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Physical mapping of fluidization regimes—the EMMS approach
Chemical Engineering Science 57 (2002) 3993 – 4004
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Physical mapping of #uidization regimes—the EMMS approach Wei Ge, Jinghai Li ∗ Multi-Phase Reaction Laboratory, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100080, People’s Republic of China Received 19 March 2001; accepted 29 January 2002
Abstract The existence of multiple regimes of distinctive #ow structure is a remarkable characteristic of #uidization, which is far from being physically interpreted under a uni5ed approach. The energy minimization multi-scale model (Particle-Fluid Two-Phase Flow, the Energy Minimization Multi-Scale Method, Metallurgical Industry Press, Beijing, 1994) is potentially such an approach in which the inclusion of stability criteria enables the prediction of heterogeneity and non-linear behaviors in #uidized beds. However, fully analytical solution of the model is impossible so far, and numerical solutions have resorted to general optimizing software. Therefore, the detailed characteristics of the solutions and their theoretical implications have not been fully explored. In this paper, we have achieved this by a rigorous numerical approach and by retrieving all missing roots, which leads to physical mapping of #uidization regimes. The model is also extended to unsteady conditions with acceleration and simpli5ed by employing a single stability criterion, which identi5es choking as a jump between two branches of the stable solution. Calculations based on this version are in reasonable agreement with measurements on bench, pilot and commercial scale circulating #uidized beds. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Choking; Fluidization; Hydrodynamics; Multi-scale; Regime transition; Stability
Despite its wide applications, #uidization is not well understood for its multi-scale structure and multiple regime transitions. These remarkable characteristics are shaped by the stability that has been presented as the extrema of energy dissipation terms in the energy minimization multi-scale (EMMS) model (Li & Kwauk, 1994; Li, Wen, Ge, Cui, & Ren, 1998). Therefore, the model has predicted the general #ow structure and regime transition reasonably. However, the non-linear nature of the model equations has prohibited its full analytical solution, so calculations have resorted to general-purpose optimization software (Li, 1987) or simpli5ed numerical and analytical schemes (Xu, 1996). So far, the mathematical character of the model is not understood thoroughly, which has limited its application and physical improvement. This work is therefore devoted to a complete numerical solution of the model to explore some new features. ∗
Corresponding author. Tel.: +86-10-62558318; fax: +86-10-62558065. E-mail address: [email protected] (J. Li).
1. Formulation The EMMS model is currently a 0-D description of the time-averaged behavior of #uidization, but its principle is equally applicable to the elements in two-#uid models (TFM) (e.g. Gidaspow, 1994), and it can be expected to improve the accuracy of TFM by describing the signi5cant heterogeneity within the element that is neglected now. However, in elements particle weight is not always equal to the drag acting on it. Therefore, as a 5rst step, we extend the model to where the particles have equal acceleration. To facilitate the discussion, we use a general equation for the momentum balance in uniform suspension of mono-sized spherical objects (refer to the Notation for the de5nitions of the symbols), i.e., 1 − CD0 (Us d= ) 2 1 d f Us2 (=6)d3 4:7 4 2 =(1 − )( p − f )(a + g)k; which can be rewritten to Us d 4 p − f Us2 = CD0 (a + g)d4:7 k; 3 f
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 2 3 4 - 8
(1) (2)
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W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
where Us is de5ned as Ug − Ud =(1 − ). We denote this equation as a function Us (d; ; k; a). For the dilute phase, all eJective particle weight is balanced by #uid drag, i.e., Gf = (1 − f )( p − f )(g + a) ⇒ Usf = Us (dp ; f ; 1; a);
(3)
where Usf = Uf − Udf f =(1 − f ). For the dense phase, eJective particle weight is partially supported by the dense phase #uid #ow, and the rest is supported by the bypassing dilute phase #uid #ow, termed here as interphase (dilute-to-dense phase) drag, i.e., Gi + Gc = (1 − c )( p − f )(g + a):
(4)
As the dense phase is assumed to occur as spherical clusters, we may de5ne kc = Gc =(Gc + Gi ) and write Usc = Us (dp ; c ; kc ; a);
(5)
Usi = Us (l; 1 − f; (1 − c )(1 − kc ); a);
(6)
where Usc = Uc − Udc c =(1 − c ); Usi = (1 − f)(Uf − Udc f =(1 − c )). Next, the dense phase pressure drop is balanced by that of the dilute phase plus the interphase, i.e., fGi Gf + (7) = Gc = G = ( p − f )(1 − )(g + a): 1−f From Eqs. (4) and (7), we have kc = (1 − )=(1 − c ) and Usi = Us (l; 1 − f; − c ; a). Besides the force balances, the continuity of the #uid and the solids requires Ug = fUc + (1 − f)Uf ;
(8)
Ud = fUdc + (1 − f)Udf :
(9)
Finally, the correlation for cluster diameter is ( p − f )(g + a)Ud = p (1 − max ) − Nst; mf l = ; dp Nst − Nst; mf
(10)
where by de5nition Nst = Wst =(1 − ) p and Wst = fGc Uc + (1 − f)(fGi + Gf )Uf :
(11)
It can be derived from the foregoing equations that p − f f − Nst = Ug − ; (12) f(1 − f)Uf (g + a) 1− p p − f Ud mf Umf + (g + a): (13) Nst; mf = p 1 − mf The constraints of the model consist of Eqs. (3), (4), (7) – (10), but they have 8 independent variables: Uf ; Uc ; Udf ; Udc ; f ; c ; f and l, which needs the optimization of Nst to close the model as stability criterion. 2. Physical background Previous results from this model with a = 0 can be summarized as follows: For conditions below choking, i.e.,
Ug ¡ Upt (Gs ) or Gs ¿ Gs∗ (Ug ), the stability criterion is assumed to be Nst → min which results in the most distinct two-phase structure of c = mf and f → 1. Beyond choking, Nst → max is assumed, leading to uniform suspension in theory. And the criterion determining choking is proposed to be (Wst )Nst →min = (Wst )Nst →max|c =mf :
(14)
Despite its agreement with experimental phenomena, the results above are imperfect in two aspects, i.e., the dilute phase should also have particles (f ¡ 1), and weak heterogeneity should still exist beyond choking. For the 5rst problem, we notice that the physical mechanism of particle aggregation is in some ways comparable to the condensation of steam from air. Steam condenses to liquid when their inter-molecular attraction can overcome their heat movement, and particles aggregate when their random motion cannot resist the non-uniform drag on them and the non-contacting forces between them. It follows that, in the presence of the aggregated dense phase, the dilute phase would remain at the critical concentration for aggregation, just as the saturated steam. This is the rational background of the correction suggested by Li et al. (1999) that f = uni (Ug ; Gs∗ (Ug ));
(15)
which means the dilute phase is always at the critical state between heterogeneity and homogeneity. However, this correction is not easily implemented because Gs∗ (Ug ) itself is to be calculated from the model, so global iterations are needed to determine f , and the algorithm could be very costly and unstable. On the other hand, we may 5nd that max , which is obtained from kinetic analysis (Chen, Wu, Li, & Kwauk, 1994), is indeed corresponding to this critical concentration, and it can be calculated from Ug and Gs directly (in Chen’s treatment, Ug is derived as Ud p =( p − f )=(1 − ), but that is an approximation valid only when Ug and are high enough). It should be noted, however, that max is Not the lowest voidage that homogeneity can exist, though it is the highest voidage that heterogeneity can exist. For many aggregative systems, max is very close to unity, but a particulate regime can exist at voidages near mf . The reason behind is, that particle–#uid systems are, after all, more complicated than steam in air. Once a cluster forms, it will have meso-scale interactions with the dilute phase on their interface, and exert additional constraints on the formation of the clusters. The EMMS model is capable of describing this hydrodynamic eJect, as we can see later, but the kinetics of particle suspension are not incorporated previously, which has led to the prediction of f = 1. Now we may say that f at the beginning of clustering would be max , or the upper limit of f is max . The solution to the second problem of heterogeneity beyond choking requires a reexamination of the numerical methods of the model.
