Chemical Engineering Journal Advances

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Numerical model of an ultrasonically induced cavitation reactor and application to heavy oil processing

Item Type Article

Authors Guida, Paolo;Viciconte, Gianmaria;Ceschin, Alberto;Colleoni, Elia;Hernandez Perez, Francisco;Saxena, Saumitra;Im, Hong G.;Roberts, William L.

Citation Guida, P., Viciconte, G., Ceschin, A., Colleoni, E., Hernández Pérez, F. E., Saxena, S., Im, H. G., & Roberts, W. L. (2022). Numerical model of an ultrasonically induced cavitation reactor and

application to heavy oil processing. Chemical Engineering Journal Advances, 100362. https://doi.org/10.1016/j.ceja.2022.100362 Eprint version Publisher's Version/PDF

DOI 10.1016/j.ceja.2022.100362

Publisher Elsevier BV

Journal Chemical Engineering Journal Advances

Rights © 2022. The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://

creativecommons.org/licenses/by-nc-nd/4.0/) Download date 2024-01-16 21:56:51

Item License https://creativecommons.org/licenses/by-nc-nd/4.0/

Link to Item http://hdl.handle.net/10754/679675

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Chemical Engineering Journal Advances 12 (2022) 100362

Available online 13 July 2022

Numerical model of an ultrasonically induced cavitation reactor and application to heavy oil processing

Paolo Guida

^*

, Gianmaria Viciconte, Alberto Ceschin, Elia Colleoni,

Francisco E. Hern ´ andez P ´ erez, Saumitra Saxena, Hong G. Im, William L. Roberts

Clean Combustion Research Center, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

A R T I C L E I N F O Keywords:

Ultrasonically induced cavitation Oxidative desulfurization OpenFOAM

Heavy fuel oil

A B S T R A C T

This study describes a numerical approach to model ultrasonically induced cavitation (UIC) reactors. UIC forms vapour-filled cavities in a liquid medium due to an applied acoustic field and their eventual collapse. UIC reactors are characterized by the presence of a vibrating probe that generates pressure waves by high-frequency oscillations (>20 kHz), which control the formation, dynamics, and eventual collapse of the vapour cavities.

Those vapour cavities eventually enhance mixing and favour the occurrence of gas-liquid reactions. The zones of high mixing and reactivity coincide with the presence of the bubble cloud, which depends on the shape of the vessel and sonotrode. The development of advanced computational fluid dynamics (CFD) models is crucial to optimizing UIC processes’ geometry and operation parameters. A new algorithm for modelling UIC has been implemented within the OpenFOAM framework in the present study. The volume-of-fluid (VoF) method employs a diffuse interface approach for the volume fraction transport equation. The bubble dynamics are solved with sub-grid models, and the coupling between the main flow field and the sub-grid scales is performed through source terms in the transport equations. The source terms are de-coupled from convective and diffusive components of the volume fraction equation. The history of the bubbles is considered to consist of nucleation, oscillations, and collapse. The oscillations are resolved via the Rayleigh–Plesset equation. The concluding part of the work demonstrates the application of the algorithm to simulate the operation of an UIC reactor, which was designed to desulfurize fuels using the oxidative (ODS) process.

1. Introduction

Ultrasonically induced cavitation (UIC) is a physical phenomenon that entails the formation and subsequent fate of vapour-filled cavities in a liquid irradiated by a source of acoustic waves. In a standard UIC configuration, a vibrating probe, called sonotrode, generates pressure waves by high frequency (>20 kHz) oscillations. Those pressure waves induce nucleation of bubbles, oscillations, and eventual collapse. The latter is of particular interest because of the peculiar thermo-physical conditions it creates, leading to extreme temperature and pressure in localized hotspots. By using spectrometric methods and additional experimental techniques, Suslick and Flannigan [1], Didenko et al. [2], 3] quantified maximum temperatures in the order of 4300K in regions where the bubbles collapse take place. In these spots, sonochemistry may lead to the formation of unstable species such as hydroxyl (OH*) and hydrogen (H*) radicals when UIC is applied to an aqueous solution.

Several applications of UIC have been developed for wastewater treatment as well as for pharmaceutical, medical, and food industries [4–10]. UIC is also widely applied in the petroleum and biofuels industry. The low-value feedstocks, such as heavy oils and biomass, can be upgraded using UIC. Kapusta [11] studied the influence of ultrasound in sewage sludge pre-treatment before the hydrothermal liquefaction process, obtaining a maximum increase of 19% in bio-oil yield for son- icated sludge. UIC is used in the ultrasound-assisted oxidative desulfurization process in the oil industry to recover a significant fraction of sulfur not removed by the standard desulfurization process such as hydrodesulfurization. In the presence of an oxidative agent, typically hydrogen peroxide, UIC leads to the formation of sulfones that the fuel can easily extract [12]. Other applications of UIC include oil sand extraction, viscosity reduction, de-metallization, and upgrade of heavy oil (see, e.g., the review by Avvaru et al. [13]). One of the characteristic features of UIC reactors is the potential spatial heterogeneity of

* Corresponding author.

E-mail address: [email protected] (P. Guida).

Contents lists available at ScienceDirect

Chemical Engineering Journal Advances

journal homepage: www.sciencedirect.com/journal/chemical-engineering-journal-advances

https://doi.org/10.1016/j.ceja.2022.100362

Received 31 May 2022; Received in revised form 27 June 2022; Accepted 7 July 2022

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temperature and species concentration. The temperature distribution in the reactor is strongly dependent on the presence of the sonotrode, which generally has a higher temperature than the surrounding liquid.

Also, the collapse of the bubbles generates heat which reflects into a higher temperature in the zones where a bubble cloud is present.

Additionally, concentration gradients are expected if the mixture has a wide boiling range since the species will preferentially make bubbles with the lowest volatility.

Given the relevance and complexity of UIC, the development of advanced computational fluid dynamics (CFD) models is crucial to optimizing an ultrasonic reactor’s geometry and process parameters. In the past, some authors devoted their attention to this purpose and tried to solve the problem by increasing the model complexity. However, the nature of the relevant features in a UIC-CFD code strongly depends upon the physical processes one desires to describe and model. Van Wijn- gaarden [14] proposed one of the first approaches to solve the problem by modelling it as a one-dimensional unsteady flow in bubble-fluid mixtures. Later, Caflisch et al. [15] refined the concept by including other relevant effects such as surface tension, viscous dissipation, and heat transfer to describe more appropriately the micro-scale phenomena.

Real applications, however, require the mathematical model to be flexible and fast enough to be applied to complex geometries. To this end, Schnerr and Sauer [16] formulated a simple cavitation model to be used within a volume-of-fluid (VoF) framework, where the formation, oscillation, and collapse of the bubbles are accounted for through a source term in the volume fraction equation. Following the Rayleigh–- Plesset (RP) equation [17,18], the source term is proportional to the square root of the difference between the saturation and actual pressures of the liquid. Znidarˇ ˇciˇc et al. [19] demonstrated that, although the Schnerr–Sauer model gives good results for hydrodynamic cavitation, it has poor efficacy in describing UIC. In hydrodynamic cavitation, the time-scale of pressure changes is much larger than bubble dynamics, which is not the case in UIC, where the pressure field rapidly fluctuates.

In an attempt to overcome the limitations of the previous models, Znidarˇ ˇciˇc et al. [19] introduced a correction by including the acceleration term of the RP equation. Nevertheless, the model could not predict the nucleation of the cavitation bubbles.

Louisnard [20] demonstrated that a correct evaluation of the wave attenuation in a bubbly liquid requires a realistic estimation of the power dissipated by the oscillation of each bubble within the pressure field and employed the Caflish model [15] to describe the propagation of acoustic waves in a bubbly liquid. Furthermore, the complete RP equation was also used to solve the bubble dynamics and calculate the energy dissipation by heat transfer between the vapour bubbles and the surrounding liquid. Louisnard also reported the importance of the intense convective transport inside the fluid domain, and the effect was accounted for by adopting the Bjerknes force acting on the bubbles [21].

More recently, Lebon et al. [22] proposed a non-linear acoustic model capable of predicting the modification of the ultrasonic wave propagation in the liquid due to cavitating vapour bubbles. The source term in the vapour phase transport equation was obtained through the solution of the complete RP equation. This is because the values of the dynamic viscosity and the surface tension between the gas and liquid interface are considerable in the liquid metal ultrasound process.

As the literature survey indicates, existing models for UIC are built upon assumptions that limit their predictive capabilities and applica- bility. Further developments are needed to study the processing of complex liquids with higher fidelity using large scale simulations.

A more comprehensive approach to describe UIC is proposed in this work, and a new solver is developed based on the OpenFOAM framework (version 2006) [23]. The volume-of-fluid (VoF) method is used, with the two-phase flow being treated as a homogeneous mixture, and only one set of equations is adopted for the entire domain. The bubbles dynamics is solved within each computational cell. The coupling between the flow field and micro-scale processes introduces a source term

in the volume fraction transport equation, which is de-coupled from the advection step using the operator splitting technique. A correlation is proposed to estimate the bubble density in a given computational cell as a function of the pressure. The correlation allows the existence of bubbles only within the boundaries established by the Blake threshold [24], as proposed by Louisnard [25], 26] and Lebon et al. [22]. In this way, the present approach estimates the surface area between bubbles and the liquid medium, a key parameter in a gas-liquid reactive system, in describing the dynamics of a single bubble.

The induced pressure waves in the reactor were simulated by adopting a moving mesh that replicated the oscillation of the sonotrode.

The solver was validated against experimental results published by ˇZnidarˇciˇc et al. [19] and Campos-Pozuelo et al. [27]. Subsequently, the code was used to study a large scale UIC reactor simulating the formation and motion of bubbles in a complex setup. The case study was a UIC reactor for oxidative desulfurization (ODS) of fuels. The simulation allowed us to identify the bubbles size distribution, and, hence, the expected reactive region of the system. Furthermore, the method was demonstrated to be efficient for large scale simulations.

2. Model description

The proposed approach for predicting UIC phenomena is based on the VoF method. The transport equation of the liquid volume fraction is coupled with the continuity and momentum conservation equations to solve for the flow field. The advantage of the VoF method is that it allows for tracking both liquid and gas phases with a single equation for each variable of interest. The bubbles dynamic is described by the Ray- leigh–Plesset equation, which is solved at each computational cell and coupled to the flow field through source terms in the transport equations.

The interface between bubbles and bulk flow is not fully resolved.

Instead, a diffuse interface approach is used. A diffuse interface approach is justified because the solver must be efficient for real applications involving large computational domains. A geometric VoF would result in a significantly higher computational cost. The geometric VoF requires multiple cells to model the interface between liquid and gas, which is unreasonable considering the size of actual cavitation bubbles. Note that a fully Lagrangian approach is not considered because of the difficulty of addressing the time history of each bubble. In the following, the components of the model are described in detail.

2.1. Bubble sub-models

UIC involves the dissipation of mechanical energy in a rather peculiar and unique way. The mechanical energy provided by the sonotrode, through high-frequency vibration, is converted for the nucleation and oscillation of the vapour bubbles. During the collapse, the energy stored in the bubble is released into a minimal volume, leading to local hot spots characterized by high pressure and temperature. Heat is trans- ferred from the hot spots to the continuous phase leading to a temperature increase in the bulk liquid, which is lower by magnitude than the hot spots. The evolution of a bubble consists of three steps: nucleation, oscillation, and collapse.

2.1.1. Nucleation

Comprehending the complex phenomenon of nucleation is essential to building a physical model that can predict the density and size of the vapour bubbles accurately during ultrasonic cavitation. In the classical context of hydrodynamic cavitation, when the bulk pressure of the liquid goes below the saturated vapour pressure threshold, bubble nucleation may occur in discrete points of the fluid domain, and the cavitation intensity is a function of the bulk pressure and velocity of the flow field. From a thermodynamic point of view, bubble nucleation is induced by localized weak points within the liquid [28].

In the present work, a combination of the approaches proposed by

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Louisnard [25,26] and Vanhille [29] is used to model cavitation inception. Louisnard [26] defines the domain of existence of the bubbles using the Blake threshold [30–32]. Accordingly, in the present CFD code, the bubble inception as the onset of cavitation is triggered when the pressure amplitude within a computational cell exceeds the Blake pressure [24]. The bubbles density function is then described with the hyperbolic tangent function proposed by Vanhille [33] expressed as:

N=

⎧⎪

⎨

⎪⎩ Nmaxtanh

(

| P− PBlake

Psat− PBlake

| )

if|P| ≥PBlake,

0 if|P|<PBlake,

(1)

where PBlake corresponds to the cavitation threshold of the medium that was proposed by Vanhille in his formulation [29], and Nmax is the maximum bubble density that can be reached within the computational cell. The latter was calibrated based on experimental observations [22, 27,34,35] and set to 1×10¹², resulting in a minimum number of bubbles per unit volume in the order of 1×10¹⁰m⁻³. The saturated vapor pressure (Psat) is the highest pressure value that is able to modify the bubble density, yielding a value of bubble density equal to Nmax. The cavitation threshold (Blake pressure, PBlake) is defined according to the formulation proposed by Blake [24] and also used by Louisnard [26]:

PBlake=1+ (4

27 S³ 1+S

)

12. (2)

In the equation above, S represents the dimensionless form of the Laplace tension, given by

S= 2σ PRC

, (3)

where σ is the surface tension of the liquid phase, which depends strongly on temperature, P is the ambient pressure (neglecting the contribution given by the hydrostatic pressure inside the fluid), and RC is the equilibrium radius of the bubbles within the computational cell. In the present model, the bubble nucleation is assumed to happen instan- taneously when the condition described above is satisfied in the computational cell. The equilibrium radius of the nucleated bubbles is obtained from the Young–Laplace equation

RC= 2σ

PG− P. (4)

Equation (4) represents the balance of the forces acting on a single bubble, having RC as an equilibrium radius [28], where PG is the pressure within the bubble and P is the liquid pressure outside the bubble corresponds to the pressure within the computational cell. Moreover, assuming that the bubbles contain only vapour, the pressure inside the bubble PG equals the saturated vapour pressure Psat(T). This nucleation model implies that all the bubbles nucleating within a computational cell have the same equilibrium radius RC.

2.1.2. Bubble dynamics

After the nucleation step, vapor bubbles oscillate around their equilibrium position, defined by the equilibrium radius RC. The Ray- leigh–Plesset (RP) equation describes the bubbles’ dynamics. This equation is derived by assuming spherical symmetry and applying the Navier–Stokes equations projected onto the radial coordinate. It is written in complete form as [28]:

3 2

(dR dt )

2+Rd²R

dt² =PG(t) − P(t) ρ_L ⁻

4νL

R dR

dt− 2σ

ρ_LR. (5)

This differential equation expresses the bubble radius as a function of time (R(t)) when the vapour bubble is subject to an external pressure field, P∞(t). Assuming that the bubble contains only vapour, PG is equal to the saturated vapour pressure Psat(T)at the liquid temperature, while ρ_L, νL and σ are, respectively, the density, kinematic viscosity and surface

tension of the liquid. These parameters depend strongly on the liquid temperature TL, and thereby, the Rayleigh–Plesset equation is coupled with the energy equation. Note that TL and P(t)are given by the local values of temperature T and pressure P in each computational cell. The terms on the left-hand side represent the inertial contributions to momentum along the radial direction, while the terms on the right-hand side represent the instantaneous tension or driving force due to the pressure difference [28], and contributions of the kinematic viscosity and surface tension to the bubble dynamics.

In the present work, all the terms are preserved. This choice comes from the need to have the most general and accurate representation of bubbles’ dynamics. Moreover, the terms related to viscosity and surface tension become particularly important when dealing with the UIC of heavy fuels. An example of the numerical solution for a single bubble within a fluctuating pressure field and having the thermodynamic properties of diesel is shown in Fig. 1 (temporal variation of bubble radius and pressure field) and 2 (temporal variation of bubble temperature). Fig. 1 shows that bubble oscillations and pressure fluctuations have two different time scales [1,30], which justifies the choice of de-coupling the flow field and the bubbles’ dynamics in the numerical algorithm. The RP equation is solved via a semi-implicit method with the time step set to be smaller than 1×10⁻⁸s. Following the domain of existence of bubbles proposed by Louisnard [26], the collapse of the bubbles is assumed to happen when the pressure amplitude falls below the Blake threshold pressure [24], which is the minimum pressure that guarantees the existence of cavitation bubbles. The collapse is represented by simply assuming the instantaneous disappearance of gas bubbles and imposing the volume fraction of liquid equal to one. From a physical point of view, the energy stored in the oscillating bubbles is released in a tiny portion of space during the bubbles’ collapse. The energy associated with the oscillation of the bubbles is expressed using the Rayleigh–Plesset equation, multiplied by the time derivative of the Fig. 1.Temporal variation of bubble radius under a fluctuating pressure field.

The frequency of the fluctuating field is 20 kHz.

Fig. 2. Temperature variation of the oscillating bubble. The temperature fluctuates following the changes in the radius of the bubble. The bubble is assumed to have the same thermodynamic properties as diesel fuel.

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bubble volume, dV/dt, yielding an equation for the radial kinetic energy [26]:

3 2ρ_L

(dR dt )

2dV

dt +ρ_LRd²R dt²

dV

dt = (PG(t) − P∞(t))dV dt

− ρ_L⁴νL

R dR

dt dV

dt− 2σ R

dV dt.

