Mechanical Design for an Elliptical Shaped Nozzle Exposed for Thermal and Pressure Loads

Transcription

1 MASTER S THESIS 007:159 CIV Mechanical Design fo an Elliptical Shaped Nozzle Exposed fo Themal and Pessue Loads Caoline Pettesson Caoline Pettesson Luleå Univesity of Technology MSc Pogammes in Engineeing Mechanical Engineeing Depatment of Applied Physics and Mechanical Engineeing Division of Compute Aided Design 007:159 CIV - ISSN: ISRN: LTU-EX--07/159--SE

2 Peface This thesis has been pefomed as the final pat of the Maste of Science pogam in Mechanical Engineeing at Luleå Univesity of Technology. This thesis wok was conducted at Volvo Aeo Copoation, Nozzles and combustion chambes (6660) duing the autumn and winte I would like to thank my Supeviso at Volvo Aeo Copoation, Eva Stenstöm, fo all he help and suppot duing this wok. I would also like to show my gatitude to Fedik Hallstensson and Matin Hansson fo thei all help and infomation they povided. My examine at Luleå Univesity of Technology, Pof. Mats Nässtöm at the Depatment of Mechanical Engineeing, I thank him fo all his help and guidance thoughout the wok. Lastly, I would thank the staff at the depatment Nozzles and combustion chambes fo all thei help egading poblem solving duing the wok. Tollhättan, Januay 007 Caoline Pettesson i

3 Abstact This thesis aims to descibe a new design of low signatue nozzles offeing bette accuacy against pessue and themal loads. Low signatue nozzles ae used in, fo example, militay aicaft to lowe the signatue, defined as the contast to the backgound. The wok begins with a theoetical pat and continues with a pactical pat whee simulations in ANSYS ae done. The goal and pupose of this thesis wok ae to give an undestanding of the behaviou of cicula and elliptical outlets and find a way to make the stuctue stonge when needed without inceasing the weight. The theoetical pat gives a mathematical undestanding of the poblem that is necessay fo the simulations, conducted in the following pactical pat. Chapte 3 focuses on the pogam and model. Diffeent methods and element types wee tied and thee elements wee eventually used in the pogam. Two ae shell elements used fo stuctual foces and one is a themal element. To solve the themal load, a layeed element was used. Seveal evaluations wee done to confim that the pogam was coect. These evaluations showed that the model hee is not sensitive to the numbe of elements used in the simulation. Thee diffeent analyses wee done, all with an elastic model. The diffeent loads used ae a pessue load, a themal load and a combination of both loads. All simulations wee done with a cicula and an elliptical geomety. Fom the simulations, the conclusions can be divided in two goups, depending on the geomety. When a cicula geomety is exposed to pessue, the stesses become of a membane kind, and when the load is themal, the stesses ae of a bending kind. To lowe the membane stesses is an inceased thickness the solution. Fo the bending stesses the themal gadient, ΔT, should be loweed. An inceased ΔT inceases the stesses, but not the thickness. When the thicknesses incease, ΔT emains the same, which can be elated to how the load is applied. The elliptical geomety has mainly bending stesses fo both loads; to lowe the stesses the solution is the same as fo the cicula geomety. These conclusions ae useful when the thickness is unifom. When the thickness vaies and the load is pessue an inceased thickness is still the solution. Fo a themal load, the conclusion about ΔT emains tue. The esult gives only the stess level, the position is the weakest point in the stuctue. The highest stess level is found in the thinnest pat of the stuctue. ii

4 Table of contents Peface... i Abstact... ii Table of contents...iii Nomenclatue... v 1. Intoduction Volvo Aeo Copoation Backgound Aim of the thesis.... Theoy Shell theoy Thin walled shell Thick walled shell Cicula geomety Pipe theoy Thick walled pipes Thin walled pipe Elliptical geomety Conclusions ANSYS building the model Pogam desciption Cicula and elliptical geomety Geomety modelling Equal length in one diection Cuve though splines Pola coodinates The command cicle Scaled cicle Element selection Beam elements Link elements Plane elements Shell elements Constaints Loads Analysis Conclusions The simulations Model evaluation Visual evaluation Theoy evaluation Sensitivity evaluation Optimization paametes Pessue load Tempeatue load... 0 iii

5 4.4.1 The simulations Combined tempeatue and pessue Conclusions Results Intenal pessue Theoy Simulations Mateial model Tempeatue load Theoy Simulations Mateial model Combined tempeatue and pessue The final design Conclusions Geomety Load Conclusion Section Recommendations fo futhe wok Refeences Appendix A- Pipe theoy...i Appendix B- Thick walled pipes...i Appendix C - Geneal theoy in an elliptical shell... I Appendix D - Code fo the pessue load case... I Appendix E - Code fo the tempeatue load case... I Appendix F - Code fo the combined load case... I Appendix G - Deflection evaluation...i Appendix H - Geomety oveview...i Appendix I - Pessue esults...ii Appendix J - Tempeatue esults...i Appendix K - Tempeatue evaluation...i Appendix L- Tempeatue and pessue esults... I iv

6 Nomenclatue σ: Stess σ : Stess in -diection σ φ : Stess in φ-diection, cicumfeential in cylindical coodinate system. σ x : Stess in x-diection' σ y : Stess in y-diection σ z : Stess in z-diection σ m : Bending stess, Appendix C σ h : Cicumfeential stess, Appendix C [σ 1 ]: The maximum total stess in y-diection [σ ]: The maximum total stess in x-diection τ xy : Sheaing stess. N : Membane foce, (nomal foce). N φ : Membane foce, (nomal foce). N x : Nomal foce in x-diection. N φ : Sheaing foce in φ-diection. N φθ : Sheaing foce in φθ-diection. N θφ : Sheaing foce in θφ-diection. P: A unifom intenal pessue load T(): A unifom themal load a: Radius in x-diection fo an elliptic shape, aveage adius in some sections b: Radius in y-diection fo an elliptic shape. : Radius and the aveage adius in some sections. t: Thickness φ One diection in the coodinate system. K : Volume foce K 1: Constant fo nomalized bending moment. K : Constant fo nomalized bending moment. K x : Constant fo nomalized bending moment. ν: Poisson s constant (appoximate 0,3) E: E-module ε : Stain in -diection. ε φ : Stain in φ-diection. ε x : Stain in x-diection. ε y : Stain in y-diection. ε z : Stain in z-diection. δ: Deflection δ a : Deflection of the intenal adius (thick pipes) δ b : Deflection of the oute adius (thick pipes) δ x : Deflection in x-diection δ y : Deflection in y-diection k: Constant A: Constant Appendix B. B: Constant Appendix B. d: Diamete v

7 α: Heat expansions constant C 1 : Constant fo nomalized deflection (Appendix C), constant (Appendix B) C : Constant fo nomalized deflection (Appendix C), constant (Appendix B) D: Flexual igidity of a plate x: A constant and a diection in the coodinate systems. y: Diection in the coodinate systems. ΔT: Change in tempeatue. u: One coodinate point. A 1 : Integations constant, Appendix B. A : Integations constant, Appendix B. F el : Complete elliptical integal. E el : Complete elliptical integal. M 1 : Moment in x-diection, Appendix C M : Moment in y-diection, Appendix C M: Moment c: Distance fom the cente to the oute and. I: Moment of inetia. UX: Degee of feedom. UY: Degee of feedom. UZ: Degee of feedom. ROTX: Rotational degee of feedom. ROTY: Rotational degee of feedom. ROTZ: Rotational degee of feedom. vi

8 1. Intoduction This fist chapte pesents and intoduces the thesis wok. The focus is to pesent elevant backgound to the poblem. The main points of this wok ae found hee and will povide the necessay infomation to undestand this thesis wok. 1.1 Volvo Aeo Copoation Volvo Aeo Copoation manufactues and develops components fo commecial/militay aicaft and space populsion subsystems. Volvo Aeo Copoation is one pat of the Volvo Goup and is located in Stockholm, Malmö and Tollhättan, whee the headquates is also located; fo futhe infomation, see [1]. This thesis was conducted at Volvo Aeo Copoation in Tollhättan at the depatment of nozzles and combustion chambes. 1. Backgound Man has always looked admiably upon bids fo thei capability to fly. A common appehension of the concept militay aicaft is that it has flouished with the development of new technologies, though it is also impotant to undestand that militay use of flying technologies is not a novel phenomenon. Accoding to Gunston [], the discovey of hydogen in 1766 was the fist step towads building the fist hydogen balloon. This could be seen as the fist militay aibone tanspotation, since the govenment of Fance decided to establish a militay company of 30 hydogen balloons in Due to both technological and economical development, militay ai technology has come a long way in about 00 yeas. One obvious diffeence is the invention of militay aicaft, which Volvo Aeo has eseach and development pojects in. Due to, fo example, technical impovements in ada systems, eseach in low signatue outlet nozzles has been pioitized. Volvo Aeo is cuently involved in the Neuon poject, a Euopean collaboation fo development of an unmanned aicaft. Pictue 1. A possible design fo the neuon aicaft. Figue 1. A coss section of the plane in pictue 1. The ing maks the low signatue nozzle. 1

9 In militay aicaft, it is impotant that outlet nozzles have low signatue. The signatue can be descibed as the contast to the backgound. If the aicaft looks like the backgound it will not be detected. Low signatue includes factos such as ada, IR (infaed light), acoustic, visual and magnetic. If the aicaft has low signatue the detection distance fo sensos is shotened. This thesis focuses on the low IR-signatue equiement and how this affects the design accoding to the stength in the stuctue. In Figue 1, the low signatue nozzle position in the aicaft is maked. 1.3 Aim of the thesis The pupose of this thesis is to give an undestanding of the behaviou of cicula and elliptical outlets and to find an optimal shape of the outlet aea egading to mechanical integity. This thesis will also give an inceased undestanding fo how the outlet nozzle should be designed. The main goal is to find a way to make the stuctue stonge when needed, without inceasing the weight, and still have the desied stiffness in the stuctue. The paametes of inteest ae those that change the defomation, stess (lifetime of the poduct) and weight. The optimizing paametes ae stiffness and diffeent stesses. Figue. The gaphic desciption of the poblem. A completely cicula shape will esult in low weight and a shape with good stiffness but bad IR. An elliptical shape will give good IR but bad stiffness, which will incease the weight, see Figue. Studies will be conducted though FE analysis to find the optimal shape due to stiffness, weight and IR-signatue, both theoetical and expeimental. The defomation and stess in simple shapes such as cicles and ellipses will be theoetically studied using linea theoy. The loads acting on the outlet, which will be studied, ae intenal pessue and themal gadient though the thickness of the outlet wall. When the theoetical study is finished, FE-calculations in ANSYS will be pefomed to confim the theoetical esults.

10 . Theoy To find the optimized stuctue, the stess distibution and the defomation in the stuctue must fist be theoetically studied. The stuctue that is analysed is the walls in a low signatue nozzle, with o without stiffenes. Two diffeent types of coss sections wee analysed, a cicula and an elliptical geomety. The analysis was done with espect to two diffeent loads, intenal pessue and tempeatue..1 Shell theoy The intoduced shell theoy can be used fo all shells. Hee, shells with cicula o elliptical geomety of inteest. The infomation pesented is geneal; moe specific infomation about cicula o elliptical geomety can be seen in sections. and.3. Shells can be divided into two goups, thick and thin walled shells, which ae pesented below. In this section themal load is not discussed. The themal analysis is only done fo the specific geometies. The shell theoy can be used with a vaying thickness t. Figue 3. The components woking on an element in equilibium. The figue above is an example of how an element in 3D can be descibed..1.1 Thin walled shell Foce equilibium is fomulated to analyse the geneal stesses and deflections in a thin walled shell. The shell is assumed to be unde membane condition, which simplifies the calculations since no bending stesses will occu. The shell is exposed to an intenal pessue and the ends ae fee. When the system of the equilibium equations is found, the system can be solved and the stesses calculated. Equations (1)-(3) ae used to calculate the stesses. [3], [4] and [5] ae the equations found and the used assumptions. σ = x N x t ( 1 ) σ = y N y t ( ) 3

11 N yz τ = yz t ( 3 ) In the equations above, σ x, and σ y ae the stess components in the indicated diection and τ yz the shea stess. N x, N y and N yz, the unit is foce pe length, ae the nomal components in the same diection and t is the thickness. The emaining components ae equal to zeo, since plane stess is used..1. Thick walled shell Analysis of a thick walled shell is simila to the study of the thin. Fo a thick walled shell, the stesses in the cicumfeential diection ae not assumed to be unifom though the thickness and the stess in the adial diection cannot be neglected. Fomulas ae detemined in the same way as fo the thin shell. σ = x N x t ( 4 ) σ = y σ = z N y t N z t ( 5 ) ( 6 ) τ = yz N yz t ( 7 ). Cicula geomety The cicula geomety can be divided into thin and thick walled pipes, which ae only special cases of shell theoy. Assumptions made to ceate these special cases ae pesented in the section below...1 Pipe theoy Cetain assumptions ae made to study the pipes (see sections.. and..3), which ae tue fo both thin and thick walled pipes. The geomety is assumed to be cicula and symmetic and the loads ae unifom. The analysis is done in a cylindical system. Cylindical coodinates ae illustated in Appendix A, whee the pipe equations ae also explained. This infomation can also be found in [6]. The desciption is only geneal and is not sepaated into thick and thin walled pipes. Plane stess condition is assumed fo both thin and thick shells, the pessue P φ =0, the stesses σ, σ φ and τ φ ae constant though the thickness, and σ z =0. If a membane condition occus, the sheaing foces ae also N φθ = N θφ =0. Cylindical symmety is one of the keys to undestanding why a cicula geomety is a pefect shape when, fo example, the load is intenal pessue. Fo cylindical symmety, the stesses in the φ- and z-diections ae assumed constant. This leads to a moe simplified stess distibution and no sheaing stesses have to be consideed. 4

12 The two diffeent loads ae almost analysed the same, but fo now only the equation fo the diffeent loads is discussed. In geneal, when a pipe o a tube is exposed to intenal pessue the only acting stess is membane stess. Bending stess occus if a unifom themal load T() is applied. All stengths and defomations will be symmetical... Thick walled pipes A thick pipe is only a special case of a thick cylindical shell. In this analysis the two loads intenal pessue, P, and the themal load, T(), ae analysed sepaately. The theoy is biefly descibed hee, but a moe detailed desciption can be found in Appendix B. Moe infomation about the equations can be found in [6] and [7]. The loads ae unifomly distibuted in both cases. Figue 4. Coss section of a thick cicula pipe with adii and thickness maked. Using Hook s geneal law togethe with the equations fo the stains, two expessions fo the stesses can be found, see Appendix B. The equations below expess the stesses fo a thick walled pipe exposed to an intenal pessue, P. P a P a b 1 σ = ( 8 ) b a b a P a P a b 1 σ = + ( 9 ) ϕ b a b a P b ( 10 ) τ xϕ = ( b a ) The geneal equation to calculate the deflection in a cicula geomety is the same as fo a od, and the equation, see [5], becomes the following: k δ = σ ( 11 ) E The constant k is the adial distance in the undefomed condition fom the cente and δ is the deflection. The elastic modulus E is a mateial popety. By using the stess in the 5

13 cicumfeential diection fom (8), the deflections fo the adii a and b can be calculated in equation ( 11 ). P b a δ = a E ( b a ) ( b + a ) ( ) + b a P a δ = b ν ( 13 ) E The second case analysed is a pipe exposed to an axi-symmetic themal load T(), which gives axi-symmetic stesses and defomations. In this case, it is assumed that σ z =0 and ε z =0. The theoy is set up using the same methodology as the intenal pessue load. The equations become simila to equation ( 8 ) and ( 9 ). B E α σ = A T () d ( 14 ) d / B 1 σ ϕ = A + + E α T () d T () ( 15 ) d / P a A = b a B = P a b b a A and B in the equations above ae constants and ae given in equations ( 16) and ( 17). α is the heat tansfe constant and d stands fo the diamete. The deived fomulas ae good appoximations fo small tempeatue gadients. In linea theoy, the total stess may be found by using the pinciple of supeposition. The deflection can be calculated using equation ( 11 ). ( 1 ) ( 16 ) ( 17 )..3 Thin walled pipe A pipe may be egaded as thin walled when the thickness to adius atio is small. The thin walled pipe will only be analysed with pessue load. Fo a thin walled pipe, the tempeatue gadient is vey small; hence, the themal stess is vey small. Thus, as fo the thick walled pipe, it is assumed that the stess σ z =0 and the pessue is unifom. The pessue distibution is the same as in equations (B. 13) and (B. 14) in Appendix B. The theoy is detailed in [6]. When the mateial is thin, the stess though the thickness can be neglected. Such stuctues ae often called membane, since thee only ae stesses in the cicumfeential diection, φ-diection. A figue of the geomety is found below. 6

14 Figue 5. Geomety of a thin walled pipe. If the stuctue is in equilibium, the state of the stess can be descibed by the equations below, whee a is the aveage adius and t is the thickness as shown in Figue 5 above. σ = 0 ( 18 ) P a σ ϕ = ( 19 ) t The deflection in a thin walled pipe stuctue is given by the same equation as fo the thick pipe. a δ = σ ( 0 ) E When a shell is exposed to intenal pessue, pevious studies ae sufficient, but to study the themal load a wide study is necessay. If the shell exposed to the themal load is fee in both ends no themal stess occus. This poblem has aleady been solved. When one end is clamped, bending stesses will occu and a study of this condition is necessay. In this thesis, only themal and intenal pessues ae discussed, and only if the ends ae clamped is this infomation of inteest..3 Elliptical geomety In this section, the issues with elliptical geomety ae discussed and the infomation found is discussed and used as a platfom fo futhe discussions. Since the displacement cannot be assumed small in this analysis, the theoies to be used ae moe geneal. Since the membane condition cannot be defined though the geomety in an elliptical stuctue exposed to an intenal pessue, the effect of bending stesses must be consideed. See [8] and Appendix C fo futhe infomation. To study this moe geneal theoy, some simplifications and assumptions ae made. The theoy in [8] is only tue fo small deflections, and axial and stability effects ae not 7

15 consideed. The wall thickness and pessue ae unifom. The stesses and deflections that ae found in Appendix C ae: 6 K1 Pz a Pz a σ = + ( 1 ) 1 max t t 6 K Pz a Pz b σ = + ( ) max t t 4 C1 Pz a δ y = ( 3 ) D 4 C1 Pz a δ x = ( 4 ) D Note that these equations can only be used when the deflection is assumed to be small and linea. When the deflection is lage, the stuctue will eact diffeently. If lage deflections ae included in an FE analysis (ANSYS), the esults indicate that the elliptic stuctue at maximum becomes moe cicula. This is because the bending stiffness is too small to conseve the elliptical shape, which is the most ealistic case when the stuctue is exposed fo high loads. The delimitation is due to the stuctue becoming stiffe when it is defomed to a moe cicula geomety. The diffeence between cicula and elliptical geomety is the bend-shea foces that appea in the elliptical geomety duing defomation. Since the adii a and b diffe, the aeas that the pessue acts upon ae not the same. The bend-shea-foces can take some of the diffeence between both pessues and the stuctue becomes stiffe. No theoetical equations fo this behaviou wee found in the theoetical study of this thesis. This statement is only based on FE simulations..4 Conclusions This chapte has discussed diffeent theoies that can be used when analysing a cicula and an elliptical stuctue. Diffeent methods ae studied because it is useful when analysing simulation esults. These esults can be compaed to diffeent theoies that give infomation about how the stuctue woks. Diffeent theoies may be accuate fo diffeent simulations, depending on how the simulation is pefomed. The analysis of the two geometies is based on equation ( 5 ). Theefoe, it can be assumed that some of the assumptions may be tue in both cases. The simulation esults ae compaed to the theoy. 8

16 3. ANSYS building the model This chapte is about the witten scipt and how it is used to veify the theoetical esult. The pogam codes used to make the simulations and the diffeent choices made ae discussed. The diffeent methods analysed ae discussed moe in detail to povide a wide explanation and undestanding to the method used in the analysis. To ceate a pogam and make the coect choice, [9] was a helpful souce fo infomation, much of which is used in this chapte. 3.1 Pogam desciption To veify the theoetical esults, simulations in the FE code ANSYS 10.0 wee done. ANSYS is an FE pogam that can be used to simulate, e.g. stength poblems. A witten pogam code was used within ANSYS to simplify the simulations. The pogam included the geomety, bounday conditions and loads, and was witten in a geneal way. A pogam that can poduce both a cicula and an elliptical geomety was also witten Cicula and elliptical geomety The pogam, based on the equation descibing the geomety, is found in [10]. The same equation can be used to poduce both geometies, in fact the cicula geomety is a special case of the equation, see below. The paametes that allow this ae the adii a and b. If they ae equal, a cicle is poduced, in any othe case the geomety is an ellipse. x y + = 1 ( 5 ) b a One possibility is to use this equation to poduce the shape, though in this pogam scaling is used instead. The diffeent methods consideed ae detailed futhe on. The pogam was witten in such way that the need to change constants and paametes between diffeent simulations was minimized. Appendixes D-F pesents the pogam code, with a shot summay done hee to explain what the pogam does. How the choice was made is pesented late in this chapte, as well as discussions. Because of symmety, only one-fouth of the coss section is needed to pefom the analysis. This assumption can be used fo both cicula and elliptical geomety, which ae only supposed to have deflections, as in Figue 6 below. 9

17 Figue 6. Desciption of bounday condition and expected deflection diections. How the diffeent pats wee chosen and evaluated ae in sections 3. and 3.3, and the simulations and esults ae found in chaptes 4 and Geomety modelling This section will discuss the diffeent analysed methods used to poduce the cicula and the elliptical geomety. Section mentions the scaling method as the one chosen, but this and othe methods ae possible. This section will discuss these issues. The method s goal is to poduce a cuve that can be cut into equally sized pats of a desied length. The cuve will also be used to povide boundaies and loads. The following section descibes how a poblem can be solved in many ways. The poblem is to find the most suitable solution to the analysed application Equal length in one diection The fist method to be analysed was based on equation ( 5 ). The idea was to poduce a cuve fom the definition, since it made poducing both geometies possible, depending on what values adii a and b wee given. To poduce elements some sot of division was needed. The node was calculated though a loop in the pogam that was based on the x- coodinates. The esult was a nodal cuve whose distance in the x-diection was always the same. The method could be used to poduce both cicula and elliptical geomety. Howeve, the poblem with this method was that the element length inceased as the slope in the y-diection inceased. At this position it is desiable with a shot element to get good esult fom the simulations. Hence, this method was not used. 3.. Cuve though splines The second method was a esult fom the fist analysis. The goal was to find a method that could give elements with a smalle length at the end point. By using the same method as befoe to poduce nodes, key points ae instead ceated to complete the geomety. The distance is still equal in the x-diection, but now is a spline poduced fom the key points. With this method a spline can be poduced fom a maximum of 00 key points. This method allowed the cuve to sepaate into two diffeent pieces, whee each spline could be ceated fom 00 key points. This inceased the amount of elements in the aea of inteest. 10

18 The poblem with this method is the position whee the two splines ae linked togethe. Since the slopes fom both splines ae not a pefect match at the intesection point, a small discontinuity appeas, esulting in intenal otations and stess concentations. The stess concentation at the connection point distubs the esult. The esult is also had to intepet at the bounday place at the x-axis, since much distubance comes fom the intesection point. Hence, this method was not chosen. The second analysis was pefomed with a spline cuve to ceate the cuve as befoe with the key points and then use the ANSYS command LESIZE. This esulted in a cuve whee the elements wee of the desied equal length. Still, the spline is only an appoximation of the coect shape and the discontinuities emained Pola coodinates Anothe method that was simultaneously analysed and built on the equation that defined the cicula and elliptic cuves was an attempt to set the distance between the nodes diffeently. The geomety can be descibed fom pola coodinates. The idea was to set the distance with decided degees o adians in the cicumfeential diection. The esult would have been exact fo the cicula and almost as good fo the elliptical geomety. This method was not used because the pogam code would have been unnecessaily complicated fo the ellipse and simple methods exist The command cicle A simple way to attain a good esult fo the cicula geomety is to use the ANSYS command CIRCLE. This command poduces a pefect cicula ac, but thee is no simila command to ceate an ellipse Scaled cicle The last tested method scaled a unit cicle using the constants a and b, which coesponded to the adius in the x- and y-diections. This method gave a pefect cuve by using the command LESIZE to set the equal division along the cuve. This esulted in a modelling method that allowed the obtaining of equal sized elements. The command also pemits changing simply fom a cicle to an ellipse. This method was chosen, since it fulfilled both wishes. 3.3 Element selection In the pocess of witing the pogam, much enegy and time was used to find and select the most suitable element. To be fulfilled wee the demands that the element could handle pessue and tempeatue loads. No othe delimitations than the load desciption wee made. The matte of how the geomety was illustated in the coodinate system poved to be a impotant delimitation. The cuve was defined in the xy-plane, esulting in poblems with the use of cetain elements. 11

19 3.3.1 Beam elements Figue 7. The geomety of the Beam3 element. The fist type of elements consideed was beam elements. Most of the elements in this categoy can be used fo the desied loads. In the analysis, an element called Beam3 was used, one of the simplest beam elements to define. This element is defined by two nodes and can epesent a two-dimensional geomety. The geomety can be made fom the cuve and no othe thicknesses than those in the element ae equied. The element has thee degees of feedom, UX, UY and ROTZ, which can be used to set the bounday conditions. The esult of the analysis was not assumed to be coect. When a simulation with this element was analysed, intenal otations wee found and the deflections wee way too huge. The theoetical study evealed a numeical value to compae with, though the esult fom the analysis was not a match. One eason could be the lack of degees of feedom in the z-diection and in the otational diection. The degees of feedom could pehaps be necessay to minimize the otations Link elements One idea was to use link elements to study how a simple element would eact on the load, though it is not possible since link element cannot allow pessue loads o tempeatue loads nomal to the suface. This led to an expanded study whee plane and shell elements wee consideed Plane elements Figue 8. The geometic desciption of a Plane4 element. 1

20 To analyse plane elements, the geomety had to be modified, since the plane element needs at least fou nodes to be defined. The geomety was made with scaling and LESIZE, but two cuves wee made with the same distance as the thickness, t. Fom this geomety, an aea was ceated. Thee elements wee noticed though the thickness. An element called Plane4 was used in this analysis, epesenting a section in a twodimensional space. The element is quite simple, is defined by fou nodes, and has only two degees of feedom, UX and UY. Afte the investigation, the esult fom the simulation was disappointing, epesenting appoximately only 75% of the theoetical value. Also, it is possible hee that the poblem is dependent on the low numbes of possible degees of feedom. Also consideed was how the thee layes of elements affect the deflection. Can this modelling method poduce the coect esult with the intenal pessue o does the load not affect the element on the oute bounday coectly? Since the diection of the deflections seemed coect, it is impossible that the laye of elements ceates a poblem. An analysis whee the elements though the thickness wee set to one element did not impove the esult. This esulted in the plane element no longe being consideed and that the degees of feedom that might have caused the poblem Shell elements This sot of elements was constantly consideed, but seveal diffeent shell elements had to be analysed befoe finding one to choose. The fist thee shell elements types that wee consideed, Shell51, Shell61 and Shell08, had some sot of axi-symmety. The names of the studied element ae Shell51, Shell61 and Shell08. The elements all epesent twodimensional geomety and can be defined fom two nodes. The diffeence between them is the numbe of degees of feedom. Shell51 and Shell61 have fou degees of feedom (UX, UY, UZ and ROTZ), while Shell08 has only thee (UX, UY and ROTZ). Hee, the poblem was not the degees of feedom, but athe in which plane the geomety was placed. The axi-symmety line fo all consideed elements was the y-axis. The geomety was subsequently placed in the xy-plane and the geomety was otated ound the y-axis; hence, no cicula o elliptical shells wee ceated. The esult of the otation is a half sphee o a half ellipsoid. This explained why the simulation did not give the coect esult. One attempt to place the geomety in the zx-plane was made, but this did not yield any usable esult. The geomety was made as befoe, but the key points wee placed in the zx-plane. The poblem, howeve, was whee to place the load. The element did not have any face in the desied diection to place the load on. This might be possible to solve, but the shell elements in this case that ae not axi-symmetic wee instead chosen to be analysed. 13

21 Figue 9. The geomety of the 3D element, Shell43. The element chosen was called Shell43, since this element woked out in the desied manne duing the study. This element epesents the geomety in thee dimensions and needs fou nodes to be defined. The element has six degees of feedom, UX, UY, UZ, ROTX, ROTY and ROTZ. To use this element type, the geomety needed to be modified. Instead of a simple cuve used fo the othe shell elements, a stipe with the thickness t in the z-diection was made. The method to pefom the acs is as befoe with scaling and LESIZE. To pefom this simulation, the hadest pat was to set the bounday conditions, which is descibed in moe detail in section The esult of this analysis was satisfying. The esult pefectly matched the theoetical values fo a cicula geomety. With the esult fom the analysis of the cicula geomety, it was assumed that the element type could be used to analyse the elliptical geomety. Figue 10. The gaphic desciption of the themal 3D element, Shell131. A themal element simila to the stuctual shell element used fo the pessue analysis should be chosen. The easiest way to find this element was to use the functions in ANSYS, which change one element type fo one load type to an equal element that can handle the new load. This investigation and the eading in [9] indicated that the most suitable element fo the themal simulation was Shell131. Since the tempeatue changes though the thickness, obtaining an element that included the needed layes and Shell43 was not possible. This is the main diffeence between the two element types; othewise the elements substitute each othe well. Shell131 is a themal element with fou nodes 14

22 and no degees of feedom, but can be used in 3D. Its layes pemit studying the stesses at the top and bottom of the shell. A layeed element is also needed to apply the themal load with a linea vaiation fom one side to the othe. To pefom the themal simulation a mateial model was needed, fo which the Hastalloy X model was used with mateial popeties that vaied with tempeatue. To pefom the themal simulation two scipts needed to be developed. Figue 11. The geomety fo the stuctual 3D element, Shell181. Since the pogam is divided into one themal and one stuctual pat, the same element type is not used. To use the esult fom the themal simulation, a layeed stuctual element, Shell181, was used. A commando in ANSYS automatically changes the themal element into an equal stuctual element. This method was used to find the coect element fo this application. Biefly, this element is the stuctual vaiant of the themal. The element has fou nodes, is used in 3D and has six degees of feedom UX, UY, UZ, ROTX, ROTY and ROTZ. 3.4 Constaints The boundaies ae poduced of seveal diffeent components, e.g. the font and back side of the simulated model coupled in the x- and y-diections. The geomety is shown in Figues The faces 4 and 6 epesent the font and back side in this analysis, since the geomety shall acts as a stiff stuctue and the deflection should be the same in the stuctue fo a specific value at the x-coodinate. Because only one of the cicumfeential boundes is fixed, the stess in z-diection becomes small and can be neglected. This seems coect. If the bounday conditions at the end of the stipes ae set coectly, the deflection is only pefomed in the x- and y-diections. When this esult is pefomed fo both the cicula and the elliptical geomety the pogam is assumed coect. 3.5 Loads The load used in the fist case is an intenal pessue and in the second case, a tempeatue load. The pessue load is applied as a unifom load placed on evey element. The themal load is applied in two ways. Duing the fist simulation, the tempeatue is applied and then the equilibium is found. The tempeatues on the inne and oute 15

23 suface become diffeent depending on the thickness of the stuctue. Afte the themal evaluation is done the themal load is applied diffeently. The tempeatue is applied as a fixed tempeatue on the sufaces and is maintained duing the simulation. 3.6 Analysis The geomety and themal load ae found in the fist pogam. The second pogam uses the model fom pogam one and changes the themal load into a stuctual load. The bounday conditions ae the same as fo the pessue load. In this simulation the layeed element used is Shell131. The pessue load can be applied in the same pogam as in all othe opeations, since this is a stuctual foce and the analysis is stuctually pefomed. The tempeatue is handled slightly diffeently. The pogam is similaly constucted. The themal esult in a second pogam code is then tansfomed into stuctual foces that ae solved as befoe. Fo the last simulation whee the themal load is combined with the pessue, the pocedue is the same as fo the themal case. The diffeence is that the pessue is added in the second pogam. 3.7 Conclusions It is clea fom the section on how pogams ae witten that the assumption about the common equation ( 5 ) is of inteest. Since the same equation can be used to descibe two diffeent geometies, cicula and elliptical, assuming that the same element types and bounday conditions can be used is possible. Because of how the elements wee defined, using the same elements fo tempeatue and pessue load was not possible. In the pogam the same bounday conditions ae used. Radii a and b ae the only vaiables that ae changed togethe with a load change, depending if the load is pessue o tempeatue. The numbe of tested methods to establish the pogam code demonstates that discening which method will wok best is not easy. This is also a esult of the fact that in pogamming, even though thee ae many ways of attacking the poblem, the diffeence lies in how good the esults ae. Diffeent methods ae also geneally complicated to pefom and often, the easy way is the best way. As well, many diffeent element types wee tested. Because choosing an element is not an easy task, testing is necessay. This gives a deepe undestanding of how the pogam ANSYS woks. The completed compute pogams wok well and equie a minimum of changes. If the pogam is assumed to compute coectly, the bad esult must depend on something else, fo example the input values o how the esults ae intepeted. 16

24 4. The simulations This pat of the thesis focuses on the simulations, and declaes how and why they wee done. Model evaluation is the fist pesented followed by a section that discusses optimization paametes. The chapte ends with which loads this study focuses upon. 4.1 Model evaluation When the pogam code seems to wok popely, cetain tests must be conducted to evaluate the pogam because of the fact that the theoy fo elliptical geomety can only be used fo small defomations. To evaluate that the pogam woks popely, some simulations with a cicula geomety wee done, the esults of which wee compaed to the theoetical calculations and how they matched fo both small and lage deflections Visual evaluation A visual test was initially pefomed and only the behaviou that could be seen when the deflections wee studied was discussed. Small defomations fo the elliptical geomety only wok when the intenal pessue is low. When the pessue eaches a cetain level the deflection becomes unealistic and the geomety is stetched moe than the oiginal length. This occus because thee ae no delimitations to indicate that this is not possible. The simulations wee then pefomed with lage defomations. The gaphic esult now looked as expected. Afte moe specific delimitations wee done, the diffeence in geomety cannot deflect moe than a cicula geomety Theoy evaluation The second evaluation was done to compae the simulation esults with the gaphs found in [11], and ae only tue fo small deflections when linea theoy can be used. The theoy cannot be used if the simulation esults in lage defomations, but if the same gaphs show the same esults, the esults can be compaed. The gaphs also show when the theoy can be used. The gaphs fom these simulations, built on the theoy in [8], ae found in Appendix G. The gaphs show that the theoy can only be used fo linea poblems, i.e. the deflections and defomations must be small. Fom the gaphs in this case, it is obvious that with the desied pessue at 0.5 ba, the deflections cannot be assumed to be small and linea theoy cannot be used. It was not possible to find a theoy that included the nonlinea effects duing this thesis wok, which would be vey difficult to develop. Hence, it is concluded that the theoy cannot be used and the esults ae only veified in the simulations Sensitivity evaluation The thid evaluation was conducted to investigate how sensitive the model and the simulation esults ae to the numbe of elements used. The simulations wee planned with n=4,000, i.e. the line is going to be cut into 4,000 pieces of equal length. Though the small thickness in the global z-diection the pogam makes thee elements, leading to a 17

25 default numbe of elements of 1,000. If the sensitivity in the model is to be investigated, simulations with moe and less elements must be pefomed. The two diffeent cases with n=,000 and n=6,000 will be used to detemine the sensitivity in the model. When n=,000, the numbe of elements becomes 6,000 and when n=6,000 the numbe of elements becomes 18,000. These two cases will be compaed with the default model. The simulations and compaison ae done fo two thicknesses, t=0.00 and t= Whethe the sensitivity vaies with the mateial model will also be investigated. The two models ae line elastic (no mateial model) and kinematical hadened model. The esult is seen in gaphs1 and below. As mentioned befoe no mateial model is defined, since no mateial has yet been decided upon fo the constuction. The linea elastic model does not include any delimitation, such as yield stength and tensile stength. This means that the mateial model does not include plastic effects. In a tue mateial, this state can neve occu; studying how this affects the esult would be inteesting. The kinematical hadened mateial model includes the yield stength and tensile stength and does not epesent any eal mateial in this study. The values chosen σ s =00 MPa and σ b =600 MPa ae ealistic fo mateials that ae possible in this application. In Figue 7 below ae the two mateial models gaphically descibed. Figue 1. Gaphic desciption of the mateial model whee (a) epesents the linea elastic model and (b) descibes the kinematical hadened model. 18

26 Sensetiveness analyze Sensetiveness analyze kinhmtl Stess [Pa] 1.40E E E E E E+08.00E E E E+08 stess_x t=0,00 n=4000 stess_x t=0,00 n=000 stess_x t=0,00 n=6000 Stess_x t=0,007 n=4000 stess_x t=0,007 n=000 stess_x t=0,007 n=6000 Stess [Pa] 4.00E E+08.00E E E E E+08 stess_x t=0,00 kinhmtl n=4000 stess_x t=0,00 kinhmtl n=000 stess_x t=0,00 kinhmtl n=6000 stess_x t=0,007 kinhmtl n=4000 stess_x t=0,007 kinhmtl n=000 stess_x t=0,007 kinhmtl n=6000 Gaph 1. The stesses fo sensitivity analysis with a linea elastic mateial model. Gaph. The stesses fo sensitivity analysis with a kinematical hadened mateial model. The fist paamete studied is stess. Fo both mateial models, the same esult is obscued, and the stess levels ae not dependent of the numbe of elements. This is shown fo both thicknesses and mateial models. The biggest diffeence occued between the mateial models. The esult shows that the choice of mateial model geatly influences how the stuctue should be optimized. This point is detailed in section 4.3. and chapte 5. Sensetiveness analyze Sensetiveness analyze kinhmtl Displacement [m] displm_sum t=0,00 n=4000 displm_sum t=0,00 n=000 displm_sum t=0,00 n=6000 displm_sum t=0,007 n=4000 displm_sum t=0,007 n=000 displm_sum t=0,007 n=6000 Displacement [m] displm_sum t=0,00 kinhmtl n=4000 displm_sum t=0,00 kinhmtl n=000 displm_sum t=0,00 kinhmtl n=6000 displm_sum t=0,007 kinhmtl n=4000 displm_sum t=0,007 kinhmtl n=000 displm_sum t=0,007 kinhmtl n=6000 Gaph 3. The displacement fo sensitivity analysis with a linea elastic mateial model. Gaph 4. The displacement fo sensitivity analysis with a kinematical hadened mateial model. The second paamete investigated was deflection. The esult showed that the deflection between simulations with diffeent numbes of elements was small. The mateial model only changed the size of the deflection, not the shape of the cuve. This is one diffeence compaed to stesses, which not only changed in size, but also in shape. Afte these two paamete investigations, it becomes clea that the model is not sensitive accoding to the numbe of elements used. This ends the evaluation of the model, which is now assumed to wok as desied. Chapte 5 pesents the esults fom simulations with two diffeent mateial models. The simulations ae based on the wost case scenaio fom ealie simulations when the thickness was unifomly 0.00 m and the pessue 0.5 ba. 4. Optimization paametes When the pogam woks as planned and is evaluated, it is time to begin the last pat. This pupose of this thesis was to find a moe systematic way to place the stiffenes. To 19