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
3. Numerical methods Although g is replaced with g + a, the mathematical characteristics of model are not changed. From Eq. (12), we may 5nd that Nst → max can be reached only when f = c or f = 0 or f = 1, where Nst = NT = (g + a)Ug ( p − f )= p . In fact, they all correspond to the homogeneous state. For Nst → min, if c and f are given, the model is closed for solving the other 6 variables, which is called the sub-model. Then the whole model can be solved by searching the minimum of Nst among all pairs of c and f within the range of [mf ; max ]. Before going to details we present the general results 5rst. The model may have two, one or no solution(s), as exempli5ed in Fig. 1. It is interesting that the #ow structures 1.0
3995
for diJerent solutions seem to include all varieties for #uidized beds. To explain this accordance, we must start with the sub-model, for which Xu (1996) and Meynard (1997) have developed speci5c numerical schemes separately. For simplicity, we use Meynard’s scheme, i.e., 1. 2. 3. 4.
Calculate Usf from Eq. (3). Select a trial value for f. Calculate kc and then calculate Usc from Eq. (5). With the de5nitions of Usf and Usc ; Uf ; Uc and Udc can be solved from Eqs. (8) and (9). 5. Calculate l from Eq. (10). 6. Calculate Usi from its de5nition and Eq. (6), respectively, denote the diJerence as PUsi . 1.0
min
min 0.9
0.9
Ug=5m/s Nst=0.756(blue)~1(red) one root at right top
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Ug=0.01m/s, no solution for any combination of c and f, Nst undefined
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c
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min
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low Fig. 1. Variation of Nst with c and f for the FCC=air system (Li, 1987, p. 168) in Table 2 (Gs = 50 kg=m2 s).
1.0
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W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
Fig. 2. Roots of the sub-model for the FCC=air system in Table 2 at Ug = 0:0631 m=s and Gs = 0:0631 kg=m2 s. The PUsi (f) curve in each region has a unique type of shapes and a unique set of roots, labeled as, e.g., 2r1 .
7. If PUsi is not small enough, select a new f that can reduce this diJerence and go back to step 3. Unfortunately, a thorough study on the variation of PUsi with f is so far absent; therefore, application of common approaches for solving non-linear equations, such as dichotomizing search, has not been justi5ed for step 7. Though it has brought about reasonable results, this scheme fails to 5nd the existence of multiple roots. Now by examining a large number of cases with various parameters, we found that the PUsi (f) curves can be classi5ed into 5 types, and the distribution of these types among possible combinations of c and f at given conditions are illustrated by a typical case in Fig. 2: 1. Drop or then rise, all above zero, no root. 2. Drop and then rise, a middle section below zero, two roots. 3. Drop or then rise, a right section below zero, one root. 4. Below zero except a middle section, two roots. 5. All below zero with no root. But under some other conditions, the types may be less. A notable feature here is that the boundary between types 3 and 4 coincides with f = uni (Ug ; Gs ); Types 1–3 seem to have diJerent nature from types 4 and 5 with PUsi (0)
approaching +∞ and −∞, respectively. The physical meaning behind this is yet to be explained. Now with this simple judgement in mind, we can speed up the calculation greatly by treating the 5rst 3 and last 2 types with dichotomizing search separately. Then with the new scheme, we con5rm that Nst → min always requires the maximization of f , which is f = max instead of f = 1 now (ref. Fig. 1). And from now on, we imply f = max when we talk about the variation of Nst . The new 5nding is that Nst does not always increase monotonously with c , it may have one local minimum at c = mf and another at a higher c which we denote as o . This is seen more clearly in Fig. 3 where Nst is plotted against c only. In this case, with the continuous variation of Ug or Gs , there is a value where Nst (mf ) = Nst (o ):
(16)
It is about Ug = 3:4 and 0:6 m=s for Figs. 3a and b, respectively, and o is about 0.96 and 0.997. The #ow structures corresponding to c = mf and c = o are distinct, but they must coexist in the system. Then we realize that if this critical state is understood as choking with Eq. (16) being its criterion in place of Eq. (14), several improvements can be made simultaneously. Namely, the remaining heterogeneity observed beyond choking can be retrieved; a more clear interpretation of choking appears, which is simply the shift
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
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Fig. 3. Choking explained as coexistence of bi-stable states for the FCC=air system in Table 2, the #ow structure corresponding to each state is sketched near its curve.
between two stable states; and Nst → min can serve as the uni5ed stability condition for all heterogeneous regimes in #uidization. To support this modi5cation, we want to explain that the stability criterion of a single phase in the system, as assumed by Li and Kwauk (1994), is easy to understand: the particles always seek minimum gravitational potential; As a result, is minimized, and the #uid phase seeks least resistance which minimizes Wst . When the particles are #uidized, both conditions are subject to the constraints from the other. Therefore, the overall stability criterion should embody the in#uence of both phases. So Nst → min seems to be a reasonable combination, for it requires the minimization of both Wst and . The physical meaning of this criterion can be demonstrated more explicitly if we notice that it is equivalent to Nd = NT − Nst → max, since NT is constant under the given conditions. From Eq. (11), we have
Nd = fgi Uf , where gi = Gi =(1 − )= p is the eJective speci5c weight of the particles supported by interphase drag. We also notice that as the dense phase is assumed to occur as clusters, higher f means larger phase interface; and as Ug is given, larger Uf means sharper diJerence of #uid velocity between the two phases. In summary, Nd → max is to establish a most distinct interface between the dense and dilute phases. Interface has been the focus of stability studies on many other phenomena (Drazin & Reid, 1981), and there seem to be no exception here. 4. Regime-mapping Now what is the relation between the original and modi5ed versions of the model and how can they be uni5ed? Fig. 4 shows the solution of the new version for each point
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
Gs(kg/m s)
3998 2
(Usi (f) always positive)
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d) ize rt l a e o nid nsp (no te tra u heterogeneous solutions dil
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id ea U lize si ( f) d di al lu wa te ys tra ne ns ga po tiv rt e)
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Fig. 4. Calculated #ow structure of the FCC=air system in Table 2.
