research article model-based control of a continuous...

Research ArticleModel-Based Control of a Continuous Coating Line for ProtonExchange Membrane Fuel Cell Electrode Assembly

Vikram Devaraj1 Luis Felipe Lopez1 Joseph J Beaman1 and Serge Prudhomme2

1Department of Mechanical Engineering University of Texas at Austin Austin TX 78712 USA2Departement de Mathematiques et de Genie Industriel Ecole Polytechnique de Montreal Montreal QC Canada

Correspondence should be addressed to Vikram Devaraj vikramutexasedu

Received 16 October 2014 Accepted 19 December 2014

Academic Editor Bhaskar Kulkarni

Copyright copy 2015 Vikram Devaraj et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Themost expensive component of a fuel cell is the membrane electrode assembly (MEA) which consists of an ionomer membranecoated with catalyst material Best-performing MEAs are currently fabricated by depositing and drying liquid catalyst ink on themembrane however this process is limited to individual preparation by hand due to the membranersquos rapid water absorption thatleads to shape deformation and coating defects A continuous coating line can reduce the cost and time needed to fabricate theMEAincentivizing the commercialization and widespread adoption of fuel cells A pilot-scale membrane coating line was designed forsuch a task and is described in this paper Accurate process control is necessary to prevent manufacturing defects from occurringin the coating line A linear-quadratic-Gaussian (LQG) controller was developed based on a physics-based model of the coatingprocess to optimally control the temperature and humidity of the drying zones The process controller was implemented in thepilot-scale coating line proving effective in preventing defects

1 Introduction

The direct methanol fuel cell (DMFC) is one of the mostresearched proton exchange membrane (PEM) fuel cellsystems Their low operating temperature and high energydensity make them an attractive alternative for the electronicdevice market [1] In spite of these advantages the adoptionand commercialization of DMFC fuel cells have been slowmainly because of the high manufacturing costs of themembrane electrode assembly (MEA) the most expensivecomponent of direct methanol fuel cells

The catalysts used in the MEA consist of either platinumor platinum alloys which are historically expensivematerialsIn addition to the cost of materials manufacturing of MEAsis still performed with techniques developed for small-scalemanufacturing resulting in high production costs It wouldgreatly benefit the fuel cell industry if alternative materialsand cost-effective defect-free large-scalemanufacturing tech-niques were developed for the MEA [2]

In PEM fuel cells the catalyst layer is very thin (inthe order of a few microns thick) and is too delicate tobe manufactured separately from other components of the

cell This layer is usually formulated as liquid ink and canbe deposited by a variety of coating techniques Amongthese techniques coating the catalyst layer directly on thePEM exhibits best performance and durability in addition tohaving the fewest manufacturing steps It can be done in asingle process per electrode side [3 4]

Although direct catalyst coating of PEM is advantageouswith respect to performance it is the most challengingprocess among existing methods The most significant issueis the water absorption of Nafion the most common PEMused in DMFCs The tendency to absorb water causes thePEM to swell and distort causing wrinkles when the coatingis applied on it Figure 1 shows the wrinkles that are formedwhen ink is applied directly using a tape-casting process

This swelling tendency is the main challenge in fabrica-tion of MEA by the direct catalyst coating process Nafioncan swell up to 70 larger than its original volume whenexposed to moisture The wrinkles appear when the directcatalyst coated PEM is dried [5] The wrinkling of Nafion isthe single largest reason for coating defects and the majorreason preventing large-scale manufacturing and thus thewidespread adoption of fuel cells [6] It drives the total cost of

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2015 Article ID 572983 11 pageshttpdxdoiorg1011552015572983

2 International Journal of Chemical Engineering

Figure 1 Nafionmembrane exhibiting distortion upon contact withliquid ink

the MEA up because of wasted catalyst and membrane bothexpensive components in a DMFC in addition to thwartingthe fabrication of larger area fuel cells

These difficulties in manufacturing have forced the coat-ing of the PEM to be done by hand in small batch processes[7] Small pieces of the membrane are held down with avacuum table and the coating is directly sprayed or brushedon This process is repeated until the desired thickness of thecatalyst layer is achieved

To improve and automate the fabrication of theMEA it iscritical to understand the behavior of themembrane during atypical coating process which includes exposure to transientnonuniform water heat and mechanical stress conditions[8] Many investigators have modeled the membrane in a fuelcell operating environment [9ndash11] but there are fewermodelsthat attempt to model the membrane in a manufacturingenvironment Among the existing models many are steady-state models (not suitable for process control) and most ofthem ignore the relationship between swelling and the mem-branersquos water content In addition water transport within themembrane is often ignored as well [12ndash14] Silverman et al[8] was one of the first researchers that attempted a transientnonuniform model describing water transport stress andswelling in a membrane in a manufacturing environment

The contribution of this paper is two-fold First itdescribes a pilot-scale continuous coating line for Nafionmembranes that was designed for controlled water absorp-tion and desorption Humidity and temperature are con-trolled throughout the process to prevent wrinkles Secondit presents a low-order model of the swelling and drying phe-nomena and its application in a linear-quadratic-Gaussian(LQG) controller to prevent swelling defects This approachproved effective in preventing wrinkles and holds potentialsfor application in large-scale manufacturing as discussed atthe end of this article

2 Pilot-Scale Membrane Coating Line

A pilot-scale coating line was designed to accept rolls ofuncoated Nafion advance them to a preswelling section to acoating section and finally to a drying section in a continuousfashionThemotivation for this designwas that a significantlylower number of coating defects have been observed whenthe PEM is preswelled or saturatedwithwater prior to coating[15 16] Once the membrane is fully saturated with water it

cannot absorb any more and its water content stays constantPreswelling by immersion in water is used in this applicationfor rapid diffusion into the membrane

The manufacturerrsquos recommended storage conditions forNafion are 50 relative humidity at 23∘C It is desiredthat the coated membrane is brought to these temperatureand humidity conditions before it is stored [17] Howeverif the ambient temperature and humidity were set at therecommended storage conditions it would require a coatingline that would have to be either extremely long or slow

Previous work concluded that during the drying of thefreshly coated saturated membrane the coating rehydratesthe membrane keeping it fully hydrated as long as water ispresent in coating Even when using some potentially harshdrying conditions to remove water from the freshly coatedmembrane no undesirable effects have been reported in themembranersquos shape as it remains hydrated by the wet coating[8]This result suggests that it is possible to employ two-stagedrying where a first stage rapidly removes the water from thecoating and a second stage that brings the membrane to therecommended storage conditions

Previous research on two-stage drying has shown thatcontrolling the time spent in a hotdry zone and the timing ofthe transition from the hot and dry zone to a second coolerzone is extremely important Transitioning too early resultsin the coating not being completely dry while transitioningtoo late would mean that the membrane is over-dried [8]

A pilot-scale membrane coating line for producingcatalyst-coated membrane in a continuous fashion wasdesigned constructed and tested The pilot plant wasdesigned to accommodate Nafion ionomeric membraneavailable in 10 cm wide roll form An overview of the pro-posed pilot scale membrane coating line is shown in Figure 2It should be noted that the Nafion web moves in a clock-wise direction through the machine In this machine bothuncoated and coated rolls of Nafion are stored in the leftmostchamber

This machine has the capability to change independentlythe unwind and rewind tensions in thewebThere is a tractionroller that friction-feeds the membrane web and advances itthrough the machine at a set velocity The membrane web issupported throughout the machine by aluminum idle rollersIn addition to this the machine has a preswelling sectionand seven independent temperature-humidity controllabledrying chambers Although a doctor blade is employed tocoat ink it can be changed to any other method of inkapplication in future designs Since this line deals withaqueous swelling and wet coating of ionomeric membranesmost of the critical components were procured and built towithstand prolonged moisture exposure

21 Uncoated and Coated Membrane Storage Since it isbest to store the coated and uncoated Nafion rolls at therecommended storage conditions it is essential to havea start and end block that is temperature and humiditycontrolled as shown in Figure 3 This can prevent damageto the roll when left in the machine for extended periodsof time A stand-alone PID controller is used to control

International Journal of Chemical Engineering 3

18m1

m

(a)

(b)

Figure 2 Overview of the membrane coating machine

Figure 3 Coated and uncoated membrane roll storage

the humidity and temperature of the start block This sectionhas a dedicated heater humidifier and supply of dry air tocontrol the temperature and humidity

22 Web Tension and Velocity Pneumatically actuatedexpanding chucks secure the rolls of coated and uncoatedNafion On one side of the storage chamber there are twobrushedDCmotorsmounted on the side that apply opposingtorques for the unwind and rewind chucks These torquestranslate as tensions in the web during manufacturing Tomeasure the unwind and rewind tensions in the web duringmanufacturing there are two tension transducers (polishedrollers in Figure 3) There is another brushed DC motor thatis geared to a rubber traction roller for moving the Nafion ata controlled speed This traction roller is positioned just in

Figure 4 Two DC motors employed to apply unwind and rewindtorques to the chucks The third motor drives the traction roller

Figure 5 The preswelling tank

front of the chuck on which the coated Nafion roll is grippedFigure 4 shows the three DC motors

23 Design of the Preswelling Section The design of thepreswelling section is critical for the performance of the coat-ing line It is the first location in the process workflow wherewrinkles becomepermanentThewrinklingmodel developedby Devaraj [18] was used to set the design parameters of theswelling tank and associated rollers shown in Figure 5

In the preswelling section of the machine a piece ofNafion equilibrated at 50 RH and 23∘C is dunked in liquidwater The membrane immediately changes shape and startsto buckle at the membrane-water-air interface This elasticbuckling propagates along the direction of webmotion beforevanishing after some distance from the membrane-water-airinterface Idler rollers are used to support the membrane andto manipulate the direction of the web throughout the entirecoating process Buckling due to the sudden swelling willbecome permanent only after it passes over the idler rollers

Nafion enters the swelling tank from the top on theleft side and leaves the tank on the right side There is anidler roller made from polycarbonate that is submerged inwater The formation of permanent wrinkles depends on


the distance between the submerged roller and the roller onthe top left the level of water above the submerged rollerand the force used to pull the web through the preswellingsectionDue to space constraints on themachine the distancebetween the two rollers was set to 032 meters

Simulation results of the wrinkling model for differentwater levels in the tank suggest that too little water in thetank will cause permanent deformations at the submergedroller and too much water in the tank may cause permanentdeformations at entry roller This means that the membrane-water-air interface must be at a minimum distance fromboth the rollers to prevent permanent defects Based on thesimulations that were performed for a 032 times 01m web thepreferred water height in the tankwas defined at 019m abovethe submerged roller

24 Ink Application As mentioned before a doctor bladecoating system is used to apply the coating on the membraneThis coating is applied on a flat PTFE-coated glass plate toreduce friction A peristaltic pump feeds the wet catalyst inkinto the doctor blade and helps maintain a uniform head ofink The ink flow rate may be adjusted with the peristalticpump depending on doctor blade gate height coating speedand so forth

When Nafion exited the swelling tank some dropletsof water were found to cling onto the membranersquos topand bottom surfaces This interferes with uniform coatingapplication so a wiping step was added before coating Asimple absorbency-based pad was used to wipe the surfacedroplets of water

25 Humidity and Temperature Controlled Drying Zones Themembrane-coating pilot scale line has seven independentlycontrollable temperature-humidity drying zones Althoughin this paper we used only two distinct zones to test theeffectiveness of a model-based controller the seven zoneswere constructed to accommodate future research Figure 6shows a drying zone viewed from the upstream and down-stream sides

Each drying zone dries a span of web that is approxi-mately 30 cm long Each zone forces heated humidified airin the transverse direction and contains an axial blower aresistive heater an ultrasonic humidifier and water reservoirThe temperature and humidity of the eight zones can becontrolled independently using PID controllers

3 Modeling the MembraneSwelling and Coating

Computational modeling was used to study the physicalprocesses involved in the coating of the MEA and for processoptimization This was made possible by the 3-dimensionalmultiphysics model developed by Devaraj [18] The modelincludes water transport heat transfer and solid mechanicsin a set of differential equations that are solved for watercontent in the membrane 119898

0 temperature 119879 and strain

120576 The model although accurate is not suitable for real-time applications due to its high computational cost and

(a)

(b)

Figure 6 Upstream (a) and downstream (b) of a temperature-humidity controlled drying zone

a simplified version of it is required for process controlpurposes

31 Reduced-Order Membrane Swelling Model The reduced-order swelling model is a simplified version of the high-fidelitymodel developed byDevaraj [18]Thismodel includesonly water transport and heat transfer processes A rectangu-lar piece of membrane with length 119897 width119908 and thickness 119905is considered In thismodel119881 is the volume of themembraneand 119860 is the surface area over which drying is consideredThe water content of the membrane represented by 119898

0 is

defined as the number of moles of water per unit mass of themembrane The lumped water transport equation for 119898

0is

derived as

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905= minus

119860

119881

120572119888(1198880minus 1198880eq) (1)

where 1205880is the density of water 120588

119898is the dry density of

membrane1198720is the molar mass of water and 120572

119888is the mass

transport barrier determined for a concentration drivingforce [19] Here 119888

0is the concentration of water calculated

with 1198880= 11989801205881198981+120576V and 120576V = 119898

011987201205881198981205880Themembranersquos

water concentration at equilibrium 1198880eq is obtained from a

thermodynamic model [20]A reduced-order differential equation describes the evo-

lution of the membrane temperature 119879 considering heattransfer due to convection and advection as described by

120588119888119901

d119879d119905

= minus

119860

119881

[ℎ (119879 minus 119879surr) + 119902st120572119888 (1198880 minus 1198880eq)] (2)


where 120588 and 119888119901refer to the effective density and specific heat

of the membrane respectively It should be noted that 120588 and119888119901depend on the water content of the membrane In this

equation ℎ refers to the convection heat transfer coefficientfor themembranersquos drying conditions119879surr is the temperatureof the surroundings and 119902st is the heat of sorption obtainedfrom vapor sorption experiments [21]

