numerical model for heat transfer...
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CHAPTER - VII
NUMERICAL MODEL FOR HEAT TRANSFER
Background
The objective of the chapter is to develop a predictive procedure
with a combination of CFD analysis and numerical model and to
compare the so obtained numerical heat transfer coefficients with that
of experiment and correlations.
An integral equation for heat transfer coefficient is established
using analytical model based on separated flow approach with
annular regime as the physical model. Using turbulent Prandtl
number, momentum and heat transfer equations are coupled. The
pressure gradient and wall shear stress needed in the establishment
of numerical model is taken from CFD simulations. A code in ‘C’
language is developed to evaluate the two phase or local heat transfer
coefficient. The so obtained heat transfer coefficients are compared
with the experimental data and correlations.
7.1 Analytical and Numerical Modeling
It is difficult to model condensing flows using CFD analysis.
Hence the CFD simulations are performed under adiabatic conditions
to predict the flow regimes and pressure drop. The contours obtained
in the CFD simulations show an excellent agreement with the
predictions of Thome et al. flow regime maps for all refrigerants
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considered in the present study. Also, the pressure gradient obtained
from CFD simulations predicted the experimental data better than the
correlations available in the literature. Hence, the pressure gradient
obtained from CFD analysis is used in modeling two phase heat
transfer.
As mentioned in Chapter – III, most of the analytical
correlations for heat transfer coefficient are developed for annular flow
regime as it occupies 60 – 70% of the heat exchanger length
depending on the mass flux. Fig 7.1 represents the physical model
based on which numerical model is developed.
From the extensive literature survey as mentioned in Chapter –
III, it is observed that the correlations for heat transfer coefficient are
developed by modifying the existing correlations using the
experimental data or using homogeneous model or using two phase
multiplier approach. In the later method, single phase Dittus – Boelter
equation or other similar equations are used and the two phase
Fig 7.1 Physical Model Representing Annular Flow Regime
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multiplier is determined using experimental data. The heat transfer
coefficient correlations obtained using these methods have limited
applications to the extent that they can predict the experimental data
only for the operating conditions based on which they are developed.
In addition, Park et al. [2008] had reported that the widely used and
recently developed correlations viz., Cavallini et al. [2002], Thome et
al. [2003a, 2003b], Shah [1979] and Dobson et al. [1998] etc.,
exhibited a deviation up to 300% with the experimental data of
ammonia. Jiang et al. [2006] had reported that most of the
correlations exhibited higher deviations from their experimental data
obtained at high reduced pressures. This represents the semi
empirical nature of the correlations and their limited applications.
The chapter focusses on the development of numerical
procedure based on CFD simulations. Since the CFD simulations will
take into account different operating conditions and properties of
different fluids, the resulting numerical model will also predict the
heat transfer coefficient for any fluid and at any operating pressure.
In general, two types of analytical models are used to predict the
heat transfer coefficient for condensing flows using seperated flow
model. The first one is evaluation of eddy viscosity using Prandtl
mixing length theory in combination with Van Driest’s hypothesis and
using analogy, heat transfer coefficient is evaluated. Chitti and Anand
[1995] used this method, together with an iterative procedure to
evaluate liquid film thickness, eddy viscosity and hence local heat
transfer coefficient. Sarma et al. [2005], used their own eddy viscosity
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expression they developed in the lines of Van Driest model while
investigating the phenomenon of transition from laminar to turbulent
boundary layers for external flows and reported analytical model for
condensing flows.
The second method is a laborious process involving the
evaluation of liquid film thickness, shear stress and eddy viscosity
distribution using Von Karman universal velocity profiles in the liquid
film. Based on this data, using turbulent Prandtl number, heat
transfer coefficient is evaluated. Li et al. [2000] used this method, to
evaluate condensation heat transfer coefficient. They used Lockhart
and Martinelli correlation for the prediction of pressure drop which
exhibited a deviation of more than 100% with the experimental
pressure drop data as reported in the previous chapters. In the
present study, the second method is used to evaluate the heat
transfer coefficient using the pressure drop and wall shear stress
obtained from CFD simulations. The analytical model is presented as
follows.
7.1.1 Analytical Model
The physical model shown in Fig 7.1 represents the annular
flow regime. Uniform liquid film along the circumference of the tube is
assumed. The entrainment of liquid droplets in the vapor core is
neglected, thus assuming a smooth vapor – liquid interface. The sub
cooling of the liquid film is neglected and liquid and vapor properties
are considered to be constant corresponding to the condensing
temperature. Axial heat conduction is neglected. The flow of liquid
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film is considered to be steady and turbulent. The momentum and
energy equations used are same as that of single phase internal
turbulent boundary layer flows [1999].
The momentum and energy equations are written for the
differential element, of the liquid film as shown in Fig 7.1. For the
two dimensional turbulent flow as shown in Fig 7.1, the total shear
stress and heat transfer rate are made up of sum of molecular and
turbulent contribution [1999], given by Eqs. (7.1) and (7.2).
(7.1)
(7.2)
where, is the heat transfer rate per unit area perpendicular to heat
transfer in direction. Introducing turbulent Prandtl number,
given by Eq. (7.3), the heat transfer rate is written as Eq. (7.4).
(7.3)
(7.4)
where is the circumferential area of the differential element at a
radius, . Seperating the variables in Eq. (7.4) and integrating for the
thickness of liquid film, Eq. (7.5) is obtained.
(7.5)
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The condensation heat transfer coefficient is defined by Eq. (7.6).
(7.6)
Substituting Eq. (7.6) in Eq. (7.5)
(7.7)
In Eq. (7.7), eddy viscosity is needed to evaluate heat transfer
coefficient. From Eq. (7.1), expression for eddy viscosity can be written
as,
(7.8)
Introducing shear velocity, , the dimensionless flow
velocity and distance, are written as,
and (7.9)
Substituting Eq. (7.9) in Eq. (7.8) and in Eq. (7.7)
(7.10)
(7.11)
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Using Von Karman universal velocity distribution for liquid layer,
given by Eq. (7.12), expression for the eddy viscosity is obtained as
Eq. (7.13).
for (7.12a)
for (7.12b)
for (7.12c)
and for (7.13a)
for (7.13b)
for (7.13c)
In Eq. (7.13), the wall shear stress, is taken from the CFD
simulations. The local shear stress, is obtained by doing force
balance to the shaded portion shown in Fig. 7.1.
(7.14)
where and are the cross sectional area of the liquid film at a
given radius, and cross sectional area of the liquid film respectively.
and are perimeter at the interface and perimeter at the given
radius, respectively. From Eqs. (7.14) and (7.9), expression for local
shear stress, is written as Eq. (7.15).
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(7.15)
The interfacial velocity, is given by Soliman et al. [1968] in their
analytical model as represented by Eq. (7.16).
(7.16)
The expression for wall shear stress is obtained from Eq. (7.15) by
substituting, .
(7.17)
Rearranging Eq. (7.17), expression for interfacial shear stress, is
obtained.
(7.18)
The phase velocity of liquid and mass flow rate are obtained using Eq.
(7.19a), substituting , from Zivi void fraction formula [1964] given by
Eq. (7.19b).
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; (7.19a)
(7.19b)
Substituting Eq. (7.19) in Eq. (7.15), the expression for local shear
stress, is obtained.
(7.20)
Similarly, substituting Eq. (7.19) in Eq. (7.18), expression for
interfacial shear stress is obtained.
(7.21)
With known wall shear stress and pressure gradient from CFD
simulations, the only unknown quantity in Eqs. (7.20) and (7.21) is
dimensionless liquid film thickness, . This is obtained from the
known mass flow rate as represented by Eq. (7.22) using Eq. (7.9).
(7.22a)
(7.22b)
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And from the definition of liquid Reynolds number, , Eq.
(7.22) is written as Eq. (7.23).
(7.23)
Substituting from Eq. (7.12), the relation between liquid Reynolds
number and dimensionless liquid film thickness is obtained.
for (7.24a)
for (7.24b)
for (7.24c)
From Eq. (7.24), with the known liquid Reynolds number, is
evaluated. Hence, numerical procedure is established as follows.
7.1.2 Numerical Procedure
The local heat transfer coefficient is obtained by performing
numerical integration across the thickness of liquid film. The
sequence of steps involved in the numerical procedure is as follows.
1. From the CFD simulations, the wall shear stress data
is obtained and hence shear velocity, is calculated.
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2. Using , the ranges of liquid Reynolds number from
Eq. (7.24) are obtained.
3. With the known , appropriate expression for is
selected among Eqs. (7.24) and hence dimensionless liquid
film thickness, is calculated.
4. From the CFD simulations, pressure gradient,
and wall shear stress, is obtained.
5. With known and , local and interface shear
stress is calculated using Eqs. (7.20) and (7.21).
6. With known and , eddy viscosity is obtained at
different locations across liquid film using Eq. (7.13).
7. Eq. (7.11) is numerically integrated and hence local
heat transfer coefficient is evaluated. The turbulent Prandtl
number is assumed unity.
