# Analysis of the Thermal Characteristics of Surrounding Rock in Deep Underground Space

Authors: Liu Chen; Jiangbo Li; Fei Han; Yu Zhang; Lang Liu; Bo Zhang
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

With the development of society, the economy, and national security, the exploitation of deep underground space has become an inevitable trend in human society. However, high-temperature-related problems occur in deep underground spaces. The high temperature of deep underground space is essentially influenced by the thermal characteristics of the surrounding rock. According to the mathematical model of heat transfer of the surrounding rock in deep underground space, similar criteria numbers are established. Experiments were carried out to investigate the thermal characteristics of the surrounding rock. The distribution characteristics of temperature were determined by the Fourier number (Fo) and Biot number (Bi), and the effects of heat transfer time, airflow velocities, and air temperature and radial displacement on the distribution characteristics of temperature were studied. The results indicate that the surrounding rock temperature decreases with long heat transfer times, high airflow velocities, and low air temperatures.


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## Body

## 1. Introduction

As an important part of urban space resources, underground space is an important way to promote urban sustainable development ability [1, 2]. These uses of underground space include the installation of transportation infrastructure, public utilities, disposal of waste, energy utilization facility, storage of substances, and exploitation of minerals [3–9]. Resources in shallow underground spaces have gradually become exhausted. Therefore, exploiting deep underground space is becoming increasingly important [10].The challenges associated with deep underground spaces include high temperatures, which are harmful to human health, and decreased production efficiency [11, 12]. In deep underground space, the major source of high temperatures is geothermal energy. Geothermal energy continuously transfers from surrounding rocks to deep underground spaces. It is essential to understand the thermal characteristics of surrounding rock in deep underground spaces. Many researchers have examined the thermal characteristics of surrounding rock by theoretical, experimental, and numerical modelling approaches. The unsteady heat transfer equation of the rocks surrounding deep roadways with constant air volume was solved using a variable separation approach. The solution was an infinite series including Bessel functions [13]. Numerical simulation was utilized to study the effect of heat generated after burying radioactive waste on the stability of underground space and the surrounding rock strata for 16 years at a constant heat flux [14]. The random temperature fields of a tunnel in a cold region were obtained by numerical simulation and analysed with stochastic boundary conditions and random rock properties [15]. The steady heat transfer of surrounding rocks in roadway ventilation was numerically analysed. The heat flux was approximately uniformly distributed in a ring shape. The heat flux of the rock near the roadway wall was greater than that in the far side roadway wall [16].The heat characteristics of the surrounding rock in subway tunnels were analysed by experimental testing for 17 years. The heat characteristics of the surrounding rock varied with depth, and heat storage and release are noted [17]. Heat transfer characteristics in a single rock fracture were investigated. The experimental results indicated that the roughness of rock improves overall heat transfer, whereas lithology has little influence on heat transfer [18].Although extensive research on the heat characteristics of surrounding rock in deep underground spaces has been conducted using theoretical, numerical, and field test methods, there are only a few experimental studies, most of which are focused on steady heat transfer rather than unsteady heat transfer.This study aims at experimentally investigating the unsteady thermal characteristics of surrounding rock in deep underground space. The thermal properties of surrounding rocks and boundary conditions of different deep underground spaces have significant differences. Similarity experiments were carried out to investigate the thermal characteristics of surrounding rock, and the effects of the main parameters on the temperature distribution characteristics are discussed.


