ECE 333 – Green Electric EnergyECE_333_Lecture 3_Energy_Conservation Created Date 9/18/2017 11:15:09 PM

ECE 333 © 2002 – 2017 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1

ECE 333 – Green Electric Energy 3. Energy Conservation Principle

George Gross

Department of Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign


q Energy is, to some extent, an abstract term; for

our purposes, we view energy as “work”

q An unalterable characteristic of energy is its

invariance: the total energy in the universe

remains unchanged over time

ENERGY CONSERVATION PRINCIPLE


q The principle of energy conservation underlies all

natural physical, chemical, or biological

processes; the principle is, essentially, a very

general physical law that is based on the work of

the British physicist Joule

q Indeed, for a purely mechanical system, we can

derive the energy conservation principle directly

from the application of Newton’s laws of motion



q We examine a very simple example in which a mass m, initially on the ground at time 0 –, is thrown vertically upwards with a speed v 0 at t = 0

q We determine the relationship between v0 and the speed v(t h) at the time t h, when the mass attains the height h


t = 0 – t = 0 t = t h

m m v 0

m v (t h)

h


q We start with Newton’s Second Law to consider

the relationship

q Clearly,

q Let z be the vertical distance traveled by mass m

and so its acceleration is and velocity is


=F ma

= −F mgmmg

2

2

d zdt

dzdt


q At each instant t,

so that

q The second order differential equation has initial conditions


2

2

d zm F mgdt

= = −

2

2

d z gdt

= −

z 0( ) = 0 and dz

dt t=0

= v 0

( )∗

( )∗

( )∗∗


q Since

then

q We obtain the solution for v(t) by integrating


( )=dz v tdt

2

2

dv d dz d z gdt dt dt dt

⎛ ⎞= = = − ⎜ ⎟⎝ ⎠

( )∗∗

dvdt

⎛⎝⎜

⎞⎠⎟0

t

∫ dt = dv v 0( )

v t( )∫ = v(t)− v(0) = − gt ( )∗∗∗


q But

and upon integration

q At and so


( ) 0dzv t v gtdt = = −

( ) ( ) 20

102

z t z v t gt − = −

( )∗∗∗∗[ ] 2012h hh v t g t

= −

( )∗∗∗∗∗

t = t h , z t h( ) = h

( ) 0h hv t v gt = −


q From

so that becomes


( )∗∗∗∗

( )∗∗∗∗∗

t h =

v 0 − v(t h )g



( ) ( ){ }

−= − −⎡ ⎤⎣ ⎦

00 02

2h

h

v v tv v v t

g

( ) ( )

−= +⎡ ⎤⎣ ⎦

002

hh

v v tv v t

g

( )

⎡ ⎤= − ⎡ ⎤⎣ ⎦⎣ ⎦

220

12 hv v tg

h = v 0

v 0 − v t h( )g

− 12

g iv 0 − v t h( )⎡⎣ ⎤⎦ v 0 − v t h( )⎡⎣ ⎤⎦

g i g


q We rearrange and obtain

and multiply by m to get


( )22

01 12 2 hgh v v t

= − ⎡ ⎤⎣ ⎦

( )2 2

01 12 2hm v t mv mgh

= −⎡ ⎤⎣ ⎦ ( )†


q We associate the relationship with an energy

interpretation since the kinetic energy of mass m

at speed v (t) at time t is given by ; also,

the potential energy of mass m at height h is mgh

q Therefore, we can restate as


( )

⎡ ⎤⎣ ⎦21

2m v t

12

m v 0 2 = 1

2m v t h( )⎡⎣ ⎤⎦

2+ mgh ∗†( )

( )†


q In words, at an arbitrary height h

and so for two arbitrary values of h, say and

q We derived in a straightforward way that total

energy of mass m is invariant


kinetic energy h

+ potential energy h

= kinetic energy 0

constant! "## $##

' ' '' '' + = +

h h h hkinetic energy potential energy kinetic energy potential energy

'h ''h


q The energy conservation principle holds whenever we

convert energy from one form into another form

q We consider a hydroelectric system where a

reservoir stores water behind a dam at some

height h: the water flows through a penstock and

drives a turbine, whose rotor is connected through

a mechanical shaft to the electrical generator

APPLICATION: ENERGY CONVERSION


HYDROELECTRIC ENERGY GENERATION

h


q The water at the height h – typically, called the head h – has potential energy, which is converted into mechanical energy as the water flows through the penstock; the mechanical energy drives the turbine and is converted into electric energy by the generator

q Each unit volume of water in the reservoir has mass ρ, where ρ is the density of water, and so has potential energy ρgh



q As each unit traverses the penstock, its potential energy is converted into kinetic energy due to the speed at which it arrives at the turbine; each unit of volume of water has kinetic energy

q We assume that the pressure energy is negligibly small and there are no losses in the system – in effect, a frictionless penstock – so that the energy conservation law for the mass of the unit volume of fluid results in


12ρ v 2

ν


q The energy conservation law applies to every

energy conversion process; however, each

process undergoes losses due to inefficiencies of

the process and so some of the energy is

converted into such losses


ρ g h = 1

2ρ v 2


q As we shall see, the wind speed air mass has

kinetic energy which rotates the wind turbine,

which connects to the rotor of an electric

generator to convert that kinetic energy into

electricity

q Similar notions hold, for example, for a steam

generation plant



FOSSIL – FUEL FIRED STEAM GENERATION PLANT

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ECE 333 – Green Electric EnergyECE_333_Lecture 3_Energy_Conservation Created Date 9/18/2017 11:15:09 PM