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Research ArticleAnalytical Solutions of Heat Transfer and Film
Thickness withSlip Condition Effect in Thin-Film Evaporation for
Two-PhaseFlow in Microchannel
Ahmed Jassim Shkarah,1,2 Mohd Yusoff Bin Sulaiman,1 and Md.
Razali bin Hj Ayob1
1Faculty of Mechanical Engineering, Universiti Teknikal Malaysia
Melaka (UTeM), Melaka, Malaysia2Department of Mechanical
Engineering, Thi-Qar University, 64001 Nassiriya, Iraq
Correspondence should be addressed to Ahmed Jassim Shkarah;
[email protected]
Received 20 June 2014; Revised 24 November 2014; Accepted 25
November 2014
Academic Editor: Salvatore Alfonzetti
Copyright © 2015 Ahmed Jassim Shkarah et al. This is an open
access article distributed under the Creative Commons
AttributionLicense, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is
properlycited.
Physical and mathematical model has been developed to predict
the two-phase flow and heat transfer in a microchannel
withevaporative heat transfer. Sample solutions to the model were
obtained for both analytical analysis and numerical analysis. It
isassumed that the capillary pressure is neglected (Morris, 2003).
Results are provided for liquid film thickness, total heat flux,
andevaporating heat flux distribution. In addition to the sample
calculations that were used to illustrate the transport
characteristics,computations based on the current model were
performed to generate results for comparisons with the analytical
results ofWang etal. (2008) andWayner Jr. et al. (1976).The
calculated results from the current model match closely with those
of analytical results ofWang et al. (2008) andWayner Jr. et al.
(1976).This work will lead to a better understanding of heat
transfer and fluid flow occurringin the evaporating film region and
develop an analytical equation for evaporating liquid film
thickness.
1. Introduction
Over the last decade, micromachining technology has
beenincreasingly used to develop highly efficient heat sink
coolingdevices due to advantages such as lower coolant demandsand
smaller machinable dimensions. One of the most impor-tant
micromachining technologies is the ability to
fabricatemicrochannels. Hence, the studies of fluid flow and
heattransfer in microchannels which are two essential parts ofsuch
devices have attracted attention due to their broadpotential for
solving both engineering andmedical problems.Heat sinks are
classified as either single-phase or two-phase according to whether
liquid boiling occurs inside themicrochannels. The primary
parameters that determine thesingle-phase and two-phase operating
regimes are the heatflux through the channel wall and the coolant
flow rate. For afixed heat flux (heat load), the coolantmaymaintain
its liquidstate throughout the microchannels. For a lower flow
rate,the liquid coolant flowing inside themicrochannel may reachits
boiling point, causing flow boiling to occur, resulting in
a two-phase heat sink [1–3]. Since the initial conception
ofmicroheat pipes in 1984 by Cotter [4], a number of analyticaland
experimental investigations have been conducted. Mostof these
investigations have concentrated on the capillary heattransport
capability. With rapid advancements in microelec-tronic devices,
their required total power and power densityare significantly
increasing. Thin-film evaporation plays animportant role in these
modern highly efficient heat transferdevices. When thin-film
evaporation occurs in the thin-filmregion, most of the heat
transfers through a narrow areabetween the nonevaporation region
and intrinsic meniscusregion as shown in Figure 1. The flow
resistance of the vaporphase during thin-film evaporation is very
small comparedwith the vapor flow in the liquid phase in a typical
nucleateboiling heat transfer configuration. In addition, the
superheatneeded for the phase change in the thin-film region is
muchsmaller than that for a bubble growth in a typical
nucleateboiling, in particular, at the initial stage of the bubble
growth.The heat transfer efficiency of thin-film evaporation is
muchhigher than the nucleate boiling heat transfer. Because
Hindawi Publishing CorporationMathematical Problems in
EngineeringVolume 2015, Article ID 369581, 15
pageshttp://dx.doi.org/10.1155/2015/369581
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2 Mathematical Problems in Engineering
thin-film evaporation occurs in a small region,
increasingthin-film regions andmaintaining its stability are very
impor-tant for heat transfer enhancement. It is known that
thethermodynamic properties of the liquid thin-film region arevery
different from those of the macroregion. The effect ofthe
intermolecular forces between the liquid thin film andwall can be
characterized by the disjoining pressure, whichcontrols
thewettability and stability of liquid thin film formedon the wall.
A better understanding of the evaporationmechanisms governed by the
disjoining pressure in the thin-film region, especially for high
heat flux, is very importantin the development of highly efficient
heat transfer devices.As early as 1972, Potash Jr. and Wayner Jr.
[5] expanded theDerjaguin-Landau-Verwey-Overbeek (DLVO) theory [6]
todescribe evaporation and fluid flow from an extendedmenis-cus.
