Double pinch criterion for optimization of regenerative rankine cycles
A pinch, post-heating technique applied in the field of double pinch criteria for optimization of regenerative Rankine cycles
Active Publication Date: 2014-12-03
MASSACHUSETTS INST OF TECH
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Although the introduction of steam creates a small loss in turbine output...
Generally speaking, existing in the optimization process to regenerative Rankine cycle is that main problem is that there are three variables in each regenerative steam flow, i-exhaust steam pressure ii-exhaust flow iii- Switching load from steam flow to stream in a feedwater heater and a non-convex objective function with respect to this set of variables. Both by analytical validation and the presented numerical results it can be concluded that the conventional practice is sub-optimal. This results in inefficient operations and significant losses. In the solution disclosed by the present invention, two of the three variables are excluded, which greatly simplifies the optimization problem and improves the performance at the same time. Another advantage is that both the switching load and the exhaust flow are specified by expressions for the exhaust pressure, thereby eliminating the need for an expensive spatial distribution model to further simplify the optimization problem.
 The present invention discloses a system and method for optimizing a regenerative Rankine cycle including a double pinch criterion. In some embodiments, the efficiency of the Rankine cycle is enhanced by selecting operational variables such as exhaust pressure and exhaust flow rate such that a feedwater heater achieves a double pinch. In particular, the first pinch is taken at the onset of exhaust steam condensation and the second pinch is taken at the exhaust steam outlet of the feedwater heater. The minimum temperature difference at the first pinch point is substantially equal to the minimum temperature difference at the second pinch point. The present invention discloses a system of regenerative Rankine cycles associated with a double pinch criterion, a method of operating the system and a method of optimizing the operation of the system.
 Theorem 1 (Double Pinch Necessity): Consider a regenerative Rankine cycle with no reheat but positive isentropic efficiency of the turbine. The inlet temperature, inlet pressure, outlet temperature and outlet pressure Po of most turbines are fixed, that is, they are not affected by exhaust steam. Consider an arbitrary feedwater heater with an arbitrary but definite feed flow. Feed inlet temperature TF.i and heat transfer duty assume that the feed pressure Pf is chosen such that the feed vapor remains subcooled. It is assumed that the feedwater heater can be modeled as a counterflow heat exchanger with a minimum temperature difference ΔMITAT and no drop in air pressure. It is assumed that the state of the exhaust steam is saturated or superheated, that is, the exhaust steam temperature is not less than the saturation temperature of the exhaust steam pressure. It is assumed that the drain water is sent to the condenser. A set of exhaust steam flow and exhaust pressure PB is selected to optimize the performance of the cycle. In the case of assumptions 1 to 7, a double pinch point occurs, that is, the minimum internal temperature difference occurs at the beginning of the condenser and at the exhaust outlet at the same time.
 The optimization criteria proposed by the present invention were compared with existing design practices for micro-subcooling of drainage. First, for each closed feedwater heater (FWH1, 2, 4 & 5), the bleed flow is optimized based on a 2K minimum internal temperature difference criterion, and the drain is subcooled by 2K. The area of each open feedwater heater was then determined and used to optimize flow for three of the four closed feedwater heaters by the double pinch method described in the present invention. The double pinch is not optimal and the drain is subcooled by 2K because the drain from the last BWH is pumped upwards. Furthermore, the bleed is a two-phase liquid (not superheated) so a pinch occurs at the inlet of the bleed. The above results are shown in Table 2. The standard described in t...
Systems and methods are disclosed herein that generally involve a double pinch criterion for optimization of regenerative Rankine cycles. In some embodiments, operating variables such as bleed extraction pressure and bleed flow rate are selected such that a double pinch is obtained in a feedwater heater, thereby improving the efficiency of the Rankine cycle. In particular, a first pinch point is obtained at the onset of condensation of the bleed and a second pinch point is obtained at the exit of the bleed from the feedwater heater. The minimal approach temperature at the first pinch point can be approximately equal to the minimal approach temperature at the second pinch point. Systems that employ regenerative Rankine cycles, methods of operating such systems, and methods of optimizing the operation of such systems are disclosed herein in connection with the double pinch criterion.
Engine fuctionsSteam regeneration +6
Operating variablesEngineering +2
- Experimental program(1)
 The present invention discloses a system and method for optimizing a regenerative Rankine cycle incorporating a double pinch criterion. In some embodiments, the efficiency of the Rankine cycle is improved by selecting operating variables such as exhaust steam pressure and exhaust steam flow rate such that a feedwater heater achieves a double pinch point. In particular, the first pinch point is taken at the start of exhaust steam condensation and the second pinch point is taken at the exhaust steam outlet of the feedwater heater. The minimum temperature difference at the first pinch point is substantially equal to the minimum temperature difference at the second pinch point. The present invention discloses a system of a regenerative Rankine cycle associated with a double pinch criterion, a method of operating the system, and a method of optimizing the operation of the system.
 Some exemplary embodiments will now be described to provide an overview of the invention.
 Based on an understanding of the principles of structure, function, manufacture, and use of the disclosed systems and methods. Examples of one or more embodiments are shown in the accompanying drawings. It should be understood by those skilled in the art that the systems and methods described herein and illustrated in the accompanying drawings are not intended to limit the scope of the invention. Exemplary embodiments and the scope of the invention are defined only by the claims. Features described in one exemplary embodiment may be used in combination with features in other exemplary embodiments. Such modifications and variations should also fall within the protection scope of the present invention.
 figure 1 An exemplary embodiment of a power generation system 100 using a recuperative Rankine cycle is shown. The cycle may be subcritical or supercritical. As shown, the system 100 includes a steam generator 102 , a turbine 104 , a condenser 106 , a pump 108 , and a feed water heater (FWH) 110 .
 The steam generator 102 includes a heat source and a substantially closed vessel containing a working fluid within which is heated by the heat source. Exemplary heat sources include geothermal systems, nuclear energy systems, gas or coal fired burners, and other similar systems. A wide variety of working fluids can be used in the system, such as water, ammonia, heptane, toluene, isobutane, or a combination of the above. The steam generator 102 may be or may include a subcritical boiler or a supercritical steam generator.
 The turbine 104 includes a composite rotor having a shaft to which one or more turbine blades are attached. As the superheated steam produced by the steam generator 102 flows through the turbine 104, the steam expands and acts on the blades to rotate the shaft, thereby producing useful work. An output shaft of the turbine 104 may be connected to a power generator such that electrical power is generated as the turbine 104 rotates.
 As the working fluid exits the turbine 104, it flows into the condenser 106 and is cooled by a heat exchanger. The heat exchanger is provided with a cooling fluid for cooling the fluid exiting the turbine 104 to condense it into a subcooled liquid. A variety of cooling fluids can be used with the condenser 106, such as ambient air, water, and the like.
 The subcooled liquid from the condenser 106 is supplied to the feedwater heater 110 by a mechanical pump 108 . The feedwater heater 110 may also be supplied by other fluids, which may be from a degasser, upstream feedwater heater, or other intermediate elements of the feedwater heater 110 and condenser 106 . The feedwater heater 110 includes a heat exchanger that transfers heat from the steam stream 112 from the turbine 104 to the subcooled liquid feed from the condenser 106 . The feedwater heater 110 realizes the recuperation function of the system 100 by preheating the material supplied back to the steam generator 102 for circulation.
 Although in figure 1 Only a single steam generator, turbine, condenser, steam flow, and feedwater heater are shown and described, but this is for brevity of presentation only, and the system 100 may include any number of such elements. Also, while the term "feedwater heater" is used to describe the various elements of the system 100, it should be understood that the material may be or include other working fluids than water as described above.
 It should be appreciated that the system 100 may include various valves, pipes, lines, sensors, controllers, and other elements that assist in monitoring and adjusting various operating parameters of the system 100 . For example, an adjustable valve may be installed on the steam stream 112 for adjusting the flow rate of the steam stream 112 entering the feedwater heater 110 . The system 100 may also include a network of valves, conduits, etc. that may be adjusted to select where the vapor flow 112 exits the system 100 . This allows the exhaust steam pressure to be varied, such as by diverting the steam flow 112 from different turbine stages of the turbine 104 .
 figure 2 is a pinch point diagram of the feedwater heater 110, which shows the functional relationship between temperature and thermal energy conversion rate. As shown in the figure, there are two locations where pinch points may occur. The solid line 114 represents a feedwater with a mass flow rate M of 100.0 kilograms per second (kg/s) and a pressure P of 100 bar. Dashed line 116 represents a feedwater with a mass flow rate M of 30.0 kilograms per second (kg/s) and a pressure P of 15 bar. The dotted line 118 represents a feedwater with a mass flow rate M of 25.5 kilograms per second (kg/s) and a pressure P of 25 bar.
 Because the feed water is a subcooled liquid, the temperature of the feed water follows a smooth curve between the inlet (right side of the pinch graph) and outlet (left side of the pinch graph) of the heat exchanger rise. The vapor stream enters the heat exchanger as superheated steam, is converted to subcooled liquid and exits the heat exchanger. The steam stream thus has a superheated region during which the temperature of the steam stream decreases along a smooth curve. The gas stream also has a condensation zone during which the temperature of the vapor stream remains substantially constant. Once the vapor stream is completely cooled to a subcooled liquid, the temperature of the vapor stream again decreases along a smooth curve in the subcooled region.
 The condensation zone forms a horizontal line of the steam flow, such as the enthalpy drop at a constant temperature, if it is assumed that the heat capacity of the material in both the superheated and subcooled regions is greater than the heat capacity of the steam flow. If it is higher, the slope of the remaining two curves corresponding to the steam flow is higher than the slope of the curve corresponding to the material. Therefore, in the airflow direction from the inlet to the outlet, the superheated area and the subcooled area form a convergence between the vapor flow curve and the material curve, while in the condensation area, it can be seen that the two curves are divergent .
 Therefore, when a pure substance is used as the working fluid, there are only two places where a pinch point can occur, namely the point where the temperature difference between the feed water and the steam stream is minimal. One possible pinch point, "pinch point p", may exist at the onset of condensation in the gas stream condenser. A second possible pinch point, "pinch point o", may exist at the cold end of the heat exchanger (eg at the exhaust steam outlet of the feedwater heater). If the vapor inlet is in the two-phase region, the onset of condensation of the condenser just coincides with the inlet.
