Question

A compressible fluid flows through a compressor that increases the density from 1 kg/m3 to 5 kg/m3. The cross-sectional area of the inlet pipe is 3 m2 and that of the discharge pipe is 1 m2. The relation between the discharge volume flow rate and the inlet volume flow rate is

Answer 1

The relation between the discharge volume flow rate and the inlet volume flow rate is
`(1)/(5)`.

Step-by-step explanation:

No matter if fluid is compressible or not, mass throughout compressor, a device that works at steady state, must be conserved according to Principle of Mass Conservation:

`\dot m_(in)-\dot m_(out) = 0` (Eq. 1)

Where
`\dot m_(in)` and
`\dot m_(out)` are mass flows at inlet and outlet, measured in kilograms per second.

After applying Dimensional analysis, we expand the equation above as follows:

`\rho_(in)\cdot \dot V_(in) - \rho_(out)\cdot \dot V_(out) = 0` (Eq. 2)

Where:

`\rho_(in)`,
`\rho_(out)` - Fluid densities at inlet and outlet, measured in kilograms per cubic meter.

`\dot V_(in)`,
`\dot V_(out)` - Volume flow rates at inlet and outlet, measured in cubic meters per second.

After some algebraic handling, we find the following relationship:

`\rho_(out)\cdot \dot V_(out) = \rho_(in)\cdot \dot V_(in)`

`(\dot V_(out))/(V_(in)) = (\rho_(in))/(\rho_(out))` (Eq. 3)

If we know that
`\rho_(in) = 1\,(kg)/(m^(3))` and
`\rho_(out) = 5\,(kg)/(m^(3))`, then the relation between the discharge volume flow rate and the inlet volume flow rate is:

`(\dot V_(out))/(\dot V_(in)) = (1\,(kg)/(m^(2)) )/(5\,(kg)/(m^(3)) )`

`(\dot V_(out))/(\dot V_(in)) = (1)/(5)`

The relation between the discharge volume flow rate and the inlet volume flow rate is
`(1)/(5)`.