A method for calculating the amplitude of shock wave of electric water hammer effect taking into account the rise time

Abstract

The present invention discloses a method for calculating the amplitude of shock waves of electric water hammer effect taking into account the rise time. The method applies nanosecond pulse voltage to both ends of the electrode, calculates the ionization generation term at the initial moment based on the field ionization Zener equation; establishes an arc plasma fluid dynamics model, performs iterative operations, and calculates the electric field intensity distribution and particle concentration distribution between electrodes at any time; uses the data after the model iterative operation to calculate the instantaneous current, instantaneous power, and deposited energy of the plasma arc channel; and realizes the calculation of the amplitude of the shock wave generated in the main discharge stage of the streamer. Through the present invention, modeling and analysis of nanosecond pulse discharge in water can be achieved, which can not only reflect the starting and development characteristics of arc plasma, but also quantitatively calculate the current, instantaneous power, deposited energy and shock wave amplitude after breakdown. Deepening the understanding of the discharge process and shock wave transmission process of nanosecond pulse discharge in water has guiding significance for revealing the connection between the two processes.

Description

Technical Field

[0001] The invention relates to the technical field of liquid dielectric discharge monitoring under high electric field, and in particular to a method for calculating the amplitude of electric water hammer effect shock wave taking into account rise time. Background Art

[0002] When a high-pressure pulse in water occurs in a liquid, a strong explosion will occur in the liquid, and the impact pressure can reach 10 2 -10 4 Mpa, this is the so-called electric water hammer effect. The phenomenon of discharge between electrodes in water in a very short time to form a plasma channel, and the discharge process is divided into two stages, namely: pre-breakdown stage and main discharge stage. This effect usually produces a strong shock wave in the main discharge stage. The mechanical effect produced by the shock wave is widely used in rock fracturing, resource mining, in vitro rock crushing and other fields. The intensity of the shock wave is closely related to the development process of the streamer in the pre-breakdown stage.

[0003] However, most scholars currently use high-speed photography, voltage probes, current probes, shock wave probes, oscilloscopes and other instruments to conduct research, which can only describe the characteristics of the plasma formation stage and the shock wave transmission stage from a macroscopic perspective, but cannot provide an essential explanation from the discharge mechanism, that is, the formation and development of microscopic particles until breakdown and then the formation of shock waves.

[0004] Therefore, it is necessary to use simulation methods to model and analyze nanosecond underwater discharges, which can not only reflect the initiation and development characteristics of arc plasma, but also quantitatively calculate the current, instantaneous power, deposition energy and shock wave amplitude after breakdown. Deepening the understanding of the discharge process and shock wave transmission process of underwater discharge is of guiding significance for revealing the connection between the two processes. Summary of the invention

[0005] The purpose of the present invention is to provide a method for calculating the shock wave amplitude of the electric water hammer effect taking into account the rise time, establish a physical model for the plasma arc discharge process in water, describe the field ionization, electric field distortion, particle diffusion, particle adsorption, and particle recombination processes, quantitatively calculate the plasma current, power, energy, and shock wave amplitude, deepen the understanding of the discharge process and shock wave transmission process of nanosecond pulse discharge in water, and have guiding significance for revealing the connection between the two processes.

[0006] The implementation scheme of the present invention is as follows:

[0007] S1, applying nanosecond pulse voltage to both ends of the electrode, and calculating the ionization generation term at the initial moment based on field ionization;

[0008] S2, establish the arc plasma fluid dynamics model, perform iterative operations, and calculate the electric field intensity distribution and particle concentration distribution between electrodes at any time;

[0009] S3, using the data after model iteration operation, calculate the instantaneous current, instantaneous power and deposition energy of the plasma arc channel;

[0010] S4, realizes the calculation of the amplitude of the shock wave generated during the main discharge stage of the streamer.

[0011] Optionally, in the step S1, a nanosecond pulse voltage is applied to both ends of the electrode, and an ionization generation term at the initial moment is calculated based on field ionization, specifically including:

[0012] The nanosecond pulse voltage can use the double exponential pulse lightning wave equation, and the voltage equation, the empirical formula of the field strength at the tip of the needle electrode and the field ionization calculation equation can be used to calculate the ionization generation term at the initial moment. The formulas are as follows:

[0013] u(t)=U c (e -αt -e -βt )

[0014]

[0015]

[0016] Among them, U c is the amplitude of the pulse voltage, α is the wavefront coefficient, β is the wavetail coefficient. r1 is the tip radius of the needle electrode, L is the electrode spacing, q is the electron charge, n0 is the number of ionizable water molecules per unit volume, h is Planck's constant, a is the molecular spacing, m * is the effective electron mass, and Δ is the ionization energy of water.

