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					2 2.1

Gas turbines Applications of gas turbines transport industries

• gas turbines are widely used in the manufacturing, power generation and • aircraft engines:
1. jet propulsion for civil and military aircraft 2. shaft power for turboprop and helicopter engines 3. lightweight: GT gives ~0.25 kg/kW cf. IC engine ~0.6kg/kW

• high speed ships:
1. hovercraft, patrol boats, frigates 2. propeller or water jet

• power generation:
1. load topping because of rapid startup ability 2. high efficiency when used in conjunction with a steam turbine in a ‘combined cycle plant’, which are at present the most efficient form of large scale power generation available

• road & rail: some locomotives, possible future trucks and passenger vehicles
although at present it appears unlikely 2.2 Layout of gas turbines

2.2.1 Aircraft propulsion

• cut-away view of the Rolls-Royce RB211-524, used on all Qantas 747’s:

• in cross section, this looks like:

• note the high pressure (HP), intermediate pressure (IP) and low pressure (LP)
‘spools’. A spool is defined as one compressor, turbine and shaft set. The LP turbine drives the fan, the IP turbine drives the IP compressor and the HP turbine drives the HP compressor. The three drive shafts are coaxial: the LP shaft sits within the IP shaft, which sits within the HP shaft and each spool rotates at a different speed.

• as the cutaway view shows, the compressor and turbine feature annular rows of
airfoils. For example, a radial cross section of a typical low pressure turbine blade looks like:

• of course, numerous other types of gas turbine exist. Those discussed above are
referred to as ‘axial’ machines since the massflow through them is roughly parallel to the axis of rotation. ‘Radial’ gas turbines are also very common, especially in smaller applications such as microturbines, helicopter engines and turbochargers.

2.2.2 Power generation

• modern gas turbines used for power generation often look very similar to those
used for propulsion. The most significant difference is that the low pressure turbine does not drive the fan (which provides thrust via by bypass stream) but, instead, provides shaft work to the generator. 2.3 1. 2. 3. 4. Pros/cons of gas turbines small size and mass for a given power output ie. high W/m3 or W/kg low vibration: parts rotating and not reciprocating (cf. IC engines) reliable: low dynamic stresses from near constant ω operation low cost per kW at larger sizes

• positives

• negatives
1. thermal efficiency is usually lower than diesel engines • the thermal efficiency is directly related to the maximum temperature in the cycle, which is limited by material considerations. Considerable research effort is therefore spent on increasing material limits (high temperature alloys, ceramics, blade cooling, etc) 2. relatively poor dynamic performance – less suitable for cars and other ‘stopstart’ applications 2.4 Common types of gas turbine

2.4.1 Basic open cycle gas turbine

• where C denotes the compressor, CC the combustion chamber and T the turbine
2.4.2 Open cycle gas turbine with separate power turbine

• T1 drives the compressor only, T2 drives the output shaft

2.4.3 Open cycle gas turbine with heat exchanger

2.4.4 Open cycle gas turbine with ‘reheat’

• gas turbines typically operate at very high air/fuel ratios (very low equivalence
ratios). More fuel can be burnt without exceeding the turbine inlet temperature limits by using reheat. ‘Afterburning’ in a jet engine is a similar principle. 2.4.5 Open cycle, compounded gas turbine

• each compressor/turbine pair are coaxial and referred to as ‘spools’. The spools
speeds can be different and relative spool speeds may be changed for improved off design performance • two spools very common in aircraft propulsion (see earlier).

2.4.6 Closed cycle gas turbine

• combustion is external to the fluid within the cycle • the working fluid is typically not air: freon, benzene, butane, etc. • can use low grade fuel, since the fuel does not pass through the compressor and
turbine 2.4.7 Jet propulsion

• most simple configuration is the turbojet (eg. Concorde):

• instead of the turbine exhaust driving a power turbine, the high velocity exhaust
provide thrust to the aircraft • the earlier images were of high bypass ratio, turbofan engines, which are more complex

2.5

Analysis of simple cycles

2.5.1 Ideal, constant pressure gas turbine: ‘Joule/Brayton’ cycle

• assumptions:
1. 2. 3. 4. compression and expansion (turbine) are isentropic (hence primed numbers) no pressure losses constant specific heat C P air is the working fluid

2.5.1.1 Thermal efficiency

• the thermal efficiency ηth is defined as the work output wOUT achieved for a given
heat input qIN :

ηth =

wOUT ( wt − wc ) = qIN qIN

• where wc and wt are the compressor and turbine specific work respectively. • since the compression and expansion are both isentropic, they are by definition
adiabatic (isentropic means adiabatic and reversable). If follows from the SFEE that: wc = ∆ht c = C P (Tt 2' − Tt1 )

• and

wt = ∆ht t = C P (Tt 3 − Tt 4' )

• no work is done on the fluid during heat addition. Therefore, from the SFEE:
q = ∆ht CC = C P ( Tt 3 − Tt 2 ' )

• and:

⎡ CP (Tt 3 − Tt 4' ) − CP (Tt 2' − Tt1 ) ⎤ ⎦ ηth = ⎣ CP (Tt 3 − Tt 2' )

• define the stagnation pressure ratio as rP = Pt 2 / Pt1 . It follows that:

Tt 2' T = rP γ = t 3 Tt1 Tt 4 '

γ −1

• and:
1−γ 1−γ ⎡ ⎛ ⎞ ⎛ γ ⎢Tt 3 ⎜1 − rP ⎟ − Tt 2' ⎜1 − rP γ ⎟ ⎜ ⎢ ⎜ ⎠ ⎝ ⎣ ⎝ ηth = (Tt 3 − Tt 2' )

⎞⎤ ⎟⎥ ⎟⎥ ⎠⎦

• collecting like terms, this becomes:
1−γ

ηth = 1 − rP γ

• so, the thermal efficiency of the ideal gas turbine depends only on the pressure
ratio rP and the ratio of specific heats γ . We will see later that the thermal efficiency of the non-ideal cycle is dependent on other parameters as well.

2.5.1.2 Specific power output

• the specific power output ( w ) is the power output per unit mass flow rate through
the gas turbine ( Ws / kg )

• from the SFEE:

w = wt − wc ⎛ T ⎞ ⎛T ⎞ = CPTt 3 ⎜1 − t 4' ⎟ −CPTt1 ⎜ t 2' − 1⎟ ⎝ Tt 3 ⎠ ⎝ Tt1 ⎠
1−γ ⎛ = CPTt 3 ⎜1 − rP γ ⎜ ⎝

⎞ ⎛ γ γ−1 ⎞ ⎟ −CPTt1 ⎜ rP − 1⎟ ⎟ ⎜ ⎟ ⎠ ⎝ ⎠
γ −1

• where, as before:
Tt 2' T = rP γ = t 3 Tt1 Tt 4 '

• let:
x = rP
γ −1 γ

• the expression for the specific work then becomes:
⎛ 1⎞ w = CPTt 3 ⎜1 − ⎟ −CPTt1 ( x − 1) ⎝ x⎠

• and the maximum specific work occurs when: • i.e.
dw CPTt 3 = 2 − CPTt1 dx x

x2 =

Tt 3 Tt1

• since:
γ −1

x = rP γ =

Tt 2' Tt 3 = Tt1 Tt 4 '

• it follows that:
Tt 3 ⎛ Tt 2' ⎞ ⎛ Tt 3 ⎞ =⎜ ⎟⎜ ⎟ Tt1 ⎝ Tt1 ⎠ ⎝ Tt 4' ⎠
x x

• giving: • or:

Tt 2' = Tt 4' ⎛T ⎞ rP = ⎜ t 3 ⎟ ⎝ Tt1 ⎠
2(γ −1)

γ

• in terms of the T − s diagram, the condition that Tt 2' = Tt 4' for maximum specific
work determines the ‘shape’ of the cycle for a given pressure ratio. Remember that, for an ideal cycle, the thermal efficiency ηth depends only on rP and γ . Thus, ‘long skinny’ ideal cycles and ‘short fat’ ideal cycles on the T − s diagram will have the same thermal efficiency if their pressure ratio is the same. However, if we want to produce work compactly (i.e. in a small space and cheaply), we wish to maximise w ( Ws / kg ) by letting Tt 2' = Tt 4' . This fixes the shape of the cycle in T − s space.

2.5.1.3 How can the Joule / Brayton cycle be a heat engine?

• starting from Gibbs’ equation:
⎛ Tt b ⎞ ⎛ pt b ⎞ sb − sa = CP ln ⎜ ⎟ − R ln ⎜ ⎜ Tt a ⎟ ⎜ pt a ⎟ ⎟ ⎝ ⎠ ⎝ ⎠ • along a line of constant pressure, we obtain ⎛ Tt b ⎞ Tt b s −s / C = e( b a ) P sb − sa = CP ln ⎜ ⎟ ⇒ ⎜ Tt a ⎟ Tt a ⎝ ⎠ • thus, lines of constant pressure are exponential curves in the T − s plane. These curves become more widely spaced as the entropy increases.

• also, for the cycle diagram above
Tt 4' Tt 1 = e( s4 ' − s1 ) / CP = e( 3 ⇒ Tt 4' Tt 3 =
s − s2 ) / CP

=

Tt 3 Tt 2 '

Tt1 Tt 2'

• which can also be determined from the adiabatic pressure-temperature relations. • now, considering the turbine work:
⎛T ⎞ wt = CP (Tt 3 − Tt 4' ) = CPTt 4' ⎜ t 3 − 1⎟ ⎝ Tt 4' ⎠ ⎛T ⎞ = CPTt 4' ⎜ t 2' − 1⎟ ⎝ Tt1 ⎠ T = t 4' CP (Tt 2' − Tt1 ) Tt1
>1 wc

• ie. w = wt − wc > 0 • thus, the Joule / Brayton cycle is a heat engine purely by virtue of the divergence
of the lines of constant pressure in the T − s plane.

