Method for predicting unfrozen water content of frozen soil body by considering supercooling effect

Abstract

The invention relates to a method for predicting the unfrozen water content of a frozen soil body by considering a supercooling effect, which comprises the following steps of: (1) carrying out a soil pore size analysis test to obtain a pore size distribution curve of a soil sample; (2) performing a soil component determination experiment to obtain soil volumetric water content, soil sample size and dry density; and (3) calculating to obtain the change relation of the unfrozen water content in the soil freezing process through the calculation method provided by the invention. The method comprises the following steps: on the basis of a nucleation theory and considering an occurrence environment of pore water of a soil body, constructing an unfrozen water content expression of a supercooling section; determining an unfrozen water content change relation in a water freezing stage according to the concentration change of different components through a diffusion law; and finally, constructing a soil body freezing characteristic model covering a supercooling stage and a moisture freezing stage. The method is clear in influence factor and definite in physical significance, and can be used for calculating and predicting the unfrozen water content change relation of the soil body system in the freezing process. In addition, theoretical support can be provided for establishing a high-precision freezing characteristic curve model, and theoretical reserve is provided for perfecting a cold region engineering numerical calculation model.

Description

Technical Field

[0001] The invention relates to the technical field of frozen soil engineering, and in particular to a method for predicting the unfrozen water content of a frozen soil body taking into account supercooling effects. Background Art

[0002] Many projects will be carried out in seasonally frozen areas. The freeze-thaw cycle causes adverse geological phenomena such as soil frost heave, thaw settlement and surface salt uplift. The study of freezing characteristic curves can more effectively reflect the relationship between unfrozen water content and temperature. The supercooling characteristic can effectively reflect the problem of low but unfrozen soil. Therefore, the relationship between the change of unfrozen water content considering the supercooling effect needs to be solved urgently. Existing research on the supercooling effect is often based on experimental results, lacking the physical meaning and model to explain this water change; and the supercooling research with a theoretical basis is often based on a certain characteristic point, lacking a quantitative analysis of the change of unfrozen water in the entire freezing process.

[0003] In summary, with the urgent need for research development and engineering construction, it is urgent to propose a physical model of freezing characteristic changes considering supercooling effect to fill the theoretical gap. In-depth research on the changes of unfrozen water during freezing will help provide a solid theoretical basis for related experiments, and provide a theoretical basis for studying the freezing characteristic curve considering supercooling effect, and find the nodes of low and unfrozen phenomena, so as to protect the frozen soil in various regions from engineering diseases derived from frequent freeze-thaw cycles caused by meteorological changes and day-night temperature differences. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a method for predicting soil moisture content during freezing process based on classical nucleation theory and Fick's diffusion theory, which takes into account factors such as pore size, soil composition, moisture content and soil sample size.

[0005] In order to solve the above problems, the method for predicting the unfrozen water content of frozen soil considering the supercooling effect of the present invention comprises the following steps:

[0006] (1) Conduct soil pore size analysis test to obtain the pore size distribution curve of the soil sample.

[0007] (2) Conduct soil composition determination experiments to obtain soil volume water content, soil sample size and dry density.

[0008] (3) The calculation method of the present invention is used to calculate the change relationship of the unfrozen water content during the freezing process of the soil.

[0009] The freezing process of pore water is divided into the crystallization of supercooled water and the growth of crystal nuclei. There is a "critical nucleation phenomenon" in the crystallization process of supercooled water. The reason for this phenomenon is that it corresponds to the maximum potential energy barrier in the freezing process. It is analyzed by classical nucleation theory: In classical nucleation theory, the free energy of supercooled water is composed of:

[0010] ΔG(T)=ΔG V +ΔG S (1)

[0011] The free energy of supercooled water consists of the volume term on the left and the surface term on the right. During the nucleation process, phase change occurs, causing the van der Waals force field energy in the ice embryo to be significantly higher than the van der Waals force field energy in the pore solution. The heat released reduces the molecular chemical potential in the ice nucleus, which reduces the free energy of the volume term. When the average lattice volume of the ice molecule is v i When the free energy change caused by nucleation is ΔG V It can be calculated as follows:

[0012]

[0013] Among them, μ i is the potential energy of ice molecules, μ w is the potential energy of water molecules, n is the number of ice nucleus molecules, r is the radius of the nucleus, S i It is the stable vapor pressure ratio of water and ice crystals at the same temperature and pressure.

[0014] As the nucleus is generated and slowly grows, the ice-water interface is also expanding. The total tension term at the ice-water interface is affected by the temperature and the nucleus radius r. germ Therefore, the ice-water interface energy term can be written as:

[0015]

[0016] Therefore, the free energy expression of supercooled water is combined with equations 1 to 3 to obtain:

[0017]

[0018] The volume term on the left and the surface term on the right constitute the supercooled water free energy, in which the molecular chemical potential decreases with decreasing temperature, causing the volume free energy to decrease with decreasing temperature; the interfacial tension increases with decreasing temperature, causing the surface energy part to increase with decreasing temperature, resulting in the monotonic interval and total maximum value of the system free energy change. During the freezing process, the increase in surface energy in the monotonic increasing section is greater than the decrease in volume energy, resulting in an increase in the potential energy barrier; the decrease in volume energy in the monotonic decreasing section is greater than the increase in surface energy, resulting in a decrease in the potential energy barrier.

