It can be realized that in the present case the nonlinear terms are of the rather simple uZ form, so that very simple symmetry rules for the decom- position polynomials can be used.
Ifwe denote L == djdt, the formal solution of (18) may be put in the form
Ni(t) =Ni(O)+L-1(biNi +'faijNiNj),
)=1
i = 1,2, ...,m (27)
where L-1 == f~[ã]dt. According to the decomposition method an expansion of the following form is assumed
00
Ni(t) = L Nin,
n=O i =1,2, ...,m (28)
Substituting (28) into (27) gives for i =1,2, ... , m
or after rearranging the products
(29)
(30)
for i =1,2, ... , m. The solution is ensured by requiring
i=1,2, ...,m (31)
Ni2 = L-1(biNil + f aijt N ik N j (l-k)), j=1 k=O
i =1,2, ...,m
i=1,2, ...,m
(32)
(33)
i = 1, 2, ... ,m (34)
N in = L-1(bi N i(n-1)+ f aij'fN ik N j (n-k-l)), J=1 k=O
After carrying out the integrations, the following solution is obtained
where
i=1,2, ...,m
i=1,2, ...,m
(35)
(36) and the general term is defined through the following recurrence relation for i = 1,2, ...,m
m n-l
. _ b. . ( _ )' ' " ' " ..Cik Cj(n-k-l) ( )
Cm - t Ct(n-l) + n 1.~t:oatJ k! (n_ k_ I)!' n ~ 1 37
The decomposition method does not assure, on its own, existence and uniqueness of the solution. In fact, it can be safely applied when a fixed point theorem holds. A theorem proved in [4] indicates that it is hopeless to look for solutions globally in time. On the other hand, the decomposition method can be used as an algorithm for the approximation of the dynamical response in a sequence of time intervals [0,tI),[tI,t2), ... ,[tn-l,T) such that the condition at tpis taken as initial condition in the interval [tp, tp+l) which follows. This method has the following advantages:
1. In each time-interval one can apply a theorem proved in [4], which states that the solution obtained by the decomposition method con- verges to a unique solution as the number of terms in the series becomes infinite.
2. The approximation in each interval is continuous in time and can be obtained with the desired approximation corresponding to the desired number of terms.
The current solution of the Lotka-Volterra has now been validated against many cases, and proves to be fast and accurate, see, e.g. [3]. A similar solution was derived for a broader class of equations [5].
5. Methods for Analyzing the Potential Impact of Pumped Storage on the Lower Reservoir Aquatic Ecology
An existing pumping facility which conveys water of the Sea of Galilee to a storage cistern at Mevo Hamma, was exploited to investigate the impact of forces exerted during pumping on plankton. Comparisons were made between water samples taken close to the lake intake with samples collected in the cistern.
It was observed that most algal cells were not damaged mechanically by passage through the pumping machinery. The dinoflagellate, Peridinium, which dominated the phytoplankton biomass neither decreased numerically nor was cell motility affected by the pumping action or water column pres- sure. The physiological capability of phytoplankton did not seem to be damaged by the pumping. This was rather surprising as these dinoflag- ellates are known to be sensitive to mechanical perturbation. We assume that if Peridinium survived this treatment, most of the other phytoplankton will also not be harmfully affected. On the other hand microzooplankton, particularly ciliates, were damaged with the passage through the pumping facility. This treatment has a more severe impact in the case of the macro- zooplankton (crustaceans and rotifers). A large proportion of these classes of organisms was disintegrated by the mechanical impact of pumping [6].
It is planed to use the controlled pressure chambers at the Amiad Water Filtration factory for further testing of pressure and turbulence effects.
We propose two theoretical methods for analyzing the potential impact of PES on the Sea of Galilee. The first one is a two species predator (zoo- plankton) - prey (phytoplankton) model. The various coefficients of autoin- crease and interaction will be determined from experiments as an inverse problem. In these experiments the populations of the phytoplnakton and zooplankton will be measured at different time intervals. The derivatives in the Lotka-Volterra equations can be obtained by differencing, and the noted coefficients computed from the resulting algebraic system equations.
This procedure will be repeated for different time intervals. If the coeffi- cients turn out to be almost constant, we have a justification of the model.
Ifnot, this model can still be used for relatively short periods of time, where the coefficients are almost constant. The model is then run in a predictive manner with the computed coefficients using the present solution method.
The first model can be extended to more than two species by employing the multispecies Lotka-Volterra equations. Its coefficients can be obtained in a manner similar to the one outlined above. The new solution method ensures that, even with many simultaneous equations, an accurate solution is obtained.
6. Concluding Remarks
A combined experimental and theoretical approach was proposed in this article to study the potential impact of a pumped energy storage facility on the lower reservoir aquatic ecology. The experimental part consists of testing the impact of mechanical forces on the water plankton in exist- ing pumping and water filtration facilities. Starting from relatively simple evolution models, some basics of the deterministic theory of population dynamics were covered here, with an extension to the multispecies Lotka- Volterra equations for either competition or predator-prey relationship. A new semi-analytical solution to these equations was derived, which enables their fast and accurate solution. The coefficients of autoinc.rease and inter- action, needed as input to this model, should be obtained on the basis of experimental observations.
References
1. Rescigno A. and Richardson I.W. (1973) The Deterministic Theory of Population Dynamics, in R. Rosen (ed.) Foundations of Mathematical Biology3, Academic Press, New York.
2. Adomian G. (1988) A Review of the Decomposition Method in Applied Mathematics, J. Math. Anal. Appl. 135, 501-544.
3. Olek S. (1994) An Accurate Solution to the Multispecies Lotka-Volterra Equations, SIAM Review,36, 480-488.
4. Repaci A. (1990) Nonlinear Dynamical Systems: On the Accuracy of Adomian's Decomposition Method, Appl. Math. Lett. 3, 35-39.
5. Vadasz P. and Olek S. (1998) Transitions and Chaos for Free Convection in a Rotating Porous Layer, Int. J. Heat Mass Transfer41, 1417-1435.
6. Yacobi Y.Z., Serruya S., Nishri A. and Berman T. (1991) Feasibility Study: Potential Ecological Impacts of a Pumped-Storage Facility on the Sea of Galilee, Israel Oceano- graphic f3 Limnological Research Ltd.,Yigal Alon Kinneret Limnological Laboratory Progress Report No. T3/91, Tiberias, Israel, 7.
AND THEIR IMPACT ON THE ENVIRONMENT
D.G.KROGER
University ofStellenbosch
Department ofMechanical Engineering Private Bag Xl
Matieland, 7602, South Africa
1. Introduction
In any power generating or refrigeration cycle, heat has to be discharged. This is also true in many chemical and process cycles, internal combustion engines, computers and electronic systems. Most of the energy contained in the fuel of a modem automobile engine or fossil-fired power plant is rejected to the environment in the form of heat.
The hydrosphere has in the past been the commonly used heat sink at industrial plants. The simplest and cheapest cooling method was to direct water from a river, dam or ocean to a plant heat exchanger and to return it, heated, to its source. Inmore recent times the task of choosing the source of cooling for large industrial plants has however become increasingly more complex. Dwindling supplies of cooling water and adequate plant sites, rapidly rising water costs usually at well beyond inflation rates in most industrialized countries, noise restrictions [1] and other environmental considerations and proliferating legislation, all contribute to the complexity [2, 3].
2. Cooling Towers
Because of restrictions on thermal discharges to natural bodies of water, most new power generating capacity or large industries requiring cooling, will have to make use of closed cycle cooling systems. Evaporative-, or wet-cooling systems (cooling towers) generally are the most economical choice for closed cycle cooling where an adequate supply of suitable water is available at a reasonable cost to meet make-up water requirements of these systems. A cooling tower is a device that uses a combination of heat and mass transfer to cool water. Examples of mechanical draft and natural draft cooling towers are shown in Figs. 1 and 2 respectively.