• an aside: for lines of constant ρ = 1/ v :
Tt b Tt a = e( b
s − sa ) / CV

• since CP − CV = R , lines of constant density are steeper in the T − s than those of
constant pressure. 2.5.2 Isentropic efficiencies

• the isentropic efficiency of the compressor and turbine relates the actual work to
the ideal work for a given stagnation pressure ratio:

compression

expansion

• note that for a compression, the actual work done on the fluid is greater than the
ideal work done. We therefore define the isentropic efficiency of the compression ηc as:

ηc ≡

C (T − T ) T − T ideal work = P t 2' t1 = t 2' t1 actual work CP (Tt 2 − Tt1 ) Tt 2 − Tt1

• for an expansion (the turbine), the ideal work done on the fluid is greater than the
actual work done. Thus, ηt is defined as:

ηt ≡

actual work CP (Tt 3 − Tt 4 ) Tt 3 − Tt 4 = = CP (Tt 3 − Tt 4' ) Tt 3 − Tt 4 ' ideal work

• do not confuse the two definitions! If in doubt, think of their T − s diagrams.
2.5.3 Non-ideal cycle

• the most simple non-ideal cycle includes the effect of compressor and turbine
inefficiency. We will still assume no pressure losses during combustion, although in reality, there are combustion pressure losses of roughly 2-3%.

• using the SFEE, the specific power output is: • and the heat addition by combustion is:

w = CP (Tt 3 − Tt 4 ) − CP (Tt 2 − Tt1 )

q = CP (Tt 3 − Tt 2 )

2.5.3.1 Thermal efficiency

• the thermal efficiency ηth is once again defined as the work output achieved for a
given heat input:

ηth =

w CP (Tt 3 − Tt 4 ) − CP (Tt 2 − Tt1 ) = q CP (Tt 3 − Tt 2 )

• including the definitions of the compressor ηc and turbine ηt isentropic
efficiencies and the stagnation pressure ratio rP : 1 ηt (Tt 3 − Tt 4 ') − (Tt 2' − Tt1 ) ηc ηth = Tt 3 − Tt 2
1−γ ⎛ ⎞ Tt1 ⎛ γ γ−1 ⎞ γ ηtTt 3 ⎜1 − rP ⎟ − ⎜ rP − 1⎟ ⎜ ⎟ ηc ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = Tt 3 − Tt 2

• let α = Tt 3 / Tt1 . ηth then becomes:
1−γ ⎛ ηtα ⎜1 − rP γ ⎜ ⎝ ηth =

⎞ 1 ⎛ γ γ−1 ⎞ ⎟ − ⎜ rP − 1⎟ ⎟ ηc ⎜ ⎟ ⎠ ⎝ ⎠ Tt 2 α− Tt1

• since:
γ −1 ⎞ Tt 2 Tt 2 − Tt1 1 Tt 2' − Tt1 1⎛ γ = +1 = + 1 = ⎜ rP − 1⎟ + 1 ⎟ ηc Tt1 ηc ⎜ Tt1 Tt1 ⎝ ⎠

• it follows that:
1−γ ⎛ ⎞ 1 ⎛ γ γ−1 ⎞ γ ηtα ⎜ 1 − rP ⎟ − ⎜ rP − 1⎟ ⎜ ⎟ ηc ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ηth = ⎛ γ γ−1 ⎞ 1 α − ⎜ rP − 1⎟ − 1 ⎟ ηc ⎜ ⎝ ⎠

• which, when rearranged, gives:

1−γ ⎛ ⎞ ηcηtα rP γ −1⎟ ⎜ ⎜ ⎟ ⎠ ηth = ⎝ ηc (α − 1) −1 ⎛ γ γ−1 ⎞ ⎜ rP − 1⎟ ⎜ ⎟ ⎝ ⎠

• note:

1. in comparing this result with that of the ideal cycle, we see that ηth is no longer independent of Tt 3 . This means that the turbine material temperature limits now limit the maximum thermal efficiency achievable. 2. the loss due to ηc < 1 is partially compensated for by reduced q , so ηc appears in both the numerator and the denominator 3. all the loss due to ηt < 1 is lost from the system and hence the term ηc appears only in the numerator.

