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Set of equations

  1. Feb 3, 2006 #1
    I'm trying to solve a set of two equations, one of which is an ODE. They are

    m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)


    Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)

    I want to solve this set for [itex] T_M [/itex] and [itex] T_R [/itex], but I'm not sure about the procedure, because of the diff. Any help will be appreciated :smile:
  2. jcsd
  3. Feb 3, 2006 #2


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    Science Advisor

    Are we to assume that Tm, TR, and T0 are functions of t? If T0 is also unknown, then you don't have enough equations. If T0 is a known function of t, then from
    [tex]Q_P + \rho \cdot C_P \cdot \dot V\left( {T_O - T_R } \right) = U \cdot A\left( {T_R - T_O } \right)[/tex]
    [tex]Q_P - \rho \cdot C_P \cdot \dot V\left( {T_R - T_O } \right) = U \cdot A\left( {T_R - T_O } \right)[/tex]
    [Tex]\left(U\cdot A+ \rho \cdot C_P \cdot \dot V\right)\left(T_R- T_O\right)= Q_P[/tex]
    [tex]T_R- T_O=\frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}[/tex]
    [tex]T_R= T_O+ \frac{Q_P}{U\cdot A+ \rho \cdot C_P \cdot \dot V\right}[/tex]
    Now put that function into
    [tex]m \cdot C_P \cdot {{dT_M } \over {dt}} = U \cdot A\left( {T_R - T_M } \right)[/tex]
    and solve the differential equation.
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