A method for obtaining the colloidal deposition rate within a single pore based on an axisymmetric contractile tube model

Abstract

The present invention discloses a method for obtaining the colloid deposition rate in a single pore based on an axisymmetric contraction tube model, specifically: determining the geometric dimensions of the single pore contraction tube model according to the porosity of the porous medium and the particle size of the medium particles; calculating the medium wall distance field, introducing the hydrodynamic delay effect when the colloid approaches the medium wall, and correcting the drag and Brownian force on the colloid; simulating the flow field and colloid trajectory in the pore based on the computational fluid dynamics theory, and calculating the colloid deposition rate obtained by the Lagrangian method in the axisymmetric contraction tube model. The colloid deposition rate obtained by the present invention is more accurate and reliable, and is applicable to a wider range of parameter conditions. The present invention further reduces the error between theoretical predictions and experimental results.

Description

Technical Field

[0001] The invention belongs to the technical field of environmental pollution, and in particular relates to a method for obtaining a colloid deposition rate in a single pore based on an axisymmetric contraction tube model. Background Art

[0002] With the development of social economy and engineering technology, some complex geotechnical environment and safety issues have gradually received more attention, such as site contamination risk assessment and remediation, geological disposal of nuclear waste, dam foundation infiltration and siltation, etc. It is important to have a deep understanding of the coupling mechanism between porous media seepage and tiny particles. Colloidal particles can carry heavy metals, radionuclides and other pollutants for rapid transmission, and the long-term deposition of colloidal particles can lead to changes in the pore structure and permeability of the rock mass. The colloid deposition rate is used to quantitatively characterize the proportion of colloids in a single void that are deposited on the surface of the medium due to gravity, convection and diffusion. It is a key parameter of the classical colloid filtration theory (CFT) and is of great significance for understanding the migration and deposition behavior of colloids in porous media in the initial state.

[0003] The main methods for obtaining the colloid deposition rate are analytical formula method, numerical simulation method and experimental method. The analytical method adopts the linear additivity assumption and directly superimposes the colloid diffusion, interception and gravity sedimentation rates, resulting in large prediction errors when the colloid size is small; the numerical simulation method obtains the colloid deposition rate under different conditions through parameter examples; the experimental method uses indoor sand column tests to obtain the average colloid deposition rate. Due to the limitations of experimental conditions, it is not suitable for low flow rates and small colloid particle sizes. The numerical simulation method has become the main method due to its low cost and wide coverage of parameter range. Numerical simulation often uses a single medium sphere geometry and is simulated based on the Euler or Lagrangian method. Among them, the Euler method regards the colloid as a solute and cannot simultaneously consider the diffusion and interception coupling mechanisms of colloid deposition; while the Lagrangian method is very computationally intensive, especially for small particle colloids. Summary of the invention

[0004] Purpose of the invention: In order to solve the problems existing in the above-mentioned prior art, the present invention provides a method for obtaining the colloid deposition rate in a single pore based on an axisymmetric shrinkage tube model.

[0005] Technical solution: The present invention provides a method for obtaining the colloid deposition rate in a single pore based on an axisymmetric shrinkage tube model, comprising the following steps:

[0006] Step 1: Based on the measured porosity ε of the porous medium and the particle radius r of the medium c , determining the size of the shrink tube geometric model, and constructing the shrink tube geometric model; the size of the shrink tube geometric model includes a maximum radius and a minimum radius;

[0007] Step 2: Determine the separation distance h between the colloidal particles and the medium surface in the shrink tube geometry model;

[0008] Step 3: Based on the separation distance h, the drag force and Brownian force in the forces acting on the colloidal particles are corrected;

[0009] Step 4: Simulate the motion trajectory of the colloidal particles based on the modified drag force and Brownian force;

[0010] Step 5: Calculate the colloid deposition rate based on the simulated motion trajectory of the simulated colloid particles.

[0011] Furthermore, the inlet cross-sectional radius r of the shrink tube geometric model in step 1 is max =r c (1-ε) -1 / 3 The radius of the narrowest throat section is r min =r c (1-ε) -1 / 3 -r c .

