Agenda
• Kinetic theory of ideal gases,
• Work and heat: First Law of TD
• Equation of State
• Reversible processes (expansion/Compression)
• Carnot cycles, entropy
• Steady flow energetics
• TD of real gases
• Steam engines
• Carnot and Rankine cyclic engines
• Otto motor cycle
• Turbines, combined cycle thermal plants
ESTS 486 2019 Jane Doe
Sol
ar P
roj
2
Land lease/rental in NWNY: 1 acre @ $11,000/a
Mechanical Energy: Work and Heat
Ideal Gas: Diluted ensemble of N structure-less, independent particles with macroscopic, observable properties described by Equation of State.
→ (Sole interactions: Elastic scattering)
Enclosed in a volume V, IG exerts macroscopic (kinetic) pressure p on container walls.
→ Compression or decompression of (N=const.) changes its (kinetic-) energy content
U = capacity to perform work w. (Additional work types wel , rxns).
Heating or cooling = transfer of disorganized energy = “heat” q (motion of particles in container walls).
|0 :
( )0 :
ext
V Gas Expansionp Force Area p F h
V GasComw p
pres oV
si n
D = = D + D =
D = − D
Sign Convention: Work and heat are counted positive (w, q > 0), when they increasethe internal energy U of the gas, and negative when done or emitted by the gas.
Dh
Vacuum Pump
1. Law of Thermodynamics(Conservation of Energy in isolated system):
...dU dq dw dq p dV= + = − +
w p h= − D
=
,
,
extensive variables V Np V N R T
intensive variables p T
real gases
“inexact” differentials
F
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty4
Random (IG) Velocity/Kinetic-Energy Spectra
( ) ( )
( )
− −= =
→→ = =
3 3/2
2
24
22
3
BTk
B
B
Maxwell
Boltz
dP dP uue
d m d u k T
Mean thermal energy U Nm
u
ann
k
m
T
( ) ( )
( )
= −
→ = =
3 2 2
2
0
2( ): 4 exp
2
8
B B
B
dP u m umIG particles mass m u
du k T k T
dP u kMean thermal speed du u
du
Tu
m
Plausibility of EOS (Gas Law) :N =# of particles colliding with a wall <u> → momentum transfer to wall per particle <u>→ pressure on wall p <u>2 T (see EOS)
dP(u
)/du
(a.u
.)dP(
)/d
(a
.u.)
Bp V N k T =
Bound Lattice low T Lattice→Fluid/Real Gas
Unbound Gas, T=300K
Lattice
Real Gas
Ideal Gas i
(t)/
kB
i (t)/kB (i=1,..,1000)
Velocity u (km/s)
Kinetic Energy (10-20 J)
Transition
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty5
Heat Transfer
Heat (Q) conduction, flux=current density through area A
( )
= − = − + +
' :
( )
q
T T TFourier s Law j T r i j k
x y z
Thermal conductivity W mK
Heat convection: Heat transfer via mass flow
( )= − −
2
'
( )
ambient
dQNewton s Law of cooling A T T
dt
Heat transfer coefficient W mh K
h
Heat radiation: Heat transfer via elm. photons (like light)
( )
−
−
= −
=
− =
4 4
8 2 4
( 1)
5.6703 10
SB
SB
Q ambient
Stefan Boltzmann Law
Radiated thermal flux j T T
Emissivity often
Stefan Boltzmann constant W m K
( ) /qj dQ dt A=
Th Tl
Thermometers
6
The Ideal-Gas Equation of State
{p, V, T}
A
State “functions:” {p, V, T} (N=const.). Molar {p, V, T} hyper-plane (monotonic) contains all possible gas states A. Set of “independent coordinates.” There are no other states of the gas. → All state functions can be expressed as {p, V, T}.
Ideal Gas Constant R
R = 0.0821 liter·atm/mol·K
R = 8.3145 J/mol·K
R = 8.2057 m3·atm/mol·K
R = 62.3637 L·Torr/mol·K or
L·mmHg/mol·
Boltzmann Constant kB
kB= 1.381·10-23 J/K
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
p(V,T)= n R T/V
p·V = n·R·T; n=# moles; equivalent: p·V = N·kB·T (N=# particles)
Interacting only via elastic scattering, no bonding →→ Only gas phase!
,rando
U Cm
m iT
nC N
ot o
7
Transitions Between States
AB
State functions p, V, T,… describe the system states but not the processes connecting states. Two states A, B can be reached by different processes representing different pathways on the {p,V,T} hyperplane. The two processes and Energy transfers A→B differ by relative magnitudes of heat absorbed vs. work done.
Process 1 A→ B
along Path 1
Process 2 A→ B
along Path 2
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
Two different states (A, B), same gas.
1A B⎯⎯→ 2A B⎯⎯→
1 1
2 2
B A
q wU U U
q w
D + D D = − =
D + D
,U C T C N
8
Of interest for cyclic machines.“Slow,” equilibriumprocesses A → B,
subject to boundary conditions of:
1. Dp = 0 (isobaric)
2. DV = 0 (isochoric)
3. DT = 0 (isothermal)
4. q = 0 (adiabatic)
follow well-defined, constrained routes in the {p, V, T} hyper-plane of states. Can easily be inverted →reversible
processes.
Reversible Processes
T
q=0
Reversibility is not guaranteed for all processes involving an ideal gas.
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
Examples of reversible IG processes
9
w = - area under curve p(V)Total work (1 →2, T = const.) :
Reversible Isothermal Expansion/Compression
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
2 2
1 1
1
2
1
( )
ln
0 ( )
0
0
0
V V
V V
p V R T
w
w implies system loses internal energy
by doing work on surr
But
Use for expanding mole
dVp V d
q absorbs heat
compensates energy loss by he
V R TV
VR T
ound
at ab
U
s
T
o
ing
V
Us
rptio
= − = − =
=
D
=
→
→
D
=
.Needs contact to heat bath T
n
const→ Intersection of {p,V,T} hyper-plane with plane T=const.
1
0 2
ln1. 0:Law of Thermodynamics Isotherm expansV
q U w wion R TV=
=D − = − = −
higher T more
w
r
T
wo k
→
10
Compress 1 mole at p=const.
Heat transfer
Total heat transfer (1 →2)
Reversible Isobaric Compression
1
2
1
2
2 1
( ) :
( )
0: ,
50
2
( ) :
5
[ ] 0
(
.
.
0
2p
V
p
VV
p
V
system has to be coole
p VT
R
Work done on system
p V dV
p V Shaded Area
T emitting
w
Enthalpy change
emit
w p d
ted he
R
a
d
q C
C T C
V
H q
f
p
o
>0R T
r
t i
p
const
T T
const
= − =
= − D =
D
= = −
=
D =
−
D = − =
−
=
D
=
D
D
2 1
( )
[ 0
)
]
( )p
V
U q
nternal energy
C R T
T T
w H
C
−D = + = D
= −
D
Internal energy change
DV
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
Inverse process: heating at constant p, e.g., p=patm , leads to expansion, V2 → V1>V2 → drives piston out of its cylinder.
EOS
11
Isochoric (V = const.) decompression → of 1 mole w =-pDV=0
Heat transfer
Total energy change (1 →2)
Reversible Decompression
2 1
1
2 1
2
0
0,
1.
0
( ) ( )
[ ]
[
( )
( )
[ ]
]
:
V
V
V
p
V
p
Work done on system w
But U system emits heat
Law of Thermodynamics :
q w
pV R
q C T C T T heat ba
qU C T T
U C
th
since p
Enthalpy change
H T
con
C T T always H C
stB
<
U
0
T
T
H q
=
D →
= + = =
D = + D = + D
=
= D =
D
− D
−
=
D
D
D
→
−
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
0Vq C T= D
Inverse process: heating at constant V, leads to increased temperature and pressure.
12
1) Isothermal expansion at T1=const.
2) Isochoric decompression at V2=const.,
3) Isothermal compression at T2 =const.
4) Isochoric compression V1=const.,
Energy balance:
1) gas does work w1 = - q1; DU = 0
2) gas emits heat q < 0; DU < 0
3) gas receives work w2 = - q2; DU = 0
4) gas absorbs heat q > 0; DU > 0
Total energy change: DU = 0 (cyclic)
Total work done: w = w1+ w2 < 0
Total heat absorbed: q = q1+ q2=-w > 0
Expansion-Compression Cycles
In one cycle the gas absorbs net heat energy and does net work,
w = w1 + w2 = -q = CV∙[T2-T1]
Not all absorbed heat is converted, some has to be dumped as waste heat.
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
V1 V2
work dw=-p·dV
1
2
3
4T1
T2
Observation: IG systems absorbing external (random) heat can produce mechanical work
on surroundings (engine). Continuous operation requires cyclic process (in p-V-T space).
Needs contact to 2 heat baths T1 and T2.
13
Make an arbitrary cyclic process out of
elementary isothermal and isochoric
processes →
Heat energy q1 is absorbed at a high
temperature(s) T1, and partially
dumped, |q2| < |q1|, at a lower
temperature(s) T2.
The difference (q1 + q2)= q1-|q2| is
converted into useful work w < 0 done
on surroundings by the gas.
Thermal Engines: Principle of Operation
p
V
Net work done by gas
T2
T1
Random heat energy is converted into orderly collective energy (work, pushing a piston, turning a wheel) !!!!!!! → Practical use
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
q1
q2
Horizontal paths traveled in both directions do not contribute net work → Area within closed p-V paths = total work done in cyclic process.
16
Isothermal expansion at Th=T1
Adiabatic expansion Th→ Tc=T2
Isothermal compression at Tc <Th
Adiabatic compression Tc→ Th,
Carnot Cycles
1
1 2
2 12
1 2
:
" "
:
0
(
.
)
.
A BA B
Entropy S S
For any process
sign for reve
Entropy is conserved i
S state function descripto
n reversible
cycli
q
c processe
rsible
q
T T
qS
T
A B o
r
s
nly
S S
→→
D + D =
= −D = D
= →
→ =
= −
D Reversible adiabatic expansion or compression: DS = q/T= 0 since q= 0.Irreversible adiabatic exp./compr.: DS 0.
V
p
(p2, V2)
(p3, V3)
(p4, V4) work
(p1, V1)
T1
T2
q=0
q=0
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
−=
=
=1( 0)
p V
T V conA sdiabatic q Eo
c
tS
c
2
1
1
4
32
2
1
4
3
2
1
2
1 1 ln 0
ln 0
V
VRT V
V
V
V
w p dVq T
q
R
w p dV RV
T
V
V=
=− = =
=−
= =
Energy balance: w = q1 + q2 > 0on isothermal portions:
Adiabatic expansion/compression →V4/V1= V3/V2 → V4/V3= V1/V2
→ Adiabatic works cancel
17
Example: Adiabatic (q=0) expansion A’ → B’
into partial vacuum (both A’, B’ are legitimate states)
→ Free expansion A’→ B’, p → pext=0, wirrev = 0
occurs spontaneously. Actually, it is a non-equilibrium transport process eliminating p-differential, more probable configuration.
Equilibration A’→B’ → wirrev =0 .This work is smaller in magnitude than reversible
work wirrev ≤ 0, | wirrev | < | wrev | done by gas in equilibrium with environmentWhy can process A’→B not be reversed?
>>> Reversal would de-randomize thermal motion of gas @ equilibrium.A “free contraction” Vin → Vfin< Vin with fewer
occupied particle states would require a correlated motion of many particles towards a point in space.
Free expansion occurs only Vin → Vfin > Vin
Irreversible/Spontaneous Processes
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
AA
DT=0rev expansion
V
pgas
irrev
decom
pre
sio
n
A
B
A’
B’
pext
pext
pext
irrev
rev
B
A
A’Opening valve → Process.
X
18
Efficiency of Carnot Engines
Theoretical Carnot efficiency
Tc
Th
-w = qh+qc=DS·(Th- Tc)
qh= DS·Th
qc= -DS·Tc
11 1h
cC
h
h cC
h h
c
T
h
q qw
q q
q
Tq
T
→
+−= =
+ = − ⎯⎯⎯→=
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
In practice, Th depends on fuel heating value (max temperature Tad). Transfer from fuel to hot reservoir: ( ) ( )F ad h ad cT T T T = − −
1 c ad hC F
h ad c
T T T
T T T
−= = −
− → Effective Carnot efficiency:
( )
( )
, . U const
h ch c h
h
h ch
h
Process at p T const q H
T Tw T T S T S
T
T TH T
T
== ⎯⎯⎯→ = D →
−=− − D = − D
−= − D
DS = const
Heat Bath
Cold Sink
19
Entropy Flow in Carnot Engines
Entropy DS from the hot reservoir enters the engine with a heat energy of DS·Th,
produces work and leaves it again with a heat energy of DS·Tc, which is dumped into the cold sink.
Tc
Th
-w = qh+qc== DS·(Th -Tc)
qh= DS·Th
qc= -DS·Tc
Analog: Stream of water DM from a reservoir carries energy DM∙g∙h1 , enters a hydro-turbine, produces work, and leaves with an energy DM∙g∙h2 , which is dumped into the river.
Hydrodynamic Power Plant
DM∙g∙h1
DM∙g∙h2
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty
Turbine
Inlet
Outlet
Reservoir
.M
S
Mass flow j dM dt
Entropy flow j dS dt
Ideal Otto Cycle
W. Udo Schröder, 2013
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty2
0
1) Intake stroke ( 1→2 ), gasoline vapor and air drawn into engine.
2) Compression stroke (2→3) . p,T increase.
3) Combustion (spark) (3→4), short time, V= constant. Heat
absorbed from high-T “reservoir”.
4) Power stroke: expansion (4→5).
5) Valve exhaust: Valve opens, gas can escape.
6) Emission of heat (5→6) to low-T reservoir.
7) Exhaust stroke (2→1), piston evacuates cylinder.
crank shaft
cams
fly wheel
fuel intake
Energetics of Otto Cycle
W. Udo Schröder, 2013
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty2
1
( ) ( )
( )
( )
( )
4 3 6 53 4 5 6
3 4 3 4 4 3
5 6
4 3
:
1
. 1V VR c
V V
V
R c
c T T c T Tq qwEfficiency
q q c T T
T T
T T
Adiabatic EoS T V co rnst
→ →
−
→ →
− + −+= = =
−
−
= −
= −−
= → Effic
iency
Compression Ratio r
=Cp/Cv
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty2
2
Work in Steady-Flow Processes
1. Law of Thermodynamics (Conservation of total energy in isolated system):
21
2= + + potE U M V
( ) ( ) = = =
:
A Ai oi o
Mass conservation
dM dt dM dt
, ,i i i
u dVi
dQ≈0
dW<0
M= mass, = flow velocity, Vpot =potential
energy (often ≈0)Mass density = m (kg/m3), assume homogeneous = M/VInternal energy density u (J/m3)Enthalpy density H/V =: h = u + pDifferentials
( )
= =
=
=
=
=
2
;
(A dx )
dV A dx
1 2 (A dx )
(A dx )
i i i i
i i i
i i i i i
i i i i
i input o output
Internal energy dU u
with
Kinetic energy dK
Mechanical work dW p
dxi
dxo
W. Udo Schröder, 2019
Ene
rgy:
Sci
enc
e, T
ech
nolo
gy, S
ocie
ty2
3
Work in Steady-Flow Processes
1. Law of Thermodynamics (Conservation of total energy in isolated system):
21
2= + + potE U M V
= +
=
+
+ +
/
2
2
0
1(A dx ) (A dx )
2(A dx )
(A dx )1
(A dx ) (A dx )
,
2
Exp comp workFlow energyInternal
i i i i i i i i
o o o o o o o
i i i
o o o
incom
Carried by mass flow through
Additional
sys
mec
p
h
ing
dE u
outg
anical work
oing
dE
outpu
u p
t m
t dW
e
( )− = +
: in out
d E E dW dQ
dtSteady sta
h
tedt d
eat out d
t
put Q
= + − +
+ 2 21 1A A
2 2i i i i i o o o o o
dQ dWh h
dtd d
dW
t t
( ) ( ) = = =
:
A Ai oi o
Mass conservation
dM dt dM dt
+ − +
22
2 2o oi i
i o
hhdW dM
dt dt
, ,i i i
u
, ,o o o
u
Power Fuel Flow
dVi
dVo
if well insulated
dQ≈0
dW<0