Lilich, L. S., Shalygin, V. M., Vestn. Leningrad Univ. 21 (22), Ser. Fiz. Chim. No. 4, 104 (1966). Mascherpa, G., Rev. Chim. Minirale 2, 379 (1965). Potier, A., Ann. Fac. Sci. Univ. Toulouse 20, 1 (1956). Redlich, O., Rosenfeld, P., "Landolt-Bornstein Tabellen," 3rd Suppl., p. 2145, Springer, Berlin, 1935. Robinson, R. A., Baker, 0. J., Proc. Roy. Sod. New Zealand 76, 260 (1946).
Vandoni, M. R., Laudy, M., J . Chim. Phys. 49, 99 (1952). RECEiVED for review August 17, 1967 ACCEPTEDJanuary 11, 1968 Twelfth in a series on Thermodynamics of Solutions. For the convenience of the reader, references to the preceding papers follow.
I to IV. 0. Redlich and A. T. Kister, Znd. Eng. Chem. 40, 341, 345 (1948); J . Chem. Phys. 15,849 (1947); J . Am. Chem. SOC.71,505 (1949).
V. 0. Redlich and J. N. S. Kwong, Chem. Reus. 44,233 (1949). VI. 0. Redlich, A. T . Kister, and C. E. Turnquist, Chem. Eng. Prog. Symp. Ser. 48, No. 2, 49 (1952). VII. 0. Redlich and A. T. Kister, J . Chem. Phys. 36, 2002 (1962). VIII. 0. Redlich and A. K. Dunlop, Chem. Eng. Prog., Symp. Ser. 59, No. 44, 95 (1963). IX. F. J. Ackerman and 0. Redlich, J . Chem. Phys. 38, 2740 (1963).
X. G. W. Lundberg, J . Chem. Eng. Data 9, 193 (1964). XI. 0. Redlich, F. J. Ackerman, R. D. Gunn, Max Jacobson, and S. Lau, IND.ENC.CHEM.FUNDAMENTALS 4,396 (1965).
SURFACE DIFFUSION OF CHEMISORBED HYDROGEN ON NICKEL CHARLES N. SATTERFIELD AND H l R O S H l IINOl Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass.
02139
From steady-state flux measurements through a porous nickel disk, the surface diffusion coefficient for hydrogen on nickel at 37" C. was measured directly and found to be about 3 X lO-'sq. cm. per second. 40 microns The surface diffusion flux was a substantial portion of the total flux only at low pressures-e.g, of Hg. The value for the diffusion coefficient agrees with the results of Gomer et al. obtained by field emission microscopy but is much smaller than that predicted by the indirect thermodynamic calculations of Rideal and Sweett.
catalytic reaction involves a t least one species and many investigators postulate that migration or surface diffusion of chemisorbed species on the catalyst plays an important role in their reaction kinetics. Knowledge of the magnitude of the flux of chemisorbed species is essential to an understanding of many aspects of contact catalysis. Surface diffusion of physically adsorbed molecules has been extensively studied-see, for example, the recent reviews by Dacey (1965) and Field et al. (1963). Little has been reported on surface diffusion of chemisorbed species, although they would be expected to be bound more strongly on the surface and hence to be less mobile than physically adsorbed species. T h e diffusion of hydrogen on nickel was chosen for study here because the chemisorption is reversible and the system is important in heterogeneous catalysis. T h e only previous direct study was that of Gomer et al. (1957), using field emission microscopy. Rideal and Sweett (1960) report some indirect conclusions based on thermodynamic calculations from adsorption isotherms. T h e method used here involved measurement of the steady flow of hydrogen through a porous disk of metallic nickel formed by pressing very fine nickel powder under high pressure. The difference between the observed flow and that predicted to occur by Knudsen diffusion (the latter determined from measurements with a nonadsorbing gas, helium) is attributed to surface diffusion. I t is necessary to control the properties of the porous disk and to choose experimental condiHETEROGENEOUS
A chemisorbed
Present address, Matsuyama Plant, Teijin, Ltd., Kita-yoshida, Matsuyama-shi, Ehime-ken, Japan 214
I&EC FUNDAMENTALS
tions carefully, so that the surface diffusion flux is a n appreciable fraction of the total. T h e surface diffusion flux, N,, may be expressed by:
where
T,
accounts for the tortuous path of surface diffusion and
Ds
= aZye-EaIRT = Doe-EaIRT
(2)
T h e Knudsen diffusion flux, N K , is expressed by:
(3) where
DK
=
re =
9700 r , d T / M 20
PA1
- 0)S,
T h e ratio of surface to Knudsen flux is then:
From Equation 7 it is seen that surface diffusion is favored by low porosity, high surface area per unit weight, and steep slope of the adsorption isotherm. Low pressure favors surface
* 4
i
Y!
