Fall 2003 Math 308/501–502
3 Mathematical Models
3.3 Heating and Cooling [of Buildings]
Wed, 17/Sep
c 2003, Art Belmonte
Summary
Newton’s Law of Cooling/Heating
The rate of change
d Q
/
dt
of the temperature of an object is
proportional to the difference between the temperature
M
(
t
)
of
the surrounding medium and the temperature
Q
(t) of the object
itself; i.e.,
d Q
/
dt
=
K
(
M

Q
)
, where
K
is a proportionality
constant. In applications of this law,
M
is often constant.
Heating and Cooling of Buildings
A general model is
d Q
dt
=
K
(
M
(
t
)

Q
(
t
))
+
H
(
t
)
+
U
(
t
)
.
•
M
(
t
)
is the temperature of the surrounding medium.
•
Q
(
t
)
is the temperature inside the building.
•
H
(
t
)
is the rate of increase in inside temperature due to
people, lights, machines, etc.
•
U
(
t
)
is the rate of increase/decrease in inside temperature
due to heating/air conditioning, respectively.
•
The reciprocal 1
/
K
is known as the
time constant
for the
building.
Note on nomenclature
MATLAB on all platforms is case insensitive. That is, it internally
regards
T
and
t
as different variables. The TI89, like DOS or
Windows, does not. Accordingly, I have chosen to use
Q
for
T
above, so as to pick something that will work on all platforms.
(Of course, this is
q
on the ‘89.)
Hand/MATLAB Examples
107/4
A red wine is brought up from the wine cellar, which is a cool
10
◦
C, and then left to breathe in a room of temperature 23
◦
C.
If it takes 10 minutes for the wine to reach 15
◦
C, when will the
temperature of the wine reach 18
◦
C and be ready to drink?
Solution
Let
Q
(
t
)
be the temperature of the wine
t
minutes after being
brought up from the cellar. Newton’s Law gives
d Q
dt
=
K
(
23

Q
),
Q
(
0
)
=
10
.
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 Spring '08
 comech
 Thermodynamics, Rate Of Change, Absolute Zero, Heat, Black body, Kelvin

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