A pressure drop model for condensation accounting for non-equilibrium effects

International Journal of Heat and Mass Transfer 126 (2018) 421–430

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A pressure drop model for condensation accounting for non-equilibrium effects Jiange Xiao a, Pega Hrnjak a,b,⇑ a b

University of Illinois at Urbana-Champaign, Department of Science and Engineering, Urbana, IL, USA Creative Thermal Solutions, Urbana, IL, USA

a r t i c l e

i n f o

Article history: Received 20 February 2018 Received in revised form 27 April 2018 Accepted 30 April 2018

Keywords: In-tube condensation Pressure drop model Non-equilibrium

a b s t r a c t A mechanistic pressure drop model is proposed in this paper for condensation in horizontal smooth round tubes in order to account for the non-equilibrium effects. The model makes use of a flow regime map and void fraction correlation as well as a mechanistic heat transfer model that are all developed for condensation of superheated vapor in a vapor-compression system. The model provide seamless transition between single-phase and two-phase regions including the superheated, condensing-superheated, two-phase, condensing-subcooled and subcooled regions. Diabatic flow visualizations are used to analyze the effects on pressure drop from the formation of waves. An enhancement factor to represent the frequency and magnitude of the waves is established using Kelvin-Helmholtz and Rayleigh-Taylor instability. The two-phase pressure drop is modeled based on the single-phase pressure drop correlations, the flow regimes, void fractions as well as the enhancement factor. Data obtained from R134a, R32, R1234ze(E), R1233zd(E) and R245fa with mass fluxes from 100 kg m
2 s
1 to 400 kg m
2 s
1 and heat fluxes from 5 kW m
2 to 15 kW m
2 inside two different tubes of 4 and 6 mm are used to validate the model. Ó 2018 Published by Elsevier Ltd.

1. Introduction When modeling the pressure drop of condensation inside a smooth horizontal round tube, two-phase flow is usually assumed between bulk quality 1 and 0. The frictional pressure drop is calculated using parameters like the fluid properties, friction factors, mass fluxes and the dimensionless numbers. The gravitational pressure drop is zero in a horizontal tube. The deceleration pressure gain is usually calculated in terms of the velocity difference between the inlet and outlet of the control volume. To find the overall pressure drop, the frictional pressure drop and the deceleration pressure gain is added together as if the two contributions can be determined separately and superimposed properly without losing any important mechanisms. Although it is true that the heat flux can bring up the deceleration effect, a more vital mechanism is almost always conveniently ignored because of the thermal equilibrium assumption, which does not ever hold in a real condenser. In a vapor-compression system, the refrigerant is superheated before entering the condenser. As soon as heat exchange starts, temperature gradient becomes necessary. As a result, the first ⇑ Corresponding author at: University of Illinois at Urbana-Champaign, Department of Science and Engineering, Urbana, IL, USA. E-mail addresses: [email protected] (J. Xiao), [email protected] (P. Hrnjak). https://doi.org/10.1016/j.ijheatmasstransfer.2018.04.158 0017-9310/Ó 2018 Published by Elsevier Ltd.

droplet forms as soon as the wall temperature drops to the saturation temperature even though the bulk enthalpy is still above that of quality 1 [1–9]. The formation of the first liquid indicates the entrance of condensing superheated (CSH) region after the superheated (SH) region and before the two-phase (TP) region. The droplets quickly form partial films, and partial films merge into liquid films around the entire circumference of the tube wall. This process, in spite of its two-phase nature, is almost never modeled as a two-phase flow because the bulk enthalpy says it to be superheated vapor. Similarly, when the bulk quality of the flow becomes 0, where the TP region ends, the condensation continues because the condensate is subcooled. This indicates the entrance of condensing subcooled (CSC) region, which is right before the subcooled (SC) region where no vapor exists. In both CSH and CSC regions, where two-phase flow exists, a conventional two-phase pressure drop model cannot give any prediction because the state of the flow is out of the scope of the vapor dome. If a convention single-phase model is used, however, it does not cover the twophase mechanism and thus cannot give accurate predictions. Therefore, the non-equilibrium effects on the pressure drop is not only the pressure gain due to deceleration (which is typically very small in an application in a vapor-compression system). More importantly, It changes the heat transfer, flow regime and void fraction throughout the condensation process. This means that

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Nomenclature SH CSH CSC Re We Fr K HTC PD A D E e d T P G Q h

q

ơ u

l g x

e

superheated condensing superheated condensing subcooled Reynolds number Webber number Froude number wave-enhancement number heat transfer coefficient (W m
2 K
1) pressure drop (Pa m
1) area (m2) diameter (m) energy (J) surface roughness (m) thickness (m) temperature (K) pressure (Pa) mass flux (kg m
2) heat flux (W m
2) specific enthalpy (J kg
1) density (kg m
3) surface tension (N m
1) velocity (m s
1) dynamic viscosity (kg m
1 s
1) acceleration of gravity (m s
2) quality void fraction

the concept of where the two-phase pressure drop should be predicted should be changed. The issue for heat transfer coefficient is addressed by Xiao and Hrnjak [8] in the heat transfer model that is based on flow regime map [6] improved from El Hajal et al. [10], and it traces the development of the liquid film from the real beginning of the condensation to the end. A pressure drop model that perform the same task, however, is absent in the previous work, and is the objective of this study.

