Cylindrical control volume

The cylindrical coordinate system and cylindrical control volume are illustrated in Figure 2.6. There are some differences in the development of a mass balance equation on a cylindrical control volume. Primarily, the rdOdx side of the control volume increases in area as r increases. For the control volume of Figure 2.6, the area normal to the r-coordinate would be... 

Fig. 3.11 The cylinder on the left is filled with a gas at pressure p and bounded by two pistons that can move with velocity u. The long cylindrical annulus on the left is filled with a fluid. The center rod is fixed, but the outer cylindrical shell moves upward at a constant velocity. Under these circumstances a steady state-velocity distribution will develop in the fluid as illustrated u(r), with the zero velocity at the inner-rod wall and the wall velocity at the shell surface. A cylindrical control volume with its zrz shear stresses is illustrated.

Referring again to Fig. 4.3, consider the energy balance from the point of view of a cylindrical control volume. The conductive heat flow dQ/dt crossing the control surfaces into... 

Based on a differential cylindrical control volume, derive steady-state momentum balances for the axial and circumferential directions, i.e., the Navier-Stokes equations. 

Again, based on the differential cylindrical control volume, derive the total energy equation. 

The fluid is assumed to consist of two components, a volatile component with concentration and a nonvolatile component with concentration C. Mass balances on a cylindrical control volume on the disk yield ... 

The cylindrical control volume is a shell with an inside radius r, thickness Ar, and length Ax. At steady state the conservation of momentum, Eq. (2.8-3), becomes as follows sum of forces acting on control volume = rate of momentum out — rate of momentum into volume. The pressure forces become, from Eq. (2.8-17),... 

Figure 3.4 Illustration and terminology of the cylindrical control volume used for the purpose of integration.

Figure 7.4 Cylindrical control volume for heat conduction.

For a lake or ocean surface, consider a cylindrical control volume of depth h that moves with the mean velocity of the tracer cloud containing two gas tracers, designated A and B, that have different rates of gas transfer. Using the cylinder as our control volume, the transport relation for each of the gas tracers can be written as... 

Fig. 32. (a) Sequence of transformations l->2->3->4to place bottom front cylindrical particle (b) Midplane cross-section of the WS packed with cylinders, showing control volumes found by selection algorithm, marked as darkest cells. 

The balance is made with respect to a control volume which may be of finite (V) or of differential (dV) size, as illustrated in Figure 1.3(a) and (b). The control volume is bounded by a control surface. In Figure 1.3, m, F, and q are mass (kg), molar (mol), and volumetric (m3) rates of flow, respectively, across specified parts of the control surface,6 and Q is the rate of heat transfer to or from the control volume. In (a), the control volume could be the contents of a tank, and in (b), it could be a thin slice of a cylindrical tube. 

Solution A fixed control volume is selected as a cylindrical shape with its bases at the burner and flame tip respectively, as shown in Figure 3.11. The radius of the CV cylinder is just large enough to contain all of the flow and thermal effects of the plume. This is illustrated in Figure 3.11 with the ambient edge of the temperature and velocity distributions included in the CV. 

Apply the conservation of energy, Equation (3.40). Since the control volume is fixed the pressure work term does not apply. The shear work (v x shear force) is zero because (a) the radius of the control volume was selected so that the velocity and its gradient are zero on the cylindrical face and (b) at the base faces, the velocity is normal to any shear surface force. Similarly, no heat is conducted at the cylindrical surface because the radial temperature gradient is zero, and conduction is ignored at the bases since we assume the axial temperature gradients are small. However, heat is lost by radiation as... 

The diffusive flux rates would be treated similarly. The area of the control volume changing with radius is the reason the mass transport equation in cylindrical coordinates, given below - with similar assumptions as equation (2.18) - looks somewhat different than in Cartesian coordinates. 

The terminology of computational techniques is descriptive, but one needs to know what is being described. Table 7.1 lists some common terms with a definition relative to mass transport. Most computational techniques in fluid transport are described with control volume elements, wherein the important process to be computed is the transport across the interfaces of small control volumes. The common control volumes are cubes, cylindrical shells, triangular prisms, and trapezoidal prisms, although any shape can be used. We will present the control volume technique. 

Fig. 2.13 Two views of a cylindrical differential element, showing the positive components of the stresses on the control-volume faces.

In Section 2.8.4 a general vector analysis is used to determine the net force exerted on a control volume by virtue of stresses acting on the control surfaces. In this section forces are considered on each face of a cylindrical differential control volume. The objective is the same as in the previous section, that is, to determine the force per unit volume on a differential control volume. Here, however, by explicitly considering a particular control volume, the intent is to make more clear the physical meaning of the result. 

In the most general case, stresses on any of the six control-volume faces can potentially contribute to a force in any direction. In a cartesian coordinate system, only stresses in a certain direction can contribute to a force in that direction. In cylindrical coordinates and other noncartesian systems, the situation is more complex. As an example of this point, consider Fig. 2.15, which is a planar representation of the z face of the cylindrical differential element. Notice two important points that are revealed in this figure. One is that the the area of the 0 face varies from rdO on one side to (r + dr)dO on the other. Therefore, in computing net forces, the area s dependence on the r coordinate must be included. Specifically,... 

Consider the net mass flow through the cylindrical differential element illustrated in Fig. 3.6. The following analysis makes no explicit reference to the scalar product of the flux vector and the outward normal, j ndA. Rather, it is based on a more direct observation of how mass diffuses into and out of the control volume. It is presumed that the spatial components of j are resolved into spatial components that are normal to the control-volume faces, jk,z, jk,r, and jk,e Further it is presumed that a positive value for a spatial component of jk means that the corresponding flux is in the direction of the positive coordinate. The components of the diffusive mass flux are presumed to be continuous and differentiable throughout the fluid. Therefore the flux components can be expanded in a first-order Taylor series to express the local variations in the flux. The net mass of species k that crosses the control surfaces diffusively is given by the incoming minus the outgoing mass transport. Consider, for example, transport in the radial direction ... 

Cylindrical Differential Control Volume While the foregoing discussion works in a general vector setting, it is instructive to look more narrowly at a control volume in a particular coordinate system. In this setting it is easier to see the physical interpretation than it is in the more general vector setting. 

Figure 3.9 illustrates the spatial components of the heat-flux vector in spatial components that align with the cylindrical coordinates. Because the heat flux is a continuous, differentiable function, its variation throughout the control volume can be represented as a Taylor series expansion. In a procedure that is analogous to that in Section 3.6.2, the net heat conducted across the control surfaces is... 

Fig. 3.9 A cylindrical differential control volume showing conductive heat fluxes.

Work on a Cylindrical Differential Element Consider the cylindrical differential control volume such as the one illustrated in Fig. 3.9. A two-dimensional projection of this element is illustrated in Fig. 3.10. Recall the discussion in Section 2.8.2 on the sign convention for the stress components—the sign conventions are important. At z, r, and 0 the rates of work done on the near control-volume faces are... 

Fig. 3.10 Two-dimensional projection of the z-plane of a cylindrical differential control volume, showing the surface stresses and the velocities.

Fig. 4.12 A long cylinder rotates within an outer cylindrical shell. The inner cylinder suddenly begins rotating with angular velocity Q, with the fluid in the annulus initially at rest. Also shown is a control volume illustrating the pressure and shear stresses...

Fig. 16.2 Illustration of a cylindrical, straight-channel, plug-flow reactor. Also shown is the differential control volume by which one structures the governing-equation derivation.

Consider a cylindrical elemental control volume of dimensions Ar, rA0, and Az in the r, 0 and z directions, respectively. Derive the continuity equation in cylindrical coordinates. 

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

---