A numerical prediction method for stability characteristics of a vertical riser with coarse particles and liquid under internal flow excitation

Abstract

The application discloses a numerical prediction method for stability characteristic of a rough particle solid-liquid two-phase inner flow excited vertical lifting pipeline, and belongs to the technical field of deep-sea mining flow pipeline. In order to solve the technical problem that current numerical prediction of the stability of the flow pipeline generally only considers single-phase inner flow or gas-liquid two-phase flow, and it is difficult to effectively apply to the solid-liquid two-phase flow situation of deep-sea mining, the application adopts the slip velocity to describe the relationship between the movement speed of the solid particles in the vertical lifting pipeline and the flowing speed of the fluid in the pipeline, so as to determine the local concentration in the pipeline during stable conveying, and obtain the solid phase mass and the liquid phase mass per unit length in the pipeline during stable conveying; based on the modified Hamilton principle, a numerical prediction model for vibration of the vertical lifting pipeline under the excitation action of the solid-liquid two-phase flow is established; high-order partial differential equations which change with time and space are reduced to a series of ordinary differential equations which only contain time items; based on the eigenvalue method, a matrix eigenvalue equation is established, and the pipeline stability is determined by solving the eigenvalue.

Description

Technical Field

[0001] This invention belongs to the field of deep-sea mining pipeline technology, specifically relating to a numerical prediction method for the stability characteristics of vertical lifting pipelines. Background Technology

[0002] Developing marine resources and utilizing marine space are two core themes of human ocean development in the 21st century. The seabed contains extremely rich mineral resources, the effective development of which has significant strategic importance and enormous commercial potential. Pipeline-lift mining systems are currently recognized internationally as the most practical and promising mining systems in the field of deep-sea mineral resource development. The vertical lift pipeline, with its large aspect ratio, is the weakest and most technically challenging part of this system, but also its most crucial core component. Ensuring the stable and safe transport of ore within long-distance vertical lift pipelines is one of the key issues urgently needing to be addressed in deep-sea mining system research. The vertical lift pipeline system connects to the mining vessel at the top and to the intermediate storage compartment at the bottom, primarily playing a vital role in transporting mineral particles. When coarse-grained solid-liquid two-phase internal flow moves from the bottom of the vertical lift pipeline to the top, the internal flow generates centrifugal force and Coriolis force on the pipeline structure, potentially leading to pipeline instability.

[0003] Existing studies on the stability of pipeline systems under internal flow excitation primarily focus on single-phase or gas-liquid two-phase internal flow, with the flow pattern mainly being top-down discharge. However, in actual deep-sea mining engineering, the internal medium of vertical lifting pipelines is a coarse-particle solid-liquid two-phase internal flow. Since the density of the solid particles is greater than that of the liquid, they cannot maintain consistency with water during vertical upward transport, exhibiting obvious coarse-particle flow characteristics. Furthermore, the flow pattern of the coarse-particle two-phase internal flow within vertical lifting pipelines is bottom-up suction. This difference in internal medium flow characteristics and flow pattern leads to a significantly different stability characteristic of vertical lifting pipelines under coarse-particle solid-liquid two-phase internal flow excitation compared to those under single-phase or gas-liquid two-phase internal flow excitation. Therefore, to ensure the stable and safe operation of vertical lifting pipelines, it is essential to establish an accurate and reliable method for predicting the stability of vertical lifting pipelines and to conduct stability studies based on this method under solid-liquid two-phase flow excitation. This research can provide technical support for the early rational design of vertical lifting pipelines and their safe operation during service. Summary of the Invention

[0004] This invention addresses the problem that current numerical predictions for pipeline stability generally only consider single-phase internal flow or gas-liquid two-phase flow, making them difficult to apply effectively to solid-liquid two-phase flow scenarios in deep-sea mining.

[0005] A numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation includes the following steps:

[0006] The slip velocity is used to describe the relationship between the velocity of solid particles and the velocity of fluid flow inside the vertical lift pipe, in order to determine the local concentration C inside the pipe during stable delivery. vl Thus, the mass of solid phase m per unit length of pipe during stable transport can be obtained. s and liquid phase mass m f Based on the modified Hamiltonian principle, a numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation is established.

[0007] Then, based on the Galerkin method, the high-order partial differential equations that vary with both time and space are reduced to a series of ordinary differential equations containing only time terms; based on the eigenvalue method, the matrix characteristic equation is established, and the pipeline stability is determined by solving the eigenvalues.

[0008] Furthermore, the slip velocity U slip =U f -U s U s And U f The solid phase velocity and liquid phase velocity inside the pipe are respectively d s ω represents the diameter of the solid particles, d represents the inner diameter of the pipe, and ω represents the diameter of the solid particles. s C represents the settling velocity of a single particle in an unbounded fluid. v This indicates the initial feed concentration in the pipe.

[0009] Furthermore, the initial feed concentration in the pipe V so and V fo These represent the volumes occupied by the solid and liquid phases in the tube during the initial feed.

[0010] Furthermore, the local concentration of C within the pipe during stable delivery vl as follows:

[0011]

[0012] Where, ρ s ρ represents the density of solid particles. f C represents fluid density. v This indicates the initial feed concentration in the tube; g represents the acceleration due to gravity. U is the slip factor. slip U is the sliding velocity. f Liquid phase velocity inside the pipe.

[0013] Furthermore, the solid phase mass m s and liquid phase mass mf as follows:

[0014]

[0015] Where d represents the inner diameter of the pipe.

[0016] Furthermore, based on the modified Hamiltonian principle, the process of establishing a numerical prediction model for the vibration of a vertically lifting pipeline under solid-liquid two-phase flow excitation includes:

[0017] Based on the modified Hamiltonian principle, the internal fluid is represented as a solid-liquid two-phase flow, and the Lagrangian function is derived from it. Expressed as the difference between the system's kinetic energy T and potential energy V, and combined with the work done by non-conservative forces in the modified Hamiltonian principle, a numerical prediction model for the vibration of a vertically lifting pipeline under solid-liquid two-phase flow excitation is obtained:

[0018]

[0019] Where, “·” and “··” represent taking the first and second partial derivatives with respect to time t, and “′”, “″”, and “″″” represent taking the first, second, and fourth partial derivatives with respect to spatial coordinate z; m p The mass per unit length of the pipe structure, m e Let m be the mass of the end-concentrated mass block. a The additional fluid mass per unit length of pipe; δ2(Lz) is the Dirac function; A e and A i These represent the areas of the outer and inner cross-sections of the pipe, respectively; EI is the pipe bending stiffness; κ is the pipe curvature; P f (z) and P i (z) represents the fluid pressure outside and inside the pipe, respectively; U s And U f The solid phase velocity and liquid phase velocity inside the pipe are respectively; L is the pipe length; μ is the dynamic viscosity coefficient of the surrounding seawater; A1 is the linear damping coefficient; A2 is the nonlinear damping coefficient; and D is the outer diameter of the pipe.

[0020] Then, the numerical prediction model of vertical lifting pipeline vibration under solid-liquid two-phase flow excitation is dimensionless to obtain the final numerical prediction model of vertical lifting pipeline vibration under solid-liquid two-phase flow excitation.

[0021] Furthermore, the system kinetic energy T includes the pipeline structure kinetic energy T0. p Solid kinetic energy T inside the pipe s and the kinetic energy T of the liquid phase inside the pipe f The system potential energy V includes the pipe structure potential energy V. p Solid potential energy V inside the pipe sand the liquid phase potential energy V inside the pipe f ;

[0022] The kinetic energy T of the pipeline structure p Solid kinetic energy T inside the pipe s and the kinetic energy T of the liquid phase inside the pipe f as follows:

[0023]

[0024] Where, m p The mass per unit length of the pipe structure, m e Let δ2(Lz) be the mass of the end-concentrated mass block, and let δ2(Lz) be the Dirac function.

[0025] The potential energy V of the pipeline structure p Solid potential energy V inside the pipe s and the liquid phase potential energy V inside the pipe f as follows:

[0026]

[0027] The potential energy V of the pipeline structure p Including the gravitational potential energy V caused by pipes and concentrated mass blocks p1 and the elastic potential energy V caused by the bending deformation of the pipe p2 :

[0028]

[0029] Among them, A e and A i These represent the areas of the outer and inner cross-sections of the pipe, respectively; EI is the pipe bending stiffness; κ is the pipe curvature; P f (z) and P i (z) represents the fluid pressure outside the pipe and inside the pipe, respectively.

[0030] Furthermore, the process of dimensionless processing for the numerical prediction model of vertical lifting pipeline vibration under solid-liquid two-phase flow excitation includes:

[0031] For a cantilever pipe that is fixed at the top and free at the bottom, a dimensionless variable is introduced: η represents dimensionless displacement, ξ represents dimensionless coordinate position, and τ represents dimensionless time; thus, a dimensionless numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation is obtained:

[0032]

[0033] Where κ1 is the dimensionless solid phase mass, κ2 is the dimensionless liquid phase mass, κ3 is the dimensionless liquid phase buoyancy, κ4 is the dimensionless additional mass, κ5 is the dimensionless end mass, α is the dimensionless pipe mass, and u f χ is the dimensionless liquid flow rate, and χ is the slenderness ratio. γ is a dimensionless linear damping coefficient, and γ is a dimensionless nonlinear damping coefficient.

[0034] Furthermore, based on the Galerkin method, the process of reducing the order of high-order partial differential equations that vary with both time and space to a system of ordinary differential equations containing only time terms includes:

[0035] Based on the Galerkin method, the dimensionless displacement η of the structural vibration is discretized, and the first N mode shapes are extracted, as shown below:

[0036]

[0037] Where, φ i (ξ) represents the i-th mode shape of the conveying pipeline under cantilever boundary conditions. It is the i-th generalized coordinate;

[0038] For a cantilever pipe structure, its modal shape is represented as follows:

[0039] φ i (ξ)=cosh(β i ξ)-cos(β i ξ)-σ i [sinh(β i ξ)-sin(β i ξ)],

[0040]

[0041] Where, β i The eigenvalues ​​are the characteristic values ​​of the i-th mode shape;

[0042] Substituting η(ξ,τ) into the dimensionless numerical prediction model of vertical lifting pipeline vibration under solid-liquid two-phase flow excitation, and multiplying both sides of the equation by the mode shape function φ, we obtain the following: i (ξ), and perform a definite integral on the interval [0,1], then further rewrite the lumped mass block function described by the Dirac function as:

[0043]

[0044] Therefore, we get:

[0045]

[0046] Furthermore, based on the eigenvalue method, a matrix characteristic equation is established, and the process of determining pipeline stability by solving the eigenvalues ​​includes:

[0047] Rewrite formula (34) in matrix form:

[0048]

[0049] Among them, the corresponding matrix elements M in the mass matrix M, damping matrix C, and stiffness matrix K ij C ij and K ij It can be represented as follows:

[0050]

[0051] Transform the matrix form into a first-order differential equation:

[0052]

[0053] Y = -B -1 E,

[0054] We then obtain (λI-Y)A=0, where I is the identity matrix and A is the coefficient matrix; further, we obtain det(Y-λI)=0, and by solving the eigenvalues ​​of matrix Y, we can determine the influence of internal flow on structural stability.

[0055] Beneficial effects:

[0056] This invention not only considers the solid-liquid two-phase flow scenario in deep-sea mining but also effectively introduces the slip effect between coarse particles and the fluid. It establishes a multi-factor analysis model that comprehensively considers the influence of parameters such as particle size, feed concentration, and pipe slenderness ratio on pipe vibration stability. This model can efficiently evaluate the dynamic response of pipe stability under solid-liquid two-phase flow excitation and provides a rapid stability prediction method, enabling rapid prediction of the stability of vertical lifting pipes in deep-sea mining under solid-liquid two-phase flow excitation. Through comparative studies under different operating conditions, the operational stability of vertical lifting pipes can be accurately assessed, further ensuring the safety and sustainability of deep-sea mining operations. Attached Figure Description

[0057] Figure 1 This is a simplified diagram of the calculation model for a vertical lifting pipeline under solid-liquid two-phase flow excitation.

[0058] Figure 2 For different feed concentrations C v Slip ratio β and local concentration C vl With dimensionless liquid flow rate u f The change curve.

[0059] Figure 3For different particle sizes d s Slip ratio β and local concentration C vl With dimensionless liquid flow rate u f The change curve.

[0060] Figure 4 For different feed concentrations C v Argand plots of the first four dimensionless complex characteristic frequencies.

[0061] Figure 5 Argand plots of the first four dimensionless complex characteristic frequencies under different slenderness ratios χ. Detailed Implementation

[0062] Currently, numerical prediction of pipeline stability generally only considers single-phase internal flow or gas-liquid two-phase flow, which is insufficient to effectively address the technical gap in solid-liquid two-phase flow scenarios in deep-sea mining. Furthermore, if the influence of particle concentration and particle size variations with flow velocity on pipeline stability under solid-liquid two-phase flow conditions is not fully considered, significant errors will occur in the numerical prediction of the characteristics of vertical lifting pipelines in deep-sea mining. Therefore, this invention proposes a rapid stability prediction method for vertical lifting pipelines in deep-sea mining under coarse-particle solid-liquid two-phase flow excitation. This invention effectively introduces the slip effect between coarse particles and fluid, and establishes a multi-factor analysis model that comprehensively considers the influence of parameters such as particle size, feed concentration, and pipeline slenderness ratio on pipeline vibration stability. This model can efficiently evaluate the dynamic response of pipeline stability under solid-liquid two-phase flow excitation and provides a rapid stability prediction method, effectively supporting the structural design and operational safety of deep-sea mining pipeline systems. The following examples illustrate this method.

[0063] This embodiment presents a numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation, mainly including the following steps:

[0064] Step 1: By introducing a slip model and based on the modified Hamiltonian principle, a numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation is established.

[0065] (1) Introduce the slip model for solid-liquid two-phase flow:

[0066] To reasonably describe the internal flow characteristics of the transmission pipeline, Figure 1 The coupling model between particles and fluid inside the pipe makes the following assumptions: ① The solid particles are assumed to be ideal spheres; ② Collisions between solid particles and between particles and the pipe wall are ignored; ③ The solid particles are assumed to be uniformly distributed in the fluid inside the pipe. Based on these three assumptions, the relationship between the velocity of the solid particles and the velocity of the fluid inside the pipe can be described by the slip velocity model, as follows:

[0067]

[0068] In equation (1), U slip The sliding velocity can be expressed as: U slip =U f -U s U s And U f These represent the solid phase velocity and liquid phase velocity inside the pipe, respectively, d s ω represents the diameter of the solid particles, d represents the inner diameter of the pipe, and ω represents the diameter of the solid particles. s The settling velocity of a single particle in an unbounded fluid can be expressed as follows according to Stokes' equations:

[0069]

[0070] In equation (2) ρ s ρ represents the density of solid particles. f Let C represent the fluid density, g represent the acceleration due to gravity, and C represent the acceleration due to gravity. Ds This represents the drag coefficient of the solid particles. In equation (1), C v The initial feed concentration in the tube can be represented as follows:

[0071]

[0072] In equation (3), V so and V fo These represent the volumes occupied by the solid and liquid phases in the tube during the initial feed.

