A method for predicting heat and mass transfer performance of a medium-deep buried pipe based on a semi-analytical model

Abstract

The application discloses a kind of based on semi-analytical model's middle-deep layer buried pipe heat transfer mass transfer performance prediction method, belong to ground source heat pump technical field.The application first inputs the structural parameters of middle-deep layer buried pipe, operating parameter etc., based on analytical theory analysis middle-deep layer buried pipe wall, backfill and rock-soil heat transfer resistance, establish the heat transfer mass transfer equation of working medium considering the change of working medium thermophysical property, and then construct middle-deep layer buried pipe semi-analytical heat transfer mass transfer model, wherein, working medium thermophysical property is related to temperature and pressure;Then, the semi-analytical heat transfer mass transfer model is iteratively solved, the distribution of working medium temperature and pressure with space and time is calculated, and then the heat transfer mass transfer performance of middle-deep layer buried pipe is calculated.The middle-deep layer buried pipe heat transfer mass transfer performance prediction method proposed in the application adopts semi-analytical model with simple modeling and short calculation time, and considers the change of working medium thermophysical property with temperature and pressure, and can be used to accurately and efficiently predict the heat transfer mass transfer performance of middle-deep layer buried pipe.

Description

Technical Field

[0001] This invention belongs to the field of ground source heat pump technology, specifically relating to a method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model. Background Technology

[0002] Medium-deep ground-source pipes offer advantages such as high heat transfer intensity, small footprint, and low requirements for annual heating and cooling load balance. Medium-deep ground-source heat pumps based on medium-deep ground-source pipes provide a low-carbon, clean heating solution with promising application prospects. Accurate and efficient prediction of the heat and mass transfer performance of medium-deep ground-source pipes is of great significance; however, existing technologies have shortcomings in prediction accuracy or computation time.

[0003] In September 2023, Bai Mengmeng, Wang Fenghao, and Liu Jun disclosed an invention patent entitled "A Method for Selecting Heat Transfer Fluid for Medium-Deep Coaxial Shell-Type Buried Pipes" (application number: 202311168854.6). This patent aims to further improve the heat extraction performance of medium-deep coaxial shell-type buried pipes, effectively reduce the heat transfer fluid circulation pump consumption, and increase carbon emission reduction. It considers natural working fluids such as water, CO2, and ammonia as candidate heat transfer fluids, takes into account the real-time changes in the physical properties of the heat transfer fluids under any operating conditions, and calculates the long-term circulation pump consumption of the buried pipe under the same long-term heating conditions to determine the carbon emission reduction. Then, it combines the carbon emission reduction of water, CO2, and ammonia as heat transfer fluids to select the best heat transfer fluid, thereby deriving a comprehensive optimization technical solution that is both green and energy-saving and economical.

[0004] In April 2024, the journal *Heating, Ventilating & Air Conditioning*, Volume 54, Issue 4, pages 113-119, published an article titled "Experimental Study on the Operational Characteristics of Medium-Deep Buried Pipe Heat Pump Heating System." This article, through extensive engineering field studies, summarized the operational characteristics of medium-deep buried pipe heat pump heating technology. Medium-deep buried pipes are characterized by high outlet water temperature, large heat extraction power, and stable long-term operation, providing a high-quality low-temperature heat source for heat pump systems to improve their operational performance. Simultaneously, medium-deep buried pipes possess intermittent heat storage characteristics, effectively meeting the changing heating demands and flexible energy use of buildings. Variable frequency drive (VFD) heat pump units are more suitable for operation with medium-deep buried pipes, resulting in significant energy savings and emission reductions. The heat source-side water system features low flow rate, high head, and resistance that varies significantly with flow rate. In addition, the article proposes a fitting formula for the pressure drop of medium-deep buried pipes.

[0005] In June 2025, Yang Biao, Zhang Huifeng, Wang Ming, Qu Yifei, Wang Fenghao, and Cai Wanlong published an article entitled "Analysis of Factors Affecting the Heat Transfer Performance of Medium-Deep Buried Pipe Heat Exchangers and Design Optimization" in Volume 53, Issue 6, Pages 7-15 of *Building Energy Conservation (Chinese and English)*. This article mainly analyzes the energy transfer law of the inner pipe section in the heat extraction process of medium-deep buried pipe heat exchangers, selects typical inner pipe diameter design parameters, clarifies the influence of numerous design parameters on their heat extraction performance, and lays the foundation for the design of inner pipe parameters for medium-deep buried pipe heat exchangers. The results show that reducing the inner pipe diameter has an effect on improving the outlet temperature level of medium-deep buried pipe heat exchangers. However, at the same time, the reduction in inner pipe diameter leads to a significant increase in friction resistance, increased power consumption of the water pump, and a sharp increase in water pump operating costs during the heating season. The higher the thermal conductivity of the inner tube, the greater the impact of changes in the inner tube diameter on the heat extraction performance of the buried tube; the deeper the buried tube, the greater the impact of changes in the inner tube diameter on the heat extraction performance of the buried tube. Based on the above research and analysis results, for buried tube heat exchangers with different depths and different inner tube thermal conductivity, when the outer tube diameter of the buried tube heat exchanger changes, an optimal inner tube design diameter exists in all cases using the synergistic analysis method of heat exchanger resistance and heat transfer capacity.

