MODELLING SLOSHING IN LNG TANKS

Transcription

1 Seventh Internatonal Conference on CFD n the Mnerals and Process Industres CSIRO, Melbourne, Australa 9-11 December 2009 MODELLING SLOSHING IN LNG TANKS Murray RUDMAN and Paul W. CLEARY CSIRO Mathematcal and Informaton Scences, Prvate bag 33 Clayton South, Vctora 3169, AUSTRALIA Correspondng author, E-mal address: Murray.Rudman@csro.au ABSTRACT The Smoothed Partcle Hydrodynamcs (SPH) method s appled to the problem of modellng sloshng n a twodmensonal water model that s a representaton of a scaled LNG tank. One confguraton, whch s a transverse slce through a membrane type tank, s studed here. Two fll levels (20, 70%) and two dfferent oscllaton ampltudes (10 and 20% of the tank dmenson) are consdered. Predcted pressure sgnals are compared to expermental measurements. The peak pressure values predcted n the smulatons are generally lower than the expermental values, although they are the correct order of magntude. The stochastc nature of the oscllatons n practce means that an exact match between smulaton and experment s not feasble. Ensemble averaged pressure traces and standard devatons of the pressure from the smulaton results are also presented. They show generally hgher varablty n the pressure sgnals for low fll rato compared to hgh fll. The magntude of fluctuatons s also senstve to the sensor locaton. SPH provdes results for the peak pressures that are the correct order of magntude on average, although the hghest peaks are under-predcted. It s a natural technque for such coupled flud-structure problems wth large free surface deformatons. INTRODUCTION Sloshng n partally flled LNG tanks can arse under dfferent ocean wave condtons. Examples nclude when loadng and/or unloadng LNG from a tanker shp, when tankers must dsengage part way through offshore loadng due to adverse weather condtons or n FPSO s under normal operatng condtons. Sloshng may resonate wth structural frequences and those of wave-nduced shp motons. Ths can subsequently affect shp stablty and, of partcular mportance here, can produce large loads on the nternal tank membranes. In turn ths can lead to structural damage to tank membranes and nsulaton, leakage and potentally to tank rupture. Small-scale physcal experments can be undertaken wth water or other fluds and then scaled up to full sze to predct tank loadngs under dfferent assumed wave condtons and fll levels. However t s hghly desrable to have robust computatonal tools that provde accurate estmates of loadngs under dfferent condtons. Not only do computatonal methods allow a quck turn around for nvestgatng dfferent tank geometres, wave condtons and fll levels, t s also possble to apply the correct equatons of state for lqud LNG when modellng. The use of LNG n experments s problematc due to the dffculty n makng measurement at very low temperatures as well as the sgnfcant safety ssues when usng lqud LNG. The non-lnearty of flud motons when the forcng ampltude becomes large combned wth the possblty of free surface overturnng, fragmentaton and entranment of a gaseous phase all ndcate that smplfed computatonal approaches (e.g. potental flow methods) are nadequate n the general sloshng case. Thus an approprate numercal method must be able to handle arbtrary, complex free surface behavour. There are two man classes of methods able to handle such complexty: Interface capturng technques (examples are Volume-of-Flud (VOF) orgnally developed by Hrt and Nchols (1981) and Level Set methods, Sussman et al. (1994)); and Smoothed Partcle Hydrodynamcs (SPH) orgnally developed by Gngold and Monaghan (1977) and then extended to free surface ncompressble flows by Monaghan (1994). An ncreasng number of numercal nvestgatons of sloshng have appeared n recent years. A selecton of these are the papers of Wemmenhove et al. (2007), Jung et al. (2008), Schreer and Paschen (2008), Sngh et al (2008) and von Berghem and Thagaraan (2008). The maorty of these studes use ether commercal software (CFD/Fluent/MSc.Dytran) coupled to a VOF technque or purpose bult VOF codes such as the ComFlow code (Gerrts and Veldman, 2003). In ths paper we apply the Smoothed Partcle Hydrodynamcs technque to model the two-dmensonal sloshng systems specfed n Km et al. (2009). These systems were thn-slce scaled water models that were nstrumented to allow measurement of pressure sgnals n the tank and a number of dfferent locatons. SPH s a computatonal technque that has been wdely appled to ndustral and envronmental flows (e.g. Cleary 1998, Cleary and Prakash 2004). It has more recently been appled to oceanc and offshore hydrodynamcs (see for example Gomez-Gestera 2005, Shao 2006, Cleary and Rudman 2009). MODEL DESCRIPTION Unlke most numercal technques for Computatonal Flud Dynamcs, SPH does not utlze a fxed nodal grd. Instead, the grd s replaced by a set of movng ponts (or partcles ) on whch the dscretsed equatons are solved. Each partcle carres mass, momentum and energy and moves wth the local flud velocty. There s no explct connectvty of the partcles whch means, for example, that partcles that are close neghbours at one nstant n Copyrght 2009 CSIRO Australa 1

2 tme can be qute dstant from each other at a later tme. Also, because the partcles are transported wth the local flud velocty, the non-lnear terms that usually appear n the momentum equatons n grd-based methods are replaced by tme dervates followng the partcle moton. A detaled descrpton of the method can be found n Monaghan (1994) and an outlne of the mplementaton for sloshng s provded n Rudman et al. (2009). A mnmal descrpton of the method s ncluded here for completeness. Interpolaton SPH nterpolaton allows a contnuous functon of the spatal varables to be defned as an nterpolant of the dscrete values at the partcle postons. Ths s approxmated by a summaton over the nearby partcles as: F F ( r) = m W ( r r, h) (1) ρ Here m s the mass of partcle, F the assocated dscrete value of the feld F, and ρ the partcle densty calculated from a transport equaton descrbed below. The smoothng length, h, s chosen such that the ntegraton s performed over a volume that has a radus larger than the mean partcle spacng by a factor of two or more. The dscrete form of the spatal dervatve of an nterpolated quantty needed n the equatons of moton s: F F ( r) = m W ( r r, h). (2) ρ The choce of nterpolaton kernel W(r-r,h) s dscussed n Monaghan (1992). Equatons of Moton The equaton for mass transport s wrtten n SPH form as dr v v = v m W (3) dt 0.5( ρ + ρ ) (where W =W(r -r,h)). The partcle densty ρ s calculated from the transport equaton: d ρ = m ( v v ) W (4) dt The equaton for momentum transport s wrtten d dt P P v = m + + Π W (5) 2 2 ρ ρ where Π s the vscous term: Π = ρ ρ ξ 4. (6) μ μ v r ( μ + μ ) r r Here r = r -r, v = v -v and ξ s a factor that e vares wth dmenson and the detals of the kernel (see Cleary, 1998 for detals). SPH utlzes an equaton of state to defne the pressure and here we use the stff equaton of state ρ P = P0 1 ρ. (7) o wth γ chosen here to be equal to 7. Ths choce results n densty varatons n the flud that are typcally less than 1% of the flud densty. The Courant number (defned γ usng the speed of sound and mean partcle spacng) s 0.5. Based on typcal flud flow veloctes ths corresponds to a Courant number of Tme-steppng s by use of a second-order Improved Euler scheme. Convergence of the method wth changes n partcle spacng has not been addressed for ths confguraton, however prevous work (Monaghan 1992) ndcates that the method s 2 nd order n space. Pressure predctons The walls of the LNG tank are dscretsed usng a specal set of SPH boundary partcles whose postons are fxed relatve to each other. Ths ensemble of partcles prescrbes a path defned by the desred tank moton (roll, ptch, sway, etc). The boundary partcles repel any flud partcles that approach them usng a Lennard-Jones potental (see Cleary 1998). The pressure on the tank wall s determned by calculatng the normal force appled to the boundary partcles and dvdng by the partcle s equvalent surface area. Because the boundary force s only ever repulsve, when flud partcles move away from the zone of nfluence of the wall, they apply no force to the wall. Importantly, n cases where there could be sucton on the wall and hence negatve pressure, the SPH wall force wll be calculated as zero, as wll the wall pressure. Thus t s seen below that the SPH pressure sensor values never become negatve, unlke the expermentally measured values. Ths aspect of the modellng requres mprovement n future to