32 Modeling the Coating A zero-dimensional model wasdeveloped for the coating on the membrane assuming thatit is applied uniformly and is itself homogenous As waterdries from it the volume change due to the water lost needsto be considered The model is defined by the lumped mass-transfer and heat-transfer governing equationsThe system issolved for water content expressed as molality in the coating1198980119888and the temperature of the coating 119879

119888

The fuel cell coating considered here consists mainly ofwater with some ionomer catalyst and negligible amountsof alcohol Although some ionomer is present in the coatingthat will absorb and hold water its relative concentrationcompared to the other components is low and it is ignoredin the model Additionally since the coating has such a highwater content it is assumed that the membrane remainsfully hydrated until all liquid water from the coating is lostThe coating acts like a reservoir of water that can moveinto the membrane or evaporate into the surrounding airThis assumption is implemented in the shared boundarycondition when the membrane swelling and coating modelsare combined allowing us to piecewise consider first thedrying of the coatingwith themembrane remaining saturatedand then the drying of the membrane itself

Since the coating is about 95 water by volume thevolume changes resulting from evaporation are significantand need to be addressed It should be noted that whencoating molality 119898

0119888goes to zero the coating is completely

dry of liquid water The lumped water transport governingequation for119898

0119888is given by

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

d1198980119888

d119905= minus

119860119888

119881119888

120572coat (1198880119888 minus 1198880119888surr) (3)

where 120588119904is the density of the dry coating 119860

119888is the area over

which the drying is considered in the coating and 119881119888is the

volume of the coatingTheparameter120572coat is themass transfercoefficient for the coating and 119888

0119888surr is the surroundingrsquosliquid water concentration The parameter 119888

0119888surr was set tozero in the simulations because it is assumed that the watercontent in the surrounding air is always negligible comparedto the amount of water present in the coating An extra termwill be added to the differential equation to account for watertransport to the membrane when both systems are studiedtogether

The water concentration of the coating is defined as 1198880119888=

11989801198881205880120588119904(1205880+ 11987201198980119888120588119904) The density of the dry coating is

calculated based on themass and volume of carbon black andNafion

The mass transfer coefficient 120572coat is assumed to bedependent on temperature and Reynolds number of themoving air that is used for drying the coating It is obtained

from convection mass transfer correlations [22] For a crossflow with a turbulent boundary layer the mass transfercoefficient 120572coat can be modeled as

120572coat =0037D

0air

119871119888

(

119871119888V

]119886

)

45

(

]119886

D0air

)

13

(4)

where D0air and ]

119886 respectively refer to the diffusion

coefficient of water in air and kinematic viscosity of air Allthese properties are evaluated at the coatingrsquos temperature 119871

119888

is the length of the coating in the direction of air flow and Vis the airrsquos free stream velocity

Convection and advection are the two modes of heattransfer included in the lumped differential equation wherethe coatingrsquos temperature is referred to as 119879

119888

120588coat119888119901coat119881119888d119879119888

d119905

= minus119860119888[ℎcoat (119879119888 minus 119879surr) + ℎ

119891119892120572coat (1198880119888 minus 119888

0119888surr)]

(5)

where 120588coat and 119888119901coat refer to the effective density and specificheat of the coating respectively It should be noted that120588coat and 119888

119901coat depend on the water content of the coatingℎcoat refers to the convection heat transfer coefficient for thecoating 119879surr is the temperature of the surroundings and ℎ

119891119892

refers to the heat of vaporization of water as given in [22]Determination of the effective density of the coating

requires considering the masses and volumes of the drycoating as well as the water present in it 120588coat = 120588

0(120588119904+

11987201198980119888120588119904)(1205880+ 11987201198980119888120588119904) The effective specific heat of

coating can be computed with a weighted average of thespecific heat of the dry coating and the water

The convection coefficient for heat transfer is calculatedas follows [22]

ℎcoat =0037119896

119886

119871119888

(

119871119888V

]119886

)

45

(

119888119901119886120583119886

119896119886

)

13

(6)

33 Combining the Membrane Swelling and Coating DryingModels During the preswelling process the Nafion mem-brane is completely saturated with water or some other solu-tion This is represented in the model as a set of appropriateinitial conditions In the proposed approach the problem issimplified by considering only a single sided coating

As mentioned before the coating is assumed to be areservoir of water for the membrane as long as there is waterpresent in it This means that the membrane stays hydratedas long as the coating is wet and starts to lose water only afterall the liquid water has completely dried from the coatingHence themodel decomposes themembrane coating processinto two steps In the first it tracks the water content in thecoating and in the second it solves for thewater coating in theNafion membrane while solving for temperature 119879 in bothcases It should be noted that the membrane and coating areassumed to be at the same temperature which means 119879

119888and

119879 are one and the same In other words the model solves for1198980119888and 119879

119888while the coating is still wet and for 119898

0and 119879 as

soon as the coating has dried


331 Coating Is Wet When the coating is wet two differentcontrol volumes are used for the heat and mass transferanalysis In the case of water transport only the coating isincluded in the analysis while for heat transfer both thecoating and the membrane are included The various fluxesused to solve for water content in the coating are shown inFigure 7 The convection mass transfer from the top of thecoating is accounted for and it is considered that water islost from the saturated Nafion membrane from the bottomAlthough water is lost from the bottom surface of Nafion itis replenished from the coating instantaneously keeping themembrane water concentration constant

As explained above themodel ignores the variation in thewater content of Nafion membrane while the coating is wetassuming that it remains saturated but account for the waterlost from the coating to Nafion membrane while the coatingis drying

119881119888

1205882

0

120588119904

120588119904+11987201205881199041198980119888

d1198980119888

d119905

= minus119860119888[120572coat (1198880119888 minus 119888

0119888surr) + 120572119888(1198880minus 1198880eq)]

(7)

In the heat transfer equation a term is added to accountfor the water lost to the membrane

119881eff120588eff119888119901effd119879d119905

= minus119860eff [2ℎ (119879 minus 119879surr) + ℎ119891119892120572coat (1198880119888 minus 119888

0119888surr)

+119902st120572119888 (1198880 minus 1198880eq)]

(8)

In the above equation it can be seen that heat loss dueto convection is considered from the top of the coating andbottom of the Nafion membrane The heat loss associatedwith advection is also considered from the top of the coatingand bottom of the membrane Since the convection heat losshas the same driving force (119879 minus 119879surr) for top and bottom ofthe system it is combined and expressed as 2ℎ The terms120588eff and 119888

119901eff refer to the coating-membrane systemrsquos densityand specific heat capacity respectively It should be noted thatconvection heat transfer coefficient ℎ is the considered to beidentical as the amount of heat lost due to convection is thesame from the membrane and the coating due to identicaldrying conditions

332 Coating Is Dry After the molality of water in thecoating119898

0119888has approached zero the membrane will begin to

lose water Hence the combined equations described in thissection solve for 119898

