340
Ind. Eng. Chem. Fundam. 1981, 20, 340-346
of its kinetics is presented. We show that for the time course of a reaction to be described by the sum of two or more exponential terms it is not necessary to have stable intermediates (or precursors) in a deactivation sequence. This follows if more than one bond is required to be broken for loss of activity to occur. The concept is extended to include, in general, the involvement of n intermediates with the breaking of m identical bonds required for each step. Finally, a simple “flow-chart” type approach is suggested to model enzyme deactivation data which is different from fmt-order (overall) kinetics but involves, in general, though not necessarily, the deactivation sequence shown in eq 10 and 14b involving one or more than one (identical) bond breakage per step. Nomenclature a = fraction of original activity remaining E, El, E2,..., E,; E,; Eo = concentrations of active, active precursor(s), deactivated and initial enzyme respectively (units/cm3) k l , k2 = probability of rupture of a bond per unit time for a single-step deactivation and deactivation involving an intermediate kd = deactivation velocity constant, s-l m = number of bonds to be broken in a particular step
M = normalized concentration of enzyme precursor El/Eo; see eq 11 n = number of intermediates (precursors) in type of reaction sequence given in eq 14b N = total number of exponential terms in activity-time relationship defined by eq 16 and 17 t = time, s Greek Letters Pi, yi = constants; see eq 14a L i t e r a t u r e Cited Anflnsen, C. B. C. R. Trav. Lab. CarIsbergSer. CMm. 1958, 30, 13. Anfinsen, C. B.; Redfleld, R. A&. Proteln Chem. 1956, 77, 1. Atwood, K. C.; Norman, A. Proc. Net/. Aced. Sci. 1949, 35, 695. Joly, M. “A Fhyslcochemlcal Approach to Denatwatlon of Protelns”; Academlc Press: New York, 1965; p 228. Kalnitsky, G.; Rogers, W. J. BlocMm. Bbphys. Acta. 1965, 20, 376. Lapange, S. “Physico-chemlcal Aspects of Protein Denaturation"; Wiley: New York, 1978; Chapter 5. Relner, J. M. “Behavior of Enzyme Systems”, 2nd ed.; Van Nmtranddeinh o b New Yo&, 1969. Relner, J. M. ”Behavior of Enzyme Systems”; Burgess Publlshlng Co.: Mlnnesota, 1959; p 293. Sadana, A. Bbfech. Lett. 1980, 2(6), 279. Sela, M.; White, F. H.,Jr.; Anflnsen, C. B. Science 1957, 725, 691. Tanford, C . A&. Profeln Chem. 100, 23, 223. Wlseman, A. ”Topics In Enzyme and Fermentatbn Technology”; Wlseman, A., Ed.; Ellis ticwood: Sussex, England, 1978; Chapter 6.
Received for reuiew December 15, 1980 Accepted May 29, 1981
Inelastic Flow from a Tube into a Radial Slit Albert Co’ Cwparrtmenf of Chemical Engineering and Rheobgy Research Center, Universky of Wisconsin, Madison, Wisconsin 53706
A theoretical study has been conducted on the flow of inelastic fluids from a tube into a radial slit between two parallel disks. The entry tube is extended to infinity, at which plane the flow is considered to be fully developed, and the radii of the disks are assumed to be infinitely large. The flow system is dMded Into three flow regions: tube flow, stagnation flow, and radial flow. Numerical calculations are done by orthogonal collocation, in which the stream function and the stresses are approximated by trial functions. The results for Newtonkn f/u&lsshow that fluid inertia induces a second peak in the axial velocity profile near the tube exit and reduces pressure drop in the radial flow region. For the generalized Newtonian fluM with Carreau vlscosky function, the calculated radial velocity profile in the radial flow region is in close ageement with that from the power-law analysis; however, the calculated radial flow pressure profile is somewhat higher than the analytical power-law predlctlon.
Introduction
In this paper we consider the flow of a purely viscous, inelastic fluid from a central tube into a radial slit between two parallel disks. Viscoelastic fluids are examined in a separate paper (Co and Stewart, 1982). The tube-and-disk flow system is represented schematically in Figure 1. The fluid enters the system at a constant flow rate through the tube connected to the upper disk and then flows radially outward in the gap between the two parallel disks. Both disks have an outer radius rl and the distance between them is 2h. The circular tube has an inner radius rW Theoretical studies of steady viscous radial flow are summarized in Table I. These theoretical studies dealt primarily with the radial flow between the parallel disks and did not consider the flow in the tube nor in the
* Current address: Department of Chemical Engineering, University of Maine, Orono, Maine 04469. 0196-4313/81/1020-0340$01.25/0
stagnation region. In the present work these flow regions are included. In the experimental studies cited in Table 11,the fluid flow rate was maintained at a steady value and the pressures at various radial positions on the upper or lower disk were measured. Dye tracer experiments were also performed in some investigations. Note that in these experiments rl/rOis about 10 and r l / h lies between 90 and 2600. This flow system is of interest for the following reasons: (1)It can be used to evaluate rheological models, since the fluid is both sheared and stretched in this flow field. (2) It can be used as a viscometer to measure the zeroshear-rate viscosity (Schwarz and Bruce, 1969) and the second-order constants (Co and Bird, 1977). (3) It has applications in the design of air diffusers (which involve Newtonian fluids) and hydrostatic bearings (Woolard, 1957). (4) It resembles the arrangement in injection molding. However, it should be emphasized that this investigation deals only with isothermal steady radial flow. 0 1981 American Chemical Soclety
,
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 0 = VOLUMETRIC FLOW RATE
341
Table 11. Experimental Studies of Steady Viscous Radial Flow researcher fluids studied McGinn (1956) water Chen (1966) Comolet (1957) air Morgan and Saunders
rllr, 290-300 13 470-2160 36,72 300-1670 10 200-600 -r,lh
(1960)
Figure 1. The tube-and-diskflow geometry. Keep in mind that we take rl
>> h.
Table I. Theoretical Studies of Steady Viscous Radial Flowd method of researcher fluid model solution
(1974) fluid inertia
90-900 190-750
6 48
600-2600
10
(1965b) Laurencena and Williams
Comolet (1957) Newtonian integral approach included Woolard (1957) Livesey (1960)' Moller (1963) Hunt and Torbe series expansion
sucrose in glycerine, carboxypolymethylene solutions, hydroxyethyl cellulose solution
'Tracer experiments were also performed. Here the modified pressure P includes the effect of gravitational forces. Taking the curl of this equation, one gets [V X p Dv/Dt] = -[V X [ V * T ] ] (4)
(1962) Peube (1963) Savage (1964) Jackson and Symmons
The pressure term drops out since the curl of the gradient of a scalar field is zero.
(1965a) Cogswell and Lamb
Moller (1963) Jackson and Symmons
power law analytical upon simplification
neglected
(1970)' Laurencena and Williams
(1974) Winter ( 1975)c Na and Hansen Sisko
numerical
included neglected
(1967)
* N o comparisons with experimental data were made. Hunt and Torbe (1962)included fluid inertia in their series expansion for velocity but neglected it for the pressure distribution. An error in their expansion was later pointed out by Jackson and Symmons (1965a). Winter $1975)also included normal stresses and viscous heating. None of these studies included the flow in the tube and in the stagnation region.
Mold filling, which is transient and nonisothermal, is not considered in this work. Equations of Change In this investigation, the following assumptions are made: (1) The flow has reached steady state. (2) The fluid is incompressible;Le., its density is constant. (3) The flow is symmetrical around the z axis; hence, all variables are independent of the 0 coordinate. (4) There is no motion in the 0 direction; Le., Ug = 0. (5) Viscous heating is negligible and the flow remains isothermal. (6) No phase change occurs. Since the flow is assumed to be axisymmetric, the velocity components can be expressed as derivatives of a stream function 9, that is (Bird et al., 1960) u p = +(l/r)(W/W (1) u, = -(l/r)(W/dd
(2)
The above equations automatically satisfy the equation of continuity for an incompressible fluid, i.e., (V-v) = 0. For incompressible fluids, the conservation of momentum gives p Dv/Dt = -VI) - [VV] (3)
Generalized Newtonian Fluids (GNF) As summarized in Table I, the power-law (Ostwald, 1925; de Waele, 1923) and Sisko (1958) models were examined in earlier studies. These models fall under a class of fluids that are known as the generalized Newtonian fluids. For this class of fluids, the stresses can be described by 7
= -q+
(5)
where 7 is the fluid viscosity function and depends on the scalar invariants of the rate of deformation tensor 3. or the stress tensor T . Equation 5 can describe the shear-dependent viscosity, but not the normal-stress and timedependent phenomena of polymeric liquids. In this study we use the Carreau viscosity function (Carreau, 1968; Bird et al., 1977) q = qo(l +
Xu2j,2)("')/2
(6)
in which qo is the zero-shear-rate viscosity, h, is a time constant, n is a dimensionless parameter, and 9 [(l/ 2)(+:+)]1/2,a scalar invariant of At high 9, eq 6 reduces to the power law q = mj"-' with m = qOXun-l. As + approaches zero, eq 6 leads to the zero-shear-rate viscosity, which neither the power law nor the Sisko mode1 can describe. Dimensionless Form of the Governing Equations To transform the basic equations into dimensionless form, we introduce the following dimensionless variables R = r/ro (7)
+.
Z =z/h
(8)
9 = 2*(9 - +') / Q
(9)
J ! = 2?rr,,h2w/Q
(10)
T = 2*rdt2~,/qoQ
(11)
M = 2*rdt2(7,, - T d / a o Q N = 2*r1&~(7, - TM)/OOQ
(12) (13)
342
Ind. Eng. Chem. Fundam., Vol. 20, No. 4,
i" = 2rr,,h2i./Q H = 77/70 and the following dimensionless groups Ro = ro/h Re = 2PQ/.lrrm A, = qo&/pr8
In eq 10 the quantity w = dv,/dz - au,/ar is the 0 component of the vorticity vector [V X VI; in eq 9 the quantity p is the stream function at an arbitrary reference position, and Q is the total volumetric flow rate through the system. The resulting dimensionless equations are summarized in Table 111. Equations A to D of Table I11 are solved simultaneously for #, T,M,and N . After these have been computed, the radial component of the equation of motion (eq E of Table 111) is used to obtain the gradient dP/aR, in which P is defined as P = 2rr&'[(P + - (P+ T ~ , ) ~ I / ~ O Q(19) and vanishes at a reference position (R = R", Z = 2"). With R" specified, one can then integrate the computed dP/dR to get the P distribution at Z = 2". Boundary Conditions In order to obtain a well-posed, realistic problem statement, we include the inlet tube in the analysis and make the following assumptions: (1)The tube is sufficiently long so that at some distance from the inlet to the parallel disks, the flow is fully developed. (2) The exit effects at the outer edge of the parallel dish are negligible. This is a reasonable assumption since the ratio rl/h is rather large in the available experiments on this flow system (see Table 11). (3) The tube is connected to the upper disk with a square edge (see Figure 1). Using the first assumption, we extend the tube to infinity and use the fully developed tube flow conditions as the boundary conditions at -1/Z = 0. Using the second assumption, we extend the parallel disks to infinity and use the Newtonian creeping flow solution to obtain the boundary conditions at 1 / R = 0. The boundary conditions used are summarized in Table IV. Here we have used the stream function at the tube wall and upper disk as the reference value p. Results and Discussion The flow system is divided into three flow regions: tube flow, stagnation flow, and radial flow. The partial differential equations are solved numerically by orthogonal collocation, in which the stream function and the stresses are approximated by trial functions. The numerical methods are described in Co (1979) and Co and Stewart (1982). Here we discuss some of the calculated velocity and pressure profiles. Newtonian Fluids. The velocity and pressure profiles for a Newtonian fluid (A, = 0, n = 1) at Re = 0.01 are shown in Figures 2A to E. Figure 2A indicates that the dimensionlessaxial velocity profile (rr&&r)/Q) in the tube agrees quite well with the parabolic profile of Newtonian fully developed tube flow, even at the tube outlet (2 = -1). The dimensionless axial and radial components of velocity at various locations in the stagnation flow region are shown in Figures 2B and C, respectively. At low Reynolds numbers, the dimensionless axial velocity profile (rr&Jr,z)/Q) is symmetric along the center line ( R = 0), and the dimensionless radial velocity profile (4rhrovr(r,z)/Q) is symmetric about the midplane between the parallel disks
Table 111. Dimensionless Equations for Steady Viscous Radial Flow vorticity : (UR,,) ( a [ ( l / R ) a ( R T ) / a R ] / a R )+ aZM/aRaZ- ( l / R ) (aZ(RN)/aRaZ)- R,(a2T/aZz)- (Re/4)[(a(a / R ) / a R ) ( a s / a Z ) - ( l / R ) ( a s / a R ) ( a s / a Z ) ]= 0 (A) stresses: T = - H [ ( l / R ) ( a2 s / a Z z ) - ( l / R o 2 ) ( [a( l / R ) ( a* / a R ) l / a R ) ] M=
(B)
+ ( ~ / R , ) H[ ( l / R ) ( a z s / a R a Z )+ ( l / R z ) ( a s / a Z ) ]
N = - ( ~ / R , ) H [ ( 1 / R ) ( a 2 s / a R a z )- (2/R2)(a*/aZ)]
(C)
(D) motion (R-component): a P / a R = aM/aR - ( l / R ) ( a ( R N ) / a R ) - R , ( a T / a Z ) (Re/4)(1/R2)[(au / a Z ) (a % / a R a Z - (1/R)(a* /aZ)) - (a* /aR)(a 2q/a22)] (E) Note: The dimensionless variables a,rz, and H are determined by n = ( l / R o Z a) [( ( l / R ) ( a s / a R ) ] / a R )+ ( l / R ) ( a 's/azz) (F) I: = [ ( I/R )( a 2s /a zz) - ( i / ~a [(I ~/ R ~)(a)s( / a R ) ]/a R I + (4/R,*)[( l / R z ) (a 2 s/ a RaZ)' - ( i / R 3 ) (a s /aZ)( a %/ aRaZ) + ( 1 / R 4 ) ( a s / a Z ) z ] (GI H = (1 + ( ~ e ~ , 2 ~ ~ / 4 ) 2 r ~ ) ( n - 1 ) ' 2 (HI Table IV. Boundary Conditions for the Disk-Tube System at tube wall ( R = 1;-1 < 1 / Z < 0 ) : s=o a s / a R = 0 (no-slip condition) at axis (R = 0; Z < 1) s=1 a [ ( l / ~ ) ( a s / a ~ ) =] /oa (symmetry) ~ at upper disk ( 0 < 1/R < 1;Z = -1)
s=o a s / a Z = 0 (no-slip condition) at lower disk (R > 0; Z = 1)
s=1
a s / a 2 = 0 (no-slip condition) at (infinite) tube inlet (0 < R < 1;-1/Z = 0) a s / a Z = 0 ; aT/aZ = o aM/aZ = 0 ; a N / a z = o (fully developed tube flow) at (infinite) disk outlet (1/R= 0; -1 G Z G 1) s = ( 2 + 32 - Z3)/4 T = O ; M = 0; N = 0 (Newtonian creeping flow) (2 = 0). Note that the radial velocity first increases with R and then decreases because of the increase in the flow
cross section. Figure 2D shows the dirnensionles radial velocity profile (4rhrvr(r,z)/Q)in the radial flow region. The profile agrees very well with the Newtonian creeping flow solution, even at the inlet to the radial flow region (R = 1). The pressure gradient profile in the radial flow region is plotted in Figure 2E. At Reynolds numbers as small as this, the pressure gradient aP/aR is essentially independent of 2 and agrees with the Newtonian creeping flow solution. The velocity and pressure profiles for Re = 100 are shown in Figures 3A to E. Figure 3A shows how the axial velocity profile in the tube changes as the fluid flows toward the tube outlet. Due to pressure gradients induced by the stagnation surface and the sharp corner, a second velocity peak appears near R = 1. This feature of the axial velocity profile is retained in the stagnation flow region, as shown in Figure 3B. The radial velocity profile in the stagnation flow region is shown in Figure 3C. The symmetry observed at low Reynolds numbers is absent here. Both Figures 3B and C show negative velocity components near the corner (R = 1 , Z = -11, which indicate the pres-
Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 343 N
W
N
m
lYW 0 \
3m
a (D
ii. I
*t
Iw
*
m -1.0 - . 6
-.2
.2
.6
1.0
1.0
.8
.6
.4
.2
0
R-’
7
Figure 2. Newtonian fluid at Re = 0.01 and Ro = 60: (A) axial velocity profile in tube flow region; (B) axial velocity profiles in stagnation flow region; (C)radial velocity profiles in stagnation flow region; (D)radial velocity profile in radial flow region; (E)radial pressure gradient profile in radial flow region. Symbols designate collocation points; dashed lines represent collocation polynomials. m
0
.2
.6
,4
.8
-1.0
1.0
-.6
-2 . 2
.6
1.0
2
R
..
(D
-
N
7 3 W
ii. I
ct,
w
m -1.E
A
-.6
R-’
:. 0 6 7 , . 5
-.2
.2
.6
1.0
2
Figure 3. Newtonian fluid at Re = 100 and Ro = 6 0 (A) axial velocity profies in tube flow region; (B) axial velocity profilea in stagnation flow region; (C)radial velocity profies in stagnation flow region; (D) radial velocity profiles in radial flow region; (E) radial pressure gradient profdes in radial flow region. Symbols designate collocation points, dashed lines represent collocation polynomials.
344 Ind. Eng. Chem. Fundam., Vol. 20, No. 4, 1981 s
5.
:.@ i
.8
.6
.4
.2
2
?-I
and Ro = 60: (A) axial velocity profides in tube Figure 4. Generalized Newtonian fluid with A" = 817.7 and n = 0.4741 at Re = 1.096 X flow region; (B) axial velocity profiles in stagnation flow region; (C) radial velocity profiles in stagnation flow region; (D) radial velocity profiles in radial flow region; (E) radial pressure gradient profiles in radial flow region. Symbole designate collocation points; dashed lines represent collocation polynomials.
ence of a small vortex there. Figure 3D shows the change in dimensionless radial velocity profile in the radial flow region. At the inlet to the radial flow region (R = l),the velocity profile is not symmetric about the midplane 2 = 0. Fluid particles in the region 0 I2 I1 (near the bottom disk) move at a higher velocity than those in the region -1 I2 I0 (near the top disk). This is due to the presence of the vortex near the corner. From Figure 3B,we see that the axial velocity at R = 1 and 2 > -1 is negative. This indicates that the fluid particles are moving toward the upper disk (i.e,, in the negative 2 direction). As shown in Figure 3D, the radial velocity profile becomes essentially symmetric at R-' 5 0.93 (somewhat downstream from the tube outlet) and agrees with the Newtonian creeping flow solution at R-I I0.50. Figure 3E shows the pressure gradient profile for Re = 100. At large R (Le., far from the inlet), the profile can be described by the Newtonian creeping flow solution. However, at small R, the pressure gradients are lower than the creeping flow prediction because of the significant recovery of kinetic energy of the fluid. We also see that the pressure gradient at small R does depend on 2. This may be explained by the asymmetric velocity profile near the inlet (R = 1). The calculated results are not compared with experimental data since the published measurements on Newtonian fluids are at very high Reynolds numbers, where turbulence plays an important role. Our steady-state computations cannot describe turbulent flow. We were able to obtain solutions for Reynolds numbers up to 500. Generalized Newtonian Fluids with the Carreau Viscosity Function. The velocity and pressure profiles for a generalized Newtonian fluid with A" = 817.7 and n
= 0.4741 at Re = 1.096 X and Ro = 60 are shown in Figures 4A to E. These values of the dimensionless parameters corresponds to a fluid with rlo = 23.7 Pa-s, A, = 38.29 s, n = 0.4741,and p = 1.14 X 109 kg/m3; a geometry having ro = 3.12 cm and h = 0.052 cm; and a flow rate of 1.117 cm3/s. The parameters in the Caireau viscosity function were obtained by fitting the model to the viscoeity data of a polyacrylamide solution (Co and Stewart, 1982). The dimensionless axial velocity profile in the tube, shown in Figure 4A,is flatter than that of a Newtonian fluid with the same zero-shear-rate viscosity. However, due to inclusion of the zero-shear-rate viscosity in our model, the profile is not as flat as that predicted by a power-law model with the same exponent n. Note that the axial velocity profile in the tube does not change with 2, even at the tube outlet (2= -1). The axial and radial components of velocity at various positions in the stagnation flow region are shown in Figures 4B and C, respectively. We see that the radial velocity profile is symmetric about the midplane between the two parallel disks. Because of the changes in the radial flow volume and flow cross section with increasing R, the radial velocity first increases with R and then d e c r e k . These featurea of the stagnation flow region resemble those for Newtonian creeping flow. The dimensionless radial velocity profile in the radial flow region is shown in Figure 4D. In contrast to the tube flow region, the velocity profile is quite close to the power-law analytic solution. At the midplane 2 = 0, the calculated radial velocity is only 1% liigher than that for the power law. Tbe effect of the zero-shear-rate viscosity in the radial flow region is much smaller than that in the tube flow region because of the higher 4 in the radial flow region.
Ind. Eng. Chem. Fundam:, Vol. 20,
No. 4, 1981 945
Table V. Typical Values of the Product h,; For GNF Modela Tube Flow Region
R 0.259 0.707 0 0.354 1.178 -0.146 0.356 1.178 -0.854’ 0.413 1.175 Stagnation Flow Region 2-1
0.966 1.924 1.918 1.968
R
z 1
2
3 4 5 6 R
1Q
Figure 5. Radial flow pressure profiies at the lower disk. The flow conditions are similar to those of Figuie.4. The pressure reference point is at R = 9.77. The dashed line represents our GNF predictions; the solid line represents theanalytical rwult of Laurencena and Williams (1974)for a power-law fluid with entrance effects neglected.
Figure 4E shows the pressure gradient profile in the radial flow region. The numerical results are in close agreement with the power-law analytic solution, which gives satisfactory predictions far inelastic non-Newtonian fluids (Laurencena and Williams, 1974). The radial pressure gradient at R-’ = 0.9755 is only 1 to 2% higher than that for the power-law solution. The agreement with the power-law solution is still good at R’= 0.0245. As R-’ approaches zero, one expects the radial pressure gradient to be better described by the Newtonian creeping flow solution, since the flow is very slow there. From the numerical results, one also sees that the radial pressure gradient al?laR is nearly independent of 2. Integrating the radial flow pressure gradients in Figure 4E gives the pressure profiles in Figure 5. Here the pressure profiles for our GNF calculation is somewhat higher than the analytical result of Laurencena and Williams (1974) for a power-law fluid with entrance effects neglected. This difference can be attributed to the higher pressure gradient near R = 1 for the GNF calculations. In the GNF calculations the product A?, was evaluated at various locations. These are shown in Table V. In tube flow the transition region (from Newtonian to power law) of the viscosity curve plays an important role since A?, = 1. Its effect on the velocity profile in the tube can be seen in Figure 4A. In the stagnation and radial flow regions, the transition region of the viscosity curve is not important since A?, >> 1 in most parta of the system. As can be seen from Figure 4D, the transition has little effect on the radial velocity profile in the radial flow region. At smaller values of A,, the results should approach those for Newtonian creeping flow. Numerical results for A, = 0.10, n = 0.4741, and Re = 1.096 X do show good ag-reement with those for Newtonian creeping flow (see Co, 1979). This is possible because the zero-shear-rate viscosity is included in the model. Since most polymeric fluids are fairly viscous, no calculations are done for high Reynolds numbers. Conclusions Our calculations for Newtonian fluids show several phenomena when the Reynolds number is increased from 0.01 to 100 (1) a bimodal velocity profile near the tube exit, (2) a small vortex near the corner, and (3) a reduced dimensionless pressure drop across the radial flow region. For the generalized Newtonian fluid with Carreau viscosity function, the calculated velocity profile in the radial
-0.951 -0.588 0 0.588 0.951
0.195 530 204 57.7 203 528
0.556 1340 483 43.2 485 1344
0.832 1598 573 23.8 576 1604
0.981 1553 523 18.9 525 1558
Radial Flow Region
R-’ Z
-0.924 -0.383 0.383 0.924
0.976 1406 219 219 1406
0.794 0.500 0.206 0.0245 1143 720 297 35.1 179 112 46.3 5.57 179 112 46.3 5.57 35.1 1143 720 297
a These values of the product A,,+ were calculated for the GNF model with A,, = 817.7,n = 0.4741at Re = 1.096 X lo-’, and R , = 60.
flow region is close to that from the power-law analysis, but the calculated radial flow pressure profile is slightly higher., The calculated shear rate distribution indicates that the low-shear-rate (or “Newtonian region”) of the viscosity curve plays a minor role in this flow system. Acknowledgment The author wishes to express his gratitude to Professor R. Byron Bird, whose guidance and encouragement are invaluable, and to Professor Warren E. Stewart, whose advice and suggestions on numerical methods were vital to this work. The author also wants to thank Professors Millard W. Johnson and Arthur S. Lodge for many helpful and stimulating discussions. For financial support the author is indebted to the National Science Foundation (Grant No. ENG-78-06789) and the Vilas Trust Fund of the University of Wisconsin, which provided support under Professor Bird, and to the Research Committee of the Graduate School of the University of Wisconsin at Madison. Nomenclature g = gravity vector h = half-thicknese of the gap between the two disks m = constant in power-law viscosity function M = a dimensionless normal-stress difference, defined in eq 12 n = dimensionless exponent in power-law and Carreau viscosity function (eq 6) N = a dimensionless normal-stressdifference, defined in eq 13 p = pressure P = dimensionless modified pressure, defined in eq 19 P = modified wessure. defined bv P = v - og-r = volumetril,flow rate in the radial flbw &tern r = radial coordinate ro = inner radius of circular tube connected to the upper disk rl = outer radius of the disks r = position vector R = dimensionleas radial position, r / r o Ro = geometric ratio, ro/h
Ind. Eng. Chem. Fundam. 1981, 20, 346-349
346
Re = Reynolds number defined in eq 17 T = dimensionless shear stress, defined in eq 11 v = velocity vector with components vi z = axial coordinate Z = dimensionless axial position, z/h Greek Symbols
4 = [(1/2)(*:4)]l/2,a scalar invariant of 4 4 = rateofdeformation tensor, V v + (Vv)+,with components 9.. l” =vdimensionless measure of 9, defined in eq 14 = viscosity function qo = zero-shear-rate viscosity H = dimensionless viscosity function, q / q o 0 = tangential coordinate A, = time constant in the Carreau viscosity function, eq 6 A, = dimensionless time constant (corresponding to A,), defined in eq 18 p = fluid density T = extra stress tensor with components ~ i ’ $ = stream function, defined in eq 1 and d \k = dimensionless stream function, defined in eq 9 o = av,/dz - av,/ar, 6 component of the vorticity vector [V x v! D = dimensionless measure of w , defined in eq 10 Mathematical Operations V = “del” aperator D / D t = substantial derivative, a/at
O
+
= reference value = transpose of a tensor
Literature Cited Bird, R. 8.; Armsbong, R. C.; Hassager, 0. “Dynamlcs of Polymeric Llqulde. Vd. 1: Fluid Mechanics"; Why: New York, 1977. Bird, R. B.; Stewart, W. E.; LlgMtoot, E. N. “Transport phenomena”; WHey: New Yo& 1980. Carreau, P. J. Ph.D. Thesls, Unhrerslty of Wisconsin: Madison, 1988. Chen, C-P. J. h&?. 1 9 0 , 5, 245. CO, A. Ph.D. Thesis; Unhrerslty of Wlsconsln: Medleon. 1979, Co, A.; Bird, R. B. Appl. Scl. Res. 1977, 33, 385. Co, A.; Stewart, W. E. A I W J. 1982, in press. Cogswell, F. N.; Lamb, P. Plast, pdLm. 1970, 38, 331. Como(et, R. Bull: Soc. Fr. 1957, 23,7. de Wade, A. ON cdw chem. Assoc. J. 1923, 6.33. Hunt, J. B.; Twbe, L. Int. J. Mech. Sc/. 1982, 4, 503. Jackson, J. D.; S y m s , G. R. Inf. J. A&&. Scl. 19Wr, 7. 239. Jackson, J. D.; Symmons. G. R. Appl. Sd. Res. ( A ) l W b , 75, 59. Laurencena, B. R.; Williams, M. C. Trans. Soc. R b o l . 