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The optimum efficiency and net work of the regenerative cycle with turbine extractions, using steam as the working fluid, have been simulated and analyzed. The cycle is simulated with until five feed water heaters in a numeric method and can be easily used in solar power plants. The general expression for each component is realized through the balance of energy, collectors, turbine, condenser, pumps and feed water heaters. One analytical method is developed considering constants of the difference of enthalpy through feed water heaters as also between them. The results show that the analytical method is unsatisfactory because the optimum efficiency depends on some parameters such as evaporating temperature and superheating temperature showing in numeric method. The increase of optimum efficiency increases when the number of feed water is increased as well as evaporating and superheating temperature, for the net work presents a maximum value along evaporating temperature, decreasing with number of feed water heaters and increasing when the superheating temperature is increased. The pressure of extraction of turbine is also analyzed, varying about 5% along of evaporation temperature. This analysis is important to motivate the use in solar plants that it is considerate in this paper, just analyzing the cycle.

The regenerative cycle is a modified Rankine cycle which intends to improve the efficiency of the system. Baumann (1930) [

In the regenerative cycle, the water entering in solar collectors is hotter than the Rankine cycle and becomes better for solar power plants. The condensate of Rankine cycle at low temperature has greater irreversibility when the water is mixing in the collectors and this decreases cycle efficiency. The steam extracted from the turbine at different stages to the feed water heater with small fractions of vapor released by the turbine, reduced the irreversibility associated with the exchange of energy in the feed water heaters. The vapor or water through feed water heaters increases the average temperature of the heat source. The conditions of steam bled for each heater are selected so that the saturated steam temperature will be at 4˚C to about 100˚C higher than final condensate temperature, depending on the number of turbine extraction. It is analyzed that the thermodynamic cycle is associated to the collector’s field, looking for a way of improving its efficiency. The system is composed with the field of solar collectors, turbine, condenser, feed water heaters and pumps, as showing in _{n}) are the mass relation between the mass flow extracted from the turbine and the mass fraction that crosses the solar collectors (y_{n} = W_{n}/m_{n}).

According to the capacity of the turbines, it can be divided in three categories: a) Medium capacity turbines that do not use more than 3 feed heaters; b) High pressure high capacity that does not use more than 5 to 7 feed heaters and c) Supercritical turbines that use between from 8 to 9 heaters.

There are some advantages of regenerative power cycle compared of Rankine cycle: 1) The heating process in the collectors tends to become reversible; 2) The thermal stresses in the collectors are minimized because of the water being hotter; 3) The steam condenser size can be reduced; 4) The turbine efficiency increases decreasing the turbine damage. But also there are some disadvantages as the plant become more complicated, more expensive and increase the number of maintenance.

In a regenerative cycle with superheated steam, the irreversibility of the feed water heaters are derived from mixing an undercooled liquid with two-phase

fluid or saturated or superheated fluid. According to Bejan (1988) [

Ying (1999) [

Weir (1960) [

Fraidenraich (2013) [

exchanger is showed by 3 points: 1, 2 and 3. Point 1 indicates the inlet of the fluid extracted from the turbine which may be in the two-phase, saturated or superheated condition and which will leave at point 3, saturated liquid. Point 2 is undercooler liquid after passing through a pump. The net work cycle is represented by the area a-b-c-d-a and the heat absorbed in the collectors by area a-b-c'-d'-a. This area, a-b-c'-d'-a, would be larger without the presence of the feed water heaters of the cycle, and consequently the lower the efficiency that is the ratio of these two areas (η = (area a-b-c-d)/(area a-b-c'-d')).

The proposed equation model is based on the energy and mass balance of each component of the regenerative cycle (_{t} [kJ/kg]) is given by:

where _{t} the power of the turbine [W], H_{n}_{ }is the enthalpy at the output of the extractions, H_{b}_{ }the enthalpy at the collectors output and H_{c} the enthalpy at the condenser inlet. The mass fraction of the “i” order extraction of the turbine is y_{i} (i = 1 is the first extraction of the turbine) with equation:

where H_{i} represents the enthalpy of i-order extraction from the turbine, h_{B}(i-1) is the enthalpy at the output of each pump and h_{i} is the enthalpy at the output of the heater. The first extraction of turbine is given by:

The heat per unit mass that enters the collectors (q_{b} [J/kg]) and leaves the condenser (q_{c} [J/kg]) is directly proportional to the enthalpy difference between the inlet and outlet of the collectors, given by:

and

The efficiency of the cycle with “n” turbine extractions, defined by the ratio of the net work by heat entering the collectors, is calculated by the following equation:

The pumps work (W_{Bi}) is the difference of output enthalpy and input enthalpy.

Simulations were realized with two superheating temperature (T_{sup}) of 400˚C and 500˚C. It was analyzed for the maximum possible efficiency and its respective work per unit of mass. When there are many extraction of turbine it can have different efficiencies, depending of the positions of the extractions in turbine. For find maximum efficiency was simulated in an interactive program that searches the maximum value in a Regenerative cycle. The software used was EES―Engineering Equation solver. The hypotheses used in the simulations are: The condensing pressure is 10 kPa for all tests; the condition of water going out of the condenser and the all feed water heaters is saturated. There is not loss pressure along of the tubes and the expansion of turbine is isentropic. If you need a non-isentropic expansion just multiply the turbine work by its isentropic efficiency which will not change the behavior of the turbine work.

_{ev}) and superheating temperature (T_{sup}) is increased. The increasing of efficiency is getting smaller when the number of extraction is increased. This shows that it can be economically impracticable more than three or four extractions; the increase of efficiency is much small above four extractions.

One analytical analysis for the regenerative cycle is realized to compare with numerical analysis. For the cycle of _{C}) equal 1, 0 is given by

It can be consider as Haywood (1949) [

If “t” and “r” is considered constant, it could get the maximum of efficiency. The balance of energy for the second heater is given by:

Then, the mass flow for heater number “n” can be calculated as:

The efficiency of the cycle is calculated with the work express as the difference between the collector heat and the condenser heat that is rejected by heat absorved of collectors or:

If define a preheating factor as_{b} is the collector enthalpy in the saturation condition, the heat absorbed in collector can be given by about:

where

and

If dQ/dz = 0 to find the maximum value of Q, it can find the following expression:

Then, for n extractions of the turbine the preheating factor for the maximum efficiency is given by:

That is, the efficiency of the cycle will be maximum when the preheating factor z is equal to n/(n + 1). For example, if five (5) heat exchangers were installed in the regenerative, the preheating factor would be 5/6 of the total enthalpy (h_{b} − h_{c}). For an infinite number of heaters in the cycle the preheating factor (z) tends to 1. Then it can replace the Equation (16) in the equation of efficiency, Equation (11), getting:

Introducing_{b} − h_{n}) for known values (t, n, and R_{b}), it finds the following expression for the efficiency that optimized regenerative cycle with n stages’ extraction:

Considering that R_{b} and t are well defined values, independent of n can be studied as the efficiency of the cycle varies according to this parameter. In the case of preheating without regeneration (n = 0), the efficiency turns out to be equal:

And for a number (n) tending to infinity, the equation of efficiency is:

If considering a case that R_{b}/t = 0.461, the efficiency for infinite extractions is

Comparing the _{i} − h_{i}” and “r= (h_{i} − h_{c})/n” for the numerical analysis, where “i” is the order of the feed water heater. It is also present the results of

The evaporating temperature that represents the situation of _{b}/t = 0.461. As seen in the graph of

The pressure of extractions is also analyzed. The outlet pressure at each extraction varies with evaporation temperature (T_{ev}) as observed in

It is possible to observe that, at each extraction point, the pressures extracted from the turbine have a pressure ratio not very variable along the evaporation temperature. This pressure ratio does not vary more than 5% along the evaporation temperature for the same position of extraction pressure. The pressure ratio can increase or decrease slightly along the evaporating temperature, respecting the 5% variation. This information is important for the time of simulations along the evaporating temperature.

T_{ev} [˚C] | t_{1} | t_{2} | t_{3} | t_{4} | t_{b} | r_{1} | r_{2} | r_{3} | r_{4} | r_{b} | |
---|---|---|---|---|---|---|---|---|---|---|---|

200 | 2172 | 2229 | 2283 | 2342 | 2402 | 123.3 | 130.3 | 132.1 | 135 | 139.9 | 0.289 |

220 | 2108 | 2163 | 2207 | 2254 | 2299 | 135.5 | 145.7 | 152.5 | 155.7 | 162.3 | 0.341 |

240 | 2047 | 2098 | 2133 | 2164 | 2188 | 150.8 | 161.5 | 172.2 | 176.3 | 184.7 | 0.397 |

260 | 1987 | 2033 | 2057 | 2068 | 2067 | 164.4 | 178.7 | 193 | 198 | 208.5 | 0.461 |

280 | 1926 | 1965 | 1976 | 1964 | 1933 | 180.1 | 195 | 212 | 221.8 | 235.4 | 0.534 |

300 | 1863 | 1892 | 1888 | 1850 | 1782 | 194.1 | 211.9 | 231 | 248.8 | 266.4 | 0.621 |

320 | 1794 | 1811 | 1789 | 1720 | 1606 | 208.1 | 229.7 | 252 | 277.7 | 301.8 | 0.727 |

340 | 1716 | 1717 | 1671 | 1564 | 1392 | 225.7 | 249.2 | 274.7 | 308.7 | 343.7 | 0.869 |

360 | 1619 | 1598 | 1521 | 1368 | 1103 | 245.5 | 270.3 | 299 | 341.3 | 412.9 | 1.088 |

The regenerative cycle is a contribution to the Rankine cycle which intends to improve the efficiency of the thermodynamic cycle and warm the water entering in solar collectors, and become attractive for solar power plants. The maximum efficiency and turbine work of the regenerative cycle have been simulated and analyzed. The regenerative cycle uses feed water heaters to warm the water returned to the collectors. It analyzed cycle with until five feed water heaters using numerical method and compared with one analytical method. The analytical method presents an expression for maximum efficiency that can be simulated until infinite number of feed water heaters.

The results of numeric method show that the maximum efficiency increases when the number of heaters and the superheating temperature increase. The turbine works present one maximum value that depends on number of heaters, evaporating and superheating temperatures. It is also observed that the improvement of efficiency increases when the number of feed water increases.

The analytical method considers constants of the difference of enthalpy along of feed water heaters as well as between them. The results of this analytical method are unsatisfactory compared with numeric method which is more accurate. The difference of two methods can be more than 10% of difference, for example the efficiency of numerical method is 48% while the analytical method is 35.9%.

Analyzing the extractions pressure, it can be concluded that the outlet pressure at each extraction varies around five percent along the evaporation temperature. This conclusion helps the simulations to converge faster.

The authors would like to thank the Foundation of Support to Science and Technology of PE (FACEPE) for financial support.

Da Cunha, A.F.V., Fraidenraich, N. and De Souza Silva, L. (2017) Optimum Efficiency Analysis of Regenerative Cycle with Feed Water Heaters. Journal of Power and Energy Engineering, 5, 45-55. https://doi.org/10.4236/jpee.2017.58004