Fletcher, R., Powell, M. J. D., Comput. J . 6 (2)’ 163 (1963). Hill, W. J., Hunter, W. G., University of Wisconsin, Department of Statistics Technical Report 69 (1966). StaHogg, R. v.7 Craig, A. T.,L‘lntroduction to tistics,” 2nd ed, Macmillan, New York, N. Y., 1965. Kittrell, J. R., Hunter, W. G., Watson, C. C., A.I.Ch.E. J . 12, 5 (1966). Lindgren, B., McElrath, G., ‘‘Introduction to Probability and Statistics,” 1st ed, Macmillan, New York, N. Y., 1967. Reilly, P. M., Can. J. Chem. Eng. 48 (2), 168 (1970).
Shapiro, N. Z., Shapley, L. S., J . SOC.Indust. Appl. Math. 13 (2), 353 (1965). Tucker, E. C., Farnham, S. B., Christian, S. D., J . Phys. Chem. 73, 3820 (1969). White, W. B., Johnson, S. &, Dantzig, I., G , B., J , Chem. Phys. 2 8 , 751 (1958). Wilde, D. J., Beightler, C. S., “Foundations of Optimization,” Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. Zoutendijk, G., J . SOC.Indust. Appl. Math. 4 (I), 194 (1966). RECEIVED for review April 21, 1971 ACCEPTED April 27, 1972
A Model for Predicting the Permeation of Hydrogen-Deuterium-Inert through Palladium Tubes
Gas Mixtures
Frank J. Ackerman” and George J. Koskinas Lawrence Livermore Laboratory, University of California, Livermore, Calif. 94550
This paper describes a mathematical model of the permeation of hydrogen-deuterium-inert gas mixtures flowing through long palladium-silver alloy tubes. Permeation experiments using Dz-He, Hz-Dz, and Hz-DzHe mixtures agree well with the model. The permeation experiments were performed with a Pd-25 wt % Ag tube at 400°C under a total driving pressure of 1000 psia and a total back pressure of 20 psia. The model can readily be extended to other conditions.
Palladium has been used to produce high-purity hydrogen and its isotopes in the laboratory since the early 1920’s, but the hydrogen-palladium permeation process has been used commercially only in the last 10 years. A major cause of this delay has been the structural weakening of the palladium by a phase change in the hydride. Now, however, the development of palladium-silver alloys has overcome this problem. Even with such a long history of laboratory application, few investigators have reported hydrogen or deuterium permeation studies at common industrial operating pressures (above 50 psig) (deRosset, 1960; Hunter, 1965; Rubin 1966). However, many commercial uses for this process have been suggested. They fall into three categories (Darling, 1958; Serfass and Silman, 1965) : (1) purification of the hydrogen supplied to a process; (2) recovery of hydrogen from process streams for recycling or as a by-product; and (3) isotope separation. I n order to be practical, processes for purifying or recovering hydrogen must be designed to recover most of the hydrogen. No studies involving the substantial recovery of hydrogen from gas mixtures have been found. deRosset (1960) has studied the permeation of H2-Hz and H2-CH4 mixtures] but under conditions where most of the hydrogen remains in the mixture. The use of palladium for separating hydrogen isotopes has been studied by several investigators (Lewis, 1967). In these studies, the separation is accomplished by gas chromatographic techniques that use the difference in the solubilities of the isotopes to accomplish the separation. This is essentially 332
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
