PVP THE DYNAMIC LOAD FACTOR OF PRESSURE VESSELS IN DEFLAGRATION EVENTS

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1 Proceedings of he ASME Pressure Vessels and Piping Division Conference PVP July 7-,, Balimore, Maryland, USA PVP-57 THE DYNAMIC LOAD FACTOR OF PRESSURE VESSELS IN DEFLAGRATION EVENTS Yu u DuPon Engineering Wilmingon, DE 985 yu.xu@usa.dupon.com Kuao-John Young DuPon Engineering Wilmingon, DE 985 k-john.young@usa.dupon.com ABSTRACT There are vessels ha could be subjeced o rapidly rising pressure load during deflagraion evens. The dynamic load facor (DLF) which accouns for dynamic load effecs in such evens can be used o bridge he gap beween ransien and saic pressure loads. This paper addresses he issues in esimaing he DLF for a deflagraion even. A unified mehodology for esimaing he DLF is firs presened. The mehodology is hen validaed wih exising references. Key parameers deermining DLFs are idenified during he validaion. Case sudies are used o illusrae how o use he mehodology. Finally, his paper is concluded wih he findings and a suggesed procedure for esimaing DLFs in deflagraion evens. INTRODUCTION DuPon plans operae many low o high pressure vessels ha may be subjeced o dynamic pressure loads due o occasional deflagraion evens []. Toal ime for pressure rise and decay in such evens can las from ens of milliseconds o a few seconds. In such cases, esimaing DLFs, which accouns for dynamic load effecs and can be used o calculae equivalen saic pressure load, is essenial for boh design and finess-for-service assessmens of hese vessels. ASME BPVC Secion VIII Division 3 [] provides some guidance on esimaing dynamic load effecs for deonaion evens in Code case 564 [3], which occur much faser han deflagraion evens, bu provides no guidance on how o evaluae dynamic load effecs in deflagraion evens. ASME BPVC Secion VIII Division Appendix H [4] provides design guidance for deflagraion evens by referring o NFPA- 69 [5] and NFPA-68 [6]. Similar guidance can also be found in ASME BVPC Secion VIII Division Annex 4.D [7]. NFPA- 68 [6] suggess ha wihou a deailed srucural dynamic analysis, a DLF of.5 should be used o cover he dynamic load effecs in a vening deflagraion even. NFPA-68 [6] provides no guidance on how o esimae such a DLF, bu indicaes ha using a DLF of.5 is conservaive. Lieraure review showed ha deonaion evens insead of deflagraion evens were he focus of research involving dynamic pressure loads in pressure vessels or piping sysems. For example, Duffey, Rodriguez and Romero [8] highlighed he difference beween he ransien impulsive and quasi-saic pressure loading in a vessel under high explosive deonaion evens; Leishear [9] sudied he hoop sresses caused by a shock wave fron in a gas-filled cylinder, and found ha he criical velociy of he shock in he cylinder plays an imporan role in relaing ransien o saic hoop sresses; Yip and Haroldsen [] applied ASME BPVC Secion VIII Division 3 Code Case 564 [3] o life assessmen of he sainless seel explosive desrucion sysem (EDS) vessels. These works provide insigh o undersanding dynamic load effecs in deonaion evens. To mee he need of accuraely esimaing he DLF in a deflagraion even, a unified mehodology has been developed a DuPon. The unified mehodology simplifies he pressurized vessel as a single degree-of-freedom (DOF) massspring sysem, assumes a rapezoid shape pressure-ime curve in deflagraion evens, and evenually relaes he DLF wih he naural frequency of he vessel breahing mode and he deflagraion pressure-ime curve in an analyical soluion. The mehodology has been validaed and applied o various vessels experiencing deflagraion evens. Sudies revealed ha he raio of he pressure rise ime o he naural period of he vessel breahing mode is he conrolling parameer deermining DLF. This paper summarizes he work done a DuPon: he developmen of he unified mehodology for esimaing he DLF in a deflagraion even; validaion of he developed mehodology; findings; case sudies; and a suggesed procedure for accuraely esimaing DLFs. Copyrigh by ASME

