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Effect of temperature on rheological properties of suspensions
Journal of Non-Newtonian Fluid Mechanics, 26 (1987) 175-183 Elsevier Sciince PublishersB.V., Amsterdam - Printed in The Netherlands
175
EFFECT OF TEMPERATURE ON RHEOLOGICAL PROPERTIES OF SUSPENSIONS ATSUSHI TSUTSUMI and KUNIO YOSHIDA Department of Chemical Engineering, University of Tokyo, 7-3-l Hongo, Bunkyo-ku, Tokyo 1I3 (Japan) (Received November 14,1985; in revised form June 14,1987)
A parameter, the product of viscosity of the suspension medium and applied shear rate, is introduced to describe the effect of temperature on the rheological properties of suspensions, considering the mechanism of agglomeration in suspensions. It is found that, by plotting rheograms of the shear stress versus this parameter, a single master curve can be obtained independently of temperature. The dependence of viscosity on temperature and the flow of suspensions are shown to be well described.
1. Introduction Although the rheology of suspensions has been studied extensively, there has been little theoretical study of the effect of temperature upon the flow properties of dispersed systems. The temperature dependence of the viscosity of suspensionshas been studied by a number of investigators [l-4], most frequently by using an Arrhenius-type equation to treat the experimental data. The activation energy at constant shear rate (E+), and that at constant shear stress (ET) can both be determined. For non-Newtonian fluids, E? # E, also because, for the same temperature variation, different viscosity changes are obtained when the liquid undergoes a constant unit shear stress or a constant shear rate [4,5]. There have been many investigations that represent the parameters involved in the rheological equations as functions of temperature. Maron and Pierce [6] have applied the Powell-Eyring equation to the data on the flow behavior of latex, and examined the temperature dependence of the parameters involved in it. Weymann et al. [7] have presented a numerical analysis 0377-0257/87/$03.50
0 1987 Elsevier Science Publishers B.V.
176
for the viscosity of thixotropic clay-water suspensions and stated that two parameters in the assumed model were functions of temperature and concentration. Thurgood et al. [8] stated that, for coal-solvent slurries in a coal liquefaction preheater, both the consistency factor and the flow behavior index of the power-law equation are strongly dependent on temperature. Yamaguchi et al. [9] studied the effect of temperature on the flow properties of suspensions, and divided the total energy dissipation of flow into two terms, E, due to the interaction of flow units (floe), and E,, the viscous energy, by using energy-dissipation theory. By analysis of E, with respect to the shear rate and the effective floe concentration, they obtained a parameter which is independent of shear rate and of solids concentration, and they stated that this factor reflects the work required to separate two floes. Results from these studies, however, are not generally applicable to suspensions because of the unclear effect of particle-particle interaction on the parameters involved. The objective of this paper is to establish a method for understanding the temperature dependency of the viscosity of suspensions when considering the mechanism of agglomeration in suspensions. 2. Theoretical considerations The formation of agglomerates in a slurry takes place by some interaction between particles [lO,ll] or by liquid bridges [12] when an immiscible carrier liquid exists in the main suspending liquid. Then, the apparent viscosity of the slurry increases because the solvent is trapped into particle groups, resulting in the reduction of the effective volume concentration. In addition, a large amount of energy is dissipated due to attractive interactions between particles, resulting in an increase in apparent viscosity. In an agglomerated suspension undergoing shear with increasing shear rate, the coagulation rate increases due to the increase in collision probability between particles, while the agglomerates are subject to destruction by larger shearing forces into smaller groups. Mason [13] and Reich and Vold [14] studied the effects of agitation upon agglomerate size, and proposed that a dynamic equilibrium is established between agglomerate growth and destruction at any shear rate. When the coagulation rate is sufficiently rapid, there is the establishment of this equilibrium. High shear rates shift the equilibrium into the direction of better dispersion. However, the interactive forces between particles are as strong at high shear rates as they are at low shear rates, so even at higher shear rates the flow units may collide, but the newly-created couplet is soon destroyed by the hydrodynamic force which arises from the shear field acting on the couplet. The energy is then dissipated in this process. Thus, the
177 flow properties of suspensions depend upon the degree of agglomeration as a result of the dynamic equilibrium which is determined by the hydrodynamic force and interactive forces between particles. Hence, the rheological properties of suspensions must be examined by considering the hydrodynamic force between flow units as a parameter. Goren [15] estimated the maximum hydrodynamic force between two touching spheres in Couette flow as Fh = 6.12r/~,R~y, where p,, is the solvent viscosity, R is the radius of flow unit and T is the shear rate. It is shown in eqn. (1) that the hydrodynamic force per unit area is directly proportional to pOq. Apart from the dependence of temperature on the interactive force between particles, the degree of agglomeration seems not to vary at a constant ~~9 because it depends on the dynamic equilibrium. It can be considered that at a given shear rate an increase in temperature results in an increase in agglomeration because of the reduction of the hydrodynamic forces, and therefore the relative viscosities of suspensions increase with temperature. Consequently, without the dependence of temperature on the interactive forces between particles, one would expect that shear stress 7-T data on the same suspension at different temperatures would superimpose when plotted as r versus ~~9 and that only one master curve would be given. In case the interactive forces between particles vary with temperatures, the T-P~~ curves would not be superimposed because of the shifting of the dynamic equilibrium. It can be considered that the variation of the height of flow curves plotted as T versus pop at different temperatures would be attributable to that of interaction with temperature. Hence, in order to examine the flow properties of suspensions at different temperatures the value of pOq must be taken in the same range. 3. Results and discussion The flow curves for suspensions of CaCO, dispersed in ethylene glycol observed by Yamaguchi et al. [9] are shown in Fig. 1, the different curves being for suspensions of different temperatures. It is shown that the suspension is non-Newtonian, having taken on pseudoplastic characteristics. At a constant shear rate the apparent viscosity of the suspension decreases with temperature while the relative viscosity increases, as shown in Fig. 2. Such a change is caused by the growth of agglomeration, because hydrodynamic forces decrease with temperature. The relative viscosity is not a unique function of shear rate and the volume fraction. Let us attempt to re-plot these flow curves between r and 9 into the relation between r and pop, as shown in Fig. 3. It is noticeable that only one
178
0
300
600 3
IO
c s-11
Fig. 1. Flow curves for CaC03-ethylene
glycol suspensions given by Yamaguchi et al. [9].
master curve is obtained by introducing CL,,?.This fact supports the theory previously mentioned that by plotting T against pLoqthe flow curves should be superposed. This indicates that the interactive forces between particles are less dependent on the temperature, and that there is no variation of relative viscosity with temperature, which is represented by the ratio of ordinate to abscissa, at constant p,,?. Therefore, the viscosity of suspensions may be evaluated only from the viscosity of the solvent within a certain temperature range. Yamaguchi et al. [9,16] found that the effective volume fraction of floes obtained from Thomas’ equation [17] increased with tem-
Temperature
C ‘C 1
Fig. 2. Change of relative and apparent viscosity with temperature at constant shear rate for CaCO,-ethylene glycol suspensions. A, A, 84 s-‘; 0, m 164 s-‘; 0, l, 811 s-* and V, ethylene glycol. Open and solid marks show relative viscosity and apparent viscosity, respective.ly.
179
Au09 C Nem-*I Fig. 3. Rheograms between r and aof for CaCO,-ethylene
glycol suspensions.
perature. This is quite reasonable, because the lower hydrodynamic forces due to the increase of temperature should shift the equilibrium in the direction of better agglomeration. The data on kaolin suspension obtained by Ohgaki and Matsuo [18] are similarly replotted in the form of r vs. pOy, as shown in Fig. 4. In spite of high pseudoplasticity, superposition of rheograms is found by correlating into a single curve. In this manner, rheograms plotting T against p,+ are well superposed as long as there is no change in the interactive force between particles. Using the master curve thus obtained, the viscosity of the
0
0.5
1.0 J&O*
1.5
[ N.m-*I
Fig. 4. Rheograms for suspensions of kaolin; data given by Ohgaki and Matsuo [X3].
180
-0
1
2 409
C X10-3
3 N-m-*]
Fig. 5. Rheograms for coal-water slurries; data given
by Marlow
Rowell P91.
suspension can be estimated at any temperature, because the viscosity of the solvent is already known. At the same time, if a disagreement is found at a certain temperature in the relation between these two factors as shown in Fig. 5, it is considered that a change of the interactive force takes place. This figure shows the data on coal-water slurries by Marlow and Rowe11 [19]. Graziano et al. [20] studied the effect of temperature on rheological properties of concentrated suspensions of carbon black and found that increasing the temperature resulted in a progression from concave upward to concave downward flow curves in the plotting of 7 and p. They supposed that an increase in temperature corresponds to a shift to lower effective shear rates according to the concepts used in establishing time-rate-temperature reduced variables [21] and that a generalized Ostwald curve must be considered for any given material. Their supposition is confirmed by the trend shown &Pig. 6 in which is replotted the data of Graziano et al. in the relation between,\7 and CL,,?. It is shown that an increase in temperature reduces the hydroslynamic force, resulting in a shift to lower effective shear rates. In addition, ‘a generalized Ostwald curve is obtained, as far as the shape of the curves is concerned. Thus, within the same range of ~~9, rheological properties of suspension are determined at different temperatures. The height of curves decreases at higher temperature. This may be the result of an decrease in the interactive forces between particles with temperature or a disappearance of water bridges due to their evaporation. The rheograms replotting 7 against p,,y for the suspensions of unpurified bentonite are shown in Fig. 7, and those of purified bentonite in Fig. 8. Original data were obtained from the experiment of Weymann et al. [7]. It is interesting to note that only a single master curve is obtained at each
181
PO+ C N.m-*I Fig. 6. Rheograms for carbon black-naphthenic al. [20].
oil suspensions; data given by Graziano et
concentration except for the 4.26 wt.% purified bentonite slurry for which the height of the curves decreases with temperature. It is speculated that this exception may arise from the difference in the structure of the particle aggregates. Rheograms relating r to /.Q? which are newly introduced in this study are shown to be quite effective in making clear the change of interactive forces in particles. Also, this method would be expected to be applicable for estimating rheological properties of suspensions in different suspending fluids. Hoffman [22] studied the discontinuous and dilatant viscosity behav-
60
Fig. 7. Rheograms for suspensions of unpurified bentonite; data given by Weymann et al. [7].
IO+
C N.m-*I
Fig. 8. Rheograms for suspensions of purified bentonite; data given by Weymann et al. [7].
ior in concentrated suspensions and concluded that the occurrence of the discontinuity point for suspensions in various suspending fluids can be correlated reasonably well by use of the parameter pa?. The dilatant and discontinuous behavior is generally observed in closely packed suspensions (see e.g. Sherman [23]). When they are sheared, the packing geometry becomes looser due to an initial increase in volume before the individual particles can move past one another. Considering that the dilatant flow is a destruction process of the suspension structure, the procedure described in this paper explains that in the case where there is no variation of particle-particle interaction in different suspending medium, the discontinuous and dilatant behavior occur at the same point of pOv. The effect of medium viscosity has also been discussed by using another scaling parameter, dimensionless shear stress, which governs the balance between Brownian and shearing forces [24,25]. This scaling law is successful for colloidal suspensions, but not applicable to flocculated suspensions or systems with large particles because of relatively weak Brownian motion. When Brownian force is negligible in comparison to shearing and interpartitle forces, the proposed scaling law in this paper is useful. 4. Conclusions On the basis of theoretical considerations that the hydrodynamic equilibrium is established between agglomeration growth and destruction at any shear rate, the rheological data previously published were analysed by introducing a parameter pOq which is considered to be proportional to the
183 hydrodynamic force. Rheograms plotting the relation between r and pap were shown to elucidate the effect of temperature on rheological properties and of suspensions. When there is no change of particle-particle particle-solvent interactions, a single master curve can be obtained and be utilized to estimate the viscosity of a suspension at other temperatures by using only solvent viscosity values. On the other hand, in the case where the interactive forces between particles vary with temperature the difference of the height of the rheograms shows the variation due to the change of temperature. This method can distinguish the effect of temperature on the interactive forces between particles from that of variation of viscosity of the suspension medium with temperature in the rheological properties. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
S.H. Maron and S.M. Fok, J. CoIIoid Sci., 10 (1955) 482. W.H. Fischer, W.H. Bauer and S.E. Wiberley, Trans. Sot. Rheol., 5 (1961) 221. R.S. Porter and J.F. Johnson, J. Appl. Phys., 34 (1963) 55. K. Umeya, Kagaku Kogaku, 31 (1967) 1147. V.V. Jinescu, Int. Chem. Eng., 14 (1974) 397. S.H. Maron and P.E. Pierce, J. CoIIoid Sci., 11 (1956) 80. H.D. Weymann, M.C. Chuang and R.A. Ross, Phys. Fluids, 16 (1973) 775. J.R. Thurgood, R.W. Hanks, G.E. Ostwald and E.L. Youngblood, A.I.Ch. E. J., 28 (1982) 111. K. Yamaguchi, M. Senna and H. Kuno, J. CoIloid Interface Sci., 70 (1979) 584. A.S. Michaels and J.C. Bolger, Ind. Eng. Chem. Fundam., 1 (1962) 153. B.A. Firth and R.J. Hunter, J. Colloid Interface Sci., 57 (1976) 248. S.V. Kao, L.E. Nielsen and C.T. Hill, J. CoIIoid Interface Sci., 53 (1975) 367. S.G. Mason, Pulp Paper Mag. Can., 49 (13) (1948) 99. I. Reich and R.D. Vold, J. Phys. Chem., 63 (1959) 1497. S.L. Goren, J. CoIloid Interface Sci., 36 (1971) 94. K. Yamaguchi, M. Senna and H. Kuno, J. ColIoid Interface Sci., 72 (1979) 170. D.G. Thomas, J. CoIloid Sci., 20 (1965) 267. S. Ohgaki and T. Matsuo, Doboku Gakkai Ronbunhokokushu (Trans. Civil Eng. Sot. Jpn.), 210 (1973) 21. B.J. Marlow and R.L. Rowe& 3rd Int. Symp. on Coal-Gil Mixture Combustion Proc., Vol. 2, CONF-810498, 1981. F.R. Graziano, R.E. Cohen and AI. Medalia, Rheol. Acta, 18 (1979) 640. J.D. Ferry, Viscoelastic Properties of Polymers 3rd edn., John Wiley & Sons, New York, 1980. R.L. Hoffman, J. Colloid Interface Sci., 46 (1974) 491. P. Sherman, Industrial Rheology, Academic Press, London, 1970, p.11. I.M. Krieger, Adv. CoIloid Interface Sci., 3 (1972) 111. W.B. Russel, J. Rheology, 24 (1980) 287.
175
EFFECT OF TEMPERATURE ON RHEOLOGICAL PROPERTIES OF SUSPENSIONS ATSUSHI TSUTSUMI and KUNIO YOSHIDA Department of Chemical Engineering, University of Tokyo, 7-3-l Hongo, Bunkyo-ku, Tokyo 1I3 (Japan) (Received November 14,1985; in revised form June 14,1987)
A parameter, the product of viscosity of the suspension medium and applied shear rate, is introduced to describe the effect of temperature on the rheological properties of suspensions, considering the mechanism of agglomeration in suspensions. It is found that, by plotting rheograms of the shear stress versus this parameter, a single master curve can be obtained independently of temperature. The dependence of viscosity on temperature and the flow of suspensions are shown to be well described.
1. Introduction Although the rheology of suspensions has been studied extensively, there has been little theoretical study of the effect of temperature upon the flow properties of dispersed systems. The temperature dependence of the viscosity of suspensionshas been studied by a number of investigators [l-4], most frequently by using an Arrhenius-type equation to treat the experimental data. The activation energy at constant shear rate (E+), and that at constant shear stress (ET) can both be determined. For non-Newtonian fluids, E? # E, also because, for the same temperature variation, different viscosity changes are obtained when the liquid undergoes a constant unit shear stress or a constant shear rate [4,5]. There have been many investigations that represent the parameters involved in the rheological equations as functions of temperature. Maron and Pierce [6] have applied the Powell-Eyring equation to the data on the flow behavior of latex, and examined the temperature dependence of the parameters involved in it. Weymann et al. [7] have presented a numerical analysis 0377-0257/87/$03.50
0 1987 Elsevier Science Publishers B.V.
176
for the viscosity of thixotropic clay-water suspensions and stated that two parameters in the assumed model were functions of temperature and concentration. Thurgood et al. [8] stated that, for coal-solvent slurries in a coal liquefaction preheater, both the consistency factor and the flow behavior index of the power-law equation are strongly dependent on temperature. Yamaguchi et al. [9] studied the effect of temperature on the flow properties of suspensions, and divided the total energy dissipation of flow into two terms, E, due to the interaction of flow units (floe), and E,, the viscous energy, by using energy-dissipation theory. By analysis of E, with respect to the shear rate and the effective floe concentration, they obtained a parameter which is independent of shear rate and of solids concentration, and they stated that this factor reflects the work required to separate two floes. Results from these studies, however, are not generally applicable to suspensions because of the unclear effect of particle-particle interaction on the parameters involved. The objective of this paper is to establish a method for understanding the temperature dependency of the viscosity of suspensions when considering the mechanism of agglomeration in suspensions. 2. Theoretical considerations The formation of agglomerates in a slurry takes place by some interaction between particles [lO,ll] or by liquid bridges [12] when an immiscible carrier liquid exists in the main suspending liquid. Then, the apparent viscosity of the slurry increases because the solvent is trapped into particle groups, resulting in the reduction of the effective volume concentration. In addition, a large amount of energy is dissipated due to attractive interactions between particles, resulting in an increase in apparent viscosity. In an agglomerated suspension undergoing shear with increasing shear rate, the coagulation rate increases due to the increase in collision probability between particles, while the agglomerates are subject to destruction by larger shearing forces into smaller groups. Mason [13] and Reich and Vold [14] studied the effects of agitation upon agglomerate size, and proposed that a dynamic equilibrium is established between agglomerate growth and destruction at any shear rate. When the coagulation rate is sufficiently rapid, there is the establishment of this equilibrium. High shear rates shift the equilibrium into the direction of better dispersion. However, the interactive forces between particles are as strong at high shear rates as they are at low shear rates, so even at higher shear rates the flow units may collide, but the newly-created couplet is soon destroyed by the hydrodynamic force which arises from the shear field acting on the couplet. The energy is then dissipated in this process. Thus, the
177 flow properties of suspensions depend upon the degree of agglomeration as a result of the dynamic equilibrium which is determined by the hydrodynamic force and interactive forces between particles. Hence, the rheological properties of suspensions must be examined by considering the hydrodynamic force between flow units as a parameter. Goren [15] estimated the maximum hydrodynamic force between two touching spheres in Couette flow as Fh = 6.12r/~,R~y, where p,, is the solvent viscosity, R is the radius of flow unit and T is the shear rate. It is shown in eqn. (1) that the hydrodynamic force per unit area is directly proportional to pOq. Apart from the dependence of temperature on the interactive force between particles, the degree of agglomeration seems not to vary at a constant ~~9 because it depends on the dynamic equilibrium. It can be considered that at a given shear rate an increase in temperature results in an increase in agglomeration because of the reduction of the hydrodynamic forces, and therefore the relative viscosities of suspensions increase with temperature. Consequently, without the dependence of temperature on the interactive forces between particles, one would expect that shear stress 7-T data on the same suspension at different temperatures would superimpose when plotted as r versus ~~9 and that only one master curve would be given. In case the interactive forces between particles vary with temperatures, the T-P~~ curves would not be superimposed because of the shifting of the dynamic equilibrium. It can be considered that the variation of the height of flow curves plotted as T versus pop at different temperatures would be attributable to that of interaction with temperature. Hence, in order to examine the flow properties of suspensions at different temperatures the value of pOq must be taken in the same range. 3. Results and discussion The flow curves for suspensions of CaCO, dispersed in ethylene glycol observed by Yamaguchi et al. [9] are shown in Fig. 1, the different curves being for suspensions of different temperatures. It is shown that the suspension is non-Newtonian, having taken on pseudoplastic characteristics. At a constant shear rate the apparent viscosity of the suspension decreases with temperature while the relative viscosity increases, as shown in Fig. 2. Such a change is caused by the growth of agglomeration, because hydrodynamic forces decrease with temperature. The relative viscosity is not a unique function of shear rate and the volume fraction. Let us attempt to re-plot these flow curves between r and 9 into the relation between r and pop, as shown in Fig. 3. It is noticeable that only one
178
0
300
600 3
IO
c s-11
Fig. 1. Flow curves for CaC03-ethylene
glycol suspensions given by Yamaguchi et al. [9].
