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A study on a droplet impact on a fiber during coalescence-separation: Phenomena and models
Journal Pre-proofs A Study on a droplet impact on a fiber during coalescence-separation: phenomena and models Wenquan Gu, Shenglin Yan, Zhishan Bai PII: DOI: Reference:
S0009-2509(19)30827-9 https://doi.org/10.1016/j.ces.2019.115337 CES 115337
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
2 August 2019 19 October 2019 2 November 2019
Please cite this article as: W. Gu, S. Yan, Z. Bai, A Study on a droplet impact on a fiber during coalescenceseparation: phenomena and models, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces. 2019.115337
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A Study on a droplet impact on a fiber during coalescence-separation: phenomena and models
Article Reference: CES_115337 Authors: Wenquan Gu, Shenglin Yan, Zhishan Bai* Author affiliations: School of Mechanical and Power Engineering, East China University of Science and Technology Correspondence information: Zhishan Bai, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China. Tel.: +86-021-64253693, Fax: +86-021-64253693, E-mail: [email protected]
A Study on a droplet impact on a fiber during coalescence-separation: phenomena and models Abstract Droplets impact on fibers occurred in liquid phase plays a vital role in the coalescenceseparation applications, however, the mechanism and details are still unclear. Here we investigated the impact process through energy conversion and impact dynamics and two models are established to determine the threshold velocities of capture. The adhering phenomenon is observed when the impact velocity is smaller than the minimum threshold velocity Vc1. The droplet may be captured by fibers when the impact velocity is larger than Vc1 and smaller than the maximum threshold velocity Vc2. These two velocities limit a range in which the droplet may be captured by fibers are significant to facilitate the optimization of coalescence separation. This study opens up opportunities for understanding the details of coalescence separation as well as improving efficiency of industrial applications.
Keywords: coalescence; droplets impact; fibers; threshold velocity
1 Introduction
The technology of coalescence separation has been widely used in modern industry for separating oil-in-water (O/W) or water-in-oil (W/O) emulsion and of great significance in petrochemical industry, metal processing, energy purification and other fields [1]. Benefiting from its high efficiency, low energy consumption and eco-friendly, fibrous media applied in coalescence has attracted great interests [2]. Researchers have made great efforts to improve separation efficiency and found that the structure of coalescer [3-7] (bed height, wettability, bed geometry, the ration of length to width etc.), operating conditions (superficial velocity, emulsion concentration, etc.) and emulsion conditions (droplets sizes and surfactant effects, etc.) have a significant influence on separation performance [8-13]. However, the mechanism of coalescence separation is still unclear and the design of coalesce principally relys on experimental experiences which result in a huge waste of resources. The impact between droplets and fibers is the most common process during coalescence, and it may be helpful for understanding the mechanism by exploring the impact. The impact between droplets and fibers are widely exist in nature and industries such as rain droplets captured by spider webs [14], collecting water from foggy air in some deserts [15], and filters used to capture the liquid fraction of an aerosol [16]. The researches of droplets impact on fibers are developed from the researches of impact between droplets and non-plane surfaces [17-19]. And the first research of impact between droplets and fibers are investigated by Hung and Yao [20]. They classified the typical experiment phenomena as disintegration and dripping and established a relationship between phenomena and Weber number, Bound number and the ratio of diameters.
Besides, further studies showed that wettability also played an important role and the impact between droplets and fiber meshes was studied further [21]. Lorenceau et al [22] focused on the dynamics of the impact and found that the droplet cannot be captured by fibers when the impact velocity is larger than a critical value and this critical value is termed as the threshold velocity of capture and investigated theoretically and experimentally. The following investigations showed that off-center impact and incline fibers may significantly improve the capture ability of fibers
[23, 24].
Liang et al
[25, 26]
explored the impact between droplets and wetted fibers and some special phenomena such as rebound and spreading were observed. Comtet et al [27] and Dressaire et al [28] investigated the impact between droplets and thin flexible fibers respectively, and found that an optimal fiber length can maximize the efficiency of capture which may enhance capture ability. Kim et al
[21]
presented a regime map which identifies the typical
outcomes of impact into three: capturing, single drop falling, and splitting and explained the regime boundaries based on scale analysis of forces. This research focus on the impact occurred in liquid ambient that previous researches paid little attention. The mechanism and phenomena may be different when the ambient fluid changes form air to liquid. We investigate the energy conversion and dynamics during the impact process and two models are established. A threshold velocity below which the droplet may not be captured by fibers and another threshold velocity above which the droplet may not be captured either are proposed according to the models. These threshold velocities limit a range in which the droplet may be captured by fibers are of great importance to the industrial applications and understand
the coalescence separation process.
2 Experiment Setup and Materials 2.1 Experiment Setup The schematic diagram of the experimental apparatus is shown in Fig.1. The main facilities contain an air compressor, a pressure adjuster, an injector and an injection pump. High-speed cameras and a computer are used to record the impacting process. Besides, a light and a diffuser are necessary for the experiment, and the fiber is fixed on a cross slippery platform in a water tank.
