CN114186365A

CN114186365A - Method for determining safe load of pressure-bearing cylinder with temperature difference

Info

Publication number: CN114186365A
Application number: CN202111519339.9A
Authority: CN
Inventors: 刘朝阳; 张松; 朱瑞林
Original assignee: Shandong University
Current assignee: Shandong University
Priority date: 2021-12-13
Filing date: 2021-12-13
Publication date: 2022-03-15

Abstract

The invention discloses a method for determining the safe load of a pressure-bearing cylinder with temperature difference in order to ensure the safe design, manufacture and operation of a pressure container, save materials and reduce cost, and limits each key stress of the pressure-bearing cylinder which bears the pressure and the temperature difference load at the same time to yield strength sigma_yThe following (when engineering is applied, only need to be the one with sigma)_yDivided by a safety factor to obtain the allowable stress [ sigma ]]) The safety of the pressure container is fully ensured, and parameters such as safe operation or design pressure, safe operation or design temperature difference and the like, and parameters such as optimal operation (design) pressure, temperature difference and the like are derived. The invention provides a calculation formula and a chart of the safety and optimal parametersAnd methods of use thereof. The method according to the invention determines the operating (design, manufacturing) parameters which ensure a safe and economical pressure vessel.

Description

Method for determining safe load of pressure-bearing cylinder with temperature difference

Technical Field

The invention relates to a method for determining the safe load of a pressure-bearing cylinder with temperature difference, belonging to the technical field of high-end equipment manufacture.

Background

Pressure vessels are key equipment widely used in many industrial sectors, such as the mechanical, chemical, energy (including nuclear, etc.), aviation, aerospace, weapons, materials, pharmaceuticals, light industry, food, metallurgy, petroleum, construction, etc. The main body of the pressure vessel is mostly a cylinder which bears internal pressure, namely the pressure-bearing cylinder of the invention. The housings of any pressure-bearing equipment are mostly cylinders, such as hydraulic cylinders and air cylinders in the mechanical industry, aircraft engine housings, pressure vessel housings in the petrochemical industry, gun tubes, nuclear reactor housings, and the like. The pressure-bearing cylinder is a key component for bearing working load, the safety of the pressure-bearing cylinder is very important, and once an accident is caused by insufficient strength, the shutdown and the production halt result in great economic loss and casualties.

It is known that the distribution of the elastic stresses in the walls of a cylinder is very uneven when the cylinder is subjected to operating pressures, and that the load-bearing capacity is very low and the wall thickness is very large if the pressure-bearing cylinder is designed for maximum stress. The large wall thickness not only wastes materials, resources and funds, increases the cost, but also has potential safety hazards. It is necessary to try to reduce the stress in the vessel wall to increase the strength of the vessel, save materials, reduce costs, and increase the safety of the pressure vessel. In order to reduce the stress and improve the stress distribution to improve the carrying capacity, it is an effective method to introduce additional stress. Conventionally, prestress is generally introduced by a mechanical method, i.e. a large mechanical pressure is applied to a container before the container is put into use, so that the inner layer part material is plastically deformed, the outer layer part material is still in an elastic state, and after the mechanical pressure is removed, mechanical prestress (residual stress) is generated in the container wall: the material of the inner layer is compressive stress, and the material of the outer layer is tensile stress. The mechanical prestress can be reduced by adding the stress caused by the operation pressure of the container. When the inner wall and the outer wall of the pressure-bearing cylinder have temperature difference, temperature difference stress or thermal stress is generated in the wall of the cylinder due to expansion with heat and contraction with cold. The temperature difference between the inner wall and the outer wall of the pressure-bearing cylinder is sometimes artificially applied for improving stress distribution and sometimes inevitably exists in the production process (such as chemical pressure vessels, hydrogenation reactors, ultrahigh pressure food sterilization kettles, nuclear reactors, fusion reactors and the like).

When the pressure-bearing cylinder simultaneously bears mechanical load (i.e. operating pressure) and thermal load (i.e. temperature difference between inner wall and outer wall), the total stress is more complex, and new mechanical phenomena and laws can be generated. Therefore, when the pressure-bearing cylinder simultaneously bears the thermal load caused by the mechanical pressure load and the temperature difference, how to effectively and reasonably utilize the thermal stress (namely the temperature difference stress), how to scientifically and reasonably determine the safe operation or design parameter range, and how to determine the optimal operation parameter or optimal design parameter, thereby achieving the purposes of safety and economy, and being a technical problem to be solved urgently.

Disclosure of Invention

The invention aims to provide a method for determining the safe load of a pressure-bearing cylinder with temperature difference so as to comprehensively ensure the safety of the pressure-bearing cylinder.

