Abstract
In order to study the cooling effect of the crushed-rocks embankment, the permeability and the inertial resistance coefficient were measured by the wind tunnel test of spheres with a diameter of 20 cm, and then the stabilities of the closed crushed-rocks embankment with the wide pavement, the closed crushed-rocks embankment with the narrow pavement, and the duct-ventilated and closed crushed-rocks embankment were calculated. In the next 50 years, assuming that the temperature in Qinghai-Tibet plateau will rise by 2.6°C condition, the cooling effects of these three special high-grade embankment structures were studied. The test results and the numerical calculation results show that the relationship between pressure gradient and seepage velocity in the spheres layer diverges completely from Darcy’s law, and it shows a nice quadratic nonlinear relationship. Stabilities of those two closed crushed-rock embankments without the duct-ventilated structure could be destroyed because of the high permafrost temperature under embankments. The duct-ventilated and closed crushed-rocks embankment can cool down the permafrost effectively and raise the permafrost table and ensure the long-term thermal stability of permafrost under road.
1. Introduction
The crushed-rocks embankment has been widely applied in permafrost road engineering. Existing field tests and practical engineering applications show that the crushed-rocks embankment can provide a better cooling effect [1, 2], and large amount of test data provides reference for the design [3–5] and understanding of the cooling characteristics of the engineering structure [6–8]. Indoor model tests of various scales and structures have helped understand the cooling mechanism of the crushed-rocks. Forced convective heat transfer mostly occurs in open crushed-rocks, which, when in cold seasons, due to high density of cold air, squeezes hot air out of the ventilation tunnel under the effect of gravity and wind, constantly taking away heat from surrounding soil, effectively protecting the permafrost. Closed crushed-rocks, which mainly experience natural convective heat transfer, when in warm seasons, cause hot air to move up due to surface heat. Closed crushed-rocks act as a effective thermal barrier, make the air inside of it at a thermally conductive state, and prevent heat penetrating into the deeper permafrost. Internal air experiences different densities due to temperature upwards higher than that of downwards in cold seasons, which cause natural convective heat transfer of the air inside, eventually cooling down the embankment [9].
Currently, mathematical models about closed crushed-rocks embankment are primarily based on continuity equation in porous materials, energy equation, non-Darcy momentum equation, and Boussinesq approximation. Half-empirical formulae are often used to calculate the permeability and the inertial friction coefficient in non-Darcy momentum equation [10]; however, significant differences may occur, compared with observed data. In practice, this could change the convective properties of air, further affecting the temperature field distribution characteristics of the whole embankment. This paper used wind tunnel test and obtained the permeability and the inertial friction coefficient of cubic packed spheres and applied them into the numerical analysis for crushed-rocks embankment along Qinghai-Tibet high-grade road. The research work to study the cooling effect and the temperature field characteristics of the crushed-rocks embankment under asphalt road on the condition of climate warming provides scientific evidence for the construction of high-grade road in Qinghai-Tibet plateau permafrost areas.
2. Mathematical Model and Indoor Experiments
Closed crushed-rocks area of embankment can be considered to be a porous medium, where, when external temperatures start to change, the air inside rocks could experience natural convection. Porous medium experiences a nonisothermal and nonsteady natural convection, so the continuity equation, momentum equation, energy equation, and Boussinesq approximation [11–13] were listed as follows: continuity equation: momentum equation: energy equation: where is the velocity vector, , , , respectively, are velocity components, , , , respectively, are coordinate components, is the pressure, is the dynamic viscosity for air, is the permeability, is the inertial resistance coefficient of air, , , respectively are the instantaneous temperature and the initial reference temperature, , , respectively, are the instantaneous air density and the initial reference air density, is the specific heat at constant pressure, and is the instantaneous thermal conductivity.
Based on apparent heat capacity method, obtained the valid heat conduction coefficient and the heat capacity. This is a 3-dimensional nonlinear problem; therefore discrete equations based on finite-volume method were established, and the numerical solution is got through using SIMPLE algorithm to solve the discrete equations [14].
Crushed-rocks have uncommon shapes and are usually randomly placed; hence, it is difficult to measure its permeability and inertial resistance coefficient. To find the values for a 20 cm crushed-rock, our research makes use of 20 cm cement sphere in a wind tunnel experiment, where the boundary wind speed condition is given based on the average wind speed in Qinghai-Tibet permafrost regions. The wind tunnel works vertically downwards, consisting of initiation, rectification, experimentation, and extension section. The tunnel is 55.00 m long, with experiment section being 20.00 m long, and the cross sectional area is 1.3 m (width) × 1.45 m (height). Wind speed is variable from 4 m/s to 40 m/s. Detail setup of the apparatus is shown in Figures 1(a) and 1(b).
(a)
(b)
Figure 2 shows the relationship between the pressure gradient and seepage velocity of sphere layer, for spheres of 20 cm. Their exact relationship is as follows:where is pressure, is the experiment section length, and is measured velocity. It can be seen from the equation that the relationship between pressure gradient and seepage velocity has completely deviated from Darcy’s law, showing a nice second-order nonlinear relationship, correlating with the relationship proposed by Forchheimer [15]. Cubic packed spheres have a large porosity and larger seepage velocity. It is insufficient to take account of only viscous friction, hence requiring us to add inertial friction force. Thus, (4) can be described aswhere is the dynamic viscosity for air, , and is air density [16]. The ambient temperate and pressure can be measured during the experiment, and the experimental air density can be calculated according to the air state equation. The air state equation was described as
Through the calculation, the permeability is 3.25 × 10−5 m2 for 20 cm (diameter) cement spheres, and the inertial resistance coefficient 28.9 m−1 was obtained. As you know, the porosity of spheres layer for the regular arrangement is a given number; that is, ; thus, if the shape factors for spheres layer and crushed-rocks layer were approximated as a constant under the same diameter and average diameter, the permeability and the inertial resistance coefficient for in situ crushed-rocks layer (the average diameter is 20 cm) could be derived as follows [17]:where is the measured porosity for in situ crushed-rocks layer and (the average diameter of crushed-rocks layer is 20 cm); thus, and , respectively, are 1.6316 m2 and 51.0874 m−1.
3. Area of Calculation and Boundary Conditions
The test area of the Western Project Program of the Chinese Academy of Sciences, at an altitude of approximately 4700 m, was used as the model for calculation. The domain of calculation is shown in Figures 3, 4, and 5. Assume an initial annual average temperature of −4.0°C; if we consider the temperature increase 2.6°C within the next 50 years, in the Qinghai-Tibet permafrost region, then, from the adherent layer theory [18], temperature for asphalt roads increases 6.5°C. According to observation data from Qinghai-Tibet railway ventilation duct test [19], the average central temperature inside the ventilation duct is 1.8°C higher than ambient average. From the relative conclusions of previous researches [5, 7, 20, 21], we confirm the following boundary conditions.
Air temperature is governed by
Temperatures of natural ground AB and EF are governed by
Temperatures of slope BC and DE are governed by
Temperature of asphalt pavement CD is governed by
Temperature inside ventilation duct is governed bywhere is the temperatures (°C) and is the time (h).
The convective heat transfer coefficient between the air and the ventilation duct wall is W·m−2·°C−1 [22]. Heat flux density of the terrestrial heat through IH is W·m−2 and AJI and FGH can be considered adiabatic and assume no flow through the flow boundaries.
For an altitude of 4700 m, the isobaric heat capacity for air is J·kg−1·°C−1, conduction coefficient W·m−1·°C−1, density kg·m−3, and viscosity kg·m−1·s−1.
