Efficiencies. Damian Vogt Course MJ2429. Nomenclature. Symbol Denotation Unit c Flow speed m/s c p. pressure c v. Specific heat at constant J/kgK

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1 Turbomachinery Lecture Notes Efficiencies Damian Vogt Course MJ49 Nomenclature Subscrits Symbol Denotation Unit c Flow seed m/s c Secific heat at constant J/kgK ressure c v Secific heat at constant J/kgK volume h Enthaly J/kg n Polytroic exonent - Pressure Pa R H Reheat factor - s Entroy J/kgK T Temerature K v Secific volume m 3 /kg Ratio of secific heats - (also referred to as isentroic exonent Efficiency - Total 1 Start change of state End change of state Polytroic s Isentroic tt Total-to-total ts Total-to-static

2 Turbomachinery Lecture Notes Efficiencies In the turbomachinery context a large number of efficiencies are defined such as thermodynamic or mechanical efficiency. In the sections below the focus is ut on the thermodynamic efficiencies. For a given change of state of a fluid the efficiency is defined as the ratio between actual change in energy to ideal change in energy in case of exansion or the inverse in case of comression Exansion actual change in energy Eq. 1 ideal change in energy Comression ideal change in energy Eq. actual change in energy The symbol for efficiencies is the Greek letter (say eta ). For adiabatic rocesses the efficiency lies between and 1. Isentroic Efficiency Deending on which rocess is taken as ideal rocess efficiencies are referred to as isentroic or olytroic efficiencies. In case of an isentroic efficiency the ideal rocess is reresented by an isentroic change of state from start to end ressure, i.e. the same ressures as for the real rocess. This is illustrated in figure 1 for an exansion rocess by means of an enthaly-entroy diagram (h-s diagram). h [kj/kg] Ideal rocess 1 h 1 Real rocess s h h s s [kj/kgk] Figure 1. Exansion rocess

3 Turbomachinery Lecture Notes In the above deicted rocess the changes in total energy are referred to, which is exressed by indexing the efficiency by tt, i.e. total-to-total. Recall that the total energy is defined as follows: c h h + Eq. 3 The total-to-total isentroic efficiency (exansion) is thus given by actual change in energy h1 h tt Eq. 4 ideal change in energy h h s 1 s In case of a comression rocess the situation is as follows: h [kj/kg] h h s Ideal rocess Real rocess s 1 h 1 s [kj/kgk] Figure. Comression rocess Total-to-total isentroic efficiency (comression) ideal change in energy s hs h 1 tt Eq. 5 actual change in energy h h 1 Note: For adiabatic real rocesses the entroy must always increase during the change of state Due to this increase in entroy the real change in energy is smaller than the ideal during exansion. In other words, you get out less energy from the real rocess than you could have from an ideal one For the comression rocess the increase in entroy signifies that you need to ut in more energy to comress a fluid than you would have in an ideal rocess

4 Turbomachinery Lecture Notes Therefore the efficiency is always smaller or equal to unity The only way to reduce entroy would be to cool a rocess. However in such case we do no longer look into adiabatic rocesses In certain cases the kinetic energy that is contained in the fluid (i.e. the amount of energy that is due to the motion) can not be used at the end of a rocess. An examle for such a rocess is the last stage of a energy roducing turbine where the kinetic energy in the exhaust gases is not contributing to the total energy roduced. In such case a so-called total-to-static isentroic efficiency is used, identified by indexing the efficiency by ts, i.e. total-to-static. An exansion line is drawn in figure 3. Note that it is necessary to include total and static states in this case. h [kj/kg] c h 1 h 1 Real rocess Ideal rocess h c h 1 -h h 1 -h s h h s s [kj/kgk] Figure 3. Exansion rocess; total-to-static efficiency The total-to-total isentroic efficiency (exansion) is thus given by actual change in energy h1 h ts Eq. 6 ideal change in energy h 1 hs c s + By reformulating the above exression a relation between total-to-total and total-to-static efficiency can be obtained as follows 1 c 1 1 s + 1 s c c ts + + Eq. 7 tt This relation shows that for values of the total-to-total efficiency. c > the total-to-static efficiency is always smaller than

