WO2019091084A1 - 双电极直流电熔镁炉操作电阻计算方法 - Google Patents

双电极直流电熔镁炉操作电阻计算方法 Download PDF

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WO2019091084A1
WO2019091084A1 PCT/CN2018/087694 CN2018087694W WO2019091084A1 WO 2019091084 A1 WO2019091084 A1 WO 2019091084A1 CN 2018087694 W CN2018087694 W CN 2018087694W WO 2019091084 A1 WO2019091084 A1 WO 2019091084A1
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arc
model
electrode
resistance
raw material
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PCT/CN2018/087694
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French (fr)
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张颖伟
薛晓光
李永旭
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东北大学
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    • H—ELECTRICITY
    • H05—ELECTRIC TECHNIQUES NOT OTHERWISE PROVIDED FOR
    • H05B—ELECTRIC HEATING; ELECTRIC LIGHT SOURCES NOT OTHERWISE PROVIDED FOR; CIRCUIT ARRANGEMENTS FOR ELECTRIC LIGHT SOURCES, IN GENERAL
    • H05B7/00—Heating by electric discharge
    • H05B7/18—Heating by arc discharge
    • H05B7/20—Direct heating by arc discharge, i.e. where at least one end of the arc directly acts on the material to be heated, including additional resistance heating by arc current flowing through the material to be heated
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R27/00—Arrangements for measuring resistance, reactance, impedance, or electric characteristics derived therefrom
    • G01R27/02—Measuring real or complex resistance, reactance, impedance, or other two-pole characteristics derived therefrom, e.g. time constant
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R27/00—Arrangements for measuring resistance, reactance, impedance, or electric characteristics derived therefrom
    • G01R27/02—Measuring real or complex resistance, reactance, impedance, or other two-pole characteristics derived therefrom, e.g. time constant
    • G01R27/08—Measuring resistance by measuring both voltage and current
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/13—Differential equations
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C20/00—Chemoinformatics, i.e. ICT specially adapted for the handling of physicochemical or structural data of chemical particles, elements, compounds or mixtures
    • G16C20/10—Analysis or design of chemical reactions, syntheses or processes

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  • the invention belongs to the technical field of industrial fused magnesium smelting, and particularly relates to a method for calculating the operating resistance of a two-electrode direct current fused magnesium furnace.
  • the industrial fused magnesia furnace is mainly used to produce fused magnesia.
  • the production process is to first break the solid fused magnesia into powder, then add it to the fused magnesia furnace, insert the electrode, and rely on the electrode arc heat after electrification.
  • the fused magnesia is melted, and after the smelting is completed, the electrode is lifted off, and after the fused magnesia is cooled, it is removed from the fused magnesium furnace and naturally crystallized.
  • the overall composition and working principle of the two-electrode fused magnesium furnace equipment are shown in Fig. 1.
  • 1 is a transformer
  • 2 is a short net
  • 3 is a thyristor circuit
  • 4 is a graphite electrode
  • 5 is a furnace shell
  • 6 is a car body
  • 8 is the raw material
  • 9 is the arc
  • 10 is the molten pool.
  • the arc is the key to smelting. It is the most direct supplier of the energy of magnesium oxide melting. There is a certain relationship between the length of the arc and the voltage of the arc.
  • the arc will be affected by the magnetic field, and the arc voltage will be generated. The effect is also to consider the arcing effect when setting the electrode spacing.
  • the raw material resistance and the molten pool resistance are collectively referred to as the operating resistance in practical applications, and the raw material resistance occupies a part of the energy of the electrical model, and the generated resistance heat plays a large role in the uniform melting of the raw material.
  • the constant current mode and the constant resistance mode are frequently used means.
  • the essence of the constant current regulation is to ensure the stability of the arc length by ensuring the current stability, thereby ensuring the continuous progress of the melting process.
  • the power model can be well mastered by controlling the resistance, and then the process is controlled. Therefore, it is of great significance to study the operating resistance in fused magnesium furnaces.
  • FIG. 3 shows the circuit diagram of the two-electrode fused magnesium process.
  • Figure 2 shows the model of the graphite electrode position during the smelting process.
  • the present invention provides a method for calculating the operating resistance of a two-electrode DC fused magnesium furnace.
  • a method for calculating the operating resistance of a two-electrode DC fused magnesium furnace includes:
  • the raw material model is simplified into a cylindrical model centered on the electrode, and the electric field strength of each point in the electric field formed by the green layer around the electrode in the cylindrical model is determined, and two electric field strengths are calculated according to the electric field in the electric field.
  • the solution of the raw material resistance includes:
  • the raw material model is simplified into a cylindrical model centered on the electrode according to the depth of the electrode insertion raw material and the distance between the center lines of the two electrodes in the two-electrode DC fused magnesium furnace;
  • the electric field intensity at the center line x of the distance electrode in the cylindrical model is integrated from half of the distance between the electrode surface and the center line of the two electrodes to obtain half of the raw material voltage between the two electrodes, and the raw material between the two electrodes is calculated. The voltage, and then the resistance of the raw material.
