Variations of Viscosity

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977) 

Ec[uation (4.91) is written using the components of the rate of deformation tensor D as  

MODELLING OF STEADY-STATE VISCOMETRIC FLOW 127 in Cartesian x, y) coordinate system 

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The viscosity of a hydrocarbon mixture, as with all liquids, decreases when the temperature increases. The way in which lubricant viscosities vary with temperature is quite complex and, in fact, charts proposed by ASTM D 341 or by Groff (1961) (Figure 6.1) are used that provide a method to find the viscosity index for any lubricant system. Remember that a high viscosity index corresponds to small variation of viscosity between the low and high... 

Although Eq. (4.58) describes the variation of viscosity over a wide range of conditions quite well, it tends to break down toward lower temperatures, precisely where we wish to apply it. 

Fig. 8. Variation of viscosity with temperature for selected petroleum oils.

Fig. 9. Variation of viscosity with temperature of gas turbine fuels (14). V = specified limit for freezing (wax) point.

This diagram essentially reflects the main features of variation of viscosity properties of a liquid when solid particles of a filler is introduced into it. It remains to impart a quantitative form to these features. This form is based on a great number of theoretical and experimental papers, which made it possible to formulate fundamental concepts in this field. 

Experimental values of h for viscous oils are greater than those given by equation 9.81 for heating and less for cooling. This is due to the large variation of viscosity with temperature and the correction introduced for turbulent flow may also be used here, giving ... 

As we already know, the viscosity of oil near to the solid wall is limited to a hnite magnitude (whose value approaches the value of the relevant solid). Thus, the variation of viscosity along the direction normal to the wall plane can be described by a function as follows [31]... 

An effective viscosity rp has been introduced in the Reynolds equation to describe the non-Newtonian lubricant properties. Ignoring the variation of viscosity across the film thickness, one may evaluate the effective viscosity via the following rheological model that considers a possible shearthinning effect [19],... 

In liquid chromatography, equation (1.14) can be integrated directly, neglecting the variation of viscosity with pressure and the compressibility of the mobile phase... 

Transport Properties. Because the feed is primarily air and because substantial amounts of N2 and 02 are present in the effluent stream, we will assume that the fluid viscosity is that of air for purposes of pressure drop calculations. For the temperature range of interest, the fluid viscosity may be taken as equal to 320 micropoise. The pressure range of interest does not extend to levels where variations of viscosity with pressure need be considered. The effective diffusivities of naphthalene and phthalic anhydride in the catalyst pellet may be evaluated using the techniques developed in Section 12.2. 

Note. The variation of viscosity with presssure is small for most gases. For gases given on this page, the correction of viscosity for pressure is less than 10% for pressures up to 500 psi. 

On cooking maize starch the viscosity increases when the starch begins to gelatinise. As the temperature rises towards 95°C the viscosity falls. When the paste is cooled the viscosity rapidly increases. The variation of viscosity with temperature is characteristic for each different origin of starch. Potato starch, for example, has a lower gelatinisation temperature than maize starch but has a higher maximum viscosity. When cooled the viscosity of potato starch rises less. Once again amylopectin starches do not show this behaviour as they do not gel. 

The known information regarding the influence of various factors is given below. Most of the investigators have tried to study the effect of one variable at a time, but often minor variations of other variables also occur alongside. For example, variation of viscosity by the use of glycerol solution varies the surface tension by a few dynes per centimeter. 

Polyisobutylene. The solution viscosity of an irradiated polyisobutylene block was measured in CC14 at 30 °C. to determine the degree of degradation (5). The variation of viscosity-average molecular weight, Mv, with the dose, r, is shown in Figure 3. Nitrous oxide reduced the... 

Stress-strain relationship from a concentric-cylinder viscometer Capillary viscometers versus concentric-cylinder viscometers Inherent viscosity at low volume fractions Extent of hydration from intrinsic viscosity measurements Empirical determination of the Mark-Houwink coefficients Variation of viscosity with polymer configuration... 

Notice the very strong temperature dependence and the great variation of viscosities among different types of liquids. The inset shows the viscosity of water on a linear scale, emphasizing the strong temperature dependence. 

A detailed study of the viscosity dependence of the rate has been carried out for a large number of thermodynamic state points. A subsequent analysis reveals that over a large variation of viscosity, the rates can indeed be fitted well to Eq. (316) and the exponent a is found to depend strongly on the barrier frequency (ft) ). 

Figure 9.1 Variation of viscosity during the heating of a thermosetting polymer. 

Figure 2-13 Variation of viscosity with temperature and relationship to API gravity.

Figure 6-7 Variation of viscosity with asphaltene plus resin content of feedstocks.

Some fine review articles dealing with various aspects of photochromic polymers are reported in the literature,11-91 and several photoresponse effects have been described. These include light-induced conformational changes, photostimulated variations of viscosity and solubility, photocontrol of membrane functions, and photomechanical effects. Here we provide an overview of the photoresponse effects... 

As mentioned above, the viscosity is an important physical parameter. Table 14.7 presents comparatively the values recommended by the standards for diesel fuels in the USA and Europe. It can be observed that they are in a small range, but the upper limit is somewhat higher for biodiesel with respect to petrodiesel. Actually, the fatty esters exhibit larger variation of viscosity due to their chemical nature compared with hydrocarbon molecules. Influencing factors are chain length, position, number, and nature of double bonds, as well as the nature of the... 

Fig. 8. Variations of viscosity parameter with expansion factor. Experimental data of Schulz (229) (circles) and Krigbaum and Carpenter (146) (crosses) Theoretical curves according to Eqs. (55) and (56) for homogeneous polymers (broken curve and chain curve) and for broad fractions (dotted curve and full curve)...

Figure 9. Diagram of possible variation of viscosity of pyrolys-ing systems with temperature of pyrolysis. System A pyrolysis of coal to give coke of fine-grained mosaics, 1 pm diameter ...

An important dimensionless relationship between viscosity and concentration was found by Papkov et al. (1974) and reproduced in Fig. 16.30, where the variation of viscosity with polymer concentration for different molecular weights, expressed as intrinsic viscosities, is shown (left). The reduced viscosity t]/if vs. the reduced concentration c/c is shown on the right. The viscosity of the solution jumps down rapidly above the critical concentration as the nematic mesophase forms. The dimensionless relationship is remarkable. The relationship between the viscosity at the maximum and the intrinsic viscosity (see inset) appears to be r/max = 5.5b/]1 5, where rj is expressed in Ns/m2 and [77] in m3/kg. 

The electroviscous effects are observed as variations of viscosity upon application of outer electric fields, and as build-up of potential gradients upon flow of such fluids. See also -> electroconvection, electrorheological... 

The variation of viscosity with shear can be of three types ... 

Viscosity index, the customary basis for comparison of change of viscosity in hydrocarbon oils, becomes less satisfactory when applied to silicone oils, because the viscosity index varies with viscosity as well as with the temperature coefficient. In silicone oils the variation of viscosity with temperature is too small in relation to the viscosity itself. A true viscosity-temperature coefficient (VTC) has been proposed as a more satisfactory criterion [see Wilcock, Mechanical Engineering 66, 739 (1944)1. 

Experiments with different pressures in the chambers provide data for simultaneous determination of the diffusivities in and permeability B0 of a porous particle (Chapter 3). Allawi and Gunn [8] have described such experiments and the evaluation procedure, accounting for the variation of viscosity with composition. The permeability of the porous particle can also be found in separate experiments on single gas flow through the particle under a pressure difference. In this case the molar flux is given by the equation... 

One unique capability of the QCM-D technique is its successful extraction of quantitative information about a film s viscoelasticity. Figure 8.9 shows the variation of viscosity and shear elastic moduli of a pectin layer during pectin adsorption on the BSA surface from the Voigt model. It is noted that the shear elastic moduli is much... 

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