The technical scheme adopted by the invention is as follows:

the invention provides a method for determining the safe load of a pressure-bearing cylinder with temperature difference, wherein the load comprises the pressure and the temperature difference born by the pressure-bearing cylinder, and the method comprises the following steps:

obtaining the outer radius r of a pressure-bearing cylinder_oInner radius r_iObtaining the diameter ratio k of the pressure-bearing cylinder;

obtaining the temperature t of the inner wall of the pressure-bearing cylinder_iTemperature t of the outer wall_oObtaining the temperature difference delta t ═ t_i-t_o；

Obtaining the initial yield pressure p when the pressure-bearing cylinder bears the pressure_eObtaining the total yield load p when the pressure-bearing cylinder bears the pressure_y；

When the diameter ratio k of the pressure-bearing cylinder is more than or equal to k_bThe safe pressure range is according to p_e≤p≤2p_eDetermining, while at the same time a safe temperature difference range in delta t₁≤Δt≤Δt₂Determining;

when the diameter ratio k of the pressure-bearing cylinder_c<k<k_bThe pressure and temperature difference were determined in two cases, A, B, as follows: A. safe pressure range in p_e≤p≤σ_yp_fDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₂Determining; B. safe pressure range is as followsσ_yp_f≤p≤2p_eDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₃Determining;

when the diameter ratio k of the pressure-bearing cylinder_d<k≤k_cThe pressure and temperature difference were determined in two cases, C, D, as follows: C. safe pressure range in p_e≤p≤σ_yp_fDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₂Determining; D. safe pressure range in sigma_yp_f≤p≤p_yDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₃Determining;

when the diameter ratio k of the pressure-bearing cylinder is less than or equal to k_dThe safe pressure range is according to p_e≤p≤p_yDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₃Determining;

wherein: k is_bIs given by equation (2 k)²-1)(k²-1)＝(k⁴-k²+1)lnk²The determined value; k is_cFrom equation k²-1＝k²lnk is determined; k is_dIs represented by the equation (2 k)⁴+1)lnk²＝(4k²-1)(k²-1) the determined value;

the p is the uniform distribution pressure born by the pressure-bearing cylinder;

the Δ t₁Is composed of

The Δ t₂Is composed of

Said p is_fIs composed of

The Δ t₃Is composed of

Wherein mu is the Poisson's ratio of the pressure-bearing cylinder material, E is the elastic modulus of the pressure-bearing cylinder material, and alpha is the linear expansion coefficient or thermal expansion coefficient of the pressure-bearing cylinder material; sigma_yThe yield strength of the pressure-bearing cylinder material.

Further, the invention also provides a method for determining the optimal load of the pressure-bearing cylinder, which comprises the following steps:

(1) when the diameter ratio k of the pressure-bearing cylinder is less than or equal to k_cWhen the optimum pressure is p ═ p_y＝σ_ylnk determined, optimal temperature differential

Determining;

(2) when the diameter ratio k of the pressure-bearing cylinder is more than or equal to k_cAt the optimum pressure

Determining the optimum temperature difference

Determining; the meaning of each symbol is the same as before.

Determining a safe pressure and temperature difference range and an optimal pressure and temperature difference by a graph method; with the parameter p/sigma_yIs an abscissa and a parameter Δ t is an ordinate, said Δ t comprising Δ t₁、Δt₂、Δt₃(ii) a The method comprises the following specific steps:

(1) when the diameter ratio k of the pressure-bearing cylinder is more than or equal to k_bTime, safe load point (p/sigma)_yΔ t) lies within the trapezoid abcd, in which the pressure range is p_e≤p≤2p_eTemperature difference range of Δ t₁≤Δt≤Δt₂(ii) a The four vertex coordinates of the trapezoid abcd are:

point c is the optimum load point;

(2) when the diameter ratio k of the pressure-bearing cylinder_c<k<k_bTime, safe load point (p/sigma)_yΔ t) is located within the pentagonal abcgf, wherein the coordinates of points a, b, c are congruent, and the coordinate of point g is

The abscissa of the point f is

The ordinate of point f is the number of points p_fΔ t substituting claim 1₂Or Δ t₃The value obtained by the calculation formula (the calculation results are equal); point c is the optimum load point;

(3) when the diameter ratio k of the pressure-bearing cylinder_d<k≤k_cTime, safe load point (p/sigma)_yΔ t) is located within the quadrilateral abef, wherein the coordinates of the points a, b, f are the same as before and the coordinate of the point e is

Or

Point e is the optimum load point;

(4) when the diameter ratio k of the pressure-bearing cylinder is less than or equal to k_dTime, safe load point (p/sigma)_yΔ t) is located within the triangle hbe, where the coordinates of points b, e are the same as before and the coordinate of point h is

Point e is the optimum load point.

The invention has the advantages that:

based on the technical idea of comprehensively ensuring the safe design, manufacture and use of the pressure-bearing cylinder, the formula for accurately, scientifically and reasonably calculating the parameters such as pressure, temperature difference and the like is provided, and a visual and clear graphical method for determining safe operation (design) parameters is established. The pressure-bearing cylinder designed and manufactured according to the parameters determined by the invention can ensure safety (safety start)First) and in particular, pressure-bearing cylinders designed and manufactured according to optimum parameters, not only safe, but also material-saving, and having sigma in the entire wall_θ-σ_r≡σ_yI.e., the optimum operating (design) point determined according to the method of the present invention, an equal strength effect can be achieved, which is very beneficial for weight reduction, cost reduction, economic improvement, etc.

