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Newton�s Law of Cooling and Conduction in Thermodynamics

Introduction

Newton�s Law of Cooling states that the rate of he= at 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 functi= ons, likely in an algebra course. Deeper analysis is possible with the introduct= ion of implicit differentiation in a calculus course. The teaching and learning materials, assessments, and resources in this document are intended to be u= sed in mathematics courses included by not limited to college algebra, precalcu= lus, and single-variable calculus.

The focus of these activities is on the function

��= �� ,�� (1)<= o:p>

where

��= ��

A basic background in thermal conduction is necess= ary 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 i= nto 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.<= o:p>

Most any undergraduate differential equations text= book highlights Newton�s Law of Cooling, but a recommended Open Educational Reso= urce 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 function= s.

Learning Outcomes

Upon completion of this activity, students will be able to

-          Distinguish between constants, depende= nt variables, and independent variables in an applied mathematical system.

-          Solve algebraic equations for an unkno= wn variable.

-          Construct a written description of wha= t is occurring in an algebraic system.

Discussion Suggestion

The instructor begins a discussion by asking stude= nts �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. N= ext, 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<= /i> represents the change in time.]� The students independently map all suggestions as either

Assessment Questions

1)&n= bsp;   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 c= onstant, dependent variable, or independent variable.� (see solution A1)=

2)&n= bsp;   Suppose we placed an object outside but forgot to measure its temperature initially. After 5 minutes, the temperatu= re of the object was 58�F. After 10 minutes, it was 49�F. The ambient temperat= ure outside was a constant 40�F. What was the initial temperature of the object= ?� Let �be the constant = of proportionality. (see solution A2)

3)&n= bsp;   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.� Su= ppose we insulated a given object so that heat transferred more slowly. Would this increase or decrease the value of k? Explain your reasoning fully. <= i>(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 tran= sfer of heat through conduction when studying physics or a related discipline. In future mathematics courses, such as calculus, relating a rate with a separa= te 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.<= /span>

Activity: Calculus

The content in this section is most appropriate af= ter 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 exponent= ial 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 gr= aphs based on similar traits. The goal is to address concavity, horizontal asymptotes, and tangent lines.

Next, come back to the influential factors provide= d by the students. Discuss how each one would affect the graph of temperature wi= th respect to time. Finally, display equation (1) to the students.<= /span>

Assessment Questions=

1)&n= bsp;   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 ar= ise from your solution. (see solution C2)

3)&n= bsp;   Suppose that the initial temperature o= f 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)&n= bsp;   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 enro= lled in STEM programs. These activities and assessments prepare them for instanc= es 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 ot= her 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 c= onstant, dependent variable, or independent variable.

��= �� Solution:

A2)�� Suppose we placed an object outside = but forgot to measure its temperature initially. After 10 minutes, the temperat= ure of the object was 49�F. We know the constant of proportionality is . The ambient temperature outside was a constant 40�F. Wh= at 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.� Su= ppose 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 de= crease. 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 decre= ase.

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 cal= led the particular solution.

Solution:<= /span>

Substitute �and �into �and simplify to = solve for C.

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

C2)�� Le= t be the particular solution found from Question 1. = Find the following limit:

. Explain in your own words what physical implications ar= ise from your solution.

�� Solution:

�� Since , we use limit properties to find its limit at infinity:<= o:p>

��

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 temper= ature of an object is 120�F when it is placed in the refrigerator, which maintain= s 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:<= /span>

�� 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:<= /span>

Given , we can differentiate to obtain �and again to obt= ain .� Since the exponential factor of the first derivative test is always positive, we conc= lude that �is always negative.� By the first derivative = test, we conclude that T is strictly decreasing. Furthermore, we determine that �is always positi= ve, so the concavity of T is always upward. We know from given information = that �and from solutio= n 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:
http= s://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 Auth= ored and Edited

Books & CDs<= span style=3D'font-size:12.0pt;line-height:107%;font-family:"Times New Roman",se= rif; color:black;background:white'>. 8. http= s://digitalcommons.trinity.edu/mono/8.

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