H H* HE P
= enthalpy = enthalpy of ideal gas
experimental enthalpy pressure regression function universal gas constant molar density = absolute temperature = statistical weighting function = compressibility factor = experimental compressibility factor = calculated compressibility factor = = = = =
Q
R D
T w j
z
ZE ZC literature Cited
Barner, H. E., Schreiner, W. C., Hydrocarbon Process. 45, No. 6, 161 (1966). Benedict, hl., Webb, G. B., Rubin, L. C., J . Chem. Phys. 8 , 334 (1940). Benedict, M., Webb, G. B., Rubin, L. C., Friend, L., Chem. Engr. Proar. 47. 419 11951). Bono,OJ. L.,’M.S.’Ch.E. thesis, University of Oklahoma, 1968. Bono, J. L., Starling, K. E., Can. J . Chem. Engr. 48, 468 (1970). Cox, K. W., M.S. Ch.E. thesis, University of Oklahoma, 1968. Douslin, D: R., Harrison, R. H., Moore, R. T., McCullough, J. P., J . Chem. Eng. Data 9, 358 (1964). Hoover, A. E., Ph.D. thesis, Rice University, 1965. Jones, hl. L., Jr., Mage, D. T., Faulkner, R. C., Jr., Katz, D. L., Chem. Eng. Progr. Symp. Ser. 59, No. 44, 52 (1963).
Martin, J. J., “Progress in International Research on Thermodynamic and Transport Properties,” pp. 93-99, Academic Press, New York, N. Y ., 1962. Matthews, C. S., Hurd, C. O., Trans. A.Z.Ch.E. 42, 55 (1946). Starling, K. E., SOC.Pet. Eng. J . 6 (4), 363, (Dec 1966). Starling, K. E., “Use of Multiproperty Thermodynamic Data in Equation of State Development,’’ Research Proposal to NSF, March 1967. Starling, K. E., unpublished data 1968. Starling, K. E., Wolfe, J. F., “Nkw Method for Use of Mixture Thermodynamic Data in Equation of State Development,” Paper Presented at A.1.Ch.E. 50th National Meeting, Detroit, 1966. van Itterbeek, A,, Verbeke, O., Staes, K., Physica 29, No. 6, 742 (1963). Venn& ’A. J., Ph.D. thesis, Rice University, 1966. Wolfe, J. F., J. Pet. Tech. 18, No. 3, 364 (1966). Yesavage, V. F., Ph.D. thesis, University of Michigan, 1968. Zudkevitch. D.. Kaufmann, T. G.. A.1.Ch.E. J . 12, No. 3, 577 (1966). Zwanzig, R. W., J . Chem. Phys. 22, 1420 (1954). I
,
RECEIVED for review January 7, 1970 ACCEPTEDFebruary 1, 1971 Presented at the Symposium on Research Results in I & E Chemistry, Division of Industrial and Engineering Chemistry, 160th National hleeting of the American Chemical Society, Chicago, Ill., Sept 13-18, 1970. Work supported by National Science Foundation (Grant GK-2211).
Velocity Relaxation in Viscoelastic Liquids E. A. Uebler Engineering Technology Laboratory, E. I . du Pont de Nemours and Company, Wilmington, Del. 19898
Viscoelastic “liquids” are not capable of sustaining shearing stresses in equilibrium. While in motion, however, such liquids can b e strained from the preferred configuration which would b e assumed if motion ceased. The linear theory of viscoelasticity provides a good representation of the stresses when strains and strain rates are small. Using this theory, the equations describing a particular, experimentally convenient flow are solved. The specific flow studied is the velocity relaxation which occurs when a cylinder containing liquid and rotating at a steady rate is suddenly stopped. A Deborah number containing an explicit liquid elastic modulus determines the nature of the relaxation process. For small N B e exponentially decreasing velocities are predicted, while for large N D , damped oscillations are predicted. These predictions are confirmed quantitatively b y experiments on Newtonian and viscoelastic liquids. The experiments provide a simple scheme for measuring liquid elastic moduli.
