Question

Water at \(15^{\circ} \mathrm{C}\) is flowing through a 5 -cm-diameter smooth tube with a length of \(200 \mathrm{~m}\). Determine the Darcy friction factor and pressure loss associated with the tube for (a) mass flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) and \((b)\) mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\).

Short answer

Question: Calculate the Darcy friction factor and the pressure loss associated with the tube for two different mass flow rates: (a) 0.02 kg/s and (b) 0.5 kg/s.

Step-by-step solution

Calculate average water velocity

Calculate the average water velocity (v) in the tube using the mass flow rate (ṁ) and the cross-sectional area (A) of the tube. The formula to calculate average water velocity is: \(v = \frac{ṁ}{\rho A}\) where ρ is the water density at \(15^{\circ} \mathrm{C}\), which is about \(1000 \mathrm{~kg/m^3}\), and A is the cross-sectional area of the tube calculated as: \(A = \pi \frac{D^2}{4}\) where D is the diameter of the tube.

Calculate the Reynolds number

Calculate the Reynolds number (Re) using the average water velocity (v), the tube diameter (D), and the kinematic viscosity (ν) of the water at \(15^{\circ} \mathrm{C}\). The formula for Reynolds number is: \(Re = \frac{vD}{\nu}\) The kinematic viscosity of water at \(15^{\circ} \mathrm{C}\) is about \(1.14 \times 10^{-6} \mathrm{~m^2/s}\).

Determine the Darcy friction factor

Determine the Darcy friction factor (f) associated with the tube using the Reynolds number (Re). For a smooth tube, the Darcy friction factor can be obtained using the Blasius equation: \(f = 0.079 \times Re^{-1/4}\)

Calculate the pressure loss

Calculate the pressure loss (ΔP) in the tube using the Darcy friction factor (f), the tube length (L), the average water velocity (v), and the tube diameter (D). The pressure loss formula is: \(ΔP = f \frac{L}{D} \frac{1}{2} \rho v^2\) Now, let's find the Darcy friction factor and pressure loss for both mass flow rates. (a) Mass flow rate (\(ṁ = 0.02 \mathrm{~kg/s}\)) 1. Calculate average water velocity 2. Calculate Reynolds number 3. Determine the Darcy friction factor 4. Calculate the pressure loss (b) Mass flow rate (\(ṁ = 0.5 \mathrm{~kg/s}\)) 1. Calculate average water velocity 2. Calculate Reynolds number 3. Determine the Darcy friction factor 4. Calculate the pressure loss

Key concepts

Fluid Mechanics

Fluid mechanics is the branch of physics that deals with the behavior of fluids (liquids and gases) in motion and at rest. It encompasses various phenomena, including flow through pipes, air resistance, and water waves. In engineering, fluid mechanics is crucial for designing systems like pipelines, hydraulic machines, and even aircraft.

Understanding fluid mechanics helps predict how fluids will behave under different conditions of pressure, temperature, and force. For example, knowing the flow characteristics of water through a pipe allows engineers to design systems that minimize energy consumption and prevent damage. In this context, the exercise focuses on predicting pressure loss and determining the Darcy friction factor, which are essential factors in fluid mechanics applications.

Reynolds Number

The Reynolds number (Re) is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is crucial in fluid mechanics because it indicates whether the flow will be laminar or turbulent.

• Laminar flow: Smooth, orderly. Lower Re values.
• Turbulent flow: Chaotic, mixed. Higher Re values.

Calculating the Reynolds number involves using the formula: \[Re = \frac{vD}{u}\] where \(v\) is the velocity, \(D\) is the diameter of the pipe, and \(u\) is the kinematic viscosity.

The transition between laminar and turbulent flow typically occurs around \(Re \approx 2300\). By determining the Reynolds number, engineers can adapt their designs to either maintain laminar flow for efficiency or allow for turbulent flow when mixing effects are desired.

Pressure Loss

Pressure loss occurs when there is a decrease in pressure as a fluid flows through a pipe or duct. This is due to friction and turbulence and can affect the efficiency of a fluid transport system.

Factors influencing pressure loss include:

• Diameter and length of the pipe.
• Surface roughness of the pipe.
• Flow velocity.

The formula used to calculate pressure loss, also known as head loss, is: \[ΔP = f \frac{L}{D} \frac{1}{2} \rho v^2\] where \(f\) is the Darcy friction factor, \(L/D\) is the length-to-diameter ratio, and \(\frac{1}{2} \rho v^2\) is the dynamic pressure.

Understanding pressure loss is vital for maintaining optimal flow conditions and energy efficiency in pipelines, preventing issues such as leakages or pipe failures.

Kinematic Viscosity

Kinematic viscosity is a measure of a fluid's resistance to flow under gravity. It is expressed as the ratio of dynamic viscosity to the fluid's density. In simpler terms, it describes how "thick" or "thin" a fluid is.

Expressed in units of square meters per second (\(m^2/s\)), kinematic viscosity helps determine the Reynolds number, which influences flow patterns. The lower the kinematic viscosity, the easier it is for a fluid to flow, and vice versa.

During exercises like the one described, knowing the kinematic viscosity of water allows accurate computations of the Reynolds number, and subsequently, the Darcy friction factor, both critical in analyzing fluid flow in pipes.