- Email: [email protected]
Stability of periodic travelling waves in trickle beds
Chcmid Printed
Engineeting Scimcu. in Great Britain.
Vol.
47.
No.
13114.
pp. 3257-3264.
1992. 0
STABILITY
OF PERIODIC
TRAVELLING
D.C. DANKWORTH+
WAVES
IN TRICKLE
ooo9-2509&G $.5.00+0.00 1992 Pergamon Press Lid
BEDS
and S. SUNDARESAN
Department of Chemical Engineering. Princeton University Princeton, New Jersey 08544. U.S.A.
ABSTRACT Iliquid flow together through the interstices of a random array of solid packing. A Intricklebedreactors,agasauda flOW Dauern is obtained at high gas and liquid fluxes. A macroscopic volume averaged model can be used to &the%aticaily represent the hydrodynamics of two-phase flow in packed beds. In one dimension, the onset and characteristics of pulsing flow can be modelled as periodic travelling waves. In this paper, the stability of the onedimensional travelling waves is computed in the context of one-dimensional and two-dimensional perturbations of the full model. The 1-D waves are imposed as solutions of a pseudo spectral discretization of the 2-D equations. FIoquet theory is applied to analyze the linear stability of these time-periodic solutions. The results show marked differences between the stability of fulIy developed waves and the corresponding uniform flow solutions when subjected to the same pertmbations. The analysis also shows that two-dimensional flow pattems are likely to evolve from the early stages of pulse growth, and not from breakup of fully-developed 1-D waves. DUkiDP
KEYWORDS Trickle Bed Reactor, Pulsing Flow, Travelling Waves, Hopf Bifurcation, Stability INTRODUCTION ¤t downflow of a gas and a liquid through a packed bed is widely encountered in chemical and petroleum processing. At low fluxes of liquid and gas, one obtains a steady flow regime, known as trickling. At high gas and liquid flows, a time-dependent flow referred to as pulsing is observed. Since processes such as catalytic hydrodesulphurization do operate in the pulsing regime. the conditions for onset and details of pulsing flow have been the subjects of mauy studies, and understaucling the manner in which these flows scale up is both interesting and important. We have been addressing this problem from a theoretical point of view, through an analysis of the volume averaged equations of motion. These basically consist of continuity and momentum balances for each of the phases, along with some reasonable constitutive models for some quantities such as drag force experienced by the Rowing phases. A linear stability analysis (LSA) of the uniform state in an infinite bed can be readily performed. The loss of stability has been previously identified as the onset of pulsing (Grosser cc al., 1988). The LSA reveals that the loss of stability leads to travelling waves moving down the column. Focussing on one space dimension, we constructed in an earlier study (Dankworth er al.. 1990) a map of the time-dependent regime based on certaiu global bifurcations in the model. Such a map is shown in Fig. 1. where local and global bifurcation boundaries are plotted in terms of volumetric fluxes UT and VI_. The solid curve denotes the transition to unstable uniform flow, as found by Grosser er al. (1988). The dashed curve is the locus of double hetemclinic connection (DHC) plateaus. These plateaus represent the maximum and minimum saturation of a high amplitude square-wave solution to the model for a given UT_ The square wave solution forms the terminus for all families of travelling waves evolving from unstable uniform states. The square waves themselves are born at low amplitude from a “TKBH” singularity on the stability transition locus (Dankworth. 1991). The focus of this paper will be an analysis of the stability of these families of periodic travelling waves in a two-dimensional model.
* D-C. Dankworth’s current address is Exxon Research and Engineering Co., P-0. Box 101, Florham Park, NJ 07932, USA 3257
D. C.
3258
and S.
DANKWORTH
A2
SUNDARESAN
1.0 _____-w--c__4---
.Q
e-
_4--
/@
.a
/
l
.7
l
l
.6 .ei
l
0
.4
l
l
.3
m
l
.2 .I cl -05
-06
-07
.02
TOTAL Fig. I. predicted trickling-pulsing regime map for gas-liquid flow in a packed bed. Solid curve indicates loss of stability of uniform flow; dashed curve denotes “square wave” amplitude for pulsing flow (see Dankworth et ul., 1990). A single set of bed and fluid properties was used for all stability cahzulations reported in this paper. correspond to a flow of air and water through a bed of 3 mm diameter spheres. TWO DIMENSIONAL
LINEAR
STABILITY
OF THE?UNIFORM
These
STATE
The stabihty analysis given for one-dimensional uniform flow (Dankworth et al., 1990) can be easily extended to a two dimensional flow. In Cartesian coordinates, the equations of motion in two dimensions are:
?js+a?p+e$!i
=o,
i=l,g
1 ai
ca%-
ahiz aui, a%2 s] Piei1 at+Uiz az +uiya*ay =-eix+eiPig+ %+“iE *+ a2uiy auau* api Fiy+EiF az,Y a3 PiQ[ $+Uiz*+Uiy ay 1 =-Y&t E a22 + ayz 1
0)
(2) (3)
Constitutive relations for the where ai and Eli are the volume fraction and velocity of phase i. respectively. interphase drag forces @i), the capillary pressure (pl-pg) and all parameter values used in these calculations can be found in Dankworth et uf., (1990). If normal mode disturbances of the form
xl=texf
+ j(w,
2 +
ay
Y)
are applied to these equations linearized about a ypiform solution, a characteristic second degree polynomial in the complex growth rate, x. is obtained. Here x’ and x denote disturbance and amplitude for any dependent variable.
.. A critical wavenumber combination is a wave vector (ox,ccy) at which the real part of the growth rate is zero. The locus of critical wavenumber combinations is shown in Fig. 2 for several values of liquid flux and U~=0.1. The critical wavenumber locus in each case begins at the origin, and ends at the 1-D critical wavenumber on the wz axis. The region of unstable wavenumbers inside the curve initially increases as the liquid flux increases, then decreases as the liquid flux approaches the high-flux stability transition.
A2
3259
Stability of periodic travelling waves in trickle beds
3000
-
-
i
2500 -
z. bl Q) eoO0 “E 2
isoo-
g $
1000
-
A
0
0
500
1000
1500
BOO0
z-wavenumber Fig. 2. Critical wavenumber loci for VpO.15 flux.
3500
9000
3500
4000
(m-l)
m/s and several values of liquid mass
Several conclusions can be reached from the 2-D linear stability analysis of unifom flow. We simply note thatthe capillary pressure apparently has a similar stabilizing effect in all directions; there are no new modes of instability leads to a less unstable for the uniform solution in the transverse direction: any variation in the y-dire&on condition. The maximum growth rate will occur along the 02 axis. If oy is varied while keeping oz constant, the growth rate is maximum at oy=O, and decreases monotonically as ‘oy increases. As will he seen later. the conclusionsare very differentwhen fully-developednon-uniform solutions are analyzed for transverse stability.
v-Varvmn Modes It is important to understandthe nature of the 2-D perturbationsthat are applied in (4). The normal mode disturbancewhich has both z and y components does not have the form of a travelling-z standing-y wave, but is instead a diagonal wave which travels obliquely rather than vertically. Therefore, the nonuniform fully developed solutions bifurcating from a critical wavenumber combination off of the CO,-axis will take the form of havelling waves which move at an angle to the direction of gravity. By changing the ratio of coJ_ this angle is changed. The waves are still uniform in the directiontransverseto their motion. A 2-D travellingwave analysis could be performedLOfollow the fate of these oblique travelling waves. A standing-y travelling-z wave will result from the superposition of normal modes having the same 0~ but one with This superposition eliminates the diagonal structure and leads to waves which move +wy and the other with -oy vertically, but have variations in the transverse directions. The supeqosed disturbance has the form: x* = t etst + j %
z ) cos(~y
y)
(17)
The above disturbance mode, when insertedinto the 2-D linearstabilityanalysis, yields exactly the same expression
for the complex growth rate. Thus the stability to standing-y perturbationsis identical to the stability to oblique perturbations. This means that a family of vertically travelling, y-standing waves can also bifurcate from the uniformsolutionalong the criticalwavenumberlocus. STABlLlTY OF l-DIMENSIONAL WAVES.
The algorithm for analysis of stability of the 1-D travelling waves using a pseudospectral discretization was implemented computationally. Details of the techniquecan be found in Dankworth (1991). First, the desired travelling wave solutionto (l-3) is calculatedaccuratelyat many evenly spaced collocationpoints. These values are thenused to constructthe Jacobianof the moving (discretized)steadystate. The eigenvalues and eigenvectors of the Jacobianmatrixare convertedto FloquetMultiplierform, and the most significant@ater thana specifiedthreshold) of thesearereportedas output.
D. C.
3260
1.001
1.~1I
4 h,=
DANKWORTH
and S. SUNDARESAN
A2
5.0 mm
- UT= 0.07 m/s 0.035 m/s Amplitude=O. u,= 13242
Fig. 3. Growth rate versus cey for a fully developed travelling wave solution near the TKBH.; Flux conditions and saturation amplitude are given on the figure. Critical and maximum growth wavenumbers are labeled. ve Solutions In the discretized equations, the wavelength, or box size, is a natural parameter to be varied in search of stability changes. Variation in wavelength corresponds to variation in the minimum wave number of resulting solutions. We expect from the linear stability analysis of the I-D model that the uniform flow solution will be stable for wave numbers greater than the critical wave number. At the box size corresponding to the critical wave number, the fundamental mode of the uniform state becomes unstable, and a branch of limit cycle solutions can be found to emanate from the change in stability. These periodic solutions can be continued to higher amplitudes by increasing the axial box size. The speed of the waves is given by the ratio of the box size to their period. Other Hopf bifurcations from the uniform solution occur at integer multiples of the critical box size. These bifurcations are excitations of the second. third. fourth, etc. harmonics of the spectral approximation. Branches of periodic solutions (travelling waves) emanate from these secondary Hopf bifurcations also. These periodic solutions have the same shape as the waves on the original branch, but with the waveform repeated in the spatial dimension.
To understand the full range of behavior possible in the cocurrent downllow regime map, a systematic study of travelling wave stability was performed. The regime map in Fig. 1 shows in flux coordinates the region where travelling wave solutions can exist. Several points in the interior of the “pulsing” region were chosen for further study. These points are shown as filled circles on Fig. 1.
The primary (or unimodal) branch of travelling waves is the one which evolves from the Hopf bifurcation of the fundamental mode (i.e. at the first crossing of the critical wavenumber). These waves are distinguished from the other, multimodal branches by their appearance of having a single “hump”. If the liquid flux is constrained to a constant average value, the primary branch forms a one-parameter family of solutions, the parameter being axial wavelength. At high amplitude, as the limiting “square wave” solution is approached, all the primary branches are stable for the U~==0.15 case. Notice on the Regime Map shown in Fig. 1 that the double heteroclinic connection at this value of UT has both plateaus in the stable uniform flow region of the map. It is possible (as for UTc 0.075 in Fig. 1) for the limiting square wave solution to have one or more plateaus in the unstable uniform region. One would not expect that the infinite-period solution would be stable in such a case. This was impossible to check with the
A2
Stability of periodic travelling waves
3261
in trickle beds
600
500 -
‘; 2
400
k 2
300
c iE $ 200 L 100
-0
200
400
600 600 z-wavenumber
1000 (m-l)
1200
1400
1600
Fig. 4. Comparison of critical loci for fully developed and uniform state solutions.; Flux conditions are +-=0.07 U~=O.035 m/s. Solid curve is critical locus for uniform, dashed curve is critical locus for fully developed travelling waves (decreasing zwavenumber corresponds to increasing amplitude). Squares denote points where calculations were performed for the travelling wave case. present implementation, bifurcation.
