Consider the Control Volume as developed in **Section
a)**, however with the heat being
transferred from a thermal source at temperature T_{H}
which is higher than the surroundings temperature T_{0}.

**Energy (First Law):**

**Entropy Generated (Second Law):**

Since heat is exchanged with a temperature source
different from the dead space temperature T_{0} the entropy
generated is:

We now consider a **reversible** process between
the same inlet (i) and exit (e) states and heat transfer q. This
requires that reversible heat transfer q_{0,rev} will occur
from the surroundings to the control volume such that the entropy
generated s_{gen} = 0.

Thus:

Adding this new reversible heat source to the energy equation (1) above, and substituting equation (9) we obtain:

You may be confused as to how we can justify
transferring heat q_{0} reversibly from the surroundings at
T_{0} to the control volume at a higher temperature. This
unique approach due to Sonntag and Borgnakke (Introduction to
Engineering Thermodynamics, Wiley, 2001) is done for convenience in
order to validate the derivation of equation (10).

In order to justify this, consider the equivalent
system shown below, in which the hot source at T_{H} is used
as the heat source of a reversible heat engine, which in turn drives
a heat pump to deliver the required heat q to the control volume. We
will show that this system gives rise to the identical equation (10)
as above.

From the energy equation for the reversible heat
engine (recall **Chapter
5**) we have:

From the diagram we see that some of this work (w_{HP})
is used to drive the reversible heat pump. Since the temperature T of
the control volume varies from the inlet to the outlet, we consider
the differential energy equation for the work *into* the heat
pump:

Integrating across the entire control volume (inlet to outlet) we obtain:

From the diagram above we see that the total reversible work available from this system is given by:

Substituting from equations (11), (12) and the initial energy equation (1) above, we have:

Simplifying equation (13) leads to equation (10) as
above - **QED**. (*Quad Erat Demonstratum* - Latin for "which
was to be proved" used smugly by math gurus whenever they
successfully conclude a proof - usually accompanied by a
condescending smile)

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Engineering Thermodynamics by Israel
Urieli is licensed under a Creative
Commons Attribution-Noncommercial-Share Alike 3.0 United States
License