

When analyzing a system in fluid mechanics, we usually take one of two approaches: (1.) consider the flow of properties on a particlebased scale at every point in a given fluid (differential approach), or (2.) specify a fixed region and determine how its fluid properties vary with flow (control volume approach). Depending on your main interests, each approach offers its own advantages and disadvantages  along with different information. For example, consider diagram 1 which depicts a chamber containing two fluid inlets and one fluid outlet. Further, suppose you are interested in assessing how this device accumulates mass of a given chemical species. If you are a chemical engineer focused on production, you may concentrate on the flow of fluid properties inside the chamber  as opposed to how each individual particle flows. In that case, a control volume analysis would be more appropriate. The way in which fluid properties vary over the red, dotted region  which is fixed and formerly known as a control volume  would be of most interest. Conversely, if you are a researcher responsible for product development, you may be more interested in how the inlets/outlets affect flow and pressure distributions. In this scenario, the flow of individual particles as they enter/exit the chamber would offer more valuable information, and a differential approach becomes more appropriate. So, when studying fluid mechanics, keep in mind that each approach offers its own unique perspective and is tailored to specific situations.
Diagram 1: Schematic of a mixing chamber used to demonstrate how different analytical approaches 1.) provide specific sets of information and 2.) lead to certain outcomes more efficiently than others.
In this lesson, the goal is to first lay the
foundations for a control volume approach  which is most commonly used in
engineering studies and industrial applications. Such an approach will also be
useful when developing the differential analysis in subsequent lessons. Before
beginning, however, let's first recall how the most common conservation laws 
such as conservation of mass, momentum, energy, etc  are used in solid
mechanics. This will be useful for considering the flow of properties over our
control volume and extending these laws to fluid mechanics.
When dealing with solid mechanics, conservation
laws are developed over an entire system of particles. Such a system is
defined by a fixed quantity of mass enclosed in distinct, fixed boundaries.
For example, consider a rigid, cylindrical piece of aluminum. This solid body
contains a fixed mass enclosed in a cylindrical region or boundary. To apply
any conservation law to this rod, we would have to consider how properties are
conserved over the entire system inside the specified boundaries. For instance,
conservation of mass asserts that no mass is created or destroyed within the
system's boundaries:
Similarly, conservation of momentum  which leads to Newton's
familiar second law  dictates that any exterior forces exerted on the system's
boundaries will incite motion:
In either case, the rigid body's system  along
with its boundaries  is fixed as time progresses, and the system ultimately
remains unchanged  even though it may move. In other words, the rigid bodies
studied in solid mechanics generally possess a fixed volume and mass. This is
very beneficial when applying conservation laws  since the volume of interest always
coincides with the total mass.
Conversely, when considering a fluidbased system, this is usually not the case; the system's boundaries can change with time as the fluid evolves. For example, if you have a bucket of water, that system will initially contain a specific amount of mass in a given volume. However, unlike the solid rod discussed above, turning that bucket upside down will change the system's volume, and the boundaries no longer remain constant. This makes the application of conservation laws a bit more complex and challenging. Unlike the equivalent case in solid mechanics, setting the volume of interest equivalent to that inside the system's boundaries is no longer practical. Not only would this volume continually change in an unpredictable manner, but we are generally interested in conservation laws over a given, fixed region. For instance, in the example used for diagram 1, the chemical engineer was interested in the flow of properties inside the mixing chamber  not within the system's changing boundaries. This is where the control volume approach comes into play and addresses this dilemma. By focusing on a fixed region, the control volume approach allows us to once again analyze a volume with definite boundaries  as is done in solid mechanics. However, since the conservation laws apply to a system, the main challenge resides in translating these systembased laws over to a control volume basis. This is accomplished using the Reynolds Transport Theorem (RTT)  the derivation of which forms the basis of this lesson.
