Transport Properties of Liquid Mixtures (n-Paraffins) and the Theorem of Corresponding States Manh H. Doan and Jean Brunet* Department oj Chemical Engineering, McGill University, Montreal, Province de Qudbec, Canada
Prigogine, et a/. (1957), developed theorems of corresponding states for thermodynamic properties. Their approach has recently been extended to include transport properties of mixtures of spherical molecules (Brunet and Doan, 1970) as well as pure r-mers (Doan and Brunet, 1972). In the present paper the method i s extended to mixtures of r-mer molecules. The technique has been tested for the n-alkanes using the available experimental data: viscosity, 12 sets of data (six systems); thermal conductivity, two sets of data (one system); and diffusivity, 13 sets of data (two systems). For these systems transport properties have been predicted within 1-2% of the experimental values and this has been achieved without any mixture parameters. The chief advantages of the present theory lie in the fact that it accounts for the temperature and density (pressure) dependence of the mixture transport properties, and that it i s fully predictive and requires no mixture parameters. In the present theory the only quantities required are pair elemental interactions between alike species. In principle, pure component data are not required since a generalized correlation i s available from the work of Doan and Brunet (1 972). The present work amounts to an extension of earlier works of Brunet and Doan (1970), Tham and Gubbins (1969), Preston, et a/. (1967), and Thomaes (1 959).
I n view of the relatively recent success of the molecular theorem of corresponding states (MTCS) in predicting transport properties of simple liquids and liquid mixtures as well as those of pure r-mer molecules, any extension of this principle to include transport properties of mixtures of chain molecules (r-mer) is appropriate. The main advantages of the method we are about to develop are as follows: (a) no adjustable mixture parameters are required; hence the present method is predictive; (b) capability of describing both temperature and density (pressure) dependence of the mixture transport properties; (e) applicability to all transport properties; and, (d) ease with which the present method can be extended to mixtures of more than two components. Finally it should be pointed out that the present theory reduces properly to that of spherical molecules (r = I) presented earlier by Brunet and Doan (1970). The statistical thermodynamic theory of liquid mixtures whose components are open-chain r-mers of different lengths has been developed over several years. The quasi-lattice model for solutions or r-mers has been discussed by various authors, including Porter (1920), Hildebrand (1929), Guggenheim (1935), and Hildebrand and Scott (1949). The most complete and rigorous version of the theory is due to Guggenheim (1952). This model, however, does not provide a complete understanding of all the thermodynamic functions characterizing r-mer solutions. .4 more refined model was subsequently developed by Prigogine, et al. (1957), who combined the ideas of the quasi-lattice model for polymer molecules with those of the cell model developed earlier by LennardJones and Devonshire (1937, 1938). The main assumption behind the quasi-lattice model is that all molecules present in the solution can be arranged on a quasi-lattice, with each of the r elements of the r-mer molecule occupying r neighboring 356 Ind.
Eng. Chem. Fundam., Vol. 11,
No. 3, 1972
sites. The main idea behind the cell model consists in taking into account the nature of the force field of an element at a distance r from its lattice point. Based on these ideas, Hermsen and Prausnitz (1966) formulated a AITCS which permits one to correlate in a most remarkable way excess thermodynamic properties. Recently, Doan and Brunet (1972) have shown that transport properties of pure r-mers could be correlated within experimental error (7' RXS deviation = 2.8) using Prigogine's method of parameterization of r-mer molecules. This is done with the help of two elemental potential parameters E?, uT, and three structural dimensional numbers: (a) the number of segments (elements) per molecule r (r = 0.5 (n 1)); (b) the number of external degrees of freedom 3C,; and (c) the number of intermolecular contacts q7z which is calculated using eq 1.
+
g,z = r ( z
- 2)
+2
(1)
The above parameters have been calculated from experimental viscosity and density data for pure r-mers a t atmospheric pressure and are tabulated in the publication of Doan and Brunet (1972). Their technique has been tested over a wide range of temperature and pressure and has been found successful. MTCS for Mixtures of Liquid r-Mers
Consider a mixture of N a r-mer molecules of type i along with N , r-mer molecules of type j . Molecules of type k (i or j ) are distributed on rkk sites of a quasi-lattice and are characterized by 3Ckkexternal degrees of freedom. The intermolecular potential interactions between the elements of
molecules of types i and j are assumed to be adequately described by a two-parameter pair potential function
E
(2)
= ep(R/u)
where R is the distance of separation between two point centers. As in the case of simple liquid mixtures, we shall first calculate the average composition-dependent parameters (r, C/q, e, and u) for the mixture, and then develop suitable master equations from which mixture transport properties could be calculated. The procedure for mixtures of r-mers is then analogous to that employed for simple liquid mixtures, the details of which are reported in the work of Brunet and Doan (1970).
pure r-mer molecules to include their mixtures, it is also necessary to define an average composition-dependent elemental mass, as well as average structural parameters for the mixtures. I n the present model these were calculated from
+SP,~ Stria + SjrJj
ms = Simil TS =
(11) (12)
while the parameter C / q was computed using eq 13 to 15.
The Average Pair Potential Parameters
As we have already mentioned, the determination of the average composition mixture dependent interaction parameters ES and u s is to be carried out in exactly the same manner as was done in the case of mixtures of simple liquids. Following Prigogine, et al. (1957), we shall introduce the surface fractions Siand Sj.These are defined by
S , and S , are convenient concentration scales especially for polymer solutions where rJ,, for instance, is much larger than Tlf.
For solution of monomers (T = 1, q = I ) , the surface fractions reduced properly to the mole fraction of each monomer. Prigogine has presented two methods by which average pair potential parameters could be calculated, these two models being the crude model and the refined model. The calculations we have made for r-mers as well as the earlier results obtained by Brunet and Doan indicate that the refined model is by far superior. I n the refined model two distinct average elemental potentials are considered for segments of types i and j in the mixture. Assuming the mixture to be random, considering only first-neighbor interactions, the average potential energy suffered by an element of type i is given by Et
=
Si,Eaa
+ SjEii
Equations 11 to 15 were selected on the basis that they ensure that the mixture-reduced properties (which will be defined in the next section) map correctly into the corresponding expressions for the pure component as the proper surface fraction (or mole fraction) is set equal to unity, Transport Properties.
Master Equations
I n the work of Doan and Brunet (1972) a theorem of corresponding states was proposed for pure r-mers. Their theorem was stated through two equations of state. M7e shall assume that these are applicable for r-mers.
I n the above equations pii stands, respectively, for the reduced viscosity Gill thermal conductivity Xi(,or self-diffusivity Dit of the pure components
(4)
Assuming t h a t the various potential interactions obey a 6-12 Lennard-Jones potential law, we have
(5)
while
p it
vi,
--
T U , ~
The quantities e l and u t so obtained are composition-dependent average potential parameters for an element of an r-mer molecule of type i in the mixture. The unlike pair potential parameters and uij are calculated using the usual combining rules, eq 7 and 8
dG = 0 . 5 ( ~ i+ i
ejj
~
i
=
j
~ j j )
(7) (8)
The corresponding quantities for the mixture are given by
+ Sj‘j us = s i u t + s j u j ES
=
St€(
(9)
(10)
I n order to complete the extension of the N T C S valid for
As in the case of simple molecules, the corresponding states equation valid for mixtures of r-mer molecules is assumed to be of the same functional form as the one for pure r-mers (eq 17), i.e. lnFs
=
F(l/Fs, l/ps)
(24)
The defining equations for the mixture-reduced quantities (Fs,Bs,Ps) are eq 21 and 22, as well as eq 18 to 20, in which the pure component parameters e l , u,, m,, r , and ( C l q ) , are Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
357
1' r i i t blip
INPUT DATA i i = i,2
'1'
mi i (C/q)ii
1 q i i ; Equation 1 Equation 3
Sii
i
ms
; Equation I I
rs
I Equation 12
Vii
I Equation 21
Til
i
Pii
; Equation
Equation 22
23
questionable and can be corrected easily by using the analysis of Prigogine, et al. (1957). Finally it should be noted that the method presented here is fully predictive. The only adjustable parameter involved is the coordination number z. However, it was found that the present theory is not sensitive to changes in the number of nearest neighbors. Calculations have been carried out for z = 8 and z = 12. The results were found to be almost the same in both cases. The maximum difference in percentage RMS deviation was 0.6% for the system C6-Cls, while for others the differences were much smaller. I n the present work, only results based on z = 12 are reported. I n the next section the present theory will be evaluated. Since we are primarily interested in evaluating the theory on mixtures, we shall not use the general correlation obtained previously for pure r-mer molecules. Therefore the experimental data of the pure components will be used unless otherwise indicated.
REFINED MODEL
Comparison with Experiments and Theory
es, vs, ( C l q ) s
Available Data. I n order to evaluate the present theory, accurate experimental data are required. A survey of available experimental data revealed that, while there are a large amount of viscosity data and some thermal conductivity data for pure n-alkanes, those for mixtures are extremely limited and only data a t atmospheric pressure seem to be available. The available mixture transport property data were compared to those predicted by the present theory. Treatment of Experimental Data. I n the present analysis pure component data are required as a function of temperature and density in order to calculate the functions. When these were not available, the generalized correlations obtained by Doan and Brunet (1972) were used to obtain the f i i M s N functions. Whenever one uses the available transport property data (instead of the generalized correlations), one is confronted with the fact that the (F,T,V) data are not available. I n the present work the (F,T,P = 1) data were fitted first a t constant pressure using an equation such as
Equations 5-10, 13-15 F ~ Equations :
25-26
C a l c u l a t e Fs, Equations 18-20
Figure
1 . Method of calculation
replaced, respectively, by the mixture molecular parameters ES, U S ,ms, rs, and ( C / q ) , defined above. We should point out a t this stage that the chief advantage of equations such as (17) and (24) lies in the fact that, when expanded in a Taylor series in 1/P and 1/P, the resulting equation converges rapidly. The equations we have presented so far are merely a generalization of those presented earlier by Brunet and Doan (1970). I n the present work the mole fractions are being replaced by the corresponding surface fractions. Therefore we shall not pursue any further development and shall only quote the equations one would obtain if the analysis presented earlier by the above authors were repeated using the equations we have presented above. I n the case of the refined model, a master equation is easily obtained for the transport properties of mixtures of r-mers
EiIMvN
In
FfI = A1 + AzTii-'
+ A3P,i-2
(28)
Using the above equation, the [aMIn Fti/d(l/Tii)M]p,,were calculated. These were then used to calculate the EliM" using suitable identities. (In the present work we have restricted ourselves to the ti,l+I,[ i i 2 , 0 , f i $ i l functions.) This is illustrated for M = 1 only.
where
I n the above equations the titM" functions are defined as
These are properties of pure component i. Methods of obtaining these functions from experimental data will be discussed in the next section. The method by which the whole of the calculations can be performed is described in Figure 1. While deriving eq 26 we have assumed for simplicity that the molar volume of species in solution V iwas roughly equal to the molar volume of pure i, Vi+ This approximation is 358 Ind.
Eng. Chem. Fundam., Vol. 1 1 , No.
3, 1972
As seen from the above equation the [ i t l , O can be calculated exactly only if the thermal pressure coefficients for each pure component are available. The next step in the calculation is to calculate the densitydependent terms [ i ~ ~ NThis . is done by converting the (F,T,P)data into the ( F , T , V )set. For the first-order densitydependent term E,$,' we have
~
~
~~~~~
Table I. Comparison with Experiments. Mixture Viscosity Data
Table II. Comparison with Experiments. Mixture Thermal Conductivity Data
%
Temp, 'C
25.0 25.0 60.0 25.0 35.0 25.0 20.2 25.0 30.0 25.0
Viscosity reference
Heric and Brewer (1967) Van Geet and Adamson (1964) Van Geet and Adamson (1964) Bidlack and Anderson (1964b); Shieh and Lyons(1969) Shieh and Lyons (1969) Heric and Brewer (1967) Thor and Anderson (1958) Bidlack and iinderson (1964b) Thor and Anderson (1958) Heric and Brewer (1967); Bidlack and Anderson (1964a) j l v % RMS for 114 data points
RMS dev refined model
0.7 0.6 0.7 0.9-1.1
Temp, OC
Reference"
% RMS dev refined model
llukhamedzyanov 3.3 et al. (1964) 77.32 Mukhamedzyanov 1.9 et al. (1964) AV % ' RlIS 2.6 a Pure components data are taken from Mukhamedzyanov, c&16
26.89
et al. (1963).