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W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
in a wide range on the Ug −Gs plane for a given system with a = 0, where a panorama of #ow regimes appears. When Ug is below Umf , the PUsi (f) curve in Fig. 2 is always positive, which means the #uid #ow cannot provide enough drag to counter-balance the weight of any amount of ascending solids, this is also true if Gs is too high when Ug ¿ Umf . In fact, they correspond to 5xed beds (Gs =0) and moving beds (Gs = 0). Then if Gs is too low when Ug ¿ Umf , two-phase solutions are absent too. But in this case, homogeneous solutions from the closed constraints of force balance are available, where stability criteria play no role and Nst =NT . Therefore, the new version of Nst → min and the old versions of Nst → max are consistent here. Though no structural diJerence is predicted, calculations show that this region can be divided into a right part of ¿ max where homogeneity results from particle kinetics, and a left part of ¡ max where no stable cluster can ful5ll the hydro-dynamic constraints in the EMMS model. Compared with engineering practice, they should belong to particulate #uidization and idealized dilute transport respectively. However, for the particulate and bubbling regimes, Gs is actually not an independent parameter, it is mainly a function of Ug (for mono-sized spherical bed materials). That means, for a given system, the particulate states will be found on a curve in the Ug − Gs plane (e.g., the dash line in Fig. 4), and its intersection with the upper boundary of the particulate regime determines Umb . Unfortunately, despite this physical interpretation, the EMMS model now is unable to give the value of Umb because the macro-scale heterogeneity and dynamic process involved, especially that of the transition zone, are not considered yet. The region of two-phase solutions is partitioned into two parts by the choking line which starts from Ug = Ut roughly. The right part with loose dense phase corresponds to dilute transport, which it not discriminated from idealized dilute transport in the old version. The left part corresponds to bubbling, turbulent and fast #uidization regimes. No jumps in #ow structure between these three regimes are captured by the model, but they can be mapped relatively from left to right in order of increasing or decreasing l and f. Therefore, depending on the magnitude of Umb , the particulate regime may develop to any of the three regimes or even to (idealized) dilute transport directly as in particulate systems. However, we have to explain here, that the model seems to have low accuracy at the bottom of the two-phase region. This may be evidenced on the transition line where the calculated l approaches dp . Physically, it means that the two-phase structure will continuously degrade to homogeneous solutions, but this is not echoed by the distribution, which displays signi5cant discontinuity. Among the underlining reasons, is possibly the validity of Eqs. (6) and (10) and the presumption, that dense phase occurs as discrete cluster at such extreme conditions. Anyway, it is a subject for further study and improvement. When a = 0, we 5nd that higher a results in larger and denser clusters. As shown in Fig. 5, for a given point on the Ug − Gs plane, both and c decrease with a while l=dp
3999
increases with a. This structural change seems to counterbalance the increase of both #uid resistance and particle potential when the particles are accelerated, and it is a reasonable reaction from the point of view of the EMMS model. 5. Computational results and discussion Considering the simplicity of our model and the uncertainties in both modeling and measurements, no quantitative agreement with experiments can be expected now, but qualitative consistency is widely observed in quite diJerent systems and regimes. We think this is actually a more convincible validation of the model. In Fig. 6 and Table 1, we compare the results from the EMMS model with previous measurements on an #uid cracking catalyst (FCC)/air system (Li & Kwauk, 1994, p. 143), its speci5cations are in Table 2. In theory, choking is a jump from one state to the other and only on this point can the two states coexist, but as any practical system goes, visible diJerence between the upper and lower section of the system is observed in the measurements for a fairly broad range. As a support for our explanation of choking, we 5nd that for each Nst (c ) curve, the distinctness of its non-monotonicity in calculation is in accordance with that of the two-state structure in measurements. Here we may also get a more precise understanding of this stability. That is, the possibility of the appearance of the state is actually determined by the shape of the curve near this state rather than the absolute value of Nst . This is something like the stability of a rolling rock on the hill. It is not necessarily more stable at a lower altitude, but rather in a deeper valley. So, we 5nd the two states before and after the S-shaped transition in the upper insets roughly corresponding to the solution at c = mf and c = o (especially their tendency of variation with operating conditions), rather than the intermediate solutions which may have lower Nst than one of them. Similarly, a practical choking de5nition conforming to its theoretical counterpart (Eq. (16)) could be the point where the coexistent states are most distinctive and the transition between them is most dramatic. As a major improvement, the c beyond choking, which was somewhat arbitrarily assumed as mf in the old version to compensate for its loss of heterogeneity there, now agrees qualitatively with diJerent experimental data (Li, Ge, Guo, & Chen, 1996; Kruse & Werther, 1995), both are closer to max than mf . Our choking prediction has also been extended to larger scale CFBs with encouraging agreement to in situ measurements, as shown in Fig. 7. Still, it may be more important to recognize that simple hydrodynamic models with stability considerations can explain the general characteristics of the system, though the detailed #ow structure is so complicated that it is virtually unpredictable even for the most intensive CFD approaches. And this may be common to other complex systems also. A program has now available on http://www.chinweb.com. cn/%7Exyli/test/datainput.htm, where one can specify the
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W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
1.0
εf
voidages
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a (m/s ) 100 98
l/d p
96 94 92 90 88 86 -4
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2 2
a (m/s ) Fig. 5. EJect of particle acceleration of on #ow structure for the FCC=air system in Table 2, Ug = 3:4 m=s, Gs = 50 kg=m2 s.
material and operational parameters of the #uidized bed and get its #ow structure from our model. For instance, if you input the properties of the FCC=air system in Table 2, and specify Ug = 2:1 m=s; Gs = 32 kg=m2 s, you get f = 0:999742; c = 0:5 and = 0:81034. The computation takes 5 min on average. 6. Future work As discussed above, a better correlation of l that can accommodate the variety of meso-scale heterogeneity and which links smoothly with single particle dynamics is desir-
able. Considering the sensitivity of l on max , more elaborate kinetic theories are needed to predict max more accurately. However, it is worth reminding that l has little geometrical sense anyway, it is mainly a characteristic parameter of heterogeneity and interphase friction. For the application of the EMMS model in TFM simulations, a practical way is to take the from TFM calculations as an additional given condition and 5nd a with the EMMS model. It is slightly more complicated than the case we have discussed at given a, but it can still be solved in a similar way. Firstly, since NT and Nst are both functions of a, the stability criterion should be expressed as Nst =NT → min. And secondly, Ug and Gs will be replaced with
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
4001
Fig. 6. Experimental evidence of the bi-stable states in the FCC=air system in Table 2. (a) Gs = 14:3 kg=m2 s, Ug = 1:52 m=s, (b) Gs = 26:6 kg=m2 s, Ug = 1:52 m=s, (c) Gs = 48:2 kg=m2 s, Ug = 1:52 m=s.
vectors in elements, but we can 5nd that only slip velocities, which have counterparts in elements, are signi5cant to the hydrodynamic constraints. In fact, the absolute values of Ug and Gs that are relative to the walls are only necessary for l and Nst because they depend on the overall #ow pattern. (Wall eJect is considered in this way, otherwise the model
will give diJerent results for the same system in diJerent reference frames.) So it may be possible to 5nd a Uge for each element based on the EMMS calculation of the whole bed with modi5cations according to its location in the #ow 5eld. Then for each element, we can write the EMMS model in the direction of local averaged slip velocity with
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W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
Table 1 Calculated vs. measured voidages for the FCC=air system in Table 2
Ug (m=s)
Gs (kg= m2 s)
a Calculated
a Measured
∗ Calculated
∗ Measured
1.52 1.52 1.52 2.1 2.1 2.1 2.1 2.6 2.6 2.6 2.6
14.3 26.6 48.2 24.1 32 64.2 96.3 42.8 64.2 96.3 192.7
0.81 0.77 0.73 0.83 0.81 0.76 0.74 0.83 0.80 0.78 0.73
0.82 0.81 0.80 0.84 0.84 0.83 0.81 0.90 0.88 0.86 0.82
0.99 — — 0.99 0.98 — — 0.98 — — —
0.98 — — 0.97 0.96 — — 0.94 — — —
Table 2 Speci5cations of the cited systems
50
Chalmers (Zhang, Tung, & Johnsson 1991; Zhang, 1995) 2600 0.32 4.832 260 1.78 0.43
ham
Name References p (kg= m3 ) f (kg= m3 ) " (10−5 kg= m s) dp (m) dt ( m ) mf
C hat
lines: calculated dots: measured
F C C /a
ir
40
rs
Chatham
FCC=air (Li & Kwauk, 1994) 929.5 1.1795 1.8872 54 0.09 0.5
However, in this way the structural information obtained in the EMMS model is not used in TFM simulations. In long term, the favorable form to incorporate the two models may be called the two-phase two-#uid model. That is, the particles and #uid in the dense and dilute phase are all formulated as continuums separately, with similar but doubled equations of the TFM now, and the interaction between each pair of them are determined in a further generalized version of the model.
