Advanced Materials Research Vol. 445 (2012) pp 893-898 © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMR.445.893 A Method of Analysis to Estimate Thermal Down-shock Stress Profiles in Hollow Cylinders When Subjected to Transient Heat/Cooling Cycle 1 2 M.A. Ali , S.T.Hasan , and D.P. Myriounis 1,2,3: 3 Faculty of Arts, Computing, Engineering and Sciences, Sheffield Hallam University, Sheffield, United Kingdom. [email protected], [email protected], [email protected] Keywords: Thermal shock, down-shock, thermal stresses, cylindrical shell. Abstract. An empirical solution for the thermal shock stresses in cylindrical shell presented when cylinder is subjected to heating or re-heating case and down-shock cooling by forced air case. Linear equations are developed to describe the severity of thermal shock loading. When thermal gradient and time period are in consideration, it is shown the equations displays good approximation for major characteristics of the thermal shock stress profiles. Introduction Many industries deal with components which are subjected to higher loads at elevated temperatures due to the increasing performance requirements. The difficult prediction of a material to be well chosen for its performance under transient thermal cycling become a reason of primary damage under a large changes in temperature over a period of time. Specially, the most interesting cases are the heating or re-heating and cooling cycles during start-up and shut-down operations which cause thermal stresses superimposed with mechanical loadings. For example in land-based turbines, pressure vessel and heat exchanger in conventional and nuclear power plants, thermal fatigue may occur as a result of plasticity at the crack tip and associated stresses arising from the thermal gradients. Research effort has been concentrating on the stresses under transient cooling condition; H. F. Nied, has observed that the transient thermal stresses of a hollow cylinder under rapid cooling are dependent on the time, inner-to-outer radius and the coefficient of diffusivity of the material,. G. A. Kardomateas, has presented the transient thermal stress distribution with time through the wall thickness of the hollow cylinder subjected to a constant temperature at one side and convection heat transfer at different temperature in the other side, . For dynamic wave propagation studies, H. Cho, has studied the elastodynamic solution for thermal shock stress in thick cylinder shell subjected to rapid thermal loading showing the wave formation of the elastic stresses and the important value of the thermal shock distribution throughout the shell wall similar works due to heat transfer into a medium, . H. Cho, has presented the dynamic fluctuation of the thermal shock stresses employing practical inertia quantity and material properties through the orthotropic thick cylinder shell made of glass/epoxy, . G. A. Kardomateas, has observed the fluctuation of the transient thermal stresses with time through the thickness of composite hollow cylinder affected by a constant temperature at one side and heat transfer into a medium under various temperature conditions at the other side, . This paper produces the estimation of the transient temperature and thermal stresses in hollow cylinder which takes place as in start-up and shutdown conditions in nuclear power plant by classifying them to heating or re-heating and downshock cooling cases. All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, www.ttp.net. (ID: 188.8.131.52-04/01/12,12:31:20) 894 Materials and Manufacturing Technologies XIV Transient Temperature Formulation To analyse the hollow cylinder subjected to transient thermal loading in heating and down shock cooling cases, a temperature distribution is estimated to evaluate the thermal stresses induced due to the thermal loading. During the transient thermal loading the temperature gradients are introduced across the hollow cylinder wall thickness in which results vary with position on the shell wall thickness and time. The transient temperature distributions for a hollow cylinder geometric model are given by employing the solution of the general conduction heat equation given by: , , 1 , = + , < < , > 01 Where D is the coefficient of diffusion (i.e., = . ; , , are being the thermal conduction coefficient of the cylinder material, the mass density, and the specific heat. Carslaw and Jaeger, has modelled a hollow cylinder using the application of Laplace transformation in a cylindrical region and the Bessel's function of the first and second kinds employing the boundary conditions of each case studied, . I- Heating Case In this case the outer surface of the hollow cylinder is subjected to heating flux and the inner surface will be kept at a constant temperature and the initial temperature assumed to be at a selected temperature. The initial and general boundary conditions are: , = = 200! ° , = 02 , = # = 400! ° , = , > 03 , & = , = , > 04 The distribution of temperature , through the shell wall in this case is : , = '( + ' ln + + ',- ./ 0- . 1 0- − 1 0- . / 0- 35 ∞ -5( Where ,# are the initial and inner surface temperatures, F is the heat flux supplied to the outer surface of the hollow cylinder, a, b are inner and outer radii and '( , ' , ',- are constant shown in appendix A. Also, 0- are the positive roots of the equation: ./ 0- . 1( 0- − 1 0- . /( 0- 3 = 06 II- Down-shock Cooling Case The outer surface of the hollow cylinder will be considered at constant temperature and the inner surface will be down shock cooled by forced air and the initial temperature assumed to be at room temperature. Advanced Materials Research Vol. 445 895 The initial and general boundary conditions are: , = = 25! ° , = 07 89:,; 8 < = − # − #= , = , > 08 , = ? = 600! ° , = , > 09 The distribution of temperature , through the shell wall in this case is: , = A( + A ln + + A,- ./ 0- . 1( 0- − 1 0- . /( 0- 310 ∞ -5( Where ,# ,#= ,? are the initial, inner surface, forced air and outer surface temperatures, H is the coefficient of surface heat transfer when the inner surface is exposed to cooling by forced air depending on the air velocity and the diameter of the pipe and A(, A, A,- are constant shown in appendix A. Also, 0- are the positive roots of the equation as follow: ./( 0- . 1 0- − 1( 0- . / 0- 3 = 011 Thermal Stress Formulation An elastic relation for resultant thermal stresses have been derived by Timoshinko and Goodier, induced in rectangular plate of uniform thickness and hollow cylinder due to non-symmetrical temperature distribution, . The general thermal redial stresses formulas for a hollow cylinder is given by: σBCDEFBGHI B αE 1 r − a Q = N Trdr − N Trdr12 1 − ν r b − a H H Where E, α, and ν are the young's modulus, linear coefficient of thermal expansion and poisons' ratio respectively also, a, b are inner and outer radii. From the substitutions of the temperature distribution in the general formulas in the cases studied the thermal stresses estimation is given: I- Thermal Stress of Heating Case The final radial stress for the heating case is given by: R = R( + R + R, 13 Where R( , R STR, are the solutions of Eq.5.for this case. II- Thermal Stress of Down-shock Cooling Stress Case The final radial stress for the heating case is given by: R = R( + R + R, 14 Where R( , R STR, are the solutions of Eq.10 for this case. 896 Materials and Manufacturing Technologies XIV Results and Discussions The thermal shock loading are estimated in this research, in a hollow cylinder shell subjected to heating and down shock cooling cases. The Laplace transformation and Bessel's functions are used for solving the transient temperature under boundary conditions of each case and the integral transformation have been applied for solving the thermal stresses in cylindrical coordinates. A numerical program is designed to carry out the estimate of the transient temperature distribution and the thermal stresses for a hollow cylinder geometry made of AISI 410 Martensitic stainless steel employed in the heating and down shock cooling cases. In the heating case, the temperature distribution passing through the shell wall as the boundary conditions shown in Fig.1. Transient temperature appears as a maximum when the time period reaches close to zero which is mostly clear in Fig.2. From the results the temperature gradients are affected by a period of time and then relax later with increasing time. The radial stress quantities are maximum when the temperature distribution is minimum through the shell wall and also effected by time as illustrated in Fig.3. Thermal loads are affected by the time periods, the diffusivity property of the material, supplied heat flux and the ratio of the inner and outer radii. In the cooling case, the transient temperatures calculated are of relevance with a period of time as shown in Fig.4.The roles of the convection coefficient of the forced air with the diffusivity property of the material, in addition to the inner and outer radii have a very important implicit effect in the thermal loading results As well as, the effects of the thermal expansion factor and the stiffness property of the material on the thermal stresses. The maximum radial stress is mostly clear due to the effect of sudden cooling from inner side and heating from the outer and it will experience the opposite with the transient temperature gradient becoming highest when the time reaches close to zero as shown in Fig.5. When the cooling load increased by increasing the convection coefficient of the forced air the curves tends toward the outer surface in transient temperature distribution and thermal stresses through the shell wall and vice versa when the temperature of the outer surface increased. The description of Figs. 6 and 7, indicate the important effect of the forced air on the temperature distribution which is increased due to the sudden downshock cooling when the air temperature decreased. The thermal loading results appear very close because of the small thickness of the hollow cylinder and the thermal property of the material selected. In the two cases the values of the stresses which occurs at (t=0), has the expected values of VWX 600,6 800 600,5 700 600,4 600,3 600,2 t = 0.5 sec 600,1 t = 0.7 sec 600 599,9 0,004 t = 1 sec 0,0045 0,005 r - Radius ( m) T - Temperature ( ºC ) T - Temperature ( º C ) U (CZY[ positive for radial thermal stress. 600 500 400 r=0.004 (m) 300 200 r=0.0045 (m) 100 0 0 1 2 3 t- Time ( sec ) Figure 1. The temperature distribution with time through the shell wall in heating case. Figure 2. The temperature distribution with time in heating case. 5,00E+11 700 0,00E+00 600 -5,00E+11 0 2 T- Temperature ( º C ) σr - Radial stresses ( N/ m2) Advanced Materials Research Vol. 445 4 -1,00E+12 r=0.004 (m) r=0.0045 (m) -1,50E+12 -2,00E+12 897 500 400 r = 0.004 m r = .0045 m r = 0.005 m 300 200 100 -2,50E+12 0 0 -3,00E+12 1 2 3 4 t- Time (sec ) t -Time ( sec ) Figure 4. The temperature distribution with time in down shock cooling case. Figure 3. The radial stresses distribution with time in heating case. 2,00E+09 599,9 2 4 r = 0.004 m -4,00E+09 r = .0045 m -6,00E+09 r = 0.005 m -8,00E+09 -1,00E+10 Temperature - ( º C ) 0 -2,00E+09 599,8 599,7 599,6 Ta= 0 ºC 599,5 Ta = 50 ºC Ta= 100 ºC 599,4 599,3 0,004 0,0042 0,0044 0,0046 0,0048 0,005 t- Time (sec ) Figure 5. The radial stresses distribution with time in down shock cooling case. Figure 6. The effect of the forced r - Radius (m) air on the temperature distribution through the shell wall at t=0.5 sec. 2,00E+06 σr - Radial stresses ( N/m2 ) σr - Radial stress ( N/m2) 0,00E+00 0,00E+00 0,004 -2,00E+06 0,0042 0,0044 0,0046 0,0048 0,005 -4,00E+06 -6,00E+06 -8,00E+06 -1,00E+07 Ta = 0 ºC Ta = 50 ºC -1,20E+07 -1,40E+07 r - Radius (m) Figure 7. The effect of the forced air on the radial stresses distribution through the shell wall at t= 0.5 sec. Ta= 100 ºC 898 Materials and Manufacturing Technologies XIV Conclusion The thermal shock loading are obtained from this research in a hollow cylinder shell subjected to heating and down shock cooling cases. The combined effects of the time, material properties and the ratio of the inner and outer radii of the hollow cylinder on the transient temperature gradient are clearly domenstrated.Hence; their functions on the thermal stresses must be taken in consideration. Also, the thermo-elastic stresses are mostly influenced by the thermal expansion factor and stiffness properties of the material. In the heating case, the temperature gradient solutions affected by a period of time and its maximum appearance when the time reaches close to zero, on the contrary, the radial thermal stress minimum. In the cooling case, the down shock is clear depends on the convection coefficient effect and the temperature of the forced air which is appeared very close because of the small thickness of the shell wall and the material property selected. Appendix A '( = + # ,' = ',- = −]^ C_`a . b ?\ b &. / 0- d 0- / 0- − /( 0- /( 0- . c0- # /( 0- − #< # − #= ,e l jY ?`a .k 9n C9no: .jY ?`a C`a 9p jq #`a r A( = + ?, A = − A,- = ]^ C_`a . m = 5.5 × 10Cg . h.i . T C. `a jq b #`a CjY b ?`a References  H.F. Nied and F. Erdogan, in: Transient Thermal Stress Problem for a Circumferentially Cracked Hollow Cylinder, submitted to Journal of Thermal Stresses, Vol. 6 (1983), p. 1-14.  G.A. Kardomateas, in: The Initial Phase of Transient Thermal Stresses due to General Boundary Thermal Loads in Orthotropic Hollow Cylinder, submitted to Journal of Applied Mechanics, Vol. 57(1990), p. 719.  H. Cho, G. A. Kardomateas and C. S. Valle, in: Elastodynamic Solution for the Thermal Shock Stresses in an Orthotropic Thick Cylinder Shell, submitted to Journal of Applied Mechanics, Vol. 65(1998), p. 184.  H. Cho and G. A. Kardomateas, in: Thermal Shock Stresses due to Heat Convection at a Bounding Surface in a Thick Orthotropic Cylindrical Shell, submitted to International Journal of Solids and Structures, Vol. 38 (2001), p. 2769-2788.  G. A. Kardomateas, in: Transient Thermal Stresses in Cylindrically Orthotropic Composite Tubes, submitted to Journal of Applied Mechanics, Vol. 56 (1989), p. 411.  H. S. Carslaw and J. C. Jaeger, in: Conduction of Heat in Solids, edited by Oxford Science Publications, Oxford (1959).  S. P. Timoshenko and J. N. Goodier, in: Theory of Elasticity, edited by McGraw-Hill, Kogakusha, Ltd (1970).