Conversely, when considering a fluidbased system, this is usually not the case; the system's boundaries can change with time as the fluid evolves. For example, if you have a bucket of water, that system will initially contain a specific amount of mass in a given volume. However, unlike the solid rod discussed above, turning that bucket upside down will change the system's volume, and the boundaries no longer remain constant. This makes the application of conservation laws a bit more complex and challenging. Unlike the equivalent case in solid mechanics, setting the volume of interest equivalent to that inside the system's boundaries is no longer practical. Not only would this volume continually change in an unpredictable manner, but we are generally interested in conservation laws over a given, fixed region. For instance, in the example used for diagram 1, the chemical engineer was interested in the flow of properties inside the mixing chamber  not within the system's changing boundaries. This is where the control volume approach comes into play and addresses this dilemma. By focusing on a fixed region, the control volume approach allows us to once again analyze a volume with definite boundaries  as is done in solid mechanics. However, since the conservation laws apply to a system, the main challenge resides in translating these systembased laws over to a control volume basis. This is accomplished using the Reynolds Transport Theorem (RTT)  the derivation of which forms the basis of this lesson.
As mentioned in the previous section, the
lesson's main goal is to reformulate systembased conservation laws such that
they may be applied over a fixed, control volume. This will be accomplished by
first considering a simplified, straightforward example  onedimensional duct
flow. Then, after establishing the basic concept, the new conservation law will
be extended to a generalized case. Focusing first on the simplified case,
suppose that fluid is flowing through a rectangular duct with onedimensional
flow. In other words, the velocity only changes along the duct's length in the
xdirection (V = V(x)); no change is observed along the y and zdirections. Depicted
in Diagram 2, the fluid originally contains three separate systems (I, II, and
III) at an initial time t. Furthermore, the selected control volume  which is
depicted in red and lies between points a and b  originally encloses all of
system II, and systems I and III initially reside outside of the control
volume. Now, as time progresses, you can imagine that all three systems will flow
to the right, and system II will begin to leave the fixed control volume. Similarly,
system I will also begin to enter the control volume. This is shown in Diagram
2 (b.) for a time t + dt  where dt is a small, incremental change in time. You
can clearly see that system I has entered the control volume (signified by the
green region), while a portion of system II has left (represented by the orange
area). What we are really concerned with though is not how the separate systems
evolve. Rather, we want to know how fluid properties change in the control
volume's fixed region and how that relates to a given system.
Diagram 2: Sketches of the simplified example broken down into three separate systems. At time t when the analysis begins (sketch a), the specified control volume initially coincides with System II. However, as time progresses by an amount dt (sketch b), a portion of this system leaves the control volume  while a fraction of System I enters.
To do this, we must first distinguish between extensive
and intensive fluid properties. Traditionally, extensive properties are
those which are contingent on the extent  or overall size  of a given
system. Typical examples of extensive properties are energy, mass, and momentum
 quantities proportional to a system's total amount of matter. For example, if
you take a bucket of water, dumping half of the contents into two separate
buckets will change the extensive properties. The two new buckets will end up
with half of the original bucket's total mass and energy. Conversely, intensive
properties are independent of a given system's size; they are inherent
bulk properties describing the entire system  regardless of its total amount
of matter. Common examples of intensive properties are density and temperature;
knowledge of the entire system is not needed to define each quantity. Working
with the previous example above, the water density and temperature found in
both the initial and two subsequent buckets would most likely be equivalent. Changing
the system's size (or extent) has no effect upon the overall property. Before
moving on, it is important to quickly note a special subset of intensive
properties: specific properties. These properties are identical to intensive
properties but are defined on a permass basis. Some common examples include
specific volume (m3/kg), enthalpy (kJ/kg), and internal energy
(kJ/kg). As will be shown shortly, this subset of properties is particularly
useful for computing a given region's extensive properties.
