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Conservation of Mass

February 18, 2013

Keywords

Conservation of Mass Equation
Mass Conservation

According to the law of mass conservation mass can neither be created
nor destroyed. Therefore in the give system, inflow, outflow or any change
must be balance. A system at constant volume can show mass flow in and
out for a limited increment of time and can be expressed as:

dM = ρi vi Ai dt – ρo vo Ao dt

Where; dM = Change in mass of the system in kg

ρ = Density in kg/m3

v = Speed in m/s

A = Area in m2

dt = Time in sec

The magnitude of dM would be negative if the outflow is higher compare to
inflow while would be positive if inflow is more than the outflow. The fluid
mechanics, Equation of Continuity and the Bernoulli Equation is entirely
based on this law. Let’s discuss an example of conversation law.

The velocity of water in a pipe of 50 mm diameter is 2 m/s and density
is 1000 kg/m3. However outflow of water through a pipe of diameter 30
mm with a velocity of 2.5 m/s. Calculate the change in content after 40
minutes.

Since; dM = ρi vi Ai dt – ρo vo Ao dt

dM = (1000)(2)(3.14 (0.05 )2/4) (40 x 60 )

– (1000)(2.5)(3.14 (0.03 )2/4)(40 x 60)

Change in content after 40 minutes would be = 2590.5 kg

The conservation equations are based on two basic principles; one is
conservation laws and another is constitutive relations. There are mainly
two conservation laws; law of conservation of mass given by Conservation
of Mass Equation and Newton’s law for the conservation of momentum,
which shows the equivalence of the rate of change of momentum is equal to
the sum of the applied forces.

For fluid systems, its bit complicated to apply both laws as fluids are
transported with the mean flow.
Let’s first derive the equation of mass conservation for a sphere of radius r,
mass M and density of material present is ρ. The mass equation shows the
relation between these three parameters; mass, density and radius. Let’s

assume a thin spherical shell inside the main sphere of inner radius r and
outer radius r+δr. The volume of shell will be:
Surface area x thickness= 4πr2. δr
Since mass = density x volume
=4πr2. ρ. δr
The mass difference between both spheres of radius r and r+δr would be;
δM= M r+δr -Mr = (dM/dr) δr
Hence dM/dr = 4πr2. ρ ……………(1)
Equation (1) is called as equation of mass conservation.

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