Calculation method and calculation equipment for thickness of titanium layer of cathode roller

Abstract

The invention discloses a method and equipment for calculating the thickness of a titanium layer of a cathode roller, and relates to the technical field of hydrometallurgy. In the method, a computing device determines a target contact pressure for achieving a stable contact state between a cathode roll shaft and a titanium sleeve through a preset contact resistance test experiment; the calculation device calculates the radial interference magnitude generated in the cold shrinkage process of the titanium sleeve based on the thermal expansion coefficient of the titanium sleeve and the heating temperature range of the hot charging process; the calculation equipment performs simulation analysis through finite element simulation analysis software based on the elastic modulus and the Poisson's ratio of the titanium sleeve to obtain the equivalent elastic modulus and the equivalent Poisson's ratio of the cathode roll shaft; based on the target contact pressure, the radial interference magnitude, the equivalent elastic modulus and the equivalent Poisson's ratio, the calculation equipment performs reverse solution through a preset contact pressure calculation formula to obtain the outer radius of the cathode roll shaft; and the calculation equipment takes the difference value between the outer radius of the titanium sleeve and the outer radius of the cathode roller shaft as the thickness of the titanium layer.

Description

Technical Field

[0001] This application relates to the field of hydrometallurgical technology, and in particular to a method and apparatus for calculating the thickness of the titanium layer on a cathode roller. Background Technology

[0002] As a key component that directly supports the electrodeposition process, the titanium cylinder of the cathode roller must simultaneously withstand multiple harsh conditions such as internal cooling medium pressure, foil peeling tension, and corrosion from the acidic electrolytic environment. Therefore, the design of the titanium cylinder thickness has become a core factor affecting the manufacturing cost of the cathode roller (titanium material accounts for more than 30% of the cost), operating energy consumption, and product quality.

[0003] When designing the thickness of titanium sleeves, related technologies typically estimate the titanium layer thickness by referring to the formula that the thickness increases by 1mm for every 100mm increase in diameter. At the same time, assembly is completed according to the traditional hot fitting process (heating the titanium sleeve to a certain temperature and then fitting it into the shaft body, using cold shrinkage to achieve an interference fit).

[0004] However, the titanium layer thickness design methods of related technologies are not accurate enough when facing the precise conductivity requirements of carrying large currents. Summary of the Invention

[0005] This application provides a method and apparatus for calculating the thickness of the titanium layer in a cathode roller, thereby improving the design accuracy of the titanium layer thickness in the cathode roller.

[0006] Firstly, a method for calculating the titanium layer thickness of a cathode roller is provided, applied to a computing device. This method includes: the computing device determining the target contact pressure for achieving a stable contact state between the cathode roller shaft and the titanium sleeve through a preset contact resistance test experiment; the computing device calculating the radial interference generated by the titanium sleeve during the cooling and shrinking process based on the thermal expansion coefficient of the titanium sleeve and the heating temperature range of the hot-fitting process; the computing device performing simulation analysis using finite element simulation software based on the elastic modulus and Poisson's ratio of the titanium sleeve to obtain the equivalent elastic modulus and equivalent Poisson's ratio of the cathode roller shaft; based on the target contact pressure, radial interference, equivalent elastic modulus, and equivalent Poisson's ratio, the computing device performing inverse calculation using a preset contact pressure calculation formula to obtain the outer radius of the cathode roller shaft; and the computing device using the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller shaft as the titanium layer thickness.

[0007] By adopting the above technical solution, the computing device determines a pressure value—the target contact pressure—that enables a stable and sufficiently low contact resistance between the cathode roller and the titanium sleeve through a preset contact resistance test experiment. Next, the computing device uses a preset contact pressure calculation formula to perform a reverse calculation to obtain the outer radius of the cathode roller. The difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller is then used as the titanium layer thickness. This determines a minimum titanium layer thickness that satisfies the prerequisite of sufficiently low contact resistance, thereby reducing the total resistance of the cathode roller (composed of the titanium layer volume resistance and the contact resistance of the mating surface) and the operating energy consumption of the cathode roller.

[0008] In conjunction with some embodiments of the first aspect, in some embodiments, a preset contact pressure calculation formula is included, specifically including: Where p is the target contact pressure. The radial interference is denoted as , where 'a' is the outer radius of the cathode roller shaft and 'b' is the outer radius of the titanium sleeve. and These are the elastic moduli of the cathode roller and the titanium sleeve, respectively. and These are the Poisson's ratios of the cathode roller and the titanium sleeve, respectively.

[0009] By adopting the above technical solution, the preset contact pressure calculation formula provides a specific mathematical tool for reverse calculation of the outer radius of the cathode roller shaft, improving the accuracy of the derivation process from target pressure to geometric dimensions.

[0010] In conjunction with some embodiments of the first aspect, in some embodiments, the step of the computing device determining the target contact pressure for achieving a stable contact state between the cathode roller and the titanium sleeve through a preset contact resistance test experiment specifically includes: the computing device controlling the contact test fixture to gradually increase the contact pressure according to a preset pressure gradient, the test fixture being fitted with prefabricated titanium-copper, titanium-steel, and steel-copper materials; the computing device collecting contact resistance values ​​of different material combinations at each pressure point and recording the corresponding pressure values ​​to obtain a contact resistance-pressure curve; the computing device performing fitting analysis on the resistance-pressure curve to identify the inflection point of the contact resistance changing with pressure; the computing device calculating the contact resistance change rate corresponding to a preset number of consecutive pressure points after the inflection point; when the absolute value of the contact resistance change rate is less than a preset change rate threshold, the computing device determining that the contact resistance has reached a stable state; the computing device using the pressure value corresponding to reaching the stable state as the stable contact pressure for different material combinations, and selecting the corresponding stable contact pressure as the target contact pressure based on the actual material combination of the cathode roller.

