Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 31 1-320
support is gratefully acknowledged. Registry No. KOH, 1310-58-3;2-ceUulose,9004-34-6;tetralin, 119-64-2; 3-propanol, 67-63-0; palladium, 7440-05-3; nickel, 7440-02-0.
Literature Cited Appell, H. R.; Fu, Y. C.; Friedman, S.; Yavorsky, P. M.; Wender, I. "Converting Organic Wastes to Oil: A Replenishable Energy Source", U S . Bureau of Mines, Pittsburgh, Report of Investigations 7560, 1971. AppeW, H. R.; Fu, Y. C.; Illig, E. 0.;Steffgen, F. W.; Miller, R. D. "Conversion Of Cellulosic Wastes to Oil", US. Bureau of Mines, Plttsburgh, Report of Investlgatlons 8013, 1975. Boocock. D. G. B.; Mackay, D.; McPherson, M.; Nadeau, S.; Thurier, R. Can. J. Chem. Eng. 1979, 57, 98. Boocock, D. G. 8.; Mackay. D.; Franco, H. ACS Symp. Ser. 1980, No. 130, 363-368. Boomer, E. H.; Argue, G. H.; Edwards, J. Can. J. Res. 1935, 1313, 343. Braude, E. A.; Linstead, R. P.; Wooldridge, K. R. H. J. Chem. SOC.1954, 3566. Chin, Lih-yen; Engel, A. J. "Hydrocarbon Feedstocks from Algae Hydrogenation", paper presented at the 3rd Symposlum on Blotechnology in Energy Production and Conservation, Gatllnburg, TN, May 13, 1981.
311
Curran, G. P.; Struck, R. T.; Gorln, E. Ind. Eng. Chem. Process D e s . D e v . 1987, 6 , 166. Elliott, D. C. I n "Fuels from Biomass and Wastes", Klass, D. L.; Emert. G. H., Ed.; Ann Arbor Science Publishers, Inc.; Ann Arbor, MI, 1981; pp 435-450. Gupta, D. V.; Kranich, W. L.; Weiss, A. H. Ind. Eng. Chem. Process D e s . Dev. 1978, 15, 256. Klelderer, E. C.; Kornfeld, E. C. J. Org. Chem. 1948, 13, 455. Lindemuth, T. E. I n "Biomass Conversion Processes for Energy and Fuels", Sofer, S. S.;Zaborsky. 0. R., Ed.; Plenum Press: New York, 1981; pp 187-200. Oshima, M. "Wood Chemistry Process Engineering Aspects", Chemical Process Monograph Series No 11, Noyes Development Corp., New York, 1965. Ross, D. S.;Blessing, J. E. "Alcohols as H-Donor Media in Coal Conversion", paper presented at the 173rd National Meeting of the American Chemical Society, New Orleans, March 20-25, 1977. Vernon, L. W. Fuel 1980, 59, 102. Wen, C. Y.; Lee, E. S.,Ed.; "Coal Conversion Technology"; Addison-Wesley Publishing Co.; Reading, MA, 1979.
Received for review July 18, 1983 Revised manuscript received April 26, 1984 Accepted May 8, 1984
Corotating Disk Pumps for Viscous Liquids Zehev Tadmor,' Pradlp S. Mehta,' Leflerls N. Valsamls, and Jan-Chin Yangt Farrel Company, Division of USM, Ansonia, Connecticut 0640 1
A corotating disk pump for pumping viscous liquids is presented. Its principles of operation, design configurations, design equations, and experimental performance are presented and discussed.
Introduction Pumping of viscous and very viscous liquids, such as heavy oils, coal slurries, molasses, and polymeric solutions and melts, is frequently encountered in industry. Among the pumps which can handle viscous liquids are rotary positive displacement type pumps, such as gear pumps, lobe pumps, vane pumps, eccentric cam pumps, peristaltic pumps, and intermeshing counterrotating twin screw pumps, certain reciprocatingpositive displacement pumps, and "viscous drag" type pumps such as single-screw pumps, nonintermeshing twin screw pumps, and various intermeshing corotating twin-screw pumps. An increasing viscosity and a requirement for stable nonfluctuating output considerablyreduces the selection. Moreover, most positive displacement pumps, though very energy-efficient, cannot handle abrasive slurries. Foreign objects in the liquid may cause severe damage, and they cannot operate against closed discharge without damage except if equipped with relief valves. The common "viscous drag" type pumps are rather bulky, heavy, and energy consuming. In this paper, a novel corotating disk pump (Tadmor, 1979, 1980) is discussed, which is compact and energy efficient. It is a "viscous drag" type pump which can handle viscous liquids up to the most viscous polymeric melts. The corotating disks concept, in addition to viscous liquid pumps, is also being developed into a broad range
* Department of Chemical Engineering, Stevens Institute of Technology, Hoboken, N J 07030. t Celanese Engineering Resins, Corpus Christi, TX 78469. tMobil Chemical Co., Plastics Division, Canadaigua, NY 14424. 0196-4305/85/1124-0311$01.50/0
of polymer processing machines (Diskpack, trademark, Farrel Co.) for plasticating, mixing, and stripping (devolatilizing) of polymeric materials (Tadmor et al., 1979a,b, 1983a). Geometrical Configuration The geometrical building block of the corotating disk pump is a pumping chamber formed by two disks attached to a shaft placed within a closely fitted barrel, as in Figure 1. These surfaces may have a radial thickness profile imparting the chamber cross section with a radial gap separation profile. Figure 1C shows the optimum wedge-shaped profiie for pumping. The difference between barrel and shaft radii is generally much larger than the separation between the disks. An inlet or feeding port and a discharge port are formed by openings through the barrel. These are separated by a channel block closely fitted into the chamber. Multiple disks can be attached to the same shaft generating many pumping chambers. These may be connected in parallel, in series, or in any parallel-series combination as in Figure 2. Connection in parallel increases the pumping rate, whereas connection in series increases the discharge pressure. Disks and shaft surfaces may be temperature-controlled by making them hollow and circulating a temperature-controlling fluid through the shaft. Principle of Operation Pumping in "viscous drag" type pumps, such as screw extruders, is accomplished by the viscous drag imposed on the liquid by a surface moving parallel to its plane. Consider a single moving plate relative to a stationary plate (SMP) as in Figure 3b. For steady fully developed iso0 1985 American Chemical Society
312
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
Table I. A Comparison of Flow Rates, Pressure Rises, and Specific Power Consumptions for the JMP and SMP Models at ODtimum GaD SeDaration. n = 1” n = 0.75 n = 0.5 n = 0.25 n = 0.125 @,,(JMP)/@Ul,(SMP) 8.00 7.85 8.48 11.89 19.63 4.16 7.25 14.10 2.83 3.25 0.50 0.43 0.33 0.20 , 0.11 0.111 0.04 0.012 0.25 0.184 a
Newtonian. ,Channel
Block
,Separation Parallel Radial-thickness Profile
Between Plates (HI
Wedge Radial-thickne I S S Profile
-vo
A.
Suriace Velocity
DISKPACK
\ Moving Plate -vo
(b)
(a)
(C)
Figure 1. (a) Processing chamber of corotating disk pump; (b) and (c) two possible radial-thickness profiles. PARALLEL
SERIES
PARALLEL/SERIES COMBINATION
B.
SINGLE SCREW EXTRUDER
Figure 3. Melt velocity profiles in jointly moving plate (JMP) geometry (a) vs. single moving plate (SMP) geometry (b).
by two jointly moving surfaces. Consider a material particle located anywhere in the pumping chamber (Figure 1). This particle is confined between the two rotating disks which drag the liquid tangentially from inlet to discharge with a consequent pressure buildup. Taking a constant radius plane in the chamber and neglecting curvature, the simplest model of the corotating pump is two jointly moving parallel plates (JMP) as in Figure 3a. Here the flow rate per unit width is related to pressure rise via (A)
(E)
(C)
Figure 2. Multiple disk assembly: (a) parallel configuration; (b) series configuration; (c) parallel-series combination.
thermal flow of a Newtonian liquid, the flow rate per unit width is related to the pressure rise via
where H is the separation between the plates, Vois the plate velocity, L is its length, AP is the pressure rise (Pdischarge - Pinlet), and p is the viscosity. Rearrangement of eq 1yields
Clearly, for any given net flow rate q, the pressure rise over the pump is linearly proportional to viscosity. Hence in pumps based on drag, an increase in liquid viscosity results in an increase in discharge pressure. Equations 1 and 2 further indicate that pumping rate for a given pressure rise is a linear function of plate velocity, that pressure rise is linear with length L, and that there is an optimum gap separation for a maximum pressure rise for a given flow rate or vice versa. In single-screw extruders the two “plates” are the barrel surface and the root of the screw channel. Plate velocity is tantamount to frequency of screw rotation and L to helical length of the screw channel. Ample further discussion on screw extruders can be found in the literature (McKelvey, 1962; Tadmor and Klein, 1970; Tadmor and Gogos, 1979). The salient feature of the corotating disk pump is that viscous drag is induced
q=
-( -)
v()H- W A P 12p
L
(3)
Equation 3 is very similar to eq 1except the first term on the right hand side, expressing the drag flow, is doubled. This increase in drag flow rate has a profound effect on performance. Tadmor et al. (1979a,b) showed that in the jointly moving plate (JMP) model, the pressure rise for a given flow rate is 8 times higher than that in the single moving plate (SMP) model and the flow rate for a given pressure rise is 81/2times higher. This occurs for Newtonian fluids and for optimized gap separations in the respective models. For non-Newtonian fluids which follow the power law model, it can be shown (Appendix) that the ratio of maximum flow rate for a given pressure rise in a JMP model to its SMP counterpart is given by (4) where n is the power law exponent and s = l/n. Table I shows that this ratio goes from 81/2= 2.83 for Newtonian fluids to 4.16 for n = 0.5 and reaches a value of 7.25 for n = 0.25. The ratio of maximum pressure rise for a given flow rate of JMP to SMP models is given by (5) which has a value of 8 for Newtonian fluid. The ratio exhibits a minimum at n = 0.801. Its value rises from 7.84 at the minimum to 11.89 for n = 0.25, indicating a sig-
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 313 1 .o
1 .o
0.8
0.8
f a , a:I .a Fd, m i . 7
f a . m:.6
0.6
Fd. .:.l
0.6 Fd or F p f
0.4
..