2. Literature review Despite the important insights provided by an analytical model for two-phase pressure drop in conventional round tubes such as homogenous model, the models in the literature are based heavily upon empirical methods. For instance, Friedel [11] is one of the most mentioned correlation that predicts well in a conventional tube. The use of the two-phase multiplier with four dimensionless group make it fairly easy to use. Another example would be the model by Lockhart and Martinelli [12] based on a separated flow model. The Lockhart-Martinelli parameter that reflects the ratio of superficial pressure drop between liquid and vapor is so useful that it is almost universally adopted in both heat transfer and pressure drop models until now. Another competent model is by Muller-Steinhagen and Heck [13], who made use of the asymptotes by single-phase liquid and vapor flow. The strong preference of empirical methods is understandable. First of all, the empirical methods do not require only curve-fit of the existing data. This basically refrains the burden on the experimenters to go to the more complicated methods to understand the mechanisms behind the data. Secondly, the structure of the models are also very simple. Usually a couple of dimensionless numbers could suffice a pretty accurate model. This is especially true in the use of LockhartMartinelli parameter. A new model can be conveniently proposed by simply changing the empirical parameter C in the model. Thirdly, empirical models have more freedom in terms of curve-

h f H J1 J2

stratification angle frictional factor height of the liquid pool dimensionless number to calculate friction factor dimensionless number to calculate friction factor

Subscripts b bulk sat saturated sup superficial l liquid v vapor s single phase d dissipated trans transition crit critical min minimum crit critical upper upper part of the tube lower lower part of the tube onset onset of condensation end end of condensation Superscript ⁄ dimensionless

fitting, making them almost always statistically accurate over the range of the data they curve-fit upon. Publications [14,15] usually finding good agreements with those models is a good indication. However, having those advantages comes with a price. By ignoring the mechanism involved in the process simplifies the task for researcher but provides less directions to the future study. Easier construction of the model generates lots of correlations that are parallel in the reasoning but only different in their data bank. This suggests that even the models are accurate in their own data bank, they are usually not generalizable. A remedy for the lack of physical insights in the empirical method is to base the model on a specific flow regime. Hashizume et al. [16] proposed pressure drop model in annular flow by developing the velocity profile. Macdonald and Garimella [17] separated the pressure drop into the single-phase pressure drop and interaction between liquid and vapor for the intermittent flow. For stratified flow, Agrawal et al. [18] developed the model by focusing on the pressure drop in the vapor while assuming the velocity profile for liquid. Even though most of those models still involve empirical constants, they are considered more general and the mechanisms described are very helpful in guiding the development of better models. Recently, flow regime based model is preferred by many researchers because it takes advantages of the existing models that are developed for a specific flow regime. Cavallini et al. [19] choose Friedel correlation as the building brick for their flow regime based model. A large data bank is used for the development of the model. In the flow regimes where Friedel makes good prediction, it is left unchanged. In the other flow regimes, modifications are made to improve the performance of Friedel correlation. Another very interesting pressure drop model is by Quiben and Thome [20] inspired by the heat transfer model by Thome et al. [21]. The model assigns equations to different flow regimes based on the flow characteristics. Special care is taken to make sure the transition is continuous and the asymptote is the single-phase pressure drop. The flow regime based pressure drop models captures the mechanisms while do not lose the statistical accuracy offered in the

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empirical models. This paper presents the flow regime based model that accounts for the non-equilibrium effects and makes an effort in elaborating the dominant mechanisms in each flow regime so that it also provides useful insights into the future study. 3. The new pressure drop model 3.1. The real beginning and end of the condensation In a vapor-compression system, superheated vapor is fed into the condenser. The condemnation begins when the temperature of the tube wall becomes saturation temperature of the refrigerant even though other refrigerant is at a higher temperature [1–9]. After the first droplet, the entire flow starts to get cooler as condensation proceeds until the highest temperature inside the tube gets lower than the saturation temperature, which corresponds to the end of the condensation. A detailed process of determining the real onset and end of condensation is introduced by Xiao and Hrnjak [8] in the heat transfer model that accounts the effects of non-equilibrium. The equations are listed below as Eqs. (1) and (2). Xiao and Hrnjak [8] also proposed the definition of superficial quality using Eq. (3) to calculate the fluid properties and to serve as an input parameter in the heat transfer model. Basically, to calculate the onset of the condensation, the temperature of the tube wall should be saturation temperature. By assuming a bulk temperature and calculate the HTC of the superheated vapor at that temperature, the bulk temperature corresponding to the onset of condensation can be attained through iterations. The calculations for the end of condensation follows the same logic, except that an empirical constant 0.33 is introduced.

