32 Al-khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol.3, No2,pp32 -48 (2007) Effect of the Mechanical and Thermal Stresses of Rotating Blades Assist. Prof. Dr. Nabeel K. AL-sahib Malik M. Ali Baghdad University / AL-Khwarizmi College of Engineering (((RRR eee ccc eee iii vvv eee ddd111111DDD eee ccc eee mmm bbbeee rrr 222000000666 ;;; aaacccccc eee pppttteee ddd 222555 AAA ppprrriii lll 222000 000 777 ))) Abstract: Rotating blades are the important parts in gas turbines. Hence, an accurate mathematical estimation (F.E.M) of the stresses and deformations characteristics was required in the design applications to avoid failure. In recent year’s there are researchers interest in the effect of temperature on solid bodies has greatly increased, The main of this study investigated the thermal and rotational effects. So, the thermal stresses due to high pressure and temperature are studies, also determine the steady state stresses and deformations of rotating blades due to mechanical effect. Many parameters such as thickness and centre of rotating are investigated in this paper. The study results can ensure good recommendation for the effect of the mechanical and thermal stresses of rotary blades. Keywords: Thermal Stresses, Finite Element Method, Mechanical Stresses , Blade Material , Design . Introduction Rotating blades was one of the common structural elements used in several practical machines such as turbomachinery system, turbojet engines, turbofan, and helicopter rotors. Because turbomachinary blades are assumed as a shell, the numerical technique should be capable of representing this complicated structure based upon the shell theory. In this paper the type of rotating blades were used is rectangular blade, the geometry for this type was show in figure (1). Ramamurti and Sreenivasamurthy [1] studied the dynamic stress analysis of rotating twisted and tapered blades. The finite element method was used to determine the stresses and deformations. Three-dimensional, twenty-node isoparametric elements have been used for the analysis. Extensive analysis has been done for various pre-twist angles, skew angles, breadth to length ratios, and breadth to thickness ratios of the blades. Experiments were carried out to determine the stresses for the verification of the numerical results. Lee [2] determined the frequencies and mode shapes of turbomachinery blade having both camber and twist. The body forces were considered to be centrifugal forces generated within the shell due to rotation of the blade. Omprakash and Ramamurti This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)33 33 metal. But for high temperature gas turbines, “Nimonic” alloy series are recommended. They are basically Nickel-Chromium alloys, which withstand oxidation, mechanical stresses, and creep at working temperatures in excess of 1000o C. in the middle part of the turbine, the temperature and stress conditions are moderate, but corrosion is likely. Brass is probably the most suitable material. In the final stages, and especially in large machines, the blades are long and heavily stressed. Fatigue action due to vibration may exist. Further than this, anti corrosion properties are also desirable. Brass is a suitable material where the stress conditions allow of its use, but when the stresses are high, 5% Nickel Steel, Stainless Steel, Monel metal or Phosphor Bronze may be used. Turbine Blade Design Turbine blades operate at 1400 0C and are required to have a service life of 10000- 20000 hours. The techniques used to meet these requirements are: 1) The use of nickel based superalloys (containing many precipitates, solid solution atoms and having high oxidation resistance) 2) The blades are cast a single crystal. This means that there are no grain boundaries within the structure and thus minimizes creep see Fig. (2) 3) The blades are internally cooled to allow increased operating temperatures. Typical cooling design features are shown in Fig. (3) 4) The blades are coated to increase the oxidation resistance as shown in Fig. (4). A timeline of techniques to raise engine- operating temperatures is shown in Fig. (5 &6) [3] carried out the steady