Conservation Laws for Control Volumes

A control volume is a volume specified in transacting the solution to a problem typically involving the transfer of matter across the volume s surface. In the study of thermodynamics it is often referred to as an open system, and is essential to the solution of problems in fluid mechanics. Since the conservation laws of physics are defined for (fixed mass) systems, we need a way to transform these expressions to the domain of the control volume. A system has a fixed mass whereas the mass within a control volume can change with time. 

We will take a general and mathematical approach in deriving the conservation laws for control volumes. Some texts adopt a different strategy and the student might benefit from seeing alternative approaches. However, once derived, the student should use the control volume conservation laws as a tool in problem solving. To do so requires a clear understanding of the terms in the equations. This chapter is intended as a reference for the application of the control volume equations, and serves as an extension of thermochemistry to open systems. 

Fundamentals of Fire Phenomena James G. Quintiere 2006 John Wiley Sons, Ltd ISBN 0-470-09113-4 

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Consider any region of space that has a hnite volume and prescribed boundaries that unambiguously separate the region from the rest of the universe. Such a region is called a control volume, and the laws of conservation of mass and energy may be applied to it. We ignore nuclear processes so that there are separate conservation laws for mass and energy. For mass. 

The basic conservation laws, as well as the transport models, are applied to a system (sometimes called a control volume ). The system is not actually the volume itself but the material within a defined region. For flow problems, there may be one or more streams entering and/or leaving the system, each of which carries the conserved quantity (e.g., Q ) into and out of the system at a defined rate (Fig. 1-2). Q may also be transported into or out of the system through the system boundaries by other means in addition to being carried by the in and out streams. Thus, the conservation law for a flow problem with respect to any conserved quantity Q can be written as follows ... 

The approach is to formulate the entire burning problem using conservation laws for a control volume. The condensed phase will use control volumes that move with the vaporization front. This front is the surface of a regressing liquid or solid without char, or it is the char front as it extends into the virgin material. The original thickness, l, does not change. While the condensed phase is unsteady, the gas phase, because of its lower density, is steady or quasi-steady in that its steady solution adjusts to the instantaneous input of the condensed phase. 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

For the case of a sphere, the control volume is given by a thin spherical shell of thickness dr and radius r. If we assume that the complex diffusion process inside the porous structure can be represented by Fick s first law, and we additionally suppose that the volume change due to reaction is negligible (i.e. the total number of moles is constant), we arrive at the following form of the mass conservation law for the reacting species i ... 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

As discussed in Chapter 1, the basic principles that apply to the analysis and solution of flow problems include the conservation of mass, energy, and momentum in addition to appropriate transport relations for these conserved quantities. For flow problems, these conservation laws are applied to a system, which is defined as any clearly specified region or volume of fluid with either macroscopic or microscopic dimensions (this is also sometimes referred to as a control volume ), as illustrated in Fig. 5-1. The general conservation law is... 

This is the relationship we seek. It says that the rate of change of a property / defined over a system volume is equal to the rate of change for the control volume plus a correction for matter that carries / in or out. This follows since v w is the relative velocity of matter on the boundary of the control volume. If v — w = 0, no matter crosses the boundary. As we proceed in applying Equation (3.12) to the conservation laws and in identifying a specific property for/, we will bring more meaning to the process. We will consider the system to be composed of a fluid, but we need not be so restrictive, since our analysis will apply to all forms of matter. 

The conservation of momentum or Newton s second law applies to a particle or fixed set of particles, namely a system. The velocity used must always be defined relative to a fixed or inertial reference plane. The Earth is a sufficient inertial reference. Therefore, any control volume associated with accelerating aircraft or rockets must account for any differences associated with how the velocities are measured or described. We will not dwell on these differences, since we will not consider such noninertial applications. 

We apply the conservation laws to two control volumes enclosing these regions. Since there is no change in area, conservation of mass (Equation (3.15)) gives, for the unbumed mixture (u) and burned product (b),... 

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... 

Overall our objective is to cast the conservation equations in the form of partial differential equations in an Eulerian framework with the spatial coordinates and time as the independent variables. The approach combines the notions of conservation laws on systems with the behavior of control volumes fixed in space, through which fluid flows. For a system, meaning an identified mass of fluid, one can apply well-known conservation laws. Examples are conservation of mass, momentum (F = ma), and energy (first law of thermodynamics). As a practical matter, however, it is impossible to keep track of all the systems that represent the flow and interaction of countless packets of fluid. Fortunately, as discussed in Section 2.3, it is possible to use a construct called the substantial derivative that quantitatively relates conservation laws on systems to fixed control volumes. 

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. 

A control volume drawn around a plane wall with three layers is shown in Fig. 1.2. Three different materials, M, N and P, of different thicknesses, AxM, AxN and AxP, make up the three layers. The thermal conductivities of the three substances are kM, kN and kp respectively. By the conservation of energy, the heat conducted through each of the three layers have to be equal. Fourier s law for this control volume gives... 

The mathematical description of simultaneous heat and mass transfer and chemical reaction is based on the general conservation laws valid for the mass of each species involved in the reacting system and the enthalpy effects related to the chemical transformation. The basic equations may be derived by balancing the amount of mass or heat transported per unit of time into and out of a given differential volume element (the control volume) together with the generation or consumption of the respective quantity within the control volume over the same period of time. The sum of these terms is equivalent to the rate of accumulation within the control volume ... 

The law of conservation of mass for fluids in flow processes is most conveniently] written so as to apply to a control volume, which is equivalent to a thermodynamic] system as defined in Sec. 2.3. A control volume is an arbitrary volume enclosed] by a bounding control surface, which may or may not be identified with physical boundaries, but which in the general case is pervious to matter. The flow processes] of interest to chemical engineers usually permit identification of almost the entire control surface with actual material surfaces. Only at specifically provided entrances and exits is the control surface subject to arbitrary location, and heie it is universal practice to place the control surface perpendicular to the direction of flow, so as to allow direct imposition of idealizations 1 and 2. An example of a control volume with one entrance and one exit is shown in Fig. 7.1, The actual... 

In Chap. 2 the first law of thermodynamics was applied to closed systems (nonflo processes) and to single-stream, steady-state flow processes to provide specifi equations of energy conservation for these important applications. Our purpos here is to present a more general equation applicable to an open system or to control volume. 

Clausius/Clapeyron equation, 182 Coefficient of performance, 275-279, 282-283 Combustion, standard heat of, 123 Compressibility, isothermal, 58-59, 171-172 Compressibility factor, 62-63, 176 generalized correlations for, 85-96 for mixtures, 471-472, 476-477 Compression, in flow processes, 234-241 Conservation of energy, 12-17, 212-217 (See also First law of thermodynamics) Consistency, of VLE data, 355-357 Continuity equation, 211 Control volume, 210-211, 548-550 Conversion factors, table of, 570 Corresponding states correlations, 87-92, 189-199, 334-343 theorem of, 86... 

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