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Chapter 1 Introduction
By James E. Brannan and William E. Boyce
Assignment II (Heating and Cooling of a Building)
𝒅𝒖
= −𝒌(𝒖 − 𝑻(𝒕))
𝒅𝒕
Where k is proportional to the temperature differences u
(from Newton’s law of cooling) and T0 is the initial temperature.
temperature
100
90
80
70
60
50
40
30
0.0
time
0.5
1.0
1.5
2.0
2.5
3.0
We illustrate these equations with a dynamic graphic:
http://demonstrations.wolfram.com/ADifferentialEquationForHeatTransferAccordingToNewtonsLaw
OfC/
We illustrate this cooling law with a figure similar to the one given in 1.1.7 with u representing the
internal temperature. The variation of the directions fields is given by T(t) = 60 + 15 Sin(2θt}.
Where 60 degrees is the mean temperature and the proportionality constant k is 1.5. k is a factor that
will be proportional to the temperature difference and is established in accordance to a number of
factors to include the insulation of the building. In this example, you can modify the proportionality
constant and mean temperature. Several specific solutions, on either side of the mean, are calculated
with different initial temperatures and are shown on the graph as solid blue curves. The resulting
direction fields are correspondingly calculated in this dynamic example as the values are adjusted.
Notice that the daily oscillations are affected by the proportionality constant. In other words the
insulation of the building will affect the daily fluctuations. You can also release either of the two sliders
and watch the solution change. Using this figure you see the importance of insulation and temperature
variations in building cooling. Although this is a simple example, we see that energy management of
buildings is possible using mathematical models.
1.
When we increase the proportionality constant k the frequency of the fluctuations?
a)
Decrease.
b)
Increase.
c)
Remain unchanged.
d)
None of the above.
2.
The short term solution to the differential equation is influenced by?
a)
The temperature function T(t) only.
b)
The homogeneous solution and the temperature function T(t).
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c)
d)
e)
The initial temperature condition.
Answers b and c.
None of the above.
3.
The Long term behavior of the system is due to?
a)
The temperature function T(t) only.
b)
The homogeneous solution and the temperature function T(t).
c)
The initial temperature condition.
d)
Answers b and c.
e)
None of the above.
4.
The Amplitude of the fluctuations is determined by the?
a)
The mean temperature
b)
Is given by T(t) = 60 + 15 Sin(2θt).
c)
Is due to proportionality constant k.
d)
Answers a and c.
5.
The variation of the external temperature influences?
a)
The mean temperature.
b)
The amplitude of the fluctuations.
c)
The long term behavior of the system.
d)
None of the above.
6.
When can the temperature inside the building u exceed the temperature outside the building?
a)
During each temperature cycle.
b)
Only at the initial temperature.
c)
Never.
d)
Always.
7.
The building is better insulated when k is?
a)
Near 1.
b)
Near 3.
c)
k has no effect.
d)
When T0 is high.
8.
The temperature inside the building oscillates with?
a)
Greater amplitude than outside the building.
b)
The same amplitude as outside the building.
c)
Lesser amplitude than outside the building.
d)
Never oscillates.
9.
The temperature oscillates inside the building occur?
a)
After the temperature oscillations outside the building (delay).
b)
At the same time.
c)
Before the temperature oscillations outside the building.
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d)
10.
Randomly.
Can the temperature solutions for u ever cross?
a)
Yes.
b)
No Never.
c)
Only at large values of time t.