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

  1. #1
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    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:

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

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

    . . .[tex]k(x)\, =\, x^3[/tex]

    . . .[tex]I(t)\, =\, 1\, +\, \dfrac{1}{1\, +\, e^{-t}}[/tex]

    . . .[tex]T(x)\, =\, \log\left(2x^2\, +\, 3x\, +\, 1\right)[/tex]

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

    . . .[tex]\bar{q}(x)\, =\, c_0\, +\, c_1\, (x\, -\, 4)\, +\, c_2\, (x\, -\, 4)^2[/tex]

    for some real coefficients [tex]c_0,\, c_1,[/tex] and [tex]c_2.[/tex] Compute [tex]\bar{q}(3).[/tex]
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    Last edited by stapel; 03-14-2018 at 02:33 PM. Reason: Typing out the text in the graphic; creating useful subject line.

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

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

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

    . . .[tex]k(x)\, =\, x^3[/tex]

    . . .[tex]I(t)\, =\, 1\, +\, \dfrac{1}{1\, +\, e^{-t}}[/tex]

    . . .[tex]T(x)\, =\, \log\left(2x^2\, +\, 3x\, +\, 1\right)[/tex]

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

    . . .[tex]\bar{q}(x)\, =\, c_0\, +\, c_1\, (x\, -\, 4)\, +\, c_2\, (x\, -\, 4)^2[/tex]

    for some real coefficients [tex]c_0,\, c_1,[/tex] and [tex]c_2.[/tex] Compute [tex]\bar{q}(3).[/tex]


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    Last edited by stapel; 03-14-2018 at 02:33 PM. Reason: Copying typed-out graphical content into reply.
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