Hydrodynamic behaviors of the falling film flow on a horizontal tube and construction of new film thickness correlation

International Journal of Heat and Mass Transfer 119 (2018) 564–576

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

Hydrodynamic behaviors of the falling film flow on a horizontal tube and construction of new film thickness correlation Chuang-Yao Zhao a, Wen-Tao Ji b, Pu-Hang Jin b, Ying-Jie Zhong a, Wen-Quan Tao b,⇑ a b

College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou 310014, PR China Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China

a r t i c l e

i n f o

Article history: Received 31 July 2017 Received in revised form 31 October 2017 Accepted 16 November 2017

Keywords: Falling film Horizontal tube Film thickness Numerical simulation Laminar flow

a b s t r a c t The laminar liquid film falling on a horizontal smooth tube is studied numerically. The instantaneous hydrodynamic characteristics of falling film flow, the importance of surface tension in calculation and the effects of film flow rate, tube diameter, liquid distributor height and inlet liquid temperature on the flow field and film thickness are elucidated. The results indicate that: (1) The surface tension is important in the calculations of falling film flow on a horizontal tube; (2) The film falling on a circular tube has obvious instantaneous behaviors; (3) The film thickness increases with increase of film flow rate, while decreases with increase of the tube diameter, liquid distributor height and liquid temperature, respectively; (4) The film distribution along the peripheral angle is unsymmetrical, and the minimum thickness appears in 110–150° of peripheral angle depending on the working conditions. Furthermore, new correlations of falling film thickness on a horizontal tube based on the present data are established, which fit 97% of 84 data in h = 2–15° within ±20%, 90% of 632 data in h = 15–165° within ±20%, and 73% of 112 data in h = 165–178° within ±30%. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction The falling liquid film outside the horizontal tubes is widely encountered in the evaporation [1], condensation [2,3] and cooling [4,5] processes of desalination [6], refrigeration and petrochemical industries, mainly owing to the advantages in better heat/mass transfer performance while lower liquid charge [7–9]. However, the database regarding hydrodynamic characteristics of the liquid film falling on the horizontal tube is insufficient. The falling liquid film flowing over a single horizontal tube is influenced by various factors such as film flow rate, liquid temperature, liquid distributor type/height and tube geometries. These factors adjust the applications of gravity, viscous shearing, surface tension and wall adhesion, then change the film flow patterns and film distributions, and eventually affect the heat and mass transfer performances. When the liquid film falls on a horizontal tube, the heat is transferred from the tube wall, across the thin liquid layer, and then to the liquid gas interface, during which the film flow field and film thickness play important roles in the thermal resistance. In previous investigations, the film thickness evolution on

the horizontal tube has been studied with analytical [10–13], numerical [14–18] and experimental methods [19–25]. The published analytical and numerical studies concerning the liquid film thickness on the horizontal tube are most often based on many hypotheses. To the authors’ knowledge the study of Nusselt [10] is the pioneering work on the prediction of falling film thickness. He obtained an analytical solution of the condensing film thickness on a vertical plate with neglecting the surface tension and inertial force. By replacing the vertical plate with an inclined one, the local film thickness can be predicted by the following equation for various inclination angles 1=3

d ¼ ð3ll C=q2l gsinðpb=180ÞÞ

where b (°) is the inclination angle of the plate. Thus, the film thickness of the liquid film falling on a horizontal tube can be calculated by Eq. (1) with replacement of the inclination angle b by the peripheral angle h (°). Based on Nusselt’s solution, Rogers [11] and Chyu and Bergles [12] correlated the film thickness on a horizontal tube, as expressed


d¼

⇑ Corresponding author. E-mail address: [email protected] (W.-Q. Tao). https://doi.org/10.1016/j.ijheatmasstransfer.2017.11.086 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

ð1Þ

3ll C ql ðql  qv Þgsinðph=180Þ

1=3 ð2Þ

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Nomenclature Archimedes number, Ar ¼ q2l gD3 =l2l diameter of tube, mm source term in momentum equation gravity acceleration, m s2 liquid distributor height, m pressure, Pa film Reynolds number, Re ¼ 4lC tube spacing, m time, s temperature, °C component of velocity, m s1 coordinates, m 2 modified Webber number, We ¼ p2 Cq Dr

