A method for calculating thermal conductivity of crushed stone foundation considering thermal radiation

Abstract

The present invention belongs to the field of foundation protection technology, specifically to a method for calculating the thermal conductivity of a rubble foundation that takes into account the effects of thermal radiation. The method comprises: collecting air, ground temperature, and wind speed data in the project area; determining the physical parameters of the foundation soil and fill; measuring foundation cross-sectional parameters and the physical parameters of the rubble; calculating the structural parameters and thermal conductivity of the rubble layer; calculating the thermal radiation thermal conductivity and effective thermal conductivity of the rubble layer; establishing a rubble layer model, calculating the impact of the rubble layer on the permafrost of the foundation, taking into account the effects of thermal radiation, and monitoring the relevant calculation results. The present invention provides a theoretical basis for the design and calculation of rubble foundations, solving the problem of not considering the effects of thermal radiation heat transfer during the heat transfer process of the rubble layer. It can more accurately analyze the impact of the cooling characteristics of the rubble layer on the thermal state of the permafrost in the foundation, providing theoretical support and scientific basis for engineering construction in cold regions.

Description

Technical Field

[0001] The present invention relates to the technical field of foundation protection, in particular to a heat conduction calculation method for a crushed stone foundation taking into account the effect of heat radiation. Background Art

[0002] Permafrost refers to a special geological formation at a depth below the Earth's surface where temperatures remain persistently below 0°C and frozen for two or more years. Permafrost is widely distributed in China, covering 22.3% of the country's land area, and is predominant at high altitudes, particularly on the Qinghai-Tibet Plateau. With climate warming and intensified human activities, accelerated permafrost degradation poses a significant threat to the integrity of infrastructure, resulting in complex and diverse engineering hazards. It is projected that by 2050, approximately 30-50% of existing projects in permafrost zones in the Northern Hemisphere will be located in permafrost areas at high risk of degradation, and nearly 70% of existing infrastructure in these areas will be affected by permafrost degradation. To address the impacts of climate change and human-induced structural changes on permafrost foundations, a range of active cooling methods have been proposed, such as block crushed stone layers, ventilation pipes, heat pipes, and their composite structures. Block crushed stone layers, due to their excellent cooling performance and ease of construction, have become one of the most widely used active cooling methods, accounting for approximately 60% of cooling methods used in permafrost zones.

[0003] Massive rubble layers, with their significant cooling effects, have been widely used to protect engineering foundations in cold regions from thawing permafrost. As a highly permeable, porous medium with a defined thickness, the pore air in the massive rubble layer forms an unstable density gradient due to temperature fluctuations, leading to convective heat transfer within the massive rubble layer. Current research on the cooling mechanism of massive rubble layers primarily considers heat transfer processes through conduction and convection. However, field studies have shown that thermal radiation can also occur in gravel or coarser materials. Measurements of the effective thermal conductivity of massive rubble samples (a combination of conduction and radiation) have shown that the effective thermal conductivity at ambient temperature is 97%-209% higher than that of pure conduction. However, insufficient analysis has been conducted on the impact of massive rubble layer cooling characteristics on permafrost in foundations, even when considering thermal radiation. Therefore, a method for calculating the thermal conductivity of massive rubble foundations that considers thermal radiation is urgently needed, capable of accurately analyzing the impact of massive rubble layers on permafrost in foundations after accounting for thermal radiation. Summary of the Invention

[0004] In view of the above-mentioned deficiencies in the prior art, it is necessary to add the thermal radiation effect that has not been considered in the heat transfer process of the crushed stone layer. The present invention provides a thermal conductivity calculation method for crushed stone foundation considering the thermal radiation effect. The calculation method includes the following steps:

[0005] Step S1, collecting air, ground temperature, and wind speed data in the project area, determining the physical parameters of the foundation soil and fill, and measuring the foundation cross-section parameters and the physical parameters of the block gravel;

[0006] S101, collect air, temperature and wind speed data in the project area through the meteorological station, and fit the collected temperature and wind speed parameters into a periodic boundary condition formula;

[0007] S102, determining the physical parameters of the foundation soil and fill according to the engineering design documents, and calculating and determining the thermal conductivity, volume heat capacity, and phase change latent heat of the foundation soil and fill in frozen and unfrozen states;

[0008] S103, considering that the soil layer is infinite in the longitudinal direction, ignoring the non-uniform effect in the longitudinal direction, simplifying it to a two-dimensional isotropic problem in the cross-sectional direction, measuring the cross-sectional dimensions of the foundation, including width, height, and slope ratio;

[0009] S104, determining the physical parameters of the block and gravel material according to the engineering design documents, including density, thermal conductivity, specific heat capacity, equivalent particle size, permeability, and inertial resistance coefficient.

