A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 40 — 6(x² + y² + 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 ff, 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/(mK)). (Use symbolic notation and fractions where needed.) -K Incorrect Il vu VwdS = 19200T kW
A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 40 — 6(x² + y² + 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 ff, 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/(mK)). (Use symbolic notation and fractions where needed.) -K Incorrect Il vu VwdS = 19200T kW
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![A solid material has thermal conductivity \( K \) in kilowatts per meter-kelvin and temperature given at each point by
\[ w(x, y, z) = 40 - 6(x^2 + y^2 + z^2) \, ^\circ \mathrm{C}. \]
Use the fact that heat flow is given by the vector field \( \mathbf{F} = -K \nabla w \) and the rate of heat flow across a surface \( S \) within the solid is given by
\[ -K \iint_S \nabla w \, dS. \]
Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper
\( (K = 400 \, \text{kW/(m \cdot K)}). \)
(Use symbolic notation and fractions where needed.)
\[ -K \iint_S \nabla w \, dS = \]
\[ \boxed{19200\pi} \]
kW
*Incorrect*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77548912-c51c-4c9d-8b51-f3905a3bec75%2F21307398-509c-4abb-b963-dc16c60c50b2%2Fpqmbrr8_processed.png&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
\[ w(x, y, z) = 40 - 6(x^2 + y^2 + z^2) \, ^\circ \mathrm{C}. \]
Use the fact that heat flow is given by the vector field \( \mathbf{F} = -K \nabla w \) and the rate of heat flow across a surface \( S \) within the solid is given by
\[ -K \iint_S \nabla w \, dS. \]
Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper
\( (K = 400 \, \text{kW/(m \cdot K)}). \)
(Use symbolic notation and fractions where needed.)
\[ -K \iint_S \nabla w \, dS = \]
\[ \boxed{19200\pi} \]
kW
*Incorrect*
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