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+. CV includes the fluid stream only, no solid part. Lecture 6.2.1: C-Energy for A Steady, Incompressible, Fluid Stream: Conservation of Mechanical Energy and Mechanical Energy Loss (ME Loss). - PowerPoint PPT Presentation
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abj 1
1. A Fluid Stream and Energy Transfer To/From A Fluid Stream in Incompressible Flow Machines
2. Convection Transport and N-Flux
• A Fluid Stream and Convection Transport
3. Convection Flux of Total Energy = Convection Flux of Thermal Energy + Convection Flux of Mechanical
Energy
4. Conservation of (Mechanical) Energy and Mechanical Energy Loss (ME Loss) for A Steady, Incompressible, Fluid
Stream (with no applied heat transfer)
• Applications to 1) pump, 2) turbine, and 3) piping section
5. Hydraulic Power: MV and CV Interpretations
6. Hydraulic Head as Potential Height Equivalence for Hydraulic Power
7. Overall Efficiency of Hydraulic (Incompressible Flow) Machines
8. Conservation of (Mechanical) Energy for A Steady, Incompressible, Fluid Stream (with no applied heat transfer)
[Working Form for Piping System]
Lecture 6.2.1: C-Energy for A Steady, Incompressible, Fluid Stream: Conservation of Mechanical Energy and Mechanical Energy Loss (ME Loss)
Time
NAdVNm
A
sf ,:Fluxand:fluxMass /
0:,)()(
12
1
1
2
2
QmudmudEEWmdmemdme
AA
lossmelossmef
ME
A
ME
A
CV includes the fluid stream only, no solid part.
0:,
/,,
)()(
1212
QmudmudEE
ttsW
gHm
psWp
gHmmdmemdmeAA
lossmelossmetp
AA
Q+
W
abj 2
Very Brief Summary of Important Points and Equations [1]
1. A steady, incompressible fluid stream:
1. the energy of a fluid stream, N = Energy,
2. the convection of energy (through a cross section A, from one place to another):
of a fluid stream, and
3. the addition of energy to (pump), and the extraction of energy from (turbine), a fluid stream.
2. The decomposition of energy into two forms:
Thermal Energy (TE, U) + Mechanical Energy (ME)
3. The convection of energy of a fluid stream at any one cross section can be decomposed into
1) the convection of thermal energy (u) + 2) the convection of mechanical energy (me):
gzVpme
ME
mdme
TE
mudAdVmeu
FluxEnergyMechanical
A
A
FluxEnergyThermal
A
AA
sf 2
throughenergymechanicalofConvection
throughenergythermalofConvection
/ 2
1v,)()(
Time
EnergyAdVmeu
A
sf ,)( /
1 2N-Flux at 1 N-Flux
at 2
abj 3
Very Brief Summary of Important Points and Equations [2]
Q+
W
4. C-(ME)-Energy and ME Loss for A Steady, Incompressible, Fluid Stream (with no applied heat transfer)
Physical Interpretation:
• The above C-Energy equation can be viewed as a
conservation of mechanical energy (ME) of a fluid stream.
• The rise in mechanical energy/power of the fluid stream as it flows from 1 to 2
= ME added to the fluid stream as work of all surface forces (excluding flow
work)
- ME Loss in CV (Irreversible conversion of ME to TE)
0::
][,)()(
12
12
QmudmudE
PowerEWmdmemdme
AA
lossme
lossmef
AA
CV includes the fluid stream only, no solid part.
fW Work of all surface forces (excluding the flow work, pv, at inlet and exit),
e.g., work of shear at CS, works of pressure and shear at pump and/or turbine rotating impeller surface.
abj 4
Very Brief Summary of Important Points and Equations [3]
5. Hydraulic Power and Hydraulic Head
Potential head/height equivalence
(at the same mass flowrate ), HmH
m
m
1
)(1
A
mdmeME 2
)(2
A
mdmeME pump
12:PowerHydraulic MEME gHmeEquivalencPowerPotential=
Hydraulic Head
where gHmmdmemdmeMEMEAA
:)()(PowerHydraulic
12
12
LengthH:
Hydraulic Power
PowermdmemdmeMEMEAA
12
)()(: 12 [CV Viewpoint]
:= Increase/Decrease in ME Flux of the fluid stream from CV inlet 1 to CV exit 2
[MV Viewpoint]
:= the time rate of change of mechanical energy of the fluid stream [ENERGY STORED] as it
flows through CV [from CV inlet 1 to CV exit 2].
