Internal friction coefficient

Internal viscosity (Section 4) provides another possible source of shear-rate dependence. For sufficiently rapid disturbances, a spring-bead model with internal viscosity acts like a rigid body for sufficiently slow disturbances it is flexible and indefinitely extensible. The analytical difficulties for coupled, non-linear spring-bead systems are equally severe in linear spring-bead systems with internal viscosity. Even the elastic dumbbell with internal viscosity has only been solved exactly in the limit of small e (559), where e is the ratio of internal friction coefficient to molecular (external) friction coefficient Co n. For this case, the viscosity decreases with shear rate. 

The internal viscosity force is defined phenomenologically by equations (2.26) formulated above. Various internal-friction mechanisms, discussed in a number of studies (Adelman and Freed 1977 Dasbach et al. 1992 Gennes 1977 Kuhn and Kuhn 1945 Maclnnes 1977a, 1977b Peterlin 1972 Rabin and Ottinger 1990) are possible. Investigation of various models should lead to the determination of matrices Ca/3 and Ga and the dependence of the internal friction coefficients on the chain length and on the parameters of the macromolecule. 

In this formulae, ( is a friction coefficient of a particle in a monomer liquid, while non-dimensional phenomenological quantities B and E are measures of the increase in the external and internal friction coefficients due to the neighbouring macromolecules. 

As a general rule, the viscosity of a fluid is an intrinsic fluid property which measures the resistance of the fluid to movement. The resistance is caused by friction between the fluid and the boundary wall and internally by the fluid layers moving at different velocities, and hence it can be defined as an internal friction coefficient of the liquid. Historically, the set-up used by Isaac Newton to measure viscosity was made of a two coaxial cylinders separated by an oil film having a thickness d. A torque was applied to the external cylinder, which moved... 

Table 6.2 Polymer/polymer (internal) friction coefficient at 7 rpm for a range...

The silo contains bulk material material of the following properties viscous shear modulus /i = 1.0x10 Pa, viscous bulk modulus k — 9.0x10 Pa, internal friction coefficient (j> = 32.9° and a small cohesion value is assumed, c = 50 Pa. William-Warnke plasticity function is considered. The height of the container is 1.76 m, the width 0.64 m. The quart-circular outlet of radius 0.21 m is placed in the one of lower corners as pointed out in Figure 1 (left). The domain is discretized using ten-nodes tetrahedral unstructured mesh with 342 elements and 704 nodes as pointed out in Figure 1 (left), the mesh is biased in the lower part of domain and close to the outlet. 

The material properties of the considered container are as follows shear modulus /i= 1.0x10 Pa, bulk modulus k. = 9,0x10 Pa, internal friction coefficient 4> = 32.9 , the cohesion is assumed as c = 3000 Pa. Drucker-Prager plasticity function is taken into account. 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

Correlations for Convective Heat Transfer. In the design or sizing of a heat exchanger, the heat-transfer coefficients on the inner and outer walls of the tube and the friction coefficient in the tube must be calculated. Summaries of the various correlations for convective heat-transfer coefficients for internal and external flows are given in Tables 3 and 4, respectively, in terms of the Nusselt number. In addition, the friction coefficient is given for the deterrnination of the pumping requirement. 

Here, [L is the coefficient of internal friction, ( ) is the internal angle of friction, andc is the shear strength of the powder in the absence of any applied normal load. The yield locus of a powder may be determined from a shear cell, which typically consists of a cell composed of an upper and lower ring. The normal load is applied to the powder vertically while shear stresses are measured while the lower half of the cell is either translated or rotated [Carson Marinelli, loc. cit.]. Over-... 

This formula is another variation on the Affinity Laws. Monsieur s Darcy and VVeisbach were hydraulic civil engineers in France in the mid 1850s (some 50 years before Mr. H VV). They based their formulas on friction losses of water moving in open canals. They applied other friction coefficients from some private experimentation, and developed their formulas for friction losses in closed aqueduct tubes. Through the years, their coefficients have evolved to incorporate the concepts of laminar and turbulent flow, variations in viscosity, temperature, and even piping with non uniform (rough) internal. surface finishes. With. so many variables and coefficients, the D/W formula only became practical and popular after the invention of the electronic calculator. The D/W forntula is extensive and eomplicated, compared to the empirieal estimations of Mr. H W. 

Of major importance is the fact that the specific character of polymer chains of a given type enters the relationship (17) only through the effective size of one of its beads as indicated by the ratio f/970. Even the effect of this factor vanishes when the total internal resistance to flow is sufficiently large. Hence, in this limit, which will include nearly all actual cases of interest (see Sec. 4), the molecular frictional coefficient should depend only on the size /s and not otherwise on the nature of the polymer. Accordingly, we choose to let... 

The viscosity coefficients at dislocation cores can be measured either from direct observations of dislocation motion, or from ultrasonic measurements of internal friction. Some directly measured viscosities for pure metals are given in Table 4.1. Viscosities can also be measured indirectly from internal friction studies. There is consistency between the two types of measurement, and they are all quite small, being 1-10% of the viscosities of liquid metals at their melting points. It may be concluded that hardnesses (flow stresses) of pure... 

Intrinsic resistance to dislocation motion can be measured in either of two ways direct measurements of individual dislocation velocities (Vreeland and Jassby, 1973) or by measurements of internal friction (Granato, 1968). In both cases, for pure simple metals there is little or no static barrier to motion. As a result of viscosity there is dynamic resistance, but the viscous drag coefficient is very small (10" to 10" Poise). This is only 0.1 to 1 percent of the viscosity of water (at STP) and about 1 percent of the viscosity of liquid metals at their... 

This approximation is equivalent to assuming that the differences in internal densities and, consequently, in solvent draining, between a branched chain and the homologous linear chain, when included in their corresponding mean sizes, can describe both the friction coefficient and the viscosity. Besides these theoretical considerations, an empirical correlation in terms of a log-log fit of h vs f was employed by Roovers et al. [51]. Kurata and Fukatsu [48] and Ptitsyn [82] performed a more general Kirwood evaluation of the friction coefficient for different types of ideal branched molecules (uniform and randomly distributed stars, combs and random-branched structures). Their results for different structures are included within the limits l[Pg.60]

The fact that the velocity of a fluid changes from layer to layer is evidence of a kind of friction between these layers. The layers are mathematical constructs, but the velocity gradient is real and a characteristic of the fluid. The property of a fluid that describes the internal friction or resistance to flow is the viscosity of the material. Chapter 4 is devoted to a discussion of the measurement and interpretation of viscosity. For now, it is enough for us to recall that this property is quantified by the coefficient of viscosity 77 of a material. The coefficient of viscosity has dimensions of mass length-1 time-1, kg m-ls-1 in SI units. In actual practice, the cgs unit of viscosity, the poise (P), is widely used. Note that pure water at 20°C has a viscosity of about 0.01 P = 10-3kgm-ls-1... 

The number of beads in the model macromolecule is n, and is the Stokes law friction coefficient of each bead. The are to be evaluated for each macromolecule in its own internal coordinate system, with origin at the molecular center of gravity and axes (k = 1,2,3) lying along the principal axes of the macromolecule. The coordinates of the ith bead in this frame of reference are (x ]),-, (x2)i, and (x3)f. The averaging indicated by < > is performed over all macromolecules in the system. Thus, < i + 2 + 3) is simply S2 for the macromolecules. The viscosity is therefore identical, for all free-draining models with the same molecular frictional coefficient n and the same radius of gyration, to the expression from the Rouse theory ... 

In the case of internal flows extensive experimental data are available for turbulent pipe flow. The study of turbulent-friction coefficients in pipe flow has brought forth a number of effects displayed by flowing polymer solutions. Furthermore, many hydro-dynamic investigations in pipe flow have been made to elucidate the flow behavior (laminar and turbulent) of Newtonian fluids. Thus, the pipe is one of the most investigated and traditional pieces of test apparatus and one can easily compare the flow behavior of Newtonian fluids and polymer solutions under constant boundary conditions. 

Coefficient is calculated from the total amount of energy H that is fed into the product by heat transfer through the heating wall and by internal friction between the rotor blades and the process fluid. (F indicates the surface area, and Tw — T the temperature difference between heating wall and product). 

A force balance over a differential element (Fig. 4.5) simply using pressure P instead of the compressive stress, with shear stress at the wall xw = aw tan fiw + cw, where [lw is the angle of internal friction and cw is the coefficient of cohesion at the wall... 

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