Creeping Fronts and Travelling Waves Dedicated to Professor Dr. , Dr. h. c. Ewald Wicke on his 6Sh birthday

Some of the ramifications of Wicke's work on creeping fronts in fixed beds are reviewed in this paper. His work laid the experimental foundations for model building by Vortmeyer and Jahnel on the one hand and by Amundson and Rhee on the other. These workers showed how slowly moving waves could arise in various models of these systems. The relation between such a creeping front and a travelling wave of fixed form is further discussed and some of the more recent biological applications are mentioned.

1 Introduction

Among themeasures ofscientificstature is thecentrality ofthe problems which the scientist has considered and the breadth of vision and depth ofpenetration he has brougth to them. It is seen not only in the teacher's influence on his pupils but also in the stimulation he has provided to his colleagues. By these, as indeed by any other standards, Professor Wicke is one of the few outstanding figures in that tract of natural philosophy which stretches between physical chemistry and chemical engineering and demands attention alike from the laboratory scientist and practicing engineer. Of his contribution to the understanding of the role of diffusion in catalysis I have written elsewhere, [I], here I would like to survey some of the problems connected with slowly moving reactions zones and, without going into great detail, describe some of the cognate problems that are of current interest in other fields. My starting point is Wicke's own review of 'physical phenomena in catalysis and in gas-solid surface reactions' given at Washington in 1970 [2]. This review admirably illustrates a mature balance between theory and experiment, not conceived in petty terms of one-for-one but frank to avow its personal preference for the methods and results of 'experimental investigations' while aware of the value and need for theoretical insight. I mention this for it opened the way for the work of others, amongst whom was a friend and colleague of Wicke's, a man of comparable maturity and complementary balance, N . R . Amundson, to whom Wicke had earlier told his results. These results were part of the ongoing doctoral studies of Fieguth [3] who was working with an adiabatic packed bed in which carbon monoxide was being oxidized over a platinum- aluminacatalyst. In themiddleofthebedwasapellet with two thermocouples, one at the center and one at the surface of the pellet, measuring T, and Ts. At the same level in the bed a suction tube and thermocouple allowed T,, the ambient gas temperature, and C,,, , the concentration of carbon dioxide, to be measured. By adjusting the flow rate and inlet

* Prof. Dr. R . h i s , Department of Chemical Engineering and Materials Science, 151 Amundson Ha11,421 Washington Avenue S. E. University of Minnesota, Minneapolis, M N 55 455.

Uber langsam vorwartswandernde Reaktionsfronten und wandernde Reaktionswellen. Im folgenden werden einige Teilaspekte der Arbeiten von Prof. E. Wicke uber sich langsam fortbewegende Reaktionsfronten behandelt. Seine experimentellen Untersuchungen waren die Grundlage fur die Aufstellung von mathematischen Modellen durch Vort- meyer und Hahnel sowie Amundsen und Rhee. Diese Autoren zeigten, wie langsam wandernde Reaktionswellen in den verschiedenen Modellen entstehen konnen. Die Beziehung zwischen einer derartigen langsam vorwiirtswandernden Reaktionslront und einer Wanderwelle mit konstantem Profil wird diskutiert, weiterhin werden einige neuere biologische Anwendungen erwahnt.

Rutherford Ark*

temperature, the reaction zone could be made to move so slowlypast themeasuringpoint that thecompletedetail ofthe very steep profile could be obtained. The steepness of these profiles can be seen in Fig. 1 (Fig. 2. of Wicke [2]) where a temperature rise of 100°C over a single pellet diameter is possible and the reaction is completed in the space of two or

- 3.0

- 2.0

- 1.0

I I I - 0 1 2 3 crn 4

Fig. 1 . Profiles ofsurface ( T J , centre (T,) and gas temperature ( T g ) and CO, concentration in a creeping front. (after Wic ke [ I ] , Fig. 2).

three particles. These observations were widely influential in forming the current understanding of the nature of packed beds for they showed that:

a) the pellets were nearly isothermal but appreciably hotter

b) the longitudinal dispersion of mass is smaller than that of

c) there can be appreciable transfer ofheat against the stream

than the gas;

heat ;

to particles in which as yet there is little reaction.

