A method for dynamic analysis and life prediction of nano-bubble stability

Abstract

The application relates to the cross field of nanofluid mechanics and interface chemistry, and specifically discloses a kind of nanobubble stability dynamic analysis and life prediction method, establishes double hyperbolic pressure balance equation considering inter-ring electrostatic repulsion, constructs concentric ring dense packing ion distribution model, deduces the series expression of surface hydroxyl ion dynamic adsorption;Propose the diffusion layer Prandtl number correction method based on Brown motion correction, realize the accurate calculation of dynamic mass transfer parameter;Establish the multi-dimensional coupled nanobubble radius change rate equation, establish the critical stable radius criterion and the closed solution of life prediction. The application breaks through the plane hypothesis limitation of traditional Young-Laplace equation, solves the technical problem of large deviation of existing nanobubble model theoretical prediction through multi-physical field coupling modeling and experimental parameter joint debugging.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of nanofluid mechanics and interface chemistry, and particularly relates to a method for dynamically analyzing stability and predicting life of nanobubbles. BACKGROUND

[0002] Nanobubbles have important application value in the fields of energy, environment, and biomedicine due to their unique physical and chemical properties, but their ultra-high surface area to volume ratio leads to instability affected by interface multi-field coupling effects.

[0003] Traditional theories rely on the classical Young-Laplace equation to describe the mechanical balance dominated by surface tension, but do not fully consider two core factors: the contribution of electrostatic repulsion caused by ion adsorption on the bubble surface to the interfacial pressure difference, and the coupling regulation effect of ion dynamic distribution and Brownian motion on gas diffusion in the solution.

[0004] Existing research has significant limitations in key model construction:

[0005] (1) Electrostatic repulsion modeling: methods based on the planar double-electric-layer or uniform charge assumption can simplify calculations but ignore curvature effects, leading to large prediction errors for sub-micron bubble pressure differences. For the hexagonal lattice model with ion dense packing, the ring spacing-charge density quantitative mapping relationship cannot be established due to the lack of ring geometric symmetry and nonlinear coupling mechanism of force balance.

[0006] (2) Adsorption kinetics analysis: the traditional Smoluchowski model can describe dynamic adsorption, but ignores the nonlinear correction of ion potential well on the adsorption rate of high-curvature interface, and does not consider the micro-convection disturbance induced by Brownian motion.

[0007] (3) Stability criterion establishment: the phenomenological model based on gas diffusion flux (such as the dissolution-re-equilibrium theory) does not couple the regulation of surface charge density on the concentration gradient, resulting in a large order of magnitude deviation between the predicted critical radius and experimental data. Life prediction relies more on semi-empirical formulas, and lacks closed analytical solutions for multi-field coupling. SUMMARY

[0008] The present application aims to provide a method for dynamically analyzing stability and predicting life of nanobubbles, to solve the technical problem of insufficient characterization of multi-field coupling and large theoretical prediction deviation caused by simplification assumptions in traditional models in current nanobubble stability analysis.

[0009] To solve the above technical problems, the present application specifically provides the following technical solutions:

[0010] A method for dynamically analyzing stability and predicting life of nanobubbles, comprising the following steps:

[0011] Step 100, based on the assumption that the surface hydroxyl ions of nanobubbles are arranged in a dense packing of concentric circular rings, an explicit relationship model of charge density is constructed, a three-level ring coupling mechanism is introduced in the explicit relationship model, a normal component of inter-ring coulomb repulsion is solved, an analytical solution of the equivalent repulsive force of the surface hydroxyl ions of nanobubbles is obtained by using Hankel transform and Debye shielding correction, the modified Young-Laplace equation is obtained by using the equivalent repulsive force analytical solution, and a bubble internal and external net pressure difference calculation model is obtained;

[0012] Step 200, based on the Smoluchowski diffusion control model, the Brownian motion and the diffusion coupled Prandtl number are introduced to modify the nanobubble boundary layer thickness, and the differential equation of the nanobubble surface charge density is constructed, and the low coverage approximate analytical solution of the nanobubble surface charge density is solved in real time;

[0013] Step 300, based on the Langevin equation describing the nanobubble motion by ignoring the inertia term, the effective diffusion coefficient of the surface hydroxyl ions of nanobubbles is solved, and the effective diffusion coefficient is used to rewrite the modified nanobubble boundary layer thickness, and the adsorption rate constant of the surface hydroxyl ions of nanobubbles in the low coverage approximate analytical solution is updated;

[0014] Step 400, through mass conservation, the bubble internal and external net pressure difference calculation model is used to establish the change rate equation of the nanobubble radius with time, the critical stable radius of the nanobubble is solved, and the life prediction closed solution is obtained by integrating the change rate equation.

[0015] As a preferred scheme of the application, in step 100, the surface hydroxyl ions are assumed to be arranged in a dense packing of concentric circular rings, the first Ring radius , the tangential spacing ;

[0016] The explicit relationship of charge density is established by minimizing the total potential energy, and the calculation formula is:

[0017] ;

[0018] ;

[0019] Wherein, The base ring spacing is represented by The charge density is represented by The charge amount carried by the surface hydroxyl ions of nanobubbles is represented by The dielectric constant is represented by The Boltzmann constant is represented by The thermodynamic temperature is represented by The ring density parameter is represented by The first The relevant radius of the ring.

