Mathematical Modeling of Tube-Support Fouling in Nuclear Steam
Generators
Y. Rihan
Atomic Energy Authority, Anshas, Egypt
Abstract:
Particle deposition on steam generator tubes and supports is a complex phenomenon
involving many mechanisms that interact. The approach taken in this paper was to
characterize the two-phase flow patterns existing designs, correlate these with fouling
patterns observed in the field, and develop a support fouling mathematical model to better
understand the dynamics of the fouling mechanisms. A computer program was coded to
predict deposit thickness in and near a support. This model has the same classic particle
deposition models as in the larger codes, but considers additional factors such as stagnation
zones and surface normal to the flow.
Keywords: Fouling, Nuclear, Steam generator, Modeling.
1. Introduction:
Steam generators (SG) in power plants based on pressurized water reactors (PWRs) transfer
heat from a primary coolant system (pressurized water) to a secondary coolant system.
Primary coolant water is heated in the core and passes through the steam generator, where it
transfers heat to the secondary coolant water to make steam.The steam then drives a turbine
that turns an electric generator. Steam is condensed and returns to the steam generator as
feed water.
A general schematic view of a PWR plant is shown in figure 1. Notice that the steam from the
steam generator flows out of the containment structure and it must, therefore, be pure and not
contain any radioactive materials. Since the primary fluid does contain radioactive material,
one must preserve the complete separation between these two fluids, i.e. the integrity of the
tubes which carry the primary fluid through the SG must be maintained. This is the primary
concern in the design, construction, operation and maintenance of PWR SG.
Fouling is the accumulation of unwanted materials on heat transfer surfaces. In many
industrial heat transfer equipment, fouling is an unavoidable by-product of the heat transfer
process. These deposits are usually poor thermal conductors, and as they accumulate there
is a decline in the thermal and hydrodynamic performance of the heat transfer equipment.
The economic penalty of fouling can be attributed to: lower production capacity due to energy
losses, higher capital expenditure through oversized units to compensate for the effects of
fouling, and costs associated with periodic cleaning. In addition, fouling in steam generators
can lead to local hot spots, and ultimately it may result in mechanical failure of the heat
transfer surface, and hence unscheduled shut down of the equipment.
Commercial PWR steam generators have experienced reliability problems within the first
decade of operation associated with material degradation, one of the causes of which is
particle deposition and tube fouling. As a result steam generators often require costly outages
for inspection and cleaning of fouling deposits. Knowledge of locations where sludge has
accumulated in the steam generator can aid in planning and targeting locations for cleaning
and removal of deposits [1].
Fig. 1: General view of a PWR power plant.
Eddy current inspections have revealed that heavy deposit formation is also present on the
free span region between the tube support plates and in some cases, bridging the gaps
between tubes. Heavy deposit formation on tubes and elsewhere in the steam generator
appears to be a pre-cursor for subsequent material degradation.
Heat transfer fouling is an undesirable process in which unwanted materials with low thermal
conductivity deposit on heat transfer surfaces. Several forms of fouling exist based on the
chemical/physical conditions under which the deposit layer forms. One of the most severe
forms of fouling occurs during pool boiling heat transfer [2].
For steam generators, the fouling mechanism in the combustion chamber area is different
from that in the convection heat transfer areas. In combustion chambers, deposits grow
mainly in the form of very porous 1-10 cm pieces of ash called pyroclastic lumps. Wynnyckyi
and Rhodes [3] found that pyroclasts attach strongly to the walls of the lower areas of the
combustion chamber while they detach easily from the upper section. In the convection heat
transfer areas fouling mainly takes place as particle adhesion on the external tube walls.
Fouling modeling needs knowledge about the particle attach and detach mechanisms as well
as heuristic knowledge of the influence of geometric characteristics of the system on fouling
effects [4].
Fouling of heat transfer surfaces causes significant cost penalties, due to both reduction in
heat transfer coefficient and increases in boiler metal wall temperature in order to provide the
increased temperature differential necessary to overcome the fouling resistance. Increasing
boiler metal temperature also results in a rapid utilization of the life of the components with
possible creep rupture, if the temperature increase is too high. As a result, such devices often
require costly shutdowns for inspection and removal of fouling deposits.
Knowledge of locations in tube banks, where deposit has accumulated can aid in planning
and targeting locations for cleaning and removal of deposits. Such information is linked to the
understanding of the mechanisms of the fouling process on tube bundles during pool boiling.
Nevertheless, published studies to-date has been undertaken for single tubes or wires, while
the authors are not aware of any systematic information on fouling of tube bundles [5-14].
A computer code called SLUDGE was developed in the late 1980's to predict the deposition
of particles through the SG [15]. The local thermal-hydraulic conditions were computed by the
THIRST code [16] and fed into the SLUDGE code. This macroscopic code modeled the
supports simply in terms of porosity. A tool was needed to more accurately predict local
deposition.
The design of some supports is believed to actively contribute to blockage problems by
creating a thermal-hydraulic environment favorable to deposition. The aim of this paper was
therefore to better understand the role that thermal-hydraulics plays in the fouling of SGs, and
of supports in particular. This amounts to assessing the fouling propensity of different support
designs and to recommending methods of predicting and mitigating support fouling.
2. The Mathematical Model:
2.1 The basic equations governing particle behavior
A computer program was developed to predict the rate of deposition on supports analytically.
The deposit thickness for turbulent flow can be expressed as:
f
K