Fig. 1 The schematic diagram of the experiment apparatus (a. The fiber is fix on a transparent resin platform which is fix on two orthogonal cross slippery platforms; b. Oil droplets are generated on the top of the needle)
In this experiment, the size and velocity of the droplet can be adjusted through the pressure adjustor and the injection pump. The position of fiber can be fine adjustment by these cross slippery platforms to ensure droplets impact on center of the fiber. Two high-speed cameras are adapted in perpendicular directions, the radial direction is a Phantom (Fastcam SA-X2) high-speed camera with the capacity of 5000fps and equipped with a 10× Navita camera lens aiming at recording the impact process of front
view and the axial direction is an IDT (NX4-S3) high-speed camera with the capacity of 3000fps aiming at ensure the droplet rising directly. Take advantage of the synchronous trigger to ensure these two cameras start at the same time. The cold light source is provided by a light with a power of 500W, and a light diffuser is applied to ensure the light is distributed uniformly on the fiber. The fiber is fixed on the center of the viewfinder and the process from approaching to detachment can be fully recorded. 2.2 Materials The materials chosen in this experiment are isooctane droplets and polypropylene fibers that are widely used in petrochemical industry. Isooctane is a crucial component of gasoline and of great importance to modern industry, and polypropylene fibers are pervasive used in coalescence process. According to these facts the impact between isooctane droplets and polypropylene fibers are explored in this study. The properties of isooctane and water in this study are shown in Table 1. Benefit from the large viscosity and density, the isooctane droplet could move vertically and the impact between droplet and fiber is centered. Table.1 The physical property parameters (20℃) Isooctane
Water
Density (kg/m3)
692
998
Surface Tension (mN/m)
20.5
72.0
Dynamic Viscosity (Pa·s)
5.04 104
8.95 10 4
3 Result and discussion The industrial coalescence separations exist an optimal operating condition which requires a suitable flow rate, and the separation efficiency decreases sharply when the
operating condition is beyond the optimal operating condition. As the interaction between a droplet and a fiber is the basic process during the coalescence separation, we assume the droplet may not be captured by fibers when the impact velocity is out of a range. The lower limit of the range is termed as the minimum threshold velocity of capture Vc1, and the upper limit of the range is termed as the maximum threshold velocity of capture Vc2. The theoretical analyses and experiment verification are discussed below. 3.1 The minimum threshold velocity of capture The process of droplets impact on fibers is complex due to the moving and shape of droplets are irregular. For convence, we investigate the energy conversion between two states during the impact process: before impact and the equilibrium. The essential of droplets captured by fibers is new generated droplet-fiber interface replaces the former continuous-fiber interface, and there are three phases involved in the capture process. In this study the continuous phase is termed as phase 1, the fiber is termed as phase 2 and the droplet is themed as phase 3 and shown in Fig. 2.
Fig.2 The morphology changes of droplets before and after the impact (a. Before impact, the droplet appears spherical shape with a velocity V; b. The equilibrium, thedroplet appears barrel state after impact)
There are several energies involved during the impact process: kinetic energy (Ek),
potential energy (Ep, including gravity and buoyancy), surface energy (Es) and dissipations (Ed), and according to the energy conservation law, the global energy keeps constant before and after the impact:
Ek Es1 E p1 Es 2 E p 2 Ed
(1)
Hence, the kinetic energy can be written as:
Ek Es E p Ed
(2)
The energy conversion during the capture process is kinetic energy converses to surface energy, potential energy and dissipation according to Eq. 2. The expression of each energy is discussed as follow. The change of surface energy is related to the interfacial areas between phases, A1 (green lines) represents the interfacial area between phase 2 and 3, A2 (blue lines) represents the interfacial area between phase 1 and 3, and A3 (red lines) represents the interfacial area between phase 1 and 2. Besides, γ12, γ23, γ13 represent the interfacial tensions respectively. The change of interfacial energy of the system can be calculated by:
Es A1b ( 23 12 ) ( A2b A2 a ) 13
(3)
The change of potential energy is caused by the displacement of the droplet. The change of potential energy is determined by the displacement and can be written as: E p Fb Fg x
(4)
Where Δx is the displacement of droplet mass center, Fb and Fg are buoyancy and gravity acting on the droplet. The dissipation is mainly caused by viscous friction and the drag force during the impact process. The viscous friction causes the change of internal energy (Ei) which result in the change of system temperature and the dissipation
caused by drag force can be marked as: Ed Fd ds Ei
(5)
Where Ei is the internal change, Fd is the drag force acting on the droplet and ds is the path droplet moving through. In this study we assume the temperature keep constant and therefore Ei=0. When the droplet reaches equilibrium, the change of surface energy and potential energy are certain due to the morphology and displacement are fixed. Therefore, the kinetic energy in Eq. 2 reaches minimum when the dissipation reaches minimum ( Ed min Fd dsmin )and the impact velocity at this time is the minimum threshold velocity of capture Vc1.
2 A1b 23 12 A2b A2 a 13 Fb Fg x Fd dsmin Vc1 M
12
(6)
This threshold velocity is unusually hard to calculate due to the irregular morphology of droplets and the random movement. However, it may be get through numerical calculation under some certain circumstances. Here we discuss the condition that the captured droplets appear symmetric barrel shape, and the geometric relationship of a barrel state droplet is shown in Fig. 3.
Fig. 3 The geometric relationship of a barrel shape droplet (The origin point is the center of droplet. The droplet length Ld, droplet height Rh and contact angle θ are shown above)
The calculation of interfacial areas is the key point of the change of surface energy. According to Carrol
[30],
the Laplace excess pressure P on arbitrary point P (x, z) of
the droplet surface is constant and the interfacial area A1 can be written as:
A1 4 R f aR f F , k Rh E , k
(7)
The interfacial area A2 of the droplet and ambient phase can be calculated by:
A2 4 (aR f Rh ) Rh E , k
(8)
Where F , k and E , k represent the elliptic integral of first class and second class, and a, k , are the parameters related to the droplet height Rh, fiber radius Rf and contact angle θ:
a
Rh cos R f Rh R f cos
， k 1
a 2 R 2f Rh2
R 2f 1 ， sin 2 (1 2 ) k Rh 2
(9)
Under this condition, the displacement of droplet is the radius of droplet and the minimum dissipation energy can be calculated through the mean drag force. The drag force is caused by fibers and fluid and can be written as Fd=(2RfRd+πRd2/2) ρ1CdV2, Cd is the drag force coefficient. Hence, the potential energy and the dissipation energy can be written as: 4 E p Rd4 g 1 3 3
Ed Rd3 Rd2 R f 4
(10)
2 1CdV
(11)
Where ρ1 and ρ3 are the densities of ambient fluid and droplet, V is the impact velocity. Hence, the minimum threshold velocity of capture Vc1 can be written as: 1 Vc1 3 3 2 R R d 3 ( d Rd2 R f ) 1Cd 3 4
12
4 R f [aR f F ( , k )+Rh E ( , k )]( 23 12 ) 2 (4 (aR f Rh Rh ) Rh E ( , k ) 4 Rd ) 13 4 4 Rd g ( 1 3 ) 3
(12)
This is the minimum threshold velocity of capture when the captured droplet state barrel state. For a given system, this velocity can be calculated numerically by a MATLAB program. 3.2 The maximum threshold velocity of capture When the flow rate increases above a certain velocity the separation efficiency drops sharply which means the droplet detaches from the fiber after captured. This certain velocity is the regarded as the maximum threshold velocity of capture Vc2. Forces acting on a droplet during the impact process are gravity, buoyancy, surface tension and drag force is shown in Fig. 4.
Fig. 4 Forces acting on the droplet during the impact process
The gravity (Fg=Mg) and buoyancy (Fb=ρ1Mg/ρ3) keep constant during the impact and the drag force is caused by the fiber and ambient fluid and can be written as Fd=(2RfRd+πRd2/2) ρ1CdV2 as discussed before. The surface tension acting on the droplet is complex and according to previous researches [22] the surface tension can be regarded as along the normal direction of the contact point between droplets and fibers and the value is Fs total 2 13 R f . Hence, the total surface tension acting on the droplet in vertical direction can be written as Fs 4 13 R f sin . The angle α changes from 90° to 90° which means the direction changes from upward to downward and the value
decreases from maximum to zero and then increases to maximum again. A theoretical model was established by Lorenceau [23] to calculate this threshold velocity when the droplet impact on fiber in air surroundings and get good arrangement. However, their model neglect the change of surface tension and the drag force caused by the surroundings which may cause great errors when the impact happened in a liquid phase due to the liquid viscosity is much larger than air. In this study, we investigate the dynamics of detaching process and donate x as the axis pointing upward and set its origin point at the position of the center of the fiber. Hence, the droplet contacts the fiber at the position x=-Rd, and as we assumed the droplet keeps spherical during the detaching process, the droplet detaches at the position x=Rd. For the changing surface tension we assume Fs changes linearly with the displacement of droplet and can be written as Fs 4 13 R f x Rd .
Fig. 5 Displacements of droplet before and after the impact on a fiber
For the detaching process, according to Newton’s second law, the velocity of a droplet with mass M changes as follow: Fb Fg Fs Fd M
dV dt
(13)
The term of the right hand side of the above equation can be regarded as the inertia force (Fi) acting on a droplet. Combing forces acting on the droplet with Eq. (13), the
equation can be written as:
4 x 13 R f 1Mg dV Mg 2 R f Rd Rd2 1CdV 2 M 2 3 Rd dt
(14)
The velocity of droplet can be solved as: 1 b V 2 Ce ax (c bx ) a a
Where
a
2 g ( 1 3 ) 31Cd 2 R f Rd / 2 6 R ， b 13 4f ， c 2 3 Rd 23 3 Rd
(15) are
the
parameters that related to the system physical properties. Under the condition the droplet detach from the fiber, the detach velocity V(x=Rd) need to be equal to or larger than 0 and when the detach velocity satisfy V(x=Rd)=0, Eq. 14 can be written as: 1 1 b b V 2 (c bRd )e a ( Rd x ) (c bx ) a a a a
(16)
This is the threshold condition and the impact velocity is the maximum threshold velocity of capture and satisfy V(x=-Rd)=Vc2, hence, the threshold velocity can be calculated as: 1 b 1 b Vc 2 [ (c bRd ) (c bRd )e 2 aRd ]1 2 a a a a
(17)
This velocity is regarded as the maximum velocity of capture and termed as the upper threshold velocity of capture. This threshold velocity can be easily get through calculation for a given system. 3.3 Experiment results and verifications To verify the theoretical deductions above, we experimented the droplet impact on a fiber. For the isooctane-polypropylene-water system, we choose the droplet diameter is 120μm and the fiber diameter is 20 μm. According to Eq. 12 and Eq. 17, the threshold velocities is calculated and the value is Vc1≈19.6mm/s, Vc2≈2.1m/s. Due to the limit of
our experiment setup, the impact velocity is in a range of 3mm/s-60mm/s which means the detach phenomenon may not be observed. We set the impact velocity larger than and less than the threshold velocity Vc1 and the impact phenomena is shown in Fig. 6.
