filter does occur. Accurate information on this point has not been obtained. However, as a first approximation, which should not be used for full scale plant design, but may be used, for lack of anything better, in designing a pilot plant, the rate of pressure drop increase may be considered to be inversely proportional to the &eight mean particle diameter. This assumption must be confirmed by future experimental work. The actual form of equipment which is used for full scale plant work involves the use of multiple units of the exact type used for pilot plant work. Such a unit is shown in Flgure 8. These units can be provided in multiple lengths to furnish areas up t o 40 square feet per unit. The basic unit itself, however, consists of an oval-shaped assembly nhich is 13/4inches wide by inch thick by 24 inches long. Design is such that a catalyst can fall freely from all surfaces. The porous bayonet itself is permitted to expand freely, as compared with the central core, but is prevented from vibrating hy means of the expansion control deviczs a t the end of each bayonet unit The individual filter units are tested a t 250 pounds prr square inch differential from the outside in; their use in practicc is not recommended a t more than about 5 pounds per square inch differential. Thus, a very substantial amount of weakening of the porous material as a result of oxidation may occur before failure is actually observed. The bayonets themselves arc strong enough that a 200-pound load applied a t the center of one of the baronets will not damage it. The life of such an installation a t 900" F. is estimated t o be adequate for long-term industrial use; installations used for almost 2 years a t 900' F. have shown no serious corrosion and appeared to be suitable for many additional years of service. I n reports obtained on some thirty pilot plant installations, corrosion has not been a problem in any atmosphere, except as 3, result of hydrogen sulfide corrosion in reducing atmosphere. Where a partial pressure of 1atmosphere of hydrogen sulfide exists the maximum temperature of usage must be reduced from 900' to 500" F. The problem of providing a metal or alloy which will successfully withstand hydrogen sulfide a t higher temperatures has not as yet been completely solved; Type 309 stainless steel offers promise in this direction.
As a precautionary word, the filters cannot be used in reactors where coking may occur to plug the reactor surface. They can be used under all conditions in regenerators, particularly since sulfur dioxide, as contrasted with hydrogen sulfide, does not appear to cause serious corrosion. The filters can be used in the reactors in applications mhere coking does not occur. Khere tempeI atures exceed those permitted, particularly where the filters are used in regenerators, a simple solution to the problem is available By introducing liquid water spray in the upper part of the regenerator, the gas tempwature can be lowered by several hundred degrees. Coincidentally, owing to reduced gas volume and viscosity, pressure drop across the filters is actually reduced. This method is stronglv recommended where temperatures over 900" F. are obtained. The porous stainless steel filters have been used in a large number of pilot plant installations, in nearly all cases with good success. No failures due to breakage, other than those directly traceable to use of excessive use of temperature or to hydrogen sulfide atmosphere, have been reported where the bayonet type of unit was used. Other types of units developed earlier have also been used with almost uniform success. Two 1000-square foot installations are due to go on stream in 1953. one for the catalytic oxidation of a chemical and the second in the regenerating section of a synthesis plant. Another smaller regenerator application is also in process of construction. ACKNOWLEDGMENT
The assistance of Leon Laaare, of the American Cyanamid Corp., in connection with the derivation of the conversion formula, is gratefully acknovledged. LITERATURE C I T E D
(1) Pall, D. B., U. S. Patent 2,554,343 (1951). ( 2 ) Micro Metallic Corp., Glen Cove, N. Y.,Porous Stainless Steel Release, No. 204. (3) Vanadium-Alloys Steel Co., Prealloyed Steel Powder, Bull. No. 2. (4) Ibid., Bull. Xo. 3. RECEIVED for review
March 9,
1953.
ACCEPTEDApril 13, 1953.
Diffusion in a hid Moving at in a Tube ADRIAAN KLINKENBERG N.V. D E B A T A A F S C H E P E T R O L E U M M A A T S G H A P P I J , T H E H A G U E
H.
