Advanced Transport Phenomena Momentum Transport Steady Laminar Flow

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

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

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

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

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

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

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

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

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

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

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

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

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

FLOW IN A STRAIGHT DUCT OF CIRCULAR CROSS-SECTION 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 Ø Wall friction coefficient (non-dimensional): Ø Above Re = 2100, experimentally-measured friction coefficients much higher than laminar-flow predictions Ø Order of magnitude for Re > 20000 Ø Due to transition to turbulence within duct Ø Causes Newtonian fluid to behave as if non- Newtonian Ø Augments wall transport of axial momentum to duct

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

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

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

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

STEADY TURBULENT FLOWS: DISCHARGING JETS Ø Laminar round jet of incompressible Newtonian fluid: Far. Field Ø Schlichting BL approximation Ø PDE’s governing mass & axial momentum conservation in r, q, 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 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 Total mass-flow rate past any station z far from jet mouth yielding i. e. , mass flow in the jet increases with downstream distance Ø By entraining ambient fluid while being decelerated (by radial diffusion of initial axial momentum)

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

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