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 \(\displaystyle \bar{q}(3).\)
. . .\(\displaystyle q(x,\,t)\, =\, -k(x)\, I(t)\, \dfrac{dT}{dx},\quad x\, \geq\, 0,\, t\, \geq\, 0\)
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
Last edited by a moderator: