Q. 4.24 - An Introduction to Thermal Physics

Question

Calculate the efficiency of a Rankine cycle that is modified from the parameters used in the text in each of the following three ways (one at a time), and comment briefly on the results: (a) reduce the maximum temperature to $500°\text{C}$; (b)reduce the maximum pressure to 100 bars; (c)reduce the minimum temperature to $10°\text{C}$.

Step-by-Step Solution

Answer

(a) The efficiency of Rankine cycle is $0.463$when the maximum temperature is reduced to $500°\text{C}$.

(b) The efficiency of Rankine cycle is $0.453$ when the maximum pressure is reduced to $100$ bars.

(c) The efficiency of Rankine cycle is $0.490$ when the minimum temperature is reduced to $10°\text{C}$ .

Step 1. (a) Given information

A Rankine cycle operates between temperatures $20°\text{C}$ and $500°\text{C}$ at maximum pressure 300 bars.

The efficiency of Rankine cycle is given by

$\text{e}=1-\frac{{\text{H}}_4{\text{-H}}_1}{{\text{H}}_3{\text{-H}}_1}$

Here, ${\text{H}}_1$ is enthalpy at point 1, ${\text{H}}_3$ is enthalpy at point 3, and ${\text{H}}_4$ is enthalpy at point 4 in the below Rankine cycle.

The entropy of the superheated steam is expressed as follows:

${\text{S}}_4\text{=x}\left({\text{S}}_{water}\right)\text{+}\left(1-\text{x}\right){\text{S}}_{steam}$

where,

$$\begin{gathered} {\text{S}}_{water}\text{=Entropy of saturated water} \\ {\text{S}}_{steam}\text{=Entropy of saturated steam} \\ \text{x=fraction of water } \end{gathered}$$

So, $\text{x=}\frac{{\text{S}}_4{\text{-S}}_{\text{steam}}}{{\text{S}}_{water}{\text{-S}}_{\text{steam}}}$

Similarly enthalpy at point 4: $H_4\text{=x}\left({\text{H}}_{water}\right)\text{+}\left(1-\text{x}\right){\text{H}}_{steam}$.

where

$$\begin{gathered} {\text{H}}_{water}\text{=Enthalpy of saturated water} \\ {\text{H}}_{steam}\text{=Enthlapy of saturated steam} \\ \text{x=fraction of water } \end{gathered}$$

Step 2. Calculation

Here,

$$\begin{gathered} {\text{S}}_4{\text{=5.971 KJ·K}}^{-1}{\text{·kg}}^{-1} \\ {\text{S}}_{water}{\text{=0.297 KJ·K}}^{-1}{\text{·kg}}^{-1} \\ {\text{S}}_{steam}{\text{=8.667 KJ·K}}^{-1}{\text{·kg}}^{-1} \end{gathered}$$

So, 

$$\begin{gathered} \text{x=}\frac{5.971-8.667}{0.297-8.667} \\ \text{=0.344} \end{gathered}$$

Now,

$$\begin{gathered} \text{x=0.344} \\ {\text{H}}_{\text{water}}\text{=84 KJ/kg} \\ {\text{H}}_{steam}\text{=2538 KJ/kg} \end{gathered}$$

So,

$$\begin{gathered} {\text{H}}_4\text{=}\left(0.344\right)\left(84 \text{KJ/kg}\right)\text{+}\left(1-0.344\right)\left(2538 \text{KJ/kg}\right) \\ \text{=1693.8 KJ/kg} \end{gathered}$$

Hence,

$$\begin{gathered} \text{e=1-}\frac{1693.8-84}{3081-84} \\ =0.463 \end{gathered}$$

Step 3. Conclusion

The efficiency of Rankine cycle is 0.463 when the maximum temperature is reduced to $500°\text{C}$.

Step 4. (b) Given information

A Rankine cycle operates between temperatures  and  at maximum pressure 300 bars.

The efficiency of Rankine cycle is given by

Here,  is enthalpy at point 1,  is enthalpy at point 3, and  is enthalpy at point 4 in the below Rankine cycle.

The entropy of the superheated steam is expressed as follows:

where,

So,

Similarly enthalpy at point 4: .

where