## 2. Heat Transfer Model and Similar Criteria of Surrounding Rock in Deep Underground Space

### 2.1. Basic Assumptions

To model the heat transfer of surrounding rock in deep underground space, the following simplified assumptions are made:(1)
The inner boundary of the surrounding rock is a hollow cylinder with an infinite outer diameter(2)
The surrounding rock mass is homogeneous and isotropic(3)
The seepage effect of water in the surrounding rock is negligible(4)
The initial temperature of the surrounding rock is equivalent to its original rock temperature


### 2.2. Mathematical Model

Based on the above assumptions, which conform to one-dimensional unsteady thermal conduction, the mathematical description of cylindrical coordinate form regardless of the variation in the axial temperature of the surrounding rock is adopted.The mathematical equations governing the heat transfer of surrounding rock in deep underground space are shown as follows:(1)∂T∂τ=a∂2T∂r2+1r∂T∂r.Initial condition:(2)Tr,ττ=0=T0.Boundary condition:(3)λ∂T∂rr=r0=hT−Tf,Tr⟶∞=T0,τ>0.


### 2.3. Similarity Analysis

Dimensionless equation (1) is obtained where θ0 = T0 − Tf is selected as the temperature measuring scale, the radial radius r0 is selected as the length measuring scale, r02/α0 is selected as the time measuring scale, the average thermal conductivity coefficient λ0 is selected as the thermal conductivity coefficient measuring scale, the average thermal diffusivity α0 is selected as thermal diffusivity measuring scale, and the average convective heat transfer coefficient h0 is selected as the convective heat transfer coefficient of surrounding rock and the airflow measuring scale.Dimensionless equation (4) can be expressed as follows:(4)θ0r02/a0∂θ/θ0∂a0τ/r02=a0aa01r01/r1/r0θ0r0∂θ/θ0∂r/r0+θ0r02∂2θ/θ0∂r/r02.The dimensionless temperature, dimensionless radius, and dimensionless thermal diffusion coefficient are given as follows:(5)Θ=θθ0,R=rr0,A=aa0.Ignoring the change of thermal properties of surrounding rocks with temperature,A is equal to 1; then Equation (4) can be converted into(6)a0θ0∂Θ∂a0τ/r02=a0θ0r021R∂Θ∂R+∂2Θ∂R2.The initial conditions can be converted as follows:(7)a0τr02=0,Θ=1.The boundary conditions can be converted to(8)R=1,λ0θ0r0∂Θ∂R=h0θ0Θ,R⟶∞∂Θ∂R=0,a0τr02>0.The dimensionless equations governing the heat transfer of surrounding rock in deep underground space are shown as follows:(9)∂Θ∂Fo=1R∂Θ∂R+∂2Θ∂R2.The dimensionless initial condition is as follows:(10)Fo=0,Θ=1.The dimensionless boundary condition is as follows:(11)R=1,∂Θ∂R=BiΘ,R⟶∞,∂Θ∂R=0,Fo>0.From the derivation of the above dimensionless Equation (9), the dimensionless initial condition (12), and the dimensionless boundary conditions (11), it can be seen that the dimensionless temperature Θ of surrounding rock is a function of Fo, Bi, and R, that is,(12)Θ=fFo,Bi,R.Therefore, the conclusion is obtained as follows: the Fourier numberFo and Biot number Bi are the similar criteria numbers of surrounding rock in deep underground space.


## 2.1. Basic Assumptions

To model the heat transfer of surrounding rock in deep underground space, the following simplified assumptions are made:(1)
The inner boundary of the surrounding rock is a hollow cylinder with an infinite outer diameter(2)
The surrounding rock mass is homogeneous and isotropic(3)
The seepage effect of water in the surrounding rock is negligible(4)
The initial temperature of the surrounding rock is equivalent to its original rock temperature


## 2.2. Mathematical Model

Based on the above assumptions, which conform to one-dimensional unsteady thermal conduction, the mathematical description of cylindrical coordinate form regardless of the variation in the axial temperature of the surrounding rock is adopted.The mathematical equations governing the heat transfer of surrounding rock in deep underground space are shown as follows:(1)∂T∂τ=a∂2T∂r2+1r∂T∂r.Initial condition:(2)Tr,ττ=0=T0.Boundary condition:(3)λ∂T∂rr=r0=hT−Tf,Tr⟶∞=T0,τ>0.


