Lesson 5 - Principles of Fluid Mechanics 2 SAQ

** Bright beer is flowing down a pipe with an internal diameter of 125 mm. If the density and viscosity are 1,005 kg/m³ and 2 cP respectively, and the flow velocity is 2 m/s, confirm that the flow is turbulent. What internal pipe diameter would be required to ensure the flow was laminar, whilst achieving the same volumetric flowrate?

Response:

We calculate the Reynolds number to confirm if the flow is turbulent:

Hence, Re > 4,000, which confirms the flow is turbulent.

Laminar flow is characterised by a Reynolds number below 2100. To ensure our flow is laminar, we would require Re < 2,100, however, we cannot select a lower pipe diameter to reduce the Re as doing so would result in a lower volumetric flowrate (for the same velocity), and we are required to achieve the same flowrate. We therefore need to use a higher diameter pipe, with lower velocity, to achieve Re < 2,000 for the same flowrate. So first, let us work out what the flowrate was in the original 125 mm pipe:

To set a Re = 2,100 requires the following:

Rearranging for u:

Consequently, we cannot solve directly for u or d. To solve this for these two parameters, we need another equation involving u and d.  For this, we can turn to the requirement for the new pipe diameter to not only have laminar flow but achieve the same volumetric flowrate as previously. So expressing the flowrate as:

Equating this expression with the previous one for u gives:

And hence u:

Let us just double-check that this combination of d and u gives Re = 2,000 and Q = 0.024 m³/s:

(There is some rounding error in the calculations.)

And Q:

What should be obvious in this problem, in this case if we wish to transport bright beer, is that under laminar flow conditions, at matched flowrates, very large diameter pipes are required.