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Annular flow stability within small-sized channels
International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
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Annular flow stability within small-sized channels Niccolò Giannetti ⇑, Daisuke Kunita, Seiichi Yamaguchi, Kiyoshi Saito Department of Applied Mechanics and Aerospace Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
a r t i c l e
i n f o
Article history: Received 19 May 2017 Received in revised form 11 September 2017 Accepted 25 September 2017 Available online 17 October 2017
a b s t r a c t An analytical study based on a variational thermodynamic principle is presented to evaluate the influence of surface tension on the stability of annular flow within small-sized channels. The model introduces phenomenological assumptions in the interfacial structure of the flow regime and theoretically draws the equilibrium transition line from an annular regime to the initiation of the partial wetting condition on the inner surface. By including surface tension, this model expands previous theories and identifies the stable flow configuration in terms of void fraction and interfacial extension. The significant influence of a higher surface tension and smaller diameter (i.e. lower Weber number) are responsible for a lower stable void fraction and higher slip ratio. A complete screening of the main influential parameters is conducted to explore the descriptive ability of the model. This analysis aims at contributing to the understanding of the stability of two-phase flow regimes and can be extended to the transition between other neighbouring regimes, including wall friction as well as liquid entrainment phenomena. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Multiphase flows are used as transport media in a variety of technical systems, such as nuclear reactors, power plant boilers, internal combustion engines, heat transfer and storage devices, cooling equipment for electronic hardware, and food processing systems. Given the obvious differences in these examples with regard to their scale, transport phenomena, and operating conditions, these systems have been generally researched individually via the development of models and heuristic correlations having restricted applicability [1]. The co-presence of multiple phases substantially affects the operability of a system in terms of heat transfer performance, pressure drops, and flow stability, which, respectively, alter the capacity of the system, the power required to steadily circulate the fluid, and its reliability [2]. Even though phase change processes have been observed and technically used for centuries, owing to a flawed understanding of the concepts of heat, energy, and temperature, the theoretical background needed to model these phenomena was not consistently established until the 17th century. The presence of dynamic and deformable phase-interfaces and the related discontinuities of fluid properties are the main reasons for the high degree of complexity in writing the governing transport laws and physical principles. However, in spite of this difficulty, the theory of multiphase systems relies on the classical laws of thermodynamics, fluid mechanics, and principles of heat and mass transfer. Liquid–vapour flow is the ⇑ Corresponding author. E-mail address: [email protected] (N. Giannetti). https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.098 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
most recurrent as well as complex flow regime class. Owing to the presence of deformable interfaces, different neighbouring flow patterns can be encountered in the same system [3,4]. Numerous theoretical, numerical, and empirical approaches have been developed to predict the characteristics of liquid–vapour flows [5,6]. The most widely accepted map describing the adiabatic transition between different flow patterns was semi-theoretically obtained by Taitel and Dukler [7]. The demand for high performance and reliable electronic devices and miniaturized heat exchangers for refrigeration plants requires heat transfer devices to be able to extract the given heat load within a limited temperature range and certain overall dimensions. The structural characteristics of the interface in two-phase flows constitute the criteria by which they are classified into flow regimes. However, the concept of classifying flow regimes, which assumes a certain degree of similarity between corresponding flow structures, is based on the definition of a given volume or lengthscale. Therefore, regime dependent models may yield results contingent to the selected length-scale. Furthermore, multi-phase transport phenomena occurring at different scales may vary significantly [2]. Most of the studies related to liquid–vapour flow regime transitions have been carried out for conventionally sized passages, in which surface tension effects can typically be neglected. However, inside mini- or micro-channels, surface tension effects can significantly affect the way in which heat, mass, and momentum are transferred [8]. Therefore, it has not yet been possible to theoretically predict the characteristics of two-phase flows within small-sized channels with the accuracy needed for reliable system design and control.
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Nomenclature etot E m S
r u A0
q b x
a d r hd G
energy rate per unit cross-sectional area (W m2) energy rate (W) mass flowrate (kg s1) phase interface (m2) surface tension (J m2) fluid velocity (m s1) total cross-sectional area of the channel (m2) density (kg m3) interfacial area concentration (m1) vapour quality (–) steam void-fraction (–) thickness of the liquid flow (m) inner radius of the channel (m) dry angular-portion of the channel (rad) total mass flowrate per unit cross-sectional area (kg m2 s1)
In a previous work [9], evident differences were reported between various empirically proposed regime maps for microchannels, starting with the pioneering work of Suo and Griffith [10] on the transition boundaries in heptane and water flow, through the graphical map of Triplett at al. [11] for an air–water mixture in a 1.1 mm channel, to the experimental work of Revellin et al. [12] on the flow transition of the refrigerant R134a. The significant differences in the thermo-physical properties of the refrigerants were considered responsible for the large deviations between the results obtained by the different authors. This suggested a need for designing generalized maps, based on first principles instead of utilizing one-of-a-kind flow pattern maps, which would take into account the wide range of operating conditions and be applicable to fluids commonly used in the refrigeration field. Akbar et al. [13] used the Weber number to summarize the data for an air–water flow on a common flow regime map. For a comprehensive literature review, please refer to Ref. [14]. The present research effort applies a variational thermodynamic approach to idealized two-phase flow structures to investigate the effect of surface tension on their stability and suggests a phenomenological transition boundary. Particularly, the focus of the study is directed towards an annular flow regime and breaking of the liquid film leading to partial or complete dryout of a channel. 2. Fundamental equations Regarding initiation of the dryout of a channel, previous theoretical analyses mostly assume that it occurs when the thickness of the liquid film in an annular flow becomes zero. However, earlier experimental works [15,16] and modelling efforts [17,18] have shown that the dryout, i.e., the transition to partial wetting conditions, occurs when the liquid film is relatively thin, but not zero, and the corresponding thickness is referred to as critical film thickness. To develop mechanistic criteria to predict the characteristics and stability of multiphase flow structures, Zivi [5] applied the principle of minimum entropy production and, recently, in Ref. [19], the thermodynamic equilibrium conditions of their Helmholtz potentials have been explored. A corresponding approach has been implemented in Refs. [20,21] to establish the minimum stable thickness and characteristics of a falling liquid film. The principle of minimum entropy production is an approximate restatement of the second law of thermodynamics for characterising the steady state of open systems kept away from the thermodynamic equilibrium,
Subscripts g vapour/gas-phase gl vapour–liquid interface i internal energy k kinetic energy s Solid-phase sg vapour–solid interface sl solid–liquid interface tot total r surface tension energy Superscripts ⁄ dimensionless form 0 reference value
when all the fluxes that cross the system are constant, and under the assumption that the rate of entropy production is governed by linear phenomenological laws [22–24]. Following the approach of Zivi [5], the present paper formulates the energy flux of two different flow structures in terms of void fraction and liquid-vapour interface extension to search for the values of these parameters which minimise the entropy production rate of the process under analysis. Particularly, the thickness of the liquid film and extension of the liquid–vapour interface is critical information for predicting the heat and mass transfer rates; please refer, for instance, to the model developed by Younes et al. [25]. By doing so, it is tacitly assumed that the energy of the flow is gained within the channel, and that the work required for accelerating and extending the interface between the two phases is the largest part of the total flow work. Under this point of view, the principle of minimum entropy production indicates that the stable flow configuration is achieved through a process that requires the minimum work. Then, comparing the energy rate of the two flow structures at corresponding conditions, the theoretical transition line between them is drawn and analysed with respect to the main parameters at play. This procedure further implies that the transition between the two flow patterns is triggered by disturbance that qualifies as a zero-energy interaction. Previous results applied to hypothetical two-phase structures [5] have shown a remarkable agreement with time averaged values of the main significant parameters of steady two-phase flows. 2.1. Evaluation of the flow energy contents The open thermodynamic system under consideration is a channel with finite length, defined radius and a smooth inner surface, and emerging into a receiver, where liquid and vapour are removed separately, at a constant rate corresponding to the rate of inlet supply, but with very low effluent velocities. In this manner, the kinetic and surface tension energy fluxes are almost entirely dissipated at the outlet. As pointed out by Zivi [5] this circumstance is commonly matching technical applications. The energy rate per unit cross-sectional area of the channel can be quantified by Eq. (1), where Ek, EG, Ei and Er represent the kinetic, potential, internal, and surface tension contributions, respectively; A and S are the cross-sectional area of the channel and the contour between neighbouring phases, respectively; and r is the surface tension.
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N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162 Table 1 Dimensionless variables. S0 ¼ 2pr
q0 ¼ ql
u0 ¼ qG
A0 ¼ pr 2
r0 ¼ rgl
Dh ¼ 4AS0 L
l 0 0
qk ¼ qqk0
Ak ¼ AAk0
yG;k ¼
Si;j ¼
ri;j ¼ rri;j0
U k ¼ UUk0
1_ 1 _ Etot ¼ Ek þ E_ r þ E_ i þ E_ G A A 1 1 1 _ g u2g þ m _ l u2l þ Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ m _ g Ug m ¼ A 2 2 _ l Ul þ m _ g gyG;g þ m _ l gyG;l þm
etot ¼
ð1Þ Further developing the expression shown above (Eq. (1)), and introducing the representative (identified by the apex 0 ) and dimensionless variables (identified by the superscript ⁄) listed in Table 1, Eq. (2) can be obtained when the energy rate per unit area is divided by the inertial term q0 u0 3.
etot
q0 u03
1 3 3 ¼ ðqg Ag ug þ ql Al ul Þ 2 S0 u 0 r 0 þ 0 0 0 02 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl qAuu gy0 U0 þ 02 qg Ag ug U g þ ql Al ul U l þ 02G qg Ag ug yG;g þ ql Al ul yG;l u u ð2Þ
The extracted dimensionless numbers can be defined as follows:
We ¼
02
0
q Dh u r0
ð4Þ
U0 u02
ð5Þ
Eq. (6) highlights the impact of the main physical effects at play on the dimensionless overall energy rate per unit cross-sectional area of the channel.
e¼ ¼
etot
q0 u03 1
3 g Ag ug
l Al ul
q
l ¼ llk0
etot
q0 u03
1 3 3 qg Ag ug þ ql Al ul 2 4 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ We o 1 n 3 3 qg aug þ ql ð1 aÞul ¼ 2 4 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ We
¼
ð7Þ
The void fraction a and vapour quality x are defined by Eqs. (8) and (9), respectively. Additionally, Al⁄ = Al/A0 = 1 a, Sgl⁄ = bD/4, qg⁄ = qg/ql, and ql⁄ = 1.
Ag ¼
Ag ¼a A0
ð8Þ
_g m _ m
ð9Þ
A steady liquid–vapour flow with a vapour quality x and a defined mass velocity G within the small-sized channel is considered to be at thermal saturation for a pressure p. Furthermore, it is assumed that the velocity fields of the vapour and liquid can be described by unique constant average values ul and ug, and that the surface tension components rsl, rgl, and rsg are constant along the three interfaces. Introducing the definition of the vapour void fraction as a = Ag/A, the liquid and vapour velocities are calculated via Eqs. (10) and (11), respectively.
ul ¼
3
þq 4 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ We þ H qg Ag ug U g þ ql Al ul U l 1 þ 2 qg Ag ug yG;g þ ql Al ul yG;l Fr 2
e¼
x¼
u Fr ¼ pffiffiffiffiffiffiffiffi gDh
uk ¼ uuk0
surface tension over gravitational forces when the flow is confined within small-sized channels. The liquid entrained into the vapour and the energy dissipation owing to the wall friction are preliminarily neglected. Accordingly, the energy rate per unit crosssectional area of the channel (e) can be quantified using these terms as the sum of the rates of kinetic and surface tension energy of the flow (Eq. (7)).
