7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A simple model of annular flow and dryout in minichannels
Dariusz Mikielewicz, Jan Wajs, Michal Gliiski, AbdulBaset R.S. Zrooga
Gdansk University of Technology, Faculty of Mechanical Engineering, Heat Technology Department
Narutowicza 11/12, 80952 Gdansk, Poland
Email: Dariusz.Mikielewicz(ipg. gda.pl, jwais@imech.pg.gda.pl
Keywords: dryout, annular flow, flow boiling
Abstract
In the paper a theoretical model of dryout at high vapour quality is presented. The model is based on the analysis of the film
evaporation process. It resulted in formulation of three mass balance differential equations for liquid in the film and in the
form of droplets in the core, as well as and equation of the mass balance of vapour in the core of the annular flow. The mass
balance equation for the liquid film contains a modified evaporation term. Modified is the heat flux used in evaporation of
liquid film. The solution of these three differential equations requires knowledge of the rates of deposition, as the entrainment
is neglected in these calculations. The relations for determination of deposition rate are taken from Okawa et al. (2003).
Introduction
The phenomenon of critical heat flux occurs in various
technical applications just to mention nuclear reactors,
steam generators, evaporators in refrigeration technology or
airconditioning. Microchannel heat sinks offer also a huge
area for possible applications, Qu and Mudawar (2003).
From amongst a number of mechanisms triggering the
critical heat flux the two can be named more precisely. The
first case is when the vapour film can be formed between
the continuous liquid phase and the heating wall. Such case
is often called the departure from nucleate boiling (DNB).
The second mechanism is the disappearing liquid film at the
wall due to its evaporation. In the latter case the wall is
exposed to vapour phase and such a mode of critical heat
flux is named the dryout. That type of critical heat flux will
be considered in the paper. In both mentioned above cases
wall temperature suddenly increases and a severe
impairment of heat transfer coefficient is observed as the
wall is covered merely by vapour, Kandlikar (2001).
Annular flow is one of the most often encountered
twophase flow patterns in which a wide range of vapour
quality can be found. Transition from annular flow to the
mist flow in most cases involves passing through the dryout
conditions. For that reason accurate method of prediction of
the dryout process is of particular importance from the
engineering point of view. That is especially the case for
minichannels, where the approaches to model that
phenomenon are still at an early stage of development. The
two models which are known to authors at the moment are
the models due to Qu and Mudawar (2003) and Revellin et
al. (2008). These models do not consider entrainment of
droplets in case of flows in minichannels, as it has been
experimentally proven that such assumption is acceptable.
Bergles and Kandlikar (2005) reviewed the existing studies
on critical heat flux in microchannels. They concluded that
singletube CHF data are not available for microchannels.
The film flow analysis is one of the successful prediction
methods of the dryout in flow boiling. Such approaches
have mostly been developed for conventional size channels.
Most approaches so far used the mass balance equation for
the liquid film with appropriate formulations for the rate of
deposition and the rate of entrainment. The first approach to
such modeling was postulated by Whalley et al (1974).
Other implementations of that approach have been
postulated by Okawa et al. (2003, 2004) and Celata (2001).
The one equation model for balance of mass in the liquid
film constitutes from the contributions coming from the
deposition of droplets on the film, entrainment of droplets
from the film and evaporation of liquid from the film. Such
equation can be integrated from the point of the onset of
annular flow (assumed arbitrarily). It must be
acknowledged that any discrepancy in determination of
deposition and entrainment rates, together with
crosscorrelations between them, renders loosing of
accuracy of model predictions. Sedler and Mikielewicz
(1981) postulated a twoequation analytical model where
the balance equation in the film is supplemented with the
mass balance equation for entrained droplets in the core.
Such approach is more general, as two variables, namely the
mass velocity of film and mass velocity of droplets are
solved by two independent equations. In many approaches
to modeling dryout, which are based on a mass balance
equation of the film, there can be found a term in which the
ratio of entrained droplets with respect to the amount of
liquid is used. Such parameter is adjusted based on
experimental data. The approach presented by Sedler and
Mikielewicz can, amongst the others, model directly such
term. The original model due to Sedler and Mikielewicz
(1981) provided analytical solutions of liquid film
distribution as it was assumed that the rate of deposition D
is proportional to the flow rate of entrained droplets in the
core, whereas the rate of entrainment E was also
proportional to the flow rate of liquid in the film. The
modification to that model by D. Mikielewicz et al. (2007)
utilized functions describing more advanced terms for
entrainment and deposition terms postulated by Okawa
(2003), (211114).
In the present paper a threeequation model for mass
balance of film, droplets and gas is used to model the dryout
in annular flow. Additionally, in the present work the
amount of heat used for evaporation of film has been
modified to yield a more precise distribution as only the
interface heat flux takes part in the evaporation process. The
predictions using the model have been compared with
experimental data due to Sedler and Mikielewicz (1981) for
conventional channels and own data collected for
minichannels, Mikielewicz et al. (2010).
Nomenclature
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
ThreeEquation Model of the Dryout
Proposed three equation model (denoted as 3R in further
part of paper) bases on the mass balance of fluid in the film
and in the core and also vapour mass balance in the flow
(Fig. 1). This model is intended to predict the first critical
heat flux phenomenon of second type (dryout). Such critical
heat flux occurs at high vapour quality both in the channels
of conventional diameters and in the minichannels.
The model is developed on the following assumptions:
initial parameters of annular flow are known,
the overall balance of mass flux in the flow can be
written as: G = G+ Gk+ G,.
For the flow in the minichannels the additional assumptions
were made:
film flow is laminar,
interface is stable which means that the surface tension
influence is significant,
there is no entrainment.
C droplet concentration in core
D tube diameter (m)
f friction factor
g gravitational constant (ms1)
G mass velocity (kg m2 ')
h enthalpy (N m kg1')
i iteration number
j superficial velocity (m s 1)
k mass transfer coefficient (m s1)
P pressure (bar)
q heat flux (W m2)
R twophase multiplier
u velocity (m s 1)
x vapour quality
z length (m)
Xt Martinelli's parameter
Re Reynolds number
We Weber number
Greek letters
6 thickness of liquid film (m)
p density (kg m3)
Pu dynamic viscosity (Pa s)
v cinematic viscosity, (m2s )
7r nondimensional number
o surface tension (N m'1)
T shear stresses (N m 2)
j drag coefficient
Subscripts
c core
D deposition
E entrainment
f liquid film
i interface
k droplets
kr critical
1 liquid
Iv liquidvapour
t initial
v vapour
w wall
0 beginning of annular flow
C0i SD
C.6 G D tD
OG
G + dz
G,+ dz
O z
/ ///////////////
,______a z________
Figure 1: Schematic view of annular flow with boiling
The model presented below essentially differs from the
model proposed by Sedler and Mikielewicz (1981). In the
3R model it was assumed that the liquid film flows on the
channel wall and the flow core consist of twophase mixture
of vapour and droplets. Additionally in this approach the
arbitrary division of liquid mass flux GL on the mass flux in
the film Gf and in the core Gc was eliminated. It is
substantial progress in the critical heat flux phenomenon of
dryout type modeling.
The model of mass balances in the film, in the core and in
the vapour phase can be written in a form of three
differential equations:
dG, 4 (
dz D
h,
dG = 4 GD + GE
dz D h )
dGv
dz
4 q,
D h1
where: q,= q(1 2
It should be remembered that in the model
minichannels the term connected with
entrainment from the film is omitted (GE
recommended by Qu and Mudawar (2003).
