4. GEOMETRY OF CONSIDERED PRESSURE VESSELS

Transcription

1 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini STRESS ANALYSES OF CYLINDRICAL VESSEL WITH CHANGEABLE HEAD GEOMETRY Dražan Kozak1, Ivan Samardžić2, Antun Stoić3, Željko Ivandić4, Darko Damjanović5 1,2,3,4,5 Joip Juraj Stromayer Univerity of Oijek, Mechanical Engineering Faculty in Slavonki Brod, Trg Ivane Brlić-Mažuranić 2, HR Slavonki Brod, Croatia Abtract: The main objective of thi paper i numerical analye of cylindrical preure veel with changeable head geometry (emi-elliptical and hemipherical head) and comparion of reult in mean of preciion and time needed for getting the olution. Manufacture technology of preure veel cylinder and head are decribed, and contruction geometry of variou preure veel head i preented. Dicretiation tyle of finite element model, boundary condition, and reult are detailed hown and explained. Baed on analytical olution for membrane tre tate of preure veel, numerical analyi of it i performed alo. Comparion of harveted numerical reult for above mentioned preure veel are hown tabular, with value of time needed for getting the olution too. And comparion of analytical and numerical reult for preure veel with hemipherical head i hown too. Key word: cylindrical preure veel, hemipherical head, emi-elliptical head, finite element method, ANSYS. 1. INTRODUCTION Preure veel are inevitably element of thermal ytem, hydro ytem, chemic ytem, water upply ytem etc. It i preumed that preure veel i every tank with volume of 3 V 1,5 m, inner preure bigger than 0,05 MPa, or negative preure maller than 0,08 MPa, and accumulated energy p V 0,02 MPa m3. 2. MANUFACTURE TECHNOLOGY OF PRESSURE VESSEL Part of cylindrical preure veel are manufacturing eparately, cylinder and head. Cylinder i made from egment which are welded in one unit, and head are joining with cylinder by welding too, a hown on Fig. 1. weld egment head cylinder Fig. 1 Welded cylindrical preure veel [1] 93

2 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini Manufacture of cylinder i beginning with preparing heet metal in mean of cutting on right dimenion and preparation of edge which mut be welded longitudinal. Afterward, before final bending, heet metal mut be pre-bended in purpoe of correct longitudinal joint (after final bending) for welding, what i condition for good quality of welded joint. After welding the cylinder, it mut be calibrated to eliminate poible oval hape of cylinder. Then, header edge for joint of head mut be prepared for welding too. Head are manufacturing with procedure of egment preuring. Manufacture i coniting from following operation: prepare of material, egment preuring (Fig. 2), getting final form of head (Fig. 3), preparing edge for welding (Fig. 4). After that, head i finihed and prepared for welding to cylinder. Fig. 2 Segment preuring of preure veel head [2] Fig. 3 Getting final form of head [2] Fig. 4 Preparing edge for welding [2] a da R=0,8 da i,5 d R=0 h 3 h 3,5 Fig. 5 Flanged head [3] 54 1, =0 di R=da r di d 0,1 r= di da di Figure from Fig. 5 to Fig. 8 are howing different type of preure veel head. Fig. 6 Shallow head [3] Fig. 7 Deep head [3] Fig. 8 Hemipherical head [3] 3. MEMBRANE SHELL THEORY Shell can be thin-walled and thick-walled. Thin-walled hell are retricted with ratio of plate thickne and inner or outer radiu of hell, a hown in following equation: 1, Ri 20 1 Ro 20 (1) Shell with larger value of ratio from equation (1) are thick-walled hell. Fig. 9 how tre ditribution per plate thickne. That ditribution i per hyperbolic tate for thick-walled hell (Fig. 9, a), and per linear tate for thin-walled hell (Fig.9, b). 94

3 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini hyperbole a) traight line b) Ro N= h Fig. 9 Stre ditribution per plate thickne [4] 3.1 PRESSURE VESSEL WITH HEMISPHERICAL HEADS M= f / 6 2 l Ri l h / 2 l / 2 h l For calculating Von Mie equivalent tre, hoop and longitudinal tree are needed, which have to be calculated from the hoop and longitudinal force. a) Preure veel cylinder - Hoop force: N h p Rm, where p i inner preure and Rm i average radiu of curvature: Rm Ri 2 - Longitudinal force: p Rm. Nl 2 - Hoop tre: N h h, - Longitudinal tre: N l l. (2) (3) (4) (5) (6) After hoop and longitudinal tree are calculated, it i poible to calculate Von Mie equivalent tre [5]: eq h2 l2 h l. b) Preure veel head (7) In cae of preure veel head, hoop and longitudinal force are the ame: Nh Nl p Rm, 2 95 (8)

4 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini Hoop and longitudinal tree are ame too, and becaue of that, maximal Von Mie tre of preure veel head i equal to hoop and longitudinal tree [5]: Nh Nl h l, eq h l. (9) c) Radial diplacement -Preure veel cylinder -Preure veel head 2 p Rm R cylinder 2 (10) 2 E cylinder d) Condition for membrane tre tate 2 p Rm R head 1 (11) 2 E head For acquiring a membrane tre tate in preure veel, plate thickne of cylinder mut be greater than plate thickne of head. Adverely, on preure veel joint of cylinder and head, apart from membrane tree, bending tre will appear too, o plate thickne of cylinder mut be greater than plate thickne of head, according to following equation [5]. 2 1 cylinder head 4. GEOMETRY OF CONSIDERED PRESSURE VESSELS. (12) Preure veel conidered in thi paper are completely the ame, only difference i in type of head. There are two cae: preure veel with hemipherical head (Fig. 10), and preure veel with emi-elliptical head (Fig. 11). In both cae, internal preure i p = 1,6 MPa, inner diameter i d i = 5170 mm, cylinder length of preure veel i L = 8000 mm and plate thickne i = 30 mm. O p = 1.6 MPa R 2585 O p = 1.6 MPa 4000 Fig. 10 Preure veel with hemipherical head ,5 Fig. 11 Preure veel with emi-elliptical head

5 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini 5. ANALYTICAL CALCULATION OF PRESSURE VESSEL WITH HEMISPHERICAL HEADS Analytical calculation i performed for preure veel with hemipherical head. Dimenion and internal preure are hown on Fig. 10. In thi calculation, only Von Mie equivalent tre for preure veel cylinder i calculated, becaue from membrane hell theory (chapter 3.1) it i obviou that maximal Von Mie equivalent tre will appear on preure veel cylinder, and not on head. Radial diplacement of preure veel cylinder and head, and condition for acquiring a membrane tre tate in preure veel are calculated too, becaue of afterward numerical analyi. Reult are following: 120,089 MPa; R 1, 4593 mm; R 0, 6008 mm; 72, 857 mm eq cylinder head cylinder 6. NUMERICAL ANALYSIS WITH FINITE ELEMENT METHOD Analyi of preure veel in thi paper i realied with oftware ANSYS 11. Three type of finite element are ued: SOLID 95, PLANE 183 and SHELL 181, which are decribed in table 1. Material i et a iotropic, linear elatic with Young' modulu of elaticity E = MPa, and Poion' ratio = 0,3. Becaue of total ymmetry of preure veel - geometry and load, only 1/8 of preure veel i modeled, and becaue of that, numerical analyi i much impler and time for getting olution i much horter. table 1. Decription of element ued in analyi [6] SOLID 95 PLANE 183 SHELL 181 Decription of element SOLID 95 i defined by 20 node having three degree of freedom per node: tranlation in the nodal x, y, and z direction. The element may have any patial orientation. SOLID95 ha platicity, creep, tre tiffening, large deflection, and large train capabilitie. It can tolerate irregular hape without a much lo of accuracy. SOLID 95 element have compatible diplacement hape and are well uited to model curved boundarie. PLANE 183 i defined by 8 node or 6 node having two degree of freedom at each node: tranlation in the nodal x and y direction. The element may be ued a a plane element (plane tre, plane train and generalized plane train) or a an axiymmetric element. Thi element ha platicity, hyperelaticity, creep, tre tiffening, large deflection, and large train capabilitie. It i a higher order 2-D, 8-node or 6- node element. PLANE 183 ha quadratic diplacement behavior and i well uited to modeling irregular. SHELL181 i uitable for analyzing thin to moderately-thick 97

6 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini hell tructure. It i a 4-node element with ix degree of freedom at each node: tranlation in the x, y, and z direction, and rotation about the x, y, and z-axe. (If the membrane option i ued, the element ha tranlational degree of freedom only). SHELL 181 i well-uited for linear, large rotation, and/or large train nonlinear application. Change in hell thickne i accounted for in nonlinear analye. 6.1 NUMERICAL ANALYSIS OF PRESSURE VESSEL WITH HEMISPHERICAL HEADS Fig. 12 and Fig. 13 are howing finite element meh and boundary condition for preure veel with hemipherical head. Becaue of limited length of thi paper, only finite element meh for element SOLID 95 i hown (Fig. 12). For all type of finite element, meh i generated automatically, expect on joint of cylinder and head, where meh i et up with more denity. Fig. 13 i howing boundary condition. Red arrow are repreenting inner preure of 1,6 MPa, and blue ymbol are repreenting propoed ymmetry of preure veel. Becaue of totally ymmetry of preure veel, ymmetry i et up on all three edge of preure veel part. Fig. 14 i howing reult for Von Mie equivalent tre for preure veel with hemipherical head mehed with finite element SOLID 95. A expected, the maximal equivalent tre i appearing on preure veel cylinder with amount of 122,997 MPa for inner preure of p = 1,6 MPa. Fig. 12 Finite element meh for preure veel with hemipherical head Fig. 13 Boundary condition for preure veel with hemipherical head 98

7 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini eq, MPa Fig. 14 Von Mie equivalent tre for inner preure p=1,6 MPa - hemipherical head Fig. 15 i howing parallel comparion of equivalent tre reult for all three type of finite element. It i preented only detail of preure veel joint of head and cylinder. A figure how, tre ditribution for all three type of element i almot identical. In cae of element PLANE 183 and SOLID 95 it i poible to ee tre ditribution per plate thickne of preure veel, but in cae of element SHELL 181 it i not poible, and that i diadvantage in uing of thi element for numerical analyi. Fig. 15, c) i howing tre ditribution on inner wall of preure veel. a) b) c) Fig. 15 Parallel comparion of equivalent tre reult - hemipherical head; a) PLANE 183, b) SOLID 95, c) SHELL

8 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini NUMERICAL ANALYSIS OF MEMBRANE STRESS STATE For acquiring a membrane tre tate in preure veel, plate thickne of cylinder mut be greater than plate thickne of head. Analytical calculation i performed and it i obtained that plate thickne of cylinder mut be 72,857 mm. With numerical analyi it i acertain that with increae thickne of cylindrical plate from 30 to 72,857 mm, there i till a bending. It i obviou from thi example that i impoible to acquire membrane tre tate, becaue membrane tre tate neither don't exit, becaue bending will appear alway, in bigger or maller amount. From Fig. 16 it i obviou that equivalent tre value around joint of cylinder and head are almot the ame with amount of 68,5 MPa. Away from joint of cylinder and head, according to cylinder, tre value are decreaing for about 20 MPa and become contant in continue. Away from joint, according to head vertex, tree are contant. So, membrane tre tate i prevailed in mot of preure veel cylinder and head, but not in joint of cylinder and head. 72, , eq, MPa R, mm Fig. 16 Stre ditribution (Von Mie) through plate thickne for preure veel with hemipherical head, and with inner preure of p=1,6 MPa Fig. 17 Radial diplacement for preure veel with hemipherical head, and with inner preure of p=1,6 MPa Fig. 17 i howing radial diplacement regarding to axi of ymmetry of preure veel. A figure how, radial diplacement around joint of cylinder and head are almot the ame with value of 0,56 mm. Away from joint of cylinder and head, according to cylinder, diplacement value i decreaing for very mall amount, and become contant in continue. Away from joint, according to head vertex, diplacement value i approaching value 0. That i logic, becaue in axi of ymmetry of preure veel, there i no radial diplacement. 6.2 NUMERICAL ANALYSIS OF PRESSURE VESSEL WITH SEMI- ELLIPTICAL HEADS In thi cae of preure veel with emi-elliptical head will be hown only finite element meh and boundary condition for element SHELL 181, becaue for element PLANE 183 and SOLID 95 it i ame a in cae of hemipherical head, only difference i of coure in geometry of head. Fig. 18 and Fig. 19 are howing finite element meh and boundary condition for preure veel with emi-elliptical head. A in cae of preure veel with hemipherical head, in all cae of finite element, meh i generated automatically, expect on joint of cylinder and head, where meh i et up with more denity. In thi cae of head, meh i dener on head vertex too, becaue of mall hole which hould be created there. 100

9 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini Fig. 19 i howing boundary condition. Red arrow are repreenting inner preure of 1,6 MPa, and blue ymbol are repreenting propoe ymmetry of preure veel. A appoed in earlier cae of hemipherical head, in thi cae there are orange ymbol too, which repreent contraint of rotation. That contraint mut be et, becaue element SHELL 181 ha rotational degree of freedom around certain axi too. Reaon for the mall hole placing at the head vertex lie in fact that ingularity of tre appear at thi place. Becaue of ingularity, value of tre on head vertex i varying with number of finite element, during value of tree in all ret preure veel i the ame. So, to avoid thi appearance, mall hole on head vertex hould be created, and on edge of that hole, contraint of diplacement in radial direction mut be et. Thi contrain i jutified to et, becaue created hole i aume a endle mall, like a point on axi of ymmetry of preure veel, and on axi of ymmetry, there i no radial diplacement. In uch of et boundary condition, ingularity of tree on head vertex i avoided, and tre ditribution of preure veel i correct. Fig. 18 Finite element meh for preure veel with emi-elliptical head Fig. 19 Boundary condition for preure veel with emi-elliptical head Fig. 20 i howing the ditribution of equivalent tre for preure veel with emi-elliptical head mehed with finite element SHELL 181. A expected, the maximal equivalent tre i appearing on minimal meridian curvature of preure veel head with amount of 206,081 MPa for inner preure of p = 1,6 MPa. eq, MPa Fig. 20 Equivalent tre field for inner preure p=1,6 MPa - emi-elliptical head 101

10 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini Fig. 21 i howing parallel comparion of equivalent tre reult for all three type of finite element. It i preented only detail of minimal meridian curvature of preure veel head, becaue that i the place of maximal value of tre. A figure how, tre ditribution for all three type of element i almot identical. Fig. 21, c) i howing tre ditribution on inner urface wall of preure veel. a) b) c) Fig. 21 Parallel comparion of equivalent tre reult-emi-elliptical head; a) PLANE 183, b) SOLID 95, c) SHELL COMPARISON OF NUMERICAL RESULTS OBTAINED BY USING OF DIFFERENT TYPES OF ELEMENTS FOR BOTH HEADS GEOMETRY Table 2 and 3 are howing comparion of numerical reult for both cae of preure veel head and for all three type of element. Table 2 i howing comparion of reult for preure veel with hemipherical head, and table 3 i for preure veel with emielliptical head. table 2. Comparion of numerical reult for the preure veel with hemipherical head Element type Maximal equivalent tre eq, MPa SOLID ,997 PLANE ,867 SHELL ,051 Meh parameter, calculating time Element: Node: Time: 9 min, 31 Element: Node: Time: 2 Element: Node: Time: 10,8 102

11 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini table 3. Comparion of numerical reult for the preure veel with emi-elliptical head Element type Maximal equivalent tre eq, MPa SOLID ,438 PLANE ,073 SHELL ,081 Meh parameter, calculating time Element: Node: Time: 9 min, 27 Element: Node: Time: 2,7 Element: Node: Time: 12,9 A table how, reult for maximal equivalent tree for all three type of element and in both cae of head type are almot identical. Numerical analyi with finite element SOLID 95 require much more element, becaue it i need to meh a volume with plate thickne of 30 mm. And becaue of bigger number of element, time for calculating i much longer too. In cae of element SHELL 181 it i needed to meh a hell, o number of element and time for calculating i much maller. And in cae of element PLANE 183 it i need to meh only a plane, and in thi cae number of element and time for calculating i the mallet of coure. Table 4 i howing comparion of numerical and analytical reult for preure veel with hemipherical head, and percent dicrepancy of reult. For numerical reult i taken only thoe calculated with element PLANE 183, and reult for analytical calculating i given in chapter 5. Type of calculating table 4. Comparion of numerical and analytical reult for preure veel with hemipherical head, percent dicrepancy Maximal equivalent tre eq, MPa Radial diplacement R, mm Cylinder ( R cylinder ) Head ( R head ) Analytical olution 120,089 1,4593 0,6008 Numerical reult 122,867 1,488 0,6084 Percent dicrepancy, % 2,26 1,92 1,25 A table how, all reult dicrepancy i in allowed boundary of 5%, what mean that reult are acceptable! 8. CONCLUSION In thi paper a numerical analyi of preure veel with hemipherical and emi-elliptical head i performed, with three type of element: SOLID 95, PLANE 183 and SHELL 181. It i concluded that in both cae of preure veel head, uing of PLANE 183 element 103

12 BULETIN ŞTIINŢIFIC, Seria C, Volumul XXIII, Facicola: Mecanică, Tribologie, Tehnologia Contrucţiilor de Maşini preent the bet approach, becaue of minimal number of element for mehing, hortet calculation time, inight into the tre ditribution per plate thickne and obtained reult which are cloet to the analytical one. Thi type of axiymmetric element could be recommended in uch cae, when the total ymmetry of model i conidered (geometry and load). Analyi of cylindrical preure veel with different head type i performed in purpoe of comparion of value of maximal equivalent tree. It i concluded that maller value of equivalent tree are appearing in preure veel with hemipherical head, and equivalent tre ditribution i advantageou too in that cae of head geometry. Neverthele, in exploitation, there i much more preure veel with emi-elliptical head, becaue manufacturing of hemipherical head i much more complicated and expenive. Maximal equivalent tre for conidered veel geometry in cae of hemipherical head i appear in preure veel cylinder wall with amount of 123 MPa, and in cae of emi-elliptical head, maximal equivalent tre i almot double (209 MPa) and it i appear on minimal meridian curvature of preure veel head, what i expecting from hell of revolution theory. Again, neverthele preure veel with emi-elliptical head are much more manufacturing, becaue of earlier mention reaon. It i analytically calculated that for acquiring a membrane tre tate in conidered preure veel, plate thickne of cylinder mut be h cylinder = 72,857 mm. Afterward numerical analyi i approved that completely membrane tre tate in veel i impoible to acquire, becaue bending will appear alway, in bigger or maller amount. 9. REFERENCES [1] Đurić, V.: Parni kotlovi, Beograd, Beogradki izdavačko-grafički zavod, 1973 [2] S.I.F.E.A.SpA - Società Italiana Fondi bombati e Affini; URL: [3] Norma: HRN M.E2.252: Proračun podnica izvrgnutih unutarnjem ili vanjkom tlaku [4] Alfirević, I.: Nauka o čvrtoći II, Zagreb, Golden marketing, 1999 [5] Brnić, J., Turkalj, G.: Nauka o čvrtoći II, Rijeka, Zigo, 2006 [6] ANSYS, Releae 11: Help Topic 104