F'd 0.4
0.2
0.2 0
0
0.2
0.4
0.6
H Rd-Rs
0.8
1.0
0
0
0.2
0.4
0.6
0.8
1 .o
Figure 4. Shape correction factors for parallel-shaped chamber. Hd/Rd
nificant gain in the pressurization capability of the JMP model during pumping of highly shear thinning fluids. The ratio of total minimum specific power inputs is given by ( P ~ J M P-- 1 -(6) bw)SMP (l + s, indicating half the power consumption in the JMP model for Newtonian fluids, and reduction of power consumption to *I6when n = 0.25. However, if the power required for pressure rise is subtracted from the total power leaving only the power consumption dissipated into heat, the ratio of wasted power of the J M P to SMP models is (P*~)JMP 1 (7) b*w)SMP (1 -k S)2 indicating that the JMP model wastes into heat a quarter of the power dissipated by the SMP model for Newtonian fluids and of the power dissipated by the SMP model for fluids with n = 0.25. Further advantages of the JMP model become apparent when the shear rate distributions are analyzed. In the JMP model the shear rate is independent of plate velocity (it depends only on pressure gradient), whereas in the S M P model it is a linear function of plate velocity (eq A-3 and A-19, respectively, in the Appendix). This places upper bounds on plate velocity (hence on pumping rate) in the SMP model, for shear and heat sensitive materials. The JMP model is void of such inherent functional constraint. Design Equations A mathematical model for flow in the corotating disk pump chamber was derived by Tadmor et al. (1979a,b), assuming steady, fully developed laminar, isothermal flow of an incompressible non-Newtonian power law model liquid. Correction factors for the stationary barrel surface were derived by Edelist (1982). The flow was solved for both parallel disks and wedge-shaped disks. The latter was shown to be the optimum configuration for pumping, whereas the former has advantages when there is a need for mixing the liquid. The pumping rate of the corotating disk chamber can be obtained by integrating the flow rate per unit width, obtained from eq A-7 of the Appendix, over the radial gap profile of the chamber of Figure 1. The resulting model also provides equations for shaft power consumption, torque, and adiabatic temperature rise. The final design equations are listed in Table 11, and the shape correction factors are given in Figures 4, 5, and 6. The correction factors were derived for Newtonian fluid, but
Figure 5. Shape correction factor F b for wedge-shaped chamber. 1 .o
0.8
0.6 FP 0.4
0.2
0
0.2
0.6
0.4
0.8
1 .o
Hd/Rd
Figure 6. Shape correction factor F 6 for wedge-shaped chamber.
based on our experience in screw pump design and analysis, they can be used for non-Newtonian fluids with relatively minor error. The equations for pumping in the optimum wedge shaped chamber in Table I1 can be rewritten in dimensionless form as
Q -- 1 - G s _
(8)
Qd
where shape factors are neglected, s = l / n , and G is a dimensionless pressure gradient given by
and Qd is the maximum drag capacity of the chamber. Dimensionless pump characteristic curves given by eq 8 with the power law exponent as a parameter are shown in Figure 7. At closed discharge a finite maximum pressure is obtained (G = 1). As discharge pressure is reduced, flow rate increases, reaching the maximum drag flow rate (Le., plug flow) at open discharge conditions, where discharge pressure vanishes. The actual flow rate and discharge pressure is obtained at the cross-over point between the pump characteristic curve and the flow resistance curve of the pipe assembly
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
314
Table 11. Design Equations for Corotating Disk Pumps parallel chamber volumetric flow rate drag flow rate
wedge chamber Q=Q'd-QfP
pressure flow rate pressure flow for Newtonian fluids angular pressure gradient pressure rise
pressure t o drag flow ratio shaft power specific energy
Pw -
3
=
torque
a"-
AP 2
0')
AP 3
total rate of viscous heat dissipation adiabatic temperature rise mean residence time
geometry
where q is the non-Newtonian viscosity which is a function of i , the magnitude where m and n are the power law model of the tensor y [ y = ( 1 / l ~ : - y ) 1 ' 2 ] , r) = parameters and s = l i n . For simple shear flow, y is the shear rate. For Newtonian fluids n = 1 and q = m = p ,
= 71 r
fluid model
a E
= fractional fill of the chamber circumference.
downstream of the pump. Figure 7 also shows an increasing deviation from linearity with increasing nonNewtonian behavior of the liquid (i.e., decreasing exponent
4. The design equation for volumetric flow rate in Table
I1 can be used to derive expressions for optimum gap size for any of the following conditions (assuming F d = Fp = 1): (a) maximum flow rate for a given pressure rise, (b) maximum pressure rise for a given flow rate, (c) minimum disk frequency of rotation for a given flow rate and pressure rise, and (d) minimum disk radius for a given flow rate and pressure rise. The optimum gap size in all the foregoing cases is obtained when the following condition is satisfied QP
- =Qd
1
S + 2
(10)
where
8, = Qd
-
Q
(11)
The equation holds for both parallel and wedge shape. The physical reason for the existence of an optimum gap size stems from the fact that drag flow is proportional to gap size, whereas pressure flow is proportional to gap size to the power of (2 + s). Example. Recently, 50-70% viscous suspensions of finely pulverized coal in oil and water are being increasingly utilized as a substituted liquid fuel. A coal slurry pump was designed using the pump design equations for transporting 100 gpm of a typical coal oil slurry which developed 100 psi discharge pressure. The slurry chosen is shear thinning and follows power law model 7
= -m+l+l"-l
with m = 16 N-s"/m2 and n = 0.82 at 60 "C (Bouchez et al., 1982), the density is 10 lb/gal, and specific heat is 1 Btu/lb OF. A single-stage pump with 10 parallel chambers was selected with typical a value of 0.5. Further, it was assumed
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 315 Table 111. Coal Slurry Pump Design variable disk speed radius, Rd,mh gap size, Hopt disk top velocity, V, power, P," torque, 70 adiabatic temperature rise, AT,
\
results
units rPm m cm m/s
30 0.354 0.796 1.11 6311.gb 2009 0.063
W
N-m "C
Does not include disktop and seal contribution.
60 0.253 0.778 1.59
90 0.208 0.767 1.963
120 0.181 0.760 2.278
150 0.163 0.755 2.556
1004.6
669.7
502.3
401.8
-
.60
.70
180 0.149 0.750 2.807 6311.9 334.86 0.063
4.705 hp. .OS0 .045 .040 .035
.030 .025 .020 .015
.010
.005 .O
.10
0
0.2
0.4
0.6
0.8
.20
.30
.40
.50
.80
.90
1.0
1.o Rd(m)
DIMENSIONLESS FLOW RATE Q/Od
Figure 7. Dimensionless pump characteristic curves with the power law exponent n as a parameter.
Figure 8. Plot of Hd vs. R d with disk speed N as a parameter for the design of a "coal slurry pump". Inlet
that the fraction of circumference filled with liquid (E) is 0.75; that is inlet, outlet, and channel block occupy 25% of the circumference. Utilizing the design at optimum condition (eq 10) for each of the 10 wedge-shaped chambers in parallel, the optimum gap size H d and corresponding disk radius Rd were calculated a t various disk speeds as shown in Table 111. The table also calculates the power consumption and adiabatic temperature rise AT,, of the fluid being discharged whkh- ig ~ccntg b,ey ms-ima.
The results indicate that with increasing disk speed, pump diameter decreases, disk top tangential 'velocity increases (hence power loss over disktops and seals will increase, and torque drops. However, the optimum gap size remains relatively unaffected. The final selection would involve a more detailed economic analysis accounting for pump and energy cost. A reasonable choice in the absence of such a study would be to operate at about 90-120 rpm giving rise to a pump diameter of 362 to 416 mm. The overall dimensions of the selected 100-gpmcoal slurry pump are 360 mm (-14 in.) in diameter and about 250 mm ( N 10 in.) in axial length. The significance of the optimum gap size can be appreciated from Figure 8 which, using the design equation, plots H a vs. Rd with N as a parameter. In addition, AT, curves are also plotted. The foregoing example demonstrates the use of the pump design equations but does not account for shape factors. Referring to Figures 5 and 6, a t H d / & = 0.042 and a = 0.5, F b = 0.97 and F6 = 0.94. The flow rate with the shape factors is 98.5 gpm, that is a 2.5% reduction in capacity. The maximum discharge pressure a t closed discharge condition is 17.78 X lo5 N/m2 (258 psi) and the mean residence time is 0.54 s.
u u
v
Outlet
Figure 9. Single-stage multichamber corotating disk pump.