state, dynamic stress and deformation analysis of high pressure stage turbomachinery bladed discs taking into account all the geometric complexities involved and included the contributions due to initial stress and membrane behavior, which used a triangular shell element with six degrees of freedom per node. Yoo [4] studied the vibration analysis of rotating pre-twisted blade with a concentrated mass. The blade has an arbitrary orientation with respect to the rigid hub to which it is fixed. The equations of motion are derived based on a modeling method that employs hybrid deformation variables. The resulting equation for the vibration analysis was transformed in the dimensionless parameters on the model characteristics of the rotating blades were investigated through numerical analysis Blade Materials The elimination of blade corrosion and designing against creep and fatigue, to a large extent, is a problem of finding suitable materials. Another factor, especially in high-pressure work, is the working temperature of the blades. Under high temperatures, some materials tend to disintegrate, and certain mechanical properties, e.g. hardness (in the sense of resistance to abrasion), are affected adversely. The properties, which are desirable for example in turbine blading, vary according to the conditions of operation and according to the location of the blading in the turbine [5]. At high- pressure end, the temperature is high, and the blades are sometimes subjected to erosive action. Great strength is not required as the blades are usually short. The material should be hard at high temperatures. Suitable materials are 5% Nickel Steel, Stainless Steel, and Monel This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)34 34 Finite Element technique The difficulties in evaluating the stress and deformations of rotating blades comes from the complex geometry of blades due to asymmetry of cross-section, pre-twist, and mounting the blades at a skew angle to the rotating disk. The pre-twist and skew of the blade cause coupling in both bending directions, and the asymmetry of the cross- section causes coupling with the tensional motion of the blade. For the above complexities, all previous investigations in this field had declared that plate or shell analysis for blades is suitable and reliable. Therefore, the shell theories with finite element provide a useful tool to analyze such structures. A superparametric shell element will be used to investigate the thermal and mechanical stresses of rotating blades. This element consists of four corner and four midside nodes as shown in Fig. (7). The degrees of freedom considered at The s are the three translations u, v, w of the midsurface and two rotations α and β of the normal to the midsurface. The Cartesian coordinates of any point of the shell and the curvilinear coordinates can be written in the form:                                   h n m l f z y x f z y x i i i i i i i i 3 3 3 2  ……(1) where l3i, m3i and n3i are the direction cosines of the normal to the middle surface. Here fi is a function taking a value of unity at the node i and zero at all other nodes, it is called as “shape function “(4).as shown in table.(1). In the kinematics formulation two assumptions are imposed: 1- Nodal fiber is inextensible. 2- Only small rotations are considered. The displacements at any point (ξ, η, ζ) can be expressed in terms of the nodal displacements as                               i i i i i i i i i i i f h N w v u f w v u    2 8 1 8 1 ... …. (2) In this formula the symbol i denote the following matrix:               ii ii ii i nn mm ll 12 12 12  Column 1 in this array contains negative values of the direction cosines of the second tangential vector V2i , and column 2 has the direction cosines for the first tangential vector V1i. These vector are orthogonal to the vector V3i and to each other. The strain in the direction the normal to the mid-surface is assumed to be negligible ( z  ) The normal to the mid-surface of the shell element will remain normal to the mid- surface after deformation. The displacement shape functions may be cast into the matrix form: [fi]=[fAi]+ζ [fBi] (i=1,2…8) ………(3) where [fAi]=           00100 00010 00001 , fi & [fBi]=              ii ii ii nn mm ll 12 12 12 000 000 000 2 ih fi …..… (4) The 3 X 3 Jacobian matrix required in this formulation is: This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007) 35 35 [J]=              ,,, ,,, ,,, zyx zyx zyx ……………………(5) where “,” indicate differentiation with respected to the symbol following the coma. The derivatives in matrix [J] can be found from equation (1) i h i i i h i ii i i i h i ii i i lfx lfxfx lfxfx i i i 32 8 1 , 32 8 1 , 8 1 ,, 32 8 1 , 8 1 ,,               And so on For this element, six types of non-zero strains exit, as follows:                                              zx yz xy z y x zx yz xy z y x uw wv vu w v u ,, ,, ,, , , ,        …….…. (6) The stress-resultant vector in the local coordinate system is, {f′}={f x′f y′f x′y′ Q y′ Q x′ M x′ M y′ M x′y′ } T………..(7) Where Q x′ was the shear stress per unit length in the x and y direction. The relationship between the stress resultants and the generalized strains can be stated as follows, {f′}=[D′]{ε′} ………………………(8) where [D′] is the rigidity matrix. A typical rigidity matrix for flat plate shell is given                                   24 )1( 1212 1212 2 1 2 1 2 1 2 2 22 22 0000000 000000 000000 0000000 0000000 0000000 0000001 0000001 1          h hh hh k kEhD ..(9) k =shear shape factor (assumed k=1.2) (5). Numerical Solution Two cases were studied, steady state and thermal stresses as follows: Steady -State Analysis In this section the steady state stresses and deformations were calculated, The blades considered were made of steel; and the material properties are: modules of elasticity = 207 * 103 Mpa, Poisson’s ratio = 0.3, Density = 7850 kg/m3. Other information are: Speed of rotation = 5000 r.p.m, Disk radius/width = 4, width = 0.05m. In this respect, a numerical study for a rectangular cross-section blade is given in details because it is a common model. Comparison of the results A comparison with experimental published results is achieved between the present work and the experimental work [1]. In this comparison, a rectangular steel blade is considered, having the following dimensions and material properties: Length = 100 mm, Width = 25mm, Thickness =3mm, Skew angle = 90o , Radius to root = 125mm, and maximum speed of rotation = 2500 r.p.m, Modulus of elasticity = 207 * 103 Mpa, Poisson’s ratio =0.3, Density = 7850 kg/m3 . Strain gauges were used as transducers and the results are shown in Table 2 Effect of Rotational speed The variation of stresses and deformations with eight speeds of rotation (250, 500, 750, 1000, 1250, 1750, and 2000 r.p.m) for three aspect This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)36 36 ratios (3, 4, and 5) are investigated and shown in fig. (9,10, and 11) show the variations of v-deflection, xx- stresses, yy- stresses respectively with speed of rotation. It was observed that stresses and deformations increase that when the speed of rotation increased too. Effect of thickness In order to investigate the thickness of blade, eight thickness (1.5, 2, 2.5, 3, 3.5, 4, 4.5 and 5 mm) and three aspect ratios (3, 4 and 5) were selected for this investigated. Figs (12, 13, and 14) show the variations of v- deflection, xx-stresses, yy-stresses respectively with thickness. It can be noted that increases of the blade thickness leads to decreases in the stresses and deformations (i.e. thin blades give greater stresses and deformations than the thick blades) and that due the higher reduction in the structural stiffness. Also, it can be seen that when the aspect ratio increases the reduction in stresses and deformations increases. Blade Deflection due to Thermal Stresses The blade is relatively thin, and thus the thermal gradient through the thickness of the blade is assumed negligible. Therefore, the blade is assumed to have a uniform temperature equal to the surface temperature. This simplifies the thermal stress analysis and results in a simple linear relation:[12] u=αTL ………………………….