Fluid flow viscosity correction

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

A correction for fluid viscosity must be applied to the flow coefficient (Cv) for liquids other than water. This viscosity correction factor (Fv) is obtained from Fig. 10-23 by the following procedure, depending upon whether the objective is to find the valve size for a given Q and AP, to find Q for a given valve and AP, or to find AP for a given valve and Q. 

The z directed fluid velocity was determined by modeling the complete flow pattern in the fluid region and then defining a turnover parameter. The effective z velocity was assumed to be linearly related to this parameter times a relative viscosity correction. The details of these computations are presented in the Appendix. 

Stokes law is an analytic solution of the Navier-Stokes equation for the simplified flow case with solid particles and creeping flow. If the particles are fluid and in the absence of surface-active components, internal circulation inside the particle will reduce the drag. (Note that this is not necessarily valid for small fluid particles, but these are irrelevant in gravity separation.) The viscosity correction term for this case is given in Eq. (9). From this equation it can be seen that, for large viscosity differences between the dispersed and continuous phases, the settling will approach the Stokes velocity or 3/2 Stokes velocity (the two limiting... 

Here is fluid kinematic viscosity and t is the relaxation time for a single particle in an unbounded fluid. The first (linear) hydrodynamic drag force term in Equation 3.2 with F (( )) from Equation 7.2 approximately describes the hydraulic resistance of fine particles. This term behaves correctly at low concentrations and does not have singularities in the whole concentration range. The derivation of this term is commented on in more detail in reference [25]. The second (quadratic) term of Equation 3.2 describes the hydraulic resistance of large particles. The expression for Fj(( )) cited in Equation 7.2 follows from the model of jet flow around large particles in a concentrated disperse system. This expression was derived by Goldstik [38]. 

Based on this approach, the apparent viscosity of the polymer solution, uapp corrected if the apparent viscosity of the corresponding hypothetical Newtonian fluid flowing in the same capillary with the same total pressure drop is known. There are two procedures to determine the apparent viscosity of such a Newtonian fluid. The direct experimental procedure is to measure the apparent viscosity of the appropriate Newtonian fluid in the high-shear capillary viscometer. This experimental calibration technique was employed by Graham and co-workers (20). Although this experimental technique is direct, in practice it is difficult to perform. It is difficult to find a Newtonian fluid with the identical rheological properties as exhibited by the polymer solution at low-shear rates. 

Capillary viscometers are the most extensively used instruments for the measurement of viscosity of liquids because of their advantages of simphcity of construction and operation. Both absolute and relative instruments were constracted. The theory of these viscometers is based on the Hagen-Poiseuille equation that expresses the viscosity of a fluid flowing through a circular tube of radius r and length L in dependence of the pressure drop AP and volumetric flow rate Q, corrected by terms for the so-called kinetic-energy and end corrections ... 

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. 

Figure 3-56. Viscosity performance correction chart for centrifugal pumps. Note do not extrapolate. For centrifugal pumps only, not for axial or mixed flow. NPSH must be adequate. For Newtonian fluids only. For multistage pumps, use head per stage. (By permission. Hydraulic Institute Standards for Centrifugal, Rotary, and Reciprocating Pumps, 13th ed.. Hydraulic Institute, 1975.)...

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