Interaction between surface adsorption and transport of gases in porous media

Interaction between Sudace Adsorption and Transport of Gases in Porous Media G. M A R R O Q U I N , R O B E R T

W. C O U G H L I N , AND W. E. S C H I E S S E R

Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania Received August 13, 1970; accepted November 30, 1970 Transient solutions are presented for nonlinear, partial differential equations describing simultaneous gas-phase diffusive transport and the kinetics of physical adsorption and desorption within an idealized pore. The influence of heat of adsorption upon rate of desorption and its dependence on extent of surface coverage has been accounted for in the equations. Surface diffusion has been assumed negligible in the formulation. The solutions suggest that rates of sorption into porous materials and transient rates of transport through porous media may be significantly increased by adsorption. Also evident is significant dependency of these increases upon the form of variation of heat of adsorption with extent of surface coverage. In light of these findings it appears that transient transport rates in porous media may be influenced significantly by adsorption as well as by surface diffusion. Care must be taken in interpreting experimental results, therefore, to properly allocate the effects of adsorption as well as surface diffusion.

INTRODUCTION Important processes like separation by gas absorption into porous solids, migration of gases through porous media in gas reservoirs, and transport of gaseous reactants and reaction products within porous catalysts present situations in which diffusive transport in the gas phase, adsorption on the surface of the solid, and surface migration may take place simultaneously. The discussion presented here is restricted to gas-phase transport and physical adsorption; it neglects surface migration and assumes completely isothermal conditions. Reviewing diffusion in porous media Barter (3) discusses the influence of adsorption b y assuming instantaneous equilibrium between gas and solid as expressed b y a H e n r y ' s law constant. The present de~ velopment makes no such assumption as to instantaneous equilibrium. Moreover the present t r e a t m e n t also includes the dependence of adsorption heat on surface coverage and thereby shows the phenomenological effect on the transient process

not only of adsorbate-adsorbent interactions but also of adsorbate-adsorbate interactions. B a r e t (5, 6) has discussed, but not solved, the problem treated h e r e - - t h e combined kinetics of simultaneous diffusion and adsorption with no equilibrium a s s u m p t i o n - - a s described b y similar differential equations. Here we present numerical solutions of this problem for the onedimensional case. Model. Consider a cylindrical pore open at only one end as shown in Fig. I(A), or a similar pore of length 2L but open at both ends. These situations are mathematically identical since they are described by the same differential equations and the same set of boundary and initial conditions (see Fig. I ( A ) ) . A material balance over a differential thickness dx of the pore, including the rate (moles/cm 2 -- see) of adsorption klC(1

-

0)

and the rate of desorption OLle -Q/Rr, leads to the following equations for the Journal o] Colloid and Interface ~cience, Vol. 35, No. 4, April 1971

601

602

MARROQUIN, COUGHLIN, AND SCHIESSER (A)

(B)

7#

O (O,x)=O

.

~'

C(t ,0)= Cs

C(t,L)=O

P

C(O,x)=O

x

8(O,x)=O

/

C(t,O)= Cs

C(O,×)=O

I.

~_C=O

8x x=L PzG. 1. (A) Case I. (B) Case II. 1.0 ~

0.5 T: 2.0

o~ p-. o s % . . \

~

~

~

o,4 ,o

o.6

\

~

~

o.

- ~ - . ~ ~

-%_

----

~.~

0.2

---J

C

~

o6-oz

~-0"

2 ~ 0.1

I

0.2

0.4

0.6

0.8

1.0

Z:X/L

FIG. 2. Dimensionless concentration and fractional surface coverage profiles for Case I. With Q = 18000 cal/gmmole, f - - ; 0 ..... . isothermal

(temperature

~j _ 0.~ + h. e Or OZ 2 a0 = h[ha(1 -- O)f Or

Here

T) case: ha (1 -- 0 ) f

h~ =-- 2L2k_le-~/Rr/C, Dro , hdO]

h =-- C~roSm/2,

with the boundary and initial conditions: f(O, r) = 1,

f ( Z , O) = 0;

Of

O(Z, O) = O.

= O,

h~ ~ 2L2kl/Dro,

Case I

Journal of Colloid and Interface Science, VoL 35, No. 4, April 1971

and the other symbols are defined in the notation table. The boundary conditions above (Case I) are related to a physical situation in which a porous solid absorbs a gas. Another situation (Case II) of importance is that in which a gas is transported through a porous solid to a point where its

SURFACE ADSORPTION AND GASES IN POROUS MEDIA

1,0 ~

603

I 0.5 "r -= 1,0

o.8tf=

\

\

~

C/Cs

o.e

-1°,4

o.s

I

e 0.5

o.+I

t

0.2

! I

~ - ~ - ~ ---_ ~ - : - - - ---_-~ 0.2- - - -"~i ~

_o. s T:I.0

~

0.1 0.2

0.4

- 0.~

~

0 0

--

0.6

- - ~ 0.8

~

0 1.0

Z=X/L

FIG. 3. Dimensionless concentration and fractional surface coverage profiles for Case I. With Q = 18000-9000 0 cM/gmmole, f ; 0 ..... . concentration is maintained effectively zero, either b y dilution or pumping it away very quickly. Figure I(B) shows this situation, and it too is described b y the previous differential equations, but with the following set of boundary and initial conditions:

f(O, r) = 1,

f(Z, 0) = 0;

/(1, ~) = o,

o(z, o) = o.

Case I I

For all computations the parameters in the differential equations have been taken a s : h = ha = l a n d hd = 1.07 × 10 '3 exp -Q(O)/RT with T = 300°K and Q(O)= 18,000 eel/ gin-mole exp ( - C 2 0 ) - C10. Here C1 and C2 have been assigned various values to simulate eases in which the heat of adsorption Q is independent of coverage (C1 = C~ = 0), linearly dependent on coverage (C2 = 0, C1 > 0), or exponentially dependent on coverage (C1 = 0, C2 > 0). Note t h a t for CI = C2 -- 0 or for 0 = 0 (a bare surface) Q = 18,000 eal/gm-mole and he = 1. Independence, linear dependence, or exponential dependence of Q on 0 correspond, respectively, to Langmuir, Temkin, or Freundlich type isotherms.

The differential equations have been solved for the stated boundary conditions using a "Distributed Systems Simulator" (1, 2) software package which uses implicit finite difference approximations. The solutions presented here have a m a x i m u m absolute error of 0.5% for f and 0. For cases of exponential dependence of Q on O, it was necessary to quasi-linearize the original equations in order to obtain stable solutions. I n eases where the solutions obtained to the original equations were stable without quasi-linearization, using quasilinearization caused no significant difference in the results. Case I: Sorption into Porous Body. Transient profiles of f and 0 versus Z are given in Fig. 2 for the ease of heat of adsorption independent of 0 (CI = C2 = 0), in Figs. 3 and 4 for the case of linear dependence of Q on 0 (C2 = 0) with C1 = 9000 and 18000 eal/gm-mole, respectively, and in Figs. 5 and 6 for the ease of exponential dependence ( C 1 - - 0 ) with C2 = 0.693 (ta make Q -+ 9000 eel/gin-mole as 0 ~ 1.0} and with C2 = 4.0 (to make Q--~ ~-~306. eel/gin-mole as 0 ~ 1). I n Fig. 7 concentration profiles are compared for the eases: Journal of Colloid and Interface Science,

Vol. 35, No. 4, April 197i'

MARROQUIN,

~04

COUGHLIN,

AND

SCHIESSER 0.5

0.8

0 . 8

f=clCs

0.4

~ .

0.6

~

- 0.3

0.4 I-

\

021

"-~-- " ~ - '

-IO.Z

o

~

0.4

~

0

8

"-~- ~

] 0 2

0

,

I 0 4

0.1

"-~

~

,

-~-0.6

---T ~ 1

0.2

, 0.8

1.0

Z=X/L

FIG. 4. Dimensionless c o n c e n t r a t i o n a n d fractional surface coverage profiles for Case I. W i t h Q 18000-18000 0 cal/gmmole, f ; 9 ..... .

0.5

1.0

T=I.O

D 00O 8'0.~~ 3.04.

C o.8

f =

ICs

0.E

0.2

o.4

0.2 __L~ . ,

.

.

.

T=I.O

~_

I 0

0.2

~0.4

~

----~ 0

- ~ ,

I

o,I

Zh.E_

---,,

~

0.4.

~-

0.2

~ 1 ~ 0.6

0.I

~

-~-

-

-

I

-

0

-

0.8

1.0

Z=X/L

FIG. 5. Dimensionless c o n c e n t r a t i o n a n d fractional surface coverage profiles for Case I. W i t h Q 18000 e-°.693 e c a l / g m m o l e , f -; 8 . . . . . . Journal of Colloid and Interface Science, Vol. 35, No. 4, April 1971

S U R F A C E A D S O R P T I O N A N D GASES I N P O R O U S M E D I A 1.0 ~

605

] 0,5 lr =1.0 0,8

0,8

0.4

f=C/C$

0.6 h

\

-~o.3

~

0,4

0.2

o.z t-

~

=

0

~-~-=io. I

~

~___

~

t

"r = 1 . 0

I 0.2

0

,

I~ _ 0,4

I 0.6

L.~--__~---___ 0.8 1.0 Z = X/L

i '~

FIG. 6. Dimensionless c o n c e n t r a t i o n a n d fractional surface coverage profiles for Case I. W i t h O 1800 e- # cal/gmole, f • ; 6 ..... .

=

1.0 FOR

"r= 1.0

02

I

2

3

4

f= C/Cs 0.6

FOR

"t": 0 . 2

I DIFFUSION 4~ON[ ~ 2 Q=I8000 e

0.4

3

Q:ISO00

-

I 2 3

180000

4

4- Q= 18 OOO 0.2

0

0.2

0.4

0.6

O .8

1.0

Z= X / L

FIG. 7. C o m p a r i s o n of c o n c e n t r a t i o n profiles for Case I. w i t h various dependencies of Q on 0 a n d w i t h no adsorption. Journal of ~oUoid and I~t~rface Science, Vol. 35, No. 4, April 1971

606

MARROQUIN, COUGHLIN, AND SCHIESSER

1.8

1.6 I 2

|.4 -

Q = 18000 Q= ~8000 - 9 0 0 0 6'

3

Q= 1 8 0 0 0 - 18000 ~

4

Q= 1 8 0 0 0 e x p ( - 0 6 9 3 e )

5

Q :18000

6

DIFFUSION

exp(-4~

)

ALONE

1.21

1.0

0.8

0.6

0.4

0.2

0

,

0

L

0.2

,

I

L__,

0.4

0.6

I

,

0.8 •r = t D / L 2

I

1.0

FZG. 8. T i m e d e p e n d e n c e of t r a n s p o r t r a t e . Case I.

of pure diffusion, constant Q, and Q as a function of 0; the influence of adsorption is to lower the concentration profiles compared to the case of diffusion alone. This influence appears to be greatest for constant Q at the larger values of dimensionless exposure time r. Figure 8 shows the concentration gradient at the entrance of the pore (proportional to sorption rate) as a function of time. Again the influence of surface adsorption and the dependence of dourna~ of Colloid and Inferfacc Science, Vol. 35, No. 4, April 1971

Q on 0 is quite evident. The general effect of adsorption is to increase the transient rates of transport, this increase being the greater the less Q is influenced by 0.