[0011] By adopting the above technical solution, the computing device identifies the critical point from rapid improvement to stabilization of the contact state by analyzing the inflection point and subsequent rate of change of the contact resistance-pressure curve. This allows for the determination of a target contact pressure value that ensures excellent conductivity without excessively increasing assembly stress, thus improving the accuracy of determining the target contact pressure.

[0012] In conjunction with some embodiments of the first aspect, in some embodiments, after the calculation device uses the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller shaft as the titanium layer thickness, the method further includes: the calculation device establishing an electric field model, a thermal field model, and a force field model of the cathode roller based on the outer radius of the cathode roller shaft, the outer radius of the titanium sleeve, and the titanium layer thickness; the calculation device applying the actual working current density of the cathode roller to the electric field model, and calculating the current distribution and Joule heat distribution on the mating surface of the cathode roller based on the contact resistance-pressure curve; the calculation device using the Joule heat distribution as a volume heat source in the thermal field model, and simultaneously based on the internal... The convection heat transfer boundary conditions of the cooling medium and the heat transfer boundary conditions of the external electrolyte are used to solve the three-dimensional temperature field distribution of the cathode roller during steady-state operation. The calculation equipment applies the assembly stress generated by the radial interference in the force field model and the thermal stress generated by the three-dimensional temperature field, and calculates the total contact pressure distribution of the cathode roller mating surface after the superposition of the two. The calculation equipment calculates the dynamic average contact pressure based on the total contact pressure distribution. When the difference between the dynamic average contact pressure and the target contact pressure is greater than the preset threshold, the calculation equipment adjusts the radial interference and re-executes the simulation analysis until the dynamic average contact pressure converges to the target contact pressure.

[0013] By adopting the above technical solution, the computing device further incorporates electro-thermal-mechanical multiphysics coupling simulation analysis to verify and iteratively optimize the initial design. This method can simulate the thermal stress generated by the cathode roller under actual operating current due to the Joule heating effect and the mismatch in material thermal expansion, and calculate the total contact pressure under dynamic conditions. By comparing this dynamic pressure with the target pressure and iteratively adjusting it, the final designed titanium layer thickness not only meets the static assembly requirements but also improves stability under complex operating conditions.

[0014] In conjunction with some embodiments of the first aspect, in some embodiments, the step of the computing device calculating the dynamic average contact pressure based on the total contact pressure distribution specifically includes: the computing device dividing the mating surface of the cathode roller into regular grid cells according to the circumferential and axial directions; the computing device extracting the normal contact pressure and the corresponding contact area of ​​each grid cell; and the computing device performing integral calculation on the total contact pressure distribution using an area-weighted average method to obtain the dynamic average contact pressure.

[0015] By adopting the above technical solution, the computing device transforms the complex and uneven contact pressure distribution obtained from the simulation into a single scalar value that can represent the overall pressure level, providing a clear evaluation basis for subsequent comparison with the target pressure and iterative convergence, and improving the degree of automation.

[0016] In conjunction with some embodiments of the first aspect, in some embodiments, after the step of adjusting the radial interference and re-executing the simulation analysis when the difference between the dynamic average contact pressure and the target contact pressure is greater than a preset threshold, until the dynamic average contact pressure converges to the target contact pressure, the method further includes: the computing device applying a time-varying transient current load simulating the cathode roller's cold start-up to steady-state operation in the electric field model; the computing device using time-varying Joule heat as a transient body heat source in the thermal field model to solve the transient temperature field time history distribution of the cathode roller mating surface throughout the start-up process; the computing device calculating the transient contact pressure time history distribution of the cathode roller mating surface during the start-up process by superimposing the thermal stress and assembly stress generated based on the transient temperature field time history distribution in the force field model; the computing device identifying the transient valley pressure with the lowest pressure from the transient contact pressure time history distribution; and when the transient valley pressure is less than a preset transient safety pressure threshold, the computing device adjusting the radial interference and re-executing the simulation analysis until the transient valley pressure is not less than the transient safety pressure threshold.

[0017] By adopting the above technical solutions, the computational equipment, based on steady-state optimization, further considers the more stringent transient condition of starting the cathode roller from a cold state. Through transient analysis, it predicts and captures transient pressure loss phenomena caused by differences in the thermophysical properties of materials, ensuring that the minimum valley pressure remains above the safety threshold. This improves the safety redundancy and operational reliability of the design, thereby reducing the potential risk of equipment damage due to poor contact or arcing during startup.

[0018] In conjunction with some embodiments of the first aspect, in some embodiments, the steps of adjusting the radial interference and re-executing the simulation analysis by the computing device specifically include: when the dynamic average contact pressure is less than the target contact pressure, the computing device increases the radial interference by a preset increment ratio; when the dynamic average contact pressure is greater than the target contact pressure, the computing device decreases the radial interference by a preset reduction ratio; the computing device recalculates the outer radius of the cathode roller based on the adjusted radial interference and the preset contact pressure calculation formula, and updates the geometric parameters of the electric field model, thermal field model, and force field model.

[0019] By adopting the above technical solution, the interference fit is increased or decreased proportionally according to the deviation direction between the dynamic average contact pressure and the target pressure, and the model geometric parameters are updated in conjunction with the simulation for a new round of simulation, thereby improving design efficiency.

[0020] In a second aspect, embodiments of this application provide a computing device comprising: one or more processors and a memory; the memory is coupled to the one or more processors and is used to store computer program code, the computer program code including computer instructions, wherein the one or more processors invoke the computer instructions to cause the computing device to perform the method described in the first aspect and any possible implementation thereof.

[0021] Thirdly, embodiments of this application provide a computer program product containing instructions that, when the computer program product is run on a computing device, cause the computing device to perform the method described in the first aspect and any possible implementation thereof.

[0022] Fourthly, embodiments of this application provide a computer-readable storage medium including instructions that, when executed on a computing device, cause the computing device to perform the method described in the first aspect and any possible implementation thereof.

[0023] It is understood that the computing device provided in the second aspect, the computer program product provided in the third aspect, and the computer storage medium provided in the fourth aspect are all used to execute the methods provided in the embodiments of this application. Therefore, the beneficial effects they can achieve can be referred to the beneficial effects in the corresponding methods, and will not be repeated here.

