‘Smart’ transonic atomization and heating of a pulsating non-Newtonian liquid sheet

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‘Smart’ Transonic Atomization and Heating of a Pulsating Non-Newtonian
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Liquid Sheet
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D. M. Wilson1, W. Strasser1, R. Prichard1
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1School of Engineering, Liberty University, Lynchburg, VA 24515, USA
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ABSTRACT
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We introduce proportional integral derivative (PID) controls into transonic pulsatile steam-assisted non-
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Newtonian slurry heating and disintegration. The purpose is to ensure consistent, reliable atomization during
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generic process upset scenarios, while this implementation involves a sudden pronounced slurry property shift.
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The uniquely interrelated physical responses of phase interfacial atomizer instabilities require continuously
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coupled PID controllers, the first of which automates slurry flow based on slurry pressure drop. The second
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compensates for the variable phase momentum ratio and sets a new heating steam flow based on the targeted
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droplet size. Three tests with increasing rigor were conducted to demonstrate successful coupled controller
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adaptability. During controller compensations, slurry and steam flows were significantly altered and drastically
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changed atomization characteristics. For a 100-fold increase in slurry viscosity, however, the controllers
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successfully maintained consistent droplet size and slurry flow resistance. The control methodology was
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shown to be mesh-independent and to operate across multiple atomization regimes.
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Keywords: CFD, multiphase, atomization, PID, viscosity, AI
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The author to whom correspondence may be addressed: wstrasser@liberty.edu
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1. INTRODUCTION
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There is no shortage of computational and experimental atomization studies, but academic atomization
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processes typically entail liquids that have spatially and temporally uniform properties. Those studies of
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traditional atomization are outside the scope of this article. Consistent and reliable atomization becomes
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challenging when liquid properties like viscosity and surface tension are inconsistent. Industrial slurries, gel
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fuel, and manure slurry, for example, could have widely varying (temporally or spatially) viscosity, which
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degrades atomization quality and exacerbates pumping requirements. High viscosity restricts the flow,
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straining the pump; viscosity’s restraining force prevents smaller ligaments and droplets from forming, thereby
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reducing atomization quality. How can consistent and efficient atomization of a liquid be achieved under these
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circumstances? To the authors’ knowledge, there has been no study of heating and atomization with temporally
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or spatially varying slurry properties.
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An important application for this question is manure slurry atomization (highly variable feedstock) for waste-
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to-energy conversion. Ever increasing energy demands and scarcity of resources drive the need for alternatives
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to conventional energy production. Manure can already be harnessed for energy by processes like digestion,
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gasification, and pyrolysis.1 An alternative, and potentially more efficient, means of energy conversion is direct
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spray injection of a concentrated manure slurry (non-Newtonian) into energy harvesting equipment, such as a
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steam boiler. This method could reduce processing costs (such as drying) to increase conversion efficiency.
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Additionally, it is known that shear has cleansing effects for manure slurries, and atomization is generally a
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high-shear process.2
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Control strategies of various types provide critical support for industrial processes.3-5 Recent examples which
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entail incorporating automatic controls into computational fluid dynamics (CFD) for stable chemical reactor
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operation are given by Turman and Strasser6,7. For unit operations involving atomization technology, closed-
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loop feedback control could provide more robust spray processes for a wide range of applications. This is
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particularly true when feedstocks vary but processing consistency is sought in the face of limited resources.
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For example, active fuel spray control is of great interest for internal combustion engines (ICEs).8 Despite past
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studies of control for combustion instabilities,9-11 a 2019 review article points to the potential technology leap
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for ICEs and the need to develop new atomization control methods.8 Most recently, Osuna-Orozco et al.
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demonstrated closed-loop feedback control of an air-water coaxial atomizer.12,13 A desired spray structure,
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defined as a certain liquid distribution, was acheived by adjusting swirl ratio and gas flow.12 Electrostatic
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actuation can provide a third input to produce a goal spray liquid distribution.13 The focus of these methods is
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adjusting certain input parameters (like swirl ratio) to produce a preferred spray state.
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Processing a non-Newtonian slurry with variable properties in a twin-fluid atomization system leads to
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complex, interrelated two-phase physics that demand an innovative two-part control approach. Novelty is
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provided by introducing a new “Smart” atomization framework that incorporates proportional integral
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derivative (PID) control algorithms into (CFD) simulations to adjust for dynamically changing slurry viscosity.
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PID is an old science, but PID use in CFD, and especially with atomization, is far from it. For this paper, a
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variable slurry viscosity is considered, but alternative variables could just as easily be addressed within this
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framework. The two controllers operate continuously to account for real-time varying coupled process
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fluctuations. Hereafter, Smart atomization and heating with PID control will be referred to simply as “Smart.”
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PID involves closed-loop feedback control widely used in industrial processes and has been incorporated into
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CFD models for testing.13 Various airblast nozzle designs are available for non-Newonian slurry atomization,
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such as the three-stream atomizer studied by Strasser.15,16 The PID control is herein applied to a novel WAVE
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(Wave-Augmented Varicose Explosions) atomizer design. The unique wave-inducing inner design of the
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pulsing twin-fluid WAVE nozzle utilizes a central hot gas flow (inverse of typical twin-fluid atomizer) to
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provide mechanisms for wave-augmented slurry disintegration.17
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WAVE is designed for heating and disintegration of viscous, non-Newtonian slurries with a low gas-liquid
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mass ratio (GLR) of 2.7% (high efficiency).18 An atomizer geometry of this type was first shown to be useful
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for wastewater treatment sludge atomization.19 In short, annular slurry (cold) waves periodically “rise” into
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the central heating steam flow (see Figure 1) before crashing just outside the nozzle in the form of radial
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explosions. The high-blockage-ratio waves significantly contract the exit flow area, accelerating the steam to
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transonic states and causing temperature and pressure fluctuations at the steam-slurry interface which create a
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self-sustaining feedback loop of instabilities. Traditional effervescent atomizers and/or annular liquid
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devices20-23 do not involve high-blockage transonic steam acceleration which produces macroscopic waves
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responsible for the primary atomization mechnism. The non-Newtonian nature of slurry creates additional
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instabilities as viscosity variation results in wavelength and stress spatial variability. Breakup of slurry sheets
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extending from the nozzle is facilitated by steam heating, wave impact momentum, and windward pressure
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buildup, while fine droplets are stripped away from the penetrating wave. The physics of WAVE atomization
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for banana puree have been extensively and exclusively documented,17,18,24 but the benefits of Smart
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atomization have never been explored.
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Figure 1 Side view of the WAVE atomizer geometry with a conceptual sketch of the instantaneous position
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of the traveling wave. The cold slurry flows in from an outer annulus into the central hot steam flow, creating
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strong steam blockage and acceleration. The two phases form waves to enhance the slurry disintegration
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process.
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Two independent controllers are incorporated into our CFD model. The first, henceforth referred to as “C1”,
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adjusts the cold slurry flow based on the atomizer pressure drop. Its objective is to to maintain the consistency
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of this pressure drop given broadly variant liquid viscosity, the success of which ensures consistent slurry-feed
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pumping requirements. The second, henceforth referred to as “C2”, compensates for the newly resulting gas-
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liquid momentum ratio (GLMR) and adjusts the hot gas flow based on droplet size. GLMR is defined as
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

, where  is the gas density,  is the liquid density,  is the gas bulk velocity, and  is the liquid
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bulk velocity. While this description might imply the two controllers operate in series, they actually are
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working constantly in tandem. C2’s objective is to maintain consistent droplet sizes while the atomizer
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experiences widely varying 1) viscosity and 2) liquid flows resulting from C1 influences. C2 thus serves two
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roles, compensating for both C1 liquid flow (and thus momentum ratio) adjustments and liquid viscosity
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fluctuations. In short, a coupled controller system is used to protect atomization quality while ensuring slurry
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feed pump reliability.
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To our knowledge, this research represents the first tests of pulsating transonic heating and atomization using
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continuous closed-loop feedback control for variable viscosity feedstock with a coupled, continuous PID
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controller system. In contrast to the work of Osuna-Orozco et al,12,13 the focus of this present work is
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continuous, real-time process control to simultaneously maintain consistent atomization quality and pumping
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requirements in the presence of significant viscosity (orders of magnitude) fluctuations. Here, to the contrary,
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continous coupled adjustments are made for two measured variables, both droplet diameter and pressure drop
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in a coupled PID system. The controllers are shown to work against one another in some respects, invoking a
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nonlinear search for a new equilibrium. Additionally, our transonic pulsatile atomizer is uniquely designed for
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globally unsteady viscous slurry heating and disintegration, rendering coupled PID control challenging.
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Finally, various interfacial pressure contributions are explored for this uncommon high-blockage variable
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property non-Newtonian annular wave generator.
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A series of three Smart tests will be used to evaluate the efficacy of the coupled controller system for PID-
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assisted atomization. Test results will also provide guidance for improving the Smart approach. All three tests
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capture the system response to a pronounced step increase in viscosity, and test model rigor increases
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progressively. The hot assisting gas is steam for all three tests. Test 1 uses a basic mesh with a Newtonian
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slurry. Test 2 still uses a Newtonian slurry but with a more refined mesh and greater modeled azimuthal domain
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extent. For Test 3, Smart is applied to a WAVE model with an even finer mesh, where the impact of viscosity
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transition and feed rate changes on shear-thinning banana puree WAVE atomization is observed.
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2. METHODS
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2.1 Computational Methods
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The overarching CFD method employed here is not new and has been outlined in the cited works, but the coupled
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CFD/controls method proposed here is only captured in this document. An overview of the general computational
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approach for all three controller tests is provided here; key differences between tests are discussed in Section 2.3.
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We note again that test model rigor increases progressively from Test 1 to Test 3. For computational efficiency,
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the full, azimuthally symmetric atomizer geometry was reduced to a wedge with periodic boundary conditions on
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both azimuthal bounding faces. Almost all cells are hexahedral and swept in the general flow direction. The steam
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pipe and slurry annulus inlets (left of Figure 1) both have PID-controller-varied mass flow boundary conditions,
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with inlet temperatures of 393 K and 304 K, respectively. The outlet to the right of Figure 1 was set to atmospheric
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pressure. The nozzle exit and steam pipe diameters are both 0.016 m, and the slurry annular gap is 0.0098 m. The
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nozzle geometry is identical across all three tests, though the computational domain extent varies.
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Equations 1-3 present the governing Navier-Stokes equations for multiphase flow. They are formulated in
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vector notation, and all properties are arithmetically phase-averaged ( is phase volume fraction). Fluid
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properties are represented as follows:  is density,  is constant pressure heat capacity,  is laminar conductivity,
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 is the static temperature, and  is pressure. Other symbols are here defined:  is time,  is the velocity vector, 
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is gravity,  is the surface tension force vector,
 is the turbulent viscosity,  is the turbulent Prantdl number,
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 is the laminar shear stress tensor, and  is turbulent shear stress tensor.
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
 󰇛󰇜 
󰇛󰇜  (1)
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
 󰇛󰇜   
  
