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22Journal of Siberian Federal University. Engineering & Technologies, 2018, 11(5), 604-618
y^K 624.042.12
Compensation of Thermal Deformations of Heat Supply Network with Radial Expansion Bends
Yury L. Lipovka and Vitaly I. Belilovets*
Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia
Received 28.02.2018, received in revised form 07.04.2018, accepted 06.05.2018
A methodology for structural analysis of compensation of thermal expansions used for channel or aboveground sections of heat network pipelines with radial expansion bends has been proposed. The methodology is based on the methods of structural mechanics and allows to find technically based dimensions of radial expansion bends. Increased flexibility of the heat pipe angles and stress-raisers of additional bending stresses have been taken into account, the functional and graphical dependencies of the stress caused by thermal deformations of the heat pipe section with the U-shaped radial expansion bends on the temperature of the coolant have been presented. The authors developed and presented the computer program used for calculating radial expansion bends.
Due to the program analyzing the model of a heat pipe section with a radial U-shaped expansion bend has been calculated. Comparison of the calculation results with the results obtained by means of the well-known software system "Start" developed by the scientific-and-technological enterprise "Pipeline" has shown a sufficient convergence of results, which confirms the efficiency of the presented methodology and computer program.
Keywords: heat supply network, thermal expansions of heat pipes, compensation of thermal pipe deformations, radial expansion bends.
Citation: Lipovka Yu.L., Belilovets V.I. Compensation of thermal deformations of heat supply network with radial expansion bends, J. Sib. Fed. Univ. Eng. technol., 2018, 11(5), 604-618. DOI: 10.17516/1999-494X-0059.
© Siberian Federal University. All rights reserved Corresponding author E-mail address: lipovka.j.l@gmail.com
Компенсация температурных деформаций теплопроводов с радиальными компенсаторами
Ю.Л. Липовка, В.И. Белиловец
Сибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79
Предложена методика расчета компенсации температурных расширений для канальных или надземных участков трубопроводов тепловых сетей с радиальными компенсаторами. Методика основана на методах строительной механики и позволяет подобрать технически обоснованные габариты радиальных компенсаторов. Учтены повышенная гибкость отводов теплопроводов и концентраторы дополнительных изгибных напряжений, изложены функциональные и графические зависимости напряжения температурных деформаций участка теплопровода с П-образным радиальным компенсатором от температуры теплоносителя. Представлена разработанная авторами статьи компьютерная программа для расчета радиальных компенсаторов, с использованием которой проведен расчет модели участка теплопровода с радиальным П-образным компенсатором. Сравнение результатов расчета с результатами, полученными при помощи известной программной системы «Старт», разработанной научно-техническим предприятием «Трубопровод», показало достаточную сходимость результатов, что подтверждает работоспособность представленной методики и компьютерной программы.
Ключевые слова: тепловая сеть, температурные расширения теплопроводов, компенсация температурных деформаций труб, радиальные компенсаторы.
Introduction
For heat supply network the most significant loading factor is thermal expansion of pipes. Incorrect calculation of compensation of thermal expansion may cause heating main leaks and breakdown of equipment connected to heat supply networks. A correct solution of the issue concerning compensation for thermal expansion of heat supply networks guarantees its reliability and durability. An estimate of reliability of urban heating networks from the point of view of calculating heating mains for compensation of thermal expansions is presented in [1]. The issues concerning reliability of engineering systems are considered in [2].
Thermal expansion of pipelines of heating networks appears to be the main loading factor, regardless of the way of laying. However, the calculation of stresses based on thermal expansion for trench and trenchless heating mains varies considerably. The calculation techniques used for trench or aboveground sections of heat network pipelines are unacceptable for calculation used in terms of trenchless heating mains. Some theoretical aspects of calculating the thermal deformations of underground trenchless heating mains have been discussed in [3].
Modern technologies used for calculating main pipelines have been described in [4]. In order to make calculations heat network pipelines are typically simulated as rod structures. In the case of thermal expansion of heat supply networks the loss of pipe stability seems to be permissible. The loss of stability is another important task that should be taken into consideration in case of calculating thermal expansion of heat network pipelines. The solution of this problem has been considered in [5-8].
Computational simulation with a georeference of networks location, without which it is impossible to calculate the operating modes of district heating pipelines correctly, has been the main concern of the following papers [9-11]. The solution of new, rather complex tasks in the field of transportation of energy sources can contribute to cyber-physical systems [12] and cloud-based computing [13].
Scalable parallel computation of finite element models can be also an effective tool for managing network operation modes in complex geological conditions of modeling used in geotechnical engineering [14].
Optimizing route location of urban power grids is certain to be one of the most important tasks, ensuring a reduction in capital investments in district heating networks [15].
Remote-controlled regulating devices allow to communicate with power suppliers on-line, but this requires a radical rethinking of existing control algorithms. In [16-18], the results of simulating the distributed systems in terms of transportation of power for heating needs, ventilation and air conditioning (HVAC), have been presented.