2.5.3.2 Specific power

• the specific power output ( w ) is also defined in the same way as for the ideal
cycle i.e. it is the power output per unit mass flow rate through the gas turbine

• from the SFEE:

w = wt − wc = CP (Tt 3 − Tt 4 ) −CP (Tt 2 − Tt1 ) ⎛ T ⎞ ⎛T ⎞ = CPTt 3 ⎜ 1 − t 4 ⎟ −CPTt1 ⎜ t 2 − 1⎟ ⎝ Tt 3 ⎠ ⎝ Tt1 ⎠

• including the definitions of ηc , ηt and rP :
1−γ ⎛ w = CPTt 3ηt ⎜1 − rP γ ⎜ ⎝

γ −1 ⎞ ⎞ T ⎛ −CP t1 ⎜ rP γ − 1⎟ ⎟ ⎟ ⎟ ηc ⎜ ⎠ ⎝ ⎠

• this expression can be rearranging into non-dimensional form as:
⎡ ⎤ ⎛ γ γ−1 ⎞ ⎢ ηtα 1 ⎥ w = ⎜ rP − 1⎟ ⎢ γ −1 − ⎥ ⎟ CPTt1 ⎜ ⎝ ⎠ ⎢ r γ ηc ⎥ ⎣ P ⎦

• sketching the derived relations for ηth and w for non-ideal cycles versus rP :

• note:

1. a maximum in ηth at finite rP now exists. This maximum is at different conditions to that for maximum w . 2. raising ηc and ηt : some increase in ηth and a large increase in w 3. increasing Tt 3 : very large increase in w and some increase in ηth 4. increasing ηt has a greater effect on ηth than increasing ηc Other cycles

2.6

• it has been shown that the both ηth and w of the Joule/Brayton cycle are
functions of rP , Tt 3 , ηc and ηt : 1. rP and Tt 3 are limited by material considerations 2. ηc and ηt are already roughly 90% and are not expected to increase substantially in the future.

• further, significant performance improvements are therefore hard won and
alternative gas turbine configurations may be desirable in certain cases. 2.6.1 Reheat

• the turbine work is increased by the addition of a second combustor and second
turbine:

• the turbine work:

wt = CP ⎡(Tt 3 − Tt 4 ) + (Tt 5 − Tt 6 ) ⎤ & w56 > w4 a ⎣ ⎦

• heat addition:

q = CP ⎡(Tt 3 − Tt 2 ) + (Tt 5 − Tt 4 ) ⎤ ⎣ ⎦

• note that the specific power output is increased since w56 > w4a
2.6.2 Compressor intercooling

• two stage compression with cooling between each stage.

• and the T − s diagram looks like:

• the compressor work is now:

wC = CP ⎡(Tt 2 a − Tt1 ) + (Tt 3 − Tt 2b ) ⎤ & wC < CP (Tt x − Tt1 ) ⎣ ⎦ q = CP (Tt 4 − Tt 3 ) > CP (Tt 4 − Tt x )

• and the heat addition is:

• the effect of intercooling is to increase the specific power ouput w since wc is
reduced.

2.6.3 Regeneration

• a heat exchanger is used to transfer heat from the turbine exit to the compressor
exit:

• and the T − s diagram looks like:

• where the heat exchanger effectiveness ε is:
ε=
the turbine exit.

T2b − T2 a T4 − T2 a

Tt 2b − Tt 2 a Tt 4 − Tt 2 a

• since the Mach number is intentionally low throughout the heat exchanger and at • assuming an ideal compressor and turbine, the specific power output is:
w = wT − wC = CP (Tt 3 − Tt 4 ) −CP (Tt 2 a − Tt1 )

• and the heat addition is:

q = CP (Tt 3 − Tt 2b ) = wt = CP (Tt 3 − Tt 4 )

• the thermal efficiency is then:

ηth =

wt − wc q w = 1− c q T −T = 1 − t 2 a t1 Tt 3 − Tt 4 ⎛ Tt 2 a ⎞ − 1⎟ ⎜ Tt1 ⎝ Tt1 ⎠ = 1− Tt 3 ⎛ Tt 4 ⎞ ⎜1 − ⎟ ⎝ Tt 3 ⎠ = 1− Tt1 rP − 1 1−γ Tt 3 1 − rP γ
γ −1 γ −1 γ

• which becomes:
T ηth = 1 − t1 rP γ Tt 3

• cf. ideal gas turbine without regeneration: ηth = 1 − rP γ

1−γ

• note:
1. regenerative cycles can have higher thermal efficiency than the basic cycle. The regenerative cycle is most efficient at high Tt 3 and low rP (i.e. top left corner of the graph, and far from the peak in w ). • this is why regenerative cycles are popular in microturbines for distributed power generation, where high ηth is most important and low w can be tolerated. 2. Improvement in ηth reduces with increased rP . 3. the improvement in ηth is also strongly dependent on the performance of the heat exchanger. The effect of the heat exchanger effectiveness on ηth is algebraically complex and not discussed here.


				
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posted:7/19/2009
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