Stress of Vertical Cylindrical Vessel for Thermal Stratification of Contained Fluid49 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
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＞5Fig. 8. The maximum bending stress
00
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＞5Fig. 10. The maximum membrane stress


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


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＞5Fig. 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


(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)

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

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 Fluid55
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

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

–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

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




 , (6)
where
c
K – concentration coefficient,
f
q – characteristic value of heat flux intensity q ,
01f – parameter, which characterized the irradiation intensity distribution in a plane
normal to the laser beam axis. For
1f

the normal (gaussian) distribution and for 0f 
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)

2
cf
KBa

 . (7)
The numerical factor
f
B in the Eq. (7) can be found from the condition below (Hector &
Hetnarski, 1996):

1

B roots of Eq. (11) with respect to the
f
parameter is nearly linear:
0
(1 )
f
BB
ff

, where
0
2.1462B

is the value of
f
B at

Heat Analysis and Thermodynamic Effects

60
0f  . At 1f  from Eq. (6) is received the obvious result 1
f
B

(Rykalin et al., 1975). By
comparizing the irradiation intensity of uniform distribution (5) with the irradiation
intensity of general case distribution (6) it was found

0
.

rewritten in the form:

22
22
1
TTTT










,
0, 0, 0


, (14)

(,,0) 0T



, 0, 0,

 (15)


ff
qBf fBe



 , (see Fig. 2). (18)

00.
5
11.522.
5
0
0.2
0.4
0.6
0.8
1

q*

Fig. 2. Laser irradiation distribution – a function
*
()q

for three different
f
parameter values
(solid line corresponds to the value of
1f


(,,) ()(,,) ( )TA Jd

    




, 0, 0, 0


, (20)
where

2
2
4
*
0
0
1
() () ( ) (1 )1 , 0
24
f
B
f
qJ d f f e
B


     

  






. (22)
During transient heating of the massive body, the maximum value of temperature on the
body surface is achieved at the moment
s
tt

(
s



) – switching laser system off, whereas
inside the body at
hs
tt t

 (
hs



 , where
2


 ) at 0.6
s


(Yevtushenko et
al., 2009).

Heat Analysis and Thermodynamic Effects

62
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.

00.
5
11.
5
22.
5
0
0.1
0.2


0 0.3 0.6 0.9 1.
2
1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
T*
Z

Fig. 5. Evolution of dimensionless temperature
*
T
along symmetry axis
0


at
0.6
s

 (Yevtushenko et al., 2009).
Axi-Symmetrical Transient Temperature Fields and Quasi-Static
Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body





 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
(,,) () ( ) , 0, 0TAeJd

    







eJ d






and
2
4
2
0
0
()
f
B
eJ d


 




with use of integrals table (Prudnikov et al., 1998) the solution for stationary temperature on
semi-infinite surface was received in the following form:

2
222
22

   


  




  




. (27)
In similar way the distribution of the stationary temperature along axis 0


from the (23)
solution was found:

2
(0)
2
1
(0, , ) (1 ) (1 ) (1 ) erfc( ) .
22 2
f
f
B
f

(0,0, )
4
f
f
TAB




. (29)
For normal distribution of irradiation intensity (
1f

,
1
f
B

) from Eq. (29) the following
result is obtained (Bardybahin & Czubarov, 1996) :

(0)
(0,0, ) 0.8862
2
TAA


  (30)
and for doughnut mode structure distribution ( 0, 2.1462)
f


) distribution of laser heat flux intensity, the maximum
temperature is achieved on the surface in the centre of the heated zone. For laser systems
working in the continuous generation regime, from solutions (20)-(23) at
s
t (
s

) is
derived in general

(0)
0
(0,0, ) ( ) ( )TAerfd

   




, (32)
In case of uniform distribution (5) the function
1
() ()/J



(Matysiak et al., 1998) and
then (32) formula becomes


2
0
21
()
8
dT
AJ e d e I
d















. (34)
According to work (Abramowitz & Stegun, 1979):
Axi-Symmetrical Transient Temperature Fields and Quasi-Static
Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body 65


. (36)
By integration (36) formula along

variable with condition consideration
(0)*
(0,0,0) 0T  ,
the formula of dimensionless temperature evolution in the centre of the heated zone
(0, 0)


is derived:

1
(0)*
4
1
(0,0, ) 2 1
2
TAeerfc
















(38)
Dimensionless temperatures
(0)*
(0,0, )T

, (37) and (38), increase in monotone mode with
increase of irradiation time and reach the asymptotes respectively: 1 and
/2 0.8862


(see Fig. 6).
Fig. 6. Evolution of dimensionless temperature
(0)*
/TA in the centre of heated zone for
laser systems working in the continuous generation regime (Yevtushenko et al., 2009).
012345
0
0.15
0.3
0.45
0.6
0.75

flux intensity
0
q , laser beam radius a and effective absorption coefficient A . If all the
values are known then from Eq. (39) the dimensionless boundary time point of melting
beginning can be found. By choosing for a specific material the appropriate laser beam
parameters, the melting start point can be calculated and damage of a surface as a result of
intensive melting can be avoided.
In work (Rozniakowski, 2001), for sample of steel St45:


33.5 /( )KWmK
,
62
15 10 /kms

 , 1535
m
TC

,

heated by the Nd:YAG laser beam in system KWANT 15 ( 0.64amm

radius, laser pulse
duration
2
s
tms
),


2.5 Determination of laser irradiation effective absorption coefficient method
It should be underlined that maximum temperature value (19) on the body surface is
achieved at the moment of laser switiching off
s
tt


