Geothermal snow melting system and design method thereof

Abstract

The invention belongs to the technical field of road snow melting, and particularly provides a terrestrial heat snow melting system and a design method thereof.The terrestrial heat snow melting system comprises an underground heat extraction circulation pipeline and a concrete pre-buried circulation pipeline, and the underground heat extraction circulation pipeline comprises an ascending fluid outer pipe, an air gap layer isolation pipe and a descending fluid inner pipe which are sequentially arranged from outside to inside; the upper end and the lower end of the space between the air gap layer isolation pipe and the descending fluid inner pipe are closed to form a closed air gap layer, an overflowing gap is formed between the lower end of the descending fluid inner pipe and the bottom of the ascending fluid outer pipe, and an annular flow channel is formed between the ascending fluid outer pipe and the air gap layer isolation pipe. A liquid inlet pipe and a liquid outlet pipe are arranged at the upper end of the ascending fluid outer pipe and the upper end of the descending fluid inner pipe respectively and connected with the two ends of a concrete pre-buried circulating pipeline respectively, the circulating pipeline is filled with heat-conducting media, and the concrete pre-buried circulating pipeline is pre-buried in a pavement layer. The snow melting system can directly utilize geothermal energy to melt accumulated snow on the ground without depending on an external power supply, and pump-free operation can be achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of road snow melting, and particularly relates to a geothermal snow melting system and its design method. Background Art

[0002] In some regions of China, snowfall occurs every year, which often leads to road icing, frequent traffic accidents, and personal injuries. At present, there are mainly two methods for removing snow: mechanical snow removal and using chemical snow melting agents. Mechanical snow removal may cause traffic jams, while the use of chemical snow melting agents may pollute the durability of concrete, as well as the surrounding soil and water quality. Geothermal energy, as a clean and renewable energy source, can melt snow by utilizing geothermal energy, which not only does not pollute the environment but also does not cause traffic congestion. CN116242043A discloses a pump-free geothermal energy extraction snow melting system and a snow melting method, which uses natural circulation to complete energy transportation. This solution has the characteristics of no need for external strong power access, normal pressure operation, and rapid system response, and is a technical solution worthy of popularization in the future. However, the design of this system still has the problem that the temperature difference between the pipelines on both sides of the ground is small, the circulation speed of the heat conduction medium slows down, and the snow melting efficiency will be greatly affected. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a geothermal snow melting system and its design method, which can directly utilize geothermal energy to melt the snow on the ground, without relying on external power supply, and can achieve pump-free operation.

[0004] To solve the above technical problem, the technical solution adopted by the present invention is: a geothermal snow melting system, including an underground heat extraction circulation pipeline and a concrete embedded circulation pipeline. The underground heat extraction circulation pipeline includes an ascending fluid outer pipe, and an air gap layer isolation pipe and a descending fluid inner pipe are sequentially arranged from outside to inside in the ascending fluid outer pipe. The upper and lower ends of the space between the air gap layer isolation pipe and the descending fluid inner pipe are closed to form a closed air gap layer. The upper and lower ends of the ascending fluid outer pipe are both closed. There is a flow-through gap between the lower end of the descending fluid inner pipe and the bottom of the ascending fluid outer pipe. An annular flow channel is formed between the ascending fluid outer pipe and the air gap layer isolation pipe. The upper ends of the ascending fluid outer pipe and the descending fluid inner pipe are respectively provided with a liquid inlet pipe and a liquid outlet pipe, and the liquid inlet pipe and the liquid outlet pipe are respectively connected to both ends of the concrete embedded circulation pipeline. The annular flow channel, the concrete embedded circulation pipeline, and the descending fluid inner pipe form a circulation pipeline, and a heat conduction medium is filled in the circulation pipeline. The concrete embedded circulation pipeline is embedded in the road surface layer that needs to be snow melted.

[0005] In a preferred solution, a plurality of positioning fins are provided between the ascending fluid outer pipe and the air gap layer isolation pipe, and between the air gap layer isolation pipe and the descending fluid inner pipe.

[0006] In a preferred solution, the closed end at the lower end of the ascending fluid outer pipe is a "U" - shaped structure.

[0007] In a preferred embodiment, a flexible hose is provided between the liquid inlet pipe and the concrete embedded circulation pipeline, and between the liquid outlet pipe and the concrete embedded circulation pipeline.