ð3Þ
0
H¼
Si;j S0
yG;k y0G
ug ¼
_ m
ql Al
¼
_g m
q g Ag
Gð1 xÞ
ql ð1 aÞ
¼
Gx
qg a
ð10Þ
ð11Þ
Their equivalent dimensionless forms are as follows:
ð6Þ
If an adiabatic horizontal channel is studied at low-liquid Weber numbers (high-surface tension fluids or small-sized passages), it is assumed that the internal energy and gravitational potential energy have a marginal effect on the achievement of a stable flow configuration. In contrast, the energies acquired through the work needed to accelerate the two phases to their ultimate velocities, as well as to increase the interfacial area between the two phases to the value of a steady configuration against the interfacial tension, are considered the two main contributions to the total flow energy. Optical measurements conducted by Revellin et al. [12] demonstrated that the relatively obvious symmetry of the flow configuration is a confirmation of the dominant effect of
ug ¼
ug qx ¼ l u0 qg a
ð12Þ
ul ¼
ul 1 x ¼ u0 1 a
ð13Þ
Remarkably, even though in most of the applications of conventional-sized channels a turbulent liquid flow is observed, the flow inside small-sized channels often falls within the laminar regime. On the other hand, the irregularities in the channel surface with size at the same length-scale of the liquid flow characteristics may be responsible for the dissimilar and unpredictable flow regime transitions [9]. Assuming that the shear stress balance at the liquid–vapour interface is as given in Ref. [26], the interfacial velocity ui,j (Eq.
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(14)) is a viscosity-weighted linear function of the average velocities in the radial direction:
ui;j ¼
li /j ui þ lj /i uj li /j þ lj /i
ð14Þ
The no-slip condition at the interface of a rigid body can be obtained as the limit for li >> lj and is applied to solid–liquid and vapour–solid boundaries. Therefore, uls = usg = 0, and Eq. (15) is obtained.
o 1n 4 3 3 S u r e ¼ qg aug þ ql ð1 aÞul þ 2 We gl gl gl
ð15Þ
2.2. Annular flow A purely annular flow (with no liquid entrained in the vapour) in a channel of small diameter D on steady, thermal saturation is represented in Fig. 1. In general, the steam void-fraction a and the interfacial area concentration b constitute the main flow parameters that define the extension of the liquid–vapour interfaces. These can be theoretically interrelated once the flow geometry, vapour quality, and mass flux are defined. For instance, the geometry represented in Fig. 1 could be used for defining the steam void fraction (Eq. (8)) to make explicit the direct relationship between a and Sgl (Eqs. (16) and (17)).
d ¼ rð1
pffiffiffi aÞ
ð16Þ pffiffiffi
Sgl ¼ 2pðr dÞ ¼ 2pr a
pffiffiffi Sgl;a pD a pffiffiffi ¼ 0 ¼ ¼ a pD S
ð19Þ
pffiffiffi ll 3=2 ql ugl;a lg a ð1 xÞ þ qg ð1 aÞ 1 a x h ¼ pffiffiffi pffiffiffii u0 að1 aÞ ll a þ 1 a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1a ; d¼r 1 1 WD
ð21Þ
1 a 6 WD 6 1
ð22Þ
For this flow geometry, the liquid–gas interface Sgl in its dimensional and dimensionless forms is expressed in Eqs. (23) and (24) as follows:
Sgl ¼ ð2r 2dÞðp hd Þ þ 2d
¼
Sgl;s ¼ S0 1
p
þ
ð23Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h a D 1 þ ðpWD 1Þ 1 1 WD
ðpWD 1Þ
p
pD rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a 1 WD
ð24Þ
The interfacial velocity between the liquid and gaseous phases and its dimensionless form are expressed, respectively in Eqs. (25) and (26).
ugl ¼
ð20Þ
lg
2rðp hd Þ p hd ¼ 2r p p
Careful attention must be directed to the well-established knowledge of the importance of the value of the contact angle between solid and liquid phases on the interfacial geometry and, consequently on the value of the final results. On the other hand, due to the chaotic behaviour of the liquid interface, the precise measurement of the flow configuration shape becomes nearly unachievable. Being the contact angle dependent on the specific solid-liquid pair as well as on the specific operative conditions a more detailed geometry is disregarded for this first general analysis as it would not affect the method of analysis, while making the mathematical problem formulation more complex [27]. From the definition of the vapour void fraction,
ð18Þ
ll ðr dÞul þ lg dug ll ðr dÞ þ lg d
ugl;a ¼
WD ¼
Sgl;s ¼
Additionally, the interfacial velocity between the liquid and gaseous phases and its dimensionless form are defined, respectively, in Eqs. (19) and (20).
ugl ¼
A simplified geometry of an alternative flow configuration with a partially wetted perimeter is shown in Fig. 2, and the fundamental parameter to describe the wet part of the solid wall WD is given by Eq. (21).
ð17Þ
Therefore, Eq. (18) can be expressed as follows:
Sgl;a
2.3. Partial wetting
ugl;s
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aþl u 1 1 g g WD qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a ðll lg Þ 1 WD þ lg
ll ul lg ug
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ll 1x ql x a þ ql 1 1 ugl;s lg 1a qg a WD qg ¼ 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ll u aþ1 1 1 l 1 WD
ð25Þ
x
a
ð26Þ
3 mm
g
Fig. 1. Annular flow in a 3 mm channel (a) and its simplified geometry (b).
Fig. 2. Partial wetting flow of ammonia within a 3 mm channel (a) and its simplified geometry (b).
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Energy rate per unit cross secon area
a
7 6.3 0.7
5.6 4.9
0.65
4.2
0.6
3.5 2.8
0.55
2.1
0.5 0.45
1.4
X 0.4
0.7 0.9
0.92
0.94
0.96
0.98
1
Void fracon Fig. 3. Energy rate per unit cross-sectional area of an annular flow as a function of the vapour void-fraction a for different inlet vapour qualities x; this work (continuous lines), results from Ref. [5] (dashed lines); We = 0.01, D = 5 mm, q* = 1655, and l* = 42.84.
6 *
5 * * *
10 2 *
*
1 1/2 * 1/3 * 1/4 * 1/5 *
0.1 0.7
a
Given certain flowrate and quality, in addition to the previously defined assumptions for simplifying the geometry of the flow, the thermodynamic approach of the principle of minimum entropy production can be used as the basis to establish stable characteristics of the annular configuration and the commencement of the dryout of the liquid–solid interface at a critical value of the flow quality. Since the outlet void fraction and the liquid-vapour interfacial extension are constrained only by their definition range this variational principle implies that void fraction and liquid-vapour interface will be such as to minimise the energy flux into the receiver. Successively, these simplified flow structures are compared to define the transition line between them. Figs. 3 and 4a and b illustrate the behaviour of the dimensionless energy rate per unit cross-sectional area as a function of the vapour void fraction when the inlet vapour quality x, density ratio q⁄ of the liquid and gaseous phases (Fig. 4a), and viscosity ratio l⁄ (Fig. 4b) are varied as parameters. In general, a local minimum is always observed and a higher value of these parameters yields higher energy rates per unit cross-sectional area. In addition, higher values of q⁄ move the minimum to higher void-fractions, whereas an opposite trend is noted for higher values of l⁄. According to the mathematical formulation of the problem (Eq. (15)), an increase in the relative impact of the kinetic energy contribution of the gaseous phase will shift the minimum to a higher void fraction. Therefore, at an equivalent mass flux, a lower vapour quality (Fig. 3), similar to a lower density ratio q⁄ (Fig. 4a), leads to a lower slip ratio and lower void fraction corresponding to the minimum of the energy rate per unit cross-sectional area of the channel. In contrast, a lower ratio of viscosity l⁄ of the two phases yields a higher interfacial velocity ugl⁄ and for a given value of the Weber number We, moves the minimum to higher void fractions (Fig. 4b). Reference values of these parameters, namely, q⁄ = 1655 and l⁄ = 42.84, are obtained from the thermo-physical properties of saturated water at atmospheric pressure. Lower values represent, for instance, the circumstance of a higher operative pressure, and both q⁄ and l⁄ approaching unity at the characteristic critical pressure. The dashed lines in Fig. 3 are the results from the 1964 model of Zivi [5], which targets channels of a conventional size and neglects the contribution of surface tension. The latter condition shifts the minimum to higher values of the stable void fraction and is unaffected by the viscosity ratio of the liquid–vapour phase.
100
3
Energy rate per unit cross secon area
3.1. Flow stability
Energy rate per unit cross secon area
a
3. Results
2.8
0.75
0.8
0.85
0.9
0.95
1
Void fracon
(a)
1/20 *
2.6
2.4
1/16 *
2.2
1/12 *
2
1/8 *
1.8
1/4 *
1.6 *
1.4 1.2 1 0.8
0.85
0.9
Void fracon
0.95
1
(b)
Fig. 4. Effect of (a) density ratio and (b) viscosity ratio of the phases on the energy rate per unit cross-sectional area of an annular flow as a function of the vapour void-fraction a at atmospheric pressure for reference values of the thermo-physical properties of water (q* = 1655 and l*= 42.84); We = 0.01, D = 5 mm, x = 0.4.
If the inlet Weber number increases (ceteris paribus, corresponding to a larger diameter of the channel, lower surface tension or a higher refrigerant flowrate), the energy rate per unit crosssectional area decreases and the minimum moves to higher values of the void-fraction (Fig. 5). The results from Zivi [5] agree for a particular case of this model, which as a limit for an infinite Weber number. Under this thermodynamic approach, the steady vapour voidfraction of an ideal annular flow (Eq. (29)) will assume a value that minimises the energy per unit cross-section area of the channel (Eqs. (27) and (28)). Analytical expressions of the derivatives are reported in the Appendix A. The stable vapour void fraction of an idealized annular flow a0 is plotted in Fig. 6a for different inlet qualities. From a0, the corresponding stable slip ratio R0 (Fig. 6b) can be calculated by using the liquid and vapour average velocities from Eqs. (10) and (11). As the Weber number (flowrate or channel diameter, once the refrigerant has been chosen and a reference condition selected) is varied, it can be observed that the stable void fraction of an idealized annular flow within a small diameter channels is bracketed between a maximum value and a minimum value, representing, respectively, the predominance of inertial and surface tension effects. This result is consistent with experimental results from previous literature [28], and is recognized to have an effect even on the flow in conventional sized channels. Additionally, the same observation can be attributed to a stable slip-ratio of the flow, so that a higher vapour quality is related to a higher maximum value of R0, while the minimum is not altered. In practice, for a given thermodynamic condition and a certain fluid, the Weber number
N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
100
Energy rate per unit cross secon area
a
1158
We 4*10-6
10-5
3*10-5
10
7*10-5 2*10-4 5*10-4
0.0012
1 0.8
0.85
0.9
0.95
1
Void fracon Fig. 5. Energy rate per unit cross-section area of an annular flow (logarithmic vertical axis) as a function of the vapour void-fraction a for different values of the Weber number We; x = 0.4, D = 5 mm, reference values of the thermo-physical properties of water are q* = 1655 and l* = 42.84.
