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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.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 2
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
Figure 5.2 Arbitrarily-curved surface with two radii of curvature
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 3
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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 = ∆ =
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 4
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
From the geometry of Fig. 1(5.5)
or(5.6)
Similarly(5.7)
5.2 Surface Tension
II Rx
dzRdxx =
++
IRdzxdx =
IIRdzydy =
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 5
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 6
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
5.2.2 Interface Shapes at Equilibrium
5.2 Surface Tension
Figure 5.3 Shape of the liquid-vapor interface near a vertical wall.
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 7
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 8
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 9
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 10
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 11
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
5.2 Surface Tension
Figure 5.4 Two parallel plates in bulk water.
2y
water
y
z
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 12
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 13
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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 − −= = × × = × =
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 14
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 15
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
≠
−
=
∂ ∂ = + ∂ ∂ ∑
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 16
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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 = −
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 17
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
Figure 5.5: Cellular flow driven by surface tension gradient.
hδ
5.2 Surface Tension
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 18
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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′ ′= + =
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 19
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 20
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
Figure 5.6 Stability plane for the onset of cellular motion
5.2 Surface Tension
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 21
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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=
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 22
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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 =
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 23
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 24
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 25
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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
Transport Phenomena in Multiphase Systems by A. Faghri & Y. Zhang 26
Chapter 5: Solid-Liquid-Vapor Phenomena and Interfacial Heat and Mass Transfer
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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