allow negatve pressures to be predcted when applcable. Computatonal detals Detals of the tank geometry and sensor locatons can be found n Km et al. (2009) and wll not be repeated here. Schematcally, the tank confguraton s shown n Fgure 1 and s the octagonal or transverse model of Km et al. It s an approxmaton of a 2-D slce through a scaled 3D membrane type LNG tank. The mean partcle spacng n the smulatons here was chosen to be h = 10 mm and materal parameters are those for water. In ths prelmnary study, the partcle spacng was not vared to check on the convergence of the method wth resoluton, and ths aspect of the smulatons must be addressed n the future Fgure 1: Schematc of tank geometry and sensor locatons. The arrow ndcates the drecton of ntal tank moton. The tanks s oscllated n the x-drecton (mmckng sway moton n a real LNG carrer) wth the x-component of the tank velocty s gven by v = 2π f Asn(2π f t). (8) Copyrght 2009 CSIRO Australa 2

3 Here, f and A are the oscllaton frequency and moton ampltude (n metres). Detals are gven n Table 1 where t s noted that the two dfferent ampltudes for tank correspond to 10 and 20% of the tank dmenson n the drecton of oscllaton. Table 1 Parameters for smulatons A (metres) %Fll f (Hz) and Tme-averagng was used durng pressure data collecton wth a tme-scale equal to approxmately three acoustc tme-scales (τ=h/c S ) of the method. The equvalent smulaton data acquston frequency s approxmately 1 khz, whch s lower than the expermental measurement frequency (20 khz), hence we expect to mss the larger peaks that can be detected n the experments. Ths sgnal stll contaned a level of nose and was further smoothed wth one pass of an 11-pont runnng average flter n tme. In addton to the tme varyng pressure sgnals, an ensemble-averaged (or phase-averaged) pressure sgnal was calculated from all data collected after the frst fve oscllaton perods. Ths average llustrates the general behavour of the pressure although n t wll not reveal the maxmum peak values that wll be smoothed by averagng. From ths ensemble average, a standard devaton of the pressure trace was also calculated. Lke the ensemble average, ths s a tme-trace over one perod, wth the devaton at any tme n the phase calculated wth respect to the ensemble average at that tme. The sloshng system consdered here contans a sgnfcant stochastc component and we do not expect to get perfect agreement between experment and smulaton. However the ensemble-average pressure and standard devaton provde statstcal nformaton that could be used to compare smulaton and experment n a statstcal way. Expermental pressure traces At the tme of wrtng, the expermental data was unavalable for drect comparson. A select number of expermental pressure traces were manually dgtzed from the plots presented n Km et al. (2009) n order to compare the wth the smulaton results. As a consequence, they are mprecse. However, they correctly show the phase of the expermentally measured sgnal and the maxmum pressures measured as well as provdng an approxmately correct shape. RESULTS 20% fll level Sequences of flud dstrbuton for the two dfferent ampltude oscllatons (151 and 303 mm) for one half of a typcal perod are shown n Fgure 2 for the 20% fll level. Start-up pressure sensor readngs for sensor #9 are compared to expermental traces n Fgure 3. Ensembleaveraged pressure traces and the standard devaton of the ensemble for sensors #9 and 10 are shown n Fgure 4. (Note that sensors #5, 6, 9 and 10 were the only sensors used n the experments and hence smulatons.) The behavour for the two dfferent ampltudes wth 20% fll are qualtatvely smlar (Fgure 2). At maxmum horzontal dsplacement to the left (top row) the flud on the bottom of the tank has mpacted the left-hand wall and has ust started formng a vertcal et (151 mm ampltude) or the et s well formed and s approachng the flet on the tank wall (303 mm mpact). Note that the acceleraton toward the rght s maxmum at ths tme. The pressure of mpact combned wth a near maxmum n wall acceleraton leads to the peak pressure occurrng at ths phase of the oscllaton. As the tank moves to the rght, the acceleraton reduces and ths et starts to collapse. The pressure at sensors #9 and #10 then begns to drop. At around the pont of zero mean dsplacement (and maxmum velocty), the collapsng column forms a small overturnng wave before mpactng on the rght-hand wall (not shown) and the sequence s repeated. Fgure 2: Flud partcle postons for 20% fll (151 mm ampltude left column and 303 mm rght column). The frst half of the oscllaton perod s shown as the tank moves from left to rght. The maxmum velocty (red) s assocated wth the wave tp and s equal to 2.25 ms -1 for 151 mm ampltude and 3.0 ms -1 for 303 mm ampltude. For the low ampltude oscllaton (151 mm), the flud only occasonally reaches sensors #5 and 6 (on the lower sde of the upper flet) and only ust reaches them regularly for the hgh ampltude case (303 mm) nducng pressures of the order 1 kpa, hence detals of these traces are not dscussed. Despte the strct perodcty of the tank oscllaton, the predcted pressure peak vares sgnfcantly from perod to perod. Ths can be seen n the traces shown n Fgure 3 and the ensemble-averaged sgnals n Fgure 4. The pressure sgnal for ths case s qute nosy, wth pressure maxma at locaton #9 typcally around the 3-5 kpa (4-8 kpa) for 151 mm (303 mm) ampltude. The peak pressures predcted by SPH are n reasonable agreement wth those measured expermentally, although the stochastc nature of the sloshng results n predctons for ndvdual peaks that have dfferent magntude to the Copyrght 2009 CSIRO Australa 3

4 measurements. For the low ampltude oscllaton, the predcted peaks are approxmately 10-20% lower than the measurements. For the hgh ampltude the predcted are generally a lttle lower than measured except for the frst 3-4 peaks. The phase of the pressure peak s generally n good agreement wth the experment although there are some dscrepances. The most notceable dfference between smulaton and experment s the absence of negatve pressures n the smulaton results. The reasons for ths have been dscussed above n the secton on the computatonal method. 2 kpa for sensor #9 and 5.5 kpa for #10 both oscllaton ampltudes. The standard devaton n the peak value s predcted to be 60-70% of the mean, ndcatng hgh varablty. Hgh devaton before the tme of mean peak pressure s suggestve of varaton n the tmng of the flud mpact whereas hgh devaton wth a narrow peak suggests hgh varaton n peak magntude. The maxmum devaton occurs around the tme of the mean peak pressure and drops rapdly after the mean peak (to approxmately 30% of the mean pressure), suggestng that the pressure reducton after the peak follows a smlar pattern after each mpact. Ths s consstent wth the shape of the pressure traces seen n Fgure 3. 70% fll level Results for the case of 70% fll n the transverse tank are presented n Fgure 5 to Fgure 7. Data was collected at all sensors #1-8. As wth the 20% fll level, the results for the two dfferent ampltudes are qualtatvely qute smlar (Fgure 5). As the tank reaches maxmum dsplacement to the left, the flud has ether ust mpacted the tank wall (151 mm ampltude), or has ust formed a et that moves along the upper flet (303 mm) (frst row n Fgure 5). Fgure 3: Pressure sgnal at start-up for at sensor locatons #9 for 20% fll level n the transverse tank: 151 mm ampltude (top) and 303 mm ampltude (bottom). SPH results (sold lne) are compared to expermental results (dashed lne). Fgure 5: Flud partcle postons for the 70% fll level (wth the 151 mm ampltude n the left column and the 303 mm ampltude n the rght column). The frst half of the oscllaton perod s shown as the tank moves from left to rght. The maxmum velocty (red) s assocated wth the et and breakng wave tp and s 4.0 ms -1 for 151 mm and 5.4 ms -1 for 303 mm. Fgure 4: Ensemble averaged pressure trace and standard devaton for sensors #9 and 10 for the transverse tank wth 20% fll: Sold lne s the ensemble-averaged pressure trace and the dashed lne s the standard devaton. Top s 151 mm ampltude (top) and 303 mm ampltude (bottom). The ensemble average of these sgnals shown n Fgure 4 ndcates a predcted mean peak pressure of order As the tank starts acceleratng back to the rght, the et that has formed on the left-hand wall soon mpacts the tank celng because of the hgh fll level. As the tank contnues to move to the rght, addtonal flud bulds up on the left-hand wall and the hgh pressure at the tank celng creates a et that starts to move along the celng, however t soon falls under gravty. The general pcture s smlar to that of a breakng shore wave. For the 151 mm ampltude oscllaton, the breakng wave mpacts the surface of the flud n the tank at about the ¾ of maxmum Copyrght 2009 CSIRO Australa 4

