New Modeling Strategies Evaluate Bubble Growth in Systems of Finite

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Thermodynamics, Transport, and Fluid Mechanics

New Modeling Strategies Evaluate Bubble Growth in Systems of Finite Extent: Energy and Environment Implications Javad Sayyad Amin, Simin Tazikeh, Sohrab Zendehboudi, and Ioannis Chatzis Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00138 • Publication Date (Web): 04 Apr 2018 Downloaded from http://pubs.acs.org on April 4, 2018

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Industrial & Engineering Chemistry Research

New Modeling Strategies Evaluate Bubble Growth in Systems of Finite Extent: Energy and Environment Implications Javad Sayyad Amin1, Simin Tazikeh1, Sohrab Zendehboudi2,*, Ioannis Chatzis3b 1

Department of Chemical Engineering, University of Guilan, Rasht, 41996-13769, Iran

2

Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL, Canada 3

Department of Petroleum Engineering, Faculty of Engineering and Petroleum, Kuwait University, Kuwait * Corresponding Author’s Email Address: [email protected]

ABSTRACT Growth of a new phase is experienced in several chemical engineering processes/operations such as polymer foaming, oil/gas transportation, bubbly columns, distillation towers, and petroleum recovery. Thus, it seems vital to understand the dynamics of transport phenomena and new phase (e.g., bubbles) evolution over the corresponding processes to efficiently design and operate the plant equipment. The present study introduces new analytical and approximate solutions for the growth of a new phase in a medium of finite extent. The governing conservation equations are solved by an effective mathematical approach through combination of variables and enhanced homotopy perturbation method (EHPM) where it is assumed that the mass transfer controls the rate of growth through both convection and diffusion mechanisms. The modeling outcome confirms the importance of the convection mass transfer in growth of the new phase in finite extent. The results show that the radius of the new phase is strongly dependent on the diffusivity, temperature, and initial void fraction. It is also found that the bubble evolution follows the power-law model. To examine the practical effectiveness of the developed mathematical models, the growth of water bubbles, hydrate particles (as a new phase), and bubbles (or gas) growth in oil reservoirs are studied. The models’ results are compared with some experimental data, exhibiting a very good agreement (e.g., error percentage < 5%). This research work offers systematic strategies for investigation of bubble expansion/shrinkage phenomena where the bubble growth dynamics are observed in various processes/ systems. Keywords: Bubble Evolution; Finite Extent; Void Fraction; Mathematical Modeling; Transport Phenomena

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1. INTRODUCTION Investigation of the growth of a new phase formed in a two-phase system is central in several engineering and science fields; including, materials technology, plastic foam manufacturing, oil and gas production, separation equipment, unconventional reservoirs, hydrate formation, food industries, geophysics, physiology, and pharmaceutical production laboratories.1-6 In the oil and gas industry, the new phase growth in the form of bubbles can be found in heavy oil reserves and hydrate gas reservoirs at specific temperatures and pressures where a great deal of attention has been attracted. Under certain thermodynamic conditions, the solvent can become supersaturated with its solute gases. In this case, the gaseous solute will be released and gas bubbles will be formed.1-3 The main thermodynamic conditions such as temperature and pressure strongly affect the formation and growth of a new phase (bubbles) in gas–liquid solutions when the liquid solvent is saturated with its solute gas.5-7 Natural gas-hydrates are considered as a promising unconventional source of energy. It is estimated that there are over 1.5×1016 m3 of hydrated gases across the globe, both on the land and offshore.8, 9 Formation of gas hydrates occurs under particular conditions in terms of temperature and pressure. Gas hydrates appear in the form of crystalline compounds, in which guest molecules such as CH4, C2H6, and CO2 are trapped in host cages, which are composed of hydrogen-bonded water molecules.9-11 The general formula for gas hydrates is Mn(H2O)p, where n represents the number of gas molecules, and p stands for the number of water molecules in the hydrate. 12,13 Gas hydrates can block transportation (or transmission) components and equipment, plug blowout preventers, collapse tubing and casing, jeopardize the foundations of deep water platforms and pipeline systems, and lower the performance of compressors, process heat exchangers, control systems, and measurement devices/tools.9,10 All the unfavorable consequences listed here are normally affected by the growth and thermodynamic behaviors of hydrates at various conditions (e.g., temperature, pressure, and composition). As mentioned earlier, the bubble evolution phenomenon takes place in several chemical, petroleum, and mechanical engineering processes. Hence, it is beneficial to study the generation and growth of a new phase in an ambient phase through experimental and theoretical approaches. The equation of motion of a new phase (spherical bubble growth) was first presented by Rayleigh where the effect of inertia forces was dominant.14 Plesset and Zwick introduced an asymptotic solution for bubble growth by considering the thermal diffusion as a controlling factor, but neglecting influences of the liquid inertia and thin thermal boundary layer.15 Mohammadein and Gad-Elrab made a few modifications to the solutions obtained by Plesset and Zwick.15,16 They investigated the growth of vapor bubbles in superheated water in a finite system to obtain a new mathematical model. They then 2 ACS Paragon Plus Environment

