Non-uniform distribution of two-phase flows through parallel identical paths.

Grace, John R. ; Cui, Heping ; Elnashaie, Said S.E.H. 等

INTRODUCTION

Two- and three-phase flows are common in engineering applications. Equipment for these operations can take many physical forms. In some cases where steady-state continuous operations are carried out, there are multiple matching branches in the flow path, with the flow expected to pass in a uniform manner through the different branches. This can arise, for example, when particles are pneumatically or hydraulically conveyed and distributed to multiple feed points in a reactor, or when there are two or more cyclones in parallel at the exit of fluidized bed reactors, rotary kilns or particle dryers.

When a single-phase fluid flowing continuously at steady state is split into N parallel flow paths, which then recombine at a common destination, as illustrated schematically in Figure 1, the requirement that the pressure drop be identical through each of the paths results in a uniform distribution of the fluid, i.e., we expect a fraction of (1/N) to pass through each of the separate channels. However, when there are two or more phases present in the flow, it may be possible for there to be non-uniform distributions, which still equalize the pressure drops through the parallel paths. For example, in liquid-solid flows through a two-path bifurcation as shown in Figure 1, one could imagine more liquid travelling through one side and more solid particles through the other, giving identical pressure drops. This paper suggests that not only can this occur, but that in practice there are many instances where it does occur and where the non-uniform distribution is in fact a stable outcome. As discussed below, this non-uniformity may be of considerable practical importance in such cases.

We also note in passing that there are other situations in fluid mechanics where, despite uniform and symmetrical boundary conditions, non-linearity can result in non-uniformity and asymmetry. Examples include the Coanda effect, some flows through sudden expansions, and flows past circular pipes or spheres at certain Reynolds numbers. Other flows, such as opposing jets, can be symmetric on a time-mean basis, but instantaneously asymmetric.

We concentrate in this paper on cases where there are two phases, one composed of solid particles and the other a gas. We believe, however, that similar non-uniform flow situations may also arise in other two-phase contactors (e.g. gas-liquid flows in nuclear reactors or fuel cells) or in three-phase systems. We begin by considering evidence of the non-uniformity in actual flow situations involving gas-solid suspensions.

[FIGURE 1 OMITTED]

NON-UNIFORMITY OF GAS-SOLID FLOWS THROUGH PARALLEL PATHS

Parallel Channels inside Fluidized Beds

Bolthrunis et al. (2004) describe difficulties which plagued early fluidized bed reactors designed and operated for the production of ethylene oxide by direct oxidation of ethylene, hydrocarbon synthesis via the Fischer-Tropsch process, and phthalic anhydride manufacture. In each of these cases, early reactors featured multiple open vertical heat transfer tubes of equal length and diameter, suspended within fluidized beds. It was intended that the fluidized suspension would pass uniformly through each of the parallel passages. However, in practice there was substantial and recurring maldistribution. The description provided by the authors is instructive: "In some alternate paths the flow may be almost free of solids and friction losses may predominate; in others there may be almost no gas flow and static head losses prevail. The system is inherently unstable. In extreme cases, some tubes will plug with solids and others will operate with high velocity and low catalyst loading. In a situation where the tubes also act as a heat exchanger, the rate of heat removed will be neither stable nor predictable." Evidently, early operators experienced serious problems associated with parallel chambers and learned empirically to avoid such configurations. No amount of correcting what could have been small differences between the various flow paths was able to avoid the non-uniformity. Only by adopting tubes with the coolant inside, so that the fluidized particles circulated outside, rather than the inside, the tubes, were the operators able to solve the serious non-uniformity problem that existed with the parallel path geometries of these processes.

Boyd et al. (2005) encountered similar problems in operating an internally circulating fluidized bed, where parallel vertical membrane panels were suspended within a draft box to create a series of parallel equal vertical slots through which gas and particles were supposed to circulate equally, producing hydrogen by catalytic steam methane reforming. In practice, some slots experienced much more flow than others, with the result that the overall performance suffered and operation was difficult. Cold modelling (Boyd et al., 2007) showed that by opening up perforations or gaps between the adjacent chambers, the nonuniformity was alleviated. Similarly, in an alternative fluidized bed membrane reactor geometry shown schematically in Figure 2 (Grace et al., 2006), opening up communication between adjacent fluidization channels solved the problem of maldistribution which plagued earlier geometries where parallel vertical chambers of equal dimensions were isolated from each other over their entire height.

Cyclones in Parallel

In industrial-scale reactors, dryers and other process equipment involving solid particles, it is common to require downstream separators to remove entrained particles from the gas or liquid. Given their low capital and operating costs and the lack of moving parts, cyclones are often the separators of choice. For large units, rather than build a single cyclone, two or more cyclones are often installed in parallel. In designing these arrays of cyclones, it is generally assumed (implicitly or explicitly) that the approaching particle-bearing fluid stream will split itself evenly among the individual cyclones in parallel, so that each one will operate under the design conditions. This is of importance, both for operational reasons and because both gas cyclones and hydrocyclones show a maximum efficiency with increasing fluid volumetric flow rate, and the cyclone design is intended to ensure that each cyclone operates at or near this optimum operating condition.

In practice, however, Stern et al. (1955) reported that parallel operation of cyclones results in problems not encountered when each cyclone is operated independently. Equalizing gas and dust-load distribution among the cyclones presents a major problem. When these authors compared efficiencies of cyclones in parallel with those of individual cyclones at the same dust loading and gas flow per unit, those in parallel gave lower collection efficiencies, with the decrease in efficiency tending to increase as the number of cyclones in parallel increased. The likely cause of the decrease is that when linked together, the parallel cyclones experienced different flow conditions, one or more operating below the condition corresponding to the optimum efficiency, and the others above.

[FIGURE 2 OMITTED]

Smellie (1942) tested three identical cyclones in parallel and found that the amount collected were in the ratio of 2:1.5:1 as a result of non-uniform distribution for the individual units. Koffman (1953) tested various cyclones for engine air cleaning. The test results again showed a reduction in overall efficiency when the individual units were combined into a set, with the efficiency dropping from 96% for an individual cyclone to 92.2% when 14 small cyclones were operated in parallel with a common hopper. Broodryk and Shingles (1995) simulated industrial two-cyclone and three-cyclone geometries in cold model experimental units. Preferential flow patterns occurred in many cases, even leading to backflow and blockage of individual cyclones. The maldistribution improved, but did not disappear, at higher gas velocities and hence at higher pressure drops.

Similar findings have been reported for industrial scale cyclones. For example, measurements with water-cooled probes in a 235 M[W.sub.e] circulating fluidized bed (CFB) boiler in Poland where there are two cyclones in parallel suggest some asymmetry of the flow at the top of the unit near the cyclone outlets. Kim et al. (2006, 2007) found markedly different wear patterns in the exit region of a large CFB combustor of 5 m x 10 m cross-section and 29 m height after extended runs lasting several years, despite the fact that the two exits were located symmetrically at opposite ends of the combustor. The wear pattern suggested that the solids flow had been significantly greater through one exit than the other. It is also notable that numerical simulation of a large CFB furnace equipped with 3 cyclones (Flour and Boucker, 2002) predicted marked differences in volumetric solids fractions in the entrance pipes to the individual cyclones.

Pneumatic Feeding of Solid Particles

Economic savings can be realized in blast furnaces by having a single particle dispenser and a suspension flow divider serving all coal feed injection tuyeres rather than separate dispensers for each one. However, there are serious difficulties in achieving uniform flow to the various tuyeres (Kuznetsov et al., 1997).

Giddings et al. (2004) experimentally and numerically studied the splitting of gas-solids flow in connection with the uniformity of pneumatic injection of coal-air mixtures into power stations. For bifurcations the mass flow split varied from 42:58% to 49:51%, whereas at a trifurcation the split ranged from 16:26:58% to 17:38:45%. Schneider et al. (2002) studied a similar geometry but with a riffle box included at the root of the split. In Lagrangian tracking they were unable to obtain a uniform split of particles along the separate branches. Kuan and Yang (2005) found non-uniformities in computational fluid dynamic (CFD) predictions of the gas-solid flow of conveyed gas-solid suspensions into a bifurcation. Although the gas flows were predicted to be almost equal for the two branches, they predicted 5.7 and 9.2% more solids flow to one leg than the other for 66 and 77 [micro]m particles, respectively. In measurements in a very large Polish circulating fluidized bed boiler (Hartge et al., 2005), major differences in temperatures suggest that different amounts of coal are being fed to different symmetrically located feed points.

THEORETICAL CONSIDERATIONS

Cyclones in Parallel

To simplify the problem, consider first two cyclones in parallel, paths 1 and 2, (e.g. see Figure 1, letting the two shaded rectangular regions represent two identical cyclones). Suppose that there is a total gas mass flow rate of [m.sub.gT] and a total solids mass flow rate of [m.sub.sT] to be accommodated by the pair of cyclones. We can therefore write:

[m.sub.g1] + m.sub.g2] = [m.sub.gT] (1)

[m.sub.s1] + [m.sub.s2] = [m.sub.sT] (2)

Let [m.sub.g1] = (0.5 + [alpha])[m.sub.gT] so that [m.sub.g2] = (0.5 - [alpha])[m.sub.gT] (3)

When [alpha] > 0, there is more gas flow to branch 1, whereas for [alpha] < 0, there is a disproportionate flow of gas to branch 2. For [alpha] = 0.5 all gas would flow through branch 1.

Let [m.sub.s1] = [m.sub.sT] /2 + [DELTA][m.sub.s] so that [m.sub.s2] = [m.sub.sT]/2 - [DELTA][m.sub.s] (4)

Positive [DELTA][m.sub.s] means that more solids go to branch 1, whereas negative [DELTA][m.sub.s] denotes a greater proportion of the solids passing through branch 2.

We require that the pressure drops through the two cyclones be equal, i.e.,

[DELTA][P.sub.1] = [DELTA][P.sub.2] (5)

There are various expressions in the literature for pressure drops through cyclones. Assuming turbulent gas flow, then the pressure drop contribution from the gas is usually assumed to be proportional to the square of the gas flow through a cyclone. For very low solids flows, the contribution of the solids flux to the pressure drop is sometimes ignored. However, more generally, as outlined and explained by Chen and Shi (2006), the pressure drop for a given gas flow usually decreases at first with increasing solids flow and then goes through a minimum, thereafter increasing with increasing solids flow rate. Hence, we can write the pressure drop through each cyclone as:

[DELTA][P.sub.i] = [Cm.sup.2.sub.gi] + K f([m.sub.si]) (6)

where C and K are constants. Note the non-linearity. From Equations (3) to (6),

c[(0.5 + [alpha]).sup.2] [m.sub.gt.sup.2] + K f([m.sub.sT]/2 + [DELTA][m.sub.s]) = (7)

c[(0.5 - [alpha]).sup.2] [m.sub.gt.sup.2] + k f ([m.sub.sT]/2 - [DELTA][m.sub.s])

Rearranging this equation leads to:

2[alpha][Cm.sub.gT.sup.2] = K{f([m.sub.sT]/2 - [DELTA][m.sub.s])} = (8)

Clearly if [DELTA][m.sub.s] = 0 , then the right-hand is 0 and hence [alpha] = 0. Hence the (desirable) uniform distribution case, where [m.sub.g1] = [m.sub.g2] = 0.5[m.sub.gT] and [m.sub.s1] = [m.sub.s2] = 0.5[m.sub.sT], constitutes, as expected, one solution of the above equations. However, this solution is by no means unique.

For small deviations from the uniform distribution, we can rewrite the right-hand side of Equation (8) by employing the Taylor series expansion. This leads to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

This shows that for the initial (low ms) case where the pressure drop decreases with increasing solids flux, a small increase in solids flow through either cyclone results in a small increase in gas flow to that cyclone in order to balance up the pressures. On the other hand, if the derivative in Equation (9) is positive, i.e., where the pressure drop across the cyclone increases with increasing solids flow, then a small increase in solids flow through one of the cyclones causes a decrease in gas flow in order to maintain the pressure balance.

Overall, since there are four unknowns (two gas mass flows and two solids flows) and only three equations (1, 2 and 5), the first two arising from continuity and the third from equality of pressure drops through the two branches, the problem has an extra degree of freedom, and there are many solutions.

