Diamond Jet Hybrid HVOF Thermal Spray: Gas-Phase and ParticleBehavior Modeling and Feedback Control Design
Mingheng Li, Dan Shi, and Panagiotis D. Christofides*
Department of Chemical Engineering, University of California, Los Angeles, California 90095-1592
This paper focuses on the modeling and control of an industrial high-velocity oxygen-fuel (HVOF)thermal spray process (Diamond Jet hybrid gun, Sulzer Metco, Westbury, NY). We initiallydevelop a fundamental model for the process that describes the evolution of the gas thermaland velocity fields and the motion and temperature of particles of different sizes and explicitlyaccounts for the effect of the powder size distribution. Using the proposed model, a comprehensiveparametric analysis is performed to systematically characterize the influence of controllableprocess variables such as the combustion pressure and oxygen/fuel ratio, as well as the effect ofthe powder size distribution, on the values of the particle velocity, temperature, and degree ofmelting at the point of impact on the substrate. (These are the variables that directly influencecoating microstructure and porosity, which, in turn, determine coating strength and hardness;see the second article of this series for details.) A feedback control system that aims to controlthe volume-based average particle velocity and melting ratio by directly manipulating the flowrates of fuel, oxygen, and air at the entrance of the HVOF gun is developed and applied to adetailed mathematical model of the process. Closed-loop simulations show that the feedbackcontroller is effective in driving the controlled outputs to the desired set-point values and isalso robust with respect to various kinds of disturbances in the operating environment.
1. Introduction
The past two decades have witnessed the wide ap-plication of high-velocity oxygen-fuel (HVOF) thermalspray technology (see Figure 1 for a schematic of thisprocess) as a means for depositing coatings of cermets,metallic alloys, and composites to modify the surfaceproperties of a base material (substrate). Using thethermal energy produced by the combustion of fuel withoxygen to heat and propel the powder particles, theHVOF thermal spray provides a highly efficient way tomodify the surface properties of a substrate to extendproduct life, increase performance, and reduce mainte-nance costs. Recently, there has been increasing interestin the HVOF thermal spray processing of nanostruc-tured coatings whose grain size is less than about 100nm.1 This interest has been motivated by severalfactors, including (1) the cost-effective production ofhigh-quality nanosize powders; (2) the superior qualitiesof coatings made with the HVOF process;2 and (3) thediscovery that nanostructured coatings exhibit superiorqualities over traditional counterparts (made of materi-als with micro-sized grains) in several aspects includinghardness, strength, ductility, and diffusivity.1,3
Over the past decade, the need to optimally designand operate thermal spray processes has motivatedsignificant research on the development of fundamentalmathematical models to capture the various physico-chemical phenomena taking place during thermal sprayprocesses and to describe the dynamic behavior ofvarious process components. Specifically, fundamentalmodels have been developed describing the gas dynam-ics and particle in-flight behavior inside the gun and inthe free jet;4-6 molten droplet deposition, solidification,and microstructure development;7,8 and the relationship
between coating microstructure and mechanical proper-ties.9 In addition, research has been carried out on theintegration of the detailed models of the aforementionedcomponents to develop general simulators that describethe behavior of the entire thermal spray processes.8,10
To reduce product variability and to improve robust-ness with respect to variations in the operating condi-tions in industrial HVOF thermal spray processes, it isimportant to implement excellent real-time processdiagnosis and control that can lead to the fabrication ofcoatings with microstructures that yield the desiredproperties. Despite the recent progress on the modelingof the various phenomena that affect droplet motion,deposition, solidification, and microstructure develop-ment in HVOF thermal spray processes, no systematicframework currently exists for the integrated on-linediagnosis and control of the HVOF thermal sprayprocess that is capable of achieving precise regulationof the microstructure and ultimate mechanical andthermal properties of the sprayed coatings. In addition,incorporation of advanced real-time diagnosis and con-trol schemes into thermal spray processes is expectedto reduce operating costs and environmental impactsand allow for the deposition of nanostructured andcomplex (multimaterial) coatings with very low vari-ability. Because the application of optimization andcontrol techniques to spray casting processes has beenreported to lead to significant improvements in theiroperation and performance,11,12 it is important to de-velop real-time computer control systems for thermalspray processes by integrating fundamental models thataccurately describe the inherent relationships betweenthe coating microstructure and the processing param-eters with on-line state-of-the-art diagnostic techniquesand control algorithms. Recent efforts in this directionhave mainly focused on diagnostics and control ofplasma thermal spray;13 the reader can refer to Moreau
* To whom correspondence should be addressed. Tel.: (310)-794-1015. Fax: (310)206-4107. E-mail: [email protected].
and Leblanc14 for a discussion of various process opti-mization and control issues. In our previous work,15,16
we performed a comprehensive control-relevant para-metric analysis and proposed a novel formulation of thecontrol problem that accounts for the important effectof the powder size distribution for an HVOF process inwhich air is used as the oxidant and propane is used asthe fuel gas.
The objective of the present research is to develop acomputational methodology to precisely control thecoating micro- or nanostructure that determines thecoating mechanical and physical properties by manipu-lating macroscale operating conditions such as the gasflow rate and spray distance. The major challenge onthis problem lies in the development of multiscalemodels linking the macroscopic-scale process behavior(i.e., gas dynamics and particle in-flight behavior) andthe microscopic-scale process characteristics (evolutionof coating microstructure) and the integration of models,measurements, and control theory to develop measure-ment/model-based control strategies.17 The underlyingmultiscale behavior of the HVOF process is shown inFigure 2. On one hand, the microstructure of thermallysprayed coatings results from the deformation, solidi-fication, and sintering of the deposited particles, whichare dependent on the substrate properties (e.g., sub-strate temperature), as well as the physical and chemi-cal state (e.g., temperature, velocity, melting ratio, andoxidant content, etc.) of the particles at the point ofimpact on the substrate. On the other hand, the particlein-flight behavior is coupled with the gas dynamics,which can be manipulated by adjusting operating condi-tions such as the total gas flow rate and the fuel/oxygenratio. Whereas the macroscopic thermal/flow field can
be readily described by continuum-type differentialequations governing the compressible two-phase flow,the process of particle deposition is stochastic anddiscrete in nature, and thus, it can be best describedby stochastic simulation methods.18
This article is the first in a series of two articlesfocusing on the modeling and control of an industrialhigh-velocity oxygen-fuel (HVOF) thermal spray process(Diamond Jet hybrid gun, Sulzer Metco, Westbury, NY).We initially develop a fundamental model for theprocess that describes the evolution of the gas thermaland velocity fields and the motion and temperature ofagglomerate particles of different sizes and explicitlyaccounts for the effect of the powder size distribution.In addition to providing useful insight into the in-flightbehavior of different-size particles, the model is usedto perform a comprehensive parametric analysis of theHVOF process. This analysis allows us to systematicallycharacterize the influence of controllable process vari-ables such as the combustion pressure and oxygen/fuelratio, as well as the effect of the powder size distribu-tion, on the values of the particle velocity and temper-ature at the point of impact on the substrate. Specifi-cally, this study shows that the particle velocity isprimarily influenced by the combustion pressure andthe particle temperature is strongly dependent on thefuel/oxygen ratio. These findings are consistent withexisting experimental studies and set the basis for theformulation of the control problem for the HVOFthermal spray process. To develop a feedback controllerthat can be readily implemented in practice, the controlproblem is formulated as one of regulating volume-basedaverages of the melting ratio and velocity of the particlesat the point of impact on the substrate (these are the
Figure 1. Schematic of the Diamond Jet hybrid HVOF thermal spray process.
Figure 2. Multiscale character of the HVOF thermal spray process.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3633
variables that directly influence coating microstructureand porosity, which, in turn, determine coating strengthand hardness) by directly manipulating the flow ratesof fuel, oxygen, and air at the entrance of the HVOFthermal spray gun. A feedback control system is devel-oped and applied to a detailed mathematical model ofthe process. Closed-loop simulations demonstrate thatthe particle velocity and melting ratio at the point ofimpact on the substrate reach the desired set-pointvalues in a short time, which validates the feasibilityof real-time implementation of feedback control on theHVOF thermal spray system. It is also shown that theproposed control problem formulation and the feedbackcontrol system are robust with respect to disturbancesin spray distance and particle injection velocity, as wellas variations in the powder size distribution.
In the second article of this series,19 we presenta stochastic model that uses information about theparticle velocity, temperature, and degree of melting atthe point of impact on the substrate from the modeldeveloped in the present paper to predict coatingporosity and microstructure.
2. Modeling of Gas Thermal and Flow Fields
2.1. Process Description and Modeling Proce-dure. Figure 1 shows a schematic diagram of theDiamond Jet hybrid gun. The premixed fuel (typicallypropylene or hydrogen) and oxygen are fed from theannular gap to the air cap (also referred to as aconvergent-divergent nozzle, whose dimensions areshown in Table 1), where they react to produce high-temperature combustion gases. The exhaust gases,together with the air injected from the annular inletorifice, expand through the nozzle to reach supersonicvelocity. The air cap is water-cooled to prevent it frommelting. The powder particles are injected from thecentral inlet hole using nitrogen as the carrier gas.Consequently, rapid momentum and heat transferbetween the gas and the powder particles leads toacceleration and heating of the powder particles. Themolten or semimolten particles exit the air cap andmove toward the substrate. The particles hit the sub-strate, cool, and solidify, forming a thin layer of a denseand hard coating. In the remainder of this section, wepresent the procedure that we follow for modeling, aswell as the equations describing the gas flow andthermal fields.
Roughly speaking, three major physicochemical pro-cesses are involved in the HVOF thermal spray pro-cess: transformation of chemical energy into thermalenergy by the combustion of the fuel, conversion ofthermal energy into kinetic energy of the burning gasesby passage through the nozzle, and transfer of momen-tum and heat from the gases to the powder particles.These processes occur simultaneously and make thefundamental modeling of the HVOF process a verydifficult task. For example, detailed fundamental mod-
eling of the gas flow and thermal fields requires state-of-the-art computational fluid dynamics (CFD) meth-odologies and leads to complex two- or three-dimensionaltime-dependent partial differential equations.6,20,21 Forthe purposes of control system design and implementa-tion, a compromise between model complexity, compu-tational cost, and model ability to capture the dominant(from a control point of view) phenomena occurring inthe process is essential. To simplify the analysis, theprocess model used in this paper is based on the one-way coupling assumption, i.e., the presence of theparticles has a minimal influence on the gas dynamics,whereas the particle in-flight behavior is dependent onthe gas thermal/flow field. This assumption is reason-ably accurate because the particle loading in the HVOFprocess, which is defined as the ratio of the mass flowrate of the particles to that of gases, is typically lessthan 4%.5 In addition, the assumptions of instantaneousequilibrium at the entrance of the HVOF gun and frozenisentropic flow during passage through the nozzle aremade. These assumptions were initially proposed andjustified by Swank et al.22 on the basis of a comparisonof numerical simulations and experimental results andlater also recommended by Cheng et al.2 Comparisonsof simulation results and experimental data shown laterin this paper further substantiate the validity of theseassumptions (see discussion in subsections 4.1 and 4.2).
Regarding the role of the air stream, it is especiallydifficult to predict what portion of the air takes part inthe reaction. Whereas the air has been treated as acoolant solely to isolate the wall from the high-temper-ature flame gases in some references,23,24 in others9,20,21
it has been assumed that all of the oxygen coming fromthe air participates in the reaction. The latter assump-tion is employed here, as it was pointed out clearly byGourlaouen et al.9 that the airflow mixing with theoxygen/propylene mixture should be more effective inthe currently used Diamond Jet hybrid gun (which isthe process under consideration in this work) than thepreviously used Diamond Jet gun, as implied by the“water-cooled” (not “air-cooled”) nozzle. Other assump-tions in the model are as follows: (1) All gases obey theideal gas law. (2) The combustion gases behave as aperfect gas during isentropic compression and expan-sion, and the specific heat ratio is nearly constant. (3)The effects of friction and cooling water along the nozzleand barrel are negligible, so that the laws of isentropicflow of compressible fluids apply.
Because the flow is chocked at the throat of the nozzle,the convergent part of the air cap and the divergent partcan be solved separately.21 The modeling procedure thatis followed in the simulation is based on the sequentialmodular method. Specifically, given the mass flow ratesof each stream (fuel, oxygen, air, and carrier gas) and apostulated combustion pressure, the temperature andgas composition at the entrance of the nozzle is calcu-lated using an instantaneous equilibrium model, andthen the nozzle flow is solved using standard isentropiccompressible flow relationships. The total mass flow rateat the throat of the nozzle is then calculated andcompared with that at the entrance. The combustionpressure is then adjusted using the shooting method25
until the discrepancy between the calculated and speci-fied values of the total mass flow rate falls below thespecified tolerance. After the gas properties at the nozzlethroat are determined, the divergent part is solved using
Table 1. Dimensions of the Air Cap
parameter value
inlet diameter (mm) 14nozzle diameter (mm) 7.16outlet diameter (mm) 11inlet half-angle (deg) 12outlet half-angle (deg) 2length of convergent part (mm) 16length of divergent part (mm) 54
3634 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
isentropic flow relationships. The external thermal/flowfield in the free jet is described by empirical formulas.
2.2. Modeling of the Gas Thermal and FlowFields inside the Gun. To calculate the equilibriumcomposition and temperature of the combustion gases,the method of minimization of the Gibbs free energyunder adiabatic conditions is employed. This approachis advantageous compared to the equilibrium constantmethod because it can track a large number of speciessimultaneously without specifying a set of reactions apriori.26 Under the assumption of adiabatic combustionunder constant pressure, the calculation of equilibriumtemperature and composition can be formulated as anoptimization problem of the following form
where G is the Gibbs free energy of the product mixture(including inert gases and excess reactants); pr standsfor products; êi, Hi, and µi correspond to the stoichio-metric coefficient, enthalpy, and chemical potential ofspecies i, respectively; Teq is the equilibrium tempera-ture (subscript eq represents equilibrium); aij is thenumber of elements i in species j; l is the total numberof chemical elements involved in the system; and thesuperscript L stands for standard conditions. bi
0 )∑k∈reaikêk is the number of moles of elements i perkilogram of reactants (subscript re represents reac-tants), h0 ) ∑k∈reêkHk
L(Tin) is the enthalpy per kilogramof reactants (subscript in represents inlet), and E0 )∑k∈re
1/2êkMkvk2 is the kinetic energy per kilogram of
reactants (Mk is the molecular weight and vk is thevelocity of species k). For a gas obeying the ideal gaslaw, the chemical potential can be determined by thefollowing expression
where R is the gas constant and P is the pressure. Bycontinuity, the gas velocity after the combustion reactionis given by
where m̆g is the total mass flow rate of the gas, Ain isthe cross-sectional area at the inlet of the air cap, Mpris the average molecular weight of the product mixture,and êT ) ∑j∈prêj. From eq 3, it follows that
Defining f ) G + ∑i)1l λi(∑j∈praijêj - bi
0), where λirepresents the so-called Lagrangian multipliers, theoptimal solution of the optimization problem of eq 1 canbe determined by solving the following nonlinear alge-braic equations
The variables to be determined are the equilibriumcompositions êj ( j ) 1, ..., s), the Lagrangian multipliersλi (i ) 1, ..., l), the total number of moles êT, and theequilibrium temperature Teq. The set of s + l + 2nonlinear algebraic equations of problem 5 are solvedusing the descent Newton-Raphson method. The cen-tral idea of the Newton-Raphson method is to applymultivariable Taylor series expansion to a nonlinearvector function, truncate all terms that contain deriva-tives of second order and higher, and then use theresulting expression to build an iterative formula thatcan be used to compute the solution given an initialguess that is close to the solution; the reader is directedto ref 25 for details.
Because êj, êT, and T should be positive numbers, toavoid taking the logarithm of negative numbers in theiteration procedure, we have chosen ∆ ln êj ( j ) 1, ..., s),∆ ln êT, ∆ ln T, and - λi/RT (i ) 1, ..., l) as the solutionvariables at each iteration step. In the above equations,the thermodynamic data, such as the heat capacity,enthalpy, entropy, and chemical potential of each spe-cies, are calculated by the following equations26
where a1-a9 are constants for a given species. About10 iterations are usually required to obtain a convergentsolution.
Under the assumption of isentropic frozen flow, theproperties of the gas phase during passage through the
0 ) µjL + RTeq ln(P/PL) + RTeq ln(êj/êT) +
∑i∈ l
λiaij (j ) 1, ..., s)
0 ) ∑j∈pr
aijêj - bi0 (i ) 1, ..., l )
(5)
0 ) ∑j∈pr
êj - êT
0 ) ∑j∈pr
êjHjL(Teq) +
m̆g2êT
2R2Teq2
2P2Ain2
- h0 - E0
cpL(T)R
)a1
T 2+
a2
T+ a3 + a4T + a5T
2 + a6T3 + a7T
4
HL(T)RT
) -a1
T 2+
a2
Tln T + a3 +
a4
2T +
a5
3T 2 +
a6
4T 3 +
a7
5T 4 +
a8
T (6)
SL(T)R
) -a1
2T 2-
a2
T+ a3 ln T + a4T +
a5
2T 2 +
a6
3T 3 +
a7
4T 4 + a9
µjL
RT)
HL(T)RT
-SL(T)
R
min G ) ∑j∈pr
µiêi
s.t.
0 ) ∑j∈pr
aijêj - bi0 (i ) 1, ..., l )
(1)(mass balance)
0 ) ∑j∈pr
êjHjL(Teq) +
1
2veq
2 - h0 - E0
(energy balance)
µi(T) ) µiL(T) + RT ln
P
PL+ RT ln
êi
∑i∈pr
êi
(2)
veq )m̆g
FAin
)m̆gRTeq
PAinMpr
)m̆gêTRTeq
PAin
(3)
12veq
2 )m̆g
2êT2R2Teq
2
2P2Ain2
(4)
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3635
nozzle (both the convergent part and the divergent part)can be solved using the following equations (which arederived by solving the conservation equations governingcompressible flow)27
where A is the cross-sectional area perpendicular to theflow direction, F is the gas density, γ is the specific heatratio calculated by the expression γ ) cjp/(cjp - R) wherecjp ) ∑iêicp
i/∑iêi, and M is the mach number definedby the ratio of the gas velocity to the local sonic velocity(a ) xγP/F). At the throat of the nozzle, where themach number is 1,27 the mass flow rate can be calcu-lated using the formula
where the subscript g stands for gas and the subscriptt stands for throat. With a postulated combustionpressure, the calculated mass flow rate at the throat isusually different from that at the entrance of the gun.The shooting method is then applied to adjust thecombustion pressure until these two flow rates match.
We note that the isentropic relationships (eqs 7-10)are valid only if there is no shock inside of the nozzle.This can be guaranteed as long as the following inequal-ity holds
where Pb is the back-pressure (ambient pressure) andPe is the gas pressure at the exit of the nozzle. For theexperimental Diamond Jet hybrid gun system, ourcalculations show that the right-hand side of the aboveequation is about 5, and therefore no shock will everoccur inside of the nozzle as long as the exit pressure islarger than one-fifth of the back-pressure, a conditionthat is always satisfied under industrial operatingconditions.
We have applied the above modeling procedure andequations to analyze the Diamond Jet HVOF process,21
which is similar to the one shown in Figure 1 but whosenozzle has only a convergent part. The combustionproducts considered in our numerical simulation are Ar,CO, CO2, H, H2, H2O, NO, N2, O, O2, OH, etc. It is worthpointing out that C3H6 is not one of the products undernormal operating conditions. For the given four differentoperating conditions, the combustion pressures pre-dicted by the above procedure are all within 6% of theexperimentally measured values (see Table 2); thisresult is more accurate than the one obtained with the
two-step chemical kinetics model21 and implies that thecombustion model should take into account the dissocia-tion of the combustion products.
2.3. Modeling of the Gas Thermal and FlowFields outside the Gun. At the exit of the nozzle, theReynolds number based on the diameter and the gasvelocity is about 3 × 104, and the flow is fully turbulent.Depending on the magnitudes of the gas pressure at theexit of the air cap and the back-pressure, the flowoutside the nozzle might be underexpanded, ideallyexpanded, or overexpanded. Usually, the velocity andtemperature of the gas in the free jet are lower thanthose at the nozzle exit.28,29 From the exit of the nozzleto a position whose distance is not larger than thepotential core length (Lpc), the gas velocity and temper-ature can be considered almost constant.30 Furtherdownstream, the gas velocity and temperature decayrapidly because of the entrainment of the surroundingair. This decay of the gas velocity and temperature canbe described by the empirical formulas30
and
where x is the axial distance from the exit of the gunbarrel (x > Lpc) and R and â are parameters obtainedfrom experimental measurements. Lpc is a function ofthe mach number at the exit of the gun barrel (Me) andthe barrel diameter (D) according to30
3. Modeling of Particle Motion andTemperature
The particle trajectories and temperature histories inthe gas field are computed by the momentum- and heat-transfer equations. Because the acceleration and decel-eration of the particles in the moving gas in the HVOFprocess are dominated by the drag force,31 other forcesapplied on the particles can be neglected, and theparticle motion can be adequately described by thefollowing two first-order ordinary differential equations
where mp is the mass of the particle, vp is the axialvelocity of the particle, Ap is the projected area of theparticle on the plane perpendicular to the flow direction,Fg is the density of the gas, CD is the drag coefficient,
A2
A1)
M1
M2{1 + [(γ - 1)/2]M22
1 + [(γ - 1)/2]M12}(γ+1)/2(γ-1)
(7)
T2
T1)
1 + [(γ - 1)/2]M12
1 + [(γ - 1)/2]M22
(8)
P2
P1) {1 + [(γ - 1)/2]M1
2
1 + [(γ - 1)/2]M22}γ/(γ-1)
(9)
F2
F1) {1 + [(γ - 1)/2]M1
2
1 + [(γ - 1)/2]M22}1/(γ-1)
(10)
m̆g ) FtvtAt ) xγPtFtAt (11)
Pb
Pe< 2γ
γ + 1Me
2 - γ - 1γ + 1
(12)
Table 2. Comparison of Computational andExperimental Results for a Diamond Jet Process21
and xp is the position of the particle. The absolute signin the relative velocity between the particle and the gasimplies that a particle is accelerated if its velocity isless than that of the gas and decelerated otherwise. Totake into consideration the fact that many powders usedin the HVOF process are not spherical, a formula forthe drag coefficient CD that accounts for the particleshape using the concept of sphericity φ (defined as theratio of the surface area of a sphere with equivalentvolume to the actual surface area of the particle) is usedin this paper; it has the following form32,33
where K1 and K2 are two sphericity-related factors. Thelocal Reynolds number (Re) for this two-phase flowproblem is defined on the basis of the relative velocityRe ) dp|vg - vp|Fg/ηjg, where dp is either the particlediameter if the particle is spherical or an appropriateequivalent diameter if the particle is not spherical andηjg is the gas viscosity.
In the HVOF process, the Biot number of the particles(Bi ) hL/λp, where h is the heat-transfer coefficient, Lis a characteristic dimension defined by the ratio of theparticle volume to its surface area, and λp is the thermalconductivity of the particle) is typically less than 0.1,32
which means that the particles are heated with negli-gible internal resistance and that the temperaturegradients inside the particles can be neglected.34 Con-sequently, the equation describing the heat transferbetween a single particle and the gas reduces to a first-order ordinary differential equation. Depending on thevalue of the particle temperature, different equationsare used. With the assumption of negligible particlevaporization, the particle heating can be described by
where Tp is the temperature of the particle; A′p is thesurface area of the particle; Tm is the melting point ofthe particle; ∆Hm is the enthalpy of melting; and fp isthe melting ratio, or the ratio of the melted mass to thetotal mass of the particle (0 e fp e 1). The heat-transfercoefficient h is computed by the Ranz-Marshall empiri-cal equation35
where the Prandtl number (Pr) is calculated accordingto Pr ) cjpgηjg/λhg.
