SIMULTANEOUS ABSORPTION OF CARBON DIOXIDE: AND AMMONIA IN WATER T
.
F
.
HATC H
, J
R
.
,l
A ND R , L
.
P I G F0 R D
,
University of Delaware, .Vewark, Del.
The process o f gas absorption accompanied by chemical reaction has been studied, using the system CO1"3-water. The two gases were mixed and contacted with a smooth laminar jet of water; after dissolution they reacted with each other by a bimolecular reaction. Rates of absorption were determined by chemical analysis of the liquid leaving the iet absorption device. The rate of absorption of COS, significantly influenced b y reaction with an excess of N H 3 in the liquid, agreed with the prediction based on the penetration theory for absorption of two gases followed by second-order, irreversible reaction. The high solubility of NH3 in water, resulting in a large gas-side resistance to its absorption, made it impossible to determine the effect of the rate of reaction on its rate of absorption because of the difficulty in knowing the NH3 concentration at the gas-liquid interface. The confirmation of the diffusion-reaction theory was therefore limited to the observations on C 0 2 absorption, its validity depending on the accuracy of the NH3 gas-phase resistance measurements.
HE PRACTICAL IMPORTANCE O f
absorption processes in which
Tboth diffusion and chemical kinetic mechanisms affect the rare of interphase mass transfer is demonstrated by several examples? such as the washing of acidic gases with alkaline solutions, the manufacture of HiYOl, and the contacting of gaseous and liquid reagents in general. Quantitative expression of the effect in these processes of diffusion and reaction rates has been developed (76) largely through the penetration theory, in which the partial differential equations of diffusion in stagnant or uniformly flowing liquid have to be solved with reaction rate terms included. The mathematical problems involved are orten difficult, but numerical methods have been resorted to where necessary. As a result, a considerable literature has now built up from which the effect of a reaction process of known kinetic behavior can be calculated. None of the previous work has extended, however, to those situations in which two chemically active gases are dissolved simultaneously in a liquid, diffusing and reacting rapidly near the interface after solution occurs. This is despite a t least two important practical exainples: the washing of a gas containing both COS and NHl with water in the Solvay process and the simultaneous removal of H,S and COZfrom a gas by washing with a n alkaline liquid in the production of synthesis gas. In these cases, the rate of absorption of each substance is affected by the rate of the other, for the concentration distribution near the interface of one substance is affected by its reaction with the other. Roper, Hatch, and Pigford (72) have given some approximate solutions to the transient diffusion equations of the penetration theory applicable to these cases, and it was of interest to test out the theoretical results. Unfortunately, existing empirical data from tests of operating absorbers are not satisfactory for testing such theories, for the rate of absorption per unit of surface area is never known accurately. Experimentaj. studies suitable for such a study have to be made carefully so that flow conditions will be well known and reproducible. The work described here, therefore, involves the idealized, laboratory-scale absorption apparatus of hfanogue ( 9 ) and Scriven and Pigford ( 7 4 , in which the liquid
Present address, Department University, Cambridge, Mass.
