Binary Diffusion of Gases in Capillaries in the Transition Region between Knudsen and Molecular Diffusion Ronald R. Remickl and Christie J. Geankoplis" T h e Ohio State University, Columbus, Ohio 4%lO
The transition region of diffusion for a binary gas mixture was experimentally investigated in an isobaric open system of fine glass capillaries in parallel. Results were obtained over a 675/1 pressure range from 0.444 to 300.2 mm of Hg absolute in the transition region between Knudsen and molecular (ordinary) diffusion and confirm the theoretical equation for the flux in true capillaries in an open system. This equation has been derived by others using a momentum balance with various assumptions, such as additivity of the momentum transfer rates for the Knudsen and molecular regions, etc. These assumptions seem to b e confirmed. This appears to be the first known experimental confirmation of this equation for capillaries over a wide pressure range. Experimental flux ratios obtained in the transition region confirm the predicted inverse square root relationship with the molecular weights. Data for counterdiffusion in an open system near the Knudsen region tend to confirm the Knudsen equation and data in the molecular region also substantiate the Stefan-Maxwell relations.
D i f f u s i o n of gases in fine pores and capillaries is important ill many areas of the physical and biological sciences, especially in heterogeneous catalysis involving porous solids. X major portion of the important industrial catalytic reactions incorporate solid catalysts (Satterfield, 1970) which are generally porous. Furthermore, excluding surface diffusion, gas diffusion ill the pores of these solids is often in the transition region between Knudsen and molecular (bulk or ordinary) diffusion (Cunningham and Geankoplis, 1968; Satterfield, 1970; Scott and Dullien, 1962). Several iiivestigators have studied transition region diffusion. Generally their work has been related to binary sjfsteins in porous solids (Brown, et al., 1969; Cunningham and Geankoplis, 1968; Evans, et al., 1961; Henry, et al., 1967; Horak and Schneider, 1971; Johnson and Stewart, 1965; Rothfeld, 1963; Satterfield and Cadle, 1968; Scott and Dullien, 1962; Wakao and Smith, 1962). Diffusion experiments in pores are generally performed using a n open or a closed system. -4closed system is characterized by a closed volume a t each end of the diffusion path. I n a n open system, the closed volumes are replaced by gas streams flowing by each end of the diffusion path. Theoretical equations for open system diffusion of gases in a straight capillary in the transition region have been derived by Scott and Dullien (1962) and Rothfeld (1961). Evans, et al. (1961), derived similar equations for porous solids. These equations reduce to appropriate expressions in the true Knudsen and molecular regions. For porous catalysts, however, a model for the nonideal pore structure ivhich usually contains adjustable geometric parameters is needed in combination with these theoretical transition region capillary equations in order bo predict diffusion rates in these solids. Since the pore model adds considerable uncertainty in testing the basic theoretical equations, the experimental verification of the Present address, Atlantic Richfield Co., Harvey, Ill. 214
Ind. Eng. Chem. Fundam., Vol. 12, No. 2; 1973
theory and assumptions inherent in these capillary equations is best carried out using straight capillaries of uniform circular cross section. This is a n ideal system of pores compared with the irregular and contorted pore network of a typical porous solid. llistler, et al. (1970), and Visner (1951) have conducted closed system self-diffusion experiments in the transition region in capillaries. However, as far as has been determined, no known experimental work has been performed in the area of isobaric open system transition region diffusion with two or more components in true capillaries t o test rigorously the transition region equations for fluxes and flux ratios where no adjustable parameters are used. I n the present work the transition region of diffusion for a binary gas system was experimentally investigated in a n isothermal, isobaric open system of fine glass capillaries in parallel. The total pressure was varied to cover most of the transition region while keeping capillary diameter and temperature constant. Experimental fluxes, flux ratios, and effective diffusivities were compared with theoretical values predicted using the capillary transition region equations of Scott and Dullien (1962) and Rothfeld (1961). Equations for theoretical binary concentration profiles were derived. Surface diffusion was minimized by using low pressures, and the total pressure was maintained the same a t both ends of the capillaries to eliminate forced flow effects. To the authors' knowledge, this is the first known experimental confirmation of the theoretical binary transition region diffusion equations in true capillaries in an open system over a wide pressure range of 675/1. literature Review and Theory
A. Knudsen Diffusion in Capillaries. I n diffusion in the Knudsen region, molecule-to-wall collisions predominate over molecule-to-molecule collisions because of the rela-
tively large value of the mean free p a t h of the gas compared with the tube diameter. Consequently, the diffusion of each species is assumed t o be independent of the number of species present. Pollard and Present (1948) discuss in detail the derivation of Knudsen (1909) for Knudsen diffusion or flow in a capillary at steady state with constant diameter and temperature. Knudsen’s approach was to determine the rate of molecules crossing a given section of the circular tube due to reflection from a n arbitrary element of tube wall surface. The net transport was then obtained by integrating over the entire surface. The result for component A in a binary open system of X and B at constant total pressure is given by N.4
=
DKA ~
RTL
( pAO
-
PAL)
DKAP RTL
= -(XAO -
X*L)
(1)
When the value of a~ from kinetic theory is used, eq 2 becomes
DICA= 9.7 X 1 0 3 r d T / J 1 ~
The theoretical capillary transition region equation for a n isothermal, isobaric open system has been derived by Scott and Dullien (1962) and Rothfeld (1961) using a n overall momentum balance approach. The method is to consider a momentum balance on species i, which includes momentum transferred by i to the wall as well as to other molecules in the bulk gas phase. The partial pressure difference expressed as momentum loss between the ends of the capillary for a given component is considered to be the sum of the momentum transferred to the wall plus the momentum transferred t o other molecules. The result for component A in a binary gas system of A and B is given by
(2)
Inherent in eq 2 is the assumption of completely diffuse reflection of a molecule after striking the wall so that the molecule is reflected in a random manner. For the possibility that a fraction f of the molecules is diffusely reflected and the fraction (1 - f ) is specularly reflected, Smoluchowski (1910) generalized the results in eq 1 and 2 to
Experiments primarily with pure gases (Knudsen, 1909; Loeb, 1934; Present, 1958) and with cocurrent binary gas flow (hdzumi, 1937) generally indicate that close to 100% diffuse reflection occurs, or f = 1, and eq 1 is followed. However, in some cases, particularly with metal capillaries (Xdzumi, 1939; Hanley and Steele, 1964; Huggill, 1952; Lund and Berman, 1958) data are not in agreement with eq 1. Further confirmation of this is needed especially in metal capillaries. B. Molecular Diffusion. I n the molecular region of diffusion, molecule-to-molecule and not molecule-to-wall collisions provide t h e resistance to diffusion of a species SO t h a t diffusion is affected by the number and types of species present. I n this case the mean free p a t h is very small compared with t h e pore diameter. Beginning with the momentum transfer method of Stefan and Maxwell (Rothfeld, 1963; Scott and Dullien, 1962), one can arrive a t the well-known binary molecular region diffusion equation for Aia t constant diameter, temperature, and total pressure given by
C. Transition Diffusion. For transition region diffusion, both molecule-to-wall collisions (or Knudsen diffusion) and molecule-to-molecule collisons (or molecular diffusion) are important. Transition diffusion occurs mainly in t h e range 0.01 < YK,, < 10 (Geankoplis, 1972). T h e Knudsen number, NK,,, is defined b y
X similar result has also been obtained by Evans, et al. (1961), using a “dusty gas” model for porous solids. IntegraOion of eq 8 a t steady state results in -Y.4
=
DABP ~
aARTi,
111
[
1-
CY.4SAL
-1
+ DAB/IDKA + D A B ~ D K A (10)
1- aa9~o
Several important assumptions are inherent in the development of eq 8-10 in the momentum balance approach. These are as follows. (1) The assumed form of momentum transfer to the wall in the pure Knudsen region is that of Knudsen’s equation wit,h j = 1. This is shown by decreasing the pressure until D S A = D K . in ~ eq 9, and the resulting integrated form of eq 8 is identical with eq 1. (2) The assumed form of momentum transfer among molecules in the pure molecular region is that of the Stefan-Maxwell method. This can be seen by raising the pressure until D S A = DAB/(^ - a ~ S . 4 )in eq 9, and the resulting integrated form of eq 8 is identical with eq 6. (3) The net axial momentum transferred to the wall per unit’ area per unit time is negligible. (4) The ideal gas law and Ilalton’s law of partial pressures hold. (5) The rate of moment’uni transfer as calculated by the pure Kriudsen region cquatioii maintains its value over the entire pressure range from pure Knudsen to pure molecular diffusion. (6) The rate of momentum transfer a5 calculated by the pure molecular region equation maintains its value over the entire pressure range from pure Knudsen to pure molecular diffusion. ( 7 ) I n Rothfeld’s analysis (1961) , the assumption was made that the square of the drift velocity is negligible compared t o the mean square thermal velocity. Issumptions (5) and (6) allow for the simple addition of the Krludsen result and the Stefati-Maxwell result to obtain the overall momentum transfer. For a binary system in bhe transition region, t’he local value of an effect’ivediffusivity is defined as
Integration of eq 11 over the length of the tube a t steady state yields
x
S K n = -
d
Elimination of N A between eq 10 and 12 gives the theoretical Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1 9 7 3
21 5
expression for the integral value of De as
e
- In
[’
+ +
1
1 - ~ A X A LD A B / D K A - ~ A X A O DAB/DKA
DAB
D --
aAXAL1
1
1- ~AXAO
(13)
It can also be shown by starting with NB
=
-De’P dX’B - a g X g ) dz
RT(l
that Det = De in eq 13. This relation is useful in comparing experimental data for components A and B in a mixture. At high pressures in the pure molecular region using eq 8 and 11, the value of De becomes
De
=
DAB
(15)
and at low pressures in the pure Knudsen region
De =
~ A D K A P A-OXAL) l“
[: T
been studied experimentally by several investigators (e.g., Cadle, 1966; Cunningham and Geankoplis, 1968; Foster, et al., 1967; Henry, et al., 1967; Rothfeld, 1963; Scott and Dullien, 1962; Wakao and Smith, 1962). The observed flux ratios follow the square root relationship given by eq 17 for simple gases over a wide pressure range. Deviations are generally attributed to surface diffusion (Foster, et al. 1967; Satterfield, 1970). E. Derivation of Binary Concentration Profiles. Development of the equations for t h e concentration profiles in the Knudsen, transition, and molecular regions, which have not been given previously, is as follows. For the Knudsen region, eq 1 is written in differential form as
Upon integration using the boundary condition that X A = X A Oat z = 0, the following is obtained which relates the mole fraction of A to the distance z
(16)
] : :A:
D. Flux Ratios. T h e equation for the theoretical flux ratio for a binary gas in a n open system for all three regions of diffusion has been derived b y Hoogschagen (1955), Rothfeld (1961), Dullien and Scott (1962), Scott and Dullien (1962), Evans et al. (1961), and Mason, et al. (1967), using various approaches and is (in the absence of chemical reaction)
The theoretical value of N A from eq 1 is used in eq 19. The corresponding concentration of B at the same distance is given by
XB
=
1 - XA
(20)
For the molecular region of diffusion, eq 6 can be written in differential form as
(17) Equation 21 can be written in the form A simpler approach t o derive eq 17 is to rewrite eq 8 and 9 for B, form the ratio NB,”A, and then substitute eq 3 written for A and B into this ratio. The approach used by Hoogschagen (1955) and Scott and Ihllien (1962) to derive eq 17 is to use elementary kinetic Integrating and using the boundary condition that X A = theory momentum transfer considerations. This is done by XAO a t z = 0 results in the final equation for the mole fraction assuming diffuse reflection and setting the net axial momenof A with distance z in the molecular region as tum transferred to the wall equal to zero. Scott and Dullien (1962) point out that a zero net axial momentum transfer to the wall is not rigorously true, but is a good approximation as the capillary tliameter becomes sufficiently small. The corresponding concentration of B a t the same distance is Mason, et al. (1967), and Evans, et al. (1961), criticize the given b y eq 20. The theoretical value of N A given by eq 6 is momentum transfer derivation of the flux ratio in eq 17 on the used in eq 23 so i s to satisfy the remaining boundary condibasis that, although the technique is correct in principle, intion that XA = XALa t z = L. Also, the theoretical value of correct averages were used. I n their “dusty gas” model, the CYAis used in eq 23. assumption of zero net force at the wall is not needed, and After differentiating eq 23 and equating the result to zero, they rigorously show the existence of a second-order correction no minimum or maximum in the concentration profile is found. factor which depends in a complex way on composition. They Hence, no maxima or minima are possible in the binary moindicate that this correction is generally small so that the lecular region, unlike the ternary system studied by Remick first-order approximation given by eq 17 from the momentum and Geankoplis (1970). transfer developments is essentially correct. For the transition diffusion region, eq 8 can be written in As the capillary diameter becomes larger in a n open system the form (or, equivalently, as the pressure increases) in the molecular dX.4 region of diffusion, Dullien and Scott (1962) show-~ a smooth Ct’XA c3’ = 0 (24) dz theoretical curve progressing from N B / N A = - ~ ~ ‘ A /a tX B smaller diameters to,NB/NA = -L I I ~ / Lat l flarger ~ diameters. Solving eq 24 and applying the boundary condition at z = 0 Evans, el al. (1961), have derived the same result for the larger results in diameters. I n a closed system, unlike a n open system, the fluxes are equal but opposite in direction as pointed out in the work by Evans, et al. (1961). The binary transition region of diffusion in a n open system in various porous solids (but not in straight capillaries) has ~
21 6 Ind.
Eng. Chem. Pundam., Vol. 1 2,
No. 2, 1973
+
+
Table 1.
Experimental Data and Comparison with Predicted Theoretical Values 7 0 Dev of exptl
Experimental
P , mm Hg abs.
0,444 0.704 1,128 2.292 3.96 9.26 22,54 96 72 300,21
XAO
NP XAL
0,9641 0,9741 0.9836 0,9644 0,9115 0,8308 0.8840 0,9390 0.9445
0,0090 0.0064 0.0049 0,0088 0,0286 0.0459 0.0553 0,0235 0.0240
Mole Fraction 1,
O C
28.9 27.7 29.8 25.1 26.5 27.4 28.3 28.3 27.0
Theoretical
x
from theoret
106
NN*
(DJN~
(DJH~
NNX ~ 106
De
1.379 1.965 2.893 4.932 7.212 9.320 12 47 15.42 15.68
2.790 2.946 2.872 2.834 2.666 2.542 2.306 2.484 2.569
98.77 87.32 80.15 68.26 61.75 37.76 20.25 5.279 1.717
104.2 97.26 87.02 73.14 62.25 36.28 17.65 4.959 1.665
1.338 2.056 3.090 5,173 6.778 9.086 11.98 15.54 16.23
95.90 91.38 85.60 71.59 58.03 36.81 19.45 5.322 1.777
N~~
PJN~ 3.0 -4.4 -6.4 -4.6 6.4 2.6 4.1 -0.8 -3.4
-"e/
NN~
(Deb
8.6 6.4 1.7 2.2 7.3 -1.4 -9.2 -6.8 -6.1
5.5 11.4 8.6 7.1 0.8 -3.9 -12.8 -6.1 -2.9
Again, eq 20 applies and the theoretical value of N A from eq 10 is used in eq 25 to satisfy the boundary condition at z = L. The theoretical value of CYAis also used in eq 25. After differentiating eq 25 and setting the result equal to zero, no maximum or minimum is found in the binary transition region concentration profile. Experimental Methods
CAPILLARY DIFFUSION
A. Experimental Apparatus. T h e Wicke-Kallenbach
(1941) type of apparatus and t h e process flow to measure t h e experimentd countercurrent fluxes N A and A'B of nitrogen and helium, respectively, through a parallel bundle of straight glass capillaries are shown in Figure 1. Details a r e given b y Remick (1972). The system was allowed to reach steady state with pure Nz from G2 flowing past one side of the capillary diffusion cell and pure H e from G1 past the other, both sides being a t constant and equal total pressures. The pressures were measured with NcLeod gauges at low pressures and with H g manometers a t higher pressures. Flows were measured using precision-bore glass capillary flow meters , (FM) which were previously calibrated using a soap bubble film meter. The emf outputs from Gow-Mac thermal conductivity cells were monitored for a n indication of steady-state conditions. h i oil manometer not shown in Figure 1 was connected between each side of the capillary diffusion cell in order to ensure a negligible pressure difference across the cell and, thus, to minimize forced flow. Stopcocks V-1, V-3, V-4, V-6, V-7, V-8, V-9, and V-10 were open during a diffusion run and V-2 and V-5 were closed. Gas samples mere taken in sample tubes S-1 and S-2 by simultaneously closing V-7, V-8, V-9, and V-10, and they were analyzed by means of mass spectrometry. Total pressures of experimental runs ranged from 0.444 to 300.2 mm H g abs., or a range of 675/1. Temperatures were those a t ambient conditions. B. Capillary Flow Meters. Uncertainty exists a s t o the extent of saturation of the gas in the soap bubble calibration buret used. T o check this the vapor pressure of the soap solution was experimentally found to be approximately t h a t of pure water. Calculations assuming a wettedwall column indicated t h a t essentially total saturation was obtained after the gas had passed through the first 3% of the length of the buret. I n other experiments the actual amount of water vapor in the exit gas from calibration burets of various lengths was measured, and the results confirmed the wetted-wall predictions. Details are given elsewhere (Remick, 1972). A number of experimeutal diffusion runs were made at conditions such that the flow regime in the capillary flow
VP
Figure 1. Process flow diagram: FM, capillary flow meter; G, gas storage cylinder; MV, throttle (needle) valve; P, McLeod gauge or Hg manometer; S, sample tube; TC, thermal conductivity cell; VI stopcock; VP, vacuum pump; VR, vacuum pressure regulator
meters was that of "slip flow," in which the continuum restriction for the use of the laminar Hagen-Poiseuille equation is not satisfied. The data cited by Brown, et al. (1946), were reevaluated and re-correlated to yield
F
=
I
+ 6.6773(X)
(X< 0.10)
(26)
The predicted Hagen-Poiseuille flow value is multiplied by F to obtain the true flow under noncontinuum conditions. The expression in eq 26 for F differs from that obtained by Brown, et al., by 1% or less for the flows in the present work. C. Capillary Diffusion Cell. h total of 644 glass capillaries manufactured b y Corning Glass Works having a n average length of 0.96 ern and a n average inside diameter of 0.00391 cm with a n average deviation of 10.00012 cm were used in t h e capillary diffusion cell. The total cross sectional area of the 644 capillaries was 7.72 X em2. The capillaries were placed and sealed in holes which were drilled in a n aluminum disk having a diameter of 3.5 em. This disk containing the capillaries was sealed in the capillary diffusion cell shown in Figure 1. Details are given by Remick (1972). Experimental Diffusion Data and Calculations
The experimental diffusion data are given in Table I. The experimental fluxes for nitrogen and helium, IVA and A'B, were calculated from material balances using the measured Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
217
1000.0
10.0 0. I
I .O
10.0
P, mm
1000.0
100.0 HO
molecular diffusivity used for all runs was 0.706 cm2/sec at 1 atm. This diffusivity was corrected for temperature by using the fact that the diffusivity is proportional to T1.I5 (Geankoplis, 1972). Correction for pressure was made using the fact that the molecular diffusivity is inversely proportional to the pressure. The theoretical limiting values of De in the pure molecular region and in the pure Knudsen region were calculated using eq 15 and 16, respectively. An average temperature of 28.O"C for all nine runs was assumed and average values of concentrations for all runs given by X A O= 0.9329 ahd XAL= 0.0229 were used to calculate the limiting values.
Figure 2. Experimental and predicted effective diffusivities Results and Discussion
0
2.2
I
I
0.1
I I IIlll
I
I
'
111111
1.0
I I111111
10.0
P,
la0
'
1 1 1 1
IOOM
mm HE
Figure 3. Experimental and predicted theoretical flux ratios
flow rates and compositions of the gas streams. The experimental flux for PI;* is shown in Table I and the experimental flux for He can be obtained by using the experimental Nz flux and the experimental flux ratio shown. The experimental diffusivity based on the material balance for component i , Le., (De)rin Table I, was calculated using eq 12 with the experimental flux N , and the theoretical value of cy1. from eq 17. The theoretical flux ratio is -2.645 for smaller diameter capillaries. The per cent deviations of the effective diffusivities given in Table I are the same as the deviations in the fluxes, as seen by eq 12. Theoretical values for the nitrogen flux and the effective diffusivity were calculated using eq 10, 13, and 17 and the experimental concentrations, temperatures, and pressures in Table I. The concentration driving forces for each of the nine runs in Table I deviate from the average for all runs by a maximum of approximately 11%. However, the maximum deviation for all runs of the theoretical De calculated using the actual driving force compared to that using the average driving force for all runs is about 3%. Hence, the theoretical value of D, is only slightly affected by concentration. I n addition, using a n average temperature for all runs to predict Deinstead of using the actual run temperature produces only a small deviation of approximately 1%. Also in the prediction of the theoretical value for De, a value for the binary molecular diffusivity DAB of the N2-He mixture is needed. There is a n effect of concentration on the diffusivity in binary mixtures called the second approximation (Geankoplis, 1972). For a n N2-He mixture a t 1 a t m and 298"K, this concentration effect produces a n increase of approximately 5.3% in the binary diffusivity in going from nearly pure He with a diffusivity of 0.688 cm2/sec to nearly pure Nz with a diffusivity of 0.724 cm2/sec. This increase is based on data by Walker and Westenberg (1958) and Giddings and Seager (1962). I n Table I the average value of XA in each run is approximately 0.5. Hence, the average binary 218 Ind.