however,
due to the difficulty
in obtaining accutate wave profiles
near the global
The blmodal (two-hump) branch was also examined. Since the bimodal branch evolves from a secondary Hopf bifurcation of the first harmonic of the overall wavelength, the resulting limit cycles are always initially unstable. This is due to the instability of the fundamental mode which was excited at a smaller wave number. Thus, the bimodaI travelling waves (which have the spatial pattern repeat4 twice) are expected to be unstable at low amplitude. This is indeed confiied by our calculations. STABILITY
OF 1-D WAVES
IN 2-DIMENSIONS
Having developed tbe basic framework for calculation of tlte stability of travelling periodic 1-D solutions using the travelling wave ODE and a spectral discrctization, it is a straightforward extension to also consider the stability of the 1-D waves to perturbations in both the axial and transverse directions. This section will include details of the results of the analysis. A full description of the analysis technique can be found in Dankworth (1991). The important differences between the 2-D and 1-D stability calculations are that there are more equations (PDEs) and variables in the 2-D system, and both the axial and lateral box size enter as parameters. v of a Smele Wave to v Pe The stability of a given travelling wave solution to transverse perturbations was found to depend on the transverse box size. The axial box size is fixed by the periodic nature of the travelling wave solution, and remains constant_ Figure 3 shows the variation of the dominant Floquet multiplier with y-wavenumber for a particular solution along a wave having U~O.07, U~=O.035 and amplitude of 0.13242. The Bow condition corresponding to this wave can be found on Fig. 1. Notice that Fig. 3 represents conditions very near the TKBH singularity. At low wavenumber (large box size), the wave is marginally stable to transverse perturbations. As the wavenumber increases, the The waves eventually become stable FIoquet multiplier increases above 1.0, and exhibits a maximum at 160 m-l. to y-perturbations as the wavenumber exceeds a critical value (235 m-l). Compare the above results with those obtained from the linear stability analysis of the uniform solution discussed earlier. In that case, if the transverse wavenumber was varied from zero while the axial wavenumber remained fixed,
3262
A2
D. C. DANKWORTH and S. SUNDARESAN
. . . . . . . . .
. . . . * . . . .
. . . . . . . . .
* . _ . _ . . .
. . . . . . . .
. . . . . . .
. . . _ .
. . . . . . .
. . . . . . .
_ . . . _ . .
9 .
1
,’
I
....... .......
S2
.
,,‘l+b~I .
.
,,
,y i
.
.
.
.
.
.
.
.
.
. .
. .
. .
. .
. .
. .
_
-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‘,:
. . .
+
C
.
. .
. . . . . . .
. . . . . . .
. . . . . . .
. * . _ * .
. . . . . .
. . . . . .
_ _ . .
. . . . . .
. . . . .
. . . . _ *
Fig. 5. Schematic diagram depicting the bypassing mechanism of transverse instability.; Liquid saturation is indicated by degree of shading. Gas prefers to bypass the high saturation region (Sl) by flowing through the less saturated region (S2). This tendency is dampened by the extra distance the gas must travel through the gas-rich region.
the growth rate was found to be a monotonically decreasing function of the transverse wavenumber. with a maximum at zero wavenumber. This implies, as expected, that the 2-D linear analysis of the uniform state does not hold valid for nonuniform solutions. There is evidently a different mechanism of destabilization which occurs in the nonuniform case which does not enter into the uniform analysis. of Waves to v-Pem Building on the results for a single wave presented iu the previous section, the analysis can be extended to study the changes in the growth rate vs. wavenumber relationship as the travelling wave solutions are followed for increasing amplitude (and Lz) along a constant flux branch. Starting at very low amplitude, near the critical z-wavenumber Hopf bifurcation, the critical y-wavenumber is large. as the uniform analysis suggests. Higher amplitude (lower zwavenumber) solutions initially have larger critical y-wavenumbers. At some amplitude, this trend reverses, and the critical y-wavenumber starts to decrease with increasing amplitude. The maximum growth rate also decreases. As the limiting amplitude of the square wave is approached, the solutions become more and more stable and eventually, at a finite z-wavenumber, become stable to transverse perturbations. It is possible to construct a critical wavenumber locus for the stability of fully developed solutions similar to those constructed for the uniform analysis shown in Fig. 2. For the fully-developed waves, the z-wavenumber axis represents solutions of different amplitude, with the high amplitude square-wave solution occurring at cnz=O_ Figure 4 shows such a locus for a family of waves at Q-=0.07 m/s. U~=O.035 m/s. The corresponding critical locus for the uniform solution having the same UT and UL is also shown for comparison. The critical locus for the fully developed waves does not return to the origin as the axial wavenumber goes to zero. It intersects the oz axis at a finite wavenumber. For some flux conditions, nearthe TKBH point, the travelling waves are entirely stable to 2-D perturbations (e.g. L+O.O65. UL=O.O39). Several constant flux families were studied in this way, from U~=O.065 to U~=0.15. All of the families exhibited the same qualitative behavior. From these results, it can be concluded that while there is a new destabilizing effect present when the stability of a fully developed finite amplitude wave is considered, there is also a stabilizing effect which invariably makes the very large amplitude waves stable to all small y-perturbations. A possible physical interpretation of these effects is as follows. Let us start with a 1-D solution mapped into a 2-D box, as in Fig. 5, so that the darker region is the liquid-rich portion of the pulse. Let us suppose that we open up the liquid-rich segment so that there are regions where there is a lower liquid saturation (S2). The velocity of the gas in the gas-rich segment is in general different from that of the wave itself. When we make the perturbation in liquid saturation, the gas has two options. It can either go straight through,or take a circuituousroute through the liquidlean path. If the gas bypasses. the gas flux in the liquid-lean segment increases and this contributes to spread the liquid there even further. Thus bypassing is destabilizing. This is counteracted by the effect of lateral variation in
A2
Stability of periodic travelling waves in trickle beds
3263
Fig, 6. Comparison of the maximum rates of 1-D coalescence and initial transverse growth for two families of travelling waves.; UT”o.10, U1_=0.05, and U1==0,07, U~=O.O35 mls. the axial veIocity of the liiuid Such a variation leads to a deviatoric stress in the llfodel which would act to restore the region to uniform conditions. DISCUSSICBN
AND
SUMMARY
In the initial section of this paper, it was stated that the unifurm state is most tmstabie to Imrely vertical travelling wave disturbances, although other Z-dimensional instabilities were also present. This suggests that some segregation of the gas and liquid may cccur in the axial direction before any lateral segregation patterns will be observed, It seems reasonable to expect that this occurs through repeated growth and coalescence of of short wavelength, low amplitude waves. The 2-D linear stability analysis of the fully developed travelllng waves showed that low amplitude waves are also unstabie in the transverse direction, which suggested that formation of lateral segregation patterns is possible. In one-dimension, tutveIling waves are found to be unstable at low amplitude, but stable at high amplitude. PhysicaI ubservatlons of pulsing flow, however, suggest that localized pulsing and laterally segregated flow patterns da arise (Chrlstenseu er al., 1986). At some stage in the inception of a macroscopic pulse, the tendency for transverse growth must be comparable to the tendency for axial coalescence. Figure 6 shows a comparison of the growth rates for axial coalescence and tmusverse growth as a function of the ampfit82de of travelhug waves alon two matam fkx families of solutious. The growth rates are taken as the value of the dominant eigenvalues ( in s’5 ) for the two mechanisms. The two families have the same ratio of IJ@lTW but different UT. At low amplitude (high wavenumher) the coalescence rate is faster than the rate of growth of transverse patterns for both families. Corner of the two rates as solitude increases suggests that different behavior might be observed at the two flux conditions. For the U-I=O. IO m/s family, the ~~e~w~~~~ while the coalescence rate dt9zaxs, and eventually the two rates are similar in magnitude. This occurs at an amplitude of 0.15. The wavelength of the fully developed waves at this amplitude is still very close to the critical wavelength, on the order of a particle diameter. For waves of amplitude greater than 0.15, the transverse growth dominates. Eventually, as the maximum amplitude (square wave) solution is approached (at an amplitude of 0.43), both the growth rates diminish to zero. Since the transverse growth daminates over coalescence at such an early stage in the growth, it is likely that fully developed pulsing flow
D: C.
3264
DANKWORTH
A2
and S. SUNDARESAN
growth dominates over coalescence at such an early stage in the growth, it is likely that fully developed pulsing flow at these fluxes will exhibit localized 2-D structure at all stages of growth, rather than fast forming a 1-D pattern then breaking into 2-D structures. Experimental observations reported by Christensen et al. (1986) support this conclusion. The pulses they observed in a 2-D column were already smaller than the cmss section of the bed when fmt observed. No further breakup of tbe macroscopic pulses in the lateral direction was encountered. For UT=O.O7 m/s, both rates are lower, and the coalescence rate remains substantially greater than the maximum transverse growth rate at all amplitudes where the wave is unstable. The two curves cross only after the waves have become stable to both coalescence and transverse growth. This suggests that pulsing flow at these conditions should not tend to have a Z-D structure if initiated uniformly across the bed. From the above comparison of two flux conditions, one may conclude that there is a trend to greater 2-D instability as the total flux is increased away horn the TKBH singularity. In summary, a 2-D linear stability analysis around the uniform “trickling” flow solution suggests that lateral perturbations have a stabilizing effect. In contrast, 2-D LSA of fully developed 1-D waves suggests that lateral perturbations can be destabilizing. A mechanistic reasoning for this is presented, In the high amplitude limit, however, the 1-D waves were found to be marginally stable to lateral perturbations. These results lead us to conclude that the evolution of 2-D flow patterns occurs quite early in the wave growth process. ACKNOWLEDGEMENT The support of the Fannie and John Hertz Foundation through a Graduate Fellowship gratefully acknowledged.
for D.C. Dankworth is
NOMENCLATURE Fix, i = 1.g
Xi* i = 1s i; UT Y z
i= 1.g
volume fraction of phase i in the
drag force per unit bed volume experienced by phase i in the axial direction. acceleration due to gravity pressure in phase i time local mean velocity of phase i superficial velocity of the liquid combined superficial velocity of the liquid and gas phases transverse distance axial distance
5
V
bed wavelength density of phase i effective nscosity interfacial tension axial and transverse components of the 2-D wavenumber real part of the complex growth rate imaginary part of the complex growth rate complex growth rate double heteroclinic connection Talcens-Bogdanov-Hysteresis (double xem + cusp)
singukuity
REFERENCES Christensen, G.. S.J. McGovern, S. Sundaresan (1986). Cocunent downflow of air and water in a two-dimensional packed column. MChE J., 2, 1677-1689 Dankworth, D.C. (1991). Macroscopic Structure of Time-Dependent Two-Phase Flow Regimes in Packed Beds. Ph.D. Dissertation, Dept. of Chemical Engineering, Princeton University Dankworth, DC.. Kevrekidis, LG.. S. Sundaresan (1990) Dynamics of pulsing flow in trickle beds. AfC!rE J.. s. 605-62 1 Grosser, K.A., R.G. Carbonell. S. Sundaresan (1988). Onset of pulsing in two-phase cocurrent downflow through a packed bed. AfChl? J. ,B, 1850-1860
Engineeting Scimcu. in Great Britain.