In a modem coal-fired power plant 1.6 to 2.5 f of cooling water are required to generate 1 kWh of electricity (e.g. ten 100 W light bulbs burning for one hour).
Cooling water lost due to evaporation and drift has to be replaced. In addition, a portion of the circulating cooling water must be systematically discharged as waste in order to limit the build-up of dissolved salts in the circulating water. In a 600 MW(e) coal-fired plant a 10 000 m3 waste stream or blowdown may have to be disposed of per day. To
163
A. Sejan et al. (eds.), Energy and the Environment, 163-174.
©1999Kluwer Academic Publishers.
reduce this volume there is a tendency to accept higher concentrations of impurities in the circulating water. This in tum may require more costly chemical treatment or other treatment to avoid excessive fouling of thefill.
\ Ai r I
out c::::=:::::><:=Fan
ColdlIater I"Qter
Drift eliminators
Hot
~ ~i~::-:~~~S:R~r~a~Y~S~~~~iii~rr..,aAir inFiII ,--, / t er
, -iR '- //
louvers "./' ain zane "-
~w..J.J",_-..:.;..;..c...:..:..-....:...::...::..c....:..:...:=--~~ wã / bas i n :::::._::::__:::::::._::::__:__::::
\ Air I
out
(a) Crossflow (b) Counterflow
FigureJ. Mechanical draft cooling towers.
\ Air I
out
\ Airout I
'Water basin Drift
Cold lIater Sprays
FiII
JI•• ~*t7/Hot
(a) Crossflow (b) Counterflow
Figure2. Natural draft cooling towers.
Drift losses are reduced by installing effective drift eliminators downstream of thefill.
Modem drift eliminators as shown in Fig. 3, are shaped to minimize drift losses at the
liminator no. I liminator no. 2 Eliminator no. 3
')(
1'><:
r-,.;
Figure3. Modem drift eliminators.
Figure4. Plastic fills: (1) and (3) film, (2) trickle grid, (4) splash.
lowest possible airflow resistance.
Modem plastic film type fills, examples of which are shown in Fig. 4 are being developed with a view improving performance and reducing fouling. The latter is achieved by encouraging turbulent liquid film flow resulting in improved rinsing. Splash type fills requiring more volume than film type fills for the same performance are used where fouling is likely to be more serious. In some zero-discharge plants, wastewater is actually evaporated in the cooling tower before discharging the remaining relatively small but highly impure stream to an evaporation pond. Improved plastic fill materials, that
can be recycled, are being developed.
Environmental requirements for limiting temperature rise of surface water and maximum temperature limit of returning cooling water has resulted in greater use being made of so-called once-through helper-cooling towers i.e. river or other surface water may be passed through a surface condenser to achieve the required cooling before being cooled in a helper-cooling tower and returned to its source. Depending on the seasonal availability of cooling water and environmental considerations, plants incorporating helper-cooling towers may be operated in an open circuit requiring no towers, a closed circuit relying on the cooling towers or an open circuit in which the cooling tower functions as a helper-tower [4].
Inthe case of wet-cooling towers other factors including the changes in micro climate, corrosion of equipment, piping and structural steel, emission of chemicals, poor visibility and freezing of ground or road surfaces located near cooling tower plumes as well as potential health hazards [5, 6] (legionnaires' disease) in poorly maintained systems, cannot be ignored in practice. The impact of all these factors on the comparative economics of alternative heat rejection systems will depend upon the unique circumstances of each particular application.
10
I.Mixing elemenls 2.Noise ottenuoters 3.Fons, dry section 4.Finned tubes, dry section 5.FiIl, wet section
6.Fans, wet section
operotion
10
7.Cold woter piping. dry section a.Main pumps
9.Condensers
1a.Ho water piping, wet section 11 .Hot water piping. dry section 12.Booster pumps
Figure5. GKN hybrid cooling tower.
