CN112434472B - Method for calculating additional mass of narrow slit gap of multilayer coaxial cylinder of reactor - Google Patents
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Abstract
The invention belongs to the technical field of fluid mechanics, and particularly relates to a method for calculating the additional mass of a narrow gap of a multilayer coaxial cylinder of a reactor, which comprises the following steps of firstly, giving model parameters, and supposing and approximating; establishing a fluid equation, and expanding the fluid pressure along the circumferential direction and the axial direction; step three, selecting a vibration mode function of the shell; expressing boundary conditions and pressure as expressions meeting the shell vibration mode function; step five, solving the additional mass; the method can realize the calculation of the additional mass of the three-dimensional beam type and various three-dimensional shell vibration modes. Compared with a formula of a two-dimensional method given by the American society of mechanical engineers, the calculation strategy provided by the invention can greatly reduce the additional mass of beam vibration mode and solve the problem of over conservation; the problem that the coaxial cylinder body with the single-end simple support boundary condition has no additional mass analysis solution of the shell vibration mode is solved, and support and basis are provided for reactor design and safety analysis.
Description
Technical Field
The invention belongs to the technical field of fluid mechanics, and particularly relates to a method for calculating additional mass of a narrow gap of a multilayer coaxial cylinder of a reactor.
Background
The pool type reactor, such as a pool type fast reactor, has the characteristics of large size and thin wall, so the rigidity is relatively low, and the design, analysis and verification of the reactor earthquake resistance are important concerns for the safety evaluation of the reactor. Inside the reactor is a very complex system, with many complex structures such as the core, heat exchangers and main pumps, all surrounded by multiple layers of metallic heat shields in order to protect the equipment from excessive temperatures. A slight gap exists between the heat shields and the apparatus and is filled with coolant. Vibration caused by earthquake or fluid flushing causes fluid-solid coupling action to occur between the equipment and the supporting cylinder, and the inherent vibration characteristics of the structure are changed, which is important for the equipment safety of the reactor.
In structural seismic design, a method of adding mass to fluid is generally adopted to replace complex fluid-solid coupling effect. The method is based on potential flow theory, the fluid force suffered by the structure is simplified into inertial force related to the movement acceleration of the structure, the coefficient of the inertial force is called as additional mass, and the mass is added on the structure to perform earthquake-proof design.
The American Society of Mechanical Engineers (ASME) gives an infinitely long cylinder immersed in a fluid an additional mass formula that is widely used in industrial design, but the method derives based on two-dimensional theory, the result being too conservative and applicable only in the case of beam vibration, especially not in the case of structural three-dimensional vibration of narrow gap fluids.
In order to solve the problem of three-dimensional vibration, researchers in the us arches laboratory were first led to a method for calculating the additional mass of beam and shell type vibrations of a double-ended multi-layered shell. The three-dimensional method can remarkably reduce the problem that the additional mass is too conservative, but the method is only suitable for the problem of double-end simple support, so that the method cannot be used for vibration of single-end simple (cantilever beam type) multi-layer narrow slit gap cylinders such as a fast reactor main pump support cylinder, a heat shield, a main container, a heat shield and the like. The invention provides a method for calculating the additional mass of fluid in a narrow gap of a multilayer coaxial cylinder of a reactor, which aims to solve the problem of single-end simple support three-dimensional vibration.
Disclosure of Invention
Aiming at the defects, the invention aims to provide a method for calculating the additional mass of the gap of the multi-layer coaxial cylinder of the reactor, which solves the problem that the existing two-dimensional theory is too conservative for the additional mass estimation of the gap fluid of the three-dimensional single-end simply supported coaxial cylinder, thereby improving the economy of the design of the reactor.
