Lecture Notes: Fluid Mechanics - Chapter 8

General Announcements

• Lecturer was late due to personal reasons.
• Current chapter: Chapter 8.

Overview of Chapter 8

• Focus on viscous flow in pipes, specifically for chemical engineers.
• Problems selected are relevant for exams.
• Emphasis on applying formulas, not deriving them.

Recap of Previous Chapters

• Chapter 1: Viscous flow, shear stress, viscosity.
• Chapter 2: Pressure variation with height/depth.
• Chapter 3: Ideal flow, Bernoulli's equation, continuity.

Viscous Flow in Pipes

• Targeted towards chemical engineers.
• Internal flow (in pipes, valves) vs external flow (aerospace).

Laminar vs Turbulent Flow

• Laminar Flow: Orderly, can be mathematically described (e.g., Newton's second law).
• Turbulent Flow: Disorderly, challenging to predict precise flow velocities.

Reynolds Number

• Key to distinguishing flow regimes.
• Formula: Reynolds number = (Inertial forces) / (Viscous forces).
• Components:
  • ( \rho ): Density
  • ( u ): Average velocity
  • ( d ): Inner diameter of the pipe
  • ( \mu ): Viscosity

Thresholds

• Laminar: ( 0 < Re < 2000 )
• Turbulent: ( Re > 4000 )
• Transitional: ( 2000 < Re < 4000 ) (avoid designing in this range)

Calculations

• Hydraulic Diameter:
  • ( D_h = \frac{4A}{P} )
  • Useful for non-circular cross-sections.

Entrance Flow Region

• Non-viscous core gradually becomes fully viscous.
• Length ( L_e ) differs for laminar and turbulent.
  • Laminar: ( L_e = 0.06 \cdot Re \cdot d )
  • Turbulent: ( L_e = 4.4 \cdot Re^{0.16} \cdot d )

Predicting Flow Velocity

• Force balance in x-direction leads to a solution for flow velocity.
• Velocity Profile:
  • Derived using Newton's Second Law.
  • ( u = \frac{\Delta p}{16 \mu L} \left(1 - \left(\frac{2r}{D}\right)^2\right) )
  • Maximum velocity at the center, ( u_{max} = \frac{\Delta p D^2}{16 \mu L} )

Modified Bernoulli's Equation

• Extended to account for viscosity and energy input/output.
• Additional terms:
  • ( h_{pump} ) for added energy.
  • ( h_L ) for energy loss due to friction.
  • ( \alpha ) (correction factor):
    • Laminar: 2
    • Turbulent: 1

Application Examples

• Calculate Reynolds number and entrance flow region for different scenarios.

Exam Preparation

• Upcoming mock exam, not compulsory but recommended for practice.
• Focus on understanding formula application and concepts over memorization.

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Note: This lecture includes important formulas and distinctions crucial for exams. Practice problems provided align with the lecture content. Ensure familiarity with Reynolds number calculations and modified Bernoulli applications.