It is known in the field of
76
railway tunnel ventilation that, in addition to the natural winds from the entrance or exit of a tunnel, mechanical
77
ventilation and train-induced winds are present inside the tunnel and cannot be ignored when the tunnel is in use.
78
Thus, this paper primarily examines the influence of natural, mechanical ventilation and train-induced winds on
79
the temperature fields in a cold-region tunnel based on both theoretical finite-difference calculations and the results
80
of a field investigation. In addition, the effects of mechanical ventilation and train-induced winds on the freezing
81
scale in a cold-region tunnel are analyzed.
82
2. Tunnel temperature field model based on the coupled problem of
83
the convective exchange between the air and tunnel lining and the
84
heat transfer in the lining structures and surrounding rock
85
2.1. Fundamental assumptions
86
The primary purpose of the current study was to determine how the mechanical ventilation wind and train-induced
87
wind in a tunnel influence the tunnel’s temperature field distribution. Thus, for simplicity, the following basic
88
assumptions are made regarding the transient heat transfer:
89
(1) The calculated cross section along the longitudinal direction of the tunnel is circular [6,7,17], and the
90
equivalent hydraulic radius is obtained in the computation.
4
Page 4 of 25
91 92
(2) At the beginning of the calculation, the temperature of the tunnel structure is equal to the initial temperature
of the surrounding rock, which is constant in the radial direction.
93
(3) The air is laminar and incompressible, and the air pressure does not vary with temperature [10,15].
94
2.2. Governing equations
95
The process of heat transfer in the surrounding rock-lining-wind system primarily involves the heat transfer
96
between the surrounding rock and the tunnel lining, the convective heat transfer between the tunnel wall and the
97
air, and any thermal radiation that occurs. The thermal radiation is ignored in this analysis because of its relatively
98
small magnitude.
99
The transmission of heat is described by the Fourier law [18] as follows:
100
(1) is the temperature gradient in the direction normal to the area A, λ is the heat transfer rate, and φ is the
101
where
102
heat that passes through a given area in unit time.
103
The temperature field of a tunnel consists of a set of temperatures at various points at every moment. The
104
temperature function in a three-dimensional rectangular coordinate system can be expressed as
105
(2)
106
According to the fundamental assumptions stated in Section 2.1, the governing equation of the longitudinal
107
transient temperature field of the surrounding rock and lining in cylindrical coordinates [10,17] is
108 109
(3)
The boundary conditions are
110
(4)
111
(5)
112
where T is the temperature of the tunnel lining and surrounding rock,
is the initial temperature of the
5
Page 5 of 25
113
surrounding rock, Tf is the wind temperature, r is the radial distance, a (
114
coefficient, t is time, ρ is the density of the tunnel lining and surrounding rock, and h is the convective heat transfer
115
rate.
116
The wind in the tunnel is regarded as a non-viscous, incompressible fluid exhibiting a laminar flow, non-viscous,
117
incompressible fluid, and the heating effect from the equipment in the tunnel is considered. The control equation
118
for the transient temperature of the axial wind in the tunnel [17] is
119 120
) is the thermal diffusion
(6)
The boundary conditions are
121
(7)
122
where
123
effect in the tunnel, and
124
At the frost front position s(t), the continuity condition and the conservation-of-energy requirement should be
125
satisfied, i.e.,
is the temperature of the tunnel wall, U is the perimeter of the cross section, qs is the equipment heating is the constant-pressure specific heat capacity of the surrounding rock.
126
(8)
127
Suppose that the phase change occurs in the temperature region denoted by Tm+
128
frozen regions ( ), the heat capacity of unfrozen regions ( ), the heat transfer rate in frozen regions (λf) and the
129
heat transfer rate in unfrozen regions (λu) do not depend on the temperature (T). Then, the following definitions
130
may be assumed [8]:
131
132
and that the heat capacity of
(9)
and
133
(10)
6
Page 6 of 25
134
2.3. Finite-difference discretization
135
Numerically calculating partial differential equations requires two steps. First, the continuous calculation range is
136
divided into finite and discrete point sets, and the partial differential equations and definite conditions are
137
transformed into algebraic equations at the corresponding points. Then, the computed results for the partial
138
differential equations at these points are obtained. Because there must be a smooth change between the computed
139
results for adjacent discrete points, approximate solutions over the entire calculation range can be obtained via
140
interpolation. Clearly, the accuracy of the computed results strongly depends on the number of discrete points
141
considered. Finally, the time domain should also be discretized for transient heat conduction problems.