Ar D F g H p Re s t T u, v x, y We

Greek

a d h

l q r

l

The thickness of the liquid film falling on a horizontal tube has also been measured in several studies with intrusive or nonintrusive techniques. Rogers and Goindi [19] measured the film thickness of laminar water film on a circular tube with large diameter (132 mm) using three dial point gauges, and built a film thickness correlation 1=3

d=d ¼ 1:186Re1=3 ðArsinðph=180ÞÞ

ð3Þ

Xu et al. [20] measured the falling film thickness on a horizontal tube with micro-measuring instrument and observed that the film thickness increases with the film flow rate and the average film thickness almost remains constant when tube diameter varies from 20 to 40 mm. By using an optical method, Gstoehl et al. [21] measured the falling film thickness of water, reagent grade ethylene glycol, and a water–glycol mixture (50%–50% by mass) on a horizontal tube with different film flow rates and liquid distributor heights. Mohamed [22] investigated the effect of fluted surface on the film thickness, and found that the fluted structure facilitates the thinning of liquid film especially for smaller flute pitch. Hou et al. [23] found that the tube diameter has little influence on film thickness, and that the tube spacing, s, should be considered for the lower part of the tube (h > 90°). They also developed a piecewise correlation based on Nusselt’s solution [10] with considering the effect of s/D, as follows


d¼C

3ll C ql ðql  qv Þgsinðph=180Þ

1=3 ðs=DÞn

ð4Þ

where C = 0.9754, n = 0.1667 for 0° < h  90°, and C = 0.84978, n = 0.16479 for 90° < h < 180°. Recently, Narváez-Romo and Simões-Moreira [26] evaluated the intrusive methods of film thickness measuring techniques. They inferred that the deviations between their intrusive measurement and the Nusselt film thickness model [10] were related to two reasons: (1) Nusselt expression was developed for a liquid film flow over a vertical flat plate without interfacial waves; (2) the probe or needle tip of the measuring instrument gave rise to waves and crest at the liquid surface. They also modified the Nusselt’s film thickness correlation based on their data, as follows


d¼

volume fraction liquid film flow rate on one side of the tube per unit length, kg m1 s1 film thickness, m peripheral angle from the upper stagnation point, °(degree) dynamic viscosity, kg m1 s1 density, kg m3 surface tension coefficient, N m1

C

3ll C ql ðql  qv Þgsinðph=180Þ

1=3 !1:041 ð5Þ

In recent years, with the development of computer, the falling film flow behaviors are studied by numerical simulations, which is generally based on a few assumptions while can avoid the interference of measuring elements on the film surface. Min and Choi [14] used marker and cell (MAC) method to track the phase

interface during the absorption of the LiBr solution on a horizontal tube with self-developed Navier–Stokes procedure. Besides, the VOF method is widely employed to capture the free surface of the liquid film on a horizontal tube [15–17,27–30]. Fiorentino and Starace [5] simulated the flow patterns and the film thickness on the triangular tube bundle with a 2-D model. And found that both tube arrangement and film flow rate have effects on the flow patterns. More recently, Ji et al. [31] calculated the falling film flow of LiBr solution over a hydrophilic horizontal tube and modified the Nusselt equation with considering the asymmetric feature, as follows


d¼

3l l C ql ðql  qv Þgsinð0:75ðph=180ÞÞ

1=3 ð6Þ

The falling film thickness can be obtained by above analytical, numerical and experimental methods. In general, the accurate measurement of film thickness is very difficult due to the free surface waviness and the small order of film thickness. For that reason, the reliable numerical simulations are required to help us understand the hydrodynamic characteristics of the thin liquid film flow. However, the previous studies always introduced oversimplifications of transport phenomena in falling films, such as neglecting inertial force and surface tension. Additionally, there is no universal correlation of falling film thickness over a horizontal tube to consider these forces and the aforementioned factors. In this paper, the instantaneous hydrodynamic characteristics of the falling film flow and the effects of aforementioned factors are taken into account, and new falling film thickness correlations are also constructed. The rest sections are arranged as follows: firstly the numerical method and procedure are introduced; then the results and discussion are presented, including the role of surface tension in calculation, instantaneous film flow characteristics, and the effects of five factors on the film flow field and film thickness, as well as the construction of the falling film thickness correlations, and finally some conclusions are drawn.

2. Numerical simulation approach 2.1. Physical model This study focuses on a horizontal smooth tube in a squarepitch tube bundle, as shown in Fig. 1(a), where D is the tube diameter, s the tube spacing, and H the liquid distributor height. Fig. 1(b) illustrates the schematic liquid film distribution on the horizontal tube. One half of the tube and inter-tube space is served as the calculation domain due to symmetry. The scopes of parameters in the calculations are listed in Table 1.

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VOF formulations, the momentum equations (Eqs. (8) and (9)) are shared by the liquid and gas phases. The continuity equation is modified to be solved for one phase, which can be expressed as !

