Ind. Eng. Chem. Res. 1992, 31,440-445
440
The Solution of Transcendental Trigonometric Characteristic Equations Enio Kumpinskyt Du Pont Polymers, E. I . du Pont de Nemours & Company, 4200 Camp Ground Road, Louisville, Kentucky 40216-4698
A method is proposed for the solution of transcendental trigonometric characteristic equations. The nontrivial roots (eigenvalues) are obtained by means of inverse trigonometric functions and by taking advantage of the fact that computers only calculate the principal form of these functions. With this technique, a single equation possessing an infinite number of solutions is transformed into an infinite number of equations having a limited number of positive solutions, and, in simple cases, only one solution. Conversion is quick and independent of the initial guess when there is only one root. This method has its main application in the solution of partial differential equations of boundary value problems. Examples with different levels of difficulty are provided to illustrate the procedure. Introduction In many industrial applications, the engineer faces the prospect of solving partial differential equations in Cartesian coordinates that approximately describe chemical and physical phenomena. In particular, boundary value problems are of great interest in the chemical industry. They have use, for example, in the solution of mass and energy balances across membranes or solid walls and vibration of mechanical systems. These problems tend to become very difficult, unless limiting assumptions are made for the boundaries. One of the greatest obstacles in dealing with boundary value problems is the solution of the characteristic equation of the associated eigenvalue problem. Without the limiting assumptions for all boundaries, this equation, which has an infinite number of solutions, turns out to be transcendental and it requires a precise guess to converge to each solution. This task becomes monumental, considering that in many cases the engineer has to perform parametric studies to choose, among others, wall thickness, materials of construction, and process conditions. Every time one of these parameters is changed, a new educated guess has to be made for the eigenvalues. Carslaw and Jaeger (1978) provide the first six eigenvalues with four digits after the decimal point for simple transcendental trigonometric equations, and they are useful in that a trend is formed after the fourth or fifth eigenvalue. In the author’s experience, many problems need hundreds or even thousands of serial terms with their associated eigenvalues and eigenfunctions, each eigenvalue requiring accuracy of six to eight digita. Interpolation from tables, then, becomes unworkable since this search may have to be done every time a parameter is changed. For many boundary value problems with practical interest, tables are not even available. Due to the adverse conditions, it is not unusual for the practitioner to give up on analytical methods and look for numerical solutions. As mentioned by Ybarra and Eckert (19851, the evaluation of eigenfunctions and eigenvalues may not be trivial, but better accuracy is exp e d than when the original partial differential equation is numerically solved. The inverse trigonometry method that is now presented greatly reduces the effort in the search for eigenvalues. For simple transcendental equations, such as those listed by Carslaw and Jaeger (1978), the solutions can be obtained regardless of the initial guess. This is because the single equation with an infinite number of solutions is converted into an infinite number of equations, each having only one t Current address: Research and Development Department Ashland Chemical Co., P.O.Box 2219,Columbus, OH 43216.
positive nontrivial solution. For difficult eigenvalue problems, there can be more than one positive nontrivial solution for each of the equations, but they can still be handled by this method. A good guess can be automated in the computer program, for any choice of parameters, because of the nature of the inverse trigonometric equations. The scope of this work is not to find a solution to the entire PDE but, rather, to show how to solve the characteristic equation of the associated eigenvalue problem. The author has used the inverse trigonometry method with a problem having a fairly simple transcendental characteristic equation (Kumpinsky, 1987) but which would be otherwise difficult to solve using the traditional methods of graphic search or table interpolation. The principles used in that publication are now extended and generalized.
The Method The proposed method for the search of eigenvalues involves splitting the characteristic equation into two parts, the leading term and the characteristic function, and then applying inverse trigonometry to both terms. The leading term is an elementary trigonometric function, such as sine, cosine or tangent, while the characteristic function can be a combination of different functions, trigonometric or not. If the characteristic function is not identically zero, it is necessary to solve a transcendental equation. In symbols we can write trig SA'/^) = q ( x ) (1) The leading term is on the left-hand side of eq 1. Since computers calculate only the principal inverse form of trigonometric functions, we can rewrite eq 1 as = ((n- 1 )+~arctrig [q(X,)])/s (2) When the characteristic function is relatively simple, there is only one positive nontrivial solution associated with each n,that is, the subscript n uniquely binds eq 2 to a particular eigenvalue. In this case, m = n and convergence to takes place regardless of the initial gum. When the characteristic function is elaborate, there may be more than one eigenvalue m ~ssociatedwith each n. A trial and error procedure can usually be directly applied to eq 2. Very rarely will a technique like Newton-Raphson be required to attain convergence. It is important to understand eq 2 in order to pick the leading term. In simple cases there exists only one trigonometric function and the choice is obvious. In cases of higher complexity, however, the characteristic equation m have more than one trigonometric function and the leading term must be selected among them. The method works best when the characteristic function cycles less often than
0888-5885/92/2631-0440$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 1, 1992 441 BC:
alu’(a) - a2u(a) = 0; &u’(b)
cc:
= u(6,);
U(6J
alu’(6-) =
- &u(b) = 0
(8)
PlU’(6,)
(9)
The solution of the eigenvalue problem, eqs 7-9, yields the following characteristicequation, with the leading term already isolated: tan (z,X’/~) = tan A = g(AJ3)
(10)
C ~ X “-~(c2X - c3) tan ( Z ~ X ’ / ~ ) @;(A) = g(AJ3) = Al/Z
the leading term, in order to decrease the probability of multiplicity of solutions for each n. Furthermore, tangent and cotangent are usually preferred over sine and cosine as the leading term. That is because the former two are defined in the (-a,+-)range, causing no problems in a trial and error procedure. Since the latter two are confined to the (-l,+l) range, a poorly guessed or calculated eigenvalue will result in an error message. This method is a very good tool for the search of eigenvalues of trigonometric characteristic equations. Four examples are presented here to illustrate the usefulness of the technique.
Examples Example 1. Consider the design of a new jacketed cylindrical chemical reactor for the high-pressure exothermic synthesis of a polymer. The reactor is made of carbon steel, and it is glass-lined to prevent corrosion of the steel and freezing of the polymer on the wall. A refrigerant circulates through the jacket, and because of the nature of this polymerization, process temperature excursions are possible. The engineer has to design the thickness of the steel as the best compromise between safety (thicker wall) and heat transfer (thinner wall). Similarly, concessions have to be made for the glass-lining thickness between strength (thicker glass) and heat transfer (thinner glass). A lumped overall heat-transfer coefficient cannot be assumed until the solution of the distributed system proves this is reasonable, from design and process control viewpoints. The wall curvature can be neglected because the reactor under consideration is large. A two-layer transport equation without chemical reaction, with unmixed boundary conditions, good contact between the layers, and constant physical properties can now be written: a b, two-layer problem y o ( x ) = initial condition, example 1 y o , l ( x )= initial position of baffle, m Y ~ , ~ ( X=) initial velocity of baffle, m s-l z, = defined after eq 11, s1/2 zg = defined after eq 11, s1/2 z, = ~ 1 / ~ (-6a), s1/2 2,
= v'/Z(b - 6), 9112
Greek Letters
= parameters of first layer, as defined in example 1, i = 1-3 fli = parameters of second layer, as defined in example 1, i = 1-3 7 = M/V 6 = location of the interface in examples 1, 2, and 4, m X = eigenvalue or characteristic value, s-l
ai
p
445
= [(EIlg)/(wS)]1/2, m2 s-l [(EZ&)/(WS)]'/~, m2 s-l
v =
Literature Cited Bauld, N. R., Jr. Typical Properties for Some Common Materials. Mechanics of Materials, 2nd ed.; PWS Engineering: Boston, 1986; Appendix A, p 632. Carslaw, H. S.; Jaeger, J. C. The Roota of Certain Transcendental Equations. Conduction of Heat in Solids, 2nd ed.; Oxford University: Oxford, U.K., 1978; Appendix IV, pp 491-493. Courant, R.; Hilbert, D. Vibration and Eigenvalue Problems. Methods of Mathematical Physics, 1st ed. (English);Interscience: New York, 1953; Vol. 1,Chapter V, pp 275-396. Edwards, M. F. A Review of Liquid Mixing Equipment. In Mixing in the Process Zndwtries; Harnby, N., Edwards, M. F., Nienow, A. W.,Eds.; Butternorth: London, 1990; Chapter 7, pp 113-118. Friedman, B. Operational Methods for Separable Differential Equations. In Modern Mathematics for the Engineer; Beckenbach, E. F., Ed.; McGraw-Hill: New York, 1961; Second Series, Chapter 2, pp 51-67. Kumpineky, E. Water Vapor Uptake by Ceramic Micrcepheres. Znd. Eng. Chem. Res. 1987,26,1597. Lam, P. Vibration Analysis. In ASM Handbook of Engineering Mathematics; Belding, W. G., Ed.; ASM Materials Park, OH, 1989; Chapter 16, pp 313-332. R a d k h n a , D.; Amundson, N. R. The Mathematical Understanding of Chemical Engineering Systems. In Selected Papers of Neal R. Amundson; Aris, R., Varma, A., Eds.;Pergamon: New York, 1980, pp 774-786. Wylie, C. R. Partial Differential Equations. Advanced Engineering Mathematics, 4th ed.; McGraw-Hik New York, 1975; Chapter 8, pp 321-366. h a , R. M.; Eckert, R. E. Transport Eigenvalue Problems-Effect of Order of Approximation and Step Size on Solution Accuracy. AIChE J. 1985,31,1755.
Received for review May 24, 1991 Accepted August 23, 1991
RESEARCH NOTES Separation of m - and p -Xylene via Selective A1kylation/Dealkylat ion-Tr ansa1kylation The separation of close-boiling rn-xylene and p-xylene via selective alkylation and subsequent dealkylation or transalkylation was studied. The alkylation was carried out with isobutylene or diisobutylene with concentrated sulfuric acid as a catalyst at -10 to 20 "C,and Filtrol-24 acid clay at 80 to 130 "C,respectively. rn-Xylene reacts very selectively (Filtrol-24 gives selectively B-tertbutyl-rn-xylene), and the alkylated products can be dealkylated at higher temperatures, in the presence of Filtrol-24 catalyst, to give relatively pure rn-xylene and isobutylene which can be recycled. Introduction The manufacture of pure p-xylene from mixtures with m-xylene, required for the manufacture of fiber grade terephthalic acid or dimethyl terephthalate, is practiced on a large scale. However, these methods based on adsorption or crystallization do not permit pure m-xylene to be obtained and all the methods of manufacturing pure m-xylene appear to involve separation through reactions based on alkylation, sulfonation, etc. Pure m-xylene is required for isophthalic acid, isophthalonitrile, etc. The separation process based on alkylation/dealkylation with isobutylene has been known for a long time, but
quantitative details and kinetia of &lation/dealkylation have not been reported. All the earlier reporta are based on sulfuric acid as a catalyst which has to be necessarily operated at temperatures below 0 "C to avoid oligomerization of isobutylene. In this work the kinetics of alkylation has been studied with m- and p-xylene mixtures and with pure isomers. Further alkylation with isobutylene using acid-activated clays like Ffitrol-24 was conducted at higher temperatures in the range of 80-130 "C. It was thought that the use of activated clay may also allow diisobutylene to be used as this is expected to deoligomerize to isobutylene at higher temperatures and tert-butyl derivatives may be obtained
0888-5885/92/2631-Q445$Q3.00/00 1992 American Chemical Society