## 2.3. Similarity Analysis

Dimensionless equation (1) is obtained where θ0 = T0 − Tf is selected as the temperature measuring scale, the radial radius r0 is selected as the length measuring scale, r02/α0 is selected as the time measuring scale, the average thermal conductivity coefficient λ0 is selected as the thermal conductivity coefficient measuring scale, the average thermal diffusivity α0 is selected as thermal diffusivity measuring scale, and the average convective heat transfer coefficient h0 is selected as the convective heat transfer coefficient of surrounding rock and the airflow measuring scale.Dimensionless equation (4) can be expressed as follows:(4)θ0r02/a0∂θ/θ0∂a0τ/r02=a0aa01r01/r1/r0θ0r0∂θ/θ0∂r/r0+θ0r02∂2θ/θ0∂r/r02.The dimensionless temperature, dimensionless radius, and dimensionless thermal diffusion coefficient are given as follows:(5)Θ=θθ0,R=rr0,A=aa0.Ignoring the change of thermal properties of surrounding rocks with temperature,A is equal to 1; then Equation (4) can be converted into(6)a0θ0∂Θ∂a0τ/r02=a0θ0r021R∂Θ∂R+∂2Θ∂R2.The initial conditions can be converted as follows:(7)a0τr02=0,Θ=1.The boundary conditions can be converted to(8)R=1,λ0θ0r0∂Θ∂R=h0θ0Θ,R⟶∞∂Θ∂R=0,a0τr02>0.The dimensionless equations governing the heat transfer of surrounding rock in deep underground space are shown as follows:(9)∂Θ∂Fo=1R∂Θ∂R+∂2Θ∂R2.The dimensionless initial condition is as follows:(10)Fo=0,Θ=1.The dimensionless boundary condition is as follows:(11)R=1,∂Θ∂R=BiΘ,R⟶∞,∂Θ∂R=0,Fo>0.From the derivation of the above dimensionless Equation (9), the dimensionless initial condition (12), and the dimensionless boundary conditions (11), it can be seen that the dimensionless temperature Θ of surrounding rock is a function of Fo, Bi, and R, that is,(12)Θ=fFo,Bi,R.Therefore, the conclusion is obtained as follows: the Fourier numberFo and Biot number Bi are the similar criteria numbers of surrounding rock in deep underground space.


## 3. Experimental Method

### 3.1. Prototype and Model

Surrounding rock in a deep underground tunnel was taken as the prototype. The depth of the deep underground tunnel centre is 965 m. Taking the rock surrounding a tunnel as a three-dimensional model, it was simplified as cuboids. The tunnel space is located in the middle of the surrounding rock, and the equivalent diameter of the tunnel is 5 m. The surrounding rock of the prototype is 25 m long, 25 m wide, and 62.5 m high. A 1 : 25 scale model was built to carry out small-scale model experiments to investigate the thermal characteristics of surrounding rock in deep underground space.


### 3.2. Experimental Device

Figure1 shows the schematic diagram of the experimental set up. The experimental set up consists of a surrounding rock model, a space model, air conditioning equipment, a thermal boundary system, and a data acquisition system. The surrounding rock model is enclosed by stainless steel and covered with an insulating layer to ensure a constant boundary temperature. The air-conditioning equipment, which consisted of cooling coil, an electric heater, a steam humidifier, and a process fan, is able to supply air under various conditions to the space model. Figure 2 is a photograph of the air conditioning equipment. The thermal boundary system is composed of a heating belt and a temperature controller that are uniformly attached to the outer wall of the surrounding rock to guarantee the fixed thermal boundary temperature requirements were satisfied. The data acquisition system collects and controls the air temperature, air humidity, air speed, and internal temperature of the surrounding rock. There are arranged temperature measuring points in the surrounding rock. The layout of the measuring points in the surrounding rock model is shown in Figure 3. Each temperature in radius displacement r is taken from the average of the vertical and horizontal direction temperature. The photograph of the experimental set up is shown in Figure 4.Figure 1
Schematic diagram of the experimental set up.Figure 2
Photograph of the air-conditioning equipment.Figure 3
Layout of measuring points in the surrounding rock model.Figure 4
Photograph of the experimental set up.The measurement parameters used for the experimental set up and the type and accuracy of the sensor are shown in Table1. The sensor position is shown in Figures 1 and 3.Table 1
Measured parameters.
Measured parameter
Type
Measurement range
Accuracy
Temperature
PT 100
−40°C∼150°C
±0.5°C
Relative humidity
Capacitive
0%∼100% RH
±2%
Airflow velocity
Hot wire
0 m/s∼15 m/s
±0.2 m/s


