Chapter8, Lesson A - Heat, Work and Entropy Generation

8A-2 :	Heat, Work and Entropy Generation						5 pts

A reversible power cycle R and an irreversible power cycle I operate between the same two reservoirs. Each receives QH from the hot reservoir. The reversible cycle develops work WR, while the irreversible cycle develops work WI. The reversible cycle discharges QC to the cold reservoir, while the irreversible cycle discharges Q'C.

a.)	Evaluate the Sgen for cycle I in terms of WI, WR, and temperature TC of the cold reservoir only.
b.)	Demonstrate that WI < WR and Q'C > QC.


Read :	Start with the equation for the entropy generated and do an energy balance on both the reversible and irreversible cycles. Put the equations together and simplify to get a equation in the desired terms.


Given :	A reversible power cycle R and an irreversible power cycle I operate between the same two reservoirs.


Find :	Part (a)	Evaluate Sgen for cycle I in terms of WI, WR, and TC.
	Part (b)	Show that:		WI < WR	and	Q'C > QC.

Diagram :









Assumptions :
	1 -	The systems shown undergo power cycles. R is reversible and I is irreversible.
	2 -	Each system receives QH at a constant temperature region at TH from the hot reservoir and rejects heat, QC, at a constant temperature region at TC to the cold reservoir.


Equations / Data / Solve :

Part a.)	Let's begin with the defintion of entropy generation:						Eqn 1

	We can solve Eqn 1 for Sgen :						Eqn 2

	Since we are dealing with a cycle, ΔS = 0 and Eqn 2 becomes:
							Eqn 3

	In the irreversible process, the system receives heat, QH, at a constant temperature, TH , and rejects heat, Q'C, at a constant temperature, TC. Because the temperatures are constant, they can be pulled out of the integral in Eqn 3 leaving :



							Eqn 4

	The 1st Law for cycles is:						Eqn 5

	We can apply Eqn 5 to both the reversible and the irreversible cycles, as follows :

			Eqn 6				Eqn 7

	We can combine Eqns 6 & 7 to obtain :						Eqn 8

	Now, solve Eqn 8 for Q'C :						Eqn 9

	Next, we can use Eqn 9 to eliminate Q'C from Eqn 4 to get :
	Next, we can use Eqn 9 to eliminate Q'C from Eqn 4 to get :						Eqn 10


	We can rearrange Eqn 10 slightly to make it more clear how to proceed :
							Eqn 11

	Because R is a reversible cycle and we use the Kelvin Temperature Scale :
							Eqn 12

	Eqn 12 can be rearranged to help simplify Eqn 11 :						Eqn 13

	This yields :						Eqn 14


Part b.)	Because irreversibilities are present in cycle I :						Eqn 15

	Rearranging Eqn 15 gives us :						Eqn 16

	Finally, we can rearrange Eqn 9 to help us determine whether Q'C or QC is larger :
							Eqn 17

	Since Eqn 16 tells us that WR > WI, Eqn 17 tells that :						Eqn 18

Verify :	The assumptions made in this solution cannot be verified with the given information.


Answers :	Part a.)				Part b.)

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