ARARAT — A computer code for thermal design of cooling towers

NUCLEAR ENGINEERING AND DESIGN 24 (1973) 57-70 NORTH-HOLLAND PUBLISHING COMPANY

ARARAT

-

A COMPUTER

CODE FOR THERMAL

DESIGN OF COOLING

TOWERS

Miodrag M MESAROVIC ENER GOPROJEKT, Belgrade, Yugoslavm

Received 24 July 1972

A computer program is presented for thermal and hydraulic design ot coohng towers Options have been provided for the evaluation of coohng tower size and performance curves by applying a basic physical model of heat and mass transfer The solutmn is conducted by multiple lteratmn, m which lteratmn loops are mutually inclusive Both film and spray-filled coohng towers are considered with either induced or natural air circulation Numerical solutmns are presented to a number of natural draft cooling towers which serve present nuclear or conventlonal power plants

1 Introduction Coohng water supply relates both to thermal poilutmn limits and to the stze and number of thermal power plants As larger generating faclhtles are installed, condenser cooling water supply becomes more critical and often a major consideration of the plant design Since adequate water for a conventional once-through condenser arrangement is becoming scarce, the designer must provide means to supplement cooling, or an alternate cooling system, such as a conventional condenser with a cooling p o n d or cooling tower (induced draft, natural draft, dry type-closed) or a split vacuum condenser with any of the above arrangements Cooling towers may be applied to mrculatlon systems as well as to once-through systems With a closed circulation system the fresh water requirement is reduced to about 2 5%, or to zero m a true dry cooling system Depending on the applicable design parameters, the cooling air is supplied by fans (induced-draft cooling tower IDCT) or by thermal uplift (natural-draft cooling tower - NDCT) Once-through cooling towers are used when coohng water is abundant, but ecological reasons impose hmlts on the return water temperature IDCTs for large generating stations require Installation of multiple cells arranged to minimize ground foggrog and reclrculatlon of warm exhaust air Although the imtlal costs are lower than for a NDCT, the total

cost may be higher due to increased land area, piping, incremental power and maintenance, etc NDCTs with asbestos or plastic fall, or packing, have long life, low maintenance costs and low amortlsatlon rate Compared to an IDCT it is free of ground fogging and reclrculatlon of warm moist air and requires far less space The mare limitation of NDCTs IS high initial cost, sometimes double of an equivalent IDCT NDCTs become relatively more attractwe for large installations as overall capacity is increased The closed system dry-cooling tower conserves water and removes heat by sensible heat transfer, while open w e t - c o o h n g towers remove approximately 3 of the heat by evaporation of water and ~ by sensible heat transfer The mstalled cost of a dry cooling tower IS several times the cost of an open coohng tower, and consequently its apphcataon can be expected mainly in areas of serious water shortage, when the minimum supply is inadequate to provide wet cooling tower makeup requirements It is evident that optimum towers for large conventional or nuclear units, under normal financial con&tlons, on a fully evaluated basis, tend to be NDCTs However, for most cases an optimum solution should be found among the possible alternatives by applying rehable calculatlonal procedures, as well as further Investlgatmns of advanced systems Whtle It IS true that a c~rculation system wall reduce

58

M M Mesarovtd 4 R A I G t T

the fresh water requirements, the daily consumption of large modern power stations is still in excess of 10 5 m 3 To mlmmlze th~s enormous water requirement, a new system of combined wet and dry cooling is apphed, partially maintaining the advantages of both wet and dry systems General Electric ts developing a new low-silhouette NDCT [8], whach would permit multlunlt power stations to propel thermal exhaust thousands of feet upwards, where the waste heat would be dispersed into the upper atmosphere, by providing a massive slowrising plume of warm moist air that could move up through reversion layers Due to continuous growth of thermal pollution of water, coohng towers are facing a new era of wide apphcatlon and extensive development and, therefore, a better understanding of all related topics becomes an Important need for all designers, manufacturers and exploiters in this field It would permit the designer to remove numerous safety factors presently in u s e , cover a variety of uncertainties m predicting heat and mass transfer processes, hydrauhcs, etc Basic theory for the heat and mass transfer calculation was developed by Merkel [5] in 1925 The principal assumptions he introduced led to a practical handcalculation procedure Later development has been directed mainly towards a more accurate numerical solution of MerkeFs principal equation [4], out

,ff = L x F _ f Ga m

dh h~-h "

(1)

(Fuller [2] In 1956 and others), as well as to the improvement of his approximate solution In this respect, Spangemaher [7] Introduced a very accurate correction by means of an evaporation parameter Berman began along the same line in 1938 and 1941, but his latter work (m 1957) turned towards the basic theory of cooling towers, from wtuch he developed two sxmphfled solutions [ 1], similar to those of the former literature However, his basic model could also be used by designers to obtain more sophisticated solutions A straight-forward analytical solution of heat and mass transfer is difficult, especially so in turbulent flow. Consequently, for quantitative assessment of these processes in practical applications certain emplrl-

a c o m p u t e r code

cal relations are employed, such as heat and mass transfer coefficients, dlffusivlty, friction losses, et~ Development of coohng towers will also be prornoled by recent advances m rocket and reactor techniques and full-scale investigations of heat and mass transfer by phase change Besides, calculatlonA models are available for the heat and mass transter analyses in the boundary layer Itself, [3] It is, therefore possible to apply a more appropriate metht)dology of coohng tower design, based on the present status ol computing technique In this respect a computer code has been developed for thermal design of both IDCTs and NDCTs with film or spray cooling of water The principal analytical model on which the ARARAT code has been based is developed from the laws of mass, energy and nromenturn conservation as apphed to heat and mass transfer processes and hydraulics in tower fill The above laws are solved analytically by introducing a number of assumptions, whach are the same as those of Berman

[11 2 Analytical model of heat and mass transfer in cooling tower f'dl

2 1 Space-dependent conditions o] heat and mass transfer It is desirable to estabhsh a uniform space distribution of water loading at a given axial postlon of fill in order to maintain a continuous film or flow stream However, this proved to be extremely difficult, especially when water flow has to be reduced and/or some sections of the fill by-passed at low wet bulb temperatures The same apphes to the air flow distribution, which is affected both by Inlet configuration of the tower, including fans, etc, and water flow Therefore, evaporative coohng of water within a tower fill must be calculated for space-dependent condatlons of heat and mass transfer and hydraulics This IS particularly so because moxsture saturation of air takes place wathln the fill, and consequently water in the respective regions is cooled by physically different mechanisms Of course, analytical representation of the heat and mass transfer and hydraulic processes IS difficult, and requires the use of digital computers to obtain their space distribution

59

M M MesarovIF, ARARAT - a computer code

Thus, water loading g, air loachng ga and heat loading q are represented respectively by complex functions g(x ~v,z), ga(X,y #) and q(x~v,z) or g(pco), ga(O, co, z) and q(o,co, z), depending on whether rectangular or cylindrical coordinates are more appropriate m a particular case To take into account spatial varmtlon of heat and mass transfer conditions, the basic equations of mass, energy and momentum conservation are solved for a space element ds, equal either to the surface element cV for fdm-fdled tower, or to the volume dement dv for spray-filled tower

vapour pressure difference between the boundary layer and bulk air Thus, the heat qc transferred by convection and conduction from the space element ds IS qc = (tf -- 0) 0~,

(5)

The heat qe transferred by evaporation is defned by the mass of water converted to steam within space element ds, dG/ds, and the content heat of vaporization, rf l¢

ds = { dx dz and dy dz, or O dco dz and do dz, dI, = dx dy dz or O do dco dz

(2)

In the present analysis heat and mass are considered to transfer from water surface to the mowng air Irrespective of which way ds is formed The heat and mass transfer coefficients are defined accordingly [~f for film coohng, X = [ % for spray cooling, ;fv for film cooling, E =

tt

The partial pressure difference, pf - p, can be replaced by the respective moisture concentration or weight content differences If the mass transfer coefficient is calculated for another driving force, the corresponding conversion is applied For example, l f E refers to the moisture content, the conversion o f E X to the pressure related Ep is Ra

(3) S

for spray coohng

These coefficients are calculated from empirical relatmns, developed from the limited experimental data, and do not take into account the effects of the fill configuration and the state of the air and water in the space element ds This fact could justify assuming these coefficients as space independent 2 2 Bastc phystcal model o f heat and mass transfer Evaporative cooling of water in a cooling tower IS performed through several physmally different processes, convection, conduction and radiation heat transfer and surface or volume evaporation The efficiency of these processes depends on the state of the water and air Although the heat and mass transfer are mutually interrelated, overall heat transfer from the water to the air may be assumed in two separate parts (neglecting radaation) dQ/ds = q = qc + qe + qr ~- qc + qe

(6)

qe = rf dG/ds = rf E (pf - p)

(4)

Heat transfer IS governed by the temperature difference between water and air, and mass transfer by the

S

S

In general, however, Ap IS a more realistic driving force than ~ g Analytical simulation of the physical phenomena in a tower is necessarily based on the wet air properties, the prlnmpal laws and conservation equations of heat and mass transfer, similarity criteria and empirical relations for heat and mass transfer coefficients and friction losses Due to space variation of water and air propertles, the analytical model of heat and mass transfer is written in a differential form and solved rather by numerical integration or In an explicit form through further development Because atmospheric air is a mtxture of dry air and steam, it may be considered to behave as an ideal gas [6] Dalton's law for a mixture of two ideal gases yields Pb=Pa +Ps =Pa + p

(8)

By applying the equations of state for dry air and water vapour, the moisture content X is X = RaP/RsPa = Gs/G a

(9)

X" IS calculated from the above equation by mtrc~ ducing p " instead of p

60

M M Mesarovtk, ~IRARA T

Heat and mass transfer vary with the aar and water flow ratxo L, whtch is subject to a continuous change wath weather conchtlons Since both saturated and unsaturated wet air condmons are present, the respectwe options of the model should be solved simultaneously The total rate of heat transfer from the water per unit s d.Qw d(Gh)_ dG dh ds = ds - h-~-- + G ~ -

(10)

is equal to that received by the air dQa

d(Gah a + Gshs) -

ds

ds

dh dG s ~ Ga---~+ h s ds

(12)

Ttus coohng IS assumed driven by enthalpy gradient, as derived from eqs. (5), (10) and (12) dh ~--= (qc + qe )/G

(13)

The heating of the wet air also uses enthalpy gradtent, as obtained from eqs (5), (11) and (12) Bearing m mind that dG s ds

(14)

- dG/ds,

d(p/P a) ds

Rs qe Ra Garf

(16)

Apparently, eqs (13), (15) and (16) include all the parameters necessary for sophisticated calculation of heat and mass transfer from water to unsaturated wet aJr Under saturated wet air condmons the heat and mass are also transferred by evaporation, convection and conduction, but the evaporation process IS caused by the wet air temperature rise only Bemdes, an amount of steam is condensed m the wet air. which is either carried over by the air stream, or mixed with the water, thus reducing its temperature The co~esponchng mass balance is

(11)

As G a lS constant, the right hand su:le o f e q (11) does not contain its derivative G s dhs/ds is also omitted since it is neghglble For unsaturated wet air con&tlons, the water coolmg is performed by convection and conduction as well as by evaporation dQw dG dG ds - q c + r f ~ - + h ~ -

a computer code

dGs ds

dG ds

dGc ds '

(14')

and the total heat transferred from the water dQw ,, dG ds - q c + h f ~ - -

h' dGc s ds

(12')

From eqs (5), (9), (10), (12'), and (14') the water enthalpy Is dh 1 Fq c [ h's] , Rad(p/Pa)q d s - G,_ +qe [ 1 - ~ ) - h s R~ ]

(13')

By neglecting the mfluence of steam condensation on water temperature, eqs (12') and (13') would be reduced to eqs (12) and (13) respectively [1] However, the wet air energy balance is affected by this condensation process significantly Thus, from eqs (5), (9), (12') and (14') dha 1 f s ,, , [qe Ra d (Is - Ga qc + - - r f qe + (hs - hs) - G a Rs

pa)

Fr

ds - G a

c

rf

qe

(15)

The second term on the right hand side o f e q (15) is ormtted m [1], presuming that heat transfer Is already gwen m a Since eq (15) alone Is msufficmnt to represent the state of the wet mr, as the vapour component m it is superheated, an addmonal differential equation on vapour pressure is obtained from eqs (5), (9) and (14), m the form

(15') Eq (15') is sufficient to describe the behavlour of moisture-saturated air since the vapour pressure Is determined by saturation condltmns. As vapour enthalpy t, h s is very close to h i' at the boundary layer on the water surface, the middle term in the parenthesis of eq (15') has been omitted m [1]

61

M M Mesarovtk, A R A R A T - a computer code 2 3 Analyncal development

To evaluate the necessary fill stze S (either surface F or volume V), as well as to calculate the cooling effects, of a glven tower, the distribution of temperatures and pamal pressures have to be solved The main difference between the authors hes just in solving the basic heat and mass transfer equations Usual assumptions lead to a variety of solutions, differing both as to accuracy and practical applicability The basic model of section 2 2 can be solved in two ways Firstly, an accurate solution can be reached by direct numerical Integration, even on a small digital computer, without further sImphficatlon Such a solution may be rather time-consuming as it includes a great deal of xteratlve calculations in which several iteration loops are mutually Inclusive Secondly, further development of this model can be undertaken, to solve it analytically The necessary solution to evaluate cold water temperature and two outlet air parameters, for example, (given the flow and boundary conditions), must Include a number of addmonal assumptions That is usually done in two steps, first being an analytical solution wtuch is practically apphcable by the aid of computers only Because an explicit solution is very difficult, further simplifying assumptions are Introduced in order to get a model as simple as feasible for hand calculation The degree of simplicity depends on the assumptions on which the model is based However, simple techniques may prove to be Insufficiently accurate, as they Involve a large number of safety margms for covering the uncertainties due to simplifying assumptions On the other hand, it is clear that, as the number and size of cooling towers increase, particularly of these with natural draft air cIrculanon, every safety margin revolves a considerable cost penalty, and should therefore be ehmlnated by a more accurate calculatlonal procedure As a compromise, an exphclt analytical solution is adopted for the computer code ARARAT, developed from the present heat and mass transfer equations with only the Indispensable assumptions These are part of the assumptions of Berman [1], as listed below The heat and mass transfer coefficients a and E, and the specific heats C, C a and Cs are assumed constant The pamal pressure of water vapour p is small m comparison with the atmospheric pressure Pb, therefore the partml pressure of the air, Pa, is assumed constant

The amount of water evaporated is small compared to the total water flow G, so G is assumed constant The water surface temperature tf may be considered equal to the mean water temperature t, so that rf, h'f'and p~'can refer to t instead Owing to a relatively small temperature variation, the relationship between saturation vapour pressure p" and temperature t is represented by a straight hne p" = m + n t .

