A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 15 - 9(x? + y? + z2) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K s Vw dS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m K)). (Use symbolic notation and fractions where needed.)
A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 15 - 9(x? + y? + z2) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K s Vw dS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m K)). (Use symbolic notation and fractions where needed.)
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
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![A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by
15 – 9(x2 + y + z?) °C. Use the fact that heat flow is given by the vector field F
w(x, y, z)
-KVw and the rate of heat
flow across a surface S within the solid is given by -K s Vw dS.
Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper
(K
400 kW/(m · K)).
(Use symbolic notation and fractions where needed.)
-K
Vw dS =
kW](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33d65e5b-41b1-4586-a5a9-78bc9a97b2d1%2Fafcb5146-a643-414c-aab8-085306af255f%2Ftnq1ch_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by
15 – 9(x2 + y + z?) °C. Use the fact that heat flow is given by the vector field F
w(x, y, z)
-KVw and the rate of heat
flow across a surface S within the solid is given by -K s Vw dS.
Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper
(K
400 kW/(m · K)).
(Use symbolic notation and fractions where needed.)
-K
Vw dS =
kW
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