A code is developed in the programming language, ‘C’ and is
enclosed in Appendix VI. The resulting local heat transfer coefficient
data is compared with the experimental data and correlations.
7.2 Comparison with Experimental Data
The heat transfer coefficient obtained from the numerical model
for R22, R134a and R407C is compared with the experimental data
and presented in Figs 7.2 and 7.3.
The comparison in general shows that the numerical model
predicts the experimental data well for high pressure refrigerants, R22
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and R407C as shown in Figs 7.2 a) and 7.2 c) compared to a low
pressure refrigerant, R134a as shown in Fig 7.2 b). This is primarily
due to R134a being a low pressure refrigerant, its reduced pressure
values are lower than R22 and R407C as given in Appendix III. The
separated flow approach considered in the present numerical model
as mentioned in Chapter – II, gives better results for fluids with high
reduced pressure. Similar results are observed with annular flow
based Shah correlation that takes the reduced pressure of fluids into
account while comparing the experimental data with correlations in
Chapter V.
a)
c)
b)
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Fig 7.2 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment for a) R22 b) R134a and c) R407C
a)
b)
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Fig 7.3 b) shows that most of the experimental data falls within
a deviation band of ±20% including that of R407C. Figs 7.2 a) and 7.3
a) shows that for R22, the numerical model closely follows the
experimental heat transfer coefficient at the mass fluxes considered,
exhibiting a deviation of 8% at low mass flux, 4 - 9% at medium and
high mass flux.
For R134a, the numerical model over predicts the experimental
data for low and medium mass fluxes with a deviation of 11% and
22% respectively as shown in Fig 7.2 b) and 7.3 a). The numerical
model under predicts upto medium qualities for high mass flux with a
deviation of 15%.
As shown in Fig 7.2 c), for R407C the numerical model over
predicts at medium and high mass fluxes with a deviation of 14% at
medium mass flux and 3% at high mass flux as shown in Fig 7.3 a). It
is observed that even for a mixture refrigerant, the numerical model
predicted the heat transfer coefficients well.
7.3 Comparison with Correlations
Fig 7.3 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment for a) Deviation Graph b) Parity Graph
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The numerical heat transfer coefficients are also compared with
the correlations to evaluate their relative performance in predicting
the experimental data from present study.
7.3.1 Comparisons for R22
As shown in Fig 7.4 a), the heat transfer coefficients predicted
using numerical model are as good as that of Cavallini et al.
correlation at a low mass flux of 200 kg/m2s, both exhibiting a
deviation of 8% from the experimental heat transfer coefficients, while
Dobson et al correlation exhibits lowest deviation of 3% among the
other correlations as shown in Fig 7.4 d).
a)
f)
e)
c)
b)
d)
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At a medium and high mass fluxes, the numerical model
predicts the experimental data better than correlations with a
deviation in the range of 4 – 9% as shown in Figs 7.4 e) and 7.4 f).
Cavallini et al. correlation under predicts with a deviation of 9%, while
the numerical model over predicts the experimental data as shown in
Figs 7.4 b) and 7.4 c). Other correlations exhibited further larger
deviations.
In general for R22, the predictions of numerical model are better
than the correlations used for comparison. The numerical model over
predicts the experimental data with an average deviation of 5 - 9%,
where as Cavallini et al. correlation under predicts the experimental
data with an average deviation of 9%.
7.3.2 Comparisons for R134a
As shown in Fig 7.5 a), all the correlations under predict the
experimental data at low mass flux while the numerical model over
predicts at medium and high qualities. Cavallini et al. correlation
Fig 7.4 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment and Correlations for R22 at a) G= 200 b) 400 andc) 600 kg/m2s and Corresponding Deviation Graphs
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exhibits a minimum deviation of 5%, followed by the numerical model
and other correlations with a deviation of 11%.
A minimum deviation of 4% is exhibited by Cavallini et al.
correlation at medium mass flux and by Shah correlation at high
mass flux as shown in Figs 7.5 e) and 7.5 f). The deviation of
numerical model is higher than that of correlations at medium mass
flux with 22%, while Shah and Dobson et al. correlations deviate by
15 – 16%.
a)
f)c)
e)
d)
b)
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At high mass flux, numerical model is as good as Cavallini et al.
correlation showing a deviation of 15%, while Dobson et al. correlation
exhibits better predictions with a deviation of 10% from the
experimental data as shown in Figs 7.5 b) and 7.5 c).
For R134a, on an average Cavallini et al. correlation exhibits a
deviation of 9% followed by Shah correlation with a deviation of 10%
and numerical model with a deviation of 15% from the experimental
data.
