Transcript Slide 1

Lecture 6.2: Conservation of Energy (C-Energy), and
Energy Transfer as Work of (Surface) Forces

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
LHS:
(TE + ME + Others)
•
Decomposition of Energy Transfer:
•
Energy Transfer As Work (of A Force)
Heat + Work + [Others, if any]
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
•
abj
•
+
[Modes of] Energy Transfer
•

Q
(Heat/TE + Work/ME + Others)
2. Forms of Energy Stored

W
•
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.
1
Very Brief Summary of Important Points and Equations
Q
W
+
Q  W  ......

C-Energy for A MV

Time Rate of Energy Transfer
to MV from its surroundin gs
in various modes (as heat and work, etc.)
dEMV
dt


Time Rate of Change/Increase in Energy Stored
in MV in various forms (TE and ME)
C-Energy (Working Forms) for A CV



d 
Q  W   e( dV )   (e  pv)( V f / s  dA);

dt  CV
 CS

:
e-pv - form

e  pv

V2
eu
 gz
2
V2
u  pv 
 gz,
2



me
:
:
:
abj
:
u-me - form
h - form
ho - form
V2
 gz  pv  ke  pe
2

u  me,
me : pv 

V2
h
 gz,
2
h : u  pv
ho  gz,
V2
ho : h 
= stagnation enthalpy
2

W : W shaft  W shear  W others
2
Energy as A Conserved Quantity/Scalar
• Conservation of Energy for An Isolated System
• Conservation of Energy for A MV (Closed System)
abj
3
“According to Classical Mechanics”
Let’s say, the universe – that we are a part of - is an isolated system.
Conservation of Mass
 According to classical mechanics, there are
249 689 127 954 677 702 907 942 097 982 129 076 250 067 682 009 482 730 602 701 620 707 616 740 576 190 705
687 196 070 561 076 076 104 051 876 549 701 707 617 048 651 671 076 017 057 901 710 461 765 379 480 547 610
707 617 019 641 127
kg of mass in the universe.
Also
Conservation of Energy
 According to classical mechanics, there is a total of
580 140 804 219 884 603 733 864 586 354 599 887 940 543 537 431 687 943 187 603 734 360 687 465 465 075 940
408 562 545 546 454 651 326 406 306 302 135 543 067 654 987 651 861 684 616 846 516 516 576 516 546 165 131
986 543 074 921 975 970 297 249 027 290 579 540 410 434 573 805 706 076
J of energy in the universe.
 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.]
abj
 According to classical mechanics, energy – like mass – is a conserved scalar/quantity.
4
Energy as A Conserved Quantity/Scalar
Conservation of Energy for
An Isolated System
Universe (Isolated System)
EU = Constant (Conserved)
abj

dEU = 0
5
Relation Between Changes of Various Parts
U = MV + Surroundings
Universe (Isolated System)
EU = Constant (Conserved)
EU = EMV + ESur

An Isolated System
Surroundings, ESur
dEU = 0
EMV
= Constant
MV (Closed System)
Total Amount
- dESur
Universe (Isolated System)
EU = Constant (Conserved)

- dESur =
abj
changes
dEU = 0
dEMV
dEU = dEMV + dESur = 0
Relation between
various parts
Surroundings
dEMV
of
Change/Increase in Energy Stored
The amount of energy transferred to a
system must come from its surroundings.
6
Energy as A Conserved Quantity/Scalar
Conservation of Energy for
A MV (Closed System)
Focus on a MV (closed system) as a part of the Universe
Universe (Isolated System)
EU = Constant (Conserved)
- dESur

dEU = 0
Energy Transfer
to MV from its surroundings in
various modes
dEMV
dEU = dEMV + dESur = 0
Change/Increase in Energy Stored
in MV in various forms
Surroundings
dESur

