abj 1 Energy as A Conserved Quantity • Conservation of Energy for An Isolated System • Conservation of Energy for A MV (Closed System) 1. Modes of Energy Transfer (Heat/TE + Work/ME + Others) 2. Forms of Energy Stored (TE + ME + Others) LHS: [Modes of] Energy Transfer • Decomposition of Energy Transfer: Heat + Work + [Others, if any] • Energy Transfer As Work (of A Force) • Decomposition of Work of Surface Force: Pressure + Shear Finite Control Volume Formulation of Physical Laws: C-Energy Conservation of Energy (Working Forms) Basics and Various Cases of Energy Transfer as Work of (Surface) Forces [Surface Force = Normal/Pressure Force + Shear Force] Example of Energy Transfer as Work of (Surface) Forces: Pump and Turbine Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces • Here we limit ourselves to an observer in an inertial frame of reference (IFR) only. • Note that kinetic energy (KE) – being defined from velocity - is frame of reference dependence, i.e., Lecture 6.2: Conservation of Energy (C-Energy), and Energy Transfer as Work of (Surface) Forces Q + W
Transcript
abj 1
Energy as A Conserved Quantity
• Conservation of Energy for An Isolated System
• Conservation of Energy for A MV (Closed System)
1. Modes of Energy Transfer (Heat/TE + Work/ME + Others)
2. Forms of Energy Stored (TE + ME + Others)
LHS: [Modes of] Energy Transfer
• Decomposition of Energy Transfer: Heat + Work + [Others, if any]
• Energy Transfer As Work (of A Force)
• Decomposition of Work of Surface Force: Pressure + Shear
Finite Control Volume Formulation of Physical Laws: C-Energy
Conservation of Energy (Working Forms)
Basics and Various Cases of Energy Transfer as Work of (Surface) Forces
[Surface Force = Normal/Pressure Force + Shear Force]
Example of Energy Transfer as Work of (Surface) Forces: Pump and Turbine
Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as
Work of Forces• Here we limit ourselves to an observer in an inertial frame of reference (IFR) only.
• Note that kinetic energy (KE) – being defined from velocity - is frame of reference dependence, i.e., observers moving relative to each
other observe different amount of KE for the same mass.
Lecture 6.2: Conservation of Energy (C-Energy), and Energy Transfer as Work of (Surface) Forces
Q+
W
abj 2
Very Brief Summary of Important Points and Equations
Q+
W
C-Energy for A MV
ME) and (TEformsvariousStoredEnergy
etc.) work,and heat (asmodesvarious
TransferEnergyinMVin
rease inChange/IncofRateTime
MV
ingssurroundinitsfromMV to
ofRateTime
dt
dEWQ ......
C-Energy (Working Forms) for A CV
othersshearshaft
oo
me
CS
sf
CV
WWWW
Vhhgzh
puhgzV
h
pekepgzV
pmemeu
gzV
pupe
gzV
ueAdVpedVedt
dWQ
::
2:,:
v:,2
:
v2
v:,:
,2
vv:
2;)(v)()(
2
2
2
2
2
/
= stagnation enthalpy
u-me - form
h - form
ho - form
e-pv - form
abj 3
Energy as A Conserved Quantity/Scalar
• Conservation of Energy for An Isolated System
• Conservation of Energy for A MV (Closed System)
abj 4
“According to Classical Mechanics”
Let’s say, the universe – that we are a part of - is an isolated system.
Of course, the numbers are not real (I made them up, obviously), but you get the idea of the concept
of conservations of mass and energy. [Both are conserved scalar/quantity.]
According to classical mechanics, energy – like mass – is a conserved scalar/quantity.
abj 5
Energy as A Conserved Quantity/Scalar Conservation of Energy for
Universe (Isolated System)
EU = Constant (Conserved) dEU = 0
An Isolated System
abj 6
Relation Between Changes of Various Parts U = MV + Surroundings
Universe (Isolated System)
EU = Constant (Conserved) dEU = 0
EU = EMV + ESur = Constant EMV
MV (Closed System)
Surroundings, ESur
Total Amount
dEMV
Change/Increase in Energy Stored
- dESurUniverse (Isolated System)
EU = Constant (Conserved) dEU = 0
dEU = dEMV + dESur = 0
- dESur = dEMV
Surroundings
Relation between changes of various parts
The amount of energy transferred to a system must come from its surroundings.
