Minimizing and maximizing compressor and turbine ork respectively
Reversible steady-flo ork In Chapter 3, Work Done during a rocess as found to be W b dv Work Done during a rocess It depends on the path of the process as ell as the properties at the end states.
Work Done During a steady state process In a steady state process, usually there are no moving boundaries It ould be useful to be able to express the ork done during a steady flo process, in terms of system properties Recall that steady flo systems ork best hen they have no irreversibilities 3
4 Consider general form of the Energy Balance for steady flo steady state processes ( ) ( ) + + + + + e e e e i i i i z g V h m z g V h m W Q dpe dke dh q rev rev + + δ δ pe ke vd rev
For devices dealing ith compressible fluids, like turbines and compressors, v is not constant, but the KE and E are negligible. Hence rev vd ke pe rev vd In order to integrate, e need to kno the relationship beteen v and.
Important observation Note that the ork term is smallest hen v is small, so for a pump (hich uses ork) you ant v to be small. For a turbine (hich produces ork) you ant v to be large. rev vd 6
Minimizing the Compressor Work The best ay, is to keep the specific volume as lo as possible during the compression process, by cooling it Maximizing the turbine Work The best ay, is to keep the specific volume as high as possible during the expansion process, by heating it 7
Effect of cooling the compressor To understand ho the cooling affects the ork, let us consider three processes: Isentropic process (No cooling) olytropic process (some cooling) Isothermal process (maximum cooling) Assume also that all three processes Have the same inlet and exit pressures. Are internally reversible The gas behaves as an ideal gas Specific heats are constants. 8
- Isothermal process rev, vd in Consider an ideal gas, at constant T v RT rev RT ln, in Remember, this is only true for the isothermal case, for an ideal gas 9
- Isentropic process Isentropic means reversible and adiabatic (Q0) i.e. No cooling is alloed Recall from isentropic relations for an ideal gas v k rev, in C krt k v C k k ( k ) k plug in and integrate rev, vd Remember, this equation only applies to the isentropic case, for an ideal gas, assuming constant specific heats in 0
3- olytropic process rev, in vd v n C Back in Chapter 3 e said that in a polytropic process v n is a constant This is exactly the same as the isentropic case, but ith n instead of k!! rev, in rev, in v v nrt n n ( n ) ( T ) nr( T T ) R T n n n
Let us plot the three processes on a -v Diagram for the same final and initial pressures The area to the left of each line represents the ork, vd rev, in vd Note, that it takes the maximum ork in isentropic compression hile it takes minimum ork for an isothermal compression
So as an engineer, you should compress gas isothermally, in order to consume minimum ork. Hoever, for a turbine, e need to produce the maximum ork. So, a turbine should expand isentropically ( adiabatically and reversibly). That is hy e assume Q 0 in the st lo analysis of a turbine. 3
Multistage compression ith intercooling One common ay is to use cooling jackets around the casing of the compressor. Hoever, this is not sufficient in some cases. Instead, multistage compression is more common, ith cooling beteen steps. The gas is compressed in stages and cooled to the initial temperature after each stage. This is done by passing it a heat exchanger called intercooler. Multistage cooling is attractive in high pressure ratio compression. 4
To stage Compressor The colored area on the -υ diagram represents the ork saved as a result of to-stage compression ith intercooling. 5
Minimizing the ork input for a to stage Compressor The size of the colored area (the saved ork input) on previous slide varies ith the value of the intermediate pressure x. The total ork input for a tostage compressor is the sum of the ork inputs for each stage of compression. W comp,in W comp I,in nrt n + W x comp II,in ( n ) / n ( n ) + nrt n x / n 6
The only variable is x. The x value that ill minimize the total ork is determined by differentiating the above expression ith respect to x. And setting the result to zero. This gives x x That is to minimize the compression ork during to stage compression, the pressure ratio a cross each stage of the compressor must be the same. comp I,in comp II,in 7