UDC 532.5:532.135
L.V. Ravichev, V.Ya. Loginov, A.V. Bespalov
STUDY OF THE VISCOSITY OF SUSPENSIONS OF SPHERICAL PARTICLES
A study has been carried out and a mathematical model has been proposed that makes it possible to determine the viscosity of suspensions of spherical particles with a diameter of 30 to 800 μm in the concentration range from 1 to 30% by volume with a different nature of their size distribution and at a shear rate from 0.1667 to 437.4 s"1 .
Mathematical model, polymer, suspension viscosity L.V. Ravichev, V.Y. Loginov, A.V. Bespalov RESEARCH OF VISCOSITY OF SUSPENSIONS OF SPHERICAL PARTICLES
Research is carried out and the mathematical model, allowing to calculate viscosity of suspensions of spherical particles in diameter from 30 to 800 microns in a range of concentration from 1 to 30% volume is offered at various character of their distribution in the sizes and at speed of shift from 0.1667 to 437.4 s"1.
Mathematical model, polymer, viscosity of suspensions
To effectively control the process of processing concentrated suspensions, it is necessary to know the dependence of the viscosity of such suspensions on temperature, shear rate, filler concentration, and filler particle size distribution. In addition, it has been experimentally established that by changing the fractional composition of the filler, it is possible to significantly increase the total filling in the suspension, while maintaining its viscosity at a level sufficient for processing into a finished product.
As a model mass for the study and mathematical modeling of the viscosity properties of suspensions, the following composition was used: "polymer (spherical solid particles) - glycerin ("inert" suspension medium)". To study the rheological properties of polymer suspensions in glycerol, five fractions of a spherical polymer with particle diameters of 30, 70, 150^200, 400^500, 700^800 µm were selected. The studies were carried out using a rotational viscometer "Reotest-2".
Instead of the effective viscosity n, it is more convenient to use the concept of relative viscosity Potn = Ts/Tssr, where wsr is the viscosity of the suspension medium.
The use of relative viscosity makes it possible to compare the results of experiments carried out at different temperatures. We write the equation for the dependence of the viscosity of glycerol on temperature in the form:
Vcp = a -10-8 ■ exp ^b j, (1)
where the coefficients a = 1.07979, b = 6069.70 determine the dependence of the viscosity of glycerin on temperature.
The system of equations for the mathematical description of the viscosity properties of a suspension of solid particles in an "inert" medium can be written in general form as follows:
P-I (Per, Kvz, Fm, F) 'Per - 1 (T) ' K vz \u003d 1 (] '¥) ' Fm \u003d 1 (yA ¥, ■>) ' (2)
where A - fractional composition of the filler;) - shear rate; Kvz - coefficient taking into account the interaction of solid particles with the suspension medium and among themselves; d is the particle diameter; T - temperature; F is the volume concentration of the filler; Fm is the maximum volumetric concentration of the filler; n is the effective viscosity of the suspension; psr - effective viscosity of the suspension medium; - shape factor (for a ball y = 1, for non-spherical particles
shapes 0< 1^< 1).
As the main equation for calculating the viscosity of the system "spherical solid particles -" inert "suspension medium", the Mooney equation was chosen, proposed for calculating the viscosity of concentrated suspensions and giving good agreement with experimental data:
P-PSR exp
An analysis of the Mooney equation shows that the viscosity of suspensions is largely determined by the maximum concentration of the filler Fm. The larger the value of Fm, i.e. the denser the particles of the suspension can be packed, the less viscosity the entire system will have at a given concentration Ф, or at high concentrations of the filler the suspension will retain the ability to flow. In this regard, the maximum concentration of the filler Fm acquires fundamental importance for characterizing the technological qualities of the suspension and predicting its rheological properties.
The maximum filler concentration can be expressed in terms of the porosity of a layer containing the same particles and in the same ratio as the suspension:
Fm -1 ~£, (4)
where B is the porosity of the layer of suspension particles - the proportion of voids in the layer formed from the particles of the suspension, with their most dense packing. Often expressed in terms of the coefficient of porosity n:
which is the ratio of the volume of voids in the layer to the volume of particles.
The paper presents the ratios that allow calculating the porosity of a polyfractional mixture, if the coefficients of porosity of the fractions u, constituting the polyfractional mixture, the equivalent particle diameters of the fractions yr, and the volume fraction of the fractions xr (fractional composition) are known:
ir+) „ % ■ (1 + 2%) % ■ (3 + %)
W th, ’ K ": w, ■ (1 + 2 w,) + (1 - w,) 2’ Ki, ‘ w, ■ (3+ w,) + (1 - w (6)
A, \u003d K "p ', \u003d K2, ■ ", +1) -1, i \u003d 1, 2, ..., M - 1, \u003d 1, 2, ..., M - i
Az \u003d E (x, ,), i \u003d 2, 3, ..., M, (7)
A4 - 2 (X A2y-1), * = 1, 2, ..., M - 1, (8)
P - A3 + x* n° + A4, r = 1, 2, ..., M, (9)
As the actual value of the coefficient of porosity, the maximum value of pg is taken, which allows, taking into account relations (5, 4), to calculate the maximum
filler concentration.
On fig. Figure 1 shows the dependence of the relative viscosity of monofractional suspensions of a spherical polymer on the volume concentration of the solid phase. On fig. It can be seen from Fig. 1 that the viscosity of suspensions depends not only on the concentration of the solid phase, but also on the diameter of the particles of the suspension, and this is most pronounced in the region of low shear rates (Fig. 2). A sharp increase in viscosity is observed for suspensions containing particles with a diameter of less than 100 μm.