One of the most important tasks of optimizing the transportation system of power sources is certain to minimize the power consumption of pumps and heat losses in the pipeline network. The solution of this problem by means of the nonlinear programming algorithm (NPA) and the genetic algorithm (GA) has been described in [19]. The research results of the dynamic characteristics of the thermal regimes of district heating systems due to dynamic modeling method have been presented [20].
Methods
Methods of structural mechanics allow to perform structural analysis of the heat supply network to loads and imposed deformations, including compensation for thermal expansions. The most commonly used method is the force method. In order to calculate heat pipes for thermal expansion there is a modification of the force method, called the elastic center method. Structural analysis of expansion bends for thermal effects by the elastic center method are considered in [21, 22]. It should be noted that, in terms of structural mechanics, the heat pipe is a statically indeterminate system. The assigned task is to disclose this static indeterminacy. In case of using the force method, the disclosure of static indeterminacy occurs due to eliminating of primary system from redundant constraints and their replacement by a statically determinate primary system. Unknown reactions of eliminated redundant constraints are determined by forces and moments. Displacements for each of eliminated redundant constraints are equal to zero. Primary unknowns are forces in eliminated redundant constraints, which are calculated on the basis that corresponding displacement to each constraint are equal to zero.
Let us consider a section of a heat pipe bounded by two fixed supports (Fig. 1a). In terms of structural mechanics, this section is a three times statically indeterminate system. The eliminated fixed support is placed at the origin of an XY-coordinate system. The second fixed support is shown as B. Reactions of the eliminated redundant constraints X1, X2, X3 are positive.
Generally speaking, for a common flat section of a heat pipe, primary system is determined by eliminating constraints for one of the fixed and all sliding supports. The eliminated redundant constraints are replaced by unknown forces and moments. Subsequently it is required to find generalized forces. In the meantime, it should be assumed that displacements caused by the generalized forces in primary system do not violate restrictions imposed by redundant constraints on given system.
X3_
X B
XI
X2 a)
Fig. 1. Design piping diagrams
Problem conditions under consideration can tie expressed ;as tlsa followinp system of equations, calledthesystem ob oanonical aquntsons ef tlie force method
(finXj + <yi2!X0^ + 51 -i- ••• + + A i = 0;
jPoi^i ii- p22:x2 -f <J2fX3 ff • + ¿f2kXXk + A2= 0; (1)
, 5-— + <5,2X2 + 5° XX00 + ••• + fXk + (3, where Sir, n1lt...,Sir,..., 02i, S22, are ahe ueir displacemeces of psimary system; Xi, X2, X3, Xk —
ehr tlie primary unknownsi Os, St2i ...,Ai a- are tte !-re^ Oerms of tOa ennoarrel enuaiions.TPeso tecms representneneraiieeeOispleoemTntr inpnimaieisysiam,which correspond to the generalized force X;. Externalloading, tnmperaSurediffeerneost disfdocemenS oii miesu^urts, pss-tensiomng oJ arererponeiMe for chesn dir^acemeses.
nee etas terms era Crterminedoniho brsss oo ehh S'ollasvetnh dsjisnt^eteces
Ao = A^ e- AoM m A^ -o Aop;
A. d hit -a hcM Se hco a hiP; p^t d, Ace a Aap, (2)
wheseAt, n3 -- air Ohe Otsplocements of appiicaiion poinir of ihe cerresponOingunknnwnforces X1, Xt. X- in the direction ef theme fsirces tFlg la); Alt; n)t - sea iie tanrmal expansions af pipas; A1M, A2M - reduced to the origin coordinates components of the displacement in the direction of the forces X1, X2 from pre-compression (for pre-tension, displacements are negative); A10, A20, Aa0 - are the linear and angular displacements, which are caused by displacement of supports; A1p, A2p, Aap - are thedisalacements that correspond to unknown Xt arising en paimary ^^^t^^m ai aresultof aaternal londing.
The sleermal sxpanskms on iestscs arecaieuinted sr fotlnws
A.^ -Lx(xAt; A2r= -LycAa, (3)
where Lx, Ly - are the projections on the x, y axis of distance between end supports (Fig. 1a); a - is the linear expansion coefficient of pipe material for specified temperature of pipe wall; At - is the temperature iiffeeence.
Thencgative sign in equations (3) is responsible for process of the thermal expansion of the pipeline. With decreasing temperature, equations (3) are positive, since the reverse process occurs. The point of reference of the change in temperature is the temperature at which the installation of the heatpipe iscompleted.
The linear and angular displacements caused by displacement of supports are determined by the following equationr
A10 = Aa - A] -i yBaB; = Aa - AtT -i- Aa0 = Ac^ + Aa)T (4)
whfru A !T ^20- Wfcti - arethe line ar and angula- bisglacements caused by displacement of the supports; Un , U- , Uct- - are tPs positive Tisnktcsmeets o) th- eliminated support A tF°g. lcs) in 1-e direciion of thg unknownt X1, X2, X3, respectively; Afo, tS^s, Aa® - are the positive displacements of the support B in the same directions; xB, yB - are the coordinates ofthesupportB.
