11.6 Two-Phase Flow and Heat Transfer Chapter 11: Two

Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 1

Chapter 11: Two-Phase Flow and Heat Transfer

11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels11.6.1 Two-Phase Flow Patterns

11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels

• Conventional channels: 3mmhD > • Minichannels: 200μm 3mmhD< ≤ • Microchannels: 10μm 200μmhD< ≤ • Transitional channels: 0.1μm 10μmhD< ≤

Transitional microchannels: 1μm 10μmhD< ≤ Transitional nanochannels: 100nm 10μmhD< ≤

• Molecular nanochannels: 100 mhD n≤




Figure 11.19 Flow patterns in horizontal minichannels




Figure 11.20 Flow patterns in horizontal microchannels




11.6.2 Flow Condensation

Figure 11.21 Flow pattern of two-phase condensation flow in a capillary horizontal tube




Figure 11.22 Flow map for

condensation in miniature/micro

channels



Since the heat transfer capacity of a single miniature/micro channel is very low, multiple channels inevitably will be necessary for practical applications.

Riehl and Ochterbeck (2002) studied convective condensation heat transfer of methanol in square channels with dimensions of 0.5, 0.75, 1.0 and 1.5 mm; the corresponding channel numbers are 14, 12, 10 and 8, respectively.

The experiments were performed at two different saturation temperatures: 45 °C and 55 °C. The experimental results were correlated in the following form:

(11.202)


JaNu We Re Prh

YD

−=



(11.203)

(11.204)

(11.205)

(11.206)

V is working fluid velocity (m/s), and L is the channel length. Equation (11.202) correlated over 95% of experimental results with a relative error of less than 25%.


2

We V Lρσ

= l

( )p sat w

v

c T TJa

h−

= l

l

Reh

hD

GDµ

=l

1.3 Re 65

0.5 1 Re 652

h

h

D

hD

h

Y DD

≤= − >



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature ChannelsAnnular Flow Condensation in a Miniature Tube The model condensation in miniature tube differs from traditional

models for film condensation in conventionally sized tubes, due to the following features:

1. The disjoining pressure and the interfacial resistance can affect both the liquid and vapor flow for extremely small channels and film thickness. Therefore the terms containing the disjoining pressure and interfacial resistance should be included in the model.

2. The effect of the surface tension on the fluid flow is included in the model and the two principal radii of the liquid-vapor interface curvature are used in the Laplace-Young equation.

3. Liquid subcooling is accounted for in the model, since it can result in a significant variation of the wall temperature under convective cooling conditions.

4. The liquid momentum conservation equation in cylindrical coordinates is utilized and the corresponding expression for the shear stress at the liquid-vapor interface is accounted for.



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels A steady-state mathematical model of condensation

leading to complete condensation was developed by Begg et al. (1999). The model is based on the following simplifying assumptions:

1. The vapor is saturated and there is no temperature gradient in the vapor in the radial direction.

2. Heat transport in the thin film is due to conduction in the radial direction only.

3. Inertia terms can be neglected for the viscous flow in the liquid films with low Reynolds numbers.

4. Force on liquid due to surface tension is much greater than the gravitational force, thus the gravitational body force is neglected. Therefore the liquid is distributed onto the walls in an axisymmetric film.

5. The solid tube wall is infinitely thin, so that its thermal resistance in the radial direction, as well as the axial heat conduction, can be neglected.




Figure 11.23 Description of the physical model for annular

film condensation in a miniature tube: (a) structure of two-phase flow for complete

condensation, and (b) coordinate system and

conventions for film condensation model (Begg et

al., 1999).



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels Within the assumptions considered above, the mass and energy

balances for the liquid film shown in Fig. 11.23(a) yield

(11.207) Q(z) is the rate of heat through a given cross-section, due to phase

change for z>0, and is defined as follows: (11.208)

The heat flux at the wall is

(11.209) From eqs. (11.208) and (11.209), we obtain the following equation

for the heat rate rejected per unit length of the tube:

(11.210)

( ) ( ),

12

R

inR v

Q zrw r dr m

hδ π ρ−

= −

∫ l l

l l

&

( ) ( ), , , , ,02

z

w p p in in inQ z R q dz c m T c m Tπ ′′= − −∫ l l l l l l& &

( )ln /w

wT Tq k

R R Rδ

δ−′′ =

− l

( ) ( ),2ln /

wp

T TdQ dk c m Tdz dzR R

δπδ

−= −

− l l l l&



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels For consideration of subcooling in the condensed liquid, is found

from an area average given by

(11.211)where is the assumed liquid film temperature profile given by the

temperature distribution in a cylindrical wall.

(11.212) Substituting eq. (11.212) into eq. (11.211) and integrating results in

an expression for the derivate of in the axial is approximated by

(11.213)

Tl

( )( ) 22

2R

RrT r dr

TR R

δ

δ−=− −

∫ ll

( )T rl

( ) lnln

wT T rT r T R RR

δδ δ

δ

−= +

−−

l

( )( )

1

22

22

2ln

1ln2 2 4

wdT dT dTdT Rdz dz dz dz R R R

RR RR

δ δ δδ

δδ

−− = + − − −

− × − + −

l



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels The axial momentum conservation for viscous flow in a liquid film in

which the inertia terms are assumed to be negligible is (11.214)

The axial momentum conservation for viscous flow in a liquid film in which the inertia terms are assumed to be negligible is

(11.215)

(11.216)(11.217)

where is defined as the interfacial shear stress in the absence of phase change and is given by

(11.218)

0r Rw = =l

( ),

1v

r R

dTw d Er dT dz

δ

δ

στµ= −

∂ = − − ≡ ∂ l

ll

1 1 sinw dpr gr r r dz

ρ ϕµ

∂∂ = + ∂ ∂ l l

ll

( ), , 0 / exp 1v v a aτ τ ′ ′= − l l

, 0vτ l

( ) 2, 0 , ,0.5v v v vf w w δτ ρ= −l l l




In equation (11.218), pv is the vapor density, wv is the axial vapor velocity and a’ is the ratio of the local condensation mass flow rate to the vapor mass flux rebounding from the interface, as approximated by

(11.219)

(11.220) From an experimental study for a simulated change of phase the

vapor friction factor, fv, is given as follows: (11.221)

Rer is found using the vapor suction velocity at the liquid-vapor interface due to condensation, vv,δ, which is defined through the condensing mass flux.

(11.222)

( ), ,/ v v v vdQa Rh f w wdz δπ ρ ′ = − − l l l

, 1 3602v vf fR

δ = + l

( ) ( )16 1.2337 0.2337exp 0.0363Re exp 1.2Ma / Rev rf = − − ×

( ),1

2vv v

dQvdz R hδ π δ ρ

=− l



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels Solving eqs.(11.214) – (11.216), the velocity profile is expressed as

follows:(11.223)

Substituting eq. (11.223) into eq. (11.207), we obtain the following equation for the axial pressure gradient in the liquid:

(11.224)

where(11.225)

( ) ( ) ( )2

2 21 1 ln ln4 2

Rdp r rw R r E Rdz R R

δδ

µ

−= − − + + −

ll

l

( )

( ) ( )

,

12 24 2

1sin2

16 2 8 4

inv

dp Qg m E R Fdz h

R RR RF

ρ ϕ µ δπ ρ

δ δ−

= + − + −

− − × + + −

ll l l

l l

&

( ) 2 21ln2 2 4

R R RFR

δδ

− = + − −




The pressure difference between the vapor and liquid phases is due to capillary effects and disjoining pressure, pd,

(11.226) The term with cosine on the right-hand side of this equation is due

to the second principal radius of interfacial curvature. Introducing an additional variable

(11.227)eq. (11.226) can be rewritten as follows:

(11.228) The integral equations of mass conservation for the vapor and liquid

flows take the following form:(11.229)

3222

211 cos atanv d

d d dp p pdz R dzdz

δ δ δσδ

− − = + + − − l

ddz

δ = ∆

( ) ( )3/ 22 cos atan 1 v dp p pd

dz Rσ δ ∆ − +∆ = + ∆ − −

l

( ) ( ), , , /v v v v in v in v in vA w z w A Q z hρ ρ= + l




(11.230) The compressible quasi-one-dimensional momentum equation for

the vapor flow is modified to account for nonuniformity of the vapor cross-sectional area of the liquid-vapor interface

(11.231)

The perfect gas law is employed to account for the compressibility of the vapor flow,

(11.232) Therefore,

(11.233)

( ) ( ), , , /in in in vA w z w A Q z hρ ρ= −l l l l l l l

( ) ( )

( ) ( )

2 2

2,

1sin

2 sin atan

vv v v v v v v v

v

v v

dp dg w A f w Rdz A dz

R v δ

ρ ϕ β ρ ρ π δ

π δ ρ

= + − − −+ − ∆

vv

g v

pR T

ρ =

21 1v v v v

g v v

d dp p dTdz R dz T dzTρ

= −




The saturated vapor temperature and pressure are related by the Clausius-Clapeyron equation, which can be written in the following form:

(11.234) Seven boundary conditions are set forth at z = 0.

(11.235)(11.236)

(11.237)(11.238)(11.239)

(11.240)(11.241)

2g vv v

v v

R TdT dpdz dz p h

=l

inδ δ=0∆ =

,2

v in din

p p pR

σδ

= − +−l

0Q =( ), , ,v v in v sat v inp p p T≡ =

,,

,

v inv in

g v in

pR T

ρ =

,v v inT T=



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels The parameter should be found using a constitutive condition

at the entrance of the condenser (11.242)

The interfacial resistance is defined as

(11.243) The relation between the saturation vapor pressure over the thin

condensing film, affected by the surface tension, and the normal saturation pressure corresponding to Tδ is given by the following equation

(11.244)

It follows form eq. (11.209) and (11.243)(11.245)

, /in t in vm m Q h= −l l& &

,inml&

( )22 2

satv v

g v

ph pqR T T

δδ

δ

αα π

′′ = − − − l

( ) ( ) ( ) ( )exp sat sat d

sat satg

p p T K pp p T

R Tδδ

δδδ

σρ

− − +=

l

( )( )2ln

2 2satv v

wg v

ph pR RT Tk R R R T T

δδ

δ

αδ δ α π

= + × − − − − l

l



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels If the convective heat transfer coefficient at the outer tube wall, ho,

and the cooling liquid temperature are known, the local wall temperature can be defined using an energy balance

(11.246)

The additional boundary condition is given by (11.247)

( ) 12o w

dQh T TR dzπ∞− =

,0, w w inz T T= =




Figure 11.24 Annular film condensation in a

circular tube with constant wall temperature

boundary condition. R = 1.55mm, Tw = 363 K, g/s. (a) Film thickness

versus position, (b) vapor pressure versus position, (c) liquid pressure versus position, (d) cumulative

heat rejected versus position (Begg et al.,

1999).



11.6.3 Flow Evaporation and BoilingOnset of Nucleate Boiling in Microchannel Flow Hsu proposed the ONB heat flux correlation based on the minimum

superheat criterion for pool boiling

(11.248) The effect of contact angle on the ONB heat flux was considered by

Davis and Anderson (1966), who proposed the following correlation

(11.249) The superheat equation for the bubble nucleus can be obtained

from equilibrium theory (11.250)


2( )12.8v v w sat

ONBsat

k h T TqT

ρσ

−′′ = l l

2( )8(1 cos )

v v w satONB

sat

k h T TqT

ρθ σ

−′′ =+

l l

2 bb sat

v v b

TT Th R

σρ

− =l




Hc

Figure 11.25 Geometric configuration of the microchannels.

wc ww

W

L






Since the bubble shape is a truncated sphere, we have (Fig. 11.26)(11.251)(11.252)

Substituting eq. (11.251) into eq. (11.250) and solving for Tb, one obtains

(11.253) The liquid temperature near the wall of the microchannel can be

assumed as(11.254)

