5.2 Surface Tension Chapter 5: Solid-Liquid-Vapor

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

Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer

5.2 Surface Tension5.2.1 Capillary Pressure: The Young-Laplace Equation

5.2 Surface Tension

Vapor

Liquid

Fo

FsFs

Fi

Figure 5.1 Origin of surface tension at liquid-vapor

interface.



5.2 Surface Tension5.2.1 Capillary Pressure: The Young-Laplace Equation

5.2 Surface Tension

Figure 5.2 Arbitrarily-curved surface with two radii of curvature



Displace the surface, the change in area is(5.1)

If dxdy≈0, then(5.2)

The energy required to displace the surface(5.3)

Energy attributed to generating this pressure(5.4)

5.2 Surface Tension

( ) ( )A x dx y dy xy∆ = + + −

A y dx x dy∆ = +

( )W xdy y dxδ σ= +

capW p xy dz p xy dzδ = ∆ =



From the geometry of Fig. 1(5.5)

or(5.6)

Similarly(5.7)

5.2 Surface Tension

II Rx

dzRdxx =

++

IRdzxdx =

IIRdzydy =



For equilibrium, the two expressions for energy must be equal

(5.8)

(5.9)

The pressure difference Δp=(ρv-ρl) between the two phases becomes

(5.10)

This expression is called the Young-Laplace equation, and it is the fundamental equation for capillary pressure.

5.2 Surface Tension

dzxypdxydyx ∆=+ )(σ

dzxypRdzxy

Rdzxy

III

∆=

+σ

+=∆=

IIIcap RR

pp 11σ



5.2.2 Interface Shapes at Equilibrium

5.2 Surface Tension

Figure 5.3 Shape of the liquid-vapor interface near a vertical wall.



The first principal radius of curvature RI

(5.11)

The Young-Lapace equation becomes(5.12)

At a point on the interface that is far away from the wall, the liquid and vapor pressures are the same, i.e.,

(5.13) The liquid and vapor pressures near the wall are

(5.14)(5.15)

The relationship between the pressures in two phases can be obtained by combining eqs. (5.13) – (5.15), i.e.,

(5.16)

5.2 Surface Tension

[ ] 2/32

22

)/(1/1dydzdyzd

RI +=

/v Ip p Rσ− =l

( )v vp p g zρ ρ− = −l l

( ,0) ( ,0)vp p∞ = ∞l

( ) ( ,0)v v vp z p gzρ= ∞ −( ) ( ,0)p z p gzρ= ∞ −l l l



Combining eqs. (5.15-5.16), the equation for the interface shape becomes

(5.17)

Multiplying this equation by dz/dy and integrating gives

(5.18)

Since at y→∞ both z and dz/dy equal zero, C1 = 1. The boundary condition for equation (5.18) is

(5.19)

5.2 Surface Tension

3/ 22 2

2

( ) 1 0vg z dz d zdy dy

ρ ρσ

− − − + =

l

1/ 222

1( ) 1vg z dz C

dyρ ρ

σ

− − + + =

l

∞→= 0)/( ydydz



Equations (5.18) and (5.19) can then be solved for z(0)

(5.20)

Using (5.20) as a boundary condition, integrating eq. (5.18)

(5.21)

where(5.22)

5.2 Surface Tension

2/1

0 )(2)0(

−

==g

zzvρρ

σ

1 1/ 2 1/ 21 2 202 2

0

2 2cos h cos h 4 4c c

c c c

L L zy zL z z L L

− − = − + + − + 2/1

)(

−

=g

Lv

c ρρσ



5.2 Surface Tension

Example 5.1 Two parallel plates are put into bulk water at the bottom

end (see Fig. 5.4). Estimate the minimal distance between the plates at which the central point of the liquid-air interface between the plates is not elevated from the bulk water level at equilibrium condition. Assume that water completely wets the material of the plates. The system temperature is 20 °C, σ = 0.07288 N/m, and = 999 kg/m.ρ l



5.2 Surface Tension

Figure 5.4 Two parallel plates in bulk water.

2y

water

y

z



5.2 Surface Tension

Solution:Since the density of vapor is much less than that of the liquid –

– eq. (5.22) can be simplified to obtain

The rise of the liquid surface at y=0 is obtained from eq. (5.20):

vρ ρ l=1/ 2 1/ 2 1/ 2

30.07288 2.728 10 m( ) 999 9.8c

vL

g gσ σ

ρ ρ ρ− = = = × − × l l

B

1/ 2 1/ 2

0

1/ 23

2 2(0)( )

2 0.07288 = 3.859 10999 9.8

vz z

g g

m

σ σρ ρ ρ

−

= = = −

× = × ×

l l



5.2 Surface Tension

The value of y for z = 0 can be found by using eq. (5.21), i.e.,

i.e.,This value of y is one-half the minimal distance between the plates.

Therefore, the minimal distance is about 5.96 mm.

1 1/ 2 1/ 21 2 202 2

0

1 13 3

3

23

3

2 2cos h cos h 4 4

2 2.728 10 2 2.728 10cosh cos h03.859 10

3.859 1042.728 10

c c

c c c

L L zy zL z z L L

− −

− −− −

−

−

−

= − + + − +

× × × ×= − ×

×+ + ×

1/ 2 1/ 22

304

2.728 10

0.909

−

− + × =

3 30.909 0.909 2.728 10 2.48 10 m 2.48mmcy L − −= = × × = × =



5.2.3 Effects of Interfacial Tension Gradients Since surface tension depends on

temperature, permanent nonuniformity of temperature at a liquid-vapor interface causes a surface tension gradient.

This may in turn establish a steady flow pattern in the liquid.

5.2 Surface Tension



5.2 Surface Tension

The surface tension of a multicomponent liquid that is in equilibrium with the vapor is a function of temperature and composition of the mixture, i.e.,

(5.23) The change of surface tension can be caused by either

change of temperature or composition, i.e.,

(5.24)

1 2 1( , , , , )NT x x xσ σ −= L

1

1 ,i j i

N

ix ii T x

d dT dxT xσ σσ

≠

−

=

∂ ∂ = + ∂ ∂ ∑



Curve-fit equations for surface tension are almost linear

(5.25)

This motion of a liquid caused by a surface tension gradient at the interface is referred to as the Marangoni effect.

The most well-known example of surface-tension-driven flow is Bernard cellular flow, which occurs in a thin horizontal liquid layer heated from below.

5.2 Surface Tension

0 1C C Tσ = −



Figure 5.5: Cellular flow driven by surface tension gradient.

hδ

5.2 Surface Tension



Boundary condition at the interface(5.26)

Local temperature(5.27)

Velocities(5.28)

(5.29)

5.2 Surface Tension

yx xx x

u Tx T xδ

δ δ

στ µ=

= =

∂ ∂ ∂ = = ∂ ∂ ∂ l

( )wT T T T y Tζ′ ′= + = − +

v v v v′ ′= + =

u u u u′ ′= + =



Substituting eqs. (5.27)-(5.29) into the continuity, momentum and energy equations, and subtracting the corresponding base flow equations

(5.30)

(5.31) The case of marginal stability

(5.32)

where Ma is the Marangoni number, where α=2π/λ

(5.33)

5.2 Surface Tension

( ) i x tT y e α βθ +′ =( ) i x tv V y e α β+′ =

)ˆ/()ˆcoshˆ8(ˆsinhˆcoshˆ)ˆcoshˆsinhˆ)(ˆsinhˆcoshˆ(ˆ8

2533 ααααααααααααα

+−−−+=

BoCrBiMa

2( / )d dTMa ζ σ δα µ

=l l



Figure 5.6 Stability plane for the onset of cellular motion

5.2 Surface Tension



Wave number(5.34)

Biot number(5.35)

Bond number(5.36)

Crispation number(5.37)

5.2 Surface Tension

hBikδ δ=l

2( )g gBo

ρ ρ δσ

−= l

Cr µ ασ δ

= l l

α̂ α δ=



The liquid film is stable if its Marangoni number is below that predicted by eq. (5.32).

The marginal stability predicted by eq. (5.32) for some typical combinations of parameters is illustrated in Fig. 5.6.

Since the Fourier components of all wavelengths can be contained in a random disturbance, the system becomes unstable when it is unstable at any wavelength.

For a system with and , Fig 5.6 indicates that the critical Marangoni number is about 80 and the associated dimensionless wave number

The Marangoni effect can have an important influence on the evaporation of a falling film, and cause vapor bubbles in a liquid with a temperature gradient to move toward the high temperature region during boiling.

5.2 Surface Tension

410Cr −< 0, 0Bi Bo→ →

ˆ 2.α =



5.2 Surface Tension

Example 5.2 A 0.3-mm-thick water film sits on a surface held at a

temperature of Tw = 80 °C. The top of the liquid film is exposed to air at a bulk temperature of TG = 20 °C, and the convective heat transfer coefficient between the liquid film and the air is = 10 W/m2K. Determine whether the water film is stable.

hδ



5.2 Surface Tension

Solution: Since the liquid film is very thin, the temperature drop across the liquid

film will be very small. The properties of the liquid film can be determined at the wall temperature of 80 °C, i.e., = 0.67 W/m-K,

= 971.9 kg/m3, = 1.64x10-7 m2/s, = 351.1x10-6 N-s/m2, σ = 0.0626 N/m, and dσ/dT = -1.7x10-4 N/m °C.

The Biot number is

At steady-state, the surface temperature of the interface, Tδ, satisfies

klρ l α l µ l

310 0.3 10Bi 0.004250.67

hkδ δ −× ×= = =l

( )wG

T Tk h T Tδδ δδ

−= −l



5.2 Surface Tension

i.e.,

which demonstrates that the temperature drop across the liquid film is minimal. The Bond number is obtained by using eq. (5.37), i.e.,

The Crispation number Cr is

3o

3

11

80 4.25 10 20 79.75 C1 4.25 10

w Gw G

T BiTh hT T Tk k Biδ δ

δδ δ

−

−

+= + + = +

+ × ×= =+ ×

l l

2 2 2( ) 971.9 9.8 0.0003Bo 0.0140.0626

g g gρ ρ δ ρ δσ σ

− × ×= = =l lB

6 76351.1 10 1.64 10Cr 3.07 10

0.0626 0.0003µ α

σ δ

− −−× × ×= = = ×

×l l



5.2 Surface Tension

which is less than 10-4. The critical Marangoni number, Mac, below which the liquid film is stable, can be obtained from Fig. 5.4; its value is 80 at .

The Marangoni number of the system can be obtained from eq. (5.33), i.e.,

Therefore, the system is unstable and Marangoni convection will occur.

ˆ 2α =

2 2( / ) ( / )wT Td dT d dTMa δζ σ δ σ δα µ δ α µ

−= =l l l l

4 2

c7 679.75 80 ( 1.7 10 ) 0.0003 221.4 Ma

0.0003 1.64 10 351.1 10

−

− −− − × ×= × = >

× × ×

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5.2 Surface Tension Chapter 5: Solid-Liquid-Vapor