Thedisplacements -lia- correspona Cr Olie unknown Xi m primary pysitene from ex-eanac leadmg ate caloukiBc til by the Modp 'o mTfi.21i?Alo',)as^.
l'os o sat litîiit pine eonststine of 01 oiemeots, -we; can wrntefde OolSewiog sî^usil)<0]oi-
e v. sL.MiMp „ „ fL. M2Mp,, l „ fL. tt 3Mp t(
aaip = En J0 lta-Ep^d^; aa2p = .-^dL; a<xp = IEn J0 k-¡^dL, (5 )
wPsct Mg, Ma, M3 - are tine bendmr mements en an asCitrarysectcon ef pemsey system,caused br acriin ofltoe unttfooces X^ X3 respectively; Mp - is the bending moment in the same seel ion, paused by exOomal Ooading ; k - is rhs e le xM Hty factor; EI - is the pipe stiffness factor.
Tho numbor of canonical eqaotions noreorponds Oo tOe of system static indeterminacy.
The crtffiqfenlt ob ad: efuaOioesd^ o ontrne tota aituS the displacements of application points of the unknown forces m Ohe dirartioo oS odeir i-trse caused by nni) forpes or moments. The first of the displarement indicea Sd nnpoasenis tde unknown fore; e Bums ti wlkicli giken genosulized displacement cfrrcepondt to; aie aeconW dndsx is the numhen of un-t forae ihat would cause tills displacement. The dirplbcemont nilt oorec^o nds to tont foece ita wOen X;S° is ohe wsrk of ihc Oorci X; on the displacement Sik. The coefficient is tlie displacement of the application poln! of the force Xi in the direction of its action, which i s caused by external loading, temperature differences, displacement of the supports and pre-ieneion of tpe hsbipipc.
Ifwe de not\ckemtokccuunteffectofeengitudinarandtofnskernefoiees os; the displacement, onty tteJbtiog ttsl<f cncourt hmding andiorséon, Mohr'eOormnin fer tht ihreocdimensional section of the piaittiioc, conrtItiog oS n-demente, rcilS htve ihc hollowing eorm
Sis = Zfei J0Li k»dLj + Zf=1 J0Li HGHf dLj, (6)
wheoo Mj-k Shn fendi ng moment in an arbitrary section of primary system from the action of the uoit generaliztd force Xi = 1; Mk - is the bending moment in the same section from Mi; Hi - is the torooe inanarbilraib section of primary system from the force Xi = 1; Hk - is the torque in the bomeiection faomthefaeceXk = 1; E - is the modulus of elasticity of pipe material; G - is the shear modulus of pipe material; I - is the moment of inertia of pipe cross-section; Ip - is the polar moment of inertia of pipe cross-section; L - is the length of the pipeline element; k - is the pipe flexibility fontbe
For otlot boat piae, in tine obsence of toroion,formnla (61 can be transformed to the following
form
skfentfktttedL,, (7)
For acommon flathort pipe section, tlie coefficients ofthe enoonire1 equations are determined by OOe sum of tho Vhehe's ^oih^gi^alo over the lenglh of oach ciemnnO TOe equationo under doniiOeraticn con be oxpresseO see followr
L L L
Sn = £ / Hg dL; 522 = / 4dL; 533 = £ / |dL;
000
s13 = -a /0rkfdL;523 = -a ^dL^i = ai/crip1mic^iL. ^
If the hsat vpe ¡scrtttioti ins;luclt;s: pijt^s of diffosont cress-sestions, sho stiffnessvahie EI will differ fer fach of the e-otion ектеШп TTcil^ingtllLis; ^nio crcousit, it iss necessary to introduce notion of the reduced flexibilityfdctor.This flexibility factor is denote d by the sy mbol r) andexpressed as follows
П = C- (9)
w=are is - is tiie flexibrlity factsr; EI0 --is theelemdntwith tliefreatect s^^Jffnem value; EI - is the stiffness value oS given element.
Considoring thi, tire deiermmedintegrels informuias (8) tageghe meaning of the geometric tgaraoeerisiecs o1 lis r^^^ciecl le^td af lilt te^sm dm dor heat glpe eremeni^a. The free terms of the canonteal ngiua^toiis d)etem wit etnjr^^s^ ;ae foltown
— = Д^; — rc — r= /E3^I0. (10)
Tire reduced moments of inprtia relotiee to ths s^^^ bb^C, Y orecaiculated by formulas
Jx,i pr= /,Р У2^; Jy,b Pr = /0 x2T|de (11)
The reducedt nertia pro duct
g xy,i pr = jJ^y^cllL. (12)
Xhc reduecd sintin sci)meiits; relative tcthe X, Y axes
Sx,i ip1- = jg У1^9 ; Sy, j,^ = £ xr|dL (13)
Tfe eeduced ^<^i^gitciof' elie nenierimeoflhe eiei=e—
= /0LTld Sn "Ч^?!" (14)
Ш exueostiens (ll) с (if t dC-flays tfe elemehinumber,L,f the Ocnxth, ei;f- reduced the flexibility factor.
