20°C water flows through a 2.0-m-long, 6.0-mm-diameter pipe. What is the maximum flow rate in L/min for which the flow is laminar?

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1. Determine the condition for laminar flow using the Reynolds number. The flow is laminar if the Reynolds number \( Re \) is less than 2000. The Reynolds number is given by \( Re = \frac{\rho v D}{\mu} \), where \( \rho \) is the density of the fluid, \( v \) is the flow velocity, \( D \) is the diameter of the pipe, and \( \mu \) is the dynamic viscosity of the fluid.
2. Rearrange the Reynolds number formula to solve for the maximum velocity \( v_{\text{max}} \) that ensures laminar flow: \( v_{\text{max}} = \frac{Re \cdot \mu}{\rho \cdot D} \). Use \( Re = 2000 \) as the threshold for laminar flow.
3. Look up the properties of water at 20°C: the density \( \rho \) is approximately \( 998 \; \text{kg/m}^3 \), and the dynamic viscosity \( \mu \) is approximately \( 1.002 \times 10^{-3} \; \text{Pa·s} \). Substitute these values, along with the pipe diameter \( D = 6.0 \; \text{mm} = 0.006 \; \text{m} \), into the equation for \( v_{\text{max}} \).
4. Once \( v_{\text{max}} \) is calculated, determine the maximum volumetric flow rate \( Q_{\text{max}} \) using the formula \( Q = v \cdot A \), where \( A \) is the cross-sectional area of the pipe. The area is given by \( A = \pi \cdot \left( \frac{D}{2} \right)^2 \). Substitute \( v_{\text{max}} \) and \( A \) into the equation for \( Q \).
5. Convert the volumetric flow rate \( Q_{\text{max}} \) from \( \text{m}^3/\text{s} \) to \( \text{L/min} \) by multiplying by \( 1000 \; \text{L/m}^3 \) and \( 60 \; \text{s/min} \). This gives the maximum flow rate in \( \text{L/min} \) for which the flow remains laminar.

Key Concepts

Laminar Flow

Laminar flow is a type of fluid motion characterized by smooth, parallel layers of fluid that do not mix. In this regime, the flow is orderly, and the fluid moves in straight lines, which minimizes turbulence. The flow is typically laminar when the Reynolds number, a dimensionless quantity, is less than 2000.

Reynolds Number

The Reynolds number (Re) is a dimensionless value that helps predict flow patterns in different fluid flow situations. It is calculated using the formula Re = (density × velocity × characteristic length) / viscosity. A low Reynolds number indicates laminar flow, while a high number suggests turbulent flow, making it crucial for determining the flow regime in pipes.

Flow Rate

Flow rate is the volume of fluid that passes through a given surface per unit time, commonly expressed in liters per minute (L/min). It is influenced by factors such as pipe diameter, fluid velocity, and viscosity. Understanding flow rate is essential for determining how much fluid can move through a pipe under specific conditions, particularly in assessing whether the flow remains laminar.