J. KRAJENBRINK
AND
H. A. LAUWERIER
ROYAL DUTCH S H E L L LABORATORY, AMSTERDAM, HOLLAND
T
HE present study deals with the concentration distributions
caused by diffusion in a fluid moving in a cylindrical tube a t uniform velocity. The solute emerges from a continuous source in a point on the axis of the tube. The amount of solution introduced in this source is so small that it does not upset the uniformity of the velocity. Axial and radial diffusivity are not assumed to be necessarily equal. The stationary state is assumed to have been reached. A general equation has been derived for such nonisotropic diffusion. The solution is obtained with the aid of two-sided Laplace integrals, which are made to fit a t the injection point. Later on D, may be equated either to D, or to zero. The results are computed numerically. Corresponding equations are given for diffusion in the infinite
1202
space (absence of the cylinder wall). These must be applicable when the solute has not yet reached the wall. It is examined when they lose validity. The boundary conditions in this problem correspond to the experimental arrangement used by Bernard and JTilhelm ( I ) in their determination of eddy diffusion constants. RADIAL AND AXIAL EDDY DlFFUSlVlTY
In order to explain the practical importanct of the above mathematical problem, it is necessary t o discuss its phyeical background. It is often stated that a velocity which is uniform over the cross section is realized in the case of flow through a packed bed. This, of course, refers t o macroscopic observations only. This
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 6
”
I n one experimental technique eddy diffusion constants are derived from concentration patterns produced by a point source of solute on t h e axis of a packed tube. It was desired t o include inequality of radial and axial diffusivity in t h e treatment and t o compute t h e concentration curves. T h e concentration distribution was calculated by t h e method of t h e two-sided Laplace integrals. T h e results are presented in form of graphs covering variation of all three variables. T h e applicability of simplifications (absence of axial diffusion, equali t y of radial and axial diffusivity, diffusion in infinite space) was examined. . T h e results should aid in evaluating data on diffusion, primarily eddy diffusion. T h e mechanism of axial eddy diffusion was analyzed and t h e importance of this phenomenon i s stressed. A
macroscopic, mean velocity is used in our fundamental equation in the term that accounts for convection. The transports of solute described as diffusion will be of two kinds-viz., movement of solute with respect to solvent (true, molecular diffusion) and movement of solute due to the microscopic fluctuations of the flow. The latter process is, macroscopically seen, called eddy diffusion. It is, however, not a t all self-evident that the transport caused by the above fluctuations should be describable by diffusion laws. The name “eddy diffusion”in this case is a misnomer, as there may
not be any eddies. However, the process bears a certain resernblance to the eddy diffusion as observed in the absence of packing material in fluids in turbulent flow. Eddy diffusion is often much more important than molecular diffusion, especially in liquids. In random packings the molecular diffusion will be isotropic and there is no need to distinguish radial (or lateral) and axial diffusivity. However, the processes whose effects are termed eddy diffusion are not the same radially and axially. Radial eddy diffusion is caused by subdivision and reconibination of the streams around particles, and, a t high flow rates, possibly by eddy formation. Axial eddy diffusion is ascribable to various causes which may be discussed most convenientIy in the one-dimensional case with a peak of solute introducud a t the plane z = 0 (the bed entrance) a t zero time. The solute molecules after a certain time will then be found not to be all in one plane. The followingmechanismsare held responsible for thiR phenomenon:
1. I n each ore the liquid passing near the wall moves slower &an the liquid passing near the axis. 2. I n any real packing, other than the closest packing of equally sized spheres, there will be wide and narrow pores in parallel, with consequent differences in flow rate. 3. There may be differences in pore structure if one compares regions of packing which are large with regard to particle size. 4. At high flow rates there will be turbulence.] 2
P
P Figure 1.
