WATER POLLUTION CONTROL RESEARCH SERIES • 150-80-EPL 04/70
SPREADING AND MOVEMENT OF OIL SPILLS
US DEPARTMENT OF THE INTERIOR • FEDERAL VVAiE/i HOMJUl'ION CONTROL ADMINISTRATION
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WATER POLLUTION CONTROL RESEARCH SERIES
The Water Pollution Control Research Reports describe the
results and progress in the control and abatement of pollu-
tion of our Nation’s waters. They provide a ceatral source
of information on the research, developnent and demonstra-
tion activities of the Federal Water Pollution Control
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to requesters as supplies permit. Requests should be sent to
the Planning and Resources Office, Office of Research and
Developii nt, Federal Water Pollution Control Administration,
Department of the Interior, Washington, D. C. 2O2I 2.
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SPREADING AND MOVEMENT OF OIL
SPILLS
FEDERAL WATER POLLUTION CONTROL ADMINISTRATION
DEPARTMENT OF THE INTERIOR
by
Henry G. Schwartzberg
Chemical Engineering Department
New Tork University
Program Number 150 80
Contract Number WP 013U2-01A
March 1970
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FWPCA Review Notice
This report has been reviewed by the Federal
Water Pollution Control Administration and
approved for publication. Approval does not
signify that the contents necessarily reflect
the views and policies of the Federal Water
Pollution Control Administration, nor does
mention of trade names or commercial products
constitute endorsement or recommendation for
use.
ii
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ABSTRACT
The spreading and rixwement of oil spills on water were investigated.
Areas for spills tiic i form lenses were measured and correlated -
i quations (8) and ( l). Most crades tested formed thin films, not
lenses; but lens formation could be induced and spreading greatly
reduced by surfactant treatment.
Spreading rates for small spills were measured and correlated with spill
volume, oil density and water viscosity - Equation (S3). Field data and
energy conservation, however, indicates Equation ( 3) cannot be scaled
up to large spill volumes.
Wind-water basin tests indicated that on quiet open water oil should
drift windward at 3.66 + .l7 of the wind velocity. The percentage
drift was not significantly affected by oil or water properties, depth,
and wind speed, and agrees fairly well with field data. Waves caused
significant reductions in wind drift, but in the shallow basin used
did not induce significant drift themselves. Since wind causes waves,
and deep water waves cause drift, further investigation of wind and
wave drift interaction is recommended.
Wind drift was found to be confined to a thin surface layer. The use
of 1 - 1 1/2 inch deep oil-confining drogues markedly reduced wind
drift. Investigation of the use of nets of such drogues and of lens
formation to reduce oil spread and novement is recommended.
Combined wind and current drifts were found not to be directly additive,
and were roughly correlated by Equation (72).
This report was submitted in fulfillment of project 9P Ol3 2-OU
between the Federal ?ater Pollution Control A±dnistration and New
York University.
KEY DRDS
Oil Spills Oil Lenses
Cii on Water Wind Drift
Oil Spill Areas Wave Drift
Oil Spill Spreading Rates Combined Current Wind Drift
Oil Spill Drift Wind Wave Drift Interactions
111
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CONTENTS
Page
Abstract iii
Section 1. Conclusions and Reconuendations 1
Section 2. Introduction
Section 3. Body of Report 7
Analysis of Prior State of Knowledge 7
Spread 7
Spreading Rates 18
Drift-Wir I Effects 22
Drift-Current Effects 26
Drift—Wave Effects 27
Experimental 32
Section b. Discussion of Results 147
Lenses 147
Crude Oils
Crude Oil Spreading Rates 63
WjndDrjft 78
Wave Drift 90
Current Drift 92
Con,bjned Wind-Current Drift 93
References 96
Notation 100
iv
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TABLES
o. Title Page
1 Spill Volumes and Distances Traveled by
Recent Ci i Spills S
2 Oil Drift vs Wind Speed 23
3 Estimated Wave Drift for Fully Developed
Wind Waves 29
L . Lens Characteristics 5l- 2
Characteristics of Crudes 60
6 Area Spreading Rates as a ux tion of
Oil Density 67
7 Area Spreading Rates vs Temperature for
100 cm - 3 Pools of Fullerton Crude 72
8 Relative Drift Rates 89
9 Combined Wind a d Current Drift 9 1 L
V
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FIGURES
No. Title Page
I. Lens Cross Section 7
I I . Neumann’s Triangle 10
III. Lens Angle U
IV. Test Basin Layout 37
V. Wind Duct Support Systan 39
VI. Transition Piece Arrangement 141
VII. Wave }Iachins 142
VIII. Wind-Water Tunnel
IL Different Lens’ Shapes 147
X. Log Volume vs Log Diameter for Light
Paraffin Oil on Water 149
XI. Log Volume vs Log Diameter for Water
on Dibutyl Phthalate
XII. Generalized Value vs Diameter Correlation
for Lenses
XIII. Pool Area vs Time for 100 cm 3 Pools of
Various Crudee 6
XIV. Spreading Rate vs Density Factor 66
XV. Oil Spreading Rates (parameter volume of spill) 69
XVI. Initial Spreading Rate vs Pool Volume 70
XVII. Spreading Rates for 100 cm 3 Pools of
Fullerton Crude vs Viscosity of Underlying
Water 73
XVIII. Disc Model of Spreading Pool
XIX. Percent Drift vs Wind Speed 76
XX, Velocity Profile of Air-Water System 81
XXI. Draggers 93.
vi
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SECTION 1
CONCLUSIONS AND RECONMENDATIONS
1. The diameter of floating pools of oil thich form lens (those for
which °L - - > 0) can be approximately predicted as function
of spill volume through the use of quation (8), derived from
Hardy’s asymptotic lens thickness correlation.
2. Lens diameter vs volume can be more accurately predicted using
Equation (53) a simple but fairly accurate approximation developed
from Langmuir’s more exact but complicated equations for lenses.
3. A fair nwt er of crudes form lens like pools on water on which
an oil film has previously been deposited, but most crudes
will not form lenses on clean water. Crude oils which do not
form lenses will ultimately form fume roughly 0.013 to O.O l mm
(0.0005 to 0.0016 in.) thick.
14. Cr ides can be induced to form lenses by treating the surrounding
water with a surfactant which sufficiei tly lowers the water
surface tension aT without excessively lowering the interfacial
tension ij, (i.e. which causes L — - to become positive).
Such lenses will be forty to one hundred times thicker than the
thin films which would develop in the absence of lens formation,
and the area of coverage and pollution will be reduced by a
similar factor. Surf actant induced lens formation may represent
a useful technique for minimizing the spread and area of pollution
of oil films.
5. Small scale crude oil spills, will after an initial induction
period (in which energy conversion limits the rate of spread),
spread so that the pool area increases at a constant rate with
respect to time. This constant rate is a function of oil and
water density, spill volume, and water viscosity namely
r-# p 1 ,8
L
dO iL L/
This constant spreading rate is ultimately followed by a lower
spreading rate which has not yet adequately been correlated. For
oils that are susceptible to the induction of lens forxi tion the
area of spread may approach a constant value.
-l
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The preceding equation predicts excessively rapid spreading
rates for large volume spills) and another correlation -
perhaps that characterizing the lower secondary spreading rate
mentioned above - must apply to larger volume spills.
6. Wind will cause oil pools to drift on relatively calm water in
the direction of the wind at a velocity equal to 3.66% ± .l7
of the wind velocity - a figure which is in good agreenEift
with those reported for the Torrey Canyon drift. Waves will
cause significant reductions in the rate of wind induced drift,
but this wave wind-drift interaction has not been adequately
correlated yet.
7. Though negligible wave induced drift was noted in the resent test
work, on the basis of past work by other investigators it appears
that significant wave induced drift is likely to occur in deep
water. The n asurenent and correlation of wave drift and wave-
wind drift interactions will require testing in deep water basins.
8. Wind induced drift and current induced drift are not simply
additive. The data obtained to date is roughly correlated by
the equation
T W + o. 6C
there T is the total drift, W the wind drift and C the current
drift. Though this equation must break down as W goes to zero,
no significant change in the coefficient for C was observed then
W was varied. The above correlation is suspect.
9. Ci ]. tends to drift at a slightly slower speed than solid object
floating in a current in the absence of wind. In the case of
wind induced drift, oil will drift slightly faster than solid
objects floating on the surface.
10. dnd induced drift is ccnflned to a thin layer near the surf ace of
the water, and iaUow floats which confine the oil and have drag
surfaces penetrating one or two inches down into the water will
substantially reduce (e. g. by O%) wind induced oil pooi drift.
Floating nets of such draggera n y represent a feasible means
of greatly reducing the drift and also the spread of oil spills.
It is reco sended that:
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1. The wind drift correlation be substantiated by testing in a longer
(e.g. 14.0 ft. long) wind-water test basin of improved design. This
should insure the attainment of steady conditions, the detection
of fetch effects, and increased accuracy.
2. Wave drift and wave—wind drift interactions be investigated using
a deep water (e.g. 6 ft. deep) wind wave test basin. Correlation
of such drift and interaction appears necessary in order to
adequately predict oil pool drift.
3. The drift correlations developed herein be verified by compari son
with available field data.
I . Efforts be made to systematically obtain records of the wind, wave,
current and tide conditions in the area of accidental spills and
the corresponding travel and spread of such spills. This field data
should be used for further improvement and verification in correla-
tions for spreading and movement.
5;. That the feasibility of surfactant induced lens formation, and
drag networks for limiting the spread and movement of spilled oil
be investigated.
6. That spreading rate measurements be made on spiiis of larger volume
than tested in this work and that a spreading rate correlation
suitable for large volume spills b& developed.
-3-
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SECTICN 2
Th TRCrUCTICN
The spillage of oil at sea and in estuarine and coastal waters has
occurred with increasing frequency in recent years. To a certain
extent deliberate spillage due to dumping, the pumping out of bilges
and tank bottoms, has been curtailed by international convention (h2)
and the use of slop collection tanks at refineries and terminals.
However the advent and increasing use of large tankers and off-shore
drilling has greatly increased the probability of accidental large
scale spillage.
Table I represents a list of major spills and some typical minor spills
which have occurred in recent years. Also listed are the estimated
spill volumes, and, where available, the longest distance which the
spill traveled while still in a reasonably coherent or identifiable
form. Detectable spills of some magnitude occur alii st every week along
the Northeast Coast of the United States.
Not only have very large spills occurred but these spills have often
traveled considerable distances. In addition to those spills listed
in Table I, spill travels of SCO miles have been reported by Egyptian
authorities, O miles by Canadian authorities (1i2), and Stroop (71)
has tracked oil slicks for up to ninety miles from the noint of spillage.
In the case of very large spills oil concentration remains at a noxious
level even when the spill is dispersed over a very large area. As the
area of spread becomes larger, dispersion by the turbulent diffusive
action of the sea becomes progressively less effioiei t; and greatly
increased time is required for the oil concentrations to fall to a non-
noxious level. Thus large spills persist as identifiable entities for
long distances and times.
Until recently the use of detergents to induce and aid dispersion and to
clean contaminated shorelines has been the primary method of combatting
spills. For large spills this practice has proved increasingly in-
effective. The oil, while dispersed into droplets or emulsified, persists
at higher than tolerable concentrations until spatially dispersed by
turbulent diffusion, For large spills this occurs so slo r1y that clean
shorelines are frequently polluted or repolluted by redeposition of oil
washed off previously fouled shores. In other instances the use of
detergents led to the formation of persistent water in oil emulsions
- b -
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Table I. SpiU Volume and Distance Traveled for Recent Oil Spills
axiimiy
Spill Volume Distance
Date SpiU Source Location Gallons of Travel
3/18/67 Torrey Canyon (tanker) Cornwall, +
England 9,200,0O0 3 0 miles
3/7/68 General Colocotronis Eleuthra Island, 610,000 to
(tanker) Bahan s 1,250,000 10 miles
3/3/68 Ocean Eagle (tanker San Juan, 1,000,000 to
Puerto Rico 2,800,000
3/ 7 Tampico Naru. (tanker) Baja,
California 1,680,000
2/1/69 Union Oil Well Santa Barbara 0
to 14/69 aiannel 7,000,000 30 nd.].es
3/16/69 !bbil—Newimnt Oil Louisiana
Well Coast 5 miles
7/23/69 Unknown Ship Atlantic City,
New Jersey 145 miles
1/21/69 Unknown Ship New London,
Connecticut ? 20 miles
19% Gerd Maerslc
(tanker) North Sea near
Germany 1,900,000
+ Gross distance traveled by oil which came ashore at Pointe du Raz
and the Crozon peninsula on 5/19/67 and 5/20/6? 52 to 55 days after
3/26—29/69 the probable dates of release from the Torrey Canyon.
O My estimate based on the reported area ultimately covered by the
spill.
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(umoussesU) and actually retarded dispersion. Further in many instances
detergents have proved to be toxic and more injurious to marine life than
the oil itself.
These facts clearly imply the need for new or revised techniques for
combatting oil spills, for developing criteria as to which technique
or techniques iould be used, and for efficiently deploying the re-
sources needed to implement these techniques.
The choice of technique and the efficient deployment of resources for
coinbatting spills depends in a large measure on the rate and extent of
spread and the direction and velocity of spill movement. Further, from
the point of view of law enforcement, detection of the source of a spill
of unknown origin can be greatly facilitated if rates of spread and
movement are known and can be predicted.
Until recently very little was kno n regarding spread and movemant. A
few experiments (71) have been carried out on the rates of spread and
movement, but most of these were qualitative and inconálusive.
The extent of spread of pure substances, particularly those forming
either coherent monomolecular films or thick stable lens like pools can
be predicted. Crudes and fuel oils, however, are complex mixtures and the
spread correlations for pure substances cannot readily be used except to
provide a theoretical. frameimrk for analysis.
Field records and observations for a number of major spills have recently
become available (after the start of this investigation). Based on these
records certain qualitative predictions can be made regarding spill spread
and movement (66). Bat these predictions are rough and based on average
values. Deviation from these average values are large and the causes for
deviation are not readily apparent.
It there! ore appears that, in spite of this new field data, considerable
uncertainty remains regarding spiU spread and movement, and that re-
liable predictive correlations are not yet available for such spread and
movement. The work described herein aims at developing such correlations.
It is directed towards empirically identifying, and at least roughly
correlating all major factors influencing spread and movement. It is
based on small scale test basin werk, and in common with all scaled down
experiments involves questions as to the validity of scale-up. 1’&ierever
possible attempts were made to resolve such questions experimentally or
theoretically, but in many instances the ultimate test of validity will
have to be made on the basis of comparison with available field data.
Such comparison Is recommended for future mork.
-6
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SECTION 3
BODY OF REPORT
Analysis of Prior State of Knowledge
Spread
The spreading of a i ire liquid or liquid mixture of stable composition
on the surface of another liquid with which it is immiscible (e.g.
oil on water) is determined primarjly by the surface and interfacia].
tension imbalance set up between the liquids (32). If one considers
a half cross section of a pool of floating liquid and its substrate,
such as shown in Figure I, the nature of the surface tension imbalance can
readily be seen. At some distance from the pool the surface of the lower
layer is horizontal, and ci, its surface tension acts horizontally. At
the center of the pool , the surface tension of -the uPper layer and
the interfacial tension also act horizontally.
crj acts to expand the pool, and a and oj to contract it. Thus if the
surface tension imbalance (oj, - - a 1 ), sometimes also called the
spreading force, is positive the pool will spread. If it is negative
the pool will contract until the surface tension forces are counteracted
by hydrostatic pressure, thus producing a pool which looks like a doubly
convex lens. Such pools are therefore called lenses.
Lenses
The force balance which determines the equilibrium thickness of a lens
irovides insight into the type of forces which are operative in controlling
both the equilibrium spread and the rate of spread of oil pools of all
types, both lens forming and free spreading, and will therefore be con-
sidered in some detail.
(Y
Ii
4
Figure I
—7 —
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In general a pressure difference exists across curved interfaces (3 ).
This pressure difference is given by the equation
p— crf -+ 3 (1)
ithere is the appropriate surface or interfacial tension and R 1 and
R 2 the local radii of curvature of the interface. For relatively large
pools, the surfaces at the center may be considered horizontal planes.
In suth a case, at the center line Ri . and R 2 are infinite and pressures
are the same on both sides of the air-lens interface and also the lens-
substrate interface. Thus p gt hydrostatic pressure above the lens-
substrate interface is equal to o 1 ,gh the pressure below the interface;
i.e. pj Ii oat, ithere p , is the density of the lower layer (substrate),
o the density of the upper layer (lens), g the gravitational acceleration,
t the lens thickness at the center line and h is the submergence of the
bottom of the lens at its center line below the undisturbed surface of the
substrate.
If one considers a thin slice centered about the median plane of a
lens so large that curvature in the horizontal plane can be neglected,
a free-body force balance in the horizontal direction can be set up.
The free body can be visualized as consisting of a thin slice parallel
to the cross. section shown in Figure I. The left hand edge of the slice
terminates at the center line of the lens, and the right hand edge in the
T substrateata point iere the substrate surface is horizontal. The bottom
of the slice is horizontal and just touches the bottom of the lens.
The net surface tension force acting to the left per unit length of lens
perimeter is -F 8 . The hydrostatic pressure forces per unit length are
tt
1 1 gp t 2
JP 11 .dz J gp zd = 2 2
acting to the right, and similarly
h 2
FpL — gPL (3)
acting to the left. In the above, the local hydrostatic pressure P in
eath layer has been simply equated, the density p of the layer times the
depth z and the gravitational acceleration g.
—8 —
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The net pressure force acting to the right is
F?x ( Qut 2 p h 2 )
2
But since p h p t we can substitute ( ) t 2 for h yielding
2
2
0 C 1 .L P ’°L )
( )
( )
] quating the pressure force acting to the right to the surface tension
force acting to the left and solving for t there is obtained
1/2
(6)
This assyinptotic relationship for very large lens was derived by
Hardy (32). This solution essentially applies to a lens so large that
it can be treated as a two dimensional case. The above value of t is
sometimes denoted to,
For large lenses, the upper and lower surfaces are flat for a considerable
portion of the lens diai ter and the average lenè thickness t should
be n derately well approximated by t 0 , . In such a case a given spill
volume V should be related to the equilibrium lens diameter D by the
following approximation
• ¶TD 2
14
Substituting for to, and solving for D yields
D ç
8 V 2
.
gp (l Pu 4
-
i/b
(7)
(8)
Thus if a spill tends to form lenses, as a first approximation the
lens diameter8 should be proportional to the square root of the
- 2 F 8
—9
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volume of oil contained in the lens. Alternatively stated, for
moderately large lenses, the lens thickness should be appro d.mate]
independent of the lens volume or diameter.
Real lenses of even moderately large size deviate somewhat from these
assyraptotic relationships. The diameter equation is intrinsically less
subject to de 1ation than the thickness equation. As will be seen later,
even for lens volumes as small as 1-cm 3 the actual diameter is usually
only 2O larger than predictedby &iuation (8) and even in the worst
cases is only 33% larger.
