Volume control

A control volume is an open system that involves mass flow in and out. The conservation of mass statement for a control volume takes account of all mass flow in and mass flow ouf as well as change in any mass inside fhe confrol volume. The sfafemenf is derived as 

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Now Integrate equations (11.1) through the shaded control volumes indicated in Figure 11.1, with the result... 

Fig. 16. Nonstaggered grid system showing the velocity and scalar control volumes (cv). The scalar quantities are calculated at the nodes (eg, P, N, S, E, W)...

Fig. 21. Control volumes for appHcation of the integral equations of motion where 1, 2, and 3 are the location of control surfaces and Sy (a) general,...

To illustrate the use of the momentum balance, consider the situation shown in Figure 21c in which the control volume is bounded by the pipe wall and the cross sections 1 and 2. The forces acting on the fluid in the x-direction are the pressure forces acting on cross sections 1 and 2, the shear forces acting along the walls, and the body force arising from gravity. The overall momentum balance is... 

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... 

Multiplication by the time lequiced for a unit mass of fluid to enter or leave the control volume gives equation 112, where each term is for a unit mass of fluid. 

The process is internally reversible within the control volume. 

Heat transfer external to the control volume is reversible. 

For the special case of a single stream flowing through the control volume,... 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

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. 

Mass balance Apphed to the control volume, the principle of consei vation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981)... 

Simplified forms of Eq. (6-8) apply to special cases frequently found in prac tice. For a control volume fixed in space with one inlet of area Ai through which an incompressible fluid enters the control volume at an average velocity Vi, and one outlet of area Ao through which fluid leaves at an average velocity V9, as shown in Fig. 6-4, the continuity equation becomes... 

Momentum Balance Since momentum is a vector quantity, the momentum balance is a vector equation. Where gravity is the only body force acting on the fluid, the hnear momentum principle, apphed to the arbitraiy control volume of Fig. 6-3, results in the following expression (Whitaker, ibid.). 

Here g is the gravity vector and tu is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integr on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors Vi and V9, the momentum equation becomes... 

FIG. 6-4 Fixed control volume with one inlet and one outlet. 

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-... 

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

Contributing to f are losses for the entrance to the pipe, the three sections of straight pipe, the butterfly valve, and the 90 bend. Note that no exit loss is used because the discharged jet is outside the control volume. Instead, the V v2 term accounts for the kinetic energy of the discharging stream. The Reynolds number in the pipe is... 

Tittlebaum, Marty E., Roger Seals, Frank Cartledge, Stephanie Engels, Louisiana State University. State of the Art on Stabilization of Hazardous Organic Liquid Wastes and Sludges. Critical Reviews in Environmental Control, Volume 15, Issue 2,1985. 

Grid generation by subdividing the domain into a number of smaller non-overlapping subdomains. This creates a grid (or mesh) of cells (or control volumes or elements). 

Integrating the governing equations of fluid flow over all the finite control volumes of the solution domain. 

A balanced equation for every extensive property in each control volume may be written as ... 

Elementary single-component systems are those that have just one chemical species or material involved in the process. Filling of a vessel is an example of this kind. The component can be a solid liquid or gas. Regardless of the phase of the component, the time dependence of the process is captured by the same statement of the conservation of mass within a well-defined region of space that we will refer to as the control volume. 

In this chapter we will apply the conservation of mass principle to a number of different kinds of systems. While the systems are different, by the process of analysis they will each be reduced to their most common features and we will find that they are more the same than they are different. When we have completed this chapter, you will understand the concept of a control volume and the conservation of mass, and you will be able to write and solve total material balances for single-component systems. 

The net rate of mass accumulation within a control volume is equal to the rate at which mass enters the control volume by any process minus the rate at which it leaves the control volume by any process. 

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