MSc Dissertation Thesis
MSc in Sustainable Energy Systems
Self-Embedding and
Self-Removing Sand Anchor
Maja Persson
S1033273
August 2014
i
Mission Statement
Background
One of the challenges with wave energy devices is the attachment to seabed. This is often solved
if the specific seabed has certain features for instance greater rocks that the Wave Energy
Converter (WEC) can be anchored to. However this is not always the case; it may be desired to
install a WEC where the seabed just consists of sand. Installing permanent hardware in the
seabed is unlikely to be accepted and so a possible design of an anchoring device has been made
that needs model testing to analyse its capability.
It is made out of three hollow legs that enable the anchor to float if they are filled with air and to
sink when a small amount of water is pumped in. The more the legs are filled with water, the
more the anchor is embedded into the seabed. A vacuum is created in the three legs once the
anchor is fixed into place. It is self-removing as well, which is useful when the WEC has reached
its end of life. Letting the hollow legs fill up with air again enables this. But this is rather
problematic when sand grains are involved, and so the water in the anchor legs is first forced
out, creating a fluidised bed around the anchor and the fluid-behaving sand and water mixture
then enables the anchor to move.
Objectives

To see if the friction created between the anchor legs and the sand grains is enough to
hold the force of the waves.

To see if fluidising the seabed allows the anchor to both sink and float back.
Aims
This will involve testing a representative model leg of the anchor in a water butt filled with sand
and water working as the seabed.

For the tests, a vacuum will be created through the anchor leg using difference of height
instead of using a pump as with the real model.

Friction force of the anchor leg will be tested when it is fully embedded. A crane with a
load cell will be used as well as a scale under the water butt.

It will be investigated if the anchor can suck the water and sand mixture up and if it can
be installed and removed by fluidising the bed.

A scale will be placed underneath the water butt to inform how much water there is
between the sand grains in the seabed.

The water flow and the vacuum pressure will be measured.
ii
Interim Targets
By having output data in flow rates, holding power and positioning, the practical experiments
will give an insight of the effectiveness of this possible WEC anchor design. If proven successful,
then it will be very useful for future WECs since the features of the seabed will then no longer be
a limitation. This will also aid in further understanding of fluidised beds and an alternative
usage of it for the student.
iii
Abstract
The purpose of this project is to investigate the feasibility of a new design of an anchor for
mooring a wave powered desalination device in a sandy seabed and where permission requires
future removal. It consists of a tripod made out of three hollow post-tensioned concrete legs
joined by a steel shell. It is buoyant for easy towing then flooded to sink. It embeds itself easily
by pumping water to fluidise the seabed then pulls a vacuum to force the surrounding particles
to travel towards it due to pressure difference, creating a strong grip that can resist the
maximum wave. The tripod can be removed from the sand by pumping air to get buoyancy and
then re-fluidising the seabed.
The project tested a model of one of the tripod’s legs. The sand grip was measured. Scaling rules
show that the 40 MN force from a 100-year Atlantic wave acting on 12 metre diameter 25 metre
desalination unit could be taken by a roughly 3 metres diameter anchor leg, where 12 metres
length is appropriate considering water depth of 40 metres.
The suction flow rates were also measured. From these investigations, it was found that the
design has the potential to be a successful mooring for the desalination project. There is some
concern about high flow rates, but it is expected that the build up of plankton and fine sediment
will cause these to fall.
iv
Declaration of Originality
I declare that this thesis is my original work, except where stated otherwise. This thesis has
never been submitted for any degree or examination at any other University.
………………………………………………………………………(Signature)
v
Symbols
A = Area (m2)
a=Acceleration (m/s2)
Cp=Discharge coefficient
d= Diameter (m)
g= Acceleration due to gravity (m/s2)
𝑔𝑐 = Newton’s law proportionality factor (=1 for SI units [1])
F = Force (N)
𝐻 = Height (m)
L = Length of bed (m)
m=Mass (kg)
Δ𝑝 = Pressure drop N/m2
P = Pressure (Pa)
Q=Volumetric flow rate (m3/s)
q=Angle (degrees)
sp = Surface area of single particle (m2)
t= Wall thickness (m)
U=Velocity (m/s)
̅𝑈̅̅𝑜̅ = Superficial velocity (m/s)
̅̅̅̅̅̅
𝑈𝑂𝑀 = Minimum fluidising velocity (m/s)
V=Volume (m3)
v=Velocity (m/s)
xi= Weight fraction of particle size
𝛾 = Specific gravity
𝜀 = Porosity/void fraction
𝜂 = Friction coefficient
𝜇 = Absolute viscosity (Pa*s)
ρ= density (kg/m3)
𝜎𝜃 =Hoop stress (Pa)
Φ𝑠 = Surface-volume ratio
Subscript
anch=Anchor leg
atm=Atmospheric
b=Buoyancy
d=Drag
i=Internal measurement
f=Fluid/gas
M=Minimum
m=sand and water mixture
o=Outside measurement
p=Particle
s=Solid
sc=Sand crystal
si=Silica
T=Total
V=Void space
vac=Vacuum
w=Water
wv=Water in voids
vi
Table of Contents
1. Introduction ................................................................................................................................. 1
2. Literature Review ....................................................................................................................... 2
2.1 Mooring Systems ................................................................................................................................. 2
2.1.1 Uplift-Resisting Anchors ........................................................................................................................... 2
2.1.2 Self-Embedding Anchor Concepts......................................................................................................... 3
2.1.3 Suction Caisson Anchors ........................................................................................................................... 7
2.2 Fluidised Beds ...................................................................................................................................... 8
2.3 Sand Anchor Concept .......................................................................................................................10
2.3.1 Reasons for Developed Design ............................................................................................................ 10
2.3.2 The Tripod ................................................................................................................................................... 11
2.3.3 Forces on Tripod and Forces According to Scaling..................................................................... 12
3. Design Methodology ................................................................................................................ 13
3.1 Designing the Experiment Layout ...............................................................................................13
3.1.1 Experiment Diagram ............................................................................................................................... 14
3.1.2 Logistics of Assembling Certain Parts .............................................................................................. 15
3.2 Pressure Flow Rates .........................................................................................................................16
3.3 Sand Investigations ..........................................................................................................................19
3.3.1 Choice of Sand: Fluidising Properties, Availability and Health Risks.................................. 19
3.3.1 Investigational Experiments with Sharp Sand ............................................................................. 20
3.3.2 Void Fraction Comparison .................................................................................................................... 22
3.3.3 Sand Conclusions ...................................................................................................................................... 22
3.4 Minimum Fluidising Velocity for Anchor Leg Model ............................................................22
3.5 Risk Assessment ................................................................................................................................23
3.6 Anchor Leg Modifications ..............................................................................................................25
3.6.1 Machining Parts ......................................................................................................................................... 25
3.6.2 Alterations on the Model ....................................................................................................................... 27
4. Experiments ............................................................................................................................... 34
4.1 Set-up.....................................................................................................................................................34
4.2 Results ...................................................................................................................................................37
4.2.1 Change of Grip during a One Hour Period ...................................................................................... 37
4.2.2 Change of Flow Rate during a One Hour Period........................................................................... 38
4.2.3 Further Testing .......................................................................................................................................... 40
vii
5. Discussion ................................................................................................................................... 44
5.1 Comments on the Experiments ......................................................................................... 44
5.2 Further Calculations from Experiment Results .....................................................................45
5.2.1 Size Related Force..................................................................................................................................... 45
5.2.2 Creating Boundary Conditions for the Real-Life Model ............................................................ 46
5.2.3 Stokes Law ................................................................................................................................................... 46
5.3 Error Analysis and Recommendations for Further Work ..................................................47
6. Conclusions ................................................................................................................................. 50
Acknowledgements ...................................................................................................................... 50
Bibliography ................................................................................................................................... 51
Appendices ...................................................................................................................................... 54
Appendix A .................................................................................................................................................55
Appendix B .................................................................................................................................................56
Appendix C ..................................................................................................................................................58
Appendix D .................................................................................................................................................60
Appendix E ..................................................................................................................................................61
Appendix F ..................................................................................................................................................62
Appendix G..................................................................................................................................................64
Appendix H .................................................................................................................................................65
Appendix I ...................................................................................................................................................68
Appendix J ...................................................................................................................................................69
Appendix K .................................................................................................................................................70
Appendix L ..................................................................................................................................................71
Appendix M.................................................................................................................................................72
1
1. Introduction
The world depends on water, yet fresh water scarcity is already affecting every continent
making it an increasing problem for many societies. It is crucial for healthy ecosystems and
socio-economic development. By 2025 it has been estimated that 1.8 billion people will be in
areas of absolute water scarcity and approximately two thirds of the world’s population could
experience water stress according to UN’s World Water Day report [2]. Considering the world’s
population is expected to reach 8.9 billion by 2050, water scarcity will become a harder
problem to solve, where it is already unevenly divided leaving economically weaker countries
suffering [3]. In California, USA, there has been a noticeable decline of water levels in ground
water, especially since 2010. This has led to more water wells being drilled both during dry and
wet periods to maintain water supplies [4]. With the increasing need of water for various
purposes and evidently decreasing fresh water supply, providing water is becoming an
economic and practical issue. Desalination of seawater could solve fresh water scarcity.
Instead of having desalination plants driven by electrical power, S. Salter designed a Wave
Energy Converter (WEC) in the 1970s that can directly use the wave energy for vapourcompression desalination process inside the device [5]. This creates a more sustainable
desalination process, since source of energy is an important factor in today’s society, where
energy demand is high and environmental impact must be considered.
However, a common problem that WECs have to face is mooring of the device, since permanent
foundations in seabed may reduce chance of project approval. This creates a difficult situation if
there are not any rocks, cliffs or other seabed features that the WEC may be attached to instead.
An anchor has therefore been designed, which could provide the solution for a sandy seabed. In
short, it has been developed to float when filled with air and sink as water enters into it. When it
approaches the seabed, it uses water to fluidise the bed to easily embed itself and finally, pull a
partial vacuum to force the sand particles towards it due to pressure difference. This creates a
strong grip. This project is aiming to test a simpler model of the design, to find if it is suitable to
bring forward to next design stage.
2
2. Literature Review
Creating a self-embedding and self-removing sand anchor for WECs is a new concept that lacks
previous experience. Relevant research was therefore required to gain sufficient amount of
knowledge needed to construct a successful initial model. This model would then lead to insight
of the idea and contribute significantly to the final design. The requirements of the sand anchor
are that it will be able to fluidise the seabed appropriately for both embedment and removing,
create a strong grip during vacuum and finally be able to float when filled with air.
2.1 Mooring Systems
Anchoring techniques vary in a wide range of sizes and designs depending on their desired
performance. The traditional drag-based anchor increases the drag the further it is moved along
the seabed. Such anchor would not be practical for a WEC since it is not secured enough. It
needs to be embedded firmly but temporarily. There are unique designs that may be learnt
from, both old uplift-resisting concepts and self-embedding inventions.
2.1.1 Uplift-Resisting Anchors
Uplift-resisting anchors were partly developed due to the increase of operations and
constructions located at deeper waters, where there was a demand for a more advanced
anchoring system. The requirements of these anchors were to have characteristics such as
‘highly efficient, reliable, and light weight’ as R. J. Taylor (1975) states, where major advantages
compared with the drag anchor was that they used significantly less scopes of line and
accessories [6]. The scope for an anchor is the ratio of length of anchor rode to the water depth,
where rode is the anchor line [7]. It should be stressed that amount of line scope makes a great
impact on costs in deeper waters; consequently a design using less rode is favoured.
There are many categories for uplift–resisting anchors where a few will briefly be discussed [6].
One concept is assembled by anchor-projectile (including a piston and fluke), gun and reaction
vessel working together to be propelled into the seabed at a high velocity. It can usually just be
recovered at water depths less than 100 m, which is the main disadvantage. However, it is a
compact and light weighted assembly making it easy to handle [6].
3
Vibrated direct-embedment anchors are slender designs. They embed themselves into the
seabed through vibration. They can penetrate layered seafloors and have reasonable holding
capacity, but they are difficult to handle and the seabed may not slope more than 10° [6].
The driven anchor is continually forced into the seabed through impulsive forces, where its
specific design may vary. It has advantages in a sandy seabed and its movement is negligible
compared to the maximum capacity reached. However, it is limited to roughly 300 m depth due
to the surface hammers used to cause the impulsive forces and they require surface support [6].
Finally, deadweight anchors are ‘dense, heavy and resistant to deterioration in water’ as stated
in [6]. These are simple designs that may alter depending on their specific operation at sea. They
are economical, can be used for a wide range of seabeds and have a predictable upliftresistance. However, they are not practical for anchoring anything beyond a few hundred kg [6].
Majority of these anchors are more ‘explosive’ and permanent. They are not temporarily lifted
out from seabed easily, because they are designed to automatically resist it. It should however
be noted that their characteristics such as efficient and reliable are qualities that are desired by
this project’s sand anchor. Like the deadweight anchor, the tripod should use its own weight to
sink into the seabed, but it also needs to be removed.
2.1.2 Self-Embedding Anchor Concepts
It is known that self-embedding sand anchors have been developed historically. Although this is
only one of the three requirements of this project’s anchor, a great deal may be learnt from
these simpler designs. C. H. Howland speaks of a self-embedding anchoring system for his
invention ‘temporary floating breakwater and causeway with simulated beach and kelp’ from
2011; which protects areas in remote locations in various actions such as remove oil from water
during an oil spill [8]. He emphasizes that anchors should hold a mooring capacity of
approximately fifteen times its weight according to navy ratings and that self-embedding as one
of its features is not new engineering. The patent also mentions that jetted or screw anchor
designs may result in reduced mass required and using a vacuum creates a high strength
lightweight system [8].
4
Patent in [9] by D. L. Tanner reviews an invention for a ‘mobile anchor and a method for
embedding same’ from 1977. This hollow anchor has locking arms that are pivoted outwards
when desired. The anchor may embed itself easily when arms are locked in by letting
pressurised water enter in one end of the tube and jetting out at the other. D. L. Tanner
discusses the displacement of the sand grains is what enables the embedment, which relates to
this project’s fluidising feature [9]. See Figure 1 for a clearer view of the design.
Figure 1 Schematic of the mobile anchor developed by Tanner, D L [9]
5
Furthermore, patent [10] from 1973 consisted of an idea where a heavy block would be
positioned on the seabed with a blade attached that could be pivoted along with an actuator. P.
Rhodes, the inventor, designed it such that the actuator would enable the block to be swung to
engage with the seabed and thereby bury itself deeper into the sand. Such motion meant that
the model did not have to be dragged along the seabed to be embedded unlike most common
anchors [10], but instead influences the sand like this project’s anchor to sink in.
Finally, the RoboClam, published April 2014, is one of the newest inventions that have shown
impressive results found in [11]. A. G. Winter discusses the inspiration of the design was the
Atlantic razor clam which manipulates the seabed to burrow itself down as deep as 700 mm
using approximately 10 N force. The soil manipulation reduces the energy required and
burrowing drag to such extent that without it, the clam would only dig a depth up to 20 mm
[11]. In [11], the clam’s actions were studied to be imitated by the RoboClam, where test results
showed the amount of energy required to locally fluidise the seabed was higher than pushing
the design downwards. However, energy was more constant, and the higher energy was
assumed to be due to tests carried out at shallow water where energy required opening and
closing the valves at the end effector of the design would be larger than when in deeper water.
The design was initially developed to burrow deep into the seabed to anchor miniature
submarines, but may potentially have more uses [11]. The comparison of digging cycles between
the Atlantic Razor Clam and the RoboClam are shown in Figure 2 and Figure 3, where it is
evident how the behaviour of the clam has been used as inspiration. Figure 3 illustrates how the
RoboClam pumps out water to fluidise the seabed and dig itself as deep as the fluidised sand
(light grey area) will allow.
6
Figure 2 Atlantic Razor Clam digging cycle kinematics in [11]
Figure 3 RoboClam digging cycle kinematics from [11]
7
2.1.3 Suction Caisson Anchors
This anchoring system has increased its popularity in the oil and gas industry for deep-water
operations due to the costs and challenges in installation equipment for driven piles that have
previously been used. Suction caisson is simple, reliable and provides better control during
installation as B. Sukumaran states in [12]. The anchoring system is first penetrated into the
seabed through its own weight. This initial embedment is substantial enough to create a
satisfactory seal to start the suction process. A submersible pump, located at the top of the
sealed caisson, pumps water out from the inside of the caisson, creating a strong suction force
due to a created pressure differential. This has clear similarities with the project’s anchor,
proving that a strong grip can be achieved by creating a large enough pressure difference. The
installation sequence is illustrated in Figure 4.
Figure 4 Installation sequence of suction caisson from [12]
Investigating existing mooring systems showed that the concept of localised fluidisation to
create quicksand-type behaviour has been used before to cause an anchor to sink easily using
its weight. However, the tripod requires to be embedded for a few months at a time, to
afterwards be removed unlike some uplift-resisting sand anchors. This complicates procedures,
where the anchor legs must implant themselves deep enough to be secured for several months
at minimum and additionally pull a partial vacuum successfully to create a strong grip. It
therefore differs by combining many of the various features that past concepts have.
8
2.2 Fluidised Beds
A bed of sand can be expanded to have many properties similar to a liquid by continuously
adding a fluid. The process is known as fluidisation; where R. H. Perry and D. W. Green highlight
that particle sizes between 1 μm and 60 mm in diameter can be fluidised depending on the
velocity of the liquid or gas [13]. The method has historically been used in a wide range of
applications, for instance in chemical reactions where homogeneous non-catalytic reaction uses
fluidisation to ensure temperature control and mixing gases. Such procedure may be carried out
for oxidation of gaseous fuels [13]. Fluidising a dense seabed will enable the sand to behave as a
fluid and thereby an object will either sink into it or, if already embedded, be easily removed
depending on the characteristics of the object.
Fluidising solid particles is only possible if the fluid added is travelling at a velocity high enough
to suspend the weight of the solid bed. This is the minimum fluidising velocity. If using gas, or if
the solid particles are very small when using a liquid, then this velocity may be calculated using
the following equations [13].
𝑅𝑒𝑚𝑓 = (11.35.7 + 0.0408𝐴𝑟)0.5 − 33.7
(2-1)
Where:
𝑅𝑒𝑚𝑓 =
𝐴𝑟 =
̅̅̅𝑝 =
𝑑
̅̅̅̅
𝑑𝑝 𝜌𝑓 𝑈𝑀
𝜇
̅̅̅̅
3 𝜌 (𝜌 −𝜌 )𝑔
𝑑𝑝
𝑓 𝑠
𝑓
𝜇2
1
𝑥
Σ( 𝑖 )
(2-2)
(2-3)
(2-4)
𝑑𝑝
See page (v) for explanation of the notifications.
When the velocity of the liquid or gas travelling through the bed is very low, the particles
remain stationary. At this point there will be a pressure drop found using the Ergun equation,
which covers the full range of flow rates. It assumes however viscous- and kinetic energy losses
are additive [14]. The Ergun equation is shown in equation (2-5) where it may be noticed that
first part is dependent on viscosity and second part on density of the fluid.
9
Δ𝑝
𝐿
=
2
̅̅̅̅
150𝜇𝑈
𝑜 (1−𝜀)
𝑔𝑐 Φ2𝑠 𝐷𝑝2 𝜀 3
+
2
̅̅̅̅
1.75𝜌𝑈
0 (1−𝜀)
𝑔𝑐 Φ𝑠 𝐷𝑝 𝜀 3
6⁄
𝑑𝑝
Φ𝑠 = 𝑠𝑝
⁄𝑣𝑝
𝑉
𝜀 = 𝑉𝑉
𝑇
(2-5)
(2-6)
(2-7)
See page (v) for notifications.
Ergun equation can be re-arranged to find the minimum fluidisation velocity when the bed of
sand is fluidised with a fluid and the grain size of the sand is too big to use the formula stated by
R. H. Perry and D. W. Green. The derivation is found in Appendix A, where the final equation is
(2-8) for Reynolds number < 1 [14]. Details of new notifications are on page (v).
̅̅̅̅̅̅
𝑈𝑂𝑀 ≈
3
𝑔(𝜌𝑝 −𝜌𝑓 ) 𝜀𝑀
Φ2 𝐷 2
150𝜇 1−𝜀𝑀 𝑠 𝑃
(2.8)
The change of minimum fluidisation velocity depending on particle diameter for air is seen in
Graph 1 for a range of pressure flow rates. Once fluidised, the pressure will remain constant,
however bed height should increase as the flow of the fluid is increasing. The bed height will not
decrease until the flow rate is reduced, and at this point, the pressure should still be unchanged
[14].
10
Graph 1 Minimum fluidisation velocity with air at 20° and 1 atm from [14]
2.3 Sand Anchor Concept
2.3.1 Reasons for Developed Design
A rocky seabed is preferred for anchoring WECs, since it is then easy to create a secure
attachment. The project’s sand anchor was developed by S. Salter as a solution to mooring for
the wave power desalination project at locations consisting of a sandy seabed. Such seabeds
require a secure mooring created, adding further issues when the device, and possibly the
mooring system, might be removed for maintenance reasons or end of life. Due to the natural
motion of the waves, the anchoring must enable the device maximum movement, 180° about
the vertical axis. Finally, a permanent foundation into the seabed may not be allowed in a worstcase scenario, and so a heavy and hard-to-handle anchor was not desired, since these are often
left in seabed as seen in 2.1.1 [5].
11
2.3.2 The Tripod
The self-embedding and self-removing sand anchor design, discussed in [5], consists of three
straight anchor legs forming a tripod as shown in Figure 5. The hollow legs have a wall
thickness thin enough to enable the anchor to float when filled with air. By adding water into
the hollow legs, the anchor will sink until it reaches the seabed. At the ends of each anchor leg
there will be water jets to create a localised fluidisation, resulting in a quicksand-behaviour of
the seabed so that the anchor embeds itself easily by its own weight. The water flow direction
will be reversed once embedment is completed to remove water from the voids between the
sand grains located close to the anchor leg. A vacuum is needed for the desalination process,
which will be achieved using a vacuum pump for the full-scale anchor. Creating a partial vacuum
will produce high radial forces towards the surface of each leg, resulting in a strong grip,
securing the tripod in a set location. At the point of removing the sand anchor, the vacuum will
be released and air will fill the hollow legs. By re-fluidising the seabed, the anchor may float up.
A cast steel shell joins the tripod’s legs and has a hollow design big enough to enable workers to
carry out tension and post-tension checking [5].
Figure 5 The Tripod Design by S. Salter [5]
12
The legs should be post-tension concrete tubes that may be created by slip forming. This is a
process where the set framework is raised vertically in a continuous manner. The framework
consists of three platforms: storage and distribution area, main working platform and finally
concrete finishing platform. It can be used for any regular shape, and is ideal for various high
cylindrical towers since it creates robust and economical solutions [15]. An example of slip
forming is seen in Figure 6. Concrete is not good with tension, but has excellent compressive
properties. By using post-tensioned concrete, where strong bars between the two ends cause
constant compression, tensile forces on the anchor legs in various directions will be contoured
by the pre loading of the wires. It will only vary in amount of compression as explained in [16].
Figure 6 Slip forming a concrete tower found in [17]
2.3.3 Forces on Tripod and Forces According to Scaling
The tripod is designed to be able to hold against a 100-year Atlantic wave, which is roughly 40
MN [5]. It was found during slamming tests in 1978 for the desalination model, that the
maximum forces produced were at a 45° angle below the forward horizontal [18]. Each anchor
leg should therefore be designed to individually be able to resist 40 MN at 45°.
This can be applied to smaller models by including influence of scaling. For a model that is the
inverse of a scale, the force it can resist should be multiplied by cube of scale to achieve the
equivalent force by the full-scale model [18].
13
3. Design Methodology
The design process contributed to a big part of this project. It was decided to model and test one
anchor leg on these initial experiments as crucial knowledge could be gained. As this was a
concept that had not been tested, calculations had to be carried out to justify selecting sand
grain size, hole sizes along the anchor leg etc. Furthermore, some calculations required
educated guesses at first attempt to later do adjustments. This chapter explains the wide range
of areas that had to be covered during the design process before the tests could be carried out.
3.1 Designing the Experiment Layout
The experiment was carried out using a large selection of parts that are mass-produced and
could therefore be easily purchased. The following items created the needed combination for
the model anchor leg to be tested appropriately:

Water butt

Hosepipes

Hose lock with tap connector

Hose lock dual tap connector (called Y-tap in report)

Hose locks with hose end connector

Hose locks (male connector)

Hose lock four way tap connector

Hose locks with Y-connector (called Y-piece in report)

Ballcock
Some of the items are clarified in Appendix B. Filling the water butt with water and sand would
create the model seabed for the testing. The water butt was approximately 8 m above ground
level, where the difference in height was utilised so the anchor leg would syphon water from the
tank removing the need of vacuum pump. This was achieved by having a hose directly
connected to the anchor leg running all the way down to a drain at ground level.
Apart from two long lengths of hose pipes running from the top level down the 8 m to the drain,
one for inducing a vacuum and one for the overflow, many smaller pieces of hose were cut up
for various functions. The wide range of hose locks was then used to connect the hoses. The
ballcock was used to stop over-filling of the water butt.
14
3.1.1 Experiment Diagram
Figure 7 Diagram of experiment design
15
Figure 7 illustrates the schematic layout of the experiment. The image clearly highlights the
different settings the y-tap connected to the anchor leg may be set to, which are explained in
point 1.
3.1.2 Logistics of Assembling Certain Parts
The water butt purchased has outer walls that curve in slightly at the top. This means that the
overflow exit might not be able to drain out the exceeding water if the tank would become too
full. This would only be a risk if the hose’s path connected to the overflow goes higher than the
water butt’s opening. The water butt with the hose was therefore modelled using the CAD
software SolidEgde to achieve a further insight regarding this potential issue. Measurements
were taken of the water butt and used to produce a model. The highest point of the hose bend
was found to be 52mm from the centerline of the connection. The assembled parts are shown
below in Figure 8 with the water butt, hose, hose lock with hose end connector, hose lock with
male connector, o-ring and overflow-exit modelled. Figure 9 shows the hose would not reach
higher up than the top of the water butt and would therefore be draining water in worst-case
scenario, where the water butt would be completely full. Note that the red line highlights this
critical height. The choice of position of the overflow was to enable maximum water and sand in
the water butt for the experiments and is at a flat dented-in circular section, ideal for drilling the
overflow hole.
Figure 8 Assembled CAD parts for insight of hose at overflow exit
16
Figure 9 Assembled CAD parts in a drawing, focusing on height reached by hose
3.2 Pressure Flow Rates
Void fraction, also known as porosity, is calculated by using total volume of a substance and the
volume taken by either gas or liquid between the solid particles. The void fraction varies
depending on material, shape and size of particles, where nearly spherical particles would lead
to a void fraction in the range of 0.40-0.45 [19]. To simplify calculations and fluidisation, nearly
spherical particles are easier to evaluate theoretically, so sand with void fraction within this
range would be favoured.
Referring back to Ergun equation (2-5) in section 2.2 for fluidised bed, the change in pressure
may be considered for this project to see how it is affected by the void fraction and help
deciding on sand grain size. Since nearly spherical sand grains are preferred, this was
considered for surface area – volume ratio, which is 1 for a perfect sphere. The change of
pressure depending on void fraction was plotted for various sand grain sizes seen in Graph 2
and Graph 3 where the surface area - volume ratio is first assumed to be 0.83 and then 0.6 for
comparison. 0.83 was chosen from [14] since it is the sphericity of rounded sand. The
calculations are found in Appendix C. It may also be seen in Appendix C that the first part of the
Ergun equation, which is in terms of viscosity, is the dominant part of the final pressure flow
rate.
17
Graph 2 Change of pressure for surface area - volume ratio=0.83
18
Graph 3 Change of pressure for surface area - volume ratio=0.6
It was clear, especially for particle size of 0.05 mm, that the change on surface area - volume
ratio has a big impact on the change in pressure. The values are nearly doubled at void fraction
of 0.4 when the ratio is 0.6. The larger diameters result in more constant pressure flow rates.
This makes these grain sizes better for the experiments if it is difficult to get a precise value for
the void fraction. Superficial velocity was however calculated from assuming maximum possible
cross-sectional area of fluidisation, so the entire cross-section of the water butt and the flow
rate directly from the water tap. This was due to difficulty of knowing these exact numbers and
therefore the maximum possible values had to be used. Note that superficial velocity is the
hypothetical fluid velocity assuming there is only one flowing phase in the cross-sectional area.
19
3.3 Sand Investigations
3.3.1 Choice of Sand: Fluidising Properties, Availability and Health Risks
The sand grains for marine seabeds are assumed to most likely be nearly spherical if they are
fine and had a chance to be rubbed against each other to create more rounded surface; resulting
on a favourable void fraction.
It would therefore be ideal to use very fine sand, since it is both better for fluidisation and in
relation to the model anchor leg that is a lot smaller than the actual anchor would be. Sand grain
sizes were categorised in 1922 by Wentworth, C K, seen in Table 1, where silica sand is between
0.1-0.5 mm [20]. Silica sand is industrial and can therefore be found in various sizes depending
on use.
Table 1 The canonical definition of sediment grain size from [21]
20
Very small particles of sand may enhance health risks such as silicosis (lung disease). The
investigation of purchasing smallest possible grained sand led to researching silica flour, which
is very fine silica sand. It ranges from 3 to 10 𝜇m in particle size [22]. It was then discovered that
there is definite risk of silicosis present up to 5 𝜇m diameter particles and health risk can occur
up to 10 𝜇m [23]. Silica flour was therefore ruled out completely.
Fine sand is more difficult to find, especially when the grain size must to be known. A small
amount of sharp sand, which is used for building, was therefore experimented with to see if this
could be used for the anchor leg.
3.3.1 Investigational Experiments with Sharp Sand
For this experiment, only simple measures had to be made to conclude if the sharp sand could
be used. By filling a transparent plastic box half way with the sharp sand and adding water until
the sand was completely covered, a simplified fluidisation experiment could be carried out with
a hose being the anchor leg. Figure 10 and Figure 11 illustrate the experiment. The fluidised
sand is clearly highlighted in the drawn box for Figure 11.
21
Figure 10 Fluidisation set-up with sharp sand
Figure 11 Fluidised sharp sand
22
3.3.2 Void Fraction Comparison
A small amount of Kiln sand, silica of 0.5 mm, was purchased to compare the two void fractions.
Void tests were carried out where it was measured how much water had to be added to roughly
fill all voids for each type when the grains were dry. It was found that for 178 mL of dry silica
sand, 79.76 mL of water was needed, giving a void fraction of 0.447. In comparison, 200 mL of
dry sharp sand only needed 60.12 mL of water, resulting in void fraction of 0.301. From
fluidisation investigation, it is known that the bigger void fraction was preferred. See to
Appendix D for calculations.
3.3.3 Sand Conclusions
Figure 11 shows that sharp sand may be fluidised. There were however some concerns
regarding the sand grain size. According to the contents data, the particles may be up to 4 mm,
which is very large. There was a risk that the holes on the anchor leg may be blocked easily if
the sand grains were this size. Also the fluidisation might not be as smooth, which was not
desired when investigating the potential of the anchor leg as a concept. Kiln sand had more
favourable void fraction. From these remarks, it became clear that the silica sand would be used
for the experiments.
3.4 Minimum Fluidising Velocity for Anchor Leg Model
Referring back to section 2.2 about fluidised beds, the derived Ergun equation could be used to
calculate the minimum fluidising velocity for the sand anchor. By assuming that Reynolds
number is less than 1 and that the minimum void fraction is the same as void fraction found
experimentally, the minimum fluidising velocity was able to be calculated. For rounded sand
giving surface area - volume ratio of 0.83, this resulted in minimum fluidising velocity of 1.1
mm/s. If the ratio would be 0.6, then the minimum fluidising velocity nearly halved to 0.6 mm/s.
See appendix E for detailed calculations.
23
3.5 Risk Assessment
The following risks were identified for this particular project.