Die 500 1Meter] 70; ; D- e1 part *> - 1 700-80 itz: 50; □ 0 µm O o "* SP ■ 1-" 7
Rice. Fig. 1. Dependence of the relative viscosity of suspensions of spherical polymer particles of various diameters on the concentration of the solid phase. Shear rate 1 s-1
Rice. 2. Dependence of the relative viscosity of suspensions of spherical polymer particles of different diameters on the shear rate. Suspension concentration 30% vol
An analysis of the results of our own experimental studies and published experimental data of other researchers shows that the relative viscosity of spherical polymer suspensions depends not only on the maximum concentration of the filler, but also significantly depends on the particle size and decreases sharply when the particle diameter is less than 100 μm.
A review of the literature experimental data on the porosity and maximum concentration of the filler (various materials: steel, quartz sand, MaCl, glass, titanium dioxide, cellulose nitrate, pyrocollodium, titanium; particle shape: spherical, cylindrical, cubic, angular, acute-angled) showed that that Fm depends on the particle size and sharply decreases when the equivalent particle diameter jeq is less than 100 μm (Fig. 3). For particles with a diameter of more than 100 µm, the average Fm value is 0.614, for particles with a diameter of less than 100 µm, the maximum concentration of the filler depends significantly on the particle diameter.
The analysis of the experimental data (Fig. 3) shows that this dependence is well approximated by an equation of the form
FM \u003d Vo + B1 + B2 ■ -GG '(10)
where B0 = 0.6137; Vx = - 4.970; B2 = 18.930.
Based on the results of our own experimental studies of the viscosity of mono- and polyfractional suspensions of a spherical polymer in glycerol, Kvs values were found in the shear rate range of 0.1667^437.4 s-1. The obtained Kvz values fit into one generalizing dependence (Fig. 4). It is characteristic that the extrapolation of the obtained dependence
into the region of infinitely small shear rates gives the value of the interaction coefficient close to 2.5. those. to the value defined by Einstein.
^(Jeq), µm
Rice. 3. Dependence of the maximum filler concentration on the equivalent diameter
Rice. 4. Dependence of the interaction coefficient on the shear rate
Di e 500 ameter] ■70; ; D- "s frequent O- 1 700-80 itz: 50; □ 0 microns) 30; - 40
Particle diameters: C80; n>-70! A 160: p - 400-:
5 (Yu; d- 700-80) microns
Rice. Fig. 5. Dependence of the relative viscosity of suspensions of spherical polymer particles of different diameters on the shear rate. The concentration of suspensions is 30% vol. Experimental points are given. Dashed line - model calculation
Rice. Fig. 6. Dependence of the relative viscosity of three-fraction suspensions of a spherical polymer on the shear rate. Concentration - 30% vol. Experimental points are given. Dashed line - model calculation
The dependence Kvz = /(/£(/)) is well approximated by an equation of the form:
Kvz \u003d a + ax + ax2 + ax3,
where x = ^(]); a0 = 2.344; a1 = 0.290; a2 = 0.204; a3 = 0.067.
Thus, finally, the system of equations for the mathematical description of the viscosity properties of a suspension of spherical particles takes the form:
where m is the number of fractions of filler particles.
A comparison of the experimental and calculated values of the viscosity of suspensions of mono- and polyfractional suspensions of spherical polymer particles in glycerin shows their good agreement (Fig. 5, 6).
It should be noted that the model obtained makes it possible to calculate the viscosity of suspensions not only in the case when the filler is spherical particles, but also when the filler is particles of irregular shape. In this case, the equivalent particle diameter is calculated, which is defined as the diameter of a sphere having the same volume as the given particle.
The developed mathematical model makes it possible to calculate the viscosity of suspensions containing spherical particles of various fractional compositions (diameter from 30 to 800 μm) in a wide range of shear rates (from 0.1667 to 437.4 s-1) and concentrations of solid particles from 1 to 30% about. with different character of their size distribution.
1. Mooney M. The viscosity of concentrated suspensions of spherical particles // Journal of Colloid Science. 1951.V.6. No. 2. R.162.
2. Smith T.L., Bruce C.A. The viscosity of concentrated suspensions // J. Colloid and Interface Sci.1979.V.72. No. 1. P.13.
3. Wickovski A., Strk F. Porovatosc cial sypkich. Mieszaniny wieloskladnikowe // Cem. sto-sow. A. 1966. 4B. S. 431-447.
4. Ravichev L.V., Loginov V.Ya., Bespalov A.V. Modeling of viscosity properties of concentrated suspensions // Theoretical foundations of chemical technology.. 2008. V.42. No. 3. S. 326-335.
5. Einstein A. Uber die von der molekularkinetischen Theorie der Warme geforderte
Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen // Annalen der Physik. 1905, 322(8). P.549-560.
Ravichev Leonid Vladimirovich -
Candidate of Technical Sciences, Associate Professor of the Department of Management of Technological Innovations of the Russian Chemical Technical University named after. D.I. Mendeleev
Loginov Vladimir Yakovlevich -
Candidate of Technical Sciences, Programmer of the Department of Licensing and Accreditation of Educational Programs of the Russian Chemical-Technological University. DI. Mendeleev
Bespalov Alexander Valentinovich -
Doctor of Technical Sciences, Professor of the Department of General Chemical Technology of the Russian Chemical-Technological University. DI. Mendeleev
ANNOTATION
The rheological properties (density and viscosity) of carbonate-containing suspensions formed during the conversion of calcium nitrate with ammonium carbonate were studied at the ratio of circulating solution (CR) : CaCO 3 - 3: 1 ÷ 8: 1, temperature - 20-60 ° C and N a: N n (ratio of ammonia nitrogen to nitrate) - 0.2 ÷ 1.0. It has been established that with an increase in the ratios of CR: CaCO 3 , N a: N n and temperature, the density of the suspension decreases monotonously and rectilinearly. At a concentration of the sum of salts of 30% with an increase in temperature from 20 to 60 ° C, CR: CaCO 3 3: 1 and 4: 1 and N a: N n from 0.2 to 1.0, the density of the suspension decreases from 1.458, 1.447 to 1.293 , 1.272 g/cm 3 and from 1.429, 1.420 to 1.272, 1.249 g/cm 3 respectively. With similar changes in the parameters, the values of the viscosities of the suspensions also decrease. With variation of the listed parameters, the values of the density of the suspension fluctuate in the ranges of 1.272-1.415 and 1.368-1.502 g/cm %.