27 do so, the esults fom a pefomed analysis must be studied in detail. Which paametes ae of impotance and how should this infomation be used? This section ties to answe these two questions. Depending on what the optimization is fo, diffeent paametes ae of inteest. Must the stuctue withstand fatigue, weight, deflection o stesses? Befoe this question can be answeed, a basic undestanding of the stuctue and its eaction to loads must be fomulated. No final solutions ae suggested in this thesis, only infomation that gives the base fo futhe wok. To find the best way to place the stiffenes, the diffeent optimization paametes must fist be defined and then weighted against each othe. The optimization paametes then give the factos that decide what to focus upon. To find the optimized stuctue, a good idea is to ceate a pogam that calculates the solution, depending on which values and paametes ae given, and consides the most dangeous paametes. Unfotunately, this thesis did not have the time necessay fo this wok. 4.3 Pessue load The fist case to analyse involves intenal pessue. Duing the wok to find the optimized stuctue and an undestanding of how the stuctue is defomed, seveal simulations wee done. A common assumption duing this test was that the pessue was held at 0.5 ba and the adii a=0.,5 m and b=0. m. In this fist case, the simulation was pefomed as befoe, without any specified mateial model. The mateial was only supposed to be linea elastic because no mateial was as yet decided upon fo the low signatue nozzle. The simulations included only the intenal pessue; its pupose was to gaphically show how the hoop stesses (stess_x) vay along the x-coodinate in the model fo diffeent coss sections. The coss section only showed how the thickness in the stuctue vaied. Thee diffeent kinds of simulations wee made, whee the thickness was unifomed in the fist, the thickness vaied linea fom one end to the othe in the second, and vaied linea fom the ends into the middle in the thid. Fo moe detail, see Appendix M. The load, the intenal pessue, is applied as nodal foces and is unifomly distibuted ove the inside of the stuctue. Of inteest fom these simulations is how the stesses and deflection vay along the x- coodinate. All esults fom this pat of the wok ae found in the diections of the elements. To get this infomation a local coodinate system at the element is used. This is done because the infomation needed is in the cicumfeential diection of the stuctue. If the infomation is instead fom the global coodinate system, the esult would not be epesentative, since the element s diection vaies. Close to the global y-axis, the global and local coodinate systems ae in the same diection. If the distance to the global y-axis inceases the diffeences between the coodinate systems becomes moe visible. This esults in the local coodinate system being the coect one. 4.4 Tempeatue load This section discusses the simulations that only include a tempeatue load. A themal static analysis is pefomed to calculate the tempeatue distibution on the nozzle. The themal load is assumed to be calculated in advance in, e.g., a CFD analysis and is given 0

28 as convection loads descibed by bulk tempeatues and heat tansfe coefficients on the inside and outside of the nozzle wall. A linea tempeatue gadient is assumed though the thickness of the elements. Since the themal load is constant, the tempeatue gadient will vay with the thickness of the mateial. The tempeatue field is ead into the stuctual analysis. All simulations ae pefomed with the linea elastic mateial The simulations The geomety is oientated along the x-axis, as befoe, and the thickness vaiation is shown in the coss section. Thee diffeent kinds of simulations ae conducted as befoe, i.e. unifomed and diffeent vaiations of linea thickness. In this case a possible mateial, Hastalloy X (a Ni based alloy), is used in the FE model. The data vaies with tempeatue. When the simulations descibed above wee pefomed, a second issue handled what popety deives the stess in the stuctue: Does the thickness o the geomety deive the stess levelling in the mateial? To answe this question a second set of simulations was needed. The simulation was based on using the same tempeatues fom the fist simulation. The tempeatue on the sufaces depends on what case is studied. In this case, only the unifomed thick stuctue is studied. The mm and 15 mm stuctues ae chosen. The point is to foce the stuctue to have a cetain tempeatue on the suface that will give the same gadient though the thickness fo both analysed stuctues. Since the tempeatues ae given in the stuctual pogam, thee is no need in this case fo a themal simulation. Only the esulting tempeatues fom the fist case ae needed. Appendix K pesents the esult. 4.3 Combined tempeatue and pessue The thid load case that was simulated included both a tempeatue load and an intenal pessue. The simulation was based on the same pogam used to pefom the othe two simulations. Also, the same assumptions wee made, specifically that the pessue is still 0.5 ba and the tempeatue field was ead fom the themal static. The adii fo the elliptical geomety ae a=0.5 m and b=0. m. The analysed stuctues ae educed to two vaying thicknesses. The fist has t= mm at the left end and t=15 mm at the ight end. The second geomety is just mioed. 4.4 Conclusions Fom the evaluations descibed in sections to 4.1.3, the following conclusions wee discoveed. All simulations, pessue, must be pefomed with lage defomations, othewise the esult will not be coect fo the used load, 0.5 ba in this analysis. The theoy found in [8] can only be used fo linea poblems, i.e. the deflections and defomations must be small. In this analysis, small defomations cannot be assumed. The evaluated sensitivity showed that the model not was sensitive to the numbe of elements used in this case. 1