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

A Modeling Method for Dynamic Flow Potential in Porous Media Converters Download PDF

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CN111382498B
CN111382498B CN202010045217.XA CN202010045217A CN111382498B CN 111382498 B CN111382498 B CN 111382498B CN 202010045217 A CN202010045217 A CN 202010045217A CN 111382498 B CN111382498 B CN 111382498B
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王美玲
王思劢
明丽
宁可
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Beijing Institute of Technology BIT
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Abstract

本发明公开了一种多孔介质转换器动态流动电势的建模方法,在建立多孔介质转换器的动态流动电势模型时,考虑了双电层厚度对电流和电势分布的影响,由于溶液的种类和浓度是影响双电层厚度的重要因素,因此该模型能够适用于不同溶液性质下的分析;本文建立了动态流动电势模型,这使得可以利用该模型分析动态特性,例如不同频率输入时,流动电势耦合系数的变化,这对于使用该多孔介质转换器作为重要敏感元件的传感器性能分析具有重要意义;该模型涉及到多孔介质转换器的多个参数,例如毛细管等效平均半径,因此可以利用该模型分析参数改变对多孔介质转换器流动电势耦合系数的影响。

Figure 202010045217

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.

Figure 202010045217

Description

一种多孔介质转换器动态流动电势的建模方法A Modeling Method for Dynamic Flow Potential in Porous Media Converters

技术领域technical field

本发明涉及到多孔介质技术领域,尤其涉及到一种多孔介质转换器动态流动电势的建模方法。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.

背景技术Background technique

多孔介质由固体微珠烧结而成,内部结构包括微珠骨架部分与微珠之间的空隙部分,具有空隙尺寸微小、比表面积大的特点,流体在其内部以渗流的方式进行运动。根据界面双电层效应,当固、液表面接触时,固液两相之间会形成电量相等、电性相反的电荷,当流体流过时,孔内由于界面双电层效应而产生的电荷随着流体运动会形成流动电势和流动电流,该过程称为动电效应,因此多孔介质材料可以作为转换器将流体的压力变化转化为电信号,并利用流动电流、流动电势以及流动电势耦合系数进行定量分析。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.

目前针对多孔介质转换器中微流体与动电效应的研究集中在稳态模型的研究,然而,对于动态流场下动电效应的分析较少。且目前的研究大都基于薄双电层假设,即假设双电层厚度远小于多孔介质孔径,然而,当溶液特性与多孔介质特征参数改变时,双电层厚度可能会增大,此时薄双电层假设不再适用,需要考虑双电层的厚度对电荷分布的影响。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.

发明内容SUMMARY OF THE INVENTION

有鉴于此,本发明的目的是提供一种多孔介质转换器动态流动电势的建模方法,能够适用于不同双电层厚度下的分析。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.

一种多孔介质转换器动态流动电势建模方法,包括如下步骤:A method for modeling dynamic flow potential of a porous medium converter, comprising the following steps:

步骤1、采用一束弯曲的圆形毛细管对多孔介质转换器进行等效,即多孔介质转换器的毛管束模型,具体为: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:

根据多孔介质转换器的微珠粒径分布计算出非均匀分布的毛管束模型的半径分布规律,然后计算均匀分布的毛细管等效平均半径rc,从而将毛管束模型简化为一束弯曲的、具有相同半径的毛细管,等效半径的计算公式如下: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:

Figure GDA0003340409320000011
Figure GDA0003340409320000011

其中,r表示一根毛细管的半径,n(r)表示毛细管半径的密度分布函数,rmin和rmax分别表示毛细管半径的最大值与最小值,E(·)表述数学期望;同时,根据毛管束模型的孔隙率公式(2),推导得到等效平均毛管数Nc,如式(3)所示: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):

Figure GDA0003340409320000021
Figure GDA0003340409320000021

Figure GDA0003340409320000022
Figure GDA0003340409320000022

其中,V和Vf分别表示转换器的体积和毛管束内流体的体积,Lc表示等效毛细管的长度,L表示转换器的厚度,A表示转换器的截面面积;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;

步骤2、判断单一毛细管为薄双电层还是厚双电层:Step 2. Determine whether a single capillary is a thin electric double layer or a thick electric double layer:

考虑到双电层厚度对毛细管内电荷分布的影响,采用毛细管半径r和双电层厚度d的比值a作为评价标准: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:

Figure GDA0003340409320000023
Figure GDA0003340409320000023

若a>50,则认为该多孔介质转换器内双电层结构为薄双电层;若a≤50,则认为双电层结构为厚双电层;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;

双电层厚度d的计算公式由式(5)给出:The formula for calculating the thickness d of the electric double layer is given by equation (5):

Figure GDA0003340409320000024
Figure GDA0003340409320000024

其中,e0表示基本电量,ε表示介电常数,kB为玻尔兹曼常数,T和n0分别表示温度和离子数,v表示溶液的离子价,溶液的离子数可以利用溶液浓度Cf和阿伏伽德罗常数NA进行计算,即n0=1000NACfAmong 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 ;

步骤3、在动态情况下,对毛管束内部流动电流和流动电势进行分析:Step 3. Under dynamic conditions, analyze the flowing current and flowing potential inside the capillary bundle:

首先根据Navier-Stokes式(6)以及外加电场E=0,求解得到毛细管中的流体流速u在径向上的分布情况u(r1),如式(7)所示: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):

Figure GDA0003340409320000031
Figure GDA0003340409320000031

Figure GDA0003340409320000037
Figure GDA0003340409320000037

其中,K2=iωρ0/μ,i表示虚数单位,ω表示角速度,ρ0和μ分别表示溶液密度和溶液粘度,Ik(·)表示k阶第一类修正贝塞尔函数,I0(·)表示0阶第一类修正贝塞尔函数,z表示毛细管轴向坐标,P表示动态压力差,r1为毛细管内的径向坐标,ρe表示净电荷密度;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;

考虑双电层的扩散分布,电荷与电势的关系如式(8)和式(9)所示:Considering the diffusion distribution of the electric double layer, the relationship between charge and potential is shown in equations (8) and (9):

Figure GDA0003340409320000032
Figure GDA0003340409320000032

Figure GDA0003340409320000033
Figure GDA0003340409320000033

其中,ρe(r1)和Ψ(r1)分别表示r1处的电荷密度和电势;where ρ e (r 1 ) and Ψ(r 1 ) represent the charge density and potential at r 1 , respectively;

步骤4、计算薄双电层的毛细管内的电势分布和流动电流:Step 4. Calculate the potential distribution and flowing current in the capillary of the thin electric double layer:

双电层的厚度远远小于毛细管道半径,忽略不计,此时认为双电层电荷主要集中在靠近壁面的流体中,其电荷与电势分布可以分别表示为式(10)与式(11):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:

Figure GDA0003340409320000034
Figure GDA0003340409320000034

Figure GDA0003340409320000035
Figure GDA0003340409320000035

其中ζ表示固液相表面的zeta电势,自变量r1为径向坐标;where ζ represents the zeta potential of the solid-liquid surface, and the independent variable r 1 is the radial coordinate;

根据电荷分布计算毛细管内的流动电流Isp_thin(r),由电流的定义得:Calculate the flow current I sp_thin (r) in the capillary from the charge distribution, which is defined by the current:

Figure GDA0003340409320000036
Figure GDA0003340409320000036

步骤5、计算厚双电层的毛细管内的电势分布和流动电流:Step 5. Calculate the potential distribution and flowing current in the capillary of the thick electric double layer:

对于厚双电层结构,将其分为两个区域即高电势区和低电势区,其电势分布表示为式(13)所示的分段函数: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):

Figure GDA0003340409320000041
Figure GDA0003340409320000041

其中,对径相坐标进行无量纲化处理R=r1/d,阈值R*是使得Ψ*(R*)=1的值;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;

Figure GDA0003340409320000042
的表达式用一个分段函数进行表示,由式(14)给出:
Figure GDA0003340409320000042
The expression of is represented by a piecewise function, which is given by Eq. (14):

Figure GDA0003340409320000043
Figure GDA0003340409320000043

其中,

Figure GDA0003340409320000044
ΨS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,
Figure GDA0003340409320000044
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);

Figure GDA0003340409320000045
Figure GDA0003340409320000045

Figure GDA0003340409320000046
Figure GDA0003340409320000046

(16)式中,上角标“′”表示求导;In formula (16), the superscript "'" indicates the derivation;

将电势的表达式(13)代入式(9),得到厚双电层情况下电荷分布表达式: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:

Figure GDA0003340409320000051
Figure GDA0003340409320000051

根据电荷分布计算毛细管内的动态流动电流Isp_thick(r),对高电势区和低电势区两部分对应的动态流动电流Isp_L(r)和Isp_H(r)分别进行计算: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 :

Figure GDA0003340409320000052
Figure GDA0003340409320000052

Figure GDA0003340409320000053
Figure GDA0003340409320000053

Figure GDA0003340409320000054
Figure GDA0003340409320000054

对于式(20),当C<0时,表示为式(21):For formula (20), when C<0, it is expressed as formula (21):

Figure GDA0003340409320000055
Figure GDA0003340409320000055

步骤6、计算基于毛管束模型的多孔介质转换器整体的流动电势:Step 6. Calculate the overall flow potential of the porous medium converter based on the capillary bundle model:

首先,利用等效半径的毛细管束模型,计算转换器整体的流动电流Isp,如式(22)所示: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):

Isp=NcIsp(rc) (22)I sp =N c I sp (rc ) (22)

其中,Isp(rc)在薄双电层情况下公式(12)表示在半径为rc的毛细管中流动电流值;或者为在厚双电层情况下公式(18)表示半径为rc的毛细管中流动电流值;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;

假设传导电流Ic与流动电流相互平衡,转换器的总电导表示为∑,则根据欧姆定律,转换器两端的流动电势Esp的计算公式为: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:

Figure GDA0003340409320000056
Figure GDA0003340409320000056

用流动电势耦合系数Csp表征多孔介质的流动电势模型:The streaming potential model of porous media is characterized by the streaming potential coupling coefficient Csp :

Figure GDA0003340409320000061
Figure GDA0003340409320000061

其中,电导∑由双电层表面电导∑S和溶液电导率σ0计算:where the conductance Σ is calculated from the double layer surface conductance Σ S and the solution conductivity σ 0 :

∑=(E(r20+2·E(r)∑s)/rc 2=(rcσ0+2·∑s)/rc (25)∑=(E( r 20 +2·E( r )∑ s )/rc 2 =(rc σ 0 +2·∑ s )/ rc (25)

将上述计算得到的流动电流表达式代入,薄双电层和厚双电层毛管束模型下多孔介质转换器的流动电势耦合系数,即得到多孔介质的流动电势模型如式(26)及式(27)所示。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.

Figure GDA0003340409320000062
Figure GDA0003340409320000062

Figure GDA0003340409320000063
Figure GDA0003340409320000063

本发明具有如下有益效果:The present invention has the following beneficial effects:

本发明在建立多孔介质转换器的动态流动电势模型时,考虑了双电层厚度对电流和电势分布的影响。由于溶液的种类和浓度是影响双电层厚度的重要因素,因此该模型能够适用于不同溶液性质下的分析。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.

本文建立了动态流动电势模型,这使得可以利用该模型分析动态特性,例如不同频率输入时,流动电势耦合系数的变化。这对于使用该多孔介质转换器作为重要敏感元件的传感器性能分析具有重要意义。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.

该模型涉及到多孔介质转换器的多个参数,例如毛细管等效平均半径,因此可以利用该模型分析参数改变对多孔介质转换器流动电势耦合系数的影响。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.

附图说明Description of drawings

图1为多孔介质转换器的毛管束等效模型。Figure 1 shows the equivalent model of the capillary bundle of the porous medium converter.

图2为薄双电层和厚双电层的示意图。Figure 2 is a schematic diagram of a thin electric double layer and a thick electric double layer.

图3为界面双电层的结构图。FIG. 3 is a structural diagram of an interface electric double layer.

图4为毛细管中的动电效应原理示意图。FIG. 4 is a schematic diagram of the principle of the electrokinetic effect in the capillary.

图5为转换器结构参数改变对动态流动电势耦合系数的影响曲线。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.

具体实施方式Detailed ways

下面结合附图并举实施例,对本发明进行详细描述。The present invention will be described in detail below with reference to the accompanying drawings and embodiments.

