Wong, K . F , Eckert, C. A,, lnd. Eng. Chem.. 62 (9), 16 (1970a). Wong, K F , Eckert, C. A,, Trans. FaradaySoc., 66. 2313 (1970b) Wong, K . F , Eckert, C. A,, J. Chem. Eng. Data, 16, 56 (1971). Woodward, R . 6. Katz. T. J.. Tetrahedron, 5, 70 (1959)
Received for reuieu: May 16,1973 Accepted February 28,1574 T h e authors gratefully acknowledge the financial support of the National Science Foundation.
Mechanism of the Radiofrequency Ozonizer Discharge Takaaki Aiba’ and Mark P. Freeman*2 Massachusetts Institute of Technology, Cambridge, Massachusetts
The radiofrequency (1-20 MHz) ozonizer discharge or “chemical corona” has been compared in its operation in oxygen to the same device operated at 60 Hz. Comparable chemical, energetic, and electrical data were taken. I n the high-frequency regime where the behavior of the discharge is dominated by a steady-state accumulation of excess positive ions in the gap, the best energy efficiency is several orders of magnitude worse than at low frequencies. A simple model, assuming minimization of the rate of production of entropy, relating the electrical parameters and the observed contraction of the discharge is seen to account for this difference in chemical efficiency. Regions of E / p sufficiently high to support the underlying ionization avalanche mechanism and to create ozone molecules are seen to be confined to layers comparable in thickness to the Debye length adjacent to the electrode (barrier) and thin compared to the aerodynamic boundary layer which probably accounts for the remarkable insensitivity of this discharge to flow rate.
Introduction The high-pressure (ca. atmospheric) barrier discharge, otherwise known as the ozonizer, chemical corona, or silent electric discharge, has long been employed for the commercial production of ozone (Kirk and Othmer, 1952), for which it is uniquely well suited. In its usual configuration, it consists of a high-voltage ac (ca. 15,000 V; 5010,000 Hz) gas breakdown phenomenon occurring between conducting electrodes separated by a gas discharge region (2-3 mm) and one or two dielectric (quartz) barriers. Despite long familiarity with the operational aspects of the device, it is only recently that the discharge mechanism (El Bakkal and Loeb, 1962) and the consequent peculiar chemistry (Coffman and Brown, 1965) have begun to be understood. In brief, when the voltage across the gap between two barriers (or between an electrode and a barrier) due to residual charges on the surfaces(s) and polarization of the bulk of the barrier(s) reaches a certain potential, often called the sparking potential, a pulse of electrons appears to leave the instantaneous cathode surface and cross the gap in a very short period of time. The sparking fields and mean free paths are such that the electrons pick up sufficient energy so that some sort of inelastic collision is possible every mean free path. Evidently conditions are optimal for the formation of ozone, because in a typical commercial ozonizer, if the rate of charge transfer across the gap is compared to the rate of ozone formation, it is clear that about 500 ozone molecules are formed by each electron crossing the gap; hut a t the same time distances and elastic collision cross sections are such that in the direction of the field, the direction of nonrandom energy input, the same electron will not have traversed more than 2-3000 mean free paths. Thus, the commercial suc-
’
Kureha Chemical Industry Company Ltd., Nishiki-Cho, lwaki City Fukushima Pref. 974, Japan. Address correspondence to this author at Dorr-Oliver Incorporated, 77 Havemeyer Lane, Stamford, Conn. 06904.
cess of this device may be attributed to the fortuitous coincidence of the electron energies required for ozone formation and for gas breakdown. In actual fact the “sheet” of electrons is only an energetic equivalent to the true process which must involve a space-wise ionization avalanche across the gap. The growth of this avalanche, which starts with a negligible number of electrons a t the cathode, will be less than exponential because of space charge limitations and the electronegativity of oxygen. For the same reasons, a t any instant of time the electrons will be confined to a more or less two-dimensional sheet which will arrive a t the instantaneous anode in 5-10 nsec while the residual ions, mainly positive, will migrate slowly to their respective destinations. Because of irregularities in charge buildup on the barrier(s), these pulses might start a t slightly different times in different parts of the reactor; however, for a pulse starting from a given homogeneous area, the entire pulse mechanism is over in 10-100 psec. All ions are then out of the gap and, because of the low capacitance of the barrier(s), the charge transferred by the pulse will depress the field so that no new pulses will start until the continually increasing applied voltage repolarizes the barrier(s) enough to restore the sparking potential. As a consequence of this process we see that the number of pulses per half cycle of the applied voltage will depend mainly on the magnitude of the voltage rather than the frequency. Thus, power input (sparking voltage times current transported per unit time) and rate of ozone formation, which might be expected to be in simple proportion to the total number of pulses per unit time, would also be expected to be proportional to the frequency, which is in fact observed (Kovaliv and Briner, 1952; Starke, 1923). The question arises, what happens in such a discharge when the frequency is raised another factor of 100 into the megahertz regime? Certainly were the same proportional relationship to hold, the power input density would be I n d . E n g . C h e m . , F u n d a m . , Vol. 13, No. 3, 1974
179
disruptively high, Although no one has as yet reported a systematic study on this, in a few isolated instances chemical experiments have been performed in equipment in which such discharges probably obtained (Blaustein and Fu, 1967; Epple and Apt, 1962; Lunt, 1925). Characteristically, the discharge tends to contract into a picturesque spot pattern. The power densities, though high, are one to two orders of magnitude lower than the values extrapolated from low frequency. Furthermore, discharge chemistry seems to be unique. Specifically, hydrogen and carbon monoxide are reported to form methane and water as sole products (Epple and Apt, 1962), whereas both a t higher (glow discharge) frequencies and a t lower (ozonizer) frequencies many other products are observed as well. There are thus both physical and chemical indications that the radiofrequency ozonizer constitutes a somehow different type of discharge. Clearly, we might expect any differences to depend importantly on the fact that the low mobility of the positive ions precludes their leaving the gap within the time allowed by a half cycle of the applied voltage, excepting immediately contiguous to the walls, so that some sort of accumulation might be expected to occur. It is the purpose of this work to compare the electrical, energetic, and ozone synthetic characteristics of an ozonizer reactor operating in the 6-22-MHz range with those of the same reactor operating a t 60 Hz to attempt to understand, and if possible, to quantify the factors giving the radiofrequency device its unique character. Although the low-frequency device itself cannot as yet be said to be completely understood, we may nonetheless carry through the analysis by differential comparison of the two regimes.
Experimental Section A cutaway view of the reactor configuration employed is shown in Figure 1. The mean gap between the 1-mm quartz barrier and the 2.18 cm diameter stainless steel inner electrode is just 2.24 f 0.2 mm. The area of the copper screen used in all 60-Hz measurements and in most of the chemical measurements a t high frequency was 55.5 cm2, 6 cm in length. Because of the high displacement current through the reactor a t radiofrequencies, it was found necessary to shorten this electrode to 1 cm (9.25 cm2) to perform electrical measurements. This still exceeded the typical maximum area of the discharge, apparently the same with the small electrode as with the large, although when the smaller is used the visible contraction of the discharge assumes the form of a homogeneous illuminated area instead of the complex spot pattern formed with the large electrode. (Note that as nearly as could be determined, the glow appeared to fill the gap; it certainly could not be assigned to either electrode.) The small electrode is calibrated with 1 cm2 markings to facilitate area estimation. Areas of the spot patterns of the large electrode (see filled points Figure 3) are estimated by drawing the pattern roughly to size on paper and measuring the area of the sketch. Insofar as the 60-Hz discharge emits no visible radiation, it is assumed that in this case the discharge fills the entire area available. The entire reactor is mounted vertically in a calibrated flow of kerosine so that a t high frequencies the power input, A T , is readily obtained calorimetrically by measuring the temperature rise of the kerosine. (The bar over an extensive group denotes a lumped measureable.) This heat balance technique neglects energy lost by radiation and the endotherms of chemical reactions, but the former effect is presumably small and the latter both small and readily calculable from the chemical analysis. Although details of physical arrangement and circuit values necessarily vary with the frequency regime, the 180
I n d . Eng. C h e m . , F u n d a m . , Vol. 13,No. 3,1974
2.24mm
-
A TRUNCATED ELECTRODE SHOWN
Figure 1. Reactor configuration
7’ I
I
LOW FREQ. HIGH FREQ.
