Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979 Greener, J., Middleman, S.,Polym. f n g . Sci., 14(11), 791 (1974). Greener, J., Ph.D. Dissertation, University of Massachusetts, Amherst, 1978. Greener, J., Middleman, S.,submitted to Polym. Eng. Sci., 1978. Greener, J., Middleman, S.,submitted to J . Rheol., 1979. Hintermaier, J. C., White, R. E., Tappi, 48(11), 617 (1965). Hoffman, R. D., Myers, R. FI., Trans. SOC. Rheol., 6, 197 (1962). Hopkins, M. R.. Brit. J . Appl. Phys., 8, 442 (1957). McKelvey, J. M., "Polymer Processing", Wiley, New York, N.Y., 1962. Middleman, S.,"Fundamentalls of Polymer Processing", McGraw-Hill, New York, N.Y., 1977. Miller, J. C., Myers, R. R., Trans. SOC.Rheol., 2, 77 (1958). Myers, R. R., Hoffman, R. EL, Trans. SOC.Rheol., 5, 317 (1961). Myers, R. R., Miller, J. C., Zettlemoyer, A. C., J . ColloidSci., 14, 287 (1959).
41
Padday, J. F., Surf. Colloid Sci., 1, 39 (1969). Pinkus, O., Sternlicht, B., "Theory of Hydrodynamic Lubrication", McGraw-Hill, New York, N.Y., 1961. Pitts, E., Greiller, J., J . Fluid Mech.. 11, 33 (1961). Savage, M. D., J . Fluid Mech., 80(4), 743 (1977). Tanner, R. I., Int. J . Mecb., I, 206 (1960). Williamson, A. S..J . Fluid Mech., 52, 4 (1972).
Received for review March 10, 1978 Accepted October 24, 1978 This work was supported by the Eastman Kodak Company.
A Derivation of the Field Equations for Slow Viscous Flow through a Porous Medium Florian K. Lehner' Water Resources Program, Department of Geological and Geophysical Sciences, Princeton University, Princeton, New Jersey 08540
The macroscopic field equations for slow viscous flow of an incompressible fluid through a rigid porous medium are derived with the help of well-known averaging techniques. A general reciprocal theorem, which holds for macroscopically nonuniform filter flows in the presence of a body force, is implied by an energy balance and serves to establish the existence of a symmetric and positive-definite resistance or permeability tensor. Macroscopic vorticity is shown to diffuse into a homogeneous porous medium from its boundary but decays steeply enough to justify its neglect and the consequent reduction of the field equation to Darcy's law in most applications.
Introduction The macroscopic field equations governing the transport of mass and energy in slow viscous flow through porous media are commonly set up by combining general balance laws with constitutive relations obtained from experiments on porous continua. .4lternatively, they are deduced from conceptual models which seek to exploit analogies between porous media flow and certain well-understood flows such as Poiseulle flow, for example. From a theoretical as well as a practical point of view, this status of the macroscopic field equations has long seemed unsatisfactory. For here, unlike in other branches of continuum mechainics, the physics on the next smaller scale, i.e., the scale of an individual pore, is still continuum physics and presumably is adequately described in terms of well-established field equations. In applications, however, one frequently must rely on extrapolation from experiments and deal with the shortcomings of conceptual models. For these reasons much attention has been given in recent years to analytical methods by which the macroscopic field equations, are derived from the equations that govern the transport of mass and energy on the pore scale. The viewpoint taken in these analyses is that a macroscopic relationship like Darcy's law, for example, expresses a property of spatial averages of velocity and pressure fields which, on a microscopic scale, are governed by the linearized Stokes equations. This is motivated by postulating that the porous medium be intrinsically disordered and statistically homogeneous on a local scale to such a degree as to make velocities and stresses in the fluid stationary 'Address correspondence to the author at the Division of Engineering, Brown University, Providence, R.I. 02912. 0019-7874/79/1018-0041$01.0010
random functions of position over distances of the order 1 >> d , where d corresponds to the average pore size and 1 is still small in comparison with the typical linear extent of the macroscopic flow domain. By this, one also ascertains the equality of integral averages taken with respect to one or more spatial coordinates over distances of the order of 1. In particular, one is assured of the equality of volume and surface averages, the former being convenient for analytical purposes and the latter being usually measured. One can seek results through this approach of spatial averaging that are of two kinds. Of the first kind are the macroscopic equations themselves, their form as well as their range of validity. Of the second kind would be precise definitions of the phenomenological material parameters that appear in the macroscopic equations, in terms of geometrical and dynamical characteristics of the particular microscopic phenomenon under consideration or, more likely, in terms of certain averages of such characteristics. An example of this kind from the related field of suspension flow is Saffman's elegant derivation of Einstein's viscosity law for dilute suspensions (Saffman, 1971). For porous media, however, very few results of sufficient generality have been obtained in this direction of characterizing phenomenological parameters. The present study deals with what might be considered the first problem of the first kind, namely the derivation of the macroscopic field equations for the slow flow of an incompressible Newtonian fluid through a porous medium. We shall deduce them as an immediate consequence of a general reciprocal theorem which may be seen as the corresponding statement for porous media flows of a property expressed by Lorentz's well known reciprocal theorem. Inspired by the work of Happel and Brenner (1965), our approach is thus similar to that of Poreh and 0 1979
American Chemical Society
42
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
Elata (1966), who have derived Darcy’s law with the help of Lorentz’s theorem. However, we obtain a more general reciprocity relation than theirs, allowing for nonuniform macro flows and the presence of a body force. The derivation of this relation proceeds directly from a balance of mechanical energy and makes it an obvious consequence of the neglect of kinetic energy and of the linearity of the constitutive equation. It is of some general interest to note that one arrives at the macroscopic field equations not by a straightforward averaging of the micro-scale field equations, but through averaging an energy balance and exploiting the symmetry properties of the associated energy integral. The averaged balance of mechanical energy obtained here also suggests the definition of a macroscopic dissipation function for porous media flow, which is equal to the phase average of Rayleigh’s dissipation function, but expressible in terms of macroscopic quantities. Finally, we present a general argument justifying the assumption of vorticity-free macro flows and the corresponding simplification of the field equations to Darcy’s law through deletion of the term p@lVz( v ) . This argument also furnishes = p ( v).K-‘.( v ) as the appropriate macroscopic dissipation function for porous media flow. The Reciprocal Theorem for Slow Viscous Flow of an Incompressible Fluid through a Rigid Porous Medium We start by considering the balance of mechanical energy for a moving fluid mass which occupies a region Vf in space at a given instant of time. We think of Vf as the interconnected pore space associated with a representative volume element V of a porous medium. Let T denote the Cauchy stress tensor and let D be the deformation rate tensor defined by D = l/z[V v + (Vv ) ~ ] in terms of the velocity gradient. Then for an arbitrary material region Vf with closed boundary S the rate of change of the kinetic energy K = .fv, p v. v d V is given by (cf. Serrin, 1959, p 138) dK -= pEv dV + bvdS T:D dV (1) dt S Vf Here p is the density of the fluid, f is a body force, tis the traction, and the integration is performed over fixed configurations of V, and S. For incompressible flow we have tr D = div v = 0 (2)
+
1,
$
1
and the constitutive equation for the stress is T = -PI 2fiD
S,which comprises the intersection of the circumference of V with the voids. Consequently the equation
is equivalent with (4) under (6). Notice here that the adherence condition (6) expresses the fact that the fluid remains at rest along Sf,with respect to a frame attached to the rigid porous medium. While v in condition (6) is therefore a relative velocity, this distinction need not be made explicit as long as the rigid body motion of the porous matrix is without dynamical significance. To ensure this, we postulate that the solid matrix shall experience only such motions as will keep acceleration forces transmitted to the fluid negligible, thus ensuring the validity of the step from eq 1 to eq 7. When considering the behavior of v under a change of frame, its definition as a velocity difference must be kept in mind. Formally, the objectivity of vis then readily deduced upon observing that only such frames are admitted as are attached to the rigid solid matrix. Next, let us define phase averages by the following volume integrals, taken over Vf 1 v dV, etc. (v) = v/ These averages will form the variables of the desired macroscopic description. They are to be considered as point functions in “macro-space”, the points of which are in a one to one correspondence with the centroid positions, in “micro-space”, of representative elementary volumes (REV’S). The various aspects of this macroscopization procedure by spatial averaging have been discussed extensively in the literature to which the reader may here be referred (see, e.g., Whitaker, 1969; Slattery, 1972; Gray, 1975. A theorem of central importance derives from the rule for differentiating a definite integral with respect to a parameter (cf. J. Hadamard, “Cours d’Analyse”, Vol. 2, p 504, 1927). Adapted to the present situation (Slattery, 1972; Gray and Lee, 19771, the theorem states that
v
Here J/ may be any component of a tensor of arbitrary rank and niis the ith component of the outer unit normal to
s,.Application of (8) and (9) to eq 7 now provides us with
+
(3) Consequently T:D = -pI:D + 2pD:D = -p tr D + 2pD:D = 2pD:D. Moreover, in creeping viscous flow one neglects kinetic energy and assumes that the terms on the righthand side of eq 1 balance exactly. We shall thus be considering the following balance of mechanical energy pBvdV+
$ b v d S = 1vi @ dV
pP( v)
S
(10)
where the body force was assumed to remain constant. We may consider the fields T and v as composed of a mean and a fluctuating part and define
(4)
in which
+ div(T.v) = (a)
T = ( T ) f +T;
(T) = 0
(11)
v = (v)’+ k
( e )= 0
(12)
in terms of “intrinsic phase averages” defined by (T)f= @-I(T)etc., where = V,/V is the porosity. With these definitions substituted into eq 10 we obtain # IJ
@ = 2pD:D
represents Rayleigh’s dissipation function. The pore walls make up the portion S, of the boundary S of Vp Here the adherence condition v(x) = 0; x E s,, (6) will be assumed to hold. One may therefore replace the surface integral in (4)by an integral over the part S, of
p k ( v)
+ div((T)f.(v)) + div(T.i.)
=
(a)
(13)
The third term on the left-hand side of this equation is nonzero only in macroscopically nonuniform flows. From a scaling consideration, similar to one applied by Batchelor (1970) in the statistical theory of suspensions, one may conclude, however, that this term is negligible. In fact, the orders of magnitude of i. and T are 1 ( v )I and pI ( v ) I / d near
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
the walls of a typical pore of diameter d and smaller elsewhere so that the ratio of the third term to ~ l v)I2/d2, ( the magnitude of ((a), is never larger than the ratio d / l of the pore size to the distance over which the macroscopic flow changes appreciably. Therefore the third term in eq 13 may be neglected under the conditions stipulated above. The macroscopic lbalance of mechanical energy thus becomes pP( v) -I- div((T)f-(v)) =
(14)
((a)
and is therefore identical in form with the local energy balance for the microscopic flow as implied by (4). If we expand the second term in (14) with the help of the identity div((T)f-(v ) ) = div(T)f.(v ) (T)f:(D), obtained by making use of (6), we get
+
(pf+
div(T)f).(v)
+ 2p(D)f:(D)+ ( @ )
(15)
where we have once substituted the averaged constitutive equation (T)f= -(p)fI + 2p(D)ffor ( T ) f a n dtaken into account the incompressibility condition I : ( D ) = tr ( D ) = 0. The argument that leads us directly to the desired reciprocal theorem no’wis the following. By neglecting the kinetic energy in the original balance (1)we have, of course, set the expression on the right-hand side of (1)identically equal to zero and have thereby assumed that
for a n y velocity field. It is therefore the same thing to consider any arbitrary pair (v’,v’?, including the pair (v,v), and assume that
Jv/ p v d’v”. dV ~ =
Sv,pv”--ddtv’ dV
=0
(17)
If one substitutes, fior example, into the first of these identities, the local rnomentum balance pdv“/dt = p f + div T”, one is led to the equation pP( v’) (div(T”.v?) = 2p(D”:D’) (18)
+
Using Gauss’ theorem together with (6) and (91, one may now proceed, taking the same steps as in the derivation of eq 15, and arrive at (pf
+ div(T”)f).( v’) + 2p(D”)f:(D’) = 2p(D”:D’)
(19)
A corresponding equation results upon exchange of indices. Observing the symmetry of the terms that involve the deformation rates, one thus deduces the reciprocity relation (pf
+ div(T”)’).( v’) = ( p f +
div(T’)f).(v”)
(20)
It is clear from our derivation that the property of macroscopic flows expressed by (20) is a consequence of the neglect of kinetic energy as well as of the linearity of the constitutive equation. The reciprocity relation (20) thus has its origin in the same facts as underlie Lorentz’ reciprocal theorem (cf. Happel and Brenner, 1965) for slow viscous flow. A special case of (201, valid only for macroscopically uniform flows in the absence of body forces, was indeed derived from Lorentz’ theorem by Poreh and Elata (1966). These ;authors also took the important last step and deduced Darcy’s law and the symmetry of the permeability tensor from their result. However, only the present derivation establishes the reciprocity property of porous media flows in full generality, enabling us to deal with macroscopically nonuniform flows and to admit the presence of a body force.
43
T h e Resistance Tensor a n d t h e Dissipation Function From the reciprocity relation (20) and from the fact that (v’) and ( v”) are arbitrary we conclude that the vectors ( v ) and ( p f + div(T)f) must be linearly related. Accordingly we put div(T)f + p f = pR.( v) = 0
(21)
The second-order tensor R thus defined may, as will be seen, appropriately be called the Stokes resistance of the porous medium. The factor p has been added for dimensional reasons and ensures that R is an intrinsic property of the porous medium. Upon substitution of (21) in (20) it is found that ( v’) *R* ( v”) = ( v”) -R-( v’) (22) from which follows that R is symmetric. That R is also positive definite can be shown by substitution of (21) in (15) which produces p( v).R.( V) = ( @ ) - 2p(D)f:(D) = 2p(D:D) > 0 (23) The last relation may also serve as a defining relation for a macroscopic dissipation function \k furnished by the phase average of the Rayleigh dissipation function \k
( @ ) = p( v)*R.(V)
+ 2p(D)f:(D)
(24)
The first term on the right-hand side makes \k formally distinct from @ and in fact will be found to comprise most of the energy dissipated in slow viscous flow through a porous medium. It is clear that (21) is an equation of equilibrium for the forces acting on the pore fluid. The nature of the apparent body force pR.( v ) is revealed upon noticing that div(T)f = (divT)f -
$
Tsn d S = -pf--
1
Vf
Tends SP
so that by (21)
which, since n was directed outwardly, is seen to represent the force per unit of pore volume exerted by the solid on the moving pore fluid. Clearly, if the-fluid is hydrostatically at rest, T = (-po + pf.x)I and T = p ( f - S ) I and the integral in (25) vanishes for a porous medium which is statistically homogeneous on the scale of a REV. T h e Macroscopic Field Equations To obtain field equations of the simplest form in the measurable variables ( v ) and ( p ) f ,let us assume macroscopic homogeneity of the porous medium. Taking incompressibility into account we get div(T)f = div(-p I
+ 2pD)f = - grad(p)f + &-lo2(v)
Substitution in (21) then yields the field equation p4-’V2( v) - pR.(v) = grad(p)f - p f
(26)
which together with the condition div( v) = 0 (27) governs the macroscopic flow. For a body force field that is derivable from a potential such that f = - grad U one may define P = ( p ) f+ pU p o in terms of which (26) becomes p - l V 2 ( v)
-
pR-( v) = grad P
(28)
44
Ind. Eng. Chem. Fundam., Vol. 18, No.
1, 1979
From (27) and (28) it follows of course that P satisfies Laplace’s equation V2P = 0 (29) The field equation (28) differs in form from Stokes’ equation by the Darcy term -pR.( v ) which, as we have seen, embodies a specific resistance force exerted by the porous medium on the fluid. An equation of this general kind was proposed by Brinkman (1947) on heuristic grounds. More recently, Slattery (1969,1972) and Saffman (1971) have used averaging techniques to arrive a t eq 28; their analyses fall short, however, of establishing the fundamental resistance transformation (25). It may be noted here that Slattery’s results admit of an inappropriate dependence of the resistance tensor on the direction of the mean filter velocity ( v ) with respect to the medium. The correct result-our eq 25-may be recovered by Slattery’s approach only upon requiring a linear dependence of the resistance force Vcl Is, T-n d S upon ( v ) , but this of course amounts to directly postulating (25), which is the step chosen by Saffman. In Slattery’s result there remains therefore an unnecessary empirical element, calling for a superfluous experimental verification of the constancy of the resistance tensor R. Darcy’s Law It remains to be shown that in most practical situations the Darcy term will actually dominate the leading term on the left-hand side of (28) so that the latter may be disregarded for most practical purposes. This is obviously a matter of the magnitude of R and scaling arguments have been employed in the past to justify the neglect of the term (cf. Slattery, 1972, Chapter 4.3.10). The following argument seems most satisfactory, however. Upon taking the curl of eq 28 and putting w = curl ( v ) , one obtains the equation
V2w - @R-w= 0
(30)
which describes a process of stationary diffusion of vorticity accompanied by decay. From the properties of R it follows that for homogeneous boundary conditions this equation can only have the trivial solution w = 0. This implies that the macroscopic vorticity w must be diffused into a region from its boundary, a property that obviously parallels a well-known fact of viscous incompressible flow (cf. Serrin, p 248). Typically, the vorticity decay will be exponential with the linear distance from the boundary, the magnitude of R determining the steepness of the decay. Consider, for example, an isotropic porous half space z 2 0 and assume plane flow in the x,z plane with the only nonzero component wy = a(u,)/az of w varying in z direction. Let there be a constant value of wy prescribed at the free surface z = 0, maintained by a surface stream, say. Then wy satisfies the equation a2wy -@Rwy=O; @R>O az2
subject to the condition w (0) = wo as well as to the condition that wy be boundecf at infinity. The appropriate solution is then wy
=
woe-z(6/m”*
where K = R-’ is the permeability of the isotropic porous medium. In major applications, dealing with the seepage through concrete, dams or subterraneous formations, the range of permeability values is such that in the present example
any vorticity will have decayed over distances of the order of a grain diameter. It is thus confirmed that the neglect of vorticity, implying the neglect of the term p@-’V2(v ) = - p4-I curl w in eq 28, is entirely justified in such applications. Equation 28 is reduced under this condition to Darcy’s law ( v ) = - p-’K.grad P
(31)
where K = R-’ is the permeability tensor which, of course, is symmetric and positive definite. As a further consequence of the deletion of the term p@-lV2( v ) we note that the traction exerted on the fluid within V f over the portion Sff of the boundary of V will be ascribed entirely to the normal pressure in the surrounding fluid. We have 1 0 = p V 2 ( v) = 2p div(D) = 21Den d S =
v
s
Sff
and hence
a result which is obviously exact for uniform macroflows. Finally, we observe that the dissipation function \k, defined previously through (24), also acquires a simplier form in the absence of macroscopic vorticity. For in slow incompressible flow the identity ( D ) : ( D )= ‘/2 (curl( v ) ) ~ holds and therefore \k = p ( v)-K-’.( V )
(33)
is the appropriate form of the macroscopic dissipation function in Darcy flow. Expression 33 may thus be substituted for the phase averaged dissipation term in a macroscopic energy balance (cf. Slattery, 1972) as long as compressibility effects are negligible. Nomenclature D deformation rate tensor f external force per unit mass I identity transformation K permeability tensor n outward unit normal vector pressure P reference or ambient pressure Po R resistance tensor closed bounding surface of V f Sf portion of Sf which coincides with pore walls SfS portion of Sf which lies in bounding surface of V Sff t time T Cauchy stress tensor velocity measured in frame attached to rigid porous V medium averaging volume V fluid saturated interconnected pore space within Vf V Cartesian coordinates xi X position vector
Greek Letters viscosity density @ porosity 9 Raleigh’s dissipation function phase average of 9, dissipation function for filter \k flows w vorticity vector p p
Ind. Eng. Chem. Fundam., Vol. 18, No. 1, 1979
Special Symbols phase average Over volume v of quantity ,J, asso($) ciated with fluid; see eq 8 intrinsic phase average over volume V, of quantity ($)f $ associated with fluid; ($)' = @-l($) fluctuation about intrinsic phase average $ Literature Cited Batchelor, G. K. J . Nuid Mech. 1970, 41, 545-570. Brinkman, H. C. Appl. Sci. Res. 1947, A I , 27-34. Gray, W. G. Chem. Eng. Sci. 1975, 30, 229-233. Gray, W. G.; Lee, P. C. Y. Dit. J . Mulriphase Now 1977, 3 , 333-340. Happei, J.; Brenner, H. "Low Reynolds Number Hydrodynamics", Prentice-Hall: Englewood Cliffs, N.J., 1965.
45
Poreh, M.; Elata, C. Isr. J . Techno/. 1966, 4(3), 214-217. Saffman, P. G. Stud. APPi. Math. 1971, 50(2), 93-101. Serrin, J. In "Handbuch der Physik", Bd. VIIIII, Springer: Berlin, 1959; pp 125-263. Slattery, J. C. AIChE J , 1969, 15(6),866-872. Slattery, J. C. "Momentum, Energy and Mass Transfer in Continua", Mc&aw-Hill: New York, 1972. Whitaker. S. Ind. Eng. Chem. 1969, 61(12), 14-28.
Received for review March 29, 1978 Accepted September 28, 1978 This research W a s supported by grant number NSF-ENG75-16072 from the National Science Foundation.
Accelerated Settling by Addition of Buoyant Particles Ralph H. Welland' and Rodney R. McPherson Departrnent of Chemical Engineering, University of Queensiand, St. Lucia, Queensiand, Australia 406 7
Buoyant particles, when added to a settling suspension, cause all heavy and light material to quickly segregate laferal'y into separate fast-moving streams. These streams convect particles vertical& at up to six times normal settling rates. Enhancement is greatest with the most concentrated suspensions.
Introduction The settling of suspensions of small particles is a notoriously slow process requiring large diameter equipment. A means of speeding it up would be beneficial in reducing the size of new settlers and thickeners and increasing the handling capacity of existing ones. Most recent technology has been concerned with improving settling rates by flocculation of the suspended particles. However, in attempting to study the interactions between particles falling in a concentrated suspension, Whitmore (1955) found that just by adding neutrally buoyant particles to the slurry, he could greatly increase its settling rate for at least part of the settling period; the lighter particles werle not entrapped in the thickened sludge. Despite its potential benefits, this surprising result does not appear to have been explored further. If one were to contemplate adding buoyant particles to a sedimenting suspension, the common-sense expectation would be greatly increased hindrance and a marked reduction in settling rates. However, the benefits found when particles of neutral density were used lead one naturally to enquire about the possibility of achieving even greater enhancement by the use of positively buoyant particles (which might be separated from the slurry at the same time as the heavy ones) and to enquire about the mechanism responsible. This is the motivation for the current work. We have investigated the effect of varying the concentration of both settling and buoyant particles under two different density driving forces. Concentrated suspensions usually settle out with a rather sharp interface separating the clear fluid which lies above the settling mass; there may also be a zone below the suspension in which settled solids consolidate into an arrangment of closer packing. The addition of buoyant particles to such a suspension results in quite a different picture. We now have the possibility of two interfaces, one falling and the other 0019-7874/79/1018-0045$01.00/0
rising, as shown schematically in Figure 1. Between the two interfaces is a mixture of both heavy and light particulates undergoing separation and settling. In the space above the upper interface, which would normally contain clear fluid, one now finds a mass of buoyant rising particles, while below the lower interface, the light particles have departed. Here one finds only heavy particles falling through the supporting fluid, just as in normal settling. As sedimentation continues, the two interfaces move progressively closer together until eventually they pass through each other. In so doing, they cease to demarcate a suspension of two solids from one of only heavy or light particles. Instead we find a settling suspension of heavy material at the lower end of the vessel, separated from a rising suspension of light material at the top by a region of clear fluid. The heavy and light particles now continue to move as they would in normal settling, the situation at the top end of the vessel being an inverted image of the lower end. Consolidation may occur at both the top and bottom. Experimental Section A glass cylinder 25 mm in inside diameter and 315 mm long was used for containing the suspensions. It was graduated in 1-mm divisions and was made vertical with the aid of a cathetometer and plumb bob (Whitmore reports that displacement from the vertical of 1 in 200 causes a just-discernible difference in settling rate). The stirrer was made from 1-mm brass rod, twisted into a loop at the bottom and passing through a rubber stopper at the top. With the tube completely filled with fluid, air entrainment was prevented during stirring. The tube was illuminated by a fluorescent lamp, but at concentrations of the rising particles greater than 1070,it was found necessary to use the more intense lighting of a projector type of microscope lamp. In general it was not possible to see the top (or bottom) interface which formed between the region of rising solids 0 1979 American Chemical
Society
