1. © Fluent Inc. 8/10/2015G1 Fluids Review TRN-1998-004 Heat Transfer

2. © Fluent Inc. 8/10/2015G2 Fluids Review TRN-1998-004 Outline  Introduction  Modes of heat transfer  Typical design problems  Coupling of fluid flow and heat transfer  Conduction  Convection  Radiation

3. © Fluent Inc. 8/10/2015G3 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.

4. © Fluent Inc. 8/10/2015G4 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 q convection q conduction q radiation

5. © Fluent Inc. 8/10/2015G5 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?

6. © Fluent Inc. 8/10/2015G6 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)

7. © Fluent Inc. 8/10/2015G7 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

8. © Fluent Inc. 8/10/2015G8 Fluids Review TRN-1998-004 Fourier’s Law  “heat flux is proportional to temperature gradient”  where k = thermal conductivity  in general, k = k(x,y,z,T,…) hot wallcold wall 1 temperature profile x heat conduction in a slab units for q are W/m 2

9. © Fluent Inc. 8/10/2015G9 Fluids Review TRN-1998-004 Generalized Heat Diffusion Equation  If we perform a heat balance on a small volume of material…  … we get: thermal diffusivity T heat conduction in heat conduction out heat generation rate of change of temperature heat cond. in/out heat generation

10. © Fluent Inc. 8/10/2015G10 Fluids Review TRN-1998-004 Boundary Conditions  Heat transfer boundary conditions generally come in three types: T = 300K specified temperature Dirichlet condition q = 20 W/m 2 specified heat flux Neumann condition q = h(T amb -T body ) external heat transfer coefficient Robin condition T body

11. © Fluent Inc. 8/10/2015G11 Fluids Review TRN-1998-004 Conduction Example  Compute the heat transfer through the wall of a home: shingles k=0.15 W/m 2 -K sheathing k=0.15 W/m 2 -K fiberglas insulation k=0.004 W/m 2 -K 2x6 stud k=0.15 W/m 2 -K sheetrock k=0.4 W/m 2 -K T out = 20° FT out = 68° F Although slight, you can see the “thermal bridging” effect through the studs

12. © Fluent Inc. 8/10/2015G12 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  often, we want to know the heat transfer coefficient, h (next page) flow over a heated block

13. © Fluent Inc. 8/10/2015G13 Fluids Review TRN-1998-004 Newton’s Law of Cooling T body average heat transfer coefficient (W/m 2 -K) q

14. © Fluent Inc. 8/10/2015G14 Fluids Review TRN-1998-004 Heat Transfer Coefficient  h is not a constant, but h = h(  T)  Three types of convection: Natural convection fluid moves due to buoyancy Forced convection flow is induced by external means Boiling convection body is hot enough to boil liquid Typical values of h: 4 - 4,000 W/m 2 -K 80 - 75,000 300 - 900,000 T hot T cold T hot T cold T hot

15. © Fluent Inc. 8/10/2015G15 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 y velocity boundary layer edge thermal boundary layer edge

16. © Fluent Inc. 8/10/2015G16 Fluids Review TRN-1998-004 Nusselt Number  Equate the heat conducted from the wall to the same heat transfer in convective terms:  Define dimensionless quantities:  Then rearrange to get: Nusselt number “dimensionless heat transfer coefficient” conductivity of the fluid

17. © Fluent Inc. 8/10/2015G17 Fluids Review TRN-1998-004 Energy Equation  Generalize the heat conduction equation to include effect of fluid motion:  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 temperature distribution

18. © Fluent Inc. 8/10/2015G18 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:  where the Prandtl number is defined as:  Typical values are: Pr = 0.01 for liquid metals Pr = 0.7 for most gases Pr = 6 for water at room temperature

19. © Fluent Inc. 8/10/2015G19 Fluids Review TRN-1998-004 Convection Examples  Developing flow in a pipe (constant wall temperature) x bulk fluid temperature heat flux from wall T

20. © Fluent Inc. 8/10/2015G20 Fluids Review TRN-1998-004 Convection Examples  Natural convection (from a heated vertical plate) u T TwTw gravity 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: The dimensionless group that governs natural convection is the Rayleigh number:

21. © Fluent Inc. 8/10/2015G21 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

22. © Fluent Inc. 8/10/2015G22 Fluids Review TRN-1998-004 Surface Characteristics q W/m 2 (incident energy flux)  q (reflected)  q (transmitted)  q (absorbed) absorptance reflectance transmittance translucent slab

23. © Fluent Inc. 8/10/2015G23 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:  This is Stefan-Boltzmann law;  is the Stefan-Boltzmann constant (5.6697e-8 W/m 2 -K 4 )

24. © Fluent Inc. 8/10/2015G24 Fluids Review TRN-1998-004 “Real” Bodies  Real bodies will emit less radiation than a black body:  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 emissivity (between 0 and 1)

25. © Fluent Inc. 8/10/2015G25 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 CO 2, H 2 O  Radiation heat exchange is difficult solve (except for simple configurations) — we must rely on computational methods

26. © Fluent Inc. 8/10/2015G26 Fluids Review TRN-1998-004 Heat Transfer — Summary  Heat transfer is the study of thermal energy (heat) flows: conduction convection radiation  The fluid flow and heat transfer problems can be tightly coupled 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?