Y IAQ + V I
IbJ
X
T
-
-Y
‘
IAQJ
V
:
I
M
E
Figure 2. ( a ) Circuit used to measure the electrical parameters of the discharge; (b) and (c) typical wave forms a t 60 Hz;(d) typical wave forms a t l o 7 Hz.
electrical circuit of Figure 2a is used to determine the electrical parameters of the discharge. It may be readily shown that if Cs and R are adjusted so that the voltage signal X - Y is identically zero under conditions of no discharge, then the amplitudes of the stairstep waveforms that appear across C1 when the discharge starts (the integrated current pulses are shown for a single period comprising the discharge, in Figure 2c) m c s u r e the charge transferred across the gap per half-cycle, A Q
+
(1) Y)(1 CJCh + c,/c,,c, where C, and Cb are the respective capacitances of the gap and barrier. (At high frequencies the balances with and without discharge are slightly different, due no doubt
AQ = ( X -
Table I. Typical Ozone Yieldsa
Hz
Press., Torr
Flow rate, mol(STP) imin
60 60 60 7 . 4 2 (6) 7 . 3 5 (6) 6 . 6 5 (6)b 7 . 4 2 (6) 7 . 4 1 (6) 7 . 3 9 (6) 2 . 1 5 (7) 2 . 1 4 4 (7) 2 . 1 1 1 (7)
751 751 76 796 787 -800 400 200 76 792 400 76
0.94 (-2) 0 . 9 9 (-1) 0 . 9 4 (-2) 0.94 (-2) 0 . 9 9 (-1) 2 . 0 5 (0) 0.94 (-2) 0 . 9 4 (-2) 0 . 9 4 (-2) 0.94 (-2) 0.94 (-2) 0 . 9 4 (-2)
Frequency,
Power, W
Ozone production, mol(STP)/min
Energy required, eV/molecule
0.95 0.78 0.52 85.6 91.6 334.6 57.9 22.3 25.5 274 401 475
5 . 7 4 (-5) 5 . 9 8 (-5) 1 . 3 7 (-5) 3 . 9 (-7) 7 . 7 (-6) 6 . 1 (-5) 2 . 0 (-7) 1 . 3 (-7) 6 . 9 (-6) 7 . 0 (-8) 1 . 8 (-7) 2 . 8 (-7)
10.3 8.1 23.7 1 . 3 6 (5) 7 . 4 (3) 3 . 4 1 5 (3) 1 . 8 (6) 1 . 1 (5) 2 . 2 3 (3) 2 . 4 (6) 1 . 4 (6) 1.1(6)
Electron yield, molecules of Od electron 482 619 61
... ...
(0.87)c ... , . . , . .
... ... ...