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compared the results attained from their new developed model with the real data and the outcomes from other available theoretical and empirical strategies. Szekely and Fang studied the impacts of viscous forces, liquid inertia, surface kinetics, and surface tension on the bubble volume and growth rate.17 Scriven introduced a formulation for the bubble growth due to the heat or/and mass transfer, where the controlling mechanism/stage is the heat transfer.18 Venerus developed and evaluated transport phenomena models of diffusion-induced bubble growth in finite and infinite viscous liquids so that the bubble radius and pressure were determined over a certain range of process conditions.19 The findings were compared with the results attained from the works by Amon et al., implying an acceptable agreement.20,21 As expected, the results of the finite and infinite models in the early stage of bubble growth are almost the same. However, a considerable difference between the models in terms of bubble radius is noticed over time (after the early stage), while the equilibrium bubble diameter for the finite model is attained. At intermediate stages, a reasonable match is noticed for relatively low growth rates. In addition, Sta Maria and Eckmann investigated gas expansion in blood systems. Their theoretical approach predicted that the small gas bubbles experiences rapid and indefinite expansion where the bubbles are trapped in very tiny blood vessels.22 It can be concluded that the size of capillaries limits the diameter of the bubbles. Mohammadein and Mohammed also proposed an analytical solution for the bubbles growing in the blood and other tissues, where different conditions such as variable pressure and constant pressure throughout the decompression process were taken into account.4 As a result, concentration distribution around the bubbles was obtained. In another work, Mohammadein and Mohammed developed a new model for the growth of a vapor bubble in a superheated liquid in terms of surface tension and viscosity between two finite boundaries.23 Their model resulted in an analytical solution through using the modified version of Plesset and Zwick method.15. Hashemi and Abedi introduced an analytical solution for the new phase growth in infinite extent. In addition to the analytical equation, they also presented an approximate method and compared the results of both analytical and approximate methods.6 Recently, Zendehboudi et al. obtained an analytical model for CO2 droplets shrinkage during the ex-situ dissolution process where the turbulent bubbly flow regime is established under steady state conditions.24 In another work, a mathematical approach for CO2 droplets evolution in the ex-situ dissolution technique was proposed by Zendehboudi et al.25 The droplet break-up process and transient mass transfer were considered in their model. Their modeling results showed a very good agreement with the experimental data.25 Growth of a new phase in an environment is strongly dependent on whether the liquid surrounding the bubble is infinite or finite extent.14-18 For the case of infinite extent, the new phase grows immediately 3 ACS Paragon Plus Environment


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because an unlimited amount of solute is available. It has been claimed that the bubble growth in liquids with finite extent provides a more appropriate model for polymer foaming, hydrate formation, physiology, and pharmaceutical production as a given bubble surrounded by other bubbles contains a finite magnitude of solute.1-5, 26 It is obvious that the mass transfer is an important factor affecting the growth of the new phase. While most of previous research studies have ignored the effect of convective mass transfer and assumed that the only controlling mechanism is diffusion, the current research investigation highlights the impact of both factors where a new phase growth in finite extent is experienced. The growth of a new phase is a combination of momentum, heat, and mass transfer phenomena so that the bubble growth is generally affected by various factors such as thermal conductivity, diffusion coefficient, viscosity, surface tension, and extent of super-saturation. Mass transfer plays an important role in development of a new phase in many areas such as hydrate particles, heavy oil in petroleum reservoirs, and bubbles in blood. There are various mathematical methods to solve the governing differential equations of the transport phenomena cases such as bubble formation and growth. The homotopy perturbation method (HPM), as an approximate solution, was introduced by He (1999).27-29 This technique can be utilized to solve a variety of linear and nonlinear equations (differential and integral) in various science and engineering disciplines.27-29 In fact, this mathematical strategy, which is resulted from the combination of classis perturbation techniques and homotopy in topology, offers approximate analytical solutions for complex equations through a convenient/straightforward and reliable manner. HPM and its extended (or advanced) versions yield a quick convergence of the solution in most cases so that precise solutions are generally attained after a few iterations.28-31 There are two main issues with asymptotical methods such as conventional HPM; namely, convergence and stability. To eliminate these drawbacks, advanced versions of HPM (or enhanced homotopy perturbation method (EHPM)) are proposed through making suitable modifications on the original version of HPM. For instance, a linear part is incorporated in the HPM to maintain the inherent stability.28-31 It has been proved that EHPM is a more efficient and powerful tool, compared to conventional HPM, to find solutions for differential equations, particularly nonlinear fractional models.28-31 In this study, a mathematical modeling is conducted to further understand the new phase evolution in a finite environment where the mass transfer controls the rate of growth, considering both convection and diffusion mass transfer mechanisms. To obtain instantaneous size of the new phase, the governing mass conservation equation is solved by an analytical method and enhanced homotopy perturbation method 4 ACS Paragon Plus Environment

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(EHPM) for a finite system. The governing conservation equation is first solved by an analytical method where both convection and diffusion mass transfer control the rate of bubble growth. To further highlight the importance of convective mass transfer in the finite extent, the EHPM is employed. This new technique is introduced to show high capability of an approximate analytical method to find acceptable solutions for linear and non-linear differential equations. This would be beneficial particularly when finding an exact analytical solution for a highly non-linear equation is not feasible.2831

The extended version of homotopy perturbation method also provides a solution with acceptable

accuracy through a fairly straightforward procedure. The homotopy perturbation method (HPM) and EHPM solutions are generally generated non-iteratively with a minimal number of terms in a simpler form in comparison to analytical solutions.28-31 This is our motivation to include this new powerful tool (e.g., EHPM) to solve real world non-linear problems such as bubble evolution cases. Introducing new approximate analytical solutions also adds further novelty to the manuscript. To assess the theoretical calculations, a series of experimental data is presented for the comparison purposes with the aid of an appropriate statistical analysis.

2. MATHEMATICAL MODELING APPROACH Figure 1 illustrates a simple schematic of the growth of a new phase. Growth of a new phase in a solution is normally composed of two main steps: 1) The core is formed with a radius of R0, 2) A new phase with an initial radius R0 grows. Hence, the new phase formed in the solution has an initial radius and initial concentration of R0 and C0, respectively. The medium of finite extent has a radius of Rm and concentration of Cm (see Figure 1).



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Figure 1: Schematic of a new phase evolution model including initial and boundary conditions

The growth of the new phase in finite extent is studied, considering the following assumptions: •

The new phase is assumed to have a spherical geometry.

•

The process is isothermal. Thus, mass transfer (not heat transfer) controls the new phase growth (the growth of a new phase in a solution is resulted from the solute transfer occurring from the solution phase to the new phase).

•

Pressure and temperature distribution inside the new phase is assumed to be uniform.

•

Equilibrium condition exists at the interface between the solution and the new phase.

•

The mass diffusivity is constant.

•

The viscosity and surface tension of the fluids are constant.

•

The density is constant.

The mathematical model is obtained through combination of the continuity equation and mass transfer equation. The continuity equation is given below:

  .V t

(1)

in which, ρ stands for the density, t represents the time, and V is the velocity vector.


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For the spherical symmetry, constant density (e.g., temporal and spatial variations of density are negligible), and negligible angular velocity components in θ and ϕ directions, Equation (1) for the solvent surrounding the new phase can be written as follows:

1  2 r u   0 r 2 r

(2)

In Equation (2), r denotes the radius, and u signifies the velocity in r direction. The mass balance equation for multi-component systems with a constant density is presented as follows:

DCi  .J i  ri Dt

(3)

where DCi Ci   .CiV Dt t

(4)

in which, Ji, ri, and Ci are the mass flux, rate of chemical reaction, and concentration of component i, respectively. For a binary system in the absence of chemical reaction, the mass balance equation for component 1 is expressed based on Equation (3) as follow: C1  .VC1  .J1 t

(5)

For the spherical symmetry, constant mass density, only velocity in r direction, no mass fluxes other than ordinary diffusion, and constant diffusivity, Equation (5) is reduced to the following expression:18

C C   2C 2 C   D 2  u t r r  r  r

(6)

In Equation (6), D represents the diffusion coefficient.

The mass balance equation for the solute in the new phase on the surface in the absence of reaction can be written as follows:



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Variation of solute mass with respect to time or [Final mass – Initial mass] = [Outlet solute mass flow rate] – [Inlet solute mass flow rate] (7)

The mass transfer phenomenon for the solute includes both convective and diffusion mass transfer. Considering the relationship between the density, concentration, and sphere volume for the solute in the •

new phase, the surface of the new phase moves with the velocity R and the solution surrounding the new phase moves with the velocity u(R) such that the mass balance equation for the solute in the new phase on the surface based on Equation (7) is obtained by the following relationship:   •  C (r , t )  d 4 3  2    R m n   4R C ( R, t ) R  u ( R)  D    dt  3 r r  R      

