Uneven and non-stationary temperature field is formed in the reservoir wall while loading pressure reservoir with a liquefied gas in cooling state, at a temperature lower than the saturation temperature at given pressure. Mathematical model of the temperature field is developed by setting and solving differential equation of heat diffusion in the spherical wall in real conditions of the wall getting cold. The model enables computation of the temperature gradient attaining very high values at the liquid level. A simplified procedure for calculating the wall temperature increase in the narrow area above the liquid level was also developed. Understanding of the temperature field in the reservoir wall allows further heat stresses calculation. For this purpose, a temperature discontinuity should be approximated by a continuous function. Approximation was applied by Fourier\u27s orders, and coefficients of an order are determined by the electronic computer program. The model application procedure has been illustrated by the example of spherical pressure reservoir for liquefied carbon dioxide.Tijekom punjenja tlačnog spremnika ukapljenim plinom, pothlađenim ispod temperature zasićenja za zadani tlak, nastaje u stijenci spremnika nejednoliko i nestacionarno temperaturno polje. Matematički model temperaturnog polja razvijen je postavljanjem i rješavanjem diferencijalne jednadžbe provođenja topline u sfernoj stijenci u realnim uvjetima ohlađivanja stijenke. Model omogućava izračunavanje temperaturnog gradijenta koji na visini razine kapljevine poprima vrlo visoke vrijednosti. Razvijen je također i pojednostavljeni postupak za izračunavanje iznosa porasta temperature stijenke u uskom području iznad razine kapljevine. Poznavanje temperaturnog polja u stijenci spremnika omogućava dalje izračunavanje toplinskih naprezanja. U tu svrhu diskontinuitet temperature treba aproksimirati kontinuiranom funkcijom. Primijenjena je aproksimacija pomoću Fourierovih redova, a koeficijenti reda se određuju pomoću programa za elektroničko računalo. Postupak primjene modela ilustriran je na primjeru sfernog tlačnog spremnika za ukapljeni ugljični dioksid

Drago Kraljević

Pavo Baličević

Željko Ivandić

Hrčak - Portal of scientific journals of Croatia

1IntroductionUvodSpherical vessels are widely used as pressure reservoirsfor liquefied petroleum gas (LPG), carbon dioxide,ammonium, and many other technical gasses. Operatingpressure and temperatures the reservoir walls are exposed todepend on the stored gas type i.e. its saturated-liquid curveparameters. In some types gasses are stored at anenvironment temperature at atmospheric or increasedpressure. For example, calculated temperature in designingpressure vessels for LPG is regulated at 40 °C by theCroatian standards. Associated saturation pressure i.e. thelowest pressure at which the gas is allowed to convert into aliquid state at a given temperature is accepted as a calculatedpressure. The other way is storing gas in cooling conditionsi.e. at temperatures lower than the environment ones. Forexample, carbon dioxide has, at environment temperatures,high saturation pressures (at 25 °C 64,3 bars). This gascan be obtained in a liquefied state even at lower pressures,if kept in a cool state (at –25 °C 16,8 bars), but not lowerthan 5,18 bars since then sublimation occurs. The cold stateis desirable to be maintained with ammonia although itsstoring in a liquefied state, at an environment temperature,does not require too high pressures (at 25 °C 10 bars).Namely, ammonia is very dangerous for health condition,thus, protection measures from unexpected outflow andspreading into atmosphere are very important. Thus,accident danger is considerably lower if cold ammonia iskept at atmospheric pressure, liquefied at –31°C.In certain operating conditions the vessels are exposedto significant temperature change effects. Temperature field→→→31P. Baličević, Ž. Ivandić, D. KraljevićISSN 1330-3651UDC/UDK 621.642.036 : 536.5 : 519.6TEMPERATURE TRANSITIONAL PHENOMENA IN SPHERICAL RESERVOIR WALLPavo Baličević, Željko Ivandić, Drago KraljevićUneven and non-stationary temperature field is formed in the reservoir wall while loading pressure reservoir with a liquefied gas in cooling state, at atemperature lower than the saturation temperature at given pressure. Mathematical model of the temperature field is developed by setting and solvingdifferential equation of heat diffusion in the spherical wall in real conditions of the wall getting cold. The model enables computation of the temperature gradientattaining very high values at the liquid level. A simplified procedure for calculating the wall temperature increase in the narrow area above the liquid level wasalso developed. Understanding of the temperature field in the reservoir wall allows further heat stresses calculation. For this purpose, a temperaturediscontinuity should be approximated by a continuous function.Approximation was applied by Fourier's orders, and coefficients of an order are determined bythe electronic computer program. The model application procedure has been illustrated by the example of spherical pressure reservoir for liquefied carbondioxide.Keywords: liquefied gas, spherical reservoir, temperature fieldOriginal scientific paperTijekom punjenja tlačnog spremnika ukapljenim plinom, pothlađenim ispod temperature zasićenja za zadani tlak, nastaje u stijenci spremnika nejednoliko inestacionarno temperaturno polje. Matematički model temperaturnog polja razvijen je postavljanjem i rješavanjem diferencijalne jednadžbe provođenjatopline u sfernoj stijenci u realnim uvjetima ohlađivanja stijenke. Model omogućava izračunavanje temperaturnog gradijenta koji na visini razine kapljevinepoprima vrlo visoke vrijednosti. Razvijen je također i pojednostavljeni postupak za izračunavanje iznosa porasta temperature stijenke u uskom području iznadrazine kapljevine. Poznavanje temperaturnog polja u stijenci spremnika omogućava dalje izračunavanje toplinskih naprezanja. U tu svrhu diskontinuitettemperature treba aproksimirati kontinuiranom funkcijom. Primijenjena je aproksimacija pomoću Fourierovih redova, a koeficijenti reda se određuju pomoćuprograma za elektroničko računalo. Postupak primjene modela ilustriran je na primjeru sfernog tlačnog spremnika za ukapljeni ugljični dioksid.Ključne riječi: sferni spremnik, temperaturno polje, ukapljeni plinIzvorni znanstveni članakTemperaturne prijelazne pojave u stijenci sfernog spremnikaTemperaturne prijelazne pojave u stijenci sfernog spremnikaTechnical Gazette 17, (2010), 311 -34gradient can, at some points of the vessel wall, reach highvalues due to the effect of non-homogeneous and non-stationary temperature field. Stresses and deformationscaused by this gradient have dynamic character in thetransferable period and may significantly change originalimage of the stress field. In order to analyse temperaturestresses in the vessel walls it is necessary to anticipatecritical temperature distribution per thickness and per wallsurface as well as dynamics of its change in the vesseldesigning phase. It means that non-stationary temperaturefield, expected to occur in real conditions of the vesselexploitation, should be modelled.Figure 1Slika 1.Spherical shape pressure reservoir for liquefied gasTlačni spremnik sfernog oblika  za ukapljeni  plinGenerally, especially while talking about fast temperaturechanges or high speeds of deforming, both temperature fieldand the deformation field are coupled quantities. Thus,mathematical methods of their determination are verycomplex and are studied in a coupled dynamic theory ofthermo elasticity [1]. Such approach is necessary instudying the phenomena of thermo-elastic dissipation ofenergy. Since elastic body deformation causes slight changeof its temperature, mutual action effect of the temperatureand deformation field is not significant for problems oftemperature stresses determination in common conditions.In this case the problem can be reduced to quasi static taskof thermo-elasticity where non-stationary temperature fieldis considered not coupled with shifts field. In this waytemperature field can be defined prior to determination ofdeformation field.The topic of the consideration will be the problem ofmodelling the temperature field occurring in the sphericalreservoir wall in the course of loading the reservoir withliquefied gas. It is characterized