Lefebvre A.H., Ballal D.R. Gas Turbine Combustion: Alternative Fuels and Emissions

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320 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition

The beam length, l

, is determined by the size and shape of the gas vol-

ume. For most practical purposes, it is given to sufcient accuracy [5] by the

expression

(volume)/( surfacearea).= 34.

(8.10)

For tubular systems, the above expression yields values of beam length, l

ranging from 0.6 D

to 0.9 D

, depending on the length/diameter ratio of the

liner. For annular combustors, l

, is 1.0 D

for the inner liner and 1.2 D

for

the outer liner.

8.3.2 radiation from Luminous gases

When a hydrocarbon fuel is burned in a combustion chamber, soot particles

are formed; these particles have an important effect on the nature of the

radiation from the ame. At atmospheric pressure, the soot particles are too

small in number and size to radiate appreciable energy. However, some of

the radiation from these hot, glowing particles falls in the visible spectrum

and gives rise to the name “luminous ame.” With increasing pressure,

the luminous radiation increases in intensity, and the banded spectra from

water vapor and carbon dioxide become less pronounced. At the high levels

of pressure encountered in modern gas turbines, the soot particles can attain

sufcient size and concentration to radiate as black-bodies in the infrared

region, and the ame is then characterized by a predominance of luminous

radiation. It is under these conditions that severe radiant heating and its

attendant problem of liner durability are encountered.

According to Lefebvre and Herbert [2], the inuence of luminosity on gas

emissivity may be accounted for by including a luminosity factor L into

Equation 8.9 to obtain

gbg

=− −

(

)









−

1290

exp.

PL ql T

(8.11)

The original equation [2] for L is

LCH=−

(

)

75355

084

./ .,

(8.12)

which was later modied to

LCH=−

(

)

352

075

/.,

(8.13)

where C/H is the carbon to hydrogen ratio of the fuel by mass.

Kretschmer and Odgers [6] recommend the following expression:

Heat Transfer 321

LCH=−

(

)

0 0691182

271

./..

(8.14)

Lefebvre [7] has correlated modern engine combustor data using:

LH= 336

(8.15)

where Η is the fuel hydrogen content (by mass) in percent.

For any given fuel type, the luminosity factor L may be calculated from

one of these equations and inserted in Equation 8.11 to obtain the luminous

ame emissivity ε

. Substituting this value of ε

into Equation 8.7 gives the

radiation heat ux from the ame to the liner wall.

8.4 External Radiation

The radiation heat transfer, R

, from the liner wall to the outer casing can

be approximated by assuming gray surfaces with emissivities ε

and ε

and

assuming that

and T

are approximately uniform in the axial direction.

The net radiation heat transfer from the liner is then given by

AAF

111

wwwwwc c

−

(

)

−++−

εε ε()//()/εε

(8.16)

where A

is the surface area of the liner wall, A

is the surface area of the

casing, and E

is the geometric shape factor between liner and casing.

The amount of heat transferred from the liner to the casing is usually quite

small compared with C

, the external convective heat transfer [2]. Its signi-

cance increases with liner-wall temperature, and at low values it can often be

neglected. It can be estimated only approximately because of a lack of accu-

rate knowledge of wall emissivities. For this reason, it is sufcient to use the

cooling-air temperature, T

, in place of the unknown temperature of the outer

casing. Also, for radiation across a long annular space, the geometric shape

factor can be assumed to be equal to unity. The expression for the net radia-

tion ux then reduces to:

+−

(

)

(

)

−

(

)

εε

εε ε

cw cwc

(8.17)

For a tubular chamber, A

is equal to the ratio of liner diameter to cas-

ing diameter at the section considered. For tuboannular systems, in which

the depth of the annulus varies from point to point around the liner, an

322 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition

average value of 0.8 should be used. For an annular chamber, the ratio A

is slightly greater than unity for the inner liner, and slightly less than unity

for the outer liner.

