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Entropy

- Second Law Characteristic

Entropy

- Clausius Inequality
- For internally reversible cycles

Entropy

- Entropy is defined as dS (dQ/T)int rev

(kJ/K) - S is entropy per unit mass (kJ/kgK)
- Entropy is a property of a state not a process
- Change of entropy

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Entropy

- Increase of Entropy Principle
- The entropy of an isolated system during a

process always increases or, in the limiting case

of a reversible process, remains constant. - dS (dQ/T) where T is the thermodynamic

temperature at the boundary where dQ is

transferred to the surroundings

Entropy

- Process can occur in one direction only, not in

any direction. Must proceed in direction of

least increase in entropy. Sgen 0 - Entropy is a non-conserving property.
- Performance is degraded by irreversibilities,

entropy generation is a measure of the magnitude

of the irreversibilities during the process

Entropy

- Sgen gt 0 irreversible processes
- 0 reversible processes
- lt 0 impossible processes

Entropy

- Entropy of a fixed mass can be changed by
- Heat transfer
- Irreversibilities
- If no change, Isentropic
- Reversible, adiabatic process

Entropy Change of Pure Substance from tables

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Isentropic Processes

- Isentropic process internally reversible,

adiabatic, entropy remains constant ?s 0 or

s2 s1

Entropy

- Essentially Isentropic Processes
- Pumps
- Turbines
- Nozzles
- Diffusers

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Property Diagrams

- Temperature-entropy (T-s) diagrams
- Area under process curve on a T-s diagram equals

heat transfer during an internally reversible

process

Property Diagrams

- Isentropic processes are a vertical line on T-s

diagrams

Property Diagrams

- Enthalpy-entropy diagram, h-s diagram, Mollier

diagram - Change in h is a measure of work
- Change in s is a measure of irreversibilities

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Entropy

- Entropy is measure of molecular disorder,

molecular randomness

Entropy

- Boltzmann equation
- S k ln(p)
- where k 1.380610-23 J/K
- p thermodynamic probability, number of

possible microscopic states of system

Third Law of Thermodynamics

- The entropy of a pure crystalline substance at

absolute zero temperature is zero, since there is

no uncertainty about the state of the molecules

at that instant - Provides an absolute reference point for

determining entropy

Entropy

- There is no entropy transfer associated with

energy transferred as work - Irreversibilities (friction) will case entropy

increase

Entropy

- Work changed to heat increases entropy

Entropy

- During heat transfer net entropy increases

Entropy

- To find the change in entropy, need to do the

cycle integral of dQ/T. - If isothermal, only need the function for Q
- If not isothermal, need functions for Q and T

Entropy

Entropy

Entropy

- Can find entropy by integration of either

equation - Need to know the relationship between du or dh

and temperature - For ideal gases
- du cv dT
- Or dh cp dT
- And PvRT

Entropy Changes of Liquids and Solids

- Liquids and solids are incompressible
- dv 0
- Also cv cpc and du c dT

Entropy Change of Ideal Gases

- In the basic equation, substituting du cv dT

and P RT/v - Substituting dh c dT and v RT/P

Entropy Change of Ideal Gases

- Need the relationship between the specific heats

and temperature - Assume constant specific heats, simpler

integration, approximate analysis - Work with variable specific heats, use tables,

exact analysis

Constant Specific Heats (Approximate Analysis)

Variable Specific Heats (Exact Analysis)

- If temperature change is large
- Specific heats are non-linear with temperature
- Need accurate relationships
- Calculate integrals with respect to reference

entropy (at absolute zero)

Variable Specific Heats (Exact Analysis)

- So we can find
- And substituting into
- Get

Variable Specific Heats (Exact Analysis)

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Isentropic Processes of Ideal Gases

- Process that has
- No change in entropy, ?s 0
- Is internally reversible
- Is adiabatic

Isentropic Processes of Ideal Gases (Approximate

Analysis)

- Assume constant specific heats, so
- Where k cp /cv

Isentropic Processes of Ideal Gases (Approximate

Analysis)

- These equations can be stated as
- Since the specific heat ratio k, varies with

temperature, the average k should be used

Isentropic Processes of Ideal Gases (Exact

Analysis)

- Set up to use tables
- Working with
- Establish new dimensionless quantities from the

equations

Isentropic Processes of Ideal Gases (Exact

Analysis)

- The quantity exp(s/R) is defined as relative

pressure Pr - P is a function of T only, so can be tabulated

against T.

Isentropic Processes of Ideal Gases (Exact

Analysis)

- So using table A-17 for air

Isentropic Processes of Ideal Gases (Exact

Analysis)

- When specific volume ratios are given instead of

pressure - The quantity T/Pr is a function of T only,

defined as relative specific volume vr

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Reversible Steady-Flow Work

- Work done during process depends on
- Properties at the end states
- Path between the end states
- Quasi-equilibrium work interactions
- Max work output
- Min work input

Reversible Steady-Flow Work

- Energy balance for steady-flow, internally

reversible process - Substituting

Reversible Steady-Flow Work

- For incompressible fluids, volume remains

constant - When the device has no work interactions (nozzle,

pipe sections), w 0 - The larger the specific volume, the larger the

reversible work produced or consumed

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Minimizing Compressor Work

- Minimized when internally reversible
- Compressor work required
- Methods to minimize work
- Reduce irreversibilities
- Keep specific volume as small as possible
- Keep gas temperature as low as possible

Minimizing Compressor Work

- Look at three processes
- Isentropic process, no cooling
- Polytropic process, some cooling
- Isothermal process, maximum cooling

Minimizing Compressor Work

- Assume, same pressures, ideal gases, constant

specific heats, internally reversible

Minimizing Compressor Work

- Methods of removing heat
- Cooling jackets on compressor
- Multistage compressor with intercooling

Minimizing Compressor Work

- For multistage compression
- To minimize compressor work in 2-stage

compressor, the pressure ratio across each stage

must be the same

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Isentropic Efficiencies of Steady-Flow Devices

- Ideal model for turbines, compressors, nozzles,

etc. - Adiabatic, no irreversibilities, isentropic

process - Measure actual devices against ideal model

Isentropic Efficiency of Turbines

- Turbine efficiency is the ratio of the actual

work output to the work output of an isentropic

process between the same states

Isentropic Efficiency of Turbines

- Actual efficiencies run from 90 for large

turbine to 70 for smaller turbines

Isentropic Efficiency of Compressors and Pumps

- Isentropic efficiency of a compressor is ratio of

the isentropic work required to raise the

pressure of a gas to the actual work input

Isentropic Efficiency of Compressors and Pumps

Compressor Efficiency

- Isothermal efficiency

Entropy Balance

- The entropy change during a process is equal to

the net entropy transfer through the system

boundary and the entropy generated in the system

Entropy Change of a System

- For a system
- Entropy is a property so entropy does not change

unless the state of the system changes

Mechanisms of Entropy Transfer

- Heat transfer

Mechanisms of Entropy Transfer

- Mass Flow

Mechanisms of Entropy Transfer

Entropy Generation

- Irreversibilities such as
- Friction
- Mixing
- Chemical reactions
- Heat transfer (finite temperature difference)
- Unrestrained expansion
- Non quasi-equilibrium compression or expansion
- Causes entropy increase or generation

Entropy Generation

Entropy Generation, Closed Systems

Entropy Generation, Open Systems

Entropy Generation, Open Systems

Entropy Generation Associated with Heat Transfer

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