(17)

The error introduced by eq (17) is small enough and can be reduced even to none by a simple correction term 6p", as defined by Berman By applying the above assumptions to the basic physical model of section 2 2 , bearing in mind those already mentioned, eqs (13), (15) and (16) for nonsaturated wet air conditions will yield respectively dt

1

ds - CG [ ~(t - O) + rE(p" - p) 1,

dO I d~ C a [~ + C E ( p a

rr

(18)

I~

- p)] (t - O) ~ U ~ ( t

a

a

- 0),

a

(19) d p / ds = E R s P a ( p " - p ) / G a R a

(20)

For saturated wet air condmons the vapour pressure is calculated from eq (17) both for water surface and bulk wet air conditions (0 and t), by Introducing m t and n t and m o and n o coefficients respectwely For these conditions and preceding assumptions eqs (13') and (15') are rewritten as follows dt/ds =

dO

(E/CG)

CGRs

dt

~ = G.(CaRs& + Ramo) ds

(22)

In eq (22) the latent heat of evaporation is considered to be the same at the water surface as m the wet air In order to solve the preceding differential equations explicitly, with t, 0 and p as Independent variables, no further assumptions are requtred The integration is done analytically with coordinate s referring either to the

MM Mesarovlc 4RARA F a compute1 {ode

62

surface f or to the volume v according to the type ot fill selected Such a solution contains Integration constants by which the boundary conditions are taken Into account * The option of the model developed for nonsaturated wet air conditions forms a system of unsaturattaneous first order differential equations with constant coefficients, eqs (18), (19) and (20) This system is solved by a double differentiation with a gradual elimination of u n k n o w n variables other than those whose spatial distribution IS sought In such a manner three third order differential equations without free terms are obtained, as In [1], which contain differentials o1 only one variable, t, 0 or p These prove to be all in the same form

0 = K 4 exp (E u-~s/(, } + h ~,

L( t = K4

Rar"o~

Ca + ~ j e x p (tzu3s/G) /{sPa /

K5(a E

+ - -

{ 27 }

1

+ rno )

r(mt ~ mo)

(281

edf 1 + r/to K 4 and K 5 are obtained from the respective boundary conditions, while u 3 is the root of characteristic equation ofeqs (21) and (22) As mentioned before, the vapour pressure p is calculated from the satmatlon conditions

3 Description of computer code ARARAT

dis3

{:Is2 G d~od£

-

I-d1

1 ~ RsPa

el

~aL- + ~-J rnl=O,

3 1 Prlnclpalfeatures (23)

-co

a

where,=t, 0 orp The solution of eq (23) of the most practical Interest IS as follows 0 = K 1 exp (Eu 1s/G ) + K 2 exp(Eu 2 s/G ) + K 3 , (24) t = KI( 1 + ECaLUl/O 0 exp (EUlS/G)

(25) + K2(1 + ECaLU2/a) exp (Eu2s/G) + K 3, p = p , , + K 1 Ul r I [ C a L ( I + E U l / O 0 _ I ] exp(EUlS/G ) (26)

+ K2u2r-i [CaL( 1 + Eu2/a}-l]ex p (Eu2s/G) /x I , K 2 and K 3 are the integration constants and u 1 and u 2 the non-zero roots of the characteristic equation to eq ( 2 3 ) , p " is defined by eq (17) The option of the model for the moisture saturated wet air conditions, ordinary first order differential equations (21) and (22), is solved directly

* A similar model of heat and mass transfer in a crossflow cooling tower fill can be derived by the same approach Its solutmn, however, can be obtained through a numerical lntegratmn only, because the respective boundary conditions are inappropriate for an analytmal development

The ARARAT code is developed from the exphcit solutions of the basic model, eqs (24) to (28), for counterflow coohng towers Two options of the program have been provided. ARARAT-1 for the evalu~ tlon of the overall tower design, and ARARAT-2 for calculating tower performance curves For this purpose the respective boundary conditions are introduced, which refer both to the inlet and outlet conditions of water and air and the tower fill geometry The solution is conducted along the axial coordinate only and the boundary between the saturation and nonsaturation zones determined by the relative air humidity of 100%, which IS calculated from the ratio of the vapour pressure over saturation pressure Owing to the physical analogy between the heat and mass transfer parameters, the heat transfer coefficient a and mass transfer coefficient E are in a very close relationship their ratio varies very httle in the range of temperatures and humidities of interest in cooling tower operation Since this ratio may be considered constant for the practical purposes [ 1], only one of them ought to be solved The mass transfer coefficient is adopted for the purpose of the ARARAT code and the heat transfer coefficient calculated subsequently from their ratio The mass transfer coefficient is calculated with reference either to the surface or to the volume of the fill for either film or spray cooling Surface mass transfer coefficient, relative to the pressure difference, is

M M Mesarovtk, A R A R A T

calculated from a number of empmcal relations alternatwely, wtuch are functaons of the fill geometry, Nusselt number Nu d and chffuslon coefficmnt o d, so that Efdm = Ep,f (geometry, Nu d,

Od)

(29)

The spray coohng volume mass transfer coefficient, relatwe to the momture content, m calculated from a number of relanons m the form Espray

=

E X,v (ge ome try, g, L),

(3 0)

and subsequently converted to the mass transfer coefficmnt, relatwe to the pressure difference, Ep, v, accordmg to eq (17) As tower hydrauhcs m mutually hnked w~th the heat and mass transfer, it is calculated by the ARARAT code mmultaneously. For this purpose friction losses are determined through the following empmcal relatmns for the pressure drop coefficients [ START

START]

I

I NPUT DATA G, t!n, @xn,~xn] TYPE OF FILL, r OR V, TOWERDIMENSIONSI FANt ETC. T [ FNITIAL tou t [INITIAL L [ i ADJUST L I----I~,ITI^L t. I i [ CALCUI.,ATE B1 /NONSATURATED/ I / CALCULATEt~ @, p,~, VALUES AT ZONESBOUNDA~ ][~" 1 NO [ CNEeMIkl, AGAINST Vz/IF~I00% t