󰇛  󰇜 
      
 (2)
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
 󰇛󰇜 
󰇟󰇛󰇜󰇠  
 󰇣󰇡  
󰇢
󰇤 (3)
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The computational approach to CFD Smart tests is much the same as that outlined in 24, and these methods
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have been extensively validated for transonic, non-Newtonian airblast atomization over the last decade.15, 16, 25-
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31 A summary of validation exercises and representative pictures of meshes with varying degrees of refinement
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have been previously published.17 Experimental results have been reproduced with CFD both qualitatively and
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quantitatively, including the globally pulsing characteristic of the system (qualitative), the accoustic signature
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(quantitative), and the axial droplet size profile (quantitative). Additionally, the primary atomization ligament
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wave positions quantitatively matched experimental measurements, and the trajectory of a droplet after
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exposure to a normal shock followed a known empirical approximation. These validation exercises provide
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confidence in the solver’s capability for capturing important physical mechanisms in the particular context of
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a non-Newtonian slurry in an airblast atomizer.
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As recorded in past documentation, the framework used to solve the Reynolds-averaged Navier-Stokes and
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volume-of-fluid (VOF) equations in commercial CFD solver ANSYS Fluent 2020 R1 are briefly outlined as
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follows. A k-ω turbulence model (“standard” or SST, depending on the controller test) and SIMPLE pressure-
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velocity coupling scheme were utilized. PRESTO! was used for pressure; second order upwind for momentum,
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energy, and density; and first order upwind for turbulence quantities. Geometric reconstruction (also known as
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“piecewise linear interface capturing” or PLIC)26,32 was used for the volume-of-fluid (VOF) interface. Slurry
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was modeled as incompressible. Time step varied depending on the model and mesh size, but Courant number
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was generally kept below 1. Typical time step sizes were astoundingly low at 󰇛󰇜.
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We remind the reader that RANS does not resolve any turbulent structures, and true atomization mesh
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independence is likely impossible. Based on spatiotemporal integral length scale (ILS) estimates discussed in
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prior works,17,18 the largest turbulent structures are likely to be too small to control interfacial deformation of
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the wave and droplets we seek to resolve. It is certainly possible that our ILS estimates are incorrect, which
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could affect droplet physics downstream of the waves. The scope of this study is limited to wave formation
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and primary atomization physics, where our proposed control methodology is relevant, and we are ignoring
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the influence of unresolved turbulent structures on smaller droplets and secondary breakup. The mesh
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nomenclature is as follows: the coarsest is a “Base” mesh, which is then refined. The only cells not refined were
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upstream in the steam pipe and slurry annulus, far from any steam-slurry interfacial development. “Ref-1” refers
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to a single refinement, cutting each cell edge length in half in all three dimensions to increase the cell count by a
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factor of approximately 8. “Ref-2” refers to a second refinement, reducing cell size (and increasing cell count) yet
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again.
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2.2 PID Control Algorithm Implementation
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A PID control algorithm was implemented as user-defined functions (UDFs) in ANSYS Fluent 2020 R1 for
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the two controllers. Equation 4 is the discrete velocity PID algorithm used, where  is the controlled variable,
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 is the discrete time stamp,  is the sampling time, and , , and  are the controller’s proportional,
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integral, and derivative gain constants, respectively, and  is the controller error. The error is the difference
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between the measured variable and setpoint (not an error between CFD and experimental results), where
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measured values are time-averaged within each  sampling period. The sampling time was on the order of
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1×10-3 s. A physical implementation of this system would require a pump to respond at this time scale, which
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could present challenges; however, the basic premises of our methodology can be adapted accordingly. Figure
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2 illustrates the PID controller feedback loop in the form of a block diagram, where the process (controlled
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variable) is adjusted to keep the output (measured variable) at setpoint. Note that in all cases, setpoints were
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arbitrarily chosen based on quasi-steady-state (QSS) results of the baseline tests, and those will be mapped
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out below. Other demonstrations of setpoints could just have easily been carried out. Proportional gain helps
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immediately and linearly adjust the controlled variable to reduce the error, integral gain considers the
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combination of how long the error has been in effect plus to what extent, and the derivative gain takes into
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account how quickly the error is approaching zero from either the positive or negative directions. In concert,
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they each contribute components to each controller which drive the respective errors to zero as quickly as is
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physically possible. It can be thought of as a primitive form of artificial intelligence which does not require
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training. Processes, which can be thought of as individual physics manifestations within a given control
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volume, have unique natural response timescales. Controller algorithms should be tuned to accommodate the
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governing processes of the system.
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    󰇣󰇛 󰇜

 󰇛   󰇜󰇤 (4)
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Figure 2 PID controller feedback loop illustrated with a block diagram. The process is adjusted based on three
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controller constants,  (proportional),  (integral), and  (derivative) to keep the output at setpoint. Each
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controller behaves independently based on each desired set of inputs and outputs.
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For C1, the measured variable is liquid pressure drop, and the controlled variable is liquid mass flow. Pressure
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at the liquid inlet represents pressure drop, as the outlet is constant at 1 atmosphere, absolute. Since “pressure”
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and “pressure drop” are functionally the same here, they are used interchangeably in referring to the measured
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variable for C1 in the following discussions. For C2, the measured variable is droplet size, and the controlled
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variable is steam mass flow. Droplet size is captured by a UDF as the volume-averaged Sauter mean diameter
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(SMD) in a series of 10 volumes spaced axially away from the injector. A rigorous and documented verification
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process showed that the algorithm is able to reproduce the theoretical SMD for test objects in the domain.25
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For each test, the PID input variable is one of these 10 volume-averaged SMD measures, about halfway
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between the nozzle exit and the end of the computational domain. Future efforts could record the entire droplet
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distribution, which would make available other droplet statistics besides SMD. For each test, SMD is measured
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about halfway between the nozzle exit and the end of the computational domain. Two protections are added
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to each controller to maintain a stable and practical system: 1) mass flow rates of either phase cannot be set
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below zero and 2) measured values outside of three standard deviations (computed instantaneously) are not
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included in the averaging process. In the future, an operating range for GLR could be included as a third
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protection measure to keep the system within practical limits. The steam pipe and slurry annulus inlets both
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have PID-controller-varied mass flow inlets. For the Smart tests, one, none, or both controllers can be engaged
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to observe various responses to a step increase in viscosity. “C0” means no controllers are engaged, “C1”
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implies only C1 is engaged, “C2” designates that only C2 is engaged, and “C12” means both C1 and C2 are
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engaged. Rather than utilizing C12 against viscosity fluctuations, an alternative use of these controllers for a
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fuel with constant properties would be to use C2 to hit a target SMD setpoint while C1 maintains a constant
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air-fuel ratio. In response to a user change in SMD setpoint, the controllers would modify flow rates to reach
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a new QSS.
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Since the physics of the system drive the need for dual interactive controls, more clarity surrounding aspect of
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the physics is necessary to setup discussions of C1 and C2 controller mechanics. In the context of pressure
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drop, the subscript “s” will be used for the liquid phase (slurry), and the subscript “g” will be used for the gas
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phase (steam). The subscript “i” will be used a placeholder for either phase. The total slurry pressure drop
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() is made up of two components,  and ,.  is the pressure drop as it flows along the slurry
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annulus, and  is the interfacial pressure drop at the nozzle exit, i.e., the force required to break through
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the tangentially flowing steam. Each contributes to , but they are affected by different mechanisms.
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Changes to slurry viscosity, as well as changes to slurry mass flow, will affect both  and  (that is,
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). Changes to steam mass flow rates will only affect . Steam also has a  where it is forced to
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interact with the slurry wave but not a significant . Each  at the interface includes both normal
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(breakthrough) forces and tangential (frictional) forces as the steam and slurry interact while exiting the nozzle.
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Furthermore, since the steam is compressible, changes in  at the nozzle exit will feed pressure back to
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the steam inlet, which will alter steam velocity (and thus the phase momentum ratio) even if the steam mass
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flow were fixed.
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While the C1 and C2 control algorithms are functionally independent, physical interaction between variables
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couples the controllers. Figure 3 illustrates the relationship between the two controllers and the interrelated
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physics. As C1 adjusts the slurry mass flow rate (󰇗 ) to affect , droplet size (SMD) and  are
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also impacted. Changes in 󰇗  cause a shift in the phase momentum ratio, which is important for the
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atomization process. Similarly, when C2 adjusts the steam mass flow rate (󰇗 ) to affect SMD,  (but
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not ) is also impacted. Changes to either 󰇗  or 󰇗  will alter  and consequently the steam
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inlet velocity because of pressure feedback (since the steam is compressible). Thus, coupled physics cause
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both C1 and C2 to affect each other’s outcomes in a coupled controller system. Not shown in Figure 3 is the
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internal physics feedback loop between temperature- and shear-dependent viscosity and interfacial instabilities.
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Variable steam and slurry flows leads to variable interfacial heating dynamics, which affects  and .
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Figure 3 Despite independent control algorithms, the C1 and C2 controllers are coupled by physical
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interactions. The total slurry pressure drop () includes the pressure drop in the slurry annulus () and
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the interfacial pressure drop at the nozzle exit (). Changes in slurry mass flow rate (
󰇗) affect not
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only  and  but also Sauter mean diameter (SMD), while changes in steam mass flow rate (
󰇗)
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affect both SMD and . Both C1 and C2, then, affect each other’s outcomes via coupled physics. Not
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shown in this figure is the internal thermal-instability feedback loop at the phase interface.
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2.3 Smart Atomization and Heating Test Descriptions
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The CFD model for Test 1 employs an unconventionally coarse Base mesh, simply as an initial proof-of-
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concept. It is understood that this model is not for predicting accurate droplet sizes or wave generation physics.
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The full 360° axisymmetric atomizer geometry was reduced to a 1/32nd (11.25°) wedge with a total
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computational cell count of 250,000. The slurry is Newtonian, and the steam is modeled as incompressible.
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The slurry annulus is considerably longer than the steam pipe, extending approximately 10 times farther back.
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This “long slurry entry” strengthened the correlation between slurry viscosity and pressure drop in the annulus,
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creating a more realistic scenario for PID control testing, i.e. creating . The computational domain extends
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10 orifice diameters beyond the atomizer orifice in the axial direction.
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Test 1 includes the response of all four combinations of controllers to a viscosity shift. For the PID input, SMD
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is measured 5 nozzle diameters away from the orifice. Slurry viscosity is initially set to a constant 0.05 kg/m-
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s. The resulting QSS pressure and SMD are used as setpoint values for the controllers. All tests began with
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steam and slurry flow rates necessary to maintain these “baseline” setpoint values. Viscosity is then suddenly
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changed to 5 kg/m-s for a 100-fold step increase. After the viscosity shift, four separate simulations model the
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following scenarios: C0, C1, C2, C12. Each test follows this pattern: a baseline simulation models the system
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at QSS with a baseline (lower) viscosity value. After the viscosity is step-increased to a higher value, a
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response simulation is run for each combination of controllers under investigation. The C12 response is of
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greatest interest since this represents the coupled controller system, but comparisons among the C0/C1/C2
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responses also provide valuable insights.
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Mesh resolution and azimuthal domain extent are increased for Test 2. The mesh is refined once in the entire
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atomization domain (called “Ref-1”). The wedge angle is increased from 11.25° to 45° for a total of 2.3 million
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computational elements. The same Newtonian slurry and steam of Test 1 are used again here, and both are
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modeled as incompressible. To increase the correlation between the slurry viscosity and pressure drop in the
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annulus even further than Test 1, the long slurry entry is shortened and includes a section of porous media.
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The porous media is a proxy for  over a much longer slurry feed piping system, and this pressure drop is
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linear with viscosity in laminar flow based on the well-known Darcy’s Law. The computational domain beyond
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the nozzle orifice is shortened, and SMD for PID input is measured about 2.5 nozzle diameters away from the
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orifice. Since the computational domain is shortened with each test to reduce computational cost, the SMD
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measure location moves successively closer to the nozzle. Like Test 1, the baseline viscosity of 0.05 kg/m-s
300
was increased by 100x to 5 kg/m-s. Then, three separate simulations model C0, C2, and C12 responses. The
301
C1 response was not included to minimize computational expenses and run time.
302
303
Finally, Test 3 represents PID-assisted WAVE atomization. The mesh is refined twice (called “Ref-2”), and
304
the wedge angle is kept at 45°, with 8.3 million computational elements. Increasing to 90° does not
305
significantly alter droplet sizes.18 While full mesh independence was established for Ref-3, the Ref-2 axial
306
SMD profile is sufficiently close to mesh independence for controller testing.18 The full non-Newtonian banana
307
puree viscosity UDF, which uses the Herschel-Bulkley model, is incorporated into the CFD model.24, 33, 34 The
308
Herschel-Bulkley model says the puree is shear-thinning beyond a yeild stress and also temperature-thinning.
309
The UDF computes viscosity from strain rate magnitude and temperature and was implemented using a
310

13
validated methodology.34 We conducted additional verification for this particular UDF, showing the %
311
difference between CFD and hand-calculated viscosity values to be less than 0.005%.24
312
313
As with Test 2, a porous media section in the puree annulus creates a direct linear correlation between viscosity
314
and  for Test 3. Unlike Tests 1 and 2, the steam is modeled as compressible (ideal gas with temperature
315
and pressure-dependent density). The SMD for C2 measured value is sampled 1 nozzle diamter away from the
316
orifice. The baseline viscosity, modeled by the standard viscosity UDF, increased by a factor 10, after which
317
the response of C12 is captured. A slight change in the control algorithm for Test 3 made the sampling time
318
 a function of the convective time scale. This is important, because atomization dynamics are suppressed as
319
feed flows are adjusted for the higher viscosity. Table 1 provides an overview of the differences across Smart
320
tests.
321
322
Table 1 Summary of setup and differences between the three Smart tests. “SMD Control Point” is the distance
323
from the orifice (in orifice diameters) at which the SMD is measured for PID input. Model rigor is increasing
324
from Test 1 to Test 3.
325
326
Test Setup
Test 1
Test 2
Test 3
Mesh Resolution
Base
Ref-1
Ref-2
Wedge Angle
11.25°
45°
45°
Annulus
Long
Porous Media
Porous Media
Liquid
Newtonian Slurry
Newtonian Slurry
Non-Newtonian Puree
Steam
Incompressible
Incompressible
Compressible
SMD Control Point
5 diameters
2.5 diameters
1 diameter
Viscosity Shift
100x
100x
10x
Response
C0, C1, C2, C12
C0, C2, C12
C12
327
328
3. RESULTS AND DISCUSSION
329
3.1 Test 1
330
Figure 4 shows the measured and controlled variables for each scenario across the viscosity change at
331
normalized flow time = 0. All data are non-dimensionalized in this manner: pressure and SMD data are
332
normalized by their respective setpoints (equivalent to starting QSS values), and mass flows are normalized
333
by the starting value just before the viscosity change. Flow time is normalized by the convective time scale for
334

14
slurry droplets to travel roughly five diameters axially away from the nozzle exit (where SMD is measured for
335
controller Test 1). The bulk velocity is around 150 m/s, making the convective time scale is around   
336
s.
337
338
The slurry inlet pressures for C1 and C12 behave similarly, successfully adjusting to the dramatically higher
339
viscosity level and remaining much lower than those for C0 and C2. Pressure initially doubles in response to
340
the viscosity increase but is driven back down to the setpoint as the slurry mass flow is decreased. For C0 and
341
C2, pressure initially increases by a factor of about 2.5. The C0 pressure then remains constant, while the C2
342
pressure continues to increase because of the increasing steam flow. Table 2 further elucidates the controller
343
responses with time-averaged (TA) pressure and temporal standard deviation statistics. C1 and C12 pressures
344
experience no change in mean and a slight decrease in standard deviation through the viscosity shift (the
345
transition period has been removed for these statistics). C0 and C2 stand in stark contrast; the mean and
346
standard deviation increase dramatically for both. C2 represents the largest change with a 320% increase in
347
mean and a 530% increase in standard deviation. Interestingly, the C1 and C12 slurry flows begin to diverge
348
after beginning near one another. The C12 slurry flow is increasing because of decreasing steam flow, lowering
349
slurry pressure and causing C1 to increase the slurry flow. This phenomenon is part of the interplay between
350
controllers illustrated in Figure 3.
351
352
C2 and C12 maintain lower and more stable SMD values compared to C0 and C1. While the C2 and C12 TA
353
SMDs remain unchanged through the viscosity change, the C0 and C1 TA SMDs increase by 20% and 40%,
354
respectively (Table 2). While appreciable, we expected stronger effects by the viscosity change on the TA SMD
355
without controller intervention. Our preliminary results indicate a surprisingly robust system in terms of average
356
SMD. The most notable change in SMD, as is visually evident from Figure 4, is the large increase in variation for
357
C0 and C1. While the SMD standard deviation for C2 and C12 increases moderately, it increases by 450% and
358
520% for C0 and C1, respectively.
359
360

15
C2 and C12 adjust the steam flow in different ways because of the difference in sludge flow. C2’s behavior is
361
expected: the steam flow increases to decrease the SMD resulting from the changes in momentum ratio. The C12
362
steam flow swings both above and below the constant C1 value. We thus arrive at a useful and obvious conclusion:
363
engaging C2 allows not only for increased steam usage to maintain the SMD setpoint but also for decreased steam
364
usage as necessary. In other words, the coupled controller system is free to use only what steam it requires to
365
maintain atomization quality. It is clear, however, that the C12 steam flow has not reached QSS. The
366
aforementioned interplay between two controllers seems to be at work and will dictate how the steam flow adjusts
367
moving forward.
368
369
370
Figure 4 Test 1 controller response plots showing (a) slurry pressure (equivalent to pressure drop), (b) slurry
371
mass flow rate, (c) Sauter mean diameter (SMD), and (d) steam mass flow rate. A viscosity step increase of
372
100x occurs at normalized flow time = 0. Test 1 provides an initial indication that the coupled controller system
373
(C12) is effective and that both controllers are important (used with permission from Begell House35).
374
375

16
376
Table 2 Descriptive statistics for SMD and pressure in Test 1 presented as the ratio of a value after the 100x
377
slurry viscosity change to that before the viscosity change (excluding transition regions). Thus, a value of 1
378
indicates no change through the 100x viscosity increase.
379
380
Ratio of Means
Ratio of Standard Deviations
Case
SMD
Pressure
SMD
Pressure
C0
1.4
2.5
5.5
1.8
C1
1.2
1.0
6.2
0.9
C2
1.0
4.2
1.5
6.3
C12
1.0
1.0
1.8
0.9
381
In summary, Figure 4 and Table 2 provide an initial demonstration of 1) the efficacy of the coupled controller
382
system and 2) the need for both controllers. The C1 controller is clearly necessary to maintain a constant slurry
383
pump requirement. Both C0 and C2, individually, would be problematic, as the increase in pressure may be
384
unacceptable. The C2 pressure increases and varies most dramatically; therefore, C2 alone is not a preferred
385
option despite its successfully maintaining an acceptable SMD. The dramatic decrease in slurry flow for C12
386
resulted in less than the expected increase in steam flow, demonstrating the flexibility of the C2 controller.
387
388
Before moving to Test 2, which involves a finer mesh than Test 1, mesh resolution effects are elucidated.
389
Slurry atomization instantaneous snapshots with various meshes are illustrated in Figure 5, revealing
390
atomization differences and similarities for three mesh refinement levels (Base, Ref-1, and Ref-2). Figure 5 is
391
a cross-section, equivalent to the perspective in Figure 1, which shows representative contours of slurry (red)
392
and steam (black). Each case was run with the baseline viscosity of 0.05 kg/m-s. The only distinction between
393
these three models and that used for Test 1 is that these have a 90° azimuthal angle. It must be emphasized that
394
even Ref-1 and Ref-2 do not necessarily predict mesh-independent droplet sizes. These models are intended
395
for preliminary assessment of mesh sufficiency and atomization characteristics.
396
397
A comparison of the Base, Ref-1, and Ref-2 cases in Figure 5 show that the general characteristics of WAVE
398
atomization remain relatively unchanged as the mesh is refined. The Base case (left in Figure 5) represents the
399
mesh element size used for Test 1. While finer meshes do result in production of smaller droplets, the general
400

17
atomization characteristics are largely the same through 2 refinement levels. Of particular importance to our
401
work is this: if a controller configuration is effective or ineffective for the Base case, it should perform
402
comparably for the Ref-1 or Ref-2 case. In other words, the Base mesh is sufficient for an initial Smart proof-
403
of-concept.
404
405
406
Figure 5 Representative contours of slurry (red) and steam (black) with a slurry viscosity of 0.05 kg/m-s,
407
demonstrating the change in droplet resolution as the mesh is refined. The approximate computational cell
408
counts from left to right are 1, 8, and 66 million, representing Base, Ref-1, and Ref-2. The leftmost picture
409
represents the mesh element size range used for Test 1. While refining the mesh does enable increasingly
410
smaller droplet resolution, the general characteristics of atomization remain largely unaltered through each
411
level of refinement (used with permission from Begell House34).
412
413
3.2 Test 2
414
The results of Test 2, though similar in some ways to Test 1, brought differences. An important factor in these
415
differences is the strengthened relationship between pressure drop and viscosity. For Test 2, a porous media section
416
replaced the long slurry annulus to create a significantly higher  and sensitivity of pressure to viscosity
417
changes. Pressure, SMD, and mass flow rates are given in Figure 6 in the same normalized manner as Test 1. A
418
slightly different convective time scale of    s is used to normalize the flow time, as the location where
419
SMD is measured is closer to the nozzle exit in Test 2.
420
421
After the viscosity shift, the pressure initially increases by over 5000%, which is a much greater jump than was
422
observed for Test 1. Again, this was expected because of the porous media addition to the model. However, the
423
pressure only increased by about half the viscosity increase factor of 100 because the viscosity primarily affects
424
. Pressure contours for the Test 2 baseline in Figure 7 show the two contributing pressure drops ( and
425
) were roughly equal. As the interface is dynamic, along with , Figure 7 only represents only a single
426

18
realization. Since the porous media only accounted for half of the total pressure drop, a 100x change in viscosity
427
only created a 50x change in pressure drop. More will be highlighted on  in the next section. The pressure then
428
returns back to setpoint for C12 but remains relatively constant for C0 and C2, as they are not designed to address
429
pressure spikes. Table 3 provides TA pressure and SMD statistics. Note in particular the substantial increase in
430
pressure standard deviation for C0 and C2 (by factors of 4.9 and 17, respectively). C12, on the other hand, actually
431
experiences a decrease in pressure variation after the viscosity shift.
432
433
434
Figure 6 Test 2 controller response plots showing (a) slurry pressure (equivalent to pressure drop), (b) slurry
435
mass flow rate, (c) Sauter mean diameter (SMD), and (d) steam mass flow rate. A viscosity step increase of
436
100x occurs at normalized flow time = 0. While Test 2 generally maintains pressure and SMD setpoint targets
437
with C12, SMD hangs slightly below setpoint while the steam flow plummets.
438
439
After an initial increase, the SMD is brought back to setpoint for both C2 and C12 as designed. The TA SMD
440
increases by about 180% for C0 with a significant increase in variability (see Table 3), indicating the necessity of
441

19
C2 to maintain droplet size. The steam flows for C2 and C12 are briefly similar before diverging. As expected, the
442
C2 steam flow increases to drive down SMD. In contrast, the C12 steam flow begins to plummet and drops below
443
the C0 steam flow. The SMD is slightly below setpoint, even as the steam flow continues its downward trajectory.
444
The low SMD occurs because of the changing system dynamics as the slurry flow is significantly reduced
445
(more on this in the next section). By the end of the test, steam flow had bottomed out at its lower limit,
446
highlighting the need for controller adjustments to accommodate these circumstances.
447
448
Table 3 Descriptive statistics for SMD and pressure in Test 2 presented as the ratio of a value after the 100x
449
slurry viscosity change to that before the viscosity change (excluding transition regions). Thus, a value of 1
450
indicates no change through the 100x viscosity increase.
451
452
Ratio of Means
Ratio of Standard Deviations
Case
SMD
Pressure
SMD
Pressure
C0
2.8
54
13
4.9
C2
1.1
55
3.2
17
C12
0.7
1.0
4.3
0.7
453
454
455
Figure 7 Side view of instantaneous pressure contours for half of the axisymmetric geometry (top). The porous
456
media section in the slurry anulus is highlighted, which ensures the sensitivity of pressure drop to viscosity,
457
simulating a longer slurry feed piping system. The total pressure drop from the slurry inlet (left) to the domain
458
outlet (right) includes two roughly equal contributions: the pressure drop across the porous media () and
459

20
the pressure drop at the nozzle exit (). A close-up view of the nozzle exit (bottom) includes the 3D gas-
460
liquid interface (gray) for a 45° azimuthal angle.
461
462
463
464
3.3 Test 3
465
Test 3 included the complete WAVE atomization model with shear-thinning banana puree as the slurry and
466
compressible steam. For this test, the viscosity was increased by a factor of 10 rather than a factor 100 like Tests
467
1 and 2, and only the C12 response is captured. Once again, pressure, SMD are normalized in the same manner as
468
Tests 1 and 2. Rather than convective time scale, the flow time is normalized by the “wave time” of the system.17
469
Wave time, which is equivalent to the pulsing time scale and around 0.001 s (about 10 convective times), reflects
470
the frequency at which slurry waves form regularly inside the nozzle, roll up over the nozzle exit lip, and crash
471
back into the proceeding thin slurry film. The wave time changes during the course of Test 3; 0.001 s reflects the
472
value at QSS before the viscosity shift.
473
474
The C12 response plots are presented in Figure 8. In response to the 10x shift in viscosity, pressure initially
475
increases by about 400%. This increase in pressure is comparable to that for Test 2 (1/10th the viscosity increase
476
produced approximately 1/10th the pressure increase). Thus, modeling the slurry as non-Newtonian roughly
477
maintains the sensitivity of  to viscosity changes. Pressure returns to setpoint as the slurry flow is reduced.
478
As with Test 2, slurry flow rate is reduced dramatically, ending at just 2.5% of its original value. To keep the
479
pressure moving steadily towards setpoint during this transition, the proportional gains for both controllers were
480
reduced by a factor of 4 at normalized flow time = 24. We suggest that variable gains could improve the controller
481
system; fuzzy logic could provide an effective means of tuning the gains in real-time.36
482
483
The SMD initially increases by 125%. In response, the steam flow increases, and the SMD fluctuates around
484
setpoint. Then the SMD drops far below setpoint, and the steam flow plummets. The flat line towards the end of
485
the test is where the steam flow reaches the lower limit (as imposed in the control algorithm) of 0.0003 kg/s. The
486
reader will notice that the time between steam flow adjustments gradually increases towards the end of the test
487

21
before the steam flow bottoms out. This is a consequence of the variable sampling time implemented in the control
488
algorithm for Test 3, where sampling time adjusts with the convective time scale. As flow rates reduce towards
489
the end of the test and the convective time scale lowers, the sampling time increases.
490
491
492
Figure 8 Test 3 controller response plots showing (a) slurry pressure drop, (b) slurry mass flow rate, (c) Sauter
493
mean diameter (SMD), and (d) steam mass flow rate. A viscosity step increase of 10x occurs at normalized
494
flow time = 0. The reduction in slurry flow rate significantly changed atomization characteristics, which caused
495
the steam flow to bottom out. A third control mechanism could provide a more robust system.
496
497
The outcome for C12 is best explained by the reduction in slurry flow rate. When the pressure increases by 400%,
498
the slurry flow rate plunges down to 2.5% of its original value. Consequently, the atomization characteristics
499
change, as illustrated by Figure 9. The top of Figure 9 provides an instantaneous snapshot of WAVE atomization
500
as it has been studied.16 The bottom of Figure 9 provides a snapshot of the atomization by the end of Test 3. There,
501
globs of puree burst more slowly from the nozzle. Meanwhile, small droplets are being stripped off, which explains
502

22
why SMD stays so far below setpoint. It appears that under the Test 3 circumstances, particularly the fact that we
503
are imitating a long slurry feed piping system, the pressure and SMD setpoints are not simultaneously achievable
504
while operating within the bounds of the current controller configuration. A shorter slurry supply line (where 
505
is not such a dominant factor, as in Test 1) should be easier to manage. We propose that to accommodate such
506
circumstances as are presented in Test 3, an additional control knob is needed.
507
508
509
Figure 9 Representative snapshots from Test 3 showing atomization before the viscosity shift and at the end
510
of the test. The flow rates are dramatically reduced, changing the atomization characteristics of the system.
511
512
Important dimensionless numbers from before the viscosity shift (start) and the end of Test 3 are listed in Table
513
4. The numbers are defined as follows: Weber number ()  
 and Reynolds number () 
514
. The Ohnesorge number (  ) is around    before and after. Steam Mach
515
number is reported at the steam inlet. Here  is the average density across both phases,  is steam density, 
516
is the slurry density,  is the bulk velocity (both phases), 
 is the steam velocity at the inlet,  is surface
517
tension,  is the slurry dynamic viscosity, and  is the orifice diameter.  decreases by two orders of
518

23
magnitude, and  decreases by one order of magnitude, both largely driven by the reduction of slurry and
519
steam velocities. The Mach number at the steam inlet is reduced to 1/5th the original value. Steam Mach number
520
at the nozzle exit varies within a given wave cycle as the slurry blockage changes; it starts in the transonic
521
range and even touches supersonic with a slurry-reduced exit area but then falls to below Mach 0.1. During
522
the test, the GLR rises from 2.7% to 12% because of the fall in slurry flow rate (although both flows fall
523
significantly). Meanwhile, the gas-liquid momentum ratio increases from 52% to 1500%.
524
525
Table 4 Dimensionless numbers describing the flow before the viscosity increase (start) and at the end of Test
526
3. The steam Mach number is reported for the steam inlet and reduces to 1/5th the original value. Reynolds
527
number () and Weber number () both fall by at least an order of magnitude. Here GLR is gas-liquid
528
mass ratio, and GLMR is the gas-liquid momentum ratio.
529
530
Start
End
We
  
60
Re
  
  
Mach
0.22
0.043
GLR
2.7%
12%
GLMR
52%
1500%
531
532
Figure 10 illustrates the change in wave character from the beginning to the end of Test 3. By the end of the
533
test, some waves are still observed, but they are smaller and less frequent. The annular slurry sheet that stretches
534
out from the nozzle before the wave crashes largely disappears. While a pulsing characteristic is maintained, the
535
frequency is lower. As the wave blockage ratio decreases, steam acceleration above the wave and pressure buildup
536
behind the wave both decrease as well. Two important atomization mechanisms are abated: wave impact
537
momentum and pressure buildup. Additionally, droplets are primarily stripped from the waves as opposed to the
538
flicking, buckling, and breaking of wave crests in the steam flow.17,18,24
539
540

24
541
Figure 10 Representative instantaneous snapshots from Test 3 showing a side view of wave formation inside
542
the nozzle. Slurry is yellow, steam is purple, and flow is from left to right. The top picture is before the viscosity
543
shift, and the bottom picture is at the end of the test. The reduction in steam and slurry flow rates significantly
544
alter the wave character, producing smaller, less frequent, low blockage ratio waves.
545
546
During the course of the test,  becomes the dominant contributor to the total . At the same time, 
547
and  both fall by an order of magnitude. Figures 11 and 12 illustrate changes in the interfacial pressure drop
548
() for slurry and steam, respectively. The left of each figure is  before the viscosity shift, and the right is
549
 at the end of Test 3. The “before” (left) slurry and steam driving pressures are roughly equal, and four distinct
550
wave cycles are observed. The “after” (right) slurry and steam pressures are again roughly equal, though an order
551
of magnitude lower. The slurry is having to work less to move into the steam flux (i.e.,  falls), and the steam
552
is not being blocked any longer by large slurry waves (i.e.,  falls).  falls slightly more due to the drop
553
in steam density. As flow rates are reduced, wave frequency falls to less than a quarter of the original frequency.
554

25
555
Figure 11 Slurry interfacial pressure drop () at the nozzle exit before the viscosity shift (left) and at the
556
end of Test 3 (right). For both data sets, flow time is normalized by the original wave time from before the
557
viscosity shift. Four clear wave cycles are observed before the viscosity shift. By the end of the test,  is
558
an order of magnitude lower, and the wave cycle time is less than a quarter of the original.
559
560
561
Figure 12 Steam interfacial pressure drop () at the nozzle exit before the viscosity shift (left) and at the
562
end of Test 3 (right). For both data sets, flow time is normalized by the original wave time from before the
563
viscosity shift. Four clear wave cycles are observed before the viscosity shift. By the end of the test,  is
564
an order of magnitude lower, and the wave cycle time is less than a quarter of the original.
565
566
Figure 13 shows an unraveled view of the wave with representative snapshots before the viscosity shift (left,
567
Multimedia online animation view) and at the end of Test 3 (right, Multimedia online animation view). The
568
annular wave, whose shoreline is curved, has been unraveled onto a flat shoreline, similar to waves on a beach.
569
Both pictures in Figure 13 show instantaneous realizations where the wave is roughly peaking. The slurry
570
surface is colored by directional derivative of pressure normal to the surface (
  
