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Optimal energy-efficient combustion process control in heating furnaces of rolling mills
References
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2. Kazakevitch V.V., Rodov A.B. The automatic optimization systems. Moscow: Energy, 1977, 288 p.
3. Parsunkin B.N., Andreev S.M. Processing control optimization of fuel combustion in heating furnaces workspace. Steel. 2000, no. 5, pp. 48-52.
4. Parsunkin B.N., Andreev S.M., Obukhova T.G. Study of optimum energy efficient fuel combustion process in metallurgical furnaces workspace. Vestnik Magnitogorskogo gosudarstvennogo tehnicheskogo universiteta im. G.I. Nosova. [Vestnik of Nosov Magnitogorsk State Technical University]. 2005, no. 4, pp. 28-36.
5. Parsunkin B.N., Bushmanova M.V., Andreev S.M. Calculations of automatic systems of processing optimization in metallurgy: Textbooks. Magnitogorsk: Nosov Magnitogorsk State Technical University, 2003, 267 p.
6. Sayrov A.M. Optimization of thermal management in heating furnace workspace. Automated technologies and production: Collection of Scientific Papers. Ed. Parsunkin B.N. Magnitogorsk: Nosov Magnitogorsk State Technical University, 2013. no. 5, pp. 296-301.
7. Parsunkin B.N., Andreev S.M. Ways to improve the efficiency and noise immunity of processing control automatic optimization. Automated technologies and production: Collection of Scientific Papers. Ed. Par-sunkin B.N. Magnitogorsk: Nosov Magnitogorsk State Technical University, 2013. no. 5, pp. 277-290.
8. Rumyantsev M.I., Shubin I.G., Nosenko O.U. Model designing to calculate the temperature of low-alloy steels in hot rolling. Vestnik Magnitogorskogo gosudarstvennogo tehnicheskogo universiteta im. G.I. Nosova. [Vestnik of Nosov Magnitogorsk State Technical University]. 2007, no. 1, pp. 54-57.
9. Zadonskaya T.A., Shvetsova E.S., Koptsev V.V. Firing of high-speed streams of natural gas. Vestnik Magnitogorskogo gosudarstvennogo tehnicheskogo universiteta im. G.I. Nosova. [Vestnik of Nosov Magnitogorsk State Technical University]. 2009, no. 3, pp. 67-68.
Antsupov A.V., Antsupov A.V. (jun), Antsupov V.P.
DESIGNED ASSESSMENT OF MACHINE ELEMENT RELIABILITY DUE TO EFFICIENCY CRITERIA
Abstract. The universal method of reliability assessment of mechanical system loaded elements at the design stage as a sequence of steps within the procedure of constructing physical and probabilistic models of parametric failure formation based on various criteria is suggested. The methodology of forecasting durability of parts by kinetic strength is represented and an example of its implementation is shown.
Keywords: methodology, forecasting, reliability, dependability, durability, failure, damage susceptibility, gamma-percent life.
The main problem of the reliability theory is behavior prediction of mechanical system parts and components in supposed conditions of external loading, when it becomes possible to evaluate their reliability and durability in early stage design. In this case, the assessment of system element behavior and their parameters changing over time in future running is carried out on the dynamic, physical and probabilistic models [1].
A single, universal methodological approach to probability forecasting of trouble-free operation and resource characteristics of loaded elements of mechanical systems according to various criteria of their performance was stated in this paper, on basis of mathematical formaliza-tion of reliability theory basic concepts of engineering objects (GOST 27.002-89), and general concept of their gradual failure formation [2-4].
To describe theoretically the objective formation of technical product failures during their damageability (degradation) under external affecting, the suggested approach is stated as a series of rules of their parameter reliability dynamic models designing.
This approach is presented in a probabilistic form, and is a combination of the following steps.
I. Selection of object state basic parameter.
Parameter Xt (a random variable) is selected for the testing product type, according to the standard (GOST 20911-89) definition of «object state». Variable changing over time simulates the parameter behavior (state changing) during the entire operation period under certain external affecting conditions.
II. The equation formulation of object state.
Random function (dependency) elaboration or choosing, that describes parameter Xt increasing (+) or decreas-
ing (-) changing over time, and models the product state changes in aging (degradation) during the operation can be written as the following:
t
Xt = X0 ±J X • dt, (I)
0
where X0 is Xt parameter distribution at time T = t0 characterizing the initial object state; Xt = dXt / dt denotes random variable current distribution of object damageability rate at time T = t;
If a random variable of object damageability rate does not change over time - Xt = X = const, then the conditions (I) can be written as follows:
Xt = X0 + X ■ t (I.a)
Equations (I) simulate object damageability over time.