Inordertocalculate byformulas(11) -(14), the centerlineoftheheat pipe should be divided into elements with constant bending stiffness and curvature. Further on we need to define the initial and final coniUmdiet of eat- ekmeni, ire reduced АехШШСу factnc and the ecdneed kngth.
infig. tC, ticecen-ral ares oCihc element (a, -3) eec drawn paraltel to ChegivunaxesX, Y. Therefore
themomentsofinertiafor anyelementrelativetothe axesX,Y can becalculated by formulas
_2 _2
Jx,i pr —,i pr "" L"i prjlj i —,i pr Ja,ipa "" Liprxi ■ (15)
The reducedproductofinertia ofthei-th element isdeterminedbytheformula
- 609 -
Jxy,i pr Jap,i pr ^ Li prxiyi>
ei6)
where 1-i pr - is the reducsd le^j^je^le of the element, m; if - is the cooey/wote oh tlid straighrelement center of gravity retatinr to telle X axis, m; - is the coordinate of the steaight eleme nt oenths of g ravity relative do the Y axis, m; Ja i pr, Jpj pr - are the reduced moments of inertia of the i-th element relative to thteigtncentral axes a, P; Jaisir - is the reluced procieci^ ii^^iria of the i-th element relative to the eigen c e ntral axes a, p.
The to osdinstee o:f thon senter of graeity relative to the X, Y axes (Fsg. 1 b) are sletermined my ihe fonowinp arpretsJoes
xi = ^ (xi,i + xo,i); Yi=x (yi,i +" yo()-The raok) .or tfie reduce0 ef nheh oof tliiE; heat ptpe1 s as follow;
Lipr = npjOxi,i - xo,i)2 + (y(,i - yo,i)^-
(17)
(18)
rriiL^ relireed momontr of imitie relative tee the eiron tentit axes tin XeOerminee be the following dep rndoeroe
i ( : 11 2 Je,ipo = —tLipti.xl,i "h xO,i) ; Ji,i (t = —( fi jy1,i y y0,ill ■ (19)
TXa fic^tlitciice ^riacliJL^t of htttie retante to tho eioen c-ntal xeot id <eo^r^eLit^ineil by the following foemnle
Jfiipr — ni I-ip^X^ yo,0-
(20)
nhus, rnven geometric ataiaiteriseiti for egir ilra^he ete nxeit (if ehe heat pbtearc dtteemined as fellows
tree reduaed momeais if inereia oi the i-th element
Jx,l fir
Jy,l(r
= Li
= L
pr
(r
Jyi,i-yo,i)2 L pyi,i+yo,i)2
1 ( 4
(x(,l —i,0 (-(,l + -0,l)
(21)
12
Ol^^rereuci^d drohnci oil ineitir olt itlio Rit element
Jxy,ipr L"i
(xu-XohXyu-yo,-) , (x:L,i+Xo,i)(y:L,i+yo,-)l
)o
■ + ■
(22)
The reduceb static moments of the i-th straight element can be determinedby the following for mutas
Sx,ipr = -i p ryi = Li + y0,i); Sy,ipr = Li prXi = 3 1. pr (xii + x0,i)- (23)
Fig. 1c shows the curvilinear element of the heat pipe in the XY coordinate system. The coordinates of the center of curvature of the element are denoted by Xc, Yc. The radius of curvature is denoted by R. The central angle of the element is denoted by 9. The angle of inclination of the initial tangent to the positive directionoftheXaxisis denotedbya.
- 610-
The reduced moment of inertia and product of inertia of the i-th curvilinear element located arbitrarily are determine d by thr fo llowing equations
W = Ripr^/Ci - ZI^C" -I- R^CC); (24)
Jy,ipr = ot(ccc,i2)U + RRixc,iC3 "12" fi Q);
.xy.i pr = Ri piCxc,iyc,i Cl _ Rixc,iC2 + fiyc iCc Ri'Ce^
where xrib ycj - are Che coondinetes od eurvatuse center of the lonotudinad oxis of the i-th cerv/'l/'aear element, m; Rf - iis the radius of curvature of the longitudi nal axis of the i-th curvilinear element, m; hd pr - iis? She reduced radus e:f carvatf ne
RiprfR: (25)
TSie rciluced li^nrffli iis c alcolatedbn fofmulo
Lipr rr Ri prCl- (25)
The redueedteetic itio mesntia aed cckuited by formubts
pr = Ri pr(yc,iC) _ RA); Vipr rrpr = Frji pr(xCiC) + R|C3). (27) Tho vhluus of ihh cocfficiente Q are determtned tty the Jfo^l^w^^j« expresuionr
C^ = (p; C2, = 2sin^cos (a -I- ; C: = -sin^son (a + ; (28) C- = 0,5[c- + sinqo cos(2a -I- tp)]; C5 = 0,5[cp — sinqo cos(2a + 9)]; C6 = 0^!5s^ni2 sm(=a + (f>). On (28X tilte; anfkr a nnd qt are considered in radlens.