Fig.6 The impact phenomenon of a droplet impact on a fiber in liquid ambient (a. Adhering phenomenon, the droplet diameter is 120μm and the impact velocity is 17.5mm/s; b. Capture phenomenon, the droplet diameter is 120μm and the impact velocity is 41.5mm/s)
When the impact velocity is less than Vc1, the phenomenon is shown in Fig. 6(a). The impact velocity of this droplet was 17.5mm/s which is smaller than Vc1 (≈19.6mm/s), there was no obvious deformation during the impact and the droplet finally adhere on the fiber surface. Besides, the droplet reached the equilibrium once impacted on the fiber and adhered on the fiber surface for a long time. The typical classification of this phenomenon is droplet adhering on the fiber surface without obvious deformation and this phenomenon is regarded as adhering phenomenon. With the increase of impact velocity, the capture phenomenon was observed and is shown in Fig. 6(b). The impact velocity of this droplet was 41.5mm/s and the fiber immersed in the droplet after the impact and the droplet deformed sharply. The typical classification of this phenomenon is droplet spread on the fiber with obvious deformation and state barrel state.
The detaching phenomenon is not observed in this study due to the limit of our experiment. As the viscosity of liquid is much larger than air, the droplet velocity decreases sharply after generated and reaches a constant according to Stocks equation. Hence, it’s hard to generate a droplet with the impact velocity larger than Vc2. However, from the tendency in Fig. 6(b) we can see that, the droplet was tend to detach from the fiber (t=12.6ms) and the position of droplet is above the equilibrium position. The droplet may detach from the fiber with the increase of impact droplet. Therefore, the impact phenomena occurred in liquid ambient may be regarded as adhering phenomenon, capturing phenomenon and detaching phenomenon.
4 Conclusion The impact between a droplet and a fiber occurred in liquid ambient is investigated and two models are established. One model is established through the energy conservation law and to investigate the minimum threshold velocity of capture Vc1. When the impact velocity is smaller than Vc1 the adhering phenomenon was observed by experiment and the capture phenomenon was observed when the impact velocity is larger than Vc1. Another model is established by the forces analyses of the impact dynamics to study the maximum threshold velocity Vc2. The droplet may not be captured by fibers when the impact velocity is larger than Vc2 and the droplet may detach from the fiber. This model is an improvement of previous researches and modified to fit the liquid ambient environment. Combing the theoretical analysis and experiment results, the typical outcomes of a droplet impact on a fiber in liquid ambient can be classified as
adhering phenomenon, captureing phenomenon and detaching phenomenon. These threshold velocities are of great importance to industrial coalescence separation applications. They can be used to optimize the structure of devices by calculating the property impact velocity and choose the fiber size. The small/pilot scale tests may be reduced and may bring great economic benefit.
Nomenclature Rd
m
Radius of a droplet
Rf
m
Radius of a fiber
Dd
m
Diameter of a droplet
Df
m
Diameter of a fiber
ρ1
kg/m3
Density of phase 1
ρ2
kg/m3
Density of phase 2
ρ3
kg/m3
Density of phase 3
γ12
N/m
Interfacial tension between phase 1 and 2
γ23
N/m
Interfacial tension between phase 2 and 3
γ13
N/m
Interfacial tension between phase 1 and 3
θ
°
Contact angle
A1
m2
Interfacial area between phase 2 and 3
A2
m2
Interfacial area between phase 1 and 3
A3
m2
Interfacial area between phase 1 and 2
ΔEs
J
x
m
Displacement of a droplet
Fb
N
Buoyancy acting on a droplet
Fg
N
Gravity acting on a droplet
Fs
N
Fd
N
Drag force acting on a moving droplet
M
kg
Mass of a droplet
V
m/s
Impact velocity
Vc1
m/s
The threshold velocity of impact also termed as
The change of interfacial energy after and before the impact
Surface tension acting on a droplet in vertical direction
the lower limit of capture velocity The threshold velocity of capture also termed as
Vc2
m/s
μd
Pa·s
Dynamic viscosity of a droplet
μl
Pa·s
Dynamic viscosity of ambient fluid
μ*
1
The ratio of μd to μl
the upper limit of capture velocity
Acknowledgments This work was supported by the National Natural Science Foundation of China, China (51578239) and Scientific Research Projects of Shanghai, China (17DZ1202802).