Let’s denote the LHS instead by ET.(=
- dESur)
E
T
Energy Transfer
to MV from its surroundin gs
in various modes
abj

dEMV
dEMV




Change/Increase in Energy Stored
in MV in various forms
7
Conservation of Energy for A MV (Closed System)
1.
Modes of Energy Transfer
1.
2.
3.
2.
Q, Q , Thermal Energy Transfer)
 , Mechanical Energy Transfer)
Energy Transfer As Work (W , W
Other Modes of Energy Transfer ( E )
T
Energy Transfer As Heat (
Forms of Energy Stored
1. Thermal Energy (TE)
2. Mechanical Energy (ME)
3. Other Forms of Energy Stored
abj
8
Conservation of Energy for
Modes of Energy Transfer
Energy Transfer
to MV from its surroundings in
A MV (Closed System)
and
Forms of Energy Stored
ET
various modes
dEMV
Change/Increase in Energy
Surroundings
Stored
in MV in various forms
MV (Closed System)
E
T

Energy Transfer
to MV from its surroundin gs
in various modes
dEMV




Change/Increase in Energy Stored
in MV in various forms
Modes of Energy Transfer
Forms of Energy Stored
 Energy Transfer as Heat Q = Thermal energy transfer
 Thermal energy
 Energy Transfer as Work W = Mechanical energy transfer
 Mechanical energy ME
 Other modes of energy transfer ET (e.g., electromagnetic radiation,
 Other forms of energy stored ( e.g.,
etc.)
electrical, chemical, etc.)
KEY:
abj
TE
(= U)
(= KE)
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)
9
Conservation of Energy for
In most of our problems of interest, only
are excited/changed
A MV (Closed System)
1) Thermal Energy (TE) and 2) Mechanical Energy (ME)

Energy Transfer ET  Q  W  ET ,others
Key:
If some other forms of energy are
also excited/changed, they must
be
taken
into
accounted
according to the conservation of
energy.

to MV from its surroundings in
dEMV  d (TE  ME) MV
various modes
Change/Increase in Energy
Surroundings
Stored
in MV in various forms
MV (Closed System)
E
T

Energy Transfer
to MV from its surroundin gs
in various modes
Q
W
+
dEMV




Change/Increase in Energy Stored
in MV in various forms
Q
 W
 ET 

dEMV
Energy
Q
 W
 ET 

dEMV
dt
 Energy
 Time 


ET = Q + W + [ET ]
Q
= Heat
W = Work
abj
EMV  TE + ME [+ Other forms]
= Thermal energy transfer
TE
= Thermal energy
= Mechanical energy transfer
ME
= Mechanical energy
ET = Other modes of energy transfer
…... = Other forms of energy stored
10
C-Energy for A MV (Closed System)
Time Rate of Energy Transfer

Q  W  Wothers (if any)

to MV from its surroundings in
dEMV
dt
various modes
W
Q
Time Rate of Change/Increase in Energy
+
Surroundings
Stored
in MV in various forms
MV (Closed System)
Q  W  .....


Time Rate of Energy Transfer
to MV from its surroundin gs
in various modes (as heat and work, etc.)
abj

dEMV
dt


Time Rate of Change/Increase in Energy Stored
in MV in various forms (TE and ME)
11
Work
Body
mgof mg
and Potential Energy [1]
Scratch of
Note:
ProofForce
of Work
z
 
W mg  mg  V

ˆ
 m( gk )  V
 mgVz
dz
dt
d (mgz)

dt
 mg
d ( PE) MV
W mg  
,
dt
abj

g   g kˆ
y
x

mg

V
PE : mgz
12
The Two Forms of C-Energy for A MV (Closed System)
(according to where we put the work of mg / potential energy)
Q  W
Form 1

dEMV
,
dt
:
1
E  TE  ME  
U  KE  U  mV 2

2
TE
Form 2
ME
:
W must includework of mg,
:
1

E  TE  ME  
U  ( KE  PE)  U   mV 2  mgz

 

2

TE
:
abj
ME
W must not includework of mg,
Q  W
Q  W  
W  ...  W mg  ...

