Znd. Eng. Chem. Res. 1991,30, 1977-1981
1977
Effect of Shear Friction on Solid Flow through an Orifice Ji-Yu Zhang Znstitute of Coal Chemistry, Academia Sinica, Box 165, Taiyuan, China
Victor Rudolph* Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland, Australia 4072
A model to predict the flow rate of particulate solids through an orifice is developed based on force and momentum balances at the stress-free surface (SFS),which occurs near the orifice. It is shown that shear friction between flowing and nonflowing particles around edges of the bottom orifice in standpipes is an important factor to be taken into account. The model utilizes a simplified description of the effects of shear friction between flowing and stagnant particles. The resulting equation has a similar form to Beverloo’s equation but with an added term for avoiding the orifice size correction inherent in the Beverloo form. Introduction The familiar Beverloo equation (Beverloo et al., 1961) has been recommended as a common and convenient form for the prediction of solid flow rate for gravity discharge through an orifice (Nedderman et al., 1982). The equation is semiempirical, requiring an arbitrary coefficient that artificially reduces the size of the orifice, purportedly to account for the “empty annulus” reported for solids discharge through orifices. The coefficient is essentially a fitting parameter, and has variously taken values ranging from 1 to 3d, for different investigators (Beverloo et al., 1961; Williams, 1977). The lack of a convincing physical explanation for this coefficient, as well as questions as to why the value should vary for different size particles having the same (round) shape or for different materials when the particle sizes are similar, casts suspicion on the general applicability of the equation. In order to provide an explanation to these questions, a theoretically based force analysis on the flow of gas-solid mixtures through an orifice is proposed in the present paper. In many practical instances solids discharge through an orifice is accompanied by a counterflow of gas. Examples include unventilated hopper flow or the many industrial processes with circulating fluidization operations, such as sand cracking and carbonization of oil shale (Kunii and Levenspiel, 1969), where granular moving beds of solids flow down through bottom-restricted standpipes. To generalize the force analysis, the drag force arising from the negative pressure gradient is included. Force Analysis on Stress-Free Surface When particles flow down through an orifice restriction, they pass through a stress-free surface (SFS)just spanning the edges of the orifice, where the stresses on the particles become zero (Zhang et al., 1989a,b). In order to develop a model for predicting the solid flow rate through an orifice, it is necessary to focus attention on resolving the problem of solid motion at the SFS. The SFS in the orifice region is critically important to flow in the whole standpipe. Assuming constant bulk density above the SFS, &,, the first step is to consider an elementary volume (dAdr) of a stream tube just approaching on the SFS,in radial downflow and at coordinates (r, e), described in Figure la. It may be supposed that particles in the stream tube have a uniform average solid velocity, Us, through any surface (Aro)orthogonal to a radius vector, r , at angle, 8.
* To whom correspondence should be addressed.
The continuity equation of solids is UsArepb= constant (1) For three-dimensional flow in a cone Ad, which is proportional to r2, will decrease toward the apex, which is a converging point sink somewhat below the orifice. Thus eq 1 becomes U, = X/r2 (2) where the proportionality constant h is independent of 0 because of the assumption of a uniform U,. If N is the number of particles flowing vertically per unit area per second through an elementary area dA and ml the mass of an average particle, then conservation of numbers flowing through dA is given by N = (pb/mi)u, (3) The mass flow rate of solid, W,, through the SFS of the spherical cap of radius, R,, with half-angle 0 is
W , = x a m l N dA = x a m l N ( 2 * R , ) sin 0 dl
(4)
where dl is the arc length between 0 and 0 + dB and dl = R , de. Substituting eqs 2 and 3 into 4 and integrating = 2?rpb(1 - cos 0 ) h (44 Second, consider the force equilibrium on the whole SFS area, Af,, with an elementary thickness, dr. The sign conventions here are that clockwise and centripetal directions are taken as positive. Since the additional gas pressure between particles can be regarded as an extra body force to incorporate into the body force stemming from gravity on the particles (Hancher and Jury, 1959; Brandt and Johnson, 1963), momentum conservation illustrated in Figure l b is given by I
gravity shear friction acceleration where re, is the shear stress between flowing and nonflowing particles at the edges of the orifice; c, is the circumferential perimeter of the corrected orifice diameter, i.e., De = (Do- d,), so c, = *(Do- Dp). The correction is explained below. Logically, a uniform particle residing on an edge of the orifice is in a stable situation if it is more than d,/2 distant from the orifice open edge. As a result, the limiting distance around the orifice edge, corresponding to a particle diameter, d,, may be taken as a corrected term for char-
@ 1991 American Chemical Society oaaa-5aa5pi ~2630-19~1~02.5~/0
1978 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
p = */4 - &/2
(8) If & is taken to have values typically between 20° and 50° and (1 - l/(pbg)(dp/dz)) is replaced by a modified gravity factor, cgp, eq 7 may be rewritten as
L's
w.=
(a)
Stream-tube in continuum
(b) Force analysis on SFS a t fl
(c)
Shear force on perimeter of SFS
Figure 1. Force analysis on stress-free surface (SFS).
