MATHEMATICS O F ADSORPTIOK I N BEDS
1153
calculated. From the entropies, an explanation of the different rates a t different dilutions is proposed. A chain mechanism of the denaturation of virus and other proteins is proposed which qualitatively accounts for much of the experimental data. REFEREXCES
F. : d d v a u c e s iri Colloid Science, pp. 380-1. Interscience Publishers, Inc., S e w York (1942). (2) E Y R I N GH, E N R Y :J . Chem. Phys. 3, 107 (1935). (3) EYRING,H E N R YASD , STEARN, A . E . : Chem. Revs. 24, 253 (1939). (4) L a MER, V. K.: Science 86, 614 (1937). ( 5 ) L.4UFFER, Max A , : J. Biol. Chem. 126, 483 (1938). ( 6 ) LAUFFER, M a x A , , A N D PRICE, W .C . : J. Biol. Chem. 133, 1.(19-10). L . : Proc. Natl. h a d . Sci. U.S. 22, 439 (1936). (7) hIimIi'i, A . E., A N D PAULIXG, (8) NEURATH, H., GREENSTEIN, J . P., P r r x . m , F. W., A N D EHICKSON, J. 0 . : Chem. Revs. 34, I57 (1944). (9) SPONBLER, 0. L . : ?"he Cell arid Protoplasm, 11. 171. The Science Press, Lancaster, Pennsylvania (1940). (10) STEINHARDT, J . : Kgl. Danske S'idenskab. Selskab Mat.-fys. M e d d . 14, S o . 11 (1937). (11) WYCXOFF,R . W. G., A X D COREY,R. B . : J. Biol. Chem. 116, 51 (1936). (1)
~ X D E R S O N ,THobras
A XOTE ON THE MATHEbIATICS OF ilDSORPTIOX IT\' BEDS
s r x R.ANUKDSOS ,9chr1ol 0.1"Chonzstri/. 1 ) i i i s z o n ojChctiiica1 f ? ' n g z i i e e i z n q . 1 - i i i i ! e i Fzty of l l i n r i e s o t a , 117 11 ti ra pol i 14 l l z i l r ( t colt1 ~
l ~ l l ~ l l f lI ll t 1 I t l l
10 lR48
The mathematical solution of the adsorption of solutes from liquids and gases on solids in beds is complicated to the extent that of the many rnechanisms which can be imagined, all have some basis in physical fact. The mathematical complications hare their origins in our inability to cope u i t h the non-linear partial differential equations which arise. Certain problems have been solved completely, while others in which more complex mechanisms are assumed have been solved only partially, by either linearizing approximations or numerical methods. Two recent review articles by Thiele ( 5 ) and Klotz (3) cover the literature carefully. It is the purpose of this note to supply a complete niathematical solution of one particular problem, i t . , that in which the adsorption occurs irrereryibly a t a local rate of removal described by equation 2 belov-. If we assume that n-e have a bed of unit cross section n here depths aie measured as z, and if c
=
7%
= =
V
concentration in moles per unit volume of adsorbate in the fluid stream, amount of adsorbate on bed in moles per unit volume of bed, velocity of fluid through the interstices of the bed in length per second,
1154
S E A L R. AJIUNDSON
a = fractional void volume of the bed, and t = time in seconds,
then on making a material balance over an elemental thicltnrss of tied, the equation (see Klotz (3)) :
\VP
oh! :iin
If the solute is adsorbed irreveribly, Bohsrt :md .klams (1) and later Th15h(11wood (2) assumed that the local rate of removal is governed by
\\here -YOis the saturation capacity of ‘L unit volume of bed. T o equations 1 and 2 v e must append ceitain auxiliary conditions. The first states that until the fluid has traversed a distance x in the bed, the bed at point x must contain the amoiint of adsorbate it had before adsorption began, i.e., n(2, f) =
no(,.), whcn
1
2
2 7 ,
t 20
0)
If a fresh bed of adsorbent u c i ~present :It thc beginning, then n0(2) = 0. ‘rh(> second condition stateb that a t the bet1 entrance the concentration of adsorbate in the fluid may vary with time, i.e.,
c(x, t )
= c o ( t ) , \\hm z =
0
(4) t ) and c(x, t ) ,
I n order t o obtain s solution of the problem t n o functions, I L ( X , must be found xhich will satisfy equations 1, 2 , 3, and 4. Partial solutions have been obtained by the pieviouily mentioned vriters (1, 2 ) . In order to favilitatc tht. sohilion n c make thc following changes of v:iriahl(,. Let
then equations 1 , 2 , 8, nntl 4 hccomc
1155 where
IZ~(.C)=
no(T7.r), cl(y) = co(y/a).
Xote that equation 5 says
-am_ ay
r3c
ax
which implies that there exists a functionf(2, y) such that
df
=
indx
+ cdy
and
Substituting in equation 6 we obtain
In order to fiiicl \ \ hirli is it lion-linear hyperbolic partial differential equation. wlutions of this equation n-e make the substitution used by Thomas (6) :
Equation 10 becomes
and equations 9 become
I a log 4 c =I: d y From these t n o equations, it is seen that the auxiliary conditions 7 and 8 are N o -
n1h)
=
dz,0) -I,1. a log8.2:
S o t e that ttiese are ordinary differential equations which can be integrated to log +(x, 0)
=
1':SO - n1M1
d[
0
Since an arbitrary constant added to log
+ CI
+ is immaterial, x-e can define +(O,
0)
1156
NEAL R. AhILTNDSON
= 1 so that
NOJTthe gencral solution of equation 11 is
So, from equations 12 and 13
Therefore