1.1 1.4 5.0 4.7 3.2 3.9-4.8 2.0
As seen from the aboveequation the are easily calculated through the use of the 7-P data along with the coefficient of isothermal compressibility K . It should be pointed out t h a t in practice only first-order density correction is possible, since in general the dependence of K on pressure is not available. Having calculated the [ functions, the calculation of the mixture transport properties proceeds as described in Figure 1 using the potential and structural parameters tabulated elsewhere by Doan and Brunet (1972). Comparison with Experiments. Viscosity a n d Thermal Conductivity. Using t h e corresponding states theorems developed for mixtures of r-mers, the viscosity of binary systems of liquid n-alkanes was calculated. The percentage RlIS deviations between experimental and calulated results are presented in Table I. For the calculations appearing in this table, only the following functions were used: [ , i M , o , (M = &3), and ~ i r o ~ l . As can be seen in Table I, the viscosities of mixtures of liquid n-alkanes were predicted within 2.001,,using the present model. In the case of systems where the chain lengths of the pure components are not significantly different (C14-C16, CCCI,) , terms including are not necessary. The discrepancy between theory and experiment is seen to increase with the difference in chain lengths (as was to be expected). These errors may be attributed to a certain number of factors. 1. Errors may arise from the approximate nature of the combining rules used t o calculate the mixture potential and structural parameters. 2. Part of the discrepancy may be attributed to the method we have used to evaluate the functions t i p These functions were calculated using eq 30 along with the values of the isothermal compressibility coefficient K reported by Weast (1968). For many of the pure components involved in the present calculations, high-pressure viscosity data were not always available, and it was therefore necessary to use the generalized correlations presented previously by Doan and Brunet (1972). I n addition, extrapolations both with respect to temperature and to chain length were required whenever experimental data for K were not available.
"
System
For instance, in the case of the system c7-c16 a t 20.2"C, where greater discrepancy is noted, the experimental data for K of cg, c7, cs,Cl2,and Cls were extrapolated with respect to temperature to obtain corresponding values of K a t 20.2OC. The resulting values of K were then plotted against n, the chain length, and extrapolated to obtain the corresponding values of K for c16 a t 20.2"C. For the same system a t 3OoC, where extrapolat'ion with respect to temperature was not required, the discrepancy between theory and experiment is indeed much smaller, which is, t o say t'he least, reassuring. Attempts have also been made to predict the thermal conductivity of mixtures of n-CTH16 and n-C16H34,for which a few experimental data are available. The results are shown in Table 11. I n these calculations, only funct'ioiis [ii"*o (-If = 0-2) were considered. The lack of meaningful high-pressure data prevents one from including the [iio+v functions. The predicted values of the thermal conductivity are very close to the experimental values; thus t'he results obtained so far are promising. Comparison with Experiments. Diffusivity. Complete diffusion-composition data are not plentiful, especially for chain molecules. Recently some diffusivity data (mutual and tracer) have been published (Table 111). We shall use these data to test the present theory. T17hereas in the case of viscosity and thermal conductivit,y the meaning of pure component data was self-explanatory and the data readily available, this is not the case with diff usivity. For t'he purpose of the present discussion, we shall define t'he pure component data Dii as the value or the diffusion coefficient a t infinite dilution of the other speciesj; thus
It should be pointed out that of the above four possible limiting values for Dii (DitMD,D i t s D ) ,two of them (DiisD) could in principle be obtained from the generalized principle of corresponding states for pure components. Unfortunately such a correlation could not be obtained by Doan and Brunet (1972) due to a lack of data on pure component self-diffusion coefficients. Furthermore, because of the above lack of data a t infinite dilution, we had to limit ourselves to systems for which diffusivities were available a t more than one temperature. I n all cases we had to restrict ourselves to a firstorder correction in temperature (t;,,lJ') and in no case were we able to perform a density correction (eq 29, 30). Again the results we have obtained are interesting, even though all the data necessary for carrying out the computation were not available. Comparison with Other Theories
With regard to the transport properties of mixtures of liquid hydrocarbons, many equations have been proposed in Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
359
0.7
It is seen that they yield a satisfactory prediction of the viscosity only for systems in which the sizes of the components are similar. I n all'other cases, the errors involved are quite large. I n the last column of Table IV we also included the results obtained recently by Coursey and Heric (1969), who used the restricted principle of congruence. Their results were selected to illustrate a method which is known to yield very good results when applied to chain molecules. As seen from this table, our predicted results are as good as those obtained by the principle of congruence. It should, however, be pointed out that in their approach the mixture viscosities of a base system are required in order to predict viscosity of other systems, while in our analysis no mixture data are required. Furthermore, as it stands now, the method of Coursey and Heric is restricted to viscosity of mixtures ranging from C6 to C16 and is applicable only at atmospheric pressure. The method we have presented here does not suffer from these limitations. Furthermore, because we have provided a MTCS for pure r-mer molecules, no pure component data are required. The advantage of the method presented thus lies in the fact that it is fully predictive and is applicable to all transport properties. The situation with respect to diffusivity is about the same as that for viscosity. Methods for predicting diffusion have been proposed recently by Leffler and Cullinan (1970) and Cullinan (1971). For the systems CB-CI~and C,-CI2 they report percentage RMS deviations which are similar to those presented in Table IV. The main advantage of the theory we have presented lies in its simplicity, accuracy, and applicability to all transport properties under a wide range of temperature and pressure.
