Hydrodynamics of Microbubble Suspension Flow ... - ACS Publications

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Hydrodynamics of Microbubble Suspension Flow in Pipes Rajeev Parmar and Subrata Kumar Majumder* Department of Chemical Engineering, Indian institute of Technology Guwahati, Guwahati-781039, Assam, India S Supporting Information *

ABSTRACT: Microbubbles exhibit excellent gas-dissolution abilities owing to their larger gas−liquid interfacial areas and longer residence times compared to conventional larger bubbles. Hence, it is expected that microbubbles should increase the efficiency of gas−liquid contact devices for various applications in chemical and biochemical processes. In most of these applications, it is necessary to understand the hydrodynamics, such as the rheology, pressure drop, and friction factor, associated with microbubble flow in devices. This study investigates the hydrodynamic characteristics of the flow of a microbubble suspension in a surfactant solution through a pipe. A mechanistic model has been developed to analyze the interfacial stress of microbubble suspension flow in a pipe by considering bubble formation, drag at the interface, and loss of energy due to wettability. A correlation between the intensity factor of interfacial stress and the friction factor based on energy loss due to wettability has been developed. The functional form of the correlation appears to predict the hydrodynamics satisfactorily for the flow of a microbubble suspension in a pipe. The present study might be helpful in further understanding multiphase flow for industrial applications.

1. INTRODUCTION Dispersed bubbly flow plays an important role in the chemical industries. Recently, the flow of dispersed microbubbles has gained a great deal of because of to its spectacular uses. Microbubbles are miniature gas bubbles with diameters of less than 100 μm in water. Microbubbles, which mostly contain oxygen or air, can remain suspended in water for an extended period. Gradually, the gas within the microbubbles dissolves into the water, and the bubbles disappear. It has been reported that microbubbles produced in water and surfactant solutions contain stable bubbles with a narrow and reproducible size distribution, exhibit high stability, separate easily from the bulk liquid phase, and can be pumped easily through pipes while keeping their structure unchanged.1,2 Microbubbles in surfactant solutions exhibit the same physical phenomena as those in foams, including Gibbs−Marangoni effects, interbubble gas diffusion, and liquid drainage.3,4 There is still ambiguity regarding the formation and structure of microbubbles in surfactant solutions. The structure suggested by Sebba1 is still the most widely accepted structure. Sebba proposed that bubbles have a multilayered shell as shown in Figure 1, but no direct observation or physical models have demonstrated such a structure. The main supporting supporting factors were (i) the absence of bubble coalescence and (ii) the fact that hydrophobic globules attached to the surface of the bubbles. Sebba1 did not report any details about the thickness of the microbubble shell. Later, Amiri and Woodburn5 studied the shell thickness of microbubbles in a surfactant medium. They reported that the thickness of the soapy shell is 750 nm for the cationic surfactant cetyltrimethylammonium bromide (CTAB), whereas Bredwell and Worden6 found a shell thickness of 200− 300 nm for the nonionic surfactant Tween-20. More recently, Jauregi et al.7 employed freeze fracture with transmission electron microscopy (TEM) and X-ray diffraction to study the structure of the soapy shell. They imaged and measured the thickness of the shell and found it to be 96 nm. They argued © 2014 American Chemical Society

Figure 1. Microbubble with a surfactant film on its shell.1

that the shell does not provide room for a finite inner water phase as proposed by Sebba.1 Microbubbles are advantageous over conventional bubbles because of their large interfacial area, high mass-transfer rate, and characteristic to adsorb fine particles at their surface.8 Microbubbles find a large number of applications in the engineering and medical fields.9−12 Some of their common applications include wastewater treatment, oil separation, recovery of fine particles, friction reduction, fermentation, and materials synthesis.13−17 Microbubbles in surfactant solutions, being of extremely small size, are characterized by having Received: Revised: Accepted: Published: 3689

August 27, 2013 February 3, 2014 February 8, 2014 February 9, 2014 dx.doi.org/10.1021/ie402815v | Ind. Eng. Chem. Res. 2014, 53, 3689−3701

