Condensation - Nano and Microscale Thermophysical Engineering

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Condensation in MicrochannelsYongping Chen

a, Mingheng Shi

a, Ping Cheng

b& G. P. Peterson

c

aDepartment of Energy and Thermal Science, School of Energy and

Environment, Southeast University, Nanjing, Jiangsu, P.R. China

b School of Mechanical and Power Engineering, Shanghai Jiaotong

University, Shanghai, P.R. Chinac

Department of Mechanical, Aerospace and Nuclear Engineering,

Rensselaer Polytechnic Institute, Troy, New York, USA

Published online: 10 Jul 2008.

To cite this article: Yongping Chen , Mingheng Shi , Ping Cheng & G. P. Peterson (2008) Condensation

in Microchannels, Nanoscale and Microscale Thermophysical Engineering, 12:2, 117-143, DOI:

10.1080/15567260701866702

To link to this article: http://dx.doi.org/10.1080/15567260701866702

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CONDENSATION IN MICROCHANNELS

Yongping Chen,1 Mingheng Shi,1 Ping Cheng2, and G.P. Peterson3

1Department of Energy and Thermal Science, School of Energy and Environment,

Southeast University, Nanjing, Jiangsu, P.R. China2School of Mechanical and Power Engineering, Shanghai Jiaotong University,

Shanghai, P.R. China3Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer

Polytechnic Institute, Troy, New York, USA

Condensation in microchannels has applications in a wide variety of advanced microthermal

devices. Presented here is a review of both experimental and theoretical analyses of con-

densation in these microchannels, with special attention given to the effects of channel

diameter and surface conditions on the flow regimes of condensing flows occurring in these

channels. This review suggests that surface tension, rather than body or buoyancy forces, is

the dominant force that governs the condensation and two-phase flow in these microchannels.

Recent experimental results indicate that with decreases in the channel diameter, the domi-

nant condensing flow pattern is intermittent injection/slug/bubble flow, as opposed to strati-

fied or annular flow, which is typically found in two-phase flows in larger one-g channel flows.

As a result, existing annular flow condensation models cannot be used to accuratelyrepresent

or predict the actual physical mechanisms that occur in these condensing flows in micro-

channels. This therefore necessitates the use of semitheoretical models or correlations basedupon experimental data. Since wettability and surface roughness play an important role in

the condensing flow in microchannels, an optimization of these effects may provide a

mechanism by which very high condensation heat fluxes can be achieved.

KEY WORDS: condensation, heat transfer, microchannel, capillary

INTRODUCTION

In addition to applications in specific devices, such as micro heat pipes, micro

fuel cells, and microthermal control systems for spacecraft, two-phase flow in micro-

channels is of importance in a wide variety of applications in the chemical processing,pharmaceutical, and biomedical fields. In micro heat pipes, the vaporization and

condensation cycle results in a high effective thermal conductivity and a high degree

of temperature uniformity, making these devices especially applicable to the

Nanoscale and Microscale Thermophysical Engineering, 12: 117143, 2008

Copyright Taylor & Francis Group, LLC

ISSN: 1556-7265 print / 1556-7273 online

DOI: 10.1080/15567260701866702

Address correspondence to Yongping Chen, Department of Energy and Thermal Science, School of

Energy and Environment, Southeast University, Nanjing, Jiangsu 210096, P.R. China. E-mail:

[email protected]

The authors gratefully acknowledge the support provided by the NASA, the Key Project of the

Chinese Ministry of Education No. 105082, Fok Ying Tung Young Teacher Education Foundation

No.101055, and Outstanding Young Teacher Foundation at Southeast University. The partial support of

this work by Natural National Science Foundation of China through grant No. 50536010 is also gratefully

acknowledged.

117


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microelectronic cooling and biomedical fields [15]. In fuel cell applications, mini- or

micro-proton exchange membrane (PEM) fuel cells have been developed to provide a

portable high-power density source of energy [68]. In both of these applications,

system optimization requires an understanding of two-phase flows in general and

condensation in microchannels in particular. Because of the relative impact of surface

tension, this is of particular importance for applications in microgravity

environments.

Although there have been a number of theoretical analyses on condensingflow in microgravity environments, there is relatively little experimental data

available, making it difficult to determine the validity of these models. Because

of the relatively small effect of gravity in mini- and microchannels in one-g

conditions, it may be possible to utilize ground-based experimental data for

condensing flows to reliably predict the behavior in a microgravity environment,

provided the characteristic diameters are sufficiently small. By carefully consid-

ering the resulting forces and their relative magnitudes, condensation in micro-

channels in one-g environments can reasonably simulate an equivalent system

in microgravity environments. This approach has been utilized previously in the

determination of two-phase flow patterns occurring in horizontal capillary tubes

[9]. In addition, the thermocapillary effects have been found to be very sensitive

NOMENCLATURE

A areaa,a0 factorb width of the heat transfer surfacec,c0 factorcp specific heat at constant pressurecv specific heat at constant volumed, D diameter

f Darcy friction factorg gravity accelerationhfg latent heatK1 2/3K2 0.5L lengthm mass fluxNu Nusselt numbern, n0 factorPr Prandtl number

p pressureq heat transfer rater, R radiusR0 inner radius of the channelRe Reynolds numberRg ideal gas constantSLW wet wall lengthSV perimeter of vapor regimeT temperatureU velocity

v velocityw velocity

Xtt Martinelli parameter

Greek Letters void fractionx heat transfer coefficient0 accommodation factor specific heat ratio ( cp / cv) liquid film thicknesse roughness thermal conductivity dynamic viscosity specific volume friction factor density surface tension shear stress inclination angle

SubscriptsG gasl, L liquidsat saturatev,V vaporW wall

118 Y. CHEN ET AL.


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for highly wetting fluids associated with phase change in low-gravity environ-

ments [10, 11].

For two-phase flow in microchannels, it has been suggested that the dominant

force is surface tension [5]. As a result, it is reasonable to expect that the flow regimes

and heat transfer coefficients for condensation in microchannels may very well bedifferent from what has been observed in macrochannels. For example, the instabil-

ities associated with condensing flow will be more dramatic as the channel diameter

is decreased. Unlike single-phase flow or flow boiling in microchannels, which have

been extensively studied [1217], investigations on the fundamental phenomena of

condensation in microchannels is rather limited [5, 18]. Although the present review

is focused on flow patterns, pressure drop, and heat transfer in condensing flow in

microchannels, the related problem of condensation in minichannels is also

discussed.

FLOW AND HEAT TRANSFER EXPERIMENTS ON CONDENSATIONIN MICROCHANNELS

It is well known that the pressure drop and heat transfer for condensing flow in

channels are strongly dependent upon the liquid/vapor flow patterns. These flow

patterns are typically described using some sort of two-phase flow map to describe

various types of flow and the transition regions between each type of flow. Suo and

Griffith [19] were among the first to study adiabatic two-phase flow in horizontal

capillary tubes and observed long bubble flow connected by smaller liquid slugs. This

resulted in a correlation that relates the density and thickness of the liquid film around

bubbles under different flow conditions.

Early experimental investigations of two-phase flow under microgravity condi-tions were conducted in drop towers and Learjet trajectories. For example, Dukler

et al. [20] investigated gas-liquid flow in tubes with diameters of 9.52 and 12.7 mm

under microgravity conditions and observed slug, annular, and bubbly flows in these

tubes. However, the stratified wavy flow normally occurring in macrochannels under

one-g environments was noticeably absent. Based upon the experimental data, a map

of the flow patterns for two-phase flow under microgravity was developed. However,

since the diameters of the tubes used in the tests were similar, the influence of the tube

diameter on the flow regime was not presented. Velocity models were also developed

to explain the flow map in the absence of gravity. Neglecting the local relative velocity

between the liquid and gas, in bubble to slug pattern, the relation ofUL, UG, which are

the superficial velocities of liquid and gas phase, respectively, was expressed as:

UL

UG 1

1

where is the area average void fraction, which is 0.52 for small bubbles in a cubicarray and 0.5 for large spherical bubbles or, as determined by Duckler [21], generally

equal to 0.45. It therefore follows that

UL 1:22UG 2

In slug to annular flow pattern, the relation ofUL, UG is

CONDENSATION IN MICROCHANNELS 119


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UG

UL UG c0 3

where c0 ranges between 1.15 and 1.30, depending on the flow rates of the phases.