Corotating Pump Design Configurations and Considerations The single-stage multichamber corotating disk pump, as in Figure 9, is the simplest design configuration. Liquid is fed in parallel to all chambers. If feeding pressure is low and the viscous liquid cannot be fed into the relatively narrow chambers, an undercut in the housing downstream the feeding port may be incorporated into the design. The relative motion between disk tops and housing generates pressure which in turn feeds the liquid into the chamber. For a gravitationally fed pump, a two-stage design as in Figure 1OA can be used. Here liquid feeds gravitationally into relatively wide chambers, which only generate a limited pressure rise to feed the narrow chambers. A special 2-stage pump design configuration (Valsamis, 1980) is shown in Figures 10B and lOC. Here the wide, low-pressure chambers are outboard and the narrow, high-pressure chambers are inboard. This design not only relaxes out-
316
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 Channel Blocks ide Chamber
t
C
IShaft
Outlet
Figure 10. Gravitationally fed two-stage corotating disk pump.
.,
Inlet
Outlot
Outlet
Figure 11. Multi-stage pressure booster corotating disk pump.
board sealing problems, but is characterized by an exceptionally stable output flow rate because the flow rate in the wide outboard chamber exceeds that in the the narrow inboard chambers which after being stuffed full of liquid are reexposed to the (atmospheric) inlet pressure. The excess flow rate is recycled into the liquid reservoir in the feeding hopper. Thus, pumping rate and pressurization in the narrow chambers are disengaged from the instantaneous overall feeding rate, and the pump acts as a metering pump. It is worth noting that in 2-stage pump design, a substantial level of mixing also takes place in the wide chamber. This may be an important advantage in pumping slurries and suspensions. A multistage pressure booster pump, where the chambers are connected in series, is shown in Figure 11, although most discharge pressure needs can be met by a single high-pressure stage. For a successful practical implementation of the corotating disk concept, a number of inherent design problems had to be satisfactorilysolved. Outstanding among these are (a) chamber sealing, and (b) chamber to chamber material transfer. The need for outboard sealing to prevent material leakage is obvious. However, in two- or multistage designs, inboard sealing is also needed to prevent excessive leakage from a high-pressure chamber to its neighboring low-pressure chamber. A comprehensive seal study carried out in these laboratories on both theoretical and experimental levels resulted in the selection and optimization of dynamic (viscous) seals (placed on disk tops) for inboard sealing and a combination of viscous and corotating disk seals (Hold and Tadmor, 1980) for outboard sealing. By
Figure 12. A 7.5-diameter single chamber experimental corotating disk pump.
these optimized dynamic seals, outboard leakage can be completely eliminated and inboard leakage kept- (purposely) within preset ranges. It is worthwhile to note that among the alternative design solutions, viscous seals were selected because unlike in some other machines, in corotating disk pumps (and plastics processors) they perform with outstanding effectiveness. The reason is that every chamber has an angular pressure profile. Hence, individual channels of the viscous seals on the disk tops “see” a discharge pressure which fluctuates with each revolution from a very low chamber inlet pressure to a maximum value at the channel block. Consequently, viscous seals in corotating disk pumps are “breathing” seals. That is, in one part of each revolution some liquid flows from the chamber into the sealing channel, whereas in the other part of it, the liquid is being discharged back into the pumping chamber. This “breathing” effect minimizes shear degradation of the liquid pumped (this is important for shear sensitive polymeric liquids and solutions), eliminates seal clogging after prolonged use, and provides for containing (sealing) pressures much higher than an equivalent nonbreathing seal would. Chamber to chamber material transfer was effectively solved by machining transfer grooves into the inside surface of the barrel. By cutting the grooves helically, channel blocks do not have to be staggered and the “tops” of the disks separating neighboring chambers help “drag” (pump)
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 317
2000 1800 1600
--:
1400
In iL
1200 1000
: 800 600 400
200
0 0
20
40
60
80
100
120 140 160 183 2 0 0 220 240 260 280 300 320 340 360
Angular Position (Degrees)
Figure 13. Characteristic linear pressure profiles for molten polystyrene in the pump of Figure 12, obtained a t different exit valve settings. Closing the valve initially fills up the pump with little throughput reduction. Then throughput drops rapidly, and discharge temperature increases due to viscous dissipation, bringing about a drop in discharge pressure (Tadmor et al., 1979a).
