Integral Transform Method Versus Green Function Method
in Electron, Hadron or Laser Beam - Water Phantom Interaction
259




2
0
ij
ij
ij
Kx
kx K x
xx









(4)
Equation (4) provides a series of positive eigenvalues
,
j
jN



ij
ii i
ii
k x mx k x mx
Kx AJ BY
j
N
mm

   

  


(5)
Where:
iii
c
and J
0
and Y
0
are the Bessel and Weber functions respectively.
After the application of the integral operator K
i
(x) equation (4) becomes:





x
ut TxtKxdx
C

 





1
1
,,,
i
i
x
j
jj
j
x
f
t
f
xtK x dx
C

 

(,)
kk
Ky
and
(,)
ll
Kz
, which satisfy the
equations:






2
2
2
2
2
2
,
,0
,
,0
kk
kk k
ll
lll
Ky

ˆ
(,,,)
(,,,)
ji j k l ki j k l li j k l
ij
kl
i
ii
TtTtTt
Tt
fxyzt
c
t

     




(9)
Where from it follows:

2
13
23
1
()()()
/2
/2
/2 /2

22 2 22 2
0
22 2
() ()()
1
0
()
111
(,,,) [1 (1 )( )]
(,,) (,) (,) (,)
jl k jl k
jl k
ttt
i
jkl
jkl ijj kk ll
Txyzt e e ht t
gKxKyKz

        



 
     

(11)
where:

2

 
  

 
(12)

stands here for the thermal diffusivity. We point out that our semi-analytical solution
becomes analytical, if we observe the, after 10 iterations, the solution becomes convergent
(we have values of temperature less than 10
-2
K for: i>10;j>10 and k>10). Under these
conditions equation (10) becomes:

22 2 22 2
0
22 2
10 10 10
() ()()
1
0
()
111
(,,,) [1 (1 )( )]
(,,) (,) (,) (,)
jl k jl k
jl k
ttt
i
jkl
jkl ijj kk ll


Fig. 1. Thermal field distribution in case of 1 MeV proton beam irradiation of a water
phantom, for 120 sec.
Fig. 2. Thermal field in water submitted to cw CO
2
laser irradiation for 50 sec.

Heat Analysis and Thermodynamic Effects
262
The power of the cw CO
2
laser beam was P= 1W.
It is known from experience [8] that proton therapy is more efficient in the “presence” of a
laser beam. We plotted in figure 2 the thermal gradient in water produced by cw CO
2
laser
irradiation for 50 sec. (P = 1W). In Fig. 3 we presented the temperature field in water
produced by an electron beam, when the “steady - state” is achieved. The white color
corresponds to an increase of temperature, and the black color represents a decrease of
temperature. We have use sub-domains of 0.25 cm. The thickness of the water phantom was
0.25 cm, and was contained in a plastic cube with a mass density close to 1 g/cm
3
. Figure 3
was obtained using eq. (13). Fig. 3. Temperature field in water produce by an electron beam, when the “steady- state” is

( ) 83.2337 18.6522 15.1080 4.1417 0.3506Dx x x x x


(16)
Integral Transform Method Versus Green Function Method
in Electron, Hadron or Laser Beam - Water Phantom Interaction
263
Here x stands for the direction of electron propagation. We will consider the radiation
(electron beam) normal to water surface.
From the standard theory of Green function applied to multi-layer structures, we have:

12
12 12
12 12
n
ll
ll ll
nn
kk k kk k
nn
ll l
l
K
 

   

(17)
where l
i

3
1
22
0
[1 ( )]
() () ( )
0
()
PRT
KT
f
d








(20)
where:
K
K



and:

()KKT

264
Let us considering the following relations:

123
(,,,) (,,,) (,,,) (,,,)Axyzt A xyzt A xyzt A xyzt
(23)
Therefore:

123
(,,,) (,,,) (,,,) (,,,)Txyzt T xyzt T xyzt T xyzt
(24)