  • the solving arc resistance relational model includes:
  • the arc magnetic field model and the arc deflection trajectory model are established by assuming the shape and position of the arc during the operation of the fused magnesium furnace;
  • the arc deflection trajectory model is as follows:
  • R a is biased trajectory arc radius
  • L A is the actual arc length
  • L is the ideal arc length, i.e. the distance to the surface of the bath electrode.
  • the arc magnetic field model and the arc deflection trajectory model are established by the assumption of the shape and position of the arc during the operation of the fused magnesium furnace, and the specific method is:
  • the theoretical model assumption of the actual arc is made, and the arc magnetic field model is established, that is, the sum of the magnetic field generated by the electrode and the magnetic field generated by the arc;
  • the trajectory of the arc on the X-Y plane that is, the arc deflection trajectory following the circular path, is determined to establish an arc deflection trajectory model.
  • the solution of the molten pool resistance includes:
  • the molten pool model is simplified into a hemispherical model with the center of the bottom end of the electrode as the center of the sphere and the radius from the bottom end of the electrode to the surface of the molten pool;
  • the arc Since the length of the arc and the voltage of the arc have a certain relationship, the arc is affected by the magnetic field, and there is an arcing effect, which has an influence on the arc voltage. Therefore, the arcing effect needs to be considered when setting the electrode spacing.
  • the raw material resistance and the molten pool resistance are collectively referred to as the operating resistance in practical applications, and the raw material resistance occupies a part of the energy of the electrical model, and the generated resistance heat plays a large role in the uniform melting of the raw material. There is a close relationship between resistance and power. Therefore, the present invention can control the power output by adjusting the resistance, and then control the working process of the fused magnesium furnace.
  • Figure 1 is a schematic view showing the structure of an fused magnesium furnace
  • 1 is a transformer
  • 2 is a short net
  • 3 is a thyristor circuit
  • 4 is a graphite electrode
  • 5 is a furnace shell
  • 6 is a car body
  • 7 is an electrode holder
  • 8 is a raw material
  • 9 is an arc
  • 10 is a molten pool
  • 11 is an electrode
  • 12 is an electrode
  • 13 is a deflection arc
  • 14 is an electrode
  • the shadow part of the bottom layer is a well-conducting melt layer, which is considered to be a smelting product, and the resistivity is small, which is not considered; the end of the electrode is a hemispherical surface with a radius of r 0 ;
  • E 1 -E 3 are transformer secondary side voltages
  • r 1 -r 3 are transformer internal resistances
  • L 1 -L 3 are transformer inductances
  • T is a thyristor circuit
  • R n is the short-circuit equivalent resistance
  • R l is the raw material resistance
  • R h1 -R h2 is the arc resistance
  • R c1 -R c2 is the molten pool resistance
  • Figure 4 is a raw material model of a specific embodiment of the present invention.
  • Figure 5 is a molten pool model of a specific embodiment of the present invention.
  • FIG 6 is an arc deflection trajectory model specific embodiments of the invention.
  • L a is the actual arc length
  • ⁇ a is the distance of the arc from the electrode connection position to the centerline of the plating surface, is a measure of the deflection of the arc;
  • Figure 7 is a theoretical voltage versus electrode height curve.
  • a method for calculating the operating resistance of a two-electrode DC fused magnesium furnace includes:
  • Step 1 Solving the raw material resistance: Simplify the raw material model into a cylindrical model centered on the electrode, and determine the electric field strength of each point in the electric field formed by the green layer around the electrode in the cylindrical model, according to the electric field at each point in the electric field. Intensity calculates the raw material voltage between the two electrodes, thereby obtaining the raw material resistance between the two electrodes;
  • Step 1.1 The raw material model is simplified into a cylindrical model centered on the electrode according to the depth h l of the electrode inserted raw material and the distance ⁇ e between the center lines of the two electrodes in the two-electrode DC fused magnesium furnace.
  • the raw material resistance that is, the resistance of the unmelted charge zone, radiates the current flowing from the circumferential side of the electrode and passes through the electrical resistance to become thermal energy.
  • the size of the raw material resistance is mainly related to the raw material composition, the depth of the electrode inserted into the charge, the electrode spacing, and the temperature of the raw material zone. Under normal conditions, the raw material resistance is greater than the resistance of the molten pool, so only a small portion of the current of the electrode flows through the raw material resistance.
  • a cylindrical model centered on the electrode 11 as shown in Fig. 4 was established, and the electrode diameter was d in units of cm.
  • Step 1.2 Applying the microscopic form of Ohm's law to determine the electric field strength at each point in the electric field formed by the green layer around the electrode in the cylindrical model;
  • R l be the raw material resistance between the two electrodes
  • R c be the resistance of the reaction zone at the lower end of the electrode, that is, the resistance of the molten pool; when the fused magnesium furnace is working, an electric field is formed in the layer around the electrode.
