Power and Refrigeration Cycles: Method

Power and Refrigeration Cycles: the Cycles, the Property Models and How the Results Are Computed

What the tool does

The tool runs one of five idealised cycles state by state: the Brayton gas-turbine cycle, the Otto spark-ignition cycle, the Diesel compression-ignition cycle, the Rankine steam-power cycle and the vapour-compression refrigeration cycle. For each it tabulates every state (pressure, temperature, specific volume, enthalpy, entropy and, where a two-phase state occurs, the quality), writes each process equation with the substituted numbers, and reports the specific work and heat of every process, the net work, the thermal efficiency or coefficient of performance, the back-work ratio, the mass flow that delivers an entered net power or cooling capacity, and the Carnot bound between the cycle's extreme temperatures. The T-s diagram of the computed case is drawn (with the saturation dome for steam and refrigerants) and, for the gas cycles, the p-v diagram as well. The gas cycles are computed with variable specific heats and the cold-air-standard constant-k closed forms of the textbooks are printed beside them, so the reader sees both the handbook value and the better one.

Conventions

All work and heat quantities are per unit mass of working fluid and positive in the direction named (compressor work is work done on the fluid, turbine work is work done by it, heat added is heat received, heat rejected is heat given up). Temperatures are entered as plain numbers (°C in SI mode, °F in USCS mode; the unit switch does not convert them) and the state tables show them both in that unit and as absolute temperatures (K or °R), which are what the equations use. Pressures, powers and mass flows go through the site's unit dictionary; specific enthalpies, entropies and volumes are plain numbers in kJ/kg, kJ/(kg·K) and m³/kg (SI) or Btu/lb, Btu/(lb·°R) and ft³/lb (USCS), with 1 Btu/lb = 2,326 J/kg and 1 Btu/(lb·°R) = 4,186.8 J/(kg·K). State numbering follows the usual textbook order (1 the start of compression, 2 its end, 3 the start of expansion, 4 its end), with the isentropic end states marked 2s and 4s when a component efficiency moves the real state away from them.

Ideal-gas properties with variable specific heats

The gas cycles treat the working fluid as an ideal gas, p v = R T, whose specific heat depends on temperature. The dependence is taken from the NASA Glenn seven-coefficient fits of McBride, Zehe and Gordon (NASA/TP-2002-211556, 2002), the data file of NASA's chemical-equilibrium program CEA, for air (the NASA "Air" reactant record, 78.084 % N₂, 20.9476 % O₂, 0.9365 % Ar, 0.0319 % CO₂ by mole), nitrogen, oxygen, argon, carbon dioxide and water vapour. In each of two temperature intervals (200–1,000 K and 1,000–6,000 K) the dimensionless molar heat capacity, enthalpy and entropy are

c_p/R = a₁ T⁻² + a₂ T⁻¹ + a₃ + a₄ T + a₅ T² + a₆ T³ + a₇ T⁴,

H/(RT) = −a₁ T⁻² + a₂ (ln T)/T + a₃ + a₄ T/2 + a₅ T²/3 + a₆ T³/4 + a₇ T⁴/5 + b₁/T,

S/R = −a₁ T⁻²/2 − a₂ T⁻¹ + a₃ ln T + a₄ T + a₅ T²/2 + a₆ T³/3 + a₇ T⁴/4 + b₂,

with R = 8.314510 J/(mol·K) as the report uses and the entropy at the standard-state pressure of 1 bar. The tool divides by the molar mass of the record, refers the enthalpy to 298.15 K (h = 0 there; only differences enter a cycle), takes u = h − RT and s(T, p) = s°(T) − R ln(p/1 bar). The fits reproduce the report's Table B1 values at 298.15 K to the printed digits and the NIST-JANAF values at 1,000 and 2,000 K within 0.15 % for air, nitrogen, oxygen, argon and carbon dioxide; the water-vapour record (Woolley's 1987 partition function) lies 1.1 % above the 1998 JANAF heat capacity at 2,000 K, a known difference between the two compilations. Below 200 K or above 6,000 K the nearest interval is extrapolated and the state is flagged.

An isentropic process between two pressures is solved from s°(T₂) = s°(T₁) + R ln(p₂/p₁), which is the relative-pressure method of the gas tables (p_r = exp(s°/R)) without the table; between two specific volumes it is solved from s(T₂, v₂) = s(T₁, v₁), the relative-volume method. Both are monotone in T and are inverted by bisection to 10⁻¹⁰. A constant-volume heat addition uses u, a constant-pressure one uses h, and a temperature given an enthalpy or internal energy is found by the same bisection.

