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  # Flow In Pipes
  
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  ### Flow In Pipes
  
  1. 1. FLOW IN PIPES
  2. 2. Laminar and Turbulent Flows
  3. 3. Laminar and Turbulent Flows  Critical Reynolds number (Recr) for flow in a round pipe Re < 2300 ⇒ laminar 2300 ≤ Re ≤ 4000 ⇒ transitional Re > 4000 ⇒ turbulent  Note that these values are approximate  For a given application, Recr depends upon:  Pipe roughness  Vibrations  Upstream fluctuations, disturbances (valves, elbows, etc. that may disturb the flow) Definition of Reynolds number
  4. 4. Laminar and Turbulent Flows  For non-round pipes, define the hydraulic diameter Dh = 4Ac/P Ac= cross-section area P = wetted perimeter  Example: open channel Ac = 0.15 * 0.4 = 0.06m2 P = 0.15 + 0.15 + 0.5 = 0.8m Don’t count free surface, since it does not contribute to friction along pipe walls! Dh = 4Ac/P = 4*0.06/0.8 = 0.3m What does it mean? This channel flow is equivalent to a round pipe of diameter 0.3m (approximately).
  5. 5. The Entrance Region  Consider a round pipe of diameter D. The flow can be laminar or turbulent. In either case, the profile develops downstream over several diameters called the entry length Lh. Lh/D is a function of Re. Lh
  6. 6. Fully Developed Pipe Flow  Comparison of laminar and turbulent flow There are some major differences between laminar and turbulent fully developed pipe flows Laminar  Can solve exactly (Chapter 9)  Flow is steady  Velocity profile is parabolic  Pipe roughness not important It turns out that Vavg= 1/2Umax and u(r)= 2Vavg(1 - r2 /R2 )
  7. 7. Fully Developed Pipe Flow Turbulent  Cannot solve exactly (too complex)  Flow is unsteady (3D swirling eddies), but it is steady in the mean  Mean velocity profile is fuller (shape more like a top-hat profile, with very sharp slope at the wall)  Pipe roughness is very important  Vavg 85% of Umax (depends on Re a bit)  No analytical solution, but there are some good semi-empirical expressions that approximate the velocity profile shape. See text Logarithmic law (Eq. 8-46) Power law (Eq. 8-49) Instantaneous profiles
  8. 8. Fully Developed Pipe Flow Wall-shear stress  Recall, for simple shear flows u=u(y), we had τ = µdu/dy  In fully developed pipe flow, it turns out that τ = µdu/dr slope slope Laminar Turbulent τw τw τw,turb > τw,lam τw = shear stress at the wall, acting on the fluid
  9. 9. Fully Developed Pipe Flow Pressure drop  There is a direct connection between the pressure drop in a pipe and the shear stress at the wall  Consider a horizontal pipe, fully developed, and incompressible flow  Let’s apply conservation of mass, momentum, and energy to this CV (good review problem!) 1 2 L τw P1 P2V Take CV inside the pipe wall
  10. 10. Fully Developed Pipe Flow Pressure drop  Conservation of Mass  Conservation of x-momentum Terms cancel since β1 = β2 and V1 = V2
  11. 11. Fully Developed Pipe Flow Pressure drop  Thus, x-momentum reduces to  Energy equation (in head form) or cancel (horizontal pipe) Velocity terms cancel again because V1 = V2, and α1 = α2 (shape not changing) hL = irreversible head loss & it is felt as a pressure drop in the pipe
  12. 12. Fully Developed Pipe Flow Friction Factor  From momentum CV analysis  From energy CV analysis  Equating the two gives  To predict head loss, we need to be able to calculate τw. How?  Laminar flow: solve exactly  Turbulent flow: rely on empirical data (experiments)  In either case, we can benefit from dimensional analysis!
  13. 13. Fully Developed Pipe Flow Friction Factor  τw = func(ρ, V, µ, D, ε) ε = average roughness of the inside wall of the pipe  Π-analysis gives
  14. 14. Fully Developed Pipe Flow Friction Factor  Now go back to equation for hLand substitute f for τw  Our problem is now reduced to solving for Darcy friction factor f  Recall  Therefore  Laminar flow: f = 64/Re (exact)  Turbulent flow: Use charts or empirical equations (Moody Chart, a famous plot of f vs. Re But for laminar flow, roughness does not affect the flow unless it is huge
  15. 15. Fully Developed Pipe Flow Friction Factor  Moody chart was developed for circular pipes, but can be used for non-circular pipes using hydraulic diameter  Colebrook equation is a curve-fit of the data which is convenient for computations (e.g., using EES)  Both Moody chart and Colebrook equation are accurate to ±15% due to roughness size, experimental error, curve fitting of data, etc. Implicit equation for f which can be solved using the root-finding algorithm in EES
  16. 16. Types of Fluid Flow Problems  In design and analysis of piping systems, 3 problem types are encountered: 1. Determine ∆p (or hL) given L, D, V (or flow rate) Can be solved directly using Moody chart and Colebrook equation 1. Determine V, given L, D, ∆p 2. Determine D, given L, ∆p, V (or flow rate) Types 2 and 3 are common engineering design problems, i.e., selection of pipe diameters to minimize construction and pumping costs However, iterative approach required since both V and D are in the Reynolds number.
  17. 17. Types of Fluid Flow Problems  Explicit relations have been developed which eliminate iteration. They are useful for quick, direct calculation, but introduce an additional 2% error
  18. 18. Minor Losses  Piping systems include fittings, valves, bends, elbows, tees, inlets, exits, enlargements, and contractions.  These components interrupt the smooth flow of fluid and cause additional losses because of flow separation and mixing  We introduce a relation for the minor losses associated with these components • KL is the loss coefficient. • Is different for each component. • Is assumed to be independent of Re. • Typically provided by manufacturer or generic table (e.g., Table 8-4 in text).
  19. 19. Minor Losses  Total head loss in a system is comprised of major losses (in the pipe sections) and the minor losses (in the components)  If the piping system has constant diameter i pipe sections j components
  20. 20. Piping Networks and Pump Selection  Two general types of networks:  Pipes in series  Volume flow rate is constant  Head loss is the summation of parts  Pipes in parallel  Volume flow rate is the sum of the components  Pressure loss across all branches is the same
  21. 21. Piping Networks and Pump Selection  For parallel pipes, perform CV analysis between points A and B  Since ∆p is the same for all branches, head loss in all branches is the same
  22. 22. Piping Networks and Pump Selection  Head loss relationship between branches allows the following ratios to be developed  Real pipe systems result in a system of non-linear equations. Very easy to solve with EES!  Note: the analogy with electrical circuits should be obvious  Flow flow rate (VA) : current (I)  Pressure gradient (∆p) : electrical potential (V)  Head loss (hL): resistance (R), however hL is very nonlinear
  23. 23. Piping Networks and Pump Selection  When a piping system involves pumps and/or turbines, pump and turbine head must be included in the energy equation  The useful head of the pump (hpump,u) or the head extracted by the turbine (hturbine,e), are functions of volume flow rate, i.e., they are not constants.  Operating point of system is where the system is in balance, e.g., where pump head is equal to the head losses.
  24. 24. Pump and systems curves  Supply curve for hpump,u: determine experimentally by manufacturer. When using EES, it is easy to build in functional relationship for hpump,u.  System curve determined from analysis of fluid dynamics equations  Operating point is the intersection of supply and demand curves  If peak efficiency is far from operating point, pump is wrong for that application.