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Numerical Simulation on Thermal Stresses in An Annular Fin Made of Isotropic Material

Year 2018, Volume: 33 Issue: 3, 67 - 80, 30.09.2018

Ali Yıldırım Durmuş Yarımpabuç , Kerimcan Çelebi

https://doi.org/10.21605/cukurovaummfd.500573

Abstract

In this study, temperature distribution and thermal stresses of an axisymmetric thin annular fin with rectangular profile made of isotropic and homogeneous material are determined by using Complementary Function Method and Pseudospectral Chebyshev Method. Both of the methods are compared with analytical results obtained using Bessel functions and thermoelastic theory. Error analysis of these numerical methods at different partition points is shown using Euclidean norm. It is observed that pseudospectral Chebysev method approximates to the analytical results more rapidly with increasing the number of points. 

Keywords

Annular fin Thermal stresses Complementary function method Pseudospectral Chebysev method

References

  • 1. Biswas, G., Mitra, K., Fiebig, M., 1994. Heat Transfer Enhancement in Fin-Tube Heat Exchangers by Winglet Type Vortex Generators, Int. J. Heat Mass Transfer, 37(2), 283-291.
  • 2. Kraus, A.D., Aziz, A., Welty, J.R., 2001. Extended Surface Heat Transfer, John Wiley and Sons, New York.
  • 3. Incropera, F., Dewıitt, D.P., Bergman, T.L., Lavine, A.S., 2007. Fundamentals of Heat and Mass Transfer, John Wiley and Sons, New York .
  • 4. Mallick, A., Ghosal, S., Sarkar, P.K., Ranjan, R., 2015. Homotopy Perturbation Method for Thermal Stresses in an Annular Fin with Variable Thermal Conductivity, Journal of Thermal Stresses, 38(1), 110-132.
  • 5. Wu, S.S., 1997. Analysis on Transient Thermal Stresses in an Annular Fin, Journal of Thermal Stresses, 20, 591-615.
  • 6. Gardner, K.A., 1945. Efficiency of Extended Surface, Trans. ASME, 67, 621-631.
  • 7. Brown, A., 1965. Optimum Dimensions of Uniform Annular Fins, International Journal of Heat and Mass Transfer, 8, 665-662.
  • 8. Yang, J.W., 1972. Periodic Heat Transfer in Straight Fins, Journal of Heat Transfer, Trans. ASME, 94, 310-314.
  • 9. Aziz, A., 1975. Periodic Heat Transfer in Annular Fins, Journal of Heat Transfer, Trans. ASME, 97, 302-303.
  • 10. Aziz, A., Enamul Huq, S.M., 1975. Perturbation Solution for Convecting Fin with Variable Thermal Conductivity, Journal of Heat Transfer, 97, 300-301.
  • 11.Krane, R.J., 1976. Discussion on a Previously Published Paper by Aziz A. and Enamul Hug S.M., Journal of Heat Transfer, 98, 685-686.
  • 12.Muzzio, A., 1976. Approximate Solution for Convective Fins with Variable Thermal Conductivity, Journal of Heat Transfer, 98, 680-682.
  • 13. Razelos, P., Imre, K., 1980. Optimum Dimension of Circular Fins with Variable Thermal Parameters, Journal of Heat Transfer, Trans. ASME, 102, 420-425.
  • 14. Ullmann, A., Kalman, H., 1989. Efficiency and Optimized Dimensions of Annular Fins of Different Cross-section Shapes, Int. J. Heat Mass Transfer, 32(6), 1105-1110.
  • 15. Campo, A., Harrison L., 1994. Prediction of Safe Tip Temperature in Uniform Annular Fins for the Design of Thermal Exchnage Equipment Via Sympolic Mathematics, Int. Commun. Heat Mass Transfer, 21(4), 531-538.
  • 16. Zubair, S.M., Al-Garni, A.Z., Nizami, J.S., 1996. The Optimal Dimensions of Circular Fins with Variable Profile and Temperaturedependent Thermal Conductivity, Int. J. Heat Mass Transfer, 39(16), 3431-3439.
  • 17. Campo, A., Stuffle, R.E., 1996. Symbolic Mathematics for Calculation of Thermal Efficiencies and Tip Temperatures in Annular Fins of Uniform Thickness, Int. J. Heat Mass Transfer, 40(2), 490-492.
  • 18.Kundu, B., Das, P.K., 2001. Performance Analysis and Optimization of Annular Fin with a Step Change in Thickness, Journal of Heat Transfer, Trans. ASME, 123(3), 601-604.
  • 19. Chiu, C.H., Chen, C.K., 2002. A Decomposition Method for Solving the Convective Longitudinal Fins with Variable Thermal Conductivity, International Journal of Heat and Mass Transfer, 45, 2067-2075.
  • 20.Mokheimer, E.M.A., 2002. Performance of Annular Fins with Different Profiles Subject to Variable Heat Transfer Coeficient, Int. J. Heat Mass Transfer, 45(17), 3631-3642.
  • 21. Bertola, V., Cafaro, E., 2003. Cooling Fin Design, Journal of Thermophysics and Heat Transfer, 17(4).
  • 22. Arslantürk, C., 2004. Performance Analysis and Optimization of a Thermally Nonsymmetric Annular Fin, Int. Comm. Heat Mass Transfer, 31(8), 1143-1153.
  • 23.Kang, H.S., Look, D.C., 2007. Optimization of a Thermally Asymmetric Convective and Radiating Annular Fin, Heat Transfer Engineering, 28(4), 310-320.
