5.2-Control Volume Approach, fluid mech
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Control Volume Approach
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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Control Volume Approach
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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Control Volume Approach
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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Control Volume Approach
(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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Control Volume Approach
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.
Copyright ¨ 2009 John Wiley & Sons, Inc. All rights reserved.
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