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Numerical simulation for steady annular condensation flow in triangular microchannels
International Communications in Heat and Mass Transfer 35 (2008) 805–809
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Numerical simulation for steady annular condensation flow in triangular microchannels☆ Yongping Chen a,⁎, Jiafeng Wu a, Mingheng Shi a, G.P. Peterson b a b
School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China Office of the Chancellor, University of Colorado at Boulder, Regent Administrative Center — 17 UCB, 2 Boulder, CO 80309-001, USA
A R T I C L E
I N F O
Available online 24 April 2008 Keywords: Condensation Microchannels Numerical simulation
A B S T R A C T A steady one-dimensional model for annular condensation flow in triangular microchannels is developed. The curvature radius distribution of the condensate stream along the channel has been determined numerically. The results indicate that the curvature radius of the liquid phase would increase rapidly at the beginning, and then as the condensation process progresses along the length of the microchannels, the radius increase would proceed more slowly. At the end of the condensation flow, the radius increases rapidly again. A smaller contact angle and heat flux or a larger hydraulic diameter and steam pressure will all result in a longer condensation length. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Condensation flow in microchannels has been widely used in a variety of advanced microthermal devices, such as microheat pipe and microfuel cell. Although there have been a number of investigations on boiling flow in microchannels, there are relatively a little experimental data and theoretical analysis of this condensation process available in the literature, making it difficult to understand the mechanisms of condensation in microchannels. Garimella [1] presented an overview of the visualization study of the condensation of refrigerants in mini-channels. Experiments for condensation in round, square and rectangular tubes with hydraulic diameters ranging from 1–5 mm were reported and twophase flow regimes and patterns in mini-channels were presented. It was noted that the surface tension, not gravity, played the dominant role in microchannels having a hydraulic diameter less than 1 mm. Médéric et al. [2] also provided sufficient evidence that the capillary force was dominant in channels having a diameter of less than 1 mm. Chen and Cheng [3] carried out a visualization study on condensation of steam in a microchannel with a hydraulic diameter of 75 μm, and found that droplet flow existed upstream and intermittent flow existed downstream of the microchannel. Subsequently, Wu and Cheng [4] performed an experiment on condensation of steam in parallel microchannels having a hydraulic diameter
☆ Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. Tel.: +86 25 8379 4526(8118). E-mail address: [email protected] (Y. Chen). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.03.001
of 83 μm, and found that droplet flow, annular flow and injection flow existed in the microchannels. Similarly, Chen et al. [5] conducted an investigation of condensation in triangular silicon microchannels with diameters less than 250 μm. The results indicated that the channel scale, heat flux, and mass flux all have significant influence on the condensation flow patterns, and the droplet flow, annular flow and injection flow etc. were all observed. These were then combined into a condensation flow map for microchannels. Of particular interest here was the new observation of injection flow at a constant frequency, which has received increasing attention in the past several years. While there have been a number of experimental studies, there is little analytical work that has been done on condensation in microchannels [6,7]. In the current work, a steady one-dimensional model of triangular microchannels with constant heat flux was developed. It was used to describe the characteristics of annular condensation flow, which is a very typical flow pattern for condensation in microchannels. The results will do help for the understanding of the mechanisms of annular condensation in microchannels. 2. One-dimensional model 2.1. Momentum equations Peles and Haber [8] proposed a one-dimensional model of boiling two-phase heat transfer in triangular microchannels. Here, we would like to investigate the opposite process, condensation flow in triangular microchannels. As suggested by Peterson and Ma [9], it was assumed that the condensate was concentrated at the corners of the channel because of the surface tension, and the liquid film
806
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
2.2. Expressions of flow characteristics in vapor regime
Nomenclature A b c D hfg L l m p q R Re S v x X̄
In the vapor momentum equation, Eq. (1), the term on the left side of the equation stands for the momentum variety of the vapor control volume, while the other two terms at the right side stand for the pressure drop and the influence of the shear stress of the vapor respectively. Under the constant heat flux boundary condition, the vapor velocity in the vapor phase is:
cross-section area perimeter of heat exchange cross-section coefficient hydraulic diameter latent heat of vaporization approximate condensation length length at the end of condensation mass flow pressure heat flux radius of curvature Reynolds number cross-section perimeter velocity coordinate nondimensional parameter
vv ¼
mv qbðL−xÞ ¼ ρv Av ρv Av hfg
ð3Þ
where mv is the vapor mass flux, q is the heat flux, b is the heat transfer cross-section perimeter of a single channel, and hfg is the latent heat. And the pressure in the vapor phase is presented by: "
# q2 b2 qbηv − ðf ReÞv x2 −2Lx : pv ¼ pv0 − ρv A2v h2fg ρv D2v Av hfg
ð4Þ
pv0 is the vapor pressure at the inlet, ζv is the shear stress factor, ηv is the viscosity of the vapor, Dv is the vapor hydraulic diameter, and (ζRe)v depends on the vapor cross-section shape.
Greek symbols α coefficient, constant β half included angle η viscosity θ contact angle ρ density σ surface tension coefficient τ friction, shear stress ζ friction factor, shear stress factor (ζRe) Poiseuille number
2.3. Expressions of flow characteristics in liquid regime In the liquid momentum equation, Eq. (2), i represents the corner number, each corner of the triangle should be calculated separately, except for the case of an equilateral triangle, in which the condensation streams in each corner are the same. As shown in Fig. 1(b), the liquid momentum equation, Eq. (2), describes the balance of the pressure variation in the liquid, shear stress of the vapor, friction of the wall, momentum transfer by phase change and the change in liquid momentum. Since the surface tension coefficient is assumed to be constant and the pressure drop in vapor is much smaller than the pressure drop
Subscripts i number l liquid v vapor w wall 0 inlet
between the corners was too thin to be considered in the momentum equation. The approximate distribution of the vapor and liquid phase in the triangular microchannels is shown in Fig. 1(a). The vapor phase flows in the core of the channel, while the condensate stream is flowing in the corners. The liquid cross-section area would increase along the channel. Condensation also occurs in l section, but it can be negligible, because l is much shorter than L. All the physical parameters in vapor and liquid phase are assumed to be constant, and the influences of gravity and buoyancy are both neglected. The momentum equations are: Vapor regime: d ρv v2v Av ¼ −Av dpv −Sv τv dx:
ð1Þ
Liquid regime: −Ali dpli þ τvl dAvli −τlwi dAlwi þ vv dmli ¼ dðmli vli Þ
ð2Þ
where ρv, vv, pv, Av, Sv and τv represent for the density, velocity, pressure, cross-section area, cross-section perimeter and shear stress at the vapor boundary in the vapor phase respectively, and vli, pli, mli, Ali, Avli, Alwi τvl and τlwi represent the velocity, pressure, mass flux, cross-section area, liquid-vapor interface area, liquid-wall interface area, shear stress at liquid-vapor interface and friction at liquid-wall interface in the ith corner, respectively.
Fig. 1. Schematic of condensation in triangular microchannel and liquid control volume: (a) distribution of the vapor and liquid, (b) liquid control volume.
