I

Cold Reservoir

Fig. 4.8 Diagram to demonstrate that the Carnot efficiency is the highest possible for cycle A than for the Carnot Cycle C operating as a heat pump, then shows that this supposition violates the Second law.

Cycles A and C are set to exchange equal quantities of heat with the hot reservoir, which therefore experiences zero net heat transfer. Because cycle A is presumed to be more efficient than Carnot cycle C, the work WA is greater than the work WC needed to operate cycle C. Overall, cycles C and A taken as a single system within the dashed box in the drawing receive an amount of heat net QLC - QLA from the cold reservoir and produce an equal amount of net work, WA - WC. This is accomplished without rejecting heat to another reservoir, thus constituting a violation of the Kelvin-Planck statement of the Second law. This result demonstrates that the initial premise was false; heat engine A cannot have an efficiency greater than the Carnot efficiency.

Example: Calculate the overall efficiency of a Carnot engine placed between two identical blocks, one initially at temperature Tin and the other at temperature Ttq

Ti t2

Because the blocks do no work, the 1s law for each is:

The 2nd law for the heat engine is: SQh/Ti = SQl/T2 , or, with the 1st law: dTi/Ti = -dT2/T2 Integrating from the initial temperatures to a common final temperature Tf gives:

The heat delivered to the engine from the hot source and from the engine to the cold sink is:

Qh = Cv(Tio - Tf); Ql = Cv(Tf - T20) From which the overall engine efficiency follows:

The table below shows several features of this process for T20/T10 = 0.67. The middle column gives results for the heat exchange between the two blocks without the heat engine (Fig. 1.16). The entropy changes for each block are computed using Eq (3.23)

Property

w/o engine

w/ engine

Tf/T10

Getting Started With Solar

Getting Started With Solar

Do we really want the one thing that gives us its resources unconditionally to suffer even more than it is suffering now? Nature, is a part of our being from the earliest human days. We respect Nature and it gives us its bounty, but in the recent past greedy money hungry corporations have made us all so destructive, so wasteful.

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