Proceedings of the World Congress on Mechanical, Chemical, and Material Engineering (MCM 2015)
Barcelona, Spain – July 20 - 21, 2015
Paper No. 302
Numerical Analysis of the Effect of Filling Ratio on the
Transient Behaviour of a Two-Phase Closed Thermosyphon
Davoud Jafari, Sauro Filippeschi, Alessandro Franco, Paolo Di Marco
DESTEC / Università di Pisa
Largo Lucio Lazzarino 2, 56122 Pisa, Italy
d.jafari@studenti.unipi.it
Abstract –The use of the two-phase closed thermosyphons (TPCTs) is increasing for many heat transfer
applications. In this study, the development of a numerical model is presented with the purpose of determining the
transient characteristics of a TPCT. The model includes the heat transfer through the wall, vapour core, liquid pool
and the falling condensate film. The complete two-dimensional conservation equations for mass, momentum, and
energy are solved using finite volume scheme for the vapour flow and pipe wall. The liquid film is modelled using a
one dimensional quasi-steady Nusselt type solution. The model is validated by comparison with existing
experimental data. The effect of filling ratio (volume of working fluid to evaporator section) for a copper
thermosyphon has been investigated. Water is used as a working fluid for different filling ratio from 20 to 100%. The
investigations are aimed to determine working thermal performance of a TPCT at different filling ratio for solar
energy applications.
Keywords: Numerical simulation, two phase closed thermosyphon, filling ratio
1. Introduction
Heat pipes have emerged as an interesting and cost effective technology for the thermal control
solution (Faghri, 1995). A special heat pipe where the condensed liquid moves to the condenser by gravity
is two phase closed thermosyphon (TPCT), without capillary structure. The TPCT technology has found
increasing interest of the researchers in many industrial applications (Vasiliev and Kakac, 2013) and
applications like solar (Esen and Esen, 2005; Akbarzadeh and Wadowski, 1995).
The widely spread use of TPCTs has led to increase the demand for analytical and numerical
mathematical models. Considering lumped capacity modelling, Dobran (1985) provided a model aiming
to determine the steady-state characteristics and stability thresholds of a TPCT. Farsi et al. (2003)
improved the model proposed by (Dobran, 1985) in order to analyse the transient regime of a TPCT.
Ziapour and Shaker (2010) presented a simplified thermal network model for a TPCT. Concerning
analytical and numerical approaches modelling, El-Genk et al. (1997) and Pan (2001) and developed a
condensation model of a TPCT by considering the interfacial shear due to mass transfer and interfacial
velocity. Jiao et al. (2008) developed a steady state model to investigate the effect of the FR on the
distribution of the liquid film and liquid pool. Jiao et al. (2012) developed the previous model to utilize the
criteria for dryout, flooding and boiling limits to investigate the effects of FR. Fadhl et al. (2013) built a
CFD modelling to simulate the operation of a TPCT including the pool boiling in the evaporator section
and the condensed liquid film in the condenser section. The volume of the fluid (VOF) model in FLUENT
was used for the simulation. Harley and Faghri (1994) presented a numerical model of which coupled a
general quasi-steady Nusselt type solution of the falling film without considering the liquid pool.
Recently, Shabgard et al. (2014) numerically developed a 2-D model to simulate the transient operation of
a TPCT with various filling ratios (FRs), based on the numerical model for the condensate film developed
by (Harley and Faghri, 1994).
Although considerable studies had been accomplished to investigate the behaviour of the TPCT in
steady operating conditions, there still exists considerable uncertainty in the description of the complex
302-1
physical phenomena of TPCTs in transient mode. The heat and mass transfer through a TPCT is, in fact,
significantly affected by different design variables such as: geometry, inclination angle, FR and working
fluid. The effect of filling ratio in performance of the TPCT considered in present paper which has not
been well investigated in the available literatures due to lack of the comprehensive model to analyse the
effects of filling ratio. In this study, a transient numerical model of TPCT is presented that account for
heat transfer through the wall, liquid pool and the falling condensate film.
2. Mathematical Model of the TPCT
The operation of a TPCT is easily understood by using the physical scheme, as illustrated in Fig.1.
Basically, TPCTs consist of three different zones: evaporator, condenser and adiabatic regions. The heat is
supplied to the pipe in the evaporator region, which causes the liquid contained in the pool to evaporate.
The generated vapour then moves upwards to the condenser section. In the condenser, heat is removed,
condensing the vapour, and the resulting liquid returns to the evaporator (by gravity). The present
numerical model consists of: the transient 2-D conservation equation for the vapour flow and the pipe
wall, a 1-D quasi-steady liquid film and a liquid pool. This approach is based on Faghri and Harley (1994)
and considered in some recent paper (Shabgard et al., 2014). The assumptions in this analysis are:
curvature effect of liquid film is negligible, the vapour is an ideal gas, the condensate flow is 1-D, the
condensate flow occurs at the working fluid saturation temperature, all thermophysical properties are
constant and expansion of the liquid pool due to the formation of vapour bubbles is neglected.
outer wall:
T