[0017] Optionally, in step S2, an arc plasma fluid dynamics model is established, and iterative operations are performed to calculate the electric field intensity distribution and particle concentration distribution between electrodes at any time, specifically including:

[0018]

[0019]

[0020]

[0021]

[0022] Among them, ε r represents the relative dielectric constant of liquid water medium, ε0 represents the dielectric constant of vacuum, n e The three symbols represent H + OH - , the charge density of e, D e are the diffusion rates of the three particles, μ e They represent the mobility of the three particles, τ is the electron-water molecule adsorption time, Respectively represent H + The recombination rate of and e, H + and OH - The compound rate;

[0023] Based on a large amount of experimental data fitting, it is found that μ e The empirical formula for calculation is as follows:

[0024]

[0025]

[0026] The specific method for calculating τ is as follows:

[0027]

[0028]

[0029]

[0030] Among them, k t represents the heat transfer coefficient of water, λ e represents the electron decay length, t1 represents the rising edge time of the applied voltage, t2 represents the falling edge time of the applied voltage, C v is the specific heat capacity of water.

[0031] Combine the continuity equations of hydrogen ions, hydroxide ions and electrons to describe the generation, recombination and adsorption mechanisms of the three particles during the flow process;

[0032] The drift and diffusion movement of charged particles will change the original electric field distribution. The Poisson equation is used to express the coupling relationship between the electric field and the space charge, which can specifically express the interaction between the space charges and the distortion effect of the space charge on the electric field. The above formulas are coupled and calculated to obtain the updated electric field intensity distribution and particle concentration distribution.

[0033] Optionally, in step S3, the instantaneous current, instantaneous power and deposition energy of the plasma arc channel are calculated using the data after the model iteration operation, which specifically includes:

[0034]

[0035] P(t)=u(t)I(t)

[0036] Epl =∫p(t)dt

[0037] Among them, the streamer radius r2 after the plasma arc breakdown can be obtained from the radial particle concentration distribution, N A is Avogadro's constant;

[0038] The arc channel deposition energy can be obtained from the above arc channel current equation and arc channel instantaneous power equation;

[0039] Optionally, the step S4 is to calculate the amplitude of the shock wave generated during the main discharge phase of the streamer, as follows;

[0040] The relationship between deposition energy and shock wave amplitude was obtained by fitting the experimental data of hydroelectric effect in water, as shown in the following formula:

[0041] P=k×E pl 0.54

[0042]

[0043] Where L is the distance between electrodes. k is the streamer polarity coefficient in water, which is related to the streamer polarity.

[0044] According to the above equation, the shock wave amplitude can be derived from the deposited energy.

[0045] The beneficial effect of the present invention is that through the present invention, modeling and analysis of nanosecond pulse discharge in water can be realized, which can not only reflect the initiation and development characteristics of arc plasma, but also quantitatively calculate the current, instantaneous power, deposition energy and shock wave amplitude after breakdown. It can deepen the understanding of the discharge process and shock wave transmission process of nanosecond pulse discharge in water, and has guiding significance for revealing the connection between the two processes. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Attached Figure 1 The present invention is a flow chart of a method for calculating the amplitude of an electric water hammer effect shock wave taking into account the rise time.

[0047] Attached Figure 2 2 is a waveform diagram of an externally applied voltage in an embodiment of the present invention.

[0048] Attached Figure 3 This is the underwater needle-plate electrode model in the embodiment of the present invention.

[0049] Attached Figure 4 Graph showing the temporal and spatial distribution of the electric field intensity in an embodiment of the present invention.

[0050] Attached Figure 5 This is a morphological diagram of the water flow breakdown in an embodiment of the present invention. DETAILED DESCRIPTION

[0051] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0052] like Figure 1 As shown, the present invention is used for a method for calculating the amplitude of an electric water hammer effect shock wave taking into account the rise time, comprising the following steps:

[0053] S1, applying nanosecond pulse voltage at both ends of the electrode, and calculating the ionization generation term at the initial moment based on field ionization;

[0054] S2, establish the arc plasma fluid dynamics mathematical model, perform iterative operations, and calculate the electric field distribution and particle concentration distribution between electrodes at any time;

[0055] S3, using the data after model iteration operation, calculate the instantaneous current, instantaneous power and deposition energy of the plasma arc channel;

[0056] S4, realizes the calculation of the amplitude of the shock wave generated by the main discharge process.

