Heat Conduction
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Fluids Review TRN-1998-004 Heat Transfer G1 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Outline Introduction Modes of heat transfer Typical design problems Coupling of fluid flow and heat transfer Conduction Convection Radiation G2 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Introduction Heat transfer is the study of thermal energy (heat) flows Heat always flows from “hot” to “cold” Examples are ubiquitous: heat flows in the body home heating/cooling systems refrigerators, ovens, other appliances automobiles, power plants, the sun, etc. G3 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Modes of Heat Transfer Conduction - diffusion of heat due to temperature gradient Convection - when heat is carried away by moving fluid Radiation - emission of energy by electromagnetic waves qconvection qradiation qconduction G4 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Typical Design Problems To determine: overall heat transfer coefficient - e.g., for a car radiator highest (or lowest) temperature in a system - e.g., in a gas turbine temperature distribution (related to thermal stress) - e.g., in the walls of a spacecraft temperature response in time dependent heating/cooling problems - e.g., how long does it take to cool down a case of soda? G5 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Heat Transfer and Fluid Flow As a fluid moves, it carries heat with it -- this is called convection Thus, heat transfer can be tightly coupled to the fluid flow solution Additionally: The rate of heat transfer is a strong function of fluid velocity Fluid properties may be strong functions of temperature (e.g., air) G6 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Conduction Heat Transfer Conduction is the transfer of heat by molecular interaction In a gas, molecular velocity depends on temperature hot, energetic molecules collide with neighbors, increasing their speed In solids, the molecules and the lattice structure vibrate G7 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Fourier’s Law “heat flux is proportional to temperature gradient” T T Q q kT k A x y units for q are W/m2 where k = thermal conductivity in general, k = k(x,y,z,T,…) temperature profile heat conduction in a slab dT dx 1 hot wall cold wall x G8 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Generalized Heat Diffusion Equation If we perform a heat balance on a small volume of material… heat conduction in T q … we get: heat conduction out heat generation T 2 c k T q t rate of change of temperature heat cond. heat in/out generation k thermal diffusivity c G9 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Boundary Conditions Heat transfer boundary conditions generally come in three types: q = 20 W/m2 specified heat flux Neumann condition T = 300K specified temperature Dirichlet condition Tbody G10 q = h(Tamb-Tbody) external heat transfer coefficient Robin condition © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Conduction Example Compute the heat transfer through the wall of a home: Tout = 20° F Tout = 68° F Although slight, you can see the “thermal bridging” effect through the studs 2x6 stud k=0.15 W/m2-K sheetrock k=0.4 W/m2-K shingles k=0.15 W/m2-K fiberglas insulation sheathing k=0.004 W/m2-K k=0.15 W/m2-K G11 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Convection Heat Transfer Convection is movement of heat with a fluid E.g., when cold air sweeps past a warm body, it draws away warm air near the body and replaces it with cold air flow over a heated block often, we want to know the heat transfer coefficient, h (next page) G12 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Newton’s Law of Cooling T q Tbody q h (Tbody T ) h T h average heat transfer coefficient (W/m2-K) G13 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Heat Transfer Coefficient h is not a constant, but h = h(T) Three types of convection: Natural convection 1 4 h T , h T Typical values of h: fluid moves due to buoyancy 1 3 Thot 4 - 4,000 W/m2-K Forced convection flow is induced by external means h const Tcold Tcold Thot 80 - 75,000 Boiling convection body is hot enough to boil liquid h T Tcold Thot 2 G14 300 - 900,000 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Looking in more detail... Just as there is a viscous boundary layer in the velocity distribution, there is also a thermal boundary layer thermal boundary layer edge velocity boundary layer edge y T , U T ( y) t Tw G15 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Nusselt Number Equate the heat conducted from the wall to the same heat transfer in convective terms: conductivity of the fluid T kf h(Tw T ) y Define dimensionless quantities: Tw T T Tw T y y L Then rearrange to get: T T w Tw T hL Nu kf y L G16 Nusselt number “dimensionless heat transfer coefficient” © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Energy Equation Generalize the heat conduction equation to include effect of fluid motion: T c u T k2T q t Assumes incompressible fluid, no shear heating, constant properties, negligible changes in kinetic and potential energy Can now solve for temperature distribution in boundary layer Then calculate h using Fourier’s law: From calculated q k T temperature h distribution T T T T y w w G17 y 0 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Correlations for Heat Transfer Coefficient As an alternative, can use correlations to obtain h E.g., heat transfer from a flat plate in laminar flow: Nu x 0.332 Re 0x.5 Pr 0.333 where the Prandtl number is defined as: c Pr k momentum diffusivit y thermal diffusivit y Typical values are: Pr = 0.01 for liquid metals Pr = 0.7 for most gases Pr = 6 for water at room temperature G18 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Convection Examples Developing flow in a pipe (constant wall temperature) Tw T T Tw T Tw T T Tw Tw bulk fluid temperature heat flux from wall G19 x © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Convection Examples Natural convection (from a heated vertical plate) T (T ) Tw As the fluid is warmed by the plate, its density decreases and a buoyant force arises which induces flow in the vertical direction. The force is equal to: ( ) g u gravity The dimensionless group that governs natural convection is the Rayleigh number: T , Ra G20 gTL3 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Radiation Heat Transfer Thermal radiation is emission of energy as electromagnetic waves Intensity depends on body temperature and surface characteristics Important mode of heat transfer at high temperatures Can also be important in natural convection problems Examples: toaster, grill, broiler fireplace sunshine G21 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Surface Characteristics q W/m2 (incident energy flux) q (reflected) q (absorbed) translucent slab q (transmitted) 1 absorptance transmittance reflectance G22 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Black Body Radiation A “black body”: is a model of a perfect radiator absorbs all energy that reaches it; reflects nothing therefore = 1, = = 0 The energy emitted by a black body is the theoretical maximum: q s T4 This is Stefan-Boltzmann law; s is the Stefan-Boltzmann constant (5.6697e-8 W/m2-K4) G23 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 “Real” Bodies Real bodies will emit less radiation than a black body: q s T 4 emissivity (between 0 and 1) Example: radiation from a small body to its surroundings both the body and its surroundings emit thermal radiation the net heat transfer will be from the hotter to the colder Qnet As (Tw4 T4 ) Qnet G24 T q qw Tw A © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 When is radiation important? Radiation exchange is significant in high temperature problems: e.g., combustion Radiation properties can be strong functions of chemical composition, especially CO2, H2O Radiation heat exchange is difficult solve (except for simple configurations) — we must rely on computational methods G25 © Fluent Inc. 3/19/2016 Fluids Review TRN-1998-004 Heat Transfer — Summary Heat transfer is the study of thermal energy (heat) flows: The fluid flow and heat transfer problems can be tightly coupled conduction convection radiation through the convection term in the energy equation when properties (, ) are dependent on temperature While analytical solutions exist for some simple problems, we must rely on computational methods to solve most industrially relevant applications Can I go back to sleep now? G26 © Fluent Inc. 3/19/2016
