A method for predicting friction pressure drop in settling slurry pipelines

Abstract

The present invention relates to the technical field of slurry pipeline transportation, and discloses a method for predicting friction pressure drop in a sedimentation slurry pipeline. First, the boundary conditions of the water flow in the model are determined, and a prediction model for the friction pressure drop in the sedimentation slurry pipeline is constructed. The prediction model for the friction pressure drop in the sedimentation slurry pipeline includes a continuity equation, a momentum equation, a friction law equation, a particle support equation, and a solid transportation equation, and an analytical solution is obtained through the solid transportation equation. The prediction model for the friction pressure drop in the sedimentation slurry pipeline considers the influence of static sediments on the friction pressure drop prediction model of the slurry pipeline, and the model is suitable for slurry pipeline transportation with sediments. The model has a simple structure, is suitable for engineering practice, has a lower calculation cost, and can accurately predict the hydraulic gradient and riverbed thickness, thereby completing the prediction of the friction pressure drop in the sedimentation slurry pipeline.

Description

Technical Field

[0001] The invention relates to the technical field of slurry pipeline transportation, and in particular to a method for predicting friction pressure drop in a settling slurry pipeline. Background Art

[0002] Pipeline transportation has become the fifth largest mode of transportation after highway, railway, water transport and aviation. It has developed rapidly in recent years and has played a huge role in promoting the development of the transportation industry. In particular, it has provided a set of scientific and feasible solutions for the transportation of mine concentrates in complex terrain in remote mountainous areas, which is of great significance to promoting the development and comprehensive utilization of mineral resources.

[0003] In actual industrial sites, design and operation engineers want to keep their slurry pipelines free of sediments, as sediments may cause pipeline blockage. However, in some special cases or for other reasons, the slurry flow rate is too low to avoid the formation of a particle bed at the bottom of the pipeline. Therefore, it is of great practical significance to predict the friction pressure drop in slurry pipelines containing static sediments. The existing prediction models are mainly used to predict the friction pressure drop in slurry pipelines without static sediments, which is not suitable for slurry pipeline transportation with sediments. In addition, the models are too complex and not suitable for engineering practice. There is a lack of models covering this type of sedimentation slurry flow. Summary of the invention

[0004] 1. Technical issues to be resolved

[0005] In view of the shortcomings of the prior art, the present invention provides a method for predicting the friction pressure drop in a sedimentation slurry pipeline. The model is suitable for slurry pipeline transportation with sediments. It has the advantages of simple model structure, suitability for engineering practice, lower calculation cost, and can accurately predict the hydraulic gradient and bed thickness, thereby effectively completing the prediction of the friction pressure drop in the sedimentation slurry pipeline, thereby solving the above-mentioned technical problems.

[0006] (II) Technical solution

[0007] To achieve the above object, the present invention provides the following technical solution: a method for predicting friction pressure drop in a settling slurry pipeline, comprising the following steps:

[0008] S1. Collect the parameters of water flow in the pipe, including the cross-sectional area A of the flow area above the bed. a , the cross-sectional area A of the flow region above the bed a It is divided into two sub-areas, namely, sub-area A associated with the pipe wall aw Sub-area A associated with the top of the bed ab , and determine the boundary conditions, including the perimeter of the flow area at the top of the riverbed O w and deposited on the top of the riverbed width O band the bed thickness is y b ;

[0009] S2. Based on the parameters collected in step S1, a prediction model of the friction pressure drop in the settling slurry pipeline is constructed, wherein the prediction model of the friction pressure drop in the settling slurry pipeline includes a continuity equation, a momentum equation, a friction law equation, a particle support equation, and a solid transport equation, and an analytical solution is obtained through the solid transport equation;

[0010] The continuity equation is used to convert the cross-sectional area of ​​the entire pipe A to the cross-sectional area of ​​the flow area above the bed A a In conjunction, the continuity equation can be expressed as follows:

[0011] V m A=v a ·A a

[0012] Where A represents the area of ​​the entire pipe cross section, V m Indicates the average slurry velocity in the entire pipe section, A a represents the cross-sectional area of ​​the flow region above the bed, v a It represents the average velocity of the slurry in the flow area above the bed, and calculates the velocity Q of the slurry flowing through the flow area s The expression is as follows:

[0013] Q s =C vd Q m =C vd ·v a ·A a

[0014] Among them, Q s represents the average volume flow rate of solids, C vd represents the average transport concentration of the slurry, Q m represents the average volume flow rate of slurry, v a It represents the average velocity of the slurry in the flow area above the bed;

[0015] The momentum equation is expressed as follows:

[0016] ρ f ·g·i m ·A a =τ w ·O w +τ b ·O b

[0017] ρ f ·g·i m ·A aw =τ w ·Ow

[0018] ρ f ·g·i m ·A ab =τ b ·O b

[0019] Among them, ρ f represents the density of the slurry, i m represents the hydraulic gradient, g represents the acceleration of gravity, τ w represents the shear stress in the flow region at the top of the riverbed, τ b represents the shear stress in the flow area at the bottom of the riverbed, O w represents the perimeter of the flow area at the top of the riverbed, O b Indicates the width of the sediment on the top of the riverbed, A a represents the cross-sectional area of ​​the flow region above the bed, A aw represents the sub-region associated with the tube wall, A ab represents the sub-region associated with the top of the bed, and the shear stress τ at a specific boundary w and τ b The calculation formula is as follows:

[0020] τ w =ρ f ·g·R hw ·i m

[0021] τ b =ρ f ·g·R hb ·i m

[0022] Among them, i m represents the hydraulic gradient, R hw Represents the sub-area A associated with the pipe wall aw The hydraulic radius, R hb Indicates area A associated with the top of the bed ab The hydraulic radius of

[0023] The friction law equation includes two aspects of calculation, specifically:

[0024] S2.a1, calculation of pipe wall friction;

[0025] S2.a2, bed surface friction calculation;

[0026] The particle support equations include the following operations:

[0027] S2.b1, based on the stratification of particle support mechanism, establish a pressure drop model and analyze the passing particles;

[0028] S2.b2, contact load friction calculation;

[0029] S2.b3, calculate the stratification ratio and its relationship with flow parameters;

[0030] The solid transport equation is used to relate the solid flux to the dimensionless bed shear stress and is expressed as follows:

[0031]

[0032] Among them, q s is the average solid flux per unit width of the flow area, α and β represent empirical coefficients, and θ b,cr is the Shields parameter θ for the particles at the top of the bed to start b The threshold value, g represents the acceleration due to gravity, d 50 is the median diameter of the solid particles, S S is the specific gravity of the solid, θ b represents the Shields parameter for the start of the particles at the top of the bed, and q s The expression is as follows:

[0033]

[0034] Among them, q s is the average solid flux per unit width of the flow area, O b represents the width of the sediment on the top of the riverbed, Q s represents the average volume flow rate of solids, C vd is the input parameter of the slurry pipeline hydraulic gradient prediction model, v a It represents the average velocity of the slurry in the flow area above the bed, R ha represents the hydraulic radius of the discharge area, A a Representing the cross-sectional area of ​​the flow region above the bed, the analytical solution to the general equation for solid flux is:

[0035]

[0036] Among them, q s is the average solid flux per unit width through the flow region, y is the vertical position above the reference plane, y0 is the initial vertical position of the solid particle velocity and concentration profile, c(y) represents the local concentration of solid particles, and u s (y) is the local velocity of the solid particles and H is the height of the flow zone.

[0037] As a preferred technical solution of the present invention, the calculation formula of step S2.a1 is as follows:

[0038]

[0039] Among them, λ w Re represents the friction coefficient of the pipe wall, w represents the Reynolds number, v f represents the kinematic viscosity of the liquid, f n is the correlation function between the tube wall friction coefficient and the Reynolds number, R hw Represents the sub-area A associated with the pipe wall aw The hydraulic radius, v a It indicates the average velocity of the slurry in the flow area above the bed.

[0040] As a preferred technical solution of the present invention, the calculation formula of step S2.a2 is as follows:

[0041]

[0042] Among them, λ b represents the bed friction coefficient, R hb Indicates area A associated with the top of the bed ab The hydraulic radius, k s represents the hydraulic roughness of the bed top, and ln* represents the logarithmic function.

