Question

A power cycle operates between hot and cold reservoirs at 600 K and 300 K, respectively. At steady state the cycle develops a power output of 0.45 MW while receiving energy by heat transfer from the hot reservoir at the rate of 1 MW.
(a) Determine the thermal efficiency and the rate at which energy is rejected by heat transfer to the cold reservoir, in MW.
(b) Compare the results of part (a) with those of a reversible power cycle operating between these reservoirs and receiving the same rate of heat transfer from the hot reservoir.

Answer 1

Concepts and reason

Thermal efficiency of the cycle:

It is defined as the ratio of the work done by the cycle to the amount of heat supplied to the cycle. It is denoted by and it is expressed in terms of percentage.

The thermal efficiency of a power cycle is calculated as the ratio of rate of work output from the cycle to rate of heat added to the cycle.

The rate of heat transfer to the cold reservoir is calculated by subtracting rate of power output from rate of heat transfer from the hot reservoir.

Fundamentals

Heat:

It is a form of energy exchanged between two systems and responsible for temperature difference between the systems.

Work:

Work is the amount of energy transferred between two systems or from a system to its surroundings.

Reversible process:

It refers to the thermodynamics process for which the direction of process can be reversed i.e. the initial and final conditions of a process can be inversed.

First law of thermodynamics:

First law of thermodynamics correlates the energy conservation law which states that “Energy can neither be created nor be destroyed. It can only be transformed from one form to other”.

For any system, the internal energy change will be the difference in heat that is supplied to the system and the work output gained from the system.

Thermodynamic cycle:

It refers to the series of thermodynamic processes for which the initial and final properties are identical. When the supplied heat energy is transformed to purposeful work output, the cycle is referred as power cycle. When the conventional direction of heat flow is inversed using work in a cycle, the cycle is referred as heat pump cycles.

The formula to calculate the thermal efficiency $\left( \eta \right)$ of a power cycle is as follows:

…… (1)

Here, the rate of work output from the cycle is and the rate of heat added to the cycle is .

The formula to calculate the rate of work output $\left( {\dot W} \right)$ from the cycle is as follows:

…… (2)

Here, the rate of heat rejected from the cycle is .

The formula to calculate the thermal efficiency of a reversible power cycle $\left( {{\eta _{rev}}} \right)$ operating between different temperatures is as follows:

…… (3)

Here, the higher temperature of the reversible cycle is and the lower temperature of the reversible cycle is .

a)

Calculate the thermal efficiency $\left( \eta \right)$ of the power cycle.

Here, the rate of power output is and the rate of heat transfer from the hot reservoir is .

Substitute for and for .

Calculate the rate of heat transfer to the cold reservoir.

Here, the rate of heat transfer from the cold reservoir is .

Substitute for and for .

b)

Calculate the thermal efficiency of the reversible power cycle.

Here, the temperature of hot reservoir is and the temperature of cold reservoir is .

Substitute for and for .

Calculate the rate of work output from the reversible power cycle.

Here, rate of work output from the reversible power cycle is ${\dot W_{rev}}$ .

Substitute for and for .

Calculate the rate of heat transfer to the cold reservoir.

Substitute for and for .

Ans: Part a.1

The thermal efficiency of the power cycle is .