Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
273
500 1000
1500 2000
2500 3000
3500 4000
4500 5000
10 20
30 40
50 60
70 Q W
h
ev
W m
².K
500 1000
1500 2000
2500 3000
3500 4000
4500 5000
h
c
W m
². K
Thermosyphon position Anti-gravity position
Horizontal position T
sf
= 40 °C
Fig. 13. Evaporation and condensation heat transfer coefficient Vs. Q, for different orientations
IV. FMHP Modeling
The section of the FMHP is illustrated by Fig. 2 square microchannels with D
g
= W
g
= d. The liquid accumulates in the corners and forms four meniscuses
Fig. 14. Their curvature radius, r
c
, is related to the difference of pressure, between vapor and liquid phase,
by the Laplace-Young equation.
Fig. 14. Evolution of the curvature radius along a microchannel
In the evaporator and adiabatic zones, the curvature radius, in the parallel direction of the microchannel axis,
is lower than the one perpendicular to this axis. Therefore, the meniscus is described by only one
curvature radius. In a given section, r
c
is supposed constant. The axial evolution of r
c
is obtained by the differential of the Laplace-Young equation. The part of
wall that is not in contact with the liquid is supposed dry and adiabatic.
In the condenser, the liquid flows toward the microchannel corners.
There is a transverse pressure gradient, and a transverse curvature radius variation of the meniscus.
The distribution of the liquid along a microchannel is presented in Fig. 14.
The microchannel is divided into several elementary volumes of length, dz, for which, we consider the
Laplace-Young equation, and the conservation equations written for the liquid and vapor phases as it follows:
Laplace-Young equation:
2 v
l c
c
dP dP
dr dz
dz dz
r σ
− = −
11 Liquid and vapor mass conservation:
1
l l
l v
d w A
dQ dz
h dz ρ
= ∆
12
1
v v
v v
d w A
dQ dz
h dz
ρ
= − ∆
13 Liquid and vapor momentum conservation:
2
l l
l l l
il il
lw lw
l l
d A w dz
dz d A P
dz A
A dz
g A sin dz
ρ τ
τ ρ
θ
= =
+ +
+ −
14
2 v
v v
v v
il il
vw vw
v v
d A w dz
dz d A P
dz A
dz A
g A sin dz
ρ τ
τ ρ
θ
= = −
− +
− −
15
Energy conservation:
2 2
1
w w
w sat
w w
T h
dQ T
T t
l t dz
z
λ
∂ −
− = −
× ∂
16 The quantity dQdz in equations 12, 13, and 16
represents the heat flux rate variations along the elementary volume in the evaporator and condenser
zones, which affect the variations of the liquid and vapor mass flow rates as it is indicated by equations 12 and
13. So, if the axial heat flux rate distribution along the microchannel is given by the following Eq.17:
1
a e
e a
e e
a e
a a
e a
t b
c b
Q z L z
L Q
Q L
z L
L L
L z
Q L
L z
L L
L L
≤ ≤
⎧ ⎪
= +
⎪ ⎨ ⎛
⎞ + −
⎪ +
+ ≤ ≤ −
⎜ ⎟
⎪ −
⎝ ⎠
⎩
17
We get a linear flow mass rate variations along the microchannel.
In equation 17, h represents the heat transfer coefficient in the evaporator, adiabatic and condenser
sections. For these zones, the heat transfer coefficients
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
274 are determined from the experimental results section 3.
Since the heat transfer in the adiabatic section is equal to zero and the temperature distribution must be represented
by a mathematical continuous function between the different zones, the adiabatic heat transfer coefficient
value is chosen to be the average between the evaporator and condenser heat transfer coefficients in order to
compute the temperature distribution corresponding to the transition from the evaporator zone to the adiabatic
one, and from the adiabatic section to the condenser one.
