J . Phys. Chem. 1985, 89, 2852-2855
2852
The Perlstaltic Effect on Chemical Oscillations K. Bar-Eli Chemistry Department, Tel-Aviv University, Tel-Aviv 69978, Israel (Received: August 27, 1984; In Final Form: November 30, 1984)
The effect of the periodicity of the peristaltic pumps used for experimentation with chemical oscillators is examined. The Field-Koros-Noyes (FKN) mechanism for the Beiousov-Zhabotinskii oscillating reaction is used for this purpose. Solution of the appropriate differential equations shows that the oscillation pattern and period change dramatically as the pumping period is changed. It is shown that only at short pumping periods can one retrieve the same pattern and period of oscillations as in continuous pumping. The relevance of these findings to experimental results is discussed.
Introduction Most experiments in chemical oscillators are done in a CSTR-i.e. a continuous-flow stirred tank reactor-in which the chemical species are continuously fed into the reactor as the products are pumped out of it, thus keeping the volume of the reaction mixture constant. Two implicit assumptions are made: (a) mixing of the incoming species is perfect and infinitely faster than any chemical reaction and (b) the feed of materials is continuous. This last assumption is certainly not valid since the usual apparatus for feeding the chemical species is a peristaltic pump. This pump pushes the material, usually in a liquid form, from a reservoir to the reactor via elastic tubes. The push is not done continuously but periodically-pushstoppush-thush-thus introducing the material into the reactor dropwise, hence the name peristaltic. In most calculations this periodicity has been neglected, since usually it is of the order of seconds compared to the oscillatory period, which is of the order of minutes. Geiselerl and Bar-Eli and Geiseler* checked experimentally the NFT oscillator3by working with a peristaltic pump and by pushing the material continuously. They obtained oscillations in both cases, but they did not check the effect of the peristaltic periods on the oscillations. In this work we examine the effect of the period of the pump on the oscillations. The examination is done by calculating the effect of pump periodicity on a typical chemial oscillator, namely the Belousov-Zhabotinskii (BZ) reaction4 We have not challended in this work the first assumption, namely the fast mixing of the components in the CSTR. This point was examined experimentally by Sorensen,Io who obtained different oscillations for different mixing rates in the BZ and similar systems.
The mechanism consists of the following seven chemical reactions.
+ 2H+ * HBr0, + HOBr
Br03- + Br-
kl = 2.1 M-3 s-l HBr02
k2 = 2
k-] = 1 X lo4 M-' s-I
+ Br- + H+ * 2HOBr
lo9 M-,
X
k-, = 5
s-l
Br03- + HBrO,
k4 = 6.5
X
k5 = 4
X
~t
HOBr
lo7 M-I s-I
Ce4+
Ce4+
lo7 M-I
(3) s-l
+ HBr0, X
(4)
lo7 M-I s-I
+ Br03- + H+
k-s = 2
X
(5)
M-,
s-I
+ MA e gBr- + Ce3+ + prod k6 = 0.53 g = 0.4615
Ce4+ + Br0,.
k7 = 9.6 M-'
s-I
(6)
+ HzO 2 Ce3+ + Br03k-7 = 1.3 X
M-I
(7) s-l
The relevant differential equations are C = R ( c ) + ko(co - C )
(8) where c is the vector concentrations of the various species, R(c) is the vector of chemical mass action rates derived from the above mechanism, co is the vector concentrations of the inflowing species, and ko is the volume flow rate divided by the reactor volume p/Vo. Its reciprocal is referred to as the residence time. If a different mechanism is used, R(c) should be changed accordingly. In all previous work ko was taken to be a constant. In this work it is taken to change with time as follows: ko(t) = Eo(sin ut + 1) (9)
Calculations The BZ reaction is modeled by the Field-Koros-Noyes (FKN) mechanism which was shown by DeKepper and Bar-Eli6 and Showalter et al.' to give fairly good agreement with experiments conducted in CSTR and by Bar-Eli and Haddad* to agree with batch experiments.