W mg

d ( PE) MV
Q  W  
dt

Q  W 

d (U  KE ) MV
dt
d (U  KE ) MV
,
dt
d (U  KE ) MV
,
dt
d (U  KE  PE) MV
dt
W  ...  W mg  ...
W  W   W mg
d ( PE) MV
W mg  
dt
13
Sign Conventions for The Energy Equation
Energy input into a system causes increase in energy of the system.
Energy extracted from a system causes decrease in energy of the system.
W
Q
+
W
Q
+
dE
Equation: Q  W  MV
dt
dE
Equation: Q  W  MV
dt
Equation: positiveQ

Q - input
Physics:
causes positivedEMV / dt



Equation: positiveW

Physics:
W - input
causes positivedEMV / dt



Equation: positiveQ

Q - input
Physics:
causes positivedEMV / dt



Similar can be said for
abj
Q
+
causes
causes
E-increase
E-increase
E-increase
Equation: positiveW causes negativedEMV / dt



W - output
Physics:
W
causes
W
causes
E-decrease
Q
+
14
C-Energy for A MV (Closed System)
Q  W  ......


Time Rate of Change/Increase in Energy Stored
in MV in various forms (TE and ME)
Time Rate of Energy Transfer
to MV from its surroundin gs
in various modes (as heat and work)
LHS:
abj
dEMV
dt


[Modes of] Energy Transfer
1. Energy Transfer as Heat
Q
[Thermal Energy Transfer]
2. Energy Transfer as Work
W
[Mechanical Energy Transfer]
15
Modes of Energy Transfer on The LHS
LHS = Energy Transfer to MV
dEMV (t )
dt



Q

W 




Time rate of energy transfer
to MV (t ) as heat and work
Time rate of change
of energy of MV (t )
Thermal Energy Transfer
Mechanical Energy Transfer
 )
(as Heat Q
 )
(as Work of Forces W
Like
Recall in C-Mom
Keys


F
1.
Recognize various types of forces.
2.
Know how to find the resultant of
various types of forces (e.g.,
pressure, etc.).
3.
abj
 Energy
 Time 


Energy Transfer in Other Modes

F
W others
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.
Keys: Energy Transfer to MV
Q W  ...
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.
Sum all the external forces.
16
W
Energy Transfer Modes
(between a system and its surroundings)
Heat 

Q
+
Work
( Q  q  dA )
If any other
Work of Forces
 
Other Modes
W F  V  dF
of Energy Transfer
(input-positive)
Work of Surface Force/Stress
Work of Body Force/mg
 
  
W S  V  dFS  V  T dA


 
W B  V  dFB  V  BdV
 

Stress vector T  Tnormal  Ttangential
Normal (Pressure)

Tnormal   peˆn
Tangential (Shear)


Ttangential  Tshear
W others
Through a finite surface S :
 
Q    q  dA
(input-positive) S
abj


 
W p    pv V  dA
S
(input-positive)
W shear  



 
Tshear V dA
S
(input-positive)
Work of mg is
later accounted
for as potential
energy
(input-positive)
e.g. electrical,
electromagnetic, etc.
17
If there are other body forces besides mg, all
must be accounted for.
Energy Transfer As Work of A Force W F
[Mechanical Energy Transfer]
abj
18
W
: Energy Transfer as Work (Mechanical Energy Transfer)
Q  W



 .....
Time rate of energy transfer
to MV (t ) as heat and work
Pressure p
Coincident CV(t) and MV(t)
CV(t)
MV(t)
Shear t

Fi
FBD
Volume/Body Force


gdm  g ( dV )
 Work is the mode of (mechanical) energy transfer.
 Work is work of a force,
W F
 In order to apply C-Energy,

W
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
abj
Forces in Fluids
and
FBD
19
Recall 1: Recall all and various types of forces.
W
must be the sum of the works of all the forces on MV(t).

F


and Free-Body Diagram (FBD) for the Coincident CV(t) and MV(t)
Net external force
Coincident CV(t) and MV(t)
CV(t)
MV(t)
Pressure p
2. Distributive Surface Force
Shear t
(in fluid part)
1.
Concentrated/Point Surface Force

Fi

Net Surface Force

FBD

FS
1. Concentrated/Pointed Surface Force
Volume/Body Force


gdm  g ( dV )

F 


Fi
2. Distributive Surface Force in Fluid [Pressure p + Friction t ]
abj

FS 


FB
Net Volume/Body Force

mg 


FB

 g (dV )
CV  MV
20
Recall 2: Energy Transfer as Work of A Force (Mechanical Energy Transfer)

Work of A Force F ( WF , W F )

F
Concept

V
 


WF  F  V
 
dS  Vdt
Work
=
Force
x
 Energy
 Time 


Displacement in the direction of the force
(per unit time)
Particle
WF
 
 F  dS
 


WF  F  V
abj
Energy
 Energy
 Time 


21
Energy Transfer as Work of A Force (Mechanical Energy Transfer)
Particle VS Continuum Body

Work of A Force F , WF , W F

F

V
 


WF  V  dF
 Energy
 Time 


Particle
 
WF  F  dS
 


WF  F  V
 
W FB  V  dFB  V  BdV

  
   V  dF  V  T dA
abj
W
S
FS

V

dFB
 Energy
 Time 


Continuum Body
Energy
Same concept, just that
 Energy
 Time 



 
W F   (V  B)dV
B
V


V
Work = Force x Displacement in the
direction of the force
 


WF  F  V


dFS
Same Concept
 
dS  Vdt


dA

  
W F   V  T dA
S
S
1)
there are more types of forces to be accounted
for: Surface force and Body force (and…)
2)
Each type is described differently

 
dFS  T dA ,


dFB  BdV ,
 Force
T
Area
 Force
B
Volume
3) As before, how to sum them all.
22

Work of All Forces W
Coincident CV(t) and MV(t)
CV(t)
MV(t)
Pressure p
2. Distributive Surface Force
Shear t
(in fluid part)
1.
W 

Note
abj

Fi
Concentrated/Point Surface Force
W S ( Surface force)

W p ( pressure )  W s ( shear )
W s (shaft )
=
FBD
Volume/Body Force


gdm  g ( dV )

W B ( Body

W mg
 Energy
 Time 


force)
 W
concentrated surface force
 W others (if any)
Shaft work is work due to shear stress (surface force) at the cross section
of a shaft.
23

Work of Surface Forces:
1) Pressure Force (Flow Work), 2) Shear Force

dA
Recall the coincident CV(t) and MV(t)



dFS  dFpressure  dFshear
CV(t)
W
Q

V
MV(t)
Surroundings
+
S



dFS  dFpressure  dFshear


 ( pdA)  dFshear
Work of pressure force on CS/MS:
Work of shear force on CS/MS:
•
•
Infinitesimal work of pressure force:



 

 
W p  V  dF p  V   pdA   pV  dA
 
  pv(V  dA)



Infinitesimal work of shear stress:


W shear  V  dFshear 
input into MV  positive
input into MV  positive
1.
Rate of work (power) done on a finite closed
surface S:

 
 
W p   pv V  dA
abj
1.
Rate of work (power) done on a finite closed
surface S:

W p  W p
S

 
V  Tshear dA

 

W shear  W shear

S
input into MV  positive
S
W shear 


 
V  Tshear dA
S 



input into MV  positive
24
Finite Control Volume Formulation of Physical Laws
C-Energy
abj
26
Finite CV Formulation of Physical Laws: C- Energy
Recall the coincident CV(t) and MV(t)
W
Q
Q
Surroundings
dEMV/dt
+
W
Material Volume (MV)
Energy transfer as heat
Energy transfer as work of forces
p, t
CV(t), MV(t)
C-Energy: N  E ,   e
Physical Laws
Q  W

