1
THEORETICAL
1
3.32
L
I
0
t 0
4 O
HELIUM
co2
2
4
6
8
IO
12
14
c t
0 0
0
16
2
I
I
I
I
4
6
8
IO
P, A T M .
Figure 1. pressure
T =
- 19.0' C.
12
14
16
P, A T M .
C02-He counterdiffusional flux as a function of Alumina pellet:
1
HELIUM CO2
Figure 2. pressure
C02-He counterdiffusional flux as a function of
p = 1.01
Alumina pellet:
T = 0' C. p = 1.01
most simply pictured as due to an actual reduction in pore area for diffusional transport arising from the thickness of the adsorbed COz layer on the pore walls. I t is difficult to make a quantitative assessment of the differing effects of this in micropore and macropore structures, hov ever. Certainly the dimension of an adsorbed molecule of carbon dioxide is not negligible when compared to the average micropore radius (about 18 A,) reported by Rivarola and Smith, but there is reason to believe that the range of relative pressures of CO2 involved in the experimentation corresponds to multilayer coverage where the thickness of the adsorbed multilayer perceptibly decreases area available for transport in the macropores and essentially blocks completely the micropores. In any event, with adsorbed species on the pore Malls one does not know the correct radius or porosity for predicting gas phase transport. Thus, it is our belief that the subtraction
step used by Rivarola and Smith is open to some question and that it may not be possible to calculate the surface transport rates by simple subtraction of diffusion flow (calculated with a model) from a measured total flow.
SIR: Foster, Bliss, and Butt question the subtraction step for evaluating surface diffusion rates because of the unknown change in porosity and pore radius due to adsorption. However, the experimental data in Table I11 of Rivarola and Smith ( 7 ) show that this effect is not significant a t the operating conditions employed. T h e question is whether the adsorbed species reduces the pore radius sufficiently to affect the volume diffusion rate. This may be ascertained by comparing diffusion rates for hydrogen in the HrN2 and Ha-CO2 systems shown for pellet o-b in Table 111. Hydrogen does not absorb, so that its rate is a measure of the volume diffusion. This quantity is noted to be consistently less in the CO2 system than in the nitrogen case and the reduction may well be due to a decrease in effective radius due to adsorption of CO2. Nevertheless, the decrease is but 270, a figure approaching the reproducibility limit for such measurements.
The reason for the insignificance of the effect proposed by Foster, Bliss, and Butt in the results of Rivarola and Smith is probably the low surface coverage of adsorbed COn (10 to 30% of a monomolecular layer). However, a simple method exists for correcting the subtraction method so that it would be applicable even though the surface coverage were large. This is to measure the diffusion rate of a nonadsorbing gas, first in a binary system with the adsorbable gas and second in a binary system with a nonadsorbing gas. Comparison of the two rates indicates whether the effect is important. If it is, the rate of the nonadsorbing gas in the adsorbing system can be used to calculate the volume contribution of the adsorbing gas by the molecular weight relationship. Thus if the rate of the nonadsorbing gas is N A , the volume diffusion rate of the adsorbing component, B , is
580
l&EC F U N D A M E N T A L S
literature Cited
(1) Butt, J. B., Foster, R. N., Nature 211, 284 (1966). ( 2 ) Foster, R. N., Ph.D. dissertation, Yale University, 1966. ( 3 ) Rivarola, J. B., Smith, J. M., IND.ENC.CHEM.FUNDAMENTALS 3, 308 (1964). (4)LVicke, E., Kallenbach, R., Kolloid Z. 97, 135 (1941).
Richard N . Foster Harding Bliss J o h n B. Butt Yale C'niversity h'ew Haven, Conn.
given Rivarola and Smith-that tionable when: Alternatively, the measured value of i l ’ ~could be used, along with a pore model, to predict a revised pore radius which would take into account the decrease caused by the adsorption of B . Either approach xvould lead to a correct volume contribution for the adsorbable gas, which then could be subtracted from the observed total to obtain the surface contribution. These remarks are not intended to imply that the subtraction method will always lead to accurate surface migration rates. Rather they reaffirm that the serious restrictions are those
is, the method \vi11 be ques-
1. Adsorption-desorption rates are not rapid lvith respect to diffusion, so that equilibrium cannot be assumed between gas phase and surface concentrations. 2. T h e surface contribution is small, so that subtracting two relatively large quantities will not give an accurate difference. Literature Cited (1) Rivarola, J. B., Smith, J. M., IXD.ENG.CHEM.FUXDAMENTALS 3, 308 (1694).
J . M . Smith Uniuersity of Cahyornia Davis, Calif.
DYNAMICS OF FLOW-FORCED SYSTEMS IO
SIR: I n two recent publications (7, 2), solutions to the
O B
partial differential equations describing a flowforced heat exchanger and a plug flow tubular reactor were presented. T h e particular equations solved, using the nomenclature from the previous works, are
0 6 0 4
02
0 IO
0 08
for the heat exchanger (condensing steam as heating medium), and for the reactor,
0
+
0 06 0 04
a K
p W
002
3
k z
2
=
001
0008
0 006 0 004
These are nonstationary partial differential equations which can be solved by several methods. A solution to Equation 1 that can be obtained by manipulation of results presented earlier (7, 2) or by a n alternate method shown a t the end of this note is
T h e solutions to Equations 2 and 3 are similar to this. result is valid for arbitrary flowforcing functions r ( t ) .
0 002 02
0 4 0 6
I O
2
FREOUENCY
Figure 1 .
4
,
6
IO
20
100
4 0 6 0
cycles per minute
Bode plot for flow-forced heat exchanger r ( t ) = 0.1 12 cos ( u t )
The
Results
Some numerical results obtained for these systems have been reported (7, 2). Because much experimental work has been done on frequency rcsponse of such systems (4, 5), further numerical results were obtained here for the case where r ( t ) = A cos ut
2
0 06
c a
004
K W
3
002
-c (5)
T h e theoretical frequency response was obtained on a digital computer (less than 1 minute per case) by solving for 0 a t many values of t a t the outlet of the heat exchanger, using data corresponding to experimental work (3). T h e magnitude ratios so obtained are presented as functions of frequency and compared with experimental results (3) in Figures 1 and 2. As can be seen, the analytical results are in good agreement with the experimental data. T h e numerical solution of
f =
001
0 008
0 006 0 004
0 002 0 2
0 4 0 6
2
10
4
6
10
20
40
60
100
FREOUENCY, cycles per minute
Figure 2.
Bode plot for flow-forced heat exchanger r ( t ) = 0.523 cos (ut) VOL. 5
NO. 4
NOVEMBER 1 9 6 6
581