Case II: Transport through A Porous Medium. Transient profiles of concentration and coverage along the pore length are also shown for this case in Fig. 9 for constant Q = 18000 cal/gm-mole and in Figs. i0 and Ii for linear dependence of Q on o. Figure 12 shows the relative rate of trans-

S U R F A C E A D S O R P T I O N A N D GASES IN POROUS M E D I A 10.5

I.O

i

r:2.0

fo;

f = c/cs

607

o

"

o.6-

, ~ _ _ . _ _ ~ _ _ ~ o.2

\\\

\

0.4

\

\ \

\

~

~__~ 20 0.2

\

0.2

\ 0.2

0.5

\ \

\ 0.4

-~0.4

\\

.

•

\~

0

0.2

\\

\~\~

-lo,

0

0 0.4

0.6

0.8

1.0

Z = X/L

FIG. 9. Dimensionless concentration and fractional surface coverage profiles for Case II. With Q 18000 ca]/gmmole,

f

----;

O .....

=

.

1.0

0.5

0.5 0.8

0.3

0.4-

0.2 f .= C /Cs

0.1

0.6 ~

0.5

0,2. i ~..-

.

~

•

---_ ---_ -----

0

0.2

0.4

----~ ~

0.6

0.[

~

~

~-%\\ _ ~

~--~

0.8 Z :

o

1.0 XIL

Fzo. 10. Dimensionless concentration and fractional surface coverage profiles for Case I1. With Q = 18000-9000 o cal/gmmole, f - - - - ; ~ . . . . . . Journal of Colloid and Interface Science, Vol. 35, No. 4, April 1971

608

MARROQUIN, COUGHLIN, AND SCHIESSER 1.0

0.5

0.8

0.4

f= C/Cs 0.6

0.3

0.4

0.2

0.2

O. I

0

0 O

0.2

O .zl

0,6

0.8

I.O

Z= X / k

Fro. 11. Dimensionlessconcentration and fractional surface coverage profiles for Case II. With Q = 18000-18000 0. f ; 0 ..... . port (--[Of/OZ ]z=0) as a function of time for the various dependencies of Q on O. It is clear from these graphs that adsorption on the walls also increases the transient transport rate for this case as well. The magnitude of the heat of adsorption and the dependence of Q on 0 also influence the transient concentration profiles and the rates of transport through porous media. Figure 13 provides a comparison of transient concentration profiles without adsorption and with adsorption. The influence of adsorption again is evident as in Case I. The influence of the dependence of Q on 0 is also apparent. DISCUSSION

It is clear from the results above that transient transport into and through porous media may be substantially influenced by physical absorption on the porous solid. Such absorption tends to increase transient transport rates into the material and the increase in rate is not accounted for by ordinary diffusion theory. Including the effects of adsorption together with fluid phase diffusive transport leads to nonJournal o/Colloid and Interface Science, Vol. 35, No. 4, April 1971

linear differential equations that must be solved by numerical techniques. It should be mentioned that the values of the parameters chosen here (ha = h = 1.0) correspond approximately to 100A pores, 75A2 molecular cross-sectional area, 10 atm pressure, room temperature, 1 mm pore length, diffusivity about 0.2 cm2/sec, and a sticking coefficient of about 10-8. The values of ha, he, and h were chosen equM to unity solely for computational convenience. More realistic situations corresponding to larger values of sticking coefficient, and therefore, to larger values of ha should produce an even more marked influence of adsorption on transport. As mentioned earlier, the model considered here takes no account of surface migration which can, of course, occur together with adsorption and gas-phase transport. Barrer (3) and Satterfield (4) have discussed the role of such surface diffusion in contributing to transport in porous media and the latter author points out that even when surface mobility is unimportant, accumulation of material by adsorption may be of major significance

SURFACE ADSORPTION AND GASES IN POROUS MEDIA

609

1.8

1.6

/\\\ \\\\

d~zf z= o

I

Q =18000

~ ~

°-- , . o o o - ~ o o o e o; ,~ooo_ ,~ o o o ~

1.4

I

1.2

1.0 ¸

0.8

0.6

04

0.2

0

I

0

0-2

,

0.4

[

0.6

,

t

,

0.8 t-=tD/L

[

1-0 2

FTG. 12. Time dependence of transport rate. Case II.

