Coefficients friction factor

Here we no longer have molecular diffusion operating exclusively ( ) and the diffusion coefficient, frictional factor and chemical potential are no longer Interrelated. Also, the energy dissipation function is probably no longer quadratic. For simplicity, at unit rejection, for the steady state... 

Figure 30. Dependence of cuttings drag coefficient (friction factor fc) on cuttings Reynolds number. (Data from reference 112, plot symbol X reference 116, plot symbol + and reference 114.)...

Stoichiometric coefficient for y-th reaction and i-th component Mole fraction Axial coordinate Heat transfer coefficient Friction factor Membrane thickness Density... 

Cy, resistance coefficient (friction factor) of the packing, (obtained from Fig. 2-77) and the Reynolds number Reg of the gas phase... 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

Before pursuing the diffusion process any further, let us examine the diffusion coefficient itself in greater detail. Specifically, we seek a relationship between D and the friction factor of the solute. In general, an increment of energy is associated with a force and an increment of distance. In the present context the driving force behind diffusion (subscript diff) is associated with an increment in the chemical potential of the solute and an increment in distance dx ... 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

The convective heat-transfer coefficient and friction factor for laminar flow in noncircular ducts can be calculated from empirically or analytically determined Nusselt numbers, as given in Table 5. For turbulent flow, the circular duct data with the use of the hydrauhc diameter, defined in equation 10, may be used. 

Fanning friction factor /i for inner wall and / 2 for outer wall of annulus /l for ideal tube bank sldn friction drag coefficient Dimensionless Dimensionless... 

The heat transfer coefficient is best based on the Reynolds-Colbum analogy using a modified friction factor/ ... 

Once the diameter is known, the Reynolds number. Re can be computed and the friction factor f obtained from Figure 21. Assuming a straight length of pipe for L = 5(X) ft, N (line resistance factor) can be calculated. Next Gj, is calculated based on the downstream pressure and G/G j evaluated. From Figure 20 the ratio Pj/Pfl can be obtained. Since Pj is known, Pq can then be calculated. The pressure at the inlet of the knock-out drum is given by Pq -I- 0.5 psi. Table 6 provides typical values of resistance coefficients for various pipe fittings. 

To achieve these high coefficients it is necessary to expend energy. With the plate unit, the friction factors... 

For flow in a smooth pipe, the friction factor for turbulent flow is given approximately by the Blasius equation and is proportional to the Reynolds number (and hence the velocity) raised to a power of -2. From equations 12.102 and 12.103, therefore, the heat and mass transfer coefficients are both proportional to w 75. 

Because most applications for micro-channel heat sinks deal with liquids, most of the former studies were focused on micro-channel laminar flows. Several investigators obtained friction factors that were greater than those predicted by the standard theory for conventional size channels, and, as the diameter of the channels decreased, the deviation of the friction factor measurements from theory increased. The early transition to turbulence was also reported. These observations may have been due to the fact that the entrance effects were not appropriately accounted for. Losses from change in tube diameter, bends and tees must be determined and must be considered for any piping between the channel plenums and the pressure transducers. It is necessary to account for the loss coefficients associated with singlephase flow in micro-channels, which are comparable to those for large channels with the same area ratio. 

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 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. 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

The Y-axis represents the magnitude of the friction signal force and the X-axis is the load. The slope of the trend line is dehned as the friction factor (friction force signal/load) which is used to express the relative friction coefficient (friction force/load). Experiments that have been done in the same monolayer L-B him but different scan ranges give similar results as shown in Fig. 24 and Fig. 25. The friction factors of this monolayer L-B him, 0.0265 and 0.0203, are similar. The topographies of these two areas are shown in Fig. 26. 

Models of the polymer coil are based on the end-to-end distance, which is generally not directly available as a quantitative feature. Coils in dilute solution can be characterized in terms of the radius of gyration, Rg, which is a statistical measure of the distribution of mass about the center of gravity or in terms of the hydrodynamic radius, Rh, that is usually determined through the use of Stokes law and a measurement of a drag coefficient or friction factor, /drag/ for the coil,... 

Because the friction loss and wall stress are related by Eq. (5-47), the loss coefficient for pipe flow is related to the pipe Fanning friction factor as follows ... 

A commercial steel (e = 0.0018 in.) pipeline is 1 in. sch 40 diameter, 50 ft long, and includes one globe valve. If the pressure drop across the entire line is 22.1 psi when it is carrying water at a rate of 65 gpm, what is the loss coefficient for the globe valve The friction factor for the pipe is given by the equation... 

---