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Performance evaluation of dry-cooling systems for power plant applications
~
Applied Thermal Engineering Vol. 16, No. 3, pp. 219-232, 1996 Copyright © 1995 Elsevier Science Ltd 1359-4311(95)00068-2 Printed in Great Britain. All rights reserved 1359-4311/96 $15.00 + 0.00
Pergamon
P E R F O R M A N C E E V A L U A T I O N OF D R Y - C O O L I N G SYSTEMS F O R P O W E R P L A N T A P P L I C A T I O N S A. E. Conradie and D. G. Krrger Department of Mechanical Engineering, University of Stellenboseh, Stellenbosch, 7600, South Africa
(Received with revisions 2 May 1995) Abstract--The performance evaluation of dry-cooling systems rely heavily on the ability to model the physical phenomena of the system. A sophisticated equation-based model, consisting of the conservation equations (energy, mass, momentum) and engineering design relations, is presented to model the dry-cooling systems performance for power plant applications. The implementation of this model gives rise to practical and reliable design methods and system performance evaluation throughout its entire operating life. Small modifications, as well as improved and reliable design practices, may result in major savings in capital and running costs over the operating life of these often large and costly plants. Keywords--Dry-cooling systems, power plant, performance evaluation, natural draught, forced draught. NOMENCLATURE A cp d E e F f G
g
H h ifs K k L m N n P p Ap Q
R
T t U V v W z
area (m 2) specific heat at constant pressure (J k g - ~K - t) diameter (m) energy (J) effectiveness (%) correction factor, or force (N) friction factor mass velocity (kg s -I m -2) gravitational acceleration (m s -2) height (m) heat transfer coefficient (W m -2 K - I ) latent heat (J kg- J) loss coefficient thermal conductivity ( W i n -~ K -I) length (m) mass flow rate (kg s- i ) revolutions per second (s -I) number pitch (m) or power (W) pressure (N m -s) pressure differential (N m -2) heat transfer rate (W) universal gas constant (J k g - t K - ~) temperature (°C or K) thickness (m) overall heat transfer coefficient (W m -2 K -~) volume flow rate (m 3 s -I) velocity (m s - t ) width (m) elevation (m)
Greek letters • c kinetic energy coefficient surface roughness (m) efficiency (%) 0 angle (°) # dynamic viscosity (kg m - t s - l ) p density (kg m -3) a area ratio z time (s) Subscripts air, or based on airside aux auxiliary
a
area 219
220 b c ct ctc cte d D do e em F f Fb Fr fr h he i id iso 1 m n o p r s sd t T tb tg ts up v V w wp 0
A. E. Conradie and D. G. Kr6ger bundle, or bellmouth contraction, or casing, or condensate cooling tower cooling tower contraction cooling tower expansion diagonal, or downstream drag, or D'Arcy, or densimetric downstream electricity,or effective, or expansion electric motor fan fin, or friction fan bays fan rows frontal hub heat exchanger inlet, or inside ideal isothermal longitudinal mean net, or normal outlet, or outside pump root, or row static,or shell, or stream stream duct throat, or tube, or transversal; or total, or turbine, or tower, or tip temperature tubes per bundle turbo-generator tower support, or tube cross-section upstream vapour volumetric water, or walkway, or wall water passes oblique,or inclined
INTRODUCTION M o s t industrial processes require the rejection o f low-quality waste heat. In particular, steam-electric plants reject heat at approximately twice the rate at which electricity is generated. F o r a long time, designers found once-through and evaporative cooling an efficient means to reject waste heat at low cost. However, water shortages and stringent environmental regulations forced designers to consider less efficient and more expensive air-cooling, or dry-cooling systems as they are often termed. Dry-cooling systems rely on the convective heat transfer which is governed by the drybulb air temperature. Dry-cooled plants offer potential economic and collateral advantages due to plant siting flexibility. Future trends suggest that the need for dry-cooling will increase [1]. It is thus very important to improve existing designs by understanding and modelling all the relevant fundamental physical p h e n o m e n a mathematically. The fundamentals o f the design and construction o f dry-cooling systems for power plant applications are discussed in refs [2]-[6]. M o n t a k h a b [7] present a survey o f dry-cooling system technology for power plant applications. Experience gained over m a n y years on previously erected smaller units and progress in dry-cooling technology p r o m p t e d the design and construction o f larger and m o r e sophisticated units [8-10]. The heat rejection performance o f the dry-cooling system under varying ambient weather conditions and the thermodynamic performance characteristics o f the turbine are closely interrelated. The plant capability and efficiency are based on the performance o f the turbine and the auxiliary loads (e.g. fan or p u m p i n g power), both o f which vary with the performance o f the cooling system. Two methods o f dry-cooling system thermal design can be identified from the above literature. In the first method, a fixed operating point is specified, while the second m e t h o d concerns the design over a range o f ambient and steam temperatures. The former design m e t h o d provides an effective means to obtain
Evaluationof dry-coolingsystemsfor powerplants
221
an order of magnitude estimation. Because of the seasonal variations in the ambient drybulb air temperature, the design air temperature is of great economic importance. In the latter method, the frequency of ambient air temperatures, together with the turbine characteristics, are taken into account to provide a more reliable method for selecting a cooling system design for any potential plant site. The gaining of extensive operational experience, as well as a better understanding of the relevant physical phenomena of these systems over the last few years, extensively improved dry-cooling technology. The results of modern experimental and theoretical work are included in this paper to model all the physical phenomena of these systems. These engineering design relations provide reliable methods to design dry-cooling systems, according to the methods above, to ensure their effective performance in power plant applications. The investigation concerns both forced-draught direct air-cooled condensers and hyperbolic natural-draught indirect dry-cooling towers with surface condensers, as shown schematically in Fig. 1.
PERFORMANCE EVALUATION OF DRY-COOLING SYSTEMS The performance prediction of dry-cooling systems are based on the fundamental conservation equations, namely the momentum equation (draught) and the energy equations (heat transfer). These equations are derived by using physical laws and engineering principles. The dry-cooling system is also analyzed as an integral part of a power plant.
Natural-draught &direct dry-cooling tower Consider the example of a hyperbolic natural-draught dry-cooling tower, as shown in Fig. 2. The heat exchanger bundles consist of one or more rows of finned tubes. The bundles are assembled
(a)
/
Processfluid L (stun)
[ ~ 1 ~ 1 . fluid
_~(water),
F
Naturaldraught coolingtower
~,,~~
Heatexchanger
-~
bundles
Surface
condenser Condensate Circulating pump
(water)
Circulatingpump ~/ ~
(b)
~
Steam
~ ~
Ex~ust steam duct Supply header
Heatexchanger bundles
Condensate header
Condensate
/ pump
Condensate I storage tank
f 9
Fig. 1. (a)Natural-draughtdry-coolingtowerwithsurfacecondenser(indirectsystem).(b) Forced-draught direct air-cooledcondenser(A-framearrangement).
222
A.E. Conradie and D. G. Kr6ger
r
d5
Tower shdl
F
Hs
ds
Ht Heat exchanger bundles
IIIIIIlilUlllllllllllllllt
Tower support
J
Airflow
Fig. 2. Natural draught dry-cooling tower with horizontal heat exchanger.
in the form of A-frames and located horizontally at the inlet cross-section of the tower. The density of the heated air inside the tower shell is less than that of the atmosphere outside the tower, with the result that the pressure inside the tower is less than the external pressure at the same elevation. The pressure differential causes air to flow through the tower at a rate which is dependent on the various flow resistances encountered, the cooling tower dimensions and the heat exchanger characteristics. The heat transfer rate from the condenser cooling water to the air stream is
Q.
=
maCpam(Ta4 -- Ta3 ) -
UAFT[(Twi- Ta4 ) - - ( T w o -
Ta3)]
{ n [(Twi - Ta4)/(Two - Ta3)]
= mwCpwm(Twi - Two) = Qw,
(1)
where (2)
UA = (1/haeAa q'- 1/hwAw)-'
and Aw is the total waterside surface area. For round, bimetallic, extruded, round-finned tubes, as shown in Fig. 3, the effective airside thermal conductance may be expressed as haeAa = [l/h,,erA, + fn(do/di)/(2~ktLtntbnb) + dn(d,./do)/(2gkfLtntbnb)] -l.
(3)
According to Briggs and Young [11], the airside heat transfer coefficient through bundles of radially finned tubes in an equilateral, triangular, tube layout is given by h a -- 0.134 Re °68' Pr°333k./dr[2(Pf- t r ) / ( d f - dr)]°2((Pf- tf)/tf) °1'34,
A
Section A-A Fig. 3. Extruded finned tube.
(4)
Evaluation of dry-cooling systems for power plants
223
where Re~ = G¢dr/#a = madr/tTldaAfr
and
Pr a =
PaCpa/ka.
The effectiveness of the circular finned surface is expressed in terms of the fin efficiency, i.e. ef = 1 -- At(1 - r/f)/h a .
(5)
According to Schmidt [12], the fin efficiency for radial fins of uniform thickness can be determined approximately from r/r = [tanh(bdr 4/2)]/(bd~ 4 / 2 ) ,
(6)
where 4 = ( d r / d r - 1)[1 + 0.35fn(dr/dr)]
and
b = [(2ha)/(tfkf)] °5.
The Reynolds number of the water flowing inside the heat exchanger tubes is Rew = Pwm Vw di/]~wm
=
4mw nwp/(~di r/tb nb/2win )-
(7)
The friction factor inside the tube for Er/di > 10 -4 is [13] fDw = 0"3086[Iog,0 {6.9/Rew + ((Er/d~)/3.7) L,l}] -2.
(8)
The frictional pressure drop inside the tubes per unit length is mpfw ~
2 0.5fi~wPwmVw/di .
(9)
The total water pumping power is __
2
Pw - ~zdi 6pfwVwLw/(4rlprlem),
(10)
where Lw is the total equivalent flow length, based on the heat exchanger tubing geometry, to make provision for additional flow resistances (e.g. bends, headers, valves). The waterside heat transfer coefficient is [14] kwm(fow/8)(Rew - 1000)Prwm[l + ( d i / t t ) 0"67] 0.67 dill + 12.7(fDw/8)0.5 (Prwm -- 1)]
hw =
( 1l)
The logarithmic mean temperature difference correction factor for cross-flow conditions can be expressed as [15] 4
FT = 1 -- ~
4
~ ai~(l
-
43) k
sin[2i arctan(4~/4z)],
(12)
i=lk=l
where the values of aik are individual to each heat exchanger configuration [16] and 4, = (Tw~- Two)/(Tw~- Ta~) 42 = (T.o - T a J / ( T w ~ - Ta~)
43 = ( 4 ~ - 42)/(
(13)
In practice the D A L R rarely occurs in the first l0 m above ground level, Tat is obtained by extrapolating the measured D A L R to the ground level and will usually differ from the actual
224
A.E. Conradie and D. G. Kr6ger
measured value. At elevation 6, which corresponds to the top of the cooling tower, the ambient air temperature is Ta6 = Tal
-
-
0.00975/-/5.
(14)
The approximate temperature at the inlet to the heat exchanger bundles is Ta3 ~ Ta~ - 0.00975H3.
(15)
To derive the draught equation, consider the variation with elevation of the pressure in the atmosphere external to the dry-cooling tower in a gravity field, i.e.
dp, = - pag dz.
(16)
For a perfect gas the following relation holds:
Pa =pa/RZa •
(17)
Substitute equations (13) and (17) into equation (16) and integrate to find the pressure difference between point 1 and a point at elevation z external to the cooling tower (refer to Fig. 2):
pa~--paz=pa~[1 --(1 --O.O0975z/Ta~)~°2564*/X]~pa~[1 --(1 --O.O0975z/Ta~) 35] with g = 9.8 m s-: and R = 287.08 J kg -~ K
-1.
(18)
The pressure external to the tower at section 6 is
Pa6 = Pal (1 -- O.O0975Hs/Tdl )3.5.
(19)
Stagnant ambient air at 1 accelerates and flows across the tower supports at 2 before flowing through the heat exchanger bundles from 3 to 4. The flow is essentially isentropic from 4 to 5. In most practical towers, the change in kinetic energy between sections 4 and 5 is normally approximately an order of magnitude smaller than the corresponding change in potential energy. A total pressure balance between 1 and 5 yields Pal -- [Pa5 -]- O~es(ma/As)2/(2pas)] = (Kts + Kct + Kctc+ Khe + Kcte)he(ma/Afr)2/(2pa34) --bPal [1 -- { 1 -- 0.00975(//3 + Ha)/(2Ta, )}3.5] +Pa4[1 -- {1 -- 0.00975(H5 -- H3/2 - H4/2]/Ta,}3"5].
(20)
For easy comparison of all the flow losses, all the loss coefficients K, are based on the frontal area of the heat exchanger and the mean air density through it. The frontal area, Art, is the projection of the effective finned surface as viewed from the upstream side. Obstructions located up against the finned surface which impede the flow through the heat exchanger must be considered when evaluating the effective frontal area. Du Preez and Kr6ger [17] studied the velocity and pressure distribution in the outlet plane of hyperbolic natural-draught cooling towers. They find that for 1/Fro ~< 3, the velocity distribution is almost uniform, i.e. ~e5 ~ 1 for dry-cooling towers where the heat exchangers are located in the cross-section near the base of the tower. The mean pressure at the outlet plane is found to be slightly less than that of the ambient air at the same elevation, i.e.
Pa5 = Pa6 + Apa56= Pa6 "b Kto (ma/A5 )2/(2pa5).
(21)
For a hyperbolic tower with a cylindrical outlet the loss coefficient is given by gt ° = Apa56/(pasvas/2)2 = 2pasApa56/(ma/As)2 = --0.28 Fr~ I + 0.04 Fr~ 1~,
(22)
where Fro
= (ma/As)2/[Pas(Pa6
--
Pas)gds].
This equation is valid for 0.5 ~< ds/d3 <~0.85 and 5 ~< Khe ~<40. The approximate temperature at the tower outlet is Ta5 ~ Ta, - 0.00975(H~ -/-/4).
(23)
Evaluation of dry-cooling systems for power plants
225
From the perfect gas relation it thus follows for Pa5 ~ Pa6, that the density at the outlet of the tower is P,5 = Pa6/[R { Ta4 - 0.00975(//5 -/-/4)}].
(24)
The density of the ambient air at elevation 6 is Pa6 =
(25)
Pa6/RZa6.
If dynamic effects are neglected, an approximate expression for Pa4 may similarly be obtained: Pa4 '~Pal [1 -- 0.00975(/-/3 + H4)/(2Tat)] 35 -- (Kts + K~t + K¢~¢+ Kho +
gcte)he(ma/Afr)2/(2pa34).
(26)
Substitute equations (19), (21) and (26) into equation (20) and find, with ~es = 1, p,~ [{1 - 0.00975(/-/3 + H4)/(2T~ )}35{1 - 0.00975(//5 - / / 3 / 2 - H4/2)/Ta4} 35 - (1 - O.O0975Hs/Tal)3"5] = (Ku + Kct + Kct¢+ Khe +
gcte)he(ma/Afr)2/(2pa34)
x [1 - 0.00975(H 5 - 1-13/2
-
n4/2)Ta4] 3"5"~ (1 + Kto)(ma/As)2/(p~,5).
(27)
This equation is known as the draught equation for a natural-draught dry-cooling tower where the heat exchangers are arranged horizontally in the base of the tower. If the heat exchangers are arranged in the form of A-frames or V-arrays, Khe (non-isothermal) is replaced by Kot. In determining the dry air density after the heat exchanger, the specified pressure at ground level can be employed in the perfect gas relation, i.e. P a 4 ~'~ Pal/(RZa4)"
(28)
The approximate air density at section 3 is Pa3 ~Pal/(RZa3).
(29)
The harmonic mean density through the heat exchanger is given by P~34~ 2pa~/[R(T,3 + T~,)].
(30)
The loss coefficient of the tower supports Kts, based on conditions at the heat exchanger, is gtshe ~
2 ,Oa34Cots Lts dts n~sArr/[P,~ (r~d3/-/3)3],
(31)
where Cots is the drag coefficient of the tower supports, tts is the support length, dts is its effective diameter or width and nts is the number of supports. Due to separation at the lower edge of the tower shell and distorted inlet flow patterns, a cooling tower loss coefficient, KCt, based on the tower cross-sectional area at 3, can be defined to take these effects into consideration. For dry-cooling towers where Kh~ 1> 30 and 5 ~< d3/H3 <~ 10, Geldenhuys and Kr6ger [18] recommend the following expression based on conditions at the heat exchanger: Kct = (Pa34/Pa3) (Afr/A3)2[O.O72(d3/1-13)2_ 0.34(d3/113) + 1.7].
(32)
Only a portion of the available area in the cooling tower base is effectively covered by heat exchanger bundles due to their rectangular shape. The reduction in effective flow area results in contraction and subsequent expansion losses. The contraction losses can be approximated by loss coefficients based on the effective reduced flow area, Ae3 [19]: get c =
1 --
2#re + I/a~.
(33)
The contraction coefficient, at, is given by ac = 0.6144517 + 0.04566493a~3 - 0.3366510"23 + 0.4082743a 33+ 2.672401 a~3 -- 5.963169a~3 + 3.558944a63.
(34)
The expansion losses can be approximated by K~o = (1 - A o 3 / A 3 ) 2 = (1 - #03) 2.
(35)
226
A, E. Conradieand D. G. Kr6ger
The effective area, A,3, corresponds to the frontal area of the heat exchanger bundles if they are installed horizontally. In the case of an array of A-frames, Ae3 = Aft s u b 0 b and thus corresponds to the projected frontal area of the bundles. Based on the conditions at the heat exchanger, the above expressions become K~tche= Ket~(P~34/Pa3) (A fr~he3 )2
(36)
gctehe= gcte(Pa34/Pa4)(aft/Ae3 )2.
(37)
and
For non-isothermal oblique flow through an array of V-bundles, the following relation holds for the loss coefficient [20]:
¥-- 0a4/
= Ko,,o +
1 ( 1;in ) ( s0m in0rn
1
_1._2K0.5~( 2pa4 .~..]_gd~. ?Pa3 ~, (38) //,Pa3 "Ji-Pa4/ ,Pa3 -I- Pa4/
where a is the ratio of the minimum free flow area through the heat exchanger bundle to the free stream flow (frontal) area. For radially finned tubes the heat exchanger loss coefficient under normal non-isothermal flow conditions can, in general, be expressed as [21]
ghe : ~ [EH ..~_(Pa3- Pa4X)l= ~ [ Apapare (Pa3= Pa4X)l" \Pa3 "}-,Oa4/_] L G~ + kP~3 + P~4,//
(39)
Robinson and Briggs [22] propose the following correlation for the pressure drop across the bundles with an equilateral, triangular, tube layout: (~2 / p ~-0.927/ ]9 \0.515
Ap a = 18.93n r Rea0.316 - ¢ i - t |
[~ t |
Pa ~ r J
~gdJ
,
(40)
where Rea = G~dr/#~
and
Pd = [(Pt/2) 2 + (P02] °5 (diagonal pitch).
The entrance contraction loss coefficient K~ ~ 0.05 for many round industrial finned tube heat exchangers. The mean flow incidence angle may be expressed by the following empirical relation
[20]: 0m = 0.00190~ + 0.91330b -- 3.1558,
(41)
where the heat exchanger bundle semi-apex angle, 0b, is given in degrees; Kd is the downstream loss coefficient that includes the jetting and kinetic energy losses, and can be expressed in terms of the following relation [20]: Kd = exp(5.488405 -- 0.21312090b + 3.533265 x 1 0 - 3 0 5 - 0.2901016 x 10 -403).
(42)
Forced-draught direct air-cooled condenser Consider the forced draught air-cooled condenser shown schematically in Fig. 4. In this configuration the heat exchanger bundles are arranged in the form of an A-frame. The air-cooled heat exchanger may consist of one or more rows of finned tubes, each row having a different fin pitch. The airflow across the heat exchangers is created by means of axial flow fans. The heat transfer rate from the condensing steam to the air stream is Qa = maCpam(Ta6 -
Tas) = mcifg=
Qc-
(43)
If the geometry of the finned tubes changes in consecutive rows, the heat transfer is
nr iffil
nr
macvam(,)(Tao(O-Tai(i))= £
nr
mc(i)ifg(i)= E iffil i=l
m.C~m(o(T~-(o- Tai(i))e(i),
(44)
where the effectiveness for each tube row is [23] e(o = 1 - exp[-(1/ha~(oA~( o + 1/hc(oA,~o)-l/(m~cpam(o)], A¢(0 is the inside tube area of tube row (i) exposed to the condensing steam.
(45)
Evaluation of dry-cooling systems for power plants
227
Steam header
Y
Windwall Heat exchanger Condensate
drain
i
Walkway
L --1' ~ - - - " J
Lw
Fan Safety screen I-I 3
Supports
Fig. 4. Air-cooled condenser unit.
