pair potential function transport property Roltzmann constant mass of a molecule chain length pressure first-neighbor pairs of a n r-mer number of sites occupied by a n r-mer intermolecular distance surface fraction, defined by eq 3 absolute temperature molar volume mole fraction number of first neighbor on the quasi-lattice GREEKLETTERS pair potential parameter, characteristic energy absolute viscosity thermal conductivity derivative of transport property pair potential parameter, characteristic length universal potential function isothermal compressibility species in the mixture pure i, j , k , respectively for mixture for pure r-mer SUPERSCRIPTS = for mutual diffusion coefficient VD = for tracer diffusion coefficient SD = reduced property N
literature Cited
Bidlack, D. L., Anderson, D. K., J . Phys. Chem. 68,206 (1964a). Bidlack, D. L., Anderson, D. K., J. Phys. Chem. 68,3790 (196413).
Bingham, E. C., “Fluidity and Plasticity,” McGraw-Hill, ?Jew York. N. Y.. 1922. Brunet,’J., Ddan, M. H., Can. J . Chem. Eng. 48, 441 (1970). Coursey, B. M., Heric, E. L., Can. J . Chem. Eng. 47, 410 (1969). Cronaner, D. C., Rothfus, R. R., Kermore, R. I., J . Chem. Eng. Data 10, 131 (1965). Cullinan, H. T., Can. J.Chem. Eng. 49, 130 (1971). Doan, M. H., Brunet, J., Can. J . Chem. Eng. in press (1972). Guggenheim, E. A., +. Roy. SOC.,Ser. A 148, 304 (1935). Guggenheim, E. A., Mixtures,” Oxford University Press, 1952. Heric, E. L., Brewer, J. G., J . Chem. Eng. Data 12, 574 (1967). Hermsen, R. W., Prausnitz, J. M., Chem. Eng. Sci. 21, 802 (1966). Hildebrand, J. H., J . Amer. Ch;(m. SOC.51, 69 (1929). Hildebrand, J. H., Scott, R., Solubility of Non-Electrolytes,” Reinhold, New York, N. Y., 1949. Kendall, J., Monroe, X. P., J. Amer. Chem. SOC.39, 1787 (1917). 9, 84 Leffler, J., Cullinan, H. T., IND.ENG.CHEM.,FUNDAM. (1970). Lennard-Jones, J. E., Devonshire, A. F., Proc. Roy. SOC.,Ser. A 163,63 (1937). Lennard-Jones, J. E., Devonshire. A. F.. Proc. Rou. Soc.. Ser. A 164, l(1938). ’ Mukhamedzyanov, G. Kh., Usmanov, A. G., Tarzimanov, A. A., Izv. Vyssh. Ucheb. Zaved., Neft Gaz. 6, 75 (1963). Mukhamedzyanov, G. Kh., Usmanov, A. G., Tarzimanov, A. A., Izv. Vyssh. Ucheb. Zaued., Neft Gaz. 7, 70 (1964). Porter, A. W., Trans. Faraday SOC.16, 35 (1920). Preston, G. T., Chapman, T. W., Prausnitz, J. M., Cryogenics 7.274 (1967). Prigogine, I., Bellemans, A., Marhot, V., “The Molecular Theory of Solutions,” North-Holland Publishing Co., Amsterdam 1957. Reid, R. C., Chem. Eng. Progr. 61,58 (1965). ENG.CHEM.,FUNDAM. 8, 791 Tham, M. J., Gubbins, K. E., IND. (1969). Thomaes, G., J . Mol. Phys. 2, 372 (1959). Thor, A. B., Anderson, K., Acta Chem. Scand. 12, 1367 (1958). Shieh, J. C., Lyons, P. A., J . Phys. Chem. 73, 3258 (1969). Van Geet, A. L., Adamson, A. W., J . Phys. Chem. 68, 238 (1964). Weast, R. C., Ed., “Handbook of Chemistry and Physics.” Chemical Rubber Co., Cleveland, Ohio, 1968. RECEIVED for review June 11, 1971 ACCEPTED February 28, 1972
Perturbation Solution of the Steady Newtonian Flow in the Cone and Plate and Parallel Plate Systems Raffi M. Turian Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, N . Y . 13.220
The steady incompressible flow of a Newtonian fluid in the cone and plate i s analyzed by perturbation. Asymptotic expansions for the velocity are developed in which the ordering i s based on two small parameters: an appropriately defined Reynolds number and a function of the small cone angle. The terms in these expansions are complete to the appropriate order, and at each stage of the approximation the solutions are given explicitly in closed form. Expressions are developed for interpreting torque and normal force data using the cone and plafe system, and these are in excellent agreement with available experimental data. The present theory i s also applicable to the small gap parallel plate system, and it forms the basis for extension of the analysis to non-Newtonian fluids.
T h e cone and plate geometry has been widely used in rheological testing, and its appeal resides in the fact that, t o a first approximation, the purely tangential main flow can be described without the need t o prescribe the rheological characteristics of the test fluid. Moreover, the stress and the strain rate associated with this main flow are, in the limit
of vanishingly small cone angle, constant in the fluid gapa particularly desirable feature for testing non-Newtonian fluids. However, the general flow in this geometry is inherently three-dimensional, and despite the fact that cone angles are typically small, ranging from ‘/3 to 6”, the secondary flows do affect the rheological measurements. These are Ind. Eng. Chem. Fundom., Vol. 11, No. 3, 1972
361
particularly important as far as the normal force data are concerned, and a t sufficiently high rotational speeds, torque measurements are also affected. The proper interpretation of rheological data from the cone and plate, therefore, depends on the ability t o account quantitatively for these effects. The motion in the cone and plate has been examined before. Bhatnagar and Rathna (1963) employed a straightforward linearization of the equations of motion based on the idea of viewing the secondary motions as disturbances on the primary tangential flow. However, because they did not exploit the small angle property from the outset (although they eventually had to invoke it t o justify their infinite series approximations), their solutions are in the form of infinite series in the small angular variable. In order to utilize their results it is necessary to determine the coefficients in these series anew for each new set of operating conditions, and hence these results are not of great help to the experimentalist. In Bhatnagar and Rathna’s scheme the linearization is accomplished by considering the tangential velocity t o consist of a primary part and a disturbance, while the remaining two velocity components are taken to be disturbances entirely. Second-order terms in the disturbances are neglected systematically, and the Reynolds number is carried through as a free parameter. It is difficult to carry out the approximations beyond the first-order terms in the disturbances, because the ordering process becomes confused. Giesekus (1965, 1967) used the same linearization scheme as Bhatnapar and Rathna, but he refrained from invoking the small angle approximation, because one of his primary aims was to provide qualitative explanations for the flow patterns observed by him and also by Hoppmann and Miller (1963) in large cone angle systems. However, in deriving solutions t o the linearized partial differential equations the separation of the variables, accomplished by prescribing the radial dependence, implies that the solutions pertain to cones of infinite extent since for a finite, large cone angle system the influence of the edge is significant. A quantitative analysis of Hoppmann and Miller’s large cone-angle flows must include an accounting of the influence of the cylindrical boundary used to contain the fluid. The primary aim of the present work is to establish results for interpreting rheological data using the cone and plate or the narrow gap parallel plate rheometer. The solutions developed here do provide descriptions of the flow patterns, but these, for the present work, are incidental. The flow in the cone and plate is governed (excluding the surface tension effect for the moment) by two parameters: an appropriate Reynolds number and a suitably defined function of the small cone angle. Approximate solutions in the forms of asymptotic expansions in these parameters are derived. All the terms in these asymptotic expressions are complete to the appropriate order; i.e., no redundant terms are included, and all of the solutions, including results for interpreting torque and normal force data, are given explicitly in closed form. These results are useful to the experimentalist and are in excellent quantitative agreement with available torque and normal force data. A very desirable feature of the ordering scheme, which is the basis of the asymptotic expansions developed here, is that it can be used to extend the solutions to account for edge effects and also for non-Newtonian effects. I n both cases, in fact, the present Newtonian solutions form the basis for such extensions. Edge effects are included by considering the present solutions to be the straightforward expansions in a singular perturbation scheme, in which the region near the cone and plate periphery is treated as a boundary layer. The 362 Ind. Eng.
