Ordinary Differential Equations[FDM 1023]

APPLICATIONS OF ODE

Chapter 7

Overview

Chapter 7: APPLICATIONS OF ODE

7.1. Exponential Growth and Decay

7.2. Newtons Law of Cooling / Heating

7.3. Mixture of Solutions

Learning Outcome

At the end of this section, you should be ableto:

Solve first-order ODE from models ofNewtons Law of Cooling / Heating

7.2. Newtons Law of Cooling / Heating

Newtons Law of Cooling states that the rate ofchange of the temperature T of an object isproportional to the difference between T andthe (constant) temperature of thesurrounding medium, called the ambienttemperature.

The mathematical formulation of this statement is:

where k is a constant.

=

7.2. Newtons Law of Cooling / Heating

General Solution

= +

If the initial temperature of the object is 0 = , then

=

7.2. Newtons Law of Cooling / Heating

=

Thus, the temperature of the object at any time t is given by

= +

Example 1

A metal bar with initial temperature 25oC is droppedinto a container of boiling water (100oC). After 5seconds, the temperature of the bar is 35oC.

i. What will the temperature of the bar be after 1 minute?

ii. How long will it take for the temperature of the bar to be within 0.5oC of the boiling water?

7.2. Newtons Law of Cooling / Heating

Example 2

A thermometer is taken from a room where thetemperature is 72oF to the outside where thetemperature is 32oF. After 1/2 minute, thethermometer reads 50oF.

i. What will the thermometer read after it has been outside for 1 minute?

ii. How many minutes does the thermometer have to be outside for it to read 35oF?

7.2. Newtons Law of Cooling / Heating

Example 3

A metal ball at room temperature 20oC is droppedinto a container of boiling water (100oC). Given thatthe temperature of the ball increases 2oC in 2seconds.

i. What will the temperature of the ball after 6seconds in the boiling water?

ii. How long it will take for the temperature of the ball to reach 90oC?

7.2. Newtons Law of Cooling / Heating

Example 4

Suppose Bob made a cup of coffee with boilingwater at 100oC. Bob likes to drink his coffee at70oC. He left his coffee on a table with constantroom temperature of 26oC. After 1 minute, thecoffee temperature had dropped to 95oC. By usingmodel,

How long will it take for Bob to start drinking hiscoffee?

7.2. Newtons Law of Cooling / Heating

=