Determined from parallel experiments a t lower flow rate. Truncated a ExDonential notation: 7.42 (6) = 7.42 x 106, etc. screenelectrode, 9.25 cm*; discharge area -3 cm2. to the reactance of the reactor altering under discharge conditions, so in this case CS is adjusted for minimum amplitude of the error signal X - Y. A small residual imbalance due to the flow of displacement current through the circuit resistance is known as a function of the voltage a t Y with no discharge and this is subtracted from the minimum amplitude obtained with discharge.) At high frequencies, the steps are observed using direct deflection on the vertical deflection plates of a Tektronix Model 502 oscillograph tube. The steps, shown in Figure 2a for two periods of the applied voltage, through recognizable, lack the perfection of those obtained a t low frequencies due to the relative crudeness of the instrumentation and to the continuous unpulsed contribution of the ion current. At 60 Hz the power, too small to be determined calorimetrically, is determined to within 10L70 from the area of the parallelogram resulting when the X and Y voltage components are displayed on the oscillograph 90" out of phase (Figure 2b). The 60-Hz power supply is a 115-11,000 V rms transformer with Variac input from the normal line voltage, while the radiofrequency power supply is a Lepel 2.5-kW rf generator Model T-2.5-1-MC with filtered (&5%) B voltage supply. The high voltages required to start and maintain the discharge are obtained by a simple rf transformer arrangement. Chemical analysis is done by standard iodimetric techniques (Byers and Saltzman, 1959). The principal impurities in the oxygen are argon, 0.159'0, and nitrogen, 0.1%. No attempt is made to distinguish between ozone and nitrogen oxides formed from the contaminant nitrogen.
Results Chemical analyses are presented in abbreviated (Aiba, 1968) form in Table I. It is clear that the radiofrequency ozonizer is quite valueless as a practical source of ozone. That the discharge itself is regular and reproducible is shown by a plot of input power AW us. discharge area A data in Figure 3, and A W u s . the total charge transported per half-cycle A T in Figure 4. Note the implication that Q and W , though frequency and pressure dependent, do not change with power level (derived variables are defined in Table 11). In both frequency regimes, there is some decomposition of the product ozone in the discharge, which can be reduced by increasing the gas throughput. In Table I1 typical data and characteristics for the atmospheric pressurelimiting yields are compared in greater detail for the two cases. The following analysis will attempt to explain the obvious large discrepancy.
I
I
I
I
I
I
8
4w
300
-2 1%
2w
100
06.6MHZ
A6.6 MHZ
i 760T0RR 76 TORR
0 2
4
6
8
10
AREA lcmZ)
Figure 3. Measured discharge power, A W , LIS. the luminous area of the discharge, A . Filled points estimated from large electrode.
Discussion As indicated in the Introduction, the obvious difference between the radiofrequency and the conventional regimes is that while electrons have sufficient mobility to cross the gap many times in the period of a megahertz half-cycle, positive ions do not. Indeed, as shown below. they can scarcely move and yet, by the postulated mechanism of the discharge, they must be produced in equal number to the electrons (and negative ions) produced in every pulse. Thus, the behavior of the discharge might be expected to be considerably modified by the presence of the resulting space charge, which will act to strengthen the field a t the instantaneous cathode and weaken that a t the instantaneous anode. We postulate that the avalanche process starts when the field due to the voltage difference across the gap (now considerably less than the sparking potential) plus the space charge field a t the surface of the instantaneous cathode reaches some critical value E,, probably not very different from that due to the sparking potential a t low frequencies. The resulting electron avalanche will be essentially two-dimensional as before, but presumably the avalanche process will stop shortly after it leaves the instantaneous cathode because of the rapidly decreasing contribution of the ion space charge to the field. This is supported by the data of Table I1 where Q / n Irf, the charge per unit area involved in the current sheet as it arrives a t the instantaneous anode, is only about the value a t 60 Hz. Neglecting negative ions, if Q / n ( n is the number of pulses in a half-cycle) electrons are in the sheet when it reaches the instantaneous anode, then a t steady state the same number of positive ions must someInd.