The pressure losses in the turbine exhaust steam duct and at the inlet to the finned tubes are expressed as Apsd = 0.5pvi(l)Vvi(l)(1 2 -- 0-2 q- Kc + K~),
(46)
where 0-c is the tube inlet contraction area ratio and the contraction loss coefficient, K¢ ~ 0.6 (sharp inlet), K~ is the loss coefficient for the steam duct system. The mean saturation pressure of the steam at the inlet of the finned tubes, Pv~, is calculated by subtracting Ap~ from the mean saturation steam pressure at the turbine outlet. The inlet conditions (thermophysical properties) of all the tube rows are thus assumed to be equal. Due to pressure changes along the finned tube and in the direction of air flow, the condensation process will not take place at a constant temperature. The mean static pressure in the finned tubes can be determined from the correlations given by Groenewald and Kr6ger [24]: Pyre,) : P v i
0.1 58 2/.t ~i L t . . . . .
- - ~
Pvi ae Kevi(i)
1~ _2.75
1.75
2
2
to.zo/a~
(47)
The coefficients at and a2 are functions of the suction Reynolds number, Revn. For round tubes, these coefficients are al¢o = 1.0046 + 1.719 x 10 -3 Revn,) -- 9.7746 x 10 -6 Re~n<0
a2
(48)
and the suction Reynolds number is expressed as R e vn(i) = (Pvi Uvi(i)di/Pvi ) ( d i / 4 L t ) = Revi(i)(di/4Lt
).
(49)
It is assumed that condensation occurs at the mean steam temperature, Tvm<~),corresponding to the mean steam pressure inside the tubes. It should be noted that the pressure drop in the different tube rows is usually not identical, with the result that backflow of steam will occur [6]. To avoid this and the corresponding accumulation of non-condensables, a dephlegmator is usually installed after the condenser. F o r round tubes, the mean condensation heat transfer coefficient for inclined tubes according to Schulenburg (25) is used, i.e. h~i) = 1.197(sin 0 b) 0.175k era(i)/ d i(Pcm(i)~vm(i)/P~m(i)~cm(i)) 0.5 Re 0.325 vi(i) •
(50)
228
A.E.
Conradie and D. G. Kr6ger
Kr6ger [1] presents a detailed derivation of the draught equation for the air-cooled condenser shown in Fig. 4, i.e. Pa~
[{1 -
0.00975(H 7 --
n6)/Ta6 }3.5 _
{1 - - 0 . 0 0 9 7 5 ( H 7 - -
H6)/Tal
}3.5]
gts (m a/A 2)2/(2pa I ) -'b gup ( m a / A Fe)2/(2pa3 ) - - gFs (ma/AFc)2/(2pa3 )
"q-gdo(ma/AFe)2/(2pa3) "b got(ma/Afr)2/(2pa56).
(51)
The choice of a suitable axial flow fan must be such that it will efficiently deliver a cooling air flow rate that will guarantee the desired heat transfer rate. An example of performance curves of a large fan is shown in Fig. 5. In actual installations the fan operating efficiency is a function of the fan and drive system design, system configuration and operating conditions [1, 26, 27]. Fan control to vary the air mass flow rate through the cooling system's operating range has great economic advantages. I n large cooling plants covering a considerable area and including numerous fans, some of the fans may be subjected to significant cross-flow, which tends to distort inlet conditions to the fan, resulting in a corresponding reduction in performance. The effect of the inlet losses is correlated by Salta and Kr6ger [28] for more than two fan rows as a volumetric effectiveness, i.e.
ev = VF/VFid
=
0.985 - exp[-(1
+
45/n)H3/6.35dr],
(52)
where n = nvr for a freestanding fan platform and n = 2nFr for a non-freestanding fan platform. This correlation is applicable for Hb/dE = 0.19 and WE~de = 1.27, where Hb is the height of the bellmouth fan inlet from the fan platform and WE is the fan pitch; net is the number of fan rows (number of fans per bay), nFb is the number of fan bays and the total number of fans are nF = nrr nFb; Vr/VFid correlates the effect of the inlet losses on the entire fan system. The actual airside heat transfer rate can be obtained by multiplying the maximum ideal airside heat transfer rate by the fan volumetric effectiveness ev. The operating point of the fixed-geometry dry-cooling system is defined as the combination of operating variables that will simultaneously satisfy the draft and heat transfer equations for specified process fluid and ambient air conditions. The geometrical variables represent the physical dimensions of the system components and the operating variables represent the operating conditions.
Combined performance of dry-cooling systems and steam turbo-generators The dry-cooling systems can be connected to the low-pressure side of specially designed high back-pressure steam turbines. The turbo-generator-condenser characteristics, as shown in Fig. 6, are expressed in terms of the turbine back pressure, Ptv, or the corresponding vapour temperature, Tt,. The turbine operates between fixed back-pressure limits to prevent the occurrence of
Z
300 !
220
250
210
200
200
e,
<~
150
J
e~
o ,-
100 -
Fan diameter = 9.145 m ~ Blade angle = 16 °
190 O
A i r density = 1.2 kg/rn 3 - Speed = 125 r p m
50
L~
170
1 1
0 200
300
400
500
600
160 700
Volume flow rate, Vr, ma/s Fig. 5. Performance of an axial flow fan.
800
900
Evaluation of dry-coolingsystems for power plants
229
240
365
235
360
230
~'~
355
225
.0
c5 O
~220
.4 0 o
.2
215
"O
~ 210
/ l
205
1
l
45
50
200 40
J
330 325
55
60
65
70
75
80
85
90
95
Turbine exhaust saturated steam temperature, Try,*C Fig. 6. Performancecharacteristics of a turbo-generator. flow-induced vibrations and choking [29]. It is assumed that the power needed for the auxiliaries other than cooling water pumps or axial flow fans is already subtracted from the generated power. For indirect dry-cooling the temperature of the heated water that leaves the surface condenser and the condensing turbine exhaust steam differ with the value of the terminal temperature difference (TTD). Typical values for T T D range from 2 to 4°C for surface condensers. For a fixed condenser design and off-design conditions, the relationship between the heat rejection rate and the T T D can be approximated as T T D ~ TTDdesign Q/Qdesign.
(53)
In the case of the direct condensing system the temperature difference in the condensing and exhaust steam is caused mainly by frictional effects. Atmospheric conditions, especially changes in the ambient temperatures, influence the performance of dry-cooling systems, resulting in a wide fluctuation of turbine back-pressure. An example of the annual mean hourly frequency of ambient temperatures, as shown in Fig. 7, is used for selecting cooling system design and operating conditions. The operating point of the turbo-generator is determined by matching the operating point of the dry-cooling system and the performance characteristics of the turbo-generator at a specific ambient air temperature selected from the annual ambient temperature frequency set. This calculation involves the selection of turbine exhaust conditions such that the heat to be rejected by the heat exchanger after the turbo-generator equals the heat absorbed by the air (heat rejected 8OO 700
Drybulb temperature
g 6oo 500 400
300
200
k
100 0 -5
0
5
10
15
20
25
30
Temperature, T, °C Fig. 7. Frequency of ambient dry- and wetbulb air temperatures.
35
230
A.E. Conradie and D. G. Kr6ger
by the dry-cooling system), while satisfying the draught equation. At this point the power requirement of the cooling water pumps or the axial flow fans, as well as the generator power output, are known. Subtract the total auxiliary power consumed by the pumps or fans from the generator power output to find the net power output of the plant. The net power output is multiplied by the corresponding number of operating hours to give the net energy output for this period. These calculations are repeated for each of the ambient temperatures listed to obtain the total annual net energy output: E, = ~ i=1
(Ptg(i)-
(54)
eaux(i))'~(i) .
The performance of dry-cooling systems deteriorates with increasing ambient air temperatures, resulting in an increased turbine back-pressure and a decrease in power generation. Air humidification [30] may be considered to increase the cooling capacity during these conditions. NUMERICAL EXAMPLES In this section we present two practical examples to illustrate the performance evaluation of dry-cooling systems with particular reference to power plants. The details of the existing dry-cooling systems used are listed in Appendix A. The first example considers a natural-draught indirect dry-cooling tower, while the second deals with a forced-draught direct air-cooled condenser. In both examples, the operating point of the dry-cooling system is computed at the design atmospheric and process fluid conditions. The annual net power output is also computed when the dry-cooling systems are coupled to turbo-generator sets. The draught and energy equations are simultaneously solved to determine the performance characteristics of the dry-cooling systems [31]. The results of each problem are listed below.
Example 1: natural-draught indirect dry-cooling tower Operating point results: m a = 11,020 kg s-1 Pw=211.5kW
Ta4 = 45.3°C
Q =331.063MW
Two = 43.4°C 1/FrD=2.841.
When the cooling tower is coupled to the turbo-generator, an annual net power output of E, = 2074.337 GWh is generated, while 2886.389 GWh heat is dissipated. The turbine operates in the following region: 45.87°C ~< Try ~< 83.08°C.
Example 2: forced-draught direct air-cooled condenser Operating point results (the values shown are for a typical unit as shown in Fig. 4): ma = 540.821 kg s -I ev = 92.665% Tvm(l) -~-
59.122°C
VF = 529.381 m 3 s -l
Ta6 = 37.30°C Tvm(2) =
PF = 101.007 kW
Tvi = 59.372°C
59.143°C
mc = 4.675 kg s i.
For the entire air-cooled condenser consisting of 30 units, the heat dissipated is 331.08 MW. When the air-cooled condenser is coupled to the turbo-generator, an annual net power output of En = 2058.351 G W h is generated, while 2877.138 GWh heat is dissipated. The turbine operates in the following region: 44.42°C ~< Try ~< 78.34°C. The above examples illustrate that the operating point and power generation calculations provide the feature to generate detailed information on the performance of the dry-cooling systems. The performance of dry-cooling systems decreases at high ambient temperatures and the auxiliary power requirements vary accordingly. The interaction of the dry-cooling system and the turbo-generator unit forms an integral part of the design process. Designing a dry-cooling system in this way is different from the conventional means of design that is purely based on fixed ambient and process fluid conditions. Finding the best design over a range of operating conditions will definitely result
Evaluation of dry-cooling systems for power plants
231
in a more realistic design. However, this design method will involve a considerable amount of computation. A practically similar procedure was followed in the design of the indirect dry-cooling system of the Kendal Power Station [10]. Operating point calculations can still be performed to analyze the cooling system's performance characteristics at any prescribed conditions. The illustration is by no means complete and the programs can be used in a variety of other applications concerning design calculations, performance evaluation and cost estimation. CONCLUSIONS
Dry-cooling systems are a n e n v i r o n m e n t a l l y s o u n d alternative to wet-cooling systems. T h e number and size of these installations will probably increase in future. However, due to their high capital and operating costs, reliable design practices are of the utmost importance to ensure their effective performance. In this study we present a practical design procedure for dry-cooling systems with particular reference to power plant applications. When properly implemented, this procedure provides a reliable practical tool with which realistic answers and trend information can be obtained. In general, dry-cooling system designs were usually based on a fixed operating point. However, the only correct way to compare different cooling systems and assess them from an economic point of view is to design them by taking the turbo-generator characteristics, the annual frequency of ambient air temperatures, the planned number of operating hours and the internal power consumption into account. A standard dry-cooling system design will not necessarily be optimal, therefore each plant and site should have an optimized dry-cooling system. One of the major components of any engineering optimization study is system modelling or simulation. The authors have extended the above study to perform cost-optimal designs of dry-cooling systems. Acknowledgement--The financial support from the Water Research Commission is gratefully acknowledged.
REFERENCES 1. D. G. Kr6ger, Fan performance in air-cooled steam condensers. Heat Reeot:ery Systems & CHP 14(4), 391-399 (1994). 2. J. P. Rossie and E. A. Cecil, Research on Dry-type Cooling Towers fi~r Thermal Electric Generation: Parts I and 11. EPA Report, Water Pollution Control Research Series 16130 EES, November (1970). 3. E. C. Smith and M. W. Larinoff, Power plant siting, performance, and economics with dry cooling tower systems. Proc. American Power Con/i 1970. Vol. 32, pp. 544 572 (1970). 4. H. Heeren and L. Holly, Dry cooling eliminates thermal pollution. Energie Z3 (1971). 5. E. S. Miliaras, Power Plants with Air-cooled Condensing Systems. MIT Press, Cambridge, Massachusetts (1974). 6. M. W. Larinoff, W. E. Moles and R. Reichhelm, Design and specification of air-cooled steam condensers. Chem. Engng, 22 May, 86 94 (1978). 7. A. Montakhab, A survey of dry cooling tower technology for power generation application. In Heat Exchangers, ThermaI-Hv~h'aulic Fundamentals and Design (Edited by S. Kakac, A. E. Bergles and F. Mayinger), pp. 799-816. Hemisphere, New York (1981). 8. A.J. Ham and L. A. West, ESCOM's advance into dry cooling. VGB Conf., 9 13 November 1987, Johannesburg, South Africa, pp. 33 43 (1987). 9. H. Knirsch, Design and construction of direct dry cooling units. VGB Conf. 9 13 November 1987, Johannesburg, South Africa, pp. 54 69 (1987). 10. B. Trage and F. J. Hintzen, De,Sgn and construction of indirect dry cooling units. VGB Conf., 9-13 November 1987, Johannesburg, South Africa, pp. 70-79 (1987). I 1. D. E. Briggs and E. H. Young, Convection heat transfer and pressure drop of air flowing across triangular pitch banks of finned tubes. Chem. Engng Prog. Syrup. Series 59(41), 1-10 (1963). 12. T. E. Schmidt, La production calorifique des surfaces munies dailettes. Annexe Du Bulletin De L'Institut International Du Froid, Annex G-5 (1945-1946). 13. S. E. Haaland, Simple and explicit formulas for the friction factor in turbulent pipe flow. Trans. ASME J. Fluids Engng 105(3), 89 90, March (1983). 14. V. Gnielinski, Forsch. lng. Wesen 41(1) (1975). 15. W. Roetzel, Berechnung yon W/irmefibertragern, VDl-W6rmeatlas, 4.Auflage, VDI, D/isseldorf, Cal-Ca31 (1984). 16. W. Roetzel and F. J. L. Nicole, Mean temperature difference for heat exchanger design--a general approximate explicit equation. J. Heat Transfer, Trans. A S M E 97(1), February (1975). 17. A. F. Du Preez and D. G. Kr6ger, The influence of a buoyant plume on the performance of a natural draft cooling tower. Paper presented at the 9th IAHR Cooling Tower Conf., September 1994, Brussels. 18. J. D. Geldenhuys and D. G. Kr6ger, Aerodynamic inlet losses in natural draft cooling towers. Proe. 5th IAHR Cooling Tower Workshop, Monterey (1986). ATE 163
D
232
A . E . Conradie and D. G. Kr6ger
19. W. M. Kays, Loss coefficients for abrupt changes in flow cross-section with low Reynolds number flow in single- and multiple-tube systems. Trans. A S M E 72(8), 1067-1074 (1950). 20. J. C. B. Kotz6, M. O. Bellstedt and D. G. Kr6ger, Pressure drop and heat transfer characteristics of inclined finned tube heat exchanger bundles. Proc. 8th International Heat Transfer Conf., San Francisco (1986). 21. D. G. Kr6ger, Performance characteristics of industrial finned tubes presented in dimensional form. Int. J. Heat Mass Transfer 29(8), 1119-1125 (1986). 22. K. K. Robinson and D. E. Briggs, Pressure drop of air flowing across triangular pitch banks of finned tubes. Chemical Engng Progress Symp. Series 62(64), 177-184 (1966). 23. J. P. Holman, Heat Transfer. McGraw-Hill, New York (1986). 24. W. Groenewald and D. G. Kr6ger, Effect of suction on turbulent friction inside ducts (submitted for publication). 25. F. J. Schulenberg, W~irmeiibergang und Druck~.nderung bei der Kondensation von str6mendem D a m p f in geneigten Rohren, Dr-Ing. thesis, Universit/it Stuttgart (1969). 26. R. C. Monroe, Minimizing fan energy costs, Part 1. Chem. Engng, 27 May, 141-142 (1985). 27. R. C. Monroe, Minimizing fan energy costs, Part 2. Chem. Engng, 24 June, 57-58 (1985). 28. C. A. Salta and D. G. Kr6ger, Effect of inlet flow distortions of fan performance in forced draught air-cooled heat exchangers. Heat Recovery Systems & CHP, 15, 555-561 (1995). 29. Z. Szab6, Why use 'Heller System'? Circuitry, characteristics and special features. Paper presented at the Symposium on Dry Cooling Towers, Tehran (1991). 30. T. A. Conradie and D. G. Kr6ger, Enhanced performance of a dry-cooled power plant through air precooling. ASME Paper 91-JPGC-Pwr-6, presented at the ASME International Joint Power Conf., October 1991, San Diego (1991). 31. A. E. Conradie, Performance optimization of engineering systems with particular reference to dry-cooled power plants, Ph.D dissertation, Department of Mechanical Engineering, University of Stellenbosch, Stellenbosch, South Africa (1995).
APPENDIXA:
NUMERICAL
EXAMPLES
Example 1: natural-draught indirect dry-cooling towers A natural-draught hyperbolic concrete cooling tower, as shown schematically in Fig. 2, has the following dimensions: H 5 = 120 m, H3 = 13.67 m, d 5 = 58 m and d 3 = 82.958 m. The tower shell is supported by 60 tower supports of length Lts = 14.547 m with an effective diameter of dt~ = 0.5 m. The drag coefficient of each support is Cot~= 2. It is further assumed that the tower shell has a uniform thickness of lets = 0.25 m. Extruded, bimetallic, finned tubes, as shown in Fig. 3, are used. The steel core has an inside diameter of di = 21.6 mm and an outside diameter of do = 25.4 mm. The aluminium fins have a diameter of df = 57.2 mm with a root diameter of d r = 27.6mm, a mean thickness of tr= 0.5 mm and a pitch of P r = 2.8 mm. The density of the steel core tube is Pt = 7850 kg m -3 and its thermal conductivity is k t = 50 W m - i K - 1. The aluminium fins have a density of Pf = 2707 kg m -3 and a thermal conductivity of k r = 204 W m -~ K - L The tube inside surface roughness is 0.0115 mm. The finned tubes bundles are arranged radially in the form of V-arrays at bundle semi-apex angles of 0b = 30.75 ° in the base of the tower shell. Due to geometrical considerations, only 52.4% of the tower inlet cross-sectional area is covered by a total o f n b = 142 heat exchanger bundles. Each bundle is L t = 15 m long and Wb = 2.262 m wide. Each bundle contains ntb = 154 tubes arranged on a triangular pitch with Pt = 58 mm and Pt -- 50.22 mm. There are n r = 4 tube rows with ntr = 39 tubes per row and nwp = 2 water passes. The total equivalent flow length is twice the length of the finned tubes. Hot water at a mass flow rate of mw = 4390 kg s-~ is pumped through the finned tubes. The design inlet water temperature is Twi = 61.45°C.
Example 2: forced-draught direct air-cooled condensers A forced-draught direct air-cooled condenser consists of an array of nrr x nrb = 5 × 6 = 30 A-frame units as shown schematically in Fig. 4. Each unit has an axial flow fan of dr = 9.145 m diameter with performance characteristics as shown in Fig. 5. The fan hub diameter is dFh = 1.4 m and the fan casing diameter is dvc= 9.17 m. Each fan rotates at N v = 100 rpm and a fan drive efficiency of 90% is assumed. The upstream and downstream loss coefficients due to obstacles close to the fan are Kup = 0.297 and Kdo = 0.391, respectively. The non-freestanding fan platform is H 3 = 30 m above the ground level. It is supported by steel beams with an effective loss coefficient of Kts = 1.5 based on the frontal area of the heat exchanger. Walkways of Lw = 0.4 m width are located between adjacent A-frames and the condenser is surrounded by windwalls. The condenser consists of two rows of extruded, bimetallic, finned tubes, as shown in Fig. 3. The steel core has an inside diameter of di = 35.1 mm and an outside diameter of do = 38.1 mm. The aluminium fins have a diameter of dr= 69.9 mm with a root diameter of dF = 40.3 mm, a mean thickness of tr = 0.35 mm. Each tube row has different performance characteristics such that approximately the same amount of steam condenses in each tube row to ensure that non-condensables are not trapped in the condenser. The fin pitches are P~) = 3.63 mm and P~2) = 2.54 mm, respectively. The density of the steel core tube is Pt = 7850 kg m -3 and its thermal conductivity is k t = 50 W m - t K - ~. The aluminium fins have a density of pf = 2707 kg m -3 and a thermal conductivity of k r = 204 W m - J K -~. There are n b = 2 heat exchanger bundles above each fan. The finned tubes are L t = 10 m long and arranged at a bundle semi-apex angle of 0b = 30 °. The tube rows have nt~o) = 152 and ntr(2 ) = 153 tubes, respectively, that are arranged on a triangular pitch with Pt = 76.2 m m and P~ = 65.991 mm and the bundle width is Wb = 11.659 m. Saturated steam at a design temperature of 60°C is supplied to the condenser by means of a steam header with an effective diameter of dsh = 1.25 m. The steam ducting loss coefficient is K~ = 0.6. In both examples, the cooling systems operate under a design ambient temperature of T~ = 15.6°C and a design atmospheric pressure ofp~ I = 86,400 N m -2. The systems operates 8760 h per annum and their operating lifetime is assumed to be 30 yr. The cooling systems forms part of a power plant that is erected at the location corresponding to the design atmospheric pressure state above. The corresponding annual temperature distribution (drybulb and wetbulb temperatures) is shown in Fig. 7. The cooling systems are coupled to turbo-generator sets, the performance characteristics of which are shown in Fig. 6. The design point terminal temperature difference for the indirect dry-cooling system is 2.5°C at a heat transfer rate of 327.63 MW.
