Research Journal of Applied Sciences, Engineering and Technology 3(3): 210-217, 2011
ISSN: 2040-7467
© Maxwell Scientific Organization, 2011
Received: January 29, 2011
Accepted: February 23, 2011
Published: March 30, 2011
Predicting the Mean Liquid Film Thickness and Profile along the Annular Length
of a Uniformly Heated Channel at Dryout
1
V.Y. Agbodemegbe, 1C.Y. Bansah, 1N.A. Adoo, 1E. Alhassan and 1,2E.H.K. Akaho
1
National Nuclear Research Institute, Ghana Atomic Energy Commission,
P.O. Box. LG 80, Legon, Accra-Ghana
2
School of Nuclear and Allied Sciences, University of Ghana,
P.O. Box. AE1, Kwabenya, Accra-Ghana
Abstract: The objective of this study was to predict the mean liquid film thickness and profile at high shear
stress using a mechanistic approach. Knowledge of the liquid film thickness and its variation with two-phase
flow parameters is critical for the estimation of safety parameters in the annular flow regime. The mean liquid
film thickness and profile were predicted by the PLIFT code designed in Fortran 95 programming language
using the PLATO FTN95 compiler. The film thickness was predicted within the annular flow regime for a flow
boiling quality ranging from 40 to 80 % at high interfacial shear stress. Results obtained for a laminar liquid
film flow were dumped into an excel file when the ratio of the actual predicted film thickness to the critical
liquid film thickness lied within the range of 0.9 to unity. The film thickness was observed to decrease towards
the exit of the annular regime at high flow boiling qualities and void fractions. The observation confirmed the
effect of evaporation in decreasing the film thickness as quality is increased towards the exit of the annular
regime.
Key words: Annular flow regime, droplets entrainment, film evaporation, flow boiling quality, interfacial
shear stress, void fraction
is in-turn dependent on the precision with which the
thickness of the liquid film attached to the wall of the
coolant flow channel is predicted. The liquid film models
employed for CHF prediction are either complete
disappearance of the liquid film or the critical film
thickness models. The flow of liquid film of a certain
critical thickness however ensures cooling of the surface
otherwise; heat transfer is controlled by steam flow which
worsens cooling (Kolev, 2006) and hence leads to the
deterioration of the stability of the equipment.
The interface between the liquid film and the gas core
is also one other major consideration during these
predictions. While some researchers consider a highly
dynamic and irregular liquid film interface, others use
mean film thickness where the waves at the interface are
considered non-existent and thus making the interface
regular. The interfacial disturbances however enhance
heat transfer from the interface to the steam and also do
not allow the liquid film to become very thick
(Kolev, 2005). In this study the regular film interface
model is investigated into to serve as basis for the analysis
of wavy liquid film interface.
The objective of this study was to predict the liquid
film thickness and profile at high shear stress using a
mechanistic approach. The supplied heat flux becomes the
dry out heat flux when the liquid film thickness predicted
INTRODUCTION
The annular two-phase flow regime: The annular twophase flow regime is the predominant flow pattern
observed in evaporators, many different types of boilers,
natural gas pipelines and general steam generating
systems which include nuclear reactors. It is characterized
by a fast flowing central gas core surrounded by a liquid
film attached to the wall of the tube or duct through which
flow takes place. The gas core may or may not contain
droplets and mass exchange in the form of droplets
entrainment and re-deposition is frequently present
(Edwards et al., 1998).The thin wavy liquid film is
dragged along the channel wall by a shear force exerted
by the gas phase. The structure and morphology of the
interface between the liquid film and the gas core which
is intrinsically time-dependent, strongly affects all
transport processes taking place between the phases and
hence play a central role in the overall transport
mechanisms and fluid dynamics of annular flow
(Cioncolini et al., 2009).
In view of the industrial importance of the regime, a
number of models have been developed to predict the
upper limit of safe and efficient heat transfer in the regime
marked by a safety parameter commonly known as
Critical Heat Flux CHF. The estimation of this parameter
Corresponding Author: V.Y. Agbodemegbe, National Nuclear Research Institute, Ghana Atomic Energy Commission, P.O. Box.
LG 80, Legon, Accra-Ghana
210
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
is equal to the critical liquid film thickness. Other
conditions assumed were, laminar film flow, regular film
interface, liquid film is small as compared to the channel
diameter, steady state operation and uniform heating of
channel.
THEORY
Estimating the liquid film flow rate: Equation (4) and
(5) were solved for the liquid and gas flow velocities
within the respective boundary conditions of 0#y#*m and
*m# y#Rin.y, *m and Rin are the distance from the channel
wall, the average (mean) liquid film thickness and the
inner tube radius respectively. U, P and g are velocity,
pressure and acceleration due to gravity respectively.
The liquid film flow rate per unit width of channel QL
in Eq. (1) and (2) was determined by integrating the liquid
velocity obtained from Eq. (4). Thus:
FILM THICKNESS MODELS
The mean liquid film thickness is the most
fundamental parameter for liquid film flow. Many
correlations therefore exist as stated in (Fukano and
Furukawa, 1998) for its prediction.
For regular film interface model, the liquid film
thickness for both high and negligible interfacial shear
stress were given respectively by the equations
(Butterworth and Hewitt, 1977):
y =δ m
QL =
(1)
δN
(2)
where, the thermal properties such as dynamic viscosity
(:L), liquid density (DL), vapor density (Dg), specific heat
capacity (CpL) as a function of temperature are
approximated by the general polynomial expression:
∑ aT
Φ (T ) =
∑ bT
j
2
Rin − δm ) ⎡ dP
(
⎤
+ ρg g ⎥
τi = −
⎢
2( Rin − δm ) ⎣ dz
⎦
j
(3)
j
Evaluating the overall pressure drop within the flow
regime: The overall pressure drop dP/dz in the two-phase
flow channel was expressed as the sum of friction, gravity
and acceleration pressure drop terms. Thus:
M(T)denotes any of the specified property at temperature
T(ºC). ai and bj are coefficients.
dP ⎛ dP ⎞
=⎜
⎟
dz ⎝ dz ⎠
Velocity profile in the liquid film and vapor core: The
second order ordinary differential equations that the
velocity profile must satisfy in both the liquid film and the
gas core are respectively defined by the equations
(Drosos et al., 2006):
⎛ d 2U L ⎞ dp
⎟ =
− ρL g
2
⎝ dy ⎠ dz
(4)
⎛ d 2U g ⎞ dP
⎟
− ρg g
2 ⎟ =
⎝ dy ⎠ dz
(5)
µL ⎜
µg ⎜⎜
(7)
The overall pressure drop dP/dz is negative for flow in the
positive axial direction (Todreas and Kazimi, 1798).
i
i i
(6)
The force per unit area on the liquid film exerted by
the central gas core causes a shear at the liquid film-vapor
core interface. The interfacial shear stress was determined
from a one dimensional equation for momentum
conservation of the gas-liquid droplets mixture and
yielded assuming no entrainment is present in the gas core
(Wongwises and Kongkiatwanitch, 2001):
1
⎛ 2Q µ ⎞ 2
=⎜ L L⎟
⎝ τi ⎠
L dy
1 ⎡⎛
dp ⎞ ⎛ δ m3 ⎞ ⎛ τ i δ m2 ⎞ ⎤
⎟⎥
⎢ ⎜ ρL g −
⎟⎜ ⎟ − ⎜
=
µ L ⎢⎣ ⎝
dz ⎠ ⎝ 3 ⎠ ⎝ 2 ⎠ ⎥⎦
1
⎛ 3Q µ ⎞ 3
δH = ⎜ L L ⎟
⎝ ρL g ⎠
∫U
y=0
⎛ dP ⎞
⎛ dP ⎞
+⎜
+⎜
⎟
⎟
⎝ dz ⎠ grav ⎝ dz ⎠ acc
fric
(8)
The form pressure drop term which represents the
pressure gradient due to change of the channel crosssectional area was in this analysis ignored since the
system under study was a uniform one with no changes in
cross section. The frictional pressure drop was predicted
by separated flow models as applied by Lockhart and
Martinelli and stated in (Yao and Giaasiaan, 1996) as:
⎛ dP ⎞
⎜
⎟
⎝ dz ⎠
211
fric
⎛ dP ⎞
= Φ i2 ⎜
⎟
⎝ dz ⎠ i
(9)
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
where x is the flow boiling quality which in annular flow
regime was considered in the range of, (0.4# x #0.8). X
is the Lockhart-Martinelli parameter for flow of fluids and
for the case under study was expressed as:
where, Φ i2 is the two phase multiplier for either phase
(i ) flowing alone and (dP/dZ)i is the single-phase pressure
drop for either phase flowing separately:
[
]
⎛ dP ⎞
= g ρgα + ρ L (1 − α )
⎜
⎟
⎝ dz ⎠ grav
(10)
⎡ (1 − x )2
x2 ⎤
⎛ dP ⎞
2
G
−
=
⎥
⎢
⎜
⎟
⎝ dz ⎠ acc
⎣⎢ (1 − α )ρ L αρg ⎦⎥
(11)
⎛ µ L ⎞ ⎛ ρg ⎞ 2
⎟⎜
⎟
X 2 = (1 − α )⎜⎜
⎟
⎝ µg ⎠ ⎝ ρL ⎠
(
⎡
⎛
⎢ ⎛ Rin − δ m ⎞ ⎜ m L ρ g Rin − δ m
⎜
⎟
⎢⎝ δ
mg ρ L δ m
⎠ ⎜⎝
m
⎢⎣
The total mass flux and the vapor core flow rate per
unit width of the channel were evaluated respectively by
the expressions:
3⎤
(18)
⎥
⎟ ⎥
⎠ ⎥⎦
" is the void fraction expressed in this study as:
xρ L
α=
⎤
mg
7 ⎡ mL
⎥
⎢ 2 +
G=
22 ⎢ Rin (δm − Rin ) 2 ⎥
⎦
⎣
) ⎞⎟
⎛ 1 ⎡
⎞
2
xρ L + ρg ⎜⎜
(1 − x )3 + x − x 2 ρL ⎤⎥ ⎟⎟
⎢
⎦⎠
⎝ ρg ⎣
(
(13)
)
1
2
(19)
y =δ m
Qg =
∫U
g dy
=
y=0
(
⎡
⎛
dP ⎞ ⎜ Rin − δ m
1 ⎢⎛
⎜
⎟
ρ g−
µg ⎢⎝ g
dz ⎠ ⎜
3
⎝
⎢⎣
)
Criterion for critical heat flux in annular flow regime:
Dryout occurred at saturated state when the liquid film
thickness is thinner than the critical liquid film thickness.
This approach is physically appropriate due to the
possibility of instantaneous disappearance of liquid film
as the liquid film gets very thin. The concept was assessed
in order to increase the accuracy of dryout calculations
and also enable accurate prediction of dryout positions
especially in non-uniform heating situations. In the
present study, dryout was assumed to occur when the ratio
of the predicted actual liquid film thickness (Eq. 23) to the
critical film thickness lied in the range of 0.9 to 1.0. The
critical liquid film thickness expression employed is that
of Chun as expressed in (Ji-Han et al., 2003, Ji-Han and
Un-Chul, 2008) and for this study was given by the
equation:
⎞ ⎤ (14)
⎛
⎟ ⎜ τ g ( Rin − δm) ⎟ ⎥
⎟−⎜
⎟⎥
2
⎠ ⎥⎦
⎠ ⎝
3⎞
2
Evaluating the friction pressure drop: The friction
pressure drop is the most significant contributing factor to
the overall pressure drop and its accurate estimation
specific to the case studied was relevant. For the
estimation of the friction pressure drop term, the singlephase pressure drop for liquid film flowing alone at the
channel wall was expressed as:
28µ L mL
⎛ dP ⎞
⎜
⎟ =
⎝ dz ⎠ Lo 11ρ L Rin4
(15)
⎛
⎞
φ
⎟⎟
δ crit = ⎜⎜
⎝ 2π H fg Rin ρ L QL ⎠
The Lockhart-Martinelli correlation for the two-phase
multiplier as stated in was used due to its specific nature
with flow configuration in the annular flow regime. For
the laminar-liquid, laminar-vapor configuration
considered:
1.75 ⎡
5
1 ⎤
⎢1 + X + X 2 ⎥
⎣
⎦
Φ i2 = (1 − x )
⎛ µg ⎞
8.8⎜ ⎟
⎝ µL ⎠
5
1 ⎤ ⎛ 28µ L mL ⎞
⎢1 + X + X 2 ⎥ ⎜⎝ 11ρ R 4 ⎟⎠ (17)
⎣
⎦
L in
1.75 ⎡
= (1 − x )
fric
v fg µ L2
σ
0.617
×
(20)
N is the heat flux supplied to the channel at critical liquid
film thickness, vfg = vg-vL is the specific volume at
evaporation and F is the surface tension which for liquid
rising up in tubes was approximated by the expression
(Basmadjian and Farnood, 2007):
(16)
Substituting Eq. (15) and (16) into (9) yielded:
⎛ dP ⎞
⎜
⎟
⎝ dz ⎠
0.35
σ=
212
ρ L gRin Lann
2
(21)
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
Table 1: Applicable range of Eq. (20)
Parameter
Range
Pressure, MPa
0.5-12.0
Mass Flux, kg/m2s
100-2000
Flow quality, %
0.1-0.9
Ji-Han et al. (2003) and Ji-Han and Un-Chul (2008)
Table 3: Parameter range for present prediction model
Parameter
Range
Tube diameter, mm
1.02-37.5
Dimensionless tube length, L/D
7.69-800
Pressure, MPa
0.5-12.0
Inlet sub-cooling, K
0.00-210
100-2000
Mass Flux, kg/m2s
Flow quality, %
0.4-0.8
Table 2: Parameter range of critical heat flux data for water
Parameter
Range
Number of data
4375
Tube diameter, mm
1.02-37.5
Dimensionless tube length, L/D
7.69-800
Pressure, MPa
0.103-19.0
Inlet velocity, m/s
0.0103-25.0
Inlet subcooling, K
0.00-210
Critical heat flux experimental, MW/m2
0.0347-21.4
Okawa et al. (2003)
Table 4: Simulation Input Parameter Values
Parameter
Temperature of fluid at saturation, ºC
Enthalpy of fluid at saturation, kJ/kg
Inlet fluid enthalpy, kJ/kg
Latent heat of fluid at saturation, kJ/kg
Specific volume of liquid, m3/kg
Specific volume of steam, m3/kg
Channel inner radius, m
Channel length, m
Heat flux supplied, kW/m2
Mean film thickness (initial), m
Specific gas velocity, m/s
Flow quality (initial)
Liquid mass flow rate, kg/s
Vapor mass flow rate, kg/s
Lann is the annular length expressed as (Okawa
et al., 2003):
[
⎛ GRin ⎞
Lann = z − ⎜
⎟ × Hsub + xH fg
⎝ 2φ ⎠
]
(22)
Value
100
419
251
2257
0.001044
1.673
0.035
15
107174
0.0008
10
0.3
0.43
1.72
The actual liquid film thickness for the uniform heat
flux supplied was estimated for high shear stress in the
ELFT-SUB (Estimation of the Liquid Film Thickness
Subroutine) by modified forms of Eq. (1) as:
Equation (20) is applicable within the parameter
range specified in Table 1. Parameter range of critical
heat flux data for water is also tabulated in Table 2.
DEVELOPING THE PLIFT CODE FOR FILM
THICKNESS PREDICTION
1
⎡
1 ⎛ dP 3τ i ⎞ ⎤ 3
δ Hp = δ m ⎢1 −
−
⎟ ⎥ (23)
⎜
⎢⎣ ρ L g ⎝ dz 2δ m ⎠ ⎥⎦
Features and flow chart of the PLIFT code: The PLIFT
(Prediction of Liquid Film Thickness) code was
developed in Fortran 95 programming language using the
PLATO FTN95 compiler to predict mean film thicknesses
at high interfacial shear stress in the annular flow regime
using the regular liquid film interface model. The code
was subdivided into four features or subroutines that
perform specific calculations.
The EFTP-SUB (Evaluation of Fluid Thermal
Properties Subroutine) calculates the thermal properties of
the fluid (liquid, gas) such as density, viscosity, specific
heat capacity and thermal conductivity of both phases at
prevailing temperature. These results were then used in
the other subroutines. Equation (3) was used in this
subroutine to estimate the densities of liquid and gas, the
viscosity of liquid and the specific heat capacity of liquid.
The EOPD-SUB (Evaluation of Overall Pressure
Drop Subroutine) employs Eq. (8) to calculate the overall
pressure drop. Equation (10), (11) and (17) were
substituted into 8 to obtain a relation for the overall
pressure drop. The overall pressure drop obtained from
EOPD-SUB was used to estimate the interfacial shear
stress by Eq. (7) and also substituted into Eq. (6) and (14)
in EPFR-SUB (Estimation of Phase Flow Rate
Subroutine) to estimate the individual phase flow rates of
the liquid film and vapor core.
The flow chart of the PLIFT code is depicted in Fig. 1.
Code parametric range and simulation: The code is
applicable within the parameter range specified in
Table 3. The PLIFT code was run for water entering a
channel of radius 0.035 m and length 15 m at a
temperature of 60ºC until it reached saturation. The flow
quality was initialized at 0.3 and iterated within the range
of 0.4# x #0.8. The mean film thickness was also
initialized at 0.0008 m and iterated for each flow quality
until the predicted liquid film thickness is thinner than the
critical film thickness. At this point, the supplied heat flux
was the critical heat flux and the results were dumped into
an excel file. The simulation input parameters are as
depicted in Table 4.
RESULTS AND DISCUSSION
Void fraction increased (Fig. 2) towards the exit of
the annular regime as flow boiling quality is increased.
This resulted in a decrease in the mean liquid film
thickness (Fig. 3 and 4) through the processes of
evaporation and entrainment from the liquid film
interface.
213
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
START
δ m
INITIALIZE: , x = x
δ
δ
m
m
i
+ ∆ x
= δ
m i
*δ
= 0 . 999
EFTP:
Φ (T ) =
EOPD/EISS: ∑
∑
QL =
i
a iT
j
b jT
i
j
(Rin − δ m )2 ⎡ dP + ρ g ⎤
g
⎥⎦
2(R in − δ m ) ⎢⎣ dz
∫ULdy=
y=0
ELSE
m i
τi = −
y=δ m
EPFR: If, X < 0.8
x
1 ⎡⎛
dP⎞⎛ δm3 ⎞ ⎛τiδm2 ⎞⎤
⎟⎥
⎢⎜ρLg − ⎟⎜⎜ ⎟⎟ − ⎜⎜
µL ⎢⎣⎝
dz⎠⎝ 3 ⎠ ⎝ 2 ⎟⎠⎥⎦
1
ELFT:
δ
p
H
⎡
1 ⎛ dP 3τ i ⎞⎤ 3
⎜⎜ −
⎟⎟⎥
= δm ⎢1−
⎣ ρL g ⎝ dz 2δ m ⎠⎦
δm
PRINT OUT RESULTS
If, 0.9 =HFR =1.0
x
ELSE
END
Fig. 1: Liquid film thickness prediction flow diagram
Mass transfer from the film interface by processes of
evaporation and entrainment contribute on one side to
reduce the film thickness at high supplied heat and also
high shear exerted by the steam at the interface between
the gas core and the liquid film. On the other hand
condensation of the liquid vapor and re-deposition of the
entrained droplets which also occur simultaneously
counter affect the decrease in film thickness caused by the
first two processes. The rate at which these sets of
processes occur is a significant factor that determines the
liquid film profile along the annular length.
Near the exit of the annular flow regime, flow quality
and steam velocity tends to increase resulting in increased
evaporation and liquid droplet entrainment from the film
214
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
9.995E-01
Void fraction
9.999E-01
9.985E-01
9.980E-01
9.975E-01
9.970E-01
3.20E-01
4.20E-01
7.20E-01
6.20E-01
5.20E-01
Flow boiling quality
8.20E-01
8.20E-01
1.92E-06
2.12E-06
Fig. 2: Plot of void fraction versus flow boiling quality
9.00E-01
Flow boiling quality
8.00E-01
7.00E-01
6.00E-01
5.00E-01
4.00E-01
3.00E-01
2.00E-01
1.00E-01
0.00E-01
9.17E-07
1.12E-06
1.32E-06
1.52E-06
1.72E-06
Mean film thickness (m)
Fig. 3: Plot of flow boiling quality versus mean film thickness
9.995E-01
Void fraction
9.990E-01
9.985E-01
9.980E-01
9.975E-01
9.970E-01
8.96E-07
1.10E-06
1.30E-06
1.50E-06
1.70E-06
Mean film thickness (m)
1.90E-06
2.10E-06
Fig. 4: Plot of void fraction versus mean film thickness
surface and hence thinning of the liquid film (Fig. 2)
(Tong and Tang, 1997).
The profile in Fig. 5 indicates a continuous layer of
liquid film which is in direct contact with the channel wall
as observed by Levy (1999). The thickness of the
continuous liquid film layer decreased with increased
liquid flow rate (Fig. 6) towards the exit of the annular
regime. The continuous film thickness reached a
maximum at a point assumed to be the onset of liquid
entrainment and significant evaporation. Above this point,
the mean film thickness decreased until the onset of mist
flow. When the film becomes so thin, droplets
entrainment from the film surface is reduced leading to a
more even and sustained thickness.
215
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
1.494E+01
Annular length (m)
1.492E+01
1.490E+01
1.488E+01
1.486E+01
1.484E+01
1.482E+01
1.480E+01
9.38E-07
1.14E-06
1.34E-06
1.56E-06
1.74E-06
1.94E-06
2.14E-06
Mean film thickness (m)
Fig. 5: Plot of annular length versus mean film thickness
Mean film thickness (m)
2.50E-06
2.00E-06
1.50E-06
1.00E-06
5.00E-07
0.00E-00
9.20E-07
9.22E-07
9.24E-07
9.28E-07
9.26E-07
Volumetric liquid flow rate (m 3/s)
9.30E-07
9.32E-07
1.92E-06
2.12E-06
Fig. 6: Plot of mean film thickness versus volumetric liquid flow rate
1.40E+04
Interfacial shear stress (Pa)
1.20E+04
1.00E+04
8.00E+04
6.00E+04
4.00E+04
2.00E+04
0.00E+00
9.17E-07
1.12E-06
1.32E-06
1.52E-06
1.72E-06
Mean film thickness (m)
Fig. 7: Plot of interfacial shear stress versus mean film thickness
The decrease in film thickness as interfacial shear
stress increased (Fig. 7) could be attributed to the effect of
the vapor pressure at the liquid-vapor interface. The shear
exerted by the gas tends to create a drag force which drags
the film along the channel length reducing its thickness.
For a constant annular length, the drag created at the
interface increased towards the end (exit of annular
regime) due to increasing pressure drop with flow boiling
quality and hence the film thickness is decreased.
CONCLUSION
The PLIFT code was designed and used to predict the
liquid film thickness and profile. The thickness was
216
Res. J. Appl. Sci. Eng. Technol., 3(3): 210-217, 2011
estimated at high shear stress by Eq. (23) which was the
modified form of Eq. (1).
Generally the film thickness was observed to
decrease towards the exit of the annular regime at high
flow boiling quality and void fraction. The observation
confirmed the effect of evaporation in decreasing the
thickness of the film as quality is increased towards the
exit of the annular flow regime.
Increase in steam velocity and volumetric flow rate
resulted in increased droplets entrainment from the film
surface and hence thinning of the liquid film.
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ACKNOWLEDGMENT
The authors wish to commend all scientific staff and
colleagues at the National Nuclear Research Institute
(NNRI) who in one way or the other made very relevant
inputs to the design, implementation, analysis and
interpretation of the output data of the PLIFT code.
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