MODELING AND SIMULATION OF A FUEL CELL REFORMER FOR CONTROL APPLICATIONS

Transcription

1 MODELIG AD SIMULAIO OF A FUEL CELL REFORMER FOR COROL APPLICAIOS By MOHUA AH A HESIS PRESEED O HE GRADUAE SCHOOL OF HE UIVERSI OF FLORIDA I PARIAL FULFILLME OF HE REQUIREMES FOR HE DEGREE OF MASER OF SCIECE UIVERSI OF FLORIDA 7 1

2 7 Mohua ath

3 o my paents fo thei love, sacifice and steadfast suppot 3

4 ACKOWLEDGMES he completion of this wok would not have been possible without the immense contibution of my committee, D. William Lea, D. Osca Cisalle and D. James Fletche. hey have guided, advised and encouaged me with a lot of patience and suppoted me in evey step duing my mastes pogam. I would like to take this oppotunity to expess my deepest gatitude to them. I would like to thank the most impotant people in my life, my paents and siste whose faith in me have bought me hee. I would also like to mention my fiends Alpana Agawal, Jaya Das, Gauav Malhota, adeem Islam and Champak Das fo being like a family to me and ceating a home away fom home. A special mention goes to Daniel Betts fo shaing his knowledge of fuel cells with me and poviding me with valuable suggestions. Last but not the least; I would like to thank Rana Dutta fo motivating me to make this last leap possible. ou ae the light at the end of the tunnel. 4

5 ABLE OF COES ACKOWLEDGMES...4 LIS OF ABLES...7 LIS OF FIGURES...8 ABSRAC...9 CHAPER 1 IRODUCIO...11 page 1.1 Fuel Cells Phosphoic Acid Fuel Cells Polyme Electolyte Fuel Cells (PEMFC Alkaline Fuel Cells (AFC Molten Cabonate Fuel Cells (MCFC Solid Oxide Fuel Cells (SOFC Hydocabons as Indiect Fuel Fuel Refoming Autothemal Refoming (AR Patial Oxidation Refoming Steam Refoming...18 BACKGROUD AD LIERAURE REVIEW Refome Liteatue eview Langmui Hinshelwood Model akagaki Coelation MODELIG OF A PACKED BED REFORMER Oveview Backgound of hemal Model Methanol Steam Fuel Cell Refome Geneal Desciption of the Refome Patial Diffeential Equation Finite Diffeence Method Discetization Initial and Bounday Conditions Diichlet s Bounday Conditions eumann s Bounday Conditions Initial conditions

6 eatment of an undetemined bounday condition Special Case of Fictitious odes at eumann s Bounday Conditions Model Validation on-dimensional Analysis RESULS AD DISCUSSIO COCLUSIOS...6 APPEDIX A MALAB CODES...63 B ES BED BUS- COROL LOGIC...8 C DESCRIPIO OF COROL SCHEME...91 LIS OF REFERECES...18 BIOGRAPHICAL SKECH

7 LIS OF ABLES able page 4.1 Constants used fo finding solution to the efome model

8 LIS OF FIGURES Figue page.1 Genealized Fuel Cell Schematic Genealized efome schematic Discetization along length and adius of the efome Discetization along length and adius of the efome and along time incements Fictitious nodes empeatue pofile of efome afte 3 mins empeatue plots at mid-adius of efome afte 3 minutes empeatue pofile accoding to analytical solution Refome tempeatue with zeo bounday conditions (umeical solution empeatue pofile of mid-adius accoding to Analytical Solution ansient analytical solution fo axial tempeatue gadient ansient numeical solution fo axial tempeatue gadient Analytical heat tansfe solution fo tempeatue as a function of axial position and time at adial distance =.75R umeical solution at diffeent location along x-axis at =.75 m. he tempeatue pofiles ae shown at diffeent time instances on dimensional tempeatue pofile using analytical method Compaison of tempeatue pofile of mid-adius accoding to umeical Solution with and without heat geneation empeatue pofile of efome afte 3 mins (with heat geneation Compaison of tempeatue pofile of mid-adius accoding to numeical solution with and without convection

9 Abstact of hesis Pesented to the Gaduate School of the Univesity of Floida in Patial Fulfillment of the Requiements fo the Maste of Science MODELIG AD SIMULAIO OF A FUEL CELL REFORMER FOR COROL APPLICAIOS Chai: William Lea Cochai: Osca Cisalle Majo: Mechanical Engineeing By Mohua ath Decembe 7 he limited success in the hydogen stoage and distibution technology has diven the need fo the development of an effective fuel pocesso. he dynamic pefomance of a efome is of citical impotance fo the successful commecialization of hydogen as fuel fo stationay and tanspotation applications. he efoming technology is of paticula inteest to utilities that equie a clean and efficient method of geneating electicity fom fuel cells. As an effot to achieve a bette contol of the fuel pocesso paametes, a dynamic model of a genealized efome is built. he model successfully pedicts the tansient tempeatue gadient acoss a efome catalyst bed and the efomate exit tempeatue and is capable of pedicting the esponse of the efome to distubances and load fluctuations. he eaction and heat tansfe in the catalyst packed bed was analyzed numeically using a genealized physical model. hese esults povide valuable insight into the tansient esponse of a efome. he numeical output was compaed with analytical esults which ageed well with each othe. his confimed the validity of the numeical method. Also as a pat of this eseach, the contol logic of a methanol poweed fuel cell bus was studied. he oveall contol scheme shows that the catalyst bed tempeatue plays an impotant 9

10 ole detemining the fuel flow into the efome. he successful pediction of efome paametes can thus be utilized to eventually design a efome capable of quick stats and faste tansient esponse. 1

11 CHAPER 1 IRODUCIO he gowing wold economy and the limited esouces of nonenewable fuels emphasize the need fo aggessive development of altenative fuels. Advanced powe geneation technology utilizing altenative fuels can become a facto in educing emission of geenhouse gases, impoving uban ai quality and educing dependency on foeign oil. Hydogen can be used in fuel cell which is a pomising technology, poviding efficient and eliable souce of enegy fo a wide ange of applications [1,6]. Although hydogen is abundantly found in natue, extaction of hydogen fom its compounds emains a challenge befoe it can be commecially viable as a fuel. Hydogen can be used fo multiple applications, anging fom powe geneation to tanspotation applications, o intenal combustion engines to fuels cells. Refoming hydogen fom any hydocabon caying fuel such as natual gas, biomass, coal o ammonia povides an attactive solution fo hydogen stoage fo potable applications [1,]. he paticula case of efoming to poduce hydogen fo use in fuel cells fo tanspotation applications is selected hee fo specification of geometic and themodynamic paametes to be used in this study. 1.1 Fuel Cells Fuel cells ae electochemical devices that poduce diect cuent electical enegy fom chemical enegy. Fuel and an oxidant ae continually supplied to the fuel cell fo the eaction to take place []. he main components of a fuel cell ae an electolyte, catalyst and a poous anode and cathode. In the pesence of a catalyst, the fuel, paticulaly hydogen, splits into a poton and an electon. he electons flow though an extenal cicuit geneating electicity and theeafte combines with potons and oxidants to fom by-poducts at the cathode. Oxygen, acting as an 11