[0024] One or more technical solutions provided in the embodiments of this application have at least the following technical effects or advantages: 1. The computing device determines a pressure value—the target contact pressure—that enables a stable and sufficiently low contact resistance between the cathode roller and the titanium sleeve through a preset contact resistance test experiment. Next, the computing device uses a preset contact pressure calculation formula to perform a reverse calculation to obtain the outer radius of the cathode roller. The difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller is then used as the titanium layer thickness. This determines a minimum titanium layer thickness that satisfies the prerequisite of sufficiently low contact resistance, thereby reducing the total resistance of the cathode roller (composed of the titanium layer volume resistance and the contact resistance of the mating surface) and the operating energy consumption of the cathode roller.

[0025] 2. The computational equipment further incorporates electro-thermal-mechanical multiphysics coupling simulation analysis to verify and iteratively optimize the initial design. This method can simulate the thermal stress generated by the cathode roller under actual operating current due to the Joule heating effect and material thermal expansion mismatch, and calculate the total contact pressure under dynamic conditions. By comparing this dynamic pressure with the target pressure and iteratively adjusting it, the final designed titanium layer thickness not only meets the static assembly requirements but also improves stability under complex operating conditions.

[0026] 3. Based on steady-state optimization, the computational equipment further considers the more stringent transient condition of starting the cathode roller from a cold state. Through transient analysis, it predicts and captures transient pressure loss phenomena caused by differences in the thermophysical properties of materials, ensuring that the minimum valley pressure remains above the safe threshold. This improves the design's safety redundancy and operational reliability, thereby reducing the potential risk of equipment damage due to poor contact or arcing during startup. Attached Figure Description

[0027] Figure 1 This is a flowchart illustrating a method for calculating the thickness of the titanium layer on a cathode roller according to an embodiment of this application.

[0028] Figure 2 This is another flowchart illustrating a method for calculating the thickness of the titanium layer on a cathode roller according to an embodiment of this application.

[0029] Figure 3 This is a schematic diagram of the physical device structure of a computing device in the embodiments of this application. Detailed Implementation

[0030] The terminology used in the following embodiments of this application is for the purpose of describing particular embodiments only and is not intended to be limiting of this application. As used in the specification and appended claims of this application, the singular expressions “a,” “an,” “the,” “the,” “the,” and “this” are intended to include the plural expressions as well, unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used in this application refers to and includes any or all possible combinations of one or more of the listed items.

[0031] Hereinafter, the terms "first" and "second" are used for descriptive purposes only and should not be construed as implying or suggesting relative importance or implicitly indicating the number of indicated technical features. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature, and in the description of the embodiments of this application, unless otherwise stated, "multiple" means two or more.

[0032] This application provides a method and apparatus for calculating the thickness of the titanium layer in a cathode roller, thereby improving the design accuracy of the titanium layer thickness in the cathode roller.

[0033] Please see Figure 1 This is a flowchart illustrating a method for calculating the thickness of a cathode roller titanium layer in an embodiment of this application.

[0034] S101. The calculation device determines the target contact pressure for achieving a stable contact state between the cathode roller and the titanium sleeve through a preset contact resistance test experiment.

[0035] The pre-designed contact resistance test is a standardized physical experiment designed in advance to simulate the electrical characteristics of the internal interface of the cathode roller, used to quantify the contact resistance of different materials under different pressures. The cathode roller shaft represents the core internal component of the cathode roller, typically made of highly conductive copper or a steel-copper composite structure that combines strength and conductivity, used to bear structural loads and conduct most of the current. The titanium sleeve refers to the titanium alloy layer wrapped around the cathode roller shaft, its main function being to provide a surface resistant to electrolyte corrosion. A stable contact state refers to a physical state where, after the pressure applied to the two contact surfaces increases to a certain level, the interfacial contact resistance no longer decreases significantly, tending towards a stable low value. This state indicates that the physical contact and electrical pathway between the interfaces have reached saturation and are excellent. The target contact pressure is the minimum interfacial normal stress value required to achieve the above stable contact state; it is a key design input parameter directly related to whether the cathode roller can maintain low energy consumption and high-efficiency current transmission during operation.

[0036] Specifically, in the initial design phase of the cathode roller, the computing device first invokes or controls a contact resistance testing system. This system includes a contact test fixture capable of accurately applying and measuring pressure, and a four-probe measuring instrument capable of accurately measuring minute resistance values. Before the experiment, standard samples with the same material and surface roughness are prepared according to the actual material composition of the cathode roller (e.g., titanium sleeve and copper base, or titanium sleeve and steel mandrel). At the start of the experiment, the computing device controls the test fixture to gradually apply pressure to the sample from zero using a preset pressure gradient, for example, increasing by 0.5 MPa in each step. At each pressure level, the pressure is kept stable, and the current contact resistance value is collected and recorded by the resistance measuring instrument. Through multiple measurements, a series of pressure-resistance data points are obtained, thereby plotting a contact resistance-pressure curve. Because only a few micro-protrusions are in contact on the contact surface in the low-pressure region, the contact resistance is high. As the pressure increases, the contact area increases, the conductive channels increase, and the resistance decreases rapidly. When the pressure reaches a certain value, plastic deformation causes the contact area growth to tend to saturate, and the resistance value also stabilizes. The computing device performs nonlinear fitting on the curve and calculates its slope or first derivative, identifying the inflection point region where the curve's slope decreases and flattens out. To quantitatively determine the steady state, the computing device further analyzes the contact resistance change rate corresponding to multiple consecutive pressure points (e.g., 5) after the inflection point. When the absolute values ​​of these change rates are all less than a preset change rate threshold (e.g., 1%), the contact resistance is considered to have reached stability. At this point, the computing device determines the pressure value corresponding to the initial pressure point that leads to the steady state as the stable contact pressure for this material combination and stores it as the target contact pressure required for subsequent calculations.

[0037] S102. The calculation device calculates the radial interference generated by the titanium sleeve during the cooling and shrinking process based on the thermal expansion coefficient of the titanium sleeve and the heating temperature range of the hot fitting process.

[0038] The coefficient of thermal expansion refers to the degree of change in length or volume of a material under a unit temperature change; it is an inherent thermophysical property of the material. Thermal fitting is an assembly method that utilizes the principle of thermal expansion and contraction to achieve a tight fit between parts. Typically, the outer ring part (sleeve) is heated to expand its inner hole, then fitted onto the inner shaft part. After cooling, the sleeve contracts, generating a significant clamping force. The heating temperature range refers to the interval within which the titanium sleeve is heated from ambient temperature to the target peak temperature in the thermal fitting process. The upper limit of this range is limited by material phase transformation, oxidation, and equipment capabilities. Radial interference refers to the dimensional difference between the inner diameter of the titanium sleeve and the outer diameter of the cathode roller shaft at room temperature. This geometric interference is the fundamental source of contact pressure between the mating surfaces.

[0039] Specifically, this step is performed after the target contact pressure is determined. Its purpose is to initially transform the abstract target contact pressure into a specific geometric dimension, namely the radial interference. The calculation equipment first retrieves the linear thermal expansion coefficient (α) of the titanium alloy within the corresponding temperature range from the material database. Simultaneously, based on the factory's process specifications or safe operating procedures, the heating temperature (T_heat) and the ambient temperature (T_ambient) during assembly are determined, thus obtaining the total temperature difference ΔT = T_heat - T_ambient. The titanium ring for heat fitting, after previous smelting and plastic processing, achieves a fine-grained network structure suitable for electrical conductivity. To ensure that heating does not change the grain size and distribution, the heating temperature (T_heat) must be controlled to be less than the safe heating temperature (T). Theoretically, in order for a sleeve with an inner diameter of d to be smoothly fitted onto a shaft with an outer diameter of d+δ, the inner diameter expansion (Δd) of the sleeve after heating must be at least equal to the interference δ. Based on the linear expansion formula Δd = dαΔT, the calculation equipment can initially estimate an interference that matches the heat fitting process. However, in practice, to ensure sufficient clearance during assembly to avoid scratches and jamming, an assembly clearance (Δc) is usually reserved. Therefore, the radial interference δ calculated by the calculation equipment is based on the inner diameter of the sleeve after expansion at the heating temperature, which should be larger than the outer diameter of the shaft by a safe assembly clearance.

[0040] S103. The calculation equipment uses finite element simulation analysis software to perform simulation analysis based on the elastic modulus and Poisson's ratio of the titanium sleeve to obtain the equivalent elastic modulus and equivalent Poisson's ratio of the cathode roller.

[0041] Among them, the elastic modulus (also known as Young's modulus) is used to represent the material's ability to resist elastic deformation; the larger the value, the stiffer the material. Poisson's ratio is used to represent the absolute value of the ratio of lateral deformation to axial deformation when a material is under uniaxial tension or compression. Finite element simulation analysis software refers to commercial or open-source computer-aided engineering (CAE) tools such as ANSYS and ABAQUS, which can discretize complex continuum structures into a finite number of elements and approximate the mechanical response of the structure by solving a system of algebraic equations. In this context, the cathode roller specifically refers to a composite structure composed of multiple layers of materials (such as a steel core and copper sleeve). The equivalent elastic modulus and equivalent Poisson's ratio are virtual material parameters calculated using specific methods to simplify the complex multilayer roller into a homogeneous solid body in subsequent analytical formulas, and are equivalent to the original composite structure in terms of macroscopic mechanical response.

[0042] Specifically, classic formulas for contact pressure in thick-walled cylinders (such as Lamé's formula) typically assume that both mating components are homogeneous and isotropic materials. However, the shaft of a high-performance cathode roller is often a composite structure of a steel core and a copper sleeve to balance strength and conductivity, and its mechanical behavior cannot be simply represented by the material parameters of steel or copper. To continue using concise and efficient analytical formulas for reverse engineering, this step employs finite element numerical experiments to calibrate the equivalent mechanical properties of the composite shaft. The computational device first establishes an accurate two-dimensional axisymmetric or three-dimensional finite element model for only the cathode roller shaft portion (excluding the titanium sleeve). The model includes both the steel core and the copper sleeve, and assigns them their respective true elastic moduli and Poisson's ratios. Then, a known uniform virtual radial pressure P_virtual is applied to the outer surface of the copper sleeve in the model. Static structural analysis is performed to solve for and obtain the radial displacement U_r of the outer surface of the copper sleeve under this virtual pressure. Finally, the computing device treats this composite shaft as a black box, utilizing the material properties required for a single-layer thick-walled cylinder to produce the same radial displacement U_r under the same external pressure P_virtual, and calculates its equivalent elastic modulus E_eff and equivalent Poisson's ratio ν_eff using inverse kinematics. Thus, the complex composite shaft is replaced by a homogeneous solid shaft with E_eff and ν_eff properties.

[0043] S104. Based on the target contact pressure, radial interference, equivalent elastic modulus, and equivalent Poisson's ratio, the calculation device performs a reverse solution using a preset contact pressure calculation formula to obtain the outer radius of the cathode roller.

[0044] The preset contact pressure calculation formula refers to a mathematical model describing the relationship between the interference fit, material properties, geometric dimensions, and contact pressure. Reverse solving is a calculation approach opposite to conventional forward calculation (i.e., calculating pressure given all dimensions). It involves solving for unknown input parameters (geometric dimensions) given the desired output (target pressure) and some inputs (material properties, interference fit). The outer radius of the cathode roller refers to the radius of the shaft portion that mates with the inner surface of the titanium sleeve in the design target.

