Exploring the Newton\'s Cooling approximation of the Stefan-Boltzmann law: A cub

Question

Exploring the Newton's Cooling approximation of the Stefan-Boltzmann law: A cube of copper of mass 10 kg and initial temperature of 110 oC is set to radiatively cool in an environment of 100 oC. (Note: copper has an emissivity of about 0.05. Also, neglect conduction and convection as cooling agents in this problem.) a.) What is the total outside surface area of this cube? .06456 Correct: Your answer is correct. m2 b.) Before the copper has had a chance to cool, find the net rate of heat transfer of the cube to the environment. Note: to calculate the % error accurately, make sure to keep enough decimal places in both P values. Initially, Pcube = exactly: .39445 Incorrect: Your answer is incorrect. W in the Newton's cooling approximation: .3955 Incorrect: Your answer is incorrect. W size of % error in approximation= % (Answer + only.) c.) Using the approximation, find: the time constant () for this system: min the time it takes for the cube to cool within 1oC of the environment: min Also, sketch a plot of T vs. t under this approximation. d.) Revisiting part b, re-estimate the validity of the approximation for this 1oC temperature difference: Now, Pcube = exactly: W in the Newton's cooling approximation: W size of % error = % (Answer + only.)

Explanation / Answer

(a) Before the copper has had a chance to cool, find the net rate of heat transfer of the cube to the environment.
Using Pnet=Aes(to^4-ts^4), s=stefan boltzman constant, to= temp of object, ts= temp of surroundings, e=emissivity, a= surface area.
(.6456)(.05)(5.67*10^-8)(383.15^4-373.15^4)=.3.95954069438

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