6-3-2011.pdf


Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳۳

NUMERICAL ANALYSIS OF VAPOR FLOW IN A
HORIZONTAL CYLINDERICAL HEAT PIPE

Mr. Selah M. Salih Mr. Qahtan A. Abed Mr. Dhafeer M. AL-Shamkhi
Automobile Tech. Eng. Dept.

Technical College - Najaf
Automobile Tech. Eng. Dept.

Technical College - Najaf
Automobile Tech. Eng. Dept.

Technical College - Najaf

ABSTRACT
             The steady two-dimensional flow of a horizontal heat pipe in vapor region is investigated
numerically. For study of heat transfer and fluid flow behaviors of the heat pipe, the governing
equations in vapor region have been solved using a finite difference method. The numerical results
of heat transfer and fluid flow are presented for Reynolds numbers ranging of (Re =4, 10), the
Prandtl number taken is (Pr=0.00368), and the pipe dimension is taken to be (L/R =5). The results
show that the stream function at the wall increases linearly in the evaporator, decreases linearly in
the condenser and is steady in the adiabatic region because of uniform inflow and outflow boundary
conditions. Also, it can be seen that as the Reynolds number increases, the pressure distributions
shift up without considerable change in their shapes. The numerical analysis have shown that for the
low and moderate Reynolds number, the shear stress becomes zero at a point very close to the end
of the condenser. For verification of current model, the results of stream function for a heat pipe
have been compared with the previous study at the same boundary conditions and a good agreement
has been noticed.

KEYWORDS—Vapor region; heat pipe; numerical study.

. . ..

- - -
        :

.

.  في
ـي   لا.  ف

 (Re =4, 10)،(Pr=0.00368) ،( L/R =5)  .

 ".
ل.، 



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳٤

 ، "" .
،.

NOMENCLATURE
Cp  = heat capacity at constant pressure (kj/kg. K)
hfg  = latent heat of vaporization (kj/kg)
k  = thermal conductivity (W/m. K)
L = length (m)
P = pressure (N/m2)
Q = heat transfer (W)
R    = gas constant (kj/kg. K)
rv   =  Vapor radius (m)
ro  =outer radius of pipe (m)
Re  = Reynolds number
Pr   = prandtl number
T    = temperature (K)
v = radial vapor velocity  (m/sec)
u   = axial velocity (m/sec)
V   = reference velocity (m/sec)
x = axial coordinate (m)
r = redial coordinate (m)

Greek Symbols
Ѱ  = stream function (m2/sec)
ω  = vorticity (sec-1)
α   = fluid thermal diffusivity (m2/sec)
υ   = kinematics viscosity (m2/sec)
μ  = dynamic viscosity (kg/m. sec)
ρ   = density (kg/m3)
τ   = shear stress (N/m2)
θ = dimensionless temperature

Subscripts
* = dimensionless term
a   =adiabatic
c   =condenser
e   = evaporator
v   = vapor
int = interface
o   = outer
sat   = saturated

INTRODUCTION
The heat pipe is a vapor-liquid phase-change device that transfers heat from a hot reservoir to

a cold reservoir using capillary forces generated by a wick or porous material and a working fluid.
Heat pipes are the most effective passive method of transferring heat available today. Heat pipes
can transmit heat at high rates and have a very high thermal conductance.

Heat pipes have been applied to a wide variety of thermal processes and technologies. Heats
pipes have been applied in the cooling devices include generators, motors, nuclear, heat collection



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳٥

from exhaust gases, solar and geothermal energy. In general, heat pipes have advantages over many
traditional heat-exchange devices, (Doran, 1989).

The vapor flow in heat pipes has been investigated by various authors, (Rajashree and
Sankara, 1990), (Chan and Faghri, 1995), (Zhu and Vafai, 1999), and (Kim et al, 2003) have
published many techniques, theories, and experimental investigations of different heat pipe
structures. They found the pressure and velocity distributions along the heat pipe depended on the
value of radial Reynolds numbers.

(Nouri-Borujerdi and Layeghi, 2004) analyzed the vapor flow in concentric annular heat
pipe using SIMPLE algorithm and staggered grid scheme. They found the flow and heat for vapor
heat pipe was affected by increasing of radial Reynolds numbers.

(Yau, 2007) studied an 8-row thermosyphon-based heat pipe heat exchanger for tropical
building HVAC systems experimentally. This research was an investigation into how the sensible
heat ratio (SHR) of the 8-row HPHE was influenced by each of three key parameters of the inlet air
state, namely, dry-bulb temperature, and relative humidity and air velocity.

(David et al., 2008) design and test of a pressure controlled heat pipe (PCHP), testing
showed that (PCHP) was capable of maintaining a stable evaporator temperature within (0.1K)
despite wide swings in heat load and heat sink temperature.

In this paper a numerical model has been used for analysis of vapor flow in heat pipe
operation. The steady state incompressible flow has been solved in cylindrical coordinates in vapor
region. The governing equations have been solved using finite difference with collocated grid
scheme. The objective of this paper is study the heat transfer and fluid flow behavior of a
conventional heat pipe operation.

 MATHEMATICAL MODEL AND GOVERNING EQUATIONS
Figure 1 shows the simplified model and the coordinate system of the constant conductance

heat pipe (CCHP) used in the present study. The heat pipe configuration can be divided into three
radial regions, namely, vapor space, wick region and wall region .The working fluid is saturated
with wick in liquid phase. The power applied to the heater in evaporator causes the liquid in the
wick to vaporize. The vapor flows to the condenser section and releases the heat as it condenses.
The released heat is rejected through the wall to the ambient. The condensed working fluid in the
wick returns to heater section by the capillary force of the wick structure. To analysis the behavior
of flow of fluid and heat through the heat pipe by using continuity, momentum and energy
equations as flows.

Heat applied to evaporator section by an external source is conducted through the pipe wall
and wick structure, where it vaporizes the working fluid. The resulting vapor pressure drives the
vapor through the adiabatic section to the condenser, where the vapor condenses and releasing its
latent heat of vaporization to the provided heat sink. The capillary pressure created by the wick
structure, pumps the condensed fluid back to the evaporator. Therefore, the heat pipe can
continuously transport the latent heat of vaporization from the evaporator to the condenser sections.
This process will continue as long as there is sufficient capillary pressure to drive the condensate
back to the evaporator.

At the vapor–wick interface, the temperature is assumed to be saturated, corresponding to
interface pressure during heat pipe operation. Axial conduction along the wall and wick is assumed
negligible.
              The steady state two–dimensional incompressible laminar flow with constant viscosity in
cylindrical (r-x) coordinate, and no heat generation. The governing equations in vapor region are
continuity, Navier-Stokes and energy equations, (Rajashree and Sankara, 1990), are given as
follows:

0
)(1

r
rv

rx
u

 (1)



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳٦

2

21
2

2

r

u
r
u

rx

u
x
p

r
u

v
x
u

u (2)

22
21

2
2

r
v

r
v

r
v

rx
v

r
p

r
v

v
x
v

u (3)

2
21

2
2

r
T

r
T

rx
T

K
r
T

v
x
T

ucp (4)

The radial velocities at liquid-vapor interface, (Borujerdi, 2004), as following:

fgvco

c
c

a

fgveo

e
e

hLR
Q

v

v
hLR

Q
v

2

0

2
(5)

The temperature at the vapor-liquid interface of the evaporator and condenser is calculated
approximately using Clausius-Clapeyron equation, (Borujerdi, 2004):

)(

)(
ln

1
1

),(int

satTsatP
vTvP

fgh
R

satT

vrxTT (6)

The boundary conditions for vapor region are as following, (Rajashree and Sankara,
1990).
At both pipe ends are:

00
x
T

uv (7)

At pipe centerline the symmetry boundary conditions are:

0&0,0
r
T

r
u

v (8)

              Pressure gradient at the wall at a particular time is calculated from the momentum equation
and it is given by:

r
u

v
r
u

r
u

rx
p

2
1

Re
2

2

2

(9)

The shear rate is also calculated from the equation:



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳۷

r
u

Re
2

(10)

METHOD OF SOLUTION
The governing equations are discretized using a finite difference approach and the

equations are solved using upwind difference method with collocated grid scheme as shown in
Fig.2. For the numerical analysis, it is convenient to use the governing equations in stream function
and vorticity function:

r
ru (11)

x
r (12)

r
u

x
(13)

Using these and eliminating pressure the governing equations are transformed to

rrrrxr 3
1

2

2

2
1

2

2

2
1

(14)

rrrxrxxrr
3

2

2

2

21
(15)

r
T

rr

T

x

T
r
T

xx
T

rr
1

2

2

2

21
(16)

Now, to make the governing equations form in non-dimensional and the boundary
conditions using the following dimensionless quantities:

vVr

k
cp

pr
sToT

T

V

p
p

V
r

vVrVV
u

u
vr
r

r
vr
x

x

v Re;;;
2

2
1*

;*

)17(

*;*;*;*;*

By substitution the above dimensionless quantities, in the governing equation of motion
yields:



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳۸

*

*
3
*

1
2
*

*
2

2
*

1
2
*

*
2

2
*

1
* rrrrxr

  (18)

2
*

*

*

*

*

1
2
*

*
2

2
*

*
2

Re
1

*

*

*

*

*

*

*

*

*

1

rrrrxrxxrr
(19)

**

1
2
*

2

2
*

2

PrRe
1

**

*

**

*

*

1
rrrxrxxrr

(20)

A finite-difference technique is applied to solve the governing equations. These three

equations Eqs.(18), (19), and (20) are to be solved in a given region subject to the condition that the

values of the stream function, temperature, and the vorticity, or their derivatives, are prescribed on

the boundary of the domain.

          Eq.(18) can be approximated using central – difference at the representative interior point

(i , j), then it can be written for regular mesh as:

]2/[)],(ω)())1,(ψ)1,(ψ(

)))(2/(()1,(ψ)1,(ψ),1(ψ),1(ψ[),(ψ
2
*

2
**

2
*

2
*

2
***

**
2
*

2
***

2
***

)Δ(Δ)Δ(Δ
)Δ(Δ))(Δ())(Δ(

xrjijrrxjiji
jrrxxjijirjijiji

        (21)

           Also, a central – difference formulation can be used for Eqs.(19), and (20). It is known that

such a formulation may not be satisfactory owing to the loss of diagonal dominance in the sets of

difference equations, with resulting difficulties in convergence when using an iterative procedure.

          A forward – backward technique can be introduced to maintain the diagonal dominance

coefficient of (ωi,j) in Eq.(19) and (θi,j) in Eq.(20) which determines the main diagonal elements of

the resulting linear system; this technique is outlined as follows (Najdat,1987):

Set; * j,1-i
*

j,1i1 ψψB    and
*

1-ji,
*

1ji,2 ψψB

         Then approximate Eq.(19) by:

*

*

*

3
2
*

*
2

2
*

*
2

Re
1

*

*

*2
1

*

*

*2
2

*

1
rrrxrx

B

xr

B

r

Now, if

r
ff

r
f

,0B,,,
r

ff
r
f

,0 1ji,ji,1
ji,1ji,

1 ΔΔ
B

if



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲۳۹

if
x

ff
x
f

,0B,,,
x
ff

x
f

,0B ji,j,1i2
j,1iji,

2 ΔΔ

where, (f )  refers to ** ,

           To assure the diagonal dominance of the coefficient matrix for )ji,
*

ji, θω (and)( , which

depends on the sign of ( 1B ) and ( 2B ), Eqs (19) and (20) are expressed in the following difference

forms:

0Bnd,0BFor 21 a

]
2

Re)B(B3
)(2/[)]1,(ω)(),1(ω)(

),1()ω
2

Re
()1,()ω

3
2

Re
[(),(ω

*

**21

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r

rxB
rji

r
rx

r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

           (22a)

0Bnd,0BFor 21 a

]
2

Re)B(B3
)(2/[)]1,(ω)(),1(ω)(

),1()ω
2

Re
()1,()ω

3
2

Re
[(),(ω

*

**21

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r

rxB
rji

r
rx

r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                     (22b)