Accurate values for the emissivity of various materials may be obtained

from Reference [8]. However, for most practical purposes, the following

expressions, based on typical values of emissivity and diameter ratio, should

sufce:

RTT

=−

(

)

.,σ

(8.18)

for an aluminum air casing and

RTT

=−

(

)

.,σ

(8.19)

for a steel air casing.

8.5 Internal Convection

Of the four heat-transfer processes that together determine the liner tem-

perature, internal convection is the most difcult to estimate accurately. In

the primary zone, the gases involved in heat transfer are at high temperature

and are undergoing rapid physical and chemical change. Further difculty

is introduced by the existence within the primary zone of steep gradients of

temperature, velocity, and composition. Uncertainties regarding the airow

pattern, the state of boundary-layer development, and the effective gas tem-

perature make the choice of a realistic model almost arbitrary.

In the absence of more exact data, it is reasonable to assume that some

form of the classical heat-transfer relation for straight pipes will hold for

conditions inside a liner, provided the Reynolds-number index is consistent

with established practice for conditions of extreme turbulence. This leads to

an expression of the form

0 020=













−

(

)



(8.20)

where d

is the hydraulic diameter of the liner:

Cross-sectionalflowarea

Wettedperimeter

4=== D

Heat Transfer 323

Hence,

0 020=













−

(

)



(8.21)

Several difculties arise when Equation 8.21 is applied to the primary com-

bustion zone. These have been discussed in detail elsewhere [2], so reference

need only be made to the three factors that must be taken into account in

calculations of wall temperature in the primary zone.

The primary zone contains, by design, a reversal of ow, so that only in

a region adjacent to the wall does the direction of ow correspond to the

assumed pipe analogy. A mean value



across the whole section may

be obtained by summing the upstream and downstream ow components,

irrespective of sign. A further complication is that, if a swirler is used, the

local gas velocity at the wall is greater than the downstream component by

the factor (cos β)

−1

, where β is the angle that the local velocity makes with the

combustor axis.

Of equal importance is the question of which gas temperature is to be

used in Equation 8.21. The bulk gas temperature, T

, is appropriate for radia-

tion, but the conventional primary-zone radial temperature prole is delib-

erately arranged to provide lower-than-average gas temperatures near the

wall. To account for this, the value of the constant in Equation 8.21 is reduced

from 0.020 to 0.017, so that for calculations in the primary zone, Equation 8.21

becomes

0017













−

(

)



(8.22)

8.6 External Convection

In estimating this component, Re is now based on the hydraulic mean diam-

eter, D

, of the annulus air space. This diameter is given by

Cross-sectionalareaofflow

Wettedperimet

= 4

eer

which, for a tubular chamber, gives

DDD

an refL

=−,

324 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition

and for an annular chamber,

localannulusheight.=×2

Again, the contents of the annulus may be assumed sufciently stirred for

fully developed turbulent transfer to occur. Hence,

0 020













−

(

)

an a



(8.23)

The uid properties are now evaluated at the annulus air temperature, T

. In

practice, the cooling-air temperature will increase during the passage down-

stream, but normally this increase amounts to no more than a few degrees

and can reasonably be neglected.

8.7 Calculation of Uncooled Liner Temperature

Summarizing, equations for all four heat-transfer processes have now been

derived, namely,

RTTT

15 25

05 1

(

)

−

(

)

σεε

wggg

(8.7)

0020













−

(

)



(8.21)

or, in the primary zone,

0 017













−

(

)



(8.22)

RZTT

=−

(

)

(8.24)

where Ζ depends on the casing emissivity, and

0 020













−

(

)

an a



(8.23)

Heat Transfer 325

For equilibrium,

RCRC

TT K

11 22 12

+=+= −

(

)

−

(8.25)

8.7.1 Method of Calculation

The uncooled liner-wall temperature may be calculated as follows:

1. Estimate the mean fuel/air ratio for the zone under consideration.

If the maximum possible value of T

in the combustion zone is

required, the fuel/air ratio should be assumed to be stoichiometric.

