# Stirling engine

of 11 Something about stirling engines power efficiency

## Content

Lecture 11. Real Heat Engines and refrigerators (Ch. 4)

• • • • •

Stirling heat engine Internal combustion engine (Otto cycle) Diesel engine Steam engine (Rankine cycle) Kitchen Refrigerator

Carnot Cycle
- is not very practical (too slow), but operates at the maximum efficiency allowed by the Second Law.

P

1

absorbs heat
2

1–2 2–3 3–4 4–1

isothermal expansion (in contact with TH) isentropic expansion to TC isothermal compression (in contact with TC) isentropic compression to TH (isentropic ≡ adiabatic+quasistatic)

TH
4 3

rejects heat

TC V

Efficiency of Carnot cycle for an ideal gas: (Pr. 4.5)

emax = 1 −

TC TH

S
3 entropy contained in gas 2

On the S -T diagram, the work done is the area of the loop:

∫ dU = 0 = ∫ TdS − ∫ PdV
4 1

The heat consumed at TH (1 – 2) is the area surrounded by the broken line: T

TC

TH

QH = TH (S H − SC )

S - entropy contained in gas

Stirling heat engine
Stirling engine – a simple, practical heat engine using a gas as working substance. It’s more practical than Carnot, though its efficiency is pretty close to the Carnot maximum efficiency. The Stirling engine contains a fixed amount of gas which is transferred back and forth between a "cold" and and a "hot" end of a long cylinder. The "displacer piston" moves the gas between the two ends and the "power piston" changes the internal volume as the gas expands and contracts.

T T2 T1

2

3

1

4

V1

V2

Page 133, Pr. 4.21

Stirling heat engine
The gases used inside a Stirling engine never leave the engine. There are no exhaust valves that vent high-pressure gasses, as in a gasoline or diesel engine, and there are no explosions taking place. Because of this, Stirling engines are very quiet. The Stirling cycle uses an external heat source, which could be anything from gasoline to solar energy to the heat produced by decaying plants. Today, Stirling engines are used in some very specialized applications, like in submarines or auxiliary power generators, where quiet operation is important.
T T2 T1
2 3

1

4

V1

V2

Efficiency of Stirling Engine
In the Stirling heat engine, a working substance, which may be assumed to be an ideal monatomic gas, at initial volume V1 and temperature T1 takes in “heat” at constant volume until its temperature is T2, and then expands isothermally until its volume is V2. It gives out “heat” at constant volume until its temperature is again T1 and then returns to its initial state by isothermal contraction. Calculate the efficiency and compare with a Carnot engine operating between the same two temperatures.

T T2 T1

3 nR(T2 − T1 ) > 0 W12 = 0 2 2 3 V2 V2 ⎛V ⎞ dV = nRT2 ln⎜ 2 ⎟ > 0 Q23 = −W23 2-3 Q23 = ∫ PdV = nRT2 ∫ ⎜V ⎟ V ⎝ 1⎠ V1 V1 3 4 1 Q34 = CV (T1 − T2 ) = nR(T1 − T2 ) < 0 W34 = 0 3-4 2 V1 V1 ⎛V ⎞ dV V2 V1 Q41 = ∫ PdV = nRT1 ∫ = nRT1 ln⎜ 1 ⎟ < 0 Q41 = −W41 4-1 ⎜V ⎟ V ⎝ 2⎠ V2 V2 3 1 QH Q12 + Q23 2 (T2 − T1 ) + T2 ln (V2 / V1 ) T2 3 1 = = = = + > (T2 − T1 )ln(V2 / V1 ) e W Q23 − Q41 T2 − T1 2 ln (V2 / V1 ) emax
1-2

Q12 = CV (T2 − T1 ) =

Internal Combustion Engines (Otto cycle) - engines where the fuel is burned inside the engine cylinder as opposed to that where the fuel is burned outside the cylinder (e.g., the Stirling engine). More economical than ideal-gas engines because a small engine can generate a considerable power.
Otto cycle. Working substance – a mixture of air and vaporized gasoline. No hot reservoir – thermal energy is produced by burning fuel.
0 1 2 3 4 1 2 3 4 1 intake (fuel+air is pulled into the cylinder by the retreating piston) isentropic compression isochoric heating isentropic expansion 0 exhaust

ignition 2 exhaust 4 0 intake/exhaust

P

3

Patm

1

V2

V1 V

Otto cycle (cont.)
V2 - maximum cylinder volume V1 - minimum cylinder volume V2 - the compression ratio V1

The efficiency: (Pr. 4.18)

⎛ V2 ⎞ e =1− ⎜ ⎟ ⎝ V1 ⎠

γ −1

T1 =1− T2

γ = 1+2/f - the adiabatic exponent

For typical numbers V1/V2 ~8 , γ ~ 7/5 → e = 0.56, (in reality, e = 0.2 – 0.3) (even an “ideal” efficiency is smaller than the second law limit 1-T1/T3)
ignition 3

P

S
3

S1
exhaust 4

Q=0 QH

4

2

QC
Q=0
1

Patm

0 intake/exhaust

1

S2

2

V2

V1 V

V2

V1 V

Diesel engine
e =1−

δ Qc δ Qh

(Pr. 4.20)

1 → 2 : adiabatic, TV1γ −1 = T2V2γ −1 1
2 → 3: isobaric, δ Qh = f +2 nR (T3 − T2 ) 2

3 → 4 : adiabatic, T3V3γ −1 = T4V4γ −1

f 4 → 1: isochoric, V4 = V1 , δ Qc = nR (T4 − T1 ) 2

⎛ V2 ⎞ e =1− ⎜ ⎟ ⎝ V1 ⎠

γ −1

1

( )
V3 V2

V3 γ V2

−1

γ

−1

Steam engine (Rankine cycle)
hot reservoir, TH
P
water

Boiler
3

QH

Boiler
heat

Pump Turbine

2

processes at P = const, δ Q=dH Sadi Carnot Turbine

Pump

work

1
Water+steam

4

steam

heat

condense

V

Condenser
QC
cold reservoir, TC

1 → 2 : isothermal ( ≈ adiabatic )

NOT an ideal gas! H = U + PV , ⇒ (δ Q ) P = ( ΔH ) P

2 → 3: isobaric, δ Qh = H 3 − H 2 3 → 4 : adiabatic 4 → 1: isobaric, δ Qh = H 4 − H1

Steam engine (Rankine cycle)
hot reservoir, TH
P
water

Boiler
3

QH

Boiler
heat

Pump Turbine

2

processes at P = const, δ Q=dH Sadi Carnot Turbine

Pump

work

1
Water+steam

4

steam

heat

condense

V

Condenser
QC
cold reservoir, TC

H 4 − H1 H 4 − H1 e =1− ≈1− H3 − H2 H 3 − H1

( 4.12 )

Here H 2 ≈ H1 , water is almost incompressible.
gas liquid S3 = S4 ⇒ S3gas = x ⋅ S4 + (1 − x ) ⋅ S4 gas liquid H 4 = x ⋅ H 4 + (1 − x ) ⋅ H 4

Kitchen Refrigerator

A liquid with suitable characteristics (e.g., Freon) circulates through the system. The compressor pushes the liquid through the condenser coil at a high pressure (~ 10 atm). The liquid sprays through a throttling valve into the evaporation coil which is maintained by the compressor at a low pressure (~ 2 atm). P
liquid

condenser 3 2

processes at P = const, δ Q=dH
compressor

throttling valve
cold reservoir (fridge interior) T=50C

4
liquid+gas

1
gas

evaporator

V

COP =

QC H1 − H 4 H − H4 = = 1 QH − QC H 2 − H 3 − ( H1 − H 4 ) H 2 − H1
The enthalpies Hi can be found in tables.
liquid liquid gas H 3 = H 4 , ⇒ H 3 = x ⋅ H 4 + (1 − x ) ⋅ H 4

hot reservoir (fridge exterior) T=250C

S2 = S1 → T2 → H 2 ( T2 , P2 )

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