Jakob, XI., Phys. 2. 22, 65 (1921). Joule, J. P., Thomson, W., Phil. Mag. 4, 481 (1852). Koeppe, W:, Kaeltetechnik 14, 399 (1962). Koeppe, W., Proceedings of 10th International Congress on Refrigeration (Copenhagen, 1959), Vol. 1, pp. 156-63, Pergamon Press, New York, 1960. Kordbachen, R., Tien, C., Can. J . Chem. Eng. 37, 162 (1959). Lydersen, A. L., Greenkorn, R. A., Hougen, 0. A., “Generalized Thermodynamic Properties of Pure Fluids,” University of Wisconsin, College of Engineering, Rept. 4 (October 1955). Martin, J. J., Chem. Eng. Progr. Symp. Ser. 59, KO.44, 120 (1963). Martin, J. J., Ind. Eng. Chem. 59 (E),34 (1967). Onnes, H. K., Commun. Phys. Lot; Univ. Leiden, No. 23 (1896). Onnes, H. K., Keesom, W.H., Die Zustandgleichungen,” in Encyclopedia IIath. Wiss. T-10, p. 842, B. G . Teubner, Leipzig, 1912. Partington, J. R., “,4dvanced Treatise on Phyical Chemistry,” 1‘01. 1, p. 622, Longmans,, Green, London, 1949. Pitzer, K., Brewer, L., revisers, “Lewis and Randall’s Thermodynamics,” Appendix 1, p. 611, McGraw-Hill, S e w York, 1961. Porter, A. W., Phil. Mag. (6) 11, 554 (1906). Porter, A. W., Phil. Mag. (6) 19, 888 (1910).
Price, D., Ind. Eng. Chem., Chem. Eng. Data Ser. 1, 83 (1956). Redlich, O., Kwong, J. N.S., Chem. Rem. 44, 233 (1949). Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,” 2nd ed , pp. 50-2, McGraw-Hill, New York, 1966a. Reid, R. C., Sherwood, T. K., “Properties of Gases and Liquids,’’ 2nd ed., Appendix A, pp. 571-84, McGraw-Hill, NeR- York, 1966b. Roebuck, J. R., Murrell, T. A., Miller, E. E., J . Anaer. Chem. SOC. 64,400 (1942). Roebuck, J. R., Osterberg, H., J . Amer. Chem. SOC.60,341 (1938). Roebuck, J. R., Osterberg, H., J . Chem. Phys. 8 , 627 (1940). Roebuck, J. R., Osterberg, H., Phys. Ret. 46, 785 (1934). Roebuck, J. R., Osterberg, H., Phys. Rev. 48, 450 (1935). Shah, K. K., Thodos, G., Ind. Eng. Chem. 57 (3), 30 (1965). van der Waals, J. D., Proc. Sect. Sei. Kon. M e d . Akad. Wctenschap. (Amsterdam),2 , 379 (1900). Witkowski, A. K., Bull. Acad. Scz. Cracovzc (July 1898). Yen, L. C., Alexander, R. E., A.I.Ch.E. J . 1 1 , 334 (1965). RECEIVED for review September 29, 1969 ACCEPTED July 13, 1970 Work carried out under the auspices of the U. S.Atomic Energy Commission.
Characterization of Liquid-Solid Reactions Hydrochloric Acid-Calcium Carbonate Reaction Bert B. Williams,l John 1. Gidley,2 James A. G ~ i n a, n~d Robert S. Schechter4 Esso Produciion Research Co., Houston, Tex. 77001
Techniques commonly used to determine reaction rates for heterogeneous liquid-solid reactions such as the hydrochloric acid-calcium carbonate reaction are mass transportlimited and do not reflect surface kinetics. A procedure where liquid flows through a channel composed of solid reactant i s proposed for obtaining kinetic data. An exact solution for reaction rate with an arbitrary kinetic model i s obtained by numerical methods based upon Duhamel’s theory, for first- and second-order reversible and irreversible reactions. The method i s easily extended to other models. An exact boundary layer solution i s given for the first-order irreversible reaction and an approximate boundary layer technique for solution of the arbitrary reaction rate case i s developed. The approximate technique yields simple solutions which are accurate under certain prescribed conditions. Existing experimental data for the hydrochloric acid-calcium carbonate reaction are analyzed in light of the proposed kinetic models.
A c i d i z i n g of subsurface reservoirs to increase oil or gas production ha5 been practiced since the late 1800’s. In these techniques an acid, such as hydrochloric acid, is forced to flow within the pore structure of the rock matrix, or along a hydraulically induced fracture, reacting with the rock and altering reservoir characteristics. Schechter and Gidley (1969) have developed a method of computing the rate of acid spending and the effect of acid reaction on formation flow capacity for the case of matrix acidiaing. These computations, as &-ellas calculations of reaction rate in flow along a fracture, are deTo whom correspondence should be sent. address, Humble Oil & Refining Co., Houston, Tex. 77001 Present address, Auburn University, Auburn, Ala. 36830 Present address, Department of Chemical Engineering, University of Texas, Austin, Tex. 78712
pendent upon knowledge of the surface react’ion rate of the acid. It is, therefore, of practical importance to be able to obt,ain correct expressions for the surface reaction rate under a variety of conditions of temperature, pressure, and acid composition. The reaction rate is relatively fast and only carefully performed experiments, analyzed to account for diffusion correctly, are of general applicability. Reaction of an acid rvith calcium carbonate can be geiieralized as kl
2H+
l
* Present
+ CaCO3 @ka COz + Ca2+ + HzO
or
+ B @ka C + D + E icl
2-1
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
589
Therefore, the general rate expression for reaction of slightly dissociated acids will have the form
Table 1. Effective Mass Transfer Ratio for 1 to 1 Ratio of Acid Volume to Rock Area Time, Min
f
10 20 30 40 50 60 70
0.25 0.40 0.50 0.58 0.63 0.71 0.76
-r
(D/IZ)o.'
0,009 0.010 0.010 0.010 0.010 0.010 0.010 Av. 0.010
For this reaction, a general rat'e expression is
-r
=
klaAuucYaDa -
I
JiZuAu acY
,
,
aD6
(1)
where a t is the activity of the i t h species. To satisfy equilibrium requirement., the ratio of t,he rate coefficients must be equal to the equilibrium constant to a power n . This means that the exponents of the activities in Equation 1 are related by Equation 2.
Highly dissociated acids such as hydrochloric acid are most commonly used in stimulation techniques. If we assume that the activity coefficient for all reactants a t t8hesurface is 1, and the reaction is first-order in hydrogen ion activity ( a = l),the rate expression is -7
=
I;,Ca -
(3)
ksCc'12CDl:2
Experimental data show that even a t high (20,partial pressure, reaction with strong acids goes essentially to completion. Therefore, the backward rate will be negligible in most applications and the rate expression can be written
-r
=
klCa
(4)
By a similar argument, the rate expression assuming the reaction is of order il' can be written
-r
=
(5)
klCA5
Acids that are only slight,ly dissociated in aqueous solution are also used to acidize format,ions: formic, acetic, propionic, and other organic acids. Study of these acids is complicated by t,he dissociation process. A general acid (HI?) dissociates according to the relation
HP
$ H+
+ P-
wit.h equilibrium described by an equilibrium const'ant, Kd, defined as
Again assuming first-order kinetics in hydrogen ion activity and unit activity coefficients, and introducing acid dissociation, gives the rate expression
-r
=
klCH+ - k2CCO1WCa2 +liZ
=
ki(CHpKd)'Iz-
590 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
+'Iz
kl(CHpKd)OL- k2CCouCCa~tP
(8)
Historically, acid reaction rate dat'a have been determined in a stat'ic reaction test,, in which a cube of rock of known surface area is placed in a basket a t the top of a reaction vessel and a known volume of acid is placed in the bot,tom of the container. Test pressure is usually applied to the chamber by injecting iiit,rogen so as to stimulate reservoir conditions, and a test is started by inverting the equipment, thereby bringing rock and acid in contact. After a set time, the cell is righted and an analysis made t o determine the quant.ity of rock dissolved. Data from this test are normally reported as the fraction of acid spent, f, as a function of contact time for various ratios of acid volume t.0 rock surface area. Some insight into what is occurring during static tests can be obtained as follows. First, assume a model wit,h two parallel walls separated by a distance 1 with no fluid circulation, a reaction rate at the solid surface very fast, relative to the rate of acid transfer to the surface (C = 0 a t y = I ) , no alteration in diffusivity of hydrogen ion mith increase in product concentration, and a t y = 0, an inert surface where the concentration gradient is zero [dC(y = Oj/dy = 01. For this system solution of the mass-balance equation can be simplified for typical values of D , 1, and t to t'he form f = 1
-
e = (4Dt:
7rP)1:2
(9)
