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Modelling flow in an experimental marine condenser
Int. Comm. HeatMass Transfer, Vol. 24, No. 5, pp. 597~508, 1997 Copyright © 1997 Elsevier Science Ltd Printed in the USA. All rights reserved 0735-1933/97 $17.00 + .00
Pergamon
PII S 0 7 3 5 - 1 9 3 3 ( 9 7 ) 0 0 0 4 6 - 8
MODELLING FLOW IN AN EXPERIMENTAL MARINE CONDENSER
M.R. Malin CHAM Limited Wimbledon, London SW19 5AU United Kingdom
( C o m m u n i c a t e d by P.J. H e g g s )
ABSTRACT A Computational Fluid Dynamics (CFD) model is described for simulating flow and heat transfer in a condenser. A single-phase approach is used which predicts the flow of a steamair mixture within the condenser shell, and the heat and mass transfer processes are modelled using empirical correlations. The model is used to calculate the overall performance of an experimental marine condenser with a superheated steam supply. © 1997 Elsevier Science Ltd
Introduction The potential of CFD for investigating future condenser designs and improving existing condenser performance has resulted in the development of a large number of condenser models [1-18]. The method has been applied extensively to land-based power-plant condensers, but less so for marine applications where compactness is a very important design consideration.
The flow field in a condenser comprises two distinct regions, one corresponding to the shell-side, and the other to the tube side which usually consists of a single- or multi-pass parallel arrangement of tubes through which cooling water passes continuously. The shell-side flow comprises a two-phase mixture of gas and liquid condensate. The gas phase contains air, which inevitably leaks into a system under vacuum, as well as steam vapour. The liquid phase arises from the condensation of the steam onto the tubes, and may exist either as droplets or columns of fluid between the tubes, and as films on the tubes.
Almost all CFD models employ a single-phase approach whereby transport equations are not solved for the flow of condensate, but rather the condensing steam is assumed to disappear from the solution domain, and only the condensate's inundation effect on the heat transfer is modelled. Empirical correlations are employed to account for the hydraulic resistance of the tube nests, and similarly, empirically-based relationships are used in specifying the local mass condensation and heat-transfer rates between the bulk and the coolant. The method most often used for calculating the local heat and mass transfer rates is that described by Chisholm and McFarlane [19], which uses the Berman-Fuks [20] correlation for the gas-side heat-transfer coefficient. Strictly speaking, this correlation is valid only for bulk 597
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conditions close to saturation. In the CFD model the steam-air mixture is presumed close to the saturation temperature corresponding to the vapour partial pressure, thereby obviating the need for a transport equation for the mixture enthalpy. The flow in the cooling-water tubes is treated as one-dimensional and is subject to a thermal analysis only, as the flow pattern of the tube-side fluid is known.
The present paper describes the development of a basic CFD model for predicting the overall performance of an experimental marine condenser. The particular operating characteristics of the experimental rig result in superheated steam entering the condenser unit. The present model therefore requires the novelty of solving a transport equation for the mixture enthalpy.
Mathematical Model Conservation equations: The following conservation equations for the steam-air mixture on the shell-side of the condenser are solved in a Cartesian coordinate system x (= x,y,z): V. ( p U ) =- m
;
V . ( p U Ya ) = V . ( ( #/(~s + pt/Gs.t ) V Ya )
V.(pU®U)=V.(peVU)-Vp+F-mU
+V.(pe(VU)
;
Ys = 1 - Ya
(1)
2 T - ~(#eV.U)_I)
(2)
V . ( p U H ) = V . ( (#/oh + WOh.t) V H ) + qg + qd
(3)
where: p is the mixture density; U (=u,v,w) the velocity vector; m the mass condensation rate per unit volume; p is the pressure; F (=Fx, Fy. Fz) is the friction force per unit volume due to the hydraulic resistance of the tube nest; H is the mixture static enthalpy; # and Mt are the laminar and turbulent dynamic viscosities; (~h and (~h.t are the laminar and turbulent Prandtt numbers; Ya and Ys are the mass fractions of air and steam vapour; and (~s and C~s.tare the laminar and turbulent Schmidt numbers. The Prandtl and Schmidt numbers are taken as unity, and the dynamic mixture viscosity tae= p. + Mr.
Flow-resistance relations: The axial resistance force per unit volume, Fz is calc,!!ated from [21]: Fz = - 0.5 fa P w2 / dh
(4)
where dh is the hydraulic diameter of the flow passage, and fa is a friction factor given by: fa = 64/Rez
{if
Rez< 2308 } "
fa=5.5.10 "3 [1 + ( 2 . 1 0 4
fa = 1.547.10 -4 Re °67 { if
k~ +106/Rez)l/3] do
{if R e z > 4 2 0 0 }
2308 < Rez < 4200 } (5)
where kr is the sand roughness height of the tubes, taken here as zero, and the axial Reynolds number Rez=plwld~p., where Iwl is the absolute value of the axial velocity. The horizontal and vertical velocity components u and v are in the cross-flow mode over the tube bundle, and their respective frictional
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resistance forces per unit volume are estimated from the Jakob correlation for single-phase flow across banks of staggered tubes [22] : Fx=-2fcNpUcru
; Fy='2fcNpUcrv
; f c = [ 0 . 2 3 + 0 . 1 1 / ( P × / d o - 1 ) 1'°8 ] R e ~ ~5
(6)
where: N is the number of tube rows in the flow direction per unit length; Ucr is the resultant crossflow velocity; fc is the friction factor ; do is the tube outer diameter; and Px is the transverse tube pitch, which is the same as the diagonal pitch in the present tube arrangement. The velocities u and v are maximum velocities based on the minimum free area, and the Reynolds number Rec = pUcrdo/p..
Enthalpy sink terms: The mixture enthalpy equation (3) contains two sink terms which are associated with enthalpy transport from the bulk steam-air mixture to the vapour-liquid interface. The first sink term, qg, represents the transfer of heat to the interface, and the second term, qd, represents a net enthalpy flux owing to the species diffusion currents: qg = - % (T- Ti ) Aw/V
; qd = - (Cp,s - Cp.a ) m (T- T~ )
(7)
where: % is the gas-side heat transfer coefficient; T is the bulk temperature of the steam-air mixture; T~ is the interface or condensate surface temperature; Aw is the total tube surface area in the computational control volume V; %.a and Cp.sare the specific heats at constant pressure of air and steam vapour; and m is defined by m = m" Aw/V, where m" is the mass condensation rate of vapour per unit area.
Thermodynamic relations: The bulk temperature of the vapour-air mixture is determined from: H = Ya Ha + Ys Hs
;
Ha = Cp,a(T - To)
;
Hs = %,s (T - To )
(8)
where: Ha and Hs are the static enthalpies of air and steam vapour; and To is a reference temperature taken as zero K. These equations imply that the specific heat of the mixture Cn,m is given by: Cp,m = YaCp,a + Yscp,s
(9)
where here %,~=1900 J/kgK and %,a=1005 J/kgK. The foregoing relationships make use of the fact that the enthalpy of the gas phase knows nothing about the latent heat, and also that the enthalpy of the liquid phase need not be introduced into the model at all because it only has to be taken out again.
