Circumferential profiling

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

Fig. 3.12 Axial velocity and circumferential vorticity profiles in a circular channel with axisymmetric flow.

Considering an incompressible fluid with constant properties (Eq. 3.81), the equations governing the circumferential incompressible flow profile between rotating cylinders reduce to ... 

The circumferential-momentum equation is a parabolic partial differential that requires solution for the radial dependence of the circumferential velocity. With the velocity profiles in hand, the radial-momentum equation can be used to determine the resulting radial pressure dependence. 

For a nondimensional oscillation period of tp = 0.1, Fig. 4.15 shows the circumferential velocity profiles at four instants in the period. The wall velocity follows the specified rotation rate exactly, which it must by boundary-condition specification. The center velocity r — 0 is constrained by boundary condition to be exactly zero, incenter = 0. The interior velocities are seen to lag the wall velocity, owing to fluid inertia and the time required for the wall s influence to be diffused inward by fluid shearing action. 

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. 

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

The boundary-layer thickness is a function of the rotation rate and can be derived from the nondimensional velocity profiles. Boundary-layer thickness can be defined in different ways, but generally it represents the thickness of the viscous layer. Defining the boundary-layer thickness as the point at which the circumferential velocity is 1% of its surface value gives zi% = 5.45. 

The radial velocity profile is linear and the circumferential velocity is zero outside the viscous boundary layer, which indicates that the vorticity is constant in that region. Thus, for substantial ranges of the flow and rotation Reynolds numbers, the flow is inviscid, but rotational, outside the viscous boundary layer. For sufficiently low flow, the boundary-layer can grow to fill the gap, eliminating any region of inviscid flow. 

This paper discusses the impact of wind action on natural-draft cooling towers. The structure of the wind load may be divided into a static, a quasistatic, and a resonant part. The effect of surface roughness of the shell and of wind profile on the static load is discussed. The quasistatic load may be described by the variance of the pressure fluctuations and their circumferential and meridional correlations. The high-frequency end of the pressure spectra and of the coherence functions are used for the analysis of the resonant response. It is shown that the resonant response is small even for very high towers, however, it increases linearly with wind velocity. Equivalent static loads may be defined using appropriate gust-response factors. These loads produce an approximation of the behavior of the structure and in general are accurate. 11 refs, cited. 

Fig. 10.65 Experimental apparatus used by McCullough and Hilton (109). (a) Transducer circumferential locations, (b) kneading block used for the ZSK 30 barrel. [Reprinted by permission from V. L. Bravo, A. N. Hrymak, and J. D. Wright, Numerical Simulation of Pressure and Velocity Profiles in Kneading Elements of a Co-TSE, Polym. Eng. Set, 40, 525-541 (2000).]...

Figure 7.80 Fracture surface profile shows the presence of a circumferential ridge and depression.1 (Reprinted with the permission of M. Zamanzadeh, ATCO Associates, Pittsburgh,...

The fracture surface profile is shown in Figure 7.80. The presence of a circumferential ridge and depression in the cylindrical surface is to be noted in the case of both the broken and reference pins. By comparing the cylindrical parts of the broken pin with the reference pin it was concluded that the fracture of the broken pin initiated at the circumferential depression. Macroetching with 50% hydrochloric acid for 30 s enabled the identification of fracture initiation site in Figure 7.80. 

Fig. 2-12 Efficiencies of circumferential fins of rectangular profile, according to Ref 3...

A 2.5-cm-diameter tube has circumferential fins of rectangular profile spaced at 9.5-mm increments along its length. The fins are constructed of aluminum and are 0.8 mm thick and 12.5 mm long. The tube wall temperature is maintained at 200°C, and the environment temperature is 93°C. The heat-transfer coefficient is 110 W/m2 - °C. Calculate the heat loss from the tube per meter of length. 

A circumferential fin of rectangular profile is constructed of I percent carbon steel and attached to a circular tube maintained at I50°C. The diameter of the fin is 5 cm, and the length is also 5 cm with a thickness of 2 mm. The surrounding air is maintained at 20°C and the convection heat-transfer coefficient may be taken as 100 W/m2 °C. Calculate the heat lost from the fin. 

A circumferential fin of rectangular profile is constructed of aluminum and surrounds a 3-cm-diameter tube. The fin is 2 cm long and I mm thick. The tube wall temperature is 200°C, and the fin is exposed to a fluid at 20°C with a convection heat-transfer coefficient of 80 W/m2 °C. Calculate the heat loss from the fin. 

For a BSR built up of cylindrical catalyst rods (i.e., infinitely long catalyst cylinders), the boundary condition of Eq. (14) will result in circumferential variation of both the mass flux and the surface concentration. Because of the varying surface concentration, the concentration profile in the rod will not be axisymmetric, which will influence the effectiveness factor of the rod. Fortunately, the influence can be expected to be limited. This is because the strongest (relative to the flux) circumferential variations of the surface concentration are obtained for the boundary condition of constant flux, but this boundary condition corresponds to a relatively low reaction rate, which will usually correspond to small intraparticle concentration gradients. 

In Europe, as opposed to domestic practice, design and quality control of grouting are generally the province of the contractor, and field practice tends to be more precise and detailed than current domestic practice. (This statement does not apply to the Washington Metro work, which was very closely controlled and monitored. See Sec. 19.4 for discussion.) An example of this is shown in Fig. 19.5 (Ref. [8], part 1, pp. 35-36), a typical section design for grouting of a profile with various soil types. A circumferential zone of treatment is defined in heavy lines within this section are different... 

Figure 10.4 Circumferential-averaged velocity profile ait z/R = 0.1 for Swirler 504545 1 — mean axial 2 — mean tangential 3 — rms axial and 4 — rms tangential.

The contour plots of Fig. 10.3 were averaged circumferentially to produce radial profiles of the axial and tangential velocity components as shown in Fig. 10.4 for Swirler 504545 at c// = 0.1. The locations of the peak velocity of the mean and tangential components are at r/R = 0.9 and r/R = 0.95. The peaks of the rms velocity of the two components are at nearly the same locations dXr/R — 0.6, where the shear layer of the annular jet is located. The CTRZ has a radius of r/R = 0.5, which is primarily contributed by the inner swirler and partially by the intermediate swirler. 

Figure 11.5 Circumferentially- and time-averaged mean (a) and rms (6) velocity profiles (open symbols — tangential, and filled symbols — axial) at the TARS outlet plane from UC LDV data [6]. Indicated radii (R, = D,j2) characterize the approximate locations of air-flow passages, cf. Fig. 11.4 1 — 304545 2 — 304545c and 3 — 3045c45.

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