[0020] As a preferred embodiment of the present invention, the low-coverage approximate analytical solution of the surface charge density of nanobubbles obtained in real time is substituted into the explicit relationship of charge density, thereby correcting the calculation model of the net pressure difference inside and outside the bubble.

[0021] As a preferred embodiment of the present invention, a three-level ring coupling mechanism is introduced into the explicit relational model, and the method for calculating the normal component of the Coulomb repulsion force between rings specifically includes:

[0022] Based on the single-loop interaction model, the calculation of the first... Normal pressure contribution of the ring to the center point The calculation formula is:

[0023] ;

[0024] Introducing a three-ring coupling mechanism, considering the total hydroxide ion pressure on the surface of nanobubbles coupled with adjacent three rings. for:

[0025] ;

[0026] The Debye shielding effect is then introduced to correct the inter-ring interaction potential. :

[0027] ;

[0028] Debye shield length :

[0029] ;

[0030] in, Line charge density ( ); Represents the coordinate variable in the vertical direction; Represents an angle variable, used to describe the spatial angle distribution during integration; Indicates the distance between ring-shaped structural units; This represents the Debye shielding parameter, which describes the degree of shielding between charge interactions; Indicates the bulk ion concentration.

[0031] As a preferred embodiment of the present invention, based on the Smoluchowski diffusion control model, and simultaneously introducing the Prandtl number coupled with Brownian motion and diffusion, it is necessary to construct the time-varying evolution conditions of the charge density on the surface of nanobubbles, quantifying the change in charge density during dynamic adsorption, specifically including:

[0032] Weak electric field conditions, surface potential , meet the linearized Poisson-Boltzmann equation;

[0033] Symmetrical electrolyte, the positive and negative ions of the electrolyte in the solution are 1:1 type;

[0034] Quasi-steady-state diffusion layer, ignoring the influence of convection, the double layer is in dynamic equilibrium.

[0035] As a preferred scheme of the present application, in step 300, the Smoluchowski diffusion control model is constructed, and the Brownian motion and diffusion coupling Prandtl number are introduced to modify the assumed condition of the boundary layer thickness of the nano-bubble, specifically including:

[0036] Low Reynolds number condition, Ignoring the inertial term of fluid motion;

[0037] Homogeneous fluid, the viscosity of the solution , the diffusion coefficient D is independent of the spatial position;

[0038] Statistical average approximation, the Brownian motion velocity field is treated by ensemble average, and the fluctuation term is linearly related to the concentration gradient.

[0039] As a preferred scheme of the present application, the bubble radius change rate equation is established by the mass conservation equation, and the critical stable radius The calculation formula is:

[0040] ;

[0041] By integrating the mass conservation equation, the life prediction closed solution The calculation formula is:

[0042] When ;

[0043] The life prediction closed solution Degenerate into:

[0044] ;

[0045] Wherein, Indicates the gas diffusion coefficient; Indicates the initial radius; Indicates the gas constant; Indicates the Henry coefficient; Indicates the vacuum dielectric constant; Indicates the surface tension, which describes the interaction between the hydrogen and oxygen ions on the surface of the nano-bubble, so that the surface has a shrinkage tendency.

[0046] As a preferred scheme of the present application, the life prediction closed solution The correction is made, specifically including:

[0047] After the experimental time exceeds the set time, a time function is introduced ;

[0048] And / or when there is multi-bubble interference, an empirical correction factor is introduced , The number of bubbles per unit volume.

[0049] As a preferred scheme of the present application, the dynamic diffusion equation describing the motion of nanobubbles based on the Langevin equation ignoring the inertial term is modified based on the Brownian motion convection effect, the ion concentration evolution equation is obtained, and after statistical average processing, the effective diffusion equation is obtained, and then the effective diffusion coefficient is obtained;

[0050] Based on the time matching of diffusion and momentum characteristics and the association of scale, combined with the stable condition, the effective diffusion coefficient is used to rewrite the modified nanobubble boundary layer thickness, and the modified nanobubble boundary layer thickness Is:

[0051] .

[0052] As a preferred scheme of the present application, the updated adsorption rate constant of the hydroxyl ion on the surface of the nanobubble Is:

[0053] ;

[0054] Wherein, the effective diffusion coefficient ; Indicates the Prandtl number.

[0055] The present application has the following beneficial effects compared with the prior art:

[0056] The present application constructs a "concentric ring dense packing" ion arrangement model of the ion on the surface of the nanobubble, deduces the ring spacing parameter of the ring Coulomb repulsion series solution combined with the AFM actual measurement, realizes accurate calculation of the electrostatic pressure component, establishes a surface charge density implicit differential equation combined with the dynamic adsorption kinetics and Debye shielding effect, introduces a Brownian motion-diffusion coupling Prandtl number to modify the mass transfer boundary layer, solves the electro-chemical-force coupling equation, derives the critical stability criterion and the analytical solution of the life, and effectively solves the problems of insufficient description of multi-field coupling in the current nanobubble stability analysis and large deviation of the theoretical prediction caused by the simplification assumption of the traditional model. BRIEF DESCRIPTION OF DRAWINGS

[0057] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required to be used in the description of the embodiments or the prior art will be briefly introduced. Obviously, the drawings in the following description are only exemplary, and other drawings can be obtained by the provided drawings without creative labor for those skilled in the art.

[0058] Figure 1 A flowchart of a nano-bubble stability dynamic analysis and life prediction method of an embodiment of the present application is shown.

[0059] Figure 2 A schematic diagram of the interface structure of a nano-bubble and the concentric ring ion distribution of an embodiment of the present application is shown.

[0060] Figure 3 A schematic diagram of the ring structure details of an embodiment of the present application is shown.

[0061] Figure 4 A vector resolution diagram for calculation of the electrostatic repulsive force between rings of an embodiment of the present application is shown.

[0062] Figure 5 A dynamic adsorption evolution curve diagram of the surface charge density of an embodiment of the present application is shown.

[0063] Figure 6 A schematic diagram of the Brownian motion-diffusion coupling effect and diffusion layer correction of an embodiment of the present application is shown.

[0064] Figure 7 A critical stability radius-charge density phase diagram of an embodiment of the present application is shown.

[0065] Figure 8 An initial radius-life prediction model verification diagram of an embodiment of the present application is shown. DETAILED DESCRIPTION

[0066] The technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application.

[0067] As shown in Figures 1 to 8 The present application provides a nano-bubble stability dynamic analysis and life prediction method, and the specific purpose is to realize a nano-bubble stability analysis method based on electro-chemical-force multi-field coupling, including the following steps:

[0068] Step S1: Electrostatic force balance equation construction

[0069] Step S1.1: Theoretical system construction framework

[0070] For the stability problem of nanobubbles, combined with the classical Young-Laplace equation and the inter-ring electrostatic coupling effect, a modified electrostatic force balance equation is established, as follows:

[0071] (1) Establishment of modified equation: Introduce the ring-shaped electrostatic repulsion correction term in the surface tension dominated equation to construct the hyperbolic pressure balance relationship;

[0072] (2) Ion distribution modeling: Based on the concentric circular ring close packing arrangement, a radial-tangential distribution model of surface hydroxyl ions is established;

[0073] (3) Inter-ring interaction calculation: Derive the Coulomb repulsion tensor of a single ring on adjacent rings, and project it to the normal pressure component;

[0074] (4) Multi-ring coupling analysis: Linking the inter-ring distance, ring radius, and charge density, construct the series expression of .

[0075] Preferably, the step S1.1 comprises:

[0076] (1) Equation modification logic

[0077] Define the radial-tangential double balance condition:

[0078] ① Normal pressure balance: The pressure difference inside and outside the bubble needs to offset the normal component of surface tension and inter-ring electrostatic repulsion.

[0079] ② Tangential self-balancing: The tangential component of inter-ring repulsion cancels each other due to symmetry.

[0080] (2) Micro-element balance equation

[0081] For a ring-shaped micro-element , the normal force balance satisfies:

[0082] (1)

[0083] Where, is the angle between the force of the th ring and the normal direction, rad; is the electrostatic repulsion of the th ring on the micro-element, N; denotes the angle of the ring-shaped micro-element.

[0084] (3) Global integral equation

[0085] Integrate along the bubble surface to get the modified equation:

[0086] (2)

[0087] wherein, represents the internal pressure of the bubble; represents the external pressure of the nanobubble (ambient pressure); represents the surface tension of the nanobubble; represents the radius of the nanobubble;

[0088] The physical meaning of this equation is that the circumferential integral of the repulsive force between rings produces an equivalent negative pressure gradient. This equation introduces the circumferential integral effect of the repulsive force between rings, breaking through the traditional plane assumption.

[0089] Step S1.2: Construction of concentric ring ion model

[0090] Preferably, the model is established as follows:

[0091] (1) Assumed conditions:

[0092] ① Geometric arrangement

[0093] Hydroxyl ions are distributed along concentric circular rings, and the number of ions in the first ring is Ring radius ( is the base ring spacing);

[0094] The number of ions per ring , satisfying the tangential close-packed condition.

[0095] ② Energy optimization

[0096] The total potential energy of the system When the total potential energy takes a minimum value, we obtain:

[0097] (3)

[0098] wherein, is the ring density parameter, ; represents the base ring spacing (the spacing of the concentric circular rings of surface ions), in m; represents the elementary charge (the amount of electronic charge), represents the dielectric constant of the solution, in F / m; represents the Boltzmann constant; represents the thermodynamic temperature; represents the relevant radius of the ring.

[0099] (2) Calculation of geometric parameters:

[0100] Surface charge density (unit area charge amount):

[0101] (4)

[0102] Step S1.3: Inter-ring repulsion and equivalent pressure calculation

[0103] Preferably, the calculation process comprises:

[0104] (1) Single-ring action model

[0105] ① First Normal pressure component of ring on central point :

[0106] (5)

[0107] where, is the linear charge density (ρ) , C / m.

[0108] ② Considering adjacent three-ring coupling Total pressure:

[0109] (6)

[0110] (2) Thermal disturbance correction:

[0111] Introducing Debye shielding effect to correct the inter-ring interaction potential:

[0112] (7)

[0113] where,

[0114] where, is the linear charge density (ρ) ); denotes the coordinate variable in the vertical direction; denotes the angular variable, used to describe the spatial angular distribution when integrating; denotes the distance between ring-shaped structural units; denotes the Debye shielding parameter, describing the shielding degree of charge interaction; denotes the bulk ion concentration;

[0115] , , denotes the normal pressure component of the first , , ring.

[0116] (3) Analytical solution of equivalent pressure:

[0117] Obtaining series solution through Hankel transformation:

[0118] ​ (8)

[0119] where, is the second kind modified Bessel function, , and denote the second kind modified Bessel function, the first and second values corresponding to the decay oscillation characteristics, respectively,

[0120] Step S1.4: calibration and verification of experimental parameters

[0121] Preferably, the calibration process comprises:

[0122] (1) parameter acquisition

[0123] ① Use atomic force microscope (AFM) to scan the surface ion arrangement to obtain the ring spacing (accuracy ± 0.2 nm);

[0124] ② Debye length calculation: input solution ion concentration , calculate the shielding parameter: .

[0125] (2) example verification:

[0126] ① input parameters: nm (AFM measurement), , , .

[0127] ② calculation steps:

[0128] 1. from equation (8)

[0129] 2. for bubble, the correction pressure:

[0130]

[0131] 3. Compared with the traditional model, the error is reduced by 42%.

[0132] (3) verification rules:

[0133] ① applicable conditions: when , the ring model degenerates into a plane approximation;

[0134] ② switching threshold: when , the shielding effect correction term is enabled.

[0135] Step S1.5: process implementation specification

[0136] Technical features and operational requirements:

[0137] (1) Equipment requirements: high-speed atomic force microscope (HS-AFM) is used to realize dynamic monitoring of ring spacing;

[0138] (2) Formula application domain: when the bubble surface curvature radius and the solution ionic strength , activate this model;

[0139] (3) Real-time iteration: establish a feedback regulation algorithm for ring spacing d and Zeta potential.

[0140] Step S2: ion adsorption kinetics analysis and surface charge density calculation

[0141] Step S2.1: model construction background and assumption conditions

[0142] Objective: to establish the time-varying evolution equation of surface charge density , and to quantify the change of charge density in dynamic adsorption process.

[0143] Assumption conditions:

[0144] ① Weak electric field condition: surface potential , which satisfies the linearized Poisson-Boltzmann equation.

[0145] ② Symmetrical electrolyte: the positive and negative ions of the electrolyte in the solution are 1:1 type (such as NaCl or NaOH solution), and the bulk concentration is .

[0146] ③ Quasi-steady-state diffusion layer: ignore the influence of convection, and the double electric layer is in dynamic equilibrium.

[0147] Step S2.2: Debye-Hückel approximation derivation and parameter coupling

[0148] Preferably, the derivation process includes:

[0149] (1) Potential distribution model:

[0150] ① Linearized Poisson-Boltzmann equation:

[0151] (9)

[0152] Where, is the inverse of Debye length; represents the potential at a distance of from the surface; Laplacian operator (describing the spatial variation of potential); represents the Debye parameter.

[0153] 2. Boundary conditions:

[0154] When ( surface ), ; when .

[0155] 3. Analytical solution:

[0156] (10)

[0157] where is the bubble surface potential, i.e. the potential at , and is the distance from the surface.

[0158] 4. Surface charge density expression (Gauss theorem):

[0159] (11)

[0160] (2) Surface ion concentration correction:

[0161] 1. Hydroxyl ion surface concentration correction

[0162] (12)

[0163] 2. Simultaneous equations (10) and (11), eliminating , we get:

[0164] (13)

[0165] where is the ion concentration at the bubble surface.

[0166] Step S2.3: Establishing the adsorption kinetics differential equation

[0167] Preferably, the equation is constructed as follows:

[0168] Dynamic equilibrium condition: the rate of change of surface charge density is determined by the difference between the adsorption and desorption rates:

[0169] (14)

[0170] where is the maximum surface charge density, C / m 2 ; is the adsorption rate constant, m 3 / (mol·s) ; is the desorption rate constant, s -1 .

[0171] Substituting the surface ion concentration expression, we get:

[0172] (15)

[0173] Step S2.4: Brownian motion dominated adsorption rate calculation

[0174] Preferably, the calculation is based on the Smoluchowski diffusion-controlled model:

[0175] (1) Adsorption rate constant:

[0176] (16)

[0177] where, is the diffusion coefficient (Stokes-Einstein equation),

[0178] (17)

[0179] where, is the solution viscosity, is the ion effective radius.

[0180] Diffusion boundary layer thickness under no convection conditions is:

[0181] (18)

[0182] (2) The final expression for the adsorption rate constant can be obtained:

[0183] (19)

[0184] where, represents the effective radius of the ion.

[0185] Step S2.5: Steady-state analytical solution of the kinetic equation

[0186] Preferably, the steady-state solution is derived as follows:

[0187] Steady-state condition , the equation simplifies to:

[0188] (20)

[0189] Introducing the dimensionless parameter:

[0190] (21)

[0191] (dimensionless parameter) (22)

[0192] ​ Dimensionless parameter (23)

[0193] Implicit equation transformation:

[0194] (24)

[0195] Low coverage approximation solution :

[0196] (25)

[0197] where, is the dimensionless surface charge density, is the actual charge density.

[0198] Step S2.6: Experimental parameter calibration and verification procedure

[0199] Preferably, the calibration process includes:

[0200] (1) Input parameter instance:

[0201] ① Solution condition: NaOH solution, ,

[0202] ② Physical parameters: , nm

[0203] ③ Calculated value: ,

[0204] (2) Step-by-step verification:

[0205] ① Calculate the adsorption rate :

[0206]

[0207] ② Reverse the desorption rate : Assuming that the experiment measures the steady state , substitute the formula to get .

[0208] ③ Verify the surface potential: substitute , calculate , and the measured Zeta potential range error .

[0209] Step S2.7: Implementation specification and verification standard

[0210] Technical features and operational requirements:

[0211] (1) Parameter measurement requirements:

[0212] ① Ionic strength: measured using a conductivity meter , and the value is calculated

[0213] ② Surface morphology: atomic force microscopy (AFM) calibrates the effective radius of the ions

[0214] ③ Dynamic data: measure , 10, 100 s , verify the kinetic model.

[0215] (2) Abnormal handling rules:

[0216] ① Out-of-tolerance determination: when the deviation between theoretical prediction and experimental value is greater than 15%, re-calibration is required or check for surface contamination;

[0217] ② Equilibrium time verification: if the time constant is not within 10~ 100s, adjust the solution stirring conditions.

[0218] Step S3: Brownian motion-diffusion coupling effect and diffusion layer Prandtl number correction

[0219] Step S3.1: Model background and basic assumptions

[0220] Objective: Establish a coupled kinetic model of Brownian motion and diffusion mass transfer, and correct the boundary layer thickness to accurately describe the adsorption rate on the surface of nanobubbles.

[0221] Assumptions:

[0222] (1) Low Reynolds number condition: , ignore the fluid motion inertia term, and use the Stokes resistance formula;

[0223] (2) Homogeneous fluid: solution viscosity , diffusion coefficient D and spatial position are independent;

[0224] (3) Statistical average approximation: the velocity field of Brownian motion is treated by ensemble average, and the fluctuation term is linearly related to the concentration gradient.

[0225] Step S3.2: Langevin equation construction ignoring the inertia term

[0226] Preferably, the equation construction process is as follows:

[0227] (1) Simplify the motion equation

[0228] Langevin equation describes the motion of nanobubbles:​​

[0229] (26)

[0230] Neglecting the inertial term (v ), the equation reduces to:

[0231] (27)

[0232] where v represents the instantaneous velocity of the bubble; and D represents the damping coefficient (D = 3η / ρ ).

[0233] Statistical properties:

[0234] Mean velocity ;

[0235] Velocity autocorrelation function .

[0236] (2) Diffusion coefficient derivation (Einstein relation):

[0237] (28)

[0238] where R represents the bubble radius.

[0239] Step S3.3: Brownian motion-diffusion mass transfer coupled equation

[0240] Preferably, the coupled equation is constructed as follows:

[0241] The dynamic diffusion equation is modified to take into account the Brownian motion convection effect, and the ion concentration evolution equation is:

[0242] (29)

[0243] After statistical averaging, the effective diffusion equation is obtained:

[0244]

[0245] where represents the statistical average of the ion concentration; (30)

[0246] Effective diffusion coefficient :

[0247] (31)

[0248] Step S3.4: Prandtl number correction of diffusion layer thickness

[0249] Preferably, the correction process comprises:​

[0250] (1) Prandtl number definition:

[0251] Diffusion to momentum transport ability ratio:

[0252] (32)

[0253] where, is kinematic viscosity, is fluid density.

[0254] (2) Boundary layer thickness correction formula derivation:

[0255] Based on dimensional analysis and boundary layer analogy, the corrected thickness is:

[0256] (33)

[0257] Derivation logic:

[0258] ① Momentum-diffusion characteristic time matching: ;

[0259] ② Size correlation simultaneous equation, combined with steady-state conditions to derive the final formula.

[0260] Step S3.5: Adsorption kinetics parameter update

[0261] Preferably, the parameter update includes:

[0262] (1) Adsorption rate constant correction:

[0263] The original diffusion layer thickness is replaced by the corrected value, and the adsorption rate is updated to:

[0264] (34)

[0265] (2) Surface charge density dynamics equation update:

[0266] Substitute the corrected , we get:

[0267] (35)

[0268] where, represents the rate of change of surface charge density with time; represents the ion diffusion coefficient; represents the Brownian diffusion coefficient; represents Avogadro's number; represents the Prandtl number; represents the bulk concentration of ions in the solution; represents the vacuum permittivity; Maximum surface charge density (charge density at saturation adsorption); denotes the desorption rate constant.

[0269] Step S3.6: Parameter calibration and implementation verification

[0270] (1) Preferably, the calibration and verification process comprises:

[0271] ① Key parameter measurement:

[0272] Fluid property: kinematic viscosity Determined by a rotational rheometer; ion diffusion coefficient Calibrated by electrochemical impedance spectroscopy or dynamic light scattering technology.

[0273] Nanobubble parameter: radius Measured by atomic force microscopy (AFM) or dynamic light scattering (DLS).

[0274] ② Model verification step:

[0275] Theoretical calculation: calculated according to the formula , , ; and then substitute the formula to calculate .

[0276] Experimental control: measure steady-state under different , verify . Compare the fitting error of the model before and after correction to the σ(t) curve, which needs to meet the relative error ≤10%.

[0277] Step S3.7: Implementation specification and abnormality handling

[0278] Technical features and operational requirements:

[0279] (1) Parameter checking sequence:

[0280] First, measure , , , calculate and , and then calculate and .

[0281] (2) Model verification threshold:

[0282] If , check whether the low Reynolds number assumption is invalid.

[0283] If the calculated , recheck the measurement accuracy of .

[0284] (3) Exception handling rules:

[0285] The adsorption rate is abnormally high: check Are there any omissions in the calculation? . contributions.

[0286] The diffusion layer thickness is too small ( ): Remeasure ionic strength and recalculate .

[0287] Step S4: Stability Criteria and Lifetime Prediction of Nanobubbles

[0288] Step S4.1: Establishment of the equation for the rate of change of radius and definition of parameters

[0289] Preferably, the equation construction process is as follows:

[0290] (1) Construction of the mass conservation equation:

[0291] ① Relationship between bubble volume change rate and gas diffusion flux:

[0292] (36)

[0293] Indicates the concentration of dissolved gas; bubble surface ( The gas concentration gradient.

[0294] Simplifying, we get the rate of change of bubble radius over time. equation:

[0295] (37)

[0296] Parameter description:

[0297] , representing the gas concentration inside the bubble, where (including environmental static pressure) (and Laplace pressure);

[0298] Dissolved gas diffusion coefficient, calibrated through gas permeation experiments.

[0299] (2) Setting gas diffusion boundary conditions:

[0300] Surface concentration expression:

[0301] (38)

[0302] Far-field boundary conditions: Indicates the surface of the bubble ( The gas concentration of ) Gas concentration in bulk solution; measured Dissolved oxygen meter or infrared spectroscopy is used.

[0303] Step S4.2: Derivation of steady-state analytical solution of diffusion field

[0304] Preferably, the solving process comprises:

[0305] (1) Simplification of control equation:

[0306] The dissolved gas diffusion equation is:

[0307] (39)

[0308] Its general solution is:

[0309] (40)

[0310] , represents the integral constant, determined by the boundary condition.

[0311] (2) Substitution of boundary conditions:

[0312] Solving the constant and .

[0313] Thus, the interfacial concentration gradient is obtained:

[0314] (41)

[0315] Step S4.3: Derivation of radius dynamics equation

[0316] Preferably, the equation derivation is as follows:

[0317] Substitute the concentration gradient into the radius change rate equation, we get:

[0318] (42)

[0319] Key parameter calibration:

[0320] ① Static pressure : Real-time record the environmental hydrostatic pressure with pressure sensor, the accuracy requirement .

[0321] ② Surface tension : Determined by pendant drop method or surface tension meter, temperature control within ±0.5℃.

[0322] Step S4.4: Determination of stability criterion

[0323] Preferably, the criterion is constructed as follows:

[0324] (1) Analytical solution for critical radius:

[0325] make The value equals zero, so the critical radius corresponding to the stability condition is obtained:

[0326] (43)

[0327] (2) Verification method

[0328] ① Measurement of different Value of actual stable radius of bubble Verify whether it is satisfied .

[0329] ② If there is a deviation Inspection required The measurement method (e.g., replacing it with electrophoresis for retesting).

[0330] Step S4.5: Integral implementation of the lifetime prediction equation

[0331] Preferably, the integration process includes:

[0332] (1) Dimensionless treatment: Ignore the static pressure term Lifetime integral Simplified to:

[0333] (44)

[0334] (2) Derivation of the closed-loop solution:

[0335] (45)

[0336] (3) Simplified formula applicable conditions: when... At that time, it degenerates into:

[0337] (46)

[0338] Indicates the initial radius of the bubble; This represents the diffusion coefficient of the dissolved gas.

[0339] Step S4.6: Experimental Verification and Parameter Calibration Procedure

[0340] Preferably, the verification process includes:

[0341] (1) Experimental data collection:

[0342] ① Initial radius determination: using an atomic force microscope (AFM) Statistical analysis of nanobubble imaging Distribution.

[0343] ② Lifetime measurement: In the temperature-controlled cell (temperature control accuracy ±0.1℃), the bubble disappearance time Texp is recorded by a high-speed camera.

[0344] ③ Parameter synchronous recording: Each batch of experiments needs to synchronously obtain (Zeta potential instrument) and (electrochemical workstation microelectrode method).

[0345] (2) Error correction algorithm:

[0346] ① Surface charge decay correction: If the experimental duration exceeds 5 minutes, a time function needs to be introduced to correct the lifetime formula.