[0010] Step S2, calculating the empirical coefficient of the block gravel layer structure and the thermal conductivity coefficient;

[0011] S201, the structural parameters are a function of the thermal conductivity ratio of air to gravel in the gravel layer. Based on the empirical coefficients of particle shape and cementation, the structural parameters of the gravel layer are calculated as:

[0012]

[0013] Where, κ 2p is the structural parameter; s is the thermal conductivity of the solid phase; λ f is the thermal conductivity of the liquid phase; is the empirical coefficient of the structural parameter, when λ f / λ s <1 / 15, the empirical coefficient of the crushed stone particle structure parameter is 0.54, when λ f / λ s >January 15th

[0014] S202, the thermal conductivity of the crushed stone layer is calculated as:

[0015]

[0016] Where λ c is the thermal conductivity of the crushed stone layer; n is the porosity.

[0017] Step S3, calculating the thermal radiation thermal conductivity and the effective thermal conductivity of the crushed stone layer;

[0018] S301. Thermal radiation in the gravel layer occurs only within the voids of the porous medium and strongly absorbs radiation during transmission. Integrating each term in the thermal radiation transfer equation over the entire space by 4π can yield the radiation energy conservation equation for the spatial microelement. The full-wavelength radiation energy equation for the gravel layer is calculated as:

[0019]

[0020] Where: q λ,x ,q λ,y ,q λ,z is the spectral radiation heat flux density vector q λ Components in the x, y, and z coordinates.

[0021] S302, the heat radiation propagation in the gravel layer is simplified as a gray fluid confined between two infinite isothermal parallel plates of opaque material with gray diffusing surfaces. The radiation heat flux is obtained by analogy with the heat exchange between the diffusing gray bodies:

[0022]

[0023] Where ε1 and ε2 are the emissivities of different surfaces; T1 and T2 are the temperatures of different surfaces; and σ is the Stefan-Boltzmann constant, which is 5.67×10 -8 W·m -2 ·K -4 .

[0024] S303, for surfaces with similar emissivity, the equation for radiative heat flux simplifies to:

[0025]

[0026] Where, ε is the emissivity of the surface of the gravel layer; d is the effective particle size; is the average temperature.

[0027] S304: During the heat transfer process in the gravel layer, the direction of heat transfer due to thermal radiation is the same as that of heat conduction. The problem of heat transfer due to thermal radiation is simplified to the problem of heat conduction with a thermal conductivity that depends on temperature. The thermal conductivity of the gravel layer due to thermal radiation is calculated as:

[0028]

[0029] Where λ r is the thermal radiation conductivity.

[0030] S305, the energy equation for the block gravel layer is:

[0031]

[0032] Where c pis the specific heat capacity; C e ,λ e is the effective volume heat capacity and effective thermal conductivity of the gravel layer.

[0033] S306: The radiation intensity in a local area of the gravel layer is only affected by the thermal radiation from the adjacent surface, and the radiation energy can only travel a short distance before being absorbed. The radiative heat transfer within the gravel layer becomes a diffusion process. The direction of thermal radiation transfer is the same as that of heat conduction. Therefore, it can be expressed in the same form as Fourier's law using the diffusion approximation. The effective thermal conductivity of the gravel layer, taking into account the effect of thermal radiation, is:

[0034] λ e =λ c +λ r ;

[0035] Where λ e is the effective thermal conductivity of the crushed stone layer.

[0036] S307, the energy equation for the block gravel layer considering thermal radiation is:

[0037]

[0038] Step S4: Establish a block gravel layer model, calculate the impact of the block gravel layer on the permafrost of the foundation taking into account the effect of thermal radiation, and monitor the relevant calculation results.

