LINEAR SOLUTIONS OF FICK’S LAW ROBERT B. ANDERSON, JAMES BAYER, AND LAWRENCE J. E. HOFER Pittsbztryh Coal Rrsearch Crntrr, Burrau
of
.Mines. LT. S. Ihpartment of the Interior, Pzttsburqh, Pa.
As a part of Bureau of Mines investigations on the release of methane during the mining of coal, rate equations for diffusion-controlled processes were investigated. Linear forms of solutions of Fick’s law as given b y Crank for spheres, cylinders, and platelets, valid for the initial 87, 73, and 45y0of the rate process, respectively, are presented. For the remainder of the rate process the usual exponential equations may b e simplified b y dropping all exponential terms except the first. These equations may be used for resolving rate curves that are composites of curves for particles of different diffusivity or size. Rate curves for the adsorption of methane on 270- to 325-mesh Pocahontas low volatile bituminous coal at 30” C. are used as examples.
Equations for Fick’s law are compared with parabolic rate equations.
law states that the transport of the diffusing substance per unit area in a given direction is proportional to its concentration gradient in this direction. Solutions of Fick’s law for diffusional processes are found in standard monographs on diffusion and adsorption ( 7 - 1 ) in terms of a series of exponential terms. These equations have been frequently applied to rates of adsorption and solid phase reactions; however. they are difficult to apply to experimental d a t a to determine either the validity of the equation or the value of the constants. Crank (,?) also presented linear solutions for platelets, cylinders, and spheres. Nelson and Walker (.5>6) apparently were the first to apply these linear solutions to adsorption kinetics. ( I n the.present paper the term “linear equations” is used to denote equations for which suitable simple functions may be plotted as a straight line.) T h e present paper presents linear forms of diffusion equations of Crank ( 3 ) for infinite platelets, infinite cylinders, and spheres that are valid for substantial initial portions of the process. T h e term “infinite” refers to platelets with length and Liidth very much greater than thickness and cylinders with length very much greater than radius. T h e remainder of the rate process may be expressed in the exponential form using only the first exponential term. T h e linear equations may be applied to diffusional processes that are composites of two or more rate curves. These equations are applied to the rate of adsorption of methane on coal. Equations for Fick’s law are compared with parabolic rate equations. I n the present paper equations are defined for rate of adsorption and desorption problems; however, they can be readily translated into units suitable for exchange reactions in solids, etc. For sorption processes the equations presented subsequently are applicable only if the adsorption isotherm is linear or nearly linear ( 3 ) . I n the examples given the amount adsorbed increased with pressure to the 0.87 power. ICK’S
linear Forms of Fick’s l a w
For the solutions of Fick’s law considered in this paper, as applied to adsorption processes, the adsorption isotherm is assumed to be of the form Q ’ = aC. where Q ’ and C are, respectively. the amount adsorbed in moles per cubic centimeter of adsorbent and the concentration in the gas phase in moles per cubic centimeter, and a is the proportionality constant. Adsorption equilibrium is assumed to be obtained instantaneously. and the rate-controlling factor is diffusion of the
adsorbate. T h e adsorption process is assumed to occur a t constant temperature. T h e diffusivity defined as D in the present paper is the apparent diffusivity, and it is related to the truediffusivity, D,,,,, by the equation D = D t F U e / ( l a ) . Boundary conditions a t the start of the process are that the concentration of adsorbate is constant throughout the adsorbent, and in equilibrium with the concentration of the adsorbate in the gas phase. At t = 0, the concentration of adsorbate in the gas phase is instantaneously changed to a new value and adsorption equilibrium is instantaneously attained between the new concentration in the gas phase and the amount adsorbed a t the periphery of the particles. For adsorption data. the fractional approach to equilibrium, Z, is given by Z = ( Q t- Qo)/(Qm - Qo),where Q t , Q,,, and Qm are, respectively, the amounts adsorbed at time t , a t the start of the process, and at equilibrium. Solutions of Fick’s law in terms of a series of exponential terms are :
+
PLATELETS m
1 - Z = (8/nz)
[1/(2n n=O
+ 1 ) 2 ]exp - ( 2 n f 1)2(n2klt)/4 (1)
CYLINDERS m
SPHERES m
I n these equations k l = D / P , where 1 is half of the thickness of the platelet and D is the diffusivity; kz = D / a 2 , where u is the radius of the cylinder; k3 = DIr2, where Y is the radius of the sphere; j, has the following values: j~ = 2.405.jy = 5.520, j 3 = 8.654 ( 7 ) . Crank (3) has presented the following linear equations that are accurate for the first 45% or more of the adsorption process. PLATELETS [for 0
< ( k , ~ ) ” ( 4 / ~ ~ ’ ~ ) “ 2 t )-’ ’kyt ~ - ( 1 / 3 ~ ’ ~ ) ( k 2 t ) ~ ~ (5) ~ V’OL. 4
NO. 2
APRIL
1965
167
SPHERES[for 0
< (k3t)I Z
< 0.4. 0 < Z < 0.871
This leads to the equation
- 3k3t
( 6 n1 ' ) ( k 3 t ) '
(6)
At the upper limits, Equations 4. 5. and 6 are accurate to better than O.lYc, and the equations are exact for values of ( k , t ) 1 2 less than 0.35. T h e third term of Equation 5 makes only less than a 2y0 contribution to Z a t (k2t)l = 0.4, and is relatively less significant for smaller values of k j t ; thus, for most purposes this term can be ignored. Equations 5 (with third term omitted) and 6 may be transformed to
=
2{ 1 - dl - (nZ,4)
1
= 2k4''tl2,
Xo
(12 )
(7)
where n = 1: 2. and 3 for infinite platelets. infinite cylinders, and spheres, respectively. is the rate constant. and Xo is half of the thickness for platelets and the radius for cylinders and spheres. For platelets ( n = 1 ) the equation is identical to Equation 4 except for constants. The parabolic equation for cylinders ( n = 2) is virtually the same as the Fick's law Equation 8 for spheres. The rate equations based on the parabolic law model hold over the entire range 0 Z 1, whereas the linear Fick's law equations change to first-order equarions as Z increases: as given in Equations 9 to 11.
(8)
Rate of Adsorption on Composite Samples Containing Spheres Having Different Values of D' 2 / r
CYLINDERS (7rk2t)'
1 - (1 - Z)'
<