1974, 18, 331. Uvesey, J. L. Inf. J. Mech. Sd. 1960, 7, 84. McGlnn, J. H. Appl. scl. Res. ( A ) 1#6, 5, 255. Motler. P. S. Aeronaut. Q. 1963, 14, 183. Morgan, P. G.; Saunders, A. Jnt. J. Meah. Scl. 1960. 2, 8. Na, T. Y.; Hensen, A. G., Int. J. Non-Llnssr Mech. 1967, 2 , 281. Ostwald, W. K W - 2 . 1026, 38,99. Peube, J-L. J. MBC. 198% 2,377. Savage, S. B. Trans. ASME, J . Appl. Mech. lW4, 37,594. Schwarz, W. H.; Bruce, C. C h m . Eng. &I. 1969, 24, 399. SiSkQ, A. W. Id. Em. them. 1958, 50, 1789. Wlnter, H. H. polym. Eng. &I. 1975, 75, 480. Woolard, H. W. J . Appl. Mech. 1957, 24, 9, 644.
*.
+ v-V
Received for review October 24, 1980 Reoised manuscript received August 17,1981 Accepted August 17,1981
Superscripts
Correktth of the SdubHtty of Methane in Hydrocarbon Solvents Herbert M. Sebadlan, Ho-Mu Lin, and Kwangthu Chao’ School of Chemical Engineerhg, Pmlw University, West Lafayeffe, Indiana 47907
A correlation is developed for the solubHity of methane in hydrocarbon solvents at temperatures from 300 to 700 K and pressures up to 250 atm. The fugacity of dissolved methane at zero pressure is correlated as a function of SolUwRty parameter and temperature. The hi@qmsm fugacity is obtained upon appiykrg a Poynting correction for wMch the required partial volume of methane is correlated as a function of temperature. The Henry constant of methane io induded in the conelatkn. When OOmpBreld with 526 experimental data points of 16 mixture systems, the correlation shows an overall average absolute deviation of 6.6 % .
Introduction The solubility of methane is of considerable technological interest. Recent development in heavy fossil-energy technology has created a need for information at elevated temperatures and pressures. As a result, a large amount of high-pressure experimental gas-liquid equilibrium data has been determined for mixtures of methane in hydrocarbon solvents of diverse nature, including paraffins, mono- and polynuclear aromatics, naphthenes, and N, S, and 0 containing aromatics at temperatures up to 700 K. These conditions of elevated temperature and heavy solvents were not met in previous correlationssuch as those of ChaoSeader (1961)and the Grayson-Shed (1963). The data base of the Chao-Seader correlation was limited mainly to light paraffim solventa (up to Clo) with benzene and toluene as the only aromatics and cyclohexane, cyclopentane, and methylcyclohexane as the only naphthenics. The temperature range of the data base was also limited to below 510 K. Grayson and Streed modified the Chao-Seader correlation on the basis of proprietary 01964313/81/1020-0346$01.25/0
data on hydrogen + oil systems. The data were not published and no comparison of the correlation with data was made. In this work we present a correlation of the solubility of methane based on all available gas-liquid equilibrium data for methane-containing mixtures of identifiable components at temperatures of 300 to 700 K and pressures to 250 atm. The New Correlation The new correlation expresses the fugacity of dissolved methane as a function of temperature, pressure, and liquid composition. Solubility parameter is used to characterize the liquid solution. The method is similar to that we used for the correlation of the solubility of hydrogen (Sebastian et al., 1981). The logarithm of the ratio of the fugacity f of dissolved methane to its mole fraction x is expressed as the s u m of a zero pressure term and a Poynting correction In V / x ) = In (f/x),,o + pP/(RT) (1) 0 1981 Amerlcan Chemical Society
