Pipe flow April 8 and 15, 2008 Outline Pipe Flow • Laminar and turbulent flows • Developing and fully-developed flows Larry Caretto • Laminar and turbulent velocity profiles: Mechanical Engineering 390 effects on momentum and energy Fluid Mechanics • Calculating head losses in pipes – Major losses from pipe only April 8 and 15, 2008 – Minor losses from fittings, valves, etc. • Noncircular ducts 2 Piping System What We Want to Do • Determine losses from friction forces in Fundamentals of Fluid Mechanics, 5/E by Bruce straight pipes and joints/valves Munson, Donald Young, and Theodore Okiishi Copyright © 2005 by John Wiley – Will be expressed as head loss or & Sons, Inc. All rights reserved. “pressure drop” hL = ΔP/γ • Will show that this is head loss in energy equa- tion if variables other than pressure change • System consists of – Losses in straight pipes are called “major” losses – Straight pipes – Losses in fittings, joints, valves, etc. are – Joints and valves called “minor” losses – Inlets and outlets – Minor losses may be greater than major – Work input/output 3 losses in some cases 4 Pipe Cross Section The Pipes are Full • Most pipes have circular cross section • Consider only flows where the fluid to provide stress resistance completely fills the pipe • Main exception is air conditioning ducts • Partially filled pipes are considered • Consider round pipes first then extend under open-channel flow analysis to non-circular cross sections – Extension based on using same equations as for circular pipe by defining hydraulic diameter = 4 (area) / (perimeter), which is Driving force D for circular cross sections Driving force is pressure is gravity 5 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, 6 Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. ME 390 – Fluid Mechanics 1 Pipe flow April 8 and 15, 2008 Laminar vs. Turbulent Flow Laminar vs. Turbulent Flow II • Most flows of engineering interest are turbulent – Analysis relies mainly on experimentation guided by dimensional analysis – Even advanced computer models, called computational fluid dynamics (CFD) rely on “turbulence models” that have large degree • Laminar flows of empiricism have smooth • Can get some (very limited) analytical • Turbulent flows layers of fluid results for laminar flows have fluctuations Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 7 8 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Laminar vs. Turbulent Flow III Flow Development • Condition of flow as laminar or turbulent depends on Reynolds number • For pipe flows –Re = ρVD/μ < 2100 is laminar –Re = ρVD/μ > 4000 is turbulent – 2100 < Re < 4000 is transition flow • Other flow geometries have different characteristics in Re = ρVLc/μ and different values of Re for laminar and turbulent flow limits 9 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 10 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Developing Flows Developing Flows II • Entrance regions and bends create • Entrance regions and bends create changing flow patters with different changing flow patters with different head losses head losses • Once flow is “fully developed” the head • Once flow is “fully developed” the head loss is proportional to the distance loss is proportional to the distance • Entrance pressure drop is complex – Complete entrance region treated under minor losses – Will not treat partial entrance region here 11 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 12 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. ME 390 – Fluid Mechanics 2 Pipe flow April 8 and 15, 2008 Developing Flows III Fluid Element in Pipe Flow • After development region, pressure drop (head loss) is proportional to pipe length • Equations for entrance region length, ℓe – Laminar flow: l e = 0.06Re D • Look at arbitrary element, with length ℓ, l e = 4.4Re1 6 – Turbulent flow: D and radius r, in fully developed flow – Turbulent flow rule of thumb ℓe ≈ 10D • What are forces on this element? 