The liquid and vapor passage sections, A
l
, and A
v
, the interfacial area, A
il
, the contact areas of the phases with the wall, A
lp
and A
vp
, are expressed using the contact angle and the interface curvature radius by:
2 2
2 4
2
l c
sin A
r sin
θ θ θ
⎛ ⎞
= − +
⎜ ⎟
⎝ ⎠
18
2 v
l
A d
A =
−
19
8
il c
A r
dz
θ
= × × ×
20
16 2
lw c
A r sin
dz
θ
=
21 16
4 2
vw c
A d
r sin dz
θ ⎛
⎞ =
− ⎜
⎟ ⎝
⎠ 22
4
π θ
α
= −
23 The liquid-wall and the vapor-wall shear stresses are
expressed as:
2
1 2
lw l
l l
w f
τ ρ
=
,
l l
el
k f
R =
,
l l
hlw el
l
w D R
ρ µ
=
24
2
1 2
vw v
v v
w f
τ ρ
=
,
v v
ev
k f
R =
,
v v
hvw ev
v
w D R
ρ µ
=
25 where k
l
and k
v
are the Poiseuille numbers, and D
hlw
and D
hvw
are the liquid-wall and the vapor-wall hydraulic diameters, respectively.
The hydraulic diameters and the shear stresses in equations 24 and 25 are expressed as follows:
2
2 2
2
c hlw
sin r
sin D
sin
θ θ θ
θ
⎛ ⎞
× − +
⎜ ⎟
⎝ ⎠
=
26
2 2
2
2 4
2 4
2
c hvw
c
sin d
r sin
D d
sin r
θ θ θ
θ
⎛ ⎞
− − +
⎜ ⎟
⎝ ⎠
= −
×
27
2
1 2
2 2 2
l l
l lw
c
k w sin
sin sin
r
µ θ
τ θ
θ θ
= ⎛
⎞ − +
⎜ ⎟
⎝ ⎠
28
2 2
2
4 2
2 4
2
v v
v c
vw c
k w d
sin r
sin d
r sin
µ θ
τ θ
θ θ
⎛ ⎞
⎛ ⎞
− ⎜
⎟ ⎜
⎟ ⎝
⎠ ⎝
⎠ =
⎛ ⎞
⎛ ⎞
− − +
⎜ ⎟
⎜ ⎟
⎝ ⎠
⎝ ⎠
29
The liquid-vapor shear stress is calculated by assuming that the liquid is immobile since its velocity is
considered to be negligible when compared to the vapor velocity w
l
w
v
. Hence, we have:
2
1 2
v v v
il eiv
w k R
ρ τ
=
,
v v
hiv eiv
v
w D R
ρ µ
=
30 where D
hiv
is the hydraulic diameter of the liquid-vapor interface. The expressions of D
hiv
and τ
iv
are:
2 2
2
2 4
2 2
c hi
c
sin d
r sin
D r
θ θ θ
θ
⎛ ⎞
− − +
⎜ ⎟
⎝ ⎠
=
31
2 2
2
2 4
2
v c
v v
il c
k r w
sin d
r sin
θ µ
τ θ
θ θ
= ⎛
⎞ −
− + ⎜
⎟ ⎝
⎠
32
The equations 11-16 constitute a system of six first order differential, nonlinear, and coupled equations.
The six unknown parameters are: r
c
, w
l
, w
v
, P
l
, P
v
, and T
w
. The integration starts in the beginning of the evaporator z = 0 and ends in the condenser extremity z
= L
t
- L
b
, where L
b
is the length of the condenser flooding zone. The boundary conditions for the adiabatic
zone are the calculated solutions for the evaporator end. In z = 0, we use the following boundary conditions:
c c min
r r
=
33a
v l
w w
= =
33b
v sat
v
P P
T =
33c
v l
c min
P P
r σ
= − 33d
The solution is performed along the microchannel if r
c
is higher than r
cmin
. The coordinate for which this condition is verified, is noted L
as
and corresponds to the microchannel dry zone length. Beyond this zone, the
liquid doesnt flow anymore. Solution is stopped when r
c
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
275 = r
cmax
, which is determined using the following reasoning: the liquid film meets the wall with a constant
contact angle. Thus, the curvature radius increases as we progress
toward the condenser Figs. 14 a to b. When the liquid film contact points meet, the wall is not anymore in
direct contact with vapor. In this case, the liquid configuration should correspond to Fig. 14 c, but
actually, the continuity in the liquid-vapor interface shape imposes the profile represented on Figure 14 d. In
this case, the curvature radius is maximum. Then, in the condenser, the meniscus curvature radius decreases as
the liquid thickness increases Fig. 14 e. The transferred maximum power, so called capillary limit, is determined
if the junction of the four meniscuses starts precisely in the beginning of the condenser.
V. Numerical Results and Analysis