2. Obtain R

as a function of

from Equation 8.7, in which the gas

emissivity may be obtained from Equation 8.11.

3. Obtain R

as a function of

from Equation 8.24.

4. Calculate C

as a function of

from Equation 8.21. or Equation 8.22,

using the values of k and µ for combustion products at tempera ture T

5. Calculate C

as a function of

from Equation 8.23, using values of

k and µ for air at temperature T

6. Solve Equation 8.25 for

and

example

We wish to estimate the liner-wall temperature that could be expected in the pri-

mary zone of a tubular combustor if no ﬁlm cooling was used, given the following

information:

= 30 atm = 3040 kPa

= 880 Κ

Casing diameter = 0.192 m

Liner outer diameter = 0.1344 m

Liner wall thickness = 0.0012 m

Liner inner diameter = 0.132 m

= 0.4 (aluminum casing)

= 0.7 (Nimonic 75 liner material)

= 26 W/(m K)



= 7.074 kg/s



= 2.62 kg/s

= 0.0588

L = 1.7 (kerosine fuel)

The primary-zone gas temperature, T

, is determined using Equation 8.8.

A primary-zone combustion efﬁciency of 85% is assumed. Thus, the effec-

tive value of q

is 0.85 × 0.0588 = 0.050. From temperature-rise curves for

326 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition

kerosine–air mixtures, for T

= 880 K, P

= 3040 kPa, and q = 0.050, we obtain

ΔT

comb

= 1455 K.

A correction must now be made for the heat lost in evaporating the unburned

fuel and raising its temperature to that of the surrounding hot gas. This is estimated

at 55 K. Hence,

K=+ −=880 1455 55 2280 .

The beam length, l

, is obtained as

m.==06 0 0792..

For the calculation of R

, we have

σ= ×

(

)

−

5671 Wm K

824

./·,0

7,= 0.

228= 0Κ,

L = 17.,

P = 3040kPa,

q = 0 0588.,

m.= 0 0792.

Substitution of these values into Equations 8.11 and 8.7 yields

andW/m==−061 794460 0 0032

...

(8.26)

For the calculation of R

, we use Equation 8.17

+−

(

)

−

(

)

εε

εε ε

cw cLref

where

σ= ×

(

)

−

56710

./·,Wm K

7,= 0.

4,= 0.

Lref

/./. .,==0 1344 0 192 07

K==

880 .

The equation for R

thus becomes

229

100

13 715













−.,,

W/m

(8.27)

Heat Transfer 327

The calculation of C

proceeds from the Equation 8.22

0 017













−

(

)



The gas properties at T

= 2280 K and P = 3040 kPa are

W/ m·K=

(

)

0 157.,

kg ms=×

(

)

−

70510

./·,

./



kg s,= 262

m,= 0 132.

4 132 1368m=

(

)

=π/. ..000

Substitution of these values into Equation 8.22 yields

CTTT

565 1 280 500 562

,, .

pz gw w

W/m=−

(

)

=−

(8.28)

For the calculation of C

via Equation 8.23

0 020













−

(

)

an a



we have

880Κ,

553Wm K=

(

)

00./·,

kg ms=×

(

)

−

38910

./·,



kg s,. /= 7 074

(

)

(

)

=π/. –. .,40192 0 1344 0 01476

m.==0 192 0 1344 0 0576.–..

328 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition

Hence,

CT T

291 880 921 810 480

=−

(

)

=−

W/m,.

(8.29)

1–2

is calculated from Equation 8.3:

KTTTT

0 0012

21667

12 12

−

=−

(

)

=−

(

)

ww ww

Finally, substitution of the calculated expressions for R

, C

, R

, C

, and K

1–2

into

Equation 8.25 yields

KK.