I’ as Function of f and
--- c* == 2
m
June 1953
I’as function of f/2p
p
The fluid a t the wall of a pore is at a standstill, but it will exchange solute with the core of the pore by diffusion. Thie radial diffusion will reduce the spread in the times of passage of the solute to below that in the times of passage of the solvent-Le., axial diffusion is less than the above purely convective mechanisms would indicate. The progress of such radial diffusion must depend pore width X velocity and it must be on the group diffusivity small when this group is large with respect to 1. I n view of the fact that diffusivity in liquids is smaller than kinematic viscosity by a factor of at least 1000, this condition is met in liquids a t Re = 1 but-at least for noncolloidal solutes-it is no longer BO at Re = 10-8. Any of the above convective mechanisms when applied once only will not give rise to a Gaussian probability distribution of solute in the bed. A fundamental theorem in the theory of statistics due to Laplace-Ljapounoff states that a great number of independent repetitions will produce B probability distribution. This is irrespective of the kind of distribution produced by the individual step, if only a certain criterion for convergence is met. This is the justification for applying diffusion
INDUSTRIAL AND ENGINEERING CHEMISTRY
1203
laws to eddy diffusion, and even for naming it so. Ccnditions may, however, arise where diffusion laws cannot be applied, two examples of which may be mentioned. 1. If the regions of inhomogeneity of packing become comparable to the tube size, the diffusion l a m are no longer applicable. Such experiments would be discarded on account of channeling.
In the immediate vicinity of the source diffusional transport is fast with rcgard to convection, so that the convection term in Equation 1 may be dropped. Equation 1 may now be reduced to the corresponding cquation for isotropic diff usion by introducing a new length coordinate
This isotropic process is spherically symmetrical, so that Equation l can now be replaced by:
where 2 2
= 2:
+
1.2
The solution thereof is
This transformed concentration,-cx, is related to the original c by
Transforming back to the old coordinate, z , we find 4
c =
470;
+ D,r2)'
/2(D,z2
A, '
Equation 1 may be simplified by introducing the dimensionless groups
2
(0
figure 2.
for
p =
0
2 . Eddy diffusion lavs should not be applied just below the injection point. Several eddy diffusion mechanisms do not allow the presence of solute upstream of the injection point, yet diffusion laws would allow it there. This is no practical limitation to the applicability for certain results, the low value of the group diameter required for this upstream diffusion diffusivity being unattainable for eddy diffusion.
A criterion has been derived to show when to neglect axial with regard to radial diffusion. This is found to be allowed a t high values of the group tube diameter times velocity divided by geometric mean of axial and radial diffusivities. Since axial diffusivity may be much larger than radial diffusivity, it is unsafe to do so on the strength of data on radial diffusion alone. There is, in fact, a region (liquids a t low Reynolds numbers, as examined by Latinen and Wilhelm, 3 ) , where radial eddy diffusion vanishes but where axial eddy diffusion remains proportional to flow rate, so that the corresponding PBclet group remains constant. Better knowledge of axial eddy diffusion is considered important in the study of several processes carried out in beds (unsteady state heating and cooling, chromatography, chemical reactions). DERIVATION OF GENERAL EQUATIONS
The processes of diffusion and convection in a tube are governed by the partial differential equation:
and the boundary conditions: z =
-m,c=o;
r = R , -ac= O
br together with the condition that there is a source of constant strength, q (unit: mass per time), in the origin, First the condition of a source of constant strength, p, will be discussed.
1204
D!
__-'22
D:/*R
2 Cm
=
r
(radial coordinate)
= p
-
(3a)
r (axial coordinate)
(group measure for concentration)
c, denotes the concentration at
t = a ,which
(3d)
is given by
(3e)
It is to be noted that four dimensionless groups are sufficient, whereas dimensional analysis would allo57 five. Equation 1 now takes the form
p =
r=o
- 3 3
The condition for the source becomes: =
q p 2
+ f2)1/Z
+S(P,
PI
wherej ( p , c ) is finite in the origin. Using the method of two-sided Laplace transformations we Put
which transforms Equation 4 into a differential equation of Bessel type
where y= =
250.3
INDUSTRIAL AND ENGINEERING CHEMISTRY
- s2 Vol. 45, No, 6
The general solution of Equation 5 is
r
= AiIo(rp)
+ AzKdrp)
The back-transform of the IO(y p ) component remains finite in 0, 1 = 0; the back-transform of the K O( y p ) component has a singularity in the origin corresponding with the source. More precisely, the back-transform of KO( y p ) is the function p =
From this it follows that Az tion a t q = 1 it follows that AJi(y) -
=
'p.
