Newton’s
Law of Cooling and Conduction in Thermodynamics

Introduction

Newton’s Law of Cooling states that the rate of heat
loss of an object in a cooler environment is proportional to the temperature
difference between the object and the environment. The rate of heat transfer
may begin to be studied after students are introduced to exponential functions,
likely in an algebra course. Deeper analysis is possible with the introduction
of implicit differentiation in a calculus course. The teaching and learning
materials, assessments, and resources in this document are intended to be used
in mathematics courses included by not limited to college algebra, precalculus,
and single-variable calculus.

The
focus of these activities is on the function

,             (1)

where

A basic background in thermal conduction is necessary
as preparation for the instructor. Newton’s Law of Cooling comes from the
differential equation

,                     (2)

whose general solution is equation (1). Equation (2)
can be expanded to evaluate the rate of conductive heat transfer

,                (3)

where

Please note that k is not consistent in
equations (2) and (3). Instead, the k in equation (2) is separated into
components k, A, and d in equation (3) and normalized to
convert temperature (T) to power (P) for a more comprehensive
study of the effects of insulating material and relative sizes of objects.

Most any undergraduate differential equations textbook
highlights Newton’s Law of Cooling, but a recommended Open Educational Resource
is by Trench (2013). The rate of conductive heat transfer is emphasized in many
undergraduate Physics textbook, such as the OER by Ling et al. (2018).

Activity: Algebra

The content in this section is appropriately placed
when after students have a familiarity with evaluating exponential functions.

Learning Outcomes

Upon completion of this activity, students will be
able to

-
Distinguish between constants, dependent variables,
and independent variables in an applied mathematical system.

-
Solve algebraic equations for an unknown
variable.

-
Construct a written description of what is
occurring in an algebraic system.

Discussion Suggestion

The instructor begins a discussion by asking students “If
we place a hot object in a cold environment, it will cool down. What factors
determine how fast the object cools?” All suggestions should be recorded. Next,
display equation (2) to the students and identify the variables [it may be
helpful to note that dT represents the change in T and dt
represents the change in time.]  The
students independently map all suggestions as either

Assessment Questions

1)
An object that is 100°F is placed in an
environment that is 40°F. Assume we want to determine what the temperature of
the object would be after a certain amount of time. Identify each symbol in the
equation  as either a constant,
dependent variable, or independent variable.  (see solution A1)

2)
Suppose we placed an object outside but
forgot to measure its temperature initially. After 5 minutes, the temperature
of the object was 58°F. After 10 minutes, it was 49°F. The ambient temperature
outside was a constant 40°F. What was the initial temperature of the object?  Let  be the constant of
proportionality. (see solution A2)

3)
The instantaneous rate of change, , of a cooling object is calculated by evaluating , where k is the constant of proportionality, T
is the temperature of the object, and  is the ambient
temperature of the environment.  Suppose
we insulated a given object so that heat transferred more slowly. Would this
increase or decrease the value of k? Explain your reasoning fully. (see
solution A3)

Future Value to the Students

Students who achieve the specified outcomes through
the activities should be better prepared to intuitively understand the transfer
of heat through conduction when studying physics or a related discipline. In
future mathematics courses, such as calculus, relating a rate with a separate
variable is an important skill (assessment question 3). In application,
Newton’s Law of Cooling extends to the rate of conductive heat transfer. The
components of the constant of proportionality have practical use in various
industries including insulation, refrigeration, and heat pumps.

Activity: Calculus

The content in this section is most appropriate after
the students have studied differentiation including the natural exponential
function and natural logarithm function.

Learning Outcomes

Upon completion of this activity, students will be
able to

-
Evaluate the derivative of an exponential
function with an initial condition.

-
Explain the limit at infinity as it
applies to an exponential decay model.

-
Graph the function that models Newton’s
Law of Cooling.

Discussion Suggestion

The instructor asks, “If we place a hot object in a cold
environment, what factors influence how long it takes to cool down?” The
instructor should record all contributions from students. Next, the students
draw the graph of the model that describes how the temperature (T)
changes over time (t) which all are to be displayed to the class
anonymously. The instructor asks them to make reasonable assumptions about
values and label their axes. Via class discussion, the group categorizes graphs
based on similar traits. The goal is to address concavity, horizontal
asymptotes, and tangent lines.

Next, come back to the influential factors provided by
the students. Discuss how each one would affect the graph of temperature with
respect to time. Finally, display equation (1) to the students.