[0057] Wherein, in the step S1, a nanosecond pulse voltage is applied to both ends of the electrode, and the ionization generation term at the initial moment is calculated based on the field ionization, specifically including:

[0058] The nanosecond pulse voltage can use the double exponential pulse lightning wave equation, and the voltage equation, the empirical formula of the field strength at the tip of the needle electrode, and the field ionization calculation equation can be used to calculate the ionization generation term at the initial moment. The formulas are as follows:

[0059] u(t)=U c (e -αt -e -βt )

[0060]

[0061]

[0062] Among them, U c is the amplitude of the pulse voltage, α is the wavefront coefficient, β is the wavetail coefficient. r1 is the tip radius of the needle electrode, L is the electrode spacing, q is the electron charge, n0 is the number of ionizable water molecules per unit volume, is the electric field strength, h is Planck's constant, a is the molecular distance, m * is the effective electron mass, and Δ is the ionization energy of water.

[0063] The specific voltage waveform applied in this embodiment is as shown in the attached Figure 2 As shown; the electrode structure is a needle plate electrode, the specific structure is as shown in the attached Figure 3 shown.

[0064] The step S2 also includes: establishing a mathematical model of arc plasma fluid dynamics, performing iterative operations, and calculating the electric field distribution and particle concentration distribution between electrodes at any time, specifically:

[0065]

[0066]

[0067]

[0068]

[0069] Among them, ε r represents the relative dielectric constant of liquid water medium, ε0 represents the dielectric constant of vacuum, n e The three symbols represent H + OH - , the charge density of e, D e are the diffusion rates of the three particles, μ e They represent the mobility of the three particles, τ is the electron-water molecule adsorption time, Respectively represent H + The recombination rate of and e, H + and OH - The compound rate;

[0070] Based on a large amount of experimental data fitting, it is found that μ e The empirical formula for calculation is as follows:

[0071]

[0072]

[0073] The specific method for calculating τ is as follows:

[0074]

[0075]

[0076]

[0077] Among them, k t represents the heat transfer coefficient of water, λ e represents the electron decay length, t1 represents the rising edge time of the applied voltage, t2 represents the falling edge time of the applied voltage, C v is the specific heat capacity of water.

[0078] Combine the continuity equations of hydrogen ions, hydroxide ions and electrons to describe the generation, recombination and adsorption mechanisms of the three particles during the flow process;

[0079] The drift and diffusion movement of charged particles will change the original electric field distribution. The Poisson equation is used to express the coupling relationship between the electric field and the space charge, which can specifically express the interaction between the space charges and the distortion effect of the space charge on the electric field. The above formulas are coupled and calculated to obtain the updated electric field intensity distribution and particle concentration distribution.

[0080] The temporal and spatial distribution of the electric field intensity in the embodiment of the present invention is shown in the attached figure. Figure 4 As shown in the figure, the morphology of the water flow breakdown is as follows Figure 5 shown.

[0081] In step S3, the instantaneous current, instantaneous power and deposition energy of the plasma arc channel are calculated using the data after model iteration operation, which specifically includes:

[0082]

[0083] P(t)=u(t)I(t)

[0084] E pl =∫p(t)dt

[0085] Among them, the streamer radius r2 after the plasma arc breakdown can be obtained from the radial particle concentration distribution, N A is Avogadro's constant;

[0086] The arc channel deposition energy can be obtained from the above arc channel current equation and arc channel instantaneous power equation;

[0087] In the step S4, the amplitude of the shock wave generated during the main discharge process is calculated as follows;

[0088] The relationship between deposition energy and shock wave amplitude was obtained by fitting the experimental data of hydroelectric effect in water, as shown in the following formula:

[0089] P=k×E pl 0.54

[0090]

[0091] Where L is the distance between electrodes. k is the streamer polarity coefficient in water, which is related to the streamer polarity.

[0092] According to the above equation, the shock wave amplitude can be derived from the deposited energy.

[0093] Table 1 Main physical parameters of the control equation

[0094] parameter Symbols and units Numeric Electron charge q(C) <![CDATA[1.602×10 -19 ]]> Number of water molecules per unit volume <![CDATA[n0(m -3 )]]> <![CDATA[3.344×10 28 ]]> Distance between water molecules a(nm) 0.31 Planck constant h(J·s) <![CDATA[6.626×10 -34 ]]> Vacuum dielectric constant <![CDATA[ε0(F / m)]]> <![CDATA[8.85×10 -12 ]]> Electron effective mass <![CDATA[m * (kg)]]> <![CDATA[9.1×10 -32 ]]> Relative dielectric constant of water <![CDATA[ε r ]]> 83.83 Electron decay length <![CDATA[λ e (mm)]]> 1 Water heat transfer coefficient <![CDATA[k t (W / (m 2 ·K))]]> 0.599 Avogadro's constant <![CDATA[N A ]]> <![CDATA[6.02×10 23 ]]> Specific heat of water <h2 style=";text-align:left;direction:ltr"><![CDATA[C <h2 style=";text-align:left;direction:ltr"> v <h2 style=";text-align:left;direction:ltr"> (J / (kg·K))]]><h2 style=";text-align:left;direction:ltr"> <![CDATA[4.2×10 3 ]]>

[0095] The present invention obtains the values ​​of the physical quantity parameters in each control equation by consulting multidisciplinary data, as shown in Table 1. According to the values ​​of each parameter in Table 1, each formula is substituted and the formula is solved and calculated.

