Advanced Gas Turbine Cycles Episode 11 potx

In this Appendix, an analysis by Holland and Thake [l], which allows external film cooling flow through the blade surface as well as internal convective cooling flow through the interna

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PROCESS STEAM TEMPERATURE - Tp (" C) Fig 9.7 (Useful heat)/work as a function of process steam temperature (after Porter and Mastanaiah [2])

Chapter 9 The gas turbine as a cogeneration (combined heat and power) plant 179

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I80 Advanced gas turbine cycles

production to 3 5 t h Gases leave the exhaust stack at 138°C under maximum load conditions

For the first operating condition (HRSG unfired) the heat load is estimated at 7.5 MW For the second condition (HRSG fired) when 35 t/h of saturated steam is raised, the heat

load is 23 MW The values of heat to work ratios (AD) are thus

7.5

(=) = 2.34, and ($ ) = 7.19, respectively

Other parameters for the plant operating condition-f HRSG unfired (WHR) and HRSG fired (WHB)-are as follows:

Alternator power output 3.2 MW

Airmass flow rate 20.45 kg/s

Maximum temperature 890°C

Thermal efficiency 0.23

Heat recovery steam generator

A full description of this plant is given in Ref [l]

9.6.2 The Liverpool University CHP plant

A gas turbine CHP scheme which operates at Liverpool University, UK, consists of a Centrax 4 MW (nominal) gas turbine with an overall efficiency of about 0.27, exhausting

to a WHB The plant meets a major part of the University’s heat load of about 7 MW on a mild winter’s day Supplementary firing of the WHB (to about 15 MW) is possible on a cold day Provision is also made for by-passing the WHB when the heat load is light, in spring and autumn, so that the plant can operate very flexibly, in three modes viz., power only, recuperative and supplementary firing

The major performance parameters at design operating conditions are as follows:

Chapter 9 The gas turbine as a cogeneration (combined hear and power) plant 181

Exhaust gas flow (MG)

15.3 kgls

0.4 X 0.9

0.27(0.9 + 1.7 X 0.4)

F E S R Z 1 -

(&G = I .7, vc = 0.4, vc = 0.9) = 0.155

A full description of the economics of operating this plant over a complete year is given

by Horlock [I]

References

[I ] Horlock, J.H (1997) Cogeneration-Combined Heat and Power Plants, 2nd edn, Krieger, Malabar, Florida [2] Porter, R.W and Mastanaiah, K (1 982), Thermal-economics analysis of heat-matched industrial cogeneration systems, Energy 7(2) 171 - 187

Appendix A

A.1 Introduction

The stagnation temperature and pressure change in the cooling mixing process have

been shown to be dependent on the cooling air flow (w,) as a fraction of the entering gas

flow (w,), i.e on JI = wc/wg In this Appendix, an analysis by Holland and Thake [l],

which allows external film cooling (flow through the blade surface) as well as internal convective cooling (flow through the internal passages), is summarised (see also Horlock

et al [2] for a full discussion) It is based mainly on the assumption that the external Stanton number (Sr,), which is generally a weak function of the Reynolds number, remains

constant as engine design parameters (Tco, and r) are changed

A.2 Convective cooling only

A simple heat balance for a typical convectively cooled blade (as illustrated in Fig A 1 a, which shows the notation) is

It is assumed that the temperature of the coolant does not fully reach the temperature of the

metal before it leaves the blade, i.e Tc, < Thus, the concept of a cooling efficiency is introduced

so that

The exposed area for heat transfer (Asg) is then replaced on the premise that, for a set of similar gas turbines, there is a reasonably constant ratio between A,, and the cross- sectional area of the main hot gas flow Axg Thus, writing A, = hixg = Awg/p,Vg in

Eq (A3) gives

183

184 Advanced gas turbine cycles

(a) CONVECTIVE COOLING NOTATION

-

(b) FILM COOLING NOTATION

%t = 'fg (Taw- Tbl )

Fig A 1 Notation for turbine blade cooling (a) Convective cooling and (b) film cooling (after Ref [2])

so that

(WclWg) = A(cpg/c,)(hg/cp,pgVg)(T,i - TbI)/%ool(Tbl - Tci)

For a row in which the blade length is L, the blade chord is c, the spacing is s and the

where Stg = hg/(cpgpgVg) is the external Stanton number

flow discharge angle is a, the ratio h is given approximately by

h = A,,/A,, = 2Lc/(Ls COS a) = 2c/(s COS a)

With s/c = 0.8 and a = 75", the value of A is then about 10 The total cooled surface area

is found to be greater than the surface area of the blade profiles alone because of the

presence of cooled end-wall surfaces (adding another 30-40% of surface area), complex

trailing edges and other cooled components It would appear from an examination of

practical engines that h(cpg/c,) could reasonably be given a value of about 20 Eq (A4)

then provides the basic form on which a cooling model can be based

The external Stanton number is assumed not to vary over the range of conditions being

studied Considering (cp,/c,)(A,,/A,,)Stg as a constant C, Eq (A4) then becomes

$h = W c / W g = c w + = C&"/7)coo,( 1 - E"), (A5)

Appendix A Derivation of required cooling Jows

where w+ is the 'temperature difference ratio' given by

and eo is the overall cooling effectiveness, defined as

80 = (Tgi - Tbl)/(Tgi - Tci)

Tgi and Tci are usually determined from and/or specified for cycle calculation so that the

cooling effectiveness .zO implicitly becomes a requirement (subject to Tbl which again can

be assumed for a 'level of technology') If r)cool and C are amalgamated into a single

constant K, then

(A8)

l+b = K&"/( 1 - Eo),

for convective cooling, as used by El-Masri [3]

A.3 Film cooling

The model used by Holland and Thake [ 11 when film cooling is present is indicated in

Fig A.lb Cooling air at temperature Tc, is discharged into the mainstream through the

holes in the blade surface to form a cooling film The heat transferred is now

649)

where Taw is the adiabatic wall temperature and hfg is the heat transfer coefficient under

film cooling conditions The film cooling effectiveness is defined as

('410)

Qnet = Asghg(Taw - Tbl) = Wccpc(Tco - Tcih

EF = (Tgi - Taw>/(Tgi - Ted

Then a new 'temperature difference ratio' W + may be written as

w+ = (Taw - Tbl)/(Tco - Tci)

= [EO - (1 - r)cool)&F - &O&F~c0011/r)cool(l - E O ) ('41 1)

It can be argued that cF should be independent of temperature boundary conditions and

It follows from Eqs (A9) and (AlO) that

in the subsequent calculations it is taken as 0.4, based on the experimental data

l+b = (wc/wg> = (c,g/c,)(Asgs~,/A,g>~w+, (A 12)

where p = hfg/[kg( 1 + B)] in which hf, is the heat transfer coefficient under film cooling

conditions and B = hfgt/k is the Biot number, which takes account of a thermal barrier

coating (TBC) of thickness r and conductivity k

In practice, hfg increases above h,, and (1 + B) is increased as TBC is added For the purposes of cycle calculation, p is therefore taken as unity and

where C is the same constant as that used for convective cooling only

Advanced gas turbine cycles

A.4 The cooling efficiency

The cooling efficiency can be determined from the internal heat transfer If Tbl is taken

to be more or less constant, then it may be shown that

where 6 = (h,A,/w,c,) = (St,A,/A,,), St, is now the internal Stanton number, and A,

and A,, refer to surface and cross-sectional areas of the coolant flow

Experience gives values of 8 for various geometries, but Sr, is also a weak function of Reynolds number and so, in practice, there is relatively little variation in cooling efficiency

(0.6 < cool < 0.8) In the cycle calculations described in Chapter 5, cool was taken as 0.7, and assumed to be constant over the range of cooling flows considered

A S Summary

Since ‘open’ film cooling is now used in most gas turbines, the form of Eq (AI 3) was

adopted for the cycle calculations of Chapter 5, i.e

Taking (cpg/cF)(As,/Ag) = 20 as representative of modern engine practice, and

Sr, = 1.5 X a value of C = 0.03 is obtained The ratio (cpg/cF) should then increase with Tg (but only by about 8% over the range 1500-2200K) This variation was, therefore, neglected in the cycle calculations described in Chapter 5

However, it was found that the cooling flows calculated from these equations were less than those used in recent and current practices in which film cooling is employed This is for two main reasons:

(i) designers are conservative, and choose to increase the cooling flows

(a) to cope with entry temperature profiles (the maximum temperature being well above the mean) and local hot spots on the blade and

(b) locally, where cooling can be achieved with relatively small penalty on mixing loss (and hence on polytropic efficiency), so regions remote from these injection points are cooled with this low loss air;

(ii) in practice, some surfaces in a turbine blade row will be convectively cooled with no film cooling The use of Eq (A15) with Eq (AI 1) for the whole blade row assembly therefore leads to the total cooling flow being underestimated Film cooling leads to more efficient cooling, which is reflected in W + being much less than w + ; for the

NGVs of a modem gas turbine W + may take a value of about 2 but w + about 4

In the calculations described in the main text, allowance was made for such practical issues by increasing the value of the constants C by a ‘safety factor’ of 1.5 Thus, cooling

flows were determined from

Appendix A Derivation of required cooling j b w s 187

with

w + = [EO - (1 - r]cool)&F - EOEFr]~ooll/r]cool(~ -

W + = [ E O - 0.12 - 0.28~,]/0.7( I - E O )

(A 17)

in which EF was taken as 0.4 and r]cool as 0.7, so that

(A181

In any particular cycle calculation, with the inlet gas temperature Tg known together

with the inlet coolant temperature Tci, and with an assumed allowable metal temperature

Tbl, cO was determined from Eq (A7) W + was then obtained from Eq (A18) and the

cooling flow fraction $ from Eq (A16)