(8)

where m is the mass fraction of the solute in the solution, C(R,t) stands for the solute concentration in the bubble, and ρn denotes the density of the new phase. After simplification, Equation (9) is generated: •  • C (r , t ) m n R  C ( R, t )  R  u ( R)  D   r  

(9) rR

The mass balance equation for the solvent in the new phase on the surface in the absence of reaction is as follows: Variation of solvent mass with respect to time or [Final mass – Initial mass] = [Outlet solvent mass flow rate] – [Inlet solvent mass flow rate] (10) For the solvent, the mass transfer due to the diffusion is negligible (small solvent concentration gradient), compared to the convective mass transfer. Thus, the mathematical form of the above relationship (e.g., Equation (10)) is given below:

•  d 4 3  2    R   4  R  R  u ( R ) n l  dt  3     

(11)

in which, ρl represents the solvent density.


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It is assumed that the density of the new phase is constant with the time. Thus, Equation (11) is simplified to the following expression: 4n R 2

dR  4R 2  l ( R  u ( R )) dt

(12)

where dR  R dt

(13)

After rearrangement of Equation (12), the following expression is obtained:

u( R)  (1 

n  )R l

(14)

Defining a new dimensionless parameter (   1 

n ), Equation (14) turns to the following form: l

•

u ( R)   R

(15)

In the above equations,

n is the density ratio, and ε stands for a dimensionless parameter. l

Based on the assumptions made in this study, the initial condition (IC) and boundary conditions (BCs) are listed below:

IC:

C(R0,t0)=C0

(16)

BC 1: C(R,t)=Ce

(17)

BC 2: C(Rm,t)=Cm

(18)

in which, C0, Ce, and Cm are the initial concentration, the equilibrium concentration, and the corresponding concentration at the maximum radius (Rm, which is the medium radius), respectively. Using Equation (15), integration of Equation (2) results in the following relationship: • R u   R  r

2

(19)

Plugging Equation (19) into Equation (6) leads to the following equation:



• C   2C 2 C   R  C  D 2     R  t r r   r  r  r

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2

(20)

Equation (21) is also obtained when Equations (9) and (15) are combined, as shown below. •

R

D C (r , t ) mn  Ce 1    r r  R

(21)

As mentioned earlier, two different methods including analytical method and advanced homotopy perturbation method (EHPM) are proposed to solve the governing differential equation. Some information on the solution procedures are given in the following sections.

2.1. Analytical Method The method of combination of variables32 is used for solving Equation (20), where it is assumed that, C(r,t) → C(s)

when

t 

(22)

and

s

r

(23)

f (t )

where s and β are the arbitrary combined variables. If s= β, it can be concluded that r = f(t) Employing the initial and boundary conditions, the variation of radius with time is obtained as follows:

R

2 D 3

1

1 C0  03 A0 2 3 0



t  t   R 0

(24)

2 0

in which, •

A0  R0 R0

m n  C e 1    D

(25)

0 is the initial void fraction (dimensionless). 10 ACS Paragon Plus Environment

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The initial void fraction ( 0 ) is defined as the ratio of the initial bubble volume to the medium volume (, which has a radius of Rm). Thus, the definition is expressed as follows:

4  3  R03    R0  Initial Bubble Volume  3  0     Medium Volume 4 3   Rm   R  m 3 

(26)

As clear from Equation (26), the void fraction is controlled by the process initial conditions, fluids properties, and the medium radius (or maximum radius).

After rearrangement, the solution of Equation (20) is attained as follows:  3  2  C0  C ( Rm , t )  C0 ( R0 , t )  A0 exp  D      2 6      3D  2 D  R0   2erf  2    Rm      6 D   3D  R  1  0   erf  2  Rm    

  

2

   

3D 2

  3D  exp  2  

        

(27)

in which, Α and γ are constant. Further details on the mathematical solution technique and a sample of calculations can be found in Supporting Information.

2.2. Enhanced Homotopy Perturbation Method (EHPM) The homotopy perturbation method (HPM), as an approximate method27, has been successfully utilized to solve several types of linear and nonlinear functional equations. This technique is an integration of homotopy in classic perturbation and topology methods, which was introduced by He.28 A broad application of this technique in physical, mathematical, and engineering problems confirms the efficiency, simplicity, and capability of the method, compared to other mathematical methods, for solving ordinary and partial differential equations or/and systems of non-linear equations.27,28 Construction of a homotopy is conducted through using an embedding parameter q  0,1 , which is taken into account as a small parameter, compared to the traditional perturbation techniques.

To illustrate the HPM, the following general nonlinear differential equation is considered: 11 ACS Paragon Plus Environment


A( x)  f ( )  0

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  (28)

B ( x,

x )0 t

 

(29)

where Γ denotes the boundary of domain Ω, f ( ) refers to a known analytical function, B stands for a boundary operator, and A represents a general differential operator.29 The operator (A) is generally divided into two parts; namely N and L; where N is a nonlinear function, while L is linear. Equation (28) is rewritten as follows:

L( x)  N ( x)  f ( )  0

(30)

The homotopy perturbation structure is shown as follows: 30 H ( , q)  (1  q)L( )  L( x0 )  qA( )  f ( )

(31)

Rearranging Equation (31) leads to: H ( , q)  L( )  L( x0 )  qL( x0 )  N ( )  f ( )

(32)

where  ( , q) :   0,1  R and x0 is the first approximation that satisfies the boundary condition. Using Equation (31), the following expressions can be written: H ( ,0)  L( )  L( x0 )  0

(33)

H ( ,1)  A( )  f ( )  0

(34)

Changing q from zero to unity corresponds to changing x0 ( ) to x ( ) . This is called deformation. L( )  L( x0 )  0 and A( )  f ( )  0 are also called homotopic in topology. The solution of Equation

(31) can be written as a power series in terms of q, as follows: n

   0  q 1  q 2 2  ....   q i i

(35)

i 0

The best approximation for the solution is given below:

x  lim   0   1   2  ...

(36)

q1

HPM may lead to an unexpected solution when a nonlinear differential equation includes an unstable linear part. To resolve this issue, an additional term is added to the linear part and subtracted from the rest at the same time such that a linear part is stabilized. This method is called enhanced homotopy perturbation method (EHPM). 31


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The EHPM is used for solving Equation (20), considering the following two cases: 1) both convection and diffusion mass transfer control the rate of growth; 2) only diffusion mass transfer controls the rate of bubble growth.

Case 1: Equations (20) and (21), which include the effect of the convection and diffusion mass transfer, are solved through using an approximate method known as EHPM, where it is assumed that

R   tD . The following expressions represent the approximate solutions in terms of concentration and ζ: exp( t )  1 1 R R 2 C  C0   (  5 )(1  exp( t )  t exp( t )) r r r

m n  Ce (1   ) 2 tD

 4 (

2  2 exp(t )  t exp(t )  tD

5 5 2

2t D

3 2

(37)

(1  exp( t )  t exp(t ))  0

(38)

Case 2: Neglecting the convective mass transfer, Equation (20) turns to:

  2C 2 C  C   D 2  t  r r  r  

(39)

Implementing the EHPM approach, the approximate expressions for the concentration and ζ are obtained as follows:

3 C  C0   r

(3  2t 

t2 ) exp( t ) 2 r

(40) 1

  3 2 t2   3  (3  2t  ) exp( t )      2   (m n  Ce ) tD 

(41)

More details on the mathematical solution via EHPM are available in Supporting Information.



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3. EXPERIMENTAL INVESTIGATION To assess the accuracy and applicability of the mathematical solutions attained in this study, a simple case including water at 103.1 ºC and 1 atm is considered and some tests are conducted by the authors. A simple schematic of the experimental set up is depicted in Figure 2. The main elements of the laboratory set up are a beaker, lamps, a professional camera, and a thermometer. In the experimental process, the water is first boiled in a large container for about 1hr to remove most of the air in the water and to attain uniform heating. After this heating process, the water phase still would contain around 25% of the initial air. Prior to the experiments, the beaker is carefully cleaned by a detergent to remove any contaminants. The beaker with water is then heated by two 200-watt reflector type infrared lamps. In the tests, 400 cm3 of water is brought to the boiling temperature in around 15 minutes, using the lamps. The water is superheated up to a temperature of 104 oC at the atmospheric pressure to obtain bubbles. A professional camera is utilized to record the change in the bubble size at a high speed. The pictures are taken at a rate of 2000 exposures per second. The picture/video magnification is determined through placing a scale in the water and considering a piece of film in the camera. The size of bubbles is measured directly via a microscopic comparator. To check repeatability of the experiments, the diameter is measured three times and the average value is considered for the comparison purpose.

Lamp

Lamp

Figure 2: A simple schematic of the experimental set up

The same experiments were conducted by Dergarabedian under the same process conditions.33 In this study, the average magnitudes of the results for three bubbles were used. It should be noted that the 14 ACS Paragon Plus Environment

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difference between the experimental data from different research investigations (related to bubbles in superheated water) reported in this paper is minor (e.g., < 2%). Table 1 lists the physical properties and other corresponding data for the bubble/water case. The data tabulated in Table 1 are used in the exact and approximate analytical solutions to generate Figures 3 to 6. Table 1: Experimental data for a water case over the bubble growth process (modified after 16, 33)

R0

10-5m

ΔC0

1 kg.m-3

Rm

10-3m

m

1

ϕ0

10-6

D

5×10-9 m2.s-1

ρn

1.37 kg.m-3

T

103.1 ºC

ρl

937.5 kg.m-3

Ce

0.0444 kg.m-3

4. RESULTS AND DISCUSSION The growth phenomenon is generally affected by various factors such as initial concentration, diffusion coefficient, velocity of mass element flowing, concentration difference, viscosity, surface tension, and void fraction. In this study, the growth of the bubbles is investigated at a constant temperature, which occurs in several cases (conventional production/recovery methods, bubble expansion in pure and multi-component liquids after the nucleation stage). It does not mean that the effect of temperature is ignored. As expected, the physical characteristics of fluids and bubbles vary by changing the temperature, leading to a considerable effect on the bubble growth. It is important to note that we investigate the bubble or new phase growth (not bubble formation) in this research work. The growth of the new phase in finite extent is obtained through solving the integrated momentum/mass equation through the variable combination approach (exact analytical method) and EHPM (approximate method). The experimental data are obtained by the authors and from the documents/research works available in the literature

9,16,33

. The real data are used to validate the

modeling approach introduced in the current study. As seen in Equation (24), there is a relationship between the radius of the new phase, time, and diffusivity. The distribution of concentrations around a growing new phase can be also found through 15 ACS Paragon Plus Environment


Page 16 of 28

employing Equation (27) where both diffusion and conversion mass transfer affect the bubble evolution. Figure 3 represents a comparison between the analytical method, EHPM, and the real data. In the case of boiling water, air (as a dilute component) transfers from the boiling liquid (water) to the gas (vapor) inside the bubble. Thus, it is assumed that the growth of the bubble is as result of the mass transfer of air from the liquid to the gas (vapor) forming the bubble. In Figure 3, Experiment (1) corresponds with the real data obtained by the authors for the bubble/superheated water case; Experiment (2) is attributed to the experimental data collected by Mohammadein and Gad-Elrab 16 for the same bubble/superheated system; and Experiment (3) corresponds with the real data of the experimental investigation conducted by Dergarabedian33 on the bubble/superheated water case. Thus, all three experimental works in Figure 3 illustrate the actual data for the bubble/water cases where the initial conditions, physical properties, and mass transfer characteristics are almost the same for all tests (see Table 1), though the experimental data have been attained by different researchers. As clear from Figure 3, an acceptable agreement is noticed between the analytical method, EHPM, and the experimental data. The small difference between the results of these two deterministic approaches arises from the assumptions made throughout deriving the final model. For example, it is supposed that the new phase growth is controlled only by mass transfer; the effect of temperature gradient is neglected, and the physical properties of water liquid and the gas or vapor phase are assumed to be constant. All these assumptions are slightly deviated from the conditions in the real cases, leading to minor errors in the model predictions. As seen in Figure 3, there is a very good match between the predicted results and real data when both convective and diffusion mass transfer are considered (Case 1). However, a considerable deviation is experienced when only diffusion mass transfer is considered (Case 2). Therefore, both diffusion and convective mass transfer have a significant impact on the growth of the new phase in finite extent. It should be noted that the analytical method and EHPM (Case 1) are solved where it is assumed that both convection and diffusion mass transfer control the rate of bubble growth. According to Figure 3, these two different solution techniques lead to almost the same outputs in terms of bubble radius.


Page 17 of 28

Table 2 displays the results of error analysis for the mathematical solutions developed in this study. One of the parameters for error analysis is the mean squared error (MSE). When the MSE approaches zero, it means that the error of this model is very low.34-36 Based on the result obtained for MSE, these models have a desirable performance. Mean absolute error (MEAE) is another parameter, which can designate the accuracy of the developed solutions/models. Maximum absolute error (MAPE) and minimum absolute error (MIPE) are other two factors to examine the performance of the solutions (or models) introduced in this study. The smaller the error percentage, the more accurate the model is. The corresponding formulas of the above parameters (MSE, MEAE, MAPE and MIPE) for the error analysis are presented in Supporting Information. The low values of the statistical evaluation parameters clearly imply the effectiveness and acceptability of the proposed modeling strategy.34-36

0.1

Experiment 1 Experiment 2 Experiment 3 Analytical model EHPM (case 1) EHPM (case 2)

0.09 0.08

0.07

R(cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.06 0.05 0.04 0.03 0.02 0.01 0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

t(sec) Figure 3: Comparison of bubble radius values obtained from the analytical modeling, EHPM, and experimental work [bubble/water case; Initial void fraction = 10-6].

Table 2: Error analysis of the models developed in this study

Analytical Method

MSE 6.38×10-6

MEAE% 3.69

MAPE% 4.92

MIPE% 0.0289

EHPM (case 1)

6.71×10-6

7.4

14.1

0.879



The impact of void fraction (or gas hold-up) on the new phase bubble size is demonstrated in Figure 4. It is observed that increasing the initial void fraction of the bubbles lowers the new phase radius. The main reason for this behavior is that higher void fraction corresponds to a higher number of bubbles in the ambient phase, leading to the lower mass transfer (from the ambient to the new phase) and consequently smaller new phase. Figure 5 shows the influence of the diffusivity (or molecular diffusion coefficient) on the new phase evolution. As the diffusivity increases, the new phase size becomes bigger because of a greater molecular diffusion mass transfer rate. Figure 6 demonstrates the variation of the bubble radius as a function of the temperature. It is clear that the physical characteristics of the system is strongly dependent on the temperature, implying different temperature results in different bubble growth rate. As the temperature increases, the vapor pressure increases, leading to a greater bubble growth rate. It is obvious that if the temperature is constant during bubble growth process, the vapor pressure remains constant, concluding the temperature is not important over isothermal processes.

0.08 ϕ=10e-6

0.07

ϕ=1.95×10e-6 0.06 R (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 28

ϕ=4.62×10e-6

0.05 0.04 0.03 0.02 0.01 0 0

0.003

0.006

0.009

0.012

0.015

0.018

t (sec)

Figure 4: Bubble radius in boiling water case versus time for three different values of initial void fraction (ϕ0).


Page 19 of 28

0.09 0.08 0.07 0.06

R (cm)

0.05 0.04 0.03

D=2×10e-9 m2/s

0.02

D=3×10e-9 m2/s

0.01

D=5×10e-9 m2/s

0 0

0.004

0.008

0.012

0.016

0.02

0.024

t (sec)

Figure 5: Variation of the bubble size in boiling water case with diffusion coefficient (D).

0.07 0.06 0.05

R (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.04 0.03 0.02

25 ⁰C 50 ⁰C

0.01 100 ⁰C 0 0

0.005

0.01

0.015

0.02

0.025

t (sec)

Figure 6: Bubble size in boiling water case against time at various temperatures.



Page 20 of 28

Other case studies discussed in this manuscript are the formation of gas hydrates and gas/oil solution systems, which play an important role in energy storage and transport, and oil recovery where hydrate gas reservoirs and oil reserves are targeted. In general, the hydrate formation (or/and nucleation) is strongly dependent on kinetic rate, mass transfer, and heat transfer where thermodynamic conditions play an important role. It is also clear that both mass transfer and heat transfer can affect the bubble growth in hydrate formation and oil/gas systems. After the hydrate is formed, it is acceptable to assume that the expansion of the new solid phase occurs mainly due to the mass transfer of gas component to the solid phase where the temperature remains almost constant so that no considerable change is experienced in the saturation pressure. The real data for three multicomponent systems reported in the literature are tabulated in Table 3. 9, 37

Table 3: Physical properties and relevant data for methane/water, carbon dioxide/water, and gas/oil solution systems (modified after 9, 37) ρn

ρl 3

kg/m CH4 CO2 Gas/oil solution

D 3

Ce

m 3

ΔC0

ϕ0

kg/m

2

3

m /s

kg/m

919.7

1002

8.7×10-7

1.235

0.134

0.102

10-6

1100

1002

3.4×10-10

1.502

0.298

0.297

10-6

152.86

800

5×10-5

4.404

1

0.151

10-6

kg/m

Using the real input data, the growth of hydrates such as methane [ CH 4 H2O5.75 at T=2 ºC and P= 28.4 atm] and carbon dioxide [ CO2 H 2O5.75 at T=10 ºC and P=44 atm] with an initial radius of 2×10-8 m is determined as a function of time via employing the developed model, as depicted in Figure 7. The growth of bubbles in heavy oil reservoirs at a constant temperature of 50 ºC and P=24.7 atm is also described in Figure 7.


Page 21 of 28

0.04 Carbon Dioxide

0.035

Methane

0.03

R (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Oil

0.025 0.02 0.015 0.01 0.005 0 0

0.004

0.008

0.012

0.016

0.02

0.024

t (sec) Figure 7: Evolution of the new phase for CO2/H2O, CH4/H2O, and gas/oil solution cases.

As Figure 3 and Figure 7 illustrate, the developed model appears to acceptably predict the variation of bubble diameter as a function of time. It is also concluded that CH4 experiences higher bubble size as well as growth rate, compared to CO2. The main reason for this behavior is that the values of viscosity and density for CH4 are lower than those for CO2, leading to higher mass transfer rate if thermodynamic conditions are the same for both gases. Figures 3-7 convey the message that bubble evolution follows the power-law model (e.g., R(t) = btN) where both diffusion and convection mass transfer mechanisms are the limiting factors, which influence the bubble size throughout the shrinkage and expansion phenomena. It is apparent that the power N and the constant b are dependent on the fluids properties and process conditions. Figure 8 confirms this finding, as well. The developed solutions (e.g., exact analytical and approximate) and Figure 8 clearly show that b is more affected by the process characteristics, compared to N.



0.12

102.1 ⁰C [32] 103.1 ⁰C [32]

0.1

R= 2.9349t0.7747

104.5 ⁰C [32] 0.08

R (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 28

R = 2.829t0.8146

101 ⁰C [32]

0.06 R = 1.1938t0.7357

0.04 R = 0.3145t0.6598

0.02

0 0

0.005

0.01

0.015

0.02

t (sec) Figure 8: Bubble radius versus time for boiling water case while employing power law model

The current analytical and approximate modeling introduces an effective strategy to forecast bubble size with reasonable accuracy in finite systems, which can be seen in various real cases corresponding to oil and gas transport, blood circulation in human body, boilers, and hydrate formation in pipeline systems. In-depth understanding of bubble growth in finite and infinite environments assists researchers and engineers to design and operate the corresponding process equipment (e.g., nozzles, spargers, and pipeline elements) through efficient engineering and economical manners.

5. CONCLUSIONS New analytical and approximate mathematical solutions for growth of a new phase in a medium of finite extent are developed in this study. The obtained solutions consider all important parameters in the bubble evolution process under isothermal condition. The governing equations obtained in this study are based on the fact that the mass transfer controls the bubble evolution. The following conclusions can be drawn on the basis of the results attained in this research work:

1- Growth of a new phase in a finite system is strongly affected by both diffusive and convective mass transfer in the absence of heat transfer for isothermal operations. 2- It is found that the bubble growth is directly proportional to the diffusivity and temperature, but inversely varies with the initial void fraction. 22 ACS Paragon Plus Environment

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3- It is concluded from both experimental and modeling results that the power law model can appropriately describe the bubble evolution behavior (e.g., bubble diameter in terms of time) 4- The models’ results reasonably match the experimental data so that the error analysis clearly confirms the precision of the models. 5- There is a very good match between the values of bubble radius obtained from the analytical method and enhanced homotopy perturbation method (EHPM) 6- The developed mathematical solutions can have broad applications in energy, environment, and health systems where generation and growth/shrinkage of bubbles in various mixtures are experienced.

SUPPORTING INFORMATION The Supporting Information is available free of charge on the ACS Publications website. Further details on analytical method and enhanced homotopy perturbation method (EHPM); samples of calculations; and definitions of statistical parameters and their importance.

NOMENCLATURES Acronyms BC EHPM HPM IC MAPE MEAE MIPE MSE

Boundary condition Enhanced homotopy perturbation method Homotopy perturbation method Initial condition Maximum absolute error Mean absolute error Minimum absolute error Mean squared error

Variables A0 and A1 A(x)

Constants defined by Equations (S.14) and (S.24)

B

General differential operator Boundary operator

C Ce ΔC0 D

Concentration (kg.m-3) Equilibrium concentration (kg.m-3) Concentration difference (kg.m-3) Diffusivity coefficient (m2.s-1)

f ( )

h J

Analytical function Constant defined by Equation (S.25) Rate of chemical reaction 23 ACS Paragon Plus Environment


Page 24 of 28

Vector mass flux

j L m N P r R R0

Linear function Mass fraction of the solute in the new phase

Nonlinear function Pressure Radial coordinate Bubble radius (m) Initial bubble radius (m) Instantaneous new phase velocity (m.s-1) Arbitrary function defined by Equation (23) Temperature Time (sec) Radial velocity Velocity vector Approximation function Constant defined by Equation (S.21) Constant defined by Equation (S.21)

•

R

s T t u v x y z

Greek Symbols/Variables α β Г γ

Constant defined by Equation (S.23) Arbitrary variable for combining independent variables

Boundary of domain Ω

  1 n l

η ρ ϕ Ω

Constant defined by Equation (S.22) Dimensionless parameter Constant defined by Equation (S.22) Density Void fraction (dimensionless) Domain

Subscripts e l m n 0

Equilibrium Liquid phase Maximum value New phase Initial value/condition

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For Table of Contents Only

Porous Medium

0.04 Carbon Dioxide Methane Oil

0.035 0.03

R (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 28

0.025 0.02 0.015 0.01 0.005 0 0

0.004

0.008

0.012

t (sec)

S1 ACS Paragon Plus Environment

0.016

0.02

0.024

New Modeling Strategies Evaluate Bubble Growth in Systems of Finite

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