2 DLF FOR AN ARBITRARY TRAPEZOID ECITATION To derive an analyical soluion for esimaing DLFs in deflagraion evens, he following assumpions were made:. The vessel is simplified as a single DOF mass-spring sysem shown in Figure. Is characerisic parameer is he period of he vessel breahing mode, T n.. The deflagraion pressure-ime curve is assumed o be a rapezoid as shown in Figure a. The curve can be divided ino four regimes: rise, sep, decay and residual zones. 3. The derivaion of he DLF is limied o single DOF linear elasic vibraion response. This implies ha plasiciy, srain rae dependence, and wave propagaion are beyond he scope of he presen sudy. K M x F FIGURE : A SIMPLIFIED REPRESENTATION OF A VESSEL SUBJECTED TO DEFLAGRATION PRESSURES. The deformed vessel shape under a uniformly deflagraion pressure mos likely resembles he vessel breahing mode. Therefore, he sysem characerisic ime is assumed o be he period of he vessel breahing mode. Normalizaion is needed for mahemaical manipulaion. The exciaion load is normalized o ; ime is normalized o he sysem naural period ; and displacemen response is normalized o which is he displacemen a F he quasi-saic load. A normalized exciaion load-ime curve is shown in Figure b. F F F T T T T n T n FIGURE : AN ASSUMED DEFLAGRATION PRESSURE- TIME CURVE; THE NON-DIMENSIONAL PRESSURE TIME CURVE. For a given non-dimensional exciaion, a ypical nondimensional ransien response is shown in Figure 3. The DLF equals he raio of he sysem peak ransien response,, max o he sysem response a he quasi-saic load, i.e.. In oher words, DLF is he non-dimensional peak response shown in Figure 3. Non-dimensional response.5.5 x max /x F T T T Time FIGURE 3: A TYPICAL NON-DIMENSIONAL TRANSIENT TIME RESPONSE. Residual zone Decay zone Sep zone Rise zone Similar o he simulaed pressure-ime curve for a vening deflagraion even [], Figure 4 highlighs he load variaion in he even and differeniaes he dynamic load from he saic load. The operaing pressure load before he deflagraion even should be reaed as quasi-saic load. However, he rapid pressure rise and decay in a vening Copyrigh by ASME

3 deflagraion even should be considered as dynamic load. I is he dynamic load ha is magnified by DLF. For he dynamic load F shown in Figure 4, he equivalen quasi-saic load increase is F DLF. FIGURE 4: DYNAMIC LOAD AND STATIC LOAD IN A VENTING DEFLAGRATION EVENT. The sysem ransien he convoluion inegral [] response can be derived using x( ) F( ) g( ) d, () Load w here F ( ) is he exciaion load defined in Figure a, g( ) sin( n) m, () n F Saic Time is he sysem response o a uni impulse, and n is he sysem circular frequency. Using Equaions () and (), he non-dimensional ransien response can be derived as piece-wise funcions in differen ime regimes: rise, sep, decay and residual zones. I should be noed ha he ransien ime response in residual zone is he combined resuls of loads in rise, sep, and decay zones. Afer edious mahemaical manipulaion, he non- ransien ime response can be expressed dimensional as Compared o Equaion (3), he final analyical soluion of DLF is even more complicaed so ha a compuer program is needed for compuing DLFs. Wih he developed compuer program, he nondimensional ransien ime response, such as he one shown in Figure 3, can be easily obained for an arbirary exciaion shown in Figure. DLF can hen be deermined wih a known non-dimensional ransien ime response. RESPONSE SPECTRUM AND VALIDATION Based on he analyical soluion of he ransien ime response o an arbirary exciaion presened in he previous secion, he response specrum or shock specrum, in which he sysem peak response is expressed as a funcion of he sysem naural frequency [], was furher derived. The complexiy involving in he response specrum is ha he peak response could occur in any one of he rise, sep, decay or residual zones because of he ime lag beween he exciaion and response []. Similar ime lag was also found in finie elemen simulaion of cylinder shells subjeced o waer hammer dynamic loads [3]. Equaion (3) gives he ransien ime response in erms of displacemen o a rapezoid exciaion for each loading zone shown in Figure. Differeniaing he displacemen wih respec o ime yields he velociy. The maximum response occurs a zero velociy poins or sar/end poins of he ime regime []. Muliple zero velociy poins may exis for he ransien ime response in a ime zone. The larges value of he values a hese zero velociy poins and sar/end poins mus be used o consruc he sysem peak response. Wih his scheme, he non-dimensional peak response o a rapezoid exciaion, i.e. DLF, can be analyically expressed as a funcion of he sysem naural frequency. The derived response specrum was hen validaed by hree special cases wih exising analyical soluions. Firs, he exciaion is a sep impulse wih a consan slope fron shown in Figure 5a. The general loading case shown in Figure b can be reduced o his special case wih T,,. Accordingly, he general specrum is reduced o x () x sin ( ) sin sin( ) sin sin ( ) sin sin sin ( ) sin sin (3) max sin( ). (4) The peak response given in Equaion (4) is idenical o ha derived in Reference []. The peak response is ploed in Figure 5b as a funcion of, he raio of he load rise ime o he sysem naural period. 3 Copyrigh by ASME

4 Non-dimensional exciaion force.5.8 x max.6 x.4. Exciaion T 4 6 FIG URE 5: ECITATION A STEP IMPULSE WITH A CONSTANT SLOPE FRONT; THE RESPONSE SPECTRUM. As illusraed in Figure 5b, is he conrolling parameer for deermining DLF. As approaches, DLF becomes as large as. As is gr eaer han 5, DLF is less han.58. The pressure-ime curve in a non-vening deflagraion even may be simplified as he curve shown in Figure 5a. As shown in Figure 5b, he DLF in such an even varies depending on he raio of he pressure rise ime o he sysem naural period. i.e.. As is less han, DLF could be as large as. As is greaer han 5, dynamic load effecs become negligible. In he second case, represening he sep zone, he exciaion is a recangular impulse. The exciaion and response specrum for his case are shown in Figure 6. Non-dimensional exciaion force x max x Exciaion T FIGURE 6: ECITATION A RECTANGULAR IMPULSE; THE RESPONSE SPECTRUM. The general loading case shown in Figure b can be reduced o his special case wih T,,. Special reamens, aking limis a a zero, are needed o enforce nd in Equaion (3). For example, he response in he rise zone can be rewrien sin() as. Using Taylor s series and noing, he expression can be furher simplified as ( ) 3 ( )... 3 lim ( )... lim. 3 The response in he sep zone can be wrien as sin( ) cos[ ( )]. Taking limis furher reduces he expression o 4 Copyrigh by ASME

5 sin lim lim cos cos ( ). Using similar echniques, he responses in he decay and residual zones can also be reduced. The general specrum is hen reduced o max sin( ).5. (5).5 The peak response given in Equaion (5) was also derived in References [8, 4]. As indicaed in Figure 6b, DLF remains a a value of as long as he raio of he load period o he sysem naural period is greaer han.5. A symmeric riangular impulse, represening he rise and decay zones, is he hird case for he validaion. The general loading case shown in Figure b can be reduced o his special case wih,,. The peak response in he residual zone can hen be simplified o max sin( ) sin( ) T. (6) The same expression was also given in Reference [4]. Figure 7 shows a symmeric riangular impulse exciaion and is response specrum. The doed line in Figure 7b represens he peak response in he residual zone given in Equaion (6). All he oher lines represen peak responses in oher zones such as sep zone and decay zone. To auhors bes knowledge, here is no analyical soluion available in he lieraure for hose ime regimes. The upper envelope of all hese peak responses corresponding o differen ime zones consiues he response specrum. Two observaions can be made from Figure 7b:. DLF becomes less han.75 as he raio of he load rise ime o he sysem naural period is greaer han 3;. The larges value of DLF is abou.5 which occurs a he raio of.453. Non-dimensional exciaion force x max x.5 Exciaion T FIGURE 7: ECITATION A SYMMETRIC TRIANGULAR IMPULSE; THE RESPONSE SPECTRUM. Figure 7b helps gain insigh o undersanding dynamic load effecs in a vening deflagraion even. The pressure-ime curve in such an even may be simplified as he one shown in Figure 7a. Corresponding DLF varies depending on he raio of he pressure rise ime o he sysem naural period. i.e.. As is less han, DLF could be as large as.5, which is coincidenally he same as he DLF suggesed by NFPA-68 [6]. As is greaer han 3, dynamic load effecs become negligible. The validaion helped build confidence in he derived analyical soluion, given in Equaion (3), of he ransien ime response o a rapezoid exciaion. More imporanly, i helped idenify wo key parameers in deermining DLF: he shape of pressure-ime curve; he raio of he load rise ime o he sysem naural period. As his raio is greaer han 3, dynamic load effecs mos likely become insignifican. CASE STUDIES The firs case sudy is a possible deflagraion even in a high pressure vessel. In he deflagraion even, he inernal pressure rises o 93% of he operaing pressure in less han 4 5 Copyrigh by ASME