It is of interest to see how the total pressure drop varies as [alpha] and [DELTA][m.sub.s] vary from 0 above. If we let [m.sub.gt] = [m.sub.g1] and [m.sub.st] = [m.sub.s1], then substitute for [m.sub.g1] and [m.sub.s1] from Equations (3) and (4), and take the first term of the Taylor series expansion of ([m.sub.sT] + [DELTA][m.sub.s]), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

If we now replace the derivative in the last term with the aid of Equation (9), we find that:

[DELTA]P = c[(0.5[m.sub.gT]).sup.2] + Kf([m.sub.sT]/2) + C[[alpha].sup.2][m.sup.2.sub.gT] (11)

But the first two terms are simply the pressure drop, which we can call [DELTA][P.sub.0], that we would have for the equal distribution case, i.e., for [m.sub.g1] = [m.sub.g2] = 0.5[m.sub.gT] and [m.sub.s1] = [m.sub.s2] = 0.5 [m.sub.sT]. That is,

[DELTA]P = [DELTA][P.sub.0] + C[[alpha].sup.2][m.sup.2.sub.gT] (12)

We obtain exactly the same result if we calculate the pressure drop based on branch 2 rather than branch 1. Since C, [[alpha].sup.2] and [m.sup.2.sub.gt] must all be positive, then the final term must also be positive. As a result, the pressure drop for the uniform distribution case is a minimum, and each of the other solutions of the governing equations will result in a total pressure drop through the pair of cyclones greater than for the base (uniform distribution) case. Extremum principles are sometimes postulated (e.g. see Li and Kwauk, 2003) to suggest that nature will choose, for example, the solution that corresponds to the minimum pressure drop. If this condition were to be appropriate, then the equal distribution solution would be the favoured one. However, the experimental results referred to above suggest that in practice, the actual flow distribution may deviate significantly from the equal-distribution case. The theory above allows one to calculate the deviations from equal gas flows (represented by the variable [alpha]) as a function of deviations from uniformity of solids flow (represented by ?ms) providing that the functional relationship, f([m.sub.s]) is available, for example via the comprehensive cyclone pressure drop model proposed by Chen and Shi (2006).

Parallel Vertical Channels or Risers Subject to Pneumatic Conveying

Let us now consider the case where there are parallel vertical passages, fed from a common source, such as a fluidized bed, and with a common termination. Again to simplify the analysis we consider only two parallel paths. In addition, we assume that the flow is one-dimensional within each path. We further make the common assumptions that the particles are identical and, once the flow is fully developed, that the relative (or slip) velocity between the two streams is equal to [v.sub.t], the terminal settling velocity of the particles in the gas in question.

If the flow were to be evenly divided between the two legs then:

[m.sub.gT] = 2[[rho].sub.g][bar.u][bar.[epsilon]] A (13)

[m.sub.sT] = 2[[rho].sub.s][bar.v](1 - [bar.[epsilon]])A (14)

If the length of the riser exceeds the acceleration length, then:

[bar.u] - [bar.v] = [v.sub.t] (15)

With the aid of some simple algebra, one can show that the above three equations lead to:

[bar.[epsilon]] = (1 + G + S)-[[square root of (1 + G + S)].sup.2] - 4G/2 (16)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

are dimensionless gas and solids mass flow rates, respectively. Hence, Equations (16) and (17) can be utilized to predict the mean or expected voidage on both sides, and then Equations (13) and (14) are employed to obtain the corresponding average or expected gas and solids velocities, respectively. These three values are only valid, however, if the distribution is uniform, i.e., if the flow splits evenly between the two parallel paths.

[FIGURE 3 OMITTED]

In the more general case, gas and solids continuity give:

[m.sub.g1] = [gamma][m.sub.gT] = [[rho].sub.g][u.sub.1][[epsilon].sub.1]A (18)

[m.sub.s1] = [sigma][m.sub.sT] = [[rho].sub.s][v.sub.1](1 - [[epsilon].sub.1])A (19)

[m.sub.g2] = (1 - [gamma])[m.sub.gT] = [[rho].sub.g][u.sub.2] [[epsilon].sub.2]A (20)

[m.sub.s2] = (1 - sigma][m.sub.sT] = [[rho].sub.s][v.sub.2](1 - [[epsilon].sub.2])A (21)

where [gamma] and [sigma] are the fractions of the gas and solids flows, respectively, passing through riser 1. In view of the slip assumption, we can also write:

[u.sub.1] - [v.sub.1] = [v.sub.t] (22)

[u.sub.2] - [v.sub.2] = [v.sub.t] (23)

To obtain the pressure drop on both sides, we begin with an expression derived by Louge and Chang (1990) for the case where the gas density is much lower than the solids density:

dP/dz = [[rho].sub.g]g(1 - [epsilon])- [m.sup.2.sub.s]/[[rho].sub.s][A.sup.2]d/dz (1/1 - [epsilon]) (24)

Here the first term on the right-hand side accounts for the static head, and the second arises from the acceleration and resulting voidage gradient. Integrating for the case where the height, H, of each of the risers is significantly greater than the acceleration length, and requiring that the pressure drops on each side is equal, i.e., [DELTA][P.sub.1] = [DELTA][P.sub.2], results in:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Equations (18) to (23) plus Equation (25) represent seven algebraic equations, but there are eight unknowns ([gamma], [sigma], [u.sub.1], [v.sub.1], [[epsilon].sub.1], [u.sub.2], [v.sub.2] and [[epsilon].sub.2]). Hence, just as in the cyclone case considered above, there is one extra degree of freedom, unless we can invoke an extra condition, such as pressure drop minimization. Note, however, that the uniform condition (where [gamma] = [sigma] = 0.5; [u.sub.1] = [u.sub.2]; [v.sub.1] = [v.sub.2]; and [[epsilon].sub.1] = [[epsilon].sub.2]) can satisfy all of the seven equations and hence represents one possible solution.

To explore the nature of other possible solutions, the above set of non-linear algebraic equations was solved for specific values of [gamma] for two case studies. Case A corresponds approximately to conveying of Geldart type A particles in intermediate-density conditions through a pair of vertical risers of 0.2 [m.sup.2] cross-sectional area and 2 m height, whereas case B represents dense conveying of a type B solid through a pair of similar, but somewhat shorter vertical channels. The voidages at the bottom of the risers are taken as 0.50 and 0.46 in the two cases. Values of the prescribed particle properties and operating parameters, values of the other seven variables that satisfy the seven equations when [gamma] is assigned specific values, and resulting pressure drops are compiled in Table 1.

As expected, the corresponding solutions for both cases A and B yield [sigma] = 0.5; [u.sub.1] = [u.sub.2]; [v.sub.1] = [v.sub.2]; and [[epsilon].sub.1] = [[epsilon.sub.2] when we set [gamma] = 0.5. However, Table 1 shows that whenever [gamma] [not equal to] 0.50, there is a solution which shows asymmetry, sometimes corresponding to a substantial flow maldistribution, with more flow of both gas and solids travelling through one branch than through the other branch. (While this is not indicated in Table 1, each of the solutions is symmetric about [gamma]= 0.5; for example, [gamma] = 0.30 gives the same result as [gamma] = 0.70, except that the subscripts 1 and 2 are interchanged and [sigma] is an equal distance on the other side of 0.5, e.g. 0.769 rather than 0.231 for case A.)

Although based on a simplified one-dimensional model, the values appearing in Table 1 indicate that there is a very broad range of values, including those signifying highly asymmetric distributions, which can satisfy the operating conditions and model equations. The uniform distribution ([gamma] = [sigma] = 0.50, i.e., with exactly half of the gas flow and half of the solids flow passing through each of the separate paths) gives the lowest overall pressure drop. Hence, if energy minimization is applicable, one might expect the uniform flow distribution to be the solution found in practice. However, the pressure drop minimum in each case is remarkably shallow. For example, the increment in pressure drop from the even flow distribution in case B where [DELTA]P = 5,265 Pa is only about 6% for the most "maldistributed" solution shown, where the gas flow splits 30 : 70% and the solids flow splits 5.5 : 94.5%. In other words, the "penalty" in pressure drop or energy dissipation for adopting a non-uniform flow distribution, rather than a perfectly even one, is surprisingly small. The smallness of the extent of this penalty may account for the apparent finding that energy minimization does not appear to be applicable in this case, given the empirical findings, discussed above, which suggest that non-uniform distributions are widespread, perhaps even favoured, in practice.

Stability Considerations

There are a number of situations for natural and man-made systems where there are multiple steady-state solutions, but where, due to non-linearity, one or more of these solutions is unstable and therefore unlikely to be found in practice (Elnashaie and Grace, 2007). The best known of these situations in chemical engineering systems occurs in reaction engineering when an exothermic reaction is taking place in a continuously stirred reactor and it is possible for the energy and mole balances to be satisfied at multiple points. Typically there are three steady-state solutions, with the central one being unstable, whereas the outer two are stable. It is possible that the two-phase flow distribution problem considered in this paper may be similar in nature. If it is, then, for a pair of flow paths, the uniform distribution case will always be a solution. Moreover, if there are other (nonuniform) solutions, they will always be paired on either side of the uniform one, since if one non-uniform solution exists, then there must also be another with the flows through the two branches interchanged with each other. This suggests an odd number of solutions, with the central (uniform) one being unstable. In essence, one might then explain the instability in the following terms: Consider an equal distribution of flow in the two paths. Then imagine that there is a small increase in gas flow in one of the two legs; this could cause a dilution of the suspension on that side resulting in a reduction in pressure drop. This in turn would trigger more flow to that leg resulting in the paired flow moving away from the original steady state, until some new state is found where the pressure drops in the two branches are balanced.

Such instability, bifurcation and multiplicity would account for the difficulties which have been reported by a number of companies in seeking to balance multi-phase flows in multiple paths. They would also explain why great efforts to eliminate any small geometric differences between paths so that they are identical have been unsuccessful. Even if a uniform distribution could be achieved initially, small perturbations would always exist which would result in the flow distribution migrating towards one or other of the stable flow situations on either side of the (unstable) uniform distribution flow. Such bifurcation would also explain cases where a particular maldistribution has become entrenched, such as in the cyclones (discussed above) where non-uniform wear patterns suggest that a preference for flow of one phase through one branch has been maintained over many years.

While this kind of flow instability involving a limited number of alternating stable and unstable steady states is certainly possible, our analysis above for two cases--a pair of identical cyclones in parallel and a pair of vertical risers in parallel--indicates another possibility. This is that, by virtue of the flow configuration and multiple phases, there may be more variables than equations to be satisfied, resulting in a continuous array of solutions. In that case, minimization of the (shared) pressure drop over the two paths could, as noted above, be important, but the minimum may be so shallow, that in practice the flow never settles on any particular flow pattern. It is also possible that no minimization or other extremum applies in these cases.

In order to settle these questions and to understand how to minimize the extent of maldistribution, both computational fluid dynamics and careful experimentation are needed. We intend to embark on such studies to seek a better understanding of the complexity offered by these flows. For example, numerically one could begin with a uniformly distributed flow and then introduce a perturbation and allow the governing equations to determine the response. Experimentally, there are questions of whether uniform flows through multiple identical paths can be achieved and sustained under various conditions, and whether there are corrective or control measures that can make it possible under practical conditions to equalize the flows in the various branches. Ultimately, it also will be important to extend these investigations from two to three or more branches, and to consider whether other two-phase systems (e.g. liquid-solid, gas-liquid, liquid-liquid) and three-phase systems encounter similar phenomena. As the above cases illustrate, there are significant practical implications also for the operation of process equipment.

CONCLUSIONS

When two-phase suspensions are conveyed through identical parallel flow paths, the flow distribution can be significantly non-uniform in practice. Industrial examples of such nonuniformity include multiple gas-solid cyclones in parallel, vertical pipes or flow channels within gas-fluidized beds, and distributed pneumatic-feed systems.

While the uniform flow distribution is always a solution of the governing equations for cases where there are symmetric boundary conditions, this solution may be, at least in some cases, an unstable steady-state solution between stable solutions. Simple theoretical treatments for cyclones in series and for risers in parallel suggest, however, that there is a continuum of solutions since there is one extra degree of freedom, i.e., one more variable than governing equations. Multi-phase flow through flow networks consisting of identical branches requires careful analysis and experimentation to understand and predict the resulting flow patterns and flow distribution for each phase.

ACKNOWLEDGEMENT

The authors acknowledge the role of Larry Hackman and Craig McKnight of Syncrude Canada Limited in increasing the authors' interest in this problem, and the inspirational role of Jacob Masliyah with respect to multi-phase flow problems of many kinds.

NOMENCLATURE

A cross-sectional area of channel ([m.sup.2])
C constant coefficient of gas flow term in cyclone
 pressure drop relation ([k.sup.g-1][m.sup.-1])
f([m.sub.s]) function giving dependence of cyclone pressure
 drop on solids mass flow rate (-)
G dimensionless gas flow rate defined by
 Equation (17) (-)
K constant in cyclone pressure drop relationship,
 Equation (6) (Pa)
m mass flow (kg/s)
S dimensionless solids flow rate defined by
 Equation (17) (-)
[bar.u] one-dimensional gas velocity (m/s)
[bar.v] one-dimensional solids velocity (m/s)
[v.sub.t] terminal settling velocity of particles in the
 gas (m/s)
z vertical coordinate, upwards positive (m)

Greek Symbols

[alpha] dimensionless deviation from uniform gas flow,
 Equation (3) (-)
[gamma] fraction of gas flow passing through branch 1
[DELTA][m.sub.s] deviation from uniform solids distribution,
 Equation (4), (kg/s)
[DELTA]P pressure drop across entire flow path (Pa)
[epsilon] voidage, i.e., volumetric void fraction (-)
[bar.[epsilon]] average voidage over flow channel (-)
[[epsilon].sub.0] voidage at entrance of channel (-)
[rho] density (kg/[m.sup.3])
[sigma] fraction of solids flow through branch 1 (-)

Subscripts

1, 2 branch 1, 2
g gas
i ith branch or channel
s solids
T total

Manuscript received February 17, 2007; revised manuscript received March 20, 2007; accepted for publication March 20, 2007.

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John R. Grace (1) *, Heping Cui (1) and Said S. E. H. Elnashaie (2)

(1.) Department of Chemical and Biological Engineering, The University of British Columbia, 2360 East Mall, Vancouver, BC, Canada V6T 1Z3

(2.) Pennsylvania State University at Harrisburg, Middletown, PA, U.S.A. 17057-4898

* Author to whom correspondence may be addressed. E-mail address: jgrace@chml.ubc.ca

Table 1. Case studies for gas-solid flows through two risers
in parallel

 Case A

[[rho].sub.g], kg/[m.sup.3] 1.0
[[rho].sub.s], kg/[m.sup.3] 1800
[[epsilon].sub.0] 0.50
[v.sub.t], m/s 1.0
A, m 0.2
H, m 2.0
[m.sub.gT], kg/s 1.0
[m.sub.sT], kg/s 72
[gamma], - 0.50 0.40 0.30 0.20
[sigma], - 0.50 0.375 0.231 0.041
[[epsilon].sub.1], - 0.940 0.934 0.926 0.913
[[epsilon].sub.2], - 0.940 0.943 0.943 0.941
[u.sub.l], m/s 2.66 2.14 1.62 1.10
[u.sub.2], m/s 2.66 3.18 3.71 4.25
[v.sub.1], m/s 1.66 1.14 0.62 0.10
[v.sub.2], m/s 1.66 2.18 2.71 3.25
[DELTA]P, Pa 2389 2512 2669 3071

 Case B

[[rho].sub.g], kg/[m.sup.3] 1.2
[[rho].sub.s], kg/[m.sup.3] 1400
[[epsilon].sub.0] 0.46
[v.sub.t], m/s 1.25
A, m 0.2
H, m 1.6
[m.sub.gT], kg/s 0.8
[m.sub.sT], kg/s 120
[gamma], - 0.50 0.45 0.40 0.30
[sigma], - 0.50 0.394 0.285 0.055
[[epsilon].sub.1], - 0.767 0.764 0.759 0.745
[[epsilon].sub.2], - 0.767 0.770 0.772 0.772
[u.sub.l], m/s 2.17 1.96 1.76 1.34
[u.sub.2], m/s 2.17 2.33 2.59 3.02
[v.sub.1], m/s 0.92 0.71 0.51 0.092
[v.sub.2], m/s 0.92 1.13 1.34 1.77
[DELTA]P, Pa 5265 5284 5346 5597