[0012] Furthermore, in step 1, a geometric model of the geometric shrink tube is constructed in COMSOL, which specifically includes the following steps:

[0013] Step 1: Draw a shape with size r max ×2r max A rectangle with the left boundary of the rectangle as the axis of symmetry is established, and an xz rectangular coordinate system is established, with the origin of the rectangular coordinate system located at the center of the left boundary;

[0014] Step 2: Draw a radius r c The semicircle with its center at the coordinate point (r max ,0) at;

[0015] Step 3: Use difference to subtract the semicircle from the rectangle to obtain the rotation section of the shrink tube model;

[0016] Step 4: Mesh the rotation section and encrypt the mesh near the arc surface.

[0017] Furthermore, in step 2, when the surface of the porous medium is smooth, the separation distance between the colloidal particles and the medium surface is expressed as:

[0018]

[0019] Among them, x p and z p is the position coordinate of the colloidal particles in the xz section of the contraction tube geometry model, r p is the radius of the colloidal particle;

[0020] When the surface of the porous medium is rough, the expression of the separation distance between the colloidal particles and the medium surface is:

[0021] h=δw,p -r p

[0022] Among them, δ w,p is the distance between the center coordinate of the colloidal particle and the wall, δ w,p The distance field between the colloidal particle position and the wall is obtained by sampling the colloidal particle position and the wall distance field is solved by the following equation:

[0023]

[0024] in, is the normal vector pointing to the colloidal particle from the nearest wall, is the Nabla operator symbol, is the inverse of the wall distance, δ w is the wall distance, and γ s is the smoothness factor.

[0025] Furthermore, the expressions of the drag force and Brownian force corrected in step 3 and directly applicable to the Cartesian coordinate system are as follows:

[0026]

[0027] Among them, F D is the drag force; F B is the Brownian force, t' represents the corrected tangential direction, and n' represents the corrected normal direction, where n and t represent the vertical vector from the medium surface to the colloidal particles and the tangential vector along the flow direction, respectively, i = (1, 0) and j = (0, 1); as well as The expression is as follows:

[0028]

[0029] Where μ is the fluid dynamic viscosity, r p is the radius of the colloidal particle, u=u x i+u z j,v=v x i+v z j, where u x ,u z represents the components of flow velocity on the x-axis and z-axis, v x , v z represents the component of particle velocity on the x-axis and z-axis, Indicates F B The projection on the x-axis, Indicates F BThe projection on the z-axis, θ is the angle between the line connecting the center of the colloid particle and the center of the medium particle and the vertical direction; the expressions of n, t and θ are as follows:

[0030]

[0031] Among them, x p and z p is the position coordinate of the colloidal particles in the xz section of the contraction tube geometric model;

[0032] The expressions of f1(H), f2(H), f3(H) and f4(H) are as follows:

[0033]

[0034] Where H is the dimensionless separation distance, H = h / r p , r p is the radius of the colloidal particle.

[0035] Furthermore, in step 4, when the separation distance between the colloid particles and the medium surface is less than 0.5 nm, it is assumed that the position of the colloid particles is fixed; the lower boundary is the colloid free outflow boundary, and the others are collision rebound boundaries. The Langevin equation is used to simulate the motion trajectory of the colloid particles:

[0036]

[0037] Where, m is the mass of the colloidal particles, m * is the virtual mass of the colloidal particles in the fluid, F D is the corrected drag force, F B is the modified random Brownian force, F ext represents other external forces, including gravity, surface force, shear lift and pressure gradient force.

[0038] Furthermore, according to the number of colloidal particles N, the inlet boundary of the contraction tube geometric model is divided into N arc segments. The expressions of the deposition amount I and the deposition rate η in the contraction tube geometric model are:

[0039]

[0040] Among them, J cap,i represents the deposition flux of the ith colloidal particle, x i represents the horizontal coordinate of the center of the i-th colloidal particle at the initial moment; N rel represents the total amount of colloidal particles released at the inlet of the contraction tube geometry model, p i represents the deposition rate of the i-th colloidal particle; N rel and pi The expression is as follows:

[0041] N rel =u ∞ C0r max

[0042] p i =J cap,i

[0043] Among them, u ∞ is the flow rate at the inlet of the contraction tube geometry model, C0 is the concentration of colloidal particles at the inlet of the contraction tube geometry model, r max is the inlet section radius of the shrink tube geometry model.

[0044] Beneficial effects: The present invention simulates colloid motion based on the Lagrangian particle tracking method, can realize the simulation of Brownian particles and non-Brownian particles, and can also consider the gravity, convection and diffusion coupling mechanisms at the same time, while greatly reducing the amount of calculation. Experiments have proved that the colloid deposition rate obtained by the present invention is more accurate and reliable, and is applicable to a wider range of parameter conditions. The present invention further reduces the error between theoretical predictions and experimental results. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 It is the overall flow chart of the present invention.

[0046] Figure 2 Schematic diagram of the axisymmetric shrinkage tube model of the present invention, wherein (a) is a complete schematic diagram of the axisymmetric shrinkage tube model, (b) is a schematic diagram of the axisymmetric part of the axisymmetric shrinkage tube model, (c) is a rotated cross-sectional diagram of the axisymmetric shrinkage tube model, and (d) is a schematic diagram of the cross-sectional mesh division of the axisymmetric shrinkage tube model.

[0047] Figure 3 Schematic diagram of the separation distance between colloidal particles and the medium surface, where (a) is a schematic diagram of the separation distance when the medium surface is smooth; (b) is a schematic diagram of the separation distance when the medium surface is rough.

[0048] Figure 4 This is the curve diagram of the hydrodynamic delay correction function.

[0049] Figure 5 It is a comparison chart of the simulation results of the present invention with the analytical solution and the predicted value of the regression equation. DETAILED DESCRIPTION

[0050] The accompanying drawings, which constitute a part of the present invention, are used to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute improper limitations on the present invention.

[0051] This application discloses a method for obtaining colloid deposition rate using an axisymmetric shrink tube model. The specific steps and process are shown in Figure 1 The method comprises the following steps:

[0052] Step 1: Build the Geometry Model

[0053] According to the measured porosity ε of the porous medium and the particle radius r of the medium c , determine the size of the shrink tube geometry model. The shrink tube geometry model is as follows Figure 2 As shown in (a), the central axis is the rotational symmetry axis. The cross section of the model is a circular surface, and the inlet section radius is r max =r c (1-ε) -1 / 3 The radius of the narrowest throat section is r min =r c (1-ε) -1 / 3 -r c The axisymmetric geometry model of the shrink tube was constructed in COMSOL. The specific steps are as follows:

[0054] (1) Draw the size r max ×2r max The left boundary is the axis of symmetry, and an xz rectangular coordinate system is established with the origin (0,0) located at the center of the left boundary;

[0055] (2) Figure 2 As shown in (b), the radius is r c The semicircle with its center at (r max ,0);

[0056] (3) Using difference set, subtract the semicircle from the rectangle to obtain the rotation section of the shrink tube model, as shown in Figure 2 As shown in (c) in .

[0057] (4) Grid the rotating section, and the grid near the arc surface needs to be encrypted to ensure that the simulated flow field has a sufficiently high accuracy, such as Figure 2 As shown in (d) in .

[0058] Step 2: Introducing distance fields and force correction

[0059] When colloidal particles move near the surface of the medium, hydrodynamic retardation effects occur due to uneven force. In the geometric model, it is necessary to determine the separation distance between the colloidal particles and the surface of the medium, so as to determine the force correction functions (Universal correction functions, UCFs) and correction formulas that take into account the hydrodynamic retardation.

[0060] (1) The medium surface is smooth

[0061] like Figure 3 As shown in (a), when the medium surface is smooth, the separation distance can be directly calculated using the analytical formula:

[0062]

[0063] Where: h is the separation distance; x p and z p is the position coordinate of the colloidal particles in the XZ section of the contraction tube geometric model (i.e., in the xz rectangular coordinate system); r p is the radius of the colloidal particle.

[0064] Let the angle between the line connecting the center of the colloidal particle and the center of the collector and the vertical be θ, then:

[0065]

[0066] The vertical vector n pointing from the medium surface to the colloidal particles and the tangential vector t along the flow direction are:

[0067]

[0068] Where: i = (1, 0) and j = (0, 1) are the basis vectors of the Cartesian coordinate system respectively.