[0043] As a preferred technical solution of the present invention, the calculation formula of the pressure drop model in step S2.b1 is as follows:

[0044] C va ·A a =C vab ·A ab

[0045] Among them, C va In the flow area The average spatial concentration within the ab represents the sub-area associated with the top of the bed, C vab In area A ab The concentration of all particles in the

[0046] As a preferred technical solution of the present invention, the contact load friction expression of step S2.b2 is as follows:

[0047]

[0048] Among them, ρ f represents the density of the slurry, τ bs represents the bed shear stress acting as a result of the particle normal stress applied to the top of the bed by contact load particles travelling in the flow above the bed, S S Specific gravity of solid, O b represents the width of the riverbed top, g represents the acceleration of gravity, A a represents the cross-sectional area of ​​the flow region above the bed, Cc Represents the total spatial concentration C va contact part.

[0049] As a preferred technical solution of the present invention, the stratification ratio equation expression of step S2.b3 is as follows:

[0050]

[0051] Where μ represents the friction coefficient, C c Represents the total spatial concentration C va The contact part, C vd represents the input parameter of the slurry pipeline hydraulic gradient prediction model, μ *b represents the ratio of bed shear velocity, w t It represents the particle settling velocity, a dimensionless parameter used to evaluate the suitability of the flow capacity of suspended particles. K represents a constant and n represents an empirical coefficient.

[0052] Compared with the prior art, the present invention provides a method for predicting the friction pressure drop in a settling slurry pipeline, which has the following beneficial effects:

[0053] The invention considers the influence of static sediment on the friction pressure drop prediction model of slurry pipeline, and the model is suitable for slurry pipeline transportation with sediment. The model has a simple structure and is suitable for engineering practice, thereby completing the prediction of friction pressure drop in sedimentation slurry pipeline. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 It is a schematic diagram of the process of the present invention;

[0055] Figure 2 It is a schematic diagram comparing the hydraulic gradient experimental value and the predicted value of the present invention;

[0056] Figure 3 This is a schematic diagram comparing the experimental and predicted values ​​of hydraulic gradient based on the semi-empirical model;

[0057] Figure 4 It is a schematic diagram comparing the experimental value and the predicted value of the relative thickness of the fixed bed of the present invention;

[0058] Figure 5 It is a schematic diagram of the pipeline cross section of the present invention. DETAILED DESCRIPTION

[0059] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0060] The material properties of the pipeline studied are as follows: the elastic modulus of the pipeline is 68 GPa, the Poisson's ratio is 0.3, and the density of the pipeline is 2700 kg·m -3 , the outer diameter of the pipe is 46 mm, the inner diameter of the pipe is 40 mm, the length of the pipe is 2 m, and the density of the fluid is kg·m -3 The average internal pressure is 2 MPa. x1 = 260, x2 = 1.0, c3 = 2.5, c4 = 1.7, d 50 =150mm, K=1.9, n=1.0, β=1.5

[0061] See also Figure 1-5 , a method for predicting friction pressure drop in a settling slurry pipeline, comprising the following steps:

[0062] S1. Collect the parameters of water flow in the pipeline and determine the boundary conditions. When the slurry flows in the pipeline with sediment, the slurry will be restricted by two boundaries with different hydraulic roughness (pipe wall and bed top). The velocity profile in the vertical cross section is distorted, and the hydrodynamic axis moves upward, such as Figure 5 In addition, a concentration profile is formed in the discharge area, with the maximum concentration at the bottom, i.e., at the top of the discharge area at the top of the sediment, the concentration is C b , while the local concentration near the top is the smallest. The asymmetric distribution of slurry velocity and concentration in the discharge area affects the boundary friction, thereby affecting the friction pressure drop of slurry flow. The cross-sectional area A of the flow area above the bed a In the cross section of the pipe with an inner diameter of D, the perimeter O of the flow area at the top of the riverbed is w and deposited on the top of the riverbed width O b and the bed thickness is y b Restriction. The cross-sectional area A of the flow region above the bed a It is divided into two sub-areas, namely, sub-area A associated with the tube wall aw Sub-area A associated with the top of the bed ab , each sub-region is associated with a boundary, the pipe wall is considered to be a hydraulically smooth boundary, and the top of the stationary bed is a hydraulically rough boundary with a friction law; in the sub-region A associated with the pipe wall aw In the case of the flow being affected by the friction of the tube wall, in the sub-region A associated with the top of the bed ab In the case of