Recall the coincident MV(t) and CV(t)
mgzmVpVMechENgzVpmedt
tMechEd MV 2
2
1,
2
2
1v,
)(:
abj 5
Very Brief Summary of Important Points and Equations [4]
6. C-(ME)-Energy and ME Loss for A Steady, Incompressible, Fluid Stream (with no applied heat transfer)
Physical Interpretation:
• The above C-Energy equation can be viewed as a
conservation of mechanical energy (ME) of a fluid stream.
• The rise in mechanical energy/power of the fluid stream as it flows from 1 to 2
= ME added to the fluid stream at pump
- ME extracted from the fluid stream at turbine
- ME Loss in CV (Irreversible conversion of ME to TE)
0::
][,
/
)()(
12
12
QmudmudE
PowerE
ttW
gHm
pWp
gHmmdmemdme
AA
lossme
lossmetp
AA
abj 6
A Fluid Stream
A fluid stream here refers to
A stream of flowing fluid (we may choose a CV that excludes any solid part).
Its side wall can be
real solid boundary, e.g.,
a stream of fluid flowing between two sections (1 and 2) of a pipe, or
imaginary wall, e.g.,
a stream of fluid flowing in a stream tube.
1
2Pump Turbinea b c d
0:
1
/1 A
sf AdVm
Mass flux:
0:
2
/2 A
sf AdVm
Mass flux:
1 2
No mass flow through side wall
0:
1
/1 A
sf AdVm
Mass flux:
0:
2
/2 A
sf AdVm
Mass flux:
A stream tube as a fluid stream
(stream surface as an imaginary side wall) A section of a piping system as a fluid stream
abj 7
Example: Energy Transfer To/From A Fluid Stream Choices of Control Volumes for A Fluid Stream, 1) Forces and FBD, and 2) Energy Transfer as Work
of Forces
Example 2: CV includes the fluid stream only, no solid part.
1m
2m
1
2Pump Turbinea
1(pump)b
2(pump)c
1(turbine)d
2(turbine)
1 2
CV2 / MV2
fW
fW Impeller Work
Energy transfer as work at the rotating impeller surface
Example 1: 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.1m
2m
1
2Pump Turbinea
1(pump)b
2(pump)c
1(turbine)d
2(turbine)
sW Shaft Work
Energy transfer as work at the rotating shaft cross section
1 2
CV1 / MV1sW
abj 8
Pump: A Cascade of Energy Transfers as Work Through 1) Rotating Shaft and 2) Rotating Impeller, and Finally To 3) the Fluid Stream Turbine: The Direction is Reversed
Incompressible flow machines
Pump
• Add (mechanical) energy to the fluid
stream
1 2
CV1 / MV1
CV2 / MV2fW
sW
ME1 ME2
ME2 > ME1
Turbine
• Extract (mechanical) energy from the
fluid stream.
1 2
CV1 / MV1
CV2 / MV2fW
sW
ME1ME2
ME2 < ME1
abj 9
Recall Relation Between 1) Forces and FBD and 2) Energy Transfer as Work of (All) Forces
FBDand
Energy Transfer as
Work of All Forces
Coincident MV(t) and CV(t)
CV(t)MV(t)
Pressure pShear
Solid part
V
V
0
V
1. Moving solid surface
2. Stationary solid surface
3. Stationary imaginary surface
)( dVgdmg
1. C-Energy: In the application of C-Energy, we must find energy
transfer as work of all forces ( ) that act on our
coincident MV/CV.
2. FBD: We should then recognize/draw a FBD for all the
forces that act on our coincident MV/CV.
W
abj 10
Example 1: FBD and Work Piping System: CV includes the fluid stream only, no solid part.
1m
2m
1
2Pump Turbinea
1(pump)b
2(pump)c
1(turbine)d
2(turbine)
Stationary solid surface (no-slip)
• Pressure and shear
No work from both pressure and shear
0
V
0since0, VWWp
V
Moving solid surface (no-slip)
• Pressure and shear
There are works from both pressure and shear
0, WWp
Inlet and Exit (stationary imaginary surface)
• Pressure
Work of pressure is included as flow work pv.
• Shear
If velocity is to the CS – hence to shear, no work of shear.
pv
V
workflowWp
VifW
VifW
0
0
Note: This shear work is often neglected in comparison to the flow work.
abj 11
Example 2: FBD and Work Stream Tube: CV includes the fluid stream only, no solid part.
12
1m
2m
Streamlines
Stationary imaginary surface (streamlines)
• Pressure
Pressure is velocity, no work of pressure
• Shear
There is work of shear.
0
VCS is streamline
VpWp
since0
0W
Inlet and Exit (stationary imaginary surface)
• Pressure
Work of pressure is included as flow work pv.
• Shear
If velocity is to the CS – hence to shear, no work of shear.
pv
V
workflowWp
VifW
VifW
0
0
Note: This shear work is often neglected in comparison to the flow work.
abj 12
Convection Transport and N-Flux
abj 13
• The mass flux/flowrate carries with it (its own property) the flux/flowrate of N
through a cross section A.
• This results in a convection transport of physical quantities (mass, momentum,
energy, etc.) from one place to another.
Convection (Transport) and N-Flux
Time
NAdV
A
sf ,/
N flux:
Mass flux:
Time
MassAdVm
A
sf
/:
m
1 2
A
abj 14
1
2Pump Turbinea b c d
1 2
Time
NAdV
A
sf ,
1
/
0:
1
/1 A
sf AdVm
Mass flux:
N-flux:
Time
NAdV
A
sf ,
2
/
0:
2
/2 A
sf AdVm
Mass flux:
N-flux:
In this topic, we shall focus on
• the energy of the fluid stream, N = Energy,
• the convection of energy (from one place to another) of the fluid stream, and
• the addition of energy to (pump), and the extraction of energy from (turbine), the fluid stream.
abj 15
Convection Flux of Total Energy= Convection Flux of Thermal Energy
+ Convection Flux of Mechanical Energy
abj 16
Convection Flux of Energy = Convection Flux of Thermal Energy (TE) + Convection Flux of Mechanical Energy (ME)
1 2
W
11
1
1
)(:
METE
mdmeuEA
22
2
2
)(:
METE
mdmeuEA
Pump
fW sW
For an incompressible flow machines (hence, for turbine, it is often called hydraulic turbine):
• Pump adds mechanical energy (me) to the fluid stream to increase the me of the stream
via energy transfer as work through shaft and impeller
• Turbine extracts mechanical energy (me) from the fluid stream to convert it to useful shaft work
via energy transfer as work through impeller and finally shaft
12 MEME
12 MEME fWsW
gzVpme
mdmeMEA
2
2
1v:
,)(:
Mechanical Energy Flux
through a cross section A
gzVpmeTime
Energy
ME
mdme
TE
mduAdVmeu
A
A
A
AA
sf
2
throughenergymechanicalofConvection
throughenergythermalofConvection
/ 2
1v,,)()(
It is then convenient to decompose
Convection flux of total energy (E)
= Convection flux of thermal energy (TE) + Convection flux of mechanical energy (ME)
abj 17
Conservation of (Mechanical) Energy
and Mechanical Energy Loss (ME Loss)
for A Steady, Incompressible, Fluid Stream
(with no applied heat transfer)
abj 18
Recall: C-Energy
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 19
Conservation of (Mechanical) Energyfor A Steady, Incompressible, Fluid Stream (with no applied heat transfer)
Net Convection Efflux Term 12
)()()()( /
AACS
sf mdmeumdmeuAdVmeu
Net Energy Transfer as Heat Q No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss.
aQ
Net (ME) Energy Transfer as Work fW Work of all surface forces (excluding the flow work, pv, at inlet and exit), e.g., work of shear at CS, works of pressure and shear at pump and/or turbine rotating impeller surface.
1
2Pump Turbinea b c d
1 2
CV: CV includes the fluid stream only, no solid part.
Assumptions:
1. Incompressible flow ( is steady and uniform)
2. All flow properties are steady (or steady-in-mean)
3. No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss.
aQ
Q+
W
mean.)-in-steadyor steady are propertiesflow(All0)(
CV
dVedt
d Unsteady Term
C-Energy (A))()()( /
CS
sf
CV
AdVmeudVedt
dWQ
gzVpme 2
2
1v
abj 20
fW Work of all surface forces (excluding the flow work, pv, at inlet and exit),
e.g., work of shear at CS, works of pressure and shear at pump and/or turbine rotating impeller surface.
C-Energy
1212
12
)()(
)()(
)()()( /
AAAA
AA
f
CS
sf
CV
mudmudmdmemdme
mdmeumdmeuWQ
AdVmeudVedt
dWQ
Separate thermal energy from mechanical energy and group corresponding terms together:
1 2
111 : METEE 222 METEE
1
2Pump Turbinea b c d
111 : METEE
222 METEE
QmudmudWmdmemdme
AA
f
AA
1212
)()(
TE2ME2 ME1 TE1
Q+
W
abj 21
The rise in mechanical energy of the fluid stream – from 1 to 2 – as measured by the difference in
ME (me flux)
is equal to
+ (mechanical) energy added to the fluid stream as work (of surface forces)
- the thermal energy term
12
)()(12
AA
mdmemdmeMEME
fW
QmudmudAA
12
1 2
111 : METEE 222 METEE
1
2Pump Turbinea b c d
111 : METEE
222 METEE
QmudmudWmdmemdme
AA
f
AA
1212
)()(
TE2ME2 ME1 TE1
Q+
W
abj 22
Mechanical Energy Loss: Irreversible Conversion of Mechanical Energy to Thermal Energy According to the Second Law of Thermodynamics
1 2
111 : METEE 222 METEE
1
2Pump Turbinea b c d
111 : METEE
222 METEE
lossmeE
It can be proven from the second law of thermodynamics (see appendix) that, this thermal energy
term – which we shall denote it by
represents the time rate of irreversible conversion of mechanical energy to thermal energy according
to the second law of themodynamics, or mechanical energy loss (ME Loss).
012
:,
QmudmudE
AA
lossme
lossmeE
abj 23
C-Energy for A Steady, Incompressible Fluid Stream (with no applied heat transfer)
Assumptions:
1. Incompressible flow ( is steady and uniform)
2. All flow properties are steady (or steady-in-mean)
3. No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss.
aQ
1 2
111 : METEE 222 METEE
1
2Pump Turbinea b c d
111 : METEE
222 METEE
fW Work of all surface forces (excluding the flow work, pv, at inlet and exit),
Q+
W
0::
][,)()(
12
12
QmudmudE
PowerEWmdmemdme
AA
lossme
lossmef
AA
ME2 ME1
CV includes the fluid stream only, no solid part.
abj 24
1] Pump: Application of C-Energy for A Steady, Incompressible Fluid Stream
1 2
CV1 / MV1
CV2 / MV2fW
sW
ME1 ME2
ME2 > ME1
Wge
lossme
Wge
f
Wge
AA
EWmdmemdme
)20(.,.
0
)100(.,.)80(.,.
12
)()(
Here, we use CV2/MV2
12
)()(,,
AA
pfps mdmemdmeWW = Mechanical energy that the fluid stream actually receives from inlet 1 to exit 2.
ME loss at bearings, etc.
ME loss due to friction in the fluid stream itself in the pump (between 1 and 2).
• Recall the shear stress/Newton’s viscosity law in Chapter 2.
• Rubbing your hands together and they get warmer; friction converts mechanical energy (kinetic energy) to thermal energy.
• Impeller transfers (mechanical) energy as work at its surface to the fluid stream (+100 W).
• ME loss – ME being irreversibly converted to TE - in the stream as it flows from 1 to 2 (+20 W). [The loss occurs throughout the whole volume in CV.]
• The stream emerges at exit 2 with only (+80 W) more mechanical power than at inlet 1.
abj 25
2] Turbine: Application of C-Energy for A Steady, Incompressible Fluid Stream
1 2
CV1 / MV1
CV2 / MV2fW
sW
ME1 ME2
ME2 < ME1
Wge
lossme
Wge
f
Wge
AA
EWmdmemdme
)20(.,.
0
)80(.,.)100(.,.
12
)()(
Here, we use CV2/MV2
12
)()(,,
AA
tfts mdmemdmeWW = Mechanical energy that the fluid stream actually gives up from inlet 1 to exit 2.
ME loss at bearings, etc.
ME loss due to friction in the fluid stream itself in the turbine (between 1 and 2).
• Recall the shear stress/Newton’s viscosity law in Chapter 2.
• Rubbing your hands together and they get warmer; friction converts mechanical energy (kinetic energy) to thermal energy.
• The fluid stream gives up its mechanical power as it flows from 1 to 2 (-100 W).
• ME loss – ME being irreversibly converted to TE - in the stream as it flows from 1 to 2 by (+20 W) [The loss occurs throughout the whole volume in CV.]
• The impeller actually receives the transfer of energy as work at its surface from the fluid stream only (- 80 W).
abj 26
3] Pipe Section: Application of C-Energy for A Steady, Incompressible
Fluid Stream
Wge
lossme
Wge
AA
Emdmemdme
)20(.,.
0
)10080(.,.
12
)()(
• Due to ME loss – ME being irreversibly converted
to TE - in the stream as the fluid stream flows from 1
to 2, the stream emerges at 2 with (+20 W) less
mechanical power than when it enters at 1.
ME2 < ME1
ME1
ME2
0fW
abj 27
Hydraulic Power:MV and CV Interpretations
abj 28
Hydraulic Power: MV and CV
Interpretations Since we are interested in the change in mechanical energy of the fluid stream as it
flows through CV, from CV inlet 1 to CV exit 2, we define
A
mdmeME )(:
= ME flux through cross section A1 2
ME1 ME2
Hydraulic Power : = the time rate of change of mechanical energy of the
fluid stream [ENERGY STORED] as it flows
through CV [from CV inlet 1 to CV exit 2].
Hydraulic Power [MV Viewpoint]
Hydraulic Power [CV Viewpoint]
12
)()(:PowerHydraulic 12
AA
mdmemdmeMEME gzVpme 2
2
1v:
Recall the coincident MV(t) and CV(t)
The two views are equivalent, see next slide.
abj 29
MV and CV Aspect of Hydraulic Power: Another Application of RTT
The coincident MV has the time rate of change of its mechanical energy
= the net convection efflux of me
= ME2 – ME1
=: Hydraulic Power1 2
ME1 ME2
coincident MV
mgzmVpVMechENgzV
pme 22
2
1,
2v
RTT:
PowerHydraulic:)()()(
steady are properties flow all0)(
:
)()(,))(()()(
)()()(
12
)()(
/
)(
/
AA
MV
CV
tV
V
tCS
sfCVMV
tCS
sfCVMV
mdmemdmedt
tMechEd
dt
tMechEd
dVmetMechEAdVmedt
tMechEd
dt
tMechEd
AdVdt
tdN
dt
tdN
MV View CV View
abj 30
Recall The Three Regions I – II – III in the derivation of RTT
PowerHydraulic:)()()(
12
AA
MV mdmemdmedt
tMechEd
The original MV that instantaneously coincides with the CV has the instantaneous time rate of
increase of mechanical energy
=
coincident MV(t) and CV(t) at time t CV(t) at time t+dt
Original MV at some later time t+dt
MV (t+dt)
Part of the original MV flows out to Region III
New – but not yet of interest - MV flows into Region I
II
IIII
abj 31
Some Aspects of Hydraulic Power
Hydraulic Power
• is the property of the fluid stream, not that of the solid shaft.
• must be evaluated from the properties of the fluid stream (i.e., pressure of fluid p, specific
volume of fluid v, velocity of fluid V, etc.) - not that of the solid shaft.
1 2
ME1 ME2
gzVpme 2
2
1v:
12
)()(:PowerHydraulic 12
AA
mdmemdmeMEME
= the time rate of change of mechanical energy of the fluid stream
[ENERGY STORED] as it flows through CV [from CV inlet 1 to CV exit 2].
NOTE
• The term hydraulic “power” can be little misleading.
• It may cause confusion that this is the time rate of work of force (ENERGY TRANSFER). It is not.
• It is the time rate of change of mechanical energy of the fluid stream (ENERGY STORED).
abj 32
Hydraulic Power VS Impeller Power VS Mechanical Power at Shaft
Shaft Work
[Mechanical] Energy transfer
as work (between one part of
the solid shaft to another part
of the solid shaft) at the solid
cross section of a shaft.
TWs
1 2
sWsWShaft Power
Impeller work
[Mechanical] Energy transfer
as work (between the moving
solid impeller and the fluid
stream) at the moving solid
impeller surface. (Solid-Fluid
Interaction]
fW
1 2fW
fWImpeller Power
Hydraulic Power
The actual amount of mechanical energy me that the fluid
stream receives/gives up from inlet 1 to exit 2.
Hydraulic Power
12
)()(:PowerHydraulic 12
AA
mdmemdmeMEME
gzVpme 2
2
1v: Properties of fluid stream, not
those of solid shaft, e.g., pressure p of fluid, velocity V of fluid, etc.
1 2 12 MEME
abj 33
Hydraulic Headas Potential Height Equivalence for Hydraulic Power
abj 34
Hydraulic Power and Hydraulic Head H as the Potential Power Equivalence
HPotential head/height equivalence
(at the same mass flowrate ), Hm
m
m
pump1
)(1
A
mdmeME 2
)(2
A
mdmeME
For ease of grasping the physical sense of the magnitude of the hydraulic power, we
equate the hydraulic power to the amount of power that is required to lift the fluid at the same mass
flowrate to the height H (hydraulic head), i.e., m
12:PowerHydraulic MEME gHmeEquivalencPowerPotential=
Hydraulic Head
where gHmAA
mdmemdmeMEME :12
)()(PowerHydraulic 12
LengthH:
abj 35
In order to determine the hydraulic power,
the hydraulic head H must be stated together with the mass flowrate ( )
since, according to the definitions
Stating only the head H is not enough.
m
gHmmdmemdmeMEMEAA
:)()(PowerHydraulic
12
12
12: MEME Hydraulic Power and Hydraulic Head H
m
Hydraulic Head H at m
m
abj 36
Overall Efficiency of Hydraulic (Incompressible Flow) Machines
Pump and (Hydraulic) Turbine
Note: Not Gas or Steam Turbine where the flow inside is compressible.
abj 37
Overall Efficiency of Pump and (Hydraulic) Turbine
PUMPOverall Pump Efficiency
Mechanical power that the fluid
stream receives from inlet 1 to exit 2
Mechanical power input at shaft
T
gHm
W
MEME p
ps
p
,
12
:= Hydraulic Power
TurbineOverall Turbine Efficiency
Mechanical power that the fluid
stream gives up from inlet 1 to exit 2
Mechanical power output at shaft
t
ts
t
gHm
T
MEME
W
12
,
:= Hydraulic Power
abj 38
sWShaft PowerfWImpeller Power
1 2
sW
1 2fW
1 2gHm
gHmHydraulic Power
1 2
sW
1 2fW
1 2gHm
pgHm TWppfW ,
ME loss at
bearings, etc.ME loss due to friction in
the fluid stream itself in the pump (between 1 and 2).
tgHm TWttfW ,
ME loss at
bearings, etc.ME loss due to friction in
the fluid stream itself in the pump (between 1 and 2).
PUMP
Overall Pump Efficiency
p
pp
W
gHm
:
Turbine
Overall Turbine Efficiency
t
tt gHm
W
:
For simplicity in notation, later on we shall drop the ‘s’ for ‘shaft.’
abj 39
Conservation of (Mechanical) Energy
and Mechanical Energy Loss (ME Loss)
for A Steady, Incompressible, Fluid Stream
(with no applied heat transfer)
[Working Form for Piping System]
abj 40
1m
2m
1
2Pump Turbinea
1(pump)b
2(pump)c
1(turbine)d
2(turbine)
Her, we consider many CV’s and sum their equations together.
CV’s include fluid stream only, no solid part.
pipelossmetp
AA
pipelossme
AA
tfttlossmetf
AA
pipelossme
AA
pfpplossmepf
AA
pipelossme
AA
EgHmgHmmdmemdme
Emdmemdme
WgHmEWmdmemdme
Emdmemdme
WgHmEWmdmemdme
Emdmemdme
d
cd
bc
ab
a
,
,
,,,
,
,,,
,
12
2
1
)()(
)()(
)0(,)()(
)()(
)0(,)()(
)()(
CV: 1-a(pipe)
CV: a-b(pump)
CV: b-c(pipe)
CV: c-d(turbine)
CV: d-2(pipe)
+
+
+
+
LHS
= ME change from inlet to exit
CV: 1 2
abj 41
Conservation of (Mechanical) Energy for A Steady, Incompressible, Fluid Stream(with no applied heat transfer)
1m
2m
1
2Pump Turbinea
1(pump)b
2(pump)c
1(turbine)d
2(turbine)
1 2
Assumptions:
1. Incompressible flow ( is steady and uniform)
2. All flow properties are steady (or steady-in-mean)
3. No energy transfer as heat (no applied heat transfer ) except that is associated with ME Loss.
aQ
0::
][,
/,,
)()(
12
12
QmudmudE
PowerE
ttsW
gHm
psWp
gHmmdmemdme
AA
lossme
lossmetp
AA
Physical Interpretation:
The above C-Energy equation can be viewed as a
conservation of mechanical energy (ME) of a fluid stream.
The rise in mechanical energy/power of the fluid stream as it flows from 1 to 2
= ME added to the fluid stream at pump
- ME extracted from the fluid stream at turbine
- ME Loss in CV (Irreversible conversion of ME to TE)
abj 42
“Walking” (from 1 to 2) Interpretation
1m
2m
1
2Pump Turbinea
1(pump)b
2(pump)c
1(turbine)d
2(turbine)
1 2
0::
/,,
)()(
12
12
QmudmudE
E
ttsW
gHm
psWp
gHmmdmemdme
AA
lossme
lossmetp
AA
Physical Interpretation:
• The above C-Energy equation can be viewed as a
conservation of mechanical energy (ME) of a fluid stream.
• ME at 2 = ME at 1 (starting ME) + Increase in ME at pump
- Decrease in ME at turbine
- ME Loss in CV (Irreversible conversion of ME to TE, piping
sections)
abj 43
What’s coming up?
0::
/,,
)()(
12
12
QmudmudE
E
ttsW
gHm
psWp
gHmmdmemdme
AA
lossme
lossmetp
AA
Chapter 10: Turbomachine
We will learn how to find the hydraulic power and hydraulic head for idealized machines in terms of the machine’s
geometrical and kinematical parameters.
Chapter 8: Viscous Internal Flow
We will learn how to correlate ME Loss in piping components in terms of mechanical quantities (not thermal quantities).
Chapter 6: Bernoulli’s Equation
For a steady incompressible flow in a stream tube, in the absence of energy transfer as work and in the absence of friction (ME Loss), mechanical energy of the fluid stream is conserved.
For an infinitesimal stream tube, i.e., a streamline, this becomes the conservation of mechanical energy along the streamline.
abj 44
Special Case 1: No energy transfer as heat or work except heat transfer that is associated with ME Loss.
1 21
2Pump Turbinea b c d
0::
)()()()(
12
2112
QmudmudE
mdmemdmeEEmdmemdme
AA
lossme
AA
lossmelossme
AA
ME Loss is simply the difference between ME at 1 and 2.
Assumptions:
1. Incompressible flow ( is steady and uniform)
2. All flow properties are steady (or steady-in-mean)
3. No energy transfer as heat (no applied heat transfer )
except that is associated with ME Loss.
4. No energy transfer as work of surface forces,
except the flow work pv.
aQ No energy transfer as heat or work except -heat transfer that is associated with ME Loss, and - the flow work pv.
abj 45
Special Case 2: No energy transfer as heat or work. NO ME Loss,
1 21
2Pump Turbinea b c d
0lossmeE
12
2
2
1v,)()(
12
MEME
gzVpmemdmemdmeAA
In the absence of
1) energy transfer as heat or work (except the flow work pv), and
2) ME Loss,
the mechanical energy of a steady, incompressible fluid stream is conserved.
• This leads to the Bernoulli’s equation as the Conservation of Mechanical Energy Equation for such a fluid stream
In chapter 6: Incompressible,
Inviscid flow, we will derive this
Bernoulli’s equation from another
principle, the C-Mom Equation.
0lossmeE
Assumptions:
1. Incompressible flow ( is steady and uniform)
2. All flow properties are steady (or steady-in-mean)
3. No energy transfer as heat.
4. No energy transfer as work (except the flow work pv).
5. No ME Loss, . [Essentially, we assume no friction/viscous force, i.e., inviscid flow.]
No energy transfer as heat or work, except the flow work pv.
abj 46
NOTE: Conversion Between ME and TE
Two mechanisms that can cause transformation between ME and TE
ME TE1. Compressibility (Compression/Expansion work)
Reversible process, ME and TE can be converted back and forth
If the flow is incompressible, ME cannot be converted to TE via this mode, and vice versa.1. ME transfer as work of force cannot be converted to TE. ME transfer as work of force results in increase in ME of the MV only.
2. TE transfer as heat cannot be converted to ME. TE transfer as heat results in increase in TE of the MV only.
ME TE2. Friction / Viscous Dissipation
Irreversible process, convert ME to TE only
If the flow is inviscid, ME cannot be converted to TE via this mode.
abj 47
Special Case 2: C-ME Energy in Relation to the The Bernoulli’s EquationAs a result,
for a 1. steady,
2. incompressible (No reversible ME TE conversion via compressibility),
3. inviscid flow (No irreversible ME TE conversion via viscous dissipation),
• [MV view] the mechanical energy of the coincident MV in a fluid stream, or
• [CV view] the mechanical energy flux through any cross section of the coincident CV in the fluid
stream,
is conserved.
• This is true even though there is energy transfer as heat so long as the flow remains incompressible, since the
incompressibility condition suppresses TE ME via mechanism 1. There can be heat transfer, but it only results in the
increase in TE (u) in incompressible flow.
• Certainly, in order for the ME to be conserved there can be no (mechanical) energy transfer as work since that will affect the
ME of the fluid stream directly,
• Finally, in order for
• [MV view] the mechanical energy of the coincident MV in a fluid stream, or
• [CV view] the mechanical energy flux through any cross section of the coincident CV in the fluid stream,
to be conserved (in case of CV, conserved = ME flux through any cross sections remain the same), the assumptions in
Special Case 2 can be replaced by the above 3 assumptions
+ 4. no energy transfer as work.
[ 1) Drop the no energy transfer as heat assumption, and 2) replace no ME Loss by inviscid/no friction. ]
NOTE: If the fluid is a compressible fluid such as air, energy transfer as heat generally couples with the change in density, hence not an incompressible flow and the above result does not apply.
abj 48
APPENDIX (Related Proof for)ME Loss = Irreversible Conversion of ME to TE According to The Second Law of Thermodynamics
Q
+W
1
MV, CV
2
Q WA simple way to think about this:
But, if ds and q/T are to have some definite values,
their difference must equal something,
[ let that something be called Ps :
]
and that something has to be non-negative, .
0// TqdsTqds
sPTqds /
0sPIn order to consider the ME Loss, we consider the coincident MV:
1. Second law of thermodynamics:
Rewrite it in a more convenient form:
(1)
where Ps is the production in entropy and is never negative: .
2. Gibb’s equation for a simple compressible substance: (2)
3. (1) = (2):
4. For incompressible flow, dv = 0.
Thus,
With the above assumptions, this equation basically states that
the sum of the net increase in TE of an incompressible system and the net (TE) energy transfer
out as heat (-q) is never negative.
T
qds
vpdduTds
0sP
sPT
qds
sPTqpddu v
sPTqTds
vpdPTqdu s
0 sPTqdu
Recall the RTT and the relation between the coincident MV and CV.