When the I 300 MW(e) Neckarwestheim nuclear power plant was planned to be located amongst the beautiful vineyards adjacent to the Neckar river in Germany, the construction of an unsightly high cooling tower emitting a plume that would cast a shadow over the surrounding area was rejected by the community. The problem was solved by the construction of a relatively low hybrid-cooling tower located in an old quarry. The tower, as shown in Fig. 5, has a base diameter of 160 m and consists ofa
counterflow wet section supplied with air by 44 low noise fans. A dry-cooling section allows operation in the hybrid mode during periods when the visible plume is not desirable [7].
Other hybrid cooling towers have been constructed in areas where the wet plume may impair visibility or result in the formation of ice on nearby road or runway surfaces. Recently a hybrid cooling tower was commissioned in a Japanese national park. Strict plume control was specified to prevent the formation of frost on sensitive vegetation during winter.
._,.---
'..
Figure6. Flue gas and plume dispersion.
Spray d,s ribul,')n p'pes
Figure 7. Collecting troughs underfill.
In view of environmental considerations, cooling towers incorporating ducts that introduce desulphurized flue gas into the plume for better dispersion are operational at a number of power plants [8]. Due to the relatively compact structure of a cooling tower plume, good dispersion is obtained, leading to better ground level concentration patterns of flue gas expelled via the cooling tower than in the case of a conventional stack (see Fig. 6)
When wet-cooling towers are located near densely populated areas noise control is important. Noise due to falling droplets in the rain zone can be reduced by means of suitable inclined splash plates. Invery large natural draft cooling towers the entire rain zone can be eliminated by installing water collecting troughs immediately below the fill as shown in Fig. 7. This reduces noise and water pumping power [9].
3. Air-cooled Heat Exchangers
Inan air-cooled heat exchanger, or air cooler, heat is usually transferred from the process fluid to the cooling air stream via extended surfaces or finned tubes. While the performance of wet-cooling systems is primarily dependent on the ambient wetbulb temperature, the performance of air-cooled heat exchangers is determined by the drybulb temperature of the air which is usually higher than the wetbulb temperature and experiences more dramatic daily and seasonal changes.
Although the capital cost of an industrial air-cooled heat exchanger is usually higher than that of a water-cooled alternative (this need not always be the case) the cost of providing suitable cooling water and other running expenses may be such that the former is more cost effective over the projected life of the system. Other considerations are also of importance depending on the process or application [10]. In arid areas where insufficient or no cooling water is available, air cooling is the only effective method of heat rejection.
~c:. support bridge
(a) Forced draft. (b) Induced draft.
Figure8. Mechanical draft air-cooled heat exchanger bay.
Various air-cooled heat exchanger configurations are found in practice. In some situations the choice of design is however critical for the proper operation of the plant.
Air-cooled heat exchangers may be of the forced or induced draft type. In the case of the former the fans are installed in the cooler inlet air stream below the finned tube heat exchanger bundle as shown for a particular example in Fig. 8(a), with the result that the power consumption for a given air mass flow rate is less than that for the induced draft configuration. The fan drives located in the cooler air flow below the unit are also easier to maintain and the fans are not exposed to high temperatures, which makes the choice of construction material less critical.
Since the escape velocity of the air from the top of the bundle is low (2.5 mlsto 3.5 mls) the unit is susceptible to hot plume air recirculation. This problem may be accentuated by the proximity of similar heat exchangers or other structures. Anti- recirculation fences or windwalls are often fitted in such cases. Generally the air flow distribution through the heat exchanger is not as uniform as for the induced draft installation. Since the heat exchanger is open to the atmosphere the performance can change measurably due to wind, rain, hail or solar radiation. Hail screens may be required to protect the fInned surfaces. Since it is not always possible to model the effects of plume recirculation and the influence of distorted inlet flows on fan performance in the laboratory, increasing use is being made of numerical methods to solve these problems [11,12].
Heat exchanger
Fan Fan suppar and elk ey
Figure9. A-frame air-cooled condenser.
An induced draft system as shown schematically in Fig. 8(b) is less sensitive to certain changes in weather conditions. Generally the air flow distribution through the heat exchangers is more uniform than in a forced draft unit and because of the relatively high escape velocity of the air from the fan this type of system is less susceptible to crosswinds and plume recirculation. The usually higher
fan power consumption for a given air mass flow rate and the fact that the fan and its drive system are exposed to the warm air stream, are disadvantages of this configuration.
In large air-cooled condensers the finned tube bundles may be sloped at some angle up to 60° with the horizontal (A-frame) as shown in Fig. 9, in order to reduce plot area. This arrangement however generally has a higher air-side pressure drop [13].
There are basically two types of air-cooled or dry-cooling systems that [md application in power plants. In the so-called direct system, also sometimes referred to as the GEA system, the turbine exhaust steam is piped directly to the air-cooled finned tube condenser as shown in Fig. 10. The finned tubes are arranged in the form of an A-frame or delta to reduce the required land area. The steam exhaust pipe has a large diameter and is required to be as short as possible to minimize pressure losses. A forced or induced flow of cooling air through the fmned tube bundles is created by axial flow fans [14].
Condensate
Figure 10. Direct air-cooled condenser.
Figure11. Matimba power plant, South Africa.
Presently the world's largest direct air-cooled power plant, Matirnba, became fully operationalin1991 at Ellisrasinthe Republic of South Africa [15]. Large reserves of coal justified the erection of a dry-cooled plant inthis relatively arid part of the country. A
photo ofthe 6 x 665 MW(e) plant is shown in Fig. 11.
The air-cooled condenser consists of 384 heat exchanger bundles per unit, each almost 3 m wide and 9.6 m long, made up of two rows of galvanized plate finned elliptical tubes as shown schematically in Fig. 12, and arranged in the A-frame configuration with an apex angle of 56°. Air is forced through the bundles by 48 axial flow fans per unit, each 9.1 mindiameter, located underneath the bundles and about 45 m above ground level.
1 2 3 4
Figure 12. Finned tubes. (l) Integrally finned tube (skived fin); (2) extruded bimetallic finned tube; (3) helically wound galvanized steel finned tube; (4) galvanized plate finned tube; (5)
helically wound bimetallic finned tube.
Generator
Figure13. Indirect dry-cooling system.
A schematic drawing of an indirect dry-cooling tower incorporating a direct contact spray condenser is shown in Fig. 13. Recooled water from the cooling tower is introduced into the condenser via nozzles, such that the turbine exhaust steam condenses directly on the droplets or water jet [16].
A part of the condensate is returned to the boiler but most is pumped, at a positive gauge pressure, to finned tube heat exchangers located at the base of a natural draft cooling tower. The recooled water returns to the condenser via an energy recovery turbine, through which its pressure drops to below ambient conditions. This particular layout is also referred to as the Heller system, after the Hungarian engineer who originally proposed the concept [17].
The Kendal power plant in the Republic of South Africa is the largest indirect dry- cooled plant in the world and has a total of6 x 686 MW(e) turbines [18].
The six hyperbolic concrete natural draft cooling towers are each 165m high with a base diameter of 163m and each tower is equipped with 500 heat exchanger bundles arranged in concentric circles at the base of the tower as shown in Fig. 14. The helically wound galvanized elliptical finned tubes as shown in Fig. 12, have a total length of approximately 2000 kIn per tower. The circuit includes a conventional surface condenser. Compared to a wet-cooling system, approximately 50 x 106m3water is saved annually.
Figure14. Kendal cooling tower, South Africa.
Ongoing research has led to the development of improved heat transfer surfaces and finned tube geometries, resulting in better performance and lower plant cost. The improved all-aluminum Forgo-type finned tube shown in Fig. 15(a) [19] is employed in indirect dry-cooling towers, while the new larger flattened fmned tube shown in Fig.
15(b) finds application in direct air-cooled steam condensers.