Critical radius cylinder

It is widely accepted that two mechanisms contribute to the observed hysteresis. The first mechanism is thermodynamic in origin [391,392], It is illustrated in Fig. 9.14 for a cylindrical pore of radius rc. The adsorption cycle starts at a low pressure. A thin layer of vapor condenses onto the walls of the pore (1). With increasing pressure the thickness of the layer increases. This leads to a reduced radius of curvature for the liquid cylinder a. Once a critical radius ac is reached (2), capillary condensation sets in and the whole pore fills with liquid (3). When decreasing the pressure again, at some point the liquid evaporates. This point corresponds to a radius am which is larger than ac. Accordingly, the pressure is lower. For a detailed discussion see Ref. [393],... 

Heat Conduction in Cylinders and Spheres 150 Multilayered Cylinders and Spheres 152 3-5 Critical Radius of Insulation 156... 

We stait this chapter with one-dimensional steady heat conduction in a plane wall, a cylinder, and a sphere, and develop relations for thennal resistances in these geometries. We also develop thermal resistance relations for convection and radiation conditions at the boundaries. Wc apply this concept to heat conduction problems in multilayer plane wails, cylinders, and spheres and generalize it to systems that involve heat transfer in two or three dimensions. We also discuss the thermal contact resislance and the overall heat transfer coefficient and develop relations for the critical radius of insulation for a cylinder and a sphere. Finally, we discuss steady heat transfer from finned surfaces and some complex geometries commonly encountered in practice through the use of conduction shape factors. 

Note that the critical radius of insulation depends on the thermal conductivity of the insulation k and the external convection heat transfer coefficient h. The rate of heat transfer from the cylinder increases with the addition of insulation for r2 < r teaches a maximum when rj = r , and starts to decrease for 2 > Thus, insulating the pipe may actuaUy increase the rale of heat transfer from the pipe instead of decreasing it when T2 < r -... 

Figure 3. Domain-complexion diagrams (at left) and phase distribution (at right, condensate in black, vapour in blank) within the pores (sites circles, bonds cylinders) on planes of 3D porous networks for actual states of diverse sorption processes, a) Boundary ascending (BA) curve on network la, b) boundary descending (BD) curve on network 2a, c) primary ascending (PA) curve on network 3a and d) primary descending (PD) curve on network 4a. Rc is the critical radius of curvature at the present state of the sorption process and Rc is the critical radius of curvature at the point of reversal for scanning curves. Shaded areas (pores filled with condensate) delimited by full lines in the complexion diagrams represent current states of the sorption systems, broken lines delimit states at the points of reversal.

Example 4.6 Calculation of the critical radius of a hollow cylinder... 

The critical radius of a hollow cylinder according to Eq. (4.61) and the corresponding temperature profile are to be calculated. Such a hollow cylinder is used here to model pipework lagging. 

Critical thiclDiess of slab or cube and critical radius of sphere or cylinder. 

Calculated Critical Cylinder at Optimum Pitch Using Hammer Flog Option Yielded an Effective Critical Radius of 32.3 cm. 

Polymer crystallization is usually initiated by nucleation. The rate of primary nucleation depends exponentially on the free-energy barrier for the formation of a critical crystal nucleus [ 110]. If we assume that a polymer crystallite is a cylinder with a thickness l and a radius R, then the free-energy cost associated with the formation of such a crystallite in the liquid phase can be expressed as... 

The critical nucleus corresponds to p = 1 = m, and the global free-energy minimum corresponds to infinitely large dimensions of the cylinder in both length and radius. 

From Eq. (5.2a) it becomes clear that the stronger (weaker) the adhesion of the nucleus on the substrate, the smaller (larger) Gc. The role of inhomogeneities in the substrate, such as defects, impurities, etc., is not taken into account in the model but they may become relevant because they would reduce Gc. In Section 5.4 we shall deal with the case of a cylinder-shaped nucleus growing onto a substrate following the arguments described here. In this case will be replaced by the critical 2D radius R. Note that hAti — Ayy must be non-zero in order to avoid mathematical singularities and that Gc = hAy Lc/2. 

In the case K > fi, the usual diffusion determines the kinetics for any gel shapes. Here the deviation of the stress tensor is nearly equal to — K(V u)8ij since the shear stress is small, so that V u should be held at a constant at the boundary from the zero osmotic pressure condition. Because -u obeys the diffusion equation (4.18), the problem is trivially reduced to that of heat conduction under a constant boundary temperature. The slowest relaxation rate fi0 is hence n2D/R2 for spheres with radius R, 6D/R2 for cylinders with radius R (see the sentences below Eq. (6.49)), and n2D/L2 for disks with thickness L. However, in the case K < [i, the process is more intriguing, where the macroscopic critical mode slows down as exp(- Q0t) with Q0 oc K. 

However, the dislocation is practically infinitely long compared to the size of any realistic critical nucleus. If the nucleus were of uniform radius along a long length of the dislocation, AQc would be very large. A critical nucleus will form from a local fluctuation in the form of a bulge of the cylinder associated with the metastable state A, as illustrated in Fig. 19.16. The problem is thus to find the particular bulged-out shape that corresponds to a minimum activation barrier for nucleation. 

For the treatment of practical cases, it is often necessary to assess other shapes other than a slab, infinite cylinder, or sphere. In such a case, it is possible to calculate the Frank-Kamenetskii criterion for some commonly used shapes. For a cylinder of radius r and height h, the critical value of the Frank-Kamenetskii criterion is given by [7]... 

An increase in Be indicates a competition between the irreversibilities caused by heat transfer and friction. At high Reynolds numbers, the distribution of Be is relatively more uniform than at lower Re. For a circular Couette device, the Reynolds number (Re = wr2lv) at the transition from laminar to turbulent flow is strongly dependent on the ratio of the gap to the radius of the outer cylinder, 1 — n. The critical Re reaches a value 50,000 at 1 n 0.05. We may control the distribution of the irreversibility by manipulating various operational conditions such as the gap of the Couette device, the Brinkman number, and the boundary conditions. 

Using a rotating cylinder electrode is a good way to achieve high rates of mass transport. This is very different, however, from the RDE in that the flow is turbulent rather than laminar. As a result, it is not possible to derive theoretical equations that relate the rate of mass transport to the various parameters in the reaction, and one must resort to empirical correlations. These tend to be critically dependent on dimensions and on the specific configuration of the cell, hence are less reproducible. A typical equation for mass transport to a rotating inner cylinder of radius r is ... 

Problem 6-5. Honey on a Spoon . You must have observed that when you dip a spoon into honey (or other viscous fluid) withdraw it and hold the spoon horizontal, you can keep the honey from draining by rotating the spoon. If you rotate the spoon too slowly the honey will drain, but above a critical rotation speed the honey will not drain from the spoon. (Try this with water and it will not work ) We want to analyze this problem and see if we can t predict the critical speed of rotation required to keep the honey on the spoon. The honey forms a thin layer of thickness h(0) around the spoon. The spoon is modeled as a cylinder of radius R and rotates about its axis at angular velocity 2 so that the speed of the surface of the cylinder is U = HR. The force of gravity acts downward and tries to pull the fluid off the cylinder. 

A cylindrical UME can be fabricated simply by exposing a length / of fine wire with radius tq. As in the case of the band, the length can be macroscopic, typically millimeters. The critical dimension is tq. In general, the mass-transfer problem to a cylindrical UME is simpler than that to a band, but operationally, there are many similarities between a cylinder and a band. 

---