CORRESPONDENCE
The Stability of Fiber Drawing Processes SIR: A theoretical understanding of the conditions which give rise to instabilities in fiber drawing processes is very desirable. The equations describing the time-dependent behavior of small disturbances from the steady state are derived by Pearson and Matovich (1969). They study in particular the case where viscous forces predominate, and they determine the response t o periodic fluctuations in the input conditions. Denoting the input and output velocities b y Tioand VI, they show t h a t a t certain ratios Vl/Vo of which the lowest is 20.21 the small disturbance theory implies a n infinite response to fluctuations in the input of a certain period. An analysis using numerical methods is given below for the development of small initial disturbances in the fiber when the input and output conditions are held fixed. The results suggest t h a t the process is in fact unstable for ratios Vl/Vo greater than 20.21 with constant viscosity. If V‘ is the velocity and A’ the cross-sectional area the governing equations are for constant viscosity bA’
__
at
+ ax-b (V’A’)
=
0
The steady-state boundary conditions of interest here are V’ = V OA’ , = A,, a t x = 0 ; V’ = VI a t x = X. The steadystate solutions are Ti = VO(VI/VO)”~,A = A o ( V O / V I ) ~ ’ ~ . Writing V’ = V(1 b ) , A’ = A ( l a), the linearized small disturbance equations are
+
ba
at
+
ab + v-abax + Ti-= ax
certain matrix. The difficulties of the numerical approach are t h a t in general all the eigenvalues of the matrix must be obtained t o ensure that the most significant is not missed and that only some of the eigenvalues are the desired close approximations t o values of X, the rest (those where the corresponding a does not vary smoothly from point to point of the mesh) being determined rather by the difference method, T o simplify the calculation and interpretation of the eigenvalues, a difference method with the following properties has been used: (A) it is consistent n-ith a stable difference scheme for (1) and (2); (B) the variable b may be conveniently eliminated from the difference equations. Property -4 is desirable to permit easy comparison of a time-dependent calculation including nonlinear effects with the implications of the small disturbance theory. Vsing mesh points at x 4 = i X / N , i = 0 to N, and using the subscript i to denote the values of variables a t xI,(3) and (4) have been discretised as follows.
0
(3)
The boundary conditions are a. = bo = b,v = 0. Elimination of the variables bi leaves a Hessenberg matrix whose eigenvalues have been obtained using the algorithm of hIartin, et al. (1970). The finite difference method is not particularly accurate, but the results a t varying A: do support the expectation t h a t the errors in the X are roughly proportional t o 1/N. The two dimensionless values p = XX/Vo with the greatest real parts are given in Table I in terms of Y = log
(4) Solutions are required which behave with time as ex’ under the boundary conditions a = b = 0 a t x = 0 and b = 0 a t x = X. X is not expected t o be real, but the solutions of most interest are those for which X has the largest real parts, these being the most unstable or least stable solutions. Writing Xa for ba/dt in (3), finite difference methods may be used so that the X are approximated by the eigenvalues of a 534 Ind.
Eng. Chem. Fundam., Vol. 10, No.
3, 1971
Table 1.
1.o 2.0 3.0 4.0 5.0
The Most Significant Eigenvalues
-3.83 -1.03 -0.02 0.55 0.92
7.65 5.42 4.66 4.28 4.05
-6.06 -2.28 -0.85 -0.05 0.46
18.43 13.14 11.55 10.86 10,52
(VL/Vo),the values beiiig obtained by extrapolation to 1,” = 0 from computed results with N = 10, 20, and 30. These results agree well with those of Pearson and Matovich; their work implies that particular solutions are Re(p) = 0, I ( p ) = 4.66 for Y = 3.006 a n d Re(p) = 0, I ( p ) = 10.79 for Y = 3.912. The present results suggest t h a t the process is unstable in the classical sense for Y greater than 3.006, and so for V1/V0 greater than 20.21, with a n additional “normal mode’’ becoming unstable a t each of the resonances noted in the earlier work. While a demonstration t h a t the set of solutions (a, A) is complete (in the sense that a n y initial disturbance can be expanded in terms of them) is desirable, it is not essential in drawing the conclusion that the steady solutioii can only be expected t o occur in practice for V,/Voless than 20.21. This study is not immediately relevant to practical processes, which operate successfully a t much greater values of
Vl/Vo. However, the numerical approach can include the effects of increasing viscosity due to cooling, and also of gravity, surface tension, and inertia, which all become important in some circumstances. literature Cited
Martin, R. S., Peters, G., Wilkinson, J. H., Sum. Math. 14, 219 (1970). Pearson, J. R. A., hlatovich, M. A., IND. EXG.CHEM.,FUNDAY. 8, 605 (1969). D. Gelder Pilkington Bros. Ltd., Research Department Lathom, Ormskirk Lancashire, England The author wishes to thank the Directors of Pilkington Brothers Ltd., and Dr. D. S.Oliver, Director of Group Research and Development, for permission t o publish this letter
The Stability of Fiber Drawing Processes SIR:The previous authors concur with the views expressed. Gelder has pointed out a correct interpretation of what had appeared to us a paradoxical result.
John R . A . Pearson
Department of Chemical Engineering universityCambridge Cambridge,England
Determination of Start-up Conditions for Chemical Reactor Stability SIR:I n a recent letter concerning my paper (Han, 1970), Luus (1971) has stated that the problems of determining start-up conditions can be avoided altogether if one operates the reactor a t a n unstable steady state by applying the proportional control scheme proposed b y Ark and Amuiidson (1958). I n my opinion, the statement made by Luus is not correct for the following reasons. First, Luus does not appear to have correctly understood the limitation of the proportional control scheme, which was originally suggested by h i s and Amuiidsoii (1958) for controlling the locally linearized system. On the other hand, I have considered the nonlinear system in presenting a technique for determiriiiig initial-state regions for the stable operatioii of chemical reactor systems. There is no a priori guarantee that a controller designed for the locally linearized system would work for the original nonlinear system over a wide range of reactor temperatures. Therefore, Luus’ statement is not correct. Secondly, Luus has chosen an instance in which there arises little or no necessity for determining start-up conditions and then has made a n unjustified generalization. I n other words, Luus has missed a n instance of great practical importance which does necessitate the determiiiation of start-up conditions.
Consider a situation where the operatioii of a reactor a t or near the temperature of unstable steady state is already so high t h a t degradation (or decomposition) might occur of either reactant, or product, or bot’h. I n such an instance, the control of product quality is inore important than obtaining high conversion, and hence the argument given by Luus completely breaks down, because one is compelled to operate the reactor below a prescribed temperature limit, which happens to be lower than the unstable steady state. It is then obvious that, in such a n instance, the determination of startu p conditions is of very practical importance, contrary to the rather misleading argument given by Luus. It is my earnest hope that’ this correspondelice will help someone better understand the importance of determining start-up coiiditions for the stable operatioii of chemical reactor systems. literature Cited
Aris, R., .4mundson, ?;. It., Chem. Eng. Sci. 7, 121, 132 (1958). Han, C. D., IND. E N G CHEM., . FUKDAM. 9, 634 (1970). Luus, E., IXD. ESG. CHEW,Fum.or. 10, 322 (1971). Chang Dae Nan Department of Chemical Engineering Polytechnic Institute of Brooklyn Brooklyn, New York 11201
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