ABSTRACT
In this study the rheological properties (density and viscosity) of carbonate containing suspension forming during the conversion process of calcium nitrate into ammonium carbonate under the recycling solution ratio (RS):CaCO 3 equal to 3: 1 ÷ 8: 1, temperature - 20 -60°С to N a: N n (the ratio of ammoniac nitrogen to nitrate) is 0.2 ÷ 1.0., have been studied. It has been established that with increasing RS:CaCO 3 , N a: N n and temperature, the suspension’s density is decreased monotonous and straight line. When concentration 30% of salt sum, suspension' density is reduced from 1.458, 1.447 to 1.293, 1.272 g/cm 3 and from 1.429, 1.420 to 1.272, 1.249 g/cm 3 , respectively with temperature rise from 20 to 60°C , RS: CaCO 3 equal to 3: 1 and 4:1, N a: N n is from 0.2 to 1.0. The suspension's viscosity is decreased under the analogous changes of the parameters. With variation of these noted parameters of the suspension's density is varied in a range 1.272-1.415 and 1.368-1.502 g/cm 3 , and the viscosity is 1.30-3.78 and 1.49-7.09 centipoise, respectively with 30-50% of salt's total.
As rheological properties let recommending transportation of formed suspension with existing pumping devises without some limitation that is the most important during the treatment in building chalk and ammonium nitrate.
Introduction
A progressive direction in the processing of low-grade phosphate raw materials is the use of the nitric acid decomposition method, which allows the use of nitric acid not only as a means for converting insoluble phosphates into a soluble form, but also as an additional source of nutrients.
One of the most important issues in the production of phosphorus-containing fertilizers by the method of nitric acid decomposition of phosphorites of the Central Kyzylkum is the optimal choice of a method for processing a by-product - calcium nitrate. The resulting calcium nitrate has unsatisfactory physical and chemical properties, contains about 10-11% nitrogen, i.e. not suitable for use as a fertilizer. In the scientific and technical literature, there are several ways to process this product, of which the most practical interest is the conversion of calcium nitrate to calcium carbonate and ammonium nitrate using ammonium carbonate.
Previously, we have shown that during repulpation and ammonization of nitric acid extract at the stage of nitric acid decomposition of pretreated phosphorites, a suspension is formed, consisting of solutions of ammonium and calcium nitrate (ANC) and precipitate. The first part of the filtrate formed during the separation of the precipitate is returned to the stage of repulpation of the nitric acid pulp as a circulating solution of ammonium and calcium nitrate. The second part of the NAA solution can be processed using various methods:
1. Direct evaporation of the solution to obtain nitrogen-calcium fertilizer, however, this results in a product with unsatisfactory physical and chemical properties.
2. Conversion of calcium nitrate with ammonium carbonate solution to obtain ammonium nitrate and calcium carbonate according to the following equation:
Ca (NO 3) 2 + (NH 4) 2 CO 3 \u003d CaCO 3 + 2NH 4 NO 3
After separation of the sludge, the filtrate containing ammonium nitrate is sent to the pre-treatment of low-grade phosphorites as a circulating solution. The filtered washed chalk can be used as a building material or a highly effective soil liming agent.
It should be noted that during the conversion of calcium nitrate in the circulating solution, a carbonate-containing suspension is formed with different ratios of CR: CaCO 3 . In this regard, we studied the rheological properties (density and viscosity) of carbonate-containing suspensions.
experimental part
The density of the studied pulps was measured by the pycnometric method (measurement accuracy up to 10 −5 g/cm³), viscosity was measured using a VPZh-1 glass capillary viscometer (measurement accuracy ±0.1%) at temperatures of 20, 30, 40, 50, and 60°C (temperature determination accuracy ± 0.1°С). To measure the density and viscosity of suspensions, the ratio of CR: CaCO 3 varied over a wide range from 3: 1 to 8: 1 and the ratio of N a: N n (ratio of ammonia nitrogen to nitrate) in solution - from 0.2 to 1.0.
Research results
As can be seen from the experimental data (Tables 1 and 2), with an increase in the ratios of CR: CaCO 3 , N a: N n and temperature, the density of the suspension decreases. For example, with CR: CaCO 3 3: 1 and 4: 1 with an increase in N a: N n from 0.2 to 1.0, the density of the suspension decreases from 1.458, 1.447 to 1.293, 1.272 g / cm 3 at 20 ° C and from 1.429, 1.420 to 1.272, 1.249 g/cm 3 at 60°C, respectively, at a total salt concentration of 30% (Table 1). The same pattern is observed at a total salt concentration of 50% and ranges from 1.585, 1.568 to 1.386, 1.369 g/cm3 at 20°C and from 1.562, 1.543 to 1.368, 1.349 g/cm3 at 60°C, respectively (Table 2 ). With an increase in the proportion of calcium nitrates, i.e. with a decrease in the ratio of N a: N n, the density increases by 1.14 and 1.16 times, respectively, at a ratio of CR: CaCO 3 = 3:1 and 8:1 (temperature 20°C). An increase in the ratios of CR: CaCO 3 , N a: N n and the effect of temperature on the viscosity of the suspension is leveled (Tables 3 and 4), which is associated with a high fluidity of the ammonium nitrate solution.
With variation of the listed parameters, the density of the suspension changes in the ranges of 1.272-1.415 and 1.368-1.502 g/cm 3 , the viscosity - in the ranges of 1.30-3.78 and 1.49-7.09 cP, respectively, with a sum of salts of 30-50%.
