Aerosol Science and Technology, 39:304–318, 2005
c American Association for Aerosol Research
Copyright ISSN: 0278-6826 print / 1521-7388 online
DOI: 10.1080/027868290931069
Thermophoretic Deposition in Tube Flow
C. Housiadas1 and Y. Drossinos2
1
2
“Demokritos” National Centre for Scientific Research, Athens, Greece
European Commission, Joint Research Centre, Ispra (Va), Italy
Thermophoretic deposition in laminar and turbulent circularpipe flows is investigated. One-dimensional (1D) Eulerian and twodimensional (2D) Eulerian and Lagrangian models are developed.
In 1D models the importance of correct reference scales is demonstrated. A 1D universal expression for the thermophoretic deposition efficiency in a long tube is derived that is valid for laminar
and turbulent flows and that gives excellent agreement with previous empirical correlations and theoretical results. Two-dimensional
models incorporating radial-profile effects are developed to assess
the effectiveness of the 1D approach. The 2D modelling is based
on a nonstochastic Lagrangian methodology that allows the calculation of thermophoretic deposition with computationally inexpensive means. The developed models are extensively validated by
comparing their predictions to experimental results, previous numerical calculations, and theoretical results in laminar and turbulent flows. The models are also used to calculate thermophoretic
deposition in large-scale experiments simulating fission-product
behavior during a postulated severe accident at a nuclear power
plant. It is found that in laminar flow a properly constructed 1D
description provides accurate predictions. In turbulent flow 1D and
2D predictions provide the same degree of accuracy, unless large
bulk gas-to-wall temperature differences prevail where the more
detailed 2D approach offers significant improvement.
INTRODUCTION
Thermophoresis, the motion of suspended particles in a fluid
induced by a temperature gradient, is of practical importance
in a variety of industrial and engineering applications including
aerosol collection (thermal precipitators), nuclear reactor safety,
gas cleaning, corrosion of heat exchangers, and microcontamination control. Herein, thermophoretic deposition in circularpipe flow for both laminar and turbulent flows is investigated.
The deposition of aerosol particles in a general flow field
under the influence of external forces and Brownian diffusion
may be calculated either via a Eulerian or a Lagrangian description. In the Eulerian approach the particulate mass conservation
Received 16 September 2004; accepted 2 February 2005.
This work was partially supported by the European Commission
under grant FIKS-CT-1999-00009, project PHEBEN2.
Address correspondence to C. Housiadas, “Demokritos” National
Centre for Scientific Research, 15310 Agia Paraskevi, Athens, Greece.
E-mail: [email protected]
304
equation is solved with the particle velocity field expressed in
terms of the carrier-flow velocity and the drift velocities arising
from the external forces and diffusion. In a Lagrangian approach
the particle equations of motion are solved with the addition of
all forces acting on the particle. Both approaches require the
carrier-flow velocity and temperature fields that are determined
either from analytical expressions (laminar flow in simple geometries) or from computational fluid dynamics (CFD) codes
(turbulent flow, complex laminar flow geometries). The Eulerian
approach may be simplified further by rendering the calculation
one dimensional (1D); one-dimensional cross-section averaged
quantities (temperature, carrier-gas velocity, particle concentration) are used in the mass and heat balance equations. Since
these averaged quantities are mostly required in practice, the 1D
approximation is frequently made in practical applications, especially in large engineering codes like the primary-circuit codes
used to calculate particulate deposition in nuclear-power-plant
structures under simulated severe-accident conditions. Numerous 1D expressions for thermophoretic deposition in laminar or
turbulent pipe flows have been proposed in the literature (see,
for example, Romay et al. 1998, or Table 1 in the present article).
However, these expressions were derived under assumptions applicable to specific conditions, rendering difficult the choice of a
general expression to be included in a generic engineering code.
One-dimensional primary-circuit codes have been used to
simulate thermophoretic particulate deposition in experiments
from the Phebus-FP international programme, experiments aiming to investigate the in-pile transport, retention, and chemistry of fission products under severe-accident conditions in a
light water reactor (Clément et al. 2003). Even though the thermophoretic models implemented in these codes had been validated against separate-effects tests (of, for example, the TUBA
programme) considerable disagreement between code predictions and experimental results was noted. In particular, measured
retention of elements present in aerosol form at the entrance of
the model stream-generator tube was about 15% of the inlet particulate mass, whereas primary circuit codes predicted retention
of more than 30%. Thus, calculations overestimated deposition
by a factor of two, and they correspondingly underpredicted
fission-product release in the containment.
The aim of the current study is, thus, twofold: first, to derive
a 1D universal expression (i.e., independent of the nature of the
305
THERMOPHORESIS IN TUBE FLOW
TABLE 1
Total deposition efficiency formulae for convective thermophoresis in a long tube
Reference
Nishio et al. (1974)
Walker et al. (1979)
Total deposition efficiency
1
E th∞ = 1 − exp(Pr K 0.5+θ
∗)
Turbulent flow
E th∞ = Pr θK∗φ0 with
∗
θ /(1 + θ ∗ ) if Pr K = 1
φ0 =
from tabulated values if Pr K = 1
Batchelor and Shen (1985)
1
E th∞ = Pr K 1+θ
∗ (1 +
Stratmann et al. (1994)
+0.025 0.932
E th∞ = 1 − exp[−0.845( PrθK∗ +0.28
)
]
Lin and Tsai (2003)
E th∞ = 0.783( Prθ ∗K )0.94
Present work
θ
Pr K
E th∞ = 1 − ( 1+θ
= 1 − ( TTw0 )PrK
∗)
∗
flow and dependent only on dimensionless variables) for the
thermophoretic deposition efficiency in a long tube; second, to
assess the validity of the 1D approach by comparing it with
more detailed two-dimensional (2D) models, and to elucidate
the failure of 1D codes to reproduce Phebus-FP results.
Initial investigations of thermophoretic deposition in the
Phebus-FP steam-generator tube (Housiadas et al. 2001) suggested that the approximate boundary-layer treatment inherent
in 1D models becomes questionable in cases of turbulent flow
and large gas-to-wall temperature differences, where radial temperature profile effects dominate. Herein, we extend these investigations by developing and implementing Eulerian (1D and 2D)
and Lagrangian (2D) models to calculate thermophoretic deposition in pipe flows.
The 1D model was developed to simulate the results of the
primary circuit codes and to derive simple expressions for thermophoretic deposition. We found that an essential ingredient
of a correct calculation is the judicious choice of reference
scales (temperature, boundary-layer thickness). The 2D models
describe in detail (radial) temperature-profile effects on thermophoretic deposition. They were developed to assess the validity of the simple 1D calculations. A 2D Lagrangian (particletracking) model was developed for both laminar and turbulent
flow conditions. Diffusion (Brownian or turbulent) is neglected
as being very small compared to thermophoresis. In turbulent
flow, fluctuations of the thermophoretic force are also neglected.
Moreover, the effect of fluctuations of the drag force (arising
from turbulent fluid velocity fluctuations) is modeled by the tur-
1−Pr K
1+θ ∗
Comments
)
Laminar flow
Empirical fit
Laminar flow
Exact if Pr K = 1
Approximate if Pr K = 1
Laminar flow
Empirical fit
Laminar flow
Empirical fit, valid for
0.007 < Pr K /θ ∗ < 0.19
Laminar and turbulent flow
Derived from 1D heat and
mass transfer analysis
Valid for any Pr K
bophoretic force. These approximations allow us to treat both
laminar and turbulent flow conditions with the same nonstochastic formalism, rendering the 2D calculation computationally inexpensive. Also, for the laminar flow cases, 2D CFD-based Eulerian calculations including Brownian diffusion are performed
to check the 2D Lagrangian algorithm and model.
