Heat Transfer

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Heat Transfer
2
Outline

• Introduction
• Modes of heat transfer
• Typical design problems
• Coupling of fluid flow and heat transfer
• Conduction
• Convection
• Radiation

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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.

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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
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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?

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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)

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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

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Fouriers Law

• heat flux is proportional to temperature
  gradient
• where k thermal conductivity
• in general, k k(x,y,z,T,)

units for q are W/m2
temperature profile
heat conduction in a slab
1
hot wall
cold wall
x
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Generalized Heat Diffusion Equation

• If we perform a heat balance on a small volume of
  material
• we get

heat conduction in
heat conduction out
T
heat generation
rate of change of temperature
heat cond. in/out
heat generation
thermal diffusivity
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Boundary Conditions

• Heat transfer boundary conditions generally come
  in three types

q 20 W/m2 specified heat flux Neumann condition
q h(Tamb-Tbody) external heat
transfer coefficient Robin condition
T 300K specified temperature Dirichlet condition
Tbody
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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 k0.15 W/m2-K
sheetrock k0.4 W/m2-K
shingles k0.15 W/m2-K
fiberglas insulation k0.004 W/m2-K
sheathing k0.15 W/m2-K
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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
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Newtons Law of Cooling
q
Tbody
average heat transfer coefficient (W/m2-K)
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Heat Transfer Coefficient

• h is not a constant, but h h(DT)
• 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
Thot
Tcold
4 - 4,000 W/m2-K
Tcold
80 - 75,000
Thot
Tcold
300 - 900,000
Thot
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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
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Nusselt Number

• Equate the heat conducted from the wall to the
  same heat transfer in convective terms
• Define dimensionless quantities
• Then rearrange to get

conductivity of the fluid
Nusselt number dimensionless heat transfer
coefficient
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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 Fouriers law

From calculated temperature distribution
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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

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Convection Examples

• Developing flow in a pipe (constant wall
  temperature)

T
bulk fluid temperature
heat flux from wall
x
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Convection Examples

• Natural convection (from a heated vertical plate)

T
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
Tw
u
The dimensionless group that governs natural
convection is the Rayleigh number
gravity
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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

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Surface Characteristics
q W/m2 (incident energy flux)
rq (reflected)
aq (absorbed)
translucent slab
tq (transmitted)
absorptance
transmittance
reflectance
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Black Body Radiation

• A black body
• is a model of a perfect radiator
• absorbs all energy that reaches it reflects
  nothing
• therefore a 1, r t 0
• The energy emitted by a black body is the
  theoretical maximum
• This is Stefan-Boltzmann law s is the
  Stefan-Boltzmann constant (5.6697e-8 W/m2-K4)

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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)
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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

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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 (r, m) 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?