Table 1.
Density of carbonate suspension (NAC concentration -30%)
CR ratio: CaCO 3 | NA: Nn | ° WITH |
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Table 2.
Density of carbonate suspension (NAC concentration -50%)
Ratio CR: CaCO 3 | NA: Nn | Density (g / cm 3), at temperatures,° WITH |
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Table 3
Viscosity of carbonate suspension (NAC concentration -30%)
CR ratio: CaCO 3 | NA: Nn | ° WITH |
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Table 4
Viscosity of carbonate-containing suspension (NAC concentration -50%)
Ratio CR: CaCO 3 | NA: Nn | Viscosity (cP), at temperatures,° WITH |
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Conclusion
Thus, these studies indicate that the rheological properties of the suspension formed during the conversion of calcium nitrate with ammonium carbonate are quite acceptable for technological purposes, i.e. suspensions are fluid and can be easily transported by existing pumping devices without any restrictions.
Bibliography:
1. Allamuratova A.Zh., Erkaev A.U. Enrichment of high-carbonate phosphorites of the Central Kyzylkum with the help of solutions of nitrate salts // DAN Academy of Sciences of the Republic of Uzbekistan. - Tashkent, 2010. - No. 5. - S. 57-60.
2. Allamuratova A.Zh., Erkaev A.U. Technological control of the main parameters of processing of low-grade phosphorites of the Central Kyzylkum // Chemical technology. Control and management.-2010. No. 6. pp.19-23.
3. Allamuratova A.Zh., Erkaev A.U., Toirov Z.K., Reimov A.M. Technological studies of obtaining phosphorus-containing fertilizers from nitric acid extract of phosphorites of the Central Kyzylkum // Chemical industry. - St. Petersburg, 2011. - v.88. - No. 3. - S. 109-114.
4. A.S. 538990 USSR. Method for producing calcium carbonate / Goldinov A.L., Novoselov F.I., Abramov O.B., Bevzenko I.I., Tereshchenko L.Ya., Tyurin E.I., Loginov N.D., Afanasenko B. P. // No. 2029236/26; Claimed 01/31/74; Published 12/15/76. B.I. - 1976. - No. 46.
5. Goldinov A.L., Novoselov F.I., Abramov O.B., Afanasenko B.P. Two-stage method for the conversion of calcium nitrate to calcium carbonate and ammonium nitrate // Chemical industry. - 1981. - No. 2. - S. 32-33.
6. Pozin M.E. Technology of mineral fertilizers. / L.: Chemistry. - 1989. - P.340.
7. Solubility of CaCO3 in aqueous solutions of ammonium nitrate // Proceedings of GIAP.-Issue 31. -1975. - P.5-8.
8. Fridman S.D., Skum L.S., Polyakov N.N., Belyaeva N.N., Kirindasova R.Ya. Conversion of calcium nitrate to calcium carbonate and ammonium nitrate. // Proceedings of GIAP. Chemistry and technology of nitrogen fertilizers. Release. 31. - M., 1975. - S. 8-11.
9. Allamuratova A. J., Erkaev A. U., Reymov A. M. Conversion of calcium nitrate solution obtained from Kyzylkum phosphorite with ammonium carbonate // American Chemical Science Journal. – 2016. Vol. 16(4). P.1-6.
The invention can be used in alumina production, hydrometallurgical production, mining industry, etc. The method consists in measuring the viscosity of the liquid phase μ w and the suspension μ c at different shear rates S i and maintaining temperature control on at least three suspensions of different solids content (1-ε). Produce a graphical construction of functional dependencies μ zhi =ft and μ ci =fS i , (1-ε), determination of the coefficients, the value of the solid content (1-ε), as well as the values of viscosity μ si according to the established equation. The technical result of the invention is to improve the accuracy of measurements. 2 ill., 1 tab.
The invention relates to methods for determining the viscosity and rheological characteristics of Newtonian and non-Newtonian liquid media - suspensions and can be used in alumina production, hydrometallurgical production, mining industry, etc.
A known method for measuring the viscosity of a liquid medium according to the author's certificate of the USSR No. 371478, which consists in the successive passage of the liquid through two capillary tubes of the same diameter, but of different lengths, measuring the pressure drop and flow rate of the liquid, by which the viscosity value is calculated. In this way, it is possible to determine only the viscosity of the conveyed medium without measuring the shear rate, which affects the viscosity.
More perfect in relation to the above is the method for determining the rheological characteristics of viscoplastic media according to USSR author's certificate No. 520537 on a 3-channel capillary viscometer by pumping the test medium through three different systems of capillaries equipped with capillary tubes of different lengths and diameters, with measuring the pressure drop along the length capillaries of the same diameter and liquid flow.
This method makes it possible, using three parallel measurements, to calculate the pressure loss due to friction in capillary tubes of different lengths and diameters, and from these data to determine the values of the viscosity of the medium under study and the shear stress.
Disadvantages of the method: the bulkiness of the device, the need to equip the viscometer with an additional system for supplying the test medium, the inevitable measurement errors associated with the loss of pressure at the inlet to each capillary. In the case of studies on dilute aqueous suspensions with a rapidly separating solid phase in laminar flow, sedimentation is possible on horizontal capillary tubes, which will lead to additional measurement errors.
There is a simpler method for determining the viscosity of suspensions [A.N. Planovsky, V.N. Ramm, S.Z. Kagan. Processes and apparatuses of chemical technology. Goshimizdat, M., 1962, p. 294], which includes measuring the viscosity of the liquid phase, corresponding to the temperature of the suspension, and the solid content in the suspension, in which the viscosity of the suspension is determined by the empirical equation:
μ c \u003d μ f,
where μ W - viscosity coefficient of the liquid phase, cP,
ε is the proportion of the liquid phase per unit volume of the suspension, d.u.,
4.5 - reduction factor.