0and 119879 The now dry coating is modeled

to be a porous structure and the water loss through thecoating is modeled as diffusion through a porous media (seeFigure 8)

The mass transfer equation that describes the watertransport is given by

119881

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905

= minus119860 [119865120572119888(1198880minus 1198880eq) + 120572

119888(1198880minus 1198880eq)]

(9)

Coating c0c (t)

120572coat (c0c minus c0ceq)

Membrane c0120572c(c0 minus c0eq)

120572c(c0 minus c0eq)

(a) Mass transfer

Coating

hfg120572coat (c0c minus c0ceq)

Membrane

qst120572c(c0 minus c0eq)h(T minus Tsurr)

T(t)

hcoat (T minus Tsurr)

(b) Heat transfer

Figure 7 Fluxes used in heat andmass transfer when coating is wet

where 119865 is the effective diffusion coefficient factor defined by119865 = 120576119905120575120591 in which 120576

119905 120575 and 120591 are the porosity constrictivity

and tortuosity respectively It should be noted that theseconstants can be calculated experimentally for the dry coatingand are dimensionless Values of 119865 are necessarily less than 1and this shows reduced flux due to the presence of porousmedia between the membrane and air

Heat transfer is described with

120588eff119888119901effd119879d119905

= minus

119860eff119881eff

[2ℎ (119879 minus 119879surr) + 119865119902st120572119888 (1198880 minus 1198880eq)

+119902st120572119888 (1198880 minus 1198880eq)]

(10)

The lumped zero-dimensional model presented in thispaper is able to track the water content and temperature ofthe coating by solving for water molality and coating tem-perature The zero-dimensional membrane swelling modeland the zero-dimensional coating drying model must becombined for use inmodeling anMEAmanufacturing designand control strategy Since this model is defined for a fullysaturated membrane on which the coating is applied itcan be solved in two steps the first step that tracks themolality of water in the coating alongwith the systemrsquos overalltemperature that is valid when the coating is still wet and asecond step that tracks the Nafionmembranersquos molality along


Coating

c0 (t)Membrane


F120572c(c0 minus c0eq)

(a) Mass transfer

Coating

Membrane


T(t)

hcoat (T minus Tsurr) Fqst120572c(c0 minus c0eq)

(b) Heat transfer

Figure 8 Fluxes used in heat andmass transfer when coating is dry

with the system temperature that is valid after the coating hasdried

4 Linear-Quadratic-Gaussian Controller

The model presented in the previous section is far fromperfect and neither will be the measurements taken fromthe process A linear-quadratic-Gaussian (LQG) approacha combination of a Kalman Filter and a linear quadraticregulator (LQR) is chosen for real-time control under uncer-tainties The model is converted from a Lagrangian referenceframe to an Eulerian reference frame and method of lines isapplied to convert the resulting partial differential equationsinto a set of ordinary differential equationsThese differentialequations are used to obtain the nominal operating condi-tions and then these conditions are used to linearize thesystemThe linearized state-spacemodel is used to design thelinear quadratic regulator and Kalman filter

It should be remembered that the modeling of the dryingprocess had been divided into two distinct processes in thefirst only the coatingrsquos water content is computed duringdrying and in the second the water content in the membraneis tracked when the system is brought to storage condi-tions This two-staged drying is implemented in the controlstrategy The region where only the coating loses water isreferred to as zone one and the region inwhich themembranedehydrates is referred to as zone twoThe proposed approach

intents to control the transition between the two zones sothat the transition happens when the coating becomes justdry A more general description of the process would begiven by a moving boundary problem where the transitionbetween the wet coating model to the dry coating model isdetermined by the physics of the process In this simplifiedmodel it is assumed that the transition between the modelsis determined by the dimensions of the machine Developingthe equations necessary to implement control for the twodrying zones is discussed however the development of theLQG controller is shown only for the first zone and is notrepeated for the second zone because of the similarity inprocedure

41 Equations for Model-Based Control (Zone 1) The equa-tions presented in the previous section do not account for themovement of the web through the first drying zone Howeverthemovingweb causes a gradient in thewater content and thetemperature of the membrane along the direction of motionInclusion of the material derivative instead of the temporalone results in (11) In this model it is assumed that the onlycomponent of the velocity vector is in the direction of thecoating line

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff

[2ℎ (119879 + 119879surr1) + ℎ119891119892120572coat (1198880119888 minus 119888

0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)

Methods of lines is used to convert the system of partialdifferential equations to a set of ordinary differential equa-tions Application of method of lines involves the construc-tion of a numerical solution for the spatial derivatives whichare discretized while the time variable is left continuous Afinite-difference method is used to divide the control volumeinto an equispaced grid and then a first-order backwarddifference is used for the discretization

(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)

In these equations the subscripts 119909 = 119894 and 119909 = 119894 minus 1

indicate the position on the grid and Δ1199091= 1198971119899with 119897

1being

the grid length and 119899 referring to the number of elements usedin the discretization of the first drying zone (see Figure 9)


Web motion direction

x = 1 x = 2 x = 3 x = n minus 1 x = n

l1

Figure 9 Finite difference grid applied to the web in the first dryingzone

When method of lines is implemented a system ofordinary differential equations is obtained for mass transfer

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)

Similarly for the heat transfer equations consider

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)

The system of ordinary differential equations describesthe dynamics of the molality of the water and the coatingtemperature of the membrane-coating assembly at the gridpoints The equations are coupled and solved together Itshould be noted that the values of 119898

0119888|119909=1

and 119879|119909=1

arerequired as boundary conditions These values are deter-mined at the first zonersquos entry and are known becausethey correspond to the inkrsquos composition and preswelledmembranersquos temperature

By changing 1198880119888surr1 1198880eq1 and 119879surr1 the evolution of

the various molalities and temperatures can be modifiedThis can be achieved by changing the temperature 119879surr1 andwater activity 119886

0surr1 of the drying zone Unfortunately thosevalues cannot be modified instantaneously because of theirrelatively slow response As explained before PID controllersembedded in LabView are used to control the heaters andhumidifiers that affect the drying zone However the setpoints of the PID controllers can bemodified instantaneouslyand can be used as the linersquos inputs in the process controllerBecause there will be a lag between a change in the PID setpoints and the actual change in zone temperature and wateractivity it is necessary to model the relationship between theset points and the actual values in the zone


l2

x = n + 1 x = n + 2 x = n + 3 x = n + m minus 1 x = n + m

Figure 10 Finite difference grid applied to the web in the seconddrying zone

A first-order response is assumed for the drying zonersquostemperature and water activity for a change in the respectiveset points

d1198860surr1

d119905= minus120572zone1 (1198860surr1 minus 119886

0sp1)

d119879surr1d119905

= minusℎzone1 (119879surr1 minus 119879sp1)

(15)

where 120572zone1 and ℎzone1 denote the mass and heat transfercoefficients for the first frying zone and are obtained bymatching them to fit actual experimentsThe zonersquos set pointsare given by 119886