Neumann’s Triangle
In veloping volume—thickness-diameter relationships for
lenses of small to moderate volume, the angles the surfaces and inter-
faces form at the edge of the lens are significant intermediate
variables. These angles are shown in Figure II.
Since the edge of the lens is in equilibrium, the vertical and horizontal
components of the surface tension forces must satisfy the following
force balances, sometimes known as eumarJn 15 triangle.
(horizontal)
cT 7 COSAL
a 11 CO5 Au
+ 0j cos A 1
(vertical)
ajsinAj =
sin A 11
4 ar sinAj
AL
Figure II
(9)
(10)
-1.0 -
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gV
(11)
There are two equations in three unknowns, A.d, k and A 1 . A third
equation is needed to perntit a solution. Such an equation can be
pro’vided by an overafl vertical force balance for the lens
gV o + 2TrRc sinA
where V is the volume of the lens, VL is the volume of substrate
which it displaces, and the last term is the vertical component of the
surface tension of the substrate times the peri eral length over which
it is acting, R is the lens radius.
V and VL can be determined as functions of R if equations are available
or can be developed describing the shape of the liquid surfaces and
interfaces of the lens. Unfortunately these equations usually involve
angles Au, and Aj arid the solution of Equation (11) while possible is
far from simple and, except in certain limiting cases, usually involves
trial and error.
The u].tiniate solution
for a related pair of
for• angles A arid A 1 is facilitated by solving
angles shown in Fjgure III.
Here we have the somewhat simpler component relationships
005 B
u 1n B
+ cos Bj - CL
+ c 81iiBj 0
Bj being taken as a negative angle.
Figure III
(12)
(13)
- 11 -
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These can be solved 1 m i ii taneous].y to yield
cosB 1 ( L2_C2+C 2 )/2CL j) (Th)
( 2 + 2 — (72 )/2 L ) (15)
And
B -AL (16)
Aj Bj + AL (17)
Thus if AL can be determined from Equation (U) A and A 1 can be
determined through the use of Equations (iii) through (17).
Values of AL, A , and Aj have been experimentally determined by CoghIU
and Anderson (iB) and Lyons (Iii). The value of A 1 is quite small
usually 100 or less. Lyons claims that the angles do not nec’tessari]..y
satisfy neumann’s triangle, particularly for small lenses. However
it can be noted that in those cases where significant deviations from
1 enaaim’s triangle exist Lyons has used the literature values for the
surface tensions of pure liquids in his computations, rather than
measured values of surface tension when saturated with the other liquid of
the pair. These surface tension values, particularly those of water, can
be significantly lower, thereby explaining most of Lyons’ discrepancies.
The force balance used to derive Equation (6) essentially involves treating
the lens as if it were a flat disk. It considers the surface and inter-
facial tension of the horizontal surfaces only, neglecting the surface
tension forces due to the short vertical sides of the hypothetical disk.
Li proved correlations for t, t, and D can be obtained by considering a
mere rea].istic n de1 of the lens shape and cross section. Surprisingly,
while two dimensional solutions are readily available finite exact
solutions for the three dimensional surfaces of revolution produced
by the interaction of surf ace tension and gravitational forces do not
exist. Tabular numerical approximations, of great accuracy, the thforth
and Ad me tables ( 7 ) do exist, but these are out of print and very rare.
Ba zforth and Adams tables may be used for predicting lens volumes and
diameters. A procedure for doing so was developed in the present.
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investigation, but it is quite complex, difficult to apply, and
therefore will not be discussed.
Simpler modified versions of the Bashforth and Adams tables Dorsey s ( 27)
and Pbrter’ a (59) tables for sessile drops and Sugden’ s ( 72) tables for
meniscuses are used for correction purposes in surface and interfacial
tension measurements. However use of these tables apply only to surfaces
and interfaces which are vertical at their edge, and thus they cannot be
used for predicting the characteristics of lenses.
Langnhlirss Work
The most comprehensive work dealing with lenses is that of Langnndr ( 37).
A fair number of interi ’ diaté steps are left out in the derivation of
equations in Langmuir’s article. Since these omissions can prove quite
confusing, Langmuir’ s approach is recapitulated below.
In essence Langmuir uses a method of successive corrections. First he
develops equations for the cross-section,boundary curves of the surfaces
and interfaces of a lens which is so large that it can be essentially
treated as a two dimensional case (i.e. its curvature in the horizontal
plane can be neglected). Using these equations for the boundary curves,
he develops expressions for the extent to which the surface tension and
hydrostatic pressure forces for the cross section so defined deviate
from the corresponding forces in the simple disc model used to obtain
tc ,.
Langmuir calls the total deviating force, obtained by integrating the
local deviating forces over the entire width of the lens, the “linear
tension”, f. The I1j a tension” appropriate to each surface or
interface is shown to be equal to
3/2
I = .?. a fi—cos ( ) 3 (18)
3 2
where o is the surface or inte4 cia1. tension, a is the capillary
constant, 1 2 a/g ( o 2 —Pa 1 ) N ‘, P2 being the density of the lower
liquid and p the density of the upper liquid (or gas) bounding the
interface, g is the gravitational acceleration, and A is the angle from
the horizontal at which the surface intersects the edge of the lens.
-13 -
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Using values of f for the surfaces in question, Langmuir then corrects
the equations for the cross section boundary curve so that f is taken
into account. The equations for these corrected boundary curves are
then solved aji ,1taneously to obtain an equation for the thickness of
the lens at its center line, namely
r L fu fj
t go (P -o ) a
I (19)
L
ithere the subscripts L refer to lower layer, u upper layer) and i
interface.
Using the equations for the lens boundary curves Langi’tdr shows that
for soderately large lens the lens volume V is given by
V 7T B 2 — IT B (aj 2 sin A + a 2 sin A ) (20)
The various f terms are functions of the angles A. 1 , L, and AL. Thus
both t and V are functions of these angles. The exact value of these
angles cannot be determined solely from Nenniann’ a triangle, but, as
previously indicated, depends on AL the angle of the substrate surface.
Using the equations for the boundary curves, Langimdr was able to solve
for ALin the case ithere the various A were s al1 enou for terne of
order hi ier tb 1l/21 to be negligible in the series expansion for
1-cos A. In this solution L is a fairly oon 1ex function of the capillary
constants aL and a , the lens radius R and thickness t, and the liauid
densities. AL thus varies as the lens volume varies. It approaches an
assymptotic value A as B goes to infinity and t goes to t . This is
5LBU _ (PLP )J 2 to ’
(21)
+
From Langrnuir’ a experimental data it appears that AL differs from
A byr only 1% for lens as smell as 1 cm. diameter. Thus negligible
error is involved in using A for AL. His calculated values for AL
are in good agreement with those experimentally measured by Coghill and
Anderson ( 18) aM Lyons (lii).
Trial and error solution are required if one is to rigorously solve
for V , t, and t as functions of B.
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Direct approximate solutions, good for lenses of large diameter, can
be obtained by using .A , and the corresponding value A , and .Aj
in Equations (18), (19) and (20), t being given by the expression
v/n R 2 . Langimair ‘a approximate solution for t is about 9% in error
for lenses of 2 cm 3 volume, and the error rapidly decreases as the
volume increases.
Langnn]ir’s Equations provide a significant improvement over the approx-
imation t t . They are however not rigorous and limited somewhat by
assumptions made in their derivation. In some instances, particularly
those where a is greater than nj,, the neglect of the AI1/21i. term in the
series expansions for 1 -cos A can lead to an 8% error in the value of A.
For very small lenses the radius of curvature of the lens surfaces at
the center line is sufficiently small that the assumption Pu t p h
no longer holds. In Equation (19) the a/ fl correction terms which
occur in the denominator are approximations which are true only when the
respective angle A corresponding to the particular a is small. 1 ’bre
exactly the term should be a sin (A)/(2)3/ 2 sin (A/2); and when c >
this nxre exact form deviates significantly (up to 29%) from a/ J7
for the case of aj, the nxst in rtant of the correction terms. The
assumptions upon which the aj/J2 corrections in Equation (19) are
derived appear to be somewhat arbitrary. Equation (20) appears to be
strictly applicable only for large lenses, otherwise the term multiply-
ing a 4 2 sin Aj should be of the order IT (Ft - ai sin AiX 2)3/2 sin 4j12)
and tfiat lTalltiplying a 2 sin A.d of the order IT (Ft - au sin A 2)3/ Sin Au/2)
rather than IT R in each case. Despite this fault finding the errors in
question apply mainly to correction terms, and their overall effect is
probably not significant.
Lens Stability -
Very little information appears to be available on the ability of lens
to resist deformation and breakup under the influence of externally
applied stresses, such as mipht arise from wind or wave action or
turbulent eddies. Bradley (13) has derived an equation for the period T
with which a lens will oscillate when distorted horizontally from a
circular to an elliptical shape. This equation is
T — 2ff (22)
t 2 - 2 F 5 + b f/R J
where f represents the summation of the linear tensions f , ‘L’ and
Li.
-1 -
-------
From this equation it can be seen that the tiii required for a deformation
to relax increases as the lens radius increases, and for lenses of large
size this rela tion tine is proportional to it. The terms in —F 3 and t2
dominate the denominator for large values of R, and for large lenses t2
is almost directly proportional to -F 5 . Thus if we substitute for —F 0
in terms of t or vice versa it can be sho n that f or large lenses T
is inversely proportional to ti! 2 or _F 3 l/J4 Thus It appears that
deformations in large radius lenses or lenses of small thickr ss or
those with small values of -F 3 should take a long tine to relax. Since
relaxation is brought on by the action of forces opposing deformation,
lenses which take a long tine to relax must possess less deformation
resisting capability. Therefore large lenses, thin lenses, and lenses
with low values of —F 3 should be more susceptible to breakup.
Non lens Formers
Pure materials for which F 3 is positive do not form lens and in theory
will spread to form layers of near monomolecular thickness. Cils, which
are complex mixtures, exhibit more com )licated behavior but will never-
theless form films which are very thin. The few thickne ss measurements
in the literature and estimates based on spill spread areas reported by
Bloldçer (11), Sigwalt (6 ), Stroop (10) and Smith (66) indicate average
thicknesses ranging from 0.013 to .Ctil ian (o.000S to 0.O0l in) conipared
to thicknesses of the order of l.2 to l2. mm (O.o to O. O in) for
lenses.
Induced Lens Formation
If the surface tension of the water is sufficiently lowered without
lowering the interfacial tension of the oil by an equal amount
the spreading force F 3 can be changed from a positive value to a
negative value. Thus it should be possible to convert a spreading
oil into a lens former and greatly reduce the extent of spread of a
spill.
This idea in effect has been investigated by Sigwalt (63) and Garrett
(31) and briefly by Blokker (11). Garrett showed that a wide variety
of substances which tended to form monomolecular films could, when
spread on water, induce lens formation, and that the thickness of the
resultant lenses is roughly predictable by Equation (6) with (F 31 -F 3 )
substituted for F 3 , ithere F 81 is the spreading force for the oil-water
system, and 182 is the spreaaing force for the monolayer—water system.
It will be shown later that this is equivalent to lowering the surface
-16 —
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tension of the water by an aa unt equal to -F 3 ,. It can be seen
that if F 51 is positive, F 52 must also be positive and greater than
F 51 to induce lens formation.
Sigwalt showed that the spreading of oil film could be checked and the
oil film driven back by the spreading of a film of fatty acid fnich
tended to spread at a more rapid rate than the oil, but his ideas were
embodied in a less general, less quantitative form than Garrettts.
Since Garrett showed that the spreading rate for monomolecular films
tends to be proportional to the spreading force, Sigwalt’ s spreading rate
criterion is essentially equivalent to Garrettts (F 5 - F 32 ) criterion.
Blokker showed that traces of sodium alkyl sulfate by lowering the
surface tension of water induced the formation of a 1 to 1.5 mm thick
lens in a middle Eastern crude which normally spread to a thickness of
0.015 nn — i.e. reduced the area of spread by a factor of 100.
Iater in Oil Emulsions
Under certain circumstances, e.g. sloshing of oily ballast, and water-
containing oil, detergeflt treatment of spills, etc., water in oil
emulsions will form, and, if these are discharged or found at sea, they
exhibit irarkedly reduced spreading as compared to the oil from which they
are forii d. Emulsion film patches several inches thick have been
reported (71). In one instance (66) the data indicates a sea area of
roughly 1140,000 ft. 2 was covered by a emulsion layer averaging 90 umi
(3.6 in) thick. Following the Torrey Canyon incident, 100 to l O mm
(14 to 6 in) thick grease like patches, presumably emulsion, were
reported by the French Navy.
Such emulsion patches tend to gradually thin out, thin films spreading
from the edge of the patch, carrying away their patch oil content. If
the patch is broken up by sea action the thinning out proOess appears
to be accelerated. Such breakup appears to be favored by choppy waves
and not greatly influenced by swells (71).
It is interesting to note that the formation of stable emulsion patches
represents a method for keeping spilled oil pools in a compact form
if desired. Since in a water in oil emulsion, oil, the continuous
phase, can not occupy less than 26% of the volume, the oil volume per
square foot of pool area is even greater than that of a thick lens.
Based on the reports of the French Navy following the Torrey Canyon spill
such emulsion patches have persisted f or as long as two months.
-17 -
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Spreading Rates
The spreading rates of floating liquid pools has been studied by
a nui ber of investigators, Reynolds (62), Brinkman (114), Gary and
R1c3.eal (15), Ramdas (61) Wooly (79) Garrett (31), Blokker (11),
Lippok (28), Sigwalt (655, Abbott (15 and Stehr (69). Of these
only the last five are concerned with large pools such as those
resulting from spills. The remaining are concerned with the spreading
of i rnomo1ecular films from droplets.
B].okker derives a relationship for the spread of oil down a long
straight sided channel. He assumes that the instantaneous rate of
spread is proportional to excess hydrostatic pressure in the pool which
he assumes is proportional to (t - t 1 ) P (1 -
(23)
where £ is the length of the pool, Q time and K the constant of
proportionality. This is not quite according with the hydrostatic
force equations developed for lenses, according to which this expression
should be [ (t 2 - t , 2 )/2 t I p. U - DU/UL) i.e. the hydrostatic force
per unit length divided by the pool thickness. However as t becomes very
small (as Is the case in B].okker’s work) the two expressions become
exactly proportional to one another.
In Equation (23) Blokker substitutes V/b, the spill volume divided by
the channel width for t, and integrates the rate expression for the case
where t , is negligible, thereby obtaining
2 ô
—2KI1—— 1 (214)
° P EJJ b
where £ is the original pool length. Blokker verified that , 2 did
increase linearly with time. The constant K varied from system to system
though. K and the rate of spreading were not inversely proportional to the
oil viscosity. For all the mate4als tested K only varied by a factor of 3
ranging from 9,800 to 30,000 mini, even though the viscosity varied by a
factor of 600. This lack of Influence of oil viscosity on spreading rates
was also reported by Stehr (69) and lock (14h).
—18 —
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On the other hand in Sigwalt’s experiments on spreading, which were
carried out over a temperature range of 7°C to 2 °C, the rate of spreading
was found to decrease as the temperature decreased. Both the ‘viscosity
of the oil and that of the underlying water increase as the temperature
decreases, and thus the effect produced by the change of temperature
may well have been due to the concomitant change in viscosity of either
the oil or the water.
If viscosity regulates the spread rate , as in ].antnar flow, the time
required to spreading a given distance should be proportional to the
viscosity. If one looks up water viscosities corresponding to Sigwalt a
experimental temperatures it can be seen that these viscosities are
roughly proportional to the spreading times reported by Sigwalt. For
example in one instance a 1.73 fold increase in spreading time is accom-
panied by 1.60 fold increase in water viscosity. In another instance a
1.62 fold increase in spreading time is accompanied by a l.% fold in-
crease in water viscosity. Sigwalt only identifies his test oil as an
“huile grasse activee”, and therefore it is not possible to determine
its variation in viscosity. Sigwalt’ a temperature vs spreading time data
and Blokker ( 11), Stehr ( 69) and l rlo ckts ( hh) observation that
spreading time does not vary greatly with oil viscosLty appears to indi-
cate that oil, pool spreading rates are roughly inversely proportional to
the viscosity of the underlying water. Unfortunately we cannot tell
from Sigwalt’s spreading rate data whether he was working with hydro-
statically driven spreading (thick pools) or surface tension driven
spreading (films). Since the mechanism determining spreading distance
may be different for these two cases, an inference as to the influence
of water viscosity on spreading rate may be invalid or may apply only to
one form of spreading.
Though the linear increase of with time is verified by Blokker ‘a tests,
The validity of the factor (1 - OLIN Pu! cannot be regarded as experi-.
‘ b
mentally verified. Blokker varied the initial pool thickness of the oil
in several of his experiments, but it is not apparent whether he did so
by varying the volume V, or whether he used the same volume and varied
the original length ,• Thus on the basis of the data as reported in the
Blokker article we cannot decide whether the rate of increase in 2 j
truly proportional to V. Somewhat similarly because K varied significantly
in the one series of tests where B].okker varied (1 - °U 0 u without
OL
markedly varying the other factors, the proportionality of d ,°/dG to
(1 - cannot be regarded as experimentally established.
\ P 1,
Blokker on similar grounds derived an expression for the spreading of a
circular pool. For the case where the final pool thickness can be
regarded as negligible this is:
-19 -
-------
2h
D3_D 0 3= K(1_ ) p V 0 9 (2 )
This postulated behavior of the pool diameter D predicted by this
relationship i s not subjected to experimental test. Since only a
fragmentary hydrodynamic model was used by Blokker in deriving his
equations, it is by no means certain that K should be the same for the
circular po e1 case as it is for the linear spreading case. If one attempts
to apply Blokker’ s equation for the spreading of circular pools to the
spread data reported by Stroop ( 71) for the spread of non-emulsified oil
at sea Blokker ’s formula, using his largest K value, predicts values of 1)3
which are 6 times too small. Evaluating D, (i.e. taking the cube root
of 1)3) reduces the error so that the predicted values of 1) are only 14 times
too small.
It might be noted that for monomolecular fi)ins spreading fron droplets or
small, lenses floating on the films Garrett ( 31) showed that the spreading
rate i s proportional to F 3 rather than the (1 - ) o,
For such fi1u’ the spreading rates range from rough].y b to 13 cm/sec.
Based on Garret’s data it appears that the spreading rates of monomolecular
films can be roughly (i.e. within + 39%) correlated by d6 = .38 F 3
Diffusive Spread
Diffusive spread must be considered in addition to hydrostatic and surface
tension drive spread in determining the gross area of coverage of an oil
spill. Under the stresses developed by the wind and seas oil pools when
thinned out sufficiently will usually break up into smaller patches. The
probability of recoalescence of these patches will be low,and driven by
the turbulence of the sea they will tend to diffuse away from one another.