Cold-air standard beside

The cold-air-standard assumption takes c_p constant at its room-temperature value, so that c_p = kR/(k − 1) with the entered k (1.4 for air). The tool runs the same cycle code with that constant-c_p model and the same component efficiencies, which is what a textbook example with the "cold-air" tables gives, and prints the ideal closed forms as well: for the Otto cycle η = 1 − r^1−k, for the Diesel cycle η = 1 − r^1−k(r_c^k − 1)/[k(r_c − 1)], for the simple Brayton cycle η = 1 − r_p^(1−k)/k, and the Brayton pressure ratio of largest net work between fixed T₁ and T₃, r_p,opt = (T₃/T₁)^k/(2(k−1)). Because c_p of a real gas rises with temperature, the variable-c_p efficiency is a few points below the cold-air value for the same inputs (the Otto cycle at r = 8 with 1,000 kJ/kg: 51.5 % against 56.5 %) and the peak temperatures are lower; the variable-c_p numbers are the ones to quote.

Brayton cycle

Stations: 1 compressor inlet, 2 compressor exit, 3 turbine inlet, 4 turbine exit. The compressor raises the pressure by the ratio r_p; its isentropic exit state 2s follows from the entropy relation and the real exit from the isentropic efficiency, h₂ = h₁ + (h_2s − h₁)/η_c. Heat is added at constant pressure to the turbine-inlet temperature T₃; the turbine expands to p₁ with h₄ = h₃ − η_t(h₃ − h_4s); the exhaust is cooled at constant pressure back to state 1 (an open cycle rejects it to the atmosphere, which is the same energy balance). Then w_c = h₂ − h₁, w_t = h₃ − h₄, q_in = h₃ − h₂, q_out = h₄ − h₁, w_net = w_t − w_c, η = w_net/q_in and the back-work ratio w_c/w_t, typically 40–60 % in a gas turbine.

A regenerator of effectiveness ε heats the compressed air with the exhaust: h₅ = h₂ + ε(h₄ − h₂) on the cold side, h₆ = h₄ − (h₅ − h₂) on the hot side, and the combustor then supplies only h₃ − h₅. A regenerator helps only while the exhaust is hotter than the compressor delivery (low pressure ratio, high T₃); when T₄ ≤ T₂ the tool ignores it and says so. Intercooling splits the compression into two stages of equal pressure ratio √r_p with cooling back to T₁ between them, which lowers the compressor work; reheat splits the expansion likewise with reheating to T₃, which raises the turbine work. Each stage keeps the entered efficiency. On their own, intercooling and reheat lower the efficiency of a simple cycle (the added heat enters at lower temperatures) while raising the net work; with a regenerator they raise both, which is why the three appear together. With the mass flow for the entered net power the tool also prints the heat-addition rate.

Otto cycle

States: 1 bottom dead centre at the intake temperature and pressure, 2 top dead centre after the isentropic compression v₂ = v₁/r, 3 the end of the constant-volume heat addition (the peak pressure and temperature), 4 bottom dead centre after the isentropic expansion back to v₁. The heat addition is specified either as q_in per unit mass (then u₃ = u₂ + q_in is solved for T₃) or by the peak temperature (then q_in = u₃ − u₂). The pressures follow the gas law along the constant-volume lines, p₃ = p₂ T₃/T₂. The heat rejected is u₄ − u₁, the net work q_in − q_out, the efficiency w_net/q_in and the mean effective pressure MEP = w_net/(v₁ − v₂), the constant pressure that would do the cycle's net work over the piston displacement. The compression and expansion works u₂ − u₁ and u₃ − u₄ give the back-work ratio.

Diesel cycle

The compression is the same; the heat is added at constant pressure from top dead centre to the cutoff, v₃ = r_c v₂, so that T₃ = r_c T₂ by the gas law and q_in = h₃ − h₂; the expansion is isentropic from v₃ to v₁ and the rejection at constant volume. The heat addition may be fixed by the cutoff ratio, by q_in (h₃ = h₂ + q_in solved for T₃, then r_c = T₃/T₂) or by the peak temperature. The cutoff ratio must lie between 1 and the compression ratio; above it the heat addition would run past bottom dead centre, and the tool refuses the case. The expansion work counts the constant-pressure part p₂(v₃ − v₂) as well as u₃ − u₄. At the same compression ratio the Diesel cycle is less efficient than the Otto cycle (its heat enters at a lower mean temperature) but a Diesel engine runs at a far higher compression ratio, which more than makes up for it.