Asbestos: This is located close to the top of the stairs where the experiments were
carried out. Due to the potential danger of the protective layer being damaged, which
would cause harm, instructions were carefully given by the asbestos supervising officer
Gordon Duff to ensure safety. This included stressing the importance of using pig mats
between the experiment and the asbestos so that any water spilt would not be able to
reach this zone.

Tripping hazard: Due to the experiment layout illustrated in section 3.1.1, there were
many pieces of hoses as a result, where some were on the floor.

Water butt failure: The water butt used is a standard design from B&Q, 0.21 m3 volume
and made out of high-density polyethylene (HDPE) [24]. A water butt is designed to
contain mainly water with potential leaves and dirt entering, but the experiment
required a substantial quantity of sand along with the water to be held inside it. This
mixture would have a very different density to water.
Calculating the hoop stress and comparing it with the strength of HDPE material
identified the safety factor of the water butt. For a thin-walled cylinder, a simplified
water butt, the hoop stress is defined as:
𝜎𝜃 =
𝑃𝑑𝑖
2𝑡
(3.1) [25]
For this, the density of the sand and water mixture had to be calculated. It was assumed
the sand would fill ¾ of the water butt. Its density when dry would be 1602 kg/m3 from
[26]. The density of the wet silica sand had to be found since the gaps between the sand
grains would create a different density depending on if they were filled with water or
air. Very fine-grained sand, silica or silicon dioxide, has a density of 2600 kg/m3 if there
is no space between grains [27]. The mass of air was assumed to be negligible compared
to the mass of sand to find the volume occupied by the sand grains alone. Density of
water from the tap was assumed to be 998 kg/m3, since the temperature would be
roughly 15oC [28].
24
From this, two densities where calculated, one assuming the sand and water mixture
was ¾ of the water butt; the other was for a worst-case scenario where the remaining
quarter would have been filled with water. For the first case, hoop stress of 1.3 MPa was
found giving a factor of safety of over 25; and for worst-case scenario the hoop stress
would have been 1.5 MPa giving a factor of safety of roughly 22, since the tensile
strength of HDPE is 32 MPa [29]. See Appendix H for full calculations.

Air born sand particles: In section 3.3, choice of sand was investigated where in 3.3.1,
the health issues regarding the sand particle size was evaluated. It was therefore known
that the silica sand used should mainly be of particles that are too big to breath in.
However, the few small grains would still have been a risk.

Leaking water: Working with water travelling through various hoses would cause a risk
of leakage. This could have been a problem if it came in contact with the asbestos or
with electrical appliances.

Electrical shock: If any electrical appliances were used during the experiments and they
were in contact with water then electrical shock could have occurred.

Falling: There are several stairs leading up to the top floor where experiments were
carried out. Walking up and down several times during experiments to check on water
flow rate at the bottom etc. increased risk of tripping.
A risk assessment was carried out to highlight these potential hazards. The University of
Edinburgh’s official risk assessment sheet, completed for this particular project, is found in
Appendix F. It was however decided that more detailed risk assessment would be made,
including the actions that needed to be taken both to prevent the risks and procedures to carry
out if the hazards still occurred. This was organised in a table that is seen in Appendix G. The
latter was agreed to be more useful for the experiments and was agreed on by Gordon Duff.
25
3.6 Anchor Leg Modifications
The model anchor leg used for the experiments was a standard white ~56 mm diameter
plumbing pipe. Modifications then had to be made to create the model appropriate for testing.
3.6.1 Machining Parts
With a simple white draining pipe and standard garden watering supplies, some parts had to be
machined to connect the anchor leg to the garden hose and also create an end fitting the white
pipe. This was done using a lathe (see Figure 12). By machining PVC plastic and using o-rings,
two secure and sealing ends were produced; where one was also threaded to fit a hose lock
male adapter. These are seen in Figure 13 to Figure 15.
Figure 12 Machining for the anchor leg
26
Figure 13 Machined anchor end parts with o-rings
Figure 14 Threaded inside for hose lock male adapter
27
Figure 15 Fitted hose lock male adapter
3.6.2 Alterations on the Model
At first a small hole was drilled into the bottom end piece for the anchor leg for fluidising and to
create a grip while pulling a vacuum. However, it was found that the grip was not strong enough
to secure the model leg. Through careful evaluation, it became clear that a hole at the bottom
would cause the sand grains to be drawn towards the anchor leg end and so the area of gripping
would be a lot smaller. It was decided to have many smaller holes along the side of the anchor
leg since this would force the sand particles to be drawn towards its surface area and create a
sufficient grip. The difference of the two scenarios is clearly illustrated in Figure 16.
28
Figure 16 Difference of grip depending on location of holes
29
Since similar anchor had not been developed prior to the project, smallest feasible dimensions
in terms of hole-sizes according to calculations were carried out and then altered after short
tests. The holes were in ten rows of three at 120° degree angle apart so that the first row had
holes at 0°, 120° and 240°, while second row had holes at 60°, 180° and 300°. They were first 2
mm diameter since this resulted in acceptable flow rate according to calculations in Appendix I,
but these were quickly changed to 2.6 mm diameter to experimentally see if it increased the
grip further. The process of creating the holes is shown from Figure 17 to Figure 22.
Figure 17 Drilling holes into anchor leg
Figure 18 Perfectly aligned for
accurate positioning of holes
30
Figure 19 A drilled hole in the anchor leg
Figure 20 Tail stop keeping model leg in place
31
Figure 21 Index head keeping model in place
Figure 22 Holes separated by 120°
32
It was found that the small hole at the end of the anchor leg for fluidisation was not wide
enough to enable the model leg to easily sink into the sand. This was therefore altered to m5,
m8 and finally m10 taps so screws could be fitted in if the hole would be blocked. When this still
did not create a sufficient fluidisation, the hole was enlarged to 12 mm and finally 14 mm
diameter. This enabled fluidisation, but it was not effective enough, since it was too centralised
and additionally the anchor leg sucked up too much sand. A brief re-design of the leg led to
three hoses along the sides to fluidise around the end of the model and blocking the fluidising
hole. A filter made out of standard 40 denier tights was fitted to stop sand being sucked in. The
final anchor leg model is shown in Figure 23 to Figure 25. Note that Figure 25 also shows a
combination of jubilee clips and rope in a double hitch that were used to attach the leg to the
load cell, enabling the engine hoist to pull it up from the sand.
Figure 23 Anchor leg with three fluidisation hoses
33
Figure 24 Cutouts in anchor leg end to secure fluidising hoses
Figure 25 Filter around anchor leg
34
4. Experiments
4.1 Set-up
The experiments were carried out by arranging the parts identified in section 3.1 to match the
drawn diagram in Figure 7. The anchor leg was then attached to the load cell, which was further
attached to an engine hoist. The anchor leg would as designed first fluidise the wet sand to
easily be pushed in, then the y-tap would be set to both let water out of anchor leg and down the
vacuum hose to remove air. Once achieved, y-tap was changed again to only have a connection
between anchor leg and vacuum hose so water was sucked out of the water butt creating a
strong grip for the model. By maintaining water supply, water was still entering the water butt
through the ballcock as long as level was not too high. When desired, the engine hoist lifted up
the anchor leg to take a reading of uplifting force required. This was done smoothly at a
negligible speed to reduce risk of force created by acceleration.
The experiment layout is seen in Figure 26, colour-coded to follow the diagram design. The load
cell, seen in Figure 27, measured in lbs. with a 2 lbs. error, which had to be subtracted from the
value read when pumping up the engine hoist and to finally convert it to Newtons. Figure 28
shows the anchor leg in the sand pulling a vacuum, with water entering the water butt through
the ballcock in the background. Figure 29 is from when the anchor leg was pulled out of the
water and clusters of sand would stay by the holes due to the force the sand particles were
experiencing.
35
Figure 26 Experiment set-up
Figure 27 Load cell starting at 2 lbs. instead of zero
36
Figure 28 Anchor leg pulling a vacuum
Figure 29 Sand clusters by the holes
37
4.2 Results
4.2.1 Change of Grip during a One Hour Period
The main experiments carried out were to see how the maximum lifting force required
removing the anchor leg changed with time. This was done in 10 minutes steps from 0 to 60
minutes. Every time the anchor leg had been lifted and read one result, it had to re-fluidise the
sand to embed itself again before starting another test for a different time period. This was
repeated three times seen below as Test 1, Test 2 and Test 3 in Table 2 and Graph 4.
Table 2 Results from standard one-hour tests
Time (min)
Lifting Force
Lifting Force
Lifting Force
Test 1 (N)
Test 2 (N)
Test 3 (N)
0
302
240
213
10
226
293
169
20
391
338
182
30
373
262
182
40
257
169
124
50
266
160
128
60
262
177
146
38
Graph 4 Results from standard one-hour tests
4.2.2 Change of Flow Rate during a One Hour Period
For Test 2 and Test 3, the flow rates were examined. This was done by having 1 litre of water in
a bucket, taking the end of the vacuum hose from the water-filled collecting beaker to the
bucket without letting air in and measuring the amount of water after 1 minute. From this,
suction flow rate in mL/s could be calculated from which the results are displayed in Table 3
and Graph 5.
39
Table 3 Results for flow rate during one hour
Time (min)
Flow rate during test 2
Flow rate during test 3
(mL/s)
(mL/s)
0
172
153
10
178
132
20
169
162
30
185
173
40
188
154
50
169
139
60
173
148
Graph 5 Results for suction flow rate during one hour
40
4.2.3 Further Testing
Friction Test
From the results for the three one-hour tests, it was decided to test the friction of the plastic
anchor leg material to find the friction coefficient. It was found by considering all forces on the
anchor leg as it is tilted until the sand particles are only just displaced from original location.
These are shown in Figure 30.
Figure 30 Finding friction coefficient
Friction coefficient is equal to tan(q), and varied for the anchor leg depending on if the sand was
outside or inside of the leg and if the particles were wet or dry. The results are in Table 4, where
details of why friction coefficient is tan(q) and calculations for the values are in Appendix J.
Table 4 Friction coefficient
Dry Sand
Wet Sand
Inside Anchor
Outside Anchor
Inside Anchor
Outside Anchor
Leg
Leg
Leg
Leg
q (deg) η
21.670
q (deg)
0.397
27.330
q (deg)
η
0.517
28.000
q (deg)
η
0.532
33.670
η
0.666
It was expected that the friction factor would be higher on the inside of the anchor leg, due to
the force from the walls. However, when investigating further, it was noticed by scratching the
surface with fingernails that the inside of the anchor leg was a lot smoother. This contributed to
the friction factor, resulting in higher value on the outside of the anchor leg.
41
Plastic Sheet Between Sand and Water Test
The flow rates were considered to be varying too much during tests 2 and 3 as well as being too
high, so a plastic sheet was fitted on top of the sand before fluidisation to see if this would alter
the results. The aim was that the plastic would act similarly to having a thin layer of clay at the
top, which is a common situation, especially in fluidisation, since smaller particles are then
tempted to travel upwards. A layer of clay is denser and keeps the sand in place, hence why the
plastic sheet with a hole in the middle for the anchor leg could bring similar behaviour. This was
wished to create a situation where the pressure of the water would spread more to the sides
and not be centralised around the anchor leg. In theory, that would reduce the flow of the water
and could even cause it to become more constant.
The results are seen in Table 5. After one hour, the lifting force was examined, which came to
164 N. Comparing this to the previous lifting forces after one hour, it was roughly between the
results of Test 2 and Test 3.
Table 5 Change in flow rate during 1 hour for test 4
Time
Flow rate during
(min)
test 4 (mL/s)
0
171
10
162
20
157
30
154
40
158
50
152
60
158
42
Increasing Amount of Holes for Anchor Leg
Two rows of three holes were added in same pattern to investigate the influence the number of
holes have on the sand anchor, especially the suction flow rates. The plastic sheet was used
again due to its positive effect. The results are seen in Table 6 and compared with previous
values in Graph 6. The average values and independent error bars are shown in Graph 7, where
the decreased range of values caused by using the plastic sheet is evident. After one hour, the
lifting force was 169 N, a small increase compared to test 4.
Table 6 Flow rates for test 5 during one-hour period
Time
(min)
Flow rate during
test 5 (mL/s)
0
10
20
30
40
50
60
172
162
175
175
178
177
178
Graph 6 Comparison of all volumetric flow rates
43
Graph 7 Average flow rates of all tests
Gripping Force without Vacuum
A final experiment, called test 6, was carried out to see what force was needed to pull the anchor
leg out when it was not pulling a vacuum. This was done with same settings as test 5, with
plastic sheet and added holes, since it was only the natural pull of the anchor leg being tested. It
was repeated three times giving the results of 102, 89 and 98 N. This averaged a force of
approximately 96 N. Test 6 was completed by pulling a vacuum to force the sand particles
towards the leg, then releasing it to see if gripping force was improved. At this point, roughly
146 N was needed to pull it out.
44
5. Discussion
5.1 Comments on the Experiments
The results indicate that the concept for the sand anchor works. The seabed was able to be
fluidised by the use of water jets and enable the anchor leg to embed itself. Furthermore, pulling
a vacuum created a strong grip in relation to the model. The gripping force had a general
pattern of increasing until 20 minutes of pulling a partial vacuum and then decline slightly to
finally become more constant. This suggests that the anchor at sea would after some time have a
constant gripping force. With highest force achieved at approximately 391 N and lowest at 124
N, the forces varied but remained high in comparison to the model.
It is clear from Graph 2 that for every test carried out, the force curve for one hour decreased
slightly. This may be due to smaller sand particles getting through the filter and gradually
blocking the hose-system more every experiment. This assumption is due to noticing that the
water appeared cleaner for every new test, suggesting less small particles floating in the water.
The flow rates from the experiments were very high, and lower flow is preferred for the real-life
model. However, over time, the anchor would gradually get seaweed, plankton and other fine
sediment, blocking holes, which would help reducing the water flow. It was seen from Graph 6
that including a plastic sheet to force the pressure of the water to spread out changed the
suction flow rates. They became more constant in relation to test 3 and on average reduced
slightly compared to test 2. However, increasing the amount of holes led to relatively constant,
but higher, suction flow rates.
The friction tests suggested a minimum friction coefficient of roughly 0.4, which was the reading
for inside of the anchor leg for dry silica sand. Although the lower coefficients were expected to
be outside the anchor leg due to shape, the finger nail scratching investigation explained in 4.2.3
informs why this was not the case due to surface smoothness.
Another important observation was that the anchor leg sucked water too fast during the 10
minute testing for Test 2, leading to air getting into the system. This was both evident in the
water butt, where there was too little water, and in the collecting beaker where very large air
bubbles started appearing. This is likely the cause of the ‘dip’ seen in Graph 4 for Test 2. The air
in the system had a negative impact on the gripping force since the partial vacuum reduced.
45
5.2 Further Calculations from Experiment Results
5.2.1 Size Related Force
The force on the anchor leg model could be calculated by using the equation:
𝐹 =𝑃∗𝐴
(5-1)
Through derivation of this equation found in Appendix K, the force on the anchor leg while
pulling a vacuum could be defined as:
𝐹 = [(𝜌𝑠𝑖 ∗ 𝐻𝑠𝑖 ∗ 𝑔) + (𝜌𝑤 ∗ 𝐻𝑤 ∗ 𝑔) + 𝑃𝑎𝑡𝑚 − 𝑃𝑣𝑎𝑐 ] ∗ 𝜋 ∗ 𝐷𝑎𝑛𝑐ℎ ∗ 𝐿𝑎𝑛𝑐ℎ ∗ 𝜂
(5-2)
Using the equation for the 0.74 m long, 0.056 m diameter model leg, the maximum possible
gripping force was 10.4 kN for wet sand friction coefficient and 8.1 kN for dry sand friction
coefficient. These values are high but the assumption is that there is a perfect vacuum, hence 0
Pa for absolute vacuum pressure. The anchor leg would realistically be in partial vacuum; hence
the force would be lower.
Force depending on scaling must also be considered as mentioned in section 2.3.3, where for a
model being the inverse of a scale, force is found by cube of the scale. By evaluating equation (52), this is seen to be true since height H is scale to the power of one, and 𝜋 ∗ 𝐷𝑎𝑛𝑐ℎ ∗ 𝐿𝑎𝑛𝑐ℎ
resulting in the anchor leg’s surface area is the square of the scale. The model anchor leg is
roughly one hundredth of scale; therefore if the gripping force were 164 N as found in test 4,
then the full-scaled anchor leg would hold 164 MN. Calculations found on Appendix K.
46
5.2.2 Creating Boundary Conditions for the Real-Life Model
From the slamming tests carried out in [18], it was found that the maximum force was roughly 1
MN per metre of the device. Therefore, a 5 m long anchor leg, for example, would at most
experience 5 MN force. This is significantly lower than the 100-year Atlantic wave of 40 MN,
which must be considered when choosing both size of anchor and its concrete grade.
One of the main requirements of the anchor legs of the final design is that they would be able to
float when filled with air. For a cylinder to float, then the following equation must be satisfied:
𝐷𝑖
𝐷𝑜
= √1 −
1
𝛾
(5-3) [30]
For concrete, specific gravity 𝛾 is 2.4, giving a wall thickness to be 0.12 of the outer diameter to
enable floating [30]. See Appendix L for details.
For a 40 MN force produced by a 100-year Atlantic wave, the outer diameter should be at
roughly 3 m minimum. This is assuming grade 55 MPa concrete used, post-tensioned to 20%
and that the cylinder is buoyant. Calculations found in Appendix L. Furthermore, the depth of
seabed will roughly be 40 m where the desalination device will be anchored. Therefore, a length
of approximately 12 m would be good for the anchor legs. This is considering practical issues
with towing and handling it, which will be easier if it does not touch the seabed before
embedment.
5.2.3 Stokes Law
The sand particles may be investigated further to see what velocity they move in naturally in
the water. This is achieved using Stokes law, which mathematically describes the force for a
spherical particle to move through a fluid and is as follows:
𝐹𝑑 = 3𝜋𝜇𝑑𝑣
(5-4) [31]
It has already been established that the Kiln sand has nearly spherical particles. Equation (5-4)
may therefore be used, assuming they are spheres. From drawing a free body diagram of the
sphere in water, an equation for the velocity of the particle in water was derived:
𝑣=
𝑑 2 ∗𝑔∗(𝜌𝑝 −𝜌𝑓 )
18𝜇
(5-5)
47
With Kiln sand diameter being max 0.5 mm and silica crystal density 2600 kg/m 3, the velocity
becomes 0.218 m/s when the particle is in water. This leads to a Reynolds number of about 109
referred to diameter [32]. See details of these calculations in Appendix M.
As H. Schlichting shows in his graph for drag coefficient for spheres as a function of Reynolds
number, 109 suggest a drag coefficient of 1 as seen in Graph 8. This is quite a high Reynolds
number, but since the sand particles are nearly spherical, the drag coefficient would not be 1. In
reality, Reynolds number may be lower.
Graph 8 Drag coefficient of sphere depending on Reynolds number from [32]
5.3 Error Analysis and Recommendations for Further Work
There were two problems that were noticed while testing the suction flow rates. The first was
that the process of timing it might not have been accurate due to the time taking the hose to and
from the collecting bucket and possibly letting air in. More advanced flow rate measuring
equipment would have improved the measurement accuracy and experimental influences.
Secondly, small air bubbles coming from the vacuum hose were observed in collecting beaker,
suggesting a small air leak. This is seen in the drawn box in Figure 31. This would have had
negative effect on the vacuum, and furthermore the suction.
48
Figure 31 Air bubbles from vacuum hose
Additionally, the suction flow rates would be very high for a full-scale anchor leg. But the smallgrained Kiln sand would realistically be similar to small pebbles when scaled up. Flow rates are
assumed to be lower if the sand grain size was smaller to model scaling for anchor leg and sand
particles properly. However, this would have required using Silica flour, which was identified to
cause health risk.
The minimum fluidisation velocity calculation in section 3.4 was assuming Reynolds number is
less than one, however, further calculations using stokes law to find Reynolds number proved
this to be wrong. Although, the density part of the Ergun equation would have been small, this
should be included in future work. Velocity of sand particles in the water should have been
investigated to see the proportion of particles that are 0.5 mm and what proportion is less.
Assumptions regarding pressure were common during the experiments. This was wished to be
investigated further, especially to see if the pressure outside the anchor leg became more
constant after an hour to compare it with theory. Comparing the pressure at the outside of the
anchor leg to the inside would have provided an insight of the suction force due to pressure
difference. However, this could not be done due to being unable to get hold of the required
equipment such as pressure transducers. This is something that would be useful to investigate if
this project would be taken forward for more experiments.
49
An error was found, that for every experiment, more sand was clogging the system and affecting
the results. This was known from using the y-tap, where it was evident that sand was building
up since it became more difficult to change settings.
For future work, it is recommended that a small tripod would be modelled. The results indicate
that the methods works and the model tripod could be directly compared with this project.
However, more work must be focused on reducing the flow rates during suction. It is advised
that a layer of clay or at least a plastic sheet will be placed on top of sand to ensure less
variation and reduction in flow rate. A vacuum pump may also be a possible option for testing,
since suction would be more controlled.
It was also acknowledged during the experiments that a see-through water tank would achieve
a clearer view of the fluidisation and the suction process, as seen in the experimental testing of
sharp sand in Figure 11. This would have enabled an increased understanding of volume of sand
been fluidised and also what proportion of the seabed is greatly affected by the suction force.
This could have also been achieved by using water-adapted microphone to hear the difference
in the movement of sand particles at various locations.
More varieties of anchor legs should be tested, investigating the affect of changing number of
holes diameter sizes. The number of holes was briefly investigated by increasing the number,
and even though the gripping force increased to 169 N, the flow rates increased. Fewer holes
should therefore be tested to find an acceptable balance between flow rates and gripping force.
Positioning of holes should be evaluated as well.
Furthermore, for future work the navy ratings mentioned in the literature review, where a
traditional anchor should have a mooring capacity of fifteen times its weight could be compared
with this anchor. It should be stronger than these rating, and should therefore be weighed and
compared to its average gripping force.
There are various grades of concrete, where 55 MPa capacity concrete should be used on a
model for testing. It is one of the stronger grades, which would be able to resist the force of 100year Atlantic wave.
Finally, a tripod model of the best output in terms of holes sizes and number should be tested in
a real seabed for a wide range of time periods. This would test its gripping under real sources of
impact.
50
6. Conclusions
Of the three requirements of the sand anchor mentioned, two of them directly relate to the
objectives set to see if friction created was enough to hold the waves’ force and if fluidisation
enabled the anchor leg to embed itself and be removed.
The anchor leg was embedded easily into the sand and water mixture, indicating sufficient
fluidisation. With maximum force measured at 391 N and minimum at 124 N, it was clear that
the friction created is strong enough for a wave, since maximum Atlantic wave force is 40 MN, a
full-scale model will satisfy this requirement. The anchor leg alone measured 164 in the fourth
test with plastic sheet working as a thin layer of clay, which according to scaling rule would
enable a full scale model to hold roughly 164 MN. It was evident from test 6 that pulling a
vacuum increased the anchor’s grip significantly. For this project’s experiments, all fluidisation
was carried out with water, resulting in full potential of the anchor not being investigated since
the removing of anchor leg should be done with air. However, the seabed re-fluidised with
water enabled the model to be removed and calculations in 4.2.2 indicate that concrete anchor
legs will be able to float when filled with air if wall thickness is at least 0.12 of the outer
diameter.
The majority of the aims identified in mission statement were achieved apart from scale
underneath water butt to find exact volume of water in voids when filled with sand and
measuring the vacuum pressure. This was due to being unexpectedly unable to source the
required equipment, but should be considered for next stage, especially measuring pressure
using pressure transducers.
Suction flow rates were found to be high reaching up to 188 mL/s, but were slightly decreased
and less varied with a plastic sheet acting as a thin layer of clay on top of the sand.
This concept should be taken to next phase, with the recommendations of further work being
considered. It proves that the design can provide a temporary, but powerful, sand anchor for the
wave powered desalination device and for other WECs.
Acknowledgements
I would like to thank my supervisor Stephen Salter for his support and infinite ideas. I would
also like to acknowledge the support from my family, motivating me throughout the project.
51
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54
Appendices
Symbols
A = Area (m2)
a=Acceleration (m/s2)
Cp=Discharge coefficient
d= Diameter (m)
g= Acceleration due to gravity (m/s2)
𝑔𝑐 = Newton’s law proportionality factor (=1 for SI units [1])
F = Force (N)
𝐻 = Height (m)
L = Length of bed (m)
m=Mass (kg)
Δ𝑝 = Pressure drop N/m2
P = Pressure (Pa)
Q=Volumetric flow rate (m3/s)
q=Angle (degrees)
sp = Surface area of single particle (m2)
t= Wall thickness (m)
U=Velocity (m/s)
̅𝑈̅̅𝑜̅ = Superficial velocity (m/s)
̅̅̅̅̅̅
𝑈𝑂𝑀 = Minimum fluidising velocity (m/s)
V=Volume (m3)
v=Velocity (m/s)
xi= Weight fraction of particle size
𝛾 = Specific gravity
𝜀 = Porosity/void fraction
𝜂 = Friction coefficient
𝜇 = Absolute viscosity (Pa*s)
ρ= density (kg/m3)
𝜎𝜃 =Hoop stress (Pa)
Φ𝑠 = Surface-volume ratio
Subscript
anch=Anchor leg
atm=Atmospheric
b=Buoyancy
d=Drag
i=Internal measurement
f=Fluid/gas
M=Minimum
m=sand and water mixture
o=Outside measurement
p=Particle
s=Solid
sc=Sand crystal
si=Silica
T=Total
V=Void space
vac=Vacuum
w=Water
wv=Water in voids
55
Appendix A
Derivation of Ergun equation (equation 1.5) to find minimum velocity from [14]
Details of notations on page 51
Ergun equation (as explained in section 2.2):
̅̅̅̅2 (1 − 𝜀)
̅̅̅𝑜̅𝜇(1 − 𝜀)2 1.75𝜌𝑈
Δ𝑝 150𝑈
0
=
+
𝐿
𝑔𝑐 Φ𝑠2 𝐷𝑝2 𝜀 3
𝑔𝑐 Φ𝑠 𝐷𝑝 𝜀 3
At minimum fluidising velocity, the pressure drop across the bed equals the bed per unit area of
cross section. This is what allows buoyant force:
𝑔
Δ𝑝 = 𝑔 (1 − 𝜀)(𝜌𝑝 − 𝜌𝑓 )𝐿
𝑐
(A-1)
When fluidisation is first introduced to the seabed, the porosity, 𝜀, will be minimum porosity, 𝜀 M.
Δ𝑝
𝐿
=
𝑔
𝑔𝑐
(1 − 𝜀𝑀 )(𝜌𝑝 − 𝜌𝑓 )
(A-2)
Rearranging Ergun equation:
Δ𝑝𝑔𝑐
𝐿
=
2
̅̅̅̅
150𝑈
𝑜 𝜇(1−𝜀)
Φ2𝑠 𝐷𝑝2 𝜀 3
+
2
̅̅̅̅
1.75𝜌𝑈
0 (1−𝜀)
Φ𝑠 𝐷𝑝 𝜀 3
(A-3)
If A-3 is applied to A-2, where fluidisation occurs, then a quadratic equation may be obtained.
̅̅̅̅2 (1 − 𝜀)
̅̅̅𝑜̅𝜇(1 − 𝜀)2 1.75𝜌𝑈
Δ𝑝𝑔𝑐 150𝑈
0
=
+
= 𝑔(1 − 𝜀𝑀 )(𝜌𝑝 − 𝜌𝑓 )
𝐿
Φ𝑠 𝐷𝑝 𝜀 3
Φ𝑠2 𝐷𝑝2 𝜀 3
̅̅̅̅̅̅̅
150𝑈
𝑂𝑀 (1−𝜀𝑀 )𝜇
Φ2𝑠 𝐷𝑝2 𝜀𝑀 3
+
̅̅̅̅̅̅̅
2
1.75𝜌𝑈
𝑂𝑀 (1−𝜀𝑀 )
Φ𝑠 𝐷𝑝 𝜀𝑀 3
= 𝑔(𝜌𝑝 − 𝜌𝑓 )
(A-4)
When the particles are very small, then it is only the laminar-flow part that has a big impact.
𝐷𝑝 𝑣𝜌
This would be when Reynolds number, (
𝜇
), <1; Dp is particle diameter (m), v is velocity
(m/s), 𝜌 is density (kg/m3) and 𝜇 is viscosity (Pa*s). This leads to a simplified definition for
minimum fluidising velocity:
3
𝑔(𝜌𝑝 −𝜌)𝜀𝑀
̅̅̅̅̅̅
𝑈
Φ2 𝐷 2
𝑂𝑀 ≈
) 𝑠 𝑝
150𝜇(1−𝜀𝑀
(A-5)
56
Appendix B
Experiment layout items