where 6 is the mean concentration in the cell a t time t divided by the init,ial concentration. f is the fraction of the acid that has reacted. Llaiiy results obtained in the static cell can be explained with this equation in terms of mass transfer properties independent of rate of surface reaction. Values for t.he group (D;P)0.5 computed for the data of Hendrickson et al. (1960) for a 1 to 1 acid volume to rock surface area ratio are given in Table I. The average spacing from the rock saniple to a wall in the test cell is not known, but would appear to be about 2 cm. .Issuming this is representative, the effective mass transfer cniz per second, or about 20 coefficient for the test is 4 x times the molecular diffusion coefficient,. Therefore, this reaction is entirely controlled by mass transfer to the rock surface. I n these tests, the apparent mass transfer rate is much more rapid than is possible by diffusion alone, indicating that convection occurring in the aqueous phase controls reactmionrate. It appears that this test is sensitive to any change that alters fluid circulation properties and is coniplet'ely insensitive to reaction rate a t t,he rock surface. Therefore, changes in over-all reaction rate attributed to changes in kinetic rate by Chamberlain and Boyer (1939), Hendrickson et al. (1960), and van Poollen (1968) are apparent'ly caused by changes in rate of fluid convection. Since these tests are mass transfer-limited, data will indicate a first-order reaction regardless of the true order of the surface reaction. For these reasons, kinetic data from static tests cannot be used with confidence to predict rate of acid spending in field operations. I n t,his paper a procedure for determining accurate kinetic data is proposed and a theory developed for expected kinetic rate expressions. Dynamic Acid Reaction Tests
Or, in terms of the undissociated acid concentration (approximately equal to total acid concentration), -T
=
(7)
General Consideration. Experiments cited to demonstrate t h e first-order dependence of acid reaction rate on acid concentration are diffusion-controlled and therefore, are not reliable indicators of t h e true nature of the surface
reaction. Flow tests in which acid in fully developed laminar flow comes in contact with a reactive surface are well suited for experimental determination of reaction rate coefficients-for example, laminar flows through a tube or between parallel walls of calcium carbonate can be used to determine reaction rate, if the experiment is devised so t h a t the steady-state acid effluent concentration can be obtained before there is a significant change in the dimensions of the system owing to solution of the carbonate walls. An additional assumption is t h a t diffusivity for each ion is constant for all ionic concentrations and independent of concentration of other ions. Experiments satisfying the above conditions have been performed by Hoelscher and Cowhead (1965) in studying kinetics of interfacial reactions. Consider first reaction in flow between parallel walls of a reactive solid, as shown in Figure 1. Since the correct reaction rate expression a t the rock surface is not known a priori, a general solution for this problem follows for the surface chemical reaction symbolized as 86iA = 0, where 6, are stoichiometric coefficients (positive for reactants and negative for products). A mass balance for the i t h component gives
Y 7
t
=
~
CENTERLINE
AVERAGE VELOC!TY
VELOCITY
7g,,,-
X
Figure 1.
-2
=- V
Geometry for parallel plate reaction system
To solve the unit flux problem, the following change of variables is convenient: g i ( e t J q ) = uZ(et,v)
where
for laminar flow between parallel plates. The gi(ei,7) must then satisfy the differential equation
and the boundary conditions This equation must be solved subject to the boundary conditions
Solutions for mass transport problems in flowing fluids are often expressed in terms of eigenvalue expansioiis, but near the inlet ( e i = 0) such expansions converge slowly and boundary-layer solutions are more convenient. This development makes use of both solutions. To write the solution t o the general problem, the solution to the unit flux problem is denoted as uz(ti,n),where ut satisfies Equation 10 together with the boundary conditions
The solution of this problem is
where the jn(7) are eigenfunctions defined by the differential equat,ion (17) subject to the boundary conditions
The B, are determined by and
B,
=
'l1 tk
Qr
4E)dtdAfn(aMlt)d7
l1
(19)
u(7)fn2~7)d7
Then by Duhamel's theorem, The solution to t'he unit flux problem is then
Integrating this expression by parts, a form more convenient for numerical computation is obtained:
J'S,'
Qf o
UdEdX
(20)
For fully developed laminar flow where v(7) = 1 - v 2 and &, = 2/3, the eigenvalues and eigenfunctions evaluated a t the wall are tabulated in Table 11. A similar analysis for a circular tube with a reactive wall gives for the solution of the unit flux problem where all axial positions are referred to e Equation 13 provides a means of obtaining the solution for acid concentration for an arbitrary surface reaction, once the unit flux solution has been obtained. Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
591
Table II.