Following this work, extensive investigations have beenconducted to
further understand mechanisms of fluid flowcoupled with evaporating
heat transfer in thin-film region.Stephan and Busse [7] developed a
mathematical modelbased on the theoretical analysis presented by
Wayner Jr. etal. [5, 8] to investigate the heat transfer
coefficient occurringin small triangular grooves and found that the
interfacetemperature variation plays an important role in the
thin-film evaporation. Schonberg andWayner Jr. [9] developed
ananalytical model by ignoring capillary pressure and found
ananalytical solution for the maximum heat evaporation fromthe thin
film. Ma and Peterson [10] studied the thin-filmprofile, heat
transfer coefficient, and temperature variationalong the axial
direction of a triangular groove. Hanlon andMa [11] found that when
particles become smaller, thin-filmregion can be significantly
increased. More recently, Shaoand Zhang [12] considered the effect
of thin-film evaporationon the heat transfer performance in an
oscillating heat pipe.Wang et al. [13, 14] established a simplified
model basedon the Young-Laplace equation and obtained an
analyticalsolution for the total heat transfer in the thin-film
region. Inthe current investigation, amathematicalmodel is
establishedand its analytical solutions are obtained to evaluate
the heatflux, total heat transport per unit length along the
thin-film profile, thin-film thickness, and location for the
maxi-mum heat evaporation in the thin-film region andmaximumheat
transfer rate per unit length by thin-film evaporation.
2. Theoretical Analysis
Figure 1 illustrates a schematic of an evaporating thin
filmformed on a wall. For the current investigation, it is
assumedthat fluid flow in the thin-film region is two-dimensional
andpressure in the liquid film is a function of the
𝑥-coordinateonly. The wall temperature, 𝑇
𝑤, is greater than the vapor
temperature, 𝑇V. The momentum equation governing thefluid flow
in the thin film can be found by
𝑑𝑝𝑙
𝑑𝑥
− 𝜇𝑙
𝜕2
𝑢
𝜕𝑦2= 0, (1)
where
Liquid bulk
NonevaporatingEvaporating
Intrinsic meniscus
y
x
𝛿𝛿0
ṁe
ṁx
thin film
T�
Tw
Figure 1: Schematic of an evaporating thin film.
𝑢 = 𝑢wall = 𝛽𝑑𝑢
𝑑𝑥
wall
𝑦 = 0,
𝛿𝑢
𝛿𝑦
= 0 𝑦 = 𝛿.
(2)
Solving (1) with boundary condition (2), we get
𝑢 = (
1
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
)
𝑦2
2
−
𝛿
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥 1
𝑦 + 𝑢wall,
𝑢wall = 𝛽𝑑𝑢
𝑑𝑥
wall
,
(3)
where 𝛽 is the slip coefficient. If 𝛽 is equal to zero, then
noslip boundary condition is obtained.
So we can find
𝑢 =
1
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
(
𝑦2
2
− 𝛿𝑦 − 𝛽𝛿) , (4)
where
𝛽 = 𝛽0(
1
√1 − 𝛾/𝛾𝑐
) , (5)
where 𝛽0is the limiting slip length and 𝛾
𝑐represents the
critical value of the shear rate. The slip coefficient underthe
condition of this study turns out to be approximately1 ∗ 10
−9m. From (4), the mass flow rate at a given location 𝑥can be
found as
�̇�𝑥= ∫
𝛿
0
𝜌𝑙𝑢𝑙𝑑𝑦 =
𝜌𝑙
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
(
−2𝛿3
− 6𝛽𝛿2
6
) . (6)
Taking a derivative of (6),
𝑑�̇�𝑥
𝑑𝑥
=
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
(
−2𝛿3
− 6𝛽𝛿2
6
)] . (7)
But
�̇�𝑒= −
𝑑�̇�𝑥
𝑑𝑥
(8)
so the net evaporative mass transfer can be obtained as
�̇�𝑒= −
𝑑�̇�𝑥
𝑑𝑥
=
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
(
2𝛿3
+ 6𝛽𝛿2
6
)] . (9)
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Mathematical Problems in Engineering 3
The heat transfer rate by evaporation occurring at the
liquid-vapor interface in the thin-film region can be determined
by
𝑞
= �̇�𝑒ℎ𝑓𝑔= ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
(
2𝛿3
+ 6𝛽𝛿2
6
)] . (10)
But at the same time 𝑞 in (10) is equal to the heat transferrate
through the liquid thin film; that is,
𝑞
= 𝑘𝑙
𝑇wall − 𝑇𝛿𝛿
. (11)
From expanding the Clausius-Clapeyron equation, we canfind
(
𝑑𝑝
𝑑𝑇
)
sat=
ℎ𝑓𝑔
𝑇V (1
𝜌V−
1
𝜌𝑙
)
, (12)
𝑇𝛿= 𝑇V (1 +
Δ𝑝
𝜌Vℎ𝑓𝑔) . (13)
The pressure difference between vapor and liquid, Δ𝑝, at
theliquid-vapor interface is due to both the capillary pressureand
disjoining pressure and is expressed using the
augmentedYoung-Laplace equation:
Δ𝑝 = 𝑝V − 𝑝𝑙 = 𝑝𝑐 + 𝑝𝑑. (14)
The disjoining pressure for a nonpolar liquid is expressed
as
𝑝𝑑=
𝐴
𝛿3, (15)
where𝐴 is the dispersion constant and 𝛿 is the film
thickness.The capillary pressure is the product of interfacial
curvature𝐾 and surface tension coefficient 𝜎:
𝑝𝑐= 𝜎𝐾, (16)
𝐾 = (
𝑑2
𝛿
𝑑𝑥2)[1 + (
𝑑𝛿
𝑑𝑥
)
2
]
3/2
. (17)
Substituting (13) into (11) the heat flux can be rewritten
as
𝑞
=
𝑇wall − 𝑇V (1 + Δ𝑝/𝜌Vℎ𝑓𝑔)
𝛿/𝑘𝑙
. (18)
Substituting (14) into (16) the heat flux can be rewritten
as
𝑞
=
𝑇wall − 𝑇V (1 + (𝑝𝑐 + 𝑝𝑑) /𝜌Vℎ𝑓𝑔)
𝛿/𝑘𝑙
. (19)
Substituting (19) into (10) gives
𝑇wall − 𝑇V (1 + (𝑝𝑐 + 𝑝𝑑) /𝜌Vℎ𝑓𝑔)
𝛿/𝑘𝑙
= ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
𝑑𝑝𝑙
𝑑𝑥
(
2𝛿3
+ 6𝛽𝛿2
6
)] .
