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

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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

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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

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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

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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

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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

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Three-dimensional heat-conduction model for gas turbine blade

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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)

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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

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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

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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

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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

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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

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

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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

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

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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

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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

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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:

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

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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

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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

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

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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

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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

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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

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