a batch separation method. The use of palladium in a continuous separation process has not been reported. This study is concerned with the efficient recovery of Hz or Dz from gas mixtures containing one or more nondiffusing contaminants. The geometry of the system is similar to a single-tube heat exchanger. The entering mixture becomes depleted in hydrogen as it flows through the center of a heated palladium-silver tube, and pure hydrogen is withdrawn from the outside of the tube. A mathematical model of this process has been formulated to examine the influence of the process variables (composition, temperature] driving pressure, back pressure, bleed rate, and tube geometry) on the recovery of hydrogen or deuterium. The isotope separation that is obtained during the simultaneous permeation of two hydrogen isotopes can also be calculated with this model. The model has been tested with mixtures of DZ-He, Hz-Dz, and HrDz-He. Helium is a convenient nondiffusing contaminant since it is readily available in pure form and has properties similar to hydrogen and deuterium, thereby simplifying the PVT calculations. The permeation experiments were performed with a Pd-25 wt % Ag permeation tube a t nominal conditions of 4OO0C, 1000 psia of total driving pressure, and 20 psia of total back pressure. Theory
Pure-Gas Permeation. Our purpose was to develop a mathematical model for predicting the permeation of gas mix-
tures through palladium alloys. Since pure-gas permeation data are relatively easy to obtain, we wanted the model to make mixed-gas predictions based on constants obtained from pure-gas data. I t is generally agreed that the rate-controlling step in the permeation of pure hydrogen or deuterium through unpoisoned palladium is diffusion through the bulk metal (Davis, 1955; Darling, 1963). The simple integrated form of Fick’s law for the steady-state diffusion of a pure gas through a solid, assuming a constant diffusion coefficient, is given by (Bird, et al., 1960)
R
A
= X
9(Ci
- Cz)
(1)
where R is the permeation rate, A is the effective diffusion area, x is the length of the diffusion path, 9 is the diffusivity, and C1 and CZ are the concentrations of the gas in solution a t the opposite surfaces of the solid (solubility). For diffusion through the wall of a cylinder, the effective area is given by
where 1 is the cylinder length, x = rz - rl, and rl and r2 are the inner and outer radii corresponding to C1 and Cz.An empirical equation similar to eq 1 is sometimes used (Barrer, 1941; Rubin, 1966)
where a and b are constants. T is the absolute temperature of the gas, P I is the driving pressure, and PZ is the back pressure. This form lumps together the temperature dependence of both the diffusion coefficient and the solubility. The P”? relationship between solubility and pressure is Sieverts’ law (Sieverts and Krumshaar, 1910). Our earlier experiments have shown that this equation is valid up to 1000 psi for hydrogen between 400 and 500°C and for deuterium between 300 and 500°C (Ackerman and Koskinas, 1972). Gas-Mixture Permeation. Equation 3 can be used for mixtures of one diffusing gas and one or more nondiffusing gases by substituting the local partial pressure of the diffusing gas. When there are two diffusing hydrogen isotopes, the situation is somewhat more complicated since the presence of one isotope influences the solubility and the diffusion coefficient of the other. Hickman’s measurements of the solubility of HP, Dz, and H2-Dz mixtures show that the total solubility of the mixture (C) should be expressed by (Hickman, 1969)
C
=
+
P”~(XH~KH,
XDJW
(4)
where P is the total pressure of the hydrogen and deuterium, XH,and XD,are the mole fractions of the hydrogen and deuterium, respectively, and K H and ~ K D ,are constants for the hydrogen and deuterium as obtained from pure-gas solubility measurements. Using eq 4 as a guide, we can modify eq 3 to obtain the permeation rate for each isotope when both are present. On the basis of eq 4,we assume that the solubility of each component in a mixture is proportional to XfP‘”. The expression for the permeation rate of each isotope then becomes
On the driving-pressure side in this equation (subscript 1)
p1 = (Pt
+ ph1; X r
=
~
(Pi
Y
Pj1
and on the back-pressure side (subscript 2) P2
= (Pt
+ p h ; Yi
=
(Pi ~
?
PjZ
(7)
The total permeation rate for the two isotopes is given by
RT
=
R,
+ R,
(8)
The partial pressures of the two hydrogen isotopes (ptand p,) are found by assuming that all of the hydrogen is H2 and that all of the deuterium is Dz. The actual amounts of Hz, HD, and DZin the gas mixture have no effect on the permeation rate since the controlling mechanism is the diffusion of the hydrogen and deuterium atoms. In general, the partial pressure should be replaced by fugacity in the above equations, but the correction is small for the relatively ideal mixtures used in this study and so has been neglected. The constants at and b i are determined from pure-gas measurements of each isotope using eq 3. As expected, eq 5 reduces to eq 3 when only one isotope is present ( i e . , when X I = 1 and Y t = 1).The equations remain the same for mixtures of two hydrogen isotopes when other nondiffusing gases are also present. The determination of a iand b, from pure-gas measurements assumes that the diffusion coefficient of each isotope is unaffected by the presence of the other isotope. Permeation through Long Tubes. I n a real system in which significant amounts of hydrogen isotopes permeate through the walls of a tube as a gas mixture flows down the tube, the composition of the gas on the driving-pressure side depends on the sampling position chosen. Xote also that eq 5 gives the permeation rate only a t a particular point. The complete solution can be found by coupling eq 5 with the appropriate mass-balance equations. For simplicity, the solution described below is for a gas mixture flowing down a palladiumsilver tube, with the hydrogen isotopes permeating to the outside of the tube. This solution is obviously applicable to a variety of geometries in which a gas mixture flows past a permeation area. Figure 1 shows a simplified diagram of the permeation system used in both the mathematical model and the experiments mentioned above. The Hz-D2-inert gas mixture flows down the permeation tube a t a constant pressure. Permeation of the Hz and Dz causes the mixture to become depleted in these gases as it flows toward the bleed valve. d bleed valve is necessary so that the tube does not fill up with the nonpermeating component and inhibit permeation. The permeated H2 and DPare measured as they leave the system. The mathematical model makes the following assumptions. 1. The composition of the gas flowing down the tube varies with position but not with time. 2. The pressures of the supply gas and the permeated gas are constant. 3. The pressure drop in the flowing gas mixture is small and can be neglected. 4. The temperature along the tube is constant. 5. The gas mixture remains well mixed radially as it flows down the tube. A closed-form analytical solution can be found for mixtures of one permeating gas and one nonpermeating gas under certain conditions. The case of two permeating gases is considerably more complicated, so a closed-form solution has not been attempted. Instead, the equations are set up by means of finite-difference methods. I n order to solve the equations, certain boundary conditions must first be estimated. Then mass-balance calculations are made, and the resulting boundInd. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
333
p2 Figure 1.
=
Pz
we arrive a t
Schematic of the permeation system Product gas hn dn Bleed gar
Similarly, for Dz Dn-1
hn
dn =
(Hn-1
+ Dn-)(
+
Hn-1 Dn-1 Hn-i Dn-1+ I
+
(C h nCdn + ) fi] Cdn
1
1
Figure 2. model
QD[
Permeation tube as viewed by the mathematical
ary values are compared with the assumed values. The final solution is found Via an iterative procedure. The details of this solution can best be described with reference to Figure 2. The permeation tube is divided into elements, and a mass balance is made for each element. The molar flow rate into the tube (G) consists of Ho moles/min of hydrogen (as Ht), DO moles/min of deuterium (as Dz), and IO moles/min of inert gas. The molar flow rate out of the tube ( L ) consists of H N , D N , and IN moles/min of hydrogen, deuterium, and inert gas, respectively. The quantities Hn-l, Dn-l, and I are the moles/min of each gas flowing into tube element n; H n , Dn, and I are the moles/min of each gas flowing out of element n; and hn and dn are the moles/min of hydrogen and deuterium diffusing through the wall of element n. Since the system is in a steady state, the mass balance for element n is
(12)
At the start of the problem, we know the composition of the supply UH,fD, and fd, the bleed rate ( L ) ,the supply pressure (PI),the back pressure (Pz), the operating temperature (T), the tube area ( A ) , and the wall thickness (2). The unknown factors are the flow rate into the tube (G), the flow rate of each gas making up the bleed (HN, DN, and I N ) and the total rate of permeated or product gas ( C h , and I n order to start the calculations, we must assume some values for G, Y H , and YD. For simplicity, we can assume f ~ )and , YD = fD/(fH f ~ )Then . G = L,YH = fH/(fH H o = f H . G , Do = fD.G, and I = f1.G. Now, beginning with tube element 1, we compute hl and dl from eq 11 and 12. Then HI = Ho - hl and D1 = Do - dl. I n this way we proceed segment by segment until the end of the tube is reached. Then better estimates of the starting values are obtained from
Edn).