[0039] S401, after simulating and calculating the initial temperature field with the rubble layer model, the impact of the cooling characteristics of the rubble layer on the thermal state of the foundation permafrost is calculated based on the boundary temperature and boundary wind speed conditions under the climate warming condition;

[0040] S402, verifying the model by comparing the measured monitoring point temperature with the simulated monitoring point temperature;

[0041] S403, evaluate the cooling effect of the gravel layer on the permafrost of the foundation under the action of thermal radiation by using the heat flux changes between the gravel layer and the soil monitoring surface and the ground temperature distribution map after 50 years of the gravel layer's action.

[0042] The beneficial effects of the present invention lie in the following: Based on the combined effects of three heat transfer modes in the rubble layer, the present invention proposes a method for calculating the thermal conductivity of a rubble foundation that takes into account the effects of thermal radiation. By accounting for the effects of thermal radiation in the rubble layer, the thermal radiation heat transfer of the rubble layer is expressed as the effective thermal conductivity of a homogeneous material (accounting for thermal radiation), and a homogenization method that does not rely on radiation transfer equations is used. The method calculates the thermal state of the permafrost foundation in the rubble layer, taking into account thermal radiation. This method provides theoretical support and scientific basis for engineering construction in cold regions and has significant engineering application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] In order to more clearly illustrate the technical solutions and advantages of the embodiments of the present invention or the prior art, the following briefly introduces the drawings required for use in the embodiments or the prior art descriptions. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0044] Figure 1 This is a flow chart of the heat conduction calculation method of the block gravel foundation considering the effect of heat radiation of the present invention;

[0045] Figure 2 Schematic diagram of the air temperature fitting curve of the present invention;

[0046] Figure 3 Schematic diagram of the calculation model of the block gravel layer of the present invention;

[0047] Figure 4 This is a graph showing how the thermal conductivity of thermal radiation varies with temperature.

[0048] Figure 5 This is a comparison chart of the measured monitoring point temperature and the simulated monitoring point temperature of the present invention;

[0049] Figure 6 This is a diagram showing the bottom heat flux variation when the block gravel layer of the present invention is in its tenth year of action;

[0050] Figure 7 This is the ground temperature distribution diagram after 50 years of action of the block gravel layer of the present invention. DETAILED DESCRIPTION

[0051] The following will be combined with the drawings in the embodiments of this application to clearly and completely describe the technical solutions in the embodiments of this application. Obviously, the embodiments described are only part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of this application.

[0052] The present invention provides a method for calculating heat conduction of a crushed stone foundation taking into account the effect of heat radiation, the method comprising the following steps:

[0053] S1, collect air, ground temperature, and wind speed data in the project area, determine the physical parameters of foundation soil and fill, and measure foundation cross-section parameters and physical parameters of block gravel;

[0054] S2, calculate the structural parameters of the block gravel layer and the thermal conductivity coefficient;

[0055] S3, calculating the thermal radiation conductivity and effective thermal conductivity of the crushed stone layer;

[0056] S4: Establish a block gravel layer model, calculate the impact of the cooling characteristics of the block gravel layer on the foundation permafrost considering the effect of thermal radiation, and monitor the relevant calculation results.

[0057] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments:

[0058] like Figure 1 As shown, step S1 is to collect air, ground temperature, and wind speed data in the project area, determine the physical parameters of the foundation soil and fill, and measure the foundation section parameters and the physical parameters of the block gravel:

[0059] S101, collect air, temperature and wind speed data in the project area through the meteorological station, and fit the collected temperature and wind speed parameters into a periodic boundary condition formula.

[0060] Considering the boundary temperature conditions under climate warming conditions:

[0061]

[0062] Where, T is the calculated temperature; T0 is the annual average temperature; A is the temperature amplitude; t h is the time, in h; α0 is the initial phase.

[0063] Based on the collected local meteorological data, the boundary conditions required in the calculation process can be determined. The temperature function will be used as the temperature condition input, and the fitted air temperature curve is as follows: Figure 2 The boundary temperature conditions are shown below.