  • ⁇ l is the raw material resistivity, that is, the resistivity of the green layer having a depth h l between the two electrodes, the unit is ⁇ cm; J l is the current density flowing through the raw material; h l is the electrode inserted raw material The depth, in cm; I l is the current flowing through the raw layer between the two electrodes, the unit is A.
  • Step 1.3 Integrate the electric field intensity at the center line x of the distance electrode from the electrode surface to the center line of the two electrodes in the cylindrical model to obtain half of the raw material voltage between the two electrodes, and calculate the between the two electrodes.
  • the raw material micro-voltage difference dU x from the electrode center line x is as follows:
  • U l is the voltage of the center of the two electrodes relative to the center of the electrode
  • U 0 is the voltage of the electrode surface relative to the center of the electrode
  • U n between the two electrodes is as follows:
  • Step 1.4 Determine the raw material resistance R l between the two electrodes:
  • the ⁇ e /d of the fused magnesium furnace is mostly in the range of 2.2 to 2.3. If it is too small, the electrothermal conversion is insufficient, and the electrode arrangement is difficult. If the galvanic furnace is too large, the change of the influence on the electrothermal conversion is small, and has no practical significance.
  • Step 2 Solving the relationship model of arc resistance: Determine the relationship between the actual arc length and the distance from the electrode to the surface of the molten pool, and solve the relationship between the arc voltage and the actual arc length, that is, the arc resistance relation model.
  • Step 2.1 Determine the arc resistance relational model:
  • Step 2.2 According to the arc voltage and volt-ampere characteristic model proposed by Bowman, for the steady-state electrical behavior of the DC arc, the arc voltage is described according to the ideal arc length and current change:
  • Equation (9) conforms to the formula (7); the ideal arc length L is the length of the arc regardless of the arc deflection, and its value is equal to the distance from the end of the electrode to the surface of the bath.
  • Step 2.3 Establish an arc magnetic field model and an arc deflection trajectory model by assuming the shape and position of the arc during operation of the fused magnesium furnace.
  • Step 2.3.1 Make a theoretical model assumption for the actual arc and establish an arc magnetic field model, which is the sum of the magnetic field generated by the electrode and the magnetic field generated by the arc.
  • L a is the actual arc length from the arc that is projected to a point to the bath surface from the tip of the electrode
  • ⁇ a is projected from the bath surface and the arc distance between the point of the center line of the electrode, i.e., the arc of deflection Measure
  • B is the magnetic field vector
  • ⁇ 0 is the permeability of the free space (constant)
  • I is the current carried by the current carrying element
  • dl is the differential distance vector of the current carrying element
  • r is the dl from the dl to the B space
  • (x 0 , y 0 ) is the coordinate of the point A to be calculated on the arc plane;
  • B z, e , B z, a are the magnetic field generated by the electrode and the magnetic field generated by the arc , respectively.
  • B z B z, e + B z, a .
  • Step 2.3.2 Determine the trajectory of the arc on the X-Y plane, that is, follow the arc deflection trajectory of the circular path to establish an arc deflection trajectory model.
  • the differential equation of motion of the free particles of the current-carrying arc plasma gas must be solved.
  • the particles are part of a high velocity gas jet emerging from the vicinity of the arc attachment zone on the surface of the electrode and are acted upon by the magnetic field as it moves between the electrode and the molten pool.
  • dF is the force that the carrier element N having the length
  • l is defined in the direction of the current flow vector, and the coordinates of the carrier element N in the magnetic field at a certain time are (x, y).
  • v x and v y are the velocity components in the X and Y directions of the carrier element N selected in the arc
  • dm is the differential mass of the carrier element N
  • a is the acceleration vector of the carrier element N
  • r a is The radius of the flow element N
  • ⁇ a is the plasma density of the arc.
  • a and B are the constants of the integral.
  • C and D are integral constants.
  • Step 2.4 Combine the arc deflection trajectory radius R a and the arc deflection trajectory model to find the relationship between the actual arc length L a and the ideal arc arc length L.
  • the actual arc length is no longer the distance from the end of the electrode to the surface of the bath, and the arc voltage is also affected by the deflection of the arc.
  • Step 2.5 Solving the actual arc voltage and the arc length L a relationship, i.e. the arc resistance relationship model.
  • the electrode height relative to the arc radius R a deflected trajectory is quite small, the arc voltage curve of the arc remains near undeflected.
  • the curve separates, and the arc in a two-electrode furnace that exhibits significant arc deflection may be 30% higher than the Bowman model than an unbent arc.
  • Step 3 Solve the pool resistance, that is, the sum of the series resistances of the two electrodes.
  • Step 3.1 Modeling the molten pool model: simplifying the molten pool model into a hemispherical model with the center of the bottom end of the electrode as the center of the sphere and the radius from the bottom end of the electrode to the surface of the molten pool;
  • the resistance of the bath that is, the resistance of the reaction zone at the lower end of the electrode.