Fig. 4. The temperature field in water produced by a 10 MeV electron beam, when the
“steady- state” is achieved.
We suppose that for the heat transfer coefficients:
123456
hhhhhhh


. If we
consider a linear heat transfer at the sample surface (the “radiation” boundary condition
[11]), we have:
for the first laser beam , direction of propagation along x axis: 22
2






  



 


 

 
 


 

 
 

(25)
for the second laser beam, direction of propagation along y axis:
Integral Transform Method Versus Green Function Method
in Electron, Hadron or Laser Beam - Water Phantom Interaction
265
LLL
yK zK zK
 






 


 

 




 


 

 

(26)
for the third laser beam, direction of propagation along z axis:
MMM
yK zK zK
 






 


 

 











(27)
The solution of the heat equation subjected to boundary conditions (25), (26) and (27) is:

, 111


,111
(,) ( ,)
yv zw
rtvw
My
Mz
 












(28)
We have:


22
2
2
2
2
0

22
(,,) ( (1 )(,)
()) ( ,) ( ,)
i
xmn
a
mn
x
ijo i Sxi
a
ijo
bc
SS mnyj zo
bc
I
aerKx
KC C C
rxdx IKyKzdydz






     
  



(30)

Kx xhk x

 
(32)
The other formulas can be easily obtains by “rotations” of the indices.
t
-is the time and
o
t
the exposure time.
We have:
S
r
is the parameter which take care of the surface absorption and which make
sense only for one photon absorption.
Here
:
222
,,,,,
ii pp tt
   
are the eigenvalues corresponding to the eigenfunctions:
,,
,,,,, ,,,,,, , , ,,,
xyzxyzxyzxyz x y z x y z
KKKPPPLLLTTTMMMNNN
[7].
,,,,,,
ij o p r s t v
CC C C C C C C












,111
222 22 2 2 2 2
,111
11
)(,)(,)(,)
(,,)(,,,)(,)(,)(,)
(, , )(, , ,) (,) ( ,) ( ,)
xv yr zs
vq p r s
pr s vr s xv yr zs
vq p r s
tvw tvw xt yv zw
vw
LxLyLz
dtTxTyTz
eftMxMyMz






 


















(33)
In formula (33) the upper index 2 means that the corresponding values are connected with
two photon absorption. The eigenfunctions and the eigenvalues for two absorption
phenomena can be calculated in the same way like in the case of one photon absorption with
the only change that we have another absorption formula. It make no sense to take into
account three or more photons absorption phenomena because in this situations the cross
sections are very small.
In the next pages we will present three simulations, using the developed “multiple beam
irradiation”.

3
(, ,,)Txyzt are
coupled via boundary conditions. Our model can be easily generalized for the cases when:
123456
hhhh hhor


x
yz


.The model could be applied to any laser-solid
system whose interaction can be described by Beer law.
The integral transform technique has proved once again it’s “power” in resolving heat
equation problems [14-17]. Fig. 7. Temperature field when the sample is irradiated simultaneously with the two laser
beams, mentioned above (Fig.5 and Fig.6)
6. Discussions and conclusions
We developed a method for solving the heat diffusion equation- based on dividing the
whole domain into small intervals, the length of each depending on the required accuracy of
the final solution. The theory is applicable to laser, electrons and hadrons beams interaction
with human tissues (which are simulated by a water phantom). In each of the obtained
intervals the thermal conductivity function is approximated by a linear function. This
function is introduced in the heat equation associated to each interval. At the interface
between intervals, the continuity of temperature function and its first derivative are
ensured, these conditions providing the values for the coefficients obtained in the final
solution.
Integral Transform Method Versus Green Function Method

laser beam
with multi-layered structures (of the type thin films substrate) [10] or with optical
components [11].
The actual strength of the model is that it can take into account any form of the beam spatial
distribution and any stationary type of interaction. That was the starting point for
developing the semi-classical heat equation solution, which included the multi-photons
laser-sample interaction [12]. The particular case
0
i
m

(i.e. when k
i
(x)=k (x
i
) ) was
analyzed in Ref. - [13].
The “power” of integral transform technique was emphasized in references [14-17]; both in
classic and quantum physics.
Finally: a remark about figures 1-4. We mention that: dT (x, y, z, t) is in general proportional
with S (x, y, z, t). This is not always true, but in our case is valid because the small values of
the heat transfer coefficient. For a comprehensive discussion of the importance of heat
transfer coefficient, see Appendix B in reference [11].
Our model offers a first simple approximation of the temperature field in (electron, proton,
laser) beam (liquid, solid) target interaction.
The model can also describe the thermal fields for three different beams (electron, proton and
laser), which act simultaneously onto a sample along the three Cartesian coordinates axes.
Figure 3 is illustrative for the strength of our model. The simulations performed using sub-
domains of 0.25 cm were indeed in good agreement with the solutions given by the Green
function method.

for Electronic Cooling
Seok-Hwan Moon and Gunn Hwang
Electronics and Telecommunications Research Institute (ETRI)
Korea
1. Introduction

Electronic devices have been minimized, but their performance is becoming better and
better. Their heat flux has been significantly increased and has already exceeded about 100
W/cm
2
recently. The insufficient dissipating of the heat flux may lead to performance
decrease or failure of the electronic device and components. Heat flux in laptop computers
has not been questioned; therefore, only a heat sink has been applied on cooling them.
Recently, however, a more powerful cooling solution is sought for high heat flux. Solid
materials with high thermal conductivity have been mainly used in low heat flux
applications, whereas small-sized heat pipes have been utilized in high heat flux
applications. The use of small-sized heat pipes in electronic devices like laptop computers
has only been developed recently. For example, the use of heat pipes with diameter of 3–4
became common in laptop and desktop computers only during the early 2000s. Recently, as
electronic devices have started to become smaller and thinner, heat pipes with diameter of
3–4 mm have been pressed to fit the form factor to them. However, a lot of problems were
encountered in their thermal performance, thus micro heat pipes (MHPs) were developed to
solve them. Specifically, a flat plate micro heat pipe (FPMHP) with diameter of less than 1.5
mm was developed by Moon (Moon et al., 2002). FPMHPs are being used mainly in display
panel BLU applications and are being prepared to be used in the LED headlight of vehicles.
However, in spite of their thermal performance and broad applications, FPMHPs may still
show degradation in thermal performance in the case of thinner applications.
If we consider phase-change cooling devices like heat pipes that have thermal conductivity
that is 500 times larger than copper rods for small-sized and thin electronic devices, there is
a need to develop new cooling methods suitable for them.

Therefore, grooves were fabricated on the MHP envelope or the sharp corners of the
polygonal structure by reforming itself act as the wick. Tubular type MHPs with circular
or polygonal cross section are suitable for small-sized application, whereas flat plate type
MHPs are suitable for display application. Cooling solutions that may be considered for
small-sized electronic and telecommunication devices are as follows. Materials with high
thermal conductivity like copper and aluminum are cooling solutions that can be used
easily. Such materials are widely used as cooling solutions in the fields of electronic
packaging module and system levels. A liquid cooling using micro channels is suitable for
high heat flux application due to its high heat transfer rate. However, because it has
constraint in the form factor, it is not suitable for mobile application. Furthermore, overall,
the spray cooling method and thermo-electric cooling (TEC) may be considered for a
specific application.
3. Micro Heat Pipes (MHPs)
3.1 Characteristics of the MHP
Because MHPs have limited inner space compared to medium-size heat pipes, inserting
additional wick for liquid flow path is not easy. Therefore, MHPs are characterized to have
capillary structure on their wall. MHPs have small vaporizing amount with latent heat due
to their small size, therefore they are not suitable for high heat flux application. The
existence of non-condensable gas, albeit in very small amount, can lead to the decrease in
performance; therefore, high-quality fabricating process is needed.
Cotter (Cotter, 1984) was first to propose the concept of an MHP for the purpose of cooling
electronic devices. The MHP was so small a heat pipe that the mean radius of curvature of
the liquid-vapor interface is comparable to the hydraulic radius of the flow channel.
Typically, MHPs have a convex but cusped cross section with hydraulic diameters of 10–500
µm and lengths of 10–50 mm (Faghri, 1995). Since the initial conceptualization by Cotter in
1984, numerous analytical and experimental investigations have been reported so far.