@ a1 =@t þ v rða1 Þ ¼ 0

ð10Þ

where a1 is the volume fraction of phase 1. The volume fraction for phase 2, a2, can be obtained by a1 + a2 = 1. In this study, the interface where the volume fraction of liquid phase is 0.5 is designated as the liquid free surface. And the film thickness is determined according to this consideration. The Continuum Surface Force (CSF) model proposed by Brackbill et al. [33] is implemented to add surface tension to the source term of the momentum equation. In CSF model, the surface tension is expressed as

(a)




F ¼ rjn

Fig. 1. Schematics of the calculation domain and the grid system.

Table 1 Calculated condition of all cases. Physical parameter

Symbol

Calculation region

Unit

Film flow rate Tube diameter Liquid distributor height Liquid inlet temperature

C

0.025–0.284 6.35–50.8 3.0–50.8 2–104

kg m1 s1 mm mm °C

2.2. Mathematical model To simplify the physical model, a 2-D falling film flow is considered with the following assumptions: (1) the thermophysical properties of the liquid are constant and are evaluated at the inlet liquid temperature; (2) the liquid film distribution is symmetrical about the perpendicular bisector of the tube; (3) the film flow is in the laminar regime; (4) the film flow rate is large enough to avoid film breakout. The governing equations based on the above assumptions can be expressed as

@ðquÞ @ðqv Þ þ ¼0 @x @y

ð7Þ

@ðquÞ @ðquÞ @ðquÞ @ðpÞ þu þv ¼ þ rðlruÞ þ qg x þ F x @t @x @y @x

ð8Þ

@ðqv Þ @ðqv Þ @ðqv Þ @ðpÞ þu þv ¼ þ rðlrv Þ þ qg y þ F y @t @x @y @y

ð9Þ

Re ¼

4C

ð12Þ

l

where C is the film flow rate on one side per unit length of the tube. The non-uniform quadrilateral grids are employed in the present work, as shown in Fig. 1(c), in which the grid size varies from a minimum of 0.1 mm near the tube wall to a maximum of 0.72 mm at the right boundary in all cases. All calculations are performed by the commercial software FLUENT 15.0 [35]. And the boundary conditions are shown in Fig. 1(c). 3. Numerical procedure 3.1. Grid independence Four simulations have been conducted (C = 0.168 kg m1 s1, D = 25.4 mm, H = 6.3 mm) with total grid elements of 2330, 3889, 5517 and 7629, respectively, to provide a grid sensitivity analysis. The film thickness as a function of the peripheral angle of the tube 1.2

Grid of 2330 total elements Grid of 3889 total elements Grid of 5517 total elements Grid of 7629 total elements

1.1 1.0

where u and v are the velocity components, qgx and qgy are gravitational body forces in x and y coordinate, Fx and Fy are the volumetric force due to surface tension. Finite volume method (FVM) is implemented to discretize the governing equations. The 1st-order implicit scheme for the time discretization and the 2nd-order upwind scheme for momentum discretization are selected. A finite-volume method is employed and the equations of momentum and continuity are integrated over every computational cell. The pressure-implicit with the splitting of operators (PISO) method [32] is used to couple the velocity and pressure. The VOF method [32] with Geo-Reconstruct scheme is adopted to capture the free surface location of the liquid. In the

0.9 0.8

/ mm

D H Ti

ð11Þ

where r is the surface tension coefficient, n is the normal vector of the interface, and j is the curvature of the interface. A wall adhesion model [33] conjunction with the surface tension model is employed to consider the effect of the property of the tube wall, in which the contact angle is needed to adjust the body force term in the surface tension calculation. In this study, the tube and liquid are assumed as copper and water, respectively. And therefore, the contact angle is designated 10° according to [34]. The film Reynolds number, Re, is defined by the following equation

(c)

(b)



q n ; n ¼ ra; j ¼ r  jnj ðql þ qg Þ=2

=0.168kgm s D=25.4mm H=6.3mm

0.7 0.6 0.5 0.4

Water Ti=46 C

0.3 0.2

0

20

40

60

80

100

degree

120

140

160

Fig. 2. Variations of the local film thickness with grid number.

180

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1.0

are shown in Fig. 2, as indicated by these curves, the results agree well with each other. And therefore, the grid system with 7629 elements is adopted under this tube diameter and feeder height, in which the minimum and maximum size of the grid are 0.1 mm and 0.72 mm, respectively. This minimum grid size is also met four other tube diameters and feeder heights.

0.8 0.7

/ mm

3.2. Time-step and initialization

1 Dt

Z

Dt

ð13Þ

dðtÞdt

Re=574 D=25.4mm H=10mm

0.6 0.5

The film thicknesses using four time steps are shown in Fig. 3, which indicates that the results are basically independent of the time steps in the current range. In this study, an adaptive timestep is adopted with a setting of maximum Courant number of 0.15 (Co  0.15), meaning that the time-step during calculation fluctuates from 5e6 to 5e5 s approximately. The local film thickness is computed with the time averaged values, as follows

d¼

Present study with surface tension Present study without surface tension Hou et al. [23] Nusselt [10]

0.9

0.4

Water Ti=27 C

0.3 0.2

0

20

40

60

80

100

degree

120

140

160

180

Fig. 4. Comparison of the local film thickness between present results and other results.

0

1.3

where d(t) is the instantaneous film thickness, and Dt denotes the targeted time period, which is carefully chosen as 0.2–0.3 s including sufficient repetitions of the film thickness profile.

=0.088kgm =0.088kgm =0.103kgm =0.103kgm =0.168kgm =0.168kgm

1.2 1.1 1.0

3.3. Validation by comparisons with previous correlations In order to validate the reliability of the numerical method in this study, comparisons of our numerical results with the ones from the literature are conducted, as shown in Fig. 4. It can be seen that the present results agree well with the ones in [23] but disagree with Nusselt’s solution [10] due to the assumptions in Nusselt’s analysis.

mm

0.9 0.8

1

1

s without surface tension 1 s with surface tension 1 1 s without surface tension 1 1 s with surface tension 1 1 s without surface tension 1 1 s with surface tension 1

Water H=6.3mm D=25.4mm

0.7 0.6 0.5 0.4 0.3

4. Results and discussion

0.2

4.1. The role of surface tension in the calculation

1.3 1.1 1.0

=0.168kgm s D=25.4mm

0.9

/ mm

t=1e-4s t=5e-4s t=1e-5s t=5e-6s

Water Ti=46 C

1.2

20

40

1.1

60

80

100

degree

=0.088kgm =0.088kgm =0.103kgm =0.103kgm =0.168kgm =0.168kgm

1.0 0.9 0.8

mm

To reveal the role of the surface tension, calculations with C = 0.088, 0.103 and 0.168 kg m1 s1, respectively, and D = 25.4 mm, H = 6.3 mm are conducted for conditions with surface tension or not. The film thicknesses are compared in Fig. 5. As seen, there are great gaps in the region of 0–60° between two conditions. To understand the mechanisms of the surface tension effects, the free surface contours and streamlines at the regions near two stagnation points are compared between the two conditions. Fig. 6 shows the results of four film flow rates: 0.103, 0.117, 0.168 and 0.220 kg m1 s1. As seen, the results of the calculations

0

0.7

1

120

140

160

180

1

s without surface tension 1 s with surface tension 1 1 s without surface tension 1 1 s with surface tension 1 1 s without surface tension 1 1 s with surface tension 1

Water H=6.3mm D=25.4mm

0.6 0.5 0.4 0.3 0.2

0

20

40

60

80

100

120

140

160

180

degree

0.8 Fig. 5. Comparison of the local film thickness between the cases with and without surface tension.

0.7 0.6 0.5 0.4 0.3 0.2

0

20

40

60

80

100

degree

120

140

160

Fig. 3. Variation of the local film thickness with time step.

180

neglecting surface tension behave the following features: producing an obvious thinner film, failing in predicting the liquid vortex at the region near the upper stagnation point, occurring obvious flow separation and gas vortex in the region near the lower stagnation point which grows with increase of film flow rate. Instead, the model considering the surface tension can predict the film flow behaviors with a high accuracy, so the surface tension is considered in all following calculations.

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With surface tension Without surface tension

Gas vortex Liquid vortex Without surface tension With surface tension

(a) Γ = 0.103 kgm 1s

1

(b) Γ = 0.117 kgm 1s

1

(c) Γ = 0.168 kgm 1s

1

(d) Γ = 0.220 kgm 1s

1

Fig. 6. Free surface contours and streamlines for various film flow rates under with and without surface tension effect.

As known to us, the effect of surface tension on the liquid film decreases with increase in film flow rate, due to the ratio of surface tension to the inertial force decreases. As shown Fig. 5, it can also be found that the deviation of three film flow rates decreases when film flow rate increases, namely, the surface tension effect generally decreases with an increase in film flow rate.

4.2. Instantaneous film flow characteristics The liquid film flow on the horizontal tube is a transient process, so the parameters describing the flow patterns are dependent on the flowing time [17]. Instantaneous hydrodynamics of a liquid film flow on inclined plates [36,37] and horizontal tubes [21,38] have

been studied. In this section, the instantaneous phenomena of the liquid film flowing over the horizontal tube will be discussed. We calculated the liquid film flow on the tube with a diameter of 25.4 mm and a distributor height of 6.3 mm under three film flow rates of 0.088, 0.168 and 0.284 kg m1 s1. The instantaneous film thicknesses at different peripheral positions are displayed in Fig. 7. The curves show two features: (1) the variations of film thicknesses are less significant at the moderate film flow rate (C = 0.168 kg m1 s1); (2) the wavy motion is more violent at the upper part (h < 90°) than the lower part (h > 90°) of the tube at the smaller film flow rate (C = 0.088 kg m1 s1), while the opposite trend is found at the larger film flow rate (C = 0.284 kg m1 s1). The previous investigations probably focus on the instantaneous film thicknesses, which deviate largely with each other. According

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to the instantaneous behavior of liquid propagation, all film thicknesses in the following section are average values by using Eq. (13). It is worth pointing out that while a further increase in film flow rate will lead to more intensive wavy even turbulent, which may be researched in future work.

4.3. The effect of film flow rate on film distribution For two tube diameters, the variations of the film thickness with peripheral angle under different film flow rates are displayed in 10

1.4

20

40

90

140

160

1.2

/ mm

1.0 0.8 0.6 0.4 0.2 6000

8000

10000

time step

12000

14000

(a) Γ = 0.088 kgm 1s 1, D = 25.4 mm, H = 6.3 mm

Fig. 8. From these curves we can find that: the film thickness at the same peripheral position increases with an increase in the film flow rate; the difference in film thickness under various film flow rates is more obvious in the upper part than the lower part of the tube; in the region near the lower stagnation point, the film thickness varies irregularly due to the strong fluctuation of liquid film surface; the film thickness decreases dramatically at h > 90° when C < 0.168 kg m1 s1. According to Eqs. (1) and (2) the film thickness is symmetrical to the horizontal central line, and the minimum film thickness appears at h = 90°. However, the present results in Fig. 8 display a distinct asymmetrical distribution, where the minimum film thickness appears in the range of 110–150°, which is quite consistent with the results of Hou et al. [23] (90–115°), Fiorentino and Starace [5] and Ji et al. [31] (120°). The reason for the divergence is the inertial force is neglected in Nusselt’s solution, which produced smaller film flow velocity at the lower part of the tube and corresponding larger film thickness. While the film flow velocity increases along the tube due to application of the gravity, which produces thinner film on the lower part than the upper part. In addition, through comparison between the two figures in Fig. 8, we can see that the effect of film flow rate becomes insignificant with the increase of the tube diameter. The free surface contours and streamlines under different film flow rates are illustrated in Fig. 9. The inertial force increases with increasing film flow rate, which enhances the disturbance of the film flow, as seen the sizes of the vortexes near the upper and 2.0

10

1.4

20

40

90

140

160

1.6

1.2

/ mm

/ mm

0.8

1.2

1

0.135 0.168 0.220 0.284

1.0 0.8

0.6

0.6

0.4

0.4

0.2 6000

8000

10000

time step

12000

0.2

14000

0

20

10

1.4

20

40

90

140

40

60

80

100

120

140

160

180

degree

(b) Γ = 0.168 kgm 1s 1, D = 25.4 mm, H = 6.3 mm

(a) D = 12.7 mm, H = 6.3 mm 2.0

160

1

: kgm s

1.8

1.2

Water Ti=46 C

1.6 1.4

/ mm

1.0

/ mm

0.103 0.151 0.190 0.250

H=6.3mm D=12.7mm

1.4

1.0

1

: kgm s

Water Ti=46 C

1.8

0.8

0.088 0.103 0.135 0.168 0.25

=6.3mm D=25.4mm

1.2

1

0.095 0.117 0.151 0.190 0.284

1.0 0.8

0.6

0.6

0.4

0.4 0.2 6000

8000

10000

12000

time step (c) Γ = 0.284 kgm 1s 1, D = 25.4 mm, H = 6.3 mm

14000

Fig. 7. Instantaneous film thickness at different positions along the periphery angle for various film flow rates.

0.2

0

20

40

60

80

100

120

140

160

degree

(b) D = 25.4 mm, H = 6.3 mm Fig. 8. Variation of the local film thickness with film flow rate.

180

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3.0

lower stagnations enlarge with increase of film flow rate, and the upper vortex is smaller than the lower one.

2.5

4.4. The effect of tube diameter on film distribution

/ mm

6.35 19.05 34.9 50.8

=0.284kgm s H=6.3mm Ti=46 C

2.0

The relationships between the film thickness and the tube diameter from 6.35 to 50.8 mm are illustrated in Fig. 10. From these figures we can see that the film thickness generally decreases with increase of tube diameter, and that the film distribution becomes more uniform on tubes with larger diameters (34.9– 50.8 mm) because that the film velocities vary gently due to the smaller curvatures of the tubes with larger diameters. Fig. 11 displays the free surface contours and streamlines for different tube diameters. As seen, with the increase of tube diameter the flow fields are basically similar at the region near the upper stagnation point: with vortex at the region near the lower stagnation point, but the size of vortex decreases obviously because that the growing space of the vortex becomes smaller.

12.7 25.4 44.5

1.5 1.0 0.5 0

20

40

60

80

100

120

140

160

180

degree Fig. 10. Variation of the local film thickness with tube diameter.

with an increase in liquid distributor height, due to the larger liquid velocity under the application of gravity; (2) the effect of the liquid distributor height is more significant in the region of 0– 90° than the rest region; (3) at larger liquid distributor height, the liquid film is thinner near the upper stagnation point, mainly

4.5. The effect of liquid distributor height The general decreasing trends of the film thickness with increasing distributor height from 30 to 50.8 mm are illustrated in Fig. 12. It can be seen that: (1) the liquid film becomes thinner

Free surface

D: mm

Water

Streamlines

(a) Γ = 0.088 kgm 1s

1

. (b) Γ = 0.151 kgm 1s

1

liquid recirculation

(c) Γ = 0.284 kgm 1s

1

Fig. 9. Free surface contours and streamlines for various film flow rates (D = 25.4 mm, Ti = 46 °C, H = 6.3 mm).

C.-Y. Zhao et al. / International Journal of Heat and Mass Transfer 119 (2018) 564–576

571

(a) D = 6.35 mm

(b) D = 12.7 mm

(c) D = 19.05 mm

(d) D = 34.9 mm

(e) D = 50.8 mm Fig. 11. Free surface contours and streamlines for various tube diameters (C = 0.284 kg m1 s1, Ti = 46 °C, H = 6.3 mm).

because that the surface tension becomes smaller compared with the inertial force. Moreover, the film thickness at the same position decreases more sharply for smaller liquid distributor heights (3–20 mm), while decreases gently for the larger liquid distributor heights (20–50.8 mm). The effects of liquid distributor height on the free surface contours and streamlines are shown in Fig. 13. As shown, in the region near the upper stagnation point, the vortex becomes smaller and gradually diminishes with increase of liquid distributor height; in the region near the lower stagnation point, the vortex becomes larger even splits with increasing in liquid distributor height.

4.6. The effect of inlet liquid temperature Fig. 14 depicts the variation of film thickness with inlet liquid temperature ranging from 2 to 104 °C. We can see that the film thickness decreases monotonously with increase of liquid inlet temperature. It is because that both liquid viscosity and surface tension decrease with the increasing inlet liquid temperature, which facilitates spreading and thinning of the film. The free surface contours and streamlines at different liquid temperatures are compared in Fig. 15. We can see that: (1) the film flow becomes more unstable with increase of liquid temperature

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4.0

Water Ti=46°C

3.5 3.0

1

=0.284kgm s D=25.4mm

/ mm

2.5

because of decreasing in viscosity; (2) at the region near the upper stagnation point, the vortex enlarges and splits at Ti = 104 °C; (3) at the region near the lower stagnation point, there almost 0 vortex at Ti = 2 °C, 1 vortex at Ti = 46 °C (see Fig. 6(d)), 2 vortexes at Ti = 60 °C and 3 vortexes at Ti = 104 °C.

H: mm 3 10 20 30 40

1

6.3 15 25.4 35 44.5

50.8

4.7. Film thickness prediction 0.50

2.0

0.45

1.5

0.40 0.35

1.0

28

32

36

40

0.5 0

20

40

60

80

100

degree

120

140

160

180

Fig. 12. Variation of the local film thickness with liquid distribution height.

4.7.1. Dimensional analysis In this section, correlations predicting the film thickness of the falling film over a single horizontal smooth tube are constructed by dimensional analysis based on the current data. First the dimensional analysis method is adopted to obtain the related dimensionless numbers for the film thickness. According to the simulation ranges covered in this study, the film thickness d outside a single horizontal smooth tube could be influenced by the following relevant parameters: film flow rate C, liquid viscosity ll, liquid density ql, surface tension r, tube diameter D, liquid

(a) H = 3.0 mm

(b) H = 10.0 mm

(c) H = 20.0 mm

(d) H = 30.0 mm

(e) H = 50.8 mm Fig. 13. Free surface contours and streamlines for various liquid distribution heights (C = 0.284 kg m1 s1, Ti = 46 °C, D = 25.4 mm).