The developed models are extensively validated against experimental, numerical, and analytical laminar-flow results
(Montassier et al. 1991, 1990; Stratmann et al. 1994; Walker
et al. 1979). In the 2D Lagrangian calculations, the required
temperature and velocity fields were obtained with a CFD code.
Particle-tracking calculations were subsequently performed to
simulate experimental results under a variety of conditions (classic isothermal deposition experiments, e.g., Liu and Agarwal
1974; Phebus-FP experiment FPT1; test TT28 of the TUBA programme; and smaller laboratory-scale experiments, e.g., Romay
et al. 1998). Our approach relies on a CFD code only in the
determination of fluid properties: it provides a methodology
that is simple to implement, thereby avoiding the use of timeconsuming calculations to estimate particulate deposition in
cases of practical interest.
ONE-DIMENSIONAL DESCRIPTION
Model Development
The 1D equations for the axial dependence of the mean particle concentration and temperature, determined from a mass and
306
C. HOUSIADAS AND Y. DROSSINOS
heat balance over a tube section, are
where
dCm
2Cm
=−
(Vt + Vd + Vth ),
dx
Um R
[1]
dTm
2Nu
=
(Tw − Tm ).
dx
RRe Pr
[2]
In Equation (1) deposition velocities due to thermophoresis
(Vth ), Brownian or eddy diffusion (Vd ), and turbulent impaction
(Vt ) are considered. The total deposition velocity is expressed
as the linear sum of deposition velocities arising from individual deposition mechanisms, a frequently made approximation
(for example, in filtration theory) of weakly interacting deposition mechanisms. Whereas the emphasis of this work is on
thermophoresis, deposition mechanisms that act in unison with
thermophoresis are also included. In particular, in turbulent flow
the combined action of particle inertia and thermophoresis has to
be considered (Konstandopoulos and Rosner 1995; Healy 2003).
The deposition velocity due to diffusional transport is generally
very small in comparison to Vt or Vth . However, its inclusion in
an Eulerian model is straightforward, and as such it has been
included for completeness.
The thermophoretic velocity Vth depends on the local temperature and its radial gradient, as
Vth = −K
νg ∂ T
,
T ∂r
[3]
where K is the thermophoretic coefficient. This coefficient will
be determined from the Brock-Talbot expression (Talbot et al.
1980), an expression that provides an adequate approximation
over the whole range of Knudsen numbers. In a 1D formalism,
Equation (3) becomes Vth = (−K νg /T ∗ ) T ∗ /δ ∗ , where T ∗ ,
T ∗ , and δ ∗ are appropriate scales for temperature, temperature difference, and boundary-layer thickness, respectively. We
choose T ∗ = Tm , T ∗ = Tw − Tm , and δ ∗ = 2R/Nu to obtain
the 1D approximation of Equation (3):
Vth = −K
νg Tw − Tm
.
Tm 2R/Nu
[4]
Different scales for T ∗ , T ∗ , and δ ∗ can be chosen to recast
Equation (3) in 1D form. For example, T ∗ = Tw is an alternative
choice (Schmidt and Sager 2000), as is δ ∗ = R/Nu (Byers and
Calvert 1969). We shall show that the scales proposed here are
the most appropriate and consistent choices because they lead
to the correct limiting behavior at long axial distances.
The system of Equations (1) and (2) can be solved analytically
for the mean temperature and penetration to give
+
x
Tm − Tw
+
+
θm =
= exp −4
Nu(ξ )dξ ,
T0 − Tw
0
f =
Cm
= f t f d f th ,
C0
[5]
[6]
2Vt (ξ )
dξ ,
Um R
0
x
2Vd (ξ )
f d = exp −
dξ
Um R
0
0
x
0
0
= exp −4
Sh(ξ )dξ ,
f t = exp −
x
0
x+
f th = exp −4 Pr K
0
θm (ξ + )Nu(ξ + ) +
dξ .
θm (ξ + ) + θ ∗
[7]
[8]
[9]
The overall penetration f = Cm /C0 is found to be the product of
individual penetrations due to turbulent impaction, f t , diffusion,
f d , and thermophoresis, f th . This factorization is a consequence
of the decomposition of the total deposition velocity into the
sum of deposition velocities arising from individual transport
mechanisms. All deposition mechanisms predict an exponential
dependence of deposition on tube length, an exponential decay
that was measured in Phebus-FP experiments, as discussed below in the section “Nonisothermal Flow: Phebus FPT1.”
The whole set of formulae necessary to perform a complete
1D calculation under laminar and turbulent flow conditions is
given in Table 2. The suggested correlations were carefully
selected, as summarized in the following subsections.
Laminar Flow
In laminar flow deposition due to turbulent impaction is zero
and the corresponding penetration f t = 1. The deposition velocity due to Brownian diffusion may be expressed in terms of
the concentration boundary-layer thickness, an approximation
consistent with the expression of the temperature gradient in
terms of the thermal boundary-layer thickness. According to the
heat-mass transfer analogy the Nusselt number is replaced by
the Sherwood number to give Vd = ShD p /2R (Hontañón et al.
1996). With this approximation the penetration, f d , becomes
(as can be easily verified), formally identical to θm , the mean
temperature decay (along the axial direction) in convective heat
transfer (see Equations (5) and (8)). Therefore, either f d or θm
may be calculated from the Graetz infinite series solution of
convective heat or mass transfer in laminar tube flow. Care is
only required in choosing the appropriate dimensionless length
scales, namely x + = x/(2RRe Pr) for the temperature problem and x 0 = x/(2R Re Sc p ) for the particle diffusion problem.
The accurate numerical evaluation of the Graetz series requires
considerable effort (Housiadas et al. 1999). For simplicity, we
will use the approximate solution of Ingham (1975), cf. Table 2,
which reproduces the exact solution with an accuracy better than
0.5%.
The Nusselt number required in Equation (9) is determined
from the theory of convective heat transfer. Since in laminar flow
the thermal entrance region may have an appreciable length, the
307
THERMOPHORESIS IN TUBE FLOW
TABLE 2
Formulae required to perform a 1D deposition calculation (this work)
Laminar
θm =
ft
fd
f th
f th∞
Tm −Tw
T0 −Tw
+
Turbulent
+
0.819e−14.63x + 0.0976e−89.2x
+
+ 2/3
+ 0.0325e−228x + 0.0509e−125.9(x )
1
0
0
0.819e−14.63x + 0.0976e−89.2x
0
0 2/3
+ 0.0325e−228x + 0.0509e−125.9(x )
x+
+
)Nu(ξ + )
+
exp(−4 Pr K 0 θmθ(ξ
+
∗ dξ )
m (ξ )+θ
(numerical integration required)
( TTw0 )Pr K
2Nux
exp(− RRe
)
Pr
exp(− U2Vmt Rx )
dx
exp(− 2V
)
Um R
[θ
∗
+exp(−2Nux/RRe Pr) PrK
]
1+θ ∗
Total penetration is f = f t f d f th .
The Nusselt number is evaluated either from Equation (10) (laminar flow) or Equation (13) (turbulent flow).
Nusselt number depends on position; it decreases monotonically
from its high value at the inlet to reach asymptotically Nu∞ =
3.657 in the fully developed region (x + > 0.1). The function
Nu(x + ) is given by an infinite series, analogous to the Graetz
series. Again, for simplicity it will be determined from empirical
fits, as provided by Shah and London (1978):

+ −1/3
for x + ≤ 0.01,
1.077(x )
+
3
+
Nu(x ) = 3.657 + 6.874(10 x )−0.488 exp(−57.2x + )

for x + > 0.01.
