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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(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/(m .K)). (Use symbolic notation and fractions where needed.) x ff. vw d -K VwdS= 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/(m .K)). (Use symbolic notation and fractions where needed.) x ff. vw d -K VwdS= kW

Advanced Engineering Mathematics
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(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 Vwds. 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 d VwdS = kW
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(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 Vwds. 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 d VwdS = kW
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