, where  is pressure
571
and  is the slurry phase volume fraction). Positive and negative values represent gas decelerating and
572
accelerating toward the slurry, respectively.
573

26
574
Both before and after the test, the pressure gradient is largely neutral in the wave’s wake and positive its
575
windward side. In other words, the steam is decelerating as it hits the windward side of the wave. Especially
576
before the viscosity shift, a negative pressure gradient, or “suction,” appears on the leeward side of the wave
577
as it curls up. The variation in gradient direction (both positive, blue, and negative, brown) on the windward
578
side highlights the presence of Rayleigh-Taylor instabilities (RTI).24 A visual inspection shows reduced RTI
579
activity by the end of the test, with a less rippled windward wave surface and a mostly positive pressure
580
gradient. Steam density fluctuates with the wave cycle as steam is compressed and decompressed, which feeds
581
back to the inlet to affect steam pressure. Both parts of Figure 13 show a thin region of higher density steam
582
(more compressed and cooled by the slurry) over the windward surface of the wave. The expansion and cooling
583
where steam contacts the slurry phase could lead to steam condensation here, but phase change is not included
584
in our modeling approach. Future work may include a more thorough heat transfer analysis, as steam
585
condensation could be important. As slurry and steam flow rates subside and wave dynamics become less
586
influential towards the end of the test, overall steam density falls.
587
588
589
Figure 13 Representative snapshots of the unraveled wave with slurry surface colored by pressure gradient
590
normal to the surface and contours of steam density in the background. A positive pressure gradient indicates
591
steam decelerating toward the interface, while a negative pressure gradient indicates steam accelerating toward
592
the interface. Left is before the viscosity shift (Multimedia online animation view) and right is at the end of
593
Test 3 (Multimedia only animation view). By the end of Test 3, the steam density has fallen dramatically, and
594
the wave’s prominence is much diminished.
595
596

27
4. CONCLUSION
597
Demonstrations of Smart atomization, involving continuous coupled PID adaptations, for variable-viscosity
598
slurry heating and disintegration in a transonic pulsatile twin-fluid atomizer have been presented for the first
599
time. The WAVE atomizer facilitates slurry high-blockage-ratio wave production and favorably destabilized
600
interfacial dynamics. Distortion and rupture of slurry sheets extending from the atomizer results from pressure
601
fluctuations, wave impacts, and viscosity variability. Interfacial perturbations are exacerbated by instabilities
602
and self-perpetuated in a non-Newtonian instability cycle. Surface distortion and strain rate irregularity creates
603
significant viscosity and, consequently, wavelength variation, which in turn excites the instabilities. Complex
604
multi-stream, property-variant atomization physics drive the need for this novel Smart dual-control approach.
605
PID control algorithms were incorporated into CFD simulations to model the Smart coupled controller system
606
(both C1 and C2, referred to as “C12”). Smart continuously and simultaneously adjusts the cold slurry and hot
607
gas flow rates to maintain a consistent slurry pressure drop and droplet SMD. In addition to the total ,
608
changes to slurry flow alter the gas-liquid momentum ratio, which impacts the SMD. On the other hand, gas
609
flow changes impact the momentum ratio, , and ,which represents the interfacial contribution to
610
. C12 thus compensates for several interacting variables, including an internal physics-based feedback loop
611
within the slurry-steam environment. The controller actions create a nonlinear search for new equilibrium
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conditions due to the offsetting nature of pressure drop, flow rates, interfacial instabilities, and atomization
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quality.
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The initial proof-of-concept (Test 1) demonstrated the necessity and efficacy of the coupled controller system.
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An additional benefit of C2 is providing for flexibility with steam usage: rather than a fixed (and potentially
617
excessive) steam flow rate, steam is only used as necessary to keep droplet sizes at setpoint. A comparison of
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meshes showed that general atomization characteristics remain largely unchanged through refinement, even
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though droplet size decreases. Adding a porous media section in the slurry annulus (Test 2) significantly
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increased the sensitivity of pressure drop to viscosity, resulting in an initial 5000% pressure drop increase for
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a 100x increase in viscosity. The pressure drop showed a similar sensitivity to viscosity with the non-
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Newtonian slurry in Test 3. Because of this high sensitivity and the resulting reduction in slurry flow rate, the
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system’s atomization characteristics changed drastically. Small droplets being stripped off kept the SMD
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below setpoint and caused the steam flow to bottom out near zero.
625
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As presented, Smart was mesh-independent and multiphase physics-independent; however, there is room for
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improvement. In the future, we note the need for variable controller gains, perhaps using fuzzy logic, as the
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relationship between measured and controlled variables changed during Test 3. A third control knob might
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also improve results for the Test 3 scenario. Though SMD proved to be a sufficient single metric for
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atomization quality, other droplet size statistics could replace SMD as a measured variable.
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While we demonstrate Smart WAVE atomization for pronounced (orders of magnitude) viscosity changes, it
633
is broadly designed to achieve consistent, reliable atomization for viscous, non-Newtonian slurries in the
634
context of unfavorable and/or unplanned process variation. An alternative use of these controllers would be to
635
use C2 to hit a target SMD setpoint while C1 maintains a constant air-fuel ratio (AFR). On-the-fly user changes
636
to the SMD setpoint would force the controllers to modify flow rates to reach a new quasi steady state. Another
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potential use would be to adjust AFR based on fuel carbon content. The controllers would maintain droplet
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size and air (or fuel) flow in response to feedback from some composition analyser. The tool proposed in this
639
paper is limited to use within processes where two feedback data streams are available and two knobs exist to
640
adjust the two feed stream flow rates. However, derivatives and subsets of this method are readily conceivable.
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ACKNOWLEDGEMENT
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The authors thank Valda Rowe and Dr. Mark Horstemeyer for their administrative support.
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DATA AVAILABILITY
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The data that support the findings of this study are available from the corresponding author upon reasonable
647
request.
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CONFLICT OF INTEREST
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The authors have no conflicts to disclose.
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