本发明的一种多孔介质转换器动态流动电势建模方法,采用了毛管束模型对多孔介质转换器进行等效,如图1所示。多孔介质材料内部包含有骨架部分和空隙部分,结构复杂,难以直接地对内部动态流动电势进行定量分析,因此采用一束弯曲的圆形毛细管对其进行等效,即多孔介质转换器的毛管束模型,这些毛细管的半径服从一定的分布规律。进一步地,根据多孔介质转换器的微珠粒径分布计算出非均匀分布的其半径分布规律,然后计算均匀分布的毛细管等效平均半径rc,从而将毛管束模型简化为一束弯曲的具有相同半径的毛细管,等效半径的计算公式如下: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:

Figure GDA0003340409320000071
Figure GDA0003340409320000071

其中,r表示一根毛细管的半径,n(r)表示毛细管半径的密度分布函数,rmin和rmax分别表示毛细管半径的最大值与最小值,E(·)表述数学期望。同时,根据毛管束模型的孔隙率公式(式(2)),可以推导得到等效平均毛管数Nc,如式(3)所示: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):

Figure GDA0003340409320000072
Figure GDA0003340409320000072

Figure GDA0003340409320000073
Figure GDA0003340409320000073

其中,V和Vf分别表示转换器的体积和毛管束内流体的体积,Lc表示等效毛细管的长度,L表示转换器的厚度,A表示转换器的截面面积。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.

下面讨论单一毛细管内的动态流动电流和电势计算。The dynamic flow current and potential calculations within a single capillary are discussed below.

考虑到双电层厚度对毛细管内电荷分布的影响,采用毛细管半径r和双电层厚度d的比值a作为评价标准,若a>50,则认为该多孔介质转换器内双电层结构为薄双电层;若a≤50,则认为双电层结构为厚双电层。双电层的结构示意图如图2所示。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.

Figure GDA0003340409320000081
Figure GDA0003340409320000081

双电层厚度d的计算公式由式(5)给出:The formula for calculating the thickness d of the electric double layer is given by equation (5):

Figure GDA0003340409320000082
Figure GDA0003340409320000082

其中,e0表示基本电量,ε表示介电常数,kB为玻尔兹曼常数,T和n0分别表示温度和离子数,v表示溶液的离子价。溶液的离子数可以利用溶液浓度Cf和阿伏伽德罗常数NA进行计算,即n0=1000NACfAmong 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 .

为了对动态情况下,毛管束内部流动电流和流动电势进行分析,首先根据Navier-Stokes公式(式(6))以及外加电场E=0求解得到毛细管中的流体流速u在径向上的分布情况u(r1),如式(7)所示。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).

Figure GDA0003340409320000083
Figure GDA0003340409320000083

Figure GDA0003340409320000084
Figure GDA0003340409320000084

其中,K2=iωρ0/μ,i表示虚数单位,ω表示角速度,ρ0和μ分别表示溶液密度和溶液粘度,Ik(·)表示k阶第一类修正贝塞尔函数,I0(·)表示0阶第一类修正贝塞尔函数;z表示毛细管轴向坐标,P表示动态压力差,r1为毛细管内的径向坐标,ρe表示净电荷密度。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.

以上就得到了毛细管内流体流速分布情况u(r1)。如图3所示,考虑双电层的扩散分布,电荷与电势的关系如式(8)和式(9)所示: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):

Figure GDA0003340409320000085
Figure GDA0003340409320000085

Figure GDA0003340409320000086
Figure GDA0003340409320000086

其中,ρe(r1)和Ψ(r1)分别表示r1处的电荷密度和电势。where ρ e (r 1 ) and Ψ(r 1 ) represent the charge density and potential at r 1 , respectively.

对于薄双电层,双电层的厚度远远小于毛细管道半径,可以忽略不计,此时认为双电层电荷主要集中在靠近壁面的流体中,其电荷与电势分布可以分别表示为式(10)与式(11):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):

Figure GDA0003340409320000091
Figure GDA0003340409320000091

Figure GDA0003340409320000092
Figure GDA0003340409320000092

其中ζ表示固液相表面的zeta电势,(zeta电势是固相带电表面与液体之间的电势差,可以反应固相带电的强弱程度,薄双电层情况又称为低zeta电势情况,厚双电层情况又称为高zeta电势情况),自变量r1为径向坐标。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.