An Isolated System
abj 7
Energy as A Conserved Quantity/Scalar Conservation of Energy for Focus on a MV (closed system) as a part of the Universe
Universe (Isolated System)
EU = Constant (Conserved) dEU = 0
dEU = dEMV + dESur = 0
dEMV
Change/Increase in Energy Stored
in MV in various forms
- dESur
Energy Transfer to MV from its surroundings in
various modes
Surroundings
formsvarious
StoredEnergy
modesvarious
TransferEnergyinMVin
rease inChange/Inc
MV
ingssurroundinitsfromMV to
T dEE
Let’s denote the LHS instead by ET.(= - dESur)
MVSur dEdE
A MV (Closed System)
abj 8
Conservation of Energy for A MV (Closed System)
1. Modes of Energy Transfer
1. Energy Transfer As Heat ( , Thermal Energy Transfer)
2. Energy Transfer As Work ( , Mechanical Energy Transfer)
3. Other Modes of Energy Transfer ( )
2. Forms of Energy Stored
1. Thermal Energy (TE)
2. Mechanical Energy (ME)
3. Other Forms of Energy Stored
WW ,TE
QQ ,
abj 9
Conservation of Energy for Modes of Energy Transfer and Forms of Energy Stored
dEMV
Change/Increase in Energy Stored
in MV in various forms
MV (Closed System)
ETEnergy Transfer
to MV from its surroundings in
various modes
Surroundings
formsvarious
StoredEnergy
modesvarious
TransferEnergyinMVin
rease inChange/Inc
MV
ingssurroundinitsfromMV to
T dEE
KEY: Regardless of the number of modes of energy transfer and forms of energy stored, the basic idea of the conservation of energy is that
All must be accounted for so that EU is conserved or - dESur = dEMV (a simple balance law)
Forms of Energy Stored
Thermal energy TE (= U)
Mechanical energy ME (= KE)
Other forms of energy stored ( e.g.,
electrical, chemical, etc.)
Modes of Energy Transfer
Energy Transfer as Heat Q = Thermal energy transfer
Energy Transfer as Work W = Mechanical energy
transfer
Other modes of energy transfer ET (e.g., electromagnetic
radiation, etc.)
A MV (Closed System)
abj 10
Conservation of Energy for In most of our problems of interest, only 1) Thermal Energy (TE) and 2) Mechanical Energy (ME) are excited/changed
ET = Q + W + [ET ]
Q = Heat = Thermal energy transfer
W = Work = Mechanical energy transfer
ET = Other modes of energy transfer
EMVTE + ME [+ Other forms]
= Thermal energy
ME = Mechanical energy
…... = Other forms of energy stored
Time
Energy
dt
dEEWQ MV
T
EnergydEEWQ MVT
formsvarious
StoredEnergy
modesvarious
TransferEnergyinMVin
rease inChange/Inc
MV
ingssurroundinitsfromMV to
T dEE
Energy Transfer to MV from its surroundings in
various modes
Surroundings
othersTT EWQE ,
Change/Increase in Energy Stored
in MV in various forms
MVMV METEddE )(
MV (Closed System)
Key:
If some other forms of energy are
also excited/changed, they must
be taken into accounted
according to the conservation of
energy.
Q+
W
A MV (Closed System)
abj 11
C-Energy for A MV (Closed System)
ME) and (TEformsvariousStoredEnergy
etc.) work,and heat (asmodesvarious
TransferEnergyinMVin
rease inChange/IncofRateTime
MV
ingssurroundinitsfromMV to
ofRateTime
dt
dEWQ .....
Surroundings
Time Rate of Energy Transfer
to MV from its surroundings in
various modes
)( anyifWWQ others
Time Rate of Change/Increase in Energy Stored
in MV in various forms
dt
dEMV
MV (Closed System)
Q+
W
abj 12
Scratch Note: Proof of Work of mg
mgzPEdt
PEdW
dt
mgzddt
dzmg
mgV
Vkgm
VgmW
MVmg
z
mg
:,)(
)(
)ˆ(
Work of Body Force mg and Potential Energy [1]
x
y
z kgg ˆ
V
gm
abj 13
The Two Forms of C-Energy for A MV (Closed System)(according to where we put the work of mg / potential energy)
dt
PEKEUdWQ
dt
PEdW
dt
KEUd
dt
PEdWQ
WWWdt
KEUdWWQ
dt
KEUdWQ
MV
MVmg
MVMV
mgMV
mg
MV
)(
)(,
)()(
,)(
)(
......,ofworkincludenot must:
2
1)(:
......,ofworkincludemust:
2
1:
,
2
2
mg
METE
mg
METE
MV
WWmgW
mgzmVUPEKEUMETEE
WWmgW
mVUKEUMETEE
dt
dEWQ
Form 1
Form 2
abj 14
Energy input into a system causes increase in energy of the system.
Energy extracted from a system causes decrease in energy of the system.
Sign Conventions for The Energy Equation
Physics: - output causes E-decreaseW
dtdEnegativecausesWpositiveEquation MV /:
Physics: - input causes E-increaseQ
dtdEpositivecausesQpositiveEquation MV /:
Q
+
W
dt
dEWQEquation MV :
Physics: - input causes E-increaseQ
dtdEpositivecausesQpositiveEquation MV /:
Q
+
W
dt
dEWQEquation MV :
Physics: - input causes E-increaseW
dtdEpositivecausesWpositiveEquation MV /:
Q
+
WQ
+
W
Similar can be said for
abj 15
C-Energy for A MV (Closed System)
W
Q
LHS: [Modes of] Energy Transfer
1. Energy Transfer as Heat [Thermal Energy Transfer]
2. Energy Transfer as Work [Mechanical Energy Transfer]
ME) and (TEformsvariousStoredEnergy
work)and heat (asmodesvarious
TransferEnergyinMVin
rease inChange/IncofRateTime
MV
ingssurroundinitsfromMV to
ofRateTime
dt
dEWQ ......
abj 16
Recall in C-Mom
Keys
1. Recognize various types of forces.
2. Know how to find the resultant of various types of forces (e.g., pressure, etc.).
3. Sum all the external forces.
F
Keys: Energy Transfer to MV
1. Recognize various types/modes of energy transfers.
2. Know how to find the energy transfer of various types/modes (e.g., heat (TE), work (ME), electrical (EE), etc.).
3. Sum all the energy transfers to MV.
...WQ
Like in C-Mom, regardless of how it is written or
notations used, the key idea is to sum all (the modes
of) the energy transfers to MV.
F
LHS = Energy Transfer to MV
Time
Energy
dt
tdEWQ
tMV
MV
tMV
)( ofenergy of change of rate Time workandheat as )( to
ansferenergy tr of rate Time
)(
Mechanical Energy Transfer
(as Work of Forces )WEnergy Transfer in Other Modes othersWThermal Energy Transfer
(as Heat ) Q
Modes of Energy Transfer on The LHS
abj 17
Q+
W
Through a finite surface S :
(input-positive)
S
AdqQ
Heat ( )AdqQ
If any other
Other Modes of Energy Transfer
(input-positive)
e.g. electrical, electromagnetic, etc.
othersW
Work
Energy Transfer Modes (between a system and its surroundings)
Work of Forces
(input-positive)
FdVWF
tangentialnormal TTT
Stress vector
sheartangential TT
Tangential (Shear)
(input-positive)
S
shearshear AdVTW
Normal (Pressure)
nnormal epT ˆ
(input-positive)
S
p AdVpW v
Work of Surface Force/Stress
AdTVFdVW SS
Work of mg is later accounted for as potential
energy
Work of Body Force/mg
dVBVFdVW BB
If there are other body forces besides mg, all must be accounted for.
abj 18
Energy Transfer As Work of A Force[Mechanical Energy Transfer]
FW
abj 19
.....
workandheat as )( toansferenergy tr of rate Time
tMV
WQ
: Energy Transfer as Work (Mechanical Energy Transfer)W
CV(t)MV(t)
Pressure p
Shear
iF
Coincident CV(t) and MV(t)
)( dVgdmg Volume/Body Force
FBD
W
Work is the mode of (mechanical) energy transfer.