  • 24. Soliman, H.M., Elazhary, A.M., 2008. Comment on Cooling, Fin Design, Journal of Thermophysics and Heat Transfer, 22(2), 319-320.
  • 25. Iborra, A.A., Campo, A., 2009. Approximate Analytic Temperature Distribution and Efficiency for Annular Fins of Uniform Thickness, International Journal of Thermal Sciences, 48, 773-780.
  • 26.Kang, H.S., Look, D.C., 2009. Optimization of a Trapezoidal Profile Annular Fin, Heat Transfer Engineering, 30(5), 359-367. 27. Arslantürk, C., 2009. Correlation Equations for Optimum Design of Annular Fins with Temperature Dependent Thermal Conductivity, Heat Mass Transfer, 45(4), 519-525.
  • 28. Aziz, A., Fang, T., 2010. Alternative Solutions for Longitudinal Fins of Rectangular, Trapezoidal and Concave Parabolic Profiles, Energy Conversion and Management 51, 2188-2194.
  • 29. Ganji, D.D., Ganji, Z.Z., Ganji, H.D., 2011. Determination of Temperature Distribution for Annular Fins with Temperature Dependent Thermal Conductivity by HPM, Thermal science, 15, 111-115.
  • 30. Qian, J., Heat Transfer Analysis of Uniform Annular Fin on Regular Perturbation Method, Proc. Second Int. Conf. on Mechanic Automation and Control Engg, IEEE 2211, 2011.
  • 31. Peng, H. S., Chen, C. L., 2011. Hybrid Differential Transformation and Finite Difference Method to Annular Fin with Temperature-dependent Thermal Conductivity, International Journal of Heat and Mass Transfer, 54, 2427-2433.
  • 32. Darvishi, M.T., Khani, F., Aziz, A., 2016. Numerical Investigation for a Hyperbolic Annular Fin with Temperature Dependent Thermal Conductivity, Propulsion and Power Research, 5(1), 55-62.
  • 33. Roy, R., Ghosal, S., 2016. Homotopy Perturbation Method for the Analysis of Heat Transfer in an Annular Fin with Temperaturedependent Thermal Conductivity, Journal of Heat Transfer, 139(2), 1223-1231.
  • 34.Kundu, B., 2017. Exact Method for Annular Disc Fins with Heat Generation and Nonlinear Heating, Journal of Thermophysıcs and Heat Transfer, 31(2), 337-345.
  • 35.Yu, L.T., Chen, C.K., 1998. Application of Taylor Transformation to the Thermal Stresses in Isotropic Annular Fins, Journal of Thermal Stresses, 21(8), 781-809.
  • 36. Yu, L.T., Chen, C.K., 1999. Application of the Hybrid Method to the Transient Thermal Stresses Response in Isotropic Annular Fins, Journal of Applied Mechanics, 66, 340-346.
  • 37.Yang, Y.C., Chu, S.S., 2001. Transient Coupled Thermoelastic Analysis of an Annular Fin, Int. Comm. Heat and Mass Transfer, 28(8), 1103-1114.
  • 38. Lee, H.L., Yang, Y.C., Chu, S.S., 2002.. Transient Thermoplastic Analysis of an Annular Fin with Coupling Effect and Variable Heat Transfer Coefficient, Journal of Thermal Stresses, 25, 1105-1120.
  • 39. Chiu, C.H., Chen, C.K., 2002. Application of the Decomposition Method to Thermal Stresses in Isotropic Circular Fins with Temperaturedependent Thermal Conductivity, Acta Mechanica, 157, 147-158.
  • 40. Chiu, C.H., Chen, C.K., 2002. Thermal Stresses in Annular Fins with Temperature Dependent Conductivity under Periodic Boundary Condition, Journal of Thermal Stresses, 25, 475-492.
  • 41. Wang, C.C., Liao, W.J., Yang, Y.C., 2013. Hybrid Spline Difference Method for Heat Transfer and Thermal Stresses in Annular Fins, Numerical Heat Transfer Part B, Fundamentals, 64(1), 71-88.
  • 42. Tütüncü, N., Temel, B., 2013. An Efficient Unified Method for Thermoelastic Analysis of Functionally Graded Rotating Disks of Variable Thickness, Mechanics of Advanced Materials and Structures, 20, 38-46.
  • 43. Baş, H., Keleş, I., 2014. Novel Approach to Transient Thermal Stress in an Annular Fin, Journal of Thermophysics and Heat Transfer, 29(4), 705-710.
  • 44. Timoshenko, S.P. Goodier, J.N., 2003. Theory of Elasticity, McGraw-Hili, New York, 1970.
  • 45. Cengel, Y.A, Heat Transfer a Practical Appraoch, McGraw-Hill.
  • 46. Aktaş, Z., 1972. Numerical Solutions of Twopoint Boundary Value Problems, Metu, Depart. of Compt. Eng.
  • 47.Agarwal, R.P., 1982. On the Method of Complementary Functions for Nonlinear Boundary-value Problems, Journal of Optimization Theory and Applications, B36(1), 139-144.
  • 48. Roberts, S.M., Shipman, J.S., 1979. Fundamental Matrix and Two-point Boundaryvalue Problems, Journal of Optimization Theory and Applications, 28(1), 77-88.
  • 49. Gottlieb, D., 1981. The Stability of Pseudospectral-Chebyshev Methods, Mathematics of Computation, 36(153), 107-118.
  • 50. Trefethen, L.N., 2000. Spectral Methods in Matlab, SIAM, Philadelphia, PA.
  • 51.Bazan, F.S.V., 2008. Chebyshev Pseudospectral Method for Computing Numerical Solution of Convection-Diffusion Equation, Applied Mathematics and Computation, 200, 537-546.

İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi

Year 2018, Volume: 33 Issue: 3, 67 - 80, 30.09.2018

Ali Yıldırım Durmuş Yarımpabuç , Kerimcan Çelebi

https://doi.org/10.21605/cukurovaummfd.500573

Abstract

Bu çalışmada, izotropik ve homojen malzemeden yapılmış, dikdörtgen profilli eksenel simetrik olan ince bir dairesel kanatçıktaki sıcaklık dağılımı ve ısıl gerilmeler Tamamlayıcı Fonksiyonlar Metodu ve Pseudospectral Chebysev Metodu kullanılarak elde edilmiştir. Her iki sayısal yöntem de, Bessel fonksiyonları ve termoelastik teori kullanılarak elde edilen analitik sonuçlarla karşılaştırılmıştır. Bu sayısal yöntemlerin farklı bölüntü noktaları için hata analizleri Öklid normu kullanılarak gösterilmiştir. Pseudospectral Chebysev yönteminin nokta sayısının artması ile analitik sonuca daha hızlı bir şekilde yaklaştığı görülmüştür. 

Keywords

Dairesel kanatçık Tamamlayıcı fonksiyonlar yöntemi Pseudospectral Chebysev yöntemi Isıl gerilmeler

References

  • 1. Biswas, G., Mitra, K., Fiebig, M., 1994. Heat Transfer Enhancement in Fin-Tube Heat Exchangers by Winglet Type Vortex Generators, Int. J. Heat Mass Transfer, 37(2), 283-291.
  • 2. Kraus, A.D., Aziz, A., Welty, J.R., 2001. Extended Surface Heat Transfer, John Wiley and Sons, New York.
  • 3. Incropera, F., Dewıitt, D.P., Bergman, T.L., Lavine, A.S., 2007. Fundamentals of Heat and Mass Transfer, John Wiley and Sons, New York .
  • 4. Mallick, A., Ghosal, S., Sarkar, P.K., Ranjan, R., 2015. Homotopy Perturbation Method for Thermal Stresses in an Annular Fin with Variable Thermal Conductivity, Journal of Thermal Stresses, 38(1), 110-132.
  • 5. Wu, S.S., 1997. Analysis on Transient Thermal Stresses in an Annular Fin, Journal of Thermal Stresses, 20, 591-615.
  • 6. Gardner, K.A., 1945. Efficiency of Extended Surface, Trans. ASME, 67, 621-631.
  • 7. Brown, A., 1965. Optimum Dimensions of Uniform Annular Fins, International Journal of Heat and Mass Transfer, 8, 665-662.
  • 8. Yang, J.W., 1972. Periodic Heat Transfer in Straight Fins, Journal of Heat Transfer, Trans. ASME, 94, 310-314.
  • 9. Aziz, A., 1975. Periodic Heat Transfer in Annular Fins, Journal of Heat Transfer, Trans. ASME, 97, 302-303.
  • 10. Aziz, A., Enamul Huq, S.M., 1975. Perturbation Solution for Convecting Fin with Variable Thermal Conductivity, Journal of Heat Transfer, 97, 300-301.
  • 11.Krane, R.J., 1976. Discussion on a Previously Published Paper by Aziz A. and Enamul Hug S.M., Journal of Heat Transfer, 98, 685-686.
  • 12.Muzzio, A., 1976. Approximate Solution for Convective Fins with Variable Thermal Conductivity, Journal of Heat Transfer, 98, 680-682.
  • 13. Razelos, P., Imre, K., 1980. Optimum Dimension of Circular Fins with Variable Thermal Parameters, Journal of Heat Transfer, Trans. ASME, 102, 420-425.
  • 14. Ullmann, A., Kalman, H., 1989. Efficiency and Optimized Dimensions of Annular Fins of Different Cross-section Shapes, Int. J. Heat Mass Transfer, 32(6), 1105-1110.
  • 15. Campo, A., Harrison L., 1994. Prediction of Safe Tip Temperature in Uniform Annular Fins for the Design of Thermal Exchnage Equipment Via Sympolic Mathematics, Int. Commun. Heat Mass Transfer, 21(4), 531-538.
  • 16. Zubair, S.M., Al-Garni, A.Z., Nizami, J.S., 1996. The Optimal Dimensions of Circular Fins with Variable Profile and Temperaturedependent Thermal Conductivity, Int. J. Heat Mass Transfer, 39(16), 3431-3439.
  • 17. Campo, A., Stuffle, R.E., 1996. Symbolic Mathematics for Calculation of Thermal Efficiencies and Tip Temperatures in Annular Fins of Uniform Thickness, Int. J. Heat Mass Transfer, 40(2), 490-492.
  • 18.Kundu, B., Das, P.K., 2001. Performance Analysis and Optimization of Annular Fin with a Step Change in Thickness, Journal of Heat Transfer, Trans. ASME, 123(3), 601-604.
  • 19. Chiu, C.H., Chen, C.K., 2002. A Decomposition Method for Solving the Convective Longitudinal Fins with Variable Thermal Conductivity, International Journal of Heat and Mass Transfer, 45, 2067-2075.
  • 20.Mokheimer, E.M.A., 2002. Performance of Annular Fins with Different Profiles Subject to Variable Heat Transfer Coeficient, Int. J. Heat Mass Transfer, 45(17), 3631-3642.
  • 21. Bertola, V., Cafaro, E., 2003. Cooling Fin Design, Journal of Thermophysics and Heat Transfer, 17(4).
  • 22. Arslantürk, C., 2004. Performance Analysis and Optimization of a Thermally Nonsymmetric Annular Fin, Int. Comm. Heat Mass Transfer, 31(8), 1143-1153.
  • 23.Kang, H.S., Look, D.C., 2007. Optimization of a Thermally Asymmetric Convective and Radiating Annular Fin, Heat Transfer Engineering, 28(4), 310-320.
  • 24. Soliman, H.M., Elazhary, A.M., 2008. Comment on Cooling, Fin Design, Journal of Thermophysics and Heat Transfer, 22(2), 319-320.
  • 25. Iborra, A.A., Campo, A., 2009. Approximate Analytic Temperature Distribution and Efficiency for Annular Fins of Uniform Thickness, International Journal of Thermal Sciences, 48, 773-780.
  • 26.Kang, H.S., Look, D.C., 2009. Optimization of a Trapezoidal Profile Annular Fin, Heat Transfer Engineering, 30(5), 359-367. 27. Arslantürk, C., 2009. Correlation Equations for Optimum Design of Annular Fins with Temperature Dependent Thermal Conductivity, Heat Mass Transfer, 45(4), 519-525.
  • 28. Aziz, A., Fang, T., 2010. Alternative Solutions for Longitudinal Fins of Rectangular, Trapezoidal and Concave Parabolic Profiles, Energy Conversion and Management 51, 2188-2194.
  • 29. Ganji, D.D., Ganji, Z.Z., Ganji, H.D., 2011. Determination of Temperature Distribution for Annular Fins with Temperature Dependent Thermal Conductivity by HPM, Thermal science, 15, 111-115.
  • 30. Qian, J., Heat Transfer Analysis of Uniform Annular Fin on Regular Perturbation Method, Proc. Second Int. Conf. on Mechanic Automation and Control Engg, IEEE 2211, 2011.
  • 31. Peng, H. S., Chen, C. L., 2011. Hybrid Differential Transformation and Finite Difference Method to Annular Fin with Temperature-dependent Thermal Conductivity, International Journal of Heat and Mass Transfer, 54, 2427-2433.
  • 32. Darvishi, M.T., Khani, F., Aziz, A., 2016. Numerical Investigation for a Hyperbolic Annular Fin with Temperature Dependent Thermal Conductivity, Propulsion and Power Research, 5(1), 55-62.
  • 33. Roy, R., Ghosal, S., 2016. Homotopy Perturbation Method for the Analysis of Heat Transfer in an Annular Fin with Temperaturedependent Thermal Conductivity, Journal of Heat Transfer, 139(2), 1223-1231.
  • 34.Kundu, B., 2017. Exact Method for Annular Disc Fins with Heat Generation and Nonlinear Heating, Journal of Thermophysıcs and Heat Transfer, 31(2), 337-345.
  • 35.Yu, L.T., Chen, C.K., 1998. Application of Taylor Transformation to the Thermal Stresses in Isotropic Annular Fins, Journal of Thermal Stresses, 21(8), 781-809.
  • 36. Yu, L.T., Chen, C.K., 1999. Application of the Hybrid Method to the Transient Thermal Stresses Response in Isotropic Annular Fins, Journal of Applied Mechanics, 66, 340-346.
  • 37.Yang, Y.C., Chu, S.S., 2001. Transient Coupled Thermoelastic Analysis of an Annular Fin, Int. Comm. Heat and Mass Transfer, 28(8), 1103-1114.
  • 38. Lee, H.L., Yang, Y.C., Chu, S.S., 2002.. Transient Thermoplastic Analysis of an Annular Fin with Coupling Effect and Variable Heat Transfer Coefficient, Journal of Thermal Stresses, 25, 1105-1120.
  • 39. Chiu, C.H., Chen, C.K., 2002. Application of the Decomposition Method to Thermal Stresses in Isotropic Circular Fins with Temperaturedependent Thermal Conductivity, Acta Mechanica, 157, 147-158.
  • 40. Chiu, C.H., Chen, C.K., 2002. Thermal Stresses in Annular Fins with Temperature Dependent Conductivity under Periodic Boundary Condition, Journal of Thermal Stresses, 25, 475-492.
  • 41. Wang, C.C., Liao, W.J., Yang, Y.C., 2013. Hybrid Spline Difference Method for Heat Transfer and Thermal Stresses in Annular Fins, Numerical Heat Transfer Part B, Fundamentals, 64(1), 71-88.
  • 42. Tütüncü, N., Temel, B., 2013. An Efficient Unified Method for Thermoelastic Analysis of Functionally Graded Rotating Disks of Variable Thickness, Mechanics of Advanced Materials and Structures, 20, 38-46.
  • 43. Baş, H., Keleş, I., 2014. Novel Approach to Transient Thermal Stress in an Annular Fin, Journal of Thermophysics and Heat Transfer, 29(4), 705-710.
  • 44. Timoshenko, S.P. Goodier, J.N., 2003. Theory of Elasticity, McGraw-Hili, New York, 1970.
  • 45. Cengel, Y.A, Heat Transfer a Practical Appraoch, McGraw-Hill.
  • 46. Aktaş, Z., 1972. Numerical Solutions of Twopoint Boundary Value Problems, Metu, Depart. of Compt. Eng.
  • 47.Agarwal, R.P., 1982. On the Method of Complementary Functions for Nonlinear Boundary-value Problems, Journal of Optimization Theory and Applications, B36(1), 139-144.
  • 48. Roberts, S.M., Shipman, J.S., 1979. Fundamental Matrix and Two-point Boundaryvalue Problems, Journal of Optimization Theory and Applications, 28(1), 77-88.
  • 49. Gottlieb, D., 1981. The Stability of Pseudospectral-Chebyshev Methods, Mathematics of Computation, 36(153), 107-118.
  • 50. Trefethen, L.N., 2000. Spectral Methods in Matlab, SIAM, Philadelphia, PA.
  • 51.Bazan, F.S.V., 2008. Chebyshev Pseudospectral Method for Computing Numerical Solution of Convection-Diffusion Equation, Applied Mathematics and Computation, 200, 537-546.
There are 50 citations in total.