(13) for the
the droplets
= 0), what is
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Split of mass flux at the beginning of annular flow in the
particular phase components of the flow was done in the
following manner (Qu and Mudawar, 2003):
 mass velocity of droplets in the core:
Gk = 0.951 0.15 We (4)
 mass velocity of vapour in the core:
G, = G. x (5)
 mass velocity of the core, in which the vapour with
suspended liquid droplets is flowing:
G =Gk + G, (6)
 mass velocity of the liquid film on the channel wall:
G, G G (7)
The Weber number is defined as:
GzD
We = (8)
Pio
In the 3R model the expression describing velocity of
droplets deposition GD is defined as a product of the
deposition coefficient kD and the droplets concentration in
the core C. Relation defining the coefficient kD was taken
from Qu and Mudawar (2003) for microchannels:
S0147
D 47.8 q01 (9)
j, Gh, P)
The droplets concentration in the core of annular flow C
was defined in the basis of fluid mass balance in the core:
C= G G (10)
pv G, + C
P, PA
Location of annular flow origin is defined by Taitel and
Dukler's relation with application of Martinelli's parameter
(X,). The transition criterion is equal to Xt = 1.6. For the
laminar twophase flow Qu and Mudawar (2003) presented
the vapour quality xt in the beginning annular flow regime
as a function of Martinelli's parameter:
Xtt ll/ 1 Xt Ui1 (
X Xt uv
Location of the critical flux is determined based on the
liquid film thickness on the channel wall. The liquid film
disappears with the liquid evaporation on its surface and in
the place of its dryout = 0:
= f I .I.D (12)
4 v P J
This assumption was utilized in the modeling of dryout in
the channels of small diameter by many authors, e.g.
Revellin and Thome (2008), Qu and Mudawar (2003).
The thickness of liquid film at the annular flow beginning
was estimated on the basis of following expression:
where r, shear stresses on the channel wall, defined as:
8p,
In equation (14) the twophase multiplier (Ro) and the drag
coefficient (Co) are determined from the relations:
R0 =1+ PVx (15)
P1
0.316
SRe (16)
ef
Mentioned above set of equations (1+16) was solved by
iterative loop with the constant step length Az = 0.001 m.
The length, on which the liquid evaporates from the film zkr
is determined as the product of iteration number and
employed step:
kr iAz (17)
The iteration number i is found from the equality of critical
vapour quality xkr defined as equation (18) and critical
vapour quality obtained from the experiment:
xkr (18)
G
Remained critical conditions are modeled.
The 3R model (1+3) was verified on the basis of
experimental data obtained by Sedler and Mikielewicz
(1981) for the conventional channels of diameter D = 8 mm
and also on the basis of own experimental data for the
minichannels of diameters D = 1.15 mm, D = 2.3 mm and
three fluids, i.e. SES36, R134a and R123.
Comparison Between the Mathematical Model and
Experimental Data for the Conventional Diameter
Channels
As it was mentioned, the verification of 3R model for the
channels of conventional diameters was done based on the
experimental data obtained by Sedler and Mikielewicz
(1981). Their experimental investigations were carried out
with the refrigerant R21. Experimental section of internal
diameter equal to 8 mm and of 4.22 m length was made of
the stainless steel. Experimental studies were realized at the
pressures P = 5.5, 10.6, 15.0 bars and at three values of the
mass flux G = 1000, 1750 and 3500 kg/(mns).
The comparison was done for the data obtained at the
pressures 5.5 and 10.6 bars and also two values of the mass
flux, i.e. 1000 and 3500 kg/(m2s). The results are listed in
Table 1 and presented in Fig. 2 (a,b,c,d). They show the
dependence of the liquid film dryout location (zkr) on the
vapour quality in the critical flux conditions (xkr).
go Gf 2,u,
Z'.P
Table 1. The results for conventional channels
Model 3R Experiment
P G q,
Xkr Zkr Xkr Zkr
bar kg/(m2s) W/m2 m m
79720 0.63 3.24 0.63 3.24
1000 65050 0.66 4.19 0.66 4.16
52710 0.72 5.71 0.72 5.6
5.5
74990 0.22 3.5 0.22 4.21
3500 85550 0.31 4.4 0.31 5.2
82070 0.35 5.4 0.35 6.12
80670 0.51 1.85 0.51 2.32
1000 65690 0.61 2.83 0.605 3.38
106 43910 0.67 4.76 0.67 5.6
10.6
92850 0.245 2.38 0.245 4.32
3500 77440 0.29 2.98 0.29 4.81
52260 0.36 6.21 0.36 6.39
a)
6,0
5,5
50
4,5
N
4,0
3,5
3,0
25
0,60
0,65
0,70
b) G =1000kgm s. P= 10.6bar
3 [ ,1
4 ........
xkr [
...................... ....... ........ ......... ;.......... ....... ;............................. .. ...... ....
......... i...........i 4 .......... +. ....... .. i ...I ...........    ...... .......... i..........