根据电荷分布计算毛细管内的流动电流Isp_thin(r),由电流的定义得:Calculate the flow current I sp_thin (r) in the capillary from the charge distribution, which is defined by the current:

Figure GDA0003340409320000093
Figure GDA0003340409320000093

对于厚双电层结构,双电层的厚度会影响毛细管道内的电荷分布,如图2(b)所示,在靠近管壁与远离管壁的区域电势分布有明显差异,将其分为两个区域即高电势区和低电势区,其电势分布可以表示为式(13)所示的分段函数: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):

Figure GDA0003340409320000094
Figure GDA0003340409320000094

这里为了便于分析,对径向坐标进行无量纲化处理R=r1/d,阈值R*是使得Ψ*(R*)=1的值。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.

进一步地,

Figure GDA0003340409320000095
的表达式用一个分段函数进行表示,由式(14)给出:further,
Figure GDA0003340409320000095
The expression of is represented by a piecewise function, which is given by Eq. (14):

Figure GDA0003340409320000101
Figure GDA0003340409320000101

其中,

Figure GDA0003340409320000102
ΨS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,
Figure GDA0003340409320000102
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).

Figure GDA0003340409320000103
Figure GDA0003340409320000103

Figure GDA0003340409320000104
Figure GDA0003340409320000104

(16)式中,上角标“′”表示求导;In formula (16), the superscript "'" indicates the derivation;

将电势的表达式(13)代入式(9),可以得到厚双电层情况下电荷分布表达式: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:

Figure GDA0003340409320000105
Figure GDA0003340409320000105

根据电荷分布计算毛细管内的动态流动电流Isp_thick(r),需要对高电势区和低电势区两部分对应的动态流动电流Isp_L(r)和Isp_H(r)分别进行计算: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 :

Figure GDA0003340409320000106
Figure GDA0003340409320000106

Figure GDA0003340409320000107
Figure GDA0003340409320000107

Figure GDA0003340409320000111
Figure GDA0003340409320000111

对于式(20),当C<0时,可以进一步地简化为:For equation (20), when C<0, it can be further simplified as:

Figure GDA0003340409320000112
Figure GDA0003340409320000112

上述就得到了单毛细管中动态流动电势和流动电流的计算方法,该方法针对任意厚度的双电层结构均适用。接下来计算基于毛管束模型的多孔介质转换器整体的流动电势。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.

首先,利用等效半径的毛细管束模型,计算转换器整体的流动电流Isp,如式(22)所示: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):

Isp=NcIsp(rc) (22)I sp =N c I sp (rc ) (22)

其中,Isp(rc)在薄双电层情况下公式(12)表示的流动电流在半径为rc的值;或者为在厚双电层情况下公式(18)表示的流动电流在半径为rc的值;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 ;

如图4所示,假设传导电流Ic与流动电流相互平衡,转换器的总电导可表示为∑,则根据欧姆定律,转换器两端的流动电势Esp的计算公式为: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:

Figure GDA0003340409320000113
Figure GDA0003340409320000113

流动电势耦合系数Csp常用来刻画多孔介质的流动电势特性,表达式为:The flow potential coupling coefficient C sp is often used to describe the flow potential characteristics of porous media, and the expression is:

Figure GDA0003340409320000114
Figure GDA0003340409320000114

总电导∑由双电层表面电导∑S和溶液电导率σ0计算:The total conductance Σ is calculated from the electric double layer surface conductance ΣS and the solution conductivity σ0 :

∑=(E(r20+2·E(r)∑s)/rc 2=(rcσ0+2·∑s)/rc (25)∑=(E( r 20 +2·E( r )∑ s )/rc 2 =(rc σ 0 +2·∑ s )/ rc (25)

将上述计算得到的流动电流表达式代入,可以得到基于毛管束模型的多孔介质转换器的流动电势耦合系数: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:

Figure GDA0003340409320000115
Figure GDA0003340409320000115

Figure GDA0003340409320000121
Figure GDA0003340409320000121

以上,我们建立了对于任意厚度双电层都适用的多孔介质转换器动态流动电势模型Csp,该模型描述了动态情况下转换器受到的压力与其电势的关系,该模型包含了动态流场的频率、液体的特性参数、毛细管的等效半径等多个参数,因此可以利用该模型分析上述参数变化对转换器动态流动电势耦合系数的影响。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.

选取三种结构参数不同的多孔介质转换器B1、B2和B3,他们的毛细管等效平均半径rc与双电层厚度d的比值分别为251.7,348.9和506.4,因此均为薄双电层的情况。利用上述建立的模型,对不同频率下三种转换器的流动电势耦合系数进行计算,计算的理论结果如图5所示。可以看出,毛细管半径增大时,动态流动电势耦合系数会增大,但其带宽会减小。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.