Chem. Fundam., Vol. 1 1 , No. 3, 1972
more meaningful advantage of the present ordering scheme, however, derives from the fact that it precludes the need to prescribe the material coefficients in constitutive equations arbitrarily as constants or zero, as has been done in Bhatnagar and Rathna’s, and also in Giesekus’ work. For constitutive equations in which the material coefficients are implicit functions of the strain rates and the like, a strict application of the present ordering scheme can be used to reexpress these coefficients in explicit forms with coefficients which are functions only of the (constant) strain rate arising from the primary tangential flow in the limit of vanishingly small cone angle. The present approach conforms to our primary objective, which is to establish quantitative results relating to the proper interpretation of rheological data. Therefore, i t is important t o avoid preassigning the values of the material coefficients, since the very purpose of the rheometric measurements is to establish these values and their variation. These constraints are not as rigorous if the objective is merely the one of establishing the qualitative nature of the flow patterns. Solutions of the Equations of Motion
We consider the system shown in Figure 1, and take u,, ug, and u, to be the components of the velocity vector nondimensionalized with respect to (RQ),in which R is the cone radius and Q is a characteristic angular speed. The pressure is nondimensionalized with respect to p a , and the first Reynolds number is defined by Reo = R 2 Q p / p
(1)
Instead of the angular coordinate 8, we use the angle measured from the plate and defined by p = (7r/2) - 8. Accordingly, p = 0 represents the plate surface and p = PO represents the cone surface. The stream function is defined by
and we set w
=
r cos pu,
(3)
I n terms of the above definitions and the transformation sin p
sin B sin Bo
{=-=-
(4)
E
we can show that the equations of motion for an incompressible Newtonian fluid, for a flow without dependence on 6,are given by
with (7)
I n addition, the boundary conditions on w are given by
w(r,O) =
4
r2 -
Q
=
7-2 KO
(9)
in which & ! and 81 are the angular velocities of the plate and cone, respectively. The reference angular speed 8 is taken to be equal to either Q or Q1, depending upon whether the plate is rotating or the cone is rotating. Besides eq 8 and 9, we require (b$/dr) and (b+/br)to vanish a t ( = 0 and = 1. Equations 8 and 9 state that w(r,{) is of order 1 in the flow, We and we deduce from eq 5 that +(r,l) is of order Re.$. therefore define
r
x(r,l) = +(r,0/Reoe3
.
I
I
(10)
I
so that x(r,T) is O(1). Using eq 10, eq 5 and 6 are transformed into Figure 1. Cone and plate system
-
h d r ; €2)
in which Re = Reoe2. It is deduced from eq 11 and 12 and boundary conditions 8 and 9 that the appropriate parameters governing the flow are Re2 and c2. I n fact, a straightforward perturbation solution of these equations can be shown to have the form w(T,{; Re2,$)
x(r,l; Re2,4
-
w0‘(r,~;e2)
4
XO(T,~;
+ Re%(r,f;
...
(13)
+ Re2xl(r,l; 4 +
.. .
(14)
=
0; wO(T,O)
=
+KO,
w ( T , ~ )= rZ(l -
€2hil(r)
= 0 ; fo(0) =
foo”
= SZfOl”
- 2f01;
h ~ ‘ ”= -2f00s00’; /I&‘’ =
(2(2hoo”’)’
(i = 0, 1) (21)
=
fo(1) =
boo'
=
fOZ(1)
= 0
2foohoo‘
ho~’ = 0
- 5foo’hoo; fio(0)
(15) hl0
=
+ hoo’hm‘‘
=
(224
Ki
0, foi(1)
- 24hoo” - 2(foofol)‘ hoi
fio”
KO,
f02(0)
~ O = O
hlo” = -5hoohoo”’ e2)~1
+ ...
- 2foo; foi(0)
= r’foo’’
foi”
in which wo, WI, x0, and x1 can be expressed as asymptotic expansions in powers of 9. When eq 13 and 14 are substituted into eq 11 and 12, and terms of the same order in Re2 are equated, we get
Elwo
+
When eq 19-21 are substituted into eq 15-18, we get the following differential equations for the terms in the asymptotic expansion for frand hi
fo2”
+
9)
hro(l)
=
- ~ i
= 0
(22c)
(r = 0, 1)
(23a)
4p,
=
Reo
3 20
- r2
- -- _ 739_ x 20
3465
Re2
(7) (43)
in which we have dropped the term O ( e 2 ) for the reasons stated previously. I n fact, this term is negligible even for the cone and plate data of Markovitz and Brown, because their cone angle was about 2'. Markovitz and Brown combined Greensmith and Rivlin's parallel plate data together with their own cone and plate data, and found out that a factor of 0.149, instead of the '/6, gave a much more satisfactory comparison with the experimental data. The present theory predicts a value of 0.15, and this is in excellent agreement with the experimental data, as has already been discovered empirically by Markovita and Brown. The second factor in eq 42, -14.295/R2, was determined empirically by Greensmith and Rivlin, and if we use their value for the plate radius, R = 9.7 em, it assumes the value -0.152, which is in excellent agreement R-ith the present theoretical value of -0.15. It was further observed by Greensmith and Rivlin, and also by Markovitz and Brown, that for their less viscous fluids the dependence depicted in eq 42 was not obeyed. I n fact, Greensmith and Rivlin also found that the comparison worsened as the rotational speed and/or the gap width were increased. These observations are consistent with the present theory since under these circumstances the effect of the term O(Re2) in eq 43 becomes important. The term Re2 is proportional t o (gap ~ v i d t h ) ~(rotational , speed)2, and (vi~cosity)-~. The ratio of viscosities of Greensmith and Rivlin's high viscosity to low viscosity fluids was about 300, which means that under the same operating conditions thelterm O(Re2) is increased by a factor of 90,000 for the lowviscosity fluid and can no longer be assumed to have a negligible effect. Greensmith and Rivlin do not present enough details on their experimental results with less viscous fluids t o permit a more complete quantitative comparison with our theory in this case. I n the commercially available cone and plate unit the total normal force, rather than the force distribution, acting on the plate a t b = 0 is measured. Designating this force by Fo, we have
Fo
=
L1
2?rR2pQ
-
( n 'n x)lr=o rdr
(44)
For the case when the cone is rotating and the plate is stationary, K~ = 0 and K~ = 1, and eq 47 becomes
The results in this section are of considerable practical value in the interpretation of cone and plate, or parallel plate, rheological data. In fact, all of these results are applicable regardless whether the fluid is Xewtonian or not, provided that terms O(Re'), for i 2 2, are excluded for nonNewtonian fluids. The particular rheological character of the test fluid only begins to affect these results through these higher approximations. Calculation of the Torque
X convenient method for determining the torque required to maintain the motion is through calculation of the power input P. The power is given by the integral of the viscous dissipation, p@", over the fluid volume, Le., by P
s," 1''
p@vr2 COS pdpdr
=
2~
=
~ T R ~ ~ V E
l1l1(2)
(49)
r2drdP
in which 7 = Rr and (@v/Q2) is the dimensionless dissipation function. The dissipation function is calculated from
[;
(;)I2
{> [ 1
+ Re2
d
dr ' ( X .r2 )1'+2[;($)l2+2[$
G
r2
e2r
2
Xtl-
X
d -
d 1r2
t y 2
]'}
X
(50)
Using eq 50 and our asymptotic solutions for w and x we can show that the dissipation function can be ordered in the following way.