III. The formulation of object efficiency condition.
In accordance with the standard definition of «object performance capability», according to GOST 27.002-89, the condition of its performance is mathematically formulated in the form of one possible inequality:
t
Xt = X0 + J Xt ■ dt < xL or
0
t (II)
Xt = X0 -J Xt • dt > Xl ,
0
where xL is a limit value of Xt parameter, established in
AntsupovA..V, AntsupovA..V. (jun), Antsupov V.P.
technological standards (TS) or assigned under the operating experience of such objects.
If the random variables in the condition (I) are normally distributed, and Xt = X = const , then performance capability conditions (II) can be written as the following:
X, = X0 + X • t < xL and X, = X0 - X ■ t > xL . (II.a)
Centring and normalizing values Xt and xL, inequalities (II.a) can be written, using proper quantiles, as the following:
Ut < UL{t) 0r Ut > UL{t) ■
(Il.b)
Expanding the current value of parameter standard normal distribution quantile (SND), conditions (II.b) can be written as following:
2 2 2 < UL(t) 0r
X± -(x0 + x • t)
t _
Xt xo " x 't)
7
(II.B)
2 2 2 L(t)
xL - ( x0 ± x ■ t) where ullt, =—, denotes the current limit
L (t)
4
^0 + ^ * t2
are
bility is denoted using SND function F (uL(t)) =Ft(XL) or the Laplace function @(uL(t)) :
P(t) = P(Xt < xL ) = F (UL,t,) = i'
= ®(UUt)) = 0
( (- - \\ X -( X + XX • 11
_2 .2 -CT. • t
or
P(t) = P(Xt > XL) = 1 -F (Ul(t)) =
(III.a)
= 1" ^("l (t)) = 1" ^
L (t )J - ~r \\
XL (X0 X ' t)
7
CT20 +CT2 • t2
X 0 X
Equations (III.a) define the object reliability law in the integral (or differential f (t) = -dP(t) / dt) form, the law of object gradual failure formation in solving the direct task of the reliability theory. According to [2], it is asymmetric and does not obey the normal distribution.
V. The formulation of the object transition equation to the limit state (the state ofparametric failure).
According to GOST 27.002-89 standard definitions of «object performance capability» and «parametric failure», the conditions of product transition into the parametric failure state in the form of limit value achieved by Xt parameter are formulated as following:
value of SND quantile of a random variable Xt = xL
x0 = (x0max + X0min)/2; °x 0 = (X0max _ X0min )/ 6
numerical characteristics (mean and standard) of a random parameter Xt = X0 of object state at the initial instant of time T=t0; x0max, x0min are maximum and minimum values of object X0 parameter, defined as the initial conditions; x h ax are numerical characteristics of a
random parameter XX.
Equations (II) reflect the range of all possible operable object states.
IV. Elaboration of equations for object reliability assessment.
Using the basic concepts "distribution function" of the probability theory makes possible to formulate the dependencies for estimation of object failure-free operation probability - condition efficiency probability (II) for any fixed instant of future operation:
t
P (t) = P(Xt < xl ) = P((X0 + JXt ■ dt) < xl ) = F(xl ) or
0
P (t) = P(X, > XL ) = (III)
t .
= 1 -P((X0 -JXt • dt) <xl) = 1 -F(xl)'
0
If the normal distribution of Xt parameter is constant over time (Xt = X = const), the main indicator of relia-
t
Xt = X0 ±{Xt • dt = XL
(IV)
If the normal distribution of X t parameter is constant over time (Xt = X = const), the models (IV) can be expressed as
Xt X0 + X • t) _
I 2 2 2 L(t^
vr
X 0
t
Xt "(X0 " X ' f)
(IV.a)
s<
2—2 ~ UL(t). 2 • t
Equations (IV) reflect the range of all possible object limit states.