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electrical charges, and they attract suspended floating particles very effectively. This particular property has been exploited in sludge treatment by using the microbubbles to capture and float organic matter, to reduce the time required for sludge treatment. Another emergent use of microbubbles is in the area of cancer treatment. Scientists are in the process of developing a method of diagnosing cancer lesions by injecting microbubbles into the bloodstream. Many other chemical and biochemical processes have become particularly important in recent years as advances with microbubble phenomena. Most of the applications of microbubbles take advantage of (i) their large interfacial area, (ii) the adsorption of particles at microbubble interfaces, and (iii) their stability for enhanced mass transfer.6 The stability of microbubbles in a liquid can be increased by coating them with lipids or surfactants.18 The presence of the coating not only lowers the interfacial tension at the bubble surface but also slows gas diffusion, thus avoiding rapid dissolution and/or combination and disproportionation within the suspension.19 The surfactant shell encapsulates the inner gas core and serves as a barrier to gas diffusion. D’Arrigo and Imae20 reported that lipid-coated microbubbles of 1−5-μm diameter were stable in an air-saturated aqueous solution for up to one year. Soetanto et al.21 claimed that coating microbubbles with sodium laurate can reduce the rate of dissolution of the microbubbles. The durability of surfactant-coated microbubbles can be optimized by using an appropriate molar ratio of constituent surfactants.22 Recently, Mohamedi et al.23 reported that microbubbles prepared from a liquid suspension of gold nanoparticles were found to be significantly stable compared with bubbles coated with only surfactant. Katiyar et al.24 investigated the effect of encapsulation elasticity on the stability of microbubbles. Various mathematical models of the dissolution and stability of microbubbles have also been proposed.24−27 Larmignat et al.28 studied the rheology of two-phase mixtures having bubble diameters in the range of 10−100 μm and gas holdup in the range 0.6−0.75. They reported that the rheology of two-phase mixtures is affected by the surfactant concentration in the mixture and not by the pipe shape and hydraulic diameter. Tseng et al.2 studied the rheological behavior of two-phase mixtures in mini-channels having porosities in the range of 0.64−0.70. They reported that the fluid behaves as a shear thinning fluid under the experimental conditions. Other studies on rheology29−34 have mainly focused on the rheological properties of gas−liquid mixtures having high gas holdup or porosity. However, the rheological behavior of gas−liquid mixtures with high gas holdup can differ from that of gas−liquid mixtures of low gas holdup. Shen et al.35 studied the rheological behavior of concentrated monodisperse food-emulsifier-coated microbubble suspensions. They observed that the viscosity of the system decreased as the shear stress increased, indicating that microbubble suspensions are shear thinning in nature. Several studies have reported reductions of the friction factor in microbubble flow as compared to single-phase flow.36,37 Wu et al.38 investigated the effect of drag reduction phenomena of microbubble suspensions. They reported that drag reduction by microbubble suspensions is affected by the gas holdup and curvature of pipes,39,40 whereas the type of gas and bubble diameter do not affect friction reduction.41−43 Kato et al.44 reported that only bubbles smaller than certain diameters can effect the friction of microbubble-induced flow. Haapala et al.45 studied the hydrodynamic drag and rise velocity of microbubbles in the papermaking process. In most applications of

microbubble suspensions, the microbubbles are pumped through columns, pipes, or fittings, or the applications are executed by the effects of microbubble flow behavior in the devices. However, it is difficult to transport large flow rates of microbubble suspensions through these pipes as the flow rheology completely differs from that of single-phase flow.46 Most studies of microbubble technology have mainly focused on the development of new technologies that can generate microbubbles at both low price and low energy input. Although a great deal of success has been achieved in this field to date to produce low-price microbubble generators, the generation of low-cost, high-quality microbubbles has not completely solved the complex phenomena associated with flowing microbubble suspensions in pipes and tubes. Other important parameters such as gas holdup, bubble size distribution, pressure drop, bubble−bubble interaction, frictional loses, and hydrodynamic drag still require attention in this field to import this technology on a large scale.8,11,47−50 Even though the flow properties of microbubbles have been discussed qualitatively in the literature, there is still a significant knowledge gap on microbubble flow behavior in various liquid media. To the best of our knowledge, very few data on the rheology, pressure drop, and friction factor associated with microbubble flow have been reported. To achieve technical feasibility of microbubble flow behavior, a detailed study of transport phenomena of dispersed microbubbles is required. In the present study with vision of engineering importance as well as scientific attentiveness, the rheological behaviors of flow containing dispersed microbubbles in pipes with different concentrations of surfactants have been investigated.

2. EXPERIMENTAL SECTION 2.1. Microbubble Generation. Many methods of microbubble generation are available in the literature. Pressurized dissolution is one of the most useful methods of producing microbubbles.11 Gas is allowed to dissolve in a liquid by applying a pressure of about 3−4 atm, as shown in Figure 2. Liquid and gas are mixed in a static mixer and moved to the pressure chamber. By adjusting the outlet flow rate to 4−10 L/ min by microbubble output control valve, the pressure is built up in the pressure chamber to dissolve the gas into the liquid. The water with oversaturated air is released to atmospheric pressure by the output control valve. At this low atmospheric

Figure 2. Schematic representation of the microbubble generation technique used in this work. 3690

dx.doi.org/10.1021/ie402815v | Ind. Eng. Chem. Res. 2014, 53, 3689−3701


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(model AS-MKIII; Riverforest Corporation, Escondido, CA), a pressure transducer with a data logger system (model 3002 U1PD2; Integrated Electrolife System, Kolkata, India), one rotameter, valves, and a test section. Pressure drop was measured in straight horizontal pipes having diameters of 3 × 10−3, 6 × 10−3, 8 × 10−3, and 1 × 10−2 m, respectively. The length of each pipe was chosen to ensure that accurate measurements of the pressure drop could be acquired with the pressure transducers and that fully developed flow could be obtained throughout the pipe. The distance between the two pressure transducers was taken as 2.4 m for each pipe based on the condition L/d > 50 for fully developed flow. The test section was designed well to minimize losses due to contraction and expansion. The rotameter and pressure transducers were calibrated well to minimize the experimental error. Microbubbles were generated in a tank with a volume of 5 × 10−2 m3 by the microbubble generator. The volumetric flow rate of fluid flowing through the pipe was measured by the rotameter and controlled by a flow control valve. The test fluid containing dispersed microbubbles was transported from the tank to the test section by a peristaltic pump (flow rate range of 0−6 × 10−4 m3/s). The pressure transducer with a data logger was used to measure the pressure drop along the length. Each experiment was performed repeatedly and allowed to run for 5 min to attain steady-state conditions. 2.4. Estimation of Gas Holdup. Gas holdup is defined as the volume fraction of the gas phase occupied by bubbles. The gas holdup in a microbubble column can be measured by the phase isolation method.51,52 The gas holdup in microbubbleaided aeration systems can be measured as

pressure, the supersaturated gas is released into the expelled liquid, which produces microbubbles. A typical snapshot (after refinement by with Adobe Photoshop 5.5 software) of microbubbles taken by a high-resolution camera (model PCO Pixelfly-USB, resolution 1360 × 1024, 19 fps; PCO Ltd., Kelheim, Germany) is shown in Figure 3.

