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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
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)
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.
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)
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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