An Exact Solution of Navier-Stokes Equations for the Study of Vapor Flow in Cylindrical Heat Pipes M. Layeghi Department of Mechanical Engineering, Karaj Azad University, E-mail:
[email protected] Karaj, Gohardasht, Tehran, P. O. Box : 31485-313, Tel: +98 0261 4418143-6, Fax: +98 0261 4418156
Abstract An innovative exact solution of Navier-Stokes equations is derived for the prediction of vapor pressure distributions, and velocity profiles in the conventional heat pipes. The steady-state laminar and incompressible vapor flow in a cylindrical vapor space subjected to uniform injection and suction is studied. Mathematically, using an appropriate change of variable, Navier-Stokes equations are transformed to a set of nonlinear ordinary differential equations. The governing equations are solved analytically using series solution method. The vapor pressure distribution and vapor velocity profiles along the cylindrical vapor space are determined and analytical results are compared with the available data in the literature. There is a relatively small difference about 10%, in the worse case, between the analytical and previous numerical and experimental results. Keywords: Conventional heat pipes, Navier-Stokes equations, laminar and incompressible vapor flow, analytical solution
1. INTRODUCTION Heat pipes are effective heat transfer devices with many heat transfer engineering applications including: temperature and humidity control, cooling, heat recovery, and numerous other industrial applications in wide range of temperatures. Heat pipes have a very high thermal conductance usually several hundred more than metallic conductors. They have quick response time and can transfer heat with small temperature drop. A conventional heat pipe (Fig. 1) usually consists of three parts: a sealed container, a wick, and a working fluid. The wick is attached longitudinally inside the container and is usually saturated with the working fluid. A heat pipe consists of three distinct sections: an evaporator or heat addition section, a condenser or heat rejection section, and an adiabatic or isothermal section. When heat is added to the evaporator section of the container, the working fluid present in the wicking structure is heated until it vaporizes. The high temperature and corresponding high pressure in the evaporator section cause the vapor to flow to the cooler condenser section, where the vapor condenses, releasing its latent heat of vaporization. The capillary forces existing in the wicking structure then pump the liquid back to the evaporator.
Fig. 1 Schematic of a conventional heat pipe along with coordinate system
1
The analysis of the performance of a heat pipe traditionally consists of the analysis of vapor flow, liquid flow through wick, and vapor-liquid interface. The last two parts of this analysis are basically similar for many conventional heat pipes. However, the dynamics of vapor flow may be more complex due to the geometry and boundary specifications. The vapor flow in heat pipes has been investigated by various authors since 1964[1-6]. Cotter [7] developed a basic theory for vapor flow analysis in a heat pipe. Busse [8] studied the vapor flow in a conventional heat pipe using a one-dimensional model. Bankston and Smith [9] modeled the flow of vapor using the stream function and vorticity formulation and determined numerically the pressure and velocity distributions. Rohani and Tien [10] studied the influence of pressure drop on the temperature distribution in heat pipes using sodium as the working fluid. In their analysis the momentum equations were used in terms of stream function and vorticity coupled with the energy equation in terms of enthalpy. Decoupling of the liquid and vapor momentum equations can result an error when calculating the liquid flow rates and vapor pressure. However, at low and moderate temperature ranges, this error is negligible [1, 3, 11, 12]. Faghri and Chen [12] considered the effects of compressibility and viscous dissipation using a conductive model for the porous and metallic regions. Ismail and Zanardi [13] presented a solution of a coupled vapor and liquid flow in a heat pipe. Cao and Faghri [14] presented a mathematical model of a two-dimensional heat pipe, where they considered both flows including the compressibility and the viscous dissipation effects. Chen and Faghri [15] and Faghri and Buchko [16] solved the problem of heat pipes with multiple sources and their results were compared with experiments. The other works related to the copper-water heat pipes can be reviewed in [17-18]. A number of new mathematical models for heat pipe analysis have also been investigated in [19-24]. The main objective of these works has been the analysis of the non-Darcian effects, vapor-liquid interface interactions, and possible improvements in heat pipe performance. A new criterion based on critical heat flux has been also presented by Layeghi and Nouri [25] by fully one-dimensional analysis of conventional heat pipes. The possibility of the presence of an exact solution based on the physics of the vapor flow in conventional heat pipes has been usually neglected by the previous researchers except for some low radial Reynolds number flows [9]. However, there are a number of approximate analytical solutions for the other types of heat pipes in the literature. For example, Vafai et. al [26-33] presented a number of approximate analytical solutions for the vapor flow analysis in flat-plate and disk-shaped heat pipes. In this paper, an exact series solution of simplified Navier-Stokes equations is derived for the first time for the prediction of vapor pressure distributions, and velocity profiles in the conventional heat pipes. The steady-state vapor flow in a conventional copper-water heat pipe is investigated as a test case. Here, the main objective is to find exact series solutions for the analysis of vapor flow in conventional heat pipes and check the accuracy and validity of this solution at various ranges of radial Reynolds numbers. The velocity profiles and pressure drop along the heat pipe are determined and flow reversal phenomenon is also predicted. A twodimensional formulation is used, which is a simplified form of the elliptic model used by Faghri [1]. The analytical results are validated against available results [9, 10] and a number of new results are also presented. The results of the analytical model have shown relatively good agreement in all compared cases. 2 Governing Equations The axisymmetric motion of vapor in the vapor space is considered here (Fig 1). The present model is based on the following assumptions: (1) The vapor flow is laminar and incompressible and in steady-state; (2) The properties of the vapor are assumed to be constant; (3) Both evaporation and condensation are modeled as a uniform injection and suction, respectively; (4) No phase change is actually involved in the calculations; (5) The coupling between the vapor and liquid flows in the heat pipe is neglected; (6) No-slip boundary condition is assumed at the vapor-liquid interface. The vapor flow is described by the Navier-Stokes equations, together with the mass conservation equation. These equations are written in their conservative form as:
2
∂w 1 ∂ [r v ] = 0 + ∂z r ∂r
ρ w
(1)
∂ 2 w 1 ∂ ∂ w ∂w ∂ w ∂P = − r +v + µ 2 + r ∂ r ∂ r ∂z ∂r ∂z ∂ z
(2)
∂ 2v 1 ∂ ∂ v v ∂v ∂v ∂P = − r − +v + µ 2 + ∂r ∂r r ∂ r ∂ r r 2 ∂z ∂ z
ρ w
(3)
where w and v are the velocity components in the z and r directions, respectively and P is the vapor pressure. ρ and µ are the density and viscosity of the vapor, respectively. The boundary conditions are defined as follows. w(0, r ) = v(0, r ) = 0
,
w( L, r ) = v( L, r ) = 0 ,
∂w ( z ,0) = v ( z ,0) = 0 ∂r
,
w( z , R) = 0 ,
v( z , R) = V ( z )
,
P(0, r ) = 0
(4a-f)
where L is the total length of the heat pipe, R is the vapor space radius. The radial blowing and suction velocities at the inner wall, in the evaporator and condenser, are computed as
Ve =
Qe 2 π RL e ρ h fg
Vc =
,
Qc 2 π RL c ρ h fg
(5a,b)
where subscripts e and c refer to the evaporator and condenser, h fg is the latent heat of vaporization.
Qe and Qc are the heat addition and rejection at the evaporator and condenser sections, respectively.