Figure 3. Typical snapshot of microbubbles produced at Cs = 15 ppm.

2.2. Estimation of Bubble Size. Generally, the size of gas bubbles is measured by image processing using an optical microscope or video camera. Because the size of gas bubbles (currently the focus of interest) is several micrometers or smaller, the laser diffraction method has come to be used for such measurements. In the present study, the bubble size was measured with a particle size analyzer (model APA 2000; Malvern Instruments Ltd., Malvern, U.K.) by the laser diffraction technique. 2.3. Pressure Drop Experiments. The experimental setup to study pressure drop characteristics is shown in Figure 4. It consists mainly of a tank, a pump, a microbubble generator

⎛ ρ ⎞ εg = ⎜⎜1 − m ⎟⎟ ρw ⎠ ⎝

(1)

where εg is the gas holup and ρm and ρw are the densities of the gas−liquid mixture and water, respectively. In the case of microbubble-aided systems, the volume of liquid before and after bubbling does not change markedly. Therefore, the accuracy of the phase isolation method decreases as the bubble

Figure 4. Schematic representation of the experimental setup for pressure drop measurements. Legend: Ai, air inlet; CP, pump; DL, data logger unit; Li, liquid inlet; MBG, microbubble generator; MS, microbubble suspension; P1−P4, pipes (i.d. 0.003, 0.006, 0.008, and 0.010 m); R, rotameter; S, self-angle support; T1−T8, pressure transducers; TS, test section; Ts, microbubble suspension tank with cooling jacket; Tm, thermometer; V1−V11, flow control valves; Wi, water inlet; Wo, water outlet. 3691



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Figure 5. Schematic representation of the apparatus for gas holdup measurements. Legends: D, diameter of the probe; Le, separation between the electrodes; TI, temperature indicator; I, current; V, voltage.

with different concentrations of surfactant (sodium dodecyl sulfate) were found to behave as shear-thinning power-law-type non-Newtonian liquids. The densities of the fluids were calculated with a specific gravity bottle. The surface tension was measured by tensiometer (model K9-MK1, Krüss GmbH, Hamburg, Germany). The physical properties of the nonNewtonian liquid for different concentrations at a temperature of 25 ± 1 °C are listed in Table 1. The effective viscosity of the

size and liquid volume under test become much smaller. In the present study, the gas holdup was measured by the electrical conductivity method. Maxwell53 reported that the effective conductivity of a dispersion (Kl−d) is related to the volume fraction (εd) of a dispersed nonconductive phase by ⎛ 1 − εd ⎞ k l− d = k l ⎜ ⎟ ⎝ 1 + 0.5εd ⎠

(2)

Gas holdup can be calculated based on this principle as ⎛ k l − k l− g ⎞ ⎟⎟ εg = ⎜⎜ ⎝ kl + 0.5kl−g ⎠

Table 1. Physical Properties of the System at 25 ± 1 °C liquid

(3)

watera without microbubbles microbubble suspension without surfactant microbubble suspension with 5 ppm surfactant microbubble suspension with 10 ppm surfactant microbubble suspension with 15 ppm surfactant aira

where kl and kl−g are the electrical conductivities of the liquid and liquid−gas mixture, respectively. The electrical conductivities of the liquid and liquid−gas mixture were measured by digital conductivity meter (model VSI-04 ATC; VSI Electronics Pvt. Ltd., Kolkata, India). A schematic diagram of the apparatus used for gas holdup measurements is shown in Figure 5. The cell in the conductivity meter was an adaptation of a section of the “ideal” cell consisting of two infinite and parallel plate electrodes.54 The electrical conductance provided by the electrodes is described by the equation

Kl = κ

A Le

density (kg/m3)

surface tension (mN/m)

999.68 998.62

71.20 71.20

998.09

68.00

965.15

64.70

933.38

62.50 −

1.18

Conductivities of air and water are 5 × 10 respectively, at 25 ± 1 °C. a

−11

and 424 μS/cm,

liquid flowing through the pipe was calculated according to the equation

(4)

where Kl is the electrical conductance (inverse of resistance); κ is the electrical conductivity; D is the diameter of the electrode; and A and Le are the area (m2) and separation (m) of the electrodes, respectively. A/Le is referred to as the cell constant. Such a cell has been used to measure effective conductivities of dispersions.55−58 Alternating current (ac) of sufficiently high frequency (about 1000 Hz) and low voltage (∼1.5 V) was used to avoid polarization of the electrodes. 2.5. Physical Properties of the System. The properties of non-Newtonian fluids cannot be described with Newton’s law of viscosity, as the viscosities of such fluids are dependent on the rate of shear. Moreover, the viscosities of these fluids can increase or decrease due to changes in the rate of shear, which again is subject to the nature of the fluid. Unlike Newtonian fluids, non-Newtonian fluids are defined as materials that do not conform to a direct proportionality between shear stress and shear rate.59 In the present study, microbubble suspensions

n−1 ⎛ 3n + 1 ⎞n⎛ 8Vsl ⎞ ⎟ ⎟ ⎜ μe = K ⎜ ⎝ 4n ⎠ ⎜⎝ d p ⎟⎠

(5)

where μe is the effective viscosity of the suspension, n is the flow behavior index, K is the flow consistency index, V is the velocity of the microbubble suspension, and dp is the diameter of the pipe. 2.6. Uncertainty Analysis of Experimental Data. In the present work, each experiment was repeated at least five times, and the average values of the results were taken. The standard deviations (STDEV) and standard uncertainties (U) of the repeated experiments were calculated as N

STDEV = 3692

∑i = 1 (xi − x ̅ )2 N−1

(6)


Industrial & Engineering Chemistry Research U=

STDEV N

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between the microbubble wall and the liquid (El) and due to the wettability between the liquid and the pipe wall (Ew). Then, eq 13 can be represented as

(7)

where x̅ represents the mean value, which is mathematically expressed as x̅ =

1 N

ΔPl−mb = E /VmA p

N

∑ xi i=1

The energy dissipation (El) at the microbubble−liquid surface occurs as a result of the drag force exerted by the liquid on the microbubble surface.60 Equation 15 describes the amount of energy (El) irreversibly converted into thermal energy due to friction. The total energy dissipation can be calculated from the product of the force exerted by the liquid on a single bubble (Fd), the liquid velocity, and the bubble population, which can be represented as

(8)

The ranges of the means, standard deviations, and uncertainties of the experimental results for gas holdup and frictional pressure drop are reported in Tables 2 and 3 (Supporting Information), respectively.

3. THEORETICAL BACKGROUND 3.1. Properties of Microbubble Suspension Flow. Microbubble suspensions behave as non-Newtonian shearthinning (pseudoplastic) fluids.35 For pseudoplastic fluids, the most common model relating the wall shear stress (τw), wall shear rate (γw), and apparent shear rate (γa) is a power-law model, given by τw = μe γa = Kγw n

1 1 E l = Fd(Vsl /εl)Nb = Cm πdb 2 ρl (Vsl /εl)2 (Vsl /εl)Nb 4 2 (15)

where Cm is the drag coefficient of microbubble flow in the fluid, εl is the liquid holup, db is the bubble diameter, and ρl is the density of the liquid. The bubble population, Nb, in length L can be calculated as

(9)

Nb =

The apparent shear rate and wall shear rate are related as ⎛ 3n + 1 ⎞ ⎟γ γw = ⎜ ⎝ 4n ⎠ a

γa =

D h ΔP 4Z

32Q πD h 3

(11)

Ew =

(1/6)πdb3

(16)

πd pVslσl−w εl

+

Vslσl−mb aεl

(17)

where a is the specific interfacial area of a bubble, σl−w is the interfacial surface tension at the boundaries between the liquid and the wall, and σl−mb is the surface tension between the liquid and the microbubble. The specific interfacial area of a bubble can be calculated from the liquid holdup and bubble diameter61

(12)

3.2. Analysis of Interfacial Shear Stress in Microbubble Suspension Flow. The fluid dynamics of a microbubble−liquid mixture flow through a pipe is described in this section using an internal flow model. The dynamic interaction of the phases is taken into account in modeling the flow by introducing the rate of energy dissipation. The mechanical energy balance is used to calculate the pressure drop, which can be regarded as either the force per unit crosssectional area required to overcome frictional forces or the energy dissipation per unit volume. The model is presented with the following assumptions: (i) The flow is steady and isothermal with uniform gas holdup. (ii) Acceleration effect is negligible due to the absence of interphase mass transfer. (iii) Uniform frictional loss is assumed throughout the pipe. The mechanical energy balance equation for the microbubble-liquid mixture is given by ΔPl−mbA pVm − E = 0

L(1 − εl)A p

The rate of energy loss due to the wettability of the liquid with the surface of the pipe wall is the summation of the energy loss due to wettability between the liquid and the surface of the column wall and the energy loss due to the wettability between the liquid and the surface of the bubble. This can be represented as

(10)

The wall shear stress (τw) and apparent shear rate (γa) were experimentally determined from the volumetric flow rate (Q) and pressure drop (ΔP) according to the relations31 τw =

(14)

a=

6(1 − εl) db

(18)

Wettability is the affinity of a solid matrix for aqueous phases. It is normally quantified by the value of the contact angle. A contact angle of θ < π/2 indicates that the solid is wetted by the liquid, whereas θ > π indicates nonwetting conditions. The limits θ = 0 and θ = π define complete wetting and complete nonwetting, respectively. The energy loss due to the wetting of the liquid depends on the dynamic contact angle between the liquid and the solid wall.62 The dynamic contact angle (θ) is approximately equal to Ca1/3, where Ca is the capillary number, which is defined as Ca =

(13)

μe Vsl εlσl

(19)

The surface tension between the liquid and the solid wall (σls) can be calculated from Young’s equation as

where ΔPl−mb is the frictional pressure drop in the liquid− microbubble suspension, Vm is the velocity of the mixture, and Ap is the cross-sectional area of the pipe. In eq 13, the first term is energy due to liquid−microbubble flow pressure (N·m/s), and the second term (E) represents the energy dissipation per unit mixture volume due to friction

σl cos θ = σl−w

(20)

where σ represents the force needed to stretch an interface by a unit distance. According to the present study, the contact angle is less than 90° within a range of liquid velocity, and the range 3693



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f l−mb/f l0, and α″l is defined as (area of contact)l−mb/(area of contact)l0. The value of α″l is equal to the specific interfacial area in the test section of the pipe. The parameter αl is an intensity factor that signifies the intensity of interfacial shear stress. ΔPfl0 is the frictional pressure drop due to liquid when only the liquid phase flows through the pipe. The frictional pressure drop of the single liquid phase is calculated using Fanning’s equation

of the capillary number is 0.011−0.032. Therefore, eq 14 can be written as ⎛ 1 ⎞⎛ 3 πd pVslσl−w 1 − εl L ⎟⎟⎜ Cmρl Vsl 3A p ΔPl−mb = ⎜⎜ + 3 εl εl db ⎝ VmA p ⎠⎝ 4 +

Vslσl−mb ⎞ ⎟ aεl ⎠

(21)

ΔPfl0 = 2fl0 ρl Vsl 2L /d p

From the experiments, the total pressure drop can be obtained as the frictional pressure drop due to liquid flow in the absence of a vertical hydrostatic pressure drop, which can then be expressed as ΔPl−mb

The friction factor is calculated as fl0 =

⎛ 1 ⎞⎛ 3 1 − εl ⎟⎟⎜ Cmρl Vsl 3A p = ΔPfl = ⎜⎜ V A 4 εl 3 ⎝ ⎝ m p⎠

⎞ πd pVslσl−w Vσ L + + sl l−mb ⎟ db εl aεl ⎠

fl0 =

Re =

εl

3

+ (23)

Cm =

(30)

(31)

⎛ 1 ⎞⎛ 3 πd pVslσl−w 1 − εl L ⎟⎟⎜ Cmρl Vsl 3A p = ⎜⎜ + 3 εl εl db ⎝ VmA p ⎠⎝ 4

Vslσl−mb ⎞ ⎟ aεl ⎠

(32)

⎛ πd pVslσl−w ⎞⎤ Vσ 4 ⎡ αl ΔPfl0VmA p ⎢ −⎜ + sl l−mb ⎟⎥ 3 εl 3 ⎢⎣ aεl ⎠⎥⎦ εl ⎝

⎡ ⎤ εl 3db ⎢ ⎥ ⎢⎣ (1 − εl)ρl Vsl 3A pL ⎥⎦

(33)

In eq 26, except for αl, all of the parameters are known. Using the experimental pressure drop, the corresponding values of αl can be calculated for different variables from eq 26. To estimate the values of αl, the experimental data on ΔPl−mb for different operating conditions of non-Newtonian flow in the pipe were taken from the experimental results. Once the value of αl has been calculated using eq 34, the drag coefficient of microbubble flow can be calculated from eq 33. From the definition of αl, one can calculate the effective friction factor, f l−mb, in the straight pipe as

(area of contact with wall)l−mb α 1 = αl′ 2 αl″ = 2l (area of contact with wall)l0 εl εl

which implies αl ΔPfl0 (25)

which becomes ΔPl−mbεl 3 ΔPflεl 3 = ΔPfl0 ΔPfl0

(29)

This gives the hydrodynamic drag coefficient as

(24)

αl =

for turbulent flow

d pnVsl 2 − nρ ⎛ 4n ⎞n ⎜ ⎟ 8n − 1K ⎝ 3n + 1 ⎠

αl ΔPfl0

f (area of contact with wall)l−mb (V 2 ) × = l−mb sl 2l−mb (area of contact with wall)l0 fl0 (Vsl )l0

εl 3

n

n (Re)2.63/10.5

(28)

Substituting eq 25 for ΔPfl into eq 22, one obtains

(0.5fρl Vsl 2)l−mb ΔPflA t εl = ΔPfl0A t (0.5fρl Vsl 2)l0

ΔPfl =

0.079 5

fl0 = 0.125[n n (0.0112 + Re−0.3185)]

which implies

×

for laminar flow

For transitional flow, the friction factor f can be deduced from the generalized pressure loss equation64

ΔPfl(cross‐sectional area)l−mb (wall shear stress)l−mb = ΔPfl0(cross‐sectional area)l0 (wall shear stress)l0 (area of contact with wall)l−mb (area of contact with wall)l0

16 Re

where Re is the Reynolds number for non-Newtonian fluid flow.59 Re can be represented as

(22)

To determine the frictional losses due to liquid flow in the pipe, a model can be formulated on the basis of the following assumptions: (i) The friction factor for the liquid phase is a constant multiple, α′, of that for a single liquid phase without bubbles taking place in the pipe. (ii) The area of contact of the liquid phase with the wall is α″ times that of the flow of a single phase without bubbles in the pipe.63 From these assumptions, a simple overall momentum balance for liquid phase can be represented as

×

(27)

(26)

fl−mb =

where the subscript l0 refers to the single liquid phase, the subscript l−mb refers to the liquid−microbubble wall, and the subscript sl means superficial liquid. (Vsl)l−mb = (Vsl)lo/εl, αl = αl′αl″, and the cross-sectional area for the flow of liquid is equal to εl times the cross-sectional area of the pipe. αl′ is defined as

αlfl0 αl″

(34)

The value of α″l is equivalent to the specific interfacial area in the test section of the pipe, which is calculated by eq 18. The average liquid−microbubble surface interfacial shear stress (τi) is expressed as 3694


Industrial & Engineering Chemistry Research ⎛ α lf ⎞ τi = 0.5(fl−mb ρl Vsl 2)/εl 2 = 0.5⎜ l0 ρl Vsl 2⎟ /εl 2 ⎝ αl″ ⎠

Article

(35)

If there is slip at the interface, then eq 34 can be expressed as 1 τi = [fl−mb ρm (Vsl − Vis)2 ]/εl 2 (36) 2 where Vis is slip velocity.

