Frictional factor complexity

The empirical frictional factor (T fric) is independent of shear rate but increases in poor solvent this permits to account for the dependence of the scission rate constant on solvent quality. The entanglement part (r enl), as given by Graessley s theory which considers the effect of entanglement and disentanglement processes, is a complex function of shear rate ... 

More complex equations have been developed for the flow of power-law fluids under turbulent flow in pipes [85,86,90], The foregoing applies to smooth pipes. Surface roughness has little effect on the friction factor for laminar flow, but can have a significant effect when there is turbulent flow [85],... 

The most important case of this transition for chemical engineers is the transition from laminar to turbulent flow, which occurs in straight bounded ducts. In the case of Newtonian fluid rheology, this occurs in straight pipes when Re = 2100. A similar phenomenon occurs in pipes of other cross sections, as well and also for non-Newtonian fluids. However, just as the friction factor relations for these other cases are more complex than for simple Newtonian pipe flow, so the criteria for transition to turbulence cannot be expressed as a simple critical value of a Reynolds number. 

FIGURE 6 Fanning friction factor-Bingham plastic Reynolds number curves for Bingham plastic fluids. [Reproduced from Hanks, R. W. (1981). Hydraulic Design from Flow of Complex Mixtures, Richard W. Hanks Associates, Inc., Orem, UTJ... 

The real term, e r, physically means the dielectric constant and the imaginary term, e" means the loss factor. Complex permittivity is introduced to take into account the dielectric losses due to resistance, or friction due to the polarization and orientation of the electric dipoles. If it is applied, a sinusoidal field due to a sinusoidal voltage (see Figure 1.26) can be expressed as follows ... 

Figure 6.3 shows a complex-series pipeline made up of four lengths of different size pipe. Determine the equivalent length of this pipe if each size of pipe has the same friction factor. 

Find the equivalent length of each pipe. For any complex-series pipeline having equal friction factors in all the pipes, Le = equivalent length, ft, of a section of constant diameter = (actual length of section, ft) (inside diameter, in, of pipe used to express the equivalent lcnglh/insidc diameter, in, of section under consideration)5. 

Equation 2.2 shows that D is determined by the interplay of the thermodynamic factor dH/dc and the friction factor /. In general, the friction factor is expected to increase monotonically with increasing c rind decrease with rising temperature. On the other hand, as can be deduced from the known information about osmotic pressure, the thermodynamic factor as a function of c varies in complex ways with solvent quality and temperature. Thus, the concentration dependence of D for a given polymer should exhibit a variety of features depending on solvent conditions. 

In this first chapter, some of the basic properties of solids, which are important to the composition of slurries, were reviewed. Their importance will be emphasized in the next few chapters. They may iead to Newtonian as well as more complex non-Newtonian flows that require special equations to determine the friction factors, velocity of flow, pipe sizes, head, and efficiency losses in pumps. 

This expression gives the friction factor to within plus or minus 10 percent accuracy. Other relationships do exist, but they are more complex and do not provide very much greater accuracy. However, even to predict AP under turbulent conditions one needs to have evaluated k and n from laminar flow data at the same wall shear stress as that expected in the actual pipe flow operating under turbulent conditions. ... 

Here k j is a constant of order unity (1/2ti in ref. 218) X = colIRq and and C are the molar mass and friction factor per chain atom, respectively. The function with = 1, corresponds to a prediction of the Rouse theory. The theoretical estimate n = 1 is larger than the value n =112 observed for solutions at the theta temperature.Equation (204) specifies a complex dependence on c and M, i.e. 

A correct value of friction factor is required for the estimation of pressure drop. The value of the friction factor depends upon the flow characteristics. For laminar flow less than 2100), the friction factor varies inversely with the Re5molds number, whereas for turbulent flow, the friction factor has a complex relationship with the pipe diameter, roughness of the pipe, and the Reynolds number. 

Empirical attempts have been made to relate strip and grab test results, particularly for cotton fabrics, so that if one strength is known, the other can be calculated. The relationship is complex, depending on fiber strength and modulus, yam size and crimp, yam-to-yam friction, fabric cover factor, weave, weight, and other factors (19). 

An industrial chemical reacdor is a complex device in which heat transfer, mass transfer, diffusion, and friction may occur along with chemical reaction, and it must be safe and controllable. In large vessels, questions of mixing of reactants, flow distribution, residence time distribution, and efficient utilization of the surface of porous catalysts also arise. A particular process can be dominated by one of these factors or by several of them for example, a reactor may on occasion be predominantly a heat exchanger or a mass-transfer device. A successful commercial unit is an economic balance of all these factors. 

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