CN114861396A - Mathematical model considering fine sand impact compression characteristic and heat absorption phase change and modeling method - Google Patents
Mathematical model considering fine sand impact compression characteristic and heat absorption phase change and modeling method Download PDFInfo
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Abstract
The invention is suitable for the technical field of explosion and impact numerical simulation, and discloses a mathematical model considering fine sand impact compression characteristics and heat absorption phase change and a modeling method, wherein the method comprises the following steps: establishing a dual-fluid model control equation based on the continuous medium hypothesis; establishing a gas phase state equation closed model according to the explosion type; describing the impact compression characteristic of the fine sand by adopting a condensed medium state equation, determining state equation parameters according to experimental data of a material impact state equation, and establishing a solid phase state equation closed model; establishing an interphase resistance action model according to the flow state of the multiphase flow; and establishing an interphase heat exchange rate and an interphase mass transfer model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium. The invention provides a closed mathematical model for describing the interaction physical process of shock waves and a sand filling structure, and provides a theoretical model for the research on the numerical simulation technology of the wave absorption and energy absorption effects of the sand filling structure under explosive loading.
Description
Technical Field
The invention belongs to the technical field of explosion and impact numerical simulation, and relates to a double-fluid mathematical model considering impact compression characteristics and heat absorption phase change of fine sand and a modeling method thereof.
Background
The sand-filled structure is used as an economical explosion protection structure and is used for reducing the damage of explosion impact load to personnel and various targets. Through carrying out reasonable optimization to filling sand structural design form, can promote the barrier propterty of structure, reduce the high-speed motion impact effect of main structure body granule crowd to the at utmost. In the field of explosion protection technology research, numerical value research is one of the main technical means for solving the problem of scientific optimization design of a sand filling structure.
The sand filling structure is composed of fine sand, under the loading of shock waves, the fine sand presents the shock compression characteristic of the simulated fluid, the development of a two-phase flow field is subject to various flow pattern changes such as stacked particle flow, dense gas-solid two-phase flow, sparse gas-solid two-phase flow and the like, and strong coupling effect of mass, momentum and energy is generated between the high-speed high-temperature fluid and the fine sand. In the technical field of explosion and impact numerical simulation, a numerical simulation method suitable for solving the problem of interaction between an impact wave and a sand filling structure is a multi-medium Euler method. At present, the multimedia mathematical models developed under the multimedia euler coordinate system and suitable for solving the problems can be roughly divided into two types: one type assumes that a clear material interface is formed between a fluid and a fine sand main body, and a mathematical model considers the momentum and energy coupling effect at the material interface, but cannot consider the temperature rise phase change effect caused by heat transfer at the interface, such as: a multi-media Euler solver integrated by main stream commercial software LS-DYNA and AUTODYN; in another type, the media are assumed to be mutually permeated and occupied by volume, and a mathematical model fully considers the resistance action, the heat transfer and the mass transfer process of the phases, but does not consider the impact compression characteristics of the media, such as: a dual-fluid model developed in the field of gas-solid multiphase flow. In order to solve the problem that the impact compression characteristic and the heat absorption phase change effect of a fine sand main body cannot be comprehensively considered by the existing multi-medium Euler method mathematical model, a more perfect mathematical model suitable for numerical solution under an Euler coordinate system needs to be established.
Disclosure of Invention
The invention aims to provide a double-fluid mathematical model considering fine sand impact compression characteristics and heat absorption phase transition and a modeling method thereof, which can reasonably and comprehensively describe the interaction physical process between media in a two-phase flow field under the condition of shock wave loading, make up the defects of the existing multi-media Euler numerical calculation model under the interaction of shock wave and a sand filling structure, play an important role in promoting the development of the multi-media Euler numerical calculation simulation technology under the interaction of shock wave and the sand filling structure, and play an important engineering application value in developing the numerical research of the explosion protection performance of the sand filling structure.
The technical scheme of the invention is to provide a mathematical model considering fine sand impact compression characteristics and heat absorption phase change, which is characterized in that: the method comprises a double-fluid model control equation, a gas phase state equation closed model, a solid phase state equation closed model, an interphase resistance action model, an interphase heat exchange rate model and an interphase mass transfer model.
Further, the two-fluid model control equation is as follows:
gas phase conservation of mass equation:
in the formula, epsilon g Is the volume fraction of the gas phase per unit volume, p g Is gas phase density, v g Is the velocity vector of the gas phase, Γ g Is the mass exchange rate between the phases and is,
is Hamiltonian;
gas phase momentum conservation equation:
in the formula (I), the compound is shown in the specification,
is epsilon p Gradient of (a), p g Represents the partial pressure of the gas phase, n p Representing the number density of solid phase particles, F D (S) Representing the rate of exchange of momentum between phases, v p Is the velocity vector of the solid phase, Γ p Representing the mass exchange rate between phases;
gas phase energy conservation equation:
in the formula, E g Representing the total energy per unit mass of the gas phase, F D (S) Representing the rate of exchange of momentum between phases, Q p (S) Is the interphase heat exchange rate;
solid phase conservation of mass equation:
in the formula, epsilon p Volume fraction of solid phase per unit volume, p p Is solid phase density, v p Is the velocity vector of the solid phase;
solid phase x, y, z direction momentum conservation equation:
in the formula, p p Represents the solid phase partial pressure;
solid phase energy conservation equation:
in the formula, E p Representing the total energy of the solid phase unit mass;
solid phase particle number conservation equation:
the relationship satisfied between the gas and solid phase volume fractions:
ε g +ε p ＝1
the solid phase particle number density and the solid phase volume fraction satisfy the following relationship:
in the formula (d) p Representing the particle diameter.
Further, the solid phase equation of state closed model has the following formula:
total energy E per unit mass of solid phase p Internal energy e in unit mass ratio p The following relationship is satisfied:
partial pressure p of solid phase p With partial pressure p of the gaseous phase g Satisfies the following relationship:
p p ＝ε p p g +p p,K
in the formula, p p,K The collision normal stress caused by particle-particle and particle-wall collision;
describing impact compression response characteristic, p, of fine sand by using a condensed medium state equation p,K The calculation formula of (a) is as follows:
in the formula:
is the density of the solid phase particulate material; c. C 0 Is the sound velocity in the normal state of the solid phase, ε p,max Is the solid phase maximum packing volume fraction, gamma is a state equation parameter,
is the increase in solid phase to mass ratio internal energy, Δ e, under compression p,H Is the increase in internal energy, Δ e, in solid-to-mass ratio caused by impact p,Ref Is the increase of the internal energy of the solid phase mass ratio caused by temperature rise, e p,Ref Is the solid phase reference mass specific energy, T p,Ref Is the solid phase reference temperature.
Further, the interphase resistance action model is as follows:
the function of interphase resistance F is described by using a Newton resistance formula D (S) The formula of (c) is as follows:
in the formula, C D Is an interphase resistance coefficient;
C D the calculation formula is of the form:
reynolds number Re of the particles p The calculation formula is as follows:
where μ is the hydrodynamic viscosity coefficient.
Further, the interphase heat exchange rate model is as follows:
heat exchange rate between phases Q p (S) The calculation formula is of the form:
where Nusselt Nu is used to characterize the ratio of convective heat transfer to conductive heat, λ is the thermal conductivity of the fluid, T g Is the gas phase temperature, T p Is a solid phase temperature, theta p Is the emissivity, θ, of the particles g Emissivity of ambient gas, σ B Is the Stefan-Boltzmann constant.
Further, the interphase mass transfer model uses the interphase mass transfer rate gamma p Represents:
interphase mass exchange rate Γ p The formula of (2) is of the form:
in the formula: c. C p Is the constant pressure specific heat capacity of the fluid, B is the mass transfer coefficient, q e Is the latent heat of vaporization.
Further, the solid phase temperature T p The following relationship is used for calculation:
in the formula: t is m 、T e Respectively solid phase melting and vaporization temperature, q m 、q e Respectively solid phase melting and latent heat of vaporization. c. C p,s Is the specific heat capacity of the solid phase.
Further, the gas phase equation of state closed model specifically includes:
total energy E of gas phase per unit mass g Internal energy e in unit mass ratio g Satisfies the following relationship:
the gas phase thermodynamic temperature and the unit mass ratio can satisfy the following relations:
T g ＝T g (e g )
gas phase partial pressure p g The following relationship is satisfied:
γ g ＝γ g (e g )
in the formula: gamma ray g Is an adiabatic index; t is g Is the gas temperature; p is a radical of REAL Is a real gas state equation; p is a radical of JWL For describing the detonation product state equation.
The invention also provides a modeling method of the mathematical model considering the impact compression characteristic and the heat absorption phase change of the fine sand, which is characterized in that based on the continuous medium hypothesis, the following models are established according to the following steps:
step 1, establishing a dual-fluid model control equation;
describing the impact compression characteristic of the fine sand by adopting a condensed medium state equation, determining state equation parameters according to experimental data of a material impact state equation, and establishing a solid phase state equation closed model;
establishing an interphase resistance action model according to the flow state of the multiphase flow;
establishing an interphase heat exchange rate model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium;
and establishing an interphase mass transfer model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium.
Furthermore, assuming that the main fine sand forming the sand filling structure is a quasi-fluid, a double-fluid model is adopted to describe the interaction physical process of the shock wave and the sand filling structure, and a gas phase and a solid phase are described under an Euler-Euler coordinate system. The theoretical control equation of the dual-fluid model deduced in the literature monographs is very complex, and the mathematical model is difficult to seal and solve and is not suitable for engineering application, so that only key influencing factors and mechanisms are concerned in modeling, and the method specifically comprises the following steps:
(1) the main body fine sand of the sand filling structure is assumed to be composed of a single spherical inert particle medium;
(2) the gas-solid two-phase flow system is assumed to be heat-insulated, and surface heat transfer, body heating and external force acting do not exist between the gas-solid two-phase flow system and the outside;
(3) the turbulent effect of gas-solid two-phase flow is not considered;
(4) considering the heat conduction and radiation heat transfer process between the gas and the particle group;
(5) the melting and vaporization process of the inert particles caused by heat absorption is considered;
(6) the conservation of particle number of particles in the interphase mass transfer process is assumed;
(7) taking into account the inter-phase drag effects between the gas and the particle population;
(8) considering the impact compression process of fine sand;
(9) the shearing effect of the fine sand is not taken into account.
Further, the control equation of the two-fluid model established in step 1 is as follows:
gas phase conservation of mass equation:
gas phase momentum conservation equation:
gas phase energy conservation equation:
solid phase conservation of mass equation:
solid phase x, y, z direction momentum conservation equation:
solid phase energy conservation equation:
solid phase particle number conservation equation:
the relationship satisfied between the gas and solid phase volume fractions:
ε g +ε p ＝1
the solid phase particle number density and the solid phase volume fraction satisfy the following relation:
further, in the step 2, the impact compression characteristic of the fine sand is described by adopting a state equation of a condensed medium, and a solid-phase state equation closed model is established according to an impact state equation experiment of the fine sand, wherein the solid-phase state equation closed model specifically comprises the following steps:
total energy E per unit mass of solid phase p Internal energy e in unit mass ratio p The following relationship is satisfied:
partial pressure p of solid phase p With partial pressure p of the gaseous phase g The following relation is satisfied:
p p ＝ε p p g +p p,K
in the formula: p is a radical of p,K The representation represents the collision normal stress caused by particle-particle and particle-wall collision.
Describing the impact compression response characteristic, p, of the sand filling structure by adopting a condensed medium state equation p,K The calculation formula of (a) is as follows:
the first term at the right end of the above formula mainly reflects the property of the cold state and is caused by the volume deformation of the medium; the second term reflects primarily the nature of the "hot state".
Determining a state equation parameter c according to experimental data of a material impact state equation 0 γ, and function Δ e p,H (ε p ) The specific expression of (1).
Further, in step 2, an interphase resistance action model is established according to the multiphase flow state, and when the relative motion between the fluid and the particles is maintained, the interphase resistance action model is subjected to generalized interphase resistance and lateral force from the fluid, wherein the interphase resistance action model comprises the following steps: resistance, additional mass force, baseset force, the latter including: lift, Magnus and Staffman forces, etc. In the modeling, only the fluid resistance which plays a main role is considered, the interphase resistance function is described by adopting a Newton resistance formula, and the interphase resistance F in the step 4 is given D (S) The calculation formula of (2).
