Calculating Maximum Volumetric Flow Rate for Laminar Flow in a Pipe
Understanding Laminar Flow and Maximum Volumetric Flow Rate
For a 1.25 in. diameter pipe, what is the maximum volumetric flow rate at which water can be pumped and the flow will remain laminar?
Final answer: The maximum volumetric flow rate at which water can be pumped and the flow will remain laminar depends on the diameter of the pipe and the Reynold's number. The flow will remain laminar if the Reynold's number is below 2,000.
Explanation:
Poiseuille's law applies to the laminar flow of an incompressible fluid through a tube. Flow rate is directly proportional to the pressure difference and the fourth power of the radius, and inversely proportional to the length of the tube and the viscosity of the fluid. The maximum volumetric flow rate at which water can be pumped and the flow will remain laminar depends on the given diameter of the pipe.
To find the maximum flow rate, we need to determine the critical Reynolds number (NR) at which the flow transitions from laminar to turbulent. For laminar flow, NR is typically less than 2,000.
Given a diameter of 1.25 inches, we can convert it to meters (0.03175 m). The radius is half the diameter, so the radius is 0.015875 m. From Poiseuille's law, we know that laminar flow depends on the fourth power of the radius. Therefore, if the diameter were doubled to 2.5 inches, the radius would double as well to 0.03175 m. As a result, the maximum flow rate would increase by a factor of 2^4 = 16. Similarly, if the radius were halved to 0.0079375 m, the maximum flow rate would decrease by a factor of (1/2)^4 = 1/16 = 0.0625.
In conclusion, the maximum volumetric flow rate at which water can be pumped and the flow will remain laminar depends on the diameter of the pipe. The flow will remain laminar as long as the Reynolds number is below 2,000.
Question: How does the diameter of the pipe affect the maximum volumetric flow rate for laminar flow? Answer: The diameter of the pipe plays a significant role in determining the maximum volumetric flow rate for laminar flow. As explained earlier, the flow rate is directly proportional to the fourth power of the radius. This means that any changes in the diameter of the pipe will impact the maximum flow rate accordingly. A smaller diameter will result in a lower maximum flow rate, while a larger diameter will allow for a higher maximum flow rate, as long as the flow remains laminar with a Reynold's number below 2,000.