Garimella [22] presented an overview of a visualization study of the condensa-

tion of refrigerants in minichannels. Experiments for condensation in round, square,

and rectangular tubes with hydraulic diameters in the range of 15 mm and for vapor

quality 0, x, 1 were reported. Two-phase flow regimes and patterns in minichannels

are presented in Table 1. As shown from the table, annular flow occurs when the vapor

flows in the center of the channel with a few liquid droplets, and the liquid flows

around the vapor core along the tube. At relatively low vapor velocities, the gravita-

tional body force causes the liquid to flow along the bottom of the tube, while the

vapor flow occurs on the upper part of the tube. The vapor-liquid interface is often a

wavy film; hence, this regime is referred to as wavy flow. However, the wavy flow of

water and air under adiabatic conditions is different from that of condensing flow,

since condensing flows are expected to have a coating of liquid around the whole

circumference of the tube but water-air flow is not. When stable bubbles move

axisymmetrically along the channel separated by clear liquid slugs or plugs, the flow

is referred to as intermittent flow. When the vapor bubbles are dispersed in the liquid, it

is called dispersed flow. The effects of hydraulic diameter on flow patterns are pre-sented in Figures 2 and 3.

Table 1 Condensation flow map [22]

120 Y. CHEN ET AL.


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Figure 1. Micro heat pipe operation [4].

Figure 2. Effect of hydraulic diameter on the intermittent flow regime [22].



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Figure 2 illustrates the transition of intermittent condensing flow in four

square channels. As illustrated, the size of the intermittent regime increases with

decreasing channel diameter. Figure 3 illustrates that with a decrease in the hydrau-

lic diameter, the wavy flow is increasingly replaced by the annular flow regime and

completely disappears in tubes or channels with hydraulic diameters of approxi-

mately 1 mm; i.e., Dh 1 mm. This phenomenon is similar to what happens inmicrogravity environments [20], which implies that surface tension, rather than the

gravitational body force, plays the dominant role in microchannels having hydraulic

diameters less than 1 mm. In the visualization study [20, 22], it was also found thatthe size of the intermittent flow regime in a round tube is larger than that in a square

tube at lower mass fluxes, but the sizes of these regimes are similar at high mass

fluxes. However, the influence of tube shape is much weaker than that of the

hydraulic diameter.

A visualization study of condensing flow patterns in three tubes with inner

diameters of 0.56, 1.1, and 10 mm was performed by Mederic et al. [23]. High-speed

photographs showed that annular and spherical bubbles and isolated spherical

collapsing bubbles appeared in both mini and microchannels. In the 10-mm-

diameter tube, the flow was strongly stratified, due to the dominant gravity. In the

1.1-mm-diameter tube, stratification still occurred, due to the competition of the

gravitational and capillary forces, and was manifested by the liquid film being

thicker at the bottom than at the top. This effect, however, was weaker when

compared to that observed in larger diameter tubes. In the 0.56-mm-diameter

tube, a significant difference in the annular region was observed. Here, stratification

disappeared and the liquid film inside the tube had the same thickness at the top and

at the bottom, with a circular vapor core in the middle of the tube. This observation

provided sufficient evidence that the capillary force was dominant in the channels

having diameters of less than 1 mm.

In a visualization study, Mederic et al. [24] presented new measurements of the

local void fraction to replace the traditional mean void fraction for condensing flow in

capillary cylindrical tube. Based upon the observed and measured film thickness,, thevoid fraction, , for an axisymmetric flow structure can be expressed as:

Figure 3. Effect of hydraulic diameter on the annular flow regime [22].

122 Y. CHEN ET AL.


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1 2realD

2 4

However, the uncertainty of the film thickness as determined from a single picture isvery large, up to 16 m. As a result, this method requires a large number of tests andimages to provide an average value of that can be relied upon with any level ofconfidence.

Chen and Cheng [25] conducted one of the first attempts to study condensation

in silicon microchannels with diameters less than 100 m. In this investigation, avisualization study of condensation in parallel microchannels having a hydraulic

diameter of 75 m was performed. Saturated steam at different pressures flowedthrough these microchannels and was condensed in the test section, which was cooled

by natural convection of air in the room. Images of the condensation process were

taken using a high-speed digital camera. In this visualization study, it was found that

droplet condensation occurred near the inlet of the microchannel, while intermittentflow of vapor and condensate was observed downstream.

On the basis of the investigations of Chen and Cheng [25], Wu and Cheng [26]

conducted a simultaneous visualization and measurement study on condensing flow

patterns of saturated steam flowing through an array of trapezoidal silicon microchan-

nels, having a hydraulic diameter of 82.8m. It was found that the mass flux has a greatinfluence on condensing flow patterns. With decreasing mass flux and steam pressure,

different condensation regimes such as fully droplet flow, droplet/annular/injection/slug-

bubbly flow, annular/injection/slug-bubbly flow, and fully slug-bubbly flow appeared in

the microchannels.

Figure 4 presents an image for the case of droplet/annular

/injection

/slug-bubblyflow at an inlet pressure of 2.15 105 Pa. Because of the occurrence of injection flow,

different alternating flow patterns appeared in the microchannels, causing fluctua-

tions in the wall temperature. The fluctuation periods of the wall temperatures and

pressures increased with decreasing mass flux.

The measurement of the heat transfer rate in these channels provides important

data. Garimella and Bandhauer [27] performed an experimental investigation of

condensing flows in microchannels and developed a technique for the measurement

of condensation heat transfer coefficients in microchannels. The inlet and outlet

qualities were measured through an energy balance. The test section was cooled by

high-velocity cooling water and the measured heat transfer coefficients for the con-

densation of refrigerant R134a in a square microchannel (having a hydraulic diameterof 0.76 mm) were found to be in the range of 2,110 to 10,640 W/m2-K.

Yan and Lin [28] performed an experimental investigation of heat transfer and

pressure drop for condensation of refrigerant R134a in a small, horizontal circular

tube, having a diameter of 2.0 mm. It was found that the condensation heat transfer

coefficient in this mini-diameter tube was about 10% higher than that in an 8-mm-

diameter tube. The pressure drop in these mini-tubes was found to increase with

increasing mass flux but decrease with increasing heat flux.

Cavallini et al. [29] measured the heat transfer coefficient for condensation of

R134a and R410a inside multiport minichannels having a hydraulic diameter of

1.4 mm. In this investigation, the experimental data were compared with predictionsbased on the models developed by Moser et al. [30], Zhang and Webb [31], and



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Cavallini et al. [32] and found that the heat transfer coefficient was underestimated by

all three of these models.

Kim et al. [33] experimentally investigated the condensation of R134a in a singleround tube with an inner diameter of 0.691 mm. It was found that the experimental

Droplet flow (0


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heat transfer data could not be accurately predicted by the existing correlations

obtained for condensation in macrochannels, and the discrepancies became more

obvious, especially at low mass fluxes.

Begg et al. [34] conducted both flow visualization experiments and measure-

ments of the heat transfer and pressure drop in horizontal tubes with diametersranging from 1.7 to 4 mm. In this investigation, it was observed that the average

heat transfer coefficient and pressure drop increased with decreasing tube diameter.

The heat transfer enhancement resulting from capillary forces in mini-tubes was

confirmed experimentally by Yang and Webb [35], who performed an experimental

investigation of R-12 condensation when subcooled in two flat extruded aluminum

tubes with small hydraulic diameters: one smooth type with a hydraulic diameter of

2.637 mm and the other with a hydraulic diameter of 1.564 mm and micro-fins. It was

proposed that surface tension force was effective in enhancing the condensation

coefficient for vapor quality greater than 0.5, and this enhancement was particularly

strong at low mass velocities.

THEORETICAL MODELS AND CORRELATIONS FOR CONDENSATION

IN MINI AND MICROCHANNELS

Analyses or correlation equations for condensation in macroscale channels has

been investigated extensively after the first in-depth analytical study performed by

Nusselt [36]. For example, Shah [37] presented a simple dimensionless correlation for

predicting heat transfer coefficients during film condensation inside pipes. This cor-

relation was verified by experimental data with the mean deviation of 15.4%.

However, because of the influence of surface tension, recent experimental investiga-

tions [29, 33] show that the data from condensation in microchannels cannot beaccurately predicted by the available correlations based on macrochannels.

Straub [38] presented a comprehensive review of the role of surface tension for

two-phase heat and mass transfer in the absence of gravity. It was pointed out that

when the buoyancy force was reduced, the transport processes were determined by the

properties at the interface alone. Condensation in microchannels is similar to con-

densation in a microgravity environment, where the contribution of the buoyancy or

body force is drastically reduced compared that of the surface tension.

Tabatabai and Faghri [39] proposed a surface tension force domain criterion for

condensation in micro-tubes. For a unit length of the flow regime, this value was given

as shown in Figure 5 as

Figure 5. Force balance elements for a given unit length of flow [39].



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Fsurface tension > Fshearj j Fbuoyancy 5

The surface tension force per unit length of the flow regime is given by

Fsurface tension

L 2PL

Pt 21 0:5 6

r2G

r2O7

where (PL/Pt) is a dimensionless ratio of liquid regime perimeter to total perimeter, isthe surface tension, and rG and rO are the radius of the vapor bubble and the tube,

respectively. Alternatively, the shear force per unit length of the flow can be expressed as

Fshear

L fLDO Lu

2L

48

wherefL is the friction factor of the liquid flow, DO is the diameter of the tube, uL is the

liquid velocity, and L is the density of the liquid.The buoyancy force per unit length of the flow regime is given by

Fbuoyancy

L gL GA 9

where g is the gravitational acceleration,G is the density of the gas, and A is the cross-sectional area of the flow.

Alternatively, Cheng and Wu [5] proposed the Bond number, Bo, as the criteria

to distinguish between microchannels, minichannels, and macrochannels. The Bond

number is defined as

Bo g v d2

10

which is a measure of the relative importance of the buoyancy forcetosurface

tension force. Based on the theoretical work of Li and Wang [40], Cheng and Wu [5]

distinguished between microchannels, minichannels, and macrochannels as follows:

1. Microchannels: If Bo, 0.05 where the gravity effect can be neglected;

2. Minichannels: If 0.05 , Bo , 3.0 where the surface tension effect becomes

dominant and the gravitational effect is small;

3. Macrochannels: If Bo. 3.0 where the surface tension is small in comparison with

the gravitational force.

If the conditions given by Eq. (5) or Eq. (10) are satisfied, the surface tension can

alter the conditions at which flow regime transitions occurs, which is different from thecondensation in macroscale situations where the surface tension force is negligible in

126 Y. CHEN ET AL.


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comparison to the buoyancy force. This is the reason why the existing macroscale

models, which have not taken into consideration surface tension, cannot be used to

predict the condensation heat transfer in microchannels. Thus, models for condensing

flow in microscale must take into consideration the surface tension effect.

A Steady One-Dimensional Model for Condensation in Microchannels

Peles and Haber [41] proposed a simple one-dimensional model of boiling two-

phase flow and heat transfer in a single, triangular microchannel based on the Young-

Laplace equation. In their model, the liquid flow inside the microchannels was

assumed to be driven by surface tension and shear forces. Similar to this boiling

model, a typical steady-state, one-dimensional model for annular condensation in

horizontal triangular microchannels can be developed.