[0019] When the free energy of supercooled water is sufficient to cross this potential energy barrier, the crystal nucleus can grow stably, otherwise its growth trend is extinction. When the crystal nucleus reaches the stable growth stage, its growth is mainly dominated by interface phase change and diffusion movement.

[0020] The interfacial phase transition is the change of the water molecule structure: the breaking and reorganization of hydrogen bonds, and the number of molecules remains unchanged during this process. The diffusion movement of water molecules in ordinary water systems is an irregular movement affected by temperature. When water molecules are in the pores of the soil system, this paper believes that their diffusion movement is affected by the concentration of supercooled water outside the ice-water interface and the concentration gradient of ice molecules inside the interface. The concentration unit is molar concentration.

[0021] In the process of cooling the pore water components, the existence of the nucleation barrier makes the pore water temperature lower than the initial freezing temperature but does not freeze. At this time, although ice nuclei begin to gradually form in the pore water, their potential energy does not cross the nucleation barrier. Therefore, the ice nuclei formed at this stage are in a metastable state and face continued growth or direct extinction; as the temperature continues to drop and the size of some nuclei continues to grow, their potential energy is sufficient to cross the nucleation potential barrier, and the metastable state is transformed into a stable growth state and can proceed spontaneously. The lower temperature makes the supercooled water around the nucleus interface have an extremely high freezing tendency. When the ice nuclei can grow stably and spontaneously, the supercooled water molecules quickly attach and undergo a phase change. Accompanied by the release of drastic latent heat of phase change, the temperature of the soil system rises briefly and rapidly, thus producing the supercooling jump phenomenon.

[0022] There is a critical state that allows the nucleus to grow stably. At this time, the change in the free energy of the nucleus is functioned as ΔG. r The partial derivative with respect to the nucleus radius r is instantaneously 0:

[0023]

[0024] In this regard, this paper finds the minimum size of the crystal nucleus that is sufficient for stable growth, and will not die as the crystal nucleus size grows. Combining equations (1-2), the critical size of the crystal nucleus for stable pore water in the soil system is obtained, and the crystal nucleus size increases with temperature changes and water freezing.

[0025]

[0026] Where v ice (T) is the volume of water molecules in the crystal nucleus, σ iw (T) is the ice-water interfacial tension, S i represents the stable vapor pressure ratio of water and ice crystals at the same temperature and pressure. All three are functions of temperature. B The Boltzmann constant is 1.38×10 -23 J / K.v ice(T) The calculation process can use the Zobrist equation:

[0027]

[0028] in, N a refers to Avogadro's constant, ρ i0 Refers to the density of ice at zero degrees, which can be calculated by the HRPruppacher & J.D.Klett equation i0 =0.9167g / cm 3 . The empirical model of ice-water interfacial tension is:

[0029] σ iw [J / m 2 ]=2.91·10 -2 +0.18 10 -3 (T-273.15) (8)

[0030] Murphy & Koop studied the effects of different parameters on the ice-water equilibrium vapor pressure ratio and concluded that many models can achieve good prediction levels above 238.15K. For convenient calculation within the experimental temperature range (>238.15K), the following temperature fitting model was proposed:

[0031]

[0032] In the pore water system, there is a relationship between the number of molecules and the size of the crystal nucleus:

[0033]

[0034] And it is known from the above that the size of the stable growing nucleus is r germ , combined with the relationship between the number of molecules and the size of the crystal nucleus, it can be known that when the ice crystal nucleus is sufficient to grow stably, the number of water molecules consumed is:

[0035]

[0036] Therefore, this paper obtains the number of consumed water molecules in the critical state of the crystal nucleus. In the process of continuous growth of the crystal nucleus, the size parameter and the quantity parameter are functions related to the temperature. Therefore, if you want to obtain the relationship between the unfrozen water content and temperature in the freezing characteristic curve, you need to transform formula (8).

[0037] Based on formula (11), the Avogadro constant N is introduced A , the molar mass of water M w , the number of consumed water molecules is converted into the mass of water molecules, thus deducing the unfrozen water content in the supercooled jump part:

[0038]

[0039] w s is the mass of soil particles, and W0 is the initial water content.

[0040] In homogeneous nucleation systems, in order to study the growth process of crystal nuclei, some scholars believe that the time and space effects of the generation of crystal nuclei in pores are random; on the other hand, some scholars describe an ideal situation to more simply control the growth process of crystal nuclei, assuming that all crystal nuclei particles in the pores are generated at the same time and grow in the same size. Neither of these two descriptions can reasonably describe the formation and growth processes of crystal nuclei at the same time, so this paper makes a reasonable assumption: since the nucleation process is too short compared to the growth process, even if the generation of crystal nuclei is random in time and space, it is believed that all crystal nuclei have negligible similar sizes at the critical state of formation and stable growth, and experience the subsequent growth process with the same size.

[0041] When the crystal nucleus is stable enough to grow, a large number of water molecules with a tendency to freeze gather around the ice crystal nucleus. Their temperature is lower than the freezing temperature but no phase change occurs. They are called supercooled water molecules. The two-step nucleation theory holds that during the freezing process, water molecules first migrate from the liquid phase to the metastable phase where supercooled water molecules exist, and then cross the ice-water interface to undergo a phase change. This paper uses Fick's diffusion law to study the diffusion and migration of water molecules during the freezing process.

[0042] Assuming that there is an ice-water interface with a cross-sectional area of ​​A, a horizontal diffusion probability of 0.5, the number of water molecules on the left is N(x), and the number of ice molecules on the right is N(x+Δx), then the diffusion flux j passing through the cross section A in unit time t is:

[0043]

[0044] There is also volume concentration The diffusion flux is thus converted to:

[0045]

[0046] Among them, D(m 2 / s) is the diffusion coefficient; M (mol / m 3 ) is the molecular volume concentration; j (mol / m 2 ·s) is the diffusion flux of water molecules.

[0047] The volume concentration of ice molecules inside the crystal nucleus is [M] r , the volume concentration of water molecules with freezing tendency is [M] s , the volume concentration of water molecules in the outside world is [M] bIf the water molecules with freezing tendency are regarded as a transition phase, there is a diffusion relationship between the liquid phase and the transition phase. Combining formula (14), the rate of the amount of material from the liquid phase to the surface of the crystal nucleus is obtained:

[0048]

[0049] When water molecules cross the ice-water interface and undergo phase change, only the molecular structure changes, and the number of molecules does not change. Therefore, based on the conservation of the amount of matter, the rate of molecules consumed by the crystal nucleus growth is deduced from the volume change rate:

[0050]

[0051] Where k (m / s) is the reaction constant, [M] r is the volume concentration of ice molecules in the crystal nucleus,

[0052] The unfrozen water content is converted into the rate of change of the crystal nucleus size to obtain the relationship between the water molecule consumption rate and the crystal nucleus volume change rate:

[0053]

[0054] Formula (15-17) represents the transformation relationship of water molecules during the freezing process. Combining the above three relationships, the molar concentration of the supercooled water molecule phase (18) and the rate of change of the crystal nucleus size (19) are:

[0055]

[0056] Therefore, the change rate of unfrozen water content is obtained:

[0057]

[0058] At the nanoscale, the concentration of water molecules is extremely sensitive to the change of the microscopic potential energy of the soil system. Based on the Gibbs-Tomson relationship, according to the relationship between different potential energies in the soil system, for example: soil-water interfacial tension Ice-water interfacial tension Δμ iw =4πr germ 2 σ iw (T), corrected for the concentration of molecules in the soil system:

[0059]

[0060] During the freezing process of soil, the water content and freezing amount of the system change dynamically with the temperature, so the analysis of the energy field is essential to the study of the freezing characteristic curve. The various components in the aluminum can are regarded as a whole, and the freezing process is a water bath cooling. The energy field of this model is analyzed: Taking the soil system as a reference, its energy source and destination: the cold energy Q1 transmitted to it by the outside world during the water bath process; the phase change latent heat L released during the phase change process m ; The heat transfer Q2 from the soil system to the outside world; The change in the internal energy of the soil system ΔH. The above factors constitute the energy field of the water bath process, and the four of them are related:

[0061] ΔH=-Q1+Q2+L m (twenty two)

[0062] For heat transfer in an energy field, Newton's law of cooling states that when two objects of different temperatures come into contact with each other, heat is transferred from the object with a relatively high temperature to the object with a low temperature, which gradually cools itself. Mathematically, the rate of energy change is equal to the product of the temperature difference between the two objects and the contact area and the heat transfer coefficient:

[0063]

[0064] Where, T w is the water bath temperature, T a is the outside air temperature, H is the heat transfer coefficient (W / (m 2 ·K)), S is the contact area.

[0065] For the latent heat release rate of phase change during freezing, there is a relationship:

[0066]

[0067] Where L is the latent heat released by the phase change of a single water molecule, and J is the total flux of water molecules on the ice-water interface during the phase change process.

[0068] The rate of change of the internal energy of the soil system can be calculated by the specific heat of the system:

[0069]

[0070] In summary, combined with equations (22-26), the overall conversion of the energy field is obtained:

[0071]

[0072] According to formula (27), the temperature change relationship is integrated to obtain the relationship between temperature and time during the freezing process:

[0073] mxD c T=[HS(TT w )-HS(TTa )+LJ]t (28)

[0074] Substituting formula (28) into formula (22), we finally get the relationship between the change of unfrozen water content in the freezing process:

[0075]

[0076] Where W m The residual unfrozen water content is the residual unfrozen water content. When a small amount of pore water freezes, the soil pores have a strong adsorption effect on water molecules and form a layer of adsorbed water film on the surface of the soil pores, making it difficult for water molecules to diffuse and undergo phase change. Jinxiao calculated the unfrozen water content through the double electron layer model:

[0077] W m =A s ρ bw δ×10 -2 (30)

[0078] The adsorption layer thickness δ is taken as Strongly bound water density ρ bw =1.1(g / cm); A s is the specific surface area.

[0079] The present invention is based on the nucleation theory and takes into account the occurrence environment of soil pore water, constructs an expression for the relationship between the unfrozen water content and temperature in the supercooling stage of the freezing process; starting from the changes in the frozen components and the freezing mechanism during the freezing of pore water, the change relationship of the unfrozen water content in the water freezing stage is determined according to the changes in the concentrations of different components through the diffusion law; and finally constructs a soil freezing characteristic model covering the supercooling stage and the water freezing stage. The soil sample physical property parameters required by the present invention include pore size distribution curve, water content, soil composition, ambient temperature, dry density and soil sample size. In general, the present invention takes into account the objective influencing factors comprehensively, has a clear physical meaning, and can be used to predict the changes in the unfrozen water content of soil freezing. In addition, the present invention can also provide a theoretical basis for controlling the freezing process of agricultural and engineering soil in cold regions, provide theoretical support for establishing a high-precision freezing characteristic curve model, and provide a theoretical reserve for improving the numerical calculation model of cold region engineering.

[0080] Compared with the prior art, the present invention has the following advantages:

[0081] 1. The present invention takes nucleation theory and diffusion law into full consideration, establishes a SFCC physical model that takes supercooling effect into consideration, and obtains the relationship between unfrozen water content and temperature under soil cooling conditions by analyzing the formation and growth of ice nuclei and the composition of the energy field during the phase change process. The present invention finally obtains the temperature change expression with input parameters of soil pore size distribution, water content, soil system specific heat and ambient temperature.

[0082] 2. The present invention can be used to analyze and predict the changes in unfrozen water content during the freezing process, taking into full account many influencing factors such as pore size, water content, and soil physical properties. The influencing factors are clear and have clear physical meanings. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] The specific implementation modes of the present invention are described in detail below with reference to the accompanying drawings.

[0084] Figure 1 It is the critical nucleation state analysis diagram;

[0085] Figure 2 It is a diffusion diagram;

[0086] Figure 3 1:1 diagram for prediction effect;

[0087] Figure 4 Differentiate diagram for freezing process components. DETAILED DESCRIPTION

[0088] The method for predicting the unfrozen water content of frozen soil considering the supercooling effect includes the following steps:

[0089] (1) Conduct soil pore size analysis test to obtain the pore size distribution curve of the soil sample.

[0090] (2) Conduct soil composition determination experiments to obtain soil volume water content, soil sample size and dry density.

[0091] (3) The calculation method of the present invention is used to calculate the change relationship of the unfrozen water content during the freezing process of the soil.

[0092] The freezing process of pore water is divided into the crystallization of supercooled water and the growth of crystal nuclei. There is a "critical nucleation phenomenon" in the crystallization process of supercooled water. The reason for this phenomenon is that it corresponds to the maximum potential energy barrier in the freezing process. It is analyzed by classical nucleation theory: In classical nucleation theory, the free energy of supercooled water is composed of:

[0093] ΔG(T)=ΔG V +ΔG s (1)

[0094] The free energy of supercooled water consists of the volume term on the left and the surface term on the right. During the nucleation process, phase change occurs, causing the van der Waals force field energy in the ice embryo to be significantly higher than the van der Waals force field energy in the pore solution. The heat released reduces the molecular chemical potential in the ice nucleus, which reduces the free energy of the volume term. When the average lattice volume of the ice molecule is v i When the free energy change caused by nucleation is ΔG V It can be calculated as follows:

[0095]

[0096] Among them, μ i is the potential energy of ice molecules, μ w is the potential energy of water molecules, n is the number of ice nucleus molecules, r is the radius of the nucleus, S i It is the stable vapor pressure ratio of water and ice crystals at the same temperature and pressure.

[0097] As the nucleus is generated and slowly grows, the ice-water interface is also expanding. The total tension term at the ice-water interface is affected by the temperature and the nucleus radius r. germ Therefore, the ice-water interface energy term can be written as:

[0098]

[0099] Therefore, the free energy expression of supercooled water is combined with equations 1 to 3 to obtain:

[0100]

[0101] The volume term on the left and the surface term on the right constitute the supercooled water free energy, in which the molecular chemical potential decreases with decreasing temperature, causing the volume free energy to decrease with decreasing temperature; the interfacial tension increases with decreasing temperature, causing the surface energy part to increase with decreasing temperature, resulting in a monotonic interval and a maximum value of the total amount of system free energy change. During the freezing process, the increase in surface energy in the monotonic increasing section is greater than the decrease in volume energy, resulting in an increase in the potential energy barrier; the decrease in volume energy in the monotonic decreasing section is greater than the increase in surface energy, resulting in a decrease in the potential energy barrier, such as Figure 1 shown.

[0102] When the free energy of supercooled water is sufficient to cross this potential energy barrier, the crystal nucleus can grow stably, otherwise its growth trend is extinction. When the crystal nucleus reaches the stable growth stage, its growth is mainly dominated by interface phase change and diffusion movement.

[0103] The interfacial phase transition is the change of the water molecule structure: the breaking and reorganization of hydrogen bonds, and the number of molecules remains unchanged during this process. The diffusion movement of water molecules in ordinary water systems is an irregular movement affected by temperature. When water molecules are in the pores of the soil system, this paper believes that their diffusion movement is affected by the concentration of supercooled water outside the ice-water interface and the concentration gradient of ice molecules inside the interface. The concentration unit is molar concentration.

[0104] In the process of cooling the pore water components, the existence of the nucleation barrier makes the pore water temperature lower than the initial freezing temperature but does not freeze. At this time, although ice nuclei begin to gradually form in the pore water, their potential energy does not cross the nucleation barrier. Therefore, the ice nuclei formed at this stage are in a metastable state and face continued growth or direct extinction; as the temperature continues to drop and the size of some nuclei continues to grow, their potential energy is sufficient to cross the nucleation potential barrier, and the metastable state is transformed into a stable growth state and can proceed spontaneously. The lower temperature makes the supercooled water around the nucleus interface have an extremely high freezing tendency. When the ice nuclei can grow stably and spontaneously, the supercooled water molecules quickly attach and undergo a phase change. Accompanied by the release of drastic latent heat of phase change, the temperature of the soil system rises briefly and rapidly, thus producing the supercooling jump phenomenon.

[0105] There is a critical state that allows the nucleus to grow stably. At this time, the change in the free energy of the nucleus is functioned as ΔG. r The partial derivative with respect to the nucleus radius r is instantaneously 0:

[0106]

[0107] In this regard, this paper finds the minimum size of the crystal nucleus that is sufficient for stable growth, and will not die as the crystal nucleus size grows. Combining equations (1-2), the critical size of the crystal nucleus for stable pore water in the soil system is obtained, and the crystal nucleus size increases with temperature changes and water freezing.

[0108]

[0109] Where v ice (T) is the volume of water molecules in the crystal nucleus, σ iw (T) is the ice-water interfacial tension, S i represents the stable vapor pressure ratio of water and ice crystals at the same temperature and pressure. All three are functions of temperature. B The Boltzmann constant is 1.38×10 -23 J / K.v ice (T) The calculation process can use the Zobrist equation:

[0110]

[0111] in, N a refers to Avogadro's constant, ρ i0 Refers to the density of ice at zero degrees, which can be calculated by the HRPruppacher & J.D.Klett equation i0 =0.9167g / cm 3 . The empirical model of ice-water interfacial tension is:

[0112]

[0113] Murphy & Koop studied the effects of different parameters on the ice-water equilibrium vapor pressure ratio and concluded that many models can achieve good prediction levels above 238.15K. For convenient calculation within the experimental temperature range (>238.15K), the following temperature fitting model was proposed:

[0114]

[0115] In the pore water system, there is a relationship between the number of molecules and the size of the crystal nucleus:

[0116]

[0117] And it is known from the above that the size of the stable growing nucleus is r germ , combined with the relationship between the number of molecules and the size of the crystal nucleus, it can be known that when the ice crystal nucleus is sufficient to grow stably, the number of water molecules consumed is:

[0118]

[0119] Therefore, this paper obtains the number of consumed water molecules in the critical state of the crystal nucleus. In the process of continuous growth of the crystal nucleus, the size parameter and the quantity parameter are functions related to the temperature. Therefore, if you want to obtain the relationship between the unfrozen water content and temperature in the freezing characteristic curve, you need to transform formula (8).

[0120] Based on formula (11), the Avogadro constant N is introduced A , the molar mass of water M w , the number of consumed water molecules is converted into the mass of water molecules, thus deducing the unfrozen water content in the supercooled jump part:

[0121]

[0122] w s is the mass of soil particles, and W0 is the initial water content.

[0123] In homogeneous nucleation systems, in order to study the growth process of crystal nuclei, some scholars believe that the time and space effects of the generation of crystal nuclei in pores are random; on the other hand, some scholars describe an ideal situation to more simply control the growth process of crystal nuclei, assuming that all crystal nuclei particles in the pores are generated at the same time and grow in the same size. Neither of these two descriptions can reasonably describe the formation and growth processes of crystal nuclei at the same time, so this paper makes a reasonable assumption: since the nucleation process is too short compared to the growth process, even if the generation of crystal nuclei is random in time and space, it is believed that all crystal nuclei have negligible similar sizes at the critical state of formation and stable growth, and experience the subsequent growth process with the same size.

[0124] When the crystal nucleus is strong enough to grow stably, a large number of water molecules with a tendency to freeze gather around the ice crystal nucleus. Their temperature is lower than the freezing temperature but no phase change occurs. They are called supercooled water molecules. The two-step nucleation theory holds that during the freezing process, water molecules first migrate from the liquid phase to the metastable phase where supercooled water molecules exist, and then cross the ice-water interface to undergo a phase change. This paper uses Fick's diffusion law to study the diffusion and migration of water molecules during the freezing process. Figure 2 shown.

[0125] Assuming that there is an ice-water interface with a cross-sectional area of ​​A, a horizontal diffusion probability of 0.5, the number of water molecules on the left is N(x), and the number of ice molecules on the right is N(x+Δx), then the diffusion flux j passing through the cross section A in unit time t is:

[0126]

[0127] There is also volume concentration The diffusion flux is thus converted to:

[0128]

[0129] Among them, D(m 2 / s) is the diffusion coefficient; M (mol / m 3 ) is the molecular volume concentration; j (mol / m 2 ·s) is the diffusion flux of water molecules.

[0130] The volume concentration of ice molecules inside the crystal nucleus is [M] r , the volume concentration of water molecules with freezing tendency is [M] s , the volume concentration of water molecules in the outside world is [M] b If the water molecules with freezing tendency are regarded as a transition phase, there is a diffusion relationship between the liquid phase and the transition phase. Combining formula (14), the rate of the amount of material from the liquid phase to the surface of the crystal nucleus is obtained:

[0131]

[0132] When water molecules cross the ice-water interface and undergo phase change, only the molecular structure changes, and the number of molecules does not change. Therefore, based on the conservation of the amount of matter, the rate of molecules consumed by the crystal nucleus growth is deduced from the volume change rate:

[0133]

[0134] Where k (m / s) is the reaction constant, [M] r is the volume concentration of ice molecules in the crystal nucleus,

[0135] The unfrozen water content is converted into the rate of change of the crystal nucleus size to obtain the relationship between the water molecule consumption rate and the crystal nucleus volume change rate:

[0136]

[0137] Formula (15-17) represents the transformation relationship of water molecules during the freezing process. Combining the above three relationships, the molar concentration of the supercooled water molecule phase (18) and the rate of change of the crystal nucleus size (19) are:

[0138]

[0139] Therefore, the change rate of unfrozen water content is obtained:

[0140]

[0141] At the nanoscale, the concentration of water molecules is extremely sensitive to the change of the microscopic potential energy of the soil system. Based on the Gibbs-Tomson relationship, according to the relationship between different potential energies in the soil system, for example: soil-water interfacial tension Ice-water interfacial tension Δμ iw =4πr germ 2 σ iw (T), corrected for the concentration of molecules in the soil system:

[0142]

[0143] During the freezing process of soil, the water content and freezing amount of the system change dynamically with the temperature, so the analysis of the energy field is essential to the study of the freezing characteristic curve. The various components in the aluminum can are regarded as a whole, and the freezing process is a water bath cooling. The energy field of this model is analyzed: Taking the soil system as a reference, its energy source and destination: the cold energy Q1 transmitted to it by the outside world during the water bath process; the phase change latent heat L released during the phase change process m ; The heat transfer Q2 from the soil system to the outside world; The change in the internal energy of the soil system ΔH. The above factors constitute the energy field of the water bath process, and the four of them are related:

[0144] ΔH=-Q1+Q2+L m (twenty two)

[0145] For heat transfer in an energy field, Newton's law of cooling states that when two objects of different temperatures come into contact with each other, heat is transferred from the object with a relatively high temperature to the object with a low temperature, which gradually cools itself. Mathematically, the rate of energy change is equal to the product of the temperature difference between the two objects and the contact area and the heat transfer coefficient:

[0146]

[0147] Where, T w is the water bath temperature, T a is the outside air temperature, H is the heat transfer coefficient (W / (m 2 ·K)), S is the contact area.

[0148] For the latent heat release rate of phase change during freezing, there is a relationship:

[0149]

[0150] Where L is the latent heat released by the phase change of a single water molecule, and J is the total flux of water molecules on the ice-water interface during the phase change process.

[0151] The rate of change of the internal energy of the soil system can be calculated by the specific heat of the system:

[0152]

[0153] In summary, combined with equations (22-26), the overall conversion of the energy field is obtained:

[0154]

[0155] According to formula (27), the temperature change relationship is integrated to obtain the relationship between temperature and time during the freezing process:

[0156] mxD c T=[HS(TT w )-HS(TT a )+LJ]t (28)

[0157] Substituting formula (28) into formula (22), we finally get the relationship between the change of unfrozen water content in the freezing process:

[0158]

[0159] Where W m The residual unfrozen water content is the residual unfrozen water content. When a small amount of pore water freezes, the soil pores have a strong adsorption effect on water molecules and form a layer of adsorbed water film on the surface of the soil pores, making it difficult for water molecules to diffuse and undergo phase change. Jinxiao calculated the unfrozen water content through the double electron layer model:

[0160] W m =A s ρ bw δ×10 -2 (30)

[0161] The adsorption layer thickness δ is taken as Strongly bound water density ρ bw =1.1(g / cm); A s is the specific surface area.

[0162] Model results verification

[0163] The present invention uses measured data from multiple groups of scholars for verification, and the corresponding soil sample physical property parameters are summarized as shown in the following table:

[0164]

[0165]

[0166] The data in the above table include soil samples of various soil types (silt and silty clay), moisture content and ambient temperature, which are highly representative. Through the prediction method of the present invention, the change value of the unfrozen water content of the soil system can be calculated by inputting the sample parameters. Finally, by comparing with the measured data and calculating the error, the "invention prediction effect analysis diagram" ( Figure 3 ) "Actual prediction model diagram" ( Figure 4 ), for samples of different soil types, the calculation results of the present invention show high reliability.

[0167] In summary, the calculation method of the present invention is not only widely applicable, has clear physical meaning, but also has high calculation accuracy, and can meet the calculation requirements of the SFCC model. Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention is described in detail with reference to the preferred embodiments, ordinary technicians in this field should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution of the present invention, which should be included in the scope of the claims of the present invention.

Claims

1. A method for predicting the unfrozen water content of frozen soil considering supercooling effect, characterized in that: The following steps are involved: (1) Conduct soil pore size analysis test to obtain the pore size distribution curve of the soil sample. (2) Conduct soil composition determination experiments to obtain soil volume water content, soil sample size and dry density. (3) The calculation method of the present invention is used to calculate the change relationship of the unfrozen water content during the freezing process of the soil.