(20)
We can find 𝑑𝑝𝑙/𝑑𝑥 by differentiating (14) with respect to
𝑥,
and assume uniform vapor pressure, 𝑝V, along the meniscus:
𝑑𝑝𝑙
𝑑𝑥
= −(
𝑑 (𝑝𝑐+ 𝑝𝑑)
𝑑𝑥
) . (21)
Substituting (21) into (20) yields
𝑇wall − 𝑇V (1 + (𝑝𝑐 + 𝑝𝑑) /𝜌Vℎ𝑓𝑔)
𝛿/𝑘𝑙
= −ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
(
𝑑 (𝑝𝑐+ 𝑝𝑑)
𝑑𝑥
)(
2𝛿3
+ 6𝛽𝛿2
6
)] .
(22)
For the evaporating thin-film region, the disjoining pressureis
one dominant parameter, which governs the fluid flow inthe
evaporating thin-film region. And in the evaporating thinfilm
region, the absolute disjoining pressure is much largerthan the
capillary pressure especially when the curvaturevariation along the
meniscus is very small. In order to findthe primary factor
affecting the thin-film evaporation in theevaporating thin-film
region, it is assumed that the capillarypressure, 𝑝
𝑐, is neglected.
Then, (22) becomes
𝑇wall − 𝑇V (1 + 𝑝𝑑/𝜌Vℎ𝑓𝑔)
𝛿/𝑘𝑙
= −ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
(
𝑑𝑝𝑑
𝑑𝑥
)(
2𝛿3
+ 6𝛽𝛿2
6
)] .
(23)
From (15) we find 𝑝𝑑and 𝑑𝑝
𝑑/𝑑𝑥;
𝑑𝑝𝑑
𝑑𝑥
=
−3𝐴
𝛿4
𝑑𝛿
𝑑𝑥
. (24)
Substituting (15) and (24) into (23) yields
𝑇wall − 𝑇V (1 + (𝐴/𝛿3
) /𝜌Vℎ𝑓𝑔)
𝛿/𝑘𝑙
= −ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙
𝜇𝑙
(
−3𝐴
𝛿4
𝑑𝛿
𝑑𝑥
)(
2𝛿3
+ 6𝛽𝛿2
6
)] .
(25)
We can rewrite the heat flux as
𝑞
= ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙𝐴
𝜇𝑙
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)] . (26)
Simplifying (25) gives
1
ℎ𝑓𝑔
(
𝑘𝑙𝑇wall𝛿
−
𝑘𝑙𝑇V
𝛿
−
𝑘𝑙𝑇V𝐴
𝜌Vℎ𝑓𝑔𝛿4)
=
𝑑
𝑑𝑥
[
𝜌𝑙𝐴
𝜇𝑙
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)] .
(27)
We need to solve (27) to find 𝑑𝛿/𝑑𝑥.
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4 Mathematical Problems in Engineering
1
0.8
0.6
0.4
0.2
0
0 4 8 12 16 20
qt
(W/m
)
Analytical solution of Wang et al. [13]Numerical solution of
Schonberg and WaynerCurrent full numerical solutionCurrent
analytical solution
Twall − T�
Figure 2: Comparison of the total heat transfer rate through
thinfilm region with results presented byWang et al. [13] andWayner
Jr.et al. [8].
2
1.8
1.6
1.4
1.2
1
0 2 4 6
𝛿/𝛿
0
x/𝛿0
Numerical solutionAnalytical solution
Tw − T� = 0.1K
Figure 3: Dimensionless evaporative film thickness profile at
asuperheat of 0.1 K.
𝛿/𝛿
0
x/𝛿0
8
6
4
2
0
0 2 4 6
Tw − T� = 0.5K
Numerical solutionAnalytical solution
Figure 4: Dimensionless evaporative film thickness profile at
asuperheat of 0.5 K.
By multiplying 2 sides by 2 ∗ [(𝑑𝛿/𝑑𝑥)(1/𝛿 + 3𝛽/𝛿2)]
andintegrating two sides with respect to 𝑥 we get
[
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)]
2
+ 𝐷
=
2𝜇𝑙
ℎ𝑓𝑔𝐴𝜌𝑙
[
𝑘𝑙
𝛿
(𝑇V − 𝑇wall) +3𝑘𝑙𝛽
2𝛿2(𝑇V − 𝑇wall)
+
𝑘𝑙𝑇V𝐴
4𝜌Vℎ𝑓𝑔𝛿4+
3𝛽𝑘𝑙𝑇V𝐴
5𝜌Vℎ𝑓𝑔𝛿5] + 𝐶.
(28)
𝐷, 𝐶 are integration constants.So
[
𝑑𝛿
𝑑𝑥
]
2
=
1
[1/𝛿 + 3𝛽/𝛿2]2
× (
2𝜇𝑙
ℎ𝑓𝑔𝐴𝜌𝑙
[
𝑘𝑙
𝛿
(𝑇V − 𝑇wall)
+
3𝑘𝑙𝛽
2𝛿2(𝑇V − 𝑇wall)
+
𝑘𝑙𝑇V𝐴
4𝜌Vℎ𝑓𝑔𝛿4+
3𝛽𝑘𝑙𝑇V𝐴
5𝜌Vℎ𝑓𝑔𝛿5] + (𝐶 − 𝐷)) .
(29)
-
Mathematical Problems in Engineering 5𝛿/𝛿
0
x/𝛿0
20
16
12
8
4
0
0 2 4 6
Tw − T� = 1K
Numerical solutionAnalytical solution
Figure 5: Dimensionless evaporative film thickness profile at
asuperheat of 1 K.
Let
𝐵 = 𝐶 − 𝐷, (30)
𝑑𝛿
𝑑𝑥
= (
1
[1/𝛿 + 3𝛽/𝛿2]2
× (
2𝜇𝑙
ℎ𝑓𝑔𝐴𝜌𝑙
× [
𝑘𝑙
𝛿
(𝑇V − 𝑇wall) +3𝑘𝑙𝛽
2𝛿2(𝑇V − 𝑇wall)
+
𝑘𝑙𝑇V𝐴
4𝜌Vℎ𝑓𝑔𝛿4+
3𝛽𝑘𝑙𝑇V𝐴
5𝜌Vℎ𝑓𝑔𝛿5] + 𝐵))
1/2
.
(31)
To find constant 𝐵 we apply the boundary condition
𝑑𝛿
𝑑𝑥
= 0 𝛿 = 𝛿𝑜. (32)
So we get
𝐵 = −
2𝜇𝑙
ℎ𝑓𝑔𝐴𝜌𝑙
× [
𝑘𝑙
𝛿𝑜
(𝑇V − 𝑇wall) +3𝑘𝑙𝛽
2𝛿2
𝑜
(𝑇V − 𝑇wall)
+
𝑘𝑙𝑇V𝐴
4𝜌Vℎ𝑓𝑔𝛿4
𝑜
+
3𝛽𝑘𝑙𝑇V𝐴
5𝜌Vℎ𝑓𝑔𝛿5
𝑜
] .
(33)
𝛿/𝛿
0
x/𝛿0
60
40
20
0
0 2 4 6
Tw − T� = 2K
Numerical solutionAnalytical solution
Figure 6: Dimensionless evaporative film thickness profile at
asuperheat of 2 K.
From (26) and (27),
𝑞
= ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙𝐴
𝜇𝑙
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)]
= (
𝑘𝑙𝑇wall𝛿
−
𝑘𝑙𝑇V
𝛿
−
𝑘𝑙𝑇V𝐴
𝜌Vℎ𝑓𝑔𝛿4) .
(34)
So
(𝜇𝑙𝑘𝑙/ℎ𝑓𝑔𝜌𝑙𝐴) (𝑇
𝑤− 𝑇V) − (𝜇𝑙𝑘𝑙𝑇V/ℎ
2
𝑓𝑔𝜌𝑙𝜌V𝐴) (1/𝛿
3
)
𝛿
=
𝑑
𝑑𝑥
[
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)] .
(35)
Then
𝑑
𝑑𝑥
[
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)]
=
[(𝑘𝑙𝜐𝑙/ℎ𝑓𝑔𝐴) (𝑇
𝑤− 𝑇V)] − (𝑘𝑙𝜐𝑙𝑇V/ℎ
2
𝑓𝑔𝜌V) 𝛿−3
𝛿
.
(36)
Note that
𝜐𝑙=
𝜇𝑙
𝜌𝑙
. (37)
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6 Mathematical Problems in Engineering𝛿/𝛿
0
x/𝛿0
800
600
400
200
0
0 2 4 6
Tw − T� = 5K
Numerical solutionAnalytical solution
Figure 7: Dimensionless evaporative film thickness profile at
asuperheat of 5 K.
The optimum thickness of the evaporating thin film,
𝛿optimum,where the heat flux reaches its maximum, can be found
bytaking a derivative of 𝑥 for heat flux equation (34):
𝑑𝑞
𝑑𝑥
= 0
=
𝑑 (𝑘𝑙𝑇wall/𝛿 − 𝑘𝑙𝑇V/𝛿 − 𝑘𝑙𝑇V𝐴/𝜌Vℎ𝑓𝑔𝛿
4
)𝛿=𝛿optimum
𝑑𝑥
.
(38)
So
(−
𝑇wall
(𝛿optimum)2+
𝑇V
(𝛿optimum)2+
4𝑇V𝐴
𝜌Vℎ𝑓𝑔 (𝛿optimum)5) = 0.
(39)
Then
𝛿optimum =3√
4𝐴𝑇V
𝜌Vℎ𝑓𝑔 (𝑇𝑤 − 𝑇V). (40)
For the nonevaporating film region, the heat flux is
zero.Clearly, the interface temperature is equal to the wall
tem-perature. The equilibrium thickness 𝛿
𝑜can be readily found
by
𝑞
= (
𝑘𝑙𝑇wall𝛿𝑜
−
𝑘𝑙𝑇V
𝛿𝑜
−
𝑘𝑙𝑇V𝐴
𝜌Vℎ𝑓𝑔𝛿4
𝑜
) = 0. (41)
𝛿/𝛿
0
x/𝛿0
60000
40000
20000
0
0 2 4 6
Tw − T� = 10K
Numerical solutionAnalytical solution
Figure 8: Dimensionless evaporative film thickness profile at
asuperheat of 10 K.
So
𝛿0≥3√
𝐴𝑇V
𝜌Vℎ𝑓𝑔 (𝑇𝑤 − 𝑇V),
𝛿min0
=3√
𝐴𝑇V
𝜌Vℎ𝑓𝑔 (𝑇𝑤 − 𝑇V).
(42)
From (31), (33), and (34) we get
𝛿optimum =3√4𝛿𝑜. (43)
The total heat transfer rate per unit width along
themeniscus,𝑞𝑡, can be calculated by
𝑞𝑡= ∫
𝑥
0
𝑞
𝑑𝑥. (44)
From (34)
𝑞𝑡= ∫
𝑥
0
ℎ𝑓𝑔
𝑑
𝑑𝑥
[
𝜌𝑙𝐴
𝜇𝑙
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2)]𝑑𝑥. (45)
So
𝑞𝑡=
ℎ𝑓𝑔𝜌𝑙𝐴
𝜇𝑙
𝑑𝛿
𝑑𝑥
(
1
𝛿
+
3𝛽
𝛿2) . (46)
Dimensionless thickness of the evaporating region can bedefined
as
̂𝛿 =
𝛿
𝛿𝑜
, (47)
where 𝛿𝑜is the thickness of the nonevaporating region.
-
Mathematical Problems in Engineering 7
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 4 8 12 16
Tw − T� = 0.3K
Numerical solutionAnalytical solution
Figure 9: Dimensionless heat flux profile at a superheat of 0.3
K.
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 4 8 12 16
Tw − T� = 0.7K
Numerical solutionAnalytical solution
Figure 10: Dimensionless heat flux profile at a superheat of 0.7
K.
A dimensionless position also can be defined as
𝜓 =
𝑥
𝛿𝑜
. (48)
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Tw − T� = 1K
Numerical solutionAnalytical solution
Figure 11: Dimensionless heat flux profile at a superheat of 1
K.
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Tw − T� = 2K
Numerical solutionAnalytical solution
Figure 12: Dimensionless heat flux profile at a superheat of 2
K.
For dimensionless heat flux,
𝜙 =
𝑞
𝑞
𝑜
, (49)
-
8 Mathematical Problems in Engineering
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Tw − T� = 5K
Numerical solutionAnalytical solution
Figure 13: Dimensionless heat flux profile at a superheat of 5
K.
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Tw − T� = 10K
Numerical solutionAnalytical solution
Figure 14: Dimensionless heat flux profile at a superheat of 10
K.
where
𝑞
𝑜=
𝑘𝑙(𝑇𝑤− 𝑇V)
𝛿𝑜
(50)
is the heat flux at the interface temperature equal to the
vaportemperature.
x/𝛿0
𝜙
𝛽 = 0 (no slip)
0.8
0.6
0.4
0.2
0
0 2 4 6
Numerical solutionAnalytical solution
Figure 15: Dimensionless heat flux profile at a 𝛽 = 0 nm.
Substituting (36), (34), and (50) in (49) we found
𝜙 =
ℎ𝑓𝑔𝜌𝑙𝐴
𝜇𝑙
×
([(𝑘𝑙𝜐𝑙/ℎ𝑓𝑔𝐴) (𝑇
𝑤− 𝑇V)] − (𝑘𝑙𝜐𝑙𝑇V/ℎ
2
𝑓𝑔𝜌V) 𝛿−3
) /𝛿
(𝑘𝑙(𝑇𝑤− 𝑇V) /𝛿𝑜)
=
𝛿𝑜
𝛿
−
𝛿4
𝑜
𝛿4,
(51)
or
𝜙 =
1
̂𝛿
(1 −
1
̂𝛿3
) . (52)
By considering 𝑑𝜙/𝑑̂𝛿 = 0, we can determine the
maximumdimensionless heat flux 𝜙max. The local heat flux through
theevaporating thin film reaches its maximum when ̂𝛿 = 41/3.Letting
̂𝛿 = 41/3 in (52), the maximum dimensionless heatflux, ̂𝛿max, can
be found as
̂𝛿max =
3
44/3
≈ 0.473. (53)
Equation (53) indicates that the maximum heat flux occur-ring in
the evaporating thin-film region is not greater than0.473 times of
the characteristic flux heat.
-
Mathematical Problems in Engineering 9
𝛽 = 0.5 ∗ 10−9 m
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Numerical solutionAnalytical solution
Figure 16: Dimensionless heat flux profile at a 𝛽 = 0.5 nm.
Consider the heat transfer coefficient of
𝑞
= ℎ (𝑇𝑠− 𝑇V) . (54)
To solve (31) we have two methods:
(1) exact solution or analytical solution,
(2) numerical solution.