The function of interphase resistance F is described by using a Newton resistance formula D (S) The formula of (c) is as follows:
coefficient of resistance C of particle group D The invention provides a new resistance coefficient calculation formula which can reflect the influences of the volume fraction of particle groups, the Reynolds number of particles and unsteady flow characteristics in a sparse and dense two-phase flow state. C D The calculation formula is of the form:
reynolds number Re of the particles p The calculation formula is as follows:
further, the heat transfer mechanism of the interphase heat transfer and the radiation heat transfer is comprehensively considered in modeling. When a high temperature gas stream flows over the particle surface, if there is a temperature difference between the phases, then there is a temperature gradientThe heat conduction process is generated by driving the phase to the phase, the heat quantity transferred to the particles by the fluid on the particle surface in unit time is in direct proportion to the temperature difference and the particle surface area, and the phase heat transfer rate is described by a Newton heat conduction equation in modeling; besides, the fluid-particle surface can perform heat transfer among phases in a heat radiation mode, and the Stefan-Boltzmann law is adopted in modeling to describe the radiation heat flow intensity of the particles to the surrounding fluid. Finally giving the phase heat exchange rate Q in the step 5 p (S) The formula of (2) is of the form:
heat exchange rate between phases Q p (S) The calculation formula is of the form:
further, under the action of high-temperature high-speed airflow scouring, the fine sand absorbs heat to generate temperature rise, and when the temperature of the particles reaches the melting temperature to the vaporization temperature, the particles generate melting and vaporization phase change processes from the surface, and the particles and the gas phase generate a mass transfer process. In the forced convection environment, the interphase mass transfer process is described by adopting a liquid drop evaporation formula considering the forced convection effect, and the interphase mass transfer rate gamma in step 6 is given p The formula of (2) is of the form:
interphase mass exchange rate Γ p The formula of (2) is of the form:
further, in step 5 and step 6, under the action of the high-temperature gas flow, when the melting and vaporization phase change process of the solid-phase particle medium is considered, the solid-phase temperature T is p The following relationship is used for calculation:
further, step 2, establishing a gas phase equation of state closed model according to the explosion type. When the mass energy density of the explosive source area is different, a specific product state equation is needed to describe the change of thermodynamic parameters in the expansion process of the source area, and the specific steps are as follows:
total energy E of gas phase per unit mass g Internal energy e in unit mass ratio g Satisfies the following relationship:
the gas phase thermodynamic temperature and the unit mass ratio can satisfy the following relations:
T g ＝T g (e g )
gas phase partial pressure p g The following relationship is satisfied:
γ g ＝γ g (e g )。
the invention has the beneficial effects that:
the invention provides a double-fluid mathematical model considering fine sand impact compression characteristics and heat absorption phase change, which can comprehensively consider the impact compression process of fine sand under the loading of shock waves and the heat absorption phase change process caused by a radiation and convection mechanism under a unified Euler-Euler type calculation model framework, makes up the defect that the existing multi-medium Euler numerical value calculation model under the action of the shock waves and a sand filling structure cannot reasonably describe the impact compression process of the sand filling structure, has an important promotion effect on the development of a multi-medium Euler numerical value simulation technology under the interaction of the shock waves and the sand filling structure, and has an important engineering application value for developing the numerical research on the explosion protection performance of the sand filling structure.
Drawings
FIG. 1 is a flow chart of a method of an embodiment.
FIG. 2 is a schematic diagram of an embodiment of the present invention, wherein (a) is a schematic diagram of an explosive source in a pipeline, and (b) is a schematic diagram of a sand filling structure in the pipeline.
Fig. 3 shows the cloud map distribution of gas-solid two-phase flow field parameters at time t equal to 0.2ms, where (a) is the cloud map distribution of solid phase volume fractions, (b) is the cloud map distribution of gas phase and solid phase pressure, (c) is the cloud map distribution of gas phase and solid phase velocity moduli, and (e) is the cloud map distribution of gas phase and solid phase temperature.
FIG. 4 is a solid volume fraction, solid phase pressure profile cloud at times t of 5.5ms, 6.0ms, 6.5ms, 7.0ms, and 7.5ms, where (a) is the solid volume fraction profile cloud and (b) is the solid phase pressure p p And (4) distributing cloud pictures.
Fig. 5 is a flow field energy and mass conservation test, wherein (a) is an energy conservation test, (b) is a mass conservation test, (c) is a change curve of the proportion of two-phase kinetic energy to total energy, and (d) is a change curve of the proportion of two-phase kinetic energy to total energy.
Fig. 6 is a comparison of the pressure time history numerical simulation result and the experimental test result, wherein (a) is a comparison of the pressure time history numerical simulation result at 0.621m of the pipeline wall surface and the experimental test result, and (b) is a comparison of the pressure time history numerical simulation result at 0.691m of the pipeline wall surface and the experimental test result.
Detailed Description
In order to make the objects, methods for establishing mathematical models, and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Aiming at the problem that a mathematical model in the existing multi-medium Euler simulation method does not simultaneously consider medium impact response characteristics and a heat transfer mechanism, the invention provides a two-fluid mathematical model considering fine sand impact compression characteristics and heat absorption phase change, and the invention is described in detail by combining the attached drawings.
