Homework #1
Surface Tension
Surface Tension
The property of a liquid surface that allows it to resist an external force is called the surface tension (see the picture of
the water strider below). When a molecule is located in the bulk of a liquid, the interaction between the molecule and
the molecules that surround it are, on average, equal. Translating the molecule to the surface unbalances the interactions because the interactions between the molecule at the surface and the molecules above the surface are different
from the interactions between the molecule and the molecules in the bulk of the liquid. This results in a tension in the
surface that produces a certain amount of strength in the surface film.
The units on surface tension are F/length or dyne/cm, N/m, lb f ë ft. It also has the units of energy/area i.e. joules ë m2
etc. The two different units are equivalent. Surface tension is also referred to as the surface energy.
joule
kg m mës2
m2
m2
=
N
m
since a joule = kg
m2
s2
and a newton =
kg m
s2
Surface tension is given by
s=
dF
dl
where F is a cohesive force and l is a length.
Water has a surface tension of about 72 dyne/cm around room temperature. Reported values of water surface tension
vary somewhat because the surface tension is influenced by impurities in the water. A plot of water surface tension
versus temperature is shown below. As you can see, surface tension is a function of temperature.
Water Surface Tension
Surface Tension HDyneêcmL
75
70
65
60
0
20
40
60
80
100
Temperature H°CL
ü Internal Pressure in Droplets or Bubbles
When a droplet forms in the absence of a gravitational field it will be perfectly spherical. This is the result of the
surface tension trying to minimize the surface area. As a result, there is a pressure differential between the inside and
outside of a droplet or bubble.
Looking at the figures below, the surface tension acts along the circumference of the drop or bubble while the external
pressure acts on the cross-sectional area. Note: a bubble has two surfaces so the 2 p R s term must be multiplied by
two.
HomeWork1_SurfaceTension.nb
ü Droplet
2 p R s = p R2 DP where DP = Pi - Po ; Solving for DP ï Pi - Po =
2s
R
ü Bubble
2 H2 p RL s = p R2 DP where DP = Pi - Po ; Solving for DP ï Pi - Po =
4s
R
A plot of DP versus bubble diameter is shown below for bubble sizes from one nanometer to one meter. The internal
pressure on a one nanometer bubble is over 2800 atmospheres or 42,000 psi.
DP versus Diameter for a Bubble
107
105
DP HNêm2 L
2
1000
10
0.0
0.2
0.4
0.6
0.8
1.0
Diameter HmL
ü Capillary Rise
Surface tension also causes capillary rise in tubes. When the end of a tube is immersed into a liquid the liquid will rise
up into the tube for a distance that is determined by the balance of gravitational and capillary forces.
r g Ip R2 hM = 2 p R s cos f
where f is the contact
angle between the liquid and the interior tube
HomeWork1_SurfaceTension.nb
r g Ip R2 hM = 2 p R s cos f
3
where f is the contact
angle between the liquid and the interior tube surface.
h =
2s
cos f
rgR
Problems
ü Functions and constants
fh[] is a function that converts the input to a height
fh@sig_, rho_, g_, r_, angle_D :=
2 sig
rho g r
Cos@angle D;
fp@sig_, r_, p0_D := p0 + 2 sig ê r;
g = 32.2 ft ë s2 ;
gc = 32.2
lbm ft
lbf s2
;
1. A 0.013 inch in diameter glass tube is inserted into kerosene at 68 °F. The contact angle is 26°. What is the capillary
rise height?
r = 0.862 µ 62.4
s = 28
dyne
cm
f = 26 °;
lbm
;
ft3
lbf
4.4482 µ 105 dyne
2.54 cm
in
;
s
0.000159885 lbf
in
h = fhBs, r
ft3
H12 inL3
, g ê gc, 0.013 in ê 2, fF; NumberForm@h, 3D
Ï 1.42 in
2. What is the pressure inside a spherical 60 °F water drop if the diameter is
(a) 2mm
(b) 0.2 mm
(a) 2 mm
fp@65 dyne ê cm, 2 mm ê H10 mm ê cmL ê 2, 0D
Ï
1300 dyne
cm2
(b) 0. 2mm
fp@65 dyne ê cm, 0.2 mm ê H10 mm ê cmL ê 2, 0D
Ï
13 000. dyne
cm2
3.To what height will water at 10°C rise in a glass tube if the diameter is
(a) 0.5 inches
(b) 1 mm
(c) 0.1 mm
(d) 500 microns
4
3.To what height will water at 10°C rise in a glass tube if the diameter is
(a) 0.5 inches
(b) 1 mm
(c) 0.1 mm
(d) 500 microns
HomeWork1_SurfaceTension.nb
ü First transform the surface tension into the various units
m
sSI = 0.0742 Nt ê m; sAES = sSI
0.22481
lbf
; sCGS = sSI
105 dyne
m
39.37 in
Nt
Nt
100 cm
Print@"s — SI = ", sSI, "\ns — CGS = ", sCGS, "\ns — AES = ", sAESD
;
0.0742 Nt
s — SI =
m
74.2 dyne
s — CGS =
cm
0.000423696 lbf
s — AES =
in
ü Second change the units on the variables to yield the answer in the desired units and find the
capillary rise for each case.
(a) 0.5 inches
rw = 62.4 lbm ë ft3
ft3
H12 inL3
; r = 0.5 in ê 2;
fh@sAES, rw , g ê gc, r, fD
Ï 0.0843652 in
(b) 1 mm
rw = 1
cm3
gm
cm
3
H10 mmL
3
; g = 980
cm 10 mm
s
2
cm
; gc = 1
gm cm
mm
2
s dyne 10 cm
; r = 1 mm ê 2; s = sCGS
cm
10 mm
;
fh@s, rw , g ê gc, r, fD
Ï 0.272206 mm
(c) 0.1 mm
rw = 1
cm3
gm
cm3 H10 mmL3
; g = 980
cm 10 mm
s2
cm
; gc = 1
gm cm
mm
s2 dyne 10 cm
; r = 0.1 mm ê 2; s = sCGS
fh@s, rw , g ê gc, r, fD
Ï 2.72206 mm
(d) 500 micron
rw = 1000
kg
m
3
; g = 9.81
m
s
2
; gc = 1
fh@s, rw , g ê gc, r, fD 106
Ï 54 385.7 micron
micron
m
kg m
s2 Nt
; r = 500 µ 10-6 m ê 2; s = sSI;
cm
10 mm
;