acterizing the statistically empty annular zone adjacent to the orifice edge as observed by Brown and Richards (1960). In the analysis of the data for empty annuli for slots and circles in a flat floor reported by Brown and Richards (1960, 19701, the dimensionless constant, k, in the term of (Do - kd,) obviously approximates to 1, as previously recognized and used for treating experimental results in gravitational flow (Ching et al., 1966). Therefore, (Do- d ) is used as an effective orifice diameter for deriving the foiowing series of correlations. At the steady state corresponding to a given negative pressure condition, using eq 2 and letting r = R , and then integrating and rearranging eq 5 yields
(7b) Note that to predict T,, is a very complex problem as discussed in the theoretical analysis of stresses in axially symmetric hoppers by Walters (1973). Also, /3 is changed by different upward gas velocity through the SFS (Rudolph and Judd, 1989; Zhang, 1990). If the modified gravity term, Gv,namely ( p g - dp/dz), is divided into two parts, G, and G,, which are respectively in the radial and tangential directions at half-angle, p, as illustrated in Figure IC,then
G, = G, sin
=
( 2)
- sin p =
p g -
The shear stress T~~ is in direct proportion to G,; Le., ( T ~ , / G J= constant. The ratio of [(4/[pb(D0- dp)g])7ew] to G, is also a constant because Pb, Do, and d, are given in each particular case. Thus G V
I
I
(6) Since R, = (Do- d,)/(2 sin P), see Figure la, eq 6 becomes
G,
where k72 = k71 (sin p ) p g . Substituting eq 10 into 7 and 7b, respectively,and taking a shear friction factor, c, = (1 - k T 2 ) , gives r (1- cos p)"2 w,= 4(2)1/2 sin3l2p
Finally, substituting eq 6a into eq 4a gives
pbg
?dz) ) l 2 ( D 0
- dp)5/2(11)
and W , = (0.54-0.68)pbg'~2(c,c,)'~2(Do - dp)5/2 ( l l a )
If the downward solid flow is only under gravity and the effect of shear friction between flowing and nonflowing particles around the edges of the orifice is neglected, eq 7 can be simplified as
At this stage, the problem of predicting the solid flow rate is converted to that of actually determining c,. In the case of gravitational flow only, eqs 11 and l l a become
(lib)
and Equation 7a is Brown and Richards' correlation representing the average velocity over the spherical cap of the SFS based on the minimum energy theorem (Brown and Richards, 1970). However, in fact the effect of shear friction between flowing and nonflowing particles should not be neglected and must physically exist. In the above equations, the angle of approach of particles to the orifice, P, is a function of the angle of internal friction of particles, &, and can be estimated from the angle of slide (Wieghardt, 1975):
W,, = (0.54-0.68)pbg1/2(~,)1/2(D0 - dp)5/2 (11~) Also
Comparing eq I l c with Beverloo's equation (Beverloo et al., 19611, namely W,, = (0.55-0.65)pb0g'/~(D,- kdp)5/2 (12)
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1979 Table I. ComDarison between Exwrimental and Calculated Gravity Solids Flow in Different Experiments author properties of solids geometries of system Do, cm (WmL g/s (WdL, g/s (Wd),/(Wd).,o this studv elass beads 2 = 1.25 m 0.8 8.25 8.27 +0.12 D, = 3.9 cm 1.2 26.27 26.51 +0.87 d , = 0.0677b 1.4 41.26 41.18 -0.19 p, = 2.478b 1.7 70.81 71.71 +1.27 pm 1.596b 2.0 114.51 112.55 -1.70 &i = 27' 64, the effect of shear friction around the edges of the 9FS on particle flow is negligible and c, = 1. Since c, accounts for friction, it seems plausible that c, depends on the frictional properties of the powder. These are fully described by the angle of slide between flowing and nonflowing particles; see Figure IC. For no gas flow, Le., when the particles are flowing under gravity only P w = */4 + @i/2 (13) When gas flow does occur, simple corrections to eq 13 may be applied (Rudolph and Judd, 1989). We speculate that if the functional dependence of c, on P, is known, then this provides a method for predicting c, from &, a physical property of the powder. Noting on Figure 2 that the nodes represented by D o / d , = 10 and 30 give c, values of (1 - l/tan 8,) and (l/tan Ow), respectively, linear extrapolation provides
Some evidence is available (Zhang, 1990) to support eq 14, based on changing the magnitude of 3/, by imposing gas flows through the SFS.
1980 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 cp = circumferential perimeter of effective orifice (L)
modified gravity factor 2 Dtmean orifice and tube diameter, respectively (L) particle diameter (L) =
=
d, =
dp/dz = pressure gradient per unit length (M/T2 L2) g = gravitational acceleration (L/T2) G,, G, = radial and tangential component of G,, respectively (M/T2L2) G , = modified gravity force (Figure IC)(M/T2 L2) Dt
039
dP 0 0677
0 1 9 5 005, A
n
L
0
30
10
4 4
005
i
60 (WsOIe
80
100
120
(g SI
Figure 3. Comparison between ( WB& and (W&. Solid flows calculated from eqs 14 and 11, (Wso)c, are both for this compared with experimental values, ( WsO)e, and another study (O’Dea, 1988) in Table I and in Figure 3. Agreement is seen to be satisfactory. It should be noted that O’Dea’s observations were carried out in a 60’ hopper with different bottom orifices, so a hopper cone-angle correction factor, co, is necessary in the calculation of solid flow rate, i.e.
k = coefficient of empty annulus around orifice edges k,,, k = correction factors in eq 10, respectively (L2 T2/M)
an2 dimensionless m, = mass of an average particle diameter (M) N = number of particles flowing vertically per unit area per second [1/(L2TI1 r = radial distance from vertex of the orifice (L) R, = radial distance from vertex to stress-free surface (L) Uso= superficial solid velocity under gravity flow (L/T) Wso,W , = solid flow rate under gravitational and negative pressure gradient, respectively (M/T) 2 = total standpipe length (L) Subscripts c = calculated values e = experimental values Greek Symbols
= angle between vertical and shoulders of immobile material
Referring to the illustrative values suggested by Zenz and Othmer (1960), c, can be taken to have approximate values of 1.15 for D o / d , from 10 to 20 and 1.1 for D J d , from 20 to 30, respectively. Here, it is necessary to make the point again that for different materials c, can be determined by experiments and depends on particle properties such as d, and & The good agreement of the model with experiment is consistent with the assumed added relationship between solid flow and shear friction between flowing and stagnant particles around the edges of the orifice. The model provides a convenient means for predicting the solid flow rate under gravity flow and negative pressure gradient.
above orifice (rad) slide between flowing and nonflowing particles at the orifice edges (Figure IC)(rad) t, tmf = voidage in standpipe and at minimum fluidization, respectively t p , tpo = voidage in tightest packing and gravity flow, respectively 0 = polar angle for spherical coordinates (rad) Pb, pbo = bulk density in aerated and gravity flow, respectively (M/L3) ps = solid density (M/L3) T,, = shear stress between flowing and nonflowing particles around the orifice edges (M/T2 L) @ l = angle of internal friction of particulate material (rad) 4s = sphericity 4, = angle of friction between solids and walls (rad)
Conclusions 1. The problem of predicting the solid flow rate can be reduced to considering the momentum conservation of downflowing particles through the stress-free surface and determining the effects of shear friction between flowing and nonflowing particles around the edges of the bottom orifice in standpipes. 2. On the basis of this model, the correlation for calculating the solid flow rate in gravitational flow only is analogous to the Beverloo equation (Beverloo et al., 1960). The form proposed here includes a term for shear friction correction and uses an effective orifice diameter (Do- d,) based on a reasonable physical picture.
Literature Cited Beverloo, W. A.; Leninger, H. A.; Van de Velde, J. The flow of granular solids through orifices. Chem. Eng. Sci. 1961, 15, 260. Brandt, H. L.; Johnson, B. M. Forces in moving bed of particulate solids with interstitial fluid flow. AIChE J. 1963, 9 (6), 771. Brown, R. L.; Richards, J. D. Profile of flow of granular solids through apertures. Trans. Inst. Chem. Eng. 1960, 37, 243. Brown, R. L.; Richards, J. C. Principles of Powder Mechanics; Pergamon Press: New York, 1970; p 221. Ching, C. K.; Chen, H. S.; Tu, T. L.; Tsui, S. Y. The flow of solid particles through an orifice. J. Chem. Ind. Eng. (China) 1966,2, 115. Hancher, C. W.; Jury, S. H. Semicontinuous counter-current apparatus for contacting granular solids and solution. Chem. Eng.
@, = angle of
Acknowledgment
This work was supported by the Australian Commonwealth Government through the Australian Research Commission. Nomenclature Af,, At = area of stress-free surface and standpipe, respectively (L2) Ars = surface area orthogonal to a radius vector r at angle 0 (L2) c,, c, = cone-angle and shear friction correction factor, respectively
Prog. Symp. Ser. 1959, 55 (24), 87. Kunii, D.; Levenspiel, 0. Fluidization Engineering; Wiley: New York, 1969. Nedderman, R. M.; TMin, U.; Savage, S. B.; Hourly, G . T. The flow of granular materials-I: Discharge rates from hopper. Chem. Eng. Sci. 1982, 37 (ll),1597. O’Dea, D. P. Private communication, University of Queensland, Australia, 1988. Rudolph, V.; Judd, M. R. An analysis of stress fields occurring in fluidized beds of particles with particular reference to slugging systems. Chem. Eng. Commun. 1989, 80, 101. Walters, J. K. A theoretical analysis of stresses in axially-symmetric hoppers and bunkers. Chem. Eng. Sci. 1973, 28, 779. Wieghardt, K. Experiments in granular flow. Annu. Rev. Fluid Mech. 1975, 7, 89. Williams, J. C. The rate of discharge of coarse granular materials
Znd. Eng. Chem. Res. 1991,30, 1981-1989 from conical mass flow hoppers. Chem. Eng. Sci. 1977,32,241.
Zenz,F. A.; Othmer, D. F . Fluidization and Fluid-Particle Systems;
Reinhold: New York, 1960. Zhang, J.-Y.; Rudolph, V.;Leung, L. S. Non-fluidized flow of gassolids in standpipes under negative pressure gradient. In Fluidization VI; Grace, J. R., et al., Eds.; Engineering Foundation: New York, 1989a; p 154. Zhang, J.-Y.;Rudolph, V.; Leung, L. S. Standpipe flow against a
1981
negative pressure gradient. Proceedings of the Seventeenth Australasian Chemical Engineering Conference, CHEMECA 89, Aug 23-25; 1989b; p 882. Zhang, J.-Y. Non-fluidized standpipe flow under negative pressure gradient. PbD. Thesis, University of Queensland, Australia, 1990.
Receiued for review January 8, 1991 Accepted April 8, 1991
Simultaneous S02/N0, Removal and SO2 Recovery from Flue Gas by Pressure Swing Adsorption Eustathios S. Kikkinides and Ralph T. Yang* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260
T h e feasibility of employing pressure swing adsorption (PSA) for simultaneous SOz/NO, removal from flue gas and concentrating the SO2 in the desorption stream (expressed as enrichment ratio) is established by model simulation. A weak-base macroreticular resin is used as the sorbent. Concentrating the adsorptive is a new and promising application for PSA. This work also delineates the underlying principles for concentrating the adsorptive. The purge/feed ratio is a key parameter for the adsorptive enrichment ratio, which reaches a peak value a t a certain purge/feed ratio. Effects of the other PSA cycle parameters on the enrichment ratio are also illustrated. For a simulated power-plant flue gas containing 0.5% SOz, 0.13% NO2,and 18% COz, a simple Skarstrom cycle can produce a purified stream with 0.035% SO2,and 0.069% NO2,while the desorption stream contains 7% SOz. Further improvement can be obtained by employing a more complex PSA cycle by which a purified stream with 0.064% SO2, and 0.055% NO2, and a desorption stream with 9% SOz can be produced. T h e desorption stream can be readily converted t o elemental sulfur by commercial processes. Preliminary experiments performed in this laboratory using polymeric sorbents instead of macroreticular resins suggest the validity of this work and indicate that the results of the present work can be further improved by using these polymeric sorbents.
Introduction The pressure swing adsorption (PSA)process, first developed by Skarstrom (1959), has been employed in a broad range of industrial applications such as drying, air separation, hydrogen purification, normal paraffin and isoparaffin separation, etc. (Yang, 1987). In the PSA process the adsorbent is rapidly regenerated by reducing the partial pressure of the adsorbed component and/or purging at a low pressure. This results in short cycle times (minutes) together with a requirement for a minimal energy input, which gives rise to a much higher feed throughput (sorbent productivity) compared to all other cyclic processes. A variation of PSA is vacuum swing adsorption, in which a subatmospheric pressure is used in desorption (Sircar and Zondlo, 19771, also a commercialized process. In the present study PSA is considered for the first time for SOz/NO, removal and sulfur recovery, for flue gas applications. Postcombustion desulfurization and denitrification systems must be efficient, of low cost, mechanically simple, and with no serious maintenance requirements. Fixed bed adsorbers have the potential for providing desulfurization systems with the desired characteristics. However, successful practical application depends on the availability of a proper adsorbent. Macroreticular resins, which are basic polymeric sorbents, have found commercial and commercially promising applications for water conditioning and aqueous separations (Kuo et al., 1987; Garcia and King, 1989). A unique sorbent property of the resins is that some resins exhibit low selectivity toward both water and carbon dioxide. This
* Author to whom correspondence should be addressed
property has prompted research in the past two decades into their sorption properties for SOz and NO, for consideration of pollution control (Chen and Pinto (1990) and the literature cited therein). Chen and Pinto (1990) have recently reported reversible adsorption capacities of a weak-base resin, Dowex MWA-1, for SOz, NOz, and COz at two temperatures. On the basis of these data, they have suggested thermal swing adsorption for flue gas cleanup. The data reported by Chen and Pinto were based on moisture-free conditions. In contrast to the inorganic sorbents (zeolites, carbon, silica gel, and alumina) the presence of moisture usually does not decrease the resin’s capacity for acidic gases (such as SO2) and sometimes increases it (Belyakova et al., 1975). However, weak-base resins exhibit a serious limitation on the rates of sorption. Layton and Youngquist (1969) observed an attainment of 50-60% of the equilibrium capacity of SOz on Amberlyst A-21 in approximately one-half hour, followed by a very slow approach to equilibrium over a week or more. Similar behavior has been observed for the sorption of all three gases, SOz, NOz,and COz, in Dowex MWA-1, by Chen and Pinto. The data reported in the latter work reflected the useful capacity of the resin and corresponded roughly to the amount sorbed during the first 30 to 60 min. In the present work, preliminary experiments have been undertaken in our laboratory which indicated that there exist polymeric sorbents which are hydrophobic and can exhibit high selectivity of SOz over COz with much higher rates of adsorption compared to the ones reported on ion-exchange resins. The first objective of this research was to study the feasibility of simultaneous SOZ/NO2removal as well as the enrichment of SO2 in the desorption product to a con-
0888-5885/91/2630-1981$02.50/00 1991 American Chemical Society