…(10) Where L= blade length (m), α= thermal expansion coefficient (1/Co) and T=Temperature (Co). Figs (15, 16, 17, and 18) show the variations of temperature u- deflection, xx-stresses, yy-stresses respectively with x-axis and Figs (19, 20, 21, and 22) show the contour of temperature, u- deflection, xx-stresses, yy-stresses respectively. It can be noted that increases of the blade temperature leads to increase in the stresses and deformations Conclusions: The conclusions obtain from the present work can be summarized as when the increases of the blade thickness leads to decrease in the stresses and deformation, (i.e. thin blades give greater stresses and deformations than the thick blades) and that is duo to the higher reduction in structural stiffness, the generated stresses will be increases for the blades having large aspect ratios, as well as the deformations become large, since the total mass of the structure is proportional to the aspect ratio. The increases of the blade temperature leads to increase in the stresses and deformations so, shaped film-cooling and full coverage film cooling are the most useful cooling schemes for reducing the amount of cooling air and decreasing thermal stresses. This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)37 37 This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)38 38 Fig. (2) Turbine blades cast to promote different grain structures [9] Fig. (3) Blade cooling techniques [10] Fig. (4) History of turbine blade coating systems [11] This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)39 39 Fig. (5) Developments to increase engine-operating temperatures [10] Fig. (6) Developments to increase turbine metal capability [8] ξ ηζ ζ=1 ζ=-1 η=-1 η=1 ξ=1 ξ=-1 Fig. (7) Eight noded shell element This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)40 40 Serendipity 8-node element: Corner nodes: )1)(1)(1( 4 1  iiiiif  Midside nodes: )1)(1( 2 1 )1)(1( 2 1 2222   iiiiif Table (1) Shape Function for Midsurface Interpolation of Shell Elements Table 2 Comparison with experimental results. (radial stress) αo Distance Exp. (Mpa) Present work (Mpa) Ω (r.p.m) 0 0.1 L 9.1154 9.0086 2500 15 0.25 L 5.7115 5.5613 2000 z, w βi y, v Fig. (8) Degrees of freedom at a node i This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)41 41 0 1 2 3 4 5 6 0 250 500 750 1000 1250 1500 1750 2000 2250 speed of rotation (r.p.m) v_ d e fle ct io n { m m }x 1 0 ^ - 2 l/b=3 l/b=4 l/b=5 0 1 2 3 4 5 6 7 8 9 0 250 500 750 1000 1250 1500 1750 2000 2250 speed of rotation (r.p.m) xx _ st re ss e s{ M p a } l/b=3 l/b=4 l/b=5 Fig. (9) Variation of v-deflection with speed of rotation Fig. (10) Variation of xx-stresses with speed of rotation Fig. (11) Variation of yy-stresses with speed of rotation 0 3 6 9 12 15 18 21 24 27 30 33 0 250 500 750 1000 1250 1500 1750 2000 2250 speed of rotation (r.p.m) yy _ s tr e ss e s{ M p a } l/b=3 l/b=4 l/b=5 R=0.225 m, h=0.0018 m Skew angle =20o Pre-twist angle = 12o Al-alloy (5052) This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)42 42 0 1 2 3 4 5 6 7 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Thickness (mm) v_ d e fle ct io n { m m } 1 0 ^_ 2 l/b=3 l/b=4 l/b=5 0 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Thickness (mm) xx _ s tr e ss e s{ M p a } l/b=3 l/b=4 l/b=5 0 5 10 15 20 25 30 35 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Thickness (mm) yy _ s tr e ss e s{ M p a } l/b=3 l/b=4 l/b=5 Fig. (12) Variation of v-deflection with thickness Fig. (13) Variation of xx-stresses with thickness Fig. (14) Variation of yy-stresses with thickness R=0.225 m, Ω=2000 r.p.m Skew angle =20o Pre-twist angle = 12o Al-alloy (5052) This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)43 43 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 0 0 . 0 0 5 0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3 x - a x is { c m } T e m p e ra tu re {C } 0 . 0 0 E + 0 0 2 . 0 0 E + 0 2 4 . 0 0 E + 0 2 6 . 0 0 E + 0 2 8 . 0 0 E + 0 2 1 . 0 0 E + 0 3 1 . 2 0 E + 0 3 1 . 4 0 E + 0 3 1 . 6 0 E + 0 3 0 0 . 0 1 0 . 0 2 0 . 0 3 x - a x i s { c m } x x _ s tr e s s e s { M P a } Fig. (15 Variation of temperature with x-axis Fig. (18) Variation of yy-stresses with x-axis 0 . 0 0 E + 0 0 1 . 0 0 E + 0 2 2 . 0 0 E + 0 2 3 . 0 0 E + 0 2 4 . 0 0 E + 0 2 5 . 0 0 E + 0 2 6 . 0 0 E + 0 2 7 . 0 0 E + 0 2 8 . 0 0 E + 0 2 9 . 0 0 E + 0 2 1 . 0 0 E + 0 3 0 0 . 0 0 5 0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3 x - a x is { c m } y y _ s tr e s s e s { M P a } 0 . 3 5 0 . 3 9 0 . 4 3 0 . 4 7 0 . 5 1 0 . 5 5 0 0 . 0 0 5 0 . 0 1 0 . 0 1 5 0 . 0 2 0 . 0 2 5 0 . 0 3 x - a x i s { c m } u _ d e fle c tio n { m m } Fig. (17) Variation of xx-stresses with x-axis Fig. (16) Variation of u-deflection with x-axis This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)44 44 Fig. (19) Contour of temperature Fig. (20) Contour of u-deflection This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)45 45 Fig. (21) Contour of xx-stresses Fig. (22) Contour of yy-stresses This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)46 46 Reference: 1- Ramamurti, and Sreenivasamurthy; “Dynamic stress analysis of rotating twisted and tapered blades”, J. Strain Analysis, Vol. 15, No. 3, 1980 2- J. K. Lee, A. W. Leissa and A. J. Wang, “Vibration of blades with variable thickness and curvature by shell theory”, J. of Eng. For Gas Turbine and Power, Vol. 106, pp. 11- 16, 1984. 3- V. Omprakash and V. Ramamurti, “Dynamic stress analysis of rotating turbomachinery blade-disc systems”, J. of Computers and structures, Vol. 32, No. 2, pp. 477-488, 1989. 4- H. H. Yoo, J. Ykwak and J. Ghung, “Vibration analysis of rotating pre- twist blades with a concentrated mass”, J. of sound and vibration, Vol. 240, No. 5, pp. 891-908, 2001 5- Kearton, W. J. ,”Steam turbine operation”, Sir Petman & Sons, Ltd., 4th ed. 1945 6. Flower, H.M. ed., High Performance Materials in Aerospace, Chapman & Hall, London, 7. Sims, C.T., Stoloff, N.S. & Hagel, W.C. eds., Superalloys II, John Wiley & Sons, New York, 1987 8. Askeland, D.R., The Science and Engineering of Materials: Third SI Edition, Chapman and Hall, London, 1996 9. Callister, W.D., Jr, Materials Science and Engineering an Introduction: 5th Edition, John Wiley and Sons Inc, New York, 2000 10. Ashby, M.F. & Jones, D.R.H., Engineering Materials 1: An Introduction to their Properties and Applications, 2nd Edition, Butterworth Heinemann, Oxford, 1997 11. Sims, C.T., Stoloff, N.S. & Hagel, W.C. eds., Superalloys II, John Wiley & Sons, New York, 1987 12- Aeronautical Materials – Teacher Reference, School of Material Science and Engineering, University of New South Wales 2001 This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)48 48 Internet links General aerospace http://www.howstuffworks.com/sc-aviation-transportation.htm http://www.matweb.com/ http://www.howstuffworks.com/airplane.htm http://www.howstuffworks.com/search/index.htm?words=materials+engineering http://smc.larc.nasa.gov/coe/ http://as.wm.edu/Nondestructive.html ex.html http://www.grc.nasa.gov/WWW/HSR/EPMAirf.html http://www.grc.nasa.gov/WWW/HSR/ind http://www.boeing.com/commercial/aeromagazine/aero_07/corrosn.html http://www.boeing.com/commercial/747family/background.html http://www.boeing.com/commercial/747family/index.html http://www.boeing.com/news/feature/new747/passenger.html Turbines and materials http://www.gas-turbines.com/begin/index.htm#HISTORY http://www.howstuffworks.com/turbine.htm http://www.sti-tech.com/ This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/ Dr. Nabeel K. AL-sahib /Al-khwarizmi Engineering Journal ,Vol.3, No2 . PP32-48 (2007)49 49 تأثیرات االجھادات المیكانیكیة والحراریة للمراوح الدوارة مالك محمد علي عبد الصاحب نبیل كاظم. د.م.أ جامعة بغداد -كلیة ھندسة الخوارزمي :الخالصة أن التمثیل ھي من األجزاء المھمة في التوربینان الغازیة، لذلك لدوارةألمراوح أ لألجھادات والتشوھات ضروري من اجل التطبیقات ) طریقة العناصر المحدودة (الریاضي بتأثیر درجات الحرارة على الباحثین في السنوات األخیرة زاد أھتمام. التصمیمیة لتجنب الفشل جھادات الحراریة الناتجة من ، دراسة األ من ھذه الدراسة الھدف الرئیسي. األجسام الصلبة جھادات المستقرة والتشوھات للمراوح وكذلك تم أحتساب األ. الضغط ودرجة الحرارة العالیین مثل عنھا في ھذا البحث تقصيتم ال العدید من العوامل. ثیرات المیكانیكیةالدوارة الناتجة من التأ ألخ.... السمك ومركز الدوران . جھادات المیكانیكیة والحراریة للمراوح الدوارة ات جیدة لتأثیرات األصینتائج الدراسة تؤمن تو This page was created using Nitro PDF trial software. To purchase, go to http://www.nitropdf.com/ http://www.nitropdf.com/