For a smell spill this breakup and diffusion process ultimately spreads
the oil. over so much space that its concentration falls below the
nuisance level. Detergent treatment of small spills is designed to
accelerate this diffusion driven dispersive process.
Diffusion at sea has been the subject of much study (30, 149, SO, 51, 52 ).
It is a matter of considerable practical interest in that it helps regulate
gradients in salinity and t nperature and the dilution of sewage ou.tfa].ls
and radioactive wastes. Unfortunately neither the theoretical nor the
empirical, study of diffusion at sea is as yet well resolved. In particular
- 20 -
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the effective turbulent diffusivity appears to vary with time and
distance In a not whofly predictable way.
The classical (FIc cian) solution for diffusion from a point source ( li 8 )
H
(26)
where C is the concentration of the diffusing substance as a function
of r the radial distance from the source and 9 the time elapsed after
release of the source. N is the source strength and K is the effective
diffusiv ity.
If one makes various assumptions as to how the effective diffusity varies
with time and distance different solutions similar in form to Eouation ( 26)
are generated but with r and 9 raised to different powers ()49, 50, 5 1, 52).
Dye tracer studies at sea and in bays show that the Pick’s law solution
is not applicabl%. and though there is considerable scatter In the data,
solutions in WhiCh 92 or 9 . appear in the denominator of the non
exponential term fit the dye concentration data best. This means that
concentrations should decay considerably faster than predicted by the
Fickts law expression.
Typical values of K obtained from a dye tracer study, Foxworthy et a)..
( 30 ) range from 1.6 ft 2 /sec. after b90 sec. and Th ft 2 /sec. after
ttl3O sec. the lack of constancy in this case indicating the diffusion is
non-Fickian.
In any case 2 the concentration at any given time and distance is
proportional to the source strength (i.e. the oil spill volume). For
points reasonably distant from the spill, the oil concentration will be
proportional to the initial spifl volume and will thus take longer to
decay to a tolerably low concentration. The rate of decay varies with
the effective diffusivity K, but for Fickian diffusion there are two
limiting cases ithen K or t is very large and When K or t is very sinali.
In the first case the concentration will be inversely proportional to
time and in the second C wiU be inversely proportional to time squared.
For the first case the time required for oil concentration to reach a
non noxious level at a given distance r , remote from the center of the
spill, will be proportional to the spill volume and in the second case
to the square root of the spill volume.
- 21 -
-------
Close to the spill, the spill can no longer be treated as a point
source but must be treated as an area source. The sources corres-
ponding to each point in the original spill area must be added, i.e.
the sources integrated over the spill area. This source addition or
integration process leads to Slower decay of concentration near the
center of the spill. Essentially because diffusion is a “random walk
process”, when the spill concentration is uniform,oil from the edge
of the spill is just aslikely to walk towards the center as oil from
the center is to walk towards the edge. It is only when the concentration
near the center decays that the center concentration can begin to decay,
and the larger the spill the longer this process takes.
Drift-Wind Effects
Until recently very little quantitative information has been available
about the drift of oil deposited at sea.
Stroop ( 71) followed a series of test spills. Analyzing his oil drift
data it appears that oil drift over a given period of tine everaged 3. 981
of the total wind travel during the sane tine. The scatter about this
average was very great the percentage drift ranging from 1.38% to 6.58%.
It is impossible to tell from Stroop’ s reported data whether any appre-
ciable currents were present — the tests were in mid ocean near Hawaii, -
and whether the wind direction had shifted significantly during the
course of the tests. The slicks were tracked for a considerable period
of tine - between 23 and 117 hours.
Following spillage of oil from the Gerd I4aersk in l9 the German
Hydrographic Institute of Hanburg tracked the spill and concluded that
it i oved at about ! . 2% of the wind velocity (Tomczak). This movement
occurred in the shallow coastal waters off Germany and Denmark and tides
and coastal currents may have been complicating factors.
Thiring the course of the present study the book tTorrey yon’ Pollution
and Marine Life edited by J • S. Smith appeared. Chapters 8-and 9 of this
book are valuable though somewhat fragmantary sources of information
concerning the drift oi]. spilled from the Torrey Canyon. On the basis
of the drift of recognizable patches of oil which were charted using
aerial photographs taken by the Royal Air Force, Smith concluded that
the oil moved in the direction of the wind at an average velocity equal
to 3. i% of the wind velocity. The foUowing table is adapted from
— 22 —
-------
Table 26 in Smith ‘s book and u esents the data upon which be based
his conclusions.
Table 2. Oil Drift vs Wind Speed
Path Vector Distance
Se nent Traveled
(Nautical Ni].es)
,h Percent Drift (Oil Velocity/Wind Velocity) x lC O
Our corrected figures. These values appear to be in
error in Smith’s table
It can be seen that the
age, ranging as low as
not appear to correlate
over the wind has blown
of variation may be due
Smith.
Direction
of
Movement
percent drift varies widely about the aver—
2.61% and as high as IL.7 2 %. This variation does
with wind velocity or with fetch, the distance
without markedly shifting direction. The cause
to tidal and current influences not reported by
For all path segments except F, the last, the oil moved at a slightly
less clockwise angle than the wind. The oil movement for segments A
through E averaged 7.8° less clockwise than the wind.
The counterclockwise shift in oil movement may have been due to the
current in the Fkiglish Q tanne1 which though weak and rather variable
tends to run in the East North East direction (3S), the correct
Elapsed.
Time
(hrs)
-7’
Percent
Drift
Wind
Velocity
(knots)
Oil
Wind
Oil
Wind
A
23.3
831i
h8
2.79
l7.t
ii6°
1200
B
23.8
911
102
2.61
8.1
69°
? 3
C
29.9
633
1 L3
.
1. .72
IJ .8
81°
98°
D
20.3
6
97
3.6S
7
]J 5O
1S2°
E
27.1
8h9
73
3.19
11.6
63°
700
F
Total
hi.?
166.11’
1081
t 86Ii.
98
1 t 6 1
3.86
3.l l
11.1
10.6
l 3°
91°
123°
9 °
-23 -
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direction for producing the shift noted. In contrast to all the other
segments the wind direction during segment F was less clockwise than the
direction of oil movement and differed fairly markedly in angle, i.e.
by 300. However during segment F the oil was in the Gulf of St. i1alo
a region noted for large tides, and these tides may have caused the
deviant behavior.
Thus barring other influences Smith’s conclusion that oil patches will
move in the direction of the wind seeme reasonably correct (i.e. at
least within roughly- 8°).
As noted currents in the nglish Channel are weak and rather variable in
direction. This low degree of influence of current may explain part of
the success of Smith’s correlation of oil patch drift with wind velocity.
Although there is a great deal of scatter in the data, it is noteworthy
that when the oil movement direction deviates most markedly from the
wind direction, the percentage drift tends to be high. This suggests
that the directional devtation and the percentage drift increment might
both be due to current induced drift.
On the open ocean far from shore currents are usually moderately weak,
e.g. O.1 to 0.6 knots and quite variable in direction. Current induced
drifts of this magnitude, or smaller, aiding or opposing wind induced
drift could have caused the variability in percentage drift calculated
for Stroop’i open ocean work and also the difference between Stroop’s
percentage drift results and those for the Torrey Canyon. Similar
factors could explain the difference between the Gerd Naersk percentage
drift and the Torrey Canyon drift.
Smith pointed out that the movement of the oil which came ashore at
Pointe du Raz and the Crozon Peninsula 52- .5 days after its estimated
release fran the Torrey Canyon was also reasonably well predicted by
utilizing a 3.3 percentage drift figure. Here the predicted distance
moved was 20% greater than the actual movement. Smith noted that this
difference could have been caused by a slow northerly current having a
speed of only .O knots.
The percentage drift found by- Smith agrees well with measurements made
by Hughes ( 33 ) of the drift of plastic envelopes floating close to the
surface of the Atlantic Ccean. Hughes found that the envelopes drifted
in the direction of the wind at a velocity equal to 3.3’ of the wind
velocity. Hughes recommended that in the absence of direct wind velocity
data, that the wind velocity be estimated as 2/3 of the geostrophic wind
-------
obtained from isobaiic plots. Where necessary, i.e. on the open sea
in the absence of wind monitoring stations, Smith used this procedure to
calculate the wind velocities on which his percentage drift figures
re calculated. In all cases but one, where checks could be made, the
wind velocity calculated in this manner agreed with that obtained by
direct measurement of nearby wind monitoring stations.
It should be noted that though the agreement between Hughes and Smith’s
results is good, the question of wind induced surface drift is a subject
of debate in Oceanography and results deviating from those of Hughes
have been reported. For example Thorade (73 )reports that the surface drift
V (in centimeters/sec.) is given by the following equations
2.59 /V for w < 6 meters/sec. (26)
/sin
1.26 W
and V j for W > 6 meters/sec. (27)
where W is the wind speed in meters/sec. and is the latitude. It is
obvious that this relationship must break down as the Equator is
approached. If we substitute the latitude of the English Channel (50°),
the second of Thorade ‘s equations predicts a percentage drift of l.1 3%
which is much lower than that noted by Smith and Hughes.
The question of wind induced surface drift depends in large measure on
the shear stress deve] ped by the wind. in blowing over the water. This
stress in turn depends on the drag coefficient C, and the drag coefficients
for such flow are subject to a great deal of uncertainty and debate.
Figure 8.18, page 209 in Neumann and Pierson’s Principles of Ithysical
Oceanography indicates some of the highly scattered drag coefficient data
and two of the principal proposed correlations, namely: I that the drag
coefficient is independent of the wind velocity and 2 that. the drag
coefficient Is inversely proportional to the square root of the wind
velocity (1 8). There are many complicating factors, i.e. the waviness of the
sea surface and air turbulence which may account f or some of the scatter
and preclude a simple drag coefficient correlation.
Further It appears, following Ekmán, that Corio].is forces should cause
surface currents to deflect to the right of the wind in the Northern
hemisphere and to the left in the Southern hemisphere. Deflections of
about j5O have been reported for the open ocean Krummel. In shallow
- 25 -
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seas the deflection is significantly smaller and has been found to
decrease with increasing wind velocity. A typical correlation, Witting( 77 )
relates a the angle of deflection to the wind speed W (in meters/sec.)
by the following equation
30° —7. JW (28)
This) for the range of wind speeds encountered following the Torrey Canyon
incident would predict a deflection of ]h°, clockwise. Other similar
correlations, Nenm nn (1i7) indicate deflection of about 100 clockwise.
Except for the last portion of drift in the Gulf of St. I ].o the Torrey
Canyon drifts average 80 counterclockwise from the wind direction. This
noderately large 18° to 22” deviation as previously noted may have been
due to prevailing currents flowing towards the North.
Drift-Current F fects
It is comncnly assumed, either explicitly or complicitly, that oil drift
will occur at the same rate as the drift of water at the surface of the
sea. This surface drift and thus that of floating oil wifl be the sum
of drift due to wind, wave current and tide action. This assumpt1on
while plausible and providing a good working bypothesia is by no means
proven. J.E. Smith (as shown in Figure 33 page l 2 of “ Torrey Canyon ’ t
Pollution and Marine Life ) makes this assumption when ve torial1y
adding the tidal current to the calculated wind drift in an effort to
trace the path of an oil spill near the Torrey Canyon. The results in
Figure 33 qualitatively support Smith ‘a assumption, but quantitative
agreement is not de’ rnstrated - even taking into account local variations
in tidal strength which Smith neglected.
The assumption that drift vii ]. tend to occur at the same velocity as
surface currents was also used by the British 1 .nistry of Transport and
Civil Aviation (1&2) in devising boundaries for zones near the British
Isles In which the dn’ ping of oil ballast was prohibltedb
Mapa and charts presenting average current data as a function of position
in the ocean are available (3 ). These currents are fairly variable and
though the variability can in part be specified thz ugh the use of current
roses and stability iMices, the prediction of current induced drift
involves undertainty because of this variability. Current induced oil
drift is therefore best predicted using current measurements taken near
- 26 -
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the time and place of the spill rather than current charts. In most
instances ocean currents are about 0. S knots but in some instances
(e.g. near the northeastern coast of South J merica) currents of about
2 knots wi]]. occur. Since for winds of moderate intensity (i.e. 10 to
20 knots) wind induced drift should be of the order of 0.3 to 0.7
knots it can be seen that current induced drift can not be neglected in
most instances.
In shallow waters surface flow due to tides can reach rather hi
velocities, much greater than that produced by currents on the open ocean
and wind drift. Tidal amplitude tables and tidal current tables are
usually available for most coasts. \ iere tidal current data is not
available currents can be predicted from tidal amplitude and period data
and topography using the method of Defant (22 ). In general tidal flows
can be predicted with much more accuracy than ocean currents.
Drift-Wave Effects
Stokes has shown that there is a steady second order mean forward
velocity U associated with free surface wave motion. Based on the
assumption of irrotational flow and a sinusoidal wave form he developed
the following equation for this forward velocity
A 2 COBb 19t( Z h) A 2 ts 211h
— — coth ( )
U 2h (29)
, 2Tih
ere cosh is the hyperbolic cosine, auth the hyperbolic sine, coth
the hyperbolic cotangent, A the wave amplitude (1/2 the wave length),
Z the distance below the mean surface elevation, h the water height,
m the angular frequency,(i.e. ,) = 21T/T,’where T is the wave period), and
A is the wave length. For the free surface, Z = 0, and the expression
for u0 the surface drift becomes
___ 2 2”h A 2’ih
1J = 1 + coth ( —c) — —r ()
For h/A >> 1, i.e. for deep water, this reduces to
2ff A 2 W (31)
- 27 —
-------
or alternatively
( 2!T £2 )
U 0 AT (32)
For deep water, wave velocity depends on wave length and because of the
interdependence of wave velocity, frequency and wave length these
expressions reduce to
2T 3/2 £2 g 1/2 (33)
or
This would appear to imply that wave induced drift in deep water increases
as the period and wave length decrease. This is true if the amplitude A
remains constant. However the maximum value of A is limited to one-
fourteenth of the wave length and when this maximum value or its equivalent
period is substituted for A there is obtained
( )3/2 1/2
(Ag) (3S)
max
or
2TrgT
196 (36)
However it is extremaly rare that waves of n iimant amplitude are generated.
In a fully developed sea for a twenty knot wind the average wave height
is only about &0% of the maximum height corresponding to the wave period
(8 Sec.) in which must wave energy Is concentrated. In a fully developed
sea for a 30 knot wind the average wave height is only . 2 of the maximum
for the period (12 sec.) where must energy is concentrated and for a
knot wind (period 16 sec.). Thus for wii waves in a deep sea,
drifts will range between 0.27% and 16% of the maximum values.
Based on average wave heights taken from the wind wave spectra of
Pierson, Neumann, and Jamas (5 ), and the equations derived above
from Stokes formulas, the following table of estimated wave drift for
fully developed seas corresponding to various wind speeds has been coinDiled.
—28 —
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Table 3. Estimated Wave Drift for Fully Developed Wind Waves
Period Average 0 Estimated + Conditions
Wind iI ere Nost Wave Drift Speed Percent Required
Speed Energy is Height (knots) Drift for Full
(knots) Concentrated (ft.) Development
(Sec.)
Fetch Time
(nautical (hi’s.)
ITtiles )
10 14 0.9 0.0142 0.12 10 2.14
iS 6 2. 5 0.033 0.22 314 6
20 8 5 0.056 0.28 75 10
2 5 10 9 0.092 0.37 160 16
30 12 114 0.129 0.143 280 23
140 16 28 0.217 0.514 710 142
50 20 148 0.326 0.65 11420 69
o Average wave height - 2 A
+ Percent Drift (Estimated Wave Drift Speed/Wind Speed) x 100
The above table only has qualitative significance and may only be used
for very rough estimation. The distribution of wave heights and periods
in the wave spectrum was not considered - 10 of the waves have heights
roughly twice the average height, and one-third have heights roughly
1.5 to 1.6 times the average height; the wave periods in which there is
significant wave energy content range roughly from twice to or half the
period where nxst wave energy is contained. Since the drift depends on
the amplitude squared (A 2 ) and the inverse of the period cubed (1/T 3 ),
the use of simple average values for A and periods of maximum energy
content for T is not strictly valid. Further wave spectra are somewhat
variable, and somewhat different spectra, e.g. Moskowitz (145) and
Pierson and Noskowitz (514) have been reported for fully developed wind waves.
Nevertheless it appears that wave induced drift should be of significantly
lower magnitude than the direct wind induced drift (roughly 20 of the
wind induced drift or less). This is however subject to debate. Computer
studies are under way elsewhere at New York University (in the Meteorology
and Oceanography Department) in which the spectral characteristics of the
— 29 —
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waves are considered, ( 17). These studies indicate that nuich of what
is thou t to be wind induced drift may be drift lnthced by wind
generated waves. (Note: Our experimental results do not agree with
these computer based results.)
It mLght be noted that the average wind speed during Stroop t s drift
tests was ].9.S knots whereas the average wind speed during the tracking
of the Torrey Canyon oil patches was 10. knots. The effect of this
difference in wind speed on wind-wave induced drift may account for
roii 1y O.l6 of the OS7% difference between Stroop’s average 3.98%
drift and the 3.141 average drift for the Torrey Canyon oil.
Test basin results, Mitchint (tt3) and the U.S. Beach Erosion Board ( 76),
indicate that Stokes equations are valid for deep water waves. However
other tests, Coligny ( 19), U.S. Beach Erosion Board ( 76) and Bagnold,
indicate that Stokes equations do not hold for shallow water. In fact
in so instance, Bagnold ( ), backward notion has been observed at the
surface.
Longuet Higgins ( l 0) taking into account the viscosity of the fluid -
i.e. abandoning the assumption of irrotational flow - developed drift
equations which differ from those developed by Stokes. These equations
provide better agreement with shallow water test *,rk, but since in nost
instances we will be dealing with deep water waves, they will not be
discussed further.
seal waves are not sinusoidal as assumed by Stokes and Lonquet-Higgins
in their analyses of wave drift, and wind waves as previously noted
consist of a broad spectrum of wave frequencies. Chang (16 ) building
on the wave spectrum analyses of Tick and Pierson derived expressions
for drift in terms of these wave spectra. Making suitable substitutions
Qiang’ s expression for drift at the free surface can be shown to be
14 0’ g 5 ( ) d c i )
where S (w) is the spectral distribution factor of wave energy (pro-
porttonal to the square of the displacement Z from the mean surface
level) as a function of the frequency w.
Qiang obtained good agreement with this equation in tests using r*nd i
ird.xed waves in a deep water test basin. This equation reduces to Stokes
equation for waves of a single frequency.
- 30 -
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If one substitutes for SZ(w), Pierson and Moskowitz’ s generalized
wind-wave spectra expression
‘ yg 2 -B(g iw)
S(w) — e (38)
2w
where W is the wind velocity, and carries out the indicated integration
one obtains
IY .022 W (39)
for the wave induced drift velocity. This result indicates that for
fully developed wind waves the wave induced drift velocity should be 2.2%
of the wind velocity. This is substantially higher than indicated by our
previous method of estimation and substantially higher than obtained for
wind drift measurements at sea. It should be noted that there are fairly
large quantitative differences between the Pierson and Moskowitz spectrum
equation and the spectra used to obtain the amplitude data for the ca].cu-
lations in Table 3.
Both sets of wave drift results computed so far have been for fully
developed wind waves. As indicated in Table 3 It takes a considerable
fetch and period of blow for full wave development to occur at high wind
velocities. In the case of the Torrey Canyon drift arid even in Stroop’s
tests the wind velocities were low enough that full development was probably
present nest of the time.
Because winds give rise to wind waves,and wind waves give rise to wave
induced drift) it should be apparent that the separation of wind induced
drift from the drift caused by waves generated by the wind can be a
difficult problem. The problem is further complicated by the fact that
wind Induced drift should depend on the drag coefficient for wind flow
over water. This drag coefficient is very likely to be a function of the
waviness of the water. To the extent that wind drag induces waviness
(essentially turbulence) it is less likely to directly induce a nniform
directed drift. That is as greater a unts of waviness are generated) n re of
the i mentum transferred from the wind is likely to go Into maintaining the
waviness rather than directly maintaining a mean forward drift velocity.
The complexity of interaction between wind aid waves is great. Similar
interactions may exist with respect to combined wave n tion and current
imtion, and current induced drift aid wind drift, and the principle of
additivity of drifts, while initially plausible, may now be seen to be
subject to serious questions.
..31 ..
-------
The use in the present program of a relatively short wind wave test basin
precludes full wind-wave developnent, and to that extent the tests are
not representative of conditions at sea. On the other hand the relative
absence of wind waves allows a separate clear-cut evaluation of wind
induced drift without the complicating effects of wind waves.
Experimental
Lens Neasurements
The substrate liquid was generally deposited in a shallow (1 ineh deep
by 2 1 L inches long x 18 inches wide) rectangular tray made of fiberglass
reinforced melamine plastic. If the substrate liquid was expensive or
was likely to react with or dissolve the plastic, a smaller glass,
aluminum or teflon coated aluminum tra.y was used. Prior to depositing
the substrate liquid the tray was washed and rinsed ten times with tap
and then ten times with distilled water. If a non-aqueous substrate was
to be used, the tray was dried and then rinsed ten times with the substrate
lic?iid prior to use.
Various an unts of the lens forming naterial were then deposited from
calltrated pipets or burets onto the substrate surface. The pipets and
burets were previously cleaned by a procedure similar to that used in
cleaning the trays, except that they were first subjected to a wash with
H 2 3Oj , K 2 Cr 2 O 7 cleaning solution.
The diameter of the deposited lens was measured with a transparent
centimeter scale calibrated to the nearest millimeter. Care was taken to
iid.nimize parallar. The scale was generally rested on wooden blocks
alongside the tray so as to avoid disturbing the tray. lenses were
quite mobile, and gentle air currents, and vibration tended to greatly
distorb them and caused them to move about, so that measurement was
difficult. Two lens diameters at ri t angles to ons another were
measured in each instance so as to insure that lens was at equilibrium
and was truly circular.
Additional measured volumes of lens forming liquid were added to that
previously deposited and the resulting new diameters measured. Check
runs showed that the seine lens diameter was obtained using this incre-
mental method of addition as was ithen the sane total volume was added
all at once.
As the lenR diameter increased the lens became progressively more
—32 —
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susceptible to distortion by minor disturbances. Not only did small
disturbances tend to produce greater distortion, but these distortions
took a longer time to disappear, arid often a relatively minor disturbance
produoe&distortion which propagated through the lens breaking it into
ti or more parts. Thus as the lens became larger it became pro—
greasively more difficult to measure, and it became progressively more
difficult to successfully add, without lens breakup, the additional
material required to produce further increases in lens voluine.
There are slight end effects associated with the use of small trays.
If the small tray was well wetted by the substrate liquid so the
substrate meniscus curves upward at the tra y edge, the size of the lens
tended to decrease slightly. If the tray wall was poorly wetted so that
the substrate meniscus curves downward at the tray edge the size of the
lens tended to increase slightly, and further the lens tended to drift
to the side of the tray. These difficulties were circumvented by
using large trays wherever possible.
The surface tensions of the substrate and lens forming liquid and their
interfacial tension were measured using a Du Nu.oy ring tensiometer. The
raw tension measurements were corrected using Zuidema and Water’s
correction factor charts (82). In all cases the tensions were measured
using liquids which had been niutual].y saturated with one another. Diffi-
culty was experienced in measuring the interfacial tension for liquid pairs
which differed only very slightly in density. In such cases the ring
tended to pull the interface up to the top liquid surface. If the top
liquid level was raised to prevent this, it tended to wet the ring support
brace, thus interfering with the measurement. Further when the density
difference was small corrections had to be extrapolated a considerable
distance beyond the range of Zuidema and Water’ s charts. The liquid
densities re me ,pured both by the weighing of liquid samples in pyc-
nometer bottles, and by means of a Westpha]. balance.
Because of the difficulties involved in measuring large lenses attempts
were made to photograph such lens and then measure the photographs.
Because of shadows cast by the lens edges, and the continued problem of
maintaining the lens undistorted, this did not prove to be any irore
convenient than direct measurement.
Motion pictures were taken of lens which were deliberately distorted by
means of styluses and by impulse waves in the substrate fluid. These
pictures were reprojeoted, tracings made of the lens boundary, and the
lens areas measured by running a planimeter around the boundary trace.
-33 —
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preading Rate Measurements
A three inch deep by six foot square plywood test basin was constructed.
This basin was painted light gray to provide thotographic contrast, and
was lined with polyethylene film. A new sheet of polyethylene film was
used for each run, and the old sheet was discarded, thus providing a
clean basin surface for each run.
A six foot high bridge was set up over the basin. A 16 imn Kodak notion
picture camera pointing downward was mounted at the center of the bridge. Two
1 O watt floodlights each pointing downward at a 1 0 angle and at the
center of the basin were mounted on the bridge support towers.
A p onogra ih motor turning a turntable at 78 r.p.m. or 1 r.p.m.
was mounted over one corner of the pool. A radial line was inscribed
on the turntable, so that ithen pictures were taken including the rotating
turntable the rotating line provided an elapsed time reference.
The polyethylene lined basin was filled to a depth of two inches? generally
with cold tap water, though in some instances the water temperature was
varied for experimental purposes, i.e. to change the viscosity of the
water . A sign identifying the nature of the run was positioned over the
basin and a short sequence of motion pictures was taken. The sign was
renK)ved and a ring darn was inserted at the center of the basin. This darn
consisted of a vertical cylinder with an open top and bottom. The dam was
allowed to rest on the bottom of the basin, its upper edge protruding
several inches above the surface of the water. A predetermined, measured
voiu of oil (a natural crude oil) was deposited i n.qjde the ring, the
motion picture camera was started and the ring darn lifted. Pictures were
taken of the spreading pool of oil until the spring motor of the camera
ran down (38 feet or 1 2O frames could be exposed per winding). An
exposure rate of 32 fra*es/ second at f li.O was used in some instances.
In other cases a 8 frame faecond rate at f 14.0, with the shutter opened
only one quarter to provide equivalent photogra tiio exposure, was used
so as to obtain pictures of the spreading oil over a longer period of time.
Tn I, positive reversal film, Kodak type 7278 was need.
The pictures were projected, and the major and minor axis of the pool
measured usually at five, ten or twenty frame intervals. The area A
of the pool was calculated on the basis that it was an ellipse. That is
A - r Dl 02/14 iètere i and 2 are the major and minor diameters
-------
respectively. In a nuither of instances the area was checked by using
a planimeter and making a traverse around the pool’ a outline • Good
agreement was obtained between the areas as measured by plariimeter and
those calculated using the ellipse formula; even though in some cases
the periphery was locally distorted. The difference between the two
areas was less than 2% in all instances in which a check was made. It
is possib].e though 1 that poor agreement would have occurred if checks had
been nade in cases of more extren distortion.
Drift Test Basin
Tests involving wind and current drift and the influence of waves were
carried out in a 20-ft. long by 8-ft. wide wind water test basin. Because
of the thickness of wooden walls and support beams, the actual inside
dimensions of the basin were 19-ft. 1 1/2-in, long, 7-ft. 1 1/2-in, wide
by 11 3/li in. deep. The basin was lined with 10 ml ]. polyethylene film.
Return ducts were created on each side of the test basin by insta].]ing
17-ft. long by 1]. 1/2—in, deep by 3/li-in, thick redwood planks, their
long side parallel to the long walls of the basin, each plank standing
12 in. away from the neighboring wall. 18-in, spaces which formed the
entrance and exit port to the return ducts were left between each end
of the redwood planks and the ends of the test basin. The space between
the lower edge of the redwood plank and the bottom of the test basin was
sealed by a lip of polyethylene film tacked onto the redwood plank. The
redwood planks were held in place by slotted angle iron beams to which
their top edges were fastened. These angle iron beams ran the length of
the basin and were in turn fastened to the end wall of the basin. 1jjfj
the installation of the return ducts, the width of the main channel of
the test basin was reduced to 5—ft.
The bottom of the basin was formed by li-ft. by 8-ft. sheets of 3/b in.
plywood. These were supported off the floor by five evenly spaced
2 in. x 6 in. wooden beams lying flat, and running the length of the
basin. The beams provided clearance between the basin bottom and the
floor so that any leakage could drain away.
Variable speed propeller-type mixers were mounted with their propellers
in the return ducts at the downstream end of the test basin. The shafts
of these mixers were inclined at a slight angle to the horizontal. Thus
the propellers when turning would drive water down the return duct; and
the mixers functioned as high velocity - low head pumps. This pumping
action was used to create currents by return flow in the main duct. The
current speed was varied by varying the rotational speed of the propeller.
- 35 -
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The flow created by the propellers Ii i discharged into the main duct
was not uniformly distributed and created strong eddies which gave rise
to abnormal and non-reproducible drift. This condition was corrected
by the insertion of an egg-crate grid and fiber mat flow dibtributor.
The grid was 1/2-in, deep and had 3/ i-in. square openings. The fibrous
mat was made of loose random array of rubberized fibers and was roughly
2-in, thick. Though this distributor greatly improved the flow uniformity
and eliminated eddying, it tended to greatly reduce the ma.ximwu current
sreeds that could be produced by the propellers. The discharge capacities
of the prcpeflers fall off rapidly as the head (or pressure drop) they have
to overcome increases, Flow mu formity was achieved by increasing resistance
and pressure drop and flow capacity was consequently greatly reduced. No
simple way has as yet been found to circumvent this difficulty. Pumps
capable of providing high flow rates and overcoming the required pressure
drop without loss In capacity can be obtained but their cost and that of
the associated electrical installation and piping would greatly exceed the
total cost of the present test setup.
The return ducts, in addition to providing a means for creating currents
in the main duct, helped to minimize some of the main channel end effects.
Any drift in the main channel has to come to a halt at the channel’s
downstream end. It the main channel were closed off, no net flow could
occur within it. Thus any surface flow, such as might be induced by wind,
would have to be accompanied by a return flow presumably in the lower
la -ers of the water in the channel, The zone of transition between the
surface and the return flow occupies a fairly long length at the end of
the channel, preventing that length from being used for drift r ieasure ents.
These effects, which occur in a closed channel, would not occur in most
situations at sea. The t side ducts allow the drift induced flow to
return outside of the main channel, thus providing a mere normal flow
profile and minind. 1ng the length of the main channel which could not be
used because of drift termination end effects.
The overall layout of the test basin is shown in Figure IV. For clarity
the angle iron supports for the rethnod planks are not shown.
A variable height wind duct was mounted over the main channel of the test
basin. The walls and top of the wind duct were formed of 10 mu poly-
ethylene film attached by pressure sensitive tape to supporting slotted
angle-iron cross beams. The cross beams supporting the top of the duct
were in turn attached to vertical slotted angle iron pillars bolted to
the side wail of the test basin. The support beams could be fastened at
various elevations on these pillars thus changing the height of the duct.
The side walls of the wind duct were fastened at their lower edge to the
-36-
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F1 URE iv
TEST BASIN LAYOUT
p .-
P PROPE.LLOR
TYPE PUMP.&
o EGG C.RPTE FtBER
FLOW tNSTRtBUTOR
MAT
/ I,
7- 1Y 2
• I
____ RETU K DUCT
.
I
I I
I
I
I
MAIt’J I I ‘
I I
I I
t
I
I
I
r
Drr)
8’
. 4 -
I I
R TU N DUCT
I,
I I -.
4
12. ”
I
I
ii
20’ —
L
(‘7,
-------
angle iron beams supporting the redweod planks forming the return duct.
The excess width of the polyethylene film was draped over the top of the
return duct thereby- forming a dust cover for the return duct. Details of
the method of supporting the wind duct are shown in Figure V.
To permit better observation of material and drift in the test basii a
number of windows made of 1/16 -in. thick cellulose butyrate sheeting were
instafled in the wind duct. Rectangular holes were cut in the polyethylene
film, and a matching rectangular sheet of the cellulose butyrate was
fastened over the holes by i ans of pressure sensitive polyethylene tape.
The wind duct was connected to a fan by means of a transition piece. This
transition piece was essentially four plane surfaces connecting the square
cross section (roughly four feet by four feet) of the fan housing to the
rectangular cross section of the wind duct (s-ft. by 22—in, in most tests).
Two fans were used, an i/h H.P. attic fan capable of providing a free
discharge flow of 12,000 SCFM (standard cubic feet per minute) and a 3 H.P.
industrial fan capable of providing a free discharge of 30,000 SCFN. The
industrial fan had greater constancy of flow rate as the discharge pressure
drop increased. The speed of the attic fan and thus its volumetric dis-
charge rate was varied by changing the size of the drive pulleys connecting
the fan and its drive motor. The industrial fan was provided with a
variable radius drive pulley, and its speed could be varied over a rela-
tively narrow range by means of this pulley. In some instances the fan
discharge rate was varied by closing part of its Intake area, OX’ by
inserting mere than the usual number of flow distri itor grids in the fan
discharge air stream.
In the ease of the attic fan, the walls of the transition piece were made of
10 mu polyethylene sheeting supported on an angle iron frame • In the case
of the industrial fan,l0 mu polyethylene was not strong enough to stand
up against the pressure drops caused by the higher wind velocities.
Therefore,for those tests involving the industrial fan,the transition piece
was made of ],/Ij” plywood supported and reinforced by 2” x t 1 ” wooden beams
and slotted angle iron. It was also necessary to reinforce the wind duct
with slotted angle iron to prevent excessive bulging at the higher wind
speeds and higher pressure drops provided by the industrial fan..
Because of its simpler construction the plastic transition piece could
be readily altered to provide a smooth transition whenever the wind duct
height was changed. It was much mere difficult to alter the wooden
— 38 —
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IGUR E V
WIND DUCT SUPPORT SYSTEM
POLYETHYLENE PLM
4 ANGLE ROI J PiLLAR
B I WGLE IROI J CROSS SEAM
C BASU J WALL
0 REI WOOD PLA 4K
E PLANK SUPPORT BEAJPIS
F BASIM BOTtOM
FILM REIN FORCEME ’ IT BEAt4
K ETURi’J DUCT
I
—
-------
transition piece. Therefore with the wood transition piece a somewhat
different expedient was used when the duct height was altered. These
different expedients are shown in Figure VI.
In the case of the wooden transition piece, part of the air discharged by
the fan was bypassed around the wind duct when the duct height was lower
than 30-in. To prevent excessive bypassing an egg-crate grid was placed
across the discharge end of the transition piece. This grid constituted
the major flow resistance in the air stream and thus the smaller added
resistance of the wind duct did not cause too math excess flow to escape
from the open section of the grid. If the height of the discharge and of
the wooden transition piece had been reduced when the wind duct height was
reduced, higher wind velocities could have been obtained but this would
have led to unsymnietricá]. convergence of the air stream with a consequent
downward component of wind velocity at the wind entrance. Farther) higher
wind velocities could not be experimentally utilized. Winds of higher
than 30 ft./sec. velocity tended to blow excessive quantities of water out
of the test basin.
With the plastic transition piece unsyminetric convergence did occur at low
duct heights. Here, because of the lower capacity of the attic fan it was
necessary to cause all the air flow to pass through the wind duct in order
to achieve reasonably high wind velocities.
The egg-crate grid served to eliminate swirl components in the fan dis-
charge as well as to improve flow uniformity. In the case of the attic
fan.. the grids were placed immediately adjacent to the fan frame where
the air velocity was somewhat lower and not at the transition piece
discharge. This served to reduce the pressure drop produced by the grid,
a factor of importance because of the marked effect pressure drop had on
the discharge capacity of the attic fan. The grids had 3/li-in, square
openings and were 1/2 in. thick. Two grids with their openings in line
with each other were generally used, but in some instances 1 to minimize
pressure drop, only one grid was used.
A rocker-type wave machine was installed at the downstream end of the
test basin. A sketch of this machine is shown in Figure VII. In the
final version used, the slat like redwood paddle was attached to a
parallel shaft at its lower edge. This shaft could pivot in two wooden
bearings held in place near the bottom surface of the basin. Two angle
iron arms were attached to the vertical sides of the paddle. These arms
-ILO -
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IWDUSTR L. FAI4
FIGURE L
TRANSITIO J Pt CE ARRAN P4E.WT
r1
22”
L
B
D
T
D
a
1 TTIC. ;PJ
0
0
F: FRI..L B TEST BASRI
i/RIt OUCT T A 4SITtO ..L PIECE
C RLD
-------
FI&URE! [ WAVE MACHIME
ANGLE ROP4 P8LLAR
ANGLE IRON
CROSS BEAM
PLA3TIC ROOF
I
I
$
I
I
WIRE ROPE C UIDE Put
VARIABLE SPEED ECCEP
CxRAI4AM DRIVE
WU JD DUCT
REYURt’ DUCT WALL
BEARING SUPPORT FRAME
-------
projected vertically about 12-in, above the upper edge of the paddle.
A wire rope and a spring were attached to the uppsr ends of these arms.
The spring was attached to angle iron beams supporting the return duct
redwood wall about two feet in front of the paddle and thus tended
to force the paddle forward. The wire rope was attached to a rotating
eccentrio,driven by a shaft which was held in rocker bearings canti-
levered out from the rear wall of the test basin. As the eccentric
rotated it alternately pulled and slacked up on the dre rope which
in turn pulled arid slacked up on the shaft. The combined action of
the spring and wire rope thus imparted a reciprocating motion to the
wave paddle.
The eccentric shaft was turned by a belt driven, triple- step cone
pulley mounted on the shaft. The belt in turn was driven by a 10-in.
diameter pulley driven by a 1t4 H.P. Graham variable speed drive
capable of producing speeds from 0 to 60 RPM. In general the s-in.
step on the cone pulley was used and thus the paddle rocked at a
frequency whith was twice the rotational speed of the Graham drive.