0sp1 and 119879sp1A similar procedure is followed to obtain the differential

equations for the second zone

1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)

where subscript 2 refers to the quantities in the second zoneThe numbering of the grid points begins at 119909 = 119899+1 and endsat 119909 = 119899 + 119898 In these equations Δ119909

2= 1198972119898 where 119897

2is the

length and119898 is the number of elements in the second dryingzone (see Figure 10)

42 Process Controller The proposed model is able todescribe the system based on a state vector x inputs u andmeasurements y defined as shown in (17) Although theweb velocity V

119909could be controlled to optimize the use of

Nafion a fixed velocity of 2mms is used in the pilot-scaleline A 1200 Series laser displacement sensor from Laser-View Technologies was used to measure the total thickness


at the end of the first drying zone The measurement is usedto infer the coating molality 119898

0119888|119909=119899

An infrared type-Tthermocouple purchased from Omega is used to measure119879|119909=119899

A SHT15 digital temperature and humidity sensoris used to measure the zonersquos water activity 119886

0surr1 andtemperature 119879surr1

x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)

In the first zone the nominal operating conditions werechosen such that the water molality of the coating at theexit 119898

0119888|119909=119899

is just above zero The coating was applied withthe doctor blade and had an applied wetting thickness of 50microns A length of 119897

1= 062m was determined for the

drying length The nominal setpoints were defined as 1198860sp1 =

045 and 119879sp1 = 323K The set of nonlinear differentialequations were linearized about these nominal conditionsand then discretized for a constant time step to obtain

120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)

In a scale-up industrial implementation of this model-based control approach more sensors can be added toimprove accuracy It should be noted that laser thicknessmeasurement sensors have been historically more expensivewhen compared to thermocouple-based temperature sensorsAs a practical consideration it is recommended to design asystem using more inexpensive temperature measurementsand less of the more expensive thickness measurements as anindirect method of measuring the molality

The cost function used in the design of the optimalcontroller is denoted by J

J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)

The input that minimizes the cost function is given by

u (119896) = u0minus K119903120575x (119896) (20)

in which K119903is the gain matrix The system is defined

as an infinite horizon optimization problem and K119903can

be computed offline In this application both the output-weighting matrix and the input-weighting matrixQ

119903andR

119903

are defined as diagonal matrices

43 Estimator The LQR has been designed based on theassumption of full state feedback However measurements

are limited and they are corrupted with noise A discrete-time Kalman filter is used to ldquoblendrdquo the predictions from themodel and the actual measurements into an optimal estimateof the state variables denoted by 120575x

Inclusion of the process and measurement noise (w(119896)and k(119896) resp) in the system equations yields

120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)

These noise terms aremodeled as Gaussian processes thatare uncorrelated with each other and in time Matrices Q

119896

and R119896are known as the state and measurement covariance

matrices respectively

E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)

The estimate of the state vector 120575x(119896) using all measure-ments available up to time 119896 minus 1 is given by 120575x(119896 | 119896 minus 1)When the new set of measurements y(119896) becomes availablethe state estimate is updated to 120575x(119896 | 119896) using the Kalmangain K

119896

120575x (119896 | 119896 minus 1) = A120575x (119896 minus 1 | 119896 minus 1) + B120575u (119896)

120575x (119896 | 119896)

= 120575x (119896 | 119896 minus 1) + K119896[120575y (119896) minus C120575x (119896 | 119896 minus 1)]

(23)

In the sequential estimation problem the Kalman gainneeds to be computed at every iteration These gains areknown to converge to a steady state value if the system isobservable In this application a suboptimal approach wasfollowed using the steady state Kalman gain 119870 instead ofthe time-varying one Simulations show that the proposedcontroller and estimator are able to drive the system to desiredoperating conditions even when uncertainty is present inthe system These simulations were performed using 20 gridpoints when the systemrsquos initial condition was deviated fromthe nominal value in addition to the estimates displaced fromthe actual value of the state variables A plot highlighting theevolution of the some of the states is shown in Figures 11 and12

5 Implementation

Wolfram Mathematica was used to compute the matrices AB C K

119903 and K

119896 However interaction with the pilot-scale

coating linewas achieved through LabViewTheperformanceof the LQG controller was tested in the pilot-scale coatingplant by coating and drying long pieces of Nafion at theconditions discussed in the paper Wrinkle-free coated PEMswere obtained as shown in Figure 13


0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)

Figure 11 States (red) and estimates (blue) at the first drying zonersquos exit

0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)

Figure 12 States (red) and estimates (blue) at the first drying zonersquos mid point

Figure 13 Photograph showing nonideal coating of Nafion (left)next to drying controlled by a model-based LQG (right)

6 Conclusions

The ultimate aim of this work was to develop an automatedcontinuous and low-cost MEA fabrication process for PEMfuel cells A roll-to-roll membrane coating process wasselected for its ability to make the entire process continuousThe ionomeric polymer membrane a key component of aPEM fuel cell has been shown to rapidly absorb water from

the liquid ink during direct coating This rapid absorptionof water results in swelling that deforms the membranewhich in turn causes wrinkling manufacturing defects Thispaper hypothesized that a model-based optimal controlstrategy would be beneficial to the MEA fabrication result-ing in reducing the number of manufacturing defects Thecontroller was implemented in the pilot machine showingpromising results and strongly suggesting further develop-ment of this method

While this work was primarily focused on applicationto Nafion due to its current commercial popularity otherionomeric polymer membranes are available and the perfor-mance of these other ionomeric membranes in this processis currently unknown and should be investigated For costconsiderations the coating formulations used to test thisprocess did not contain any catalyst and future researchshould be done to test the validity of the presented modelswith actual catalyst containing ink

Also in the optimal control strategy that was imple-mented only the drying zonersquos water activity and temperaturewere controlled Controlling additional variables includingline speed and web tensions may yield additional flexibilityand benefit when scaling up this process for industry


Finally the proposed process control strategy can be usedin conjunction with statistical process control techniquesStatistical process control can be used to determine thelikelihood of defect-free membranes based on admissibleerror bands for process parameters The combination ofboth approaches would potentially eliminate the need forpostproduction quality control

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors gratefully acknowledge funding for this workfrom the Office of Naval Research (ONR) through MURIgrant no N00014-07-1-0758

References

[1] K Joon ldquoFuel cellsmdasha 21st century power systemrdquo Journal ofPower Sources vol 71 no 1-2 pp 12ndash18 1998

[2] V Mehta and J S Cooper ldquoReview and analysis of PEM fuelcell design and manufacturingrdquo Journal of Power Sources vol114 no 1 pp 32ndash53 2003

[3] A Lindermeir G Rosenthal U Kunz and U Hoffmann ldquoOnthe question of MEA preparation for DMFCsrdquo Journal of PowerSources vol 129 no 2 pp 180ndash187 2004

[4] Q Mao G Sun S Wang et al ldquoComparative studies ofconfigurations and preparation methods for direct methanolfuel cell electrodesrdquo Electrochimica Acta vol 52 no 24 pp6763ndash6770 2007

