Question

Consider a junction that connects three pipes A, B and C. What can we say about the mass flow rates in each pipe for steady flow?

Answer 1

Answer:

The statement regarding the mass rate of flow is mathematically represented as follows $\Rightarrow \rho \times Q_{3}=\rho \times Q_{1}+\rho \times Q_{2}$

Explanation:

A junction of 3 pipes with indicated mass rates of flow is indicated in the attached figure

As a basic sense of intuition we know that the mass of the water that is in the pipe junction at any instant of time is conserved as the junction does not accumulate any mass.

The above statement can be mathematically written as

$Mass_{Junction}=Constant\\\\\Rightarrow Mass_{in}=Mass_{out}$

this is known as equation of conservation of mass / Equation of continuity.

Now we know that in a time 't' the volume that enter's the Junction 'O' is

1) From pipe 1 = $V_{1}=Q_{1}\times t$

1) From pipe 2 = $V_{2}=Q_{2}\times t$

Mass leaving the junction 'O' in the same time equals

From pipe 3 = $V_{3}=Q_{3}\times t$

From the basic relation of density, volume and mass we have

$\rho =\frac{mass}{Volume}$

Using the above relations in our basic equation of continuity we obtain

$\rho \times V_{3}=\rho \times V_{1}+\rho \times V_{2}\\\\Q_{3}\times t=Q_{1}\times t+Q_{2}\times t\\\\\Rightarrow Q_{3}=Q_{1}+Q_{2}$

Thus the mass flow rate equation becomes $\Rightarrow \rho \times Q_{3}=\rho \times Q_{1}+\rho \times Q_{2}$