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
The nondimensional form of Eq. (6) is:
Table 1 Parameters of typical conditions Contact angle θ Hydraulic diameter D Heat flux q Inlet pressure pv0
Parameter
° Condition 1 15 Condition 2 15 Condition 3 15
μm
W/cm2
kPa
100 200 450
25 15 5
200 300 200
4 3 dR i α 2 L−x R i þ α 5 L−x R i −α 4 xR i −α 3 xR i ¼ 3 dx R −α x 2 i
expected in the liquid domain, the Young–Laplace equation can be stated as: σ σ dpli ¼ d pv − ¼ 2 dRi Ri Ri
ð5Þ
α 2 D2 α3 α4 α5 D ; α4 ¼ ; α5 ¼ ; α3 ¼ : α1 α1 α1 D α1 D
p li ¼
pli v vv ; v ¼ li ; v v ¼ : pv0 li vv0 vv0
dRi α 2 ðL−xÞR4i þ α 5 ðL−xÞR3i −α 4 xRi −α 3 xRi ¼ dx α 1 R3i −α 4 x2
x ¼ L; R i ¼ R iL ¼ b sinβi 6 cosðθ þ βi ÞD:
ðfReÞv ηv qb π −θ−βi α2 ¼ ρv Dv Av hfg 2
α4 ¼
α5 ¼
2ðfReÞl ci qbηl 8 h 92 i cos Þ cosθ > > <2 sinðββi−þθ = πþβ þθ sinβ i i 2 i ρl hfg ðθ þ β i Þ cos > > : ;
ρl h2fg
2c2 q2 b2 h i
cosðβi þθÞ cosθ
i
σ pv0 D
ð7Þ ð8Þ
ð9Þ
Three typical conditions for the simulation are shown in Table 1. Eq. (14) is solved by Runge–Kutta method numerically. Fig. 2 shows the distribution of the dimensionless curvature radius, Rī , along the x̄axis. As shown in this figure, the trends of the curvature radius distribution are nearly the same for all three conditions: at the beginning and the end of the condensation, the curvature radius would increase rapidly along the channel, while it increases more slowly in the middle. And the middle part with a smooth radius distribution takes the majority of the condensation length.
ð10Þ
ð11Þ
2.4. Nondimensionalization and boundary conditions Equilateral triangle microchannels and isosceles triangle microchannels approaching to equilateral (for example, microchannels with slope of 54.7° etched on 〈100〉 silicon wafer by KOH) are widely used, hence, the case for equilateral triangle will be simulated here. All of the parameters at the equations are nondimensionalized. Let rffiffiffi 3 b 9
ð12Þ
as the hydraulic diameter of the channel. And Ri ¼
Ri x L ;x ¼ ;L ¼ : D D D
ð18Þ
Condensation characteristics
In Eqs. (7)–(11), θ is the contact angle, βi is half of the included angle, ζl, ρl and ηl are the friction factor, the density of the liquid phase and the liquid viscosity respectively, Dli is the liquid hydraulic diameter, (ζRe)l depends on the liquid cross-section shape, and ci = 1/3 for equilateral triangle.
4A D¼ ¼ b
ð17Þ
3. Simulation results and discussion
sin βi −π2þβ i þθ
ci q2 b2 : ρv Av h2fg
ð16Þ
For an equilateral triangle channel, the boundary conditions are given by: x ¼ 0; R i ¼ R i0 ¼
ð6Þ
ð15Þ
Defining
where σ is the surface tension coefficient of the vapor liquid interface. A differential equation about the curvature radius of the liquid phase Ri to the x-axis can be acquired from Eqs. (2) and (5):
where cosðβi þ θÞcosθ π − þ βi þ θ σ α1 ¼ sinβi 2
ð14Þ
4
where
α2 ¼
α3 ¼
807
ð13Þ
Fig. 2. Distribution of dimensionless curvature radius R¯ī.
808
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
Fig. 3. Distribution of dimensionless liquid pressure p¯lī.
Fig. 6. Condensation length L versus the hydraulic diameter D. (pv0 = 200 kPa, q = 20 W/ cm2).
Fig. 3 illustrates the distribution of the dimensionless pressure along the axis. As shown in Fig. 3, the liquid pressure in the condensate stream will increase with the increase of the curvature radius. At the inlet of the channel, the liquid pressure will jump from 0 drastically, and then it will increase slowly along the channel, until the maximum value is reached. This distribution is influenced by the hydraulic diameter of the channel. In condition 1, which has the smallest hydraulic diameter, the distribution along the axis has no obvious inflexion. The hydraulic diameter in condition 2 is larger than in condition 1, an obvious inflexion occurs, the increase of the dimensionless pressure along the channel before this inflexion is more drastically than what after this inflexion.
and D = 250 μm. As shown in the figure, the condensation length will increase with the increase of the inlet vapor pressure. Fig. 5 shows the plot of the dimensionless condensation length, L ̄, as a function of the heat flux q, where pv0 = 200 kPa and D = 250 μm. As shown, L ̄ will decrease with the increase of the heat flux. Fig. 6 illustrates the condensation length, L, as a function of the hydraulic diameter, D, where pv0 = 200 kPa and q = 20 W/cm2. The condensation length, L, will be longer in larger channels. As shown in Figs. 4–6, the condensation length will be longer when the contact angle is smaller for the same condition. This trend is the same as what occurs in boiling processes [8].
Condensation length
4. Discussion
The condensation length in triangular microchannels depends on the contact angle, the hydraulic diameter of channel, the heat flux and the inlet steam pressure. In the following, the influences of the above parameters on the condensation length are presented. Fig. 4 represents a plot of the dimensionless condensation length, L ̄, as a function of the inlet vapor pressure pv0, where q = 20 W/cm2
A steady one-dimensional model is developed based on the momentum and energy equations of the condensation process in triangular microchannels. The distribution of the curvature radius of the condensate stream and the variety of pressure in the liquid phase along the channel are acquired, as well as the influence of the parameters on the condensation length. This model can predict some characteristics of condensation in triangular microchannels, and will provide help in understanding the mechanisms of this process. However, it has been proved by the recent experiments that the annular flow is just one of the flow patterns in condensation process in triangular microchannels [4,5]. Other flow patterns such as injection flow or slug-bubble flow, which cannot be described by the steady model, would possibly appear. Therefore, it is necessary to develop more accurate models to describe the other flow patterns and find the criterion in the pattern alternation. 5. Conclusions
Fig. 4. Dimensionless condensation length L̄ versus the inlet vapor pressure pv0. (q = 20 W/cm2, D = 250 μm).
Fig. 5. Dimensionless condensation length L̄ versus the heat flux q. (pv0 = 200 kPa, D = 250 μm).
A steady one-dimensional model for annular condensation flow has been developed and the curvature radius distribution of the condensate stream along the channel has been calculated. The increase of the curvature radius at the channel inlet and end of the condensation flow is more interesting than the middle region, where there is a smooth distribution of the curvature radius along a majority of the condensation length. The distribution of the pressure drop in the liquid has been obtained. The liquid pressure, which is influenced by the channel size and the heat flux, increases with the increase of curvature radius along the channel. At the inlet of the channel, liquid pressure will jump from 0 drastically, and then the rate of increase slows along the channel, until a maximum value is reached. The condensation length in triangular microchannels depends on the contact angle, the channel hydraulic diameter, the heat flux and the inlet steam pressure. A higher steam pressure, hydraulic diameter, a longer condensation length will perform, as the mass flux increases. And the condensation length will decrease when the heat flux increases or the contact angle is larger.
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
Acknowledgements The authors gratefully acknowledge the support provided by Fok Ying Tung Young Teacher Education Foundation No.101055, Outstanding Young Teacher Foundation at Southeast University, Outstanding Young Academic Leader Foundation at Jiangsu Province and the NASA-Glenn Research Center. References [1] S. Garimella, Condensation flow mechanisms in microchannels: basis for pressure drop and heat transfer models, Heat Transfer Engineering 25 (3) (2004) 104–116. [2] B. Médéric, M. Miscevic, V. Platel, Experimental study of flow characteristics during condensation in narrow channels: the influence of the diameter channel on structure patterns, Superlattices and Microstructures 35 (3–6) (2004) 573–586.
809
[3] Y.P. Chen, P. Cheng, Condensation of steam in a silicon microchannel, International Communications in Heat Mass Transfer 32 (1–2) (2005) 175–183. [4] H.Y. Wu, P. Cheng, Condensation flow patterns in silicon microchannels, International Journal of Heat and Mass Transfer 48 (11) (2005) 2186–2197. [5] Y.P. Chen, J. Li, G.P. Peterson, Influence of Hydraulic Diameter on Flow Condensation in Silicon Microchannels, 13th International Heat Transfer Conference, Australia, Sydney, 2006. [6] T.S. Zhao, Q. Liao, Theoretical analysis of film condensation heat transfer inside vertical mini triangular channels, International Journal of Heat and Mass Transfer 45 (13) (2002) 2829–2842. [7] X.Z. Du, T.S. Zhao, Analysis of film condensation heat transfer inside a vertical micro tube with consideration of the meniscus draining effect, International Journal of Heat and Mass Transfer 46 (24) (2003) 4669–4679. [8] Y.P. Peles, S. Haber, A steady state, one dimensional, model for boiling two phase flow in triangular micro-channel, International Journal of Multiphase Flow 26 (7) (2000) 1095–1115. [9] G.P. Peterson, H.B. Ma, Temperature response of heat transport in a micro heat pipe, Journal of Heat Transfer 121 (2) (1999) 438–445.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Numerical simulation for steady annular condensation flow in triangular microchannels☆ Yongping Chen a,⁎, Jiafeng Wu a, Mingheng Shi a, G.P. Peterson b a b
School of Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, PR China Office of the Chancellor, University of Colorado at Boulder, Regent Administrative Center — 17 UCB, 2 Boulder, CO 80309-001, USA
A R T I C L E
I N F O
Available online 24 April 2008 Keywords: Condensation Microchannels Numerical simulation
A B S T R A C T A steady one-dimensional model for annular condensation flow in triangular microchannels is developed. The curvature radius distribution of the condensate stream along the channel has been determined numerically. The results indicate that the curvature radius of the liquid phase would increase rapidly at the beginning, and then as the condensation process progresses along the length of the microchannels, the radius increase would proceed more slowly. At the end of the condensation flow, the radius increases rapidly again. A smaller contact angle and heat flux or a larger hydraulic diameter and steam pressure will all result in a longer condensation length. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Condensation flow in microchannels has been widely used in a variety of advanced microthermal devices, such as microheat pipe and microfuel cell. Although there have been a number of investigations on boiling flow in microchannels, there are relatively a little experimental data and theoretical analysis of this condensation process available in the literature, making it difficult to understand the mechanisms of condensation in microchannels. Garimella [1] presented an overview of the visualization study of the condensation of refrigerants in mini-channels. Experiments for condensation in round, square and rectangular tubes with hydraulic diameters ranging from 1–5 mm were reported and twophase flow regimes and patterns in mini-channels were presented. It was noted that the surface tension, not gravity, played the dominant role in microchannels having a hydraulic diameter less than 1 mm. Médéric et al. [2] also provided sufficient evidence that the capillary force was dominant in channels having a diameter of less than 1 mm. Chen and Cheng [3] carried out a visualization study on condensation of steam in a microchannel with a hydraulic diameter of 75 μm, and found that droplet flow existed upstream and intermittent flow existed downstream of the microchannel. Subsequently, Wu and Cheng [4] performed an experiment on condensation of steam in parallel microchannels having a hydraulic diameter
☆ Communicated by P. Cheng and W.Q. Tao. ⁎ Corresponding author. Tel.: +86 25 8379 4526(8118). E-mail address: [email protected] (Y. Chen). 0735-1933/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2008.03.001
of 83 μm, and found that droplet flow, annular flow and injection flow existed in the microchannels. Similarly, Chen et al. [5] conducted an investigation of condensation in triangular silicon microchannels with diameters less than 250 μm. The results indicated that the channel scale, heat flux, and mass flux all have significant influence on the condensation flow patterns, and the droplet flow, annular flow and injection flow etc. were all observed. These were then combined into a condensation flow map for microchannels. Of particular interest here was the new observation of injection flow at a constant frequency, which has received increasing attention in the past several years. While there have been a number of experimental studies, there is little analytical work that has been done on condensation in microchannels [6,7]. In the current work, a steady one-dimensional model of triangular microchannels with constant heat flux was developed. It was used to describe the characteristics of annular condensation flow, which is a very typical flow pattern for condensation in microchannels. The results will do help for the understanding of the mechanisms of annular condensation in microchannels. 2. One-dimensional model 2.1. Momentum equations Peles and Haber [8] proposed a one-dimensional model of boiling two-phase heat transfer in triangular microchannels. Here, we would like to investigate the opposite process, condensation flow in triangular microchannels. As suggested by Peterson and Ma [9], it was assumed that the condensate was concentrated at the corners of the channel because of the surface tension, and the liquid film
806
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
2.2. Expressions of flow characteristics in vapor regime
Nomenclature A b c D hfg L l m p q R Re S v x X̄
In the vapor momentum equation, Eq. (1), the term on the left side of the equation stands for the momentum variety of the vapor control volume, while the other two terms at the right side stand for the pressure drop and the influence of the shear stress of the vapor respectively. Under the constant heat flux boundary condition, the vapor velocity in the vapor phase is:
cross-section area perimeter of heat exchange cross-section coefficient hydraulic diameter latent heat of vaporization approximate condensation length length at the end of condensation mass flow pressure heat flux radius of curvature Reynolds number cross-section perimeter velocity coordinate nondimensional parameter
vv ¼
mv qbðL−xÞ ¼ ρv Av ρv Av hfg
ð3Þ
where mv is the vapor mass flux, q is the heat flux, b is the heat transfer cross-section perimeter of a single channel, and hfg is the latent heat. And the pressure in the vapor phase is presented by: "
# q2 b2 qbηv − ðf ReÞv x2 −2Lx : pv ¼ pv0 − ρv A2v h2fg ρv D2v Av hfg
ð4Þ
pv0 is the vapor pressure at the inlet, ζv is the shear stress factor, ηv is the viscosity of the vapor, Dv is the vapor hydraulic diameter, and (ζRe)v depends on the vapor cross-section shape.
Greek symbols α coefficient, constant β half included angle η viscosity θ contact angle ρ density σ surface tension coefficient τ friction, shear stress ζ friction factor, shear stress factor (ζRe) Poiseuille number
2.3. Expressions of flow characteristics in liquid regime In the liquid momentum equation, Eq. (2), i represents the corner number, each corner of the triangle should be calculated separately, except for the case of an equilateral triangle, in which the condensation streams in each corner are the same. As shown in Fig. 1(b), the liquid momentum equation, Eq. (2), describes the balance of the pressure variation in the liquid, shear stress of the vapor, friction of the wall, momentum transfer by phase change and the change in liquid momentum. Since the surface tension coefficient is assumed to be constant and the pressure drop in vapor is much smaller than the pressure drop
Subscripts i number l liquid v vapor w wall 0 inlet
between the corners was too thin to be considered in the momentum equation. The approximate distribution of the vapor and liquid phase in the triangular microchannels is shown in Fig. 1(a). The vapor phase flows in the core of the channel, while the condensate stream is flowing in the corners. The liquid cross-section area would increase along the channel. Condensation also occurs in l section, but it can be negligible, because l is much shorter than L. All the physical parameters in vapor and liquid phase are assumed to be constant, and the influences of gravity and buoyancy are both neglected. The momentum equations are: Vapor regime: d ρv v2v Av ¼ −Av dpv −Sv τv dx:
ð1Þ
Liquid regime: −Ali dpli þ τvl dAvli −τlwi dAlwi þ vv dmli ¼ dðmli vli Þ
ð2Þ
where ρv, vv, pv, Av, Sv and τv represent for the density, velocity, pressure, cross-section area, cross-section perimeter and shear stress at the vapor boundary in the vapor phase respectively, and vli, pli, mli, Ali, Avli, Alwi τvl and τlwi represent the velocity, pressure, mass flux, cross-section area, liquid-vapor interface area, liquid-wall interface area, shear stress at liquid-vapor interface and friction at liquid-wall interface in the ith corner, respectively.
Fig. 1. Schematic of condensation in triangular microchannel and liquid control volume: (a) distribution of the vapor and liquid, (b) liquid control volume.
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
The nondimensional form of Eq. (6) is:
Table 1 Parameters of typical conditions Contact angle θ Hydraulic diameter D Heat flux q Inlet pressure pv0
Parameter
° Condition 1 15 Condition 2 15 Condition 3 15
μm
W/cm2
kPa
100 200 450
25 15 5
200 300 200
4 3 dR i α 2 L−x R i þ α 5 L−x R i −α 4 xR i −α 3 xR i ¼ 3 dx R −α x 2 i
expected in the liquid domain, the Young–Laplace equation can be stated as: σ σ dpli ¼ d pv − ¼ 2 dRi Ri Ri
ð5Þ
α 2 D2 α3 α4 α5 D ; α4 ¼ ; α5 ¼ ; α3 ¼ : α1 α1 α1 D α1 D
p li ¼
pli v vv ; v ¼ li ; v v ¼ : pv0 li vv0 vv0
dRi α 2 ðL−xÞR4i þ α 5 ðL−xÞR3i −α 4 xRi −α 3 xRi ¼ dx α 1 R3i −α 4 x2
x ¼ L; R i ¼ R iL ¼ b sinβi 6 cosðθ þ βi ÞD:
ðfReÞv ηv qb π −θ−βi α2 ¼ ρv Dv Av hfg 2
α4 ¼
α5 ¼
2ðfReÞl ci qbηl 8 h 92 i cos Þ cosθ > > <2 sinðββi−þθ = πþβ þθ sinβ i i 2 i ρl hfg ðθ þ β i Þ cos > > : ;
ρl h2fg
2c2 q2 b2 h i
cosðβi þθÞ cosθ
i
σ pv0 D
ð7Þ ð8Þ
ð9Þ
Three typical conditions for the simulation are shown in Table 1. Eq. (14) is solved by Runge–Kutta method numerically. Fig. 2 shows the distribution of the dimensionless curvature radius, Rī , along the x̄axis. As shown in this figure, the trends of the curvature radius distribution are nearly the same for all three conditions: at the beginning and the end of the condensation, the curvature radius would increase rapidly along the channel, while it increases more slowly in the middle. And the middle part with a smooth radius distribution takes the majority of the condensation length.
ð10Þ
ð11Þ
2.4. Nondimensionalization and boundary conditions Equilateral triangle microchannels and isosceles triangle microchannels approaching to equilateral (for example, microchannels with slope of 54.7° etched on 〈100〉 silicon wafer by KOH) are widely used, hence, the case for equilateral triangle will be simulated here. All of the parameters at the equations are nondimensionalized. Let rffiffiffi 3 b 9
ð12Þ
as the hydraulic diameter of the channel. And Ri ¼
Ri x L ;x ¼ ;L ¼ : D D D
ð18Þ
Condensation characteristics
In Eqs. (7)–(11), θ is the contact angle, βi is half of the included angle, ζl, ρl and ηl are the friction factor, the density of the liquid phase and the liquid viscosity respectively, Dli is the liquid hydraulic diameter, (ζRe)l depends on the liquid cross-section shape, and ci = 1/3 for equilateral triangle.
4A D¼ ¼ b
ð17Þ
3. Simulation results and discussion
sin βi −π2þβ i þθ
ci q2 b2 : ρv Av h2fg
ð16Þ
For an equilateral triangle channel, the boundary conditions are given by: x ¼ 0; R i ¼ R i0 ¼
ð6Þ
ð15Þ
Defining
where σ is the surface tension coefficient of the vapor liquid interface. A differential equation about the curvature radius of the liquid phase Ri to the x-axis can be acquired from Eqs. (2) and (5):
where cosðβi þ θÞcosθ π − þ βi þ θ σ α1 ¼ sinβi 2
ð14Þ
4
where
α2 ¼
α3 ¼
807
ð13Þ
Fig. 2. Distribution of dimensionless curvature radius R¯ī.
808
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
Fig. 3. Distribution of dimensionless liquid pressure p¯lī.
Fig. 6. Condensation length L versus the hydraulic diameter D. (pv0 = 200 kPa, q = 20 W/ cm2).