6 milliseconds and hen drops back o he operaing pressure wihin milliseconds. To esimae DLF for he deflagraion even, he sysem breahing mode naural period is needed. A simplified axisymmeric finie elemen (FE) model was employed o calculae he naural frequency of he vessel breahing mode. The FE analysis consiss of wo analysis seps. In he firs analysis sep, he vessel is pressurized o he operaing pressure saically. Frequency exracion (eigenvalue analysis) is hen carried ou in he subsequen sep. The pressure-ime curve in he deflagraion even can be normalized as shown in Figure 8. Non-dimensional response Non-dimensional exciaion force.5 Normalized Simplified Pressure-Time Curve 4 FIGURE 8: THE NORMALIZED PRESSURE-TIME CURVE IN A POSSIBLE VENTING DEFLAGRATION EVENT IN A HIGH PRESSURE VESSEL. The raio of he load rise ime o he sysem breahing mode naural period is.36. Using he compuer program described in he previous secion, he ransien ime response can be obained and ploed as shown in Figure 9. The DLF in he deflagraion even is esimaed as.9 based on he ransien ime response shown in Figure 9. If operaing pressure is denoed as, hen he peak pressure.93 p in he deflagraion even can be denoed as op. Hence, he.93p dynamic pressure increase is op, and he equivalen saic peak pressure in he deflagraion even is.936 p op, i.e. p op.93p op pop. Apparenly, he dynamic load effec is negligible in his even. p op FIGURE 9: THE NON-DIMENSIONAL TRANSIENT TIME RESPONSE IN THE DEFLAGRATION EVENT BASED ON THE SINGLE DOF SYSTEM. To furher confirm ha he dynamic load effec is negligible in he deflagraion even, boh an explici dynamic analysis wih he acual pressure-ime curve and an implici saic analysis were carried ou using ABAQUS/Explici and ABAQUS/Sandard, respecively. Theoreically speaking, explici dynamic analyses can capure all dynamic load effecs including wave propagaion. The explici dynamic analysis showed ha he sysem kineic energy is negligible compared o he sysem srain energy. Comparison beween he explici dynamic analysis and he implici saic analysis demonsraed ha he difference beween wo analyses is absoluely negligible. The second case sudy is he deflagraion even ha occurred in a pressure separaor []. The simulaed pressureime curve is close o a riangle and can be simplified as he curve shown in Figure, in which he pressure has been normalized. In he case sudy, he operaing pressure is assumed o be,6 psi and he peak pressure in he deflagraion even is assumed o be 7, psi. Non-dimensional pressure.5 4 Time (second) FIGURE : A SIMULATED PRESSURE-TIME CURVE IN AN OCCURRED DEFLAGRATION EVENT. 6 Copyrigh by ASME

7 Wihou knowing he naural frequency of he separaor breahing mode, he case sudy here is inended for illusraing how o esimae he DLF using he newly developed mehodology, and he impac of he variaion of he naural frequency of he sysem breahing mode on DLFs. If he naural frequency of he separaor breahing mode is 4 Hz, hen he normalized pressure-ime curve and non-dimensional ransien ime response are as ploed in Figure. Non-dimensional pressure.5 Non-dimensional response Non-dimensional pressure FIGURE : Tn.5 Second, 5.48 THE NON- DIMENSIONAL PRESSURE-TIME CURVE; THE NON- DIMENSIONAL TRANSIENT TIME RESPONSE. The non-dimensional peak response shown in Figure b is.36, i.e. DLF. The equivalen saic peak pressure is 7,58 psi, i.e.,6 4,4.36 7,58. If he naural frequency of he separaor breahing mode is.4 Hz, hen he normalized pressure-ime curve and non-dimensional ransien ime response are as ploed in Figure. The non-dimensional peak response shown in Figure b is.476, i.e. DLF. The equivalen saic peak pressure is esimaed as 9,94 psi, i.e.,6 4, ,94. Non-dimensional response 3 3 FIGURE : Tn.5 Second,.54 THE NON- DIMENSIONAL PRESSURE-TIME CURVE; THE NON- DIMENSIONAL TRANSIENT TIME RESPONSE. The response specrum corresponding o he deflagraion pressure-ime curve shown in Figure is obained and ploed in Figure 3. DLF.5.5 T 4 FIGURE 3: THE RESPONSE SPECTRUM FOR THE DEFLAGRATION IN THE SEPARATOR. 7 Copyrigh by ASME

8 As found in he previous secion, he raio of he pressure rise ime o he naural period of he separaor breahing mode is he conrolling parameer in deermining DLFs in he deflagraion even, as illusraed in Figure 3. As he raio is greaer han 4, dynamic load effecs become negligible. This observaion implies ha dynamic load effecs can be negleced as along as he naural frequency of he separaor breahing mode is greaer han.96 Hz. Avoiding significan dynamic load effecs should be one design consideraion. If a new separaor has he same deflagraion pressure-ime curve as shown in Figure, hen Figure 3 can be used o guide he design. For example, may be needed for avoiding significan dynamic load effecs, which in urn requires ha he naural frequency of he new separaor breahing mode is greaer han.48 Hz. CONCLUSIONS The DLF in a deflagraion even can be esimaed using he new mehodology presened in his paper wih a pressure-ime curve and he naural frequency of he vessel breahing mode. The mehodology is based on a single DOF linear elasic vibraion response of he vessel so ha plasiciy, srain rae dependence and wave propagaion are beyond he scope of he presen sudy. The raio of he pressure rise ime o he naural period of he vessel breahing mode is idenified as he conrolling parameer deermining DLF. The dynamic load effecs for high pressure vessels in vening deflagraion evens may be negligible because he srucure is quie siff. A procedure for esimaing DLF in a deflagraion even is suggesed as follows:. Use he maximum value wihou known pressure-ime curves:.5 for a vening deflagraion even, and. for a non- vening deflagraion even.. Use he new mehodology presened in his paper o esimae DLF more accuraely wih known pressure-ime curves. The naural frequency of he vessel breahing mode can be deermined using finie elemen analysis (FEA). 3. Fully evaluae he dynamic load effecs using explici dynamic FEA. ACKNOWLEDGMENTS The auhors would like o hank Mr. M. Emile Dupre and Mr. Eduardo H. Perez of DuPon for providing heir valuable commens and recommendaions on his paper. The auhors also wan o hank he reviewers for heir candid and consrucive criicisms, which enable he auhors o fine-une many aspecs of he paper. REFERENCES [] Sullivan, J. F., Shannon, D. I., Decomposiion Vening in a Polyehylene Produc Separaor, Proc. of 99 ASME Pressure Vessel and Piping Conference, June 99, New Orleans, Louisiana, USA. [] ASME Boiler and Pressure Vessel Code Secion VIII Division 3, Ediion. [3] Task Group on Impulsively Loaded Vessels, 8, Code Case 564: Impulsively Loaded Pressure Vessels, Secion VIII Division 3, Cases of ASME Boiler and Pressure Vessel Code. [4] ASME Boiler and Pressure Vessel Code Secion VIII Division, Ediion, Appendix H. [5] Naional Fire Proecion Associaion (NFPA) 69, Sandard on Explosion Prevenion Sysems, Chaper 5, Deflagraion Pressure Conainmen, issue effecive wih he applicable Addenda of he ASME BPVC. [6] Naional Fire Proecion Associaion (NFPA) 68, Guide for he Vening of Deflagraions, issue effecive wih he applicable Addenda of he ASME BPVC. [7] ASME Boiler and Pressure Vessel Code Secion VIII Division, Ediion, Annex 4.D. [8] Duffey, T. A., Rodriguez, E. A., Romero, C., Deonaion- Induced Dynamic Pressure Loading in Conainmen Vessels, Par, WRC Bullein 477, The Welding Research Council, New York, NY, December. [9] Leishear, R. A., Sresses in a Cylinder Subjeced o an Inernal Shock, Proc. of 5 ASME Pressure Vessel and Piping Conference, July 7-, 5, Denver, Colorado, USA. [] Yip, M., Haroldsen, B., Life Assessmen of Full-Scale EDS Vessel under Impulsive Loadings, Proc. of ASME Pressure Vessel and Piping Conference, July 8-,, Bellevue, Washingon, USA. [] Thomson, W. T., Theory of Vibraion wih Applicaions, Prenice-Hall, Inc., 3 rd ed., 988. [] Meirovich, L., Elemens of Vibraion Analysis, McGraw- Hill Book Co., nd ed., 986. [3] Cheng, C., Widera, G. E. Oo, Dynamic Burs Pressure Simulaion of Cylindrical Shells, Proc. of 8 ASME Pressure Vessel and Piping Conference, July 7-3, 8, Chicago, Illinois, USA. [4] Ayre, R. S., Transien response o sep and impulse funcions, in Shock Vibraion Handbook, edied by Harris, C. M., McGraw-Hill Book Co., 3rd ed Copyrigh by ASME