Diffusion in Beds of Porous Solids Measurement by Frequency Response Techniques P A U L F. DEISLER, J R . ~AND , RICHARD H. WILHELM PRINCETON UNIVERSITY. PRINCETON, N. J.
Interparticle and intraparticle diffusion effects are important in determining temperature and concentration gradients i n catalytic chemical reactors. For design purposes knowledge is needed regarding various diffusional mechanisms i n beds of porous solids which, as a class, include chemical reactors. A frequency response technique is used which has t h e general advantage of simultaneous measurement of several mechanism constants. Included in present work are: axial diffusion i n t h e fluid between particles, pore diffusion within particles, and a composition equilibrium between gases within and outside of particles. Experimental variables are total gas flow rate, and changes i n phase angle and in amplitude as periodically varying binary gaseous composition waves are sent a t steady state through a test bed. Results of t h e experiments lead, through appropriate equations, t o individual diffusion constants and adsorption capacitances. It was found with porous Norton catalyst spheres and a hydrogen-nitrogen mixture t h a t the diffusion constant within spheres was only 3% of t h e normal molecular diffusion constant, the gases do not adsorb measurably, and axial diffusion between particles a t a Reynolds number between 4 and 50 is significantly larger t h a n for molecular diffusion.
D
IFFUSION rate processes in fixed-bed catalytic reactors fall into two categories: diffusion within the pores of catalytic particles and diffusion in the fluid phase between the particles. Both are measured simultaneously in the present work but have been considered individually by previous authors. Thiele (10) addressed himself to the mathematical problem of simultaneous chemical and diffusion rate processes within catalyst pellets. Wicke with Brotz (11) and with Voight (12) provided experimental verification for Thiele’s equations and extended the theory to include three diffusional mechanisms. The early paper by Wiike and Hougen ( I S ) is representative of the considerable literature on the diffusion of matter from the particles in a bed t o the fluid stream between the particles. Bernard and Wilhelm ( 1 ) have discussed transverse diffusion of matter within a fluid stream that is coursing through a packed bed, and experimental and theoretical extension has been made by Latinen (6). No previous study of diffusion within a stream in the direction of flow between particles has been found. The present contribution stresses a steady-state, dynamic experimental technique which permits simultaneous determination of several mechanism constants. For packed beds of porous particles these mechanisms are axial diffusion in the continuous phase between particles, pore diffusion within particles, and a n equilibrium relationship between concentrations of material within and outside of particles. Experimentally, binary gas mixtures, a t a constant total flow rate but varying periodically and sinusoidally in composition with time, are passed into the packed bed. The diffusion mechanisms acting in the bed cause a decrease in amplitude and a shift in the phase of the composition wave as it passes through the bed. Equations relating the influent and effluent composition waves are derived. From the measured amplitude and phase angle shifts at different total gas flow rates the diffusion and equilibrium process constants mentioned above are then evaluated through the equations. Solution of the equations depends upon having a linear system, in which, for example, the mechanism constants are independent of the mean composition of the binary mixture. Present experimental results were found to be linear. Nonlinear systems may 1
Present address, Shell Development Co., Emeryville, Calif.
be approximated, however, by linear equations when the composition amplitude is held t o narrow limits. The procedure outlined above is only one aspect of frequency response procedures. Two other examples of these methods, which bear a certain mathematical similarity to the periodic steady-state analysis used in this work, are the study of the transient state of a system and the study of the periodic steady state of a closed-loop system. A closed-loop system is one in which the output modifies the effect of the input. The present investigation is concerned with open-loop systems, for the output has no effect on the input. Generally speaking, periodic steadystate analysis and transient analysis are means to similar ends, the former tending toward relative mathematical simplicity and experimental complexity, and the latter, the reverse. The prime prior reference is that of Rosen and Winsche (9), who first adapted the theory and the mathematical methods used in alternating current theory t o the study of mass transfer and chemical interchange processes. Kreezer and Kreeaer (4, 6) have applied the methods of studying transient-state behavior t o the investigation of the response of certain luminescent bacteria to temperature changes. Rosen (7) has predicted, theoretica!ly, the time variation of composition of the output stream from a fixed bed of porous solids, the input to which has undergone a sudden change in composition. I n the case presented by Rosen, diffusion in the solid, and film transfer to and from the solid, are assumed to be the important mechanisms. THEORY
Harmonic variation with time of the composition of a binary gas mixture at total constant flow rate is ~ ( 0= ) ZM
+ A(0)
COS
wt
(1)
where z(0) is the inlet composition in mole per cent of one component, ZM is the mean mole per cent composition about which the total composition of the stream oscillates, A(0) is the amplitude of oscillation, in mole per cent, t is time, and w is the angular frequency, or angular rotation, in radians per second. When such a stream is the input to a bed packed with porous solids, the varying gas composition throughout the bed will cause 1219
diffusion in and out of the porous solids, adsorption and desorption of the gaseous components, and diffusion in a longitudinal direction under the influenre of the longitudinal concentration gradients set up by the harmonically varying composition. The combined effects of the various diffusion and adsorption mechanisms, above, will be to decrease the amplitude of the composition waves and to cause a shift in phase a i t h respect to the inlet waves of Equation 1 as the waves pass domn the bed. For some time after the harmonic variation of the composition of the input wave is started, there will exist a transient, or unsteady, state as the output stream from the bed goes from its initial condition to the final periodic steady-state condition which is of interest here. The form of the periodic steady-state outlet composition xyave is
s(L) = x,v
+ A(0)e-B cos (ut - +)
(2)
where z ( L ) is the composition of the outlet stream in mole per cent, provided that all physical processes occurring in the bed may be described by linear equations. The quantities B and $ are functions of directly measurable quantities such as flow rate, bed length, fraction void, and frequency, a s \vel1 as of the coefficients TThich it is desired to evaluate. Once the mathematical forms of B and + are known, experimental measurements of the amplitude of the inlet and of the outlet waves, A(0) and A(L) = A ( O ) e - B , andof the apparent phase angle of the outlet waves with respect t o the inlet waves make possible the calculation of B and 4 , and from them the desired coefficients mav be obtained. Forms which B and + will take will depend upon the total mechanism assumed to act within the bed. Conservation of matter within a differential segment of the packed bed a t some distance, z. from the entrance to the bed yields the equation,
for the present case. Here, D L is the diffusion coefficient in the gas phase in the longitudinal direction; U is the average actual linear flow rate in the bed, and is equal t o the volumetric flow rate divided by the product of the column cross section (empty) and the fraction voids between the packing; q is the average concentration of material in moles/(total cubic centimeters of porous solids) within the solids a t any particular time, t; e is the fraction voids in the bed, between the particles; and p is the gas density, or the total concentration of all components taken together, in moles/(cubic centimeters of gas) in the gas phase. T a m s on the left-hand side represent the transfer of the component considered by diffusion in the longitudinal direction and by flow of the gas stream, respectively, while on the right side the first term represents the accumulation of the component in the gas phase, and the second term, the accumulation in the solid phase. In deriving Equation 3 the assumptions are: no radial concentration or velocity gradients, and constancy of gas temperature and pressure.
To solve Equation 3 requires an expression for a(l - in terms of at
x, z , and t. Such an expression is obtained by solving the equation of conservation for transfer within the solids (Equation 4). Spherical pellets were considered and the transfer within the pellets was assumed to take place by diffusion only. Tt was assumed also that the porous solid could be treated as a single homogeneous medium in which diffusion obeys Fick’s Ian, and that a single coeffirient for diffusion would represent all possible diffusion mechanisms in the solid over the composition amplitude range.
(4) where D, is the diffusion coefficient of the gases in the porous solid, and cs is the composition in moles/(cubic centimeters of porous solid) a t a radial distance, r, from the center of the sphere.
1220
Rosen and Winsche give t,he solution for Equations 3 and 4 for the case of DL = 0. The present work uses a n ext,ension where DL # 0 and restricts attention t o the steady-state solution. For present purposes the method of solution is out,lined briefly, boundary condit,ions are given, and the solution is presented. Details are given elsewhere ( 2 ) . The solution employs complex variables and the principle of superposition. Equation 4 for a spherical pellet a t position z in the bed is solved to give a relation of es to r. Average concentration in the solids, q, is then obtained by integration of es over the radius of the sphere. Differentiation of the expression for q with respect t o time then gives aP - in terms of the concentration
at
in the solid a t the pellet surface, c,(R), after application of boundary conditions in the solution of Equation 4. Bn over-all equilibrium expression relating the total composition of stored or adsorbed component within the boundaries of a pellet to the ambient gas phase composition in the interstice adjacent to the pellet,, is as follows:
c8 = Kc = Kpx
(5)
It is assumed that t,he equilibrium constant, K, will serve for both gases over the composition range considered, an assumption JThich can be justified only by the experimental results. Should the amount of adsorption of the individual gases, or their diffusivities in the porous solids, be different’, then separate equations of the form of Equat,ions 3, 4, and 5 would be necessary for each gas. Proceeding with the simpler condition, Equat,ion 5 is applied to give - in terms of gas phase composition,
at
a!l from above into Equation Substitution of t
2.
3 noiv gives the
equation of t,ransfer for the bed with q eliminated. Solution of this last equation TTith Equation 1 as the inlet condition gives a solut,ion of the form of Equation 2. Boundary Conditions. Equation 3. 1. Inlet condition a t z = 0 is given by Equation 1 2. A(z) = 0 for z = 33 Equation 4. 1. c,q(R) = A’(R) COS (ut - 4) 2. _ dA’(r) _ = 0 for 1’ 0 dr
I n terms of complex variable these boundary conditions are : Equation 3. 1. A ( z ) = A ( z ) for z = 0 2. A ( z ) = 0 for z = Equation 4. 1. A ’ ( r ) = A ’ @ ) for r = R 2. _ _ - 0 for = 0 dr
Solutions. COMPLETE SOLUTION.Introduction of measured amplitude change and phase shift over a bed of length L permits calculation of experimental values of B and +. One algebraic form of the solution for Equation 3, for the entire bed, leads to the following expressions for the real and imaginary parts, Y 1 and Y z . respectively, of the complex diffusional admittance for the spherical pellets:
which permit the calculation of experimental values of Y1 and Y , from the measured values of B and 4, provided that D L is known. If DL is not known, trial and error is involved in the reduction of data. With experimental values for Y I and Y t one may proceed to an evaluation of constants D, and K from complete expressions
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 6
for these admittances which were found by Rosen and Winsche from the steady-state portion of the solution of the heat transfer analog of the spherical particle Equation 4,a s follows:
+
pR(sinh 2pR sin 2pR) cash 2pR - COS 2pR
-
11
'
(8)
and pR(sinh 2pR - sin 2pR) cash 2pR - COS 2DR
1
R
and
+=
[. + (+)
Yz]
;
(15)
These last two equations are exact, and are, in fact, the result obtained directly by Rosen and Winsche for the solution of Equation 3 combined with Equations 4 and 5 for DL = 0. For nonporous pellets in the bed, for which Ds = 0, and K = 0, Y1 and YObecome zero, and Equations 12 and 13 become
where
(16) and
The above authors presented a direct semigraphical method for the calculation of both D, and K from Y1 and YO. The method is based on the fact that, by Equations 8 and 9, the ratio YI/ Y Ois a function of pR only, and that a curve of YI/Yz may be plotted in advance against pR. By means of this curve, a n experimental value of Yl/YO yields the corresponding value of pR, from which D, may be calculated directly. Knowing D,, K is calculated from either Y1 or Ys by means of Equation 8 or 9. APPROXIMATESOLUTIONS AND SPECIAL CASES. For experi