[0063] A a =A aw +A ab

[0064] Each specific sub-area is described by its hydraulic radius:

[0065]

[0066] Among them, R hw Represents the sub-area A associated with the pipe wall aw The hydraulic radius, R hb Indicates area A associated with the top of the bed ab The hydraulic radius of

[0067] S2. Based on the parameters collected in step S1, a prediction model for the friction pressure drop in the settling slurry pipeline is constructed. The prediction model for the friction pressure drop in the settling slurry pipeline includes a continuity equation, a momentum equation, a friction law equation, a particle support equation, and a solid transport equation; the average slurry velocity V in the entire pipe section m Different from the average velocity v of the slurry in the flow area above the bed a The continuity equation for slurry flow converts the cross-sectional area A of the entire pipe into the cross-sectional area A of the flow region above the bed. a In conjunction, we can derive the following expression:

[0068] V m A=v a ·A a

[0069] Where A represents the area of ​​the entire pipe cross section, V m Indicates the average slurry velocity in the entire pipe section, A a represents the cross-sectional area of ​​the flow region above the bed, v a It represents the average velocity of the slurry in the flow area above the bed, and calculates the velocity Q of the slurry flowing through the flow area s The expression is as follows:

[0070] Q s =C va Q m =C vd ·v a ·A a

[0071] Among them, Q s represents the average volume flow rate of solids, C vd represents the average transport concentration of the slurry (input parameter of the prediction model), Q m represents the average volume flow rate of slurry, v a A represents the average velocity of the slurry in the flow area above the bed. a represents the cross-sectional area of ​​the flow region above the bed;

[0072] The momentum equation assumes that in the two partitions of the flow zone, the hydraulic gradient i m and the average flow velocity va The balance of driving force and resistance in the unit length of the pipe section is valid for both partitions, and the momentum equation is as follows:

[0073] ρ f ·g·i m ·A a =τ w ·O w +τ b ·O b

[0074] ρ f ·g·i m ·A aw =τ w ·O w

[0075] ρ f ·g·i m ·A ab =τ b ·O b

[0076] Among them, ρ f represents the density of the slurry, i m represents the hydraulic gradient, g represents the acceleration of gravity, τ w represents the shear stress in the flow region at the top of the riverbed, τ b represents the shear stress in the flow area at the bottom of the riverbed, O w represents the perimeter of the flow area at the top of the riverbed, O b Indicates the width of the sediment on the top of the riverbed, A a represents the cross-sectional area of ​​the flow region above the bed, A aw represents the sub-region associated with the tube wall, A ab represents the sub-region associated with the top of the bed, and the shear stress τ at a specific boundary w and τ b The calculation formula is as follows:

[0077] τ w =ρ f ·g·R hw ·i m

[0078] τ b =ρ f ·g·R hb ·i m

[0079] The boundary shear stress is related to the average flow velocity through λ and the Darcy-Weisbach friction coefficient of the specific boundary, that is:

[0080]

[0081] Then the combination of boundary shear stress equations is:

[0082]

[0083] Friction law equation:

[0084] Calculation of pipe wall friction, in determining the shear stress τ in the flow area at the top of the riverbed w When , it is assumed that the same wall friction laws apply to water flow without solid particles and to flow with sediment slurry. In other words, it is assumed that the presence of particles in the pipe does not change the cross-sectional area A enclosing the flow region. a The friction law of the pipe wall; related experimental experience shows that after a period of time, the pipe wall of the pipe transporting sediment slurry will be polished by particles and can be regarded as hydraulically smooth. The friction law of the smooth pipe wall converts the friction coefficient into the Reynolds number Re w (via the flow area relative to the pipe wall)

[0085]

[0086] Among them, λ w Re represents the friction coefficient of the pipe wall, w represents the Reynolds number, v f represents the kinematic viscosity of the liquid, f n is the correlation function between the tube wall friction coefficient and the Reynolds number, R hw Represents the sub-area A associated with the pipe wall aw The hydraulic radius, v a It indicates the average velocity of the slurry in the flow area above the bed.