Drawings

In FIG. 1, k is k_bSafe operating (design) parameter ranges.

FIG. 2 is k_c<k<k_bSafe operating (design) parameter ranges.

Fig. 3 k-k_cSafe operating (design) parameter ranges.

FIG. 4 is k_d<k<k_cSafe operating (design) parameter ranges.

Fig. 5 k-k_dSafe operating (design) parameter ranges.

FIG. 6 is k<k_dSafe operating (design) parameter ranges.

Detailed Description

It is to be understood that the following detailed description is exemplary and is intended to provide further explanation of the invention as claimed. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.

It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of exemplary embodiments according to the invention. As used herein, the singular forms "a", "an", and/or "the" are intended to include the plural forms as well, unless the invention expressly state otherwise, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof;

as described in the background section, when the pressure-bearing cylinder is subjected to both mechanical load (i.e., operating pressure) and thermal load (i.e., temperature difference between the inner and outer walls), the total stress is more complex, and new mechanical phenomena and laws will appear. Therefore, when the pressure-bearing cylinder simultaneously bears the thermal load caused by the mechanical pressure load and the temperature difference, how to effectively and reasonably utilize the thermal stress (namely the temperature difference stress), how to scientifically and reasonably determine the safe operation or design parameter range, and how to determine the optimal operation parameter or optimal design parameter, thereby achieving the purposes of safety and economy, and being a technical problem to be solved urgently. The invention provides a method for determining the safe load of a pressure-bearing cylinder with temperature difference, which is used for comprehensively ensuring the safety of the pressure-bearing cylinder.

The method proposed by the invention is specifically introduced below and illustrates the scientific basis of the invention:

the inner radius of the pressure-bearing cylinder is r_iOuter radius of r_o. Hereinafter, the symbols with subscripts i, o represent values on the inner and outer wall surfaces, respectively; the inner and outer wall temperatures are respectively t_i、t_oThe temperature difference is delta t ═ t_i-t_o. The temperature at any radius r in the cylinder wall is t.

When the pressure-bearing cylinder simultaneously bears mechanical pressure load and thermal load, according to related knowledge, the stress at any radius r in the wall of the pressure-bearing cylinder is shown as the formula (1).

In the formula, σ_r、σ_θAnd σ_zRadial stress, hoop stress and axial stress respectively; e is the modulus of elasticity; alpha is linear expansion coefficient or thermal expansion coefficient; μ is Poisson's (Poisson) ratio; k is a diameter ratio, and k is r_o/r_iK is a dimensionless number and reflects the wall thickness of the pressure-bearing cylinder, and the larger k is, the larger the thickness is; p is the uniform pressure acting on the inner wall surface of the pressure-bearing cylinder; x is the relative position of any point in the wall, and x is r/r_i，r_i≤r≤r_o(ii) a Order to

p_tReferred to as thermal loadingThe unit is the same as the unit of the pressure p and is MPa.

The general technical idea of the invention is to comprehensively control each key stress of the pressure-bearing cylinder. Research has shown that the stresses on the inner and outer wall surfaces of the pressure-bearing cylinder are most dangerous. Therefore, the safety of the pressure-bearing cylinder can be comprehensively ensured by the formula (2).

|σ_ri|≤σ_y，|σ_θi|≤σ_y，|σ_zi|≤σ_y，|σ_θi-σ_zi|≤σ_y，|σ_zi-σ_ri|≤σ_y，|σ_θi-σ_ri|≤σ_y

|σ_ro|≤σ_y，|σ_θo|≤σ_y，|σ_zo|≤σ_y，|σ_θo-σ_zo|≤σ_y，|σ_zo-σ_ro|≤σ_y，|σ_θo-σ_ro|≤σ_y (2)

For the inner wall surface, x is 1. Therefore, in the formula (1), the three-dimensional stress on the inner wall surface is given by x being 1 as follows.

σ_ri＝-p，

Let sigma_ri≥-σ_yTo obtain

p/σ_y≤1 (4)

As shown in the formula (3), σ_θi>σ_ziSo if σ_zi≥-σ_yMust have a_θi≥-σ_y(ii) a If σ_θi≤σ_yMust have a_zi≤σ_y. Therefore, let σ_zi≥-σ_yTo obtain

Let sigma_θi≤σ_yTo obtain

The stress difference of the inner wall surface is obtained by using the formula (1) or the formula (3) to make x equal to 1

Due to sigma_θi-σ_zi>σ_zi-σ_riAnd σ_θi-σ_ri>σ_zi-σ_riTo ensure the safety of the pressure-bearing cylinder, the following must be provided:

σ_θi-σ_zi≤σ_y，σ_zi-σ_ri≥-σ_y，σ_θi-σ_ri≤σ_y

therefore, let σ_θi-σ_zi≤σ_yTo obtain

Wherein p is_eThe pressure-bearing cylinder is subjected to an initial yield pressure in the case of a pressure p (no thermal load). Due to the fact that

Therefore, the condition p/sigma is limited_yLess than or equal to 1, namely the formula (4) is cancelled.

Let sigma_zi-σ_ri≥-σ_yTo obtain

Let sigma_θi-σ_ri≤σ_yTo obtain

For the outer wall, x ═ k. Therefore, in the formula (1), x ═ k indicates the three-dimensional stress on the outer wall surface as follows.

σ_ro＝0，

Due to sigma_θo>σ_zo>0>-σ_ySo if σ_θo≤σ_yMust have a_zo≤σ_y. Thus let σ_θo≤σ_yTo obtain

And the stress difference of the outer wall surface is obtained by the formula (1) or the formula (3) that x is k

As can be seen from formula (13), σ_θo-σ_ro>σ_zo-σ_ro>σ_θo-σ_zo>0, i.e. sigma_θo-σ_roAnd max. But sigma_θo-σ_ro＝σ_θoWhile σ has been defined above_θo≤σ_y. Thus, at the outer wall surface, σ_θo-σ_ro、σ_zo-σ_roAnd σ_θo-σ_zoAre all within the safe range.

At p/sigma_yIn abscissa, Δ t (including Δ t)₁、Δt₂、Δt₃、Δt₄And Δ t₅) For the ordinate, in the above 6 safety operating (design) conditions, i.e. in equations (5), (6), (8), (9), (10) and (12), the pressure (in dimensionless amount p/σ) is fixed when the diameter ratio k is fixed_yExpressed) and the temperature difference at is a straight-line relationship.

Straight line Δ t₄～p/σ_yAnd Δ t₁～p/σ_yThe intercepts on the vertical axis are equal and negative, the straight line Δ t₁～p/σ_yThe slope of (a) is greater. Therefore, is greater than Δ t₁Must have a temperature difference ofGreater than Δ t₄. Therefore, the formula (6) is eliminated.

Straight line Δ t₂～p/σ_yAnd Δ t₅～p/σ_yThe intercepts on the vertical axis are equal and positive, the slopes of both are positive, but the straight line Δ t₅～p/σ_yThe slope of (a) is greater. Therefore, less than Δ t₂Must be less than at₅. Therefore, the formula (9) is eliminated.

Straight line Δ t₁～p/σ_yIs positive, straight line Δ t₃～p/σ_yIs negative and its intercept on the vertical axis is positive, the coordinate of the intersection of these two lines is

Or

This intersection point is designated as point e in the drawings of the specification.

When k is less than or equal to k_cWhen the temperature of the water is higher than the set temperature,

when k is more than or equal to k_cWhen the temperature of the water is higher than the set temperature,

when k is k_cWhen the temperature of the water is higher than the set temperature,

at this time, k is 2.2184574899167 … and is an infinite acyclic decimal, and k is 2.2184574899167 k_c. Therefore, when k is less than or equal to k_cWhen the maximum pressure is p/sigma_yLnk; when k is more than or equal to k_cThe maximum pressure is the formula (8). On the other hand, if p/σ_y>lnk, then Δ t₁>Δt₃This is in contrast to the safe operating (design) condition Δ t₁≤Δt≤Δt₃Are contradictory, therefore, p/σ_yMust not exceed lnk.

Further studies have shown that there are several specific and critical diameter ratio values that determine the safe manufacturing, operating (design) parameter ranges.

1.k≥k_bTime of flight

It can be shown that as k increases, the straight line Δ t₂～p/σ_yThe slope of (a) decreases; as k decreases, the straight line Δ t₂～p/σ_yThe slope of (a) increases. I.e. as k increases, the straight line deltat₂～p/σ_yRotating clockwise; as k decreases, the straight line Δ t₂～p/σ_yRotating counterclockwise. When k is infinity, the straight line Δ t₂～p/σ_yIs a horizontal line passing through point c (see fig. 1 for point c):

referring to FIG. 1, the equation for the plumb line cd or cg is

When k is lowered to make the straight line Δ t₂～p/σ_y、Δt₃～p/σ_yWhen the three lines of the plumb line cd (or cg) intersect at a point d or g (see FIG. 1), the diameter ratio k at this time is denoted as k_bAt this time, the trapezoidal abcd range reaches the maximum. k is a radical of_bCan be obtained by the following method: let Δ t₂＝Δt₃Can solve p/sigma_yLet obtained by solution

Get an equation (2 k)²-1)(k²-1)＝(k⁴-k²+1)lnk²Or at Δ t₂、Δt₃Zhongling

Let Δ t again₂＝Δt₃The equation can also be obtained, and the equation can be solved to obtain k 2.419275 … k_b，k_bTaking k as infinite acyclic decimal_b2.419275. In FIG. 1, k is k_b2.419275 th three straight lines Δ t₁～p/σ_y、Δt₂～p/σ_y、Δt₃～p/σ_yThe positional relationship of (a). Several key points in the figure are illustrated below.

i. Point a is the plumb line ab (the equation for this line is p/σ)_y≡p_e/σ_y) And a straight line Δ t₂～p/σ_yHas coordinates of

Point b is a straight line Δ t₁～p/σ_yThe coordinate of the intersection with the horizontal axis is b (p)_e/σ_y0), i.e.

a. b the difference between the ordinate of the two points (i.e. the ordinate of the point a) is when p equals p_eThe temperature difference allowed.

