Problem 1: First-Order Transient Response (25 points)

According to Newton's law of cooling, the time rate of change of temperature Tb (t) of a body

immersed in a medium of temperature Tm(t) is proportional to the difference Tm(t) - Tb (t). That is,

dTb (t)

dt

= a (Tm(t)-Tb (t))

a) Write the transfer function for this system, treating Tm (t) as the input, Tb (t) as the output,

and a as a constant coefficient.

b) What are the time constant and static sensitivity of this system, as a function of a?

An object initially at 30°C is placed inside an oven regulated to Tm (t) = 400°C. After 20

minutes, it is found that the temperature Tb (t) of the object is 100°C. Find the value of a

(including units) and sketch the temperature response as a function of time.

c)

Hints: Treat the temperature of the oven as an input that "steps" at the instant the object is

placed inside. You will need to nondimensionalize to account for the nonzero initial

temperature.

d) What temperature will the object reach after a duration of two time constants?

Hint: This can be found without even needing to calculate the time constant.