[0006] In December 2025, Zhang Qiqiang, Wang Ye, Liu Hengjian, Zhang Luyu, Lu Zhi, and Heng Zixin published an article entitled "The Influence of Non-Uniform Depth Arrangement of Medium-Deep Geothermal Pipe Groups on Heat Extraction Performance" in Volume 44, Issue 12, pp. 6859-6870 of *Chemical Industry and Engineering Progress*. Under the condition of equal total buried pipe length, this article used COMSOL Multiphysics 6.2 for numerical simulations of both uniform and non-uniform depth arrangements, comparing and analyzing the heat extraction performance of three well pipe arrangements—cross-linear, rectangular, and circular—after 20 years of operation. The results show that, regardless of whether the arrangement is uniform or non-uniform depth, the different well pipe arrangements of DBHE have varying effects on the heat extraction performance of the buried pipes. The cross-linear arrangement results in the smallest decrease in outlet water temperature and heat extraction power, followed by the rectangular arrangement, while the circular arrangement shows the largest decrease. Targeted removal of some well casings based on the extent of the "cold accumulation" area can effectively improve the heat extraction capacity of the casing group in both constant-depth and non-constant-depth layouts. Compared with the constant-depth layout, the non-constant-depth layout after improvement shows that the heat extraction rate of rectangular arrangement can reach 24.59%, the cross-linear arrangement can reach 24.36%, while the circular arrangement only reaches 13.04%. The research results have certain theoretical guiding significance for casing group design.

[0007] However, only the first paper considered the changes in the working fluid's thermal properties with temperature and pressure, while the other papers ignored these changes, leading to lower performance prediction accuracy. Furthermore, papers 1, 3, and 4 all used numerical models to simulate the heat and mass transfer performance of medium-deep buried pipes, which suffers from drawbacks such as complex modeling and long computation time. The second paper used an empirical fitting formula to analyze the mass transfer performance of medium-deep buried pipes, resulting in low accuracy. Summary of the Invention

[0008] 1. The technical problem that the invention aims to solve Medium-deep underground pipelines are located at considerable depths, resulting in a wide range of variations in the temperature and pressure of the working fluid, leading to significant changes in its thermal properties. Furthermore, there is a strong coupling relationship between these thermal properties and the temperature and pressure of the working fluid. However, existing methods for predicting the heat and mass transfer performance of medium-deep underground pipelines either ignore these variations or rely on complex numerical models, resulting in low prediction accuracy, complex modeling, or long computation times. To address these issues, this invention proposes a method for predicting the heat and mass transfer performance of medium-deep underground pipelines based on a semi-analytical model. This method employs analytical theory to analyze the thermal resistance of the pipeline wall, backfill soil, and soil / rock within the medium-deep underground pipeline. Numerical methods are used to simulate the heat and mass transfer process considering variations in the working fluid's thermal properties, thereby constructing a semi-analytical heat and mass transfer model for the medium-deep underground pipeline. This model is then iteratively solved to calculate the spatial and temporal distribution of the working fluid's temperature and pressure, ultimately determining the heat and mass transfer performance of the medium-deep underground pipeline.

[0009] 2. Technical Solution To achieve the above objectives, the technical solution provided by the present invention is as follows: This invention discloses a method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model. First, the structural parameters, operating parameters, and related thermophysical parameters of the medium-deep buried pipe are input. Based on analytical theory, the thermal resistance of the pipe wall, backfill soil, and soil in the medium-deep buried pipe is analyzed to establish a heat and mass transfer equation considering the changes in the thermophysical properties of the working fluid. Then, a semi-analytical heat and mass transfer model for the medium-deep buried pipe is constructed, wherein the thermophysical properties of the working fluid are related to temperature and pressure. Next, the semi-analytical heat and mass transfer model is iteratively solved to calculate the spatial and temporal distribution of the working fluid temperature and pressure, thereby calculating the heat and mass transfer performance of the medium-deep buried pipe.

[0010] Furthermore, the total computation time for iteratively solving the semi-analytical heat and mass transfer model is divided into N equal parts. The model solution steps at any nth time are as follows: calculate the working fluid thermophysical property distribution at the nth time based on the working fluid temperature and pressure distribution at the (n-1)th time, substitute the obtained working fluid thermophysical property distribution into the semi-analytical heat and mass transfer model, and estimate the working fluid temperature and pressure distribution at the nth time. Then, calculate the working fluid thermophysical property distribution at the nth time based on the estimated working fluid temperature and pressure distribution at the nth time, and substitute the obtained working fluid thermophysical property distribution into the semi-analytical heat and mass transfer model again to calculate the working fluid temperature and pressure distribution at the nth time.

[0011] 3. Beneficial effects Compared with the prior art, the technical solution provided by this invention has the following advantages: (1) The present invention provides a method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model. This method comprehensively considers the changes in the thermal properties of the working fluid with temperature and pressure. It uses analytical theory to analyze the thermal resistance of the pipe wall, backfill soil and rock soil in medium-deep buried pipes, and uses numerical methods to simulate the heat and mass transfer process of the working fluid considering the changes in the thermal properties of the working fluid. Then, a semi-analytical heat and mass transfer model of medium-deep buried pipes is constructed. The model is then iteratively solved, which can accurately simulate the distribution of the temperature and pressure of the working fluid in medium-deep buried pipes with time and space. It can be used to accurately analyze the heat and mass transfer performance of medium-deep buried pipes, and its accuracy is much higher than that of models that ignore the changes in the thermal properties of the working fluid.

[0012] (2) The present invention provides a method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model. It adopts a semi-analytical model that is easy to model, has simple formulas, and has short calculation time, thus overcoming the shortcomings of other methods, such as complex modeling, complex formulas, and long calculation time. Attached Figure Description

[0013] Figure 1 This is a flowchart of the present invention.

[0014] Figure 2 This is a comparison of the outlet temperature in Embodiment 1 of the present invention.

[0015] Figure 3 This is a comparison result of the voltage drop in Example 1 of the present invention.

[0016] Figure 4 This is a comparison of the outlet temperature in Embodiment 2 of the present invention.

[0017] Figure 5 This is a comparison result of the voltage drop in Embodiment 2 of the present invention.

[0018] Figure 6 This is a comparison of the outlet temperature in Example 3 of the present invention.

[0019] Figure 7 This is a comparison result of the voltage drop in Example 3 of the present invention. Detailed Implementation

[0020] The present invention provides a method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model, the steps of which are as follows: First, the structural parameters, operating parameters, and related thermal properties of the medium-deep buried pipe are input. Based on analytical theory, the thermal resistance of the pipe wall, backfill soil, and soil in the medium-deep buried pipe is analyzed. The heat and mass transfer equation of the working fluid considering the change of the working fluid's thermal properties is established. Then, a semi-analytical heat and mass transfer model of the medium-deep buried pipe is constructed. The working fluid's thermal properties are related to temperature and pressure. The construction process of the semi-analytical heat and mass transfer model for U-shaped medium-deep buried pipes is as follows: The heat and mass transfer equations considering changes in the thermophysical properties of the working fluid are as follows:

[0021]

[0022] —The inner radius of the pipe; —Mass flow rate of the working fluid; —The temperature of the working fluid varies with depth z and time t; —Working fluid pressure; —Specific heat capacity of the working fluid; —Specific heat capacity of the working fluid; —The convective heat transfer coefficient of the working fluid; —Specific volume of the working fluid; —Equivalent gravitational acceleration, which varies with depth z:

[0023] —Acceleration due to gravity; —The angle between the downcomer and the slant pipe; —The lengths of the downcomer, slant pipe, and riser; —Length of the U-shaped medium-deep buried pipe ; —The Darcy friction coefficient of the working fluid, which is related to the Reynolds number of the working fluid. The relevant calculation formula is as follows:

[0024]

[0025] —The temperature of the inner surface of the pipe wall at any nth time (t n The expression for ) is as follows:

[0026] —Heat flow through the inner surface of the pipe wall:

[0027] —The thermal resistance of the pipe wall, backfill soil, and soil can be calculated using line heat source theory, column heat source theory, etc.