4. RESULTS AND DISCUSSION 4.1. Bubble Size Distribution and Terminal Rise Velocity. The effects and characteristics of bubbles are highly dependent on the size of the bubbles. In leading-edge areas, bubbles of several hundred micrometers in size are highly anticipated. There are apparently still many unknown points regarding the nature of microbubbles, including their size. A typical bubble size distribution measured by particle size analyzer obtained from our present experiments is shown in Figure 6. It can be seen that the microbubble size decreased Figure 7. Variation of microbubble size with surfactant concentration.

Figure 6. Typical microbubble size distribution at a surfactant concentration of 15 ppm. Figure 8. Variation of gas holdup with surfactant concentration.

with increasing concentration of surfactant, as shown in Figure 7. An increase in surfactant concentration reduces the liquid surface tension, causing a reduction in bubble size. Previous works also stated that surface-active agents reduce surface tension, produce smaller bubbles, and prevent microbubble coalescence.45,65−67 The bubble size with surfactant concentration based on our experimental result can be expressed by db = 52.05 × 106 exp( −0.03Cs)

tension of the liquid, as a result of which the interface between the phases was easily stretched. This led to the easy formation of new surface area between the phases, which caused an increase in gas bubbles per unit volume and ultimately increased the gas holdup. Pallapothu and Al Taweel68 also reported an increase in gas holdup for microbubble-induced an internal loop reactor with surfactant concentration. Other studies in conventional systems also reported an increase in gas holdup in the presence of surfactants.69−72 Anastasiou et al.73 used three different types of surfactants in their study and observed that the surfactants significantly enhanced gas holdup. The gas holdup profile was found to satisfy the following correlations developed on the basis of the present experimental data

(37)

where db is the bubble diameter and Cs is the concentration of surfactant in the liquid in parts per million. Microbubbles of about 52 μm were generated by the microbubble generator when no surfactant was present in the water. It was seen that an increase in surfactant concentration increased the gas holdup through the entrainment of more gas and increasing bubble population by decreasing bubble size. The effect of surfactant addition in the liquid is shown in Figure 8. It can be seen that surfactant strongly enhanced the gas holdup. When the gas− liquid mixture was introduced into the mixing chamber of the generator, a high shearing and smashing occurred between the two phases. The addition of surfactant lowered the surface

⎧ 7.66 × 10−3C (1 − 0.018 if 0 < C < 20 ppm s s ⎪ εg = ⎨ Cs) − 0.023 ⎪ if Cs = 0 ⎩ 0.012 (38) 3695



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4.2. Shear Stress in Microbubble Suspension Flow. The shear stress for water increases linearly with shear rate. Microbubble suspensions exhibit the power-law behavior of a pseudoplastic non-Newtonian fluid. The shear stress verses shear rate behaviors of single-phase water and microbubble suspensions with and without surfactant are shown is Figure 9.

the temperature gradient around the bubbles was negligible. The minimum bubble diameter for this effect is around 10 μm.77 From the present experimental data, it was found that the range of microbubble size was 30−50 μm, which was 3−5 times higher than the minimum value. Therefore, in the present study, it was assumed that the Marangoni effect was negligible. The variation of the effective viscosity of microbubble suspensions with shear rate is shown in Figure 10. The

Figure 9. Variation of shear stress with shear rate for a microbubble suspension. Figure 10. Variation of effective viscosity with apparent shear rate for a microbubble suspension.

The non-Newtonian nature of the microbubble suspensions was found to be dependent on surfactant concentration in the suspension. An increase in the concentration of surfactant in water increased its non-Newtonian behavior. An increase in surfactant concentration significantly decreases the surface tension, which causes a reduction in bubble size and increases in the bubble population and gas holdup.74 However, it is difficult to correlate the bubble size with the rheological properties of the fluid.75,76 The increase in gas holdup decreases the density of the fluid mixture, so that the momentum flux in the direction of the flow decreases and causes the fluid flow behavior to change. The extent of non-Newtonian behavior, however, largely depends on the physiochemical properties of the fluid. The values of the flow behavior index in all of the pipes for microbubble suspensions were found to be less than 1. It was observed that microbubble suspensions with high surfactant concentrations had the lowest of flow behavior indexes, which demonstrates the high degree of non-Newtonian behavior. However, a general trend for the flow consistency index in all of the pipes was not observed. Shen et al.35 observed the shear-thinning nature of microbubble suspensions. The shear-thinning nature of microfoam has also been reported in the literature.2,28,29 However, they observed an increase in shear stress with surfactant concentration and concluded that this contradiction arises as a result of an increase in the effective viscosity of the fluid with shear rate. This might be due to the Marangoni effect. When there is a temperature gradient around the bubble, a nonuniform steady-state distribution of surfaceactive species along the interface is created. The surface tension gradient leads to tangential stress at the bubble interface, which drives the bubble toward a warmer region or a low-surfacetension region.77 The phenomenon is known as the Marangoni effect. This effect is prominent in microbubble systems, if a temperature gradient is provided. In the present study, experiments were performed under isothermal conditions, so

effective viscosity of the microbubble suspension in pipe flow decreased with increasing shear rate, which resulted in the shear-thinning nature of the microbubble suspension. The rates of decrease of the effective viscosity with shear rate were also different for all of the microbubble suspensions. On increasing the concentration of surfactant in the suspensions, the effective viscosity decreased more rapidly. The addition of surfactant decreased the surface tension, which increased the gas holdup and decreased the bubble size. The pressure between the inlet and outlet generally decreased as the gas holdup increased, as a result of which the shear stress decreased. The decrease in shear stress could cause the effective viscosity to decrease. Shen et al.35 observed a decrease in the viscosity of microbubble suspensions with apparent shear rate. Tseng et al.2 also reported a decrease in effective viscosity with apparent shear rate in microfoams having bubble diameters in the range of 10− 100 μm. 4.3. Wall Friction Factor. The frictional pressure drop is caused by the resistance to the flow of fluid. The main factors of resistance to fluid flow are changes in kinetic energy through the pipe and fluid viscosity. Figure 11 shows the variation of the frictional pressure drop across the test section with Reynolds number based on the pipe diameter. It can be seen that the frictional pressure drop per unit length increased with increasing Reynolds number and decreased with increasing concentration of surfactant in the suspension. An increase in the Reynolds number enhances the kinetic energy of the fluid flowing. The increase in kinetic energy raises the interfacial shear stress in the pipe, which results in an increase in pressure drop. The increase in surfactant concentration leads to increases in the bubble population, which decreases the liquid holdup in the suspension, causing a reduction in the 3696



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⎛ d p ⎞−0.0439 −0.4359 Re (αl″)−1 fl−mb = 0.3958(1 + Cs)−0.0277 ⎜ ⎟ ⎝ db ⎠ (40)

The correlation coefficient and standard error of the correlation were found to be 0.998 and 0.097, respectively. 4.4. Friction Factor in Microbubble Suspension Flow Based on Energy Loss Due to Wettability. The wetting effect of the solid−liquid surface during the flow of fluid is a matter of concern because some amount of energy is lost due to the wettability. The extent of wettability is determined by an equilibrium between adhesive and cohesive forces. Surface tension, fluid velocity, and gas holdup are important factors upon which the wettability of liquid in two-phase flow depends. The variation of the friction factor with energy losses due to wettability at different surfactant concentrations is shown in Figure 13. It can be seen that the friction factor decreased with Figure 11. Variation of friction pressure drop with Reynolds number.

momentum of the fluid flowing in the pipe, as a result of which the pressure drop decreases. The variation in the friction factor with Reynolds number based on the pipe is shown in Figure 12.

Figure 13. Variations of the friction factor with energy loss due to wettability at different surfactant concentrations.

increasing energy loss due to wettability for a specific wetted area, whereas the friction factor increased with wetted area at fixed energy dissipation due to wettability. Higher surfactant concentrations in the suspensions led to a decrease in the surface tension and an increase in the gas holdup. Both of these factors decreased the energy loss due to wettability because of capillary action of the flowing fluid in the pipe. The friction factor due to wettability can be expressed as

Figure 12. Friction factor as a function of Reynolds number.

It can be observed that friction factor decreased with increasing Reynolds number based on pipe diameter. The viscous force decreased with increasing Reynolds number because the viscosity of the suspension decreased as a result of the shearthinning nature of the fluid. This caused a reduction of the friction factor with Reynolds number. The dependency of the friction factor on the Reynolds number can be presented by developing a correlation based on the present experimental data as

fl−mb =

1.5398 αl″Re 0.454

fl−mb αl″ = pEw q

(41)

where the parameters p and q are defined as p × 104 = 4.595 + 0.601Cs − 0.04Cs 2 ,

R2 = 0.978 (42)

q = 0.447 − 0.024Cs + 0.001Cs 2 ,

R2 = 0.991

(43)

4.5. Hydrodynamic Drag Coefficient in Microbubble Suspension Flow. The drag coefficient derived for the system includes the hydrodynamic drag on the microbubbles and the pipe wall. Drag on the bubble depends on the surface area of bubble (bubble size), fluid velocity, fluid density, and so on. Changing surfactant concentrations changes bubble size, which affects the drag force experienced by bubble. Similarly, changing

(39)

A more generalized correlation of the friction factor with the other independent variables can be represented as 3697



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the pipe diameter changes the fluid velocity in the pipe, which ultimately changes the drag coefficient. The effect of the Reynolds number on the drag coefficients is shown in Figure 14. At low Reynolds number, the viscous forces in the pipe are

Figure 15. Variations of the drag coefficient with Weber number at different surfactant concentrations.

4.68 × 104 ≤ Figure 14. Variations of the drag coefficient with Reynolds number at different surfactant concentrations.