[0069] (2) Rough surface of the medium

[0070] like Figure 3 As shown in (b), when the medium surface is rough, the Eikonal equation is used to obtain the distance field. w =0, and other boundaries are Modified Eikonal equation form:

[0071]

[0072] Where: is the symbol of the Nabla operator; δ w is the wall distance; is the reciprocal of the wall distance; is the normal vector pointing to the colloidal particle from the nearest wall; s is the smoothing factor, which is usually taken as 0.05.

[0073] The above formula is used to solve the colloidal particle position to wall distance field.

[0074] The final separation distance between the colloidal particles and the medium surface is expressed as:

[0075] h=δ w,p -r p

[0076] Among them, δ w,p is the distance between the center coordinate of the colloidal particle and the wall, δ w,p The wall distance field is sampled by the colloidal particle position.

[0077] After obtaining the separation distance h, the drag force and Brownian force in the colloidal particle force are corrected according to the universal correction function (UCFs). UCFs are obtained by fitting the results of experiments and simulations and have different expressions. For example, Figure 4 The expression used here is:

[0078]

[0079]

[0080] Where: H = h / r p is the dimensionless separation distance.

[0081] The expressions of the drag force and Brownian force of the colloid in the normal and tangential directions are:

[0082]

[0083] Where: the subscripts n and t represent the normal and tangent directions in the local wall coordinate system; F D is the drag force; F B is the Brownian force; μ is the fluid dynamic viscosity coefficient; u is the flow velocity, v is the particle velocity; R(0,1) is a Gaussian random number with a mean value of 0 and a variance of 1, and the Boltzmann constant k B =1.38×10 -23 J / K, T is the absolute temperature, and Δt is the time step of the numerical calculation.

[0084] Since the Stokes-Einstein diffusion coefficient is expressed as D = k B T / (6πμr p ), the Brownian force is further written as follows:

[0085]

[0086] Where: D n , D t are the normal and tangential diffusion coefficients of the wall, respectively. Let the velocity vector u = u x i+u z j, the particle velocity is vector v = v x i+v z j.

[0087] Considering that the model uses the Cartesian coordinate system, the above radial coordinate system cannot be directly applied, and the normal and tangential drag and Brownian force need to be further corrected to the Cartesian coordinate system:

[0088]

[0089]

[0090] Wherein: t' represents the corrected tangential direction, and n' represents the corrected normal direction.

[0091] The corrected force is further projected onto the xz coordinate axis to obtain the corrected component in the xz coordinate system.

[0092] The following formulas can be directly applied to the calculation of drag and Brownian forces in a Cartesian coordinate system:

[0093]

[0094] Step 3: Colloid trajectory and deposition simulation

[0095] Since the particle size of the colloid particles is small and the volume fraction is small, the effect of the colloid on the flow field is not considered. For steady laminar flow with a small Reynolds number, the Stokes equation is used to obtain the flow field in the simulation domain:

[0096]

[0097] Where: P is the fluid pressure, μ is the fluid viscosity coefficient; the upper boundary of the axisymmetric contraction tube model is set to a constant flow rate, and the lower boundary is set to zero pressure.

[0098] The flow field velocity is further transmitted to the colloid particles, and the movement of the colloid is simulated based on the Langevin equation:

[0099]

[0100] Where: m is the mass of the colloid; * is the virtual mass of the colloid in the fluid (for spherical colloids, it is about 1 / 2 of the mass of the colloid volume replacing the fluid); v is the particle velocity vector. F is the force applied to the colloid, is the corrected drag force; represents the modified random Brownian force; F ext Represents other external forces, which may include gravity, surface force, shear lift, pressure gradient force, etc.

[0101] According to different porosity and medium particle size conditions, batch scanning simulation is established to automatically modify the geometric model size. ∞ and colloidal particle size r p, the motion trajectory of colloid particles in different situations can be obtained. The upper boundary of the model releases colloid particles uniformly, and when the separation distance between the colloid and the medium surface is h<0.5nm, it is assumed that the position of the colloid particles is fixed (captured); the lower boundary is the colloid free outflow boundary, and the others are collision rebound boundaries.

