Powell, R. J., J. Chem. Eng. Data, 17, 302 (1972). Spencer, J. N.. Voigt, A. F., J. Phys. Chem., 72, 464 (1968). Stepanova, G. S.,Gazov. Deb, 1, 26 (1970). Thomson, E. S.,Gjaldbaek, J. Chr., Acta. Chem. Scand., 17, 127 (1963). Washburn, E. W., Ed., "International Critical Tables", Vol. Ill,p 255,McGaw-Hill, New York, N.Y., 1928. Weiner, P. H., Parcher, J. R., J. Chromatogr. Sci., 10, 612 (1972).
Wilhelm, E., Battino, R., J. Chem. Thermodyn., 3, 379 (1971). Wilhelm, E., Battino, R., Chem. Rev., 73, l(1973). Yeh, S.Y., Peterson, R. E., J. Pharm. Sci., 52, 453 (1963).
Receiued for reuiew April 12,1976 Accepted June 8,1976
EXP E R IMENTA1 TECHNIQUES
Measurements of Concentration Fluctuations in Gaseous Mixtures William M. Edwards, Jorge E. Zunlga-Chaves, Frank L. Worley, Jr., and Dan L u s + Department of Chemical Engineering, University of Houston, Houston, Texas 77004
A new experimental technique is presented for measuring concentrationfluctuations in gaseous mixtures using a catalytic wire on which a mass-transfer-limitedexothermic reaction occurs. This method utilizes a catalytic sensor in conjunction with a constant-temperature anemometer unit. It should be a useful tool for studies to improve the design of gas mixing equipment and of chemical reactors in which the yield and/or conversion depend on the degree of mixing.
Information about instantaneous concentration fluctuations in gaseous mixtures is very useful for the design of industrial gas mixing equipment and of reactors in which the yield and/or conversion are sensitive to the mixing of the reactants. The techniques now available for measuring these fluctuations (Corrsin, 1949; Blackshear and Fingerson, 1962; McQuaid and Wright, 1973) are either suitable only for mixtures in which the heat transfer properties (e.g., k , C,) of the two gases are significantly different, or require the simultaneous use of several sensors. Recently, optical techniques were developed which use interferometer or crossed Schlieren optical systems to detect selective index gradients for mixtures of gases with large density differences (Wilson arid Prosser, 1971). These can be related to density fluctuations and hence concentration fluctuations for isothermal gas flow. We deskribe here a novel technique for measuring rootmean-square concentration fluctuations. The method utilizes a catalytic sensor in conjunction with a constant-temperature anemometer unit. It is applicable to gaseous mixtures containing species which can react rapidly and exothermally on a platinum wire, such as hydrocarbons and oxygen. A major advantage of this technique is that it can be applied to mixtures of gases with similar physical properties.
Theoretical Background Consider a catalytic wire whose capacity parameter, defined as the ratio of the characteristic time for changes in wire temperature to the characteristic time for surface concentration changes due to changes in the rate of mass transfer, is large. When a mass-transfer-limited exothermic reaction occurs on such a wire its temperature may be described by the equation (Edwards et al., 1974).
For wires with negligible end effects, the first term in the right-hand side of (1)can be discarded (this term is normally ignored when dealing with hot wires with length-to-diameter ratios larger than 200 (Hinze, 1959)). If in addition the wire is heated by means of an electric current, eq 1 takes the form dT ApC, - = h P ( T , - T) P(-AH)k,C J12R (2) dt The temperature of the wire can be maintained a t a constant level by using a constant-temperature anemometer to control the heating current. Therefore, the time derivative in (2) vanishes. Moreover, the transport coefficients, the limiting reactant concentration, and the electric current can be expressed as the sum of a stationary time average and a fluctuating component thus enabling the reduction of (2) into
+
+
In this equation all second-order terms have been neglected and ( ) denotes a stationary time average. Time averaging of (3) yields
( h ) ( T- T g )= (-AH)( k , ) ( C ) + J ( I ) 2 R / P
(4)
When no electrical heating is used
T* - T , A (-AH)( k c ) ( C ) / ( h ) (5) and T* - T, is defined as the adiabatic temperature rise. Subtracting (4) from (3) and division by (5) yields
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
341
Table I. Extinction Concentrations and Lower Explosive Limits for Various Reactants in Air
where
T * - T , - R*-Rg m= T-T* R-R*
(7)
Extinction concn, mol %
and
Reactant
J(I)2R =1 P ( h ) ( T- T*) For flow normal to a cylindrical wire the heat and mass transfer coefficients may be approximated by (Treybal, 1968; P 63)
Ammonia 3.9 Methane 1.5 0.8 Butane Hydrogen 1.3 Values from Steere (1967).
