# Mass Continuity Formula

This principle is known as the conservation of mass, it claims that if there are no possible discharge of mass to another system, the mass in the system will remain constant at any time.

Mass entering per unit time = Mass leaving per unit time

The mass is written in terms of the density of a fluid and the volume occupied.

ρ_{E} V_{E} = ρ_{L }V_{L}_{}

Where we have:

V: Volume of the fluid that is variating and being transfer from one place to another.

ρ: Density of the fluid

Another way to write this formula is,

ρ_{E} V_{E} - ρ_{L }V_{L } = ρ_{E} A_{E }v_{E} - ρ _{L }A_{L }v_{L} = 0

v: Upstream velocity of the fluid

A: Is the transverse area of the pipe

Mass Continuity Formula Questions:

1) A fluid is with density 1 Kg/m^{3} is moving through a pipe which has a transverse area of 0.3 m^{2} in one side and 1.3 m^{2} in the other. All the fluid in the tube of 30 cm has the same density, its velocity when entering the pipe is 1 m/s. What is the mass flowing through the tube?

Answer: The total mass in one side of the tube is given by

ρ V_{E} = ρ_{ }v_{E} A_{E} Δt

where, Δt is the time I takes to the fluid to go through the pipe. The equation of the mas leaving the pipe is,

ρ V_{L} = ρ_{ }v_{L} A_{L} Δt

Using the formula of mass continuity

ρ_{E} A_{E }v_{E} = ρ_{L }A_{L }v _{L}

using the fact that the fluid is the same along the pipe, ρ_{E} = ρ _{L}, from which is obtained

A_{E }/A_{L}*v_{E} =v_{L}

0.3 m^{2} / 1.3 m^{2} * 1 m/s = 0.23 m/s.

Knowing the large of the tube and its velocity we can calculate the Δt

Δt = x/v_{L} = 0.3 m / 0.23 m/s = 1.3 s

Using,

Total mass = ρ V_{L} = ρ_{ }v_{L} A_{L} Δt

= 1 Kg/m^{3} * 0.23 m/s * 1.3 m^{2} *1.3 s

= 0.38 Kg

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