Mathematical Properties of Unit Flux Solution
Parallel Plates s
ha
fJ1)
0 1 2 3 4 5 6 7 8
0 4,2872 8.3037 12.3106 16.3145 20.3171 24.3189 28.3203 32.3214
1 - 1.2697 1,4022 -1.4916 1,5601 -1.6161 1.6638 - 1.7054 1.7425
where
s
Circular Tubes B8
- 39/280 0.17$03 -0.05173 0.02505 -0,01492 0.009969 -0.007164 0.005415 -0,004248
1
Qc =
0
w(r)d?
and the F , satisfy the differential equation
and the boundary conditions
dF,(O) dr
-
@,(I) dr
Xa
FA1 1
0 5.0675 9.1576 13.1972 17.2202 21.2355 25.2465 29,2549 33.2615
1 -0.49252 0.39551 -0.34587 0.31405 -0.29125 0.27381 - 0.25985 0.24833
D.
-7/24 0.40348 -0.17511 0.10559 -0.07328 0.05504 -0.04348 0,03560 -0.02991
artificial introduction of additional terms as was done by Solbrig (1967). By assuming a value of q ( N + l ) , the concentrations a t point &'+I) along the surface can be computed using Equation 23, and in turn, be used to compute a corrected value of p(N+l). By continuing this iterative process, new reaction rates and concentrations satisfying Equation 23 can be found. This process of calculation can be continued to yield both the concentrations and the reaction rates evaluated along the surface. The amount of acid reacted is given by
-0 (24)
The coefficients D, are determined from This quantity is easily shown to be related to the mean concentration divided by the inlet concentration according to the relationship
Using the method given below, it can be shown that the boundary-layer solution to the unit flux problem evaluated along the surface of the wall is
where F(e) is the gamma function of the argument e . I n the general case, the dimensionless reaction rate q ( e ) depends on the local value of the concentrations a t the reactive surface. Equation 13 can be used in a n iterative fashion to evaluate both the surface concentration and the reaction rate along the surface. Suppose that the rate of reaction has been established at positions e(o), dl), . . . giving the particular values q ( l ) , q ( l ) , . . . Q(N). It is desired to establish q ( d v + l ) , From Equation 13, we can write
assuming that dq/& is a constant between the nodal points. In performing the calculations, it must be remembered that the boundary layer solution, Equation 22, should be used for small values of e and then switched to Equation 20 for larger values of the argument. The solutions as computed from 20 and 22 agree to within 2% over the region e t = 10+ to c t = 7 x l o + and thus the switching point should be in this interval. This method of calculation does not require the 592
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
where
elis the average dimensionless concentration at,L*.
First-Order Irreversible Reactions
While it appears t'hat. the experiments leading to the accepted conclusion that, the rate of acid reaction with carbonate is firsborder are not valid, it is possible that t,his is, in fact, the nature of the reaction. First-order irreversible reactions are also of interest,, since the result'ing equations are linear and can be solved without using Duhamel's theorem or the unit flux solut'ions developed above. The solut,ions developed by numerical methods based on Duhamel's theorem were tested against these exact solutions and found to be in agreement within 1% over the entire range of e. Also, approximate solut'ions introduced below can be compared to t'hese exact solutions. The first-order irreversible case is studied by putting p ( ~ = ) -C1. For larger values of the dimensionless distance el, (el E e), solut,ion in terms of an infinite series using the methods of separation of variables can be found. However, in the entry region, the rate of diffusion to the reactive surface is controlled by the development of a diffusion boundary layer, the rat'e-limiting processes occur very near the wall, and the velocity distribution can be replaced by the approximation 4 7 ) = 1(1 - 7)
For this special case, the expression developed by Acrivos and Chambrk (1957) reduces to
where Table 111.
Making the substitution 2 3
= 2*
Z13 =
E
gives
Values of Coefficients in Power l a w Solution n
bl
1 2 3 4 5 6 7 8 9 10 11
-0,82699 0.60412 -0.40275 0.24980 - 0.14598 0.08110 - 0.04306 0.02143 - 0.00724 0.00047 -0.00000
The concentration a t the wall is obtained in terms of the power series m
Substituting Equation 27 into Equation 26, it is found that
_--
-
where
Equation 29 can be integrated t o give
DIMENSIONLESS LENGTH
Figure 2.
-
100
lo
L'
Surface reaction rate in circular tube Irreversible reaction
Substituting Equation 30 into Equation 28 and equating coefficients of like powers, it is found that
and DIMENSIONLESS LENGTH
Figure 3.
-
Lo
Surface reaction rate between parallel plates Irreversible reaction
The first eleven values of Pi are given in Table 111. With these values known, concentration a t the wall is given by
Reversible Reactions
The reaction of weak acids should be considered as reversible. Figures 4 and 5 show dimensionless conversion R* as a funct'ion of length in a parallel plate system for reactions having the form Using Equation 24 gives the final result
-a(€)
=
Ci -
-a(€)
=
Ci2 - KCzCa, K
KC21'2C31/2,
K
=
= 0.1
1.0
In these figures the dimensionless quantity, P, is defined as A similar analysis for a circular tube gives Equation 32, again with R*, Q f , and L* replaced by t,heir analogs for the circular case. Results predicted by combination of infinite series solution and the boundary layer solution are shown in Figures 2 and 3.
P
721
=
-
Di
and P
=
klCl -,D1 respectively
These calculations have been performed for Dzl = -0.125 and D31 = -0.375, which are approximately valid for the system HCl(CJ, CaClZ(Cz), and COZ(C3). Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
593
I
with the boundary conditions Ct(et,a )
I
I
=
Cfoand
Equation 34 can be integrated directly to yield, after applying the boundary conditions,
I
l0-2
1/
10-3
1o
Figure 4.
- ~
Surface reaction rate between parallel plates
This expression represents an approximate solution for the wall concent'ration for an arbitrary surface reaction rate. Applying this expression to the unit flux problem in which
Reversible reaction
I
10
I
3% ~-(e i,O) I
I
I
z
ar 1
- 1
and ut(et,w) =
0
yields
When compared with Equation 22, Equation 36 is in error by about 20%. The approximation improves considerably as the order of the reaction increases with the unit flux problem being of order zero. For an nth-order irreversible reaction, Figure 5.
Surface reaction rate between parallel plates
d e ) = - [C1(e,O)l"
Reversible reaction
The simple iteration scheme previously suggested did not converge in all cases for reversible reaction and it was sometimes necessary to use a direct search technique to find values for C1, CB,and Ca that satisfied Equation 23. The technique of Powell (1964) worked well in these cases. I n this tech-
and C1 is given implicitly by Equation 35. I n particular, if n = 1,
~. l + 3 A
3
[CiP+l - C 2 p ] is z minimized, where
nique, the quantity
giving for R*
2=1
C f pis the concentration of component i a t the pth iteration as found from Equation 23. Approximote Solutions
The results of an approximate boundary layer technique introduced by Solbrig (Solbrig and Gedaspow, 1967) are compared with the exact results obtained earlier. This approximate method is useful because the calculations involved are simpler than those required to solve the integral equations resulting from the exact boundary layer method. I n this procedure t is defined as the dimensionless distance measured from the reactive surface-Le., t = 1 - q (for 2 t ) . Changing variables in Equation 10 from small t , 1 - q2 el,^) to (el,Zt)where Z f = t ~ , - l ' gives ~
-
This approximate result is compared with the exact solution in Table IV. Similarly for n = 2,
Cl(€,O)
=
(33) Neglecting the term on the left-hand side of Equation 33 and defining
(+)
For the second-order reversible reaction where q(z)
=
- C12 + KC2C3and assuming t.hat the inlet concentrations of
both substances 2 and 3 vanish from Equation 35, it is found that
= ( y Z t
Equation 33 can be written
(34) 594 Ind. Eng. Cham. Fundarn., Vol. 9, No. 4, 1970
2Pr
3A
Cl(€,l) = (1 - 2BE)
+ {(2BE - 1)' - 4(1 - B E ) ( B - BE))'" 2(B - BE)
~
Table IV.
Comparison of Approximate with Exact Boundary layer Solutions for First-Order Irreversible Reactions R * ( P = 0.1)
L*
1 493 1 48s 1 471 1 439
10-4 10-3 10-2
10-1
Table V.
R*(P = 1.0) Approx.
L*
Exact
Approx.
1 492 x 10-5 I 484 x 10-4 1 465 X 1 427 X
10-a 10-2 IO-'
1 374 x 10-3 1 251 X 1 047 X 10-I
1 352 x 10-3 1 215 X 10-2 9 997 x 10-2
Exact
x x
x
x
10-5 10-4 10-3
10-2
Apparent Reaction Rate Coefficients from Data of Barron ef a/. (1962)
Plate Spacing, Inch
P
0.025 0.05 0.10 0.20
1.25 2.5 5.0 10.0
where
l-E 10-3L4 10-
10-3
I
lo-'
10-2
DIMENSIONLESS LENGTH
-
Lo
Figure 6. Experimental and theoretical data for HCI reaction between parallel walls
Similarly,
and Cz = (D2/D3)"3(D3i/D21)C3
These expressions can be substituted into Equation 24 and the dimensionless reaction rate coniput,ed. These approximate expressions have been found to represent a very accurate (to within 2 70) approximation of t'he solution for values of L* less than 0.1 for P = 0.1. Since they are much more compact and provide straightforward solutions compared to the use of Duhamel's theorem, their use in the inlet region is recommended. Dynamic Reaction Rate Tests
The only available data from which a reaction rate coefficient for hydrochloric acid can be obtained are those of Barron et al. (1962), who studied the reaction of 15% HC1 in flow between parallel walls of calcium carbonate. The results of their experiments are compared to predicted reaction rates for a first-order irreversible model in Figure 6. .It low velocity thePe data appear to fit the first-order irreversible reaction model with the approximate values of dimensionless rate given in Table 1'. At the higher velocities, observed reaction rates are larger than expected if mass transport to the rock surface is diffusionlimited. This would indicate that instability in the flow may occur even though the Reynolds number is well in t.he laminar range. These instabilities may in part be caused by heat evolut,ioii at the surface, surface roughness, or entry effects in the model. Assuming that experiments a t low ve1ocit.y represent the diffusion-controlled react ion, that' first-order kinetics are applicable, and the diffusion coefficient for HC1 is 2.6 x 10-5
em2 per second, the rate coefficient for reaction of HC1 and cm per calcium carbonate la approximately 1.0 x second. This value for the rate coefficient should be considered as qualitative until additional, carefully controlled experiments have been obtained. T o verify the order of the reaction, these experiments mubt include variation in acid concentration as M ell as flou rate and system geonietr) . Nomenclature
ai
B,
c
=
activity Coefficient for i t h component
= coefficients in unit flux solution =
dimensionless average concentration
= actual inlet concentration of component
i, moles/liter concentration of component i a t pt.h iteration as calculated from Equation 23 C , = dimensionless reactant concentration-Le., CA/CP C%O = dimensionless initial concentration of reactant-Le., =
c,o/c,o
L L*
=
1
=
n
=
molecular diffusivity for i t h component, cm2/second dimensionless diffusivity ratio, D,&/Di& fraction of spent acid distance from center line of parallel plate system to reactive wall, cm reaction rate coefficient, cm/second equilibrium coefficient', k2/k1 total svstem length in flow exgeriments, cni LD dimensionless length, __ 2LD' (cylinder) 3ah2 (plates), spacing betxeen reactive and noiireacting surface in static test, cm order of reaction with respect t o component 1
P
=
reaction rate coefficient,
D,
=
D%1 =
f h
= =
I;,
=
Ke, = =
Y
k2
h"(2:)n-1
(parallel plates),
q ( e ) = dimensionless reaction rate, r ( e ) / + l ( C ~ ~ ) ~ Q = constant, Q, for plates, Qc for cylinder r ( e ) = reaction rate a t position e, moles/cm2/second Ind. Eng. Chem. Fundam., Vol.