In an earlier version of the wave machine the paddle had been pivoted
at the top rather than the bottom. The bottom pivot arrangement was
finally resorted to because it produced greater amplitude of motion at
the water surface where motion should be the greatest if it is to
conform to the usual pattern of wave motion. Though the wave machine
was designed to produce a wide variety of wave frequencies b7 varying
the drive pulley arrangement and the speed of the Graham drive .and
amplitudes by varying the throw of the eccentrics, it functioned most
efficiently when producing with a period of about one second (i.e.
one wave per second). The wave amplitude depended on the frequency
of rocking as well as the stroke length of the paddle; and when the
frequency was reduced much below one stroke per second the wave
amplitude fell off markedly even though the stroke was lengthened by
increasing the throw of the eccentric. Frequencies significantly
higher than one cycle per second tended to overload the wave machine
drive.
Wave reflection tended to occur at the upstream end of the test basin)
particularly if the current flow distributor was not being used.
This reflection, if not minimized, caused the formation of standing
waves of large amplitude. The current flow distributor when placed
1/ wave length away from the upstream end of the test basin was most
efficacious in minimizing standing formation. Part. of the wave passed
- I 3 -
-------
throu i the distributor, part was absorbed, and part was reflected
1800 oat of phase with the incident wave. The part passing throu
the distributor changed phase 900 in traveling to the upstream wall,
1800 upon reflection from the upstream wall, and 900 in traveling
back to the distributor. Thus when it arrived back at the distributor
it was 180° out of phase with that part of the wave reflected at the
distributor. The two reflections therefore tended to cancel each
other, minimizing the total reflection and standing wave formation.
An overall view of the wind-wave-current test basin setup is shown
in Figure VIII.
Drift Test Procedure
A ring dam, a vertical circular cylinder open at both ends (roughly
—in. in diameter and b-in, high) was mounted about two to three
feet from the upstream end of the test basin. This ring darn when
partially imnersed formed a trap within tiich oil could be contained.
A known volume of oil (usually 0 or 100 cc) was fed into the ring
dam from a plastic hose connected to a separatory funnel mounted above
the basin. k atever drift producing device (i.e. fan, propellers, or
wave machine) being used was turned on and conditions in the basin
were allowed to equilibrate. By means of a projecting arm attached
to the dam, the ring dam was then lowered below the surface of the
water, thus releasing the oil. The oil moved downstream in the basin
under the influence of whatever drift action was being tested. carker
strings were strung across the basin every ti feet. The time re-
quired for the oil to travel between two of these strings, two or three
gaps apart, (i.e. four or six feet) was timed to the nearest 0.1
second using a stopwatch.
It was noted that the oil required a certain time to accelerate
after being released. Thus timing was usually not started until the
oil had reached a string three to five feet away from the point of
deposition. The flow termination which occurred at the downstream
wall caused oil drift to slow down as the downstream wall was
aporoached. To eliminate this end effect, the last five feet of the
basin were not used in making drift rate measurements.
Drift tests were carried out under wind action, current action, and
under various combinations of these actions. In certain cases the
drift rate of wooden chips or polyethylene film squares were also
measured for comparison purposes.
-hb-
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FIGURE t
U ,.
-------
In carrying out the various tests, the following data were taken where
applicable: air temperature, water temperature, oil type, oil spill
volume, wind velocity profile, depth of water in basin, location
and distance between points at which drift measurements were carried
out, elapsed times required for oil to traverse distances over which
drift was measured, surface current speed, propeller speed, wave
frequency, wave amplitude (estimted) and wave length (estimated),
type of water used in basin and degree of cleanliness of the water.
Standardized data sheets were used in most runs • A copy of these
data sheets is enclosed as an appendix.
Air velocity profiles were measured by means of a moving vane
anei meter mounted in the downstream opening of the wind duct.
Either a nine point or a fifteen point traverse was used in measuring
the profile. Velocities were measured at three heights, one at the
horizontal center line of the duct, one It-in, below the top edge of
the duct and one 14-in, above the rim of the basin. Measurements were
taken at either three (18-in, apart) or five (12-in, apart) evenly
spaced points at each of these levels, the center point at each level
being at the vertical center li of the duct.
The cross sectional area for air flow at the duct discharge opening
is less than air flow area over the water in the test basin. The
ratio of these two areas is equal to the ratio of the duct height
above the basin rim to duct height above the water surface. Thus
since the same volume of air must flow through the t areas
V H
— — (37)
VD H+T-W
where V is the average velocity of the air over the water, VD
the avJage velocity of the air in the duct opening, H is the height
of the duct opening, T the basin depth and W the depth of the water
in the basin. VD was con uted by arithmetically averaging the
velocities at the various points in the discharge traverse.
? bst of_the wind drift data_is correlated in terms of percentage
drift (Vo/V ) x 100, where V 0 is the average velocity of the oil
as determined from a series of drift rate measurements.
- 146 -
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Section I .
DISCUSSION OF RESULTS
Lenses
Eleven lens foring systems covering a wide range of surface and
interfacial tensions and liquid densities were tested. Lens diameters
were measured vs vo1ui up to volumes of 100 cc or more in most
instances. Lenses ranging in thickness from i.I imn to 12 mu and
having a variety of shapes were formed. In the case of those lenses
there the surface tension of the upper phase was greater than that
of the lower phase, the bottom of the lens curved around so the
interfacial tension aided the lower layer surface tension in opposing
the upper layer surface tension at the edge of the lens. Such a
lens,which usually occurred for water floating on a denser organic
layer,is shown in Figure IX-A.
Figure IX.
AD
In a more typical case with an organic upper layer floating i water,
the surface tension of the upper layer was smaller than that of the
lower layer and the lens formed was similar to that shown in Figure
IX-B.
The lens diameters then plotted vs volume on log-log paper form
smooth curves which ares in many instances aln st straight lines,
andy if care is not taken, may easily be mistaken for strai t lines.
Actually the slope of the curves gradually increasea as the volunE
increases until the slope reaches an asymptotic value of 0. at
- t 7 -
-------
large voiwnes - thus satisfyin Hardy’s and Langinuir’s asymptotic
r3lationship for large lenses. Some typicaa loc diameter vs log
volume curves are shown in Figure X and X I.
At a lens volume of 100 cc, the lens diameter can be moderately well
predicted by assuming that t = t and that Equation ( ) holds
1/14
D u 18 V 2 g o (1 - Pu/OL)1
— J
The measured lens diameter at a vo1w of 100 cc, the lens diameters
predicted using Equation (8) and the operties of the lens formiw
systems are presented in Table 14. The average deviation between the
predicted value of D and the observed value is L. , and the greatest
deviation is slightly less than ll; .
The deviations are usually largest wI- re -F 5 or - is small.
In each of these cases, the predicted value of D depends on taking a
small difference between t large numbers, which is a computation
quite conducive to error. In all cases but one, where the deviation
between the predicted and observed value is extremely small, the
observed diameter is larger than the ‘edicted diameter, which is as
it should be, if Lang nuir’s prediction that the thickness t should be
less than t for small lenses is correct.
The slope of the log D vs log V curve in the solwme range of 1 to
10 c m 3 is also included in Table Ii. It can be seen that in this
range this slope can deviate quite markedly from the asy-ii otic
value of 0. .
If Langmuir’s expression (E quation 20) for the lens volwr is
expanded there Is obtained
v 1.I 2 t + [ oi. aL(1 — cos 3 (Ail2)) + oj a 1 (l-cos 3 (A /2))
+ o . a (l_oos3(.AuJ2))] — 1YR (a 1 2 sin A 1 — a 2 sin A )
+ [ LaL 2 1_c05 3 il2))+ o.a 2 (1-cos 3 (A./2))+ aua2(l_cos 3 (Aa/2))] (38)
-
-------
ri
I
a’
67891
3 4 567891
3 4 o 7891
3 4 567891
3 4 5 67891
4
VoLu.vi . Ccw )
-------
3
4 5 67891
2
3 4 567891
S
4 567891
6 7891
S
I
IT:T4..p
1±iTj f: !:
4 4t
tI1
9Tm i
iffl
0
‘I’
Voi 1 [ cm’)
-------
Table 4. Lens Characteristics
System Upper L er Cf D 100 d log D t a 1
Code Phase Phase d log V —
3 Actual Predicted
(gr / cm ) (dynes/cm)
(cm) (cm)
LPO Light Water 0.8599 0.9928 29.8 69.3 50.3 10.8 18.6 17.2 .410 .368* 2.28
Paraffin
oil
HPO Heavy Water 0.8830 0.9980 30.8 69.3 47.5 9.0 17.4 17.3 .1+70 .1#26 2.15
Paraffin
oil
LPO Light Synthetic 0.8599 1.0186 29.11 66.3 49.6 12.7 18.5 17.1 .439 .373* 2.05
SW Paraffin Sea
Oil Water
0
DEC Decalin Water 0.871+4 0.9968 29.2 69.1+ 47.8 7.6 18.4 18.1 .1+16 .388 2.29
DOD Dodecane Water 0,71415 0.9965 24.3 69.5 1+7.3 2.1 30.3 28.9 .498 .138* 14.1*8
BB 0.2 Vol. Water 0.9739 0.9965 30.7 68.8 L 5,7 7,3 16.0 15.7 .391 .519° 3,99
LPO Bromobenzerie
0.8 Vol. Lt.
ParaffIn Oil
TEl Tetradecane Water 0.7569 0.9965 24.8 68.7 47.8 3.9 27.0 24.6 .1*61+ .175* 3.62
DSP Water Dlbutyl 0.9982 1.0453 52.6 32.7 21.8 1+1.7 10.2 9.6 .li32 1.3900 0.70
Phtha late
-------
Table 4. Lens Characteristics
System Upper Lower Pu aL c -F 5 D 100 d log D t
Phase Phase d log V —
Actual Predicted
(gr / crn ) (dynes / cm)
(cm) (cm)
DEM Water Dlethyl 0.9982 1.0680 1.9.0 33.4 9.6 25.2 11.80 11.85 .1.21 .905° .583
Ma I ona te
DMP Water Dimethyl 0.9952 1,181.0 l+9•5 27.4 10.3 32.1+ 15.7 14.0 .476 .651 .516
Phtha late
BB Water Bromo. - 0.9971 1.4863 68.7 35.2 22.7 56.2 15.3 14,6 .470 .591° .520
benzene
c Based on observed rather than EquatIon (6)
0
Based on Equation (6)
-------
The term is the asymptotic relationship between the volune
radius and thickness) and the other terms essentially correct this
asymptotic relationship for pools of moderate diameter. If one
examines the relative magnitude of these correction terrns,it can be
seen that for other than very small lenses the last bracketed terra
is negligibly small, and that of the remaining terms those involving
a 1 2 or c jaj are the dominant corrections.
Retaining only the dominant correction terms there is obtained after
some rearrangement
v 2t [ i+r
- ( •) ( ) A ] (39)
It can be shown by substitution that
= ( .)2 ( )
Using this expression for c /(—Fs) in the simplified expression for
V there is obtained
2 1 ,, aj 2 aj , / Aj
V=TrRt l+ 3 ‘. ;) l-cos (7
Lal a 1 1
- n—) (i—) A J i)
It can be clearly seen from this expression that the magnitude of the
correction will depend largely on a 1 / t,. It will be shown later
that if a 1 / t is large the angle A 1 will be small. For small values
of Aj it can be shown that expansion of the term 1 — cos3 ( ) will
yield a term equal to 0.18 ( ±2 ) sin A 1 . Thus if the correction
t R
- S3 -
-------
is large, the term 0.82 (a 1 /t )(a 1 /R) sin A 1 will be the dominant
correction term. That is for small A 1 the correction due to the
(1 — cos 3 (A 1 /2) term will be of opposite sign and 18 as large as
the term containing sin A 1 . If ailt is not too small and A 1 not
too large, the term containing 1 — cos 3 (Aj/2) will, because of its
smaller coefficient and the fact that it is multiplied an additional
tima by the small factor aj/t ,, , still remain smaller than the term
containing sin A 1 . Thus when A 1 = 1200, (a It .58) the term con-
taining sine Aj will be roughly 3.b times as great as that containin- ,
1 - cos 3 (A 1 /2), and at A = 150° (a /t = .52), roughly 2.0 times as
great.
For all the systems tested aj/t was never less than 0.55.
As an approximation, the terms in 1 - cos 3 (A 1 /2) and sine Aj will
be combined to yield 0.82 (a 1 /t )(a 1 /R) sin A 1 . The error in this
approximation is significant only wfien the correction is small, i.e.
-rhen aj/t , is small, and the actual error introduced by the approxima-
tion should therefore always be small. Thus we will write:
V 1 7TR t Ii — o.82( )( sin A 1 j (L 2)
The angle A 1 equals B 1 + AL (refer to Equation 16). Most of the time
AL is small, and for the purPoses of approximation itwiul be assumed
A 1 B 1 . An alternative form to Ea’iation (i1 ) for B is:
sin 2 (B 1 /2) = -F 8 ° L + - a .. )/I L i
Making the assumption that a we get as a rough approximation
(B 1 /2) f= - F 8 1. (Tj ( )
Referring back to quation (tLO) it can be seen that
- = t\ 2 (h S)
Substituting for — F 8 /a. 1 , we obtain upon taking square roots:
-514-
-------
ta, 1/2 A 1
sin( ) (• ) sth( ) ( 46)
2 a 1
A 1 o 1/2
For small angles sin A 2 sin -p-. Further ( ) rarely
differs from 1.0 by ir re thar lC’ . Therefore
ta,
sinAj (b7)
verifying our previous assumption that A is small when aj/ta, is
large. Substituting the above for sin A 1 in Equation (Li.2) there
is obtained
v 2 t lo82 (I 8)
Putting this in dimensjonless form by dividing through by t 2 there
is obtained:
(b 9)
Thus we see that the dimensionless vo{uifle v/Tr t 3 should be a function
of the dimensionless radius R/t with the dimensionless capillary
constant as a parameter. As a 1 /t goes to 0, or R/t goes to infinity
the relationship reduces to the assymptotic t volurae-ractLus equation,
as it should. Putting this in the somewhat n re convenient diameter
form
V 1 D 1.614aj1
rr t a , 3 (—i;) I.- t , — ta, J ( °)
The coefficient for aj/ta, is somewhat in error due to the
dropping of various terms and the simplifying assumptions used.
These errors were evaluated for typical lenses. Based on the
calculations it was found that these errors could be roughly corn-
pensated for by changing the coefficient of aj/ta, to l.3 , thus
yielding
- —
-------
v D D - l.3 5a
to, t 0 ,
V/11 t 3 as calculated from the above equation is plotted versus D/t,, in
Figni III for various values of the parameter a 1 /t . E erirnentally
obtained V/ i l to, 3 and D/to, points for the various systems tested are
plotted on the same curve. h iere there was a large discrepancy
between D as obtained from Equation (3) and the measured value of D
for a 100 c m 3 volume, the value of to, used was not that obtained from
:ouation (6) but rather the t, val’ie obtained by extrapolating a t
vs I plot to equals zero (after the fashion of Langmuir ( 37 ) ).
it. can be seen that this method of plotting, u.njfie5 in a single plot
a ‘wide variety of data with only a single added parameter aj/to,
being necessary to fit most of the data with fairly good accuracy.
Therefore it appears that either E uation (Si) or Figure XI I, in
conjunction with Equation (6), can be used for predicting the non-
asymptotic volume vs diameter behavior of lens fonnin systems.
From Figure XII it can be se3n that at D/t values of 20 or greater
all systems tend to approach the assymptotic D vs V behavior within
lY . This corresponds roughly to 100 cm 3 pod values for most systems,
thus explaining ‘why the relatively good agreement between D observed
at 100 cm 3 volume and that predicted by Equation (8) arid thy the
observed diameter averaged 1 .I % hi er than the predicted values.
It should be noted that the asymptotic Equation (8) and Equation
(51) for the non—asymptotic case, apply to systems based on sea
water (simulated) and oil, as well as for fresh water and oil systems.
A procedure based on the Bashforth and Adams’ tables (7 ) was
develoned for cornputinc. lens value vs diameter behavior but it is
much n re awkward and time consuming, than the previously described
method. Because of this awkwardness the procedure was not investigated
extensively.
- 56 -
-------
100S
800
Data for
a
Diameter Correlation for
Lenses
600.
; ; I II
1
00
300
200
l00
80
60..
=
-—
i& T4.
I - -h-
4T
!
-
—
.
T
r
—
-
.t
f
.
—
—U
— —
— • 4 s
—
‘T
-±1
——-
—
EF
— —— -- 1-—
I
ff ff*
. --,
—
‘
* —
— —
— —
—
-.J
I
—‘
---
‘
if
——
—
—
.
U
. -
:•
—I —
-A
H
I00
- - - .R r
I Ii
i F EE
E1 :i
E
P
‘4 I
I
I
— - —1
30
20
10
8.
6
3
2
1
-i ! l i
- .— F—
V 30
I
t11 h i
p
0.5< ‘<1.1
1 .9<+ 2.4
Figure XII
Generalized Volume vs.
.
J2o
I
r
--
ii
N
—
II -
2 3 4 6 810
- 57 —
-------
‘rests were carried out using light paraffin oil, and water containing
various levels of detergent (o. and 1.0%). These systems showed
tynical J vs V behavior, but D and t at 100 cm 3 value were markedly
aifferent from the values predicted, on the basis of measured surface
a id interfacial tensions, from Equations (8) and (6) respectively. The
observed diameters were much smaller than predicted by quation (8),
i.e. t was much thicker than predicted. The disagreement aD ,j ears to be
due to measurement of the iriterfacial tension j A plot of t .1
was extraDolated to = 0 to obtain t and then using ouation (6) this
value of t was used to obtain cij from the 1Easured values of ci ,
0 u and AT,. Usin:: this value of Oj, aj J2 gjJ(o_r u)g was calculated.
ased on this value of a 1 and the extranolation value o t the V/TT t)
vs data fit the curves of dgure X II very well.
The areas of distorted lenses were measured by planimet.er traverses of
l i.s images on motion pictures taken of relaxing deformed lens. Thr
eccentricities (major diameter/minor diameter) as great as 2:1 these
deformed lens covered areas which were only 1.6 to 1.7 greater than
the area covered by the corresoonding undeforined circular lens.
? s noted previous1 the larger the lens diameter, the more susceptible
it is to deformation and breakup. It was also noted that as -F 5
aporoaches 0, the more susceptible a lens is to breakup. ssentially
a le-z value of -F 5 jnn,fles a low t , va1ue and thus for a c iven value
à J a lens characterized by a low value of -F 3 will have a higher
dimensionless diameter n/ta, than one with a high -F 5 value. Thus it
appears that D/t , may very well be the criterion for lens stability.