Applied Thermal Engineering Vol. 16, No. 3, pp. 219-232, 1996 Copyright © 1995 Elsevier Science Ltd 1359-4311(95)00068-2 Printed in Great Britain. All rights reserved 1359-4311/96 $15.00 + 0.00
Pergamon
P E R F O R M A N C E E V A L U A T I O N OF D R Y - C O O L I N G SYSTEMS F O R P O W E R P L A N T A P P L I C A T I O N S A. E. Conradie and D. G. Krrger Department of Mechanical Engineering, University of Stellenboseh, Stellenbosch, 7600, South Africa
(Received with revisions 2 May 1995) Abstract--The performance evaluation of dry-cooling systems rely heavily on the ability to model the physical phenomena of the system. A sophisticated equation-based model, consisting of the conservation equations (energy, mass, momentum) and engineering design relations, is presented to model the dry-cooling systems performance for power plant applications. The implementation of this model gives rise to practical and reliable design methods and system performance evaluation throughout its entire operating life. Small modifications, as well as improved and reliable design practices, may result in major savings in capital and running costs over the operating life of these often large and costly plants. Keywords--Dry-cooling systems, power plant, performance evaluation, natural draught, forced draught. NOMENCLATURE A cp d E e F f G
g
H h ifs K k L m N n P p Ap Q
R
T t U V v W z
area (m 2) specific heat at constant pressure (J k g - ~K - t) diameter (m) energy (J) effectiveness (%) correction factor, or force (N) friction factor mass velocity (kg s -I m -2) gravitational acceleration (m s -2) height (m) heat transfer coefficient (W m -2 K - I ) latent heat (J kg- J) loss coefficient thermal conductivity ( W i n -~ K -I) length (m) mass flow rate (kg s- i ) revolutions per second (s -I) number pitch (m) or power (W) pressure (N m -s) pressure differential (N m -2) heat transfer rate (W) universal gas constant (J k g - t K - ~) temperature (°C or K) thickness (m) overall heat transfer coefficient (W m -2 K -~) volume flow rate (m 3 s -I) velocity (m s - t ) width (m) elevation (m)
Greek letters • c kinetic energy coefficient surface roughness (m) efficiency (%) 0 angle (°) # dynamic viscosity (kg m - t s - l ) p density (kg m -3) a area ratio z time (s) Subscripts air, or based on airside aux auxiliary
a
area 219
220 b c ct ctc cte d D do e em F f Fb Fr fr h he i id iso 1 m n o p r s sd t T tb tg ts up v V w wp 0
A. E. Conradie and D. G. Kr6ger bundle, or bellmouth contraction, or casing, or condensate cooling tower cooling tower contraction cooling tower expansion diagonal, or downstream drag, or D'Arcy, or densimetric downstream electricity,or effective, or expansion electric motor fan fin, or friction fan bays fan rows frontal hub heat exchanger inlet, or inside ideal isothermal longitudinal mean net, or normal outlet, or outside pump root, or row static,or shell, or stream stream duct throat, or tube, or transversal; or total, or turbine, or tower, or tip temperature tubes per bundle turbo-generator tower support, or tube cross-section upstream vapour volumetric water, or walkway, or wall water passes oblique,or inclined
INTRODUCTION M o s t industrial processes require the rejection o f low-quality waste heat. In particular, steam-electric plants reject heat at approximately twice the rate at which electricity is generated. F o r a long time, designers found once-through and evaporative cooling an efficient means to reject waste heat at low cost. However, water shortages and stringent environmental regulations forced designers to consider less efficient and more expensive air-cooling, or dry-cooling systems as they are often termed. Dry-cooling systems rely on the convective heat transfer which is governed by the drybulb air temperature. Dry-cooled plants offer potential economic and collateral advantages due to plant siting flexibility. Future trends suggest that the need for dry-cooling will increase [1]. It is thus very important to improve existing designs by understanding and modelling all the relevant fundamental physical p h e n o m e n a mathematically. The fundamentals o f the design and construction o f dry-cooling systems for power plant applications are discussed in refs [2]-[6]. M o n t a k h a b [7] present a survey o f dry-cooling system technology for power plant applications. Experience gained over m a n y years on previously erected smaller units and progress in dry-cooling technology p r o m p t e d the design and construction o f larger and m o r e sophisticated units [8-10]. The heat rejection performance o f the dry-cooling system under varying ambient weather conditions and the thermodynamic performance characteristics o f the turbine are closely interrelated. The plant capability and efficiency are based on the performance o f the turbine and the auxiliary loads (e.g. fan or p u m p i n g power), both o f which vary with the performance o f the cooling system. Two methods o f dry-cooling system thermal design can be identified from the above literature. In the first method, a fixed operating point is specified, while the second m e t h o d concerns the design over a range o f ambient and steam temperatures. The former design m e t h o d provides an effective means to obtain
Evaluationof dry-coolingsystemsfor powerplants
221
an order of magnitude estimation. Because of the seasonal variations in the ambient drybulb air temperature, the design air temperature is of great economic importance. In the latter method, the frequency of ambient air temperatures, together with the turbine characteristics, are taken into account to provide a more reliable method for selecting a cooling system design for any potential plant site. The gaining of extensive operational experience, as well as a better understanding of the relevant physical phenomena of these systems over the last few years, extensively improved dry-cooling technology. The results of modern experimental and theoretical work are included in this paper to model all the physical phenomena of these systems. These engineering design relations provide reliable methods to design dry-cooling systems, according to the methods above, to ensure their effective performance in power plant applications. The investigation concerns both forced-draught direct air-cooled condensers and hyperbolic natural-draught indirect dry-cooling towers with surface condensers, as shown schematically in Fig. 1.
PERFORMANCE EVALUATION OF DRY-COOLING SYSTEMS The performance prediction of dry-cooling systems are based on the fundamental conservation equations, namely the momentum equation (draught) and the energy equations (heat transfer). These equations are derived by using physical laws and engineering principles. The dry-cooling system is also analyzed as an integral part of a power plant.
Natural-draught &direct dry-cooling tower Consider the example of a hyperbolic natural-draught dry-cooling tower, as shown in Fig. 2. The heat exchanger bundles consist of one or more rows of finned tubes. The bundles are assembled
(a)
/
Processfluid L (stun)
[ ~ 1 ~ 1 . fluid
_~(water),
F
Naturaldraught coolingtower
~,,~~
Heatexchanger
-~
bundles
Surface
condenser Condensate Circulating pump
(water)
Circulatingpump ~/ ~
(b)
~
Steam
~ ~
Ex~ust steam duct Supply header
Heatexchanger bundles
Condensate header
Condensate
/ pump
Condensate I storage tank
f 9
Fig. 1. (a)Natural-draughtdry-coolingtowerwithsurfacecondenser(indirectsystem).(b) Forced-draught direct air-cooledcondenser(A-framearrangement).
222
A.E. Conradie and D. G. Kr6ger
r
d5
Tower shdl
F
Hs
ds
Ht Heat exchanger bundles
IIIIIIlilUlllllllllllllllt
Tower support
J
Airflow
Fig. 2. Natural draught dry-cooling tower with horizontal heat exchanger.
in the form of A-frames and located horizontally at the inlet cross-section of the tower. The density of the heated air inside the tower shell is less than that of the atmosphere outside the tower, with the result that the pressure inside the tower is less than the external pressure at the same elevation. The pressure differential causes air to flow through the tower at a rate which is dependent on the various flow resistances encountered, the cooling tower dimensions and the heat exchanger characteristics. The heat transfer rate from the condenser cooling water to the air stream is
Q.
=
maCpam(Ta4 -- Ta3 ) -
UAFT[(Twi- Ta4 ) - - ( T w o -
Ta3)]
{ n [(Twi - Ta4)/(Two - Ta3)]
= mwCpwm(Twi - Two) = Qw,
(1)
where (2)
UA = (1/haeAa q'- 1/hwAw)-'
and Aw is the total waterside surface area. For round, bimetallic, extruded, round-finned tubes, as shown in Fig. 3, the effective airside thermal conductance may be expressed as haeAa = [l/h,,erA, + fn(do/di)/(2~ktLtntbnb) + dn(d,./do)/(2gkfLtntbnb)] -l.
(3)
According to Briggs and Young [11], the airside heat transfer coefficient through bundles of radially finned tubes in an equilateral, triangular, tube layout is given by h a -- 0.134 Re °68' Pr°333k./dr[2(Pf- t r ) / ( d f - dr)]°2((Pf- tf)/tf) °1'34,
A
Section A-A Fig. 3. Extruded finned tube.
(4)
Evaluation of dry-cooling systems for power plants
223
where Re~ = G¢dr/#a = madr/tTldaAfr
and
Pr a =
PaCpa/ka.
The effectiveness of the circular finned surface is expressed in terms of the fin efficiency, i.e. ef = 1 -- At(1 - r/f)/h a .
(5)
According to Schmidt [12], the fin efficiency for radial fins of uniform thickness can be determined approximately from r/r = [tanh(bdr 4/2)]/(bd~ 4 / 2 ) ,
(6)
where 4 = ( d r / d r - 1)[1 + 0.35fn(dr/dr)]
and
b = [(2ha)/(tfkf)] °5.
The Reynolds number of the water flowing inside the heat exchanger tubes is Rew = Pwm Vw di/]~wm
=
4mw nwp/(~di r/tb nb/2win )-
(7)
The friction factor inside the tube for Er/di > 10 -4 is [13] fDw = 0"3086[Iog,0 {6.9/Rew + ((Er/d~)/3.7) L,l}] -2.
(8)
The frictional pressure drop inside the tubes per unit length is mpfw ~
2 0.5fi~wPwmVw/di .
(9)
The total water pumping power is __
2
Pw - ~zdi 6pfwVwLw/(4rlprlem),
(10)
where Lw is the total equivalent flow length, based on the heat exchanger tubing geometry, to make provision for additional flow resistances (e.g. bends, headers, valves). The waterside heat transfer coefficient is [14] kwm(fow/8)(Rew - 1000)Prwm[l + ( d i / t t ) 0"67] 0.67 dill + 12.7(fDw/8)0.5 (Prwm -- 1)]
hw =
( 1l)
The logarithmic mean temperature difference correction factor for cross-flow conditions can be expressed as [15] 4
FT = 1 -- ~
4
~ ai~(l
-
43) k
sin[2i arctan(4~/4z)],
(12)
i=lk=l
where the values of aik are individual to each heat exchanger configuration [16] and 4, = (Tw~- Two)/(Tw~- Ta~) 42 = (T.o - T a J / ( T w ~ - Ta~)
43 = ( 4 ~ - 42)/(
(13)
In practice the D A L R rarely occurs in the first l0 m above ground level, Tat is obtained by extrapolating the measured D A L R to the ground level and will usually differ from the actual
224
A.E. Conradie and D. G. Kr6ger
measured value. At elevation 6, which corresponds to the top of the cooling tower, the ambient air temperature is Ta6 = Tal
-
-
0.00975/-/5.