Eng. Chem. Fundam., Vol. 12, No. 2, 1973
A. Effective Diffusivities and Fluxes. The experimental effective diffusivities for all nine binary diffusion runs as shown in Table I are plotted as DeP us. P in Figure 2. As mentioned previously, the experimental flux follows the same trends and deviations from the theoretical value as does the effective diffusivity. The solid line in Figure 2 is the theoretical transition region curve based on the theoretical values of De in Table I. At 0.444 mm H g abs. the theoretical transition region curve in Figure 2 is 8.3% below the Knudsen limit line, and a t 300.21 mm i t is only 1.8% below the molecular limit line. The experimental data are seen to agree well with the theoretical binary transition region equation for De given by eq 13 and, on the average, do not deviate significantly above or below the theoretical line. Also, the experimental fluxes show the same agreement with the theoretical fluxes as do the experimental effective diff usivities compared to the theoretical effective diff usivities. From Table I the experimental diffusivities based on Nz vary from the theoretical by -6.4 to +6.4% over the pressure range studied. The range of the diffusivity deviations based on H e is -9.2 to +8.6%, The average deviation for all nine runs is =t4.0% based on Nz and =!=5.5%based on He, which is a good experimental confirmation of the theoretical binary transition region capillary diffusion equations for a n open system. This appears to be the first known experimental verification of these equations over a wide pressure range of 675/1. The lowest pressures investigated were sufficiently close to the pure Knudsen region to draw conclusions about trends in this region. The predictions near the Knudsen region can be compared with the deviations of the experimental binary effective diffusivities actually obtained near the Knudsen region as given in Table I. These deviations are calculated from the theoretical transition equation, but a t low pressures this transition equation reduces to the Knudsen diffusion equation. On the average, the three lowest pressure runs in Table I near the Knudsen region are 2.67, below the theoretical liue for Nzand 5.6% above for He. Therefore, Knudsen's equation, which implies the assumption of independence of diffusion of each species, is experimentally supported. Another consideration in the Knudsen diffusion equation is the generalization which assumes the diffuse reflection factor (2 - f)/J As a base case, the diffusivity with j = 1 can be chosen. If, instead, f = 0.9, then the diffusivity in Knudsen's equation is (2 - 0.9)/0.9 times greater or 22% larger than the base case. The experimental data near the Knudsen region shown in Figure 2 approach the Knudsen line with .f = 1 (shown as the Knudsen limit line) reasonably closely. Hence, this again supports the validity of Knudsen's equation with f essentially equal to 1 as used in the theoretical derivation of the transition region equation.
At the highest experimental run pressure the theoretical transition curve in Figure 2 is only 1.8% below the molecular limit line so that diffusion was essentially all molecular. The molecular region diffusion contribution to the transition equations is based on the Stefan-Maxwell momentum balance equations as discussed previously. At high pressures the transition diffusion equations reduce to the molecular region Stefan-Maxwell equations. In this study in a n open system a t the higher pressures (Figure 2), the experimental data essentially confirm the Stefan-Xaxwell equations since the diffusion mechanism is primarily that of the molecular region. I n the derivation of the transition region diffusion equations i t iq assumed that the net axial momentum transferred to the walls is negligible. It appears that a significant net transfer of momentum would result in the experimental effective diffusivities deviating from the predicted values over a range of pressures, which is not the case in Figure 2 . Also in the transition region, the assumption of additivity of the pure Knudsen and pure molecular diffusion expressions' has been made and, again, the experimental data in Figure 2 support this procedure. Finally, the assumption that the square of the drift velocity is much less than the mean square thermal velocity appears to be justified since experimental data check eq 10 throughout the transition region. B. Flux Ratios. T h e experimental flux ratios of helium to nitrogen as given in Table I are plotted in Figure 3 as a function of pressure. Also shown is the theoretical value of -2.645 obtained from eq 17 for smaller diameters, which is constant with pressure. T h e experimental flux ratios are seen to deviate above and below the theoretical value in a manner having no trend with pressure. The range of deviations for all nine runs is -12.8 to +11.4% and the average deviation of the experimental values from the theoretical value is only *6.6%,. Hence, good agreement with the theoretical square root relationship given by eq 17 has been obtained for a binary system in capillaries in the transition region. This again confirms the flux ratio theories of others for smaller diameters (Dullien and Scott, 1962; Evans, et al., 1961; Hoogschagen, 1955; Rothfeld, 1961) which have previously been tested and confirmed in porous solids but not in capillaries throughout the transition region. The assumption of . V H ~ / N X=~ -M N ~ / X Hwhich ~ , applies to larger diameters or higher pressures, instead of the square root relationship produces a theoretical flux ratio which is 165% greater than that using eq 17. Since the experimental deviations in Figure 3 a t higher pressures are only of the order of 7%, it is seen that the square root relationship is valid at the experimental conditions of this work.
