
Conservation of Mass


Solid Mechanics
The conservation of mass is a fundamental concept of
physics along with the conservation of
energy and the conservation of momentum.
Within some problem domain, the amount of mass remains constantmass
is neither created nor destroyed. This seems quite obvious, as long
as we are not talking about black holes or very exotic physics
problems. The mass of any object can be determined by multiplying the
volume
of the object by the density of the
object. When we move a solid object, as shown at the top of the
slide, the object retains its shape, density, and volume. The mass of
the object, therefore, remains a constant between state "a" and state
"b."
Fluid Statics
In the center of the figure, we consider an amount of a static
fluid ,
liquid or gas.
If we change the fluid from some state "a"
to another state "b" and allow it to come to rest, we find that,
unlike a solid, a fluid may change its shape. The amount of fluid,
however, remains the same. We
can calculate the amount of fluid by multiplying the density
times the volume. Since the mass remains constant, the product of the
density and volume also remains constant. (If the density remains
constant, the volume also remains constant.) The shape can change,
but the mass remains the same.
Fluid Dynamics
Finally, at the bottom of the slide, we consider the changes for a
fluid that is moving through our domain. There is no accumulation or
depletion of mass, so mass is conserved within the domain. Since the
fluid is moving, defining the amount of mass gets a little tricky.
Let's consider an amount of fluid that passes through point "a" of
our domain in some amount of time t. If the fluid passes through an
area A at velocity V, we can define the volume Vol
to be:
Vol = A * V * t
A units check gives area x length/time x time = area x length =
volume. Thus the mass at point "a" ma is simply density r
times the volume at "a".
ma = (r * A * V * t)a
If we compare the flow through another point in the domain, point
"b," for the same amount of time t, we find the mass at "b"
mb to be the
density times the velocity times the area times the time at "b":
mb = (r * A * V * t)b
From the conservation of mass, these two masses are the same
and since the
times are the same, we can eliminate the time dependence.
(r * A * V)a = (r * A * V)b
or
r * A * V = constant
The conservation of mass gives us an easy way to determine the
velocity of flow in a tube if the density is constant. If we can
determine (or set) the velocity at some known area, the
equation tells us the value of velocity for any other area. In our
animation, the area of "b" is one half the area of "a." Therefore,
the velocity at "b" must be twice the velocity at "a." If we desire a
certain velocity in a tube, we can determine the area necessary to
obtain that velocity. This information is used in the design of
wind tunnels. The quantity density
times area times velocity has the dimensions of mass/time and is
called the mass flow rate. This quantity
is an important parameter in determining the
thrust produced by a propulsion
system.
As the speed of the flow approaches the
speed of sound the density of the
flow is no longer a constant and we must then use a
compressible form
of the mass flow rate equation.
The conservation of mass equation also occurs in a differential
form as part of the
NavierStokes equations
of fluid flow.
Here's is a still version of the graphic that you can use in your own
presentation of this material:
Guided Tours

Basic Fluid Dynamics Equations:

Speed of Sound:
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