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2.0 1.8

2 30 60 82

Water =0.168kgm s D=25.4mm H=6.3mm

1.6 1.4

/ mm

distributor height H, and peripheral position represented by ph/180. The external force driving the liquid flow is gravity, represented by the gravitational acceleration g. By using the Buckingham Pi theorem to the above parameters, several dimensionless parameters are built: P1 ¼ d=D, P2 ¼ ll =C,

Ti: C

1.2

15 46 70 104

P3 ¼ qCl D2r, P4 ¼

q2l gD3 C2

, P5 ¼ H=D, P6 ¼ h. And therefore the

following dimensionless criteria can be obtained:

1.0

dimensionless film thickness d=D ¼ P1 ; film Reynolds number Re ¼ 4=P2 ¼ 4C=ll ;

0.8

modified Webber number We ¼ ðp2 P3 Þ

0.6 0.4 0

20

40

60

80

100

120

140

160

180

1

2

¼ p2Cq Dr; l

3 2 2 Archimedes number Ar ¼ P4 P2 2 ¼ ql gD =ll ; dimensionless liquid distributor height H=D ¼ P5 ; and peripheral position ph=180 ¼ pP6 =180 .

degree Fig. 14. Variation of the local film thickness with liquid temperature.

(a) Ti = 2 ºC

(b) Ti = 60 ºC

(c) Ti = 104 ºC Fig. 15. Free surface contours and streamlines for various inlet liquid temperature (C = 0.168 kg m1 s1, D = 25.4 mm, H = 6.3 mm).

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4.7.2. Correlations of falling film thickness Considering the general trend of the film thickness along the peripheral angle, correlations are constructed for three zones: 2–15°, 15–165° and 165–178°. The correlations of the three regimes are proposed as follows:

By using a regression analysis the values of C and a1–a5 are obtained. The resulted correlations for three zones, respectively, are: h = 2–15°:

2

ð14Þ

/Dpre 10

d=D ¼ CRea1 Wea2 Ara3 ðH=DÞa4 ðph=180Þa5

20 10 8 6

= 2 15 Re = 400 2485 We = 4.55 189.52 Ar = 7.17 3.67 H/D= 0.12 2

+20% 20%

4 2

d=D ¼ 3:45 2

1:67

 10 Re

0:63

We

Ar

0:87

0:74

ðH=DÞ

0:38

ðph=180Þ

ð15Þ 1 0.8 0.8 1

h = 15–165°:

d=D ¼ 0:42Re0:89 We0:14 Ar0:53 ðH=DÞ0:23 ðph=180Þ0:31

ð16Þ

97% of 84 data within 20% 2

4

6

8 10

20

2

/Dnum 10

(a) θ = 2 15º

h = 165–178°:

ð17Þ 20

–4

4.7.3. Comparison with data in literature In this section, the local film thicknesses of the liquid film flow over a single horizontal tube available in previous investigations are compared with the present correlations (Eqs. (15)–(17)), as shown in Fig. 18. Plotted in the figure are the dimensionless local film thicknesses in the literature (d/Dliterature) versus the predicted ones by the present correlations (d/Dpre). In this figure, the working fluids include pure water [21,23,39–41] and seawater [29,41], and the scopes of the entire data points are: Re from 141 to 1269, D from 19 to 25.4 mm, H from 3.2 to 40 mm and h from 19 to 170°. As seen, the present correlations predict 70% of 261 data with the deviations of ±30%. It worth noting that for the cases of H = 3.2 mm in Gstoehl et al. [21], the agreement between our prediction and test data is much worse, whose deviations exceed 50%. In their tests, Gstoehl et al. [21] observed two contrary trends of

2

10 8

/Dpre 10

where Re from 400–2485, We from 4.55  10 to 21.4  10 , Ar from 7.17  106 to 3.67  109, H/D from 0.12 to 2. The comparisons of the predictions from the present correlations with the present numerical data are shown in Fig. 16. The deviations are as follows: 97% of 84 data are within ±20% in h = 2–15°, 90% of 632 data are within ±20% in h = 15–165°, and 73% of 112 data are within ±30% in h = 165–178°. The larger divergence in h = 165–178° may be due to the effect of vortex and departure from the tube, so there is still room for the accuracy of the present correlation in this range. The comparisons between the present film thickness correlations and the ones in the literature of [10,19,23,31] are depicted in Fig. 17. In this figure the water at 46 °C is used as the working fluid, and film flow rate of 0.168 kg m1 s1, tube diameter of 25.4 mm and film distributor height of 6.3 mm are selected, and a larger liquid distributor height of 50.8 mm is used for a further comparison. It can be seen that: (1) the present correlations give the different film thickness due to the different film distributor, while the previous cannot recognize the effects of such parameters as film distributor and tube diameter; (2) the present correlations provide the largest film thickness in h = 2–110° while moderate film thickness in the rest zone; (3) the correlations in [10] and [19] provide symmetrical film distribution regarding to h = 90°; (4) the correlations in [10,19,23] overestimate while the one in [31] underestimates the film thickness in h = 110–178°; (5) the typical divergence between the minimum and the maximum prediction is around 34%.

