Heat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = – k ▿ T. which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/m-s- K . A temperature function for a region D is given. Find the net outward heat flux ∬ S F ⋅ n d S = − k ∬ S ∇ T ⋅ 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 . 42. T( x , y , z ) = 100 + x 2 + y 2 + z 2 ; D = {( x , y , z ): 0 ≤ x ≤ 1, 0 ≤ 1, 0 ≤ z ≤ 1}
Heat transfer Fourier’s Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = – k ▿ T. which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/m-s- K . A temperature function for a region D is given. Find the net outward heat flux ∬ S F ⋅ n d S = − k ∬ S ∇ T ⋅ 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 . 42. T( x , y , z ) = 100 + x 2 + y 2 + z 2 ; D = {( x , y , z ): 0 ≤ x ≤ 1, 0 ≤ 1, 0 ≤ z ≤ 1}
Solution Summary: The author calculates the outward heat flux using the divergence theorem and evaluates a triple integral.
Heat transferFourier’s Law of heat transfer (or heat conduction) states that the heat flow vectorFat a point is proportional to the negative gradient of the temperature; that is,F = –k▿T. which means that heat energy flows from hot regions to cold regions. The constant k > 0 is called the conductivity, which has metric units of J/m-s-K. A temperature function for a region D is given. Find the net outward heat flux
∬
S
F
⋅
n
d
S
=
−
k
∬
S
∇
T
⋅
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.
42. T(x, y, z) = 100 + x2+ y2+ z2;
D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ 1, 0 ≤ z ≤ 1}
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Thomas' Calculus: Early Transcendentals (14th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.