Degree of polymerization 104 103
I " ' " '' Em .-
-
u .-1
.
K
-
" A " " '
'
'
I ,
'
5 -
1 20
22
24
26
28
30
32
Elution volume (counts)
Figure 10. Molecular weight distribution under control for different limits of dose rate and reaction temperatures (batch) IH, IL, Reaction Curve radls radls temp O C K 1 2 3 4
44.5 36.1 44.5 36.1
1.4 8.3 1.4 8.3
23 23 55 60
0.50 0.50 0.30 0.25
is smaller for high inlet temperature of coolant. This is explained by the fact that the high coolant temperature reduces the rate of heat removal. In this figure the bath temperature is lower than the inlet temperature of coolant; this may be caused by heat removal from the bath through the insulator. Responses of the reaction temperature and dose rate at the set point of 23 "C for different limit values of dose rate are shown in Figure 9. The molecular weight distributions of the polymer formed a t the same conditions as in Figure 9 are shown in Figure 10. Curve 1 is the distribution curve of the polymer formed by two-position control of dose rates 44.5 and 1.39 rad/s; there are two peaks a t degrees of polymerization of 2 X lo3 and 1.7 X lo4.Curve 2 is by the dose rates of 36.1 and 8.33 rad/s, and it has only one peak a t 2.5 X lo3. This is because the peak positions of the polymer formed by continuous K
irradiation of dose rates of 36.1 and 8.33 rad/s are close to each other. Curve 3 is the distribution by the dose rates of 44.5 and 1.39 rad/s at 55 "C, and has a peak at the degree of polymerization 3.5 X lo4 and the shoulder near 2 X lo4. This curve is larger than curve 1 in the high molecular weight fraction because of small K , and shifted to a higher molecular weight region from the position of curve 1. This is explained mainly by the fact that the molecular weight increases with the reaction temperature, as the activation energy of the propagation is larger than that of termination. Curve 4 is the distribution for 36.1 and 8.33 rad/s a t 60 "C. This curve has only one peak a t 6 X lo3, and also is shifted to a high molecular weight region from the position of curve 2. This shift is due to the same reason as for the case of curve 3.
Conclusion Experimental study was carried out on the control of a reactor by two-position regulation of dose rate for the polymerization of MMA. Experiments in batch and flow systems showed that: (1)The reaction temperature responded quickly to dose rate change and was successfully controlled by dose rate regulation. (2) The control with a dead time resulted in periodic irradiation between the high and low dose rates, and the amplitude of the temperature oscillation was small to show a high accuracy of control. (3) The molecular weight distribution of the polymer varied in the polydispersity according to the control conditions. Acknowledgment We are indebted to Drs. Masaaki Takehisa and Masamitsu Washino for their valuable discussion. Literature Cited Bamford, C. H., Tompa, H., Trans. Faraday Soc., 5 0 , 1097 (1954). Hashimoto, S., Kawakami, W., Akehata, T.. ind. Eng. Chem., Process Des. Dev., 15, 244 (1976).
Tung, L. H.. J. Appl. Poiym. Sci., 10, 375 (1966).
Receiced for recieu, December 15,1975 Accepted June 7,1976
A Quantitative Design Procedure for Vertical Pneumatic Conveying Systems L. S. Leung" and Robert J. Wiles Department of Chemical Engineering, University of Queensland, St. Lucia. Australia 4067
In vertical pneumatic conveying, the task of the design engineer is, for a given rate of conveying a particular solid, to specify pipe size, air flow rate, pressure drop, and flow pattern in the riser. In this paper a quantitative procedure for tackling this design problem is outlined. The procedure is an updated version of an earlier procedure presented by Leung et al. (1971a) and is based on recent advances on prediction of flow regime and pressure drop in vertical pneumatic conveying.
