Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 3 Lecture 9 Constitutive Laws: Momentum Transfer

CONSTITUTIVE LAWS Conservation equations are necessary but not sufficient for predictive purposes, since they lack closure on:  Local state functions (thermodynamic)  Local diffusion fluxes (mass, momentum, energy)  Reaction-rate laws All must be explicitly related to “field densities”. 2

 Appropriate laws must be used for fluid mixture under consideration  Come from equilibrium chemical thermodynamics.  Mixture assumed to be describable in terms of state variables  p, T, composition ( ), e, h, s, f (= h- Ts) 3 EQUATIONS OF STATE

 Nature of relationship may differ from fluid to fluid  Perfect gases  Liquid solutions  Dense vapors, etc.  Appropriate laws must be used for fluid mixture under consideration  Come from equilibrium chemical thermodynamics 4 EQUATIONS OF STATE

CHEMICAL KINETICS  Individual net chemical species source strengths must be related to local state variables (p, T, composition, etc.)  To define reacting mixture  Info comes from chemical kinetics  Comprehensive expression based on all relevant (molecular-level) elementary steps, or  Global expressions empirically derived  Needed to size chemical reactors  Rate laws can be simple or quite complicated 5

 Rate laws must satisfy following general constraints:  No net mass production  No net charge production 6 CHEMICAL KINETICS

 Vanishing of net production rate of each chemical species at local thermo chemical equilibrium (LTCE) where (i=1,2, …, N) are calculated at prevailing T, P, chemical-element ratios 7 CHEMICAL KINETICS

DIFFUSION FLUX– DRIVING FORCE LAWS/ COEFFICIENTS  Simplest laws :  Fluxes linearly proportional to driving forces, i.e., local spatial gradients of field densities  Valid for chemically reacting gas mixtures provided state variables do not undergo an appreciable fractional change in:  A spatial region of the dimension ca. one molecular mean-free-path  A time interval of the order of mean time between molecular collisions 8

 Positive energy production  in the presence of diffusion, irrespective of direction of diffusion fluxes  Material frame invariance  same form irrespective of changes in vantage point 9 DIFFUSION FLUX LAWS– GENERAL CONSTRAINTS

 Local action (space & time)  Isotropy  transport properties are not direction- dependent  Linearity  Laws linear in local field variables and/ or their spatial gradients 10 DIFFUSION FLUX LAWS– GENERAL CONSTRAINTS

where T = “extra” stress associated with fluid motion; viscous stress = total local stress p = thermodynamic (normal) scalar pressure I = unit tensor 11 LINEAR MOMENTUM DIFFUSION VS RATE OF FLUID-PARCEL DEFORMATION

only 6 of the 9 components are independent because of symmetry 12 VISCOUS STRESS

 Each component a force per unit area  First subscript: surface on which force acts (e.g., x = constant)  Second subscript: direction of force  : normal (tensile) stress  : shear stress in y direction on x = constant surface 13 VISCOUS STRESS

 -T  rate of linear momentum diffusion  JC Maxwell: For gases, fluid velocity gradients result in corresponding flux of linear momentum  Proportionality constant: viscosity coefficient  For a low-density gas in simple shear flow: 14 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

 Stokes: For more general flows, T is linearly proportional to local rate of deformation of fluid parcel, which has two components: 15 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

16 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

 Rate of angular deformation in x-y plane:  Rate of volumetric deformation: 17 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

Stokes’ constitutive law for local extra (viscous) stress: where, in general:  Applicable for Newtonian fluid 18 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

  dynamic viscosity   kinematic viscosity (diffusivity; cm 2 /s)   bulk viscosity, neglected for simple fluids When momentum-flux law is inserted into PDE governing linear momentum conservation, Navier-Stokes equation is obtained  Basis for most analyses of viscous fluid mechanics 19 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

grad v may be decomposed into a symmetrical & anti- symmetrical part: Symmetric portion: Def v, associated with local deformation rate of fluid parcel Anti-symmetric portion: Rot v, “spin rate”, defines local rotational motion of fluid parcel 20 STOKES’ EXTRA STRESS VS RATE OF DEFORMATION RELATION

where specific enthalpy, 21 ENERGY EQUATION IN TERMS OF WORK DONE BY FLUID AGAINST EXTRA STRESS

Subtracting mechanical-energy conservation equation from above gives: Both rate of heat addition (term on RHS in parentheses) and rate of viscous dissipation contribute to accumulation rate and/ or net outflow rate of thermodynamic internal energy 22 ENERGY EQUATION IN TERMS OF WORK DONE BY FLUID AGAINST EXTRA STRESS

 T. n dA  surface force on differential area n dA associated with all contact stresses other than local thermodynamic pressure  div T  local net contact force per unit volume in the limit of vanishing volume 23 VISCOUS DISSIPATION

In Cartesian coordinates: 24 VISCOUS DISSIPATION

 Angular momentum conservation at local level leads to conclusion that T is symmetric, and T : grad v = T : Def v where local fluid parcel deformation rate is also symmetric. 25 VISCOUS DISSIPATION

Rate of entropy production due to linear- momentum diffusion T : Def v T Def v => positive entropy production for positive 26 VISCOUS DISSIPATION

 Velocities fluctuate  Contribute additive “correlation terms” to time- averaged energy equations  For incompressible turbulent flow of a constant- property Newtonian fluid: 27 VISCOUS DISSIPATION IN TURBULENT FLOWS

2 nd set of terms: viscous dissipation rate per unit mass associated with turbulent motion of fluid. 28 VISCOUS DISSIPATION IN TURBULENT FLOWS

 For steady turbulent flows (e.g., through straight ducts, elbows, valves), viscous dissipation associated with both time-mean & fluctuating velocity fields contributes to:  Net inflow rate of per unit mass flow,  Corresponding rise in internal energy per unit mass of fluid. 29 VISCOUS DISSIPATION IN TURBULENT FLOWS

 Heating associated with local viscous dissipation can strongly modify:  Local temperature field,  All temperature-dependent properties, including m itself  Especially important in high-Ma viscous flows (e.g., rocket exhausts); low-Re, low-Ma flows in restricted passages (e.g., packed beds in HPLC columns) 30 VISCOUS DISSIPATION IN TURBULENT FLOWS

 Experimentally obtained by establishing a simple flow (e.g., steady laminar flow in a pipe) & fitting observations (e.g., pressure-drop for given flow rate) to predictions based on mass & linear- momentum conservation laws & constitutive relations 31 DYNAMIC VISCOUS COEFFICIENT

 Unit: cp (centi-Poise) in cgs; kg/ (m s) , momentum diffusivity, or kinematic viscosity; m 2 /s  For low-density gases, independent of P, ~ T DYNAMIC VISCOUS COEFFICIENT

Chapman – Enskog Expression:  intermolecular potential function  dimensionless temperature  depth of potential energy “well” 33   FROM KINETIC THEORY OF GASES  intermolecular spacing at which potential crosses 0

34 INTERMOLECULAR POTENTIAL

(Volumes  cm 3 /g-mole, T  K, critical pressure  atm) 35   FROM KINETIC THEORY OF GASES

Square-root rule: For liquids, viscosity decreases with increasing temperature 36 MIXTURE & LIQUID VISCOSITY

Andrade-Eyring two-parameter law:  activation energy for fluidity (inverse viscosity) R  universal gas constant  (hypothetical) dynamic viscosity at infinite temperature 37 MIXTURE & LIQUID VISCOSITY

38 CORRESPONDING STATES CORRELATION FOR VISCOSITY OF SIMPLE FLUIDS

 No simple relations for viscosity of liquid solutions  Empirical relations specific to mixture classes employed  e.g., glass, slags, etc.  Gases & liquids in turbulent motion display augmented viscosities 39 VISCOSITY OF LIQUID SOLUTIONS, TURBULENT VISCOSITY