As Wicke remarks : 'the background of these temperature and concentration profiles in theory and experience contains the

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whole field of physical phenomena in catalysis and gas-solid reactions’. The results which Wicke reviewed in 1970 were of a piece with his work with several other collaborators, as many of the references given below will show, [4- 121. Vortmeyer [13,14] summed up this work with the observation that the velocity of the creeping reaction zone, w, was related to the gas velocity, u, and concentration of reactent CO by the empirical re1 at ionship

(1 1 = ,&.77 ~ 0 . 5 - 0 P u .

Fig. 2. shows the excellent agreement of this formula with Wicke and Vortmeyer’s experiments. By means of sample calculations Vortmeyer showed that his model (to be

Fig. 2. Reaction zone velocity (w) as function of fluid velocity (uo) and inlet CO concentration (Co). (after Vortmeyer and Jahnel [14], Fig 1).

described below) gave good agreement with these experi- mental results, the values of the exponents being independent of the kinetics and those of a and P dependent on the physical calculations in a sensible way. Before considering the elucidation of these relations let us summarize themodels that have been treated.

2 Models of the Creeping Reaction Zone

In his review [2], Wicke raised the question of whether it is better to work with cell models and difference equations or pseudo-continuous models and differential equations. It was indeed a cell-model that Amundson and Rhee first used [15]. Later they looked at a continuous model, first with complete equilibrium 1161 and then with exchange between phases 1171. On the one hand, these were not unrelated to models of the packed bed developed with Liu a decade earlier [18 - 201 and a cell-model including radiative effects [21]. On the other, they reflect Amundson and Rhee’s interest in waves of fixed form in the context of chromatography and other exchange processes [22-241. These connections - always of interest to the

student of natural philosophy - will become clearer with the publication of Amundson’s selecteana later this year [25] ; the literature would be enriched if a similar compilation of Wicke’s papers could be made. The characteristics of the several models are displayed in Table 1. The longitudinal diffusion and conduction in this

Table. 1. The Characteristics of the Several Models.

Reference 1141 [Is] [I61 1171

One phase X

Two phase x x x

Cell model X

Continuous model X x x

Equilibrium (chemical) Finite reaction rate

X

x x X

Longitudinal diffusion X

Longitudinal conduction X

X

X

x x Particle-particle conduction X

Radiation X

Transfer between phases Equilibrium (physical)

X X

X

table refer to the dispersive processes in the moving phase usually expressed by Fick’s law with an effective diffusivity, often approximated by the assumption of a Peclet number close to 2. The particle to particle conduction which introduces a Fickian term into the equation for the temperature in the fixed phase. This has been studied by Eigenberger [26,27] who showed that it resolved the difficulty ofthe infinite number ofsteady-states found in one of the Liu- Amundson models [18]. It is to be noted that high multiplicities have been found in isothermal packed beds [28] where this transfer mechanism is beside the point. Radiative transfer is treated in [14] and [21].

3 The Speed of the Creeping Front

Eq. (1) for the speed of the creeping front is a satisfactory correlator of data, experimental or calculated, but holds out little hope of having any explanatory power. The remarkable fact is that a single formula emerges out of all three models. This is of a striking simplicity and is obtained by integrating the model equations between two points far on either side of the reaction front. Let [a be this difference in the temperature and [c] the difference in concentrations of the reactant. Then [q = J [ c ] , whereJ = - AH/C, (AH is the heat of reaction and C, the volumetric heat capacity of the reacting fluid), is the concentration difference measured in units of temperature. The equilibrium theory of [16] shows that the creep speed can best be represented by its reciprocal, the ’slowness’. Thus, in Amundson and Rhee’s notation, oS = u/t’ = stream speed/ reaction front speed (NB. u in [15] and [I71 and, by implication, in [16], is -w of Eq. (1) and [14]. In the absence of reaction the slownesses of a pureconcentration wave, oc, or of a pure temperature wave, oT, are given by the ratio of the total capacities for mass and heat respectively to the capacity of the moving stream. This follows from an elementary mass balance since u (capacity of stream) [C] = amount brougth to the moving front = w (total capacity) [C] = amount left behind, where w = u/oC for a pure concentration wave; a similar argument holds, mutatis mutandis, for a temperature wave.