(14)
The approximate temperature at the inlet to the heat exchanger bundles is Ta3 ~ Ta~ - 0.00975H3.
(15)
To derive the draught equation, consider the variation with elevation of the pressure in the atmosphere external to the dry-cooling tower in a gravity field, i.e.
dp, = - pag dz.
(16)
For a perfect gas the following relation holds:
Pa =pa/RZa •
(17)
Substitute equations (13) and (17) into equation (16) and integrate to find the pressure difference between point 1 and a point at elevation z external to the cooling tower (refer to Fig. 2):
pa~--paz=pa~[1 --(1 --O.O0975z/Ta~)~°2564*/X]~pa~[1 --(1 --O.O0975z/Ta~) 35] with g = 9.8 m s-: and R = 287.08 J kg -~ K
-1.
(18)
The pressure external to the tower at section 6 is
Pa6 = Pal (1 -- O.O0975Hs/Tdl )3.5.
(19)
Stagnant ambient air at 1 accelerates and flows across the tower supports at 2 before flowing through the heat exchanger bundles from 3 to 4. The flow is essentially isentropic from 4 to 5. In most practical towers, the change in kinetic energy between sections 4 and 5 is normally approximately an order of magnitude smaller than the corresponding change in potential energy. A total pressure balance between 1 and 5 yields Pal -- [Pa5 -]- O~es(ma/As)2/(2pas)] = (Kts + Kct + Kctc+ Khe + Kcte)he(ma/Afr)2/(2pa34) --bPal [1 -- { 1 -- 0.00975(//3 + Ha)/(2Ta, )}3.5] +Pa4[1 -- {1 -- 0.00975(H5 -- H3/2 - H4/2]/Ta,}3"5].
(20)
For easy comparison of all the flow losses, all the loss coefficients K, are based on the frontal area of the heat exchanger and the mean air density through it. The frontal area, Art, is the projection of the effective finned surface as viewed from the upstream side. Obstructions located up against the finned surface which impede the flow through the heat exchanger must be considered when evaluating the effective frontal area. Du Preez and Kr6ger [17] studied the velocity and pressure distribution in the outlet plane of hyperbolic natural-draught cooling towers. They find that for 1/Fro ~< 3, the velocity distribution is almost uniform, i.e. ~e5 ~ 1 for dry-cooling towers where the heat exchangers are located in the cross-section near the base of the tower. The mean pressure at the outlet plane is found to be slightly less than that of the ambient air at the same elevation, i.e.
Pa5 = Pa6 + Apa56= Pa6 "b Kto (ma/A5 )2/(2pa5).
(21)
For a hyperbolic tower with a cylindrical outlet the loss coefficient is given by gt ° = Apa56/(pasvas/2)2 = 2pasApa56/(ma/As)2 = --0.28 Fr~ I + 0.04 Fr~ 1~,
(22)
where Fro
= (ma/As)2/[Pas(Pa6
--
Pas)gds].
This equation is valid for 0.5 ~< ds/d3 <~0.85 and 5 ~< Khe ~<40. The approximate temperature at the tower outlet is Ta5 ~ Ta, - 0.00975(H~ -/-/4).
(23)
Evaluation of dry-cooling systems for power plants
225
From the perfect gas relation it thus follows for Pa5 ~ Pa6, that the density at the outlet of the tower is P,5 = Pa6/[R { Ta4 - 0.00975(//5 -/-/4)}].
(24)
The density of the ambient air at elevation 6 is Pa6 =
(25)
Pa6/RZa6.
If dynamic effects are neglected, an approximate expression for Pa4 may similarly be obtained: Pa4 '~Pal [1 -- 0.00975(/-/3 + H4)/(2Tat)] 35 -- (Kts + K~t + K¢~¢+ Kho +
gcte)he(ma/Afr)2/(2pa34).
(26)
Substitute equations (19), (21) and (26) into equation (20) and find, with ~es = 1, p,~ [{1 - 0.00975(/-/3 + H4)/(2T~ )}35{1 - 0.00975(//5 - / / 3 / 2 - H4/2)/Ta4} 35 - (1 - O.O0975Hs/Tal)3"5] = (Ku + Kct + Kct¢+ Khe +
gcte)he(ma/Afr)2/(2pa34)
x [1 - 0.00975(H 5 - 1-13/2
-
n4/2)Ta4] 3"5"~ (1 + Kto)(ma/As)2/(p~,5).
(27)
This equation is known as the draught equation for a natural-draught dry-cooling tower where the heat exchangers are arranged horizontally in the base of the tower. If the heat exchangers are arranged in the form of A-frames or V-arrays, Khe (non-isothermal) is replaced by Kot. In determining the dry air density after the heat exchanger, the specified pressure at ground level can be employed in the perfect gas relation, i.e. P a 4 ~'~ Pal/(RZa4)"
(28)
The approximate air density at section 3 is Pa3 ~Pal/(RZa3).
(29)
The harmonic mean density through the heat exchanger is given by P~34~ 2pa~/[R(T,3 + T~,)].
(30)
The loss coefficient of the tower supports Kts, based on conditions at the heat exchanger, is gtshe ~
2 ,Oa34Cots Lts dts n~sArr/[P,~ (r~d3/-/3)3],
(31)
where Cots is the drag coefficient of the tower supports, tts is the support length, dts is its effective diameter or width and nts is the number of supports. Due to separation at the lower edge of the tower shell and distorted inlet flow patterns, a cooling tower loss coefficient, KCt, based on the tower cross-sectional area at 3, can be defined to take these effects into consideration. For dry-cooling towers where Kh~ 1> 30 and 5 ~< d3/H3 <~ 10, Geldenhuys and Kr6ger [18] recommend the following expression based on conditions at the heat exchanger: Kct = (Pa34/Pa3) (Afr/A3)2[O.O72(d3/1-13)2_ 0.34(d3/113) + 1.7].
(32)
Only a portion of the available area in the cooling tower base is effectively covered by heat exchanger bundles due to their rectangular shape. The reduction in effective flow area results in contraction and subsequent expansion losses. The contraction losses can be approximated by loss coefficients based on the effective reduced flow area, Ae3 [19]: get c =
1 --
2#re + I/a~.
(33)
The contraction coefficient, at, is given by ac = 0.6144517 + 0.04566493a~3 - 0.3366510"23 + 0.4082743a 33+ 2.672401 a~3 -- 5.963169a~3 + 3.558944a63.
(34)
The expansion losses can be approximated by K~o = (1 - A o 3 / A 3 ) 2 = (1 - #03) 2.
(35)
226
A, E. Conradieand D. G. Kr6ger
The effective area, A,3, corresponds to the frontal area of the heat exchanger bundles if they are installed horizontally. In the case of an array of A-frames, Ae3 = Aft s u b 0 b and thus corresponds to the projected frontal area of the bundles. Based on the conditions at the heat exchanger, the above expressions become K~tche= Ket~(P~34/Pa3) (A fr~he3 )2
(36)
gctehe= gcte(Pa34/Pa4)(aft/Ae3 )2.
(37)
and
For non-isothermal oblique flow through an array of V-bundles, the following relation holds for the loss coefficient [20]:
¥-- 0a4/
= Ko,,o +
1 ( 1;in ) ( s0m in0rn
1
_1._2K0.5~( 2pa4 .~..]_gd~. ?Pa3 ~, (38) //,Pa3 "Ji-Pa4/ ,Pa3 -I- Pa4/
where a is the ratio of the minimum free flow area through the heat exchanger bundle to the free stream flow (frontal) area. For radially finned tubes the heat exchanger loss coefficient under normal non-isothermal flow conditions can, in general, be expressed as [21]
ghe : ~ [EH ..~_(Pa3- Pa4X)l= ~ [ Apapare (Pa3= Pa4X)l" \Pa3 "}-,Oa4/_] L G~ + kP~3 + P~4,//
(39)
Robinson and Briggs [22] propose the following correlation for the pressure drop across the bundles with an equilateral, triangular, tube layout: (~2 / p ~-0.927/ ]9 \0.515
Ap a = 18.93n r Rea0.316 - ¢ i - t |
[~ t |
Pa ~ r J
~gdJ
,
(40)
where Rea = G~dr/#~
and
Pd = [(Pt/2) 2 + (P02] °5 (diagonal pitch).
The entrance contraction loss coefficient K~ ~ 0.05 for many round industrial finned tube heat exchangers. The mean flow incidence angle may be expressed by the following empirical relation
[20]: 0m = 0.00190~ + 0.91330b -- 3.1558,
(41)
where the heat exchanger bundle semi-apex angle, 0b, is given in degrees; Kd is the downstream loss coefficient that includes the jetting and kinetic energy losses, and can be expressed in terms of the following relation [20]: Kd = exp(5.488405 -- 0.21312090b + 3.533265 x 1 0 - 3 0 5 - 0.2901016 x 10 -403).
(42)
Forced-draught direct air-cooled condenser Consider the forced draught air-cooled condenser shown schematically in Fig. 4. In this configuration the heat exchanger bundles are arranged in the form of an A-frame. The air-cooled heat exchanger may consist of one or more rows of finned tubes, each row having a different fin pitch. The airflow across the heat exchangers is created by means of axial flow fans. The heat transfer rate from the condensing steam to the air stream is Qa = maCpam(Ta6 -
Tas) = mcifg=
Qc-
(43)
If the geometry of the finned tubes changes in consecutive rows, the heat transfer is
nr iffil
nr
macvam(,)(Tao(O-Tai(i))= £
nr
mc(i)ifg(i)= E iffil i=l
m.C~m(o(T~-(o- Tai(i))e(i),
(44)
where the effectiveness for each tube row is [23] e(o = 1 - exp[-(1/ha~(oA~( o + 1/hc(oA,~o)-l/(m~cpam(o)], A¢(0 is the inside tube area of tube row (i) exposed to the condensing steam.
(45)
Evaluation of dry-cooling systems for power plants
227
Steam header
Y
Windwall Heat exchanger Condensate
drain
i
Walkway
L --1' ~ - - - " J
Lw
Fan Safety screen I-I 3
Supports
Fig. 4. Air-cooled condenser unit.
The pressure losses in the turbine exhaust steam duct and at the inlet to the finned tubes are expressed as Apsd = 0.5pvi(l)Vvi(l)(1 2 -- 0-2 q- Kc + K~),
(46)
where 0-c is the tube inlet contraction area ratio and the contraction loss coefficient, K¢ ~ 0.6 (sharp inlet), K~ is the loss coefficient for the steam duct system. The mean saturation pressure of the steam at the inlet of the finned tubes, Pv~, is calculated by subtracting Ap~ from the mean saturation steam pressure at the turbine outlet. The inlet conditions (thermophysical properties) of all the tube rows are thus assumed to be equal. Due to pressure changes along the finned tube and in the direction of air flow, the condensation process will not take place at a constant temperature. The mean static pressure in the finned tubes can be determined from the correlations given by Groenewald and Kr6ger [24]: Pyre,) : P v i
0.1 58 2/.t ~i L t . . . . .
