Transport theorem

The Reynolds transport theorem is a general expression that provides the mathematical transformation from a system to a control volume. It is a mathematical expression that generally holds for continuous and integrable functions. We seek to examine how a function fix, y, z, t), defined in space over x, y, z and in time t, and integrated over a volume, V, can vary over time. Specifically, we wish to examine... 

Equation (3.9) is the Reynolds transport theorem. It displays how the operation of a time derivative over an integral whose limits of integration depend on time can be distributed over the integral and the limits of integration, i.e. the surface, S. The result may appear to be an abstract mathematical operation, but we shall use it to obtain our control volume relations. 

Let us see how to represent changes in properties for a system volume to property changes for a control volume. Select a control volume (CV) to be identical to volume V t) at time t, but to have a different velocity on its surface. Call this velocity, w. Hence, the volume will move to a different location from the system volume at a later time. For example, for fluid flow in a pipe, the control volume can be selected as stationary (w = 0) between locations 1 and 2 (shown in Figure 3.4, but the system moves to a new location later in time. Let us apply the Reynolds transport theorem, Equation (3.9), twice once to a system volume, V(t), and second to a control volume, CV, where CV and V are identical at time t. Since Equation (3.9) holds for any well-defined volume and surface velocity distribution, we can write for the system... 

Equal and constant specific heats, cp = constant. From the Reynolds transport theorem, Equation (3.9), the pressure terms can be combined as... 

Reynolds Transport Theorem The purpose of the Reynolds transport theorem is to provide the relationship between a system (for which the conservation law is written) and an Eulerian control volume that is coincident with a system at an instant in time. The control volume remains fixed in space, with the fluid flowing through it. The Reynolds transport theorem states that... 

Conservation Equation for the Control Volume Using the Reynolds transport theorem... 

Consider the system and control volume as illustrated in Fig. 2.2. The Eulerian control volume is fixed in an inertial reference frame, described by three independent, orthogonal, coordinates, say z,r, and 9. At some initial time to, the system is defined to contain all the mass in the control volume. A flow field, described by the velocity vector (t, z,r, 9), carries the system mass out of the control volume. As it flows, the shape of the system is distorted from the original shape of the control volume. In the limit of a vanishingly small At, the relationship between the system and the control volume is known as the Reynolds transport theorem. 

As illustrated in Fig. 2.2, At is relatively large and the system has been displaced considerably from the control volume. Such a picture assists constructing the derivation, but the Reynolds transport theorem is concerned with the limiting case At - 0, meaning that the system has not moved. It is concerned not with finite displacements but rather with the rate at which the system tends to move. 

Combining Eqs. 2.22, 2.24, and 2.26 yields the Reynolds transport theorem, which relates the time rate of change (net accumulation) of an extensive property in a flowing system to a fixed control volume that coincides with the system at an instant in time,... 

Using the Gauss divergence theorem, the Reynolds transport theorem (Eq. 2.27) can be rewritten as... 

In the case of N and p being vectors, as they are for momentum and velocity, the Reynolds transport theorem takes the primative form... 

This chapter established three important concepts that are essential for the derivation of the conservation equations governing fluid flow. First, the Reynolds transport theorem was developed to relate a system to an Eulerian control volume. The substantial derivative that emerges from the Reynolds transport theorem can be thought of as a generalized time derivative that accommodates local fluid motion. For example, the fluid acceleration vector... 

For a fluid flow, of course, one uses the Reynolds transport theorem to establish the relationship between a system (where the momentum balance applies directly) and a control volume (through which fluid flows). In terms of Eq. 3.2, the extensive variable N is the momentum vector P = mV and the intensive variable tj is the velocity vector V. Thus the fundamental approach yields the following vector equation... 

Form the volume integral in the Reynolds transport theorem for the differential spherical control volume. 

Review Section 2.3.3, where the substantial derivative is derived in the context of the Reynolds Transport theorem. Discuss the role of the continuity equation in the definition of the substantial-derivative operator and the conservative form. 

Explain why this problem requires that the Reynolds transport theorem must be used in the more primative form as... 

Considering a general differential control volume, use a conservation law and the Reynolds transport theorem to write a species conservation equation for gas-phase species A in general vector form. Considering that the system consists of the gas phase alone, the droplet evaporation represents a source of A into the system. 

Beginning with a mass-conservation law, the Reynolds transport theorem, and a differential control volume (Fig. 4.30), derive a steady-state mass-continuity equation for the mean circumferential velocity W in the annular shroud. Remember that the pressure p 6) (and hence the density p(6) and velocity V(6)) are functions of 6 in the annulus. 

Beginning with appropriate forms of the Reynolds transport theorem, derive the continuity and momentum equations. Show that they can be written as... 

Deriving the mass-continuity equation begins with a mass-conservation principle and the Reynolds transport theorem. Unlike the channel with chemically inert walls, when surface chemistry is included the mass-conservation law for the system may have a source term,... 

In the application of the steady-state Reynolds transport theorem the intensive variable is the mass fraction Yk. Evaluating the integrals on the differential control volume yields... 

Deriving the conservation equations that describe the behavior of a perfectly stirred reactor begins with the fundamental concepts of the system and the control volume as discussed in Section 23. Here, however, since the system is zero-dimensional, the derivation proceeds most easily in integral form using the Reynolds transport theorem directly to relate system and control volume (Eq. 2.27). 

Turning again to the Reynolds transport theorem, relating the flowing system to the control volume yields... 

Deriving the governing equations begins with the underlying conservation laws and the Reynolds transport theorem. Consider first the overall mass continuity, where... 

All conservation equations in continuum mechanics can be derived from the general transport theorem. Define a variable F(t) as a volume integral over an arbitrary volume v(t) in an r-space... 

The equation describing the general transport theorem can be obtained by substituting Eq. (5.7) into Eq. (5.6) as... 

---