r
AzKi(7) = 0
+
= I&
+
We choose E = 'p and p u t s = 'P (1 ui)where u is a new variable of integration. The integral in the last expression can be written as
From the boundary condi-
Hence the Laplace transform of the solution is c
*
~ I , ( W ) K ~ ( Y )I I ( y ) K o ( y p ) )
(7)
The original of Equation 7 can be found by two methods. A. By Means of the Expansion Theorem. Equation 7 can be rewritten as (8) r = L Jo(rpi)Y,(ri)- J l ( Y i ) Y O ( Y P i ) 1 2Jl(ri)
r
This in turn can be written as
and we find for
r
=
The poles are given by J 1 (+) = 0, or s n = ' p - d m
n 2 0
tn=q+dZ/q2+k?,
nkO
9e d
2
r the final expression
i
e--P
4m-F
d r n
+
l+) cos 'ppru I0( ' p p d / 1 + K l ( P d r1(e 4 1 7 )
?J"
du
f
Comparing the two results, Equations 9 and 10, we see that Equation 9 is more appropriate for large values of (-Le., far from the source. For = 0 both series Equation 9 are divergent and for small 1 they converge too slowly.
where k , represents the nth zero of J l ( k ) , ko = 0. The solution is expressed by the integral
r
r
2
IO0
4
For positive 1 the path of integration can be deformed to the left and only the s-poles contribute to the expression. For negative 1 the path will be deformed to the right and we get an expansion according to the &poles. The residues are:
s = o
IO
.L
1
In view of the W-ronski relation
we get thus two expansions 01
100
'p
Jo(pkn)efl-P
r = p ~ - 0
J3kd
+ d-1 d r n
= 0.2
(9b)
The summation variable k , in Equation 9a runs through all zeros of J l ( u ) including the trivial zero ko = 0, which is in agreement with the residue for s = 0. Equation 9b is derivable from Equation Sa by changing the direction of flow, and the direction of z, hence inverting the signs of and 1. B. By splitting up the right-hand side of Equation 7 in two parts and transforming them back separately we get June 1953
r(s,'p)for p
Figure 3.
1 5 0
On the contrarv Equation 10 furnishes the behavior of the solution for small ( and p but is of no practical use for large 1and p-Le., near the boundary of the tube and far from the source. The behavior of the solution in the vicinity of the source is in first approximation determined by the first term in the right-hand side of Equation 10.
r-
'P
exp
-d-r 2
d
INDUSTRIAL AND ENGINEERING CHEMISTRY
-t P
d p 2
+
r21
(11)
V
1205
SPECIAL CASES
1. Isotropic Diffusion, D, = D, = D. The final results, Equations Sa and Sb, remain unchanged, Equations 3b and 3c, however, simplify to:
In the absence of axial diffusion the successive layers of liquid do not interact. Therefore an observer moving with the fluid will consider the radial diffusion to be a transient process, emerging from an instantaneous line source on the axis. Carslaw and Jaeger ( 8 ) Equation 7 , page 306 can be rearranged to Equation 16. 3. Infinite Space, General Case or Isotropic Diffusion. -111 four g r o u p as used before depend on the tube radius, R. By letting R + only three independent finite products remain. The transformation cannot be carried out on Equations 9a and Sb, but the solution emerges from an alternative derivation (see Equations 6 and 10). The solution, however, is well known for isotropic diffusion. In view of the experience with the tube, it must remain the same for the general case D , # D,. This solution reads, as may be derived from %%on ( 5 ) or by rearranging Equation 11: I
p =
UR 20
In a given cylindrical tube the concentration ratio r retains sense and one may use Equation 18 as an approximation for Eiquation Sa. In the infinite field r loses meaning. In this case the left-hand member with the aid of Equationfi, 3b, 3c, and 3e is rewritten as 4rrczDr P (0
r(r,p) for p
Figure 4.