Point c is the plumb line cg (the equation for this line is p/σ)_y≡2p_e/σ_y) And a straight line Δ t₁～p/σ_yHas coordinates of

The plumb line cg is formula (8).

Point d is the plumb line cg and the line Δ t₂～p/σ_yHas coordinates of

d. c difference between ordinate of two points (value of

) When p is 2p_eThe allowable temperature difference range.

v. Point e is a straight line Δ t₁～p/σ_yAnd a straight line Δ t₃～p/σ_yHas coordinates of

Or

Point g plumb line cg and straight line Δ t₃～p/σ_yHas coordinates of

The line cg is when

Time straight line Δ t₁～p/σ_yAnd a straight line Δ t₃～p/σ_yThe distance between them. When k is k_bWhen the point g coincides with the point d, that is, when k is k_bTime, straight line Δ t₂～p/σ_y、Δt₃～p/σ_yThe three lines, the plumb line cg, cross at a point d or g.

When k is more than or equal to k_bSafe operating (design) parameter points or safe load points (p/sigma)_yΔ t) lies within the trapezoid abcd: pressure range of p_e≤p≤2p_eTemperature difference range of Δ t₁≤Δt≤Δt₂(ii) a The point c is the optimal operation (design) point, and the parameter is the coordinate of the point c; at point c, the operating (design) pressure is at its maximum, i.e. the load carrying capacity is at its maximum, and the stress distribution is at its most reasonable.

2.k_c<k<k_bTime of flight

FIG. 2 is k_c<k<k_bTime three straight line Δ t₁～p/σ_y、Δt₂～p/σ_y、Δt₃～p/σ_yThe positional relationship of (a). When the straight line Δ t₂～p/σ_yRotate counterclockwise to k_c<k<k_bIn the state (2), the safe operation (design) parameter range is located in the pentagon aIn bcgf, where point f is a straight line Δ t₂～p/σ_yAnd a straight line Δ t₃～p/σ_yOf intersection point of (a) with abscissa p_fIs that

Pressure in the range of p in pentagonal abcgf_e≤p≤σ_yp_fThe safe temperature difference range is delta t₁≤Δt≤Δt₂(ii) a Pressure range of σ_yp_f≤p≤2p_eThe safe temperature difference range is delta t₁≤Δt≤Δt₃。k_c<k<k_bWhen the load is at the optimum load point, point c is still the optimum load point.

3.k_d<k≤k_cTime of flight

As k is continuously decreased, the points c and g are closer, i.e. the segment cg is shorter. Since, when k is k_cWhen, lnk ═ k²-1)/k²Therefore, when k is k_cWhen the two are in contact, the three points c, g and e are superposed. Fig. 3 k-k_cTime three straight line Δ t₁～p/σ_y、Δt₂～p/σ_y、Δt₃～p/σ_yThe positional relationship of (a). When k is equal to k_cIn this case, the safe operating (design) parameters, i.e. the safe pressure and temperature difference range, lie within the quadrilateral abe (cg) f, point e being the optimal operating (design) or load point, where p/σ is_y＝lnk＝(k²-1)/k²0.796812 … while

At point c, the operating (design) pressure is at its maximum, i.e. the load carrying capacity is at its maximum, and the stress distribution is at its most reasonable.

As k continues to decrease, the line Δ t₂～p/σ_yContinuing to rotate counterclockwise, point a rises and point f is on a straight line Δ t₂～p/σ_yMove up and left. Since when k is<k_cWhen (k)²-1)/k²>lnk, so points c and g move to the right of point e away from point e, where point c is higher than point eg, i.e. Δ t₁>Δt₂. Therefore, when k is<k_cWhen is, p/sigma_y＝(k²-1)/k²Is not preferable, and the bearing capacity in this case must be p/sigma_yLnk. FIG. 4 is k_d<k<k_cThe case (1). In this case, the safe load parameter range is still the quadrilateral abef, and point e is the optimal load point. Within the quadrilateral abef, the pressure is in the range p_e≤p≤σ_yp_fThe safe temperature difference range is delta t₁≤Δt≤Δt₂(ii) a Pressure range of σ_yp_f≤p≤p_yThe safe temperature difference range is delta t₁≤Δt≤Δt₃。

4.k≤k_dTime of flight

Referring to FIG. 5, the equation for the plumb line ab is

When the straight line Δ t₂～p/σ_yRotated counterclockwise until points a and f coincide, i.e. straight line Δ t₂～p/σ_y、Δt₃～p/σ_yWhen the three lines of the plumb line ab intersect at a point a or f (see fig. 5), a safe load point (p/sigma)_yΔ t) lie within the triangle abe, the ratio of diameters k at this time being denoted as k_d。k_dCan be obtained by the following method: let Δ t₂＝Δt₃Can solve p/sigma_yLet obtained by solution

Get an equation (2 k)⁴+1)lnk²＝(4k²-1)(k²-1), solving this equation to obtain k-1.936876 … -k_d，k_dTaking k as infinite acyclic decimal_d＝1.936876。k＝k_dWhen 1.936876, the straight line Δ t₂～p/σ_y、Δt₃～p/σ_yThe plumb line ab crosses at a point f or a point a in a three-line mode; the safe load parameter range is changed to a range defined by a straight line delta t₁～p/σ_y、Δt₃～p/σ_yAnd a triangle f (a) be surrounded by three vertical lines ab. See fig. 5. Within triangle f (a) be, safe pressureRange is p_e≤p≤p_ySafe temperature difference range of Δ t₁≤Δt≤Δt₃(ii) a Point e is the optimum load point.

k<k_cThereafter, the points c, g are no longer control points for the safe operation (design), and the point e becomes a control point for the safe operation (design).

When k is equal to k_dAfter which k continues to decrease, line Δ t₂～p/σ_yNo longer a safety control line, i.e. k<k_dTime, safety control condition σ_zi≥-σ_yNot to be considered, see fig. 6.

k<k_dThe safe load parameter range is defined by the straight line Δ t₁～p/σ_y、Δt₃～p/σ_yA triangle hbe enclosed by three lines of the plumb line ab, as shown in FIG. 6, wherein the point h is the plumb line ab (i.e., p/σ)_y≡p_e/σ_y) And a straight line Δ t₃～p/σ_yHas coordinates of

Within the triangle hbe, the safe pressure range is p_e≤p≤p_ySafe temperature difference range of Δ t₁≤Δt≤Δt₃(ii) a Point e is the optimum operating (design) point.

In conclusion, the invention obtains the accurate calculation formula of the safe operation (design) parameter range of the pressure-bearing cylinder based on the technical idea of comprehensively ensuring the safety of the pressure-bearing cylinder, and can conveniently, accurately and intuitively determine the safe operation (design) parameter range by utilizing the simple linear variation relation. The following illustrates how this can be done. The material parameters of the examples given in the present invention are all set to E ═ 1.95 × 10⁵MPa、μ＝0.3、α＝1.2×10^-5℃^-1、σ_y＝350MPa。

Example 1 has a pressure-bearing cylinder with k-3, which determines its safe and optimal operating (design) parameters.

The coordinates of the above points are: a (0.444, 164.98 ℃), b (0.444, 0), c (0.889, 156.30 ℃), d (0.889, 173.66 ℃), e (1.099,230.05 ℃), g (0.889,246.68 ℃).

Obtaining p from formula (8)_e155.56MPa, when p is p_eWhen 155.56MPa, Δ t is obtained from formula (10)₁0 ℃ to obtain Δ t from formula (5)₂164.98 ℃. When p is 2p_eWhen 311.11MPa, Δ t is obtained from formula (10)₁Δ t from formula (5) at 156.3 ℃₂＝173.66℃。

k is 3 and belongs to k>k_bIn this case, the safe operating (design) pressure range is p_e≤p≤2p_eNamely p is more than or equal to 155.56MPa and less than or equal to 311.11 MPa. When p is 155.56MPa, Δ t₁≤Δt≤Δt₂Namely, delta t is more than or equal to 0 and less than or equal to 164.98 ℃; when p is 311.11MPa, Δ t₁≤Δt≤Δt₂Namely delta t is more than or equal to 156.3 ℃ and less than or equal to 173.66 ℃. Point c is the optimum load point, i.e. (p 311.11MPa, Δ t 156.3 ℃) is the optimum operating (design) parameter.

It can be verified that if the load point is within the trapezoid abcd, the pressure-bearing cylinder is safe, and the operating (design) point, i.e. the load point is outside this region, is unsafe. For example, at point a:

obtaining σ from formula (3)_ri＝-p＝-0.44σ_y>-σ_y，σ_θi＝-175MPa>-σ_y，σ_zi＝-350MPa＝-σ_y. And (4) safety.

Obtaining σ from formula (7)_θi-σ_zi＝175MPa<σ_y，σ_zi-σ_ri＝-194.44MPa>-σ_y，σ_θi-σ_ri＝-19.44MPa>-σ_y. And (4) safety.

Obtaining σ from formula (11)_ro＝0，σ_θo＝220.95MPa<σ_y，σ_zo＝201.51MPa<σ_y. And (4) safety.

Obtaining σ from formula (13)_θo-σ_zo＝19.44MPa<σ_y，σ_zo-σ_ro＝201.51MPa<σ_y，σ_θo-σ_ro＝220.95MPa<σ_y. And (4) safety.

As another example, when p/σ_yWhen equal to 0.7, at the straight line Δ t₃～p/σ_yTaking one point, obtaining Δ t from equation (12)₃261.66 ℃. At this time, σ is obtained from the formula (3)_ri＝-p＝-0.7σ_y>-σ_y，σ_θi＝-279.68MPa>-σ_y，σ_zi＝-555.31MPa<-σ_y. Unsafe!

Example 2 a pressure-bearing cylinder with k 2.3 was designed.

This belongs to k_c<k<k_bIn the case of (1), first, p is obtained from the formula (14)_f0.698, so p_fσ_y244.3 MPa. Obtaining p from formula (8)_e141.9187 MPa. (1) If the pressure-bearing cylinder is subjected to p/sigma_yPressure 0.6, i.e. p 210 MPa. This belongs to p_e≤p≤p_fσ_yIn the case of (1), the temperature difference range must be Δ t₁≤Δt≤Δt₂. Obtaining Δ t from the formula (10)₁At 79.37 deg.C, obtaining Δ t from formula (5)₂188.6 ℃. (2) If the pressure-bearing cylinder is subjected to p/sigma_yPressure 0.7, i.e. p 245 MPa. This belongs to p_fσ_y≤p≤2p_eIn the case of (1), the temperature difference range must be Δ t₁≤Δt≤Δt₃. Obtaining Δ t from the formula (10)₁At 120.18 ℃, Δ t from formula (12)₃＝192.08℃。

It can be verified that if the operating (design) point is within the pentagonal abcgf, the pressure-bearing cylinder is safe, and the operating (design) point is outside this area, it is unsafe. For example,

(1) at the straight line Δ t₃～p/σ_yPoint s (see figure 2 of the specification), p/sigma_y＝0.76>p_fΔ t is obtained from the formula (12)₃184.1 ℃, then:

obtaining σ from formula (3)_ri＝-p＝-0.76σ_y>-σ_y，σ_θi＝0.57MPa<σ_y，σ_zi＝-327.44MPa>-σ_y. And (4) safety.

Obtaining σ from formula (7)_θi-σ_zi＝328.00MPa<σ_y，σ_zi-σ_ri＝-61.44MPa>-σ_y，σ_θi-σ_ri＝266.57MPa<σ_y. And (4) safety.

Obtaining σ from formula (11)_ro＝0，σ_θo＝350MPa＝σ_y，σ_zo＝288.00MPa<σ_y. And (4) safety.

Obtaining σ from formula (13)_θo-σ_zo＝62.00MPa<σ_y，σ_zo-σ_ro＝288.00MPa<σ_y，σ_θo-σ_ro＝350MPa＝σ_y. And (4) safety.

(2) At the straight line Δ t₂～p/σ_yPoint n (point n is outside the safety region, see figure 2 of the specification), p/sigma_y＝0.76>p_fObtaining Δ t from formula (5)₂194.77 ℃, then:

obtaining σ from formula (11)_ro＝0，σ_θo＝363.09MPa>σ_y. Unsafe!

Obtaining σ from formula (13)_θo-σ_zo＝62.00MPa<σ_y，σ_zo-σ_ro＝301.88MPa<σ_y，σ_θo-σ_ro＝363.09MPa>σ_y. Unsafe!

(3) At the straight line Δ t₁～p/σ_yThe following point j (point j is outside the safety region, see FIG. 2 of the specification), p/σ_y＝0.76，Δt₂At 100 ℃. Obtaining σ from formula (7)_θi-σ_ri＝444.48MPa>σ_y. Unsafe!

(4) Point c is the optimum design point, when p/σ is_y＝(k²-1)/k²0.811. Obtaining Δ t from the formula (10)₁＝165.46℃。

Obtaining σ from formula (3)_ri＝-p＝-283.84>-σ_y，σ_θi＝66.16MPa<σ_y，σ_zi＝-283.84MPa＝σ_ri>-σ_y. And (4) safety.

Obtaining σ from formula (7)_θi-σ_zi＝350MPa＝σ_y，σ_zi-σ_ri＝0，σ_θi-σ_ri＝350MPa＝σ_y. And (4) safety.

Is obtained by the formula (11)σ_ro＝0，σ_θo＝335.43MPa<σ_y，σ_zo＝269.26MPa<σ_y. And (4) safety.

Obtaining σ from formula (13)_θo-σ_zo＝66.16MPa<σ_y，σ_zo-σ_ro＝269.26MPa<σ_y，σ_θo-σ_ro＝335.43MPa<σ_y. And (4) safety.

Example 3, a pressurized cylinder with k 1.7 was identified for safe and optimal operating (design) parameters.

This is k<k_dThe case (1).

(1) Point e is the optimal operating (design) point: p/sigma_yLnk 0.531. Obtained by both the formulae (10) and (12)

Obtaining σ from formula (3)_ri＝-p＝-0.531σ_y>-σ_y，σ_θi＝164.28MPa<σ_y，σ_zi＝-119.7MPa>-σ_y. And (4) safety.

Obtaining σ from formula (7)_θi-σ_zi＝283.98MPa<σ_y，σ_zi-σ_ri＝66.02MPa<σ_y，σ_θi-σ_ri＝350MPa＝σ_y. And (4) safety.