2. A method for predicting the unfrozen water content of frozen soil considering the supercooling effect as claimed in claim 1, characterized in that: The freezing process of pore water is divided into the crystallization of supercooled water and the growth of crystal nuclei. There is a "critical nucleation phenomenon" in the crystallization process of supercooled water, and the reason for this phenomenon is that it corresponds to the maximum potential energy barrier in the freezing process. It is analyzed by classical nucleation theory: In classical nucleation theory, the free energy of supercooled water consists of: ΔG(T)=ΔG V +ΔG S (1) The free energy of supercooled water consists of the volume term on the left and the surface term on the right. During the nucleation process, phase change occurs, causing the van der Waals force field energy in the ice embryo to be significantly higher than the van der Waals force field energy in the pore solution. The heat released reduces the molecular chemical potential in the ice nucleus, which reduces the free energy of the volume term. When the average lattice volume of the ice molecule is v i When the free energy change caused by nucleation is ΔG V It can be calculated as follows: Among them, μ i is the potential energy of ice molecules, μ w is the potential energy of water molecules, n is the number of ice nucleus molecules, r is the radius of the nucleus, S i It is the stable vapor pressure ratio of water and ice crystals at the same temperature and pressure. As the nucleus is generated and slowly grows, the ice-water interface is also expanding. The total tension term at the ice-water interface is affected by the temperature and the nucleus radius r. germ Therefore, the ice-water interface energy term can be written as: Therefore, the free energy expression of supercooled water is combined with equations (1 to 3) to obtain: The volume term on the left and the surface term on the right constitute the supercooled water free energy, in which the molecular chemical potential decreases with decreasing temperature, causing the volume free energy to decrease with decreasing temperature; the interfacial tension increases with decreasing temperature, causing the surface energy part to increase with decreasing temperature, resulting in the monotonic interval and total maximum value of the system free energy change. During the freezing process, the increase in surface energy in the monotonic increasing section is greater than the decrease in volume energy, resulting in an increase in the potential energy barrier; the decrease in volume energy in the monotonic decreasing section is greater than the increase in surface energy, resulting in a decrease in the potential energy barrier. When the free energy of supercooled water is sufficient to cross this potential energy barrier, the crystal nucleus can grow stably, otherwise its growth trend is extinction. When the crystal nucleus reaches the stable growth stage, its growth is mainly dominated by interface phase change and diffusion movement. The interfacial phase transition is the change of the water molecule structure: the breaking and reorganization of hydrogen bonds, and the number of molecules remains unchanged during this process. The diffusion movement of water molecules in ordinary water systems is an irregular movement affected by temperature. When water molecules are in the pores of the soil system, this paper believes that their diffusion movement is affected by the concentration of supercooled water outside the ice-water interface and the concentration gradient of ice molecules inside the interface. The concentration unit is molar concentration. In the process of cooling the pore water components, the existence of the nucleation barrier makes the pore water temperature lower than the initial freezing temperature but does not freeze. At this time, although ice nuclei begin to gradually form in the pore water, their potential energy does not cross the nucleation barrier. Therefore, the ice nuclei formed at this stage are in a metastable state and face continued growth or direct extinction; as the temperature continues to drop and the size of some nuclei continues to grow, their potential energy is sufficient to cross the nucleation potential barrier, and the metastable state is transformed into a stable growth state and can proceed spontaneously. The lower temperature makes the supercooled water around the nucleus interface have an extremely high freezing tendency. When the ice nuclei can grow stably and spontaneously, the supercooled water molecules quickly attach and undergo a phase change. Accompanied by the release of drastic latent heat of phase change, the temperature of the soil system rises briefly and rapidly, thus producing the supercooling jump phenomenon. There is a critical state that allows the nucleus to grow stably. At this time, the change in the free energy of the nucleus is functioned as ΔG. r The partial derivative with respect to the nucleus radius r is instantaneously 0: In this regard, this paper finds the minimum size of the crystal nucleus that is sufficient for stable growth, and will not die as the crystal nucleus size grows. Combining equations (1-2), the critical size of the crystal nucleus for stable pore water in the soil system is obtained, and the crystal nucleus size increases with temperature changes and water freezing. Where v ice (T) is the volume of water molecules in the crystal nucleus, σ iw (T) is the ice-water interfacial tension, S i represents the stable vapor pressure ratio of water and ice crystals at the same temperature and pressure. All three are functions of temperature. B The Boltzmann constant is 1.38×10 -23 J / K.v ice (T) The calculation process can use the Zobrist equation: in, N a refers to Avogadro's constant, ρ i0 Refers to the density of ice at zero degrees, which can be calculated by the HRPruppacher & J.D.Klett equation i0 =0.9167g / cm 3 . The empirical model of ice-water interfacial tension is: σ iw [J / m 2 ]=2.91·10 -2 +0.18·10 -3 (T-273.15) (8) Murphy & Koop studied the effects of different parameters on the ice-water equilibrium vapor pressure ratio and concluded that many models can achieve good prediction levels above 238.15K. For convenient calculation within the experimental temperature range (>238.15K), the following temperature fitting model was proposed: In the pore water system, there is a relationship between the number of molecules and the size of the crystal nucleus: And it is known from the above that the size of the stable growing nucleus is r germ , combined with the relationship between the number of molecules and the size of the crystal nucleus, it can be known that when the ice crystal nucleus is sufficient to grow stably, the number of water molecules consumed is: Therefore, this paper obtains the number of consumed water molecules in the critical state of the crystal nucleus. In the process of continuous growth of the crystal nucleus, the size parameter and the quantity parameter are functions related to the temperature. Therefore, if you want to obtain the relationship between the unfrozen water content and temperature in the freezing characteristic curve, you need to transform formula (8). Based on formula (11), Avogadro constant N is introduced A , the molar mass of water M w , the number of consumed water molecules is converted into the mass of water molecules, thus deducing the unfrozen water content in the supercooled jump part: w s is the mass of soil particles, and W0 is the initial water content. In homogeneous nucleation systems, in order to study the growth process of crystal nuclei, some scholars believe that the time and space effects of the generation of crystal nuclei