Advanced Transport Phenomena Momentum Transport Steady Compressible Fluid
Advanced Transport Phenomena Momentum Transport: Steady Compressible Fluid Flow Module 4 Lecture 12 Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras
Momentum Transport: Steady Compressible Fluid Flow
Momentum Transport Mechanisms, Rates, Coefficients
RELEVANCE OF FLUID DYNAMICS Ø Performance of a chemical reactor, separator, etc. intimately linked to: Ø Reagent contacting patterns (flows of reactants & products, recirculation, backmixing), Ø Local turbulence levels, etc. Ø Hence, to laws governing momentum transfer
CRITERIA FOR QUIESCENCE Ø Convection usually dominates diffusion in the streamwise direction Ø As a result of imposed or forced flow (e. g. , by piston, fan, motion of device) Ø Buoyancy (natural convection) Ø Criterion to neglect convection: Ø Important for stagnant fluids, but rarely encountered in practical engineering applications
CLASSIFICATION OF CONTINUUM FLUID FLOWS Ø Internal versus external: Ø Confined flows (e. g. , in ducts) are “internal” Ø Flows over surfaces (e. g. , over an airplane) are “external” Ø Sometimes, the distinction is not clear: e. g. , flow over an airfoil mounted in a wind-tunnel test section
CLASSIFICATION OF CONTINUUM FLUID FLOWS Ø Constant-Property versus Variable-Property Ø Chemically-reacting multi-component flows exhibit large spatial & temporal variations in V, T, wi, m, k, Di, etc. Ø In simple cases (e. g. , low-Ma gas or liquid flows with modest temperature & concentration non-uniformities), properties often assumed constant
CLASSIFICATION OF CONTINUUM FLUID FLOWS Ø Single-phase vs. multi-phase: Ø Single-phase continuum formulation appropriate when Ø Region contains a single phase (e. g. , gas mixture, liquid solution), or Ø Dispersed phase motion is tightly coupled to that of host phase (e. g. , high MW species in gas mixture) Ø In multiphase fluid flow (e. g. , droplets in gas, bubbles in liquid), explicit transport laws needed for each phase
INTERACTIVE ROLE OF EXPERIMENT & THEORY Ø Cost of experimentation can be minimized by proper use of conservation and constitutive laws, by: Ø Substituting “easy” measurements for more difficult ones (e. g. , T vs w) Ø Reducing number of independent measurements in full- scale configuration
INTERACTIVE ROLE OF EXPERIMENT & THEORY CONTD… Ø Designing and interpreting small-scale model experiments Ø Making analytical or numerical predictions for quantities that are not directly measurable Ø Optimizing based on models Ø Avoiding unpleasant surprises!
MECHANISMS OF MOMENTUM TRANSPORT Ø Convection Ø Diffusion
MOMENTUM CONVECTION Ø V specific linear momentum (per unit mass of fluid mixture) Ø local flux of linear momentum by convection (“bodily transport”) Ø For uniform flow of velocity U in x-direction, momentum convective flow rate per unit area
MOMENTUM DIFFUSION Ø Linear momentum flux by diffusion = where = contact stress operator Ø Part of linear momentum diffusion associated with fluid deformation rate = Ø Not present in quiescent fluid, or Ø In “solid body” motion (uniform translation, solid body rotation, etc. ) Ø Associated phenomenological coefficient Ø Function of local fluid-state variables Ø Analogous to mass and energy diffusion
CONVECTIVE MOMENTUM TRANSPORT IN GLOBALLY INVISCID FLOW Ø Often, effects of momentum diffusion are confined to a negligibly small fraction of device volume, e. g. : Ø Immediate vicinity of solid walls (fluid brought locally to rest) Ø Within abrupt fluid dynamic transitions (such as shock waves, detonations)
STEADY 1 D COMPRESSIBLE FLUID FLOW
STEADY 1 D COMPRESSIBLE FLUID FLOW
STEADY 1 D COMPRESSIBLE FLUID FLOW
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Total Mass Conservation: Ø Net convective outflow (dropping accumulation term) or which implies:
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Species Mass Conservation: Ø Neglecting streamwise diffusion compared to transverse diffusion flux (N-1 of which are independent)
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Streamwise Momentum: Ø Assuming: Ø No accumulation term Ø Streamwise momentum convection dominates streamwise momentum diffusion Ø Streamwise body force per unit mass common to all species (e. g. , gravity gz)
STEADY 1 D COMPRESSIBLE FLUID FLOW Then:
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Energy Conservation: Ø Assuming: Ø No accumulation term Ø Fluid does no work at duct surface in overcoming wall friction (streamwise velocity is zero) Ø Streamwise body force per unit mass common to all species (e. g. , gravity gz with potential f)
STEADY 1 D COMPRESSIBLE FLUID FLOW Then: so that where (local stagnation or total enthalpy)
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Entropy: Ø Assuming: Ø No accumulation term Ø Entropy production due to streamwise momentum diffusion, energy diffusion and species i mass diffusion are negligible Ø Gas mixture is thermodynamically ideal
STEADY 1 D COMPRESSIBLE FLUID FLOW Then: or
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Entropy: Ø In the absence of volumetric energy addition ( ) via radiation absorption, wall entropy diffusion flux Ø Volumetric entropy production rate due to finite-rate homogeneous chemical reactions
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Steady, frictionless flow of a nonreacting gas mixture in a duct of variable area (nozzle) without heat addition: Ø Conservation equations may be written as:
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Flow initiated from a large-area reservoir where velocity is small: (hypothetical values obtained if prevailing mixture were isentropically decelerated to rest– i. e. , local stagnation values)
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Equivalent interrelations between differentials: where, for a perfect gas mixture of molecular weight M:
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø If heat capacities (hence, ) are constant: Ø Local speed of sound (acoustic speed) in prevailing gas mixture, a, is given by: Accordingly, in this isentropic flow, locally:
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø Combining previous relationship with momentum conservation (Euler) constraint gives: where G mass flux, given by: and Ma local Mach number, defined as:
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø G increases with u for subsonic flow, decreases with u for supersonic flow, has maximum value G* where Ma = 1 (u = a *) Ø Maximum mass flow through duct where Amin minimum (“throat”) area, A* Ø Since ho = constant:
STEADY 1 D COMPRESSIBLE FLUID FLOW For a perfect gas: Therefore: and (basis for “critical orifice flowmeter”)
STEADY 1 D COMPRESSIBLE FLUID FLOW Ø For = constant (isentropic) gas flow, all local state properties can be uniquely related to their corresponding stagnation values and local Mach number, Ma: (since = constant)
STEADY 1 D COMPRESSIBLE FLUID FLOW To achieve supersonic velocities, area of duct downstream of throat must increase– “De. Laval” converging-diverging nozzles, widely used in steam turbines, gas turbines, rocket engines Steady one-dimensional isentropic flow of a perfect gas with g =1. 3
- Slides: 35