CN111382498A - Modeling method for dynamic flowing potential of porous medium converter - Google Patents
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Abstract
The invention discloses a modeling method of dynamic flowing potential of a porous medium converter, which considers the influence of double-layer thickness on current and potential distribution when building a dynamic flowing potential model of the porous medium converter, and the model can be suitable for analysis under different solution properties because the type and concentration of a solution are important factors influencing the double-layer thickness; a dynamic flow potential model is established, so that the model can be used for analyzing dynamic characteristics, such as the change of a flow potential coupling coefficient when different frequencies are input, and the method has great significance for the performance analysis of a sensor using the porous medium converter as an important sensitive element; the model relates to a plurality of parameters of the porous medium transducer, such as equivalent mean radius of the capillary tube, and therefore the model can be used to analyze the effect of parameter changes on the flow potential coupling coefficient of the porous medium transducer.
Description
Technical Field
The invention relates to the technical field of porous media, in particular to a modeling method of dynamic flowing potential of a porous medium converter.
Background
The porous medium is formed by sintering solid microspheres, the internal structure comprises a microsphere skeleton part and a gap part between the microspheres, the porous medium has the characteristics of small gap size and large specific surface area, and fluid moves in the porous medium in a seepage mode. According to the interface double layer effect, when solid and liquid surfaces are contacted, electric charges with equal electric quantity and opposite electric property are formed between the solid and the liquid, when fluid flows through, the electric charges generated in the pores due to the interface double layer effect can form flowing potential and flowing current along with the movement of the fluid, the process is called electrokinetic effect, therefore, the porous medium material can be used as a converter to convert the pressure change of the fluid into an electric signal, and the flowing current, the flowing potential and the flowing potential coupling coefficient are utilized to carry out quantitative analysis.
At present, research on microfluidics and electrokinetic effects in porous medium converters focuses on research on steady-state models, however, analysis of the electrokinetic effects under dynamic flow fields is less. Most of the current research is based on the assumption of a thin electric double layer, that is, the thickness of the electric double layer is far smaller than the pore diameter of the porous medium, however, when the solution characteristics and the characteristic parameters of the porous medium are changed, the thickness of the electric double layer may increase, and at this time, the assumption of the thin electric double layer is no longer applicable, and the influence of the thickness of the electric double layer on the charge distribution needs to be considered.
Disclosure of Invention
In view of the above, the present invention provides a method for modeling a dynamic flow potential of a porous medium converter, which can be applied to analysis under different electric double layer thicknesses.
A porous medium converter dynamic flow potential modeling method comprises the following steps:
calculating the radius distribution rule of a non-uniformly distributed capillary bundle model according to the particle size distribution of the microbeads of the porous medium converter, and then calculating the equivalent average radius r of uniformly distributed capillariescThus, the capillary bundle model is simplified to a bundle of bent capillaries with the same radius, and the calculation formula of the equivalent radius is as follows:
wherein r represents the radius of a capillary, n (r) represents the density distribution function of the radius of the capillary, rminAnd rmaxRepresents the maximum and minimum values of the capillary radius, respectively, E (·) represents the mathematical expectation; meanwhile, according to the porosity formula (2) of the capillary bundle model, the equivalent average capillary number N is obtained through derivationcAs shown in formula (3):
wherein, V and VfRespectively representing the volume of the transducer and the volume of the fluid in the bundle of capillaries, LcRepresents the length of the equivalent capillary, L represents the thickness of the transducer, a represents the cross-sectional area of the transducer;
step 2, judging whether the single capillary is a thin electric double layer or a thick electric double layer:
taking into account the effect of the electric double layer thickness on the charge distribution in the capillary, the ratio a of the capillary radius r to the electric double layer thickness d is used as an evaluation criterion:
if a is more than 50, the electric double layer structure in the porous medium converter is considered as a thin electric double layer; if a is less than or equal to 50, the double electric layer structure is considered as a thick double electric layer;
the calculation formula of the double-electrode thickness d is given by equation (5):
wherein e is0Denotes the basic electric quantity,. epsilon.denotes the dielectric constant, kBIs the Boltzmann constant, T and n0Respectively representing temperature and ion number, v represents ion valence of the solution, and ion number of the solution can be determinedWith solution concentration CfAnd the Avgalois constant NAPerform a calculation of n0＝1000NACf；