= 0.4
Equation Sa, with this significance of j- and p, has been derived by one of the authors by the.method of moving point sources (see quotation by Bernard and Wilhelm, 1). Equation Sb can be obtained in the same way. vR A high value of p will 2. High Value of (O = --2D;l20'/2 be due to a high ratio of pipe radius to packing size. Whether D , equals D, or not, both must be understood to be proportional to packing size and to velocity of flow ( I ) , so that cp is proportional to the ratio of pipe radius to packing size. I n this case a reasonable progress of the diffusion process is observed for high values of
r
=
zD1 & only. '2
The equations for this case are derived by letting p and r approach infinity, which one may also conveniently do by letting D , approach zero. The same equations are therefore obtained in the absence of axial diffusion. In this case the set of four dimensionless groups, r, p , and p, s reduces to three-viz., r, p, and -.
r,
'p
Figure 5. r(b,p) for
p =
0.6
2rp
r =f
(P,
&)
The above solution was used by Towle and Sherwood ( 4 ) ,but these authors believed it to be true only in the absence of axial diffusion. In the plane of injection, r = 0, Equation 18 becomes:
where the new group
Equations Sa and 9b now reduce to:
4. Infinite Space Equations with Large 5 and ducing r >> p Equation 18 simplifies to - 5 P 2
r=O 1206
forre ‘p Neglecting axial diffusion ( D , = 0, = m , (p = OJ, - given by 2r Equation 21) also leads to Equation 20. On the stream lines close to the axis the concentration will pass through a maximum, near the wall it does not do so. The condition for the maximum is found from Equation 20:
With the laminar mechanism of eddy diffusion visualized above, the solute could never get upstream of the injection point. Unless this mechanism has been repeated a great many times, the diffusion laws do not yet hold. One could, however, expect this behavior in the case of turbulence or for ordinary diffusion (the latter a t a very low velocity of flow).
IW
r
3=
21
1 (maximum of r a t constant p ) P2
IO
(22)
e rmax. = P2
r
h
Since r approaches unity for m , the value of be less than l--i.e., there are no maxima for p > The concentration on the axis, p = 0, is given by .-f
This concentration does not depend on v. in moving as in standing fluid (5).
rmrt.. cannot = 0.61.
‘i
_. 0.1
Thus, it is the same
‘p
r(r, (o) for ‘p
Figure 7.
=
1.0
r
Figures 2 to 7 give the concentration ratio, r, as a function of and ‘p a t various distances from the axis-viz., p = 0, 0.2, 0.4, 0.6, 0.8, and 1.0. Figure 8 shows r against for various values of p a t a constant value of ‘p = 2. One notes the maxima, occurring for low values
r
of
p.
General Case ( 0 , # D,) and Isotropic Diffusion ( D , = 0,) for High Values of ‘p. If ‘p is high for the reasons discussed above (tube diameter large with respect to particle diameter or 2.
absence of axial diffusion),
r
becomes a function of the ratio
29
only (see Equation 16 and Figure 1). I n Figures 2 to 7 the curves for constant r are then seen to approach straight lines. 3. Infinite Space Equation with Large and ‘p. As in case 2, in Figures 2 t o 7 straight lines are obtained (Equation 20). One may ask when Equation 20 (infinite space) may be used as a substitute for Equation 16 (cylinder) which for large 6 and ‘p is very accurate. Clearly, this is a matter determined by the choice of
r For low - the infinite space equation may be used. r For instance, taking - = 0.20 with p = 0 and p
P
Figure 6.
r(r,‘p)for p
p
and
b-. ‘p
(0
= 0.8
= I, one
‘p
finds NUMERICAL EVALUATION
Equation 16
General Case ( D , # Or) and Isotropic Diffusion ( D , = Dr). Both are described by Equations 9a and 9b, expressing the concentration ratio, r, as a function of the coordinates, p and and the parameter, ‘p. Figure 1 gives curves for constant P in a p-6 diagram for such a low value of the parameter ( q = 2 ) that there is considerable diffusion upstream of the injection point. 1.