12 oxidizing agent in this eaction, combines with the potons and electons at the cathode to fom a molecule of wate. he oveall eaction is as follows: 1 H + O H O 1-1 hee ae many types of fuels cells cuently unde investigation, including phosphoic acid fuel cells (PAFC, polyme electolyte fuel cells (PEMFC, alkaline fuel cells (AFC, molten cabonate fuel cells (MCFC and solid oxide fuel cells (SOFC. he classification of fuel cells is made based on the type of electolyte. Among these, the PAFCs and PEMFCs hold the most potential fo use as an altenative to intenal combustion engines fo tanspotation applications [1] Phosphoic Acid Fuel Cells Phosphoic acid fuel cells wee the fist type to be commecially investigated, othe than fo the U.S. space pogam. he electolyte is phosphoic acid, usually contained in a silicon cabide matix, and the electodes made of eflon -bonded platinum o poous polytetadluooethylene (PFE - bonded cabon, which is a polymeic binde used to hold the cabon black paticles togethe [1, 7, 5]. In the pesence of a catalyst, usually platinum, the positively chaged hydogen ions migate towads the cathode. Electons geneated at the anode tavel though an extenal cicuit towads the cathode, thus ceating electic cuent. he hydogen ions and electons combine to fom wate which is a by-poduct of this electochemical pocess. he phosphoic acid fuel cells opeate optimally between 15 C to C, as at lowe tempeatues, cabon monoxide poisoning of the anode may occu. PAFC stacks need a heat sink duing opeation, usually a coolant, eithe liquid o gas, that passes though channels integal to the membane electode assembly [1, ]. 1

13 he chemical eactions ae as below: Anode: H H + + e 1 Catode: O H + e + + H O 1 Oveall: H + O H O he PAFC stack opeates at efficiencies between 37% and 4% and since heat is a bypoduct of the electochemical pocess, the oveall efficiency of a combined pocess can each 8%. PAFCs can toleate a.5% concentation of cabon monoxide and ae minimally affected by the pesence of cabon dioxide Polyme Electolyte Fuel Cells (PEMFC Poton Exchange Membane Fuel Cells ae consideed most suitable fo tanspotation applications. A PEMFC is based on a solid polyme membane which may be a thin plastic film of sulphonated fluo-polymes that act as an electolyte [1]. he two poous electodes on eithe side of the membane ae made hydophobic by coating with a compound like PFE, thus helping the eactants to diffuse into a platinum laye that acts as a catalyst. he hydogen ions diffuse though the membane towads the cathode and the electons flow though an extenal cicuit thus poducing electic cuent. Oxygen acts as an oxidizing agent and combines with the electons and hydogen ions and foms a by-poduct of wate. he chemical eactions ae as below: Anode: H 4H + + 4e + Catode: O + 4H + 4e H O Oveall: H + O H O PEMFCs have highe volumetic enegy density than othe types of fuel cells, thus making them compact and suitable fo vehicula applications. he optimal opeating tempeatue is aound 8 C, which allows quick stat-up and faste tansient esponse. Recent developments 13

14 howeve have elevated the opeating tempeatue of PEMFCs beyond 15 C to educe the effect of CO poisoning, simplify wate and themal management and to ecove high value heat. Othe advantages ae due to the fact that the electolyte is solid theefoe thee is no spillage and coosion thus contibuting to its longe shelf life [5, 7]. One of the majo disadvantages of the PEMFC is that the membane is equied to be continually hydated to opeate optimally, thus wate management becomes a citical issue. PEMFCs use platinum as a catalyst, which has vey low toleance to CO poisoning. hey can opeate unde a maximum of 1 ppm of CO, thus equiing a clean efomate gas to be used as a fuel. Also, inceasing the fuel cell tempeatue beyond 1 C can vapoize the wate in the electolyte which is essential fo the conduction of ions, thus equiing tight contol of fuel cell tempeatue and pessue [, 7] Alkaline Fuel Cells (AFC Alkaline fuel cells use a wate-based electolyte solution of potassium hydoxide (KOH with a concentation that can vay accoding to the fuel cell opeating tempeatue, which could be fom 65 C to C [, 7]. he hydoxyl ion (OH - acts as the chage caie as in all othe fuel cell types, Howeve, the wate fomed at the anode tavels towads the cathode and egeneates to hydoxyl ions. heefoe, in this type of fuel cell the by-poduct is only heat. he chemical eaction is shown below: Anode: H + 4 OH - 4 H O + 4 e - Cathode: O + H O + 4 e - 4 OH - Oveall: H + O H O he main disadvantage of AFCs is that hydogen and oxygen have to be supplied to the cell with negligible concentations of CO, CO o CH 4 as it can poison the electolyte. his equiement makes it difficult to be used fo tanspotation applications. he advantage is that 14

15 they opeate on compaatively lowe tempeatues does not equie noble metals and have vey high efficiency Molten Cabonate Fuel Cells (MCFC hese cells use a mixtue of molten salt cabonates as an electolyte which is contained in a poous, inet matix made of ceamic. he mixtue is usually of vaied pecentages of lithium cabonate and potassium cabonate. hese fuel cells nomally opeate at a tempeatue of aound 65 C. he high opeating tempeatue indicates that these fuel cells can opeate diectly on gaseous hydocabon fuels [1, ]. At the high tempeatue of 65ºC, the alkali melt and become conductive to cabonate ions (CO - 3 which tavel towads the anode. he ions flow fom the cathode to the anode whee they combine with hydogen to poduce wate, cabon dioxide and electons. hus the by-poducts ae cabon dioxide, wate and heat. he chemical eactions ae shown below: Anode: CO H H O + CO + e - Cathode: CO + 1/O + e - CO 3 Oveall: H + 1/O + CO H O + CO he main advantage is that at the highe tempeatue, fuel efoming can take place inside the fuel cell stack itself, thus eliminating the need of an extenal efome. he disadvantage of these type of fuel cells is the time equied to obtain the high tempeatue which geatly slows the stat-up pocess and makes the esponse sluggish [7] Solid Oxide Fuel Cells (SOFC he Solid Oxide Fuel Cell (SOFC opeates unde the highest tempeatue conditions, anging fom 6ºC to 1ºC; thus it can opeate with a numbe of diffeent fuels. he electolyte is a thin, solid ceamic mateial which conducts the chage caying oxygen ions. he 15

