The Reynolds 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 

A word of caution should be raised. The formidable process of conducting such spatial integration should be viewed as principally symbolic. Although the mathematical operations will be valid, it is rare that in our application they will be so complicated. Indeed, in most cases, these operations will be very simple. For example, iff(x,y,z, t) is uniform in space, then we simply have, for Equation (3.1), 

Let us examine this more generally for f(x,y,z, t) and a specified moving volume, V(t). 

From the definition of the vector dot product, Equations (3.4) and (3.5) can be combined to give 

Since v is always more than 90° out of phase with n for Vn, the minus sign in Equation (3.3) for the integral is directly accounted for by the application of Equation (3.6) to AVn as well. It then follows that 

The Reynolds transport theorem is a convenient device to derive conservation equations in continuum mechanics. Toward derivation of the general population balance equation, we envisage the application of this theorem to the deforming particle space continuum defined in the previous section. We assume that particles are embedded on this continuum at every point such that the distribution of particles is described by the continuous density function / (x, r, t). Let i//(x, r) be an extensive property associated with a single particle located at (x, r). 

Consider an arbitrarily selected domain in the particle space continuum at some arbitrary reference time t = 0. Note that consists of a part A, in the space of internal coordinates part A in the space 

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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,... 

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... 

The physical laws of conservation of mass, momentum, and energy are commonly formulated for closed thermodynamic systems,2 and for our purposes, we need to transfer these to open control volume3 formulations. This can be done using the Reynolds Transport Theorem.4... 

Of the fluid mechanics applications used in this book, the Reynolds transport theorem is a bit sophisticated. Hence, a review of this topic is warranted. This theorem will be used in the derivation of the amount of solids deposited onto a filter and in the derivation of the activated sludge process. These topics are discussed somewhere in the textbook. Except for these two topics, the theorem is used nowhere else. The general knowledge of fluid mechanics will be used under the unit operations part of this book. 

The Reynolds transport theorem demonstrates the difference between total derivative (or full derivative) and the partial derivative in the derivation of the material balance equation for the property of a mass. It is important that this distinction be made here... 

Equation 75, however, is incorrect. Note that there is a flow Q into and out of the reactor. This means that the system is open and the full derivative, dXIdt, cannot be used. Nothing in the literature points out this mistake, however, because everything is written the same way. The Reynolds transport theorem distinguishes the difference between the total and partial derivatives, so wrong equations like the previous one will not result in any derivation if using this theorem. After the derivation below of the theorem is complete, we will come back and derive the correct material balance of the microbial kinetics of the activated sludge process. 

It is possible to convert one method of description to the other using the Reynolds transport theorem attributed to Osborne Reynolds. In other words, the theorem converts the Eulerian method of description of fluid flow to the Lagrangian method of description of fluid flow and vice versa. 

Kinetics of growth. In order to derive the correct version of Equation (75) using the Reynolds transport theorem, the kinetics of growth needs to be discussed. Let [Z] be the concentration of mixed population of microorganisms ntiUzing an organic waste. The rate of increase of [Z] fits the first order rate process as follows ... 

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