Self-Oscillation Liquid Front inside a Partially Wetted CatalystPellet under r-Methylstyrene Hydrogenation: Experiment andTheory

Alexey B. Shigarov,* Alexander V. Kulikov, and Valery A. Kirillov

Boreskov Institute of Catalysis, Pr. Acad. Lavrentiev 5, Novosibirsk 630090, Russia

Self-oscillations on a partially wetted cylindrical catalyst pellet 1% Pd/γ-Al2O3 in R-methylstyrene(AMS) hydrogenation accompanied by liquid evaporation are studied. By NMR tomography(magnetic resonance microimaging), images of the liquid-phase distribution within the porouspellet are obtained and the liquid-phase redistribution is monitored immediately during theprocess. The physical mechanism and simplified one-dimensional model are suggested to explainthe observed phenomenon. The model predicts well the experimental period (2-3 min) and theamplitudes of the temperature oscillations at the lower (dry) part of the pellet and also theliquid front oscillations at the upper part. The instability mechanism (negative feedback withtime lag) is based on complex interaction among the processes of imbibition, evaporation, andgas-phase hydrogenation.

Introduction

It is known that interaction of physical processes(imbibition, evaporation, and heat and mass transfer)with exothermic reaction can result in the formation ofvarious types of instability and critical phenomena inmultiphase catalytic systems. The investigation of suchregimes is interesting from both academic and indus-trial points of view with regards to hot spots andrunaway phenomena in trickle-bed reactors. Liquidhydrocarbon hydrogenation presents an important ex-ample on this item. It is worth mentioning here theearly research,1 where the crucial roles of gas-phasereaction and moving hot spots were experimentallyfound in a laboratory trickle-bed reactor under R-meth-ylstyrene (AMS) hydrogenation. Slow progress in theunderstanding of various intriguing phenomena ob-served in laboratory trickle-bed reactors is partly ex-plained by complex hydrodynamics of multiphase flow.So, the focus of research on a single particle scaleseemed promising. For example, AMS hydrogenationwas used to demonstrate the acceleration of drying ona single catalyst pellet.2 The multiplicity of catalystpellet liquid filling and temperature steady states undercyclohexene hydrogenation in a single-pellet reactor wasreported.3 Ignition and extinction phenomena on dif-ferent types of catalyst pellets under variation of theliquid AMS feed were experimentally studied.4,5 Theo-retical considerations are rather scarce. Careful analysisof the model of a half-wetted and partially liquid-filledcatalytic porous slab was made6 but without its experi-mental verification. Physical mechanism and math-ematical models were proposed7 that successfully ex-plained the hysteresis behavior observed earlier4,5 bythe same authors. A novel promising approach to theexperimental study is application of in situ NMRimaging,8 accompanied by accurate intrapellet thermo-couple measurements. This method allows one to obtaina picture of the liquid distribution inside the porouspellet and to in situ monitor its redistribution in time.

The first brief report on this item appeared recently.9Further development of this approach enabled one tofind experimental self-oscillations on a single cylindricalcatalytic pellet under AMS hydrogenation.10 The goalof our current work is coupling of experimental andtheoretical approaches in order to explain this unusualself-oscillating behavior.

Experimental Setup

As a model reaction, we chose AMS hydrogenation:

This reaction is widely used as a model reaction in theexperimental investigation of processes in trickle-bedreactors. Figure 1 presents a schematic of the single-pellet reactor unit we used. This setup contained areactor consisting of three vertical coaxial glass tubes.In the inner tube (9 mm i.d.), a catalyst pellet wasplaced and blown with a preheated hydrogen flow. The10-mm-high cylindrical catalyst pellet of 5 mm diametercontained 1% Pd/γ-Al2O3. The pellet was kept by achromel/copel thermocouple of 0.2 mm diameter. Theearlier experiments9 revealed that a thermocouple itselfsignificantly distorts NMR microimages of the liquiddistribution inside a porous catalyst. So, we were forcedto place the thermocouple at the bottom part of thepellet in order to avoid distortions at least for the restpellet volume. The thermocouple was introduced into acarefully drilled channel in the lateral surface of thecylinder. The channel was at a distance of ∼6 mm ofthe pellet height from the upper base of the pellet. Thedepth of the channel was equal to the cylinder radius.The gap between the inner and middle tubes was usedto control the reactor temperature by blowing with ahot air flow. The gap between the middle and outertubes was evacuated to prevent the NMR tomographfrom being heated. Liquid AMS was fed to the upperbase of the pellet through a glass capillary of 0.2 mmi.d. Simultaneously, the catalyst pellet was blown witha pure hydrogen flow. The temperature of the gas

* To whom correspondence should be addressed. E-mail:shigarov@catalysis.nsk.su.

C9H10 + H2 ) C9H12

9712 Ind. Eng. Chem. Res. 2005, 44, 9712-9717

10.1021/ie050269x CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 05/26/2005

(hydrogen) flow was measured with a second thermo-couple, which was placed upflow at 8 mm from the pellettop. The experimental setup also included hydrogen andliquid AMS supply systems, temperature control andmeasurement units, and air heaters for controlling thereactor temperature. All of the experiments were carriedout at atmospheric pressure, a hydrogen flow rate of10.9 cm3/s, and a gas and liquid inlet flow temperatureof 87 °C. The rate of feed of liquid AMS onto the pelletwas 0.8 mg/s. The original details of the NMR-micro-imaging technique can be taken from a previous re-port.10

Results of Experiments

At the first time, self-oscillations under AMS hydro-genation in a single-pellet reactor were experimentallyfound by temperature measuring (without the use ofNMR imaging). In those experiments, the cylindricalcatalytic pellets were half as short (D ) H ) 5 mm),and thermocouple was placed at the center and top ofthe particle. A typical steady-state curve7 displayed thehysteresis behavior (Figure 2). Self-oscillations appearedon the higher branch of this curve at sufficiently largeliquid flow rates. This phenomenon seems so unbeliev-able that it was decided to apply NMR imaging infurther research and to take cylindrical pellets withlarger height (H ) 10 mm).

Figure 3 presents the results of the temperaturemeasurement at the catalyst pellet bottom in the

process of self-oscillations and also two-dimensionalimages, which were obtained by NMR microimaging andcharacterized the filling of the catalyst pellet with theliquid (dark zones). The numbers at the temperaturecurve indicate the moments of time at which themicroimages were made. It is seen that liquid AMSimbibes into the porous structure from the wetted pellettop and leads to the formation of a wetted zone in thecatalyst pellet with clear-cut contours of the liquiddistribution within the porous structure. The consecu-tive magnetic resonance microimages in Figure 3 showthat the size and boundary of this zone continuouslypulsate. The rest of the pellet down to the liquid frontremains dry and filled with the gas phase (white zones).Note a time shift between the maximal temperature andthe minimal volume of the wetted zone. When the moviewas made from the set of microimages, it became clearthat oscillation of the liquid front resembles the motionof a liquid piston.

Mathematical Model

By now it seems extremely difficult just to formulatea three-dimensional model of the process. Therefore, asimplified one-dimensional model was developed thatmeanwhile includes (as we believe) an intrinsic mech-anism of self-oscillations. The basic simplification is thatliquid forms a cylindrical zone of constant radius butvariable height inside the pellet (Figure 4). This zoneis surrounded by the dry porous layer of constant thick-ness ∆rdry. The bottom of the liquid-filled zone tends tomove down owing to liquid imbibition (capillary forces)from the top. However, this tendency is counteractedby the process of evaporation. A part of the liquid evap-orates without hydrogenation from the wetted top of thepellet. Another part evaporates in a narrow front withinthe pellet at the boundary of the liquid-filled zone, andthe forming AMS vapor reacts with hydrogen in the dryporous layer surrounding this zone. The external surfaceof the pellet exchanges heat with the hydrogen flow. Theheat transfer within the pellet occurs by heat conductionin the axial (vertical) direction.

Figure 1. Schematic of a single pellet reactor unit placed in theprobe of the NMR tomograph.

Figure 2. Steady-state hysteresis curve for AMS hydrogenationon a single catalyst pellet (from ref 7) with the region of sustainedself-oscillations.

Figure 3. Experimental catalyst pellet temperature oscillationsand NMR microimages of liquid distribution inside the pellet atdiscrete moments of time (Gliq ) 0.8 mg/s).

Ind. Eng. Chem. Res., Vol. 44, No. 25, 2005 9713

The model is written as a set of two differentialequations:

The boundary conditions for eq 1 are as follows:

where

Initial conditions: t ) 0, T(h,0) ) T0, æ(0) ) æ0.At the top (h ) 0, 0 < r < Rliq) and lateral (r ) Rliq,

0 < h < Hliq) boundaries of the liquid-filled cylindricalzone, the conditions for equilibrium saturation of hy-drogen with AMS vapor at the temperature T(h,t) aremet. The AMS saturation vapor pressure and thespecific heat of evaporation are found from the empiricalcorrelations11

The evaporation rate at the lateral surface is deter-mined by the internal-diffusion mass transfer from theliquid-phase boundary through the dry layer of thick-ness ∆rdry. The internal-diffusion mass-transfer coef-ficient is calculated as

Because the coefficient DAH of molecular diffusion ofAMS vapor with hydrogen is close to that for cumenewith hydrogen, the effect of multicomponent diffusioncan be ignored and the DAH value can be taken to beequal to the coefficient of binary diffusion of vapor(mixture of AMS and cumene) with hydrogen. Thisallows one to calculate the interphase heat- and mass-transfer coefficients using the conventional empiricalcorrelations12

The thickness of the lateral dry catalyst layer wasconsidered as a single adjustable parameter of themodel, and its value was chosen to fit the experimentaldata. It is important that this value appears realistic.

The degree of the catalyst pellet liquid filling æ can beexpressed through the ratio between the height of thecylindrical liquid-phase region and the pellet itself:

The evaporation rate from the internal liquid front(across the dry porous layer) to the hydrogen flow iscontrolled by internal and/or external mass transfer.This rate is easily derived if we compare the internalmolar vapor flux (from the liquid front to the pelletsurface) to the external one (from the pellet surface tothe hydrogen flow):

and, thus, the expression for the lateral surface partialpressure of vapor is derived from eq 13:

When eq 14 is substituted into eq 13, the final expres-sion for the evaporation rate at the lateral surface isobtained:

For the dry (in our case bottom) part of the pellet, theevaporation rate is zero.

Figure 4. Schematic of physical assumptions that are adoptedin the mathematical model of the self-oscillation liquid front insidea catalyst pellet.

0 e h e H, t g 0

π(R2Fcatcpcat + Rliq

2εFliqcp

liq) ∂T∂t

) π(R2λcat +

Rliq2ελliq)

∂2T

∂h2+ 2πR[Wev

int(QrXcon - Qev) -

Rgas(T - Tgas)] (1)

πRliq2HεFliq

dædt

) Gliq - mliq(πRliq2Wev

ext +

2πR∫0

Hliq(æ)Wev

int dh) (2)

h ) 0 (pellet top)

πR2Rgas(T - Tgas) + πRliq2Wev

extQev +

cpliqGliq(T - Tliq) ) (πR2λcat + πRliq

2ελliq)

∂T∂h

(3)

Wevext ) âgas

Psat

RgasT(0,t)(4)

h ) H (pellet bottom)

Rgas(T - Tgas) + λcat∂T∂h

) 0 (5)

Psat ) 1.3367 × 1010(-Hev

RgasT),Qev ) RgasTcTb ln Pc

(Tc - T)0.38

(Tc - Tb)1.38

(6)

âpel ) (ετ) DAH

∆rdry, DAH ) 6 × 10-5( T

373)1.9(7)

Sh ) 2 + 0.6xReSc1/3, Nu ) 2 + 0.6xRePr1/3 (8)

âgas )ShDAH

deq, DAH ) 6 × 10-5(Tgas

373)1.9

(9)

Rgas )Nuλgas

deq, λgas ) 0.216(Tgas

373)0.78

(10)

∆rdry ) R - Rliq ) 0.00045 m, ∆rdry/R ) 0.18 (11)

æ ) Hliq/H (12)

Wevint ) âgas

Ps

RgasT) âpel(Psat - Ps

RgasT ) (13)

Ps

RgasT)

âpel

âpel + âgas( Psat

RgasT) (14)

Wevint ) ( âgasâpel

âgas + âpel)( Psat

RgasT), if 0 < h < Hliq (15)

9714 Ind. Eng. Chem. Res., Vol. 44, No. 25, 2005

To estimate the degree of conversion, we assume thatthe rate of AMS hydrogenation in the liquid phase canbe neglected in comparison with the rate of AMShydrogenation in the gas phase; the AMS hydrogenationin the gas phase takes place in the dry surface layer ofthe pellet; the change in the hydrogen partial pressurealong the thickness of the reaction catalyst layer isinsignificant and is equal to the equilibrium pressure:

and the hydrogenation kinetics is described by thepower law1

where the order in AMS vapor is zero.Under these assumptions, from known local values

of AMS intrapellet evaporation rate (15) and (16) andhydrogenation rate (18), one can find the thickness ∆rreacof the dry layer (surrounding the liquid-filled cylindricalzone inside the catalyst pellet) that is necessary for100% conversion of AMS vapor:

The degree of AMS conversion is calculated fromexpressions (17)-(19) as follows:

Results of Modeling

Numerical analysis of the mathematical model showedthat, in the physically realizable parameter range, thereis a unique steady-state solution. After a certain bound-ary in the space of the model parameters is crossed, thisstate can become unstable simultaneously with the“soft” birth of a self-oscillation mode (so-called Hopfbifurcation). Here we illustrate this by the influence ofthe AMS liquid flow rate. As has been mentionedalready in the Experimental Section, the steady-statestability at low AMS flow rates was found experimen-tally. The model successfully confirms this observation.The modeling results at Gliq ) 0.7 mg/s are given inFigure 5.

The phase trajectories are presented in Figure 5a.Figure 5b presents the dynamics of attainment of astable steady-state mode of the catalyst pellet. In theinitial moment of time (at t ) 0), the pellet is absolutelydry (æ ) 0) and its temperature is equal to thetemperature of gas flow of 87 °C. Oscillations emergingafter the start-up of the liquid feed are graduallydamped, and a stable steady-state mode is established.The calculated temperature of the wetted zone (Figure5b) is more than 35 °C higher than the temperature ofthe dry zone and the gas flow. The calculated degree ofthe catalyst pellet liquid filling æ is 0.28. The resultingstable steady-state temperature profile along the pelletheight is shown in Figure 5c. The maximal temperatureis achieved at the middle of the liquid-filled zone. AMSconversion for this and all other calculations wascomplete (Xcon ) 1). Taking into account that the specificheat of reaction exceeds the specific heat of evaporation

(Qr > Qev), one may conclude that a positive heat sourcearising at the lateral pellet surface (in eq 1) is distrib-uted along the whole liquid-filled zone (0 < h < Hliq).The conductive heat flux to the left-hand side of themaximum (Figure 5c) is mainly consumed by theevaporation at the wetted top of the pellet (at h ) 0).The conductive heat flux from the maximum to theright-hand side transfers heat to the dry zone (Hliq < h< H), which is cooled by hydrogen flow.

When the liquid flow rate has been increased from0.7 to 0.8 mg/s with all other parameters remainingconstant, then the new steady state became unstable.Figure 6a illustrates the development of self-oscillationsafter the start-up of the liquid feed and attainment of aperiodic mode (cycle) at Gliq ) 0.8 mg/s. It is seen thatthe trajectory in the phase space winds, approachingan attracting orbit. Figure 6b compares the calculateddynamics for the upper (h ) 2 mm) and lower (h ) 6

Wevint ) 0, if Hliq < h < H (16)

PH2) P - Psat(T) (17)

Wr ) k0PH2

0.64 exp(-E/RgasT) (18)

∆rreac ) Wevint/Wr (19)

if ∆rreac < ∆rdry, then Xcon ) 1 (20)

if ∆rreac > ∆rdry, then Xcon ) ∆rdry/∆rreac (21)

Figure 5. Calculated (a) phase trajectory, (b) dynamics ofattainment of a stable steady state, and (c) steady-state temper-ature profile at Gliq ) 0.7 mg/s.

Ind. Eng. Chem. Res., Vol. 44, No. 25, 2005 9715

mm) parts of the pellet with the experimental thermo-couple measurement. Note the satisfactory agree-ment between the calculation results and the experi-mental data on the period (2-3 min) and the amplitude(85-93 °C) of the temperature oscillations at the bottompart of the pellet. Figure 6b also demonstrates a phaselag between the oscillations of the temperature anddegree of the catalyst pellet liquid filling æ. The self-oscillations of a temperature profile along the pelletheight at the discrete moments of time are shown inFigure 6c.

It is known that self-oscillations are possible in thedynamic system in which there is delayed negative

feedback. In our case, this feedback acts through achange in the degree of liquid filling æ and the pellettemperature T. For example, if because of a randomperturbation the heat release exceeds the heat removal,the catalyst pellet temperature begins to increase. Thiscauses an increase in the partial pressure of vaporwithin the pellet and, hence, in the total evaporationrate, which leads to a decrease in the liquid filling andthe evaporation surface area. A consequence of theseprocesses is the deceleration with a time lag in theincrease in the total evaporation rate, which leads to adecrease in the partial pressure of vapor and, conse-quently, in the heat release rate. If this time lag, whichis caused by the delay in change of æ, is sufficientlylarge, then the system may “jump” over its steady stateand, furthermore, start to swing. In our case, the delayin negative feedback is ensured by the fact that thecharacteristic time it takes for the pellet to be filled withthe liquid is long enough in comparison with thethermal relaxation time of the pellet. The increase inthe amplitude of such oscillations can be held backonly by nonlinear mechanisms or additional restrictions(0 < æ < 1).

Conclusions

The self-oscillations of the temperature and liquidfraction within the partially wetted catalyst pellet inAMS hydrogenation were experimentally detected andexplained using a simplified one-dimensional model.The results of the calculations agreed with the experi-mental data on the dynamics of the temperature in thebottom part of the cylindrical pellet and the dynamicsof the liquid distribution within the pellet. This con-firmed the previously proposed mechanism7 of criticalphenomena and hysteresis of temperature and liquidfilling for a partially wetted pellet during exothermicreactions accompanied by evaporation of the liquidreactant. In the proposed mechanism, a decisive con-tribution to the total conversion rate is made by the gas-phase catalytic reaction, which is controlled by internal-diffusion evaporation. Note also the experimentallydetermined high nonuniformity of the temperature andliquid-phase distributions over a single catalyst pellet.The challenging future task is to find out how singleoscillating catalytic particles may cooperate and whethersuch self-oscillation phenomena should be avoided in anindustrial trickle-bed reactor or it may give rate en-hancement.

Acknowledgment

The financial support from NWO Grant 047.014.004is greatly acknowledged. The authors are also thankfulto Dr. I. V. Koptyug and Dr. A. Lysova for kindlypresenting NMR microimages of the intrapellet liquiddistribution.