in pores are random; on the other hand, some scholars describe an ideal situation to more simply control the growth process of crystal nuclei, assuming that all crystal nuclei particles in the pores are generated at the same time and grow in the same size. Neither of these two descriptions can reasonably describe the formation and growth processes of crystal nuclei at the same time, so this paper makes a reasonable assumption: since the nucleation process is too short compared to the growth process, even if the generation of crystal nuclei is random in time and space, it is believed that all crystal nuclei have negligible similar sizes at the critical state of formation and stable growth, and experience the subsequent growth process with the same size. When the crystal nucleus is stable enough to grow, a large number of water molecules with a tendency to freeze gather around the ice crystal nucleus. Their temperature is lower than the freezing temperature but no phase change occurs. They are called supercooled water molecules. The two-step nucleation theory holds that during the freezing process, water molecules first migrate from the liquid phase to the metastable phase where supercooled water molecules exist, and then cross the ice-water interface to undergo a phase change. This paper uses Fick's diffusion law to study the diffusion and migration of water molecules during the freezing process. Assuming that there is an ice-water interface with a cross-sectional area of ​​A, a horizontal diffusion probability of 0.5, the number of water molecules on the left is N(x), and the number of ice molecules on the right is N(x+Δx), then the diffusion flux j passing through the cross section A in unit time t is: There is also volume concentration The diffusion flux is thus converted to: Among them, D(m 2 / s) is the diffusion coefficient; M (mol / m 3 ) is the molecular volume concentration; j (mol / m 2 ·s) is the diffusion flux of water molecules. The volume concentration of ice molecules inside the crystal nucleus is [M] r , the volume concentration of water molecules with freezing tendency is [M] s , the volume concentration of water molecules in the outside world is [M] b If the water molecules with freezing tendency are regarded as a transition phase, there is a diffusion relationship between the liquid phase and the transition phase. Combining formula (14), the rate of the amount of material from the liquid phase to the surface of the crystal nucleus is obtained: When water molecules cross the ice-water interface and undergo phase change, only the molecular structure changes, and the number of molecules does not change. Therefore, based on the conservation of the amount of matter, the rate of molecules consumed by the crystal nucleus growth is deduced from the volume change rate: Where k (m / s) is the reaction constant, [M] r is the volume concentration of ice molecules in the crystal nucleus, The unfrozen water content is converted into the rate of change of the crystal nucleus size to obtain the relationship between the water molecule consumption rate and the crystal nucleus volume change rate: Formula (15-17) represents the transformation relationship of water molecules during the freezing process of water. Combining the above three relationships, the molar concentration of the supercooled water molecule phase (18) and the rate of change of the crystal nucleus size (19) are: Therefore, the change rate of unfrozen water content is obtained: At the nanoscale, the concentration of water molecules is extremely sensitive to the change of the microscopic potential energy of the soil system. Based on the Gibbs-Tomson relationship, according to the relationship between different potential energies in the soil system, for example: soil-water interfacial tension Ice-water interfacial tension Δμ iw =4πr germ 2 σ iw (T), corrected for the concentration of molecules in the soil system: During the freezing process of soil, the water content and freezing amount of the system change dynamically with the temperature, so the analysis of the energy field is essential to the study of the freezing characteristic curve. The various components in the aluminum can are regarded as a whole, and the freezing process is a water bath cooling. The energy field of this model is analyzed: Taking the soil system as a reference, its energy source and destination: the cold energy Q1 transmitted to it by the outside world during the water bath process; the phase change latent heat L released during the phase change process m ; The heat transfer Q2 made by the soil system to the outside world; The change of the internal energy of the soil system ΔH. The above factors constitute the energy field of the water bath process, and the four of them are related: ΔH=-Q1+Q2+L m (22) For heat transfer in an energy field, Newton's law of cooling states that when two objects of different temperatures come into contact with each other, heat is transferred from the object with a relatively high temperature to the object with a low temperature, which gradually cools itself. Mathematically, the rate of energy change is equal to the product of the temperature difference between the two objects and the contact area and the heat transfer coefficient: Where, T w is the water bath temperature, T a is the outside air temperature, H is the heat transfer coefficient (W / (m 2 ·K)), S is the contact area. For the latent heat release rate of phase change during freezing, there is a relationship: Where L is the latent heat released by the phase change of a single water molecule, and J is the total flux of water molecules on the ice-water interface during the phase change process. The rate of change of the internal energy of the soil system can be calculated by the specific heat of the system: In summary, combined with equations (22-26), the overall conversion of the energy field is obtained: According to formula (27), the temperature change relationship is integrated to obtain the relationship between temperature and time during the freezing process: mC c T=[HS(T-T w )-HS(T-T a )+LJ]t (28) Substituting formula (28) into formula (22), we finally get the relationship between the change of unfrozen water content in the freezing process: Where W m The residual unfrozen water content is the residual unfrozen water content. When a small amount of pore water freezes, the soil pores have a strong adsorption effect on water molecules and form a layer of adsorbed water film on the surface of the soil pores, making it difficult for water molecules to diffuse and undergo phase change. Jinxiao calculated the unfrozen water content through the double electron layer model: W m =A s r bw d×10 -2 (30) The adsorption layer thickness δ is taken as Strongly bound water density ρ bw =1.1(g / cm2); A s is the specific surface area.

3. The method for predicting the unfrozen water content of frozen soil considering the supercooling effect according to claim 2 is characterized in that: Based on the nucleation theory proposed in the present invention and taking into account the storage environment of soil pore water, an expression for the relationship between the unfrozen water content and temperature in the supercooling stage of the freezing process is constructed; starting from the changes in freezing components and freezing mechanism during the freezing process of pore water, the change relationship of the unfrozen water content in the water freezing stage is determined according to the changes in the concentrations of different components through the diffusion law; finally, a soil freezing characteristic model covering the supercooling stage and the water freezing stage is constructed.

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