Differential Equations for Heating and Cooling

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

Fig. 1 [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

.

Fig. 2 [Khamsi 02]

Therefore,

Fig. 3 [Khamsi 02]

implying

Fig. 4 [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

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

Fig. 6

Upon adding the second condition, T(5)=70°F, we have

Fig. 7

Then by inserting the k value into fig.1,

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:

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,

Fig. 15 [Expertune 02]

or the parallel algorithm,

Fig. 16 [Expertune 02]

or the series algorithm,

Fig. 17 [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.