Method of Solution The governing equations can be simplified to a set of ordinary differential equations using appropriate change of variables based on the physics of the vapor flow in conventional heat pipes. Here, we find analytical solution of the ordinary differential equations for the evaporator, adiabatic, and condenser sections, separately. Only one symmetrical half of the vapor space longitudinal section is considered in the present analysis. At the evaporator section, previous numerical, analytical, and experimental investigations and boundary conditions (4a-f) have shown that the following assumptions can be made in addition to the assumptions (1)-(6) in the previous section for various radial Reynolds numbers:
we ( z , r ) = z Ge (r ) ,
ve ( z , r ) = H e (r )
(6a,b)
Eq. (6a) can be easily deduced from the conservation of mass and boundary conditions for axial velocity. The vapor mass flow rate at section z can be written as
m& e ( z ) = 2 π R ρ Ve z = ρ π R 2 we ( z )
(7)
where we (z ) is the average vapor velocity at section z in the evaporator section. Rearranging Eq. (7) gives we ( z ) = 2 V e
z R
3
(8)
Assuming we ( z, r ) = f e ( z ) g e (r ) and using the definition of average velocity we obtain R
we ( z ) =
∫
f e ( z ) g e (r ) 2 π r dr
0
(9)
= c1 f e ( z )
π R2
where c1 is a constant value. Comparing Eqs. (8) and (9) yields f e ( z ) = c z , where c is a constant value. Therefore, we ( z , r ) can be assumed to have the form we ( z, r ) = c z g e (r ) . Finally, we ( z , r ) should have the form like that expressed in Eq. (6a). Assumption (6b) can be easily deduced from Eq. (1) by nothing that ∂ we / ∂ z is independent of z since we ( z , r ) assumed to be in the form of Eq. (6a). Substitution of Eqs. (6a,b) into Eqs. (1)-(3) gives the following set of ordinary differential equations: Ge +
1 H e + H e′ = 0 r
(10a)
∂ Pe 1 = µ G e′ + G e′′ − ρ G e 2 + H e G e′ z ∂ z r
[
]
(10b)
∂ Pe 1 1 = µ H e′′ + H e′ − 2 H e − ρ H e H e′ ∂r r r
(10c)
where ϑ is the kinematic viscosity of the fluid. Differentiating from Eqs. (10b,c) with respect to r and z , respectively and combining them by removing the pressure gives
[
]
d 1 2 µ Ge′ + Ge′′ − ρ Ge + H e Ge′ = 0 d r r
(11)
Substitution of function Ge from equation (10a) in terms of function H e into equation (11) gives
(
ϑ r 4 H e (iv ) + 2 r 3 H e′′′ − 3r 2 H e′′ + 3rH e′ − 3H e
)
(12)
= r H e′′′ H e − r H e′′ H e − r H e′′ H e′ − r H e′ − 3r 2 H e′ H e + 4rH e 2 4
3
4
3
2
Equation (12) is a fourth order nonlinear differential equation which needs four boundary conditions to be solved. The associated boundary conditions in terms of H e (r ) can be obtained from Eqs. (4c-e) which are: H e (r = 0) = 0 , 1 =0, H e + H e′ r r =R
H e (r = R) = −V 1 1 =0 − 2 H e + H e′ + H e′′ r r r =0
(13a,b) (13c,d)
Equation (12) along with boundary conditions (13a-d) can be solved using a series solution method and additional assumptions related to the physics of the problem. The solution procedure is as follows. (1) Solve Eq. (12) to find H e (r ) . (2) Use Eq. (10a) to find Ge (r ) and then obtain we ( z , r ) = zGe (r ) . (3) Find pressure drop using Eqs. (10b,c).
4
In order to solve Eq. (12), a series solution in the form: H e (r ) =
∞
∑a
n
r n is used. Substitution of this
n=0
series solution into Eq. (12) gives: H e (r ) = a1r + a3 r 3 + a5 r 5 + a6 r 6 + ... ,
(14)
a 0 = a 2 = a 4 = 0 , a1 , a3 ≠ 0 , 11a1 a 3 + 96ϑ a 5 = 0 , −6a1 a 3 + 525 ϑ a 6 = 0 , …
(15a,b)
The function Ge can then be obtained using Eq. (10a) as: Ge (r ) = −2a1 − 4a3 r 2 − 6a5 r 4 − ...
(16)
The overall second order form of the axial velocity component should be:
(
we ( z , r ) = z Ge = z − 2a1 − 4a3 r 2 − 6a5 r 4 + O(r 5 )
)
(17)
This solution can be called approximate fourth order solution. However, we can find higher order approximate solutions by taking into account the additional terms in the series solution. For the fourth order solution, using boundary conditions ve (r = R ) = −V or H e ( R) = −V and we ( z , R ) = 0 or Ge (r = R) = 0 in Eqs. (14) and (16) at r = R give : a1 R + a 3 R 3 + a 5 R 5 = − V ,
− 2 a1 − 4a 3 R 2 − 6a 5 R 4 = 0
(18a,b)
Eqs. (18a,b) along with Eq. (15a) provide a set of three nonlinear equations for a1 , a 3 , and a 5 . The results show that our problem may have more than one solution. Finally, the velocity components we , ve can be obtained as: (19a) we ( z , r ) = −2 (a1 + 2a3 r 2 + 3a5 r 4 ) z ve (r ) = a1r + a3 r 3 + a5 r 5
(19b)
The axial velocity profile at z = Le can obtained using Eq. (19a). w f (r ) = − 2 (a1 + 2a3 r 2 + 3a5 r 4 ) Le
(20)
The pressure drop can be easily obtained by the integration of Eq. (10b) along the microchannel. Substitution of functions Ge and H e into Eq. (10b) and integration from z = 0 to z = L gives:
{ (
) [
]}
Pe ( z, r ) − Pe (0, r ) = −2 4 µ a3 + 6a5 r 2 + ρ a12 + 6a1a3 + 6a3 2 + 12a1a5 r 4 + 20a3 a5 r 6 + 15a5 2 r 8 z 2 (21) At the adiabatic section, the velocity profile is assumed to be fully developed velocity profile in a pipe and pressure changes linearly along this section [1-4]. The velocity profile at the end of the evaporator section (at z = Le ) can be easily determined using Eq. (20). Here, it is assumed that this velocity profile is suddenly changed to a parabolic fully developed velocity profile in a pipe as:
r2 wa (r ) = 2 w 1 − 2 , R
w=
Qe 2
π R ρ h fg
= 2 Ve
Le R
(22)
where w is the mean velocity of the vapor. The radial velocity is assumed to be zero at the adiabatic section. The pressure variation in the adiabatic section is linear with respect to z and can be obtained using
5
Eq. (2). ∂Pa 1 d dwa =µ r ∂z r dr dr
(22)
Since, wa is only a function of r , integration of the above equation with respect to z from z = Le to z = z yields µ Ve Le ( z − Le ) (23) Pa ( z , r ) − Pa ( Le , r ) = −16 R3
At the condenser section, the following assumptions can be used at the condenser section wc ( z , r ) = ( L − z )G c (r ) ,
v c ( z, r ) = H c (r )
(24a,b)
Following the similar procedures to those used at the evaporator section, we obtain a set of ordinary differential equations: 1 − G c + H c + H c′ = 0 (25a) r
[
]
∂ Pc 1 = µ Gc′ + Gc′′ − ρ − Gc 2 + H c Gc′ ( L − z ) ∂ z r
(25b)
∂ Pc 1 1 = µ H c′′ + H c′ − 2 H c − ρ H c H c′ ∂r r r
(25c)
(
ϑ r 4 H c (iv ) + 2 r 3 H c′′′ − 3r 2 H c′′ + 3rH c′ − 3H c
)
= r 4 H c′′′ H c − r 3 H c′′ H c − r 4 H c′′ H c′ − r 3 H c′ 2 − 3r 2 H c′ H c + 4rH c 2
(26)
Equation (26) is a fourth order nonlinear differential equation which needs four boundary conditions to be solved. The associated boundary conditions in terms of H c (r ) can be obtained from Eqs. (4c-e) which are: H c (r = R) = V
H c ( r = 0) = 0 , 1 =0, H c + H c′ r r =R
(27a,b)
1 1 =0 − 2 H c + H c′ + H c′′ r r r =0
In order to solve Eq. (26), a series solution in the form: H c (r ) =
∞
∑m
n
(27c,d)
r n is used. Substitution of this
n =0
series solution into Eq. (26) gives: H c (r ) = m1r + m3 r 3 + m5 r 5 + m6 r 6 + ... , m0 = m2 = m4 = 0 , m1 , m3 ≠ 0 , 11m1m3 + 96ϑ m5 = 0 , −6m1m3 + 525ϑ m6 = 0 , …
(28) (29a,b)
The function Gc can then be obtained using Eq. (25a) as: Gc (r ) = 2m1 + 4m3 r 2 + 6m5 r 4 + ... The overall second order form of the axial velocity component should be:
6
(30)
(
wc ( z , r ) = ( L − z ) Gc = ( L − z ) 2m1 + 4m3 r 2 + 6m5 r 4 + O(r 5 )
)
(31)
This solution can be called approximate fourth order solution. However, we can find higher order approximate solutions by taking into account the additional terms in the series solution. For the fourth order solution, using boundary conditions vc (r = R) = V or H c ( R) = V and wc ( z , R) = 0 or Gc (r = R ) = 0 in Eqs. (14) and (16) at r = R give : m1 R + m3 R 3 + m5 R 5 = V ,
2 m1 + 4m3 R 2 + 6m5 R 4 = 0
(32a,b)
Eqs. (32a,b) along with Eq. (29a) provide a set of three nonlinear equations for m1 , m3 , and m5 . The results show that our problem may have more than one solution. Finally, the velocity components wc , vc can be obtained as: wc ( z , r ) = 2 (m1 + 2m3 r 2 + 3m5 r 4 ) ( L − z ) 3
vc (r ) = m1r + m3 r + m5 r
(33a)
5
(33b)
Finally, the pressure drop can be obtained by substitution of H c (r ) and Fc (r ) into Eq. (25b).
{ (
) [
Pc ( z , r ) − Pc ( L − Lc , r ) = 2 4 µ m3 + 6m5 r 2 + ρ m1 2 + 2m1m3 r 2 + 2m3 2 r 4 + 4m3 m5 r 6 + 3m5 2 r 8
[
× Lc 2 − ( L − z ) 2
]
]}
(34)
Results and Discussion
The effects of radial Reynolds numbers, as a result of various heat fluxes at the evaporator and condenser sections are investigated. Numerical calculations consist of two parts. First, for a copper-water heat pipe with design parameters given in Table 1, the normalized pressure drop is calculated for two different radial Reynolds numbers. These test cases are the same as those used by Cotter [7], Busse[8], and Faghri [1] for the low and moderate radial Reynolds numbers: Re r = 0.6 and Re r = 2.4 . It is shown that there are more than one solution at moderate radial Reynolds number: Re r = 2.4 . After that, the velocity profiles and pressure distribution at various radial Reynolds numbers are predicted. Table 1: Design Parameters of the Conventional Heat Pipe
Total Length Evaporator Length Adiabatic Length Condenser Length Pipe Inner Radius
Fluid Properties
Heat Pipe Dimensions 500 mm 100 mm 300 mm 100 mm 10 mm
Sat. Temp. = 373.15 K Kg KJ ρ = 0.597 4 3 Kg m µ = 120.3e − 7 Pa. sec
h fg = 2257
k = 0.5977
W mK
C P = 2028
J Kg K
Test Cases A computer code has been developed for predicting the velocity, and pressure fields along the conventional heat pipe. Using this code, various symmetric and asymmetric heat addition or rejection conditions can be analyzed. For a symmetric condition, Re e = Re c = Re r and Le = Lc , where Re r is the characteristic
7
radial Reynolds number in each test case. Asymmetric cases are those cases in which Re e ≠ Re c or Le ≠ Lc . The normalized pressure distribution along the conventional heat pipe for two different radial Reynolds numbers Re r = 0.6 and 2.4 are shown in Fig 2. ∆P ( pa )
∆P ( pa ) 0
0
-0.0025
-0.01
-0.005
-0.02
Re r = 0.6
-0.0075
-0.0125
0.1
0.2
ڤCotter [7] ∆ Busse [8] О Faghri [1]
-0.05 -0.06
0
Re r = 2.4
-0.04
ڤCotter [7] ∆ Busse [8] О Faghri [1]
-0.01
-0.015
-0.03
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
Z(m)
Z(m) (a) Low radial Reynolds numbers
(b) Moderate radial Reynolds numbers
Fig. 2 Normalized pressure distribution along the heat pipe There is a small difference between the various results presented in Fig. 2. However, at the adiabatic and condenser sections the differences are larger and about 10% in the worse case. This is due to the differences between various models and solution methods and approximation we made in the present work. Cotter [7] used a first-order, simplified, closed form approximation, Busse [8] used a second order approximation, and Faghri [1] developed a generalized similarity closed form approximation. No coupling between the vapor flow and the liquid flow in the wick has been actually involved in these models and also in the present work. However, in the present analysis we only used three terms of the series solution and the solution shown in Figs. (2a,b) may not be more accurate than the other models. However, the overall results of the model seems to be acceptable at least at low to moderate radial Reynolds numbers. The velocity profiles at Re r = 0.6 and Re r = 2.4 are also shown in Figs. (3a-c) and (4a-c) at various sections.
r(m)
r(m)
r(m)
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0.04 0.03 0.02 0.01 0 0
0
-0.01
-0.01
-0.01 -0.02
evaporator section
-0.03
-0.02 -0.03
-0.04
0
0.025
0.05
0.075
0.1
-0.04 0.2
-0.02
adiabatic section 0.225
0.25
Z(m)
0.275
-0.03 0.3
-0.04 0.4
condenser section 0.425
Z(m)
Fig. 3 Vapor velocity profiles in the heat pipe at Re r = 0.6
8
0.45
0.475
Z(m)
0.5
r(m)
r(m)
r(m)
0.04
0.04
0.04
0.03
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0
0
0
-0.01
-0.01
-0.01
evaporator section
-0.02
-0.02
-0.03 -0.04
-0.03 0
0.025
0.05
0.075
-0.04 0.2
0.1
-0.02
adiabatic section 0.225
0.25
Z(m)
0.275
condenser section
-0.03 0.3
-0.04 0.4
0.425
0.45
Z(m)
0.475
0.5
Z(m)
Fig. 4 Vapor velocity profiles in the heat pipe at Re r = 2.4 At Re r = 0.6 , there are two real solutions in the both evaporator and condenser sections. However, at Re r = 2.4 there are two real solutions at the evaporator section but two complex solutions at the condenser section. There are two sets of complex conjugate coefficients m1 , m 2 , and m3 which are obtained at the condenser section. Table. 2 gives the coefficients m1 , m 2 , and m3 for both cases. Table 2: Coefficients of series solutions at the evaporator and condenser sections Radial Reynolds Number( Re r ) 0.6
a1
a3
a5
m1
m3
m5
first second
-0.2292 0.92662
958.312 -22157.1
1.2495×106 1.1682×108
first
-0.8479
2459.5
1.1867×107
second
1.0017
-34532.3
1.9683×108
0.26816 0.79176 0.80182 0.45434 i 0.80182 + 0.45434 i
-1738.94 -12210.5 -1537.73 + 9086.7 i -1537.73 9086.7 i
2.6536×106 5.50114×107 -1.6476×107 – 4.5433×107 i -1.6476×107 + 4.5433×107 i
2.4
Fig. 5 shows the other solutions at the evaporator and condenser sections at Re r = 0.6 . It can be seen that these solutions represent reversed flows in the evaporator and condenser section which seems to be not physical at least at low to moderate radial Reynolds numbers. At the condenser section, the second solution is surprisingly an asymmetric solution. Mathematically speaking, the geometry and boundary conditions are symmetric but the solution is asymmetric. This phenomenon seems to be related to the presence of nonlinearity in the mathematical model which causes more than one solution to be existed.
r(m)
r(m)
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
-0.01
evaporator section
-0.02
-0.02
-0.03 -0.04
-0.03 0
0.025
0.05
0.075
-0.04 0.4
0.1
Z(m)
condenser section 0.425
0.45
0.475
Z(m)
Fig. 5 Vapor velocity profiles in the heat pipe at Re r = 0.6
9
0.5
Fig.6 shows the second real solution at the evaporator section and the imaginary part of the first solution at the condenser section of the heat pipe at Re r = 2.4 .
