Lecture 7
Entropy and the Second Law of
Thermodynamics
The Adiabatic Expansion of Gases
In an adiabatic process no heat is transferred, Q=0
= CP / CV is assumed to be constant during this
process
The pressure and volume of an ideal gas at any time
during an adiabatic process are related by
PV = constant
All three variables in the ideal gas law (P, V, T ) can
change during an adiabatic process. We get
PiVi
=
Pf V f
15/08/07
(Q=0)
Assume an ideal gas is in an equilibrium state and so
PV = nRT is valid. We get in differential form
PdV + VdP = nRdT
Since R= CPCV we have PdV + VdP = ndT
We also have dEint=nCVdT=QPdV= PdV (Q=0)
CP CV
or
ndT = -PdV/CV
Using both equations gives
Integrating gives
dP CP dV
+
=0
P CV V
CP lnV = const.
ln P + CV or PV = const.
The adiabatic curve in the PV diagram depends on For a monatomic gas CV=3/2R, CP=5/2R and = 5/3
For a diatomic CV=5/2R, CP=7/2R and = 7/5
For a polyatomic CV=3R, CP=4R and = 4/3
By finding one can determine the nature of the gas
PiVi = Pf V f
PiVi PiVi = 1 ln = ln1
Pf V f Pf V f Example: Cylinder of 50 cm3 of air at 27 ºC and 1 atm
is compressed very rapidly (adiabatically) to 10 cm3.
What is the final temperature?
PiVi
=
Pf V f
nRTi nRT f V =
Vf
Vi i
Vf
which gives
Vi T f = Ti Vf 1
TiVi 1 = T f V f 1
50 = 300K 10 0.4
= 571K
The air is heating up to 298ºC , it becomes very hot
No heat
in or out
The Heat Engine
A heat engine is a device that takes in energy by heat
and, operating in a cyclic process, expels a fraction of
that energy by means of work
A heat engine carries some working substance
through a cyclical process
The working substance absorbs energy by heat from a
high temperature energy reservoir (Qh)
Work is done by the engine (Weng)
Energy is expelled as heat to a lower
(colder) temperature reservoir (Qc)
Since it is a cyclical process, Eint = 0
V
Eint = 0 Qnet = Weng
The work done by the
engine equals the net
energy absorbed by the
engine
Qc can be thought of the
heat loss to the
environment, that is Qh is
not transformed 100% to
work
Qh
Qc
Thermal Efficiency of a Heat Engine
Thermal efficiency is defined as the ratio of the
net work done by the engine during one cycle to the
energy input at the higher temperature
=
Weng
Qh
Qh Qc
Qc
Tc
=
= 1
= 1
Qh
Qh
Th
We can think of the efficiency as the ratio of what
you gain to what you give
If Qc=0 we get =1, that is a 100% efficiency
In a similar way one defines the coefficient of
performance (COP)
Q
COP =
h
Weng
Second Law: Kelvin Form
It
is impossible to construct a heat engine that,
operating in a cycle, produces no other effect
than the absorption of energy from a reservoir
and the performance of an equal amount of work
Means
that Qc cannot equal 0
Some Qc must be expelled to the
environment
Means
that cannot equal 100%
Second Law: Clausius Form
It
is impossible to construct a cyclical machine
whose sole effect is to transfer energy
continuously by heat from one object to another
object at a higher temperature without the input
of energy by work
Energy
does not transfer spontaneously by heat
from a cold object to a hot object
Impossible Engines
Perfect Heat Engine
Perfect Heat Pump
The Carnot Engine
A theoretical engine developed by Sadi Carnot
A heat engine operating in an ideal, reversible
cycle (now called a Carnot cycle) between two
Sadi Carnot
reservoirs is the most efficient engine possible
This sets an upper limit on the efficiencies of all
other engines
No real heat engine operating between two energy
reservoirs can be more efficient than a Carnot
engine operating between the same two reservoirs
All real engines are less efficient than a Carnot
engine because they do not operate through a
reversible cycle
The Carnot Cycle
Eint = 0 for the entire cycle
Weng=|Qh| – |Qc|
Carnot showed that the efficiency of the engine
depends on the temperatures of the reservoirs
== 1 Tc
Th
Temperatures must be in Kelvins
All Carnot engines operating between the same two
temperatures will have the same efficiency
Efficiency is 0 if Th = Tc
Efficiency is 100% only if Tc = 0 K
Such reservoirs are not available, as the absolute
zero temperature cannot be reached
Efficiency is always less than 100%
The
efficiency increases as Tc is lowered and as Th is
raised
In most practical cases, Tc is near room temperature,
300 K
So generally Th is raised to increase efficiency
Theoretically, a Carnot-cycle heat engine can run in
reverse
This would constitute the most effective heat pump
available
This would determine the maximum possible COPs
for a given combination of hot and cold reservoirs
In heating mode:
COPh =
Weng
Th
=
Th Tc
In cooling mode:
COPc =
Qh
Qc
Weng
Tc
=
Th Tc
A good refrigerator should have a high COP
Typical values are 5 or 6
The Combustion (Gasoline) Engine
In a gasoline engine, six processes occur during
each cycle: for a given cycle, the piston moves up
and down twice
This represents a four-stroke cycle
The
processes in the cycle can be approximated by
the Otto cycle
OA B C D A O
NB: We are not on
the isotherms, this
process deviates
substantially from a
Carnot cycle
O A in the Otto cycle
During the intake stroke,
the piston moves
downward
A gaseous mixture of air
and fuel is drawn into the
cylinder
Energy enters the system
as potential energy in the
fuel
Intake
A B in the Otto cycle
In the compression stroke
The piston moves upward
The air-fuel mixture is
compressed adiabatically
The temperature increases
The work done on the gas is
positive and equal to the
negative area under the
curve
Compression
B C in the Otto cycle
Combustion occurs when
the spark plug fires
This is not one of the
strokes of the engine
It occurs very quickly
while the piston is at its
highest position
Conversion from chemical
energy of the fuel + O2 to
internal energy
Spark
C D in the Otto cycle
In the power stroke, the
gas expands adiabatically
This causes a temperature
drop
Work is done by the gas
The work is equal to the
area under the curve
Power
D A in the Otto cycle
Valve Opens: An exhaust valve opens
as the piston reaches its bottom position
The pressure drops suddenly
The volume is approximately constant
So no work is done
Energy begins to be expelled from the interior of the
cylinder (through the exhaust of the engine)
A O in the Otto cycle
In the exhaust stroke,
the piston moves upward
while the exhaust valve
remains open
Residual gases are
expelled to the
atmosphere
The volume decreases
Exhaust
Exhaust
If the air-fuel mixture is assumed to be an ideal gas, then
the efficiency of the Otto cycle is connected
approximately to an adiabatic process:
T1V1 1
= T2V2 1
= 1
1
(V1 / V2 ) 1
is the ratio of the molar specific heats
V1 / V2 is called the compression ratio
Typical values: Compression ratio = 8, = 1.4, = 56%
Efficiencies of real engines are 15% to 20%
Mainly due to friction, energy transfer by conduction,
incomplete combustion of the air-fuel mixture etc.
Diesel Engines
Operate on a cycle similar to the Otto cycle
without a spark plug
The compression ratio is much greater and so the
cylinder temperature at the end of the compression
stroke is much higher
Fuel is injected and the temperature is high
enough for the mixture to ignite without the spark
plug
Diesel engines are more efficient than gasoline
engines
The Future: H2 or CH3OH Fuel Cells
Proton exchange membrane fuel cell
The Toyota FCHV is a series of prototype hydrogen
fuel cell vehicle, presented in 2001.
A fuel cell is an electrochemical energy conversion
device. It produces electricity from external supplies
of fuel (on the anode side) and oxidant (on the
cathode side). The tank-to-wheel efficiency of a fuel
cell vehicle is about 45% at low loads and shows
average values of about 36% when a driving cycle
like the NEDC (New European Driving Cycle) is
used as test procedure. The comparable NEDC value
for a Diesel vehicle is 22%.
Methanol Fuel Cell