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 Sontag 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
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