We are going to solve (31) by using both methods
2.1. Analytical Solution. We can rewrite (31) in the form
𝑑𝛿
𝑑𝑥
= (Σ𝜎 (𝛿) (1 −
𝛿0
𝛿
) + Ψ𝜎 (𝛿) (1 −
𝛿2
0
𝛿2)
+ Ω𝜎 (𝛿) (
𝛿4
0
𝛿4− 1) + Π𝜎 (𝛿) (
𝛿5
0
𝛿5− 1))
1/2
,
(55)
where
𝜎 (𝛿) =
𝛿2
0
(𝛿0/𝛿 + (3𝛽/𝛿
0) (𝛿2
0/𝛿2))2,
Σ =
2𝜇𝑙𝑘𝑙(𝑇𝑤− 𝑇V)
ℎ𝑓𝑔𝐴𝜌𝑙
1
𝛿0
,
𝛽 = 1 ∗ 10−9 m
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Numerical solutionAnalytical solution
Figure 17: Dimensionless heat flux profile at a 𝛽 = 1 nm.
Ψ =
3𝜇𝑙𝛽 (𝑇𝑤− 𝑇V)
ℎ𝑓𝑔𝐴𝜌𝑙
(
1
𝛿0
)
2
,
Ω =
𝜇𝑙𝑘𝑙𝑇V
2ℎ2
𝑓𝑔𝜌V(
1
𝛿0
)
4
,
Π =
6𝜇𝑙𝑘𝑙𝛽𝑇V
5ℎ2
𝑓𝑔𝜌V
(
1
𝛿0
)
5
,
Λ =
3𝛽
𝛿0
.
(56)
Since Σ ≫ Ψ,Π,Ω so we can rewrite (55) as
𝑑𝛿
𝑑𝑥
= √Σ(
𝛿2
0
(𝛿0/𝛿 + 3 (𝛽𝛿
2
0/𝛿0𝛿2))
× [(1 −
𝛿0
𝛿
) +
Ψ
Σ
(1 −
𝛿2
0
𝛿2)
+
Ω
Σ
(
𝛿4
0
𝛿4− 1) +
Π
Σ
(
𝛿5
0
𝛿5− 1)])
1/2
.
(57)
-
10 Mathematical Problems in Engineering
𝛽 = 3 ∗ 10−9 m
x/𝛿0
𝜙
0.8
0.6
0.4
0.2
0
0 2 4 6
Numerical solutionAnalytical solution
Figure 18: Dimensionless heat flux profile at a 𝛽 = 3 nm.
We introduce ̂𝛿 = 𝛿/𝛿0and 𝑥 = 𝑥√Σ to get
𝑑̂𝛿
𝑑𝑥
= (
̂𝛿2
(1 + Λ (1/̂𝛿))
2
× [(1 −
1
̂𝛿
) +
Ψ
Σ
(1 −
1
̂𝛿2
)
+
Ω
Σ
(
1
̂𝛿4
− 1) +
Π
Σ
(
1
̂𝛿5
− 1)])
1/2
.
(58)
Since 3𝛽/𝛿0≪ 1, we can make the following approximation:
1
(1 + Λ (1/̂𝛿))
2= 1 − 2
Λ
̂𝛿
+ (
Λ
̂𝛿
)
2
. (59)
To get
𝑑̂𝛿
𝑑𝑥
= (̂𝛿2
[1 − 2
Λ
̂𝛿
+ 3(
Λ
̂𝛿
)
2
]
× [(1 −
1
̂𝛿
) +
Ψ
Σ
(1 −
1
̂𝛿2
)
+
Ω
Σ
(
1
̂𝛿4
− 1) +
Π
Σ
(
1
̂𝛿5
− 1)])
1/2
,
x/𝛿0
𝛽 = 0 (no slip)
0 2 4 6
𝛿/𝛿
0
4000
3000
2000
1000
0
Numerical solutionAnalytical solution
Figure 19: Dimensionless evaporative film thickness profile at a
𝛽 =0 nm.
𝑑̂𝛿
𝑑𝑥
= ([1 − 2
Λ
̂𝛿
+ 3(
Λ
̂𝛿
)
2
]
× [ (̂𝛿2
−̂𝛿) +
Ψ
Σ
(̂𝛿2
− 1)
+
Ω
Σ
(
1
̂𝛿2
−̂𝛿2
) +
Π
Σ
(
1
̂𝛿3
−̂𝛿2
)])
1/2
(60)
we have
𝑑̂𝛿
𝑑𝑥
= (̂𝛿2
[1 +
Ψ
Σ
−
Ω
Σ
−
Π
Σ
]
−̂𝛿 [1 + 2Λ [1 +
Ψ
Σ
−
Ω
Σ
−
Π
Σ
]]
− [
Ψ
Σ
− 2Λ] + ⋅ ⋅ ⋅
1
̂𝛿
+ ⋅ ⋅ ⋅
1
̂𝛿2
+ ⋅ ⋅ ⋅ )
1/2
.
(61)
Let
Γ = 1 +
Ψ
Σ
−
Ω
Σ
−
Π
Σ
. (62)
Because ̂𝛿 ≥ 1, we neglect 1/̂𝛿, 1/̂𝛿2, . . . . Therefore,
thesolution is accurate for ̂𝛿 ≫ 1 as
𝑑̂𝛿
𝑑𝑥
=√̂𝛿2Γ −
̂𝛿 [1 + 2ΛΓ] − [Γ − 1 − 2ΛΓ]. (63)
The general solution of (63) by using the initial condition
̂𝛿 = 1 𝑥 = 0 (64)
-
Mathematical Problems in Engineering 11
4000
3000
2000
1000
0
𝛽 = 0.5 ∗ 10−9 m
x/𝛿0
0 2 4 6
𝛿/𝛿
0
Numerical solutionAnalytical solution
Figure 20: Dimensionless evaporative film thickness profile at a
𝛽 =0.5 nm.
is
̂𝛿 =
1
4 × (Γ/ (1 + 2ΛΓ))
×
1
2 × (Γ/ (1 + 2ΛΓ)) − 1
× 𝑒−𝑥√Γ
{4
Γ
1 + 2ΛΓ
(
Γ
1 + 2ΛΓ
− 1)
+ [1 + (2
Γ
1 + 2ΛΓ
− 1) 𝑒−𝑥√Γ
]
2
} .
(65)
And from (65) we can evaluate an analytical solution for𝑞𝑡(𝛿 →
∞) as
𝑞𝑡=
ℎ𝑓𝑔𝜌𝑙𝐴
𝜇𝑙
√Σ. (66)
2.2. Numerical Solution. The solution can be readily
obtainedusing the fourth order Runge-Kutta method for the
evaporat-ing thin film profile. The governing equation (31) is
solvedwith the use of a Runge-Kutta (4) method; the
solutionprocedure is iterative. As a first guess 𝛿
0, the values from
the previous step are used, and the calculated values
arereturned from the Runge-Kutta solver and compared to theguess
values. A comparison of the guess and calculated valuesis performed
and looped until reaching convergence criteriafor both film
thicknesses. The numerical solver is coded inMATLAB. With these
initial conditions, we have
𝛿 = 𝛿0
𝑥 = 0,
𝑑𝛿
𝑑𝑥
= 0 𝑥 = 0.
(67)
𝛽 = 1 ∗ 10−9 m
4000
3000
2000
1000
0
x/𝛿0
0 2 4 6
𝛿/𝛿
0Numerical solutionAnalytical solution
Figure 21: Dimensionless evaporative film thickness profile at a
𝛽 =1 nm.
Table 1: Liquid properties and operating conditions.
Liquid Water𝐴 10−20 J𝑇V 353K
∘
𝜌V 0.083 kg/m3
𝜐𝑙
0.4996 ∗ 10−6m2/sℎ𝑓𝑔
2382700 J/kg𝑘𝑙
0.65w/m⋅K
3. Results and Discussion
3.1. Comparison of Analytical Solution and Full Model.
Aspresented above, a mathematical model for predicting evap-oration
and fluid flow in thin-film region is developed.Utilizing
dimensionless analysis, analytical and numericalsolutions are
obtained for the heat flux distribution, totalheat transfer rate
per unit length, location of the maximumheat flux, and ratio of the
conduction to convection thermalresistance in the evaporating film
region. In order to verifythe analytical solution derived herein,
results predicted byWang et al. [13] and numerical solution by
Wayner Jr. et al.[8] are used. Figure 2 shows the comparison of
analyticaland numerical results of the total heat transfer rate
throughthin-film region with results presented by Wang et al.
[13]and Wayner Jr. et al. [8]. Total heat flux is presented
infunction of the superheat temperature. Our numerical
andanalytical solutions are compared to the ones of Wangand
Schonberg and Wayner. It shows good agreement formoderate superheat
temperatures; however, our analyticalsolution tends to
underestimate total heat flux at large
-
12 Mathematical Problems in Engineering
Table 2: Comparison of previous studies on evaporating extended
meniscus.
Authors Numerical solution analytical solution Finding
analytical equation for 𝛿 Slip conditionPotash and Wayner [5] o x x
xMoosman and Homsy [15] o o x xSchonberg and Wayner [9] o o x
xStephan and Busse [7] o x x xSchonberg et al. [16] o o x
xShikhmurzaev [17] x o x xMa and Peterson [10] o x x xPismen and
Pomeau [18] x o x xCatton and Stroes [19] o o x xChoi et al. [20] o
x x oQu and Ma [21] o o x xChoi et al. [22] o x x oPark and Lee
[23] o x x xBy Morris [24] x o x xDemsky and Ma [25] o x x xJiao et
al. [26] x o x xNa et al. [27] o o x xSultan et al. [28] x o x
xWang et al. [14] o x x xMa et al. [29] o x x xWang et al. [13] o o
x xZhao et al. [30] o x x oZhao et al. [31] o x x oBenselama et al.
[32] x o x xBiswal et al. [33] o x x oLiu et al. [34] x o x xBai et
al. [35] o o x xBiswal et al. [36] o x x xThokchom et al. [37] o x
x xYang et al. [38] o x x xCurrent study o o o o
superheat temperatures. In addition, the model presentedherein
can be used to predict analytically and numericallyall of the
equilibrium film thickness, heat flux distribution,film thickness
variation of evaporating film region,maximumtotal heat transfer
rate through the evaporating film region,and ratio of the
conduction to convection thermal resistance.The following
calculations and predictions are based onthe thermal properties and
operating conditions shown inTable 1.