by considerably lowertemperature than the one of the reservoir wall. The vesselwall is supposed to be heated at initial temperature =50°C, at the beginning, due to any environment conditions.Liquefied gas, cooled to a storing temperature =–50 °C isconveyed into the reservoir. The reservoir wall is exposedto temperature changes caused by cooling during thetransient process. Cooling wall rate in the lower part of thevessel, i.e. the place where it is in contact with the liquid, isconsiderably higher compared to the cooling wall rate in thearea above the liquid surface, where the gas phase is present.This occurs due to high coefficient of heat transfer from thewall to the liquid compared to the amount of thecorresponding heat transfer coefficient from the wall to thegas phase i.e. Thus the temperature of the reservoirwall varies per meridian length. It can be expected that walltemperature bears high increase in the wall narrow bandabove the liquid surface. While the liquid level increases,this temperature increase rate decreases in the course oftime. Its dependence on the level height will be determinedin further consideration.The spherical reservoir can be seen in Fig. 1. Whilereservoir loading the reached liquid level height in the fixedtime moment is designated by a meridian angleThe reservoir is supposed to be well thermally insulated.According to [3], the value of the Biot number for the part ofthe wall of the thin-walled vessel located above the liquidlevel is very small, Therefore, it is reasonable toassume that the temperature in this wall is approximatelyuniformly distributed along the wall thickness at any timeduring the transient process, so the Lumped capacitancemethod can be used to determine its temperature.In the area where the wall is located bellow the liquidlevel, the Biot number attains very high values. In this casethe difference between the temperature of the inner surfaceof the wall and the liquid is small, As we are interested in the temperature of the inner surface only, the changeof the temperature depending on the wall thickness will notbe analized. Note that due to small wall thickness with goodinsulation and high thermal conductivity, the temperature ofthe steel wall becomes uniform shortly after sinking.->> .= = .<<1.. -2.Modelling of a temperature fieldModeliranje temperaturnog poljaTTϑ ϑ0Hα α1 20 0t tBi32Temperature transitional phenomena in spherical reservoir wall P. Baličević, Ž. Ivandić, D. KraljevićTehni ki vjesnikč , 3117, 1(2010) -34Initially, when the reservoir is still empty, the walltemperature is uniform in the whole superficial area of thevessel and equal to the environment temperature . Thereservoir loading rate by the cooled liquid was determinedby the volume flowrate . Time is required to pass so as toload the reservoir of radius with liquid to the level givenby the meridian angle :TQ tRϑ0where is a part of the liquid-filled reservoir volume.The reservoir wall was getting cool in the course of thisperiod. Taking into account that the outer side of thereservoir is insulated, heat loss via insulation can beneglected. Due to high heat transfer coefficient, the walltemperature on the liquid-touched part is somewhat higherthan that of the liquid itself. However, the wall gets coolconsiderably slower above the liquid surface. Itstemperature in this area can be determined by using out heatbalance equation. If a heat balance is considered for the wallpart above the liquid surface, that has temperature , thatwithin the short time interval d , gets cool by the rate d , isexpressed as:VTt TSSHS TT  coscos32333 		QπRQVt (1)SHS d TdcρhFtTTFα 	 (2)where designates heat transfer coefficient,  wall materialdensity, specific wall heat capacity,    wall thickness,wall area above the liquid surface.Integrating of equation (2) leads to the expression:αc h Fwhere denotes loading time determined by the expression(1). Attained expression shows that the wall temperatureabove the liquid surface decreases in the course of time bythe exponential act. Thus, according to [4], a time constantcan be introduced:tτ1 tρhcαTTTT H0HS e	 (3)αρhcτ	1 . (4)By the expression (1), at , time required for the reservoirto be fully loaded