N N =~c1
+C
~ N Npr0.31 R ~ ~
= c1
+C
~ N NR s ~ ~ ~ ~ . ~ ~(10)
NSh
(9)
where C1, C2, and n are experimentally determined constants. When the transport coefficients and the velocities in (9) and (10) are expressed as the sum of time-averaged and fluctuating quantities, and the result is substituted into (6) the following equation is obtained
where (12)
L.E.C.,” mol % 16 5
1.9 4
heating. Typical values of these bounds for several mixtures of reactants in air are reported in Table 1. Experimental System a n d Procedure A catalytic wire probe similar to that described by Edwards et al. (1974, Figure 3) was used to measure the point concentration fluctuations. The sensor consisted of a 6 mm long by 0.025 mm diameter platinum wire (lld = 240) supported by two 3.2-mm diameter brass rods. For the aspect ratio used it was assumed that the conduction end effects were small, in accordance with the normally accepted practice in hot wire anemometry. Constant wire temperatures were attained by connecting the probe to a Thermo Systems Model 1010A constant-temperature anemometer. This unit provided a convenient method for maintaining desired values of the overheat parameter, defined as
Squaring and time averaging (11) yields
m = (R* - R,)/(R - R*)
(7)
m = (R - AR - R,)/AR
(19)
overheat resistance = (R - R*)
(20)
or
where It follows from eq 14 that by operating the constant-temperature anemometer a t three different wire temperatures (values of m ) the resulting simultaneous equations can in principle be solved to determine Urrm,/(U ) , (U’C’)/ ( ( U ) ( C ) )and C’,m,l(C), where the subscript rms (rootmean-square) refers to the square root of the time averaged square values that appear in eq 14. Equation 14 can be rewritten as
where
u‘rmsc’rms The correlation coefficient RTJC is bounded between -1 and +1. Consequently, for B < 1 l-BIFIl+B
(18)
This suggests that if B can be made sufficiently small, the parameter F in eq 15 may be taken as unity and C’,,$( C ) can then be determined from a single measurement of Efrms/(E). This condition can be met if the system is operated with a large value of m. Note that the catalytic probe technique described above should not be used for mixtures in which the average reactant concentration exceeds the lower explosive limit (LEL). On the other hand, the limiting reactant concentration should not be lower than the extinction concentration below which the reaction cannot be sustained on the sensor without electric 342
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
AR
The instantaneous anemometer bridge voltage was conditioned by filtering out all frequencies above a predetermined cutoff frequency of fc with a Krohn-Hite Model 3750 low-pass filter and then recorded on magnetic tape using a HewlettPackard Model 3960 FM recorder. The signal was subsequently processed by digitation a t a sampling rate of 2fc to obtain ( E ) and EJrm,.A calibration of (E) vs. AR was utilized in interpreting the tape recorder signals. While the above procedure was found to be the most accurate, reasonably good values could also be obtained by direct measurements without tape recording. In this case the instantaneous bridge voltage was passed through a dc offset unit (two operational amplifiers in series) to remove the dc component of the original signal. The fluctuating quantity was then passed through a band-pass filter and finally connected to a Thermo Systems Inc. Model 1060 RMS voltmeter. This procedure enabled a simultaneous measurement of both ( E ) and E’,,, and these values agreed to within 6% of those determined by processing of the tape recorded data. To determine the maximum error in the concentration fluctuations prediction of eq 15 (single point procedure), it is necessary to estimate values of the parameter E via eq 16. To do this, an independent measurement of the turbulence intensity, U’,,J( U ) ,was made using the constant-temperature anemometer unit and a Thermo Systems hot film sensor. In practice this measurement can be made with the catalytic wire by operating a t a sensor temperature below the ignition temperature or with a nonreactive mixture. Initial experiments indicated that the electrical resistance of the platinum sensor could increase up to 10% after several hours of operation due to a progressive roughening of its surface. To eliminate this effect, the wire was first pretreated
The experiments indicate that the proposed technique is suitable for a rapid evaluation of mixing characteristics of gaseous streams. Its applicability to a variety obsystems, such as mixtures of hydrogeh, ammonia, and hydrocarbon species in air, should make it useful for evaluation of industrial mixers and in scale-up of reactors in which fast chemical reactions occur. 0
3 f
.05
4
2
-:e ,"I" 0
Ckvs BY ASPIRATING
PROBE, (H,-N,)
Figure 1. Comparison of C'RMS/(C)measurements by t h e catalytic probe w i t h those made by t h e aspirating probe. (T,= 24 "C, channel Reynolds n u m b e r = 4000).