7.3.3 Comparisons for R407C
Fig 7.5 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment and Correlations for R134a at a) G= 200 b) 400 andc) 600 kg/m2s and Corresponding Deviation Graphs
a)
d)
c)
b)
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All the correlations and numerical model over predict the
experimental data at a medium mass flux of 400 kg/m2s as shown in
Figs 7.6 a) and 7.6 c) with Cavallini et al. correlation exhibiting a
minimum deviation of 6 - 7% and the numerical model exhibiting a
deviation of 14%. The predictions of the numerical model are in
excellent agreement with the experimental data at high mass flux with
a deviation of 3% exhibiting an over prediction. Cavallini et al.
correlation is equally good, but exhibits under prediction. Thus, the
predictions of the numerical model exhibit excellent agreement even
with mixture refrigerant, compared to the correlations.
The maximum deviation exhibited by the numerical model is
22% for low pressure refrigerant, R134a and the minimum deviation
is 3% for mixture refrigerant, R407C. On an average, the predictions
of the numerical model are better than that of correlations for the
refrigerants considered in the present study, as the numerical model
exhibits over prediction of experimental data in general.
Fig 7.6 Comparison of Numerical Heat Transfer Coefficient with that ofExperiment and Correlations for R407C at a) G= 400 b) 600 andCorresponding Deviation Graphs
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7.4 Conclusions
The objective of the chapter is to develop a predictive procedure
using a combination of CFD analysis and numerical model to predict
the heat transfer coefficient for two phase flow of any fluid and at any
operating pressure.
The chapter describes the development of integral heat transfer
coefficient equation using separated flow approach by considering the
annular flow regime with uniform liquid film as physical model. A code
in ‘C’ language is written for the evaluation of dimensionless liquid
film thickness, interface shear stress and for the evaluation of local
shear stress at different locations in the liquid film using Von Karman
universal velocity profiles. The wall shear stress and pressure gradient
needed in the evaluation of local and interface shear stress is taken
from CFD simulations.
The heat transfer coefficients obtained from the numerical
model predicts the experimental data of all the refrigerants considered
in the present study within a deviation of ±20%. The model predicts
the heat transfer coefficients of high pressure refrigerants, R22 and
R407C with less deviation compared to R134a. This is due to the
separated flow approach used in the numerical model which gives
better results for fluids with high reduced pressure.
The numerical heat transfer coefficients predicted the
experimental data of R407C with a minimum deviation of 3% at a
mass flux of 600 kg/m2s and a maximum deviation of 14% at medium
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mass flux. The predictions of numerical model are better than
correlations that exhibited a minimum deviation of 6-7% by Cavallini
et al. correlation and a maximum deviation of 30% by Shah
correlation. These results meet the objective of the present chapter.
Cavallini et al. correlation generally under predicts the
experimental data, while numerical model over predicts with an
average deviation of 11%. Therefore numerical model is suggested as a
better option compared to Cavallini et al. correlation for conservative
thermal design of condensers.
CHAPTER - VIII
CONCLUSIONS
The objectives of the present study are achieved primarily by
developing a test facility to evaluate the two phase heat transfer and
pressure drop at high pressures using refrigerants, R22, R134a and
R407C. Secondly, by developing a predictive procedure for heat
transfer coefficient and pressure drop using a combination of CFD
analysis and numerical model.
8.1 Experimental Study
The experimental analysis is performed for mass flux of 200,
400 and 600 kg/m2s for a pressure range of 10 – 16 bar at a
condensing temperature of 400C. The experiments are conducted with
average vapor quality of test section in the range of 0.28-0.71.
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The experimental set up has shown good repeatability by
exhibiting a heat balance error of less than 1% for 98% of the
runs, in spite of higher heat duties involved for the refrigerants,
R22, R134a and R407C considered in the present study.
The uncertainty analysis performed on experimental heat
transfer coefficient results in an average uncertainty of
±10.85%. The uncertainty involved with pressure drop is
±0.065% or ±0.081 kPa.
The validation of test facility with R22 shows that the
performance parameters are following the physics involved and
shows a definite behavior for runs at low, medium and high
mass fluxes.
The experimental data points plotted on Thome et al. [2003a]
flow regime map shows that it spreads in stratified wavy, slug
(Intermittent) and annular flow regimes.
8.2 Experimental Heat Transfer Coefficient
At low mass flux of 200 kg/m2s, the variation of two phase or
local heat transfer coefficient is not significant with vapor
quality. At high mass fluxes of greater than 400 kg/m2s, a
linear variation of heat transfer coefficient with quality is
observed. These trends corroborate with the heat transfer
characteristics of the flow regimes predicted by Thome et al.