[0008] In a preferred embodiment, the liquid inlet pipe is connected to a liquid filling branch pipe, the liquid outlet pipe is provided with a liquid discharge branch pipe, and valves are provided on both the liquid filling branch pipe and the liquid discharge branch pipe.

[0009] In a preferred embodiment, the concrete embedded circulation pipeline is laid in an "S" shape.

[0010] The present invention also provides a design method for a geothermal snow melting system, comprising the following steps: Step 1: Select the design parameters to be determined: including the depth of the concrete buried pipe, the diameter of the buried pipe, the spacing of the buried pipes, the length of the buried pipes, the depth of the heat source, the height of the heat source, the diameter of the air gap isolation pipe, the diameter of the downflow inner pipe, and the diameter of the upflow outer pipe; Step 2: Node division: including the nodes of the circulation pipeline and the concrete nodes, to obtain the distribution of temperature T and the flow velocity u; Step 3: Calculate the surface heat flux q value according to the surface heat flux equation. When q is greater than q set After that, with the currently selected parameters, solve the transient equation, and then recalculate the surface heat flux q value to obtain the change of the surface heat flux q with time, q set is the set value of the surface heat flux; When q is first greater than q set At this time, take the current running time as the system response time. If the response time is less than the user's expected value, then recognize the current design parameters. Otherwise, adjust the design parameters and repeat Steps 1-3 for recalculation until the response time is less than the user's expected value.

[0011] In a preferred embodiment, in Step 2, the application equation for the nodes of the circulation pipeline is: (1); Wherein, is the fluid thermal expansion coefficient, is the acceleration of gravity, L is the depth of the heat source, and are the temperatures of the same height nodes in the upflow outer pipe and the downflow inner pipe respectively, is the resistance coefficient, is the circulation flow velocity, is the equivalent diameter of the circulation pipeline. In the downflow inner pipe section, d takes the diameter of the downflow inner pipe. In the upflow outer pipe section, d takes the difference between the diameter of the upflow outer pipe and the diameter of the air gap isolation pipe. In the concrete embedded circulation pipeline section, d takes the diameter of the concrete embedded circulation pipeline; (2); Wherein, is the density reference value, is the specific heat at constant pressure, T is the node temperature, is used to add the heat transfer boundary conditions in the geothermal heat source and the concrete buried pipe, and z represents the coordinate consistent with the fluid flow direction; For the concrete nodes above the concrete pre-buried circulation pipeline, the applied equation expression is: (3); Wherein, x represents the coordinate consistent with the ground normal direction; According to equations (1) to (3), the distribution of temperature T and the flow velocity u are obtained.

[0012] In the preferred solution, in the third step, the surface heat flux The calculation formula is: (4); Wherein, is the number of nodes of the concrete buried pipe, is the equivalent heat transfer coefficient, T A is the ambient temperature.

[0013] In the preferred solution, in the third step, the transient equations include the transient equation of the circulation pipeline nodes and the transient equation of the concrete nodes, and the expressions are: Transient equation of the circulation pipeline nodes: (5); (6); The transient equation of the concrete nodes is: (7); Solve equations (5) to (7) to obtain the change of temperature T with time.

[0014] A geothermal snow melting system and its design method provided by the present invention have the following beneficial effects: 1. The extraction process of geothermal energy does not require pumping, and medium and deep geothermal resources can be directly utilized without additional heating. The design of the whole system is simplified, and the driving force generated by the natural density difference is relied on to realize the efficient extraction of geothermal energy.

[0015] 2. The system operates at normal pressure. Compared with the pressurization and decompression filling of the heat pipe system, this system does not require additional pressurization or vacuum pumping operations. 3. The system responds quickly. Thanks to the closed air gap layer, the response start time of this system is shortened on the basis of the original natural circulation system. 4. This system only extracts heat without pumping groundwater, thus maintaining the stability of the groundwater level and preventing groundwater pollution.

[0016] 5. The provided design method does not need to consider the surface snowmelt process, simplifies the calculation, and takes into account both the steady-state performance and the dynamic characteristics of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The present invention will be further described below in conjunction with the drawings and embodiments: Figure 1 is a schematic diagram of the overall structure of the present invention; Figure 2 is the cross-section of the underground heat extraction circulation pipe; Figure 3 is the longitudinal section at the bottom of the underground heat extraction circulation pipe; Figure 4 is the connection schematic diagram of the above-ground outlet of the underground heat extraction circulation pipe; Figure 5 The surface snowmelt situation of the present invention and the prior art under the same working conditions; Figure 6 The comparison results of the flow velocity in the circulation pipe of the present invention and the prior art under the same working conditions; In the figure: underground heat extraction circulation pipeline 100, rising fluid outer pipe 110, air-gap layer isolation pipe 120, descending fluid inner pipe 130, closed air-gap layer 140, flow-through gap 150, annular flow channel 160, liquid inlet pipe 170, liquid outlet pipe 180, positioning fins 190; concrete embedded circulation pipeline 200; telescopic hose 300; liquid filling branch pipe 400; liquid drainage branch pipe 500; valve 600. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0018] Embodiment 1: As Figures 1 to 4 shown, a geothermal snowmelt system includes an underground heat extraction circulation pipeline and a concrete embedded circulation pipeline. The underground heat extraction circulation pipeline is driven into the depth of the effective temperature zone underground and adopts a nested structure. The underground heat extraction circulation pipeline includes a rising fluid outer pipe, and an air-gap layer isolation pipe and a descending fluid inner pipe are sequentially arranged from outside to inside in the rising fluid outer pipe. The upper and lower ends of the space between the air-gap layer isolation pipe and the descending fluid inner pipe are closed to form a closed air-gap layer. Specifically, the air-gap layer isolation pipe and the descending fluid inner pipe are welded at both ends to block the ends of the closed air-gap layer therebetween.

[0019] Both the upper and lower ends of the rising fluid outer pipe are closed. The lower end of the rising fluid outer pipe is closed by a circular plate or an arc plate. In this embodiment, the closed end of the lower end of the rising fluid outer pipe is a "U" - shaped structure.

[0020] A section of pipeline is provided at the lower end of the rising fluid outer pipe and is in direct contact with the underground heat source without being wrapped with thermal insulation material. The outer side of the remaining pipe sections of the rising fluid outer pipe except for the pipe section in direct contact with the underground heat source is wrapped with thermal insulation material to isolate the heat exchange between the external soil and the pipeline.

[0021] An overflow gap is provided between the lower end of the descending fluid inner pipe and the bottom of the rising fluid outer pipe. An annular flow channel is formed between the rising fluid outer pipe and the air gap layer isolation pipe. During the operation stage, the fluid flows downward from the descending fluid inner pipe and turns back into the annular flow channel between the rising fluid outer pipe and the air gap layer isolation pipe from the bottom head of the rising fluid outer pipe.

[0022] The upper ends of the rising fluid outer pipe and the descending fluid inner pipe are respectively provided with a liquid inlet pipe and a liquid outlet pipe. The liquid inlet pipe and the liquid outlet pipe are respectively connected to both ends of the concrete embedded circulation pipeline. The annular flow channel, the concrete embedded circulation pipeline, and the descending fluid inner pipe form a circulation pipeline. The circulation pipeline is filled with a heat-conducting medium. The concrete embedded circulation pipeline is embedded in the road surface layer that needs to be de-iced. The melting point of the heat-conducting medium is lower than the lowest value of the ambient temperature of the use environment.

[0023] A valve can also be provided on the liquid inlet pipe to control the start of the system.

[0024] The upper half of the concrete embedded circulation pipeline is in direct contact with the concrete, and the lower half is surrounded by fluffy thermal insulation material. On the one hand, it can prevent heat from dissipating downward, and on the other hand, it reserves expansion space for the concrete embedded circulation pipeline. The turning part of the concrete embedded circulation pipeline is wrapped with a pipe with a larger diameter to resist axial expansion. The concrete embedded circulation pipeline is laid in an "S" shape to increase the contact area with the concrete.

[0025] Preferably, a number of positioning fins are provided between the rising fluid outer pipe and the air gap layer isolation pipe and between the air gap layer isolation pipe and the descending fluid inner pipe to solve the problem of coaxiality deviation in long-distance transmission.

[0026] Positioning fins with a 120-degree radial interval are designed in the nested pipe, and one group of positioning fins is provided every 5 m.

[0027] Preferably, as Figure 4 shown, a telescopic hose is provided between the liquid inlet pipe and the concrete embedded circulation pipeline and between the liquid outlet pipe and the concrete embedded circulation pipeline. By providing the telescopic hose, it is used to buffer the thermal expansion stress.

[0028] During specific use, the telescopic hose is connected to the liquid inlet pipe, the concrete embedded circulation pipeline, or the liquid outlet pipe through a quick-connect joint or a flange.

[0029] Preferably, the liquid inlet pipe is connected to a liquid filling branch pipe, and the liquid outlet pipe is provided with a liquid discharge branch pipe. Valves are provided on both the liquid filling branch pipe and the liquid discharge branch pipe. By providing the liquid filling branch pipe, it is convenient to add the heat transfer medium in the circulation pipeline. By providing the liquid discharge branch pipe, it is convenient to discharge the heat transfer medium in the circulation pipeline, facilitating the replacement of the heat transfer medium.

[0030] During specific use, an exhaust port is provided at the highest position of the entire pipeline, and a valve is installed at the exhaust port for convenient exhaust.

[0031] Embodiment 2: A parameter calculation method for a geothermal snow melting system includes the following steps: Step 1. Select the design parameters to be determined: including the depth of the concrete buried pipe, the diameter of the buried pipe, the spacing of the buried pipes, the length of the buried pipes, the depth of the heat source, the height of the heat source, the diameter of the air gap isolation pipe, the diameter of the downflow inner pipe, and the diameter of the upflow outer pipe; In this embodiment, the depth of the heat source is the depth of the lower end of the upflow outer pipe, and the height of the heat source is the length extending upward from the lower end of the upflow outer pipe, generally 10 - 20 m. In this interval, the underground heat extraction circulation pipeline is in direct contact with the underground heat source and is not wrapped with thermal insulation materials. The outside of the upflow outer pipe in other intervals of the underground heat extraction circulation pipeline is wrapped with thermal insulation materials to isolate the heat exchange between the external soil and the pipeline.

[0032] The heat source temperature is calculated according to the following formula: ; Where C 1 is the heat factor of the ground source, generally between 0.3 and 10; T A is the ambient temperature; H is the depth of the heat source; T s is the heat source temperature.

[0033] Step 2. Node division: including the nodes of the circulation pipeline and the concrete nodes, obtaining the distribution of the temperature T and the flow velocity u.

[0034] The application equation for the nodes of the circulation pipeline is: (1); Where is the fluid thermal expansion coefficient, is the acceleration due to gravity, L is the depth of the heat source, and are the temperatures of the same height nodes in the upflow outer pipe and the downflow inner pipe respectively, is the resistance coefficient, is the circulation flow rate, is the equivalent diameter of the circulation pipeline. In the downflow inner pipe section, d is taken as the diameter of the downflow inner pipe. In the upflow outer pipe section, d is taken as the difference between the diameter of the upflow outer pipe and the diameter of the air gap isolation pipe. In the concrete embedded circulation pipeline section, d is taken as the diameter of the concrete embedded circulation pipeline.

[0035] (2); Among them, is the density reference value, is the specific heat at constant pressure, T is the node temperature, is used to add the heat transfer boundary conditions of the geothermal heat source and the concrete buried pipe. z represents the coordinate in the same direction as the fluid flow direction; At the underground heat source, the fluid absorbs heat, can be calculated according to the following formula: ; Among them, A s is the heat absorption area at the heat source, calculated based on the height of the heat source and the diameter of the upflow outer pipe.

[0036] The calculation in the embodiment can adopt .

[0037] At the concrete buried pipe, can be calculated according to the following formula: ; Among them, A c is the outer surface area of the concrete buried pipe, calculated based on the length and diameter of the concrete buried pipe. T w is the temperature of the outer wall of the concrete buried pipe and is also the temperature of the bottommost node of the concrete. k is the adjustment coefficient.

[0038] A in the embodiment c The calculation can be carried out according to the following formula: ; The function of the adjustment coefficient k is the proportion of heat transfer from the concrete buried pipe to the concrete, and it is calculated using the following formula: ; For the concrete nodes above the concrete embedded circulation pipeline, the application equation expression is: (3); Among them, x represents the coordinate in the same direction as the ground normal direction; The integration length in the x direction in the above integral solution is the depth of the concrete buried pipe.

