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Approximation Error RACKETT EQUATION RADAR RADIAL COMPRESSOR RADIAL ENERGY FLOWS RADIAL FANS RADIAL GAP SIZE RADIATION RADIATION ABSORPTION METHOD RADIATION BETWEEN PARALLEL PLATES RADIATION DIFFUSION APPROXIMATION, FOR COMBINED RADIATION AND CONDUCTION RADIATION DOSIMETRY RADIATION DRYING Radiation from semi-transparent oxide particles in thermal spraying Radiation heat transfer in a supersonic nozzle of a solid-propellant rocket engine Radiation heat transfer in solid-propellant rocket engines RADIATION IN ENCLOSURES Radiation in nanomanufacturing Radiation in production of carbon fibers Radiation of an isothermal plane-parallel layer Radiation of isothermal volumes of scattering medium: An error of the diffusion model Radiation of nonisothermal layer of scattering medium RADIATION SHIELDS RADIATION TO FURNACE TUBES Radiation transfer between the surfaces through a non-participating medium Radiation transfer in emitting, absorbing and scattering media Radiation transfer in combustion chambers Radiation transfer problems in nature and engineering Radiation transfer theory and the computational methods Radiation-turbulence interaction Radiative boundary layer Radiative cooling and solidification of core melt droplets Radiative cooling of particle flow in vacuum RADIATIVE DIFFUSION Radiative effects in semi-transparent liquid containing gas bubbles Radiative equilibrium in plane-parallel layer RADIATIVE EXCHANGE Radiative exchange between an isothermal gas and surrounding walls RADIATIVE HEAT FLUX Radiative heat transfer Radiative heat transfer in moving media RADIATIVE HEAT TRANSFER, IN POROUS MEDIA Radiative properties of gas bubbles in semi-transparent medium Radiative properties of metal particles in infrared and microwave spectral ranges Radiative properties of micro- and nanostructures Radiative properties of particles and fibers (theoretical analysis) Radiative properties of polydisperse systems of independent particles Radiative properties of semi-transparent fibers at arbitrary illumination Radiative properties of semi-transparent particles Radiative properties of single particles and fibers: the hypothesis of independent scattering and the Mie theory Radiative properties of soot particles Radiative properties of water droplets in near infrared RADIATIVE SPECTRAL INTENSITY Radiative transfer equation Radiative transfer equation: a general formulation Radiative transfer for coupled atmosphere and ocean systems Radiative transfer in combustion phenomena affected by radiation Radiative transfer in combustion systems Radiative Transfer in Coupled Atmosphere and Ocean Systems: Impact of Surface Roughness on Remotely Sensed Radiances Radiative transfer in coupled atmosphere and ocean systems: the adding and doubling method Radiative Transfer in Coupled Atmosphere and Ocean Systems: the Discrete Ordinate Method RADIATIVE TRANSFER IN COUPLED ATMOSPHERE AND OCEAN SYSTEMS: THE SUCCESSIVE ORDER OF SCATTERING METHOD Radiative transfer in glass production Radiative transfer in laminar flames Radiative transfer in laser processing Radiative transfer in medical laser treatment RADIATIVE TRANSFER IN MULTIDIMENSIONAL PROBLEMS: A COMBINED COMPUTATIONAL MODEL Radiative transfer in space applications Radiative transfer in the atmosphere Radiative transfer in turbulent flames Radiative transfer in two-phase combustion Radiative-conductive heat transfer in dispersed materials Radiative-conductive heat transfer in foam insulations RADIO FREQUENCY HEATING RADIO FREQUENCY, RF, DRYING RADIO WAVES RADIOACTIVE DECAY RADIUM RADIUS, HYDRAULIC RADON RAE RAFFINATE PHASE RAINBOW VOLUMIC VELOCIMETRY RAINFALL RAMAN SPECTROSCOPY RAMJET ENGINES RANDOM PROCESSES RANKINE CYCLE RANKINE DEGREE RANKINE VORTEX RANKINE, WJM RAOULT'S AND DALTON'S LAW RAOULT'S LAW RAREFACTION RAREFACTION