master curve is obtained by introducing CL,,?.This fact supports the theory previously mentioned that by plotting T against pLoqthe flow curves should be superposed. This indicates that the interactive forces between particles are less dependent on the temperature, and that there is no variation of relative viscosity with temperature, which is represented by the ratio of ordinate to abscissa, at constant p,,?. Therefore, the viscosity of suspensions may be evaluated only from the viscosity of the solvent within a certain temperature range. Yamaguchi et al. [9,16] found that the effective volume fraction of floes obtained from Thomas’ equation [17] increased with tem-
Temperature
C ‘C 1
Fig. 2. Change of relative and apparent viscosity with temperature at constant shear rate for CaCO,-ethylene glycol suspensions. A, A, 84 s-‘; 0, m 164 s-‘; 0, l, 811 s-* and V, ethylene glycol. Open and solid marks show relative viscosity and apparent viscosity, respective.ly.
179
Au09 C Nem-*I Fig. 3. Rheograms between r and aof for CaCO,-ethylene
glycol suspensions.
perature. This is quite reasonable, because the lower hydrodynamic forces due to the increase of temperature should shift the equilibrium in the direction of better agglomeration. The data on kaolin suspension obtained by Ohgaki and Matsuo [18] are similarly replotted in the form of r vs. pOy, as shown in Fig. 4. In spite of high pseudoplasticity, superposition of rheograms is found by correlating into a single curve. In this manner, rheograms plotting T against p,+ are well superposed as long as there is no change in the interactive force between particles. Using the master curve thus obtained, the viscosity of the
0
0.5
1.0 J&O*
1.5
[ N.m-*I
Fig. 4. Rheograms for suspensions of kaolin; data given by Ohgaki and Matsuo [X3].
180
-0
1
2 409
C X10-3
3 N-m-*]
Fig. 5. Rheograms for coal-water slurries; data given
by Marlow
Rowell P91.
suspension can be estimated at any temperature, because the viscosity of the solvent is already known. At the same time, if a disagreement is found at a certain temperature in the relation between these two factors as shown in Fig. 5, it is considered that a change of the interactive force takes place. This figure shows the data on coal-water slurries by Marlow and Rowe11 [19]. Graziano et al. [20] studied the effect of temperature on rheological properties of concentrated suspensions of carbon black and found that increasing the temperature resulted in a progression from concave upward to concave downward flow curves in the plotting of 7 and p. They supposed that an increase in temperature corresponds to a shift to lower effective shear rates according to the concepts used in establishing time-rate-temperature reduced variables [21] and that a generalized Ostwald curve must be considered for any given material. Their supposition is confirmed by the trend shown &Pig. 6 in which is replotted the data of Graziano et al. in the relation between,\7 and CL,,?. It is shown that an increase in temperature reduces the hydroslynamic force, resulting in a shift to lower effective shear rates. In addition, ‘a generalized Ostwald curve is obtained, as far as the shape of the curves is concerned. Thus, within the same range of ~~9, rheological properties of suspension are determined at different temperatures. The height of curves decreases at higher temperature. This may be the result of an decrease in the interactive forces between particles with temperature or a disappearance of water bridges due to their evaporation. The rheograms replotting 7 against p,,y for the suspensions of unpurified bentonite are shown in Fig. 7, and those of purified bentonite in Fig. 8. Original data were obtained from the experiment of Weymann et al. [7]. It is interesting to note that only a single master curve is obtained at each
181
PO+ C N.m-*I Fig. 6. Rheograms for carbon black-naphthenic al. [20].
oil suspensions; data given by Graziano et
concentration except for the 4.26 wt.% purified bentonite slurry for which the height of the curves decreases with temperature. It is speculated that this exception may arise from the difference in the structure of the particle aggregates. Rheograms relating r to /.Q? which are newly introduced in this study are shown to be quite effective in making clear the change of interactive forces in particles. Also, this method would be expected to be applicable for estimating rheological properties of suspensions in different suspending fluids. Hoffman [22] studied the discontinuous and dilatant viscosity behav-
60
Fig. 7. Rheograms for suspensions of unpurified bentonite; data given by Weymann et al. [7].