Work is work of a force,
In order to apply C-Energy,
on the LHS must be the sum of all the energy transfers as work,
i.e.,
the sum of works of all the forces.
Recall then Forces in Fluids and FBD
FW
abj 20
and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t)
forceexternalNet
F
BS FFF
1. Concentrated/Pointed Surface Force iF
2. Distributive Surface Force in Fluid [Pressure p + Friction ]
Net Surface Force SF
Net Volume/Body Force BF
MVCV
dVggm )(
CV(t)MV(t)
Pressure p
Shear
iF
2. Distributive Surface Force
(in fluid part)
1. Concentrated/Point Surface Force
Coincident CV(t) and MV(t)
)( dVgdmg Volume/Body Force
FBD
Recall 1: Recall all and various types of forces.
must be the sum of the works of all the forces on MV(t).W
abj 21
Recall 2: Energy Transfer as Work of A Force (Mechanical Energy Transfer)
EnergySdFWF
Time
EnergyVFWF
Particle
Concept
Work = Force x Displacement in the direction of the force
(per unit time)
Work of A Force ( )F
FF WW ,
Time
EnergyVFWF
VF
dtVSd
abj 22
Energy Transfer as Work of A Force (Mechanical Energy Transfer) Particle VS Continuum Body
V
Ad
SFd
V
BFd
Work of A Force ,F
FF WW ,
Same Concept
Work = Force x Displacement in the direction of the force
Time
EnergyVFWF
Time
EnergyFdVWF
EnergySdFWF
Time
EnergyVFWF
Particle Continuum Body
Same concept, just that
1) there are more types of forces to be accounted for: Surface force and Body force (and…)
2) Each type is described differently
3) As before, how to sum them all.
Volume
ForceBdVBFd B
,
Area
ForceTAdTFd S
,
S
FSF AdTVWAdTVFdVWSS
V
FBF dVBVWdVBVFdVWBB
)(
VF
dtVSd
abj 23
Work of All Forces
CV(t)MV(t)
Pressure p
Shear
iF
2. Distributive Surface Force
(in fluid part)
1. Concentrated/Point Surface Force
Coincident CV(t) and MV(t)
)( dVgdmg Volume/Body Force
FBD
)()()(
)()(
anyifWWWWW
Time
EnergyWWW
othersforcesurfaceedconcentratmgshearspressurep
forceBodyBforceSurfaceS
Note = Shaft work is work due to shear stress (surface force) at the cross section
of a shaft.)(shaftsW
W
abj 24
Work of shear force on CS/MS:
• Infinitesimal work of shear stress:
1. Rate of work (power) done on a finite closed
surface S:
positiveMV
S
shearshear
S
shearshear
AdTVW
WW
intoinput
positiveMV
shearshearshear AdTVFdVW
intoinput
Work of Surface Forces: 1) Pressure Force (Flow Work), 2) Shear Force
shearshearpressureS FdApdFdFdFd
)(
Work of pressure force on CS/MS:
• Infinitesimal work of pressure force:
1. Rate of work (power) done on a finite closed
surface S:
positive
pp
AdVp
AdVpApdVFdVW
MVintoinput
)v(
positiveMV
S
p
S
pp
AdVpW
WW
intoinput
v
Q+
W
Recall the coincident CV(t) and MV(t) Ad
S
shearpressureS FdFdFd
V
SurroundingsMV(t)
CV(t)
abj 26
Finite Control Volume Formulation of Physical Laws
C-Energy
abj 27
Finite CV Formulation of Physical Laws: C- Energy
C-Energy: eEN ,
Time
EnergyAdVe
dt
tdE
dt
tdEWQ
tCS
