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 = PfVf

Entropy and the Second Law of Thermodynamics

Lecture 7 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

Since R= CP CV we have

We also have dEint=nCVdT=Q PdV= PdV (Q=0)

or ndT = -PdV/CV

Using both equations gives

Integrating gives or PV = const.

PdV +VdP = nRdT

PdV +VdP

CP CV= ndT

dP

P+

CPCV

dV

V= 0

lnP +CPCV

lnV = 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 = PfVf PiViPfVf

= 1 lnPiViPfVf

= ln1

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?

which gives

The air is heating up to 298ºC , it becomes very hot

No heat

in or out

Tf = TiViVf

1

= 300K 50

10

0.4

= 571K

PiVi = PfVf nRTiVi

Vi =nRTfVf

Vf

TiVi1= TfVf

1

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 is defined as the ratio of the net work done by the engine during one cycle to the energy input at the higher temperature

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)

Thermal Efficiency of a Heat Engine

=Weng

Qh=Qh QcQh

= 1QcQh

= 1TcTh

COP =QhWeng

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%

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

Second Law: Clausius Form

Perfect Heat Engine Perfect Heat Pump

Impossible Engines

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

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

Sadi Carnot

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

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%

== 1TcTh

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:

In cooling mode:

A good refrigerator should have a high COP

Typical values are 5 or 6

COPh =QhWeng

=Th

Th Tc

COPc =QcWeng

=Tc

Th Tc

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

O A 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)

Exhaust

Exhaust

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

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:

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.

= 11

V1 /V2( ) 1T1V11= T2V2

1

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

Diesel Engines

The Future: H2 or CH3OH Fuel Cells

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

Proton exchange membrane fuel cell