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Could someone Please help with DIMENSIONLESS VARIABLES.

  1. Feb 7, 2010 #1
    My professor yields this equation. d^2z/d0^2+1=0. This problem has to do with heat conduction. So a plane sheet -b<=x<=+b with a constant heat source Q. The equation that needs to be solved is k(d2T/dx2+Q=0. To change this equation to dimentionless, therefore lets say 0=x/b and z=(T-T(0))/(Qb^2/K). This gives d2z/d0^2+1=0. I dont not understand where the 1 comes from can someone explain?
     
  2. jcsd
  3. Feb 10, 2010 #2
    Its a bit unfortunate to use O as a variable symbol. Btw, there is a parenthesis error in your expression k(d2T/dx2+Q=0. I'll assume k is only multiplying the derivative.

    Anyway, I'd define s := x/b, so its domain is [-1,1].

    The equation then becomes

    k/(Qb^2) * d2T(bs)/ds2 + 1=0

    where I have divided the whole equation by Q. Then define

    z(s) := k/(Qb^2) * T(bs)

    You then end up with

    d^2z/ds^2+1=0

    for s in [-1,1]. The boundary condition on z is obtained from the boundry condition on T, of course.

    Torquil
     
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