Neil Greenham

ncg11@cam.ac.uk

Exergy

Exergy

The following systems all “store” about 1 kW hr of energy

• 36,000 C of charge at a potential of 100 V

• 3600 kg of water at a height of 100 m

• 1/7 litre of petrol

• 3000 m3 of air at 1°C above room temperature

Which of these would you pay the most money for?

Energy is conserved!

So, why do we have an “energy crisis”?

When we “use” energy, we convert it from a more useful form to a less useful form

How to quantify?

Exergy p, V, S, T

p0, T0

A = U + p0 V – T0 S

Often defined relative to “dead state”, where system is in equilibrium with surroundings

A = (U - U0) + p0 (V - V0) – T0 (S – S0)

Can also include (macroscopic) kinetic and potential energy terms

Where chemical species can be exchanged with environment, add chemical potential terms, µ0N

Thermodynamics: An Engineering Approach; Y. A. Cengel, M. A. Boles Fundamental of Engineering Thermodynamics; M. J. Moran, H. N. Shapiro

(also known as “availability”)

Example

dS = dQ/T = c dT/T S = c ln(T/T0)

Heat engines

W

TH

TL

QH

QL

Coefficient of performance, η = W/QH

p, V

TH

TL

QH

QL

Carnot cycle

1-2 Isothermal expansion 2-3 Adiabatic expansion 3-4 Isothermal compression 4-1 Adiabatic compression

Maximum efficiency (reversible)

H

Lrev T

T−= 1η

Heat pump

Refrigerator

W

TH

TL

QH

QL

Coefficient of performance, η = QH/W

LH

Hrev TT

T−

=η

W

TH

TL

QH

QL

Coefficient of performance, η = QL/W

LH

Lrev TT

T−

=η

Exergy

Proof:

dU = δQ + δW = δQ – p dV = δQ – (p – p0) dV – p0 dV = δQ – δWuseful,done – p0 dV

System with U, S, p, T Surroundings at p0, T0

Remove heat –δQ, and increase volume by dV

δWuseful,done = - dU – p0 dV + δQ

−δQ used to run heat engine operating between T and T0, doing work δWHE

δWHE = - (1 - T0/T) δQ = - δQ + T0/T δQ = - δQ + T0 dS For reversible change

(maximum work) Total useful work done, δWu = δWuseful,done + δWHE = - dU – p0 dV + δQ – δQ + T0 dS = - dU – p0 dV + T0 dS

Available work = = (U – U0) + p0 (V – V0) - T0 (S – S0) = exergy ∫dead

initial

udW

Exergetic efficiency

“Second law efficiency”

revex CoP

CoP=η = 1 for ideal, reversible system

E.g. for engine

usedexergydonework

ex =η

coefficient of performance

Is setting fire to fuel to make heat a good thing?

Neil Greenham

ncg11@cam.ac.uk

Heating and Cooling

Engines

Efficiency for heating

• Reduce temperature difference • Turn the thermostat down

• Reduce heat loss • Increase CoP of heat creation

Leakiness 8 kWhr / day / °C

Heat loss = leakiness × Average temperature difference

kWhr/day

kWhr / day / °C °C

Power required = heat loss / CoP

Reduce leakiness

New leakiness 6 kWh / day / °C

Old leakiness 8 kWh / day / °C

Increase coefficient of performance - use heat pumps

http://www.ecosystem-japan.com/

EcoCute water heater CoP = 4.9

Heating without fossil fuels

Heat pumps, powered by electricity

Ground-source heat pumps

Air-source heat pumps

4 times more efficient than ordinary electric heating

Ideal heat pump performance

Combined heat and power?

“Microgeneration”, “Decentralisation”

(combined heat and power) (cogeneration) Carbon Trust on Micro-CHP

"Micro-CHP is an emerging set of technologies with the potential to provide carbon savings in both commercial and domestic environments."

Efficiency of CHP

EcoCute water heater - CoP = 4.9

Can we do better than CHP? - Heat pumps

Engines

From Cengel & Boles, Thermodynamics

Engine efficiency

Air standard Otto cycle

Carnot efficiency

E.g. r = 8, 800 kJ/kg heat supplied, T1 = 290 K, k = 1.4

T3 = 1575 K, T4 = 701 K, T2 = 666 K

Actual efficiencies only ~20%

123

14 111 −−=−−

−= kth rTTTTη

r = compression ratio = V3/V4 = V2/V1

k = cp/cv

%8213

1 =−=TT

revη

%56=thη

(If heat supplied at T3 and extracted at T1)

From Cengel & Boles, Thermodynamics

Can Carnot efficiencies be achieved?

Stirling engine

1-2 Isothermal expansion 2-3 Cool at constant volume 3-4 Isothermal compression 4-1 Heat at constant volume

Transfer heat to “regenerator”

Recover heat from “regenerator”

Regenerator must be at same temperature as gas ⇒ Reversible

From Cengel & Boles, Thermodynamics

```````

`

```````

Ideal Stirling Cycle

Hot gas expands Work out Heat in Transfer gas to

cold cylinder Heat transferred to regenerator

Cold gas contracts Work in Heat out

Transfer gas to hot cylinder Heat transferred from regenerator

From Cengel & Boles, Thermodynamics

More complex than the ideal cycle!

Alpha Stirling Engine

```````

`

```````

Beta Stirling Engine

• Single piston for compression/expansion

• Move (insulated, loose fitting) displacer piston to “move” gas from hot to cold region and vice versa

Practical points

• Real efficiencies ≤ 50% • External combustion less easy to regulate • More expensive than diesel engines

The Sun

Roughly a black body

• Temperature ~5800K • Distance 1.5×1011 m • Diameter 1.4×109 m

Stefan’s law

4TbodyblackbyemittedareaunitperPower σ=

42832

45

KmW1075152 −−−×== .

hckπσ

( )1

5

2

12−

−

=

kThchcTB

λλλ exp

Making things hot

Total power radiated by sun 24 4 sunrT πσ

Intensity (W m-2) at distance s 2

24

srT sun

sunσ

Power absorbed by object, radius r 22

24 r

srT sun

sun πσ

In equilibrium, power absorbed = power radiated 242

2

24 4 rTr

srT objectsun

sun πσπσ =

24

41

=

s

rT

T sun

sun

object

For sun, at earth, data from previous slide give ~1400 W m-2

Actual average value at noon, at equator, beyond atmosphere = 1366 W m-2

s radius rsun

radius r

Tobject = 280 K for searth-sun 256 K if 30% reflecting

Making things hotter

• Heat engines run better from high-temperature sources

• But, at high temperatures • Re-radiation increases • Things melt

Solar concentrator

• To increase temperature • Use a greenhouse • Increase the range of angles from which radiation is incident

See website for derivation of optimum temperature

Stirling Energy Systems 14 W/m2

Andasol, Spain

10 W/m2

Photo: ABB

Photo: IEA SolarPACES

Cover every south-facing roof

110 W/m2

10 m2 per person Assume 50% efficient

Solar Thermal

13 kWh per day per person

Real data

3 m2

3.8 kWh/d average

13 kWh/d for 10 m2