Micro Capillary Pumped Loop for Electronic Cooling
273
According to the previous investigations reported by Cotter (Cotter, 1984), Babin and


12
11
cvl
cc
PPP
rr


 


(1)
Where r
c1
and r
c2
are the principal capillary radii of the meniscus. For the MHP designed in
this study, the capillary radius can be considered as r
c1
r
c(x)
and r
c2
 since capillary
radius variation over axial direction occurred, but the capillary radius variation through
radial direction nearly not occurred. Fig. 3 shows the vapor and liquid pressure
distributions over axial direction of the MHP in the condition of 50 °C of operating
temperature.


Axial position x (m)

Fig. 2. Capillary radius variation over axial position

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035
0
200
400
600
800
1000
1200
1400
Vapor
Liquid
Pressure difference (Pa)
Axial position x (m)

Fig. 3. Pressure distributions of vapor and liquid over axial position
3.2 Fabrication of the MHP
The size of the MHP developed in this study is smaller than that of the miniature heat pipe
with diameter of 3–4 mm. The MHP could be manufactured using current mechanical
technology of the simple manufacturing process. Therefore, this manufacturing process of
the MHP has a good productivity. In fact, the microstructures of the MHP may be
manufactured with etching process (Gerner, 1989), but such process has disadvantages in
terms of productivity and cost. The container of the MHP was manufactured with the
drawing process, and Fig. 1 shows its cross section. This MHP does not have additional
wick installed on the inner wall of the general heat pipe, but has sharp corners made with
structural deformation of its wall, which serves as wick.


Table 1. Experimental specification of MHP
FPMHPs with cross section of rectangular, modified rectangular, and triangular types were
newly developed in the present study. The container of the FPMHPs was manufactured by
the extrusion process, which could effectively form a sharp edge and significantly enhance
the productivity of FPMHPs. Moreover, mass production of FPMHPs is possible through
this method. The FPMHPs have three types of cross section – rectangular, modified
rectangular, and triangular. Fig. 5 shows the cross section of the FPMHPs; Table 2 presents
the dimensions of the FPMHPs.

Heat Analysis and Thermodynamic Effects
276

Thickness Width Total length
Rectangular type 2 mm 12 mm 50 mm
Modified rectangular type 1.5 mm 14 mm 50 mm
Triangular type 1.5 mm 12 mm 50 mm
Table 2. Dimensions of the FPMHPs (a) Rectangular (b) Modified rectangular (c) Triangular
Fig. 5. Cross sections of the FPMHPs
3.3 Thermal performance of the MHP
The testing apparatus for the thermal performance of the MHP was composed of an MHP, a
vacuum chamber unit, a constant temperature bath for cooling the MHP, a data acquisition
system, and a DC power supplying unit, as shown in Fig. 6. The evaporator of the MHP was
heated using the electric resistance heater and DC power supply unit. The wire with 0.36
mm diameter and 10 Ω/m resistance per meter, as a heater, was wound around the copper
block with an interval of 0.5 mm for supplying the constant thermal load, which was
attached on the outer wall of the evaporator. The condenser of the MHP is cooled by the
water jacket with circulating water. Thermal grease (0.74 W/m °C) was filled between the

The experimental test was performed to investigate the thermal performance of the MHP.
The operating temperature of the MHP identical with the temperature at the adiabatic
section of the MHP was considered for the four cases – 60 °C, 70 °C, 80 °C, and 90 °C. The
temperature and the amount of coolant circulating between the water jacket and the
constant temperature bath were controlled carefully to maintain the conditions of the
constant operating temperature of the MHP. The thermal load supplied to the MHP was
increased stepwise by 0.5 W from 0.5 W. The wall temperature of the MHP was recorded at
the steady state by each thermal load step. The measurement was stopped when the wall
temperature of the evaporator of the MHP rapidly increased due to dry-out. The wall
temperatures of the evaporator, the adiabatic section, and the condenser of the MHP were
averaged in each zone. This test was measured in the chamber with vacuum condition of 10-
2–10
-3
torr. The results of the present study included errors in measurement, i.e., the
tolerance in the heat supply (
0.05 V for voltage, 0.01 A for current) and that in the
temperature measurement (
0.1 °C). The performance testing apparatus of the FPMHP is
similar to that of the circular type MHP.

3.3 3.3 7.57.5 8.38.3
10 15 25
Evaporator
(Heater)
Adiabatic
Section
Condenser
(Water jacket)
50
Vacuum chamber(10-2~10-3torr)

were averaged over 60 seconds after steady state to reduce minor temperature fluctuation
error. As shown in Fig. 7, the wall temperatures of the MHP are increased as the thermal
load is increased. This means that the thermal equilibrium, which is the isothermal property
of the MHP from the evaporator to the condenser, is well accomplished. The temperature
differences between the evaporator and the condenser were 4.3
–
9.8 °C over the thermal
loads of 0.5–4 W. However, the temperature difference of 9.8 °C between the evaporator and
the condenser in the thermal load of 1 W is higher than that in other thermal loads. This is
because the amount of latent heat to be transported toward the condenser is small by
insufficient vaporization at the evaporator, and the thermal resistance is high by a relatively
thick liquid film under low thermal load near 1 W.
Fig. 8 shows the temperature distribution by the axial length at the operating temperature of
90 °C, which equals the temperature at the adiabatic section. The tested MHP is the same as
the one in Fig. 7. It is seen that the temperature difference between the evaporator and the
condenser is increased as the thermal load is increased at the constant operating
temperature of 90 °C. This can be explained by the fact that vapor flow velocity is increased
as the thermal load is increased. Therefore, the friction force on the vapor-liquid interface
and the pressure drop in the liquid flow are increased. Because the space for the vapor flow
in the MHP is narrower than that in the conventional heat pipe, the pressure drop caused by
the friction on the vapor-liquid interface may largely affect the MHP performance.
Fig. 9 shows the effect of the inclination angle on the thermal performance of the triangular
MHP. In the figure, the negative inclination angle indicates a top heating mode in which the
evaporator is located higher than the condenser, and conversely, the positive inclination angle
means a bottom heating mode in which the evaporator is located lower than the condenser. As
shown in Fig. 9, the effect of the inclination angle on the thermal performance is small. The
thermal performance of the MHP was almost the same for the tilting mode from the horizontal
mode to the top heating mode with –90 degrees. However, there was a decrease in the thermal
performance of the MHP as the inclination of the MHP was rotated from the bottom heating
mode to the top heating mode. That is, the thermal performance of the triangular MHP was

(2)
Where T
e
and T
c
are the wall temperatures at the evaporator and the condenser of the MHP,
respectively, and Q(W) is a thermal load at the evaporator.
The tested MHP has a fill ratio of 20% to the internal total volume of the MHP. The
operating temperature is not constant, but increases as the thermal load is increased. The
heat dissipating at the condenser of the MHP was accomplished by circulating 20 °C water,
which was controlled by a constant temperature bath. As shown in Fig. 11, the heat transfer
limit of the triangular MHP is 1.6 times larger than that of the rectangular MHP. The heat
transfer limits were 4.5 W and 7 W for the rectangular MHP and the triangular MHP,
respectively. This is because the corners for the rectangular MHP are not developed sharply
compared to that for the triangular MHP and the capillary force needed for returning the
condensed liquid to the evaporator cannot be obtained sufficiently.
The property of the rectangular MHP having one additional corner than the triangular MHP
may make an advantage in its thermal performance. However, because a radius of curvature
at the corner is not sufficiently small to retain capillary pressure, the performance of the
rectangular MHP cannot be superior to that of the triangular MHP. The performance of the
MHP is largely restricted by the capillary limit.
The factor that mainly affects the capillary limit is the radius of curvature at the corner. The
radius of curvature
()r
is a function of a corner aperture angle
()

, a contact angle
()


section and a stainless steel as container material. Pure water was used as working fluid. As
shown in Fig. 13, the heat transfer limit of the present study is 1.7–2.1 times larger than that of

Heat Analysis and Thermodynamic Effects
280
Moon et al. over the operating temperature of 60–80 °C. This result shows that large capillary
limit was obtained in the present study compared to that in Moon (Moon et al., 1999). High
productivity and simple manufacturing process were considered, and enhanced performance
was obtained compared to that of Moon et al. for the future applications.
Figs. 14 and 15 show the performance test results of temperature distributions along with
the length of the FPMHP. In both figures, it can be seen that the temperatures of the
condenser instantly follow the temperature of the evaporator for overall thermal loads. This
phenomenon means that the FPMHPs have good constant temperature characteristics as a
heat pipe. The temperature differences between the evaporator and the condenser are 2.5–
6.4 °C for the FPMHP with fill ratio of 25% and 2.2–11.9 °C for the one with fill ratio of 15%,
respectively.
When the temperature of the evaporator was considered within the temperature of 120 °C,
Table 3 shows the heat transfer rate of the FPMHP with fill ratio of 20%. In the table, the
heat transfer rate of the rectangular FPMHP is higher than that of other FPMHPs. The heat
transfer rate of the modified rectangular FPMHP is as high as that of the rectangular
FPMHP is due to the capillary force of the former being higher than that of the latter.
Meanwhile, the heat transfer rate of the triangular FPMHP is lower than that of other
FPMHPs due to its characteristic of having small space for vapor flow.

0 1020304050
30
45
60
75
90

0 1020304050
84
87
90
93
96
99
102
Condenser
Adiabatic
section
Evaporator
Wall Temperature (
o
C)
Axial Length (mm)
Q
in
=4W
Q
in
=5W
Q
in
=6W
Q
in
=7W
Q
in

1000
1200
1400
Overall Heat Transfer Coefficient (W/m
2o
C)
Input Power (W)
L
total
=100mm
L
total
=50mm

Fig. 10. Overall heat transfer coefficient by total length

012345678
0
5
10
15
20
25
Thermal Resistance (
o
C/W)
Input Power (W)
Curved rectangular MHP
Curved triangular MHP


Fig. 12. Thermal performance by the operating temperature

50 60 70 80 90
2
4
6
8
10
12
Heat Transfer Limit (W)
Operating Temperature (
o
C)
Experimental data(Present study)
Experimental data(Moon et al,1999)

Fig. 13. Experimental results comparison between the present study and Moon et al.

0 1020304050
30
45
60
75
90
105
120
135
Condenser
Adi abatic
section

90
100
Condenser
Adiabatic
section
Evaporator
Wall temperature,
o
C
Axial Length, mm
Q
in
=0.5W
Q
in
=1.0W
Q
in
=1.5W

Fig. 15. Temperature distributions along with the axial length for rectangular FPMHP with
fill ratio of 15%

Heat transfer rate
Rectangular 13.66 W
Modified rectangular 13 W
Triangular 8 W
Table 3. Heat transfer rate of each FPMHP with fill ratio of 20%
3.4 Why micro CPL is needed in the future
Vapor and liquid have counter flow pattern in the heat pipe. Therefore, pressure drop on