[0028] and The thermal resistance of soil and rock calculated using the linear heat source theory and the column heat source theory, respectively:

[0029]

[0030] —The thermal conductivity of the pipe; —The thermal conductivity of the backfill soil; —The thermal conductivity of soil and rock varies with depth z; —Thermal diffusivity of soil and rock; —Outer radius of the pipe; —Drilling radius; —Integral variable; —The zeroth and first-order Bessel functions of the first kind; —The zeroth and first-order Bessel functions of the second kind; The convective heat transfer coefficient of the working fluid is calculated as follows:

[0031] —The thermal conductivity of the working fluid; — Prandtl number of the working fluid.

[0032] The working fluid temperature and pressure vary with depth z and time t, while the working fluid's volumetric specific heat capacity, specific heat capacity, specific volume, and other thermophysical parameters vary with temperature and pressure. Furthermore, the working fluid's convective heat transfer coefficient is also related to its thermophysical properties. Therefore, both the working fluid's thermophysical parameters and convective heat transfer coefficient vary with depth z and time t.

[0033] Based on the above equations, boundary conditions and initial conditions are established, and then a semi-analytical heat and mass transfer model for U-shaped medium-deep buried pipes is constructed.

[0034] The initial conditions are as follows:

[0035]

[0036] The boundary conditions are as follows:

[0037]

[0038] —Surface temperature; —Inlet temperature of the working fluid; —Initial temperature; —Geothermal gradient; — Surface pressure; —Inlet pressure of the working fluid; — Thermal output power.

[0039] The above equations are all the formulas for the constructed U-shaped medium-deep buried pipe semi-analytical heat and mass transfer model.

[0040] For casing-type medium-deep buried pipes, the construction process of its semi-analytical heat and mass transfer model is as follows: The heat and mass transfer equations for the working fluid inside the tube are as follows:

[0041]

[0042] —Inner radius of the inner tube; —Mass flow rate of the working fluid; —Acceleration due to gravity; —The Darcy friction coefficient of the working fluid in the inner tube; —The temperature of the working fluid inside the tube varies with depth z and time t; —The temperature of the working fluid in the outer tube varies with depth z and time t; —Inner tube working fluid pressure; —Specific heat capacity of the working fluid in the inner tube; —Specific heat capacity of the working fluid in the inner tube; —Specific volume of the working fluid inside the tube; —Thermal resistance between the inner tube working fluid and the outer tube working fluid:

[0043] —Outer radius of the inner tube; —Inner tube thermal conductivity; —The convective heat transfer coefficient of the working fluid in the inner tube:

[0044] —Convective heat transfer coefficient of the working fluid in the outer tube:

[0045] —The Reynolds number of the working fluid in the inner tube; — Prandtl number of the working fluid in the inner tube; —The thermal conductivity of the working fluid in the inner tube; —The Darcy friction coefficient of the working fluid in the outer tube; —The Reynolds number of the working fluid in the outer tube; — Prandtl number of the working fluid in the outer tube; —Thermal conductivity of the working fluid in the outer tube; —Inner radius of the outer tube.

[0046] The heat and mass transfer equations for the working fluid in the outer tube are as follows:

[0047]

[0048] —External working fluid pressure; —Specific heat capacity of the working fluid in the outer tube; —Specific heat capacity of the working fluid in the outer tube; —Specific volume of the working fluid in the outer tube; —The temperature of the inner surface of the outer tube, at any nth time, is expressed as follows:

[0049] —The thermal resistance of the outer pipe, backfill soil, and soil can be calculated using line heat source theory, column heat source theory, etc.

[0050] —External tube thermal conductivity; —Outer radius of the outer tube; — Thermal resistance of rock and soil calculated by the linear heat source theory; —The thermal resistance of rock and soil calculated by the column heat source theory; —Heat flow through the inner surface of the outer tube:

[0051] The thermal properties of the working fluid in the inner and outer tubes, such as volumetric specific heat capacity, specific heat capacity, and specific volume, vary with temperature and pressure. Furthermore, the convective heat transfer coefficient is also related to these thermal properties. Therefore, the thermal properties and convective heat transfer coefficient of both the inner and outer tube working fluids vary with depth z and time t.

[0052] Based on the above equations, boundary conditions and initial conditions are established, and then a semi-analytical heat and mass transfer model for casing-type deep buried pipes is constructed.

[0053] The initial and boundary conditions are as follows:

[0054]

[0055]

[0056]

[0057]

[0058]

[0059] —Initial temperature; —Surface temperature; —Geothermal gradient; — Surface pressure; —Inlet temperature of the working fluid; —Heat output power; —Inlet pressure of the working fluid; —The length of the casing-type medium-deep buried pipe; The above equations are all the formulas for the constructed semi-analytical heat and mass transfer model of the casing-type deep buried pipe.

[0060] Then, the semi-analytical heat and mass transfer model is iteratively solved to calculate the spatial and temporal distribution of the working fluid temperature and pressure, thereby calculating the heat and mass transfer performance of the medium-deep buried pipe. The solution steps at any given time are as follows: Calculate the current working fluid thermophysical property distribution based on the previous time's working fluid temperature and pressure distribution, substitute this value into the semi-analytical heat and mass transfer model, and estimate the current working fluid temperature and pressure distribution. Then, based on the estimated current working fluid temperature and pressure distribution, calculate the current working fluid thermophysical property distribution, substitute this value into the semi-analytical heat and mass transfer model, and calculate the current working fluid temperature and pressure distribution.

[0061] It is worth noting that the distribution of the working fluid's thermal properties is strongly coupled with its temperature and pressure distributions. The working fluid's thermal properties are determined by its temperature and pressure distributions at the corresponding moment, while solving for the temperature and pressure distributions depends on the working fluid's thermal properties as input conditions; the two cannot be solved independently. Therefore, this invention first estimates the current working fluid's thermal property distribution using the previous moment's temperature and pressure distributions, then substitutes this estimate into a semi-analytical heat and mass transfer model to estimate the current working fluid temperature and pressure distributions. Next, based on these estimated temperature and pressure distributions, the current working fluid's thermal property distributions are updated, and the updated distributions are substituted back into the model for calculation. This process corrects the solution for the working fluid's temperature and pressure distributions, effectively improving the solution accuracy.

[0062] For U-shaped medium-deep buried pipes, the total computation time of the semi-analytical heat and mass transfer model for U-shaped medium-deep buried pipes is divided into N equal parts. The solution steps for the model at any nth time are as follows: (1) Calculate the thermophysical property distribution of the working fluid at time n based on the temperature and pressure distribution at time n-1, that is, assign the relevant parameters at time n as follows:

[0063] (2) Based on the above assignment, the semi-analytical heat and mass transfer model of the U-shaped deep underground pipe at time n is solved to estimate the working fluid temperature distribution at time n. and pressure distribution ; (3) Based on the above working fluid temperature distribution and pressure distribution The estimated values ​​are used to calculate the thermophysical property distribution of the working fluid at time n, that is, to assign the relevant parameters at time n to values ​​related to... and Functions:

[0064]

[0065] (4) Based on the above assignment, the semi-analytical heat and mass transfer model of the U-shaped deep underground pipe at the nth time is solved to calculate the working fluid temperature and pressure distribution at the nth time.

[0066] Based on the above calculation results, the heat and mass transfer performance of the U-shaped medium-deep buried pipe at time n can be calculated, where the outlet temperature is... This reflects the heat transfer performance of medium-deep buried pipes; the higher the value, the better the heat transfer performance. Pressure drop, i.e. The value reflects the mass transfer performance of medium-deep buried pipes; the smaller the value, the better the mass transfer performance.

[0067] For casing-type medium-deep buried pipes, the total computation time of the semi-analytical heat and mass transfer model for casing-type medium-deep buried pipes is divided into N equal parts. The solution steps for the model at any nth time are as follows: (1) Calculate the thermophysical property distribution of the working fluid at time n based on the temperature and pressure distribution at time n-1, that is, assign the relevant parameters at time n as follows:

[0068]

[0069] (2) Based on the above assignment, the semi-analytical heat and mass transfer model of the casing-type deep buried pipe at the nth time is solved, and the temperature distribution of the working fluid in the inner pipe at the nth time is estimated. and pressure distribution Temperature distribution of working fluid in outer tube and pressure distribution ; (3) Calculate the thermophysical property distribution of the working fluid at time n based on the above estimated values ​​of working fluid temperature and pressure distribution, that is, assign the relevant parameters at time n to values ​​related to the working fluid temperature and pressure distribution. , , and Functions:

[0070]

[0071]

[0072]

[0073] (4) Based on the above assignment, the semi-analytical heat and mass transfer model of the casing-type deep buried pipe at the nth time is solved to calculate the working fluid temperature and pressure distribution at the nth time.

[0074] Based on the above calculation results, the heat and mass transfer performance of the casing-type deep buried pipe at time n can be calculated, where the outlet temperature is... This reflects the heat transfer performance of medium-deep buried pipes; the higher the value, the better the heat transfer performance. Pressure drop, i.e. The value reflects the mass transfer performance of medium-deep buried pipes; the smaller the value, the better the mass transfer performance.