9,No. 4, 1970 595
= radius of tube in calculations for reaction along cylindrical channel. cm = dimensionless reaction rate,
R
R*
{%
jR&(plate) ; (cylinder) Raw = average reaction rate over total reactive surface Re = Reynolds number, d v p / g t = seconds z i = average velocity, centimeter/second 4 1 ) = dimensionless velocity profile, for laminar flow, v ( 0 ) = (1 - s2) x = distance in axial direction, cm Y = distance in transverse direction, cm zi = dimensionless distance along x coordinate xD,/2ziR2 (cylinder)
6,
= stoichiometric coefficient
ei
=
1 p
= =
dimensionless position along z coordinate,
22Di (flat
3h20 plate) dimensionless position along y coordinate, y/h density, grams/cm3
literature Cited Acrivos, A,, ChambrB, P. L., I n d . Eng. Chem.49,1025 (1957). Barron, A. N.. Hendrickson. A. R.. Wieland. D. R.. Trans. AIME. 225,409 (1962). Chamberlain, L. C., Boyer, R. F., Ind. Eng.Chem. 31, 400 (1939). Hoelscher, H. E., Cowhead, c., IND.ENG.CHEM.FCND.4M. 4, 150-4 (1965). Hendrickson, A. R., Rosene, R. B., Wieland, D. R., Division of Petroleum Chemistry. 137th Meeting- ACS, Cleveland, Ohio. April 1960. Powell, J. J. D., Computer J . 7, 155 (1964). Schechter, R. S.,Gidley, J. L., A.I.Ch.E. J . 15,339 (1969). Solbrig, c. W., GidasPow, D.7 Can. J . C‘hem. Eng. 45935 (1967). van Poollen, H. K., Jargon, J. R., Oil Gas J . 66,84 (1968). RECEIVED for review Xovernber 12, 1969 ACCEPTEDJuly 15, 1970 I .
GREEKLETTERS = apparent reaction order for weak acids = coefficients in series approximation t o boundary layer
a
Pn y
r(n)
= =
Droblem dimensionless radial position, r / R gamma function of variable n , tabulated function
Flow of Single-phase Fluids through Fibrous Beds Chwan P. Kyan, Darshanlal 1. Wasan,’ and Robert C. Kintner Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Ill.
60616
A pore model for the flow of a single-phase fluid through a bed of random fibers is proposed. An effective pore number, Ne,accounts for the influence of dead space on flow; deflection number, N6,characterizes the effect of fiber deflection on pressure drop. Experimental data were obtained with glass, nylon, and Dacron fibers of 8- to 28-micron diameter and with fluids of viscosity ranging from 1 to 22 cp. A generalized friction factor-Reynolds number equation i s presented. The effects of dead space in a fibrous bed on flow and of fiber deflection on pressure drop have no parallels in a granular bed.
THE flow of fluids through porous media has been a subject of investigation for many years. A considerable amount of research has been done on the flow through granular beds and many useful results have been obtained (Brownell and Katz, 1947, 1956; Ergun, 1952; Ergun and Orning, 1949). A lesser number of investigations have been done on the phenomena of flow of fluids through fibrous media, mostly in connection with aerosol filtration. General approaches pursued by most workers on the flow of fluids through fibrous beds involved the development of theoretical pressure drop equations from either a “channel model” or a “drag model.” The former was the more extensively used. Nost workers using the channel model started with the Kozeny-Carman equation,
which in the friction factor form becomes
To whom correspondence should be sent. 596
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
I n this equation, t’he fact that k depends on fiber orientation and porosity had been observed and discussed by Sullivan and Hertel (Sullivan, 1941; Sullivan and Hertel, 1940) based on t’heir experimental work. Thus Equation 1 was inadequate for pressure drop correlations. Various workers using the channel model have elaborated upon Equat,ion 1 with modification for shape and orient,ation (Davies, 1952; Fowler and Hertel, 1940; Langmuir, 1942; Sullivan and Hertel, 1941). Most workers (Chen, 1955; Iberall, 1950; Wong et al., 1956) using t’he drag model rejected the applicability of the channel model because of the high porosity of a fibrous bed and derived a pressure drop equation by considering the drag forces due to fluid flow on the fibers. Wong et al. (1956) employed an effective drag coefficient, CD,, to account for the fiber orientation, interference of neighboring fibers, fiber ends, and nonuniformit’y of fiber distribution in the bed. They concluded that the fiber volume fraction, y, has a marked effect on Cn,. The higher the value of y, the higher is t’he neighboring fiber interference which leads to a higher C D , . They also not’iced the leveling off of the effective drag coefficient-Reynolds number plot a t Reynolds numbers greater than 6. Gunn and Aitken (1961) in their study of the mechanism