VI. Elaboration of equations for object durability assessment (service life characteristics).
In accordance with GOST 27.002-89 gamma-percent resource t^ definition, the dependences are derived for its assessment by equation (IV) solution concerning t = ty for assigned accepted value of failure-free operation probability [P (t) ] = y and the relevant distribution quantile [uL(^ ] value owing to the known law of Xt random variable. In general, this dependence can be represented by any function:
ty= f(x0, ox0, x, ok,[uL(y)]). (V)
If the normal distribution of Xt parameter is constant in time (X = X = const), gamma-percent resource is determined by equation (IV.a) solution concerning t = ty
during Xt = xL and uL(t) = [uL^ ] substitution in the form of:
tr =
(±x)• Ar, -yj(Ax,-x)2 -([Mi(r)]2-a2-x2)-([Mi(r)]2 -ArL2)
X " [UL(r)]
where the value \UL(r) ] is determined by assigned accepted value of failure-free operation probability [p (t) ] = y , and the value AxL = xL — x0 is the assessment of mathematical expectation of A Xt limit change of Xt parameter;
VII. Elaboration o/ kinetic equation o/ object dam-ageability.
On basis of any theory, concept or experimental researches, a kinetic equation is stacked up to estimate object damageability rate Xt, depending on its geometrical and microgeometrical di, At characteristics, ov material properties, power and kinematic parameters F, external impact V, time t and other factors.
In general, it can be represented in the form of any random function:
X, = f (d,, A,, F, V, a,., t,...).
(VI)
The represented methodological approach (I)-(VI) can be formulated in the form of a separate methodology of reliability assessment (prediction) of certain groups of technical objects, using one of the possible criteria such as performance conditions (II) [1-4]:
- static or kinetic strength;
- durability;
- hardness;
- bearing value;
- heat resistance and others.
When selected Xt parameter, the status of a particular product (parts, unit) and known data about:
- law of its distribution;
- boundary conditions, describing the object loading diagram, its properties, and the initial state in supposed operating conditions;
- its damageability equation (VI) for Xt assessment, designated sequence of steps produces a sequence of operations (probabilistic technique) of quantity estimation of failure-free measure p (t) = P (Xt < xL ) or p (t) = P (Xt > xL)
and durability ty of the investigated object either in its
designing, or in running.
In particular, in publications [5-11], this approach is implemented as a reliability parametric prediction strategy
of a large object group - «stationary» triboconjugation, by their component wear resistance criterion.
Beneath, on basis of suggested approach a reliability prediction strategy of different object group - machine and unit loaded elements, according to kinetic strength criterion, is formulated.
Currently, the problem of the strength of solid bodies under load is considered from the point of kinetic approach [12-13]. From this point of view, the destruction process is represented as evolutive process of material structur-(V.a) al damage accumulation in time.
In reaching material structure defect by current solidity in any local volume - its damageability, a limit value, microcrack occurrence takes place, which extends through the most loaded material volumes and leads to its division into parts (destruction).
The degree of local volume structure damage of part material at any fixed time point t is estimated quantitatively by density value of potential defect energy
ue(t) = uet, which is determined by the external loading
conditions and material properties [13].
Taking into account the aforementioned, the methodology steps of loaded part reliability and durability prediction may be formulated according to (I) - (VI), in a sequence of the following operations.
At the first stage as part state Xt parameter, in which internal stresses a emerge, affected by external loading at temperature T, we take the potential energy density of defects uet, which characterizes the actual degree of
structure damage of material part local volumes [13]. Herewith, according to the central limit theorem of probability theory, we consider normal the distribution of a
random variable Xt = uet at any time t, as a parameter,
depending upon a set of independent random factors [1]. Furthermore, to simplify mathematical expressions we
will operate with its mean value ue t.
At the second stage, we will formulate the equation of loaded element states:
■fuet ■dt .
(1)
where ue 0 is the average density of material component
part potential energy in the initial state (at t = 0), which, according to [13], can be defined as a function of hardness average value HV according to Vickers:
(0,071- HV )2:
6• G(T)•(6,47-10"6 • HV+0,12-10~2)
(1.a)
where uie t = due t / dt denotes damageability average velocity (damage accumulation) of element material
structure at time t.
e 0
AntsupovA.V., AntsupovA.V. (jun), Antsupov V.P.
At the third stage, we will state the condition of loaded component part performance efficiency:
< u„
(2)
nt = ue./ uet = ue./
i
(3)
= u„
(4)
At the sixth stage, using the expression (5), we formulate the equation, its solution for t = tL , will define the average limit resource to element failure (damage):
= ua. - u
(5)
At the seventh stage as a damageability kinetic equation for the evaluation of average velocity uet of element
damageability, which is in the loaded state, which is included in the equation (1)-(5) of its failure forming general diagram, one can use V.V. Fedorov dependency, he stacked up in the publication [13] using long-time strength thermodynamic criterion.