Eng. Chem., Fundam., Vol. 13, No.3 , 1974 181
Table 11. Operating Characteristics for Atmospheric Pressure Limiting Yields Quantity
Low frequency
High frequency
Frequency, f Area, A Pulses per half cycle, n Power, AW Power density, W Charge transport per half-cycle, A & Charge transport per unit area per half-cycle, Q Charge transport per unit time Gas flow Ozone production rate
60 Hz 5 5 . 5 cm2
6 . 6 MHz 4 cm2
3-6
2-3 125 W 31 W cm-:
0.5 W
0.011W cm-* 8 . 3 x lo-' 1 8 . 5 X 10-gC cm-2 10-4 A 0,009-0.09 mol/min 5.98 X 10 - 5 mol/min
how be collected by the instantaneous cathode in the period of time between the start of successive pulses. Note that this is still the same in terms of energetics and charge transfer as if the entire charge transfer had been due to Q/n electrons simultaneously leaving unit area of the cathode and traveling to the anode, a simplified model sometimes used to represent this sort of discharge. The complex discharge phenomena observed in this experiment must be the result of known physical phenomena such as avalanching and ion drift and diffusion processes, but it is not clear how the fundamental equations for these processes can be combined to provide an analytical description of the overall discharge. However, the visible contraction of the discharge, reminiscent of the behavior of arcs, suggests the use of a minimum principle of some sort to bypass this area of ignorance. Although the use of irreversible thermodynamics is certainly open to criticism in a device so far removed from equilibrium, we nonetheless postulate that the device is governed by the principle of minimization of the rate of entropy production, which for such a discharge device means essentially minimization of the rate of power dissipation. This provides us immediately with two functional relationships that serve to relate the critical field E,, the positive charge retained in steady state per unit area in the gaps, and the voltage difference. First, as the space charge increases, multiplication occurs over a shorter distance so that Q / n decreases and a t the same time the voltage across the gap required for a given E, decreases. Second, the model assumes a symmetric accumulation of positive ions in the gap and so the higher the space charge becomes, the lower the anode field becomes. If enough ions were to collect to cause the anode field to reverse sign, then the electrons could no longer reach the anode because of the high gas density which obviates inertial effects. A plasma would then form in the vicinity of the anode, and the number of residual free electrons available to start avalanching in the next half cycle, when the erstwhile anode becomes the cathode, would be large indeed. Under these circumstances, the discharge current would become very large and the power dissipation would again rise. (It is interesting to note that the power dissipation is in fact observed to rise when the power level is raised beyond the point where the screen electrode is filled with the discharge. A new spot pattern occurs, now more nearly resembling an arc, and barrier failure occurs very rapidly.) Thus, the condition of minimum power dissipation occurs when the space charge due to accumulated positive ions causes the anode field to vanish; that is, the discharge area will adjust itself so that the requisite avalanche current density is obtained to maintain this condition. We write for the critical field a t the surface of the cathode (shown as a function of pressure, P ) assuming a steady-state distribution of residual positive ions 182
Ind. Eng. Chem., Fundam., Vol. 13, No. 3 , 1974
Approximate high/low
-lo6 -0.1 -0.5 0 . 2 5 x 103 0.28
x
104
6 X 10-gC
-0.01
1 . 5 X 10-gC ern+ 10-l A 2.05 mol/min 6 . 1 X 10-6 mol/min
=o. 1 1000
= 100 51
where t o is the permittivity of free space, q ( x ) is the net charge density in the gap (positive ions less negative ions) which we assume to be symmetric but which in general will not be constant, and S is the area density of net charge in the gap, width d. By rearrangement of eq 2 we can see that V / d is not only the average applied field, but also the actual field a t the center of the gap, postulated here to coincide with the center of a symmetric charge distribution. Similarly, for the field a t the surface of the anode postulated to be zero because of the minimum principle
o = -vd- -
s 2to
(3)
From eq 2 and 3 (4)
aind
E,(P) = 2V/d
(5)
Thus we come to the immediate result that the effective sparking voltage V (and hence the field in the center of the gap, V / d ) is just half its value in the absence of residual space charge. It remains to determine E,(P);to the extent that it agrees with that a t low frequency (Table 11) it should be a measure of the applicability of the model. The energy dissipated per unit area in the discharge per half cycle is clearly the product of the charge transported per half cycle per unit area and the potential difference across the gap, which a t the onset of each pulse is the sparking voltage. The power per unit area is thus
