Ind. Eng. Chem. Process Des. Dev. 1904, 23 I 737-74 1
Henley, E. J.; Sea&, J. D. "Equilibrium Stage Separations in Chemical Engineering"; Wiley: New York, 1981. Ladisch, M. R.; Dyck, K. Science 1979, 205, 898. Leeper, S. A.; Wankat, P. C. Ind. €ng. Chem. Process D e s . D e v . 1982, 27, 331.
Oudejans, J. C.; Van Den Oosterkamp, P. F.; Van Bekkum, H. Appi. Catai. l.B - -8-,2 . -3 ,. 109-11s. .- - . .-. Poclus, D.; Ladlsch. M. R; Tsao. A. T. "Calcium Sulfate as a Selective Adsorbent of Water"; Paper 87d. AIChE Annual Meeting, LA, Nov 14-19, 1982.
737
Scheller, W. A. "Energy Requirements for Grain Alcohol Production"; Presented at 176th National Meeting of the American Chemical Society, Miami Beach, FL, Sept 10-15, 1978. Scheibel, E. G. Ind. Eng. Chem. 1950, 42, 1497-1508. Whitcraft, D. R.; Verykios, X. E.; Mutharasan, R. Ind. Eng. Chem. Process D e s . Dev. 1983, 22, 452-457.
Received for review March 28, 1983 Accepted November 14, 1983
Mixing Characteristics of the Positive and Negative Pressure Air Mixers Tetsuo Aklyama" and Isao Tada Department of Chemical Engineering, Shizuoka Universw, Hamamatsu, 432, Japan
This paper compares the mixing efficiency of the negative pressure air mixer (NPAM) and the positive pressure air mixer (PPAM). Experiments were carried out using five pairs of particles which were chosen from five kinds of particles. The quantity of air and work necessary to mix 1 kg of solid particles to a state of M (the degree of mixing) = 0.1 were compared. Experimental results indicated that the NPAM can become more advantageous than the PPAM in the range of practical use. This fact was then backed by a simple theoretical analysis through calculation of the work that can be done by the jetting air stream.
Introduction Mixing of particulate material is widely practiced in industry. It is often required to mix two or more kinds of particulate materials or to homogenize raw or manufactured material. Attainment of fine homogeneity is highly desired in ceramics and other industries and the rising energy cost calls for more energy-effective mixing techniques. When large quantities of solids are to be handled, an air mixer is often used. A drawback of this mixer is that it requires filter bags. To overcome this drawback, a negative pressure air mixer (NPAM) was proposed in the previous paper (Akiyama et al., 1982). The NPAM is suitable to handle smaller volumes of various kinds of particulate material which are sensitive to contaminants. Thus the NPAM was recommended for mixing of foodstuffs or medicine. A quantitative comparison of the mixing efficiency among the NPAM, the positive pressure air mixer (PPAM), and the V-shaped tumbling mixer has been reported by Akiyama et al. (1983). The study indicated that under limited conditions, the NPAM was more advantageous than the PPAM and that the V-shaped tumbling mixer was better suited for rather coarse mixing but it was unable to attain as fine a homogeneity as was possible through the use of the air mixers. The mixing characteristics of the nonuniform fluidization mixers were investigated by Ando et al. (1970) and Sano (1973), and comparisons among various types of mixers were reported by Miles et al. (1970) and Shah (1973). These studies, however, were concerned with a steady-state type of operation, and direct comparison with the NPAM was not possible. The operation of the NPAM is inherently transient because it needs repetitive air injections. The objective of the present investigation is to make a quantitative comparison of the mixing efficiency between the NPAM and PPAM over a broader range of conditions than those studied in the previous work. As criteria of comparison, the air quantity and work that are necessary 0196-4305/84/ 1123-0737$01.50/0
Table I. Physical P r o m r t i e s of Particles true bulk av diam density, density, Particle X lo3. m ke/m3 ke/m3 glass bead A 0.214 2520 1600 glass bead B 1.705 2520 1600 millet seed 1.573 841 1386 polythylene 3.70 940 563 pellet polystyrene 0.997 1070 651 pellet
void ratio 0.365 0.365 0.393 0.401
angle of repose, rad 0.262 0.209 0.611 0.436
0.392
0.436
Table 11. Physical Properties of Systems
system G G : glass bead A-glass bead B M-G millet seed-glass bead A P-G: polyethylene-glass bead A PS-G: polystyrene-glass bead A P-M: polyethylene-millet seed
diam ratio 7.97 7.35 17.3 4.66 2.35
angle of repose, rad 0.349 0.576 0.384 0.462 0.733
to secure the desired degree of mixing were experimentally determined. Then it was shown that the NPAM can be more advantageous than the PPAM. A simple theoretical analysis was also made to show that the NPAM can indeed become more advantageous than the PPAM in the range of practical interest. Experimental Section Schematics of the NPAM and PPAM are shown in m3, Figures 1and 2, respectively. A column (8.16 X 0.3 m i.d.) made of transparent cylindrical acrylic resin with a cone of 65' was used as a mixing vessel. This column was interchangable with the rest of the test equipment. We used five kinds of particles, from which five pairs weighing 7.5 kg each (15 kg in total)were chosen to carry out experiments. Physical properties of these particles are listed in Table I, and some pertinent properties of the pairs are shown in Table 11. The initial 0 1984 American Chemical Society
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Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
The PPAM. The top of the vessel was always open to the atmosphere. The compressor started to fill the air tank after valve 1was opened and the solenoid valve was closed. Then the compressor was stopped, and the solenoid valve was opened to send the compressed air into the vessel, thus mixing the particles. Pressure in the air tank was measured by using a transducer. Valve 2 was used to control the air flow rate. The air injection was repeated from 10 to 15 times for each set of experiments.
> 3 \e
Sa* "le, rDI"E-2
Figure 1. Schematic of the experimental apparatus of the NPAM.
Characteristic Values Degree of Mixing. The same index that we chose in the previous reports was used to define the degree of mixing, i.e.
M =
U / U ~
(1)
n
Figure 2. Schematic of the experimental apparatus of the PPAM.
pressure differences of air applied in mixing experiments were -33, -40, and -54 kPa for the NPAM, and 98, 147, and 196 kPa for the PPAM. The smallest pressure differences, -33 kPa and 98 kPa, are approximately the minimum values that enabled effective mixing. Atmospheric air was injected into the vessel to mix the particulate material in the case of NPAM. This air injection was repeated until the desired degree of mixing was secured. In the case of PPAM, the compressed air stored in the air tank was injected into the vessel. The volume of the air tank (0.03 m3) was so chosen as to give the amount of air that was not significantly different from that of the NPAM by one air injection. Preliminary experiments indicated that there was not much difference in sampling at depths of 0.03 and 0.06 m. Therefore, to avoid the bed disturbance due to samplings, samples were taken at the depth of 0.03 m at five points in the bed. Each sample weighed approximately 0.03 kg. Since the mixing time with one air injection was short (a few seconds), static electricity did not appear to be significant (except for the polystyreneglass bead A system), so that no precautions were taken to prevent the development of static electricity. Experimental procedures are described in the following paragraphs. The NPAM. Air was evacuated from the vessel, after the solenoid valve and valve 2 were closed and valve 3 was opened, until the air pressure was reduced to a preselected value. Then by closing valve 3 and opening the solenoid valve, nearly the same quantity of air at atmospheric pressure was permitted to enter into the vessel. Thus, the particles within the vessel were agitated. This operation of evacuation and injection of air was repeated from 10 to 15 times for each set of mixing experiments. The flow rate of air entering into the vessel was controlled by valve 1. Pressure was measured by a transducer at the top of the vessel and recorded on a magnetic oscillograph.
where n is the number of sampling points, the n values of x i are the volume ratios of smaller particles, and x, is the value when perfect mixing is achieved. The initial variance, go, is the value before the mixing operation starts. The degree of mixing, M, is unity before mixing. It starts to decrease as mixing progresses and becomes zero when the state of complete mixedness is achieved. To compare the mixing efficiencies, M = 0.1 was chosen as a critical value, and the air quantity and work that were required to reach the state of M = 0.1 were compared. The air velocity entering into the column was dependent on the pressure within the column (NPAM), or the air tank (PPAM), and also on the degree of valve opening at the inlet. We chose the average air velocity at the conical apex (cross sectional area = 0.00126 m2) as a characteristic velocity and calculated it from the following formula: (3) where p o is the atmospheric pressure, Ap refers to the gauge pressure in the column for the NPAM, or the air tank for the PPAM (The absolute value of the gauge pressure was used in the previous report: Akiyama et al. (1983)). The absolute value of Apf was approximately 6-15 kPa in both mixers. V stands for the volume of the column, minus that of particles for the NPAM, and the volume of the air tank for the PPAM. Sa = 0.00126 m2 is the cross-sectional area at the conical apex, and to represents a time span during which mixing effectively occurred, i.e., the time required for the pressure to decrease from lApil to lApfl. The air quantity which was effectively used for mixing with one air injection, wl,was calculated from
w1 = PoVIAPi - APd/Po
(4)
The work performed by air for mixing with one air injection, N,,was calculated from NI = IAPi - ApflV
(5)
(The work due to the gauge pressure was used in the previous work: Akiyama et al. (1983)). When N air injections were necessary to mix particles to a prescribed degree of mixing, the total air, w , and work, N,,are, respectively w = Nw,
(6)
Nt = NN1
(7)
These expressions were used in the evaluation of the mixing efficiency.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
739
Qo3
-a02
GGsvstm W=15kg
P I
" 0
4
5
.*
a01 -
4 2 4 6 8 10
A
...
-A
0
P
A 0.4 0 0
o", A
L
Number of air injections[-]
2
0
Figure 3. The degree of mixing vs. number of air injections.
4
6
Olmlsl
Figure 5. Air quantity required to mix 1 kg of solid particles to the state of M = 0.1 vs. the velocity.
Number of air injections 1-1
Figure 4. The degree of mixing vs. number of air injections (P-M system).
Results and Discussion The maximum pressure difference that can be imposed on the NPAM is one atmosphere. Air that enters into the vessel is one atmosphere. The air volume is that of the vessel minus particles. Thus, it can be clearly seen that there is a limit of the particle loading which can be effectively mixed, and that the limit is inherent to the column volume. The limit is not so critical in the case of the PPAM, because the volume of the air tank as well as the air pressure can be increased as much as one wishes-at least in principle. This, however, does not necessarily lead to an improvement in efficiency (Akiyama et al., 1983). Typical examples showing the behavior of the degree of mixing with the number of air injections are shown in Figure 3. The figure indicates, for all the systems illustrated in the figure, that the degree of mixing becomes less than 0.1 after several air injections and then levels off. Thus M = 0.1 was chosen as the critical value in evaluation of the mixing efficiency. The P-M system behaved defferently from the others, as is seen in Figure 4. This system was very difficult to spout, and needed higher IApil to initiate mixing. It also appeared unlikely for this system to reach the state of M = 0.1. Table I1 indicates that the P-M system has the largest angle of repose, and the G G system the lowest. Figures 3 and 4 illustrate that the P-M system is the hardest and the G G system is the easiest to mix. Thus, the angle of repose appears to be a good index in predicting the mixability of particulate material through the air mixers. In the following discussion, the P-M system will be excluded due to its exceptional difficulty to spout. To compare the mixing efficiency of the NPAM and the PPAM, the air quantity and work were considered. There are of course other physical parameters such as the moment or the kinetic energy (due to the jetting air stream) that can be compared. It would, however, be convenient to consider work when one compares the air mixers with other types of mixers, or when one considers the overall efficiency including the power consumption due to a
5 d 0
D'o
2
I 6
4
0 lmlsl
Figure 6. The work required to mix 1 kg of solid particles to the state of M = 0.1 vs. the velocity. Key A P , l k f u l System
i_
A
P-G i Ps-G
Q W=15kg 0
A
4A
1000 3
0
m
0 0
do
*
4 ~[m/s~
I
6
Figure 7. Comparison of the work required to mix 1 kg of solid particles to the state of M = 0.1.
vacuum pump or a compreasor. The air quantity and work required to mix 1 kg of solid particles to the state of M = 0.1 are plotted against the average velocity in Figures 5 and 6, respectively. These figures clearly indicate that it is possible to operate the NPAM under such conditions as to require less air and less work than the PPAM to mix solid particles. We obtained the same result under much
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Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 4, 1984
Table 111. The Work Due to the Jetting Air Stream and the Work Reouired to Obtain A D $ NPAM PPAM APi? V, kPa m3 -33 0.0757
Na, J
Nm, J
317.5
80.25
-40
0.0757
380.2
120.1
-53
0.0757
531.1
235.3
m3
Nm J
Nm, J
0.0316 0.03 0.0382 0.03 0.0522 0.03
308.1 308 377.7 375.1 522.7 512.0
151.1 158.4 182.9 224.9 249.5 397.8
Api,
kPa
V,
98 104 98 128 98 181
\
G-G Svslem
1
2
3
4
tlsl
Figure 8. The transient air velocity from the nozzle.
more limited conditions in the previous study, where experiments with only one system and one pair of Api were compared. In this study we carried out experiments with Api that covered ranges of practical interest. Although the effect of Api does not appear so certain, air seems more effectively used with smaller App This can also be seen in Figure 7, where results of other systems are illustrated with one value of Api for each mixer. If the above experimental finding, that the NPAM is sometimes more advantageous than the PPAM, could be backed with some physical principles it would be of considerable interest and of practical value. In fact, we can show by a simple theoretical analysis that the NPAM can be more advantageous than the PPAM in the sense that air is more effectively used in the former than in the latter. The analysis is described in the following text. We calculate work that can be done by the jetting an air stream from a nozzle. No interaction with solid particles is considered because it is simply the potential due to the air stream which is to be compared. We will assume here for the sake of simplicity that the flow is both frictionless and adiabatic and that the quasi-steady state holds. Then the Bernoulli equation may be used to obtain the following expressions for the instantaneous velocity of the jetting air stream from the nozzle (Bird et al., 1960).
The mass balance gives dP2
(PPAM)
The work due to the jetting air stream was calculated from N , = S , S t b 1 u dt
(NPAM)
(12)
N , = S n S0 t b 2 udt (PPAM)
(13)
0
where S, (= 2.04 X lo-* m2) is the cross sectional area of the nozzle; subscripts "1" and "2" in p and p refer to the upstream and the downstream, respectively, and y = 1.4 for air; p 1 in eq 12 was so chosen as to yield the maximum mass rate with the relation p 1 = ~ ~ 1 0 . until 5 3 p 2 reached 0.53p0, and afterward p 1 was set at p o until p z was increased to po; p 2 in eq 13 was so chosen as to give the maximum mass rate with the relation p 2 = 0 . 5 3 until ~ ~ p1 reached ~ ~ 1 0 . 5then 3 ; p2 was set at p o until p 1 was reduced to Po. As the basis of calculation, experimental conditions of the NPAM were chosen. Then, the calculation for the PPAM was carried out under the condition that equal amounts of air are being used for mixing, i.e. (PO - dvn = (pi - P O ) V ~ (14)
[m] ___..
'\
v-dPl = -Snp2u dt
V- = SnplU (NPAM) dt
(10)
where pi on the left-hand side of the above equation is the initial density in the column, whereas pi on the right-hand side is that in the air tank; thus their values are different. V , is the volume of the column minus that of particles and V pis the volume of the air tank. The above equations were used to calculate the density, pressure, mass flow rate, and work through the use of the isentropic relationship (pp' = pipi'). The calculated works are listed in Table I11 for selected parameter values. This indicates that the jetting air stream can yield larger work, Ne, in the NPAM than in the PPAM, and this tendency becomes more pronounced as the absolute value of Api is increased. The N , in Table I11 is the work that is required to obtain Api through the use of the pump or compressor and was calculated from the Bernoulli equation. I t was assumed that the flow was both frictionless and adiabatic, and that the kinetic energy was negligible. The mechanical efficiency of the pump or compressor was not taken into account. The values of N, also indicate that less work is needed in the NPAM than in the PPAM to get the desired Api (e.g., N , = 120.1 J and 182.9 J for Ap = -40 kPa and 98 kPa, respectively). Representative transient behaviors of the air velocity and the mass rate of flow are illustrated in Figures 8 and 9. These figures show clearly the different characters of the mixers. In the case of PPAM, rather high mass rates of flow are observed at the onset of the air injections, then they decrease rapidly as time elapses, whereas in the case of NPAM the air velocity or the mass rate of flow does not decrease so rapidly. It maintains relatively high values longer than the PPAM. This seems to explain why air is more effectively used for mixing in the NPAM than in the PPAM. It was reported (Akiyama et al., 1982) that the duration of mixing was generally more important than the intensity of mixing in the NPAM. This seems to apply in comparison between the NPAM and the PPAM. However, one should not forget the different basis of comparison between the transient type and the steadystate type of operations. The mixing efficiency is considered in terms of the required amount of air in the former, whereas it is in terms of the required time in the latter. In many practical situations the expenditure of air and of work is of less consequence than the time required to achieve a certain degree of mixing; then the PPAM will
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984 741
m,
I
G-G System W=15k Line AP IkPaI Vlmll
_._._ 98 a0522
1
2 t (SI
3
Figure 9. The transient mass rate of flow from the nozzle.