Heat-transfer relations: The transfer of heat from the bulk mixture to the interface comprises two components [23]; the heat released at the interface due to the condensing vapour, and the sensible heat transferred through the diffusion layer to the interface. Thus, q"=m"L+
~ct.g(T-Ti)
;
(p = ~0H](1-exp(-q)H))
;
~H=m"Cp,s/O(g
(10)
where q" is the heat flux, L is the latent heat of condensation, and ~0 is the Ackerman-correction factor [24] which accounts for the effect of mass transfer on %. The heat fluxes through the condensate film and through the wall to the coolant are given by
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M.R. Malin
q" = ~c (Ti- Two)
;
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q" = CZwt(T~,o - Ti)
(11)
where Two is the temperature of the tube outer surface, c~c is the condensate-film heat-transfer coefficient, T~ is the coolant temperature, and Ct.w~is the heat-transfer coefficient between the coolant and the tube outer wall. An effective overall heat-transfer coefficient % may be calculated from q" = c% (T - T{) once the temperature difference and the heat flux are known. Mass-transfer relations: The condensation mass flux is computed from: m"=pK,,, ,~,f~ l n ( l + p , - p , . i )
M,,,
;
p - p~
K
'"
=[Pr/Sc]Z/3o~,/(pCp,,,,)
(12)
where: Km is the mass transfer coefficient; Mm and Ms are the molecular masses of the mixture and steam vapour; Ps is the vapour partial pressure in the bulk; Ps.~is the vapour partial pressure at the interface; the Prandtl number Pr=cp.mp/k; the Schmidt number Sc=p/(pDsa); k is the mixture thermal conductivity; and Dsa is the diffusivity of the steam in air [25]: D,. = 1.38674.10 s7'~/2[1/M, + 1 / M , ] I/2 / p
(13)
where the temperature T is in K, p is the static pressure in bar, and Ma is the molecular mass of the air. The values of D~a, k and it are evaluated at the film temperature Tf=(T+Ti)/2. The bulk vapour partial pressure is determined from:
p, = p / [ I + Y , M
/(YsM
)]
(14)
The interface vapour partial pressure Ps, depends on the interface temperature T~, which is taken to be at the saturation state for the condensing vapour.
Heat-transfer coefficients: The heat-transfer coefficients c% and c~ are given by:
c¢., = [
d
"
-t
d In(d,, / d i) "
dio: I
2k
+ Rt ]4
"
oq = kt 0.023ReC/S Pr/°4 d,
(15)
where Rf is the fouling resistance taken as 1.5.10 s m2Kj'VV, d~ is the tube inner diameter, kw is the tubewall thermal conductivity taken as 27.3 W/mK, k~ is the coolant thermal conductivity, Ret=p~w~di/i.q, Pr~=Cpkp~/k~,and the coolant properties p~, it{, cp~ and k~are evaluated at the local bulk coolant temperature. The condensate-film heat-transfer coefficient is based on the equation of Nusselt [24]:
Otc = o~;F,.Ftl- ~.
;
a~ = 0 . 7 2 5 ( k ] L p ~ g
/ [ d / 2 c ( T , - T,,,,)]) '/4
(16)
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where a~ is the condensate film heat-transfer coefficient in stationary vapour, and kc, Pc and I-tc are the condensate thermal conductivity, density and dynamic viscosity, respectively. The vapour-shear factor Fv [19], superheat factor Fs [24], and inundation factor, Ft [19, 26] are given by :
n 118/N.J~
Fv = 1 + 0.0095 KeA
'
;
F, =
%,, ( r - ~ , ) L
;
FI =
"
m / m
l
(17)
where: Nu c = ~z*, d,,/kc; ReA is the approach Reynolds number of the mixture based on do and a resultant velocity based on the approach values of the cross-flow components; and Tsar is the saturation temperature evaluated at the bulk vapour partial pressure. The inundation factor represents the effect of condensate drainage, and the numerator of the F~ expression denotes the sum of the condensation rates from the uppermost to the current row of tubes, and mn is the condensation rate on the nth tube.
The gas-side heat transfer coefficient % is estimated from standard single-phase correlations for flow across staggered tube banks [22], assuming that the presence of the condensate film has no effect: 0~~ = ( k / d~,)Nu = (k / d,)[0.33 Re~ 6 Pr 'j3 ]
(18)
where k is the thermal conductivity of the steam-air mixture based on the film temperature.
Heat and mass transfer algorithm: The conditions at the liquid-vapour interface are unknown, and
consequently the local temperatures are determined by solving equation (10) for the local heat flux q" in an iterative manner. The solution proceeds from the last available values of q" and T~, and from the values of T and T~ resulting from the solution of the shell-and-tube side enthalpy equations. The interface is assumed to be a saturation condition for the vapour, and so Ps., may be evaluated from knowledge of T~. The bulk value of Ps is known from equation (14), and so once % and Km are evaluated, the condensation mass flux can be updated via equation (12). Next, ~, C~w~and ~ are computed from equations (10), (15) and (16), and q" is then updated from equation (10) expressed in the form:
q"=m"L+Oa~[T-T
t -(T~ - T
)-(T,,,,-Tt) ]
(19)
The temperature differences (T~ - Two) and (Two - T~) are evaluated using the last iterate values of q" in equations (11). These equations, together with the new value of q", are then used to update the values of Two and T~. The iterative nature of the CFD solution cycle is utilised so as to introduce successively better input values for equation (10), until a converged solution is obtained for the current cell.
The local coolant temperature T~ is calculated by application of the one-dimensional steady-flow energy equation, which can be expressed in terms of temperature changes as follows:
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Vol. 24, No. 5
ml[ c.,i,o~tTi,out - cp,l,i. TI,in ] = q" A .
(20)
where m~ is the mass flow rate of coolant flowing through the tubes in the finite-volume cell; and the subscripts in and out denote inlet and outlet values with respect to the control volume. The condenser employs a two-pass arrangement of the coolant, which is handled by assuming that at the exit from the first pass, the coolant is thoroughly mixed to give a uniform temperature into the second pass.
Turbulence model: The turbulence model employed here calculates the turbulent viscosity from: pt= C p Vslrn
(21)
where C is a constant, V s is a turbulent velocity scale and I,,, is the turbulent mixing length. In untubed regions, C=0.5, Vs is estimated at 5% of the steam inlet velocity, and Im is taken to be 10% of the minimum distance between the tube bundle and the condenser casing. In the tubed region, C=1, Im =0.1dh where dh is the hydraulic diameter of the flow passage, and V s is computed from [27]:
Vs
R o~
-
e,,, - 15,u p( P~ - d,, )
;
Rein
= p ( U . U )t/2
dh/p.
(22)
where Rein is a Reynolds number, with the resultant velocity U (=u, v, w) based on the maximum velocities associated with the minimum flow area. These velocities are calculated directly from the momentum equations because the flow areas are multiplied by the appropriate porosity factors: ~z=l-~
d~ /(PxPy)
;
!3x =]~y =(P~-do)/P×
(23)
where Py is the longitudinal pitch.
Mixture properties: The mixture density 9, laminar viscosity p ,and thermal conductivity k are calculated from [28]: 11#. p=l/(Ya/Pa+Ys/Ps)
; Y= -
-
lzy +
t~ + n O , :
' '
n +n.,Oz r
.
(l+AV2B[m)2 ~12-
" (P21 = 41__L (8[I+I/BI)
I/~
(24)
AB
where Pa and 9s are the densities of the air and steam vapour; y=# or k; Ya and Ys are the component properties; na and ns are the mole fractions; A=Ma/Ms; and B=Ms/Ma.
Steam properties: The steam vapour density is calculated from [29]:
p,=
[2pM~ I (R,,T)
;
B=aI(I+T/~z)
I + a , e ~ ( 1 - e r ) 5 / - r ln +axr
(25)
I+[I+4pBM, /(R,,T)] ]12
where T is in K, Ro is the universal gas constant, B is the second virial coefficient, t = 1500/T, a~ = 0.0015, a2 = -0.000942, a3 = -0.0004882 and o~= 104. The dynamic viscosity Ps and thermal conductivity ks of the steam vapour are computed from:
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M O D E L L I N G F L O W IN AN E X P E R I M E N T A L M A R I N E C O N D E N S E R
,u s = a , X 3 + b , u X 2 + c u X + d ~
; k =a k +bk(T-273.15)+c,(T-273.15)
2
603
(26)
where X = T/273.15, Y is in K. a~=-0.18321.10 s, b~=0.59888.10 -s, c~=0.43864.10 -5, d,=-0.43769.10 6, ak=0.0163918, bk=7.26527.10 .5 and ck=0.115507.10 -e. The saturation pressure at the liquid-vapour interface is computed from the saturation temperature by use of the following relationship [30]:
p~ = 6894.76exp(/3)
(27)
with
/3 = 73.32642 - 8.2In(T) +0.003173T- 13023.8/T ~=a+b/T+cfl,[exp(dfl~)-l]/T+eexp(ffl12
25)
{ for T < 100°C }
(28)
{for100°C
(29)
131 = T 2 - 9.51588.10 -5 ; /3, = 1165.09- T
(30)
and T is in °R, a=15.182911, b=-8310.453, c=1.7741.10 -4 , d=2.624453.10 12 , e=-1.013139.10 -2 and f=6.31141.10 .3.
The bulk saturation temperature Tsa [30] and the latent heat of condensation are evaluated from:
~
= a T +bTF+CT 1"2 +dT I'~ +Or r4
;
L = a L + b L T + c L T ~-+ c l t T 3
(31)
where T is in K, r=ln(pv), aT=4.92510 -3, bT=-2.23610 "4, CT=7.0461.10 -6, dT=-5.208410 "7, eT=1.096910 "a, aL=3.46892106, bL=-5707.4, CL=I 1.5562 and dL=-0.0133103.
Air properties: The air density 9a, dynamic viscosity ~[a and thermal conductivity ka are given by: P"
= pM, R,,T
;
I~,, = P r ( T / T r ) °76
;
k =a+b(T-273.15)
(32)
where .Ur= 1.846.10 -5, Tr = 300K, a=0.0242, b=7.49486.10 5 and T is in K.
Coolant properties: The coolant density p,, dynamic viscosity ~a,, specific heat cp.~ and thermal conductivity k~ are calculated from:
Pt = ap + b p T l "}'CpTI 2 k t = a, +b~.T t + c , Tl -~
;
I'll = au(bu + c , Tt + d~,T~2 ]
;
c j = a c +b.T t
(33) (34)
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Vol. 24, No. 5
where TI is in °C, ap=1028, %=-0.103, %=-3.22.10 -3, a~=4.133.10 -4, b~,=4.09, q,=-0.0945, d,=8.5.10 4, ac=3987, bc=0.444, ak=0.544, bk=2.225.10 3 and Ck=-I 0 -~.
Condensate Properties: The properties are taken as constant at values based on saturation conditions at the condenser operating pressure: pc=978.1 kg/m 3, kc=0.667W/mK and pc=4.10 .4 Ns/m 2.
Experimental Confi.quration The model is applied to a two-pass surface marine condenser studied experimentally by the DRA and GEC ALSTHOM The overall dimensions are 1.238m wide by 3.65m high by 2.476m long. The steam-air mixture enters the condenser through a long perforated pipe located along most of the condenser length at the top of the unit, and then impinges vertically downwards on the two circular tube bundles located in the main body of the condenser. The cooling water (CW) enters the lower bundle through a parallel staggered arrangement of tubes of 5/8" outside diameter with a mean thickness of 0.048". The CW then returns through the upper bundle and leaves at the top RH end of the unit. The tube bundles are mounted on a number of support plates through which all tubes pass, thereby partitioning the condenser into several sections and restricting the axial flow within the tube nest. The upper bundle comprises 928 tubes and the lower bundle 909 tubes, resulting in a total heat-transfer surface area of 225.84m 2. Each bundle has an air-extraction gallery which runs the full length of the bundle. The steam-air mixture enters the bottom of the extraction system passing between plates hanging from the top of each bundle, and is then removed via vent-offtake piping. The operating conditions for one half of the condenser are as follows: Steam
mass
inflow
rate=8.63
kg/s;
Air
mass
inflow
rate
including
natural
leakage=2.78g/s;
pressure=0.345 bar; Steam inlet temperature=116.5°C; Inlet saturation temp~,ature=72.5°C: Mixture mass outflow rate=13.25g/s; CW mass flow rate=207.87kg/s; and CW inlet temperature=13.07°C.
Solution Method The conservation equations are discretised using the finite-volume approach embodied in the generalpurpose CFD code PHOENICS [32]. The method uses a staggered-variable arrangement for the velocity components and scalar variables, and the effects of convection and diffusion are weighted using the hybrid differencing scheme [33]. The solutions of the discretised elliptic conservation equations are obtained using an iterative pressure-correction method based on the SIMPLE algorithm [34].
The condenser is symmetrical about the vertical centre plane, and so only one half of the flow geometry is modelled. The solution domain is discretised using 7 grid cells in the lateral (x) direction, 19 cells in the vertical (y) direction and 9 in the axial (z) direction. Grid-dependency studies revealed that a mesh with twice the present number of grid cells in each coordinate direction produced almost identical predictions of the overall performance parameters for the condenser. Grid cells totally impervious to flow, such as those located inside the inlet pipe, are represented by the use of a porosity treatment, whereby the free volumes and cell-face areas of these cells are set to zero. This practice is also used to define the
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605
vent plates of the air-extraction system. The flow restriction in the tube bundle is represented with partial area and volume porosities, which are also used for the prescription of the shape of the tube bundle and the steam inlet pipe. The air-extraction system is terminated at the top of each tube bundle, so as to define the vent outlet boundary for the steam-air mixture. At the steam-air inlet boundary, the local mass inflow rate per unit area is calculated from mi, ' = Co[2pi,,(pi,
-
pr)] ~/=, where Co is a discharge coefficient taken as 0.94,
pp is the in-cell static
pressure, and p~, and Pin are the inlet static pressure and density, respectively. The inlet resultant velocity (typically about 130 m/s) is computed from min and P~n,and then it is resolved into the x and y coordinate directions so as to define the incoming values of the velocities u and v. The inlet air mass fraction is prescribed according to calculated mass inflow rate of steam and the measured mass inflow rate of air. The inlet mixture enthalpy is calculated from equations (8). The upper and lower vent boundaries are divided into two equal lengths, resulting in four vent-outflow boundaries. At each of these boundaries, the mixture outflow per unit area is fixed according to the measured partitioning of the total mixture outflow: 5% extraction from the upper bundle, split equally between the top two vents; 95% extraction from the bottom bundle, with 2/3rd of 95% taken from the vent at the cold end, and 1/3 of 95% taken from the other lower vent.
The CW inlet temperature and mass flow rate are specified directly, and no outlet condition is required as the coolant exit temperature is an outcome of the integration performed in the direction of the CW flow. Results and Discussion
For reasons of brevity and the absence of detailed measurements, velocity and contour plots resulting from the 3D computation are not
presented, but rather attention is focused on describing the main
features of the calculation and comparing the simulated and measured performance parameters.
The main results are presented in Table 1, which presents the computed and measured principal performance parameters for one-half of the condenser. The table indicates the various inflows on the shell-side of the condenser, the amount of steam condensed, the power absorbed by the cooling water on each pass, and also the corresponding temperature rises in the coolant fluid.
An inspection of Table 1 reveals that the calculated and measured performance parameters are in excellent accord with the experimental data. The predicted overall condensation rate agrees almost exactly with the measured value, and the inflow rates of steam and air are also predicted very well by the CFD model. The results show that more heat is extracted on the average from the lower bundle, and the partitioning of the thermal duty between the upper and lower bundles is predicted to within 2% of the measured value. This
heat load is obviously reflected in the condensation rates, which are
correspondingly higher on the lower bundle.
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TABLE 1 Comparison of Measured and Predicted Performance Parameters Quantity
Prediction
Data
Total mixture inflow (kg/s)
8.66
8.63
Total air inflow (g/s)
2.79
2.78
Total condensation rate (kg/s)
8.65
8.62
CW exit temperature after 1 pass (°C)
26.65
26.79
CW average exit temperature (°C)
37.67
38.10
Total heat transfer rate to CW (MW)
20.52
20.80
% Heat transfer rate to lower bundle
56.0
54.80
The largest air mass fractions are predicted in the lower bundle, with values around 24% near the venting at the cold end, and around 18% at the hot end. The model predicts lower values of around 4 or 5% near the venting on the upper bundle. In general, the air concentrations are low in the bundle periphery, where the heat fluxes are high, and high in the bundle core where the heat fluxes are lower. The computed values of the overall heat-transfer coefficient are in the range 2.3 to 4 kW/m2K. The average value, based on the condenser saturation temperature and the mean coolant temperature, is 3.85 kW/m2K, which compares favourably with the experimentally-derived value of 3.94 kW/m2K. It is interesting to note that these values, as well as the proportion of thermal loading on each bundle and the mass fraction of air leaving the unit (21%), compare very well with those reported by Beynon [35] and Wilson [2] for an experimental condenser of very similar construction with similar operating conditions and a slightly-superheated steam supply. The condensation of superheated steam differs from that of saturated steam in so far as steam which comes into contact with the tube wails will condense only if the wall temperature Two is less than the saturation temperature Tsat. Otherwise, the tube wall will be dry and desuperheating will occur, i.e. heat transfer without condensation. If anywhere, this is most likely to occur near the hot end of the condenser, and so the CFD model detects whether Two > Tsar, and then switches to accommodate dry-tube heat transfer. However, the tube temperatures of the final solution indicated no desuperheating.
Conclusions
A CFD model has been applied to an experimental marine condenser. The validation against data revealed that the model predicted very accurately the overall condenser performance. The computed overall condensation rate was in excellent agreement with that measured, and the thermal loading on
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607
each pass was predicted correctly. Further work should include refining the modelling, and performing more varied and detailed validation exercises, especially for the saturated or wet inlet conditions usually encountered under commercial operating conditions. The pressure-drop correlation can easily be extended to account for the presence of the two phase flow of gas and condensate [7,24], and the cooling-water treatment might be refined by incorporating a log-mean temperature analysis. The Nusselt heat-transfer coefficient could be refined to account for the effect of condensate sub-cooling [36], and the inundation treatment so as to correlate the inundation factor against the local Froude number [7].
Acknowledgements The experimental information presented in this paper appears by permission of GEC-ALSTHOM in Rugby, and the Defence Research Agency (DRA), UK. The study was carried out with the support of these organisations during 1989. The author also acknowledges the input provided by GEC-ALSTHOM on the specification of the condenser boundary conditions.
References 1. J.L.Wilson, I.Chem.E.Symp., Series No.75, p3:21, (1972). 2. J.L.Wilson, NEL Report No.619, p132, Steam Turbine Condenser Meeting, NEL, UK, (1976). 3. B.J.Davidson, NEL Report No.619, p152, Steam Turbine Condenser Meeting, NEE, UK, (1976). 4. B.J.Davidson and M.Rowe, In Power Condenser Heat Transfer Technology, Ed.P.J.Marto and R.H.Nunn, p17, (1980). 5. S.A.AI Sanea, N.Rhodes, D.G.Tatchell and T.S.Wilkinson, I.Chem.E.Symp.,Series 75, p70, (1983). 6. C.Caremoli, I.Chem.E.Symp., Series No.75, p89, (1983). 7. H.L.Hopkins, J.Loughhead and C.J.Monks, I.Chem.E.Symp., Series No.75, p152, (1983). 8. S.A.AI Sanea, NRhodes and T.S.Wilkinson, 2nd Int. Conf. Multi-Phase Flow, Paper D2, p169, London, (1985). 9. L.N.Carlucci and I.Cheung, HTFS RS618, p569, (1985). 10. A.W.Bush, G.S.Marshall and T.S.Witkinson, 2nd Int. Symp. on Condensers and Condensation, p223, Univ. of Bath, UK, (1990). 11. H.Shida, M.Kuragasaki, M.Soda and T.Urate, 2nd Int. Symp. on Condensers and Condensation, p245, Univ. of Bath, UK, (1990). 12. Y.Machida and Y.Miura, 2nd Int. Symp. on Condensers and Condensation, p213, Univ. of Bath, UK, (1990). 13. R.H.Jureindini, M.R.Malin, MJ.Lord and K.K.Yau, 4th.lnt. PHOENICS Users Conf., Miami, Florida, (1991). 14. E.N.Jal and H.Tinoco, Proc. I.Mech.E., Paper C461/029/93, p93, (1993).