W
=
2fQV
(6)
If we multiply eq 6 through by the area of the discharge, it is clearthat the slope of a plot of the measureable total power A W us. the measurable total charge transported per half-cycle, A 3 (Figure 4), will yield the sparking voltage multiplied by twice the frequency. The voltages thus obtained are equivalent to those obtained a t the lower frequency by means of the power parallelogram (Figure 2b) and are thus given in Table III as V (power), together with the remarkably constant sparking potentials for the 60 Hz case obtained by inspection of the power parallelogram. Using eq 5 and the geometry of the apparatus we can immediately write E, ( P ) for the different high-frequency discharges (Table 111). The agreement between the high-pressure runs is excellent while consistency with the low-frequency critical fields is really better than could be hoped for serving as a t least semiquantitative justification for use of the minimum principle, which through eq 4 now
Table 111. Descriptive Discharge Parameters
From Figures
Power measurement Frequency, HZ
Press., Torr
17 X lo6 6 . 6 X lo6 6 . 6 X lo6 60
760 760 76 760 76
Ec (PI V m-I
9
V, V
lo6 lo6 lo6
sc i t h
m-2
C m-2
C
m-;j
~
Eq 9 6, mm
_
_
_
X 1O-j 1.27 X 1O-j 0.65 X l o m 5
S + Q -~ 2
(7)
If, as a limiting case, we assume a rectangular distribution of ion density equal to q ( d / 2 ) , that a t the cathode, being acted upon by the full field E c , it should be possible to estimate the fraction of the discharge in which ions are created. We thus calculate q ( d / 2 ) from the mobility relation q(d/2)p+E,(P) (8) For atmospheric pressure and the values of Ec determined above, the mobility of positive ions in oxygen is about 2 cm2 V-I sec-I (Samson and Weissler, 1965). Thus, we can calculate a maximum thickness of the distribution, 6 , maximum because the concentration of ions caused by the avalanche should actually increase with distance from the instantaneous cathode
6
S,.,tl, ILY,
1.2
gives the area density of net charge in the gap. This, too, is tabulated in Table 111. It remains to ascertain how these ions are distributed within the gap. Now, the fields are such that a negligible number of positive ions will go to the instantaneous anode. Therefore a t steady state, in spite of this low drift velocity, as many positive ions must be collected a t the instantaneous cathode as electrons a t the anode in each half cycle. It follows that the average value of S / 2 , the sum of residual positive ions left in the cathode space a t the start of each pulse and those generated in this space by the discharge must be such that in the time between pulses, Y ~ ( n f ) -Q~/,n ions will be driven to the instantaneous cathode by the local field which is certainly no greater than Ec(P).Hence, Q ions in time ( y 2 f - l ) . But Q is already known to be constant from the linearity of Figures 3 and 4 and indeed one can compute it for each instance readily from the slopes of the lines in these figures (Table 111). We take the average area density of positive ions in the cathode space as
2fQ
C
Eq 8 5J(d/2),
Eq 7
Q7
1.01 X 1O-j 1.1 X 10-5 1.20 0.011 1 . 6 5 X 10-5 1 . 4 6 X 10-j 0.47 0.031 0.36 X 0.50 X 10-j 0.15 0,033 Sparking potential from power parallelogram = 5000 V; E , (760) = 2.24 x 106 V m - 1 Sparking potential from power parallelogram = 1400 V; E,(76) = 0.62 x 106 V m-1 1510 1610 818
1.35 X 1.44 X 0.73 X
3 and 4,
Eq 4 S , C m-2
=
= S/(2q(r1/2)) =
S,,%,h ,,/q(d/2)
(9)
Values of q ( d / 2 ) and 6 are also tabulated in Table 111. Note that these remarkably thin sheaths, only about 1% of the gap, are well within the aerodynamic boundary layers. This probably explains the insensitivity of the observables of this discharge to the flow rate. The question naturally arises in connection with the model whether the numbers obtained in the preceding paragraph are consistent with the Debye shielding distance. Under the conditions of the discharge we expect the electron temperatures to be a t least 105"K (Brown, 1966), so that a t these ion concentrations the Debye length (Spitzer, 1964) will be about 0.014 mm, which is not small compared to the distribution and is certainly as large or larger than the thickness of the electron sheet. Thus, there would seem to be no inconsistency involved. We are now prepared to examine the chemical consequences of this model. Evidently, a t high frequencies the entire ionization avalanche occurs in less than 1% of the
1 1
0 6.6 MHZ I7'O
Id loo
A
6.6 MHZ
76 TORR
1 2
4
6
8
10
12
14
16
A T fCOUL/HALF CYCLE x
18
20
22 24
lo9)
Figure 4. Measured discharge power, A T , us. the total charge transported per half cycle, A?.
gap (5-10 mean free paths) in the space immediately contiguous to the instantaneous cathode. In this space about 10% of the ions are created that would have been created had the avalanche occurred over the entire gap as it does a t low frequencies (Table 11). Note that the field falls rapidly with distance through the ion layer so that the bulk of the ionization probably occurs in a somewhat shorter distance. The remaining 99% of the gap has a field something less than half the value obtaining in the low-frequency ozonizer (eq 5 ) where, as we noted before, conditions are apparently nearly optimum for the formation of ozone. Because of energy requirements ozone synthesis might therefore be expected to occur only in the same region in which the avalanche occurs. Thus each electron crossing the gap should only make about 1% of the ozone molecules it would make a t low frequency. Further, because the electron density in the sheet is only one-tenth the value a t 60 Hz and because there are only half as many pulses per half cycle, we expect only l/2000 as many ozone molecules to be formed per unit area per half-cycle. Again, because the rf discharge in this reactor is typically only about one-tenth of area of the low-frequency discharge, another factor of 10 arises. Finally, one might expect rather heavy wall-losses of the ozone created exclusively so close to the wall, but this can account for a t most a factor of 2 in the operationally realized limit of a rapidly flowing stream providing an ozone "sink" comparable to the wall, for diffusional forces away from the wall should be just as strong as those toward the wall. Based on the model and a frequency ratio of lo5, we therefore find that on the order of two times as many ozone molecules should be formed per unit time, whereas the discharge power consumption should have gone up a factor of 103. This is consistent with the experimental data of Tables I and 11, a t once confirming the model and a t the same time demonstrating that the best performance observed in the present work for this radiofrequency ozonizer is of the order of magnitude of its best possible theoretical performance. Note that to obtain this value it is necessary to work a t flow rates one to two orders of magnitude higher than those used at low frequencies to decrease decomInd. Eng.Chem., Fundam., Vol. 13, No. 3,1974
183
position of the ozone formed by the high temperatures and possible electron bombardment in the remaining 99% of the gap where the field and hence electron temperatures are still about one-half the value a t 60 Hz.
Conclusions The radiofrequency ozonizer discharge has been compared in its operation with the same device operated a t 60 Hz and comparable chemical, energetic, and electrical data have been taken. In comparison to the low-frequency ozonizer, the rf discharge is observed to be luminous and contracts so as to cover but a fraction of the electrode area, which is an additional datum recorded. It is clear that the best energy efficiency for the production of ozone by this device is several orders of magnitude worse a t the high frequencies than a t the low. A simple model is proposed assuming minimizaticn of the rate of production of entropy. This somewhat qucstionable procedure is used to supply relationships otherwise inaccessible because of the complexity of the avalanching process. The model nicely interrelates the measurables of the discharge: (1) power dissipation, (2) area of the luminous discharge, and (3) the total charge transported across the gap per half-cycle. This model also provides some insight into the interior of the discharge space where a steady-state ion population (area density) is maintained on the same order as the area density of electrons crossing the gap in each half-cycle. Furthermore, these "space charge" ions seem to be confined near the electrode and/or barrier(s) in layers thin compared to the aerodynamic boundary layer and comparable to the Debye length. Because the creation of ozone requires electrons of energy comparable to those creating ions in the assumed avalanche mechanism, the ozone must also be formed in these same layers. Thus, it is seen that the ozone produced is consistent with the amount expected from the model. Most of the energy of the discharge is lost in (net) nonproductive collisions in the gap. Note that the considerations of the preceding paragraph do not obviate the high-frequency ozonizer as a chemical
reactor. Even though it is clearly inefficient for the production of ozone, the electrons over most of the gap have a fie1d:pressure ratio, and hence temperature, about onehalf that required for ozone formation, but which may well be suited to some less energetic reaction (cf. Introduction). For a reaction of sufficiently modest requirements, the device should be ideal for quantity production because of its intrinsic high power input and gas throughput.