16 efficiency of the fuel cells ae aound 6% which is the highest among the fuel cell types. he by-poduct of steam can be utilized fo vaious puposes. he chemical eaction is as follows: Anode: H + O H O + 4 e - Cathode: O + 4 e - O -- Oveall: H + O H O he main application of SOFCs is lage-scale industial systems whee the demand is highe powe, and long stat-up times only minimally affect the pefomance o system equiements. Simila to MCFCs, the high tempeatue of the solid oxide fuel cells make them capable of opeating on impue fuels and efoming occus inside the fuel cell itself. SOFCs ae being developed moe than MCFCs because they have highe efficiency and ability to opeate unde highe tempeatues [, 7]. 1. Hydocabons as Indiect Fuel Hydogen acts as an enegy caie; howeve it is not a souce of enegy. hus thee is a need fo conveting othe souces of enegy to hydogen befoe it can be used diectly in a fuel cell. Electolysis of wate is a well known method of hydogen poduction, but it cannot be used fo commecial poduction of hydogen as it would be moe economical to use the electicity diectly as an enegy souce than to use it to poduce an enegy caie in the fom of hydogen. he othe method commonly used to geneate hydogen is by the efoming of hydocabons. Hydocabons in liquid fom ae easie to cay on-boad and thus can be used fo tanspotation applications. he most commonly-used pocess to poduce hydogen is the steam efoming eaction which is to eact hydocabons with wate at a high tempeatue. o impove the hydogenpoduction efficiency and emove impuities, a wate-gas shift eaction usually takes place afte 16

17 the steam efoming eaction. In this pocess, cabon monoxide poduced fom steam efoming eaction can be utilized to futhe beak down wate into hydogen. 1.3 Fuel Refoming Autothemal Refoming Among all the methods that poduce hydogen, autothemal efoming eaction is consideed to be one of the most effective pocesses as it allows faste stat-up and esponse time. AR uses liquid hydocabons as fuel that undegoes a eaction with steam o ai in a single eacto. While opeating in ideal condition, with the optimal amount of ai, fuel and steam, the eaction s efficiency can each up to 93.9%. he AR pocess is also capable of using hydocabons such as gasoline and diesel, which can make it moe commecially viable. One of the moe ecent developments is the possible eduction of opeating tempeatues fom 1, C to 65-9 C by educing oxygen to cabon atios. he main advantage of educing the opeating tempeatue is that it allows fo simple eacto design, lowes cost in tems of mateial complexity and equies less fuel duing statup conditions to pe-heat the eacto [8]. he basic eaction is as follows: CH O H + CO CH + H O 3H + CO Patial Oxidation Refoming Patial oxidation eaction is not as fequently used fo commecial puposes as the othe types. It is an exothemic eaction whee a hydocabon eacts with contolled amount of oxygen. he advantages of this eaction ae that it has a fast esponse time, high efficiency and does not equie a catalyst. In addition, the bypoduct of heat can be tansfeed via a heat exchange fo 17

18 othe applications. he disadvantages of this pocess ae that it equies a high opeating tempeatue and a high fuel /ai atio fo the combustion. he oveall eaction is as below: CH4+ 1 O H + CO Steam Refoming he majoity of hydogen poduced commecially is fom steam efoming eaction. his eaction combines steam with hydocabon feedstock in a high tempeatue and pessue eacto. his is an endothemic eaction, thus equiing an extenal souce of to maintain the tempeatue in the eacto. Geneally, a nickel catalyst bed is used to speed up the eaction and incease the efficiency of the pocess. he advantages of this eaction ae that it can achieve efficiency as high as 85% with heat ecovey and can achieve the eaction efficiency of 8% [1]. he main dawbacks of this pocess ae its plant size and slow statup. he oveall eaction is as below: CH + H O 3H + CO 4 18

19 CHAPER BACKGROUD AD LIERAURE REVIEW.1 Refome he development of a fuel efome and its optimization may hasten the advent of widespead fuel cell deployment. he limited success so fa, in the establishment of an infastuctue fo hydogen supply, can be ovecome by the altenative solution of efoming hydogen ich fuels on-boad. Reseach on stoage methods of hydogen by physical o chemical means fo fuel cell based vehicles has not yet povided a satisfactoy pactical solution, thus futhe suppoting the need of efoming a hydocabon fuel such as methanol. he fuel used fo efoming may vay accoding to the application. Wheeas methanol, gasoline o ethanol may be a pefeed fuel fo tanspotation applications, natual gas o popane may have advantages fo stationay applications. [1]. A efome system typically consists of a pemix tank, peheate, a eacto, a gas peteatment unit and a bune. A mixtue of liquid hydocabon and wate is fed fom the pemix tank to the peheate whee it is vapoized and supeheated befoe it is sent to the eacto. he enegy fo the endothemic eaction is supplied by the catalytic bune which heats up the eacto to a peset tempeatue. he efomed gas is teated in the gas pe-teatment unit to emove impuities like cabon monoxide and to futhe adapt it to meet the fuel cell equiements. he paticula type of efoming taken as an example in this study is a catalytic steam pocess. It utilizes catalytic steam efoming to pocess the fuel mixtue of methanol and wate into a hydogen ich gas. he powe geneation is accomplished by the oxidation of hydogen in the fuel cell stack. he depleted fuel mixtue is combusted in the steam efome bune befoe is the poducts ae finally libeated into the atmosphee. 19

20 Seveal engineeing issues hinde the commecialization of fuel cell-efome systems fo vaious applications. One of the most impotant issues is that the steam efome dynamic esponse is consideably slowe than that of the fuel cell stack. he esponse of the efome is limited by heat tansfe ates between the bune gases and the catalyst bed. he esponse can be defined as a coesponding incease in hydogen flow ate coesponding to a simila step change in the load at the fuel cell stack. A quick change in heat tansfe ates is equied to meet the vaiation of powe levels. In addition, the tempeatue of the catalyst bed should emain constant at design tempeatue conditions. he incease of catalyst bed tempeatue fom its design conditions can cause the pemanent degadation of the catalyst. hus an optimized efome design and obust contol scheme fo the coesponding efome is to be developed fo a quick and efficient dynamic esponse of the system. he pesent wok is to povide a foundation fo such a development. ighte contol of the efome is equied in ode fo the fuel cell pefomance to follow the load changing conditions of the vehicle. Contol of the efome has to be achieved by keeping some vaiables as close to the set point as possible, mostly the catalyst bed tempeatue, while changing the efomate flow in accodance with the demand at the fuel cell stack which is the load-following geneation of electic powe. his can be achieved by vaying the tempeatue and flow of exit gases fom the bune which is the souce of heat fo the efome endothemic eaction. Figue.1 shows the geneal schematic of a fuel cell system with an on-boad methanol efome, whee the oveall output, in the fom of electicity can be stoed in batteies o can be used to dive the powe-tain of a vehicle.

