Table
I. Rate Constants Estimated from Error-Free Data Time Points
bl
31 5 6
1 0 1 013 0 994 1 000
True H-J-R This u o r k This work
b2
0 0 0 0
5 497 485 503
br
b3
10 10 10 9
0 125 620 855
5 4 5 4
0 990 420 962
Table II. Simulations With Random Errors
True values Five-point estimates Average Range Standard deviation Six-point estimates .Sverage Range Standard deviation
bi
bp
b3
1. o
0.5
10.0
bd
5.0
10.447 0.431 0.968 5.386 7,889 3,779 0.246 0.875 -16.673 -9,190 -1.020 -0.547 0.055
0.119
3.311
2.003
0,460 9.641 0.978 4.358 0,282 7.397 0.887 3,483 ~ 1 1 . 7 6 7 -6.210 -1.047 -0.582
I n practice one should always be cautious in interpreting the estimated rate constants since observed concentrations are seldom error-free. For the present method, as well as the original method of Himmelblau, et ai., it is extremely difficult, if not impossible, to make a rigorous statistical analysis. A practical approach is to carry out a N o n t e Carlo simulation study. The results from such a study using a sample size of 9 are given in Table I1 where errors in Clr and C2k were assumed to be independent, normally distributed with means of zero, and standard deviations amount to 2.5% of the respective true concentrations. (The covariance terms are omitted from the table.) It can be seen that the estimated standard deviations for the 6-point case are reasonably small. The proposed procedure should, therefore, give good results. At the least, the estimates provide a good starting point for a much more costly nonlinear regression approach (category 3a in Himmelblau , et al.). Acknowledgment
The author is indebted to the management of the Engineering Department, Union Carbide Corporation, Chemicals and Plastics, for permission to publish this paper. Nomenclature
0,059
0.139
1.826
1.264
on a digital computer with double precision arithmetic in solving the matrix equations. Results from error-free simulations are summarized in Table 1. The rate constants obtained by Himmelblau, et al., are also included for comparison. With the added time point t = 0.1, the 6-point case yields significantly better estimates. This indicates, as noted by Himmelblau, et al., that the estimates are sensitive to errors resulting from approximating the initial integrals. The use of 5 and 6 time points are for illustration purpose. For the present problem there are two independent observations a t each time point. The total number of observations excluding the initial time are then 8 for the 5-point case and 10 for the 6-point case. Since there are 4 rate constants to be estimated, the corresponding degrees of freedom are 4 and 6, respectively.
b, = rate constant for reaction 1. C I = concentration of component i. g = a smoothing factor in spline curve-fitting. m = number of components. n = number of reactions. s t j = stoichiometric coefficient for component i in reaction j . t = time. wI = weighting factor in spline curve-fitting. literature Cited Greville, T. ?T. E., Introduction to Spline Functions, “Theory and Applications of Spline Functions,” p 1, Academic Press,
1969. Himmelblau, D. AI., Jones, C. R., Bischoff, K. B., IND.ENG. CHEM.,F U N D A M . 6 , 539 (1967).
Y. P. TANG Engineering Deparfment Cnion Carbide Corporation Chemicals and Plastics South Charleston, K-.V a . $5303 RECEIVED for review March 2, 1970 ACCEPTED November 30, 1970
CORRESPONDENCE
Determination of Start-up Conditions for Chemical Reactor Stability SIR: I wish to raise an important consideration which appears to have been overlooked by Han (1970) in the determination of start-up conditions for chemical reactor stability. Han considers the start-up of a continuous stirred-tank reactor in which a first-order irreversible reaction A + I3 occurs. There are three possible steady states and Han has chosen the low conversion steady state to be the desired state of operation, since the middle steady state is unstable and the high conversion steady state is beyond the safe temperature limit of the reactor. 322
Ind. Eng. Chern. Fundam., Vol. 10, No. 2, 1971
The middle steady state, however, can be made stable very easily by the use of proportional control with a sufficiently large proportional control constant. This mode of control as applied to a reactor is discussed and analyzed in detail first by Ark and Amundson (1958), and then by Lapidus and Luus (1967) , mho considered the stability problems associated with a series of CSTR’s. If we look a t Han’s proposed conditions of operation, it is readily seen that his chosen steady state of CAS = 0.484 with CAO = 0.50 gives a yield of only 3.2% whereas the “un-
stable" steady state of Cas = 0.270 produces a yield of 46%. T o run a reactor, with full assurance of stability, a t a substantially higher yield by simply inserting a cooling coil and making the flow rate of cold n a t e r through this coil proportional to the temperature deviation from the desired steadystate temperature (as suggested by Aris and ;Imundson) certainly is a practical way of operating the system. Much of the uncertainty of choosing the initial starting conditions can thereby be avoided altogether.
literature Cited
Aris, R , Amundson, N. R., Chem. Eng. Sci. 7 , 121, 132, 148 (1958). Han, C. D., IND.ETG.CHEM.,F C N D ~9,M634 . (1970). Lapidus, L,, Luus, R.,Itoptimal control of E~~~~~~~~~~ Processes,'' pp 409-437, Blaisdell Publishing Co., Waltham, Mass., 1967. Rein Luus Department of Chemical Engineering CniversitYof Toronto Toronto 5, Ontario, Canada
Streaming Potential Fluctuation around a Cylinder in Water SIR: We wish to point out some of our recent results regarding the effect of free stream turbulence on the flow past, circular cylinders held transversely to the main stream in the Reynolds number range 4000 to 10,000. The free stream turbulence intensity ranged from 0.5YG(clear tunnel) to 12.5%. Contrary to the findings of Liu, Binder, and Cermak (1970), we did not find any noticeable effect of free stream turbulence on the Strouhal number over the ent'ire range of Reynolds numbers arid intensities. The experiments were conducted in the 11 X 11 inch test section of a low-speed wind tunnel. Smooth Plexiglas cylinders ranging in diameter from ' 1 4 to 1 inch were used. The shedding frequency was measured b y placing a hot-wire probe in the near wake (about 2 diameters downstream and 0.5 diameter off the axis) and autocorrelat'ing the turbulence signal in real time. The accuracy of the frequency measurement is estimated to be within 2%. Bearman (1968) has also reported absence of a n y effect of free st'ream turbulence on the Strouhal number in the subcritical Reynolds number range. T h e critical Reynolds number itself, however, is a function of the stream turbulence. T h e streaming-potential-fluctuation (SPF) spect'ra of Liu, Binder, aiid Cermak do not display the Strouhal peak a t the forward st'agnat'iori point (0 = 0 " ) in the presence of a n estimated free stream turbulence level of approximately 2.75%, while our hot-wire autocorrelat,ion nieasurements in the vicinity of the forward stagnation point (l/lo diameter upstream) did exhibit the Strouhal frequency up to free stream levels of about 5.0%. .At higher free stream levels, t8hewake-induced periodicity cannot be detected around 0 = 0". Smit'h (1968) has also reported some spectral data indicating the effect of free stream turbulence on the periodicity of the flow near the forward stagnation region. I n view of the above discussion, it is suggested that t'he lowering of the Strouhal frequency when the test cylinder is located in a fully developed turbulent pipe flow is possibly
owing to extraneous sources other than the turbulence level. For instance, the effective band width of the filters, sweep rate, and the integrating t'ime constant' of t'he electronic circuitry could affect the location of the spectral peaks within a few per cent. Precise location of t'he frequency becomes especially difficult when the peak becomes broader as a result of the stream turbulence. (Compare Figure 7 with Figures 8 t o 11 of Liu et al., 1970.) Manual tuning of the spectral analyzer using narrow-band filters or use of a time-delay correlator would yield more accurate results. I n connection with their observation that t'he average mean velociby should be used in the Strouhal number computat'ion when the main stream velocity varies along the axis of the cylinder, the authors may wish to refer to the recent paper of Chen and Mengione (1960).
SIR: If the discrepancies pointed out b y Mujumdar and Douglas were indeed caused by a deficiency in t8heelectronic circuitry used in recording and/or analyzing the streaming potential fluctuations, as they suggested, it would be difficult t o understand why a correct location of Strouhal peaks was possible with the same equipment when the test cylinder was located near the pipe ent,raiice (see Figures 6 and 7 ) . but not' when it was situated in a fully developed region (Figures 8 to
11). It would be even more difficult to explain why the four runs in fully developed flows all experienced lowered Strouhal frequencies. The Strouhal frequencies in these cases were found t o be approximately 20 to 30% lower than usual; such differences exceed expected errors arising from a manual determination of t,he peak of a spectral graph. For these reasons we do not feel that' the peculiarity displayed by our signal had anything to do with the electronic
Acknowledgment
T h e authors gratefully acknowledge the financial support of the National Research Council of Canada and the Pulp aiid Paper Institute of Canada, Pointe Claire, Quebec, Canada. literature Cited
Bearman, P. W., National Physical Laboratory, Teddington, England, NPL Aero Rept. 1264 (1968). Chen, C. F., Mangione, V. J., A . I . A . A .J . 7, 1211-12 (1969). 9, Liu, H., Binder, G., Cermak, J. E., IKD.EXG.CHKY.FCXDAM. 211-16 (1970). Smith, hl. C., Phys. Fluids 11, 1618-20 (1968).
-4.S.Illujumdarl Carrier Corp. Syracuse, N . Y . 13201
IP. J . -11.Douglas
McGill Cniversity Montreal, Quebec, Canada 1
To whom correspondence shoiild be sent
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
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