If a constant heat flux, q”, is applied to the bottom, the liquid temperature at the outlet is related to that at the inlet by

(11.255)


(1 cos )b by R θ= +sinc br R θ=

21b satv v b

CT Th y

σρ

= −

l

( ) /w wT y T q y k′′= −l l

, ,0 ( )out in

c c

q WLT Tc u nw Hρ

′′= +l l

l l



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels If the convective heat transfer coefficient is uniform along the

channel surfaces and flow in the channel is fully-developed, the channel wall temperature can be obtained by

(11.256)where is the Nusselt number for fully developed flow in a three-sided

(bottom and side walls) rectangular channel (11.257)

The effective wall heat flux is related to the applied heat flux by

(11.258)

/w

wh

qT TNuk D

′′= +l

l

1 2c w

wc

w wq qH

αη α

+′′ ′′= +

2 3 4 5Nu 8.235(1 1.883/ 3.767 / 5.814 / 5.361/ 2 / )α α α α α= − + − + −



11.6 Two-Phase Flow and Heat Transfer in Micro/Miniature Channels The condition for nucleation boiling to occur is . The

necessary condition for ONB can be obtained from eqs. (11.253) and (11.254), i.e.,

(11.259)which can be rearranged as an equation of yb,

(11.260) For the roots of eq. (11.260) to be real, the following condition must

be satisfied

(11.261)

( )b bT y T≥l

21ww b sat

v v b

q CT y Th yk

σρ

′′ − = −

ll

2 2 2 0w wb w b w

v v v v b

q qC Cy T y Tk h k h y

σ σρ ρ

′′ ′′ − + + =

l l l l

22 24 0w w

w wv v v v b

q qC CT Th k k h y

σ σρ ρ

′′ ′′ + − ≥

l l l l




Eq. (11.261) can be rearranged to yield the superheat criterion

(11.262) If the liquid inlet conditions are prescribed but the heat flux is

allowed to vary, the ONB heat flux can be obtained by

(11.263)

2 ww sat

v v

qCT Th k

σρ

′′− ≥

l l

,0

1 2( ) /

c wONB

cin sat

c c h

w w qHq WLT T

c u nw H Nu k D

αη α

ρ

+ ′′ +′′ + + −ll l l

1 22c w

ONBc

v v

w w qHC

h k

αη ασ

ρ

+ ′′ + =l l




The wall superheat can be obtained from inequality eq. (11.262) as

(11.264) Substituting eq. (11.258) into eq. (11.264), the wall superheat

becomes

(11.265)

2 22w ww sat sat

v v v v

q qC CT T Th k h k

σ σρ ρ

′′ ′′− = +

l l l l

21 2

221 2

c ww sat

v v c

c wsat

v v c

w wCT T qh k H

w wCT qh k H

σ αρ η α

σ αρ η α

+ ′′− = +

+ ′′+ +

l l

l l



Flow Evaporation in Minichannel Heat Sink with Axial Grooves

New advanced cooling technologies are needed to meet the demands for dissipating extremely high heat fluxes (over 300W/cm2) from electronic components.

The performance characteristics of a flat high heat flux evaporative mini-channel heat sink with axial capillary grooves on its inner surface, as shown in Fig. 11.16, was modeled by Khrustalev and Faghri (1996).

The small axial grooves allow this heat sink to withstand high heat fluxes with a small pressure drop, low thermal resistance, and a comparatively large effective length.




In order to evaluate the advantages of this design compared to existing designs, the maximum heat flux, thermal resistance, and the pressure drop on the element will be estimated for some operational intervals of independent parameters, such as the saturated-vapor outlet temperature , the geometry of the grooved surfaces, and the effective lengths of the evaporators under consideration.


,v oT




Figure 11.27 Schematic of the flat miniature evaporator with axial capillary grooves (not to scale; Khrustalev and Faghri 1997))



(11.266)

At any axial location, the following mass conservation equation must hold over the cross-section of the evaporator modeled under steady-state conditions:

(11.267)

where , , N is the number of grooves, and the subscript “o” denotes the evaporator outlet.

(11.268)

where the inertia terms are neglected, since Khrustalev and Faghri (1994) demonstrated that this simplification is even when the liquid cross-sectional area varies along the flow.


, ,( ) ( )o v o vm z m m m z= + −l l& & & &

2 2

2 2, ,

2 2sinh h

dp w mg f fdz D D A N

ρρ φρ

+ = − ≡ −l l l ll l l

l l l l

&

v v v vm w Aρ=& m Nw Aρ=l l l l&

2 2, 1

2v v hw D

WeW

ρπ σ

≡ <l



For vapor flow in a channel with a small Mach number (Ma1), the momentum conservation equation is

(11.269)

Differential form of Laplace Young equtaion(11.270)

Mass and energy conservation balances relate the vapor mass flow rate with the heat flux at the inside surface of the vapor channel

(11.271) where


22

,

2( ) sin v vv v v v v v

h v

wd p w g fdz D

ρρ β ρ φ+ + = −

v

men

dp dpddz R dz dz

σ = −

l

22 ( ) ( 2 )v v

t wv

dm q z t tdz h

′′= −

l

&

( )v e w vq h T T′′ = −



(11.272)

which should be solved for Hmen at every point on z since , , and depend on Hmen. According to eq. (11.271), the vapor mass flow rate at the outlet is

(11.273)

For the given temperature of the vapor at the outlet, , the boundary conditions for eq. (11.269) and (11.271) at the right-hand end of the evaporator ( ) are

(11.274)

(11.275)


1/ 22 2,

, , ( ) sin2

h vo v o v

D A N dpm m m z gf dz

ρρ φ

+ − = − + l l l

l ll

& & &

,hD l Al fl/vdp dz

, 222 ( )( 2 )sL

v o v t wv

dzm q z t th

′′= −∫l

&

,( )v sat v op p T=

,v v om m=& &

,v oT

sz L=



Taking into consideration that varies along the z-coordinate due to the axial vapor temperature gradient, – estimated initially by Eq. (11.273) at – should be adjusted so as to satisfy the following condition:

(11.276)

(11.277)

which implies that the liquid and vapor flows become separated at z = 0 under the influence of the capillary forces. In cases of a given heat flux below the maximum heat flux, should be chosen so as to satisfy eq. (11.277).


vhl

,v om ,v oT

00v z

m=

=&

,min

10

cos1/ 2

men

men h w gzR t t t

θ

=

=− −

,oml&



The numerical procedure begins at and moves toward the left-hand end of the vapor zone. At some point on z – which can be denoted as z1 – the condition is reached, and, starting from this point, eqs. (11.268) – (11.271) should be solved for with the following boundary conditions (at ):

(11.278)

(11.279)

(11.280)

(11.281)


sz L=

men gt t=

, , and v men vp p R ml &1z z=

1men,min

v z zp p

Rσ

== −l

1v v z zp p

==

men men,minR R=

1v v z z

m m=

=& &







Figure 11.28 Performance characteristics of the evaporator along the axial coordinate: (a) Effective height and radius of the liquid-vapor meniscus; (b)

Vapor and liquid mass flow rates; (c) Vapor and liquid pressure variations; (d) Vapor and wall temperature

variations, and effective heat transfer coefficient



Different configurations of miniature heat sinks are needed for the varied applications.

The most important characteristics of miniature heat sinks are listed below: Heated length and external thickness Uniformity of temperature over the surface Effective heat transfer coefficients (or thermal resistance) Maximum attainable heat fluxes on the surface Pressure drop and mass flow rate Stability of operation

Operating temperature versus absolute pressure range.


11.6.3.2 Boiling Heat Transfer in a Miniature Axially-Grooved Rectangular Channel with Discrete Heat Source




Figure 11.29 Cross-section of the miniature axially-grooved heat sink

(Khrustalev and Faghri, 1997).




Figure 11.30 Test section with six thick film resistors



(11.282)

Boiling in Micro/Miniature Channels For two-phase flow in a microchannel, there is no

stratified flow; therefore, the orientation of the channel has little effect on the flow pattern.

The flow patterns for boiling in microchannels, in order of appearance from the inlet are: bubbly flow, elongated bubbles, annular, mist, and flows with partial dryout.


1/ 2

1,

( / 2) [ 2( )]vcrit v h w g

eff h

Wq h t t tL D

π σ ρ′′ = − +ll



Since the diameter of the channel is so small, the bubbly flow regime is very short as the bubbles grow to the size of the channel very quickly.

Once the bubbles grow to the size of the channel, the flow pattern becomes elongated bubble flow, which followed by an annular flow regime.

Cyclic dryout is also possible in the elongated bubble flow regime, which is also referred to as the confined bubble regime.

The flow regime shown in Fig. 11.10 is still applicable to boiling in micro channels except that the plug flow regime is replaced by the elongated bubble flow regime.




Figure 11.31 Three-zone heat transfer model for elongated bubble flow in microchannels




Vlasie et al. (2004) thoroughly reviewed the available empirical correlations for heat transfer in flow boiling in small-diameter channels. Lazarek and Black (1982) studied flow boiling of R-113 in 123- and 246-mm-long circular vertical tubes with internal diameters of 3.1 mm. They recommended the following correlation:

(11.283)

(11.284)

is boiling number.


0.857 0.714Nu 30Re Bo=

Bov

qh G

′′=

l



Equation (11.283) is obtained by analyzing experimental data performed in the following ranges: , , , and . In arriving at eq. (11.283), the effect of quality on heat transfer is neglected. Kew and Cornwell (1997) performed experiments of boiling of R-141 in a 500 mm-long, small-diameter tube ( ). While the results for 3.69 and 2.87 mm tubes follow trends similar to those observed in conventionally-sized tubes, the results for the 1.39mm tube depart from these trends significantly. To account for the effect of quality on boiling heat transfer, Kew and Cornwell (1997) proposed a modified Lazerek and Black correlation as follows:

(11.285)


2 214kW/m 380kW/mq′′< <2125kg/m -s 2725kg/m -sG< < 2 2125kg/m 725kg/mG< <

130kPa 140kPap< <

1.39 3.69mmD = −

0.857 0.714 0.143Nu 30Re Bo (1 )x −= −



Qu and Mudawar (2004) correlated 42 data points from their experimental results for water in microchannels with , as well as data for R-113 in circular tube with from their previous study.

They recommended the following empirical correlation for critical heat flux in microchannels:

(11.286)

(11.287)


380μmhD =510 μm and 2.54 mmD =

0.361.110.21max 33.43 Wev

v h

q LGh D

ρρ

−−′′

= l l

We GLσ ρ

=l




Yu et al. (2002) experimentally investigated flow boiling of water in a horizontal tube of 2.98-mm inner diameter at a system pressure of 200 kPa. They proposed the following correlations

(11.288) For nucleate boiling in five parallel rectangular channels

with a hydraulic diameter (11.289)

For convective boiling heat transfer in the same geometric configuration, they suggested:

(11.290)

6 2 0.27 0.26.4 10 (Bo We ) ( / )vh ρ ρ −= × l l

0.641.068h q′′=

0.024 0.857 0.8 1h 4.068 10 Re (1 ) e

o ee

xxx

−= × −

l




Figure 11.32 Microchannels with re-entrant cavities (Koşar et al., 2005).




Figure 11.33 Effective heat flux versus average surface temperature (Koşar et al., 2005).




Figure 11.34 Vapor back-flow

near CHF




Qu and Mudawar (2004) correlated 42 data points from their experimental results for water in microchannels with Dh = 380 µm, as well as data for R-113 in circular tube with D = 510 μm and 2.54 mm from their previous study. They recommended the following empirical correlation for critical heat flux in microchannels:

(11.291)where

(11.292)

0.361.110.21max 33.43 Wev

v h

q Lm h D

ρρ

−−′′

= ′′ l l&

We m Lσ ρ

′′=

l

&

Documents

11.6 Two-Phase Flow and Heat Transfer Chapter 11: Two