1.9
Conclusion
Table 111. Comparison with Experiments. Diffusivity (Mutual and Tracer)
L
76 Diff usivity
System
CbC12
Cs-C,n
C6Cl2
Temp,
O C
D"
25.1
D"
25.0
D"
35.0
D"
25.0
D"
60.0
DcssD
25.0
DceSD
35.0
Dc12sD 25.0
Cs-C12
DcllsD
35.0
DcssD
25.0
DcaSD
60.0
DcLlSD 25.0 DCllSD
60.0
Reference
Bidlack and Anderson (196413) Shieh and Lyons (1969) Shieh and Lyons (1969) Van Geet and Adamson (1964) Van Geet and Adamson (1964) Av % RMS for 50 data points Shieh and Lyons (1969) Shieh and Lyons (1969) Shieh and Lyons (1969) Shieh and Lyons (1969) Van Geet and Adamson (1964) Van Geet and Adamson (1964) Van Geet and Adamson (1964) Van Geet and Adamson (1964) Av % RMS for 52 data points
RMS
d ev reflned model
1.6 0.7 1.8 0.4
0.8 2.2 2.5 2.0 1.0 3.8 3.6 1.1
0.8
Table IV. Comparison with Theories. Mixture Viscosity Data q- RMS dev Bingham
(1917)
(1965)
(1969)a
0.7 11.8 15.8
0.5 1.3
This work
C16-Cl4 c14-C~ c16-C6 a
Coursey and Heric
0.7 1.3 0.7 1.4 35.8 13.9 3.9 46.9 18.4 Cle-Ce has been taken as base system.
System
(1922)
Kendall Cronaner and Monroe et al.
order to calculate mixture properties from those of the pure components. Quite often the methods are empirical and, because they lack physical meaning, they frequently give incorrect values that differ significantly from experimental data. The experimental studies of transport properties of mixtures of r-mers are not plentiful, and in a recent review Reid (1965) pointed out the need for new methods for computing these in the absence of experimental data. I n the present section the present general theory will be compared to others. We have already pointed out that a number of equations have been proposed for predicting mixture viscosities from pure component viscosity data. The more well known of these equations include those suggested by Kendall and Monroe (1917), Bingham (1922), and Cronaner, et al. (1965). The equations are compared to our predictive theory in Table IV. 360
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
An extension of the MTCS for predicting transport properties of mixtures of liquid r-mers has been presented. The present model requires no adjustable parameters and uses like pair potential parameters. The present theory predicts the viscosity, diff usivity (mutual and tracer), and thermal conductivity of liquid mixtures within experimental error. The method reduces properly to the case previously investigated by Brunet and Doan (1970) for spherical molecules and is applicable over a wide range of temperature and pressure. In summary, it provides a systematic and comprehensive method correlating transport properties for the homologous series of liquid hydrocarbons. Most important, it permits the prediction of mixture transport properties from the knowledge of the pure component properties without requiring additional parameters. Considering the limited amount of data for mixture transport properties, the present calculations are of interest. This is particularly true if the method developed for pure components is used in parallel with that developed for mixtures. Acknowledgment
The support from the National Research Council of Canada (Grant No. NRC-A-5402) is greatly appreciated. The present work was presented a t the 21st Canadian Chemical Engineering Conference, Montreal, P.Q., Canada, 1971. Nomenclature
AI, A z , A 3 = constants in eq 28 = one-third of number of degrees of freedom per C molecule D = diffusivity
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 t h e rheological characteristics of the test fluid. Moreover, t h e 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 t h a t 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