r(m)
r(m) 0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
-0.01
-0.01
-0.02
-0.04
-0.02
evaporator section
-0.03 0
0.025
0.05
0.075
-0.03 0.1
-0.04 0.4
condenser section 0.425
0.45
0.475
0.5
Z(m)
Z(m)
Fig. 6 Vapor velocity profiles in the heat pipe at Re r = 2.4 , (a) Real solution, (b) Imaginary part of the first solution It can be seen that these solutions also represent reversed flows in the evaporator and condenser sections. These solutions seems to be meaningless from the physical point of view but they are interesting solutions of the Navier-Stokes equations. At higher radial Reynolds numbers, there are also more than one solutions. This subject has been also discussed by previous researchers using various numerical and analytical models [34-37]. This is the subject of the future studies of the author. Conclusions
It is shown that the laminar and incompressible vapor flow in a copper-water heat pipe can be accurately simulated using the present model. It is also demonstrated that the model works well for the prediction of vapor flow and pressure distribution in the copper-water heat pipes and presents reasonable results in the range of low to moderate radial Reynolds numbers. However, further studies and series solutions with more terms need to be analyzed for better validation of the model. The vapor pressure distributions and velocity profiles along the vapor space are predicted for a number of test cases in the range of low to moderate radial Reynolds numbers( Re r ≤ 2.4 ). The results have been compared with the available numerical data in the literature and have shown relatively good agreement in all cases. The model shows the existence of symmetric and asymmetric reversed flow solutions as well. However, these solutions seem to be physically meaningless and need to be more studied. The present analysis can be easily extended to a more complete analysis, including phase change, heat and fluid flow through wicks and vapor-liquid interface interactions. This is the subject of the author’s current research. References [1] Faghri, A.(1995), Heat Pipe Science and Technology, Taylor & Francis, Washington, D. C. [2] Peterson, G. P.(1994), An Introduction to Heat Pipes: Modeling, Testing, and Applications, John Wiely, New York. [3] Dunn, P. D. and Reay, D. A.(1982), Heat Pipes, Pergamon, Oxford. [4] Chi, S. W.(1976), Heat Pipe Theory and Practice, Hemisphere, Washington, D.C. [5] Vasiliev L. L.(1998) “State-of-the-art on heat pipe technology in the former soviet union”, Applied Thermal Engineering, Vol. 18, No. 7, pp. 507-551.
10
[6] Silverstein, C. C.(1992), Design and technology of heat pipes for cooling and heat exchange, Hemisphere Publishing Corporation, Washington DC. [7] Cotter, T. P.(1965), Los Alamos Scientific Laboratory. [8] Busse, C. A.(1967), “Pressure drop in the vapor phase of long heat pipes”, Proc. 1967IEEE Thermionic Conversion Specialist Conf., Palo Alto, California, p. 391. [9] Bankston C. A. and Smith H. J.(1971), “Incompressible laminar flow in cylindrical heat pipes”, ASME PAPER 17-WA/HT-15. [10] Rohani, A. R. and Tien, C. L.(1974), “Analysis of the effects of vapor pressure drop on heat pipe performance”, International Journal of Heat Mass Transfer, Vol. 17, pp. 61- 67. [11] Ismail K. A. R. and Zanardi, M. A.(1996), “A steady-state model for heat pipes of circular or rectangular cross-sections”, Applied Thermal Engineering, Vol. 16, pp.753-767. [12] Faghri, A. and Chen M. M.(1989), ”Numerical analysis of the effect of conjugate heat transfer, vapor compressibility and viscous dissipation in heat pipes ”, Numerical Heat Transfer, Vol. 16, pp. 398-405. [13] Ismail K. A. R. and Zanardi, M. A.(1986), “ Two-dimensional analysis of flow and heat transfer in porous heat pipe”, Proc. VI International heat pipe Conf., Grenoble, pp. 164-167. [14] Cao, Y. and Faghri, A.(1990), “A transient two-dimensional compressible analysis for high temperature heat pipes with a pulsed heat input”, Numerical Heat Transfer, Vol. 18, pp. 483-502. [15] Chen M. M. and Faghri, A.