Introduction The task confronting the design engineer in the design of a pneumatic conveying system for conveying a given material a t a given rate is often to specify (i) the pipe size, (ii) the air 552
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
flow rate, (iii) the overall pressure drop, and (iv) the flow pattern in the pipe. Although much work has been carried out in the past on pneumatic conveying, the solution to the above problem is usually based on actual operating experience and rules of
thumb such as those offered by Zenz and Othmer (1960),the Engineering Equipment Users’ Association (1963), and Kraus (1968).Quantitative design procedures have been presented by Duckworth and Rose (1969), Kunii and Levenspiel (19691, and Leung et al. (1971). Since the last publication, a number of studies in flow transition, pressure drop, and solid velocities in vertical pneumatic conveying have been published. The purpose of this paper is to present an updated design procedure taking into account the recent advances in pneumatic conveying. Flow P a t t e r n s in Vertical Conveying Description of Important Regimes. Three main types of flow patterns may be described for vertical pneumatic conveying. As there is some confusion in the literature on the description of flow patterns, we shall define these as follows: (a) dilute phase flow (or lean phase flow), in which the solids are carried upward as an apparently evenly dispersed SUSpension with low volumetric solid concentration (generally less than about 5%);(b) dense phase flow: two types of dense phase flow can be distinguished depending if slugging occurs in a particular gadsolid system. If slugging occurs, solids are carried up by air slugs analogous to a slugging fluidized bed. This type of dense phase conveying is defined here as slugging dense p h a s e flow. For some systems, however (such as fine powders, for instance), slugging does not occur and solids are carried upward in a dense phase with considerable internal solid recirculation resulting sometimes in a negative solid friction factor due to solid flow downward near the wall. This type of dense phase flow is analogous to a recirculating fluidized bed or a “fast fluidized bed” (Reh, 1971; Yerushalmi et al., 1974). We shall define this as dense phase flow without slugging;(e) moving bed flow, in which solids are carried upward en-bloc as a packed bed a t a voidage corresponding to that of a packed bed, with hardly any relative motion between particles. Many so-called “dense phase” systems with high solid t o air ratios reported in the literature in fact operate in the dilute phase flow regime (Kraus, 1968). Moving bed flow had been referred to as dense phase flow by Sandy e t al. (1970) and as mass flow by Chari (1970). We believe our definitions above are more appropriate and they will be used in this paper. Vertical pneumatic conveying is generally carried out in the dilute phase regime and much of the published work on vertical pneumatic conveying has been restricted to this regime. For operation in the dilute phase regime it is desirable to operate a t as low an air flow rate as possible from energy requirements, pipe erosion, and particle attrition considerations. Dense phase flow is less often used because of the erratic nature of the flow, the pressure fluctuations, high pressure drops, and pipe line vibration. Moving bed flow is generally to be avoided because of very high pressure drop, the problem of blockage, and higher power requirement. Predicting Flow Pattern. (a) Choking vs. Nonchoking Systems. In the design of a vertical pneumatic conveying system, it is important to be able to predict for given velocities the flow regime to be encountered. For systems at which dense phase slugging flow occurs, the transition point from upflow of solids as a thin suspension (dilute phase) to slug flow is referred to as the choking point. The phenomenon of choking was described in detail by Zenz and Othmer (1960) and more recently by Capes and Nakamura (1973), Gau and Yousfi (19741, and Yang (1975). Until recently it was thought that choking is a clearcut phenomenon and occurs in all systems. The recent work and the observation of Kehoe and Davidson (1970) on the gradual breakdown of the slug regime in a fluidized bed shows that choking does not occur in all gas-solid systems. TWOquantitative criteria have been published on whether choking would occur in a particular gas/solid/tube
system (Gau and Yousfi, 1974; Yang, 1976a). The following criterion of Yang is adopted here as it takes into account the important effect of tube diameter: U,/(gD)l/* < 0.35 for no choking. Thus for a given system, eq 1 can be used to establish if choking may be expected to occur. (b) Prediction of Choking Velocities. For systems with a choking transition, a method of predicting choking flow rate is important in the design of vertical pneumatic conveying systems. In our previous paper we recommended the use of the Leung et al. choking correlation (1971b) which was shown to correlate all published results within about 70% (Leung e t al., 1973). A more recent empirical equation by Yang (1975) was shown to be superior, correlating 90% of published data to within f30%. Yang’s correlation, written in the following form, will be adopted here:
v, = (V, - UJ(1 -
tJ
(V, - U,)2/2gD = ~OO(C,-~.’ - 1)
(2a) (2b)
If the solid superficial velocity, V, is specified for a given riser, the superficial gas velocity and the voidage a t choking point can be solved simultaneously from eq 2a and 2b. Note that in eq 2a and 2b the superficial gas velocity V , is used instead of the correct actual gas velocity U,, an approximation that is justified when voidage is close t o 1.If tCis much lower than 1, V, in eq 2a and 2b should be replaced by U , [Le., VgICcl. The above equations refer to uniformly sized particles; in practice solids handled are generally nonuniform. For pneumatic conveying it is generally recognized (Zenz and Othmer, 1960) that the critical air flow rate a t a given solid rate for mixed size solids is higher than for a uniform-sized material of the same mean diameter. For mixed particles the in situ particle size distribution will be different from the particle size distribution of the feed solids. The difference arises as the smaller particles (with lower Ut) tend to rise faster than the larger particles. Thus the in situ particle size distribution will contain a higher fraction of larger particles compared with the feed solids. For mixed size particles we suggest, following Leung et al. (1971b), that eq 2a and 2b can be modified to give
and
where xf, = volume fraction of particles in the feed with terminal velocity of Ut, and xt, = volume fraction of particles in the riser with terminal velocity Ut,. As xfi and V, are normally specified, V,, tcr and xt, can be solved from eq 3a-3c. The reliability of this method for obtaining choking velocity for mixed size particles is uncertain as few experimental results in mixed size particles are available in the literature. Nakamura and Capes studied pneumatic conveying of binary particle mixtures and suggested particle interaction effects may be important. Equations 3a to 3c imply that particle interaction is negligible. Until further experimental results on choking velocity for mixed size particles are available to verify the equations, they are to be used with caution and an appropriate safety factor should be applied in design. An alternative method for estimating choking velocity for a mixed size particle is simply to assume
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
553
of the pressure drop in vertical pneumatic conveying may be due to the gravity head caused by solid hold-up, which has little connection with the roughness of the wall which determines a ldrge part of the pressure drop for air alone. Examples of correlation in this category are those due to Vogt and White (1948) and Metha and Smith (1957). These correlations are not recommended here for the above reason. An example of the use of the Ergun type correlation (category (b) above) was due to Wen and Galli (1971). The use of this type of equation is not recommended here as the calculated AP is extremely sensitive to voidage uncertainty. For the design procedure we shall recommend the use of type (a) correlation. In this case the pressure drop over distance L of a vertical pipe can be written in the following general form (Hinze, 1962; Stermerding, 1962).
/
LOG (Dimensionkss Gas Vebcity)
- Log
+
AP = [Pg€Ug2 p,(l - C)Us2]OL
(Vg/Vo)
+g
Figure 1. Quantitative flow regime diagram in a vertical pneumatic conveying for a choking system.
V, and tCcan then be solved using eq 4 in conjunction with eq 2a and 2b (Yang, 197613). This method is simpler to use but assumes that no segregation occurs in the tube. ( c )Fluidized Bed to Moving Bed Flow Transition. For dense phase conveying the slip velocity between gas and solid is necessarily higher than that corresponding to minimum fluidization. The equation for predicting the transition from dense phase flow (fluidized bed flow) to moving bed flow may be derived by equating the slip velocity in vertical pneumatic conveying with the slip velocity a t incipient fluidization (Leung et al., 1969). At this transition the voidage in the tube is equal to the voidage a t minimum fluidization. Taking an average value of 0.45 for this voidage the following equation for predicting transition from dense phase flow to moving bed flow in vertical pneumatic conveying may be readily derived: O.55Vg - 0.45V, = 0.55Vo
(5)
Equation 5 is applicable to uniformly sized particles and nonuniformly sized particles provided the appropriate VOis used. Equations 1 to 5 may be used for predicting quantitatively the flow pattern of a given system a t specified flow rates. Quantitative flow regime diagrams similar to those presented earlier (Leung et al., 1971a; Wiles and Leung, 1972) can be prepared using these equations (Figure 1).Quantitative prediction of flow pattern is an important part of the development of a general design procedure for vertical pneumatic conveying.
Pressure Drop Prediction Dilute Phase Flow Regime. Numerous correlations are available in the literature for predicting pressure drop in dilute phase vertical pneumatic conveying as pointed out in recent reviews (Richards and Wierma, 1972; Khan and Pei, 1973; Capes and Nakamura, 1973). The published correlations can be classified into three categories: (a) Pressure drop is considered in terms of three components: acceleration, gravity, and wall friction. Empirical equations are used to predict the wall friction component. (b) Pressure gradient is correlated in an Ergun (1952) type equation for flow through a packed bed using an appropriate slip velocity. (c) Pressure gradient is expressed as a ratio of [lPm,xture/Pa,r alone] and correlated with other variables. Richards and Wierma pointed out the invalidity of the use of the ratio in (c) above to correlate results in vertical pneumatic conveying. They stressed that a significant component 554
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
soL
[(l- e)ps
+ cpg]dz + 47L/D
(6)
The three terms on the right-hand side of eq 6 represent pressure drop due to acceleration, gravity, and wall friction. Assuming pressure drop in riser is small compared with the absolute pressure level, the acceleration section is short and the terms involving gas density are negligible, eq 6 can be simplified to
AP = p,(l - t)U,2 + (1 - t)psgL + 47L/D
(7)
T o calculate AP we deed to estimate the actual solid velocity ( U s ) voidage , ( e ) , and wall friction loss. If the superficial velocities of solid and gas are known, U s and t can be obtained by solving eq 8 and 9a simultaneously
v,= U,(1 - c) v,- us= ut
(8) (94
Equation 9a implies that voidage is close to 1;hence slip velocity is equal to the free fall velocity, Ut,and superficial gas velocity and actual gas velocity are approximately equal. A more accurate estimate of U s and hence t can be made using an equation suggested by Yang (1973).
us= u, - [(I+ fpUs2/2gD)[4(p,- pg)&t4 7 ] / 3 ~ g C ~ ~ ] 1(9b) '2 where CDS= drag coefficient of a single particle in an infinite medium, f p = friction factor defined by Yang as f p = 2DAF'fs/pmUs2L,and APf, = frictional pressure drop due t o solid particles. Equation 9b can only be used if the frictional pressure drop due to solid particles in the riser is known. An iteration procedure is required to calculate U s from (9b). Numerous correlations are available for predicting the frictional loss term in eq 7. The most recent of these are due to Knowlton and Bachovchin (1975),Yousfi and Gau (1974), Capes and Nakamura (1973), Yang (1974), Khan and Pei (1973) and Richards and Wierma (1972). The frictional pressure loss (AF'f) is generally taken to consist of two components: one due to fluid and the other one due to solid. The equation for frictional loss can be written as (Hinkle, 1953; Stermerding, 1962; Jones et al., 1967; Konno and Saito, 1969).
+
APf = 47L/D = APf, APf, = 2Lf,p,Ug2/D
+ 2Lf,(l - t)psUS21D
(10)
Knowlton and Bachovchin correlated their pneumatic conveying measurements a t pressures up to 700 psia by fs =
~ o . o ~ ~ ~ ~ ~ w , / p g u , ~ ~ -~0.03 o ~ ~(11) ~~us/u,~
While the general applicability of eq 11 has yet to be estab-
lished, it is worth pointing out that the work of Knowlton and Bachovchin is the only detailed work published in high pressure conveying. In the design of pneumatic conveying system operating a t high pressure, the use of eq 11 is recommended. For normal operating pressure Capes and Nakamura (1973) analyzed the results of a number of workers. Their analysis together with other results is summarized in Table I. Four of the correlations for fs shown in the table suggest that f s is an inverse function of the solid velocity while others indicate otherwise. Jones e t al. (1967) presented a more complex relationship. There is little doubt that f s is dependent on a number of parameters (Khan and Pei, 1974; Duckworth and Rose, 1969) not considered in the first four equations in the table. As the solid frictional loss is often a small component of the overall pressure drop in vertical conveying, we recommend the use of a simple equation by averaging the first three equations in Table I giving
Table I. Correlations for Solid Friction Factor Equation for f , ( U sin m s-l)
Reference
van Swaaij et al. f s = o.080us-1 (1970)a Reddy and Pei (1969)" f s = 0.O46Us-' Konno and Saito f 5 = 0.O25Us-' (1969)" Capes and Nakamura f s = 0.048Us-1.22 (1973)" Stermerding (1962) f s = 0.003 Yousfi and Gau (1974) f s = 0.0015 (for polystyrene particles) f s = 0.003 (for glass particles) Jones e t al. (1967) f s = [1.89 X AoOx/x1/2]pgU,2/ (I
4PSUS2
de Jong (1975)
f s = 0.0021 (for sand) fs
Yang (1974)
f5
= 0.0012 (for glass particles) = 0.0206 (1 - c ) [ ( 1 - €)Ret/
Re,] -o,869/4t3 From Capes and Nakamura (1973). Average values of f s obtained for downward conveying. Average coefficientfor the first three equations = 0.05. a
fs
= 0.05U,-1
( U sin m s-l)
(12)
For more accurate estimation the equation of Duckworth and Rose (1969) or the equation of Yang (1974) may be used. One further method of predicting pressure drop in vertical pneumatic conveying deserves mentioning. Richards and Wierma (1972) analyzed over 2000 data points from experimental risers up to 0.3-m diameter and presented the following correlation for operating gas velocities greater than 5 m s-':
P= P & l + W,/W,)V,* + PgL(l + Ws/Wg) X (1
+1 7 . 5 a )
(13)
Richards and Wierma showed that eq 13 was superior to two earlier and often used correlations: Hinkle's correlation (1953) and the EEUA correlation (1963). Currently we are carrying out a detailed comparison between eq 13 and other equations, with particular reference to their applicability to commercial scale risers such as those studied by Matsen (1975). At this stage our recommendations for pressure drop prediction in dilute phase flow are summarized as follows: (i) we recommend the use of eq 7,8,9b, and 11 for high-pressure systems: (ii) for low-pressure operation, eq 7, 8,9b, and 12 are recommended. Equation 7 implies that the acceleration length is small or that in the acceleration length, the frictional loss is similar to that for fully developed flow. A method of estimating acceleration length and acceleration pressure drop has recently been published (Yang and Keairns, 1976) and eq 7 can be extended to incorporate the refinement. However, in many practical systems, the acceleration length is considerably smaller than the actual length of the vertical riser. Such a refinement to eq 7 is often not necessary. Dense Phase Flow Regime. In the dense phase flow regime, wall frictional pressure drop is generally small compared with that due to the weight of solids in the tube (Ormiston, 1966: Matsen, 1973,1975). Often in dense phase flow intense solid recirculation occurs. This may result in solid downflow near the wall giving a "negative" wall shear stress, Le., the shear stress acting opposite to the normal direction (Van Swaaij et al., 1970; Capes and Nakamura, 1973). In this work we shall neglect wall frictional loss and assume that the overall pressure drop is equal to that due to the weight of solids in the tube, i.e.
AP
= p,(l
- €)gL
(14)
T o calculate 2we need to estimate t for a particular system. This will be considered separately for slugging flow and for dense phase flow without slugging.
(a) Dense Phase Slugging Flow. For slugging flow the density of a gas-solid mixture in the riser can be calculated from the following equation due to Matsen (1973): (1 - E ) / ( l - to) = [ u b f ws/[Ps(l
- tO)]]/(vg + Ub - VO + ws/Ps)
(15)
where Ub = bubble rise velocity in a non-flowing fluidized bed ( = 0.35for slugging conveying). Voidage in the riser for a particular system can be calculated from eq 15 and hence the pressure drop from eq 14. Under certain conditions (with coarse particles in particular), half nose slugs are formed and Ub in eq 15 can be replaced by ub = 0 . 3 5 m (Kehoe and Davidson, 1970). (b) Dense Phase Flow without Slugging. This type of flow pattern is characterized by intense particle recirculation observed a t gas velocities close to the free fall velocity of the particles. A plausible model for this flow regime was suggested by Nakamura and Capes (1973,1975). They proposed an annular flow model by dividing the flow of solids in the riser into two regions: a core with upflow of solids and an annulus with a lower solid velocity (which can be negative). Equations were presented by Nakamura and Capes for predicting the average voidage in the riser. The equations, however, contain empirical constants which have to be determined experimentally. I t is doubtful a t this stage if their equations can be applied to a system of which no experimental data are available. A second source of information for this flow regime comes from the study of circulating fluidized beds, often known as "fast fluidized beds" (Reh, 1971: Yerushalmi et al., 1974, 1976).However, a t this stage no reliable equations are available for estimating bed voidage. Much remains to be learned in this type of operation. In the absence of a predictive method, the operating voidage has to be determined experimentally in a test rig. As a first approximation, however, a voidage of about 0.6-0.8 may be assumed depending on operating velocity. Moving Bed Flow Regime. As this type of operation has little practical application in upward conveying, pressure drop estimation in this regime will only be considered briefly. For moving bed flow pressure drop can be calculated from a modified Ergun type equation using an appropriate slip velocity (Yoon and Kunii, 1971; Leung e t al., 1971a): LP,[(Ug
SPde3 = 150/Re,l+ 1.75 - C's)t12(1 - 4
Ind. Eng. Chern., Process Des. Dev., Vol. 15,No. 4, 1976
(16)
555
Design Procedure for Vertical Pneumatic Conveying Systems In the design problem we are given: the physical characteristics of the solid particles to be conveyed (i.e., p,, particle size and shape factor),the rate of solids to be conveyed, W's, and the pressure at the discharge end of the conveyor line, pd.
We shall now develop the design procedure for a vertical pneumatic conveying system as follows: (i) Fix any pipe diameter, D , and calculate the superficial solids velocity, V,
(ii) Guess a pressure drop AP for the system. The terminal falling velocity Ut and minimum fluidization velocity Vo at the average pipeline pressure ( P d * / 2 ) are calculated from known correlations (Zenz and Othmer, 1960; Wen and Yu, 1966; Kunii and Levenspiel, 1969). A. Dilute Phase Flow Operation. (A.iii) Calculate from V , and Ut the choking gas velocity and the choking voidage from eq 2a and 2b. (A.iv) Using a safety factor of 1.5, the operating gas velocity is taken as 1.5 times choking gas velocity. (A.v) From the gas and solid velocities estimate the pressure drop in the system following the recommendations listed in the last paragraphs of the section entitled Dilute Phase Flow Regimes. (A.vi) If the calculated AP is significantly different from the assumed AP, repeat procedure (ii) to (v) until calculated and assumed AP are similar. (A.vii) Repeat steps (i) to (vi) for different pipe diameters. Different combinations of pipe diameter, pressure drop, and gas flow rates are obtained from these calculations. Selection of a suitable combination may be based on the usual economic evaluation of the different possible combinations. B. Dense Phase Flow Operation. (B.iii) Calculate from V,, Ut, and Vo,the minimum gas velocity to prevent moving bed flow in the riser (eq 5 ) . (B.iv) Choose an operating gas velocity at least twice the transition to moving bed flow velocity. (B.v) From the gas and solid velocities, calculate the pressure drop in the system using eq 14 and 15 for slugging flow. (For dense phase nonslugging flow a guessed pressure drop equivalent to a voidage of 0.6-0.8 may be used.) (B.vi) If the calculated A P is significantly different from the assumed A P , repeat procedure (ii) to (v) until calculated and assumed AP are similar. (B.vii) Repeat steps (iv) to (vi) for other gas velocities. (Whereas for dilute phase flow it is generally desirable to operate at as low a gas flow rate as possible, the same is not generally true for dense phase flow.) (B.viii) Repeat steps (i) to (vii) with different pipe diameters. Selection of a suitable combination of gas flow rate, pipe diameter, and pressure drop can be made in the usual way.
+
Discussion In the above design procedure for vertical pneumatic conveying, we have not considered pressure drop in bends. Bend loss in the range of 0.5 to 1.5(Ug2p,/2g) have been recommended in the E.E.U.A. Handbook (19631, depending on bend radius, and should be used here. It is worth pointing out that often a right angle bend is more trouble free (in terms of attrition wear) than a bend with a large radius of curvature (Calcott, 1975).This has important practical implications for the conveying of abrasive materials. In the design procedure we have considered that the solid flow rate can be directly controlled (using a rotary valve feeder for instance). In some feeding device, the solid mass flow rate is not controlled independently and is dependent on the air flow rate and other parameters. One such example is the pneumatic conveyor with the bell-shaped end piece immersed in a fluidized bed (E.E.U.A., 1963; Decamps, et al., 1972). The 556
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
solid flow rate in this case depends on the rate of air flow into the conveyor via the bell-shaped feeder. The present design procedure can be extended to cover such feeding systems following a procedure outlined by Leung and Towler (1973). It should be stressed that the design procedure outlined here is intended for design engineers without access to pneumatic conveying test rig. The state of the art of pneumatic conveying has yet to reach the stage when the test rigs are completely unnecessary for design! Nor have we considered the case of horizontal conveying. In a system with vertical and horizontal lines, the minimum operating gas velocity is set by the minimum transport velocity in the horizontal section and not by a critical velocity in the vertical section. We shall discuss the case of horizontal conveying in a separate paper.
Nomenclature Ao = specific surface area of particles, m-l CDS = drag coefficient of a single particle at free fall velocity D = diameter of riser, m d = particle diameter, m f g = friction factor for gas flow only defined in eq 10 f p = solid friction factor defined as (20APf,/p,Us2L) f, = solid friction factor defined in eq 10 g = gravitational acceleration, m s+ L = lengthofriser,m P d = pressure at discharge end of conveying h e , P a A P = overall pressure drop over length of L of riser, Pa APf = component of pressure drop due to wall shear stress, Pa APf,= frictional pressure drop due to gas, Pa APf, = frictional pressure drop due to solid particles, P a U = actual velocity in riser, m s-l Ut = free fall velocity of a single particle, m s-l Ub = bubble velocity in a nonflowing fluidized bed ( = 0 . 3 5 m for slugging conveying), m sK1 V = superficial velocity, m s-l Vo = superficial minimum fluidization velocity, m s-l W = mass flow rate per unit cross-sectional area, kg s-l m-2 W' = mass flow rate, kg sK1 xt, = volume fraction of particles in riser with terminal velocity of Ut, xf, = volume fraction of feed particles with Ut, t = voidage in riser cc = voidage at choking to = voidage of a fluidized bed at minimum fluidization velocity p = density, kp m K 3 pm = density of mixture ( = p,(l - t ) pgc), kg m-3 8 = W,/W, /1. = viscosity of gas, kg m-l sK1 T = shear stress on wall of riser, P a x = shape factor of particles
+
Subscripts g = gas s = solid Literature Cited Calcott, T. G., The Broken Hill Pty Ltd., Australia, private communication, 1975. Capes, C. E., Nakamura. K. Can. J. Chem. Eng., 51, 31-38 (1973). Chari, S. S., "Pressure Drop in Horizontal Dense Phase Conveying of Air-Solid Mixtures", 63rd Annual Meeting, A.I.Ch.E., Chicago, Ill., Dec 1970. Decamps, F., Dumont, W., Goossens, W., Powder Techno/., 5, 299-306 (1972). de Jong, J. A. H., Powder Techno/., 12, 197-200 (1975). Duckworth, R . A., Rose, H.E.. Engineer, 227, 392-396, 430-433, 478-483 (1969).
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Receiued f o r reoiew December 31, 1975 Accepted M a y 19,1976
Thermodynamic Properties of Gas Mixtures Containing Common Polar and Nonpolar Components Ryoji Nakamura, Gerrit J. F. Breedveld, and John M. Prausnitr' Chemical Engineering Department, University of California, Berkeley, California 94720
A perturbed-hard-sphere equation of state is given for gaseous mixtures containing highly polar molecules (water, ammonia, sulfur dioxide) and nonpolar, or slightly polar, molecules (hydrogen, argon, nitrogen, carbon monoxide, methane, ethane, ethylene, propane, propylene, hydrogen sulfide, and carbon dioxide). Equation-ofstate constants are given for these 14 fluids and their mixtures. Because experimental data are scarce for mixtures containing one or more polar components, many of the binary-interaction constants are only approximate. The equation of state may be used to calculate densities, enthalpies, entropies, and fugacity coefficients for gaseous mixtures often encountered in chemical technology (especially in synthetic-gas processes) for a wide temperature range and for pressures to about 5000 psia.
Thermodynamic properties of gas mixtures are frequently required in chemical process design; densities, enthalpies, entropies, and fugacity coefficients are needed to design separation equipment, heat exchangers, power cycles, etc. While much attention has been given to the properties of gas mixtures containing nonpolar (or slightly polar) components, little is known about the properties of those gas mixtures which contain highly polar molecules (e.g., water, ammonia, sulfur dioxide) in addition to common nonpolar molecules such as hydrogen, nitrogen, methane, etc. However, such mixtures are increasingly encountered in chemical technology, especially in processes concerned with the production of
synthetic gases from coal and heavy fossil fuels. This work presents a semiempirical equation of state which provides estimates of thermodynamic properties for these gas mixtures. Since experimental data are rare for mixtures containing one or more polar components, it is difficult to assess the accuracy of these estimates. Since the proposed equation of state uses only a few constants, it is not able to give a highly accurate representation of volumetric properties. The work presented here provides no more than a reasonable procedure for making the best estimates possible a t this time. The equation-of-state constants given will require revision as new experimental data become available. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4,1976
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