Rankine cycle

States: 1 saturated liquid leaving the condenser at p_cond, 2 the pump exit at the boiler pressure, 3 the turbine inlet (saturated or superheated vapour at p_boil), 4 the turbine exhaust at p_cond. The pump work is v₁(p_boil − p_cond)/η_p (the liquid is nearly incompressible), the turbine work h₃ − h₄ with h₄ = h₃ − η_t(h₃ − h_4s) and h_4s at the inlet entropy, which usually lies inside the dome so that 4s is a wet mixture of quality x_4s = (s₃ − s_f)/(s_g − s_f). The heat added is h₃ − h₂, the heat rejected h₄ − h₁. With reheat the high-pressure turbine expands to the reheat pressure, the steam returns to the boiler and is reheated (to the inlet temperature unless another is entered), and the low-pressure turbine expands to the condenser; the heat added gains the reheat term and the exhaust quality rises, which is the purpose of reheat: wet steam in the last stages erodes the blades, and a quality below about 0.88 is flagged. With one open feedwater heater a fraction y of the steam is extracted at the heater pressure and mixed with the condensate pumped to that pressure; y follows from the heater's energy balance, y h_ext + (1 − y) h_{pumped condensate} = h_f(p_fwh), the mixture leaves as saturated liquid and a second pump raises it to the boiler pressure. The work and heat are then per kilogram entering the boiler, the condenser and low-pressure turbine terms being weighted by (1 − y). The thermal efficiency is w_net/q_in and the back-work ratio w_p/w_t, a few per cent for a steam plant.

Steam properties come from the IAPWS Industrial Formulation 1997 (regions 1, 2 and 4 of the shared fluid-property library of this site): the compressed-liquid, superheated-vapour and saturation equations, with the quality in the wet region from s or h. The reference state is that of IF97 (saturated liquid at the triple point), which is also the reference of the steam tables in the textbooks, so the tabulated h and s may be compared directly with them.

Vapour-compression refrigeration

States: 1 vapour leaving the evaporator and entering the compressor, superheated by ΔT_sh above the evaporator saturation temperature (0 for saturated vapour); 2 the compressor discharge at the condenser pressure, with h₂ = h₁ + (h_2s − h₁)/η_c and 2s at the inlet entropy; 3 the liquid leaving the condenser, subcooled by ΔT_sc below the condenser saturation temperature (0 for saturated liquid; the enthalpy of a subcooled liquid is taken at the saturated-liquid value of its temperature, the pressure correction being negligible); 4 the state after the expansion valve, at the evaporator pressure with h₄ = h₃ (throttling), a wet mixture whose quality is the flash-gas fraction. The refrigerating effect is q_L = h₁ − h₄, the compressor work w_c = h₂ − h₁, the condenser heat q_H = h₂ − h₃ = q_L + w_c, the coefficient of performance COP_R = q_L/w_c (as a heat pump, COP_HP = q_H/w_c = COP_R + 1), and the mass flow for an entered cooling capacity is Q̇_L/q_L, which gives the compressor power and the condenser load. The Carnot bound is T_L/(T_H − T_L) between the two saturation temperatures. The evaporator and condenser may be fixed by their saturation temperatures or by their pressures. Refrigerant properties (R-134a, ammonia, carbon dioxide, propane) come from the NIST Chemistry WebBook tables held in the shared fluid-property library, saturation line and superheated-vapour isobars, interpolated; their reference state is the IIR convention (h = 200 kJ/kg, s = 1 kJ/(kg·K) for the saturated liquid at 0 °C), which is also that of most refrigerant tables.

Carnot bound and energy balance

For every power cycle the tool prints the Carnot efficiency between the lowest and highest cycle temperatures, 1 − T_min/T_max, and the fraction of it the cycle reaches; for the refrigeration cycle the reversed-Carnot COP. No cycle can exceed these and the tool flags any input set that would. The first-law balance q_in − q_out − w_net is printed with the heat rejected; it is zero to round-off for every case, because the same property functions evaluate every term.

Reading the diagrams

The T-s diagram shows the cycle as a closed path: constant-pressure and constant-volume processes are drawn along their property lines, isentropic processes as vertical lines, and the irreversible compressions and expansions as straight segments from the inlet state to the real exit state, with the isentropic end states marked. The area enclosed on a T-s diagram of a reversible cycle is the net work; with irreversible components the path is only a guide to the eye. On the p-v diagram of a gas cycle the enclosed area is the net work per unit mass and the width at the base is the displacement v₁ − v₂, whose ratio is the mean effective pressure. For the steam and refrigerant cycles the saturation dome is drawn from the property library so the wet states can be seen inside it.

Limits

The gas cycles are air-standard cycles: the working fluid is a fixed ideal gas, there is no combustion chemistry, no change of composition or mass, no pressure loss in the heaters and no heat loss from the components; the intake and exhaust strokes of the piston engines are not modelled, and the mean effective pressure is the indicated value of the ideal cycle. The Rankine cycle has no pressure drops in the boiler, condenser or piping, one reheat and one open feedwater heater at most, and no closed heaters, moisture separators or deaerator steam. The refrigeration cycle has no pressure drops, no suction-line heat exchanger and no compressor heat loss. A real plant's efficiency is below the value printed here by the losses of these components; the cycle analysis fixes the ideal against which they are measured.