As shown in fig. 1, the two-dimensional axisymmetric mathematical model is used as an example for explanation in this embodiment, and the specific modeling process is as follows:
s101: establishing a two-dimensional axisymmetric dual-fluid model control equation based on the continuous medium hypothesis, wherein the form is as follows;
gas phase conservation of mass equation:
the momentum conservation equation of the gas phase in the radial direction R and the axial direction Z is as follows:
gas phase energy conservation equation:
solid phase conservation of mass equation:
the momentum conservation equation of the solid phase in the radial direction R and the axial direction Z is as follows:
solid phase energy conservation equation:
solid phase particle number conservation equation:
the relationship satisfied between the gas and solid phase volume fractions:
ε g +ε p ＝1 (10)
in the above formulas: the following tables g, p represent gas and solid phases, respectively; epsilon is the volume fraction of phase k (═ g, p), representing the volume fraction of phase k per volume; rho is the phase density, representing the ratio of the mass of phase k to the volume occupied by phase k; u and v are radial R and axial Z velocity components of phase k; n is p Represents the solid phase particle number density; d p Represents the particle diameter; gamma-shaped p 、F D (S) 、Q p (S) 、E p Respectively representing the mass, momentum, heat and energy exchange rate of phases; p is a radical of g Represents the gas phase partial pressure; p is a radical of p Representing the solid phase partial pressure.
The solid phase particle number density and the solid phase volume fraction satisfy the following relationship:
s102: and establishing a gas phase state equation closed model according to the explosion type.
FIG. 2 is a schematic diagram of an example of the invention, wherein: the explosive source area is a chemical charge, the explosive charges explode in the center of the pipeline, and 13 sand walls are arranged on two sides of the pipeline respectively. Given the following formal state equation:
total energy E of gas phase per unit mass g Internal energy e in unit mass ratio g Satisfies the following relationship:
the gas phase thermodynamic temperature and the unit mass ratio can satisfy the following relation:
gas phase partial pressure p g The following relationship is satisfied:
in formulae (39) to (41): gamma ray g Is an adiabatic index; t is g Is the gas temperature.
S103: and describing the impact compression characteristic of the fine sand by adopting a state equation of a condensed medium, and determining a solid-phase state equation closed model according to experimental data of the state equation of the fine sand material.
The total energy per unit mass of the solid phase and the ratio per unit mass can satisfy the following relation:
the following relationship is satisfied between the solid phase partial pressure and the gas phase partial pressure:
p p ＝ε p p g +p p,K (17)
in the formula: p is a radical of p,K Representative of the collision normal stress induced by particle-particle, particle-wall collisions.
Describing impact compression response characteristic, p, of fine sand by using a condensed medium state equation p,K The calculation formula of (a) is as follows:
in formula (19):
is the density of the solid phase particulate material; c. C 0 Is the sound velocity in the normal state of the condensed medium, and gamma is a parameter. The first term at the right end of the above formula mainly reflects the property of the cold state and is caused by the volume deformation of the medium; the second term reflects primarily the nature of the "hot state".
The sand filling structure shown in FIG. 2 is composed of fine quartz sand, and the state equation parameter c is determined by referring to the experimental test data of the state equation of dry quartz sand material given by Laine et al (2001) and Brown et al (2007) 0 γ, and function Δ e p,H (ε p ) The specific expression form of (b) is as follows:
c 0 ＝236.6m/s
the values of other material parameters of the quartz fine sand are as follows:
s104: establishing an interphase resistance action model according to the flow state of the multiphase flow, which specifically comprises the following steps:
the function of interphase resistance F is described by using a Newton resistance formula D (S) The formula of (c) is as follows:
wherein:
C D for the interphase resistance coefficient, the formula form is calculated as follows:
the reynolds number of the particle is calculated as follows:
in the formula: d p μ is the hydrodynamic viscosity coefficient for the particle diameter.
S105: and establishing an interphase heat exchange rate model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium.
Heat exchange rate between phases Q p (S) The calculation formula is of the form:
in the formula: the first item on the right end represents the heat exchange between phases caused by a heat conduction mechanism, and the second item on the right end represents the heat exchange caused by a radiation heat transfer mechanism; nusselt Nu is used to characterize the ratio of convective heat transfer to conductive heat transfer; λ is the thermal conductivity of the fluid; theta p And theta g Emissivity of the particles and the ambient gas, respectively; sigma B Is the Stefan-Boltzmann constant, typically taken as 5.67X 10 -8 W/(m 2 ·K 4 )；T k (k ═ g, p) represents the phase k temperature.
Nu is calculated using the well-known Ranz-Marshall formula, of the form:
Nu＝2.0+0.6Re p 0.5 Pr 1/3 (25)
in the formula: the Prandtl (Prandtl) number Pr characterizes the ratio of specific heat at constant pressure to specific heat at constant volume.
