Ind. Eng. Chem. Fundam. 1983, 22, 269
In our original paper the subscript i was lost in eq 3, and consequently we used the rate expression given below
r(C,C,) = kC,2 - ( k / K ) C , C
(4)
Clearly, the effect of dropping the subscript i in eq 3 is to use the current polymer concentration, instead of the initial polymer concentration, as the driving force for the reverse reaction, and, since C, < Cpi, our earlier analysis should have predicted values for the polymer concentration and degree of polymerization that are too large. Warner (1981) had, in fact, pointed out to us some time ago the incorrectness of eq 4 and on the basis of his comments we then computed solutions to the revised equations. Indeed, our revised computations yielded values for the degree of polymerization at the exit of the reactor which were smaller than the ones we reported earlier. However, i n all cases t h e qualitative trends were f o u n d t o be identical.
269
We are unable to comment on the quantitative result for the degree of polymerization at the reactor exit, E, = 52.7, given by Gupta et al., since the Gupta model incorporates an additional dimensionless parameter and is philosophically different from our model. The Gupta model takes the wiped film to be of finite thickness, whereas our model considers the film thickness to be infinite in relation to the penetration depth for mass transfer. Literature Cited Amon, M.; Denson, C. D. Ind. Eng. Chem. Fundam. 1980, 19, 415. Warner, G. Department of Chemical Engineering, University of Delaware, private communcation, April 1981.
Department of Chemical Engineering University of Delaware Newark, Delaware 19711
C . D. Denson* M. Amon
Comments on “Determination of Diffusion Coeff icients by Frequency Response in Taylor Flow” Sir: In the paper by Turner and Chong (1982),a method is suggested for the determination of diffusivity which is based on Taylor analysis of axial dispersion of concentration disturbance in laminar flow of a liquid where an introduction of continuous harmonic variations of concentration is the new element of the method. A basic motivation consists of avoiding the problems which are bound to an experimental realization of a sudden change of concentration, the response to which serves as a basis for evaluation of diffusion coefficients. This was realized in similar work by Taylor (1953,1954)Pratt and Wakeham (1974,1975). We wish to emphasize three important points with regard to the paper by Turner and Chong which are in the closest agreement with the motivation formulated by them. 1. Besides the works by Taylor and Wakeham and Pratt, the measurements of diffusion coefficients in laminar flow by methods which utilize a jump change or Dirac function-type change in concentration as an input and analyze the response to it by Taylor analysis of the resulting convective diffusion were performed by a whole series of authors in both liquids (Bailey and Gogarty, 1962; Ouano, 1972; Koniyama and Smith, 1974; Baldauf, 1979; HanEil et al., 1979) and gases (Bournia et al., 1961; Evans and Kenney, 1965). 2. In all of these works the variance of estimated diffusion coefficients was found to be smaller than that in the Turner-Chong paper. HanEil et al. used practically the same flow rate and inner diameter of the capillary with a fixed length of about 3 m. That means that the times of the experiments were approximately the same as those in the Turner-Chong paper. However, the relative deviation of the results by HanEil and co-workers was smaller than one-tenth of the value given for a single system in the Turner-Chong paper. Problems with experimental realization of a jump change (in this case with Dirac function-type pulse) were avoided by the use of a wellknown design of an injecting device similar to that which is frequently used in liquid chromatography. Minute and precise amounts of a sample can be instantaneously injected or better displaced into a laminar stream of a carrier liquid by means of a six-way valve with a sampling loop. Also, other problems with boundary effects and in situ detectors were suitably solved. Using a highly sensitive 0 196-43 13/83/1022-0269$01.50/0
differential refractometer, the response can be recorded even for minute concentration differences between the main flow of the carrier liquid and the injected sample. Hence differential diffusivities can be measured. In the same paper by HanEil and co-workers a statistical estimation of parameters is used which yields an estimate of diffusivity in the procedure occupying the IBM 370 computer for several minutes. No errors due to moment methods are encountered. Comparing experimental simplicity and the reliability of the experimental alternatives of determination of diffusion coefficients in the cited works, one can conclude that the Turner-Chong method does not compete successfully with other mentioned methods which were verified for more systems in a much broader range of experimental conditions. 3. It is also worth mentioning that the diffusivity of KC1 determined by Turner and Chong with unacceptably high relatiave standard deviation of about 29% (0.004 N, 18 “C) = 1460 pm2/s) differs more than 10% from the value which follows from Harned’s measurements as presented in the Landolt-Bernstein tables. The majority of the cited authors claim that the experimental error of the methods of this class is about 2-3%. The affirmation of Turner and Chong, “If there is either no flow or steady flow then this becomes... measurements in this system are difficult, uncertain and tedious” seems to be fair with respect to the authors of the works on the improved version of measurements of diffusivities in a stagnant liquid column in a capillary (Luk et al,, 1975; Gosman and Tuchlor, 1981). I t is necessary to say that even in these cases the results obtained were more accurate. Maximum error was about 5 % . The list of works cited above should not be considered as a critique of the completeness of the literature cited in the Turner-Chong paper, but it should show that the main motivation of the work seems to be artificially sophisticated. The motivation could be considered as at the origin of Chong’s M.S. thesis (1974), which can be regarded as a basis of the Turner-Chong work. However, in 1982 it suggests a certain underestimation of the results obtained by the cited authors. Literature Cited Bailey, H. R.; Gogarty, W. B. Proc. R . Soc.London. Ser. A lg82,259, 352.
0 1983 American Chemical Society
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Ind. Eng. Chem. Fundam. lB83, 22,270
Baldeuf, W. Bestimmung von Dlffusionkoefflcienten in Testgemischen fik die Fliissig-Fkissig Extraction, Arbeitsbericht zum DFG ForschungvorhabenKn 122/25. July 8, 1979. Baldauf, W. Ph.D. Thesis, Technlsche Unlversltlit Berlin. 1981. Bournia, A.; Coull, J.; Houghton, G. Proc. R . Soc. London, Ser. A 1961, 281, 227. Evans, E. V.; Kenney, C. N. Roc. R. SOC.London, Ser. A 1965,284, 540. Chona, M. S. M.A.Sc. Thesis, Unlversltv of Waterloo. 1974. Gosnien, A.: Tuchlor, D. Radiochem. Radioanal. Lett. 1961, 47. 19. HanEil, V.; Rod, V.; Rosenbaum, M. Chem. Eng. Commun. 1979, 3, 155. Konlyama, h.; Smith, J. M. J . Chem. Eng. Data 1974, 19, 384. “Landolt-BiKnstein Zahlenwerte und Functionen aus Physik, Chemie, Astronomie, Geophysik and Technlk: Transportphhnomene I (Viskosltlit und Diffusion)”; 5. Tell, Bandteii a; Schifer K., Ed.; Springer-Veriag, Berlin, 1969. Luk, C. Y.; Nanls, L.: Lltt, M. Ind. Eng. Chem. Fundam. 1975, 74, 92.
Pratt. K. C.; Wakeham, W. A. Roc. R. SOC.London, Ser. A 1974, 336, 393 1975, 342, 401. Ouano, A. C. Ind. Eng. Chem. Fundem. 1972, 1 1 , 268. Taylor, (3. I.R o c . R . Soc. London, Ser. A 1913, 219, 186 1954 225, 473. Taylor, G. I.Proc. h y s . Soc.1915, 47, 857. Turner, G. A.; Chong, M. S . Ind. €ng. Chem. Fundem. 1962, 27, 78.
Institute of Process Chemical Fundamentals Czechoslovak Academy of sciences Prague, Czechoslouakia
V. HanEil* V. Vacek‘
’ Institute of Inorganic Chemistry.
Response to Comments on “Determlnatlon of Diffusion Coefflclents by Frequency Response In Taylor Flow”
Sir: We thank Hancil and Vacek for their valuable comments. The paper was intended to draw attention to a method of determining diffusion coefficients which, while not new in principle, seems to have been overlooked by others who have used Taylor flow. In the authors’ eyes a sinusoid is more satisfactory than a pulse or step because the physical system can be made to correspond rigorously to the defining equations and boundary conditions. There are thus fewer uncertainties. There is also greater experimental freedom both because of the absence of a tube-length condition and because of the availability of another parameter &e., frequency) that can be adjusted to suit the circumstances. The example quoted in the paper was intended to give experimenters an idea of the apparatus and procedure to be used. Potassium chloride was chosen because its diffusivity was known and because its concentration could be easily measured and recorded by electrical means-or so it was thought. The scatter in the results was discussed in the original manuscript but excised in the interests of brevity. The main cause was thought to be the unexpectedly large drift of the measuring-cell electrodes, a fact only realized toward the end of a master’s degree course. Ideally, the measurements should have been repeated, but it was considered that the results would serve the purpose of demonstrating the technique. It should be appreciated that these measurements were spread over about five weeks and that the results represented ten combinations of five tube lengths (counting zero length), using four frequencies per length and at least three
cycles per frequency. On the other hand, if a similar structure of measurements to those of, say, Hancil, Rod, and Rosenbaum had been adopted, namely by using four cycles of one sinusoid, with nothing else changed, the scatter would have been equal to or better than that claimed by these authors. With regard to other specific comments by Hancil and Vacek and their references to the paper by Hancil et al., the following replies may be made. (a) The sensitivity of the detection system is a matter common to both sinusoids and pulses, of course. On the other hand, for a given sensitivity and noise level the ratio of concentrations, namely the maximum at the inlet to the minimum at the exit, is smaller for a sinusoid than for a pulse, and the more the input signal of a pulse approaches the theoretical ideal of a Dirac impulse the worse the comparison becomes. (b) To compensate for end and other effects in their apparatus, Hancil et al. ran a substance of known diffusivity. Back-calculation then gave an “effective* tube diameter to be used in subsequent determinations, in place of the actual diameter. It is not clear that this correction is constant under all circumstances. On the other hand, a sinusoid does not require a calibrating substance, and end effects are determined for each case. Hence the measurements are, in this sense, absolute. Department of Chemical Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3Gl
0196-4313/83/1022-0270$01.50/0@ 1983 American Chemical Society
G. A. Turner* M. S. Chong