  • the current flowing from the lower end surface of the electrode is converted into thermal energy through the resistance of the molten pool.
  • the size of the bath resistance is mainly determined by the distance from the lower end of the electrode to the bottom of the fused magnesium furnace, the size of the reaction zone at the lower end of the electrode, and the temperature of the reaction zone. Under normal circumstances, the resistance of the molten pool is small, and most of the current of the electrode flows through the resistance of the molten pool.
  • the hemispherical model is shown in Fig. 5.
  • Step 3.2 Apply the microscopic form of Ohm's law to find the electric field strength E c of the spherical surface from the spherical center r in the hemispherical model;
  • Step 3.3 Integrate the electric field intensity from the bottom end of the electrode to the surface of the molten pool in the hemispherical model to obtain a voltage drop from the bottom end of the single electrode to the surface of the molten pool;
  • Step 3.4 Calculate the sum of the series resistances of the two electrodes, that is, the molten pool resistance R c .

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Abstract

本发明提供一种双电极直流电熔镁炉操作电阻计算方法，包括：求解生料电阻：将生料模型简化成以电极为中心的圆柱形模型，确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度，根据电场中各点电场强度计算两电极间的生料电压，进而得到两电极间的生料电阻；求解电弧电阻关系式模型：确定实际电弧弧长与电极到熔池表面距离关系，求解出电弧电压与实际电弧长度关系式，即电弧电阻关系式模型；求解熔池电阻，即两根电极的熔池电阻串联之和。本发明通过对电阻进行调控可以很好的控制功率输出，进而对电熔镁炉工作过程进行控制。

Description

双电极直流电熔镁炉操作电阻计算方法 技术领域
本发明属于工业电熔镁冶炼技术领域，具体涉及一种双电极直流电熔镁炉操作电阻计算方法。
背景技术
现阶段工业电熔镁炉主要用来生产电熔镁砂，生产过程为首先将固态电熔镁砂打碎成粉，然后加入到电熔镁炉中，插入电极，通电后主要依靠电极电弧热对电熔镁砂进行融化，熔炼结束后抬出电极，等到电熔镁砂冷却后搬离出电熔镁炉，并进行自然结晶。双电极电熔镁炉设备的整体组成及工作原理如图1所示，其中1为变压器，2为短网，3为晶闸管电路，4为石墨电极，5为炉壳，6为车体，7为电极夹，8为生料，9为电弧，10为熔池。
在双直流电熔镁电气模型中，存在着诸多的电阻，包括：变压器电阻、晶闸管电阻、短网电阻、生料电阻、熔池电阻、电弧电阻。它们在氧化镁熔炼过程中起着很大的影响，有些对电熔镁炉的设计起着关键作用。变压器内阻、短网电阻以及晶闸管的波动，在恒流调控过程中，会产生很大的负面作用。因此，怎样保证他们的稳定性是重中之重。其次，电弧是冶炼的关键，它是氧化镁熔化的能量最直接的供应者，电弧的长度、电弧的电压存在着一定的关系，电弧受磁场影响会存在偏弧效应，对电弧电压又会产生影响，在设置电极间距的时候也需要考虑偏弧效应。生料电阻和熔池电阻在实际应用中被合称为操作电阻，生料电阻会占据电气模型一部分的能量，其产生的电阻热对生料的均匀熔化起到很大的作用。在对电极的自动调控过程中，恒流模式、恒阻模式是经常使用的手段，其中的恒流调控的实质就是通过保证电流稳定来保证电弧长度的稳定，从而保证熔炼过程的连续进行。电阻和功率有着紧密的关系，因此通过对电阻调控可以很好的掌握功率模型，进而对过程进行控制。因此，研究电熔镁炉中操作电阻有着重要的意义。
双电极直流电熔镁炉与三相交流电熔镁炉最大的区别就是：变压器二次侧由交流供电变为直流供电，石墨电极由三根变为两根，电弧由交流电弧变为直流电弧。图3给出了双电极电熔镁过程的电路图，为了方便对操作电阻分析，图2给出了熔炼过程中石墨电极位置的模型图。
发明内容
针对现有技术存在的问题，本发明提供一种双电极直流电熔镁炉操作电阻计算方法。
本发明的技术方案如下：
一种双电极直流电熔镁炉操作电阻计算方法，包括：
求解生料电阻：将生料模型简化成以电极为中心的圆柱形模型，确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度，根据电场中各点电场强度计算两电极间的生料电压，进而得到两电极间的生料电阻；
求解电弧电阻关系式模型：确定实际电弧弧长与电极到熔池表面距离关系，求解出电弧电压与实际电弧长度关系式，即电弧电阻关系式模型；
求解熔池电阻，即两根电极的熔池电阻串联之和。
所述求解生料电阻，包括：
根据电极插入生料的深度以及双电极直流电熔镁炉中两电极中心线间的距离，将生料模型简化成以电极为中心的圆柱形模型；
确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度；
对圆柱形模型中距离电极中心线x处的电场强度进行从电极表面到两电极中心线间的距离的一半进行积分，得到两电极间的生料电压的一半，计算出两电极间的生料电压，进而求出生料电阻。
所述求解电弧电阻关系式模型，包括：
确定电弧电阻关系式模型；
根据理想电弧长度和电流变化描述电弧电压：
通过对电熔镁炉运行过程中电弧形状与位置的假设，建立电弧磁场模型，以及电弧偏转轨迹模型；
结合电弧偏转轨迹半径以及电弧偏转轨迹模型，求出实际电弧长度与理想电弧弧长的关系；
求解电弧电压与实际电弧长度关系式，即电弧电阻关系式模型。
所述电弧偏转轨迹模型如下：
Figure PCTCN2018087694-appb-000001
其中，R a是电弧偏转轨迹半径，L a是实际电弧长度，L是理想电弧长度，即电极到熔池 表面距离。
所述通过对电熔镁炉运行过程中电弧形状与位置的假设，建立电弧磁场模型，以及电弧偏转轨迹模型，具体方法是：
对实际电弧进行理论模型假设，并建立电弧磁场模型，即电极产生的磁场和电弧产生的磁场之和；
确定电弧在X-Y平面上的运动轨迹，即遵循圆形路径的电弧偏转轨迹，建立出电弧偏转轨迹模型。
所述对实际电弧进行理论模型假设，具体包括：
假设电弧只是相互偏转，即主电弧偏转效应发生在由两个电极限定的平面内；
假设电极的长度是无限大的，即忽略最终效应；
假设电弧在偏转时遵循直线；
忽略通过熔池的电流产生的磁场对电弧的影响。
所述求解熔池电阻，包括：
将熔池模型简化成以电极底端中心为球心、以电极底端到熔池表面距离为半径的半球形模型；
求出半球形模型中距球心r处的球面的电场强度；
对半球形模型中距球心r处的电场强度进行从电极底端到熔池表面进行积分，得到单根电极底端到熔池表面的电压降；
计算两根电极的熔池电阻串联之和，即所求解熔池电阻。
有益效果：
由于电弧的长度和电弧的电压存在着一定的关系，电弧受磁场影响会存在偏弧效应，对电弧电压又会产生影响，因此在设置电极间距的时候也需要考虑偏弧效应。生料电阻和熔池电阻在实际应用中被合称为操作电阻，生料电阻会占据电气模型一部分的能量，其产生的电阻热对生料的均匀熔化起到很大的作用。电阻和功率有着紧密的关系，因此本发明通过对电阻进行调控可以很好的控制功率输出，进而对电熔镁炉工作过程进行控制。
附图说明
图1为电熔镁炉结构示意图；
其中，1为变压器，2为短网，3为晶闸管电路，4为石墨电极，5为炉壳，6为车体，7 为电极夹，8为生料，9为电弧，10为熔池；11为电极；12为电极；13为偏转电弧；14为电极；
图2为电熔镁炉的生料电阻、上层熔池电阻示意图；其中δ e为双电极直流电熔镁炉中两电极中心线间的距离，d为电极直径；h l为电极插入生料的深度(也可以称为生料层高度)，ρ l为生料电阻率；h c是电极下端到熔池表面的高度(也可以称为上层熔体高度)，ρ c为熔体层电阻率；底层阴影部分是导电性良好的熔体层，认为是熔炼产品，电阻率较小，一股不加以考虑；电极的端部为半球面，半径为r 0；
图3为本发明涉及到的电熔镁炉电路图；E 1-E 3为变压器二次侧电压，r 1-r 3为变压器内阻，L 1-L 3为变压器感抗，T为晶闸管电路，R n为短网等效电阻，R l为生料电阻，R h1-R h2为电弧电阻，R c1-R c2为熔池电阻；
图4为本发明具体实施方式的生料模型；
图5为本发明具体实施方式的熔池模型；
图6为本发明具体实施方式的电弧偏转轨迹模型；在图中，L a是实际电弧长度，δ a是从电极中心线到电镀表面的电弧连接位置的距离，是电弧偏转的量度；
图7为理论电压与电极高度曲线。
具体实施方式
下面结合附图对本发明的具体实施方式做详细说明。
一种双电极直流电熔镁炉操作电阻计算方法，包括：
步骤1：求解生料电阻：将生料模型简化成以电极为中心的圆柱形模型，确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度，根据电场中各点电场强度计算两电极间的生料电压，进而得到两电极间的生料电阻；
步骤1.1：根据电极插入生料的深度h l以及双电极直流电熔镁炉中两电极中心线间的距离δ e，将生料模型简化成以电极为中心的圆柱形模型。
生料电阻即未熔化的炉料区的电阻，从电极圆周侧面辐射流出的电流，经过该电阻而变为热能。生料电阻的大小，主要与生料组成、电极插入炉料的深度、电极间距、生料区温度有关。正常情况下，生料电阻比熔池电阻大，因此，电极的电流只有较小一部分流过生料电阻。建立如图4所示的以电极11为中心的圆柱形模型，电极直径为d，单位是cm。
步骤1.2：应用欧姆定律的微观形式，确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度；
设R l为两电极间的生料电阻，R c为电极下端反应区的电阻即熔池电阻；电熔镁炉工作时，在电极周围的料层中形成电场。
根据欧姆定律，电场中某点的电场强度(V/cm)公式为：
Figure PCTCN2018087694-appb-000002
其中，ρ为电场介质的电阻率，Ω·cm；J为该点的电流密度，A/cm 2I 0为该点电流，S 0为该点横截面积。
因此，圆柱形模型中距离电极中心线x处的电场强度E l如下：
Figure PCTCN2018087694-appb-000003
其中，ρ l为生料电阻率，即两电极间深度为h l的生料层的电阻率，单位是Ω·cm；J l为流经生料的电流密度；h l为电极插入生料的深度，单位是cm；I l为流过两电极间生料层的电流，单位是A。
步骤1.3：对圆柱形模型中距离电极中心线x处的电场强度进行从电极表面到两电极中心线间的距离的一半进行积分，得到两电极间的生料电压的一半，计算出两电极间的生料电压；
距离电极中心线x处的生料微电压差dU x如下：
Figure PCTCN2018087694-appb-000004
对圆柱形模型中距离电极中心线x处的电场强度E l进行从电极表面到两电极中心线间的距离的一半δ e/2进行积分，得到电极间距内生料电压的一半U n/2：
Figure PCTCN2018087694-appb-000005
其中，U l为两电极中间生料相对电极中心的电压，U 0为电极表面相对电极中心的电压，两电极间的生料电压U n如下：
Figure PCTCN2018087694-appb-000006
步骤1.4：求出两电极间的生料电阻R l：
Figure PCTCN2018087694-appb-000007
电熔镁炉的δ e/d多数在2.2～2.3范围，过小则电热转换不充分，而且电极配置困难，过大则对电热转换的影响的变化很小，没有实用意义。
步骤2：求解电弧电阻关系式模型：确定实际电弧弧长与电极到熔池表面距离关系，求解出电弧电压与实际电弧长度关系式，即电弧电阻关系式模型。
步骤2.1：确定电弧电阻关系式模型：
U h＝U x+bL a             (7)
其中，U h为电弧电压，单位是V；U x为熄弧电压，单位是V；b为弧压系数，单位是V/cm；L a为弧长，单位是cm。
步骤2.2：根据Bowman提出的电弧电压与伏安特性模型，对于直流电弧的稳态电气行为，根据理想电弧长度和电流变化描述电弧电压：
Figure PCTCN2018087694-appb-000008
Figure PCTCN2018087694-appb-000009
其中，ρ a是电弧电阻率，单位是Ω·cm；j a是阴极点电流密度，单位是kA.cm -2；单位是V；I电弧电流，单位是A；r k是阴极附着点半径，单位是cm；U h是电弧电压，单位是V；L理想电弧长度，单位是cm；
当电弧电流I＝1000A且理想电弧长度L＞2cm时，电弧电压与理想电弧长度的关系式化简为：
U h≈4.81L+23.8             (9)
公式(9)与公式(7)形式符合；理想电弧长度L为不考虑电弧偏转情况下的电弧长度，其值等于电极末端到熔池表面的距离。
步骤2.3：通过对电熔镁炉运行过程中电弧形状与位置的假设，建立电弧磁场模型，以及电弧偏转轨迹模型。
步骤2.3.1：对实际电弧进行理论模型假设，并建立电弧磁场模型，即电极产生的磁场和电弧产生的磁场之和。
首先，假设电弧只是相互偏转，即主电弧偏转效应发生在由两个电极限定的平面内。
其次，假设电极的长度是无限大的，即忽略最终效应(这是一个合理的假设，因为在典型的电熔镁炉设计中，电极的长度远大于其影响的电弧的长度)。
再次，假设电弧在偏转时遵循直线。
最后，忽略通过熔池的大部分电流产生的磁场不会很大地影响上述的电弧。
如图6所示，L a是实际电弧长度即从电弧与熔池表面投射点到到电极尖端的距离，δ a是从电弧与熔池表面投射点到电极中心线的距离，即电弧偏转的量度；
根据电磁学理论，有如下公式成立：