[0075] To further understand the content of this invention, a detailed description of the invention will be provided in conjunction with the accompanying drawings and embodiments.

[0076] Example 1 This embodiment is based on a field experiment of a U-shaped medium-deep buried pipe using water as the working fluid (Reference: Li C, Jiang C, Guan Y. An analytical model for heat transfer characteristics of adeep-buried U-bend pipe and its heat transfer performance under different deflecting angles[J]. Energy, 2022, 244: 122682) to verify the reliability of the semi-analytical heat and mass transfer model of the U-shaped medium-deep buried pipe constructed in this invention. Specifically, the reliability of the model's simulated heat transfer performance is verified using outlet temperature experimental data. Due to the lack of pressure drop experimental data, the Darcy-Weisbach equation (an empirical formula for calculating pressure drop) is used to verify the reliability of the model's simulated mass transfer performance. Relevant parameters are shown in Table 1, and the thermal properties of the soil and rock vary with depth; specific values ​​are given in the aforementioned reference.

[0077] Table 1. Relevant parameters from field experiments using water as the working medium for U-shaped medium-deep buried pipes.

[0078] Figure 2 and Figure 3 The results are compared with experimental and empirical formulas. The established model agrees well with the experimental results. Specifically, the average absolute error of the outlet temperature over 8 to 72 hours is 1.06 ℃, indicating that the established model can accurately calculate the heat transfer performance of U-shaped medium-deep buried pipes. If the thermophysical properties of the working fluid are ignored, the pressure drop calculated by the model is constant at 0.0918 MPa, which is very close to the value calculated by the Darcy-Weisbach equation (which also ignores the thermophysical properties of the working fluid), indicating that the established model can accurately calculate the mass transfer performance of U-shaped medium-deep buried pipes. However, considering the thermophysical properties of the working fluid, the calculated pressure drop changes over time, and the difference from the calculation result ignoring these changes is significant, indicating the necessity of considering the thermophysical properties of the working fluid. In conclusion, the established model can accurately calculate the heat and mass transfer performance of U-shaped medium-deep buried pipes using water as the working fluid.

[0079] Example 2 This embodiment is based on a field experiment of a coaxial-type deep-buried pipe using water as the working fluid (reference: Li C, Jiang C, Guan Y, et al. Development and applicability of heat transfer analytical model for coaxial-type deep-buried pipes[J]. Energy, 2022, 255:124533) to verify the reliability of the semi-analytical heat and mass transfer model for coaxial-type deep-buried pipes established in this invention. Specifically, the reliability of the model's simulated heat and mass transfer performance is verified using experimental outlet temperature data and the Darcy-Weisbach equation. Relevant parameters are shown in Table 2, and the geotechnical thermophysical parameters are the same as those in Example 1.

[0080] Table 2. Relevant parameters from field experiments on casing-type medium-deep buried pipes using water as the working fluid.

[0081] Figure 4 and Figure 5The figure shows a comparison between the established model and the experimental and empirical formulas. The outlet temperature calculated by the established model also agrees well with the experimental results. Specifically, the average absolute error of the outlet temperature over 8 to 72 hours is only 0.82℃, indicating that the established model can accurately calculate the heat transfer performance of casing-type deep underground pipes. If the thermophysical properties of the working fluid are ignored, the pressure drop calculated by the established model is 0.592 MPa, only 3.0% higher than the value calculated by the Darcy-Weisbach equation, indicating that the established model can accurately calculate the mass transfer performance of casing-type deep underground pipes. Furthermore, the difference between the calculation results considering and ignoring the thermophysical properties of the working fluid is significant. In conclusion, the established model can accurately calculate the heat and mass transfer performance of casing-type deep underground pipes using water as the working fluid.

[0082] Example 3 This embodiment is based on a case study of a U-shaped medium-deep buried pipe using CO2 as the working fluid (reference: Zhang Q, Liu F, Han Y, et al. Numerical analysis of the heat extraction performance of a U-type medium-deep borehole heat exchanger utilizing supercritical CO2 as a heat-carrying fluid[J]. Applied Thermal Engineering, 2025, 127834), to verify the reliability of the present invention for simulating the heat and mass transfer performance of a U-shaped medium-deep buried pipe using CO2 as the working fluid. The reliability of the model's simulation of heat and mass transfer performance is verified using the numerical simulation results from the aforementioned literature and the Darcy-Weisbach equation. Relevant parameters are shown in Table 3, and the geotechnical thermophysical parameters also vary with depth; specific values ​​can be found in the aforementioned literature.

[0083] Table 3. Relevant parameters of U-shaped medium-deep buried pipes using CO2 as the working fluid.

[0084] Figure 6 and Figure 7The figure shows a comparison between the results of the established model and those from literature and empirical formulas. The outlet temperature calculated by the established model agrees well with the numerical simulation results from the literature. Specifically, the average absolute deviation of the outlet temperature over 120 days is 0.18 ℃, and the absolute deviation on day 120 is only 0.1 ℃, indicating that the established model can accurately calculate the heat transfer performance of U-shaped medium-deep buried pipes. If the thermophysical property changes of CO2 are ignored, the pressure drop calculated by the established model is 0.195 MPa, which is 7.7% lower than the value calculated by the Darcy-Weisbach equation, indicating that the established model can accurately calculate the mass transfer performance of U-shaped medium-deep buried pipes. In addition, there is a significant difference between the calculation results considering and ignoring the thermophysical property changes of the working fluid. In conclusion, the established model can accurately calculate the heat and mass transfer performance of U-shaped medium-deep buried pipes using CO2 as the working fluid.

[0085] The present invention and its embodiments have been described above illustratively. This description is not restrictive, and the accompanying drawings are only one embodiment of the present invention, and are not actually limited thereto. Therefore, if those skilled in the art are inspired by this description and design similar structures and embodiments without departing from the spirit of the present invention, such designs should fall within the protection scope of the present invention.

Claims

1. A method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model, characterized in that: First, the structural parameters, operating parameters, and related thermophysical parameters of the medium-deep buried pipe are input. Based on analytical theory, the thermal resistance of the pipe wall, backfill soil, and soil in the medium-deep buried pipe is analyzed, and a heat and mass transfer equation considering the changes in the thermophysical properties of the working fluid is established. Then, a semi-analytical heat and mass transfer model of the medium-deep buried pipe is constructed, in which the thermophysical properties of the working fluid are related to temperature and pressure. Then, the semi-analytical heat and mass transfer model is iteratively solved to calculate the spatial and temporal distribution of the working fluid temperature and pressure, and then the heat and mass transfer performance of the medium-deep buried pipe is calculated.

2. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 1, characterized in that: The total computation time for iteratively solving the semi-analytical heat and mass transfer model is divided into N equal parts. The model solution steps at any nth time are as follows: calculate the working fluid thermophysical property distribution at the nth time based on the working fluid temperature and pressure distribution at the (n-1)th time, substitute the obtained working fluid thermophysical property distribution into the semi-analytical heat and mass transfer model, and estimate the working fluid temperature and pressure distribution at the nth time. Then, calculate the working fluid thermophysical property distribution at the nth time based on the estimated working fluid temperature and pressure distribution at the nth time, and substitute the obtained working fluid thermophysical property distribution into the semi-analytical heat and mass transfer model again to calculate the working fluid temperature and pressure distribution at the nth time.

3. A method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model, as described in claim 1 or 2, characterized in that: For U-shaped medium-deep buried pipes, the heat and mass transfer equations for the working fluid, considering changes in its thermophysical properties, are as follows: In the formula, The inner radius of the pipe; The temperature of the working fluid varies with depth z and time t. The mass flow rate of the working fluid; For the working fluid pressure; is the volumetric specific heat capacity of the working fluid; Specific heat capacity of the working fluid; The convective heat transfer coefficient of the working fluid; Specific volume of the working fluid; This refers to the temperature of the inner surface of the pipe wall. This is the equivalent gravitational acceleration, which varies with depth z; denoted as Darcy's coefficient of friction of the working fluid.

4. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 3, characterized in that: The inner surface temperature of the pipe wall At any nth time t n The expression is as follows: In the formula, The initial temperature; The thermal resistance of heat transfer in the pipe wall, backfill soil, and rock can be calculated using the line heat source theory and the column heat source theory. This is for the heat flow through the inner surface of the pipe wall.

5. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 4, characterized in that: Based on the heat and mass transfer equations of the working fluid considering changes in its thermophysical properties, boundary conditions and initial conditions are established, and a semi-analytical heat and mass transfer model for a U-shaped medium-deep buried pipe is constructed. The initial conditions are as follows: The boundary conditions are as follows: In the formula, It refers to the surface temperature; The inlet temperature of the working fluid; The initial temperature; The geothermal gradient; It is the surface pressure; The inlet pressure of the working fluid; This refers to the thermal output power. The length of the U-shaped medium-deep buried pipe. ; , , These are the lengths of the downcomer, inclined tube, and ascender, respectively. For integration variables; The angle between the downcomer and the tilting tube.

6. A method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model, as described in claim 1 or 2, characterized in that: For casing-type medium-deep buried pipes, the heat and mass transfer equations considering changes in the thermophysical properties of the working fluid include the heat and mass transfer equations for the working fluid in the inner pipe and the working fluid in the outer pipe; wherein... The heat and mass transfer equations for the working fluid inside the tube are as follows: In the formula, The inner radius of the inner tube; The mass flow rate of the working fluid; It is the acceleration due to gravity; The Darcy friction coefficient of the working fluid in the inner tube; The temperature of the working fluid inside the tube varies with depth z and time t; The temperature of the working fluid in the outer tube varies with depth z and time t; The pressure of the working fluid in the inner tube; The specific heat capacity of the working fluid in the inner tube; Specific heat capacity of the working fluid in the inner tube; The specific volume of the working fluid inside the tube; The thermal resistance between the working fluid in the inner tube and the working fluid in the outer tube; The heat and mass transfer equations for the working fluid in the outer tube are as follows: In the formula, The inner radius of the outer tube; The outer radius of the inner tube; The pressure of the working fluid in the outer pipe; The specific heat capacity of the working fluid in the outer tube; Specific heat capacity of the working fluid in the outer tube; Specific volume of the working fluid in the outer tube; This refers to the temperature of the inner surface of the outer tube. The convective heat transfer coefficient of the working fluid in the outer tube; is the Darcy friction coefficient of the working fluid in the outer tube.

7. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 6, characterized in that: The inner surface temperature of the outer tube The expression at any nth time is as follows: In the formula, The initial temperature; The thermal resistance of heat transfer in the outer pipe, backfill soil, and rock and soil is calculated using the line heat source theory and the column heat source theory. For the heat flow through the inner surface of the outer tube: 。 8. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 7, characterized in that: Based on the heat and mass transfer equations of the working fluid considering changes in its thermophysical properties, boundary conditions and initial conditions are established, thereby constructing a semi-analytical heat and mass transfer model for casing-type deep underground pipes. The initial and boundary conditions are as follows: In the formula, It refers to the surface temperature; The geothermal gradient; It is the surface pressure; The inlet temperature of the working fluid; This refers to the thermal output power. The inlet pressure of the working fluid; The length of the casing-type medium-deep buried pipe; It is the integral variable.

9. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 5, characterized in that: For a U-shaped medium-deep buried pipe, the total computation time of the semi-analytical heat and mass transfer model for the U-shaped medium-deep buried pipe is divided into N equal parts. The solution steps for the model at any nth time are as follows: (1) Calculate the thermophysical property distribution of the working fluid at time n based on the temperature and pressure distribution at time n-1, that is, assign the relevant parameters at time n as follows: (2) Based on the above assignment, the semi-analytical heat and mass transfer model of the U-shaped deep underground pipe at time n is solved to estimate the working fluid temperature distribution at time n. and pressure distribution ; (3) Based on the above working fluid temperature distribution and pressure distribution The estimated values ​​are used to calculate the thermophysical property distribution of the working fluid at time n, that is, to assign the relevant parameters at time n to values ​​related to... and Functions: (4) Based on the above assignment, the semi-analytical heat and mass transfer model of the U-shaped deep underground pipe at the nth time is solved to calculate the working fluid temperature and pressure distribution at the nth time.

10. The method for predicting the heat and mass transfer performance of medium-deep buried pipes based on a semi-analytical model according to claim 8, characterized in that: For casing-type medium-deep buried pipes, the total computation time of the semi-analytical heat and mass transfer model for casing-type medium-deep buried pipes is divided into N equal parts. The solution steps for the model at any nth time are as follows: (1) Calculate the thermophysical property distribution of the working fluid at time n based on the temperature and pressure distribution at time n-1, that is, assign the relevant parameters at time n as follows: (2) Based on the above assignment, the semi-analytical heat and mass transfer model of the casing-type deep buried pipe at the nth time is solved, and the temperature distribution of the working fluid in the inner pipe at the nth time is estimated. and pressure distribution Temperature distribution of working fluid in outer tube and pressure distribution ; (3) Calculate the thermophysical property distribution of the working fluid at time n based on the above estimated values ​​of working fluid temperature and pressure distribution, that is, assign the relevant parameters at time n to values ​​related to the working fluid temperature and pressure distribution. , , and Functions: (4) Based on the above assignment, the semi-analytical heat and mass transfer model of the casing-type deep buried pipe at the nth time is solved to calculate the working fluid temperature and pressure distribution at the nth time.

Images