]

~in1

YES OOqt I iO^LCU~TE 8~ ~SATU~TED Z0HE+------l ICALC0IATE OOTLET t . ^ND ~31 .~S, .__ I F NITIAL

I

[TEST FoR C O . V E R G E N C E oF Oo~t~ [SATISFACTORY

I sOLVE F0~ L A,D ~ VAL~ES

NO

]

J

I

NOO '~ 'TESTFORCONVERGENCE0F L? ] [SATISFACTORY

FILM[ FYPEOF~'LL I

ISP~Y

SOLVEFORtout BY NNON'S ]

L j

I

CALCULATE 8=B I I

[CALCULATEOUTLET @ AND p[ ITEST FOR CONVERG'ENOEOF ~@,.~.1./-.(1001"}} NO SATISFIED [

IsT°P

~ ' ~ - 7

SP~,~ '

~Jt-~a =

[ PHYSICAL CONSTRAINTS ~]

}f (Re,

}spray =

FLFRICTIONLOSSES I

INDUCED~ TYPE or CIRC~LATIONI--]HAT~L DESIGN

geometry),

(31)

NO

SATISFIED I

~OTOR, FAN, ETC ~ - - ~ T O W E R

I

Fig 2 Performance evaluaUon of coohng towers Sxmphfied flow diagram ol opUon ARARAT - 2

lET,GEOM..H~. '~'~_~dEv'Ex.~,X":~.b' I IrILL D~STGN I I

HO

[ TZST FOR CONVERGENCEOF to~,lt?] ]SATISFACTORY I CALCULATEAXIALDISTRIBUTION] I [PRINT 0trrrur I

[SOLVE FOR Wa. L ]

J

NO

ITECHNOLOGICAL CONSTRAINTS ?.] SATISFIED I

{ PRINT0UTPU~ I FST0 P

"]

METHOD, COMPARING CALCULATED TO ORIGINAL F OR V VALUE

I

FIL. I

~. [

I Ef/~-EoM.,H~d, ~j ~EJEx,v,X,X",%/

| [INPUT DATA txn, tout,G, ~in. TYPE OF TOWER AND FILL, ETC.

IIHITIALW

63

- a computer code

]

Fig 1 Overall des,gn of coohng towers Slmphfled flow diagram of option ARARAT - 1

}v(g' geometry),

(32)

m conformity wath the respective fill configuration It appears, however, that the exhaust mr condmons can be calculated without knowmg any of these empirical coefficient in advance, as seen from flow diagrams of both options of the program, figs 1 and 2 It Is sufficient, without knowing any E, F or V values, to calculate thexr rano m the form F VE B =~-Ef = G v

(33)

64

MM Mesarovw, AR4RA I

The B parameter was adopted by Berman [ 1] and proved to be very useful In practical applications In eqs (24) to (28) ItS two components are included by the exponential terms when putting s equal to the respective parts o f F or V occupied by saturated and nonsaturated wet air Thus, denoting by radices 1 and 2 nonsaturated and saturated conditions, respectively. and calculating the mass transfer coefficients from eqs (29) and (30) t-or both zones of the fill. Berman's parameter is written B =B 1 +B 2-

F1+F2 V 1 + V~ G E~G -Ev

(33')

Based on this parameter, the outlet 0 and p values are calculated from eqs (24) to (28) and thereafter their space dlstrlbutmn IS evaluated from heat and mass transfer, given the boundary condmons Following next, the required surface F or volume V of the fill. as well as the cooling effects and tower hydrauhcs are evaluated Simultaneously, a number of restrictions concernang the physical and technological nature of the processes must be satisfied, e g homogenlety of water dispersion and excessive water carry-over, which are Included in the Iteration loops as well 3 2 Overall design o¢ coohng towers ARARAT-1 is employed for the evaluation of tower sxze (overall dimensions, fan power, type of fill, etc ) as required by the design cooling capability, which is supplied at the input Principles of the method of solution adopted for this option are presented in fig 1. the slmphfied flow diagram The type of circulation and fill are chosen in advance and the respective calculation carried out by an ateratlve adjustment ol L and Wa viiuses. so that the required cooling capability is met and the imposed hmltatlons satisfied simultaneously Because the tower design IS evaluated on the basis of a minimum coohng effect criterion saturation of the wet air has not been allowed at the design wet bulb temperature Besides the tower size, a variety of working conditmns is calculated at the design point and printed out This information enables the designer to work out wide parametric studies in order to select the most economical sotutlon for every particular case 3 3 Calculating performance curves Performance curves of a g~ven tower are calculated by the option A R A R A T - 2 , fig 2, in which the cold

a computercode water temperature is deternluled fol ever> pdll ()1 ]lUmldlt) and dly bulb temperature of the air The solution is round lteratlvely by Newton's me thod (if successive approximations with reference tc~ ttlc cold W,lter temperature tout Inside this iteration loop anoiheL one IS established for calculating the respectwe all flow (L), which m turn comprises two serial loops, first lor calculating the size ot the nonsaturated zone by the adjustment of boundary t value, and second for calculating the exhaust air temperature IJut To provide an extensive insight into the perl orlnance parameters of the tower, all of those included m the model are printed In the output The axial distribution of t, 0, X, p, specific density 3' and humidity ~f ol the air is also calculated The computing time is long compared to that ol A R A R A T - 1 It may, however, be reduced slgmficantly, if feasible an a particular case by choosing a lower degree of accuracy of the Iteration loops, in conformity with the measuring errors and the purpose of the results ol calculatmn

3 4 Numerical examples The A R A R A T computer code is apphed for the e~aluatlon of IDCTs and NDCTs required to satisfy design cooling conditions, as well as for the calculation of their performance curves The size of an IDCT is hmlted by the avmlable fan sizes, and the increased heat loads must be handled by increasing the number of cells However. while the size and drive of IDCTs are standardized (small cell towers and large round coolers), the size and principal design features of NDCTs must be optimized for every pamcular case. due to their much ~ g h e r direct costs Oil the other hand, for unit capacities above 100 to 200 MW (depending on the annual utlhzataon factor). NDCTs prove to be more economical and therefore more attractive for the present thermal power stations For these reasons only counterflow NDCTs are considered in this investigation As the air flow is due to the stack action, or difference In density between the mlett and the exhaust air, the NDCTs are subject to variable air flow through the till The air flow increases during the cooler months of the year and this increase will offset the lower evaporation heat transfer due to lower wet bulb temperature (IDCTs can only increase air flow at the expense of greater fan power) Fig 3 shows the performance curves of five NDCT designs tou t , the cold water temperature, vanes direct-

M M Mesarovt(, ARARAT

65

TEMPERATURE

COLD WATER TEMpeRATURE I

1

No.

TOWER tln,°C 30

- a computer code

I

I

2

3

~

3.0x107 /

l

,0)

5

q6.8 qq,0 ql 6 kO h 39 2

-- G, kg/h

oc

Qc/Q, % 35

/

{

{

I

TOWER SIZE Db=81.3m , Dt=66.Tm , H t o t = 1 0 2

3m,

Hzn=IBm

30

IclQ

25

t in-t 20

20

20

tOUt

HW 30

Dt

,ut"~ OVERALL DIMENSIONS,m 10

o

Db

Dt

Hto t

H1n

1 2

97 98

68 66

75 87

18 18

3

9~

66 102

18

q

9~

67

115

18

5

97

69 127

1B

{ 5

I

Qc

20

10

Db HOT WATERT E M P t i n • ~1.6°C WATERFLOWG • 3.0x107 kg/h I

15

)

i

25

OUTSIDE WET BULB TEMPERATURE,°C

Fig. 3 Natural draft coohng tower (NDCT) performance curves with the wet bulb temperature r, but changes less than r because the ability of the air to absorb moisture IS reduced at lower temperatures Under summer condatlons the sensible heat gain by the air becomes less and the gain due to evaporation proportionately more, fig 4 Therefore an actual optimum selection should include complete hourly weather data for the year on wluch an economic operation with 7 is based, so that the total cost of the plant with NDCT nummlzes total year-round cost Such a selection is complicated by the fact that during the longest period of the year (spring summer and autumn) evaporation process is dominant (ca 90% and even more of the heat IS transferred by evaporation during hot summer period), but In winter, due to low ~-, convection and conduction cooling may increase to about 50%, and even 70% during the coldest period In hot dry climates the efficiency of NDCTs is reduced

0

{

I

E

~

}

10 OUTSIDE WET BULB TEMPERATURE,

20 °C

Fig 4 Cooling range, temperature approach and heat transfer performances versus outside wet bulb temperature

The size of NDCTs is very sensitive to the colder terminal temperature dafference, or "approach", tou t - r, as the design approach is narrowed, cooling surfaces greatly increase Therefore, great care must be taken in choosing the appropriate approach in order to optimize the tower size, or even the whole power plant As the calculations show, the optimum performance usually occurs when the approach is equal to or greater than the cooling range (tm - tout) The cooling range also affects the tower size, since higher ranges mean higher air temperatures and result In an increased draft and air flow, while larger ranges mean less water flow and smaller tower, but higher turbine back-pressure The selection of air velocity is of course part of optimum NDCT design As evident from fig 5, the size

66

M M Mesarovtd, d R A R A

SURFACE, 2 x 10 " 5

FILL m

T

a c o m p u t e r ~o d e

P, bars

STRUCTURE

t,Q,

°C

R T7.!~ _ m

S 1

NONSATURATED B1 :

I

I

SATURATED

ZONE

B2 =

2217 b a r -I

ZONE

2586 b a r -I 1 30

7

~0

30O

I

030

7. 25

t0

200

020

1 20

100

010

G:Sxl07kg/h, ~ n = 9

~°C, L:I 549

Wa=2 735m/s, W= 25m/s~ Pb=l 013 b I __

78 q0

0

15

DESIGN l~lg

3

2,5

5

AIR

VELOCITY,

5

mlsec

NDCT size versus design a l r velocity

of the tower, primarily its height, increases rapidly as the demgn air velocity is increased, whale the required active surface of the fill, made of corrugated vertical plates 4 mm thick, spaced 24 mm, is reduced considerably That proves that an optimum design mr velocity exists, which yields the minimum of the overall tower cost function This function is very complex, an increase in tower height to satisfy higher air velocity, for example, IS accompanied by a reduction of tower diameter, but involves drift and foggmg problems at higher velocities In adchtion, the optimum IS affected by the applied technology of building, to a great extent The results are shown for NDCTs of 3 X 104 m 3 of water per hour and three different hot water temperatures with the coohng range of

92°C

RELATIVE

DISTANCE

1 15

80 FROM THE FILL BOTTOM,

Fig 6 Axial distribution ot water and mr parameters Axial distribution of water and air parameters within the fill of the above tower, for a given size and ~ = 9 4°C, is shown in fig 6 Berman's parameter B for this case is 0 4803 bar 1 as calculated by the A R A R A T code (B 1 = 0 2217 bar -1 for the unsaturated zone andB 2 = 0 2586 bar -1 for the saturated one) and refers to the air velocity Wa = 2 735 m/s, water velocity 0 25 m/s and air flow G a = 3 838 X 107m3/h with L = 1 549 kg of air per kg of water Air saturation occurs at approximately 46% of the fill height Within the rest of the fill the mass transfer is governed by the rise of 0 only Thus, the slope of t will change and even become convex at lower r as seen from fig 10 When the relative humidity q0 increases, B l (11o11saturated zone) will decrease, but the overall Berman parameter B will increase on account of substantial rise o f B 2 (saturated zone) Though the heat and mass transfer coefficients, which keep constant across

67

M M Mesarovt~ ARARAT - a computer code

tout' 35,

Oout

°c I

I

I

I

Wa. m/s o~ L, kg/k I ., ~.0

p,bars o~ X, kg/kg

TOWER SIZE

H n: 18m

R

1 200

030

Db= 9~m, Dr= 57m, Htot= 102r

/

HOT WATER TEMPERATU

t

2$

=~1.6°C

3 0

@out

1 150

020

1S

2 0

010

I 100

.SO B2/B

.2S

1 0 000

1 050 0

0

~ 0

10

10

2(3

OUTSIDE WET BULBTEMPERATURE, °C

20

Fig 8 Vanatlon ot air parameters with outside wet bulb OUTSIDE WET BULB TEMPERATURE, °C

temperature

Fig 7 NDCT operating performances versus outside wet bulb temperature

the fill for a given r, are increasing due to Increased air velocity Wa, and L, tou t IS higher, since the overall coohng effect is reduced The vanatmn of the main water and air parameter in a NDCT wath changing wet bulb temperature is shown m figs 7 and 8, as calculated for t m = 41 6°C Evidently, the driving head is greatly Increased at lower r because the wet air density is reduced along the fill much more than at higher r, as may be concluded from 7a =

P b - ( R s - Ra) p/R s R T

(34)

a

As a consequence, the air flow (Wa, L) xs reduced during the hot weather, although p (and X) is recewlng practically the same increment over the fill

In the above example, saturanon conditions will exist at r < 18°C Naturally, the lower the wet bulb temperature, the larger portion of fill will be occupied by saturated mr, as determined by the B2/B ratio This ratio will increase at higher humidities as well, so that almost 70% of the fill will contain saturated mr at r = 0°C Saturauon will not appear at r >~ 18°C for counterflow tower in this example However, in a crossflow tower saturation may be present in part of the fill lrrespectwe of how high z is, which reduces the overall cooling effect In comparison wath that of the counterflow tower Hourly variation of cooling capability of two large NDCTs is shown m fig 9 for typical summer and autumn wet bulb temperature condmons m central Yugoslavia As mennoned earlier, the slope of t versus z depends on the proporUon of heat transfer by evaporation This is clearly demonstrated in fig 10, where

M M Mesalovt(, ~IRARA T

68

COLD WATER ~EMPERATURE C

a c o m p u t e r code

WATER ~MPERATURE I

I

1

1

I

r

l

WATER INLET CURVE No.

"~', °C

1

9 q ~.7

2 3

0 2 -~.3

SUMMER

//

/

2~

%,

oC

I

/

// /

TYPICAL

/

15

/

I

%

/

/

~" 10

1

2

Db, m

13q

133

Dr, m

0q

9q

Htot, m Hin , m

02 25

121 25

t i n , °C G, kg/h

39 9 38.7 6 10 7

TOWER No.

I

8

I

I

// TOWER SIZE

/

Db = 0~.~ •

BOUNDARY / BETWEEN

ZONES

Dt

• 6B.7

m

Nto t • 1 0 2 . 3 •

/

Hln • 17 7 1

/

/ lOJ

I

16

2~

i

/

a tl0

0

I

I

-5

00

RELATIVE DISTANCE FROM THE FILL ROTTOM,%

HOURS OF THE DAY Fig 9 Hourly varlatlon ot cooling capablht,¢ of large N D C f s

Fig

the axial dxstrlbUtlon of water temperature is represented with the wet bulb temperature as a parameter It is seen that at low wet bulb temperatures, the greater part of the heat of the water is transferred in contact with cold unsaturated air, although the moisture saturated zone is dominant m the tower fill

such methods, presently used by designers Moreover, its solution by computer makes xt possible to conduct wide parametric studies and to select the most economical design for every particular case The adoptxon of an exphclt solution similar to that of Berman, instead of a numerical integration of differential equations, was intended to approach this to well known procedures of thermal design of cooling towers To what extent the present procedure as close to such procedures It can be demonstrated by calculating the Merkel's evaporation crxteraon M by the aid of Berman's parameter B, as defined here A short-cut analysis shows that a simple relation exists between these two, le

4 Conclusion Although the physical model of heat and mass transfer is comphcated by the fact that these processes take place simultaneously with, being also mutually affected by tower hydrauhcs, It proved to be very attractwe for practical purposes By a straight-forward solutxon of ttus model, wtuch is in prmcxple the same as used m earher, simpler methods, it is expected to avoid the differences m accuracy and practical apphcabdlty of

10 A x i a l d i s t r i b u t i o n o! w a t e r t e m p e r a t u r e in t h e till

M=8~x/L~

(35)

M M Mesarovt6, A R A R A T - a computer code

Thus, the solution of the right hand side of eq (1) can also be performed by the A R A R A T code, which avoids the majority of the simplifying assumptmns usually employed in calculations based on Merkel's equation The assumptions that could not be ehmlnated in the present procedure are reduced to the flow dlstributmn and the empirical relationships for coefflments of heat and mass transfer, and friction losses They may influence the accuracy considerably and should therefore be based on thorough investigation. Special attention should be paid to ensure uniform flow &strlbution, as otherwise numerous difficulties may occur in practical solutions when designing and/or operating a coohng tower

Nomenclature B C D E

= Berman's parameter [m2/N], = specific heat at constant pressure [J/Kg°C], = diameter [m], = mass transfer coefficient [kg/m 2 s/N/m 2 or kg/m 3 s/N/m2], F,¢ = evaporation surface, net [m2], G = mass flow [kg/s], g = specific mass flow [kg/m 2 s], H = height [m], h = enthalpy [J/kg], K t _ 5 = integration constants [°C], L = air over water flow ratio [kg/kg], M = Merkel's criterion [dimensionless], m --- free term m eq (17) [N/m2], Nu = Nusselt number of heat transfer [dimensionless], n = temperature gradient of saturation pressure [N/m2°C], p = pressure [N/m2], Q= heat quantity [W], q = heat load [W/m 2 o r W / m 3 ] , R = gas constant [J/kg°C], Re = Reynolds number [dimensionless], r = latent heat of evporatlon [J/kg], $= space coordinate [m 2 or m3], T = absolute air temperature [°K], t = water temperature [°C], U l _ 3 = roots of characteristic equations [N/m2], V,v= evaporation volume, net [m3],

69

W= X = x,y,z = =

velocity [m/s], moisture content [kg/kg] rectangular coordinates [ml, heat transfer coefficient [W/m 2 °C or W/m 3 °C], 3' = specific density [kg/m3], 6p" = Berman's pressure correcnon term [N/m 2 ], 0 = air temperature [°C], = friction coefficient [dimensionless], fi = radial coordinate [m], o = diffusion coefficient [kg/m 2 s], r = wet bulb temperature [°C], ¢ = relatwe humidity [N/m2/N/m 2 ], = independent variable [°C or N/m2], co = angular coordinate [rad],

NDCT = natural-draft coohng tower, IDCT = reduced-draft coohng tower, Subscnpts

a = air, b = barometric, base, c = convection, condensation, d = diffusion, e = evaporation, f = surface, film, in = inlet, m = average out = outlet, p = pressure, r = radiation, t = water temperature, top, v = volume, spray, X = moisture contents, w(or none)= water, z = boundary of the zones, 0 = air temperature, " = saturated dry, ' = saturated wet

References [1] L D Berman, Evaporatwe Cooling of Circulating Water (in Russian Ispantel'noe ohlazhdeme clrcul'aclonnox vody) Gosenergmzdat, Moscow - Lemngrad, 1957 [2] L A Fuller, Computer Evalutes Coohng Towers, Petroleum Refiner, 35, 12 (1956) 211

70

M M MesarovlC A R A R A T

[31 J P Hartnett, Mass Transter Coohng m Turbulent Boundar', Layer, Internat Summer School on Heat and Mass Transler in Turbulent Boundary Laver, Herceg No~l, 9 21 Sept 1968 [4] M Jacob, Heat Transfer (John Wdey and Son~, New York and Chapman and Gall, London) 1957 [5] I Merkel, Evaporative Cool,ng, (in German Verdunstungskuhhng), VDI Forschungs, No 275 (VDI Verlag GmbH, Dusseldorf) 1925

a (ornputel code

161 M Mesarovl6 and B Gaber~&k, Prussure-Temerature ~Il,mslents lot Containment Design of Water-Cooled Rea~.tors, Nt.Aear Engineering and Design, 17, 3 (1971J 428 [7] K Spangemaher, Calculating C~ollng Towers and Spra~ C(~olers bv tile Aid ot an kvap~ratlon Parameter, (m (,~rman Berechnung voL~ Kuhlturmen und Emsprltzkuhlern mlt Hdfe emer Verdunstungs-Kennzahl), BreranstotlWarme-Kralt, 5, (May 1958~ [8] Coohng Towers, Ne',~ Design I eatures Low Silhouette Nuclear Ne~s, 5, (May 1971)