1.2 V4CdUU PUVP
Figure 1 .
Diffusion cell assembly
Nickel disk F i , Fp. FllaskS GI,G2. AkLeod gages M. Mercury pressure balancer N1, NP. liquid nitrogen traps SI,S2. Metal-glass seals
0.8 0
D.
diffusion because the slope of the adsorption isotherm increases greatly with decreasing pressure. Increasing temperature should decrease the surface flux, since the surface concentration, C, will decrease more rapidly than D, will increase. Mathematical analysis of the time required to reach essentially steady state (Iino, 1966) reveals the importance of working with as thin porous disks as possible. Apparatus a n d Procedure
T h e apparatus essentially consisted of a steel diffusion cell containing a porous nickel disk, two 1-liter glass flasks, two liquid nitrogen traps, two McLeod gages, and a mercury pressure balancer as shown in Figure 1. T h e diffusion cell was connected to the two fl.asks with metal-glass seals. Porous disks were made from finely divided, air-stabilized, spectroscopically pure nickel powder, supplied by the National Research Corp., Cambridge, Mass., which had a surface area of about 75 sq. meters per gram. This was reduced in hydrogen for a week a t 170' to 190' C. (after which the surface area was about 5.1 sq. meters per gram by Kr adsorption), and then compressed in the diffusion cell with two pistons made of Sparta air-hardened steel. A considerable number of disks were made, but difficulties were encountered in several with large permeability of the disk or mechanical breakage. All the results reported here were obtained with two disks designated as disk 6 and d.isk 19. Disk 6 was compressed a t 373,000 p.s.i. for 18 minures and disk 19 at 466,000 p.s.i. for 60 minutes. T h e cross-sectional area of each disk was 0.713 sq. cm. and their thickne5,ses were 0.378 and 0.373 mm., respectively. Disk 6 was exposed to air before being sealed into the diffusion system, but disk 19 was sealed without being exposed to air, by maintaining it at all times in an inert gas stream. Presumably its surface was somewhat cleaner than that of disk 6. I n each case the disk in the diffusion cell \vas subsequently heat-treated ;at 80' C. for 1 hour in hydrogen a t atmospheric pressure, evacuated, heat-treated again in the presence of hydrogen at 813' C., and evacuated again at 80' C. T h e traps were immersed in liquid nitrogen from the first evacuation of the system to the end of all the experiments. Cylinder hydrogen (prepurified grade, Airco) \vas further purified by passage through a 1-foot long piece of glass tubing packed with Pd asbestos and heated to 300' C. and then through alumina desiccant cooled to liquid nitrogen temperature. Cylinder helium (A.irco) was passed through a bed of nickel powder that had been previously reduced with hydrogen and evacuated at high temperature and then cooled down to room temperature in vacuum. T h e temperature of the disk was controlled at the desired level, but the remainder of the system except for the traps was at room temperature. Permea-
I*; 200
400
600
800
Pi F H g
Figure 2.
M e a n permeability of disk 6 a t
37' C.