[0039] According to equations (1) to (3), the distribution of temperature T and the flow velocity u are obtained.

[0040] The flow velocity can help designers evaluate the overall circulation performance, and the temperature T is a necessary condition for calculating the ground heat flux q.

[0041] Step 3: Calculate the value of the ground heat flux q according to the ground heat flux equation. The ground heat flux is calculated by the formula: (4); where is the number of nodes of the concrete buried pipe, is the equivalent heat transfer coefficient.

[0042] When q is greater than q set After that, q set is the set value of the ground heat flux. Solve the transient equation with the currently selected parameters.

[0043] The transient equation includes the transient equation of the circulation pipeline nodes and the transient equation of the concrete nodes, and the expressions are: Transient equation of the circulation pipeline nodes: (5); (6); The transient equation of the concrete nodes is: (7); Solve equations (5) to (7) to obtain the change of temperature T with time.

[0044] Recalculate equation (4) to obtain the change of the ground heat flux q with time.

[0045] When q is greater than q set At this time, take the current running time as the system response time. If the response time is less than the user's expectation value, then recognize the current design parameters. Otherwise, adjust the design parameters and repeat steps 1 to 3 for recalculation until the response time is less than the user's expectation value.

[0046] Example 3: Referring to Example 2, design the parameters of the snow melting system for the snowfall situation in Tianjin. The design process: First, select the design parameters as shown in Table 1.

[0047]

[0048] Set q set to 500 W / m 2 , and set the response time to 6 hours. According to the above calculation process, it is found after the first calculation that the steady-state result q reaches 338 W / m 2 , which is less than q set. Modify the underground heat source depth in Table 1 above to 1000 m, keep other parameters unchanged, and recalculate. The new calculation results show that q can reach 600 W / m 2 Above, it meets the requirements, and the response time is 4 hours and 13 minutes. Based on this, it is determined that the design parameters meet the target.

[0049] Example 4: To demonstrate the feasibility and superiority of the present invention, take a snowfall in Tianjin on February 28, 2021 as an example to compare the invention with the application number 2023100836031 and the present invention. The design parameters are in accordance with those given in Example 3. Figure 5 It shows the situation of surface snow and water accumulation. The road surface was dry before the snowfall. After the snowfall, the system started synchronously, and the road surface changed to a snow-covered state. After the snow-melting ability of the geothermal system was improved, part of the snow melted and became a state of coexistence of snow and water until all the snow melted, and the road surface would become wet. It can be seen that the snow-free time (that is, the time from the coexistence of snow and water to the wet road surface) of this patent is 16760 s, while the snow-free time of the comparative patent is 21830 s, more than one hour earlier.

[0050] Figure 6 The results given are the flow velocities in the circulation pipes of the comparative patent and the present invention. Due to the improved design, the flow velocity in this system starts to rise after the system starts, and the time to reach the peak is also much earlier than that of the comparative prior art.

[0051] The above embodiments are only the preferred technical solutions of the present invention and should not be regarded as limitations on the present invention. The embodiments and the features in the embodiments in this application can be arbitrarily combined with each other without conflict. The protection scope of the present invention should be the technical solutions recorded in the claims, including the equivalent replacement solutions of the technical features in the technical solutions recorded in the claims. That is, the equivalent replacement improvements within this scope are also within the protection scope of the present invention.

Claims

1. A geothermal snow melting system, characterized in that: It includes an underground heat-raising circulation pipeline and a concrete pre-buried circulation pipeline, the underground heat-raising circulation pipeline includes an outer tube for rising fluid, an air gap layer isolation tube and an inner tube for descending fluid are arranged in sequence from the outside to the inside of the outer tube for rising fluid, the upper and lower ends of the space between the air gap layer isolation tube and the inner tube for descending fluid are closed to form a closed air gap layer, the upper and lower ends of the outer tube for rising fluid are closed, a flow gap is arranged between the lower end of the inner tube for descending fluid and the bottom of the outer tube for rising fluid, an annular flow channel is formed between the outer tube for rising fluid and the air gap layer isolation tube, the upper ends of the outer tube for rising fluid and the inner tube for descending fluid are respectively provided with a liquid inlet pipe and a liquid outlet pipe, the liquid inlet pipe and the liquid outlet pipe are respectively connected to the two ends of the concrete pre-buried circulation pipeline, the annular flow channel, the concrete pre-buried circulation pipeline and the inner tube for descending fluid constitute a circulation pipeline, the circulation pipeline is filled with heat-conducting medium, and the concrete pre-buried circulation pipeline is pre-buried in the pavement layer where snow melting is required.

