Conservation of Energy for a Control Volume

This is a very important result which is significant for problems involving buoyant fluids. Finally, the last force to consider is the weight for the balloon consisting of the massless membrane and the hot air pg Vb  

The first term on the right-hand side is known as the buoyant force, the second is known as thrust. If this were just a puff of hot air without the balloon exhaust, we would only have the buoyant force acting. In this case we could not ignore (d/dt) Jjf pvxdV since the puff would rise (vx +) solely due to its buoyancy, with viscous effects retarding it. Buoyancy generated flow is an important controlling mechanism in many fire problems. 

As before, we ignore kinetic energy and consider Equation (2.18) as a rate equation for the system separating out the rate of work due to shear and shafts, Ws, and pressure, Wv. Then 

The internal energy of the system is composed of the contributions from each species. Therefore it is appropriate to represent U as 

A subtle point is that the species always occupy the same volume V(f) as the mixture. We can interchange the operations of the sum and the integration. Then, taking/ as ptUi for a system volume always enclosing species i, we need to use v, as our velocity, as was done in Equation (3.16a), to obtain 

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For a control volume shown in Figure 3.3, the first law of thermodynamics for a control volume is derived on the basis of the conservation of energy across a control volume as... 

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 ... 

Under steady conditions and in the absence of any work interactions, the conservation of energy relation for a control volume with one inlet and one exit with negligible changes in kinetic and potential energies can be expressed as... 

Note that conservation of radiant energy should not be confused with conservation of energy any loss in radiant energy within a control volume via the absorption mechanism is a gain for the overall energy balance and vice versa. 

As in the derivation of the differential equation of momentum transfer, we write a balance on an element of volume of size Ax, Ay, and Az which is stationary. We then write the law of conservation of energy, which is really the first law of thermodynamics for the fluid in this volume element at any time. The following is the same as Eq. (2.7-7) for a control volume given in Section 2.7. 

The fundamental governing equations for a control volume are the continuity equation, based on the conservation of mass, the energy equation, and the momentum equation. Because this study considers the overall effects, the... 

Because flowing viscous liquids can generate heat, we also need to consider the energy balance. In a manner similar to that used with the mass and momentum conservation relations, we can write a balance for the rate of change of internal energy over a control volume. This integral balance can be converted to a difierential balance (Bird et al., 1987, p. 9) giving... 

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. 

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... 

At x = 0, we can employ a control volume (CVm) just surrounding the surface. For this control volume, the conservation of mass is Equation (6.20). The conservation of energy becomes... 

However, by the thermally thin approximation, T(x,t) T(t) only. A control volume surrounding the thin material with the conservation of energy applied, Equation (3.45), gives (for a solid at equal pressure with its surroundings)... 

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... 

Frequently the integral form of the conservation la v of the property is particularized as total and partial mass balance and also as energy or thermal balance [3.7]. For each particularization, a control volume must be selected in order to have a form capable of permitting the computation of each integral from the relation (3.5). As an initial condition, we have to declare the property, the transport vector and the property generation rate. Figure 3.2 presents the way to obtain the equations of the differential balance of total mass, mass species and energy (heat). The... 

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 ... 

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. 

The zone fire models discussed here take the mathematical form of an initial value problem for a system of differential equations. These equations are derived using the conservation of mass or continuily equation, the conservation of energy or the first law of thermodynamics, the ideal gas law, and definitions of density and internal eneigy. The conservation of momentum is ignored. These conservation laws are invoked for each zone or control volume. A zone may consist of a number of interior regions (usually an upper and a lower gas layer), and a number of wall segments. The basic assumption of a zone fire model is that properties such as temperatures can be uniformly approximated throughout the zone. It is remarkable that this assumption seems to hold for as few as two gas layers. 

By combining Fourier s law and the requirements of energy conservation, one can derive the governing equation for conduction when the temperature varies in only one coordinate, for example, the x-axis. We do that by considering a control volume of infinitesimal size in the solid lattice (Figure 7.2). 

The law of conservation of mass is a statement of the mass balance for flow in and out and changes of mass storage of a system. Change in mass caused by any energy transfer such as in a chemical reaction or a combustion process is negligibly small and therefore is not taken into account. For analysis purposes, the conservation of mass law is presented for both the system and the control volume. 

Analytical Model. The internal heat transfer and electric charge transportation are analyzed using the thermal diffusion and charge transportation equations. The differential equations for the heat conduction at each finite element volume are solved on the basis of energy conservation. The heat and electric charge balances in a control volume are shown in Fig. 2, where q and J denote the heat and current density, respectively. 

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