[10]
The predictions of the above correlation agree within ±3% with
more accurate results. The penetration, f th , due to convective
thermophoresis is calculated by inserting θm (see Table 2) and Nu
(Equation (10)) into Equation (9) and by evaluating the resulting
integral numerically.
Turbulent Flow
The deposition velocity due to turbulent impaction can be
inferred from the “free-flight” theory of Friedlander and Johnstone (1957). Alternatively, an empirical fit to deposition measurements expresses it in terms of the dimensionless particle
relaxation time (Drossinos and Housiadas 2005)
Vt /u ∗ = min 6 × 10−4 τ p2+ , 0.1 .
[11]
Deposition due to diffusion arises primarily from eddy diffusion, molecular diffusion being negligible in comparison. It will
be approximated by the correlation of Wells and Chamberlain
(1967), as recommended in a recent experimental study (Malet
et al. 2000):
−1/8
Vd /u ∗ = 0.2Sc2/3
.
p Re
[12]
The thermal entrance region is short; thus, the fully developed
value of the Nusselt number can be used. This value is calculated
with the Gnielinski expression
Nu =
( f /2)(Re-1000) Pr
,
1 + 12.7( f /2)1/2 (Pr2/3 − 1)
[13]
where the friction factor, f , is calculated with Churchill’s
expression,
10 1/5
2
Re
1
+ 2.21 ln
.
=
10
20
1/2
f
[(8/Re) + (Re/36500) ]
7
[14]
The suggested correlations are in overall best agreement with experimental data over the entire range from low to high Reynolds
numbers (Kakac et al. 1987).
For constant (position-independent) Nu, Vt , and Vd , Equations (5), (7), and (8) simplify considerably to the formulae
presented in Table 2. The thermophoretic penetration fraction
is obtained by substituting θm into Equation (9). Tedious but
straightforward algebraic manipulations yield
∗
θ + exp(−2Nux/RRe Pr) Pr K
f th =
,
[15]
1 + θ∗
where θ ∗ is the dimensionless temperature θ ∗ = Tw /(T0 − Tw ).
Equation (15) is the same, although expressed in different form,
to the result obtained by Romay et al. (1998).
Thermophoretic Deposition Efficiency:
Model Comparisons
In this subsection we analyze the case of thermophoretic deposition acting by itself; diffusional deposition in transitional
flow is analyzed theoretically and experimentally in Shimada
et al. (1993). The quantity most frequently required is the total
deposition in a long tube. Since sufficiently far from the tube inlet the gas and wall temperatures equilibrate and thermophoresis
stops, the total thermophoretic deposition efficiency E th ∞ is the
limit of 1 − f th as x → ∞. For turbulent flow, the long-distance
308
C. HOUSIADAS AND Y. DROSSINOS
limit of Equation (15) evaluates to
E th∞ = 1 −
θ∗
1 + θ∗
Pr K
=1−
Tw
T0
Pr K
.
[16]
Interestingly, the same expression is also obtained for the case of
laminar flow: for x + → ∞, Nu ∼ Nu∞∗, θm ∼ exp(−4N∞ x + ),
∞ x/RRe Pr) Pr K
] .
and Equation (9) behaves as f th ∼ [ θ +exp(−2Nu
1+θ ∗
Hence, the long-distance limit of f th in laminar flow is identical
to the turbulent-flow limit.
Equation (16) is one of the main results of this work. It
shows that within the 1D approximation total thermophoretic
deposition, in either laminar or turbulent flow, depends on two
dimensionless parameters: the dimensionless parameter, θ ∗ , or
equivalently the dimensionless temperature ratio, Tw /T0 , and
the dimensionless product, Pr K . Previous analyses of convective thermophoresis (mainly in laminar flow) also identified Pr K
and θ ∗ as the parameters controlling deposition, and they suggested correlations for the total deposition efficiency as a function of these two parameters (see, for example, Lin and Tsai
2003; Romay et al. 1998; Batchelor and Shen 1985; Stratmann
et al. 1994; Walker et al. 1979; Nishio et al. 1974). However,
none of the previous expressions has the range of validity and
the formal rigor of the expression derived here. Table 1 summarizes these expressions. Figure 1 compares their predictions
with those obtained using Equation (16). The figure shows that
all expressions agree for large values of θ ∗ . For small values
there are some (minor) differences among them, with the exception of the expression of Lin and Tsai (2003), which is valid over
a narrow range of values of Pr K and θ ∗ (see Table 1). The excellent agreement obtained with Equation (16) justifies a posteriori
the choices made for the scales T ∗ , T ∗ , and δ ∗ . For instance,
if T ∗ = Tw was chosen instead of Tm and the analysis repeated,
the long-distance thermophoretic deposition efficiency would
be
Pr K
E th ∞ = 1 − exp − ∗ .
[17]
θ
The predictions of Equation (17) are also plotted in Figure 1;
it can be seen that they deviate markedly from the correct behavior.
The results in Figure 1 and the data in Table 1 demonstrate the
potential of Equation (16) and its clear advantage over previous
expressions. The suggested expression reproduces practically
the same results as previous ones but it is simpler, it applies to
both laminar and turbulent flows, it was derived from a formal
theoretical analysis rather than an empirical fitting procedure,
and it is valid for any value of parameters θ ∗ and Pr K . Of course,
Equation (16) is accurate inasmuch as the 1D approximation is
valid, i.e., as long as Equation (4) adequately describes the thermophoretic deposition velocity over a cross section. As shown
in the remainder of the work, this is precisely the case in any
laminar flow and in turbulent flow under moderate gas-to-wall
temperature difference (T0 − Tw ). Instead, for turbulent flow and
FIG. 1. Total deposition efficiency for convective thermophoresis in a long tube (E th∞ ) as function of parameters θ ∗ = Tw /(T0 − Tw ) and Pr K . Theoretical
predictions with Equation (16) are compared to predictions of previously suggested formulae (cf. Table 1).
309
THERMOPHORESIS IN TUBE FLOW
large T0 − Tw , a 2D description is required; Equation (16) provides less accurate results.
become
d Vx
= −Vx + Ux ,
dt
d Vr
τ p Cn
= −Vr + Vth + Vturbo .
dt
τ p Cn
TWO-DIMENSIONAL DESCRIPTION
Lagrangian Modeling
In a Lagrangian description particles are tracked during their
motion in the fluid until they either leave the tube or deposit on
the tube walls. The particle equations of motion are considered
under a number of assumptions. Diffusion (Brownian and/or turbulent) is ignored. This is a legitimate approximation because
diffusive deposition is largely overwhelmed by thermophoretic
deposition, as can be attested by comparing deposition velocities
due to diffusion and thermophoresis (Thakurta et al. 1998; Healy
2003). Turbulence effects are incorporated in the calculation of
the mean flow velocity and mean temperature. Fluctuations of
the thermophoretic force are ignored, an assumption shown to be
valid for large bulk gas-to-wall temperature gradients (Kröger
and Drossinos 2000; He and Ahmadi 1998). Inertial effects due
to turbulent eddy impaction, and consequently the effect of fluctuations of the Stokes drag on particle motion, are modeled by
adding the turbophoretic force (Reeks 1983) in the equations of
motion. Hence inertial particle motion due to fluctuating fluid
velocities is seen as an additional phoretic mechanism characterized by an appropriate macroscopic drift velocity, Vturbo , the
turbophoretic velocity.
These assumptions greatly simplify the numerical solution of
the equations of motion because they render them nonstochastic, deterministic. Under the “zero-fluctuations” boundary layer
assumption, the equations of particle motion in laminar and turbulent flow become the same, thereby allowing both cases to be
treated with the same set of equations (apart, of course, the turbophoretic velocity).1 Moreover, the absence of random walks
and thus of crossing trajectories allows the calculation of deposition in either laminar or turbulent conditions with a simple
critical trajectory method (e.g., Lin and Tsai 2003). Accordingly,
there is a critical radial position, r ∗ , such that all particles originating at r > r ∗ deposit, whereas particles originating at r < r ∗
do not reach the wall and they are transported to the tube outlet.
The critical radius defines the critical trajectory. Tube deposiR
R
tion is determined from the ratio r ∗ U0 C0 rdr/ 0 U0 C0 rdr. In
the present calculations C0 is taken constant (uniform concentration at the inlet) and the inlet axial fluid, U0 , is either parabolic
(laminar flow) or uniform (turbulent flow).
For axisymmetric flow inside a circular tube, with a zero
radial fluid velocity component, the particle equations of motion
1
To simplify notation we use the same symbols for the local temperature and local fluid velocity for both laminar and turbulent cases, but
these variables should be interpreted differently because they refer to
local values in laminar flow and local, but Reynolds-averaged, values
in turbulent flow.
[18]
[19]
The thermophoretic velocity is obtained from Equation (3) using
the local, radially dependent fluid temperature, T , and temperature gradient, ∂ T /∂r . The required velocity and temperature
fields are calculated with the CFD code ANSWER (ACRi 2001).
The turbophoretic velocity is determined from the gradient of the
turbulence intensity involving the fluid Reynolds stress (Reeks
1983; Young and Leeming 1997; Guha 1997),
Vturbo = −τ p Cn
∂
(Ur Ur ),
∂r
[20]
where = Vr Vr /Ur Ur depends on particle inertia and particle–
turbulence interactions. It may be approximated (Reeks 1977)
by = τ L /(τ p Cn + τ L ). Note that has a significant effect
only close to the wall; far from the wall (and for not very large
particles) ≈ 1. The gradient ∂Ur Ur /∂r is obtained from the
standard gradient diffusion hypothesis,
Ur Ur =
∂Ur
2 k
− Vt
.
3 ρg
∂r
[21]
Thus, Ur Ur may be calculated from variables routinely available
in a k-ε–based CFD solution, i.e., the turbulent kinetic energy,
k, the turbulent viscosity, Vt , and the mean radial flow velocity,
Ur . Because in the present case Ur = 0, Equation (21) simplifies
to Ur Ur = 2k/(3ρg ).
Turbulent Pipe Flow: Wall Region Modeling
In laminar flow the implementation of the above Lagrangian
method is straightforward. Instead, in turbulent flow, subtleties
associated with CFD modelling of the wall region have to be
dealt with. The usual k-ε turbulence model does not resolve the
viscous sublayer and buffer layer. This is a serious limitation
because precisely within these layers thermophoresis becomes
significant (in the turbulent core temperature profiles are mostly
uniform due to turbulent mixing). Of course, one can always resort to sophisticated but routinely unavailable turbulence modelling to resolve explicitly the velocity and temperature fields
in the wall region. In the present work we follow an alternative
approach (Housiadas and Drossinos 2003): we determine the
fields in the bulk of the flow with a standard CFD calculation,
and close to the wall we use laws-of-the-wall functions to match
the CFD solution obtained at the first grid level (always positioned at y + ≥ 30). The velocities and temperatures in the wall
310
C. HOUSIADAS AND Y. DROSSINOS
with the coefficients c0 , c1 , c2 , and c3 given by
c0 = −0.1 − 2.2 Pr + 0.68 ln(1 + 5 Pr),
c1 = 0.043 + 1.920 Pr −0.288 ln(1 + 5 Pr),
c2 = −5.33 × 10−3 − 0.104 Pr +3.36
× 10−2 ln(1 + 5 Pr),
−4
FIG. 2.
c3 = 1.33 × 10 + 1.6 × 10
× 10−4 ln(1 + 5 Pr).
Schematic representation of the wall region in turbulent flow.
region are determined as follows (see Figure 2):
Ux+ (y + )
,
Ux+ (y +
p)
T + (y + )
T (y + ) − Tw = (T p − Tw ) + + ,
T (y p )
Ux (y + ) = Ux, p
[22]
[23]
+
with y + = yy +
p /y p , where, y p , U x, p , and T p are obtained from
the CFD solution of the outer region.
The functions Ux+ (y + ) and T + (y + ) are approximated by standard laws-of-the-wall for fully developed turbulence. We used
Reichard’s expression for the dimensionless velocity (Kakac
et al. 1987),
−3
[28]
Pr −6.4
Besides the temperature and velocity fields, statistical information on the fluctuating fields in the wall region is required
to calculate the turbophoretic velocity. We used the following
correlations (Kallio and Reeks 1989) to calculate the particle
Reynolds stress, Ur Ur :

10

for y + < 5,

 10 + Cnτ +
p
(y + ) =
[29]

b0 + b1 y + + b2 (y + )2

+

for
y
≥
5,
Cnτ p+ + b0 + b1 y + + b2 (y + )2
2
A(y + )2
Ur Ur =
,
[30]
1 + C(y + )n
where b0 = 7.122, b1 = 0.5731, b2 = −0.001290, A = 0.005,
C
= 0.002923, and n = 2.128.
y −0.33y +
+
In summary, the Lagrangian calculation of deposition in turUx+ (y + ) = 2.5 ln(1 + 0.4y + ) + 7.8 1 − e−y /11 −
.
e
11
bulent flow consists of three parts: (1) the temperature and veloc[24] ity fields, as well as the Reynolds stresses, are calculated with the
CFD code ANSWER; (2) in the wall layer these fields are calcuThis expression, valid for all y + , continuously describes the ve- lated by extrapolating the values at the closest-to-the wall node
locity across the different zones of the boundary layer. For the using laws-of-the wall correlations as previously described; (3)
dimensionless temperature we used Kay’s expression (Kakac particles released at the inlet are tracked to determine the deposiet al. 1987), which refers to a fully developed temperature dis- tion. Obviously, in laminar flow only steps 1 and 3 are required.
tribution in a flow with turbulent Prandlt number of unity and
for fluids with Pr > 0.03, being thus appropriate for the aerosol Eulerian Modeling (Laminar Flow)
The 2D Eulerian approach is based on the solution of the mass
flows considered here. For the viscous sublayer Kay’s expresbalance
equation for the particle concentration equation with
sion is
the addition of the thermophoretic flux. As in the previously
T + = Pr y + for 0 < y + < 5,
[25] described 1D Eulerian model, particle diffusion can be easily
included. Eulerian 2D calculations were performed in laminar
flow conditions to check the Lagrangian algorithm and model, in
whereas in the turbulent core it is
particular the assumption of neglecting the diffusion of particles.
+
The mass conservation equation becomes (in cylindrical
y
T + = 2.5 ln
+ 5 Pr +5 ln(1 + 5 Pr) for y + ≥ 30. [26] coordinates)
30
∂C
1 ∂
∂C
1 ∂
+ +
+
+
U
=
D
r
−
(r Vth C).
[31]
The required wall function T (y ) and its derivative ∂ T /∂ y
x
p
∂
x
r
∂r
∂r
V
∂r
+
have to be continuous for all y to avoid unphysical discontinuities in the evaluation of the thermophoretic velocity. For As before, the thermophoretic velocity is obtained from Equathis reason Equations (25) and (26) were interpolated over the tion (3) using local values. Equation (31) may be viewed as a
interval 5 ≤ y + < 30 (buffer layer) by a polynomial func- convective diffusion equation with the extra source term S =
tion, requiring continuity of values and derivatives at the points V1 ∂r∂ (r Vth C). Commercial CFD codes allow the introduction of
y + = 5 and y + = 30. The following expression was obtained:
application-specific source terms if expressed as a function of
the dependent variables (S = S(T, C) in the present case). The
T + = c0 + c1 y + + c2 (y + )2 + c3 (y + )3 for 5 ≤ y + < 30, [27] CFD code ANSWER was used to calculate the velocity and
+
THERMOPHORESIS IN TUBE FLOW
temperature fields in the gas and to solve numerically Equation
(31) by appropriately specifying the source term in the mass
conservation equation.
LAMINAR FLOW
For model validation we chose the thermophoretic-deposition
experiments of Montassier et al. (1991). In these experiments
311
deposition measurements were performed using accurately controlled inlet conditions (uniform particle concentration and gas
temperature), accurately controlled particle sizes (monodisperse
aerosol), as well as accurately controlled flow conditions (laminar flow in a tube). The comparison is shown in Figure 3. Note
that the Eulerian model, which includes Brownian diffusion (see
Equation (31)), gives practically identical predictions with the
FIG. 3. Thermophoretic deposition in laminar tube flow: comparison of 1D and 2D model predictions with experimental data and simulation results from
Montassier et al. (1990, 1991). ṁ = 0.2 g/s, Tw = 293 K, T0 = 373 K, R = 1 cm.
312
C. HOUSIADAS AND Y. DROSSINOS
FIG. 4.
Thermophoretic deposition in Poiseuille flow: comparison of 1D and 2D calculations with available analytical and numerical predictions.
Lagrangian model. This justifies our assumption to neglect diffusion in the Lagrangian model (see Equation (19)). Very good
quantitative agreement is observed between experimental and
model-predicted values for the larger diameter particles, d p =
0.38 µm (Figure 3a). For the smaller diameter particles, d p =
0.10 µm, both numerical calculations tend to underestimate deposition (Figure 3b). However, the discrepancy can be considered as tolerable given the experimental uncertainties. Note also
that in all cases model calculations are in very close agreement
with the numerical results of Montassier et al. (1990), who used
a numerical model based on the finite-volume approach and
Patankar’s SIMPLER algorithm. The good agreement provides
the required evidence and support for the proper operation of the
developed models and justifies the neglect of Brownian diffusion
in the Lagrangian approach. Superimposed on the same figure,
the predictions of the 1D solution are also shown, obtained as
described in Table 2. There is a clear trend to underpredict experimental values. Still, as can be observed, the 1D predictions
are very close to the 2D results. Therefore, from a practical point
of view the improvement obtained with a 2D calculation does
not justify the significant additional computational effort.
Comparisons of 1D and 2D predictions were also made with
the analytical results of Walker et al. (1979) and the numerical
calculations of Stratmann et al. (1994). These authors calculated thermophoretic deposition in a laminarly flowing gas inside a tube (Poiseuille flow) whose walls were abruptly cooled.
Their analysis was 2D and Brownian diffusion was neglected.
The comparisons between their results and the predictions of the
current models are shown in Figure 4. The 2D models (both Eulerian and Lagrangian) give predictions in excellent agreement
with both analytical and numerical results. The predictions of
the 1D solution are also shown: we present results obtained as
described in Table 2, as well as those obtained under the simplifying assumption of constant Nusselt number Nu∞ (i.e., on the
basis of Equation (15); see discussion after Equation (16)). The
full 1D solution (developing Nusselt number) gives results in excellent agreement with the 2D solutions at all axial distances. As
expected, the predictions of the simplified 1D (constant Nusselt
number) solution deviate markedly from the correct answer (by
a factor of two or three) at short distances. However, at long distances from the tube inlet the simplified 1D solution predicts the
deposited fraction with very good accuracy. The excellent limiting behavior of the 1D solution demonstrates that Equation (16)
correctly predicts total deposition in long tubes. The discrepancy
between the two 1D solutions at shorter distances is due to the
developing radial temperature profile over the thermal entrance
region. Sufficiently far downstream, the temperature profile becomes fully developed and the 1D simplified solution provides
correct predictions, whereas the 1D solution with a developing
Nusselt number accommodates the development of the thermal
boundary layer and thus provides accurate predictions all along
the tube length. This result is in agreement with the conclusions
of a recent study on thermophoretic deposition in laminar flow
(Lin and Tsai 2003).
THERMOPHORESIS IN TUBE FLOW
The above comparisons show that in laminar flow a 1D description is adequate, provided correct reference scales and correct Nusselt numbers are used, i.e., an accurate boundary-layer
thickness is used. A direct consequence of this conclusion is that
Equation (16) gives accurate results in all cases in laminar flow.
TURBULENT FLOW
The Lagrangian model is used to reproduce experimental
measurements for deposition profiles and efficiency using the
wall region modelling previously described. Four cases were
chosen from independent experimental works that cover a broad
range of experimental conditions: deposition in isothermal turbulent flow (Liu and Agarwal 1974), deposition profiles in two
experiments that simulated conditions expected in nuclearreactor steam-generator tubes in case of a severe accident
(Phebus-FP experiment FPT1 and TUBA experiment TT28),
and total deposition in nonisothermal pipe flow (Romay et al.
1998).
Isothermal Flow
Liu and Agarwal (1974) presented dimensionless deposition
velocities as a function of dimensionless particle relaxation time.
These carefully conducted and well-documented experiments
are the most frequently quoted experimental results for particulate deposition in isothermal turbulent flow.
Model-calculated values and experimental results are compared in Figure 5. The calculated values were obtained from the
313
numerical solution of Equations (18) and (19) without the thermophoretic term. The dimensionless mean axial fluid velocity
was determined from Equation (24), and the turbophoretic velocity from Equation (20) with the correlations summarized in
Equations (29) and (30). Total penetration, which was calculated
in terms of the critical radius, was converted into a deposition
velocity by inverting Equation (7). Liu and Agarwal (1974) used
the same approach to determine the deposition velocity from the
experimental penetration.
The comparison shows reasonable agreement of calculated
values with experimental measurements. This is a noteworthy
success given the relative simplicity of the model. The increase
of deposition from the diffusional-deposition regime to the inertiamoderated regime (Young and Leeming 1997) is well reproduced, albeit more abruptly than the measurements suggest.
The predicted steeper rise, as well as the relative insensitivity of the calculated deposition velocity on particle relaxation
time for small τ p+ , can be attributed to the neglect of diffusional
deposition. This approximation is not expected to alter deposition results in the inertia-moderated regime since the diffusional
dimensionless deposition velocity, estimated, for example, via
Equation (12), is orders of magnitude smaller than that due to
turbulent impaction. It should be noted, however, that since diffusion was neglected the numerical results for small τ p+ depend
strongly on the choice of the boundary conditions, or equivalently on the interception distance that determines whether a
particle deposits. A similar effect was noted in the simulation
FIG. 5. Validation of nonstochastic Lagrangian method in turbulent flow; comparison with the experimental data of Liu and Agarwal (1974) on deposition in
turbulent pipe flow (Re = 9894).
314
C. HOUSIADAS AND Y. DROSSINOS
results of Pyykönen and Jokiniemi (2001), who used surface
roughness as fitting parameter. Brownian and turbulent deposition of submicron particles is analyzed in Shimada et al. (1993),
who investigated the dependence of the total deposition velocity
on the Brownian diffusion coefficient.
The aim of the comparison presented in this subsection is
not to propose an alternative method to random-walk simulations of particle deposition in isothermal turbulent flows. It
aims to determine whether the use of the turbophoretic force in
a nonstochastic Lagrangian calculation reproduces the significant increase of deposition with increasing particle size. Since
the model will be used to calculate deposition in nonisothermal turbulent flows, small τ p+ deposition will be dominated by
thermophoresis. Hence the comparison justifies the simplified
nonstochastic turbophoresis model in the deposition regime,
where both thermophoresis and turbulent impaction act simultaneously.
Nonisothermal Flow: Phebus FPT1
Particulate deposition in the second experiment of the PhebusFP programme (FPT1) was calculated to elucidate the differences between code-calculated deposition and measured values. For a comprehensive review of the experimental results
from the first experiment (FPT0) see Clément et al. (2003). The
main difference between tests FPT1 and FPT0 was the nature
of the fuel (fresh, in-pile irradiated in FPT0; used, reirradiated
in FPT1), all other conditions, including aerosol behavior, being
similar. The measured and calculated results thus apply equally
well to both experiments. Herein we concentrate on particulate
FIG. 6.
thermophoretic deposition in the steam generator. The experimental conditions were such that a 700◦ C steam–hydrogen mixture containing fission products (and core structural materials)
was conducted into an inverted U-shaped tube, the model steam
generator, whose walls were kept at 150◦ C. Measured deposition, primarily due to thermophoresis, was localized close to the
steam generator inlet. It was approximately 15% of the entering aerosol mass, but most 1D fission–product transport codes
(e.g., SOPHAEROS and VICTORIA; see Clément et al. 2003)
predicted retention of more than 30%.
We calculated deposition in the FPT1 steam generator under
a number of approximations. The steam generator geometry was
modeled as a vertical tube of 2 cm diameter. Steady-state conditions were assumed to prevail since the gas residence time, approximately 1 s, was very short with respect to the time scale that
characterized the fission product release transient. The flow in
the steam generator was in the transitional regime (Re ≈ 4000),
and the experimentally determined size distribution of aerosol
particles had an aerodynamics mass median diameter (AMMD)
between 1.5 and 2.0 µm (AMMD is taken to be 1.75 µm in
the calculations) and a geometric standard deviation of σg = 2.
The input data were as follows: mass flow rate, ṁ = 2.2 g/s;
particle density, 5500 kg/m3 ; and particle thermal conductivity,
100 W/m K.
Figure 6 shows calculated deposited fraction as a function of
distance x from the inlet. The calculations were performed with
the 1D analytical solution as described in Table 2 and with the 2D
Lagrangian method. For comparison the experimental data with
the associated uncertainties are also shown. The experimental
Comparison of model predictions with particle deposition measurements in the steam-generator hot leg in experiment Phebus FPT1.
THERMOPHORESIS IN TUBE FLOW
FIG. 7.
315
Comparison of model predictions with particle deposition measurements in TUBA experiment TT28.
data were deduced from the measurements of deposits reported
in the FPT1 Final Report (Clément et al. 2000). They indicate
that deposition along the steam-generator hot leg (rising part)
expressed as deposited mass per unit length can be described by
E = E ∞ (1 − e−x/H ),
2003). Hence, the significant overestimation of the 1D calculations coupled to the improved agreement of the Lagrangian
2D calculation indicate that a full 2D calculation that calculates
boundary-layer effects is required to reproduce accurately deposition results in turbulent flow under large temperature gradients.
[32]
where the characteristic decay length is H = 1.48 ± 0.1 m, and
E ∞ is the total retention in the steam generator. It was estimated
(Clément et al. 2000) to be E ∞ = 14.3 ± 2%, a value that
corresponds to an average for all the fission products entering
the steam generator in aerosol form, independently of chemical
speciation. The experimental uncertainty bars shown in Figure 6
were determined from Equation (32) by considering variations
in H and E ∞ .
As shown in Figure 6 the 2D-calculated profile follows, in
general, the experimental trend, although the particle-tracking
model consistently overestimates deposition. This discrepancy
can be attributed to uncertainties regarding boundary conditions,2 to aerosol mass distribution and fluid velocity at the
inlet of the tube, or to the neglect of other mechanisms that
eventually remobilize particles such as physical resuspension or
revolatilization (Jokiniemi et al. 2003). Nevertheless, total predicted retention is of the order of 25%, a significant improvement over the 1D result (∼35%). The 1D analytical calculation
accurately reproduces calculated retention by the primary circuit codes VICTORIA 92 and SOPHAEROS 1.3 (Clément et al.
2
The Phebus experiments are large-scale, integral tests performed
on a industrial-scale facility.
Nonisothermal Flow: TUBA TT28
The discrepancy between 1D and 2D results for FPT1 suggested a reconsideration of the turbulent-flow experiment TT28
of the TUBA programme that most closely emulated the experimental conditions at the entrance of the FPT1 steam generator.
Experiment TT28 was a thermophoretic deposition in turbulent
flow experiment (pipe diameter 1.8 cm), where the deposition
profile and total retention efficiency were measured. Results
from TUBA TT28 were used to validate the thermophoretic
models and 1D approximations used in primary-circuit codes
(Dumaz et al. 1993). Nevertheless, the two tests had not been
performed under identical conditions: the gas-to-wall temperature difference at the entrance of the FPT1 steam-generator tube
was more than 500 K, whereas the corresponding temperature
drop in TT28 was about 300 K; the Reynolds number in FPT1
was approximately 4000, whereas in TT28 it was approximately
5000; the carrier gas in FPT1 was primarily steam, whereas it
was air in TT28. The injected aerosol particles were CsI, and the
aerosol was polydisperse (AMMD of 1.19 µm and σg = 1.86).
The used data were as follows: ṁ = 1.95 g/s, T0 = 368◦ C, and
Tw = 39◦ C.
The results of our simulations are reported in Figure 7. Inspection confirms that the 1D model reproduces the deposition
316
C. HOUSIADAS AND Y. DROSSINOS
profile quite accurately, even though it consistently underestimates local deposition. This behavior is not unexpected; 1D
codes were validated with these experimental results. The 2D
particle-tracking simulation also gives results that are in reasonable agreement: however, they consistently overpredict deposition. Hence, either 1D or 2D model may be used to simulate
TT28, implying that for relatively small temperature differences
in transitional flow a 2D model is not required; a 1D approach
is sufficient.
Nonisothermal Flow: Total Deposition
Total deposition in nonisothermal pipe flow for various particle sizes, flow rates, and inlet-to-wall temperature differences
is reported by Romay et al. (1998). These experiments had
been performed in a pipe of 0.5 cm diameter with air as a carrier gas, and with monodisperse NaCl particles of diameters of
0.1, 0.3, and 0.7 µm. The inlet radial temperature gradient was
T ≤ 125 K. Two different flow rates were chosen, Re = 5500
and 9500. The comparison between 1D and 2D calculations
and experimental results are summarized in a compact form in
Figure 8. Results from the Phebus-FP and TUBA experiments
are also included.
Data are plotted as calculated deposition fraction versus measured deposited fraction. Romay et al. (1998) reported only
total deposition data, so each point corresponds to a different test. For Phebus-FP (FPT1) and TUBA (TT28) more than
one value is plotted per experiment, corresponding to deposited
FIG. 8.
fractions along the length of the tube, an indication of the deposition profile. Larger separation of the data points from the
diagonal implies worse agreement. High deposition values correspond to deposited fraction further down from the tube inlet
(increasing deposition with distance from the inlet) or to different experiments.
Numerical simulations of the experiments reported in Romay
et al. (1998) show that the 1D and 2D calculations are equally accurate in reproducing the experimental results. However, some
differences remain: the 1D results consistently underpredict deposition, whereas the 2D results usually overpredict. Hence, as in
the case of TT28, either calculation may be used to simulate the
experimental data. The TUBA TT28 experimental results show
low deposition, since the temperature gradient was not particularly high and the flow not very turbulent. Both 1D and 2D
calculations reproduce rather satisfactorily the measurements,
the 1D slightly underpredicting them while the 2D overpredicts
them. The FPT1 experimental data show higher deposition, possibly because the temperature gradient at the steam generator
inlet was higher. Both calculations overpredict the data, the 1D
calculation becoming worse with axial distance while the 2D
calculation, which significantly overpredicts deposition close
to the entrance, saturates at long distances, the calculated-tomeasured-data ratio remaining constant downstream. Therefore,
for large gas-to-wall temperature differences a proper calculation of boundary-layer effects via a 2D calculation is preferable
to the 1D calculation.
Overall summary and comparison of measured with calculated thermophoretic deposition under turbulent flow conditions.
THERMOPHORESIS IN TUBE FLOW
SUMMARY AND CONCLUSIONS
We analyzed thermophoretic deposition in circular pipes under both laminar and turbulent flows with particular emphasis
on the choice between 1D and 2D descriptions of particulate
deposition. We were primarily concerned with the development
of theoretically justifiable formulae and methods that allow simple, computationally tractable calculations of particle retention
in pipes that give reasonable agreement with experimental measurements. In doing so, we developed Eulerian and Lagrangian
(particle-tracking) models to compare 1D and 2D calculations.
The 2D descriptions, complementary to the Eulerian 1D description, were deemed necessary to investigate the importance of
detailed modeling of boundary-layer effects (specifically, radial
temperature profile effects) on thermophoretic deposition.
In the Eulerian 1D calculations we stressed the requirement
that appropriate reference scales for the temperature and
boundary-layer thickness be chosen. When the reference scales
are chosen as suggested in this work, the long-distance limit of
the total thermophoretic efficiency becomes a universal function (independent of the nature of the flow), dependent on two
dimensionless parameters, Tw /T0 and Pr K . The theoretically
obtained expression (Equation (16)) was shown to be in excellent
agreement with numerical and analytical results, and also with
previously proposed empirical correlations (and thus of limited
range of applicability)—see, for example, Figures 1 and 4 and
Table 1. Therefore, the analytical expression proposed in this
work is preferable to existing expressions for thermophoretic
deposition efficiency in that it is formally justifiable and it applies to both laminar and turbulent flows.
The proposed expression remains valid inasmuch as a 1D
description is accurate. Two-dimensional modelling of thermophoretic deposition in laminar and turbulent flows permitted the
assessment of the validity of the 1D description. It was shown
that for laminar flows existing heat-transfer relationships and
the correct choice of reference scales are sufficient to reproduce
accurately experimental and theoretical data with a 1D calculation: 2D calculations, more complex numerical schemes, and
empirical fits are not necessary. The assessment for the case
of turbulent pipe flow is concisely summarized in Figure 8.
Analyses of these results suggests that in turbulent flow the approximate 1D boundary-layer treatment is generally adequate,
with the exception of cases of large gas-to-wall temperature differences where radial temperature profile effects dominate: for
large temperature differences a 2D description provides significantly improved predictions. Instead, for smaller temperature
differences the usual 1D description, as adopted by engineering
codes (e.g., the primary-circuit codes in nuclear safety analyses),
is sufficient to determine deposition profiles and efficiencies.
A computationally expedient Lagrangian method was developed to determine the thermophoretic deposition in both laminar and turbulent flows with the same nonstochastic formalism.
Turbulent-flow subtleties associated with the determination of
the flow and temperature fields near the wall were addressed by
matching the k-ε–based fields in the outer region with law-ofthe-wall functions in the wall region. The Lagrangian simula-
317
tion included only deposition processes deemed to be important
under the experimental conditions: diffusion was shown to be
negligible in the presence of thermophoresis, whereas turbulence effects were incorporated by adding a turbophoretic force
dependent on Reynolds-averaged quantities only. This approach
greatly simplifies the calculation because it removes the stochastic characteristics of the Lagrangian approach. Still, the proposed
simplified approach was shown to reproduce accurately enough
deposition in the inertia-moderated regime, as determined from
comparisons with the experimental results of Liu and Agarwal
(1974).
NOMENCLATURE
Cn
C
Dp
dp
E
f
K
k
ṁ
Nu
Pr
R
r
r∗
Re
Sc p
Sh
T
U
u∗
V
Vd
Cunningham slip correction factor
particle concentration
particle diffusion coefficient
particle diameter
deposition efficiency
friction factor, penetration fraction (1− E)
thermophoretic coefficient
turbulent kinetic energy
mass flow-rate of carrier gas
Nusselt number
Prandlt number
tube radius
radial coordinate
critical radial position
Reynolds number
particle Schmidt number
Sherwood number
fluid temperature
fluid velocity
friction velocity
particle velocity
deposition velocity due to diffusion (Brownian or eddy)
Vt
deposition velocity due to turbulent impaction
Vth
thermophoretic velocity
Vturbo
turbophoretic velocity
x
axial coordinate
x + = x/(2R RePr) dimensionless axial distance (heat
transfer)
x 0 = x/(2R ReSc p ) dimensionless axial distance (mass
transfer)
y
distance from wall
Greek
ε
−Tw
θm = TTm0 −T
w
w
θ ∗ = T0T−T
w
µg
νg
ρg
ρp
turbulent kinetic energy dissipation rate
mean dimensionless fluid temperature
characteristic dimensionless temperature
fluid (gas) viscosity
kinematic fluid (gas) viscosity
fluid (gas) density
particle bulk density
318
σg
τL
τp =
τ p+ =
C. HOUSIADAS AND Y. DROSSINOS
geometric standard deviation
turbulent eddy time scale
ρ p d 2p Cn
18µg
τp
νg /(u ∗ )2
Subscripts
g
m
p
r
th
x
w
0
Superscripts
+
particle relaxation time
dimensionless particle relaxation time
fluid (gas)
bulk mean (cross-section averaged)
particle
radial component
thermophoretic
axial component
wall
inlet, initial
dimensionless in wall units, heat-transfer
dimensionless variable
fluctuating
REFERENCES
Analytic & Computational Research Inc. (ACRi) (2001). ANSWER User’s Manual. Analytic & Computational Research Inc., Bel Air, CA.
Batchelor, G. K., and Shen, C. (1985). Thermophoretic Deposition of Particles
in Gas Flowing over Cold Surfaces, J. Colloid Interface Sci. 107:21–37.
Byers, R. L., and Calvert, S. (1969). Particle Deposition from Turbulent Streams
by Means of Thermal Force, I EC Fundam. 8:646–655.
Clément, B., editor (2000). Final Report FPT1. CEA, issued by ADB Communication (available in CD-ROM), France.
Clément, B., Hanniet-Girault, N., Repetto, G., Jacquemain, D., Jones, A. V.,
Kissane, M. P., and Von der Hardt, P. (2003). LWR Severe Accident Simulation: Synthesis of the Results and Interpretation of the First Phebus FP
Experiment FPT0, Nucl. Eng. Design 226:5–82.
Drossinos, Y., and Housiadas, C. (2005). Aerosol Flows. In The Multiphase
Flow Handbook, edited by C. Crowe. CRC Press LLC, Boca Raton (in press),
FL., Chap. 6.
Dumaz, P., Drossinos, Y., Areia Capitao, J., and Drosik, I. (1993). Fission Product
Deposition and Revaporization Phenomena in Scenarios of Large Temperature Differences, In ANS Proceedings 1993 National Heat Transfer Conference. American Nuclear Society, Atlanta, GA, pp. 348–358.
Friedlander, S. K., and Johnstone, H. F. (1957). Deposition of Suspended Particles from Turbulent Gas Streams, Ind. Engng. Chem. 49:1151–1156.
Guha, A. (1997). A Unified Eulerian Theory of Turbulent Deposition to Smooth
and Rough Surfaces, J. Aerosol Sci. 8:1517–1537.
He, C., and Ahmadi, G. (1998). Particle Deposition with Thermophoresis in
Laminar and Turbulent Duct Flows, Aerosol Sci. Technol. 29:525–546.
Healy, D. P. (2003). On the Full Lagrangian Approach and Thermophoretic
Deposition in Gas-Particle Flows. PhD dissertation, University of Cambridge.
Hontañón, E., Lazaridis, M., and Drossinos, Y. (1996). The Effect of Chemical
Interactions on the Transport of Caesium in the Presence of Boron, J. Aerosol
Sci. 27:19–39.
Housiadas, C., and Drossinos, Y. (2003). Thermophoretic Deposition of Lowinertia Particles. European Aerosol Conference, Madrid, J. Aerosol Sci. 34
(Suppl. 1):S113–S114.
Housiadas, C., Ezquerra Larrodé, F., and Drossinos, Y. (1999). Numerical Evaluation of the Graetz Series, Int. J. Heat Mass Transfer 42:3013–3017.
Housiadas, C., Mueller, K., Carlsson, J., and Drossinos, Y. (2001). Twodimensional Effects in Thermophoretic Particle Deposition: The Phebus-FP
Steam Generator. European Aerosol Conference, Leipzig, J. Aerosol Sci.
32(Suppl. 1):S1029–S1030.
Ingham, D. B. (1975). Diffusion of Aerosols from a Stream Flowing Through a
Cylindrical Tube, J. Aerosol Sci. 6:125–132.
Jokiniemi, J., Auvinen, P., Bottomley, P. D., and Tuson, A. (2003). Revaporisation Tests on Samples from Phebus Fission Products. In Proceedings of the 5th
Phebus Technical Seminar, 24–26 June 2003. Calliscope, Aix-en-Provence,
France.
Kakac, S., Shah, R. K., and Aung, W. (Eds.) (1987). Handbook of Single-Phase
Convective Heat Transfer. John Wiley & Sons, New York.
Kallio, G. K., and Reeks, M. W. (1989). A Numerical Simulation of Particle
Deposition in Turbulent Boundary Layers, Int. J. Multiphase Flow 15:433–
446.
Konstandopoulos, A. G., and Rosner, D. E. (1995). Inertial Effects on
Thermophoretic Transport of Small Particles to Walls with Streamwise
Curvature—I. Theory, Int. J. Heat Mass Transfer 38:2305–2315.
Kröger, C., and Drossinos, Y. (2000). A Random-walk Simulation of Thermophoretic Particle Deposition in a Turbulent Boundary Layer, Int. J. Multiphase Flow 26:1325–1350.
Lin, J.-S., and Tsai, C.-J. (2003). Thermophoretic Deposition Efficiency in a
Cylindrical Tube Taking into Account Developing Flow at the Entrance Region, J. Aerosol Sci. 34:569–583.
Liu, B. Y. H., and Agarwal, J. K. (1974). Experimental Observation of Aerosol
Deposition in Turbulent Flow, J. Aerosol Sci. 5:145–155.
Malet, J., Alloul, L., Michielsen, N., Boulaud, D., and Renoux, A. (2000). Deposition of Nanosized Particles in Cylindrical Tubes Under Laminar and Turbulent Flow Conditions, J. Aerosol Sci. 31:335–348.
Montassier, N., Boulaud, D., and Renoux, A. (1991). Experimental Study of
Thermophoretic Particle Deposition in Laminar Tube Flow, J. Aerosol Sci.
22:677–687.
Montassier, N., Boulaud, D., Stratmann, F., and Fissan, H. (1990). Comparison between Experimental Study and Thermophoretic Particle Deposition in
Laminar Tube Flow, J. Aerosol Sci. 21:S85–S88.
Nishio, G., Kitani, S., and Takahashi, K. (1974). Thermophoretic Deposition of
Aerosol Particles in a Heat-Exchanger Pipe, Ind. Engng. Chem. Des. Develop.
13:408–415.
Pyykönen, J., and Jokiniemi, J. (2001). Model Studies of Deposition in Turbulent
Boundary Layer with Inertial Effects, In Proceedings of the VIII Finnish
National Aerosol Symposium. Report Series in Aerosol Science 54, Finnish
Association for Aerosol Research, Finland, pp. 28–38.
Reeks, M. W. (1977). On the Dispersion of Small Particles Suspended in an
Isotropic Turbulent Fluid, J. Fluid Mech. 83:529–546.
Reeks, M. W. (1983). The Transport of Discrete Particles in Inhomogeneous
Turbulence, J. Aerosol Sci. 14:729–739.
Romay, F. J., Takagaki, S. S., Pui, D. Y. H., and Liu, B. Y. H. (1998). Thermophoretic Deposition of Aerosol Particles in Turbulent Pipe Flow, J. Aerosol
Sci. 29:943–959.
Schmidt, F., and Sager, C. (2000). Deposition of Particles in Turbulent Pipe
Flows. European Aerosol Conference, Dublin, J. Aerosol Sci. 31(Suppl.
1):S847–S848.
Shah, R. K., and London, A. L. (1978). Laminar Flow Forced Convection in
Ducts. Academic Press, New York.
Shimada, M., Okuyama, K., and Asai, M. (1993). Deposition of Submicron
Aerosol Particles in Turbulent and Transitional Flow, AIChE J. 39:17–26.
Stratmann, F., Otto, E. and Fissan, H. (1994). Thermophoretic and Diffusional
Particle Transport in Cooled Laminar Tube Flow, J. Aerosol Sci. 25:1305–
1319.
Talbot, L., Cheng, R. K., Schefer, R. W., and Willis, D. R. (1980). Thermophoresis of Particles in a Heated Boundary Layer, J. Fluid Mech. 101:737–758.
Thakurta, D. G., Chen, M., McLaughlin, J. B., and Kontomaris, K. (1998). Thermophoretic Deposition of Small Particles in a Direct Numerical Simulation
of Turbulent Channel Flow, Int. J. Heat Mass Transfer 41:4167–4182.
Young, J., and Leeming, A. (1997). A Theory of Particle Deposition in Turbulent
Pipe Flow, J. Fluid. Mech. 340:129–159.
Walker, K. L., Homsy, G. M., and Geyling, F. T. (1979). Thermophoretic Deposition of Small Particles in Laminar Tube Flow, J. Colloid Interface Sci.
69:138–147.
Wells, A. C., and Chamberlain, A. C. (1967). Transport of small particles to
vertical surfaces, Brit. J. Appl. Phys. 18:1793.