[0102] Step 4: Deposition rate calculation and verification

[0103] For three-dimensional geometric models, η = N cap / N rel Directly calculate the colloid deposition rate, where N cap is the number of captured colloids, N rel is the number of colloids released from the inlet. The axisymmetric contraction tube model proposed in the present invention requires a unique method to calculate the deposition rate. Assuming that the deposition of colloid particles is an event that has already occurred, the probability of deposition of the i-th colloid particle released from the inlet is p i , then ①p i =1 means that the colloid is finally deposited on the surface of the medium; ②p i = 0 means that the colloid finally escapes from the simulation area (axisymmetric contraction tube model). Similar to the method of calculating the flux by Euler method, the inlet boundary is divided into N arc segments according to the number of colloid particles N. The total amount of particles released at the inlet of the axisymmetric two-dimensional section N rel and the deposition probability p of the i-th colloidal particle i for:

[0104] N rel =u ∞ C0r max

[0105] p i =J cap,i

[0106] Where: u ∞ is the inlet flow rate of the axisymmetric contraction tube model; C0 is the number concentration of colloid at the inlet of the axisymmetric contraction tube model; C0 is the concentration of colloid particles at the inlet of the contraction tube geometry model, J cap,i represents the deposition flux of the i-th colloidal particle.

[0107] Thus, the deposition amount I and deposition rate η in the axisymmetric model are expressed as:

[0108]

[0109] In the Lagrangian particle tracking simulation process, by recording the horizontal coordinate x of the center of the colloidal particle at the initial moment i , and the final state of the particle p i , then the above formula can be used to calculate η.

[0110] In a simulation example, assuming the inlet flow rate u ∞ =4.6×10 -6 m / s, medium particle diameter is 500μm, colloid radius is d p ∈[0.04,16]μm, release N from the inlet rel = 5000 particles. The simulation results are compared with the analytical solution and the predicted curve based on the experimental data related equation. The results are as follows Figure 5 As shown. The analytical solution predicts that η>1 when the colloid particle size is large, and there is also a lack of coupling between the various deposition mechanisms. Therefore, the analytical solution prediction is generally not reliable; the prediction formula based on the experimental data correlation equation is obtained through nonlinear regression based on limited experimental data. The experimental conditions u ∞ >2×10 -5 m / s, resulting in its regression equation in d p <1μm, the predicted result is too small, but d p When the particle size is >1μm, it is close to the simulation result. The above analysis also further confirms the correctness of the proposed model and simulation method.

[0111] It should also be noted that the various specific technical features described in the above specific embodiments can be combined in any suitable manner without contradiction. In order to avoid unnecessary repetition, the present invention will not further describe various possible combinations.

Claims

1. A method for obtaining the colloid deposition rate in a single pore based on an axisymmetric shrinkage tube model, characterized in that: The steps include: Step 1: Based on the measured porosity ε of the porous medium and the particle radius r of the medium c , determining the size of the shrink tube geometric model, and constructing the shrink tube geometric model; the size of the shrink tube geometric model includes a maximum radius and a minimum radius; Step 2: Determine the separation distance h between the colloidal particles and the medium surface in the shrink tube geometry model; Step 3: Based on the separation distance h, the drag force and Brownian force in the forces acting on the colloidal particles are corrected; Step 4: Simulate the motion trajectory of the colloidal particles based on the modified drag force and Brownian force; Step 5: Calculate the colloid deposition rate based on the simulated motion trajectory of the simulated colloid particles; In step 4, when the separation distance between the colloid particles and the medium surface is less than 0.5 nm, it is assumed that the position of the colloid particles is fixed; the lower boundary is the colloid free outflow boundary, and the others are collision rebound boundaries. The Langevin equation is used to simulate the motion trajectory of the colloid particles: Where, m is the mass of the colloidal particles, m * is the virtual mass of the colloidal particles in the fluid, F D is the corrected drag force, F B is the modified random Brownian force, F ext represents other external forces, including gravity, surface force, shear lift and pressure gradient force, and x' and y' represent the x-axis and y-axis of the Cartesian coordinate system respectively; According to the number of colloidal particles N, the inlet boundary of the contraction tube geometric model is divided into N arc segments. The expressions of the deposition amount I and deposition rate η in the contraction tube geometric model are: Among them, J cap,i represents the deposition flux of the ith colloidal particle, x i represents the horizontal coordinate of the center of the i-th colloidal particle at the initial moment; N rel represents the total amount of colloidal particles released at the inlet of the contraction tube geometry model, p i represents the deposition rate of the i-th colloidal particle; N rel and p i The expression is as follows: N rel =u ∞ C0r max p i =J cap,i Among them, u ∞ is the flow rate at the inlet of the contraction tube geometry model, C0 is the concentration of colloidal particles at the inlet of the contraction tube geometry model, r max is the inlet section radius of the shrink tube geometry model.

