Answer: Heat transfer Fourier’s Law of heat transfer (or

QUESTION:

Heal transfer Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F al a point is proportional To the negative gradient of the temperature: that is, \(\mathbf{F}=-k \nabla T\). which means that heal energy flows fmnt hot regions to cold regions. The constant k is called the conductivity which has metric units of J/m-s-K or W/m-K. A temperature function for a region D is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of D.

In some cases it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1.

\(T(x, y, z)=100+e^{-z}\);

\(D=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}\)

Text Transcription:

F = -k nabla T

iint_S F cdot n dS = -k iint_S nablaT cdot n dS

T(x, y, z) = 100 + e^-1

D = {(x, y, z): 0 leq x leq 1, 0 leq y leq 1, 0 leq z leq 1}