[0045] Specifically, the preset contact pressure calculation formula is as follows: Where p is the target contact pressure. The radial interference is denoted as , where 'a' is the outer radius of the cathode roller shaft and 'b' is the outer radius of the titanium sleeve. and These are the elastic moduli of the cathode roller and the titanium sleeve, respectively. and These are the Poisson's ratios of the cathode roller and the titanium sleeve, respectively. Solving the above formula inversely yields: ,make ,but ,get Substituting the target contact pressure, radial interference, elastic modulus and Poisson's ratio of the titanium sleeve, and equivalent elastic modulus and equivalent Poisson's ratio of the cathode roller obtained from the above steps into the formula, we can calculate a (outer radius of the cathode roller).

[0046] S105. The calculation device uses the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller as the titanium layer thickness.

[0047] Specifically, the calculation device uses the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller shaft calculated in S104 as the titanium layer thickness.

[0048] In the above embodiments, a physical performance-oriented method for designing the titanium layer thickness of a cathode roller is provided through a series of calculation steps based on idealized static conditions. This method first determines the target contact pressure to ensure excellent electrical contact through experiments. Then, combining the thermal charging process and the principle of equivalence in composite material mechanics, it uses analytical formulas to deduce the initial geometric dimensions of the cathode roller shaft and titanium layer required to meet this target pressure.

[0049] However, the titanium layer thickness obtained in the above embodiments is calculated based on an ideal state after static assembly at room temperature. During the actual operation of the cathode roller, the passage of a large current generates significant Joule heating, resulting in an uneven temperature field among the various components (titanium, copper, and steel). Due to the significant differences in the coefficients of thermal expansion of different materials, this temperature field induces secondary thermal stress, thereby altering the actual contact pressure at the mating surfaces. This dynamically changing contact pressure during operation may deviate from the initially set target contact pressure, leading to problems such as decreased conductivity, localized overheating, and even a risk of a brief drop in contact pressure (transient pressure loss) at startup due to differences in thermal response speed.

[0050] Please see Figure 2 This is another flowchart illustrating a method for calculating the thickness of the titanium layer on a cathode roller in an embodiment of this application.

[0051] S201. The calculation device determines the target contact pressure for achieving a stable contact state between the cathode roller and the titanium sleeve through a preset contact resistance test experiment.

[0052] S202. The calculation device calculates the radial interference generated by the titanium sleeve during the cooling and shrinking process based on the thermal expansion coefficient of the titanium sleeve and the heating temperature range of the hot fitting process.

[0053] S203. The calculation device uses finite element simulation analysis software to perform simulation analysis based on the elastic modulus and Poisson's ratio of the titanium sleeve to obtain the equivalent elastic modulus and equivalent Poisson's ratio of the cathode roller.

[0054] S204. Based on the target contact pressure, radial interference, equivalent elastic modulus, and equivalent Poisson's ratio, the calculation device performs a reverse solution using a preset contact pressure calculation formula to obtain the outer radius of the cathode roller.

[0055] S205. The calculation device uses the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller as the titanium layer thickness.

[0056] Step S201 is similar to step S101, step S202 is similar to step S102, step S203 is similar to step S103, step S204 is similar to step S104, and step S205 is similar to step S105, so they will not be repeated here.

[0057] S206. The calculation device establishes the electric field model, thermal field model and force field model of the cathode roller based on the outer radius of the cathode roller shaft, the outer radius of the titanium sleeve and the thickness of the titanium layer.

[0058] The electric field model is a mathematical and physical model used to describe and calculate the current density distribution, potential distribution, and energy loss inside the cathode roller. The thermal field model is a mathematical and physical model used to describe and calculate the heat generation, heat conduction, convection, and radiation processes generated by electrothermal conversion inside the cathode roller, and ultimately solve for its temperature distribution. The force field model (or structural mechanics model) is a mathematical and physical model used to describe and calculate the stress, strain, and displacement of the cathode roller under internal and external loads (such as assembly stress, thermal stress, and mechanical pressure). These three models are geometrically unified, but each has its own emphasis on physical properties and boundary conditions, and are solved in a coupled manner using finite element analysis software.

[0059] Specifically, after obtaining the initial geometric dimensions of each component of the cathode roller (outer radius a of the cathode roller shaft, outer radius b of the titanium sleeve, titanium layer thickness t=ba) through steps S201-S205, the computing device starts the finite element analysis software and constructs a three-dimensional or two-dimensional axisymmetric geometric model of the cathode roller based on these dimensional parameters. This model includes at least two parts: the titanium sleeve and the internal roller shaft (e.g., a steel core-copper sleeve composite structure). Subsequently, the computing device assigns corresponding physical properties to different parts of the model: electrical conductivity for the electric field model; thermal conductivity, specific heat capacity, and density for the thermal field model; and elastic modulus, Poisson's ratio, and coefficient of thermal expansion for the force field model. These models share the same mesh, forming a digital twin of the cathode roller.

[0060] In some embodiments, multiphysics models can be established in several ways: Optionally, step-by-step modeling and mesh mapping techniques can be employed. First, models are independently established and initially configured in the electrical, thermal, and mechanical analysis modules respectively. Second, the data transfer relationships between fields are defined using the software's coupling interface; for example, the Joule heat calculated from the electric field analysis is loaded as a load into the thermal field model. Finally, mesh mapping techniques enable precise data transfer between different physical fields in shared geometric regions (such as the entire cathode roller) or boundaries (such as mating surfaces). Optionally, a parameterized model can be established. First, key geometric dimensions (such as radial interference δ or the outer radius a of the cathode roller shaft) and operating parameters (such as current magnitude and coolant temperature) are defined as variable parameters. Second, the entire modeling, loading, solving, and post-processing process is scripted. Finally, this allows subsequent iterative optimization processes to be automated; the computing device only needs to adjust the parameter values ​​to automatically regenerate the model and perform a new round of simulation analysis.

[0061] S207. The calculation device applies the actual working current density of the cathode roller in the electric field model, and calculates the current distribution and Joule heat distribution on the mating surface of the cathode roller based on the contact resistance-pressure curve.

