Summary of Activities

Summary of Answers and Full Solutions

Additional Reynolds Number Problems

Problem 1 (5 Points): Oil Flow

Given: μ = 0.1 Pa·s, ρ = 900 kg/m³, D = 0.05 m, v = 2 m/s

Re = (ρ × v × D) / μ = (900 kg/m³ × 2 m/s × 0.05 m) / 0.1 Pa·s = 90 / 0.1 = 900

Flow Regime: Laminar

Problem 2 (5 Points): Air Flow at 25°C

Given: μ = 1.85×10⁻⁵ Pa·s, ρ = 1.184 kg/m³, D = 0.1 m, v = 3 m/s

Re = (ρ × v × D) / μ = (1.184 kg/m³ × 3 m/s × 0.1 m) / (1.85×10⁻⁵ Pa·s) ≈ 0.3552 / 1.85×10⁻⁵ = 19,200

Flow Regime:Turbulent

Problem 3(5 Points): Glycerin Solution

Given: μ = 1.2 Pa·s, ρ = 1260 kg/m³, D = 0.02 m, v = 0.5 m/s

Re = (ρ × v × D) / μ = (1260 kg/m³ × 0.5 m/s × 0.02 m) / 1.2 Pa·s = 12.6 / 1.2 = 10.5

Flow Regime: Laminar

Problem 4 (5 Points): Ethanol Flow

Given: μ = 0.0012 Pa·s, ρ = 789 kg/m³, D = 0.03 m, v = 1.5 m/s

Re = (ρ × v × D) / μ = (789 kg/m³ × 1.5 m/s × 0.03 m) / 0.0012 Pa·s = 35.505 / 0.0012 = 29,588

Flow Regime: Turbulent

Step-by-Step Reynolds Number Computations with Problem Statements

Problem 1 (5 Points)

Problem: A fluid with a viscosity of 0.001 Pa·s flows through a pipe with a diameter of 0.05 m. The fluid velocity is 0.2 m/s. Use a density of 1000 kg/m³. Calculate the Reynolds number and determine the flow regime.

Computation:

• Given: μ = 0.001 Pa·s, D = 0.05 m, v = 0.2 m/s, ρ = 1000 kg/m³
• Re = (ρ × v × D) / μ = (1000 × 0.2 × 0.05) / 0.001 = 10,000
• Flow Regime: Turbulent

Problem 2 (5 Points)

Problem: Water at 20°C flows through a pipe with a diameter of 0.1 m. The flow rate is 0.01 m³/s. Use a viscosity of 0.001 Pa·s and a density of 1000 kg/m³. Calculate the Reynolds number and identify the flow regime.

Computation:

• Given: Q = 0.01 m³/s, D = 0.1 m, μ = 0.001 Pa·s, ρ = 1000 kg/m³
• A = πD²/4 = π(0.1)²/4 = 0.00785 m²
• v = Q/A = 0.01 / 0.00785 ≈ 1.27 m/s
• Re = (1000 × 1.27 × 0.1) / 0.001 ≈ 127,000
• Flow Regime: Turbulent

Problem 3 (5 Points)

Problem: Oil with a viscosity of 0.05 Pa·s flows through a pipe with a diameter of 0.02 m. The fluid velocity is 1.5 m/s. Use a density of 900 kg/m³. Calculate the Reynolds number and classify the flow regime.

Computation:

• Given: μ = 0.05 Pa·s, D = 0.02 m, v = 1.5 m/s, ρ = 900 kg/m³
• Re = (900 × 1.5 × 0.02) / 0.05 = 540
• Flow Regime: Laminar

Problem 4 (5 Points)

Problem: Air at 25°C flows through a duct with a diameter of 0.3 m. The air velocity is 10 m/s. Use a viscosity of 1.8×10⁻⁵ Pa·s and a density of 1.1839 kg/m³. Calculate the Reynolds number and determine the flow regime.

Computation:

• Given: μ = 1.8×10⁻⁵ Pa·s, D = 0.3 m, v = 10 m/s, ρ = 1.1839 kg/m³
• Re = (1.1839 × 10 × 0.3) / (1.8×10⁻⁵) ≈ 197,317
• Flow Regime: Turbulent

Problem 5 (5 Points)

Problem: A fluid with a viscosity of 0.002 Pa·s flows through a pipe with a diameter of 0.08 m. The fluid velocity is 0.5 m/s. Use a density of 1000 kg/m³. Calculate the Reynolds number and determine the flow regime.

Computation:

• Given: μ = 0.002 Pa·s, D = 0.08 m, v = 0.5 m/s, ρ = 1000 kg/m³
• Re = (1000 × 0.5 × 0.08) / 0.002 = 20,000
• Flow Regime: Turbulent

Step-by-Step Derivation of Reynolds Number Equation - Explanation Questions (5 Points each)

• How does Newton's second law of motion apply to fluid dynamics in the derivation of the Reynolds number?
  Newton's second law (F = ma) is applied to fluid particles to evaluate the balance between inertial and viscous forces. It forms the theoretical basis for deriving dimensionless numbers like Reynolds number by comparing these forces.
• What steps are involved in calculating the inertial force for a fluid particle using its density, characteristic length, and velocity?
  Inertial force is calculated as:
  F_inertia = ρ × v² × L² (proportional estimate)
  where ρ is fluid density, v is velocity, and L is the characteristic length.
• Describe the process of determining the viscous force in a fluid using dynamic viscosity, area, and velocity gradient.
  Viscous force is estimated using:
  F_viscous = μ × A × (dv/dy)
  where μ is dynamic viscosity, A is the contact area, and dv/dy is the velocity gradient.
• Explain how the ratio of inertial force to viscous force leads to the derivation of the Reynolds number.
  Reynolds number (Re) is defined as:
  Re = (Inertial force) / (Viscous force) = (ρ × v × L) / μ
  It quantifies the relative significance of inertial effects compared to viscous damping in fluid motion.
• What is the significance of the Reynolds number in distinguishing between laminar and turbulent flow regimes in fluid dynamics?
  Reynolds number determines the flow regime:
  • Re < 2300: Laminar (smooth and orderly)
  • 2300 ≤ Re ≤ 4000: Transitional
  • Re > 4000: Turbulent (chaotic and mixed)
  It helps predict the behavior of fluid in different engineering applications.

Fully Developed Laminar Flow Problems

• Problem 1(5 Points): Re = 50,000 → Turbulent, Entry Length ≈ 300 m
• Problem 2(5 Points): Re = 450,000 → Turbulent, Entry Length ≈ 4050 m
• Problem 3(5 Points): Re = 100,000 → Turbulent, Entry Length ≈ 600 m

Fully Developed Turbulent Flow (5 Points)

Wind Turbine Blade: Reₓ = 500,000 at x = 1.5 m

Boundary Layer Thickness: δ = 0.37x / Re^(1/5) ≈ 3.7 cm