Fig. 3 illustrates the distribution of the dimensionless pressure along the axis. As shown in Fig. 3, the liquid pressure in the condensate stream will increase with the increase of the curvature radius. At the inlet of the channel, the liquid pressure will jump from 0 drastically, and then it will increase slowly along the channel, until the maximum value is reached. This distribution is influenced by the hydraulic diameter of the channel. In condition 1, which has the smallest hydraulic diameter, the distribution along the axis has no obvious inflexion. The hydraulic diameter in condition 2 is larger than in condition 1, an obvious inflexion occurs, the increase of the dimensionless pressure along the channel before this inflexion is more drastically than what after this inflexion.
and D = 250 μm. As shown in the figure, the condensation length will increase with the increase of the inlet vapor pressure. Fig. 5 shows the plot of the dimensionless condensation length, L ̄, as a function of the heat flux q, where pv0 = 200 kPa and D = 250 μm. As shown, L ̄ will decrease with the increase of the heat flux. Fig. 6 illustrates the condensation length, L, as a function of the hydraulic diameter, D, where pv0 = 200 kPa and q = 20 W/cm2. The condensation length, L, will be longer in larger channels. As shown in Figs. 4–6, the condensation length will be longer when the contact angle is smaller for the same condition. This trend is the same as what occurs in boiling processes [8].
Condensation length
4. Discussion
The condensation length in triangular microchannels depends on the contact angle, the hydraulic diameter of channel, the heat flux and the inlet steam pressure. In the following, the influences of the above parameters on the condensation length are presented. Fig. 4 represents a plot of the dimensionless condensation length, L ̄, as a function of the inlet vapor pressure pv0, where q = 20 W/cm2
A steady one-dimensional model is developed based on the momentum and energy equations of the condensation process in triangular microchannels. The distribution of the curvature radius of the condensate stream and the variety of pressure in the liquid phase along the channel are acquired, as well as the influence of the parameters on the condensation length. This model can predict some characteristics of condensation in triangular microchannels, and will provide help in understanding the mechanisms of this process. However, it has been proved by the recent experiments that the annular flow is just one of the flow patterns in condensation process in triangular microchannels [4,5]. Other flow patterns such as injection flow or slug-bubble flow, which cannot be described by the steady model, would possibly appear. Therefore, it is necessary to develop more accurate models to describe the other flow patterns and find the criterion in the pattern alternation. 5. Conclusions
Fig. 4. Dimensionless condensation length L̄ versus the inlet vapor pressure pv0. (q = 20 W/cm2, D = 250 μm).
Fig. 5. Dimensionless condensation length L̄ versus the heat flux q. (pv0 = 200 kPa, D = 250 μm).
A steady one-dimensional model for annular condensation flow has been developed and the curvature radius distribution of the condensate stream along the channel has been calculated. The increase of the curvature radius at the channel inlet and end of the condensation flow is more interesting than the middle region, where there is a smooth distribution of the curvature radius along a majority of the condensation length. The distribution of the pressure drop in the liquid has been obtained. The liquid pressure, which is influenced by the channel size and the heat flux, increases with the increase of curvature radius along the channel. At the inlet of the channel, liquid pressure will jump from 0 drastically, and then the rate of increase slows along the channel, until a maximum value is reached. The condensation length in triangular microchannels depends on the contact angle, the channel hydraulic diameter, the heat flux and the inlet steam pressure. A higher steam pressure, hydraulic diameter, a longer condensation length will perform, as the mass flux increases. And the condensation length will decrease when the heat flux increases or the contact angle is larger.
Y. Chen et al. / International Communications in Heat and Mass Transfer 35 (2008) 805–809
Acknowledgements The authors gratefully acknowledge the support provided by Fok Ying Tung Young Teacher Education Foundation No.101055, Outstanding Young Teacher Foundation at Southeast University, Outstanding Young Academic Leader Foundation at Jiangsu Province and the NASA-Glenn Research Center. References [1] S. Garimella, Condensation flow mechanisms in microchannels: basis for pressure drop and heat transfer models, Heat Transfer Engineering 25 (3) (2004) 104–116. [2] B. Médéric, M. Miscevic, V. Platel, Experimental study of flow characteristics during condensation in narrow channels: the influence of the diameter channel on structure patterns, Superlattices and Microstructures 35 (3–6) (2004) 573–586.
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