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15. N.Rhodes and C.R.Marsland, Proc. I.Mech.E., Paper C461/044/93, p109, (1993). 16. C.Zhang, A.C.M.Sousa and J.E.S.Venart, ASME J. Heat Transfer, VoI.115, p435, (1993). 17. C.Zhang and Y.Zhang, J. Energy Res. Tech., Vo1.15, p213, (1993). 18. S.J.Ormiston, G.D.Raithby and L.N.Carlucci, Num. Heat Transfer, Part B, 27, p81, (1995) 19. D.Chisholm and M.W.McFarlane, NEL Report 161, NEL, UK, (1964). 20. L.D.Berman and S.N.Fuks, Teploenergetika, Vol.6, No.7, p74, (1959). 21. P.J.Phelps and D.Kirkcaldy, CHAM Report 4690/1, CHAM, UK, (1984). 22. W.H.McAdams, Heat Transmission, McGraw Hill, (1954). 23. A.P.Colburn and O.A.Hougen, Ind.Eng.Chem., Vol.26, No.11, p1178, (1934). 24. D.G.Butterworth and G.F.Hewitt, Two Phase Flow and Heat Transfer, Oxford Univ. Press, (1978). 25. D.Q.Kern, Process Heat Transfer, McGraw Hill, (1950). 26. I.D.R.Grant and B.D.J.Osmet, NEL Report 350, NEL, UK, (1968). 27. J.Weisman and R.W.Bowring, Nuc. Sci. & Engng., Vo1.57, p255, (1975). 28. R.C.Reid and T.Sherwood, The Properties of Gases and Liquids, 2nd Edition, McGraw Hill, (1966). 29. J.B.Young, ASME J.Eng.Gas Turbines and Power, Vo1.110, pl, (1988). 30. R.W.Bain, Steam Tables, HMSO, Edinburgh, UK, (1964). 31. M.Moheban, PhD Thesis, Camb. Univ, UK, (1984). 32. D.B.Spalding, PHOENICS V1.4 Reference Manual, CHAM TR/200, UK, (1987). 33. D.B.Spalding, Int.J.Num.Meth.Engng., Vol.4, p551, (1972). 34. S.V.Patankar and D.B.Spalding, Int.J.Heat Mass Transfer, Vo1.15, p1787, (1972). 35. R.H.Beynon, NEL Report No.619, p98, Steam Turbine Condenser Meeting, NEL, UK, (1976). 36. W.M.Rosenhow, Trans. ASME 78, p1645, (1956).
Received October 25, 1996
Pergamon
PII S 0 7 3 5 - 1 9 3 3 ( 9 7 ) 0 0 0 4 6 - 8
MODELLING FLOW IN AN EXPERIMENTAL MARINE CONDENSER
M.R. Malin CHAM Limited Wimbledon, London SW19 5AU United Kingdom
( C o m m u n i c a t e d by P.J. H e g g s )
ABSTRACT A Computational Fluid Dynamics (CFD) model is described for simulating flow and heat transfer in a condenser. A single-phase approach is used which predicts the flow of a steamair mixture within the condenser shell, and the heat and mass transfer processes are modelled using empirical correlations. The model is used to calculate the overall performance of an experimental marine condenser with a superheated steam supply. © 1997 Elsevier Science Ltd
Introduction The potential of CFD for investigating future condenser designs and improving existing condenser performance has resulted in the development of a large number of condenser models [1-18]. The method has been applied extensively to land-based power-plant condensers, but less so for marine applications where compactness is a very important design consideration.
The flow field in a condenser comprises two distinct regions, one corresponding to the shell-side, and the other to the tube side which usually consists of a single- or multi-pass parallel arrangement of tubes through which cooling water passes continuously. The shell-side flow comprises a two-phase mixture of gas and liquid condensate. The gas phase contains air, which inevitably leaks into a system under vacuum, as well as steam vapour. The liquid phase arises from the condensation of the steam onto the tubes, and may exist either as droplets or columns of fluid between the tubes, and as films on the tubes.
Almost all CFD models employ a single-phase approach whereby transport equations are not solved for the flow of condensate, but rather the condensing steam is assumed to disappear from the solution domain, and only the condensate's inundation effect on the heat transfer is modelled. Empirical correlations are employed to account for the hydraulic resistance of the tube nests, and similarly, empirically-based relationships are used in specifying the local mass condensation and heat-transfer rates between the bulk and the coolant. The method most often used for calculating the local heat and mass transfer rates is that described by Chisholm and McFarlane [19], which uses the Berman-Fuks [20] correlation for the gas-side heat-transfer coefficient. Strictly speaking, this correlation is valid only for bulk 597
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M.R. Malin
Vol. 24, No. 5
conditions close to saturation. In the CFD model the steam-air mixture is presumed close to the saturation temperature corresponding to the vapour partial pressure, thereby obviating the need for a transport equation for the mixture enthalpy. The flow in the cooling-water tubes is treated as one-dimensional and is subject to a thermal analysis only, as the flow pattern of the tube-side fluid is known.
The present paper describes the development of a basic CFD model for predicting the overall performance of an experimental marine condenser. The particular operating characteristics of the experimental rig result in superheated steam entering the condenser unit. The present model therefore requires the novelty of solving a transport equation for the mixture enthalpy.
Mathematical Model Conservation equations: The following conservation equations for the steam-air mixture on the shell-side of the condenser are solved in a Cartesian coordinate system x (= x,y,z): V. ( p U ) =- m
;
V . ( p U Ya ) = V . ( ( #/(~s + pt/Gs.t ) V Ya )
V.(pU®U)=V.(peVU)-Vp+F-mU
+V.(pe(VU)
;
Ys = 1 - Ya
(1)
2 T - ~(#eV.U)_I)
(2)
V . ( p U H ) = V . ( (#/oh + WOh.t) V H ) + qg + qd
(3)
where: p is the mixture density; U (=u,v,w) the velocity vector; m the mass condensation rate per unit volume; p is the pressure; F (=Fx, Fy. Fz) is the friction force per unit volume due to the hydraulic resistance of the tube nest; H is the mixture static enthalpy; # and Mt are the laminar and turbulent dynamic viscosities; (~h and (~h.t are the laminar and turbulent Prandtt numbers; Ya and Ys are the mass fractions of air and steam vapour; and (~s and C~s.tare the laminar and turbulent Schmidt numbers. The Prandtl and Schmidt numbers are taken as unity, and the dynamic mixture viscosity tae= p. + Mr.
Flow-resistance relations: The axial resistance force per unit volume, Fz is calc,!!ated from [21]: Fz = - 0.5 fa P w2 / dh
(4)
where dh is the hydraulic diameter of the flow passage, and fa is a friction factor given by: fa = 64/Rez
{if
Rez< 2308 } "
fa=5.5.10 "3 [1 + ( 2 . 1 0 4
fa = 1.547.10 -4 Re °67 { if
k~ +106/Rez)l/3] do
{if R e z > 4 2 0 0 }
2308 < Rez < 4200 } (5)
where kr is the sand roughness height of the tubes, taken here as zero, and the axial Reynolds number Rez=plwld~p., where Iwl is the absolute value of the axial velocity. The horizontal and vertical velocity components u and v are in the cross-flow mode over the tube bundle, and their respective frictional
Vol. 24, No. 5
MODELLING FLOW IN AN EXPERIMENTAL MARINE CONDENSER
599
resistance forces per unit volume are estimated from the Jakob correlation for single-phase flow across banks of staggered tubes [22] : Fx=-2fcNpUcru
; Fy='2fcNpUcrv
; f c = [ 0 . 2 3 + 0 . 1 1 / ( P × / d o - 1 ) 1'°8 ] R e ~ ~5
(6)
where: N is the number of tube rows in the flow direction per unit length; Ucr is the resultant crossflow velocity; fc is the friction factor ; do is the tube outer diameter; and Px is the transverse tube pitch, which is the same as the diagonal pitch in the present tube arrangement. The velocities u and v are maximum velocities based on the minimum free area, and the Reynolds number Rec = pUcrdo/p..
Enthalpy sink terms: The mixture enthalpy equation (3) contains two sink terms which are associated with enthalpy transport from the bulk steam-air mixture to the vapour-liquid interface. The first sink term, qg, represents the transfer of heat to the interface, and the second term, qd, represents a net enthalpy flux owing to the species diffusion currents: qg = - % (T- Ti ) Aw/V
; qd = - (Cp,s - Cp.a ) m (T- T~ )
(7)
where: % is the gas-side heat transfer coefficient; T is the bulk temperature of the steam-air mixture; T~ is the interface or condensate surface temperature; Aw is the total tube surface area in the computational control volume V; %.a and Cp.sare the specific heats at constant pressure of air and steam vapour; and m is defined by m = m" Aw/V, where m" is the mass condensation rate of vapour per unit area.
Thermodynamic relations: The bulk temperature of the vapour-air mixture is determined from: H = Ya Ha + Ys Hs
;
Ha = Cp,a(T - To)
;
Hs = %,s (T - To )
(8)
where: Ha and Hs are the static enthalpies of air and steam vapour; and To is a reference temperature taken as zero K. These equations imply that the specific heat of the mixture Cn,m is given by: Cp,m = YaCp,a + Yscp,s
(9)
where here %,~=1900 J/kgK and %,a=1005 J/kgK. The foregoing relationships make use of the fact that the enthalpy of the gas phase knows nothing about the latent heat, and also that the enthalpy of the liquid phase need not be introduced into the model at all because it only has to be taken out again.
Heat-transfer relations: The transfer of heat from the bulk mixture to the interface comprises two components [23]; the heat released at the interface due to the condensing vapour, and the sensible heat transferred through the diffusion layer to the interface. Thus, q"=m"L+
~ct.g(T-Ti)
;
(p = ~0H](1-exp(-q)H))
;
~H=m"Cp,s/O(g
(10)
where q" is the heat flux, L is the latent heat of condensation, and ~0 is the Ackerman-correction factor [24] which accounts for the effect of mass transfer on %. The heat fluxes through the condensate film and through the wall to the coolant are given by
600
M.R. Malin
q" = ~c (Ti- Two)
;
Vol. 24, No. 5
q" = CZwt(T~,o - Ti)
(11)
where Two is the temperature of the tube outer surface, c~c is the condensate-film heat-transfer coefficient, T~ is the coolant temperature, and Ct.w~is the heat-transfer coefficient between the coolant and the tube outer wall. An effective overall heat-transfer coefficient % may be calculated from q" = c% (T - T{) once the temperature difference and the heat flux are known. Mass-transfer relations: The condensation mass flux is computed from: m"=pK,,, ,~,f~ l n ( l + p , - p , . i )
M,,,
;
p - p~
K
'"
=[Pr/Sc]Z/3o~,/(pCp,,,,)
(12)
where: Km is the mass transfer coefficient; Mm and Ms are the molecular masses of the mixture and steam vapour; Ps is the vapour partial pressure in the bulk; Ps.~is the vapour partial pressure at the interface; the Prandtl number Pr=cp.mp/k; the Schmidt number Sc=p/(pDsa); k is the mixture thermal conductivity; and Dsa is the diffusivity of the steam in air [25]: D,. = 1.38674.10 s7'~/2[1/M, + 1 / M , ] I/2 / p
(13)
where the temperature T is in K, p is the static pressure in bar, and Ma is the molecular mass of the air. The values of D~a, k and it are evaluated at the film temperature Tf=(T+Ti)/2. The bulk vapour partial pressure is determined from:
p, = p / [ I + Y , M
/(YsM
)]
(14)
The interface vapour partial pressure Ps, depends on the interface temperature T~, which is taken to be at the saturation state for the condensing vapour.
Heat-transfer coefficients: The heat-transfer coefficients c% and c~ are given by:
c¢., = [
d
"
-t
d In(d,, / d i) "
dio: I
2k
+ Rt ]4
"
oq = kt 0.023ReC/S Pr/°4 d,
(15)
where Rf is the fouling resistance taken as 1.5.10 s m2Kj'VV, d~ is the tube inner diameter, kw is the tubewall thermal conductivity taken as 27.3 W/mK, k~ is the coolant thermal conductivity, Ret=p~w~di/i.q, Pr~=Cpkp~/k~,and the coolant properties p~, it{, cp~ and k~are evaluated at the local bulk coolant temperature. The condensate-film heat-transfer coefficient is based on the equation of Nusselt [24]:
Otc = o~;F,.Ftl- ~.
;
a~ = 0 . 7 2 5 ( k ] L p ~ g
/ [ d / 2 c ( T , - T,,,,)]) '/4
(16)
Vol. 24, No. 5
MODELLING FLOW 1N AN EXPERIMENTAL MARINE CONDENSER
601
where a~ is the condensate film heat-transfer coefficient in stationary vapour, and kc, Pc and I-tc are the condensate thermal conductivity, density and dynamic viscosity, respectively. The vapour-shear factor Fv [19], superheat factor Fs [24], and inundation factor, Ft [19, 26] are given by :
n 118/N.J~
Fv = 1 + 0.0095 KeA
'
;
F, =
%,, ( r - ~ , ) L
;
FI =
"
m / m
l
(17)
where: Nu c = ~z*, d,,/kc; ReA is the approach Reynolds number of the mixture based on do and a resultant velocity based on the approach values of the cross-flow components; and Tsar is the saturation temperature evaluated at the bulk vapour partial pressure. The inundation factor represents the effect of condensate drainage, and the numerator of the F~ expression denotes the sum of the condensation rates from the uppermost to the current row of tubes, and mn is the condensation rate on the nth tube.
The gas-side heat transfer coefficient % is estimated from standard single-phase correlations for flow across staggered tube banks [22], assuming that the presence of the condensate film has no effect: 0~~ = ( k / d~,)Nu = (k / d,)[0.33 Re~ 6 Pr 'j3 ]
(18)
where k is the thermal conductivity of the steam-air mixture based on the film temperature.
Heat and mass transfer algorithm: The conditions at the liquid-vapour interface are unknown, and
consequently the local temperatures are determined by solving equation (10) for the local heat flux q" in an iterative manner. The solution proceeds from the last available values of q" and T~, and from the values of T and T~ resulting from the solution of the shell-and-tube side enthalpy equations. The interface is assumed to be a saturation condition for the vapour, and so Ps., may be evaluated from knowledge of T~. The bulk value of Ps is known from equation (14), and so once % and Km are evaluated, the condensation mass flux can be updated via equation (12). Next, ~, C~w~and ~ are computed from equations (10), (15) and (16), and q" is then updated from equation (10) expressed in the form:
q"=m"L+Oa~[T-T
t -(T~ - T
)-(T,,,,-Tt) ]
(19)
The temperature differences (T~ - Two) and (Two - T~) are evaluated using the last iterate values of q" in equations (11). These equations, together with the new value of q", are then used to update the values of Two and T~. The iterative nature of the CFD solution cycle is utilised so as to introduce successively better input values for equation (10), until a converged solution is obtained for the current cell.
The local coolant temperature T~ is calculated by application of the one-dimensional steady-flow energy equation, which can be expressed in terms of temperature changes as follows:
602
M.R. Malin
Vol. 24, No. 5
ml[ c.,i,o~tTi,out - cp,l,i. TI,in ] = q" A .
(20)
where m~ is the mass flow rate of coolant flowing through the tubes in the finite-volume cell; and the subscripts in and out denote inlet and outlet values with respect to the control volume. The condenser employs a two-pass arrangement of the coolant, which is handled by assuming that at the exit from the first pass, the coolant is thoroughly mixed to give a uniform temperature into the second pass.
Turbulence model: The turbulence model employed here calculates the turbulent viscosity from: pt= C p Vslrn
(21)
where C is a constant, V s is a turbulent velocity scale and I,,, is the turbulent mixing length. In untubed regions, C=0.5, Vs is estimated at 5% of the steam inlet velocity, and Im is taken to be 10% of the minimum distance between the tube bundle and the condenser casing. In the tubed region, C=1, Im =0.1dh where dh is the hydraulic diameter of the flow passage, and V s is computed from [27]:
Vs
R o~
-
e,,, - 15,u p( P~ - d,, )
;
Rein
= p ( U . U )t/2
dh/p.
(22)
where Rein is a Reynolds number, with the resultant velocity U (=u, v, w) based on the maximum velocities associated with the minimum flow area. These velocities are calculated directly from the momentum equations because the flow areas are multiplied by the appropriate porosity factors: ~z=l-~
d~ /(PxPy)
;
!3x =]~y =(P~-do)/P×
(23)
where Py is the longitudinal pitch.
Mixture properties: The mixture density 9, laminar viscosity p ,and thermal conductivity k are calculated from [28]: 11#. p=l/(Ya/Pa+Ys/Ps)
; Y= -
-
lzy +
t~ + n O , :
' '
n +n.,Oz r
.
(l+AV2B[m)2 ~12-
" (P21 = 41__L (8[I+I/BI)
I/~
(24)
AB
where Pa and 9s are the densities of the air and steam vapour; y=# or k; Ya and Ys are the component properties; na and ns are the mole fractions; A=Ma/Ms; and B=Ms/Ma.
Steam properties: The steam vapour density is calculated from [29]:
p,=
[2pM~ I (R,,T)
;
B=aI(I+T/~z)
I + a , e ~ ( 1 - e r ) 5 / - r ln +axr
(25)
I+[I+4pBM, /(R,,T)] ]12
where T is in K, Ro is the universal gas constant, B is the second virial coefficient, t = 1500/T, a~ = 0.0015, a2 = -0.000942, a3 = -0.0004882 and o~= 104. The dynamic viscosity Ps and thermal conductivity ks of the steam vapour are computed from:
Vol. 24, No. 5
M O D E L L I N G F L O W IN AN E X P E R I M E N T A L M A R I N E C O N D E N S E R
,u s = a , X 3 + b , u X 2 + c u X + d ~
; k =a k +bk(T-273.15)+c,(T-273.15)
2
603
(26)
where X = T/273.15, Y is in K. a~=-0.18321.10 s, b~=0.59888.10 -s, c~=0.43864.10 -5, d,=-0.43769.10 6, ak=0.0163918, bk=7.26527.10 .5 and ck=0.115507.10 -e. The saturation pressure at the liquid-vapour interface is computed from the saturation temperature by use of the following relationship [30]:
p~ = 6894.76exp(/3)
(27)
with
/3 = 73.32642 - 8.2In(T) +0.003173T- 13023.8/T ~=a+b/T+cfl,[exp(dfl~)-l]/T+eexp(ffl12
25)
{ for T < 100°C }
(28)
{for100°C
(29)
131 = T 2 - 9.51588.10 -5 ; /3, = 1165.09- T
(30)
and T is in °R, a=15.182911, b=-8310.453, c=1.7741.10 -4 , d=2.624453.10 12 , e=-1.013139.10 -2 and f=6.31141.10 .3.
The bulk saturation temperature Tsa [30] and the latent heat of condensation are evaluated from:
~
= a T +bTF+CT 1"2 +dT I'~ +Or r4
;
L = a L + b L T + c L T ~-+ c l t T 3
(31)
where T is in K, r=ln(pv), aT=4.92510 -3, bT=-2.23610 "4, CT=7.0461.10 -6, dT=-5.208410 "7, eT=1.096910 "a, aL=3.46892106, bL=-5707.4, CL=I 1.5562 and dL=-0.0133103.
Air properties: The air density 9a, dynamic viscosity ~[a and thermal conductivity ka are given by: P"
= pM, R,,T
;
I~,, = P r ( T / T r ) °76
;
k =a+b(T-273.15)
(32)
where .Ur= 1.846.10 -5, Tr = 300K, a=0.0242, b=7.49486.10 5 and T is in K.
Coolant properties: The coolant density p,, dynamic viscosity ~a,, specific heat cp.~ and thermal conductivity k~ are calculated from:
Pt = ap + b p T l "}'CpTI 2 k t = a, +b~.T t + c , Tl -~
;
I'll = au(bu + c , Tt + d~,T~2 ]
;
c j = a c +b.T t
(33) (34)
604
M.R. Malin
Vol. 24, No. 5
where TI is in °C, ap=1028, %=-0.103, %=-3.22.10 -3, a~=4.133.10 -4, b~,=4.09, q,=-0.0945, d,=8.5.10 4, ac=3987, bc=0.444, ak=0.544, bk=2.225.10 3 and Ck=-I 0 -~.
Condensate Properties: The properties are taken as constant at values based on saturation conditions at the condenser operating pressure: pc=978.1 kg/m 3, kc=0.667W/mK and pc=4.10 .4 Ns/m 2.
Experimental Confi.quration The model is applied to a two-pass surface marine condenser studied experimentally by the DRA and GEC ALSTHOM The overall dimensions are 1.238m wide by 3.65m high by 2.476m long. The steam-air mixture enters the condenser through a long perforated pipe located along most of the condenser length at the top of the unit, and then impinges vertically downwards on the two circular tube bundles located in the main body of the condenser. The cooling water (CW) enters the lower bundle through a parallel staggered arrangement of tubes of 5/8" outside diameter with a mean thickness of 0.048". The CW then returns through the upper bundle and leaves at the top RH end of the unit. The tube bundles are mounted on a number of support plates through which all tubes pass, thereby partitioning the condenser into several sections and restricting the axial flow within the tube nest. The upper bundle comprises 928 tubes and the lower bundle 909 tubes, resulting in a total heat-transfer surface area of 225.84m 2. Each bundle has an air-extraction gallery which runs the full length of the bundle. The steam-air mixture enters the bottom of the extraction system passing between plates hanging from the top of each bundle, and is then removed via vent-offtake piping. The operating conditions for one half of the condenser are as follows: Steam
mass
inflow
rate=8.63
kg/s;
Air
mass
inflow
rate
including
natural
leakage=2.78g/s;
pressure=0.345 bar; Steam inlet temperature=116.5°C; Inlet saturation temp~,ature=72.5°C: Mixture mass outflow rate=13.25g/s; CW mass flow rate=207.87kg/s; and CW inlet temperature=13.07°C.