Vol.
47.
No.
13114.
pp. 3257-3264.
1992. 0
STABILITY
OF PERIODIC
TRAVELLING
D.C. DANKWORTH+
WAVES
IN TRICKLE
ooo9-2509&G $.5.00+0.00 1992 Pergamon Press Lid
BEDS
and S. SUNDARESAN
Department of Chemical Engineering. Princeton University Princeton, New Jersey 08544. U.S.A.
ABSTRACT Iliquid flow together through the interstices of a random array of solid packing. A Intricklebedreactors,agasauda flOW Dauern is obtained at high gas and liquid fluxes. A macroscopic volume averaged model can be used to &the%aticaily represent the hydrodynamics of two-phase flow in packed beds. In one dimension, the onset and characteristics of pulsing flow can be modelled as periodic travelling waves. In this paper, the stability of the onedimensional travelling waves is computed in the context of one-dimensional and two-dimensional perturbations of the full model. The 1-D waves are imposed as solutions of a pseudo spectral discretization of the 2-D equations. FIoquet theory is applied to analyze the linear stability of these time-periodic solutions. The results show marked differences between the stability of fulIy developed waves and the corresponding uniform flow solutions when subjected to the same pertmbations. The analysis also shows that two-dimensional flow pattems are likely to evolve from the early stages of pulse growth, and not from breakup of fully-developed 1-D waves. DUkiDP
KEYWORDS Trickle Bed Reactor, Pulsing Flow, Travelling Waves, Hopf Bifurcation, Stability INTRODUCTION ¤t downflow of a gas and a liquid through a packed bed is widely encountered in chemical and petroleum processing. At low fluxes of liquid and gas, one obtains a steady flow regime, known as trickling. At high gas and liquid flows, a time-dependent flow referred to as pulsing is observed. Since processes such as catalytic hydrodesulphurization do operate in the pulsing regime. the conditions for onset and details of pulsing flow have been the subjects of mauy studies, and understaucling the manner in which these flows scale up is both interesting and important. We have been addressing this problem from a theoretical point of view, through an analysis of the volume averaged equations of motion. These basically consist of continuity and momentum balances for each of the phases, along with some reasonable constitutive models for some quantities such as drag force experienced by the Rowing phases. A linear stability analysis (LSA) of the uniform state in an infinite bed can be readily performed. The loss of stability has been previously identified as the onset of pulsing (Grosser cc al., 1988). The LSA reveals that the loss of stability leads to travelling waves moving down the column. Focussing on one space dimension, we constructed in an earlier study (Dankworth er al.. 1990) a map of the time-dependent regime based on certaiu global bifurcations in the model. Such a map is shown in Fig. 1. where local and global bifurcation boundaries are plotted in terms of volumetric fluxes UT and VI_. The solid curve denotes the transition to unstable uniform flow, as found by Grosser er al. (1988). The dashed curve is the locus of double hetemclinic connection (DHC) plateaus. These plateaus represent the maximum and minimum saturation of a high amplitude square-wave solution to the model for a given UT_ The square wave solution forms the terminus for all families of travelling waves evolving from unstable uniform states. The square waves themselves are born at low amplitude from a “TKBH” singularity on the stability transition locus (Dankworth. 1991). The focus of this paper will be an analysis of the stability of these families of periodic travelling waves in a two-dimensional model.
* D-C. Dankworth’s current address is Exxon Research and Engineering Co., P-0. Box 101, Florham Park, NJ 07932, USA 3257
D. C.
3258
and S.
DANKWORTH
A2
SUNDARESAN
1.0 _____-w--c__4---
.Q
e-
_4--
/@
.a
/
l
.7
l
l
.6 .ei
l
0
.4
l
l
.3
m
l
.2 .I cl -05
-06
-07
.02
TOTAL Fig. I. predicted trickling-pulsing regime map for gas-liquid flow in a packed bed. Solid curve indicates loss of stability of uniform flow; dashed curve denotes “square wave” amplitude for pulsing flow (see Dankworth et ul., 1990). A single set of bed and fluid properties was used for all stability cahzulations reported in this paper. correspond to a flow of air and water through a bed of 3 mm diameter spheres. TWO DIMENSIONAL
LINEAR
STABILITY
OF THE?UNIFORM
These
STATE
The stabihty analysis given for one-dimensional uniform flow (Dankworth et al., 1990) can be easily extended to a two dimensional flow. In Cartesian coordinates, the equations of motion in two dimensions are:
?js+a?p+e$!i
=o,
i=l,g
1 ai
ca%-
ahiz aui, a%2 s] Piei1 at+Uiz az +uiya*ay =-eix+eiPig+ %+“iE *+ a2uiy auau* api Fiy+EiF az,Y a3 PiQ[ $+Uiz*+Uiy ay 1 =-Y&t E a22 + ayz 1
0)
(2) (3)
Constitutive relations for the where ai and Eli are the volume fraction and velocity of phase i. respectively. interphase drag forces @i), the capillary pressure (pl-pg) and all parameter values used in these calculations can be found in Dankworth et uf., (1990). If normal mode disturbances of the form
xl=texf
+ j(w,
2 +
ay
Y)
are applied to these equations linearized about a ypiform solution, a characteristic second degree polynomial in the complex growth rate, x. is obtained. Here x’ and x denote disturbance and amplitude for any dependent variable.
.. A critical wavenumber combination is a wave vector (ox,ccy) at which the real part of the growth rate is zero. The locus of critical wavenumber combinations is shown in Fig. 2 for several values of liquid flux and U~=0.1. The critical wavenumber locus in each case begins at the origin, and ends at the 1-D critical wavenumber on the wz axis. The region of unstable wavenumbers inside the curve initially increases as the liquid flux increases, then decreases as the liquid flux approaches the high-flux stability transition.
A2
3259
Stability of periodic travelling waves in trickle beds
3000
-
-
i
2500 -
z. bl Q) eoO0 “E 2
isoo-
g $
1000
-
A
0
0
500
1000
1500
BOO0
z-wavenumber Fig. 2. Critical wavenumber loci for VpO.15 flux.
3500
9000
3500
4000
(m-l)
m/s and several values of liquid mass
Several conclusions can be reached from the 2-D linear stability analysis of unifom flow. We simply note thatthe capillary pressure apparently has a similar stabilizing effect in all directions; there are no new modes of instability leads to a less unstable for the uniform solution in the transverse direction: any variation in the y-dire&on condition. The maximum growth rate will occur along the 02 axis. If oy is varied while keeping oz constant, the growth rate is maximum at oy=O, and decreases monotonically as ‘oy increases. As will he seen later. the conclusionsare very differentwhen fully-developednon-uniform solutions are analyzed for transverse stability.
v-Varvmn Modes It is important to understandthe nature of the 2-D perturbationsthat are applied in (4). The normal mode disturbancewhich has both z and y components does not have the form of a travelling-z standing-y wave, but is instead a diagonal wave which travels obliquely rather than vertically. Therefore, the nonuniform fully developed solutions bifurcating from a critical wavenumber combination off of the CO,-axis will take the form of havelling waves which move at an angle to the direction of gravity. By changing the ratio of coJ_ this angle is changed. The waves are still uniform in the directiontransverseto their motion. A 2-D travellingwave analysis could be performedLOfollow the fate of these oblique travelling waves. A standing-y travelling-z wave will result from the superposition of normal modes having the same 0~ but one with This superposition eliminates the diagonal structure and leads to waves which move +wy and the other with -oy vertically, but have variations in the transverse directions. The supeqosed disturbance has the form: x* = t etst + j %
z ) cos(~y
y)
(17)
The above disturbance mode, when insertedinto the 2-D linearstabilityanalysis, yields exactly the same expression
for the complex growth rate. Thus the stability to standing-y perturbationsis identical to the stability to oblique perturbations. This means that a family of vertically travelling, y-standing waves can also bifurcate from the uniformsolutionalong the criticalwavenumberlocus. STABlLlTY OF l-DIMENSIONAL WAVES.
The algorithm for analysis of stability of the 1-D travelling waves using a pseudospectral discretization was implemented computationally. Details of the techniquecan be found in Dankworth (1991). First, the desired travelling wave solutionto (l-3) is calculatedaccuratelyat many evenly spaced collocationpoints. These values are thenused to constructthe Jacobianof the moving (discretized)steadystate. The eigenvalues and eigenvectors of the Jacobianmatrixare convertedto FloquetMultiplierform, and the most significant@ater thana specifiedthreshold) of thesearereportedas output.
D. C.
3260
1.001
1.~1I
4 h,=
DANKWORTH
and S. SUNDARESAN
A2
5.0 mm
- UT= 0.07 m/s 0.035 m/s Amplitude=O. u,= 13242
Fig. 3. Growth rate versus cey for a fully developed travelling wave solution near the TKBH.; Flux conditions and saturation amplitude are given on the figure. Critical and maximum growth wavenumbers are labeled. ve Solutions In the discretized equations, the wavelength, or box size, is a natural parameter to be varied in search of stability changes. Variation in wavelength corresponds to variation in the minimum wave number of resulting solutions. We expect from the linear stability analysis of the I-D model that the uniform flow solution will be stable for wave numbers greater than the critical wave number. At the box size corresponding to the critical wave number, the fundamental mode of the uniform state becomes unstable, and a branch of limit cycle solutions can be found to emanate from the change in stability. These periodic solutions can be continued to higher amplitudes by increasing the axial box size. The speed of the waves is given by the ratio of the box size to their period. Other Hopf bifurcations from the uniform solution occur at integer multiples of the critical box size. These bifurcations are excitations of the second. third. fourth, etc. harmonics of the spectral approximation. Branches of periodic solutions (travelling waves) emanate from these secondary Hopf bifurcations also. These periodic solutions have the same shape as the waves on the original branch, but with the waveform repeated in the spatial dimension.
To understand the full range of behavior possible in the cocurrent downllow regime map, a systematic study of travelling wave stability was performed. The regime map in Fig. 1 shows in flux coordinates the region where travelling wave solutions can exist. Several points in the interior of the “pulsing” region were chosen for further study. These points are shown as filled circles on Fig. 1.
The primary (or unimodal) branch of travelling waves is the one which evolves from the Hopf bifurcation of the fundamental mode (i.e. at the first crossing of the critical wavenumber). These waves are distinguished from the other, multimodal branches by their appearance of having a single “hump”. If the liquid flux is constrained to a constant average value, the primary branch forms a one-parameter family of solutions, the parameter being axial wavelength. At high amplitude, as the limiting “square wave” solution is approached, all the primary branches are stable for the U~==0.15 case. Notice on the Regime Map shown in Fig. 1 that the double heteroclinic connection at this value of UT has both plateaus in the stable uniform flow region of the map. It is possible (as for UTc 0.075 in Fig. 1) for the limiting square wave solution to have one or more plateaus in the unstable uniform region. One would not expect that the infinite-period solution would be stable in such a case. This was impossible to check with the
A2
Stability of periodic travelling waves
3261
in trickle beds
600
500 -
‘; 2
400
k 2
300
c iE $ 200 L 100
-0
200
400
600 600 z-wavenumber
1000 (m-l)
1200
1400
1600
Fig. 4. Comparison of critical loci for fully developed and uniform state solutions.; Flux conditions are +-=0.07 U~=O.035 m/s. Solid curve is critical locus for uniform, dashed curve is critical locus for fully developed travelling waves (decreasing zwavenumber corresponds to increasing amplitude). Squares denote points where calculations were performed for the travelling wave case. present implementation, bifurcation.
however,
due to the difficulty
in obtaining accutate wave profiles
near the global
The blmodal (two-hump) branch was also examined. Since the bimodal branch evolves from a secondary Hopf bifurcation of the first harmonic of the overall wavelength, the resulting limit cycles are always initially unstable. This is due to the instability of the fundamental mode which was excited at a smaller wave number. Thus, the bimodaI travelling waves (which have the spatial pattern repeat4 twice) are expected to be unstable at low amplitude. This is indeed confiied by our calculations. STABILITY
OF 1-D WAVES
IN 2-DIMENSIONS
Having developed tbe basic framework for calculation of tlte stability of travelling periodic 1-D solutions using the travelling wave ODE and a spectral discrctization, it is a straightforward extension to also consider the stability of the 1-D waves to perturbations in both the axial and transverse directions. This section will include details of the results of the analysis. A full description of the analysis technique can be found in Dankworth (1991). The important differences between the 2-D and 1-D stability calculations are that there are more equations (PDEs) and variables in the 2-D system, and both the axial and lateral box size enter as parameters. v of a Smele Wave to v Pe The stability of a given travelling wave solution to transverse perturbations was found to depend on the transverse box size. The axial box size is fixed by the periodic nature of the travelling wave solution, and remains constant_ Figure 3 shows the variation of the dominant Floquet multiplier with y-wavenumber for a particular solution along a wave having U~O.07, U~=O.035 and amplitude of 0.13242. The Bow condition corresponding to this wave can be found on Fig. 1. Notice that Fig. 3 represents conditions very near the TKBH singularity. At low wavenumber (large box size), the wave is marginally stable to transverse perturbations. As the wavenumber increases, the The waves eventually become stable FIoquet multiplier increases above 1.0, and exhibits a maximum at 160 m-l. to y-perturbations as the wavenumber exceeds a critical value (235 m-l). Compare the above results with those obtained from the linear stability analysis of the uniform solution discussed earlier. In that case, if the transverse wavenumber was varied from zero while the axial wavenumber remained fixed,
3262
A2
D. C. DANKWORTH and S. SUNDARESAN
. . . . . . . . .
. . . . * . . . .
. . . . . . . . .
* . _ . _ . . .
. . . . . . . .
. . . . . . .
. . . _ .
. . . . . . .
. . . . . . .
_ . . . _ . .
9 .
1
,’
I
....... .......
S2
.
,,‘l+b~I .
.
,,
,y i
.
.
.
.
.
.
.
.
.
. .
. .
. .
. .
. .
. .
_
-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
‘,:
. . .
+
C
.
. .
. . . . . . .
. . . . . . .
. . . . . . .
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. . . . . .
. . . . . .
_ _ . .
. . . . . .
. . . . .
. . . . _ *
Fig. 5. Schematic diagram depicting the bypassing mechanism of transverse instability.; Liquid saturation is indicated by degree of shading. Gas prefers to bypass the high saturation region (Sl) by flowing through the less saturated region (S2). This tendency is dampened by the extra distance the gas must travel through the gas-rich region.
the growth rate was found to be a monotonically decreasing function of the transverse wavenumber. with a maximum at zero wavenumber. This implies, as expected, that the 2-D linear analysis of the uniform state does not hold valid for nonuniform solutions. There is evidently a different mechanism of destabilization which occurs in the nonuniform case which does not enter into the uniform analysis. of Waves to v-Pem Building on the results for a single wave presented iu the previous section, the analysis can be extended to study the changes in the growth rate vs. wavenumber relationship as the travelling wave solutions are followed for increasing amplitude (and Lz) along a constant flux branch. Starting at very low amplitude, near the critical z-wavenumber Hopf bifurcation, the critical y-wavenumber is large. as the uniform analysis suggests. Higher amplitude (lower zwavenumber) solutions initially have larger critical y-wavenumbers. At some amplitude, this trend reverses, and the critical y-wavenumber starts to decrease with increasing amplitude. The maximum growth rate also decreases. As the limiting amplitude of the square wave is approached, the solutions become more and more stable and eventually, at a finite z-wavenumber, become stable to transverse perturbations. It is possible to construct a critical wavenumber locus for the stability of fully developed solutions similar to those constructed for the uniform analysis shown in Fig. 2. For the fully-developed waves, the z-wavenumber axis represents solutions of different amplitude, with the high amplitude square-wave solution occurring at cnz=O_ Figure 4 shows such a locus for a family of waves at Q-=0.07 m/s. U~=O.035 m/s. The corresponding critical locus for the uniform solution having the same UT and UL is also shown for comparison. The critical locus for the fully developed waves does not return to the origin as the axial wavenumber goes to zero. It intersects the oz axis at a finite wavenumber. For some flux conditions, nearthe TKBH point, the travelling waves are entirely stable to 2-D perturbations (e.g. L+O.O65. UL=O.O39). Several constant flux families were studied in this way, from U~=O.065 to U~=0.15. All of the families exhibited the same qualitative behavior. From these results, it can be concluded that while there is a new destabilizing effect present when the stability of a fully developed finite amplitude wave is considered, there is also a stabilizing effect which invariably makes the very large amplitude waves stable to all small y-perturbations. A possible physical interpretation of these effects is as follows. Let us start with a 1-D solution mapped into a 2-D box, as in Fig. 5, so that the darker region is the liquid-rich portion of the pulse. Let us suppose that we open up the liquid-rich segment so that there are regions where there is a lower liquid saturation (S2). The velocity of the gas in the gas-rich segment is in general different from that of the wave itself. When we make the perturbation in liquid saturation, the gas has two options. It can either go straight through,or take a circuituousroute through the liquidlean path. If the gas bypasses. the gas flux in the liquid-lean segment increases and this contributes to spread the liquid there even further. Thus bypassing is destabilizing. This is counteracted by the effect of lateral variation in