[0087] Bed surface friction calculation, the bed top is composed of particles and cannot be considered hydraulically smooth. Considering the size of the particles that make up the bed top, it is usually considered as a hydraulically rough boundary; according to the purpose of the proposed prediction model, the friction law of the bed top is expressed by the Nikuradze equation modified by the bed friction condition as follows:

[0088]

[0089] Among them, λ b represents the bed friction coefficient, R hb Indicates area A associated with the top of the bed ab The hydraulic radius, k srepresents the hydraulic roughness of the top of the bed, and ln* represents the logarithmic function. In this model, it is assumed that the constant values ​​in the Nikuradze equation will not be modified, that is, it is assumed that the Kármán constant (κ = 0.4) and the integral constant of the logarithmic velocity profile remain unchanged in the boundary layer when the water flows over the rough wall and the slurry flows over the top of the bed. The expression for the flow with low bed shear stress, that is, the flow with a small value of the Shields parameter, is as follows:

[0090]

[0091] Among them, ρ s is the density of the solid, ρ f represents the density of the slurry, d 50 is the median diameter of the solid particles, θ b The Shields parameter for the start of particle movement at the top of the bed, τ b Represents the shear stress in the flow area at the bottom of the riverbed.

[0092] When approaching the critical value where particles start to move, the hydraulic roughness is linearly related to the diameter d of the particles that make up the bed. The value of θ generally varies between 2 and 5. However, the typical flow conditions in slurry pipelines are high shear stress. In this flow (θ b >0.8, upper plane bed state), the top of the bed is sheared off, and a large number of particles are transported above the unsheared part. This changes the friction conditions at the top of the bed and significantly increases the hydraulic roughness. It is no longer a simple linear function of particle size, but is related to more parameters, among which the bed shear stress is the most important parameter. For pure surface flow (i.e., all transported particles almost always maintain contact with each other and the bed), is θ b k is a linear function of k. Therefore, for this type of flow, the hydraulic roughness is independent of the particle size. Unlike the particle size, the friction law for face flow requires the thickness of the shear layer (the part of the bed that is sheared away, over whose height the particle concentration and velocity are almost linearly distributed) as its length parameter. Settling slurry flows, unlike pure face flow, require more complex relationships to determine k. s The hydraulic radius R of the discharge area ha and flow rate v a and particle settling velocity w t The ratio between is used together with the Shields parameter in the prediction model. The predicted pressure drop model uses the following expression:

[0093]

[0094] Among them, x1, x2, x3, and x4 are empirical coefficients, and the cross-sectional area A of the flow area above the bed is aThe hydraulic radius R of the discharge area ha The expression is as follows:

[0095]

[0096] Particle support equations;

[0097] Based on the stratification of the particle support mechanism and following the standard method of determining bed shear stress from pipeline tests, the pressure drop model assumes that in a settling slurry flow, all transported particles contribute to the total friction through the bed shear stress. Therefore, the equation for the pressure drop model is:

[0098] C va ·A a =C vab ·A ab

[0099] Among them, C va Indicates that in the flow area A a The average spatial concentration within the ab represents the sub-area associated with the top of the bed, C vab In area A ab The concentration of all particles in the

[0100] If not all particles pass through, then:

[0101] C vab =C vabc +C vabt

[0102] Among them, C vabc represents the fractional concentration of particles in contact in the sub-region associated with the bed top, while C vabt represents the concentration of the part of particles suspended in the region. In order to determine the concentration of the contact part in the total transported particles, the stratification ratio parameter is introduced. The stratification ratio represents the proportion of particles transported as contact (the particle support mechanism is the contact between particles. The particles transfer their immersion weight to the flow boundary through mutual contact, thereby contacting the boundary and contributing to the mechanical friction of the flow boundary) and the proportion of particles transported as suspension (the main particle support mechanism is the interaction with the turbulent eddy of the flowing liquid. The particles mainly transfer their immersion weight to the carrying liquid and contribute to the viscous friction of the flow boundary). Let C c is the total spatial concentration C va If Then all particles are contact particles. Then all particles are suspended particles. In the stratification ratio, the average delivery concentration C of the slurry is usually used. vd Replace C va ; C vd is the input parameter of the slurry pipeline hydraulic gradient prediction model. Rather than When , the stratification ratio is also affected by the average velocity of the liquid and solid in the flow;

[0103] Contact load friction calculation, at the bottom of the flow, the shear stress τ applied by the flow in the bottom flow area of ​​the riverbed b The shear stress balance consists of two components and is expressed as follows:

[0104] τ b =τ bs +τ bf

[0105] Among them, τ bs is the bed shear stress acting due to the particle normal stress applied to the top of the bed by contact load particles travelling in the flow above the bed, τ bf It is the fluid-like shear stress that opposes the flow carrying the liquid and the solid particles suspended in the flow and not contributing to the contact load. The expression for the submerged weight FW exerted by the submerged body consisting of the contact load particles in the unit length of the pipe is as follows:

[0106] FW=ρ f ·(S S -1)·g·C vabc ·A ab

[0107] Among them, S S is the specific gravity of the solid, Deposits on the top of the riverbed width O b The particle normal stress σ generated at bs The expression is as follows:

[0108]

[0109] Among them, R hb Indicates area A associated with the top of the bed ab The hydraulic radius of the fixed bed is τ, and Coulomb equilibrium requires that the particle normal stress at the top of the fixed bed is related to the bed shear stress through the friction coefficient μ, so τ bs The expression is as follows:

[0110] τ bs =μ·ρ f (S S -1)·g·C vabc ·R hb

[0111] C c ·A a =C vabc ·A ab

[0112] Modifying the equation for bed shear stress yields the following equation for the predicted pressure drop model:

[0113]

[0114] Calculate the stratification ratio and its relationship with flow parameters, and the following expression is obtained from the above expression:

[0115]

[0116] Therefore, the stratification ratio Directly affects the hydraulic gradient m The shear stress term in the equation represents the viscous shear stress τ bf The effect of stratification ratio on the stratification ratio is inferred that under the upper plane bed flow state, τ bs Much higher than τ bf Therefore, we simplify it to τ b =τ bs ;

[0117] The ratio of bed shear velocity μ *b Defined as:

[0118]

[0119] And the particle settling velocity w t It is a dimensionless parameter used to evaluate the suitability of the flow capacity of suspended particles, because w t is directly related to the vertical component of the turbulence intensity, which is related to the turbulent eddies that support particles in the flowing liquid. An additional suitable parameter is the hydraulic diameter of the flow area, D ha The ratio is defined as:

[0120] D ha =4·R ha

[0121] The particle size d also affects the stratification ratio because the hydraulic diameter limits the maximum size of turbulent eddies generated in the flow. In flows over erodible sediments, v a The increase of μ *b and D ha At the same time, the existing experimental data show and There is a good correlation between the velocity ratio and the effect; in addition, the speed and item There is also a fairly strong correlation between ; therefore, under the assumption that , the modified stratification ratio equation is as follows:

[0122]

[0123] Where μ represents the friction coefficient, C c Represents the total spatial concentration C va The contact part, C vd represents the input parameter of the slurry pipeline hydraulic gradient prediction model, μ *b represents the ratio of bed shear velocity, w t It represents the particle settling velocity, a dimensionless parameter used to evaluate the suitability of the flow capacity of suspended particles. K represents a constant and n represents an empirical coefficient.