Acknowledgment The writers wish to thank Professor R. F. Baddour of M.I.T. for extending the use of the facilities of his laboratory for the experimental work. They also wish to acknowledge helpful discussions with G. Bekefi and D. Flamm during the course of the undertaking.
Literature Cited Aiba, T., M.S. Thesis, Massachusetts Institute of Technology, Cambridge, Mass. 1968. Blaustein. B. D., Fu, Y. C., Amer. Chem. SOC., Div. f u e l Chem. Prepr., 11, 243 (1967). Brown, S.C., "Basic Data of Plasma Physics," 2nd ed rev, p 105, M.I.T. Press, Cambridge, Mass. 1966. Byers, D. H., Saltzman, B. E., Advan. Chem. Ser., No. 21, 93 (1959) Coffman, J. A., Brown, W. A., Sci. Amer., 212 ( 6 ) , 90 (1965). El Bakkal, J. M., Loeb, J. B., J. Appi. Phys., 33, 1567 (1962). Epple, R . P., Apt, C. M . . American Gas Association Catalog No. 59/OR, American Gas Association, Inc., New York, N. Y.. 1962. "Encyclopedia of Chemical Technology," R. E. Kirk and D. F. Othmer, Ed.. Vol. 9, pp 735-753, Interscience. New York, N. Y . . 1952. Kovaliv, B., Briner, E., Helv. Chim. Acta, 35, 2283 (1952). Lunt, R . W.. Proc. Roy. Soc., Ser. A, 108, 172 (19'5). Samson, J. A. R.. Weissler, G. L.. Phys. Rev., A137, 381 (1965). Spitzer. L., Jr., "Physics of Fully Ionized GaAes," 2nd ed, p 23, Interscience, New York, N. Y . , 1964. Starke, A., Z.Elektrochem., 29, 358 (1923)
Receiued for reoieu J u n e 20, 1973 Accepted F e b r u a r y 19,1974 Work p e r f o r m e d at Massachusetts I n s t i t u t e o f Technology. Fin a n c i a l s u p p o r t received f r o m t h e Sloane Fund of M.I.T. a n d t h e N a t i o n a l Science F o u n d a t i o n .
Laminar Mixing of a Pair of Fluids in a Rectangular Cavity Donald Bigg and Stanley Middleman* Department of Chemical Engineering, University of Massachusetts, Amhersf, Massachusetts 0 1002
The Marker and Cell method has been used to study the laminar mixing of two viscous fluids initially stratified within a rectangular cavity, the upper surface of which is suddenly set in uniform motion. Principal parameters studied are the viscosity ratio of the two fluids and the Reynolds number of the system.
In the most common model of flow in a melt extruder (McKelvey, 1962) it is assumed that the fluid is confined in a rectangular cavity, the screw channel, one surface of which is in steady motion. Figure 1 shows the geometry usually considered. Fluid elements circulate in a transverse motion generated by the movement of the upper surface. In addition, because the upper surface also 184
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has a velocity component normal to the rectangular cross section, fluid also moves normal to the channel cross section. The net result is that fluid elements perform a helical motion down the screw channel. In a related study by Bigg and Middleman (1974), the residence time distribution of such a flow field was considered. A mathematical model was developed which was