Hose lock with tap connector

Hose lock dual tap connector (called Y-tap in report)

Hose locks with hose end connector
57

Hose locks (male connector)

Hose lock four way tap connector

Hose locks with Y-connector (called Y-piece in report)

Ballcock
58
Appendix C
Calculation pressure flow rate
Details of notations on page 51
A perfect sphere has a surface area of
𝑠𝑝 = 𝜋𝑑 2
(C-1)
The volume per spherical particle is
𝑉𝑝 =
𝜋𝑑3
(C-2)
6
This means that surface area – volume ratio for a sphere is 1 because:
4𝜋𝑟 2 3
𝑠𝑝
⁄𝑣𝑝 =
=
4𝜋𝑟 3 𝑟
3
6⁄
2𝑟𝑝
Φ𝑠 =
=1
3⁄
𝑟𝑝
A nearly spherical shape is desired, so Φ𝑠 will be assumed to be 0.83. For this, void fraction will
vary from 0.4-0.45. Particle diameters evaluated: 0.05, 0.1, 0.2, 0.3, 0.4 and 0.5 mm.
Ergun equation:
Δ𝑝
𝐿
=
2
̅̅̅̅
150𝑈
𝑜 𝜇(1−𝜀)
𝑔𝑐 Φ2𝑠 𝐷𝑝2 𝜀 3
+
2
̅̅̅̅
1.75𝜌𝑈
0 (1−𝜀)
𝑔𝑐 Φ𝑠 𝐷𝑝 𝜀 3
(C-3)
Length of bed assumed to be 0.5 m is approximation of how deep into the sand in the 0.915 m
high water butt the anchor leg will fluidise. Density is for the water, 998 kg/m3, and superficial
velocity is:
̅𝑈̅̅𝑜̅ = 𝑄
𝐴
(C-4)
59
Since the water butt is 0.56 m in diameter, maximum fluidised cross-sectional area is:
𝐴=
𝜋𝐷 2 𝜋 ∗ 0.562
=
= 0.246 𝑚2
4
4
Maximum flow rate is the flow rate directly from the water source. Measuring this resulted in a
bucket of volume 5.767*10-3 m3 filled in 35 seconds. This gives a maximum flow rate of:
𝑄=
5.767 ∗ 10−3
= 1.648 ∗ 10−4 𝑚3 /𝑠
35
Maximum possible superficial velocity for the experiment would therefore be:
̅𝑈̅̅𝑜̅ =
1.648 ∗ 10−4
= 6.691 ∗ 10−4 𝑚/𝑠
0.246
Viscosity is 0.001 Ns/m2 for water in room temperature [33], so if void fraction of 0.4 is used,
then pressure flow rate for a 0.1mm diameter particle is:
∆𝑝 =
=
̅̅̅̅2 (1 − 𝜀)
̅̅̅𝑜̅𝜇(1 − 𝜀)2
150𝑈
1.75𝜌𝑈
0
∗
𝐿
+
∗𝐿
𝑔𝑐 Φ𝑠 𝐷𝑝 𝜀 3
𝑔𝑐 Φ𝑠2 𝐷𝑝2 𝜀 3
150 ∗ 6.691 ∗ 10−4 ∗ 0.001 ∗ (1 − 0.4)2
1.75 ∗ 998 ∗ (6.691 ∗ 10−4 )2 ∗ (1 − 0.4)
∗
0.5
+
∗ 0.5
1 ∗ 0.832 ∗ 0.00012 ∗ 0.43
1 ∗ 0.83 ∗ 0.0001 ∗ 0.43
= 4.102 ∗ 104 𝑃𝑎
Finding out what part dominates in the Ergun equation:
First part with viscosity:
2
̅̅̅̅
150𝑈
𝑜 𝜇(1−𝜀)
𝑔𝑐 Φ2𝑠 𝐷𝑝2 𝜀 3
∗𝐿 =
150∗6.691∗10−4 ∗0.001∗(1−0.4)2
1∗0.82 ∗0.00012 ∗0.4 3
∗ 0.5 = 4.097 ∗ 103 𝑃𝑎
Second part with density:
2
̅̅̅̅
1.75𝜌𝑈
0 (1−𝜀)
𝑔𝑐 Φ𝑠 𝐷𝑝 𝜀 3
∗𝐿 =
1.75∗998∗(6.691∗10−4 )2 ∗(1−0.4)
1∗0.8∗0.0001∗0.4 3
∗ 0.5 = 44.158 𝑃𝑎
60
Appendix D
Void fraction calculations from tests
Details of notations on page 51
Weight of plastic cup for water measurements: 8 g
Weight of plastic cup for the sand: 3.8 g
Density of water = 998 kg/m3
Void fraction for Silica sand
Exact amount of water added:
57.2 + 28.5 + 17.9 − (3 ∗ 8) = 79.6 𝑔 = 79.6 ∗ 10−3 𝑘𝑔
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝑚𝑎𝑠𝑠
79.6 ∗ 10−3
=
= 7.976 ∗ 10−5 𝑚3
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
998
Weight dry sand:
289.6 − 3.8 = 285.8 𝑔 = 0.2858 𝑘𝑔
Density of dry silica sand = 1602 kg/m3
Volume of dry silica sand:
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝑚𝑎𝑠𝑠
0.2858
=
= 1.78 ∗ 10−4 𝑚3
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
1602
Volume of total volume is the same as volume of dry sand since this includes air filled voids
between sand grains.
𝑉
𝜀 = 𝑉𝑣
𝑇
𝜀=
(D-1)
7.976 ∗ 10−5
= 0.447 ≈ 0.45
1.78 ∗ 10−4
Void fraction for sharp sand
Exact amount of water added:
57.3 + 18.7 − (2 ∗ 8) = 60 𝑔 = 60 ∗ 10−3 𝑘𝑔
𝑉𝑜𝑙𝑢𝑚𝑒 =
𝑚𝑎𝑠𝑠
60 ∗ 10−3
=
= 6.012 ∗ 10−5 𝑚3
𝑑𝑒𝑛𝑠𝑖𝑡𝑦
998
Volume of dry sharp sand = 0.2*10-3 m3
Volume of total volume is the same as volume of dry sand since this includes air filled voids
between sand grains. Void fraction of sharp sand:
𝜀=
6.012 ∗ 10−5
= 0.301 ≈ 0.30
0.2 ∗ 10−3
61
Appendix E
Calculating minimum fluidising velocity
Details of notations on page 51
Assuming Reynolds number < 1
̅̅̅̅̅̅
𝑈𝑂𝑀 ≈
3
𝑔(𝜌𝑝 −𝜌𝑓 ) 𝜀𝑀
Φ2 𝐷 2
150𝜇 1−𝜀𝑀 𝑠 𝑃
(E-1)
g= 9.81 m/s2
𝜌𝑝 = 1602 kg/m3
𝜌𝑓 = 998 kg/m3
𝜀𝑀 = 0.447
Φ𝑠 = 0.83 or 0.6
𝜇 = 0.001 Pa*s
Dp = Up to 0.5 mm for silica sand
If surface-volume ratio is 0.83, then:
̅̅̅̅̅̅
𝑈𝑂𝑀 ≈
9.81 ∗ (1602 − 998) 0.4473
0.832 ∗ (0.5 ∗ 10−3 )2 = 1.0966 ≈ 1.1 ∗ 10−3 𝑚/𝑠
150 ∗ 0.001
1 − 0.447
If surface-volume ratio is 0.6:
̅̅̅̅̅̅
𝑈𝑂𝑀 ≈
9.81 ∗ (1602 − 998) 0.4473
0.62 ∗ (0.5 ∗ 10−3 )2 = 0.5742 ≈ 0.6 ∗ 10−3 𝑚/𝑠
150 ∗ 0.001
1 − 0.447
62
Appendix F
63
64
Appendix G
65
Appendix H
Calculating factor of safety for water butt
Details of notations on page 51
For thin-walled cylinder, hoop stress is:
𝜎𝜃 =
𝑃𝑑𝑖
2𝑡
(H-1) [25]
We know that:
𝑃 = 𝜌𝑚 ∗ 𝑔 ∗ 𝐻
(H-2)
Assuming the water butt will be filled ¾ up with sand, volume by sand in water butt is:
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑎𝑛𝑑 𝑖𝑛 𝑤𝑎𝑡𝑒𝑟 𝑏𝑢𝑡𝑡 = 𝑉𝑠𝑎
3
𝑉𝑠𝑎 = 4 ∗ 𝑉𝑇
𝑉𝑠𝑎 =
(H-3)
3
∗ 0.21 = 0.1575 𝑚3
4
Calculating mass of the sand in the water butt knowing the density of dry sand=1602 kg/m3
[26]:
𝜌𝑠𝑎 =
𝑚𝑠𝑎
𝑉𝑠𝑎
𝑚𝑠𝑎 = 𝜌𝑠𝑎 ∗ 𝑉𝑠𝑎
(H-4)
𝑉𝑠 = Volume of sand (m3)
𝑚𝑠𝑎 = 1602 ∗ 0.1575 = 252.315 𝑘𝑔
Density of the dry sand is different to the density of wet sand, since the gaps between the sand
grains are either filled with air or water, changing the density. Calculating actual volume of the
sand grains alone lets us find out the volume between grains.
Density of silica/silicon dioxide without space in-between is 2600 kg/m3 [27]. Mass of air
between sand particles is assumed negligible compared to mass of sand. Mas of dry sand may
therefore be used to find volume occupied by sand grains alone.
66
𝑉𝑠𝑐 =
𝑉𝑠𝑐 =
𝑚𝑠𝑎
𝜌𝑠𝑐
(H-5)
252.315
= 0.097 𝑚3
2600
Volume of voids between sand grains is therefore volume of sand crystals subtracted from total
volume of dry sand:
𝑉𝑉 = 𝑉𝑠𝑎 − 𝑉𝑠𝑐
(H-6)
𝑉𝑉 = 0.1575 − 0.097 = 0.061 𝑚3 .
𝑉𝑉 is the volume filled with water during the experiments. Density of water from the tap is
assumed to be 998 kg/m3 [28]. As earlier, calculating the mass from density and volume:
𝑚𝑤𝑣 = 𝜌𝑤 ∗ 𝑉𝑉
𝑚𝑤𝑣 = 998 ∗ 0.061 = 60.878 𝑘𝑔
Density of the sand and water mixture filling three quarters of the water butt is therefore:
𝜌𝑚 =
𝜌𝑚 =
𝑚𝑤𝑣 +𝑚𝑠𝑎
𝑉𝑠𝑎
60.878 + 252.315
= 1988.527 𝑘𝑔/𝑚3
0.1575
Therefore:
𝜎𝜃 =
𝑔 = 9.81
ℎ=
𝑃𝑑 𝜌𝑚 ∗ 𝑔 ∗ 𝐻 ∗ 𝑑
=
2𝑡
2𝑡
𝑚
𝑠2
3
3
∗ 𝑤𝑎𝑡𝑒𝑟 𝑏𝑢𝑡𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 = ∗ 0.915 𝑚
4
4
𝑑 = 𝑤𝑎𝑡𝑒𝑟 𝑏𝑢𝑡𝑡 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 − 2 ∗ 𝑤𝑎𝑙𝑙 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 = 0.56 − 2 ∗ 2.96 ∗ 10−3 𝑚
𝑡 = 2.96 ∗ 10−3 𝑚
𝜎𝜃 =
3
1988.527 ∗ 9.81 ∗ (4 ∗ 0.915) ∗ (0.56 − 2 ∗ 2.96 ∗ 10−3 )
2 ∗ 2.96 ∗ 10−3
= 1,252,950 𝑃𝑎 ≈ 1.3 𝑀𝑃𝑎
(H-7)
67
For worst-case scenario, water may be filled up to the top. Occurs if the anchor leg and the
water outlet fail. The mass of the water will be the mass of water between the sand grains and
the mass of the extra water filling the remaining quarter of the water butt.
1
𝑚𝑤 = 𝑚𝑤𝑣 + 𝜌𝑤 ∗ 4 ∗ 𝑉𝑇
(H-8)
1
𝑚𝑤 = 60.878 + 998 ∗ ∗ 0.21 = 113.273 𝑘𝑔
4
Density of water and sand mixture when the container is completely filled is then:
𝜌𝑚 =
𝜌𝑚 =
𝑚𝑤 +𝑚𝑠𝑎
𝑉𝑠𝑎
(H-8)
113.273 + 252.315
= 1740.895 𝑘𝑔/𝑚3
0.21
𝐻𝑜𝑜𝑝 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝜎𝜃 =
𝑃𝑑 𝜌𝑚 ∗ 𝑔 ∗ 𝐻 ∗ 𝑑
=
2𝑡
2𝑡
H is the total height of the water butt, which is 0.915 m
1740.895 ∗ 9.81 ∗ (0.915) ∗ (0.56 − 2 ∗ 2.96 ∗ 10−3 )
𝜎𝜃 =
= 1,462,559 𝑃𝑎 ≈ 1.5 𝑀𝑃𝑎
2 ∗ 2.96 ∗ 10−3
Tensile strength of HDPE is 32 MPa [29]. For worst-case scenario, factor of safety is:
32
≈ 22
1.463
68
Appendix I
Flow rate at end of anchor leg for 2 mm diameter drilled holes.
Details of notations on page 51
Velocity may be calculated by:
2𝑃
𝑈=√
𝜌
(I-1)
𝑓
Assuming pressure is 4 bar since this is a reasonable minimum pressure for standard municipal
water tap according to Prof. Stephen Salter, and density is the standard value of 998 kg/m 3, the
velocity of the water is:
2 ∗ 4 ∗ 105
𝑢=√
= 28.313 𝑚/𝑠
998
This is assuming the pressure at the anchor leg is the same as the pressure at water tap. In
reality, it would be slightly different due to distance water has travelling and with atmospheric
velocity acting on water level surface. The volumetric flow rate is then given by:
𝑄 = 𝑈𝐴𝐶𝑝
(I-2)
For a normal drilled hole with no rounded edges, discharge coefficient commonly used is 0.6.
If hole size was 2 mm, then:
𝐴=
𝜋𝑑2 𝜋 ∗ (2 ∗ 10−3 )2
=
= 3.142 ∗ 10−6 𝑚2
4
4
So flow rate would be:
𝑄 = 28.313 ∗ 3.142 ∗ 10−6 ∗ 0.6 = 5.3 ∗ 10−5 𝑚3 /𝑠
In Appendix C, flow rate directly from tap is 1.648 ∗ 10−4 𝑚3 /𝑠 which is roughly three times
more. For a single hole, this makes a high enough flow rates that means the particles of sand
would be forced towards the anchor leg.
69
Appendix J
Calculating friction coefficient
Referring to Figure 30 in section 4.2.3, the force may be split to x and y components. Let x be
down the slope and y be perpendicular to the slope. By applying Newton’s second law:
(J-1)
𝐹 = 𝑚𝑎
This may be applied in terms of the force of gravity in x- and y-directions:
𝐹𝑜𝑟𝑐𝑒𝑥 = 𝑚𝑎𝑥 = 0
𝐹𝑜𝑟𝑐𝑒𝑦 = 𝑚𝑎𝑦 = 0
𝑚𝑔 ∗ sin(𝑞) − 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 = 0
𝑁𝑜𝑟𝑚𝑎𝑙 − 𝑚𝑔 ∗ cos(𝑞) = 0
𝑚𝑔 ∗ sin(𝑞) = 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ∗ 𝑛𝑜𝑟𝑚𝑎𝑙
𝑁𝑜𝑟𝑚𝑎𝑙 = 𝑚𝑔 ∗ cos(𝑞)
This means that:
𝑚𝑔 ∗ sin(𝑞) = 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ∗ 𝑚𝑔 ∗ cos(𝑞)
𝜂 = 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 = tan(𝑞)
Outside anchor leg, following angles were measured:
Tilting angle with
Average value
Tilting angle with
Average value
dry sand (degrees)
(degrees)
wet sand (degrees)
(degrees)
22.000
20.000
25.000
21.667
32.000
23.000
28.000
27.000
Inside the anchor leg, following angles were measured:
Tilting angle with
Average value
Tilting angle with
Average value
dry sand (degrees)
(degrees)
wet sand (degrees)
(degrees)
22.000
20.000
25.000
21.667
23.000
32.000
28.000
27.000
The average values resulted in friction coefficients shown in Table 4 in section 4.2.3 where
numbers were found through same calculation of:
𝜂 = tan(𝑞)
For example: Dry sand, inside of the anchor leg
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑞 = 21.670°
𝜂 = tan(21.670°) = 0.397 ≈ 0.4
(J-2)
70
Appendix K
Force on anchor leg
Details of notations on page 51
Force may be defined as:
𝐹 =𝑃∗𝐴
(K-1)
Pressure will be the total of pressure from sand, pressure from water, atmospheric pressure
and minus vacuum pressure. A perfect vacuum has nothing inside it, hence no pressure. This
would not be the case for the anchor leg, however there was no equipment to measure the
pressure inside so must be assumed to be 0 Pa. Atmospheric pressure is 101325 Pa [34].
Therefore, by adding all pressures, multiplying it with surface area of the anchor leg and the
friction coefficient, the force is then defined by:
𝐹 = [(𝜌𝑠𝑖 ∗ 𝐻𝑠𝑎 ∗ 𝑔) + (𝜌𝑤 ∗ 𝐻𝑤 ∗ 𝑔) + 𝑃𝑎𝑡𝑚 − 𝑃𝑣𝑎𝑐 ] ∗ 𝜋 ∗ 𝐷𝑎𝑛𝑐ℎ ∗ 𝐿𝑎𝑛𝑐ℎ ∗ 𝜂
(K-2)
The water but is 0.915 m high, and was ¾ full of sand, which gives height of sand to be:
𝐻𝑠𝑎 =
3
∗ 0.915 = 0.686 𝑚
4
It was seen during experiments that water would usually be another 0.10 m above sand level.
Height of water is therefore:
𝐻𝑤 = 0.1 + 0.686 = 0.786 𝑚
Diameter of anchor leg is 0.056 m, and it is 0.74 m in length. The coefficient of friction used will
be measurements for outside the anchor leg. So when sand was wet, 𝜇 equaled 0.666, and when
sand was dry 𝜇 was 0.517. Knowing the density of silica sand is 1602 kg/m3 [26] and for water it
is 998 kg/m3 [28].
When the sand is wet, force on anchor leg is:
𝐹 = [(1602 ∗ 0.686 ∗ 9.81) + (998 ∗ 0.786 ∗ 9.81) + 101325 − 0] ∗ 𝜋 ∗ 0.056 ∗ 0.74 ∗ 0.666
= 10.4 𝑀𝑁
And when sand is dry:
𝐹 = [(1602 ∗ 0.686 ∗ 9.81) + (998 ∗ 0.786 ∗ 9.81) + 101325 − 0] ∗ 𝜋 ∗ 0.056 ∗ 0.74 ∗ 0.517
= 8.1 𝑀𝑁
Scaling equation:
𝑆𝑐𝑎𝑙𝑒 𝑜𝑓 𝑚𝑜𝑑𝑒𝑙 =
1
; 𝐹𝑢𝑙𝑙 𝑠𝑐𝑎𝑙𝑒𝑑 𝑓𝑜𝑟𝑐𝑒 = 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑓𝑜𝑟𝑐𝑒 ∗ 𝑥 3
𝑥
Example. Test 4 measured 164 N; full-scale model (if sand anchor leg is one hundredth of scale)
would have a gripping force of:
164 ∗ 1003 = 164 𝑀𝑁
71
Appendix L
Wall thickness for the concrete legs to float
Details of notations on page 51
Equation for floatation:
𝑑𝑖
𝑑𝑜
= √1 −
1
(L-1) [30]
𝛾
Specific density equal 2.4 for concrete [30], so inner-outer diameter ratio becomes:
𝑑𝑖
1
= √1 −
= 0.764
𝑑𝑜
2.4
Rewriting inner diameter in terms of outer diameter:
𝑑𝑖 = 0.764𝑑𝑜
The thickness of a hollow cylinder is:
𝑤𝑎𝑙𝑙 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 =
𝑑𝑜 − 𝑑𝑖 𝑑𝑜 − 0.764𝑑
=
= 0.12𝐷𝑑𝑜
2
2
Minimum outer diameter to both allow buoyancy and to resist 100-year Atlantic wave:
Force from 100-year Atlantic wave = 40MN
Concrete grade = 55 MPa
Assuming post-tensioned to 20% of 55 MPa, so 𝜎=0.2*55=11 MPa
𝜎=
𝐹
𝐴
→ 𝐴=
𝐹
𝜎
𝜋
𝐴 = ∗ (𝑑𝑜2 − 𝑑𝑖2 )
4
𝜋
𝐹 40
∗ (𝑑𝑜2 − 𝑑𝑖2 ) = =
= 3.64 𝑚2
4
𝜎 11
𝜋
∗ (𝑑𝑜2 − (0.764𝑑𝑜 )2 ) = 3.64
4
𝑑𝑜 = 3.34 ≈ 3 𝑚
(L-2)
(L-3)
72
Appendix M
Calculating velocity of sand particle in water
Details of notations on page 51
Stokes law for a sphere:
𝐹𝑑 = 3𝜋𝜇𝑑𝑣
(M-1) [31]
For a sphere in water, the free body diagram will be such that the force pulling it down will
equal to the drag force and buoyancy force together. In other words:
𝐹 = 𝑚𝑔 = 𝐹𝑑 + 𝐹𝑏
(M-2)
The buoyancy force will be the weight of displaced fluid, which depends on the volume taken up
by the sphere. Volume of a sphere is:
1
𝑉 = ∗ 𝜋 ∗ 𝑑3
6
(M-3)
So for buoyancy force, 𝐹𝑏 = 𝑚𝑓 ∗ 𝑔 = 𝑉𝑓 ∗ 𝜌𝑓 ∗ 𝑔, where f is for fluid. (M-2) can therefore be rewritten to:
1
𝑚𝑔 = 3𝜋𝜇𝑑𝑣 + ∗ 𝜋 ∗ 𝑑3 ∗ 𝜌𝑓 ∗ 𝑔
6
This is re-arranged in the following steps, where p is short for particle:
1
𝑉𝑝 ∗ 𝜌𝑝 ∗ 𝑔 = 3𝜋𝜇𝑑𝑣 + ∗ 𝜋 ∗ 𝑑3 ∗ 𝜌𝑓 ∗ 𝑔
6
1
1
∗ 𝜋 ∗ 𝑑3 ∗ 𝜌𝑝 ∗ 𝑔 = 3𝜋𝜇𝑑𝑣 + ∗ 𝜋 ∗ 𝑑3 ∗ 𝜌𝑓 ∗ 𝑔
6
6
𝑣=
𝑑 2 (𝜌𝑝 −𝜌𝑓 )∗𝑔
18𝜇
(M-4)
For 0.5 mm diameter kiln sand with 2600 kg/m3 density for a silica crystal, its velocity in water
where density is 998 kg/m3 and viscosity is 0.001 Pa*s is:
𝑣=
(0.5 ∗ 10−3 )2 ∗ (2600 − 998) ∗ 𝑔
= 0.218 𝑚/𝑠
18 ∗ 0.001
This means that Reynolds number is:
𝑅𝑒 =
𝑅𝑒 =
𝜌𝑣𝑑
𝜇
998 ∗ 0.2183 ∗ (0.5 ∗ 10−3 )
= 108.92 ≈ 109
0.001
(M-5) [32]