[0064] Air temperature:

[0065]

[0066] Natural surface temperature:

[0067]

[0068] Slope temperature:

[0069]

[0070] Top temperature:

[0071]

[0072] Air can pass through the block gravel structure under the forced action of wind. The wind speed boundary is applied to the PJ edge. The standard wind speed v at a height of 10m above the ground is determined based on monitoring data. 10 The expression is as follows:

[0073]

[0074] The wind speed at other heights can be approximated by the following relationship:

[0075]

[0076] The boundaries on both sides of the model are adiabatic boundaries, and the bottom has a constant heat flux of 0.06W / m 2 .

[0077] S102. Determine the physical parameters of the foundation soil and fill based on the engineering design documents, calculate and determine the thermal conductivity, volume heat capacity, and phase change latent heat of the foundation soil and fill in the frozen and unfrozen states, and determine the specific physical parameters of the foundation soil and fill as shown in Table 1.

[0078] Table 1 Physical parameters of foundation soil and fill

[0079]

[0080] In the calculation, the apparent heat capacity method is used to deal with the latent heat of phase change in the aqueous medium. It is assumed that the phase change of the aqueous medium in the model occurs in the temperature range (T m +ΔT). When establishing the equivalent volume heat capacity, the influence of the temperature interval ΔT should be considered, and it is assumed that the volume heat capacity of the medium when it is frozen and not frozen is C f and C u and thermal conductivity λ f and λ u The thermal conductivity and volume heat capacity of the foundation soil and fill in the frozen and unfrozen states are determined independently of the temperature, so the simplified construction is and The expression is as follows:

[0081]

[0082] Where, is the equivalent volume heat capacity and equivalent thermal conductivity of the medium, C f and C u is the volume heat capacity of frozen and unfrozen; λ f and λ u is the thermal conductivity coefficient when frozen and unfrozen; T m is the phase change temperature of the aqueous medium; ΔT is the phase change temperature interval of the aqueous medium; L is the latent heat of phase change per unit volume of the aqueous medium.

[0083] S103, considering that the soil layer is infinite in the longitudinal direction, ignoring the uneven effects in the longitudinal direction, simplifies it to a two-dimensional isotropic problem in the cross-sectional direction, and measures the cross-sectional dimensions, including width, height, and slope ratio. The cross-sectional dimensions obtained through on-site measurements are: width 10.4m, slope ratio 1:1.5, fill height 3.6m, ballast fill height 0.6m, block gravel slope protection fill thickness 1.5m, and the specific block gravel layer calculation model is as follows: Figure 3 shown.

[0084] S104: Determine the physical parameters of the block and gravel material according to the engineering design documents, including equivalent particle size, permeability, and inertial resistance coefficient. The equivalent particle size of the block and gravel is about 10 cm, so the porous medium permeability k and inertial resistance coefficient B are taken as 1.58×10 -6 m 2 and 840.32m -1 The equivalent particle size of ballast gravel is 5 cm, and the permeability k and inertial resistance coefficient B are 6.32×10 -7 m 2 and 1866.67m -1 , the surface emissivity of the gravel is taken as 0.9.

[0085] like Figure 1 As shown, step S2 is to calculate the empirical coefficient of the block gravel layer structure and the thermal conductivity coefficient:

[0086] S201, the structural parameters are a function of the thermal conductivity ratio of air to gravel in the gravel layer. Based on the empirical coefficients of particle shape and cementation, the structural parameters of the gravel layer are calculated as:

[0087]

[0088] Where, κ 2p is the structural parameter; s is the thermal conductivity of the solid phase; λ f is the thermal conductivity of the liquid phase; is the empirical coefficient of the structural parameter, when λ f / λ s <1 / 15, the empirical coefficient of the crushed stone particle structure parameter is 0.54, when λ f / λ s >January 15th

[0089] S202, the thermal conductivity of the crushed stone layer is calculated as:

[0090]

[0091] Where λ c is the thermal conductivity of the crushed stone layer; n is the porosity.

[0092] The measurement site is located in the Kaixinling mountainous area of the Qinghai-Tibet Plateau, which is a continuous permafrost area with an altitude of about 4,600 meters. The constant pressure specific heat and thermal conductivity of the air were determined to be 1.004 kJ·m -3 ℃ -1 and 0.02W·m -1 ℃ -1, density is ρ = 0.641 kg·m -3 , dynamic viscosity is μ=1.75×10 5 kg·m -1 ·s -1 , the physical parameters of the crushed stone materials are shown in Table 2.

[0093] Table 2 Physical parameters of crushed stone materials

[0094]

[0095] like Figure 1 As shown, in step S3, the thermal radiation thermal conductivity coefficient of the crushed stone layer and the effective thermal conductivity coefficient of the crushed stone layer are calculated:

[0096] S301. Thermal radiation in the gravel layer occurs only within the voids of the porous medium and strongly absorbs radiation during transmission. Integrating each term in the thermal radiation transfer equation over the entire space by 4π can yield the radiation energy conservation equation for the spatial microelement. The full-wavelength radiation energy equation for the gravel layer is calculated as:

[0097]

[0098] Where: q λ,x ,q λ,y ,q λ,z is the spectral radiation heat flux density vector q λ Components in the x, y, and z coordinates.

[0099] S302, the heat radiation propagation in the gravel layer is simplified as a gray fluid confined between two infinite isothermal parallel plates of opaque material with gray diffusing surfaces. The radiation heat flux is obtained by analogy with the heat exchange between the diffusing gray bodies:

[0100]

[0101] Where ε1 and ε2 are the emissivities of different surfaces; T1 and T2 are the temperatures of different surfaces; and σ is the Stefan-Boltzmann constant, which is 5.67×10 -8 W·m -2 ·K -4 .

[0102] S303, for surfaces with similar emissivity, the equation for radiative heat flux simplifies to:

[0103]

[0104] Where, ε is the emissivity of the surface of the gravel layer; d is the effective particle size; is the average temperature.

[0105] S304: During the heat transfer process in the gravel layer, the direction of heat transfer due to thermal radiation is the same as that of heat conduction. The problem of heat transfer due to thermal radiation is simplified to the problem of heat conduction with a thermal conductivity that depends on temperature. The thermal conductivity of the gravel layer due to thermal radiation is calculated as:

[0106]

[0107] Where λ r is the thermal radiation thermal conductivity;

[0108] The thermal conductivity of the crushed stone layer changes with temperature as follows Figure 4 shown.

[0109] S305, the energy equation for the block gravel layer is:

[0110]

[0111] Where c p is the specific heat capacity; C e ,λ e is the effective volume heat capacity and effective thermal conductivity of the gravel layer.

[0112] S306: The radiation intensity in a local area of the gravel layer is only affected by the thermal radiation from the adjacent surface, and the radiation energy can only travel a short distance before being absorbed. The radiative heat transfer within the gravel layer becomes a diffusion process. The direction of thermal radiation transfer is the same as that of heat conduction. Therefore, it can be expressed in the same form as Fourier's law using the diffusion approximation. The effective thermal conductivity of the gravel layer, taking into account the effect of thermal radiation, is:

[0113] λ e =λ c +λ r ;

[0114] Where λ e is the effective thermal conductivity of the crushed stone layer.

[0115] S307, the energy equation for the block gravel layer considering thermal radiation is:

[0116]

[0117] like Figure 1 As shown, in step S4, a crushed stone layer model is established to calculate the impact of the cooling characteristics of the crushed stone layer on the permafrost of the foundation considering the effect of thermal radiation and monitor the relevant calculation results:

[0118] S401, after simulating and calculating the initial temperature field using the rubble layer model, the impact of the cooling characteristics of the rubble layer on the thermal state of the foundation permafrost is calculated based on the boundary temperature conditions and boundary wind speed conditions under climate warming conditions.

[0119] S402, the model is tested by comparing the measured temperature of the monitoring point with the simulated temperature of the monitoring point. The monitored temperature of the fill area 1.5m below the left shoulder is selected for comparison with the simulated temperature. Figure 5 shown.

[0120] S403, evaluate the cooling effect of the crushed stone layer under the action of thermal radiation by evaluating the heat flux change between the crushed stone layer and the soil layer monitoring surface, and select the instantaneous heat flux change diagram of the bottom of the crushed stone layer in the tenth year of its action, as shown in Figure 6 As shown, positive values indicate heat absorption, negative values indicate heat release, and the negative sign only represents heat release.