Order of experiments indicated b y enclosed number 0 Hydrogen 0 Helium
bilities were measured by continuously evacuating the downstream side of the disk and adding gas intermittently to the upstream flask to keep the pressure essentially constant. T h e flux was calculated from the quantity of gas that had disappeared from the upstream side. Usually several measurements were made in a series with hydrogen, after which the entire system was evacuated for 24 hours at 37' C. before a subsequent series of measurements with helium. After these the system was again evacuated and saturated with hydrogen at some specified pressure before the next series of hydrogen runs were started. Details of apparatus and procedure and analysis of data are given by Iino (1 966). Fluxes are expressed in terms of the permeability, y1, defined by :
y1 is related to the effective diffusion coefficient defined by
Equation 4 by:
For comparison with the hydrogen permeability the helium permeability was multiplied by d4.003/2.016 = 1.409. V I was 1350 cc. for disk 6 and 2378 cc. for disk 19. Equation 9 applies strictly only when gas pressures are measured a t 25' C. A room temperature correction factor, f, was therefore applied, such that is independent of room temperature and is directly proportional to Deft. T h e variation off from unity was less than 3% in almost all cases.
fy
Results
Most of the studies were carried out at 37' C. At higher temperatures sintering of the disks occurred-for example, a t 62' C. the permeability changed by about 7% per day, which was too large a change ivith time to detect a small surface diffusion flux accurately. Although the porous structure could probably have been stabilized by pretreatment a t a n elevated temperature, presumably the surface area would have been so reduced as to lower the surface diffusion flux to a n insignificant fraction of the total. T h e results with disk 6 are summarized in Figure 2. T h e sequence of usable results VOL 7
NO. 2
MAY 1968
215
h
m
I =! 0
5.2
.
h
140
I
t
1
u
tJI
I 7
1
5.0
t
c J
5 Q
4.0
W
8otL
1
l Y W
a
z
'
4.6
4
2
-
4.4
0
1-h
200
400
600
800
P i uHg
Figure 3.
Mean permeobilityof disk
20
19 a t 37' C, 0
Order of experiments indicated by enclosed number
u
if
0 Helium
consisted of seven series of runs, each series consisting of from four to nine runs made under the same conditions. Helium and hydrogen were studied alternately in sequence, except that series 5 and 6 were both with helium. 'The lengths of the solid arroivs (for hydrogen) and the dotted arrows (for helium) correspond to the 95% confidence limits of hydrogen and helium permeability for each series of runs. Data points are given for each run except for those in series 7, in which they overlap too closely for representation. T h e nine runs in this series gave permeabilities varying between the extreme values of 2.03 2nd 2.31. T h e horizontal dashed lines correspond to the 95% confidence limits for the entire series of helium runs at all pressures and to those for the hydrogen runs at low pressure. This group of seven series of runs extended over a period of 4 weeks, during which the helium permeability did not change to a detectable degree, indicating that no alteration to the porous character of the disk occurred during this time. \Vhen hydrogen was admitted to one side of the evacuated system, the other side being closed, for more than a day the upstream pressure tended to decrease more rapidly and the downstream pressure increase more slowly than Ivould be calculated assuming steady state had been established with respect to adsorption in the disk. Presumably, this represented an over-all slow adsorption of hydrogen on the disk caused by the shift in gasphase partial pressure distribution through the plug affecting the surface concentration distribution. T h e "unsteady-state" data \\-ere not used in the calculation of mean permeability. Figure 3 shows the results of the permeability studies on disk 19, also a t 37' C. This disk was heated to 48' C. for 12 hours between the third and fourth series of runs and comparison of series 1 and 4 shows a 3y0 increase in helium permeability, indicating that a slight degree of sintering of nickel occurred even at this relatively low temperature. At the relatively high pressure range of 600 to 800 microns of Hg, on both disks the permeabilities of hydrogen and helium follow the l / d Z l a w , indicating that there was no observable contribution by surface diffusion to Knudsen diffusion. At the low pressure of 40 microns of Hg, however, the hydrogen permeability was significantly higher than the helium permeability, especially in the case of disk 6 (Figure 2), where the hydrogen permeability was 216
