Energy Conversion

CHE 450/550

Ideal Gas Basics and Heat Capacities - IIdeal gas:

– a theoretical gas composed of a set of non-interacting point particles.

– obeys the ideal gas law: PV=nRT• R is “gas constant” [R = 8.314 J·K-1·mol-1]• You may see Rspecific=R/MW [J·K-1·kg-1]

– At close to normal conditions most real gases behave like an ideal gas.

• Various relationships written. E.g., ConstantPV

T=

Ideal Gas Basics and Heat Capacities - IIHeat capacity “C” relates the change in temperature DT

that occurs when an amount of heat DQ is added

Usually given as per mass (specific heat capacity, c) [J.kg-1.K-1]

The conditions under which heat is added play a role:– At constant volume, cV=(du/dt)V

(no PV work performed during heating)

– At constant pressure cP=(dh/dt)P

(constant P, so as T increases, V increases: PV work performed)

– A thermally perfect gas can be shown to have cP=cV+Rspecific

(Sorry but it would take too long to go through the formal derivation of this)

C T Q=D D

Ideal Gas Basics and Heat Capacities - III• An important quantity is k=cP/cV

– known as the “adiabatic index” or “isentropic expansion factor” (you’ll also see it written as g gamma or k kappa)

• Polytropic processes: PVN=constant (N = polytropic index)N = 0 (PV0 = P) an isobaric process (constant pressure) N = 1 (PV = nRT) an isothermal process (constant temperature) 1 < N < k A quasi-adiabatic process (real process)N = k since k is the adiabatic index, this is an adiabatic process

(no heat transferred, all excess energy converted to PV work) N=∞ Equivalent to an isochoric process (constant volume)

Some key terms:Isobar – “at the same pressure”Isochore – “at the same volume”Isotherm – “at the same temperature”Isentropic – “at the same entropy”Adiabatic – “without heat exchange (with the surroundings)”

PV and TS diagrams

P

V

T

S

PV and TS diagrams – Isobar and Isochore

P

V

T

S

Isobar – “at the same pressure”Isochore – “at the same volume”

Where do those go on the PV and TS diagrams?

PV and TS diagrams – Isotherm, Isentropic and Adiabatic

P

V

T

S

Isotherm – “at the same temperature”Isentropic – “at the same entropy”Adiabatic – “without heat exchange (with the surroundings)”

Where do those go on the PV and TS diagrams?

Q T SD = D

TS diagram – Isobars with phase change

Steam quality (fraction of fluid that is steam) – 0 < X < 1 – At X = 0 we have all fluid in liquid phase– At X = 1 we have all fluid in gas phase (pure steam)

Rankine Cycle

Rankine Cycle: Common Improvements• Increase supply pressure, decrease exhaust

pressure• Superheat• Reheat• Feedwater Heater

– open/closed

Solar Thermal Power Plant Ausra (Bakersfield, CA, 10/2008)

Direct Steam Generation

Brayton Cycle

http://commons.wikimedia.org/wiki/File:Brayton_cycle.svg

Ideal Brayton Cycle Analysis

Open system energy balance based on enthalpy

Steady Flow Energy Balance

kQ W U E pE PV

H U PV

Q W H

1 2 2 1

2 3 3 2

3 4 3 4

4 1 4 1

W h h

Q h h

W h h

Q h h

Ideal Brayton Cycle Analysis 3 4 1 23 4 1 2

2 3 3 2th

p

th

h h h hW W

Q h h

C

3 4 1 2

p

T T T T

C

41

11 4

3 23 2 32

2

2 3 1 4

1

1 1

1

;

TTTT T

T TT T TTT

P P P P

Ideal Brayton Cycle Analysis

Efficiency is function of compression ratio!

1 1

1

2 3 1 4

3 31 1

2 2 4 4

3 4

2 1

1

122

2

1

;

;

1 11 1 1

k k

k

k kp

v

th

k

P P P P

CP TT Pk

T P P T C

T T

T T

TTT

PTP

Brayton Cycle: Common Improvements• Increase Compression Ratio

– Also increases air temperature coming out of compressor (bad) (Karlekar, 1983)

(Segal, 2003)

Actual processes are not isentropic

Turbines, Compressors, generators can be highly efficient (>80%)

Example:A compressor has an isentropic efficiency of 85%,

meaning that the actual work required is 1/0.85 times that of an isentropic process.

0.85 bs a

b a

h h

h h

Wcompressor

“a”“b”

Improving efficiency• Intercooling and Reheat

– Allows for higher compression ratios– Cool before compression, reheat during/between

expansion• Regeneration

– Heat the compressed air with turbine exhaust

Combined Cycle Power Plant