entropy equation in chapter 4: control mass approach 4... · 2018. 9. 13. · entropy equation in...

Energy Equation

Entropy equation in Chapter 4: control mass approach

The second law of thermodynamics

Availability (exergy)– The exergy of a system is the maximum useful work possible during a process that brings the system

into equilibrium with a heat reservoir.

– The availability function is a characteristic of the state of the system and the environment which acts as a reservoir at constant pressure (p0 ) and constant temperature (T0 ).

(4.27)

Energy Equation


Availability/exergy

Carnot cycle efficiency

,

, ,

Energy Equation



From state 1 to 2

Maximum useful work

Maximum useful work obtainable from the specified change

Energy Equation

Energy and entropy equation in Chapter 4: control mass approach


(4.27)

Energy Equation



(4.27)

the second T‐ds relation

Energy Equation



(4.27)

Energy Equation


Entropy change of ideal gas

From the first T ds relation From the second T ds relation

Energy Equation

Energy equation in Chapter 4: control volume approach

The first law of thermodynamics

In the control volume approach, there are flow in and out.

The first law

Energy Equation


Heat addition term

Work term

Work done by the mass in the control volume + work associated with mass flow

Body force

PvmmPVPW

VPxAPW

normal

normal

I

ii

I

ii gzm

dtmgzd

DtmgzDm

dtmd

DtmD

11

Energy Equation


Control mass

Control volume

For stationary control volume

(4.38)

dtdVP

dtdW

normal

Energy Equation


Control volume

Energy Equation

Entropy equation in Chapter 4: control volume approach

Entropy equation

For a stationary control volume

Energy Equation


Actual work

Useful work (useful part of the actual work)

Maximum useful work

Maximum useful workActual workUseful work

Energy Equation


Energy equation

Useful work (useful part of the actual work)

Useful work (or useful actual work)

Partial derivative? Stationary!

Energy Equation


Multiplying , then subtracting from (4.44)

Express the actual useful work with entropy

Energy Equation


Maximum useful work

Irreversibility

=0

Energy Equation


Irreversibility

Energy Equation


Example

Energy Equation


Example

(4.27)

Maximum useful work

Energy Equation


Example

Actual work

Useful work

Useful work

Energy Equation


Example

Irreversibility

Maximum useful work

Useful work

Energy Equation


Example

Irreversibility

Contents

1. Introduction

2. Nonflow Process

3. Thermodynamic Analysis of Nuclear Power Plants

4. Thermodynamic Analysis of A Simplified PWR System

5. More Complex Rankine Cycles: Superheat, Reheat, Regeneration, and Moisture Separation

6. Simple Brayton Cycle

7. More Complex Brayton Cycles

8. Supercritical Carbon Dioxide Brayton Cycles

Introduction

Summary of the working forms of the first and second laws

First law for control volume

Second law

Working forms

Contents

1. Introduction

2. Nonflow Process







Thermodynamic Analysis of Nuclear Power Plants

Analysis of NPPs

Provides the relation between the mixed mean outlet coolant temperatures (or enthalpy forBWR) of the core through the primary and secondary systems to the generation of electricityat the turbine.

Cycles used for the various reactor types and the methods of thermodynamic analysis of these cycles are described. Rankine cycle for steam‐driven electric turbines(PWR, BWR) Brayton cycle can be considered for gas cooled reactor systems Both cycles are constant‐pressure heat addition and rejection cycles for steady‐flow operation


Analysis of NPPs

Simple Brayton cycle

Maximum temperature of the Brayton cycle (T3) is set by turbine blade and gas cooledreactor core material limits far higher than those for the Rankine cycle, whichis set by liquid-cooled reactor core materials limits .


Efficiency to assess the thermodynamic performance of components and cycles

Thermodynamic efficiency (or effectiveness): The ratio between the actual useful work and the maximum useful work

Isentropic efficiency: Special case of thermodynamics efficiency Adiabatic case Important for devices such as pump, turbine, compressor

Thermal efficiency: The fraction of the heat input that is converted to net work output The cycle efficiency that we are interested in.


