Question

Your company is planning to build a pipeline to transport gasoline from the refinery to a field of storage tanks. The parameters for the prototype system are a pipe diameter of 1 m, with a flow velocity of 0.5 m/s at 25°C. The model system will use water at STP with a geometric scaling factor of 1 : 20. What fluid velocity is required in the model system to guarantee kinematic similarity in the form of equal Reynolds numbers?

Answer 1

The model system will need water flowing at a velocity of 2.07 meters per second to guarantee kinematic similarity in the form of equal Reynolds numbers.

Step-by-step explanation:

The Reynolds number (
`Re_(D)`) is a dimensionless criterion use for flow regime of fluids, which is defined as:

`Re_(D) = (\rho \cdot v\cdot D)/(\mu)` (Eq. 1)

Where:

`\rho` - Density, measured in kilograms per cubic meter.

`\mu` - Dynamic viscosity, measured in kilograms per meter-second.

`v` - Average flow velocity, measured in meters per second.

`D` - Pipe diameter, measured in meters.

We need to find the equivalent velocity of water used in the prototype system. In this case, we assume that
`Re_(D,gas) = Re_(D,w)`. That is:

`(\rho_(w)\cdot v_(w)\cdot D_(w))/(\mu_(w)) = (\rho_(gas)\cdot v_(gas)\cdot D_(gas))/(\mu_(gas))` (Eq. 2)

Where subindex
`w` is used for water and
`gas` for gasoline.

If we know that
`\rho_(gas) = 690\,(kg)/(m^(2))`,
`\mu_(gas) = 0.006\,(kg)/(m\cdot s)`,
`v_(gas) = 0.5\,(m)/(s)`,
`D_(gas) = 1\,m`,
`\rho_(w) = 1000\,(kg)/(m^(3))`,
`\mu_(w) = 0.0018\,(kg)/(m\cdot s)` and
`D_(w) = 0.05\,m`, then we get the following formula:

`57500 = 27777.778\cdot v_(w)`

The fluid velocity for the prototype system is:

`v_(w) = 2.07\,(m)/(y)`

The model system will need water flowing at a velocity of 2.07 meters per second to guarantee kinematic similarity in the form of equal Reynolds numbers.