Wall friction

Relations for transport properties such as viscosity and thermal conductivity are also required if wall friction and heat-transfer effects are considered. 

Wall Friction Angles. Wall friction values, important when characterizing the dow properties of a bulk soHd, are expressed as the wall friction angle or coefficient of sliding friction. The lower the friction, the less steep the hopper walls need to be to achieve mass dow. 

Fig. 5. Wall friction test (a) apparatus for measurement, and (b) results, where is the wall friction angle.

The following variables can affect wall friction values of a bulk soHd. (/) Pressure as the pressure acting normal to the wall increases, the coefficient of sliding friction often decreases. (2) Moisture content as moisture increases, many bulk soHds become more frictional. (3) Particle size and shape typically, fine materials are somewhat more frictional than coarse materials. Angular particles tend to dig into a wall surface, thereby creating more friction. (4) Temperature for many materials, higher temperatures cause particles to become more frictional. (5) Time of storage at rest if allowed to remain in contact with a wall surface, many soHds experience an increase in friction between the particles and the wall surface. (6) Wall surface smoother wall surfaces are typically less frictional. Corrosion of the surface obviously can affect the abiUty of the material to sHde on it. 

A material s flow function is usually measured on the same tester as the wall friction angle, although the cell arrangement is somewhat different (Fig. 6). ConsoHdation values are easily controUed, and the cohesive strength of the bulk soHd is determined by measuring interparticle shear stresses while some predeterrnined normal pressure is being appHed. 

Conica.1 Hoppers. Design charts for conical hoppers typically are plots of wall friction angles, ( ), vs hopper angle, 9. Charts such as that in... 

Turbulent flow near a wall, friction velocity = x/l Tp... 

Figure 4-114. Trend of wall frictional losses as a function of blade height.

Inclined bounding faces can cause modeling failures in orthogonal meshes, as the face is modeled as a "stair instead of a plane face. The wall friction of such a stair is sometimes not accounted for. 

Barth assumed that the pressure loss of a cyclone consists mainly of the pressure loss required to overcome the wall friction of the cyclone and the pressure drop to drive the fluid out of the cyclone outlet pipe. This leads to the following expression for the total loss factor C/ ... 

This represents the linear effect of tip speed and the square of impeller size on the flow of a specific impeller. Referring to Figure 12-46, very low flow coefficients for a specific type of centrifugal or axial flow machine cause excessive wall friction or leakage losses, and very high-flow coefficients tend to he subject to turbulence losses due to insufficient flow guiding. ... 

In this equation, r is the cyclone radius and n is dependent on the coefficient of friction. Theoretically, in the absence of wall friction, n should equal 1.0. Actual measurements, however, indicate that n ranges from 0.5 to 0.7 over a large portion of the cyclone radius. The spiral velocity in a cyclone may reach a value several times the average inlet-gas velocity. 

Fig. 5.27 Model-predicted pressure drops normalized with experimentally measured pressure drops for a circular test section (Triplett et al. 1999b). Model predictions represent the homogeneous wall friction model. Reprinted from Triplett et al. (1999b) with permission...

Fig. 18 A typical force-displacement curve. WF= work done overcoming die wall friction WD, work of elastic recovery Wp/, net work involved in tablet compact formation.

SB Tan, JM Newton. Influence of capsule dosator wall texture and powder properties on the angle of wall friction and powder-wall adhesion. Int J Pharm 64 227-234, 1990. 

The intrinsic constitutive laws (equations of state) are those of each phase. The external constitutive laws are four transfer laws at the walls (friction and mass transfer for each phase) and three interfacial transfer laws (mass, momentum, energy). The set of six conservation equations in the complete model can be written in equivalent form ... 

Knowlton has cautioned on the difference between small diameter and large diameter systems for pressure losses. The difference between these systems is especially apparent for dense phase flow where recirculation occurs and wall friction differs considerably. Li and Kwauk (1989, 1989) have also studied the dense phase vertical transport in their analysis and approach to recirculating fluid beds. Li and Kwauk s analysis included the dynamics of a vertical pneumatic moving bed upward transport using the basic solid mechanics formulation. Some noncircular geometries were treated including experimental verification. The flows have been characterized into packed and transition flows. Accurate prediction of the discharge rates from these systems has been obtained. 

For these reasons and other complex influences (e.g., large-diameter pipelines, particle-wall friction, particle shape, bends, etc.), it has been accepted that if high accuracy is needed, then some form of empiricism must be adopted. The preferred test-design procedure is listed below. 

Xb Particle-wall friction factor in bend As Particle-wall friction factor in straight pipe pu Loose-poured bulk density, kg nr3 Py Air density, kg mr3 ps Particle density, kg nr3... 

There would be a minimum of 80 data sets needed to generate this data for one temperature. Because of the time involved, usually about 10 to 15 shear rate data points are generated at each temperature. The plot of the viscosity as a function of shear rate at 270°C is presented in Fig. 3.22. The viscosity below a shear rate of 5 1/s would be best taken using a cone and plate rheometer. The wall friction for the capillary rheometer between the piston and the rheometer cylinder wall would likely cause a force on the piston of the same order as the force due to the flow stress. 

For single-phase flow the momentum balance can be written to give the static pressure drop as the resultant of acceleration, hydrostatic, and wall-friction pressure-drop terms. 

The first term on the right expresses the energy transferred from liquid to gas (Vi o being the actual relative velocity at the gas-liquid interface), and the second term the energy dissipated in wall friction. 

If now these facts are used to divide the energy loss due to irreversibilities into two parts, one part can be specified as being due to wall friction, and the other part due to relative motion between the phases. Defining this energy loss due to slip as Qo + Qz dP, we have... 

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