At large D/t values t approaches t closely and barely varies at all
with diameter. Thus for a large D/t lens, a deformation (which
effectively changes the local radius of curvature, does not change the
lens thickness very much (as also seen by our area measurements on
large deformed lenses). This smallness of change in thickness in turn
on use only a very small increase in the hydrostatic forces which act to
nush the lens into a circular form. Thus as D/t becomes large, lenses
approach a condition of neutral equilibrium with respect to deformation,
and deformation, once started tencL to continue with very little resist-
ance permitting breakup under very slight stress.
T wind drift tests it was observed thatlenses tended to break into
smallar lenses under wind action, the resultant lens size being smaller
the higi er the wind velocity. At about 10 knots, light paraffin oil
would break upintolensea of rou ily 3 or cm diameter, and at 20 knots
- -
-------
lenses of roughly 1 to 2 cm diameter would be obtained.
since over the range in question the lens diamoter varies rouqhly as
the OJi5 power of the lens volume, t varies rou -ily as V° -. This is
a breakup induced change from a 1000 cm3 pool to 1000, 1 cm 3 pools would
only cause a two-fold decrease average lens thickness, and a consequent
two-fold increase in net area of coverage. Lens breakup therefore is
not likely to be accompanied by 1ar e increase in net area of coverage.
The cross area of coverage may well increase due to turbulent diffusion
and spreading apart of the individual small lenses with respect to one
another.
Crude Oils
The spreading characteristics of four typical crudes were tested.
Some of the properties of these crudes are listed in Table .
:hese crudes vary widely in their spreading character. The ‘ul1erton
spreads out uniformly and ranidly. It quickly covers the surface of
small test basins. Upon standing . dispersed small beads form upon the
surface. This behavior may- be akin to that observed in confined
monolayer forming systems, where the excess film material gathers in
beadlike little lenses upon the monolayer, Adam ( 3 ). The bead
formation may also be due to progressive changes in composition induced
by evaDoration.
The Tia Juana,Safaniya. and Urania when initially deposited in small
quantities disperse very rapidly, covering the entire surface of a
small test basin. Subsequent additions spread more slowly often
forming a diffuse, but readily discernible pool. then still further
oil is added, irregular somewhat thicker and more deeply colored pools
form in the case of the Tia Juana and Safaniya crudes. The thickness
of such a pool of Safaniya crude will be roughly 0.013 cm. On the
other hand Urania crude upon further deposition forms a lens-like pool
on top of the previously deposited thin film. Such pools had thick-
nesses ranging from 0.IIL to 0.13 cm, larger thicknesses being obtained
at larger spill volumes. The diameters of these lens like pools vary
roughly as the 0. 1j5 power of the deposited volume, somewhat similar to
the case of regular lens formers.
The behavior of the crudes seernsto form a progression where high
viscosity, high density, high carbon to hydrogen ratio all indicate
- —
-------
Table 5. Characteristics of Crudes
Point Density . Viscosity Boi].in Point
Type of Oririn gr/cm 3 dynes/ i¾ Saybolt at Vol.
Seconds distillate
i?ullerton Texas 0.811 1 4(c) 21.i.9(a) !l)LPi’a) 1F.7(b) —1.2 1i5,8(h) 2L6(e)
0.8090(d) -2 6 .I (g)
Tia Juana Venezuela 0.8961(c) 20.9 Ia) 58.7 a) 3L 1 .3(b) —3.5 297 (h) ttOlt(e)
0.8876(d) — lLj..8(g)
Urania Louisiana 0.9260(c) 31.5(a) 58.1(a) 17.5(b) —9.1 2591(h) 51 5(e)
0.9202(d) —21. 6 (g)
Safaniya Saudi Arabia 0.8927(c) 27.3(a) 53.3(a) 30.1(b) —i .1 270(h) L 80(f)
0.8588(d)
(a) After mixing and separating oil and water.
(b) Letting oil and water quietly contact each other in Tension ter dish.
(c) At 20°C, provided by the obi1 Oil Company.
(d) At 27°C, measured in our laboratory.
(a) Henipel
(f) ASTM
(g) Based on a clean water surface tension of 70 dynes/om
(h) At 60°F
-------
a greater tendency towards ultimately displaying lens-like behavior,
with intermediate values of these factors being associated with
irregular pool formation, and low values with uniform filni formation.
The group c - - 0 L’ (i.e. -F 5 ) is negative for each of the crudes.
Each of the crudes appreciably lowers the surface tension of the
water, the reduction being most marked in the case of Fullerton crude.
This lowering is presumably produced by the extractive transfer of
rate-soluble components from the crude into the water, and therefore
may depend on the relative quantities of crude and water in contact
with each other. In a dynamic spreading situation the pool may
spread faster than such extraction can take place, and a , the
surface tension of the water may well be closer to that of pure water
uncontaminated by material extracted from the crude. Based on a
clean water surface tension of 70 dynes/crn, a set of alternative
values of —F 5 have been calculated and are also tabulated in Table S
In the case of -F calculated in this manner,Fullerton has the
greatest —F 5 value, and the -F’s values vary from system to system by
less than a factor of 2. In the case where is that for oil con-
tamiriated water, —F 5 is least for Fullerton,is significantly lower
for all systens) and varies from system to system by a factor of
almost eight.
Since —F 5 is negative for all the crudes tested, none should ordinarily
form lenses. Yet the Urania forms a lens-like pool when deposited on
a previously formed Urania film. By measuring the surface tension of
water covered with this film it was found that it decreases as the
amount of oil in the film increases and that it ultimately levels
off at a value l& dynes/cm below that of water saturated with Urania
crude. (It is perhaps a coincidence but the interfacial tension of
Urania Crude and water is also about 1.0 dynes.) In any case then
the saturated water surface tension lowered by this 13 dyries is
substituted for o-j, —F becomes positive (9.L dynes/cm) indicating
that a lens should be capable of forming. That is, since the lens
rests on this thin film, the surface tension of the film rather than
that of water should be used for o . The lens like pools that do form
are significantly larger, (29 cm diameter for a 100 cm 3 pool) than
predicted using the t anpro dmation, and the 9.L dyne/cm -F 5 value
based on maximum surface tension lowering (16.2 cm diameter for a
100 c m 3 pool). This may be because we are not using the correct
interfacial tension for oil in contact with the film. Actually we
have no means of measuring this film-oil interfacial tension
and can only measure and therefore use the water-oil interfacial
- 61 -
-------
tension. Alternatively; the water surface tension may not have been
lowered by the iiiaxixnwn amount that Urania film deposition could induce.
It was previously noted that up to a leveling off point the lowering of
the water surface tension was dependent upon the amount of film
forming material that had been deposited.
In the case of the Safaniya and Tia Juana crudes) similar lot ’ surface
tension film formation takes place. But here the value of -F 3 thou h
positive at the ITrLnimum film surface tension is small, and the tendenc ’
towards film formation is marginal — which is in qualitative accord
with the pool forming behavior of these crudes.
It appears that the spreading process for crude oil lenses and pools
may depend largely on the rapidity of the convective and diffusive
breakup and thinning of the thin oil films adjacent to these pools.
For small spills on the ocean, such thinning should be rapid and
lens-like pools are not likely to remain stable for any prolonged
period of time. For large pools however lens-like pools may persist
for considerable periods of time.
*ien traces of surface active agents capable of lowering the surface
tension of water, are added to water adjacent to the oil spill
boundary, all of the crudes tested will form lens, reducing the area
of pool coverage by a factor of 3 to 10. If the surface active agent
is added five to ten minutes after crude deposition some case
hardening of the oil pool appears to have taken place, arid while the
pool volume contracts it retains vestiges of its former shape. Often it
has shard-like jagged edges and will not form a circular lens in many
cases. Thus by adding suitable surface active agents it should be
possible to greatly reduce the area of spread of spilled oil. Though
this idea was developed independently, it was subsequently found
that Sigwalt (6S), Blokker ( 11) and Garrett ( 31) had previously
developed somewhat similar ideas. The use of external fili ts, mono-
layers or surf actant treatment to prevent pool spreading at sea may
not be feasible under unfavorable weather conditions. Sigwalt noted
that oil pool spreading was not retarded by fatty acid ilms when
winds of greater than 2 to L knots were blowing in the direction of
sDreading. Garrett noted that the ab lity of monolayers to resist
wind induced deformation denended on their spreading pressure F 5
The higher F 3 was, the more resistant the monolayer was to deformation,
but even at the largest F 3 value tested, 141.1 dynes/cm, the film could
— 62 —
-------
not resist deformation at wind speeds in excess of 10 knots. However
in both Sigwa].t’s and Garrett’s test the monolayer was restrained on
its downward side and thus could not move downwind as a thole. It
is possible that in an open sea situation, the oil spill and the
surrounding monolayer or surf actant treated surface water could move
downwind together - the zone of modified surface still continuing to
maintain the oil in lens—like pools.
It was noted that partial vaporization of the crudes tended to make
them more susceptible to lens formation. In a spill at sea the
spreading will be so rapid that very thin films will usually be ob-
tained before any evaporation induced lens forming tendencies manifest
themselves. Evaporation will also cause stable skin formation. The
oil, through loss of its more volatile components, is converted into
a tarry non-deforinable film. As spreading of fluid oil continues) the
surface area available for evaporation increases. Continued evapora-
tion, aided by the increase in avaporation area but slowed do n by the
depletion of the more volatile fractions, should ultimately lead to a
degree oi tarryness which will prevent further spreading. T’xcept for
thin hi 1y volatile oils, the time in thich stable skin formation is
induced will tend to determine the ultimate thickness and area of spread
of the spill. For a typical Nid F astern cruder due to the nearly complete
loss of the more volatile fraction; evaporation is likely to virtually
cease after thirty percent of the crude has evaporated. For large spills
Blokker has estimated that this may take from 7 to Th hours under near
quiescent wind conditions, and 2 to 1 hours with 29 knot winds.
II’ tarryness sets in at this 30 b evaporation level, it can be estimated
from Blokker’sdata that the resulting tarry films will be 0.Th to 0.23 mm
thick (.00 to .009-in. thick).
Crude Oil Spreading Rates
Using motion pictures, as described in the Procedures Section, the areas
of oil pools spreading on quiet water were measured vs elapsed time.
Ir all cases, after an unsteady state period of about 0.2 seconds, the
pool area initially increased at a constant rate. For the Fullerton
crude (the one least likely to form a lens in the presence of a film
contaminated surface’ the rate of increase in area remained constant
over the entire range of times tested (20 seconds). During this time
the pool thinned down to an average thickness of 0.017-cm (0.O0 6 —in.’
in some instances.
At the other extreme, for the Urania crude the constant rate period
persisted for only roughly l. seconds, after .inich the rate of
- 63 -
-------
expansion gradually decreased, and the pool appeared to asymptotically
approach a thickness of 0.15 cm,the thickr ss previo’isly noted en
l rania forms a lens-like pool on top of a previously deposited Urania
film
Levelin -off of the rate of increase in area also occurred for the Tia
Juana and Safaniya crudes, but the constant rate of expansion period
persisted for more tir - usually eight seconds or more. The time at
which leveling—off startedwas very variable. In one case the area
increase rate for a Tia Juana pool started leveling-off after only t
seconds; in another case only after 13 seconds. Though the rate of
increase in area gradually decreased after the onset of leveling-off,
the Tia Juana and Safaniya pools did not appear to approach an asymp -
totic thickness. Rather they appeared to approach a new but markedly
lower constant rate of exDansion. The Tia Juana and Safaniya pools were
measured until they reached thicknesses of .029-cm (0.011-in.) in some
instances
Typical pool area vs time curves are shown in Figure XIII. Though the
initial rate of increase in area is clearly a constant in any given
test, the reproducibility of the rate data in duplicate test3 is poor.
In multiple tests, at the presumably same conditions, deviations from
the average spreading rate have ranged up to Th . The cause for this
variation me.y be partly due to variation in temperature, but more
likely it is due to variations in the amount of oil released. It is
difficult to accurately measure and release a known volume of the
stickier more viscous oils. On the basis of volume vs rate of spread
correlation to be discussed later, it is anticipated that a 7. ’
difference in volume would be required to produce a l1 change in the
rate of spread.
Since the pools tested were initially quite thick, it was anticipated
that the hydrostatic pressure force, g t 2 f 1 - °u I , would be more
2
important than -F 5 in promoting initial spreading. Initial spreading
rates (d.A/dQ) 0 for 100 cm 3 pools of the four different types of crudes
were measured. These rates and the corresponding values of
1 1 - I , ( [ 1 - °u J p )2 , and (dJ /dQ) 0 are tabulated in
Table 6, and ( ) is plotted vs I - ] °u in Figure XIV.
- 6}. -
-------
280
POOL ( kEA VS TIME FOR.
100 C.M POOLS OF VAkIOUS CRUDE
o. 0 2t4- SEOl JDS/FI ME
- - 4 AREA AS TO BE AD IUSTEC FG .
SCALE OF P OJECTIO 1 OF PICTURE
FIGURE L
a6o
240
I—
LI 2c
2
/
60
‘4’
0
60
40
160 200
-------
FIGURE 2
SPREADU4& R1\TE VS DEMSITY
F CTDR (FOR %ODC)1 POOLS
OF VARUDUS CRUDE OQS)
_____ 5LQPE 2.0
9LOPE L8
2 3
4 5 b I VlO
ft 1 ____
- -; : )/°wJ c i. i
2
0.1
4 6 -T
9 -.
g.
7-
6
S.
4-
0
o 5.
2.
1
I
S
1
S
5
4
0
S
0
I
-66-
-------
Table 6. Area Spreadir Rates as a Function of Oil Density
(100 cm 3 pools)
(1 - I (1 - C u ) 2 dA/dQ dA/do
Type of L 6 Spreading Rate [ ( . — 2
Crude gr/cm 3 gr 2 / cm cm 2 /aeo.
cm 8 / gr 2 sec.
Fullerton .iSI 5 .0239 291 12,200
Safariiya .1213 .O1) 7 181.3 12,3 O
Tia Juana .09976 .00995 138.5 13,900 ‘0
Urania .073)43 .005392 69.7 12,950
-------
It can be seen that a line of slope 2.0 fits the points fairly well.
ccordingly, the group (dA/dQ) 0 / I (1 - OU ) 2 rer ains fairly
constant, the individea.l values deviating from the average value
(12, 2 g cm 8 /gram2_sec.) by 8 at n st and lL. 6 on the average.
Actually because a relatively short range of the ouantity
I ( 1. - Pu ) o . j is covered it is difficult to decide on ernnirical
01
grounds ithether the exponent for i (i — Ou ) ° J should be 2.0,
or some value close to 2.0. An exoonent of 1.8 provides a sli t1Y
better fit than 2.0, the group (dA/dQ) 0 / [ (1 - 0 U ) °u 1 L
deviating from its average value (P,230 cm7 /gram] .R —see.) by
on the average and 6.L at r st.
The effect of spill volume on spreading rate was tested. Fullerton
crude, the least viscous, and therefore the crude whose pool volume
could be controlled with the greatest precision was used in these
tests. Fullerton also had the advantage of being the least sus-
ceptible to lens formation and falling off in spreading rate, Spill
vo1ur s were varied from 2 c m 3 to l 0 cm 3 . The area vs tirre plots
for these tests are shown in Figure XV. It can be seen that the
spreading rate increases markedly as the volume increases. The area
spreading rates are plotted vs pool volume in a lo -1og plot in
Figure XVI. It can be seen that with the exception of the 2 cm 3
pool volume the points fall fairly well on a straight line of slope
1.8. It will b assumed that the 2 cm 3 point is in error. The
(dA/d ) 0 / V 1.0 values — excluding that for 2 cm 3 - deviate from
their average value . 177 /cm3.LL sec., by 2; on the average and
at most. In more limited and less accurate tests with Urania crude
a V exponent of 1.6 was obtained.
The excess hydrostatic pressure is equal to the hydrostatic force
per unit length of pool cross section o g 1 1 - 0 u J t 2 divided
2
by the pool thickness t, i.e. ou g 1 — ou ] t . 4ien the pool
2
has any given radius R, the thickness is equal to V/if R 2 where V
is the pool volume. Thus for any given pool radius the excess
— 6P -
-------
24-
2 - FIGuRE X
22 OIL SPREADI R \TES
21 -
20-
19 -
IT
LO -
16
15
14
12
11
5
2
I I I I I
1 2 3 4 5 6 7 8 9 10 11
O FRJIP-IE
I I I I I I I
13 14 15 16 17 1
( 1RE I /S Ti 4E)
WATER. TEfr P
O L ULER.TOI i CRUDE
O.O 52 Se .c/vR \fr1E
V 1TEI TEMP 15C
I
4-
.5
- 69 —
-------
F1
It Ifl1AL 5P AL ki& PJLVVE
VS. POOL JOLUME FOk
POOLS O FULLER.TDIJ CRUDE.
DEPO5 rED O&.1 WAYER..
I I I I I I
9-
8
7—
6
5-
4 ,-
3-
1-
0
7
5
4
3
2
I
0
10 30 40 60 70 90 1X
OtL PooL VOLUME (me)
- 70 -
-------
hydrostatic pressure should be proportional to (o V/2rr d 2 ) [ 1 — P /°L 1.
This would imply that the grouping Pu V r i - P /OL ] might be a
significant characteristic parameter in determining the spreading
characteristics of a spill. Based on this inference and the closeness
of the empirically determined exponents for the individual groups
°u (1 — oU/pL)and V it will be assumed that both are raised to the
same exoonent. That is the initial rate of increase in pool area will
be assumed proportional to r 0 0- - ou/OL) V ] where n is approx-
L ately 1.8. R—2, which varies as the pool spreads does not arpear
in this cz up, which therefore can be taken as a time invariant
characteristic parameter. Essentially, it has been as um d that the
area spreading rate is a function of the excess hydrostatic pressure,
which in turn varies as R- 2 Yet the area spreading rate, which is
constant, does not depend on R. Therefore, if our reasoning is
correct, there must be some other factor, as yet urideter:ained, rhich
exactly cancels out the dependenc7 on R.
The various oils differ n .rkedly in viscosity. The Urania is C
times more viscous than the Fullerton. At the sa c timo the group
°u — was varied, the viscosity also experienced a
concomitant variation; and it is difficult to wholly separate the
effects of these twc variations. However it can be noted that for
equal voiwie pools of Fullerton and Urania, the Fullerton spreads
rcughly four tinEs as fast in area, has twice as high a value of
1 1 — P /OL 1 O , and a forty—five tinEs lower viscosity in the
test temperature range. It thus appears that, if the oil viscosity
is a variable affecting the rate of spreading, it must be a very weak
one. There are two situations that could ve rise to this situation.
One is that the spreading process is turbulent in w iich case the
influence of viscosity, though not completely negligible, u1d be
weak. The second is that the flow resistance to spreading depez s
on flow induced in the water layer beneath the spreading oil. In
such a case the viscosity of the water might influence the spreading
- 71 —
-------
rate. Fullerton spreading tests were carried out at three different
water temperatures: 11CC, 1SOC and )jJ. 0C. The spreading rates
are tabulated below.
Table 7. Area Spreading Rates vs Temperature
for 100 cm 3 pools of Fullerton Crude
(d.A/d ) 0
Temperature Spreadiflg Rate Water Viscosity (d.A/dQ) x
2
cm / sec. cpa
11°C 1.271i, 700
15°c 650 1.Th5 “145
1060 0.608 61
Average 696
The spreadini rate is also plotted vs the viscosity of water on
log—log paper in Figure XVII. It can be seen that the plot has a
slope of roughly —1.0. That is the area spreading rate appears to
be inversely proportional to the viscosity of the water on which the
sp’eading is taking place. This is substantiated by the rouc
constancy of the term (dA/d ) 0 x in Table 7. This term has
an average value of 696 cm 2 /sec. cps, and the individual values
deviate from this fi; ure by only 5 on the average, and 7.5 at
st. .mpirica1ly it appears that an even better fit is obtained
usinc an exponent n of —0.8 or —0.9 for u , in which case the term
(dA/d a) 0 x varies from its average value by an average of 3%,
and at most. A relatively small range of viscosities has been
covered and like the previously established exponents for V and
— ou ) Du the exponent n is not fixed with a high degree
OLI
of accuracy. For this reason the simple correlation
_i- (52)
will be used until the exoonent for , is fixed with greater
certainty. Tests in which the viscosity of water is altered by
a thickening agent may be suitable for exter ing the range over
which LLW can be varied.
— 72 —
-------
FIG YVfl !
( 1
I
U
‘U
c 1
2
w
cr)
-J
SPREP D 1& kJ TE 100 CJ ’1 3
POOLS OF FULLE TOU CRUDE
DIL VS VISCOSITh’ OF THE
UP IDER.L 1MG W [ \TER..
I I
- .2. 4 . .e .1 £
NP TEk VISCDStTY C C24T1 POISE)
a- . £
3-
7-
5-
4-
3-
2.
1.
S
I
6
5
4
3
2
0
0
0
I . .
0
0
—‘73 -
-------
The change in temperature also causes a lesser change in the
viscosity of the Fullerton crude. Thus it might appear that one
could equally well assume that the spreading rate was a function
of the viscosity of the oil. However if this were so, (dA/d ) 0
would depend so strongly on the viscosity of the oil that
(dA/dG) 0 for the Fullerton crude i uld be many times i re than
four times larger than (dA/d ) 0 for the highly viscous Urania crude.
Thus though the spreading rate appears to depend on the viscosity
of the underlying water, not the oil. Varying the viscosity of the
water by a thickening agent should help establish this somewhat
surprising conclusion on a firmer basis.
On the basis of these empirical tests the following relationship
can be derived to predict. the spreading of suddenly released pools
of oil
( ) ( )
where K has the value when is in centipoises, o and
in grams/cm 3 , V is in cm 3 , and dA% is in cm 2 /seo. K has the
value 2.03 when V is in gallons arid d.A/dG is in square miles/hr.,
the other variables retaining the previously cited dimensions.
This predicts extremely rapid rates of spreading for large volume
spills - e.g. a 1000 gallon spill of Urania crude would according
to the equation just presented, cover 1 square mile in less than
1 second. This is manifestly incorrect - the rates predicted are
very much larger than any measured by lokker ( U) or Stroop ( 71).
Stroop in test spills noted that a 1260 gallon spill spread to cover
an area of greater than 0.5 square miles in less than three hours.
He ncted however that the original spill area was about .01 sauare
mile. Since he was spilling oil at a finite rate it wculd nave
been impossible for him to have completed the spill and have the
oil remain with a 0.01 square mile area if the above equation were
truly applicable under the circumstances.
It therefore appears that the spreading rate equation just developed
cannot be used for large volume spills. This may well be because
- 7) -
-------
larger spills involve a different hydrodynamic regime. Since the
small scale spreading rate depends strongly on the viscosity of
water it must be a viscous or streamline regime. LarFer spills
probably fall within a turbulent flow regirie, where the conditions
regulating the rate of spread are different, it would be very
desirable in future work to determine: whether the two regimes
just postulated truly exist; what the critical conditions are
which determine which regin spreading will take place; and what
spreading correlation governs large scale spills. - third flow
regin e, where F 5 the surface tension spreading force, rather than
the excess hydrostatic pressure 1’ 1 — o TI ou g t 2 , provides the
driving forces may well exist when the pool becomes very thin. The
rate of spread transition noted with Safaniya and. Tia Juana crudes
may represent the onset of surface tension driven spread. Thus
Equation (53) may no longer apply when the pooi thickness, as a
result of spreading, falls below a critical value. Equation (3)
probably no longer applies as the pool thickness approaches 0.001-in.-
the approximate minimum thickness noted for films of snilled oil.
Equation (53) is wholly empirical and dinEnsionally inconsistent
(except for the forced consistency imposed by the dimensions
assigned to the constant K). As yet no hydrodynanic model, which
might justify Equation 53 and provide insi it as to the actual
flow processes taking place, has been found. A wide variety of
models have been explored but none has proved satisfactory.
Conservation of energy limits the maximum spreading rate. An energy
balance can be readily obtained for a simple disc model of an oil
pool such as shown below:
Figure XVIII
Water
h 0
h
- 75 —
-------
Neglecting surface energy, it can be shown that when a cylindrical
disc like pool of oil of volwi V spreads from an original radius
R 0 to a radius R, thereby decreasing its thickness from to to t
arid it subn rgence from h 0 to h, the decrease in potential energy
A. P will be equal to:
APE vPu(l_ôu) (tc _t) ( )
The increase in kinetic energy t ICE can be shown to be equal to
o, V dR.2
KE ( ) ( )
where in deriving the above equation use has been made of the fact
that conservation of rtass requires the local radial velocity U
tc be related to the boundary radial velocity by
dR
r —t
d ) 6)
and that consequently
= dQ) (57)
Equating the decrease in potential energy to the increase in
kinetic energy and solving for (dR/dQ) there is obtained
2g (t - t) 1 — Du) (5B
or
f2gVIl-ou rl -l’ (59)
d ‘L J
- 76 -
-------
It can be seen from either of the last two exiressions. when time
Q = 0, R = H 0 and t = to the spreading rate should be eaual to 0,
i.e. less than the constant spreading rate predicted by quation
(53). The spreading rate thereafter rapidly increases until
it is greater than that predicted by ‘ quation (53). 1 ]hen Equation
(59), obtained by considering the potential energy from which the
kinetic energy of expansion is derived) predicts a lower velocity
than Equation (53) the ve1ocit obtained from energy conservation
should., as a first approxin tion, be valid. However when Equation
( 3) predicts a lower velocity it, when otherwise applicable, should
be valid. The critical time and the critical radius R 0 at which
the two predicted rates of spreading are equal can be obtained by
solving Equations (53) and (59) simultaneously for R 0 / and
integration of Equation (59) for Q using the previously obtair d
value of Rc/Ro and the assumed or given values of V , 0 and
Solution for Rc/Ro yields
1.8
________ IK’ 11 — \ 1 1.3
= 1+ (—1 ( —)v I (6o
( g)0•S 0 L/ J
and for c
= R r ( d 2 — R 0 ) (61)
2 V g (1 -
hen these equations are solved for Q and Ft 03 for typical values
for V, ‘ and PL it is found that R. and depend very much
on V and R 0 . For the ease of a cylindrical pool which initially
is as high as it is wide ( V = 2 r 03 ) solutions obtained for
values of V = 100 cm3 and 1000 cm 3 with o = 0.855 and = 1.0
indicate that RC/RO = 1.30 for V 100 cJ and Rc/ = 16.7 for
V = 1000 cm 3 . Thus the range at which T quation (53) becomes
applicable starts at very much larger radii as the spill volume
increases. The corresponding values of c are 0.16 seconds for
100 cm3 and 5.0 sec. for 1000 cm 3 . At 10 liters the corresponding
values are R 0 /R 0 = 338 and G = 5L seccnds. The critical time Q 0
corresponds to an induction period during which expansion is slow
and during which Equation (53) is not yet applicable. For 100 cm 3
spills the predicted Q 0 , 0.16 sec., is roughly equal to the
induction period of 0.2 seconds observed for test spills before the
— 77 —
-------
constant rate of expansion in area equation begins to beconE
applicable. For a 1000 gallon spill the induction period would
be 8 hours and Rc would be one nautical mile; but the film would
probably have stopped expanding long before these conditions were
reached. It appears likely that such a large spill would go from
an energy conservation limited spreading rate to a turbulent
resistance limited spreading rate £ or which the transitions might
be significantly different than the critical transitions just
caLculated.
Wind Drift
Fort’i r_five wind drift tests were carried out. In most of these,
approximately ten oil pool drift rates were i easured at a given set
of operating conditions. Forty of the tests were made at a stathardized
wind duct height or 22 inches. Tests involving wirx.1 speeds of less than
19 ft./sec. were made using the attic fan, those greater than 19 ft./sec.
the industrial fan. The percentage drift, (oil pool velocity/wind
velocity) x 100, for each of these tests was calculated using average
wind velocity determined as described in the procedure section. i].e
there was considerable scatter in the data it was noted that the percentage
drift for the runs involving the attic fan averaged 3.2T 1 3 while those
involving the industrial fan averaged 2.8g. Since the industrial fan
prov-ided higher wind speeds this might appear to be a velocity dependent
etfect. Inspection showed however that the two different fans provided
different wiz i velocity profiles. For the industrial fan, which had a
large hub, the velocity at the vertical center line of the duct averaged
7 lower than the mean velocity in the duct, whereas for the attic fan
the corresponding center velocity was 3. 3 higher than the mean velocity.
The oil pools moved down the center of the duct and therefore the
percentage drift should be based on the center line WLrK1 velocity.
Accordingly the average speeds for the industrial fan were multiplied
by 0.93 and the attic fan speeds by 1.033 and the corresponding drift
percentages calculated on the basis of these new velocities.
The resulting drift percentages and velocities are plotted vs each
other in Figure XIX. ‘i hile there is still a good deal of scatter
in the data, the average percentage drift appears to remain fairly
constant as the velocity varies. For the forty test the percentage
drift averages 3.09%. The average relative deviation from this
— 73 —
-------
H
‘ -3
0
H
t-1
3.0
2.0
1.0
0
—
— —
0
0
oo
801
QdII
oo
0 o
0 o
p ?
0 0
!.
!
•
t
• Urania Crude
o Light Paraffin Oil
A Heavy Paraffin Oil
— Salt Water (Overlined)
— Industrial Fan (Underlined)
Figure XIX
PERCENT DRIFT VS.
JIND SPEED
0
5
10 15
20 25
WIND SPEED (FEET PER SECOND)
-------
value is 6.2 (0.19 drift percentage units) and the maximum
relative deviation is l9 . The average percentage drift for
the high velocity industrial fan, i.e. above 19 ft./sec. wind
velocity, is 2.98 while that for the low velocity attic fan is
3.11 (2.88 at below 6 £t./sec. and 3.18 between 6 and 19 ft./
sec. wind velocity). The apparent variation in percentage drift
with wind velocity is comparable to the scatter, and it is therefore
difficult to detect whether the variation is real. The assumption
that. the test basin percentage drift remains constant at 3.O9
appears reliable within +.05, —.21 drift units over the range
tested.
The percentage drift test figures require correction to render them
aoplicable to open sea conditions. ssentially one must extra olate
drift conditions obtained usin a finite wind duct height to the
infinite duct height of the open air. In a duct a relatively large
pressure drop accompanies the flow of air. This pressure drop
causes the water level to be lower at, the upwind end of the test
basin. In essence, the water in the basin acts like a manometer
in responding to the uressure drop. The difference in water level
in turn produces two effects which oppose surface drift. One, a
slope current, flowing in the opposite direction to the wind is
created in the bulk of the liquid. In a closed end basin this slope
current provides a return flow which exactly balances the surface
drift, thereby satisfying the necessary condition of zero net mass
flux. The return ducts accommodate some of the return flow and
help ninin ze the end effects associated with return flow. The
return flow is distributed over a larger area and thus has a much
lower velocity than the surface flow, which, as will be shown later,
is confined to a very thin layer. Since the surface layer is moving
counter to the bulk flow the surface drift velocity should tend to
be son that reduced by the return flow.
Secondly ,the oil in moving on the water surface ,is moving slightly
uphill due to the difference in water level. This should also tend
to slow down the oil mov nent.
A simple theory will help provide insight as to magnitude of the
nercentage drift and how the duct height correction might be applied.
The wind in moving over the water creates a surface shear stress
which is proportional to £ °A 112/2 where f, the friction factor,
is a function of the Reynolds number characterizing the flow,
— —
-------
U is the velocity relative to the surface, and A is the density
of the air. If the air is moving over water which is not moving
below the surface (i.e. the return flow is distributed over such
a wide area that its velocity may be assumed negligible the
velocity profile for the air-water system can be represented by
the following sketch.
Figure XX
Vertical
Distance
Velocity
At the surface of the water the shear stress due to the air movement
will cause a drift velocity US to develop. Relative to the surface
of the water, the water appears as if it is moving backwards with
a velocity -U 5 , and the air forward with a velocity (UA - US). At
equilibrium the shear stresses due to these two relative flows
should be equal, i.e.
(UAUs) 2 ( Us) 2
A PA (62)
2 2
where A is the air friction factor, the water friction factor,
and ow the water density. For turbulent flow over flat surfaces,
the friction factor can be related to the Reynolds number by a
simple equation of the form
,LUpn
£ = ) (63)
ithere K is a constant, the viscosity, and n a characteristic
exponent. For high flow Reynolds numbers, n equals - 1/7 (Falkner
( 29) ). Substituting for f, using a n value of — 1/7, there is
Air
— - - — — Interface
Water
Us
- 81 -
-------
obtained upon rearrangement arid cancellation of terms which appear
on both sides the equation
= ( / [ 1 + ( ).0:7 3
(6L)
T oon substituting values for the density and viscosity of air and
water, there is obtained
US
= 0.03l (6 )
whj&t is not too far from the IJ3/UA = 0.0309 figure obtained in
the test basin. However, Equations (611) and (6 ) should apply to
open water not to air flow in a duct, and the C .03l8
fiure (or Its salt water counterpart 0.03Th) is somewhat lower
than the u 3 /u 4 = 0.03t l renorted for the Torrey Canyon drift (66)
or the slic htly higher US/UA figures reported by or calculated from
Stroop’s data (71), the Gerd I•Iaersk drift ( 7 ) and Hughes (33).
quation (6 ) predicts Us/UA values which are significantly higher
than those calculated frcm Thoradets drift equations (73) which
antear to be in error.
T’ e percentage drift is not very sensitive to the exact value of n
used. - For moderately high iteynolds numbers as contrasted to very
high teynolds number a value of —0.2 is soxi times used for n.
In such a case Us/ISA would equal 0.0311, and if n were 0, US/ISA
would equal 0.0330 (both for fresh water. This insensitivity is
due to the fact that the Reynold numbers for the air and water
differ only by a factor of 1.8.
In previously considering duct height corrections it was postulated
that these were ultimately due to air flow induced pressure drop.
Since the flow regime is turbulent for both the water and the air,
it will be assui d that the term fi.j U 3 2 /2 in Equation (62) will
have to be corrected by a term proportional to the air flow induced
— 82 —
-------
pressure drop. But this in turn is proportional to
where A’ is the appropriate friction factor for duct flow.
Substituting this proportionality in Equation (62) and neglecting
the small difference between UA 2 and (TJA-US) 2 in the left hand side
there is obtained after some rearrangement
0 A UA2 ( A - K A ) = f O j Us 2 (66)
where K is the constant of proportionality in the correction term.
Both friction factors A and should show similar functional
dependence on the appropriate Reynolds numbers but the characteristic
lengths in the Reynolds numbers should be different, D for and L
for A• Substituting for and fAt in terms of these Reynolds
numbers and assuming £ is proportional to Re 1/7 there is obtained
upon rearrangement:
U 5 (OA)0. 4 62 i 0.077 [ - K ( )l.l1&3j ).539 (67)
This is similar to Equation (61i) with the correction term
°A 0 I 62 LLA 0.077
F 1 + ( — ) ( — ) 1 in the denominator dropoed and the
ow ‘ Iw
term [ 1 - K ( ) 3 3 0.S39 which corrects for duct height added
D
as a factor. It was previously noted that use of the factor
A )0 l L 6 2 A )0.077 yielded percent drift estimates which were
slightly lower than percent drifts measured at sea. Let u.s assume
this factor requires correction arid replace it by an unknown factor N.
In our tests L the basin length was maintained constant. Introducing
M and lumping all the constant terms in Equation (67) there is
obtained:
— 83 —
-------
I
K -.
— (68)
This equation can be used to obtain both N , the open sea
percentage drift, and ‘ if reliable experimental percentage drift
data is available at t m, different duct heights. If reliable
percentage drift data is available at three different duct heights,
the validity of Equation (68) can also be checked. Assume US/IJA is
known at two different duct heights D 1 and D 2 . By dividing Equation
(68) for case 1 by Equation (68) for case 2 there is obtained upon
rearrangement
1.86 1.86
( UiluA) 1 - (us/UA) 2
(u UA) 86 _ (U uA)1 86 (69)
D Th
Equation (69)is solved for K’and K’ is in turn substituted in
Equation (68) where a particular exoerimental value of TJ 3 /u and the
corresponding value of D is used to solve for N . If the same
values of K’ and N are obtained using the different pairs of three
different duct heights and correspox ing US/IJA values Equation (6F )
would be verified.
Five tests were carried out at lower than normal duct heights. Since
the average duct height for these tests did not vary greatly, and
individual percentage drift measurements were subject to great
variation, the overall average duct and average U5/UA value for the
tests — 13.1 inches and 0.O2 9 respectively will be used in calculating
K’ and N . It can be seen that US/UA at these lower duct heights
is significantly smaller than US/UA at the 22 inch duct height, and the
duct height correction is therefore quite significant. Using 13.1 inches
for and 0.O2 9 for (Us/UA)2 and the standard duct height 22 inches
for Dj and the previously established value 0.0309 for (Us/TJA) 1 ,
Equation (69) was solved for K’. A value of 8.9l inchesl. 4
was found for K’. This value)when substituted in Equation (68) yielded
-814-
-------
a value for M of 0.0366 when appropriate values of (uS/HA) and D
were substituted. Error analysis accounting for the probable errors
in the US/ values at both the low and high duct heights indicates
that, based on this value of M, the open-ocean percentage drift
should be 3.66% * .l7 . This figure is in quite good agreement
with average field data.
It should be noted that the open ocean drift value obtained is only
moderately sensitive o the exact method of scaling up to infinite
duct height. For example ass uining TJS/UA varies linearly with l/D
and goes to N as 1/D goes to 0 yields a open-ocean percentage drift
figure of 3.82 which is not too different from the 3.66 obtained by
the present more elaborate procedure.
In contrast to wind duct height, the water depth had a negligible
effect on wind drift rates. Three tests were carried out at water
depths ranging from 3—1/2 inches to -l/2 inches - in contrast to
the normal 8 to 10-1/2 inch depth. The average percentage drift for
these three tests was.3.lL , which barely differs at all .t’rcrn the
3.09% average drift value for all the normal depth tests
Similarly it anpears that the drift rate is not greatly effected by
the viscosity of the water or the oil. The percentage drift for
heavy paraffin oil, which is much more viscous than light paraffin
oil, averaged 3.11%, which scarcely differs from the normal 3.09. value.
Two tests were carried out in which the water tenperature was varied
appreciably. In one test a temperature of e. oc was used and in the
other 1100. The percentage drift for these two tests averaged 3.21;,
which is slightly higher than the normal 3.09f value. However the
0.12 difference between these low te iperature results and the nori al
drift results (where the temperature averaged 19 to 2 CC) is less than
the 0.19% average deviation normally experienced in percentage drift
tests, and less than the C.13 deviation that would, on the basis
of error analysis, occur on the average for any random sequence of
t tests. In the range 8. °C to 22°C, water viscosity changes by a
factor of i.lj., and if water viscosity were to significantly effect
drift, the change should be much more marked and of onoosite si i
than the slight change noted quation ( 61 L) indicates the water
viscosity should affect the percentage drift. But, the predicted
effect for a 1.1j fold change in water viscosity u1d. be only 0.08
percentage drift units, which scarcely could be detected due to the
normal scatter in the drift data.
— —
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The percentage drift for crude oil (Urania) averaged 2.98 and
thus was slightly lower than the normal 3.09% value. However the
great bulk of the crude oil tests were made at high velocities using
the industrial fan and the difference can be mostly attributed to
either the higher velocity or artifacts associated with the use of
the industrial fan. Only Urania crude was used in these high
velocity tests, because the araifin oils provided less visual
contrast and broke up into small pools, and thus could not he readily
observed. In three low velocity tests in which Urania crude was used,
the percentage drift averaged 3.O8 which is virtually identical with
the normal 3.09: figure. It thus appears that differences in percentage
drift between crude oils and the paraffin oils used in most of the tests
is r gligibly snail.
Based on the Urania crude tests, the paraffin cii tests, and limited
tests with tetradecane the effect of F 5 , the surface tension spreading
pressure, on wind drift is negligibly small.
The use of salt water as opoosed to fresh water had little effect on
drift rates. In two tests carried out on 3.S salt water using Urania
crude and wind speeds close to 2b.0 ft./sec. the percentage drift
averaged 2.96%, which is virtu ily identical with the 2.98; average
drift reoorted for all tests involving Urania crude and high wind
speeds. Equation (6L ) indicates that a O.Oi reduction in percentage
drift should have occurred upon switching from fresh water to salt
water. Such a small change could not be detected in cur tests. It
appears that the test basin results which were mcstly obtained with
fresh water can equally well be applied to salt water with no sig —
nificant error.
Tests were carried out at pool vclw s different from the standard
cc volume. In two tests with paraffin oils at 100 cc volwi the
percentage drift averaged 2.99. and in one test at 200 cc vclune
2.98 . while slightly lower than 3.09%, these results are well
within the range of variation of individual tests and the deviation
limits that would be expected on the basis of error analysis to
occur for a random sequence of three tests. It thus appears that
pool vo1un has a negligibly small effect on percentage drift.
In nest tests, the test basin surface was kept fairly clean by
sweeping of f residual oil. In one wind drift test with uaraff in
-86-
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oil, however, the basin surface was deliberately dirtied with crude
oil. In this test the drift averaged 3.l9 . This value is slightly
higher than normal, but because of the large normal scatter in the
drift data it is difficult to determine whether the difference is
significant or accidental. It was observed that the oil suppressed
wave formation and the slight increase in drift effect nd ht be due
to this wave suppression action.
It was noted that wave action significantly lowered wind drift. In
four tests in which waves having a period of roughly 1 second, a
wave length of roughly b5 inches and amplitudes of roughly L inches
were used, the average percentage drift was 2.66 , as contrasted
to the normal 3.09g. Drift caused by these waves alone tras found
to be small, roughly .01 ft./sec., as determined by tests. Such
a drift could only influence the apparent percentage wind drift by
0.O7 in the range tested. Therefore the decrease in atrarent tTind
drift could not be due to an undetected wave drift, but must
instead be due to increased resistance to wind induced drift, or
less effective transfer of momentum from the wind to the oil floating
on the water. This is further substantiated by the fact that waves
moving in the direction of the wind and waves moving: counter to the
wind both produced the same reduction in wind drift.
It is believed that reduced efficiency of momentum transfer comes
about because there are drag free stagnant zones in the lee of each
wave crest. If this is so, longer waves and less sharply peaked
waves may cause less reduction in wind drift. hith such waves there
would be fewer sheltered lee zones and, because the air flow
diversion would be more gradual, the relative size of the sheltered
areas would be smaller. The testing of this hypothesis through the
use of longer period waves would be most desirable. Unfortunately
because of limitation in the range of operation of the wave machine,
and the depth of the basin, the desired longer period waves could not
be generated, and important questions relating to wave wind-drift
interaction remain unresolved.
The wind drift retarding effect of waves may exlain the slight
tendency towards lower drift at high wind speeds. ore waves, and
waves of greater amplitude were created in the test basin at these
higher wind speeds. These wind created waves in turn should
somewhat reduce the percentage wind drift - but whether the effect
can in fact be reliably detected under the test conditions used is
— 87 —
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somewhat questionable. Since wind will generate similar waves,
but of higher amplitude and greater wave length and spectral
variety on the open sea - particularly as the fetch increases -
the effects of wind and wave are inextricably bound together. This
combined effect must be resolved if reasonable drift predictions
are to be made.
Though a twenty foot test basin might appear adequately long,
significant end effects are readily apparent. In wind drift tests
using wood chips, the wood chip velocity peaked at roughly to 7
feet downstream from the point where oil usually was deposited. Up
to that distance the wood was accelerating and beyond it decelerating.
This peak velocity sometimes was as much as 9 higher than the average
velocity over the usual drift test length. With oil the peakin was
less marked, the peak velocity being roughly 3 higher than the mean
velocity in the test section. The deceleration is caused in part
by approach to the end of the test basin. Because this deceleration
starts so far upstream in the basin it is difficult to be certain
that the wood chip or oil pool has reached its steady drift velocity
prior to deceleration. A forty foot long test basin would increase
the available drift test length by a factor of three or four and permit
verification that steady state drift velocities had been reached.
Cne complicating factor which is not an end effect also might con-
tribute to drift rate reduction as the oil proceeds downstream in the
basin. The wave amplitude increases in the downstream direction, and
this increase in amplitude, because of the wave-wind drift interaction,
could also cause a reduction in percentage drift. The detection,
measurement, and correlation of this type of interaction would also
be facilitated by a longer basin.
It was observed that wood chips drifted at rates that were sig-
nificantly lower than those for oil pools at the same test conditions.
The following table lists relative drift rates for wood chips and
other drift tracers.
— 88 —
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Table 8. Relative Drift Rates
Marker Drift Velocity
Oil Pool Velocity
3/ti)’ x 3/)4” paper marker .90
Xylene, Dibutyl Ththalate, Zinc Oxide
mixture of specific gravity 1.00
floating just below surface .91
3/Li” x 3/ i L” x 3/32” thick wood chips
3/Li ” x 3/iL ” x 3/16” wood chips .78
It is somewhat remarkable that thin slips of paper floating on the
surface and droplets of specific gravity 1.00 liquid floating just
below the surface move at a lower velocity than the oil. The reason
for this is not kno .
The progressive decrease in drift rate with increasing wood chip
thickness indicates that the surface drift is confined to an extremely
thin layer. A simple model for this surface flow is a velocity
profile in which the velocity exponentially decays as the depth below
surface increases, i.e.
—kZ
UD US e (7 0\
where Z is the depth below the surface, tJD the drift velocity at
that depth, US the surface drift velocity and k the decay constant.
The wood chip data in table three indicates that for such a model k
should equal roughly 1.9 in. 1 . For this k va1ue the local drift
velocity should equal only lO of the surface drift velocity at a
water depth of 1.2 inches.
The average velocity UD over a given depth for this model is given
by the formula
- 89 —
-------
US -kZ
= j tl-e ] (71)
This formula indicates that for a floating object 1.2 inches deep
the drift velocity should be only 39% of the surface drift velocity.
This apparent reduction of drift velocity with depth suggested that
the use of shallow drogues or subsurface draggers nii ht markedly
reduce wind induced oil pool drift. Accordingly th’aggers having the
configurations shown in Figure XXI were constructed and tested.
In case A, the dragger was dropped to the center of the oil pool.
The dragger percent drift was l.63. , much less than normal oil drift,
but most of the pooi of oil pulled away from the dragger. In case 3,
a similar reduction in drift was noted but here much more of the oil
was retained by the dragger. In case C no oil was lost, and the
percent. drift was i. l%. The drift reductions are of the order of
magnitude that might be expected through the use of Thuation (71’)
with a k value of 1.9 in. ’.
These tests indicate the technical feasibility of reducing wind
drift by a factor of t or r.ior through the use of draggers. A
slight degree of confinement is apparently necessary to prevent oil
from pulling away from the dragger. The economic feasibility of
using draggers depends on developing cheap simply produced draggers
which can be readily disseminated.
It is noteworthy that configuration C also prevented crude oil from
spreading and thus effectively confined the spill to an arsa many
times smaller than it would normally occupy. It is quite possible
that a floating net like array made up of foamed plastic strands
could be used to provide a similar type of confinement for large
oil spills.
Wave Drift
It was previously noted that wave drifts of 0.01 ft./sec. were
noted for wave action alone. This is markedly less than predicted
-90-
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Ill I.S ,‘
Ifq” ROJJ JD
FIIGURE xQ
DRAC GER S
L
1
6”
4;
Ia—
WI
wtQ-
- 91
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by Stokes’ correlation, Equation (30), which indicates a wave
drift of 0.2 ft./sec. In part this discrepancy may be due to
inadequate da.mp iing of wave reflection, which could have caused
the waves to have more of a standing wave character than a
progressive wave character. But based on the appearance of the waves -
which clearly appeared to be progressive - and the fact that apparent
wind drift was reduced by roughly the same amount when the waves
travelled with the wind as when they travelled against the wind
this appears unlikely. The discrepancy is more likely to be due
to the breakdown of Stokes’ theory in shallow water. Similar dis-
creparicies in shallow water wave drift work have been cited in our
“Analysis of Prior Workil section.
It therefore appears that experimental resolution of questions
relating to wave drift, and wave, wind-drift interactions will
require testing in a deep water basin. Since,as indicated in
“Analysis of Prior Work T ’ wave drift might possibly constitute an
appreciable fraction of the total drift, and since our work indicates
highly significant wave, wind-drift interactions such deep water basin
testing is highly recommended.
Current Drift
Oil pools floating on water which was undergoing a current-like flow
drifted at slower velocities than wood chips floating in the same
current. For example, in the sane current>oil drifted at 30.6 ft./nmin.
whereas 3/li” x 3/1 ” x 1/li.” thick wood chips drifted at 36.0 ft./rnin.
This discrepancy is probably due in oart to the velocity profile
characterizing the flow. In a typical velocity profile for open
channel flow the velocity at the free surface will be about lower
than the rriaximuin velocity, which occurs a short distance below the
surface (Rouse p. 277). The thick wood chip penetrates into this
zone of maximum velocity and therefore should have a higher velocity
than a pool floating on the surface. However, the velocity profile
effect.if norma]. would only account for roughly one—third of the
discrepancy between the oil pool and wood chip velocities.
‘Iien machine made waves (having the characteristics previously
described) are superimposed on the current like flow, the oil pool
drift and the wood chip drift becoma virtually identical - the
- 92 -
-------
difference in drift being less than i% of the drift velocity.
This may well be due to wave induced uniformity in the velocity
profile. Since conditions at sea are wavy it appears valid to
assume that oil pools will experience a current induced drift
equal to surface drift associated with the current.
The surface currents are not uniform in the length direction in the
test basin. For distances moderately removed from the do nstream
end of the test basin the surface current increases in the down—
stream direction. This is probably due to a transition from the
uniform velocity profile in osed by the flow distributor grid at
the unstream end of the test basin to the normal flow profile which
is peaked near the free surface.
Combined Wind-Current Drift
Sixteen tests were carried out in which floating oil pools were
subjected to both wind and current action. The results of these
tests are tabulated in Table 9. In these tests the wind drift
was rreasured in the absence of currents, and the current drift
measured in the absence of the wind, and then the combined drift
for the same current pumping speed, and fan speed was measured
In all cases the contribition of the current to the combined
drift was less than the current drift in the absence of wind.
This result is very startling and contrary to prior expectations
and the usual assumption of the simple vector additivity of drifts.
The combined drift T is ver r rou ily correlated by the equation
T = W+O. 7C (72)
Although the average deviation from this equation was only 6.L and
the rnaxtmum deviation l3. %, the agreement is only fortuitous and
the equation can not in general be correct. It is obvious that as W
goes to zero, T should approach C, not . 57 C as indicated by the
equation. This requirement implies that the coefficient for C should
increase as W decreases, but the data provides no indication of such
an increase. The cause for the lack of additivity of the wind and
current drifts has not been determined in spite of considerable effort.
- 93 —
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Table 9. Combined Wind and Current Drift
W C W+C T
Wind Wind Current Sum of Actual W+C—T (W+C-T)/C
Velocity Drift Drift Drifts Combined
Drift
( ft./sec. ) _______ ______
7.93 .220 .291. .5114 .379 .135 .1469
10.36 .2148 .270 .518 .365 .153 .566
.1.i.b6 .293 .759 .591 .168 .572
12.55 .396 .1149 .5145 •1470 .075 .503
10.55 .321 .3149 .1470 .395 .075 .503
25.87 .6014 .322 .926 .756 .171 .528
26.62 .7714 .3142 1.116 .898 .218 .637
25.95 .83]. .327 1.158 .889 .266 .83.3
23.68 .600 .330 .930 .814.5 .085 .258
23.214 .650 .330 .980 .883. .099 .300
21.o l .597 .370 .967 .750 .217 .586
25.014 .750 .3 ,]. 1.101 1.000 .101 .288
214. 30 .639 .351 .990 .b89 .101 .288
25.23 .686 .380 1.069 1.020 .0149 .129
214.95 .697 .373 1.070 .970 .100 .268
22.20 .598 —.191 .1407 —.037 .193
Average .14)43
- 914 -
-------
There is no marked trend in the additivity behavior as the wit
velocity or wind—drift, current drift ratio varies. This combined
drift result is at such variance with the anticipated simple
additivity, and the limiting relationship that must hold then W
equals zero, that it must be viewed with suspicion.
It is possib1 that current induced turbulence could increase the
shear stress friction factor on the water side of the air-water
interface. If this were the case the wind induced drift would be
reduced arid the combined drift would be less than the sum of the
individual drift.
It s iould be noted that only one test was run with the current o osing
the ririd . In that test there was a large scatter in the current drift
data, a:d the deviation from Equation (72) was fairly large. However
even in this case the contribution of the current drift to the
co rbined drift was less than tho current drift in the absence of wind.
- 9 —
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— 99 —
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Notation
In carrying out this project, reference is made to
work in several different disciplines, arid by many different authors.
In such work, traditional commonly accepted systems of notation have
developed. In the present rk these conm nly accepted systems of
notation have been retained for the most part. This serves to prevent
confusion in comparing the present and prior work. Howeverit has led
to the same letter symbol having different meanings depending on its
context. In all cases the correct meaning should be clear from the
context involved, but to prevent any confusion there multiple definitions
are provided for the sane letter, each multiple definition in the
following table is referred back to the first equation of a given
sequence in which it is used.
A Angle at edge of lens (Figure II, Equations 9, 10)
A lave amplitude (Equation 29)
A Spill Area (Equations 52, 53)
a Capillary constant [ 2 W g ( - 1/2
B Angle at edge of lens relative to extended edge surface of water
b Channel width
C Concentration (Equation 26)
C Current Drift Velocity (Equation 72)
D Pool Diameter (Equation 8)
D Duct Hei t (Equation 66)
f Linear tension (Equation 18)
f Friction factor (Equation 62)
F Force
F 5 Spreading force ( o — - c )
g Acceleration of gravity
H Height of Wind Duct (Equation 37)
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h Lens submergence (Figure I, Equation 3)
h Water depth (Equations 29, 30)
K Effective diffusivity (Equation 26)
K Constant in friction factor relationship (Equation 63)
K’ Duct height correction factor (Equation 68)
K Spreading rate coefficient (Equations 23, 53)
k Attenuation vs depth factor for surface drift (Equation 79)
KE Kinetic energy
L Length of duct, length of wind path
P Spill length
Wind drift proportionality factor (Equation 68)
n Exponent
P Hydrostatic pressure
PB Potential energy
R Radius of curvature ( quation 1)
R Lens or pool radius (Eouations 19, 22, 55)
r Distance from center of pool (Equation 26)
.3 Spectral distribution function for wave energy
T Total drift velocity (Equation 72)
T Basin Depth (Equation 37)
T wave period (Equation 32)
T Period of oscillation (Equation 22)
t Lens thickness (Figure 1, Equation 2)
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U Water surface velocity due to wind shear (Figure XX, Equation 62)
Air velocity (Figure XX, Equation 62)
Water velocity below free surface (Equation 72)
IT Mean forward velocity due to wave motion (Equation 29)
Ur Radial. velocity (Equation S6)
V Surface drift velocity (I uation 26)
Average wind velocity over water (Equation 37)
Average wind velocity in duct (Equation 37)
Average oil velocity (page 1 l)
V Spill volume (Equations 7, 2 , 38, 3)
W Wind drift velocity ( quation 72)
W Wind speed (Equation 26)
Depth of water in basin (Equation 3?)
z Depth below free surface of liquid (Equation 2)
Z Depth below mean free surface (Equation 29)
Greek Letters
Angle between wind path and drift path
B Latitude
T ifference
Time
X Wave l gth
p Density
a Surface or interfacial tension
w Angular frequency of wave motion
Viscosity
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Subscripts
A Air
C Critical or transition between types of spreading
D Subsurface (Equation 70)
D In duct (Equation 37)
i Interfacial
L Lower layer (erig. water) in lens system (Equation 3)
L Acting to the left (i quation 3)
o Original (Figure XVIII, Equation )
o At the free surface (: ‘quation 30)
o Ci i (page I’l)
P Due to hydrcstatic pressure
R Acting to the right
r Radial
u Upper layer (e.g. oil) in lens system
z In vertical direction
w Water, over water
1 Above interface, major
2 Below interface, minor
100 For 100 c m 3 volume
Limiting case for large lenses or fully spread spill
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