0.95 0
Stable void fracon
0.7
0.85
WD
0.6
ð30Þ
0.5
0.8
0.4
0.75
(a)
14
12 10
0.7
0.6
the following can be obtained,
6
2 1.E-05
1.E-03
1.E-01
Weber number We
1.E+01
1
(b)
p 1Atm
2
@WD
> 0 and
@ 2 es >0 @ a2
ð32Þ
is proportional to the mass flux G and the annular flow that cannot be observed at excessively low flowrates.
(
)
4 pffiffiffi a We pffiffiffi l a3=2 ð1 xÞ þ q ð1 aÞ 1 a x pffiffiffi p ffiffiffi að1 aÞ l a þ 1 a 3
3 1 ð1 xÞ 2 x q 2 þ 2 a ð1 aÞ2
@ 2 ea P 0; @ a2
This finally yields,
þ
10Atm
0
0.8
Void fracon
Fig. 6. Stable values of the (a) vapour void-fraction a0 and (b) slip ratio R0 for different values of the inlet flow quality x; D = 5 mm, reference values of the thermo-physical properties of water are q* = 1655 and l* = 42.84.
ð33Þ
3Atm
0.9
@ ea ¼ 0; @a
@ 2 es
> 0;
qg lg ; ; x; Sgl ðb; DÞ; We ql ll
qg lg ; ; x; Sgl ðb; DÞ; We WD0 ¼ u2 q l ll
4
ea ¼
ð31Þ
a0 ¼ u1
0.4
0 1.E-07
!2
@ 2 es @ 2 es @ 2 es 2 @ a2 @WD@ a @WD
0.5
8
8 > @ es @ 2 es > > ¼ 0; P 0; 06a61 < @a @ a2 2 > @ es @ es > > P 0; WDmin ðaÞ 6 WD 6 1 ¼ 0; : @WD @WD2 Under the condition that,
X 0.8
16
Stable slip rao
Fig. 7 shows a first comparison of the results of the present work (continuous lines) to those obtained from the 1964 model (dashed lines). The saturation pressure is varied within a channel of 0.5 mm diameter enclosing a two-phase water flow. The corresponding lines are represented by colours to identify the operative pressures. As a rule, the inclusion of the contributions related to the interface extension, interfacial velocity, and surface tension (right-hand side of Eq. (15)) yield globally lower void fractions; and the difference between the corresponding results increases as the saturation pressure increases and approaches the critical value. Using the same approach, a stable condition for the configuration characterized by partial wetting of the solid surface is sought for by minimizing the energy content defined in Eq. (30) with respect to the void-fraction as and the parameter defining the wet part of the channel surface WD (Eq. (31)). This is to select a single value among the set of possible values of the latter variable to determine the stable geometry of this configuration.
ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1x x 1a x 4 1 ðpWD 1Þ 1 a l 1a q a 1 WD þ q a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ þ We p p WD ðl 1Þ 1 1a þ 1
X 0.8
0.9
18
ð29Þ
( ) 3 1 ð1 xÞ3 2 x es ¼ q 2 þ 2 a ð1 aÞ2
1
0.7 20
qg lg ; ; x; Sgl ðb; DÞ; We ql ll
a0 ¼ u
0.7 0.6 0.5 0.4 0.3
Zivi S.M. 1964 0.2
This work
0.1
ð27Þ
0 0
0.2
0.4
0.6
0.8
1
Vapour quality X 06a61
ð28Þ
Fig. 7. Comparison with a previous work of Zivi reported in Ref. [5]; stable vapour void fraction a0 of a saturated water flow as a function of the inlet vapour quality x for different operative pressures; D = 0.5 mm and G = 200 kg m2 s1.
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Fig. 8 presents an example of the energy rate per unit crosssectional area of the channel es as a function of the independent variables a and WD for two reference values of the inlet vapour quality. A higher quality x is associated with higher energy rates and a higher stable void fraction a0, similar to the annular flow and lower wettability WD. When the Weber number is decreased (Fig. 9), the minimum of the energy rate per unit cross-sectional area of the channel moves to lower stable void fractions a0 and higher wettability WD0. 3.2. Flow pattern transition In addition to inertial and viscous properties (l, q), identification of the flow pattern of a two-phase flow has been recognized to depend on the fluid interfacial tension forces (r), wetting characteristics, pipe geometry (D), relative velocities, and interfacial structure (ba, bs) [29,30]. Given the large number of relevant parameters, validity of most of the empirical flow maps is limited to a specific system and to the heuristic method by which they were obtained. This phenomenological model is based on the assumption of the interfacial structure and the balance of the shear stress at the interface to obtain a more comprehensive picture of the flow while minimizing the dependence on the heuristic data. By equating the steady values of the energy of the two configurations, a critical value of the flow quality below which the annular flow becomes energetically disadvantageous with respect to the
(a)
(b) Fig. 9. Energy rate per unit cross-sectional area as a function of void fraction a and WD; D = 5 mm, x = 0.4, and reference values of the thermo-physical properties of water (q* = 1655, and l* = 42.84) at (a) We = 5 * 108 (b) We = 0.0012.
corresponding steady geometry of the partially wetted flow configuration, is calculated (Eq. (35)).
ea ðaa;0 Þ ¼ es ðaa;0 ; WD0 Þ
(a)
(b) Fig. 8. Energy rate per unit cross-sectional area as a function of void-fraction as and WD; D = 5 mm, We = 0.01, reference values of the thermo-physical properties of water (q* = 1655 and l* = 42.84) (a) x = 0.4 (b) x = 0.5.
ð34Þ
Finally, we obtain,
x0 ¼ u3 q ; l ; Sgl;a ðba ; DÞ; Sgl;s ðbs ; DÞ; We
ð35Þ
A parametric analysis is performed to explore as a first screening of the model results. Figs. 10 and 11 show the transition lines obtained, respectively, when changing the density ratio q⁄ and the viscosity ratio l⁄ of the liquid and gaseous phases. A higher value of the first parameter is responsible for a delayed annular flow transition at lower Weber numbers while other parameters are fixed. In comparison, the second parameter, l⁄, has an opposite effect. The obtained transition line resembles the empirical flow transition from annular to the stratified flow of adiabatic horizontal channels [31–33]. The transition lines from Refs. [32–36] and those obtained from the present analysis for the corresponding thermo-physical conditions are plotted on a conventionally dimensional graph (Fig. 12) for a qualitative comparison. Graphs (a) and (b) refer to results obtained within small-sized channels, with 0.5 and 1.03 mm diameters, respectively, whereas (c) plots the heuristic flow pattern map of a conventional-sized channel with R134a. The relatively sharp steepness variation of the transition line from the present model
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1
1000
0.9
900 *
Mass flux G [kg·m-2s-1]
Vapourquality X
0.8
*/2
0.7
*/3 */4
0.6
*/5
0.5 0.4 0.3 0.2 0.1
800 700 Annular (A)
600
500 400 Semi-Annular (SA)
300 200
Slug+Semi-Annular (SSA)
100
0 0.01
0 0.1
1
10
0
Weber number We
0.9 0.8 0.7 0.6 1/2 *
0.5
0.4
0.6
0.8
1
(a)
1000 R236fa Tsat 31oC D 1.03mm [34] Present work
900
Mass flux G [kg·m-2s-1]
1
*
0.2
Vapourquality X
Fig. 10. Transition curves as a function of the flow Weber number for different values of the density ratio q*; D = 5 mm, and reference values of the thermophysical properties of water (q* = 1655 and l* = 42.84).
Vapourquality X
R134a Tsat 35oC D 0.5mm [35-36] Present work
2 *
800 700 600
500 400 A
300
SA
200
0.4
100
0.3
0 0
0.2
0.2
0.4
0.6
0.8
1
Vapourquality X
(b)
0.1 700
0 0.1
1
R134a Tsat 10oC D 12 mm [32-33] Present work
10
Weber number We Fig. 11. Transition curves as a function of the flow Weber number for different values of the viscosity ratio l*; D = 5 mm, and reference values of the thermophysical properties of water (l* = 42.84 and q* = 1655).
at low vapour quality is associated with the transition from a range of conditions dominated by inertial effects (high Weber number) to a region mostly subject to surface tension (low Weber number) under the assumption of the given idealized interfacial structures (Figs. 1 and 2). The comparison with experimental transition lines is not meant as a validation of the model, but rather as a reference point of view from which the results of this analysis are observed. Namely, limiting the emphasis to inertial and surface tension effects, adopting phenomenological assumptions for the interfacial structures, and adopting a thermodynamic criterion, it is possible to theoretically draw the transition lines between neighbouring two-phase flow patterns and provide general insights regarding the influence of these effects. The first comparison highlights the importance of surface tension on the transition between neighbouring twophase flow patterns within small-sized channels, and supports the method adopted. Further efforts in this direction for the refinement of the theoretical understanding of these phenomena are required.
600
Mass flux G [kg·m-2s-1]
0.01
500 Intermient
A
400 300 200 Strafied-Wavy
100 Strafied
0 0
0.2
0.4
0.6
Vapourquality X
0.8
1
(c)
Fig. 12. Comparison of the results of the present work with previous empirical studies from Refs. [31–35].
question the accuracy and direct applicability of the results extracted. Nevertheless, this simplified thermodynamic approach was helpful in providing some physical hints regarding the stability of two-phase flow patterns and effect of surface tension within small-sized channels that in turn may be useful for the enhancement of the modelling resolution and mechanistic approaches in this field. Specifically, in addition to the results of this analysis, the following main conclusions can be stated:
5. Conclusions In the analysis presented in this work, a series of major simplifications and physical assumptions was introduced, which might
– By including the effect of surface tension, small-sized passages having a stable flow configuration in terms of the vapour void fraction and extension of the interfaces were targeted. This
N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
–
–
–
–
–
model expanded the previously reported theoretical works and contributed to the background needed for an accurate modelling of two-phase flows within mini- and micro-channels. When a miniaturized adiabatic horizontal channel and a refrigerant with high surface tension were considered, the energy needed to accelerate the two phases to their ultimate velocities, as well as increase the interfacial area between them versus the interfacial tension to reach the geometry of a steady configuration, were the two main contributions to the total energy of the flow. The transition between two neighbouring regimes was evaluated in terms of the specific energy contents, and the principle of minimum entropy production was used to evaluate the characteristics of the flow at this condition. The steam void fraction a and interfacial area concentration b were theoretically quantified for a defined flow geometry, vapour quality, and mass flux. A local minimum of the dimensionless energy rate per unit cross-sectional area was always identified, and higher values of the inlet vapour quality x, density ratio q⁄ between the liquid and gaseous phases, and viscosity ratio l⁄ yielded higher energy rates per unit cross-sectional area. An increase in the relative impact of the liquid phase kinetic energy contribution (lower slip ratio related to a lower vapour quality at equivalent mass flux, or lower density ratio q⁄) caused a shift of the minimum to lower void fractions.
