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force F per cubic meter (where pu and pb are the unburnt and burnt gas density and g is the acceleration due to gravity)
F =
( p , - pb)g
(N/m3)
is remarkably constant for the limit curves having values of 11.30 and u = 0.40, 10.70 and u = 0.30, and 10.3 and u = 0.2 (N/m3) (a being the standard deviation) for determinations at 20, 100, and 150 O C , respectively.
Conclusions The peak limit for hydrogen sulfide, carbon dioxide, and air for upward propagation at 20, 100, and 150 " C have been determined. Adiabatic flame gas temperature calculations indicate that the carbon dioxide's primary activity is to bring the final flame gas temperature to a particular value for any given equivalence ratio. Above stoichiometry the carbon dioxide also acts as a weak oxidant. Acknowledgment
the Environment, UK. It is contributed by permission of the Director of Building Research Establishment. The authors would like to thank Mr. C. Finch for assistance with the experiments. Literature Cited Andrews, G. E.; Bradley, D. Fourteenth Symposium (International)on Combustion 1973, p 1119. Anthony, E. J.; Powell, M. F. Riv. Combust. 1977, 3 7 , 361. Crescitelii, S.;Russo, G.; Tufano, V.; Napolitank, F.; Tranchino, L. Combust. Sci. Techno/. 1977, 15, 201. Egerton, A.; Powling, J. Proc. R. SOC. London, Ser. A 1948, A793, 190. Gaydon, A. G.; Wolfhard, H. G. "Flames. Their Structure Radiation and Temperature", 3rd ed, Chapman and Hall: London, 1973. Grav. P.: Sherrinaton. M. E. J. Chem. Soc..Faradav Trans. 1 . 1974. 70. 2338. Herkberg, M.; Burgess, D. Twentieth Annual ISA Analysis Instrumentation Symposium, 1974. Hertzberg, M. U . S . Bur. Mines Rep. Invest. 8727, 1976. Kelly, R.; Padley, P. J. Trans. Faraday SOC. 1971, 6 7 , 740. Larsen, E. JFFIFire Ref. Chem. 1975a, 5 . Larsen, E. ACS Symp. Ser. 1975, No. 16, 376. Larsen, E. Private communication, 1978. Lovachev, L. A.; Babkim, V. S.: Bunev, V. A,; V'Yum, A. V.; Kriwlln, V. N.; Baratov, A. N. Combust. Flame 1973, 2 0 , 259. Popov, P. V.; Beuub, K. E. Trans. Sci. Fertilizers Insectofungkbs USSR 1939, 135, 92. Zabetakis, M. G. U . S . Bur. Mines Bull., 1965, No. 627.
This paper forms part of the work of the Fire Research Station, Building Research Establishment, Department of
Received for review May 8, 1978 Accepted February 22, 1979
Oxygen Probe Dynamics in Flowing Fluids V. Linek,' P. Bene5, and V. Vacek Department of Chemical Engineering, Institute of Chemical Technology, 16628 Prague, Czechoslovakia
r(
The relation between dissolved oxygen concentration changes G(t ) and the responses t ) of membrane-covered polarographic oxygen probes is investigated. This relation is described by a model considering nonsteady oxygen diffusion through a membrane and an electrolyte layer and convective oxygen transfer from the bulk of fluid to the outer membrane surface. Various methods of evaluating the process parameters from the probe response are reviewed with regard to the terms in which the function of G ( t ) is defined and with regard to the procedures available for experimental determination of the transient characteristics of the probe. The description of probe dynamics has been amplified by a treatment of those probes which exhibit a slowdown under conditions such that the probe reading is significantly influenced by hydrodynamics. The description of probes with spherical cathodes was shown to be equivalent to that of probes with planar cathodes. Problems which may arise in the application of oxygen probes in various situations are discussed.
Introduction Membrane-covered polarographic oxygen probes are used ever more frequently for the determination of kinetics and/or transport characteristics of various processes by dynamic methods. It is a unique advantage of these probes that the measurements can be taken in any medium unless the probe material is chemically attacked. The dynamic method consists of monitoring the changes of oxygen concentration G ( t ) due to a step change of oxygen concentration in one of the input inflowing streams. The probe reading is proportional to the oxygen flux to the cathode. Thus it is quite possible that some instruments will not be able to follow the rapidly changing oxygen concentration, owing to the dynamics of the probe proper. The relation between the concentration change studied, G ( t ) , and probe response r ( t )can be derived from the mechanism of oxygen transfer from bulk to the probe cathode. Probes with planar cathodes are used most often, and for these probes the analysis of the relation between G ( t )
and r(t)has advanced the most (Heineken, 1970; Benedek and Heideger, 1970; Linek et gl., 1978a). The cathode is circular (probes by Clark, Cerkasov, Hospodka, and CBslavskY, the Beckman probe, the WTW probe) or ring shaped (e.g., YSI probe) or is of a lattice form (e.g., the Borkowski-Johnson probe). Probes with spherical cathodes, having tip diameters of 0.2 to 0.6 ym, were constructed recently by Lee et al. (1978) for measuring fluctuations of dissolved gas concentration in the liquid layer adjacent to the gas phase; this concerned fluctuations induced by liquid phase mixing. In all studies published so far, the oxygen transfer to the probe cathode was considered to be a consequence of one-dimensional molecular oxygen diffusion through a membrane of uniform thickness and eventually also through an electrolyte layer between membrane and cathode (the so-called one-region, one-dimensional diffusion model). The cathode geometry is the dominant factor affecting the quality of approximation by this type of model. The deviations in behavior of real probes from
0019-7874/79/1018-0240$01.00/00 1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 3, 1979 241 bulk of f l u i d
C = G(t1
:
c=ci
MEMBRANE
u=
O
x=d
1
,"= o
h = k (ci-G(tl)
L
ax
UI
CI.