of
Mathematics, Harvard
surface is that of a laminar jet of liquid which is surrounded with the gases to be absorbed. The jet is formed in a smooth nozzle that produces a nearly flat velocity profile. The theory of purely physical absorption in such equipment has been carefully worked out (73), and observations of absorption rates, after comparison with the theory, lead to rather accurate knowledge of the contributions of the chemical reaction to the absorption rate process. The chemical system chosen for this study involves COz and SH3 as the gaseous components and distilled water as the solvent. The chemistry of this system has been studied by Faurholt (7) and others ( 7 7), who found that the gases react in solution by the following steps:
+ XH3 NHsCOOH NEIzCOOH + NH3 NH: + NHzCOONHzCOO- + HzO NHdCO; NHaCO; NH; + COiCOz
+
+
+
+
(1) (2) (3) (4)
Reaction 1 involves the formation of carbamic acid, which then picks up another NHB to form ammonium carbamate, which partially ionizes into the ammonium and carbamate ions. The latter slowly react with water to form the ammonium bicarbonate ion, which may dissociate into ammonium and carbonate ions. There are, of course, ionic equilibria involving several of the steps. Reaction 1 is rate determining in the absorption process because the second step, Equation 2, is very fast and irreversible. These two reactions taken together inper molecule volve the consumption of t\vo molecules of "3 of CO, by a net reaction: COz
+ 2NH3
+
NH:
+ NHzCOO-
(5)
According to Faurholt (7), Equation 5 follows a second-order rate law. The rate is proportional to the product of the concentrations of NH3 and COS. T h e third reaction, Equation 3> is much slower than the others and occurs mostly outside the experimental absorption apparatus. (In an industrial absorber, hobvever, in which the water phase is alloired to build up an appreciable equilibrium vapor pressure of NH3 and CO?, the rate of the third step might be of some consequence if the liquid hold-up is large.) VOL 1
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AUGUST
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209
Experimental
Apparatus. Several experiments that have been carried out to test the validity of penetration theories have involved equipment in which the interfacial surface was not smooth and uniform, a condition that is required for the strict application of the theoretical results. In the frequently used long.
wetted-wall column, for example, serious end effects and surface rippling occur, indicating the presence of turbulent mixing within the falling liquid film. Nijsing and Kramers (70) and Scriven and Pigford (74) have employed an improved absorption device in which the liquid stream issues from a carefully shaped nozzle, falls in a laminar jet through the gas, and is caught in a small-bore glass receiver. End effects at the receiver are negligible with this apparatus, and deviations from rodlike flow and constant jet diameter are small; furthermore, these deviations can be accounted for mathematically in some simple cases. Scriven and Pigford ( 7 4 ) had found this apparatus to be useful in their studies of possible interfacial resistance in the absorption of pure C 0 2 into water, and it was decided to use the same equipment in this study of absorption with accompanying reaction. The penetration theory for purely physical absorption into a laminar liquid phase leads to the well known result:
ii
=
2 4 D T t
(6)
For the jet absorber, t is the time of flight of a liquid particle in the jet from the nozzle exit to the receiver, given approximately by : t = hnd2/4q~
(7)
for a jet of perfectly uniform diameter and having a rodlike velocity distribution. For such an "ideal" jet, the total rate of absorption for the whole jet surface is:
The jet is not perfectly ideal, however, owing to the velocity boundary layer in the jet as it emerges from the nozzle and also to the accelerating effect of gravity. For the nozzle employed by Scriven and Pigford (73) and for the jet length employed in this investigation (1 1.81 cm.) a maximum deviation of only 37, from Equation 8 was calculated (74) : Figure 1.
Jet absorption apparatus @A
I
0
2
3
4
JJET LENGTH, 4 , cm!" Figure 2.
Rates of absorption of pure
COn in water Data corrected to q L = 5.02 cc. p e r second and AL = 3.26 X gram-mole p e r cc. Ideal jet, Equation 8 Predicted, Scriven and Pigford ( 1 4 )
Ai =
--_ 0 Scriven using brass nozzle X 0
210
Fields (2) using brass nozzle This work using stainless steel nozzle
I&EC FUNDAMENTALS
= 3.878
(Ai
- AI)
(9)
Equation 9 should apply to the liquid-phase resistance to absorption of both C 0 2 and NH3 when each is present alone. Figure 1 shows the jet absorber employed for this work. The nozzle produced a jet having an average diameter of 1.45 mm. and a length of 11.81 cm. At the bottom of the chamber the jet was caught in a glass capillary tube receiver (inside diameter = 2 mm.) where neither liquid spillover nor gas entrainment was permitted. The water then passed continuously through a chamber from which samples could be withdrawn for the measurement of electrical conductivity and then entered a closed, constant-level overflow device. At this point, part of the water was withdrawn from well below the surface and collected in a special sampling pipet; the rest overflowed to the drain. The distilled water used in the absorption experiments was first freed of dissolved gas in a small stripping column. After this treatment it contained 4.6 X 10-6 gram-mole per liter of dissolved C 0 2 and had a n electrical conductivity smaller than 2 micromhos per cm. at 25 O C. Before entering the jet absorber, the water flowed through copper coils to adjust its temperature to 25OC. Gas was fed to the absorption chamber tangentially a t its bottom and left through a n opening a t the top. Ammonia, C 0 2 . He, and N2 (in various runs) were taken from cylinders and metered by calibrated rotameters so that composition could be determined. Flow rates from 800 to 2200 cc. per minute \*ere employed.