[0124] Solid transport equations, which are applicable to flows carrying solid particles above a fixed bed. In a pipe, this flow occurs within a certain range of average velocities, which is limited by the minimum velocity required to start the movement of solid particles at the top of the fixed bed and the maximum velocity at which the entire bed starts to slide or disappear. The maximum velocity therefore corresponds to the sedimentation limiting velocity. The minimum velocity corresponds to the Shields criterion. For the purposes of the model, it is assumed that the equation for calculating the solid flux through the flow zone above the fixed bed has the same functional form as the equation for calculating the bed load flux in an open channel. The equation relates the solid flux to the dimensionless bed shear stress. For sediment transported in an open channel as bed load (i.e., contact load), an MPM (Meyer-Peter and Müller) type of formula is usually used:

[0125]

[0126] where q s is the average solid flux per unit width of the flow area, α and β represent empirical coefficients, and θ b,cr is the Shields parameter θ for the particles at the top of the bed to start b The threshold value, and q s The expression is as follows:

[0127]

[0128] Among them, Q s Represents the average volume flow rate of solids, the analytical solution of the general equation for solid flux:

[0129]

[0130] where y is the vertical position above the reference plane, y0 is the initial vertical position of the solid particle velocity and concentration profile, and u sis the local velocity of the solid particles, H is the height of the flow zone, and if the velocity and concentration on the shear layer are assumed to be linearly distributed, the MPM-type thin-layer flow equation can be derived;

[0131] Finally, according to Figure 2 , Figure 3 , Figure 4 As shown, the hydraulic gradient i m Relative thickness of fixed bed The comparison of the predicted values ​​with the experimental values ​​shows that the difference between the measured and predicted values ​​is within the range of about ±0.25%. Compared with the semi-empirical formula, the model can better predict the hydraulic slope pressure drop.

[0132] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for predicting friction pressure drop in a settling slurry pipeline, characterized in that: The following steps are involved: S1. Collect the parameters of water flow in the pipe, including the cross-sectional area A of the flow area above the bed. a , the cross-sectional area A of the flow region above the bed a It is divided into two sub-areas, namely, sub-area A associated with the pipe wall aw Sub-area A associated with the top of the bed ab , and determine the boundary conditions, including the perimeter of the flow area at the top of the riverbed O w and deposited on the top of the riverbed width O b and the bed thickness is y b ; S2. Constructing a prediction model of friction pressure drop in the settling slurry pipeline according to the parameters collected in step S1, wherein the prediction model of friction pressure drop in the settling slurry pipeline includes a continuity equation, a momentum equation, a friction law equation, a particle support equation, and a solid transport equation; S3, according to the equation established in step S2, the hydraulic gradient i is obtained m and fixed sediment thickness y b The continuity equation is used to convert the cross-sectional area of ​​the entire pipe A to the cross-sectional area of ​​the flow area above the bed A a In conjunction, the continuity equation can be expressed as follows: In m ·A=v a ·A a Where A represents the area of ​​the entire pipe cross section, V m Indicates the average slurry velocity in the entire pipe section, A a represents the cross-sectional area of ​​the flow region above the bed, v a It represents the average velocity of the slurry in the flow area above the bed, and calculates the velocity Q of the slurry flowing through the flow area s The expression is as follows: Q s =C vd ·Q m =C vd ·v a ·A a Among them, Q s represents the average volume flow rate of solids, C vd represents the average transport concentration of the slurry, Q m represents the average volume flow rate of slurry, v a It represents the average velocity of the slurry in the flow area above the bed; The momentum equation is expressed as follows: r f ·g·i m ·A a =t w ·O w +t b ·O b r f ·g·i m ·A aw =t w ·O w r f ·g·i m ·A ab =t b ·O b Among them, ρ f represents the density of the slurry, i m represents the hydraulic gradient, g represents the acceleration of gravity, τ w represents the shear stress in the flow region at the top of the riverbed, τ b represents the shear stress in the flow area at the bottom of the riverbed, O w represents the perimeter of the flow area at the top of the riverbed, O b Indicates the width of the sediment on the top of the riverbed, A a represents the cross-sectional area of ​​the flow region above the bed, A aw represents the sub-region associated with the tube wall, A ab represents the sub-region associated with the top of the bed, and the shear stress τ at a specific boundary w and τ b The calculation formula is as follows: t w =ρ f ·g·R hw ·i m t b =ρ f ·g·R hb ·i m Among them, i m represents the hydraulic gradient, R hw Represents the sub-area A associated with the pipe wall aw The hydraulic radius, R hb Indicates area A associated with the top of the bed ab The hydraulic radius of The friction law equation includes two aspects of calculation, specifically: S2.a1, calculation of pipe wall friction; S2.a2, bed surface friction calculation; The particle support equation is used to express the effect of the transported particles on the total friction through the bed shear stress, which specifically includes the following three operations: S2.b1, based on the stratification of particle support mechanism, establish a pressure drop model and analyze the passing particles; S2.b2, contact load friction calculation; S2.b3, calculate the stratification ratio and its relationship with flow parameters; The solid transport equation is used to relate the solid flux to the bed shear stress and is expressed as follows: Among them, q s is the average solid flux per unit width of the flow area, α and β represent empirical coefficients, and θ b,cr is the Shields parameter θ for the particles at the top of the bed to start b The threshold value, g represents the acceleration due to gravity, d 50 is the median diameter of the solid particles, S S is the specific gravity of the solid, θ b represents the Shields parameter for the start of the particles at the top of the bed, and q s The expression is as follows: Among them, q s is the average solid flux per unit width of the flow area, O b represents the width of the sediment on the top of the riverbed, Q s represents the average volume flow rate of solids, C vd is the input parameter of the slurry pipeline hydraulic gradient prediction model, v a It represents the average velocity of the slurry in the flow area above the bed, R ha represents the hydraulic radius of the discharge area, A a Denotes the cross-sectional area of ​​the flow region above the bed, the analytical solution to the general equation for solid flux is: Among them, q s is the average solid flux per unit width through the flow region, y is the vertical position above the reference plane, y0 is the initial vertical position of the solid particle velocity and concentration profile, c(y) represents the local concentration of solid particles, and u s (y) is the local velocity of the solid particles, H is the height of the flow zone, represents the integral from y0 to H.

2. A method for predicting friction pressure drop in a settling slurry pipeline according to claim 1, characterized in that: The calculation formula of step S2.a1 is as follows: Among them, λ w Re represents the friction coefficient of the pipe wall, w represents the Reynolds number, v f represents the kinematic viscosity of the liquid, f n is the correlation function between the tube wall friction coefficient and the Reynolds number, R hw Represents the sub-area A associated with the pipe wall aw The hydraulic radius, v a It indicates the average velocity of the slurry in the flow area above the bed.

3. A method for predicting friction pressure drop in a settling slurry pipeline according to claim 2, characterized in that: The calculation formula of step S2.a2 is as follows: Among them, λ b represents the bed friction coefficient, R hb Indicates area A associated with the top of the bed ab The hydraulic radius, k s represents the hydraulic roughness of the bed top, and ln* represents the logarithmic function.

4. A method for predicting friction pressure drop in a settling slurry pipeline according to claim 1, characterized in that: The calculation formula of the pressure drop model in step S2.b1 is as follows: C va ·A a =C vab ·A ab Among them, C va represents the average spatial concentration in the flow area, A ab represents the sub-area associated with the top of the bed, C vab In area A ab The concentration of all particles in the 5. A method for predicting friction pressure drop in a settling slurry pipeline according to claim 4, characterized in that: The contact load friction expression in step S2.b2 is as follows: Among them, ρ f represents the density of the slurry, τ bs represents the bed shear stress acting as a result of the particle normal stress applied to the top of the bed by contact load particles travelling in the flow above the bed, S S Specific gravity of solid, O b represents the width of the riverbed top, g represents the acceleration of gravity, A a represents the cross-sectional area of ​​the flow region above the bed, C c represents the average spatial concentration C in the flow area va contact part.

6. A method for predicting friction pressure drop in a settling slurry pipeline according to claim 5, characterized in that: The layer ratio equation expression of step S2.b3 is as follows: Where μ represents the friction coefficient, C c represents the average spatial concentration C in the flow area va The contact part, C vd represents the input parameter of the slurry pipeline hydraulic gradient prediction model, μ *b represents the ratio of bed shear velocity, w t It represents the particle settling velocity, a dimensionless parameter used to evaluate the suitability of the flow capacity of suspended particles. K represents a constant and n represents an empirical coefficient.

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