vui
a U
t
I
1
I
0
,
I
I
L
J
I
I
I
I
i
5
10
15
20
CI
= 'pv,
MEMBRANE
t,s
Figure 1. Transient characteristics of various oxygen probes. Probe without slowdownJCerkasov probe) and with slowdown (YSI probe, Model G-1678-5; CSAV probe).
that predicted by the one-region one-dimensional diffusion model are best discernible from the probe responses to a step concentration change: the last 20% of the real response are lagging behind the model response, exihibiting the so-called slowdown. The probes with high cathode circumference to area ratios exhibit a definite slowdown, e.g., YSI or the Borkowski-Johnson probe; typical responses of this kind (Linek and Benes, 1977) are shown in Figure 1. For probes with circular cathodes the tendency to slow down increase! with decreasing cathode diameter. For example, the CSAV probe with 1 mm cathode diameter exhibits considerable slowdown; see Figure 1. Slowdown can be quantified using the concept of lateral diffusion as implied in the central well model given by Kok and Zajic (1975) and, equivalently, using the idea of a membrane with nonuniform thickness according to Linek and Benes (1977). In both these models only one-dimensional diffusion is considered. The applicability of oxygen was considerably extended for measurements in flowing liquids by the model which quantitatively described the influence of the resistance against oxygen transfer in the liquid film adjacent to the outer side of the membrane (Linek and Vacek, 1977,1976). However, this model was derived only for probes with planar cathodes without slowdown. The dynamics of ultramicroprobes with spherical cathodes was derived only for stagnant media (e.g., for measurements in tissues), even though these probes are being used in flowing liquids. The aim of this work is to extend the treatment of probe dynamics, taking into account the oxygen transfer resistance in the liquid film before the membrane, so as to include probes which exhibit slowdown and to show that the description of probe dynamics derived for probes with planar cathodes holds also for those with spherical cathodes. Theory Oxygen flux through membrane to the cathode M = kODlau/azl,, is described by Fick's second law. For dimensionless probe response r(t)= M(t)/Moto an arbitrary concentration change G ( t ) it follows from the linearity of the diffusion equation (Carslaw and Jeager, 1960; Mueller et al., 1967) that ut)=
JYW d P (dtt 0
dG(t -
T)
dT=
dt
INSULATOR
I 1
b
Figure 2. Coordinate systems and concentrations for planar and spherical cathodes.
under which the concentration change G ( t ) is taking place. The properties of the pulse probe characteristics, drl(t)/dt, are mentioned first. Then relation 1is analyzed as to the possibilities of determining the parameters of the concentration change G ( t ) from the experimental probe response rE(t). Further, the problems are discussed which arise in practical determination of probe pulse characteristics as well as in solving the convolution integral (1) in specific applications. Pulse Probe Characteristics. For both the planar and the spherical cathodes (see Figure 2 for definitions of coordinates and concentrations), the analytical expression for the normalized dimensionless probe response P ( t ) to the step concentration range G ( t ) = 0 for t < 0 G ( t ) = 1 for
t I. 0
follows from the diffusion equation for the concentration profile inside the membrane, u ( z ) au =
D-a2u az2
at
with boundary and initial conditions
au az
-D-1
zd
= hL[ci(t) -
- -- -
For a planar cathode, u u, z x , zo = 0, and z d = d , whereas for a spherical cathode, u ur, z r, zo = ro, zd = ro + d. The normalized response
is of the form (Carslaw and Jeager, 1960)
T)
I'l(T)
di
(1) where P ( t ) is the transient probe characteristics determined under experimental conditions identical with those
m
rW
= 1 - C I C q n l exp(-ynlklt) n=l
(4)
For the constants used in eq 4 see Table I. The parameter L , describes the effect of liquid film resistance against
242
Ind. Eng. Chern. Fundam., Vol. 18, No. 3, 1979
Table I. Meaning of Constants q n r , ynrra n d Cra 4 nr
Lr+0 < Lr