be superio to the NPAM by this criterion. Although a number of assumptions have been made in the above analysis, the results obtained will be valid in evaluating the relative efficiency of the two mixers. There are of course other factors, such as the overall cost, physical properties of particles, and the total mass that must be taken into account when decisions are to be made in choosing a mixer. Conclusions Experimental and theoretical studies have been made regarding the mixing efficiency of the NPAM and the PPAM. Five pairs of particles were selected from five kinds of particles. The angle of repose was found to be a good indicator in predicting the mixability of particles. The air quantity and work required to mix 1kg of particles to the state of M = 0.1 were compared for the two mixers. It was found that the NPAM can become more advantageous than the PPAM in the range of practical use. Then, calculations were made to obtain the work that can be done by the jetting air stream from a nozzle, assuming that the flow was both frictionless and adiabatic and that the quasi-steady state held. The calculation revealed that the jetting air stream can in fact yield larger work in the NPAM than in the PPAM. Acknowledgment We thank H. Inagaki and Y. Matauo for their help in the experimental work. Nomenclature M = u/uo, degree of mixing
N = number of air injections N1= work performed by air for mixing with one air injection, J N , = work performed by the jetting air stream from the nozzle, J N , = work required to obtain Api, J Nt = "1, J p o = atmospheric pressure, Pa p 1 = upstream pressure, Pa p 2 = downstream pressure, Pa APi = Pi - PO,Pa APf = P f - Po, pa S, = cross-sectional area at the conical apex, m2 S , = cross-sectional area of the nozzle, m2 t = time, s to = time span during which mixing effectively occurred, s u,u(t) = air velocity from the nozzle, m/s ii = characteristic velocity calculated from eq 3, m/s V = V , or V,, m3 V , = volume of the column minus particles, m3 V , = volume of the air tank, m3 w = N u l , kg w1 = air quantity which was effectively used for mixing with one air injection, kg W = total particle weight, kg x = volume fraction of smaller particles x = average value of x for five sampling points
Greek Letters p = air density, kg/m3 p 1 = upstream air density, kg/m3 p 2 = downstream air density, kg/m3 u = variance defined by eq 2 uo = variance before mixing
Subscripts
i = initial f = final Literature Cited Aklyama, T.; Peters, L. K.; Kageyama, S.; Hosol, M.; Yokota, I.; Kono, M. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 664. Akiyama, T.; Keno, T.; Matsubara, K.; Kono, M.; Melo, J. Huntalkopku 1983. 20, 141. Ando. K.; Taho, H.; tiara, H. Huntadkogaku 1970, 7 , 356. Bird, R. 8.; Stewart, W. E.; Llghtfwt. E. N. "Transport Phenomena"; Wiley: New York, 1960 Chapter 15. Miles, J. E. P.; Schofield, C. Trans. Inst. Chem. €ng. 1970, 4 8 , T85. Shah, V. D. Process Eng. Sopt 1973, 87. Sano, Y . Hunsall973 No. 16, 23.
Received for review April 5, 1983 Accepted February 6, 1984