0Bnd,0BFor 21 a

]
2

Re)B(B3
)(2/[)]1,(ω)(),1(ω)(

),1()ω
2

Re
()1,()ω

3
2

Re
[(),(ω

*

**12

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r

rxB
rji

r
rx

r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                    (22c)

0Bnd,0BFor 21 a

]
2

Re)BB-(3
)(2/[)]1,(ω)(),1(ω)(

),1()ω
2

Re
()1,()ω

3
2

Re
[(),(ω

*

**21

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r

rxB
rji

r
rx

r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                    (22d)

Dimensionless energy equation by using upwind finite difference:



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤۰

0Bnd,0BFor 21 a

]
2

PrRe)B(B
)(2/[)]1,()(),1()(

),1()
2

PrRe
()1,()

2
PrRe

[(),(

*

**21

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r
rxB

rji
r

rx
r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                (23a)

0Bnd,0BFor 21 a

]
2

PrRe)B(B
)(2/[)]1,()(),1()(

),1()
2

PrRe
()1,()

2
PrRe

[(),(

*

**21

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r
rxB

rji
r

rx
r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                  (23b)

0Bnd,0BFor 21 a

]
2

PrRe)B(B
)(2/[)]1,()(),1()(

),1()
2

PrRe
()1,()

2
PrRe

[(),(

*

**12

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r
rxB

rji
r

rx
r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                 (23c)

0Bnd,0BFor 21 a

]
2

PrRe)B(-B
)(2/[)]1,()(),1()(

),1()
2

PrRe
()1,()

2
PrRe

[(),(

*

**21

*

*
2
*2

*
2
**

2
*

2
*

*
*

**22
**

*

*
2
*

*

**12
**

*
r

rx
r

rx
rxjixjir

ji
r
rxB

rji
r

rx
r
rxB

xji

Δ
ΔΔΔ

ΔΔΔ

                (23d)

The corresponding boundary conditions at both ends of the heat pipe which are the no slip
condition for the velocity and adiabatic condition for temperature.

0
*

;0*,*;*

)24(

0
*

;0*,0*;0*

x
r

vr
L

vr
L

xat

x
rxat

At the centerline, the symmetry conditions are applied.



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤۱

)25(0
*

;00,**;0* r
xrat

At the vapor –wick interface:

)26(*20
*1,**;1* dx

fghveL
Qv

r
xvrxrat

According to the coordinate system shown in figure 1, stream function *,** rx   is

negative in evaporator and positive in the condenser while in adiabatic regain is zero, (Borujerdi,
2004) .

)(

)(
ln

1

1
1,*

satTsatP
vTvP

fgh
R

satT
TT

x

so

   (27)

The solution procedure of the discredited equations is based on a line-by-line iteration
method in the axial and radial directions using FORTRAN program. The sequence of numerical
steps based on upwind difference is as follows:
1. Initialize the velocity, pressure, shear rate and temperature fields
( ,*,*,*,* Pvu ).

2. Solve Eqs.(11) and (12) for u and v.
3. Solve Eq. (9) for P.
4. Solve Eq. (10) for τ.
5. Solve Eq. (20) for θ .
6. Check Eq. (28) for convergence, if it is satisfied, calculations will be ended. Otherwise, replace
( ,*,*,*,* Pvu ) and return to step (2) and repeat the above procedure until convergence is

achieved.
The accuracy of the numerical solution is checked first by summation of the absolute value

of the relative errors should be equal or less than (10-4). Second, the spot value should approach a
constant value. The relative error (Err) in the numerical procedure is defined as, (Borujerdi, 2004):

4101
1

cells n
nn

Err     (28)

where superscript ( n ) refers to the previous iteration and term ( )refers to ( =u, v, P, τ, θ).
A cylindrical heat pipe with water as working fluid is selected, in which the length of the

evaporator is the same as the length of the condenser and a comparatively long adiabatic section is
considered. The pipe dimensions are taken to be (L/R =5), increment in space coordinates are
(Δx=0.05) and (Δr =0.1). The computation is done for a mesh (101×11), Reynolds number (4 and
10), and the Prandtl number taken is (Pr=0.00368). In the present analysis, the axial conduction
along the heat pipe wall is neglected. The evaporator is maintained at constant temperature over its
entire length and the condenser is cooled uniformly and is also kept at constant temperature.



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤۲

RESULTS AND DISCUSSION
A computer program has been developed for predicting the stream function, radial and axial

velocities, and temperature, pressure, and shear rate fields of vapor flow along the copper-water
heat pipe. Using this program, a symmetric heat addition and rejection conditions are observed, as
shown in Fig.1.

The stream function Fig.3 at the wall increases linearly in the evaporator, decreases linearly
in the condenser and is steady in the adiabatic region because of uniform inflow and outflow
boundary conditions. From the axial variation of temperature graph Fig.4 it is found that the
temperature increases in the evaporator section, decreases in the condenser section, and the
minimum temperature occurs in the condenser region. As the temperature in the evaporator
increases, the vapor density in this section also increases and molecular mean free path becomes
small compared to the vapor core diameter. Because of the vapor condenses and releasing its latent
heat of vaporization to the provided heat sink.

When the Reynolds number is small, the streamlines are in the direction of increasing axial
distance and the axial velocity component is positive everywhere with a transition to zero value
occurring in a thin layer at the walls which is observed from Fig.5. The axial velocity profile
becomes fully developed in a short distance and stays parabolic all along the length of the heat pipe.
Because of the length of evaporator and condenser section is equal, and then the two quantities of
evaporation and condensation of vapor are equal.

Also the radial velocity is negative in the evaporator section, zero in adiabatic section and
positive in the condenser section, because of the flow direction is assumed negative of downward
and positive of upward direction, as shown in Fig.6.

Figure 7 illustrates the pressure distribution along the heat pipe space centerline for various
Reynolds numbers. It can be seen that as the Reynolds number increases, the pressure distributions
shift up without considerable change in their overall shapes. As the Reynolds number increases the
pressure in the condenser section is more recovered. The pressure distribution in the adiabatic
section is a straight line similar to poiseuille flow results, while the profiles in the evaporator and
condenser section demonstrate the effects of pressure head absorbed or created by evaporation or
condensation, Due to the small pressure drop along the heat pipe at evaporator and condenser
sections.

The shear rate as shown in Fig.8 is found to be symmetric. Similar pattern follows for
pressure distribution and the decrease in vapor pressure occurs smoothly in evaporator section,
remains steady in the adiabatic section and then increases smoothly in the condenser section
because of uniform inflow and outflow boundary conditions are equal.

The numerical analysis have shown that for the Reynolds number (Re=4, 10), the shear
stress becomes zero at a point very close to the end of the condenser. However, as the Reynolds
number increases, the flow reversal point moves backward toward the adiabatic section. Under this
condition, the reversed flow region extends from the flow reversal point to the end of the condenser.

To verify our numerical solution we have recovered the heat pipe. A computer code,
developed in the present work based on the FORTRAN language, has been validated using the
results based on similar problem with those reported by (Rajashree and Sankara, 1990), as shown
in Fig.9 shows stream function comparison with previous study. This demonstrates that the present
numerical analysis a good agreement for predicting the stream function distribution at Reynolds
number (Re=4).

CONCLUSION
             A numerical study is investigated the flow, temperature, velocity, pressure, and shear stress
distributions for a horizontal heat pipe based on the obtained results in the present study, finding
are:

1- At low and moderate Reynolds numbers, the present analysis predicts very small vapor
temperature drop along the heat pipe. Due to the small pressure drop along the heat pipe.



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤۳

2- The results show the stream function at the wall increases linearly in the evaporator,
decreases linearly in the condenser and is steady in the adiabatic region because of uniform
inflow and outflow boundary conditions.

3- Also, it can be seen that as the Reynolds number increases, the pressure distributions shift
up without considerable change in their shapes.

4- The numerical analysis have shown that for the low and moderate Reynolds number, the
shear stress becomes zero at a point very close to the end of the condenser.

5- The results have been compared with the available numerical data which have been done in
the literature and have shown a good agreement.

REFERENCES
 [1] F.Doran, 1989, “Heat pipe research and development in the Americas,” Heat Recovery Systems
and CHP, Vol.9, pp.67-100.

[2] R.Rajashree and K. Sankara, 1990, ‘‘ A numerical Study of the Performance of Heat Pipe,’’
Indian J. pure appl. Math, 21(1), pp: 95-108.

[3] M. M. Chan and A. Faghri, 1995, “An Analysis of The Vapor Flow and Heat Conduction
Through The Liquid Wick and Pipe Wall in a Heat Pipe With Single or Multiple Heat Sources,” Int.
J. Heat Mass Transfer, Vol. 33, No. 9, pp. 194.

[4] N. Zhu and K. Vafai, 1999, “Analysis of Cylindrical Heat Pipe Incorporating the Effect of
Liquid-Vapor Coupling and Non-Darcian Transport-A Closed form Solution,” Int. J. Heat Mass
Transfer, Vol. 42, pp. 3405- 3418.

[5] S. J. Kim, J. K. Seo and K. H. Do, 2003, “Analytical and Experimental Investigation on the
Operational Characteristics and Thermal Optimization of a Miniature Heat Pipe with a Grooved
Wick Structure,” Int. J. Heat Mass Transfer, Vol. 46, pp. 2051-2063.

[6] A. Nouri-Borujerdi and M. Layeghi, , 2004, “Numerical Analysis of Vapour Flow in Concentric
Annular Heat Pipes,” Transaction of ASME: Journal of Heat Transfer, Vol. 126, pp. 442-448.

[7] Y. H. Yau , 2007, “Application of a heat pipe heat exchanger to dehumidification enhancement
in a HVAC system for tropical climates – a baseline performance characteristics study,”
International Journal of Thermal sciences , Vol. 46, pp.164 -171.
[8] David B. Sarraf, Sanjida Tamanna, and Peter M. Dussinger, 2008, “pressure controlled heat
pipe for precise temperature control,” Space Technology and Applications International, American
Institute of Physics, pp. 3-11.

[9] A. Nouri-Borujerdi , 2004 “A Numerical Analysis of Vapor Flow in Concentric Annular Heat
Pipes,” Transaction of ASME: Journal of Fluids Engineering, Vol. 126, pp.442- 448.

[10].Najdat N., 1987 “Laminar Flow Separation in Constructed Channel”, Ph.D. Thesis, Michigan

State University.



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤٤

.

Fig.(2) The discretized domain

rorv

Evaporator CondenserAdiabatic
section

Solid wallWick

Qe Qc

Vapor space
u , x

v , r

Fig. (1) Schematic of  a  cylindrical heat pipe

Jn=Jn-1+∆r

J3=J2+∆r

J2=J1+∆r

Im=Im-1+∆xI3=I2+∆xI2=I1+∆xI1=0

∆r

∆x

J1=0



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤٥

                          (a)                                                                       (b)
Fig. (3) Distribution of stream function at the wall a) Re = 4      b) Re = 10

                                     (a)                                         (b)
Fig. (4) Distribution of temperature near the wall a) Re = 4      b) Re = 10

                                  (a)                                                                        (b)
Fig. (5) Distribution of axial velocity at the wall a) Re = 4      b) Re = 10



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤٦

                                     (a)                                                                       (b)
Fig. (6) Distribution of redial velocity at the wall a) Re = 4      b) Re = 10

                                       (a)                                                                (b)
Fig. (7) Pressure distribution at the wall a) Re = 4      b) Re = 10

                      (a)                             (b)
Fig. (8) Distribution of shear rate at the wall a) Re = 4      b) Re = 10



Al-Qadisiya Journal For Engineering Sciences       Vol. 4    No. 3    Year 2011

۲٤۷

Fig. (9) The stream function calculated in this study versus that of
            (Rajashree and Sankara, 1990), at the wall for (Re = 4)