Journal of Colloid and Interface Science, Vol. ~5, No. 4, April 1971

610

MARROQUIN, COUGHLIN, AND SCHIESSER 1.0

O.e

I

DIFFUSION

2

Q :

3

Q = 18000

ALONE

18000-18000~

f : C/Cs

FOR

O.E

v = 0.5 I

0,4 2

FOR 1 : : 0 . 2

.3

I

0,2

0

0.2

0.4

0.6

0.8

1.0

Z:X/L

FIG. 13. Comparison of co~ce~ltrationprofiles for Case II with various dependencies of Q on e and with no adsorption. but is sometimes overlooked. The work presented here demonstrates the potential significance of such adsorption in transient transport within pores. It is clear from this development that, when the transient flow rate of an adsorbable gas in a porous medium is greater than predicted from the flow of a nonadsorbed gas, it may be explained on the basis of transient adsorption alone into an immobile adsorbed layer, or by surface diffusion alone within a mobile adsorbed layer that is in instantaneous equilibrium with the gas phase, or by some combination of both of these processes. The results here suggest that, whenever possible, measurement of surface diffusion rates should be based only on steady-state experiments and that the estimations of surface diffusion rates from unsteadystate experiments should be reexamined in light of the combined effect of simultaneous adsorption and surface migration. NOTATION C(t, x)--Concentration of adsorbate within the fluid phase (moles/tin3). C~Coneentration at the mouth of the pore (moles/cm3). Journal of ~olloid and Interface Science, ¥oI. 35, No. 4, April 1971

D--Diffusivity for the adsorbate (cm2/sec). E~--Activation energy of adsorption (cal/ gm-mole). Ex--Activation energy of desorption (cal/ gin-mole). f--Dimensionless concentration C/C~. h--Dimensionless factor CeroSm/2. h~--Dimensionless factor 2L2kl/Dro. hx--Dimensionless factor 2L2k - ze(--Q/RT)/ C~ Dro . kl--Adsorption rate constant (cm/sec). L1--Desorption rate constant (moles/tin 2set).

L---Length of the pore (cm). P--Pore diameter (cm). Q--Heat of adsorption (cal/gm-mole). Q0--Heat of adsorption when 0 = 0 (eal/ gm-mole). r0--Pore radius (cm). S,,--Surface area occupied by one mole of adsorbate (cm2/mole). S.S.--Steady state. t--Time (set). x--Axial distance (cm). Z--Dimensionless axial distance X / L . e--Fractional surface coverage. 7--Dimensionless time t D / L 2.

SURFACE ADSORPTION AND GASES IN POROUS MEDIA ACKNOWLEDGMENT G. M. is grateful to the National Polytechnic Institute of Mexico for its financial support. REFERENCES 1. ZELLNER, M. G., "Distributed System Simulator," Ph.D. Dissertation, Lehigh University, 1970. 2. ZELLNER, M. G., COUGHLIN, R. W., STEIN, F. P. AND SCgIESSER, W. E., Proceedings of

611

the 1970 Summer Computer Simulation Conference (ACM/SHARE/SCI), Vol. 1, pp 284294, Denver, June 1970. 3. BAnRE~, R. M., Applied Mater. Research 2 (3), 129 (1963). 4. S~TTERFIELD, C. N . , " M a s s T r a n s f e r in H e t e r ogeneous Catalysis," p 49. M.I.T. Press, 1970.

5. BAaET, J. F., J. Colloid Interface Sci. 30, 1 (1969). 6. BARET, J. F., dr. P h y s . Chem. 72, 2755 (1968).

Journal of Colloid and Interface Science, Vol. 35, No. 4, April 1971