Thus, the flow changes from zero to twice the value of the average flow E,,. The effect of smaller amplitudes of the sine function is currently being investigated. If one takes a 100% amplitude variation, the case of a dropwise feed is covered. This situation is encountered in experiments where the feed tubes are not immersed in the liquid. Different periodic functions could be taken, but the one chosen seems to adequately describe the action of the peristaltic pump. The differential equations (8) were solved by the Gear methodg and the steady states found by Newton's method.
(1) (a) W. Geiseler, Ber. Bunsenges. Phys. Chem., 86,721 (1982); (b) W. Geiseler, J . Phys. Chem., 86, 4394 (1982). (2) K. Bar-Eli and W. Geiseler, J. Phys. Chem., 87, 3769 (1983). (3) R. M. Noyes, R. J. Field, and R. C. Thompson, J. Am. Chem. Soc., 93, 7315 (1971): (4) (a) B.P. Belousov, Sb. Ref.Radiars. Med. 1958, Medgir. Moscow, 145 (1959); (b) A. M. Zhabotinskii, Dokl. Akad. Nauk SSSR, 157,392 (1969); (c) A. N. Zaikin and A. M. Zhabotinskii, Narure (London), 225, 535 (1970). (5) R.J. Field, E.Koros,and R.M.Noyes, J . Am. Chem. Soc., 94, 8649 (1972). (6) P. DeKepper and K . Bar-Eli, J . Phys. Chem., 87, 480 (1983). (7) K. Showalter, R. M. Noyes, and K.Bar-Eli, J. Chem. Phys., 69, 2514 (1978). (8) (a) K. Bar-Eli and S . Haddad, J . Phys. Chem., 83, 2944 (1979); (b) ibid., 83, 2952 (1979).
(9) (a) C. W. Gear,"Numerical Initial Value Problems in Ordinary Differential Equations", Prentice-Hall, Englewood Cliffs, NJ, 1971, pp 209-229; (b) A. C. Hindmarsh, "Gear: Ordinary Differential Equation System Solver", UCID 30001, rev 3, Dec 1974. (10) (a) P. G. Sorensen, Ber. Bunrenges. Phys. Chem., 84,408 (1980); (b) P. G. Sorensen, "Discussion Meeting Deutische Bunsengeseltschaft fur Physikalische Chemie", Aachen, West Germany, Sept 1979, p 41.
0022-3654185 , ,12089-2852%01.50/0 0 1985 American Chemical Societv I
X
k-4 = 2.4
lo5 M-I s-I ~t
X
k-3 = 2
+ BrO,. + H+
2HBr0,
(2) M-I s-l
+ H + ~t 2Br02. + H 2 0
k3 = 1 X lo4 M-, s-l Ce3+
(1)
-
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2853
The Peristaltic Effect on Chemical Oscillations TABLE I: Steady States and Stability of FKN System'
steady-state designationb
An range, s-l
SSI SSI SSII SSII SSII SSII SSIII
0.1360513 X 10-2-0.1675974X lo-' 0.1675974 X 0.1440620 X 10-4 0.1440620X 104-0.4884664 X 0.4884664 X 10-2-0.5130475X 10 0.5130475 X 10'-0.5253734 X 10 0.1360513 X 10-2-0.5253734X 10
no. of positive eigenvalues 2 0 0
2 0 2 1
lo-' M,[Ce3+lo= M, [H'], = 1.5 M. bSSI is the one with a comparatively high concentration of [Br-] and low [Ce"], while the reverse is true for SSII. SSIII is located between the other two and is always unstable. 'Constant constraints: [BrO3-Io = [MA], = 3 X
3
X
lo-' M,[Br-Jo = 60 X
10"
300b"
T ~997 3
'
0.4
I
'
0.0
I
I
12
'
1.6
i&x103(sec-l)
Figure 2. Oscillation periods as a function of average flow Eo. Constant
constraints as in Figure 1. T=5610
T.2762
1500
,
I
,
I
i I
1300-
!! II
,-
T.9973
I
T=1579
1100 -
I
I
II !\
I
~i
Io
I I
1
I (
r(sec)-
I I
900-
I
1
; I
1 11
I ;
I
700-
6
1200' 2400l 0 TIME
1200' 2400' 0 TIME
1200' 2 4 m J TIME
Figure 1. Series of oscillations calculated for the indicated peristaltic periods T = 27r/w. Constant constraints: [BrO