T b;onset ¼ T sat þ

Q HTC onset

T b;end ¼ T sat
0:33

ð1Þ

Q HTC end

ð2Þ

h
hend honset
hend

xsup ¼

ð3Þ

A sample flow regime map from [9] is provided as Fig. 1 for R134a. When the mass flux is high, condensation starts in annular flow regime where film is uniformly distributed along the tube wall. Then the flow regime becomes intermittent when wave grows enough to block the entire cross section of the tube. Eventually the vapor bubbles shrink and float at the upper part of the tube until they vanish. When the mass flux is low, condensation also starts in stratification flow regime. Then the liquid is pulled towards the bottom of the tube. The asymmetric distribution of the film thickness indicates the transition to the stratified wavy flow regime. As the condensation continues, flow velocity becomes low enough that no wave is generated and that is where fully stratified flow regime starts. To draw the flow regime map, follow the steps below. 1. Calculate the real onset and end of condensation using Eqs. (1)–(3). 2. Calculate the void fraction using Eqs. (4)–(6). 3. Calculate the stratification angle, dimensionless cross sectional area of liquid and vapor, height of the liquid pool and the perimeter of the liquid-vapor interface using Eqs. (7)–(11). 4. Find the ratio of Weber number and Froude number by Eq. (12). 5. Find Gwavy,1 by Eq. (13) and determine the minimum value and its corresponding superficial quality, denoted as Gwavy, min and xsup,min respectively. Set the values of all the points of Gwavy,1 after the minimum point to Gwavy,min. 6. Determine Gwavy,2 by Eq. (14). 7. Find the transition curve from the annular flow to the stratified wavy flow Gwavy by asymptotically adding up Gwavy,1 and Gwavy,2 through Eq. (15). 8. Find the transition curve from the stratified wavy flow to the fully stratified flow Gstrat through Eq. (16). 9. Find the transition line from the annular flow to the intermittent flow xIA through Eq. (17) 10. The flow map is completed.

( hstrat ¼ 2p
2


3.2. Void fraction correlation and flow regime map Xiao and Hrnjak [8] has summarized the development of void fraction correlations and flow regime maps for condensation during the past decades. The non-equilibrium effects on the void fraction and flow regime are also discussed. The void fraction model and flow regime map that accounts for the non-equilibrium effects [6] is used in this model because they are the best fit for the scope. The void fraction is calculated by the equations listed below based on El Hajal et al. [10].



eh ¼ 1 þ eRA ¼

xsup

qv þ

e¼




1 1
xsup qv xsup ql  1 þ 0:12ð1
xsup Þ


xsup

qv

þ

1
xsup

1:18ð1
xsup Þ½g rðqL
qV Þ0:25 Gq0:5 l

eh 
eRA  ln

ð4Þ

eh eRA



ql

#
1

Al ¼

p

Av ¼

pe

4

200


13 h i 1 3 1 p 1
2ð1
eÞ3
e3 2



ð1
eÞ½1
2ð1
eÞ½1 þ 4ð1
eÞ2 þ 4e2 

ð7Þ

ð1
eÞ

ð8Þ ð9Þ

4


 hstrat Hl ¼ 0:5 1
cos p
2

ð10Þ


 hstrat Pi ¼ sin p
2

ð11Þ


 2 We gd ql ¼ Fr L r

ð12Þ

ð5Þ Gwav y;1 ¼ ð6Þ

e

pð1
eÞ þ

8 <

16A3 v gdqv ql

:x2 p2 ½1
ð2H
1Þ2 0:5 sup l


þ 100
50e

#90:5

1:023 = We þ1 2 ; Fr l 25Hl

"

p2

2 ðx2
0:97Þ xð1
xÞ

ð13Þ

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Fig. 1. Flow regime map for R134a condensing at 30 °C in 6 mm tube with G = 100 kg m
2 s
1.


0:5 xsup
xsup;min Gwav y;2 ¼ Gwav y;min 1
1
xsup;min

ð14Þ

Gwav y ¼ Gwav y;1 ð1
xÞ þ Gwav y;2 x

ð15Þ

Gstrat ¼

" #13  226:32 A2 v Al qv ðql
qv Þll g þ 20
40x2 sup x2sup ð1
xsup Þp3

(" xIA ¼

)
1 1


1:75  1# qv ll
7 þ1 0:2914

ð16Þ

ql

lv

ð17Þ

3.3. Overview of the model As demonstrated in Fig. 2, in the SH region, pressure drop decreases as heat is rejected out of the refrigerant. It is because the density of the flow is increasing and the velocity of the flow decreases as a result. As soon as the condensation starts, waves start to generate. The interaction between liquid and vapor dissipates more energy than single-phase flow, thus causes the pressure drop to go up in the beginning of the annular flow regime. While the waves keeps growing, the flow velocity keeps decreasing, which contributes to the reduction of pressure drop. The effects of increasing waves and decreasing velocity compete with each other, forming a peak of pressure drop somewhere around bulk quality 1 (not necessarily at bulk quality 1). As the flow velocity decreases further, the waves eventually die down and pressure drop lowers until the end of condensation where the refrigerant is all in liquid phase. The process is elaborated in details in Xiao and Hrnjak [9]. Following the process mentioned above, an overview of the model is illustrated in Fig. 3. When the refrigerant enters the condenser in a vapor-compression system, it is superheated vapor. As soon as heat is exchanged into the secondary fluid such as cooler

