Frictional factor parameter

In Chap. 9 we shall discuss in considerable detail a parameter called the molecular friction factor f. For velocities that are not too great, the friction factor expresses the proportionality between the frictional force a particle experiences and its velocity ... 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

Equation (2.56) not only enables us to understand the basis for the first-power dependence of rj on M, but also presents us with a new and important theoretical parameter, the segmental friction factor. We shall see in the next chapter that it is a quantity which can also be extracted from measurements of the viscoelasticity of polymers. 

Another parameter that plays an important role in unifying viscosity, diffusion, and sedimentation is the friction factor. This proportionality factor between velocity and the force of frictional resistance was introduced in Chap. 2, and its role in interrelating the topics of this chapter is reflected in the title of the chapter. 

Thus, in laminar flow with fluid slip, the friction factor is a function of not only the Reynolds number Re, but also the non-dimensional parameter jx/a ). If the flow does not exhibit fluid slip, Eq. (3.43) gives A = 64/Re on substituting /3 —> <=o into the equation. 

In experiments of flow and heat transfer in micro-channels, some parameters, such as the Reynolds number, heat transfer coefficient, and Nusselt number, are difficult to obtain with high accuracy. The channel hydraulic diameter measurement error may play a very important role in the uncertainty of the friction factor (Hetsroni... 

In Table 6.7, C is the Martinelli-Chisholm constant, / is the friction factor, /f is the friction factor based on local liquid flow rate, / is the friction factor based on total flow rate as a liquid, G is the mass velocity in the micro-channel, L is the length of micro-channel, P is the pressure, AP is the pressure drop, Ptp,a is the acceleration component of two-phase pressure drop, APtp f is the frictional component of two-phase pressure drop, v is the specific volume, JCe is the thermodynamic equilibrium quality, Xvt is the Martinelli parameter based on laminar liquid-turbulent vapor flow, Xvv is the Martinelli parameter based on laminar liquid-laminar vapor flow, a is the void fraction, ji is the viscosity, p is the density, is the two-phase frictional... 

In this table the parameters are defined as follows Bo is the boiling number, d i is the hydraulic diameter, / is the friction factor, h is the local heat transfer coefficient, k is the thermal conductivity, Nu is the Nusselt number, Pr is the Prandtl number, q is the heat flux, v is the specific volume, X is the Martinelli parameter, Xvt is the Martinelli parameter for laminar liquid-turbulent vapor flow, Xw is the Martinelli parameter for laminar liquid-laminar vapor flow, Xq is thermodynamic equilibrium quality, z is the streamwise coordinate, fi is the viscosity, p is the density, <7 is the surface tension the subscripts are L for saturated fluid, LG for property difference between saturated vapor and saturated liquid, G for saturated vapor, sp for singlephase, and tp for two-phase. 

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. 

The Fanning friction factor/is a function of the Reynolds number Re and the roughness of the pipe e. Table 4-1 provides values of e for various types of clean pipe. Figure 4-7 is a plot of the Fanning friction factor versus Reynolds number with the pipe roughness, eld, as a parameter. For laminar flow the Fanning friction factor is given by... 

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. 

This is consistent with the Blasius type of expression used for the friction factors in deriving the Martinelli parameter. Using the value n = 0.20 and expressing the ratio of flow rates in terms of the quality... 

The friction factor per base pair y for rotation of DNA around its symmetry axis was determined from FPA studies of restriction fragments containing N+ 1 =43 and 69 bp.(109) Both fragments are sufficiently short that a substantial amplitude of C (t), and also F (t), resides in their Uniform Mode Zones. Particular values of certain parameters were assumed, namely, the rise per base pair h = 3.4 A, the hydrodynamic radius b = 12 A for transverse motion in Eqs. (4.43)-(4.47) (which are quite insensitive to b), and D, = 1.8 x 106 s-1 for 43 bp and D = 4.8 x 105 s for 69 bp. The latter values were extrapolated or interpolated from the data of Elias and Eden using an inverse cubic relation between DL and L. They are close to the values calculated using the theory of Tirado and Garcia de la Torre.(129)... 

The adjustable parameters were a, y, and rQ. The best-fit friction factors y for the 43-bp fragment are the same on all four time spans, as shown in Figure 4.9. The best-fit y values are similarly independent of time span for the 69-bp fragment.(109) The hydrodynamic radius a for azimuthal rotation was calculated from the measured friction factor for uniform azimuthal rotation of the entire filament, f = (N+ 1 )y, using the formula of Tirado and Garcia de la Torre,(129)/ii = 3Mlrjna2L(l +<5,), where t is the solvent viscosity, and dM is an end-plate correction, which they tabulate. The same value... 

At high Reynolds numbers (Re > 2500), the surface roughness is an important parameter and must be allowed for in the calculations. Friction factor charts [53] include curves relating to various values of the relative roughness, that is the ratio of the mean height of surface roughness to the tube diameter. 

Obviously, correlations of one of these friction factors with an analogous Reynolds number, or with two-phase pressure-drop, throws little light on the other variables concerned, and these quantities will appear as parameters in any proposed relationship. However, Govier and Omer point out that plots of such a form do give a systematic spread of data above the single phase lines and allow easy comparison of trends. 

Pressure-drop data obtained in these tests were correlated empirically by defining a friction-factor based on gas-phase properties and superficial gas-flow rates. This friction-factor was plotted against the group Glijll/Ggi G with the water-oil feed mass-ratio as a parameter. 

A new set of flow characteristics gradually emerges as the concentration of polymer becomes large. The solution viscosity loses its direct dependence on solvent viscosity and comes to depend on the product of two parameters a friction factor C which is controlled solely by local features such as the free volume (or alternatively the segmental jump frequency), and a structure factor F which is controlled by the large scale structure and configuration of the chains (16) ... 

Consider the fully developed steady flow of an incompressible fluid through an annular channel, which has an inner radius of r, and an outer radius of r0 (Fig. 4.27). The objective is to derive a general relationship for the friction factor as a function of flow parameters (i.e., Reynolds number) and channel geometry (i.e., hydraulic diameter Dh and the ratio f A friction factor /, which is a nondimensional measure of the wall... 

After substituting the relationship between the friction factor and the nondimensional pressure gradient, solve the nondimensional differential equation to develop an expression for the circumferential velocity profile w(r). The product Re/ should appear as a parameter in the differential equation. Assume no-slip boundary conditions at the channel walls. 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

The ambiguity of definition of Re encountered in the concentric annulus case is compounded here because of the fact that no viscosity is definable for non-Newtonian fluids. Thus, in the literature one encounters a bewildering array of definitions of Re-like parameters. We now present friction factor results for the non-Newtonian constitutive relations used above that are common and consistent. Many others are possible. 

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