In the above equations, the viscosity and thermalconductivity of each species are calculated by thefollowing formulas26
where b1-b4 and c1-c4 are constants for a specificspecies. For gas mixtures, the average viscosity andthermal conductivity are calculated by the followingmixing rules26
where the interaction coefficients ψij and ςij are obtainedfrom the following formulas
At each step, we integrate eqs 16 and 18 with a smallenough time step such that the gas velocity, gas tem-perature, and local Reynolds number can all be consid-ered constant over this interval. After one integrationstep, we update the gas velocity and gas temperatureaccording to the new particle position and then applythe same strategy for the next time step. This method-ology was proposed by Crowe and Stock36 and was foundto be computationally economical and accurate. Toaccount for the particle melting behavior, we modifiedthis approach to check the molten state of the particleat each time step and apply different formulas for theparticle heating. Specifically, the iterative formulas forparticle velocity, position, and temperature are
where τp ) 4Fpdp2/3ηjgCDRe and ωp ) Fpcpdp
2/6Nuλhg. Inthe four possible phase transition points, in which thecurrent step and the next step correspond to differentparticle molten states, the successive formulas forparticle temperature and melting ratio take the follow-ing forms:
1. T pi < Tm (f p
i ) 0), T pi+1 > Tm, from totally solid
state to partially melted state
CD
K2) 24
ReK1K2[1 + 0.1118(ReK1K2)
0.6567] +
0.43051 + 3305/ReK1K2
(17)
hA′p(Tg - Tp) ) {mpcpp
dTp
dt, Tp * Tm
∆Hmmp
dfp
dt, Tp ) Tm
(18)
hdp
λhg
) Nu ) 2 + 0.6Re1/2Pr1/3 (19)
ln(η) ) b1 ln(T) + b2/T + b3/T2 + b4
(20)ln(λ) ) c1 ln(T) + c2/T + c3/T
2 + c4
ηj ) ∑i
xiηi
xi + ∑j*i
xiψij
(21)
λh ) ∑i
xiλi
xi + ∑j*i
xiςij
ψij ) 14[1 + (ηi
ηj)1/2(Mj
Mi)1/4]2[ 2Mj
Mi + Mj]1/2
(22)
ςij ) ψij[1 +2.41(Mi - Mj)(Mi - 0.142Mj)
(Mi + Mj)2 ]
vpi+1 ) vg
i - (vgi - vp
i ) exp(-∆t/τp)
xpi+1 ) xp
i + vpi ∆t
T pi+1 ) T g
i - (T gi - T p
i ) exp(-∆t/ωp), (23)
T pi , T p
i+1 > Tm or T pi , T p
i+1 < Tm
f pi+1 ) f p
i +cpp
(Tg - Tm)
∆Hm
∆tωp
, 0 < f pi , f p
i+1 < 1
f pi+1 ) f p
i +cpp
(T pi+1 - Tm)
∆Hm(24)
T pi+1 ) Tm
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3637
2. T pi > Tm (f p
i ) 1), T pi+1 < Tm, from totally liquid
state to partially melted state
3. T pi ) Tm (0 < f p
i < 1), T pi+1 > Tm, from partially
melted state to totally liquid state
4. T pi ) Tm (0 < f p
i < 1), T pi+1 < Tm, from partially
melted state to totally solid state
To reduce the computation time and to maintainaccuracy at the same time, a time-varying time step ofthe form
is used, where ∆xmax is chosen to be 10-4 m, which isthe maximum flight distance in each time interval. Thefirst two constraints guarantee that the gas velocity andtemperature will change little in each time step.
4. Analysis of Gas and Particle Behavior
4.1. Analysis of the Gas Dynamics without Air.Initially, we included only propylene and oxygen as thefeed to the system and tested the influence of thecombustion pressure and oxygen/fuel ratio on the gastemperature, velocity, density, and momentum flux (Fu2)in the internal field. The reasons for this approach arethe following: (1) The influence of the process param-eters on the gas dynamics and the particle in-flightbehavior is apparent in this simplified case. (2) Abijection of the pressure and equivalence ratio to thefuel and oxygen flow rates is possible, and calculationstarting from either side is equivalent, which facilitatesthe numerical calculation. (3) There are many HVOFprocesses whose feed consists of only fuel and oxygen(i.e., without air); see, for example, Gu et al.37 and Yangand Eidelman.5 The process model was based on theassumptions of negligible injection gas velocity, instan-taneous equilibrium under constant enthalpy and pres-sure in the injection interface, and frozen isentropic flowduring passage through the nozzle. To account for thedissociation of gaseous products, the model includedeight species (CO, CO2, H, H2, H2O, O, O2, and OH) inthe product mixture. The equilibrium temperature andcomposition of combustion were solved by minimizingthe total Gibbs free energy, and the gas propertiesduring passage through the nozzle were determined bythe standard laws governing compressible flows de-scribed in subsection 2.2.
In Figure 3a, the combustion pressure is fixed to be9 bar, and the equivalence ratio varies from 0.5 to 1.6.In this range, a peak is observed in each of thetemperature vs equivalence ratio plots. As the equiva-lence ratio increases, the temperatures at the entrance,the throat, and the exit first increase, reaching amaximum value, and then decrease. However, theequivalence ratio associated with each peak tempera-ture is about 1.2 (indicating a fuel-rich system), whichis somewhat different from the value expected for a fuel/air system,15 whose optimal value is close to 1.05. It canalso be seen that as the equivalence ratio increases from0.5 to 1.6, the gas velocities both at the throat and atthe exit increase by about 22-23%, while the gasdensity decreases by about 33%. As a result, themomentum flux remains almost constant at these twopositions. In Figure 3b, the equivalence ratio is kept at
Figure 3. Normalized gas temperature, velocity, momentumflux, and mass flow rate in the internal field under the follow-ing operating conditions: P ) 5-15 bar and æ ) 0.5-1.6.Normalization is done with respect to the corresponding gasproperties under the following operating conditions: P ) 9 barand æ ) 1.0 (Table 3).
Table 3. Gas Properties under the Operating ConditionsP ) 9 bar and æ )1.0
1.0, and the combustion pressure varies from 5 to 15bar. In the combustion pressure range of interest, it isfound that both gas velocity and gas temperaturechange little (about 2 and 5%, respectively); however,the gas density and momentum flux change almostlinearly with respect to the combustion pressure, byabout 190 and 200%, respectively. We also computed3D profiles of gas properties under different combustionpressures and equivalence ratios, as shown in Figures4-6. Further analysis reveals the following:
1. At a fixed pressure, there is a peak in the equilib-rium temperature vs equivalence ratio plot, whose valueis about 1.2. It is worth noting that the peak flametemperature occurs not at stoichiometric, but rather atfuel-rich conditions. This is because the equilibriumtemperature is determined not only by the heat gener-ated by the exothermic reaction process, but also by theheat capacity of the product mixture. As the equivalenceratio increases to slightly above 1, the gas temperatureincreases further with the equivalence ratio; this isbecause the heat capacity of the products decreasesmore rapidly than the heat released. Beyond the equiva-lence ratio associated with the peak temperature (about1.2), the heat generated falls more rapidly than doesthe heat capacity, and the temperature decreases. Onthe other hand, when the equivalence ratio is fixed, theequilibrium temperature increases with pressure. Theprimary cause of the equilibrium temperature variationwith pressure is the product dissociation because higherpressure favors larger molecules (Le Chatelier’s prin-ciple). Further increasing the pressure results in anincrease in H2O with respect to H2 and O and helps toincrease the temperature.
2. The higher the equivalence ratio, the smaller thetotal mass flow rate is needed to achieve the samecombustion pressure to choke the flow. On the otherhand, under the same equivalence ratio, the combustionpressure increases linearly with the total mass flow rate.These observations can be explained by the followingequation
Note that the combustion process tends to increase thetotal number of moles in the product mixture and todecrease the average molecular weight. In the fuel-richcase, the total amount of dissociation becomes signifi-cant, and the molecular weight decreases continuouslyas æ increases. Referring to Figure 7, when æ increasesfrom 0.5 to 1.6 for a fixed P and a <10% variation in T,Mpr decreases by about 30% following a nearly linearfunction, and as a consequence, Mpr/T decreases mono-tonically. Therefore, m̆g decreases monotonically as æincreases. When the combustion pressure increases witha fixed æ, both Mpr and T increase slightly because the
product dissociation is suppressed. Because xMpr/Tvaries by less than 2% in the pressure range of interest,the total mass flow rate of the gas is roughly propor-tional to the combustion pressure. Equation 29 alsoindicates that the pressure can be increased by (1)increasing the total mass flow rate of the gas and (2)increasing the equivalence ratio.
3. The gas density at the nozzle throat can increaseby increasing the combustion pressure and decreasing
Figure 4. Profile of equilibrium temperature and total mass flowrate with respect to P and æ. Operating conditions: P ) 5-15 barand æ ) 0.5-1.6.
Figure 5. Profile of gas density and velocity at the throat withrespect to P and æ. Operating conditions: P ) 5-15 bar and æ )0.5-1.6.
m̆g ) FgvgA ) M PxγMpr
RTg
A ) M PγAa
(29)
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3639
the equivalence ratio. This can be explained by theexpression F ) γP/a2. However, the gas velocity at thenozzle throat, where the Mach number is one, is mainlya function of the equivalence ratio and changes littlewith the combustion pressure, as previously discussed.
4. The momentum flux at the throat of the nozzle isindependent of the equivalence ratio and is a linearfunction of the combustion pressure. The influence ofpressure on momentum flux can be explained by theequation
where γ is nearly constant and M mainly depends onthe geometrical configuration of the nozzle. As a con-sequence, the momentum flux is a nearly linear functionof the gas pressure. Equation 30 is also applicable toHVOF systems that include air in the feed stream.
Because the drag force, which is the dominant forcedetermining the motion of the particles in the gas field,is approximately proportional to the gas momentumflux, and because the difference between the gas tem-perature and the particle temperature provides thedriving force for particle heating, it follows from theabove analysis that the particle temperature and veloc-ity can be nearly independently adjusted by manipulat-ing the equivalence ratio and the combustion pressure,respectively.
4.2. Analysis of the Gas Dynamics Including theAir Stream. Motivated by the conclusions drawn fromthe above parametric analysis, we included air in thefeed stream to the HVOF process; this makes theprocess analysis more difficult. Assuming the air to be
composed of only O2, N2, and Ar, the reaction formulabecomes
where x can be any number from 0 to 16.7, correspond-ing to the cases of pure oxygen as the oxidant and pureair as the oxidant, respectively. Obviously, in this case,the equilibrium temperature is dependent not only onæ, but also on x. Furthermore, because the pressuredepends on the temperature, the average molecularweight, and the mass flow rate (eq 29), the air streamplays an important role in achieving a high pressure.
Figure 8 shows the combustion pressure and equilib-rium temperature under different total mass flow ratesand equivalence ratios for a fixed value of x ) 3.97. Itcan be seen that the process behavior is very similar tothat observed without air if x is kept constant (compareFigures 8 and 4). The pressure contour in Figure 8ashows that the total mass flow rate required to achievethe same combustion pressure decreases as the equiva-lence ratio increases. The total mass flow rate increaseslinearly with pressure when the equivalence ratio isfixed. The equilibrium temperature is significantlydependent on the equivalence ratio, but varies onlyslightly with the total mass flow rate. This also implies
Figure 6. Profile of temperature and momentum flux at thethroat with respect to P and æ. Operating conditions: P ) 5-15bar and æ ) 0.5-1.6.
Fgvg2 ) FgM 2a2 ) FgM 2(xγP/Fg)
2 ) M 2γP (30)
Figure 7. Normalized average molecular weight, temperature,and sonic velocity at the throat of the nozzle under the followingoperating conditions: P ) 5-15 bar and æ ) 0.5-1.6. Normaliza-tion is done with respect to the corresponding gas properties underthe following operating conditions: P ) 9 bar and æ ) 1.0 (Table3).
æC3H6 + 4.5O2 + x(N2 + 1/78Ar) w ∑i∈pr
êi(PR)i (31)
3640 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
that pressure variations do not significantly affect theequilibrium temperature.
Figure 9 shows the combustion pressure and equilib-rium temperature under different equivalence ratiosand oxygen/nitrogen ratios for a fixed total mass flowrate to that used under the baseline conditions. It canbeen seen that, as the fraction of air in the total reactantmixture increases, both the pressure and temperaturedrop. As x varies from 0 (pure oxygen as the oxidant) to16.7 (pure air as the oxidant), the equivalence ratiocorresponding to the peak equilibrium temperaturedecreases from 1.23 to 1.05 (see Figure 10); this resultprovides a way to optimally manipulate the relative flowrate of oxygen to air.
We tested the gas dynamics under the nine differentoperating conditions listed in Table 4. The baselineconditions are recommended by the manufacturer. Thesimulation results are reported in Table 5. Under thebaseline conditions, the pressure at the exit of the aircap calculated by the proposed procedure is 0.63 bar,which implies that the flow outside the gun is overex-panded. The manufacturer, Sulzer Metco, measured agauge pressure of -4 psig (-0.3 bar, or the absolutepressure is about 0.7 bar) at the nozzle exit under thesame operating conditions,38 which validated the modeland assumptions applied in this work. In fact, theoverexpanded flow condition gives a slightly higher gasvelocity.
In the nine different operating conditions, the equi-librium temperature is a function of the total mass flowrate, as well as æ and x. For instance, cases 1 and 6-9
have the same value of x (about 4.0), under which theequivalence ratio associated with the peak temperatureis around 1.2 according to our previous discussion.Equilibrium temperature in case 7 is the lowest because
Figure 8. Profile of pressure and equilibrium temperature withrespect to æ and m/mbl. Operating conditions: x the same as forbaseline conditions [propylene, 176 scfh (standard cubic feet perhour); oxygen, 578 scfh; air, 857 scfh; nitrogen, 28.5 scfh], totalmass flow rate varying from 0.8 to 1.2 times the baseline value(18.1 g/s), and æ ) 0.5-2.0.
Figure 9. Profile of equilibrium temperature and pressure withrespect to æ and x. Operating conditions: total mass flow rate thesame as for baseline conditions, æ ) 0.5-2.0, and x ) 0-16.7.
Figure 10. Profile of the optimal equivalence ratio correspondingto the peak temperature of the gas with respect to x. Operatingconditions: total mass flow rate the same as for baseline condi-tions, x ) 0-16.7.
its equivalence ratio is only 0.7. Although the equiva-lence ratios in cases 2 and 6 are quite different, thetemperatures are almost the same. This is because thesetwo equivalence ratios are located on opposite sides ofthe optimal equivalence ratio and the total mass flowrates differ slightly. The temperatures are quite differ-ent in cases 5 and 6 although the equivalence ratios arevery close because the mass flow rates are different.Note that a higher mass flow rate favors a higherpressure and, accordingly, a higher equilibrium tem-perature. Case 3 has the lowest x and an equivalenceratio close to the optimal one, as a consequence; itsequilibrium temperature is the highest, compared tocases 1 and 2, even when the total mass flow rate islow.
On the other hand, the combustion pressure underthe above operating conditions is roughly a linearfunction of the total mass flow rate and changes littlewith the gas composition. This is because the averagemolecular weight of the reaction product mixture is 24-28 × 10-3 kg/mol, and the temperature is 2.9-3.1 × 103
K. Consequently, the sonic velocity does not vary much,and the pressure is proportional to the total mass flowrate (which is in agreement with eq 29).
