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So, I'm given the following equation: $$\frac{∂Q}{∂t}+\frac{dm_i}{dt}(E_i+P_iV_i)=\frac{dE}{dt}+\frac{dm_{fi}}{dt}(E_f+P_fV_f)-\frac{∂W_s}{∂t}$$

Using that equation, I am supposed to then simplify it into $$\overset{.}Q+\overset{.}W=\overset{.}m(h_1-h_2)$$ However, I don't know how to start, and so far I haven't found anything similar that I can use as a basis.

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    $\begingroup$ It wouldn't hurt if you define every variable in the posted equation, too. I'm pretty sure you are using inconsistent labelling, e.g. with "f", "2" and "fi" for "final", and the derivative shouldn't be a variable. A general note: in chemistry and physics, boldface usually refers to a vector. $\endgroup$
    – andselisk
    Mar 9 '21 at 16:03
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    $\begingroup$ Please rewrite ! Write $E$ instead of $e$, $ H$ instead of $ h$. Write $dt$ instead of $d_t$. Write the indexes as proposed by Andselisk. Explain what is $m$ ! Write $dE$ instead of $E_{t+dt} $- $E_t$ Explain why the mass and the work are supposed to change in time. $\endgroup$
    – Maurice
    Mar 9 '21 at 21:11
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The first equation is the general form of the open system (control volume) version of the 1st law of thermodynamics. The 2nd equation is what this reduces to under a certain set of circumstances. In determining how this comes about, I would consider:

  1. Steady state versus transient operation

  2. Expression for e in terms of specific internal energy u, kinetic energy per unit mass, and gravitational potential energy per unit mass.

  3. Importance of internal energy change relative to kinetic energy change and gravitational potential energy change.

  4. Expression for h in terms of u and Pv.

  5. The need for a subscript of S on the W of the 2nd equation, and what this means physically.

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