A2
Stability of periodic travelling waves in trickle beds
3263
Fig, 6. Comparison of the maximum rates of 1-D coalescence and initial transverse growth for two families of travelling waves.; UT”o.10, U1_=0.05, and U1==0,07, U~=O.O35 mls. the axial veIocity of the liiuid Such a variation leads to a deviatoric stress in the llfodel which would act to restore the region to uniform conditions. DISCUSSICBN
AND
SUMMARY
In the initial section of this paper, it was stated that the unifurm state is most tmstabie to Imrely vertical travelling wave disturbances, although other Z-dimensional instabilities were also present. This suggests that some segregation of the gas and liquid may cccur in the axial direction before any lateral segregation patterns will be observed, It seems reasonable to expect that this occurs through repeated growth and coalescence of of short wavelength, low amplitude waves. The 2-D linear stability analysis of the fully developed travelllng waves showed that low amplitude waves are also unstabie in the transverse direction, which suggested that formation of lateral segregation patterns is possible. In one-dimension, tutveIling waves are found to be unstable at low amplitude, but stable at high amplitude. PhysicaI ubservatlons of pulsing flow, however, suggest that localized pulsing and laterally segregated flow patterns da arise (Chrlstenseu er al., 1986). At some stage in the inception of a macroscopic pulse, the tendency for transverse growth must be comparable to the tendency for axial coalescence. Figure 6 shows a comparison of the growth rates for axial coalescence and tmusverse growth as a function of the ampfit82de of travelhug waves alon two matam fkx families of solutious. The growth rates are taken as the value of the dominant eigenvalues ( in s’5 ) for the two mechanisms. The two families have the same ratio of IJ@lTW but different UT. At low amplitude (high wavenumher) the coalescence rate is faster than the rate of growth of transverse patterns for both families. Corner of the two rates as solitude increases suggests that different behavior might be observed at the two flux conditions. For the U-I=O. IO m/s family, the ~~e~w~~~~ while the coalescence rate dt9zaxs, and eventually the two rates are similar in magnitude. This occurs at an amplitude of 0.15. The wavelength of the fully developed waves at this amplitude is still very close to the critical wavelength, on the order of a particle diameter. For waves of amplitude greater than 0.15, the transverse growth dominates. Eventually, as the maximum amplitude (square wave) solution is approached (at an amplitude of 0.43), both the growth rates diminish to zero. Since the transverse growth daminates over coalescence at such an early stage in the growth, it is likely that fully developed pulsing flow
D: C.
3264
DANKWORTH
A2
and S. SUNDARESAN
growth dominates over coalescence at such an early stage in the growth, it is likely that fully developed pulsing flow at these fluxes will exhibit localized 2-D structure at all stages of growth, rather than fast forming a 1-D pattern then breaking into 2-D structures. Experimental observations reported by Christensen et al. (1986) support this conclusion. The pulses they observed in a 2-D column were already smaller than the cmss section of the bed when fmt observed. No further breakup of tbe macroscopic pulses in the lateral direction was encountered. For UT=O.O7 m/s, both rates are lower, and the coalescence rate remains substantially greater than the maximum transverse growth rate at all amplitudes where the wave is unstable. The two curves cross only after the waves have become stable to both coalescence and transverse growth. This suggests that pulsing flow at these conditions should not tend to have a Z-D structure if initiated uniformly across the bed. From the above comparison of two flux conditions, one may conclude that there is a trend to greater 2-D instability as the total flux is increased away horn the TKBH singularity. In summary, a 2-D linear stability analysis around the uniform “trickling” flow solution suggests that lateral perturbations have a stabilizing effect. In contrast, 2-D LSA of fully developed 1-D waves suggests that lateral perturbations can be destabilizing. A mechanistic reasoning for this is presented, In the high amplitude limit, however, the 1-D waves were found to be marginally stable to lateral perturbations. These results lead us to conclude that the evolution of 2-D flow patterns occurs quite early in the wave growth process. ACKNOWLEDGEMENT The support of the Fannie and John Hertz Foundation through a Graduate Fellowship gratefully acknowledged.
for D.C. Dankworth is
NOMENCLATURE Fix, i = 1.g
Xi* i = 1s i; UT Y z
i= 1.g
volume fraction of phase i in the
drag force per unit bed volume experienced by phase i in the axial direction. acceleration due to gravity pressure in phase i time local mean velocity of phase i superficial velocity of the liquid combined superficial velocity of the liquid and gas phases transverse distance axial distance
5
V
bed wavelength density of phase i effective nscosity interfacial tension axial and transverse components of the 2-D wavenumber real part of the complex growth rate imaginary part of the complex growth rate complex growth rate double heteroclinic connection Talcens-Bogdanov-Hysteresis (double xem + cusp)
singukuity
REFERENCES Christensen, G.. S.J. McGovern, S. Sundaresan (1986). Cocunent downflow of air and water in a two-dimensional packed column. MChE J., 2, 1677-1689 Dankworth, D.C. (1991). Macroscopic Structure of Time-Dependent Two-Phase Flow Regimes in Packed Beds. Ph.D. Dissertation, Dept. of Chemical Engineering, Princeton University Dankworth, DC.. Kevrekidis, LG.. S. Sundaresan (1990) Dynamics of pulsing flow in trickle beds. AfC!rE J.. s. 605-62 1 Grosser, K.A., R.G. Carbonell. S. Sundaresan (1988). Onset of pulsing in two-phase cocurrent downflow through a packed bed. AfChl? J. ,B, 1850-1860