. ............ 4 .. ... i .. .... i..... ... i......... ..... i .... .. ....... ... ... ... ..  .....
2 .5 ........ ........... .......... ...... ..... ... ........ .......... .... .......... .......... I.......... ..........
. ........... .... .. .. .. .. ...... ........ .. ....... i .. ..... ; .. ... .. .. .; ......... i ........ ........
i, .i i i . "i i i i i i
0,40 0,45 0,50 0,55 0,60 0,65 0,70
Xkr [1
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
7,0
C) G = 3500kg/m2s, P= 5.5bar
,5 D Model 3R ... ...............
Experiment
6 ,& .............. ................................................................. ................................ ...............
S........................................ ...............
5 . . ... ... .. ........ . .  ...... .. .. ... ......... . .... .... .. .. .. ..... .. .. . .. .. .
. o . .. ......... . .... ........ ... ... .. .......... .
4 ,& ............................... .......... .............................................. 4 ..........................
3 ,5 ........ . .: ........................................ ................. ................ ................................
3,
4,
0,20 0,25 0,30 0,35 0,40
41
1, 0 de I Mdl3
7,0
dThe results indicate that te 3R m el gives te satisfy.
 * i  * *  *  *+   **. . 
6 ,5 ............... ................ ................ ................ ................. ............... ............................
5 ,5 .............. ................ ................ ................ .............. ..... .... ... .. ................ ...............
.... ........... ................ ................ ........ ........ .... ......... ................. ...............
S 4 ,0 . ............................................... ................ .......... .............. ................................
........ ................. 
mass flux, i.e. 1000 kg/(m2S........) and at pressure 5.5 bars.
2,0 :: .... ........... :.............. ..: 350.kg/m:s, P 10.6bar
15. 5 ......... I.. . ... ...... Model 3R
Experiment
0,20 0,25 0,30 0,35 0,40
Xkr [1
Figure 2: Comparison between the results obtained from
the 3R model and the experimental data by Sedler and
Mikielewicz (1981)
The results indicate that the 3R model gives the satisfactory
correspondence with the experimental data at low values of
mass flux, i.e. G = 1000 kg/(m2s) and at pressure 5.5 bars.
Increasing mass velocity of the flow and the pressure of
medium in the channel gives some discrepancy between the
model and experiment data.
Comparison Between the Mathematical Model and
Experimental Data for the Minichannels
The verification of the 3R model was done based on the
own experimental investigations carried out in the channels
of internal diameters D = 1.15 mm and D = 2.3 mm. In
Mikielewicz et al. (2010) experimental studies for three
various working fluids were reported, i.e. SES36, R134a
and R123. In the present study tests for R123 are presented.
It should be recalled that the term describing droplets
entrainment from the liquid film on the channel wall was
omitted in the annular flow modeling connected with the
channels of small diameter (Qu and Mudawar, 2003).
... .. .. . . ........ .i
. ...... ...... .....................  .  ...................... .....................
.. ............. ....... ..... ..... .i . .................
 4
G = 1000kg/m's, P= 5.5bar
D Model 3R
: Experiment
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
The verification results at selected mass velocity (G = 300
kg/(m2s)) in the channels of diameter equal to D = 1.15 mm
and D = 2.3 mm are listed in Table 2. In Figs. 3 + 6 the
influence of the critical vapour quality on the length of
liquid film dryout is presented.
Table 2. Verification data for the working fluid of R123 at
mass velocity of G = 300 k /(m2s)
D q k Experiment Model 3R
D qw P E Gf Gk
xkr Zkr Xkr Zkr
mm W/m bar kg/ms kg/ms m m
26.6 1.1 0.15 232 41 0.87 0.235 0.87 0.443
31.3 1.11 0.18 221 52 0.84 0.192 0.84 0.363
1.15
34.91 1.13 0.205 217 56 0.82 0.169 0.82 0.318
37.43 1.16 0.225 211 61 0.8 0.153 0.8 0.286
69.26 1.32 0.08 249 22 0.95 0.251 0.95 0.367
80.57 1.39 0.095 245 25 0.93 0.237 0.93 0.307
2.3
89.82 1.38 0.105 242 28 0.92 0.214 0.92 0.272
98.05 1.43 0.12 237 33 0.9 0.191 0.9 0.242
0,50
o 4 5 . .. ....... .. ........ .. .. .. ... . .. ............. ......
.................... ... ....... .. .... .. .... ...... ... ... ... .. .. .... .. .. .... ....................
0 3 5 .................... ..................... ...................... .. .......... ...................... .....................
0 ... .. .. .. .... .. ..
0 ,2 ............................ .. .............. .. ................. ... ............ .......... .....................
0 ,1 5 . . ...... .  .. ..... .....
.R123, D = 1.15mm, G =300kg/m2s
0,05. . ................ :..................... i Experim ent
S........ ... ... .. .. Model 3R
0,40&
0,75 0,80 0,85 0,90
xkr
Figure 3: Comparison between the calculation and
experimental results for R123 medium; D = 1.15 mm,
G = 300 kg/D=), G
0,05experiment
0,4 
0 ,4 1 .................... .......................  ...................... .................. .... ........... .....................
... ................. ..................... ... ....... .............. . ...... ... ........... ... ......... .....................
0 ,3 0 ............... ................... ............. ...................... .....................
0 2 5 . ................... y, ... .......... .... ....... ... ..... .... ... ................
0 ............... ............ .... .. ......... .............. .... .... .. .... .. ..... .....................
0,75 080 085 090
0,15
0,102
R123, D = 2.3mm, G= 300kg/m2s
00 S Experiment
S. ... .. ... ... .. .. .. ........... M odel 3R
0,00  i     i  i  
0,00
0,85 0,90 0,95 1,00
Xkr [
Figure 4: Comparison between the calculation and
experimental results for R123 medium; D = 2.3 mm,
G= 300 kg/(m2s)
Figure 5: Influence of the R123 medium mass velocity on
the precision of simulations; D
2.3 mm, G = 500 kg/(m2s)
.030 ................ . .. . .............. ... ... ......... .. .......... .... ...............
0 ,3 11 . ....... ................ . ............ .. ............. ...... ... .. .... . .... .... .. ... ................
0 ............ ................ ................ ................ ................ ................................
N .. ................ .. ... . ...... .. .. .. .. .
0,1
0 1 ...... ........ ................ . ............ ................ ................ ................ ................:.................
0,1
005. .... .......................... R123, D = 2.3mm G = 700kg/ 2s
SExperiment
; A Model 3R
0,00.  i   I   i    
0,55 0,60 0,65 0,70 0,75
Xk[1
Figure 6: Influence of the R123 medium mass velocity on
the precision of simulations; D=2.3 mm, G=700 kg/(m2s)
The analysis of data obtained for R123 medium indicates
the satisfactory consistency for all considered cases of mass
flow rate. The similar consistency is for both tube
diameters.
Summary
Presented in the paper three equation model is the authors
proposal to describe the conditions of dryout critical heat
flux and to find the location of liquid film dryout in the
annular flow. The model consists of three differential
equations, which allow determination of mass fluxes of
droplets flowing in the core and of the fluid flowing in the
film. Such approach is more general, because the proposed
model can be applied in both the channels of conventional
diameter and also to the minichannels. The precision of
calculations can be improved, when the proper correlations
describing the entrainment and deposition of droplets can be
found entrainmentt in case of annular flows). Application of
three equations representing the mass balance of vapour and
droplets in the core and the mass balance in the liquid film
.. .. . . .. Z ..... .... ..
i ...........  . 
****[*.:."; :'*.....................
S.. . .
R123, D = 2.3mm, G = 500kg/m2s
Experiment
 A Model 3R
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
without division of the fluid mass flux on fluid in the film Sedler B., Mikielewicz J. A simplified model of the boiling
and in the core gives considerable progress in the modeling crisis. Int. J. Heat Mass Transfer, Vol. 24, 431438 (1981)
of liquid film dryout phenomenon on the channel wall.
Whalley P.B., Hutchinson P., Hewitt GF. The calculation of
critical heat flux in forced convection boiling. Paper B6.11,
Acknowledgments 5th International Heat Transfer Conference, Tokyo, Japan
(1974)
The work presented in the paper was partially funded from
the Polish Ministry for Science and Education research
project No. N512 459036 in years 20092011.
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