 The rate of gas stream temperature drop in the superheated steam and subcooled liquid stages can be adjusted by adjusting the mass flow rate of the gas stream. (As the mass of the vapor stream fluid entering the heat exchanger increases, the time required to cool the vapor stream increases).
 In addition, the temperature at which the vapor stream enters the heat exchanger and the temperature at which condensation begins can be adjusted by adjusting the pressure of the vapor stream. (The higher the pressure, the higher the temperature of the gas stream as it enters the heat exchanger, and the higher the temperature of the gas stream when it begins to condense).
 Therefore, even if the area of the heat exchanger and the mass flow rate and pressure of the feed water entering the heat exchanger are fixed, it is possible to adjust the relationship between the steam flow and the feed water by adjusting the mass flow rate and pressure of the steam flow. temperature difference between.
 In conventional power generation systems, operating parameters are selected that minimize the pinch point p. That is, the pressure and/or flow of the steam stream is adjusted so that the temperature at which the steam stream begins to condense in the heat exchanger is approximately equal to or very close to the feedwater temperature at that time. This passes figure 2The dashed line 116 in the figure shows that increasing the mass flow rate M of the vapor stream while decreasing the vapor stream pressure P to achieve a very small pinch point p. In this case, the temperature difference between the steam flow and the feed water at the cold end of the heat exchanger (where pinch point o may occur) becomes very large. In other words, in the conventional system, pinch p is minimized, but no attempt is made to obtain pinch o, and pinch o is rarely minimized.
 pass figure 2 The dotted line 118 in shows the operating parameters that can be selected to minimize pinch o. As shown, reducing the mass flow rate M of the vapor stream while increasing the vapor pressure P to achieve a very small pinch point o. In this case, the temperature difference between the steam stream and the feed water at the onset of condensation of the steam stream condenser (where pinch point p may occur) becomes very large. Conventional wisdom holds that this layout is suboptimal and inefficient.
 In some embodiments of the present invention, the mass flow rate M and/or the pressure P of the steam flow can be adjusted so that the temperature difference between the steam flow and the feedwater is at both pinch point p and pinch point o can be minimized. Preferably, one or all of the parameters can be adjusted such that the temperature difference between the steam flow and the feedwater at pinch point p is nearly identical to the temperature difference at pinch point o. In some embodiments, the temperature difference at pinch point p and the temperature difference at pinch point o may be equal. In some embodiments, the difference between the minimum temperature attainable at pinch point p and the minimum temperature attainable at pinch point o can be less than about 20 degrees Celsius, less than about 15 degrees Celsius, less than about 10 degrees Celsius, less than about 5 degrees Celsius , less than about 3 degrees Celsius, and/or less than about 1 degree Celsius.
 image 3 is a pinch point diagram of the feedwater heater 110, showing the functional relationship between temperature and thermal energy conversion rate when a double pinch point is achieved. As shown, both pinch p and pinch o are minimized. As noted above, conventional wisdom considers subcooling the steam stream (required for double pinch points) to be wasteful because the steam stream from the feedwater heater 110 is typically used as one or more Cascade downstream feedwater heater material. However, the description below illustrates that subcooling the steam stream in order to achieve a double pinch unexpectedly results in better operating efficiency.
 The double clamp standard for closed feedwater heaters of the regenerative Rankine cycle will be described in detail below. The feedwater heater is modeled as a counterflow heat exchanger. Therefore, as mentioned above, there are two possible pinch points on the feedwater heater: (1) at the outlet of the steam stream (exhaust) and (2) at the start of condensation of the steam stream condenser. At a given heat load of the feedwater heater, the material inlet temperature and flow, exhaust steam flow and/or pressure can be selected to achieve the same minimum approach temperature at the two possible pinch points to optimize the efficiency of the cycle. The analytical proof for a fixed pinch value is given below under a few assumptions, with the vapor stream entering the condenser. In addition, the standard numerically shows fixed pinch values and fixed heat exchange areas using the most common layout patterns: flow to condenser, flow to deaerator and cascade flow to next feedwater heating device. A similar criterion is developed below where the fluid is pumped (up or down) and mixed with the feed water. The double pinch criterion disclosed in the present invention simplifies the optimization process and results in a significant increase in the efficiency of the stationary heat exchange area. For numerical simulation and calculation reasons, pressure can be used as an optimization variable, heat load and mass flow rate can be calculated.
 term name
 The Rankine cycle is widely used in power generation, often with efficiency-enhancing features such as reheating, superheating, or regeneration. Additional information on Rankine cycles can be found in A. Bejan, Thermal Design & Optimization, John Wiley & Sons, Inc., 1996; R.W. Haywood, Analysis of Engineering Cycles, 3rd Edition, Elsevier, 1980; and M.J. Moran, H.N. Shapiro, Fundamentals of Engineering Thermodynamics, 6th Edition, John Wiley and Sons, 2007, the contents of which are hereby incorporated by reference in their entirety.
 Commercial software packages capable of high-fidelity modeling of the energy cycle are available, such as THERMOFLEX, ASPENPLUS, and GATECYCLE. These packages provide tools to perform an execution parameter and optimization study for a given cycle (process link) or to construct an energy cycle for a given application.
 All but the simplest cycles are recuperative, that is, they involve preheating of the material (returning from the condenser) by a stream of steam drawn from the turbine. This preheating process takes place in closed and/or open feedwater heaters (CFWH and OFWH). There are many different layout modes of the feedwater heater according to the position of the exhaust steam (ie, the drainage) of the enclosed feedwater heater. In general, the performance of the power cycle increases with the number of feedwater heaters, so typically, the number of feedwater heaters is determined for cost considerations, that is, for the balance between capital and operating costs . A well-known approximate design criterion is to have equal enthalpy growth for each feedwater heater in the recuperative region of the Rankine cycle. More precisely, in some ideal cases, in order to achieve the maximum efficiency of a non-reheat non-supercritical power plant, the enthalpy of the feed water rises to the saturation point, which should be approximately at all heaters and economizers. consistent.
 A closed feedwater heater must be a multiphase heat exchanger. The design of heat exchanger networks is well established and additional information can be found in: L.T. Biegler, I.E. Grossmann, A.W. Westerberg, Systematic Methods of Chemical Process Design, Prentice Hall, New Jersey, 1997; J. Cerda , A.W.Westerberg,D.Mason,B.Linnhoff,Minimum utility usage in heat-exchanger network synthesis.A transportation problem,Chemical Engineering Science38(3)(1983)373-387;J.Cerda,A.W.Westerburg,Synthesizing heat-exchanger networks having restricted stream stream matches using transportation problem formulations, Chemical Engineering Science 38(10)(1983) 1723-1740; B. Linnhoff, E. Hindmarsh, The pinch design method for heat-exchanger networks, Chemical Engineering Science 38(5)(1983 ) 745-763; S.A. Papoulias, I.E. Grossmann, A structural optimization approach in process synthesis 2. Heat-recovery networks, Computers & Chemical Engineering 7(6) (1983) 707-721; and I.C. Kemp, Pinch Analysis and Process Integration, 2nd Edition, Elsevier , 2007, the contents of which are hereby incorporated by reference in their entirety. Both the powertrain and the heat exchanger network design can be done in a simplified manner or with more rigorous models. A common simplification is the so-called pinch point analysis, where the minimum possible temperature (MITA) is selected and the inlet conditions and heat loads for each heat exchanger are calculated. The advantage of this simplified approach is the separation of capital costs and detailed design of the heat exchanger from operating costs. The implicit assumption of the pinch point method is that the desired heat exchange area is precisely characterized by MITA. Nonetheless, for economical optimization, an accurate calculation of the heat transfer area is required.
 The present invention discloses a new standard for optimizing regenerative power systems. In the discussion that follows, the criteria are applied to the two pinch point methods and optimized for a fixed thermal transfer region. The closed feedwater heaters of the present invention are typically countercurrent heat exchangers.
 Therefore, there are two possible pinch points, at the onset of condensation in the condenser and at the exhaust (drainage). The main result of the present invention is that, for most feedwater heater configurations, optimal operation can be achieved by taking two pinch points simultaneously. In some embodiments, the links between units (processes) and expansion lines (condenser temperature, turbine inlet pressure and temperature, and total flow) are assumed to be fixed. In some embodiments, the working fluid is assumed to be a pure substance with morphological changes.
 In a conventional pinch point analysis, MITA selected by economic criteria is used as a surrogate for the heat exchange area. Therefore, each closed feedwater heater has three degrees of freedom, namely heat load (via MITA as a surrogate), extraction pressure (and temperature) and extraction flow. Thus, at least in principle, the operation of the loop can be optimized numerically, taking into account simultaneous changes to these three degrees of freedom, subject to a minimum temperature difference. In a more rigorous analysis, the heat transfer region is selected by economic analysis, the remaining two free dimensions are extraction pressure and extraction flow, and MITA is free. The dual pinch design criteria described in this invention eliminate two variables for each enclosed feedwater heater. In addition, in the case of pinch point analysis, the criteria described in the present invention preclude the need to check for pinch point failure limits and the need to model the spatial distribution of feedwater heaters. Overall, the systems and methods described herein significantly simplify loop optimization. This acceleration algorithm is particularly important when the Rankine cycle is not viewed in isolation but as part of a complex process, such as an oxy-combustion system, or in other systems, such as the working fluid selection process of the organic Rankine cycle .
 Detailed information on oxy-combustion systems can be found in H. Zebian, A. Mitsos, Multi-variable optimization of pressurized oxy-coal combustion, In Press: Energy (EGY) http://dx.doi.Org/10.1016/j.energy .2011.12.043, 2011, the contents of which are hereby incorporated by reference in their entirety.
 Optimization of isolated Rankine cycles can be performed with general modeling tools. However, in the absence of the criteria described in the present invention, such "optimization" actually leads to suboptimal local optima. The method and system of the present invention allow for simpler application of approximate design criteria, such as the aforementioned isenthalpic increase for feedwater heaters.
In Chapter 2 below, the simplified rule for minimum temperature difference is studied for the simplest feedwater heater layout, ie the drain is installed in the condenser. A more precise statement is given by appropriate boundary conditions and proved analytically. In addition, it is explained how the double pinch criterion is applied in a simplified process, resulting in a significantly simplified optimization recipe relative to the method of simultaneous optimization of all variables. In Chapter 3 below, the applicability of the double pinch criterion with respect to other layout structures is discussed. In Chapters 4 and 5 below, various case studies of different feedwater heater layouts are performed, numerical calculations are performed on the simplified and constant regions, respectively, and the entropy generation of the feedwater is also discussed. The systems and methods of the present invention result in significant savings compared to current design practices, both in terms of simplified computations and rigorous computations.
 Chapter 2 - Analytical Proof of the Double Pinch in the Simplified Way
 In the following sections, a double pinch design criterion is developed and justified on a set of non-limiting assumptions. First, the pinch point diagram illustrates that there are only two points of interest, the start of condensation in the condenser and the exit of the vapor stream.
 Second, it has been shown that double pinch points are necessary to achieve optimality for a given heat load when the exhaust gas is delivered to the condenser. Additionally, there is a specific set of exhaust steam pressures and flows that can achieve a double pinch point. Therefore, double pinch points are also sufficient for optimality to be achieved. Then, a reordered set of variables is exposed according to the efficiency-optimized computational program.
 2.1 Two possibilities for generating pinch points
 Assumption 1 (Heat Capacity Ratio): The ratio of material flow to exhaust steam capacity is assumed to be high enough so that in the superheated and subcooled regions the heat capacity of the material is are higher than the heat capacity of the steam. Assumption 1 is for a typical Rankine cycle.
 The following propositions show that figure 2 There are only two points in the pinch chart shown that need to be investigated.
 Proposition 1: In the case of Assumption 1, the smallest temperature difference between the material and the steam flow can only occur at two points, namely, the condensation start of the condenser and the exhaust steam outlet.
 Proof: The feedwater heater can be modeled as a counterflow heat exchanger. The material can always be subcooled to obtain a smooth curve on the pinch plot. In contrast, the steam flow can generally comprise three regions, namely, a superheated region, a condensation region and a subcooled region, giving two varying curves. The condensation zone causes the vapor flow to form a horizontal line, ie the enthalpy decreases while the temperature remains constant. Based on Assumption 1, the other two curves corresponding to the steam flow have a higher slope than the curves corresponding to the material. Therefore, in the airflow direction from the inlet to the outlet, the superheated area and the subcooled area form a convergence between the vapor flow curve and the material curve, while in the condensation area, it can be seen that the two curves are divergent . Therefore, there are only two locations where a pinch point can occur, namely, where condensation begins and where the gas flow exits. If the vapor inlet is in the two-phase region, the onset of condensation happens to coincide with the inlet.
 As mentioned above, figure 2 An illustrative feedwater heater pinch plot is shown. There are only two possible pinch points in the thermal regeneration of a pure-substance feedwater heater: at the cold end of the heat exchanger (at the drain, the small dotted line of steam flow, marked as pinch point o) or at the gas flow condenser at the beginning of (dashed airflow line, marked as pinch p).
 2.2 Analytical proof of necessity
 In this section, a simplified method of achieving the minimum temperature difference is considered for the case where the drain water is routed to the condenser. for a given material flow Material inlet temperature T f,i and heat transfer load Except for the tedious cases, double pinch points are considered optimal. The above conclusions are based on relatively weak assumptions.
 Assumption 2 (exhaust steam saturation enthalpy): Assume the exhaust steam pressure is P B of saturated steam h g,sat The enthalpy value is not lower than h at the turbine outlet T (P o ) of the enthalpy.
 h g,sat (P B )≥h T (P o )
 where the subscript T refers to the turbine (expansion line) and the P o Refers to the outlet air pressure of the turbine.
 Assumption 2 applies to typical expansion lines. Only a very low exhaust steam pressure P is permitted only in the state of high superheat at the outlet of the turbine and the heat load is very small B This assumption can be violated. Operation in this case is suboptimal.
 Assumption 3 (Steam outlet enthalpy): Assume the enthalpy h of the steam flow at the outlet of the feedwater heater B,o not higher than the enthalpy value h at the turbine outlet T (P o ).
 h B,o ≤h T (P o )
 Assumption 3 applies to typical expansion lines and working fluids such as water and ammonia, heptane and toluene.
 Assumption 4 (exhaust steam pressure): Assume the optimal exhaust steam pressure P B Strictly higher than the pressure P at the turbine outlet o. Assumption 4 can only be violated when the heat load on the feedwater heater is particularly low and the turbine outlet is highly superheated. In other words, recuperators are not considered in this chapter.
 Assumption 5 (exhaust steam pressure): Assume the optimal exhaust steam pressure P B Strictly below the turbine inlet pressure. Assumption 5 applies to a typical Rankine cycle.
 Assumption 6 (heat capacity ratio and saturated steam): The following relationship applies to the pressure to be studied:
 ∂ h g , sat ∂ P | P B ≤ c P l ( T sat ( P B ) - Δ MITA T , P F ) ∂ T sat ∂ P | P B
 Assumption 6 applies to typical working fluids such as water, toluene and ammonia. In the case of the study, the left half of the expression is negative or slightly positive and the right half of the expression is always positive.
 Assumption 7 (positive pressure of liquid enthalpy value): the derivative of liquid enthalpy value with respect to pressure is positive:
 ( ∂ h l ∂ P ) T 0
 It should be noted that Assumption 7 applies to incompressible liquids for which it is a good approximation. Additionally, it is suitable for typical working fluids such as water, toluene and ammonia.
 In the following sections, reheating is excluded for simplification and considering a fixed expansion line, it must be assumed that the turbine isentropic efficiency is not affected by exhaust conditions.
 Theorem 1 (Double Pinch Necessity): Consider a regenerative Rankine cycle with no reheating but positive turbine isentropic efficiency. Fixed inlet temperature, inlet pressure, outlet temperature and outlet pressure P for most turbines o , that is, not affected by exhaust steam. Consider an arbitrary feedwater heater with an arbitrary but deterministic material flow Material inlet temperature T F.i and heat transfer load Assuming the material pressure P f Selected to keep the material vapour subcooled. It is assumed that the feedwater heater can be modeled to have a minimum temperature difference Δ MITA A countercurrent heat exchanger of T and the air pressure does not drop. It is assumed that the exhaust steam state is saturated or superheated, that is, the exhaust steam temperature is not less than the saturation temperature of the exhaust steam pressure. It is assumed that the drain water is sent to the condenser. Select a set of exhaust steam flow and exhaust pressure P B to optimize the performance of the loop. Under assumptions 1 to 7, a double pinch occurs, ie the minimum internal temperature difference occurs simultaneously at the beginning of the condenser and at the exhaust steam outlet.
 Proof: The following proof is done by contrast, i.e. considering a set of optimal combinations, It is assumed that no pinch points occur to conclude that the combination is not optimal. This process is done in three steps, first excluding the case where no pinch point occurs, then excluding the case where the pinch point only occurs at the beginning of the condenser, and finally excluding the case where the pinch point occurs only at the exhaust steam outlet.
 Based on the assumption that there is no reheating, the work that the exhausted steam should do in the turbine is calculated as:
 W · B = m · B ( h T ( P B ) - h T ( P 0 ) ) - - - ( 1 )
 h T (P B ) is the enthalpy value at the exhaust steam inlet of the feedwater heater, h T (P B )=h B,i. The best way to exhaust steam is to minimize the loss of work
 If the exhaust steam is saturated steam or in the two-phase region (h T (P B )≤h g,sat (P B )), the pinch point at the beginning of the condenser is tediously obtained. In addition to this, the pinch point at the outlet is assumed to be at a given pressure P BP o to minimize to minimize Therefore, it can be determined that h T (P B )>h g,sat (P B ).
 1. Assuming that the minimum internal temperature difference of the feedwater heater in the control is not reached Exhaust steam flow or pressure (or even both) is then reduced by a very small amount to allow an identical thermal conversion agreement without violating the minimum internal temperature difference. On the other hand, this reduction accounts for more energy being produced by the turbine, see equation (1), ie the initial combination is not optimal.
 2. Suppose now In the case of , a pinch point is achieved at the beginning of the condenser but not at the exhaust outlet. Maintaining a pinch point at the condenser outlet means that changes to exhaust pressure will result in changes in exhaust flow. This possibility can be achieved under assumption 4 without violating any constraints. The derivative of the exhaust mass flow rate with respect to the exhaust pressure coinciding with the pinch point at the beginning of the condenser is where the subscript p represents the pinch point p.
 Considering that the work not done by the turbine is at The partial derivative of the exhaust pressure at the pinch point at the beginning of the condenser calculated below
 ∂ W · B ∂ P B | P ‾ B = ( ∂ m · B ∂ P B ) P | P ‾ B ( h t ( P ‾ B ) - h t ( P o ) ) + m · ‾ B ∂ h t ∂ P B | P ‾ B - - - ( 2 )
 We already know that the states at the inlet and outlet of the turbine are deterministic. Optimal operation means the smallest or if we find then the combination Not optimal.
 On the right side of Equation 2, the second part is a positive value, Because the turbine does work. Therefore, if Derivative value calculated below is non-negative, we can also get or said combination Not optimal. This means we can reduce both exhaust pressure and flow and maintain the same preheat. should be noted can be directly proven.
 So we only need to consider the following cases
 ( ∂ m · B ∂ P B ) p | P ‾ B 0 - - - ( 3 )
 The following section shows that the derivative of energy loss with respect to exhaust pressure is negative, or that increasing exhaust pressure increases efficiency.
 The following lists a expression. The case of the pinch point at the beginning of the condenser is
 m · F ( h F , o - h l ( T sat ( P B ) Δ MITA T , P F ) ) = m · B ( h T ( P B ) - h g , sat ( P B ) )
 put it on P B derivation and in Calculated below
 - m · F c P ( T sat ( P ‾ B ) - Δ MITA T , P F ) ∂ T sat ∂ P | P ‾ B = ( ∂ m · B ∂ P B ) p | P ‾ B ( h t ( P ‾ B ) - h g , sat ( P ‾ B ) ) + m · ‾ B ∂ h t ∂ P B | P ‾ B - m · ‾ B ∂ h g , sat ∂ P | P ‾ B
 ( ∂ m · B ∂ P B ) p | P ‾ B ( h t ( P ‾ B ) - h g , sat ( P ‾ B ) ) = - m · ‾ B ∂ h t ∂ P B | P ‾ B + m · ‾ B ∂ h g , sat ∂ P | P ‾ B - m · F c P ( T sat ( P ‾ B ) - Δ MITA T , P F ) ∂ T sat ∂ P | P ‾ B - - - ( 4 )
 By Assumption 2 we get therefore
 h T ( P ‾ B ) - h g , sat ( P ‾ B ) ≤ h T ( P ‾ B ) - h T ( P o )
 Recalling inequality (3) we get
 ( h T ( P ‾ B ) - h T ( P o ) ) ( ∂ m · B ∂ P B ) p | P ‾ B ≤ ( h T ( P ‾ B ) - h g , sat ( P ‾ B ) ) ( ∂ m . B ∂ P B ) p | P ‾ B
 Then combining equations (2) and (4) to get
 ∂ W · B ∂ P B | P ‾ B ≤ M · ‾ B ∂ h g , sat ∂ P | P ‾ B - m · F c P ( T sat ( P ‾ B ) - Δ MITA T , P F ) ∂ T sat ∂ P | P ‾ B
 Note the condition of Assumption 6 we got It is known from Assumption 5 that it is possible to increase the air pressure, so we prove that Not optimal.
 3. Assuming that the final In the case of , we achieved a pinch point at the exhaust outlet rather than at the start of condensation.
 Similar to the previous case, we will consider holding the pinch point at the exhaust outlet That is, under the condition that the derivative of the exhaust steam mass flow rate consistent with the pinch point at the exhaust outlet remains unchanged to the exhaust steam pressure, the exhaust steam pressure is changed, wherein the subscript o represents the pinch point o.
 In a similar way to the previous case we get the equation:
 ∂ W · B ∂ P B | P ‾ B = ( ∂ m · B ∂ P B ) o | P ‾ B ( h t ( P ‾ B ) - h t ( P o ) ) + m . ‾ B ∂ h t ∂ P B | P ‾ B - - - ( 5 )
 ( ∂ m . B ∂ P B ) o | P ‾ B 0 - - - ( 6 )
 We can then see that the derivative of the lost work with respect to the exhaust is positive, meaning that the exhaust pressure should be reduced.
 Given below is a The expression of , the heat transfer load to maintain the pinch point at the exhaust steam outlet can be calculated by the following expression:
 Q · = m · B ( h T ( P B ) - h l ( T T , i + Δ MITA T , P B ) )
 It should be noted that the overall heat transfer is constant for P B The derivative of is zero. For the right half of the expression about Derivation can get:
 0 = ( ∂ m · B ∂ P B ) p | P ‾ B ( h T ( P B ) - h l ( T T , i + Δ MITA T , P B ) ) + m · ‾ B ∂ h T ∂ P B | P ‾ B - m · ‾ B ∂ h l ∂ P B | T T , i + Δ MITA T , P ‾ B
 0 = ( ∂ m · B ∂ P B ) o | P ‾ B ( h T ( P B ) - h l ( T T , i + Δ MITA T , P B ) ) = - m · ‾ B ∂ h T ∂ P B | P ‾ B + m . B ∂ h l ∂ P B | T T , i + Δ MITA T , P ‾ B
 Based on assumption 3 there is h l (T T,i +Δ MITA T,P B )=h B,o ≤h T (P o ), so we get
 h T ( P ‾ B ) - h l ( T T , i + Δ MITA T , P B ) ≥ h T ( P ‾ B ) - h T ( P o )
 Recalling inequality (6) we get:
 ( h T ( P ‾ B ) - h T ( P o ) ) ( ∂ m . B ∂ P B ) o | P ‾ B ≥ ( h T ( P ‾ B ) - h l ( T T , i + Δ MITA T , P B ) ) ( ∂ m . B ∂ P B ) o | P ‾ B
 Combine after (7)
 ( h T ( P ‾ B ) - h T ( P o ) ) ( ∂ m . B ∂ P B ) o | P ‾ B ≥ - m . ‾ B ∂ h T ∂ P | P ‾ B + m . ‾ B ∂ h l ∂ P | T T , i + Δ MITA T , P ‾ B
 and (5) can be obtained
 ∂ W . B ∂ P B | P ‾ B m . ‾ B ∂ h l ∂ P | T T , i + Δ MITA T , P ‾ B
 By assumption 7 we get It is known from Assumption 4 that it is possible to reduce the air pressure, so we prove that Not optimal.
 2.3 Image proof of uniqueness and sufficiency
 Theorem 1 proves that double pinch points are a necessary condition for optimality. In principle, there are a variety of double pinch points that apply to it, some of which may be sub-optimal. On the basis of two additional assumptions it can be shown that for a given heat load there exists a unique set of Can give double pinch points.
 Assumption 8 (weak pressure dependence of the supercooled state): the derivative of the liquid enthalpy value with respect to pressure is smaller than the derivative of the enthalpy value on the expansion line
 ∂ h l ∂ P | T , P B ∂ h T ∂ P | P B - - - ( 8 )
 In addition, for the temperature and pressure to be studied, the heat capacity rate in the subcooled region is assumed to be a function of the pressure. More precisely, for any two satisfying P B1P B2 pressure P B1 , P B2 Satisfy
 c p l ( T , P B 1 ) c p l ( T , P B 2 ) h T ( P B 1 ) - h l ( T , P B 1 ) h T ( P B 2 ) - h l ( T , P B 2 ) - - - ( 9 )
 Unless the turbine efficiency is extremely low, the conditions of Assumption 8 should be satisfied.
 Assumption 9 (decrease in enthalpy of vaporization): The derivative of enthalpy of vaporization with respect to pressure is negative
 ∂ h 1 g ∂ P 0
 Assumption 9 applies to pure substances. See B.E. Poling, J.M. Prausnitz, J.P.O'Connell, The Properties of Gases and Liquids, 5th Edition, McGraw Hill, New York, 2001, the contents of which are hereby incorporated by reference in their entirety.
 Lemma 1 (Double Pinch Uniqueness): Consider the conditions and assumptions of Theorem 1. There is a unique set of exhaust flow under additional assumptions 8 and 9 and exhaust pressure PB give double pinch points.
 Proof: Consider a set of Generates double pinch points. Consider meeting P B2 B1 The second group of P B2 , Grips are created at the exit. We will show that it violates the minimum temperature difference where condensation starts.
 consider as Figure 4 The pinch plot shown, it should be noted that there is no linearity assumption (requiring constant heat capacity in some regions). From Assumption 1, it can be known that the slope of the subcooling curve is greater than the material curve of the steam flow.
 point B 1i , and B 1o The inlet of the steam stream, the beginning of condensation, the beginning of subcooling and the outlet of the steam stream are respectively shown. also, represents the pressure P B2 The intersection of the saturation temperature of and the subcooling curve of stream 1. at last represents the pressure P B2 The saturation temperature and the crossing point of The intersection of the curves parallel to the supercooling curve.
 In order to make the steam flow 2 take a pinch point at the outlet, its outlet happens to be the same as B 1o Consistent. It should be noted that the pinch point at the exit indicates that
 Q · = m · B 1 ( h T ( P B 1 ) - h l ( T T , i + Δ MITA T , P B 1 ) )
 Q · = m · B 2 ( h T ( P B 2 ) - h l ( T T , i + Δ MITA T , P B 2 ) )
 Now we will use the two inequalities from Assumption 8. through (8) for The two conditions of Furthermore, based on the same assumptions as (9), the onset of subcooling of stream 2 is located at to the left. Finally, in conjunction with Assumption 9, the start of the arithmetic condenser is at to the left. This is to the left of the minimum temperature difference, given that the slope of the steam flow is higher than that of the material.
 Because the exhaust pressure P B1 ,P B2 is arbitrary, we can also rule out the existence of B1 higher pressure P B3 the double pinch case. In fact, suppose there is an exhaust pressure P B3 The double pinch point of , by the above argument P B1 The violation of the minimum temperature difference leads to a contradiction.
 Theorem 2: Under the assumptions of Theorem 1 and Lemma 1, the only group that obtains double pinch points is optimal. The proof of Theorem 2 is trivial and is omitted here for brevity. It should be noted that we take its existence for granted, as evidenced by the variable changes in the next section.
 2.4 Program for loop optimization
Theorems 1 and 2 prove that a double pinch is optimal for a fixed heat load. However, it is more computationally efficient to vary the exhaust pressure in the short-cut and fixed-region approach of pinch analysis.
 2.4.1 Steps of pinch point analysis
 For the pinch point analysis method, the set of exhaust flow rates that can cause double pinch points can be directly calculated and heat load
 m · B = m · F h g , sat ( P B ) - h l ( T F , i + Δ MITA T , P B ) h l ( T sat ( P B ) - Δ MITA T , P F ) - h l ( T F , i , P F )
 Q · = m · B ( h T ( P B ) - h ( T F , i + Δ MITA T , P B ) ) - - - ( 10 )
 in view of and The expression is clear that there is only one given exhaust pressure P B of double pinch points. An advantage of this independently varying variable is that it eliminates the need for a spatially discrete model of the feedwater heater by producing an unambiguous formulation of the pinch point process. However, we have not yet demonstrated whether the double pinch is optimal for a given pressure. It is therefore necessary to explain that changes in variables do not lead to convergence to a pseudo-solution when used in an optimization program. A global/local optimization for the pressure space under double pinch points will be presented below, which implies a global/local optimization of design and operation.
 Proposition 2 (uncomplicated exhaust steam pressure as independent variable): Let the superscript k=1,...,n refer to the feed water k. It is assumed that the optimization is relative to the thermal load according to and exhaust flow The exhaust flow rate obtained by equation (10) carried out, and the is the best in the set pk. Indicates the corresponding exhaust flow and heat load if the set p k including all possible exhaust pressures, the combination A global optimum of power cycle efficiency is presented. if the set p k Yes , then the combination gives a local optimum of power cycle efficiency.
 Proof: Consider the first case, the set p k Include all possible exhaust pressures. This means combining is the best of all combinations that lead to double pinch points, so the combination according to Theorem 1 is also the best of all combinations.
 Now consider the second case, the set p k Yes the inner neighborhood of . First assume that there is a single feedwater heater k=n=1. Let the solution of equation (10) be the exhaust pressure function, expressed as Local optima means
 η ( P B k , m · B k , d ( P B k ) , Q · d k ( P B k ) ) ≤ η ( P ‾ B k , m · ‾ B k , Q · ‾ k ) , ∀ P B k ∈ p k
 The continuity of the mapping of equation (10) states that the set p k Up The image is an interval. Proposition 1 guarantees that there is only one double pinch for a fixed heat load, so not on the boundary of the interval. By Theorem 1 for any We can get:
 η ( P ^ B k , m · ^ B k , Q · d k ( P B k ) ) ≤ η ( P B k , m · B k , d ( P B k ) , Q · d k ( P B k ) ) , ∀ P B k , m · ^ B k
 Therefore, there is a the inner neighborhood of which is optimal. This is the definition of a local minimum.
 Suppose now that there are multiple feedwater heaters in a local optimum. Remember to have one or more fixed temperature points in the process, such as condensers. Move in the material direction and divide the feedwater heaters in pairs between these points, if an additional feedwater heater is required. In multiple pairs of feedwater heaters, consider for the first one containing exhaust pressure in the inner neighborhood of variable, adjust the second so that the total heat load is constant, i.e. Taking a similar approach in the case of a single feedwater heater, we can use uniqueness and continuity (relative to exhaust steam pressure and material inlet temperature) to construct a containing The neighborhood of , which is internal and locally optimal. Because a given combination has a constant heat load, the same treatment is done for the individual feedwater heaters.
 2.4.2 Procedures for Fixed Areas
 The double pinch criterion can also be applied in the case of fixed heat exchange areas, but requires some iterative procedures. The optimal procedure depends to some extent on the process simulation software used. A number of reasonable options exist, and only a few exemplary options are discussed here. The generally recommended way is to change the exhaust pressure P B as the main optimization variable. Then, for a given heat transfer zone and exhaust pressure, by excluding one variable, such as exhaust flow, the double pinch criterion can fully specify the operation of the feedwater heater. It should be noted that the value of the pinch point is unknown and that what to do is implicitly determined.
 The first major option is to have the optimizer control an additional variable to satisfy nonlinear constraints, or to hide the data set from the optimizer and use it as a design criterion built into the operation of the objective function. The former can be found in L.T. Biegler, Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes, MPS-SIAM Series on Optimization, SIAM-Society for Industrial and Applied Mathematics, 2010, the contents of which are hereby incorporated by reference in their entirety. However, in some embodiments of the present invention, an embedded design specification is preferred because it avoids failures at the analog level.
 The second consideration is which set of variables and constraints to choose. In the present invention, the exhaust steam flow Adjusted to meet the double pinch criteria. This requires calculating the heat transfer on the feedwater heater for each iteration (heat exchange analysis). Once this operation has been performed, the values of the two pinch points can be obtained from the temperature distribution or by solving equation (10). Another way is to change the pinch point Δ MITA The value of T satisfies the given thermal transfer area. In this method, the exhaust gas flow and heat transfer load can be accurately calculated by equation (10) but require calculations for the heat transfer area (heat exchange design).
 3. Other water supply structures
 As previously mentioned, there are various configurations of feedwater heaters. The double pinch point is plausible for most structures, and analysis of this conclusion proves to be outside the scope of the present disclosure. as below image 3 As shown, for some structures, the standard may not be applicable.
 3.1 Exhaust steam to open feed water heater
 Drainage into the condenser is assumed in the above analytical proof. For high pressure closed feedwater heaters, the better option in some embodiments is to drain the next possible degasser or open feedwater heater, as this restores some remaining availability and alleviates the Condenser and water pump loads. The proof used in Theorem 1 assumes that the drain does not affect the temperature of the material inlet and therefore does not directly apply to the case of draining the condenser. The double pinch point standard also applies to this water supply structure. Although an analytical proof is not within the scope of this disclosure, Chapter 5 presents example calculations.
 3.2 Cascading (downstream)
 In a high-efficiency cycle with multiple closed feedwater heaters, it is customary to cascade downstream, that is, to drain to the next closed feedwater heater (reduce pressure immediately) and mix with the exhaust steam inlet. Draining an open feedwater heater in a similar fashion captures a portion of the remaining availability. Although an analytical proof is not within the scope of the present disclosure, Chapters 4 and 5 give example calculations to demonstrate that the double pinch criterion can also be applied to this structure.
 3.3 Pumps to transport materials
 like Figures 5A-5B As shown, a process structure is to use a water pump to pressurize the drainage to the material pressure and mix it with the material. like Figure 5A One way shown is mixing at the inlet of an enclosed feedwater heater, known as a back (or down) water pump. like Figure 5B Another method shown is for the pump to drain forward (or up), i.e. mix the material at the outlet of the closed feedwater heater. For any kind of pump structure, double pinch points are the preferred but not the only best choice for pinch point analysis for selecting exhaust pressure and flow. The double-grip approach is not recommended for constant regions. In general, the optimal choice is to achieve a pinch point at the onset of condensation while being sufficiently subcooled at the outlet to ensure that there are no technical difficulties in the extraction process. Similar to the double pinch criterion, this approach gives two constraints for excluding two variables from three.
 Assumption 10 (material inlet enthalpy value): Assume the enthalpy value h at the inlet of the feedwater heater F,i not higher than the enthalpy value h of the turbine T (Po)
 h F,i ≤h T (P o )
 Assumption 3 applies to typical expansion lines and working fluids such as water, ammonia, heptane and toluene.
 Theorem 3 (pinch point p of the pump structure): Consider a regenerative Rankine cycle with no reheating but positive turbine isentropic efficiency. Fix the turbine inlet temperature, inlet pressure, outlet temperature and outlet pressure P o , that is, not affected by exhaust steam. Consider an arbitrary feedwater heater with an arbitrary but deterministic material flow Material inlet temperature T F.i and heat transfer load Assuming the material pressure P f Selected to keep the material vapour subcooled. It is assumed that the feedwater heater can be modeled to have a minimum temperature difference Δ MITA A countercurrent heat exchanger of T and the air pressure does not drop. It is assumed that the exhaust steam state is saturated or superheated, that is, the exhaust steam temperature is not less than the saturation temperature of the exhaust steam pressure. Assume that drainage is fed into P f and mix with the material at the inlet or outlet of the feedwater heater. Select a set of exhaust steam flow and exhaust pressure P B to optimize the performance of the loop. Under Hypothesis 10, a minimum internal temperature difference occurs at the beginning of the condenser, furthermore, subcooling the drain water to achieve a double pinch increases the heat transfer area without leading to an increase in efficiency.
 Proof: First, we demonstrate the necessity of the pinch point at the onset of condensation by contrast. Hypothetical combination is optimal and no pinch point occurs at the onset of condensation. for any group The first law for ignoring pump power is
 m · F , i h F , i + m · B h T ( P B ) = m · F , o h F , o ,
 where the inlet state i is before mixing and the outlet state o is after mixing. Similarly, the mass balance relationship is
 m · F , i + m · B = m · F , o .
 Combining the above two equations we get
 ( m · F , o - m · B ) h F , i + m · B h T ( P B ) = m · F , o h F , o
 m · B ( h T ( P B ) - h F , i ) = m · F , o ( h F , o - h F , i )
 right differentiate and evaluate
 ( ∂ m · B ∂ P B ) T o | P ‾ B , m · ‾ B ( h T ( P ‾ B ) - h F , i ) + m · ‾ B ∂ h T ∂ P B | P ‾ B = 0
 Suppose h exists under 10 F,i ≤h T (P o ),therefore
 h T ( P ‾ B ) - h F , i ≥ h T ( P ‾ B ) - h T ( P o )
 thereby getting
 ∂ W · B ∂ P B | P ‾ B = ( ∂ m · B ∂ P B ) T o | P ‾ B ( h t ( P ‾ B ) - h t ( P o ) ) + m · ‾ B ∂ h t ∂ P B | P ‾ B - - - ( 11 )
 Similar to the proof process of Theorem 1, we get therefore Not optimal.
 We will now demonstrate that double pinch points are not recommended as far as the thermal transition region is concerned. Assuming the group is optimal and pinch points occur at the beginning of the condenser and at the drain outlet. Exhaust steam conditions are maintained and the heat exchange is divided into two stages: (i) for cooling the steam and condensing (ii) for subcooling. If we exclude the second stage, we can still get the complete condensation process: the inlet of stage (i) remains unchanged with the pump draining backwards; the material in stage (i) with the pump draining forward A drop in inlet temperature results in a higher heat transfer rate (the flow reduction has a negligible effect on the heat transfer rate). Other than that, excluding the second stage does not change the state at the material outlet (after mixing). This is evident from the first law and mass balance. Overall, the heat transfer area can be reduced without loss of performance.
 3.4 Open feed water heater
 In the case of completeness, open feedwater heaters were also taken into consideration. Open feedwater heaters are clearly not within the range of closed feedwater heater types so the double pinch criterion cannot be directly applied. On the other hand, optimization of exhaust steam pressure and flow still makes sense. There are also three optimization variables for each open feedwater heater, namely operating pressure, exhaust steam pressure and flow. According to the same process as the proof process of Theorem 3, it can be concluded that the optimal exhaust pressure is equal to the operating pressure of the deaerator.
 Furthermore, the exhaust flow is given by the desired temperature increase and the saturation requirement at the material outlet. Therefore, similar to an enclosed feedwater heater, there is only one variable that needs to be optimized.
 4 Example calculation of a simple process
 In this chapter, the validity of the described design criteria is demonstrated numerically for a single feedwater heater and for multiple feedwater heaters. Power cycles with cascaded and non-cascaded feedwater heaters are taken into account. In addition to having a predetermined minimum temperature difference, the design criteria proved in this case to have a predetermined area. For simplicity and compactness, the same cycle was used for single and multiple feedwater heaters to demonstrate the validity of the design criteria.
 like Image 6 shown, a simple Rankine cycle implemented in ASPENPLUS was used to explain the importance of the double pinch criterion. Condenser pressure 0.04 bar (bar) exiting the condenser, feedwater flow rate 100 kilograms per second (kg/s) was compressed and boosted to a boiler pressure of 100 bar (bar) before entering the feedwater heater. The condenser pressure below atmospheric pressure implies the need for a degasser, which is constructed here for simplicity, which does not affect the final result and which will be discussed in Chapter 5. The temperature of the feed water increases as thermal energy is transferred to the steam flow through the heater. The feedwater is then heated in a boiler to a fixed output temperature of 500 degrees Celsius before entering the steam expansion line where the steam turbine generates energy. The two steam streams are discharged separately in the steam expansion line. There are two exhaust configurations, cascading and non-cascading. In both configurations, the exhaust steam flow in the feedwater heater 2 is mixed with the main feedwater steam flow in the condenser. in a cascaded structure (in Image 6 Indicated as x) in the feedwater heater 1, the steam flow out of the feedwater heater 1 is mixed with the low-pressure steam flow before entering the feedwater heater 2. In contrast, in a non-cascading structure (in Image 6 marked as o) in , the steam discharged from the feedwater heater 1 goes directly to the condenser. The reported cycle efficiency is the ratio of net generated power (turbine power minus pump power) to boiler heat transfer rate. For the sake of simplicity, the reduction in air pressure is not considered and the turbine is assumed to be irreversible. It should be noted that this unrealistic assumption does not affect the qualitative results, and more realistic situations are investigated in Chapter 5.
 4.1 Single feedwater heater
 Image 6 The regeneration system shown includes two enclosed feedwater heaters. In this section only non-cascaded structures are discussed and only feedwater heater 1 (high pressure) is analyzed. In contrast, feedwater heater 2 (low pressure) is considered fixed as follows:
 P B2 =0.158bar, m · B 2 = 4.73 kg / s , Q · 2 = 9.50 MW
 Feedwater heater 2 was sized to produce a pinch value of 3 degrees Celsius. This arbitrary specification does not affect the specification in this section as it only affects the overall efficiency.
 4.1.1 Minimum temperature difference
 As mentioned earlier, a general simplification method is used for system-level analysis and optimization of pinch analysis. The proposed double pinch criterion is numerically validated by this simplified algorithm. The heat transfer load in feedwater heater 1 is fixed as This value was chosen based on the approximate optimal performance of the cycle. As previously mentioned, this heat transfer duty can be achieved by different combinations of exhaust flow and pressure to produce the desired different minimum internal temperature differences and heat transfer zones. The exhaust steam flow and pressure are respectively discrete into 200 points within the following ranges:
 P B1 ∈[13,15]bar, m · B 1 ∈ [ 22,24 ] kg / s
 Image 6 The flow shown is simulated in ASPENPLUS for each value. results in Figure 7 shown in, Figure 7 Efficiency plots are shown as a function of the two variables. This result is related to the optimization of the objective function and increases with decreasing exhaust pressure and flow. Figure 7 Also shown are optimized constraints, namely the two possible pinch points at the beginning of the condenser and at the feedwater outlet for two given minimum internal temperature differences. These pairs of lines define the feasible region (top right area) with high pressure and/or high flow and the infeasible region, i.e. operations that violate a given minimum internal temperature difference. The paired lines intersect the double pinch operation for a given minimum internal temperature difference, and the union of all intersections is shown. For a given minimum internal temperature difference, the double pinch is at a higher contour than the two pinch lines. Mathematically, it can be expressed as the slope of the objective function on a feasible cone defined by constraints . In other words, for any pinch line, the closer the double pinch point is, the higher the efficiency, and vice versa. Two other notable facts can be drawn from the figure that can be proved analytically: (i) the exhaust steam pressure of the double pinch is the minimum exhaust steam pressure such that the pinch point is at the exhaust steam outlet of the feedwater heater; and (ii) ) Exhaust flow for a double pinch is the minimum flow such that the pinch is at the beginning of the phase change.
 exist Figure 7 , the pinch lines are superimposed: pinch points with minimum internal temperature difference between 0.1°C and 2.5°C at the beginning of the condenser (dashed line labeled pinch p); Pinch for minimum internal temperature difference between 2.5 degrees Celsius (dashed line marked pinch o); double pinch (intersection of two pinch lines) with variable minimum internal temperature difference.
 The maximization of efficiency is equivalent to the overall increase in entropy or the minimization of overall exergy losses in the system. It is well known that the smallest increase in system entropy does not necessarily equate to the combination of smallest increases in entropy of each component. For example, the minimum entropy increase for a feedwater heater favors a zero heat transfer duty, but results in suboptimal overall cycle performance. Nonetheless, the numerical simulation results indicate that for such a simple process, the optimal design and operation of the feedwater heater is consistent with the minimum entropy increase of the feedwater heater for a fixed heat transfer load and minimum internal temperature difference. as Figure 8 As shown, the entropy increase rate with the aforementioned limitation is plotted. The minimum entropy increase is intuitively correct because the double pinch point appears to be the minimum average temperature difference between the material and the steam that forms. However, as figure 2 As shown (intersection of the steam flow curves), there is a trade-off between steam pressure and flow so that the proof of the effectiveness of the entropy increase is no longer difficult. Additionally, embedded in the loop simulation and optimization routine, the minimization of entropy increase inside the loop design is not practical because a constrained optimization problem would be required. For example, the cycle optimization may set/select the heat transfer duty and the entropy increase will be minimized by adjustments to exhaust pressure and flow based on minimum internal temperature differences. Embedded optimization problems are particularly challenging and non-convex problems have not been solved until recently. See A. Mitsos, P. Lemonidis, P.I. Barton, Global solution of bilevel programs with a nonconvex inner program, Journal of Global Optimization 42(4) (2008) 475-513, the contents of which are hereby incorporated by reference in their entirety. In other words, minimizing the increase in entropy is considered more complex than the original system-level optimization problem. On the other hand, the double-pinch approach eliminates the need for a spatially discrete model while eliminating the two free dimensions and satisfying the design constraints of each iteration performed by the optimizer. Finally, in the case of a fixed heat transfer zone, it will be discussed below that minimization of the entropy increase of the feedwater heater is not a very good criterion.
 exist Figure 8 shown in Image 6 Under the flow shown, the regenerative load The entropy increase rate curve of The pinch lines are superimposed: pinch points with minimum internal temperature difference between 0.1°C and 2.5°C at the beginning of the condenser (dashed line labeled pinch p); 0.1°C to 2.5°C at the feedwater heater outlet Pinch for minimum internal temperature difference between 2.5 degrees Celsius (dashed line marked pinch o); double pinch (intersection of two pinch lines) with variable minimum internal temperature difference.
 4.1.2 Fixed area
The results presented in the previous section validate the analytical proof of the simplified way of pinch analysis. This analysis generally ignores the investment costs associated with increasing the heat transfer area in the process of achieving a minimum internal temperature difference between the two locations. In order to address the issue of investment costs, the analysis is now under a given (constant) heat transfer area of the feedwater heater 2 Image 6 For the flow shown, the area is assumed to be 2516 square meters (m2). To demonstrate the generality of the results, regions were selected that were different from the regions corresponding to the minimum internal temperature difference for a given heat exchange load. Similar to the previous analysis, the exhaust flow and pressure are respectively discretized into 200 points within the following ranges:
 P B1 ∈[11,13]bar, m · B 1 ∈ [ 21,23 ] kg / s
 This area is similar but different from the previous one considering the different heat transfer areas/loads. The result is in Figure 9 shown in the efficiency curves drawn in . The efficiency is maximum when the exhaust steam pressure and flow are median. There is an exhaust flow for each exhaust pressure that maximizes cycle efficiency (the curve labeled "Maximum Efficiency Curve"), and these combinations are near-optimal. The difference is 10-5 (10-3 percentage points), a numerical difference that is negligible compared to model and/or numerical imprecision. In mathematical terms, there exists a linear manifold in the optimization variable space with very small directional derivatives. In practical applications, the loop can be optimized even if the pressure cannot be selected to arbitrary precision. This is particularly important, for example, in a possible retrofit of an existing cycle, where it is likely that only the exhaust flow can be changed without changing the exhaust pressure. Small differences in the double pinch set and fully optimal implementation mean that the retrofit measures can yield nearly the same growth results as the optimal design.
 For low exhaust pressure and high exhaust flow (top left), the temperature difference at the beginning of the condenser is much smaller than the temperature difference at the outlet, and vice versa for high exhaust pressure and low exhaust flow (bottom right) . Figure 9 Also shown is a (approximately) double-pinch line (labeled "double-pinch curve") whose efficiency differs from the aforementioned near-optimal efficiency curve by within 10-5 (10-3 percentage points). Inside. This difference is so small that it can be attributed to model/numerical inaccuracy and is insignificant from a practical point of view. It is worth noting that the optimal efficiency tends to achieve a larger pinch point at the outlet than a larger pinch point at the beginning of the condenser. This also explains the possible reason why the current design practice only slightly subcools the drain, which will be discussed in detail below. Nevertheless, the unbalanced temperature difference is still less than the balanced minimum internal temperature difference.
 Figure 10 The entropy increase of feedwater heater 2 is plotted as a function of two exhaust steam variables. and Figure 9 The same curves as in are also superimposed on this graph. Obviously, minimizing the entropy increase of the feedwater heater is not a good criterion for maximizing cycle efficiency. This is in contrast to a pinch analysis for a given heat transfer load. More specifically, the minimum increase in entropy results from low heat transfer duty, ie, low exhaust steam pressure and flow. In other words, minimizing the entropy increase for the feedwater heater ignores the benefit of increasing regeneration before entering the boiler. In contrast, the double pinch criterion described in the present invention is a good criterion for optimizing efficiency.
 exist Figure 9 Efficiency curves for a feedwater heater area fixed to 2516 square meters (m2) are shown in . The "Maximum Efficiency Curve" shows the optimal exhaust flow for a given exhaust pressure, while the "Double Pinch Curve" shows the combination that produces a double pinch. The operating points on the two lines give an efficiency difference of less than 10-5 (10-3 percentage points). The "maximum efficiency on the double pinch curve" intersection point shows the optimal solution, while the "maximum efficiency" intersection point shows the optimal double pinch point location.
 exist Figure 10 The entropy increase rate curve for a feedwater heater area fixed to 2516 square meters (m2) is shown in The "Maximum Efficiency Curve" shows the optimal exhaust flow for a given exhaust pressure, while the "Double Pinch Curve" shows the combination that produces a double pinch. The operating points on the two lines give an efficiency difference of less than 10-5 (10-3 percentage points). The "maximum efficiency on the double pinch curve" intersection point shows the optimal solution, while the "maximum efficiency" intersection point shows the optimal double pinch point location.
 4.2 Non-cascading, cascading and common practice
 A high-efficiency regenerative Rankine cycle has cascading steam flow in the feedwater heater train, where the discharge water from one feedwater heater is mixed with the inlet steam flow of the preceding feedwater heater (the next lower pressure). exist Image 6 , the exhaust steam from feedwater heater 1 is mixed with the feed steam from feedwater heater 2 (curve marked x). The advantage of this arrangement is that the exhaust stream still has a high temperature well above the temperature of the degasser or condenser that follows, so the availability of steam can be used to preheat the feedwater to reduce heating of the preceding feedwater steam flow requirements. Typically, cascaded feedwater heaters are designed and used to achieve minimum internal temperature differentials at the beginning of the condenser and to subcool the exhaust stream somewhat. This appears to reduce the area required for heat transfer without loss of performance as the vapor stream will be used later. However this analysis can be misleading because further subcooling of the steam stream means that the previous feedwater heater needs to preheat the feed to a lower temperature.
 The present invention does not give an analytical proof of the optimality of cascaded double pinch points, but according to Image 6 The procedures presented are verified by numerical simulations against the described criteria. Comparing the efficiency to the pinch analysis, it can be seen that the method clearly favors the proposed double pinch criterion due to the potentially larger thermal transfer area. Therefore, the comparison is made for an overall constant heat transfer area. The following four structures/designs are compared: (i) cascade structure with proposed double pinch criterion; (ii) non-cascaded structure with proposed double pinch criterion; (iii) for feedwater heaters 1 Existing design practice of using a small amount of subcooling at the outlet and using the proposed double pinch standard cascade structure for feedwater heater 2 (low pressure); (iv) using a small amount of subcooling at the outlet of both feedwater heaters. Cascade structures of existing design practices for cold cooling. For each case, a cycle-level optimization was performed by varying the ratio of the heat transfer area between the two feedwater heaters and the exhaust steam pressure. For the double pinch criterion, the feedwater heater is fully determined by equation (10) for a given pressure and heat transfer region. In existing design practice, the temperature at the exhaust steam outlet for a given operating pressure is determined as T sat (P B )-2K; thus the feedwater heater is completely deterministic for a given heat transfer zone. Proving this just requires a simple thought experiment, it should be noted that the inlet temperature of the material is fixed, the inlet and outlet temperatures of the steam flow are also determined, and the material flow is given, so if we choose The material flow can obtain the described heat transfer load and material outlet temperature according to the energy balance. The correlation of heat transfer leads to the calculation of the heat transfer area. In ASPENPLUS, the steam flow rate and heat transfer load of each feedwater heater are realized through design specifications embedded in the optimization method.
 Calculate the heat transfer coefficient according to different conditions, that is, for the gas phase fluid part U=0.709kW/(m 2 K), for the condensation part U=3.975kW/(m 2 K), and for the supercooled part U=1.704kW/(m 2 K), this example is derived from M.M. El-Wakil, Powerplant Technology, International Edition, McGraw-Hill, 1985, the contents of which are hereby incorporated by reference in their entirety. The value of the heat transfer coefficient actually depends on the heater geometry and flow conditions, but is treated as a constant here for brevity of presentation. In addition, the calculations should be performed for a constant overall heat transfer coefficient, but since this yields many very similar qualitative results, it is not shown here for the sake of compactness of presentation.
 Figure 11 The optimal efficiency for each of the four design procedures is plotted as a function of total feedwater heater area. As expected, all four curves are monotonically increasing, illustrating the trade-off between investment cost and efficiency, including asymptotes approaching certain finite values when the heat transfer region is infinite. Besides, as expected, the cascade process with double pinch outperforms the same non-cascade structure. The main finding was that both feedwater heaters using a double pinch cascading configuration significantly outperformed conventional practice. For the values for the larger area of heat transfer area, the structural efficiency improvement is about 2 percentage points compared to the design with a small amount of subcooling at the outlet of the two feedwater heaters, and compared to the low pressure feedwater heater The structural efficiency improvement of using the double pinch standard and a small amount of subcooling at the outlet of the high pressure feedwater heater is about 0.5 percentage points. For values with a smaller area of heat transfer area, the increase in efficiency is not as dramatic but still there. Besides, the non-cascading cycle with double pinch is better than the cascading cycle without double pinch, and is very close to the cascade structure of feedwater heater 2 with double pinch. Finally, as the heat transfer area increases, the performance difference between different design structures also becomes larger, or the design criteria described in the present invention are particularly important for the case where the heat transfer area is large.
 exist Figure 11 , the optimal efficiency of each of the four design procedures is plotted against Image 6 Image of the total area of the feedwater heaters in the flow shown: both feedwater heaters have a double-pinch standard cascade configuration (solid curve, labeled "double-pinch-double-pinch cascade"); Heater 1 uses 2K subcooling and feedwater heater 2 has a double pinch standard cascade configuration (dashed curve, labeled "traditional pinch - double pinch cascade"); both feedwater heaters have double pinch Standard non-cascading structure (x-marked curve, marked as "Double Pinch-Double-pinch non-cascaded"); 2K subcooled cascading structure for both feedwater heaters (triangular marked curve, marked with for "Legacy Grips - Legacy Grips Non-Cascading").
 5. A Numerical Case Study of Realistic Loop Design
 In this chapter, consider a Figure 12 Realistic Rankine cycle shown. The cycle includes four closed feedwater heaters and one open feedwater heater as a deaerator. The four enclosed feedwater heaters are arranged in groups of two, one above and below the deaerator. Each group of enclosed feedwater heaters uses a cascade structure, that is, the drainage of the high-temperature open feedwater heaters is mixed with the exhaust steam entering the open feedwater heaters. The lowest pressure drain is pumped upwards. Table 1 below records the specifications of the cycle. For the sake of brevity, the expansion line is considered to have constant isentropic efficiency.
 In addition, the optimization process is significantly complex.
The optimization criteria proposed by the present invention are compared with existing design practices for micro-subcooling of drainage water. First, for each enclosed feedwater heater (FWH1, 2, 4 & 5), the steam flow was optimized according to a minimum internal temperature difference criterion of 2K, and the drain was subcooled by 2K. The area of each open feedwater heater was then determined and used to optimize the flow of three of the four closed feedwater heaters by the double pinch method of the present invention. Because the drain of the last closed feedwater heater is pumped up by the pump, the double pinch is not optimal and the drain is subcooled by 2K. Furthermore, the vapor stream is a two-phase liquid (non-superheated) so a pinch point occurs at the inlet of the vapor stream. The above results are shown in Table 2. The criteria described in the present invention resulted in a significant efficiency increase of about 0.45 percentage points. It should be noted that this is done entirely by changing the steam pressure and flow, without adding any heat transfer area, adding parts, and changing connections to the flow. In addition to this, the area of each feedwater heater is selected through optimization of conventional design criteria, and features that allow for redistribution of heat transfer area can provide further savings in the criteria described in the present invention.
 Table 1: Example of a 4+1 feedwater heater Figure 12 Specifications for the process shown
 Table 2: Example of a 4+1 feedwater heater Figure 12 Results of the process shown
 6. Summary
 The above discloses a new design standard for the design and operation of feedwater heaters in a regenerative Rankine cycle. The basis is to have the same pinch point at the beginning of the condenser and at the outlet of the vapor stream. A simple structure of the criteria was demonstrated analytically and various other structures were numerically described in example studies (see Table 3). The application of this criterion results in a significant increase in efficiency with a constant heat transfer area. Furthermore, disclosed herein is a process that greatly simplifies the way a regenerative Rankine cycle is designed and optimized. In a pinch analysis, for each feedwater heater, the pinch value and exhaust steam pressure (design variables) are fixed or optimized, and the steam flow rate and heat transfer rate (manipulated variables) are adjusted to achieve double pinch points. In rigorous calculations, the exhaust steam pressure and heat transfer area (design variables) are fixed or optimized, and the steam flow and pinch values are adjusted to achieve double pinch points. The example study demonstrates that under the double pinch criteria proposed by the present invention, the performance of the cycle is not sensitive to design, and that substantial improvements in performance can be achieved only through adjustments to the manipulated variables. If a local solution is used in an optimized way, it increases the probability that the criterion will find a global optimal solution; if a global solution is used, the number of variables and constraints are reduced resulting in a significantly faster CPU time in general . The regenerative Rankine cycle is very commonly used in the industry and new energy systems, so the standards described in the present invention are of great significance to research and development.
 The double pinch criterion can be applied to different systems, such as boilers and heat recovery steam generators, and other situations where the steam exhibits a phase change. The double pinch criterion can be applied in the case of drainage separation in a cascaded or non-cascaded manner.
 Table 3: Applicability of the proposed design criteria under various structures and evidence given in the present invention
 The method detailed above describes a new method of feedwater heater operation that results in optimal operation of the Rankine cycle. For newly built power plants, this mode of operation can easily be applied, and the described mode of operation is also applicable to power plants already in operation. Furthermore, the above discloses a computational method that provides benefits to power plant modeling software for simulation and optimization in the event of errors and failures, accelerated convergence, reduced time and storage space required, by reducing the number of operating variables and actively satisfy constraints to reduce the complexity of the problem and avoid suboptimal local optimal operating conditions. This is also reflected in the tendency to use fewer procedures that require discrete heat exchange and/or number of iterations to meet the proposed criteria. By using different exhaust steam pressure and exhaust steam flow, a significant improvement is achieved compared to the traditional method, with an efficiency improvement of about 0.4% (4 percentage points). Total global electricity consumption in 2008 was 132,000TWh (billion kilowatt hours), of which 105,600TWh (billion kilowatt hours) (80%) was generated by the Rankine power cycle. At 10 cents/kWh (the average electricity price in the US), this results in a global sales price of about $11 trillion. Just a 0.01% improvement in the operation of the Rankine cycle could save a billion dollars a year (a 0.4% improvement would save nearly $40 billion a year). The analysis above attributes all electricity costs to fuel, which is inaccurate, but given that fuel is a very important component, the savings are of the same order of magnitude, in the order of billions of dollars.
