22. Heating and Cooling of Buildings

Heating and Cooling of Buildings Description of Research Many businesses seek to implement various methods to reduce both the costs of heating/cooling of their buildings and their carbon footprints. Newton’s Law of Cooling establishes the relationship between the temperature inside of a building, outside temperature, and their respective rates of change. Methodology The relationship between building temperature and ambient temperature is modeled by Newton’s Law of Cooling as a first order, linear differential equation for the temperature profile. The law states that the rate of change of the temperature of a particular object of interest is proportional to the difference of that object’s temperature and the surrounding temperature. Hence to solve this problem, compartmental analysis is used to formulate a mathematical model that describes the relationship between heat generated from the building and the outside temperature. We investigate the problem by using the theory of differential equations and the model is then implemented in MATLAB to obtain an approximate solution to the problem. Results The 24-hour temperature profile inside a building as a function of outside temperature, the heat generated inside the building, and the furnace heating or air conditioner cooling is described as a first order differential equation. The exact implicit solution to the problem is obtained. A numerical method is used to approximate the solution. Significance This model allows us to solve the problem that arises when looking at the costs of heating and cooling buildings effectively. Keywords: Temperature, Newton’s Law of Cooling, Differential Equations