In Figures 3, 4, and 5, area I is embankment filling, area II is the silty clay, and area III is slightly weathered mudstone. The thermal characteristics are shown in Table 1. Average diameter for crushed-rocks is 20 cm, permeability is m2, and inertial resistance coefficient m−1. Figure 3 shows the calculation domain for closed crushed-rocks embankment with wide pavement. Embankment is 3 m in height and 26 m in width. And the calculation domain covers 30 m from the embankment slope ends and 30 m below natural ground and crushed-rocks area is 1.4 m thick. In order to look into its temperature field characteristics, it is assumed for numerical calculation that, in the next 50 years, the temperature in Qinghai-Tibet plateau will rise by 2.6°C.
4. Numerical Results and Discussions
4.1. Temperature Field Characteristics of Closed Crushed-Rocks Embankment with Wide Pavement
Figure 6 shows the ground temperature distribution on the 50th year after construction of closed crushed-rocks embankment with wide pavement. From Figure 6(a), it can be seen that, on 15th of July, there is a clearly sunken profile of the −0.25°C isothermal line under the embankment, and the line starts to rise near the slope. The phenomena may be due to two reasons: one is that wide pavement absorbs more heat; the other is that wide pavement results in weaker natural convective heat transfer near the centre of crushed-rocks. On 15th of October, Figure 6(b), natural permafrost table is 2.35 m, the thaw depth of natural ground under central embankment is 0.69 m, and −0.25°C isothermal line drops. On 15th of January of the next year, the permafrost temperature under embankment continued to rise, compared to the previous 15th of October, and −0.25°C isothermal line continued to move downwards. Residual thawed layers emerged under natural ground; however, the embankment filling and the underneath permafrost all froze completely.
(a) 15th of July
(b) 15th of October
(c) 15th of January
The analysis above indicates that, considering a 2.6°C temperature increase in the next 50 years, permafrost temperature under closed crushed-rocks embankment with wide pavement will be relatively high, increasing to around −0.25 in the 50th year, approaching 0°C. Hence, for construction of high-grade roads in permafrost regions, it is not appropriate to use closed crushed-rocks embankment with wide pavement.
4.2. Temperature Field Characteristics of Closed Crushed-Rocks Embankment with Narrow Pavement
In the 50th year after construction of closed crushed-rocks embankment with wide pavement, permafrost under embankment has a high temperature, which may reduce stability of the embankment; hence it should be treated narrowly. Figure 4 is the calculation domain for the closed crushed-rocks embankment, in which, compared to Figure 3, only the surface width is halved, becoming 13 m; other conditions do not change. In order to analyze its temperature field characteristics, it is assumed for numerical calculation that, in the next 50 years, the temperature in Qinghai-Tibet plateau will rise by 2.6°C.
Figure 7(a) shows the ground temperature field on 15th of July, the 50th year after construction of closed crushed-rocks embankment with narrow pavement. It can be seen from the figure that the −0.35°C freezing nuclei exist in the permafrost under embankment, and soil temperature is simultaneously lower than that in closed crushed-rocks embankment with wide pavement, especially with −0.25°C isothermal line not showing a sunken profile under the embankment. This indicates that narrow pavement absorbs less heat, and there is a stronger natural convective heat transfer in central crushed-rocks of embankment. On 15th of October, Figure 7(b), natural permafrost table is 2.35 m; the thaw depth of the natural ground under the embankment central is 0.61 m. Compared with Figure 7(a), soil temperature under embankment rose to around −0.3°C, which, however, is still lower than that for embankment with wide pavement. On 15th of January, the next year, Figure 7(c), although the embankment filling and permafrost underneath have returned to their frozen state, the permafrost temperature under embankment, compared to the previous 15th of October, Figure 7(b), continues to rise, while the −0.25°C and −0.3°C isothermal lines move downwards.
(a) 15th of July
(b) 15th of October
(c) 15th of January
The above analysis shows that, considering a 2.6°C temperature increase in the next 50 years, permafrost temperature under closed crushed-rocks embankment with narrow pavement will increase to around −0.3°C, which, although still a negative temperature, is already near 0°C. To effectively ensure the thermal stability of embankment, additional protection measures are recommended, if considering the use of closed crushed-rocks embankment.
4.3. Temperature Field Characteristics of Duct-Ventilated and Closed Crushed-Rocks Embankment
For the two previous types of embankments, permafrost under the embankment has a high temperature in the 50th year. Therefore, in order to better protect the permafrost, ventilation ducts are deployed above the crushed-rocks, to further enhance the cooling effect of the embankment. Figure 5 shows the calculation domain for duct-ventilated and closed crushed-rocks embankment. Embankment is 3 m height and 13 m width and the calculation domain covers 30 m from the embankment slope ends and 30 m below natural ground and the crushed-rocks area is 1.4 m thick. Ventilation ducts are 0.4 m in diameter, with a distance of 1.2 m between centre lines of adjacent ducts, and 1.2 m away from natural ground. In order to look into its cooling effect, it is assumed for numerical calculation that, in the next 50 years, the temperature in Qinghai-Tibet plateau will rise by 2.6°C.
Figure 8 shows the temperature field of the duct-ventilated and closed crushed-rocks embankment after 50 years of construction. Comparing Figure 8(a) with Figure 7(a), permafrost temperature under the duct-ventilated and closed crushed-rocks embankment is lower by approximately 0.65°C than it is under the closed crushed-rocks embankment with narrow pavement, and −1°C freezing nuclei exist in the embankment. On 15th of January, Figure 8(b), natural permafrost table is 2.35 m, the thaw depth of the natural ground under the embankment central is 0.46 m, and soil temperature under embankment is much lower than that for closed crushed-rocks embankment with narrow pavement, Figure 7(b), and −1°C freezing nuclei still exist in the embankment. On 15th of January, Figure 8(c), the next year, the permafrost temperature under embankment, compared to the previous 15th of October, Figure 8(b), continues to rise, and residual thawed layers emerge under natural ground; however, the embankment filling and the permafrost underneath all freeze completely.
(a) 15th of July
(b) 15th of October
(c) 15th of January
The above analysis shows that, considering a 2.6°C temperature increase in the next 50 years, duct-ventilated and closed crushed-rocks embankment can effectively cool down the permafrost temperature, increasing the nature table and ensuring long term thermal stability of the permafrost under the embankment. In consideration of large air temperature rises, to ensure the stability of the permafrost and normal functions of the road, the use of such embankment for asphalt high-grade roads in permafrost regions is recommended.
5. Conclusions
Based on data collected from the wind tunnel experiment using cement spheres, 20 cm in diameter, numerical calculation is completed for crushed-rocks embankment of the Qinghai-Tibet high-grade road, and we have come to the following conclusions.(1)Relationship between the pressure gradient and seepage velocity of the internal spheres has completely deviated from Darcy’s law, suggesting a valid second-order nonlinear relationship. For cubic packed 20 cm spheres, permeability and inertial resistance coefficient are 3.25 × 10−5 m2 and 28.9 m−1, respectively.(2)Permafrost temperature is higher under closed crushed-rocks embankment with wide pavement, increasing to −0.25°C in the 50th year from now, while for closed crushed-rocks embankment with narrow pavement, the same temperature goes up to −0.3°C. Permafrost temperature is high in both situations; therefore use of such two embankments is inappropriate, when constructing high-grade roads in permafrost regions.(3)Duct-ventilated crushed-rocks (closed) embankment can effectively lower the permafrost temperature under it, increasing its permafrost table under the embankment, ensuring long-term stability of the permafrost under the embankment. In consideration of large air temperature rises, the use of such embankment for asphalt high-grade roads in permafrost regions is recommended.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (no. 41271080), the Western Project Program of the Chinese Academy of Sciences (KZCX2-XB3-19), the National key Basic Research Program of China (973 Program) (2012CB026102), and the Funding of the State Key Laboratory of Frozen Soil Engineering (SKLFSE-ZQ-35).