Now, suppose that you are interested in some extensive fluid property B (mass, momentum, energy, etc), which has a corresponding intensive property β. In this case, we will take that intensive property to also be a specific property; so, β then represents the total amount of B per unit mass of fluid (or β = dB/dm). To calculate the total amount of B in a given region (such as the control volume in Diagram 2), we can take its respective intensive property β and simply integrate over the region of interest. This is why knowledge of a system's intensive property becomes so important; it allows us to quickly calculate its cumulative extensive property with minimal effort. For any given control volume, this process is expressed mathematically as follows:
Now, suppose that you are interested in some extensive fluid property B (mass, momentum, energy, etc), which has a corresponding intensive property β. In this case, we will take that intensive property to also be a specific property; so, β then represents the total amount of B per unit mass of fluid (or β = dB/dm). To calculate the total amount of B in a given region (such as the control volume in Diagram 2), we can take its respective intensive property β and simply integrate over the region of interest. This is why knowledge of a system's intensive property becomes so important; it allows us to quickly calculate its cumulative extensive property with minimal effort. For any given control volume, this process is expressed mathematically as follows:
If
the density is constant throughout a given region, this expression may also be
further simplified. To do so, we will express each infinitesimal element of
mass dm as a function of the fluid density ρ and element volume dV. This yields
the following result  which is much easier to evaluate for a given control
volume:
Using
this expression, we can then evaluate the total amount of any extensive fluid
property found in a given region. It is also important to note that the constant density assumption
employed above is quite reasonable. While not always the case, the majority of
fluids are incompressible  especially water and other liquids; this means that density remains fairly constant, as the applied pressure increases. So, regardless of the fluid's state, it is reasonable to assert that density remains constant for all times t and throughout the domain.
Diagram 3: As the fluid flows through this example's duct, it will sweep out a volume whose length is V dt, and the crosssectional area A will remain constant; such a volume is depicted above. Thus, we may compute this sweeped volume as the product of that length and constant area (ie Volume = V A dt). Using the extensive/intensive property framework established above, this volume term can then be utilized to determine the total extensive property B flowing in or out of the control volume (or, in other words, the flux terms).
Returning to the simplified example above, we
can now define the total amount of any extensive property B in the specified
control volume. However, as explained above, fluid flow will also carry some
properties into and out of the control volume (refer to the green and orange
regions in Diagram 2 b.), respectively). Thus, in order to eventually perform a
balance, we will need to mathematically describe this process; so, let's do
that now. Fortunately, in this simplified example, we have made two important
assumptions which greatly simplify the calculations. As the fluid flows through
the duct, the crosssectional area remains constant, and there is no crosssectional
variation in the velocity profile (ie V = V(x)). This allows us to express each
flux term using constant values and sidestep any surface integrals. Under these
assumptions for a given time dt, the total volume swept through a constant area
A is then equivalent to that of a rectangular prism. Such a prism has a total
sweep length of (V dt) and encloses the volume depicted in Diagram 3. Using
this volume, we may then define the fluid property B flowing in or out of the
control volume as follows; this uses the same exact framework laid out above
for BCV:
At this point, we now have all essential
ingredients for considering the change of extensive property B over the control
volume. Remember, though; we are certainly interested in how fluid properties
change over a given control volume, but the main goal is to relate that
to changes in a given system. This will be accomplished by considering a
fluid property balance over Diagram 2's control volume. As shown in Diagram 2
a.), it is clear that system II initially coincides with the control volume.
So, at reference time t, we will assert that BCV (t) = BsysII(t)
and explore how this relation varies with time. After an incremental time dt
has passed, we already know that fluid flows into and out of the control
volume; this is captured by the green and orange areas in Diagram 2 b.), respectively.
Hence, after a time dt has passed, a portion of system II has flown out of the
control volume, while other fluid has entered. So, the extensive property B at
time t + dt may be expressed as BCV(t + dt) = BsysII(t +
dt)  Bflux, out + Bflux, in (refer to Diagram 2 (b.); I
am simply adding up the separate regions). Using this information, we can then
approximate the control volume's change of extensive property B as follows:
In
the last line above, I used the limitbased definition of a derivative to express system II's rate of
change. At this point, we now have an expression relating changes
between the control volume and system. However, as explained
above, the conservation laws are expressed on a systembasis; so, let's solve
for the system term and make a few finishing touches:
Using
this expression, we now not only know how an extensive property B changes in a given
system, but this also allows us to apply the common conservation laws  such as
conservation of mass, momentum, energy, etc  to a specified control volume. This
will be accomplished in subsequent lessons and play a huge role in extending
physical models to fluid mechanics.