 k w r ro  h (Tw  T ), (Cond. section)

 T
(adiabatic section)

ro  0,
 r
T
k
(Evap. section)
wall
r  q e ,

r o


vv  0
 u
centerline  v  0
 r
 Tv  0
 r
vv  uv  0, (no slipcondition)

(adiabaticcondition)
 z  0,
end caps  Tv
Liquid film-wall ( at   0 and L  L p ): ul  0, (no slip condition)
 kw
Tw
r
ri



(Ts  Tw ) ;   0.85 Re0f.1  6.7 10 5  l
 
v

1 kl
Liquid pool-wall: If L p  Le , for Z  Le :  k wall
Liquid film-vapour: uv  u film
T
Wall-vapour: If  , L p  0 : k w w
r
T
r
ri
 k pool

 for Cond. section and   0 for Evap. Section

T
r
ri
at r  rv
ri
 kv
Tv
r
ri
Fig. 1. Schematic of a TPCT showing the boundary conditions
2. 1. Vapor Space
The governing equations for the transient, compressible, laminar for the vapour flow of the TPCT,
assumed that the vapour is ideal gas with constant viscosity are described in cylindrical coordinates as
follows:
Mass:
v 1 
 (  v uv )

(rv vv ) 
0
t
r r
z
(1)
302-2
Momentum
(radial):
 4 1   vr  4 vv  2.vv 1  2uv 
vv
v
v
1 Pv
 vv v  uv v  
 vv 


r

2
t
r
z
 v r
z 2
3 rz 
 3 r r  r  3 r
(2)
Momentum
(axial):
 1   uv r vv  4  2uv 
uv
uv
uv
1 Pv
 vv
 uv

 vv 

g
r

t
r
z
 v z
3 z  3 z 2 
 r r  r
(3)
Energy:
( c p ) v
DTv 1  
T    Tv  DPv

 v 
 rkv v  
 kv

Dt
r r 
r  r 
z  Dt
(4)
where u and v are the axial and radial velocity components, respectively, the equation of state of ideal
gases is used to calculate the vapour density and Φ is the viscous dissipation of the working fluid:
2
2
 v  2  v  2  u  2   v
u 
u 
2 1 
  2 v    v    v     v  v   
(rvv )  v 
r 
3  r r
z 
 r 
 z    z
 r 
(5)
2. 2. Pipe Wall
The heat transfer through the heat pipe wall is transferred by conduction. The corresponding energy
equation is:
 w c pw
 1   T   2T 
T
 kw 
r

2 
t
 r r  r  z 
(6)
where cp and cpw are the density and specific heat of pipe material, respectively. The boundary
conditions describe in Fig. 1.
2. 3. Liquid Pool Model in Evaporator Section
The equations for the total mass of liquid film (Mf) and vapour (Mv) are given as
M f 
Lt
Lp
Mv   
Lt
Lp
2R     dz
(7)
R     dz
(8)
2
i
l
2
i
v
The liquid pool height (Lp) during the transient operation of the thermosyphon can be obtained as
Lp 
Mt  M f  Mv
(9)
Ri2  l
where Mt is the total mass of the working fluid within the thermosyphon. The heat conduction
equation is used to determine the temperature distribution in the liquid pool as follow:
( c p )l
 1   Tl   2Tl 
Tl
 keff 
r
 2 
t
 r r  r  z 
(10)
302-3
where the effective thermal conductivity of the pool (keff ) is calculated by considering the fact that the
heat transfer rate reaching the pool surface is equal to the heat transfer rate into the pool minus the rate of
thermal energy accumulation inside the pool:
keff
Tl
t

Lp

L p l c p,l Tpn 1  Tpn
2L
h p (Tw  Tv ) 
Rv
T

(11)
where hp is the pool heat transfer coefficient. The heat transfer mechanism at the evaporator section
includes different regimes: the natural convection, combined convection and the nucleate boiling which
depend on the filing ratio and the heat flux. There is no unique correlation or set of correlations that is
good to define the heat transfer coefficient in all thermosyphons. In order to select the appropriate heat
transfer correlation during operation, the heat transfer regime inside the pool must be specified and for this
reason the dimensionless parameter proposed by El-Genk and Saber (1998) is used to determine the
dominant pool heat transfer regime (X)

X   Ra Prl

   v
 l



0.4 

 Pv Lb q
  g h fg vl

0.35 
 Pv vl
 





0.7
(12)
1 4


 l2


 g     
l
v 

14



(13)
where  is the mixing coefficient. In this paper, the heat transfer of the pool is dominated by natural
convection if X < 106, two-phase convection if 106<X<2× 107 and nucleate boiling if X>107. For the
nucleate boiling regime, the Imura equation is selected among different nucleate boiling correlation
(Imura et al., 1983).
2. 4. Falling Film Model
In this study, the falling film along the wall has been modeled based on the quassi-steady Nusselt type
analysis, and the conservation of momentum equation applied to obtain determine the hydrodynamic
behavior of liquid film neglecting the inertial terms:
(14)
d 2 ul
1 dPl  l g


 l dz
l
dy 2
where z is distance from the condenser end cap and y is the distance from pipe wall. Since the liquid
film thickness is very thin the temperature distribution in the wick could be assumed linear. The heat
transfer across the film is obtained based on assumption of one-dimensional conduction across the liquid
film:
(15)
 2Tl
0
t
Considering small thickness of the liquid film compared to the thermosyphon length, the boundary
layer approximations is applied (dPl/dz)=(dPv/dz) and considering the equal shear stress in the liquidvapor space, the correlations of the distribution of liquid axial velocity (ul) which given the interfacial
velocity at y= is simplified to:
302-4
y   v y uv
 P
 y 2
ul ( y )   v   l g 


  v vv (u v  ul )

z
2

 l   l r rv


l
(16)
Two last terms in the right hand side of Eq. 20 describes the shear stress due to the shear stress due to
friction forces and mass transfer. By considering this point that mass inters in a fixed control volume in
the liquid film is related to the condensation/evaporation of vapor and liquid film from above, the mass
flow rate per unit width of liquid film, the vapor mass condensation/evaporation rate per unit width (  )
and the relationship between the liquid film thickness (  ), axial velocity (ul), radial velocity (vv) and
position at axial coordinate z, is yields as follows:

(17)
   l ul ( y)dy
0
m v    v, vv, dz
(18)
d
dz  m v
dz
(19)
L
z
 1 Pv  l g   l  3



 l  3
  l z
  l  2  v u v
L

  v vv (u v  ul )    v , vv , dz

z
r
2


r
v
l

(20)
Thus the film thickness can be obtained along the thermosyphon by obtained the vapor velocity field
in each iteration and solving Eq. (20). It should be noted that the falling film is defined by radial vapor
velocity. Thus, the initial vapor velocity field is set to zero and therefore there is no liquid along the pipe
and the working fluid is pooled at the bottom of evaporator section.
2. 5. Boundary Conditions, Initial Conditions and Numerical Procedures
Fig. 1 shows the boundary condition at outer wall, end caps, centerline, liquid film-wall, liquid filmvapour and wall-vapour of a TPCT. At the Liquid film-wall, in the condenser section the thermal
resistance due to the film thickness is modified to account for the entrainment effects (), see Fig. 1, which
is proposed by Jouhara and Robnison (2010). At the vapour–liquid film interface, the temperature is
assumed to be the saturation temperature corresponding to the interface pressure. Thus, the saturation
temperature can be determined by applying Clausius–Clapeyron equation. Also, the interface radial
velocity can be found through the evaporation rate required to satisfy heat transfer requirement.
 1 Rg Pv 
Ts   
ln 
 T0 h fg P0 


v  
(21)
1
Tw
h fg  r
kw
(22)
T0 and P0 are reference saturation temperature and the reference saturation pressure, and Rg is the gas
constant for the vapour. Concerning initial conditions, the temperature of the TPCT is considered as the
environmental temperature and the initial pressure is set to the saturation pressure corresponding to the
initial temperature. The governing equations along with the boundary conditions are solved by employing
the control volume finite difference approach described by Patankar (1980). A combination of direct
method and the Gauss-Siedel method is employed to solve the discretization equations. The SIMPLE
302-5
algorithm (Patankar, 1980) is used to solve the conservation equations. In this method the pressure is
considered as a dependent variable and directly apply the state equation = P/RT to obtain the vapor
density while iterating, considering the boundary and interfacial conditions. A FORTRAN program was
developed to numerically solve the problem using a structured staggered and non-uniform grid system.
Grid independence of code was investigated by varying the number of axial and radial grids and value for
grid size is considered as 14×80 (r–z). The converged solution is assumed to be reached when the relative
change of the velocity, pressure and temperature is less than 10-6.
3. Results and Discussion
In order to evaluate the model, the results have been compared with the experimental results reported
by Jouhara and Robinson (2010) for different heat input. The numerical model has been tested on a TPCT
configuration includes: a 200 mm long copper tube having an inside diameter of 6 mm and outside
diameter of 12 mm. Evaporator, adiabatic and condenser sections were 40, 100 and 60 mm long,
respectively. Fig. 3 shows the outer wall temperature of the modelled TPCT. The agreement between the
numerical and experimental results is good.
The simulation of the TPCT is established with the dimension and water as working fluid, as is shown
in Table 1 for the evaporator, adiabatic and condenser sections of 700, 50 and 250 mm long, respectively.
Table. 1. Basic specifications of the different parameters of the TPCT
Wall
Outer radius, (mm)
Thickness, (mm)
Total length, (mm)
Thermal conductivity, (W/mK)
Specific heat, (J/kgK)
Density, (kg/m3)
12.5
1.5
1000
350
380
8900
heat transfer fluid (liquid)
Thermal conductivity, 0.67
(W/mK)
Density, (kg/m3)
925
Latent heat, (kJ/kg)
2140
viscosity (Pa s)
2.2×10-4
heat transfer fluid (vapour)
Thermal conductivity, 0.0288
(W/mK)
Specific heat, (J/kgK)
2406
viscosity (Pa s)
1.35×10-5
T0 (C)
25
P0 (Pa)
3.17×103
The result of present work is explained with the aim of transient and steady state on the performance
characteristics of the TPCT for the filling ration ranged 20-100%. First, some results of the transient
behaviour of the TPCTs will be presented. The temperature distribution at the outer wall and the falling
condensate film of the TPCT for the selected filling ratio, 20%, and heating rate of 300 W are shown in
Fig. 4a and 4b, respectively. In the evaporator section, the liquid in the thin film evaporates, removing its
latent heat from the evaporator pipe wall. Following vapour condenses inside of the condenser section;
gravitational forces cause the condensate to form a thin film along the surface of the inner wall. The
relatively large vapour velocity exerts significant shear stress on the film surface at the exit of the
evaporator section, where the vapour velocity is maximized, which thereby increases the condensate film
thickness (see Fig. 4a). Fig. 4b shows the temperature distribution along the wall surface of a TPCT. The
temperature increases with time and is higher in the evaporator section and lower in the condenser section.
As evidenced in Fig. 4b the wall temperature increases and then decreases that can be described by
considering the condensate film response. In other word, the local dryout could be occurred in the region
above the liquid pool level in evaporator section. This can occur due to the effects of the high evaporation
rate and interfacial shear whereby they prevent the down flow of liquid. Under this condition, fluctuate
temperature occurs since the dryout part could be rewetted in a short time. The condensate film reaches
the bottom of the thermosyphon at t=150s. Before that, the input heat to the evaporator accumulates in the
wall and increases the temperature.
As can be seen in Fig. 5, the steady state temperature and falling condense film profile for different