()
s



(Fig. 3). In order to determine
the monochromatic effective absorption coefficient
A
, the value of the retardation time
t ()


is needed to be known (its value increases quickly with the distance from laser
irradiated surface of the body). Retardation time
t

()


can be determined from the
condition of the maximum temperature reached inside the semi-infinite body in the point of
(,)rz (,)


Differentiating equations (19)-(22) with respect to dimensionless time

we obtain

s
(0) (0)
(,,) (,,) (,, )
() ( )
s
TT T
AH H
   


 




 



 




, (41)
where







. (42)
Taking into account the form of function
()


given by (21), the Eq. (42) can be written as:

2
(0)
4
12
(,,)
[ (,) (1 ) (,)]
2
T
e
fM f M



 








, (44)
and

)
2
1
22
4
20 1
0
1
(,) 1 ( ) 1 1 (,
41414
MeJd M



 







  

f
B
B
ff f f
f
BBfB fB
T
e
B




 


















s
1
1
h
C



,
14
2
4
f
s
f
h
B
C
B




,
s
(1 4 )(1 4
3
4
)

()
(1 ) ( )( 4 )
ff
h
f
h
f
fff
fB B f B
D
fB B f B





 


   
. (49)
By applying logarithm on (47) we have the following results:

22 3 1
34 12
ln[ ( )]
f
CC СС D
 



(1 4 )
14
2
1
22
2
1
()
h
f
DD
C



 

, (51)
where
2
C is given by (48). By using relation (51), the Eq. (50) can be rewritten in the form:

22 1
34 12
ln[ ]CC СС


 ,
0, 0



. (54)
By analysing the Eqs. (53), (54) it can be found that the isotherm of maximum dimensionless
temperature for the normal (gaussian) distribution of irradiation intensity has the form of
half-ellipse (see Fig. 7) with the axes given by (53) –
h
ra


,
h
za


. In case of the toroidal
(ring) distribution of the heat flux, the isotherm takes the form of curve with maximum
shifted from axial axis. On symmetry axis 0


, the Eq. (52) has form:

21
412
ln[ ]CCC



. (55)


C and
4
C respectively (50), the following is
derived:

1/2
1
4( )
14
ln
14( )
s
h
ss s
 

















 can be
found, too. From condition
(,, )
hh
Trzt T

, where T is given from solution (19)-(22), the
following formula for determination of the effective absorption coefficient, is obtained:

*
0
h
T
AA
T
 , (57)
where

1
*
0
0
()[(,,) (,, )]( )
hs h h
AJd
      





0.37%, P–0.040%, S–0.045%. Laser
system was working in free generation regime (Nd:YAG laser type,
1.06μm


, 1.5J
i
E 
and laser pulse time duration
2ms
s
t

).

parameters material
K ,
[Wm
1

K
1

]
5
10k  ,

z
, it was also observed the
melting point starts when
82
0
8.5 10 W/mq 
where:

0
2
i
s
E
q
at


. (59)
Moreover, it was found that for
82
0
5.8 10 W/mq 
, the hardened layer depth
h
z
equals
40μm . By using Eq. (59) the radius of irradiated area can be found as 0.64mma

and
consequently the dimensionless irradiation time

parametersmaterial
5
0
10T

 ,
[K
1

]
h

,
[-]
s

,
[-]
3
10


[-]
A
experimental
A
from Eq. (57)










,
0, 0, 0


, (60)
(,,0) 0T



,
0, 0,


(61)
Axi-Symmetrical Transient Temperature Fields and Quasi-Static
Thermal Stresses Initiated by a Laser Pulse in a Homogeneous Massive Body 71

*



 ,
0


. (64)
By applying the Hankel integral transformation along the radial

(Sneddon, 1972)




0
0
,, ,,TTJd








, (65)
to the boundary heat conduction problem (60)-(63) then it is denoted

2
2

 , (68)


,0
T
BiT
Z
 




 

, (69)
where after consideration of Eq. (64)

  
2
2
4
0
0
1
2
eJ d e


   



22
2
() (),0
dT
T
d
 




 

, (72)

(,,0) 0T




. (73)
Solution of the ordinary differential equation (72) with initial condition (73) has form

Heat Analysis and Thermodynamic Effects

72

0
(,,) () (, ,)T



. (75)
By applying to the solutions (74), (75) below listed Fourier and Hankel inverted integral
transformations (Sneddon, 1972):

22
0
(,)
2
(, ,) (, ,)
N
TTd
Bi


  








, (76)

0
0
(, ,) (, ,) ( )TTJd



22
22
1
, , erfc erfc
2
22
erfc
2
Bi
Bi
ee
Bi Bi
Bi e
eBi
Bi
 



  
















 


. (80)
In case, when convection cooling does not occur on the semi-infinite surface of the body
(0Bi  ), function
(, ,)



(79) gets form (81). At

 from solution (78), (79) the
stationary temperature in the centre of heated zone 0


, 0


, is obtained:


2

73
as the result of temperature field interaction, without mass forces, can be found from
differential equations system in partial differentials (Nowacki, 1986; Timoshenko & Goodier,
1970):

,, ,
12(1)
12 12
i
jj j j
iti
uu T






. (82)
By introducing thermoelastic potential
),,(





with use of formulae (Nowacki, 1986):

11
,



, and dimensionless temperature ( , , )T



is given by (78), (79).
In case, when thermoelastic potential
),,(




is known, then respective thermal stresses
can be derived from formulae (Nowacki, 1986):
2 22
222
22 2 22 2
22122
,,,.
rr zz rz
aaaa
 



 





 




, 0,


0

 , 0

 ,
0
rz




2
0










0
r
u  0








, 0
z
u  0








,
22


. (89), (90)
has form