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The Peclet number express the ratio between the heat transferred by advection and by conduction (diffusion). If it is sufficiently high the heat transferred by conduction can be neglected with respect to the other term. In a book I've found that the first principle of thermodynamic applied to a tube, under the assumption of high Pe, is the well known formula: $$Q=m*Cp*(Tin-Tout)$$ how the first principle should became if the previous assumption is not fulfilled? Thanks

$$\frac{dE}{dt}=\sum m*(h+z+v^2/2)+Q-L$$

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  • $\begingroup$ What is the first principle of thermodynamics in your eyes? The first law of thermodynamics, i.e., $dE = \delta Q + \delta W$? $\endgroup$
    – Sanya
    Commented Sep 5, 2016 at 20:27
  • $\begingroup$ Yes, the conservation of energy... $\endgroup$
    – user116008
    Commented Sep 5, 2016 at 20:34

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Your first equation neglects axial conduction of heat at the inlet and outlet, which is a valid assumption if the Peclet number is high. But, if the Peclet number is low, there can be net axial conduction at the inlet and outlet, and this is not included in your equation (which implicitly assumes that Q as the heat passing through the pipe wall only).

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  • $\begingroup$ So, if I consider a tube and take as the boundary its external surface, the first principle balance is not always: $$Qwall=m(hin-hout)$$ Under the assumption of constant heat capacity the difference in enthalpy is written with the difference in temperature by the heat capacity itself. It seems that it should be, for Pe low: $$Qwall=m(hin-hout)+Qaxial$$ $\endgroup$
    – user116008
    Commented Sep 6, 2016 at 5:36
  • $\begingroup$ Yes. Also, if the temperature is profiled at the inlet and exit, the h's have to be based on the mixing cup average. $\endgroup$
    – Chet Miller
    Commented Sep 6, 2016 at 12:08

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