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The energy contribution due to the oscillation and collapse of the vapour bubbles is incorporated into the global energy equation through a source term. Note that interaction between bubbles is neglected as the dynamic cloud interaction parameter (B) is low for the cases considered [36]. The latter is a function of bubbles number density (Nb), bubble radius (Rb) and cloud radius (Rcloud):

B=NbRb

Rcloud

, (7)

where Rcloud is the dimension of the bubbles’ cloud, which is generally 3- 4 orders of magnitude larger than the bubbles’ average radius for the cases studied.

2.2. Flow modelling

An Eulerian volume-of-fluid (VoF) method [37] is adopted to model the two-phase flow, describing the dynamics of two-phase compressible fluids with a unified velocity u(x,t), pressure P(x,t), and temperature T(x, t). In the following, the computational domain is defined as Ω∈R³ while the two subsets Ω₁and Ω₂represent the liquid and gas phase, respectively. The intersection of the two subsets is the interface Γ∈R². The interface Γ can be seen as the boundary between two adjacent phases, Ω₁and Ω₂. The evolution of the interface is captured by a phase indicator function:

f(x,t) =

{1 x∈Ω1(t),

0 x∕∈Ω1(t), (8)

referring to the liquid phase and indicated with Ω₁. The next step consists of discretizing the computational domain by dividing it into a discrete number of control volumes (computational cells) Cl for l =1, .., Nc.

In the context of a finite volume representation, at a given time, the variable α_l(t) determines the volume fraction of liquid in a given computational cell:

αl(t) ≡ 1 Vl

∫

Cl

f(x,t)dV, (9)

where Vl is the control volume of the lth cell, note that the liquid volume fraction is a continuous variable, although it is defined as a cell-averaged quantity. The liquid volume fraction, therefore, takes on the value

α(x,t) =

⎧

⎨

⎩

1 within the liquid, ]0,1[ at the interface, 0 within the gas.

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For consistency, α1=α is defined as the liquid volume fraction, while α2=1− α is the gas volume fraction. All local physical quantities are calculated as a linear combination of the properties of the two phases. The local quantity in the liquid and gas phase also depends, in general, on the composition of the mixture. A general physical property ψ is then expressed as:

ψ=α1ψ₁+α2ψ₂, (11)

where subscripts 1 and 2 denote the liquid and gas phases, respectively.

The transport equation for the volume fraction of a compressible fluid experiencing phase change is expressed as

∂(αρ)

∂t + ∇⋅(αρu) = − mδ˙ Γ, (12)

where m ˙ is the mass exchange rate due to phase change across the interface and δΓ is the surface area density at the interface which is given by the surface of the bubbles present in the computational cell. The mass exchange flux contribution has a negative sign if the liquid evaporates and a positive sign if the vapour condenses. On the other hand, the bulk fluid motion is described by the mass and momentum conservation equations, noting that the incompressibility assumption does not hold in the bubble bursts because of the high velocities resulting from severe pressure gradients within the droplet. The continuity and momentum equations are expressed, respectively, as follows:

∂ρ

∂t+ ∇⋅(ρu) =0, (13)

∂(ρu)

∂t + ∇⋅(ρuu) = ∇⋅[

μ⁽∇u+ ∇u^T)]

− ∇P+fs+ρg, (14) where g represents the gravitational acceleration, μ is the dynamic viscosity, and fs is the surface tension force.

The surface tension force f_sis important and generally modelled with the continuum surface force (CSF) method, introduced in reference [38]:

fs=σqn, (15)

where σ is the surface tension coefficient, n is the unit normal vector to the interface and q is the curvature, expressed as the divergence of the normalized gradient of the volume fraction:

q= − ∇⋅

(∇α

|∇α| )

. (16)

Finally, the energy equation is also solved to determine temperature:

∂⁽ρcpT)

∂t + ∇⋅( ρcpuT)

= ∇⋅(k∇T) − mΔH˙ v, (17) where the last term describes the enthalpy of vaporization exchanged because of the eventual phase change, k is the thermal conductivity, cp is the heat capacity at constant pressure, and ΔHv,j is the vaporization heat of the jth species. The energy equation formulation presented above neglects the terms associated with pressure and viscous stresses, which are typically small.

By solving the RP equation, the source term m is calculated as: ˙ m˙=4α2

ρ₁ρ₂ ρR

dR

dt, (18)

R is the average radius of the bubbles present in the individual cells, the source term is then included in the volume fraction equation. An operator splitting approach is employed in its solution procedure.

2.2.1. Momentum equation

Solving the momentum equation starts from considering the following linearization:

ρ^∗uⁿ⁺¹− ρⁿuⁿ

Δt + ∇⋅(

ρu^∗uⁿ⁺¹)

= ∇⋅(

μ⁽u^∗∇ + ∇(u^∗)^T))

− ∇Pⁿ⁺¹+fs+ ∇(ρ^∗)g⋅x.

(19)

The momentum equation is then discretized as

Mu= − ∇P+fs+ ∇(ρ)g⋅x, (20)

where M is the coefficients matrix of the velocity vector. The coefficients matrix M is further decomposed, resulting in the following formulation:

Auⁿ⁺¹=H(u^∗) − ∇Pⁿ⁺¹+fs+ ∇(ρ^∗)g⋅x, (21)

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where A and H represent the diagonal and off-diagonal terms of the coefficients matrix, the superscript ^∗denotes the provisional values of the variables in the iterative solution procedure.

2.2.2. Pressure equation

The developed flow solver falls into the category of pressure-based solvers. Accordingly, the derivation of the pressure equation is briefly described, noting that the subscript i refers to the phase under consideration (one for liquid and two for gas). First, Eq. (12) is expanded to yield

ρ_i∂αi

∂t +αi

∂ρ_i

∂t +ρ_iu⋅∇αi+αiρ_i∇⋅u+αiu⋅∇ρ_i=m˙iδΓ, (22) where m˙1= − m ˙ and m˙2 =m. A pressure-dependent equation can be ˙ readily obtained from the above expression by applying the chain rule on pressure. Introducing Ptot, which denotes the sum of thermodynamic and hydrostatic pressure, Ptot = p+ρg⋅x, it is possible to obtain the following:

∂αi

∂t +u⋅∇αi= − αi

ρ_i

∂ρi

∂Ptot

(∂Ptot

∂t +u⋅∇Ptot

)

− αi∇⋅u− m˙i

ρ_i^δ^Γ^. ⁽²³⁾ Finally, by summing over the two phases, the final form of the pressure equation is obtained:

(α1

ρ1

∂ρ₁

∂Ptot

+α2

ρ2

∂ρ₂

∂P )(∂Ptot

∂t +u⋅∇P )

+ ∇⋅u− m˙ (1

ρ1

− 1 ρ2

)

δΓ=0. (24) The above equation is then linearized and solved implicitly for pressure using the velocity value obtained from the momentum equation. The linearization step reads

(α^∗₁ ρ^∗₁

∂ρ₁

∂Ptot

∗

+α^∗₂ ρ^∗₂

∂ρ₂

∂Ptot

∗)(

Pⁿ⁺¹_tot − Pⁿ_tot

Δt +u^∗⋅∇Pⁿ⁺¹_tot )

+ ∇⋅uⁿ⁺¹

− m˙^∗ (1

ρ^∗₁⁻ 1 ρ^∗₂ )

δΓ=0,

(25)

where the variables marked with the superscript ^∗ denote updated provisional values. Furthermore, the pressure equation is solved implicitly calculating the value ∇⋅uⁿ⁺¹from the following equation:

∇⋅uⁿ⁺¹= ∇⋅(

A⁻¹H(u^∗))

− ∇⋅(

A⁻¹∇Pⁿ⁺¹_tot )

, (26)

which comes from Eq. (21).

2.2.3. Solution algorithm and method

The newly developed solver was implemented in the OpenFOAM v2006 framework. The main steps of the solution procedure are listed in Algorithm 1. For the cases considered in this work, the discretized transport equations were solved using the PIMPLE method, which combines the Pressure Implicit with Splitting of Operators (PISO) and Semi-Implicit Method for Pressure Linked Equations (SIMPLE) iterative procedures [39,40]. The momentum and pressure equations are solved sequentially until convergence is achieved. In particular, the Geometric

Initialization of α, p and T fields. while t < t

end

do Solve α equation;

Solve α evaporation equation;

for i ← cell

0

to cell

N

do

if |P[cell

i

]| >|P

Blake

[cell

i

]| then if α[cell

i

]=0 then

Obtain equilibrium radius;

Obtain nucleation rate;

else

Solve Rayleigh-Plesset equation;

Obtain instantaneous radius;

Obtain instantaneous radius rate of change;

if R<R

Blake

then Impose α[cell

i

]=0;

Calculate energy source term;

end end end end

Solve Energy equation;

Solve Pressure-Velocity coupling;

Update mesh;

Check and correct mass conservation;

end

Algorithm 1.Solution algorithm of UIC solver.

Fig. 3.Boundary conditions for the configurations of Campos-Pozuelo et al. [27] (a) and ˇZnidarˇciˇc et al. [42] (b).

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Agglomerated Algebraic Multigrid (GAMG) strategy was utilized to solve the pressure equation. A second-order Gauss linear reconstruction was employed for the spatial discretization, except for the convective terms of the volume fraction and momentum equations. The mono- tonicity- and conservation-preserving scheme of van Leer [41] was applied. The time derivatives were discretized using an implicit Euler scheme. The simulations were performed on the supercomputer Shaheen-II (Cray XC40 system) at KAUST.

3. Validation

For the validation of the developed UIC modelling framework, the experimental configurations previously studied by Campos-Pozuelo et al. [27] and Znidarˇ ˇciˇc et al. [42] are considered, and the numerical predictions are compared against available data from their experiments.

Such comparison is necessary to assess the predictive capabilities of the proposed approach and its suitability for application to other UIC reactors.

3.1. Case setup and boundary conditions

The cases reported in this paper describe batch reactor configurations. The choice and implementation of the right boundary conditions were crucial to describe with high fidelity the actual physical processes of a UIC reactor. The first simulations described are the two configurations proposed by Campos-Pozuelo et al. [27] and ˇZnidarˇciˇc et al. [42].

Those two experimental setups were chosen for the validation process as they reported both spatial and temporal information. In Fig. 3, the batch reactor configurations used by Campos-Pozuelo et al. [27] and Znidarˇ ˇciˇc et al. [42] are schematically depicted through a 2D illustration (the di- mensions of the batch reactors and information on the experimental setup are available in the original works of the authors [27,42]). The rigid walls of the batch reactor and the surfaces of the ultrasonic probe must be modeled accounting for the reflective behaviour of the pressure waves. To model these boundaries in the OpenFOAM framework, a condition that assigns the pressure values via field assignment is utilized (fixedValue in Fig. 3), while the interaction between the solid walls and velocity field is modelled considering the adhesion of liquid to the wall (noSlip condition in Fig. 3).

The free surface of the liquid in the batch reactor is modelled considering a zero-gradient condition for the velocity field (zeroGradient in Fig. 3) and a value of pressure equal to the atmospheric (uni- formFixedValue in Fig. 3). Note that in the experimental setup of Campos-Pozuelo et al. [27] (configuration (a) in Fig. 3), absorbent material is used to avoid reflection of the pressure waves on the bottom wall of the batch reactor. The absorbent boundary is modelled using a zero-gradient (zeroGradient) and a wave transmissive (waveTransmissive) condition for velocity and pressure. Besides, to take into account the motion of the ultrasonic source, the existing moving mesh capability of OpenFOAM was incorporated into the solver. Indeed, this allows repli- cating the real vibration of the ultrasonic probe. The CFD simulations of the experimental configurations have been run on 2D computational domains. Indeed, the 2D domains have been extracted from the actual experimental setups. This reduces the computational cost of every

simulation and is crucial in the solver validation phase. The main parameters used for the CFD simulations of both experimental cases [27, 42] are reported in Table 1. For both the cases, the time-step was determined such that the Courant–Friedrichs–Lewy (CFL) number was less than 0.2, as suggested in Ref. [22] for a similar numerical setup.

3.2. Grid convergence

A grid convergence study is reported before presenting the results of simulations and the comparisons with experiments. This analysis aims to certify that the solver can provide a grid-independent solution. The study has been done considering ˇZnidarˇciˇc et al. [42] experimental setup (Fig. 3, Table 1). A 2D (two dimensional) computational grid (mesh) made by orthogonal hexahedral elements with similar dimension, have been considered. The cases tested, corresponding to different sizes of the grid elements, are listed in Table 2.

The parameter I used to verify grid convergence is the time- dependent integral of the vapor fraction across the whole computational domain:

I=

∫

Vtot

(1− α)dV (27)

This parameter is the most representative since, as explained in the previous chapter, the vapour volume fraction is dependent on the nucleation mechanism, bubbles dynamics, and all the other fields (pressure, velocity, and temperature). The results obtained in terms of total vapour volume fraction as a function of time are shown in Fig. 4 for each grid. Note that each value of I has been normalized according to the maximum value found: I =I/Imax. The chart shows four cycles of the probe oscillation. After the first two transient cycles, the volume fraction reaches a periodic stationary state. Since our analysis does not focus on the initial transient, the trends related to the different element sizes overlap well. Therefore, the solver can provide a grid-independent solution if the size of the mesh element is kept in the explored range.

In the next paragraphs, a comparison between CFD results and experimental data is proposed to further validate the solver.

Table 1

The main parameters are set for the simulations.

Parameter Case

[42] Case

[27] Large scale simulation Probe vibration frequency

[kHz] 20 20 20

Probe vibration amplitude [μ_m] ₁₆₄ ₇₄ ₁₆₄

Wave period [μ_s] ₅₀ ₅₀ ₅₀

End time [μ_s] ₅₀₀ ₅₀₀ ₁₀₀₀

Probe vibration cycles 10 10 20

Time step [ns] 100 100 100

Table 2

Grid convergence analysis for the configuration of ˇZnidarˇciˇc et al. [42].

Simulated cases Elements size [mm] Number of computational cells

Case 1 1 2,010

Case 2 0.5 7,880

Case 3 0.25 31,520

Case 4 0.1 197,000

Fig. 4.Grid convergence study, cell size from 1 mm down to 0.1 mm. Integral of vapor fraction across the domain. Values have been normalized by the maximum obtained. First 4 cycles are reported.

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3.3. Comparison with experimental results of Campos-Pozuelo et al. [27]

The first configuration under consideration is Campos-Pozuelo et al.

[27], schematically illustrated in Fig. 3a. They performed experiments using an ultrasonic source induced by a 20 kHz vibrating probe, having a tip diameter of 70 mm. The experiments were run by processing the liquid at room temperature and atmospheric pressure. Using a micro- phone, the pressure evolution was measured at a specific location as a function of time and different locations in the reactor.

A representative snapshot of the cavitation region that develops below the vibrating probe’s tip was also reported, as shown in Fig. 5, in which a conical bubbles structure is clearly identifiable.

Fig. 5 shows a qualitative comparison between the vapor volume fraction distribution (black iso-contours), predicted by the CFD simulation (Fig. 5b), and the snapshot of the cavitation region obtained by Campos-Pozuelo et al. [27]. In Fig. 5b the background color map corresponds to the spatial pressure distribution, for a certain instant of time, obtained with the CFD simulation. The simulation qualitatively

reproduces the conical structure of the distribution of the bubbles.

Furthermore, as shown by the high density of volume fraction iso-contours, the maximum bubbles’ density is predicted to be on the surface of the probe tip. This is in agreement with the experimental results, where the maximum bubble density (the white region in Fig. 5) is located in the region attached to the tip of the probe.

For quantitative comparison, Fig. 6 displays the pressure profile measured by Campos-Pozuelo et al. [27] using a Bruel & Kjaer (B&K) 8103 hydrophone, and the results obtained with the model. The pressure was evaluated at different depths, along the axis of the sonotrode. A good agreement with the experimental measurements is experienced.

The temporal evolution of the pressure was measured at a point located 40 mm below the tip, along the axis of the probe. The maximum pressure recorded in the experiment is 610 kPa, while the model predicted a maximum of 641 kPa within the same temporal window resulting in a 5% relative error. The model correctly predicted the frequency of pressure peaks (20 kHz), noting the presence of a sub- harmonic component in the pressure distribution at approximately 40 kHz. This is associated with forming a bubbles’ cloud, which nucleates and collapses with a characteristic frequency that differs from the vibration frequency of the ultrasonic probe.

3.4. Comparison with experimental results of ˇZnidarˇciˇc et al.

The second configuration considered for validation is the experimental data by ˇZnidarˇciˇc et al. [42], who carried out measurements of cavitation activity under different conditions by changing the following parameters: acoustic power, air saturation, viscosity, surface tension, and temperature of the liquid water. The configuration consists of a rectangular batch reactor made of glass (50 mm x 50 mm x 50 mm). A schematic representation of the reactor configuration along with the associated boundary conditions is shown in Fig. 3b. The sonotrode was submerged vertically in the reactor. The tip of the sonotrode, having a diameter of 3 mm, was located 10 mm below the free surface of the Fig. 5. Conical bubbles structure captured by Campos-Pozuelo et al. [27] (a) and volume fraction of vapor (black iso-contour lines) obtained from the simulation (b).

The background color of the numerical result corresponds to pressure distribution.

Fig. 6. Comparison of pressure profile below the sonotrode at incip- ient cavitation.

Fig. 7.Comparison of predicted cavitation zone (bottom row), identified as cells having α<0.2, and the cavitation zone captured by Znidarˇ ˇciˇc and co-workers [42]

(top row). For the numerical snapshots the solid black line corresponds to the iso-contour α=0.2 and the background color map corresponds to pressure distribution.

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water. The vibration frequency was 20 kHz, and the maximum power provided to the sonotrode was 70 W. As for the simulations, a representative case of cavitation is selected for comparison with measurements, which were performed for the following conditions: distilled water saturated with air at the temperature of 296 K, sonotrode frequency of 20 kHz, and 70% of the maximum power (49 W), which corresponds to a vibration amplitude of the ultrasonic horn of 164 μm.

Fig. 7 shows a qualitative comparison between the vapour volume fraction predicted by the CFD and the cavitation region captured through the high-speed camera by Znidarˇ ˇciˇc et al. [42], at different in- stants of time. In the CFD simulation, the cavitation region is defined by the black solid lines (Fig. 7). The black line corresponds to the iso-contour having the vapor volume fraction equal to α =0.2 implying that the area enclosed by the black line is characterized by a vapour volume fraction of α<0.2. The background colour map is related to the spatial pressure distribution. The cavitation region (α <0.2), predicted by the CFD model, (Fig. 7) is reasonably similar to that observed by the experiment (the black region in the snapshots in Fig. 7). The results thus confirm that the simulation can reproduce the region’s shape where the vapour fraction is higher than zero.

For quantitative comparison, in Fig. 8 the pressure profile recorded by the hydrophone (placed 7 mm apart from the probe tip) is compared with the pressure values provided by the simulation at the same location. The pressure peaks are predicted to occur at approximately the same frequency and with similar amplitude as the measured ones.

However, the simulation cannot correctly predict all the fluctuations recorded by the hydrophone. This is attributed to a background noise present in the hydrophone or to the real acoustic behaviour of the reactor glass walls being different from the fully reflective one as assumed in the model. Despite the observed quantitative differences, the level of agreement with the measurements is considered satisfactory, given the stochastic nature and complexity of the UIC process.

4. Case study: Ultrasonically-enhanced oxidative desulfurization

This section presents the application of the modeling framework to study a pilot-sized setup, in which Heavy Fuel Oil 380 is processed through an ultrasonically-enhanced ODS process [43]. HFO 380 has been commonly used in marine transportation, but since 2020 its use has been banned because of large sulfur amounts. In the last years, oxidative desulfurization (ODS) has been proposed to remove sulfur. The ODS process allows the modification of liquid fuel structure by selectively oxidizing and later extracting molecules containing sulfur. UIC can enhance the ODS process through improved mixing, an increased interfacial area between aqueous and oily phases, and radical formation that is believed to favor the oxidation process. Several secondary reactions, however, take place during the standard ODS process. The byproducts formation alters the physical properties of the fuel substantially. This issue and the low yield were recognized as the causes that prevented industrial applications of ODS. However, by combining

the study of flow patterns and the identification of the most active cavitation zones it is possible to prevent the occurrence of secondary reactions, such as polymerization and formation of gum, that may potentially take place and affect the effectiveness of the process [12].

Reactivity was not explicitly implemented in the reactor simulation for two reasons: 1) there is no availability of a kinetic mechanism; and 2) the proposed model does not allow to simulate two different liquids and, thereby, the reactivity between oxidizer and oil.

4.1. Case setup

The pilot-scale reactor consists of a sonotrode operating at a frequency of 20 kHz in a steel vessel. The computational domain of the pilot-scale UIC reactor is illustrated in Fig. 9. For this configuration, the original ultrasonic probe is composed of four cavitating modules.

However, in Fig. 9, only two of the four modules are reported since most of the activity occur in these two modules. The grid was built with cfMesh, a finite volume mesh generation library built on OpenFOAM. An amount of 3.3 million cells, mostly regular hexahedral, ranging from 0.5 mm up to 1 mm, were used to capture the geometry details, with special care at the sonotrode walls. It is important to note that the sizes of the elements used in the simulation, fall within the grid convergence range tested previously. Given the short time step required to solve the physics (5e-7 s), the total time of the simulation was limited to 1 ms, enough to evaluate 20 vibration cycles of the sonotrode, but not enough to solve the flow dynamics. Therefore, the process was considered as a batch, although it is intended to work continuously. As a consequence, no turbulence closure was required. Given the nature of the vessel and its qualitative similarity with the setup of ˇZnidarˇciˇc and co-workers, the boundary conditions listed in Fig. 3b are adopted. The simulation was initialized with quiescent HFO-like liquid, using the physical properties Fig. 8. Comparison of the time-varying pressure profile recorded by a probe in

the reactor and the one obtained from the simulation.

Fig. 9.Computational domain of the pilot-scale UIC reactor. The probe’s location used to sample values is indicated by the red sphere. (For interpreta- tion of the references to color in this figure legend, the reader is referred to the web version of this article.)

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of HFO. This assumption is justified because, in the ODS process, the quantity of other reagents is generally small. However, the saturation pressure was modelled by using the Antoine parameters of hydrogen peroxide, which is a commonly used oxidizer for the ODS process. In particular, temperature-dependent viscosity and surface tension were used in the RP equation and for the flow field. The fuel properties at 50

∘C were measured in previous works [44,45] and are reported in Table 3.

Given the strong temperature gradients in the system, temperature- dependent correlations for the physical properties were required. Vis- cosity was measured experimentally for HFO 380 as a temperature function, and a polynomial correlation was then extrapolated. Since mass transport and evaporation affect the local concentration of species, viscosity was also expected to be a function of mass fractions; however, in this work, the viscosity was assumed to be a function of temperature only. This assumption was necessary because mixing rules for such heavy components were not available. The surface tension was calculated using the Riazi correlation [46] for petroleum fractions. Given the absence of suitable data at the temperatures reached in the hot spot, the physical properties were then calculated at the bubble’s interface, assuming that the vicinity of the bubbles had the same temperature as the bulk liquid, i.e. the temperature in the computational cell. Addi- tional parameters set in the CFD simulation are listed in Table 1.

4.2. Results

Representative snapshots of the pressure field and vapor bubble volume fraction are displayed in Fig. 10. These fields correspond to sequential events during the 8th cycle of the sonotrode vibration. It is essential to specify that the system becomes stationary after the 4th cycle. The pressure isocontours, represented on the left of each time case (Fig. 10), clearly show the propagation of the pressure waves originating from the probe, and their reflection on the reactor walls. The vapor bubble volume fraction represented on the right of each snapshot (Fig. 10) shows that vapour bubble formation happens in toroidal regions around the main axis of the sonotrode. The colour of the bubbles maps the Bubbles’ Distribution Index (BID), a parameter formulated to reconstruct the bubbles’ cloud in the form of discrete particles in the domain. The BID is defined as:

BIDi= (1− α[celli])Vcelli, (28) where Vcelli is the volume of the ith computational cell. The regions are located at the horizontal surfaces of the ultrasonic probe, from which the pressure waves originate. The retraction of the sonotrode, at time 412 μs, led to a substantial increase in bubble size (increase in the vapour volume fraction) following the behaviour predicted by the Rayleigh–- Plesset equation. The size and number of bubbles were observed to be much more significant in the vicinity of the horizontal surfaces. Pressure waves were damped by the fluid’s viscosity and by the occurrence of bubbles that locally diminish density, making the region next to the walls poor of hot spots. This is a relevant factor to take into account when designing or optimizing a UIC reactor. For many applications, the region that does not experience cavitation can be considered dead volumes. Therefore, it is required to minimize those regions to have a ho- mogenous bubbles’ distribution within the reactor.

Pressure and vapor volume fraction (1− α)were sampled at a point Table 3

Physical properties of the HFO used in this work at 50 ^∘C.

Property Value Unit

Density 970 kg/m³

Surface Tension 0.033 N/m

Heat of vaporization 650 kJ/kg

Viscosity 380 cSt

Fig. 10. Evolution of pressure and bubbles distribution index for the pilot-scale UIC reactor during the 8th cycle (from 400 to 450 µs). In detail, from top to bottom and left to right: 400 µs, 412 µs, 425 µs and 438 µs. For each time the pressure contour is shown on the left, while the bubble volume fraction is shown on the right and represented as a Lagrangian field.

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10 mm below the sonotrode tip, along its axis (red sphere in Fig. 9). The temporal evolution of these parameters is plotted in Fig. 11a, and the resulting Fourier transform of pressure evolution is plotted in Fig. 11b.

From Fig. 11a, the vapor volume fraction peaks happen after the corresponding pressure peak, when the pressure rapidly decreases. The Fourier transform of the pressure field, in Fig. 11b, clearly indicates the characteristic frequencies of the system. The pressure signals have the same frequency of the probe vibration (20 kHz) and its multiples (40 kHz, 60 kHz and 80 kHz) [47]. Again, these characteristic frequencies are influenced by the characteristic time of collapse and nucleation of the vapor bubbles. Indeed, the oscillations of the bubbles’ clouds were observed as a function of the pressure field, resulting in a series of characteristic frequencies that are multiples of the sonotrode frequency.

The clouds of bubbles, obtained in this configuration (Fig. 10), have a peculiar toroidal structure and propagate primarily from top to bottom.

However, the formation of such vapour clouds was certainly influenced by the shape of the ultrasonic probe, which is not a straight cylinder as in the cases reported by Campos-Pozuelo et al. [27] and ˇZnidarˇciˇc et al.

[19]. The toroidal structure is supposedly formed as a consequence of the 90^∘edges present on the probe.

5. Conclusions

A new model to describe ultrasonically induced cavitation in liquid media was developed. The new model’s main features consist of the coupling between a homogeneous approach, following the volume-of- fluid formulation, and a Lagrangian-like approach, resolving bubbles’ dynamics and coupling it with the flow field using the operator splitting technique. A more refined description of the bubble dynamics allows more insights into the interaction between bubbles and flow field than previous works. Moreover, no ad hoc parameters have been used for the simulation, with the nucleation being the only empirical law. The solver was demonstrated to accurately replicate the literature’s experimental data regarding pressure fluctuations and bubble structure formed in the computational domain. The sonotrode motion was simulated through a moving mesh to replicate its influence on the pressure field. The characteristics of this solver make it particularly suitable for large scale simulations. A numerical simulation of a large-scale setup for ultrasonically-enhanced ODS was also presented. The pressure distribution, as well as the bubbles’ size distribution, were demonstrated. As expected from several experimental observations in similar systems, the most considerable bubble clouds were observed below the sonotrode and next to the second cavitating module. It is possible to conclude that CFD simulations are valuable for designing and optimising UIC reactors.

In particular, such process results in a better understanding of the reactor’s real thermodynamic and fluid dynamics conditions, which generally substantially differ from ideal model. Moreover, the bubble-

liquid interface is expected to have a crucial role in many chemical processes; therefore, estimating the latter is extremely important when processing experimental data to develop kinetic models.

Declaration of Competing Interest

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.

Acknowledgements

The work was sponsored by the Clean Combustion Research Center at King Abdullah University of Science and Technology (KAUST).

Computational resources were provided by the KAUST Supercomputing Laboratory (KSL).

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Fig. 11. Data related to probe at 10 mm away from the tip of the sonotrode in the pilot-scale reactor and collected after the initial transient. (a) Pressure and vapor fraction evolution. (b) Amplitude of modes from Fourier transform of pressure time history.