Time rate of energy transfer
to MV (t ) as heat and work

dEMV (t )
dt



Time rate of change
of energy of MV (t )

dECV (t )
dt



Time rate of change
of energy of CV (t )



e( V f / s  dA),

CS (t )
dm  dQ





 Energy
 Time 


Net convectionefflux
of energy through CS (t )
RTT
abj
27
Finite CV Formulation of Physical Laws: C- Energy
~
d
E
MV (t )
Q  W 
,
dt
~
E  U  KE ,
1
e~  u  V 2 ,
2
 W
 W
W : W p  W shear
:
shaft
 Energy
 Time ,


mg
 W others
~
dE MV (t ) 


Q W 
 Wmg ,
W : W p  W shear  W shaft
 W others
dt
d (U  KE ) MV (t ) d ( PE) MV (t )
d ( PE) MV (t )


,
W mg  
dt
dt
dt
dEMV (t )
1
Q  W 
,
N  E  U  KE  PE,
  e  u  V 2  gz
dt
2
:
W : W p  W shear  W shaft  W others
dEMV (t )
Q  W 
 W p ,
W :
dt
Apply RTT to dEMV/dt


 

 dECV (t )

 e( V f / s  dA) 

dt
CS


dm  dQ


W shear  W shaft  W others ,

CS


pv( V f / s  dA),

dm  dQ
W p  

CS


pv( V f / s  dA)

dm  dQ


dECV (t )
 (e  pv)( V f / s  dA)

dt
CS

dm  dQ


dE (t )
Q  W  CV
 (e  pv)( V f / s  dA)

dt
CS

dm  dQ




d 
 e( dV )  (e  pv)( V f / s  dA)

dt CV
 CS


dm  dQ
abj
:
 at various steps.
To save some symbols, here we redefine W
W : W shaft  W shear  W others
28
C-Energy (Working Forms)
Recall the coincident CV(t) and MV(t)
Q
Surroundings
W
dEMV/dt
Q
W
Material Volume (MV)
+
Energy transfer as heat
Energy transfer as work of forces
p, t
CV(t), MV(t)



d 


Q W 
e( dV )   (e  pv)( V f / s  dA);

dt  CV
 CS

:
e-pv - form

e  pv

V2
eu
 gz
2
V2
u  pv 
 gz,
2



me
:
:
:
abj
:
u-me - form
h - form
ho - form
V2
 gz  pv  ke  pe
2

u  me,
me : pv 

V2
h
 gz,
2
h : u  pv
ho  gz,
V2
ho : h 
= stagnation enthalpy
2

W : W shaft  W shear  W others
29
Basics and Various Cases of
Energy Transfer
as Work of (Surface) Forces
[Surface Force = Normal/Pressure Force + Shear Force]
abj
30
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.,



d 

  (e  pv)( V  dA),

Q W 
e
(

dV
)
f
/
s

 
dt  CV
 CS
:
W  W shaft  W shear  W others
1
e : u  V 2  gz
2

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.



d
Q  W    e( dV )    (e)( V f / s  dA),

dt  CV
 CS
abj
1
e : u  V 2  gz
2
31
Energy Transfer as Work of (Surface) Forces
[Surface Force = Normal/Pressure Force + Shear Force]
3. Stationary Imaginary surface
(where there is mass flow in/out.)
Pressure p
Shear t

V
Solid part
 
V 0
1. Moving solid surface
(e.g., pump impeller surface,
cross section of a rotating
solid shaft)
abj

V
2. Stationary solid surface
(e.g., pump casing)
32
Energy Transfer as Work of (Surface) Forces
[Surface Force = Normal/Pressure Force + Shear Force]
3. Stationary Imaginary surface
Work due to pressure force here is later
moved to the RHS and included as flow
work, pv, in the convection flux term:
(where there is mass flow in/out.)
In general,
W p,t
 
 V  dFp,t ,
0
Pressure p
Shear t
 
V 0


(
e

p
v)
(

V

d
A
)
f /s

CS
 
(except dF  V )

V
Note: For moving imaginary
surface, we may use the
decomposition
 
  
W p,t  V  dFp,t ,
V  Vs  V f / s
 

 (Vs  V f / s )  dFp,t
Solid part
 
V 0

V
2. Stationary solid surface
(e.g., pump casing)




W p ,t  V  dF p ,t , V  0 (no  slip)
0
1. Moving solid surface
(e.g., pump impeller surface, cross section of
a rotating solid shaft)
In general,


W p,t  V  dFp,t ,
abj
0

V  0 (no  slip)
 
(except dF  V )
33
 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
34
Various Control Volumes for A Fluid Stream,
Forces and FBD, and Energy Transfer as Work of Forces
m 1
1
Turbine
Pump
a
1(pump)
b
c
d
2(pump) 1(turbine) 2(turbine)
• CV includes the fluid stream only, no solid part.
1
2
2
m 2
• 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.
Surface Force:
Pressure and shear
on moving/rotating impeller
surface
1
2
Surface Force:
Normal and shear stress
over the moving/rotating cross section of a solid
shaft
MV
MV
Surface Force
Pressure and shear
abj
Surface Force
Pressure and shear
FBD
• Surface force: pressure/normal and shear stresses, over all surfaces. [Body force is not shown.]
35
• 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).

 dF Surroundings
  
V  r
Surroundings
  
V  r
Q  W F
MV

V
 W F
Pressure and shear stresses on the rotating
impeller surfaces act on the moving fluid 
Energy transfer as work to MV (fluid stream)
abj

dEMV
,
dt
 
 V  dF
 Energy
 Time 




dF  t dA eˆ

dF
MV

V
MV
36
W f
Energy transfer as work of forces at the surface of
the moving/rotating impeller
Surroundings
Surroundings
MV
 W F
 
 V  dF
[Pump]
• Pressure force pushes fluid,

V
W f
Energy transfer as work of force
at the rotating impeller surface
abj
MV
• Shear force drags fluid,
such that the fluid at the material
surface has velocity V .
37

Energy transfer as work of forces at the cross section of a solid shaft W
s
W s
Energy transfer as work of force at the
rotating cross section of a solid shaft.
 W F
 
 V  dF



V
W F  V  dF
MV

dF

 
 (  r )  dF
Surroundings
MV

T


  
 
Vector triple product identity: (  r )  dF    (r  dF )

 
   ( r  dF )

 
 
dT : r  dF
   dT ,


  dT

W


dF  dF  (dAzt z )eˆ
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).

shaft
shaft cross sec tion




dT
shaft cross sec tion
 
W shaft  T  

 
dF , T = External force and torque due to surroundings on our MV
abj
(Recall the concept of FBD and Newton’s Second Law)
38



dF  dF

V
MV
Surroundings
W shaft


T   T


  
Motor
Motor/Turbine
drives its
Pump/Load



V

dF

T
abj
Direction of mechanical
energy transfer as work
[Motor, Turbine]
Turbine
 0
 


Wshaft  (T   ) MV 

 0
Pump
Load
Surroundings
MV
 
 (T  ) MV  0
MV gives up its own mechanical
energy to the surroundings.
[Pump, Load]
MV receives mechanical energy
from the surroundings.
W shaft
 
 (T  ) MV  0
39
Various Control Volumes for A Fluid Stream,
Forces and FBD, and Energy Transfer as Work of Forces
W s
CV2 / MV2
CV1 / MV1
W s
W f
CV1 / MV1
CV1 / MV1 [See
W s
, but do not see
W f
.]
• [FBD] sees the shear stress at the rotating shaft cross
section,
1
W f
CV2 / MV2
• [Work] sees the energy transfer as work at the rotating
shaft cross section.
2
 , but do not see W .]
CV2 / MV2 [See W
f
s
• [FBD] sees the pressure and shear stresses on the
rotating impeller surface.
• [Work] sees the energy transfer as work at the rotating
impeller surface.
abj
40