5 ampltude to the rght (last row n Fgure 5) whereas for the 303 mm ampltude oscllaton t mpacts the opposte sde wall of the tank at the locaton of the lqud surface. Fgure 6: Pressure sgnal at start-up for at sensor locatons #5 for 70% fll level n the transverse tank: 151 mm ampltude (top) and 303 mm ampltude (bottom). SPH results (sold lne) are compared to expermental results (dashed lne). Fgure 7: Ensemble averaged pressure trace and standard devaton for sensors #2, 5 and 6 for the transverse tank wth 70% fll: Sold lne s ensemble-averaged pressure trace and the dashed lne s the standard devaton. Left s 151 mm ampltude and rght s 303 mm ampltude. The et that forms on the left sde of the tank as t comes to rest s assocated wth the hgh pressure seen n the ensemble-averaged pressure readngs for sensor #5 (see Fgure 7). The pressure peaks for sensor #5 (shown n Fgure 6) appear to be generally under-predcted by about 25% for the 151 mm ampltude sloshng and overpredcted by about the same amount for the 303 mm ampltude. The maxmum predcted ensemble-averaged pressures occur at sensor #6 on the sde wall wth values of 10 kpa (23 kpa) for 151 mm (303 mm) ampltude. Ths peak s assocated wth the vertcal wall et beng forced to change drecton between sensors #7 and #6. The second hghest mean readng s predcted to occur at sensor #2 at the celng ust around from the upper fllet, wth mean values of 9 kpa (21 kpa) for 151 mm (303 mm) ampltude. Agan, ths s assocated wth the angled wall et beng forced horzontal as t hts the tank celng. The maxmum standard devaton n pressure occurs at sensor #2 and s approxmately 40% of the mean peak pressure and occurs ust before the tme of mean mpact. At sensor #6 the standard devaton s about 30% of the mean peak pressure. At both these sensors, the standard devaton has dropped by one half by the tme of the mean peak pressure, suggestng that a sgnfcant part of the sgnal varablty s due to the tmng of the mpact. Comparson between 20% and 70% fll levels The two dfferent fll levels have dfferent oscllaton frequences and should therefore be compared wth cauton. Usng the maxmum observed smulaton velocty, V, and the nomnal heght, H, from the free surface to the celng, for the 20% fll level the dstance from the free surface to the celng s greater than V 2 /2gH for both ampltude oscllatons and thus we would not expect roof mpacts as predcted n Fgure 2. On the other hand, for the 70% fll level ths dstance s sgnfcantly smaller and we expect sgnfcant celng mpact. Indeed ths s the case as seen n Fgure 5. The peak pressures should scale lke the maxmum velocty squared. Consderng the tme traces n Fgure 3 and Fgure 6, the peak values correspond approxmately to ths scalng rato. However ths s not apparent n the maxmum ensemble averaged pressure values n whch the peak pressures are averaged out due to varablty n tmng as well as magntude. The varablty n the pressure sgnal s predcted to be hgher for the lower fll level, wth the standard devaton n mean peak pressure beng approxmately 60-70% of the mean peak for low fll case compared to 40% n the hgher fll case. CONCLUSION SPH s seen to capture the bascs of lqud sloshng n two-dmensonal water models of an LNG tank. The pressure sgnals are seen to be quas-perodc, but the flud surface changes wth each cycle and results n pressure peaks that can vary substantally between mpacts. Comparson of the pressure traces from SPH and some of those presented n Km et al. (2009) show that the phase of the pressure sgnals s n qute good agreement wth the experments (whch s to be expected), but that SPH generally under-predcts the peak pressure values by 20-30%, although n also over-predcts n some cases by smlar amounts. The free-surface profles were n reasonable qualtatve agreement for many of the dfferent sloshng cases (not shown), although entranment of bubbles n the experments (and ther absence n the modellng) makes a good comparson dffcult. Copyrght 2009 CSIRO Australa 5