[5] B P Ladewig R B Knott A J Hill et al ldquoPhysical and elec-trochemical characterization of nanocomposite membranes ofnafion and functionalized silicon oxiderdquoChemistry ofMaterialsvol 19 no 9 pp 2372ndash2381 2007

[6] B M Koraishy Continuous manufacturing of direct methanolfuel cell membrane electrode assemblies [PhD thesis] TheUniversity of Texas at Austin Austin Tex USA 2010

[7] H Tsuchiya and O Kobayashi ldquoMass production cost of PEMfuel cell by learning curverdquo International Journal of HydrogenEnergy vol 29 no 10 pp 985ndash990 2004

[8] T J Silverman J P Meyers and J J Beaman ldquoDynamicthermal transport andmechanicalmodel of fuel cellmembraneswellingrdquo Fuel Cells vol 11 no 6 pp 875ndash887 2011

[9] Z Liu Z Mao and C Wang ldquoA two dimensional partialflooding model for PEMFCrdquo Journal of Power Sources vol 158no 2 pp 1229ndash1239 2006

[10] T E Springer T A Zawodzinski and S Gottesfeld ldquoPolymerelectrolyte fuel cell modelrdquo Journal of the ElectrochemicalSociety vol 138 no 8 pp 2334ndash2342 1991

[11] S J Paddison and K S Promislow Eds Device and MaterialsModeling in PEM Fuel Cells vol 113 of Topics in Applied PhysicsSpringer 2009

[12] D-C Huang P-J Yu F-J Liu et al ldquoEffect of dispersionsolvent in catalyst ink on proton exchange membrane fuel cellperformancerdquo International Journal of Electrochemical Sciencevol 6 no 7 pp 2551ndash2565 2011

[13] R Solasi Y Zou X Huang K Reifsnider and D ConditldquoOnmechanical behavior and in-planemodeling of constrainedPEM fuel cell membranes subjected to hydration and tempera-ture cyclesrdquo Journal of Power Sources vol 167 no 2 pp 366ndash3772007

[14] R Solasi Y Zou X Huang and K Reifsnider ldquoA time andhydration dependent viscoplastic model for polyelectrolytemembranes in fuel cellsrdquo Mechanics of Time-Dependent Mate-rials vol 12 no 1 pp 15ndash30 2008

[15] I-S Park W Li and A Manthiram ldquoFabrication of catalyst-coatedmembrane-electrode assemblies by doctor blademethodand their performance in fuel cellsrdquo Journal of Power Sourcesvol 195 no 20 pp 7078ndash7082 2010

[16] C H Hsu and C C Wan ldquoAn innovative process for PEMFCelectrodes using the expansion of Nafion filmrdquo Journal of PowerSources vol 115 no 2 pp 268ndash273 2003

[17] DuPont ldquoProduct informationDupont nafionpfsmembranesrdquo2004

[18] V DevarajModeling design development and control of a pilot-scale continuous coating line for proton exchange membranefuel cell electrode assembly [PhD thesis] University of Texas atAustin Austin Tex USA 2012

[19] M B Satterfleld and J B Benziger ldquoNon-Fickian water vaporsorption dynamics by nafion membranesrdquo The Journal ofPhysical Chemistry B vol 112 no 12 pp 3693ndash3704 2008

[20] J P Meyers and J Newman ldquoSimulation of the direct methanolfuel cell II Modeling and data analysis of transport and kineticphenomenardquo Journal of the Electrochemical Society vol 149 no6 pp A718ndashA728 2002

[21] D J Burnett A R Garcia and F Thielmann ldquoMeasuringmoisture sorption and diffusion kinetics on proton exchangemembranes using a gravimetric vapor sorption apparatusrdquoJournal of Power Sources vol 160 no 1 pp 426ndash430 2006

[22] T L Bergman A S Lavine F P Incropera and D P DeWittFundamentals of Heat and Mass Transfer Wiley 7th edition2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Figure 1 Nafionmembrane exhibiting distortion upon contact withliquid ink

the MEA up because of wasted catalyst and membrane bothexpensive components in a DMFC in addition to thwartingthe fabrication of larger area fuel cells

These difficulties in manufacturing have forced the coat-ing of the PEM to be done by hand in small batch processes[7] Small pieces of the membrane are held down with avacuum table and the coating is directly sprayed or brushedon This process is repeated until the desired thickness of thecatalyst layer is achieved

To improve and automate the fabrication of theMEA it iscritical to understand the behavior of themembrane during atypical coating process which includes exposure to transientnonuniform water heat and mechanical stress conditions[8] Many investigators have modeled the membrane in a fuelcell operating environment [9ndash11] but there are fewermodelsthat attempt to model the membrane in a manufacturingenvironment Among the existing models many are steady-state models (not suitable for process control) and most ofthem ignore the relationship between swelling and the mem-branersquos water content In addition water transport within themembrane is often ignored as well [12ndash14] Silverman et al[8] was one of the first researchers that attempted a transientnonuniform model describing water transport stress andswelling in a membrane in a manufacturing environment

The contribution of this paper is two-fold First itdescribes a pilot-scale continuous coating line for Nafionmembranes that was designed for controlled water absorp-tion and desorption Humidity and temperature are con-trolled throughout the process to prevent wrinkles Secondit presents a low-order model of the swelling and drying phe-nomena and its application in a linear-quadratic-Gaussian(LQG) controller to prevent swelling defects This approachproved effective in preventing wrinkles and holds potentialsfor application in large-scale manufacturing as discussed atthe end of this article

2 Pilot-Scale Membrane Coating Line

A pilot-scale coating line was designed to accept rolls ofuncoated Nafion advance them to a preswelling section to acoating section and finally to a drying section in a continuousfashionThemotivation for this designwas that a significantlylower number of coating defects have been observed whenthe PEM is preswelled or saturatedwithwater prior to coating[15 16] Once the membrane is fully saturated with water it

cannot absorb any more and its water content stays constantPreswelling by immersion in water is used in this applicationfor rapid diffusion into the membrane

The manufacturerrsquos recommended storage conditions forNafion are 50 relative humidity at 23∘C It is desiredthat the coated membrane is brought to these temperatureand humidity conditions before it is stored [17] Howeverif the ambient temperature and humidity were set at therecommended storage conditions it would require a coatingline that would have to be either extremely long or slow

Previous work concluded that during the drying of thefreshly coated saturated membrane the coating rehydratesthe membrane keeping it fully hydrated as long as water ispresent in coating Even when using some potentially harshdrying conditions to remove water from the freshly coatedmembrane no undesirable effects have been reported in themembranersquos shape as it remains hydrated by the wet coating[8]This result suggests that it is possible to employ two-stagedrying where a first stage rapidly removes the water from thecoating and a second stage that brings the membrane to therecommended storage conditions

Previous research on two-stage drying has shown thatcontrolling the time spent in a hotdry zone and the timing ofthe transition from the hot and dry zone to a second coolerzone is extremely important Transitioning too early resultsin the coating not being completely dry while transitioningtoo late would mean that the membrane is over-dried [8]

A pilot-scale membrane coating line for producingcatalyst-coated membrane in a continuous fashion wasdesigned constructed and tested The pilot plant wasdesigned to accommodate Nafion ionomeric membraneavailable in 10 cm wide roll form An overview of the pro-posed pilot scale membrane coating line is shown in Figure 2It should be noted that the Nafion web moves in a clock-wise direction through the machine In this machine bothuncoated and coated rolls of Nafion are stored in the leftmostchamber

This machine has the capability to change independentlythe unwind and rewind tensions in thewebThere is a tractionroller that friction-feeds the membrane web and advances itthrough the machine at a set velocity The membrane web issupported throughout the machine by aluminum idle rollersIn addition to this the machine has a preswelling sectionand seven independent temperature-humidity controllabledrying chambers Although a doctor blade is employed tocoat ink it can be changed to any other method of inkapplication in future designs Since this line deals withaqueous swelling and wet coating of ionomeric membranesmost of the critical components were procured and built towithstand prolonged moisture exposure

21 Uncoated and Coated Membrane Storage Since it isbest to store the coated and uncoated Nafion rolls at therecommended storage conditions it is essential to havea start and end block that is temperature and humiditycontrolled as shown in Figure 3 This can prevent damageto the roll when left in the machine for extended periodsof time A stand-alone PID controller is used to control


18m1

m

(a)

(b)






















(a)

(b)




0 is


0is

derived as

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905= minus

119860

119881

120572119888(1198880minus 1198880eq) (1)




119888is the mass



with 1198880= 11989801205881198981+120576V and 120576V = 119898





120588119888119901

d119879d119905

= minus

119860

119881







119888





0119888is given by

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

d1198980119888

d119905= minus

119860119888

119881119888

120572coat (1198880119888 minus 1198880119888surr) (3)












120572coat =0037D

0air

119871119888

(

119871119888V

]119886

)

45

(

]119886

D0air

)

13

(4)

where D0air and ]



119888



119888

120588coat119888119901coat119881119888d119879119888

d119905


119891119892120572coat (1198880119888 minus 119888

0119888surr)]

(5)



119891119892



0(120588119904+




ℎcoat =0037119896

119886

119871119888

(

119871119888V

]119886

)

45

(

119888119901119886120583119886

119896119886

)

13

(6)



119888and



0and 119879 as





119881119888

1205882

0

120588119904

120588119904+11987201205881199041198980119888

d1198980119888

d119905

= minus119860119888[120572coat (1198880119888 minus 119888

0119888surr) + 120572119888(1198880minus 1198880eq)]

(7)


119881eff120588eff119888119901effd119879d119905


0119888surr)

+119902st120572119888 (1198880 minus 1198880eq)]

(8)









119881

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905

= minus119860 [119865120572119888(1198880minus 1198880eq) + 120572

119888(1198880minus 1198880eq)]

(9)

Coating c0c (t)




(a) Mass transfer

Coating


Membrane


T(t)


(b) Heat transfer






120588eff119888119901effd119879d119905

= minus

119860eff119881eff


+119902st120572119888 (1198880 minus 1198880eq)]

(10)



Coating

c0 (t)Membrane



(a) Mass transfer

Coating

Membrane


T(t)


(b) Heat transfer








1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff


0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)


(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)



1being




x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










18m1

m

(a)

(b)






















(a)

(b)




0 is


0is

derived as

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905= minus

119860

119881

120572119888(1198880minus 1198880eq) (1)




119888is the mass



with 1198880= 11989801205881198981+120576V and 120576V = 119898





120588119888119901

d119879d119905

= minus

119860

119881







119888





0119888is given by

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

d1198980119888

d119905= minus

119860119888

119881119888

120572coat (1198880119888 minus 1198880119888surr) (3)












120572coat =0037D

0air

119871119888

(

119871119888V

]119886

)

45

(

]119886

D0air

)

13

(4)

where D0air and ]



119888



119888

120588coat119888119901coat119881119888d119879119888

d119905


119891119892120572coat (1198880119888 minus 119888

0119888surr)]

(5)



119891119892



0(120588119904+




ℎcoat =0037119896

119886

119871119888

(

119871119888V

]119886

)

45

(

119888119901119886120583119886

119896119886

)

13

(6)



119888and



0and 119879 as





119881119888

1205882

0

120588119904

120588119904+11987201205881199041198980119888

d1198980119888

d119905

= minus119860119888[120572coat (1198880119888 minus 119888

0119888surr) + 120572119888(1198880minus 1198880eq)]

(7)


119881eff120588eff119888119901effd119879d119905


0119888surr)

+119902st120572119888 (1198880 minus 1198880eq)]

(8)









119881

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905

= minus119860 [119865120572119888(1198880minus 1198880eq) + 120572

119888(1198880minus 1198880eq)]

(9)

Coating c0c (t)




(a) Mass transfer

Coating


Membrane


T(t)


(b) Heat transfer






120588eff119888119901effd119879d119905

= minus

119860eff119881eff


+119902st120572119888 (1198880 minus 1198880eq)]

(10)



Coating

c0 (t)Membrane



(a) Mass transfer

Coating

Membrane


T(t)


(b) Heat transfer








1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff


0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)


(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)



1being




x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















(a)

(b)




0 is


0is

derived as

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905= minus

119860

119881

120572119888(1198880minus 1198880eq) (1)




119888is the mass



with 1198880= 11989801205881198981+120576V and 120576V = 119898





120588119888119901

d119879d119905

= minus

119860

119881







119888





0119888is given by

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

d1198980119888

d119905= minus

119860119888

119881119888

120572coat (1198880119888 minus 1198880119888surr) (3)












120572coat =0037D

0air

119871119888

(

119871119888V

]119886

)

45

(

]119886

D0air

)

13

(4)

where D0air and ]



119888



119888

120588coat119888119901coat119881119888d119879119888

d119905


119891119892120572coat (1198880119888 minus 119888

0119888surr)]

(5)



119891119892



0(120588119904+




ℎcoat =0037119896

119886

119871119888

(

119871119888V

]119886

)

45

(

119888119901119886120583119886

119896119886

)

13

(6)



119888and



0and 119879 as





119881119888

1205882

0

120588119904

120588119904+11987201205881199041198980119888

d1198980119888

d119905

= minus119860119888[120572coat (1198880119888 minus 119888

0119888surr) + 120572119888(1198880minus 1198880eq)]

(7)


119881eff120588eff119888119901effd119879d119905


0119888surr)

+119902st120572119888 (1198880 minus 1198880eq)]

(8)









119881

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905

= minus119860 [119865120572119888(1198880minus 1198880eq) + 120572

119888(1198880minus 1198880eq)]

(9)

Coating c0c (t)




(a) Mass transfer

Coating


Membrane


T(t)


(b) Heat transfer






120588eff119888119901effd119879d119905

= minus

119860eff119881eff


+119902st120572119888 (1198880 minus 1198880eq)]

(10)



