Moving control volume

The process of steady flame propagation into a premixed system is depicted in Figure 4.12 for a moving control volume bounding the combustion region <5r. The heat loss in this case is only considered to the duct wall. With h as the convection heat transfer coefficient, the loss rate can be written as... 

The control volume relationships that are considered are based on a moving control volume surface with velocity, w, while the medium velocity is v. The conservation of mass is given as... 

We also need to make sure that the strength of the Dirac delta has a similarity between the two solutions. In Example 2.2, the strength was a mass per unit area, or M/A For this problem, because we are following a moving control volume, we need a mass flux rate, M, and the unit area passing the outfall per unit time (an area flux rate ), Uh. Thus, similarity between the two solutions means that Uh and M/A M/(Uh), as illustrated in Figure (E5.3.2). 

Following a center-of-mass fixed coordinate system tied to an air mass, we use intrinsic coordinates to avoid artificial diffusion in the horizontal direction. Physical diffusion, therefore, is distinct and identifiable because the moving control volume can be allowed to undergo mass exchange with a neighboring air mass in a prescribed fashion. The question of horizontal spatial resolution is answered by a selection of source grid size, and the vertical resolution is set by the choice of the interval size in the z-direction. 

In this section, an analysis based on the Boltzmann equation will be given. Before we proceed it is essential to recall that the translational terms on the LHS of the Boltzmann equation can be derived adopting two slightly different frameworks, i.e., considering either a fixed control volume (i.e., in which r and c are fixed and independent of time t) or a control volume that is allowed to move following a trajectory in phase space (i.e., in which r(t) and c t) are dependent of time t) both, of course, in accordance with the Liouville theorem. The pertinent moment equations can be derived based on any of these two frameworks, but we adopt the fixed control volume approach since it is normally simplest mathematically and most commonly used. The alternative derivation based on the moving control volume framework is described by de Groot and Mazur [22] (pp. 167-170). 

If work is done on the flnid to compress it, then the density increases in a control volume that moves with the local fluid velocity. This causes an increase in internal energy in the moving control volume. If the fluid expands and does work on the surroundings, then its density decreases, which causes the internal energy to decrease in the moving control volume. These effects of compression and expansion on internal energy are consistent with the first law of thermodynamics. 

TTie derivation of the macroscopic mass balance begins with Eq. 1-1 and moves directly to Eq. 1-34. We consider an arbitrary moving control volume designated by a(t) and we integrate Eq. 1-34 over this volume to obtain... 

Rather than integrate this form of our mechanical energy equation over a fixed control volume, we perform the Integration over an arbitrary moving control volume, l a(t). An... 

The surface area of this arbitrary, moving control volume can often be represented in the following special form... 

The four terms on the right hand side of this result can be identified as the shaft work, the fiow work, the reversible work and the irreversible work (viscous dissipation). The shaft work Is a very Important quantity In many practical problems and it can only be Included in the analysis by means of a moving control volume. In addition, the use of the kinematical relation given by Eq, 2-18 has allowed us to represent the rate of work done by the body force b in terms of an accumulation and flux of potential energy. It should be clear that the concepts of kinematics and stress are essential elements of the derivation that led originally from Eq. 1-2 to the macroscopic mechanical energy balance given by Eq. 2-23. 

Integration over an arbitrary moving control volume, and use of the representation given by Eq. 2-22 leads to the general macroscopic total energy balance. 

As if in answer to Bliss editorial. Prof. Bird (1957) provided a detailed derivation of the macroscopic balances for mass, momentum and mechanical energy. These were presented for an arbitrary moving control volume, and thus captured in a rigorous manner all of the details that had been added by persuasion over the previous years. One might think that this would have ended the confusion about the macroscopic mechanical energy balance, but this has not been the case and a precise understanding of this result seems to be the exception rather than the rule. 

The nonconvective energy flux across the boundary is composed of two terms a heat flux and a work term. The work term in turn is composed of two terms useful work deflvered outside the fluid, and work done by the fluid inside the control volume B on fluid outside the control volume B, the so-called flow work. The latter may be evaluated by imagining a differential surface moving with the fluid which at time 2ero coincides with a differential element of the surface, S. During the time dt the differential surface sweeps out a volume V cosdSdt and does work on the fluid outside at a rate of PV cos dS. The total flow work done on the fluid outside B by the fluid inside B is... 

Macroscopic Equations An arbitraiy control volume of finite size is bounded by a surface of area with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitraiy control volume. 

The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundaiy. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers this work is called shaft work its rate is Ws-... 

In the IBM, the presence of the solid boundary (fixed or moving) in the fluid can be represented by a virtual body force field -rp( ) applied on the computational grid at the vicinity of solid-flow interface. Considering the stability and efficiency in a 3-D simulation, the direct forcing scheme is adopted in this model. Details of this scheme are introduced in Section II.B. In this study, a new velocity interpolation method is developed based on the particle level-set function (p), which is shown in Fig. 20. At each time step of the simulation, the fluid-particle boundary condition (no-slip or free-slip) is imposed on the computational cells located in a small band across the particle surface. The thickness of this band can be chosen to be equal to 3A, where A is the mesh size (assuming a uniform mesh is used). If a grid point (like p and q in Fig. 20), where the velocity components of the control volume are defined, falls into this band, that is... 