(1990), “An analysis of the vapor flow and heat conduction through the liquid wick and pipe wall in a heat pipe with single and multiple heat sources”, International Journal of Heat and Mass Transfer, Vol. 33, pp. 19451955. [16] Faghri, A. and Buchko, M.(1991), “ Experimental and numerical analysis of low temperature heat pipes with multiple heat sources”, ASME Journal of Heat Transfer, Vol. 113, pp. 728-734. [17] Nguyen-Chi H., Abhat A.(1979), ” Performance evaluation of gravity assisted copper-water heat pipes with liquid over fill”, AIAA Journal , Vol.17, No.9, pp. 1003-1011. [18] Tanaka, O. and Koshino H. and Kuriki, J. and Yohmatsu Y. and Harada, O. (1995), ”Heat extraction from the ground in a volcanic zone using copper water heat pipes experiment and analysis”, Experimental Thermal and Fluid Science, Vol. 11, No. 1, p. 72. [19] Tournier, J.-M. and El-Genk M. S.(1994), ”A heat pipe transient analysis model”, International Journal of Heat and Mass Transfer, Vol. 37, No. 5, pp. 753-762. [20] Khalkhali, H. and Faghri, A. and Zuo, Z. J.(1999), “Entropy generation in a heat pipe system”, Applied Thermal Engineering, Vol. 19, pp. 1027-1043. [21] Zhu, N. and Vafai, K.(1999),”Analysis of cylindrical heat pipes incorporating the effects of liquid vapor coupling and nonDarcian transport-a closed form solution”, International Journal of Heat and Mass Transfer, Vol. 42, pp. 3405-3418. [22] Alazmi, B., and Vafai, K.(2001), “Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer”, International Journal of Heat Mass Transfer, Vol.44, pp. 1735-1749. [23] Zuo, Z. J. and Faghri, A.(1998), “A network thermodynamic analysis of the heat pipe”, International Journal of Heat Mass Transfer, Vol. 41, No. 11, pp. 1473-1484. [24] Zhang, J. T. and Wang, B. X.(2002),”Effect of capillary at liquid-vapor interface on phase change without surfactant”, International Journal of Heat and Mass Transfer, Vol. 45, pp. 2689-2694. [25] Layeghi, M., and Nouri, A. (2004), ”An analysis of the vapor and liquid flows and capillary limit in conventional and concentric annular heat pipes”, under review, International Journal of Heat Exchangers. [26] Vafai, K., Wang, W. (1992), “Analysis of flow and heat transfer characteristics of an asymmetrical flat plate heat pipe”, Int. J. Heat Mass Transfer, Vol. 35, pp. 2087-2099. [27] Vafai, K., Zhu, N., and Wang, W. ( 1995), “Analysis of asymmetrical disk-shaped and flat plate heat pipes”, ASME J. Heat Transfer, Vol. 117, pp. 209-218. [28] Zhu, N., and Vafai, K. (1996), “The effects of liquid-vapor coupling and non-Darcian transport on asymmetrical disk-shaped heat pipes”, Int. J. Heat Mass Transfer, Vol. 39, pp. 2095-2113. [29] Zhu, N., and Vafai, K. (1996), “Optimization of asymmetrical disk-shaped heat pipes”, AIAA J. Thermophysics and Heat Transfer, Vol. 10, pp. 179-182. [30] Zhu, N., and Vafai, K. (1997), “Numerical and analytical investigation of vapor flow in a disk-shaped heat pipe incorporating secondary flow”, Int. J. Heat Mass Transfer, Vol. 40, pp. 2887-2900.
11
[31] Zhu, N., and Vafai, K. (1998), “Analytical modeling of the startup characteristics of asymmetrical flat-plate and disk-shaped heat pipes”, Int. J. Heat Mass Transfer, Vol. 41, pp. 2619-2637. [32] Zhu, N., and Vafai, K. (1998), “Vapor and liquid flow in an asymmetrical flat plate heat pipe: a three-dimensional analytical and numerical investigation”, Int. J. Heat Mass Transfer, Vol. 41, pp. 159-174. [33] Zhu, N., and Vafai, K. (1999), “Analysis of cylindrical heat pipes incorporating the effects of liquid vapor coupling and coupling and non-Darcian transport-a closed form solution”, Int. J. Heat and Mass Transfer, Vol. 42, pp. 3405-3418. [34] Bankston C. A. and Smith H. J.(1973), ”Vapor flow in cylindrical heat pipes”, Journal of Heat Transfer, pp. 371-376. [35] Weissberg, H. L.(1959), ”Laminar flow in the entrance region of a porous pipe”, Physics of Fluids, Vol. 2, No. 5, Sept. [36] Yuan, S. W. and Finkelstein A. B.(1955), ”Laminar flow with injection and suction through a porous wall”, Proceedings of the Heat Transfer and Fluid Mechanics Institute, Los Angeles, Calif. [37] Terrill, R. M. and Thomas, P. W. (1969),”On laminar flow through a uniformly porous pipe”, Applied Science Research, Vol. 21, Aug.
12