C ha
Gs (kg/m2s)
lm e
30
Chatham (Couturier, Doucette, & Stevens, 1991) 2500 0.32 4.832 200 4.47 0.51
7. Conclusions 20
FCC/air
10
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0 0
1
2
3
4
5
6
7
8
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Fig. 7. Choking prediction for some CFB combustors.
scalar value of Uge and Ude = (Uge − Us )(1 − )=, and transfer the solved a to a TFM calculation for the next time step.
The EMMS model is simpli5ed to assume Nst → min as the uni5ed stability criterion for all #ow regimes in #uidization, and also extended to unsteady #ows. The mathematical characteristics of the EMMS model are studied thoroughly, which 5nds the existence of two roots at given voidages and the bi-stable character of Nst . A physical mapping of #uidization regimes is proposed based on this improved model. As summarized in Table 3, the 5xed bed has no solution in the model; idealized dilute transport and particulate #uidization have homogeneous solutions under corresponding operation conditions; and diJerent two-phase structures are predicted for the rest of regimes. Especially, choking is interpreted as a condition where two stable states, i.e. the “sharply” heterogeneous regimes of bubbling to fast #uidization and the “smoothly” heterogeneous regime of dilute transport, have the same Nst and coexist most distinctively. Acceleration of
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
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Table 3 Partition of #uidization regimes suggest by EMMS
Predicted structure (No solution) Heterogeneous structure Homogeneous structure
Regimes Fixed bed Moving bed Sharply Smoothly
the solid phase results in the decrease of Gs∗ at given Ug . The general agreement of our results to measurements in diJerent scale circulating #uidized beds (CFBs) suggests that the model possesses the essential mechanism to underline the heterogeneity and regime multiplicity of particle–#uid #ow. Though it may need further supplements and improvements to become quantitatively predictive, it can be considered as a reasonable framework to work on.
Notation a d dp dt f g G Gc Gf Gi Gs Gs∗ h k kc l NT Nst Uc Uf Ud
Bubbling #uidization Turbulent #uidization Fast #uidization Dilute transport Idealized dilute transport Particulate
acceleration of the solids, m=s2 diameter of spherical objects (particle, cluster, etc.), m surface-to-volume mean particle diameter, m equivalent inner diameter of equal cross-sectional area of the CFB riser, m dense phase fraction gravitational acceleration, m=s2 eJective weight of particles in unit space, N=m3 eJective weight of particles in unit space supported by dense phase #uid #ow, N=m3 eJective weight of particles in unit space supported by dilute phase #uid #ow, N=m3 eJective weight of particles in unit space supported by interphase friction, N=m3 solids #ow rate, kg=m2 s solids #ow rate at choking, kg=m2 s height level above the distributor in the riser, m fraction in eJective particle weight fraction of dense phase eJective particle weight that are supported by dense phase #uid #ow particle diameter, m mass speci5c total energy consumption for particles, W=kg mass speci5c energy consumption for suspension and transportation of particles, W=kg dense phase super5cial #uid velocity, m=s dilute phase super5cial #uid velocity, m=s super5cial particle velocity, m=s
Ug Uge Udc Ude Udf Umb Umf Upt Us Usc Usf Usi Wst
Boundaries Gs ¿ 0
Choking ¿ max
super5cial #uid velocity, m=s equivalent super5cial #uid velocity in elements, m=s dense phase super5cial particle velocity, m=s equivalent super5cial particle velocity in elements, m=s dilute phase super5cial particle velocity, m=s minimum bubbling velocity, m=s minimum #uidization velocity, m=s super5cial #uid velocity at choking, m=s averaged super5cial slip velocity, m=s dense phase super5cial slip velocity, m=s dilute phase super5cial slip velocity, m=s interphase super5cial slip velocity, m=s volume speci5c energy consumption for suspension and transportation of particles, W=m3
Greek letters ∗ a c f max mf o uni " f p
mean voidage mean voidage in the upper dilute zone mean voidage in the bottom dense zone dense phase voidage dilute phase voidage the maximum voidage for clustering voidage at minimum #uidization c at the local minimum of Nst voidage at uniform suspension #uid dynamic viscosity, kg=m s #uid kinematic viscosity, m2 =s #uid density, kg=m3 s particle real density, kg=m3
Acknowledgements The authors are grateful to the supports on the this work from the Natural Science Foundation of China (NSFC) under the grant 20176059 and China National Key Projects for Developing Basic Sciences under the Grant G1999022103.
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References Chen, A., Wu, W., Li, J., & Kwauk, M. (1994). Particle aggregation in particle-#uid two-phase #ow. In CJF-5 committee (Eds.), Proceedings of the :fth China-Japan symposium on ;uidization Nagoya, Japan, (pp. 254 –261). Beijing: Chemical Industry Press. Couturier, M., Doucette, B., & Stevens, D. (1991). Temperature, gas concentration and solid mass #ux pro5les within a large circulating #uidized bed combustor. Large, J. F. (Ed.), Proceedings of the 12th international conference on ;uidized bed combustion San Diego, USA, (pp. 107–116). New York: ASME. Drazin, P. G., & Reid, W. H. (1981). Hydrodynamic stability. Cambridge: Cambridge University Press. Gidaspow, D. (1994). Multiphase ;ow and ;uidization, continuum and kinetic theory descriptions. San Diego: Academic Press. Kruse, M., & Werther, J. (1995). 2D gas and solids slow prediction in circulating #uidized beds based on suction probe and pressure pro5le measurements. Chemical Engineering Process, 34(2), 185–203. Li, J. (1987). Multi-scale modeling and method of energy minimization in two-phase ;ow. Doctor dissertation, Academia Sinica, China. Li, J., Cheng, C., Zhang, Z., Yuan, J., Nemet, A., & Fett, F. N. (1999). The EMMS model—its application, development and updated concepts. Chemical Engineering Science, 54(22), 5409–5425.