 s 1   s  A u *
 
2
 

  At u *
1  exp 
 f


2




(1)
In the case of laminar flow the constant A equal to zero.
In this model the effect of both drainage force and lift force are conveniently accounted for in
empirical constants such as constant A and the attachment constant Ko [17, 18].
The expression of the change of particle concentration with distance along the channel for
steady state situation is:

Dtube  Q
d

 K

dz  f Dm w  h fg

(2)
The particles are deposited on the tube and support surfaces. For turbulent flow the K term
must be modified to include the removal term. Thus, the difference in particle concentration in
the liquid over interval Δz is expressed as:
  At u * 
D z  Q
T  tube 
 2K tube  K tube/ sup exp 
 f A  h fg
  f
2


D z  Q
 L  tube 
 2K tube  K tube/ sup 
 f A  h fg



 
(3)
(4)
The empirical method by Levy [19] is used to obtain the mass quality:
  th
   th   bd exp 
  bd

 1

(5)
Where χbd is the χth at the point of bubble detachment:
 bd  
Cp f Tsub,bd
(6)
h fg
Where ΔTsub,bd can be calculated based on the work by Saha and Zuber [20]:
Tsub,bd  0.0022
Tsub,bd  153.8
QDhy
kf
Q
,
GCp f
,
Pe  70000
Pe  70000
(7)
(8)
The void fraction is calculated from the superficial liquid velocity and mass quality using the
drift flux model [21]:

jg
Co  j f  j g   Vgj
(9)
where the superficial gas and liquid velocities can be written as:
jg 
G
g
,
jf 
G1   
f
(10)
The void distribution parameter Co is set to 1.03 and the weighted mean drift velocity for this
regime in an annular channel can be expressed as [22]:
  f g  f   g 
V gj  40.25

f2


1/ 3
(11)
For two-phase friction pressure loss the Darcy friction factor is based on the all-liquid
Reynolds number [21]. Due to the discontinuity between the laminar and turbulent friction
factors and in order to avoid calculation problems, a smoothing function is applied as follow:

f  f T3  f L3

1/ 3
(12)
where,
fL 
64
Re
(13)

 Re f   / Dhy 1.11  


  
f T   1.8 log 
 
 6.9  3.7   



GDhy
Re f 
2
(14)
(15)
f
The pressure difference between node i and the previous node i-1 is a sum of pressure
changes due to static head, friction, sudden contraction, sudden expansion, area changes,
flow development, and acceleration arising from quality increase. For a given node i:
(16)
Pstatic   i gz
G2
(Pfr ) L 
2 f
 z 
 

f
1    f g  1
 D 
 

hy  

 g f

 z  

 


f
 1    f  1 1    f  1
 D 

 


hy  

 g
 
 g

 1
1 
   i 1 
Pa  G 2 


 i

g
f


G2
(Pfr ) T 
2 f
(17)
0.25
(18)
(19)
where,
 i   i 1 
QDtubez
GDtube  wwh fg
 
Gi2 
A   f
Pfl 
 i 1  i  1  
 1  
 
2 f 
A fs     g
 
2
 
  f
G2  1
Pcon  i 
 1 1  
 1  

 
2  f  Cc
    g
 
2
2
 
G 
A   f
Pexp  i 1 1  i 1  1  
 1  
 
2 f 
Ai     g
 
(20)
1.375
(21)
(22)
(23)
The expression for the overall single-phase deposition coefficient consists of the transport (K t)
and attachment (KA) coefficients working as two steps in series. The coefficients for
gravitational and centrifugal settling are added directly to the deposition coefficient for the
horizontal nodes [23-25]:
K1
1
1 
 

 Kt K A 
1
(24)
K 2  K1 
BQ
h fg 
(25)
where,
K t vertical  KTH  K D  K I
K t horizontal  KTH  K D  K I  K G  K c
(26)
(27)
The attachment deposition coefficient based on experimental results by Turner and Godin
[26]:

E
K A  K o exp 
 k Tw  KET



(28)
'
-23
-23
where Ko = 5.697, k (Boltzmann constant) = 1.3807×10 and E = 12.687×10 .
The deposition coefficient in vena contracta region is expressed as:
KV 

  f u 2f k c Ds2
s
18 f v f
1  N 1  C sin 
imp
c

zi 

z att 
(29)
Soluble such as iron and copper are known to be deposit cementing agents, as they are
typically found in the pores of very hard deposits. Impurities such as silicates and calcium
salts, which find their way into the steam generator feedwater from condenser leaks, also
harden existing deposits. Consolidation studies with artificial sludge [27] showed that only 12% of an impurity was required to significantly harden the sludge. An empirical function of the
inverse exponential function of the weight fraction of precipitate in the deposit was based on
this:

S
Ai  3  10 12 exp  1150 C
Ss


K 
1  T 
 K A 
(30)
Centrifugal settling would occur when flow is recirculating directly above the support. The
deposition coefficients for the gravitational and centrifugal settling are based on Stokes' Law
can be expressed as:
KG 
KC 


s
  f gDs2
18 f v f
s
1  N 1   N
  f u 2f k c , Ds2
18 f v f
imp
sup
1  N 1   N
imp
(31)
st
sup
st
(32)
2.2 The mathematical model validation:
A comparison between the present mathematical model and the results with those using a
model from Idel'chik [28] for thick edged orifices in a straight channel for turbulent flow at the
same conditions is shown in Fig. 2. The comparison shows that there is a good agreement
between the two models.
3500
Idel'chik
model
Present model
3000
P, Pa
2500
2000
1500
1000
500
0
0
500
1000
1500
2000
Mass flux, kg/m2.s
Fig. 2: Comparison with Idel'chik [28] model at different mass flux.
3. Results and Discussion:
Flow channel
SG tube
Suppor
t
Fig. 3: Geometry for the model.
The geometry of the model is a one-dimensional annular flow channel as shown in Fig. 3. The
inner surface of the annulus represents the heated SG tube. The outer surface gives the
proper flow channel width. The fluid is a steam/water mixture with particles flowing with the
liquid phase. The code is tested using the considered conditions given in Table 1.
Inlet pressure
Bulk temp.
Heat flux
Channel gap width
Deposit porosity
Deposit density
Table 1: Input values in the model.
4.5 MPa
Inlet magnetite concentration
o
250 C
Soluble concentration
2
120 kW/m
Tube diameter
2 mm
Free span gap width
0.5
Deposit thermal conductivity
3
5000 kg/m
Deposit surface roughness
0.5 ppm
5 ppb
12 mm
4 mm
20 w/mK
3 µm
Figure 4 give the predicted results of deposit growth on the tube in free-span region with time
at different mass flux and inlet steam quality 0.2. Deposit thickness increases as the mass
flux increases a direct result of the suppression effect, in which faster moving fluid suppresses
deposition more.
The deposit growth on the tube in the free-span region and the support region as a function of
time is shown in Fig. 5. The deposit within the support region approaches its asymptote more
quickly because of the higher velocity and hence stronger suppression term. As shown in
figure due to turbulent flow the deposit does not grow linearly but approaches a constant
thickness over time. The predicted deposit thickness decreases as the steam quality
decreases as shown in Fig. 6. The figure shows that there is little deposition when no boiling
takes place.
Deposition thickness, m
25
G=100 kg/m2.s
G=200 kg/m2.s
20
G=400 kg/m2.s
15
10
5
0
0
2
4
6
8
Time, monthes
10
12
14
Fig. 4: Deposit growth on the tube in free-span region with time at different mass flux.
Deposition thickness, m
16
14
12
10
8
6
4
Support region
2
Free-span region
0
0
2
4
6
8
Time, monthes
10
Fig. 5: Deposit growth with time.
12
14
Deposition thickness, m
30
x=0
x = 0.1
x = 0.2
x = 0.3
25
20
15
10
5
0
0
2
4
6
8
Time, monthes
10
12
14
Fig. 6: Deposit growth on the tube in free-span region with time at different steam
2
quality, G = 200 kg/m s.
Figure 7 predicts the deposit growth as a function of particle size. The figure shows that for a
2
mass flux of 400 kg/m s the particles do not impact and stick unless they are greater than 7
microns. The larger particles are generally present in the SG at lower concentration than
smaller particles. The axial pressure profile is shown in Fig. 8. There is a relatively large
pressure loss at the support inlet.
The modeling of the deposition process allows discussion of the effects of a number of
process parameters on the fouling rates. Under both constant heat flux and constant sensible
heat transfer modes, fouling is reduced by minimizing the surface temperature and the
surface-to-bulk temperature difference. The fouling model can be used to estimate optimal
cleaning cycles of the steam generators.
Deposition thickness, m
800
700
600
500
400
300
Time = 1 month
200
100
0
6
8
10
12
14
Particle size, m
2
Fig. 7: Deposit growth as a function of particle size, G = 400 kg/m s.
Relative pressure, kPa
18
Support
region
14
10
6
2
-2
-6
-10
-20
-10
0
10
20
30
40
Distance, mm
Fig. 8: Axial pressure profile.
4. Conclusions:
Particle deposition on steam generator tubes and supports is a complex phenomenon
involving many mechanisms that interact. The approach taken in this paper was to
characterize the two-phase flow patterns existing designs, correlate these with fouling
patterns observed in the field, and develop a support fouling mathematical model to better
understand the dynamics of the fouling mechanisms. A computer program was coded to
predict deposit thickness in and near a support. This model has the same classic particle
deposition models as in the larger codes, but considers additional factors such as stagnation
zones and surface normal to the flow. The deposit thickness decreases as mass flux or
quality increases. The predicted deposit thickness increases as particle size increases. The
model provides quantitative information of tubes and supports fouling on heat transfer
surfaces. The predicted results are significantly important for the design, and for formulating
operating and cleaning schedules, of any industrial heat transfer equipment subject to fouling.
Nomenclature
A
B
Cc
Co
Cp
Dhy
Dm
Ds
Dtube
f
g
G
hfg
j
k
kc
K
m
N
P
deposit removal factor
boiling coefficient, m/s
fraction of open flow area in vena contracta region
void distribution parameter
specific heat, J/kg.K
channel hydraulic diameter, m
mean diameter of annular flow channel, m
particle diameter, m
tube diameter, m
Darcy friction factor
2
gravitational acceleration, m/s
2
mass flux, kg/m .s
latent heat of vaporization, J/kg
superficial velocity, m/s
thermal conductivity, J/m.K.s
-1
inverse of streamline curvature radius, m
deposition coefficient, m/s
mass flow rate, kg/s
fraction of particles
pressure, MPa
Pe
Q
Re
S
t
u
*
u
Vgj
w
z
α
βs
βsup
χ
δ
ε
Φ
µ
ν
ρ
ρs
τ
Peclet number
2
heat flux, J/s.m
Reynolds number
rate of precipitation, kg/s
time, s
actual velocity, m/s
friction velocity, m/s
weighted mean drift velocity, m/s
width of annular flow channel, m
axial distance, m
void fraction
deposit porosity
support porosity
mass quality, kg/kg
deposit thickness, m
surface roughness, m
particle concentration, kg/kg
dynamic viscosity, kg/m.s
2
kinematic viscosity, m /s
3
density, kg/m
3
particle density, kg/m
flow development parameter
Subscripts
1Φ
2Φ
a
A
att
bd
c
C
con
D
exp
f
fl
fr
fs
g
G
i
i-1
imp
I
L
s
st
sub
sup
t
T
th
TH
w
one-phase
two-phase
acceleration
attachment
attachment
bubble detachment
centrifugal settling
cementing agent
sudden contraction
diffusion
sudden expansion
liquid
flow development
friction
free span
gas
gravitational settling
current node
previous node
impaction
intertial coasting
Laminar
solid
sticking
subcooled
support
transport
turbulent
thermodynamic
thermophoresis
wall
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ABOUT THE AUTHOR
Dr. Yasser Rihan is currently working as an Associate Professor in the Department of Mechanical
Engineering at Nuclear Fuel Tech.Dep., Atomic Energy Authority, Egypt.
RESEARCH PAPERS