### 3.3. Experimental Design

The experimental model should be guaranteed to be equal to Fourier numberFo, Biot number Bi and the single value condition of the prototype. The single value condition is the initial rock temperature, the constant temperature boundary of the infinite surrounding rock, and the convection heat transfer boundary of the space. The initial rock temperature and boundary temperature of the model are the same as those of the prototype, the inlet air temperature, and inlet air humidity. The humidity in deep underground space is generally high; accordingly, the relative humidity of the inlet air of space in this experiment is constant at 80%. Air enters the space model from the air-conditioning equipment through a section of horizontal pipe, and the length of the horizontal pipe is 40 times the diameter of the space, so it can be considered that the air in the underground space model is in the fully developed area of flow [19].


### 3.4. Thermal Properties of Surrounding Rocks

The thermal properties of the surrounding rock in deep underground space are less affected by pressure variations and temperature variations, so the thermal properties of the surrounding rock can be considered constant. A similar surrounding rock material with a mixture of cement, expanded perlite, quartz sand, aluminium powder, and water was formed. The thermal properties of the surrounding rocks measured by the apparatus in the experiment are shown in Table2.Table 2
Thermal properties of surrounding rocks.
Density (kg/m³)
Specific heat capacity (kJ/(kg·k))
Thermal conductivity W/(m·K)
772
0.804
0.139As shown in Table2, the thermal properties of the surrounding rock in the experimental model are smaller than the properties of the artificial rock. This is because r in the experimental model is small compared to r in the prototype, so with Fo being equal, smaller thermal conductivity is used in the experimental model.


### 3.5. Convective Heat Transfer Coefficient

In this experiment, the convective heat transfer coefficient was calculated. The following method was used to calculate the convective heat transfer coefficient.The heat gain of air in space is as follows:(13)Q1=Gi2−i1.The convective heat transfer between the wall surface of the space and the air is as follows:(14)Q2=hTw−TpA1.Ignoring the heat loss from airflow,Q1 should be equal to Q2. The convective heat transfer coefficient h of the wall surface of the space and the air can be obtained as follows:(15)h=Gi2−i1ATw−Tp.


## 3.1. Prototype and Model

Surrounding rock in a deep underground tunnel was taken as the prototype. The depth of the deep underground tunnel centre is 965 m. Taking the rock surrounding a tunnel as a three-dimensional model, it was simplified as cuboids. The tunnel space is located in the middle of the surrounding rock, and the equivalent diameter of the tunnel is 5 m. The surrounding rock of the prototype is 25 m long, 25 m wide, and 62.5 m high. A 1 : 25 scale model was built to carry out small-scale model experiments to investigate the thermal characteristics of surrounding rock in deep underground space.