2. A geothermal snow melting system according to claim 1, characterized in that: A plurality of positioning fins are arranged between the outer tube of the ascending fluid and the air gap layer isolation tube, and between the air gap layer isolation tube and the inner tube of the descending fluid.

3. A geothermal snow melting system according to claim 1, characterized in that: The closed end at the lower end of the outer tube of the rising fluid is a "U"-shaped structure.

4. A geothermal snow melting system according to claim 1, characterized in that: A telescopic hose is arranged between the liquid inlet pipe and the concrete pre-buried circulation pipeline and between the liquid outlet pipe and the concrete pre-buried circulation pipeline.

5. The geothermal snow melting system according to claim 1, characterized in that: The liquid inlet pipe is connected to the liquid filling branch pipe, the liquid outlet pipe is provided with a liquid discharge branch pipe, and valves are provided on the liquid filling branch pipe and the liquid discharge branch pipe.

6. The geothermal snow melting system according to claim 1, characterized in that: The concrete pre-buried circulation pipeline is laid in an "S" shape.

7. A method for designing a geothermal snow melting system according to any one of claims 1 to 6, characterized in that: The following steps are involved: Step 1: Select the design parameters to be determined: including the concrete buried pipe depth, buried pipe diameter, buried pipe spacing, buried pipe length, heat source depth, heat source height, air gap layer isolation pipe diameter, descending fluid inner pipe diameter, ascending fluid outer pipe diameter; Step 2: Node division: including circulation pipeline nodes and concrete nodes, to obtain the distribution of temperature T and flow velocity u; Step 3: Calculate the surface heat flow q value according to the surface heat flow equation. When q is greater than q set After that, the transient equation is solved with the currently selected parameters, and then the surface heat flux q value is recalculated to obtain the change of surface heat flux q with time, q set is the set value of surface heat flow; When q first becomes greater than q set When the system response time is less than the user's expected value, the current design parameters are recognized. Otherwise, the design parameters are adjusted and steps one to three are repeated to recalculate until the response time is less than the user's expected value.

8. The design method of a geothermal snow melting system according to claim 7, characterized in that: In step 2, the application equation of the circulation pipeline node is: （1）； in, is the thermal expansion coefficient of the fluid, is the acceleration due to gravity, L is the depth of the heat source, and are the temperatures of the nodes at the same height in the outer tube of the ascending fluid and the inner tube of the descending fluid, is the drag coefficient, is the circulation flow rate, is the equivalent diameter of the circulation pipeline. In the inner pipe section of the descending fluid, d is the diameter of the inner pipe of the descending fluid. In the outer pipe section of the ascending fluid, d is the difference between the outer pipe diameter of the ascending fluid and the diameter of the air gap layer isolation pipe. In the concrete pre-buried circulation pipeline section, d is the diameter of the concrete pre-buried circulation pipeline. （2）； in, is the density reference value, is the specific heat at constant pressure, T is the node temperature, Used to add geothermal heat source and heat transfer boundary conditions in concrete buried pipes, z represents the coordinate consistent with the fluid flow direction; For the concrete nodes at the upper part of the pre-buried concrete circulation pipeline, the equation expression is: （3）； Among them, x represents the coordinate consistent with the ground normal direction; According to equations (1) to (3), the distribution of temperature T and flow velocity u are obtained.

9. The design method of a geothermal snow melting system according to claim 7, characterized in that: In step 3, the surface heat flow The calculation formula is: （4）； in, is the number of nodes of the buried concrete pipe, is the equivalent heat transfer coefficient, T A is the ambient temperature.

10. The design method of a geothermal snow melting system according to claim 7, characterized in that: In step 3, the transient equation includes the transient equation of the circulation pipeline node and the transient equation of the concrete node, and the expression is: Transient equations for loop nodes: （5）； （6）； The transient equation of the concrete node is: （7）； Solve equations (5) to (7) to obtain the change of temperature T with time.

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