768 Chem.-1ng.-Tech. 51 (1979) Nr. 8, S. 767-771

For matter, this capacity ratio is 4 Waves of Fixed Form E + (1 --E)y

oc = E

where E is the void fraction of the bed and y the porosity of the particle. For heat the ratio is

(3 1 j E + (1 - E ) y) c, + (1 - E ) (1 - y ) c,

EC f oT =

where C, and C, are the volumetric heat capacities of fluid and solid respectively. The remarkable result of Amundson and Rheeis that whenanyoralloftherateprocessesarepresent,~~ that there are sharp fronts both in concentration and temperature of magnitudes [C] and [TI (in temperature units), the slowness of the front is

(4)

where A = [C]/[TJ. Because oT is of the order of magnitude of lo3, the creep velocity is always a small fraction of the fluid velocity. Elegant and simple though this formula be, it does not obviate the need for solving themodel equations for the differences [C] and [TI. Indeed it is in this way that the dependence on the dynamic parameters is felt for these (the velocity, transfer coefficients, diffusivity, conductivity and reaction rate con- stants) will all affect the solution. Amundson’s calculations with his discrete and continuous models show the same trends as the calculations of Vortrneyer and both save the experimental phenomena.

1 0.15 -

CREEP TOWAR

0.14- BED ENTRAN 0 n Y

LL CL

0.13 - v.O.0 ft/min

1 I I 0.0 100 200 300 400

INTERSTITIAL VELOCITY , u (ftlrnin)

Fig. 3. Contours of constant creep speed in the plane of flow velocity ( u ) and feed partial pressure (pp). (after Rhee, Lewis and Amundson [17], Fig. 6).

Amundson’s most complete model shows an interesting ’plateau’ in parameter space. Thus if the contours of constant creep velocity are plotted as hypersurfaces in parameter space the surface of zero creep velocity encloses a finite volume. A particular section of it by the parameters of velocity and feed concentration is shown in Fig. 3 (Fig. 6 of [17]).

Chem.-1ng.-Tech. 51 (1979) Nr. 8, S. 767-771 769

The formula (4) for the wave speed is derived from the models by assuming that the solution is a function of (x - u t ) rather than that of x and t severally. This is not usually the case for reacting systems as Feinberg helpfully insisted (cf. Acknow- ledgements in [17]). What we have in fact is a slowly moving wave of slowly changing form. Measured by the movement 01 a point of constant state the velocity would be found to depend on the state and on time. But that dependence is very weak so that the general behaviour is that of a travelling wave and it is sensible to think of a single wave speed. The difficulty is seen most clearly in the naive model of a packed bed examined by Aris and others [29,30]. If the rate of generation of heat in the stationary phase is H ( S ) , a function only the stationary phase temperature S(x, L ) , the temperatu- re in the mobile phase is T(x, t ) and h the transfer coefficient between phases, then

T, + uT, = h(S- T),

S, = h(T- S ) + H ( S ) .

A solution which is truly a travelling wave is a pair of functions

satisfying

If the solution is to pass between two states, (f+ , g+ ) far ahead (i.e. y + + c o ) and (f-,g-) far behind (i.e. y-t -a), then these two points must be critical points. But this implies that

so that there must be two distinct zeros of H ( g ) . This is not usually the case. The simplest true travelling wave is the so-called KPP equation

This was originally proposed by Fisher [31] to model the propagation of an advantageous gene and attracted the attention of Kolmogorojj, Petrouskii and Piskounou [32]. A comparable equation was then taken up by Zeldovich [33 - 351 and later by Kanel[36] in connection with combustion problems. For an excellent presentation of its setting in the context of biomathematics see Fije’s monograph 1371. The more general equation is

where f(0) =f(l) = 0. So long as f(u) has two zeros we can rescale u to make them 0 and 1 without loss of generality. A travelling wave solution of this equation is of the form

u(x,t)=U(x-ut)= U(y). (6)

It therefore satisfies

U”+vU’+f (V)=O (7)

with U-+ 1 and 5 - t - co and V-rO as 5 - 00. Fig. 4a shows the kind of solution we look for. In finite systems, this solution should be a good approximation if the

breadth of the wavefront is small compared with the length of the system and the wave is far from the ends. An alternative form of the equations is

U’= w, w= - u w - f ( U ) . (8)

Fig. 4b shows the phase plane of U and W. The trajectory in the U , W-plane which corresponds to the wave passes from (1,O) to (0,O) along a curve W ( U ) in O < U d 1, WdO. It

“ I

“t

Fig. 4. (a) A travelling KPP wave. (b) Phase plane representation. (c) Two forms of f(u). (d) Initial conditionsleading to a wave of speed

cannot cross into W > 0 for then it would have an increasing U and could not reach to origin. Thus a travelling wave is monotonic. If the curve W ( U ) satisfying

c.

- - 0 - f ( U ) ~ , W(0) = W(1) = 0 OW dU- W

is found the shape of the wave can be calculated from the first of the equations by quadratures

u (‘3 du

t = 5 1 , 2 + j w(u, I12

Because W(u) -, 0 as u + 0 or 1, the integral diverges as U -+ C or 1 and so t + c o or -a. We shall not repeat the analysis which shows that such a solution exists, but merely mention the result. Iff(u) > 0 as on the left in Fig. 4c then there is a constant c such that travelling waves exist for all u 2 c. Iff(u) d 0 for u in (0, uo) andf(u) > 0 in (uo, 1 )as shown on the right of Fig. 4c then there is a unique value of u and corresponding travelling wave. In the first case it can be shown that the minimum wave speed c gives a stable travelling wave in the following very practical sense. Consider any initial condition u(x, 0), such as is shown in Fig. 4d, for which u ( x , 0) = 1, x < x, and u(x , 0) = 0, x 2 x*, then

u(x,t)+U(x-ct+a) as t+co

where a is some constant. This constant will, of course, depend on the detail of the initial distribution and would have

to be determined from the complete solution of Eq. (5). Fije and others have proved some more general results, full reference to which will be found in [37].

5 Conclusion

It was only one topic among the many to which Professor Wicke has turned his attention which provided a starting point for this brief survey, but it is interesting to note how far ranging is its scope. Starting from Wicke’s work on packed beds we have seen how it connects with other work on combustion [S] and depends on other aspects of the reactor system [l]. This leads to a consideration of the mathematical models of packed beds and to the calculations of creeping fronts [17]. Though these are not travelling waves in the strict sense they suggest this theory with its many biological applications [37,38]. Indeed some of the mathematical work [36] has been inspired by the very combustion problems that stimulated the experimental investigation. It is in this setting of enlightened intercourse of theory and experiment that the work of Professor Wicke commands our respect and emulation.

Received : May 21, 1979 [B 42801

Literature Aris, R . : “The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts” Vol. I, pp. 41 -44. Clarendon Press, Oxford 1975. Wicke, E . : “Physical Phenomena in Catalysis and in Gas-Solid Surface Reactions”, Advances in Chemistry Series, Nr. 109, “Chemical Reaction Engineering”, Amer. Chem. SOC., Was- hington D.C. 1972. Fieguth, P.: Dissertation, Univ. Munster 1971. Padberg, G . : Dissertation, Univ. Munster 1966. Padberg, G . ; Wicke, E . : Chem. Eng. Sci. 22 (1967) p. 1035. Wicke, E . ; Padberg, G . ; Arens, H.: “Chemical Reaction Engineering” (Proceed. IV. Europ. Symp., Brussels, 1968) pp. 425 - 37. Pergamon Press, Oxford 1971. Fieguth, P . ; Wicke, E . : Chem.-1ng.-Tech. 43 (1971) p. 604. Wicke, E . ; Vortmeyer, D. : Z. Elektrochem., Ber. Bunsenges. Phys. Chem. 63 (1959) p. 145. Vortmeyer, D . : ibid 65 (1961) p. 282. Hartig, H . ; Wicke, E.: 2. Phys. Chem. N.F. 38 (1963) p. 265. Wicke, E.; Padberg, G.: Chem.-1ng.-Tech. 40 (1968) p. 1033. Wicke, E . : Chem.-1ng.-Tech. 46 (1974) p. 365. Vortmeyer, D.; Jahnel, W.: Chem.-1ng.-Tech. 43 (1971) p. 461. Vortmeyer, D.; Jahnel, W . : Chem. Eng. Sci. 27 (1972) p. 1485. Rhee, H . - K . ; Foley, D. ; Amundson, N . R . : ibid 28 (1973) p. 607. Rhee, H . -K . ; Amundson, N . R . : Ind. Eng. Chem., Fundamen. 13 (1974). Rhee, H . - K . ; Lewis, R . P . ; Amundson, N . R . : ibid. 13 (1974) p. 317. Liu, S . -L . ; Amundson, N . R . : ibid. 1 (1962) p. 200. Liu, S.-L.; Aris, R . ; Amundson, N . R.: ibid. 2 (1963) p. 12. Liu, S . -L . ; Amundson, N . R . : ibid. 2 (1963) p. 183. Vanderveen, J . W . ; Luss, D.; Amundson, N . R.: AIChE J. 14 (1968) p. 636. Rhee, H . -K . ; Bodin, B . F. ; Amundson, N . R.: Chem. Eng. Sci. 26 (1971) p. 1571. Rhee, H . -K . ; Amundson, N . R.: ibid. 27 (1972) p . 199.

[24] Rhee, H . - K . ; Amundson, N . R . : ibid. 28 (1973) p. 55. I251 Aris, R . ; Varma, A . ; (Eds.), “The Mathematical Understanding

of Chemical Engineering Systems: Selected Papers of N. R. Amundson,” Pergamon Press, Oxford 1979.

[26] Eigenberger, G.: Chem. Eng. Sci. 27 (1972) p. 1909. [27] Eigenberger, G. : ibid. 27 (1972) p. 1917. [28] Hegedus, L . L.; Oh, S . H . ; Baron, D . ; Cavendish, J . C . : ACS

[29] Schruben, D. L.; Aris, R . : Chem. Eng. J. 2 (1971) p. 179. [30] Farina, I . H . ; Aris, R . : Chem. Eng. J. 4 (1972) p. 149.

Symp. Ser. 65 (1978) p. 461.

770 Chem.-1ng.-Tech. 51 (1979) Nr. 8, S. 767-771

[31] Fisher, R. A , : Ann. Eugenics 7 (1937) p. 355. [32] Kolmogoroff; A. N.; Petrouskii, I . G . ; Piskunou, N. S . : Byull.

Moskov. Gosud. Univ. 17 (1937) p. 1. [33] Barenblatt, G. 1 . ; Zeldouich, Y. B.: Usp. Mat. Nauk. SSR 26

(1971) p. 115. Heidelberg 1979. [34] Zeldouich, Y. B.; Frank-Kamenetskii, D. A.: Dokl. Acad. Nauk.

[35] Zeldouich, Y. B.: Zh. Fiz. Khim. 22 i1948) p. 1. [36] Kanel, Y. I.: Mat. Sb. 59 (1962) p. 245; 65 (1964) p. 398. [37] Fife, P. C.: “Mathematical aspects of reacting and diffusing

systems,” Lec. Notes in Biomathematics 28, Springer Verlag,

[38] Kennedy, C . R.: Ph. D. Dissertation, University of Minnesota SSR 19 (1938) p. 693. 1979.

Drehrohrreaktoren in der Chemietechnik Herrn Prof. Dr. Dr. h. c . Ewald Wicke zum 65. Geburtstag Harald Helmrich und Karl Schugerl”

Die Rolle von Drehrohrreaktoren in der Chemietechnik wird dargelegt . Nach einer apparativen Beschreibung dieser Anlagen werden folgende Vorgange behandelt : Kornbewe- gung, Gutwanderung, axialer und radialer Feststofftrans- port, Stoff- und Warmeubergang sowie chemische Reaktio- nen. Die gebrauchlichsten Modelle werden vorgestellt . Drehrohrreaktor-Herstellungsverfahren werden exempla- risch dargestellt. Derzeitiger Stand der Regelung und Automatisierung dieser Anlagen wird diskutiert.

Rotary kiln in chemical engineering. This review stresses the importance ofrotary kilns in industry. After a briefaccount of design considerations, the following processes are treated : particle and product motion, longitudinal and transverse transfer of solids, mass- and heat transfer, as well as chemical reactions. Common models used for rotary kilns are discussed. Typical industrial processes employing rotary kilns are presented. The present status of control and automation of these plants are discussed.

1 Einfuhrung

Zahlreiche Produktionsprozesse in der chemischen und verfahrenstechnischen Industrie, im Huttenwesen und in der Industrie der Steine und Erden enthalten Gas/Feststoff- Reaktionen als Teilprozesse. Katalytisch ablaufende Reaktionen werden im allgemeinen in Festbett- oder Wirbel- schichtreaktoren durchgefuhrt, fur nichtkatalytische Gas/ Feststoff-Reaktionen konnen Schacht- oder Etagenofen, Wanderroste, Moving-bed Reaktoren, Drehrohrreaktoren, Wirbelschichtreaktoren oder Forderreaktoren eingesetzt werden [l , 21. Wahrend FlieBbett- oder Wirbelschichtreaktoren den wohl bestuntersuchten Reaktortyp darstellen - in den vergange- nen Jahrzehnten durften mehrere tausend Arbeiten veroffent- licht worden sein (vgl. z. B. Ubersicht [3]) - existieren kaum Untersuchungen uber die erwahnten weiteren zur Durchfuh- rung nichtkatalytischer Gas/Feststoff-Reaktionen geeigne- ten und bewahrten Reaktoren. Insbesondere Drehrohrreaktoren werden wegen ihrer vielsei- tigen Anwendbarkeit auch in mittelstandigen Betrieben eingesetzt. Obwohl ihr Haupteinsatzgebiet in der Zement- industrie und im Huttenwesen liegt, werden doch zahlreiche weitere chemische Reaktionen in Drehrohrreaktoren durch- gefuhrt. Sie werden z. B. verwendet, um Polyphosphate herzustellen, die u. a. zur Wasserkonditionierung, als Wasch- und Reinigungsmittel, als Lebensmittelzusatz sowie in der Leder- und Textilindustrie eingesetzt werden. Die Produk- tion weiBer und hunter Pigmente erfolgt ebenfalls in Drehrohrreaktoren. Zu den WeiBpigmenten zahlen Zink- weiD, Lithopone und Titandioxid (Rutil). Aber auch einer der Ausgangsstoffe von Lithopone, namlich Bariumsulfid, wird

* Dr. H . Helmrich und Prof. Dr. K . Schiigerl, Institut fur Technische Chemie der Universitat Hannover, Callinstr. 3 - 5 , 3000 Hanno- ver .

aus Schwerspat und Koks im Drehofen hergestellt. Gluh- phosphate, wichtige Dunger im Sortiment der Mineraldun- ger, werden in groBen Mengen im Drehrohrreaktor produ- ziert. SchlieBlich werden Katalysatoren und- tragermateria- lien sowie einige anorganische Grundprodukte wie Soda, Aluminiumoxid u. a. in Drehofen hergestellt. Trotz der vielseitigen Einsatzmoglichkeiten von Drehrohr- reaktoren in der Chemietechnik liegen kaum systematische reaktionstechnische Untersuchungen dieses Reaktortyps vor. Im vorliegenden Beitrag sol1 eine..Ubersicht uber die neueren Arbeiten gegeben werden. Eine Ubersicht uber altere Arbeiten findet man in [35]. Diesel Reaktor wird auch in absehbarer Zukunft kaum durch andere Reaktoren, z. B. FlieBbettreaktoren ersetzbar sein. Insbesondere unter dem Aspekt der Energieeinsparung wird daher eine Optimierung der in Drehrohrreaktoren durchgefiihrten Prozesse immer starker an Bedeutung gewinnen.

2 Drehrohrreaktoranlage

Ein industrieller Drehrohrreaktor besteht aus einem langen, etwas geneigten Rohr, welches von einem Antriebsmotor langsam um die Langsachse gedreht wird (Abb. 1). Das feste Reaktionsgut wird an der hoher gelegenen Seite aufgegeben und durch die Neigung und Drehung des Rohres ans andere Rohrende transportiert und dort abgezogen. Die Brenngase stromen normalerweise im Gegenstrom zum Feststoff durch den Ofen. Kennzeichnend fur den Drehrohrreaktor ist, daB die Gutbeschickung nur einen kleinen Teil des Ofenquer- schnittes fullt ;die Fullungsgrade schwanken zwischen 10 und 20%, abhangig vom durchgefuhrten ProzeD. Industrie-Drehrohrreaktoren konnen uber 100 m lang sein und Durchmesser von mehreren Metern aufweisen. Zur Erzielung hoher Standzeiten mu13 daher bei der Konstruktion der Ofen den einzelnen Baugruppen (vgl. Abb. 1) - Materialaufgabe und -entnahme

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0009-286X/79/0808-0771$02.50/0 771