– A lower ratio of viscosity of the two phases l⁄ yielded a higher interfacial velocity ugl⁄ and for a given value of the Weber number We, moved the minimum to higher void fractions. – As the Weber number was varied, the stable void fraction of an idealized annular flow within small diameter channels was bracketed between a maximum and minimum value, representing, respectively, the predominance of inertial and surface tension effects. – The results from Zivi [5] were obtained as a limiting case of this model for an infinite Weber number. – A critical value of the flow quality, below which the annular flow became energetically disadvantageous with respect to the corresponding steady geometry of the partial-wetting flow configuration, was calculated. – A higher value of the density ratio between the liquid and gaseous phases was responsible for delaying the annular flow transition at lower Weber numbers, while the viscosity ratio displayed an opposite effect. – The theoretical transition lines obtained resembled a transition to the stratified flow of an empirical flow pattern map of adiabatic horizontal two-phase flows. Appendix A Given the assumed geometry of the flow configurations, the derivatives of the energy rate per unit cross-sectional area are developed as follows:
" # 3 @ ea ð1 xÞ3 2 x þ ¼ q @a a3 ð1 aÞ3 pffiffiffi i 9 8 pffiffiffi pffiffiffi pffiffiffi h pffiffiffi að1 aÞ l a þ 1 a l ð1 xÞ 32 a þ q x 32 a 1 2p1 ffiffia > > > > > > > 2 3 > > > pffiffiffipffiffiffi > > > > 1 3 3=2 p ffiffi > > a a a þ p ffiffiffi > > 2 2 a > > 3=2 4 5 > > l a ð1 xÞ þ q ð1 a Þ 1 a x > > p ffiffiffi p ffiffiffi > > 1 ffiffi < = p þ 1Þð ð l a a l 1Þ 4 2 a ¼0 p ffiffiffi p ffiffiffi 2 > We > > > að1 aÞ2 l a þ 1 a > > > > > > > > > > > > > > > > > > > > > > : ;
ðA1Þ
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 1x q x 1a x ðl 1 > aÞ 1 WD þq a a 1a pWD1 p1 a > p ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi þ 1 þ > 2 aþ1 a WD > 2pWD ðl 1Þ 11 11 > WD WD > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q h i > > a þ 1 ðl 1x q x Þ 1 paffiffiffiffiffiffiffiffiffi ðl 1Þ 1 1 @ es 4 < 1a WD a 2WD2 11a ¼ WD > @WD We > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i > > 1x x a þ q x l 1 p1 a > ffiffiffiffiffiffiffiffiffi 1 1 >h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii l 1 a aq a WD a 2WD2 > 11 > ðpWD1Þ WD 1 1a > > þ 1 p ffiffiffiffiffiffiffiffiffi 2 : p p WD 1 a ðl 1Þ
@ es @a
¼
h
ð1xÞ3 ð1aÞ3
q
2
x3
1161
1 WD þ1
9 > > > > > > > > > > = > > > > > > > > > > ;
¼0
ðA2Þ
i
a3
8 pffiffiffiffiffiffiffiffiffi 1x q x 1a x ðl 1 > aÞ 1 WD þq a pWD1 a 1 > p ffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi þ > aþ1 a > 2pWD ðl 1Þ 11 11 > WD WD > > 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 3 > > 1a l ð1xÞ x > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i þ q 1 > 2 2 > WD ð1aÞ a þ7 < aþ1 6 ðl 1Þ 1 1 4 1x 5 WD 4 x 1 þ We l 1a q a 2WDpffiffiffiffiffiffiffiffiffi a > 11 > WD > > h i > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > l 1 pffiffiffiffiffiffiffiffiffi 1x x 1a x 1 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii l 1a q a 1 WD þ q a 2WD 11a h > > WD > 1 þ ðpWD1Þ 1 1a > 2 pffiffiffiffiffiffiffiffiffi : p p WD 1a ðl 1Þ
1 WD þ1
9 > > > > > > > > > > > > > = > > > > > > > > > > > > > ;
¼0
ðA3Þ
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References [1] M. Ishii, T. Hibiki, Thermo-Fluid-Dynamics of Two-Phase Flow, second ed., Springer Science + Business Media, New York, 2011. [2] A. Faghiri, Y. Zhang, Transport Phenomena in Multiphase Systems, first ed., Elsevier Inc., San Diego, 2006. [3] G.B. Hewitt, Multiphase fluid flow and pressure drop, Heat Exchanger Design Handbook, Begell House, New York, 1998. [4] J.R. Thome, Engineering Data Book III, Wolverine Tube Inc., Hountsville, 2004. [5] S.M. Zivi, Estimation of steady-state steam void-fraction by means of the principle of minimum entropy production, ASME J. Heat Trans. 86 (1964) 247– 251. [6] R.C. Martinelli, D.B. Nelson, Prediction of pressure drop during forced circulation boiling of water, ASME Trans. 70 (6) (1948) 695–702. [7] Y. Taitel, A.E. Dukler, A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Tubes Ph.D. Dissertation, Kansas State University, Manhattan, 1976. [8] D. Barnea, Y. Luninski, Y. Taitel, Flow pattern in horizontal and vertical two phase flow in small diameter pipes, Can. J. Chem. Eng. 61 (1983) 617–620. [9] J.R. Thome, A. Bar-Cohen, R. Revellin, I. Zun, Unified mechanistic multiscale mapping of two-phase flow patterns in microchannels, Exp. Therm. Fluid Sci. 44 (2013) 1–22. [10] M. Suo, P. Griffith, Two-phase flow in capillary tubes, J. Basic Eng. 86 (3) (1964) 576–582. [11] K.A. Triplett, S.M. Ghiaasiaan, S.I. Abdel-Khalik, D.L. Sadowski, Gas–liquid twophase flow in microchannels Part I: two-phase flow patterns, Int. J. Multiphase Flow 25 (3) (1999) 377–394. [12] R. Revellin, V. Dupont, T. Ursenbacher, J.R. Thome, I. Zun, Characterization of diabatic two-phase flows in microchannels: flow parameter results for R-134a in a 0.5 mm channel, Int. J. Multiphase Flow 32 (7) (2006) 755–774. [13] M.K. Akbar, D.A. Plummer, S.M. Ghiaasiaan, On gas–liquid two-phase flow regimes in microchannels, Int. J. Multiphase Flow 29 (5) (2003) 855–865. [14] A. Agrawal, S.G. Singh, A review of state-of-the-art on flow boiling of water in microchannel, Int. J. Microscale Nanoscale Therm. Fluid Trans. Phenomena 2 (1) (2011) 1–34. [15] V.T. Milashenko, B.I. Nigmatulin, V.V. Petukho, N.T. Tribunkin, Burnout and distribution of liquid in evaporative channels of various lengths, Int. J. Multiphase Flow 15 (1989) 393–402. [16] T. Ueda, M. Inoue, N. Nagatome, Critical heat flux and droplet entrainment rate in boiling of falling liquid films, Int. J. Heat Mass Trans. 24 (7) (1981) 1257– 1266. [17] J.H. Chun, W.J. Lee, C. Park, U.C. Lee, Development of the critical film thickness correlation for an advanced annular film mechanistic dryout model applicable to MARS code, Nucl. Eng. Des. 223 (3) (2003) 315–328. [18] M. El-Shanawany, A.A. El-Shirbini, W. Murgatroyd, A model for predicting the dry-out position for annular flow in a uniformly heated vertical tube, Int. J. Heat Mass Trans. 21 (5) (1978) 529–536. [19] D.M. Khovalyg, S.H. Predrag, A.M. Jacobi, Thermodynamic analysis of the transition between slug and annular flow in minichannels V001T04A065, Proceedings of the 13th International Conference on Nanochannels,
[20] [21]
[22] [23] [24]
[25]
[26]
[27] [28]
[29]
[30]
[31]
[32]
[33] [34]
[35] [36]
Microchannels, and Minichannels, The Fairmont San Francisco Hotel, California, 2015. J. Mikielewicz, J.R. Moszynski, Minimum thickness of a liquid film flowing vertically down a solid surface, Int. J. Heat Mass Trans. 19 (1976) 771–776. N. Giannetti, S. Yamaguchi, K. Saito, Wetting behaviour of a liquid film on an internally-cooled desiccant contactor, Int. J. Heat Mass Trans. 101 (2016) 958– 969. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, second ed., Interscience Publishers, 1961. E.T. Jaynes, The minimum entropy production principle, Ann. Rev. Chem. 31 (1980) 579–601. R. Bujakiewicz-Koronska, J. Koronski, On the principle of minimum production entropy in open systems, in: Proceedings of the XIII National Conference Application of Mathematics in Biology and Medicine, 18–22 September 2007, Institute of Mathematics Jagiellonian University, Cracow, 2007, pp. 25–30. A. Younes, L. Kadem, I. Hassan, Prediction of heat transfer coefficient in annular flow regime for flow boiling in a horizontal micro tube at a uniform heat flux, in: Proceedings of the 2nd International Conference on Fluid Flow, Heat and Mass Transfer Ottawa, Ontario, Canada, April 30–May 1, # 135, 2015. Y. Sun, C. Beckermann, Diffuse interface modelling of two-phase flows based on averaging: mass and momentum equations, Phys. D: Nonlinear Phenomena 198 (3–4) (2004) 281–308. T.S. Ng, C.J. Lawrence, G.F. Hewitt, Gravity-driven laminar flow in a partiallyfilled pipe, Chem. Eng. Res. Des. 79 (4) (2001) 499–511. M.M. Awad, Y.S. Muzychka, Bounds on two-phase flow: Part II Void fraction in circular tubes, in: Proceedings of the International Mechanical Engineering Congress and Exposition, IMECE200581543, Orlando, Florida, 2005. L. Cheng, G. Ribatski, J.R. Thome, Two-phase flow patterns and flow-pattern maps: fundamentals and applications, ASME. Appl. Mech. Rev. 61 (5) (2008) 050802–050802-28. M. Magnini, J.R. Thome, A CFD study of the parameters influencing heat transfer in microchannel slug flow boiling, Int. J. Therm. Sci. 110 (2016) 119– 136. D. Steiner, Heat transfer to boiling saturated liquids. VDI-Wa¯rmeatlas (VDI Heat Atlas), Verein Deutscher Ingenieure, VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen (GCV), Düsseldorf, Chapter Hbb, 1993. N. Kattan, J.R. Thome, D. Favrat, Flow boiling in horizontal tubes: Part I development of adiabatic two-phase flow pattern map, ASME J. Heat Trans. 120 (1998) 140–147. J.M. Mandhane, G.A. Gregory, K. Aziz, A flow pattern map for gas-liquid flow in horizontal pipes, Int. J. Multiphase Flow 1 (4) (1974) 537–553. C. Ong, J.R. Thome, Macro-to-microchannel transition in two-phase flow: Part 1 – Two-phase flow patterns and film thickness measurements, Exp. Therm. Fluid Sci. 35 (2011) 37–47. R. Revellin, J.R. Thome, A new type of diabatic flow pattern map for boiling heat transfer in microchannels, J. Micromech. Microeng. 17 (2007) 788–796. R. Revellin, J.R. Thome, Experimental investigation of R-134a and R-245fa twoPhase flows in microchannels for different flow conditions, Int. J. Heat Fluid. Fl. 28 (1) (2007) 63–71.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Annular flow stability within small-sized channels Niccolò Giannetti ⇑, Daisuke Kunita, Seiichi Yamaguchi, Kiyoshi Saito Department of Applied Mechanics and Aerospace Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
a r t i c l e
i n f o
Article history: Received 19 May 2017 Received in revised form 11 September 2017 Accepted 25 September 2017 Available online 17 October 2017
a b s t r a c t An analytical study based on a variational thermodynamic principle is presented to evaluate the influence of surface tension on the stability of annular flow within small-sized channels. The model introduces phenomenological assumptions in the interfacial structure of the flow regime and theoretically draws the equilibrium transition line from an annular regime to the initiation of the partial wetting condition on the inner surface. By including surface tension, this model expands previous theories and identifies the stable flow configuration in terms of void fraction and interfacial extension. The significant influence of a higher surface tension and smaller diameter (i.e. lower Weber number) are responsible for a lower stable void fraction and higher slip ratio. A complete screening of the main influential parameters is conducted to explore the descriptive ability of the model. This analysis aims at contributing to the understanding of the stability of two-phase flow regimes and can be extended to the transition between other neighbouring regimes, including wall friction as well as liquid entrainment phenomena. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Multiphase flows are used as transport media in a variety of technical systems, such as nuclear reactors, power plant boilers, internal combustion engines, heat transfer and storage devices, cooling equipment for electronic hardware, and food processing systems. Given the obvious differences in these examples with regard to their scale, transport phenomena, and operating conditions, these systems have been generally researched individually via the development of models and heuristic correlations having restricted applicability [1]. The co-presence of multiple phases substantially affects the operability of a system in terms of heat transfer performance, pressure drops, and flow stability, which, respectively, alter the capacity of the system, the power required to steadily circulate the fluid, and its reliability [2]. Even though phase change processes have been observed and technically used for centuries, owing to a flawed understanding of the concepts of heat, energy, and temperature, the theoretical background needed to model these phenomena was not consistently established until the 17th century. The presence of dynamic and deformable phase-interfaces and the related discontinuities of fluid properties are the main reasons for the high degree of complexity in writing the governing transport laws and physical principles. However, in spite of this difficulty, the theory of multiphase systems relies on the classical laws of thermodynamics, fluid mechanics, and principles of heat and mass transfer. Liquid–vapour flow is the ⇑ Corresponding author. E-mail address: [email protected] (N. Giannetti). https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.098 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
most recurrent as well as complex flow regime class. Owing to the presence of deformable interfaces, different neighbouring flow patterns can be encountered in the same system [3,4]. Numerous theoretical, numerical, and empirical approaches have been developed to predict the characteristics of liquid–vapour flows [5,6]. The most widely accepted map describing the adiabatic transition between different flow patterns was semi-theoretically obtained by Taitel and Dukler [7]. The demand for high performance and reliable electronic devices and miniaturized heat exchangers for refrigeration plants requires heat transfer devices to be able to extract the given heat load within a limited temperature range and certain overall dimensions. The structural characteristics of the interface in two-phase flows constitute the criteria by which they are classified into flow regimes. However, the concept of classifying flow regimes, which assumes a certain degree of similarity between corresponding flow structures, is based on the definition of a given volume or lengthscale. Therefore, regime dependent models may yield results contingent to the selected length-scale. Furthermore, multi-phase transport phenomena occurring at different scales may vary significantly [2]. Most of the studies related to liquid–vapour flow regime transitions have been carried out for conventionally sized passages, in which surface tension effects can typically be neglected. However, inside mini- or micro-channels, surface tension effects can significantly affect the way in which heat, mass, and momentum are transferred [8]. Therefore, it has not yet been possible to theoretically predict the characteristics of two-phase flows within small-sized channels with the accuracy needed for reliable system design and control.
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Nomenclature etot E m S
r u A0
q b x
a d r hd G
energy rate per unit cross-sectional area (W m2) energy rate (W) mass flowrate (kg s1) phase interface (m2) surface tension (J m2) fluid velocity (m s1) total cross-sectional area of the channel (m2) density (kg m3) interfacial area concentration (m1) vapour quality (–) steam void-fraction (–) thickness of the liquid flow (m) inner radius of the channel (m) dry angular-portion of the channel (rad) total mass flowrate per unit cross-sectional area (kg m2 s1)
In a previous work [9], evident differences were reported between various empirically proposed regime maps for microchannels, starting with the pioneering work of Suo and Griffith [10] on the transition boundaries in heptane and water flow, through the graphical map of Triplett at al. [11] for an air–water mixture in a 1.1 mm channel, to the experimental work of Revellin et al. [12] on the flow transition of the refrigerant R134a. The significant differences in the thermo-physical properties of the refrigerants were considered responsible for the large deviations between the results obtained by the different authors. This suggested a need for designing generalized maps, based on first principles instead of utilizing one-of-a-kind flow pattern maps, which would take into account the wide range of operating conditions and be applicable to fluids commonly used in the refrigeration field. Akbar et al. [13] used the Weber number to summarize the data for an air–water flow on a common flow regime map. For a comprehensive literature review, please refer to Ref. [14]. The present research effort applies a variational thermodynamic approach to idealized two-phase flow structures to investigate the effect of surface tension on their stability and suggests a phenomenological transition boundary. Particularly, the focus of the study is directed towards an annular flow regime and breaking of the liquid film leading to partial or complete dryout of a channel. 2. Fundamental equations Regarding initiation of the dryout of a channel, previous theoretical analyses mostly assume that it occurs when the thickness of the liquid film in an annular flow becomes zero. However, earlier experimental works [15,16] and modelling efforts [17,18] have shown that the dryout, i.e., the transition to partial wetting conditions, occurs when the liquid film is relatively thin, but not zero, and the corresponding thickness is referred to as critical film thickness. To develop mechanistic criteria to predict the characteristics and stability of multiphase flow structures, Zivi [5] applied the principle of minimum entropy production and, recently, in Ref. [19], the thermodynamic equilibrium conditions of their Helmholtz potentials have been explored. A corresponding approach has been implemented in Refs. [20,21] to establish the minimum stable thickness and characteristics of a falling liquid film. The principle of minimum entropy production is an approximate restatement of the second law of thermodynamics for characterising the steady state of open systems kept away from the thermodynamic equilibrium,
Subscripts g vapour/gas-phase gl vapour–liquid interface i internal energy k kinetic energy s Solid-phase sg vapour–solid interface sl solid–liquid interface tot total r surface tension energy Superscripts ⁄ dimensionless form 0 reference value
when all the fluxes that cross the system are constant, and under the assumption that the rate of entropy production is governed by linear phenomenological laws [22–24]. Following the approach of Zivi [5], the present paper formulates the energy flux of two different flow structures in terms of void fraction and liquid-vapour interface extension to search for the values of these parameters which minimise the entropy production rate of the process under analysis. Particularly, the thickness of the liquid film and extension of the liquid–vapour interface is critical information for predicting the heat and mass transfer rates; please refer, for instance, to the model developed by Younes et al. [25]. By doing so, it is tacitly assumed that the energy of the flow is gained within the channel, and that the work required for accelerating and extending the interface between the two phases is the largest part of the total flow work. Under this point of view, the principle of minimum entropy production indicates that the stable flow configuration is achieved through a process that requires the minimum work. Then, comparing the energy rate of the two flow structures at corresponding conditions, the theoretical transition line between them is drawn and analysed with respect to the main parameters at play. This procedure further implies that the transition between the two flow patterns is triggered by disturbance that qualifies as a zero-energy interaction. Previous results applied to hypothetical two-phase structures [5] have shown a remarkable agreement with time averaged values of the main significant parameters of steady two-phase flows. 2.1. Evaluation of the flow energy contents The open thermodynamic system under consideration is a channel with finite length, defined radius and a smooth inner surface, and emerging into a receiver, where liquid and vapour are removed separately, at a constant rate corresponding to the rate of inlet supply, but with very low effluent velocities. In this manner, the kinetic and surface tension energy fluxes are almost entirely dissipated at the outlet. As pointed out by Zivi [5] this circumstance is commonly matching technical applications. The energy rate per unit cross-sectional area of the channel can be quantified by Eq. (1), where Ek, EG, Ei and Er represent the kinetic, potential, internal, and surface tension contributions, respectively; A and S are the cross-sectional area of the channel and the contour between neighbouring phases, respectively; and r is the surface tension.
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N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162 Table 1 Dimensionless variables. S0 ¼ 2pr
q0 ¼ ql
u0 ¼ qG
A0 ¼ pr 2
r0 ¼ rgl
Dh ¼ 4AS0 L
l 0 0
qk ¼ qqk0
Ak ¼ AAk0
yG;k ¼
Si;j ¼
ri;j ¼ rri;j0
U k ¼ UUk0
1_ 1 _ Etot ¼ Ek þ E_ r þ E_ i þ E_ G A A 1 1 1 _ g u2g þ m _ l u2l þ Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ m _ g Ug m ¼ A 2 2 _ l Ul þ m _ g gyG;g þ m _ l gyG;l þm
etot ¼
ð1Þ Further developing the expression shown above (Eq. (1)), and introducing the representative (identified by the apex 0 ) and dimensionless variables (identified by the superscript ⁄) listed in Table 1, Eq. (2) can be obtained when the energy rate per unit area is divided by the inertial term q0 u0 3.
etot
q0 u03
1 3 3 ¼ ðqg Ag ug þ ql Al ul Þ 2 S0 u 0 r 0 þ 0 0 0 02 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl qAuu gy0 U0 þ 02 qg Ag ug U g þ ql Al ul U l þ 02G qg Ag ug yG;g þ ql Al ul yG;l u u ð2Þ
The extracted dimensionless numbers can be defined as follows:
We ¼
02
0
q Dh u r0
ð4Þ
U0 u02
ð5Þ
Eq. (6) highlights the impact of the main physical effects at play on the dimensionless overall energy rate per unit cross-sectional area of the channel.
e¼ ¼
etot
q0 u03 1
3 g Ag ug
l Al ul
q
l ¼ llk0
etot
q0 u03
1 3 3 qg Ag ug þ ql Al ul 2 4 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ We o 1 n 3 3 qg aug þ ql ð1 aÞul ¼ 2 4 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ We
¼
ð7Þ
The void fraction a and vapour quality x are defined by Eqs. (8) and (9), respectively. Additionally, Al⁄ = Al/A0 = 1 a, Sgl⁄ = bD/4, qg⁄ = qg/ql, and ql⁄ = 1.
Ag ¼
Ag ¼a A0
ð8Þ
_g m _ m
ð9Þ
A steady liquid–vapour flow with a vapour quality x and a defined mass velocity G within the small-sized channel is considered to be at thermal saturation for a pressure p. Furthermore, it is assumed that the velocity fields of the vapour and liquid can be described by unique constant average values ul and ug, and that the surface tension components rsl, rgl, and rsg are constant along the three interfaces. Introducing the definition of the vapour void fraction as a = Ag/A, the liquid and vapour velocities are calculated via Eqs. (10) and (11), respectively.
ul ¼
3
þq 4 Sls uls rls þ Ssg usg rsg þ Sgl ugl rgl þ We þ H qg Ag ug U g þ ql Al ul U l 1 þ 2 qg Ag ug yG;g þ ql Al ul yG;l Fr 2
e¼
x¼
u Fr ¼ pffiffiffiffiffiffiffiffi gDh
uk ¼ uuk0
surface tension over gravitational forces when the flow is confined within small-sized channels. The liquid entrained into the vapour and the energy dissipation owing to the wall friction are preliminarily neglected. Accordingly, the energy rate per unit crosssectional area of the channel (e) can be quantified using these terms as the sum of the rates of kinetic and surface tension energy of the flow (Eq. (7)).
ð3Þ
0
H¼
Si;j S0
yG;k y0G
ug ¼
_ m
ql Al
¼
_g m
q g Ag
Gð1 xÞ
ql ð1 aÞ
¼
Gx
qg a
ð10Þ
ð11Þ
Their equivalent dimensionless forms are as follows:
ð6Þ
If an adiabatic horizontal channel is studied at low-liquid Weber numbers (high-surface tension fluids or small-sized passages), it is assumed that the internal energy and gravitational potential energy have a marginal effect on the achievement of a stable flow configuration. In contrast, the energies acquired through the work needed to accelerate the two phases to their ultimate velocities, as well as to increase the interfacial area between the two phases to the value of a steady configuration against the interfacial tension, are considered the two main contributions to the total flow energy. Optical measurements conducted by Revellin et al. [12] demonstrated that the relatively obvious symmetry of the flow configuration is a confirmation of the dominant effect of
ug ¼
ug qx ¼ l u0 qg a
ð12Þ
ul ¼
ul 1 x ¼ u0 1 a
ð13Þ
Remarkably, even though in most of the applications of conventional-sized channels a turbulent liquid flow is observed, the flow inside small-sized channels often falls within the laminar regime. On the other hand, the irregularities in the channel surface with size at the same length-scale of the liquid flow characteristics may be responsible for the dissimilar and unpredictable flow regime transitions [9]. Assuming that the shear stress balance at the liquid–vapour interface is as given in Ref. [26], the interfacial velocity ui,j (Eq.
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N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
(14)) is a viscosity-weighted linear function of the average velocities in the radial direction:
ui;j ¼
li /j ui þ lj /i uj li /j þ lj /i
ð14Þ
The no-slip condition at the interface of a rigid body can be obtained as the limit for li >> lj and is applied to solid–liquid and vapour–solid boundaries. Therefore, uls = usg = 0, and Eq. (15) is obtained.
o 1n 4 3 3 S u r e ¼ qg aug þ ql ð1 aÞul þ 2 We gl gl gl
ð15Þ
2.2. Annular flow A purely annular flow (with no liquid entrained in the vapour) in a channel of small diameter D on steady, thermal saturation is represented in Fig. 1. In general, the steam void-fraction a and the interfacial area concentration b constitute the main flow parameters that define the extension of the liquid–vapour interfaces. These can be theoretically interrelated once the flow geometry, vapour quality, and mass flux are defined. For instance, the geometry represented in Fig. 1 could be used for defining the steam void fraction (Eq. (8)) to make explicit the direct relationship between a and Sgl (Eqs. (16) and (17)).
d ¼ rð1
pffiffiffi aÞ
ð16Þ pffiffiffi
Sgl ¼ 2pðr dÞ ¼ 2pr a
pffiffiffi Sgl;a pD a pffiffiffi ¼ 0 ¼ ¼ a pD S
ð19Þ
pffiffiffi ll 3=2 ql ugl;a lg a ð1 xÞ þ qg ð1 aÞ 1 a x h ¼ pffiffiffi pffiffiffii u0 að1 aÞ ll a þ 1 a
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1a ; d¼r 1 1 WD
ð21Þ
1 a 6 WD 6 1
ð22Þ
For this flow geometry, the liquid–gas interface Sgl in its dimensional and dimensionless forms is expressed in Eqs. (23) and (24) as follows:
Sgl ¼ ð2r 2dÞðp hd Þ þ 2d
¼
Sgl;s ¼ S0 1
p
þ
ð23Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii h a D 1 þ ðpWD 1Þ 1 1 WD
ðpWD 1Þ
p
pD rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a 1 WD
ð24Þ
The interfacial velocity between the liquid and gaseous phases and its dimensionless form are expressed, respectively in Eqs. (25) and (26).
ugl ¼
ð20Þ
lg
2rðp hd Þ p hd ¼ 2r p p
Careful attention must be directed to the well-established knowledge of the importance of the value of the contact angle between solid and liquid phases on the interfacial geometry and, consequently on the value of the final results. On the other hand, due to the chaotic behaviour of the liquid interface, the precise measurement of the flow configuration shape becomes nearly unachievable. Being the contact angle dependent on the specific solid-liquid pair as well as on the specific operative conditions a more detailed geometry is disregarded for this first general analysis as it would not affect the method of analysis, while making the mathematical problem formulation more complex [27]. From the definition of the vapour void fraction,
ð18Þ
ll ðr dÞul þ lg dug ll ðr dÞ þ lg d
ugl;a ¼
WD ¼
Sgl;s ¼
Additionally, the interfacial velocity between the liquid and gaseous phases and its dimensionless form are defined, respectively, in Eqs. (19) and (20).
ugl ¼
A simplified geometry of an alternative flow configuration with a partially wetted perimeter is shown in Fig. 2, and the fundamental parameter to describe the wet part of the solid wall WD is given by Eq. (21).
ð17Þ
Therefore, Eq. (18) can be expressed as follows:
Sgl;a
2.3. Partial wetting
ugl;s
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aþl u 1 1 g g WD qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1a ðll lg Þ 1 WD þ lg
ll ul lg ug
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ll 1x ql x a þ ql 1 1 ugl;s lg 1a qg a WD qg ¼ 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ll u aþ1 1 1 l 1 WD
ð25Þ
x
a
ð26Þ
3 mm
g
Fig. 1. Annular flow in a 3 mm channel (a) and its simplified geometry (b).
Fig. 2. Partial wetting flow of ammonia within a 3 mm channel (a) and its simplified geometry (b).
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Energy rate per unit cross secon area
a
7 6.3 0.7
5.6 4.9
0.65
4.2
0.6
3.5 2.8
0.55
2.1
0.5 0.45
1.4
X 0.4
0.7 0.9
0.92
0.94
0.96
0.98
1
Void fracon Fig. 3. Energy rate per unit cross-sectional area of an annular flow as a function of the vapour void-fraction a for different inlet vapour qualities x; this work (continuous lines), results from Ref. [5] (dashed lines); We = 0.01, D = 5 mm, q* = 1655, and l* = 42.84.
6 *
5 * * *
10 2 *
*
1 1/2 * 1/3 * 1/4 * 1/5 *
0.1 0.7
a
Given certain flowrate and quality, in addition to the previously defined assumptions for simplifying the geometry of the flow, the thermodynamic approach of the principle of minimum entropy production can be used as the basis to establish stable characteristics of the annular configuration and the commencement of the dryout of the liquid–solid interface at a critical value of the flow quality. Since the outlet void fraction and the liquid-vapour interfacial extension are constrained only by their definition range this variational principle implies that void fraction and liquid-vapour interface will be such as to minimise the energy flux into the receiver. Successively, these simplified flow structures are compared to define the transition line between them. Figs. 3 and 4a and b illustrate the behaviour of the dimensionless energy rate per unit cross-sectional area as a function of the vapour void fraction when the inlet vapour quality x, density ratio q⁄ of the liquid and gaseous phases (Fig. 4a), and viscosity ratio l⁄ (Fig. 4b) are varied as parameters. In general, a local minimum is always observed and a higher value of these parameters yields higher energy rates per unit cross-sectional area. In addition, higher values of q⁄ move the minimum to higher void-fractions, whereas an opposite trend is noted for higher values of l⁄. According to the mathematical formulation of the problem (Eq. (15)), an increase in the relative impact of the kinetic energy contribution of the gaseous phase will shift the minimum to a higher void fraction. Therefore, at an equivalent mass flux, a lower vapour quality (Fig. 3), similar to a lower density ratio q⁄ (Fig. 4a), leads to a lower slip ratio and lower void fraction corresponding to the minimum of the energy rate per unit cross-sectional area of the channel. In contrast, a lower ratio of viscosity l⁄ of the two phases yields a higher interfacial velocity ugl⁄ and for a given value of the Weber number We, moves the minimum to higher void fractions (Fig. 4b). Reference values of these parameters, namely, q⁄ = 1655 and l⁄ = 42.84, are obtained from the thermo-physical properties of saturated water at atmospheric pressure. Lower values represent, for instance, the circumstance of a higher operative pressure, and both q⁄ and l⁄ approaching unity at the characteristic critical pressure. The dashed lines in Fig. 3 are the results from the 1964 model of Zivi [5], which targets channels of a conventional size and neglects the contribution of surface tension. The latter condition shifts the minimum to higher values of the stable void fraction and is unaffected by the viscosity ratio of the liquid–vapour phase.
100
3
Energy rate per unit cross secon area
3.1. Flow stability
Energy rate per unit cross secon area
a
3. Results
2.8
0.75
0.8
0.85
0.9
0.95
1
Void fracon
(a)
1/20 *
2.6
2.4
1/16 *
2.2
1/12 *
2
1/8 *
1.8
1/4 *
1.6 *
1.4 1.2 1 0.8
0.85
0.9
Void fracon
0.95
1
(b)
Fig. 4. Effect of (a) density ratio and (b) viscosity ratio of the phases on the energy rate per unit cross-sectional area of an annular flow as a function of the vapour void-fraction a at atmospheric pressure for reference values of the thermo-physical properties of water (q* = 1655 and l*= 42.84); We = 0.01, D = 5 mm, x = 0.4.
If the inlet Weber number increases (ceteris paribus, corresponding to a larger diameter of the channel, lower surface tension or a higher refrigerant flowrate), the energy rate per unit crosssectional area decreases and the minimum moves to higher values of the void-fraction (Fig. 5). The results from Zivi [5] agree for a particular case of this model, which as a limit for an infinite Weber number. Under this thermodynamic approach, the steady vapour voidfraction of an ideal annular flow (Eq. (29)) will assume a value that minimises the energy per unit cross-section area of the channel (Eqs. (27) and (28)). Analytical expressions of the derivatives are reported in the Appendix A. The stable vapour void fraction of an idealized annular flow a0 is plotted in Fig. 6a for different inlet qualities. From a0, the corresponding stable slip ratio R0 (Fig. 6b) can be calculated by using the liquid and vapour average velocities from Eqs. (10) and (11). As the Weber number (flowrate or channel diameter, once the refrigerant has been chosen and a reference condition selected) is varied, it can be observed that the stable void fraction of an idealized annular flow within a small diameter channels is bracketed between a maximum value and a minimum value, representing, respectively, the predominance of inertial and surface tension effects. This result is consistent with experimental results from previous literature [28], and is recognized to have an effect even on the flow in conventional sized channels. Additionally, the same observation can be attributed to a stable slip-ratio of the flow, so that a higher vapour quality is related to a higher maximum value of R0, while the minimum is not altered. In practice, for a given thermodynamic condition and a certain fluid, the Weber number
N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
100
Energy rate per unit cross secon area
a
1158
We 4*10-6
10-5
3*10-5
10
7*10-5 2*10-4 5*10-4
0.0012
1 0.8
0.85
0.9
0.95
1
Void fracon Fig. 5. Energy rate per unit cross-section area of an annular flow (logarithmic vertical axis) as a function of the vapour void-fraction a for different values of the Weber number We; x = 0.4, D = 5 mm, reference values of the thermo-physical properties of water are q* = 1655 and l* = 42.84.
0.95 0
Stable void fracon
0.7
0.85
WD
0.6
ð30Þ
0.5
0.8
0.4
0.75
(a)
14
12 10
0.7
0.6
the following can be obtained,
6
2 1.E-05
1.E-03
1.E-01
Weber number We
1.E+01
1
(b)
p 1Atm
2
@WD
> 0 and
@ 2 es >0 @ a2
ð32Þ
is proportional to the mass flux G and the annular flow that cannot be observed at excessively low flowrates.
(
)
4 pffiffiffi a We pffiffiffi l a3=2 ð1 xÞ þ q ð1 aÞ 1 a x pffiffiffi p ffiffiffi að1 aÞ l a þ 1 a 3
3 1 ð1 xÞ 2 x q 2 þ 2 a ð1 aÞ2
@ 2 ea P 0; @ a2
This finally yields,
þ
10Atm
0
0.8
Void fracon
Fig. 6. Stable values of the (a) vapour void-fraction a0 and (b) slip ratio R0 for different values of the inlet flow quality x; D = 5 mm, reference values of the thermo-physical properties of water are q* = 1655 and l* = 42.84.
ð33Þ
3Atm
0.9
@ ea ¼ 0; @a
@ 2 es
> 0;
qg lg ; ; x; Sgl ðb; DÞ; We ql ll
qg lg ; ; x; Sgl ðb; DÞ; We WD0 ¼ u2 q l ll
4
ea ¼
ð31Þ
a0 ¼ u1
0.4
0 1.E-07
!2
@ 2 es @ 2 es @ 2 es 2 @ a2 @WD@ a @WD
0.5
8
8 > @ es @ 2 es > > ¼ 0; P 0; 06a61 < @a @ a2 2 > @ es @ es > > P 0; WDmin ðaÞ 6 WD 6 1 ¼ 0; : @WD @WD2 Under the condition that,
X 0.8
16
Stable slip rao
Fig. 7 shows a first comparison of the results of the present work (continuous lines) to those obtained from the 1964 model (dashed lines). The saturation pressure is varied within a channel of 0.5 mm diameter enclosing a two-phase water flow. The corresponding lines are represented by colours to identify the operative pressures. As a rule, the inclusion of the contributions related to the interface extension, interfacial velocity, and surface tension (right-hand side of Eq. (15)) yield globally lower void fractions; and the difference between the corresponding results increases as the saturation pressure increases and approaches the critical value. Using the same approach, a stable condition for the configuration characterized by partial wetting of the solid surface is sought for by minimizing the energy content defined in Eq. (30) with respect to the void-fraction as and the parameter defining the wet part of the channel surface WD (Eq. (31)). This is to select a single value among the set of possible values of the latter variable to determine the stable geometry of this configuration.
ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffi " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 1x x 1a x 4 1 ðpWD 1Þ 1 a l 1a q a 1 WD þ q a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ þ We p p WD ðl 1Þ 1 1a þ 1
X 0.8
0.9
18
ð29Þ
( ) 3 1 ð1 xÞ3 2 x es ¼ q 2 þ 2 a ð1 aÞ2
1
0.7 20
qg lg ; ; x; Sgl ðb; DÞ; We ql ll
a0 ¼ u
0.7 0.6 0.5 0.4 0.3
Zivi S.M. 1964 0.2
This work
0.1
ð27Þ
0 0
0.2
0.4
0.6
0.8
1
Vapour quality X 06a61
ð28Þ
Fig. 7. Comparison with a previous work of Zivi reported in Ref. [5]; stable vapour void fraction a0 of a saturated water flow as a function of the inlet vapour quality x for different operative pressures; D = 0.5 mm and G = 200 kg m2 s1.
N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
1159
Fig. 8 presents an example of the energy rate per unit crosssectional area of the channel es as a function of the independent variables a and WD for two reference values of the inlet vapour quality. A higher quality x is associated with higher energy rates and a higher stable void fraction a0, similar to the annular flow and lower wettability WD. When the Weber number is decreased (Fig. 9), the minimum of the energy rate per unit cross-sectional area of the channel moves to lower stable void fractions a0 and higher wettability WD0. 3.2. Flow pattern transition In addition to inertial and viscous properties (l, q), identification of the flow pattern of a two-phase flow has been recognized to depend on the fluid interfacial tension forces (r), wetting characteristics, pipe geometry (D), relative velocities, and interfacial structure (ba, bs) [29,30]. Given the large number of relevant parameters, validity of most of the empirical flow maps is limited to a specific system and to the heuristic method by which they were obtained. This phenomenological model is based on the assumption of the interfacial structure and the balance of the shear stress at the interface to obtain a more comprehensive picture of the flow while minimizing the dependence on the heuristic data. By equating the steady values of the energy of the two configurations, a critical value of the flow quality below which the annular flow becomes energetically disadvantageous with respect to the
(a)
(b) Fig. 9. Energy rate per unit cross-sectional area as a function of void fraction a and WD; D = 5 mm, x = 0.4, and reference values of the thermo-physical properties of water (q* = 1655, and l* = 42.84) at (a) We = 5 * 108 (b) We = 0.0012.
corresponding steady geometry of the partially wetted flow configuration, is calculated (Eq. (35)).
ea ðaa;0 Þ ¼ es ðaa;0 ; WD0 Þ
(a)
(b) Fig. 8. Energy rate per unit cross-sectional area as a function of void-fraction as and WD; D = 5 mm, We = 0.01, reference values of the thermo-physical properties of water (q* = 1655 and l* = 42.84) (a) x = 0.4 (b) x = 0.5.
ð34Þ
Finally, we obtain,
x0 ¼ u3 q ; l ; Sgl;a ðba ; DÞ; Sgl;s ðbs ; DÞ; We
ð35Þ
A parametric analysis is performed to explore as a first screening of the model results. Figs. 10 and 11 show the transition lines obtained, respectively, when changing the density ratio q⁄ and the viscosity ratio l⁄ of the liquid and gaseous phases. A higher value of the first parameter is responsible for a delayed annular flow transition at lower Weber numbers while other parameters are fixed. In comparison, the second parameter, l⁄, has an opposite effect. The obtained transition line resembles the empirical flow transition from annular to the stratified flow of adiabatic horizontal channels [31–33]. The transition lines from Refs. [32–36] and those obtained from the present analysis for the corresponding thermo-physical conditions are plotted on a conventionally dimensional graph (Fig. 12) for a qualitative comparison. Graphs (a) and (b) refer to results obtained within small-sized channels, with 0.5 and 1.03 mm diameters, respectively, whereas (c) plots the heuristic flow pattern map of a conventional-sized channel with R134a. The relatively sharp steepness variation of the transition line from the present model
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N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
1
1000
0.9
900 *
Mass flux G [kg·m-2s-1]
Vapourquality X
0.8
*/2
0.7
*/3 */4
0.6
*/5
0.5 0.4 0.3 0.2 0.1
800 700 Annular (A)
600
500 400 Semi-Annular (SA)
300 200
Slug+Semi-Annular (SSA)
100
0 0.01
0 0.1
1
10
0
Weber number We
0.9 0.8 0.7 0.6 1/2 *
0.5
0.4
0.6
0.8
1
(a)
1000 R236fa Tsat 31oC D 1.03mm [34] Present work
900
Mass flux G [kg·m-2s-1]
1
*
0.2
Vapourquality X
Fig. 10. Transition curves as a function of the flow Weber number for different values of the density ratio q*; D = 5 mm, and reference values of the thermophysical properties of water (q* = 1655 and l* = 42.84).
Vapourquality X
R134a Tsat 35oC D 0.5mm [35-36] Present work
2 *
800 700 600
500 400 A
300
SA
200
0.4
100
0.3
0 0
0.2
0.2
0.4
0.6
0.8
1
Vapourquality X
(b)
0.1 700
0 0.1
1
R134a Tsat 10oC D 12 mm [32-33] Present work
10
Weber number We Fig. 11. Transition curves as a function of the flow Weber number for different values of the viscosity ratio l*; D = 5 mm, and reference values of the thermophysical properties of water (l* = 42.84 and q* = 1655).
at low vapour quality is associated with the transition from a range of conditions dominated by inertial effects (high Weber number) to a region mostly subject to surface tension (low Weber number) under the assumption of the given idealized interfacial structures (Figs. 1 and 2). The comparison with experimental transition lines is not meant as a validation of the model, but rather as a reference point of view from which the results of this analysis are observed. Namely, limiting the emphasis to inertial and surface tension effects, adopting phenomenological assumptions for the interfacial structures, and adopting a thermodynamic criterion, it is possible to theoretically draw the transition lines between neighbouring two-phase flow patterns and provide general insights regarding the influence of these effects. The first comparison highlights the importance of surface tension on the transition between neighbouring twophase flow patterns within small-sized channels, and supports the method adopted. Further efforts in this direction for the refinement of the theoretical understanding of these phenomena are required.
600
Mass flux G [kg·m-2s-1]
0.01
500 Intermient
A
400 300 200 Strafied-Wavy
100 Strafied
0 0
0.2
0.4
0.6
Vapourquality X
0.8
1
(c)
Fig. 12. Comparison of the results of the present work with previous empirical studies from Refs. [31–35].
question the accuracy and direct applicability of the results extracted. Nevertheless, this simplified thermodynamic approach was helpful in providing some physical hints regarding the stability of two-phase flow patterns and effect of surface tension within small-sized channels that in turn may be useful for the enhancement of the modelling resolution and mechanistic approaches in this field. Specifically, in addition to the results of this analysis, the following main conclusions can be stated:
5. Conclusions In the analysis presented in this work, a series of major simplifications and physical assumptions was introduced, which might
– By including the effect of surface tension, small-sized passages having a stable flow configuration in terms of the vapour void fraction and extension of the interfaces were targeted. This
N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
–
–
–
–
–
model expanded the previously reported theoretical works and contributed to the background needed for an accurate modelling of two-phase flows within mini- and micro-channels. When a miniaturized adiabatic horizontal channel and a refrigerant with high surface tension were considered, the energy needed to accelerate the two phases to their ultimate velocities, as well as increase the interfacial area between them versus the interfacial tension to reach the geometry of a steady configuration, were the two main contributions to the total energy of the flow. The transition between two neighbouring regimes was evaluated in terms of the specific energy contents, and the principle of minimum entropy production was used to evaluate the characteristics of the flow at this condition. The steam void fraction a and interfacial area concentration b were theoretically quantified for a defined flow geometry, vapour quality, and mass flux. A local minimum of the dimensionless energy rate per unit cross-sectional area was always identified, and higher values of the inlet vapour quality x, density ratio q⁄ between the liquid and gaseous phases, and viscosity ratio l⁄ yielded higher energy rates per unit cross-sectional area. An increase in the relative impact of the liquid phase kinetic energy contribution (lower slip ratio related to a lower vapour quality at equivalent mass flux, or lower density ratio q⁄) caused a shift of the minimum to lower void fractions.
– A lower ratio of viscosity of the two phases l⁄ yielded a higher interfacial velocity ugl⁄ and for a given value of the Weber number We, moved the minimum to higher void fractions. – As the Weber number was varied, the stable void fraction of an idealized annular flow within small diameter channels was bracketed between a maximum and minimum value, representing, respectively, the predominance of inertial and surface tension effects. – The results from Zivi [5] were obtained as a limiting case of this model for an infinite Weber number. – A critical value of the flow quality, below which the annular flow became energetically disadvantageous with respect to the corresponding steady geometry of the partial-wetting flow configuration, was calculated. – A higher value of the density ratio between the liquid and gaseous phases was responsible for delaying the annular flow transition at lower Weber numbers, while the viscosity ratio displayed an opposite effect. – The theoretical transition lines obtained resembled a transition to the stratified flow of an empirical flow pattern map of adiabatic horizontal two-phase flows. Appendix A Given the assumed geometry of the flow configurations, the derivatives of the energy rate per unit cross-sectional area are developed as follows:
" # 3 @ ea ð1 xÞ3 2 x þ ¼ q @a a3 ð1 aÞ3 pffiffiffi i 9 8 pffiffiffi pffiffiffi pffiffiffi h pffiffiffi að1 aÞ l a þ 1 a l ð1 xÞ 32 a þ q x 32 a 1 2p1 ffiffia > > > > > > > 2 3 > > > pffiffiffipffiffiffi > > > > 1 3 3=2 p ffiffi > > a a a þ p ffiffiffi > > 2 2 a > > 3=2 4 5 > > l a ð1 xÞ þ q ð1 a Þ 1 a x > > p ffiffiffi p ffiffiffi > > 1 ffiffi < = p þ 1Þð ð l a a l 1Þ 4 2 a ¼0 p ffiffiffi p ffiffiffi 2 > We > > > að1 aÞ2 l a þ 1 a > > > > > > > > > > > > > > > > > > > > > > : ;
ðA1Þ
8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 1x q x 1a x ðl 1 > aÞ 1 WD þq a a 1a pWD1 p1 a > p ffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi þ 1 þ > 2 aþ1 a WD > 2pWD ðl 1Þ 11 11 > WD WD > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q h i > > a þ 1 ðl 1x q x Þ 1 paffiffiffiffiffiffiffiffiffi ðl 1Þ 1 1 @ es 4 < 1a WD a 2WD2 11a ¼ WD > @WD We > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i > > 1x x a þ q x l 1 p1 a > ffiffiffiffiffiffiffiffiffi 1 1 >h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii l 1 a aq a WD a 2WD2 > 11 > ðpWD1Þ WD 1 1a > > þ 1 p ffiffiffiffiffiffiffiffiffi 2 : p p WD 1 a ðl 1Þ
@ es @a
¼
h
ð1xÞ3 ð1aÞ3
q
2
x3
1161
1 WD þ1
9 > > > > > > > > > > = > > > > > > > > > > ;
¼0
ðA2Þ
i
a3
8 pffiffiffiffiffiffiffiffiffi 1x q x 1a x ðl 1 > aÞ 1 WD þq a pWD1 a 1 > p ffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffi þ > aþ1 a > 2pWD ðl 1Þ 11 11 > WD WD > > 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 3 > > 1a l ð1xÞ x > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i þ q 1 > 2 2 > WD ð1aÞ a þ7 < aþ1 6 ðl 1Þ 1 1 4 1x 5 WD 4 x 1 þ We l 1a q a 2WDpffiffiffiffiffiffiffiffiffi a > 11 > WD > > h i > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > l 1 pffiffiffiffiffiffiffiffiffi 1x x 1a x 1 > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii l 1a q a 1 WD þ q a 2WD 11a h > > WD > 1 þ ðpWD1Þ 1 1a > 2 pffiffiffiffiffiffiffiffiffi : p p WD 1a ðl 1Þ
1 WD þ1
9 > > > > > > > > > > > > > = > > > > > > > > > > > > > ;
¼0
ðA3Þ
1162
N. Giannetti et al. / International Journal of Heat and Mass Transfer 116 (2018) 1153–1162
References [1] M. Ishii, T. Hibiki, Thermo-Fluid-Dynamics of Two-Phase Flow, second ed., Springer Science + Business Media, New York, 2011. [2] A. Faghiri, Y. Zhang, Transport Phenomena in Multiphase Systems, first ed., Elsevier Inc., San Diego, 2006. [3] G.B. Hewitt, Multiphase fluid flow and pressure drop, Heat Exchanger Design Handbook, Begell House, New York, 1998. [4] J.R. Thome, Engineering Data Book III, Wolverine Tube Inc., Hountsville, 2004. [5] S.M. Zivi, Estimation of steady-state steam void-fraction by means of the principle of minimum entropy production, ASME J. Heat Trans. 86 (1964) 247– 251. [6] R.C. Martinelli, D.B. Nelson, Prediction of pressure drop during forced circulation boiling of water, ASME Trans. 70 (6) (1948) 695–702. [7] Y. Taitel, A.E. Dukler, A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Tubes Ph.D. Dissertation, Kansas State University, Manhattan, 1976. [8] D. Barnea, Y. Luninski, Y. Taitel, Flow pattern in horizontal and vertical two phase flow in small diameter pipes, Can. J. Chem. Eng. 61 (1983) 617–620. [9] J.R. Thome, A. Bar-Cohen, R. Revellin, I. Zun, Unified mechanistic multiscale mapping of two-phase flow patterns in microchannels, Exp. Therm. Fluid Sci. 44 (2013) 1–22. [10] M. Suo, P. Griffith, Two-phase flow in capillary tubes, J. Basic Eng. 86 (3) (1964) 576–582. [11] K.A. Triplett, S.M. Ghiaasiaan, S.I. Abdel-Khalik, D.L. Sadowski, Gas–liquid twophase flow in microchannels Part I: two-phase flow patterns, Int. J. Multiphase Flow 25 (3) (1999) 377–394. [12] R. Revellin, V. Dupont, T. Ursenbacher, J.R. Thome, I. Zun, Characterization of diabatic two-phase flows in microchannels: flow parameter results for R-134a in a 0.5 mm channel, Int. J. Multiphase Flow 32 (7) (2006) 755–774. [13] M.K. Akbar, D.A. Plummer, S.M. Ghiaasiaan, On gas–liquid two-phase flow regimes in microchannels, Int. J. Multiphase Flow 29 (5) (2003) 855–865. [14] A. Agrawal, S.G. Singh, A review of state-of-the-art on flow boiling of water in microchannel, Int. J. Microscale Nanoscale Therm. Fluid Trans. Phenomena 2 (1) (2011) 1–34. [15] V.T. Milashenko, B.I. Nigmatulin, V.V. Petukho, N.T. Tribunkin, Burnout and distribution of liquid in evaporative channels of various lengths, Int. J. Multiphase Flow 15 (1989) 393–402. [16] T. Ueda, M. Inoue, N. Nagatome, Critical heat flux and droplet entrainment rate in boiling of falling liquid films, Int. J. Heat Mass Trans. 24 (7) (1981) 1257– 1266. [17] J.H. Chun, W.J. Lee, C. Park, U.C. Lee, Development of the critical film thickness correlation for an advanced annular film mechanistic dryout model applicable to MARS code, Nucl. Eng. Des. 223 (3) (2003) 315–328. [18] M. El-Shanawany, A.A. El-Shirbini, W. Murgatroyd, A model for predicting the dry-out position for annular flow in a uniformly heated vertical tube, Int. J. Heat Mass Trans. 21 (5) (1978) 529–536. [19] D.M. Khovalyg, S.H. Predrag, A.M. Jacobi, Thermodynamic analysis of the transition between slug and annular flow in minichannels V001T04A065, Proceedings of the 13th International Conference on Nanochannels,
[20] [21]
[22] [23] [24]
[25]
[26]
[27] [28]
[29]
[30]
[31]
[32]
[33] [34]
[35] [36]
Microchannels, and Minichannels, The Fairmont San Francisco Hotel, California, 2015. J. Mikielewicz, J.R. Moszynski, Minimum thickness of a liquid film flowing vertically down a solid surface, Int. J. Heat Mass Trans. 19 (1976) 771–776. N. Giannetti, S. Yamaguchi, K. Saito, Wetting behaviour of a liquid film on an internally-cooled desiccant contactor, Int. J. Heat Mass Trans. 101 (2016) 958– 969. I. Prigogine, Introduction to Thermodynamics of Irreversible Processes, second ed., Interscience Publishers, 1961. E.T. Jaynes, The minimum entropy production principle, Ann. Rev. Chem. 31 (1980) 579–601. R. Bujakiewicz-Koronska, J. Koronski, On the principle of minimum production entropy in open systems, in: Proceedings of the XIII National Conference Application of Mathematics in Biology and Medicine, 18–22 September 2007, Institute of Mathematics Jagiellonian University, Cracow, 2007, pp. 25–30. A. Younes, L. Kadem, I. Hassan, Prediction of heat transfer coefficient in annular flow regime for flow boiling in a horizontal micro tube at a uniform heat flux, in: Proceedings of the 2nd International Conference on Fluid Flow, Heat and Mass Transfer Ottawa, Ontario, Canada, April 30–May 1, # 135, 2015. Y. Sun, C. Beckermann, Diffuse interface modelling of two-phase flows based on averaging: mass and momentum equations, Phys. D: Nonlinear Phenomena 198 (3–4) (2004) 281–308. T.S. Ng, C.J. Lawrence, G.F. Hewitt, Gravity-driven laminar flow in a partiallyfilled pipe, Chem. Eng. Res. Des. 79 (4) (2001) 499–511. M.M. Awad, Y.S. Muzychka, Bounds on two-phase flow: Part II Void fraction in circular tubes, in: Proceedings of the International Mechanical Engineering Congress and Exposition, IMECE200581543, Orlando, Florida, 2005. L. Cheng, G. Ribatski, J.R. Thome, Two-phase flow patterns and flow-pattern maps: fundamentals and applications, ASME. Appl. Mech. Rev. 61 (5) (2008) 050802–050802-28. M. Magnini, J.R. Thome, A CFD study of the parameters influencing heat transfer in microchannel slug flow boiling, Int. J. Therm. Sci. 110 (2016) 119– 136. D. Steiner, Heat transfer to boiling saturated liquids. VDI-Wa¯rmeatlas (VDI Heat Atlas), Verein Deutscher Ingenieure, VDI-Gesellschaft Verfahrenstechnik und Chemieingenieurwesen (GCV), Düsseldorf, Chapter Hbb, 1993. N. Kattan, J.R. Thome, D. Favrat, Flow boiling in horizontal tubes: Part I development of adiabatic two-phase flow pattern map, ASME J. Heat Trans. 120 (1998) 140–147. J.M. Mandhane, G.A. Gregory, K. Aziz, A flow pattern map for gas-liquid flow in horizontal pipes, Int. J. Multiphase Flow 1 (4) (1974) 537–553. C. Ong, J.R. Thome, Macro-to-microchannel transition in two-phase flow: Part 1 – Two-phase flow patterns and film thickness measurements, Exp. Therm. Fluid Sci. 35 (2011) 37–47. R. Revellin, J.R. Thome, A new type of diabatic flow pattern map for boiling heat transfer in microchannels, J. Micromech. Microeng. 17 (2007) 788–796. R. Revellin, J.R. Thome, Experimental investigation of R-134a and R-245fa twoPhase flows in microchannels for different flow conditions, Int. J. Heat Fluid. Fl. 28 (1) (2007) 63–71.