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Highlights 1. A model was established to calculate the minimum threshold velocity of capture. 2. Another model was improved to calculate the maximum threshold velocity of capture. 3. A new phenomenon of a droplet adhere on the fiber surface was observed. 4. Two threshold velocities limit a range map in which droplets can be captured.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0009-2509(19)30827-9 https://doi.org/10.1016/j.ces.2019.115337 CES 115337
To appear in:
Chemical Engineering Science
Received Date: Revised Date: Accepted Date:
2 August 2019 19 October 2019 2 November 2019
Please cite this article as: W. Gu, S. Yan, Z. Bai, A Study on a droplet impact on a fiber during coalescenceseparation: phenomena and models, Chemical Engineering Science (2019), doi: https://doi.org/10.1016/j.ces. 2019.115337
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier Ltd.
A Study on a droplet impact on a fiber during coalescence-separation: phenomena and models
Article Reference: CES_115337 Authors: Wenquan Gu, Shenglin Yan, Zhishan Bai* Author affiliations: School of Mechanical and Power Engineering, East China University of Science and Technology Correspondence information: Zhishan Bai, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China. Tel.: +86-021-64253693, Fax: +86-021-64253693, E-mail: [email protected]
A Study on a droplet impact on a fiber during coalescence-separation: phenomena and models Abstract Droplets impact on fibers occurred in liquid phase plays a vital role in the coalescenceseparation applications, however, the mechanism and details are still unclear. Here we investigated the impact process through energy conversion and impact dynamics and two models are established to determine the threshold velocities of capture. The adhering phenomenon is observed when the impact velocity is smaller than the minimum threshold velocity Vc1. The droplet may be captured by fibers when the impact velocity is larger than Vc1 and smaller than the maximum threshold velocity Vc2. These two velocities limit a range in which the droplet may be captured by fibers are significant to facilitate the optimization of coalescence separation. This study opens up opportunities for understanding the details of coalescence separation as well as improving efficiency of industrial applications.
Keywords: coalescence; droplets impact; fibers; threshold velocity
1 Introduction
The technology of coalescence separation has been widely used in modern industry for separating oil-in-water (O/W) or water-in-oil (W/O) emulsion and of great significance in petrochemical industry, metal processing, energy purification and other fields [1]. Benefiting from its high efficiency, low energy consumption and eco-friendly, fibrous media applied in coalescence has attracted great interests [2]. Researchers have made great efforts to improve separation efficiency and found that the structure of coalescer [3-7] (bed height, wettability, bed geometry, the ration of length to width etc.), operating conditions (superficial velocity, emulsion concentration, etc.) and emulsion conditions (droplets sizes and surfactant effects, etc.) have a significant influence on separation performance [8-13]. However, the mechanism of coalescence separation is still unclear and the design of coalesce principally relys on experimental experiences which result in a huge waste of resources. The impact between droplets and fibers is the most common process during coalescence, and it may be helpful for understanding the mechanism by exploring the impact. The impact between droplets and fibers are widely exist in nature and industries such as rain droplets captured by spider webs [14], collecting water from foggy air in some deserts [15], and filters used to capture the liquid fraction of an aerosol [16]. The researches of droplets impact on fibers are developed from the researches of impact between droplets and non-plane surfaces [17-19]. And the first research of impact between droplets and fibers are investigated by Hung and Yao [20]. They classified the typical experiment phenomena as disintegration and dripping and established a relationship between phenomena and Weber number, Bound number and the ratio of diameters.
Besides, further studies showed that wettability also played an important role and the impact between droplets and fiber meshes was studied further [21]. Lorenceau et al [22] focused on the dynamics of the impact and found that the droplet cannot be captured by fibers when the impact velocity is larger than a critical value and this critical value is termed as the threshold velocity of capture and investigated theoretically and experimentally. The following investigations showed that off-center impact and incline fibers may significantly improve the capture ability of fibers
[23, 24].
Liang et al
[25, 26]
explored the impact between droplets and wetted fibers and some special phenomena such as rebound and spreading were observed. Comtet et al [27] and Dressaire et al [28] investigated the impact between droplets and thin flexible fibers respectively, and found that an optimal fiber length can maximize the efficiency of capture which may enhance capture ability. Kim et al
[21]
presented a regime map which identifies the typical
outcomes of impact into three: capturing, single drop falling, and splitting and explained the regime boundaries based on scale analysis of forces. This research focus on the impact occurred in liquid ambient that previous researches paid little attention. The mechanism and phenomena may be different when the ambient fluid changes form air to liquid. We investigate the energy conversion and dynamics during the impact process and two models are established. A threshold velocity below which the droplet may not be captured by fibers and another threshold velocity above which the droplet may not be captured either are proposed according to the models. These threshold velocities limit a range in which the droplet may be captured by fibers are of great importance to the industrial applications and understand
the coalescence separation process.
2 Experiment Setup and Materials 2.1 Experiment Setup The schematic diagram of the experimental apparatus is shown in Fig.1. The main facilities contain an air compressor, a pressure adjuster, an injector and an injection pump. High-speed cameras and a computer are used to record the impacting process. Besides, a light and a diffuser are necessary for the experiment, and the fiber is fixed on a cross slippery platform in a water tank.
Fig. 1 The schematic diagram of the experiment apparatus (a. The fiber is fix on a transparent resin platform which is fix on two orthogonal cross slippery platforms; b. Oil droplets are generated on the top of the needle)
In this experiment, the size and velocity of the droplet can be adjusted through the pressure adjustor and the injection pump. The position of fiber can be fine adjustment by these cross slippery platforms to ensure droplets impact on center of the fiber. Two high-speed cameras are adapted in perpendicular directions, the radial direction is a Phantom (Fastcam SA-X2) high-speed camera with the capacity of 5000fps and equipped with a 10× Navita camera lens aiming at recording the impact process of front
view and the axial direction is an IDT (NX4-S3) high-speed camera with the capacity of 3000fps aiming at ensure the droplet rising directly. Take advantage of the synchronous trigger to ensure these two cameras start at the same time. The cold light source is provided by a light with a power of 500W, and a light diffuser is applied to ensure the light is distributed uniformly on the fiber. The fiber is fixed on the center of the viewfinder and the process from approaching to detachment can be fully recorded. 2.2 Materials The materials chosen in this experiment are isooctane droplets and polypropylene fibers that are widely used in petrochemical industry. Isooctane is a crucial component of gasoline and of great importance to modern industry, and polypropylene fibers are pervasive used in coalescence process. According to these facts the impact between isooctane droplets and polypropylene fibers are explored in this study. The properties of isooctane and water in this study are shown in Table 1. Benefit from the large viscosity and density, the isooctane droplet could move vertically and the impact between droplet and fiber is centered. Table.1 The physical property parameters (20℃) Isooctane
Water
Density (kg/m3)
692
998
Surface Tension (mN/m)
20.5
72.0
Dynamic Viscosity (Pa·s)
5.04 104
8.95 10 4
3 Result and discussion The industrial coalescence separations exist an optimal operating condition which requires a suitable flow rate, and the separation efficiency decreases sharply when the
operating condition is beyond the optimal operating condition. As the interaction between a droplet and a fiber is the basic process during the coalescence separation, we assume the droplet may not be captured by fibers when the impact velocity is out of a range. The lower limit of the range is termed as the minimum threshold velocity of capture Vc1, and the upper limit of the range is termed as the maximum threshold velocity of capture Vc2. The theoretical analyses and experiment verification are discussed below. 3.1 The minimum threshold velocity of capture The process of droplets impact on fibers is complex due to the moving and shape of droplets are irregular. For convence, we investigate the energy conversion between two states during the impact process: before impact and the equilibrium. The essential of droplets captured by fibers is new generated droplet-fiber interface replaces the former continuous-fiber interface, and there are three phases involved in the capture process. In this study the continuous phase is termed as phase 1, the fiber is termed as phase 2 and the droplet is themed as phase 3 and shown in Fig. 2.
Fig.2 The morphology changes of droplets before and after the impact (a. Before impact, the droplet appears spherical shape with a velocity V; b. The equilibrium, thedroplet appears barrel state after impact)
There are several energies involved during the impact process: kinetic energy (Ek),
potential energy (Ep, including gravity and buoyancy), surface energy (Es) and dissipations (Ed), and according to the energy conservation law, the global energy keeps constant before and after the impact:
Ek Es1 E p1 Es 2 E p 2 Ed
(1)
Hence, the kinetic energy can be written as:
Ek Es E p Ed
(2)
The energy conversion during the capture process is kinetic energy converses to surface energy, potential energy and dissipation according to Eq. 2. The expression of each energy is discussed as follow. The change of surface energy is related to the interfacial areas between phases, A1 (green lines) represents the interfacial area between phase 2 and 3, A2 (blue lines) represents the interfacial area between phase 1 and 3, and A3 (red lines) represents the interfacial area between phase 1 and 2. Besides, γ12, γ23, γ13 represent the interfacial tensions respectively. The change of interfacial energy of the system can be calculated by:
Es A1b ( 23 12 ) ( A2b A2 a ) 13
(3)
The change of potential energy is caused by the displacement of the droplet. The change of potential energy is determined by the displacement and can be written as: E p Fb Fg x
(4)
Where Δx is the displacement of droplet mass center, Fb and Fg are buoyancy and gravity acting on the droplet. The dissipation is mainly caused by viscous friction and the drag force during the impact process. The viscous friction causes the change of internal energy (Ei) which result in the change of system temperature and the dissipation
caused by drag force can be marked as: Ed Fd ds Ei
(5)
Where Ei is the internal change, Fd is the drag force acting on the droplet and ds is the path droplet moving through. In this study we assume the temperature keep constant and therefore Ei=0. When the droplet reaches equilibrium, the change of surface energy and potential energy are certain due to the morphology and displacement are fixed. Therefore, the kinetic energy in Eq. 2 reaches minimum when the dissipation reaches minimum ( Ed min Fd dsmin )and the impact velocity at this time is the minimum threshold velocity of capture Vc1.
2 A1b 23 12 A2b A2 a 13 Fb Fg x Fd dsmin Vc1 M
12
(6)
This threshold velocity is unusually hard to calculate due to the irregular morphology of droplets and the random movement. However, it may be get through numerical calculation under some certain circumstances. Here we discuss the condition that the captured droplets appear symmetric barrel shape, and the geometric relationship of a barrel state droplet is shown in Fig. 3.