in which Fo has dimensions of force, n is the unit normal in the positive z direction, and ?e is the dimensionless stress distribution. Accordingly
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
365
L
I
IN FIG. 3 OF CHENG ( 1 9 6 8 1 VERTICAL LINES THROUGH POINTS INDICATTCW ~ H V X I M I I E
t
+
+
+
4 2‘
CONE 44’ CONE
20
EOUATION (571
W
0
wI
I
I
10’ REYNOLDS N U M B E R SQUARED, R e 2 102
IO‘
Figure 2. Effect of secondary motion on torque value
I n the expression for the power, given by eq 49, all terms corresponding to the functions shown in eq 52-54 can be calculated. We note in this regard that although the term in f20‘ in eq 54 was not calculated previously, it does not contribute to the power because &’ is constant, and jzO(O) = f i O ( l )= 0. We substitute these equations in eq 49 and find that
3Re2
[
(Ki
-
KO(K1
K o ) ~ +
4900
-
KO)
Applications to Non-Newtonian Fluids
1260 (55)
The terms corresponding to $11 and 4m are not shown in eq 55; their determination is entirely straightforward requiring merely an unusually strong fondness for algebra. In eq 55 all the terms O(Re0) arise from the primary tangential motion, and if we define the torque corresponding to this primary motion alone as To,we have
and since P = OT, in which T is actual torque, we get finally
(T/To) =
(KI
-
+
K O ) ~
KO(KI
+ __ + 0 ( 4 ] + o ( ~ e 4 ) )
- KO)
1260
K O ~
(57)
1260
3 + 4900 -Re2
+
The comparison between eq 58 and Cheng’s data is depicted in Figure 2. Considering the large scatter in the data, which is perhaps inevitable under the extreme conditions required to 366
One of the most appealing features of the present scheme is that it can be extended to non-Newtonian fluids. I n fact, the results obtained here, relating to Newtonian fluids, can be used as the basis for such an extension. In this article we will demonstrate the procedure for carrying out such an extension in a rather brief manner, and we will not present all the details. For a nowNewtonian fluid it is convenient to split the stress tensor into “Newtonian” and “non-Newtonian” parts. Thus, we let 7tj =
Ind. Eng. Chem. Fundam., Vol. 11, No. 3, 1972
Atj
+
Qij
(59)
in which the stress tensor, T ~ and ~ , the %on-Newtonian” stress, Q f j , are made dimensionless with respect to (yon), where now the reference viscosity 90 is, say, the zero-shear value. The dimensionless strain rate, A t j , is given by Ail =
Experimental data on the effect of the secondary flows on the measured torque in the cone and plate have been reported by Cheng (1968). For these data we set K~ = 0, K~ = 1, and get
(!!‘/To) = 1
generate such high torque values, and the fact that deviations value from unity are second-order effects, the of the (?“/To) comparison is entirely satisfactory. Of course, the present theory predicts extremely well the conditions under which secondary flow effectsbecome important, but this by itself does not constitute confirmation of the theory. It is the fact that the present theory agrees with the data to within the experimental error up to values of (!!‘/TO) exceeding 2 which represents the critical test. Higher order approximations can be generated to extend the range of eq 57, but these are not of great pr:tctical value. Cheng (1968) compared his data with the theoretical treatment of Walters and Waters (1967), who expanded the equation in terms of a Reynolds number and solved the resulting equations numerically. However, aside from resulting in quite poor agreement with Cheng’s data, the theory predicts a decrease in the required torque value when this is evaluated a t the plate. This is a rather serious deficiency of the theory, because the torque value should be indifferent to whether it is measured a t the cone or the plate, and also whether one or the other, or both, are rotated. Contrary to Cheng’s implication, the fact that the ”alters and Waters theory does merely predict the point a t which deviations of the value of (“/To) from unity occur does not confirm its validity. Using the same numerical procedure as that of Walters and Waters, King and Waters (1970) extended the solution to higher values of the Reynolds number in the hope that the comparison with experiment would improve for the range when (!!‘/To) exceeded unity, but the improvement is marginal. Savins and Metzner (1970) used an approximate procedure to estimate the magnitude of the radial velocity component, but the gross approximations involved in their procedure limit its utility. As with the normal force results, the term O(Re2)in eq 56 depends on the rheological behavior of the fluid, and therefore terms to this order and higher are not applicable to non-Newtonian fluids.
ui,j
+
Vf,t
(60)
Using the same definitions for the reduced stream function X, and the tangential velocity w, one can write down the nonNewtonian counterparts of eq 11 and 12. These turn out t o be equations containing the same terms as those in eq 11 and 12, and, in addition, viscous “non-Newtonian” terms involving only the Q i 3 . In order to proceed further, a constitutive relationship must be prescribed. Because, as must by now be clear, in the present scheme the asymptotic solutions are developed as expansions, in the Reynolds number and cone angle, on the primary flow, the present method is
particularly suitable for treating non-Newtonian fluids. Constitutive relations in which the phenomenological coefficients are constants can be treated in a straightforward manner. The critical test of the present method is its effectiveness with constitutive relationships in which the material coefficients are implicit functions of the rate of strain, and other variable properties of the flow field. For example, in the Rivlin-Ericksen (1955) theory, the material coefficients are general polynomials in the traces of the so-called RivlinEricksen tensors. By repeated application of the present ordering scheme we can show that each of the material coefficients can be prescribed as an explicit algebraic form in x, w,and their partial derivatives. The coefficients in these algebraic expressions are unknown functions of only the rate of strain corresponding to the main primary flow, y . For the main primary flow the only nonvanishing component of A i l is AB+,and this can be ordered in the following manner.
Therefore, starting with a constitutive relation in which the material coefficients are implicit functions, we reduce them, through strict application of the ordering scheme, to explicit forms in which the variable coefficients are functions of y alone, i.e., a j ( y ) , etc. The constant strain rate characteristic of the vanishingly small cone angle system is given by yo = - ( K ~- K ~ ) / E , which is obviously the first term in eq 61. Now each of the coefficients q ( y ) can be expanded in a Taylor series about y = yot o give
where the notation culn(yo) = bnaj(yo)/by" has been employed. The remainder of the calculations, t o develop the stream function and the tangential velocity, can be carried out in an analogous fashion as for the Newtonian fluid, because the material coefficients have been expressed as explicit functions with constant coefficients. These coefficients are functions of yo,as they should be for a non-Xewtonian fluid. The Free Surface Effect
The inclusion of the effect of the liquid-air interface entails viewing the overall problem as a singular perturbation problem. Thus, the asymptotic solutions already obtained are assumed to be valid except in a small region where r is O(1). I n this region a new stretched variable, say (r - 1)/ X(Re,e), is defined, in which h(Re,c) is chosen in such a way that the stresses a t the interface are balanced, a t least to first order. The interfacial stress balance is established by requiring the surface tension force to equal the net force associated with the stress distributions on both sides of the interface [see, for example, Wasserman and Slattery (1964)l. This will include the mean curvature of the interface, which is, in the present case, a nonlinear function in F " ( { ) , F ' ( { ) , and F ( < ) ;F ( { ) is the value of r a t the interface, namely, the unknown shape of the interface. Thus, the problem entails also solving for the surface shape, which, of course,.is quite difficult because the equations in the fluid bulk in the region r = O(1) are coupled with the interfacial stress balance