Details

Primary Language Turkish
Subjects Architecture, Engineering
Journal Section Articles
Authors

Ali Yıldırım This is me

Durmuş Yarımpabuç

Kerimcan Çelebi

Publication Date September 30, 2018
Published in Issue Year 2018 Volume: 33 Issue: 3

Cite

APA Yıldırım, A., Yarımpabuç, D., & Çelebi, K. (2018). İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 33(3), 67-80. https://doi.org/10.21605/cukurovaummfd.500573
AMA Yıldırım A, Yarımpabuç D, Çelebi K. İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi. cukurovaummfd. September 2018;33(3):67-80. doi:10.21605/cukurovaummfd.500573
Chicago Yıldırım, Ali, Durmuş Yarımpabuç, and Kerimcan Çelebi. “İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi”. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 33, no. 3 (September 2018): 67-80. https://doi.org/10.21605/cukurovaummfd.500573.
EndNote Yıldırım A, Yarımpabuç D, Çelebi K (September 1, 2018) İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 33 3 67–80.
IEEE A. Yıldırım, D. Yarımpabuç, and K. Çelebi, “İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi”, cukurovaummfd, vol. 33, no. 3, pp. 67–80, 2018, doi: 10.21605/cukurovaummfd.500573.
ISNAD Yıldırım, Ali et al. “İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi”. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi 33/3 (September 2018), 67-80. https://doi.org/10.21605/cukurovaummfd.500573.
JAMA Yıldırım A, Yarımpabuç D, Çelebi K. İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi. cukurovaummfd. 2018;33:67–80.
MLA Yıldırım, Ali et al. “İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi”. Çukurova Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, vol. 33, no. 3, 2018, pp. 67-80, doi:10.21605/cukurovaummfd.500573.
Vancouver Yıldırım A, Yarımpabuç D, Çelebi K. İzotropik Malzemeden Yapılmış Dairesel Kanatçıklardaki Isıl Gerilmelerin Sayısal Modellemesi. cukurovaummfd. 2018;33(3):67-80.