--- title: Thermal Machines module: Thermodynamics moduleNumber: 8 lessonNumber: 6 order: 806 summary: > An engine, a refrigerator, and a heat pump are one machine read three ways: each shuttles heat between a hot and a cold reservoir while trading work at the boundary, and only the flow you call useful separates them. A heat engine turns part of $Q_h$ into work, $W=Q_h-Q_c$; a refrigerator spends work to pull $Q_c$ from the cold side; a heat pump counts the warm-side delivery instead. We measure each with its own ratio — efficiency or coefficient of performance — bound them all by the Carnot limit that reservoir temperatures alone set, and track how finite temperature differences, throttling, and friction generate entropy and pull real machines below that bound. topics: [Thermodynamics] draft: false sources: - book: Tipler & Mosca ref: "Ch. 19 — The Second Law of Thermodynamics; §§19-1–19-7" --- ## Cyclic energy balance and thermal efficiency A heat engine operates through a repeating sequence of working-fluid states. Over a complete cycle, the working fluid returns to its initial state, so its net change in internal energy is zero. If $Q_h$ enters from a hot reservoir, $Q_c$ leaves to a cold reservoir, and $W_{\rm out}$ is work delivered by the engine, the cycle balance is $$ W_{\rm out}=Q_h-Q_c. $$ The thermal efficiency is the fraction of hot-side heat input converted to useful work, $$ e=\frac{W_{\rm out}}{Q_h}=1-\frac{Q_c}{Q_h}. $$ The quantities refer to the same cycle or the same steady time interval. A turbine shaft may deliver mechanical work while generator, pump, cooling-fan, and control losses consume part of it before electricity reaches a load. The chosen system boundary determines whether those losses reduce reported output or appear as external inputs. Heat rejected to a condenser or cooling loop is not optional bookkeeping: it is required by the cyclic energy balance whenever the engine receives heat and produces less work than that heat input. Reservoirs are idealized bodies with temperatures that remain effectively constant despite the transferred heat. A boiler, combustion stream, geothermal source, river, or cooling tower approximates a reservoir only over a stated heat-load and time range. Finite heat exchangers require a temperature difference to transfer energy at a finite rate. Their working-fluid inlet and outlet temperatures differ from the reservoir temperatures, reducing the achievable device performance. A measured temperature at one metal surface is not automatically the hot or cold reservoir temperature used in an ideal comparison. ## Refrigerators and heat pumps A refrigerator uses work input $W_{\rm in}$ to remove energy $Q_c$ from a cold region and reject $Q_h=Q_c+W_{\rm in}$ to a warmer region. Its coefficient of performance is $$ \mathrm{COP}_{\rm R}=\frac{Q_c}{W_{\rm in}}. $$ A heat pump has the same physical cycle but counts warm-side delivery as useful: $\mathrm{COP}_{\rm HP}=Q_h/W_{\rm in}=\mathrm{COP}_{\rm R}+1$. COP is not bounded by one because it compares moved heat with supplied work. A heat pump with COP 3 delivers three units of warm-side heat per unit of work input; the remaining two units came from the cold-side environment. The definition does not state the device capacity, which is heat transferred per unit time, or the seasonal performance over changing outdoor conditions. | Device | Useful quantity | Energy balance | Performance measure | | :--- | :--- | :--- | :--- | | heat engine | work delivered | $Q_h=W_{\rm out}+Q_c$ | $e=W_{\rm out}/Q_h$ | | refrigerator | heat removed from cold region | $Q_h=Q_c+W_{\rm in}$ | $\mathrm{COP}_{\rm R}=Q_c/W_{\rm in}$ | | heat pump | heat delivered to warm region | $Q_h=Q_c+W_{\rm in}$ | $\mathrm{COP}_{\rm HP}=Q_h/W_{\rm in}$ | $$ % caption: Comparative energy boundaries for a heat engine and a refrigeration device. % Both panels use the same reservoir symbols and interval convention. The engine % converts part of hot-side heat to work, while the refrigeration device consumes % work to transfer heat from the cold side to the warm side. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,dashed] (4.95,.55)--(4.95,3.0); \draw[acc,thick,fill=acc!8] (.4,1.1) rectangle (1.4,2.25); \draw[black,thick] (2.0,.9) rectangle (3.1,2.45); \draw[black,thick,fill=acc!8] (3.7,1.1) rectangle (4.7,2.25); \node[acc] at (.9,1.67) {hot}; \node[black] at (2.55,1.67) {engine}; \node[black] at (4.2,1.67) {cold}; \draw[->,acc,very thick] (1.4,1.67)--(2.0,1.67); \node[acc,above] at (1.7,1.72) {$Q_h$}; \draw[->,black,very thick] (3.1,1.67)--(3.7,1.67); \node[black,below] at (3.4,1.62) {$Q_c$}; \draw[->,black,thick] (2.55,2.45)--(2.55,3.0); \node[black,right] at (2.55,2.78) {$W_{\rm out}$}; \node[black,below] at (2.55,.7) {heat engine}; \draw[acc,thick,fill=acc!8] (5.2,1.1) rectangle (6.2,2.25); \draw[black,thick] (6.8,.9) rectangle (7.9,2.45); \draw[black,thick,fill=acc!8] (8.5,1.1) rectangle (9.5,2.25); \node[acc] at (5.7,1.67) {cold}; \node[black] at (7.35,1.67) {device}; \node[black] at (9.0,1.67) {warm}; \draw[->,acc,very thick] (6.2,1.67)--(6.8,1.67); \node[acc,above] at (6.5,1.72) {$Q_c$}; \draw[->,black,very thick] (7.9,1.67)--(8.5,1.67); \node[black,below] at (8.2,1.62) {$Q_h$}; \draw[->,black,thick] (7.35,3.0)--(7.35,2.45); \node[black,right] at (7.35,2.78) {$W_{\rm in}$}; \node[black,below] at (7.35,.7) {refrigeration}; \end{tikzpicture} $$ Real-device measurements require matched control boundaries. Electrical input can include a compressor, fans, pumps, defrost heater, controls, and standby load. The cold-side heat rate can be measured from air or liquid mass flow and temperature change, from a calorimeter, or from working-fluid enthalpy differences. Each method has different calibration and heat-leak terms. Transient start-up stores energy in the heat exchangers, cabinet walls, and working fluid, so instantaneous ratios can differ from a cycle-average efficiency or COP. Record reservoir-side temperatures, heat-transfer rates, electrical power, mass flow, pressure, humidity where frost is possible, and the averaging interval. State whether reported work is compressor shaft work or total electrical draw. Pressure drops, finite exchanger temperature differences, nonideal compression, throttling, and parasitic heat leaks lower observed performance relative to an ideal cycle. These are physical transfer mechanisms with measurable boundaries, not corrections applied after an efficiency has been calculated. Performance validation closes the energy balance independently of the reported ratio. An engine requires measured heat input to be compared with shaft output plus heat rejection over one common interval. A refrigerator requires warm-side rejection to be compared with cold-side removal plus electrical input. The residual should be consistent with sensor calibration, stored energy, and unmeasured losses. A large residual identifies an omitted flow or a mismatched time basis before it can be mistaken for exceptional efficiency or COP. ## Carnot limit, reversible cycles, and entropy generation The Carnot limit compares any heat engine with a reversible engine operating between reservoirs at absolute temperatures $T_h$ and $T_c$. Its maximum thermal efficiency is $$ e_{\rm C}=1-\frac{T_c}{T_h}. $$ Only reservoir temperatures belong in this expression. The temperature of a turbine casing, a combustion flame, or a heat-exchanger wall may differ from the reservoir temperature that bounds a reversible comparison. The limit becomes zero when the reservoir temperatures are equal and approaches one only as $T_c/T_h$ approaches zero. It does not predict power output, fuel rate, heat-exchanger area, or the efficiency of a particular machine. It sets an upper bound for a device whose only net thermal contacts are the specified reservoirs. A reversible cycle is an ideal sequence with no friction, no pressure drop, no unrestrained expansion, and heat transfer across infinitesimal temperature differences. The Carnot cycle has two isothermal heat-transfer segments and two adiabatic connecting segments. It is a benchmark rather than a construction specification: infinitesimal temperature differences would require very large heat exchangers or very slow transfer for a finite heat load. Real engines operate with finite gradients and finite rates, so their efficiency is lower than the Carnot value for the same reservoir temperatures. $$ % caption: Reversible Carnot cycle on a temperature-entropy diagram. The upper and lower horizontal paths exchange heat at the hot and cold reservoir temperatures; the connecting paths are adiabatic in the ideal model, and the enclosed area equals net work per cycle. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (.45,.45)--(5.85,.45) node[right] {entropy}; \draw[->,black] (.45,.45)--(.45,3.1) node[above] {temperature}; \draw[acc,very thick] (1.5,2.5)--(4.5,2.5)--(4.5,1.0)--(1.5,1.0)--cycle; \draw[->,black,thick] (2.3,2.5)--(3.0,2.5); \draw[->,black,thick] (4.5,1.9)--(4.5,1.45); \draw[->,black,thick] (3.7,1.0)--(3.0,1.0); \draw[->,black,thick] (1.5,1.55)--(1.5,2.05); \node[black,above] at (3.0,2.5) {hot path}; \node[black,below] at (3.0,1.0) {cold path}; \node[acc] at (3.0,1.75) {net work}; \end{tikzpicture} $$ Irreversibility is tracked by entropy generation. With $Q_h$ entering an engine from the hot reservoir and $Q_c$ leaving to the cold reservoir, the combined entropy generation is $$ S_{\rm gen}=\frac{Q_c}{T_c}-\frac{Q_h}{T_h}\ge0. $$ Equality describes the reversible limit. Friction, viscous flow, combustion, electrical resistance, mixing, leakage, throttling, and heat transfer across a finite temperature difference all produce positive $S_{\rm gen}$. For fixed $Q_h$, positive generation increases the necessary rejected heat and reduces available work. The work loss is often expressed relative to an environment temperature, but that reference must be stated before assigning a numerical lost-work value. The corresponding Carnot COP limits are $\mathrm{COP}_{\rm R,C}=T_c/(T_h-T_c)$ and $\mathrm{COP}_{\rm HP,C}=T_h/(T_h-T_c)$. Their divergence as $T_h-T_c$ tends to zero does not imply unlimited real heating or cooling rate; finite devices require temperature differences for heat transfer and have finite conductance. Experimental entropy-generation estimates require complete heat and work accounts. Measure hot-side and cold-side heat rates with calibrated flow, temperature, or calorimetric methods, and assign each rate to the reservoir temperature at the transfer boundary. A single exhaust temperature cannot determine rejected heat when mass flow and heat capacity are unknown. An engine with negative estimated $S_{\rm gen}$ has a sign error, missing heat flow, or incorrect reservoir temperature. The same check for a refrigerator applies to the combined cold region, device, and warm region. The calculation should distinguish reservoir entropy change from entropy carried by working-fluid mass crossing an open control volume. Practical comparison requires measured reservoir-side temperatures and a stable operating boundary. Use absolute temperatures, common averaging intervals, and heat rates measured on both sides when possible. A real engine can appear to exceed a Carnot value if a heat input, auxiliary work input, exhaust enthalpy flow, or reservoir temperature has been omitted from the boundary. For refrigeration, include fan, pump, and defrost power when comparing a measured COP with a reservoir-based limit. Entropy generation identifies the direction of the gap from reversibility; the component-level heat and work measurements identify where that gap occurs. ## Real cycles, compressor work, and performance maps A vapor-compression refrigerator is described by four working-fluid states. State 1 is the evaporator outlet, usually vapor or slightly superheated vapor at low pressure. The compressor raises it to state 2, a higher-pressure and higher-enthalpy vapor. The condenser rejects energy until state 3 is liquid or subcooled liquid at high pressure. A throttling valve produces state 4, a low-pressure liquid--vapor mixture that enters the evaporator. The state labels refer to measured locations at component inlets and outlets. Pressure alone identifies saturation temperature only when the fluid state and composition are known. Steady flow with negligible kinetic and potential energy changes uses property data to give the specific compressor work and heat-transfer scales: $$ w_{\rm comp}\simeq h_2-h_1, \qquad q_{\rm evap}\simeq h_1-h_4, \qquad q_{\rm cond}\simeq h_2-h_3. $$ These are working-fluid quantities per unit mass. Multiplying by refrigerant mass flow estimates component rates. The ideal throttle relation is $h_4\simeq h_3$; it does not state that the pressure or temperature remains constant. Compressor isentropic efficiency compares the actual outlet enthalpy with an ideal reference at the same outlet pressure, $\eta_c=(h_{2s}-h_1)/(h_2-h_1)$. The reference state $2s$ comes from refrigerant property data and cannot be inferred from an electrical power reading alone. $$ % caption: Simplified vapor-compression loop on a pressure-enthalpy plane. The compressor raises pressure and enthalpy from state 1 to state 2, the condenser rejects heat toward state 3, the valve reduces pressure to state 4, and the evaporator absorbs heat back to state 1. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (.45,.45)--(5.85,.45) node[right] {enthalpy}; \draw[->,black] (.45,.45)--(.45,3.1) node[above] {pressure}; \draw[->,acc,very thick] (3.2,1.0)--(4.2,2.4); \draw[->,acc,very thick] (4.2,2.4)--(1.6,2.4); \draw[->,acc,very thick] (1.6,2.4)--(1.6,1.0); \draw[->,acc,very thick] (1.6,1.0)--(3.2,1.0); \draw[fill=acc!12,draw=acc,line width=.4pt] (3.2,1.0) circle (2.2pt); \draw[fill=acc!12,draw=acc,line width=.4pt] (4.2,2.4) circle (2.2pt); \draw[fill=acc!12,draw=acc,line width=.4pt] (1.6,2.4) circle (2.2pt); \draw[fill=acc!12,draw=acc,line width=.4pt] (1.6,1.0) circle (2.2pt); \node[black,below right] at (3.2,1.0) {1}; \node[black,above right] at (4.2,2.4) {2}; \node[black,above left] at (1.6,2.4) {3}; \node[black,below left] at (1.6,1.0) {4}; \node[black,above] at (2.9,2.4) {condenser}; \node[black,below] at (2.4,1.0) {evaporator}; \node[black,right] at (3.75,1.62) {compressor}; \node[black,left] at (1.6,1.7) {valve}; \end{tikzpicture} $$ Heat exchangers introduce two coupled losses. Finite temperature difference is needed to transfer a finite heat rate, and pressure drop changes the saturation temperatures seen by the working fluid. Frost, fouling, low airflow, and a restricted valve reduce evaporator capacity. High outdoor temperature, dirty condenser fins, or poor water flow raise condensing pressure and compressor work. Excessive suction-line superheat protects the compressor from liquid carryover but can reduce evaporator utilization. Subcooling at the condenser outlet can increase liquid-side enthalpy margin, yet it also depends on heat-exchanger and control conditions. $$ % caption: Qualitative performance map. At the same cooling load, a colder source or warmer sink lowers COP; increasing load can also reduce COP as pressure ratio, temperature differences, and auxiliary power rise. A map applies only to the stated equipment, refrigerant, and test boundary. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (.45,.45)--(5.85,.45) node[right] {cooling load}; \draw[->,black] (.45,.45)--(.45,3.1) node[above] {COP}; \draw[acc,thick] plot[smooth] coordinates {(.75,2.7)(1.6,2.62)(2.55,2.4)(3.5,2.0)(4.35,1.45)(5.3,.85)}; \draw[black,thick] plot[smooth] coordinates {(.75,1.95)(1.6,1.85)(2.55,1.62)(3.5,1.3)(4.35,.92)(5.3,.62)}; \node[acc,above] at (1.7,2.65) {mild source}; \node[black,below] at (1.75,1.82) {cold source}; \end{tikzpicture} $$ An operating map is an empirical summary rather than a property identity. Its axes, test temperatures, humidity, fan settings, compressor speed, refrigerant charge, and included electrical loads must be stated. Capacity and COP often change together with source temperature and part-load control. Interpolation outside the tested map can fail during frosting, defrost, cycling, or compressor speed limits. Measurement uncertainty enters both numerator and denominator of COP. Mass-flow calibration, refrigerant property interpolation, pressure-sensor offset, temperature placement, and electrical-power harmonics can each dominate a test. Average over multiple cycles after the cabinet and heat exchangers reach a repeatable condition. Close the refrigerant-side balance against independent air- or water-side calorimetry when possible; disagreement identifies heat leakage, unmeasured auxiliary power, or state-point errors before a performance map is accepted. State-point validation uses redundant measurements. The low-side and high-side pressures should agree with the saturation ranges implied by measured evaporator and condenser temperatures after accounting for pressure drop. A temperature probe on a pipe wall is not automatically the refrigerant bulk temperature; insulation contact, thermal paste, and heat exchange with ambient air affect the reading. Record sensor locations and refrigerant property source with every enthalpy calculation. If the estimated compressor work from mass flow and state points disagrees with electrical power after motor losses are considered, inspect mass-flow calibration, refrigerant charge, heat leakage, and the assumed operating steady state before adjusting the reported COP. The component energy balance must close. ## Entropy generation, exergy, and component losses The entropy balance for a steady control volume is $$ 0=\sum_j\frac{\dot Q_j}{T_j}+\sum_{\rm in}\dot m s -\sum_{\rm out}\dot m s+\dot S_{\rm gen}. $$ Heat entering the control volume is positive in this convention. The entropy generation rate $\dot S_{\rm gen}$ is nonnegative and collects irreversibility from friction, mixing, finite temperature differences, electrical resistance, pressure loss, and nonequilibrium expansion. A negative calculated value identifies an incomplete heat or mass-flow account, a sign error, or a temperature assigned at the wrong transfer boundary. The balance applies to a defined component; a whole-system balance can conceal a large local loss if another component is omitted or grouped without measurements. Exergy measures useful-work potential relative to an environment at absolute temperature $T_0$. Irreversibility destroys exergy at the rate $$ \dot X_{\rm dest}=T_0\dot S_{\rm gen}. $$ This equality does not state where heat is lost; it converts an entropy-generation rate into the corresponding lost-work scale for the selected environment. A different reference temperature changes the numerical exergy value. Component exergy destruction ranks losses in a compressor, valve, heat exchanger, or duct, provided every component is evaluated with the same reference and boundary. Finite temperature difference in a heat exchanger is a direct source of entropy generation. For heat rate $\dot Q$ transferred from a uniform hot boundary at $T_h$ to a uniform cold boundary at $T_c=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,thick,fill=acc!8] (2.15,.7) rectangle (4.45,2.8); \draw[->,acc,very thick] (.5,2.35)--(2.15,2.35); \draw[->,acc,very thick] (4.45,2.35)--(6.1,2.35); \draw[->,black,very thick] (6.1,1.15)--(4.45,1.15); \draw[->,black,very thick] (2.15,1.15)--(.5,1.15); \node[acc,above] at (1.25,2.35) {hot in}; \node[acc,above] at (5.3,2.35) {hot out}; \node[black,below] at (5.3,1.15) {cold in}; \node[black,below] at (1.25,1.15) {cold out}; \node[acc] at (3.3,1.75) {heat exchanger}; \end{tikzpicture} $$ Throttling and pressure loss create another common loss. An insulated valve with no shaft work has approximately constant working-fluid enthalpy, $h_2\simeq h_1$, but its pressure decreases and entropy increases. The valve cannot recover the pressure drop as useful work. A long pipe, filter, or partially closed valve similarly raises the required compressor pressure ratio. Pressure loss also shifts evaporating and condensing temperatures, changes heat-exchanger driving differences, and can lower capacity even when the compressor electrical input is unchanged. Quantitative diagnostics require redundant component data. Measure mass flow, inlet and outlet pressure, bulk-fluid temperature, electrical power, and heat-side flow and temperature change. Property calculations then give enthalpy and entropy differences with stated reference data. Compare a heat-exchanger heat rate from the working-fluid side with an air- or water-side calorimetric rate. Compare compressor shaft or electrical power with the enthalpy-rise estimate after motor losses are included. Uncertainty in mass flow, pressure, temperature, fluid composition, and heat leakage propagates into $\dot S_{\rm gen}$; a small residual formed by subtracting large terms can have a large relative uncertainty. Report the residual, its uncertainty, the environment temperature, and the control boundary before ranking component losses by exergy destruction. Loss allocation must avoid double counting. A compressor electrical-loss estimate belongs with the compressor boundary when its motor is included there; it should not also be assigned to a downstream heat exchanger because the rejected motor heat later appears in the warm-side energy balance. A pressure drop measured across a duct and a valve should be separated only when intermediate pressure taps establish the two losses. Heat leakage through an insulated cabinet may be reported as an external load, while fan power used to remove that load belongs to the electrical input if the reported boundary encloses the whole appliance. These bookkeeping choices change component exergy rankings without changing conservation laws. Uncertainty is especially important when a component loss is obtained as a residual. For example, a heat-exchanger entropy-generation estimate can subtract two stream-entropy changes of similar size. Temperature offsets shared by several probes then produce correlated errors that do not average away with more samples. Reversing flow where the apparatus permits, using redundant heat-rate methods, and performing zero-flow sensor checks provide diagnostic tests. A component should be ranked above another only when the difference in estimated exergy destruction exceeds their combined uncertainty over the stated operating range. ## Cyclic-process data reduction and efficiency testing Engine testing begins with a time-resolved definition of the cycle and of the measurement boundary. A pressure transducer reports pressure at a tap location; a crank-angle encoder or displacement sensor measures cylinder volume. The indicated work for one cylinder over one closed cycle is the signed loop integral $$ W_i=\oint p\d V. $$ Clockwise and counterclockwise loop directions have opposite signs under a stated coordinate convention. Compression and expansion paths should be sampled at enough crank-angle points to resolve sharp pressure changes. Use absolute pressure when the volume boundary includes atmospheric displacement work. Gauge pressure can be used only after the reference pressure and the intended work boundary are handled consistently. A low-load engine may show a pumping loop whose negative work is large relative to the positive expansion loop. Aggregating cycles before inspecting the individual loops can mask misfires, valve timing changes, and sensor drift. Pressure calibration requires traceable static points over the expected range. Volume calibration requires piston area, clearance volume, and any geometry used to convert encoder angle into volume. Phase error between the pressure and angle signals changes the enclosed area even when both sensors have correct individual values. Synchronize their clocks and test the reduction pipeline with a known pressure trace before reporting indicated work. Cycle-to-cycle averaging reduces random combustion variation, but an average loop remains invalid if the baseline pressure or volume reference changes during the acquisition. $$ % caption: Measured pressure-volume loop for one engine cycle. The signed enclosed area is indicated work; the lower pumping segment subtracts from expansion work, so its pressure reference, volume calibration, and time synchronization affect the reported result. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (.45,.45)--(5.85,.45) node[right] {volume}; \draw[->,black] (.45,.45)--(.45,3.1) node[above] {pressure}; \draw[acc,very thick] plot[smooth cycle] coordinates {(1.45,1.6)(1.85,2.5)(2.9,2.82)(3.95,2.42)(4.55,1.72)(3.35,1.55)(2.2,1.55)}; \draw[acc,thick] plot[smooth cycle] coordinates {(2.0,1.28)(2.85,1.34)(3.7,1.26)(2.85,1.02)}; \draw[->,acc,very thick] (1.62,1.95)--(1.82,2.42); \draw[->,acc,very thick] (4.35,2.0)--(4.55,1.78); \draw[->,black,thick] (3.3,1.06)--(2.55,1.04); \node[black,above] at (2.85,2.85) {expansion}; \node[black] at (2.75,1.92) {compression}; \node[black,below] at (2.85,.92) {pumping loop}; \end{tikzpicture} $$ Indicated power is $P_i=W_i n_c$, where $n_c$ is the number of measured cycles per second after accounting for the number of cylinders. Brake power is measured at the output shaft: $$ P_b=\tau\omega. $$ Here $\tau$ is shaft torque and $\omega$ is angular speed measured over the same interval as the pressure loop. The difference $P_i-P_b$ includes mechanical friction, accessory loads, pumping losses outside the selected indicated boundary, and any unaccounted transient rotational-energy change. The ratio $P_b/P_i$ is a mechanical transfer ratio, whereas thermal efficiency compares brake output with hot-side heat input. Neither ratio should be called the other. Heat input must be measured at the engine boundary, not inferred from a fuel flow alone unless fuel lower heating value, composition, unburned fuel, and enthalpy flows are included in the model. A calorimetric hot stream gives $\dot Q_h=\dot m c_p(T_{\rm in}-T_{\rm out})$ only when its heat capacity, mass flow, and heat leak are known. Exhaust, cooling-water, lubricating-oil, and charge-air flows may carry energy across a practical engine boundary. Choose either a detailed open-system balance or a calibrated fuel-input convention, state it, and use it for both the efficiency calculation and the Carnot comparison. > **Worked example.** A steady engine test gives $T_h=600\ \mathrm K$, > $T_c=300\ \mathrm K$, $\dot Q_h=12.0\ \mathrm{kW}$, indicated power > $P_i=5.0\ \mathrm{kW}$, and brake power $P_b=4.2\ \mathrm{kW}$. The brake efficiency > is $e_b=4.2/12.0=0.350$ against a Carnot bound $e_C=1-300/600=0.500$; the reversible > engine would deliver $0.500\times12.0=6.0\ \mathrm{kW}$, a shortfall of > $6.0-4.2=1.8\ \mathrm{kW}$. Heat rejection from the cycle balance is > $\dot Q_c=\dot Q_h-P_b=7.8\ \mathrm{kW}$, giving an entropy-generation estimate > $\dot S_{\rm gen}=7.8/300-12.0/600=0.006\ \mathrm{kW\,K^{-1}}$. The numbers mean > something only when the heat and work rates are averaged over the same stable > interval. Transient data require a storage term. During warm-up, the engine block, coolant, oil, exhaust hardware, and rotating shaft change internal or kinetic energy, so the steady cycle balance does not close on the measured heat and work rates alone. Reject data until reservoir temperatures, speed, load, and component temperatures meet predefined stability tolerances, or include measured storage terms in a transient balance. A short moving average can suppress random noise while still hiding a slow drift; inspect raw trends as well as the reported average. Efficiency uncertainty follows from the measured ratio. For independent brake power and heat-input estimates, $$ \left(\frac{u(e_b)}{e_b}\right)^2\simeq \left(\frac{u(P_b)}{P_b}\right)^2+ \left(\frac{u(\dot Q_h)}{\dot Q_h}\right)^2. $$ Torque calibration, speed resolution, fuel or calorimeter calibration, and shared temperature offsets can create correlated uncertainty that this expression omits. Report indicated work, brake work, heat input, reservoir temperatures, time window, cycle count, stability criterion, and uncertainty method together. A measured value above the Carnot limit is a diagnostic of incompatible boundaries or measurements, not evidence that the limit has been surpassed. Reservoir heat-rate measurement needs its own reduction path. A liquid calorimeter on the hot side gives a rate from mass flow, heat capacity, and inlet-outlet temperature difference only after heat leakage, pump work, and sensor immersion are assigned to the boundary. A fuel-flow method needs a fuel composition or heating-value calibration and an account of unburned fuel, exhaust enthalpy, and any auxiliary electrical heating. A cooling-water loop can determine rejected heat independently from its flow and temperature rise. During a steady engine test, the three rates should satisfy $\dot Q_h=P_b+\dot Q_c$ within uncertainty after all selected accessory loads are included. This redundancy is stronger than an efficiency ratio alone because it exposes a missing thermal stream even when the ratio appears plausible. Use acquisition windows long enough to include many engine cycles and short enough that load, reservoir temperatures, and calibration drift remain bounded. Store raw pressure, angle, torque, speed, flow, and temperature channels rather than only cycle averages. Outlier cycles should be retained and flagged with their cause, such as a misfire, a control transition, or a sensor dropout. Removing them without a criterion biases the reported indicated work. Estimate random uncertainty from repeat windows at the same operating point. Estimate systematic uncertainty from calibration certificates, zero offsets, reference-pressure error, heat leakage, and the reduction model. Shared clock error can correlate pressure, torque, and flow signals; it cannot be treated as three independent noise sources. The Carnot comparison also has temperature uncertainty. With $e_C=1-T_c/T_h$, its sensitivity increases when the hot and cold temperatures are close. Use reservoir temperatures in kelvin and retain their paired time histories. A local metal temperature near an exhaust port may exceed the hot-reservoir estimate, while a cooling-water outlet temperature may exceed the cold-reservoir estimate; substituting either value can create a misleadingly loose or strict bound. The final report should include the measured efficiency, Carnot bound, entropy-generation estimate, and a statement of the common test interval from which each was reduced. ## Regeneration, combined cycles, and practical operating envelopes Regeneration returns part of a cycle's internally rejected heat to the working fluid before fresh fuel or external heat is added. In a gas-turbine recuperator, hot turbine exhaust transfers energy to compressed air leaving the compressor. The combustor then requires less fuel to reach the selected turbine-inlet temperature. Recuperator effectiveness is commonly defined by $$ \varepsilon=\frac{T_{a,\rm out}-T_{a,\rm in}} {T_{g,\rm in}-T_{a,\rm in}}, $$ where the numerator is the compressed-air temperature rise and the denominator is the largest rise available from the entering exhaust stream. The definition requires the two streams to have compatible heat-capacity rates and temperature measurements at the exchanger ports. An effectiveness close to one does not guarantee a large net cycle benefit: pressure loss on the air side raises compressor work, pressure loss on the gas side reduces turbine expansion, and a large exchanger adds cost, volume, and thermal inertia. Recuperation is effective when exhaust temperature exceeds compressor-discharge temperature by a substantial margin. Reheat and intercooling alter the pressure-ratio distribution across a Brayton-type cycle. Intercooling divides compression into two stages with a cooler between them. At fixed overall pressure ratio, cooling the first-stage discharge reduces the specific volume entering the second stage and compressor work. That benefit must be compared with the heat rejected in the intercooler and the pressure loss through its ducts. Reheat divides expansion into two turbine stages with heat added between them. It can increase turbine work by restoring the gas temperature before the second expansion, but it consumes additional fuel and may raise exhaust temperature. Reheat alone does not necessarily raise thermal efficiency; the result depends on pressure ratio, maximum temperature, component losses, and whether a recuperator or bottoming cycle uses the hotter exhaust. $$ % caption: Staged compression and expansion with intercooling and reheat. Intercooling reduces the compression-work requirement before the second compressor stage, while reheat increases second-stage turbine work; both additions introduce pressure losses, heat-transfer hardware, and control variables that affect net performance. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[black,thick] (.55,2.05) rectangle (1.75,2.72); \draw[acc,thick,fill=acc!8] (2.55,2.05) rectangle (3.75,2.72); \draw[black,thick] (4.55,2.05) rectangle (5.75,2.72); \draw[->,black,thick] (1.75,2.38)--(2.55,2.38); \draw[->,black,thick] (3.75,2.38)--(4.55,2.38); \node[black] at (1.15,2.38) {comp 1}; \node[acc] at (3.15,2.38) {cooler}; \node[black] at (5.15,2.38) {comp 2}; \node[black] at (3.15,3.05) {compression}; \draw[black,thick] (.55,.62) rectangle (1.95,1.29); \draw[acc,thick,fill=acc!8] (2.75,.62) rectangle (3.95,1.29); \draw[black,thick] (4.75,.62) rectangle (6.15,1.29); \draw[->,black,thick] (.1,.95)--(.55,.95); \draw[->,black,thick] (1.95,.95)--(2.75,.95); \draw[->,black,thick] (3.95,.95)--(4.75,.95); \draw[->,black,thick] (6.15,.95)--(6.6,.95); \node[black] at (1.25,.95) {turbine 1}; \node[acc] at (3.35,.95) {reheat}; \node[black] at (5.45,.95) {turbine 2}; \node[black] at (3.15,.15) {expansion}; \end{tikzpicture} $$ A combined cycle uses the exhaust of a topping gas turbine as the heat source for a bottoming steam cycle. Hot exhaust passes through a heat-recovery steam generator (HRSG), which heats, evaporates, and often superheats water. The steam expands through a steam turbine, then condenses and returns by pump to the HRSG. The gas turbine and steam turbine have separate shaft outputs, auxiliary loads, cooling requirements, and startup constraints. A combined-cycle efficiency must include fuel input and net electrical output from both turbines after generators, pumps, cooling fans, fuel compression, and other selected auxiliary loads are included. Counting gross gas turbine power but net steam power creates a boundary mismatch. $$ % caption: Combined-cycle energy flow. Fuel produces gas-turbine work and a hot exhaust stream; the HRSG recovers part of that exhaust energy for a steam-turbine bottoming cycle, while remaining stack and condenser losses set the gap between fuel input and net plant output. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[acc,thick,fill=acc!8] (.45,1.8) rectangle (1.55,2.55); \draw[black,thick] (2.25,1.8) rectangle (3.65,2.55); \draw[acc,thick,fill=acc!8] (4.35,1.8) rectangle (5.6,2.55); \draw[black,thick] (4.35,.55) rectangle (5.6,1.22); \draw[->,black,thick] (1.55,2.18)--(2.25,2.18); \draw[->,black,thick] (3.65,2.18)--(4.35,2.18); \draw[->,acc,thick] (4.97,1.8)--(4.97,1.22); \draw[->,black,thick] (5.6,2.18)--(6.28,2.18); \node[acc] at (1.0,2.18) {fuel}; \node[black] at (2.95,2.18) {gas turbine}; \node[acc] at (4.97,2.18) {HRSG}; \node[black] at (4.97,.88) {steam turbine}; \node[black,right] at (6.28,2.18) {stack}; \node[acc,left] at (4.35,1.5) {steam}; \end{tikzpicture} $$ > **Worked example.** Take a fuel input of $100\ \mathrm{MW}$ on a lower-heating-value > basis. A simple gas turbine produces $35\ \mathrm{MW}$ net electric output. Suppose > $40\ \mathrm{MW}$ of its exhaust energy enters the HRSG after gas-side losses, and the > bottoming steam cycle converts $30\%$ of that recovered heat, contributing > $0.30\times40=12\ \mathrm{MW}$. Combined net output is > $35+12=47\ \mathrm{MW}$, a combined efficiency of $0.47$, with the remaining > $53\ \mathrm{MW}$ leaving through stack, condenser, radiation, and auxiliaries. The > incremental gain over the $35\%$ simple cycle is $12$ percentage points. This example assumes the gas-turbine output already includes its own auxiliaries, the HRSG stream is measured at one boundary, and the $12\ \mathrm{MW}$ steam value is net of steam-cycle pump and generator losses. Changing any one of those assumptions changes the reported gain. At part load, operating envelopes are constrained by component and control limits. Compressor surge margin, turbine cooling flow, minimum stable fuel flow, flame stability, emissions control, condenser vacuum, pump minimum flow, and HRSG pinch temperature can each establish a lower or upper load boundary. A recuperator can become less effective when stream heat-capacity rates shift. A combined cycle may operate the gas turbine at a load where exhaust temperature is insufficient for the desired steam conditions. Variable inlet guide vanes, variable-speed drives, bypass valves, supplementary firing, and thermal storage alter the envelope but consume power or add loss mechanisms. A map built at one ambient temperature and one fuel composition cannot be extrapolated without those control states. Start-up introduces thermal-storage and thermal-stress limits. HRSG drums, thick turbine casings, recuperator walls, piping, and steam headers must be heated at a limited rate to control temperature gradients and differential expansion. A rapid fuel ramp can raise gas-turbine output before the steam bottoming cycle reaches a repeatable state. Condensate quality, steam temperature, metal temperature, and pressure-ramp limits determine when each component is admitted to service. Treat startup and shutdown as transient energy balances with stored-energy terms rather than applying a steady combined-cycle efficiency to the entire interval. Practical performance tests therefore use separate steady windows at each load. Record fuel flow and heating value, both turbine outputs, auxiliary power, exhaust temperature and flow, steam pressure and temperature, cooling-water flow, ambient conditions, and control positions. Close the plant energy balance before comparing configurations. Uncertainty in exhaust heat recovery can dominate the incremental bottoming-cycle estimate because it is the difference between large gas-side energy flows. Report the operating envelope, excluded loads, and averaging period with every recuperation or combined-cycle efficiency claim. Off-design comparison requires corrected variables as well as raw load. Gas-turbine mass flow and compressor operating point depend on inlet temperature, inlet pressure, and humidity. A hot day can reduce air density, maximum mass flow, and available power even when fuel flow control requests the same firing condition. Steam-cycle output depends on condenser pressure, cooling-water temperature, HRSG gas flow, and the availability of heat-transfer surface. Report raw ambient data and the correction method used to compare separate test days. A quoted combined-cycle efficiency at base load does not predict net output during a hot afternoon, a cold start, or a condensing-pressure excursion. Performance accounting must also distinguish gross and net generation. Gross output is measured at generator terminals before plant auxiliary loads. Net output subtracts fans, pumps, cooling-water systems, fuel handling, emissions controls, controls, and transformer losses included within the stated plant boundary. Recuperator fans or gas-side pressure-loss penalties can appear as lower gas-turbine output, while HRSG pump power appears in the steam-cycle auxiliary account. A comparison between a simple-cycle gross value and a combined-cycle net value can create an apparent gain or loss unrelated to thermodynamic configuration. State the electrical metering locations and the lower- or higher-heating-value basis of fuel input. Uncertainty in the 47% example has an asymmetric structure. The gas-turbine output and fuel input may be measured directly, whereas recovered HRSG heat is commonly inferred from exhaust mass flow and temperature drop. Error in that inferred stream affects both the estimated bottoming input and the remaining stack loss. Repeat tests at fixed load, compare steam-side and gas-side HRSG balances, and report the largest closure residual. The incremental 12 MW bottoming contribution is credible only when its uncertainty remains smaller than the claimed difference from the simple-cycle reference. ## Refrigeration capacity, seasonal metrics, and heat-pump installation Cooling capacity is the rate at which the indoor heat exchanger removes energy from the conditioned region. A load calculation separates sensible and latent terms. An air stream has the sensible estimate $$ \dot Q_{\rm sens}=\dot m_{\rm air}c_{p,\rm air}(T_{\rm in}-T_{\rm out}). $$ Condensing water vapor contributes a latent term set by the moisture removal rate and the appropriate phase-change enthalpy. Solar gain, wall and window conduction, infiltration, occupants, lighting, appliances, duct leakage, and ventilation all contribute to the design load. A nameplate cooling capacity is meaningful only at the stated indoor and outdoor conditions, airflow, refrigerant charge, and fan setting. Selecting equipment from floor area alone omits the thermal boundary and often misstates both sensible and latent capacity. The indoor coil boundary includes room air entering and leaving the coil, condensate draining from the coil, supply-fan work if the fan is inside the selected boundary, and heat pickup in supply ducts. A cooling coil can meet the sensible load while failing to remove enough moisture if its surface temperature, airflow, or runtime is wrong. Conversely, very low airflow can increase dehumidification while reducing total delivered airflow and increasing fan or compressor limits. Capacity testing therefore records dry-bulb temperature, humidity or enthalpy, air or liquid mass flow, condensate mass where measured, and electrical power over a common interval. $$ % caption: Indoor cooling-load boundary. The evaporator removes sensible energy from the air stream and latent energy through condensate removal; fan work, airflow, entering and leaving air state, and condensate mass determine the measured indoor capacity. \begin{tikzpicture}[>=stealth,font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[acc,thick,fill=acc!8] (2.2,.8) rectangle (4.45,2.8); \draw[->,black,very thick] (.55,2.2)--(2.2,2.2); \draw[->,black,very thick] (4.45,2.2)--(6.1,2.2); \draw[->,acc,thick] (3.32,.8)--(3.32,.3); \node[black,above] at (1.38,2.2) {room air in}; \node[black,above] at (5.28,2.2) {supply air}; \node[acc] at (3.32,1.8) {indoor coil}; \node[acc,below] at (3.32,.3) {condensate}; \end{tikzpicture} $$ COP is dimensionless: it is heat moved divided by electrical or shaft work over the same interval. EER is commonly reported in British thermal units per hour per watt, so at one operating condition $$ \mathrm{EER}=3.412\,\mathrm{COP} \quad \left[\frac{\mathrm{Btu/h}}{\mathrm W}\right]. $$ EER and COP describe a specified test point. Seasonal measures aggregate delivered cooling or heating over a prescribed distribution of outdoor conditions and divide by seasonal electrical energy. Their units are often Btu per watt-hour, whereas a seasonal COP remains dimensionless. A seasonal rating cannot be converted to a single-point COP without the underlying test weighting, auxiliary loads, cycling rule, and climate assumptions. Report the metric name, units, rating condition, and whether indoor and outdoor fan power are included. Outdoor temperature changes pressure ratio, heat-exchanger temperature difference, and available capacity. In cooling mode, hotter outdoor air raises condensing temperature and compressor work. In heating mode, colder outdoor air reduces evaporator temperature, lowers capacity, and can raise the need for defrost or auxiliary resistance heat. Part-load controls can improve seasonal performance by reducing cycling losses, but compressor speed limits and minimum stable refrigerant flow define their usable range. A performance curve should show both capacity and power or COP; capacity alone does not establish energy use. Defrost is a seasonal boundary issue for air-source heat pumps. In heating mode the outdoor coil can operate below the frost point and accumulate ice, increasing air-side resistance and reducing heat transfer. A reverse-cycle or electric defrost event uses energy and can briefly interrupt indoor heating. Supplemental electric heat may maintain indoor comfort during low outdoor temperature, defrost, or capacity shortfall. Its energy must be included in seasonal heating input. Excluding defrost heaters, crankcase heaters, standby draw, or indoor fan power gives an appliance-only result, not delivered system performance. Installation changes the physical cycle. Refrigerant line length and elevation add pressure drop and heat gain or loss. Incorrect charge changes superheat, subcooling, and compressor protection. Duct leakage can return conditioned air to an attic or draw unconditioned air into the return path. Poor duct insulation adds sensible load, while high external static pressure changes blower flow and coil capacity. Outdoor unit clearance, recirculation of rejected heat, water drainage, and access for coil cleaning affect the outdoor heat-exchanger boundary. Electrical supply voltage and wire losses can change compressor and auxiliary power from rating conditions. > **Worked example.** A design calculation gives $6.0\ \mathrm{kW}$ sensible and > $1.2\ \mathrm{kW}$ latent load at the summer condition, for $7.2\ \mathrm{kW}$ total > cooling. Select equipment delivering at least $8.0\ \mathrm{kW}$ at that > indoor–outdoor condition after installation derating. At a rated-point COP of $3.20$, > the equivalent $\mathrm{EER}=3.412\times3.20=10.9\ \mathrm{Btu\,h^{-1}W^{-1}}$, and > the electrical input while delivering $7.2\ \mathrm{kW}$ is > $7.2/3.20=2.25\ \mathrm{kW}$. > **Worked example.** Over a season, the system delivers $5760\ \mathrm{kWh}$ of > cooling for $2057\ \mathrm{kWh}$ of metered electricity (fans and auxiliaries > included), a seasonal delivered COP of $5760/2057=2.80$. The seasonal value is lower than the test-point COP because it includes off-design operation, cycling, ambient variation, and the stated auxiliaries. The sizing conclusion changes if the latent load rises, ducts leak, outdoor recirculation occurs, or the rating condition differs from the design condition. Report load-calculation inputs, capacity margin, metric units, equipment boundary, electrical metering location, and uncertainty in airflow, humidity, and power before treating the example as an installation recommendation. Commissioning establishes whether the installed system can achieve the assumed capacity. Measure total external static pressure and airflow at the selected fan setting, then compare both with the blower curve and duct design. Measure supply and return air temperature and humidity at locations that avoid direct coil radiation, stratification, and outdoor-air short circuiting. Verify condensate drainage during latent-load operation. On the refrigerant side, record suction and discharge pressures, line temperatures, superheat, subcooling, compressor current, and outdoor air temperature after the system stabilizes. These measurements identify restrictions, undercharge, overcharge, inadequate airflow, coil fouling, and outdoor-air recirculation, but they do not replace a leak test or manufacturer-specific charging procedure. Seasonal data reduction uses delivered thermal energy and metered electrical energy over the same calendar or weighted test interval. Separate cooling, heating, defrost, standby, and supplemental-resistance operation when the meter resolution permits. Outdoor temperature binning shows whether poor seasonal performance occurs at high cooling ambient, low heating ambient, mild cycling conditions, or defrost events. An occupancy or ventilation change can alter the load independently of equipment performance, so compare capacity with both weather and indoor setpoint history. Missing data periods, sensor replacement, and thermostat schedule changes belong in the test record. Seasonal metrics should be reported with delivered-energy method, metering boundary, climate interval, and uncertainty from airflow, humidity, temperature, and electrical-power measurements. Installation margins have opposing costs. A small capacity margin can fail to meet a rare design load; excessive oversizing can shorten runtime, reduce latent removal, increase cycling losses, and force auxiliary heat-pump operation outside its best part-load region. Variable-capacity equipment can reduce some cycling loss, but its minimum output, duct airflow, and control deadband still constrain runtime. The selection criterion is the net delivered load across the operating envelope, not the largest catalogue capacity at a single rating point.