# Power Plant Analysis (Vapour Cycles)

This is a topic from Thermodynamics:

## Contents |

## Introduction

A common exam question which is a culmination of the material covered in the course is an analysis of the efficiency of the steam power plant, which is a form of vapour cycle. In vapour cycles the ideal gas laws do not apply because the fluid tends to change phase during the various processes which comprise the cycle such that it is not always a gas. There are many different vapour cycles which you may be asked to analyse, but two important cycles are covered in this section to exemplify the method of analysis.

## The Basic Rankine Cycle

The Rankine cycle is as follows:

There is work in during the pumping phase, heat in during the boiling phase ('vaporiser'), useful work out during the turbine phase ('expander') and heat out during the condensor phase. Recalling that thermal efficiency equals net work out divided by heat in, we need to consider the boiler heat and both the work in and work out to calculate efficiency.

## The Rankine Regeneration Cycle

This cycle is a type of Rankine cycle where the mass flow rate into the turbine is split into two components, one which passes through the high pressure turbine and into the open feedwater heater (this is path 5→6) and one which passes through the low pressure turbine and is compressed and pumped back into the open feedwater heater (this is path 5→7). A proportion of ṁ equal to y follows path 6→7→3 and the remaining proportion equal to 1-y follows path 5→7→1→2→3.

### Analysis for the Ideal Case

- The thermal efficiency is defined as
**η**_{th}= W_{net}/ Q_{in} - The heat and work flows are as follows:

- The work out of the system (Ẇ
_{out}) occurs during the expansion through the turbine. - The work into the system (Ẇ
_{in}) occurs during the two pumping phases (Ẇ_{12}and Ẇ_{34}). - The heat out of the system (Q̇
_{out}) occurs during the condensing phase. - The heat into the system (Q̇
_{in}) occurs during the boiling phase.

- The work out of the system (Ẇ

- The thermal efficiency is defined as

**Step 1: Find the value of y**

- Considering the first law of thermodynamics for the open feedwater heater:

- Q̇ - Ẇ = ṁ
_{out}(h + V^{2}/2 + gz)_{out}- ṁ_{in}(h + V^{2}/2 + gz)_{in} - 0 = ṁ
_{3}h_{3}- ṁ_{2}h_{2}- ṁ_{6}h_{6}...........(There is no work or heat flow in the open feedwater heater)

- Q̇ - Ẇ = ṁ

- Considering the proportions of the mass flow that follow each path:

- 0 = ṁh
_{3}- ṁ(1-y)h_{2}- ṁ(y)h_{6} - 0 = h
_{3}- (1-y)h_{2}- (y)h_{6}

- 0 = ṁh

- Rearranging gives:

- y = (h
_{2}- h_{3}) / (h_{2}- h_{6})

- y = (h

**Step 2: Find the value of Ẇ _{net}**

- Considering work flow rates in and out:

- Ẇ
_{net}= Ẇ_{out}- Ẇ_{in}- = Ẇ
_{turbine}- Ẇ_{12}- Ẇ_{34}

- = Ẇ

- Ẇ

- Applying the first law for Ẇ
_{turbine}gives us:

- Ẇ
_{net}= ṁ(h_{5}- h_{6}) - ṁ(1-y)(h_{6}- h_{7}) - ṁ(1-y)w_{12}- ṁw_{34}

- Ẇ

**Step 3: Find Q̇ _{in}**

- Applying the first law for the boiler gives:

- Q̇
_{in}= ṁ(h_{5}- h_{4})

- Q̇

**Step 4: Substitute and solve for η _{th}**

### Analysis for the Case with Irreversibilities

If you are told that there are irreversibilities in one of the components you will need to apply the efficiency of these components and recalculate values of enthalpy. For example if the low pressure turbine segment has irreversibilities such that the turbine efficiency η_{turbine} = 0.9 you will have to recalculate the actual enthalpy after the expansion using the definition of turbine efficiency and resubstitute to find the new, slightly reduced η_{th}.

## The Rankine Reheat Cycle

This cycle is a type of Rankine cycle where the fluid passes through the high pressure turbine, only to be reheated and pass through the low pressure turbine. There is one mass flow, but there are two processes of heat addition.

### Analysis for the Ideal Case

- The thermal efficiency is defined as
**η**_{th}= W_{net}/ Q_{in} - The heat and work flows are as follows:

- The work out of the system (Ẇ
_{out}) occurs during the expansion through the turbine (Ẇ_{34}and Ẇ_{56}). . - The work into the system (Ẇ
_{in}) occurs during the one pumping phase (Ẇ_{12}). - The heat out of the system (Q̇
_{out}) occurs during the condensing phase. - The heat into the system (Q̇
_{in}) occurs during the two boiling phases (Q̇_{23}and Q̇_{45}).

- The work out of the system (Ẇ

- The thermal efficiency is defined as

**Step 1: Find the value of Ẇ _{net}**

- Considering work flow rates in and out:

- Ẇ
_{net}= Ẇ_{out}- Ẇ_{in}- = Ẇ
_{turbine}- Ẇ_{pump} - = Ẇ
_{34}and Ẇ_{56}- Ẇ_{12}

- = Ẇ

- Ẇ

- Applying the first law for Ẇ
_{turbine}gives us:

- Ẇ
_{net}= ṁ(h_{3}- h_{4}) + ṁ(h_{5}- h_{6}) - ṁw_{12}

- Ẇ

**Step 2: Find Q̇ _{in}**

- Applying the first law for the boiler gives:

- Q̇
_{in}= Q̇_{23}+ Q̇_{45}- = ṁ(h
_{3}- h_{2}) and ṁ(h_{5}- h_{4})

- = ṁ(h

- Q̇

**Step 4: Substitute and solve for η _{th}**

*The case with irreversibilities is approached in the same way as it is with the Rankine regeneration cycle.*