I&EC FUNDAMENTALS
100
10
1
nyarogen
1000
PRESSURE IJ. Hg
Figure 4. Adsorption isotherms of hydrogen on reduced nickel powder at 37" and 62' C. Total surface area 18.26 sq. meters
40Cr, higher than would be predicted from the measurements with helium. T h e adsorption isotherms of hydrogen on reduced nickel powder were measured in separate experiments, with results shown in Figure 4. T h e powder was first reduced at 1-atm. hydrogen pressure a t 180' C. for 40 days. As has been observed by other experimenters, a rapid adsorption was followed by slow uptake of gas over a period of days and Figure 4 shows the amounts of hydrogen adsorbed after a t least 10 days. Before the adsorption studies the nickel powder was first evacuated for two days at 62' C. and 1.5 microns of Hg, so amounts adsorbed represent the incremental amount over that left by evacuation at 62' C. The adsorption study most closely related to ours is that of Kman and Kinuyama (1956) on reduced nickel powder, which covered the temperature range of 0' to 300' C. and a pressure range of about to 100 mm. of Hg. Slight extrapolation of their results indicates that the surface coverage of hydrogen on our nickel under evacuation conditions was about 407,. T h e maximum adsorption observed here, 127 pmoles, would correspond to 527, surface coverage based on a total area of 18.3 sq. meters (by BET krypton adsorption) taking 12.36 sq. A. as the surface area of nickel occupied by two hydrogen atoms. Consequently, the nickel powder was presumably approaching saturation a t about 1 mm. of Hg. T h e isoteric heat of adsorption was calculated from the van't Hoff equation to be 10.9 kcal. per mole from measurements over the range of 37' to 62' C. under conditions corresponding to high surface coverage. T h e surface diffusion coefficient divided by the tortuosity factor, Ds/rs,was calculated from the following equation :
y - -ADs p f ( i ~
A
¶
-
7 8
e)s,
dCs -
dl
-ADs
= ___ 7 8
X
Table I. Summary of Results Disk h'n . 6 79 5.2 8.3 Sg,sq. m./g. 0.114 0.0843 e 55.6 24.9 2.28 x 10-4 1.18 x 10-3 1.5 23.7 . PI, Hz, P Hg 40.9 37.7 0.4 $ 2 , Hz, P Hg 1.2 N . x 10j-8, g. mole/hr. 1.81 1.13 (Ba/r8)X lO+Q, sq. cm./se:c. 1.28 0.47
..
h', is the difference between the observed hydrogen flux and that \I.hich would be predicted from the helium flux by applying the 1 , d G law. T h e term [pt(l - O)S,C,] is the surface concentration of adsorbed species per unit volume of porous disk, values of which were estimated from the d a t a of Kinuyama and Kwan combined with hydrogen adsorption data on the disk and the adsorption isotherm on the reduced powder. Mean values of the variables of interest a t the low pressure which maximizes the surface diffusion effect are given in Table 1.
T h e value of D s / r s for disk 19 is relatively inaccurate, representing a small difference between two large numbers, so the more reliable value of B S / r sis 1.3 X 10-9 sq. cm. per second. From Kwan and Kinuyama's data (1956) the surface coverage a t the upstream and downstream faces of the plug is estimzted to be 0.56 and 0.46, respectively. Discussion
T o achieve steady-state conditions within a reasonable time it was necessary here to prepare exceedingly thin disks which had to be compressed to relatively small void fractions for mechanical strength. T h e resulting structures were probably highly anisotropic and the greatly differing values of 7 K for the t\vo disks indicate that the different methods used for their preparation must have caused the resulting pore structures to be very dissimilar. There is little basis for estimating the value of the tortuosity factor for surface diffusion, 7 8 . Barrer (1963) suggests that surface flow should follow a more tortuous path than gas phase flow and hence T~ > 7K. However, the Knudsen tortuosity factor, T ~ is, in effect a n adjustable parameter which provides for all the deviations between a n ideal pore model for Knudsen diffusion and the complexities of a n actual pore structure, including the actual pore size distribution and various anisotropic effects. I n the absence of more specific guidance we take as a n approximation, 7K = 7 5 , from which D,becomes about 3 X 10-8sq. cm. per second a t 37' C. Gomer et al. (1957) reported a value for the activation energy of surface diffusion of hydrogen on nickel of 7 k 1 kcal. per mole from field emission microscopy measurements over the range of about 258 O to 275' K. Taking their data a t 266' K. for the average distance traversed by a diffusing particle (6.4 X cm.) and time (70 seconds) and the above activation energy, the diffusion coefficient a t 37' C. (310' K.) is about 4 X 10-8 sq. cni. per second. Alternatively, if one takes the commonly used! values of 3 X 10-8 cm. and 10l2 set.