How to understand the energy involved in mass transfer into an open system?

[zenterix]

Homework Statement

Consider the flow of a constant volume of mass (a closed system) into an open system as shown in the first figure below.

Relevant Equations

The initial energy of the closed system is $(U+E_K+E_P)_i$ and the final energy is $(U+E_K+E_P)_f$.

The following is what is written in the book I am reading.

The energy required to "push" the mass into the system is

$$F\delta z=PA\delta z=PV\tag{1}$$

in which $V$ is the molar volume of the closed system, $F$ is the acting force, $A$ is the cross-sectional area, and $\delta z$ is the width of the system.

The necessary energy transferred across the boundaries of the open system is

$$E_{mt}=(U+E_K+E_P)_f=(U+E_K+E_P)_i+PV\tag{2}$$

and the net energy per mole caused by the mass transfer is

$$Net\ E_{mt}=\sum(U+PV+E_K+E_P)\tag{3}$$

To obtain the total net energy, multiply (3) by the total number of moles.

Finally we can collect all our energy transfer terms:

$$TRANS=\sum Q+\sum W+\sum (U+PV+E_K+E_P)_{mt}\tag{4}$$

My question is: how is (3) obtained?

For a bit more clarity, this book is discussing energy in the context of accounting. They introduce the "General Accounting Equation"

$$ACCumulation=TRANSfer+GENeration+CONVersion\tag{5}$$

To use this equation, w define the system, select the countable property of interest, and apply equation (5).

$TRANS, GEN,$ and $CONV$ terms are net quantities.

Transfer represents input minus output across the boundaries of the system.
Generation represents production-less destruction within the system.
Conversion represents appearance-less disappearance within the system (interchange of the countable quantity among separately identifiable forms).

Accumulation occurs over some stated period of time, the accounting period, during which the system passes from an initial state to a final state.

When we apply GAE to energy we obtain a mathematical statement of the first law.

$$ACC=E_f-E_i=\Delta (U+E_K+E_P)_{SYS}$$

The TRANS term is composed of energy transferred as heat, energy transferred as work, and energy transferred by mass.

As far as I understand, the TRANS term is the $U$ term. This is a "catch-all" term called internal energy (energy that the system possesses because it is composed of energetic particles).

Note that the notation in this book uses capital letters to denote molar properties.

The total energy of a system is then

$$nE=nU+nE_K+nE_P$$

which we can simplify to

$$E=U+E_K+E_P$$

where $E_K$ is molar kinetic energy of the system (motion of the system relative to some reference frame) and $E_P$ is molar potential energy of the system (energy due to interaction with some external field).

Considering just the $U$ term, as mentioned above, this can be changed by means of heat, work, or mass transfer.

Heat is a path function and is transferred via conduction, convection, and/or radiation.

Work is a "catch all" energy transfer mechanism: it is all the energy crossing the boundaries of the system caused by any driving force other than temperature but excluding mass transfer.

I understand calculations of heat and work, but this post is about calculations involving energy transfer via mass.