58.51 ≤

dp db

Z ≤ 7.47 × 104 , db ≤ 311.46,

1.06 × 10−2 ≤ Web ≤ 4.68,

4.95 × 102 ≤ Re ≤ 7.28 × 104

high. As the Reynolds number increases, the momentum energy of the fluid increases. Because of the shear-shinning behavior of the fluid, the effective viscosity decreases with increasing inertia of the fluid. Hence, the magnitude of the drag force decreases with decreasing viscosity of the fluid. Consequently, the drag coefficient decreases with increasing Reynolds number. It was also observed that microbubble suspensions having high surfactant concentrations resulted in low drag coefficients. This might be due to the reduction of the size of the microbubbles upon the addition of surfactants. A correlation by dimensional analysis was developed to interpret the drag coefficient based on the present experimental data with other independent variable as follows

5. CONCLUSIONS The present study mainly focused on the rheology of microbubble suspensions in a horizontal pipe and the development of a model based on wetting effects in microbubble suspensions. From the present study, the following conclusion can be made: (1) The effective viscosity of the microbubble suspensions was found to decrease with apparent shear rate, indicating that microbubble suspensions behave as shear-thinning non-Newtonian fluids. (2) An increase in surfactant concentration caused a decrease in shear stress and effective viscosity with shear rate. (3) The friction factor was found to decrease inversely with the Reynolds number to the power of 0.45, whereas it decreased without the microbubble suspension with a power of 0.25. This implies that the presence of microbubbles reduces the frictional resistance. The friction factor was found to decrease with increasing wettability. (4) As the fluid velocity increased, frictional loses decreased at the specific wetted surface area. (5) Surfactant concentration in the suspension plays an important role in increasing the gas holdup and decreasing the energy loss due to wettability. (6) The bubble diameter decreased with increasing surfactant concentration, leading to a decrease in the frictional resistance to flow and resulting in non-Newtonian flow behavior during flow of the microbubble suspension through the pipe. (7) The drag coefficient was found to decrease with increasing Reynolds number, as well as with increasing surfactant concentration in the suspension.

⎛ Z ⎞−4.11⎛ d p ⎞−1.517 Cm = 1.0639 × 10 ⎜ ⎟ Web 0.2694Re−0.796 ⎜ ⎟ ⎝ db ⎠ ⎝ db ⎠ 22

(44)

where Web is the Weber number, defined as Web =

ρl Vsl 2db σ

(45)

The correlation coefficient and standard error of eq 44 were found to be 0.987 and 0.163 respectively. The parity goodness of the correlation proposed for the drag coefficient (Cm) with the experimental value is shown in Figure 15. The correlation proposed was developed on the basis of the dynamic variable microbubble suspension velocity; the geometric variable pipe diameter; and the physiochemical properties of the system viscosity, density, and surface tension. It was found that the developed correlation is in good agreement within the ranges of variables 3698



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Article

γa = apparent shear rate (s−1) γw = wall shear rate (s−1) ΔP = pressure drop (N/m2) εd = dispersed-nonconducting-phase holdup εg = gas holdup εl = liquid holdup θ = liquid−wall contact angle (radian) κ = electrical conductivity (S/m) μe = effective viscosity (kg/m·s) μl = liquid viscosity (kg/m·s) ρl = liquid density (kg/m3) ρm = density of the gas−liquid mixture (kg/m3) ρw = density of water (kg/m3) σ = surface tension (N/m) σl−mb = surface tension between liquid and microbubble (N/ m) σl−w = surface tension at the boundary between the liquid and the pipe wall (N/m) τ i = interfacial shear stress (N/m2) τw = wall shear stress (Pa)

ASSOCIATED CONTENT

S Supporting Information *

Uncertainties for the measurement of gas holdup and pressure drop (Tables 2 and 3, respectively). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE a = specific interfacial area (1/m) A = electrode area (m2) Ap = pipe cross-sectional area (m2) Cs = concentration of surfactant (ppm) Cm = drag coefficient of the microbubble mixture Ca = capillary number [(μeffVsl)/(εσl)] db = bubble diameter (m) dp = pipe diameter (m) D = electrode diameter (m) Dh = hydraulic diameter El = energy dispersion per unit volume of test section (N·m/ s) El = energy dispersion per unit length in the pipe (N·m/s) Ew = energy loss due to wettability (N·m/s) Fd = drag force (N) f l−mb = friction factor of the liquid−microbubble system f lo = friction factor of the single-liquid system g = gravitational acceleration constant (m/s2) K = consistency of the fluid (Pa·sn) Kl = conductance of the liquid (S) kl = electrical conductivity of the liquid (S/m) kl−g = electrical conductivity of the liquid−gas mixture (S/m) L = length of the test section (m) Le = separation between the electrodes (m) MS = microbubble suspension n = flow behavior index N = number of repeated experiments Nb = number of bubbles p = parameter defined in eq 42 Pfl−mb = frictional pressure due to liquid−microbubble flow (N/m2) Pflo = frictional pressure due to the single liquid phase (N/ m2) Pl−mb = liquid−microbubble suspension pressure (N/m2) q = parameter defined in eq 43 Q = volumetric flow rate (m3/s) Re = non-Newtonian liquid Reynolds number {[(dpnVsl2−nρ)/(8n−1K)][4n/(3n + 1)]n} STDEV = standard deviation U = standard uncertainty Vis = interfacial slip velocity (m/s) Vm = gas−liquid mixture velocity (m/s) Vsl = superficial liquid velocity (m/s) Web = Weber number (ρlVsl2db/σ) xi = experimental value at i x̅ = mean of repeated experiments Z = length of the test section (m)

Subscripts

■

b = bubble f = frictional g = gas phase i = interfacial l = liquid phase l−mb = liquid−microbubble l−w = liquid−wall p = pipe s = superficial w = wettability 0 = single

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Greek Letters

αl = parameter defined in eq 26 3699



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