Viscosity scaling parameters

Fio. 29. Average steady-state size of the dispersed phase at different viscosity ratios. The solid and dashed lines represent simulations in which and /xc are held constant. Other process parameters are the same as used for Fig. 28 (except 0 = 0.05). It is clear that the magnitudes of both viscosities must be considered rather than just the viscosity ratio. The lowest viscosity in each case is 1 Pa - s and the highest 1000 Pa - s. The curves are equally spaced on a logarithmic scale for viscosity. 

The periodicities are adimensionalized with the two scale parameters of the inner layer of the turbulent boundary, the kinematic viscosity v and the friction velocity uT, by the equation ... 

The reliable characterization of molar mass, as well an availability of a series of chemical identically polymer samples with different molar masses, remain indispensable prerequisites for an efficient determination of the scaling parameters of a macromolecule. Ultimately, the scaling parameters are required to understand the physical relationships that are valid for hb polymers, and which clearly differ from those of linear polymers, taking into account their strong influence on the material s properties. The scaling laws form the basis for any characterization of the global molecular parameters of polymers, and are based on their size (mostly expressed as the radius of gyration Rg) or on their intrinsic viscosity [/j] as a function of the molar mass M ... 

For a similar system, the shear viscosity was found to follow the power law model with yield (Pal et al. 1986). Owing to the presence of yield stress, the flow of concentrated emulsion was found to be facilitated by superposition of 10 Hz oscillation on the steady-state shear flow - up to 40 % energy saving was reported (Jezequel et al. 1985). More recently, the relative viscosity of emulsions was described in terms of scaling parameters (Pal 1997). Ten principal variables were incorporated into six dimensionless groups X, k, reduced time, h = t/(r n,dV8 kB T), relative density, = pd/pm> Peclet number, Pe = ti yd /SkeT, and Reynolds number. Re = p yd /4rin,. For the steady-state flow of well-stabilized emulsions, it was argued that the relative viscosity of emulsions should depend only on two... 

In some cases, when the model and prototype use the same fluid, it may be difficult to meet the Reynolds number criterion. For gas processes, the corresponding velocities in the model can induce compression effects in which case the Cauchy number becomes a scaling parameter. For liquids, where compression is negligible, maintaining high fluid velocity requires excessive power. In this case, using a fluid of lower kinematic viscosity in the model, while meeting the Reynolds number criterion, provides scalable information [4]. 

The slopes of the log rj versus log((pc plot yield s and /t parameters. The average values of ji and s in Equations 5.10 and 5.11 are 1.04 and 0.18. These values are fairly near to the t and s values obtained for conductivity percolation. The estimated scaling parameters which are in good agreement with the experimental values, obtained with both electrical conductivity and dynamic viscosity, signify that these microemulsion systems show an interdependence of the viscosity-conductivity, especially at the stage of water percolation. 

First it is assumed that the viscosity data of all binary mixtures from among the components of a multicomponent mixture are available at the temperature of interest. In this case the prediction of the viscosity of a multicomponent mixture proceeds from an analysis of the viscosity data for each binary mixture. The first step is to estimate the ratio of cross sections A, . It turns out that this ratio is remarkably insensitive to temperature, to the intermolecular pair potential chosen for its evaluation or to the occurrence of inelastic collisions (Maitland et al. 1987 Vesovic et al. 1995). Consequently, it may be estimated from calculations for any reasonable potential model or from the correlations of the extended law of corresponding states discussed elsewhere (see Chapter 11), once a scaling parameter for energy is available. If this parameter is not listed for the system of interest (Maitland et al. 1987), it may itself be estimated with sufficient accuracy using the combination rule... 

If the pure component viscosities are not available, the procedures recommended in Section 4.2 for their evaluation should be followed. This procedure will normally then lead to values of the scaling parameters for the pure gases for use in the procedures outlined above. 

The coefficients d( are listed in Table 14.10, and B, is in units of L- mol . For the sake of consistency, the scaling parameters and a used in equation (14.57) have been chosen to coincide with the ones used for the dilute-gas viscosity representation. 

Thermoformability is a property required by the many sheet materials used in the thermoforming industry. These properties are unique for the specific forming methods used, and are best determined by actual thermoforming tests on smaU-scale equipment. The softening or drape temperature of the material, residual stress in the sheet from its manufacture, and its melt strength and viscosity are important parameters relating to this use. 

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Micro-scale variables are involved when the particles, droplets, baffles, or fluid chimps are on the order of 100 [Lm or less. In this case, the critical parameters usually are power per unit volume, distribution of power per unit volume between the impeller and the rest of the tanh, rms velocity fluctuation, energy spectra, dissipation length, the smallest micro-scale eddy size for the particular power level, and viscosity of the fluid. 

The collection efficiency of wet scrubbers is dependent on parameters such as the size and quantity of liquid droplets, the liquid/gas ratio, high wa-ter-to-particle relative velocity, wettability of dust, particle density, gas viscosity, etc. For any specific application, the design procedure is to review operating data available from the technical literature or from manufacturers for similar applications. If data are not available, it may be necessary to per form pilot scale tests, which can be used for scale-up purposes. 

Figure 2. Scaled centerline velocity as a function of the Theile modulus (j), and the viscosity parameter, a, at a dimensionless length, = 3.0.

The ratio (p/G) has the units of time and is known as the elastic time constant, te, of the material. Little information exists in the published literature on the rheomechanical parameters, p, and G for biomaterials. An exception is red blood cells for which the shear modulus of elasticity and viscosity have been measured by using micro-pipette techniques 166,68,70,72]. The shear modulus of elasticity data is usually given in units of N m and is sometimes compared with the interfacial tension of liquids. However, these properties are not the same. Interfacial tension originates from an imbalance of surface forces whereas the shear modulus of elasticity is an interaction force closely related to the slope of the force-distance plot (Fig. 3). Typical reported values of the shear modulus of elasticity and viscosity of red blood cells are 6 x 10 N m and 10 Pa s respectively 1701. Red blood cells typically have a mean length scale of the order of 7 pm, thus G is of the order of 10 N m and the elastic time constant (p/G) is of the order of 10 s. 

Figure 9. Experimental data for the effective viscosity of the foam bubble regime in Berea sandstone as a function of the foam superficial velocity. The solid line is drawn according to the scaling theory with values of the two sets of parameters e and 6 listed.

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