1640 1603==

In the early work of Lefebvre and Herbert [2], which was conﬁned to low combus-

tion pressures and low heat-transfer rates, the temperature difference across the

liner wall could be ignored and only negligible errors would result. However, as

this example shows, for modern engines of high compression ratio, neglect of this

temperature difference could give rise to signiﬁcant discrepancies in heat-transfer

calculations.

8.7.2 Significance of Calculated uncooled Liner Temperatures

The liner-wall temperatures obtained above are fundamental to the geomet-

ric design and airow distribution of the chamber. No account, however, is

taken of supplementary lm-cooling arrangements, which comprise an addi-

tional variable in any basic chamber design.

Although the calculation neglects such supplementary cooling, the result

does not necessarily represent the maximum possible metal temperature.

The calculation is based on the assumption of uniform annulus velocity,

and, in regions where the velocity is appreciably below the mean value, the

liner wall can attain exceptionally high temperatures. Severe hot spots often

arise from the combined effect of a localized low annulus velocity on one

side of the wall and a coincident breakdown in the cooling layer on the

other.

Variations in ow conditions around the liner periphery produce corre-

sponding variations in metal temperature. Nevertheless, calculated aver-

age values of liner-wall temperature, based on complete uniformity of ow

conditions, serve a useful purpose in providing a comparative measure of

the amount of supplementary cooling required. The qualitative effect of any

change in inlet conditions on metal temperature can be readily predicted,

and the operating conditions under which this temperature is a maximum

can be estimated.

Heat Transfer 329

8.8 Film Cooling

Although many methods of supplementing the removal of heat from the

liner involve a lm of cooling air on the inner surface of the liner wall, the

name lm cooling is usually reserved for those schemes that employ a number

of annular slots through which air is injected axially along the inner wall

of the liner to provide a protective lm of cooling air between the wall and

the hot combustion gases. The cool lm is gradually destroyed by turbulent

mixing with the hot gas stream, so normal practice is to provide a succes-

sion of slots at about 40–80 mm intervals along the length of the liner. At the

downstream end of the liner, the ow acceleration in the nozzle tends to sup-

press the hot stream turbulence, and the cooling lm can persist for a much

greater distance.

The main advantage of the method is that the cooling slots can be designed

to withstand severe pressure and thermal stresses at high temperatures for

periods up to several thousand hours. Moreover, the stiffness provided by

the cooling slots results in a liner construction that is both light in weight and

mechanically robust. A basic limitation of the method is that it does not allow

a uniform wall temperature. The wall is coolest near each slot and increases

in temperature in a downstream direction to the next slot. Thus, the method

is inherently wasteful of cooling air.

The most widely used lm-cooling devices are wigglestrips, stacked rings,

splash-cooling rings, and machined rings.

8.8.1 Wigglestrips

In some combustors, the static pressure drop across the liner is too low to pro-

vide the desired amount of lm-cooling air. In this situation, recourse must

be made to devices that utilize the total pressure drop across the liner. The

advantage of this approach is that it can always provide an adequate amount

of cooling air, regardless of the static pressure drop across the liner. Its basic

drawback is that variations in annulus velocity around the liner produce cor-

responding variations in the supply of cooling air.

Usually, the liner is made up of several sections, with an annular clear-

ance between each section and the next. In one early concept, the sections

overlapped each other and were joined together by “uting,” i.e., corrugating

the larger diameter and spot-welding the utes to the upstream section. This

design failed to provide a satisfactory liner life and was soon replaced by a

conguration that employs a corrugated spacer, known as a “wigglestrip,”

to connect the overlapping sections, as shown in Figure 8.2a. This method of

construction provides a strong mechanical structure, but the poor aerody-

namic quality of the cooling lm encourages hot gas entrainment. Thermal

paint tests typically indicate the presence of long hot streaks downstream

of the cooling slots. A further drawback to the wigglestrip design is that