a t the walls of the glass capillaries used) in the theoretical derivation of the trarkition equations is supported experimentally. These experimental findings support the results of other investigators using pure gases. At higher pressures the experimental data essentially confirm the Stefan-Maxwell equations since the diffusion is primarily that of the molecular region. For intermediate pressures throughout the transition region of diffusion, the additivity of the momentum transfer rate for the pure Knudsen region and the momentum transfer rate for the pure molecular region to yield the overall momentum transfer rate in the transition region has been confirmed experimentally. Acknowledgment
The analysis of mass spectrometry samples by Messrs. R. Iden and R. Livingston a t Battelle Memorial Institute, Columbus, Ohio, is appreciated. Nomenclature
- ~ A N A R T / ( D A B Pcm-l ), C~'/(-CYA),cm-' [ W D K A )f DAB) INARTIP, cm-' diameter of capillary, cm molecular diffusion coefficient for svstem A-B. cni2/sec effective diffusion coefficient defined by eq 11, cm*/sec experimental effective diffusion coefficient based on component i, cm2/sec Knudsen diffusion Coefficient for component A defined by eq 2 , cmZ/sec Knudsen diffusion coefficient for component X defined b y eq 5, cm2/sec diffusion coefficient for comnonent A defined bv" eq9, cm2/sec fraction of total number of molecules undergoing diffuse reflection at the wall correction factor t o Hagen-Poiseuille equation defined b y eq 26 1.33 (g-cm/secz)/(micron Hg-cmZ) length of capillary, cm molecular weight of component A, g/mole molar flux of component A relative t o stationary coordinates, mole/ (cm2-sec) Knudsen number defined b y eq 7 partial pressure of component A, mm or a t m total pressure, m m or a t m average total pressure in capillary flow meter, microns of H g capillary radius, cm gas constant, 082.057 (cma-atm)/ (mole-" K) temperature, C temperature, OK mean thermal velocity of component A, cm/sec mole fraction of component A mole fraction of component A a t nitrogen end of capillary bundle mole fraction of component A a t helium end of capillary bundle = correlating parameter for F where X
C1'
cz
C3'
d
DAB De (De)i DKA
Conclusions
Experimental gas diffusion data were obtained in the transition region between Knudsen and molecular diffusion for straight class capillaries over a pressure range of 0.444 to 300.2 mm. The data confirm the theoretical capillary moment u m balance equations for isothermal, isobaric open system diffusion used to predict the fluxes and effective diffusivities in a binary system of helium and nitrogen in the transition region. This appears to be the first known experimental confirmation of the theoretical binary transition region diffusion equations in true capillaries over a wide pressure range. Experimental binary flux ratios agree reasonably well with the predicted square root relationship given by N B / N A =
- v'JIA/MB. A t the lower pressures the use of Knudsen's equation with f essentially equal to 1 (ie., total diffuse molecular reflection
p/Pmr(pigc)'1%
distance along diffusion path, cm GREEKLETTERS CY(
= =
p
=
p1
=
1
+
Nj/Ni
mean free path, cm viscosity, P gas density at 1p Hg, g/(cni3-p Hg)
SUBSCRIPTS A, B = nitrogen and helium, respectively 0, L = nitrogen and helium sides of capillary, respectively Ind. Eng. Chem. Fundam., Vol. 12, No. 2,
1973 219
e
i j
K
Loeb, L. B., “The Kinetic Theory of Gases,” 2nd ed, McGrawHill, New York, N. Y., 1934. Lund, L. M., Berman, A. S., J . Chem. Phys. 28, 363 (1958). Mason, E. A., Malinauskas, A. P., Evans, R. B., J . Chem. Phys.
effective component i componentj Knudsen
= = = =
46. 3199 (1967).
Mistier, T. E., Correll, G. R., Mingle, J. O., A.I.Ch.E. J . 16, 32 (1970).
literature Cited
Adzumi, H., Bull. Chem. SOC.Jap. 12, 285 (1937). Adzumi, H., Bull. Chem. SOC.Jap. 14, 343 (1939). Brown, G. P., DiNardo, A., Cheng, G. K., Sherwood, T. K., J . Appl. Phys. 17, 802 (1946).
Brown, L. F., Haynes, H. W., Manogue, W. H., J . Catal. 14,220
Pollard, W. G., Present, R. D., Phys. Reo. 73, 762 (1948). Present, R. D., “Kinetic Theory of Gases,” McGraw-Hill, Kew York, N. Y., 1958. Remick, R. R., Ph.D. Thesis, The Ohio State University, Columbus, Ohio, 1972 Remick, R. R., Geankoplis, C. J., IND. ENG.CHEM.,FUNDAM. 9, 206 (1970).
(1969).
Rothfeld, L. B., Ph.D. Thesis, University of Wisconsin, Madison, Wis., 1961. Rothfeld, L. B., A.I.Ch.E. J . 9, 19 (1963). Satterfield, C. N., “Mass Transfer in Heterogeneous Catalysis,” R.I.I.T. Press, Cambridge, Mass., 1970. Satterfield, C. N., Cadle, P. J., Ind. Eng. Chem., Process Des.
Foster, R. N.; Butt, J. B., Bliss, H., J . Catal. 7, 179 (1967). Geankoplis, C. J., “Mass Transport Phenomena,” Holt, Rinehart, and Winston, New York, N. Y., 1972. 1, Giddings, J. C., Seager, S. L., IND.ENG. CHEM.,FUNDAM.
2076 (1961).
Scott, D. S.,Dullien, F. A. L., A.1.Ch.E. J . 8, 113 (1962). Smoluchowski, M., Ann. Phys. 33, 1559 (1910). Visner, S., Phys. Rev. 82, 297 (1951). Wakao, N., Smith, J. M., Chem. Eng. Sci. 17, 825 (1962). Walker, R. E., Westenberg, A. A., J . Chem. Phys. 29, 1139
Hanley, H. J. M., Steele, W. A., J . Phys. Chem. 68, 3087 (1964). Henry, J. P., Jr., Cunningham, R. S., Geankoplis, C. J., Chem. Eng. Sci. 22, 1 1 (1967) Hoogschagen, J., Ind. Eng. Chem. 47, 006 (1955). Horak, Z., Schneider, P., Chem. Eng. J . 2, 26 (1971). Huggill, J. A. W., Proc. Roy. SOC.Ser. A 212, 123 (1952). Johnson, 31.F. L., Stewart, W. E., J . Catal. 4, 248 (1965). Knudsen, M., Ann. Phys. 28, 75 (1909).
Wicke, E., Kallenbach, R., Kolloid 2. 2. Polym. 97, 136 (1941). RECEIVED for review June 5 , 1972 ACCEPTEDJanuary 23, 1973 Work was done under the Atomic Energy Commission Contract No. AT(11-1)-1675. A fellowship was provided during part of this study by Texaco, Inc.
Cadle, P. J., Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass. 1966. ENG.CHEM.,FUNDAM. Cunningham, R. S., Geankoplis, C. J., IND. 7, 535 (1968). Dullien, F. A. L., Scott, L). S., Chem. Eng. Sci. 17,771 (1962). Evans, R.B., Watson, G. M., Mason, E. A., J . Chem. Phys. 35,
277 (1962).
Deoelop. 7, 236 (1968).
(1958).
Experimental Determination of Hemispherical Total Emittance of Metals as a Function of Temperature Samuel H. P. Chen and Satish C. Saxena” Department of Energy Engineering, University of Illinois, Chicago, Ill. 60680
A hot-wire column type facility i s described for the measurement of hemispherical total emittance of metal wires as a function of temperature. The data on tungsten (300-2500°K) and platinum (300-1 500’K) are reported, and these ore also compared with the recommended values obtained on the basis of the available data of hemispherical total emittance in the literature.
T h e knowledge of thermal emissivity of metal surfaces as a fuiiction of temperature is quite important for a variety of practical engineering problems. The purpose of this paper is to describe a hot-wire column type facility and discuss its appropriateness for determining the hemispherical total emittance of the wire material as a function of temperature. Very briefly, the procedure involves the determination of the electrical power required to heat the hot-wire of a known length, stretched axially in a cylindrical container whose mall is maintailled a t a known fixed temperature, to different known temperatures. I n particular, we report the data for tungsten in the temperature range 300-2500°K and for platinum iii the temperature range 30&1500”K. These values are compared with the available data in the literature. Experimental Section
The principal component of the experimental arrangement is the glass column arid its typical design is as shown in Figure 220 Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
1. I t map be considered to consist primarily of three different glass sections, top, central, and bottom, assembled together. The purpose of the top and bottom sections is to provide a vacuum-tight seal to the central experimental test section as well as to provide the mounting supports and electrical connections to the axial tungsten wire. The design of the bottom section is slightly more involved in comparison to the top section because it has to account also for the expansion of the tungsten wire, which IS successively heated to the desired highest temperature. The design details which accomplish all these requirements are clearly displayed in Figure 1. The top and bottom sections are attached to the central piece through vacuum-tight standard ground-glass joints and are fused into glass t o metal seals on the other ends. Suitable metal caps are silver-soldered a t the metal ends of the seals u i t h metal electrodes to provide arrangements for outside electrical connections and appropriate extension toward the central section. At the top end this brass rod is
