Energy equation cylindrical coordinates

The transport equation for turbulent kinetic energy in cylindrical coordinates is written as ... 

Finally, the perfect-gas thermal-energy equation can be written explicitly in cylindrical coordinates. 

Table 5.5 presents the complete energy equation in the Cartesian, cylindrical and spherical coordinate systems. Table 5.6 defines the viscous dissipation terms for an incompressible Newtonian fluid. 

The equations of continuity, momentum and energy are summarized in Tables 6.1, 6.2 and 6.3, respectively. The vectorial forms in Tables 6.1 and 6.3 are provided for scholars of heat transfer that would like to go to three dimensional applications. The equations in cylindrical coordinates may be obtained from the rectangular coordinate equations by the use of the appropriate transformations. 

From the momentum equations for steady, two-dimensional, incompressible, Newtonian fluid with constant properties in cylindrical coordinates, obtain the z-momentum equation for the parallel flow (i.e., vr = 0). Obtain the corresponding energy equation. 

In terms of the cylindrical coordinate system defined earlier in this chapter, the energy equation for steady constant fluid property flow is ... 

Derive the two-dimensional energy equation in cylindrical coordinates using the control volume shown in Fig. P2.1. 

Write out the continuity, Navier-Stokes, and energy equations in cylindrical coordinates for steady, laminar flow with constant fluid properties. The dissipation term in the energy equation can be ignored. Using this set of equations, investigate the parameters that determine the conditions under which similar" velocity and temperature fields will exist when the flow over a series of axisymmetrie bodies of the same geometrical shape but with different physical sizes is considered. 

Consider fluid of particles interacting through the Lennard-Jones potential y>(r) = 4e[(energy parameters cylindrical coordinates and position the interface in the plane z = 0. The lOZ equation has the form [15]... 

The general heat conduction equation in cylindrical coordinates can be obtained from an energy balance on a volume element in cylindrical coordinates, shown in Fig. 2-23, by following the steps just outlined. It can also be obtained directly from Eq. 2-38 by coordinate transformation u ing the... 

Starting with an energy balance on a ring-shaped volume element, derive the two-dimensional steady heat conduction equation in cylindrical coordinates for T(r, z) for the caf.se of constant thermal conductivity and no heat generation./... 

Although the H2 problem for D-dimensions is separable in spheroidal coordinates, just as for D = 3, since we want to examine the nonseparable situation, we employ cylindrical coordinates. In these coordinates the nuclei are located on the z-axis at —i /2 and -t-iJ/2, respectively, and the electron is at p,z). Dimensional scaling is introduced by using units of jZ bohr radii for distance and hartrees for energy, with Z the nuclear charge and k = D — l)/2. The scaled Schrodinger equation for H then takes a simple form. 

The model is based on the standard dispersion model that describes the gas phase mass and energy transport as convective flow with superimposed radial and axial dispersion. The model equations in two-dimensional axi-symmetrical cylindrical coordinates and the boundary conditions are listed below. The following assumptions have been made in this model (although the model could be extended to include these phenomena) ... 

Since the fluid has a constant density, Eq. (5.6-14) in cylindrical coordinates will be used for the equation of energy change. For this case v, = 0 and Vg = 0. Since this will be symmetrical dTjdO and d T/dO will be zero. For steady state, dT/dt = 0. Hence, Eq. (5.6-14) reduces to... 

The governing equation for the Graetz problem may be obtained from an energy balance in cylindrical coordinates. Alternatively, one can start with the equation of energy in terms of transport properties for Newtonian fluids of... 

For a cylindrical coordinate system (as shown in Fig. 4-1), the equation of energy is ... 

On the other hand, the energy balance due to convection and conduction results from an energy balance in cylindrical coordinates. In the limit and using Fourier equation for the heat flux, q, we obtain Equation 4.36 ... 

These three results are entirely consistent with one another. Radial velocities indeed do lead to a uniform thickness. From the velocity field expressed in equation (7.61), we can calculate the energy dissipation. In a cylindrical coordinate system, we get... 

A particle-optical approach can also be used for the treatment of focusing of atoms by laser light (McClelland and Scheinfein 1991). In this approach the atoms are treated as classical particles that move in the potential field of a laser beam. This method was originally developed for charged particle optics, for calculation of trajectories in a cylindrically symmetric potential field. The equation of motion can be derived from the Lagrangian L = Mv /2 — U p, z), where U p, z) is the potential energy eqn (6.1) and 2 is the axis of symmetry. In cylindrical coordinates, the radial equation of motion is... 

The simplified energy equation in cylindrical coordinate is given as... 

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