Figure PCTCN2018087694-appb-000010
其中，B是磁场矢量；μ 0是自由空间的磁导率(常数)；I是由载流元携带的电流；dl是载流元的微分距离矢量，r是从dl到B空间中A(x 0，y 0)点的距离矢量，A(x 0，y 0)是随机选取点。
磁场矢量B仅由垂直于X-Y平面的Z分量组成，即B＝(0，0，B z)，其中的B z是标量，是垂直于X-Y平面的z分量。
B z的表达式通过式(10)分别在电极和理想电弧长度上的积分并相加来求得：
Figure PCTCN2018087694-appb-000011
Figure PCTCN2018087694-appb-000012
其中，(x 0，y 0)为在偏弧平面上要计算点A的坐标；B z，e、B z，a分别为电极产生磁场和电弧产生磁场。
将电极产生的磁场和电弧产生的磁场之和，作为建立出的电弧磁场模型：B z＝B z，e+B z，a。
步骤2.3.2：确定电弧在X-Y平面上的运动轨迹，即遵循圆形路径的电弧偏转轨迹，建立出电弧偏转轨迹模型。
为了计算电弧柱的形状，必须求解载流电弧等离子体气体的自由粒子的微分运动方程。颗粒是从电极表面上的电弧附着区附近发出的高速气体射流的一部分，并且当其在电极和熔池之间移动时被磁场作用。
根据基本电磁力学，有下列等式成立：
dF＝Idl×B            (12)
其中，dF是电弧中长度为|dl|的载流元N在电流强度为I、磁场强度为B时受到的力。l在电流流动矢量的方向上被定义，设某时刻载流元N在磁场中坐标为(x，y)。
考虑在X-Y平面上的运动有：
Figure PCTCN2018087694-appb-000013
其中，v x和v y分别是在电弧中选取的载流元N的X和Y方向速度分量，dm是载流元N的微分质量，a是载流元N的加速度矢量，r a是载流元N的半径，ρ a是电弧的等离子体密度。
从dl、B和dF的定义，可将上式变为：
Figure PCTCN2018087694-appb-000014
做如下假设：近似B z是常数，速度矢量的大小恒定并等于电弧等离子体的速度v a。
对式(14)整理得到：
Figure PCTCN2018087694-appb-000015
式(15)中的微分方程组具有谐波运动的标准形式：
Figure PCTCN2018087694-appb-000016
其中，A和B是积分的常数。
Figure PCTCN2018087694-appb-000017
由于A和B是常数，所以v a在整个运动中确实是恒定的，并且假设是合理的。
再次将(16)集成到载流元N的位置x和y作为时间的函数：
Figure PCTCN2018087694-appb-000018
其中，C和D是积分常数。
进而得出电弧偏转轨迹模型：
Figure PCTCN2018087694-appb-000019
这是X-Y平面中的圆的等式。因此，电弧偏转轨迹在电极和熔池之间行进时遵循圆形路径。这可以在电弧喷射器中在电极和浴液之间移动时产生明显的曲线。
步骤2.4：结合电弧偏转轨迹半径R a以及电弧偏转轨迹模型，求出实际电弧长度L a与理 想电弧弧长L的关系。
有了电弧偏转轨迹模型之后，实际电弧长度就不再是电极末端到熔池表面的距离了，电弧电压受电弧偏转的影响也将发生变化。
实际电弧长度L a：
Figure PCTCN2018087694-appb-000020
从式(19)，对于电弧的初始轨迹从电极垂直向下的情况
Figure PCTCN2018087694-appb-000021
有：
Figure PCTCN2018087694-appb-000022
结合式(20)，得到实际电弧长度L a与理想电弧弧长L的关系：
Figure PCTCN2018087694-appb-000023
使用L a代替式(8)中的L，能够产生双电极镁炉的理论电压与电极高度曲线。图7中示出了几个电弧偏转轨迹半径R a为不同值时的理论电压与电极高度曲线。
步骤2.5：求解电弧电压与实际电弧长度L a关系式，即电弧电阻关系式模型。
在电极移动的过程中，电极高度相对电弧偏转轨迹半径R a的相当较小时，电弧电压曲线仍然接近于未偏转的电弧。随着电极高度接近电弧偏转轨迹半径R a，曲线分离，显示出明显的电弧偏转的双电极炉中的电弧可能比未弯曲电弧可能比Bowman模型高出30％的电压。
结合步骤2.2中的电弧电压U h与理想电弧弧长L的关系式，电弧电压与实际电弧长度L a表达式如下：
U h＝6.25L a+30.9            (23)
步骤3：求解熔池电阻，即两根电极的熔池电阻串联之和。
步骤3.1：对熔池模型进行建模化简：将熔池模型简化成以电极底端中心为球心、以电极底端到熔池表面距离为半径的半球形模型；
熔池电阻，即电极下端反应区的电阻。从电极下端面流出的电流，经过熔池电阻而变成热能。熔池电阻的大小，主要决定于电极下端到电熔镁炉底部的距离、电极下端反应区的大 小及该反应区的温度。正常情况下，熔池电阻值很小，电极的电流，大部分都流过熔池电阻，半球形模型如图5所示。
步骤3.2：应用欧姆定律的微观形式，求出半球形模型中距球心r处的球面的电场强度E c；
以电极底端中心为球心的半球形模型中，距球心r处的球面的电场强度E c：
Figure PCTCN2018087694-appb-000024
距球心r处的球面的微电势差dU r可由欧姆定律的微分形式导出：
Figure PCTCN2018087694-appb-000025
步骤3.3：对半球形模型中距球心r处的电场强度进行从电极底端到熔池表面进行积分，得到单根电极底端到熔池表面的电压降；
对距球心r处的电场强度E c进行从电极底端到熔池表面进行积分，得到U c，即单根电极下端至熔池电阻两端电压降；设电极底端中心到熔池表面的压降为U 1，电极底端中心到电极底端的压降为U 0，r 0为电极半径，则：
Figure PCTCN2018087694-appb-000026
单根电极底端到熔池表面的电压降：
Figure PCTCN2018087694-appb-000027
步骤3.4：计算两根电极的熔池电阻串联之和，即所求解熔池电阻R c。
两根电极的熔池电阻串联之和，即熔池电阻R c：
Figure PCTCN2018087694-appb-000028
由该式可知：当电极无限接近熔池表面时，即h c→r 0，熔池电阻为0Ω，随着电极远离熔池表面，导出一个有指导作用的抽象概念：h c＞＞r 0时
Figure PCTCN2018087694-appb-000029
这时的熔池电阻趋于一个定值。(工业取值)
虽然以上描述了本发明的具体实施方式，但是本领域内的熟练的技术人员应当理解，这 些仅是举例说明，可以对这些实施方式做出多种变更或修改，而不背离本发明的原理和实质。本发明的范围仅由所附权利要求书限定。

Claims (7)

  1. 一种双电极直流电熔镁炉操作电阻计算方法，其特征在于，包括：
    求解生料电阻：将生料模型简化成以电极为中心的圆柱形模型，确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度，根据电场中各点电场强度计算两电极间的生料电压，进而得到两电极间的生料电阻；
    求解电弧电阻关系式模型：确定实际电弧弧长与电极到熔池表面距离关系，求解出电弧电压与实际电弧长度关系式，即电弧电阻关系式模型；
    求解熔池电阻，即两根电极的熔池电阻串联之和。
  2. 根据权利要求1所述的方法，其特征在于，所述求解生料电阻，包括：
    根据电极插入生料的深度以及双电极直流电熔镁炉中两电极中心线间的距离，将生料模型简化成以电极为中心的圆柱形模型；
    确定圆柱形模型中电极周围的生料层所形成的电场中各点电场强度；
    对圆柱形模型中距离电极中心线x处的电场强度进行从电极表面到两电极中心线间的距离的一半进行积分，得到两电极间的生料电压的一半，计算出两电极间的生料电压，进而求出生料电阻。
  3. 根据权利要求1所述的方法，其特征在于，所述求解电弧电阻关系式模型，包括：
    确定电弧电阻关系式模型；
    根据理想电弧长度和电流变化描述电弧电压：
    通过对电熔镁炉运行过程中电弧形状与位置的假设，建立电弧磁场模型，以及电弧偏转轨迹模型；
    结合电弧偏转轨迹半径以及电弧偏转轨迹模型，求出实际电弧长度与理想电弧弧长的关系；
    求解电弧电压与实际电弧长度关系式，即电弧电阻关系式模型。
  4. 根据权利要求3所述的方法，其特征在于，所述电弧偏转轨迹模型如下：
    Figure PCTCN2018087694-appb-100001
    其中，R a是电弧偏转轨迹半径，L a是实际电弧长度，L是理想电弧长度，即电极到熔池表面距离。
  5. 根据权利要求3所述的方法，其特征在于，所述通过对电熔镁炉运行过程中电弧形状与位置的假设，建立电弧磁场模型，以及电弧偏转轨迹模型，具体方法是：
    对实际电弧进行理论模型假设，并建立电弧磁场模型，即电极产生的磁场和电弧产生的 磁场之和；
    确定电弧在X-Y平面上的运动轨迹，即遵循圆形路径的电弧偏转轨迹，建立出电弧偏转轨迹模型。
  6. 根据权利要求5所述的方法，其特征在于，所述对实际电弧进行理论模型假设，具体包括：
    假设电弧只是相互偏转，即主电弧偏转效应发生在由两个电极限定的平面内；
    假设电极的长度是无限大的，即忽略最终效应；
    假设电弧在偏转时遵循直线；
    忽略通过熔池的电流产生的磁场对电弧的影响。
  7. 根据权利要求1所述的方法，其特征在于，所述求解熔池电阻，包括：
    将熔池模型简化成以电极底端中心为球心、以电极底端到熔池表面距离为半径的半球形模型；
    求出半球形模型中距球心r处的球面的电场强度；
    对半球形模型中距球心r处的电场强度进行从电极底端到熔池表面进行积分，得到单根电极底端到熔池表面的电压降；
    计算两根电极的熔池电阻串联之和，即所求解熔池电阻。
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