[0347] ② Multi-bubble interference processing: When the bubble density , an empirical correction factor (n is the number of bubbles per unit volume) is added.

[0348] Step S4.7: Implementation case

[0349] (1) Case conditions:

[0350] ① Fluid parameters: 25℃ deionized water, dissolved oxygen , ;

[0351] ② Bubble parameters: , , .

[0352] (2) Calculation process:

[0353] ① Critical radius calculation:

[0354]

[0355] ② Lifetime simplified formula calculation:

[0356]

[0357] ③ Measured verification:

[0358] The measured lifetime Texp=8.9d, the relative error is 6.7%, which meets the acceptance standard ( ).

[0359] The technical effects of the present application are reflected in three dimensions of theoretical innovation, technical breakthrough and industrial value, as follows:

[0360] (1) Theoretical innovation

[0361] ① First curved surface charge dynamic balance theory framework

[0362] Break through the traditional planarization assumption, for the first time, the electrostatic force bi-domain decomposition (normal pressure balance / tangential self-balancing), inter-ring ion migration dynamics and Debye shielding-Brownian motion coupling mechanism are included in the unified model, realizing the full coupling analysis of curved surface system parameters, and the correlation degree of model parameters reaches 95%.

[0363] ② Establishing an analytical method for ring ion distribution

[0364] An analytical algorithm for inter-ring potential energy based on Bessel function and a three-level ring coupling correction rule (main ring + double adjacent rings) are proposed, and a radial charge density gradient calculation model is developed, which greatly improves the prediction error of the diffusion layer thickness.

[0365] (2) Technical advantages

[0366] ① Precision breakthrough

[0367] Under the harsh conditions of pH 4-12, salt concentration , temperature 25-180℃, the prediction error of nanobubble stability (Traditional model error ), still maintains 12% precision advantage under extreme conditions (150℃ / 0.5M NaCl).

[0368] ② Multi-scenario adaptability

[0369] Curved surface charge representation: Analytical curvature radius of nanobubble / microbubble surface ion distribution;

[0370] Dynamic response analysis: Support level ring spacing dynamic change simulation (based on real-time feedback data of HS-AFM).

[0371] ③ Cross-scale expansion capability

[0372] By introducing the dimensionless ring density parameter , the unified modeling of bulk nanobubbles ( ) and surface nanobubbles ( ) is realized, which improves the application range of the model.

[0373] (3) Industrial application value

[0374] ① Oil and gas production efficiency

[0375] Through ring spacing regulation Optimize the electrostatic repulsion-viscous force balance of oil displacement nanobubbles in high temperature and high salt reservoirs (salinity ) in the oilfield, which is 2.3 percentage points higher than the traditional model.

[0376] ② Energy saving in wastewater treatment

[0377] Based on the dynamic compensation algorithm of Debye shielding effect, the life of nanobubbles in the air flotation process is accurately controlled:

[0378] In electroplating wastewater treatment, the bubble survival time is extended from 4.2 h to 6.8 h;

[0379] The energy consumption per unit of processing capacity is reduced by 18.7% (actual data: from 2.4 kWh / m 3 to 1.95 kWh / m 3 ).

[0380] ③ Microelectronic manufacturing innovation

[0381] Using the ring model to predict the collapse energy threshold of nanobubbles in wafer cleaning, the particle removal rate on the surface of silicon wafer is improved to 99.997% (traditional method ).

[0382] The above examples are only exemplary embodiments of the present application and are not intended to limit the present application. The protection scope of the present application is defined by the claims. Those skilled in the art can make various modifications or equivalent replacements to the present application within the spirit and protection scope of the present application, and such modifications or equivalent replacements shall also be considered to fall within the protection scope of the present application.

Claims

1. A method for dynamic analysis of the stability and lifetime prediction of nanobubbles, characterized in that, The method comprises the following steps: Step 100, based on the assumption that the surface hydroxyl ions of the nanobubbles are arranged in a concentric ring dense packing, an explicit relationship model of the charge density is constructed, a three-level ring coupling mechanism is introduced into the explicit relationship model, a normal component of the inter-ring Coulomb repulsion is solved, a Hankel transform and a Debye shielding correction are used to obtain an analytical solution of the equivalent repulsive force of the surface hydroxyl ions of the nanobubbles, the analytical solution of the equivalent repulsive force is brought into a modified Young-Laplace equation to obtain a model for calculating the net pressure difference inside and outside the bubble; Step 200, based on the Smoluchowski diffusion control model, a Brownian motion and diffusion coupled Prandtl number is introduced to correct the thickness of the nanobubble boundary layer, and a differential equation of the surface charge density of the nanobubble is constructed to solve the low coverage approximate analytical solution of the surface charge density of the nanobubble in real time; Step 300, based on the Langevin equation describing the nanobubble motion by ignoring the inertial term, the effective diffusion coefficient of the surface hydroxyl ions of the nanobubble is solved, and the effective diffusion coefficient is used to rewrite the corrected nanobubble boundary layer thickness to update the adsorption rate constant of the surface hydroxyl ions of the nanobubble in the low coverage approximate analytical solution; Step 400, through mass conservation, the net pressure difference inside and outside the bubble is calculated by using the obtained model to establish the rate equation of the nanobubble radius with time, the critical stable radius of the nanobubble is solved, and the life prediction closed solution is obtained through the integration of the rate equation.

2. The nanobubble stability dynamic analysis and life prediction method according to claim 1, characterized in that: In step 100, assuming the surface hydroxyl ions are arranged in a close-packed concentric ring, the first Ring radius , satisfying the tangential spacing ; The explicit relationship of the charge density is established by minimizing the total potential energy, and the calculation formula is: ； ； wherein, represents the base ring spacing; identifies the charge density; represents the amount of charge carried by the hydroxide ions on the surface of the nanobubbles; represents the dielectric constant; represents the Boltzmann constant; represents the thermodynamic temperature; ring density parameter; represents the first ring's associated radius.

3. The nanobubble stability dynamic analysis and life prediction method according to claim 2, characterized in that: The low coverage approximate analytical solution of the surface charge density of the nanobubble is solved in real time, which is substituted into the explicit relationship of the charge density to further correct the model for calculating the net pressure difference inside and outside the bubble.

4. The nanobubble stability dynamic analysis and life prediction method according to claim 2, characterized in that: The three-level ring coupling mechanism is introduced into the explicit relationship model, and the method for solving the normal component of the inter-ring Coulomb repulsion specifically includes: Based on the single ring action model, the normal pressure contribution of the ring to the center point is calculated Ring normal pressure contribution to the center point The calculation formula is: ； The total ion pressure of the hydroxyl radical on the surface of the nanobubble is considered by introducing a three-level ring coupling mechanism Is: ； The Debye screening effect is introduced again to correct the inter-ring interaction potential : ； Debye screening length : ； where is the linear charge density ( ); is the coordinate variable in the vertical direction; is the angular variable, used to describe the spatial angular distribution when integrating; is the distance between ring-shaped structural units; is the Debye screening parameter, describing the degree of screening of charge interactions; is the bulk ion concentration.

5. The nanobubble stability dynamic analysis and life prediction method according to claim 1, characterized in that: Based on the Smoluchowski diffusion control model, the Brownian motion and diffusion coupled Prandtl number is introduced to construct the time-varying evolution condition of the surface charge density of the nanobubble, and the change of the charge density in the dynamic adsorption process is quantified, which specifically includes: Weak field conditions, surface potential , satisfy linearized poisson-boltzmann equation; Symmetrical electrolyte, the positive and negative ions of the electrolyte in the solution are 1:1 type; Quasi-steady-state diffusion layer, ignoring the influence of convection, the double electric layer is in dynamic equilibrium.

6. The nanobubble stability dynamic analysis and life prediction method according to claim 1, characterized in that: In step 300, the assumption condition for constructing the Smoluchowski diffusion control model while introducing the Brownian motion and diffusion coupled Prandtl number to correct the thickness of the nanobubble boundary layer specifically includes: Low Reynolds number condition, Re 1, neglecting the fluid motion inertia term; Homogeneous fluid, solution viscosity , the diffusion coefficient D is independent of the spatial position; Statistical average approximation, Brownian motion velocity field is treated by ensemble average, fluctuation term is linearly related to concentration gradient.

7. The method of claim 1, wherein the method further comprises: determining the stability of the nanobubbles in the solution based on the dynamic analysis. The bubble radius change rate equation is established by the mass conservation equation, and the critical stable radius is solved The calculation formula is: ； The closed solution of lifetime prediction is obtained by integrating the mass conservation equation The calculation formula is as follows: When 1; Life prediction closed-form solution Degenerate to: ； wherein, represents a gas diffusion coefficient; represents an initial radius; represents a gas constant; represents a Henry coefficient; represents a vacuum dielectric constant; represents a surface tension, a physical quantity describing the interaction between the hydrogen and oxygen ions on the surface of the nanobubble, making the surface have a shrinkage tendency.

8. The method of claim 1, wherein the method further comprises: determining the stability of the nanobubbles in the solution based on the dynamic analysis. Closed solution for lifetime prediction by error correction correction, specifically comprising: Introducing a time function after the experimental time exceeds a set time ; and / or introducing an empirical correction factor in the presence of multi-bubble interference , N is the number of bubbles per unit volume.

9. The method of claim 1, wherein the method further comprises: determining the stability of the nanobubbles in the solution based on the dynamic analysis. The dynamic diffusion equation describing the motion of nanobubbles based on the Langevin equation ignoring the inertial term is modified based on the convection effect of Brownian motion, and the ion concentration evolution equation is obtained. After statistical average processing, the effective diffusion equation is obtained, and then the effective diffusion coefficient is obtained; Based on the time matching of diffusion and momentum characteristics and the scale correlation, combined with the stability condition, the effective diffusion coefficient is used to rewrite the corrected nanobubble boundary layer thickness, and the corrected nanobubble boundary layer thickness is: 。 10. The method of claim 1, wherein the method further comprises: determining the stability of the nanobubbles in the solution based on the dynamic analysis. The updated adsorption rate constant of the surface hydroxyl ions of the nanobubbles Is: ； where the effective diffusion coefficient ; represents the Prandtl number.

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