References

[ I ] Holland, M.J and Thake T.F (1980) Rotor blade cooling in high pressure turbines, AlAA J Aircraft 17(6), [2] Horlock, J.H., Watson, D.E and Jones, T.V (2001) Limitations on gas turbine performance imposed by [3] El-Masri, M.A (1987) Exergy analysis of combined cycles: Part 1 Air-cooled Brayton-cycle gas turbines,

412-418

large turbine cooling flows, ASME J Engng Gas Turbines Power 123(3), 487-494

ASME J Engng Power Gas Turbines 109.228-235

Appendix B

ECONOMICS OF GAS TURBINE PLANTS

B.l Introduction

The simplest way of assessing the economics of a new power plant is to calculate the unit price of electricity produced by the plant (e.g $/kWh) and compare it with that of a conventional plant This is the method adopted by many authors [1,2] Other methods

involving net present values may also be used [3,4]

B.2 Electricity pricing

The method is based on relating electricity price to both the capital related cost and the

03.1)

where PE is the annual cost of the electricity produced (e.g $ p.a.), Co is the capital cost of

plant (e.g $), P(i,N) is a capital charge factor which is related to the discount rate (i) on

capital and the life of the plant (N years) (see Section B.3 below), M is the annual cost of

fuel supplied (e.g $ p.a.), and (OM) is the annual cost of operation and maintenance (e.g $

p.a.)

recurrent cost of production (fuel and maintenance of plant):

PE = Pco + M + (OM),

The ‘unitised’ production cost (say $kWh) for the plant is

W H W H W H W H

where

&$/kWh), the rate of supply of energy in the fuel &kW) and the utilisation, H , i.e

is the rating of the plant (kW) and H is the plant utilisation (hours per annum)

The cost of the fuel per annum, M , may be written as the product of the unit cost of fuel

Thus the unitised production cost is

where (v0) = W/F is the overall efficiency of the plant Alternatively, the unit cost of fuel

4‘may be written as the cost per unit mass S (say $/kg) divided by the calorific value [CV],

189

190 Advanced gas turbine cycles

(kWh/kg), so that

In a comparison between two competitive plants, one may have higher efficiency (and hence lower fuel cost) but may incur higher capital and maintenance costs These effects have to be balanced against each other in the assessment of the relative economic merits of two plants

B.3 The capital charge factor

The capital charge factor (P) multiplied by the capital cost of the plant (CO) gives the cost of servicing the total capital required Suppose the capital costs of a plant at the beginning of the first year is CO and the plant has a life of N years so an annual amount

must be provided which is (Coi + B) The first term (COi) is the simple interest payment

and the second (B) matures into the capital repayment after N years (i.e interest added to the accumulated sum at the end of each year), thus

+ ( I + i ) + ( l + i ) 2 + + ( 1 + i ) N - ' ] = ~ 0 ,

so that

C0 i

B =

(1 + i ) N - 1

where it has been assumed that the annual payments are made at the end of each year Hence the total annual payment is

where the capital charge factor P is sometimes referred to as the annuity present worth factor and is given as

In arriving at an appropriate value of p, the choice of interest or discount rate (i) is crucial It depends on:

the relative values of equity and debt financing;

whether the debt financing is less than the life of the plant;

tax rates and tax allowances (which vary from one country to another);

inflation rates

In comparing two engineering projects the practice is often to use a 'test discount rate', applicable to both projects

An American approach has been outlined by Williams [l] He elaborates the simple

expression for P to take account of many other factors beyond a simple single interest (or

Appendix E Economics of gas turbine plants 191

discount) rate He defines a discount rate as

where ae, ad are the fractions of investment from equity and debt, re, rd are the corresponding annual rates of return and T is the corporate tax rate

B.4 Examples of electricity pricing

In the unit price of electricity (YE) derived in Section B.2, the dominant factors are the capital cost per kilowatt (Co/m, which generally decreases inversely as the square root of the power (i.e as Win), the fuel price [, the overall efficiency T ~ , the utilisation (H hours

per year) and to a lesser extent the operational and maintenance costs (OM)

Fig B 1 shows simply how YE, minus the (OM)/WH component, varies with Co/W and

m, for H = 4ooo h and 6 = 1 c k w h Horlock [4] has used this type of chart to compare

three lines of development in gas turbine power generation:

(i) a heavy-duty simple cycle gas turbine, of moderate capital cost, with a relatively low pressure ratio and modest thermal efficiency (e.g 36%);

(ii) an aero-engine derivative simple cycle gas turbine, usually two-shaft and of high pressure ratio, the capital cost per kilowatt of this plant being surprisingly little different from (i) in spite of it being derived from developed aero-engines, but thermal efficiency being slightly higher (e.g 39%);

(iii) a heavy-duty CCGT plant, based on (i), which has a high thermal efficiency but

0 zoo0 4ooo m 1 m 12OOo 14000 18ooo

HEAT RATE (kJkWh)

Fig B 1 Electricity price as a function of capital cost and plant efficiency (after Ref [4])