Newtonian friction factor

Edwards and Smith [10] suggested that the well-established Newtonian friction factor-Reynolds number relation can be used for non-Newtonian flow as... 

The transition from laminar to turbulent flow occurs at Reynolds numbers varying from ca 2000 for n > 1 to ca 5000 for n = 0.2. In the laminar region the Fanning friction factor (Fig. 2) is identical to that for Newtonian fluids. In the turbulent region the friction factor drops significantly with decreasing values of producing a family of curves. 

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length AP/L to the pipe diameter D, density p, and average velocity V through two dimensionless groups, the Fanning friction factor/and the Reynolds number Re. 

Scale reference, 16 Symbols, 19-22 Types, 4 Utility, 6, 11 Fluid flow, 52 Fluids, Newtonian, 52 Non-Newtonian, 52 Free air, 461 Friction factor, 55-132 Chart. 55 Fanning, 55 Friction factor, 68,... 

For a Newtonian fluid, the data for pressure drop may be represented on a pipe friction chart as a friction factor = (R/pu2) expressed as a function of Reynolds number Re = (udp/n). The friction factor is independent of the rheological properties of the fluid, but the Reynolds number involves the viscosity which, for a non-Newtonian fluid, is... 

Thus, the pipe friction chart for a Newtonian fluid (Figure 3.3) may be used for shearthinning power-law fluids if Remit is used in place of Re. In the turbulent region, the ordinate is equal to (R/pu2)n 0 fn5. For the streamline region the ordinate remains simply R/pu2, because Reme has been defined so that it shall be so (see equation 3.140). More recently, Irvine(25j has proposed an improved form of the modified Blasius equation which predicts the friction factor for inelastic shear-thinning polymer-solutions to within 7 per cent. 

Yoo, S, S.i Ph.D. Thesis, University of Illinois, Chicago (1974). Heat transfer and friction factors for non-Newtonian fluids in circular tubes. 

This is valid for any Newtonian fluid in any (circular) pipe of any size (scale) under given dynamic conditions (e.g., laminar or turbulent). Thus, if the values of jV3 (i.e., the Reynolds number 7VRe) and /V, (e/D) for an experimental model are identical to the values for a full-scale system, it follows that the value of N6 (the friction factor) must also be the same in the two systems. In such a case the model is said to be dynamically similar to the full-scale (field) system, and measurements of the variables in N6 can be translated (scaled) directly from the model to the field system. In other words, the equality between the groups /V3 (7VRc) and N (e/D) in the model and in the field is a necessary condition for the dynamic similarity of the two systems. 

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

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. 

This is the power law equivalent of the Hagen-Poiseuille equation. It can be written in dimensionless form by expressing the wall stress in terms of the friction factor using Eq. (6-5), solving for /, and equating the result to 16/ N t (i.e., the form of the Newtonian result). The result is an expression that is identical to the dimensionless Hagen-Poiseuille equation ... 

For the Bingham plastic, there is no abrupt transition from laminar to turbulent flow as is observed for Newtonian fluids. Instead, there is a gradual deviation from purely laminar flow to fully turbulent flow. For turbulent flow, the friction factor can be represented by the empirical expression of Darby and Melson (1981) [as modified by Darby et al. (1992)] ... 

The model for turbulent drag reduction developed by Darby and Chang (1984) and later modified by Darby and Pivsa-Art (1991) shows that for smooth tubes the friction factor versus Reynolds number relationship for Newtonian fluids (e.g., the Colebrook or Churchill equation) may also be used for drag-reducing flows, provided (1) the Reynolds number is defined with respect to the properties (e.g., viscosity) of the Newtonian solvent and (3) the Fanning friction factor is modified as follows ... 

These values can now be used to calculate the smooth pipe friction factor from Eq. (6-100). Excluding the vVDe term gives the friction factor for the Newtonian solvent (/s), and including the, VDe term gives the friction factor for the polymer solution (fp) under the same flow conditions ... 

The procedure is essentially identical to the one followed for the Newtonian fluid, except that Eq. (7-40) is used for the Reynolds number in step 2 and Eq. (6-44) is used for the pipe friction factor in step 3. 

Evaluation of each term in Eq. (15-51) is straightforward, except for the friction factor. One approach is to treat the two-phase mixture as a pseudo-single phase fluid, with appropriate properties. The friction factor is then found from the usual Newtonian methods (Moody diagram, Churchill equation, etc.) using an appropriate Reynolds number ... 

Although it is unnecessary to use the friction factor for laminar flow, exact solutions being available, it follows from equation 1.65 that for laminar flow of a Newtonian fluid in a pipe, the Fanning friction factor is given by... 

Friction factor chart for Newtonian fluids. (See Friction Factor Charts on page 349.)... 

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the generalized Reynolds number is used. 

On substituting for tw in equation 2.10, the Fanning friction factor for laminar non-Newtonian flow becomes... 

Figure 3.10 shows a friction factor - Reynolds number chart for Bingham plastics at various values of the Hedstrom number. The turbulent flow line is that for Newtonian behaviour and is followed by some Bingham plastics with low values of the yield stress [Thomas (1962)]. 

For Newtonian fluids, the Fanning friction factor / is usually defined (M4) ... 

Equation (11) states that the conventional Fanning friction factor, which may be used through Eq. (10) to calculate pipe-line pressure drops, is a unique function of two dimensionless groups for Bingham-plastic fluids. Newtonian fluids represent that special case for which r , and hence the second dimensionless group, is equal to zero. 

Fig. 5. Friction factor-Reynolds number diagram for non-Newtonians-low range. Taken from reference (Mil) with permission.

The utility of this generalized Reynolds number has been shown (Mil) by the correlation of all available literature data on flow of non-Newtonian fluids on the conventional friction factor-Reynolds number diagram which is reproduced in Figs. 5, 6, and 7. The curves shown are not drawn through the data points but rather represent the conventional... 

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