Example 8A - 2: Heat, Work and Entropy Generation

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

Two power cycles operate between the same two thermal reservoirs, as shown below. Cycle R is reversible and cycle I is irreversible.

They each absorb the same amount of heat from the hot reservoir, Q_H, but produce different amounts of work, W_R and W_I, and reject different amounts of heat to the cold reservoir, Q_C and Q'_C.

a.) Derive an equation for S_gen for the irreversible cycle in terms of W_I, W_R, and T_C only.
b.) Show that W_I W_R and Q'_C Q_C.

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 an 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 S_gen for cycle I in terms of W_I, W_R, and T_C.

Part (b)

Show that:

W_I < W_R

and

Q'_C > Q_C.

Diagram:

The diagram in the problem statement is adequate.

Assumptions:

1 -

The systems shown undergo power cycles. R is reversible and I is irreversible.

2 -

Each system receives Q_H at a constant temperature region at T_H from the hot reservoir and rejects heat, Q_C, at a constant temperature region at T_C 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 S_gen :

Eqn 2

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

Eqn 3

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