All runs in this investigation were made at a water flow of 5.02 1 0 . 0 3 ml. per second, equivalent to a surface jet velocity of about 300 cm. per second and a contact time of about 0.04 second. The conductivity of the solutions produced in the absorber was observed with a ,Serfass conductivity bridge. The conductivity cell constants were carefully determined with standard solutions of KCI, using the data of MacInnes (7) and Shedlovsky and hlacInnes (8, 75) to find their equivalent conductances. The use of these measurements for the computation of the ionic content of the solutions is described below. Absorption of COz. Measurements of the rate of absorption of CO2 in water h,id been made previously with this same apparatus (73), but they were repeated to make sure that the apparatus was working properly and that some of the analytical methods were sound. In these preliminary runs, the water entering and leaving tLle absorber was analyzed for dissolved COz by releasing samples from a pipet underneath the surface of an excess of standard h-aOH to which had been added a few milliliters of BaCl solution to precipitate BaC03. T h e excess alkali was then ti.trated with si.andard acid. The results of these measurements are shown in Figure 2: where they agree closely with the line representing Scriven's data (73) and the penetration theory according to Equation 9. Absorption of NH3. No previous experimental studies of the gas-phase resistance to mass transfer in jet absorbers having been reported, it was necessary to determine this resistance by observing rates of absorption of KH3 from NH3-Nz and "3H e mixtures. The liquid leaving the jet was analyzed for dissolved N H 3 by releasing samples beneath the surface of an excess of standard HCl and back-titrating to a methyl red end point. The flow rates of gas through the absorber were always small in these runs, so that the: flow regime must have been essentially laminar in every case, Although the flow conditions in the gas phase did not correspond exactly to those for which mathematical solutions of the diffusion eqmtion have been obtained, owing to the unknown gas velocity field around the swiftly moving jet and to the spiral flow in the bulk of the gas, it appeared very likely that dimensionless groups of physical quantities similar to those employed in standard problems should apply here. This suggested that a plot of: PB1 PBi
I -
.-
PH1
PBi
- k*CBRTxhd 4.
against the Peclet number, q,,/D,,h, should represent the emwhich is based o n the pirical results. I n evaluating k*,,, initial partial pressure difference, p~~ - psi,the small back
pressure, p ~ , ,was computed from the observed absorption rate, the known Henry's law coefficient for NH3, and Equation 9 to find Bt. As suggested by the dimensional analysis, the absorption data were plotted in the form k*GBRT?rhd/quus. q,/D,Bh. I t was found, however, that an improved correlation could be obtained by multiplying the first term by P L ~ / P . Figure 3 shows the final plot of the results. The straight line on the figure correlates the data for both NHS-Nz and NH3-He, in which the NH3 diffusion coefficient was nearly four times greater. The empirical equation for the line is :
indicating that the Stanton number for diffusion in the gas phase was only slightly influenced by flow, diffusivhy, or jet length. T h e physical dimensions of the apparatus were not varied, nor was the velocity of the jet changed, so that the work does not indicate whether these quantities are accounted for correctly in the empirical formula, Equation 10. In fact, it seems very likely that the jet speed should appear. O n the other hand, these variations were not pertinent in the further work described in this report, in which h and q L were constant. Analysis of NH3-C02-H20 Mixtures. The effluent liquid from the jet absorber was analyzed for total dissolved NH3 and C O Z by making two measurements: T h e electrical conductivity of the solution was determined, and a known volume of the solution was added to a n excess of standard acid, the COZdriven off as a gas, and the solution back-titrated to the equivalence point with standard base. From these results, plus known values for ionic conductances and equilibrium constants, the concentrations of hydrogen, bicarbonate, carbonate, and ammonium ions were determined. T h e decomposition of ammonium carbamate to ("4) 2co3 was essentially complete in a few minutes after its formation, according to the studies of Faurholt (7): who measured the following values of equilibrium constants :
The largest concentration of dissolved NH3 observed in this study was 0.0036 gram mole per liter, indicating that according to Equation 11 the carbamate concentration was only about 2% of the bicarbonate concentration when equilibrium had been achieved. A more recent determination of Kw. by Shokin and Solov'eva (77) gave Kw. = 2.2 at 25" C., corresponding to (NH2COO-)/(XH3) = 0.004 a t equilibrium.
Figure 3. Rates of absorption of NH3 from NH3-N2 and NH3-He mixtures
- Equation 10 0 NHI-Nz
0
VOL 1
NH3-He
NO. 3
AUGUST 1962
211
Furthermore, a t the high N H 4 concentrations occurring in the experiments in which both NH3 and COz were absorbed, carbonate and bicarbonate ions were present a t about equal concentrations. Therefore, at equilibrium the carbamate ion made up no more than 1% of the total dissolved COZ. Faurholt also determined the velocity of decomposition of the carbamate ion, indicating that a t 18" C. the time required for 99% decomposition should be 17 minutes. Lower concentrations of NH3 and higher temperatures imply still speedier decomposition. I t was observed in this work that 10 minutes were sufficient for steady conductivity readings to be obtained, showing that equilibrium must have been achieved in that time. Actually, 15 to 20 minutes were allowed before final readings uere taken. The electrical conductivity of a n aqueous solution depends on the type and concentrations of the ions present. In dilute solutions. such as those analyzed in this study, the contributions of individual ionic species to the total conductivity are independent of the other dissolved materials; that is. there is essentially no interaction between ions. The total conductivity is then given by
where the c's represent equivalent ionic conductances. The accuracy of this equation \vas tested by measuring the conductivities of standard solutions of KH4HC03. Using reported values of ionic conductances (3-5). agreement between expected and observed conductivities was within 1%. The following equivalent conductances were employed : Equivalent Conductance, ( M h ~ ) ( C m . ) / ( G r aEguiv.) m
Ion H+ OH NH: HCO; CO,
c, = 62 = cj = c4 = cs =
10-3 gram-ion per liter of total dissolved C0;- and 1.206 gram-moles per liter of total NH3 was analyzed by the above procedure, which was found to yield 0.6811 X 10-3 and . 1.222 x 10-3 gram-moles per liter, respectively. Results
The effect of the chemical reaction in the liquid on the rates of absorption of COz and NH3 simultaneously was studied by absorbing from COz-NH3-N2 mixtures. Owing to its greater solubility, the mole fraction of NHI in the feed gas was limited to 0.04 to 0.10. The mole fraction of COZ varied from 0.15 to 0.47; the molar ratio of NH3 to CO2 in the gas ranged betiyeen 0.12 and 0.61. Liquid leaving the absorber was analyzed according to the scheme just described. The composition of the gas entering the absorption chamber was computed from flow measurements with calibrated rotameters. It was found necessary to dilute the gas mixture with h-2 to avoid the formation of a solid reaction product, probably (SH,)?COs, on the walls of the apparatus and in the upstream tubing. M'hen this occurred, it became difficult to observe the jet through the cloudy walls to make sure that it entered the receiver properly; the tubing also became clogged and the gas rates were affected. Drying the gases by passing them through CaSOl did not alleviate the trouble. Even when h-2 was present as a diluent, the chamber walls became cloudy when the NH3 to CO1 ratio was much above 0.6, except when the total concentration of these two gases was kept very low. In the latter case, however, the amounts dissolving were too small for reliable analysis. No precipitate was observed during the runs for which data are reported. The penetration theory of diffusion plus reaction applicable to this situation, presented in a companion article (72), indi-
349.8 198.6 73.4 44.5 69.3 2.5
After the COz has been expelled, the net acid titer of the solution remaining is represented by the equation : T
=
(HCO,)
+ 2(CO,-) + (NHIOH) + (OH-) - ( H + )4- 5' (14)
where S = 2 x 10-6 gram-ion/liter, the concentration of hydrogen ions needed to bring the neutral solution to p H = 5.7, the end point employed in the analysis. Equations 13 and 14 are supplemented by the mass action equilibrium expressions, with constants evaluated at 25°C. :
K,
=
( H + ) ( O H - ) = 1.008
x
(gram-ion/liter)? (15)
lO-'4
m U E 0
a
2.0
f0 n
8
w-
Y
KB
=
(NH,+)(OH-)/(NH40H) = 1.774 X 10-5 (gram-ion/liter) (16)
I Y
K1
=
(H+)(HCO-)/(H2C03) = 4.45
I s
>(
10-7 (gram-ion/liter) (17)
>
1.5
I1
0
The symbol (HzC03) in the denominator of Equation 17 refers to the total dissolved C 0 2 that is either in the free, dissolved form or in the hydrated form. K2
= (H+)(CO-)/(HCO-) = 4.69 X lo-"
(gram-ion/liter) (18'1
From Equations 13 to 18, four equations for the unknowns (H+),(HCO,), (COJ, (NH:) were found. These were solved for each absorption run with a digital computer, using a manual trial-and-error procedure. T o test the accuracy of the analysis, a solution known to contain 0.6738 X 212
I&EC FUNDAMENTALS
1.0
2
I
I
Je Figure 4. Predicted absorption coefficients for two gases undergoing irreversible, bimolecular reaction in solution
cates that the average rate of absorption divided by the rate that Xvould occur without the chemical reaction, defined here as Q, is dependent on t h e e dimensionless groups: R = DB/D,, m = va4,jB,, and e =: kB,t. Figure 4 shows the results of these calculations. Empirically it is found that for R = 1.19, corresponding to liquid diffusivities of 1.97 x 10-5 and 2.34 X 10-6 sq. cm. per second for COSand NH3 in water, respectively, at 25’ C.. the effect of m on Q can be. represented by:
There are two major difficulties in applying Equation 19 in the present study: The interfacial concentration, B i , of NH3 was not constant, as assumed in the theory, owing to the large resistance to diffusion of NH3 through the gas; and Bi was not known reliably, so the theoretical effect of reaction on the rate of absorption of NH3 could not he tested thoroughly. T h e two difficulties spring Yrom the same source, the much greater solubility of NH3 in wa.ter than COS. Since rn is much smaller than unity, the concentration of NH3 throughout the portion of the liquid phase where COS is also present for the reaction is essentially equal to the interfacial concentration of XH3: the liquid diffusion effect being of little significance so far as SH3 is concerned. The principal effect of diffusion on the SH3 concentration occurs in the gas phase and results in a time-dependence of Ei,not allowed for in the elementary penetration theory. The key computation required for the application of Equation 19 to the data is the estimation of Bi. This was accomplished by using Equation 10 to estimate the gas-phase mass transfer coefficient, them employing Equation 9 to get the resistance to purely physical absorption in the liquid phase. The interfacial partial pressure of NH3 was, therefore, determined by dividing the total driving force in the proportion of the tw.0 resistances, according to :
using A Q A = QA(0) - Q A ( l ) read from the theoretical curves of Figure 4 a t the experimental e values. Discussion
The absorption rate for CO? appears to correspond closely to the penetration theory when the effect of reaction is allowed for Jvith the aid of known reaction mechanisms. Some doubt may be cast on the application of the theory, however, because of the variation of Bi along the interface of the jet. I n fact, to use the theory, it was necessary to use a n average value of Bi, and Equation 21 was employed for this purpose on the assumption that the gas-phase resistance was essentially constant. Under these conditions, Bi should be nearly an exponential function of distance along the jet surface. This is certainly nqt the boundary condition used in the theory for Equation 19. Is it possible that the average value of Bi chosen by Equation 21 is a good one in the sense that its use in the penetration theory, instead of the truly constant value assumed there, approximates the right answer? To answer this question a “film theory”-i.e.: a theory of steady state diffision-was worked out for the condition of exponentially Lrarying interfacial concentration, B,(i) = constant X e-Tz. It turns out (6) that when the number of gas-phase transfer units, rh! is small: the introduction into the steady state diffusion problem of an interfacial concentration that varies across the surface of the jet as a boundary condition produces only a small correction to the procedure adopted in computing the average value of Bi as described above. It may be concluded, therefore! that the penetration theory is confirmed by the experiments described here, at least for conditions closely approximating pseudo first-order reaction in the fluid.
or 2.5
The value of B , in the COZ absorption runs was then obtained with the Henry’s law {constant for KH3, HB. Naturally, this method for estimating B , precludes attaching any significance to the measured absorption rates of KH3. insofar as their dependence on the chemical reaction is concerned. The interfacial partial pressure of COS, @ A I , was found by subtracting the equilibrium vapor pressure of water from the total pressure and assuming that the remainder was divided between COZ m d NS according to their relative proportions in the bulk of ihe gas. The ratio of mass transfer coefficients for CO,, Qa, was calculated for each run as follows:
h
0 II
E 0 c U
g
2.0
0 Q
t
s
Y
whence, from Equation 9 : Q ~ e x p . =:
d / p L / D a h A P - Ai -__ 3.878
--
Ai
- Ai
The results of the C02-NH3 absorption runs are shown in Figure 5. Experimental values of QA from Equation 23 were corrected to truly first-order conditions (m = 0) before plotting by means of Equation 19: Qa(0) = Qa(expt1.) $- m4”aQ.4
(24)
._ 2.0
1.0
5.0
4 Figure 5. Rates of absorption of COz from COZNHy-N2 mixtures VOL. 1
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AUGUST 1962
213
Nomenclature
Y
A,, Bi
$AO
