2016 IEEE International Conference on Power and Renewable Energy

Energy Production from Buoyancy-Gravity Forces Effect on an Object in a Fluid

v. Labre, J. Bouffier, M. V. Cancelier, E. Gazay, T. Grandin

ECE Paris School of Engineering
Paris, France
e-mail: vlabreOO@gmail.com

A bstract-Nowadays it is necessary to develop alternative storage for compensating their unpredictability [14]. Thus, it
systems using renewable energies able to respond to consumer is difficult to obtain continuous generation of a large amount
unpredictable demand. To this aim, present study explores the of electric power through them.
feasibility of energy production and/or storage from action of A different, relatively inexpensive to exploit, and clean
antagonistic buoyancy-gravity forces. The system is composed method for generating large amounts of electric power is
of an object immersed in a fluid which moves up and down,
hydroelectric power generation [15-16]. Its use is however
depending on overtaking of buoyancy or gravity force, and
limited by nearby availability of large quantities of water and
which transfers its kinetic energy to an associated mechanical
the feasibility of constructing a large dam to store the water.
or direct inductive system transforming object kinetic energy
Geological sites where the required conditions for
into electrical energy. System theoretical model has been
hydroelectric power generation can be satisfied are few and
parametrically developed to describe its behaviour and to
calculate overall energy balance and response time. It is
far away from each other. Independent of their limitative
concluded that proposed system could be a viable system
greenhouse impact [17-21] and possible risks in their
concerning energy issues.
exploitation [22-23], because of inherent limitations of most
power generation methods with respect to the demand, it is
Keywords-archimedes principle; drag forces; buoyancy; particularly interesting to develop alternative power
density; gravity generation method that could provide power with the
benefits of hydroelectric generation, but without its
I. INTRODUCTION
geographical restrictions. It should on the other hand be
adapted to highly distributed and small size demand with
In last years, the evolution of energy demand toward large autonomy in its functioning.
more dispersed and less predictable consumption is making An elementary form to satisfy these constraints is to
traditional large size energy sources less adapted to this new combine antagonistic physical forces to create a source like
type of demand, both in terms of energy flow management falling water in conventional dams which, in fluids, is
and control in a too big network and of operating costs. It provided by gravity and buoyancy forces. Such is the system
orientates providers toward new forms of production discussed in the following, where antagonistic gravity and
characterized by high flexibility and exhibiting a large Archimedes forces are successively activated to produce up
degree of dispersion to fit as closely as possible the local and down motion of an immersed object in a fluid which is
needs. For this reason a large panel of possible sources based converted into electricity by mechanical or direct
on different physical effects have been and still are under electromagnetic inductive transformation creating electrical
study to document the possible parameter domain where they current. The aim of present study is to determine, if any, the
could be considered as reliable alternative energy sources [1- parameter domain where such system can work and deliver a
6], aside conventional fossil and nuclear sources. positive power output.
Roughly two main groups can be distinguished: 1) the so­
called "renewable" sources, including wind, solar and II. SYSTEM PRINCIPLE
hydraulic submarine ones, aimed at competing with Fig. 1 describes the system principle which can be
traditional large size units, themselves of relatively large size divided into two main parts: the cycle and the energy
(up to hundreds of MW), and 2) smaller, almost individual conversion.
units (tenths of KW and smaller), useful for isolated The object of mass M and volume V follows a four-step
consumers, which are often based on innovative physical cycle. At initial state, it is filled with a fluid and denser than
principles [7-12]. Wind and solar power are environmentally it. During first phase, the object is put into motion under its
clean and are expected to be relatively inexpensive at full own weight. Then, when the desired depth is reached the
development [13]. However, a large-scale utilization of these second step starts and compressed air is injected inside the
sources for electric power generation is still not in practice object. It will evacuate a fixed amount of fluid away and
because of several limitations inherent in these methods, decrease object volumetric mass density so that it becomes
such as a disproportionately large surface area for a large­ lighter than the fluid. Archimedes thrust then overcomes the
scale operation due to their inherently weak specific power weight and the object will start its rise in third phase. Once it
delivery. These methods are also not fully reliable because of reaches the top, the fourth and last step consists in releasing
their dependence on weather conditions and require large

978-1-5090-3068-2/16/$31.00 ©2016 IEEE 635

the air and recuperating its energy. It will instantly be system in (p,V) plane. Simple estimate is to take the
replaced by the fluid so the system returns to the initial state isentropic expansion corresponding to previous compression
and a new cycle may begin.
During its up and down motion, the immersed object will (3)
give its kinetic energy in case of simple mechanical
transformation to a wheel attached to an alternator which corresponds to the work without losses (heat, friction)
transforming mechanical energy into electrical energy during of a piston in a cylinder related by a duct to volume V 2.
its up and down displacements. Other trajectories can as well be considered leading to more
general expression Wdep = P2V2<1>(P). To establish system
energy balance, total output energy Wtot = Wfr + Wdep has to
be compared with input energy Weomp' Using P2 = P + PegH,
Jnitial State I
Wtot transforms to Wtot = [<I>(p) + (411,/3)(1 - P)]P2V2 with A, =
=
Map/(Pe V2)' Because Map (Pr - Pe)V one gets
Cycle
Rise Fall
(4)
Injection of
pressurized air
when desired depth
is reached
withpac = p- J/YPair Supposing that the absolute difference of
density between the object and the fluid is the same during
the rise and the fall one gets Pr - Pe = Pe - Pr from which
Activanon of the wheel Pf - Pr = 2(Pr - Pe) and A, = 0.5(Pe- Pac) Pe-I
+ So after reduction comparison of Wtot and Wcomp fmally
Creation of
amounts to study the sign of
mechanical energy

=
Energy �W(p) P2Vd<l>(p) +
Conversion
Activation o(the generator (2/3)(1- P)(Pe- Pac) Pe-I - a-I (1- pU)] (5)
+
Creation of
with 0 < P < 1, and there is energy release for all p such that
dearie,,! energy
�W(P) > O.
In general <1>(0) is bounded, <1>(1) = 0 and <1>'(1) < 0 so
�W (O) = P2V2 [<1>(0) + 2/3(Pe- Pac) Pe-I - a-I] < 0 and �W(1)
Figure 1. Basic system cycle and energy conversion.
=
0 with �W'(1) < O. So the curve �W(P) presents a positive
III. SYSTEM ENERGY BALANCE ANALYSIS arch for Pcrit< P < 1 with �W (p crit) = 0 see Fig.2. The
maximum energy recovery corresponds to the value p = Psup
Considering that absolute difference of volumetric mass
density with the fluid is the same during rise and fall motions, for which �W'(Psup) = O. With (3) for instance where <I>(p) =
the work done by the object is similar during the two phases. (aYrl(1 - pU), the overall energy balance of complete down­
Therefore the maximum work Wfr during one cycle by object up cycle without dissipation is positive if and only if p > Pcrit
=
fall and rise displacement is given by 0.0288 corresponding to a depth Hcrit = 348m, and the
largest energy recovery is occurring for Psup = 0.0757 for
(1) which Hsup = 126m, showing that power and energy recovery
where Map is apparent object mass and H the amplitude of do not follow the same constraints.
object displacement. The coefficient value 4/3 = 2.(2/3) for
each way will appear later in power analysis section as a
result of hydrodynamic friction effect. The work delivered
by isentropic air compression during full cycle (admission, Llsup

compression, expulsion) to compress and inject air at the

right pressure in lowest position is given by
o erH Psup p

(2)
with V2 = (Pr - Pr)(Pe - PacrlV, P2 = P + PegH, a = (y - 1)/y,
I
Pr is object density during fall part, Pr its density during rise
part, Pe is fluid density, Pac the density of compressed gas
and p = P/P2. On the other hand, one should take advantage
of work Wdep recoverable from state (P2,V2) of compressed
gas when the object is returned to upper initial state. Wdep Figure 2. Sketch of Curve ""W(p) vs p
depends on thermodynamic trajectory followed by the

636
 IV. SYSTEM D YNAMIC S Without power call the velocity continues to
monotonically rise to limit value Vlim according to (8).
When immersed in a fluid a free object with volume V
and mass M obeys Newton law

P + Par + F = MA, (6)

where P is (downward) object weight, Par is (upward)

Archimedes thrust, A is resulting object acceleration, and F
2
= - kv is friction force with drag coefficient k = 0.5peSC"
where S is object surface perpendicular to displacement and
Cx the nondimensional drag coefficient, which is
experimentally obtained and depends on object shape. Some
values are collected in Table I for different shapes. The drag
coefficient is strongly modified depending on flow
turbulence level. A sphere in water can be considered in a
very turbulent flow from a radius of 20cm and a density of °0 8,80" 1 8o ! NOffNIIIZ�Time
<'l
1100kg/m3. (a), ,(I).,
(I) (2)
(b)"
Figure 3. Object Nonnalized Velocity 11(8) vs Nonnalized Time 8
TABLE I. DRAG COEFFICIENT Cx VS OBJECT SHAPE AND FLOW
11s is the Normalized Velocity Corresponding to Papl and 80 the Actual
EUipsoiJI. Normalized Time at which Papl is Put on.
Sphere 1:2 ratio 1:4 ratio
Equation (10) gives the available power from the system
Turbulelllflow
Re < 100000,200000 0.45,0.47 0.27 0.25 for corresponding stationary object velocity. This is a cubic
Very turbulsnt flow
which only positive arch (11 > 0, IT > 0) is to be considered.
Re> 250 000, 300 000 0.2 0.13 0.1 The maximum positive value of this arch is reached at {l1max
ll2 3/2
= r , ITmax = 2.r }, see Fig. 4. It can be verified that IT(1)
= 0, i.e. that full asymptotic speed is only reached without
lI2
With normalizing variables XI = Mlk, Vlim = (k-1Mapg) , tl any power demand.
= XI/Vlim (6) simply writes
0.45

(7)
0.4
Tr ............................................
m
where 11 = V/Vlim and' = d/d8 with 8 = tltl . 11' is 0 when 11 =
035
I, i.e. the object reaches limit velocity Vlim. System free
dynamics are for initial conditions (x(O) = 0, v(O) = 0) given 03

by
11(8) = th(8) ; �(8) = In[ch(8)], (8)
02
with � = X/XI' The object down-up cycle, in the case where it
is connected to a wheel system of mass Mwh to collect the 0.15

power delivered by gravity in downward cycle part and

Archimedes force in upward cycle part, is now described by
(7) for downward motion where M is replaced by Mr = M +
0.5Mwh and by
(9) 0.4 1J.\l.6
Normalized Velocrty

with /l = M/M and IT = PaplMapgvlim the normalized power Figure 4. Deliverable Normalized Power n vs Normalized Object
demand after time to where power demand Papl is applied Velocity 11.
onto the system, see Fig. 3.
Any power value IT between 0 and ITmax is reached for
During first phase object velocity increases according to two nonnalized velocity values l1s and l1st l1s is the minimal
(8) until Vs is reached at time ts. The power Papl can be called normalized velocity necessary for the object to deliver ITapl.
at any time t > ts. If called at to > t5 the object does not If this velocity is not reached when IT ITapl is demanded, =

continue to follow (8) (blue curve on Fig. 3) and switches at t the object slows down and stops, and no power will be
= to to the solution of (9) (green curve on Fig. 3) which tends provided. l1st represents the fmal velocity value at which the
toward stationary value l1st = V5/Vlim solution of object will fall, after complete transitory state for a power
demand ITapl' It can be observed that ITmax = (2/3)l1max
(10) although the expectable power at velocity Vmax is equal to

637
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ACKNOWLEDGMENT
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The authors are very much indebted to ECE Paris School Electricity Generation. A Comparison of the Environmental impacts

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