Newton's Law of Cooling as a Differential Equation dT 4 = k(M – T) dt !! where T'is the temperature of the object at a given time t, M is the temperature of the surrounding medium, and k is a positive constant. From this we note that if M>T, we have heating, since M – T > 0 thus dT/dt > 0, which means an increasing T, heating up! If M

Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
icon
Related questions
Question
Newton's Law of Cooling as a Differential Equation
dT
k(м-т)
dt
where Tis the temperature of the object at a given time t, M is the temperature of
the surrounding medium, and k is a positive constant. From this we note that if
M>T, we have heating, since M – T > 0 thus dT/dt > 0, which means an increasing T,
heating up! If M<T, we have cooling, as since M – T < 0 thus dT/dt < 0, which means
a decreasing T, cooling down!
Now, working from the differential equation of Newton's law of cooling, solve the
following situation:
Mary and Bob each bring a cup of coffee from the dining hall at Christchurch to their
class, the classroom that is kept at a constant temperature of 70 °F. Assume the
temperature of the coffee at the beginning of class is 200 °F. Mary has a well-
insulated coffee mug; her coffee is 194 °F after 5 minutes of class. Bob has a paper
cup coffee mug, it is not well insulated; his coffee is 150 °F after 5 minutes.
Calculate the temperature of Mary's coffee at the end of class.
Calculate the temperature of Bob's coffee at the end of class.
Comment on the value of the constant usually referred to as k in the differential
equation, how does this value relate to the heat transfer between the coffee and
surrounding medium?
Transcribed Image Text:Newton's Law of Cooling as a Differential Equation dT k(м-т) dt where Tis the temperature of the object at a given time t, M is the temperature of the surrounding medium, and k is a positive constant. From this we note that if M>T, we have heating, since M – T > 0 thus dT/dt > 0, which means an increasing T, heating up! If M<T, we have cooling, as since M – T < 0 thus dT/dt < 0, which means a decreasing T, cooling down! Now, working from the differential equation of Newton's law of cooling, solve the following situation: Mary and Bob each bring a cup of coffee from the dining hall at Christchurch to their class, the classroom that is kept at a constant temperature of 70 °F. Assume the temperature of the coffee at the beginning of class is 200 °F. Mary has a well- insulated coffee mug; her coffee is 194 °F after 5 minutes of class. Bob has a paper cup coffee mug, it is not well insulated; his coffee is 150 °F after 5 minutes. Calculate the temperature of Mary's coffee at the end of class. Calculate the temperature of Bob's coffee at the end of class. Comment on the value of the constant usually referred to as k in the differential equation, how does this value relate to the heat transfer between the coffee and surrounding medium?
Expert Solution
steps

Step by step

Solved in 4 steps with 4 images

Blurred answer
Knowledge Booster
Convection
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Elements Of Electromagnetics
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY