CN111382498B - A Modeling Method for Dynamic Flow Potential in Porous Media Converters - Google Patents

Abstract

The invention discloses a modeling method for the dynamic flow potential of a porous medium converter. When establishing the dynamic flow potential model of the porous medium converter, the influence of the thickness of the electric double layer on the current and potential distribution is considered. Concentration is an important factor affecting the thickness of the electric double layer, so the model can be applied to the analysis of different solution properties; this paper establishes a dynamic flow potential model, which makes it possible to use this model to analyze dynamic characteristics, such as the flow potential when different frequencies are input. The change of the coupling coefficient is of great significance for the performance analysis of the sensor using the porous medium converter as an important sensitive element; the model involves multiple parameters of the porous medium converter, such as the equivalent average radius of the capillary, so the model can be used Analysis of the effect of parameter changes on the coupling coefficient of flow potential in porous media converters.

Description

A Modeling Method for Dynamic Flow Potential in Porous Media Converters

technical field

The invention relates to the technical field of porous media, in particular to a method for modeling the dynamic flow potential of a porous media converter.

Background technique

The porous medium is sintered by solid microbeads, and the internal structure includes the void part between the microbead skeleton and the microbeads. It has the characteristics of small void size and large specific surface area. According to the interfacial electric double layer effect, when the solid and liquid surfaces are in contact, electric charges with equal and opposite electric charges will be formed between the solid and liquid phases. Flowing potential and flowing current will be formed with the fluid movement, this process is called electrokinetic effect, so the porous medium material can be used as a converter to convert the pressure change of the fluid into an electrical signal, and use the flowing current, the flowing potential and the flowing potential coupling coefficient for quantification. analyze.

At present, the research on microfluidics and electrokinetic effects in porous media converters focuses on the steady-state model. However, there are few studies on the electrokinetic effects under dynamic flow fields. And most of the current research is based on the assumption of thin electric double layer, that is, it is assumed that the thickness of the electric double layer is much smaller than the pore size of the porous medium. However, when the characteristics of the solution and the characteristic parameters of the porous medium change, the thickness of the electric double layer may increase. The electric layer assumption is no longer applicable, and the influence of the thickness of the electric double layer on the charge distribution needs to be considered.

SUMMARY OF THE INVENTION

In view of this, the purpose of the present invention is to provide a modeling method for the dynamic flow potential of a porous medium converter, which can be applied to the analysis under different thicknesses of the electric double layer.

A method for modeling dynamic flow potential of a porous medium converter, comprising the following steps:

Step 1. Use a bundle of curved circular capillaries to be equivalent to the porous medium converter, that is, the capillary bundle model of the porous medium converter, specifically:

According to the particle size distribution of the microbeads in the porous medium converter, the radius distribution law of the non-uniformly distributed capillary bundle model is calculated, and then the equivalent average radius _rc of the uniformly distributed capillary tube is calculated, thereby simplifying the capillary bundle model into a bundle of curved, For capillaries with the same radius, the formula for calculating the equivalent radius is as follows:

Among them, r represents the radius of a capillary, n(r) represents the density distribution function of the capillary radius, r _min and r _max represent the maximum and minimum values of the capillary radius, respectively, and E( ) represents the mathematical expectation; From the porosity formula (2) of the tube bundle model, the equivalent average capillary number N _c is derived, as shown in formula (3):

Among them, V and V _f represent the volume of the converter and the volume of the fluid in the capillary bundle, respectively, L _c represents the length of the equivalent capillary, L represents the thickness of the converter, and A represents the cross-sectional area of the converter;

Step 2. Determine whether a single capillary is a thin electric double layer or a thick electric double layer:

Taking into account the influence of the thickness of the electric double layer on the charge distribution in the capillary, the ratio a of the capillary radius r and the thickness d of the electric double layer is used as the evaluation standard:

If a>50, the electric double layer structure in the porous medium converter is considered to be a thin electric double layer; if a≤50, the electric double layer structure is considered to be a thick electric double layer;

The formula for calculating the thickness d of the electric double layer is given by equation (5):

Among them, e ₀ represents the basic charge, ε represents the dielectric constant, k _B is the Boltzmann constant, T and n ₀ represent the temperature and the number of ions, respectively, v represents the ion valence of the solution, and the number of ions in the solution can be determined by the solution concentration C _f and Avogadro's constant NA are calculated, that is, n ₀ =1000N _{A C f} ;

Step 3. Under dynamic conditions, analyze the flowing current and flowing potential inside the capillary bundle:

Firstly, according to Navier-Stokes formula (6) and the applied electric field E=0, the distribution of the fluid velocity u in the capillary in the radial direction u(r ₁ ) is obtained by solving, as shown in formula (7):

where, K ² =iωρ ₀ /μ, i represents the imaginary unit, ω represents the angular velocity, ρ ₀ and μ represent the solution density and solution viscosity, respectively, I _k (·) represents the k-order modified Bessel function of the first kind, and I ₀ ( ) represents the 0th-order modified Bessel function of the first kind, z represents the axial coordinate of the capillary, P represents the dynamic pressure difference, r ₁ represents the radial coordinate in the capillary, and ρ _e represents the net charge density;

Considering the diffusion distribution of the electric double layer, the relationship between charge and potential is shown in equations (8) and (9):

where ρ _e (r ₁ ) and Ψ(r ₁ ) represent the charge density and potential at r ₁ , respectively;

Step 4. Calculate the potential distribution and flowing current in the capillary of the thin electric double layer:

The thickness of the electric double layer is much smaller than the radius of the capillary tube and can be ignored. At this time, it is considered that the electric double layer charge is mainly concentrated in the fluid near the wall, and its charge and potential distribution can be expressed as equations (10) and (11), respectively:

where ζ represents the zeta potential of the solid-liquid surface, and the independent variable r ₁ is the radial coordinate;

Calculate the flow current I _{sp_thin} (r) in the capillary from the charge distribution, which is defined by the current:

Step 5. Calculate the potential distribution and flowing current in the capillary of the thick electric double layer:

For the thick electric double layer structure, it is divided into two regions, namely the high-potential region and the low-potential region, and its potential distribution is expressed as a piecewise function shown in equation (13):

Wherein, the dimensionless processing R=r ₁ /d is performed on the radial phase coordinates, and the threshold value R ^* is a value that makes Ψ ^* (R ^* )=1;

的表达式用一个分段函数进行表示，由式(14)给出：

The expression of is represented by a piecewise function, which is given by Eq. (14):

Ψ_S为毛细管的表面电势，e为自然常数；无量纲半径的阈值R^*可以由附加边界条件式(15)和式(16)解出；in,

Ψ _S is the surface potential of the capillary, and e is a natural constant; the threshold R ^* of the dimensionless radius can be solved by the additional boundary conditions (15) and (16);

In formula (16), the superscript "'" indicates the derivation;

Substitute the expression (13) of the electric potential into the formula (9) to obtain the expression of the charge distribution in the case of a thick electric double layer:

Calculate the dynamic flow current I sp_thick (r) in the capillary according to the charge distribution, and calculate the dynamic flow current I _{sp_L} (r) and I _{sp_H} (r) corresponding to the two parts of the high potential region and the low potential region _respectively :

For formula (20), when C<0, it is expressed as formula (21):

Step 6. Calculate the overall flow potential of the porous medium converter based on the capillary bundle model:

First, use the capillary bundle model of equivalent radius to calculate the overall flow current I _sp of the converter, as shown in equation (22):

I _sp =N _c I _sp (rc ₎ (22)

where I _sp ( _rc ) in the case of a thin electric double layer, formula (12) represents the value of the current flowing in a capillary with a radius of _rc ; or formula (18) in the case of a thick electric double layer, with a radius _rc The value of the current flowing in the capillary;

Assuming that the conduction current I _c and the flowing current balance each other, and the total conductance of the converter is expressed as ∑, then according to Ohm's law, the formula for calculating the flowing potential E _sp across the converter is:

The streaming potential model of porous media is characterized by the streaming potential coupling coefficient _Csp :

where the conductance Σ is calculated from the double layer surface conductance Σ _S and the solution conductivity σ ₀ :

∑=(E( _r ² )σ ₀ +2·E( _r )∑ _s )/rc ² =(rc σ ₀ +2·∑ _s )/ _rc (25)

Substitute the flow current expression obtained by the above calculation into the flow potential coupling coefficient of the porous medium converter under the thin electric double layer and thick electric double layer capillary bundle models, that is, the flow potential model of the porous medium is obtained as formula (26) and formula ( 27) shown.

The present invention has the following beneficial effects:

In the present invention, when establishing the dynamic flow potential model of the porous medium converter, the influence of the thickness of the electric double layer on the current and potential distribution is considered. Since the type and concentration of the solution are important factors affecting the thickness of the electric double layer, the model can be applied to the analysis of different solution properties.

In this paper, a dynamic flow potential model is established, which makes it possible to use the model to analyze dynamic characteristics, such as the change of the flow potential coupling coefficient when different frequencies are input. This is of great significance for the sensor performance analysis using the porous media converter as an important sensitive element.

The model involves several parameters of the porous medium converter, such as the equivalent average radius of the capillary, so the effect of the parameter change on the coupling coefficient of the flow potential of the porous medium converter can be analyzed by using this model.

Description of drawings

Figure 1 shows the equivalent model of the capillary bundle of the porous medium converter.

Figure 2 is a schematic diagram of a thin electric double layer and a thick electric double layer.

FIG. 3 is a structural diagram of an interface electric double layer.

FIG. 4 is a schematic diagram of the principle of the electrokinetic effect in the capillary.

Fig. 5 is the influence curve of the change of the structural parameters of the converter on the coupling coefficient of the dynamic flow potential.

Detailed ways

The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

A method for modeling the dynamic flow potential of the porous medium converter of the present invention adopts the capillary bundle model to perform the equivalent of the porous medium converter, as shown in FIG. 1 . The porous medium material contains skeleton part and void part, and the structure is complex, and it is difficult to directly quantitatively analyze the internal dynamic flow potential, so a bundle of curved circular capillaries is used for its equivalent, that is, the capillary bundle of the porous media converter. model, the radius of these capillaries obeys a certain distribution law. Further, according to the particle size distribution of the microbeads of the porous medium converter, the non-uniform distribution of the radius distribution is calculated, and then the equivalent average radius _rc of the uniformly distributed capillary is calculated, thereby simplifying the capillary bundle model into a bundle of curved For capillaries of the same radius, the formula for calculating the equivalent radius is as follows:

Among them, r represents the radius of a capillary, n(r) represents the density distribution function of the capillary radius, r _min and r _max represent the maximum and minimum values of the capillary radius, respectively, and E( ) represents the mathematical expectation. At the same time, according to the porosity formula of the capillary bundle model (Equation (2)), the equivalent average capillary number N _c can be derived, as shown in Equation (3):

Among them, V and V _f represent the volume of the converter and the volume of the fluid in the capillary bundle, respectively, L _c represents the length of the equivalent capillary, L represents the thickness of the converter, and A represents the cross-sectional area of the converter.

The dynamic flow current and potential calculations within a single capillary are discussed below.

Considering the influence of the thickness of the electric double layer on the charge distribution in the capillary, the ratio a of the capillary radius r and the thickness d of the electric double layer is used as the evaluation standard. If a>50, the electric double layer structure in the porous medium converter is considered to be thin. Electric double layer; if a≤50, the electric double layer structure is considered to be a thick electric double layer. The schematic diagram of the structure of the electric double layer is shown in Figure 2.

