Plane wall geometry

Consider steady one-dimensional heat conduction through a plane wall, a cylindrical shell, and a spherical shell of unifonn thickness with constant thcimophysical propenies and no thermal energy generation. The geometry in which the variation of temperature in the direction of heal transfer will be linear is... 

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. 

So far, we have considered heat ttajisfer in simple geometries. such as large plane walls, long cylinders, and. spheres, Tliis is because heat tiansfer in such geometries can be approximated as one-dimensional, and simple analytical solutions can be obtained easily. But many problents encountered in practice are two- or three-dimensional and involve rather complicated geometries for which no simple solutions ate available. 

Consider a plane wall of thickness 2L, a long cylinder of radius r , and a sphere of radius r, initially at a nnifonn temperature T,-, as shown in Fig. 4—11. At time t = 0, each geometry is placed in a large medium that is at a constant temperature T and kepi in that medium for t > 0. Heat transfer lakes place between these bodies and their environments by convection with a uniform and constant heal transfer coefficient A. Note that all three ca.ses possess geometric and thermal symmetry the plane wall is symmetric about its center plane (,v = 0), the cylinder is symmetric about its centerline (r = 0), and the sphere is symmetric about its center point (r = 0). We neglect radiation heat transfer between these bodies and their surrounding surfaces, or incorporate the radiation effect into the convection heat transfer coefficient A. 

This completes the analysts for the solution of one-dimensional transient heat conduction problem in a plane wall. Solutions in other geometries such as a long cylinder and a sphere can be determined using the same approach. The results for all three geometries arc summarized in Table 4—1. The solution for the plane wall is also applicable for a plane wall of thickness L whose left surface at, r = 0 is insulated and the right surface at.t = T. is subjected to convection since this is precisely the mathematical problem we solved. 

The transient temperature charts in Figs. 4-15, 4-16, and 4-17 for a large plane wall, long cylinder, and sphere were presented by M. P. Heisler in 1947 and are called Heisler charts. They were supplemented in 1961 with transient heal transfer charts by II. Grober. There are three charts associated with each geometry the first chart is to determine the temperature Tj at the center of the... 

Using the appropriate nonditnensional temperature relations based on the one-term approximation for the plane wall, cylinder, and sphere, and performing the indicatetl integrations, we obtain the following relations for the fraction of heat transfer in those geometries ... 

That is, the solution for Ihe two-dimensional short cylinder of height a and radiu.s r is equal to the product of the noiidimcusionalized solutions for the oue-dimensional plane wall of thickness a and the long cylinder of radius r , which are the two geouieiiies whose intersection is the short cylinder, as shown in Fig. 4—35. We generalize this as follows the solution for a multidimensional geometry is the product of the solutions of the one-dimensional geometries whose intersection is the multidimensional body. 

Then WG determine the dimensionless heat transfer ratios for bolh geometries. For the plane wall, it Is determined from Fig. 4-15c to be... 

When the lumped system analysis is not applicable, the variation of temperature with position as well as time can be determined using the transient temperaiure charts given in Figs, 4-15,4-16, 4 17, and 4-29 for a large plane wall, a long cylinder, a sphere, and a semi-infinite medium, respectively. These charts are applicable for one-dimensional heal transfer in those geometries. Therefore, their use is limited to situations in which the body is initially at a uniform temperature, all surfaces are subjected to the same thermal conditions, and the body docs not involve any heat geiieiation. Tliese charts can also be used to determine the total heat transfer from the body up to a specified lime I. 

Figure 5-9. A sketch of the geometry and coordinate system for analysis of the motion of a sphere toward a plane wall under the action of an applied force F.

This equation is only precise for the case of a plane wall. For the technically important case of heat transfer through tubes (cylindrical geometry), the different geometry must be taken into account by a mean tube surface area for heat transfer (Equations 2.3.1-15 and 2.3.1-16 Eigure 2.3.1-lb) ... 

Of particular importance is the assumption of thin-walled geometry. From Eq. (7.3) we see that the pressure is independent of the z coordinate. Consequently, the finite element utilized for pressure calculation need have no thickness. That is, the element is a plane shell—generally a triangle or quadrilateral. This has great implications for users of plastics CAE. It means that a finite element model of the component is required that has no thickness. In the past this was not a problem. Almost all common CAD systems were using surface or wireframe modeling and thickness was never shown explicitly. The path from the CAD model to the FEA model was clear and direct. 

A warped wall is produced by out-of-plane loads acting at high angle to the wall (Fig. 2b). Warping can be to one or both sides. Thick walls made of Roman concrete can behave this way, while thin walls collapse in arcuate form (Fig. 2a). Generally, there is no change of wall geometry at foundation level. 

The flexural strength of masonry assemblages subjected to out-of-plane bending relates to the resistance of walls submitted to lateral loads from wind, earthquakes, or earth pressures. Depending on the boundary conditions and wall geometry, the bending can develop about vertical axis, about the horizontal axis, or about both directions. Thus, the tensile strength is referred to the... 

In Fig. 18, flow path lines are shown in a perspective view of the 3D WS. By displaying the path lines in a perspective view, the 3D structure of the field, and of the path lines, becomes more apparent. To create a better view of the flow field, some particles were removed. For Fig. 18, the particles were released in the bottom plane of the geometry, and the flow paths are calculated from the release point. From the path line plot, we see that the diverging flow around the particle-wall contact points is part of a larger undulating flow through the pores in the near-wall bed structure. Another flow feature is the wake flow behind the middle particle in the bottom near-wall layer. It can also be seen that the fluid is transported radially toward the wall in this wake flow. 

Fig. 25. Wall-segment geometry for 1-hole particles orthographic projections showing (a) flow path lines for particles released from vertical planes close to the tube wall (b) flow path lines for particles released from the bottom horizontal plane.

In our thought experiment we can force fluid to flow through this die at a certain volume per unit time or volumetric flow rate Q and allow no drag with the walls. This is a total slip condition so no shear will be applied to the fluid. The fluid will travel faster as the geometry contracts so the velocity V(l) in a plane across the geometry simply depends upon the area at any length / ... 

Another consequence of pore geometry is that for crystalline electrodes, other crystal planes are exposed to the electrolyte at the pore tip than at the pore walls. The dependence of pore growth on crystal orientation of the silicon electrode is discussed in Chapters 8 and 9. 

In contrast to porosity, the pore density and the specific surface area are quantities directly related to the actual size of pores and pore walls. The pore density NP is defined as the number of pores per unit area and it usually refers to a plane normal to the pore axis. For (100) oriented substrates this plane is parallel to the electrode surface, but for other orientations or strongly branched pores, there is no preferred plane orientation and NP refers to an average of the pore density of different planes. For arrays of straight pores the pore density can be directly calculated from the array geometry. For cylindrical pores of diameter d orthogonal to the electrode surface, for example, the average pore density NP is given by ... 

Apart from the short beam shear test, which measures the interlaminar shear properties, many different specimen geometry and loading configurations are available in the literature for the translaminar or in-plane strength measurements. These include the losipescu shear test, the 45°]5 tensile test, the [10°] off-axis tensile test, the rail-shear tests, the cross-beam sandwich test and the thin-walled tube torsion test. Since the state of shear stress in the test areas of the specimens is seldom pure or uniform in most of these techniques, the results obtained are likely to be inconsistent. In addition to the above shear tests, the transverse tension test is another simple popular method to assess the bond quality of bulk composites. Some of these methods are more widely used than others due to their simplicity in specimen preparation and data reduction methodology. 

In semi-infinite planar geometry, the electrode occupies the x = 0 plane and transport occurs perpendicularly to that plane from a limitless unimpeded medium as shown in Fig. 29. To prevent radial diffusion to the edge of the electrode, it is necessary to have walls of some kind to constrain the transport direction to be normal to the electrode. Because of this requirement, electrodes with precise semi-infinite planar geometry are difficult to fabricate and are, in fact, rather rare in practice. Nevertheless, because theoretical derivations are simplest for this geometry and because many practical geometries closely approximate the semi-infinite planar one as a limiting case, the geometry of this section is of paramount importance. 

Adsorption in ultramicroporous carbon was treated in terms of a slit-potential model by Everett and Powl51 and was later extended by Horvath and Kawazoe.52 They assumed a slab geometry with the slit walls comprised of two infinite graphitic planes. Adsorption occurs on the two parallel planes, as shown in figure 2.7. 

For the spherical pore geometry (see Figure 6.21), the interaction between a single adsorbate molecule and the inside wall of the spherical pore cavity of radius R consisting of a single lattice plane is given by [19]... 

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