The technical scheme of the invention is as follows:
a method for calculating the additional mass of the gap between multi-layer coaxial cylinder body and narrow slit of reactor features that the combination of axial Liang Hanshu and circumferential trigonometric function approaches the vibration mode function of cylindrical shell to obtain the additional mass of equipment,
step one, giving model parameters, assumptions and approximations;
establishing a fluid equation, and expanding the fluid pressure along the circumferential direction and the axial direction; writing a wave equation of a gap fluid pressure field, and applying a factor decomposition method, wherein n is a circumferential wave number, and k is an axial wave number along the axial direction of the cylindrical coordinate system;
step three, selecting a vibration mode function of the shell; selecting a vibration mode function of the cylinder under a fixed-free boundary condition;
expressing boundary conditions and pressure as expressions meeting the shell vibration mode function; giving boundary conditions on the fluid-solid contact surface to obtain a distribution formula of a gap fluid pressure field;
step five, solving the additional mass; writing a motion equation of vibration, solving the surface density of the additional mass, and obtaining the additional mass;
step one, giving model parameters, assumptions and approximations;
the bottoms of the two cylinders are simply supported, the tops of the two cylinders are unconstrained, and the radius R of the inner cylinder a is equal to a Length L a The method comprises the steps of carrying out a first treatment on the surface of the Radius R of outer cylinder b b Length L b The method comprises the steps of carrying out a first treatment on the surface of the The level L of the interstitial fluid, the fluid density ρ and the sound velocity C; assuming that the b cylinder is rigid;
the basic assumption is that the fluid is non-viscous; neglecting gravity effects; the fluid flow rate is less than the speed of sound; the considered frequency is lower than the coherence frequency, i.e. the coherence frequency is defined as the frequency at which the wavelength of the sound waves in the fluid medium is equal to the axial bending wavelength of the cylindrical shell;
the approximation is set as single-mode approximation, namely the coupling effect between the main frequency of the cylinder vibration and the high-order mode frequency is ignored, the actual displacement is dominated by the main frequency, and the coupling effect between different modes of the cylinder is ignored;
establishing a fluid equation, and expanding the fluid pressure along the circumferential direction and the axial direction;
wave equation for gap fluid pressure field:
wherein,the Laplace operator is that C is the sound velocity in water, and t is time; p is the pressure of a certain point inside the fluid, r is the radial position of a certain point inside the fluid, θ is the circumferential angle of a certain point inside the fluid, and z is the axial height of a certain point inside the fluid;
by applying the factorization method, the gap fluid pressure field is written to be expanded along the circumferential direction n-order of the cylindrical coordinate system, wherein n is a circumferential wave number, and expanded along the axial direction k-order of the cylindrical coordinate system, wherein k is an axial wave number, and the expression is as follows:
wherein e is natural logarithm, i is imaginary sign, ω is vibration angular frequency; n is a positive integer from 0, and k is a positive integer from 1; other parameters are defined as follows:
wherein A is nk And B nk Is a constant related to boundary conditions, L is the level of interstitial fluid;
α k the definition is as follows:
wherein l k The characteristic value obtained by bringing the formula (2) into the formula (1);
selecting a vibration mode function of the shell;
is the generalized coordinate of the normal direction of the a cylinder, +.>Is the generalized coordinate of the normal direction of the b cylinder, +.>Is the alpha-order vibration mode of the cylinder body a,the beta-order vibration mode of the cylinder b;
selecting a general vibration mode function of a cylinder body under a fixed-free boundary condition, wherein the vibration mode function of a cylinder body a is as follows:
wherein ψ is n As a circumferential vibration mode function, ψ k Is an axial vibration mode function and adopts the following expression:
wherein,
selecting the vibration mode function of the cylinder a to obtain the vibration mode of the cylinder b
Expressing boundary conditions and pressure as expressions meeting the shell vibration mode function;
boundary conditions on fluid-solid interface:
wherein ρ is the fluid density, w a For normal displacement of cylinder a, w b Is the normal displacement of the cylinder b; will w a And w b The following expressions are developed on the vibration modes of the cylinder a and the cylinder b, respectively:
will vibrate the typeAnd->With Θ in formula (3) n e k As a basis function, do gamma term expansion:
wherein,is of the vibration type +.>And->According to the basis function theta n e k Expanded item gamma->Is of the vibration type +.>And->Inner volume of (A) (I)>Is of the vibration type +.>And->Is the inner product of:
wherein the integration region Ω is a interstitial fluid region;
the formula (2) is also referred to as Θ n e k The gamma term expansion for the basis function:
wherein,
p γ (r)＝A γ I γ (r)+B r K γ (r) (11)
carrying the formula (10) and the formula (7) into the formula (6) to derive A nk And B nk And then carrying out the expression (11) to obtain a pressure expression of the development of the gamma term:
wherein,
wherein I is γ Correction of Bessel function, K for the first class corresponding to the gamma term γ Modified Bessel function for the second class corresponding to the gamma term, I γ ' is I γ Derivative of K γ ' is K γ Is a derivative of (2); carrying out the step (12) into the step (10) to obtain a distribution formula of the gap fluid pressure field;
step five, solving the additional mass;
the cylinder b is rigid, and the motion equation of the alpha-order vibration of the cylinder a is as follows:
wherein,for the areal density of cylinder a, +.>For generalized stiffness->Is generalized pressure>Fluid load +.>Is represented by the expression:
wherein R is a Is the radius of the inner cylinder;
the additional mass is that the pressure of the fluid gap is equivalent to the additional virtual mass of the cylinder, and the motion equation of the cylinder after the additional mass is added is as follows:
wherein,additional mass area density corresponding to alpha-mode of a cylinder,>generalized coordinates of the corresponding equation after attaching mass to its structure;
and (3) carrying out comparison by taking the formula (12) into the formula (14) and comparing with the formula (16) to obtain an analytical solution of the additional mass area density:
the invention has the beneficial effects that:
according to the calculation strategy of the additional mass of the gap between the coaxial cylinder bodies of the reactor, which is suitable for the single-end simple branch boundary condition, the calculation of the additional mass of the three-dimensional beam type and various three-dimensional shell vibration modes can be realized. Compared with a formula of a two-dimensional method given by the American society of mechanical engineers, the calculation strategy provided by the invention can greatly reduce the additional mass of beam vibration mode and solve the problem of over conservation; the problem that the coaxial cylinder body with the single-end simple support boundary condition has no additional mass analysis solution of the shell vibration mode is solved, and support and basis are provided for reactor design and safety analysis.
Detailed Description
The technical solutions of the embodiments of the present invention will be clearly and completely described below in conjunction with the embodiments of the present invention, and it is apparent that the described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
The method is used for calculating the approximate mass of the fluid in the narrow gap of the multilayer coaxial cylinder of the reactor. Taking a single-end simply supported double-layer cylindrical shell device as an example, adopting a combination of an axial Liang Hanshu and a circumferential trigonometric function to approach a vibration mode function of the cylindrical shell, and finally obtaining the additional mass of the device, wherein the method is implemented as follows:
step one, giving model parameters, assumptions and approximations;
the bottom ends of the two cylinders are simply supported, and the top ends of the two cylinders are unconstrained. Radius R of inner cylinder a a Length L a The method comprises the steps of carrying out a first treatment on the surface of the Radius R of outer cylinder b b Length L b The method comprises the steps of carrying out a first treatment on the surface of the The level L of the interstitial fluid, the fluid density ρ and the sound velocity C.
Basic assumption is that: the fluid is non-viscous; neglecting gravity effects; the fluid flow rate is less than the speed of sound; the considered frequency is lower than the coherence frequency, i.e. the coherence frequency is defined as the frequency at which the wavelength of the sound waves in the fluid medium is equal to the axial bending wavelength of the cylindrical shell;
approximation setting: the single mode approximation, i.e. neglecting the coupling between the main frequency and the higher order mode frequency of the cylinder vibration, the actual displacement is usually dominated by the main frequency, and the coupling between the different modes of the a-cylinder is negligible.
In this embodiment, l=l a ＝L b ＝1.117m，R a ＝0.28m，R b ＝0.308m，ρ＝1000kg/m 3 Fluid sound speed c=1400 m/s. And assuming that the b cylinder is rigid.
Establishing a fluid equation, and expanding the fluid pressure along the circumferential direction and the axial direction;
write the wave equation for the gap fluid pressure field:
wherein,the Laplace operator is that C is the sound velocity in water, and t is time; p is the pressure at a point within the fluid, r is the radial position of a point within the fluid, θ is the circumferential angle of a point within the fluid, and z is the axial height of a point within the fluid.
By applying the factorization method, the gap fluid pressure field is written to be expanded along the circumferential direction n-order of the cylindrical coordinate system, wherein n is a circumferential wave number, and expanded along the axial direction k-order of the cylindrical coordinate system, wherein k is an axial wave number, and the expression is as follows:
wherein e is natural logarithm, i is imaginary sign, ω is vibration angular frequency; n is a positive integer from 0, and k is a positive integer from 1; other parameters are defined as follows:
wherein A is nk And B nk Is a constant related to boundary conditions, L is the level of interstitial fluid.
α k The definition is as follows:
wherein l k To bring equation (2) into the eigenvalue obtained by equation (1).
Step three, selecting a vibration mode function of the shell;
reference to plate and shell theory of vibration (Cao Zhiyuan. Plate and shell theory of vibration [ M ]]Beijing, china railway Press 1989.311-313),is the generalized coordinate of the normal direction of the a cylinder, +.>Is the generalized coordinate of the normal direction of the b cylinder, +.>Is the alpha-order vibration type of the cylinder a, < >>Is the beta-order vibration mode of the cylinder b.
For the fixed-free cylindrical shell in this embodiment, a general vibration mode function of the cylinder of the fixed-free boundary condition is selected, and the vibration mode function of the cylinder a is:
wherein ψ is n As a circumferential vibration mode function, ψ k Is an axial vibration mode function and adopts the following expression:
wherein,
selecting the vibration mode function of the cylinder a to obtain the vibration mode of the cylinder b
Expressing boundary conditions and pressure as expressions meeting the shell vibration mode function;
boundary conditions on the fluid-solid contact surface are given:
wherein ρ is the fluid density, w a For normal displacement of cylinder a, w b Is the normal displacement of the cylinder b. Will w a And w b The following expressions are developed on the vibration modes of the cylinder a and the cylinder b, respectively:
will vibrate the typeAnd->With Θ in formula (3) n e k As a basis function, do gamma term expansion:
wherein,is of the vibration type +.>And->According to the basis function theta n e k Expanded item gamma->Is of the vibration type +.>And->Inner volume of (A) (I)>Is of the vibration type +.>And->Is the inner product of:
wherein the integration region Ω is the interstitial fluid region.
The formula (2) is also referred to as Θ n e k The gamma term expansion for the basis function:
wherein,
p γ (r)＝A γ I γ (r)+B r K γ (r) (11)
will (10)And (7) formula (6), export A nk And B nk And then carrying out the expression (11) to obtain a pressure expression of the development of the gamma term:
wherein,
wherein I is γ Correction of Bessel function, K for the first class corresponding to the gamma term γ Modified Bessel function for the second class corresponding to the gamma term, I γ ' is I γ Derivative of K γ ' is K γ Is a derivative of (a). And (3) carrying out the step (12) into the step (10) to obtain a distribution formula of the gap fluid pressure field.
Step five, solving the additional mass;
for this embodiment, the b cylinder is rigid, and the equation of motion for the alpha-order vibration of cylinder a can be written as:
wherein,for the areal density of cylinder a, +.>For generalized stiffness->Is generalized pressure>Fluid load +.>Is represented by the expression:
wherein R is a Is the radius of the inner cylinder.
The additional mass is equivalent to the pressure of the fluid gap as the additional virtual mass of the cylinder. Specifically, the equation of motion of the cylinder after adding the additional mass is:
wherein,additional mass area density corresponding to alpha-mode of a cylinder,>generalized coordinates of the corresponding equation after adding mass to its structure.
And (3) carrying out comparison by taking the formula (12) into the formula (14) and comparing with the formula (16) to obtain an analytical solution of the additional mass area density:
according to the parameters given in this example, the following results can be obtained:
the disclosed embodiments of the present invention relate only to methods related to the disclosed embodiments, other methods may refer to general designs, and the same embodiment and different embodiments of the present invention may be combined with each other without conflict;
the foregoing description of the preferred embodiments of the invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (1)
1. The method for calculating the additional mass of the narrow gap of the multilayer coaxial cylinder of the reactor approximates the vibration mode function of the cylindrical shell by adopting the combination of the axial Liang Hanshu and the circumferential trigonometric function, thereby obtaining the additional mass of the equipment, and is characterized in that:
step one, giving model parameters, assumptions and approximations;
establishing a fluid equation, and expanding the fluid pressure along the circumferential direction and the axial direction; writing a wave equation of a gap fluid pressure field, and applying a factor decomposition method, wherein n is a circumferential wave number, and k is an axial wave number along the axial direction of the cylindrical coordinate system;
step three, selecting a vibration mode function of the shell; selecting a vibration mode function of the cylinder under a fixed-free boundary condition;
expressing boundary conditions and pressure as expressions meeting the shell vibration mode function; giving boundary conditions on the fluid-solid contact surface to obtain a distribution formula of a gap fluid pressure field;
step five, solving the additional mass; writing a motion equation of vibration, solving the surface density of the additional mass, and obtaining the additional mass;
step one, giving model parameters, assumptions and approximations;
the bottoms of the two cylinders are simply supported, the tops of the two cylinders are unconstrained, and the radius R of the inner cylinder a is equal to a Length L a The method comprises the steps of carrying out a first treatment on the surface of the Radius R of outer cylinder b b Length L b The method comprises the steps of carrying out a first treatment on the surface of the The level L of the interstitial fluid, the fluid density ρ and the sound velocity C; assuming that the b cylinder is rigid;
the basic assumption is that the fluid is non-viscous; neglecting gravity effects; the fluid flow rate is less than the speed of sound; the considered frequency is lower than the coherence frequency, i.e. the coherence frequency is defined as the frequency at which the wavelength of the sound waves in the fluid medium is equal to the axial bending wavelength of the cylindrical shell;
the approximation is set as single-mode approximation, namely the coupling effect between the main frequency of the cylinder vibration and the high-order mode frequency is ignored, the actual displacement is dominated by the main frequency, and the coupling effect between different modes of the cylinder is ignored;
establishing a fluid equation, and expanding the fluid pressure along the circumferential direction and the axial direction;
wave equation for gap fluid pressure field:
wherein,the Laplace operator is that C is the sound velocity in water, and t is time; p is the pressure of a certain point inside the fluid, r is the radial position of a certain point inside the fluid, θ is the circumferential angle of a certain point inside the fluid, and z is the axial height of a certain point inside the fluid;
by applying the factorization method, the gap fluid pressure field is written to be expanded along the circumferential direction n-order of the cylindrical coordinate system, wherein n is a circumferential wave number, and expanded along the axial direction k-order of the cylindrical coordinate system, wherein k is an axial wave number, and the expression is as follows:
wherein e is natural logarithm, i is imaginary sign, ω is vibration angular frequency; n is a positive integer from 0, and k is a positive integer from 1; other parameters are defined as follows:
wherein A is nk And B nk Is a constant related to boundary conditions, L is the level of interstitial fluid;
α k is defined as：
Wherein l k The characteristic value obtained by bringing the formula (2) into the formula (1);
selecting a vibration mode function of the shell;
is the generalized coordinate of the normal direction of the a cylinder, +.>Is the generalized coordinate of the normal direction of the b cylinder, +.>Is the alpha-order vibration type of the cylinder a, < >>The beta-order vibration mode of the cylinder b;
selecting a general vibration mode function of a cylinder body under a fixed-free boundary condition, wherein the vibration mode function of a cylinder body a is as follows:
wherein ψ is n As a circumferential vibration mode function, ψ k Is an axial vibration mode function and adopts the following expression:
wherein,
selecting the vibration mode function of the cylinder a to obtain the vibration mode of the cylinder b
Expressing boundary conditions and pressure as expressions meeting the shell vibration mode function;
boundary conditions on fluid-solid interface:
wherein ρ is the fluid density, w a For normal displacement of cylinder a, w b Is the normal displacement of the cylinder b; will w a And w b The following expressions are developed on the vibration modes of the cylinder a and the cylinder b, respectively:
will vibrate the typeAnd->With Θ in formula (3) n e k As a basis function, do gamma term expansion:
wherein,is of the vibration type +.>And->According to the basis function theta n e k Expanded item gamma->Is of the vibration type +.>And->Is used for the internal product of (a),is of the vibration type +.>And->Is the inner product of:
wherein the integration region Ω is a interstitial fluid region;
the formula (2) is also referred to as Θ n e k The gamma term expansion for the basis function:
wherein,
p γ (r)＝A γ I γ (r)+B r K γ (r) (11)
carrying the formula (10) and the formula (7) into the formula (6) to derive A nk And B nk And then carrying out the expression (11) to obtain a pressure expression of the development of the gamma term:
wherein,
wherein I is γ Correction of Bessel function, K for the first class corresponding to the gamma term γ Modified Bessel function for the second class corresponding to the gamma term, I γ ' is I γ Derivative of K γ ' is K γ Is a derivative of (2); carrying out the step (12) into the step (10) to obtain a distribution formula of the gap fluid pressure field;
step five, solving the additional mass;
the cylinder b is rigid, and the motion equation of the alpha-order vibration of the cylinder a is as follows:
wherein,for the areal density of cylinder a, +.>For generalized stiffness->Is generalized pressure>Fluid load +.>Is represented by the expression:
wherein R is a Is the radius of the inner cylinder;
the additional mass is that the pressure of the fluid gap is equivalent to the additional virtual mass of the cylinder, and the motion equation of the cylinder after the additional mass is added is as follows:
wherein,additional mass area density corresponding to alpha-mode of a cylinder,>generalized coordinates of the corresponding equation after attaching mass to its structure;
and (3) carrying out comparison by taking the formula (12) into the formula (14) and comparing with the formula (16) to obtain an analytical solution of the additional mass area density:
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