Heat Conduction in Solids: Temperature Fields & Energy Transfer Rates

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

2. TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER RATES/ COEFFICIENTS • 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”

3. 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

4. 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

5. 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

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

7. TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION Three-dimensional heat-conduction model for gas turbine blade

8. 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)

9. TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION • 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

10. TEMPERATURE FIELDS AND SURFACE ENERGY TRANSFER FOR QUIESCENT MEDIA OF UNIFORM COMPOSITION • Criteria for quiescence: • Forced convection with imposed velocity U: • vref≡ U • Forced convection negligible if wherePeclet number; in terms of Re & Pr: Hence, criterion for neglect of forced convection becomes

11. 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)

12. 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) Grashof number = ratio of buoyancy to viscous force

13. 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: Nuh = 1

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

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

16. STEADY-STATE HEAT CONDUCTION ACROSS SOLID Heat diffusion through a composite planar slab (piecewise constant thermal conductivity)

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

18. 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:

19. 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 Tw, of diameter dw (= 2 aw) in quiescent medium of distant temperature T∞ • Spherical symmetry => energy-balance equation ( ) reduces to 2nd-order linear, homogeneous ODE:

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

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

22. STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE • Generalization I: Temperature-dependent thermal conductivity: fT(T)  heat-flux “potential” When: (e between 0.5 and 1.0), then:

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

24. 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 r= constant:

25. 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 (Nuh,0 = 2)

26. STEADY-STATE HEAT CONDUCTION OUTSIDE ISOTHERMAL SPHERE • Wall “suction” when vw 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 26