A (COz), B (SH3) in liquid A I , B1 = initial concentrations of A, B in liquid = bulk-average concentration of A in jet emerging A2 from absorption chamber Dd,DS = diffusivities of dissolved gases A, B in liquid = diffusivity of B in gas DvB d = jet diameter h = jet length K,, KB, K1, K2 = mass action equilibrium constants k = second-order reaction rate constant for reaction between A and B in liquid phase = cumulative average mass transfer coefficient for A without chemical reaction = cumulative time-average mass transfer coefficient TLA for A = average mass transfer coefficient for B k*GB m = vA,/B, = average rate of absorption of A in j e t Z A = average rate of absorption of a nonreacting gas in JVAO jet P = total pressure = arithmetic average partial pressure of inert gas PAW = partial pressure of B p~~ PA;, p~~ = interfacial partial pressures of A and B = average absorption rate with reaction/rate without Q reaction = ratio of mass transfer coefficients for A QA qL = volumetric flow rate of liquid stream through nozzle = volumetric flow rate of gas mixture into apparatus
F r t
z
e
= interfacial concentrations of
= stoichiometric coefficient = absorption rate for whole jet
Literature Cifed
(1) Faurholt, C., J . Chem. Phys. 22, 1 (1925). 2) Fields, M. C., M. Ch. E. thesis, Univ. of Delaware, 1958. 3 Harned, H. S., Davis, R., J . Am. Chem. SOC.65,2030 (1940). 4 Harned, H. S., Robinson, R. A., Trans. Faraday SOC.36, 977 (1940). (5) Harned, H. S., Scholes, S. R., J . Am. Chem. SOC.63, 1706 (1941). (6) Hatch, T., M. Ch. E. thesis, Univ. of Delaware, 1958. (7) MacInnes, D. A., “The Principles of Electrochemistry,” p. 339, Reinhold, New York, 1939. (8) MacInnes, D. A., Shedlovsky, T., Longsworth, L. G., J . Am. Chem. SOL.54, 2758 (1932). (9) Manogue, W. H., Ph.D. thesis in Chem. Eng., Univ. of Delaware, 1953. (10) Nijsing, R. A. T. O., dissertation, Delft, 1957; Nijsing, R. A. T. O., Kramers, H., Chem. Eng. Sci.8, 81-9 (1958). Pearson, L., Roughton, F. J. \V., Trans. (11) Pinsent. B. R. W., ‘ Faraday SOC. 52, 1594 (1956). (12 Roper, G. H., Hatch, T. F., Jr., Pigford, R. L., IND.ENG. &EM., FUNDAhlENTALS 1, 144-52 (1962). (13) Scriven, L. E., Pigford, R. L., A . I. Ch. E. Journal 5, 397-402 (1959). (14) Zbid., 4, 439 (1958). (15) Shedlovsky, T., MacInnes, D. A., J . Am. Chem. SOG.57, 1705 (1935). (16) Sherwood, T. K., Pigford, R. L., “Absorption and Extraction,” 2nd ed., McGraw-Hill, New York, 1952. (17) Shokin, I. N., Solov’eva, A. S., Zhur. Priklad. Khim. 26, 584 (19 53).
Ii
RECEIVED for review January 26, 1960 RESUBMITTED December 1, 1961 ACCEPTEDFebruary 20, 1962
= DB/DA = constant in equation for interfacial concentration
26th Annual Chemical Engineering Symposium, Division of Industrial and Engineering Chemistry, ACS, Baltimore, Md., December 1959. Work supported by a National Science Foundation postgraduate fellowship to T. F. Hatch, Jr., 1956-58.
= elapsed time of exposure of liquid to gas = distance along surface of jet
= kBit
CANONICAL FORMS FOR NONLINEAR KINETIC DIFFERENTIAL EQUATIONS W.
F. AMES
Department of Mechanical Engineering, Uniiiersity of Delaware, Newark, Del.
The evaluation of relative rate constants is difficult for complex chemical reactions because of the nonlinearity of the describing system of differential equations. A useful “matrix” method is described for the development of a canonical form for the kinetic differential equations. Three examples are drawn from the literature. The advantages of the procedure are simplification of the system of equations; automatic determination and elimination of redundancies; applicability to relative rate constant determination; and reduction of computation complexity.
PREVIOUS
paper (2) considered the problem of evaluating
A ratios of rate constants for systems of differential equations arising as the mathematical model of chemical reactions. The technique described was, generally, a n implicit procedure whereby the ratios of the rate constants were obtained in the form of implicit algebraic functions. These functions usually require solution by numerical means. While the problems solved (2) were real problems and the methods used are capable of generalization, no unifying pro214
I&EC FUNDAMENTALS
cedure for attacking another problem of similar type was included. This paper presents a useful general procedure for developing a canonical form for the system of differential equations. In turn this canonical form is of great use in evaluation of rate constant ratios. The motivation has been the application of matrix theory in the development of “normal” coordinates for the vibrations of linear elastic systems with many degrees of freedom, The normal coordinate procedure is based on the reduction to diagonal form of the coefficient matrix of