Ah thh co^rffieuientEi rf Ulie; canonicc° i^^ii^CiLOinis wore muMphed Iby EI0 = conot It ftslloaris thai EIt(i)ik = Sik* Thus, Uhe formuea oU the cnnonnel eruhtions roelihclento by given geomeltlcol chcruotsrirtiri oU r srnuie fla) heub push irwonctiosisd o:lF ii niumenls wcn bo ar doilowi
Eii = ULiJxa pr ; Ef2 = H L 1 -y,i prr E33 :== J-'t pr ; (29)
5,
n 11 n
= ""' Jxy,i pr ; 523 == """^yipr ; 531 == 1 S)i,ipr-
J12
i= 1 i=l i=1 "."li^ system of ciiifinjtticural equetiono for lha caktullfted seetion of the heat pppe be transposed to the followingform
f5n*Xi -I- ô:L2*X2 -I- ô-L3*X3 -X = 0;
| + 022% -I- Ô2 * X3 -X a^ïX. (30)
Al*Xl + Ô32*X2 "t" 5"3*X3 + A3 = 0, Applyi ng Crameaiformalas, solution of syetom (3 U) can representedas follows
¡ = = -^X3 = -?X (31)
where D - is the determinant of the coefficient matrix of the system (30); D^ D2, D3 - is the determinants obtained from D by replacing the corresponding column of the coefficients Sik* by free terms column.
The determinants D1( D2, D3 are defined by the following formulas
Di = Ajai + A2a4 + A3a5; = Aja4 + A^ -I- AU-^; D3 = A^ -I- A^a- + A^
(32)
where <i\, a2, a3, a4, 35, &6 — are the constants.
Tha determtnant ID is cakulated by formula
D = 5n "ai + -I- 3*05. ^3)
The constants ai, a2, a3i i, a^ e (re denermined by thefollowing focmnlaa
* * *2 * * *2 * ( *2
ai = 822 (>3;, - 833 ; a;- = 5n .533, r- 5B ; 1-3 = 5n 822 - ; (34)
a4 = 5-3 S32 — 312 533 ; as = (ill;) <5»23 — 513 322 ;
= n2i Sp3 _ S3 *ii ■
h is neceisosy to consideo aniu^^ii^iis oi2 the pige flexibilty racton m more detail. In tins beginning of tOe VO" eentuty tiiv c2nsuicai solution -on beeOtnocurved ftpes (bands) was published by T. Karman. Iui Oliis SKrlioteLon, oe energo epproath -v^stts lieee ^tit)^ecoueio^ ehlutiovr ol tke problem by the Ritz method way used. The solution wvs a thironemeheic: srs-is. 1) thi nag(ectitog of all membeoioftiie cerins, pxcejet hoo the firsi, a fsnrnda tHe flnxibieky factos (fJi^arir^an'd formefa)for 3urvod ^i3ec in 3enriee was Pksived , Thss formula liar tPe in° form
2 laow2
k =-m (h5
10 +t2im ( '
wis ne k- ts the flifxsbi^ity factor of the curved pipe (bend); X-is 1he aeomeOric characteristic (rfthe curved p(pei ^<r.ilned as fol lowe
(Dn-S)2' V i
where R — is the radius of curvature of the curved pipe, mm; Dn — is the outer diameter of the curved pipe,mm; 5-rs ehw wall lli-ckness ofthecurvedpipe, mm.
In we neglect hll hi- memloers of tliiv teriek exoeph Ulie eice^et; iovo, wn oOeain the Karman formula in Oha lecend eponoxkmation tor caiculrtlbg tOe i^lstuil—teec^thsrl^ei^, Isaving the form
aka tr-h-s. (37)
105f413trf4800:C4 v '
Tie Ohrnoe3a inthe t^lTLird. approximations defined similarly
. e+e080X0+e09e76X4+2820400X6 .„„.
k =-0-4-6- (38)
Y50+7e910Io+0446176I4+0800400I6
When deriving the formula for the flexibility factor, T. Karman made the following assumptions: 1) the radius of curvature of the central line of a curved pipe is much larger than the radius of a pipe itself; 2) the wall thickness of a curved pipe is small in comparison with the radius of a pipe; 3) the displacement of the neutral axis was not taken into account; 4) the Poisson's ratio was not taken into account; 5) the ways of fixing curved pipes with straight pipes were not taken into account; 6) the bending moment does not change along the entire length of a curved pipe; 7) the influence of internal excesspressurewasnottaken intoaccount.