+
+
N
N
G = L +
C h n + n=l
dn n=l
I N = OUT
and
and In-l
=
In
N
= I0 - 1
The overall mass balance, then, is N
G
=
L
+ n=lhn +
y D =
N
dn
n=l
(9)
The permeation of hydrogen out of each element having a length A1 and an area AA is given by eq 5.
or hn
=
Q H ( X H ~-& YH
d p Z )
(10)
where QH = a H exp(bH/T) (AAlz).By substituting the following identities into eq 7
334 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
,L dn n=l N Chn n=l
+ Cdn n= 1
This procedure is repeated until the new starting values agree with the preceding starting values to within 0.1%. Convergence is usually obtained in less than 20 iterations. In order to facilitate the calculations, a digital computer code has been written. This code has been coupled with an analysis of the experimental data by another code to permit rapid comparison of the model’s calculations with experimental data. The model we have set up should predict hydrogen and deuterium permeation through a given system for any mixture of hydrogen, deuterium, and inert gases, using constants determined from pure-gas measurements. I n addition, the computation procedure gives the partial pressures of the gases as a function of location in the tube. Permeation measurements on hydrogen-deuterium-helium mixtures have been used to develop and test the model. The apparatus for making the measurements as well as the plan of the experiments are discussed in the following section.
Experimental Procedure
A diagram of the experimental equipment used to test the model is shown in Figure 3. The flow rate of the gas entering the permeation cell is obtained by measuring the changing pressure and temperature in the supply vessel. Pressurecontrol valves are used to set both the driving pressure (supply side) and back pressure (product side) on the permeation tube. Bleed and product flow rates are each measured by a hot-wire anemometer mass flowmeter and a wet-test meter. The bleed flow rate is set by a micro-needle valve (throttle). Details of the equipment and the procedures are reported elsewhere (Ackerman and Koskinas, 1972). I n the present tests, mixtures of hydrogen, deuterium, and helium were used. The composition of supply, bleed, and product samples taken during the experiments was determined by mass spectrographic analysis. Wet test Permeation tube
Distance from tube entrance Bleed
P
Product
e= FIOW
Vkmpl*
meter’
Figure 4. Calculated deuterium distance from tube entrance
- in.
partial
pressure vs.
From Figure 4, we see that three experiments with tubes 150, 200, and 270 in. long (A, B, and C) effectively traced the steepest part of the curve. Since it is difficult to change tubes, we used a single tube with three different bleed rates to accomplish the same purpose.
bttler
Figure 3. Schematic of the experimental apparatus ( P i = pressure transducers, Ti = thermocouples)
The plan of the experiments can be shown with reference to Figure 4. This figure shows the calculated partial-pressure profile for an experiment with a typical deuterium-inert gas mixture. The most severe test of the model is not how well it calculates the quantity of permeated gas, but rather how well it reproduces the actual longitudinal partial-pressure profile. A calculation of the partial pressure of the gas leaving the bleed can be in error by more than loo%, even though an accompanying calculation of the quantity of the permeated gas is only slightly in error. This is because most of the permeation takes place in the high-partial-pressure section of the tube.
Results and Discussion
Pure-Gas Permeation. Pure-gas permeation measurements were made with six different Pd-25 wt% Ag tubes. The dimensions of these tubes are shown in Table I. Some data were obtained for all of the tubes, but several of them developed leaks before many experiments had been completed. Most of the pure-gas permeation measurements were made with tube 2. As determined from the tube-2 measurements, constants a and b from eq 3 are a = 3.85
x
mole cm 10”
min cm2 -
g----; b = -794/’K psia
Table 1. Summary of Pure-Gas Permeation Measurements Tube number
Tube dimensions Length, in. Wall thickness, mils Outside diameter, mils Hydrogen Number of experiments Relative permeation rate’ a t 400°C Deuterium Number of experiments Relative permeation rate’ a t 400°C DJHz ratio at 400°C
1
2
66- 4 5.0 50.0
299.3 4.95 49.9
4 1.03 ...
...
63 1.00 33 1.00
4
5
6
27.32 5.0 49.39
300.3 5.0 49.60
298.8 5.0 49.85
8
7
3
28.42 5.0 49.75 3
11
1.00
1.08
0.90
0.95
...
7
4
4
...
1 .oo
0.90
0.91
... 0.698 ... 0.653 0.68 0.66 a Ratio of constant a as determined from the permeation experiment with the specified tube to constant a as determined for tube 2.
Ind. Eng. Chem. Fundam., Vol. 11,
No. 3, 1972 335
for H2, and a = 1.68 X loF6
mole cm
-
~
min cm2
~
psia
7 b = -479/"K ;
for D2. These constants were determined a t driving pressures from 100 to 1000 psia, back pressures from 20 to 100 psia, and temperatures from 400 to 5OOOC for hydrogen and from 300 to 500°C for deuterium. The permeation data on the other tubes were taken a t 400"C, driving pressures of 500 and 1000 psia, and back pressures of about 20 psia. Table I gives a comparison of the pure-gas results from all six tubes. The purpose of this table is to show the consistency of the permeation measurements from one tube to another. The relative permeation rate is the ratio of the value of constant a as determined from each tube to the value of constant a as given above for tube 2. The deviation of this ratio from 1.0 is partly due to the difficulty in accurately determining the area-to-thickness ( A / z )ratio of each tube. It is interesting to note that the measured D2/H2permeation ratio comes quite close to the theoretical value of 0.707 predicted from the square root of the respective atomic weights (Barrer, 1941). Gas Mixtures. The permeation data on gas mixtures were obtained with tube 6 a t a driving pressure of 1000 psia, a back pressure of 20 psia, and a temperature of 400°C. The results from pure-gas measurements were used to determine values of constant a for use in mixture calculations. The values derived are a = 3.66 X
mole cm
-
min em2
-6
for Hz, and
a = 1.53 X lob6
mole cm min cmz
-6
for D2.The values of constant b were the same as those found from the tube-2 data. For mixtures, the model's calculations are sensitive to the composition of the supply gas. For this reason, the results of the gas-mixture experiments were checked for consistencyby an overall mass balance and a component mass balance based on an analysis of samples of the supply, bleed, and product gases. The errors in the component and overall mass balances were always less than 5%. Deuterium-Helium Mixtures. Deuterium-helium gas mixtures were studied to see how accurately the model calculates hydrogen-isotope permeation when an inert gas is also present. The results are given in Table 11. The permeation of one mixture was studied a t three different bleed rates, and duplicate experiments were performed a t each rate. The experimental results are compared with the model calculations in Table 11. The agreement between the experiments and the calculations is quite good for both the rate of deuterium permeation and the composition of the bleed gas. The two experiments with tube 6 a t low bleed rates (3A and 3B) show a higher bleed concentration of deuterium ( 5 and 4%) than was calculated (2.3 and 2.7%). This may be due t o incomplete radial mixing of the gas stream as it became depleted in deuterium. Incomplete mixing is to be expected under these conditions. The calculated partial pressures of the deuterium as a function of tube length are shown in Figure 5 . The three curves were calculated from input data supplied by experiments l A , 2A, and 3A. Our purpose was to test the model over the steep 336
Ind. Eng. Chem. Fundam., Vol. 1 1 ,
No. 3, 1972
Table II. Comparison of Experimental Results with Model Calculations for Deuferium-Helium Gas Mixturesa Bleed a a r Rate, mole/ rnin
Expt
Permeated gas
% D2 Exptl
Calcd
Rate, mole/rnin
% erroP D2
1A 0.177 61.2 60.4 0.383 - 0 . 1 1B 0.192 62.8 6 2 . 5 0.384 - 0 . 3 2A 0.085 37.0 f 2.0' 34.3 0.362 - 0 . 3 2B 0.086 35.0 i 2.0' 34.4 0.374 -0.03 3A 0.045 5.0 zk 1.5' 2 . 7 0.278 - 2 . 8 3B 0.046 4 . 0 i. 1.5' 2 . 3 0.266 -1.6 = Supply-gas composition: 87.0 i 1.0% D Pand 13.0 f 1.0% He. Experimental parameters: temperature = 400 f 25"C, driving pressure = 1000 f 40 psia, and back pressure = 20 zt 2 psia (=t values represent maximum offset of any experiment from the average of all experiments).8%error = 100 X (exptl - calcd)/ exptl. c Average of two samples.
part of the partial-pressure curves. Comparison of the experimental results with the model calculations shows good agreement. It is apparent that the helium acts only as a diluent, permitting the deuterium to permeate as if it were alone a t its own partial pressure. Calculations over the large change in partial pressure between 50 and 900 psia are a severe test of the permeation-rate equation as well as the calculation model. Hydrogen-Deuterium Mixtures. Hydrogen-deuterium mixtures %ereused to examine the simultaneous permeation of two hydrogen isotopes, The results of four experiments with two different mixtures are given in Table 111.All of the experiments show similar fractionation of the hydrogen between the bleed gas and the product gas, with the ratio of the hydrogen t o the deuterium being increased in the product gas and decreased in the bleed gas. This is to be expected since hydrogen has a higher permeation rate than deuterium. The model accounts for this difference in permeation rate and gives results for the product and bleed gases that agree well with the experimental data. We interpret these results as demonstrating that the interaction between simultaneously permeating
100
-
\'\o 3A
-
Table 111. Comparison of Experimental Results with Model Calculations for Hydrogen-Deuterium Gas Mixturesa Permeated gar
Bleed gas
% Hz in Expt
supply garb
% HZb
Rate, mole/min
Exptl
Total rate, moIe/min
Calcd
9.6 8.9 0.048 411 18.5 9.3 7.9 4B 18.1 0,048 60.9 60.5 0.048 58 77.3 61.1 60.1 0.050 5B 77.3 Experimental parameters: temperature = 400 i 8"C, driving pressure mainder of the gas is DZ. c % error = 100 X (exptl - calcd)/exptl.
70 errorc
% Hpb
Hz
Dz
19.1 -1.3 0.98 0.466 19.1 0.64 0.32 0.474 78.6 -1.5 -1.1 0.581 78.8 -0.2 -1.1 0.583 = 1000 A 15 psia, and back pressure = 22 =t1psia. * The re-
Table IV. Comparison of Experimental Results with Model Calculations for Hydrogen-Deuterium-Helium Gas Mixturesa Bleed gar Rate, Expt
moIe/min
%H2
Exptl
Permeated gar
% Dz
Calcd
Exptl
70 errorb
Total rote, Colcd
% Ha
mole/min
HP
Dt
$0.6 51.5 39.5 0.475 40.9 20.8 21.6 6A 0.193 49.9 -2.1 39.5 0.464 21 4 21.9 40.3 6B 0.193 0.451 49.9 27.4 26.6 $2.1 11.8 11.8 7A 0.100 f2.0 28.3 0.445 50.1 13.0 28.6 7B 0.104 12.4 48.4 $2.9 2.8 0.378 8A 0.050 2.2 1.7 2.8 1-3.3 2.1 3.4 3.8 0.385 48.4 8B 0.052 2.6 Supply-gas composition: 42.8 A O . l % , 45.9 f 0.1% Dz, and 11.3 A O.lyoHe. Experimental parameters: temperature driving pressure = 1000 f 23 psia, and back pressure = 21 i 1 psia. % error = 100 X (exptl - calcd)/exptl. Q
hydrogen and deuterium is adequately described by the solubility term in eq 5 and that no further corrections are needed. Hydrogen-Deuterium-Helium Mixtures. Mixtures of hydrogen, deuterium, and helium were used for the most stringent tests of the model. In these experiments, we had two simultaneously permeating hydrogen isotopes mixed with a nonpermeating gas. The results are given in Table IV. Here, one mixture was used a t three different bleed rates as described above. Duplicate experiments were performed a t each rate. The agreement between the experimental data and the model calculations is excellent. The composition of the bleed gas in the experiments a t low bleed rates (8A and 8B) does not show the possible mixing effect indicated by the deuterium-helium experiments. The partial-pressure profiles calculated for these experiments are shown in Figure 6. The close agreement between the bleed-gas compositions derived from the experiments and the model indicates that the calculated partialpressure profiles are close to actual conditione. The curves in Figure 6 show some interesting features that that can be ascribed to the interaction of the permeating hydrogen isotopes. The deuterium partial pressure actually increases in the first part of the tube. This is because in this section the hydrogen permeates a t a sufficiently faster rate than the deuterium to cause the mole fraction of deuterium in the gas to increase. On the other hand, the partial pressure of the hydrogen decreases as the latter moves down the tube, causing the hydrogen permeation rate to drop until it approaches the deuterium permeation rate. At this point, the partial pressures of both isotopes decrease as the gases proceed down the tube, and if the tube is long enough, the hydrogen and deuterium partial pressures reach equilibrium with the gas outside the tube a t the same location. The curves in Figure 6 also show that the maximum separation of Hz from Dz in the bleed gas was achieved a t the highest bleed
\ L = 0.052 mo~e/min
=
+1.2 f3.9 $1.3 +1.7 +3.1 $3.3 415 i 8"C,
rL=0.104mole/min L = 0.193 mole/min
i
e
b
L
200 150
L = 0.193 mole/min L = 0.104 mole/min
-
-
L = 0.052 mole/min
___
Hydrogen
0
50
50
Deuterium Measured values from corresponding experiments
100 150 200 Distance from tube entrance
250
- in.
300
350
Figure 6. Model calculations of hydrogen and deuterium partial pressure vs. distance from tube entrance for hydrogen-deuterium-helium mixtures corresponding to experiments 68,7B,and 88 (I = bleed rate)
rate. It is apparent from these curves that the same separation can be achieved a t a lower bleed rate with a shorter tube. Conclusions
A mathematical model has been developed to predict the permeation of hydrogen-deuterium-inert gas mixtures Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
337
through palladium-silver tubes. The constants for the model are determined from pure-gas permeation measurements. Extensive comparison of the model with experimental data shows close agreement over a wide range of compositions. The experiments used to test the model were all nominally conducted at 4OO0C, 100 psia of driving pressure, and 20 psia of back pressure. The model should be valid over the same range of conditions for which eq 3 is valid when the pure-gas constants determined for the range of interest are provided. Within the limits imposed by the assumptions, the model is also adaptable to a variety of geometries. Although helium was used as the nondiffusing contaminant in our experiments, the results should be the same for other nonpermeating gases or gas mixtures providing that they do not poison the palladium surface or react with the hydrogen (Serfass and Silman, 1965; van Swaay and Birchenall, 1960). A particular process can readily be optimized by using the model to examine the influence of the process variables on the hydrogen-isotope permeation rate. Nomenclature
A = a = b = C = Cl,z =
effective diffusion area, cm2 constant, mole cm/(min cm2 - v‘/psia constant, mole cm/(min cmz - d z ] total solubility of a gas mixture concentrations of a gas in solution a t the opposite surfaces of a solid 3 = diffusivity G = molar flow rate into a tube, mole/min K = constant obtained from pure-gas solubility measurements L = molar flow rate out of a tube, mole/min 1 = length of a cylinder, cm P = total pressure of a gas mixture, psia P1 = driving pressure, psia Pz = back pressure, psia
pt,,
R
= =
rl,z =
T
=
X
=
2
= =
Y
partial pressures of hydrogen isotopes i and j , psia permeation rate of a gas, moles/min cylinder radii corresponding t o CI and CZ,cm temperature of a gas, O K mole fraction of a hydrogen isotope on the drivingpressure side length of a diffusion path, cm mole fraction of a hydrogen isotope on the backpressure side
literature Cited
Ackerman, F. J., Koskinas, G. J., J. Chem. Eng. Data 17, 51 (1972). Barrer, R. M., “Diffusion In and Through Solids,” pp 144-203, Cambridge University Press, New York, N. Y., 194;. Bird, R. B., Stewart, W. E., Lightfoot, E. N., Transport Phenomena,” p 502, Wiley, New York, N. Y., 1960. Darling, A. S., P l a p m Metals Rev. 2, 16 (1958). Darling, A. S., in Proceedings of the Symposium on the Less Common Means of Separation, Birmingham, England, April 1963,” p 103, Institute of Chemical Engineers, London, 1963. Davis, W. D., “Diffusion of Gases Through Metals. 11. Diffusion of Hydrogen Through Poisoned Palladium,” Report KAPL1375, Knolls Atomic Power Laboratory, Schenectady, N. Y., 1985. deRosset, A. J., Ind. Eng. Chem., 52, 525 (1960). Hickman, R. G., J. Less-Common Metals 19, 369 (1969). Hunter, J. B., U. S. Patent.2,773,561 (Dec 11, 1965). Lewis, F. A., The Palladium Hydrogen System,” p 130, Academic Press, New York, N. Y., 1967. Rubin, L. R., “Permeation of Deuterium and Hydrogen Through Palladium and 75 Palladium 25 Silver at Elevated Temperatures and Pressures,” p 55, technical bulletin, Engelhard Industries, Newark, N. J., 1966. Serfass, E. J., Silman, H., Chem. Eng. (London) No. 192, CE266 (1965). Sieverts, A,, Krumshaar, W., Ber. Deut. Chem. Ges. 43, 893 (1910). Birchenall, C. E., Trans. A I M E 218,285 (1960). van Swaay, M., RECEIVED for review April 22, 1971 ACCEPTED March 30, 1972 Work performed under the auspices of the U. S. Atomic Energy Commission.
Countercurrent Backmixing Model for Fluidized Bed Catalytic Reactors. Applicability of Simplified Solutions Colin Fryer and Owen E. Potter* Department of Chemical Engineering, Monash University, Clayton, Victoria 3168, Australia
The countercurrent backmixing model for gas-fluidized catalytic reactors is studied using the assumptions of Kunii and Levenspiel concerning gas transfer between the bubble, cloud-wake, and dense phases. The various simplifications which have been made previously in the solution of this model are examined and their regions of applicability determined. The solutions obtained indicate several areas in which experimental study of reacting systems will be useful in assessing the validity of the model; in particular, concentration prdiles in the bed will be useful in distinguishing between the countercurrent backmixing model and others.
T h e existence of gas backmixing in beds fluidized with gas has long been recognized (Gilliland and Mason, 1949, 1952). Simple two-phase models such as those reviewed and developed by Davidson and Harrison (1963) and Partridge and Rowe (1966a,b) do not take account of such backmixing. 338
Ind. Eng. Chem. Fundam., Vol. 1 1 ,No. 3, 1972
However, May (1959) and van Deemter (1961) have incorporated axial diffusion in the dense phase. It is now clear that this axial diffusion is dependent on the bubbles. Stephens, et al. (1967), pointed out that the upward flow of solids with the bubbles would lead to a downflow of solids