IO+
C N.m-*I
Fig. 8. Rheograms for suspensions of purified bentonite; data given by Weymann et al. [7].
ior in concentrated suspensions and concluded that the occurrence of the discontinuity point for suspensions in various suspending fluids can be correlated reasonably well by use of the parameter pa?. The dilatant and discontinuous behavior is generally observed in closely packed suspensions (see e.g. Sherman [23]). When they are sheared, the packing geometry becomes looser due to an initial increase in volume before the individual particles can move past one another. Considering that the dilatant flow is a destruction process of the suspension structure, the procedure described in this paper explains that in the case where there is no variation of particle-particle interaction in different suspending medium, the discontinuous and dilatant behavior occur at the same point of pOv. The effect of medium viscosity has also been discussed by using another scaling parameter, dimensionless shear stress, which governs the balance between Brownian and shearing forces [24,25]. This scaling law is successful for colloidal suspensions, but not applicable to flocculated suspensions or systems with large particles because of relatively weak Brownian motion. When Brownian force is negligible in comparison to shearing and interpartitle forces, the proposed scaling law in this paper is useful. 4. Conclusions On the basis of theoretical considerations that the hydrodynamic equilibrium is established between agglomeration growth and destruction at any shear rate, the rheological data previously published were analysed by introducing a parameter pOq which is considered to be proportional to the
183 hydrodynamic force. Rheograms plotting the relation between r and pap were shown to elucidate the effect of temperature on rheological properties and of suspensions. When there is no change of particle-particle particle-solvent interactions, a single master curve can be obtained and be utilized to estimate the viscosity of a suspension at other temperatures by using only solvent viscosity values. On the other hand, in the case where the interactive forces between particles vary with temperature the difference of the height of the rheograms shows the variation due to the change of temperature. This method can distinguish the effect of temperature on the interactive forces between particles from that of variation of viscosity of the suspension medium with temperature in the rheological properties. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
S.H. Maron and S.M. Fok, J. CoIIoid Sci., 10 (1955) 482. W.H. Fischer, W.H. Bauer and S.E. Wiberley, Trans. Sot. Rheol., 5 (1961) 221. R.S. Porter and J.F. Johnson, J. Appl. Phys., 34 (1963) 55. K. Umeya, Kagaku Kogaku, 31 (1967) 1147. V.V. Jinescu, Int. Chem. Eng., 14 (1974) 397. S.H. Maron and P.E. Pierce, J. CoIIoid Sci., 11 (1956) 80. H.D. Weymann, M.C. Chuang and R.A. Ross, Phys. Fluids, 16 (1973) 775. J.R. Thurgood, R.W. Hanks, G.E. Ostwald and E.L. Youngblood, A.I.Ch. E. J., 28 (1982) 111. K. Yamaguchi, M. Senna and H. Kuno, J. CoIloid Interface Sci., 70 (1979) 584. A.S. Michaels and J.C. Bolger, Ind. Eng. Chem. Fundam., 1 (1962) 153. B.A. Firth and R.J. Hunter, J. Colloid Interface Sci., 57 (1976) 248. S.V. Kao, L.E. Nielsen and C.T. Hill, J. CoIIoid Interface Sci., 53 (1975) 367. S.G. Mason, Pulp Paper Mag. Can., 49 (13) (1948) 99. I. Reich and R.D. Vold, J. Phys. Chem., 63 (1959) 1497. S.L. Goren, J. CoIloid Interface Sci., 36 (1971) 94. K. Yamaguchi, M. Senna and H. Kuno, J. ColIoid Interface Sci., 72 (1979) 170. D.G. Thomas, J. CoIloid Sci., 20 (1965) 267. S. Ohgaki and T. Matsuo, Doboku Gakkai Ronbunhokokushu (Trans. Civil Eng. Sot. Jpn.), 210 (1973) 21. B.J. Marlow and R.L. Rowe& 3rd Int. Symp. on Coal-Gil Mixture Combustion Proc., Vol. 2, CONF-810498, 1981. F.R. Graziano, R.E. Cohen and AI. Medalia, Rheol. Acta, 18 (1979) 640. J.D. Ferry, Viscoelastic Properties of Polymers 3rd edn., John Wiley & Sons, New York, 1980. R.L. Hoffman, J. Colloid Interface Sci., 46 (1974) 491. P. Sherman, Industrial Rheology, Academic Press, London, 1970, p.11. I.M. Krieger, Adv. CoIloid Interface Sci., 3 (1972) 111. W.B. Russel, J. Rheology, 24 (1980) 287.