The advection-dispersion equation follows directly from the transport laws already presented in this chapter, and the divergence principle. The latter states that the time rate of change in the concentration of a component depends on how rapidly the advective and dispersive fluxes change in distance. If, for example, more of component i moves into the control volume shown in Figure 20.1 across its left and front faces than move out across its right and back, the component is accumulating in the control volume and its concentration there increasing. The time rate of... 

In the previous chapter (Section 20.3), we showed the equation describing transport of a non-reacting solute in flowing groundwater (Eqn. 20.24) arises from the divergence principle and the transport laws. By this equation, the time rate of change in the dissolved concentration of a chemical component at any point in the domain depends on the net rate the component accumulates or is depleted by transport. The net rate is the rate the component moves into a control volume, less the rate it moves out. 

The control volume depicted in Figure 1.3 is for one fixed in position (i.e., fixed observation point) and of fixed size but allowing for variable mass within it this is often referred to as the Eulerian point of view. The alternative is the Lagrangian point of view, which focuses on a specified mass of fluid moving at the average velocity of the system the volume of this mass may change. 

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... 

To consider the control volume form of the conservation of mass for a species in a reacting mixture volume, we apply Equation (2.14) for the system and make the conversion from Equation (3.12). Here we select/ = pt, the species density. In applying Equation (3.13), v must be the velocity of the species. However, in a mixture, species can move by the process of diffusion even though the bulk of the mixture might be at rest. This requires a more careful distinction between the velocity of the bulk mixture and its individual components. Indeed, the velocity v given in Equation (3.13) is for the bulk mixture. Diffusion velocities, Vi, are defined as relative to this bulk mixture velocity v. Then, the absolute velocity of species i is given as... 

Here T is the uniform temperature in the CV. Equations (3.45) and (3.48) are all equivalent under the three approximations, and either could be useful in problems. The development of governing equations for the zone model in compartment fires is based on these approximations. The properties of the smoke layer in a compartment have been described by selecting a control volume around the smoke. The control volume surface at the bottom of the smoke layer moves with the velocity of the fluid there. This is illustrated in Figure 3.10. 

The energy conservation is written for a control volume surrounding a moving premixed flame at velocity Sa into a fuel-air mixture at rest. The equation is given below ... 

Let us consider a case of steady evaporation. We will assume a one-dimensional transport of heat in the liquid whose bulk temperature is maintained at the atmospheric temperature, 7 X. This would apply to a deep pool of liquid with no edge or container effects. The process is shown in Figure 6.9. We select a differential control volume between x and x + dx, moving with a surface velocity (—(dxo/df) i). Our coordinate system is selected with respect to the moving, regressing, evaporating liquid surface. Although the control volume moves, the liquid velocity is zero, with respect to a stationary observer, since no circulation is considered in the contained liquid. 

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. 

Control Volume Mass Balance. We can now combine equations (2.1), (2.5), (2.6), (2.11), and (2.13) into a mass balance on our box for Cartesian coordinates. After dividing hyV = dx dy dz and moving the diffusive flux terms to the right-hand side, this mass balance is... 

Consider a cyhndrical control volume of depth h that moves with the mean velocity of the tracer cloud containing two gas tracers, designated A and B. Using the cylinder as our control volume, the transport relation for each of the gas tracers can be written as... 

The study of fluid mechanics is facilitated by understanding and using the relationship between a system and a control volume. By definition, a system is a certain mass of fluid, that can move about in space. Moreover the system is free to deform as it moves. As a result it is practically impossible to follow and account for a particular mass of fluid in a flowing process. Nevertheless, because many of the basic physical laws are written in terms of a system (e.g., F = m ), it is convenient and traditional to take advantage of the notion of a system. 

Fig. 2.2 The relationship between a system and a control volume in a flow field. The control surface has an outward-normal-pointing vector, called n. The system moves with fluid velocity V, which flows through the control surfaces.

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. 

Figure 2.2 calls particular attention to how a fluid system moves relative to a fixed control volume that is, it illustrates convective transport. It is very important to note that an extensive property of the system can change owing to molecular transport (e.g., a... 

Fig. 2.19 A rectangular system and control volume, with the system moving with a velocity field toward the upper right.

Turn now to the work term dW/dt. The stress tensor causes forces on the surfaces of a control volume, through which fluid is moving, and the result is work. 

Fig. 3.11 The cylinder on the left is filled with a gas at pressure p and bounded by two pistons that can move with velocity u. The long cylindrical annulus on the left is filled with a fluid. The center rod is fixed, but the outer cylindrical shell moves upward at a constant velocity. Under these circumstances a steady state-velocity distribution will develop in the fluid as illustrated u(r), with the zero velocity at the inner-rod wall and the wall velocity at the shell surface. A cylindrical control volume with its zrz shear stresses is illustrated.

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