## 3.2. Experimental Device

Figure1 shows the schematic diagram of the experimental set up. The experimental set up consists of a surrounding rock model, a space model, air conditioning equipment, a thermal boundary system, and a data acquisition system. The surrounding rock model is enclosed by stainless steel and covered with an insulating layer to ensure a constant boundary temperature. The air-conditioning equipment, which consisted of cooling coil, an electric heater, a steam humidifier, and a process fan, is able to supply air under various conditions to the space model. Figure 2 is a photograph of the air conditioning equipment. The thermal boundary system is composed of a heating belt and a temperature controller that are uniformly attached to the outer wall of the surrounding rock to guarantee the fixed thermal boundary temperature requirements were satisfied. The data acquisition system collects and controls the air temperature, air humidity, air speed, and internal temperature of the surrounding rock. There are arranged temperature measuring points in the surrounding rock. The layout of the measuring points in the surrounding rock model is shown in Figure 3. Each temperature in radius displacement r is taken from the average of the vertical and horizontal direction temperature. The photograph of the experimental set up is shown in Figure 4.Figure 1
Schematic diagram of the experimental set up.Figure 2
Photograph of the air-conditioning equipment.Figure 3
Layout of measuring points in the surrounding rock model.Figure 4
Photograph of the experimental set up.The measurement parameters used for the experimental set up and the type and accuracy of the sensor are shown in Table1. The sensor position is shown in Figures 1 and 3.Table 1
Measured parameters.
Measured parameter
Type
Measurement range
Accuracy
Temperature
PT 100
−40°C∼150°C
±0.5°C
Relative humidity
Capacitive
0%∼100% RH
±2%
Airflow velocity
Hot wire
0 m/s∼15 m/s
±0.2 m/s


## 3.3. Experimental Design

The experimental model should be guaranteed to be equal to Fourier numberFo, Biot number Bi and the single value condition of the prototype. The single value condition is the initial rock temperature, the constant temperature boundary of the infinite surrounding rock, and the convection heat transfer boundary of the space. The initial rock temperature and boundary temperature of the model are the same as those of the prototype, the inlet air temperature, and inlet air humidity. The humidity in deep underground space is generally high; accordingly, the relative humidity of the inlet air of space in this experiment is constant at 80%. Air enters the space model from the air-conditioning equipment through a section of horizontal pipe, and the length of the horizontal pipe is 40 times the diameter of the space, so it can be considered that the air in the underground space model is in the fully developed area of flow [19].


## 3.4. Thermal Properties of Surrounding Rocks

The thermal properties of the surrounding rock in deep underground space are less affected by pressure variations and temperature variations, so the thermal properties of the surrounding rock can be considered constant. A similar surrounding rock material with a mixture of cement, expanded perlite, quartz sand, aluminium powder, and water was formed. The thermal properties of the surrounding rocks measured by the apparatus in the experiment are shown in Table2.Table 2
Thermal properties of surrounding rocks.
Density (kg/m³)
Specific heat capacity (kJ/(kg·k))
Thermal conductivity W/(m·K)
772
0.804
0.139As shown in Table2, the thermal properties of the surrounding rock in the experimental model are smaller than the properties of the artificial rock. This is because r in the experimental model is small compared to r in the prototype, so with Fo being equal, smaller thermal conductivity is used in the experimental model.


## 3.5. Convective Heat Transfer Coefficient

In this experiment, the convective heat transfer coefficient was calculated. The following method was used to calculate the convective heat transfer coefficient.The heat gain of air in space is as follows:(13)Q1=Gi2−i1.The convective heat transfer between the wall surface of the space and the air is as follows:(14)Q2=hTw−TpA1.Ignoring the heat loss from airflow,Q1 should be equal to Q2. The convective heat transfer coefficient h of the wall surface of the space and the air can be obtained as follows:(15)h=Gi2−i1ATw−Tp.


## 4. Results and Discussion

### 4.1. Fourier NumberFo∗

The experimental conditions were as follows: original rock temperature, 50°C; inlet air temperature, 12°C; inlet air relative humidity, 80%; and inlet airflow velocity, 3.5 m/s. The dimensionless temperature distribution of the surrounding rock was tested by varyingFo′ from 0 to 0.4.Figure5 shows that the higher the Fo∗ is, the lower the Θ is. When Fo∗ is greater than 0.3, the dimensionless temperature of the surrounding rock remained substantially constant. As shown in Figure 5, three stages can be distinguished in terms of change in the dimensionless temperature: initial stage (Fo∗ = 0–0.15), transition stage (Fo∗ = 0.15–0.3), and stabilization stage (Fo∗ > 0.3). The influence of Fo∗ on Θ mainly occurs in the initial stage, while Fo∗ has an influence in the stabilization stage. This indicates that when Fo∗ is greater than 0.3, the dimensionless temperature distribution of the surrounding rock depends on R and Bi independently of Fo∗. Therefore, when Fo∗ is greater than 0.3, the unsteady heat transfer model of surrounding rock in deep underground space could be simplified to a steady heat transfer model for simplification of the calculations.Figure 5
Effect ofFo∗ on Θ.


### 4.2. Biot NumberBi

The experimental conditions were as follows: original rock temperature, 30°C; inlet air temperature, 15°C; inlet air relative humidity, 80%; and inlet airflow velocity, 3.5 m/s (Bi = 9.87) and 6.0 m/s (Bi = 15.2). The dimensionless temperature distribution of the surrounding rock was tested by varying Fo.As shown in Figure6, comparing the dimensionless temperatures under two different Bi, the results show that Θ decreases with increasing Bi, but the influence of Bi decreases with increasing dimensionless displacement R. A major reason for the decrease in Θ is that higher Bi leads to a thinner boundary layer. Since convective heat transfer between the internal surface of surrounding rock and air was strengthened, the closer the measuring point to the internal surface is, the lower the Θ is.Figure 6
Effect ofBi on Θ. (a) Fo = 0.067. (b) Fo = 0.201. (c) Fo = 0.268. (d) Fo = 0.355.
(a)
(b)
(c)
(d)It can be seen in Figure6 that there is little effect on Θ as Fo increases, and when Fo is greater than 0.355, Bi has no substantial effect on Θ. Furthermore, as a result, when Fo is greater than 0.355, the unsteady heat transfer model of surrounding rock in deep underground space is significantly not influenced by convection heat transfer boundary conditions.


### 4.3. Radial DisplacementR

The experimental conditions were as follows: original rock temperature, 50°C; inlet air temperature, 12°C; inlet air relative humidity, 80%; and inlet airflow velocity, 3.5 m/s. The dimensionless temperature distribution of the surrounding rock was tested by varyingR.Figure7 shows that when Fo is equal to 0, the heat transfer starts, and the temperature of the surrounding rock maintains the original rock temperature in all radial displacements. The results verified the accuracy of the experiment.Figure 7
Effect ofR on Θ.Figure7 shows the trends of Θ against R. When R changes from 1.5 to 4.5, Θ increases. The greater the Fo is, the higher the temperature difference between measured points or the temperature gradient is. Θ remains almost constant and equal to 1 when R is equal to 4.5. A major reason for the increase in Θ is that the original temperature field of the surrounding rock is disturbed by air from near R to far R. The wall surface near the underground space (R = 1.5) is affected by the ventilation of the deep underground space, and the temperature is slightly lower, while the bottom of the surrounding rock (R = 4.5) is affected by the original rock temperature, and the temperature is higher.As shown in Figure7, the temperature gradient decreases with increasing R; that is, the heat flux density decreases with increasing R according to Fourier’s law. This is because the geothermal heat in the deep part of the surrounding rock continuously transfers to deep underground space while the airflow passes. The airflow has not yet been able to disturb the temperature field there, or the heat carried by the airflow is less than the heat transmitted to it; that is, the temperature range of the surrounding rock that can be spread into the underground space is from near R to far R.As shown in Figure7, the curve of an Fo value of 0.403 and the curve of an Fo value of 0.470 nearly overlap. The results show that when Fo is more than 0.403, the dimensionless temperature distribution tends to be stable, as does the heat flux density distribution.


### 4.4. Inlet Air TemperatureTf

The experimental conditions were as follows: original rock temperature, 40°C; inlet air relative humidity, 80%; and inlet airflow velocity, 5 m/s. The dimensionless temperature distribution of the surrounding rock was tested by varying the inlet air temperature from 16°C to 25°C.Figure8 shows the trends of Θ against Tf. When Tf changes from 16°C to 25°C, Θ increases. A major reason for the increase in Θ is that higher Tf leads to a smaller temperature difference between the internal surface of surrounding rock and air, which decreases heat convection.Figure 8
Effect oftf on Θ. (a) Fo = 0.067. (b) Fo = 0.201. (c) Fo = 0.268. (d) Fo = 0.355.
(a)
(b)
(c)
(d)As shown in Figure8, Tf affected the closer measuring points more than the further measuring points. When R was equal to 4.5, Θ was nearly equivalent to 1.0 with any value of Fo. This also indicates that the influence of air temperature on the heat transfer range is negligible.


## 4.1. Fourier NumberFo∗

The experimental conditions were as follows: original rock temperature, 50°C; inlet air temperature, 12°C; inlet air relative humidity, 80%; and inlet airflow velocity, 3.5 m/s. The dimensionless temperature distribution of the surrounding rock was tested by varyingFo′ from 0 to 0.4.Figure5 shows that the higher the Fo∗ is, the lower the Θ is. When Fo∗ is greater than 0.3, the dimensionless temperature of the surrounding rock remained substantially constant. As shown in Figure 5, three stages can be distinguished in terms of change in the dimensionless temperature: initial stage (Fo∗ = 0–0.15), transition stage (Fo∗ = 0.15–0.3), and stabilization stage (Fo∗ > 0.3). The influence of Fo∗ on Θ mainly occurs in the initial stage, while Fo∗ has an influence in the stabilization stage. This indicates that when Fo∗ is greater than 0.3, the dimensionless temperature distribution of the surrounding rock depends on R and Bi independently of Fo∗. Therefore, when Fo∗ is greater than 0.3, the unsteady heat transfer model of surrounding rock in deep underground space could be simplified to a steady heat transfer model for simplification of the calculations.Figure 5
Effect ofFo∗ on Θ.


## 4.2. Biot NumberBi

The experimental conditions were as follows: original rock temperature, 30°C; inlet air temperature, 15°C; inlet air relative humidity, 80%; and inlet airflow velocity, 3.5 m/s (Bi = 9.87) and 6.0 m/s (Bi = 15.2). The dimensionless temperature distribution of the surrounding rock was tested by varying Fo.As shown in Figure6, comparing the dimensionless temperatures under two different Bi, the results show that Θ decreases with increasing Bi, but the influence of Bi decreases with increasing dimensionless displacement R. A major reason for the decrease in Θ is that higher Bi leads to a thinner boundary layer. Since convective heat transfer between the internal surface of surrounding rock and air was strengthened, the closer the measuring point to the internal surface is, the lower the Θ is.Figure 6
Effect ofBi on Θ. (a) Fo = 0.067. (b) Fo = 0.201. (c) Fo = 0.268. (d) Fo = 0.355.
(a)
(b)
(c)
(d)It can be seen in Figure6 that there is little effect on Θ as Fo increases, and when Fo is greater than 0.355, Bi has no substantial effect on Θ. Furthermore, as a result, when Fo is greater than 0.355, the unsteady heat transfer model of surrounding rock in deep underground space is significantly not influenced by convection heat transfer boundary conditions.


## 4.3. Radial DisplacementR

The experimental conditions were as follows: original rock temperature, 50°C; inlet air temperature, 12°C; inlet air relative humidity, 80%; and inlet airflow velocity, 3.5 m/s. The dimensionless temperature distribution of the surrounding rock was tested by varyingR.Figure7 shows that when Fo is equal to 0, the heat transfer starts, and the temperature of the surrounding rock maintains the original rock temperature in all radial displacements. The results verified the accuracy of the experiment.Figure 7
Effect ofR on Θ.Figure7 shows the trends of Θ against R. When R changes from 1.5 to 4.5, Θ increases. The greater the Fo is, the higher the temperature difference between measured points or the temperature gradient is. Θ remains almost constant and equal to 1 when R is equal to 4.5. A major reason for the increase in Θ is that the original temperature field of the surrounding rock is disturbed by air from near R to far R. The wall surface near the underground space (R = 1.5) is affected by the ventilation of the deep underground space, and the temperature is slightly lower, while the bottom of the surrounding rock (R = 4.5) is affected by the original rock temperature, and the temperature is higher.As shown in Figure7, the temperature gradient decreases with increasing R; that is, the heat flux density decreases with increasing R according to Fourier’s law. This is because the geothermal heat in the deep part of the surrounding rock continuously transfers to deep underground space while the airflow passes. The airflow has not yet been able to disturb the temperature field there, or the heat carried by the airflow is less than the heat transmitted to it; that is, the temperature range of the surrounding rock that can be spread into the underground space is from near R to far R.As shown in Figure7, the curve of an Fo value of 0.403 and the curve of an Fo value of 0.470 nearly overlap. The results show that when Fo is more than 0.403, the dimensionless temperature distribution tends to be stable, as does the heat flux density distribution.


## 4.4. Inlet Air TemperatureTf

The experimental conditions were as follows: original rock temperature, 40°C; inlet air relative humidity, 80%; and inlet airflow velocity, 5 m/s. The dimensionless temperature distribution of the surrounding rock was tested by varying the inlet air temperature from 16°C to 25°C.Figure8 shows the trends of Θ against Tf. When Tf changes from 16°C to 25°C, Θ increases. A major reason for the increase in Θ is that higher Tf leads to a smaller temperature difference between the internal surface of surrounding rock and air, which decreases heat convection.Figure 8
Effect oftf on Θ. (a) Fo = 0.067. (b) Fo = 0.201. (c) Fo = 0.268. (d) Fo = 0.355.
(a)
(b)
(c)
(d)As shown in Figure8, Tf affected the closer measuring points more than the further measuring points. When R was equal to 4.5, Θ was nearly equivalent to 1.0 with any value of Fo. This also indicates that the influence of air temperature on the heat transfer range is negligible.


## 5. Conclusions

(a)
The heat transfer between surrounding rock and deep underground space is a complicated, nonstationary process. A 1-dimensional nonstationary heat transfer model was established to model the heat transfer of surrounding rock and deep underground space. The temperature distribution of the surrounding rock was a function of the radial displacement (R), Biot number (Bi), and Fourier number (Fo) by dimensional analysis.
(b)
Experiments were carried out to investigate the thermal characteristics of the surrounding rock. In this experiment, Fourier number (Fo) and Biot number (Bi) were adopted as the criteria of similarity, and the model was made at a 1/25 geometric scale. Through simulation experiments, the temperature field of the surrounding rock is obtained, and the effects of Fo, Bi, R, and Tf on Θ are analysed.
(c)
The experimental results indicate that three stages can be distinguished in terms of change in dimensionless temperature: initial stage (Fo′ = 0–0.15), transition stage (Fo′ = 0.15–0.3), and stabilization stage (Fo′ > 0.3). To facilitate engineering applications, the nonstationary heat transfer of surrounding rock in deep underground space could be simplified as stationary heat transfer in the stabilization stage. When R is less than or equal to 4.0, the higher the Fo and Bi are, the lower the dimensionless temperature with the same R is, while the higher the Tf is, the higher the dimensionless temperature with the same R is. When R is equal to 4.5, Θ is nearly equivalent to 1.0 with any value of Fo, Bi, and Tf. Thus, when Fo is less than or equal to 0.47, and R is greater than or equal to 4.5, and the temperature field of the surrounding rock is less affected by the underground space.



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*Source: 2130943-2019-01-16.xml*