142
Based on the actual structure of the tunnel, discrete processes in the spatial domain were adopted when
143
implementing the temperature field calculations in this study. The tunnel structures in a transverse section were
144
assumed to be circular with hydraulic radii[6,7,17], and the gridding was assumed to be centrosymmetric. A model
145
with equidistant gridding in the tunnel's axial direction was adopted. The mesh generated for the model used to
146
calculate the temperature field in the tunnel is detailed in Fig. 1 and Fig. 2.
147
Fig. 1 and Fig. 2 depict the horizontal and vertical grids, respectively. Finite-difference equations were established
148
for the heat transmission between the layers of tunnel lining, between the tunnel lining and the surrounding rock,
149
and inside the surrounding rock. By contrast, partial differential equations were used to model the heat convection
150
between the air and the tunnel wall. In combination with the law of energy conservation, the entire asynchronous
151
differencing scheme for the transient heat transfer calculation was established as follows [17].
152
The formula for approximating the temperature of a grid node within the surrounding rock is
153
(11)
154
where
155
cross-sectional plane;
is the temperature of node i,j at time n;
is the radial distance corresponding to the step size in the
is the longitudinal step size; F0 is the Fourier value,
; and
is the time step.
7
Page 7 of 25
156
The formula for approximating the temperature of a node at a convection boundary is
157 158
(12)
159
where
160
The formula for approximating the temperature of a node at an intersection point between materials is
is the wind temperature of node i at time n.
161 162
(13)
The longitudinal approximation formulas for the wind temperature in the tunnel are
163
(14)
164
where
165
the tunnel.
166
The approximation formulas given above for the temperatures of the wind and the surrounding rock include both
167
heat conduction and heat convection, and both distance and time steps are considered. Furthermore, the wind
168
temperature and wind speed are taken into account in Tf and v.
169
3. Finite-difference analysis of a cold-region railway tunnel
170
The Nan Shan railway tunnel, located in northeastern China, is analyzed here as an example. The natural wind,
171
mechanical ventilation wind and train-induced wind in the tunnel are the major factors considered in calculating
172
the temperature field using the finite-difference method presented in this paper.
173
3.1. Introduction to the Nan Shan tunnel
174
The Nan Shan railway tunnel is a cold-region tunnel in Jilin Province, China. The coldest monthly average
175
temperature in the region is -12.5°C, and the extreme low monthly average temperature is -17°C; the altitude is
is the wind temperature at node i at time n+1, A is the cross-sectional area, and v is the wind speed in
8
Page 8 of 25
176
approximately 1900 m. The Nan Shan tunnel is located in an area with a humid temperate continental monsoon
177
climate with long cold snowy winters and cool wet summers.
178
3.2. Model, conditions and parameters
179
The Nan Shan tunnel has a length of 7,566 m (Bai-He to Long-He), a downward slope of 0.95%, and an elevation
180
ranging from 749.84 m to 821.72 m. The calculated cross-sectional area is 32.43 m2, the wetted perimeter is 21.76
181
m, and the equivalent diameter of the tunnel is 5.96 m, which includes a 0.45-m secondary lining and a 0.25-m
182
inner lining, as shown in the cross section in Fig. 4.
183
A longitudinal section of the Nan Shan tunnel is shown in Fig. 3. From knowledge of the buried depth and local
184
climate data of the tunnel, the initial temperature of the surrounding rock can be obtained by taking the
185
temperature gradient to be 3°C/100 m and the local average soil temperature to be equal to the subsurface
186
temperature [10].
187
The average measured natural wind speed in the tunnel is 2.5 m/s, with the wind blowing from Bai-He to Long-He.
188
Train-induced wind is also considered in this paper. This latter wind speed is calculated as follows:
mm
-
189 190 191 192
m
mm
where
n n n
(12)
is the train-induced wind;
;
-l m
;
d
n
d
;
m
;
is the train speed;
-
is the train length; N is the resistance coefficient, where
; λr is the frictional resistance coefficient of the tunnel wall; and
is the blockage ratio, where
f
, with
193
fT being the cross-sectional area of a train and A being the cross-sectional area of the tunnel.
194
Three pairs of freight trains and one pair of passenger trains, all of which are driven by diesel locomotives, travel
195
through the Nan Shan tunnel on a daily basis. Using Equation 12 and a train speed of 60 km/h, the train-induced
196
wind speeds are found to be 6.8 m/s and 6.1 m/s when the train is moving with and against the direction of the
197
natural wind, respectively, as shown in Fig. 6. 9
Page 9 of 25
198
As shown in Fig. 4 and Fig. 5, the Nan Shan tunnel ventilation system consists of 10 jet fans and 4 axial flow fans
199
located at the Long-He end. The axial flow fans are installed at equal distances in two air holes at DK76+424 and
200
DK76+354, and the jet fans are hung at the exit of the tunnel at DK76+454. According to the relevant design data
201
and the literature, the mechanical ventilation time required to exhaust dusty air is 33 minutes after a train passes
202
through the Nan Shan tunnel during operation. After a train passes through the tunnel during actual operation, the
203
corresponding fans are activated to create a reverse wind. Four pairs of trains pass through the tunnel on a daily
204
basis, and the measured mechanical ventilation wind speed is 1.5 m/s from the exit to the entrance of the tunnel.
205
The average specific heat capacities, densities, average thermal conductivity coefficients and other parameters of
206
the secondary lining, inner lining and surrounding rock can be obtained from the relevant experimental data given
207
in Table 1. The convective heat transfer rate is found to be h=3.06v+9.55
208
between the wind velocity and the convective heat transfer rate discussed by Zhang et al. [19]. According to the
209
climate data recorded at the Nan Shan tunnel over the past 20 years, the periodic entering wind temperature is
210
/(m2·°C), according to the relationship
(°C), where i represents time (in days).
211
3.3. Results and discussion
212
The train-induced wind and the mechanical ventilation wind are both considered in the calculation. The
213
train-induced wind and mechanical ventilation wind primarily influence the speed and direction of the overall wind
214
in the tunnel, which in turn will influence the convective heat transfer between the air and the tunnel wall.
215
Therefore, the effects of the wind speed and wind temperature on the temperature distributions in the lining and
216
surrounding rock are studied below.
217
3.3.1. Temperature distributions in the lining and surrounding rock in the presence of
218
natural, train-induced and mechanical ventilation winds
219
The tunnel section DK73+139 (4211 m from the tunnel entrance) was selected for the analysis of the
10
Page 10 of 25
220
cross-sectional temperature distribution, as shown in Fig. 7. The natural wind speed is 2.5 m/s from Bai-He to
221
Long-He; the mechanical ventilation speed is 1.5 m/s from Long-He to Bai-He; the train-induced wind speeds are
222
6.8 m/s and 6.1 m/s when moving with and against the direction of the natural wind, respectively; and the wind
223
temperature at the tunnel entrance is
224
distributions at various radial points in section DK73+139 follow the same sine distribution rule as the inlet wind
225
temperature when the natural wind, train-induced wind and mechanical ventilation wind are all considered. Lesser
226
changes in temperature amplitude and greater time lags compared with the sine-wave profile of the entering wind
227
temperature are predicted for regions farther from the secondary lining surface. The changes in temperature
228
amplitude observed during the first cycle are 11.02, 8.58, 5.87, 1.38 and 0.29°C, with cycles of approximately 400,
229
410, 430, 540 and 640 days at points at distances of 0.45, 0.7, 17.6, and 27.6 m from the surface of the secondary
230
lining.
231
The temperature distributions in regions at different radial depths along the entire tunnel, under the natural wind,
232
mechanical ventilation wind, train-induced wind and temperature conditions given in Section 3.1, are shown in Fig.
233
8. The figure shows that different radial nodes generally follow the same distributional trends in the longitudinal
234
direction. After 300 days of the simulation, from the surface of the secondary lining to a distance of 17.7 m from
235
its surface, the longitudinal temperature begins to approximate the initial temperature of the surrounding rock. The
236
temperature distribution at a distance of 17.7 m from the surface of the secondary lining after 300 days of the
237
simulation is approximately identical to the original temperature distribution along the tunnel in the longitudinal
238
direction, except in the entrance and exit regions.
239
3.3.2. Influence of mechanical ventilation wind on temperature distributions in the
240
lining and surrounding rock
241
The temperature distributions along the entire tunnel at the surface of the secondary lining and at a distance of 0.7
(°C). The figure shows that the temperature
11
Page 11 of 25
242
m from its surface after 300 days of simulation, with the mechanical ventilation wind both considered and not
243
considered as described in section 3.3.1, are shown in Fig. 9. The entrance region (Bai-He) and the exit region
244
(Long-He) are found to follow opposite temperature distribution rules with and without mechanical ventilation
245
wind. Within 600 m of the entrance region, the average longitudinal temperature on the surface of the secondary
246
lining in the presence of mechanical ventilation wind is 1.1054°C higher than the average longitudinal temperature
247
without mechanical ventilation wind. Within 600 m of the exit region, the average longitudinal temperature on the
248
surface of the secondary lining in the presence of mechanical ventilation wind is 1.7954°C lower than the average
249
longitudinal temperature without mechanical ventilation wind. In addition, these temperature differences become
250
0.7722 and 0.4702°C, respectively, for points at a distance of 0.7 m from the surface of the secondary lining. Thus,
251
because of the direction of the mechanical ventilation fans (from Long-He to Bai-He), the exit region of the Nan
252
Shan tunnel is more sensitive to mechanical ventilation effects.
253
3.3.3. Influence of train-induced wind on temperature distributions in the lining and
254
surrounding rock
255
The temperature distributions along the entire tunnel at the surface of the secondary lining and at a distance of 0.7
256
m from its surface after 300 days of simulation, with the train-induced wind both considered and not considered as
257
described in section 3.3.1, are shown in Fig. 10. The train-induced wind travels in the same direction as the train;
258
thus, this wind amplifies the heat convection between the air and the tunnel wall without affecting the temperature
259
distribution rule governing the tunnel as a whole. The temperature distribution in the exit region exhibits a
260
substantial difference depending on whether the train-induced wind is considered because the train-induced wind
261
from Long-He to Bai-He pulls cold air into the tunnel, strongly cooling the lining and surrounding rock.
262
3.3.4. Influence of wind speed on temperature distributions in the lining and
263
surrounding rock 12
Page 12 of 25
264
To provide a quantitative analysis of the influence of different wind speeds on the temperature distributions, a case
265
study is performed in this section. The temperature distributions in the lining and surrounding rock in a section at a
266
distance of 1620 m from the tunnel entrance (Bai-He) after 10 days of simulation at different wind speeds,
267
calculated under the conditions presented in section 3.3.1, are shown in Fig. 11. The figure shows that when the
268
wind speed is less than 5 m/s, an increase in the wind speed has a large influence on the temperature distribution.
269
By contrast, when the wind speed is higher than 10 m/s, the effect is very small, especially in regions at a depth of
270
greater than 3 m. The main reason for this difference in behavior is that the thermal exchange between the air and
271
the lining is stable when the wind speed is higher than 10 m/s [15]. Furthermore, Fig. 11 shows that the wind speed
272
has a greater influence on the temperature distribution at shallower depths.
273
3.3.5. Influence of wind temperature on temperature distributions in the lining and
274
surrounding rock
275
For the calculations presented here, the wind speed at the tunnel entrance (Bai-He) was fixed at 2.5 m/s and the
276
other conditions presented in section 3.3.1 were used. The temperature distributions in the lining and surrounding
277
rock in a section at a distance of 1620 m from the tunnel entrance (Bai-He) after 30 days of simulation at different
278
wind speeds are shown in Fig. 12. The figure illustrates that the wind temperature strongly influences the
279
temperature distribution in the lining and surrounding rock, primarily because a large temperature difference
280
between the air and the tunnel wall will promote heat convection.
281
3.3.6. Discussion
282
During tunnel operation, the temperatures of the lining and surrounding rock vary over time because of the
283
temperature differences that form between the air and the tunnel wall under the influence of periodic variations in
284
wind temperature. The temperatures of the lining and surrounding rock vary with the same periodic behavior as the
285
wind temperature but at a relative time delay. The temperature distributions of nodes at different radii generally
13
Page 13 of 25
286
follow the same distributional trends in the longitudinal direction.
287
The entrance region (Bai-He) and the exit region (Long-He) are found to follow opposite temperature distribution
288
rules with and without mechanical ventilation wind because of the reverse wind direction of the fans in the exit
289
region compared with the natural wind direction. Train-induced wind travels in the same direction as the train; thus,
290
train-induced wind amplifies the heat convection between the air and the tunnel wall without affecting the
291
temperature distribution rule governing the tunnel as a whole. The temperature distribution in the exit region
292
differs substantially depending on whether train-induced wind is considered because the train-induced wind from
293
Long-He to Bai-He pulls cold air into the tunnel, strongly cooling the lining and surrounding rock. Every railway
294
tunnel has its own properties of mechanical ventilation and train scheduling; thus, the direction and speed of the
295
ventilation fans are not common to all tunnels. Therefore, both mechanical ventilation and train-induced winds
296
should be considered when calculating the temperature field of a railway tunnel.
297
To provide a quantitative analysis of the influence of different wind speeds and wind temperatures on the
298
temperature distribution in a tunnel, two case studies were performed. The results show that both wind speed and
299
wind temperature considerably influence the temperature distribution because of their influence on the heat
300
convection between the air and the tunnel wall. A high wind speed results in the introduction of more cold air into
301
the tunnel and leads to a large convection coefficient; at the same time, a large temperature difference between the
302
air and the tunnel wall will also promote heat convection between the air and the wall.
303
3.4. Validation of the calculated result against the field-measured frozen
304
length
305
The Nan Shan tunnel opened to traffic in 2006, and a field investigation of this tunnel during the winter of 2009
306
showed that the frozen length at the entrance (Bai-He) reached a maximum of 1,472 m. The freezing sites included
307
the side drainage ditches and tunnel linings (see Fig. 13).
14
Page 14 of 25
308
To avoid damage from cold temperatures, the predicted scale of a tunnel structure under freezing temperatures is
309
important and useful in determining the appropriate scope of antifreezing measures, such as insulation layers in the
310
tunnel. The temperature field in the Nan Shan tunnel was calculated using the finite-difference model presented in
311
Section 3.3.1 given the actual wind speed and wind temperature conditions. Fig. 14 shows a maximum calculated
312
longitudinal length at temperatures below 0°C in the area between the inner lining and the surrounding rock in the
313
entrance region of the Nan Shan tunnel within the first year of operation of approximately 1,440 m, increasing to
314
approximately 1,500 m during the second year and subsequently remaining at approximately the same value,
315
which is consistent with the results of the field investigation conducted in 2009. Thus, the proposed
316
finite-difference model could be successfully applied to accurately calculate the basic temperature field
317
distribution of the Nan Shan tunnel.
318
4. Conclusions
319
Unsteady-state finite-difference equations for heat transfer and heat convection were established to calculate the
320
temperature field of a cold-region railway tunnel considering natural wind, train-induced wind and mechanical
321
ventilation wind. The temperature distributions in the tunnel structures and surrounding rock under natural wind,
322
train-induced wind and mechanical ventilation wind as well as the influence of wind temperature, wind speed and
323
wind direction on the temperature distribution were studied. The following conclusions were drawn based on a
324
combination of finite-difference calculations and the results of an in situ investigation.
325
(1) The wind temperature and wind speed inside a tunnel significantly influence the tunnel’s temperature
326
distribution. The temperature distributions at different radial points follow the same distribution rule as that
327
governing the natural wind temperature at the tunnel entrance,
328
radial distances experience smaller changes in temperature amplitudes and greater time lags with respect to the
329
wind temperature.
(°C). Points at greater
15
Page 15 of 25
330
(2) Train-induced wind and mechanical ventilation wind generally amplify the heat convection between the air and
331
the tunnel structures. Train-induced wind travels in the same direction as the train; thus, it amplifies the heat
332
convection between the air and the tunnel wall without significantly affecting the temperature distribution rule
333
governing the tunnel as a whole. Because of the reversed direction of the fans in the exit region compared with that
334
of the natural wind in the Nan Shan tunnel, the entrance and exit regions follow opposite temperature distribution
335
rules when mechanical ventilation effects are considered. These ventilation effects cannot be ignored when
336
predicting tunnel temperature fields.
337
(3) The accuracy of the temperature field calculation method for cold-region railway tunnels based on the
338
ventilation effect was verified by comparing the calculated results with the data from an in situ investigation of the
339
Nan Shan tunnel. The finite-difference model proposed in this paper can be used to guide the design of insulation
340
layers for cold-region railway tunnels, thereby overcoming the difficulties encountered because of the complexity
341
of modeling and stringent hardware requirements of general finite-element software.
342 343 344 345 346 347 348 349 350 351
Nomenclature
352 353 354 355 356 357 358 359 360 361
A
cross-sectional area [m2] temperature gradient in the direction normal to the area A
λ
heat transfer rate [W/m·K]
φ
heat passing through a given area in unit time [W/m2·s]
T
temperature of the tunnel lining and surrounding rock [°C] initial temperature of the surrounding rock [°C]
Tf
wind temperature [℃]
r
radial distance [m]
a
thermal diffusion coefficient (
t
time [s]
ρ
density [kg/m3]
h
convective heat transfer rate [W/m2·K]
U
cross-sectional perimeter [m]
qs
equipment heating effect in the tunnel [W]
)
tunnel wall temperature [℃]
constant-pressure specific heat capacity [J/m3·K] temperature of node i,j at time n [°C] radial distance corresponding to the step size in the cross-sectional plane [m] longitudinal step size [m] 16
Page 16 of 25
362
F0
363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
Fourier value, time step [s] wind temperature at node i at time n [°C] wind temperature at node i at time n+1 [°C]
v
wind speed in the tunnel [m/s]
i
time [d] train-induced wind [m/s] train speed [m/s] train length [m]
N r
resistance coefficient of a train [1/m] frictional resistant coefficient of the tunnel wall blockage ratio
fT
cross-sectional area of a train [m2] heat capacity in frozen regions [J/m3·K] heat capacity in unfrozen regions [J/m3·K]
λf
heat transfer rate in frozen regions [W/m·K]
λu
heat transfer rate in unfrozen regions [W/m·K]
References 379
[1]
380 381
Heat Mass Transf. 16 (1973) 1825-1832.
[2]
382 383
[3]
Y.M. Lai, Z.W. Wu, Y.L. Zhu, L.N. Zhu, Nonlinear analysis for the coupled problem of temperature,
seepage and stress fields in cold-region tunnels, Tunnelling Undergr. Space Technol. 13 (1998) 435–440.
[4]
386 387
G. Comini, S. Del Guidice, R.W. Lewis, O.C. Zienkiewicz, Finite element solution of non-linear heat
conduction problems with special reference to phase change, Int. J. Numer. Methods Eng. 8 (1974) 613-624.
384 385
C. Bonacina, G. Comini, A. Fasano, M. Primicerio, Numerical solution of phase-change problems, Int. J.
Y. Lai, Z. Wu, Y. Zhu, L. Zhu, Nonlinear analysis for the coupled problem of temperature and seepage fields
in cold-region tunnels, Cold Regions Sci. Technol. 29 (1999) 89–96.
[5]
C.X. He, Z.W. Wu, Preliminary prediction for the freezing –thawing situation in rock surrounding
388
DabanShan tunnel, in: Proceedings of the Fifth National Conference on Glaciology and Geocryology,
389
Culture Press of Gan Su, Lan Zhou, 1996, pp. 419–425.