Thermodynamic efficiency (or effectiveness):

Maximum useful work No entropy generation

Fixed, non‐deformable control volume with zero shear work Negligible kinetic and potential energy differences


Isentropic efficiency:

Maximum useful work

For steady‐state condition


Isentropic efficiency:

Useful actual work for an adiabatic control volume with zero shear and negligible kinetic andpotential energy differences

For steady state


Thermal efficiency:

Cycle efficiency

For adiabatic systems, it is not useful.


Example problem (a)

Thermal efficiency:

Thermodynamic efficiency (or effectiveness): 100

70

30

30100 30%

, 1 1300600 · 100 50

(a) (b)

, 1 13001000 · 100 70

3050 60%

3070 42.9%


Example problem (b)

Maximum useful work no entropy generationEntropy in = entropy out

Entropy decrease in hot reservoirs = entropy increase in cold reservoir


Example problem (b)

1275 16%

, 35.2

, 39.8

1239.8 30.2%

Contents

1. Introduction

2. Nonflow Process







Thermodynamic Analysis of A Simplified PWR System

First Law Analysis of a Simplified PWR System

One component with multiple inlet and outlet flow streams operating at steady state and surround it with a non‐deformable, stationary control volume

Energy equation

Steady Negligible Non‐deformable No shear

Negligible kinetic energy



One component with multiple inlet and outlet flow streams operating at steady state and surround it with a non‐deformable, stationary control volume

Energy equation

Mass conservation equation

• Turbine 0, 0 Left side > 0

• Condenser 0, 0 Left side > 0

• Pump 0, 0 Left side < 0

• Steam generator 0, 0 Left side < 0



Turbine

Shaft work, no heat addition adiabatic turbine

Actual turbine work is related to the ideal work

by the component isentropic efficiency.

Isentropic efficiency

Pump

Shaft work, no heat addition adiabatic pump

,

Recirculating steam generator



Steam generator

Heat transfer between the primary and the secondary

Pinch point temperature

– Minimum temperature difference between the primary and the secondary

– Inversely related to the heat transfer area

Small temperature difference: large heat transfer area

Large temperature difference: small heat transfer area

– Directly related to the irreversibility of the SG

Large temperature difference: large irreversibility

– Important design choice based on tradeoff between

cost and irreversibility



Steam generator

Reactor plant

Actual work

Positive or negative?



Reactor plant

Maximum useful work

– In section 6.4.2.3

Maximum useful work expressed

with the secondary side properties

Thermodynamic efficiency or effectiveness:

Single turbine case

Review

First law analysis in a simplified PWR system Stationary, non‐deformable control volume operating at steady state

Turbine and pump

Steam generator and condenser

=0

Review

First law analysis in a simplified PWR system Stationary, non‐deformable control volume operating at steady state

Thermodynamic efficiency and cycle thermal efficiency



Assume that the turbine and pump have isentropic efficiencies of 85%.



Example 6.4: Thermodynamic Analysis of a Simplified PWR Plant

Draw the temperature‐entropy (T‐s) diagram for this cycle.

7.75 MPa

6.89 kPa

15.5 MPa




Compute the ratio of the primary to secondary flow rates.




Compute the nuclear plant thermodynamic efficiency.

– Calculate the enthalpies of each state!

State 1




State 2

– Isentropic efficiencies of 85%.

– Isentropic process

– ,




State 3

–

State 4

– Isentropic efficiency

– Isentropic process

1 · ·

1 · ·




Compute the nuclear plant thermodynamic efficiency.




Compute the cycle thermal efficiency.

–

– ,

Thermal efficiency (cycle thermal efficiency)

= plant thermodynamic efficiency



Improvements in thermal efficiency for this simple saturated cycle

Pressure

– Increase the pump outlet pressure/decrease turbine outlet pressure



Improvements in thermal efficiency for this simple saturated cycle

Pressure

– Increase the pump outlet pressure/decrease turbine outlet pressure

Added heat

Rejected heat

Net work

Efficiency

entropy equation in chapter 4: control mass approach 4... · 2018. 9. 13. · entropy equation in...

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