Notations

cpcat ) 1070 J/(kg K), specific heat capacity of the material

of the pellet (Al2O3)cp

liq ) 2000 J/(kg K), specific heat capacity of liquid AMSDAH ) coefficient of binary diffusion of AMS vapor with

hydrogen, m2/sdeq ) 0.005 m, equivalent pellet diameterE ) 3.78 × 104 J/mol, activation energy of hydrogenation

of AMS vapor on a Pd/γ-Al2O3 catalyst

Figure 6. (a) Calculated phase trajectory approaching a closedorbit, (b) the comparison of the calculated and experimentallymeasured dynamics of the catalyst pellet temperature, and (c)oscillating temperature profiles at Gliq ) 0.8 mg/s.

9716 Ind. Eng. Chem. Res., Vol. 44, No. 25, 2005

Gliq ) (0.7-0.95) × 10-6 kg/s, liquid AMS mass flow rateh ) axial coordinate for a cylindrical pellet, mH ) 0.01 m, catalyst pellet heightHliq ) height of the cylindrical liquid-phase region within

the pellet, mmliq ) 0.118 kg/mol, molar weight of AMSP ) 105 Pa, total pressurePH2 ) hydrogen partial pressure in the reaction zone, PaPsat ) AMS saturation vapor pressure, PaPs ) vapor partial pressure on the pellet external surface,

PaPc ) 33.6 bar, AMS critical pressureQev(T) ) specific heat of AMS evaporation, J/molQr ) 1.16 × 105 J/mol, specific heat of reaction (hydrogena-

tion of AMS vapor) at 110 °CR ) 0.0025 m, catalyst pellet radiusRgas ) 8.31 J/(mol K), ideal gas constantRliq ) 2.05 × 10-3 m, radius of the cylindrical liquid-phase

region inside the pellet, mT(h,t) ) temperature of the catalyst pellet, °C or KTb ) 438.5 K, AMS boiling temperatureTc ) 654 K, AMS critical temperatureTgas ) 87 °C, inlet gas (H2) flow temperatureTliq ) 87 °C, inlet liquid (AMS) flow temperaturet ) time, sWr ) rate of hydrogenation of AMS vapor per unit catalyst

pellet volume, mol/(m3 s)Wev

int ) specific rate of AMS evaporation from the lateralsurface of the cylindrical catalyst pellet, mol/(m2 s)

Wevext ) specific rate of AMS evaporation from the wetted(top) surface of the cylindrical catalyst pellet, mol/(m2 s)

Xcon ) degree of conversion of AMS vapor

Greek Symbols

Rgas ) 132 W/(m2 K), coefficient of heat transfer betweenthe pellet surface and the gas flow

âgas ) 0.042 m/s, coefficient of mass transfer between thepellet surface and the gas flow

âpel ) internal-diffusion mass-transfer coefficient, m/s∆rdry ) 0.45 × 10-3 m, thickness of the dry subsurface

catalyst pellet layer surrounding the lateral surface ofthe liquid-filled cylindrical zone

∆rreac ) thickness of the dry subsurface catalyst pellet layersurrounding the liquid-filled cylindrical zone that isnecessary for complete conversion of AMS vapor, m

æ(t) ) degree of the catalyst pellet liquid fillingε ) 0.5, coefficient of intrapellet porosityλcat ) 0.2 W/(m K), thermal conductivity coefficient of the

material of the pellet (γ-Al2O3)λliq ) 0.2 W/(m K), thermal conductivity of liquid AMSFcat ) 800 kg/m3, pellet densityFliq ) 900 kg/m3, liquid AMS densityτ ) 3, coefficient of pore tortuosity

Dimensionless Groups

Nu ) 3.1, Nusselt numberPr ) 0.7, Prandtl numberRe ) 4.4, Reynolds numberSc ) 2.8, Schmidt numberSh ) 3.8, Sherwood numberSubscripts and Superscripts

ev ) evaporationext ) externalint ) internalpel, cat ) catalyst pelletliq ) liquids ) steady-state value

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Received for review February 28, 2005Revised manuscript received April 14, 2005

Accepted May 2, 2005

IE050269X

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