-' for j u m p length ancl j u m p frequency in Equation 2 with a n activation energy of 7 kcal., D, becomes about 10-8 sq. cm. per second. These values agree remarkably closely with ours. Rideal and Sweett (1960) calculated the fraction of mobile adsorbate thermodynamically from adsorption isosters for the HZ-Ni system. Their calculation involves two major assump-
tions: no thermodynamic interaction between adsorbate and adsorbent and a n energetically uniform surface. They report the fraction a t 50y0 of surface coverage to be 0.064 a t 25' C. and 0.071 a t 37' C. Trapnell and Hayward (1964) calculated the activation energy of surface diffusion from Rideal and Sweett's data and report values of 3.4 to 3.0 kcal. per mole at the fraction coverage in our studies. T h e exponential term becomes about 0.005 a t 37' C. Taking the pre-exponential term as about cm. per second, the diffusion coefficient itself becomes about 5 X 10-6 sq. cm. per second a t 37' C., a value much greater than that found here. T h e assumptions involved in the calculation make this a highly uncertain result and less reliable than the values determined by direct measurement. Surface diffusion coefficients for physically adsorbed species to 10-6 are generally reported to be of the magnitude of sq. cm. per second. Sladek (1967) recently found surface diffusion coefficients for hydrogen chemisorbed on platinum to be about 10-6 to 10-7 sq. cm. per second a t 50' to 110' C. T h e few previous studies of boundary-free diffusion of chemisorbed species by field emission microscopy show values in the sq. cm. per second for systems general range of 10-9 to such as hydrogen or oxygen on tungsten at temperatures a t which this form of diffusion is observed. These are about 240' to 290' K. for hydrogen on tungsten (Gomer et al., 1957) and in the range of 500' K . for oxygen on tungsten (Gomer and Hulm, 1957). With chemisorbed species having a surface diffusion coefficient of about 10-8 sq. cm. per second a t 37' C. as found here, and activation energy of 7 kcal. per gram mole, surface diffusion can make no significant contribution to the intraparticle flux in catalyst particles of typical dimensions of 0.1 to 1 cm., except when the vapor pressure of the diffusing substance is very low, conditions seldom encountered in industrial catalysis. I n multifunctional catalysis, however, where path lengths may be of the order of 10 to 100 A., the gas-phase concentration of intermediates may be extremely low and surface transport can predominate over gasphase diffusion. Ac knowledgment
We acknowledge the generous financial support of Teijin, Ltd., Japan, of Hiroshi Iino. T h e National Research Corp., division of Norton Co., kindly supplied the air-stabilized nickel powder. Nomenclature
A
= cross sectional area for diffusion, sq. cm.
j u m p length, cm. gas-phase concentration, g. mole/cc. surface concentration, g. mole/sq. cm. DO pre-exponential factor for diffusion coefficient (Equation 2), sq. cm./sec. Deff = effective diffusion coefficient, sq. cm./sec. D K = Knudsen diffusion coefficient for round capillary, sq. cm./sec. D, = surface diffusion coefficient, sq. cm./sec. 0.for mean value E, = activation energy for surface diffusion, cal./g. mole L = thickness of disk, cm. 1 = diffusion length, cm. M = molecular weight N K = Knudsen diffusion flux, g. mole/sec. N , = surface diffusion flux, g. mole/sec. p = pressure, 1.1 H g R = gas constant, cal./(g. mole)(' K.) = equivalent radius of pore, cm. re S, = total surface area of porous disk, sq. cm./g. T = absolute temperature, ' K. a
C, C,
= = = =
VOL 7
NO, 2
M A Y 1968
217
t
VI y1
e
pt rK T~
time, sec., min., or hr. volume of upstream side of apparatus, cc. = permeability, see Equation 9 = void fraction = true density of solid (in nonporous form), g./cc. = tortuosity factor for Knudsen diffusion = tortuosity factor for surface diffusion = =
SUBSCRIPTS 1, 2 = upstream and downstream conditions Literature Cited
Barrer, R. M., Appl. Materials Res. 2, 129 (1963). Dacey, J. R., Ind. Eng. Chem. 57, No. 6, 27 (1965). Field, G. J., Watts, H., Weller, K. R., Rev. Pure Appl. Chem. 13, 2 (1963).
