Advanced Thermal & Fluid Engineering Project I

5/9/2019
ADVANCED THERMAL & FLUID
ENGINEERING
PROJECT I

PART I: SIMPLE INITIAL VALUE PROBLEM
INTRODUCTION
The vibration energy harvesting proposal has been developed over the years. There are several
forms of obtaining energy from vibrating systems. There is motion energy harvesting where the
electrical generators obtain energy from the direct force devices. Further, there are energy
harvesters which obtain energy from inertial devices. A force is effected on a given transducer
system. Using the force-distance integral, energy is generated. The transducers employed in the
harvesting technique are the piezoelectric, electrostatic, or electromagnetic. There are a number
of system dynamics that dominate the driving force to generate the energy.
Figure 1 Energy Extraction system using an inverted cylinder connected to an electric generator
For the very large driving forces, the damping coefficient needs to be quite large. This is a key
design concern. The other technique for the inertial generators uses a transducer that damps the
motion and outputs electrical power which propels the electric generator. The system maximizes
the power dissipation in the damper hence the damping force in its magnitude is able to limit the
mass travel. The energy extraction device, in this design, uses a damping constant whose flow
velocity is obtained as a blend of the steady and oscillatory flow,
u ( t )=U0 +Um sin ( 2 πt
T )
The KC figure is defined as,
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KC =U m T
D
The electricity generator is modelled such that the obstacle that extract energy from the motion
of the cylinder with a constant damping constant of c. The structural damping is considerably
negligible. The energy is obtained by computing the motion or movement of the damper which
constitutes the electricity generator. The energy is given as,
P= 1
T ∫
0
T
c V 2 dt
Using numerical simulation, the power is computed using the equation below,
P= 1
N ∑
n =1
N
c ( V n )2
The power must be calculated using stable numerical solution. The hydrodynamic force on the
cylinder can be predicted using the Morison equation,
The system parameter table contains the system payload values as shown in the table 1 below,
j=8 ;Um= 1.8 m
s ; U0=0.08 m/s
The diameter of the cylinder is given as,
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D=10.8 cm
Stable vibrations are obtained while performing the simulations. It is expected that the cylinder
oscillates freely and in the direction transverse to the incoming flow. The cylinder is connected
to the damper and spring subsystem where the damper represents the energy harnessed by the
electrical generator. The system is mainly used in ocean engineering where energy is harnessed
from the flow of water or difference in pressure of water in different sections of the ocean. The
design does not consider an inviscid flow where there are no added mass effects considered. For
the additional added mass forces, there are viscosity drag forces where the separation and
boundary layer friction occurs. The forces induced on the inclined cylinder are obtained from the
moving currents in a planar oscillatory flow.
PROBLEM STATEMENT
An energy extraction device includes a cylinder elastically mounted on a spring and the
electricity generator. The cylinder vibrates only in a direction parallel to the flow and drives an
electricity generator. The electricity generator extracts the energy from the cylinder in the same
way as a damper. It is modelled as a damper with a damping constant c when the vibration is
studied. The flow velocity is a combined steady and oscillatory flow.
METHODOLOGY & SYSTEM DESIGN
Section I
To determine the equation of motion of the cylinder. The Morrison equation considers both the
inertial term and the drag term. The force is provided in the x-direction on the body following an
unsteady flow with the velocity such that,
Fx (t )=D ( t )=ρ Cm ∀U + 1
2 ρ Cd AU ∨U∨¿
The magnitude of the force on the cylinder body provides several merits to the Morrison
equation and can be used when an appropriate wave theory is selected.
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For the inclined cylinder considering the diameter, d, and the larger length, l , at a given angle
where there is unsteady flow, the Morrison equation is applied. The inflow velocity is divided
into two components namely, Um ∧Ut. Therefore,
The mass and drag coefficients of the inclined and non-inclined cylinder are of normal velocity
as they are determined by the diameter of the cylinder. The friction coefficient is used in the
tangential case instead of drag coefficient since the drag results from the flow along tangential to
the cylinder. These components are decomposed into the normal and tangential velocity with
regards to the inclined cylinder such that,
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Taking into consideration the forces acting on the inclined cylinder, with the assumption that a
linear wave forms the flow wave theory, the inertial force is obtained as dependent of the mass
coefficient and it is determined by integrating the force acting over the height of the structure.
Exerting the maximum inertial force the equations are obtained as,
The effects of the inertial and drag components of force are compared as,
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