The main disadvantage of the method is that it does not take into account the effect of the speed of movement of the suspension. For Newtonian liquid media, as the speed of movement increases, the value of the coefficient μ c increases, and for non-Newtonian, on the contrary, it decreases. Therefore, the above equation is not suitable for determining the viscosity of suspensions, in which, during their movement - mixing or pumping - the thixotropy inherent in non-Newtonian media is manifested.
The last of the considered methods, as the closest in essence to the claimed, is taken as a prototype.
The objective of the invention is to take into account the shear rate of the suspension when determining its viscosity using a standard viscometer, which can measure the shear rate and temperature control of the stirred suspension, which will improve the accuracy of determining the viscosity of the suspension.
The technical result is achieved by the fact that the method for determining the viscosity of the suspension includes measuring the viscosity of the liquid phase μ w and suspension μ s at different shear rates S i and maintaining temperature control on at least three suspensions of different solid content (1-ε), graphical construction of functional dependencies μ zhi \u003d ft and μ ci \u003d fS i , (1-ε), determination of the coefficients of the value of the solid content (1-ε) and the viscosity values \u200b\u200bof μ ci according to the equation:
where t is the suspension temperature, °С,
Coefficient taking into account the influence of the relative shear rate and solid content on the change in the structure of the suspension and (1-ε),
K t - temperature coefficient (K t \u003d 1 at t≤60 ° C, K t \u003d 1.07 at t \u003d 61-90 ° C),
K OS - reduction factor (K OS ≠1, 10).
Studies of the rheological characteristics of suspensions were carried out on a rotational viscometer of the Brookfield system (Brookfield 2005 Catalog. Viscometers, Rheometers; Texture Analyzers for Laboratory and Process Applications). On this device, the viscosity is determined by measuring the torque that occurs on the spindle shaft immersed in the investigated medium - suspension. During measurements, it is possible to change the spindle speed (n sp) by switching the toggle switch, as well as select the spindle diameter (d sp). The suspension is placed in a thermostated beaker with a slightly larger diameter (D st) and, if necessary, mixed in the beaker with a magnetic stirrer. Spindle speed is converted to shear rate (S) using the formula:
where r w, R st - the radii of the spindle and the glass, respectively.
To determine the coefficients included in the equation for determining the viscosity, measurements are performed when changing the parameters of the suspension: the content of solid T/L or (1-ε), μ W and temperature t, as well as S (at least 3 measurements on each parameter).
On the example of suspensions of red mud with T/W=1.2; 1.0; 0.5 and 0.33 (1-ε=0.257; 0.224; 0.126 and 0.087, respectively), and the concentration of the solution for Na 2 O=2.5 g/l and Al 2 O 3 =2 g/l, thermostated at t =25-60°C and 90°C (μ W =0.7 and 0.4, respectively) dynamic viscosity coefficients μ ci were measured on a rotational viscometer at shear rates S=0.8-1.61-4 s -1 (the mode corresponds to the movement of the suspension in the thickener), S=8.05-16.6-34.7 s -1 (with stirring in chain mixers) and S=80.8-159 s -1 (with hydrotransport in the pipe).
The measurement results μ ci are presented in figure 1 in the form of a functional dependence μ ci =fT/W for the above values of S i , t and μ W:
S=0.8-4 s -1 , curve 1 (t≤60°C), curve 2 (t=90°C),
S=8.05-34.7 s -1 , curve 3 (t≤60°C), curve 4 (t=90°C),
S=80.8-159 s -1 , curve 5 (t≤60°C), curve 6 (t=90°C).
| Table | |||||
| Estimated values of the coefficients included in the equation | |||||
| Parameter name | The value of the coefficients, c.u. | ||||
| S/W (1-ε) | 1,2 (0,257) | 1,0 (0,224) | 0,5 (0,126) | 0,33 (0,087) | |
| K S1 | t, °C | μ f | S=0.8-4.0 s -1 (during thickening) | ||
| 60 | 0,7 | 4,3 | 4,24 | 4,18 | 4,12 |
| 90 | 0,4 | ||||
| K S2 | 60 | 0,7 | S=8.05-34.7 s -1 (with stirring) | ||
| 4,04 | 3,93 | 3,77 | 3,56 | ||
| 90 | 0,4 | ||||
| K S3 | 60 | 0,7 | S=80.8-159 s -1 (for hydrotransport) | ||
| 3,96 | 3,71 | 3,23 | 3,01 | ||
| 90 | 0,4 | ||||
| K os =14 around, K t =1 at t≤60°C, K t =1.07 at t=61-90°C around |
To find the intermediate values of the coefficients K S built according to the table graph in figure 2 dependencies K S =fS for:
1. T/W=1.2 or (1-ε)=0.257,
4. 0,33 (0,087).
The suitability of the equation using the coefficients of the table was checked on the example of the calculation below.
Example. In a suspension of red mud, the viscosity μ С =3000 cP was measured on a rotational viscometer at a shear rate S=1.61 s -1 , the content of solid T/L=0.33 or (1-ε)=0.087 and the phase), for which the value of μ W =0.7 at a temperature of 25°C. Substituting the values of the coefficients from the table corresponding to the measurement conditions, we determine the calculated value of the viscosity of this suspension according to the equation proposed above:
μ ci \u003d 0.7 (1 + 0.087 14 4.12 1) \u003d 0.7 + 0.061 14 4.12;
lg(μ ci -0.7)=lg0.061+4.12 lg14=-1.215+4.12 1.146=-1.215+4.72=3.505;
μ ci \u003d 0.7 + 3200 \u003d 3200.7 cP.
According to the viscometer μ s \u003d 3000 cP. Therefore, the relative measurement error will be:
Δ max =(3200.7-3000) 100/3000=6.69%.
On a thicker suspension with T/W=1.2, the measured value of viscosity, other things being equal, was 12000 cP, and the calculated value was 12284 cP, for which Δ min = 2.37%.
Thus, the calculation error according to the equation was in the range of 2.4-6.7%, which is quite acceptable when measuring this type of suspension with a rotational viscometer.
A method for determining the viscosity of a suspension, including measuring the viscosity of the liquid phase μ w and suspension μ c at different shear rates S i and maintaining temperature control on at least three suspensions of different solid content (1-ε), graphical plotting of functional dependencies μ w i \u003d ft and μ ci =fS i , (1-ε), definition of coefficients 
, solids content (1-ε) and viscosity values μ ci according to the equation
where t is the temperature of the suspension;
Coefficient taking into account the influence of the relative shear rate and solids content on the change in the structure of the suspension and (1-ε);
K t - temperature coefficient (K t \u003d 1 at t≤60 ° C, K t \u003d 1.07 at t \u003d 61-90 ° C),
*r - emulsion density; cream fat density 0.8887 g / cm 3
503. The table below shows the results of a study of the viscosity of suspensions of glass beads (average diameter 65 μm) in an aqueous solution of zinc iodide (a composition that prevented sedimentation of the beads during measurements):
Use these data to plot the reduced viscosity versus the latex volume fraction and determine the intrinsic viscosity [at f ® 0]. Is it equal to the theoretical coefficient of Einstein's equation for suspensions?
505. Calculate the molar mass of polyvinyl alcohol using the viscometric method: intrinsic viscosity 0.15 m 3 /kg, constants of the Mark-Houwink equation K MH = 4.53 × 10 –5 L/g and a = 0.74.
506. Calculate the molar mass of ethylcellulose in aniline using the experimental data of the viscometric method (constants: K MH = 6.9 × 10 –5 L/g, a = 0.72):
507. For several fractions of cellulose nitrate in acetone, the viscosity was measured at 25 °C and the intrinsic viscosities were calculated:
Calculate the coefficients of the Mark-Houwink equation for this system.
508. The table below shows the results of viscometric measurements of solutions of poly(g-benzyl-L-glutamate) in dimethylformamide. Determine the coefficients of the Mark-Houwink equation from them.
509. The table below shows the results of viscometric measurements of solutions of several polystyrene fractions in methyl ethyl ketone at 22 °C:
Find the coefficients of the Mark-Houwink equation for the given system.
510. For several preparations of polycaprolactam, molar masses were established and the intrinsic viscosities of their solutions in m-cresol at 25 °C were determined:
Use these data to find the coefficients of the Mark-Houwink equation for the polycaprolactam/m-cresol system.
511. Calculate the molar mass of polyvinyl acetate in acetone using the data of the viscometric method (the Mark-Houwink equation constants K MH = 4.2 × 10 –5 L/g, a = 0.68):
512. Determine the intrinsic viscosity and Huggins coefficient for poly(g-benzyl-L-glutamate) in chloroform from the following data:
513. Determine the coefficients of the Mark-Houwink equation for polyisobutylene in cyclohexane at 30 °C using the following data:
514. Solutions of several samples of polypropiolactone in trifluoroethanol (TFE) were studied viscometrically at 25 °C and the following dependence of intrinsic viscosity on molar mass was obtained:
515.
| With, weight % | 20.0 | 16.0 | 12.0 | 8.0 | 4.0 | |
| r*, g/cm3 | 0.970 | 0.975 | 0.979 | 0.983 | 0.988 | 0.993 |
| h, cps | 0.986 | 0.857 | 0.697 | 0.612 | 0.532 | 0.476 |
*r - emulsion density; cream fat density 0.8887 g/cm3)
Build on these data a graph of the dependence of the reduced viscosity on the volume fraction of fat and determine the intrinsic viscosity [at f ® 0]. Is it equal to the theoretical coefficient of Einstein's equation for suspensions?
516. Solutions of several samples of polypropiolactone in chloroform (trichloromethane, CHCl 3) were studied viscometrically at 30 °C and the following dependence of intrinsic viscosity on molar mass was obtained:
Calculate the coefficients of the Mark-Houwink equation.
517. It has been established that at 20 °C the relationship between the intrinsic viscosity of a polyisobutylene solution and its molar mass M is described by the formula [h] (l/g) = 3.60×10 –4 × M 0.64 . Determine the molar mass of the polyisobutylene fraction in the solution, the intrinsic viscosity of which is 1.80 m 3 /kg.
518. Measurements of the inherent viscosity of solutions of several fractions of polyisobutylene with known molar masses in diisobutylene led to the following results:
Calculate the coefficients of the Mark-Houwink equation.
519. Calculate the molar mass of polystyrene from the intrinsic viscosity of its solution of 0.105 l/g. Solvent - toluene; constants of the Mark–Houwink equation for given conditions: K MH = 1.7 × 10 –5 L/g, a = 0.69.
520. Calculate the molar mass of polyvinyl acetate in benzene, if the intrinsic viscosity of its solution is 0.225 l/g, the constants of the Mark–Houwink equation K MH = 5.7 × 10 –5 L/g and a = 0.70.
522. Determine the molar mass of polyvinyl acetate in chloroform using the following data: [h] = 0.340 L/g, Mark–Houwink equation constants K MH = 6.5 × 10 –5 L/g and a = 0.71.
521. The table below shows the results of measurements of the viscosity of mixtures of cream with skimmed (skimmed) milk and distilled water as a function of fat concentration at 64 ° C
| With, weight % | 20.0 | 16.0 | 12.0 | 8.0 | 4.0 | |
| r*, g/cm3 | 1.021 | 1.029 | 1.037 | 1.045 | 1.053 | 1.061 |
| h, cps | 2.506 | 2.047 | 1.739 | 1.490 | 1.270 | 1.134 |
*r - emulsion density, cream fat density 0.8887 g/cm 3)
Build on these data a graph of the dependence of the reduced viscosity on the volume fraction of fat and determine the intrinsic viscosity [at f ® 0]. Is it equal to the theoretical coefficient of Einstein's equation for suspensions?
523. Determine the molar mass of nitrocellulose if the intrinsic viscosity of its solution in acetone is 0.204 m 3 /kg, the constants of the Mark-Houwink equation K MH = 0.89 × 10 –5 L/g and a = 0.9.
524. For solutions of several samples of polypropiolactone in butyl chloride, the following dependence of intrinsic viscosity on molar mass at 13 °C was obtained:
Calculate the coefficients of the Mark-Houwink equation.
525. Determine the molar mass of ethylcellulose in toluene using the data of the viscometric method (constants: K MH = 11.8 × 10 –5 L/g, a = 0.666):
526. At 25 °C, the intrinsic viscosity of solutions in tetrahydrofuran of several fractions of polystyrene with known molar masses was determined:
Calculate the coefficients of the Mark-Houwink equation.
Appendix 1. Units of measurement of physical quantities
A physical quantity is a product of a numerical value (number) and a unit of measure. In SI (official name : Le System International d "Unites) seven basic units of measurement and two additional ones are defined (Table 1.1). All other physical quantities are derived from the main ones using multiplication or division in accordance with physical laws (formulas). For example, the linear speed of movement is determined by the equation v= dl/d t. It has a dimension (length/time) and an SI unit (derived from the base SI units) m/s. Some of the derived units have their own names and designations (Table 1.2).
For convenient handling of large or small numerical values, the SI uses standard decimal prefixes that define multiple and submultiple decimal derivatives. (The most commonly used ones are listed in Table 1.3). For example, 1 nanometer (denoted as 1 nm) means 10–9 of a meter, that is, 1 nm = 10–9 m. 1 millipascal (1 MPa) means 10–3 pascals. The basic unit of mass “kilogram” already has the prefix kilo-. In this case, any other decimal derivatives are derived from the decimal derivative of "grams". For example, 1 milligram, 1 mg, means 10 -3 g or 10 -6 kg. (The gram is the basic unit of mass in the cgs and the SI decimal submultiple). If a mathematical operation is performed on a unit of measure with a decimal prefix, such as exponentiation, then the action applies to the entire designation. For example, 1 dm 3 means 1 (dm) 3 , but not 1 d (m) 3 .
Table 1.1 Basic and additional SI units
* SI definition: " A mole is the amount of a substance that contains as many named units as there are atoms in 0.012 kg of the 12 C isotope."In other words, a mole is the amount of a substance that contains N A (Avogadro number) units of matter, which must be clearly stated. For example, formula units AlCl 3 , 1/3AlCl 3 , ions, electrons, micelles, particles of lyophobic sol, aerosol, emulsion, etc. can be considered as units of a substance.
Tab. 1.2 Some derived SI units with their own names
| magnitude | SI unit | expression through other units. SI | ||
| Name | designation | main | other derivatives | |
| electric potential, emf, voltage, | volt | IN | kg × m 2 / (A × s 3) | J/C; W/A; |
| power | watt | Tue | m 2 × kg / s 3 | j/s |
| frequency | hertz | Hz | s -1 | |
| energy, work, heat | joule | J | kg × m 2 / s 2 | N m, Pa m 3 , V K |
| amount of electricity | pendant | cl | s×A | J/V |
| force | newton | H | kg×m/s 2 | j/m; Pa × m 2; C×W/m |
| electrical resistance | ohm | Ohm | kg × m 2 / (A 2 × s 3) | B/A |
| pressure | pascal | Pa | kg / (m × s 2) | N/m 2 ; J / m 3 |
| electrical conductivity | Siemens | Cm | A 2 × s 3 / (kg × m 2) | A/B; Ohm -1; f/s |
| email capacity | farad | F | A 2 × s 4 / (kg × m 2) | CL/V; CL 2 /J; J/W 2 |
Table 1.3 Some decimal (multiple and multiple) prefixes to SI units
According to SI grammatical rules, the decimal prefix symbol and the initial unit symbol are written together and are not accompanied by a dot as an indication of the name abbreviation, but the punctuation mark must be present if required by the grammatical rules of the text in which the symbol occurs. For example, if the designation centimeter, cm, is at the end of the sentence, then the period should be as usual, cm .
The product of two different units can be written in the following three ways (using viscosity as an example): Pa×s, Pa·s, Pa s (with a space between the factors). The ratio of two units can be written either as a fraction (for example, N/m) or as a product in three ways: N×m–1, Nm–1, and Nm–1. The ratio of three or more units of measurement must be written in accordance with the usual rules of mathematics (three-story fractions are not allowed, the denominator must be clearly defined, using brackets if necessary).
SI is the recommended and most convenient system of units in theoretical calculations and in communications (transmission of information) in the field of exact sciences. However, in many particular cases it is convenient to use other units of measurement. For example, in experimental studies using high pressures, it is convenient to use the unit of measurement “bar”, and when using a vacuum, “millimeter of mercury” (similar to how when calculating a person’s age, not seconds or gigaseconds, but years are used, while for similar purposes centuries are applied in social history). According to the SI definitions, some of these units are allowed for “temporary” use, and are actually used (see Table 1.4). Many units from past practice are not recommended for use and, in fact, are hardly used in modern measurements, but they are also useful to know, since many sources of information (encyclopedias, reference books, other publications) use them. For example, in most reference books on physical chemistry, the viscosity of liquids is indicated in centipoise, and not in SI units Pa s. The most important of these units are listed in Table. 1.5.
