Dr. Goran Jovanovic - Classes

OREGON STATE UNIVERSITY
CBEE
CHE 331
Transport Phenomena I
Dr. Goran Jovanovic
Surface Tension Forces I
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CHE 331
Fall 2015
Surface Tension
There are two kinds of surface forces that create most of the liquid
surface phenomena.
•  The cohesive forces are attractive forces among alike liquid
molecules. The cohesive forces are responsible for the
phenomenon known as Surface Tension (σ ).
•  The adhesive forces are attractive forces among unlike molecules. The relative strength between adhesive and
cohesive forces gives rise to hydrophobicity/hydrophilicity of
a given liquid on a particular solid surface.
CHE 331
Fall 2015
For example the adhesive forces between water molecules and the
walls of a glass tube are stronger than the cohesive forces of water
molecules. This creates a water meniscus at the walls of the vessel
and contributes to capillary action.
Cohesion Forces and Surface Tension
The cohesive forces between molecules inside a liquid are shared
with all neighboring atoms.
Atoms on the surface have less neighbors of its own kind. They
exhibit stronger attractive forces upon their nearest atoms on the
surface. This enhancement of the intermolecular attractive forces at
the surface contributes to interface pressure difference.
CHE 331
Fall 2015
Cohesion Forces and Surface Tension
Consider a neighborhood of molecules on a flat surface:
Fs
Fs
If we would like to tear (rupture) this surface along a line “l” we would
have to apply a force Fs. The Surface Tension is defined as:
CHE 331
Fall 2015
FS ⎡ N ⎤
σ = ⎢ ⎥
l ⎣m⎦
Cohesion Forces and Surface Tension
If the surface is curved the surface tension forces give rise to
interface pressure difference.
But first consider the concept of curvature. The curvature of a line
at a given point, A, is defined as:
1
κ=
R
RB
A
RA
B
CHE 331
Fall 2015
Cohesion Forces and Surface Tension
Similarly, for curved surfaces there are two characteristic
radii that define the curvature of the surface at a given
point.
CHE 331
Fall 2015
Cohesion Forces and Surface Tension
Now, consider a differential element of the surface : ds1 × ds2
CHE 331
Fall 2015
Cohesion Forces and Surface Tension
CHE 331
Fall 2015
The following relationships are well known
in trigonometry:
ds1 = R1dθ1 and ds2 = R2 dθ 2
⎛1
⎞ 1
⎛1
⎞ 1
sin ⎜ dθ1 ⎟ ≅ dθ1 and sin ⎜ dθ 2 ⎟ ≅ dθ 2
⎝2
⎠ 2
⎝2
⎠ 2
CHE 331
Fall 2015
Cohesion Forces and Surface Tension
⎛1
⎞
dF2,n = dF2 ⋅sin ⎜ dθ1 ⎟
⎝2
⎠
⎛1
⎞
dF2,n = σ ⋅ ds2 ⋅ ⎜ dθ1 ⎟
⎝2
⎠
dF2
1
dθ1
2
dF2,t
CHE 331
Fall 2015
dF2,n
⎛ 1 ds1 ⎞
dF2,n = σ ⋅ ds2 ⋅ ⎜
⎝ 2 R ⎟⎠
1
σ ⋅ ds2 ds1
dFn2 = 2 ⋅ dF2,n =
R1
Cohesion Forces and Surface Tension
Similarly for the force pulling in the other directions we may write:
σ ⋅ ds1ds2
dFn1 = 2 ⋅ dF1,n =
R2
Finally, the total surface tension force acting on the differential surface area in
the direction of the surface vector “n”:
σ ⋅ ds1ds2 σ ⋅ ds2 ds1
dFσ = dFn1 + dFn2 =
+
R2
R1
CHE 331
Fall 2015
⎛ 1
1⎞
dFσ = σ ⋅ ds1ds2 ⎜ + ⎟
⎝ R1 R2 ⎠
Po
dF2
dFσ
Pi
−Po ⋅ ds1ds2 + dFσ + Pi ⋅ ds1ds2 = 0
( Po − Pi ) ds1ds2 = dFσ
⎡1
1 ⎤
( Po − Pi ) ds1ds2 = σ ⎢ R + R ⎥ ds1ds2
⎣ 1
2 ⎦
CHE 331
Fall 2015
⎡1
1 ⎤
ΔP = σ ⎢ + ⎥
⎣ R1 R2 ⎦
Liquid Rise in a Capillary
A narrow capillary tube is dipped into a liquid that wets the tube. We observe
that the liquid rises in the tube, above the level of the free surface. We want
to derive a mathematical model that relates the height of capillary rise 'h' to
the surface tension ‘σ' and the inner diameter of the capillary Dc. This model
will suggest the method for measuring the surface tension.
CHE 331
Fall 2015
Patm − ΔPσ + ρ gh = P1 = Patm
ΔPσ = ρ gh
⎡1
1 ⎤
σ ⎢ + ⎥ = ρ gh and if R1 = R2
⎣ R1 R2 ⎦
2σ 4σ
4σ
=
= ρ gh ⇒ h =
R
Dc
ρ gDc
Also σ =
Fσ
FS
CHE 331
Fall 2015
Patm
Pinside
hρ gDc
4