 The disclosed operation is based on a double pinch standard for closed feedwater heaters. The steam exiting the turbine is superheated steam or two-phase fluid and exits the closed feedwater heater as a subcooled liquid. The feedwater on the other hand is generally a subcooled liquid. The feedwater heater can be modeled as a counter-flow heat exchanger, whereby there are two possible points on the feedwater heater, i) at the outlet of the steam stream and ii) the condenser the beginning of . For a given heat load in a feedwater heater, the material inlet temperature and flow rate, and the exhaust steam flow rate and pressure can be selected to achieve the same minimum temperature difference at the two possible pinch points. This result has been analytically verified for a simplified design with a preset minimum temperature difference based on the necessary assumptions about the physical properties of the working fluid and the behavior of the expansion line. The criteria for the simplified design and the design with a preset heat exchange area have been numerically demonstrated. It is easy to incorporate double pinch points within the optimization routine, as is the well-known criterion for giving feedwater heaters the same enthalpy increase. In the case of optimization, the pressure analysis as optimization variables calculates the heat load and mass flow rate. Changes in independent variables lead to significantly faster computational performance without introducing local minimum pseudo-solutions. The standard applies to the enclosed feedwater heater in the following configuration:
 Non-cascading steam: The steam at the outlet of a particular feedwater heater is injected directly into the degasser or condenser.
 Cascading steam: The steam at the outlet of a particular feedwater heater is mixed with the steam at the low pressure continuous feedwater heater before entering the latter.
 The respective steam streams of the feedwater heaters can be extracted and combined with the feedwater of the preceding feedwater heater. From a specification of minimum temperature difference, efficiency is independent of the degree of subcooling and the double pinch criterion is an appropriate design criterion. Nevertheless, in order to save heat exchange area, the steam flow can be drawn from the feedwater heater with a few degrees of subcooling.
 For the case where the steam stream at the outlet of the feedwater heater is extracted and mixed with the feedwater at the outlet of the heater, the operation is optimal when this mixing is isothermal. In other words, the temperature of the extracted steam stream exiting the feedwater heater and thus slightly below the original saturation temperature of the steam stream before being extracted is equal to the feedwater temperature. Therefore, the steam (or cascaded steam) exiting a feedwater heater of this configuration is typically a few degrees cooler.
 The criteria disclosed in the present invention are not only beneficial to the recuperation of the Rankine cycle. For example, for non-single or pure substance working fluids, this criterion can make a significant contribution to heat exchange and heat integration. The working fluid such as flue gas, waste heat recovery steam generators in waste heat recovery steam generators, waste heat recovery steam generators based on Rankine cycle, waste heat recovery steam generators using Rankine cycle working fluid multi-stage pressure, etc. working fluid. Chemical plants and chemical processes can also benefit from this thermal integration process.
 In general, the main problem in the existing optimization process of the regenerative Rankine cycle is that there are three variables for each regenerative steam flow, i-exhaust steam pressure ii-exhaust flow iii-in the feed water heating The switching load from vapor to stream in the boiler and a non-convex objective function for this set of variables. Both the analytical validation and the numerical results presented indicate that the conventional practice is suboptimal. This results in inefficient operations and significant losses. In the solution disclosed in the present invention, two of the three variables are eliminated, which greatly simplifies the optimization problem and improves the performance. Another advantage is that the switching load and exhaust flow are both specified by expressions for exhaust pressure, thereby eliminating the need for an expensive spatial distribution model to further simplify the optimization problem.
 m · B = m · F h g , sat ( P B ) - h l ( T F , i + Δ MITA T , P B ) h l ( T sat ( P B ) - Δ MITA T , P F ) - h l ( T F , i , P F )
 Q · = m · B ( h T ( P B ) - h ( T F , i + Δ MITA T , P B ) )
 In some cases, the exhaust pressure may not be accurately selected. For example, for retrofitting of existing units or other technical difficulties (eg extraction steam is only allowed within a fixed interval). As mentioned above, however, even if the exhaust pressure cannot be precisely optimized, near-optimal performance can result in exhaust mass flow rate giving double pinch points. This near-optimal performance is a significant improvement over the performance achieved by existing design practices.
 The method of operation disclosed in the present invention can be used in any instance of operating the regenerative Rankine cycle and its variants such as the Carina cycle or the organic Rankine cycle, as well as in the design and construction of power plants. In some embodiments of the present invention, the disclosed standards can achieve immediate efficiency gains without additional investment costs. The savings in this efficiency improvement are considerable, on the order of at least billions of dollars per year.
 Furthermore, the disclosed standards can be used in chemical plants employing thermal regeneration and thermal integration.
 The disclosed criteria and optimization methods can be used in simulation software to determine optimal operating parameters for a power plant or other system using the Rankine cycle. The criteria can enable the design of the plant to be optimized, while the optimization method can provide a proof of convergence to an optimal solution free of errors and failures.
 The methods, optimization methods and calculation methods disclosed in the present invention can be performed by one or more computer systems. Figure 13 An example structure of a computer system 200 is shown. Although an exemplary computer system 200 is described herein, it should be considered for generality and convenience only. In other embodiments of the present invention, the computer system may differ in structure and operation from the system described herein.
The computer system 200 includes a processor 202 that controls the operation of the computer system 200 by, eg, executing an operating system (OS), device drivers, application programs, and the like. The processor 202 may comprise any type of microprocessor or central processing unit (CPU), including programmable general purpose or special purpose microprocessors and/or any proprietary or commercial single or multiple processing system. The computer system 200 also includes a memory 204 that provides temporary or permanent storage for code executed by the processor 202 or data processed by the processor 202. The memory 204 may include read only memory (ROM), flash memory, one or more random access memory (RAM), and/or a combination of storage technologies. Various elements of the computer system 200 are integrated into a bus system 206 . The bus system 206 is an abstraction representing one or more separate physical buses, communication lines/interfaces, and/or multipoint communication or point-to-point communication connectors through appropriate bridges, adapters, and/or controllers. connect.
 The computer system 200 also includes a network interface 208 , an input/output (I/O) interface 210 , a storage device 212 and a display controller 214 . The network interface 208 enables the computer system 200 to connect to remote devices (eg, other computer systems) over a network. The input/output (I/O) interface 210 facilitates communication connections between one or more input devices, between one or more output devices, and between various other elements of the computer system 200 . The storage device 212 may include any conventional non-volatile and/or non-transitory media for storing data. The storage device 212 can thus hold data and/or instructions on a persistent basis (ie, the values can be retained in the event of a loss of power to the computer system 200). The storage device 212 may include one or more hard drives, flash drives, USB drives, optical drives, various media disks or cards, and/or combinations of the foregoing, and may be connected directly or remotely (eg, via a network) ) to other elements of the computer system 200 . The display controller 214 includes a video processor and a video memory, and generates images for display on one or more displays in accordance with instructions received by the processor 202 .
 Various functions performed by the computer system may be logically described as being performed by one or more modules. It should be understood that such modules may be implemented in hardware, software, or a combination thereof. It should also be understood that when implemented in software, the modules may be part of a single program or one or more separate programs, and may be implemented in a variety of contexts (eg, part of an operating system, a device driver, a stand-alone applications, and/or combinations thereof). Additionally, software including one or more modules may be stored as an executable program on one or more non-transitory computer-readable storage media. The functions disclosed herein to be performed by one particular module may also be performed by any other module or combination of modules, and the computer system may contain more or fewer modules than those described herein.
 In some embodiments of the invention, the computer system includes an input module configured to receive design or operating parameters of one or more power generating devices. Exemplary input parameters may include flow of the main steam flow through the turbine, steam generation temperature, steam generation pressure, and condenser operating pressure. The input module may be further configured to store the received input parameters in memory. The input parameters to the computer system may be ideal operating parameters or models of the power plant, rather than instantaneous values. This reduces convergence time and eliminates the need to obtain a lot of remote sensing and information from the loop and its model.
 The computer system may further include an optimization module configured to calculate the exhaust pressure, exhaust flow and heat exchange area of the feedwater heater used by the power generation system according to the input parameters received by the input module according to the above method. of at least one group. The exhaust steam pressure, exhaust steam flow and heat exchange area are thus calculated by the optimization module to create a pinch point on the feedwater heater.
 The computer system may also include an output module configured to display or present the result of the operation to a user. The output module may also be configured to control the operation of the power generation system to enable the computer system to automatically adjust a number of operating parameters such as exhaust steam pressure and flow and exhaust steam flow to set the power generation system to achieve what the optimization module achieves. Calculated value.
 Figure 14 The flowchart shown is an exemplary method of designing or optimizing a power generation system. Although the methods described herein may be associated with one or more flowcharts, it should be noted that the order of method steps presented in the flowcharts and descriptions is not intended to limit the order in which the methods shown are performed. Rather, the implementation steps of each method described herein may be performed in any order. Furthermore, since the described flow diagram is merely an exemplary embodiment, many methods comprising more or fewer steps than the described method are also considered to be within the scope of the present invention.
 The described method, performed by a computer system in some embodiments, begins at step S1400, where input parameters for one or more power generation systems are received. As described above, exemplary input parameters may include the flow of the main steam flow through the turbine, the steam generation temperature, the steam generation pressure, and the condenser operating pressure. Then the execution program proceeds to step S1402, in which at least the exhaust steam pressure, exhaust steam flow rate and heat exchange area of the feedwater heater used in the power generation system are calculated according to the input parameters received by the input module in step S1400 according to the above method. One group. The exhaust steam pressure, exhaust steam flow rate and heat exchange area are then calculated in step S1402 to generate a pinch point on the feed water heater. Execution then proceeds to step S1404, where an output from the optimization of step S1402 is generated. The output may include displaying or otherwise presenting the result of the operation to the user. This result can be used to design a power generation system that achieves double pinch at the feedwater heater, or manually adjust an existing power generation system to achieve double pinch at its feedwater heater. The output may also include a control signal that activates one or more valves, controllers, or other components of an existing power generation system to cause the power generation system to automatically operate with a double pinch standard on the feedwater heater. Thus, using the method described, the design and operation of the power generation system can be optimized by the double pinch criteria disclosed in the present invention.
 Although the invention has been described with reference to specific embodiments, it should be understood that many changes can be made within the spirit and scope of the invention as described. Therefore, the present invention is not limited to the described embodiments, and the full scope of the present invention is defined by the following claims.
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