Lecture 7 The Classical Second Law of Thermodynamics (2)

Classical Second Law of Thermodynamics (2)

(Lecture 7)

2021

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1

^학기

열역학

(M2794.001100.002)

송한호

5.5 The Carnot Cycle

If the efficiency of all heat engines is less than 100%, what is the most efficient cycle we can have?

è Carnot cycle is the most efficient, reversible cycle that can operate between two constant-temperature reservoirs.

(Heat engine that operates on Carnot cycle) 1 : Boiler, Reversible isothermal process

(Q_H is transferred to the system)

2 : Turbine, Reversible adiabatic process (W output, Working Fluid : T_H→T_L)

3 : Condenser, Reversible isothermal process (Q_L is rejected from the system)

4 : Pump, Reversible adiabatic process (W input, Working Fluid : T_L→T_H)

Classical Second Law of Thermodynamics

: Heat engine : Refrigerator

è First Proposition

“It is impossible to construct an engine that operates between two given reservoirs and is more efficient than a reversible engine (or Carnot engine) operating between the same two reservoirs.”

rev

irr

h

h <

Classical Second Law of Thermodynamics

5.6 Two Propositions Regarding the Efficiency of a Carnot Cycle

L

L Q

Q if

<

>

'

rev, irr

h h

è Second Proposition

“All engines that operate on the Carnot cycle between two given reservoirs have the same efficiency, independent of working

substance.”

Classical Second Law of Thermodynamics

) ,

(

_{H L}

rev

= f T T

h

Reversible

engine 1 Reversible engine 2

5.7 The Thermodynamic Temperature Scale

è From the equality of the efficiencies of Carnot cycles,

Here, the efficiency of the heat engine using a Carnot cycle is given as,

è Define the thermodynamic scale of the absolute temperature as,

Then, this leads to

Classical Second Law of Thermodynamics

) ,

(

_{H L}

rev

= f T T h

(for Carnot cycle only)

è Use of constant-volume gas thermometer of an ideal gas:

l Let the gas bulb be placed in the T measurement location.

l Let the mercury column be adjusted so that the level of mercury stands at the reference mark A.

l Read the height L to figure out pressure, thus leading to temperature.

water) of

pressure point

triple NOT

K, 273.16 at

pressure gas

:

K 273.16 water

of erature point temp

triple :

(where

16

. 273

const, when

,

,

, ,

,

P t

P t

P t P

t P

t

P

T

P T P

P P T

T

v RT Pv

=

÷÷ ø ö çç

è

= æ Þ

=

=

=

Classical Second Law of Thermodynamics

5.8 The Ideal-Gas Temperature Scale

Constant-volume gas thermometer T measurement

Used as the reference point of the ideal-gas temperature scale

è To evaluate the ideal-gas temperature by using a real gas,

l Conduct a series of the same T measurement at different pressures.

l The ideal-gas temperature can be calculated by extrapolation of the curve to zero pressure.

Classical Second Law of Thermodynamics

Constant-volume gas thermometer T measurement

Gas

è Let’s demonstrate that the ideal-gas temperature scale (from ideal gas EOS) is identical to the thermodynamic temperature scale (from Carnot cycle and the second law).

Classical Second Law of Thermodynamics

Carnot cycle in P-v diagram

dT C

du v dv

Pdv RT w

v dv dT RT

C w

du q

v v

0 0

, : gas ideal

for law

st 1

÷ = ø ç ö

è

= æ

=

÷ ø ç ö

è + æ

= +

= d

d d

!

Ideal gas T

①→②: Reversible isothermal heat addition

Classical Second Law of Thermodynamics

v dv dT RT

C

q _v ÷

ø ç ö

è +æ

= ₀

d

1 2 12

2 1 2

1 0

2

1 ln

v RT v

q q

v dv dT RT

C

q _v ÷ ® _H = = _H

ø ç ö

è + æ

=

ò ò

ò

^d

②→③: Reversible adiabatic expansion

③→④: Reversible isothermal heat rejection

④→①: Reversible adiabatic compression

2 3 3

2 3 0

2 3

2 3 0

2 0 ln

v R v T dT

dv C v

dT R T

C T

q _v ÷ ® = _v +

ø ç ö è + æ

=

ò ò ò

ò

^d

3 4 34

4 3 4

3 0

4

3 ln

v RT v

q q

v dv dT RT

C

q _v ÷ ® - _L = = _L

ø ç ö

è + æ

=

ò ò

ò

^d

1 1

1

1 q C æ R ö C v

ò ò

ò

ò

^d

Classical Second Law of Thermodynamics

è Here,

è Thus,

è Finally,

2 3 0

2 3 0

2 3 3

2

0 ln ln ln

0 v

R v T dT

C v

R v T dT

C v

R v T dT

C ^H

L L

H

T T T v

T v

v + ® = - ® =

=

ò ò ò

1 4 0

4 1 0

4 1 1

4

0 ln ln ln

0 v

R v T dT

C v

R v T dT

C v

R v T dT

C ^H

L H

L

T T T v

T

v

v + ® = - ® =

=

ò ò ò

1 2 4

3 1

4 2

3 or

v v v

v v

v v

v = =

L H L

H

L H

T T v

RT v

v RT v

q q

ln ln

4 3 1 2

=

=

Ideal gas T

Same definition as the thermodynamic temperature scale

è The thermal efficiency or coefficient of performance of a real device is always lower than that of a Carnot cycle system.

Classical Second Law of Thermodynamics

5.9 Ideal versus Real Machines

(Carnot Cycle)

(Real Device)