the liquid forward in the transfer grooves. This, of course, reduces or even eliminates pressure losses in transfer grooves and in addition, provides for a "self-cleaning"effect. A few more design considerations are worth mentioning. Housing, as well as the shaft and disks, can be temperature controlled by circulating temperature controlling liquids. The relatively simple geometrical configuration of the pump enables the use of a variety of materials of construction. Finally, pump parts which may wear, such as channel blocks, are easily replaceable. For pumps handling abrasive slurries, in addition, to channel blocks, parts located in high shear area such as seals can also be designed to be replaceable. Experimental Results The theoretical model predicts a linear angular pressure profile. This was verified with some polymeric melts in a number of pumps including a 7.5-in. diameter singlestage single-chamber experimental machine equipped with the pressure transducers at various angular positions as in Figure 12, and in a number of 350-mm pumping chambers. Typical pressure profiles are shown in Figure 13. It is seen here that on increasing the discharge head at constant disk speed, the chamber starts filling up (higher angular coverage) until the chamber capacity is reached. On further increasing the head pressure, throughput would decline unless either disk speed is increased or an inlet pressure head is available for maintaining the throughout. In addition, a detailed pumping study with various polymeric melts has indicated that the design equations listed in Table I1 predict actual performance within resonable engineering accuracy. Predicted vs. measured pressure gradients for three polymeric melts are shown in Figure 14 (which include shape correction factors). Theoretical predictions underestimate experimental results by 040%. The disagreement is believed to be partly due to nonisothermal effects since disk temperatures were slightly lower than inlet melt temperatures. This also brings about a lower experimental temperature rise as reported by Tadmor et al. (1979a,b). A further possible reason for discrepancy is the stress overshoot phenomenon characteristic to viscoelastic melts.
1.2
1.0
.e
.(I
0 100
200
300
400
(dP/d.bEX
Figure 14. Predicted vs. measured pressure gradients for LDPE, PS. and PP.
The pump design equations were obtained by integrating the velocity profiles in the corotating disk geometry. Direct experimental verification of the velocity profiles with Newtonian fluids was obtained by Edelist and Tadmor (1983), who measured the velocity vector field by use of a photogrammetric technique. Further experimental results were obtained in a 350-mm two-stage pump shown schematically in Figure 10B. The actual pump setup is shown in Figure 15. Plastic melt from a 6-in. extruder was gravitationally fed to this corotating disk pump. In order to attain pumping rates far exceeding that of the extruder, a large jacketed hopper for temporary melt storage was built over the pump. This hopper also enabled long runs at high rates by recycling the melt from the pump to the hopper. Low density polyethylene melt with melt index ranging from 0.2 to 20 was used in the experiments. Pump characteristic curves are shown in Figure 16. Analysis of the experimental results indicated that a third narrow high-pressure pumping chamber should be incorporated into the design. A similar size pump with two outboard chambers feeding three inborad chambers would have, at the same speed and
318
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
steady, fully developed laminar flow of power law model fluid, is given by
(1 - F+l)
(A-1)
where 4 = 2y/H, Vo is the plate velocity, H is plate separation, and (APIL) is the pressure rise per unit length. The shear rate profile is given by
(A-2)
i$c =
where qw is the shear rate a t the moving wall given by
(A-3)
Figure 15. A 350-mm two-stage corotating disk pump.
pressure, a 50% higher throughput. Such a pump is under design in these laboratories. Conclusion The corotating disk pump is a novel pump configuration inherently suitable for pumping viscous liquids. This geometrical configuration provides the designer with a great deal of flexibility for adapting the pump to specific needs. It shares with other drag type pumps (e.g., screw extruders) their advantages, yet they are compact in size, are energy efficient, and are free of their disadvantages. The main inherent disadvantage of this pump-the need for sealing the pumping chambers-has been resolved by a combination of specifically designed viscous seals in combination with corotating disk seals. Appendix. A comparison of SMP and JMP Models for Non-Newtonian Fluids The JMP Model. The velocity profile between jointly moving infinite parallel plates, assuming isothermal,
'Ooo 6000
I
E
Hence the shear stress a t the wall is HAP rw = 2L
(A-4)
and the power consumption per unit width is Pw= 2Lvorw = V a A P = q d u
where q d is the drag flow rate per unit width, and the specific power is P w = (qd/q)@ (A-6) Finally, the flow rate per unit width is obtained by integrating eq A-1, and it is given by q=
va--
H2
2(s
+ 2)
(HAPr 2mL
9
2000
1000
E l r 0
200
400
600
(A-7)
The maximum pressure rise a t given flow rate, and the maximum flow rate for given pressure rise, can be obtained from eq A-7 by taking the appropriate derivative with respect to H. In both cases the following relationship is obtained
Ibs/hr)
3000
(A-5)
800 P(PSI1
1000
1200
Figure 16. Characteristic curves of the gravitationally fed two-stage melt pump for 20 MI LDPE.
1400
1600
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
(A-8) Hence for a fixed flow rate q, eq A-7 implies that the optimum gap size is
(A-9)
319
and the specific power is given by
(A-22) Finally, the flow rate per unit width is obtained from Eq A-15 to give 4=
and by substituting eq A-9 into eq A-7, the following expression for the maximum pressure rise is obtained
(A-23) (A-10) Similarly, the optimum gap size for a fixed pressure rise and maximum flow rate is
It can be shown that the optimum conditions for the SMP model are obtained at X = 0 (i.e., when the shear rate a t the stationary wall vanishes). Equation A-23 reduces to
(A-24) and by substituting eq A-11 into eq A-7, the following expression is obtained for the maximum flow rate
Combining eq A-24 with eq A-16 a t X = 0 gives the optimum gap size for constant flow rate
(A-25)
The minimum power is obtained by substituting eq A-8 into eq A-5 to give p w
= (&lap s+2
(Z4.p s+2
mLvon( (1 + s ) v O ) l + n APmax = - -(1 + s) (2 + s ) 4
(A-13)
(A-14)
-n)
(A-27)
The SMP Model. The velocity profile in the SMP model for the same simplifying assumption is given by (Tadmor and Gogos, 1979)
with C;'=y/H, and X is to be determined from the following equation
where G is a dimensionless pressure gradient given by G = -H1+n ( Y ) (A-17) 6m Vo"
It can be shown that X I0 if G 1 ( l + ~ ) ~ and / 6 X C 0 if G C (1/6) ( l + ~ The ) ~ .shear rate profile is given by (A-18) Where the shear rate a t the moving plate is given by - A(' 9, = -(6G)'(1 VO (A-19) H The shear stress a t the moving wall is given by T,
= m( ;).(6G)ll
- XI = L - XI (A-20)
The power per unit width is given by P, = VOLTw = VCHAPp - XI
(A-26)
Similarly, the optimum gap size for maximum flow rate and constant pressure rise is obtained from eq A-16 with X = 0 and eq A-17
and the minimum specific power is given by pw =
and the maximum pressure rise is obtained by substituting eq A-25 into eq A-17, together with eq A-16 a t X = 0
(A-21)
and the maximum flow rate is obtained by substituting eq A-27 into A-25
Finally, the minimum power is obtained for eq A-21 by setting A = 0 and substituting eq A-25 into eq A-21 (A-29) P w , m i n = (2 + s)q@ and the minimum specific power is Pw,min
= (2 + s ) @
(A-30)
Equations 4,5, and 6 in the text of the paper are obtained by dividing the JMP model equations by their corresponding SMP counterparts. Equation 7 in the text is obtained by dividing eq A-14 with eq A-30, subsequent to subtraction from each of the equations the quantity qAP/q = AP,which is the specific power input just for the pressure rise. Nomenclature E, = total rate of viscous dissipation in pumping chamber,
JIB
Fd = shape factor for drag flow in parallel disk pumping F h = shape factor for drag flow in wedge shaped pumping chamber Fp = shape factor for pressure flow in parallel disk pumping characters F b = shape factor for pressure flow in wedge shaped pumping chamber G = dimensionless pressure gradient defined by eq 9 H = gap between parallel disks (plates), m
Ind. Eng. Chem. Process Des. Dev. 1985, 24, 320-325
320
Hd = gap between wedge shaped disks at R = Rd, m L = length of parallel plates, m m = power law model parameter, N sn/m2 N = frequency of disk rotation, rev/s n = power law model parameter (exponent) P = pressure, N/m2 pw = specific power, J/m3 Pw = power, J/s q = volumetric flow rate per unit width, m2/s Q = volumetric flow rate, m3/s Qd = drag flow rate in parallel disk chambers, m3/s Qh = drag flow rate in wedge shaped chamber, m3/s Q = pressure flow rate in parallel disk chambers, m3/s QFP = pressure flow rate in wedge shaped chambers, m3/s R = radius, m R , = shaft radius, m Rd = disk top radius, m s = l/n f = mean residence time, s V, = plate velocity, tangential disk top velocity, m/s (Y = R , / & i. = rate of deformation tensor, AP = pressure rise, N/m2 (discharge pressure less inlet pressure) AT, = adiabatic temperature rise, K t = fraction of chamber circumference filled with liquid = my”-l, non-Newtonian viscosity, N s/m2 0 = angular coordinate, rad
p T T
= Newtonian viscosity, N s/mz = torque, N-m = shear stress tensor, N/mZ
Literature Cited Bouchez, D.; Faure, A.; Scherer, G.; Tranie, L. A.; Antonini, G. “COM: The French Program; Preparation, Stabilization and Handllng of COM”, 4th International Symposium on Coal Slurry Combustion, Orlando, FL, 1982. Edelist. Y. M.S. Thesis, Department of Chemical Engineering, Technion, Israel Institute of Technology. Haifa, Israel, 1982. Edelist, Y.; Tadmor. 2. folym. Process €178. 1983, 1 , 1. McKelvey, J. M. “Polymer Processlng”. John Wiley 8 Sons, New York, 1962. Hold, P.; Tadmor, 2. US. Patent 4207004, 1980. Tandmor, 2. US. Patent 4 142 805, 1979. Tadmor, 2. U.S. Patent 4 194841, 1980. Tadmor. 2.; Gogos, C. G. ”Principles of Polymer Processing”, 1st ed.;Wifey: New York, 1979. Tadmor, 2.; Hold, P.; Valsamis, L. “A Novel Polymer Processing MachineTheory and ExperimentalResults”; 37th Annual Technical Conterence of the Society of Plastic Englneers, New Orleans, 1979a; pp 193-204. Tadmor, 2.; Hold, P.; Valsamis, L. “Plastics Englneering”, Part I, Nov 1979b; pp 20-25; Part 11, Dec 1979b; pp 34-37. Tadmor, 2.; Klein, I. ”Engineering Principles of Piasticating Extrusion”; Van Nostrand-Relnhold Co.: New York, 1970. Tadmor, 2.;Vaisamis, L.; Mehta, P.; Rapetski, W. “Mixing of Coal Slurries in Plastics and Rubber Processing Machinery, Principles and Applications”; SME-AIME Annual Meetlng: 5th International Symposium on Coal Slurry Fuels, Atlanta. 1983b; pp 75-99. Tadmor, 2.; Vaisamis, L.; Yang, J. C.; Mehta, P. S.;Duran, 0.; Hinchcliff. J. C. folym. Eng. Rev. 19830. 3(1), 29. Valsamiu. L. U S . Patent 4 213 709, 1980.
Received for review September 2 , 1983 Accepted May 7 , 1984
Influence of Compositfon Modulation on Product Yields and Selectivity in the Partial Oxidation of Propylene over an Antimony-Tin Oxide Catalyst Peter L. Sllvedon’ and Michel Forrlsler Institut
de Recherches sur L Catalyse, Centre NaNonal de /a Recherche Scientifique, Villeurbanne, Lyon, France
Composition modulation has been studied by the oxidation of propylene to acrolein over a catalyst composed of a solid solution of antimony oxide in tin oxide. Modulation consists of periodically varying the partial pressures of propylene and oxygen in a square wave pattern so as to maintain a constant time-averaged feed composition. Half-periods were of equal duration. Selectivity to acrolein was found to increase substantlalty, but dependence of the increase on amplitude and period coukl not be ascertained. Yield of acrolein decreased with,composition modulation compared with steady-state operation at the same mean feed composition. Amplitude of the square wave was found to be important, but period affected the results just slightly. Product changes with time following a composition switch are used to comment critically on reaction pathway and rate-controlling step proposals in the literature.
Introduction The object of the exploratory research described in this note was to examine a reaction system yielding a variety of products probably via different pathways to see if composition modulation could alter rates and selectivity. Composition modulation is the intentional periodic variation of feed composition to the reactor. The reaction chosen for our study was the partial oxidation of propylene (e3=) to acrolein (primary product) over a tin-antimony (Sb-Sn) oxide catalyst. Oxidation to acrolein over this *Addresscorrespondence to this author at the Department of Chemical Engineering,University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. 0196-4305/85/1124-0320$01.50/0
catalyst is thought to proceed via a redox mechanism (Trifiro et al., 1971; Crozat and Germain, 1973; Sala and Trifiro, 1976; Sokolovskii and Bulgakov, 1977). Unni et al. (1973) and Abdul-Kareem et al. (1980) have demonstrated that composition modulation raises the activity of vanadia catalysts in single reactions believed to proceed via redox mechanisms. Indeed, Belousov and Gershingorina (1968) working with the partial oxidation of C3=over various metal oxides demonstrated that catalyst activity was higher in a pulsed reactor than in a reactor operating at steady state. Trifiro et al. (1971) reported a similar observation with an Sb-Sn oxide catalyst. Up to the initiation of our work, selectivity effects of composition modulation had not been explored for complex catalytic reactions proceeding via redox pathways. Renken et al. 0 1985 American Chemical Soclety
