Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 5 Lecture 19 Energy Transport: Steady-State Heat Conduction

 Conservation Equation Governing T-field, typical bc’s, solution methods  Possible complications:  Unsteadiness (transients, including turbulence)  Flow effects (convection, viscous dissipation)  Variable properties of medium  Homogeneous chemical reactions  Coupling with coexisting “photon phase” TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER RATES/ COEFFICIENTS

 Neglecting radiation, we obtain: T:grad v  scalar; local viscous dissipation rate (specific heat of prevailing mixture) Simplest PDE for T-field: Laplace equation TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER RATES/ COEFFICIENTS

 Boundary conditions are of 3 types:  T specified everywhere along each boundary surface  Isothermal surface heat transfer coefficients  Can be applied even to immobile surfaces that are not quite isothermal  Heat flux specified along each surface  Some combination of above TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER RATES/ COEFFICIENTS

 Solution Methods:  Vary depending on complexity  Most versatile: numerical methods (FD, FE) yielding algebraic solution at node points within domain  Simple problems: analytical solutions of one or more ODE’s  e.g., separation of variables, combination of variables, transform methods (Laplace, Mellin, etc.)  Steady-state, 1D => ODE for T-field TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER RATES/ COEFFICIENTS

 Examples:  Inner wall of a furnace  Low-volatility droplet combustion  Water-cooled cylinder wall of reciprocating piston (IC) engine  Gas-turbine blade-root combination TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION

Three-dimensional heat-conduction model for gas turbine blade

TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION Heat diffusion in the wall of a water-cooled IC engine (adapted from Steiger and Aue, 1964)

 Criteria for quiescence:  Stationary solids are quiescent  Viscous fluid can exhibit forced & natural convection  Convective energy flow can be neglected if and only if or TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION

 Criteria for quiescence:  Forced convection with imposed velocity U:  v ref ≡ U  Forced convection negligible if where Peclet number; in terms of Re & Pr: Hence, criterion for neglect of forced convection becomes TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION

 Criteria for quiescence:  Natural convection:  Heat-transfer itself causes density difference  Pressure difference causing flow = If we now write where (thermal expansion coefficient of fluid) TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION

 Criteria for quiescence:  Natural convection:  Criterion for neglect of natural convection becomes where (Rayleigh number for heat transfer) TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION Grashof number = ratio of buoyancy to viscous force

STEADY-STATE HEAT CONDUCTION ACROSS SOLID  Constant-property planar slab of thickness L  ODE for T(x): (degenerate form of Laplace’s equation)  (dT/dx) is constant, thus:  Heat flux at any station is given by: Nu h = 1

STEADY-STATE HEAT CONDUCTION ACROSS SOLID Heat diffusion through a planar slab with constant thermal conductivity

 Composite wall:  (L/k) l  thermal resistance of l th layer  Thermal analog of electrical resistance in series  – voltage drop  Heat flux – current  Reciprocal of overall resistance = overall conductance = U STEADY-STATE HEAT CONDUCTION ACROSS SOLID

Heat diffusion through a composite planar slab (piecewise constant thermal conductivity)

 Non-constant thermal conductivity: Here, still applies, but k-value is replaced by mean value of k over temperature interval STEADY-STATE HEAT CONDUCTION ACROSS SOLID

 Cylindrical/ Spherical Symmetry:  e.g., insulated pipes (source-free, steady-state radial heat flow)  1D energy diffusion  = constant (total radial heat flow per unit length of cylinder)  For nested cylinders, in the absence of interfacial resistances: STEADY-STATE HEAT CONDUCTION ACROSS SOLID

STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE  Procedure in general:  Solve relevant ODE/ PDE with BC for T-field  Then evaluate heat flux at surface of interest  Then derive relevant local heat-transfer coefficient  Special case: sphere at temperature T w, of diameter d w (= 2 a w ) in quiescent medium of distant temperature T ∞  Spherical symmetry => energy-balance equation ( ) reduces to 2 nd -order linear, homogeneous ODE:

 That is, total radial heat flow rate is constant: with the solution: STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE

Corresponding heat flux at r = d w /2: If d w  reference length, then Nu h = 2 Since conditions are uniform over sphere surface: (surface-averaged heat-transfer coefficient ) STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE

 Generalization I: Temperature-dependent thermal conductivity:  T (T)  heat-flux “potential” When: (  between 0.5 and 1.0), then : STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE

Temperature-averaged thermal conductivity:  Extendable to chemically-reacting gas mixtures STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE

 Generalization II: Radial fluid convection  e.g., fluid mass forced through porous solid; blowing or transpiration to reduce convective heat-transfer to objects, such as turbine blades, in hostile environments  Convective term:  Conservation of mass yields: If  = constant: STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE

 Solving this convective-diffusion problem, the result can be stated as: or where the correction factor, F (blowing) is given by: where (Nu h,0 = 2) STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE

26  Wall “suction” when v w is negative  Energy transfer coefficients are increased  Effective thermal boundary layer thickness is reduced  Effects opposite to those of “blowing”  Blowing and suction influence momentum transfer (e.g., skin-friction) and mass transfer (e.g., condensation) coefficients as well STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE