reactant concentration m d analyzing the outlet stream withopt resorting to specially designed apparatus In conclusion. some analytical verifications will be given. Let us consider a cascade of n ideally mixed continuous tank reactors u i t h equal volumes (and equal temperature). The transient behavior corrrsponding to a stepwise change of inlet reactant concentration from c, (system initially in steady conditions) to c l is given by (5)
where 0 is the average residence time of a single reactor. and Q= 1 k 0 . The response of the same system to an impulsive reactant injection in the inlet stream is easily found to be
+
C"
=
Qt tn-' - G- I+ - e M e ___ Q" 0" (n - l ) !
(13)
Now by applying Equations 10 and 11, respectively, to Equations 12 and 13 we have immediately
Nomenclature
reactant concentration entering and leaving, respectively (steady condition) c z ( t ) , c ( t ) = reactant concentration entering and leaving, respectively (unsteady condition) €2, c = Laplace transforms of cp(t) and c ( t ) ci(t) = function representing both steady and unsteady reactant concentration of entering stream f(r) = distribution function for residence time k = first-order reaction-velocity constant L = Laplace transform operator L -1 = inverse Laplace transform operator M = amount of reactant injected per unit of flow rate S = complex variable t = time 6(t) = unit impulse function e = mean residence time 7 = residence time cl, c,
=
literature Cited
(1) Asbyornsen, 0. A , , Chem. Eng. Sci.14, 211 (1961). (2) Foraboschi, F. P., Atti V o Corso Fond. Donegani, Accademia dei Lincei, Rorna, 1960. (3) Lelli, U.; Salvigni? S., Ing. Chim. I d . 1, 15 (1965). (4) MacDonald, R. W., Piret, E. L., Chem. Eng. Progr. 47, 363 (1951). (5) Mason, D. R., Piret, E. L., Ind. Eng. Chem. 43, 1210 (1951). (6) Pearson, J. R. A., Richards, G. M., Skoczylas, H., Chem. Eng. Sct. 19, 82 (1964).
_ _T
e f(T)
=
BTn-1
8" (n
-
I)!
UGO LELLI Istituto di Impianti Chimici C'niversiti2 di Bologna Bologna, Italia
which is the well known equation of residence time distribution for a cascade of n ideally mixed tank reactors.
RECEIVED for review October 27, 1964 ACCEPTEDMarch 31, 1965
COM MU N ICATION
MASS TRANSFER I N HORIZONTAL ANNULAR GAS-LIQUID FLOW The momentum-mass transfer analogy appears to be applicable to the Con-water data of Bollinger and the "8-water data of Anderson, Bollinger, and Lamb for horizontal annular flow. Average absolute are obtained between calculated and experimental data for the Con-water and deviations of 17 and 1 1 "3-water systems.
yo
Bollinger, and Lamb ( 7 ) have recently presented data and a n analysi,j for gas-phase controlled mass transfer in horizontal annular flow. Data were obtained for the absorption of NH3 from air to water flowing in a 1-inch pipe.
A
NDERSON,
Their analysis indicated that an interchange model could be applied to these data. Bollinger (2) has reported data for the absorption of CO2 from air to water in annular flow in a 1-inch pipe. These data represent liquid-phase controlled mass transfer. This paper shows that the momentum-mass transfer analogy can be applied to the correlation of both the gas-phase and liquid-phase controlled experimental data.
Theory
The equation for mass transfer between gas and liquid phases with resistance in both phases is of the form ( 6 ) 1 1 H - koa (1) &a kLa or 1 1 1 - - _ (2) KLa kLa + %a
+-
For either cocurrent or countercurrent flow of gas and liquid in a pipe these equations are VOL. 4
NO. 3
AUGUST 1965
361
1
1
H
K c a t - kcat
+
k,;l
(3)
or
1
1
1
KLZ - kLZ
(4)
+
If holdup in gas-liquid flow is defined by RL for the liquid and RGfor the gas phase
Substituting in Equations 3 and 4
RG Re _ -- _ KG kc
'I
10 KL (experimental)
Figure 1.
COSabsorption data
100
+-HRL k~
(5)
The holdups can be calculated from an empirical correlation (3). The momentum-mass transfer analogy can be used to calculate the mass transfer coefficients, k G and k L . A friction factor is determined for the gas phase from the irreversible energy loss contribution to gas-liquid pressure drop ( 4 ) .
Mass transfer coefficients can then be estimated by the method of Kropholler and Carr (5). For the gas
and for the liquid
Q is a function of the Reynolds and Schmidt numbers. Reynolds number is defined as
ReL
=
The
DCVLPL ~
WL
---Experimental data -Calculated with kL --Calculated without kL
for the liquid phase and
for the gas phase. Analysis of Experimental Data
KO (experimental)
Figure 3. 362
NH3 absorption data
l&EC FUNDAMENTALS
Equation 6 reduces to KL = kL for the C O r w a t e r system because of the large Henry's law constant; then Equations 7 and 9 are applicable to Bollinger's data with KL = kL. Values of K L obtained with these equations show an average absolute deviation of 17% from the 16 values of the experimental data. Figure 1 compares the experimental arid calculated data. The NH3-water system represents a gas-phase controlled system but the liquid-phase resistance is not always negligible. Equations 7 and 9 can be used to calculate kL and Equations 7 and 8 for ko. KG is then calculated from Equation 5. Figure 2 compares calculated and the experimental data. of Anderson et ul. for ReL = 3500. Calculated values are also shown for the case in which the liquid-phase resistance is neglected-i.e., Kc = kc. It is apparent that the liquid-phase resistance should not be neglected. An average absolute deviation of 11% is obtained between calculated and experimental values of KG for the 32 values of the experimental data. Figure 3 shows a comparison of these data.
Nomenclature
SUBSCRIPTS
-1, a
G L
= = = =
a‘ ~
il
DC -
= =
fG
= =
i?C
H Kc
= =
KL
=
k
