Heat Analysis and Thermodynamic Effects Part 3 docx

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 s

Stress of Vertical Cylindrical Vessel for Thermal Stratification of Contained Fluid 49

0.5 0.6 0.7 0.8 0.9 1

Tf　→

Fig 6 Vessel temperatures for ramp-shaped fluid temperature

-0.5-0.4-0.3-0.2-0.100.1

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

00.10.20.3

Fig 8 The maximum bending stress

00.511.5

0.5b/β=11.5234568b/β=10b/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

00.10.20.30.40.5

Fig 10 The maximum membrane stress

0 0.2 0.4 0.6 0.8

0.5b/β=11.5234568b/β=10203050100b/β=∞

Location of Shm,max

Fig 11 Location of the maximum membrane stress

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

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 , Szb , Shm , Sn

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

00.10.20.30.40.50.6

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( L2.5 Rt) 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

00.511.5

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

Proposed method Sn(b/β,βL)≦0.508 b/β

Fig 14 Comparison of the proposed method and the conventional method

Method Proposed method FEM analyses method (38)(39) Conventional

Case Component (MPa) σ max (mm) Δz (MPa) σ max (mm) Δz (MPa) σ max (mm) Δz

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, 2nd 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

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,

4

Axi-Symmetrical Transient Temperature Fields and Quasi-Static Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body

Aleksander Yevtushenko1, Kazimierz Rozniakowski2

and Мalgorzata Rozniakowska-Klosinska3

Faculty of Mechanical Engineering

Information Technology and Applied Mathematics

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 108 –1012 W/m2 (104 –1018 J/m2 or

1023 fotons/cm2) 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

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 4 8 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 ( ,r z ) with the beginning in the centre of heated surface,

can be considered in the form:

Axi-Symmetrical Transient Temperature Fields and Quasi-Static

Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body 59

0

and for the mixed distribution – multimodal one, the heat flux intensity can be expressed by

(Hector & Hetnarski, 1996):

where K c – concentration coefficient, q – characteristic value of heat flux intensity q , f

0  – parameter, which characterized the irradiation intensity distribution in a plane f 1

normal to the laser beam axis For f 1 the normal (gaussian) distribution and for f 0

doughnut – toroidal distribution, is obtained

Fig 1 Laser irradiation heating model and area shape visualization of phase transition for

metals

Both distributions of laser irradiation intensity (5) and (6) are related by the following

concentration coefficient (Rykalin et al., 1975)

where Q – total irradiation power, Q – irradiation power which arrives to the circle of f

a radius and can be easily derived after taking into consideration the distribution (6),

consequently these two values can be substituted into Eq (8) As the result the non-linear

equation will be received for B in the form: f

A numerical analysis shows that dependence of the B roots of Eq (11) with respect to the f

f parameter is nearly linear: B f B0(1 f) , where f B 0 2.1462 is the value of B at f

f  At f 1 from Eq (6) is received the obvious result B  (Rykalin et al., 1975) By f 1

comparizing the irradiation intensity of uniform distribution (5) with the irradiation

intensity of general case distribution (6) it was found

s

q a kt

and using relations (7) and (12) the boundary value heat conductivity problem (1)-(4) can be

rewritten in the form:

 q*

Fig 2 Laser irradiation distribution – a function q*( ) for three different f parameter values

(solid line corresponds to the value of f 1, dashed line to f 0.5, dot-dashed line to f 0)

Axi-Symmetrical Transient Temperature Fields and Quasi-Static

Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body 61

2.2 Problem solution

The solution of the boundary value heat conduction problem (14)-(17), which is obtained by

applying the integral Hankel transforms with respect to the radial variable r and Laplace

transform with respect to time t , has the following form:

( , , ) ( ) ( , , ) ( )

T     A     J  d , 0,0, , (20) 0where

2

* 0 0

During transient heating of the massive body, the maximum value of temperature on the

body surface is achieved at the moment t t (s   ) – switching laser system off, whereas s

inside the body at t h   (t s t hs  , where    k t / aΔ 2) When laser system is

switched off, then laser heating source is simply cut off – as a consequence the fast cooling

of the body takes place

0 0.1 0.2 0.3 0.4 0.5 0.6

Fo

T*

Z=0

0.4

Fig 3 Evolution of dimensionless temperature T*T/(AT0) on the surface in the center of

heated zone (0,  ) and inside the body (0 0, 0.4) at s0.6 (Yevtushenko et

al., 2009)

Independently from the heat source intensity distribution, the retardation time t ( ) is increasing fast with the distance from the heated surface For a fixed value of depth from the

working surface this retardation time t  decreases with the increase of the f parameter

0 0.1 0.2 0.3 0.4 0.5

0.6 T*

r a(see Fig 4)

0 0.1 0.2 0.3 0.4 0.5

0.6 T*

Axi-Symmetrical Transient Temperature Fields and Quasi-Static

Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body 63

The effective depth of heating (the distance from the boundary body surface for which

temperature value equals 5% of the maximum value of temperature achieved on the surface)

is independent on the form of intensity distribution of the incident heat flux and for

dimensionless retardation time s0.6 is equal in approximation 1.5a (see Fig 5) At work

(Ashcroft & Mermin, 1986) was shown that for laser systems working in the continuous

regime the above quantity does not exceed the 5a value (Matysiak et al., 1998) This

conclusion confirms the results shown in the present chapter thanks to the numerical

calculations done with use of formulas (19)-(22) at s  and 100 (dimensionless time,

when stationary temperature is achieved)

2.3 Special cases of the obtained solution

At t s (s  from Eq (19) is received that ) T( , , )   T(0) ( , , )   and formula (20)

corresponds with solution presented in paper (Ready, 1997) for laser operated in the

continuous regime and irradiation of semi-infinite surface If additionally t    , ( )

then from Eq (22) follows that ( , , ) e      and then stationary temperature can be

derived from the relation:

(0)

0 0

T     A  e J   d    (23)

By substituting to the Eq (23)   , the stationary temperature on surface for semi-infinite 0

surface can be found

(0)

0 0

semi-infinite surface was received in the following form:

In similar way the distribution of the stationary temperature along axis  from the (23) 0

solution was found: