Thin Circular Disk

The case x - oo (that is, a - 0 and b = const) corresponds to diffusion to the surface of a thin circular disk of radius b perpendicular to a uniform translational Stokes flow. 

Let us indicate two essential differences between the diffusion flux distribution on the surface of a disk and that on the surface of a sphere at high Peclet 

As Pe -4 oo, the dimensionless total diffusion flux to the front part of the disk is described by the asymptotic formula [166] 

The results of numerical calculations of the mean Sherwood number at various Peclet and Reynolds numbers are given in [281], 

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The thickness-shear mode (TSM) resonator, widely referred to as a quartz crystal microbalance (QCM), typically consists of a thin disk of AT-cut quartz with circular electrodes patterned on both sides, as shown in Figure 3.2. Due to the piezoelectric properties and crystalline orientation of the quartz, the application of a voltage between these electrodes results in a shear deformation of the crystal. The crystal can be electrically excited in a number of resonant thickness-shear modes. 

Bond strength between PC overlays and portland cement concrete substrate was measured using the pull-out test method [7], The tensile bond test is illustrated in Fig. 1. Specimens were thin overlays, about 1/2-in. (1.3-cm) thick, cast directly (without the use of a primer) on sandblasted portland cement concrete slabs. Circular grooves (4-in./10. 8-cm diameter) were cored through the overlays and into the portland cement concrete substrate. Circular steel disks... 

As single-hole nozzle plates, circular orifice disks, typically 200 pm thick, with the nozzle hole in the center, are equally suitable as thin metal sheets, as used, e.g., for spatial filtering in optical setups. Thicker disks are best suitable as orifice plates if the hole exhibits a tapered (inlet) region on one side and has its nominal diameter only on a short part of the plate thickness. This feature ensures low flow resistance and good directional stability of the jet. Figure 26.9 shows typical shapes of nozzle cross section profiles [41]. 

Determine the mass moment of inertia for the following objects a thin disk, circular cylinder, and a sphere. 

Film thinning is modeled using a modified Reynolds equation [41] for the flow between two circular parallel disks, which gives the rate of film thinning as ... 

The overall sample geometry is governed by unidirectional heat flow. The only two practical geometries are either a slab-shaped solid with two parallel faces and heat flow perpendicular to these faces (Fig. 2a), most commonly used for polymers, or a right circular cylinder with heat flow in the radial direction and perpendicular to the axis (Fig. 2b), which is little used for pol3rmers. Because of the low thermal conductivity of polymers, the slab (or radius of the cylinder) is usually thin, so that heat losses in directions perpendicular to the desired heat-flow direction are minimized and the temperature drop AT is not excessive. Therefore, the preferred specimen shape is usually a thin disk with parallel faces and less commonly a long, thin rod or coaxial cylinder. 

Figure 2.7. Scattering factors for common shapes. From the top down a) randomly oriented circular rods with the length (L) to diameter (d) ratio L/d = 8, where x = 4jiLsin(0/2)/X b) randomly oriented infinitely thin disks where x = 4jtdsin(0/2)/X with d being the diameter c) spheres where x = 4ndsin(0/2)/X with d being the diameter and d) random coils where x = 4itRgSin(0/2)/X, with Rg being the radius of gyration.

Plate pr esses. Sometimes called sheet filters, these are assemblies of plates, sheets of filter media, and sometimes screens or frames. Thev are essentially modified filter presses with practically no cakeholding capacity. A press may consist of many plates or of a single filter sheet between two plates, the plates may be rectangular or circular, and the sheets may lie in a horizontal or vertical plane. The operation is similar to that of a filter press, and the flow rates are about the same as for disk filters. The operating pressure usually does not exceed 138 kPa (20 psig). The presses are used most frequently for low-viscosity liqmds, but an ordinaiy filter press with thin frames is commonly used as a clarifier for 100-Pa s (1000-P) rayon-spinning solution. Here the filtration pressure may be 6900 kPa (1000 psig). 

Rupture Disks A rupture disk is a device designed to function by the bursting of a pressure-retaining disk (Fig. 26-15). This assembly consists of a thin, circular membrane usually made of metal, plastic, or graphite that is firmly clamped in a disk holder. When the process reaches the bursting pressure of the disk, the disk ruptures and releases the pressure. Rupture disks can be installed alone or in combination with other types of devices. Once blown, rupture disks do not reseat thus, the entire contents of the upstream process equipment will be vented. Rupture disks are commonly used in series (upstream) with a relief valve to prevent corrosive fluids from contacting the metal parts of the valve. In addition, this combination is a reclosing system. 

C.7. Coaxial thin rods, minimum separation Z, length a, Hamaker form C.7.a. Cross-sectional areas Ai, A2 C.7.b. Circular cylinders, Ai = jrRf, A2 = xRl C.8. Circular disks and rods... 

The dynamic method for surface force measurement [14,155,228] consists in deducing the dependence of the surface force n on film thickness h from film thinning experiments. If a liquid film (viscosity rj and radius r) is squeezed by a pressure difference Ap between two circular flat solid disks, then the Reynolds flow is applicable to film thinning (Section 3.2). 

It may be noticed that some of the filters discussed are operated continuously and some are not. For example, the rapid sand filter, the slow sand filter, the pressure filter, and the rotary vacuum filter are all operated continuously. The plate-and-frame press is operated as a batch. Thus, filters may also be classified as continuous and discontinuous. Only the plate-and-frame press is discussed in this chapter as a representation of the discontinuous type, but others are used, such as the shell-and-leaf filters and the cartridge filters. The first operates in a mode that a leaf assembly is inserted into a shell while operating and retracted out from the shell when it is time to remove the cake. The second looks like a cartridge in outward appearance with the filter medium inside it. The medium could be thin circular plates or disks stacked on top of each other. The clearance between disks serves to filter out the solids. 

A more realistic approximate theory for the SECM with a tip shaped as a cone or spherical segment was presented in Ref. 9. The surface of the nonplanar tip electrode was considered to be a series of thin circular strips, each of which is parallel to the planar substrate. The diffusional flux to each strip was calculated using approximate equations for a disk-shaped tip over a conductive or an insulating substrate. The normalized current to the nonplanar tip was obtained by integrating the current over the entire tip surface. Two families of working curves for conical tips over conductive (Fig. 8A) and insulating substrates (Fig. 8B) illustrate the effect of the tip geometry. 

A thin circular-disk earring 5.00 cm in diameter is plated... 

J. H. Blackwell, Transient Heat Flow from a Thin Circular Disk Small-Time Solution, Journal of the Australian Mathematical Society, Vol. XIV, pp. 433-442,1972. 

E. J. Normington and J. H. Blackwell, Transient Heat Flow From Constant Temperature Spheroids and the Thin Circular Disk, Quarterly Journal of Mechanics and Applied Mathematics Vol. 17, pp. 65-72,1964. 

In a dilute suspension of thin circular disks of radius R, the contributions by individual particles scattering independently can be derived in a manner similar to that for thin rods, but here we will simply present the result (see Kratky and Porod11 for the derivation) ... 

Figure 5.3 The Debye function, Equation (5.31), for a random coil chain is plotted and compared with the independent scattering intensity function for a thin rod and a thin circular disk.

A volume V of a liquid of density p and surface tension a is contained between two parallel, concentric circular disks that are oriented horizontally and separated from each other by a distance a. A thin circular capillary tube of radius a is connected to the upper disk at its center and oriented vertically. The liquid level in the tube measured upward from the center of the space between the disks (all) to the meniscus is H (the meniscus height itself is taken to be small). The distance the liquid extends radially outward between the disks, measured from the centerline of the... 

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