As pointed out by Peterson and Ma [42], the condensate liquid film in noncir-

cular microchannels will flow into the corners due to the surface tension, and the filmin the region between the corners is much thinner than the meniscus in the corners.

Therefore, the axial flow in the film region between the corners can be neglected. The

condensing flow regime is illustrated in Figure 6.

Liquid regime:

For the liquid control volume shown in Figure 7, the momentum equation can

be written as

ALdpL VLdAVL LWdALW mLc

dvL 11

where pL is the pressure of liquid phase; VL is the shear stress on the liquid-vaporinterface; LW is the wall friction; mL is the mass flux of liquid phase; vL is the liquidvelocity; AL, AVL, ALWare the liquid cross-sectional area, liquid-vapor interface area,

and wet wall surface area at every corner, respectively; and c is the corner number,

which is equal to 3 for triangular cross sections.

vapor liquid

dx

x

Figure 6. Condensing flow regime in microchannel.



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Based on the Young-Laplace equation, the pressure difference between thevapor and liquid phases is given by:

pV pL R

12

where R is the radius.

Assuming the surface tension and vapor pressure to be constant, the above

equation becomes

dpL d pV

R R2 dR 13

The shear stress at the liquid-vapor interface, VL, is

VL 12VVv

2V 14

where V is the vapor friction factor, V is the vapor density, and vV is the vaporvelocity.

The liquid friction on the wall,LW, is

LW 12LLv

2L 15

where L is the vapor friction factor, and L is the vapor density.The hydraulic diameter for the liquid regime is

DLW 4ALSLW

16

where SLW is the wet wall length at every corner, and

dALW SLWdx 17

pLAL x+dx

pLAL x|

VLSVL

LWSLW

Figure 7. Liquid control volume.

128 Y. CHEN ET AL.


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where x is the distance along the axis. For every corner,

AL

cos cos

sin

2

!R2 18

dAVL 2 2

Rdx 19

SLW 2R cos sin

20

Since the vapor pressure is assumed to be constant, the energy equation can be

written as:

mL qbxhfg

21

where mL is the liquid mass flux, q is the heat flux, and b is the width of the heat

transfer surface.

The liquid velocity can be expressed as:

vL mLcLAL

22

Vapor regime: The combined mass and energy conservation equation is

mV qbL xhfg

23

where L is the entire condensate length and the vapor velocity is

vV mVVAV

24

Assuming the vapor velocity is much larger than liquid phase velocity, themomentum equation for the vapor can be expressed as

dVv2VAV AVdpV SVVdx 25

where AV is the cross-sectional area of the vapor regime, SV is the perimeter of

the vapor regime, and V is the shear stress on the vapor regime interface, which isgiven by

V 1

2 VVv2

V 26



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Introducing Eqs. (24) and (26) into Eq. (25), and considering that the contribu-

tion from the variation of the vapor cross-sectional area along the x-axis is very small,

yields,

2mV

VA2V

dmV

dx dpV

dx m

2VV

2VA3VV

SV 27

which can be integrated to give

pV pV0 qbL x2

hfgVA2V qb&V

2VA3VSV L

2x Lx x3

3

28

Other Theoretical Models and Correlations

Begg et al. [43] developed a physical and mathematical model of annular film

condensation in a miniature circular tube, taking into consideration the liquid-vapor

interface temperature, heat flux, shear stress, and pressure jump conditions due to the

surface tension effects. The shape of the liquid-vapor interface along the condenser

and the length of the two-phase flow region can be predicted by this model.

The velocity profile of the liquid is expressed as follows:

wL

1

L

dpL

dz 1

4

R20

r2 R0 2

2

lnr

R0 E

R0

lnr

R0 29

@wL@r

rR

E 30

dpL

dz Lg sin L 1

2L

Q

hfg mL;in

ER0 F

!

R40

16 R0

2

2F R0

2

8 R

20

4

" #1 31

F R0 2

2ln R

0

R0 12

R20

432

where wL is liquid axial velocity, L is the dynamic viscosity, R0 is the inner radius ofthe channel, is the inclination angle, r is the radial coordinate, Q is axial heat flowdue to phase change, and is the liquid film thickness.

The liquid-vapor interface temperature, T, is

T

TW

R0

LR0 ln

R0

R0 20

2 0 hfgffiffiffiffiffiffiffiffiffiffiffi2Rg

p pVffiffiffiffiffiffiTVp

psat

ffiffiffiffiffi

Tp ! 33

130 Y. CHEN ET AL.


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where TW is the wall temperature, L is the thermal conductivity of liquid, 0 isthe accommodation factor, Rg is gas constant, and TV is the vapor temperature.

Wang and Du [44] also noted that surface tension cannot be neglected in a mini-

tube of diameter less than 3 mm, especially in low-vapor-quality zones. In this

investigation, the influences of gravity, vapor shear along the axial direction, andsurface tension on the condensate film layer were analyzed and an analytical model

proposed. Both the analytical and experimental results indicate that the effect of

gravity on the flow condensation in mini-tubes decreases with decreases in the

hydraulic diameter. A comparison with the experimental data indicated that the

proposed analytical model could predict flow condensation heat transfer in mini-

diameter tubes reasonably well.

Zhang and Faghri [45] used the volume of fluid (VOF) method to investigate

condensation in a capillary groove. A governing equation applicable to both liquid

and vapor phases was developed. In this study, the following problems were success-

fully solved numerically: (1) film condensation on the top of the fin; (2) condensationat the liquid-vapor interface meniscus; and (3) fluid flow in the capillary groove. The

effects of cooling temperature, contact angle, surface tension, and fin geometry on

condensation were also investigated.

Zhao and Liao [46] presented an analytical model for predicting film condensa-

tion for vapor flow inside a vertical mini-triangular channel. The condensing flow field

was divided into three zones: the thin liquid film zone on the sidewall, the condensate

meniscus zone in the corners, and the vapor core zone. The effects of the capillary

forces induced by the free liquid film curvature variation, interfacial shear stress,

interfacial thermal resistance, gravity, axial pressure gradient, and saturation tem-

peratures were all considered and evaluated in the model. It was pointed out that the

heat transfer coefficient for steam condensing inside a triangular channel was alwayssubstantially higher than that observed for flow inside a round tube having the same

hydraulic diameter with similar inlet Reynolds numbers and levels of inlet subcooling.

At the same inlet Reynolds number, the steam velocity was higher in smaller channels,

leading to larger interfacial shear stresses. It was found that the two-phase pressure

drop increases with decreasing channel size; the smaller the channel diameter, the

higher the heat transfer coefficient in the entry region.

Du and Zhao [47] presented a theoretical investigation of film condensation heat

transfer in a vertical circular micro-tube with a thin metal wire welded on its inner

surface as shown in Figure 8. On the basis of simplified mass and energy conservation

and minimum energy principles, both the radial and the axial distributions of con-densate liquid along the tube wall and over the meniscus zone were determined. The

influences of the contact angle between the condensate liquid and the channel wall as

well as the wire diameter on the condensate distributions and the heat transfer

characteristics were examined. It was shown that the Nusselt number was smallest

for the case without the wire welded and was highest with the largest diameter wire

welded to the surface. Thus, it can be confirmed that the wire welded on the inner

surface of the tube can significantly enhance condensation heat transfer in the tube by

thinning the condensate thickness.

Wang and Rose [48] presented a theoretical model for film condensation heat

transfer in noncircular microchannels. The model was based on the laminar conden-

sate flow assumption, taking into consideration surface tension, interfacial shearstress, and gravity. As shown in Figure 9, the condensing flow is divided into two



17/28

segments: condensation on sidewalls and condensation toward the corners. In this

model, the local heat transfer coefficient is defined as:

For condensation on the sidewalls:

0 x xa; xb x xc; and xd x xm for square section;xa x xb and xc x xm for triangular section

x q=TS TW 11 L=

L

34

14 1 TS

ffiffiffiffiffiffiffiffiffiRTSp

=fg=h2fg 35

where 0.665 0.003, q is the local heat flux, is the ratio of the principalspecific heat capacities, TS is the saturation temperature of vapor, R is the

specific ideal gas constant, vfg is the difference between the vapor and liquid

specific volumes, is the film thickness, and hfg is the specific enthalpy ofevaporation.

Figure 8. Condensation in a tube with a thin metal wire welded to the inner surface. (I) Thin liquid film

inside tube wall; (II) meniscus zone; (III) vapor flow zone [63].

132 Y. CHEN ET AL.


18/28

For condensate flow toward the corners:

xa x xb and xc x xd for square sections;0 x xa and xb x xc for triangular section

x q=TS TW 1

1 LrW lnrW=rW

LrW lnrW=rW 36

Figure 9. Physical model and coordinates for horizontal microchannels [64].



19/28

where rW is the radius of curvature of the channel surface.

Similar to the conclusions of Zhao and Liao [46], the results from this model also

demonstrated significant heat transfer enhancement by surface tension in the channel

entrance. The magnitudes of the calculated heat transfer coefficients are in general

agreement with experimental data.Based on an earlier experimental investigation [35], Yang and Webb [47] pro-

posed a semi-empirical model to predict the condensation coefficient inside small

hydraulic diameter, extruded aluminum tubes having microgrooves. In this model,

the effects of the vapor shear stress and surface tension forces were evaluated. The flow

was separated into two parts consisting of the surface tension and vapor shear

controlled regimes, respectively, and the models are combined in the form of an

asymptotic equation. The prediction based on this model was in good agreement

with 95% of the experimental data, with a relative difference within 16%.Garimella et al. [48, 49] developed an experimentally validated unit cell model

for pressure drop during intermittent slug/bubble flow of condensing refrigerantR134a in horizontal circular microchannels. The unit cell utilized in this investigation

is shown in Figure 10. Two-phase pressure drops were measured in five circular

channels ranging in hydraulic diameters from 0.5 to 4.91 mm, with fluid qualities

varying from 100% vapor to 100% liquid.

In this model, the total pressure drop is

ptotal pfriction pfilm=slug 37

The pressure drop due to frictional losses is

pfriction Ltube dpdx

f=b

1 LslugLslug Lbubble

dp

dx

slug

Lslug

Lslug Lbubble

" #38

where L is the length and

dp

dx

slug

0:3164Re0:25slug

LU2slug

2Dh39

Figure 10. Unit cell for intermittent flow [48].

134 Y. CHEN ET AL.


20/28

dp

dx

f=b

0:3164Re0:25bubble

VUbubble Uinterface24Rbubble

40

with Re denoting the Reynolds Number and Uthe velocity.

The pressure drop associated with the flow of liquid between the film and the

slug is

pfilm=slug NUCpone 41

NUC aReslugn 42

where a

2.4369, n

0.5601, andpone is the pressure loss associated with a single

film-slug transition, which is given by Dukler and Hubbard [21] as

pone L 1 RbubbleRtube

2 Uslug Ufilm

Ububble Ufilm 43

where R is the radius.

The results of the above model were on average within 13.4% of the measureddata for circular tubes. The model was validated by two-phase pressure drop data in six

different noncircular channels, having hydraulic diameters ranging from 0.424 to 0.839

mm. Garimella et al. [50, 51] extended this model to noncircular tubes. The predictedpressure drop in noncirculartubes showedgood accuracy with an average error of 16.5%.

In the model for noncircular tubes,

dp

dx

slug

fReslug;"=DhLU

2slug

2Dh44

dp

dx

f=b

fRebubble; "=Dh VUbubble Uinterface2

4Rbubble45

where " is the roughness, Dh is the hydraulic diameter, and f is the Darcy frictionfactor, which is given by Churchill [52] as

fRe;"=Dh

8 8Re

12 2:457ln 17=Re0:90:27"=Dh

" #16 37530

Re

160@1A1:5

8

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Condensation - Nano and Microscale Thermophysical Engineering