equations. This system of equations appropriate to the region when r is near 1 requires boundary conditions fort heir solution. These are provided by matching with the straightforward asymptotic expansions already calculated on the one hand, and by appropriate boundary conditions on the surface shape gradient, F ' ( { ) , based upon the contact angle of the particular fluid a t the plate and cone surfaces. However, in the very important case when the contact angle is zero, F' is infinite a t both { = 0 and { = 1, and, therefore, vast disparities in the magnitude of F' are encountered as { ranges from 0 to 1. This represents an additional complication since now some intricate matching schemes must also be devised in the { direction in this region near the interface. (The effect of surface tension is incorporated in the Weber number, a suitable function of which becomes an additional parameter governing the flow.) Finally, an overall statement of mass conservation, simply based on the idea that the total volume of fluid remains constant regardless of the surface shape, serves to overcome the problem that the contact angle boundary condition can only define the surface shape t o within an arbitrary constant. The foregoing scheme applies when the boundary a t the cone and plate periphery is free. If, however, this region is bounded by a cylindrical surface used to contain the test fluid, then appropriate no slip conditions must replace the surface stress balances in the singular perturbation scheme. Conclusions
The basis of the present scheme of calculation is the system of equations given by (11) and (12), and the transformations which lead to them. Written in this form, these equations allow us to identify the appropriate perturbation parameters governing the flow, thereby rendering the mathematical problem tractable. The present approach is particularly well suited to the interpretation of experimental data, because it is possible to develop explicit solutions, which are complete for the given stage of the approximation considered. Furthermore, it is possible to use this approach to develop various extensions of the present results, using these very results as the basis for such extensions. In our proposed scheme for extending the solutions to non-Newtonian fluids, for example, we have shown that it is possible to cast the problem into one in which the approximations are about the state of constant shear strain in the fluid gap. Aside from the fact that this permits us to exploit one of the main features of the cone and plate rheometer, it leads to solutions in which only a minimum number of prior assumptions regarding the constitutive relationship need be made. Since the flow is nonviscometric, this is quite important. The effectiveness of the present scheme is further confirmed by the results of the comparison with experiments already discussed. The exact equations governing the flow considered here are nonlinear, and one might suspect that multiple solutions could exist. In the case of the parallel plate system the numerical solutions of Mellor, et al. (1968), confirm this, but in the linearization procedure, which is the basis of the treatment presented here, the ability to predict multiple solutions is somewhat forfeited. The principal aim of the present work is to provide results for correcting, and interpreting unambiguously, rheometric data. On this basis the main results of practical interest are embodied in eq 41, 43, 47, and 48, and also eq 57 and 58. It is clear from these results that normal force measurements are more sensitive to the effect of the secondary motions than is the torque value; in the former they manifest themselves as a primary effect, whereas in the latter they merely exert a secondary influence. Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 3, 1972
367
Nomenclature
Ai/
=
+ 1 d2
ui,j
e
uj,t,
r2 - bcz -_ +
,2
rate of strain tensor, dimensionless - SL” dimensionless second-order
(g2
?),
rz 3c2