firstly, according to Navier-Stokes formula (6) and an applied electric field E ═ 0, the distribution u (r) of the fluid flow speed u in the capillary in the radial direction is solved and obtained1) As shown in formula (7):
wherein, K2＝iωρ0Mu, i denotes the unit of imaginary number, omega denotes the angular velocity, p0And μ denotes the solution density and solution viscosity, Ik(. -) represents a first class of modified Bessel function of order k, I0(. cndot.) represents a first class modified Bessel function of order 0, z represents the capillary axial coordinate, P represents the dynamic pressure differential, r1As radial coordinate in the capillary, peRepresents the net charge density;
considering the diffusion distribution of the electric double layer, the relationship of the charge to the potential is shown by the formulas (8) and (9):
where ρ ise(r1) And Ψ (r)1) Respectively represent r1The charge density and potential of (d);
and 4, calculating the potential distribution and the flowing current in the capillary of the thin electric double layer:
the thickness of the electric double layer is far smaller than the radius of the capillary channel, neglected, and then the electric double layer charges are considered to be mainly concentrated in the fluid close to the wall surface, and the charge and potential distribution can be respectively expressed as formula (10) and formula (11):
where ζ represents the zeta potential of the surface of the solid-liquid phase, the independent variable r1Is a radial coordinate;
calculating the flowing current I in the capillary according to the charge distributionsp_thin(r), defined by the current:
and 5, calculating the potential distribution and the flowing current in the capillary of the thick electric double layer:
for the thick electric double layer structure, it is divided into two regions, i.e., a high potential region and a low potential region, and its potential distribution is expressed as a piecewise function shown in formula (13):
wherein, the radial phase coordinate is subjected to non-dimensionalization processing R ═ R1D, threshold R*Is such that Ψ*(R*) A value of 1;
wherein,
ΨSis the surface potential of the capillary; threshold value R of dimensionless radius*Can be solved by the additional boundary conditions of equation (15) and equation (16);
(16) in the formula, the upper corner mark "'" represents derivation;
substituting expression (13) of the potential for expression (9) gives an expression of charge distribution in the case of a thick electric double layer:
calculating the dynamic flow current I in the capillary from the charge distributionsp_thick(r) dynamic flow current I corresponding to both high potential region and low potential regionsp_L(r) and Isp_H(r) separately calculating:
for formula (20), when C < 0, expressed as formula (21):
step 6, calculating the whole flowing potential of the porous medium converter based on the capillary bundle model:
firstly, the flowing current I of the whole converter is calculated by utilizing a capillary tube bundle model with equivalent radiusspAs shown in formula (22);
Isp＝NcIsp(rc) (22)
wherein, Isp(rc) In the case of a thin electric double layer, the formula (12) is expressed at a radius rcThe value of the flow current in the capillary of (2); or in the case of a thick electric double layer, the radius r is expressed by the formula (18)cThe value of the flow current in the capillary of (2);
assuming a conduction current IcIn balance with the flowing current, the total conductance of the converter is expressed as Σ, and the flowing potential E across the converter is then according to ohm's lawspThe calculation formula of (2) is as follows:
coupling coefficient C by streaming potentialspFlow potential model characterizing porous media:
wherein, the conductance sigma is from the surface conductance sigma of the dual-electrode layerSAnd solution conductivity σ0And (3) calculating:
and substituting the calculated flowing current expression into the flowing potential coupling coefficient of the porous medium converter under the thin double-electric-layer and thick double-electric-layer capillary bundle models to obtain the flowing potential models of the porous medium as shown in the formulas (26) and (27).
The invention has the following beneficial effects:
the invention considers the influence of the thickness of the double electric layers on the current and potential distribution when establishing a dynamic flow potential model of the porous medium converter. Since the kind and concentration of the solution are important factors affecting the thickness of the electric double layer, the model can be adapted to analysis under different solution properties.
A dynamic streaming potential model is established that allows the model to be used to analyze dynamic characteristics such as changes in streaming potential coupling coefficients at different frequency inputs. This is of great significance for the performance analysis of sensors using the porous medium transducer as an important sensitive element.
The model relates to a plurality of parameters of the porous medium transducer, such as equivalent mean radius of the capillary tube, and therefore the model can be used to analyze the effect of parameter changes on the flow potential coupling coefficient of the porous medium transducer.
Drawings
FIG. 1 is a capillary bundle equivalent model of a porous media converter.
Fig. 2(a) is a schematic diagram of a thin electric double layer, and fig. 2(b) is a schematic diagram of a thick electric double layer.
Fig. 3 is a structural view of an electric double layer at an interface.
Figure 4 is a schematic diagram of the electrokinetic effect principle in a capillary.
FIG. 5 is a graph of the effect of changes in structural parameters of a transducer on the dynamic streaming potential coupling coefficient.
Detailed Description
The invention is described in detail below by way of example with reference to the accompanying drawings.
The dynamic flow potential modeling method for the porous medium converter adopts a capillary bundle model to perform equivalence on the porous medium converter, and is shown in figure 1. The porous medium material contains a skeleton part andthe gap part has a complex structure, and the internal dynamic flow potential is difficult to be directly and quantitatively analyzed, so that a bundle of bent round capillaries is adopted for carrying out equivalence on the capillary, namely a capillary bundle model of the porous medium converter, and the radiuses of the capillaries obey a certain distribution rule. Further, the radius distribution rule of the non-uniform distribution is calculated according to the particle size distribution of the microbeads of the porous medium converter, and then the equivalent average radius r of the capillary tubes which are uniformly distributed is calculatedcThus, the capillary bundle model is simplified to a bundle of bent capillaries with the same radius, and the calculation formula of the equivalent radius is as follows:
wherein r represents the radius of a capillary, n (r) represents the density distribution function of the radius of the capillary, rminAnd rmaxDenotes the maximum and minimum values of the capillary radius, respectively, E (-) expresses the mathematical expectation. Meanwhile, according to the porosity formula (2)) of the capillary bundle model, the equivalent average capillary number N can be obtained through derivationcAs shown in formula (3):
wherein, V and VfRespectively representing the volume of the transducer and the volume of the fluid in the bundle of capillaries, LcDenotes the length of the equivalent capillary, L denotes the thickness of the transducer, and a denotes the cross-sectional area of the transducer.
Dynamic flow current and potential calculations within a single capillary are discussed below.
Considering the influence of the thickness of an electric double layer on the charge distribution in the capillary, taking the ratio a of the radius r of the capillary and the thickness d of the electric double layer as an evaluation standard, and if a is larger than 50, determining that the structure of the electric double layer in the porous medium converter is a thin electric double layer; if a is less than or equal to 50, the electric double layer structure is considered to be a thick electric double layer. The schematic structure of the electric double layer is shown in fig. 2(a) and (b).
The calculation formula of the double-electrode thickness d is given by equation (5):
wherein e is0Denotes the basic electric quantity,. epsilon.denotes the dielectric constant, kBIs the Boltzmann constant, T and n0Respectively, temperature and ion number, and v represents the ion valence of the solution. The ion number of the solution can be utilized by the solution concentration CfAnd the Avgalois constant NAPerform a calculation of n0＝1000NACf。
In order to analyze the flowing current and the flowing potential inside the capillary bundle under the dynamic condition, the distribution u (r) of the fluid flow speed u in the capillary in the radial direction is obtained by solving according to the Navier-Stokes formula (6)) and the external electric field E-01) As shown in formula (7).
Wherein, K2＝iωρ0Mu, i denotes the unit of imaginary number, omega denotes the angular velocity, p0And μ denotes the solution density and solution viscosity, Ik(. -) represents a first class of modified Bessel function of order k, I0(. cndot.) represents a first class modified Bessel function of order 0; z represents the capillary axial coordinate, P represents the dynamic pressure difference, r1As radial coordinate in the capillary, peRepresenting the net charge density.
Thus obtaining the capillary tubeInner fluid flow velocity distribution u (r)1). As shown in fig. 3, considering the diffusion distribution of the electric double layer, the relationship of the charge to the potential is shown by the equations (8) and (9):
where ρ ise(r1) And Ψ (r)1) Respectively represent r1The charge density and the potential at (c).
For thin electric double layers, the thickness of the electric double layer is much smaller than the capillary radius, which is negligible, and the electric double layer charge is considered to be mainly concentrated in the fluid near the wall surface, and the charge and potential distributions can be expressed as formula (10) and formula (11), respectively:
zeta represents the zeta potential of the solid-liquid phase surface, (zeta potential is the potential difference between the solid-phase charged surface and the liquid and can reflect the intensity of the solid-phase charge, the thin electric double layer condition is also called low zeta potential condition, the thick electric double layer condition is also called high zeta potential condition), and independent variable r1Are radial coordinates.
Calculating the flowing current I in the capillary according to the charge distributionsp_thin(r), defined by the current:
for a thick double layer structure, the thickness of the double layer affects the charge distribution in the capillary channels, and as shown in fig. 2(b), the potential distribution in the region near the tube wall and far from the tube wall is significantly different, and is divided into two regions, i.e., a high potential region and a low potential region, and the potential distribution can be expressed as a piecewise function shown in formula (13):
for the convenience of analysis, the radial coordinate is subjected to non-dimensionalization processing R ═ R1D, threshold R*Is such that Ψ*(R*) A value of 1.
Further, the air conditioner is provided with a fan,
is expressed by a piecewise function given by equation (14):
wherein,
ΨSis the surface potential of the capillary. Threshold value R of dimensionless radius*This can be solved by the additional boundary conditions of equation (15) and equation (16).
(16) In the formula, the upper corner mark "'" represents derivation;
by substituting expression (13) of the potential for expression (9), an expression of charge distribution in the case of a thick electric double layer can be obtained:
calculating the dynamic flow current I in the capillary from the charge distributionsp_thick(r) dynamic flow current I corresponding to the high potential region and the low potential region is requiredsp_L(r) and Isp_H(r) separately calculating:
for equation (20), when C < 0, it can be further simplified as:
the above results in a calculation method for the dynamic streaming potential and streaming current in a single capillary, which is applicable to any thickness of the electric double layer structure. The flow potential of the porous medium converter as a whole based on the capillary bundle model is then calculated.
Firstly, the flowing current I of the whole converter is calculated by utilizing a capillary tube bundle model with equivalent radiusspAs shown in equation (22).
Isp＝NcIsp(rc)(22)
Wherein, Isp(rc) The flowing current represented by formula (12) in the case of a thin electric double layer has a radius rcA value of (d); or the radius r of the flowing current represented by the formula (18) in the case of a thick electric double layercA value of (d);
as shown in FIG. 4, assume that a conduction current IcIn balance with the flowing current, the total conductance of the converter can be expressed as Σ, and the flowing potential E across the converter according to ohm's lawspThe calculation formula of (2) is as follows:
streaming potential coupling coefficient CspThe method is commonly used for describing the flowing potential characteristic of the porous medium, and the expression is as follows:
total conductance Σ from the surface conductance Σ of the electric double layerSAnd solution conductivity σ0And (3) calculating:
and substituting the calculated flow current expression into the flow potential coupling coefficient of the porous medium converter based on the capillary bundle model, wherein the flow potential coupling coefficient is as follows:
in the above, we have established a dynamic streaming potential model C of the porous medium converter suitable for any thickness of the electric double layerspThe model describes the relationship between the pressure and the potential of the converter under the dynamic condition, and the model comprises a plurality of parameters such as the frequency of a dynamic flow field, the characteristic parameter of liquid, the equivalent radius of a capillary tube and the like, so that the influence of the parameter change on the dynamic flow potential coupling coefficient of the converter can be analyzed by using the model.
Three porous medium converters B1, B2 and B3 with different structural parameters are selected, and the equivalent average radius r of capillaries of the porous medium converterscThe ratios to the double layer thickness d are 251.7, 348.9 and 506.4, respectively, and are therefore the case for thin double layers. Using the above-mentioned established model to make three kinds of frequency responsesThe streaming potential coupling coefficient of the converter was calculated and the theoretical result of the calculation is shown in fig. 5. It can be seen that as the capillary radius increases, the dynamic streaming potential coupling coefficient increases, but its bandwidth decreases.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (1)
1. A porous medium converter dynamic flow potential modeling method is characterized by comprising the following steps:
step 1, carrying out equivalence on the porous medium converter by adopting a bundle of bent round capillary tubes, namely a capillary tube bundle model of the porous medium converter, specifically comprising the following steps:
calculating the radius distribution rule of a non-uniformly distributed capillary bundle model according to the particle size distribution of the microbeads of the porous medium converter, and then calculating the equivalent average radius r of uniformly distributed capillariescThus, the capillary bundle model is simplified to a bundle of bent capillaries with the same radius, and the calculation formula of the equivalent radius is as follows:
wherein r represents the radius of a capillary, n (r) represents the density distribution function of the radius of the capillary, rminAnd rmaxRepresents the maximum and minimum values of the capillary radius, respectively, E (·) represents the mathematical expectation; meanwhile, according to the porosity formula (2) of the capillary bundle model, the equivalent average capillary number N is obtained through derivationcAs shown in formula (3):
wherein, V and VfRespectively representing the volume of the transducer and the volume of the fluid in the bundle of capillaries, LcRepresents the length of the equivalent capillary, L represents the thickness of the transducer, a represents the cross-sectional area of the transducer;
step 2, judging whether the single capillary is a thin electric double layer or a thick electric double layer:
taking into account the effect of the electric double layer thickness on the charge distribution in the capillary, the ratio a of the capillary radius r to the electric double layer thickness d is used as an evaluation criterion:
if a is more than 50, the electric double layer structure in the porous medium converter is considered as a thin electric double layer; if a is less than or equal to 50, the double electric layer structure is considered as a thick double electric layer;
the calculation formula of the double-electrode thickness d is given by equation (5):
wherein e is0Denotes the basic electric quantity,. epsilon.denotes the dielectric constant, kBIs the Boltzmann constant, T and n0Respectively representing temperature and ion number, v represents ion valence of the solution, and the ion number of the solution can be determined by the solution concentration CfAnd the Avgalois constant NAPerform a calculation of n0＝1000NACf；
Step 3, analyzing the flowing current and the flowing potential inside the capillary bundle under the dynamic condition:
firstly, according to Navier-Stokes formula (6) and an applied electric field E ═ 0, the distribution u (r) of the fluid flow speed u in the capillary in the radial direction is solved and obtained1) As shown in formula (7):
wherein, K2＝iωρ0Mu, i denotes the unit of imaginary number, omega denotes the angular velocity, p0And μ denotes the solution density and solution viscosity, Ik(. -) represents a first class of modified Bessel function of order k, I0(. cndot.) represents a first class modified Bessel function of order 0, z represents the capillary axial coordinate, P represents the dynamic pressure differential, r1As radial coordinate in the capillary, peRepresents the net charge density;
considering the diffusion distribution of the electric double layer, the relationship of the charge to the potential is shown by the formulas (8) and (9):
where ρ ise(r1) And Ψ (r)1) Respectively represent r1The charge density and potential of (d);
and 4, calculating the potential distribution and the flowing current in the capillary of the thin electric double layer:
the thickness of the electric double layer is far smaller than the radius of the capillary channel, neglected, and then the electric double layer charges are considered to be mainly concentrated in the fluid close to the wall surface, and the charge and potential distribution can be respectively expressed as formula (10) and formula (11):
where ζ represents the zeta potential of the surface of the solid-liquid phase, the independent variable r1Is a radial coordinate;
calculating the flowing current I in the capillary according to the charge distributionsp_thin(r), defined by the current:
and 5, calculating the potential distribution and the flowing current in the capillary of the thick electric double layer:
for the thick electric double layer structure, it is divided into two regions, i.e., a high potential region and a low potential region, and its potential distribution is expressed as a piecewise function shown in formula (13):
wherein, the radial phase coordinate is subjected to non-dimensionalization processing R ═ R1D, threshold R*Is such that Ψ*(R*) A value of 1;
wherein,
ΨSthreshold value R of dimensionless radius for surface potential of capillary*Can be solved by the additional boundary conditions of equation (15) and equation (16);
(16) in the formula, the upper corner mark "'" represents derivation;
substituting expression (13) of the potential for expression (9) gives an expression of charge distribution in the case of a thick electric double layer:
calculating the dynamic flow current I in the capillary from the charge distributionsp_thick(r) dynamic flow current I corresponding to both high potential region and low potential regionsp_L(r) and Isp_H(r) separately calculating:
for formula (20), when C < 0, expressed as formula (21):
step 6, calculating the whole flowing potential of the porous medium converter based on the capillary bundle model:
firstly, the flowing current I of the whole converter is calculated by utilizing a capillary tube bundle model with equivalent radiusspAs shown in formula (22);
Isp＝NcIsp(rc) (22)
wherein, Isp(rc) In the case of a thin electric double layer, the formula (12) is expressed at a radius rcThe value of the flow current in the capillary of (2); or in the case of a thick electric double layer, the radius r is expressed by the formula (18)cThe value of the flow current in the capillary of (2);
assuming a conduction current IcIn balance with the flowing current, the total conductance of the converter is expressed as Σ, and the flowing potential E across the converter is then according to ohm's lawspThe calculation formula of (2) is as follows:
coupling coefficient C by streaming potentialspFlow potential model characterizing porous media:
wherein, the conductance sigma is from the surface conductance sigma of the dual-electrode layerSAnd solution conductivity σ0And (3) calculating:
Σ＝(E(r2)σ0+2·E(r)Σs)/rc 2＝(rcσ0+2·Σs)/rc(25)
and substituting the calculated flowing current expression into the flowing potential coupling coefficient of the porous medium converter under the thin double-electric-layer and thick double-electric-layer capillary bundle models to obtain the flowing potential models of the porous medium as shown in the formulas (26) and (27).
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