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, signifi-
cantly 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 surface is that of a laminar jet of liquid which is surrounded
Tboth diffusion and chemical kinetic mechanisms affect the with the gases to be absorbed. The jet is formed in a smooth
rare of interphase mass transfer is demonstrated by several ex- nozzle that produces a nearly flat velocity profile. The
amples? such as the washing of acidic gases with alkaline solu- theory of purely physical absorption in such equipment has
tions, the manufacture of HiYOl, and the contacting of gaseous been carefully worked out (73), and observations of absorption
and liquid reagents in general. Quantitative expression of the rates, after comparison with the theory, lead to rather accurate
effect in these processes of diffusion and reaction rates has been knowledge of the contributions of the chemical reaction to the
developed (76) largely through the penetration theory, in absorption rate process.
which the partial differential equations of diffusion in stagnant The chemical system chosen for this study involves COz and
or uniformly flowing liquid have to be solved with reaction rate SH3 as the gaseous components and distilled water as the sol-
terms included. The mathematical problems involved are vent. The chemistry of this system has been studied by
orten difficult, but numerical methods have been resorted to Faurholt (7) and others ( 7 7), who found that the gases react in
where necessary. As a result, a considerable literature has solution by the following steps:
now built up from which the effect of a reaction process of
known kinetic behavior can be calculated.
+ XH3 NHsCOOH
COz + (1)
None of the previous work has extended, however, to those NEIzCOOH + NH3 NH: + NHzCOO-
+ (2)
NHzCOO- + HzO
situations in which two chemically active gases are dissolved
NHdCO; + (3)
simultaneously in a liquid, diffusing and reacting rapidly near
the interface after solution occurs. This is despite a t least two NHaCO; NH; + COi-
+ (4)
important practical exainples: the washing of a gas containing Reaction 1 involves the formation of carbamic acid, which
both COS and NHl with water in the Solvay process and the then picks up another NHB to form ammonium carbamate,
simultaneous removal of H,S and COZfrom a gas by washing which partially ionizes into the ammonium and carbamate
with a n alkaline liquid in the production of synthesis gas. ions. The latter slowly react with water to form the ammo-
In these cases, the rate of absorption of each substance is nium bicarbonate ion, which may dissociate into ammonium and
affected by the rate of the other, for the concentration distri- carbonate ions. There are, of course, ionic equilibria involv-
bution near the interface of one substance is affected by its ing several of the steps. Reaction 1 is rate determining in the
reaction with the other. Roper, Hatch, and Pigford (72) have absorption process because the second step, Equation 2, is very
given some approximate solutions to the transient diffusion fast and irreversible. These two reactions taken together in-
equations of the penetration theory applicable to these cases,
volve the consumption of t\vo molecules of "3 per molecule
and it was of interest to test out the theoretical results.
of CO, by a net reaction:
Unfortunately, existing empirical data from tests of operating
absorbers are not satisfactory for testing such theories, for the COz + 2NH3 + NH: + NHzCOO- (5)
rate of absorption per unit of surface area is never known ac-
According to Faurholt (7), Equation 5 follows a second-order
curately. Experimentaj. studies suitable for such a study have
rate law. The rate is proportional to the product of the con-
to be made carefully so that flow conditions will be well known
centrations of NH3 and COS. T h e third reaction, Equation 3>
and reproducible. The work described here, therefore, in-
is much slower than the others and occurs mostly outside the ex-
volves the idealized, laboratory-scale absorption apparatus of
perimental absorption apparatus. (In an industrial absorber,
hfanogue ( 9 ) and Scriven and Pigford ( 7 4 , in which the liquid
hobvever, in which the water phase is alloired to build up an
appreciable equilibrium vapor pressure of NH3 and CO?, the
Present address, Department of Mathematics, Harvard rate of the third step might be of some consequence if the liquid
University, Cambridge, Mass. hold-up is large.)

VOL 1 NO. 3 AUGUST 1962 209

Experimental wetted-wall column, for example, serious end effects and surface
Apparatus. Several experiments that have been carried rippling occur, indicating the presence of turbulent mixing
out to test the validity of penetration theories have involved within the falling liquid film. Nijsing and Kramers (70) and
equipment in which the interfacial surface was not smooth Scriven and Pigford (74) have employed an improved absorp-
and uniform, a condition that is required for the strict applica- tion device in which the liquid stream issues from a carefully
tion of the theoretical results. In the frequently used long. 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 ap-
paratus to be useful in their studies of possible interfacial re-
sistance 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 = 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 con-
tinuously 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
0 I 2 3 4 dissolved C 0 2 and had a n electrical conductivity smaller than 2
LENGTH, 4 , cm!"
JJET micromhos per cm. at 25 O C. Before entering the jet absorber,
the water flowed through copper coils to adjust its temperature
Figure 2. Rates of absorption of pure
to 25OC.
COn in water
Gas was fed to the absorption chamber tangentially a t its
Data corrected to q L = 5.02 cc. p e r second and bottom and left through a n opening a t the top. Ammonia,
Ai = AL = 3.26 X gram-mole p e r cc.
--_ Ideal jet, Equation 8 C 0 2 . He, and N2 (in various runs) were taken from cylinders
Predicted, Scriven and Pigford ( 1 4 ) and metered by calibrated rotameters so that composition
0 Scriven using brass nozzle could be determined. Flow rates from 800 to 2200 cc. per
X Fields (2) using brass nozzle
0 This work using stainless steel nozzle minute \*ere employed.

210 I&EC FUNDAMENTALS

 All runs in this investigation were made at a water flow of pressure, p ~ , ,was computed from the observed absorption
5.02 1 0 . 0 3 ml. per second, equivalent to a surface jet velocity rate, the known Henry's law coefficient for NH3, and Equation
of about 300 cm. per second and a contact time of about 9 to find Bt. As suggested by the dimensional analysis, the
0.04 second. absorption data were plotted in the form k*GBRT?rhd/quus.
The conductivity of the solutions produced in the absorber q,/D,Bh. I t was found, however, that an improved correla-
was observed with a ,Serfass conductivity bridge. The con- tion could be obtained by multiplying the first term by P L ~ / P .
ductivity cell constants were carefully determined with standard Figure 3 shows the final plot of the results.
solutions of KCI, using the data of MacInnes (7) and Shedlov- The straight line on the figure correlates the data for both
sky and hlacInnes (8, 75) to find their equivalent conductances. NHS-Nz and NH3-He, in which the NH3 diffusion coefficient
The use of these measurements for the computation of the ionic was nearly four times greater. The empirical equation for the
content of the solutions is described below. line is :
Absorption of COz. Measurements of the rate of absorp-
tion of CO2 in water h,id been made previously with this same
apparatus (73), but they were repeated to make sure that the indicating that the Stanton number for diffusion in the gas
apparatus was working properly and that some of the analytical phase was only slightly influenced by flow, diffusivhy, or jet
methods were sound. In these preliminary runs, the water length. T h e physical dimensions of the apparatus were not
entering and leaving tLle absorber was analyzed for dissolved varied, nor was the velocity of the jet changed, so that the
COz by releasing samples from a pipet underneath the surface work does not indicate whether these quantities are accounted
of an excess of standard h-aOH to which had been added a for correctly in the empirical formula, Equation 10. In fact,
few milliliters of BaCl solution to precipitate BaC03. T h e it seems very likely that the jet speed should appear. O n the
excess alkali was then ti.trated with si.andard acid. The results other hand, these variations were not pertinent in the further
of these measurements are shown in Figure 2: where they agree work described in this report, in which h and q L were constant.
closely with the line representing Scriven's data (73) and the Analysis of NH3-C02-H20 Mixtures. The effluent liquid
penetration theory according to Equation 9. from the jet absorber was analyzed for total dissolved NH3
Absorption of NH3. No previous experimental studies of and C O Z by making two measurements: T h e electrical con-
the gas-phase resistance to mass transfer in jet absorbers having ductivity of the solution was determined, and a known volume
been reported, it was necessary to determine this resistance by of the solution was added to a n excess of standard acid, the
observing rates of absorption of KH3 from NH3-Nz and "3- COZdriven off as a gas, and the solution back-titrated to the
H e mixtures. The liquid leaving the jet was analyzed for equivalence point with standard base. From these results,
dissolved N H 3 by releasing samples beneath the surface of an plus known values for ionic conductances and equilibrium
excess of standard HCl and back-titrating to a methyl red end constants, the concentrations of hydrogen, bicarbonate, car-
point. bonate, and ammonium ions were determined.
The flow rates of gas through the absorber were always small T h e decomposition of ammonium carbamate to ("4) 2co3
in these runs, so that the: flow regime must have been essentially was essentially complete in a few minutes after its formation,
laminar in every case, Although the flow conditions in the according to the studies of Faurholt (7): who measured the
gas phase did not correspond exactly to those for which mathe- following values of equilibrium constants :
matical 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: The largest concentration of dissolved NH3 observed in this
study was 0.0036 gram mole per liter, indicating that accord-
PB1 I- PH1 - k*CBRTxhd ing to Equation 11 the carbamate concentration was only
PBi .- PBi 4.
about 2% of the bicarbonate concentration when equilibrium
against the Peclet number, q,,/D,,h, should represent the em- had been achieved. A more recent determination of Kw.
pirical results. I n evaluating k*,,, which is based o n the by Shokin and Solov'eva (77) gave Kw. = 2.2 at 25" C.,
initial partial pressure difference, p~~ - psi,the small back 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 NH3-He

VOL 1 NO. 3 AUGUST 1962 211

Furthermore, a t the high N H 4 concentrations occurring in the 10-3 gram-ion per liter of total dissolved C0;- and 1.206
experiments in which both NH3 and COz were absorbed, gram-moles per liter of total NH3 was analyzed by the above
carbonate and bicarbonate ions were present a t about equal procedure, which was found to yield 0.6811 X 10-3 and
concentrations. Therefore, at equilibrium the carbamate ion 1.222 x 10-3 gram-moles per liter, respectively. .
made up no more than 1% of the total dissolved COZ.
Faurholt also determined the velocity of decomposition of Results
the carbamate ion, indicating that a t 18" C. the time required
The effect of the chemical reaction in the liquid on the rates
for 99% decomposition should be 17 minutes. Lower con-
of absorption of COz and NH3 simultaneously was studied by
centrations of NH3 and higher temperatures imply still speedier
absorbing from COz-NH3-N2 mixtures. Owing to its greater
decomposition. I t was observed in this work that 10 minutes
solubility, the mole fraction of NHI in the feed gas was limited
were sufficient for steady conductivity readings to be obtained,
to 0.04 to 0.10. The mole fraction of COZ varied from 0.15
showing that equilibrium must have been achieved in that
to 0.47; the molar ratio of NH3 to CO2 in the gas ranged
time. Actually, 15 to 20 minutes were allowed before final
betiyeen 0.12 and 0.61. Liquid leaving the absorber was
readings uere taken.
analyzed according to the scheme just described. The
The electrical conductivity of a n aqueous solution depends
composition of the gas entering the absorption chamber was
on the type and concentrations of the ions present. In dilute
computed from flow measurements with calibrated rotameters.
solutions. such as those analyzed in this study, the contributions
It was found necessary to dilute the gas mixture with h-2
of individual ionic species to the total conductivity are in-
to avoid the formation of a solid reaction product, probably
dependent of the other dissolved materials; that is. there is
(SH,)?COs, on the walls of the apparatus and in the upstream
essentially no interaction between ions. The total conductivity
tubing. M'hen this occurred, it became difficult to observe
is then given by
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
where the c's represent equivalent ionic conductances. The
was present as a diluent, the chamber walls became cloudy
accuracy of this equation \vas tested by measuring the con-
when the NH3 to CO1 ratio was much above 0.6, except when
ductivities of standard solutions of KH4HC03. Using re-
the total concentration of these two gases was kept very low.
ported values of ionic conductances (3-5). agreement be-
In the latter case, however, the amounts dissolving were too
tween expected and observed conductivities was within 1%.
small for reliable analysis. No precipitate was observed during
The following equivalent conductances were employed :
the runs for which data are reported.
The penetration theory of diffusion plus reaction applicable
Equivalent Conductance, to this situation, presented in a companion article (72), indi-
Ion ( M h ~ ) ( C m . ) / ( G r aEguiv.)
m
H+ c, = 349.8
OH - 62 = 198.6
NH: cj = 73.4
HCO; c4 = 44.5
CO, cs = 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 m

hydrogen ions needed to bring the neutral solution to p H = U
E
5.7, the end point employed in the analysis. 0 2.0
Equations 13 and 14 are supplemented by the mass action
a
f0
equilibrium expressions, with constants evaluated at 25°C. : n
K, = ( H + ) ( O H - ) = 1.008 x lO-'4 (gram-ion/liter)? (15) 8
w-
Y

KB = (NH,+)(OH-)/(NH40H) = 1.774 X 10-5 (gram-ion/liter)

I Y
K1 = (H+)(HCO-)/(H2C03) = 4.45 >(
(16)
10-7 (gram-ion/liter)
>
I s 1.5
I1
(17)
0
The symbol (HzC03) in the denominator of Equation 17
refers to the total dissolved C 0 2 that is either in the free, dis-
solved form or in the hydrated form.
K2 = (H+)(CO-)/(HCO-) = 4.69 X lo-" (gram-ion/liter)
(18'1 1.0
I 2 I
From Equations 13 to 18, four equations for the unknowns
(H+),(HCO,), (COJ, (NH:) were found. These Je
were solved for each absorption run with a digital computer, Figure 4. Predicted absorption coefficients for two
using a manual trial-and-error procedure. T o test the ac- gases undergoing irreversible, bimolecular reaction
curacy of the analysis, a solution known to contain 0.6738 X in solution

212 I&EC FUNDAMENTALS

cates that the average rate of absorption divided by the rate using A Q A = QA(0) - Q A ( l ) read from the theoretical curves
that Xvould occur without the chemical reaction, defined here of Figure 4 a t the experimental e values.
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
Discussion
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 The absorption rate for CO? appears to correspond closely
10-6 sq. cm. per second for COSand NH3 in water, respectively, to the penetration theory when the effect of reaction is allowed
at 25’ C.. the effect of m on Q can be. represented by: 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
There are two major difficulties in applying Equation 19 in
Bi, and Equation 21 was employed for this purpose on the
the present study: The interfacial concentration, B i , of NH3
assumption that the gas-phase resistance was essentially con-
was not constant, as assumed in the theory, owing to the large
stant. Under these conditions, Bi should be nearly an ex-
resistance to diffusion of NH3 through the gas; and Bi was not
ponential function of distance along the jet surface. This is
known reliably, so the theoretical effect of reaction on the
certainly nqt the boundary condition used in the theory for
rate of absorption of NH3 could not he tested thoroughly. T h e
Equation 19. Is it possible that the average value of Bi
two difficulties spring Yrom the same source, the much greater
chosen by Equation 21 is a good one in the sense that its use in
solubility of NH3 in wa.ter than COS. Since rn is much smaller
the penetration theory, instead of the truly constant value
than unity, the concentration of NH3 throughout the portion
assumed there, approximates the right answer?
of the liquid phase where COS is also present for the reaction
To answer this question a “film theory”-i.e.: a theory of
is essentially equal to the interfacial concentration of XH3:
steady state diffision-was worked out for the condition of
the liquid diffusion effect being of little significance so far as
exponentially Lrarying interfacial concentration, B,(i) =
SH3 is concerned. The principal effect of diffusion on the
constant X e-Tz. It turns out (6) that when the number of
SH3 concentration occurs in the gas phase and results in a
gas-phase transfer units, rh! is small: the introduction into the
time-dependence of Ei,not allowed for in the elementary
steady state diffusion problem of an interfacial concentration
penetration theory.
that varies across the surface of the jet as a boundary condition
The key computation required for the application of Equa-
produces only a small correction to the procedure adopted in
tion 19 to the data is the estimation of Bi. This was accom-
computing the average value of Bi as described above.
plished by using Equation 10 to estimate the gas-phase mass
It may be concluded, therefore! that the penetration theory
transfer coefficient, them employing Equation 9 to get the re-
is confirmed by the experiments described here, at least for
sistance to purely physical absorption in the liquid phase.
conditions closely approximating pseudo first-order reaction in
The interfacial partial pressure of NH3 was, therefore, deter-
the fluid.
mined by dividing the total driving force in the proportion of
the tw.0 resistances, according to :

or

2.5

The value of B , in the COZ absorption runs was then obtained h

with the Henry’s law {constant for KH3, HB. Naturally, this 0
II
method for estimating B , precludes attaching any significance E
to the measured absorption rates of KH3. insofar as their 0
c
dependence on the chemical reaction is concerned. U
The interfacial partial pressure of COS, @ A I , was found by g 2.0
0
Q
subtracting the equilibrium vapor pressure of water from t
the total pressure and assuming that the remainder was s
Y
divided between COZ m d NS according to their relative pro-
portions in the bulk of ihe gas.
The ratio of mass transfer coefficients for CO,, Qa, was cal-
culated for each run as follows:

whence, from Equation 9 :

d / p L / D a h A P - Ai
Q ~ e x p . =: -__
3.878
--
Ai - Ai
The results of the C02-NH3 absorption runs are shown in ._
1.0 2.0 5.0
Figure 5. Experimental values of QA from Equation 23
were corrected to truly first-order conditions (m = 0) before 4
plotting by means of Equation 19:
Figure 5. Rates of absorption of COz from COZ-
Qa(0) = Qa(expt1.) $- m4”aQ.4 (24) NHy-N2 mixtures

VOL. 1 NO. 3 AUGUST 1962 213

Nomenclature Y = stoichiometric coefficient
$AO = absorption rate for whole jet
A,, Bi = interfacial concentrations of A (COz), B (SH3)
in liquid Literature Cifed
A I , B1 = initial concentrations of A, B in liquid
A2 = bulk-average concentration of A in jet emerging (1) Faurholt, C., J . Chem. Phys. 22, 1 (1925).
2) Fields, M. C., M. Ch. E. thesis, Univ. of Delaware, 1958.
from absorption chamber 3 Harned, H. S., Davis, R., J . Am. Chem. SOC.65,2030 (1940).
Dd,DS = diffusivities of dissolved gases A, B in liquid
DvB
d
= diffusivity of B in gas
= jet diameter
Ii
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
h = jet length (1941).
K,, KB, K1, K2 = mass action equilibrium constants (6) Hatch, T., M. Ch. E. thesis, Univ. of Delaware, 1958.
k = second-order reaction rate constant for reaction (7) MacInnes, D. A., “The Principles of Electrochemistry,”
between A and B in liquid phase p. 339, Reinhold, New York, 1939.
(8) MacInnes, D. A., Shedlovsky, T., Longsworth, L. G., J . Am.
= cumulative average mass transfer coefficient for A Chem. SOL.54, 2758 (1932).
without chemical reaction (9) Manogue, W. H., Ph.D. thesis in Chem. Eng., Univ. of
TLA = cumulative time-average mass transfer coefficient Delaware, 1953.
for A (10) Nijsing, R. A. T. O., dissertation, Delft, 1957; Nijsing,
k*GB = average mass transfer coefficient for B R. A. T. O., Kramers, H., Chem. Eng. Sci.8, 81-9 (1958).
m = vA,/B, (11) Pinsent. B. R. W., -
Pearson, L., Roughton, F. J. \V., Trans.
= average rate of absorption of A in j e t ‘ Faraday SOC. 52, 1594 (1956).
Z A
JVAO = average rate of absorption of a nonreacting gas in (12 Roper, G. H., Hatch, T. F., Jr., Pigford, R. L., IND.ENG.
&EM., FUNDAhlENTALS 1, 144-52 (1962).
jet (13) Scriven, L. E., Pigford, R. L., A . I. Ch. E. Journal 5, 397-402
P = total pressure (1959).
PAW = arithmetic average partial pressure of inert gas (14) Zbid., 4, 439 (1958).
p~~ = partial pressure of B (15) Shedlovsky, T., MacInnes, D. A., J . Am. Chem. SOG.57, 1705
PA;, p~~ = interfacial partial pressures of A and B (1935).
Q = average absorption rate with reaction/rate without (16) Sherwood, T. K., Pigford, R. L., “Absorption and Extrac-
reaction tion,” 2nd ed., McGraw-Hill, New York, 1952.
= ratio of mass transfer coefficients for A (17) Shokin, I. N., Solov’eva, A. S., Zhur. Priklad. Khim. 26, 584
QA
(19 53).
qL = volumetric flow rate of liquid stream through
nozzle RECEIVED
for review January 26, 1960
= volumetric flow rate of gas mixture into apparatus RESUBMITTED December 1, 1961
F
r
= DB/DA
= constant in equation for interfacial concentration
ACCEPTEDFebruary 20, 1962
26th Annual Chemical Engineering Symposium, Division of Indus-
t = elapsed time of exposure of liquid to gas trial and Engineering Chemistry, ACS, Baltimore, Md., December
z = distance along surface of jet 1959. Work supported by a National Science Foundation post-
e = kBit graduate fellowship to T. F. Hatch, Jr., 1956-58.

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 non-
linearity 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 cedure for attacking another problem of similar type was in-
A ratios of rate constants for systems of differential equations cluded. This paper presents a useful general procedure for
arising as the mathematical model of chemical reactions. The developing a canonical form for the system of differential equa-
technique described was, generally, a n implicit procedure tions. In turn this canonical form is of great use in evaluation
whereby the ratios of the rate constants were obtained in the of rate constant ratios. The motivation has been the applica-
form of implicit algebraic functions. These functions usually tion of matrix theory in the development of “normal” co-
require solution by numerical means. ordinates for the vibrations of linear elastic systems with many
While the problems solved (2) were real problems and the degrees of freedom, The normal coordinate procedure is based
methods used are capable of generalization, no unifying pro- on the reduction to diagonal form of the coefficient matrix of

214 I&EC FUNDAMENTALS