5 Turbomachinery Lecture Notes Calculating with Isentroic Efficiencies Next the focus is drawn towards the calculation of efficiencies and states. For erfect gases with constant secific heat the enthalies are only a function of temerature as follows c h c T Eq. 8 Furthermore the gas law for a erfect gas relates temeratures and ressures for an isentroic rocess as given below 1 T const. Eq. 9, where is non-dimensional and stands for the ratio of secific heats c cv Eq. 1 The two states 1 and s at the same entroy are thus related by T T 1 s 1 s Eq. 11 By exressing T s by 1 s T s T Eq. 1 1 the isentroic enthaly difference 1 s can be written as s 1 s cδt1 s c( T1 Ts ) ct1 1 Eq To obtain the real change in enthaly the efficiency must be accounted for as shown above yielding s 1 tt ct1 1 Eq Note that the above equation reresents a rather common roblem; very often the inlet state to a gas turbine is given by (,T), e.g. from conditions after a combustion chamber. Furthermore the exit ressure of the turbine might be set. As aroximation it can also be assumed that s. By knowing (or assuming) the efficiency the real change in enthaly can thus easily be calculaed.

6 Turbomachinery Lecture Notes Polytroic Efficiency As for the isentroic efficiency the olytroic efficiency relates a real rocess to an ideal one. The main difference however is that the ideal rocess in this case is not taken as the single isentroic change of state but rather the flow work, which is defined as follows y vd Eq. 15 Thus the definition of olytroic efficiency is given by actual change in energy Eq. 16 ideal change in energy y The flow work is not easily visualized in the h-s diagram. It can be understood as infinite number of infinitesimal small isentroic changes of state that follow the real exansion line like a saw tooth curve, see figure 4. This consideration also leads to the olytroic efficiency sometimes being referred to as small-stage efficiency. h [kj/kg] Ideal rocesses 1 h 1 s,1 Real s, h h s s,n s [kj/kgk] Figure 4. Illustration of flow work Note that the sum of all these infinitesimal isentroic changes is greater than the single isentroic change from 1 to s. s, i > s Eq. 17 This is due to the fact that the isobars are sread aart with increasing entroy, which in turn is due to the sloe of the isobars being roortional to the temerature as follows h s const. T Eq. 18

7 Turbomachinery Lecture Notes The growing sreading of the isobars is an indication for increased energy content of the fluid at the same ressure due to increased entroy (hint: see the entroy as measure for disorder; higher temerature leads to greater disorder). By aroximating the flow work by the aforementioned infinitesimal isentroic changes a henomenon known as reheat gets aarent; due to the reheating henomenon art of the heat generated due to losses (i.e. efficiency smaller than unity) is fed back to the fluid as energy and can be used during the rocess. Following this consideration a reheat factor is defined as follows s, i R H > 1 Eq. 19 s For an exansion the isentroic and olytroic efficiencies can now be related by this reheat factor following Δ hs, i RH s 1 tt R H Eq. As the reheat factor is larger than unity the olytroic efficiency is smaller than the isentroic efficiency. In case of a comression the olytroic efficiency yields from h s i RH h s Δ, Δ RHtt Eq. 1, which leads to the olytroic efficiency being greater than the isentroic efficiency. From that oint of view it is aarent that the olytroic efficiency reflects a different asect of a change of state of a erfect gas as it takes into account the effect of reheating. By knowing the olytroic efficiency it is ossible to aly the gas law as introduced further above to olytroic changes by reformulating 1 nt n const. Eq. The coefficient n is thereby referred to as olytroic coefficient and is related to the isentroic exonent as follows Exansion rocess n 1 n Eq. 3 Comression rocess n 1 1 n Eq. 4

8 Turbomachinery Lecture Notes Two states 1 and, which do not need to be at the same entroy, are now in case of an exansion rocess related by T T 1 1 Eq. 5 Going back to the exansion sketched in figure 1 a relation between the isentroic and the olytroic efficiency and thus the reheat factor can be derived. The total-to-total isentroic efficiency was given by 1 tt Eq. 6 s 1 s It has also been shown that the isentroic change in enthaly could directly be determined from the gas law for a erfect gas by 1 c( T1 T ) ct1 1 s s s Eq. 7 1 By alying the olytroic relation the actual change in enthaly can be obtained directly from 1 c( T1 T) ct1 1 Eq. 8 1 Note the resence of the olytroic efficiency in the exonent reflecting the olytroic coefficient. By substituting these exressions into Eq. 6 and under assumtion of s we obtain tt 1 1 Eq For small ressure ratios 1 1 the olytroic and isentroic efficiencies therefore differ very little. With increasing ressure ratio this difference also increases. The reheat factor in case of an exansion rocess is then obtained from Eq. as follows 1 R tt H 1 1 Eq