Coating

c0 (t)Membrane



(a) Mass transfer

Coating

Membrane


T(t)


(b) Heat transfer








1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff


0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)


(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)



1being




x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119888





0119888is given by

1205882

0

120588119904

1205880+11987201205881199041198982

0119888

d1198980119888

d119905= minus

119860119888

119881119888

120572coat (1198880119888 minus 1198880119888surr) (3)












120572coat =0037D

0air

119871119888

(

119871119888V

]119886

)

45

(

]119886

D0air

)

13

(4)

where D0air and ]



119888



119888

120588coat119888119901coat119881119888d119879119888

d119905


119891119892120572coat (1198880119888 minus 119888

0119888surr)]

(5)



119891119892



0(120588119904+




ℎcoat =0037119896

119886

119871119888

(

119871119888V

]119886

)

45

(

119888119901119886120583119886

119896119886

)

13

(6)



119888and



0and 119879 as





119881119888

1205882

0

120588119904

120588119904+11987201205881199041198980119888

d1198980119888

d119905

= minus119860119888[120572coat (1198880119888 minus 119888

0119888surr) + 120572119888(1198880minus 1198880eq)]

(7)


119881eff120588eff119888119901effd119879d119905


0119888surr)

+119902st120572119888 (1198880 minus 1198880eq)]

(8)









119881

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905

= minus119860 [119865120572119888(1198880minus 1198880eq) + 120572

119888(1198880minus 1198880eq)]

(9)

Coating c0c (t)




(a) Mass transfer

Coating


Membrane


T(t)


(b) Heat transfer






120588eff119888119901effd119879d119905

= minus

119860eff119881eff


+119902st120572119888 (1198880 minus 1198880eq)]

(10)



Coating

c0 (t)Membrane



(a) Mass transfer

Coating

Membrane


T(t)


(b) Heat transfer








1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff


0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)


(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)



1being




x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119881119888

1205882

0

120588119904

120588119904+11987201205881199041198980119888

d1198980119888

d119905

= minus119860119888[120572coat (1198880119888 minus 119888

0119888surr) + 120572119888(1198880minus 1198880eq)]

(7)


119881eff120588eff119888119901effd119879d119905


0119888surr)

+119902st120572119888 (1198880 minus 1198880eq)]

(8)









119881

1205882

0

120588119898

1205880+11987201205881198981198982

0

d1198980

d119905

= minus119860 [119865120572119888(1198880minus 1198880eq) + 120572

119888(1198880minus 1198880eq)]

(9)

Coating c0c (t)




(a) Mass transfer

Coating


Membrane


T(t)


(b) Heat transfer






120588eff119888119901effd119879d119905

= minus

119860eff119881eff


+119902st120572119888 (1198880 minus 1198880eq)]

(10)



Coating

c0 (t)Membrane



(a) Mass transfer

Coating

Membrane


T(t)


(b) Heat transfer








1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff


0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)


(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)



1being




x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Coating

c0 (t)Membrane



(a) Mass transfer

Coating

Membrane


T(t)


(b) Heat transfer








1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

1205971198980119888

120597119905

+ V119909

1205971198980119888

120597119909

]

= minus

119860119888

119881119888

[120572coat (1198880119888 minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

120588eff119888119901eff [120597119879

120597119905

+ V119909

120597119879

120597119909

]

= minus

119860eff119881eff


0119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

(11)


(

d1198980119888

d119909)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

(

d119879d119909

)

10038161003816100381610038161003816100381610038161003816119909=119894

asymp

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

(12)



1being




x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











x = 1 x = 2 x = 3 x = n minus 1 x = n

l1



1205882

0

120588119904

1205880+11987201205881199041198982

0119888

[

d1198980119888|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199091

]

= minus

119860119888

119881119888

[120572coat ( 11988801198881003816100381610038161003816119909=119894

minus 1198880119888surr1) + 120572

119888(1198880minus 1198880eq1)]

119894 = 2 3 119899

(13)


120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199091

]

= minus

119860eff119881eff

[2ℎ (119879|119909=119894

+ 119879surr1) + ℎ119891119892120572coat ( 1198880119888

1003816100381610038161003816119909=119894

minus 1198880119888surr1)

+119902st120572119888 (1198880 minus 1198880eq1)]

119894 = 2 3 119899

(14)


0119888|119909=1

and 119879|119909=1






l2




d1198860surr1


0sp1)

d119879surr1d119905


(15)




1205882

0

120588119898

1205880+11987201205881198981198982

0

[

d1198980|119909=119894

d119905+ V119909

1198980119888|119909=119894

minus 1198980119888|119909=119894minus1

Δ1199092

]

= minus

119860

119881

[119865120572119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2) + 120572

119888(1198880

1003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

120588eff119888119901eff [d119879|119909=119894

d119905+ V119909

119879|119909=119894

minus 119879|119909=119894minus1

Δ1199092

]

=

119860eff119881eff

[2ℎ (119879|119909=119894

minus 119879surr2) + 119865119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)

+119902st120572119888 (11988801003816100381610038161003816119909=119894

minus 1198880eq2)]

119894 = 119899 + 1 119899 + 2 119899 + 119898

(16)


2= 1198972119898 where 119897

2is the







0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0119888|119909=119899




x =

[[[[[[[[[[[[[

[

1198980119888

1003816100381610038161003816119909=2

1198980119888

1003816100381610038161003816119909=119899

119879|119909=2

119879|119909=119899

1198860surr1119879surr1

]]]]]]]]]]]]]

]

u = [

1198860sp1119879sp1

] y =[[[

[

1198980119888

1003816100381610038161003816119909=119899

119879|119909=119899

1198860surr1119879surr1

]]]

]

(17)


0119888|119909=119899





120575x (119896 + 1) = A120575x (119896) + B120575u (119896)

120575y (119896) = C120575x (119896) (18)



J =infin

sum

119896=1

120575y119879 (119896)Q119903120575y (119896) + 120575u119879 (119896)R

119903120575u (119896) (19)


u (119896) = u0minus K119903120575x (119896) (20)




119903andR

119903





120575x (119896 + 1) = A120575x (119896) + B120575u (119896) + w (119896)

120575y (119896) = C120575x (119896) + k (119896) (21)


119896



E [w (119896)w (119896 + 119892)119879

] =

Q119896 if 119892 = 0

0 otherwise

E [k (119896) k (119896 + 119892)119879

] =

R119896 if 119892 = 0

0 otherwise

(22)


119896


120575x (119896 | 119896)


(23)


5 Implementation


119903 and K




0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 100 200 300 400 5003195

3200

3205

3210

3215

3220

3225

Time (s)

Tx=20(K

)

(a)

0

10

20

30

40

50

0 100 200 300 400 500

Time (s)

m0cx=20

(b)


0 100 200 300 400 500

Time (s)

Tx=10(K

)

3035

3040

3045

3050

3055

(a)

0 100 200 300 400 500

Time (s)

90

91

92

93

m0cx=10

(b)



6 Conclusions









Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgment


References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article model-based control of a continuous...

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