filling ratio, the temperature distributions of the all filling ratios are similar except for the lower part of the
evaporator section that is because of liquid pool. It can be seen that with increasing filling ratio from 20 to
40 % the temperature of hottest section increases, but that temperature decrease from filling ratio of 40 to
60%. Moreover, for all filling ratios, the condenser temperature is almost uniform with the region adjacent
302-6
to the adiabatic section slightly colder due to a thicker condensate film which induces larger temperature
drops. Considering film thickness, there is not a difference in the behaviour of the liquid film of the
different filling ratios, which is expected regarding the equality of input heat transfer rates and cooling
conditions at the condenser section. Fig. 6 shows the maximum transient wall and vapour temperature for
different filling ration in comparison with steady state conditions. As filling ratio increase up to 50% the
wall and vapour temperature decreases due to the condensate film response. This can be explained by
larger interfacial area between the evaporator section wall and working fluid for higher filling ratio. It is
obvious that the there is no a difference between the maximum transient temperature of wall (vapour) and
steady state pool temperature for filling ratio more than 50%. Further, as can be seen, filling ration has a
little effect on the temperature of adiabatic (condenser) section (less than 2°C). Fig. 7 shows the effects of
filling ratios on thermal resistance of the TPCT. The results show that the filling ratios have minimal
influence on thermal resistance for filling ratio ranged 50-70%. The thermal resistance of the TPCT for
lower filling ratio (20-40%) is slightly lower than that of higher filling ratio.
Experimental data given by:
Jouhara and Robinson (2010)
Working fluid: water
FR=159%
413
Temperature (K)
393
373
353
333
Experimental data-Q=220 w
Experimental data-Q=145 w
Experimental data-Q=100 w
Present model-Q=220 w
Present model-Q=145 w
Present model-Q=100 w
313
293
273
0
0.05
0.1
0.15
Axial position (m)
0.2
Fig. 3. Axial distribution of the wall temperatures of TPCT obtained from the experimental data compared with the
present simulation
420
380
360
(a)
pool
340
320
300
Evap.
t=10 s
t=45 s
t=120 s
t=135 s
t=180 s
steady state
60
Film thickness (μm)
400
T (K)
70
t=10 s
t=45 s
t=120 s
t=135 s
t=180 s
steady state
50
40
Pool
30
20
Evap.
10
Cond.
(b)
280
Cond.
0
0
0.2
0.4
0.6
Axial position (m)
0.8
1
0
0.2
0.4
0.6
Axial position (m)
0.8
Fig. 4. Transient (a) condensate film thickness profiles and (b) temperature profiles at the outer wall
302-7
1
Pool
60
140
50
120
Max T_wall-Transient
40
340
30
20
FR=20
FR=40
FR=60
FR=80
FR=100
335
330
0
0.2
0.4
0.6
Axial position (m)
0.8
Temperature (C)
345
Film thickness (μm)
Temperature (K)
350
Max T_vapor-Transient
T_pool-steady state
100
T_adiabatic-steady state
T_condenser-steady state
80
60
10
40
0
20
20
1
Fig. 5. steady state temperature profiles and condensate
film thickness for different filling ratio
30
40
50 60 70 80
Filling ratio (%)
90 100
Fig. 6. Maximum transient temperature and
steady state temperature of different regions
0.025
Thermal resistance (C/W)
0.020
0.015
0.010
0.005
0.000
20
30
40
50
60
70
Filling ratio (%)
80
90
100
Fig. 7. The variation of the thermal resistance for different filling ratios
4. Conclusion
In this study, a transient 2-D numerical model of TPCT is presented that account for conjugate heat
transfer through the wall, liquid pool and the falling condensate film. The numerical model has been
preliminary validated using some experimental results available in the literature. The effect of filling ratio
on the TPCTs performance is investigated with specific dimensions which could be used for solar
collector application. It is found that the lower filling ratio provides the smallest thermal resistance. It is
found that the filling ratio has a slight effect on wall temperature after filling ratio of 50% however; up to
filling ratio of 50% the wall temperature increases due to local dryout.
In authors opinion, investigation of critical filling ratio and, therefore, the liquid pool dries out and the
length of the dried surface in the evaporator could be future work which unfortunately, not much is known
about the detailed relationship between the filling ratio and heat input.
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