tCSQdmd
sf
tCV
CV
tMV
MV
tMV
,)()()(
)(ough energy thr ofefflux convectionNet
)(
/
)( ofenergy of change of rate Time
)( ofenergy of change of rate Time workandheat as )( to
ansferenergy tr of rate Time
Physical Laws
RTT
Recall the coincident CV(t) and MV(t)
Q
+
WQ
WMaterial Volume (MV)
dEMV/dtSurroundings
CV(t), MV(t)
Energy transfer as heat
Energy transfer as work of forces
p,
abj 28
Finite CV Formulation of Physical Laws: C- Energy
othersshearshaft
CSQdmd
sf
CV
CSQdmd
sfCV
CSQdmd
sfCV
CSQdmd
sfp
CSQdmd
sf
CSQdmd
sfCV
othersshaftshearpMV
othersshaftshearp
MV
MVmg
MVMV
othersshaftshearpmgMV
othersmgshaftshearp
MV
WWWW
AdVpedVedt
d
AdVpedt
tdEWQ
AdVpedt
tdE
AdVpWAdVpAdVedt
tdE
WWWWWdt
tdEWQ
WWWWW
gzVuePEKEUENdt
tdEWQ
dt
tPEdW
dt
tPEd
dt
tKEUd
WWWWWWdt
tEdWQ
WWWWWW
Time
EnergyVueKEUE
dt
tEdWQ
::
)()v()(
)()v()(
)()v()(
)(v,)(v)()(
,:,)(
::
2
1,,
)(
)()(,
)()()()(
:,)(
~::
,,2
1~,~
,)(
~
/
/
/
///
2
2
To save some symbols, here we redefine at various steps.W
Apply RTT to dEMV/dt
abj 29
C-Energy (Working Forms)
Q+
W
Recall the coincident CV(t) and MV(t)
othersshearshaft
oo
me
CS
sf
CV
WWWW
Vhhgzh
puhgzV
h
pekepgzV
pmemeu
gzV
pupe
gzV
ueAdVpedVedt
dWQ
::
2:,:
v:,2
:
v2
v:,:
,2
vv:
2;)(v)()(
2
2
2
2
2
/
= stagnation enthalpy
u-me - form
h - form
ho - form
e-pv - form
Q
WMaterial Volume (MV)
dEMV/dtSurroundings
CV(t), MV(t)
Energy transfer as heat
Energy transfer as work of forces
p,
abj 30
Basics and Various Cases of
Energy Transfer
as Work of (Surface) Forces[Surface Force = Normal/Pressure Force + Shear Force]
abj 31
Basics and Various Cases of Energy Transfer as Work of (Surface) Forces
[Surface Force = Normal/Pressure Force + Shear Force] Later on, we will be writing the C-Energy in various specialized forms, e.g.,
Here, we will first focus and emphasize the basic idea of energy transfer as work of (surface)
forces first.
So, let us step back one step by moving the flow work term (pv) back to the LHS.
gzVueAdVedVedt
dWQ
CS
sf
CV
2
/ 2
1:,)()()(
othersshearshaft
CS
sf
CV
WWWW
gzVueAdVpedVedt
dWQ
:
2
1:,)(v)()( 2
/
abj 32
Energy Transfer as Work of (Surface) Forces[Surface Force = Normal/Pressure Force + Shear
Force]
Pressure pShear
Solid part
V
V
0
V2. Stationary solid surface
(e.g., pump casing)
1. Moving solid surface
(e.g., pump impeller surface, cross section of a rotating solid shaft)
3. Stationary Imaginary surface
(where there is mass flow in/out.)
abj 33
Energy Transfer as Work of (Surface) Forces[Surface Force = Normal/Pressure Force + Shear
Force]
Pressure pShear
Solid part
V
V
0
V
2. Stationary solid surface
(e.g., pump casing)
0
)(0,,,
slipnoVFdVW pp
1. Moving solid surface
(e.g., pump impeller surface, cross section of a rotating solid shaft)
In general,
)(0
)(0,,,
VFdexcept
slipnoVFdVW pp
3. Stationary Imaginary surface
(where there is mass flow in/out.)