A
polymeric melt or solution usually responds to imposed deformation in a manner normally associated with liquids; Le., it flows. However, for certain short-time processes, the same melt or solution at the same temperature can respond purely elastically in the sense of a n elastic solid. Such behavior was described by Metzner (1966, 1968), who showed that the type of response to be expected depends on the relative magnitudes of a characteristic relaxation time of the material and a characteristic deformation time of the process involved. The ratio of material relaxation time to appropriate process deformation time is termed the Deborah number, N D e . For large values of N D , , the material responds elastically and, for small N D e , it flows as a liquid. It follows that the concept of such a 250 Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
material being a “solid” or a “liquid” really has no meaning until the time scale of the deformation process is considered. The term “liquid” is generally applied to materials which take on the shape of a containing vessel and which cannot sustain a shearing stress (or strain) in equilibrium. Therefore, it is usually postulated that in steady flows a fluid has no one preferred configuration from which it is strained, and it follows that the response of the fluid cannot be influenced by finite strain. On the other hand, in (Eulerian) unsteady flows, strain from a particular reference configuration must affect the response of the fluid. Consider the following thought experiment. A “fluid” is flowing at a constant rate in a pipe under the
influence of a constant pressure gradient. Suddenly the pressure gradient is removed and the fluid particles move from some initial position to some final equilibrium position. The equilibrium position attained is dependent on the fluid properties and, moreover, this equilibrium position is the preferred reference configuration which the particles seek upon removal of all forces. The fluid response t o strain from this reference system (as well as the determination of the reference frame itself) must be considered in analyzing such flows. I n any flow problem, if the strains and rates of strain are small, the linear theory of viscoelasticity should be a good representation of the stresses in terms of the kinematic variables. I n what follows, a particular (Eulerian) unsteady process is considered, and the equations describing the process, assuming linear viscoelastic behavior, are solved. The equilibrium configuration is simply that configuration assumed by the particles as time becomes infinite and is determined as one result of the analysis. The particular flow considered, principally for experimental convenience, is the velocity relaxation which occurs when a cylinder containing a liquid in steady rotation is suddenly stopped. Experimental data were obtained to verify the analysis. The experimental technique can be used t o measure elastic moduli of liquids. Problem Statement
Polymer fluids, either melts or solutions, are known to respond nearly elastically under certain rapid deformation circumstances (Metzner, 1966, 1968). I n general, constitutive theories of fluid response, however, do not allow for a purely elastic response to shearing, which would require that the fluid possess a natural or preferred configuration from which it is strained. Fluids do possess a “preferred” configuration, however, which is simply the arrangement that the fluid particles take on upon the removal of any driving forces acting on the fluid. It is well known experimentally that different fluids seek different equilibrium configurations, depending on their physical properties, when the driving forces causing motion are removed. I n steady motion, this preferred state is a material reference frame imbedded in and moving with the fluid. As long as the motion is steady, this preferred state cannot influence the material response and is translated with the fluid. However, in unsteady motions, strains from the preferred state can significantly affect the response. Consider an unsteady flow in the Eulerian sense and let a material coordinate system be denoted by x. Let 2; represent the preferred configuration which the fluid particles take on when in equilibrium. The motion of the material can be described by the functional forms
or, equivalently
5.
=
X(L,(X,t)
(2)
I n a relaxation process, in order t o remove all strains in equilibrium
x ( 2 ; , m ) = 2;(x,m)
(3)
The equations of balance of mass and linear momentum written with respect to the x system are, assuming the materials to be incompressible and neglecting body forces V,kk
=X,kk
T,,km =
=
pirk
0
(4) (5)
in which v denotes the velocity vector and T denotes the stress tensor. T o relate the components of T to the motion of the material, it is assumed that Boltzmann’s equation of linear viscoelasticity is sufficient. This limits the analysis to motions in which the strains are small. Thus, for the relaxation process considered in which the strains from equilibrium are small for all past times s, 0 5 s 5 a ,the stress can be represented by (Coleman and Noll, 1961)
T
=
--PI
+ 2E(E(t)(t)}+ 2G(O){E(0(0) +
‘9
{E([,(t- s ) ) d s (6)
2
in which E and G(s) are linear tensor transformations and E(,,(t - s) represents the history of the relative infinitesimal is given by strain tensor. The strain tensor E($)
+ (F(o - 1IT1
E(cj = ‘/z[(F(e)- 1)
(7)
in which
Fco
=
V2;
(8)
where F([)is the relative deformation gradient matrix. The term p represents the equilibrium hydrostatic pressure. I n addition, G(s), called the shear relaxation modulus, has the property that lim G(s)
=
0
(9)
8-m
Expanding the hereditary strain measure about s
=
0
Now if rapidly fading memory is assumed such that terms of order sz and greater are negligible, then substituting Equation 10 into Equation 6 results in
Integrating and substituting Equation 9 into Equation 11 yields
The viscosity of the material, 7, is given by (13) Equation 12 together with Equation 5 will be used to solve an example problem for which the mathematical formulation becomes extremely simplified and which, in addition, is experimentally convenient to investigate. Consider the rotational flow illustrated in Figure 1. A cylindrical vessel filled with liquid is rotating steadily about its axis until some arbitrary time zero when the vessel rotation is suddenly stopped. Provided the motion was started long ago and transients have died out the velocities in the coordinate directions T , 8, and z prior to zero time are
+=o;
e = Q .
i = o
(14)
in which T , 8, and z are components of the material frame x. Provided Q is small, all of the assumptions required in the above analysis are valid. Let the components of the fixed Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971 251
Data were obtained forTs0.51 NRe = 110.
Figure 1 . Cylinder rotating at angular velocity ( Q ) prior to time zero. The velocity profile of material inside cylinder is w ( r ) = $2. At time zero, the cylinder rotation is suddenly stopped, and w = w(r,t)
reference frame c be denoted by v, 5, and r corresponding to r , e, and z, respectively. The system is determined by solving the equations
dr dt
dB 0; dt
=
=
dz w(r,t); dt
=
0.4
Points denote data, and line is Ea. 28.
0. 2
e(r, 4 measured = 11.6 rad. U r , 4 from Eq. 32 = 12. 1 rad.
0.0
0
T;
$. =
e+
15
20
a
-
0
t
Figure 2. Velocity relaxation experiment-Newtonian
with the equilibrium condition, Equation 3. These solutions are v =
10
5
w(r,s)ds;
= z
(16)
1.4
liquid
t
I n physical components, the deformation gradient matrix, the relative strain matrix, and its time derivative are Data were obtained in repeated runs for P = 0. 50,NRe = 19.1.
a i
Points denote data and line , denotes solution to Eq. 28 for E = 4.0 dynesicm? I n the experiments, a zero-shear viscosity of 6.83 poise and a density of 1.0 gm/cm3 were measured. Blr,ml measured = 1.95 rad. tVr, m) from Eq. 32 = 1.73 rad.
i 8
0.4
0. 2
bW
rb
0 dE(t) --
dt
-
’/*
0
0
6
4
8
10 12 14 16 18 20
t
&
rb
2
0
Figure liquid
0
3.
Velocity
relaxation
experiment-viscoelastic
Letting Substituting Equations 17, 18, and 19 into Equation 12 and the result into Equation 5 , the equations of motion for this process reduce to and differentiating Equation 20 with respect to time results in
subject to the following initial and boundary conditions in which
(1) w(r,O) (2) w(R,t)
=
(3) w(0,t)
=
a
0; t
=
(21)
>0
finite
(22) (23)
The initial condition was obtained from the steady-state solution to the problem. 252 Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
NRe =
RZQp (Reynolds number) 7
and, by definition
NDe =
E -
,a
(Deborah number)
The Deborah number as defined is the ratio of a characteristic process time l/Q t o a characteristic material time q/E. When No, is large, a damped wave equation results, whereas when No, is zero (Newtonian fluids) Equation 25 is of the diffusion type. Different solutions can therefore be expected for the same material, solutions which depend not only on the material parameters but also on the process variables. The solution t o Equation 25 is
6
5
4
T
3
20
..
-
w
L ’
0
2
in which 0
=
Jl(pn) n
=
1, 2 , . . .
(29)
1
and Re denotes “the real part of” the expression for G. This is the quantity measured in the experiments to follow. 0 Experimental Details
An experimental tracer technique was employed to measure particle paths as a function of radius and time. It was more convenient to measure positions than velocities, and since
-
w
=
Figure 4.
-
=
o(r,o)
t(r,m)
8
4
12
-
16
=
Velocity relaxation for different values of E
dt
+
sb watt
(31)
The fixed equilibrium position [ ( r , m ) fixes the origin of coordinates, since e(r,m) =
0
t
dB
the relative displacement from the equilibrium position is
e(r,t)
-1
e(r,o)
+
lm Gdt’
(32)
The origin of coordinates can arbitrarily be set equal t o zero, or e(r,O) = 0, and the displacements measured relative to the initial position. Thus, in dimensionless form
When E = 4.0 dyn/cm*, agreement in Figure 3 between predicted and experimental results is considered good. Deviations are due to the crudeness of the experiments. Figure 4 shows velocity relaxation curves corresponding to values of E of 1.0 and 40.0 dyn/cm2, overdamped and underdamped cases. Finally, Figure 5 shows predicted velocities us. time for both the Newtonian fluid and the polymer solution (for E = 4.0 dyn/cmz). These plots are useful for qualitative comparisons only, since the two curves correspond to completely different processing conditions. Summary
(33)
The required data were obtained in the following way. A cylindrical tube of 14-cm i.d. and 45-em height, open at one end, was mounted on a turntable equipped with a mechanical stopping device. The fluid to be studied was placed in the cylinder and a small colored bead of the same density as the fluid was placed at a specified radial position halfway between the cylinder ends within the fluid. No appreciable settling of these beads took place during a given run. The turntable was started and run for sufficient time for the system to reach steady state after which the turntable was suddenly stopped. A movie camera, positioned above the open end of the cylinder, was started just prior to stopping the turntable, and motion pictures were obtained of the position of the bead. The results of these experiments for a Newtonian corn syrup solution are shown in Figure 2 together with Equation 28. Agreement is good. Similar results for a viscoelastic solution of Separan@ AP30 in water are shown in Figure 3. Separan h P 3 0 is a partially hydrolyzed polyacrylamide manufactured by the Dow Chemical Co.
Recoil is predicted for linearly viscoelastic liquids under certain well defined conditions for which the Deborah number is large. This phenomenon depends not only on the physical properties of the fluid but also on the process variables which
Elastic
13
-1.oL
I
I
I
I
0 2
4
6
8
I
I
’
I
I
I
10 12 14 16 18
t
Figure 5.
Predicted velocity relaxations
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
253
affect the deformation. The analysis includes the material response to explicit strain in liquids, strain from the preferred configuration which the liquid particles seek upon removal of forces. When the Deborah number is small an exponential relaxation of the initial velocities is predicted. Quantitative experimental verification of the above responses was obtained for a Newtonian corn syrup solution and for a viscoelastic solution of Separan AP30. The experimental technique provides a scheme for measuring elastic moduli of liquids; Le., for the Separan solution tested, E was 4.0 dyn/cm2. As a result of this analysis the terms “solid” and “liquid” have little meaning in discussing viscoelastic materials unless the time scale of the process is also included.
t
T X
x
V
ND,
N R ~ Pn
’I
E
Acknowledgment
V,
The assistance of Dr. N. M. Howe in computing the position coordinates shown and in providing many useful suggestions in the analysis is gratefully acknowledged. Assistance in obtaining the experimental data was given by C. Lawrence, Jr.
X
P
E
E(1)
F(1 )
= deformation gradient matrix = relative deformation gradient matrix
=
r
xc 1 )
V
= infinitessimal strain tensor = relative strain tensor
E F
f,
4r,t) Q
Nomenclature
elastic modulus
G(t) = shear relaxation modulus J&), Jl(p) = zero- and first-order ordinary Bessel functions of the argument p, respectively n = numerical index of terms, n = 1, . . ., a P = arbitrary constant indicating hydrostatic pressure r, 8, = cylindrical coordinate system R = vessel radius
dummy time variable time stress tensor, contravariant components Ti’ spatial coordinate system, vector = spatial time derivative of the coordinate position x with components vi, Le., velocity. 1 indicates the spatial acceleration of the coordinate position x = velocity, vector. The components of v are vk (contravariant) and its spatial gradient is denoted in component form by u,jk = Deborah number, dimensionless, expressing the ratio of a characteristic process time to a characteristic material time. = Reynolds number = eigenvalues = viscosity coefficient = preferred coordinate system, vector = components of E = density = function which maps the material coordinates into soatial coordinates. the motion of the body,’vector = function which maps the spatial coordinates into the material coordinates, the motion of the body, vector = angular velocity, function of r and t = angular velocity, constant = gradient operator = unit matrix = transpose of A = total material time derivative of A = = = =
S
1 AT
-
A
literature Cited
Coleman, B. D., Noll, W., Rev. Mod. Phys. 33, No. 2, 239 (1961). Metzner, A. B., et al., Chem. Eng. Progr. 62, No. 12, 81 (1966); A.I.Ch.E. J. 12, 863 (1966). See also Metzner, A. B., Trans. Soc. Rheol. 12, No. 1 (1968). RECEIVED for review June 11, 1968 RESUBMITTED June 2, 1970 ACCEPTEDFebruary 25, 1971
Continuous Foam Drainage and Overflow Fang-Shung Shih’ a n d Robert Lemlich2 Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221
Stationary, overflowing, and refluxing columns of liquid foam are examined with systems of various surface viscosity. Bubble sizes are measured photographically, and foam density is measured via electrical conductivity. Liquid content, internal coalescence, and overflow rate are determined a t steady state. The experimental results are generally in accord with the theory for foam drainage and overflow presented earlier b y the authors’ group.
O v e r the years, a number of relationships have been proposed for predicting the rate of foam drainage and/or overflow (Brady and Ross, 1944; Haas, 1965; Haas and Johnson, 1967; Jacobi, et al., 1956; Miles, et al., 1945; Morgan, 1969; Ross, 1943; Rubin, et al., 1967). However, these relationships 1 2
Present address, Dow Chemical Co., Midland, Mich. To whom correspondence should be sent.
254 Ind.
Eng. Chem. Fundam., Vol. 10, No. 2, 1971
are based on highly simplified models for the drainage channels (such as vertical cylinders or parallel planes) and generally involve empirical constants peculiar to the particular system. Accordingly, several years ago, Leonard and Lemlich (1965a) proposed a more fundamental theory for foam drainage - and overflow. This theory is based on interstitial Newtonian flow through randomly oriented capillaries (Plateau borders) bounded by convexly curved gas-liquid surfaces