Obtaining σ from formula (11)_ro＝0，σ_θo＝350MPa＝σ_y，σ_zo＝251.74MPa<σ_y. And (4) safety.

Obtaining σ from formula (13)_θo-σ_zo＝98.26MPa<σ_y，σ_zo-σ_ro＝251.74MPa<σ_y，σ_θo-σ_ro＝350MPa＝σ_y. And (4) safety.

In practice, operating or designed at optimum points, there is a not only the inner and outer walls, but also the entire wall_θ-σ_r≡σ_y. I.e. the optimum operating (design) point determined according to the method of the invention, equal strength results can be achieved, which are material saving, weight reductionCost, etc.

(2) Points within hbe are all safe operating (design) points, and others are unsafe.

The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method of determining the safe load of a pressure-bearing cylinder having a temperature difference, said load comprising the pressure and temperature difference to which the pressure-bearing cylinder is subjected, characterized by:

when the diameter ratio k of the pressure-bearing cylinder_c<k<k_bThe pressure and temperature difference were determined in two cases, A, B, as follows: A. safe pressure range in p_e≤p≤σ_yp_fDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₂Determining; B. safe pressure range in sigma_yp_f≤p≤2p_eDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₃Determining;

when the diameter ratio k of the pressure-bearing cylinder_d<k≤k_cThe pressure and temperature difference were determined in two cases, C, D, as follows: C. safe pressure rangeAccording to p_e≤p≤σ_yp_fDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₂Determining; D. safe pressure range in sigma_yp_f≤p≤p_yDetermining a safe temperature difference range as delta t₁≤Δt≤Δt₃Determining;

the Δ t₁Is composed of

The Δ t₂Is composed of

Said p is_fIs composed of

The Δ t₃Is composed of

2. A method of determining the safe load of a pressure-containing cylinder having a temperature differential as claimed in claim 1, characterized in that:

when the diameter ratio k of the pressure-bearing cylinder is less than or equal to k_cWhen the optimum pressure is p ═ p_y＝σ_ylnk determined, optimal temperature differential

And (4) determining.

3. A method of determining the safe load of a pressure-containing cylinder having a temperature differential as claimed in claim 1, characterized in that:

when the diameter ratio k of the pressure-bearing cylinder is more than or equal to k_cAt the optimum pressure

Determining the optimum temperature difference

And (4) determining.

4. A method of determining the safe load of a pressure-containing cylinder having a temperature differential as claimed in claim 1, characterized in that the safe pressure and temperature differential range and the optimum load point are determined graphically as follows: with the parameter p/sigma_yIs an abscissa and a parameter Δ t is an ordinate, said Δ t comprising Δ t₁、Δt₂、Δt₃(ii) a When the diameter ratio k of the pressure-bearing cylinder is more than or equal to k_bTime, safe load point (p/sigma)_yΔ t) lies within the trapezoid abcd, in which the pressure range is p_e≤p≤2p_eTemperature difference range of Δ t₁≤Δt≤Δt₂(ii) a The four vertex coordinates of the trapezoid abcd are:

point c is the optimum load point.

5. A method of determining the safe load of a pressure-containing cylinder having a temperature differential as claimed in claim 1, characterized in that the safe pressure and temperature differential range and the optimum load point are determined graphically as follows: with the parameter p/sigma_yIs an abscissa and a parameter Δ t is an ordinate, said Δ t comprising Δ t₁、Δt₂、Δt₃When the diameter ratio k of the bearing cylinder_c<k<k_bTime, safe load point (p/sigma)_yΔ t) is located within the pentagonal abcgf, wherein the coordinates of points a, b, c are congruent, and the coordinate of point g is

The abscissa of the point f is

The ordinate of point f is the number of points p_fΔ t instead of claim 1₂Or Δ t₃P/sigma in the calculation_yThe resulting values (both calculated to be equal); point c is the optimum load point.

6. A method of determining the safe load of a pressure-containing cylinder having a temperature differential as claimed in claim 1, characterized in that the safe pressure and temperature differential range and the optimum load point are determined graphically as follows: with the parameter p/sigma_yIs an abscissa and a parameter Δ t is an ordinate, said Δ t comprising Δ t₁、Δt₂、Δt₃When the diameter ratio k of the bearing cylinder_d<k≤k_cTime, safe load point (p/sigma)_yΔ t) is located within the quadrilateral abef, wherein the coordinates of the points a, b, f are the same as before and the coordinate of the point e is

Or

Point e is the optimum load point.

7. A method of determining the safe load of a pressure-containing cylinder having a temperature differential as claimed in claim 1, characterized in that the safe pressure and temperature differential range and the optimum load point are determined graphically as follows: with the parameter p/sigma_yIs an abscissa and a parameter Δ t is an ordinate, said Δ t comprising Δ t₁、Δt₂、Δt₃When the diameter ratio k of the pressure-bearing cylinder is less than or equal to k_dTime, safe load point (p/sigma)_yΔ t) is located within the triangle hbe, where the coordinates of points b, e are the same as before and the coordinate of point h is

Point e is the optimum load point.