By Newton's law of cooling the rate of change of the temperature of an object is proportional to the difference between the temperature of the object and its environment. In other words the object's temperature T as a function of time t satisfies a differential equation dt
dT
=k( Temperature difference ) where k is the coefficient of thermal conductivity and the temperature difference is the temperature of the environment minus T(t). In this problem we use Newton's cooling law to analyze the changing temperatures in a typical home with an attic, basement and insulated main floor. Let t be time in hours since noon and z(t)= Temperature in the attic y(t)= Temperature in the main living area x(t)= Temperature in the basement
Assume the outside temperature is constantly 35 ∘
F and the basement earth temperature is 45 ∘
F, and at t=0 the temperature on each level in the home matches its surrounding temperature. At noon a small electric heater is turned on which provides a 20 ∘
F rise per hour. We introduce the constants of thermal conductivity as follows: k 1
,k 2
,k 3
are the thermal conductivity constants for the floor, walls, and main floor ceiling respectively. k 4
is the thermal conductivity constant of the attic walls and ceiling, k 0
is the thermal conductivity constant of the basement walls and floor. (a) Set an initial value problem for x(t),y(t) and z(t) using Newton's law of cooling. (b) Assume k 0
=k 1
=k 4
=0.5 and k 2
=k 3
=0.25 reflect the insulation quality. Find the equilibrium solution as well as its stability. Discuss in the context of the heating problem. (c) Find the temperature of the main floor at 1pm and 5pm