Overview
The lecture covers the calculation of flow rate and pressure drop in pipes, introduces the friction factor, and explains the use of the Moody Diagram for laminar and turbulent pipe flow.
Flow Rate and Velocity in Pipes
- Flow rate, ( Q ), is found by integrating velocity over the differential area of a pipe's cross-section.
- Differential area in cylindrical coordinates is ( 2\pi r , dr ).
- Average velocity, ( \bar{v} ), is ( Q/\text{area} ) or ( Q/(\pi R^2) ).
- For laminar flow, average velocity can be calculated as ( \Delta P D^2 / (32 \mu L) ).
Pressure Drop and Friction Factor
- Pressure drop (( \Delta P )) in a pipe can be related to fluid density, velocity, pipe length, and diameter.
- The friction factor, ( f ), simplifies pressure drop calculations: ( \Delta P = f (L/D) (\rho \bar{v}^2/2) ).
- For laminar flow, friction factor is ( f = 64/\text{Re} ), where Re is Reynolds number.
- Head loss ( h_f ) is ( f (L/D) (\bar{v}^2/2g) ), and is used in the energy equation (modified Bernoulli).
Moody Diagram and Flow Regimes
- Moody Diagram plots friction factor (f) vs. Reynolds number (Re) and relative roughness (( \epsilon/D )).
- Four flow regions: laminar (Re < ~2100), critical (transition), fully turbulent, and transition regions.
- Laminar region: straight line, use ( f = 64/\text{Re} ), independent of roughness.
- Turbulent region: f depends on both Re and relative roughness; use Moody Diagram or formulas like the modified Colebrook equation.
- Do not design for the critical region (uncertain behavior).
- For smooth pipes (PVC, copper), use the "smooth pipe" line.
Practical Tips for Calculations
- Always calculate Reynolds number first to determine the flow regime.
- For turbulent flow, find f using Moody Diagram or equation such as modified Colebrook.
- The Moody Diagram is a dimensionless chart; both f and Re are dimensionless.
- Reading a log-log chart (like Moody) requires care—use proportional distances, not linear scaling.
Key Terms & Definitions
- Flow Rate (Q) — Volume of fluid passing through a section per unit time.
- Differential Area (( dA )) — Small element of area used in integration, ( 2\pi r , dr ).
- Friction Factor (f) — Dimensionless number representing pipe friction, used in pressure drop calculations.
- Reynolds Number (Re) — Dimensionless number characterizing flow regime; ( \text{Re} = VD/\nu ).
- Head Loss (( h_f )) — Energy loss due to pipe friction, measured in length units (m or ft).
- Relative Roughness (( \epsilon/D )) — Ratio of pipe surface roughness to diameter.
- Moody Diagram — Graphical tool to find friction factor for various flow conditions.
Action Items / Next Steps
- Practice reading and using the Moody Diagram for different pipe materials and diameters.
- Complete any assigned homework on pipe flow and friction factor calculation.
- Review energy equation from chapter five for understanding head loss applications.
- Be ready to discuss minor losses in the next lecture.