WAVE RAREFIED GAS DYNAMICS RATE-CONTROLLED CONSTRAINED EQUILIBRIUM Ray effects and false scattering Ray optics and wave effects in radiation propagation Rayleigh equation, for bubble growth Rayleigh equation, for droplet formation Rayleigh formula Rayleigh law of scattering Rayleigh number Rayleigh scattering Rayleigh, Lord (1842-1919) Rayleigh-Gans scattering Rayleigh-Taylor instability REACTING GAS FLOW REACTION TURBINES REACTIVE CONTAMINANT TRANSPORT REACTOR PHYSICS Real gaseous spectra REATTACHMENT REATTACHMENT, OF BOUNDARY LAYER Reaumur Degree REBOILERS RECIPROCATING COMPRESSOR RECIRCULATION RECIRCULATION BOILERS RECONSTRUCTED WAVEFRONTS RECOVERY COEFFICIENT RECOVERY TEMPERATURE RECTANGULAR CHANNEL RECTANGULAR CYLINDERS RECTANGULAR DUCTS RECTANGULAR STENOTIC MODELS RECUPERATIVE HEAT EXCHANGERS REDLICH-KWONG EQUATION REDOX REACTIONS REDUCED GRAVITY CONDITIONS REDUCED INSTRUCTION SET COMPUTER, RISC REDUCED PROPERTIES REFINING REFLECTANCE REFLECTION COEFFICIENT (REFLECTANCE) REFLECTION COEFFICIENTS FOR EARTH'S SURFACE REFLECTIVITY REFLOOD REFLUX CONDENSATION REFLUX CONDENSER REFLUX RATIOS REFORMING REFRACTION REFRACTIVE INDEX REFRACTIVE INDICES FOR GASES AND LIQUIDS REFRACTORY MATERIALS, FOR ELECTRIC FURNACES REFRIGERANTS REFRIGERATION REGENERATIVE BURNER REGENERATIVE FEED HEATING REGENERATIVE GAS TURBINE REGENERATIVE HEAT EXCHANGERS REGULAR REGIME OF DRYING REHEATING REICHARDT'S FORMULA, FOR VELOCITY DISTRIBUTION IN TUBES REIMANN'S INTEGRAL REINER-RIVLIN FLUID RELATIVE HUMIDITY RELATIVE MOLAR MASS RELATIVE PERMEABILITY RELATIVE POWER DEMAND, RPD RELATIVE ROUGHNESS RELAXATION TIME RENEWABLE ENERGY RENEWABLE ENERGY SOURCES RESIDUAL ENTHALPY RESIDUAL GIBBS ENERGY RESINS RESISTANCE HEATING RESISTANCE THERMOMETERS RESISTANCE THERMOMETRY RESISTANCE, ELECTRICAL RESISTIVITY, ELECTRICAL RESONANCE FLUORESCENCE RETENTATE RETENTION INDEX RETROGRADE CONDENSATION RETURN TO NUCLEATE BOILING REVERSE OSMOSIS REVERSED HEAT ENGINE CYCLES REVERSIBILITY PRINCIPLE REVERSIBLE PROCESSES REWETTING REWETTING OF HOT SURFACES REYNOLDS ANALOGY Reynolds Number REYNOLDS NUMBER, CRITICAL, IN TUBES REYNOLDS STRESS REYNOLDS STRESS TRANSPORT MODELS REYNOLDS' AVERAGING REYNOLDS' EQUATIONS REYNOLDS, OSBORNE (1842-1912) RHEOLOGY RHEOMETERS RHEOPEPTIC FLUIDS RHODAMINE RICCATTY-BESSEL FUNCTIONS RICHARDSON NUMBER RIDEAL-ELEY MODEL, FOR HETEROGENEOUS CATALYSIS RIEDEL-PLANK-MILLER EQUATION RIEMAN WAVES RIGHT-ANGLE TRIANGULAR ENCLOSURE RIGID-WALLED CHANNEL RISC. REDUCED INSTRUCTION SET COMPUTER RISK ANALYSIS TECHNIQUES RISK ASSESSMENT ROASTING ROCKET PROPELLANTS ROCKETS ROD BAFFLES ROD BUNDLE TESTS ROD BUNDLES, FLOW IN ROD BUNDLES, HEAT TRANSFER IN ROD BUNDLES, PARALLEL FLOW IN ROD CLIMBING ROD-STABILIZED LAMINAR PREMIXED FLAME RODRIGUES FORMULA ROHRSCHNEIDER CONSTANT ROLL MOMENT ROLL WAVES ROOTS TYPE COMPRESSOR ROSIN-RAMMLER ROSIN-RAMMLER SIZE DISTRIBUTION ROSSBY NUMBER ROSSELAND COEFFICIENT ROTAMETERS ROTARY ATOMIZERS ROTARY DRYERS ROTARY KILNS ROTARY REGENERATORS ROTATED TUBE BANKS ROTATING CHANNEL WITH RIBS ROTATING CYLINDERS, CRITICAL SPEED ROTATING CYLINDERS, FLOW BETWEEN ROTATING CYLINDERS, FLOW OVER ROTATING DISC CONTACTOR ROTATING DISC SYSTEMS, APPLICATIONS ROTATING DISC SYSTEMS, BASIC PHENOMENA ROTATING DUCT SYSTEMS, ORTHOGONAL, HEAT TRANSFER IN ROTATING DUCT SYSTEMS, PARALLEL, HEAT TRANSFER IN ROTATING FLOW IN A POROUS LAYER ROTATING FLOW PASSAGE ROTATING PIPE FLOW ROTATING SURFACES ROTATIONAL DISCONTINUITIES Rotational Rayleigh number ROTATIONAL REYNOLDS NUMBERS ROUGH CHANNELS, FRICTION FACTOR IN ROUGH SURFACE FRICTION FACTORS ROUGH SURFACES ROUGH TUBES ROUGH TUBES, FLOW IN ROUGH TUBES, HEAT TRANSFER IN ROUGHNESS FACTORS ROYAL ACADEMY OF ENGINEERING, RAE ROYAL SOCIETY OF CHEMISTRY ROYAL SOCIETY, RS RS RSC RUBBER RUMFORD, COUNT, BENJAMIN THOMPSON RUSHTON TURBINE
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RANKINE CYCLE

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Basic Cycle

The Rankine cycle is the fundamental operating cycle of all power plants where an operating fluid is continuously evaporated and condensed. The selection of operating fluid depends mainly on the available temperature range. Figure 1 shows the idealized Rankine cycle.

The pressure-enthalpy (p-h) and temperature-entropy (T-s) diagrams of this cycle are given in Figure 2. The Rankine cycle operates in the following steps:

  • 1-2-3 Isobaric Heat Transfer. High pressure liquid enters the boiler from the feed pump (1) and is heated to the saturation temperature (2). Further addition of energy causes evaporation of the liquid until it is fully converted to saturated steam (3).

  • 3-4 Isentropic Expansion. The vapor is expanded in the turbine, thus producing work which may be converted to electricity. In practice, the expansion is limited by the temperature of the cooling medium and by the erosion of the turbine blades by liquid entrainment in the vapor stream as the process moves further into the two-phase region. Exit vapor qualities should be greater than 90%.

  • 4-5 Isobaric Heat Rejection. The vapor-liquid mixture leaving the turbine (4) is condensed at low pressure, usually in a surface condenser using cooling water. In well designed and maintained condensers, the pressure of the vapor is well below atmospheric pressure, approaching the saturation pressure of the operating fluid at the cooling water temperature.

  • 5-1 Isentropic Compression. The pressure of the condensate is raised in the feed pump. Because of the low specific volume of liquids, the pump work is relatively small and often neglected in thermodynamic calculations.

Rankine cycle.

Figure 1. Rankine cycle.

T-s and p-h diagrams.

Figure 2. T-s and p-h diagrams.

The efficiency of power cycles is defined as

(1)

Values of heat and work can be determined by applying the First Law of Thermodynamics to each step. The steam quality x at the turbine outlet is determined from the assumption of isentropic expansion, i.e.,

(2)

where is the entropy of vapor and Si* the entropy of liquid.

Inefficiencies of Real Rankine Cycles

The efficiency of the ideal Rankine cycle as described in the previous section is close to the Carnot efficiency (see Carnot Cycle). In real plants, each stage of the Rankine cycle is associated with irreversible processes, reducing the overall efficiency. Turbine and pump irreversibilities can be included in the calculation of the overall cycle efficiency by defining a turbine efficiency according to Figure 3

(3)

where subscript act indicates actual values and subscript is indicates isentropic values and a pump efficiency

(4)
Turbine efficiency.

Figure 3. Turbine efficiency.

If ηt and ηp are known, the actual enthalpy after the compression and expansion steps can be determined from the values for the isentropic processes. The turbine efficiency directly reduces the work produced in the turbine and, therefore the overall efficiency. The inefficiency of the pump increases the enthalpy of the liquid leaving the pump and, therefore, reduces the amount of energy required to evaporate the liquid. However, the energy to drive the pump is usually more expensive than the energy to feed the boiler.

Rankine cycle with vapor superheating.

Figure 4. Rankine cycle with vapor superheating.

Even the most sophisticated boilers transform only 40% of the fuel energy into useable steam energy. There are two main reasons for this wastage:

  • The combustion gas temperatures are between 1000°C and 2000°C, which is considerably higher than the highest vapor temperatures. The transfer of heat across a large temperature difference increases the entropy.

  • Combustion (oxidation) at technically feasible temperatures is highly irreversible.

Since the heat transfer surface in the condenser has a finite value, the condensation will occur at a temperature higher than the temperature of the cooling medium. Again, heat transfer occurs across a temperature difference, causing the generation of entropy. The deposition of dirt in condensers during operation with cooling water reduces the efficiency.

Increasing the Efficiency of Rankine Cycles

Pressure difference

The net work produced in the Rankine cycle is represented by the area of the cycle process in Figure 2. Obviously, this area can be increased by increasing the pressure in the boiler and reducing the pressure in the condenser.

Regenerative feed liquid heating.

Figure 5. Regenerative feed liquid heating.

Superheating and reheating

The irreversibility of any process is reduced if it is performed as close as possible to the temperatures of the high temperature and low temperature reservoirs. This is achieved by operating the condenser at subatmospheric pressure. The temperature in the boiler is limited by the saturation pressure. Further increase in temperature is possible by superheating the saturated vapor, see Figure 4.

This has the additional advantage that the vapor quality after the turbine is increased and, therefore the erosion of the turbine blades is reduced. It is quite common to reheat the vapor after expansion in the high pressure turbine and expand the reheated vapor in a second, low pressure turbine.

Feed water preheating

The cold liquid leaving the feed pump is mixed with the saturated liquid in the boiler and/or re-heated to the boiling temperature. The resulting irreversibility reduces the efficiency of the boiler. According to the Carnot process, the highest efficiency is reached if heat transfer occurs isothermally. To preheat the feed liquid to its saturation temperature, bleed vapor from various positions of the turbine is passed through external heat exchangers (regenerators), as shown in Figure 5.

Ideally, the temperature of the bleed steam should be as close as possible to the temperature of the feed liquid.

Combined cycles

The high combustion temperature of the fuel is better utilized if a gas turbine or Brayton engine is used as "topping cycle" in conjunction with a Rankine cycle. In this case, the hot gas leaving the turbine is used to provide the energy input to the boiler. In co-generation systems, the energy rejected by the Rankine cycle is used for space heating, process steam or other low temperature applications.

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