The formula for calculating the thickness d of the electric double layer is given by equation (5):

Among them, e ₀ represents the basic charge, ε represents the dielectric constant, k _B is the Boltzmann constant, T and n ₀ represent the temperature and the number of ions, respectively, and v represents the ion valence of the solution. The ionic number of the solution can be calculated using the solution concentration _{C f} and Avogadro's constant NA , ie n ₀ =1000N _{AC f} .

In order to analyze the flow current and flow potential inside the capillary bundle under dynamic conditions, firstly, the distribution u of the fluid velocity u in the capillary tube in the radial direction is obtained according to the Navier-Stokes formula (Equation (6)) and the applied electric field E=0. (r ₁ ), as shown in formula (7).

where, K ² =iωρ ₀ /μ, i represents the imaginary unit, ω represents the angular velocity, ρ ₀ and μ represent the solution density and solution viscosity, respectively, I _k (·) represents the k-order modified Bessel function of the first kind, and I ₀ (·) represents the 0th-order modified Bessel function of the first kind; z represents the axial coordinate of the capillary, P represents the dynamic pressure difference, r ₁ represents the radial coordinate in the capillary, and ρ _e represents the net charge density.

The flow velocity distribution u(r ₁ ) of the fluid in the capillary is obtained above. As shown in Figure 3, considering the diffusion distribution of the electric double layer, the relationship between the charge and the potential is shown in equations (8) and (9):

where ρ _e (r ₁ ) and Ψ(r ₁ ) represent the charge density and potential at r ₁ , respectively.

For the thin electric double layer, the thickness of the electric double layer is much smaller than the radius of the capillary, which can be ignored. At this time, it is considered that the electric double layer charge is mainly concentrated in the fluid near the wall, and its charge and potential distribution can be expressed as formula (10 ) and formula (11):

Among them, ζ represents the zeta potential of the solid-liquid surface, (zeta potential is the potential difference between the solid-phase charged surface and the liquid, which can reflect the strength of the solid-phase charge. The thin electric double layer is also called the low zeta potential, and the thick The electric double layer case is also called the high zeta potential case), and the independent variable r ₁ is the radial coordinate.

Calculate the flow current I _{sp_thin} (r) in the capillary from the charge distribution, which is defined by the current:

For the thick electric double layer structure, the thickness of the electric double layer will affect the charge distribution in the capillary tube. As shown in Figure 2(b), there is a significant difference in the potential distribution near the tube wall and the area far away from the tube wall, which is divided into two parts. These regions are the high-potential region and the low-potential region, and their potential distribution can be expressed as a piecewise function shown in equation (13):

Here, for the convenience of analysis, the radial coordinates are subjected to dimensionless processing R=r ₁ /d, and the threshold value R ^* is a value such that Ψ ^* (R ^* )=1.

的表达式用一个分段函数进行表示，由式(14)给出：further,

The expression of is represented by a piecewise function, which is given by Eq. (14):

Ψ_S为毛细管的表面电势，e为自然常数。无量纲半径的阈值R^*可以由附加边界条件式(15)和式(16)解出。in,

Ψ _S is the surface potential of the capillary, and e is a natural constant. The threshold value R ^* of the dimensionless radius can be solved by the additional boundary conditions (15) and (16).

In formula (16), the superscript "'" indicates the derivation;

Substituting the expression (13) of the potential into the formula (9), the expression of the charge distribution in the case of a thick electric double layer can be obtained:

To calculate the dynamic flow current I sp_thick (r) in the capillary according to the charge distribution, it is necessary to calculate the dynamic flow current I _{sp_L} (r) and I _{sp_H} (r) corresponding to the two parts of the high potential region and the low potential region _respectively :

For equation (20), when C<0, it can be further simplified as:

The calculation method of the dynamic flow potential and flow current in a single capillary is obtained above, and the method is applicable to the electric double layer structure of any thickness. Next, the flow potential of the porous media converter as a whole based on the capillary bundle model is calculated.

First, use the capillary bundle model of equivalent radius to calculate the overall flow current I _sp of the converter, as shown in equation (22):

I _sp =N _c I _sp (rc ₎ (22)

where I _sp ( _rc ) is the value of the flowing current expressed by formula (12) at the radius _rc in the case of a thin electric double layer; or is the value of the flowing current expressed by formula (18) at the radius of the thick electric double layer is the value of _rc ;

As shown in Figure 4, assuming that the conduction current I _c and the flowing current balance each other, the total conductance of the converter can be expressed as ∑, then according to Ohm's law, the calculation formula of the flowing potential E _sp at both ends of the converter is:

The flow potential coupling coefficient C _sp is often used to describe the flow potential characteristics of porous media, and the expression is:

The total conductance Σ is calculated from the electric double layer surface conductance _ΣS and the solution conductivity _σ0 :

∑=(E( _r ² )σ ₀ +2·E( _r )∑ _s )/rc ² =(rc σ ₀ +2·∑ _s )/ _rc (25)

By substituting the flow current expression calculated above, the flow potential coupling coefficient of the porous medium converter based on the capillary bundle model can be obtained:

Above, we established the dynamic flow potential model C _sp of the porous medium converter applicable to any thickness of the electric double layer. This model describes the relationship between the pressure and the potential of the converter under dynamic conditions. Frequency, characteristic parameters of the liquid, equivalent radius of the capillary and other parameters, so the model can be used to analyze the influence of the above parameter changes on the dynamic flow potential coupling coefficient of the converter.

Three kinds of porous dielectric converters B1, B2 and B3 with different structural parameters are selected, and the ratios of their equivalent average radius rc of capillary to the thickness _d of the electric double layer are 251.7, 348.9 and 506.4, respectively, so they are all thin electric double layers. Happening. Using the model established above, the flow potential coupling coefficients of the three converters at different frequencies are calculated, and the theoretical results of the calculation are shown in Figure 5. It can be seen that when the capillary radius increases, the dynamic flow potential coupling coefficient increases, but its bandwidth decreases.

To sum up, the above are only preferred embodiments of the present invention, and are not intended to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims (1)

1. a porous medium converter dynamic flow potential modeling method, is characterized in that, comprises the steps: Step 1. Use a bundle of curved circular capillaries to be equivalent to the porous medium converter, that is, the capillary bundle model of the porous medium converter, specifically: According to the particle size distribution of the microbeads in the porous medium converter, the radius distribution law of the non-uniformly distributed capillary bundle model is calculated, and then the equivalent average radius _rc of the uniformly distributed capillary tube is calculated, thereby simplifying the capillary bundle model into a bundle of curved, For capillaries with the same radius, the formula for calculating the equivalent radius is as follows:

Among them, r represents the radius of a capillary, n(r) represents the density distribution function of the capillary radius, r _min and r _max represent the maximum and minimum values of the capillary radius, respectively, and E( ) represents the mathematical expectation; From the porosity formula (2) of the tube bundle model, the equivalent average capillary number N _c is derived, as shown in formula (3):

Among them, V and V _f represent the volume of the converter and the volume of the fluid in the capillary bundle, respectively, L _c represents the length of the equivalent capillary, L represents the thickness of the converter, and A represents the cross-sectional area of the converter; Step 2. Determine whether a single capillary is a thin electric double layer or a thick electric double layer: Taking into account the influence of the thickness of the electric double layer on the charge distribution in the capillary, the ratio a of the capillary radius r and the thickness d of the electric double layer is used as the evaluation standard:

If a>50, the electric double layer structure in the porous medium converter is considered to be a thin electric double layer; if a≤50, the electric double layer structure is considered to be a thick electric double layer; The formula for calculating the thickness d of the electric double layer is given by equation (5):

Among them, e ₀ represents the basic charge, ε represents the dielectric constant, k _B is the Boltzmann constant, T and n ₀ represent the temperature and the number of ions, respectively, v represents the ion valence of the solution, and the number of ions in the solution can be determined by the solution concentration C _f and Avogadro's constant NA are calculated, that is, n ₀ =1000N _{A C f} ; Step 3. Under dynamic conditions, analyze the flowing current and flowing potential inside the capillary bundle: Firstly, according to Navier-Stokes formula (6) and the applied electric field E=0, the distribution of the fluid velocity u in the capillary in the radial direction u(r ₁ ) is obtained by solving, as shown in formula (7):

where, K ² =iωρ ₀ /μ, i represents the imaginary unit, ω represents the angular velocity, ρ ₀ and μ represent the solution density and solution viscosity, respectively, I _k (·) represents the k-order modified Bessel function of the first kind, and I ₀ ( ) represents the 0th-order modified Bessel function of the first kind, z represents the axial coordinate of the capillary, P represents the dynamic pressure difference, r ₁ represents the radial coordinate in the capillary, and ρ _e represents the net charge density; Considering the diffusion distribution of the electric double layer, the relationship between charge and potential is shown in equations (8) and (9):

where ρ _e (r ₁ ) and Ψ(r ₁ ) represent the charge density and potential at r ₁ , respectively; Step 4. Calculate the potential distribution and flowing current in the capillary of the thin electric double layer: The thickness of the electric double layer is much smaller than the radius of the capillary tube and can be ignored. At this time, it is considered that the electric double layer charge is mainly concentrated in the fluid near the wall, and its charge and potential distribution can be expressed as equations (10) and (11), respectively:

where ζ represents the zeta potential of the solid-liquid surface, and the independent variable r ₁ is the radial coordinate; Calculate the flow current I _{sp_thin} (r) in the capillary from the charge distribution, which is defined by the current:

Step 5. Calculate the potential distribution and flowing current in the capillary of the thick electric double layer: For the thick electric double layer structure, it is divided into two regions, namely the high-potential region and the low-potential region, and its potential distribution is expressed as a piecewise function shown in equation (13):

Wherein, the dimensionless processing R=r ₁ /d is performed on the radial phase coordinates, and the threshold value R ^* is a value that makes Ψ ^* (R ^* )=1;

The expression of is represented by a piecewise function, which is given by Eq. (14):

in,

Ψ _S is the surface potential of the capillary, e is a natural constant, and the threshold R ^* of the dimensionless radius can be solved by the additional boundary conditions (15) and (16);

In formula (16), the superscript "'" indicates the derivation; Substitute the expression (13) of the electric potential into the formula (9) to obtain the expression of the charge distribution in the case of a thick electric double layer:

Calculate the dynamic flow current I sp_thick (r) in the capillary according to the charge distribution, and calculate the dynamic flow current I _{sp_L} (r) and I _{sp_H} (r) corresponding to the two parts of the high potential region and the low potential region _respectively :

For formula (20), when C<0, it is expressed as formula (21):

Step 6. Calculate the overall flow potential of the porous medium converter based on the capillary bundle model: First, use the capillary bundle model of equivalent radius to calculate the overall flow current I _sp of the converter, as shown in equation (22): I _sp =N _c I _sp (rc ₎ (22) where I _sp ( _rc ) in the case of a thin electric double layer, formula (12) represents the value of the current flowing in a capillary with a radius of _rc ; or formula (18) in the case of a thick electric double layer, with a radius of _rc The value of the current flowing in the capillary; Assuming that the conduction current I _c and the flowing current balance each other, and the total conductance of the converter is expressed as Σ, then according to Ohm's law, the formula for calculating the flowing potential E _sp across the converter is:

The streaming potential model of porous media is characterized by the streaming potential coupling coefficient _Csp :

where the conductance Σ is calculated from the double layer surface conductance _ΣS and the solution conductivity _σ0 : Σ=(E( _r ² )σ ₀ +2·E( _r )Σ _s )/rc ² =(rc σ ₀ +2·Σ _s )/ _rc (25) Substitute the flow current expression obtained by the above calculation into the flow potential coupling coefficient of the porous medium converter under the thin electric double layer and thick electric double layer capillary bundle models, that is, the flow potential model of the porous medium is obtained as formula (26) and formula ( 27) shown

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