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.