The bending of curved pipes was investigated by R. Clark and I. Reissner. A solution of this problem was obtained, by analyzing the differential equations, when considering the bending of
curved pipes from the; pointofview ofthe theory of thin-walled shells. By the method of asymptotic integration,thefollowing Oormula for the flexibility factor
k = 2VV12(1 - r) , (39)
where! - is thegeometriccharacteristic of the curved pipe; ^ - is the Poisson's ratio.
If weassume aPoiston radio to bc equat to 0.3, then the Clarke and Reissner formula can be
represented as follows
k = , (40)
The R. Clark and I. Reissner formula gives a more precise value of the flexibility factor of curved pipesthantheT.Karman'sformulasforX <0.3.
At the present day, a great number of different works has been devoted to solving the problems of calculating pipe bends. For example, in [23-24] analytical solutions for the bending of pipes are considered. In [25] the solution for the bending of pipes u sing the finite element method is presented.
ResultsandDiscussion
The Karman's formula and formula of Clark and Reissner were investigated. The graph of the dependences of the change in theheribilfty fmctoron thegeometric characteristic of a curved pipe for three Karman aippsoximations andthe Claree andReissner formulas are constructed.
Analysisof the graphs in Fig. 2 gives thn follnwing results. Firstly, the larger is the wall thickness ofa rurvedpipe, tlieSlgRintlidy factor si itie closer le S. Secondly, the Karman's approximations are iacaioecety used in deiurnnm ng die floxifilidf fnctor of curvilinear elements of heat pipes, since they are thm-waOedstrucdures fom wluah their geematrKi chmracteristics will have small values. If we take rhnfe omefriceharaetdrisfingff cui/ed pipe tobe zero,lhen the flexibility factor for all the Karman's afprofimaSionswill have noneeeo vaiuer. Thif ir wet lrue, since with the value of the geometric charnntnrisRcof a eeevedRpeara eijualtozesc, tiisflagibility factor of this pipe is similarly equal to zero, ee foltowstnatlheKisman'nnppeonimationt incorrectly used to determine the flexibility factor of igick-waiigd pigii undrt h^e^Sr^i^^^^iso^a^i^ers ^^^^ssrre. If a heat supply network is considered, then the Clark and Reissner formula for determining the flexibility factor should be used.
Wiee bendinn eumad fipelinef umier the inhe^^ner of forces that flatten their cross section rignificanfaocafsreeeses irise.iniaelevgidwainaO sirertf s found in the conventional bending theory aee denotes b; c, ili^n tie maximum lonmiaudinal stresses can be calculated as follows
oinpx =s loo, (41)
wdsse p s-is tiis tongianninel sneees concentration factor in the curved pipe
io = -27-, (42)
X '3
where X - isthe geometric characteristic of the flexibility of the curved pipe.
It should be remembered that the local stress concentration factor is present in any places of sharp changes in the geometry of a pipeline. To these places, besides bends, you can include tees and transitions from one diameter to another. For each case, there is a definite formula for calculating this factor.
Fig. 2. Dependence of theflexibilityfactor onthegeometryof the pipe
The authors ofthis articlehave developedand registered the computerprogram" Calculation of theexpansionbends ofuheatiupeiy neiwork" [b6t.Chlculat)on en the progtam irljiie^cioec tt^e methods of sSrurdxsc^l mtehanlcs. Tho programmaJret i^tj^os^tbS^So cetcelata thermal stresses for sections of a heat tupply nitwotk iy mcetis ted nifferetet scha mer af thpensiou bends thai are poS olamped by the soil. Thi program provtdctcalIg)ratidnoO botPexpansionbeids and original aogtesof banding of a heat supply network. Based on input information entered in the input fields, calculation is performed and information on the stresses in the main sections of the configuration of the heat pipe section is displayed in the result fields. This allows you to select the optimal size of expansion bends.
Fig. 3 shows the interface of the program. As you can see, the window is divided into 3 blocks: configuration selection, initial data and results. The configuration selection block allows you to select one of the 8 circuits of a heat pipe section with an expansion bend or an original angle of bending for subsequent calculation. Thus, it is possible to calculate the symmetrical and asymmetrical scheme of U-shaped and L-shaped expansion bends, Z-shaped and L-shaped expansion bends with angles of bendmg pf p0 os mere ttegiefs. T°c wpole program ° buik ftr ihe teas of methods of structural mechanics, applied to a heat supply network. Accordingly, the program take s into accountthe i ncreased flexibility oe heat pipetendS"
Let us consider comparison between the results of calcu latign on the rmal expansionsof a section of a heat pipe with tha U-thapen axpension neiidiloig), repetpedby means of the presented program, and the results of the program system "Start". The section is bounded by fixed supports. We take the following initial data: the pipeline 159x4.5 mm, steel 09G2S; the installation temperature minus 40 °C; the temperature of the heat transfer plus 150 °C; the working pressure 1.6 MPa; the density of heat transfer 1000 kg/m3. The section has the following dimensions: the radius of bend 240 mm; the loop shoulders length 8 m; the loop legs length 4 m; the loop back width 4 m. After the calculations, the following results were obtained: the minimum stresses on the section (in the loop shoulders) was 8.9 MPa for the author's program and 9.5 MPa for the Start; the stress in the loop legs was 23.8 MPa for
Fig. 3. Interface of the programCalculationofexpansionbends
6 8 10 Sections of the heat conductor
Fig. 4. Temperature dependencesofaheat pipe section withtheU-shaped expansion bend
the author's program and 24.5 MPa for the "Start"; the maximum stress in the loop back was 47.9 MPa for the author's program and 48.2 MPa for the "Start". Due to this experiment, we can conclude that the calculation in the computer program "Calculation of the expansion bends of a heat supply network" is correct.
Let us consider the temperature dependences of a heat pipe section with an expansion bend. The results will be presented graphically.
Fig. 4 shows the temperature curves for an example of a heat-pipe section. The numbers opposite the line color show the temperature values. As we can see from the graph, the largest and smallest
stresses from thermal expansions, at any considered temperature, are in sections 6, 9 (largest) and 4, 11 (smallest). The sections with the highest stresses are in the loop back, the sections with the smallest stresses in the loop shoulders.
Conclusions
Due to the force method, a methodology for calculating the stresses based on thermal expansions for sections of heat network trenchless pipelines with radial expansion bends has been developed. In the methodology the increased flexibility of the bends has been taken into account. The optimal formula for calculating the flexibility factor of heat pipe bends has been determined. Due to developed methodology, a computer program for calculating radial expansion bends of heat networks has been developed. The efficiency of this program has been confirmed by means of comparing the results of a numerical experiment with the model of the U-shaped expansion bend with the software system "Start" developed by the scientific-and-technological enterprise "Pipeline". The experiment used the design model of the U-shaped expansion bend made it possible to draw the following conclusions: 1) the nature of temperature stresses distribution along the expansion bend sections for any temperature of the coolant coincides; 2) the angles of the expansion bend back are the most crucial elements where the greatest stresses are observed; 3) in case of calculating the radial expansion bends, it is necessary to take into account the increased flexibility of the bends and the concentration of local stresses in them, since their ignoring will cause large inaccuracies in the results.
References
[1] Белиловец В.И., Липовка А.Ю., Липовка Ю.Л. Оценка надежности городских тепловых сетей с точки зрения прочностных расчетов. Город, пригодный для жизни: материалы II Международной научно-практической конференции «Современные проблемы архитектуры, градостроительства, дизайна». Красноярск: СФУ, 2015, 376-378 [Belilovets V.I, Lipovka A.Yu., Lipovka Yu.L. Reliability Analysis of Urban Heating Networks in Terms of Strength Calculations. City fit for life: Proceedings of the II International Scientific and Practical Conference "Modern Aspects of Architecture, Urban Planning, Design. Krasnoyarsk: SibFU, 2015, 376-378 (in Russian)]
[2] Абсалямов Д.Р. Повышение надежности инженерных систем методом формализации поиска отказов. Инженерно-строительный журнал, 2012, 2(28), 39-47 [Absalyamov D.R. Improving the engineering systems reliability by formalizing the failure search. Magazine of Civil Engineering, 2012, 2(28), 39-47 (in Russian)]
[3] Липовка Ю.Л., Белиловец В.И. Некоторые теоретические стороны расчета температурных деформаций подземных бесканальных теплопроводов. Журнал СФУ. Техника и технологии, 2016, 9(4), 546-562 [Lipovka Yu.L., Belilovets V.I. Some theoretical aspects of temperature deformation analysis in underground trenchless heat supply pipelines. J. Sib. Fed. Univ. Eng. Technol, 2016, 9(4), 546-562 (in Russian)]
[4] Лалин В.В., Яваров А.В. Современные технологии расчета магистральных трубопроводов. Инженерно-строительный журнал, 2010, 3, 43-47 [Lalin V.V., Yavarov A.V. Modern technologies of calculating main pipelines. Magazine of Civil Engineering, 2010, 3, 43-47 (in Russian)]
[5] Тюкалов Ю.Я. Функционал дополнительной энергии для анализа устойчивости пространственных стержневых систем. Инженерно-строительный журнал, 2017, 2(70), 18-32 [Tyukalov Yu.Ya. The functional of additional energy for stability analysis of spatial rod systems. Magazine of Civil Engineering, 2017, 2(70), 18-32 (in Russian)]
[6] Сергеев О.А., Кисилев В.Г. Сергеева С.А. Общая потеря устойчивости и оптимизация стержневых конструкций со случайными несовершенствами при ограничениях на вероятность безотказной работы. Инженерно-строительный журнал, 2013, 9(44), 30-41 [Sergeev O.A., Kisilyov V.G., Sergeeva S.A. Overall stability and optimization of bar sterucrures with random defects in case of constraints on faultless operation probability. Magazine of Civil Engineering, 2013, 9(44), 30-41 (in Russian)]
[7] Кирсанов М.Н. Статический расчет и анализ пространственной стержневой системы. Инженерно-строительный журнал, 2011, 6(24), 28-34 [Kirsanov M.N. Static calculation and analysis of spatial rod system. Magazine of Civil Engineering, 2011, 6(24), 28-34 (in Russian)]
[8] Лалин В.В., Розин Л.А., Кушова Д.А. Вариационная постановка плоской задачи геометрически нелинейного деформирования и устойчивости упругих стержней. Инженерно-строительный журнал, 2013, 1(36), 87-96 [Lalin V.V., Rozin L.A., Kushova D.A. Variational functional for two-dimentional equilibrium and stability problems of Cosserat-Timoshenko elastic rods. Magazine of Civil Engineering, 2013, 1(36), 87-96 (in Russian)]
[9] Kudinov I.V., Kolesnikov S.V., Eremin A.V., Branfileva A.N. Computer models of complex multiloop branched pipeline systems. Thermal Engineering (English translation of Teploenergetika), 2013, 60 (11), 835-840
[10] Finney K.N., Sharifi V.N., Swithenbank J., White S., Ogden S. Developments to an existing city-wide district energy network. Part I: Identification of potential expansions using heat mapping. Energy Conversion and Management, 2012, 62, 165-175
[11] Kaliatka A., Valincius M. Modeling of pipe break accident in a district heating system using RELAP5 computer code. Energy, 2012, 44(1), 813-819
[12] Yoo H., Shon T. Challenges and research directions for heterogeneous cyber-physical system based on IEC 61850: Vulnerabilities, security requirements, and security architecture. Future Generation Computer Systems, 2016, 61, 128-136
[13] Jararweh Y., Al-Ayyoub M., Darabseh A., Benkhelifa E., Vouk M., Rindos A. Software defined cloud: Survey, system and evaluation. Future Generation Computer Systems, 2016, 58, 56-74
[14] Zhang Y.L., Tan F., Zhang L.R., Shi M.M. Scalable parallel computation for finite element model with hundreds of millions of elements in geotechnical engineering. Rock and Soil Mechanics, 2016, 37(11), 3309-3316
[15] Morvaj B., Evins R., Carmeliet J. Optimising urban energy systems: Simultaneous system sizing, operation and district heating network layout. Energy, 2016, 16, 619-636
[16] Maasoumy M., Nuzzo P., Sangiovanni-Vincentelli A. Smart buildings in the smart grid: Contract-based design of an integrated energy management system. Power Systems. 2015, 79, 103-132
[17] Radhakrishnan N., Su Y., Su R., Poolla K. Token based scheduling for energy management in building HVAC systems. Applied Energy, 2016, 173, 67-79
[18] Reppa V., Papadopoulos P., Polycarpou M.M., Panayiotou C.G. A Distributed Architecture for HVAC Sensor Fault Detection and Isolation. IEEE Transactions on Control Systems Technology, 2015, 23(4), 1323-1337
[19] Zhou S., Zhao Y., Zhang C., Zhang L. Operational optimization and algorithm of Hot-Water district heating system. ICIC Express Letters, 2015, 9(5), 1519-1524
[20] Zhou S., Tian M., Zhao Y., Guo M. Dynamic modeling of thermal conditions for hot-water district-heating networks. Journal of Hydrodynamics, 2014, 26(4), 531-537
[21] Lipovka Yu.L., Belilovets V.I., Lipovka A.Yu. The influence of slenderness ratio and stress concentration in taps on load calculations to thermal expansion in П - shaped compensators of thermal network. J. Sib. Fed. Univ. Eng. Technol, 2015, 8(1), 11-32
[22] Belilovets V.I. The Influence Factors of Flexibility and Stress Concentration in the Taps to the Calculation of Cooling Strain of Radial Compensators in Heat Supply Network. Prospekt Svobodny -2015: Proceedings of the scientific conference dedicated to the 70th anniversary of the Great Victory, Krasnoyarsk, SibFU, 2015, 11-15
[23] Radchenko S.A. Analytical and numerical solution for а elastic pipe bend at in-plane bending with consideration for the end effect. International Journal of Solids and Structures, 2007, 44, 1488-1510
[24] Kolesnikov A.M. Large bending deformations of pressurized curved tubes. Arch. Mech, 2011, 63, 507-516
[25] Fonseca E.M.M., FJM. Q. de Melo and Madureira M.L.R. A Multi-nodal Ring Finite Eleinent for Analysis of Pipe Deflection. International Journal of Manufacturing Science and Engineering, 2011, 2(2), 109-114
[26] Свидетельство о государственной регистрации программы для ЭВМ № 2014615580. Расчет радиальных компенсаторов тепловой сети. Белиловец В.И., Липовка Ю.Л., дата поступления 31.03.2014. Дата регистрации 29.05.2014 [Certificate of state registration of the computer program No. 2014615580. Calculation of the radial expansion bends of heat supply network. Belilovets V.I., Lipovka Yu.L., the date of receipt of application is 31/03/2014. Date of registration: 29.05.2014 (in Russian)]