[0121] S404, the ground temperature distribution map after 50 years of the rubble layer is used to evaluate the cooling effect of the rubble layer under the action of thermal radiation. Usually, the maximum seasonal melting depth in permafrost areas occurs in October. The ground temperature distribution map on October 15, the 50th year, is selected, as shown in Figure 4. Figure 7 shown.

[0122] This paper presents a method for calculating the thermal conductivity of a rubble foundation, taking into account the effects of thermal radiation. It proposes a method for expressing the thermal radiation heat transfer of the rubble layer as the effective thermal conductivity of a homogeneous material (accounting for thermal radiation). The thermal impact of the rubble layer on the underlying permafrost is calculated, taking into account thermal radiation. The thermal radiation effect during heat transfer within the rubble layer significantly increases with increasing temperature, causing the rubble layer to absorb additional heat in the summer and transfer it to the permafrost in the foundation. Therefore, when designing rubble layers for projects in permafrost areas, the heat transfer process should include thermal radiation to account for the additional summer heat absorption.

[0123] It should be understood that the various forms of the processes shown above can be used to reorder, add, or delete steps. For example, the steps described in this application can be performed in parallel, sequentially, or in a different order, as long as the desired results of the technical solutions disclosed in this application can be achieved. This is not a limitation herein.

[0124] The above specific embodiments do not constitute a limitation on the scope of protection of this application. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions may be made based on design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application shall be included within the scope of protection of this application.

Claims

1. A method for calculating thermal conductivity of a crushed stone foundation taking into account the effect of thermal radiation, characterized in that: The following steps are involved: S1, collect air, ground temperature, and wind speed data in the project area, determine the physical parameters of foundation soil and fill, and measure foundation cross-section parameters and physical parameters of block gravel; S2, calculate the structural parameters of the block gravel layer and the thermal conductivity coefficient; The structural parameters of the crushed stone layer are calculated as follows: Where, κ 2p is the structural parameter of the gravel layer; s is the thermal conductivity of the solid phase; λ f is the thermal conductivity of the liquid phase; is the empirical coefficient of the structural parameter, when λ f / λ s <1 / 15, the empirical coefficient of the crushed stone particle structure parameter is 0.54, when λ f / λ s >January 15th The thermal conductivity of the crushed stone layer is calculated as: Where n is the porosity; S3, calculating the thermal radiation conductivity and effective thermal conductivity of the crushed stone layer; The full wavelength radiation energy equation vector of the gravel layer is calculated as: Where: q λ,x ,q λ,y ,q λ,z is the spectral radiation heat flux density vector q λ Components in x, y, z coordinates; The radiative heat flux of the gravel layer is obtained by analogy with the heat exchange between the diffuse ash bodies: Where ε1 and ε2 are the emissivities of different surfaces; T1 and T2 are the temperatures of different surfaces; and σ is the Stefan-Boltzmann constant, which is 5.67×10 -8 W·m -2 ·K -4 ; The equation for the radiative heat flux of the rubble layer simplifies to: Where, ε is the emissivity of the surface of the gravel layer; d is the effective particle size; is the average temperature; Thermal radiation conductivity of crushed stone layer: Where T is temperature; The effective thermal conductivity of the crushed stone layer after considering the effect of thermal radiation is: l e =λ c +λ r ; Where λ c is the thermal conductivity of the crushed stone layer, λ r is the thermal radiation thermal conductivity; S4, establish a block gravel layer model, calculate the impact of the block gravel layer on the foundation permafrost considering the effect of thermal radiation, and monitor the relevant calculation results.

2. The method for calculating heat conduction of a crushed stone foundation considering the effect of heat radiation according to claim 1, characterized in that: The energy equation of the block gravel layer is: Where c p is the specific heat capacity, C e ,λ e is the effective volume heat capacity and effective thermal conductivity of the gravel layer.

3. The method for calculating heat conduction of a crushed stone foundation considering the effect of heat radiation according to claim 2, characterized in that: The energy equation considering thermal radiation in the gravel layer is:

Images