= 15 165 Re = 400 2485 We = 4.55 189.52 Ar = 7.17 3.67 H/D= 0.12 2

+20% 20%

6 4

2

1 0.8 0.8 1

90% of 632 data within 20% 2

4

6

2

/Dnum 10

8 10

20

(b) θ = 15 165º

20

2

–4

/Dpre 10

d=D ¼ 2:78Re0:56 We0:44 Ar0:31 ðH=DÞ0:09 ðph=180Þ0:06

10 8

= 165 178 Re = 400 2485 We = 4.55 189.52 Ar = 7.17 3.67 H/D= 0.12 2

+20% 20%

6 4 2 1 0.8 0.8 1

55% 112 data within 20% 73% 112 data within 30% 2

4

6

8 10 2

/Dnum 10

20

(c) θ = 165 178º Fig. 16. Comparison of calculated and predicted local film thickness.

the film thickness with decrease of H: the film become thicker in H from 19.4 to 6.4 mm while become thinner in H from 6.4 to 3.2 mm, and they thought that the latter trend is inexplicable and so could not provide any logical explanation.

C.-Y. Zhao et al. / International Journal of Heat and Mass Transfer 119 (2018) 564–576

Eqs. 14-16, H=6.3mm Eqs. 14-16, H=50.8mm [23], H=6.3mm

1.4 1.2

mm

peripheral angle of 110–150° depending on the working conditions. (6) The flow fields near the two stagnation points are strongly dependent on the film flow rate, tube diameter, liquid distributor height and liquid temperature. (7) Correlations of falling film thickness are constructed based on the present calculation data. These correlations fit 97% of 84 data in h = 2–15° within ±20%, 90% of 632 data in h = 15–165° within ±20%, and 73% of 112 data in h = 165–178° within ±30%.

[6], H=6.3mm [19], H=6.3mm [31], H=6.3mm

Water =0.168kgm s D=25.4mm

1.0

575

0.8 0.6

Conflict of interest

34% 0.4 0.2

We declare that we have no conflict of interest.

0

20

40

60

80

100

120

140

160

180

/ degree Fig. 17. Comparison of the present film thickness correlations and the ones in literature.

[21],W,Re=574,H=19.4 [21],W,Re=574,H=6.4 [21],W,Re=744,H=19.4 [21],W,Re=744,H=6.4 [23],W,Re=574,H=10 [23],W,Re=574,H=33 [29],SW,Re=594,H=12.5 [40],W,Re=141-1269 [41],SW,Re=276,D=19 [41],SW,Re=368,D=25.4

4.5 4.0 3.5

[21],W,Re=574,H=9.5 [21],W,Re=574,H=3.2 [21],W,Re=744,H=9.5 [21],W,Re=744,H=3.2 [23],W,Re=574,H=16 [23],W,Re=574,H=40 [39],W,Re=574,D=19 [41],SW,Re=184,D=19 [41],SW,Re=368,D=19 [41],W,Re=368,D=19

3.0 Water, Seawater 2.5

Re = 141 1269 H/D= 0.16 1.58

2.0 1.5 1.0 +30%

30% 50%

0.5 0.5

1.0

1.5

2.0

70% 261 data within 30% 2.5

3.0

3.5

4.0

4.5

Fig. 18. Comparisons of present correlations with previous data in literature.

5. Conclusions In this paper, the laminar liquid film falling on a horizontal smooth tube is studied numerically. The hydrodynamic characteristics of the falling film flow is elucidated, and new correlations of falling film thickness based on the current data are established. The following conclusions can be drawn: (1) The numerical results are in good agreement with the ones in previous studies. (2) The surface tension is necessary in the current calculations, especially for the regions near two stagnation points. (3) The liquid film propagation outside a horizontal tube has obvious instantaneous behaviors. (4) The film thickness increases with increase in film flow rate, while decreases with increases in tube diameter, liquid distributor height and inlet liquid temperature. (5) The film distribution along the peripheral angle is unsymmetrical, and the minimum thickness appears in

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