Solution Method The conservation equations are discretised using the finite-volume approach embodied in the generalpurpose CFD code PHOENICS [32]. The method uses a staggered-variable arrangement for the velocity components and scalar variables, and the effects of convection and diffusion are weighted using the hybrid differencing scheme [33]. The solutions of the discretised elliptic conservation equations are obtained using an iterative pressure-correction method based on the SIMPLE algorithm [34].
The condenser is symmetrical about the vertical centre plane, and so only one half of the flow geometry is modelled. The solution domain is discretised using 7 grid cells in the lateral (x) direction, 19 cells in the vertical (y) direction and 9 in the axial (z) direction. Grid-dependency studies revealed that a mesh with twice the present number of grid cells in each coordinate direction produced almost identical predictions of the overall performance parameters for the condenser. Grid cells totally impervious to flow, such as those located inside the inlet pipe, are represented by the use of a porosity treatment, whereby the free volumes and cell-face areas of these cells are set to zero. This practice is also used to define the
Vol. 24, No. 5
MODELLING FLOW IN AN EXPERIMENTAL MARINE CONDENSER
605
vent plates of the air-extraction system. The flow restriction in the tube bundle is represented with partial area and volume porosities, which are also used for the prescription of the shape of the tube bundle and the steam inlet pipe. The air-extraction system is terminated at the top of each tube bundle, so as to define the vent outlet boundary for the steam-air mixture. At the steam-air inlet boundary, the local mass inflow rate per unit area is calculated from mi, ' = Co[2pi,,(pi,
-
pr)] ~/=, where Co is a discharge coefficient taken as 0.94,
pp is the in-cell static
pressure, and p~, and Pin are the inlet static pressure and density, respectively. The inlet resultant velocity (typically about 130 m/s) is computed from min and P~n,and then it is resolved into the x and y coordinate directions so as to define the incoming values of the velocities u and v. The inlet air mass fraction is prescribed according to calculated mass inflow rate of steam and the measured mass inflow rate of air. The inlet mixture enthalpy is calculated from equations (8). The upper and lower vent boundaries are divided into two equal lengths, resulting in four vent-outflow boundaries. At each of these boundaries, the mixture outflow per unit area is fixed according to the measured partitioning of the total mixture outflow: 5% extraction from the upper bundle, split equally between the top two vents; 95% extraction from the bottom bundle, with 2/3rd of 95% taken from the vent at the cold end, and 1/3 of 95% taken from the other lower vent.
The CW inlet temperature and mass flow rate are specified directly, and no outlet condition is required as the coolant exit temperature is an outcome of the integration performed in the direction of the CW flow. Results and Discussion
For reasons of brevity and the absence of detailed measurements, velocity and contour plots resulting from the 3D computation are not
presented, but rather attention is focused on describing the main
features of the calculation and comparing the simulated and measured performance parameters.
The main results are presented in Table 1, which presents the computed and measured principal performance parameters for one-half of the condenser. The table indicates the various inflows on the shell-side of the condenser, the amount of steam condensed, the power absorbed by the cooling water on each pass, and also the corresponding temperature rises in the coolant fluid.
An inspection of Table 1 reveals that the calculated and measured performance parameters are in excellent accord with the experimental data. The predicted overall condensation rate agrees almost exactly with the measured value, and the inflow rates of steam and air are also predicted very well by the CFD model. The results show that more heat is extracted on the average from the lower bundle, and the partitioning of the thermal duty between the upper and lower bundles is predicted to within 2% of the measured value. This
heat load is obviously reflected in the condensation rates, which are
correspondingly higher on the lower bundle.
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TABLE 1 Comparison of Measured and Predicted Performance Parameters Quantity
Prediction
Data
Total mixture inflow (kg/s)
8.66
8.63
Total air inflow (g/s)
2.79
2.78
Total condensation rate (kg/s)
8.65
8.62
CW exit temperature after 1 pass (°C)
26.65
26.79
CW average exit temperature (°C)
37.67
38.10
Total heat transfer rate to CW (MW)
20.52
20.80
% Heat transfer rate to lower bundle
56.0
54.80
The largest air mass fractions are predicted in the lower bundle, with values around 24% near the venting at the cold end, and around 18% at the hot end. The model predicts lower values of around 4 or 5% near the venting on the upper bundle. In general, the air concentrations are low in the bundle periphery, where the heat fluxes are high, and high in the bundle core where the heat fluxes are lower. The computed values of the overall heat-transfer coefficient are in the range 2.3 to 4 kW/m2K. The average value, based on the condenser saturation temperature and the mean coolant temperature, is 3.85 kW/m2K, which compares favourably with the experimentally-derived value of 3.94 kW/m2K. It is interesting to note that these values, as well as the proportion of thermal loading on each bundle and the mass fraction of air leaving the unit (21%), compare very well with those reported by Beynon [35] and Wilson [2] for an experimental condenser of very similar construction with similar operating conditions and a slightly-superheated steam supply. The condensation of superheated steam differs from that of saturated steam in so far as steam which comes into contact with the tube wails will condense only if the wall temperature Two is less than the saturation temperature Tsat. Otherwise, the tube wall will be dry and desuperheating will occur, i.e. heat transfer without condensation. If anywhere, this is most likely to occur near the hot end of the condenser, and so the CFD model detects whether Two > Tsar, and then switches to accommodate dry-tube heat transfer. However, the tube temperatures of the final solution indicated no desuperheating.
Conclusions
A CFD model has been applied to an experimental marine condenser. The validation against data revealed that the model predicted very accurately the overall condenser performance. The computed overall condensation rate was in excellent agreement with that measured, and the thermal loading on
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each pass was predicted correctly. Further work should include refining the modelling, and performing more varied and detailed validation exercises, especially for the saturated or wet inlet conditions usually encountered under commercial operating conditions. The pressure-drop correlation can easily be extended to account for the presence of the two phase flow of gas and condensate [7,24], and the cooling-water treatment might be refined by incorporating a log-mean temperature analysis. The Nusselt heat-transfer coefficient could be refined to account for the effect of condensate sub-cooling [36], and the inundation treatment so as to correlate the inundation factor against the local Froude number [7].
Acknowledgements The experimental information presented in this paper appears by permission of GEC-ALSTHOM in Rugby, and the Defence Research Agency (DRA), UK. The study was carried out with the support of these organisations during 1989. The author also acknowledges the input provided by GEC-ALSTHOM on the specification of the condenser boundary conditions.
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Received October 25, 1996