Carnot’s Theorem
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Carnot’s Theorem. heat pump. heat engine. T h. T h. T c. T c. We introduced already the Carnot cycle with an ideal gas. Now we show:. Energy efficiency of the Carnot cycle is independent of the working substance. 1.

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T c

Carnot’s Theorem

heat pump

heat engine

Th

Th

Tc

Tc

We introduced already the Carnot cycle with an ideal gas

Now we show:

Energy efficiency of the Carnot cycle is independent of the working substance

1

Any cyclic process that absorbs heat at one temperature, and rejects heat at

one other temperature, and is reversible has the energy efficiency of a Carnot cycle

2

Remark:

reversible

Note:P>1

Textbook:

coefficient

of performance


T c

We can design the engine X such that

Let’s combine a fictitious heat engine X with

with a heat pump

realized by a reversed Carnot cycle

heat engine X

heat pump

Th

Th

X

C

Tc

Tc

with

Let’s calculate


T c

heat pump

heat engine X

with

>

Th

Th

>0

X

C

Tc

Tc

We can design the engine X such that

If X would be a Carnot engine it would produce the work

However:


T c

False

Let X be the heat pump and the Carnot cycle operate like an engine

False

Any cyclic process that absorbs heat at one temperature,

and rejects heat at one other temperature,

and is reversible has the energy efficiency of a Carnot cycle.

2

Energy efficiency of the Carnot cycle is

independent of the working substance.

1

Why

Because: X can be a Carnot engine

with arbitrary working substance


T c

heat engine X

heat pump

Th

Th

X

C

Tc

Tc

Carnot’s theorem:

No engine operating between two heat reservoirs is

more efficient than a Carnot engine.

Again we create a composite device

Proof uses similar idea as before:

We can design the engine X such that

operates the Carnot refrigerator


T c

Rudolf Clausius

(2.1.1822 -24.8.1888)

Let’s assume that

Note: this time engine X can be also work irreversible like a real engine does

>

My statement

holds man

Heat transferred from the cooler to the hotter reservoir

without doing work on the surrounding

Violation of the Clausius statement


T c

Applications of Carnot Cycles

We stated:

Any cyclic process that absorbs heat at one temperature, and rejects heat at one other temperature, and is reversible has the energy efficiency of a Carnot cycle.

- gas turbine

- Otto cycle

Why did we calculate energy efficiencies for

Because:

they are not 2-temperature devices, but accept and reject heat at a

range of temperatures

Energy efficiency not given by the Carnot formula

But:

It is interesting to compare the maximum possible efficiency of a Carnot cycle

with the efficiency of engineering cycles with the same maximum and minimum

temperatures


T c

Consider the gas turbine again

(Brayton or Joule cycle)

Efficiency

2

3

Heating the gas

(by burning the fuel)

2

3

Ph

cooling

Maximum temperature:

4

1

@

:

T3

3

adiabates

Minimum temperature:

4

:

@

T1

Pl

1

1

with


T c

Efficiency of corresponding Carnot Cycle

With

Unfortunately:

Gas turbine useless in the limit

Because:

0

Heat taken per cycle

Work done per cycle

0


T c

Absolute Temperature

A temperature scale is an absolute temperature scale if and only if

,

where

and

are the heats exchanged by a Carnot cycle

operating between reservoirs at temperatures T1 and T2.

T2

T1

We showed:

Energy efficiency of the Carnot cycle is

independent of the working substance.

Definition of temperature independent of any material property

Measurement of

Temperature ratio


T c

As discussed earlier, unique temperature scale requires fixed point

or

Tfix =Ttripel=273.16K

Kelvin-scale:

It turns out:

empirical gas temperature

proportional to thermodynamic Temperature T

Why

Because:

Calculation of efficiency of Carnot cycle based on

yields

a=1

With


T c

From definition of thermodynamic temperature

If any absolute temperature is positive all other absolute temperatures

are positive

there is an absolute zero of thermodynamic temperature

when the rejected heat

0

however

T=0 can never be reached, because this would violate the Kelvin

statement


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