4.3. Analysis of Particle Velocity and Tempera-ture. In the fabrication of nanostructured coatings, itis crucial to maintain a high particle temperature atthe point of impact on the substrate and, at the sametime, to prevent particles from being superheated,because it is precisely the small grain size that contrib-utes to the superior qualities of nanostructured coat-ings.32 It is also of great importance to maintain a highparticle velocity at the point of impact on the substratebecause the higher the particle velocity, the denser thecoating. We simulated the Diamond Jet hybrid HVOFprocess model under the baseline operating conditionsgiven in Table 4 for nickel (particle properties are givenin Table 6), and the results for the in-flight particlevelocity, temperature, and melting ratio are shown inFigure 11. Particles of small sizes can reach very highvelocities during flight; however, their velocities dropmore sharply than those of larger particles because oftheir smaller momentum inertias. Furthermore, theycan be heated to the melting point in a short time andcan be fully melted during flight; however, they mighteventually be in a coexistence state of liquid and solid
or even in a solid state after a long enough distance.Smaller particles tend to change their temperatureseasily because of their smaller thermal inertias. Forparticles of large sizes, however, the periods for ac-celeration and heating are both longer, and their veloc-ity (or temperature) profiles become nearly flat as theyapproach the same velocity (or temperature) as the gas.In addition, large particles might not reach the meltingpoint and be in the solid state during the entire flight.However, particles of medium sizes might becomepartially melted during flight.
To further understand the behavior of the particlesin the HVOF process, we also plotted the velocity,temperature, and melting ratio at the 0.254-m standoffas a function of particle size, as shown in Figure 12.(Note that the configuration of each figure might varywith different spray distances.) Under the baselineoperating conditions, particles in the size range of 9-30µm hit the substrate as liquid droplets. Particles of sizeslarger than 49 µm or less than 5 µm are in the solidstate at the point of impact on the substrate. Otherparticles, however, are in a semimolten state when theyreach the substrate (where both liquid and solid coexist).It is worth pointing out that, although both very smallparticles and very large particles hit the substrate in apartially molten state or even in the solid state, theirmicrostructures are not the same because the formerhave been fully melted during flight.
Figure 13 shows the influences of the particle injectionvelocity and spray distance on the profiles of the particlevelocity, temperature and flight time. It is shown thatboth disturbances have a minimal effect on the particlevelocity. However, their influence on the particle tem-perature and melting behavior cannot be neglected. Thisbehavior can be explained by the changes in theresidence time of the particles in the gas flow field. Anincrease in the particle injection velocity will result ina decrease in the particle residence time, especially inthe high-temperature zone. This is why larger particlesare affected to a greater extent. An increase in the spraydistance, however, has a greater influence on thetemperature of smaller particles. This is because largerparticles have greater thermal inertia and do not changetheir temperature very much after they reach the gastemperature.
4.4. Modeling of the Powder Size Distribution.The fact that particle temperature and velocity at thepoint of impact on the substrate depend strongly onparticle size implies that the particle size of the feed-stock is one of the key parameters determining coatingquality. This property, together with the significantpolydispersity of most powders used in the thermalspray process, motivates an attempt to account for theeffect of the powder size distribution on the processmodel, the control problem formulation, and the control-
Table 5. Gas Properties for Different Operating Conditions
ler design. Previous experimental work39,40 has shownthat log-normal functions can adequately describe thesize distribution of many powders used in the HVOFprocess. To this end, a log-normal function is used inthis paper to describe the powder size distribution withthe following form41
where f(dp) is the size distribution function and µ andσ2 are two dimensionless parameters corresponding to
the mean and the variance, respectively, of ln dp, whichobeys a normal distribution. For particles that are log-normally distributed, µ and σ can be determined usingthe following formulas15
Figure 11. Profiles of particle velocity, temperature, and meltingratio along the flow field (x ) 0 corresponds to the nozzle exit).
f (dp) ) 1x2πσdp
exp[-(ln dp - µ)2
2σ2 ] (32)
Figure 12. Velocity, temperature, and melting ratio at the0.254-m standoff (point of impact on the substrate) as a functionof particle size.
µ ) ln x3d10d50d90 - 1.831(ln xd90
d10)2
(33)
σ ) 0.781 ln xd90
d10
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3643
where d10, d50, and d90 are three characteristic diam-eters that can be obtained experimentally.1 Particlecoagulation in the HVOF thermal spray process has notbeen reported in the literature, which can be explainedby the following argument. The average distance be-tween individual particles in the HVOF thermal sprayprocess can be estimated from the analysis of Crowe etal.42 Specifically
where Ld is the distance between two particles and κ isthe ratio of the particle loading to particle/gas densityratio. Usually the particle loading is about 4%, and thedensity ratio is about 103-104; therefore, Ld/dp is about
20-50, which implies that the individual powder par-ticles can be considered to be isolated from each other.Therefore, in this work, we assume that particle coagu-lation is negligible and that the powder size distributiondoes not change during flight.
There are many ways to define average powderproperties. For example, they can be averaged withrespect to particle number or particle volume. In thiswork, the average powder properties (PP) are calcu-lated on the basis of the particle volume because largerparticles have a stronger influence on the coatingproperties than smaller ones. Volume-based averagepowder properties can be computed as follows:
5. Feedback Control of the HVOF ThermalSpray Process
5.1. Control Problem Formulation and Control-ler Design. On the basis of model predictions andavailable experimental observations, the control prob-lem for the HVOF process is formulated as the one ofregulating the volume-based averages of the meltingratio and particle velocity at the point of impact on thesubstrate (these are the variables that directly influencethe coating microstructure and porosity; see the model-ing and analysis discussion in the second paper of thisseries19) by manipulating the flow rates of propylene,oxygen, and air at the entrance of the HVOF thermalspray gun. From the analysis in the previous sections,it follows that the gas momentum flux, which is ap-proximately proportional to the drag force, and the gastemperature, whose difference from the particle tem-perature provides the driving force for particle heating,can be almost independently adjusted by manipulatingthe combustion pressure and the equivalence ratio. Todevelop a feedback controller that can be readily imple-mented in practice, the manipulation of the combustionpressure and equivalence ratio is realized by adjustingthe flow rates of propylene, u1(t); oxygen, u2(t); and air,u3(t). Because of the almost decoupled nature of themanipulated input/controlled output pairs, two propor-tional integral (PI) controllers are used to regulate theprocess. Specifically, the controllers have the form
where yspi is the desired set-point value and yi is thevalue of the output obtained from the measurementsystem (y1 is the volume-based average of the particlevelocity and y2 is the volume-based average of theparticle melting ratio). u′1 is the combustion pressure,and u′2 is the equivalence ratio. Kci is the proportionalgain, and τci is the integral time constant of the ithcontroller. The third equation makes use of the processmodel. To keep the problem simple, the ratio of air tooxygen (or x) is fixed. We note that the relationshipbetween the gas temperature and the equivalence ratiois not monotonic. Above the optimal equivalence ratio
Figure 13. Effect of disturbances in the operating conditions(particle injection velocity and spray distance) on the particlevelocity, temperature, and flight time.
Ld
dp) (π6 1 + κ
κ )1/3(34)
PP )∫0
∞ 16
πdp3PP(dp) f (dp) d(dp)
∫0
∞ 16
πdp3f (dp) d(dp)
(35)
ú̇i ) yspi- yi, úi(0) ) 0, i ) 1, 2
u′i ) Kci[(yspi- yi) + 1
τci
úi] + u′0i, i ) 1, 2 (36)
{u1, u2, u3} ) f (u′1,u′2,x)
3644 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
(about 1.2 for x ) 3.97), the gas temperature decreasesas the equivalence ratio increases. Therefore, Kc2 andτc2 should be replaced by -Kc2 and -τc2 when theequivalence ratio is above the optimum value. Thedesign of a model-based feedback control system em-ploying nonlinear control techniques for particulateprocesses,43-49 as well as applications of the controlsystem to an experimental system, will be the subjectof future work.