Fig. 3 The geometric relationship of a barrel shape droplet (The origin point is the center of droplet. The droplet length Ld, droplet height Rh and contact angle θ are shown above)
The calculation of interfacial areas is the key point of the change of surface energy. According to Carrol
[30],
the Laplace excess pressure P on arbitrary point P (x, z) of
the droplet surface is constant and the interfacial area A1 can be written as:
A1 4 R f aR f F , k Rh E , k
(7)
The interfacial area A2 of the droplet and ambient phase can be calculated by:
A2 4 (aR f Rh ) Rh E , k
(8)
Where F , k and E , k represent the elliptic integral of first class and second class, and a, k , are the parameters related to the droplet height Rh, fiber radius Rf and contact angle θ:
a
Rh cos R f Rh R f cos
， k 1
a 2 R 2f Rh2
R 2f 1 ， sin 2 (1 2 ) k Rh 2
(9)
Under this condition, the displacement of droplet is the radius of droplet and the minimum dissipation energy can be calculated through the mean drag force. The drag force is caused by fibers and fluid and can be written as Fd=(2RfRd+πRd2/2) ρ1CdV2, Cd is the drag force coefficient. Hence, the potential energy and the dissipation energy can be written as: 4 E p Rd4 g 1 3 3
Ed Rd3 Rd2 R f 4
(10)
2 1CdV
(11)
Where ρ1 and ρ3 are the densities of ambient fluid and droplet, V is the impact velocity. Hence, the minimum threshold velocity of capture Vc1 can be written as: 1 Vc1 3 3 2 R R d 3 ( d Rd2 R f ) 1Cd 3 4
12
4 R f [aR f F ( , k )+Rh E ( , k )]( 23 12 ) 2 (4 (aR f Rh Rh ) Rh E ( , k ) 4 Rd ) 13 4 4 Rd g ( 1 3 ) 3
(12)
This is the minimum threshold velocity of capture when the captured droplet state barrel state. For a given system, this velocity can be calculated numerically by a MATLAB program. 3.2 The maximum threshold velocity of capture When the flow rate increases above a certain velocity the separation efficiency drops sharply which means the droplet detaches from the fiber after captured. This certain velocity is the regarded as the maximum threshold velocity of capture Vc2. Forces acting on a droplet during the impact process are gravity, buoyancy, surface tension and drag force is shown in Fig. 4.
Fig. 4 Forces acting on the droplet during the impact process
The gravity (Fg=Mg) and buoyancy (Fb=ρ1Mg/ρ3) keep constant during the impact and the drag force is caused by the fiber and ambient fluid and can be written as Fd=(2RfRd+πRd2/2) ρ1CdV2 as discussed before. The surface tension acting on the droplet is complex and according to previous researches [22] the surface tension can be regarded as along the normal direction of the contact point between droplets and fibers and the value is Fs total 2 13 R f . Hence, the total surface tension acting on the droplet in vertical direction can be written as Fs 4 13 R f sin . The angle α changes from 90° to 90° which means the direction changes from upward to downward and the value
decreases from maximum to zero and then increases to maximum again. A theoretical model was established by Lorenceau [23] to calculate this threshold velocity when the droplet impact on fiber in air surroundings and get good arrangement. However, their model neglect the change of surface tension and the drag force caused by the surroundings which may cause great errors when the impact happened in a liquid phase due to the liquid viscosity is much larger than air. In this study, we investigate the dynamics of detaching process and donate x as the axis pointing upward and set its origin point at the position of the center of the fiber. Hence, the droplet contacts the fiber at the position x=-Rd, and as we assumed the droplet keeps spherical during the detaching process, the droplet detaches at the position x=Rd. For the changing surface tension we assume Fs changes linearly with the displacement of droplet and can be written as Fs 4 13 R f x Rd .
Fig. 5 Displacements of droplet before and after the impact on a fiber
For the detaching process, according to Newton’s second law, the velocity of a droplet with mass M changes as follow: Fb Fg Fs Fd M
dV dt
(13)
The term of the right hand side of the above equation can be regarded as the inertia force (Fi) acting on a droplet. Combing forces acting on the droplet with Eq. (13), the
equation can be written as:
4 x 13 R f 1Mg dV Mg 2 R f Rd Rd2 1CdV 2 M 2 3 Rd dt
(14)
The velocity of droplet can be solved as: 1 b V 2 Ce ax (c bx ) a a
Where
a
2 g ( 1 3 ) 31Cd 2 R f Rd / 2 6 R ， b 13 4f ， c 2 3 Rd 23 3 Rd
(15) are
the
parameters that related to the system physical properties. Under the condition the droplet detach from the fiber, the detach velocity V(x=Rd) need to be equal to or larger than 0 and when the detach velocity satisfy V(x=Rd)=0, Eq. 14 can be written as: 1 1 b b V 2 (c bRd )e a ( Rd x ) (c bx ) a a a a
(16)
This is the threshold condition and the impact velocity is the maximum threshold velocity of capture and satisfy V(x=-Rd)=Vc2, hence, the threshold velocity can be calculated as: 1 b 1 b Vc 2 [ (c bRd ) (c bRd )e 2 aRd ]1 2 a a a a
(17)
This velocity is regarded as the maximum velocity of capture and termed as the upper threshold velocity of capture. This threshold velocity can be easily get through calculation for a given system. 3.3 Experiment results and verifications To verify the theoretical deductions above, we experimented the droplet impact on a fiber. For the isooctane-polypropylene-water system, we choose the droplet diameter is 120μm and the fiber diameter is 20 μm. According to Eq. 12 and Eq. 17, the threshold velocities is calculated and the value is Vc1≈19.6mm/s, Vc2≈2.1m/s. Due to the limit of
our experiment setup, the impact velocity is in a range of 3mm/s-60mm/s which means the detach phenomenon may not be observed. We set the impact velocity larger than and less than the threshold velocity Vc1 and the impact phenomena is shown in Fig. 6.
Fig.6 The impact phenomenon of a droplet impact on a fiber in liquid ambient (a. Adhering phenomenon, the droplet diameter is 120μm and the impact velocity is 17.5mm/s; b. Capture phenomenon, the droplet diameter is 120μm and the impact velocity is 41.5mm/s)
When the impact velocity is less than Vc1, the phenomenon is shown in Fig. 6(a). The impact velocity of this droplet was 17.5mm/s which is smaller than Vc1 (≈19.6mm/s), there was no obvious deformation during the impact and the droplet finally adhere on the fiber surface. Besides, the droplet reached the equilibrium once impacted on the fiber and adhered on the fiber surface for a long time. The typical classification of this phenomenon is droplet adhering on the fiber surface without obvious deformation and this phenomenon is regarded as adhering phenomenon. With the increase of impact velocity, the capture phenomenon was observed and is shown in Fig. 6(b). The impact velocity of this droplet was 41.5mm/s and the fiber immersed in the droplet after the impact and the droplet deformed sharply. The typical classification of this phenomenon is droplet spread on the fiber with obvious deformation and state barrel state.
The detaching phenomenon is not observed in this study due to the limit of our experiment. As the viscosity of liquid is much larger than air, the droplet velocity decreases sharply after generated and reaches a constant according to Stocks equation. Hence, it’s hard to generate a droplet with the impact velocity larger than Vc2. However, from the tendency in Fig. 6(b) we can see that, the droplet was tend to detach from the fiber (t=12.6ms) and the position of droplet is above the equilibrium position. The droplet may detach from the fiber with the increase of impact droplet. Therefore, the impact phenomena occurred in liquid ambient may be regarded as adhering phenomenon, capturing phenomenon and detaching phenomenon.
4 Conclusion The impact between a droplet and a fiber occurred in liquid ambient is investigated and two models are established. One model is established through the energy conservation law and to investigate the minimum threshold velocity of capture Vc1. When the impact velocity is smaller than Vc1 the adhering phenomenon was observed by experiment and the capture phenomenon was observed when the impact velocity is larger than Vc1. Another model is established by the forces analyses of the impact dynamics to study the maximum threshold velocity Vc2. The droplet may not be captured by fibers when the impact velocity is larger than Vc2 and the droplet may detach from the fiber. This model is an improvement of previous researches and modified to fit the liquid ambient environment. Combing the theoretical analysis and experiment results, the typical outcomes of a droplet impact on a fiber in liquid ambient can be classified as
adhering phenomenon, captureing phenomenon and detaching phenomenon. These threshold velocities are of great importance to industrial coalescence separation applications. They can be used to optimize the structure of devices by calculating the property impact velocity and choose the fiber size. The small/pilot scale tests may be reduced and may bring great economic benefit.
Nomenclature Rd
m
Radius of a droplet
Rf
m
Radius of a fiber
Dd
m
Diameter of a droplet
Df
m
Diameter of a fiber
ρ1
kg/m3
Density of phase 1
ρ2
kg/m3
Density of phase 2
ρ3
kg/m3
Density of phase 3
γ12
N/m
Interfacial tension between phase 1 and 2
γ23
N/m
Interfacial tension between phase 2 and 3
γ13
N/m
Interfacial tension between phase 1 and 3
θ
°
Contact angle
A1
m2
Interfacial area between phase 2 and 3
A2
m2
Interfacial area between phase 1 and 3
A3
m2
Interfacial area between phase 1 and 2
ΔEs
J
x
m
Displacement of a droplet
Fb
N
Buoyancy acting on a droplet
Fg
N
Gravity acting on a droplet
Fs
N
Fd
N
Drag force acting on a moving droplet
M
kg
Mass of a droplet
V
m/s
Impact velocity
Vc1
m/s
The threshold velocity of impact also termed as
The change of interfacial energy after and before the impact
Surface tension acting on a droplet in vertical direction
the lower limit of capture velocity The threshold velocity of capture also termed as
Vc2
m/s
μd
Pa·s
Dynamic viscosity of a droplet
μl
Pa·s
Dynamic viscosity of ambient fluid
μ*
1
The ratio of μd to μl
the upper limit of capture velocity
Acknowledgments This work was supported by the National Natural Science Foundation of China, China (51578239) and Scientific Research Projects of Shanghai, China (17DZ1202802).
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Highlights 1. A model was established to calculate the minimum threshold velocity of capture. 2. Another model was improved to calculate the maximum threshold velocity of capture. 3. A new phenomenon of a droplet adhere on the fiber surface was observed. 4. Two threshold velocities limit a range map in which droplets can be captured.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: