Mass and Energy Balances for Open Systems

Although the focus of the preceding sections has beenon closed systems, the concepts presented find far more extensive application. The laws of mass and energy conservation apply to all processes, to open as well as to closed systems. Indeed, the open system includes the closed system as a special case. The remainder of this chapter is therefore devoted to the treatment of open systems and thus to the development of equations of wide applicability. 

Open systems are characterizedby flowing streams, for which there are four common measures of flow  

The area for flow A is the cross-sectional area of a conduit, and p is specific or molar density. Although velocity is avector quantity, its scalar magnitude m is used here as the average speed of a stream in the direction iiomial to A. Flowrates m, n, and q represent measures of quantity per unit of time. Velocity u is quite different in nature, as it does not suggest the magnitude of flow. Nevertheless, it is an important design parameter. 

The region of space identified for analysis of open systems is called a control volume it is separated from its surroundings by a control surface. The fluid witliin the control volume is the themiodynamic system for wliich mass and energy balances are written. The control volume shown schematically in Fig. 2.5 is separated from its surroundings by an extensible control surface. Two streams with flow rates rh i and m2 are shown directed into the control volume, and one stream with flow rate m3 is directed out. Since mass is conserved, the rate of change of mass witliin the control volume, dm ldt, equals the net rate of flow of mass into the control volume. Tire convention is that flow is positive when directed into the control volume and negative when directed out. Tire mass balance is expressed mathematically by  

The difference operator A here signifies the difference between exit and entrance flows and the snbscript fs indicates that the term applies to all flowing streams. 

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Mass, Energy, and Entropy Balances for Open Systems. 4-14... 

The thermodynamics of flow encompasses mass, energy, and entropy balances for open systems, i.e., for systems whose boundaries allow the inflow and outflow of fluids. The common measures of flow are as follows ... 

MASS, ENERGY, AND ENTROPY BALANCES FOR OPEN SYSTEMS... 

Mass and energy balances for an open system are written with respect to a region of space known as a control volume, bounded by an imaginary control. surface that separates it from the surroundings. This surface may follow fixed walls or be arbitrarily placed it may be rigid or flexible. 

The system will be taken to be the gas contained in the compressor. The differential form of, . the molar mass and energy balances for this open system are... 

A system of fixed mass is called a closed system and a system that involves mass transfer across its boundaries j.s called an open system or control volume. The first law of therrtiody-nnmics or the energy balance for any system undergoing any process can be expressed as... 

Summary of Equations of Balance for Open Systems Only the most general equations of mass, energy, and entropy balance appear in the preceding sections. In each case important applications require less general versions. The most common restrictedTcase is for steady flow processes, wherein the mass and thermodynamic properties of the fluid within the control volume are not time-dependent. A further simplification results when there is but one entrance and one exit to the control volume. In this event, m is the same for both streams, and the equations may be divided through by this rate to put them on the basis of a unit amount of fluid flowing through the control volume. Summarized in Table 4-3 are the basic equations of balance and their important restricted forms. 

The designation reversible arises from the following observation. Consider the change in state of a general system open to the flow of mass, heat, and work, Between two equal tiTne intervals, 0 to t and to f2, where t2 = 2t. The mass, energy, and entropy balances for this system are, from Eqs. 2.2-4, 3.1-6, and 4.1-9,... 

We begin with the application of the first law of thermodynamics first to a dosed system and then to an open system. A system is any bounded portion of the universe, moving or stationary, which is chosen for the application of the various thermodynamic equations. For a closed system, in which no mass crosses the system boundaries, the change in total energy of the system, dE, is equal to the heat flow to the system. 8Q. minus the work done by the system on the surroundings. W. For a closed sy.sreni. the energy balance is... 

The steady-state ma.ss balance equation for the open system consisting of the device and its contents is cIN/cIi = 0 = /Vj. = /V -f, Vi + N, . Since, from the problem statement. /Vi = yV = — j/V. mass is conserved. The steady-state energy balance for this device is... 

The interface itself has negligible mass compared to the masses of the phases, and during processes, states of the interface may be undefined or undefinable. We will treat the interface as an open system and interpret each phase as a "port" for the other phase that is, the open-system energy and entropy balances from 2.4 will apply. In what follows, we first derive the combined first and second laws ( 7.2.1). Then we find limits on the directions ( 7.2.2) and magnitudes ( 7.2.3) of mass and energy transfers between phases a and p. 

The CSTR is an open system exchanging mass and energy with its environment. It is said to achieve a steady state whenever its state variables are invariant in time this is also often referred to as a fixed-point solution. Under such conditions, the transient balance for each chemical component in the... 

The fluxes of mass and energy required in expression 8.7 can be obtained from the equation of continuity per volume unit and energy balance, respectively, for an open unsteady state multicomponent system. Let (a) be a given phase, then the fundamental equation of continuity is given by convective, diffusive and chemical reaction contributions. 

Work of a flowing fluid against pressure. One additional flow of energy for systems open to the flow of mass must be included in the energy balance equation it is more subtle than the energy flows just considered. This is the energy flow that arises from the fact that as an element of fluid moves, it does work on the fluid ahead of it, Snd the fluid behind it does work on it. Clearly, each of these work terms is of the PAV type. To evaluate this energy flow term, which occurs only in systems open to the flow of mass, we. will compute the net work done as one fluid element of mass (M)i enters a system, such as the valve in Fig. 3.1-1, and another fluid element of... 

Notice that in the closed-system analysis the surroundings are doing work on the system (the mass element) at the inlet to the compressor, while the system is doing work on its surroundings at the outlet pipe. Each of the.se tenns is a / Pr/V-type work term. For the open. system this work term has been included in the energy balance as a, P V A/V/ term, so that it is the enthalpy, rather than the internal energy, of the flow streams that appears in the equation. The e.xplicit J P dV term that does appear in the open-system energy balance represents only the work done if the system boundaries deform for the choice of the compressor and its contents as the system here this term is zero unless the compressor (the boundary of our system) explodes. B... 

Based on the laws of conservation of mass, energy and momentum, balance equations are set up [1.1] -[1.5]. For a general open system... 

Some models assume that a system reaches a steady state rather than equilibrium. Equilibrium is defined by the principle of detailed balance, which requires that the forward and reverse rates are equal and that each step along the reaction path is reversible. The forward and reverse rates of steady-state processes are equal but the process steps that produce the forward rate are different from those that produce the reverse rate. At steady state, the state variables of an open system remain constant even though there is mass and/or energy flow through the system. The steady-state assumption is especially useful for processes that occur in a series, because the concentrations of intermediates that are formed and subsequently destroyed are constant. Perturbation of a steady-state system produces a transient state where the state variables evolve over time and approach a new steady state asymptotically. 

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