In general,
)(0
0,,,
VFdexcept
VFdVW pp
,/
/,,
)(
,
psfs
sfspp
FdVV
VVVFdVW
Note: For moving imaginary surface, we may use the decomposition
CS
sf AdVpe )(v)( /
Work due to pressure force here is later moved to the RHS and included as flow work, pv, in the convection flux term:
abj 34
Example of Energy Transfer as Work of (Surface) Forces:
Pump and Turbine
Various Control Volumes for A Fluid Stream,
Forces and FBD, and Energy Transfer as Work of Forces
abj 35
Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces
1m
2m
1
2Pump Turbine
a1(pump)
b2(pump)
c1(turbine)
d2(turbine)
• CV includes the fluid stream only, no solid part. • CV includes the fluid stream, the solid impeller, and a section of the
solid shaft. • It cuts through the cross section of a solid shaft.
FBD• Surface force: pressure/normal and shear stresses, over all surfaces. [Body force is not shown.]
1 2
Surface Force:Pressure and shear
on moving/rotating impeller surface
Surface ForcePressure and shear
MV
1 2
Surface ForcePressure and shear
Surface Force:Normal and shear stress
over the moving/rotating cross section of a solidshaft
MV
abj 36
• Energy transfer as work of (surface) forces occurs at moving material surfaces where there are surface forces act. There can be no energy transfer as work of forces at a stationary material surface.In order to have energy transfer as work of forces (in this case, surface forces),
• the point of application of the force must have displacement (in the direction of the force).
rV
eAdFd ˆ
Fd MV
Fd
Surroundings
rV
Surroundings
Pressure and shear stresses on the rotating impeller surfaces act on the moving fluid
Energy transfer as work to MV (fluid stream)
Time
Energy
dt
dEWQ MV
F ,
FdVWF
VMV
V
MV
abj 37
Energy transfer as work of forces at the surface of the moving/rotating impeller
FdVWF
[Pump]
• Pressure force pushes fluid,
• Shear force drags fluid,
such that the fluid at the material
surface has velocity .V
fW
Surroundings
V
MV
MV
Surroundings
fWEnergy transfer as work of force at the rotating impeller surface
abj 38
Energy transfer as work of forces at the cross section of a solid shaft
FdVWF
TW
Td
TdW
FdrTdTd
Fdr
FdrFdr
Fdr
FdVW
shaft
tioncrossshaft
tioncrossshaft
shaft
F
sec
sec
:,
)(
)()(:identityproduct tripleVector
)(
Shear stress at a cross section of a solid shaft.
• It is due to the other section of the shaft (surroundings) acting on our section of the shaft (MV).
edAFdFd zz ˆ)(
V
Fd
T
MV
Surroundings
TFd
, = External force and torque due to surroundings on our MV
(Recall the concept of FBD and Newton’s Second Law)
sW
MV
sW Energy transfer as work of force at the rotating cross section of a solid shaft.
abj 39
V
Fd
T
MV
Surroundings
V
TT
MVSurroundings
FdFd
Motor/Turbine drives its
Pump/Load
0
0)( MVshaft TW
[Pump, Load]
MV receives mechanical energy from the surroundings.
[Motor, Turbine]
MV gives up its own mechanical energy to the surroundings.
Motor
Pump
Turbine
Load
Direction of mechanical energy transfer as work
0)( MVshaft TW
0)( MVshaft TW
abj 40
Various Control Volumes for A Fluid Stream, Forces and FBD, and Energy Transfer as Work of Forces
CV1 / MV1 [See , but do not see .]
• [FBD] sees the shear stress at the rotating shaft cross section,
• [Work] sees the energy transfer as work at the rotating shaft cross section.
fWsW
CV2 / MV2 [See , but do not see .]
• [FBD] sees the pressure and shear stresses on the rotating impeller surface.
• [Work] sees the energy transfer as work at the rotating impeller surface.