Li, J., Ge, W., Guo, Y., & Chen, C. (1996). Non-linear behavior in particle-#uid systems and its computer simulation. Chinese Science Bulletin (Chinese ed.), 41(supplement), 10–21. Li, J., & Kwauk, M. (1994). Particle-;uid two-phase ;ow, the energy minimization multi-scale method. Beijing: Metallurgical Industry Press. Li, J., Wen, X., Ge, W., Cui, H., & Ren, J. (1998). Dissipative structure in concurrent-up gas-solid #ow. Chemical Engineering Science, 53(9), 3367–3379. Meynard, F. (1997). An analysis of the energy minimization multi-scale model for circulating #uidized beds. Unpublished. Xu, G. (1996). Hydrodynamic modeling for heterogeneous ;ow structure in circulating ;uidized beds. Doctor dissertation, Institute of Chemical Metallurgy, Chinese Academy of Sciences. Zhang, W. (1995). Fluid dynamics of the transport zone of circulating ;uidized beds–with application to boilers. Doctor Dissertation, Chalmers University of Technology, Goeteborg, Sweden. Paper 2:9. Zhang, W., Tung, Y., & Johnsson, F. (1991). Radial voidage pro5les in fast #uidized beds of diJerent diameters. Chemical Engineering Science, 46(12), 3045–3052.
www.elsevier.com/locate/ces
Physical mapping of #uidization regimes—the EMMS approach Wei Ge, Jinghai Li ∗ Multi-Phase Reaction Laboratory, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100080, People’s Republic of China Received 19 March 2001; accepted 29 January 2002
Abstract The existence of multiple regimes of distinctive #ow structure is a remarkable characteristic of #uidization, which is far from being physically interpreted under a uni5ed approach. The energy minimization multi-scale model (Particle-Fluid Two-Phase Flow, the Energy Minimization Multi-Scale Method, Metallurgical Industry Press, Beijing, 1994) is potentially such an approach in which the inclusion of stability criteria enables the prediction of heterogeneity and non-linear behaviors in #uidized beds. However, fully analytical solution of the model is impossible so far, and numerical solutions have resorted to general optimizing software. Therefore, the detailed characteristics of the solutions and their theoretical implications have not been fully explored. In this paper, we have achieved this by a rigorous numerical approach and by retrieving all missing roots, which leads to physical mapping of #uidization regimes. The model is also extended to unsteady conditions with acceleration and simpli5ed by employing a single stability criterion, which identi5es choking as a jump between two branches of the stable solution. Calculations based on this version are in reasonable agreement with measurements on bench, pilot and commercial scale circulating #uidized beds. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Choking; Fluidization; Hydrodynamics; Multi-scale; Regime transition; Stability
Despite its wide applications, #uidization is not well understood for its multi-scale structure and multiple regime transitions. These remarkable characteristics are shaped by the stability that has been presented as the extrema of energy dissipation terms in the energy minimization multi-scale (EMMS) model (Li & Kwauk, 1994; Li, Wen, Ge, Cui, & Ren, 1998). Therefore, the model has predicted the general #ow structure and regime transition reasonably. However, the non-linear nature of the model equations has prohibited its full analytical solution, so calculations have resorted to general-purpose optimization software (Li, 1987) or simpli5ed numerical and analytical schemes (Xu, 1996). So far, the mathematical character of the model is not understood thoroughly, which has limited its application and physical improvement. This work is therefore devoted to a complete numerical solution of the model to explore some new features. ∗
Corresponding author. Tel.: +86-10-62558318; fax: +86-10-62558065. E-mail address: [email protected] (J. Li).
1. Formulation The EMMS model is currently a 0-D description of the time-averaged behavior of #uidization, but its principle is equally applicable to the elements in two-#uid models (TFM) (e.g. Gidaspow, 1994), and it can be expected to improve the accuracy of TFM by describing the signi5cant heterogeneity within the element that is neglected now. However, in elements particle weight is not always equal to the drag acting on it. Therefore, as a 5rst step, we extend the model to where the particles have equal acceleration. To facilitate the discussion, we use a general equation for the momentum balance in uniform suspension of mono-sized spherical objects (refer to the Notation for the de5nitions of the symbols), i.e., 1 − CD0 (Us d= ) 2 1 d f Us2 (=6)d3 4:7 4 2 =(1 − )( p − f )(a + g)k; which can be rewritten to Us d 4 p − f Us2 = CD0 (a + g)d4:7 k; 3 f
0009-2509/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 2 3 4 - 8
(1) (2)
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where Us is de5ned as Ug − Ud =(1 − ). We denote this equation as a function Us (d; ; k; a). For the dilute phase, all eJective particle weight is balanced by #uid drag, i.e., Gf = (1 − f )( p − f )(g + a) ⇒ Usf = Us (dp ; f ; 1; a);
(3)
where Usf = Uf − Udf f =(1 − f ). For the dense phase, eJective particle weight is partially supported by the dense phase #uid #ow, and the rest is supported by the bypassing dilute phase #uid #ow, termed here as interphase (dilute-to-dense phase) drag, i.e., Gi + Gc = (1 − c )( p − f )(g + a):
(4)
As the dense phase is assumed to occur as spherical clusters, we may de5ne kc = Gc =(Gc + Gi ) and write Usc = Us (dp ; c ; kc ; a);
(5)
Usi = Us (l; 1 − f; (1 − c )(1 − kc ); a);
(6)
where Usc = Uc − Udc c =(1 − c ); Usi = (1 − f)(Uf − Udc f =(1 − c )). Next, the dense phase pressure drop is balanced by that of the dilute phase plus the interphase, i.e., fGi Gf + (7) = Gc = G = ( p − f )(1 − )(g + a): 1−f From Eqs. (4) and (7), we have kc = (1 − )=(1 − c ) and Usi = Us (l; 1 − f; − c ; a). Besides the force balances, the continuity of the #uid and the solids requires Ug = fUc + (1 − f)Uf ;
(8)
Ud = fUdc + (1 − f)Udf :
(9)
Finally, the correlation for cluster diameter is ( p − f )(g + a)Ud = p (1 − max ) − Nst; mf l = ; dp Nst − Nst; mf
(10)
where by de5nition Nst = Wst =(1 − ) p and Wst = fGc Uc + (1 − f)(fGi + Gf )Uf :
(11)
It can be derived from the foregoing equations that p − f f − Nst = Ug − ; (12) f(1 − f)Uf (g + a) 1− p p − f Ud mf Umf + (g + a): (13) Nst; mf = p 1 − mf The constraints of the model consist of Eqs. (3), (4), (7) – (10), but they have 8 independent variables: Uf ; Uc ; Udf ; Udc ; f ; c ; f and l, which needs the optimization of Nst to close the model as stability criterion. 2. Physical background Previous results from this model with a = 0 can be summarized as follows: For conditions below choking, i.e.,
Ug ¡ Upt (Gs ) or Gs ¿ Gs∗ (Ug ), the stability criterion is assumed to be Nst → min which results in the most distinct two-phase structure of c = mf and f → 1. Beyond choking, Nst → max is assumed, leading to uniform suspension in theory. And the criterion determining choking is proposed to be (Wst )Nst →min = (Wst )Nst →max|c =mf :
(14)
Despite its agreement with experimental phenomena, the results above are imperfect in two aspects, i.e., the dilute phase should also have particles (f ¡ 1), and weak heterogeneity should still exist beyond choking. For the 5rst problem, we notice that the physical mechanism of particle aggregation is in some ways comparable to the condensation of steam from air. Steam condenses to liquid when their inter-molecular attraction can overcome their heat movement, and particles aggregate when their random motion cannot resist the non-uniform drag on them and the non-contacting forces between them. It follows that, in the presence of the aggregated dense phase, the dilute phase would remain at the critical concentration for aggregation, just as the saturated steam. This is the rational background of the correction suggested by Li et al. (1999) that f = uni (Ug ; Gs∗ (Ug ));
(15)
which means the dilute phase is always at the critical state between heterogeneity and homogeneity. However, this correction is not easily implemented because Gs∗ (Ug ) itself is to be calculated from the model, so global iterations are needed to determine f , and the algorithm could be very costly and unstable. On the other hand, we may 5nd that max , which is obtained from kinetic analysis (Chen, Wu, Li, & Kwauk, 1994), is indeed corresponding to this critical concentration, and it can be calculated from Ug and Gs directly (in Chen’s treatment, Ug is derived as Ud p =( p − f )=(1 − ), but that is an approximation valid only when Ug and are high enough). It should be noted, however, that max is Not the lowest voidage that homogeneity can exist, though it is the highest voidage that heterogeneity can exist. For many aggregative systems, max is very close to unity, but a particulate regime can exist at voidages near mf . The reason behind is, that particle–#uid systems are, after all, more complicated than steam in air. Once a cluster forms, it will have meso-scale interactions with the dilute phase on their interface, and exert additional constraints on the formation of the clusters. The EMMS model is capable of describing this hydrodynamic eJect, as we can see later, but the kinetics of particle suspension are not incorporated previously, which has led to the prediction of f = 1. Now we may say that f at the beginning of clustering would be max , or the upper limit of f is max . The solution to the second problem of heterogeneity beyond choking requires a reexamination of the numerical methods of the model.
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
3. Numerical methods Although g is replaced with g + a, the mathematical characteristics of model are not changed. From Eq. (12), we may 5nd that Nst → max can be reached only when f = c or f = 0 or f = 1, where Nst = NT = (g + a)Ug ( p − f )= p . In fact, they all correspond to the homogeneous state. For Nst → min, if c and f are given, the model is closed for solving the other 6 variables, which is called the sub-model. Then the whole model can be solved by searching the minimum of Nst among all pairs of c and f within the range of [mf ; max ]. Before going to details we present the general results 5rst. The model may have two, one or no solution(s), as exempli5ed in Fig. 1. It is interesting that the #ow structures 1.0
3995
for diJerent solutions seem to include all varieties for #uidized beds. To explain this accordance, we must start with the sub-model, for which Xu (1996) and Meynard (1997) have developed speci5c numerical schemes separately. For simplicity, we use Meynard’s scheme, i.e., 1. 2. 3. 4.
Calculate Usf from Eq. (3). Select a trial value for f. Calculate kc and then calculate Usc from Eq. (5). With the de5nitions of Usf and Usc ; Uf ; Uc and Udc can be solved from Eqs. (8) and (9). 5. Calculate l from Eq. (10). 6. Calculate Usi from its de5nition and Eq. (6), respectively, denote the diJerence as PUsi . 1.0
min
min 0.9
0.9
Ug=5m/s Nst=0.756(blue)~1(red) one root at right top
0.8
c
c
0.7
0.7
0.6
0.6
0.5
0.5
0.0
0.2
0.4
0.6
(f-c)/(max- c)
Ug=3.4m/s Nst=0.747(blue)~1(red) two roots at right top and right bottom
0.8
0.8
min 0.0
1.0
0.2
0.4
0.6
0.8
(f-c)/(max- c)
1.0
1.0 1.0
0.9 0.9
Ug=1m/s Nst=0.411(blue)~1(red) one root at right bottom
0.8
c
Ug=0.01m/s, no solution for any combination of c and f, Nst undefined
0.8
c
0.7
0.7
0.6
0.6
min
0.5 0.0
0.2
high
0.4
0.6
( f- c)/(max- c)
0.8
1.0
0.5 0.0
0.2
0.4
0.6
(f-c)/(max- c)
0.8
low Fig. 1. Variation of Nst with c and f for the FCC=air system (Li, 1987, p. 168) in Table 2 (Gs = 50 kg=m2 s).
1.0
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W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
Fig. 2. Roots of the sub-model for the FCC=air system in Table 2 at Ug = 0:0631 m=s and Gs = 0:0631 kg=m2 s. The PUsi (f) curve in each region has a unique type of shapes and a unique set of roots, labeled as, e.g., 2r1 .
7. If PUsi is not small enough, select a new f that can reduce this diJerence and go back to step 3. Unfortunately, a thorough study on the variation of PUsi with f is so far absent; therefore, application of common approaches for solving non-linear equations, such as dichotomizing search, has not been justi5ed for step 7. Though it has brought about reasonable results, this scheme fails to 5nd the existence of multiple roots. Now by examining a large number of cases with various parameters, we found that the PUsi (f) curves can be classi5ed into 5 types, and the distribution of these types among possible combinations of c and f at given conditions are illustrated by a typical case in Fig. 2: 1. Drop or then rise, all above zero, no root. 2. Drop and then rise, a middle section below zero, two roots. 3. Drop or then rise, a right section below zero, one root. 4. Below zero except a middle section, two roots. 5. All below zero with no root. But under some other conditions, the types may be less. A notable feature here is that the boundary between types 3 and 4 coincides with f = uni (Ug ; Gs ); Types 1–3 seem to have diJerent nature from types 4 and 5 with PUsi (0)
approaching +∞ and −∞, respectively. The physical meaning behind this is yet to be explained. Now with this simple judgement in mind, we can speed up the calculation greatly by treating the 5rst 3 and last 2 types with dichotomizing search separately. Then with the new scheme, we con5rm that Nst → min always requires the maximization of f , which is f = max instead of f = 1 now (ref. Fig. 1). And from now on, we imply f = max when we talk about the variation of Nst . The new 5nding is that Nst does not always increase monotonously with c , it may have one local minimum at c = mf and another at a higher c which we denote as o . This is seen more clearly in Fig. 3 where Nst is plotted against c only. In this case, with the continuous variation of Ug or Gs , there is a value where Nst (mf ) = Nst (o ):
(16)
It is about Ug = 3:4 and 0:6 m=s for Figs. 3a and b, respectively, and o is about 0.96 and 0.997. The #ow structures corresponding to c = mf and c = o are distinct, but they must coexist in the system. Then we realize that if this critical state is understood as choking with Eq. (16) being its criterion in place of Eq. (14), several improvements can be made simultaneously. Namely, the remaining heterogeneity observed beyond choking can be retrieved; a more clear interpretation of choking appears, which is simply the shift
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
3997
Fig. 3. Choking explained as coexistence of bi-stable states for the FCC=air system in Table 2, the #ow structure corresponding to each state is sketched near its curve.
between two stable states; and Nst → min can serve as the uni5ed stability condition for all heterogeneous regimes in #uidization. To support this modi5cation, we want to explain that the stability criterion of a single phase in the system, as assumed by Li and Kwauk (1994), is easy to understand: the particles always seek minimum gravitational potential; As a result, is minimized, and the #uid phase seeks least resistance which minimizes Wst . When the particles are #uidized, both conditions are subject to the constraints from the other. Therefore, the overall stability criterion should embody the in#uence of both phases. So Nst → min seems to be a reasonable combination, for it requires the minimization of both Wst and . The physical meaning of this criterion can be demonstrated more explicitly if we notice that it is equivalent to Nd = NT − Nst → max, since NT is constant under the given conditions. From Eq. (11), we have
Nd = fgi Uf , where gi = Gi =(1 − )= p is the eJective speci5c weight of the particles supported by interphase drag. We also notice that as the dense phase is assumed to occur as clusters, higher f means larger phase interface; and as Ug is given, larger Uf means sharper diJerence of #uid velocity between the two phases. In summary, Nd → max is to establish a most distinct interface between the dense and dilute phases. Interface has been the focus of stability studies on many other phenomena (Drazin & Reid, 1981), and there seem to be no exception here. 4. Regime-mapping Now what is the relation between the original and modi5ed versions of the model and how can they be uni5ed? Fig. 4 shows the solution of the new version for each point
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
Gs(kg/m s)
3998 2
(Usi (f) always positive)
1
n tio a z idi flu
10
c
(0.5~1)
-1
10
t fas
tur bu len t
10
-2
10
-4
Umf
-3
(a)
10
Ut
-2
10
-1
0
10
f
0.999~1
1
1
10
0
0
10
Gs (kg/m s)
10 2
2
Gs (kg/m s)
10
2
10
10
-1
10
-2
-1
10
-2
10
10
-3
-3
10
10
-4
-4
10
-3
10
-2
10
(b)
-1
10
0
10
2
10
1
10
-2
10
(c)
f 0~1
2
10
1
10
1
0
Gs (kg/m s)
2
2
-1
10
-2
-1
10 Ug(m/s)
0
10
1
10
l/dp 1~140000 (lg scale)
10
10
0
10
-1
10
-2
10
10
-3
-3
10
10
-4
-4
10
-3
10
Ug(m/s) 10
Gs (kg/m s)
1
10
0.5~1
(d)
high
Ug(m/s)
2
10
low
(
late ticu r a p
fixed bed
=
10
max
homogeneous solution
Umb -3
10
c>>mf
d) ize rt l a e o nid nsp (no te tra u heterogeneous solutions dil
bub bli ng
mo vin gb ed
0
ch o
kin g
c=mf
no solution
id ea U lize si ( f) d di al lu wa te ys tra ne ns ga po tiv rt e)
2
10
10 -3
10
-2
10
-1
10
Ug (m/s)
0
10
1
10
(e)
-3
10
-2
10
-1
10
Ug (m/s)
Fig. 4. Calculated #ow structure of the FCC=air system in Table 2.
0
10
1
10
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
in a wide range on the Ug −Gs plane for a given system with a = 0, where a panorama of #ow regimes appears. When Ug is below Umf , the PUsi (f) curve in Fig. 2 is always positive, which means the #uid #ow cannot provide enough drag to counter-balance the weight of any amount of ascending solids, this is also true if Gs is too high when Ug ¿ Umf . In fact, they correspond to 5xed beds (Gs =0) and moving beds (Gs = 0). Then if Gs is too low when Ug ¿ Umf , two-phase solutions are absent too. But in this case, homogeneous solutions from the closed constraints of force balance are available, where stability criteria play no role and Nst =NT . Therefore, the new version of Nst → min and the old versions of Nst → max are consistent here. Though no structural diJerence is predicted, calculations show that this region can be divided into a right part of ¿ max where homogeneity results from particle kinetics, and a left part of ¡ max where no stable cluster can ful5ll the hydro-dynamic constraints in the EMMS model. Compared with engineering practice, they should belong to particulate #uidization and idealized dilute transport respectively. However, for the particulate and bubbling regimes, Gs is actually not an independent parameter, it is mainly a function of Ug (for mono-sized spherical bed materials). That means, for a given system, the particulate states will be found on a curve in the Ug − Gs plane (e.g., the dash line in Fig. 4), and its intersection with the upper boundary of the particulate regime determines Umb . Unfortunately, despite this physical interpretation, the EMMS model now is unable to give the value of Umb because the macro-scale heterogeneity and dynamic process involved, especially that of the transition zone, are not considered yet. The region of two-phase solutions is partitioned into two parts by the choking line which starts from Ug = Ut roughly. The right part with loose dense phase corresponds to dilute transport, which it not discriminated from idealized dilute transport in the old version. The left part corresponds to bubbling, turbulent and fast #uidization regimes. No jumps in #ow structure between these three regimes are captured by the model, but they can be mapped relatively from left to right in order of increasing or decreasing l and f. Therefore, depending on the magnitude of Umb , the particulate regime may develop to any of the three regimes or even to (idealized) dilute transport directly as in particulate systems. However, we have to explain here, that the model seems to have low accuracy at the bottom of the two-phase region. This may be evidenced on the transition line where the calculated l approaches dp . Physically, it means that the two-phase structure will continuously degrade to homogeneous solutions, but this is not echoed by the distribution, which displays signi5cant discontinuity. Among the underlining reasons, is possibly the validity of Eqs. (6) and (10) and the presumption, that dense phase occurs as discrete cluster at such extreme conditions. Anyway, it is a subject for further study and improvement. When a = 0, we 5nd that higher a results in larger and denser clusters. As shown in Fig. 5, for a given point on the Ug − Gs plane, both and c decrease with a while l=dp
3999
increases with a. This structural change seems to counterbalance the increase of both #uid resistance and particle potential when the particles are accelerated, and it is a reasonable reaction from the point of view of the EMMS model. 5. Computational results and discussion Considering the simplicity of our model and the uncertainties in both modeling and measurements, no quantitative agreement with experiments can be expected now, but qualitative consistency is widely observed in quite diJerent systems and regimes. We think this is actually a more convincible validation of the model. In Fig. 6 and Table 1, we compare the results from the EMMS model with previous measurements on an #uid cracking catalyst (FCC)/air system (Li & Kwauk, 1994, p. 143), its speci5cations are in Table 2. In theory, choking is a jump from one state to the other and only on this point can the two states coexist, but as any practical system goes, visible diJerence between the upper and lower section of the system is observed in the measurements for a fairly broad range. As a support for our explanation of choking, we 5nd that for each Nst (c ) curve, the distinctness of its non-monotonicity in calculation is in accordance with that of the two-state structure in measurements. Here we may also get a more precise understanding of this stability. That is, the possibility of the appearance of the state is actually determined by the shape of the curve near this state rather than the absolute value of Nst . This is something like the stability of a rolling rock on the hill. It is not necessarily more stable at a lower altitude, but rather in a deeper valley. So, we 5nd the two states before and after the S-shaped transition in the upper insets roughly corresponding to the solution at c = mf and c = o (especially their tendency of variation with operating conditions), rather than the intermediate solutions which may have lower Nst than one of them. Similarly, a practical choking de5nition conforming to its theoretical counterpart (Eq. (16)) could be the point where the coexistent states are most distinctive and the transition between them is most dramatic. As a major improvement, the c beyond choking, which was somewhat arbitrarily assumed as mf in the old version to compensate for its loss of heterogeneity there, now agrees qualitatively with diJerent experimental data (Li, Ge, Guo, & Chen, 1996; Kruse & Werther, 1995), both are closer to max than mf . Our choking prediction has also been extended to larger scale CFBs with encouraging agreement to in situ measurements, as shown in Fig. 7. Still, it may be more important to recognize that simple hydrodynamic models with stability considerations can explain the general characteristics of the system, though the detailed #ow structure is so complicated that it is virtually unpredictable even for the most intensive CFD approaches. And this may be common to other complex systems also. A program has now available on http://www.chinweb.com. cn/%7Exyli/test/datainput.htm, where one can specify the
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1.0
εf
voidages
0.9
ε
0.8
0.7
0.6
εc
0.5
-4
-2
0
2
4
6
8
10
4
6
8
10
2
a (m/s ) 100 98
l/d p
96 94 92 90 88 86 -4
-2
0
2 2
a (m/s ) Fig. 5. EJect of particle acceleration of on #ow structure for the FCC=air system in Table 2, Ug = 3:4 m=s, Gs = 50 kg=m2 s.
material and operational parameters of the #uidized bed and get its #ow structure from our model. For instance, if you input the properties of the FCC=air system in Table 2, and specify Ug = 2:1 m=s; Gs = 32 kg=m2 s, you get f = 0:999742; c = 0:5 and = 0:81034. The computation takes 5 min on average. 6. Future work As discussed above, a better correlation of l that can accommodate the variety of meso-scale heterogeneity and which links smoothly with single particle dynamics is desir-
able. Considering the sensitivity of l on max , more elaborate kinetic theories are needed to predict max more accurately. However, it is worth reminding that l has little geometrical sense anyway, it is mainly a characteristic parameter of heterogeneity and interphase friction. For the application of the EMMS model in TFM simulations, a practical way is to take the from TFM calculations as an additional given condition and 5nd a with the EMMS model. It is slightly more complicated than the case we have discussed at given a, but it can still be solved in a similar way. Firstly, since NT and Nst are both functions of a, the stability criterion should be expressed as Nst =NT → min. And secondly, Ug and Gs will be replaced with
W. Ge, J. Li / Chemical Engineering Science 57 (2002) 3993– 4004
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Fig. 6. Experimental evidence of the bi-stable states in the FCC=air system in Table 2. (a) Gs = 14:3 kg=m2 s, Ug = 1:52 m=s, (b) Gs = 26:6 kg=m2 s, Ug = 1:52 m=s, (c) Gs = 48:2 kg=m2 s, Ug = 1:52 m=s.
vectors in elements, but we can 5nd that only slip velocities, which have counterparts in elements, are signi5cant to the hydrodynamic constraints. In fact, the absolute values of Ug and Gs that are relative to the walls are only necessary for l and Nst because they depend on the overall #ow pattern. (Wall eJect is considered in this way, otherwise the model
will give diJerent results for the same system in diJerent reference frames.) So it may be possible to 5nd a Uge for each element based on the EMMS calculation of the whole bed with modi5cations according to its location in the #ow 5eld. Then for each element, we can write the EMMS model in the direction of local averaged slip velocity with
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Table 1 Calculated vs. measured voidages for the FCC=air system in Table 2
Ug (m=s)
Gs (kg= m2 s)
a Calculated
a Measured
∗ Calculated
∗ Measured
1.52 1.52 1.52 2.1 2.1 2.1 2.1 2.6 2.6 2.6 2.6
14.3 26.6 48.2 24.1 32 64.2 96.3 42.8 64.2 96.3 192.7
0.81 0.77 0.73 0.83 0.81 0.76 0.74 0.83 0.80 0.78 0.73
0.82 0.81 0.80 0.84 0.84 0.83 0.81 0.90 0.88 0.86 0.82
0.99 — — 0.99 0.98 — — 0.98 — — —
0.98 — — 0.97 0.96 — — 0.94 — — —
Table 2 Speci5cations of the cited systems
50
Chalmers (Zhang, Tung, & Johnsson 1991; Zhang, 1995) 2600 0.32 4.832 260 1.78 0.43
ham
Name References p (kg= m3 ) f (kg= m3 ) " (10−5 kg= m s) dp (m) dt ( m ) mf
C hat
lines: calculated dots: measured
F C C /a
ir
40
rs
Chatham
FCC=air (Li & Kwauk, 1994) 929.5 1.1795 1.8872 54 0.09 0.5
However, in this way the structural information obtained in the EMMS model is not used in TFM simulations. In long term, the favorable form to incorporate the two models may be called the two-phase two-#uid model. That is, the particles and #uid in the dense and dilute phase are all formulated as continuums separately, with similar but doubled equations of the TFM now, and the interaction between each pair of them are determined in a further generalized version of the model.
C ha
Gs (kg/m2s)
lm e
30
Chatham (Couturier, Doucette, & Stevens, 1991) 2500 0.32 4.832 200 4.47 0.51
7. Conclusions 20
FCC/air
10
Chalmers
0 0
1
2
3
4
5
6
7
8
Ug (m/s)
Fig. 7. Choking prediction for some CFB combustors.
scalar value of Uge and Ude = (Uge − Us )(1 − )=, and transfer the solved a to a TFM calculation for the next time step.
The EMMS model is simpli5ed to assume Nst → min as the uni5ed stability criterion for all #ow regimes in #uidization, and also extended to unsteady #ows. The mathematical characteristics of the EMMS model are studied thoroughly, which 5nds the existence of two roots at given voidages and the bi-stable character of Nst . A physical mapping of #uidization regimes is proposed based on this improved model. As summarized in Table 3, the 5xed bed has no solution in the model; idealized dilute transport and particulate #uidization have homogeneous solutions under corresponding operation conditions; and diJerent two-phase structures are predicted for the rest of regimes. Especially, choking is interpreted as a condition where two stable states, i.e. the “sharply” heterogeneous regimes of bubbling to fast #uidization and the “smoothly” heterogeneous regime of dilute transport, have the same Nst and coexist most distinctively. Acceleration of
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Table 3 Partition of #uidization regimes suggest by EMMS
Predicted structure (No solution) Heterogeneous structure Homogeneous structure
Regimes Fixed bed Moving bed Sharply Smoothly
the solid phase results in the decrease of Gs∗ at given Ug . The general agreement of our results to measurements in diJerent scale circulating #uidized beds (CFBs) suggests that the model possesses the essential mechanism to underline the heterogeneity and regime multiplicity of particle–#uid #ow. Though it may need further supplements and improvements to become quantitatively predictive, it can be considered as a reasonable framework to work on.
Notation a d dp dt f g G Gc Gf Gi Gs Gs∗ h k kc l NT Nst Uc Uf Ud
Bubbling #uidization Turbulent #uidization Fast #uidization Dilute transport Idealized dilute transport Particulate
acceleration of the solids, m=s2 diameter of spherical objects (particle, cluster, etc.), m surface-to-volume mean particle diameter, m equivalent inner diameter of equal cross-sectional area of the CFB riser, m dense phase fraction gravitational acceleration, m=s2 eJective weight of particles in unit space, N=m3 eJective weight of particles in unit space supported by dense phase #uid #ow, N=m3 eJective weight of particles in unit space supported by dilute phase #uid #ow, N=m3 eJective weight of particles in unit space supported by interphase friction, N=m3 solids #ow rate, kg=m2 s solids #ow rate at choking, kg=m2 s height level above the distributor in the riser, m fraction in eJective particle weight fraction of dense phase eJective particle weight that are supported by dense phase #uid #ow particle diameter, m mass speci5c total energy consumption for particles, W=kg mass speci5c energy consumption for suspension and transportation of particles, W=kg dense phase super5cial #uid velocity, m=s dilute phase super5cial #uid velocity, m=s super5cial particle velocity, m=s
Ug Uge Udc Ude Udf Umb Umf Upt Us Usc Usf Usi Wst
Boundaries Gs ¿ 0
Choking ¿ max
super5cial #uid velocity, m=s equivalent super5cial #uid velocity in elements, m=s dense phase super5cial particle velocity, m=s equivalent super5cial particle velocity in elements, m=s dilute phase super5cial particle velocity, m=s minimum bubbling velocity, m=s minimum #uidization velocity, m=s super5cial #uid velocity at choking, m=s averaged super5cial slip velocity, m=s dense phase super5cial slip velocity, m=s dilute phase super5cial slip velocity, m=s interphase super5cial slip velocity, m=s volume speci5c energy consumption for suspension and transportation of particles, W=m3
Greek letters ∗ a c f max mf o uni " f p
mean voidage mean voidage in the upper dilute zone mean voidage in the bottom dense zone dense phase voidage dilute phase voidage the maximum voidage for clustering voidage at minimum #uidization c at the local minimum of Nst voidage at uniform suspension #uid dynamic viscosity, kg=m s #uid kinematic viscosity, m2 =s #uid density, kg=m3 s particle real density, kg=m3
Acknowledgements The authors are grateful to the supports on the this work from the Natural Science Foundation of China (NSFC) under the grant 20176059 and China National Key Projects for Developing Basic Sciences under the Grant G1999022103.
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