[0073] Because solid particles often accumulate during vertical upward transport, the local concentration C within the pipe during stable transport can vary. vl It should be slightly higher than the initial feed concentration C in the pipe. v To more accurately describe the flow characteristics of the fluid within the pipe during steady-state delivery, a local concentration is used to describe it, introducing a local concentration C. vl It can be represented as:

[0074]

[0075] In equation (4), U m The average velocity of the solid-liquid two-phase flow inside the pipe during steady-state transport can be expressed as follows based on the law of conservation of momentum:

[0076]

[0077] In equation (5), m so and m fo These represent the mass of the solid phase and the mass of the liquid phase entering the pipe at the initial moment.

[0078] To facilitate the description of the solid phase velocity U sWith liquid phase velocity U f The relationship between them, introduced by the slip factor β, is expressed as follows:

[0079]

[0080] Substituting equations (3), (5), and (6) into equation (4) yields the local concentration C. vl The expression is as follows:

[0081]

[0082] The local concentration C inside the tube is obtained based on equation (7). vl Then, the mass of the solid phase and the mass of the liquid phase per unit length of the pipe during stable transport can be obtained, as shown below:

[0083]

[0084] (2) Based on the modified Hamiltonian principle, a numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation is established:

[0085] Since the classical Hamilton's principle is not applicable to non-conservative cantilever pipe systems, a modified Hamilton's principle will be used to establish the pipe vibration equations. When there is mass and momentum exchange between the free end and the surrounding environment, the modified Hamilton's principle can be expressed as follows:

[0086]

[0087] On the left side of equation (9), L is the system's Lagrangian function, W is the work done by nonconservative forces, δ1 is the variational operator, and t1 and t2 represent the system's start and end times, respectively. On the right side of equation (9), M is the total mass of the flow in the pipe, U is the flow velocity in the pipe, and r L Let τ be the position vector of the free end of the pipe. L Let r be the spatial tangent vector at the free end of the pipe. The origin o is taken as the top of the pipe; the z-direction coincides with the pipe's axis when it is not deformed, and is defined downwards as the positive direction; the x-direction coincides with the pipe's vibration direction, and is defined to the right as the positive direction; the x and z coordinate directions constitute a two-dimensional Cartesian coordinate system in which the structural model vibrates. When only considering the pipe's vibration in the xoz plane, r... L and τ L It can be represented as follows:

[0088] r L =xi+zk,

[0089] In equation (10), s is the curvilinear coordinate along the pipe axis, and i and k represent the unit vectors in the x and z axes, respectively. It is worth noting that equation (9) represents the modified Hamiltonian principle expression for common single-phase internal flow. When the internal fluid is a solid-liquid two-phase flow, equation (9) can be rewritten as follows:

[0090]

[0091] It is worth noting that in formula (11), the solid phase velocity and the liquid phase velocity are respectively taken as -U s and -U f This is a mining pipeline under study. Both the solid and liquid phases flow upwards within the pipeline. Figure 1 The z-axis, as defined in the equation, is opposite in the positive direction.

[0092] L is the system's Lagrangian function, which can be expressed as the difference between the system's kinetic energy T and potential energy V, as follows:

[0093] T = T p +T s +T f V=V p +V s +V f (12)

[0094] In equation (12), the total kinetic energy T of the system consists of three parts: the kinetic energy of the pipe structure T p Solid kinetic energy T inside the pipe s and the kinetic energy T of the liquid phase inside the pipe f The total potential energy V of the system also consists of three parts: the potential energy of the pipe structure V. p Solid potential energy V inside the pipe s and the liquid phase potential energy V inside the pipe f .

[0095] The magnitude of the absolute flow velocity of the solid phase inside the pipe, v s And the magnitude of the absolute flow velocity of the liquid phase, v f The velocity is related to the pipe vibration velocity and the flow velocity of each phase relative to the pipe, respectively. Assuming the structure undergoes small deformation vibration, we can assume s≈z. When only the lateral vibration of the pipe is considered, v s and v f It can be represented as follows:

[0096]

[0097] Solid kinetic energy T inside the pipe s and the kinetic energy T of the liquid phase inside the pipe fIt can be expressed as an integral along the pipe axis, and by combining equation (13), we get the following expression:

[0098]

[0099] It is worth noting that, for ease of expression, in equation (14), “·” represents the partial derivative with respect to time t, and “′” represents the partial derivative with respect to spatial coordinate z. This simplified expression will be used throughout for the sake of convenience. For example… Figure 1 As shown, when a concentrated mass block is attached to the end of the pipeline structure, the kinetic energy of the pipeline structure can be expressed as follows:

[0100]

[0101] In equation (15), L is the pipe length; m p The mass per unit length of the pipe structure, m e Let δ²(Lz) be the mass of the end-concentrated mass block, and let δ²(Lz) be the Dirac function in units of L. -1 Its specific expression is as follows:

[0102]

[0103] In equation (12), the solid-phase gravitational potential energy V inside the pipe is... s and the gravitational potential energy V of the liquid phase inside the pipe f It can be represented in the following form:

[0104]

[0105] In equation (12), the potential energy V of the pipeline structure p It consists of two parts: the gravitational potential energy V caused by the pipe and the concentrated mass block. p1 And the elastic potential energy V caused by the bending deformation of the pipe. p2 , means as follows:

[0106]

[0107] In equations (18) and (19), A e and A i Here, EI and P represent the areas of the outer and inner cross-sections of the pipe, respectively; EI is the pipe bending stiffness; κ is the pipe curvature, which, under the assumption of small deformation vibration, is approximately equal to x″; P f (z) and P i (z) represent the fluid pressure outside and inside the pipe, respectively. When the pipe is in still water and is a suction pipe, it is assumed that the pressure is linearly distributed with the pipe length, and the pressure difference between the inside and outside of the pipe structure is equal. Equation (19) can be simplified to:

[0108]

[0109] For a cantilever mining pipe in still water, the nonconservative force f(z,t) of the system consists of two parts: the additional mass force f caused by the fluid surrounding the pipe. in (z,t) and fluid damping force f d (z,t) is represented as follows:

[0110]

[0111] In equation (21), m a The additional fluid mass per unit length of pipe can be expressed as: m a =C a ρ f A e , where C a The additional mass coefficient is usually taken as 1. μ is the dynamic viscosity coefficient of the surrounding seawater, A1 is the linear damping coefficient, A2 is the nonlinear damping coefficient, and D is the outer diameter of the pipe. Since the nonconservative forces in equation (21) (including the additional mass force and the fluid damping force) do negative work during the pipe vibration process, the work W they do can be expressed as follows:

[0112]

[0113] Substituting equations (14) to (22) into equation (11) and performing a series of variational operations, we obtain the following expression:

[0114]

[0115] like Figure 1 As shown, for a cantilever pipe that is fixed at the top and free at the bottom, its boundary conditions can be written in the following form:

[0116] x(,)=0 and x′(,t)=0 at z=0, x″(z,t)=0 and x″′(,t)=0 at z=L (24)

[0117] To better observe the dynamic response of the pipeline model, the following dimensionless variables are introduced to make equation (23) dimensionless:

[0118]

[0119] In equation (25), η is the dimensionless displacement, ξ is the dimensionless coordinate position, and τ is the dimensionless time.

[0120] Substituting equation (25) into equation (23), we obtain the following dimensionless equation:

[0121]

[0122] In equation (26), each dimensionless coefficient can be expressed as follows:

[0123]

[0124] In equation (27), κ1 is the dimensionless solid phase mass, κ2 is the dimensionless liquid phase mass, κ3 is the dimensionless liquid phase buoyancy, κ4 is the dimensionless additional mass, κ5 is the dimensionless end mass, α is the dimensionless pipe mass, and u f χ is the dimensionless liquid flow rate, and χ is the slenderness ratio. γ is a dimensionless linear damping coefficient, and γ is a dimensionless nonlinear damping coefficient.

[0125] Similarly, the boundary conditions given in equation (24) can also be written in dimensionless form as follows:

[0126] η(ξ,τ)=0 and η′(ξ,τ)=0 at ξ=0, η″(ξ,τ)=0 and η″′(ξ,τ)=0 at ξ=1(28)

[0127] Step 2: Based on the Galerkin method, the higher-order partial differential equations that vary with both time and space are reduced to a series of ordinary differential equations containing only time terms.

[0128] Based on the Galerkin method, the dimensionless displacement η of the structural vibration is discretized, and the first N mode shapes are extracted, as shown below:

[0129]

[0130] In equation (29), φ i (ξ) represents the i-th mode shape of the conveying pipeline under cantilever boundary conditions. It is the i-th generalized coordinate.

[0131] For a cantilever pipe structure, its mode shape can be expressed as follows:

[0132] φ i (ξ)=cosh(β i ξ)-cos(β i ξ)-σ i [sinh(β i ξ)-sin(β i ξ)],

[0133] In equation (30), β i β is the eigenvalue of the i-th mode shape. i It satisfies the following characteristic equation:

[0134] cos(β i cosh(β)i )+1=0 (31)

[0135] By solving equation (31), the β corresponding to the first 8 modal modes can be obtained very easily. i The values ​​are: 1.8751, 4.6941, 7.8548, 10.9955, 14.1372, 17.2788, 20.4204 and 23.5619, respectively. Substituting equation (29) into equation (26), both sides of the equation are multiplied by the mode shape function φ. j (ξ), and by performing a definite integral over the interval [0,1], we can obtain the following expression:

[0136]

[0137] To account for the influence of the end lumped mass block on the structural mode shape, the lumped mass block function described by the Dirac function in equation (32) is further rewritten as follows:

[0138]

[0139] Substituting equation (33) into equation (32), we get:

[0140]

[0141] Step 3: Based on the eigenvalue method, establish the matrix characteristic equation, and determine the pipeline stability by solving the eigenvalues;

[0142] When analyzing the stability characteristics of a pipe in still water, the amplitude of the pipe at the beginning of instability is very small, and the influence of the nonlinear damping term on the stability of the pipe can be ignored. Equation (34) can be rewritten in matrix form as follows:

[0143]

[0144] In equation (35), the corresponding matrix elements M in the mass matrix M, damping matrix C, and stiffness matrix K are... ij C ij and K ij It can be represented as follows:

[0145]

[0146] Equation (35) can be transformed into a first-order differential equation as follows:

[0147]

[0148] In equation (37),

[0149] Y = -B -1 E,

[0150] Suppose that the expression for Z(τ) is written as Z(τ) = Ae λτ Substituting it into equation (37), we get:

[0151] (λI-Y)A=0 (39)

[0152] In equation (39), I is the identity matrix and A is the coefficient matrix;

[0153] From equation (39), we can further obtain:

[0154] det(Y-λI)=0 (40)

[0155] Equation (40) shows that the influence of internal flow on structural stability can be determined by solving the eigenvalues ​​of matrix Y. The complex eigenvalues ​​λ are a series of complex numbers, and there is a constant relationship between the complex eigenvalues ​​λ and the complex characteristic frequency ω: λ ≡ iω, where i represents the imaginary part of the complex function. For eigenvalue problems, the real part of the complex characteristic frequency (or the imaginary part of the complex eigenvalue) reflects the natural frequency of the system; while the imaginary part of the complex characteristic frequency (or the real part of the complex eigenvalue) reflects the vibration stability characteristics of the system. If there exists a case where the imaginary part of the complex characteristic frequency is less than 0 (or the real part of the complex eigenvalue is greater than 0), then the structure is considered to have flutter instability.

[0156] Step 4: Select design parameters and, based on the above analysis methods, perform calculations and analyses on the slip model and pipeline stability:

[0157] Table 1 Calculation parameters for solid-liquid two-phase flow mining riser

[0158]

[0159]

[0160] (1) Numerical prediction of slip model for solid-liquid two-phase flow:

[0161] To fully understand the dynamic response characteristics of cantilever pipes caused by coarse particle solid-liquid two-phase flow, the flow characteristics of coarse particle solid-liquid two-phase flow inside the pipe are first studied. Based on the slip model proposed in the previous section, the effects of parameters such as flow velocity, feed concentration, and particle size inside the pipe on the slip factor and local concentration distribution during the steady-state conveying stage are discussed. Figure 2 Figure (a) shows the different feed concentrations C during the steady-state conveying phase. v Slide factor β as a function of dimensionless inflow velocity u f The relationship between the slip factor β and the feed concentration C when the internal flow velocity remains constant. v The overall trend is upward due to the increase in C, but when C vAfter exceeding 0.3, this growth trend tends to moderate. When the feed concentration remains constant, the slip factor β increases with u. f It increases monotonically with the increase of u. It should be particularly noted that in u... f In the low-speed stage (≤0.4), the slip factor becomes negative, indicating that solid particles fail to rise with the liquid phase and instead settle within the pipe. As u... f The increase of u f When the value is ≥0.4, the slip factor increases rapidly and gradually approaches 1, which indicates that the liquid phase has a significantly enhanced carrying capacity for the solid phase.

[0162] Figure 2 (b) shows different feed concentrations C during the steady-state conveying phase. v Local concentration C in the lower tube vl With dimensionless internal flow velocity u f The trend of change. The results show that in the low-speed stage (0 f ≤1), local concentration varies with u f The concentration increases rapidly as the liquid flow rate decreases. This phenomenon can be explained by the fact that when the liquid flow rate is low, its ability to carry solid particles weakens, making the particles more likely to deposit and aggregate inside the pipe, thus leading to a significant increase in local concentration. f The local concentration C further increases to 1. vl The values ​​from top to bottom are 0.584, 0.482, 0.38, 0.275, and 0.158, indicating that when the value increases to u... f ≥1 Its local concentration gradually approaches its respective feed concentration C v This indicates that the movement of solid particles tends to be uniformly distributed, and the transport process tends to be stable.

[0163] Figure 3 (a) in the figure gives the particle size d of different solid particles. s Slide factor β and dimensionless inflow velocity u f The relationship between the changes in the feed concentration C in the pipe at this time v Let's take 0.2. As can be seen from the graph: when the internal flow velocity u... f When constant, the slip factor β varies with the particle size d s The increase of d shows an overall downward trend; however, when d s After >0.3 days, this decreasing trend gradually slowed down. When the particle size remained constant, the slip factor β increased with u f It increases monotonically as it grows. Figure 3 (b) shows the particle sizes d of different solid particles. s Below, the local concentration of C during pipeline transportation. vl With dimensionless internal flow velocity u f The changing trend of the feed concentration C in the pipe at this time v ​Let's take 0.2. As can be seen from the graph: with u... f The increase in local concentration C vl Gradually approaching the feed concentration C v As particle size increases, the critical flow rate required for local concentration to reach a stable value also increases. This indicates that when particle size is large, a higher liquid flow rate is required to maintain a relatively stable and uniform solid-liquid two-phase flow transport.

[0164] (2) The influence of different parameters on the stability of vertical lifting pipelines:

[0165] like Figure 4 The horizontal and vertical axes represent the real and imaginary parts of the complex characteristic frequency, respectively. The real part represents the pipe's natural frequency; the imaginary part represents the pipe's damping characteristics during vibration. The system damping ratio ζ is usually defined as: ζ = Im(ω) / Re(ω). When one branch of the Argand diagram crosses the horizontal axis (i.e., Im(ω) = 0), the imaginary part of the system's complex characteristic frequency Im(ω) changes from a positive value to a negative value (e.g., ...). Figure 4 (As shown at point H), at this point the damping ratio of the structural system is less than 0, and the structure will continuously obtain energy from the internal flow, which will further lead to flutter instability of the structure.

[0166] Figure 4 Four different feed concentrations (i.e., C) are given. v Argand plots of the complex characteristic frequencies of the pipe structure as a function of internal flow velocity at ω = 0, 0.1, 0.2, and 0.4, from... Figure 4 As can be seen, when C v When four different values ​​are taken, the first three modes of the pipe structure all exhibit flutter, and the flutter instability velocity u corresponding to each mode is... cr different. Figure 4 (a) indicates that when C v When = 0, the critical flow velocity u corresponding to the first three modes of the pipeline cr The values ​​are 13.1, 7.96, and 16.2 respectively; when C v When = 0.1, the critical flow velocity u corresponding to the first three modes of the pipeline cr The values ​​are 11.35, 6.74, and 12.66 respectively. Figure 4 (b) indicates that when C v When ω = 0.2, the critical flow velocity u corresponding to the first three modes of the pipeline cr The values ​​are 9.99, 5.84, and 10.69 respectively; when C v When ω = 0.4, the critical flow velocity u corresponding to the first three modes of the pipeline is... cr The values ​​were 8.06, 4.63, and 8.30, respectively. It can be seen that for the four different feed concentrations C... vUnder these conditions, the pipeline structure first exhibits flutter instability in the second mode, indicating that the dynamic characteristics of the second mode play a dominant role in system stability. Furthermore, when C... v When the flow rates are 0, 0.1, 0.2, and 0.4, the critical flow velocity u at which the pipeline first begins to become unstable is... cr The values ​​were 7.96, 6.94, 5.84, and 4.63, respectively. Therefore, it can be seen that as the feed concentration C... v The increase of the critical flow velocity u at which pipeline flutter instability occurs. cr The trend shows a clear downward trend. This phenomenon is mainly due to the increase in solid density and total mass inside the pipe caused by the increase in feed concentration, which in turn enhances the centrifugal force and Coriolis force generated by the internal flow of the system. Among them, the increase in centrifugal force weakens the structural stiffness of the pipe, while the Coriolis force, as a non-conservative excitation source, will enhance the energy input of the system under suction conditions, making the structure more prone to flutter.

[0167] Figure 5 The feed concentration C is given. v Argand plots of the complex characteristic frequencies of the pipe structure as a function of internal flow velocity under the condition that χ = ​​0.2, for four different slenderness ratios (i.e., χ = 300, 600, 900, and 1200). Figure 5 It can be seen that when χ = 300, 600, 900, and 1200, the critical flow velocity u for pipeline flutter instability is... cr The values ​​were 19.25, 16.35, 12.24, and 9.19, respectively, indicating that the critical velocity for pipe instability decreases with increasing pipe slenderness ratio. (Comparison) Figure 5 (a) and Figure 5 (b) It can be observed that when χ = 300 and 600, the mode order in which the pipe first experiences flutter instability is order 1; however, as the slenderness ratio increases, when χ = 900 and 1200, the mode order in which the pipe first experiences flutter instability changes to order 2. This mode exchange phenomenon reflects that the increase in slenderness ratio significantly changes the dynamic characteristics of the structure and the dominant instability path. The larger the slenderness ratio, the more flexible the pipe becomes, and the higher-order modes of the structure are more easily affected by internal flow excitation. The higher-order mode frequencies decrease at a relatively faster rate, causing the system to enter the flutter instability mode first.

[0168] By applying the aforementioned methods, researchers can rapidly predict the stability of vertical lifting pipelines in deep-sea mining under solid-liquid two-phase flow excitation. Through comparative studies of different operating conditions, the operational stability of vertical lifting pipelines can be accurately assessed, further ensuring the safety and sustainability of deep-sea mining operations.

[0169] This invention may have other embodiments. Without departing from the spirit and essence of this invention, those skilled in the art can make various corresponding changes and modifications according to this invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation, characterized in that: Includes the following steps: The slip velocity is used to describe the relationship between the velocity of solid particles and the flow velocity of the fluid inside the vertical lift pipe, in order to determine the local concentration inside the pipe during stable delivery. C vl Thus, the mass of solid phase per unit length of pipe during stable transport can be obtained. and liquid phase mass Based on the modified Hamiltonian principle, a numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation is established. Then, based on the Galerkin method, the higher-order partial differential equations that vary with both time and space are reduced to a series of ordinary differential equations containing only time terms. Based on the eigenvalue method, a matrix characteristic equation is established, and the stability of the pipeline is determined by solving the eigenvalues. The slip speed , U slip = U f - U s , U s as well as U f These represent the solid phase velocity and liquid phase velocity inside the pipe, respectively. d s Indicates the diameter of the solid particles. d Indicates the inner diameter of the pipe. ω s This represents the settling velocity of a single particle in an unbounded fluid. C v This indicates the initial feed concentration in the pipe. Initial feed concentration in the tube , V so and V fo These represent the volumes occupied by the solid and liquid phases in the tube during the initial feed, respectively. Local concentration within the pipe during stable delivery C vl as follows: ， in, ρ s This represents the density of solid particles. ρ f Indicates fluid density, C v This indicates the initial feed concentration in the pipe. g Represents gravitational acceleration; The slip factor, For sliding speed, U f Liquid velocity inside the pipe; The solid mass and liquid phase mass as follows: ， in, d Indicates the inner diameter of the pipe; The process of establishing a numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation, based on the modified Hamiltonian principle, includes: Based on the modified Hamiltonian principle, the internal fluid is represented as a solid-liquid two-phase flow, and the Lagrangian function is derived from it. Represented as system kinetic energy T With system potential energy V The difference, combined with the work done by non-conservative forces in the modified Hamiltonian principle, yields a numerical prediction model for the vibration of a vertically lifting pipeline under solid-liquid two-phase flow excitation: ， in, Indicates time t Find the first and second partial derivatives. Represents spatial coordinates z Find the first, second, and fourth partial derivatives; m p The mass per unit length of pipe structure, m e The mass of the end-concentrated mass block. m a The additional fluid mass per unit length of pipe; δ 2( L - z ) is the Dirac function; A e This refers to the outer cross-section of the pipe. EI For pipe bending stiffness; U s as well as U f Separate the solid phase velocity and liquid phase velocity within the pipeline; This refers to the length of the pipe. μ Let be the dynamic viscosity coefficient of the surrounding seawater. A 1 represents the linear damping coefficient. A 2 represents the nonlinear damping coefficient. D The outer diameter of the pipe; Then, the numerical prediction model of vertical lifting pipeline vibration under solid-liquid two-phase flow excitation is dimensionless to obtain the final numerical prediction model of vertical lifting pipeline vibration under solid-liquid two-phase flow excitation.

2. The numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation as described in claim 1, characterized in that: The system kinetic energy T Including the kinetic energy of the pipeline structure T p Solid phase kinetic energy inside the pipeline T s and the kinetic energy of the liquid phase inside the pipeline T f The system potential energy V Including pipeline structural potential energy V p Solid potential energy inside the pipeline V s and the potential energy of the liquid phase inside the pipe V f ; The kinetic energy of the pipeline structure T p Solid phase kinetic energy inside the pipeline T s and the kinetic energy of the liquid phase inside the pipeline T f as follows: ， in, m p The mass per unit length of pipe structure, m e The mass of the end-concentrated mass block. δ 2( L - z ) is the Dirac function; The potential energy of the pipeline structure V p Solid potential energy inside the pipeline V s and the potential energy of the liquid phase inside the pipe V f as follows: ， ， The potential energy of the pipeline structure V p Including gravitational potential energy caused by pipes and concentrated mass blocks V p1 and the elastic potential energy caused by the bending deformation of the pipe V p2 : ， in, A e and A i These are the areas of the outer and inner cross-sections of the pipe, respectively. EI For pipe bending stiffness; κ For the curvature of the pipe, P f ( z )and P i ( z ) represent the fluid pressure outside the pipe and inside the pipe, respectively.

3. The numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation as described in claim 1, characterized in that: The process of dimensionless processing for the numerical prediction model of vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation includes: For a cantilever pipe that is fixed at the top and free at the bottom, a dimensionless variable is introduced: , For dimensionless displacement, Dimensionless coordinate position Dimensionless time; thus, a dimensionless numerical prediction model for the vibration of a vertical lifting pipeline under solid-liquid two-phase flow excitation is obtained: ; ； in, κ 1 represents the dimensionless mass of the solid phase. κ 2 represents the dimensionless mass of the liquid phase. κ 3 represents the dimensionless liquid phase buoyancy. κ 4 represents dimensionless added mass. κ 5 represents the dimensionless end mass. For dimensionless pipe mass, The dimensionless liquid flow rate For the slenderness ratio, Infinite linear damping coefficient It is a dimensionless nonlinear damping coefficient.

4. The numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation as described in claim 3, characterized in that: Based on the Galerkin method, the process of reducing a high-order partial differential equation that varies with both time and space to a system of ordinary differential equations containing only time terms includes: Based on the Galerkin method, dimensionless displacement of structural vibration is calculated. Discretize and truncate the first part. N The first-order mode shape is represented as follows: ； in, For the first flow transmission pipeline under cantilever boundary conditions First-order mode shape, It is the first Generalized coordinates; For a cantilever pipe structure, its modal shape is represented as follows: ， in, β i For the first eigenvalues ​​of the mode shape; Substituting the dimensionless solid-liquid two-phase flow excitation into the numerical prediction model of vertical lifting pipeline vibration, and multiplying both sides of the equation by the mode shape function, we obtain the following: And perform a definite integral on the interval [0, 1], and then further rewrite the lumped mass block function described by the Dirac function as follows: ； Therefore, we get:

5. The numerical prediction method for the stability characteristics of a vertical lifting pipeline under coarse-particle solid-liquid two-phase internal flow excitation as described in claim 4, characterized in that: Based on the eigenvalue method, the process of establishing the characteristic equation of a matrix and determining pipeline stability by solving the eigenvalues ​​includes: Rewrite formula (34) in matrix form: ； Among them, the corresponding matrix elements in the mass matrix M, damping matrix C, and stiffness matrix K as well as It can be represented as follows: ； Transform the matrix form into a first-order differential equation: ； And thus obtain I is the identity matrix, and A is the coefficient matrix; further, we obtain The influence of internal flow on structural stability can be determined by solving the eigenvalues ​​of matrix Y.

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