CONTROL VOLUMES
I am teaching Engineering Thermodynamics using the textbook by Cengel and Boles. This set
of slides overlap with Chapters 5 and 7.
Some figures in the slides are taken from that book, and some others are found online.
Similar figures can be found in many places.
I went through these slides in two lectures, each 90 minutes.
Zhigang Suo
The play of thermodynamics
ENTROPY
energy
space
matter
charge
• Fix space.
• Let energy and matter flow.
2
An open system exchanges energy, space
and matter with the rest of the world
• Open system: the content inside the piston-cylinder device.
• When the wall is not thermally insulated, the fire transfers energy to the system by
heat.
• When the piston moves, the system exchanges space with the rest of the world,
and the weights transfer energy to the system by work.
• When the valve opens, the system exchanges matter with the rest of the world.
weights
open system
gas
valve
liquid
fire
3
Control volume
We can choose any volume to be a control volume
4
Plan
•
•
•
•
•
•
Conservation of mass
Conservation of energy
Generation of entropy
Steady-flow devices
Isentropic efficiency of steady-flow devices
Reversible work of steady-flow devices
5
Isolated system
When confused, isolate.
Isolated
system
IS
Isolated system conserves mass over time:
dmIS
dt
=0
6
Control volume
We are accountants.
min
CV
CV
mout
mCV
æ change of mass in CV ö æ mass entering CV ö æ mass leaving CV ö
ç
÷=ç
÷-ç
÷
Dt
Dt
Dt
è
ø è
ø è
ø
dmCV
dt
=
åm - å m
in
out
7
Draw free-body diagram!
Draw control-volume diagram!
inlet (faucet)
CV (tub)
outlet (sink)
It’s complicated to construct an isolated system.
What is the boundary of an isolated system?
8
Plan
•
•
•
•
•
•
Conservation of mass
Conservation of energy
Generation of entropy
Steady-flow devices
Isentropic efficiency of steady-flow devices
Reversible work of steady-flow devices
9
Isolated system
When confused, isolate.
Isolated
system
IS
Isolated system conserves energy over time:
dEIS
dt
=0
10
Control volume
We are accountants.
Ein
CV
CV
Eout
ECV
æ change of energy in CV ö æ energy entering CV ö æ energy leaving CV ö
ç
÷=ç
÷-ç
÷
Dt
Dt
Dt
è
ø è
ø è
ø
dECV
dt
=
åE - å E
in
out
11
Flow work
Work required to push matter into a control volume of a fixed boundary
Work done by the external force:
Pressure and volume:
Substitution:
Flow work per unit mass:
W flow = FL
P = F / A, V = AL
W flow = PV
w flow = W flow / m = Pv
12
Transfer energy by matter
flowing into a control volume of a fixed boundary
(
E flow = m Pv + u + ke+pe
)
P,v,u
CV
Enthalpy
h º u + Pv
æ
ö
2
V
E flow = m ç h +
+ gz ÷
ç
÷
2
è
ø
P,v,u
CV
13
Methods to transfer energy
between a control volume and the rest of the world
Be an honest accountant. Do not double count.
CV
P,v,u
CV
transfer energy by work
dECV
dt
transfer energy by heat
transfer energy by matter
é
é
æ
öù
æ
öù
2
2
V
V
= êW + Q + m ç h +
+ gz ÷ú - êW + Q + m ç h +
+ gz ÷ú
ç
÷ú
ç
÷ú
ê
ê
2
2
è
øû out ë
è
øû
in ë
å
å
14
Plan
•
•
•
•
•
•
Conservation of mass
Conservation of energy
Generation of entropy
Steady-flow devices
Isentropic efficiency of steady-flow devices
Reversible work of steady-flow devices
15
Entropy is additive
entropy = log (number of quantum states)
Each subsystem is in a state of equilibrium,
but the subsystems may not in equilibrium with each other:
Entropy is additive:
S1 ,S2 ,S3 ,...
SIS = S1 + S2 + S3 +...
16
Isolated system
When confused, isolate. Recall the fundamental postulate.
Isolated
system
IS
Isolated system increases entropy over time:
Define entropy generation:
Define more words:
dSIS
dt
dSIS
dt
³0
= Sgen , Sgen ³ 0
ì > 0, irreversible process
dSIS ïï
í =0, reversible process
dt ï
ïî <0, impossible process
17
Internal and external reversibility
18
Transfer entropy by heat
We are accountants.
Isolated system
weights
weights
water
water
Q
reservoir of energy, TR
fire
Reservoir of energy has a fixed temperature: TR
Reservoir transfer energy to the water by heat: Q
Reservoir reduces entropy (Clausius-Gibbs equation): DSreservoir = -
Q
TR
Isolated system increases entropy over time: DSwater + DSreservoir ³ 0, DSwater + DSreservoir = Sgen
Q
Q
. DSwater =
+ Sgen
Clausius inequality. Define entropy generated by the water: DSwater ³
TR
TR
Define entropy transferred into the water by heat:
Sin =
Q
, DSwater = Sin + Sgen
TR
19
Transfer entropy by work
We are accountants.
Isolated system
weight
Weight has a fixed entropy:
Isolated system increases entropy over time:
Combine the above two statements:
Define entropy generated by the water:
The work (weight) does not transfer entropy into the water:
DSweight = 0
DSwater + DSweight ³ 0
DSwater ³ 0
DSwater = Sgen
Sin = 0, DSwater = Sin + Sgen
20
Control volume
We are accountants.
Externally reversible
Sin
CV
CV
Sout
SCV , Sgen
æ change of entropy in CV ö æ entropy entering CV ö æ entropy leaving CV ö æ entropy generated in CV ö
ç
÷=ç
÷-ç
÷+ç
÷
Dt
Dt
Dt
Dt
è
ø è
ø è
ø è
ø
dSCV
dt
=
å S - å S + Sgen
in
out
21
Methods to transfer entropy
between a control volume and the rest of the world
Transfer entropy by mass
Transfer entropy by heat
Smass = ms
Sheat =
dSCV
dt
å
Q
Tb
Transfer entropy by work
Swork = 0
æ
æ
Qö
Qö
= çç ms + ÷÷ - çç ms + ÷÷ + Sgen
Tb ø
Tb ø
in è
out è
å
å
22
Plan
•
•
•
•
•
•
Conservation of mass
Conservation of energy
Generation of entropy
Steady-flow devices
Isentropic efficiency of steady-flow devices
Reversible work of steady-flow devices
23
Steady flow
dmCV
dt
= 0,
dECV
dt
= 0,
dSCV
dt
=0
Hot water in
CV
Warm water out
Cold water in
Conservation of mass:
åm = å m
in
Conservation of energy:
Generation of entropy:
out
é
é
æ
öù
æ
öù
2
2
V
V
êW + Q + m ç h +
+ gz ÷ú = êW + Q + m ç h +
+ gz ÷ú
ç
÷ú
ç
÷ú
ê
ê
2
2
è
øû out ë
è
øû
in ë
å
å
æ
æ
æ
æ
Qö
Qö
Qö
Qö
çç ms + ÷÷ ³ çç ms + ÷÷, Sgen = çç ms + ÷÷ - çç ms + ÷÷
Tb ø
Tb ø
Tb ø
Tb ø
out è
in è
out è
in è
å
å
å
å
24
Look up for h1 , s1
Turbine
converts flow to rotation
P = 2 MPa
T
Look up for h2, s2
state 1
P = 15 kPa
state 2
s
Conservation of energy:
Generation of entropy:
æ
ö
æ
ö
2
2
V
V
m ç h1 + 1 + gz1 ÷ = Wout + m ç h2 + 2 + gz2 ÷
ç
÷
ç
÷
2
2
è
ø
è
ø
s2 ³ s1
25
Compressor
uses external work to compress fliud
Look up for h1, s1
P = 600 Pa
state 2
T
Look up for h2, s2
P = 100 kPa
state 1
s
Conservation of energy: Win +mh1 = Qout +mh2
Generation of entropy:
Q
ms2 + out ³ ms1
Tb
26
Nozzle and diffuser
A nozzle increases the velocity of a fluid at
the expense of pressure.
A diffuser increases the pressure of a fluid
by slowing it down.
Conservation of energy:
Generation of entropy:
h1 +
V12
2
s2 ³ s1
= h2 +
V22
2
27
Throttling valve
restricts flow and causes pressure to drop, often accompanied by a
drop in temperature
State 1
T1, P1
State 2
T2, P2
Conservation of energy:
h2 = h1
Generation of entropy:
s2 ³ s1
28
Mix hot and cold waters in a shower
Hot water in
m1
CV
mCV
Warm water out
m3
Cold water in
m2
Conservation of mass:
m1 + m2 = m3
Conservation of energy:
m1h1 + m2h2 = m3h3
Generation of entropy:
m3s3 ³ m1 s1 + m2s2
29
Heat exchanger
allows two fluids exchange energy by heat without mixing
state 3
state 2
state 1
state 4
Conservation of energy:
Generation of entropy:
m Ah1 + mBh3 = m Ah2 + mBh4
m A s2 + mB s4 ³ mA s1 + mB s3
30
Conduction
Qin
Qout
Tin
Conservation of energy:
Generation of entropy:
CV
Tout
Qin = Qout
Sgen =
Qout
Tout
-
Qin
Tin
31
Plan
•
•
•
•
•
•
Conservation of mass
Conservation of energy
Generation of entropy
Steady-flow devices
Isentropic efficiency of steady-flow devices
Reversible work of steady-flow devices
32
Isentropic process
Conservation of energy:
Isentropic process:
mh1 = mh2 +Wout
s1 = s2
State 1
Given: P1 = 5 MPa, T1 = 450 C
Look up: h1 = 3317.2 kJ/kg, s1 = 6.8 kJ/kg K
State 2
Given: P2 = 1.4 MPa, s2 = s1
Look up: h2 = 2967.4 kJ/kg
33
Isentropic efficiency of a turbine
For given inset and outlet pressures
hturbine =
h1 - h2a
h1 - h2s
s2 = s1
34
Isentropic efficiency of a compressor
For given inset and outlet pressures
hcompressor =
h2s - h1
h2a - h1
35
Isentropic efficiency of a nozzle
For given inset and outlet pressures
hnozzle =
h1 - h2a
h1 - h2s
36
Plan
•
•
•
•
•
•
Conservation of mass
Conservation of energy
Generation of entropy
Steady-flow devices
Isentropic efficiency of steady-flow devices
Reversible work of steady-flow devices
37
Reversible work required to move a piston
Work done by the external force to the fluid:
dW = -Fdz
Change in volume:
dV = Adz
Pressure in the fluid
(quasi-equilibrium process, reversible process):
P=F/A
Express work done by external force in terms of
thermodynamic properties of the fluid:
dW = -PdV
state
reversible
process
weights
state
closed
system
gas
liquid
F
fire
z
38
Flow work
Work required to push matter into a control volume of a fixed boundary
Work done by the external force: W
flow = FL
Pressure and volume:
Substitution:
Express work done by external force in terms of
thermodynamic properties of the fluid:
P = F / A, V = AL
W flow = PV
W flow / m = Pv
39
Reversible steady-flow work (shaft work)
Work done by a steady-flow device to fluid, also known as the shaft work
dwshft
(
)
m h + ke+pe
ms
(
) (
m ( s + ds)
CV
dqin
) (
)
méë h + dh + ke + dke + pe + dpe ùû
dqin
T
Conservation of energy: dqin + dwshaft = dh + dke + dpe
Reversible process does not generate entropy: ds =
dqin
T
Gibbs equation dh = Tds + vdP
Reversible steady-flow work
Express the shaft work in terms of thermal dwshaft = vdP + dke + dpe
dynamic properties of the fluid:
Neglect kinetic energy and potential energy: dw
shaft = vdP
40
Reversible work done by external force on fluid
Piston work
External force pushes a piston
Shaft work
External force rotates a shaft in a steady flow
dw piston = -Pdv
dwshaft = vdP
41
Compress a substance
In liquid phase (incompressible)
In gas phase (compressible)
42
Shaft work to compress an ideal gas
Shaft work:
wshaft =
P2
ò vdP
P1
Isothermal process: Pv = NkBT = constant
k
Isentropic process: Pv = constant
k = c p / cv > 1
43
Bernoulli equation
Incompressible fluid, no work
P1 ,V1 ,z1
Reversible steady-flow work:
No steady-flow device:
Incompressible fluid:
Integration between two ends:
P2 ,V2 ,z2
CV
dwin = vdP + dke + dpe
dwin = 0
v = constant
(
1
V2 -V1 + g z2 - z1 = 0
2
) (
v P2 - P1 +
) (
)
44
Summary
• An isolated system conserves mass, conserves energy,
but generates entropy.
• Translate the above statement by labeling part of an
isolated system as a control volume.
• Steady-flow devices
• Isentropic efficiency of steady-flow devices
• Reversible work of steady-flow devices (shaft work)
45