Model measurements with Newtonian fluids

All models for turbulent flows are semiempirical in nature, so it is necessary to rely upon empirical observations (e.g., data) for a quantitative description of friction loss in such flows. For Newtonian fluids in long tubes, we have shown from dimensional analysis that the friction factor should be a unique function of the Reynolds number and the relative roughness of the tube wall. This result has been used to correlate a wide range of measurements for a range of tube sizes, with a variety of fluids, and for a wide range of flow rates in terms of a generalized plot of/ versus /VRe- with e/D as a parameter. This correlation, shown in Fig. 6-4, is called a Moody diagram. 

When must model experiments be carried out exclusively with the original material system Where the material model system is unavailable (e.g., in the case of non-Newtonian fluids) or where the relevant physical properties are unknown (e.g., foams, sludges, slimes), the model experiments must be carried out with the original material system. In this case measurements must be performed in models of various sizes (cf. Example 4). 

Scale Up. The design of pipelines and other equipment for handling non-Newtonian fluids may be based on model equations with parameters obtained on the basis of measurements with viscometers or with pipelines of substantial diameter. The shapes of plots of tw against y or 8V/D may reveal the appropriate model. Examples 6.9 and 6.10 are such analyses. 

Luciani et al. (1998) critically examined the experimental methods used for the measurements of the interfacial coefficient in polymer blends as well as the theoretical models for its evaluation. A new working relation was derived that makes it possible to compute the interfacial tension from the chemical structure of two polymers. The calculations involve the determination of the dispersive, polar, and hydrogen-bonding parts of the solubility parameter from the tabulated group and bond contributions. The computed values for 46 blends were found to follow the experimental ones with a reasonable scatter of +/— 36 %. The authors mentioned also that since many experimental techniques have been developed for low-viscosity Newtonian fluids, most were irrelevant to industrial polymeric systems. For their studies, two were selected capillary breakup method and a newly developed method based on the retraction rate of deformed drop. 

The Blake-Kozeny and modified Blake-Kozeny models have been verified by comparing observed pressure drops with computed pressure drops for flow of non-Newtonian fluids through packed beds. Agreement between predicted and measured pressure drops is within 20%. Thus, it is possible to predict flow behavior through some porous materials from rheological properties and characteristics of the porous media. 

Metzner and Otto (1957) developed the best-known definition of shear rate in an agitated vessel. They measured the power number for a variety of impellers in the laminar regime in Newtonian fluids and then repeated the measurements with shear-thinning fluids. They assumed that the power number was unaffected by the fluid s non-Newtonian behavior and that the Newtonian viscosity and shearthinning apparent viscosity were equal for equal power number and Reynolds number. Once an estimate of the apparent viscosity is made, eq. (9-22) can be rearranged and the shear rate can be calculated from the power law model ... 

The correlations represented by Eqs. 5.26a through 5.26e can be extended to interpolate for polymer concentrations between 1,000 and 2,000 ppm by use of a correlation based on the modified Blake-Kozeny model for the flow of non-Newtonian fluids. 62 Eq. 5.27 is an expression for A bk derived from the Blake-Kozeny model. Note that all parameters are either properties of the porous medium or rheological measurements. Eq. 5.27 underestimates A/ by about 50%. However, Hejri et al. 6 were able to correlate pBK and A for the unconsolidated sandpack data with Eq. 5.28. Eqs. 5.27 and 5.28, along with Eq. 5.24, predict polymer mobility for polymer concentrations ranging from l.,000 to 2,000 ppm within about 7%. 

There are two difficulties in estimating Because polymers are non-Newtonian fluids, the viscosity varies with shear rate, as discussed in Sec. 5.4,3. Viscosities can be easily measured at various shear rates with a standard viscometer. A relationship is needed between shear rate and frontal-advance rate to convert viscometric data to equivalent core data, Eq. 5.168 is an empirical expression for shear rate during flow in porous media. Although this model has been widely used for computations, it does not estimate shear rates correctly for most polymer/rock systems of practical interest. At best, Eq. 5.168 may be correlated against experimental data to find the shear rate that yields the apparent viscosity observed in the rock when the frontal-advance rate is specified. 

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