综上所述,以上仅为本发明的较佳实施例而已,并非用于限定本发明的保护范围。凡在本发明的精神和原则之内,所作的任何修改、等同替换、改进等,均应包含在本发明的保护范围之内。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)

1.一种多孔介质转换器动态流动电势建模方法,其特征在于,包括如下步骤:1. a porous medium converter dynamic flow potential modeling method, is characterized in that, comprises the steps: 步骤1、采用一束弯曲的圆形毛细管对多孔介质转换器进行等效,即多孔介质转换器的毛管束模型,具体为: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: 根据多孔介质转换器的微珠粒径分布计算出非均匀分布的毛管束模型的半径分布规律,然后计算均匀分布的毛细管等效平均半径rc,从而将毛管束模型简化为一束弯曲的、具有相同半径的毛细管,等效半径的计算公式如下: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:
Figure FDA0003382996560000011
Figure FDA0003382996560000011
其中,r表示一根毛细管的半径,n(r)表示毛细管半径的密度分布函数,rmin和rmax分别表示毛细管半径的最大值与最小值,E(·)表述数学期望;同时,根据毛管束模型的孔隙率公式(2),推导得到等效平均毛管数Nc,如式(3)所示: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):
Figure FDA0003382996560000012
Figure FDA0003382996560000012
Figure FDA0003382996560000013
Figure FDA0003382996560000013
其中,V和Vf分别表示转换器的体积和毛管束内流体的体积,Lc表示等效毛细管的长度,L表示转换器的厚度,A表示转换器的截面面积;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; 步骤2、判断单一毛细管为薄双电层还是厚双电层:Step 2. Determine whether a single capillary is a thin electric double layer or a thick electric double layer: 考虑到双电层厚度对毛细管内电荷分布的影响,采用毛细管半径r和双电层厚度d的比值a作为评价标准: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:
Figure FDA0003382996560000014
Figure FDA0003382996560000014
若a>50,则认为该多孔介质转换器内双电层结构为薄双电层;若a≤50,则认为双电层结构为厚双电层;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; 双电层厚度d的计算公式由式(5)给出:The formula for calculating the thickness d of the electric double layer is given by equation (5):
Figure FDA0003382996560000021
Figure FDA0003382996560000021
其中,e0表示基本电量,ε表示介电常数,kB为玻尔兹曼常数,T和n0分别表示温度和离子数,v表示溶液的离子价,溶液的离子数可以利用溶液浓度Cf和阿伏伽德罗常数NA进行计算,即n0=1000NACfAmong 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 ; 步骤3、在动态情况下,对毛管束内部流动电流和流动电势进行分析:Step 3. Under dynamic conditions, analyze the flowing current and flowing potential inside the capillary bundle: 首先根据Navier-Stokes式(6)以及外加电场E=0,求解得到毛细管中的流体流速u在径向上的分布情况u(r1),如式(7)所示: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):
Figure FDA0003382996560000022
Figure FDA0003382996560000022
Figure FDA0003382996560000023
Figure FDA0003382996560000023
其中,K2=iωρ0/μ,i表示虚数单位,ω表示角速度,ρ0和μ分别表示溶液密度和溶液粘度,Ik(·)表示k阶第一类修正贝塞尔函数,I0(·)表示0阶第一类修正贝塞尔函数,z表示毛细管轴向坐标,P表示动态压力差,r1为毛细管内的径向坐标,ρe表示净电荷密度;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; 考虑双电层的扩散分布,电荷与电势的关系如式(8)和式(9)所示:Considering the diffusion distribution of the electric double layer, the relationship between charge and potential is shown in equations (8) and (9):
Figure FDA0003382996560000024
Figure FDA0003382996560000024
Figure FDA0003382996560000025
Figure FDA0003382996560000025
其中,ρe(r1)和Ψ(r1)分别表示r1处的电荷密度和电势;where ρ e (r 1 ) and Ψ(r 1 ) represent the charge density and potential at r 1 , respectively; 步骤4、计算薄双电层的毛细管内的电势分布和流动电流:Step 4. Calculate the potential distribution and flowing current in the capillary of the thin electric double layer: 双电层的厚度远远小于毛细管道半径,忽略不计,此时认为双电层电荷主要集中在靠近壁面的流体中,其电荷与电势分布可以分别表示为式(10)与式(11):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:
Figure FDA0003382996560000026
Figure FDA0003382996560000026
Figure FDA0003382996560000031
Figure FDA0003382996560000031
其中ζ表示固液相表面的zeta电势,自变量r1为径向坐标;where ζ represents the zeta potential of the solid-liquid surface, and the independent variable r 1 is the radial coordinate; 根据电荷分布计算毛细管内的流动电流Isp_thin(r),由电流的定义得:Calculate the flow current I sp_thin (r) in the capillary from the charge distribution, which is defined by the current:
Figure FDA0003382996560000032
Figure FDA0003382996560000032
步骤5、计算厚双电层的毛细管内的电势分布和流动电流:Step 5. Calculate the potential distribution and flowing current in the capillary of the thick electric double layer: 对于厚双电层结构,将其分为两个区域即高电势区和低电势区,其电势分布表示为式(13)所示的分段函数: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):
Figure FDA0003382996560000033
Figure FDA0003382996560000033
其中,对径相坐标进行无量纲化处理R=r1/d,阈值R*是使得Ψ*(R*)=1的值;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;
Figure FDA0003382996560000034
的表达式用一个分段函数进行表示,由式(14)给出:
Figure FDA0003382996560000034
The expression of is represented by a piecewise function, which is given by Eq. (14):
Figure FDA0003382996560000035
Figure FDA0003382996560000035
其中,
Figure FDA0003382996560000036
ΨS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,
Figure FDA0003382996560000036
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);
Figure FDA0003382996560000041
Figure FDA0003382996560000041
Figure FDA0003382996560000042
Figure FDA0003382996560000042
(16)式中,上角标“′”表示求导;In formula (16), the superscript "'" indicates the derivation; 将电势的表达式(13)代入式(9),得到厚双电层情况下电荷分布表达式: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:
Figure FDA0003382996560000043
Figure FDA0003382996560000043
根据电荷分布计算毛细管内的动态流动电流Isp_thick(r),对高电势区和低电势区两部分对应的动态流动电流Isp_L(r)和Isp_H(r)分别进行计算: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 :
Figure FDA0003382996560000044
Figure FDA0003382996560000044
Figure FDA0003382996560000045
Figure FDA0003382996560000045
Figure FDA0003382996560000046
Figure FDA0003382996560000046
对于式(20),当C<0时,表示为式(21):For formula (20), when C<0, it is expressed as formula (21):
Figure FDA0003382996560000047
Figure FDA0003382996560000047
步骤6、计算基于毛管束模型的多孔介质转换器整体的流动电势:Step 6. Calculate the overall flow potential of the porous medium converter based on the capillary bundle model: 首先,利用等效半径的毛细管束模型,计算转换器整体的流动电流Isp,如式(22)所示: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): Isp=NcIsp(rc) (22)I sp =N c I sp (rc ) (22) 其中,Isp(rc)在薄双电层情况下公式(12)表示在半径为rc的毛细管中流动电流值;或者为在厚双电层情况下公式(18)表示半径为rc的毛细管中流动电流值;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; 假设传导电流Ic与流动电流相互平衡,转换器的总电导表示为Σ,则根据欧姆定律,转换器两端的流动电势Esp的计算公式为: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:
Figure FDA0003382996560000051
Figure FDA0003382996560000051
用流动电势耦合系数Csp表征多孔介质的流动电势模型:The streaming potential model of porous media is characterized by the streaming potential coupling coefficient Csp :
Figure FDA0003382996560000052
Figure FDA0003382996560000052
其中,电导Σ由双电层表面电导ΣS和溶液电导率σ0计算:where the conductance Σ is calculated from the double layer surface conductance ΣS and the solution conductivity σ0 : Σ=(E(r20+2·E(r)Σs)/rc 2=(rcσ0+2·Σs)/rc (25)Σ=(E( r 20 +2·E( rs )/rc 2 =(rc σ 0 +2·Σ s )/ rc (25) 将上述计算得到的流动电流表达式代入,薄双电层和厚双电层毛管束模型下多孔介质转换器的流动电势耦合系数,即得到多孔介质的流动电势模型如式(26)及式(27)所示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
Figure FDA0003382996560000053
Figure FDA0003382996560000053
Figure FDA0003382996560000054
Figure FDA0003382996560000054
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