Figure 1 Simulation structure diagram for a turbine cold-end system. approximated by the two fourth degree polynomial equations as follows: The turbine "cold-and" system generally consists of turbine exhaust and condenser, and cooling tower (Fig. 1). For a 250 MW unit, the turbine performance data with the maximum steam throttle flow can be NHR NEW NHR Turbine and NHR = −45.19(CP)* + 420(CP)® – 1442(CP)2 + 2248(CP) +6666 (a) NKW = 4,883(CP)* – 44,890(CP)3 + 152,600(CP)2 – 231,500(CP) + 383,400. (b) The condenser and mechanical-draft cooling tower have the performance equations, respectively, CP 1.6302-0.50095 x 10-1 (CWT) OP Coupling 1 Condenser CR WFR CWT Coupling 2 WST Cooling Tower CR System Boundary 2 (The circulating water flow is assumed to flow at the rate of 145,000 gpm.) In addition to the given performance equations give above, we also need two coupling equations to complete the mathematical model for this cooling system. The first equation is the coupling between the condenser and the cooling tower. The heat load for these two must be the same. The coupling equation is CR-2000 (HL) (WFR) (e) The second equation is the coupling between the steam turbine and the condenser. When minor heat losses from the turbine system are neglected, the turbine waste heat rejection must take place entirely in the condenser. That is, the waste heat rejection is also the amount of heat removed from the condenser. In equation form, this can be expressed as HL= (NHR-3412)(NKW) 10% The four performance equations and two coupling equations constitute the mathematical model for the turbine "cold-end" system. There are six equations for six unknowns CP, CR, CWT,NHR,NKW, and HL. Since the system is not of sequential type, one must solve these equations simultaneously, employing, for example, the root-finding methods discussed in class, Use the approach based on the Newton-Raphson method to estimate the following quantities: +0.55796 × 10-³ (CWT)² + 0.32946 × 10-³ (HL) -0.10229 × 10¯*(HL)(CWT) + 0.16253 x 10-6 x (HL)(CWT)²+0.42658 x 10-6(HL) -0.92331 x 10-8 (HL)²(CWT) +0.71265 × 10-10 x(HL)²(CWT)² for WFR = 145,000 GPM (c) CWT = -0.10046 × 10² + 0.22801 x 10-³ (WFR) +0.85396(CR) +0.18617 x 10-5 (CR)(WFR) +0.10957 x 10(WBT) -0.22425 x 10-5 (WBT)(WFR) -0.11978 x 10-1(WBT)(CR) +0.14378 x 107(WBT)(CR)(WFR). (d) (1) Heat load (2) Condenser pressure (3) Turbine net heat rate (4) Turbine net output (5) Tower approach, and (6) Tower cooling range under various ambient wet-bulb temperatures. Nomenclature: NKW turbine net output NHR = turbine net heat rate CP = condenser pressure CWT cold water temperature HL = heat load WBT = ambient wet-bulb temperature RH ambient relative humidity CR = cooling range WFR = cooling water flow rate

Figure 1 Simulation structure diagram for a turbine cold-end system. approximated by the two fourth degree polynomial equations as follows: The turbine "cold-and" system generally consists of turbine exhaust and condenser, and cooling tower (Fig. 1). For a 250 MW unit, the turbine performance data with the maximum steam throttle flow can be NHR NEW NHR Turbine and NHR = −45.19(CP)* + 420(CP)® – 1442(CP)2 + 2248(CP) +6666 (a) NKW = 4,883(CP)* – 44,890(CP)3 + 152,600(CP)2 – 231,500(CP) + 383,400. (b) The condenser and mechanical-draft cooling tower have the performance equations, respectively, CP 1.6302-0.50095 x 10-1 (CWT) OP Coupling 1 Condenser CR WFR CWT Coupling 2 WST Cooling Tower CR System Boundary 2 (The circulating water flow is assumed to flow at the rate of 145,000 gpm.) In addition to the given performance equations give above, we also need two coupling equations to complete the mathematical model for this cooling system. The first equation is the coupling between the condenser and the cooling tower. The heat load for these two must be the same. The coupling equation is CR-2000 (HL) (WFR) (e) The second equation is the coupling between the steam turbine and the condenser. When minor heat losses from the turbine system are neglected, the turbine waste heat rejection must take place entirely in the condenser. That is, the waste heat rejection is also the amount of heat removed from the condenser. In equation form, this can be expressed as HL= (NHR-3412)(NKW) 10% The four performance equations and two coupling equations constitute the mathematical model for the turbine "cold-end" system. There are six equations for six unknowns CP, CR, CWT,NHR,NKW, and HL. Since the system is not of sequential type, one must solve these equations simultaneously, employing, for example, the root-finding methods discussed in class, Use the approach based on the Newton-Raphson method to estimate the following quantities: +0.55796 × 10-³ (CWT)² + 0.32946 × 10-³ (HL) -0.10229 × 10¯*(HL)(CWT) + 0.16253 x 10-6 x (HL)(CWT)²+0.42658 x 10-6(HL) -0.92331 x 10-8 (HL)²(CWT) +0.71265 × 10-10 x(HL)²(CWT)² for WFR = 145,000 GPM (c) CWT = -0.10046 × 10² + 0.22801 x 10-³ (WFR) +0.85396(CR) +0.18617 x 10-5 (CR)(WFR) +0.10957 x 10(WBT) -0.22425 x 10-5 (WBT)(WFR) -0.11978 x 10-1(WBT)(CR) +0.14378 x 107(WBT)(CR)(WFR). (d) (1) Heat load (2) Condenser pressure (3) Turbine net heat rate (4) Turbine net output (5) Tower approach, and (6) Tower cooling range under various ambient wet-bulb temperatures. Nomenclature: NKW turbine net output NHR = turbine net heat rate CP = condenser pressure CWT cold water temperature HL = heat load WBT = ambient wet-bulb temperature RH ambient relative humidity CR = cooling range WFR = cooling water flow rate

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.1: Systems Of Equations

Problem 25E

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