r,
June 1953
r r
= 1
4-1.419 + 0.081
= 1-
0.572
0.000 + . . . = 2.500 a t the axis + 0.024 +- 0.000 + . . . = 0.452 a t the wall
Equation 20
r r
= 2.500 a t the axis = 0.205 a t the wall
The ratio of concentrations at axis and wall in this case is 5.5.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1207
NOMENCLATURE
Bernard and Wilhelm ( 1 ) state the infinite space equation to be valid for
Taxis ~
mall
>
r for - up to a
6-i.e.,
value slightly below
‘p
0.20. I n the above example the behavior at the axis is well represented by Equation 20; a t the wall there is considerable devi-
r might be set somewhat lower.
ation and the limit for
-
A2 = coefficients c = concentration (unit: mass per volume), AIL-3 CI = transformed concentration, ML-3 e m = concentration a t z = m, U L - 3 D , = diffusion coefficient, axial, L2T-1 D, = diffusion coefficient, radial, L2T-l 81,
i I,( - )
= Bessel function, first kind, n t h order. imaginary argu-
J7&(- )
= Bessel function, first kind, n t h order
K,( - )
=
z
42.2
ment
‘p
Bessel function, second kind, nth order, imaginary argument k , = nth root of J l ( k ) = 0 L = length 21 = mass q = Etrength of source, MT-1 R = radius of tube, L r = distance from axis, L s = variable used in Laplace transformation sn, ta = values of s for J1 (yi) = 0 T = time u = integration variable ZJ = velocity, LT-1 =
Y,( - ) z ZI
I’ y
+
r2
Bessel function, second kind, nth order = axial coodinate, measured in direction of flow, distance frominjection plane, L = see Equation 2 = Laplace transform of r = coefficient in Equation 5 =
Dimensionless Groups
r
r r(p,r) for
Figure 8.
‘p
~
raxla < 3.
Sample Data Sheet I (tube of radius 2.625 em. filled iyith 1-mm. spheres)
R
For p = 0;
= 11.6;
r Consequently
-
‘p
= 7.5
P = 7.5 one reads
< 0.2;
‘p
= 170
infinite field Equation 20 is applicable
Sample Data Sheet I1 (tube of radius 2.628 em. filled with 3/~-inch cylinders) For
R
= 6.13; on the axis
For p = 0;
Consequently
1208
= 6.13;
v
r
p = -
‘p
4. Range of Variables, Important in Experiments by Bernard and Wilhelm. In both numerically worked out examples the simplifications for high and ’p are valid.
= 11.6; on axis
r
= 2.2
= 2.2 one reads p
=
’
Z
ruall
For
CCZ
r = - R- DD:: l/ s=
= 2
I n the above example the 2nd order terms of the expansion could not be ignored. According t o Bernard and Rilhelm ( I ) , this is allowed if
= dimensionless concentration
=
=
i dinlensionless coordinates
r
R
I
J
vR --~__
20: I2D: i 2 dimensionless parameter LITERATURE CITED
(1) Bernard, R. A , , and Wilhelm, 1%.H., Chem. Eng. Progr., 46,
233 (1950). and Jaeger, J. C., “Conduction of IIeat in Solids,” (2) Carslaw, H. S., London, Oxford University Press, 1947. (3) Latinen, G. A,. and Wilhelm, R. H., “Interparticle Lateral Fluid Phase Mixing,” 19th Annual Chemical Engineering Symposium, Division of Industrial and Engineering Chemis. SOC.,New Haven, Conn., January 1953. try, i 4 ~ fCHEY. ( 4 ) Towle, W, L., and Sherwood, T. K., ISD. EKG.CHEAT., 31, 457 (1939). (5) Wilson, H. A., Proc. Cambridge Phil. Soc., 12, 406 (1904).
27
= 0.23; finite field Equation 16 must be used.
RECEIVED for review January 3, 1953.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTEDMarch 16, 1953.
Vol. 45, No. 6
