Rates of transfer of thermal energy

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Rates of transfer of thermal energy from one body
to another In the preceding exercises, it was
possible to find out how MUCH thermal energy was
transferred from a hot body to a cold one. How
FAST this energy transfer takes place has not yet
been discussed. This question of the RATE of
thermal energy transfer is important if we want
to understand the issues associated with keeping
buildings warm efficiently in cold climates.
Therefore we will discuss it next.
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Ways of transferring thermal energy from one body
or material to another Conduction Energy
passes through a barrier but no massive material
flows and no electromagnetic waves (such as
light) are involved Convection The thermal
energy is carried from one place to another with
the flow of massive material which carries the
thermal energy with it. Radiation The energy
is carried from one body to another by
electromagnetic waves (such as light).
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Thermal energy losses in buildings In buildings
and homes in US climates, a major use of energy
coming through the economic system in the form of
fossil fuels burned to convert chemical
energy into thermal energy is used for
keeping the interiors of buildings at a
temperature above the outside temperature.
(About 45 of household energy use by the chart
on the next page.)
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1. Suppose you have a window in your house which
has area 1mx1m. The glass is ½ cm thick. The
inside temperature is 65F and the outside
temperature is 10F. The thermal conductivity of
glass is 1.28 watts/meter (degree Kelvin). How
much energy are you losing from this window by
conduction, per hour? a.1.28x3600x55x2joules/hour
b.1.28x55x200/(9/5) joules/hour c.1.28x3600x55/((
9/5).005) joules/hour d.1.28x3600x9/5 joules/hour
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Answer c. Q/tAk(Thot-Tcold)/d (1m2 x1.28
joule/secoC x(65-10)(5/9) oC/.005m) 7822
joules/sec (watts)x 3600 sec/hr 28.16 x 106
joules/hr. In hour 28.16 x 106 joules/3.6 x
106 joules/kwhr 7.82 kwhrx0.058/kwh0.45 (5/9)
converts Fahrenheit to Centigrade differences.
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OK I got 2.82 x 107 joules/hour. If energy is
costing 80 cents per therm 100,000 btu, how
much is this energy Flow adding to your gas
bill? (1055 joules 1btu) a.0.282x.80/1.055
/hr b.2.82x1055x100x.8 /hr c.2.82x.80/1.055
/hr d.2.82x.8 /1055 /hr
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Answer a. ((2.82x107joules/hr /1055joules/btu)/10
5btu/therm) x.8 /therm .22/hr
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It is both environmentally and economically import
ant to minimize this energy use which consumes a
lot of fossil fuels and produces a lot of CO2 To
understand how to do this, we have to think
about the rate at which energy flows from the
heating system to the outside of the house.
Thermal energy is flowing out of the house, in a
direction tending to make the inside temperature
the same as the outside temperature. Thermal
energy flows into the house from the furnace to
balance this outflow and keep the inside
temperature fixed.
Tinside
Toutside
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• 2.To minimize the environmental and economic
• cost, we want to minimize the rate at
• which the furnace has to pump energy into the
• house.
• Which of the following would be the most
• reasonable way to try to do this
• Use an electrical heating system
• Use insulation to slow down the rate at which
• thermal energy flows out the walls.
• c. Use natural gas instead of oil in the furnace.
• d. Use coal instead of oil in the furnace.

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Answer b. The others have no effect.
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R-values for insulation. To rate insulation it
is convenient to define a quantity related to
thermal conductivity but also including
information about the thickness of the material.
Recall the definition Q/t k x A x
(T2-T1)/d This can be rearranged as Q/t
Ax(T2-T1)/R Where Rd/k Here Q is heat
transferred, t is time, A is area d is insulation
thickness and k is thermal conductivity.
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R stands for resistance. The bigger R is, the
less heat gets through the insulation per
second.
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Table 5-1, p. 133
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Table 5-2a, p. 134
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3. What is the correct unit for R in the
metric system? a.m2 oK second/watt b.moK
second/joule c.m2 oK second/joule d.m2 second/oK
joule
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Answer c. Q/t A(Th-Tc)/R So R has units of
areaxdegrees/energy/time or in Metric units
m2 oKseconds/joules
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4.How many m2 oK s/joule are in 1 ft2
0Fh/btu? 1ft.3048m, 1btu1055joules
a.(.3048)2(5/9)(3600)/1055 b.(.3048)2(9/5)(3600)
/1055 c.(.3048)2(5/9)60/1055 d.
.3048(5/9)3600/1055
Table 5-2b, p. 134
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Answer a. 1 ft2 0Fh/btu (1ft2) (.3048 m/ft)2
(1F)(5/9 C/F)(1hr)(3600 sec/hr) /(1btux1055
joule/btu) .58 m2K s/joule
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• 6. Suppose you have a window with R-value
• .88 (British) and area 10ft2 in a wall with
• R-value 13 and an area 100ft2. What fraction of
• the total conductive energy loss takes place
• through the window?
• 13/.8814.8
• .88/13.06
• 1/(1.88/1.3)0.60
• 1/(11.3/.88)0.40

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Answer c. 10/.88/(10/.88100/13) (The
temperature difference cancels out)
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T1
T3
T2
R2
R1
5.Suppose you have two kinds of material in your
walls, say concrete block with R value R1 and
fiberglass with R value R2. If the difference
between the inside and outside temperatures is
T1-T3 then what is the rate Q/At of heat transfer
out of the building per unit wall
area? a.(T1-T3)/(R1R2) b. (T1-T3)/(R1-R2) c.
.(T1-T3)(1/R11/R2) d. .(T1-T3)(1/R1-1/R2)
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Answer a. The same energy flows through R1 and
R2
T1
T3
T2
R2
R1
Q/t(A/R1 )(T1 - T2) Q/t (A/R2)(T2
T3) Multiply first by R1/A and second by R2/A and
add ((R1R2)/A)(Q/t)(T1 T3) or Q/t(A/(R1R2))(
T1 T3)
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Q/tA(T1-T2)/(R1R2)
1
2
series
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Q/t(T1-T2)(A1/R1 A2/R2)
2
parallel
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Thermal energy transfer by radiation. Recall
that this was one of the three ways to transfer
thermal energy between two objects, the other
two being conduction and convection. This way of
transferring energy is distinguished from the
other two because It can take place through a
vacuum (a region of space in which there is no
matter) and The energy is carried through space
from one object to the other in the form of
electromagnetic waves.
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What are electromagnetic waves?. These waves, on
which much of modern technology depend, consist
of electric and magnetic fields. The fields, in
turn can exert electromagnetic forces on
matter. The fields themselves do not have
any mass and should not be considered to
be matter in that sense. However they can
carry energy because the forces exerted by the
fields can do work on the matter with which they
interact.We will talk more about electric and
magnetic fields later. We need to also say a
little bit about the wave like aspect of
electromagnetic waves.
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Waves occur in several physical systems. For
example Sound waves are waves in matter such as
air or your ear. Sound waves are not
electromagnetic waves. Water waves occur on the
surface of water. Also not electromagnetic
waves. Waves on a string. Not
electromagnetic. What is characteristic of all
these waves and of electromagnetic waves.
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In all waves, some disturbance is varying
in space and time as illustrated for a wave on a
string above. The peaks are separated in space by
a WAVELENGTH and move (to the right In the
picture) with a velocity v (for a
traveling wave). If you stand in one place and
measure the time between passing peaks you get
?/v called the PERIOD
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The FREQUENCY of the wave,denoted f, is the
inverse of the period fv/?
These features are characteristic of all waves.
Whats different is whats waving
material density (sound),
the surface of the water (water waves)
the position of the string
electric and magnetic fields (electromagnetic
waves) The wave velocities are
also very different. The velocity of
electromagnetic waves is the highest
known and is about 3 x 108m/s
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Electromagnetic waves are all around us and
appear in many of our technologies. They have
different names in different frequency and
wavelength regimes (but always the same
velocity). These names include Wave
wavelength X rays
10-10 m Visible light
10-6 m Infrared
10-5 m Microwaves 10-1
m Radio Waves 101 m
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In describing the energy economy we will mainly
be concerned with visible light, which dominates
in the electromagnetic radiation from the sun,
and infrared radiation, which dominates the
emissions from the earth and objects on its
surface (such as houses) which are at ambient
temperatures of around 0 to 100 C.
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