6 Suggested n ths work s that the generaton of hgh wall pressure s related to the mpngement of lqud ets on the tank walls and most mportantly by wall ets encounterng a sudden change n geometry (e.g. a tank knuckle or corner). These hgh speed ets are caused by the general flud mpact arsng from sloshng, although the prmary sloshng mpacts n themselves do not necessarly result n the hghest pressures. Ths aspect of the modellng needs to be nvestgated further wth a closer dstrbuton of pressure sensors needed to better check the full dstrbuton around the tank walls. Despte the determnstc nature of the harmonc moton of the tank, the sloshng moton and consequent pressure traces contan a sgnfcant stochastc component. The orgn of the stochastc behavour s the hgh Reynolds number, free surface nature of the flow. It s unstable and hence hghly rregular at small tme and length scales. Hence, the suggeston that pressure traces resultng from mpact could be numercally predcted wth a 1:1 correspondence to expermental results s not realstc. However, what does seem feasble s that the stochastc nature of the sloshng mpacts could be captured statstcally, and the ensemble-averaged sgnals and assocated standard devatons are one way n whch a comparson could be made. Ensemble-averaged pressure sgnals from the smulaton results are presented here. They provde useful nformaton about the pressure, although the peak nformaton s lost n the process. There s seen to be hgher varablty n the peak pressure (.e. hgher standard devaton) for the 20% fll case as compared to the 70% fll. Ths hgher varablty can be caused both by hgher varablty n the peak pressures and n the exact tmng of the maxmum pressure. The predcted ensemble averaged pressure sgnals deally need to be compared to smlar expermental averages. Ths would then provde an ndcaton of whether a computatonal technque s able to capture the mportant features of sloshng, especally relevant nformaton on the maxmum pressures and ther statstcal, spatal and temporal dstrbuton. The smulatons here have only consdered the presence of the lqud phase and the gas phase has been gnored. Includng ths n the computaton s the man mprovement to the model physcs that s requred. Its ncluson wll modfy both the flud flow and predcted maxmum pressures, and s currently under development n the SPH framework. REFERENCES CLEARY, P. W., Modellng confned mult-materal heat and mass flows usng SPH, Appl. Math. Modellng, 22, , (1998). CLEARY, P.W., and PRAKASH, M., Smooth Partcle Hydrodynamcs and Dscrete Element Modellng: potental n the envronmental scences, Phlosophcal Transactons A, 362, , (2004). CLEARY, P.W. and RUDMAN, M. Extreme wave nteracton wth a floatng ol rg, Progress n Comput. Flud. Dyn (2009) GERRITS, J, and VELDMAN, A.E.P., Dynamcs of lqud flled spacecraft, J. Eng. Mathematcs (2003). GINGOLD, R.A. and MONAGHAN, J.J. Mon. Not. Roy, Ast. Soc (1977). GOMEZ-GESTEIRA, M., CERQUEIRO, D., CRESPO, C. and DALRYMPLE, R.A. Green water overtoppng analyzed wth a SPH model, Ocean Eng (2005). HIRT, C.W. and NICHOLS, B.D., Volume of flud (VOF) methods for the dynamcs of free boundares', J. Comput. Phys. 39, (1981). JUNG, J.-J., Lee, H.-H., PARK, T.-H. and LEE, Y.-W., Expermental and numercal nvestgaton nto the effects of flud-structure nteracton on the sloshng mpact loads n membrane LNG carrer, OMAE2008, 27 th Internatonal Conference on Offshore Mechancs and Arctc Engneerng 15 th -19 th June 2008, Estorl Portugal. 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Numercal predcton of sloshng loads n shp tanks, OMAE2008, 27 th Internatonal Conference on Offshore Mechancs and Arctc Engneerng 15 th -19 th June 2008, Estorl Portugal. M. SUSSMAN, P. SMEREKA and S. OSHER, A level set approach for computng solutons to ncompressble two-phase flow, J. Comput. Phys., 114, (1994) VON BERGHEIM, P and THIAGARAJAN, K.P. The ar-water sloshng problem: parametrc studes on exctaton magntude and frequency, OMAE2008, 27 th Internatonal Conference on Offshore Mechancs and Arctc Engneerng 15 th -19 th June 2008, Estorl Portugal. WEMMENHOVE, R., LUPPES, R., VELDMAN, A.E.P and BUNNIK, T., Numercal smulaton of sloshng n LNG tankes wth a compressble two-phase model, OMAE2007, 26 th Internatonal Conference on Offshore Mechancs and Arctc Engneerng 10 th -15 th June 2007, San Dego, Calforna. Copyrght 2009 CSIRO Australa 6