Tab. 1.4 Units of measurement not included in the SI, but used along with SI units
| magnitude | Name | designation | conversion to SI |
| time | minute | min | 60 s |
| hour | h | 3600 s | |
| day | day | 86400 s | |
| pressure | bar | bar | 10 5 Pa |
| length | angström | Å | 10–10 m, 0.1 nm |
| weight | atomic mass unit | a.u.m. | 1.66054×10 -27 kg |
| dalton | Da | 1.66054×10 -27 kg | |
| ton | T | 10 3 kg | |
| volume | liter | l | 10 -3 m 3, 1 dm 3 |
| milliliter | ml | 10 -6 m 3, 1 cm 3 | |
| temperature | degrees Celsius | °С | (T– 273.15) K |
| flat corner | degree | ° | (p/180) glad |
| minute | ¢ | (p/10800) rad | |
| second | ² | (p/648000) rad | |
| energy | electron-volt | eV | 1.60219×10 -19 J |
Tab. 1.5 Some units of measurement used in physical chemistry in the past and not included in the SI
To convert physical quantities from one unit of measurement to another, remember the definition : a physical quantity is the product of a number and a unit of measurement. It is recommended to take this definition literally and treat physical quantities according to the usual rules of mathematics. Let's look at some examples.
Example 1 Calculate how many meters are in 2 µm (micrometer).
Let's represent the length l = 2 µm as l = 2×µm (although this notation is not accepted). Referring to Table. 1.3 we learn that the prefix "mk" means micro-, a factor of 10 -6. Therefore, we write l = 2×μm = 2×(10 –6 ×m) = 2×10 –6 ×m. Thus, 2 µm = 2×10 -6 m (two micrometers contains 2×10 -6 m).
Example 2 Calculate how much m 3 is contained in 2 dm 3.
Let's imagine the volume V\u003d 2 dm 3 as V\u003d 2 × dm 3. According to the table. 1.3, the prefix "d" means "deci-", a factor of 10 -1. Therefore, we can write 2 × dm 3 = 2 × (10 -1 × m) 3 = 2 × 10 -3 × m 3 = 0.002 × m 3. That is, 2 dm 3 \u003d 0.002 m 3 (2 dm 3 contains 0.002 m 3).
Example 3 The concentration given is 2 g/L. Express it in kg / m 3.
From Table. 1.3, we learn that the “kilo-” unit of mass measurement “kilogram” means a multiplier of 10 3, that is, 1 kg = 10 3 g or 1 × kg = 10 3 × g. Solving the last equation for "g", we get 1×g = 10 –3 × kg. On the other hand, from Table 1.4 it follows that 1 l \u003d 10 -3 m 3. Therefore, the following transformations can be made:
2 g/l = = =
Thus, 2 g / l \u003d 2 kg / m 3.
Example 4 express pressure R= 2 kPa in atmospheres.
From Table. 1.5 it follows that 1 atm = 101325 Pa, and from table. 1.3 - that the prefix "k" (kilo-) means a factor of 10 3 . Thus, R\u003d 2 × kPa \u003d 2 × 10 3 × Pa, that is R\u003d 2 × 10 3 Pa. Dividing both sides of the equation (1×atm = 101325×Pa) by 101325, we find 1×Pa = 9.8692×10–6×atm. Substitute this quantity into the equation for R :
R\u003d 2 × 10 3 × Pa = 2 × 10 3 × (1 × Pa) \u003d 2 × 10 3 × (9.8692 × 10 -6 × atm) \u003d 1.9738 × 10 -2 atm.
Mixtures consisting of one substance in the form of small solid, liquid or gaseous particles, dispersed randomly in another liquid substance, are quite common in nature and in industry. The term "suspension" usually refers to a system of small solid particles in a liquid, although from a dynamic point of view the nature of both media is of little importance, and we will use this term also for a system of solid particles in a gas, a system of droplets of one liquid dispersed either in another liquids (emulsions), either in gas, and systems of gas bubbles in liquid. It is interesting to find out how such suspensions will behave when the boundaries move and forces are applied. If the characteristic scale length of the suspension motion is large compared to the average distance between the particles, and we will assume that this is the case, then the suspension can be considered as a homogeneous liquid with mechanical properties,
different from the properties of the surrounding liquid in which these particles are suspended. The random distribution of spherical particles does not have any property depending on the direction of motion in the medium (particles in the form of long rods can create such properties due to their tendency to be located in a certain direction relative to the local velocity distribution, although the Brownian motion of suspended particles tends to exclude any such preferential direction ). Therefore, if the environment is a "Newtonian" homogeneous fluid, then an equivalent suspension of approximately spherical particles is also Newtonian and is characterized by shear viscosity (and possibly also bulk viscosity).
In this section, we will determine the effective viscosity of an incompressible fluid containing suspended particles of such small linear dimensions that a) the influence of gravity and inertia on the motion of the particle is not taken into account (therefore, the particle moves locally together with the surrounding fluid) and b) the Reynolds number of the perturbed motion, arising due to the presence of one particle is small compared to unity. Let us assume for the sake of simplicity that the particles have a spherical shape; in the case of small-radius liquid or gaseous particles, surface tension tends to keep the particles spherical despite the deforming effect of fluid motion, so the shape assumption is only needed for solid particles. Finally, we will assume that the suspensions are so diluted that the average distance between the particles is large compared to their linear dimensions.
Under these conditions, the main motion of the surrounding fluid, which is superimposed by the perturbed flow created by the presence of one particle in it, can be considered to consist of uniform translational, rotational, and purely deformation motions. The particle moves forward and rotates together with the fluid surrounding it, so that the perturbation is associated only with a purely deformation motion (shear). The perturbation of the deformation motion arising from the particle is apparently inevitably accompanied by an increase in the total dissipation rate, and the effective viscosity of the suspension (shear or bulk) must be greater than the viscosity of the fluid surrounding it; we will see later that this is the case.
To begin with, assume that the particles are incompressible, so the slurry also behaves like an incompressible medium, and only the effective shear viscosity value needs to be determined. This requires an explicit representation of the perturbed flow created by a single incompressible particle, and therefore we will consider the corresponding flow problem with negligibly small inertial forces.