[0096] The beneficial effects of the present invention are as follows: through the present invention, modeling and analysis of nanosecond pulse discharge in water can be realized, which can not only deeply understand the initiation and development characteristics of arc plasma, but also quantitatively calculate the current, instantaneous power, deposition energy and shock wave amplitude after breakdown. From a microscopic perspective, the discharge process and shock wave transmission process of nanosecond pulse discharge in water are deeply understood, providing a new perspective for revealing the connection between the two processes.

Claims

1. A method for calculating the amplitude of the shock wave of the electric water hammer effect taking into account the rise time, characterized in that: The following steps are involved: S1, applying nanosecond pulse voltage to both ends of the electrode, and calculating the ionization generation term at the initial moment based on the field ionization Zener equation; S2, establish the arc plasma fluid dynamics model, perform iterative operations, and calculate the electric field intensity distribution and particle concentration distribution between electrodes at any time; S3, using the data after model iteration operation, calculate the instantaneous current, instantaneous power and deposition energy of the plasma arc channel, as follows: P(t)=u(t)I(t) E pl =∫p(t)dt Among them, the streamer radius r2 after the plasma arc breakdown can be obtained from the radial particle concentration distribution, N A is Avogadro's constant, μ e are the mobility of positive ions and electrons, respectively. n e Respectively represent H + , the charge density of e, is the electric field strength, q is the electron charge; The arc channel deposition energy can be obtained from the above arc channel current equation and arc channel instantaneous power equation; S4, realizing the calculation of the amplitude of the shock wave generated during the main discharge stage of the streamer, specifically including: The empirical expression of deposition energy and shock wave amplitude is obtained by fitting a large number of experimental data obtained from the hydroelectric effect, as follows: P=k×E pl 0.54 Where L is the distance between electrodes, k is the streamer polarity coefficient in water, which is related to the streamer polarity, t1 is the rising edge time of the applied voltage, t2 is the falling edge time of the applied voltage, and U c is the amplitude of the pulse voltage, E pl To deposit energy; According to the above equation, the shock wave amplitude can be derived from the deposited energy.

2. The method for calculating the amplitude of the electric water hammer effect shock wave taking into account the rise time according to claim 1, characterized in that: In the step S1, a nanosecond pulse voltage is applied to both ends of the electrode, and the ionization generation term at the initial moment is calculated based on the field ionization Zener equation, which specifically includes: The nanosecond pulse voltage can use the double exponential pulse lightning wave equation, and the voltage equation, the empirical formula of the field strength at the tip of the needle electrode, and the field ionization calculation equation can be used to calculate the ionization generation term at the initial moment. The formulas are as follows: u(t)=U c (e -αt -e -βt ) Among them, U c is the amplitude of the pulse voltage, α is the wavefront coefficient, and β is the wavetail coefficient; is the electric field strength, r1 is the radius of the needle electrode tip, L is the electrode spacing, q is the electron charge, n0 is the number of ionizable water molecules per unit volume, h is Planck's constant, a is the molecular spacing, m * is the effective electron mass, Δ is the ionization energy of water; According to the above formula, the electric field intensity distribution between electrodes at the initial moment and the initial term of field ionization can be calculated to describe the generation of space charge.

3. The method for calculating the amplitude of the electric water hammer effect shock wave taking into account the rise time according to claim 1, characterized in that: In step S2, an arc plasma fluid dynamics model is established, and an iterative operation is performed based on the following equations to calculate the electric field intensity distribution and particle concentration distribution between electrodes at any time, specifically including: Among them, ε r represents the relative dielectric constant of liquid water medium, ε0 represents the dielectric constant of vacuum, n e The three symbols represent H + OH - , the charge density of e, D e are the diffusion rates of the three particles, μ e They represent the mobility of the three particles, τ is the electron-water molecule adsorption time, Respectively represent H + The recombination rate of and e, H + and OH - The compound rate.

4. The method for calculating the amplitude of the electric water hammer effect shock wave taking into account the rise time according to claim 3 is characterized in that: The arc plasma fluid dynamics model is established. μ e , The calculation method of τ is as follows: Among them, k t represents the heat transfer coefficient of water, λ e represents the electron decay length, t1 represents the rising edge time of the applied voltage, t2 represents the falling edge time of the applied voltage, C v Represents the specific heat capacity of water.

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