[2003a] flow regime map, viz., stratified wavy and annular.
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The experimental heat transfer coefficients of R134a are almost
same of that of R22 as their liquid property combination
parameter, which is the non dimensional combination of
thermal and transport properties, is approximately same for
both the fluids.
The experimental heat transfer coefficients of R407C are
approximately 13% lower compared to that of R22 due to
dependency of vapor heat transfer coefficient on its two phase
heat transfer coefficient though its liquid property combination
parameter is 16% higher than that of R22.
8.2.1 Comparison with Correlations
The comparison of experimental data obtained in the present
study with Shah [1979], Dobson et al. [1998] and Cavallini et al.
[2002] correlations resulted into following conclusions.
Cavallini et al. [2002] correlation predicted the experimental
data falling in stratified, slug and annular flow regimes with a
minimum deviation of 5% and a maximum deviation of 18%.
Other correlations predicted the experimental data with good
agreement for only a particular flow regime. Shah correlation
exhibited better predictions for annular regime and for fluids
with high reduced pressure. Similarly Dobson et al. correlation
exhibited better predictions for stratified wavy regime.
8.3 Experimental Pressure Drop
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Low pressure refrigerant, R134a exhibits high pressure drop
penalty as its liquid to vapor density ratio and liquid viscosity
being higher compared to the other two refrigerants.
On an average, the pressure drop of R407C is almost same as
that of R22, as both have same value of liquid to vapor density
ratio.
8.3.1 Comparison with Correlations
The comparison of experimental pressure drop data with
Lockhart-Martinelli [1947], Grönnerud [1979], Chisholm [1973],
Friedel [1979] and Müller-Steinhagen and Heck [1986] correlations
resulted into following conclusions.
Lockhart and Martinelli correlation which is widely used in the
modeling of condensing flows exhibits a deviation of more than
100% from the experimental data of all the three refrigerants
considered in the present study.
Among the pressure drop correlations considered for
comparison, only Friedel correlation predicts the experimental
pressure drop with good agreement with a minimum deviation
of 5% and maximum deviation of 22%.
All most all the pressure drop correlations exhibited higher
deviations for low pressure refrigerant, R134a, thus leaving
scope for the development of better predictive procedure for
pressure drop at high pressures.
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8.4 CFD Analysis
The fulfillment of the objectives of CFD analysis lies in proper
selection of multi phase model, appropriate mesh model of the test
section as the conditions near wall are to be captured and solver
controls according to the type of simulations.
The VOF model in the commercial CFD software, FLUENT
perfectly tracked the geometry of vapor – liquid interface of
refrigerants as the resulting mixture density contours are in
excellent agreement with the flow regimes predicted using
Thome et al. [2003a] flow regime map.
The work of Schepper et al. [2008] for air-water and gas-oil
mixtures at atmospheric pressures is successfully extended to
vapor-liquid flow of refrigerants at high pressures as all the flow
regimes are simulated in the CFD analysis.
Pressure drop data estimated using VOF model is in good
agreement with the experimental data compared to the pressure
drop correlations with a minimum deviation of 3% and a
maximum deviation of 16%.
8.5 Numerical Heat Transfer Model
The objective of developing the numerical model in combination
with the CFD analysis is to predict the heat transfer coefficient and
pressure drop for any fluid and at any pressure.
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The numerical model over predicts the experimental data with
an average deviation of 11%.
The comparison of the heat transfer coefficient predictions by
the numerical model and by the correlations establishes that
numerical model is a better tool for conservative thermal design
of condensers.
8.6 Scope for Future Work
The test facility used in the present study is designed with
flexibility to use different tube geometries for the inner tube of the test
section. In addition, the scroll compressors used with different
displacement volumes and pressure ratios will enable the testing of
performance parameters of different future refrigerants. The scope for
the future work using the test facility and using CFD analysis is
presented as follows.
To test the performance of non azeotropic blends of HFCs and
HCs by modifying the test facility to visually observe the flow
regimes
To use different enhancement methods on refrigerant side
and to experimentally evaluate enhancement factors. Also to
develop a CFD model for enhanced surfaces and observe the
corresponding flow regimes
To experimentally evaluate the volumetric void fractions as
provision is made to isolate the test section from the rest of
experimental setup and tap the refrigerant
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Prediction of flow regimes under diabatic conditions using
CFD analysis.
CFD analysis of double pipe heat exchanger with
condensation inside the inner tube.
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