21 he example system consideed hee fo the sake of this study is a 5kW, 3ft Fuel Cell Powe est Bed Bus, often efeed as BB, at the Fuel Cell Reseach Laboatoy at the Univesity of Floida [9]. A 175 cell PACF stack is used in the BB. Ai at slightly above ambient pessue is supplied to the cathode to povide oxygen. Hydogen ich gas, i.e. efomate, is geneated by an onboad fuel pocesso which convets methanol and wate to H, CO and CO. Duing nomal opeating conditions, the fuel cell stack is povided excess efomate to ensue sufficient hydogen is available to eact at the electodes. Appoximately 8% of the hydogen in the efomate is consumed and the emaining % is supplied to the efome bune poviding heat to continue with the endothemic efoming eaction. Fo a pe-mix of methanol and wate as eactants, the endothemic enegy equiement is aound 131 kj/mole at 98 C. Methanol is initially deliveed by a pump to fuel the stat-up bune, which in tun povides heat to bing the fuel cell stack and efome to an initial opeating tempeatue. Once the setpoint tempeatue is eached in the efome, the pemix fuel is pumped into the efome whee the steam efoming eaction begins to take place. In this type of indiect methanol system, the fuel flow into the efome is vaied accoding to the load demands at the fuel cell stack output. In the fist pat of the eaction inside the efome, the hydocabon, in this case methanol, undegoes a cacking eaction to decompose into cabon monoxide and hydogen. his is an endothemic eaction whee enegy is continuously absobed fom the suoundings. As the eactants that ente the efome ae aleady pe-heated, thee is an initial dop in the eactant tempeatues, afte which the heat equied to maintain the eaction is povided solely by the hot gases buned at the catalytic bune. Cabon monoxide poduced by the pocess of steam efoming may poison the noble metal catalysts in the fuel cell stacks. o educe the concentation of cabon monoxide in the efomate, a wate gas shift eaction is used to educe 1

22 the cabon monoxide concentation and additionally incease the hydogen yield. he oveall eaction is as follows: CH OH CO+ (Cacking eaction (-1 3 H CO + H + (Shift eaction (- O CO H his pocess poduces hydogen ich efomate which is sent diectly to the anode of the fuel cell to be oxidized. he unused hydogen-containing gas, called the flue gas, is etuned via a feedback loop to a bune that buns the emaining hydogen, poviding the heat equied fo the endothemic efome eaction. In case of inceasing powe demands fom the fuel cell stack, the efome must flow moe eactants into the chambe and poduce moe hydogen. his esponse is usually slow as the constaints ae the convective heat tansfe between the bune gases and efome walls and conductive heat tansfe fom the walls to the catalyst inside the efome [3]. Accompanied by an incease in fuel flow inside the efome, and a subsequent demand fo heat to the efome walls, the fuel flow inside the bune is also inceased. his helps to maintain the tempeatue inside the efome, keeping the ate of eaction constant, as the pemix fuel is maintained at unifom concentation. Since these pocesses have a lage lag time, the esult may be low quality efomate being diected to the fuel cell duing tansient conditions. On the othe hand, duing amp-down conditions of low powe demand, the pemix fuel flow is educed to adjust with the low hydogen equiement at the fuel cell stack. Subsequently, the fuel flow to the bune is also educed as the hydogen flue gas etuning fom the anode is adequate to povide heat to the eacto [9]. Due to the time delay in the efome due to eaction kinetics, heat diffusion time and convection, the effect of the contol action is not measuable fo a peiod of time. Sometimes this

23 long esponse time can exceed the quick-changing powe demands at the fuel cell stack. his causes the feedback loop to be sluggish whee feedback signal is cucial to automatic contol of the efome. he contol action is thus inadequate since a change in the fuel flow will delive the equied hydogen in the stack only afte a cetain time delay, thus aleady ceating a hydogen stavation at the stack o depleted hydogen at the anode flue gas, again causing hydogen stavation at the bune. Ai Combustion poducts Anode Flue Gas Refomate Ai Anode Cathode Refome Steam Excess bune Refome ai and wate Heat exchange plate Combustion poducts Vapoize Heat Exchange eat methanol Wate methanol pemix Ai Stat-up bune Figue.1. Genealized Fuel Cell Schematic Modeling is an essential tool to undestand the component level inteactions of the efome with the fuel cell stack and its implications on the oveall system pefomance. A easonable epesentation of the tansient esponse will enable futue development of design and contol achitectues fo the efome. he efome model is to pedict the concentation of species in the efomate and the efome catalyst bed. he hydogen output with espect to time will detemine the tansient esponse and time delay of the efome. he hydogen flue gas etuning to bune also detemines the methanol fuel flow ate to the bune. 3

24 o meet the needs of futue contol-oiented studies, anothe objective of modeling the efome is to obtain a set of diffeential equations which will epesent a state space equation. he state space equation will enable integation of the model into a simulation envionment in ode to numeically pedict the behavio of the system unde vaying opeating conditions.. Liteatue eview Accoding to Helms and Haley, a quick stating and fast tansient esponse ae the most impotant chaacteistics of a fuel cell powe plant fo tanspotation applications. By fa the most popula efoming technology fo on-boad tanspotation application is catalytic steam efoming [13]. Geye et al indicated that the methanol steam efoming technology is supeio in its steam efoming technology. Howeve, steam efoming tansient esponse is slow in compaison to othe components of a fuel cell system, thus limiting the oveall effectivity of a plant in tems of dynamic esponse [17]. hus it is cucial to incease the dynamic esponse of a fuel cell efome. he fist step towads impoving the design and esponse chaacteistics of a efome is to develop a model that can be utilized to study efficiency and themodynamics of the system. he model should also be able to be used in conjunction with a contolle design that can help achieve faste esponse to the system dynamics and distubances. Kuma et al, Vandebogh et al., and Geye et al developed steady state models of the steam efomes, which do not pedict the steam efomes tansient behavio [16, 18]. Ohl et al developed a fist pinciples dynamic model of a steam efome fo use in paametic design studies. he efome was assumed to be a well mixed tank eacto. A seies of ate eactions fo each constituent of the efomate and fuel mixtue consisted of the state vaiables of the system. he heat tansfe equation consisted of the change in specific enthalpy due to the eaction kinetics within the efome. hese equations wee then epesented in a state 4

25 equation fom to descibe a dynamic model of a steam efome. Model esults wee then coupled with an optimization pocess to detemine the design paametes of a steam efome [8,, 1]. he study howeve, does not take into consideation the heat tansfe between the walls of the efome and any extenal heating agent such as bune hot gases o electic heates. Convective heat tansfe between the gases and the efome wall tempeatues ae neglected. Ahmed et al developed a pefomance model of a efome that pedicted the pefomance and tempeatue of a efomate gas mixtue. A complete convesion of fuel was assumed as thee was a lack of chemical kinetics equation fo the patial oxidation eaction fo which the model was built. he esult of the analysis showed that thee existed a linea elation between the exit gas tempeatue and the inlet tempeatue of the fuel gas mixtue. his model was howeve specific to only patial oxidation eaction and was not a geneal efome model that could be applied fo othe types of fuel efoming [14]. Choi et al showed the esults of kinetics of methanol decomposition along with methanol steam efoming and the wate gas shift eaction. A non linea least squaes optimization method was used to obtain expessions fo ate of eactions [15]. umeical analysis concluded sepaate ate of eactions fo methanol steam eaction, wate gas shift eaction and CO selective oxidation. he thee eactos wee then integated and modeled in MALAB. his study helped in obseving the behavio of the eacto by changing its volume and tempeatue. Choi s study howeve did not take into account the heat tansfe in the eacto o the dynamic esponse of the efome...1 Langmui Hinshelwood Model Ohl et al developed a dynamic model of the methanol efome using the Langmui Hinshelwood (LH eaction ate fo the methanol decomposition eaction. he use of this eaction ate is paticulaly advantageous because of the wide ange of pessues that it coves. 5

26 As the pessue inceases the adsoption by the suface of catalyst bed and poducts slow down the eaction. his eaction ate takes all this into account. Accoding to B. A Pepply, at constant tempeatue the ate of eaction inceased with opeating pessue although the final convesion deceased with pessue at themodynamic equilibium [17]. LH eaction kinetics is one of the simplest eaction mechanisms and it descibes most catalytic suface eactions. he LH mechanism assumes that all eactants ae adsobed befoe the actual eaction can take place. Reactions occu between the adsobed molecules following a fast diffusion pocess. he adsobed molecules then ae desobed. he LH mechanism consists of many eaction steps taken togethe, anging fom and may extend upto 3. Each step is assumed to be an elementay step, meaning that the eaction is supposed to occu exactly as it is witten. Although the geneal success of the LH mechanism has been accepted, inaccuacies can appaently occu due to cetain eactions that may be autocatalytic. Since the wate gas shift eaction is itself an autocatalytic eaction, the appopiateness of the use of this ate equation fo the wate gas shift eaction is debatable. In addition to it, thee ae speculations that the wate gas shift eaction may not occu in the pesence of methanol on the catalyst bed. Since the LH mechanism assumes that one step is the ate limiting step while the othes ae at equilibium, thus it educes the ate of the oveall eaction defined by the ate of the decomposition of methanol. Ohl expessed the methanol decomposition ate as follows: m cη m = 1 + b m k p m b m p + b w m p w (-3 6

27 whee is the eaction ate fo the decomposition of methanol, m c is the mass of the catalyst bed, η is the effectiveness facto, k m is the methanol decomposition ate, b m and b w ae the methanol and wate adsoption equilibium constants espectively, p m and p w ae the patial pessues of methanol and wate, espectively... akagaki Coelation Anothe way to expess the ate of eaction has been developed by akagaki et al at oshiba Powe Systems. hey caied out tests fo evaluating the eaction ate of methanol decomposition with vaying mass flow ates. Results showed that the diffusion esistance wee moe significant fo lowe mass fluxes thus the eaction ate vaied at lowe mass fluxes [11]. Howeve it was found to be constant at highe mass fluxes, geate than.14kg/sm Also a coelation was found between pessue and the eaction ate. It was found that the eaction ate became lowe at highe pessues, thus they expessed the ate of eaction as powe law of the total pessue. Dependence of eaction ate to tempeatue was found to adhee to Ahenius s law fo tempeatues, <513 K. Howeve fo tempeatues, >513 K, conclusive esults wee not dawn. Based on these expeimental obsevations, akagaki et al., deived the eaction ate of decomposition on Methanol, m on Zn/Cu catalyst to be: l m E / R m = k P ( / 513 e CH 3 OH (-4 whee, k o = 1.35 X 1 6 mol / (gcat.s.atm l =.13 E = 1 X 1 5 J/mol = 1.3 M = -1 if > 513 K else = 7

28 CHAPER 3 MODELIG OF A PACKED BED REFORMER 3.1 Oveview his chapte descibes the basic pinciples behind the modeling of physical efomes. he theoy govening the equations and the application of initial and bounday conditions ae exploed. In addition, the text explains the method of finite diffeences adopted to solve numeically the undelying patial diffeential equation. An independent analytical solution is pesented to solve the heat tansfe equation fo a special case and thus validate the numeical model. he ate of eaction and the convesion of hydocabons to hydogen depend on the tempeatue of the catalyst bed of the efome. Consequently, the catalyst bed tempeatue and the outlet tempeatue of the poduct gases ae the most citical pocess vaiables affecting the pefomance of the efome [4]. he pupose of building the numeical model is to pedict the tempeatue pofile and efomate compositions as a function of time and space fo a eacto of any cylindical geomety unde diffeent opeating conditions. his model should be useful fo developing a contolle that will manipulate the equied pocess paametes of a methanol fuelcell efome o a biomass gasifie eacto to optimize thei pefomance Methanol Steam Fuel Cell Refome 3. Backgound of hemal Model he convesion of a methanol and steam mixtue to hydogen and othe poduct gases takes place inside the efome. his efome is usually a cylindical packed-bed eacto with Cu/Zn catalyst pellets [1]. he fuel mixtue is peheated in a supeheate befoe being injected into the efome with a feed pump. he heat fo the endothemic eaction is supplied though the 8

29 walls of the efome by a catalytic bune. he ate of eaction o decomposition of methanol depends on the tempeatue of the efome and the concentation of methanol. he motivation fo developing a themal model fo a methanol fuel-cell efome is to povide a tool fo designing a contolle which will maintain the efome at an optimal opeational tempeatue. he themal model consists of a heat tansfe equation with conduction and convection tems. 3.. Geneal Desciption of the Refome Due to the similaity between methanol efomes and othe hydocabon efomes and the common goal of developing a contolle, a single model of a geneal efome is poposed. A cylindical packed-bed efome whee the eactions take place on the catalyst pellets is assumed [19]. Figue 3.1 shows the mass and enegy flows, whee and x ae espectively the adial and longitudinal axis of the efome. he total length of the efome in the axial diection is L and that in the adial diection is R. Extenal heating is supplied though the walls of the efome. he flow of gas mixtue takes place in the axial diection and the adial mass flow is ignoed. heat x eactants efomate Figue 3.1. Genealized efome schematic. 9

30 3..3 Patial Diffeential Equation A cylindical coodinate system is chosen in this poblem to convet fom theedimensions to two-dimensions, as the heat conduction and convection is symmetical about the axis. he patial diffeential equation elevant to this model in the cylindical coodinate system is ρ C e e t = k e + k e + k e x. d E dv g + uρ gas εc g x (3- whee = (x,,t and whee ρ is the effective mass density of the contol volume, e C e is the effective specific heat of the contol volume, is the tempeatue inside the efome, k is the effective conductivity of e the contol volume, is the adial axis, x is the axial axis, d is the ate of heat geneated in. E g the eaction, V is the total volume of the efome, u is the velocity of gas, ρ gas is the density of the gas, and the void factoε is the atio of mass of gas to the total mass of solid inside the contol volume[3]. Fo simplification of the numeical analysis, the convection tem and heat geneation tem. d E dv g + uρ ε C gas g x ρ ε u gas Cg x is initially omitted. 3.3 Finite Diffeence Method Discetization he fist ode deivatives of a function f(x+h, in tems of the discete diffeences can be expessed using aylo seies expansion as 3

31 ' f k +1 f k f ( x = (3-4a h o, ' f k f k 1 f ( x = (3-4b h Adding (3-4a and (3-4b yields a thid possible appoximation, namely f f f ' 1 ( x = h k+ k 1 (3-4c Equations (3-4a, (3-4b, and (3-4c ae finite diffeence appoximations using backwad, fowad and cental diffeentiation, espectively. Using the cental diffeence scheme in the diffeential equation (3-, the fist ode discetization can be witten in the fom = i, j i, j 1 (3-5a Similaly (3-4a second ode discetization is of the fom i, j + i, j = ( i, j 1 (3-5b If the adius of the efome is divided into M equal discete elements of size, then j epesents a node on the adial axis,. Similaly if the length of the efome is discetized into elements of size x, then i epesents a node on axial diection, x. Figue 3- shows the discetization of a coss-section of the uppe half of the efome. Only the uppe half of the cylinde is consideed as the heat tansfe is symmetical about the -axis of the efome. Figue 3-3 intoduces anothe axis with time,t, as an independent vaiable. hough while simulating, discetization in time is not caied out, we will futhe poceed to make a dynamic model of the efome with the help of SIMULIK which utilizes an explicit method to find the tansients of 31

32 the tempeatue pofile evey th n instant of time, denoted as n t, whee t is a finite time inteval. x (,M (,j i-1,j i, j+1 i, j i+1,j i,j-1 (, (i, (, x Figue 3-. Discetization along length and adius of the efome. i, j+1 i, j+1 i, j i+1, j i, j 1 n x n+1 Figue 3-3. Discetization along length and adius of the efome and along time incements. 3

33 hus the govening PD (3- can be discetized in the and x E, diections into and M numbe of nodes and witten in the ODE fom. Fo simplicity of numeical analysis, initially the heat geneation tem has been excluded. ( ( 1, 1,, 1,, i j, 1,, 1, n n n n n n n n i j i j i j i j i j i j i j i j e e e eff eff d k C k k dt x ρ + + = + + (3-6 he above geneal equation fom can be witten as a set of ODEs by invoking the value and and the esulting set can be e-aanged in tems of a banded pentadiagonal matix of co-efficient, as follows: i...,3,1, = M j...,3 =,1, M l t l t l t dt d M M K K K K M M = M M M l t l t s l t s l t s l t s s l t s l s l dt d dt dt d M M K M M M M M M M K K K K K K M M M M M (3-7a whee, = similaly, l t s l s l d dt d M M M s l d dt M 1 M M ]...,,,,..,,,, [ ],...,,...,, [,, 3,, 1,,1 3,1,1 1, = + 33

34 d1 d d [,, dt dt dt d1,1 d,1 [,, dt dt 3 d d + 1 dm,...,,... ] dt dt dt d3,1 d,1 d1, d3,,..,,,, dt dt dt dt d =, dt d,... dt he nodes epesenting i = and j = have not been included in this matix because i= is the bounday egion at the entance of the efome. Since at this point, the bounday condition is assumed to be constant always, which is equal to the inlet flow gas tempeatue, it will have a fixed numeical value. Stoing numbe of numeical values fo evey computational iteation will take up a lot of compute stoage space, thus the nodes ae not taken into consideation in this matix, which was built solely fo iteation pupose. All the nodes epesenting i= is epesented by one value in the compute pogam and is used evey time the bounday nodes ae equied to calculate the tempeatue of its adjacent node. Similaly fo j= which epesents the adius of the cylindical efome, it is assumed that thee is a no flux condition acoss the adius of the efome, these adial nodes will have the same value as its adjacent nodes. In ode to save compute space by stoing duplicate values, these nodes ae not included in the matix while doing the iteations. he teatment of these bounday conditions will be dealt with moe detail in a late section. Let the x M pentadiagonal matix in equation (3-7 be denoted as P. hen,, M ] d dt = P (3-8 whee, his equation can be integated to yield [ ] =,..., 1 M 1 M t n+ 1 n n i = i d i + i P (3-9 34

35 whee is a vecto of initial conditions = 1,... M 1, M hus a new vecto of tempeatue values as a function of time can be found using vaious algoithms and iteation methods. SIMULIK has been used to solve the above equations using an explicit Runge-Kutta method Initial and Bounday Conditions Bounday conditions and initial conditions ae pescibed fo a paticula case of inteest, so as to define appopiate poblems with unique solutions. he initial condition gives the specific tempeatue distibution in the system at time zeo, and the bounday conditions specify the tempeatue o the heat flow at the boundaies of the medium Diichlet s Bounday Conditions he Diichlet s Bounday Conditions ae fixed bounday conditions on the efome wall and the inlet of the efome, whee the tempeatue is held constant by electic heates supplying heat into the system though the walls o by the inflow of pe-heated gas and. his is expessed mathematically as follows: ( x,, t = wall, at = R, x L and t ( x,, t = in, at x =, R and t eumann s Bounday Conditions. eumann Bounday Conditions ae natual bounday conditions in the outlet whee thee is no heat flux acoss the adius because of symmety and the exit of the efome is assumed to be insulated so that zeo heat flux can be assumed in that plane. his is expessed mathematically as follows: 35

36 ( x,, t =, at =, x L and t x ( x,, t =, at x = L, whee L is the length of the efome, R and t whee ( xt,, = and x( xt,, = x Initial conditions he efome is heated to a citical tempeatue by the stat-up bune. he initial condition is assumed to be unifom thoughout the efome. his is expessed mathematically as follows: ( x,, t =, at t =, x L, R init whee is a constant value. init eatment of an undetemined bounday condition Let us conside the govening equation (3- again afte neglecting the convection tem, namely ke de ρ ece = + keff + k eff t x dv. g (3-1 Fo the fist-ode tem in, the eumann s bounday conditions at = do not apply hee because of a tem / that becomes undefined. Applying L Hospital s ule yields 1 = (3-11 whee the ight hand side can be witten as, = (3-1 Replacing this tem in (3-1 yields, de& g = + t x dv ρece keff k eff (

37 Futhemoe, whee =, the tem de & g dv is neglected. While building a numeical simulation, this special case is accounted fo sepaately to obtain the coect tempeatues of the nodes at the axis of the efome Special Case of Fictitious odes at eumann s Bounday Conditions While implementing cental diffeence scheme, the tempeatue of a node depends on the tempeatues of its adjoining nodes. hus the nodes at the exit edge of the efome equies tempeatues of a fictitious node, n + 1, j. Since thee is zeo heat flux acoss the exit nodes of the efome, we make use of this to assign a value to the fictitious nodes. n, j + 1 n 1, j n + 1, j n, j 1 Figue 3.4. Fictitious nodes he no-flux condition is used to find the value of the fictitious nodes. Fist the x-diection deivative is witten as = x + 1, j 1, j x (3-14 which can be solved fo + 1, j to yield, 37

38 j j x x + = +,, 1 1 (3-15 ow at the zeo-flux condition, x =, so it follows that 1, 1, j j + = (3-16 Hence the tempeatue at fictitious node at 1, ( j + can be eplaced by the tempeatue at. A simila elationship can be developed at the othe edge having fictitious nodes. Finally afte implementing all the bounday conditions, the govening pentadiagonal n the fom, 1, ( j which lie at = matix 3-6 can be e-witten i M M M l dt 1 ( 1 1 ( 1 1( K K K K K M = l d 1 l M M + s 1 M l l M M l t l d dt M t l t t l t t s t s l t l t s l t s s s l s l s l s dt d d dt d dt d dt dt d ( ( 1 6 M M M M M M M M M M K K K K K K M M M M (3-7b 38

39 In the above equation, the fictitious node +1 and its multiples, which epesent the exit end of the efome has been included in the pentadiagonal matix. Its value is calculated exactly as the value of -1 node and its multiples. he nodes along the adius of the efome fom the undetemined bounday conditions. Hee, at the value of j=, the equation (3-13 is implemented. hus all along the adius, the node tempeatues have diffeent co-efficients than that of the est of the discetized efome. As shown above, in the initial fist nodes of the efome this bounday condition is imposed. his equation epesents the govening matix that was used to numeically model the genealized efiome. 3.4 Model Validation o validate the accuacy of the numeical esults, a compaison is made by solving the patial diffeential equation (3- using an analytical method. he physical model is simplified by neglecting the convection tem to yield, = x α t (3-17 whee, R, x L and t >, = init, and whee init is the initial tempeatue. Equation (3-17 is subject to the bounday conditions, =, at = R, x L (3-18a = at x =, R (3-18b x =, at x = L, R (3-18c he poblem (3-17 can be solved analytically by applying bounday conditions (3-18a to (3-18c to yield ( x,, t C R ( β x( η x e α β + η = m= 1 p= 1 mp m p ( m p t (

40 whee the value of constant C mp is detemined by using the pinciple of othogonality. hus multiplying both sides by R R ( β m d and x(η p, x dx yields L Substituting (3- into (3-17 yields C R L 1 = R ( β m X ( η p x Fddx β ( η (3- mp ( m p = x= α βm+ ηp t R L e ( x,, t R( βm X( ηpx F R ( βm X( ηpx ddx (3-1 ( β ( η = ( m= 1 p= 1 m p = x= ow, it is necessay to find the eigenfunctions R ( β, the nom β, and the m eigenvalue β. Fom able 3-1, Case 3 in efeence [ 1], it follows that fo the bounday m condition in question R ( β m J ( β m = (3-a and ( m 1 = ( β m R J ( β R m (3-b whee β ae the positive oots of J ( β R m, and J is the Bessel function of ode. o find m = the eigenfunctions X ( η p x, the nom ( η p, and the eigenvalue η p, we efe to able -, Case 8 in [1] to find that and X( η x = sin( η x (3-3a p P 1 = ( η p L (3-3b 4

41 whee η ae the positive oots of cos ( η =. Substituting (3-a (3-3b into (3-1 yields p L p α( βm+ ηp t R L e ( x,, t J ( β sin( η x FJ ( β sin( η x ddx = m p m p m= 1 p= 1 ( βm ( ηp = x= Sepaating the integals poduces the equation (3-4 α( βm+ ηp t R L e ( x,, t F J ( β sin( η x J ( β d sin( η x dx (3-5 = m p m p m= 1 p= 1 ( βm ( ηp = x= Integating analytically the ightmost integal, and using the expessions (3-a, (3-b, (3-3a and (3-3b yields α( βm+ ηp t R e 1 ( x,, t = F J ( β sin( η x J β d LR J β R ( ( 4 ' m p m m= 1 p= 1 m η = p (3-6 α( βm+ ηp t R 4e 1 ( x,, t F J ( β sin( η x J ( β d = ' m p m m= 1 p= 1LR J ( βmr η = p (3-7 whee fom [1], 1 J β m d = J β ( 1( β m (3-8 Substituting (3-8 into (3-7, the genealized final analytical solution educes to, ( x,, t α( βm+ ηp t βm ηp 1 βm m= 1 p= 1 LRJ1 ( βmr βmηp = 4Fe J ( sin( x J ( R (3-9 41

42 3.5 on-dimensional Analysis A non-dimensional analysis is pesented hee to detemine the heat tansfe pocess in the efome. he fomulation of the poblem in tems of non-dimensional tems helps in studying the physics of the pocess and should povide a useful tool fo data analysis and compaison. Conside the geneal patial diffeential equation 1 1 t x α = + + In ode to non-dimensionalize along adius and axial diection, let R / = (3-3a L x -3b and x = (3 wall = (3-3c Substituting these new vaiables into the govening equation yields t x x x x R wall wall wall = + + wall α (3-31 o ( ( ( ( ( t x L R R = + + α (3-3 aking L t t α = equation (3-3 can be expessed as t x R L R L = (3-33 4

43 Using the method of sepaation of vaiables (, x, t = ψ (, x Γ( t (3-34 Equation (3-3 can be epesented as 1 ψ 1 ψ ψ 1 d Γ( t ψ z x Γ( t d t + z + = = λ (3-35 whee, z = ( L / R.hen, taking the sepaated equations Γ ( t d + Γ ( t λ = d t (3-36a and ψ z ψ ψ z λ ψ = (3-36b x Again invoking sepaation of vaiables (, x = ψ R ( X ( x equation (3-35 becomes 1 1 d R + z dr d X z + + = λ (3-37a R d d X dx Sepaating in R and X, 1 X d X dx = η (3-37b and 1 d R z R d z dr v d leads to two equations that can be solved individually to yield + = β (3-37c d X + η X = X ( η, x:sin ( ηx and cos( ηx dx (3-38a 43

44 d R z dr z + + R = β (3-38b d d Γ ( t d + Γ( t λ = (3-38c dt ote: as equation has no φ dependence, it follows that, v= hus the solutions of (3-38,a-c taken as, can be expessed as e λ t, R ( β m,, X ( η p, x (, x, t C R = m= 1 p= 1 hus the final solution to equation (3-3 is mp ( β, m λ t X ( η, x e p (3-39 (, x, t = λ t e F ( β ( η m= 1 p= 1 m p R ( β, m X ( η, x p 1 1 = x= R ( β, X ( η p, x dx d m (3-4 ow to find the eigenfunctions fom efeence [1], R ( β m = J ( β m 1 = ( β R J ( β mr whee, β ae the positive oots of J ( β, R m m m = X ( η x = sin ( η x p 1 = ( η p L p (3-41a (3-41b (3-4a (3-4b whee, η ae the positive oots of cos ( η =, whee R and L ae assumed to be 1. p Substituting equations (3-41a to (3-4b in (3-4, p L 44

45 dx d x J x J R J LR F e t x p m x p m m p m t, sin (, (, sin (, ( ( 4,, ( η β η β β λ = = = = = ion in non-dimensional fom, (3-43 Solving the emaining integals gives the final solut, ( ( 4,, ( m m p p m m t J J F e t x β η β β λ = = = (, sin ( 1 m p J x β η (

46 CHAPER 4 RESULS AD DISCUSSIO his chapte summaizes the computational esults obtained by solving the physical models of the efome numeically and patially validating the esults with analytical solution. he steady state solution of the numeical model equation is solved using Matlab. he tansient solution is obtained by using SIMULIK, which uses the explicit method to find solution in the next time instant. A majo pupose of this study was to be able to develop a model that would coectly pedict the efome behavio in tems of tempeatue at any paticula location. his tempeatue would then act as an input to a contolle that would detemine the flow of pemix fuel into the system. he validity of the numeical model is established as will be shown in the subsequent sections. Fo the sake of the study the value of the efome constants wee taken same as the values of a methanol efome located at UC-Davis. he following table shows the values that wee estimated to match the UC-Davis efome paametes. able 4.1 Constants used fo finding solution to the efome model. (Souce: able 4.1, Daniel Betts, 5 ame Value Used How it was detemined Catalyst Bed Heat Capacity, C e 9 J/kg-K Infeed Catalyst Bed hemal Conductivity, k e 5. W/m-K Estimated fom Data Catalyst Bed Density, ρ 1983 kg/m 3 Measued e he catalyst heat capacity was assumed to be equal to an aveage of the heat capacities of coppe and zinc. he void facto used was 5% he catalyst density was measued via a wate displacement method fo a single pellet 46

47 Figue 4-1 shows the numeical solution of (3-7 to find the tempeatue pofile fo a simple case when the efome boundaies ae kept at constant tempeatue. he uppe bounday o the cylindical wall of the efome is kept at 58 K wheeas the left side o the face of the efome is kept at 5 K. Radius of [m] Radius of efome, [m] Length of efome [m] Length of efome, x [m].5 Figue 4-1 empeatue pofile of efome afte 3 mins he tempeatue at diffeent location inside the efome is shown with colo vaiation. he initial tempeatue of the efome was 5 K. As the time inceases the tempeatue inside the efome gadually inceases as can be seen fom the plot. he pofile shown above is only fo one half coss section of the efome. ow if this tansient fomulation is un fo longe peiod of time, it gives a steady state solution. Figue 4- shows such time histoies fo diffeent location along the cente-line of the efome. Figue 4- A shows the tempeatue plot with espect to time at the inlet of the efome. It shows constant tempeatue at the beginning and then it ises and attains a steady state tempeatue of 567 K wheeas fo figue 4- B and C the tempeatue ises fom the beginning 47

48 (A empeatue [K] (B ime [s] x empeatue [K] (C ime [s] x empeatue [K] ime [s] x 1 4 Figue 4- empeatue plots at mid-adius of efome afte 3 minutes at (A Inlet (B Cente (C Exit. 48

49 Plotting equation (4-3 in Matlab, we get a tempeatue pofile at =.75, and at diffeent location of distance (x axially along the efome. empeatue, [K] ime, t [s] Figue 4-3 empeatue pofile accoding to analytical solution Since fo the analytical solution we used the Diichlet s bounday conditions as K, a numeical simulation with simila bounday conditions was caied out by shifting the wall and inlet gas tempeatue conditions to. he initial efome tempeatue is kept at 5 K. As seen fom the plot, as the axial length inceases the tempeatue appoaches towads the steady state of K at much slowe ate. Figue 4-4 shows the tempeatue distibution plotted afte 3 seconds with zeo bounday conditions. Figue 4-5 shows the tempeatue pofile along the cente line of the efome fo the analytical solution. he tempeatue pofile fo the inlet shows constant tempeatue at the beginning befoe gadually falling and eaching a steady state tempeatue of K. 49

50 .5 Refome Radius [m] Radius of efome, [m] Length R efome of efome, length [m] x [m].5 Figue 4-4 Refome tempeatue with zeo bounday conditions (umeical solution 5

51 A empeatue [K] 3 1 (B ime [s] 5 4 empeatue [K] 3 1 (C ime [s] empeatue [K] ime [s] Figue 4-5 empeatue pofile of mid-adius accoding to Analytical Solution at (A inlet (B cente (C exit Figue 4-6 shows the plot of tempeatue at =.75 and diffeent locations along the axial cente of the efome with time. his plot epesents the analytical solution obtained diectly fom equation 3-1 wheeas figue 3-11 shows the same tempeatue plot at simila location along the axial diection fo finite diffeence solution. he plots match closely with each othe when plotted togethe. Hee they ae plotted diffeently fo bevity. 51

52 6 5 x =.1 x =.5 x =.5 x =.75 empeatue, [K] ime, t [s] 14 x 1 4 Figue 4-6 ansient analytical solution fo axial tempeatue gadient x=.1 x=.5 x=.5 x=.75 x=.9 [k] 3 x inceasing t [s] x 1 4 Figue 4-7 ansient numeical solution fo axial tempeatue gadient. Figue 4-8 shows the tempeatue plot at diffeent axial location along the efome when solved analytically. As the axial distance inceases, the tempeatue flattens out. 5

53 empeatue, [K] Length of efome, x [m] Figue 4-8 Analytical heat tansfe solution fo tempeatue as a function of axial position and time at adial distance =.75R empeatue, [K] Length of efome, x [m] Figue 4-9 umeical solution at diffeent location along x-axis at =.75 m. he tempeatue pofiles ae shown at diffeent time instances 53

54 Figue 4-1 shows the esults of the non-dimensional analysis using the analytical method. αt he x-axis epesents the time t =, the y-axis epesents the scaled efome tempeatue. L empeatue, [K] Length of efome, x [m] Figue 4-1 on dimensional tempeatue pofile using analytical method. d E g So fa the esults wee obtained without taking into consideation the heat geneation tem and convection tem. Howeve, it is impotant to include the heat of eaction in the cuent analysis as the efoming eaction is endothemic and apat fom the heat tansfe between the fluid and efome wall, enegy is consumed to maintain the eaction. he ate at which methanol is consumed is calculated using the akagaki ate of eaction as is discussed in detail in Chapte. he following esults show the compaison between the efome bed tempeatues by addition of the heat geneation tem. 54

55 A (B Figue 4-11 Compaison of tempeatue pofile of mid-adius accoding to umeical Solution with and without heat geneation at (A exit (B cente (C inlet 55

56 (C Figue Continued Figue 4-1 shows the tempeatue pofile of the efome with the heat geneation tem. he uppe bounday o the cylindical wall of the efome is kept at 58 K wheeas the left side o the face of the efome is kept at 5 K. As shown in Figue 4-11 and Figue 4-1, it can be seen that due to the endothemic eaction the efome eaches a steady state at a lowe tempeatue. hus it can be concluded that the efome exhibits tempeatue fluctuations with the change in methanol flow ate. Since the ate of eaction is diectly popotional to the numbe of moles of methanol consumed, any incease of flow of pe-mix fuel will cause a substantial dop in efome catalyst bed tempeatue. hus thee is an immediate equiement to amp up the heat supply into the efome in tems of bune hot gases. his condition will be especially noticeable in a steep incease in the load demand at the fuel cell stack. 56

57 Radius of efome, [m].5 Length of efome, x [m].5 Figue 4-1 empeatue pofile of efome afte 3 mins (with heat geneation Betts indicated that the convection tem was small compaed to the conduction in the heat tansfe equation. he authos conclusion was backed by obtaining expeimental data fom UC- Davis efome which showed that the convection was below.4% of the value of the conduction tem. hese esults ae expected as the axial tempeatue gadient is much lesse than the adial tempeatue gadient and most of the flow occus in the axial diection [4]. he cuent analysis includes the convection tem and as can be seen fom Figue 4-1, the compaison shows negligible contibution to tempeatue gadient by the convective heat tansfe. 57