Stress of Vertical Cylindrical Vessel for Thermal Stratification of Contained Fluid 49 0.5 0.6 0.7 0.8 0.9 1 4812z／t (Tm－To)／ΔT Bi=6.97(FEM) Bi=6.97(simple) Bi=2.16(FEM) Bi=2.16(simple) Bi=0.72(FEM) Bi=0.72(simple) Bi=0.23(FEM) Bi=0.23(simple) Tf　→ Fig. 6. Vessel temperatures for ramp-shaped fluid temperature -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 4 8 12 16 20 24 z／t S＝σ／EαΔT Szb(FEM) Szb(simple) Shm(FEM) Shm(simple) Shb(FEM) Shb(simple) Bi=2.16, L=8t Fig. 7. Thermal stresses for ramp-shaped fluid temperature charts were developed for b/β>0.5 and βL<5. When b/β approaches 0, S approaches 0. S is approximately inversely proportional to L for βL>5. The maximum stress location, Δz, represents the outward distance from either end of the stratified layer. In the cold section, the maximum tensile (positive) stress occurs on the inner surface at z=-Δz, while in the hot section, the maximum compressive (negative) stress occurs on the inner surface at z=L+Δz. In addition, by substituting z into Eq. (32), we can calculate the wall-averaged temperature, which is applicable to the reference temperature for material properties in structural design. Heat Analysis and Thermodynamic Effects 50 0 0.1 0.2 0.3 012345 βL Szb,max b/β=0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 b/β>20 S∝1/L for βL＞5 Fig. 8. The maximum bending stress 0 0.5 1 1.5 012345 βL βΔZ 0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 b/β>20 Location of Szb,max Fig. 9. Location of the maximum bending stress Stress of Vertical Cylindrical Vessel for Thermal Stratification of Contained Fluid 51 0 0.1 0.2 0.3 0.4 0.5 012345 βL Shm,max b/β=0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 20 30 50 100 b/β=∞ S∝1/L for βL＞5 Fig. 10. The maximum membrane stress 0 0.2 0.4 0.6 0.8 0123 βL βΔZ 0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 20 30 50 100 b/β=∞ Location of Shm,max Fig. 11. Location of the maximum membrane stress Heat Analysis and Thermodynamic Effects 52 The maximum bending stress, S zb,max and its generating location, βΔz, is shown in Fig.8 and Fig.9, respectively. The maximum membrane stress, S hm,max and its generating location, βΔz, is shown in Fig.10 and Fig.11, respectively. The maximum stress intensity, S n,max (=σ SI,max ／EαΔT) and its generating location, βΔz, is shown in Fig.12 and Fig.13, respectively. The stress intensity (Tresca's stress σ SI ) becomes the maximum value at the outer surface, where σ z and σ h have opposite signs.   ,, SI z h z h Max    (40) A small prominence observed in Fig.13 suggests the transition from the case that S n,max occurs near the location of S hm,max to the case that S n,max occurs near the location of S zb,max . The comparisons of FEM analyses, the proposed charts and the conventional method, Eq.(38) and (39), for 2 cases, (L=8t, Bi=6.97) and (L=4t, Bi=2.16), are shown in Table 1. The parameters and S values read out from the charts for the two cases are listed below. L , Bi , b , β , b/β , βL , S zb , S hm , S n 8t, 6.97, 28.96, 2.48 , 11.7 , 0.99 , 0.27 , 0.24 , 0.39 4t, 2.16, 22.41, 2.48 , 9.03 , 0.50 , 0.28 , 0.30 , 0.43 0 0.1 0.2 0.3 0.4 0.5 0.6 012345 βL Sn,max b/β=0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 20 30 50 b/β>100 S∝1/L for βL＞5 Fig. 12. The maximum stress intensity It has been demonstrated that the proposed charts are sufficiently accurate. On the other hand, the conventional method leads to an overestimation. The main error is caused by the use of the formulas beyond the applicable range, βL>π( 2.5LRt ). The comparison of the proposed method and the conventional method is shown in S n -chart, Fig.14, and the above 2 cases results are plotted on the charts. Stress of Vertical Cylindrical Vessel for Thermal Stratification of Contained Fluid 53 0 0.5 1 1.5 012345 βL βΔZ 0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 20 30 50 100 b/β=∞ Location of Sn,max Fig. 13. Location of the maximum stress intensity We often need to evaluate thermal stresses for observed thermal stratification phenomena in an engineering field. In most cases, axial temperature profile of interface between stratified fluid layers can be approximated by exponential curve or parabolic curve as shown in Fig.15 (Moriya et al., 1987; Haifeng et al., 2009; Kimura et al., 2010). We propose the effective width for such cases as following equation. 0 0.5 1 1.5 012345 βL Sn,max＝σ SI,max ／(EαΔT) Conventional method βL→0, Sn(βL)→∞ Proposed method Sn(b/β,βL)≦0.508 b/β Fig. 14. Comparison of the proposed method and the conventional method Heat Analysis and Thermodynamic Effects 54 Method Proposed method FEM analyses Conventional method (38)(39) Case Comp onent σ max (MPa) Δz (mm) σ max (MPa) Δz (mm) σ max (MPa) Δz (mm) L=8t Bi=6.97 S zb 177 149 175 150 300 0 S hm 157 8 157 10 165 0 S n 255 28 256 30 375 0 L=4t Bi=2.16 S zb 184 234 183 225 600 0 S hm 196 36 193 35 330 0 S n 282 73 282 75 750 0 Table 1. Comparison of stress evaluation results  2 12 h c T e ff f med f T LTTzdT T    (41) L eff is nearly equal to the axial width corresponding to 90% of ΔT as shown in Fig.15. It is found that the thermal stress evaluations using the proposed charts and L eff are rather conservative and good evaluation, through comparisons with the FEM analyses. z T f T h T c T med 0 L eff 90% of ΔT observed T f data Fig. 15. Effective width of interface between stratified layers 5. Conclusion To improve the accuracy of design evaluation methods of thermal stress induced by thermal stratification, this study have performed the theoretical analyses and FEM ones on steady-state temperature and thermal stress of cylindrical vessels, and obtained the following results. 1. The theoretical solution of steady-state temperature profiles of vessels and the approximate solution of the wall-averaged temperature based on the temperature profile method have been obtained. The wall-averaged temperature can be estimated with a high precision using the temperature attenuation coefficient, b. Stress of Vertical Cylindrical Vessel for Thermal Stratification of Contained Fluid 55 2. The shell theory solution for thermal stress based on the approximate solution of the wall-averaged temperature has been obtained. It has been demonstrated that the non- dimensional thermal stress, S=σ/EαΔT exclusively depends on the ratio of coefficients, b/β, and the non-dimensional interface width between stratified layers, βL. 3. Easy-to-use charts has been developed to estimate the maximum thermal stress and its generating location using the characteristic described in (2) above. In addition, a simplified thermal stress evaluation method has been proposed. 4. Through comparison with the FEM analysis results, it has been confirmed that the proposed method is sufficiently accurate to estimate the steady-state temperature and thermal stress. 5. It has been demonstrated that the conventional simple evaluation method using the shell stress solution, which assumes axial temperature profile consisting of a straight line with the maximum fluid temperature gradient, often leads to an overestimation. 6. For the convenient application of the proposed method to engineering problems, we proposed the effective width of interface between stratified layers. The thermal stress evaluation using the proposed charts with the effective width gives slightly conservative estimations. The proposed method enables simple evaluations of steady-state thermal stress induced by thermal stratification taking the relaxation mechanism of thermal stress into account. This method would contribute to the reduction of design cost and to the rationalization of design. 6. References Bieniussa, K.W. and Reck, H. (1996). Piping specific analysis of stresses due to thermal stratification, Nuclear Engineering and Design, Vol.190, No.1, pp. 239-249, ISSN:0029- 5493. Carslaw, H.S. and Jeager, J.C. (1959). Conduction of heat in solids, 2 nd edition, pp. 166-169, Oxford University Press. CRC Solutions Corp. & Japan Atomic Energy Agency (2006). FINAS User’s Manual version 18.0, (in Japanese). Furuhashi, I., Kawasaki, N. and Kasahara, N. (2007). Evaluation Charts of Thermal Stresses in Cylindrical Vessels Induced by Thermal Stratification of Contained Fluid, (in Japanese), Transactions of the Japan Society of Mechanical Engineers, Series A, Vol.73, No.730, pp. 686-693. Furuhashi, I., Kawasaki, N. and Kasahara, N. (2008). Evaluation Charts of Thermal Stresses in Cylindrical Vessels Induced by Thermal Stratification of Contained Fluid, Journal of Computational Science and Technology, Vol.2, No.4, pp. 547-558. Furuhashi, I. and Watashi, K. (1991). A Simplified Method of Stress Calculation of a Nozzle Subjected to a Thermal Transient, International Journal of Pressure Vessels and Piping, Vol.45, pp. 133-162, ISSN:0308-0161. Haifeng, G. et al. (2009). Experimental Study on the Fluid Stratification Mechanism in the Density Lock, Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol.46, No.9, pp. 925- 932, ISSN:0022-3131 Kimura, N. et al. (2010). Experimental Study on Thermal Stratification in a Reactor Vessel of Innovatic Sodium-Cooled Fast Reactor – Mitigation Approach of Temperature Gradient across Stratification Interface -, Journal of NUCLEAR SCIENCE and TECHNOLOGY, Vol.47, No.9, pp. 829- 838, ISSN:0022-3131 Heat Analysis and Thermodynamic Effects 56 Katto, Y. (1964), Conduction of Heat, (in Japanese), (1964), p.38, Yokendo. Moriya, S. et al. (1987). Effects of Reynolds Number and Richardson Number on Thermal Stratification in Hot Plenum, Nuclear Engineering and Design, Vol.99, pp. 441-451, ISSN:0029-5493. Morse, P.M. and Feshbach, H. (1953). Methods of Theoretical Physics, Part.1, pp. 710-730, McGraw-Hill. Timoshenko, S.P. and Woinowsky-Krieger, S. (1959). Theory of plates and shells, 2nd edition, pp. 466-501, McGraw-Hill. 4 Axi-Symmetrical Transient Temperature Fields and Quasi-Static Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body Aleksander Yevtushenko 1 , Kazimierz Rozniakowski 2 and Мalgorzata Rozniakowska-Klosinska 3 1 Bialystok University of Technology Faculty of Mechanical Engineering 2 Technical University of Lodz Faculty of Technical Physics Information Technology and Applied Mathematics 3 Technical University of Lodz Poland 1. Introduction In the present chapter the model of a semi-infinite massive body which is heated through the outer surface by the precised heat flux, is being under study. This heat flux has the intensity directly proportional to the equivalent laser irradiation intensity. Heating of materials due to its surface irradiation by the high-power energy fluxes, which takes place during working of the laser systems, can be modelled in a specific conditions as the divided surface heat source of defined power density or heat flux of defined intensity (Rykalin et al., 1975). Laser systems are an unusual source of electromagnetic irradiation of unique properties. These properties differ essentially from the relevant characteristics of irradiation generated by traditional sources, natural and artificial one. Laser irradiation has specific, distinguishing features: high level of spatial and time coherence, high level of monochromaticity, low divergence, high spectral intensity and continuous or impulse emission process. High level of spatial coherence gives possibility to focusing laser irradiation on the surfaces of a few to several dozens squared micrometers in a size, which correspond to very high values of power intensity even 10 8 –10 12 W/m 2 (10 4 –10 18 J/m 2 or 10 23 fotons/cm 2 ). Effectivity of local surface heating mentioned above depends on: laser pulse duration, laser pulse structure (shape) and on irradiation intensity distribution. Three specific laser pulse structures are usually under consideration: rectangular-shape pulse, triangular-shape pulse and pulse shape approximated by some defined function. Likewise to the laser pulse structure, the spatial pulse structure (distribution of laser irradiation in a plane normal to the beam axis) is also complex and challenging for precised analytical description. In approximation the spatial distribution of laser irradiation can be described by the following relations: gaussian distribution (takes place during the working of laser beam in the single-mode regime), mixed (multi-modal) or uniform distribution. In addition, laser heat source shape can be changed by the electromagnetic or optical methods. Hence, the Heat Analysis and Thermodynamic Effects 58 optimization of the source shape problem appears on the basis of various optimisation criteria as well as the minimal losses on apparatus criterion. In the former industrial practice, mainly the gaussian or uniform intensity distribution were applied. Various in nature thermal effects are present during industrial laser materials processing such as: laser hardening, laser surface modifications of metals and alloys. Nowadays the most significant role in the technological operations plays such formed laser beam which maximum power is achieved not in the centre but close to the edge of the heated zone (Hector & Hetnarski, 1996). That is why, in the emerging process of the new effective laser technologies, it is strongly reasonable to determine the analytical solutions and to conduct numerical analysis for the boundary value problem of transient heat conduction and quasi-static thermal stresses, which are crucial in calculations of:  the effective absorption coefficient,  the specific time point when surface melting occurs due to laser beam heating,  the heating velocity and cooling,  the controlled laser thermo cracking process (Lauriello & Chen, 1973; Yevtushenko et al., 1997),  the other features of initiated temperature and thermal stresses fields. 2. Influence of intensity spatial distribution of laser beam on a temperature field in the irradiated massive body (semi-infinite) 2.1 Problem statement Laser irradiation interaction of 48 2 10 10 W / m power intensity on metals is equivalent to heating them by heat flux of defined intensity (Rykalin et al., 1975). If the following conditions are fulfilled:  the power intensity generated by the laser is not sufficient to melt and evaporate the superficial layer,  the losses because of heat emission and convection from a surface body are negligible  the thermo-physical properties do not depend on temperature, then the axisymmetrical boundary value problem of heat conduction for semi-infinite body in cylindrical coordinates system ( ,rz) with the beginning in the centre of heated surface, can be considered in the form: 22 22 11 TTTT rr kt rz        , 0, 0, 0rzt, (1) (,,0) 0Trz  , 0, 0rz , (2) () ( ), 0, 0, 0 s T KAqrHttrzt z     , (3) 22 (,,) 0, , 0Trzt r z t. (4) The uniform distribution of heat flux intensity in a circle of a radius, can be described by the formula: [...]... cooling of the body takes place 0.6 T* 0.5 0.4 Z=0 0 .3 0.2 0.4 0.1 0 0 0.5 1 1.5 2 2.5 Fo 3 Fig 3 Evolution of dimensionless temperature T *  T /( AT0 ) on the surface in the center of heated zone (   0,   0 ) and inside the body (   0,   0.4 ) at  s  0.6 (Yevtushenko et al., 2009) 62 Heat Analysis and Thermodynamic Effects Independently from the heat source intensity distribution, the retardation... of Eqs ( 13) , (39 ) and numerical results on Fig 6 From Eq ( 13) the T0  0, 133  10 5 K was determined When the right side of Eq (39 ) reaches the value Tm /( AT0 )  0. 230 8 , then on Fig 6 the dimensionless time   0.0 43 for the uniform distribution and respectively   0.047 for the normal distribution can be found These dimensionless time values have respective real time values: t  1.17 ms and t ... mode with increase of irradiation time and reach the asymptotes respectively: 1 and  / 2  0.8862 (see Fig 6) 0.9 T* 0.75 0.6 0.45 0 .3 0.15 0 0 1 2 3 4 Fo 5 Fig 6 Evolution of dimensionless temperature T (0)* / A in the centre of heated zone for laser systems working in the continuous generation regime (Yevtushenko et al., 2009) 66 Heat Analysis and Thermodynamic Effects By making assumption that at... experimental and analytical values gathered in Table 2, (Rozniakowski, 2001) parameters material steel 45 T0  10 5 , [K 1 0.111 s , [-]   10 3 [-] A experimental A from Eq (57) 0.062 ] h , [-] 0.0 73 0 .33 30 %  50% 41.8% Table 2 Comparison of the experimental and analytical values of the effective absorption coefficient A 3 Quasi-static thermal stresses caused by laser irradiation heating 3. 1 Non-stationary... following results: 3 C 3  2  C 4 2  ln[С 1  С 2  D f (  )]1   0,   0 (50) 68 Heat Analysis and Thermodynamic Effects Eq (50) defines the isotherm of maximum temperature for a given values of  s and  (see Fig 7) For the case of normal distribution of irradiation intensity ( f  1 , B f  1 ), the D f (  ) function (49) is independent on dimensionless radial variable  and can be expressed... continuous generation regime, from solutions (20)-( 23) at ts   (  s   ) is derived in general  T (0) (0,0, )  A   ( ) erf (  ) d , (32 ) 0 In case of uniform distribution (5) the function  ( )  J 1 ( ) /  (Matysiak et al., 1998) and then (32 ) formula becomes  T ( 0 )* (0,0, )   0 J1 ( )  erf (  ) d (33 ) By differentiation of (33 ) solution along    variable, the following... stationary temperature along axis   0 from the ( 23) solution was found:     1   B f Bf  2 T (0) (0,  ,  )  A  (1  f )   (1  f )  (1  f ) 2  e erfc( B f  ) 2  2 2    (28) 64 Heat Analysis and Thermodynamic Effects From relations (27) and (28) follows, that stationary temperature on the body surface, in the center of heated zone, reaches the value: T (0) (0,0,  ) ... Characteristic features of steel 45 type Steel sample was heated locally by laser irradiation beam in 10 different points with changing irradiation intensity Subsequently, the metalographic cross-section were done for 70 Heat Analysis and Thermodynamic Effects the irradiated areas By using the EPITYP-2 metalographic microscope and SEM TESLA BS300 the maximum hardened layer depth zh were measured (see... of the heated zone reaches the melting temperature of material, then: T (0)* (0,0, )  Tm / AT0 , (39 ) where temperature T (0)* (0,0, ) is derived from formula (37 ) or (38 ) The right side of the Eq (39 ) includes material characteristic features such as: thermal conductivity K and melting temperature Tm , the laser beam characteristic parameters: heat flux intensity q0 , laser beam radius a and effective... , d  (72)  T  ( ,  ,0)  0 ( 73) Solution of the ordinary differential equation (72) with initial condition ( 73) has form 72 Heat Analysis and Thermodynamic Effects   T  ( ,  , )   ( )  0 ( ,  ,  ) , (74) where   0 ( ,  ,  )    ( 2  2 )   1  e       2 2 (75) 2 By applying to the solutions (74), (75) below listed Fourier and Hankel inverted integral transformations . S zb 177 149 175 150 30 0 0 S hm 157 8 157 10 165 0 S n 255 28 256 30 37 5 0 L=4t Bi=2.16 S zb 184 234 1 83 225 600 0 S hm 196 36 1 93 35 33 0 0 S n 282 73 282 75 750 0 Table 1 NUCLEAR SCIENCE and TECHNOLOGY, Vol.47, No.9, pp. 829- 838 , ISSN:0022 -31 31 Heat Analysis and Thermodynamic Effects 56 Katto, Y. (1964), Conduction of Heat, (in Japanese), (1964), p .38 , Yokendo 0.99 , 0.27 , 0.24 , 0 .39 4t, 2.16, 22.41, 2.48 , 9. 03 , 0.50 , 0.28 , 0 .30 , 0. 43 0 0.1 0.2 0 .3 0.4 0.5 0.6 01 234 5 βL Sn,max b/β=0.5 b/β=1 1.5 2 3 4 5 6 8 b/β=10 20 30 50 b/β>100 S∝1/L for