S106: and establishing an interphase mass exchange rate model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium.
Interphase mass exchange rate Γ p The formula of (2) is of the form:
in the formula: q. q.s e Is the latent heat of vaporization; b is the mass transfer coefficient; c. C p Is the constant pressure specific heat capacity of the fluid.
S105 and S106: under the action of high-temperature airflow, when the melting and vaporization phase change processes of solid-phase particle media are considered, the solid-phase temperature T p The following relationship is used for calculation:
in the formula: t is m 、T e Respectively melting and vaporisation temperatures, q m 、q e Respectively the latent heat of fusion and vaporization.
Other thermodynamic parameters of the quartz sand are shown in table 1:
TABLE 1 thermodynamic parameters of quartz sand
Fig. 3 shows the distribution of gas-solid two-phase flow field parameter cloud charts at the time when t is 0.2ms, as shown in the figure: by utilizing the developed two-dimensional axisymmetric gas-solid two-phase numerical simulation program, a detailed numerical simulation result of parameters such as the volume fraction of a sand wall, the gas-solid two-phase pressure, the two-phase speed, the two-phase temperature distribution and the like in a flow field at any moment can be obtained. Driven by the blast shock wave, the discrete particle medium finally generates stacking action at the tail end of the container under the action of the drag force of the high-speed airflow. Fig. 4 (a) shows a cloud of the solid-phase volume fraction distribution at the end of the container at the time t is 5.5ms to 7.5ms, as shown in the figure: the solid volume fraction undergoes a trend of increasing first and then decreasing: reflecting the movement process that the discrete sand grains move to the tail end of the container to be compressed and thinned under the action of the gas phase medium. Fig. 4 (b) shows a solid phase pressure distribution cloud chart during the discrete sand compression and thinning process at the time t is 5.5ms to 7.5ms, as shown in the figure: the compressive normal stress of the solid phase medium will be changed along with the change of the solid phase stacking state, and the compressive normal stress of the solid phase medium will be changed along with the change of the compressive normal stress or the strengthening or the attenuation. Fig. 4 illustrates that the current solid phase pressure model can describe the change process of the internal compressive normal stress of the discrete particles during the compression process.
Fig. 5 (a) shows the time-dependent curves of the total energy of the flow field, the total energy of the solid phase, and the total energy of the gas phase. Fig. 5 (b) shows the time-dependent change curves of the total mass of the flow field, the total mass of the solid phase, and the total mass of the gas phase. As shown in the figure: the total energy and the total mass of the flow field keep good conservation. Fig. 5 (c) is a curve showing the change of the ratio of the total energy of the gas phase to the total energy of the solid phase over time, and fig. 5 (d) is a curve showing the change of the ratio of the total energy of the two phases over time, and the energy absorption effect of the sand-filled structure can be estimated according to the curves of the changes of the two internal energy and the kinetic energy.
Fig. 6 (a) to (b) show the comparison between the results of numerical simulation and experimental tests of the pressure-time history curves of the shock waves at 0.621m and 0.691m on the wall surface of the pipeline. As shown in the figure: the overall distribution trend of a pressure time history curve obtained from the numerical simulation result is basically consistent with the experimental test result, the pressure reaches the standard static numerical simulation result and is basically consistent with the actual measurement result, and the numerical simulation result can capture the detailed details in the wave system propagation and evolution process.
The above description is only for the purpose of illustrating the present invention and the appended claims are not to be construed as limiting the scope of the invention, which is intended to cover all modifications, equivalents and improvements that are within the spirit and scope of the invention as defined by the appended claims.
Claims (17)
1. A mathematical model considering impact compression characteristics and heat absorption phase change of fine sand is characterized in that: the method comprises a double-fluid model control equation, a gas phase state equation closed model, a solid phase state equation closed model, an interphase resistance action model, an interphase heat exchange rate model and an interphase mass transfer model.
2. The mathematical model considering fine sand impact compression characteristics and endothermic phase transition as claimed in claim 1, characterized by the following two-fluid model control equation:
gas phase conservation of mass equation:
in the formula, epsilon g Is the volume fraction of the gas phase per unit volume, p g Is gas phase density, v g Is the velocity vector of the gas phase, Γ g Is the inter-phase mass exchange rate, and is a Hamiltonian;
gas phase momentum conservation equation:
wherein ∈ - p Is epsilon p Gradient of (a), p g Represents the partial pressure of the gas phase, n p Representing the number density of solid phase particles, F D (S) Representing the rate of exchange of momentum between phases, v p Is the velocity vector of the solid phase, Γ p Representing the mass exchange rate between phases;
gas phase energy conservation equation:
in the formula, E g Representing the total energy per unit mass of the gas phase, F D (S) Representing the rate of exchange of momentum between phases, Q p (S) Is the interphase heat exchange rate;
solid phase conservation of mass equation:
in the formula, epsilon p Volume fraction of solid phase per unit volume, p p Is solid phase density, v p Is the velocity vector of the solid phase;
solid phase x, y, z direction momentum conservation equation:
in the formula, p p Represents the solid phase partial pressure;
solid phase energy conservation equation:
in the formula, E p Representing the total energy of the solid phase unit mass;
solid phase particle number conservation equation:
the relationship satisfied between the gas and solid phase volume fractions:
ε g +ε p ＝1
the solid phase particle number density and the solid phase volume fraction satisfy the following relationship:
in the formula (d) p Representing the particle diameter.
3. The mathematical model considering impact compression characteristics and endothermic phase transition of fine sand as claimed in claim 2, wherein the solid phase equation of state closed model is as follows:
total energy E per unit mass of solid phase p Internal energy e in unit mass ratio p The following relationship is satisfied:
partial pressure p of solid phase p With partial pressure p of the gaseous phase g Satisfies the following relationship:
p p ＝ε p p g +p p,K
in the formula, p p,K The collision normal stress caused by particle-particle and particle-wall collision;
describing impact compression response characteristic, p, of fine sand by using a condensed medium state equation p,K The calculation formula of (a) is as follows:
in the formula:
is the solid phase particulate material density; c. C 0 Is the sound velocity in the normal state of the solid phase, ε p,max Is the solid phase maximum packing volume fraction, gamma is a state equation parameter,
is compressionIncrease in solid phase to mass energy content, Δ e p,H Is the increase in internal energy, Δ e, in solid-to-mass ratio caused by impact p,Ref Is the increase of the internal energy of the solid phase mass ratio caused by temperature rise, e p,Ref Is the solid phase reference mass specific energy, T p,Ref Is the solid phase reference temperature.
4. The mathematical model considering impact compression characteristics of fine sand and endothermic phase transition as claimed in claim 3, characterized by the following inter-phase resistance effect model:
the function of interphase resistance F is described by using a Newton resistance formula D (S) The formula of (c) is as follows:
in the formula, C D Is an interphase resistance coefficient;
C D the calculation formula is of the form:
reynolds number Re of the particles p The calculation formula is as follows:
where μ is the hydrodynamic viscosity coefficient.
5. The mathematical model considering fine sand impact compression characteristics and endothermic phase transition as claimed in claim 4, characterized in that the interphase heat exchange rate model is as follows:
heat exchange rate between phases Q p (S) The calculation formula is of the form:
where Nusselt Nu is used to characterize the ratio of convective heat transfer to conductive heat, λ is the thermal conductivity of the fluid, T g Is the gas phase temperature, T p Is a solid phase temperature, theta p Is the emissivity, θ, of the particles g Emissivity of ambient gas, σ B Is the Stefan-Boltzmann constant.
6. The mathematical model considering impact compression characteristics of fine sand and endothermic phase transition as claimed in claim 5, wherein the interphase mass transfer model is based on the interphase mass transfer rate Γ p Represents:
interphase mass exchange rate Γ p The formula of (2) is of the form:
in the formula: c. C p Is the constant pressure specific heat capacity of the fluid, B is the mass transfer coefficient, q e Is the latent heat of vaporization.
7. The mathematical model considering impact compression characteristics and endothermic phase transition of fine sand as claimed in claim 5 or 6, wherein the solid phase temperature T is p The following relationship is used for calculation:
in the formula: t is m 、T e Respectively solid phase melting and vaporization temperature, q m 、q e Respectively solid phase melting and latent heat of vaporization. c. C p,s Is the specific heat capacity of the solid phase.
8. The mathematical model considering impact compression characteristics and endothermic phase transition of fine sand as claimed in claim 7, is characterized by a gas phase equation of state closed model, specifically:
total energy E of gas phase per unit mass g Internal energy e in unit mass ratio g Satisfies the following relationship:
the gas phase thermodynamic temperature and the unit mass ratio can satisfy the following relations:
T g ＝T g (e g )
gas phase partial pressure p g The following relationship is satisfied:
γ g ＝γ g (e g )
in the formula: gamma ray g Is an adiabatic index; t is g Is the gas temperature; p is a radical of REAL Is a real gas state equation; p is a radical of JWL For describing the detonation product state equation.
9. A modeling method of a mathematical model considering impact compression characteristics and endothermic phase transition of fine sand as claimed in claim 1, characterized in that based on continuous medium assumption, the following model is established according to the following steps:
step 1, establishing a dual-fluid model control equation;
step 2, establishing a gas phase state equation closed model according to the explosion type;
establishing a gas phase state equation closed model according to the explosion type;
describing the impact compression characteristic of the fine sand by adopting a condensed medium state equation, determining state equation parameters according to experimental data of a material impact state equation, and establishing a solid phase state equation closed model;
establishing an interphase resistance action model according to the flow state of the multiphase flow;
establishing an interphase heat exchange rate model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium;
and establishing an interphase mass transfer model according to the convection heat exchange characteristics of the flow field and the melting and vaporization characteristics of the particle medium.
10. A modeling method according to claim 9, characterized in that the continuous medium assumes, in particular:
(1) the fine sand main body is assumed to be composed of a single spherical inert particle medium;
(2) the gas-solid two-phase flow system is assumed to be heat-insulated, and surface heat transfer, body heating and external force acting do not exist between the gas-solid two-phase flow system and the outside;
(3) the turbulent effect of gas-solid two-phase flow is not considered;
(4) considering the heat conduction and radiation heat transfer process between the gas and the particle group;
(5) the melting and vaporization process of the inert particles caused by heat absorption is considered;
(6) the conservation of particle number of particles in the interphase mass transfer process is assumed;
(7) taking into account the inter-phase drag effects between the gas and the particle population;
(8) considering the impact compression process of fine sand;
(9) the shear failure effect of the fine sand is not considered.
11. The modeling method of claim 10, wherein the two-fluid model control equation established in step 1 is as follows:
gas phase conservation of mass equation:
gas phase momentum conservation equation:
gas phase energy conservation equation:
solid phase conservation of mass equation:
solid phase x, y, z direction momentum conservation equation:
solid phase energy conservation equation:
solid phase particle number conservation equation:
the relationship satisfied between the gas and solid phase volume fractions:
ε g +ε p ＝1
the solid phase particle number density and the solid phase volume fraction satisfy the following relationship:
12. the modeling method of claim 11, wherein the impact compression characteristic of the fine sand is described in step 2 by using a condensed medium equation of state, specifically as follows:
total energy E per unit mass of solid phase p Internal energy e in unit mass ratio p The following relationship is satisfied:
partial pressure p of solid phase p With partial pressure p of the gaseous phase g Satisfies the following relationship:
p p ＝ε p p g +p p,K
describing impact compression response characteristic, p, of fine sand by using a condensed medium state equation p,K The calculation formula of (a) is as follows:
13. the modeling method according to claim 12, wherein the interphase resistance action model is established according to the multiphase flow state in the step 2, and specifically as follows:
the function of interphase resistance F is described by using a Newton resistance formula D (S) The formula of (c) is as follows:
C D the calculation formula is of the form:
reynolds number Re of the particles p The calculation formula is as follows:
14. the modeling method according to claim 13, wherein in the step 2, according to the convective heat transfer characteristics of the flow field and the melting and vaporization characteristics of the granular medium, an interphase heat transfer rate model is established as follows:
heat exchange rate between phases Q p (S) The calculation formula is of the form:
15. the modeling method according to claim 14, wherein in step 2, an interphase mass transfer model is established according to the convection heat transfer characteristics of the flow field and the melting and vaporization characteristics of the particle medium, specifically as follows:
interphase mass transfer model based on interphase mass exchange rate gamma p Expressed, in the form:
16. two-fluid mathematical model taking into account impact compression characteristics of fine sand and endothermic phase transition as claimed in claim 14 or 15, characterized in that under the action of high temperature gas flow, melting and vaporization phase transition of solid phase particle medium is taken into accountAt process time, solid phase temperature T p The following relationship is used for calculation:
17. the modeling method of the dual-fluid mathematical model considering the impact compression characteristic and the endothermic phase transition of the fine sand according to claim 16, wherein the step 2 is to establish a gas phase equation of state closed model according to the explosion type, specifically:
total energy E of gas phase per unit mass g Internal energy e in unit mass ratio g Satisfies the following relationship:
the gas phase thermodynamic temperature and the unit mass ratio can satisfy the following relations:
T g ＝T g (e g )
gas phase partial pressure p g The following relationship is satisfied:
γ g ＝γ g (e g )。
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