Assessment Questions

1)
The differential equation that models Newton’s
Law of Cooling is , which has a general solution . Evaluate the general solution for  and
. In doing so, solve for C and substitute back into
the solution. (see solution C1)

2)    Let
be the particular solution found from Question 1. Find the
following limit:

. Explain in your own words what physical implications arise
from your solution. (see solution C2)

3)
Suppose that the initial temperature of an
object is 120°F when it is placed in the refrigerator, which maintains a
constant 36°F ambient temperature. If the temperature at the object reduces to
78°F after 3 minutes, find k. (see solution C3)

4)
Graph the solution found in Question 3.
Include accuracy in terms of increasing and decreasing intervals, concavity,
domain and range, and endpoint behavior. (see solution C4)

Future Value to the Students

Many students who enroll in calculus are also enrolled
in STEM programs. These activities and assessments prepare them for instances
where they will need to utilize actual data and form mathematical models.  The models themselves will be studied in
depth in courses such as differential equations (ordinary and partial),
calculus-based physics, and numerical analysis. Outside of the physics
application displayed in this document, numerical models are utilized in other
sciences. Applications including by not limited to chemical reactions,
biological processes, population dynamics, and econometrics are areas where
experience working with mathematical models is important.

Solutions
to Assessment Questions

Algebra Solutions

A1)      An object that is 100°F is placed in an
environment that is 40°F. Assume we want to determine what the temperature of
the object would be after a certain amount of time. Identify each symbol in the
equation  as either a constant,
dependent variable, or independent variable.

Solution:

A2)      Suppose we placed an object outside but
forgot to measure its temperature initially. After 10 minutes, the temperature
of the object was 49°F. We know the constant of proportionality is . The ambient temperature outside was a constant 40°F. What
was the initial temperature of the object?

Solution:

Given information is
provided: , , and . We evaluate  to have . This simplifies:

So
the initial temperature was 76°F

A3)      The instantaneous rate of change, , of a cooling object is calculated by evaluating , where k is the constant of proportionality, T
is the temperature of the object, and  is the ambient
temperature of the environment.  Suppose
we insulated a given object so that heat transferred more slowly. Would this
increase or decrease the value of k? Explain your reasoning fully.

Solution:

The value of k would decrease.
Explanations will vary. However, the instructor should focus on whether or not
the following were identified:  will always be
positive and  will always be
negative (since the object is cooling). Therefore k
must be positive. If the rate of decrease in temperature gets closer to 0,
then k must also get closer to zero. Therefore, k would decrease.

Calculus Solutions

C1)      The differential equation that models
Newton’s Law of Cooling is , which has a general solution . Evaluate the general solution for  and
. In doing so, solve for C and substitute back into
the solution.  Your solution is called
the particular solution.

Solution:

Substitute  and  into  and simplify to solve
for C.

Then substitute  into  to obtain the
particular solution, .  Notice that this is
Equation (1).

C2)      Let be the particular solution found from Question 1. Find
the following limit:

. Explain in your own words what physical implications arise
from your solution.

Solution:

Since
, we use limit properties to find its limit at infinity:

Explanations of the
physical implications of the limit at infinity will vary but should include
that over a long period of time, the temperature of the object will cool to be
the same as the temperature of the environment.

C3)
Suppose that the initial temperature
of an object is 120°F when it is placed in the refrigerator, which maintains a
constant 36°F ambient temperature. If the temperature at the object reduces to
78°F after 3 minutes, find k. Then, express the particular model in
terms of time t.

Solution:

From , we substitute  and  to get  which simplifies to . Since , we have . We solve for k to obtain . We now substitute  to obtain  .

C4)      Graph the solution found in Question 3.
Include accuracy in terms of increasing and decreasing intervals, concavity,
domain and range, and endpoint behavior.

Solution:

Given , we can differentiate to obtain  and again to obtain .  Since the
exponential factor of the first derivative test is always positive, we conclude
that  is always
negative.  By the first derivative test,
we conclude that T is strictly decreasing. Furthermore, we determine
that  is always positive, so
the concavity of T is always upward. We know from given information that
and from solution C2
that .  Therefore, we also
have a horizontal asymptote at . This information is utilized to graph T, as shown
below:

Graph
obtained using DESMOS Online Calculator

References

Desmos Graphing Calculator.
2018. Desmos Graph. [online] Available at:

https://www.desmos.com/calculator.

Ling, Samuel J., Sanny,
Jeff, & Moebs, William, “University Physics
Volume 2” (2018).

Openstax.
https://openstax.org/details/books/universityphysics-volume-2.

Trench, William F.,
"Elementary Differential Equations" (2013). Faculty Authored
and Edited

Books & CDs. 8. https://digitalcommons.trinity.edu/mono/8.