Solving heat flux equation: q(x,t) = -k(x) I(t) (dT/dx), for x>=0, t>=0

lfc02

New member
Joined
Mar 13, 2018
Messages
1
The heat flux through an infinitely-long cable at time t is given by:

. . .\(\displaystyle q(x,\,t)\, =\, -k(x)\, I(t)\, \dfrac{dT}{dx},\quad x\, \geq\, 0,\, t\, \geq\, 0\)

where \(\displaystyle k(x)\, >\, 0\) is the heat conductivity function of the cable, \(\displaystyle T(x)\) is the temperature profile, and \(\displaystyle I(t)\) is the heat dissipation as a function of time. Consider the definition of the flux \(\displaystyle q(x,\, t)\) above at a fixed time instant \(\displaystyle t\, =\, 1\) and:

. . .\(\displaystyle k(x)\, =\, x^3\)

. . .\(\displaystyle I(t)\, =\, 1\, +\, \dfrac{1}{1\, +\, e^{-t}}\)

. . .\(\displaystyle T(x)\, =\, \log\left(2x^2\, +\, 3x\, +\, 1\right)\)

Suppose that a valid approximation of the flux at \(\displaystyle t\, =\, 1\) near \(\displaystyle x\, =\, 4\) is given by:

. . .\(\displaystyle \bar{q}(x)\, =\, c_0\, +\, c_1\, (x\, -\, 4)\, +\, c_2\, (x\, -\, 4)^2\)

for some real coefficients \(\displaystyle c_0,\, c_1,\) and \(\displaystyle c_2.\) Compute \(\displaystyle \bar{q}(3).\)
 

Attachments

  • Screen Shot 2018-03-13 at 15.20.51.jpg
    Screen Shot 2018-03-13 at 15.20.51.jpg
    15.5 KB · Views: 16
Last edited by a moderator:
The heat flux through an infinitely-long cable at time t is given by:

. . .\(\displaystyle q(x,\,t)\, =\, -k(x)\, I(t)\, \dfrac{dT}{dx},\quad x\, \geq\, 0,\, t\, \geq\, 0\)

where \(\displaystyle k(x)\, >\, 0\) is the heat conductivity function of the cable, \(\displaystyle T(x)\) is the temperature profile, and \(\displaystyle I(t)\) is the heat dissipation as a function of time. Consider the definition of the flux \(\displaystyle q(x,\, t)\) above at a fixed time instant \(\displaystyle t\, =\, 1\) and:

. . .\(\displaystyle k(x)\, =\, x^3\)

. . .\(\displaystyle I(t)\, =\, 1\, +\, \dfrac{1}{1\, +\, e^{-t}}\)

. . .\(\displaystyle T(x)\, =\, \log\left(2x^2\, +\, 3x\, +\, 1\right)\)

Suppose that a valid approximation of the flux at \(\displaystyle t\, =\, 1\) near \(\displaystyle x\, =\, 4\) is given by:

. . .\(\displaystyle \bar{q}(x)\, =\, c_0\, +\, c_1\, (x\, -\, 4)\, +\, c_2\, (x\, -\, 4)^2\)

for some real coefficients \(\displaystyle c_0,\, c_1,\) and \(\displaystyle c_2.\) Compute \(\displaystyle \bar{q}(3).\)
attachment.php


W
hat are your thoughts?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/announcement.php?f=33
 
Last edited by a moderator:
Top