- - ~
Pvi ae Kevi(i)
1~ _2.75
1.75
2
2
to.zo/a~
(47)
The coefficients at and a2 are functions of the suction Reynolds number, Revn. For round tubes, these coefficients are al¢o = 1.0046 + 1.719 x 10 -3 Revn,) -- 9.7746 x 10 -6 Re~n<0
a2
(48)
and the suction Reynolds number is expressed as R e vn(i) = (Pvi Uvi(i)di/Pvi ) ( d i / 4 L t ) = Revi(i)(di/4Lt
).
(49)
It is assumed that condensation occurs at the mean steam temperature, Tvm<~),corresponding to the mean steam pressure inside the tubes. It should be noted that the pressure drop in the different tube rows is usually not identical, with the result that backflow of steam will occur [6]. To avoid this and the corresponding accumulation of non-condensables, a dephlegmator is usually installed after the condenser. F o r round tubes, the mean condensation heat transfer coefficient for inclined tubes according to Schulenburg (25) is used, i.e. h~i) = 1.197(sin 0 b) 0.175k era(i)/ d i(Pcm(i)~vm(i)/P~m(i)~cm(i)) 0.5 Re 0.325 vi(i) •
(50)
228
A.E.
Conradie and D. G. Kr6ger
Kr6ger [1] presents a detailed derivation of the draught equation for the air-cooled condenser shown in Fig. 4, i.e. Pa~
[{1 -
0.00975(H 7 --
n6)/Ta6 }3.5 _
{1 - - 0 . 0 0 9 7 5 ( H 7 - -
H6)/Tal
}3.5]
gts (m a/A 2)2/(2pa I ) -'b gup ( m a / A Fe)2/(2pa3 ) - - gFs (ma/AFc)2/(2pa3 )
"q-gdo(ma/AFe)2/(2pa3) "b got(ma/Afr)2/(2pa56).
(51)
The choice of a suitable axial flow fan must be such that it will efficiently deliver a cooling air flow rate that will guarantee the desired heat transfer rate. An example of performance curves of a large fan is shown in Fig. 5. In actual installations the fan operating efficiency is a function of the fan and drive system design, system configuration and operating conditions [1, 26, 27]. Fan control to vary the air mass flow rate through the cooling system's operating range has great economic advantages. I n large cooling plants covering a considerable area and including numerous fans, some of the fans may be subjected to significant cross-flow, which tends to distort inlet conditions to the fan, resulting in a corresponding reduction in performance. The effect of the inlet losses is correlated by Salta and Kr6ger [28] for more than two fan rows as a volumetric effectiveness, i.e.
ev = VF/VFid
=
0.985 - exp[-(1
+
45/n)H3/6.35dr],
(52)
where n = nvr for a freestanding fan platform and n = 2nFr for a non-freestanding fan platform. This correlation is applicable for Hb/dE = 0.19 and WE~de = 1.27, where Hb is the height of the bellmouth fan inlet from the fan platform and WE is the fan pitch; net is the number of fan rows (number of fans per bay), nFb is the number of fan bays and the total number of fans are nF = nrr nFb; Vr/VFid correlates the effect of the inlet losses on the entire fan system. The actual airside heat transfer rate can be obtained by multiplying the maximum ideal airside heat transfer rate by the fan volumetric effectiveness ev. The operating point of the fixed-geometry dry-cooling system is defined as the combination of operating variables that will simultaneously satisfy the draft and heat transfer equations for specified process fluid and ambient air conditions. The geometrical variables represent the physical dimensions of the system components and the operating variables represent the operating conditions.
Combined performance of dry-cooling systems and steam turbo-generators The dry-cooling systems can be connected to the low-pressure side of specially designed high back-pressure steam turbines. The turbo-generator-condenser characteristics, as shown in Fig. 6, are expressed in terms of the turbine back pressure, Ptv, or the corresponding vapour temperature, Tt,. The turbine operates between fixed back-pressure limits to prevent the occurrence of
Z
300 !
220
250
210
200
200
e,
<~
150
J
e~
o ,-
100 -
Fan diameter = 9.145 m ~ Blade angle = 16 °
190 O
A i r density = 1.2 kg/rn 3 - Speed = 125 r p m
50
L~
170
1 1
0 200
300
400
500
600
160 700
Volume flow rate, Vr, ma/s Fig. 5. Performance of an axial flow fan.
800
900
Evaluation of dry-coolingsystems for power plants
229
240
365
235
360
230
~'~
355
225
.0
c5 O
~220
.4 0 o
.2
215
"O
~ 210
/ l
205
1
l
45
50
200 40
J
330 325
55
60
65
70
75
80
85
90
95
Turbine exhaust saturated steam temperature, Try,*C Fig. 6. Performancecharacteristics of a turbo-generator. flow-induced vibrations and choking [29]. It is assumed that the power needed for the auxiliaries other than cooling water pumps or axial flow fans is already subtracted from the generated power. For indirect dry-cooling the temperature of the heated water that leaves the surface condenser and the condensing turbine exhaust steam differ with the value of the terminal temperature difference (TTD). Typical values for T T D range from 2 to 4°C for surface condensers. For a fixed condenser design and off-design conditions, the relationship between the heat rejection rate and the T T D can be approximated as T T D ~ TTDdesign Q/Qdesign.
(53)
In the case of the direct condensing system the temperature difference in the condensing and exhaust steam is caused mainly by frictional effects. Atmospheric conditions, especially changes in the ambient temperatures, influence the performance of dry-cooling systems, resulting in a wide fluctuation of turbine back-pressure. An example of the annual mean hourly frequency of ambient temperatures, as shown in Fig. 7, is used for selecting cooling system design and operating conditions. The operating point of the turbo-generator is determined by matching the operating point of the dry-cooling system and the performance characteristics of the turbo-generator at a specific ambient air temperature selected from the annual ambient temperature frequency set. This calculation involves the selection of turbine exhaust conditions such that the heat to be rejected by the heat exchanger after the turbo-generator equals the heat absorbed by the air (heat rejected 8OO 700
Drybulb temperature
g 6oo 500 400
300
200
k
100 0 -5
0
5
10
15
20
25
30
Temperature, T, °C Fig. 7. Frequency of ambient dry- and wetbulb air temperatures.
35
230
A.E. Conradie and D. G. Kr6ger
by the dry-cooling system), while satisfying the draught equation. At this point the power requirement of the cooling water pumps or the axial flow fans, as well as the generator power output, are known. Subtract the total auxiliary power consumed by the pumps or fans from the generator power output to find the net power output of the plant. The net power output is multiplied by the corresponding number of operating hours to give the net energy output for this period. These calculations are repeated for each of the ambient temperatures listed to obtain the total annual net energy output: E, = ~ i=1
(Ptg(i)-
(54)
eaux(i))'~(i) .
The performance of dry-cooling systems deteriorates with increasing ambient air temperatures, resulting in an increased turbine back-pressure and a decrease in power generation. Air humidification [30] may be considered to increase the cooling capacity during these conditions. NUMERICAL EXAMPLES In this section we present two practical examples to illustrate the performance evaluation of dry-cooling systems with particular reference to power plants. The details of the existing dry-cooling systems used are listed in Appendix A. The first example considers a natural-draught indirect dry-cooling tower, while the second deals with a forced-draught direct air-cooled condenser. In both examples, the operating point of the dry-cooling system is computed at the design atmospheric and process fluid conditions. The annual net power output is also computed when the dry-cooling systems are coupled to turbo-generator sets. The draught and energy equations are simultaneously solved to determine the performance characteristics of the dry-cooling systems [31]. The results of each problem are listed below.
Example 1: natural-draught indirect dry-cooling tower Operating point results: m a = 11,020 kg s-1 Pw=211.5kW
Ta4 = 45.3°C
Q =331.063MW
Two = 43.4°C 1/FrD=2.841.
When the cooling tower is coupled to the turbo-generator, an annual net power output of E, = 2074.337 GWh is generated, while 2886.389 GWh heat is dissipated. The turbine operates in the following region: 45.87°C ~< Try ~< 83.08°C.
Example 2: forced-draught direct air-cooled condenser Operating point results (the values shown are for a typical unit as shown in Fig. 4): ma = 540.821 kg s -I ev = 92.665% Tvm(l) -~-
59.122°C
VF = 529.381 m 3 s -l
Ta6 = 37.30°C Tvm(2) =
PF = 101.007 kW
Tvi = 59.372°C
59.143°C
mc = 4.675 kg s i.
For the entire air-cooled condenser consisting of 30 units, the heat dissipated is 331.08 MW. When the air-cooled condenser is coupled to the turbo-generator, an annual net power output of En = 2058.351 G W h is generated, while 2877.138 GWh heat is dissipated. The turbine operates in the following region: 44.42°C ~< Try ~< 78.34°C. The above examples illustrate that the operating point and power generation calculations provide the feature to generate detailed information on the performance of the dry-cooling systems. The performance of dry-cooling systems decreases at high ambient temperatures and the auxiliary power requirements vary accordingly. The interaction of the dry-cooling system and the turbo-generator unit forms an integral part of the design process. Designing a dry-cooling system in this way is different from the conventional means of design that is purely based on fixed ambient and process fluid conditions. Finding the best design over a range of operating conditions will definitely result
Evaluation of dry-cooling systems for power plants
231
in a more realistic design. However, this design method will involve a considerable amount of computation. A practically similar procedure was followed in the design of the indirect dry-cooling system of the Kendal Power Station [10]. Operating point calculations can still be performed to analyze the cooling system's performance characteristics at any prescribed conditions. The illustration is by no means complete and the programs can be used in a variety of other applications concerning design calculations, performance evaluation and cost estimation. CONCLUSIONS
Dry-cooling systems are a n e n v i r o n m e n t a l l y s o u n d alternative to wet-cooling systems. T h e number and size of these installations will probably increase in future. However, due to their high capital and operating costs, reliable design practices are of the utmost importance to ensure their effective performance. In this study we present a practical design procedure for dry-cooling systems with particular reference to power plant applications. When properly implemented, this procedure provides a reliable practical tool with which realistic answers and trend information can be obtained. In general, dry-cooling system designs were usually based on a fixed operating point. However, the only correct way to compare different cooling systems and assess them from an economic point of view is to design them by taking the turbo-generator characteristics, the annual frequency of ambient air temperatures, the planned number of operating hours and the internal power consumption into account. A standard dry-cooling system design will not necessarily be optimal, therefore each plant and site should have an optimized dry-cooling system. One of the major components of any engineering optimization study is system modelling or simulation. The authors have extended the above study to perform cost-optimal designs of dry-cooling systems. Acknowledgement--The financial support from the Water Research Commission is gratefully acknowledged.
REFERENCES 1. D. G. Kr6ger, Fan performance in air-cooled steam condensers. Heat Reeot:ery Systems & CHP 14(4), 391-399 (1994). 2. J. P. Rossie and E. A. Cecil, Research on Dry-type Cooling Towers fi~r Thermal Electric Generation: Parts I and 11. EPA Report, Water Pollution Control Research Series 16130 EES, November (1970). 3. E. C. Smith and M. W. Larinoff, Power plant siting, performance, and economics with dry cooling tower systems. Proc. American Power Con/i 1970. Vol. 32, pp. 544 572 (1970). 4. H. Heeren and L. Holly, Dry cooling eliminates thermal pollution. Energie Z3 (1971). 5. E. S. Miliaras, Power Plants with Air-cooled Condensing Systems. MIT Press, Cambridge, Massachusetts (1974). 6. M. W. Larinoff, W. E. Moles and R. Reichhelm, Design and specification of air-cooled steam condensers. Chem. Engng, 22 May, 86 94 (1978). 7. A. Montakhab, A survey of dry cooling tower technology for power generation application. In Heat Exchangers, ThermaI-Hv~h'aulic Fundamentals and Design (Edited by S. Kakac, A. E. Bergles and F. Mayinger), pp. 799-816. Hemisphere, New York (1981). 8. A.J. Ham and L. A. West, ESCOM's advance into dry cooling. VGB Conf., 9 13 November 1987, Johannesburg, South Africa, pp. 33 43 (1987). 9. H. Knirsch, Design and construction of direct dry cooling units. VGB Conf. 9 13 November 1987, Johannesburg, South Africa, pp. 54 69 (1987). 10. B. Trage and F. J. Hintzen, De,Sgn and construction of indirect dry cooling units. VGB Conf., 9-13 November 1987, Johannesburg, South Africa, pp. 70-79 (1987). I 1. D. E. Briggs and E. H. Young, Convection heat transfer and pressure drop of air flowing across triangular pitch banks of finned tubes. Chem. Engng Prog. Syrup. Series 59(41), 1-10 (1963). 12. T. E. Schmidt, La production calorifique des surfaces munies dailettes. Annexe Du Bulletin De L'Institut International Du Froid, Annex G-5 (1945-1946). 13. S. E. Haaland, Simple and explicit formulas for the friction factor in turbulent pipe flow. Trans. ASME J. Fluids Engng 105(3), 89 90, March (1983). 14. V. Gnielinski, Forsch. lng. Wesen 41(1) (1975). 15. W. Roetzel, Berechnung yon W/irmefibertragern, VDl-W6rmeatlas, 4.Auflage, VDI, D/isseldorf, Cal-Ca31 (1984). 16. W. Roetzel and F. J. L. Nicole, Mean temperature difference for heat exchanger design--a general approximate explicit equation. J. Heat Transfer, Trans. A S M E 97(1), February (1975). 17. A. F. Du Preez and D. G. Kr6ger, The influence of a buoyant plume on the performance of a natural draft cooling tower. Paper presented at the 9th IAHR Cooling Tower Conf., September 1994, Brussels. 18. J. D. Geldenhuys and D. G. Kr6ger, Aerodynamic inlet losses in natural draft cooling towers. Proe. 5th IAHR Cooling Tower Workshop, Monterey (1986). ATE 163
D
232
A . E . Conradie and D. G. Kr6ger
19. W. M. Kays, Loss coefficients for abrupt changes in flow cross-section with low Reynolds number flow in single- and multiple-tube systems. Trans. A S M E 72(8), 1067-1074 (1950). 20. J. C. B. Kotz6, M. O. Bellstedt and D. G. Kr6ger, Pressure drop and heat transfer characteristics of inclined finned tube heat exchanger bundles. Proc. 8th International Heat Transfer Conf., San Francisco (1986). 21. D. G. Kr6ger, Performance characteristics of industrial finned tubes presented in dimensional form. Int. J. Heat Mass Transfer 29(8), 1119-1125 (1986). 22. K. K. Robinson and D. E. Briggs, Pressure drop of air flowing across triangular pitch banks of finned tubes. Chemical Engng Progress Symp. Series 62(64), 177-184 (1966). 23. J. P. Holman, Heat Transfer. McGraw-Hill, New York (1986). 24. W. Groenewald and D. G. Kr6ger, Effect of suction on turbulent friction inside ducts (submitted for publication). 25. F. J. Schulenberg, W~irmeiibergang und Druck~.nderung bei der Kondensation von str6mendem D a m p f in geneigten Rohren, Dr-Ing. thesis, Universit/it Stuttgart (1969). 26. R. C. Monroe, Minimizing fan energy costs, Part 1. Chem. Engng, 27 May, 141-142 (1985). 27. R. C. Monroe, Minimizing fan energy costs, Part 2. Chem. Engng, 24 June, 57-58 (1985). 28. C. A. Salta and D. G. Kr6ger, Effect of inlet flow distortions of fan performance in forced draught air-cooled heat exchangers. Heat Recovery Systems & CHP, 15, 555-561 (1995). 29. Z. Szab6, Why use 'Heller System'? Circuitry, characteristics and special features. Paper presented at the Symposium on Dry Cooling Towers, Tehran (1991). 30. T. A. Conradie and D. G. Kr6ger, Enhanced performance of a dry-cooled power plant through air precooling. ASME Paper 91-JPGC-Pwr-6, presented at the ASME International Joint Power Conf., October 1991, San Diego (1991). 31. A. E. Conradie, Performance optimization of engineering systems with particular reference to dry-cooled power plants, Ph.D dissertation, Department of Mechanical Engineering, University of Stellenbosch, Stellenbosch, South Africa (1995).
APPENDIXA:
NUMERICAL
EXAMPLES
Example 1: natural-draught indirect dry-cooling towers A natural-draught hyperbolic concrete cooling tower, as shown schematically in Fig. 2, has the following dimensions: H 5 = 120 m, H3 = 13.67 m, d 5 = 58 m and d 3 = 82.958 m. The tower shell is supported by 60 tower supports of length Lts = 14.547 m with an effective diameter of dt~ = 0.5 m. The drag coefficient of each support is Cot~= 2. It is further assumed that the tower shell has a uniform thickness of lets = 0.25 m. Extruded, bimetallic, finned tubes, as shown in Fig. 3, are used. The steel core has an inside diameter of di = 21.6 mm and an outside diameter of do = 25.4 mm. The aluminium fins have a diameter of df = 57.2 mm with a root diameter of d r = 27.6mm, a mean thickness of tr= 0.5 mm and a pitch of P r = 2.8 mm. The density of the steel core tube is Pt = 7850 kg m -3 and its thermal conductivity is k t = 50 W m - i K - 1. The aluminium fins have a density of Pf = 2707 kg m -3 and a thermal conductivity of k r = 204 W m -~ K - L The tube inside surface roughness is 0.0115 mm. The finned tubes bundles are arranged radially in the form of V-arrays at bundle semi-apex angles of 0b = 30.75 ° in the base of the tower shell. Due to geometrical considerations, only 52.4% of the tower inlet cross-sectional area is covered by a total o f n b = 142 heat exchanger bundles. Each bundle is L t = 15 m long and Wb = 2.262 m wide. Each bundle contains ntb = 154 tubes arranged on a triangular pitch with Pt = 58 mm and Pt -- 50.22 mm. There are n r = 4 tube rows with ntr = 39 tubes per row and nwp = 2 water passes. The total equivalent flow length is twice the length of the finned tubes. Hot water at a mass flow rate of mw = 4390 kg s-~ is pumped through the finned tubes. The design inlet water temperature is Twi = 61.45°C.
Example 2: forced-draught direct air-cooled condensers A forced-draught direct air-cooled condenser consists of an array of nrr x nrb = 5 × 6 = 30 A-frame units as shown schematically in Fig. 4. Each unit has an axial flow fan of dr = 9.145 m diameter with performance characteristics as shown in Fig. 5. The fan hub diameter is dFh = 1.4 m and the fan casing diameter is dvc= 9.17 m. Each fan rotates at N v = 100 rpm and a fan drive efficiency of 90% is assumed. The upstream and downstream loss coefficients due to obstacles close to the fan are Kup = 0.297 and Kdo = 0.391, respectively. The non-freestanding fan platform is H 3 = 30 m above the ground level. It is supported by steel beams with an effective loss coefficient of Kts = 1.5 based on the frontal area of the heat exchanger. Walkways of Lw = 0.4 m width are located between adjacent A-frames and the condenser is surrounded by windwalls. The condenser consists of two rows of extruded, bimetallic, finned tubes, as shown in Fig. 3. The steel core has an inside diameter of di = 35.1 mm and an outside diameter of do = 38.1 mm. The aluminium fins have a diameter of dr= 69.9 mm with a root diameter of dF = 40.3 mm, a mean thickness of tr = 0.35 mm. Each tube row has different performance characteristics such that approximately the same amount of steam condenses in each tube row to ensure that non-condensables are not trapped in the condenser. The fin pitches are P~) = 3.63 mm and P~2) = 2.54 mm, respectively. The density of the steel core tube is Pt = 7850 kg m -3 and its thermal conductivity is k t = 50 W m - t K - ~. The aluminium fins have a density of pf = 2707 kg m -3 and a thermal conductivity of k r = 204 W m - J K -~. There are n b = 2 heat exchanger bundles above each fan. The finned tubes are L t = 10 m long and arranged at a bundle semi-apex angle of 0b = 30 °. The tube rows have nt~o) = 152 and ntr(2 ) = 153 tubes, respectively, that are arranged on a triangular pitch with Pt = 76.2 m m and P~ = 65.991 mm and the bundle width is Wb = 11.659 m. Saturated steam at a design temperature of 60°C is supplied to the condenser by means of a steam header with an effective diameter of dsh = 1.25 m. The steam ducting loss coefficient is K~ = 0.6. In both examples, the cooling systems operate under a design ambient temperature of T~ = 15.6°C and a design atmospheric pressure ofp~ I = 86,400 N m -2. The systems operates 8760 h per annum and their operating lifetime is assumed to be 30 yr. The cooling systems forms part of a power plant that is erected at the location corresponding to the design atmospheric pressure state above. The corresponding annual temperature distribution (drybulb and wetbulb temperatures) is shown in Fig. 7. The cooling systems are coupled to turbo-generator sets, the performance characteristics of which are shown in Fig. 6. The design point terminal temperature difference for the indirect dry-cooling system is 2.5°C at a heat transfer rate of 327.63 MW.