13 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 14 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Fully Developed Flow Extend Relation to Wall No change in momentum Flow Direction F = πr 2 p −πr 2 ()p − Δp −τ 2πr = 0 ∑ x 1 1 l • Have Δp= 2τℓ/r for any r: 0 < r < R = D/2 2τ • Pressure drop is due • For wall r = R = D/2 and τ = τ = wall Δp = l w r to viscous stresses shear stress: Δp= 2τwℓ/R = 4τwℓ/D Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 15 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 16 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Fully Developed Laminar Flow Fully Developed Laminar Flow II 2 • Can get ⎡ ⎛ r ⎞ ⎤ exact u = uc ⎢1− ⎜ ⎟ ⎥ ⎢ ⎝ R ⎠ ⎥ equation for ⎣ ⎦ R R pressure • Laminar shear drop stress profile 128μlQ found from Δp = uc du πD4 τ = μ • Laminar 2 ⎡ ⎛ r ⎞ ⎤ dr du 2r 8μuc Fundamentals of Fluid Mechanics, 5/E by Fundamentals of Fluid Mechanics, 5/E by τ = μ = μuc = r Bruce Munson, Donald Young, and Theodore velocity u = uc ⎢1− ⎜ ⎟ ⎥ Bruce Munson, Donald Young, and Theodore 2 2 Okiishi. Copyright © 2005 by John Wiley & R Okiishi. Copyright © 2005 by John Wiley & dr R D Sons, Inc. All rights reserved. profile ⎣⎢ ⎝ ⎠ ⎦⎥ Sons, Inc. All rights reserved. 17 18 ME 390 – Fluid Mechanics 3 Pipe flow April 8 and 15, 2008 Fully Developed Laminar Flow III Effect of Velocity Profile • What is centerline velocity, uc? • Momentum and kinetic energy flow for R R ⎡ 2 ⎤ mean velocity, V 2 ⎛ r ⎞ Q = VA = VπR = udA = u2πrdr = uc ⎢1− ⎜ ⎟ ⎥2πrdr –Flow = m & V = ρVAV = ρV2(πR2) ∫ ∫ ∫ ⎢ ⎝ R ⎠ ⎥ Momentum A 0 0 ⎣ ⎦ 2 2 3 2 –FlowKE = m& V /2 = ρVAV /2 = ρV (πR )/2 RR R ⎡ r3 ⎤ ⎡r 2 r 4 ⎤ R2 • Accurate representation uses profile Q = 2πu ⎢ rdr − dr⎥ = 2πu ⎢ − ⎥ = 2πu c ∫∫2 c 2 c 2 ⎢ R ⎥ ⎢ 2 4R ⎥ 4 R ⎡ 2 ⎤ ⎣00 ⎦ ⎣ ⎦0 ⎛ r ⎞ 4 2 Flow = ρudAu = ρ⎢u ⎜1− ⎟⎥ 2πrdr = ρV A Momentum c ⎜ 2 ⎟ ∫ ∫ ⎢ R ⎥ 3 R2 2Q 2VA 2VπR2 A 0 ⎣ ⎝ ⎠⎦ 3 Q = πuc ⇒ uc = = = = 2V 2 R ⎡ 2 ⎤ 3 2 R2 R2 R2 u 1 ⎛ r ⎞ V π π π FlowKE = ρudA = ρ⎢uc ⎜1− ⎟⎥ 2πrdr = 2ρA ∫ 2 2 ∫ ⎜ R2 ⎟ 2 A 0 ⎣⎢ ⎝ ⎠⎦⎥ Centerline uc is twice the mean velocity, V 19 20 Turbulent Flow Turbulent Flow Quantities Velocities at one point • For laminar and turbulent flows, the as a function of time velocity at the wall is zero – This is called the no-slip condition – Momentum is maximum in the center of the flow and zero at the wall u(t) = instantaneous • Laminar flows: momentum transport from wall t0 +T velocity to center is by viscosity, τ = μdu/dr 1 u = u(t)dt u’ = velocity • Turbulent flows: random fluctuations exchange T ∫ eddies of high momentum from the center with t0 fluctuation = u – u low momentum flow from near-wall regions 21 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 22 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Momentum Exchange Turbulence Regions/Profiles turbulent eddy viscosity, η du τ = ()μ + η Fundamentals dr of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. • Very thin viscous sublayer next to wall Laminar flow – Turbulent flow – Copyright © 2005 by John random eddies have Wiley & Sons, – 0.13% of R = 3 in for H20 at u = 5 ft/s structure Inc. All rights molecular motion reserved. • Flat velocity profile in center of flow Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 23 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 24 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. ME 390 – Fluid Mechanics 4 Pipe flow April 8 and 15, 2008 Fundamentals of Fluid Mechanics, 5/E by Bruce Profile Effect of Velocity Profile Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John 1 n Wiley & Sons, Inc. All u ⎛ r ⎞ • Analysis similar to one used for laminar rights reserved. = ⎜1− ⎟ Vc ⎝ R ⎠ flow profile Turbulent – Determine momentum and kinetic energy flow for mean velocity velocity – Correction factor multiplies average V profiles 2 3 4 results to give integrated u and u values n = 6: Re = 1.5x10 ; Vc/V = 1.264 with n a n = 8: Re = 4x105;V/V = 1.195 c function of n Re Momentum KE n = 10: Re = 3x106;V/V = 1.155 c Reynolds 6 1.5x104 1.027 1.077 Laminar: Vc/V = 2 V = Q/A number 8 4x105 1.016 1.046 6 25 10 3x10 1.011 1.031 26 Pipe Pipe Roughness roughness effects in • Effect of rough walls on pressure drop viscous may depend on surface roughness of sublayer pipe affects Fundamentals of Fluid Mechanics, 5/E by Bruce • Typical roughness values for different Munson, Donald Young, pressure and Theodore Okiishi. materials expressed as roughness Copyright © 2005 by John Wiley & Sons, Inc. All drop in length, ε, with units of feet or meters rights reserved. turbulent • Only turbulent flows depend on flow roughness length, laminar flows do not No effect on laminar flow 27 28 Use this Energy Equation table (p • Energy equation between inlet (1) and 433 of outlet (2) text) to p V 2 p V 2 z + 2 + 2 = z + 1 + 1 + h − h find ε 2 γ 2g 1 γ 2g s L • Previous applications allowed us to compute the head loss from all other Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and data in this equation Theodore Okiishi.