2. The method for obtaining the colloid deposition rate in a single pore based on the axisymmetric shrinkage tube model according to claim 1 is characterized in that: The inlet cross-sectional radius r of the shrink tube geometric model in step 1 max =r c (1-ε) -1 / 3 The radius of the narrowest throat section is r min =r c (1-ε) -1 / 3 -r c .

3. The method for obtaining the colloid deposition rate in a single pore based on the axisymmetric shrinkage tube model according to claim 1, characterized in that: In step 1, a geometric model of a geometric shrink tube is constructed in COMSOL, which specifically includes the following steps: Step 1: Draw a shape with size r max ×2r max A rectangle with the left boundary of the rectangle as the axis of symmetry is established, and an xz rectangular coordinate system is established, with the origin of the rectangular coordinate system located at the center of the left boundary; Step 2: Draw a radius r c The semicircle with its center at the coordinate point (r max ,0) at; Step 3: Use difference to subtract the semicircle from the rectangle to obtain the rotation section of the shrink tube model; Step 4: Mesh the rotation section and encrypt the mesh near the arc surface.

4. The method for obtaining the colloid deposition rate in a single pore based on the axisymmetric shrinkage tube model according to claim 1, characterized in that: In step 2, when the surface of the porous medium is smooth, the separation distance between the colloidal particles and the medium surface is expressed as: Among them, x p and z p is the position coordinate of the colloidal particles in the xz section of the contraction tube geometry model, r p is the radius of the colloidal particle; When the surface of the porous medium is rough, the expression of the separation distance between the colloidal particles and the medium surface is: h=δ w,p -r p Among them, δ w,p is the distance between the center coordinate of the colloidal particle and the wall, δ w,p The distance field between the colloidal particle position and the wall is obtained by sampling the colloidal particle position and the wall distance field is solved by the following equation: in, is the normal vector pointing to the colloidal particle from the nearest wall, is the Nabla operator symbol, is the inverse of the wall distance, δ w is the wall distance, and γ s is the smoothness factor.

5. The method for obtaining the colloid deposition rate in a single pore based on the axisymmetric shrinkage tube model according to claim 1, characterized in that: The expressions of the drag force and Brownian force corrected in step 3 and directly applicable to the Cartesian coordinate system are as follows: Among them, F D is the drag force; F B is the Brownian force, t' represents the corrected tangential direction, and n' represents the corrected normal direction, where n and t represent the vertical vector from the medium surface to the colloidal particles and the tangential vector along the flow direction, respectively, i = (1, 0) and j = (0, 1); as well as The expression is as follows: Where μ is the fluid dynamic viscosity, r p is the radius of the colloidal particle, u=u x i+u z j,v=v x i+v z j, where u x ,u z represents the components of flow velocity on the x-axis and z-axis, v x , v z represents the component of particle velocity on the x-axis and z-axis, Indicates F B The projection on the x-axis, Indicates F B The projection on the z-axis, θ is the angle between the line connecting the center of the colloid particle and the center of the medium particle and the vertical direction; the expressions of n, t and θ are as follows: Among them, x p and z p is the position coordinate of the colloidal particles in the xz section of the contraction tube geometric model; The expressions of f1(H), f2(H), f3(H) and f4(H) are as follows: Where H is the dimensionless separation distance, H = h / r p , r p is the radius of the colloidal particle.

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