Calculation of Viscosity

The viscosity of the polymer fluid can be calculated from Equations 8.32 and 8.34  

For polymer fluids that follow the power law relationship, Equation 8.40 can be rearranged to  

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Interpretation of data obtained under the conditions of uniaxial extension of filled polymers presents a severe methodical problem. Calculation of viscosity of viscoelastic media during extension in general is related to certain problems caused by the necessity to separate the total deformation into elastic and plastic components [1]. The difficulties increase upon a transition to filled polymers which have a yield stress. The problem on the role and value of a yield stress, measured at uniaxial extension, was discussed above. Here we briefly regard the data concerning longitudinal viscosity. 

Densities required for the calculation of viscosity were obtained in three ways. The densities of the salt solutions were measured by Archimedean displacement (9) of a borosilicate glass bob weighed in air, water, and the solvents and solutions. The results were expressed as linear functions of the molar concentration of salt, and the slopes obtained are recorded in Table II. Densities of the solvent mixtures were taken from published tables (10,11). Densities of the stock acetone and methanol were measured by a conventional pycnometer to greater precision than the salt solutions, and they were compared with the literature values for analysis and with the values obtained by displacement as a check. The two methods agreed within 0.0005 g/ml. 

Let us now turn to the calculation of viscosity in an inhomogeneous solvent. This calculation is different from (almost inverse of) diffusion. The viscosity of the solvent is the weighted average of the local viscosity in the different domains. Hence it can be written as... 

In the petroleum industry a dimensionless number termed the viscosity index has been used to describe the temperature dependence of a fluid s kinematic viscosity. The calculation of viscosity index involves the use of published look-up tables [388], In terms of relative changes, a higher viscosity index represents a smaller decrease in viscosity with increasing temperature. 

Newtonian and non-Newtonian calibration fluids were used to determine the necessary calibration constants for the impeller method. It has been previously determined that the impeller method is only valid for a Reynolds number (Re) <10. Impeller rotational speed and torque data from Newtonian calibration fluids of known viscosity were employed to determine the Newtonian calibration constant, c. Cone-and plate-viscometer data from non-Newtonian calibration fluids were used to determine a viscosity vs shear rate relationship. Impeller rotational speed and torque data of the non-Newtonian calibration fluids combined with a determined viscosity vs shear rate correlation were utilized to calculate the shear rate constant, k. The impeller method calibration constants allow the calculation of viscosity, shear rate, and shear stress data of non-Newtonian suspensions. Metz et al. (2) have thoroughly discussed the equations utilized in the impeller method. 

The calculation of viscosities of electrolyte mixtures can be accomplished with the method of Andrade (see Ref. [40]) extended with the electrolyte correction by Jones-Dole [44]. First, the pure component viscosities of molecular species are determined by the three-parametric Andrade equation, which allows a mixing rule to be applied and the mixture viscosity of an electrolyte-free liquid phase to be obtained. The latter is transformed into the viscosity of the liquid phase using the electrolyte correction term of Jones and Dole [44], whereas the ionic mobility and conductivity are used as model parameters. 

The critical analysis of the results on foam rheology, proposed by Heller and Kuntamukkula [16], has shown that in most of the experiments the structural viscosity depends on the geometrical parameters of the device used to study foam flow. This means that incorrect data about flow regime and boundary conditions, created at the tube and capillary walls, etc., are introduced in the calculation of viscosity (slip or zero flow rate). Most unclear remains the problem of the effect of the kind of surfactant and its surface properties on foam viscosity and on the regime of foam flow (cross section rate profile and condition of inhibition of motion at the wall surface). 

Formally more strict is the approach to the calculation of viscosity increase based on the growth of molecular mass. For linear polymers viscosity is connected with molecular mass by the following relation [21] ... 

Equation 3B.16 is the basis for calculation of viscosity of a Newtonian fluid using glass capillary viscometer. Itshouldalso be recognized that(4g/jrro) = ilQ/nEf) gives the shear rate for Newtonian fluids but not for non-Newtonian fluids and it is called pseudo shear rate. Additional steps are required to obtain an expression for the true shear rate. 

The value of k may be measured experimentally for a given emulsion. The emulsion degradation rate decreases as pipe diameter increases. A conservative assumption for calculation of viscosities for pipeline design is that the degradation rate is proportional to the surface-to-volume ratio (1/d), where d is the pipe diameter. 

Vasu (1972a,b) used the results of the above-mentioned works for the calculation of contact correlation function in molten salts on the basis of a double hard core model, which describes better the real situation in molten salts. This contact correlation function was applied in the calculation of viscosity and electrical conductivity of molten alkali... 

I.6. Calculation of viscosity from the measured time intervals... 

If it is assumed that the local velocity gradient decreases linearly perpendicular to the interface (Newtonian profile at each point X), we can compare the average three-dimensional viscosity for a film thickness of E = 50 A and E = 5 X 106 A. In the former case we assume that the monolayer and oriented substrate are only 50 A thick the latter number is based on a much larger oriented subphase (9,15). In the first case the calculated viscosity varies between 7.07 X 10"5 and 5.41 X 10"4 cps in the second case it varies from 54 to 8 cps (with increasing distance from the barrier). From the calculation of viscosity, the assumption that E = 5 X 106 A is more reasonable. 

Computer) Repeat the calculation of viscosity of PD3 shown in Figure 3-19 using the data PIB-Rel-1. TXT or PIB-Rel-2. TXT (either one) in the CD. Use both a KWW and an exponential function to extrapolate data to long times. Comment on differences, if any between these two methods. 

Because neat solvent viscosity is so important in regulating viscosity of resin solutions, the calculation of viscosity of blends of solvents is a useful tool in formulating solvent systems for coatings. As part of a computer program to formulate solvent blends. Nelson et al. (41) calculated viscosity of blends by use of the following equation ... 

ASTM Method D 2501-66, Calculation of Viscosity-Gravity Constant (I GC) of Petroleum Oils, ASTM Standards Book, Part 17—Petroleum Products, American Society for Testing and Materials, Philadelphia. 

ASTM D2501, Standard Test Method for Calculation of Viscosity-Gravity Constant of Petroleum, ASTM Annual Book of Standards, vol. 05.01 (West Consho-hocken, PA American Society for Testing and Materials). 

The viscosity of a vegetable oil depends on its chemical composition (iodine value and saponification value) and the temperature of measurement. Equations have been derived which permit the calculation of viscosity from the knowledge of the other three parameters. ... 

Empirical expressions for calculation of viscosity of the system of a given type, which permit more precise calculations, are given in monograph. ... 

Calculation of viscosity profile ( .g., with Arrhenius equation) Determination of the impregnation performance... 

When the property calculated is a single particle property, such as the velocity autocorrelation function, or the particle s mean squared displacement, averaging over the particles does help a lot. Averaging over even a moderate number of particles of about 1000 decreases the error by more than an order of magnitude. Unfortunately, this is not possible for all properties. For instance, the calculation of viscosity calls for the stress correlation function [Eq. (67)], which is not a single particle property. This is why self-diffusion coefficients are usually estimated with much better accuracy than viscosity. 

Early calculations of viscosity (and also of self-diffusion, D) of liquid metals employed the concept of an activation energy experimental data were often compared to Andrade s Equation,... 

This energy-based method of calculation of viscosity rj is due to Einstein [87], who considered hydrod)mamic dissipation in a very dilute suspension of non-interacting spheres. Tanaka and White [86] base their calculations on the Frankel and Acrivos [88] cell model of a concentrated suspension, but use a non-Newtonian (power law) matrix. The interaction energy is considered to consist of both van der Waals-London attractive forces and Coulombic interaction, i.e. 

To determine the thermal conductivity of liquid Freon-20 at elevated pressures, the use of the generalized equation (0.46) is recommended. The coefficients of this equation are determined in Refs. [0.13, 1.4] from the experimental data for Freon-21, -22, and -23 and are given in Table 5. The generalized equations obtained in Refs. [1.3, 1.4] are suitable for the calculation of viscosity and thermal conductivity of liquid freons of the methane series at pressures up to 50-60 MPa. 

The thermodynamic tables for gaseous and liquid Freon-21 were computed from the equations indicated in Sec. 2.2 and cover the region T = 303-473 K and p = 0.01-20 MPa (Tables 22 and 23). The system of equations given in Sec. 2.3 for the calculation of viscosity and thermal conductivity of Freon-21 is applicable to all the experimentally investigated regions of state. But the accurate values of thermodynamic functions (p and c ), which are essential for the determination of V, a, and Pr are available only at pressures up to 20 MPa and at temperatures higher than NBP- Therefore, the recommended tables of transport properties (Tables 24 and 25) cover the same region of state as in thermodynamic tables. 

For the calculation of viscosity of gaseous and liquid Freon-22 at elevated... 

It is also useful to elaborate on the forms of the calculation equations. In several references (for example, [3.6, 3.20, 3.29]) for the calculation of viscosity of compressed Freon-22, equations of type (0.33) are recommmended. These equations were derived by using the experimental data of C. T. Butierskaya. In a later investigation, Eq. (0.33) at = 4 was used to calculate tables. However, at low temperatures the tabular data were obtained, not by analytical means, but on the basis of graphical analysis of experimental data. 

Calculation of Viscosity-Gravity Constant (VGC) of Petroleum Oils ... 

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