17
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390
[6]
391 392
in the surrounding rock wall of a tunnel in permafrost regions, Sci. China Ser. D 29 (1999) 1–7.
[7]
393 394
[8]
X.F. Zhang, Y.M. Lai, W.B. Yu, S.J. Zhang, Nonlinear analysis for the three-dimensional temperature fields
in cold region tunnels, Cold Regions Sci. Technol. 35 (2002) 207-219.
[9]
397 398
Y.M. Lai, S.Y. Liu, Z.W. Wu, W.B. Yu, Approximate analytical solution for temperature fields in cold
regions circular tunnels, Cold Regions Sci. Technol. 13 (2002) 43–49.
395 396
C.X. He, Z.W. Wu, L.N. Zhu, A convection – conduction model for analysis of the freeze – thaw conditions
X.F. Zhang, Y.M. Lai, W.B. Yu, S.J. Zhang, J.Z. Xiao, Forecast analysis for the re-frozen of Kunlunshan
permafrost tunnel on Qing-Tibet railway in China, Cold Regions Sci. Technol. 38 (2004) 12–22.
[10]
Y.M. Lai, X.F. Zhang, W.B. Yu, S.J. Zhang, Z.Q. Liu, J.Z. Xiao, Three-dimensional nonlinear analysis for
399
the coupled problem of the heat transfer of the surrounding rock and the heat convection between the air and
400
the surrounding rock in cold-region tunnel, Tunnelling Undergr. Space Technol. 20 (2005) 323-332.
401
[11]
402
Q. Feng, B.S. Jiang, Analytical calculation on temperature field of tunnels in cold region by LaPlace integral
transform, J. Min. Saf. Eng. 29 (2012) 391-395.
403
[12]
K. Okada, Lcile prevention by adiatic treatment of tunnel lining, Jpn. Railw. Eng. 26 (1985) 75-80.
404
[13]
K. Okada, F. Toshishige, S. Tomoyasu, K. Katsuhisa, I. Shigehiro, Adaptability of icicle prevention work by
405
adiabatic treatment in thin-earth-covering tunnel: Part. 1 (Tunnel cross section model and cyclic changes in
406
temperature), Bull. Sci. Eng. Res. Inst. 17 (2005) 1-8.
407
[14]
K. Okada, F. Toshishige, S. Tomoyasu, K. Katsuhisa, I. Shigehiro, Adaptability of icicle prevention work by
408
adiabatic treatment in thin-earth-covering tunnel: Part. 2 (Temperature responses in tunnel cross section and
409
adaptability of icicle prevention work), Bull. Sci. Eng. Res. Inst. 18 (2006) 10-23.
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410
[15]
X. Tan, W. Chen, D. Yang, Y. Dai, G. Wu, J. Yang, H. Yu, H. Tian, W. Zhao, Study on the influence of
411
airflow on the temperature of the surrounding rock in a cold region tunnel and its application to insulation
412
layer design, Appl. Therm. Eng. 67 (2014) 320-334.
413
[16]
414 415
X.F. Zhang, J.Z. Xiao, Y.N. Zhang, S.X. Xiao, Study of the function of the insulation layer for treating water
leakage in permafrost tunnels, Appl. Therm. Eng. 27 (2007) 637-645.
[17]
416
X.H. Zhou, Study on the influence of ventilation on frost resistance and reasonable range of resistance for
tunnel in cold region, Master dissertation, Southwest Jiaotong University, 2012, pp. 29-31 (in Chinese).
417
[18]
Z.R. Zhang, Heat Transfer, High Educational Press, Beijing, 1989.
418
[19]
J.R. Zhang, Z.Q. Liu, A study on the convective heat transfer coefficient of concrete in wind tunnel
419
experiment, China Civil Eng. J. 9 (2006) 39-42 (in Chinese).
Table and Figure Captions 420
Fig. 1. The partition schematic of the calculated tunnel cross section, indicating the numbering of the
421
computational nodes
422 423
Fig. 2. The distribution of the internal nodes in the tunnel along the axial direction
424
19
Page 19 of 25
425 426
Fig. 3. A longitudinal section of the Nan Shan tunnel with mileage and elevations, including the 600-m entrance
427
region from DK 68+928 to DK69+048 and the 600-m exit region from DK75+894 to DK76+574
428 429 430
Fig. 4. A cross section of the tunnel
431 432 20
Page 20 of 25
433
Fig. 5. The mechanical ventilation system of the Nan Shan tunnel, including 10 jet fans and 2 axial flow fans
434 435 436
Fig. 6. The changes in wind speed and direction in the tunnel throughout the day caused by train-induced wind
437 438 439
Fig. 7. The temperatures at various radial nodes in section DK73+139 as a function of time (considering natural
440
wind, train-induced wind and mechanical ventilation wind)
Surface of secondary lining 0.45 m (radial direction) 0.6 m (radial direction) 17.6 m (radial direction) 27.6 m (radial direction)
12 10
Temperature /℃
8 6 4 2 0 -2
0
100 200 300 400 500 600 700 800 900 100011001200130014001500160017001800
Time/day
441 21
Page 21 of 25
442
Fig. 8. The calculated temperatures at various axial nodes after 300 days (considering natural wind, train-induced
443
wind and mechanical ventilation wind)
Temperature /℃
10 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -14
surface of the secondary lining 0.45 m away from the secondary lining 0.7 m away from the secondary lining 17.7 m away from the secondary lining original temperature 0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Longitudinal distance along the tunnel /m
444 445 446
Fig. 9. The calculated temperatures at various axial nodes after 300 days (with and without mechanical ventilation
447
wind)
10 8 6 4
Temperature/℃
2 0 -2 -4 -6 -8
0.7m away from the surface of the secondary lining (under mechanical ventilation wind) 0.7m away from the surface of the secondary lining (no mechanical ventilation wind) the surface of the secondary lining (with mechanical ventilation wind) the surface of the secondary lining (no mechanical ventilation wind)
-10 -12 -14
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Longitudinal distance along the tuunel/m
448 449 450
Fig. 10. The calculated temperatures at various axial nodes after 300 days (with and without train-induced wind)
22
Page 22 of 25
6 4 2
temperature/℃
0 -2 -4 -6 -8 0.7m away from the surface of the secondary lining (under train-induced wind) 0.7m away from the surface of the secondary lining (no train-induced wind) the surface of the secondary lining (with train-induced wind) the surface of the secondary lining (no train-induced wind)
-10 -12 -14
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000
Longitudinal distance along the tunnel/m
451 452
Fig. 11. Temperature distributions in the lining and surrounding rock in a section at a distance of 1620 m from the
453
tunnel entrance for various wind speeds
8
temperature/℃
7
1 m/s 5 m/s 10 m/s 15 m/s 20 m/s
6
5
4
3
0
1
2
3
4
5
6
7
8
depth/m
454 455 456
Fig. 12. Temperature distributions in the lining and surrounding rock in a section at a distance of 1620 m from the
457
tunnel entrance for various wind temperatures
23
Page 23 of 25
20 15
temperature/℃
10 5 0 -5
-10 -15 0
1
2
3
4
5
6
-30℃ -20℃ -10℃ 0℃ 10℃ 20℃ 30℃ 7
8
depth/m
458 459 460
Fig. 13. Illustration of freezing sites in the Nan Shan tunnel
461 462 463
Fig. 14. The maximum calculated frozen lengths in the entrance region of the Nan Shan tunnel under actual
464
operating conditions (between the inner lining and the surrounding rock, below 0°C)
24
Page 24 of 25
1520
Frozen length/m
1500
1480
1460
1440
1420
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time/year
465 466 467
Table 1. Experimental thermal and physical parameters of dielectric materials at approximately 5°C Materials
Thickness (m)
Density
Specific heat capacity at constant
Thermal conductivity
pressure (J/(kg·℃))
( /(m·℃))
3
(kg/m )
surrounding rock
—
2400
850
2.5
initial liner
0.25
2500
1046
1.74
second lining
0.45
2500
1046
1.74
air flow
—
1.2
1005
—
468 469
25
Page 25 of 25