Gomer, R., Hulm, J. K., J . Chem. Phys. 27, 1363 (1957). Gomer, R., Wortman, R., Lundy, R., J . Chem. Phys. 26, 1147 (1957). Iino, Hiroshi, Sc. D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1966. Kwan, T., Kinuyama, T., J. Res. Inst. Catalysis, Hokkaido Uniu. 4, 199 (1956). Rideal, E. K., Sweett, F., Proc. Roy. SOC. A257, 291 (1960); “Actes du Deuxime Congres International de Catalyse, Paris, 1960,” p. 175, Editions Technip, Paris, 1961. Sladek, K. J., Sc. D. thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, 1967. Trapnell, B. M. JV., Hayward, D. D., “Chemisorption,” p. 301, Butterworths, LVashington, 1964. Wortman, R., Gomer, R., Lundy, R., J. Chem. Phys. 27, 1099 (1957). for review February 20, 1967 RECEIVED ACCEPTED November 30, 1967
EFFECT OF COMBUSTION TIME ON THE APPEARANCE OF VIBRATING DIFFUSION FLAMES B.
E. L . D E C K K E R A N D R . A . S R l N l V A S
Mechanical Engineering Department, University of Saskatchewan, Saskatoon, Canada
Empirical equations for the time required for the preparation and complete burning of an element of fuel in low frequency vibrating flames of butane-air and natural gas-air are presented. Using the equation for natural gas-air flames in a quasi-steady analysis, the appearance of a flame during a cycle of vibration has been predicted satisfactorily. Four examples are given of a flame vibrating-without separation, on the threshold of separation, with separation into two parts, and with separation into three parts-the fuel flow being distorted b y excitation of the higher harmonics of the fuel supply system. The appearance of low frequency vibrating diffusion flames is strongly dependent on the total time for combustion. The stably vibrating flows observed satisfy Rayleigh’s criterion for the phasing of heat release and pressure in the combustion chamber.
use of a time lag in theoretical studies of unstable comT h e concept was first invoked by Rayleigh (1945) as a necessary condition for the existence of stable vibrations of a hydrogen-air diffusion flame. I t has been used in studies of unstable combustion in liquid propellant rocket motors (Crocco, 1951; Gunder and Friant, 1950; Osborn and Zucrow, 1957), from which a more precise definition of the term combustion lag, or combustion time, has emerged. The term is now generally understood to include not only the period of induction-that is, the time required for preparation of the fuel for reaction-but also the time required to complete the reaction. The role of combustion time in unstable laboratory diffusion flames was first discussed by Barr (1953). H e showed qualitatively that the appearance of a n enclosed vibrating laminar diffusion flame, as revealed by photography, depended on the combustion time and, in particular, on the relative magnitude of the difference in the combustion times when the flame was a t the two extreme positions of vibration and one half the periodic time. At the time, the concept of combustion time in a laboratory diffusion flame was unknown and no data were HE
Tbustion is well established.
218
l&EC FUNDAMENTALS
available on which his hypothesis could be tested. I t was not until some years later that experimental techniques were devised by which the time taken for an element of fuel to burn completely in a diffusion flame could be measured (Barr and Deckker, 1961). Subsequently, the results of a large number of experiments using steady laminar diffusion flames of butaneair were correlated by the equation: T
= 0.025 F f 1.375 ( F / A )
(1 )
where r is the combustion time in seconds, and F and A are the volumetric flow rates of fuel and air, respectively, in cubic centimeters per second. A simple expression of this form was deliberately chosen since, while there was some theoretical justification for treating F and F / A as the leading independent variables (Gaydon and Wolfhard, 1960), the relative importance of the chemical and physical properties of the fuel could not be readily assessed. T h e coefficients in Equation 1, therefore, are a measure of the over-all effect of these properties on the combustion time. During a n investigation of low frequency, self-excited vibrating diffusion flames (Deckker and Srinivas, 1966) it was
