The Second Law of Thermodynamics Entropy and Work Chapter 7c

Work Done during a Process  In Chapter 4 we found the work done in a closed system due to moving boundaries and expressed it in terms of the fluid properties  In the processes describing the behavior of many engineering devices there are no moving boundaries

For example, a steam turbine does not have any moving boundaries Steam Turbine Work Steam turbines do produce work

Work Done During a Process  It would be useful to be able to express the work done during a steady flow process, in terms of system properties  Recall that steady flow systems work best when they have no irreversibilities

Energy Balance for a steady flow device Often the change in kinetic energy and potential energy is 0

“All” we have to do now is integrate!! In order to integrate, we need to know the relationship between v and P

For solids and liquids v is constant

Steady flow of a liquid through a pipe or a nozzle There is no work!! Bernoulli’s equation

Steady Flow of a Liquid through a pump or a turbine Note that the work term is smallest when v is small, so for a pump (which uses work) you want v to be small. For a turbine (which produces work) you want v to be big. Or..

Compressor Work We integrated this equation for v = constant, which is good for liquids – but what about gases? Consider an ideal gas, at constant T Remember, this is only true for the isothermal case, for an ideal gas

Compressor Work Another special case is isentropic We derived the isentropic relationships earlier in this chapter Rearrange to find v, plug in and integrate Now its “just” algebra, to rearrange into a more useful form

Remember, this equation only applies to the isentropic case, for an ideal gas, assuming constant specific heats

Compressor Work Another special case is polytropic Back in Chapter 4 we said that in a polytropic process Pv n is a constant This is exactly the same as the isentropic case, but with n instead of k!!

The area to the left of each line represents the work, vdP Note, it takes less work for an isothermal process compress isothermallyisentropic turbine You should compress isothermally, and you should use an isentropic process in a turbine!! Pv Diagram for Isentropic, Polytropic and Isothermal compression, for the same final and initial pressures Inlet Exit

How do you keep a compression process isothermal?  The gas will heat up as it is compressed, so it needs to be cooled  Intercooling is difficult  Instead, multistage compression is more common, with cooling between steps

Two stage Compressor

How do you decide how to break up the compression load?  You save the most work by intercooling, when each compressor carries the same load Since you cool back to T 1 between stages, the only things that change in this equation are the P’s

For the work done by each stage to be equal, the pressure ratio must be equal Or…

Isentropic Efficiency of Steady Flow Devices

Efficiency  We’d like a measure of efficiency to compare real devices to the best we can do  There are always irreversibilities that downgrade performance

 Most steady flow devices are intended to operate under adiabatic conditions  If a device is reversible and adiabatic, it is isentropic  Real devices are never really isentropic

Isentropic Efficiency  Lets compare how well real devices work to how well comparable isentropic devices work Same inlet conditions Same outlet conditions Turbine, Compressor and Nozzle

Turbines

Remember, the work done in a turbine can be found from the energy balance

Isentropic Efficiencies of Compressors and Pumps  Ratio of the work required to raise the pressure of a gas to a specified value, in a isentropic manner, to the actual work Note that this equation is arranged so that it is always less than one!!

Remember, the work done by a compressor can be found from the energy balance

Applies to both gases and liquids Isentropic work for a liquid Only applies to a liquid

Sometimes compressors are cooled intentionally – Why?  Cooling reduces the specific volume, resulting in less work required for compression  For compressors that are intentionally cooled, the isothermal model is more realistic

Isentropic Efficiency of Nozzles  The objective of a nozzle is to increase the kinetic energy of the gas  Usually, the inlet velocity is low enough that we can consider it to have zero kinetic energy

Entropy Balance

 There is no such thing as the conservation of entropy  Entropy of the universe always increases  Real processes always generate entropy

Often called the entropy balance

Remember, entropy is a state function. It doesn’t change unless the state of the system changes!!!

How Does Entropy Enter and Leave a System?  Heat Transfer S=Q/T (T=constant) If T is not constant S=Q k /T k  There is no entropy transfer with work!!  Mass Flow S mass =ms  Entropy Generation

Closed Systems There is no mass transfer in a closed system In an isolated system there is no heat transfer, so the equation becomes…

The Universe is an Isolated System

Control Volumes Simplify this equation, depending on the process

Surface area for heat transfer is 30 m^2 Thermal Conductivity is 0.69 W/(m C) Inside T= 27 C Outside T=0 C Inside Surface T = 20 C Outside Surface T = 5 C Consider Example 7-17 Entropy Generation in a Wall

Determine the rate of heat transfer through the wall

Determine the rate of entropy generation in the wall 0 The state of the system does not change with time

Determine the rate of entropy generation for the process 0 Consider an extended system – wall plus the surrounding air

Where is entropy generated?

Reducing the Cost of Compressed Air  Skim  Repair Air Leaks  Install High Efficiency Motors  Use a small motor at high capacity, instead of a large motor at low capacity  Use outside air for compressor intake  Reduce the air pressure setting

Summary Steady Flow work for a reversible process

Summary For incompressible substances The work term is smallest when v is small, so for a pump (which uses work) you want v to be small. For a turbine (which produces work) you want v to be big.

Summary We looked at three special cases of the work equation Isothermal Isentropic Polytropic

Summary Isothermal Compression

Summary Isentropic Compression

Summary Polytropic Compression

Summary The work input to a compressor can be reduced by using multistage compression with intercooling. For maximum savings from the work input, the pressure ratio across each stage of the compressor must be the same.

Summary Most steady-flow devices operate under adiabatic conditions, and the ideal process for these devices is the isentropic process.

In the relations above, h 2a and h 2s are the enthalpy values at the exit state for actual and isentropic processes, respectively. Summary Isentropic or Adiabatic Efficiency Actual turbine workw a h 1 - h 2a Isentropic turbine workw s h 1 - h 2s = = = ~ Isentropic compressor work w s h 2s - h 1 Actual compressor work w a h 2a - h 1 = = = ~ Actual KE at nozzle exit V 2a h 1 - h 2a Isentropic KE at nozzle exit h 1 - h 2s 2 V 2s = = = ~ 2

Summary Entropy Balance

Summary Entropy Balance – Rate form

Summary Entropy Balance – Steady-flow