In general, this equation is the following:
uet = f( ^ T, uef t
(6)
In carrying out engineering calculations we use one of the simplified versions of this dependency to determine
the time constant value u„. = ii„:
ue = MR • kl-a2 • /(6• G-v),
(6 a)
where MR =|(1 + r )2 +(1 - r )2 ^4 is a equivalency coefficient of non-stationary tension state (non-stationary tension state with a coefficient of skewness r = CTmin / CTmax transition to the equivalent steady state with voltage
where U^ * — U* U^ denotes the critical density of structure defects energy of material local volumes of loaded component part [13]; u* is the critical density of material internal energy (critical energy intensity), which is equal to its melting enthalpy in solid AHS or liquid AHL
T
state; UT = J p ■ cdT denotes the thermal component of
0
the material internal energy density of loaded component part at a given temperature T; p , c are density and heat capacity of the component part material.
At the fourth stage we will write the expression for loaded element reliability evaluation at time t, using as an indicator the reliability safety factor (safety margin), calculated according to the average value Uet of state parameter [1]:
are minimum, maximum and
peak cycle stresses; ka = l/(6,47-10"6 • HV + 0,12-10"6)
denotes an overstress coefficient of interatomic bonds; G and v are modulus of rigidity and coefficient of internal energy distribution irregularity over the loaded component part, the value of which is selected referring to [13];
Coefficient A0 of stress tensor ball portion influence on interatomic bond damage activation energy, is expressed according to [13]:
eXp
U T) R ■ T
(6.b)
At the fifth stage we will formulate the equation of loaded component part transition to the marginal state (parametric failure state):
where h is the Planck constant; N0 is the Avogadro number; R is the universal gas constant; U (&, T) is activation energy of interatomic bond damage process at the given voltage a and temperature T:
U ,r)= U0 -&U (T)-
-K. k;/ (18 <->
where U0 is free energy of process activation at T = 0
and a = 0; AU (T ) = — -a0 ■ K 1 • T is activation energy
quantity, determined by the temperature; K=E(3-(i-2-
is all-round material compression index; a0, fJ., E are linear thermal expansion coefficient; the Poisson's ratio and material elastic modulus.
Complex equations (6.a-c) denote a mathematical model of machine element and construction stationary damageability process, in which internal static or dynamic tensions a emerge, affected by external loading and constant temperature T.
For a loaded element stationary damageability process (with constant rate uet = ue, defining by models
(6.a-c)), the safety margin average value at any time t is determined by the simplified expression (3):
n, = ue./uet = ue J(ue0 + ùe ■ t)
(3.a)
and the ultimate resource is determined by the simplified expression solution (5) - iie • tL = (ue* — Ue0)/ respectively to t = tL as:
tL = (ue . - ue 0)/ile
(5.a)
The complex of equations (3.a), (5.a)-(6.a-c) denotes a physical probabilistic model of the formation process of
0
0
0
gradual failures of machine elements, in which internal static or dynamic tensions <j emerge, affected by external loading and constant temperature T.
The sequence of mathematical operations, based on this model in order to estimate the expected average resource tL , defines their durability forecasting method and is the academic analogue of S.N. Zhurkov famous exper- 1 imental equation [12, 13].
The example of the suggested method implementation is shown in figure. A graphical interpretation of the equation solution (5.a), taking into account (6.a-c) for a spindle from 25 steel grade with mechanical properties o> = 230 MP a, aB = 420 MPa and HV = 2020MPa, exposed to static uniaxial tension at different tempera- 3 tures, is represented. In particular, at 0 = 142,9 MPa
and temperature T = 40°C its forecast limit time -average limited endurance under these conditions, comprises tL « 3,7 • 106 s « 1,43mo (see marked point 4. ordinate in figure).
Nomograph for loaded element durability estimation
The suggested approach, in our opinion, allows to forecast different loaded parts durability, taking into account the expected values of maximum tension a, temperature T and physical and mechanical material properties, as well as to analyze the possible ways of their life improving at the design and operational stage.
For example, as can be seen from the nomograph, spindle durability can be greatly prolonged by its temperature reduction, physical and mechanical properties changing, as well as other parameters, included in the equations (5.a)-(6.a c).
References
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2. Antsupov A. V., Antsupov A.V. (Jr.), Antsupov V.P. et al. Ocenka dolgovechnosti nagruzhennyh detalej po kineticheskomu kriteriju prochnosti. [Evaluation of durability stressed parts on the kinetic strength criterion]. Materialy 70-j nauchno-tehnicheskoj konferencii: Sb. dokl. [Proceedings of the 70th Scientific Conference: Sat Reports]. Nosov Magnitogorsk State Technical University, 2012, vol. 1, pp. 137-141.
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