Regarding the practical implementation of the pro-posed control system (see Figure 14 for a schematic) onthe Diamond Jet hybrid HVOF thermal spray, we notethat the chamber pressure and the equivalence ratio canbe readily manipulated in real time by adjusting themass flow rates of fuel, oxygen, and air. The velocitiesand temperatures of individual particles can be mea-sured experimentally using nonintrusive optical tech-niques, such as laser doppler velocimetry,50 particleimaging velocimetry,51,52 and two color pyrometry.53-55
However, it is not possible to directly measure thedegree of melting of individual particles and conse-quently, the average degree of melting of the entireparticle size distribution. To overcome this limitation,one needs to use an estimation scheme based onmodeling equations that describe the evolution of theparticle temperature, velocity and degree of particlemelting coupled with the available gas-phase measure-ments to estimate average particle melting ratio at thepoint of impact on the substrate. The estimates obtainedby this model can be further improved through com-parison with the particle temperature measurementsat various locations across the free jet. In the simulationsection (section 5.2, below), we include the results of aclosed-loop simulation in the presence of measurementerrors to evaluate the effect of such errors on closed-loop performance; the detailed development of an esti-mation scheme for the particle melting ratio is thesubject of future work. The controller then obtainsinformation from the measurement system, and makesdecisions, which are sent to the controlled valves (totalflow of gases to the process and oxygen/fuel ratio), toadjust the manipulated input variables until the devia-tion of the controlled outputs from their correspondingset-point values falls within a given tolerance. One ofthe great advantages of feedback control is that it cancompensate for the effect of disturbances in the processoperating conditions.
5.2. Simulation Results of the HVOF ProcessModel under Feedback Control. In this subsection,simulation runs of the closed-loop system are presented.The outputs y1(t) and y2(t) are computed by averagingthe individual particle velocity and melting ratio dataobtained from the process model. To account for thepowder size distribution, we first fit a log-normaldistribution and calculate the size range to capture morethan 99 wt % of the particles. We then divide this sizerange into 100 intervals to perform the integration.(Further increasing the number of discretization inter-vals did not change the accuracy of the computedresults.)15,16 This requires that 400 ordinary differentialequations be solved simultaneously for each processsimulation. The parameters used in the closed-loopsystem simulations are listed in Table 7.
Several simulation runs of the process model underthe feedback controller were performed to evaluate theability of the controller to (a) drive the melting ratiosand velocities of the particles at the point of impact onthe substrate to desired set-point values, (b) attenuatethe effects of disturbances on process operating condi-tions, and (c) compensate for the effects of measurementerrors. The first simulation studies the behavior of theclosed-loop system in the presence of changes in the setpoint. Initially, the process is assumed to operate at thebaseline conditions, and at time t ) 10 s, the averageparticle velocity set-point value increases by 5%, andthe average particle melting ratio set-point value de-creases by 5%. Figure 15 shows how the controlledoutputs and manipulated inputs, as well as the totalmass flow rate and the equivalence ratio, respond inthe case of requesting such changes in the set-point
Figure 14. Schematic of the proposed feedback control system.
Table 7. Process and Controller Parameters Used in theClosed-Loop Simulation
parameter value
Kc1 5 × 10-3
Kc2 0.1τ1 5 × 10-2
τ2 5 × 10-2
d10 (µm)a 15d50 (µm) 35d90 (µm) 77φ 1.0
a Powders are assumed to be log-normally distributed, and d10,d50, and d90 are three characteristic diameters whose correspond-ing cumulative weight function values are 0.1, 0.5, and 0.9,respectively.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3645
values. The feedback controller drives the controlledoutputs to the new set-points in about 10 s. (Note that10 s is the time needed for the controlled outputs toreach the new set-point values, not the time for theparticles to hit the substrate, which is on the order of10-3 s.) In a typical HVOF process, the powder feed rateis in the range 20-80 g/min, and the thickness of acoating is about 100-300 µm.31 The deposition efficiencyof the HVOF process is around 70%.56 Considering the
deposition of a coating on a 0.5 m × 0.5 m substrate,the deposition time can be estimated as
For a coating with a larger area, the deposition time iseven longer, which indicates that the controller is quiteeffective (compared with the typical time needed for afull coating) and validates the feasibility of the imple-mentation of feedback control on the HVOF process.Note that, in the first 0.8 s, the liquid ratio increaseswhen the equivalence ratio decreases. This is because,in this time period, the mass flow rate and the pressureincrease by 22 and 21%, respectively, while the equiva-lence ratio decreases by 6%. As a result, the increasein gas temperature resulting from the increased pres-sure outweighs the decrease in temperature resultingfrom the lower equivalence ratio.
To demonstrate that the proposed formulation of thecontrol problem, which explicitly accounts for the effectof the powder size distribution, leads to a solution ofthe control problem that is superior (with respect to thecontrol action needed to achieve the desired controlobjectives) to a solution that assumes a monodispersepowder size distribution, the two PI controllers werealso implemented on the process model using the samecontrolled outputs but assuming that the velocity andtemperature measurements are based on a singleparticle whose size is taken to be dp ) 35 µm, which isequal to the d50 value of the powder size distributionused in our simulation. The corresponding controlledand manipulated variables are displayed in Figure 16.The results show that the desired objectives of 5%changes in the set-point values are not achieved (cf. thecontrolled output profiles of Figure 15, where thedesired set-point change is achieved); this occurs be-cause, as has been previously shown,16 the behavior ofan individual particle is insufficient to represent thebehavior of the entire powder size distribution. Thismakes clear the need to account for the effect of thepowder size distribution in the control problem formula-tion and solution.
To test the robustness of the proposed control problemformulation and of the feedback controller, the problemof controlling the HVOF process in the presence ofdisturbances was studied. Figures 17 and 18 show thecontrolled output and manipulated input profiles in thepresence of a disturbance (20% increase and 20%decrease, respectively) in the spray distance occurringat t ) 10 s. Without control, the process jumps to a newsteady state in a very short time (owing to the very shorttime of particle flight); the particle velocity in each casedrops instantaneously. The reason for this result is thatthe particles are usually accelerated first and thendecelerated in the external field, so that there is anoptimal spray distance. Nevertheless, the disturbancesin the spray distance do not have a significant effect onthe particle velocity because the velocity profile of theparticles is almost flat as they reach the gas velocity.However, the melting ratio of the particles at the pointof impact on the substrate decreases in the former caseand increases in the latter case, which can be explainedby the change in residence time of the particles in thegas flame. Such variations in the molten state of theparticle can have a detrimental effect on the coating
Figure 15. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratiounder the request of a 5% increase in the average particle velocityand a 5% decrease in the melting ratio. The control problemformulation accounts for the effect of the powder size distribution.
t ) 200 × 10-6 m × 0.5 m × 0.5 m × 8900 kg/m3
50 × 10-3 kg/min × 0.7)
12.7 min (37)
3646 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
microstructure evolution. Under feedback control, themanipulated inputs drive the process outputs to theiroriginal steady-state values in 10-25 s. It is alsointeresting to see how the controller responds to com-pensate for this velocity decrease. Whereas it is intu-itively expected that the mass flow rate increases in thelatter case to increase the particle velocity, the totalmass flow rate in the former case decreases to drive the
particle velocity to its original value. This is becausethe equivalence ratio continuously increases and thepressure increases even as the total mass flow ratedecreases.
Figures 19 and 20 show the controlled output andmanipulated input profiles in the presence of distur-bances (10% increase and 20% decrease, respectively)in the initial particle velocity at t ) 10 s. Withoutcontrol, the system jumps to a new steady state in a
Figure 16. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratiounder the request of a 5% increase in the average particle velocityand a 5% decrease in the melting ratio. The control problemformulation does not account for the effect of the powder sizedistribution.
Figure 17. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratioin the presence of a disturbance (20% increase) in the spraydistance.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3647
very short time. Whereas the particle velocity changeslittle in both cases, the particle melting behavior variesa lot. The changes in the particle temperature in bothcases can be explained by the residence time of theparticles in the flame gas, which is caused by thevariation in the particle velocity along the flight,although the particle velocity at the point of impactremains nearly the same. Under feedback control, themanipulated inputs drive the process outputs to theiroriginal steady-state values in about 20 s.
Another source of disturbance to the process opera-tion, especially in an industrial environment, is thevariation in the size distribution of the powder duringthe operation of the HVOF process. According to theanalysis of the previous sections, this might have asignificant influence on the particle velocity and particletemperature at the point of impact on the substrate .In the following simulation, it is assumed that theprocess is at steady state in the first 100 s and thenthe powder size distribution changes gradually. (Specif-
Figure 18. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratioin the presence of a disturbance (20% decrease) in the spraydistance.
Figure 19. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratioin the presence of a disturbance (10% increase) in the particleinjection velocity.
3648 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
ically, in the following calculation, µ increases accordingto the expression µ ) µ0[1 + 0.03(1 - e-t/100)], and σ2 iskept constant.) Figure 21 shows the controlled outputsand the manipulated inputs, as well as the total massflow rate and the equivalence ratio, in the presence ofsuch a variation in the powder size distribution. Underfeedback control, both the particle velocity and meltingratio fluctuate in a very narrow range around thedesired set-point values. We note that æ changes rathersharply compared to the change in m̆Fuel and m̆O2
because both m̆Fuel and m̆O2 have an influence on æ(æ ) m̆Fuel/m̆O2 × 4.5). For example, æ will go up sharplyif m̆Fuel increases while m̆O2 decreases and go downsharply in the opposite case. When no control is used,in which case the flow rate of each stream is keptconstant, both the velocity and melting ratio of theparticles decrease with time, which might have anundesirable effect on the resulting coating properties.
To demonstrate that the proposed formulation of thecontrol problem is robust with respect to measurement
Figure 20. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratioin the presence of a disturbance (20% decrease) in the particleinjection velocity.
Figure 21. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratioin the presence of variations in the powder size distribution.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3649
errors, we implemented the developed control systemon the process model under the request of a 5% increasein the average particle melting ratio set-point value anda 5% decrease in the average particle velocity set-pointvalue at time t ) 10 s, assuming that there are errorsin the values of the average velocity and degree ofmelting used in the controller. In the closed-loop simu-lation, we assume that the estimation errors follow anexponentially decaying function with an initial error of10%. The corresponding controlled and manipulatedvariables are shown in Figure 22. The results show thatthe desired control objective of a 5% change in the set-point values is eventually achieved (cf. the controlledoutput profiles of Figures 15 and 22); this demonstratesthat the proposed formulation of the control problem isrobust with respect to measurement errors.
6. Conclusions
This article presents a fundamental model and afeedback control system for an industrial high-velocityoxygen-fuel (HVOF) thermal spray process (DiamondJet hybrid gun, Sulzer Metco, Westbury, NY). Theprocess model describes the evolution of the gas thermaland velocity fields and the motion and temperature ofagglomerate particles of different sizes and explicitlyaccounts for the effect of the powder size distribution.In addition to providing useful insight into the in-flightbehavior of different-size particles, the model was usedto make a comprehensive parametric analysis of theHVOF process. This analysis allowed for the systematiccharacterization of the influence of controllable processvariables such as the combustion pressure and oxygen/fuel ratio, as well as the effect of the powder sizedistribution, on the values of the particle velocity andtemperature at the point of impact on the substrate.Specifically, the study shows that the particle velocityis primarily influenced by the combustion pressure andthat the particle temperature is strongly dependent onthe fuel/oxygen ratio. These findings are consistent withexisting experimental studies and set the basis for theformulation of the control problem for this HVOFprocess. To develop a feedback controller that can bereadily implemented in practice, the control problemwas formulated as one of regulating volume-basedaverages of the melting ratio and velocity of the particlesat the point of impact on the substrate (these are thevariables that directly influence the coating microstruc-ture and porosity, which, in turn, determine the coatingstrength and hardness) by directly manipulating theflow rates of fuel, oxygen, and air at the entrance of theHVOF gun. A feedback control system was developedand applied to the process model. Closed-loop simula-tions demonstrated that the particle velocity and melt-ing ratio at the point of impact on the substrate reachedthe desired set-point values in a short time, whichvalidates the feasibility of real-time implementation offeedback control on the HVOF thermal spray system.It was also shown that the proposed control problemformulation and feedback control system are robust withrespect to disturbances in the spray distance andparticle injection velocity, as well as variations inpowder size distribution.
In the second article of this series,19 we presenta stochastic model that uses information about theparticle velocity, temperature, and degree of melting atthe point of impact on the substrate from the model
developed in the present paper to predict the coatingporosity and microstructure.
Acknowledgment
Financial support from a 2001 Office of Naval Re-search Young Investigator Award, program manager Dr.Lawrence Kabacoff is gratefully acknowledged.
Figure 22. Profiles of controlled outputs (average particle velocityand melting ratio), manipulated inputs (flow rates of propylene,oxygen, and air), and total mass flow rate and equivalence ratiounder the request of a 5% increase in average particle velocityand a 5% decrease in the melting ratio. Closed-loop simulation inthe presence of measurement error.
3650 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Notation
a ) sonic velocity (m/s)a1-a9 ) coefficients of the polynomial expression for heat
capacityaij ) number of element j in species iA ) cross-sectional area perpendicular to the flow direction
(m2)Ap ) projected area of a particle on the plane perpendicular
to the flow (m2)A′p ) surface area of the particles (m2)b1-b4 ) coefficients of the polynomial expression for
viscositybj ) total number of elements j in each molecule of
reactants (mol/kg)Bi ) Biot numberc1-c4 ) coefficients of the polynomial expression for
thermal conductivitycp ) heat capacity at constant pressure (J mol-1 K-1 for
gas or J kg-1 K-1 for particle)CD ) drag coefficientd ) particle diameter (m)D ) diameter of the gun barrel (m)E ) kinetic energy (J)fp ) liquid fraction or melting degree of the particlesF ) cumulative volume or weight functionG ) Gibbs energy (J/kg)h ) heat-transfer coefficient (W m-2 K-1)H ) enthalpy (J/mol)Kc ) proportional gainK1, K2 ) factors used in eq17l ) total number of chemical elements involved in the
systemLd ) average distance between particles (m)Lpc ) potential core length of the supersonic free jet (m)m ) mass (kg)M ) molecular weight (kg/mol)M ) Mach numberNu ) Nusselt numberP ) pressure (Pa)PP ) particle propertiesPr ) Prandtl numberR ) gas constant (8.314 J mol-1 K-1)Re ) Reynolds numberT ) temperature (K)t ) time (s)u ) manipulated inputv ) velocity (m/s)x ) coefficient of air in the reaction formulaxp ) axial distance (m)y ) controlled output
Greek Letters
R, â ) factors used in eqs 13 and 14æ ) equivalence ratioê ) stoichiometric coefficient (mol/kg)ú ) errorµi ) chemical potential of species i (J/mol)µ ) mean of Gaussian distributionγ ) adiabatic constant, ratio of the heat capacity at
constant pressure to the heat capacity at constantvolume
F ) density (kg/m3)φ ) sphericity, defined as the ratio of the surface area of a
sphere with equivalent volume to the actual surface areaof a particle
η ) viscosity (Pa‚s)τc ) integral time constantτp ) characteristic time for particle motion (s)κ ) ratio of the particle loading to particle/gas density ratioλ ) thermal conductivity (J m-2 K-1)
λi ) Lagrangian multiplierωp ) characteristic time for particle heating (s)
Superscripts and Subscripts
a ) atmospheric conditionsb ) back conditionsbl ) baseline conditionse ) exit conditionsg ) properties related to the gasi, j ) indicesin ) inletl ) liquidp ) properties related to the particlespr ) productsre ) reactantsst ) stoichiometric conditionst ) throatT ) totalL ) standard conditions(‚h) ) average
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Received for review July 3, 2003Revised manuscript received October 9, 2003
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