Question

A power cycle receives QH by heat transfer from a hot reservoir at TH = 1200 K and rejects energy QC by heat transfer to a cold reservoir at TC = 400 K. For each of the following cases, determine whether the cycle operates reversibly, operates irreversibly, or is impossible.

a.QH = 900 kJ, Wcycle= 450 kJ
b. QH = 900 kJ, Qc = 300 kJ
c. Weycle = 600 kJ, Qc= 400 kJ
d. η = 75%

Answer 1

Answer:

a) Irreversible, b) Reversible, c) Irreversible, d) Impossible.

Explanation:

Maximum theoretical efficiency for a power cycle ($\eta_{r}$), no unit, is modelled after the Carnot Cycle, which represents a reversible thermodynamic process:

$\eta_{r} = \left(1-\frac{T_{C}}{T_{H}} \right)\times 100\,\%$ (1)

Where:

$T_{C}$ - Temperature of the cold reservoir, in Kelvin.

$T_{H}$ - Temperature of the hot reservoir, in Kelvin.

The maximum theoretical efficiency associated with this power cycle is: ($T_{C} = 400\,K$, $T_{H} = 1200\,K$)

$\eta_{r} = \left(1-\frac{400\,K}{1200\,K} \right)\times 100\,\%$

$\eta_{r} = 66.667\,\%$

In exchange, real efficiency for a power cycle ($\eta$), no unit, is defined by this expression:

$\eta = \left(1-\frac{Q_{C}}{Q_{H}}\right) \times 100\,\% = \left(\frac{W_{C}}{Q_{H}} \right)\times 100\,\% = \left(\frac{W_{C}}{Q_{C} + W_{C}} \right)\times 100\,\%$ (2)

Where:

$Q_{C}$ - Heat released to cold reservoir, in kilojoules.

$Q_{H}$ - Heat gained from hot reservoir, in kilojoules.

$W_{C}$ - Power generated within power cycle, in kilojoules.

A power cycle operates irreversibly for $\eta < \eta_{r}$, reversibily for $\eta = \eta_{r}$ and it is impossible for $\eta > \eta_{r}$.

Now we proceed to solve for each case:

a) $Q_{H} = 900\,kJ$, $W_{C} = 450\,kJ$

$\eta = \left(\frac{450\,kJ}{900\,kJ} \right)\times 100\,\%$

$\eta = 50\,\%$

Since $\eta < \eta_{r}$, the power cycle operates irreversibly.

b) $Q_{H} = 900\,kJ$, $Q_{C} = 300\,kJ$

$\eta = \left(1-\frac{300\,kJ}{900\,kJ} \right)\times 100\,\%$

$\eta = 66.667\,\%$

Since $\eta = \eta_{r}$, the power cycle operates reversibly.

c) $W_{C} = 600\,kJ$, $Q_{C} = 400\,kJ$

$\eta = \left(\frac{600\,kJ}{600\,kJ + 400\,kJ} \right)\times 100\,\%$

$\eta = 60\,\%$

Since $\eta < \eta_{r}$, the power cycle operates irreversibly.

d) Since $\eta > \eta_{r}$, the power cycle is impossible.