(activated) for a period of 8-10 h in a stream of 3 % hydrogen ~ in air. During this period the resistance reached an asymptotic value and did not change for several weeks. Experimental Results In order to test the validity of the technique and its accuracy, measurements of concentration fluctuations were made in a mixture of hydrogen and air. These results were compared with those for the nonreactive hydrogen in nitrogen mixture. The latter measurements were carried out with a Thermo Systems Model 1441 aspirating probe in conjunction with a Thermo Systems Model lOlOA constant-temperature anemometer. The aspirating probe has been described by Blackshear and Fingerson (1962) and a discussion of its applicability was presented by Edwards (1973). The measurements were carried out in the rectangular (25 X 152 mm) flow channel described by Edwards et al. (1974). A wide range of hydrogen concentration intensities, C$,'( C) , were obtainable by using eight different angles of hydrogen injection relative to the main (air or nitrogen) stream, as shown by Edwards et al. (1974). The hydrogen was injected into the main stream ( N R=~4000, T , = 24 "C) at a rate corresponding to a final average mixture concentration of 2 . 5 % ~ for all experiments. After measuring the hydrogen in nitrogen concentration fluctuations, a stream of air was substituted for the identical flow conditions ( N R=~ 4000, T , = 24 "C), and the aspirating probe was replaced by the catalytic wire probe. The chemical reaction was then initiated on the catalytic sensm and a series of measurements were made at each of the injector angles used above. In all the experiments the signal was conditioned by filtering out all frequencies above 50 Hz. Values of the hydrogen concentration fluctuation intensity were calculated using eq 15 with F = 1. The results of these two sets of measurements are summarized in Figure 1, and they indicate that there was good agreement between the two independent measuring techniques. Independent measurements of the turbulence intensity at the catalytic wire station with no hydrogen injection showed U ) = 0.11, and that values off = 0.31 and n = 0.5 that UfrmS/( could be employed for estimating the error term B , eq 16. The factor CY was calculated from eq 13 using C1 = 0.43 and C2 = 0.532 (Treybal, 1968), in conjunction with the experimentally chosen values of m. When the values of B were calculated for each experiment it was found that the largest value of B was 0.12, and the average of B for all experiments was 0.07. Thus the average error in Crrn8/( C ) due to the assumption that F = 1,was of the order of 7%.
Nomenclature A = wire cross-sectional area, cm2 B = parameter defined by eq 16 C = reactant concentration in fluid, g-mol/cm3 C, = heat capacity, cal/g K d = wire diameter, cm D A B = binary diffusion coefficient, cm2/s E = bridge voltage, V f = parameter defined by eq 12 F = parameter defined by eq 15 h = heat transfer coefficient, calls cm2 K AH = heat of reaction, cal/g-mol I = wire current, A J = conversion factor, cal/s W k = thermal conductivity, cal/s cm K k , = mass transfer coefficient, cm/s m = parameter defined by eq 7 n = empirical velocity exponent in eq 9 N N " = Nusselt number, hdlkf Np, = Prandtl number, C,fvf/kf N R =~ Reynolds number, UdlvfNsc = Schmidt number, U ~ / D A B N s = ~ Sherwood number, k,d/DAB P = wire perimeter, cm R = wire resistance, ohms Ruc = correlation coefficient (eq 17) t = time, s T = temperature, K U = gas velocity, cm/s x = distance along the wire, cm
Greek Letters cy = parameter defined by eq 13 p = viscosity, g/cm s u = kinematic viscosity, cm2/s p = wire density, g/cm3 Subscripts f = fluid g = gas phase rms = root-mean-square w = wire Superscripts = fluctuating component * = with reaction only L i t e r a t u r e Cited Blackshear, P. L., Fingerson, L.. ARS J., 1709 (1962). Corrsin, S.,NACA Technical Note 1864 (1949). Edwards, W. M., Zuniga-Chaves, J. E., Worley, F. L., Luss, D., AIChEJ., 20, 571
(1974). Edwards, W. M., Ph.D. Thesis, University of Houston, 1973. Hinze, J. O.,"Turbulence", McGraw-Hill, New York, N.Y., 1959. McQuaid, J.. Wright, W., Int. J. Heat Mass Transfer, 16, 819 (1973). Steere. N. V., Ed., "Handbook of Laboratory Safety", The Chemical Rubber Go.. Cleveland, Ohio, 1967. Treybal, R. E., "Mass Transfer Operations", McGraw-Hill, New York, N.Y.,
I 968. Wilson, L. N., Prosser, D. W., Proceedings of Symposium on Turbulence Measurements in Liquids, 197 l.
Receiued for reuzeu! October 10,1975 Accepted M a y 24, 1976 T h e financial support of t h e N a t i o n a l Science F o u n d a t i o n through grant GK 18226 is gratefully acknowledged.
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