[0062] The actual operating current density refers to the total current received and conducted from the external power source by the cathode roller under rated production conditions, divided by the conductive contact area; it is the core load for electric field analysis. The contact resistance-pressure curve refers to the functional relationship or data table obtained in step S101, describing the change in unit area resistance of the mating surface with normal pressure. Current distribution refers to the non-uniform distribution of current flowing through the mating surface of the titanium sleeve and roller shaft, depending on the magnitude of local contact resistance. Joule heat distribution refers to the distribution of heat power density generated at various points on the cathode roller, determined by the non-uniform current and resistance (including volume resistance and contact resistance).

[0063] Specifically, the computational device first applies assembly stress generated by the initial radial interference δ to the force field model, thus obtaining an initial contact pressure distribution at room temperature. Then, in the electric field analysis, the device uses this pressure distribution as input and assigns a specific contact resistance value to each mesh element on the mating surface by consulting or interpolating the contact resistance-pressure curve. Simultaneously, a total operating current is applied as a boundary condition to the external conductive region of the model. The finite element solver then solves the electric field equations to obtain the potential distribution and current density vector field of the entire cathode roller. In particular, on the mating surface, the current tends to flow towards areas with higher pressure and lower contact resistance, forming a non-uniform current distribution. Based on Joule's law, the computational device calculates the Joule heat generated by the combined volume resistance and contact resistance, which is also non-uniformly distributed, typically concentrated in areas with higher contact resistance or higher current density.

[0064] S208. The computing device uses the Joule heat distribution as the volume heat source in the thermal field model. At the same time, based on the convective heat transfer boundary conditions of the internal cooling medium and the heat transfer boundary conditions of the external electrolyte, it solves the three-dimensional temperature field distribution of the cathode roller during steady-state operation.

[0065] In this context, the volumetric heat source refers to the thermal power loaded as a heat generation term into the volume of each element in the model, as defined in the heat conduction equation. The convective heat transfer boundary condition is a mathematical expression describing the heat transfer between a solid surface and a fluid, typically defined by the convective heat transfer coefficient and the fluid temperature. Steady-state operation refers to a state where the temperature at each point on the cathode roller no longer changes with time, i.e., the heat generation rate and the heat dissipation rate are in equilibrium. The three-dimensional temperature field distribution refers to the set of temperature values ​​at every point in the cathode roller space under steady-state conditions, usually presented in the form of a contour map.

[0066] Specifically, this step aims to determine the final temperature state of the cathode roller after long-term operation. The computing device applies the Joule heat distribution calculated in S207 as a thermal load to the corresponding element of the thermal field model. Simultaneously, heat dissipation boundary conditions are set according to the actual cooling method and operating environment of the cathode roller. For example, a convective heat transfer boundary representing internal cooling water or oil is applied to the inner surface of the roller shaft, characterized by a high heat transfer coefficient and a low fluid temperature. On the outer surface of the titanium sleeve, a convective heat transfer boundary representing the external electrolyte is applied, characterized by a moderate heat transfer coefficient and a high electrolyte temperature. After setting these conditions, the computing device solves the steady-state heat conduction equation. Because titanium has much lower thermal conductivity than copper and steel, heat is mainly conducted through the highly thermally conductive roller shaft to the internal cooling medium, resulting in a significant temperature gradient between the titanium sleeve and the roller shaft. Finally, the computing device obtains a detailed three-dimensional temperature field distribution.

[0067] S209. Calculate the assembly stress generated by the radial interference applied to the force field model and the thermal stress generated by the three-dimensional temperature field, and calculate the total contact pressure distribution of the cathode roller mating surface after the two are superimposed.

[0068] Among them, assembly stress refers to the internal stress generated by the initial interference fit at room temperature, which puts the sleeve in a tensile state and the shaft in a compressive state. Thermal stress refers to the internal stress generated when there is a temperature gradient inside an object or when a combination of different materials is heated / cooled as a whole, due to the constraint of the free thermal expansion and contraction of each part. Total contact pressure distribution refers to the final normal pressure distribution pattern on the mating surface under steady-state working conditions, generated by the combined action of the initial assembly stress and additional thermal stress.

[0069] Specifically, in the force field model, the computing device first applies an initial interference δ to obtain the assembly stress field. Then, the three-dimensional temperature field distribution obtained from S208 is applied as a thermal load to the entire model. Since the titanium sleeve, copper layer, and steel core have different coefficients of thermal expansion, they will exhibit inconsistent deformation tendencies under the influence of the non-uniform temperature field. For example, the titanium sleeve, with its higher temperature, attempts to expand more than the roller shaft with its lower temperature. This inconsistency in deformation is constrained by the connection relationship of the mating surfaces, thus generating enormous thermal stress within the structure. The computing device calculates this thermal stress field by solving the mechanical equilibrium equations and linearly superimposes it with the initial assembly stress field. Finally, the normal stress components on the mating surfaces are extracted, yielding the total contact pressure distribution that considers the complete electro-thermal-mechanical coupling effect.

[0070] S210, The calculation device calculates the dynamic average contact pressure based on the total contact pressure distribution.

[0071] The total contact pressure distribution refers to the pressure numerical map obtained in S209, which shows the pressure variation with position on the mating surface. The dynamic average contact pressure refers to a single scalar value that represents the overall level of this complex pressure distribution map, calculated mathematically, in order to compare it with the design target.

[0072] Specifically, the computing device first discretizes the mating surface of the cathode roller (a cylindrical surface) into a series of tiny mesh elements along the circumferential and axial directions. Then, it extracts the normal contact pressure value P_i at the center point of each element and the area A_i of that element from the simulation results. To obtain an average value that fairly reflects the overall pressure level, the computing device uses an area-weighted average method for integral calculation. The specific formula is: P_avg=Σ(P_i*A_i) / Σ(A_i), and the summation applies to all mesh elements on the mating surface. This calculated P_avg is the dynamic average contact pressure, which integrates the combined effects of the initial assembly pressure and the thermal effects during operation, and is a key indicator for evaluating the performance of the current design under actual working conditions.

[0073] S211. When the difference between the dynamic average contact pressure and the target contact pressure is greater than the preset threshold, the calculation device adjusts the radial interference and re-executes the simulation analysis until the dynamic average contact pressure converges to the target contact pressure.

[0074] The preset threshold is a tolerance used to determine whether the calculation result meets the design requirements, such as ±5% of the target contact pressure. Adjusting the radial interference refers to strategically modifying the initial design parameters based on the deviation between the current calculation result and the target. Convergence refers to the process of iteratively adjusting and calculating until the calculated value of the dynamic average contact pressure gets closer and closer to the target contact pressure, eventually stabilizing within the preset threshold range.

[0075] Specifically, the computing device compares the dynamic average contact pressure P_avg calculated in S210 with the target contact pressure P_target determined in S201. If |P_avg - P_target| > the preset threshold, it indicates that there is a deviation in the initial design based on the normal temperature static model. At this time, the computing device automatically adjusts the radial interference δ according to the direction of the deviation: if P_avg < P_target, it means that the thermal effect during operation causes excessive pressure loss, and the initial clamping force needs to be increased. Therefore, δ is increased according to a preset increment ratio (such as increasing by 2%); conversely, if P_avg > P_target, it means that the thermal effect causes excessive pressure, which may bring excessive stress risks. Therefore, δ is reduced according to a preset decrement ratio. After adjusting δ, the computing device starts to re-execute the entire process from step S204 based on the new δ value: recalculate the roller radius a, update the model geometric parameters in S206, and execute the full set of electro-thermal-mechanical coupling simulations from S207 to S210 again. This cycle of calculation - comparison - adjustment - recalculation will continue until P_avg converges within the preset threshold range of P_target. At this time, the obtained titanium layer thickness is the final optimized design verified by the dynamic working conditions.

[0076] In addition, to further ensure the safety of the cathode roller under harsh transient working conditions, after the above steady-state optimization cycle is completed, the method may further include a transient analysis of the startup process. Specifically, the computing device applies a transient current load that changes with time and simulates a cold startup based on the model that has passed the steady-state verification. In the transient thermal analysis, the time-varying Joule heat is used as the transient body heat source to solve the time history distribution of the transient temperature field of the cathode roller during the entire startup process. Subsequently, in the transient force field analysis, the dynamic thermal stress generated based on this temperature time history is superimposed with the assembly stress to calculate the time history distribution of the contact pressure on the mating surface that changes with time. The computing device identifies the transient valley pressure with the lowest pressure from this pressure time history curve. If this valley pressure is less than a preset transient safety pressure threshold (this threshold ensures that no arc or contact failure occurs even at the lowest pressure point), it is considered that there is a risk of transient pressure loss in the design. At this time, the radial interference needs to be adjusted (usually increased), and the steady-state and transient simulation analyses are re-executed until the steady-state average pressure meets the standard and the transient valley pressure is not lower than the safety threshold.

[0077] In the above embodiments, a complete closed-loop iterative optimization design process was formed by constructing an electro-thermal-mechanical multiphysics coupling model and combining steady-state and transient simulation analyses. This method not only verified the performance of the initial design under steady-state operating conditions but also, through iterative adjustment of the radial interference, ensured that the dynamic average contact pressure converged to the target value. Furthermore, by performing transient analysis on the startup process, it ensured that a safe contact state could be maintained even at the lowest pressure trough. The final titanium layer thickness design scheme is a comprehensive optimal solution that simultaneously satisfies static assembly requirements, steady-state operating performance, and transient startup safety, thereby improving the reliability of the cathode roller throughout its entire working cycle.

[0078] The above describes a method for calculating the thickness of the titanium layer on a cathode roller in the embodiments of this application. The exemplary computing device 300 provided in the embodiments of this application is described below.

[0079] Figure 3 This is a schematic diagram of an exemplary hardware structure of a computing device 300 provided in an embodiment of this application. In some embodiments, the computing device 300 is a computer device. The computer device includes a processor, a memory, and a network interface connected via a system bus. The processor of the computer device provides computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system, computer programs, and a database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage medium. The database of the computer device stores data. The network interface of the computer device is used to communicate with other external terminals or servers via a network connection. In some embodiments, the network interface can be a wired network interface; in some embodiments, the network interface can also be a wireless network interface. When the computer program is executed by the processor, it implements a method for calculating the thickness of a cathode roller titanium layer according to an embodiment of this application.

[0080] Those skilled in the art will understand that Figure 3 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.

[0081] In some embodiments of this application, a computer-readable storage medium is also provided, including instructions that, when executed on the computing device 300, cause the computing device 300 to perform a method for calculating the thickness of a cathode roller titanium layer according to an embodiment of this application.

[0082] The above-described embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit it. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of this application.

[0083] As used in the above embodiments, depending on the context, the term "when..." can be interpreted as meaning "if...", "after...", "in response to determining...", or "in response to detecting...". Similarly, depending on the context, the phrase "when determining..." or "if (the stated condition or event) is interpreted as meaning "if determining...", "in response to determining...", "when (the stated condition or event) is detected", or "in response to detecting (the stated condition or event)".

[0084] In the above embodiments, implementation can be achieved entirely or partially through software, hardware, firmware, or any combination thereof. When implemented using software, it can be implemented entirely or partially in the form of a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., solid-state drive), etc.

[0085] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. This program can be stored in a computer-readable storage medium, and when executed, it can include the processes described in the above method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as ROM or random access memory (RAM), magnetic disks, or optical disks.

Claims

1. A method for calculating the thickness of the titanium layer on a cathode roller, characterized in that, Applied to a computing device, the method includes: The computing device determines the target contact pressure for achieving a stable contact state between the cathode roller and the titanium sleeve through a preset contact resistance test experiment. The computing device calculates the radial interference generated by the titanium sleeve during the cooling and shrinking process based on the coefficient of thermal expansion of the titanium sleeve and the heating temperature range of the hot fitting process. The computing device performs simulation analysis based on the elastic modulus and Poisson's ratio of the titanium sleeve using finite element simulation analysis software to obtain the equivalent elastic modulus and equivalent Poisson's ratio of the cathode roller. Based on the target contact pressure, the radial interference, the equivalent elastic modulus, and the equivalent Poisson's ratio, the calculation device performs a reverse solution using a preset contact pressure calculation formula to obtain the outer radius of the cathode roller. The computing device uses the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller as the titanium layer thickness.

2. The method according to claim 1, characterized in that, The preset contact pressure calculation formula specifically includes: Where p is the target contact pressure. The radial interference is denoted as , where 'a' is the outer radius of the cathode roller shaft and 'b' is the outer radius of the titanium sleeve. and These are the elastic moduli of the cathode roller and the titanium sleeve, respectively. and These are the Poisson's ratios of the cathode roller and the titanium sleeve, respectively.

3. The method according to claim 1, characterized in that, The calculation device determines the target contact pressure for achieving a stable contact state between the cathode roller and the titanium sleeve through a preset contact resistance test experiment, specifically including: The computing device controls the contact test fixture to gradually increase the contact pressure according to a preset pressure gradient. The test fixture is equipped with prefabricated titanium-copper, titanium-steel, and steel-copper materials. The computing device collects contact resistance values ​​for different material combinations at each pressure point and records the corresponding pressure values ​​to obtain a contact resistance-pressure curve. The computing device performs fitting analysis on the resistance-pressure curve to identify the inflection point of the contact resistance as a function of pressure. The computing device calculates the contact resistance change rate corresponding to a preset number of consecutive pressure points after the inflection point. When the absolute value of the change rate of the contact resistance is less than the preset change rate threshold, the computing device determines that the contact resistance has reached a stable state. The computing device uses the pressure value corresponding to the stable state as the stable contact pressure for different material combinations, and selects the corresponding stable contact pressure as the target contact pressure based on the actual material combination of the cathode roller.

4. The method according to claim 3, characterized in that, After the step of the computing device using the difference between the outer radius of the titanium sleeve and the outer radius of the cathode roller as the titanium layer thickness, the method further includes: The computing device establishes an electric field model, a thermal field model, and a force field model of the cathode roller based on the outer radius of the cathode roller shaft, the outer radius of the titanium sleeve, and the thickness of the titanium layer. The computing device applies the actual working current density of the cathode roller to the electric field model and calculates the current distribution and Joule heat distribution on the mating surface of the cathode roller based on the contact resistance-pressure curve. The computing device uses the Joule heat distribution as the volume heat source in the thermal field model, and solves the three-dimensional temperature field distribution of the cathode roller during steady-state operation based on the convective heat transfer boundary conditions of the internal cooling medium and the heat transfer boundary conditions of the external electrolyte. The computing device applies the assembly stress generated by the radial interference and the thermal stress generated by the three-dimensional temperature field to the force field model, and calculates the total contact pressure distribution of the cathode roller mating surface after the superposition of the two. The computing device calculates the dynamic average contact pressure based on the total contact pressure distribution; When the difference between the dynamic average contact pressure and the target contact pressure is greater than a preset threshold, the computing device adjusts the radial interference and re-executes the simulation analysis until the dynamic average contact pressure converges to the target contact pressure.

5. The method according to claim 4, characterized in that, The step of calculating the dynamic average contact pressure based on the total contact pressure distribution by the computing device specifically includes: The computing device divides the mating surface of the cathode roller into regular grid cells according to the circumferential and axial directions; The computing device extracts the normal contact pressure and corresponding contact area of ​​each grid cell; The computing device uses an area-weighted average method to perform integral calculation on the total contact pressure distribution to obtain the dynamic average contact pressure.

6. The method according to claim 4, characterized in that, After the step of adjusting the radial interference and re-executing the simulation analysis when the difference between the dynamic average contact pressure and the target contact pressure is greater than a preset threshold, until the dynamic average contact pressure converges to the target contact pressure, the method further includes: The computing device applies a time-varying transient current load to the electric field model, simulating the cathode roller's transition from a cold start to steady-state operation. The computing device uses the time-varying Joule heat as a transient body heat source in the thermal field model to solve the transient temperature field time history distribution of the cathode roller mating surface throughout the entire startup process; The computing device, in the force field model, superimposes the thermal stress generated by the transient temperature field time history distribution with the assembly stress to calculate the transient contact pressure time history distribution of the cathode roller mating surface during the startup process; The computing device identifies the transient valley pressure with the lowest pressure from the transient contact pressure time history distribution; When the transient valley pressure is less than the preset transient safety pressure threshold, the computing device adjusts the radial interference and re-executes the simulation analysis until the transient valley pressure is not less than the transient safety pressure threshold.

7. The method according to claim 4, characterized in that, The steps of adjusting the radial interference and re-executing the simulation analysis by the computing device specifically include: When the dynamic average contact pressure is less than the target contact pressure, the computing device increases the radial interference according to a preset increment ratio; When the dynamic average contact pressure is greater than the target contact pressure, the calculation device reduces the radial interference according to a preset reduction ratio; The computing device recalculates the outer radius of the cathode roller based on the adjusted radial interference and the preset contact pressure calculation formula, and updates the geometric parameters of the electric field model, thermal field model and force field model.

8. A computing device, characterized in that, The computing device includes: one or more processors and a memory; the memory is coupled to the one or more processors, the memory is used to store computer program code, the computer program code including computer instructions, and the one or more processors invoke the computer instructions to cause the computing device to perform the method as described in any one of claims 1-7.

9. A computer program product containing instructions, characterized in that, When the computer program product is run on a computing device, the computing device performs the method as described in any one of claims 1-7.

10. A computer-readable storage medium comprising instructions, characterized in that, When the instructions are executed on a computing device, the computing device performs the method as described in any one of claims 1-7.

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