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Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 4 - Lecture 15 Momentum Transport: Steady Laminar Flow

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PDEs governing steady velocity & pressure fields: (Navier-Stokes) and (Mass Conservation) “No-slip” condition at stationary solid boundaries: at fixed solid boundaries STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID

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Special cases: Fully-developed steady axial flow in a straight duct of constant, circular cross-section (Poiseuille) 2D steady flow at high Re-number past a thin flat plate aligned with stream (Prandtl, Blasius) STEADY LAMINAR FLOW OF INCOMPRESSIBLE NEWTONIAN FLUID

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FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION Cylindrical polar-coordinate system for the analysis of viscous flow in a straight circular duct of constant cross section

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Wall coordinate: r = constant = a w (duct radius) Fully developed => sufficiently far downstream of duct inlet that fluid velocity field is no longer a function of axial coordinate z From symmetry, absence of swirl: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Conservation of mass ( = constant): v z independent of z implies: PDEs required to find v z ( r), p(r,z) Provided by radial & axial components of linear- momentum conservation (N-S) equations: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Pressure is a function of z alone, and if p = p(z) and v z = v z ( r), then: i.e., a function of z alone (LHS) equals a function of r alone (RHS) Possible only if LHS & RHS equal the same constant, say C 1 FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Hence: New pressure variable P defined such that: and, hence P varies linearly with z a s: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Integrating the 2 nd order ODE for v z twice: Since v z is finite when r = 0, C 3 = 0 Since v z = 0 when Hence, shape of velocity profile is parabolic: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Since is a negative constant- i.e., non- hydrostatic pressure drops linearly along duct: and FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Total Flow Rate: Sum of all contributions through annular rings each of area Substituting for v z (r) & integrating yields: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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(Hagen – Poiseuille Law– relates axial pressure drop to mass flow rate) Basis for “capillary-tube flowmeter” for fluids of known Newtonian viscosity Conversely, to experimentally determine fluid viscosity FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Total Flow Rate: Average velocity, U, is defined by: Then: i.e., maximum (centerline) velocity is twice the average value, hence: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): w wall shear stress C f dimensionless coef (also called f Fanning friction factor) Direct method of calculation: and FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): Hence: equivalent to: Holds for all Newtonian fluids Flows stable only up to Re ≈ 2100 FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): Experimental and theoretical friction coefficients for incompressible Newtonian fluid flow in straight smooth-walled circular duct of constant cross section FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): Same result can be obtained from overall linear- momentum balance on macroscopic control volume A z: Axial force balance (for fully-developed flow where axial velocity is constant with z): Solving for w and introducing definition of P : FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): Configuration and notation: steady flow of an incompressible Newtonian fluid In a straight circular duct of constant cross section FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): Above Re = 2100, experimentally-measured friction coefficients much higher than laminar-flow predictions Order of magnitude for Re > Due to transition to turbulence within duct Causes Newtonian fluid to behave as if non- Newtonian Augments transport of axial momentum to duct wall FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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In fully-developed turbulent regime (Blasius): C f varies as Re -1/4 for duct with smooth walls C f sensitive to roughness of inner wall, nearly independent of Re FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Wall friction coefficient (non-dimensional): Effective eddy momentum diffusivity Can be estimated from time-averaged velocity profile & C f measurements Hence, heat & mass transfer coefficients may be estimated (by analogy) For fully-turbulent flow, perimeter-average skin friction & pressure drop can be estimated even for non- circular ducts by defining an “effective diameter”: FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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where P wetted perimeter Not a valid approximation for laminar duct flow FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION

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Circular jet discharging into a quiescent fluid Sufficiently far from jet orifice, a fully-turbulent round jet has all properties of a laminar round jet, but, intrinsic kinematic viscosity of fluid jet axial-momentum flow rate Constant across any plane perpendicular to jet axis STEADY TURBULENT FLOWS: JETS

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Laminar round jet of incompressible Newtonian fluid: Far- Field Schlichting BL approximation PDE’s governing mass & axial momentum conservation in r, , z coordinates admit exact solutions by method of “combination of variables”, i.e., dependent variables are uniquely determined by the single independent variable: STEADY TURBULENT FLOWS: DISCHARGING JETS

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Streamline pattern and axial velocity profiles in the far-field of a laminar (Newtonian) or fully turbulent unconfined rounded jet (adapted from Schlichting (1968)) STEADY TURBULENT FLOWS: DISCHARGING JETS

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Total mass-flow rate past any station z far from jet mouth yielding i.e., mass flow in the jet increases with downstream distance By entraining ambient fluid while being decelerated (by radial diffusion of initial axial momentum) STEADY TURBULENT FLOWS: DISCHARGING JETS

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Near-field behavior: z/d j ≤ 10 Detailed nozzle shape important “potential core”: within, jet profiles unaltered by peripheral & downstream momentum diffusion processes Swirling jets: Tangential swirl affects momentum diffusion & entrainment rates Predicting flow structure huge challenge for any turbulence model TURBULENT JET MIXING

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Additional parameters: Initially non-uniform density, viscous dissipation, chemical heat release, presence of a dispersed phase, etc. Add complexity; far-field behavior can be simplified TURBULENT JET MIXING

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