air or water, the bulk temperature as well as the tube wall temperature continues to reduce. Once the wall temperature drops to the saturation temperature, the first liquid droplet forms and the condensate quickly merges into a very thin film due to the fact that most surfaces used in a condenser are hydrophilic. The film covers the entire circumference of the tube wall assuming the cooling happens everywhere on the tube. The surface tension holds a certain amount of condensate from either being pulled by the gravity to the bottom of the tube or entrained by the shear from the core vapor. Hence, in every circumstance the first flow regime in a condensation process is annular. The annular flow is treated as a uniform ring of liquid surrounding the vapor core. As condensation proceeds, two paths can be taken by the refrigerant depending on the mass flux and fluid properties. When the mass flux is low, the gravitational force prevails over the shear force. The excessive condensate tends to be pulled from the upper part of the tube to the lower part of the tube along the tube wall. This creates a liquid pool at the bottom of the tube. The stratification angle, calculated from Eq. (42), is used to represent the portion of the tube that is occupied by the liquid pool. Fig. 3 presents a simplification of reality (dotted). The division between the annulus at the top and the liquid pool at the bottom is important because they behave differently in both the thermal and the hydraulic sense. In the opposite situation when the mass flux is high, the shear force wins over gravitational force. The waves grow as condensate accumulates and flow velocity increases. Following the direction Thome et al. [21] has proposed to deal the waviness, Kelvin-Helmholtz and Rayleigh-Taylor instability both contribute to the wave generation, and an enhancement factor due to the waviness Ki will be introduced in the following sections. Regarding the flow regime, as we increase the mass flux and decrease the superficial quality, at some point, the waves will grow so high that they block the entire cross section of the tube. Intermittent flow results from this process. Fig. 3 demonstrates the difference between the intermittent flow and the stratified flow in that the former divide the tube longitudinally along the tube while the latter does it across the tube

J. Xiao, P. Hrnjak / International Journal of Heat and Mass Transfer 126 (2018) 421–430

425

Fig. 2. Pressure drop data by Xiao and Hrnjak [9].

Fig. 3. Illustration of condensation process in this model.

cross section. The intermittent flow regime starts when the waves hit the top of the tube. The portion of the tube occupied by the liquid slug increases as the condensation advances while vapor bubble become shorter. The model takes into account the importance of each mechanisms (liquid-only and annular). 3.4. Wave-enhancement factor The wave formation has long been linked to the transition of flow regimes e.g. from annular to intermittent. Taitel and Dukler

[22] attributes the reason of transition from stratified flow to intermittent flow to the growing wave (in adiabatic flow). The increase of flow velocity triggers the Kelvin-Helmholtz instability, causing the wave to wash to the top of the tube when there is enough liquid load. Although in adiabatic condition this is a very useful insight into the formation of intermittent flow. In diabatic conditions, the heat transfer adds another degree of complexity to the problem. The variation of quality during condensation decreases the flow velocity while adding the liquid load. However, to have intermittent flow, the velocity needs to go up and the amount of liquid have to be

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J. Xiao, P. Hrnjak / International Journal of Heat and Mass Transfer 126 (2018) 421–430

sufficient. This means that the reduction of the velocity is actually acting against the generation of waves as the condensation proceeds. This phenomenon is pointed out in Xiao and Hrnjak [9]. The competition between reducing velocity and void fraction is the key factor to describe the pressure drop data in the study. In addition, rather than having stratified flow in the beginning, condensation always starts in the annular flow regime. It has two inferences. One is during condensation, the Kelvin-Helmholtz instability not only generates waves from the bottom to top, but also from top to bottom because there is liquid film on top as well. The other is that Rayleigh-Taylor instability is present during the condensation process too because the upper part of the tube is always generating liquid that sits upon the vapor core and tends to fall down. Thome et al. [21] has a touch on the issue where they combine the effects from the two instabilities into two dimensionless groups. Although sufficient in their heat transfer model to account for the convection effects, the authors of the current study attempt to follow along the same direction but try to make it more realistic and accurate for pressure drop enhancement, especially because the phenomenon explains the experimental data so well. The way to treat the effects of the wave formation on the pressure drop is the waveenhancement factor presented below. For Kelvin-Helmholtz instability, velocity difference and fluid properties directly indicate the frequency and the strength of the waves. It is shown by Xiao and Hrnjak [9] in the visualizations that the waves die down at the end of the condensation process because the velocity difference between vapor and liquid gets smaller. The parameter that links the fluid properties to the velocity difference is the minimum velocity to trigger the KelvinHelmholtz instability as derived in Carey [23]. The equation to find the minimum velocity required is listed below as Eq. (18). Rayleigh-Taylor instability is similar to the Kelvin-Helmholtz instability in that the properties of the fluid determine the critical wavelength. The difference is that the mechanism that triggers the instability is not the velocity difference but the gravitational force, which is directly linked to the film thickness. The critical wavelength is calculated using the method derived in Carey [23] and Eq. (19) is responsible for it. The wave-enhancement factor Ki is proposed as Eq. (20) where velocity of liquid and vapor can be determined using Eqs. (21) and (22). The film thickness can be calculated by Eq. (23). Three constants C, p, q are empirically determined to be 2.9, 0.25, and 0.41 respectively.

 umin ¼

2ðql
qv Þ

0:5 

ql



rðql
qv Þg q2v

rg ql
qv

Ki ¼ 1 þ C


p
q uv
ul d dcrit umin

ul ¼

ð18Þ

0:5

dcrit ¼ 2p

uv ¼

0:25

Gxsup

qv e Gð1
xsup Þ ql ð1
eÞ

d ¼ 0:5Dð1
e0:5 Þ

ð19Þ

ð20Þ

this paper does not preclude the use of other single-phase correlations when facing applications where Churchill might be insufficient. Here Churchill correlation is chosen due to its good agreement with the single-phase pressure drop data in [9] and its asymptotic nature that connects the predictions between laminar and turbulent flow. It is written below.

(

"


J1 ¼


2:457 ln

J2 ¼


16 37530 Re "


f ¼8

#) 0:9  e  16 7 þ 0:27 Re d

1:5 #121 12
8 1 þ Re J1 þ J2

ð25Þ

ð26Þ

2

PDs ¼

fG 2q D

ð27Þ

3.6. Model for annular flow Annular flow is always the first flow regime to be expected when condensation happens. Meanwhile, the CSH region is the first two-phase region during the condensation and the region that conventional approaches mistaken as single-phase region. Since the current model attempts to include the effects from nonequilibrium into the pressure drop prediction, the mechanism that dictates the annular flow is the most important factor to evaluate. Fig. 4 shows the conceptual picture of an annular flow in this model. The annular flow is considered as a flow where the vapor core is surrounded by a uniform ring of the liquid. The film thickness d is calculated from the void fraction using Eq. (23). The pressure drop in the flow, is taken as the energy dissipated in the liquid ring and vapor core which is represented in Eq. (28). After expressing the overall dissipated energy in terms of the overall pressure drop and the volumetric flow rate, Eq. (29) is derived from the simplification of Eq. (28). Eq. (30) is the thermodynamic density of a two-phase flow. Eqs. (31) and (32) are the calculation of pressure drop in vapor and liquid separately. The pressure drop by vapor assumes the interface between liquid and vapor to be the new ‘‘tube wall” whose diameter is smaller by the magnitude of the film thickness. The pressure drop by liquid employs two enhancement factors. One is the wave-enhancement factor proposed above in Section 3.4. The other is the enhancement factor for annular flow used by Xiao and Hrnjak [8] in the heat transfer model with a constant m, which is empirically determined to be 0.3 in this model. The single-phase pressure drop correlation is selected to be Churchill correlation for all cases. Note that the pressure drop in the liquid and vapor may not be the same, which is

ð21Þ

ð22Þ ð23Þ

3.5. Model for single-phase flow For single-phase flow, Churchill correlation [24] is used in the current model. It should be noted that the approach presented in

ð24Þ

Fig. 4. The conceptual picture of an annular flow.

J. Xiao, P. Hrnjak / International Journal of Heat and Mass Transfer 126 (2018) 421–430

ð28Þ

ities of liquid and vapor are calculated from the superficial velocities at the given superficial quality. The lower part of the tube is considered annular flow with thicker film. The film thickness is determined by calculating the equivalent film thickness deq as if the liquid pool is spread out to form a liquid ring in the lower part. The stratification angle is calculated through iteration from Eq. (42). When the stratification angle reaches 180 degrees and above, the film thickness of the lower part of the tube is set to half of the diameter.

ð29Þ

PD ¼ PDupper

potentially problematic, but the approach is not aiming at determining the pressure of vapor and liquid separately. The model merely calculates the instantaneous pressure gradient at a given segment. The resulting pressure of liquid and vapor will adjust themselves to the overall pressure drop, which is the output of the model and the parameter to be used in a condenser designing process.

E_d ¼ E_dl þ Ed_ v PD

q

¼ PDv


q¼

xsup

qv

xsup

þ PDl

qv þ

1
xsup

1
xsup

ql 
1

ql

PDv ¼ PDs K i
PDl ¼ PDs

ð30Þ ð31Þ

m

D 2d

Ki

427

ð32Þ

hstrat 2p
hstrat þ PDlower 2p 2p

PDupper ¼ PDv ;upper

PDv ;upper ¼ PDs;upper K i;upper


Compared to the annular flow, the stratified flow differs mainly in its liquid pool at the bottom of the tube. The liquid pool is the result of the gravitational force pulling the liquid downwards. The liquid pool not only change the flow regime, but the heat transfer coefficient and pressure drop that are closely affected by it. For heat transfer, the liquid pool creates barrier between the liquid-vapor interface and the tube wall, thus makes the heat transfer coefficient much lower than that of the annular flow. A widely accepted way of handling the liquid pool is to treat the perimeter of the tube occupied by it as if it is already in singlephase, started from Chato and Dobson [25], furthered by Thome et al. [21], Macdonald and Garimella [17] and eventually the method which includes the non-equilibrium effects by Xiao and Hrnjak [8]. For pressure drop model, the transition from annular to stratified flow regime is quite tricky. Unlike heat transfer, where liquid film thickness can be directly connected with the thermal resistance, pressure drop was not related to film thickness in most pressure drop models. The wave-enhancement factor mentioned above in the Section 3.4 fills the gap. Therefore, after the flow transitions from annular to stratified, the tube is divided into upper and lower parts to separately calculate pressure drop as shown in Fig. 5. The corresponding equations are listed below. The pressure drop for the upper part, which is annular flow with thin film, is determined by the transition film thickness dtrans. The film thickness for the upper part of the tube will be assumed constant throughout the entire stratified flow. At the same time, the veloc-

Fig. 5. Stratified flow is divided into the annular flow with thinner film at the top and thicker film at the bottom.

D 2dtrans

PDl;upper ¼ PDs;upper PDlower ¼ PDv

3.7. Model for stratified flow

q q x þ PDl;upper ð1
xsup Þ qv sup ql

ð33Þ ð34Þ ð35Þ

m K i;upper

q q x þ PDl ð1
xsup Þ qv sup ql

ð36Þ ð37Þ

  sinð2p
hstrat Þ dlower ¼ 0:5d 1
2p
hstat

ð38Þ

PDv ;lower ¼ PDs;lower K i;lower

ð39Þ


PDl;lower ¼ PDs;upper

q¼


xsup

qv

1
e¼

þ

D 2deq

1
xsup

m K i;lower

ð40Þ


1

ql

1 ½ð2p
hstrat Þ
sinð2p
hstrat Þ þ hstrat ð1
etrans Þ 2

ð41Þ

ð42Þ

3.8. Model for intermittent flow As illustrated in Fig. 3, the refrigerant can transition from annular to intermittent if the mass flux is high. The intermittent flow also affects both heat transfer and pressure drop. Rather than taking a portion of the tube with single phase liquid, intermittent flow fills up the whole tube with the least amount of vapor intermittently. When the tube is occupied by both liquid and vapor, the flow is nothing but annular flow because the vapor has its free path within the liquid ring. When a big wave blocks the tube cross section, the vapor cannot go through. At the same time, the liquid slug is a wave that gets constrained in a tube. The front of the liquid slug is picking up the condensate from the liquid film on the circumference of the tube wall towards the center of the tube to fill up the tube, while the rear of the liquid slug sheds liquid because the vapor is digging through and the majority of the liquid has moved forward already. The liquid slug resembles a single-phase liquid flow in the tube but it is by no means fully developed. The elongated vapor bubble between the liquid slugs cannot be treated as annular flow either. Due to its complexity, the intermittent flow has a history of being treated very differently in different models. For instance, Thome et al. [21] simply handles the intermittent flow as annular flow, which has some grounds because the way void fraction performs typically delay the formation of long liquid slug until low quality. Before the liquid slug becomes too long to be ignored, the intermittent flow and annular flow are very similar. In another model by Macdonald and Garimella [17], they treat the

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intermittent flow to be the summation of single-phase vapor pressure drop, single-phase liquid pressure drop and the interaction between them. The single-phase liquid pressure drop does make sense especially when the liquid slug is long. The interaction between liquid and vapor is also insightful in that the interaction between liquid and vapor does dissipates more energy than a smooth interface and thus increase the pressure drop. The singlephase vapor part, however, has no physical ground. Not only for intermittent flow, but during the entire process of condensation, dry wall (single-phase vapor flow) never happens because the condensate keeps forming on the tube wall from the very start. A better way of looking at intermittent flow is by Quiben and Thome [20]. They use void fraction to weigh the importance of annular flow and saturated liquid flow. They claim the model to give continuous prediction and correct limit if non-equilibrium is not considered. However, a deeper meaning is not elaborated in the paper. The use of void fraction works because it takes into account the dominant mechanism for an intermittent flow. The nature of intermittent flow dictates that when the liquid slug passes, the dissipation of energy is much smaller than when the vapor bubble passes. How much a tube is occupied by each form in an intermittent flow is determined by the void fraction while it determines the pressure drop. There are three defects in their approach. One is that the correct limit is not saturated liquid but the subcooled liquid at the end of condensation, which is the end of the CSC region. The second is that the velocity of the liquid slug should not be assumed to be constant as in the liquid-only flow, which gives a fixed pressure drop for liquid slug. For example, a liquid slug at superficial quality 0.5 moves much faster than the liquid does at superficial quality of 0. The pressure drop of a liquid slug should be a changing value that can be calculated from the fluid properties and working conditions. It is the same for annular flow. The current model combines the advantages by the abovementioned approaches and it has its own merits. It uses void fraction to calculate how much of the flow can be viewed as annular and how much can be viewed as liquid slug as in Fig. 6. It took the superficial velocity to calculate the pressure drop. It includes the interaction between vapor and liquid in the annular portion of the flow. More importantly, it extends the range of application to the real end of the condensation. One thing to note is that the film thickness at the transition between annular and intermittent flow regimes dtrans is taken as the film thickness for the annular part of the calculation throughout the intermittent flow.

PD ¼ bPDannular þ ð1
bÞPDliquid PDannular ¼ PDv ;annular

ð43Þ

q q x þ PDl;annular ð1
xsup Þ qv sup ql

PDv ;annular ¼ PDs;annular K i;annular
PDl;annular ¼ PDs;annular

D 2dtrans

ð44Þ ð45Þ

m K i;annular

ð46Þ

Fig. 6. Intermittent flow is divided into the annular flow and single-phase liquid flow.

PDliquid ¼ PDs;liquid

q¼ b¼


xsup

qv

þ

1
xsup

ð47Þ 
1

ql

e etrans

ð48Þ ð49Þ

3.9. Implementation The pressure drop model could be implemented in the following way: 1. Determine the real onset and end of condensation with Eqs. (1) and (2). 2. Find the superficial qualities from Eq. (3). 3. Define the flow regimes using Eq. (4)–(16). 4. Determine the superficial enthalpy of the transition between the annular flow and stratified flow and between annular flow and intermittent flow from the flow regime map. 5. Before the transition occurs, apply the annular flow model with Eqs. (18)–(27), (29)–(32). 6. After the transition, apply the stratified flow model with Eqs. (18)–(27), (29)–(42) or the intermittent flow model with Eqs. (18)–(27), (29)–(32), (43)–(49). 7. Before and after the onset and end of the condensation, apply the single-phase model with Eqs. (24)–(27). 4. Validation of the model Fig. 7 is an illustration of the comparison between the new pressure drop model and conventional models against the experimental data at one condition. The largest difference between the two approaches are in the CSH region where the new pressure drop model has already adopted the two-phase mechanism while the conventional approach still assumes single-phase flow. The underestimation of the conventional approach towards the experimental data in CSH is improved by recognizing that the wavy structure dissipates energy, thus raises the pressure drop. Also in Fig. 7 both current and existing (Friedel and Churchill) methods provide a continuous curve. Friedel [11] and Churchill [24] only asymptotically approach each other when the Blasius equation is substituted by Churchill’s friction factor in the application of Friedel correlation. This implies that if single-phase and two-phase correlations for pressure drop are randomly selected based on criteria like ‘‘the most statistically preferred according to some data bank”, then it will be very hard to avoid discontinuities at the transitions between the models. This makes coding for condenser design problematic and inaccurate. The new model, resolves this issue because it is based on single-phase correlation (in this case Churchill) with the wave-enhancement factor Ki asymptotically approaching the correct value at the transitions. Therefore, in case there will be a better suited single-phase correlation for other applications, the new approach still integrate each region into a seamless curve and thus eliminate the discontinuity problem. Another implication of Fig. 7 is about heat transfer. According to heat and momentum analogy, higher pressure drop in the CSH region means higher heat transfer coefficient comparing to single-phase correlation. It is indeed the case. As is mentioned above, many publications [1–9] have presented the effect of nonequilibrium on heat transfer especially in the CSH region. The findings support the results in this paper. Fig. 8 is an overall comparison between the prediction from the current model and the experimental data from Xiao and Hrnjak [9] including R32, R134a, R1234ze(E), R1233zd(E) and R245fa at heat fluxes of 5

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429

Fig. 7. Comparison between experimental data and models.

the desuperheating process is important. There are several features that are novel in this model. First of all, the model traces the development of the liquid film formation throughout the process when energy is rejected out of the superheated vapor. Thus this model finds the real onset and the end of the condensation rather than mistakenly setting them to be bulk quality 1 and 0. Additionally, the prediction of the model starts from single-phase superheated region, goes through the regions where the two-phase flow exists, and eventually ends in the single-phase subcooled region with one curve seamlessly connecting each two adjacent regions. Moreover, the model is based on the flow regime map and void fraction correlation that is developed from diabatic flow visualizations and film thickness measurements. In other words, the nonequilibrium effects are included in the very foundation of the model. Besides, the model attempts to capture the most important mechanisms in each different flow regimes. In this manner the model tries to be as general as it could and provide insights into the modeling of similar scenarios in the future studies. Last but not least, the model is validated by experimental data from two different tubes and a wide range of refrigerants under different working conditions. Fig. 8. Comparison between the predictions from the current model to the experimental data.

to15 kW m
2 and mass flux of 100 to 400 kg m
2 s
1. As far as the authors realize, the data from [9] is the only study that includes the non-equilibrium effects on the pressure drop in a condenser of a vapor-compression system. From Fig. 8 it can be seen that there are more deviations at lower pressure drop range. This is probably due to certain systematic errors during the experiments. Despite these discrepancies, most predictions of the new model fall into the ±10% deviations of the experimental data, indicating statistically good predictability.

Acknowledgement The authors thankfully acknowledge the support provided by the Air Conditioning and Refrigeration Center at the University of Illinois at Urbana-Champaign, and technical support from Creative Thermal Solutions, Inc. (CTS). Conflict of interest We have no conflict of interest. References

5. Conclusion A new pressure drop model for condensation inside a horizontal smooth round tube accounting for the non-equilibrium effects is presented in this paper. The model is especially useful for a condenser in a vapor-compression system where the modeling of

[1] C. Kondou, P.S. Hrnjak, Heat rejection from R744 flow under uniform temperature cooling in a horizontal smooth tube around the critical point, Int. J. Refrig. 34 (3) (2011) 719–731. [2] C. Kondou, P.S. Hrnjak, Condensation in presence of superheated vapor for near-critical CO2 and R410A in horizontal smooth tubes, in: The 23rd IIR International Congress on Refrigeration, Prague Czech Republic, 2011, paper ID. 389.

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[3] C. Kondou, P.S. Hrnjak, Condensation from superheated vapor flow of R744 and R410A at subcritical pressures in a horizontal smooth tube, Int. J. Heat Mass Transfer 55 (2012) 2779–2791. [4] R. Agarwal, P.S. Hrnjak, Condensation in two phase and desuperheating zone for R1234ze(E), R134a and R32 in horizontal smooth tubes, Int. J. Refrig. 50 (2015) 172–183. [5] M. Meyer, P.S. Hrnjak, Flow regimes during condensation in superheated zone, Int. J. Refrig. 84 (2017) 336–343. [6] J. Xiao, P.S. Hrnjak, A new flow regime map and void fraction model based on the flow characterization of condensation, Int. J. Heat Mass Transfer 108 (2017) 443–452. [7] J. Xiao, P.S. Hrnjak, Heat transfer and pressure drop of condensation from superheated vapor to subcooled liquid, Int. J. Heat Mass Transfer 103 (2016) 1327–1334. [8] J. Xiao, P.S. Hrnjak, A heat transfer model for condensation accounting for nonequilibrium effects, Int. J. Heat Mass Transfer 111 (2017) 201–210. [9] J. Xiao, P.S. Hrnjak, Pressure drop of R134a, R32 and R1233zd(E) in diabatic conditions during condensation from superheated vapor, Int. J. Heat Mass Transfer 122 (2018) 442–450. [10] J. El Hajal, J.R. Thome, A. Cavallini, Condensation in horizontal tubes, Part 1: Two-phase flow pattern map, Int. J. Heat Mass Transfer 46 (18) (2003) 3349– 3363. [11] L. Friedel, Improved friction pressure drop correlation for horizontal and vertical two phase flow, 3R Int. 18 (7) (1979) 485–491. [12] R.W. Lockhart, R.C. Martinelli, Proposed correlation for isothermal twophase, two-component flow in pipes, Chem. Eng. Prog. 45 (1) (1949) 39–48. [13] H. Muller-Steinhagen, K. Heck, A simple friction pressure drop correlation for two-phase flow in pipes, Chem. Eng. Prog. 20 (1986) 297–308.

[14] C. Tribbe, H.M. Muller-Steinhagen, An evaluation of the performance of phenomenological models for predicting pressure gradient during gas-liquid flow in horizontal pipelines, Int. J. Multiphase Flow 26 (2000) 1019–1036. [15] M.B. Ould Didi, N. Kattan, J.R. Thome, Prediction of two-phase pressure gradients of refrigerants in horizontal tubes, Int. J. Refrig. 25 (2002) 935–947. [16] K. Hashizume, H. Ogiwara, H. Taniguchi, Flow pattern, void fraction and pressure drop of refrigerant two-phase flow in a horizontal pipe. Part II: Analysis of frictional pressure drop, Int. J. Multiphase Flow 11 (1985) 643–658. [17] M. Macdonald, S. Garimella, Hydrocarbon condensation in horizontal smooth tubes: Part II – Heat transfer coefficient and pressure drop modeling, Int. J. Heat Mass Transfer 93 (2016) 1248–1261. [18] S.S. Agrawal, G.A. Gregory, G.W. Govier, An analysis of horizontal stratified two phase flow in pipes, Can. J. Chem. Eng. 51 (1973) 280–286. [19] A. Cavallini, G. Censi, D.D. Col, L. Doretti, G.A. Longo, L. Rossetto, Condensation of halogenated refrigerants inside smooth tubes, HVAC R Res. 8 (4) (2002) 429–451. [20] J.M. Quiben, J.R. Thome, Flow pattern based two-phase frictional pressure drop model for horizontal tubes, Part II: New phenomenological model, Int. J. Heat Fluid Flow 28 (2007) 1060–1072. [21] J.R. Thome, J. El Hajal, A. Cavallini, Condensation in horizontal tubes, part 2: new heat transfer model based on flow regimes, Int. J. Heat Mass Transfer 46 (18) (2003) 3365–3387. [22] Y. Taitel, A.E. Dukler, A model for predicting flow regime transitions in horizontal and near horizontal gas–liquid flow, AIChE J. 22 (1) (1976) 47–54. [23] V.P. Carey, Liquid-Vapor Phase-Change Phenomena, CRC Press, 2008. [24] S.W. Churchill, Friction-factor equation spans all fluid-flow regimes, Chem. Eng. 7 (1977) 91–92. [25] M.K. Dobson, J.C. Chato, Condensation in smooth horizontal tubes, J. Heat Transfer 120 (1) (1998) 193–213.