is as follows:ϑ=0QRτ	34 3V. (5)If a volume flowrate value determined by this expressionis implied in the expression (1) and ratio is introduced we can get the expression for the reservoir loadingtime at the level given by the angleQϑ-:01Vk ττ	03010 coscos324k  	τt . (6)Value of the ratio can be calculated if geometrical physicalparameters are known or determined experimentally bymeasuring time constant and time required for loading thek33Technical Gazette , 3117, 1(2010) -34P. Baličević, Ž. Ivandić, D. Kraljević Temperaturne prijelazne pojave u stijenci sfernog spremnikafull reservoir .After the reservoir loading time is calculated by theexpression (6) to the given level, the wall temperatureincrease rate can be determined by the expression (3) at thereached liquid level in a certain moment:τVwhere designates heat generation rate, Laplace operator and thermal diffusivity.Due to the spherical shape of the vessel wall theequation (9) is the most suitable for use in the sphericalcoordinate system. Since the temperature field is axi-symmetric and unchangeable per wall thickness, Laplaceoperator is reduced to the form:-= /( )Δa cλ ρ·  1H0HSS eτtTTTTT		 (7)as well as the sub-temperature of the reservoir wall part(compared to the referential temperature) located aboveliquid surface: 	10H0S e1τtTTTT . (8)Wall temperature and temperature differences defined bythe expressions (7) and (8) are graphically presented in Fig2.Heat rejected by conducting through the reservoir wallinto the meridian direction was neglected in the equationsetting (2). This is justified if there is no temperaturegradient in the wall, which is achieved for the part of thewall further from the liquid level. In the narrow area abovethe liquid level the wall experiences a prompt temperatureincrease in the meridian direction. This temperaturegradient can be determined using the Fourier differentialequation of heat diffusion through isotropic body, accordingto [5]. This equation gives:Figure 2Slika 2.a) Change of temperature differences in the course of time, b) Wall temperature increase at fixed times.a) Promjena temeraturnih razlika tijekom vremena, b) Porast temperature u stijenci u fiksiranim trenucima.Heat generation rate in the expression (9) presents heatsupplied from the outside to the wall element per volumeunit. It, in this example, refers to heat alternating on theinternal wall:ρcqTatT	 iSS(9)iq	sinsin12R. (10) SHi TThαq 	 . (11)By using the operator (10) and implementingexpression (11) into (9), Fourier's equation is as follows:	tan1λ S2S22STTRρctT SH TThρcα(12)/)=1.Values of the temperature gradient in the wallat the place of the liquid level are determined using thenumerical methods of solving differential equation (12).These values are calculated in [3] and are being used furtherto approximate the function of the wall temperatureincrease.Of all possible states in the course of the liquid levelincrease, a calculation has been conducted for five chosenlevels. Values of the ratio were determined by theexpression (6) and substituted into expression (7) for fixedliquid level heights (given by the angle and withexpected value In this way pertaining temperaturedifferences , representing temperature increase of theinternal wall surface at fixed liquid level heights, werecalculated. Their values are presented in Table 1. They wereused for drawing approximate temperature distributionfunctions per reservoir meridian length and presented inFig. 2b). Temperature distribution curves were designatedby the circled numbers. They respond to timing order ofsetting up corresponding liquid level. For example, a curvedesignated no. 1 in Fig. 1 represents approximatetemperature distribution of internal wall surface along thereservoir meridian at the moment when the liquid level3Approximation of the wall temperature increaseAproksimacija porasta temperature stjenketkT0 10SτϑΔ ST.34change function per reservoir meridian length, in reality,does not experience jump as in Fig. 2b, but it variescontinuously as in Fig 3. Gradient of the function should bedetermined either by experimental methods, measuringwall temperature in the critical area or by numericalmethods of solving differential equation (12). Taking intoaccount the above mentioned it is enough to find out whattemperature gradient values belong to certain temperaturejumps i.e. determine dependence . Thesevalues do not significantly depend either on liquid type orlevel height they are measured at but only on the vessel wallthickness. This results from the assumption .Furthermore, spherical reservoirs usually have highcapacity (e.g. from 1000 to 5000 m ), thus, while beingloaded the liquid level increases relatively slowly. This isthe reason why the wall temperature starts to rise just abovethe liquid level, as it is shown in Fig. 3.Determined temperature distribution functions alongthe reservoir meridian were approximated by Fourier'sorders of forms (13), whose coefficients can be seen in Table1. The functions are used as a basis for further temperaturestresses analysis that will be presented in the next paper.d( )/d = ( )>>Δ ΔT f Tϑ S1 2α α35ReferencesReference[1] Chadwick, P. Thermoelasticity The dynamical theory, I.N.Smeddon & R. Hill, Progress in Soled Mechanics, North-Holl. Publ. Co,Amsterdam, 1964., Zagreb,1999.[5] Incropera, F. P.; DeWitt, D. P. Fundamentals of Heat Transfer,John Wiley and Sons, NewYork, 1981.[7] Reynolds, W. C. Thermodynamics, McGraw – Hill BookCompany, NewYork, 1968.[2] Bošnjaković, F. Nauka o toplini II, Tehnička knjiga, Zagreb,1976.[3] Baličević, P. Analiza temperaturnih naprezanja utankostjenim spremnicima, Magistarski rad, FSB[4] Galović,A. Nauka o toplini II, FSB, Zagreb, 1997.[6] Koвaленко, A. Д. Основы термоупругости, Науковадумка, Киев, 1970.Temperature transitional phenomena in spherical reservoir wallTehni ki vjesnikč , 3117, 1(2010) -34P. Baličević, Ž. Ivandić, D. Kraljevićreaches the height given by the angle . Temperatureincrease presented as a discontinuity by this diagram occursat a certain small meridian length in real situation. Due tothe aforesaid these discontinuous functions areapproximated by Fourier's orders in form:=150°ϑ04ConclusionZaključakUnderstanding of non-stationary temperature field inthe reservoir wall allows temperature stresses calculation[6]. The method presented provides values of walltemperature increase in a parallel circle area with the liquidlevel. However, to analyse stress, it is not enough to knowonly the wall temperature increase ratio since temperaturestresses depend on a temperature gradient. Temperature...2cosDcosDD 2100 	 TT (13)Fourier's orders were selected for approximation of thetemperature distribution functions for mathematicalreasons. Namely, such substitution enables obtainingexplicit solution of corresponding differential equation forcalculating temperature stresses. Coefficients of theFourier's orders are calculated by means of the electroniccalculator program. They can be seen in Table 1.Figure 3 shows graphical representation of temperaturedistribution function approximation at the moment whenthe liquid level reaches the height of .=150°ϑ0Table 1Tablica 1Coefficients of Fourier's order for temperature distribution. Koeficijenti Fourierovog reda za funkcije raspodjeletemperature0 50 70 100 130 150Ts 40 °C 48 °C 69 °C 92 °C 99 °CD0 –90,444 –83,466 –67,800 –42,759 –28,500D1 17,256 26,858 43,180 53,716 47,525D2 12,381 12,437 4,359 –19,470 –29,144D3 6,229 –1,023 –12,542 –7,200 9,128D4 0,801 –6,586 –3,710 12,300 4,209D5 –2,496 –4,180 5,700 –3,062 –8,095D6 –3,314 0,900 2,847 –4,925 5,310D7 –2,301 3,367 –2,673 4,468 –0,764D8 –0,593 1,994 –1,994 0,255 –2,038D10 0,786 –0,732 1,182 –2,670 2,294Figure 3 Sub-temperature of the internal wall surface at liquid'slevel of ϑ. Podtemperatura unutarnje površine stijenke pri razinikapljevine ϑ00=150°=150°Slika 3Authors' addressesAdrese autoraDoc. dr. sc.Prof. dr. sc.Pavo BaličevićŽeljko IvandićJ  J  Strossmayer University of OsijekFaculty of AgricultureTrg Sv. Trojstva 331000 OsijekJ  J  Strossmayer University of OsijekFaculty of Mechanical Engineering. ., Croatia. .Trg Ivane Brlić Mažuranić 235000 Slavonski Brod, CroatiaMr. sc. Drago KraljevićJ  J  Strossmayer University of OsijekFaculty of AgricultureTrg Sv. Trojstva 331000 Osijek. ., Croatia

Temperaturne prijelazne pojave u stijenci sfernog spremnika

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