Chapter 3: Heat Transfer

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Chapter 3 Heat Transfer
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Basic Approach to solving heat flow problems
Find the equivalent electrical circuit

• R is the thermal resistance, analogous to
  electrical resistance units 0 K/w

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Additional equations

• Thermal resistivity

Heat flow per unit area
Thermal conductance
In the U.S., r is known as the R-value a
practical measure of the effectiveness of
insulation
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Mechanisms of Heat Transfer

• Conduction Transfer of heat by vibration of
  atoms molecules, with no mass movement of the
  medium the only mechanism in opaque solids
• Convection (Free or Forced) Transfer of heat
  by mass movement of the medium the dominant
  mechanism in fluids
• Radiation Transfer of heat by EM radiation,
  which is the only mechanism in vacuum
• Often all three will be present, but sometimes
  one or two dominates.
• Goal Find expressions for thermal resistance in
  each case can then easily analyze problem

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Example of finding the equivalent electrical
circuit
convection
convection
conduction
radiation
radiation
Outside wall
Energy flow mechanisms Note that air is a very
poor conductor of heat
Physical situation hot tank of water in a cool
room with still colder air outside only one
outside wall
Equivalent Circuit
Sink
Source
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Finding R r for conduction case
Th
A
Tc
So
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Simple example to better understand meaning of
R, r U
Using k-values given in appendix C for glass
glass fibers. As we will see later, the
effective resistance of a glass pane is
significantly greater than values computed
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Factors affecting r (R-value) of windows

• Type of glazing material
• Number of layers of glass
• Size of the air space between panes of glass
• Use of argon instead of air in space
• Thermal resistance of conductance of the frame
  and spacer materials (avoid metal)
• The "tightness" of the installation (air leaks?)

Is large or small space better?
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R-values various window types
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Convection in fluids(liquids gases)

• Much more rapid transfer than conduction
• Free convection movement of fluid caused by
  heat itself (spontaneous)
• Forced convection Outside source (pump or fan)
  drives the fluid motion
• Which case is present when a room is heated by
  heating coils in
• floor? Why?
• In ceiling? Why?

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Flow of fluid molecules above a surface

• simulation

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Boundary layer approximation

• Velocity profile of fluid above a flat surface

v v (y) T T (y)
Imagined velocity profile
Width of boundary layer (fictitious)
X
Right at surface v 0 (why?) Above some distance
v constant
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Idea of boundary layer approximation

• Assumption of a stationary boundary layer gives
  answers correct to perhaps 10
• Rationale for this approach?
• Can find the heat leaving the surface based on
  conduction through the stationary boundary layer

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Nusselt number

• A dimensionless scaling factor used in the
  analysis of all bodies having a given shape see
  formulas in appendix C for various shapes

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Convective heat transfer depends on

• Properties of the fluid (mainly density
  specific heat)
• Speed nature of flow Lamilar or turbulent)
• Shape size of surface (which determine
  thickness of stationary boundary layer)

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Heat transfer by mass transport of a fluid
Definition of specific heat
T1
T2
Mass flow rate times specific heat
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Fluid flow with a phase change

• Example of where this is used?

Latent heat of vaporization
T2
T1
Note different meaning of 2 Ts here
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Radiative heat transfer

• Electromagnetic radiation is emitted by hot
  bodies (and absorbed by cold ones) e.g., the
  filament in your toaster or a light bulb or the
  sun.

Radiant flux density (RFD) per unit wavelength
interval units are w/m3
Radiant flux in an interval
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The electromagnetic spectrum
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Radiation falling on a surfaceamount known as
irradiance,
Incident
Reflected
Absorbed
Transmitted
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Finding

• Assume we know the fraction absorbed at each
  wavelength the shape of incident spectrum

Given
Alpha calculator
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Black Bodiesblackbody surface that absorbs
all radiation at all wavelengthsAlternative
black body is an entity whose emitted radiation
follows the blackbody (Planck) spectrum derived
by Max Planck (1905) for the RFD for a blackbody
heated to absolute temperature T. This
derivation started the quantum
revolutionExamples of approximate
blackbodies?
heated filament in a light bulb, toaster or
stovethe sun starsthe whole Earth seen from
spacea small hole in the wall of an oven or
furnace (best)you in a dark roomouter space!
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Black Body Spectrum
T is absolute (Kelvin) temperature
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Peak of Black Body Spectrum
Locus of maxima for each T
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Total RFD for all wavelengthsStefan-Boltzmann Law
Recall this is RFD or power per unit area emitted
by an ideal black surface. What would be the
total power emitted by a real surface of area A?
emissivity
Be sure to always use absolute temperatures
How would emitted power change if the absolute
temperature of an object doubles?
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Taking into account absorption of heat from
surroundings-- absorption is the inverse process
from emission
F12
T1 and T2 are the absolute temperatures of the
body and the environment. What does the sign of
Pnet tell us?
Book includes another shape-dependent exchange
factor F12 in above equation, which accounts for
the exchange of heat between two objects, and
their two emissivities e1 and e2
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Finding resistance for radiative heat transfer
approximation
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Sample use of resistance equation

• Two parallel plates of area A 1.0 m2 have
  emittances e1 0.9 e2 0.2. If their
  temperatures are T1 350 T2 300 K, what is the
  resistance the net power lost by the hotter one?

For parallel plate geometry use eq C.18
Rr 0.66 0K/w
Under what conditions can we assume F12 1?
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Thermal capacitance
Electrical capacitance is the amount of charge a
pair of plates can hold (per unit volt potential
difference). How would you define thermal
capacitance?
Amount of heat energy an object posses per unit
degree temperature it is above its surroundings
mc
Consider a hot water tank at temperature T1 in a
room at temperature T0
T1
T2
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Electrical circuit Thermal circuit
t RC time constant
t Rmc time constant
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Resistance in various cases
Basic problem-solving approach Find equivalent
electric circuit Combine Rs in series parallel
to solve problem

• Conduction
• Convection
• Mass transport
• (one phase)
• Mass transport
• with phase change
• Radiative

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Heat exchangers
Animation showing how they work

• Uses of heat exchangers
• Shed heat example?
• Generate steam example?
• Generate cold example?