5.2-Control Volume Approach.pdf

Control Volume Approach

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5.2 Control Volume Approach

The control volume (or Eulerian) approach is the method whereby a volume in the flow field is identified and

the governing equations are solved for the flow properties associated with this volume. A scheme is needed that

allows one to rewrite the equations for a moving fluid particle in terms of flow through a control volume. Such

a scheme is the Reynolds transport theorem introduced in this section. This is a very important theorem because

it is used to derive many of the basic equations used in fluid mechanics.

System and Control Volume

A system is a continuous mass of fluid that always contains the same matter. A system moving through a flow

field is shown in Fig. 5.6. The shape of the system may change with time, but the mass is constant since it always

consists of the same matter. The fundamental equations, such as Newton's second law and the first law of

thermodynamics, apply to a system.

Figure 5.6 System, control surface, and control volume in a flow field.

A control volume is volume located in space and through which matter can pass, as shown in Fig. 5.6. The

system can pass through the control volume. The selection of the control volume position and shape is problem

dependent. The control volume is enclosed by the control surface as shown in Fig. 5.6. Fluid mass enters and

leaves the control volume through the control surface. The control volume can deform with time as well as

move and rotate in space and the mass in the control volume can change with time.

Intensive and Extensive Properties

An extensive property is any property that depends on the amount of matter present. The extensive properties of

a system include mass, m , momentum, m v (where v is velocity), and energy, E . Another example of an extensive

property is weight because the weight is mg .

An intensive property is any property that is independent of the amount of matter present. Examples of intensive

properties include pressure and temperature. Many intensive properties are obtained by dividing the extensive

property by the mass present. The intensive property for momentum is velocity v , and for energy is e, the energy

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per unit mass. The intensive property for weight is g .

In this section an equation for a general extensive property, B , will be developed. The corresponding intensive

property will be b . The amount of extensive property B contained in a control volume at a given instant is

(5.10)

where dm and are the differential mass and differential volume, respectively, and the integral is carried out

over the control volume.

Property Transport Across the Control Surface

When fluid flows across a control surface, properties such as mass, momentum, and energy are transported with

the fluid either into or out of the control volume. Consider the flow through the control volume in the duct in

Fig. 5.7. If the velocity is uniformly distributed across the control surface, the mass flow rate through each cross

section is given by

The net mass flow rate out * of the control volume, that is, the outflow rate minus the inflow rate, is

The same control volume is shown in Fig. 5.8 with each control surface area represented by a vector, A , oriented

outward from the Control volume and with magnitude equal to the crosssectional area. The velocity is

represented by a vector, V . Taking the dot product of the velocity and area vectors at both stations gives

because at station 1 the velocity and area have the opposite directions while at station 2 the velocity and area

vectors are in the same direction. Now the net mass outflow rate can be written as

(5.11)

Equation (5.11) states that if the dot product ρ V · A is summed for all flows into and out of the control volume,

the result is the net mass flow rate out of the control volume, or the net mass efflux. If the summation is

positive, the net mass flow rate is out of the control volume. If it is negative, the net mass flow rate is into the

control volume. If the inflow and outflow rates are equal, then

Figure 5.7 Flow through control volume in a duct.

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Figure 5.8 Control surfaces represented by area vectors and velocities by velocity vectors.

In a similar manner, to obtain the net rate of flow of an extensive property B out of the control volume, the mass

flow rate is multiplied by the intensive property b :

(5.12)

To reinforce the validity of Eq. (5.12) one may consider the dimensions involved. Equation (5.12) states that the

flow rate of B is given by

Equation (5.12) is applicable for all flows where the properties are uniformly distributed across the area. If the

properties vary across a flow section, then it becomes necessary to integrate across the section to obtain the rate

of flow. A more general expression for the net rate of flow of the extensive property from the control volume is

thus

(5.13)

Equation (5.13) will be used in the most general form of the Reynolds transport theorem.

Reynolds Transport Theorem

The Reynolds transport theorem, fundamental to the control volume approach, is developed in this section. It

relates the Eulerian and Lagrangian approaches. The Reynolds transport theorem is derived by considering the

rate of change of an extensive property of a system as it passes through a control volume.

A control volume with a system moving through it is shown in Fig. 5.9. The control volume is enclosed by the

control surface identified by the dashed line. The system is identified by the darker shaded region. At time t the

system consists of the material inside the control volume and the material going in, so the property B of the

system at this time is

(5.14)

At time t + t the system has moved and now consists of the material in the control volume and the material

passing out, so B of the system is

(5.15)

The rate of change of the property B is

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(5.16)

Substituting in Eqs. 5.14 and 5.15 results in

(5.17)

Rearranging terms yields

(5.18)

The first term on the right side of Eq. (5.18) is the rate of change of the property B inside the control volume, or

(5.19)

The remaining terms are

These two terms can be combined to give

(5.20)

or the net efflux, or net outflow rate, of the property B through the control surface. Equation (5.18) can now be

written as

Substituting in Eq. (5.13) for and Eq. (5.10) for B cv results in the most general form of the Reynolds

transport theorem :

(5.21)

This equation may be expressed in words as

The left side of the equation is the Lagrangian form; that is, the rate of change of property B evaluated moving

with the system. The right side is the Eulerian form; that is, the change of property B evaluated in the control

volume and the flux measured at the control surface. This equation applies at the instant the system occupies the

control volume and provides the connection between the Lagrangian and Eulerian descriptions of fluid flow.

The application of this equation is called the control volume approach. The velocity V is always measured with

respect to the control surface because it relates to the mass flux across the surface.

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Figure 5.9 Progression of a system through a control volume.

A simplified form of the Reynolds transport theorem can be written if the mass crossing the control surface

occurs through a number of inlet and outlet ports, and the velocity, density and intensive property b are

uniformly distributed (constant) across each port. Then

(5.22)

where the summation is carried out for each port crossing the control surface.

Interactive Application: Reynolds Transport Theorem

An alternative form can be written in terms of the mass flow rates:

(5.23)

where the subscripts i and o refer to the inlet and outlet ports, respectively, located on the control surface. This

form of the equation does not require that the velocity and density be uniformly distributed across each inlet and

outlet port, but the property b must be.

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