operator normal force on plate, dynes asymptotic term in w of O(Re2iezj) asymptotic term in x of O(ReZt&) power required to maintain motion, dyn-cm/sec pressure nondimensionalised with reference to pQ ambient pressure nondimensionalized with reference to pQ dimensionless “non-Newtonian” part of stress tensor cone radius, cm R O p l p , first Reynolds number Reot*,second Reynolds number f / R , radial coordinate, dimensionless radial coordinate, cm torque corresponding to primary tangential flow, dyn-cm actual torque, dyn-cm r component of velocity nondimensionalised with respect to RQ @ component of velocity nondimensionalised with respect to RQ @ component of velocity nondimensionalized with respect to RQ r(cos;O)u+
wi
= asymptotic term in expansion for w of O(Rezi)
GREEKLETTERS = cone angle, radians p = ( ~ / 2 )- e, angular variable = rate of strain arising from primary tangential motion, y dimensionless y o = - (KI - K ~ ) / C ,asymptotic limit of y for small E, dimensionless
Po
{ KO KI
w T,,
p T~~
x
xZ
+
Q
fil
= sin BO .-
sin P/sin Po = sin P/e dimensionless rotational velocity of plate = Ql/fi, dimensionless rotational velocity Of cone = viscosity of test liuuid. P = the total stress tensor nondimensionalised with respect to I s 2 = test liquid density = the extra stress tensor nondimensionalized with r e spect to pQ = +/Reoe3, dimensionless stream function = asymptotic term in expansion for x of O(Rezt) = dimensionless stream function defined by eq 2 = reference rotational speed, radians/sec = rotational velocity of plate, radians/sec = rotational velocity of cone, radians/sec = =
%/fi,
literature Cited
Bhatnagar, P. L., Rathna, S. L., Quart. J . Mech. Appl. Math. 16, 329 (1963). Cheng, D. C.-H., Chem. Eng. Sci. 23,895 (1968). Giesekus. H.. Rheol. Acta 4.85 (19651. Giesekus; H.; Rheol. Acta 6,’339.(1967). Greensmith, R. W., Rivlin, R. S., Proc. Roy. SOC.,Ser. A 245, 399 (1953). Hoppmann, W. H., 11, Miller, C. E., Trans. SOC.Rheol. 7, 181 (1963). King, M. J., Waters, N. D., Rheol. Acta 9, I64 (1970). Markovits, H., Brown, D. R., Trans. SOC.Rheol. 7, 137 (1963). Mellor, G. L., Chapple, P. J., Stokes, V. K., J. Fluid Mech. 31, 95 (1968). Rivlin, R. S., Ericksen, J. L., J. Rational Mech. Anal. 4,323 (1955). Savins, J. G., Metsner, A. B., Rheol. Acta 9,365 (1970). Walters, K., Waters, N. D., “Polymer Systems-Deformation and Flow,” Wetton, R. E., Whorlow, R. W., Ed., pp 211-235, London, 1967. Wwerman, M. L., Slattery, J. C., Proc. Phys. Soc. 84, 795 (1964). RECEIVED for review June 28, 1971 May 8, 1972 ACCEPTED
Growth of Magnesium Sulfate Heptahydrate Crystals from Solution Norvin A. CIontzf1Roger T. Johnsonf2Warren 1. McCabe, and Ronald W. Rousseau* Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27607
A flow system was used to investigate the resistances to growth for a single crystal magnesium sulfate heptahydrate, system on crystal faces of both the 1 1 0 and 1 1 1 forms. By gradually increasing the velocity of the solution flowing past the crystal, the growth rate of the face being examined reached a limiting value, indicating that the resistance to mass transfer had been effectively eliminated. The resistance to surface incorporation in these regions was then studied. The surface reaction coefficient was found to follow an Arrhenius relationship with no measurable difference in the coefficients among crystal faces of the 1 1 0 form. For the velocities investigated the surface reaction resistance of the 1 1 1 form appeared to be unimportant when compared to the mass transfer resistance.
F o r a crystal to grow from solution it must overcome two resistances: the first, to diffusion of the solute molecule to the crystal face1 and the second, to incorporation of that molecule 1 Present address, Deering Milliken Research Corp., LaGrange, Ga. 30240. zD-,.n,...+ -,lA.,..ri,.,...... ?7-1l--LmnL 0(orp., San Leandro,
368
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
into the face of the crystal. The diffusional resistance is a function of the solution properties as well as the nature of the fluid movement around the crystal. On the other hand, the incorporation resistance appears to be a function of the properties of the crystal face as well as the crystal temperature. Furthermore, over limited ranges, and in particular at low