3.2. Comparison of Analytic Equation for 𝛿 with PreviousStudies.
It was seen from Table 2 that, in many studies on awide range of
time, we did not find any researcher who foundthe analytical
equation for 𝛿 even only approximately, but wefound that they have
numerical studies. So according to this,we can say that (65) is the
first analytical equation for 𝛿 or atleast approximately.
3.3. Distribution of Evaporative Film Thickness. Figures 3,4, 5,
6, 7, and 8 compare our numerical and analyticalpredictions for the
dimensionless film thickness as functionof the dimensionless
position for 0.1 K, 0.5 K, 1 K, 2 K, 5 K, and10K superheat
temperatures.They clearly show that the errorof the analytical
approximation is decreasing with increasingsuperheat temperature.We
can see that for large positions thesolution is dominated by
exponential growth.
Figures 9, 10, 11, 12, 13, and 14 compare our numericaland
analytical predictions for the dimensionless heat flux asfunction
of the dimensionless position for 0.3 K, 0.7 K, 1 K,2 K, 5 K, and
10K superheat temperatures. Similar propertycan be seen; that is,
the error of the analytical approximationis decreasing with
increasing superheat temperature. Themaximum of the heat flux is
predicted correctly for allvalues of the superheat temperature
analytically; however, thelocation of themaximum is predictedwith
significant error atlow superheat temperatures.
-
Mathematical Problems in Engineering 13
𝛽 = 3 ∗ 10−9 m
4000
3000
2000
1000
0
x/𝛿0
0 2 4 6
𝛿/𝛿
0
Numerical solutionAnalytical solution
Figure 22: Dimensionless evaporative film thickness profile at a
𝛽 =3 nm.
Figures 15, 16, 17, and 18 show that nondimensional heatflux is
presented as function of the nondimensional positionfor different
values of 𝛽: 0, 0.5, 1 and 3×10−9. As 𝛽 gets larger,the error in
the analytical heat flux approximation is gettinglarger.
Figures 19, 20, 21, and 22 compare our numerical andanalytical
predictions for the dimensionless film thickness asfunction of the
dimensionless position for different values ofthe slip coefficient,
𝛽: 0, 0.5, 1 and 3×10−9. Comparing the fig-ures indicates similar
conclusions as previously mentioned:smaller slip coefficient yields
smaller error in the analyticalapproximation.
4. Conclusions
This paper presents a mathematical model for
predictingevaporation and fluid flow in thin-film region. The
thin-film region of the extended meniscus is delineated.
Utilizinganalytical solutions were obtained for heat flux
distribution,total heat transfer, and liquid film thickness in the
evapo-rating film region. The mathematical model also developsan
analytic equation for 𝛿. So according to this, we can saythat (65)
is the first analytical equation for 𝛿 or at leastapproximately.
Also we can conclude that there is small effectof slip condition𝛽
on the thin-film evaporation for two-phaseflow in microchannel. The
dimensionless heat flux throughthin-film region is a function of
dimensionless thickness.Also the results showed the assumption that
neglecting thecapillary pressure is acceptable.
Highlights
(i) Analytical two-phase flow formicrochannel heat sinkis
studied.
(ii) Numerical two-phase flow formicrochannel heat sinkis
studied.
(iii) New evaporating film thickness equation is devel-oped.
Nomenclature
𝐴: Dispersion constant (J)ℎ𝑓𝑔: Heat of vaporization (J/kg)
𝑘: Conductivity (W/m⋅K)�̇�𝑒: Interface net evaporative mass
transfer (kg/(m2s))
�̇�𝑥: Mass flow rate (kg/ms)
𝑝𝑐: Capillary pressure (N/m2)
𝑝𝑙: Liquid pressure (N/m2)
𝑝V: Vapor pressure (N/m2)
Δ𝑝: 𝑝1− 𝑝2(N/m)
𝑞: Heat flux (w/m2)𝑞
𝑜: Characteristic heat flux (w/m2)
𝜙: Dimensionless heat flux𝜙max: Maximum dimensionless heat
flux𝑞tot: Total heat transfer rate per unit width (W/m)𝑇:
Temperature (K)𝑢: Velocity along 𝑥-axis (m/s)𝑥: 𝑥-coordinate (m)𝜓:
Dimensionless 𝜓-coordinate𝑦: 𝑦-coordinate (m).
Greek Symbols
𝛿: Film thickness (m)𝛿0: Equilibrium film thickness or
characteristic thickness (m)̂𝛿: Dimensionless film
thickness𝛿optimum: Optimum thickness corresponding to the
maximum heat flux (m)𝜇: Dynamic viscosity (N s/m2)𝜐: Kinematic
viscosity (m2/s)𝜌: Density (kg/m3)𝜎: Surface tension (N/m)𝛽: The
slip coefficient (m).
Subscripts
⋅: Time rate of change𝑙: Liquidmax: Maximum quantitytot: TotalV:
Vapor𝑤: Wall.
-
14 Mathematical Problems in Engineering
Conflict of Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
Acknowledgments
This work is supported by Universiti Teknikal MalaysiaMelaka
(UTeM) andThi Qar University.
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