Differential Equations for Heating and
Cooling
Wesley Day
Auburn Walker
16 July 2002
Differential equations are very important in providing comfort. They are all
around us and engineers regularly make use of them to improve our lives. One
way that they do this is through the advent of heating, ventilation, and air
conditioning. Heating, ventilation, and air conditioning involve many differential
equations. It is obvious that a well air conditioned car left off for five minutes on a
hot summer day will not have noticeably increased in temperature, but if left for
five hours; the temperature will become excruciatingly hot. Newton’s Law of
Cooling is the first key to understanding how the temperature of an object is
connected to the temperature of its surroundings through time.
Newton’s Law of Cooling describes the rate of change of an object’s
temperature through time. It is dependent on the object’s initial temperature and
the temperature of its surroundings. The development of the equations follows
below. An example is presented assuming that a building is a homogeneous
point source without wind in its homogeneous fluid surroundings of constant
temperature.
The internal temperature change of a building can be calculated using
Newton’s Law of Cooling. The simplest form is the first order differential equation
dT
dt
Fig. 1
 k (T  TS ),
[Khamsi 02]
where T(t) is the internal temperature of the building, S is the temperature of the
surroundings, and k is a growth constant. If we assume the initial condition
T(t0)=T0, after integration and substitution, we arrive at
T (t )  TS  (T0  TS )e  kt .
Fig. 2
[Khamsi 02]
Therefore,
T ( t f )TS
T0 TS
Fig. 3
e
 k ( t f t0 )
,
[Khamsi 02]
implying
k (t f  t 0 )   ln(
Fig. 4
T ( t f )TS
T0 TS
).
[Khamsi 02]
Thus making it possible to find k with a second condition, T(tf)=Tz. Through
substitution of this second condition and after rearranging, we have
k  ( (t f 1t0 ) )  ln( TT0z TTsS ).
Fig. 5
[Khamsi 02]
Now we can use the constant k to find the temperature at any given time, by
inserting it into Fig.2.
For example, a building with a constant surrounding temperature of 85°F
has an initial condition, T(0)= 75°F. Inserting this information into fig. 2-4, we get
T (t )  85  (75  85)e  kt ,
T ( t f ) 85
7585
e
 k ( t f 0 )
k (t f  0)   ln(
,
T ( t f ) 85
7585
)
Fig. 6
Upon adding the second condition, T(5)=70°F, we have
85
k (5  0)   ln( 70
7585 ),
k  ( 15 )  ln( 32 )
k  .08109
Fig. 7
Then by inserting the k value into fig.1,
T (t )  85  (75  85)e.08109t
Fig. 8
we are able to apply the equation to further possibilities. If the surrounding
environment and internal conditions are varying, then differentiating the general
equation yields:
d 2T
dt
dTS
 k ( dT
dt  dt ).
Fig. 9
This demonstrates the acceleration change in temperature of the system. This
cooling/heating effect can be countered with an HVAC system with a good
control system.
Heating, ventilation, and air conditioning systems usually utilize automated
controls. The best common automated control scheme is the proportional
integral differential ideal algorithm [Abdou 95]. The following control circuit
implements the algorithm detailed above:
Fig. 10
[Williams 02]
[Haines 98]
The equation which describes its operation is:
Fig. 11
[Williams 02]
Proportional integral differential systems are often tuned by experience using the
Ziegler-Nichols Method. The set point control is calibrated [Gupta 96]. The
proportional part of the system, P, is the gain for the controller. As the gain
increases, the control becomes unstable and overshoots the target temperature
farther. Increased gain allows for faster response [Coffin 92]. P control is
demonstrated below:
Fig. 12
[Williams 02]
[Coffin 92]
Proportional with differential control improves the situation. The differential part
adds damping. A large amount of damping makes the response slow, so critical
damping is desired.
Fig. 13
[Williams 02]
[Coffin 92]
The addition of the integral control fixes the drift from the ideal temperature. The
following results:
Fig. 14
[Williams 02]
[Coffin 92]
Proportional integral differential controllers can use the ideal algorithm,
W  K c (e(t ) 
Fig. 15
1
de(t )
e(t )dt  D
)

I
dt
[Expertune 02]
or the parallel algorithm,
W  K p e(t ) 
1
de(t )
e(t )dt  D

I
dt
Fig. 16
[Expertune 02]
or the series algorithm,
W  K s (e(t ) 
Fig. 17
1
d
e(t )dt )(1  D )

I
dt
[Expertune 02]
These variations may be used in control systems depending upon the
requirements and cost of implementation. These control processes are designed
to operate on systems with constant temperature in the surroundings. They alter
the internal temperature and move it toward the ideal point. A heating,
ventilation, and air conditioning system is designed to counteract the effects of
Newton’s Law of Cooling (which applies when the control system is turned off)
[Haines 98]. Differential equations are useful for modeling much more in this
field.
Various forms of proportional integral differential control systems exist,
differing by the accuracy they are able to maintain. This is often done by sending
the information back to the controller. The systems are made to allow for comfort
by correcting the changes in temperature. The second order differential equation
(Fig. 9) explains how the proportional integral differential control system interacts
with Newton’s Law of Cooling. The ability to maintain temperatures in particular
environments allows for comfort in our lives. Mathematicians and Engineers
study differential equations and science in order to solve these important
problems and make our world a better place.
References
Abdou, Ossama and Francis Sando. Journal of Architectural Engineering, Sept.
1995.
Coffin, Michael. Direct Digital Control for Building HVAC Systems. Van
Nostrand Reinhold: New York, 1992.
Expertune. www.expertune.com. 2002.
Gupta, Madan and Naresh Sinha. Intelligent Control Systems. IEEE Press:
Piscataway, 1996.
Haines, Roger and C. Lewis Wilson. HVAC Systems Design Handbook.
McGraw-Hill: Washington, D.C., 1998.
Khamsi, Mohamed. www.sosmath.com. MathMedics, LLC, 2002.
Williams, Charles. http://newton.ex.ac.uk/teaching/CDHW/Feedback/index.html.
University of Exeter, 2002.