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

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

The heat flux through an infinitely-long cable at time t is given by:

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

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

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

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

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

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

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

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

2. Originally Posted by lfc02
The heat flux through an infinitely-long cable at time t is given by:

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

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

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

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

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

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

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

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

W

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