Ind. Eng. Chem. Fundam. 1982, 21, 76-83
76
Suppose sl,s2, ..., qW are the mechanistic steps in any such system, and suppose zl,...,zc is a basis for the space of cycles. Then there exists a nonsingular S X S matrix (17) which defines the change of the basis sl,s2, ...,ss into 0 0 (17)
7c1 YCZ"'
TC(S-C)
YC(S-C+I)"'
YCS
the basis sl, ..., qs4),zl,..., zc. Obviously the C X C submatrix in the lower right comer is nonsingular, which shows that any maximal cycle-free subsystem such as sl, s2, ...,sS4 can be constructed by the method described in section 4. Conversely, any subsystem constructed by this method is cycle-free, because the nonsingularity of the C X C submatrix guarantees that every nonzero cycle contains some of the mechanistic steps which are not in the
subsystem. Therefore, no cycle can be formed from the steps of the subsystem. Literature Cited Ark, R.; Mah, R. H. S. Ind. Eng. Chem. Fundam. 1963, 2 , 90. Blijmbom, P. H. AIChE J . 1977, 2 0 , 285. Happel, J.; Suzuki, J.; Kokayeff, P.; Fthenakls. V. J . Catal. 1980, 65, 59. Horlutl, J.; Ikuslma, M. h e . Imp. Aced. (Jpn.) 1939, 15, 39. Horiuti, J.; Nakamura, T. 2. W y s . Chem. (Frankfurt am &In) 1957, 1 1 , 358. Horluti, J.; Nakamura, T. A&. Catel. 1967, 77, 1. Horlutl, J. Ann. N . Y . Aced. Scl. 1973, 213, 5. Lyuberskll, G. D. MI. Nauk 1956, 110, 112. Mllner, P. C. J . Electrochem. Soc. 1964, 7 1 1 , 228. Mlyahara, K. J . Res. Inst. Catal. W a M O Unlv. 1969, 17, 219. Sellers, P. H. Arch. Ration. Me&. Anal. 1971, 4 4 , 23. Sellers, P. H. Arch. RaHon. Mech. AM/. 1972, 4 4 , 376. Smith, W. R. Ind. €ng. Chem. Fondem. 1880, 79, 1. Smith, W. R. In "TheoreticalChemistry. Advances and Perspectives"; Vol. 5, Academic Press: New York, 1980. Ternkin, M. I . Int. Chem. Eng. 1971, 1 7 , 709. Temkin, M. I. Ann. N . Y . Aced. Sei. 1973, 273. 79. Ternkin, M. I. A&. Catal. 1979, 28, 173.
Received for review April 17,1981 Accepted October 1, 1981
EXPERIMENTAL TECHNIQUES Determination of Mfusion Coefftcients by Frequency Response in Taylor Flow Godfrey A. Turner' and Mathew S. Chong Department of Chemical Englneering, University of Waterloo, Waterloo, Ontario, Canada, N2L 301
The determination of diffusion coefficients in iiqulds is facilitated by measuring the longitudinal dispersion coefficient in flow under Taylor-Aris conditions, the relation between these coefficients being then exact. However, the use of a sudden change of concentration at the inlet gives rise to difficulties, namely: (a) often both a long tube and a long time are needed; (b) the concentration varies widely during a run; (c) measurement errors are magnified in the computation of moments; (d) truncatbn errors arise from the finiteness of the time; (e) the influences of an imperfect pulse, boundary effects, and in situ detectors are difflcuit to overcome. In contrast, a continuous harmonic concentration variation not only overcomes the above difficulties but also eases the Taylor-Aris conditions. In particular, there is no restrictkn on tube length. To demonstrate the method, the diffusion coefficientof potassium chlorlde in water at 18 O C was measured. Mean and standard deviation were 1.46 X and 0.422 X respectively, in units of cm2 s-'.
Introduction Parameter Determinations. The determination of the value of a mutual diffusion coefficient is here taken to be an example of finding the value of a quantity in a mathematically defined, deterministic system by imparting a disturbance to it and measuring the response. The system is assumed to involve a fluid, and the experimental process might be either of two: in one the fluid flows, in the other it does not. Since molecular diffusion in liquids is extremely slow, the use of a flowing medium can, when coupled with appropriate hydrodynamic conditions, give rise to the much larger dispersion coefficient. Measurement of the latter is the subject of this paper. It is simple in concept but not in practice and is an example of a 0196-4313/82/1021-0076$01.25/0
general phenomenon, namely that of traveling waves. In the present case the disturbance that sets off the wave is of concentration but there are close analogies to other, more familiar, traveling waves, such as those of sound or electromagnetism (Moore, 1960; Turner, 1972). Thus the disturbance is a function of time and distance, of the form f(z f u t ) , while attributes of waves such as attenuation, reflection and transmission at an interface, and the impedance of the medium are also common to all wave processes. General Process. The general procedures of deducing values of a parameter from measurements of the propagation of concentration waves start from the assumed mathematical statement of the process. Analyses subse0 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Voi. 21, No. 1, 1982 77
B : Boundary
: Transducer
1
0 2E n =
Flow TDZZ
Source
z,
I
l
l
I
l
l
22
2,
Measurement
26
Figure 1. (a) Physical arrangement of a generalized measuring system. (b)Ekpivalent traveling-wave system. Numbers correspond to those in Figure 2. (2= longitudinal impedance; s m-l)
b (To mtxerM.Fig2)
Figure 3. Schematic of sinusoidal concentration generator. (Letters correspond to those in Figure 2).
Region 0-1 i s sine-wave aenerator I Fia. 31 a
Head adjustable
To Droin
head of liquid
1
2s
24
I
I I
Detector In Cl
Capillary TS
Detector out CZ
To Drain
Figure 2. Schematic of the experimental flow system. (Numbers correspond to those in Figure 1;letters correspond to those in Figure 3).
quently possible have been summarized by Turner (1972). Thus, curve fitting may be done in the time, Laplace, or frequency domains, and the values of n parameters found from n measurements in any of these domains. Alternatively these parameters can be found from n moments, calculated from a distribution of concentration vs. either distance or time, usually from an initial change that approximates either a step change or to a Dirac impulse. A third method uses the frequency response at n frequencies. (Thisresponse may be obtained by using either an impulse or a steady cyclic disturbance). Apart from the question of which method is most desirable mathematically-a matter of some debate, (e.g., Himmelblau, 1970)- the magnitudes of experimental errors must be taken into account. Whatever the method of analysis, an apparatus must be set up to correspond as closely as possible to the mathematical equations that can be solved; a general apparatus is shown schematically in Figure la. It would be desirable if this were, longitudinally, both infinite and homogeneous. In practice there must be entry and exit sections, and generators and detectors of concentration changes. In the main section between boundaries B3 and B4, although many physicochemical proceases may go on, only molecular diffusion and Poiseuille fluid flow are assumed here. The different zones are delineated both in Figure l a and in the equivalent Figure lb. The latter treats the concentration as a wave traveling in a wave-guide made up of different sections (labeled to correspond to Figure la) of varying values of the longitudinal impedance 2. This impedance is a (complex) ratio of complex amplitudes of what, in the jargon of system dynamics, are termed through variable and across variable. The latter, in the present case, are respectively concentration and mass flux and the dimensions of these impedances are [ r ] [Ll-l. Use of the impedance allows concepts of wave travel to be
taken over directly and applied to this diffusional process, resulting in economy in the analysis of such things as boundaries and concentration distributions. In the system of Figure l a the physicochemicalprocesses of diffusion, dispersion, and fluid flow exist. Of these the average velocity can be found from the volumetric flow rate, while the value of the diffusion coefficient can be found, once the dispersion coefficient is known,from the Taylor relation. Hence only the dispersion coefficient has to be measured. That is to say, n = 1 here and so measurements at one frequency would suffice. If there is either no flow or steady flow then this becomes a special case of the above with regard to parameter determination, but measurements in this system are difficult, uncertain, and tedious. (Glack’s (1924) observations took 12 days). Measurements in a Flowing Medium. Measurements of physico-chemical parameters, ranging in number from one to five, have been made in flow systems over at least the past 30 years. However, most have been made in packed beds or porous solids, and thus any flow would be unorganized and often turbulent, while measurements in laminar-flow Taylor diffusion have been far fewer. In the first and best-known example, Taylor (1953) derived the relation between dispersion and diffusion coefficient and measured the diffusion coefficient of potassium permanganate in a straight capillary tube. Concentration was measured by light absorption, the input was assumed to be a Dirac impulse, and the parameter was determined by curve-fitting the distribution of concentration vs. distance. This distribution is symmetrical, as opposed to a concentration-time curve, which is skewed, but the latter is experimentally easier. In a more recent example Pratt and Wakeham (1974) used an impulse to measure the diffusion coefficients of alcohols in water, using Taylor flow in a 13.645-m coiled capillary tube. Subsequently, Pratt and Wakeham (1975) reexamined the many assumptions made, particularly of the effect of coiling the tube and of disregarding the skewness of the distribution. Experimental measurements were made of the diffusion coefficients of propanol and water. The input impulse was not monitored and the measurements of concentration were made by an off-line refractometer. Effects of impedance mismatches at inlet and exit were ignored. Turner (1958,1959) used Taylor diffusion to determine structural parameters in more elaborate systems. In one (model 1) a flow-channel had a distribution of dead-end pores of various volumes and lengths communicating with
78
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982
it. In the other (model 2 ) there was a distribution of flow channels of various lengths and diameters in parallel. The purpose was to determine these two distributions of length vs. diameter, and so a solution of nitrobenzene in kerosene was used, the concentration of which could be measured in situ by its dielectric constant. The molecular diffusion coefficient had been determined separately and the radial Peclet number calculated for the appropriate conduit. These were a rectangular slot for model 1 and a cylinder for model 2. The perturbation was sinusoidal and both inlet and outlet signals were monitored to determine the overall dispersion, while the attenuating effects of impedance mismatches and measuring cells were obviated by measuring the responses between inlet and outlet cells, both with and without the test section between them (McHenry and Wilhelm, 1957; Turner, 1959, 1971). Uncertaintiesin Flow Systems. Apart from the error arising from the fact that the initial perturbation may not correspond to the postulated one, there is also the consideration that, in the case of an impulse, the finding of the values of the second (and higher) moments requires measurements over infinite time or infinite distance. In practice those values would have to be truncated, and noise would be significant as the signal decreases. Furthermore, errors of measurement would be much magnified by having to be multiplied by the square, cube...of either the time or the distance. The use of measurements in the Laplace or Fourier domains may obviate the latter errors but introduces other disadvantages (Turner, 1972, p 174). The uncertainty due to the lack of ideality of an impulse could in principle be removed by measuring its moments before and after it is affected by the dispersive process. However, in addition to the above difficulties of using moments it may be difficult to get a sufficiently fast measuring device. Furthermore, if it is an in situ sensor then the perceived impulse will degenerate for those two reasons. The situation is thus one in which the value of a diffusion coefficient is to be deduced from the comparison of concentration measurements separated by time or distance. Furthermore, boundary effects a t longitudinal discontinuities as for instance between the impedances in Figure l b have to be recognized. In traveling-wave terminology these are called impedance mismatches. Many measurements of physicochemical quantities, including diffusion coefficients, have been made in which neither comparison of inlet and outlet wave shapes nor the effect of discontinuities were taken i n b account. The errors so introduced may have been small but they would need to be shown to have been so. Continuous Harmonic Waves. In contrast to the above, sine waves of concentration have many advantages. These are, in general, as follows. (i) If the waves are impure, harmonics can be removed by a low-paas filter, viz., a mixing device in the flow. (ii) The effects of the measuring sensors and (wave) discontinuities in the system can be removed easily by a supplementary measurement discussed below. (iii) Concentration changes can be kept small but the mean concentration can be maintained at any desired value. (iv) The measuring process is of short duration. For the measurement of one parameter, the diffusion coefficient,the mathematical model is that of Taylor flow. There is steady, one-dimensional laminar flow in a cylindrical conduit at an average velocity u. The longitudinal dispersion coefficient D* is expressible in terms of the conduit dimensions, velocity and molecular diffusion coefficient. Radial concentration is uniform. In addition,
impedance discontinuities exist at z = O* (at boundary B3) and at L+ (at boundary B4). The negative and positive subscripts indicate locations just upstream and downstream, respectively, of the boundary. A concentration change exists a t z = 0-, having been both injected and measured (or assumed) upstream of this location. At z = O+ the transmitted portion of this signal progresses, the magnitude being the vector difference between incident and reflected waves. Similarly, the wave at z = L+ is a vector difference between incident and reflected waves at boundary B4, and the measurement of the traveling wave is then made downstream of L+. The statement of what happens at either boundary, z = O+, L+, is here taken to be Wehner and Wilhelm's (1956) boundary conditions, which can also be related to the first and second telegrapher's equations of traveling waves. From these the fraction of the wave reflected can be computed (Turner, 1971, 1972). For the concentration c(z, t ) there arises an equation which can be considered either a wave equation or a continuity equation for flow with an average velocity u. D*(d2c/dz2)- u(&/&)
- (&/at) = 0
(1)
There can be changes of cross-section and of system properties longitudinally. That is, the values of both D* and of u may vary with z but are radially uniform. Boundary Conditions. Atz=O ~C-CC- - D*o-,(a~,/az) uo+co+ - D*o+(iko+/az) ( 2 ) Atz=L uL-cL- - D*L-(&L-/&) = uL+cL+ - D*L+(&L+/&) (3)
Solution. A forcing function c(z, t ) = C(z) cos wt in eq 1 gives a solution in the steady cyclic state
C = A exp(ylz) + B exp(y2z)
(4) A and B are arbitrary constants and y1 and y2are wave propagation coefficients. They are functions of the attentuation coefficient a and the phase coefficient 6. The two latter quantities are "1
= (u/2D*)(1- [(F + 1)/2]1/2)
(5)
"2
= ( ~ / 2 D * ) (+l [ ( F + 1)/2]'12)
(6)
/3 = (u/2D*)[(F- 1 ) / 2 ] 1 / 2
(7)
where
F = [l + ( ~ w D * / u ~ ) ~ ] ' / ~
(8)
and so y1 = a1-
y2 = a1
i B = u/2D*
-
[ ( u / ~ D *+) i~~ / D * ] l (9) /~
+ i/3 = u/2D* + [ ( u / ~ D *+) i~~ / D * ] l / ~(10)
See Turner (1972),pages 12 and 101. Subscript 1 indicates the positive-going wave (i.e,, with the fluid flow), while subscript 2 indicates the opposite, negative-going, wave. In eq 4 the arbitrary constants A and B will depend on the system, particularly on the nature of the boundaries at B3 and B4. The disturbance, still being treated as a traveling wave, will give rise to reflections a t these boundaries and the vector magnitude of the reflections will depend on the relative impedances on both sides of the boundaries (Turner, 1971) at z = 0, L. Thus the sinusoidal concentrations, as measured upstream of z = 0 and downstream of z = L, will be affected both by the presence of the sensors and by these boundaries; the magnitudes of the effects are generally unknown. However, the effects
Ind. Eng. Chem. Fundam., Vol. 21, No. 1, 1982 70
of both may be removed by the simple expedient of measuring the responses over two different lengths L1and L2 and using the vector difference over a length (L2- L l ) (McHenry and Wilhelm, 1957; Turner, 1959). Hence if Lz - L1= (AL)and C2and C1are the complex amplitudes at L2 and L1,involving the scalar values C1and C2 a t those locations and a phase angle 4 between them C2/C1 = expyl(AL) = expa(AL) expi@(AL) (11) provided the reflected wave from either of the downstream boundaries does not reach the upstream one, only al,denoted now by a,will now be considered. This is an easy condition to satisfy for a flow system (Turner (1972), p log), because the negative-going wave dies out in a very short distance. The components of the vector ratio C2/C1in eq 11can be expressed in terms of the properties of the system (Turner (1972), pp 101,177). These components are II (E In (Cl/CJ) and 4, the phase angle between waves at L2 and L1.Then for this system In (C2/C1) II = (P”/2)(1 - [([l + 4 ( u V / P r ) z ] 1 /+2 1)/2]’/’) (12)
Table I. Relations between Dispersion and Diffusion Coefficients, with Conditions [Aperiodic] expression for D*/D
conditions
L/R > P17.3 L/R S P / 4 P S 6.9 1 t P2/48 LIR > PI15 P2/48 P > 100 P2/48
Dt/R’ 2 0.8 1 t P2/48 P < 100 Dt/R* > 1.0
references Taylor (1953) Taylor (1954) Aris (1956) Ananthakrishnan et al. (1965) Reejhsinghani et al. (1966) Ananthakrishnan et al. (1965) Reejhsinghani et al. (1966)
a Symbols: P = radial Peclet number = uR/D; R = tube radius; L = a length in direction of flow, defined by the cited authors; t = time, defined by the cited authors.
Table 11. Relations between Dispersion and Diffusion Coefficients, with Conditions [Periodic] expression for D*/D
conditions
references
and
4 = Pr/2([(1 + 4 [ 2 N / P r ] 2 ) 1 /-2 1]/2)1/2 (13) where N = frequency number = ALw/u and P” = a longitudinal Peclet number, viz., P” = ALu/D*. If the values of w , AL,and u are known, then either or both of eq 12 and 13 can be used to calculate P’. If, on the other hand, these values are not known they can be found by taking measurements at an increased number of frequencies (Turner (1972), p 176). The expressions for the response of the sine wave are always the same, viz., eq 12 and 13. Even when the only unknown is the diffusion coefficient, measurements at different frequencies will allow leasbsquares regressions to be made. Concentration changes in sine waves can usually be made so small that there are no appreciable changes of density or average velocity. The above equations allow the value of D* to be found. Now the relation between D and D* is needed. The Relation Between Dispersion Coefficient,D*, and the Diffusion Coefficient, D . Stemming from Taylor’s (1953,1954) papers on the expression for D* for laminar flow in a cylindrical tube, a number of authors have examined the relations and the conditions required for them to obtain. Thus Aris (1956) generalized Taylor’s paper to conduits of arbitrary cross section and dealt with the momenta of a degenerate impulse, in both one-dimensional space and in time. Aris (1958) dealt with the case of a conduit communicating with cylindrical dead spaces (Turner Model 1). Ananthakrishnan et al. (1965), Gill and Ananthakrishnan (1966), and Reejhsinghani et al. (1966)dealt with the conditions in a cylindrical conduit in more detail. All of these authors considered only the case where the initial perturbation was a Dirac 6-impulse. Their conclusions are summarized in Table I, which gives both the relations between D* and D and the authors’ conditions for these relations to hold. In contrast, the case where the perturbation is a steady cyclic one has aroused much less interest. Carrier (1956), Philip (196% b) and Chatwin (1973) have examined it and their results are given in Table 11. Determination of the Diffusion Coefficient, D. In this present case of Taylor flow, the expression that will be used is, from Table I1 D = u2R2/(48D*) (14)
LPIR > 0.1 LPIR S (1112.85) for n -t 0 a n d P - t
Philip (1963a) Philip (1963b) Philip (1963a,b)
oa
( 1/48)P2
a t 1
Chatwin (1973)
wD/u’ 4 1 = wR’/D; a, = eigenvalue of the zeroth a Symbols: term in the eigenfunction expansion for the periodic concentration.
This, however, has to be subject to certain conditions which are given now and in Table 11. Conditions for Taylor Flow. The condition that the direct contribution of D to D* may be ignored is condition 1:
P >> 6.9
(15)
Again, the condition ensuring that radial diffusion renders the radial concentration gradient very small is, from Table I condition 2a (for an impulse):
P
