Thermal Cycles

Otto Cycle

1-2             Reversible adiabatic compression of air (W_IN)

2-3             Addition of heat at constant volume increases pressure (Q_IN)

3-4     Reversible adiabatic expansion of air (W_OUT)

4-1     Rejection of heat at constant volume reduces pressure (Q_OUT)

§         From first law 

§         Thermal efficiency of a cycle usually found .  Also  and  hence 

§         Compression and expansion of air are Polytropic process with  hence  Where the compression ratio 

§         Hence thermal efficiency is only dependant on compression ratio 

---

Diesel cycle

1-2             Reversible adiabatic compression of air (W_IN)

2-3             Addition of heat at constant pressure increases volume (Q_IN)

3-4     Reversible adiabatic expansion of air (W_OUT)

4-1     Rejection of heat at constant volume reduces pressure (Q_OUT)

§         From first law 

§         As with Otto  but as heat is added at constant pressure 

§         Assume heat addition 2-3 is ideal gas:  but  so  hence the cut-off ratio 

§         Thermal efficiency is dependant on compression AND cut-off ratio 

---

Real gas turbines

§         Approximated by joule cycle. 

§         Air drawn into compressor.

§         Fuel injected and ignited.

§         Hot gasses pass through turbine, providing mechanical power.

§         Exhaust gasses expelled.

---

Joule Cycle

§         Approximates gas turbine, like aircraft engines.

§        

Open system so use SFEE!

1-2             Reversible adiabatic compression of air (W_IN)

2-3             Reversible addition of heat at constant p increases T (Q_IN)

3-4     Reversible adiabatic expansion of air (W_OUT)

4-1     Reversible rejection of heat at constant p reduces T (Q_OUT)

§         From first law 

§         For heat transfers at constant pressure  and 

§         For compressor and turbine work  and 

§         Compression and expansion is Polytropic with  such that 

§         Here we have the pressure ratio 

§         Thermal efficiency dependant only on the pressure ratio 

---

Otto cycle

The Otto cycle is a first approximation to model the operation of a spark-ignition engine, first built by Nikolaus Otto in 1876, and used in many cars, small planes and small power systems (below say 200 kW) down to miniature engines. This is a reciprocating internal combustion gas engine, in contrast to the, at that time master, reciprocating or rotodynamic external combustion steam engine. The Otto engine is sketched in Fig. 5.2, where the typical terms are introduced for engine-geometry characteristics (stroke, bore, displacement and compression ratio); other terms for engine-operation characteristics are shaft speed, mean pressure, power, fuel consumption, torque, volumetric efficiency, energy efficiency, etc.

Fig. 5.2. Sketch and nomenclature for reciprocating engines: a) 4-stroke, b) 2-stroke (uni-flow).

In the ideal air-standard Otto cycle, the working fluid is just air, which is assumed to follow four processes (Fig. 5.3): isentropic compression, constant-volume heat input from the hot source, isentropic expansion, and constant-volume heat rejection to the environment.

Fig. 5.3. The ideal Otto cycle in the T-s and p-V diagram, and a practical p-V trace of four-stroke and two-stroke engines.

The main parameters of both ideal and real Otto cycles are:

·        Size, measured by the displacement volume (the volume swept by the piston, V₁−V₂), usually less than 0.5 litres per cylinder, to avoid self-ignition.

·        Speed, more precisely crankshaft speed, n, with a typical operation range n=1000..7000 rpm (n=20..120 Hz). The maximum value may be in the range n_max=6000..8000rpm for four-stroke engines. Two-stroke motorcycle engines run quicker (n_max=13 000 rpm), the quickest (n_max=20 000 rpm) being the smallest engines (two stroke), used in aircraft modelling.

·        Compression ratio, r=V₁/V₂, with a typical range of r=8..10 (up to 14 in direct-injection spark-ignition engines), limited by the 'knock' or self-ignition problem.

·        Mean effective pressure, p_me, defined as the unit work divided by the displacement, with a typical range of 0.2..1.5 MPa (the full-load value may range from p_me=1.2 MPa in two-stroke motorcycle engines, to p_me=1.7 MPa in the largest turbocharged engines). Maximum pressure may have a typical range of 4..10 MPa. Performance maps of reciprocating engines are usually presented on a p_me-n diagram, i.e. mean effective pressure versus engine speed.

Real reciprocating engines may be two-strokes or four-strokes in a cycle; the simple to model is the four-strokes engine, where one stroke is used to fill the cylinder with the air-fuel mixture through the inlet valve (the inside pressure is a little below atmospheric), a second stroke to compress the mixture with valves closed and exploding it at a precise point with a spark, a third stroke to let the product gasses to expand and do work on the crank-shaft, and a fourth stroke to force the burnt gasses out through the exhaust valve, and start a new cycle (two crank-shaft turns have elapsed). In two-stroke engines there is an overlap between intake and exhaust, with the inlet stream used to sweep the burnt gasses (throwing unburnt gases to the tailpipe!), and the whole cycle is performed in one turn of the crankshaft. In spite of the higher fuel consumption and pollutant emissions, and the difficulties for their lubrication (oil is added to the fuel), two-stroke engines have very high specific power, and thus they are used for small engines (hand-held and garden engines, motorcycles and outboard boats).

The cold-air-standard model takes as working fluid air with constant properties (those at the inlet, i.e. cold), what renders the analysis simple. The energy exchanges for the trapped control mass, m, are W₁₂/m=c_v(T₂-T₁), Q₁₂=0, W₂₃=0, Q₂₃/m=c_v(T₃-T₂), W₃₄/m=c_v(T₄-T₃), Q₃₄=0, W₄₁=0, Q₄₁/m=c_v(T₁-T₄), and the energy and exergy efficiencies are:

                     and    (5.2)

Real spark-ignition engines do not recycle the working fluid; they are all internal combustion engines where ambient air is pumped in, and petrol fuel (gasoline) is added  to prepare a homogeneous reactive mixture that at a certain point (cinematically or electronically controlled) is ignited by an electrical spark. Fuel addition can be in a carburettor by Venturi suction, or in the individual inlet duct by injection, or recently by direct injection inside the cylinder as in a diesel engine (but with a much lower injection pressure, less than 10 MPa). Gaseous fuels like coal-gas, liquefied petroleum gas (LPG), and natural gas are also used, as well as other liquid fuels like methanol, ethanol and ethers, usually added to gasoline up to a 20%. Typical energy efficiencies are low, 25% to 35% when running at nominal power, much lower at partial load, but the engine is light, very powerful and not expensive.

Diesel cycle

The Diesel cycle is a first approximation to model the operation of a compression-ignition engine, first built by Rudolf Diesel in 1893, and used in most cars, nearly all trucks, nearly all boats, many locomotives, some small airplanes, and many large electric power systems and cogeneration systems. It is the reference engine from 50 kW to 50 MW, due to the fuel used (cheaper and safer than gasoline) and the higher efficiency. This is a reciprocating internal combustion engine with the same sketch as in Fig. 5.2, but substituting the spark-plug by the fuel injector (the fuel is injected at very high pressures, up to 200 MPa, to ensure immediate vaporisation). One of its key advantages compared to Otto engines is the great load increase per cylinder associated to the higher pressures allowed (the mixture of fuel and air would detonate in Otto engines at high compressions), and the further load increase associated to charging previously-compressed air (turbocharging). Another advantage is the much better performance at part load, since there it is achieved by injecting less fuel instead of by throttling, and the torque (power divided per angular speed, M=P/w), changes less with angular speed.

In the ideal air-standard Diesel cycle, the working fluid is just air, which is assumed to follow four processes (Fig. 5.4): isentropic compression, constant-pressure heat input from the hot source, isentropic expansion, and constant-volume heat rejection to the environment.

Fig. 5.4. The ideal Diesel cycle. Practical p-V diagrams are as in Fig. 5.3.

Similarly to the Otto cycle, the main parameters of ideal and real Diesel cycles are also the size, measured by the displacement volume (that may reach more than 1 m³ per cylinder in large marine engines), the compression ratio, r=V₁/V₂ (with a typical range of 16..22, limited just by strength), the cut-off ratio, r_c, or the mean effective pressure (in the range 1..2 MPa), or the maximum pressure (in the range of 3 MPa to 20 MPa), and the speed (with a typical range of 100..6000 rpm). The energy efficiency can be expressed as:

                                                                                                                     (5.3)

                  

Real compression-ignition engines take ambient air (often after a first stage compression) and compress it (inside the cylinder) so much, rising the temperature accordingly, that the fuel burns as it is injected (after a small initial delay due to vaporisation and combustion kinetics). The external compression is performed in a centrifugal compressor driven by a centrifugal turbine moved by the exhaust gasses (turbocharger); the air is cooled after external compression (inter-cooler) before further compression within the cylinders, to increase the efficiency, as explained in Multistage compression.

The higher pressures in Diesel than in Otto engines require a robust engine-frame and delicate fluid injection hydraulics (with injection pressure up to 200 MPa), but the wider range of fuels (from gas-oils to fuel-oils), the better fuel control, longer durability and economy are making Diesel engines to displace traditional gasoline-engines markets (for heavy-duty applications it has always been unrivalled). Typical energy efficiencies are from 30% to 54% (based on the lower heating value of the fuel), the latter, the largest of any single thermal engine, being achieved in large two-stroke marine low-speed engines with bores larger than 0.5 m: first, because the thermal losses decrease with size, and second, because the very low speed (some 100 rpm instead of the typical 3000 rpm for a car engine) allows for a more complete combustion (more time to burn, and burning nearly without volume change) and decrease friction losses (in spite of the fact that the mean piston speed stays at some 6..7 m/s for the whole range of reciprocating engines: from the 1 cm³ 50 W model, to a 1 m³ 5 MW 'three-store castle', per cylinder). Most large Diesel engines are supercharged, i.e. fed with compressed air instead of atmospheric air, usually by means of a turbocharger (a small compressor shaft-coupled to a small turbine driven at high speeds by the exhaust gases; some 10% of the fuel energy goes through that shaft), with an intermediate cooling of the compressed air before intake to the cylinders (intercooler). The two-stroke cycle is better suited to Diesel engines, since only air is used to sweep the burnt gases (scavenging), and not fresh mixture, but, because of the difficulty in lubricating, it is only used in the largest marine engines (10 MW..100 MW), where residual fuel (must be preheated to flow) can be used.

Mixed cycle

The mixed (or dual, or Sabaté, or Seiliger, or 5-point) cycle, sketched in Fig. 5.5, is a refinement to both Otto and Diesel cycles, at the expense of an additional parameter, the heat-addition pressure ratio, r_p=p₃/p₂.

  

Fig. 5.5. The dual or Sabaté cycle.

The energy efficiency can be expressed as:

                                                                                                         (5.4)

Brayton cycle

The Brayton cycle, named after the American engineer George Brayton (that built a two-stroke reciprocating engine in 1876 and advanced combustion chambers at constant pressure), is a good model for the operation of a gas-turbine engine (first successfully tested by F. Whittle in 1937, and first applied by the Heinkel Aircraft Company in 1939), nowadays used by practically all aircraft except smaller ones, by many fast boats, and increasingly been used for stationary power generation, particularly when both power and heat are of interest.

In the ideal air-standard Brayton cycle, the working fluid is just air, which is assumed to follow four processes (Fig. 5.6): isentropic compression, constant-pressure heat input from the hot source, isentropic expansion, and constant-pressure heat rejection to the environment. Contrary to reciprocating engines, the gas turbine is a rotatory device working at a nominal steady state (it can hardly work at partial loads); spark ignition is used to start up, since air compressor output temperature is not high enough to inflame the fuel).

Fig. 5.6. The ideal Brayton cycle in the T-s and p-V diagram, and the regenerative Brayton cycle.

Most gas turbines are internal combustion engines where the working fluid must be renovated continuously as sketched in Fig. 5.7, but some gas turbines use a closed-loop working fluid.

Fig. 5.7. Open and closed cycle gas turbine.

The main parameters of ideal and real Brayton cycles are the turbine-inlet temperature, T₃, the compressor pressure ratio, p=p₂/p₁ (with a typical range of 4 to 30), the compressor and turbine efficiencies, and the size, measured by the air mass flow rate. The energy and exergy efficiencies for the ideal Brayton cycle (compressor and turbine efficiencies of 100%), can be expressed as:

                      and                                 (5.5)

Real engine efficiencies are comparatively low, from 25% to 38%, but in combination with a bottoming vapour cycle, they reach 50..59% based on LHV; combined cycle power plants are the present standard in electricity generation. Contrary to reciprocating and steam engines, the gas turbine can only work with fine-tuned components, since it gives no net power if the compressor and turbine efficiencies fall below say 80% (modern gas turbines can have compressor efficiencies of 68% to 88%, and turbine efficiencies of 88% to 90%). It also needs high turbine-inlet temperature (modern aircraft gas turbines with blade cooling operate at up to 1700 K and at pressure ratios up to 30:1). It is easy to prove that for fixed extreme temperatures (ambient and turbine-inlet) there is a pressure ratio that maximises the work per unit mass flow rate, thus rendering the smallest engine for a given power, this optimum value being:

                                                                                                                        (5.6)

where h_C and h_T are the compressor and turbine isentropic efficiencies. .

Several improvements to the simple Brayton cycle are in use. Besides the multistage compression and expansion, the main variant is the regenerative cycle (Fig. 5.6), where heat from the exhaust gasses is used (from point 4 to 5) to heat up air before entering the combustion chamber (from point 2 to ideally up to point 2' in Fug. 5.6, although in practice the heat exchanger efficiency will limit this value). The heat recovery from the exhaust gasses may be also performed externally to the cycle, e.g. generating vapour in a heat exchanger (boiler), that may be directly used for heating applications or may even get expanded in a vapour turbine to produce further work (combined Brayton and Rankine cycles).

Stirling cycle and other gas cycles

Practical gas engines reduce to reciprocating engines and gas turbines, whose processes best represented by the dual cycle and the Brayton cycle, respectively, but there are more cycles of academic interest, as the Stirling cycle that models Stirling engines, invented in 1816 by the Scottish clergyman Robert Stirling, that, although faded away in the XIX c. against the steam and later gas engines, has got considerable attention lately.

The Stirling engine is an external combustion machine (all petrol and diesel engines, and practically all gas turbine engines, are of the internal combustion type, although, as said in the study of their cycles, they could operate as external combustion machines, in principle). The working substance may be just air or better helium, and the four processes followed are (Fig. 5.8): isothermal compression with heat input from the hot source, constant-volume compression by heat input from a regenerator (a porous solid matrix; an auxiliary displacer forces the gas through), isothermal expansion with heat rejection to the environment, and constant-volume heat rejection to the regenerator. Similar to the Stirling cycle is the Ericson cycle, where heat regeneration is isobaric instead of isometric as in the Stirling one (the regenerative Brayton cycle with infinitesimal multistage compressions and expansions approaches to the Ericson cycle). Notice that there is nothing essential to the four-process engines, and a three-process cycle (Fig. 5.8, named after J.J.E. Lenoir, a pioneer of two-strokes engines in the XIX c.) is used to model the operation of a pulse-reactor.

Fig. 5.8 The ideal Stirling cycle in the T-s and p-V diagram, and the ideal Ericson and Lenoir cycles.

Energy and exergy efficiencies of 36% and 50% respectively have been reached with prototype Stirling engines of up to 10 kW, the main problems being the regenerator loss of efficiency at high speeds (>30 Hz, i.e. >1800 rpm), the radiant heat looses at high temperature (>1000 K), incomplete exchange of gas between the hot and cold zones, and leaking at high pressure (>5 MPa).

Vapour power cycles

Rankine cycle

Most large electricity generating plants (central power stations), and very large ship engines, use water vapour (steam) as working fluid, following some variation of the basic Rankine cycle (named after the Scottish inventor William Rankine, that in 1859 wrote the first book on Thermodynamics), the only vapour power cycle in practical use since 1840 until in 1984 Alexander Kalina patented in the USA the cycle named after him.

The heat source for the boiler is usually the combustion products of a fuel (mainly coal) and air, or the primary refrigerant of a nuclear reactor, and the heat sink in the condenser is usually a water loop, open like in a river, or closed like in a cooling tower (as explained in Chapter 8). Thomas Newcomen is credited with the invention of the steam engine in 1705 for the purpose of driving the pumps used in clearing groundwater from mine shafts. Although the work-producing element was initially reciprocating cylinder-piston devices, in 1882 Gustav de Laval introduced the vapour turbine that has taken over.

Fig. 5.9. Carnot cycle within the two-phase region, basic Rankine cycle in the T-s and p-V diagram, and sketch of a vapour plant.

The four processes in a simple Rankine cycle are: isentropic compression of the liquid from 1 to 2 (Fig. 5.9), isobaric heating of the liquid, boiling and super-heating the vapour (from 2 to 3), isentropic expansion from 3 to 4, and isobaric heat rejection until full condensation of the vapour. The Rankine cycle is less efficient than the Carnot cycle (Fig. 5.9), but it is more practical since the compression is not in the two-phase region and only requires a small work, and the expansion is mainly in the gaseous phase (high-speed droplets erode turbine blades). Water is not the ideal working substance because it changes phase at relatively low temperatures (below the critical point at 647 K), generating a lot of entropy in the heat transfer from typical high-temperatures heat-sources: 1000 K in nuclear reactors up to 2000 K in conventional combustion plants. Nevertheless, water is practically the only working substance used, because of its good thermal properties and availability. A caution note is that the right-hand-side end of the two-phase region, the saturated vapour line, that for water in the T-s diagram has the shape shown in Fig. 5.9, may be more vertical and even have negative slope at low temperatures for heavier molecular substances, naturally avoiding the problem of wet-vapour at the turbine.

Since, at ambient temperature (the heat sink), water change phase at lower-than-atmospheric pressure (e.g. 5 kPa at 33 ºC) the condenser must operate under vacuum and a so-called 'deaerator' is needed to remove non-condensable gasses from the feed water or infiltration; moreover, removal of oxygen and carbon dioxide in feed water is always desirable to avoid corrosions in the circuit. Gas solubility in a liquid decreases near the pure-liquid vapour saturation curve, thus deaeration may be achieved by heating the liquid at constant pressure (e.g. adding some vapour) or by making vacuum at constant temperature (e.g. with a small jet of vapour by Venturi suction).

Maximum temperature in a steam power plant is limited by metallurgical constraints to less than 900 K (some 600 ºC), and the maximum pressure depends on the variations to the simple Rankine cycle used, with typical values of 10 MPa (supercritical Rankine cycles surpass 22 MPa). For a simple Rankine cycle with an ideal turbine exiting just at the vapour saturation point, the energy efficiency, with the perfect substance model, is:

                                                                                                        (5.8)

where c_p=2 kJ/(kg×K) is an average isobaric thermal capacity of water vapour, and h_lv1=2400 kJ/kg the enthalpy of phase change at T₁ (Fig. 5.9).

The main variants of the simple Rankine cycle are reheating (a multistage expansion) and regeneration (bleeding some vapour from the middle of the turbine, and before reheating if used, to heat the feed water). These feed-water heaters may be of the open or closed type (Fig. 5.10). In an open feed-water heater, steam extracted at some turbine stage is added to the main feed water stream (that must be previously pressurised to avoid boiling). In a closed feed-water heater, the extracted steam goes through the shell of a shell-and-tubes heat-exchanger and discharges in a lower-pressure heater or the condenser. The mass fraction of vapour to be extracted is designed to be able to heat the main feed-water stream until the saturation temperature of the extracted steam. 

Fig. 5.10. Open and closed feed-water heaters.

Steam turbines (notice that the term is used indistinguishable for the roto-dynamic device and for the whole power plant) are the largest thermal power plants, typically limited to 1000 MW per unit in nuclear power stations, with typical efficiencies from 30% to 40%, although supercritical power plants reach 45% (based on LHV). An advantage of steam turbines, extensive to all external combustion engines, is that any kind of fuel or other heat source may be used, contrary to internal combustion engines, where only fluid fuels, either residual to petroleum distillation but most of the times distillate fluids, can be used. The isentropic efficiency of the turbine is typically 85%, and the electromechanical efficiency of the alternator 98%. The energy and exergy balances in a typical steam power plant are presented in Table 1.

Table 1. Energy and exergy balances in a typical steam power plant

Component	Energy output	Exergy use
Combustion chamber	0	0.30
Boiler tubes heat transfer	0	0.30
Exhaust gasses (chimney)	0.15	0.01
Turbine	0	0.05
Condenser	0	0.03
Water cooling (condenser)	0.55	0.01
Shaft	0.30	0.30
	1	1

Kalina cycle

Most heat input/output to/from a plant's working fluid is from variable temperature heat sources/sinks, as the hot combustion gasses and the cold cooling water streams in the normal Rankine cycle. If, instead of using a pure fluid, a mixture were used in a Rankine cycle, due to its variable boiling/condensing temperature, the phase-change heating/cooling could better match the temperature rise/fall in the heating/cooling streams.

The only successful vapour-mixture cycle developed has been the Kalina cycle, proposed by Alexander Kalina in 1984 (the first Kalina power plant, of 3 MW, opened in 1991). Its characteristics are:

·         An ammonia-water mixture (70%-NH₃ and 30%-H₂O) is used (a large know-how from absorption-refrigeration cycles existed).

·         The turbine exit goes through a distillation and the heavy fraction (40%-NH₃ and 60%-H₂O) is condensed, and, once pressurised, the two fractions mix before entering the boiler again.

·         Because temperature jumps across heat exchangers can be more uniform, the efficiency increases some 10% for a normal power station, but for special low-temperature applications more than 30%.

·         Because ammonia lowers the boiling point, the Kalina cycle is better suited to low-temperature applications than the Rankine cycle, as in bottoming cycles (see below), geothermal power plants, and so on.

 

 

Combined power cycles

In a Rankine cycle, one single substance, like water, cannot easily match the high-temperature side (e.g. at the temperature of the combustion gasses, 1500 K to 2000 K, it is very difficult to transfer heat to water vapour), and the low-temperature side (to condense water vapour at ambient temperature is difficult because of the very low pressures and densities). The use of two Rankine cycles with different substances has been tried without success (an experimental plant was built with mercury for the top cycle and water for the bottom cycle).

The combination that has reached considerable success is the Brayton-Rankine combined cycle, where the exhaust gasses from a gas turbine are used to supply the heat in the boiler of a vapour turbine operating at not too-high temperatures. The Brayton-Kalina combination may be particularly successful in this respect. Natural-gas-fuelled combined power stations are the rule nowadays because of their low installation cost (some 450 $/kW against 1100 $/kW for coal stations), short-time operations start-up (2 years vs. 3.5 years for coal), and lower environmental impact (nuclear, coal and hydroelectric stations are on hold in Europe and USA), although wider fluctuation in gas price make the choice risky.

The turbodiesel engine can also be considered a combined Diesel-Brayton cycle (Fig.5.11).

Fig. 5.11. Combined power cycles: a) Brayton-Rankine, b) Diesel-Brayton.

There are other combined cycles, still in the developing stage, that show great promise from both the energetics and the environmental aspects, like Graz cycle in Fig. 5.12 (proposed in 1985 by Prof. H. Jericha from Graz University, Austria), where pure oxygen is used as oxidiser instead of air (to avoid NO_x-emissions and to have a pure exhaust gas), which on combustion with a fuel (e.g. natural gas) yields only water vapour and carbon dioxide as products, easily separated in a water condenser, ready to CO₂ capture, a major concern in the fight against global warming by anthropogenic greenhouse gases. Of course, CO₂ capture can be done with all traditional power cycles, e.g. by selective chemical absorption from the exhaust, but it is not competitive with oxy-fuel cycles like Graz's. Application of these air-independent propulsion concepts to submarines is evident.

Fig. 5.12.       The Graz cycle, a new combined gas-steam power cycle burning fuel with pure oxygen, and capturing carbon dioxide.

Propulsion

To propel or impel is to force a body to move forwards, and it requires a push (a force) and an energy expenditure (motive power) to overcome the drag imposed in practice by the nearby objects (if there were no interactions with the surroundings, a body could move without propulsion, as in outer space). Most propulsive systems aim at moving a body over land or through a fluid at constant speed, but all propulsive systems need to accelerate the body at some stage. The power required to propel a body of mass m (e.g. a car) may be expressed as:

                                                                                                   (5.9)

with:

                              (5.10)

where a is the acceleration applied, v the actual speed, q the slope of climbing, c_R a tyre-rolling coefficient (typically 10^-2), c_D a fluid-drag coefficient (typically of order 1, but may drop to 0.1 for streamlined bodies), A_F the frontal area projected by the body in the direction of motion, and r the density of the fluid medium.

The idealisation of a long rope being hauled in absence of any other interaction may be a good propulsion paradigm; because of force reciprocity, propulsion may be thought both as a pushing the environment (the rope, in the example) to the rear, and as pulling (the rope) from the front.

As a first model, for a constant-mass vehicle, the momentum balance in the direction of motion may be written as , m being the mass of the body,  its acceleration, Tthe thrust force (pull or push on the rope example) and D the drag force resisting the motion (friction with the rope, frictional and pressure drag in a fluid, etc.). For these vehicles, the drag can be measured by towing at constant speed, and the thrust by breaking. For steady motion at speed v, the energy dissipated just by the motion is Dv=Tv, and a propulsion efficiency may be defined to compare this energy with the energy supplied by the engine that cause the motion,  (because it is usually through a shaft), in the way:

                    (shaft-propeller efficiency)                                                                               (5.11)

Notice that all engines consume more energy that what they supply: electrical engines, elastic engines, etc., but particularly thermal engines, where this internal energy conversion efficiency, h_e, is of the order of 30% to 50% (in electric motors it may be typically 95%).

Most propulsion systems may be considered in this way just power plants that apply a torque, M, to a shaft at a certain rotation speed, w, producing a power =Mw. The shaft then is mechanically coupled to the propeller itself, which may be the friction wheels in land vehicles, or the pressure-reaction blade-propeller in sea and air vehicles.

But there are some propulsion systems based on the change of momentum of a working fluid, notably rockets, jet-engines and water jets, which have inlets and outlets through which some fluid flow. The momentum balance for an open system is :

                                                                                    (5.12)

where m is the mass of the body being propelled (may be changing),  is the sum of externally applied volumetric forces (gravitational or electromagnetic),  the sum of externally applied surface forces at the wall, and  and p_e the exit velocity and pressure through any opening of area A_e with an outward normal vector .

Rocket propulsion

For steady horizontal motion of a body with just one exit (rocket) the momentum balance, in the direction of motion, reduces to (see Eq. 5.20):

                                                                                                                       (5.13)

where p₀ is the pressure in the surrounding media (acting on the impermeable walls), that may be zero in outer space, and the propulsion efficiency, h_p, is defined as the ratio of thrust power, Tv, to whole power emanating from the body, that is the sum of thrust power, Tv (communicated to the surrounding media) and residual kinetic power of the jet relative to the surrounding media,  (communicated to the jet); i.e.:

                                                             (5.14)

where the simplification of considering p_e=p₀ is always acceptable (it is true for subsonic exits, and for supersonic exits with 'adapted nozzle', and very approximate even for large pressure departures). Equation 5.14 is shown in Fig. 5.13.

Fig. 5.13. Rocket propulsion efficiency vs. flight speed to exit speed ratio.

Most rockets are heat engines because they transform the heating power of a fuel/oxidiser mixture into mechanical energy (the kinetic energy of the jet), and thus an energy efficiency may be defined by:

                                                                                                                                        (5.15)

where h_P is the heating power of the fuel/oxidiser mixture per unit mass of reactants. Notice that high energy-efficiency demands high exit speed, demanding also high flight speed to keep a reasonable propulsion efficiency; i.e., rocket propulsion is inefficient at low speed. Compressed-gas rockets are also heat engines, but new magnetohydrodynamic and electrodynamic rockets are not.

Air-breathing propulsion

For steady horizontal motion of a body with one inlet and one exit (like a jet engine) the momentum balance, in the direction of motion, reduces to (see 5. 13):

                                                                                     (5.16)

where the approximation to neglect the additional mass flow rate of fuel is introduced. The propulsion efficiency, defined as for rockets, becomes:

                                                    (5.17)

where the approximation to consider the inlet speed equal to the flight speed has been introduced. Notice that the thrust, , may be obtained with a large Dv and small flow rate or vice versa, but the propulsion efficiency, h_p=2/(2+Dv/v), shows that it is better to have a small Dv and a large , what has been implemented in practice, where turbofans of high bypass have superseded simple turbojets.

Similarly to chemical rockets, air-breathing engines (turbojets) are heat engines that transform the heating power of a fuel into mechanical energy (the kinetic energy of the exiting jet minus the one at the inlet), and thus an energy efficiency may be defined by:

                                                                                                                           (5.18)

where h_P is the heating power of the fuel/oxidiser mixture per unit mass of fuel. Notice that the energy efficiency is based on thermal-to-mechanical conversion, and the propulsion efficiency is based on mechanical-to-propulsive ratio, thus, the efficiency using the fuel for propulsion is h_eh_p. The overall energy balance of the air-breathing engine may help to identify every term; in axes fixed to the surroundings:

                                                                                                                  (5.19)

with:

                                            (5.20)

 being the chemical enthalpy released,  the propulsive power,  the kinetic energy of the exit jet, and  the thermal enthalpy of the exit jet.

It is important to compare air-breathing propulsion with rocket propulsion for the same fuel expenditure (notice that for rockets , whereas for turbojets  is of the order of 2% of ). The quotient of respective thrusts is:

                                                                             (5.21)

showing that air-breathing engines are much better than rockets (if there is air!) because typical values are v_e,turbojet~2×10² m/s, and v_erockett~5×10³ m/s.

 

Cogeneration

Cogeneration is the term coined for the combined generation of work and heat (and, by extension, also of cold). Any heat engine must reject a lot of heat to the environment, but this energy has no exergy (i.e. utility) in the ideal heat engine. Practical engines however reject useful heat in the exhaust gasses and in the cooling circuit, that could be directly used (as when heating the car cabin with the engine cooling water), or the engine modified to provide more useful heat at a small expense in work output.

Without cogeneration, the work needed was taken from the electrical grid, and the heat needed was generated locally in a boiler (heat is much more difficult to transport than work). Cogeneration is provided by using a heat engine (a steam turbine, a gas turbine or a reciprocating engine) to generate work and, at the same time, heat (usually in a exhaust gasses boiler). The great advantage of cogeneration is the energy saving; an additional advantage may be the autonomy gained by self-production, and the main drawback is that the ratio of work-to-heat generation is rather stiff, none can be easily accumulated, and the actual need of work and heat may vary a lot with time (that was one of the main advantages of central production: the levelling of the averaged demand). To stiffness of a cogeneration plant is relaxed by the possibility to send excess work-power to the electrical grid, a convenience that has been enforce by public authorities on account of the social benefit that saving in primary energy resources (fossil fuels) cogeneration brings.

Efficiencies of cogeneration plants are high, but care must be paid not to mix work and heat values (everybody knows that work is usually two or three times more expensive than heat, i.e. the output should be consistently measured in exergy, so that the, instead of W+Q, the output value is W+Q(1-T₀/T), where T is the temperature at which heat is delivered.

Reciprocating engine cogeneration

The useful heat-to-work ratio is around 0.6. It is only used for direct heating with the exhaust gasses or the cooling water, but not to generate vapour.

Gas turbine cogeneration

The useful heat-to-work ratio is around 2. It does not require changes in the power plant, it is a versatile cogeneration system, and its use is expanding in most industries: textile, chemical, food. Because the exhaust gasses are relatively hot (some 700 K), it is suitable for high temperature applications.

Vapour turbine cogeneration

The useful heat-to-work ratio is around 7 (suitable for the iron industry, but still low for the ceramic and glass industry). It directly provides useful vapour, either by extracting some of it at an intermediate stage in the turbine, or by not expanding to the low pressures of ambient temperatures but to higher than atmospheric pressures (it is called a back-pressure turbine), to render the whole turbine outlet vapour useful.

 

Efficiency in power generation with heat engines and other generators

Power cannot be generated; only converted from one form to another; what is implied is the generation of some useful power at the expense of some other less-convenient power. But recall that we have restricted here the term ‘power’ to mechanical or electrical power (i.e. excluding thermal power), so that power generation can be seen as the generation of electricity (or equivalent power) from any other energy source: thermal, chemical, nuclear, etc.

Power generation efficiency can be defined as “useful output power divided by input power”, but it is not rigorous enough since at least two choices exist for the evaluation of input power: a) heat-equivalent power, and b) work-equivalent power. Table 2 presents typical values of power generation efficiencies using the raw input power (choice a), the most commonly used, although the net input power criterion, i.e. the exergy or available energy of the raw energy source, would give a sounder measure of the ‘technological efficiency’ of the power plant.

Table 2. Some power generation efficiencies.

Energy source	Typical efficiency [%]	Typical range [%]
Photovoltaic	10	5..15
Solar thermal	15	10..25
Gas turbine	30	15..38
Spark Ignition ICE	30	25..35
Nuclear	33	32..35
Steam turbine	33	25..39
Wind turbine	40	30..50
Compression Ignition ICE	40	35..49
Fuel cell	45	40..70
Combined GT-ST	50	45..60
Hydroelectrical	85	70..90

It is without saying that power generation efficiency is not the only criterion, neither the most important, to quantify engine excellence. The generic goal of maximum power at minimum cost, should include the cost of design (e.g. new technologies), cost of manufacture (e.g. new materials), cost of implementation (e.g. size and weight), cost of operation (e.g. specific fuel consumption, but also pollutant emissions and noise level), cost of maintenance (reliability), and even the cost for disposal.

Problem 1:

Consider a piston-cylinder device, holding initially 500 cm³ of ambient air, which is to be quickly compressed to 50 cm³. We aim at analysing the four following cyclic processes (find the energy exchanges and the energy efficiency):

a)      After the compression, 1 kJ of heat is added at constant volume, and then a quick expansion follows, ending with a constant volume heat release.

b)      After the compression, 1 kJ of heat is added at constant volume, and then a quick expansion follows, ending with a constant pressure heat release.

c)      After the compression, 1 kJ of heat is added at constant pressure, and then a quick expansion follows, ending with a constant volume heat release.

d)      After the compression, 1 kJ of heat is added at constant pressure, and then a quick expansion follows, ending with a constant pressure heat release.

Solution

a)      After the compression, 1 kJ of heat is added at constant volume, and then a quick expansion follows, ending with a constant volume heat release.

            Let us first work out the initial compression, common to all four cycles.

      From 1 to 2. It is a control-mass problem, since in air-standard cycles the mass of air is assumed to remain all the time within the cylinder. The energy balance is thus DE=W+Q. The enclosed mass, assuming p₀=10⁵ Pa and T₀=288 K, is =10⁵∙500∙10^-6/(287∙288)=0,61 g. After the isentropic compression, the state is V₂=50 cm³,p₂=p₁(V₁/V₂)^g=10⁵∙(500/50)^1.4=2.5 MPa, T₂=T₁(V₁/V₂)^g-1=288∙(500/50)^0.4=723 K. The work the gas receives is W₁₂=DE=mc_v(T₂-T₁)=0.61∙10^-3∙ (1000-287)(723-288)=190 J, and the heat is null, Q₁₂=0. Notice that W₁₂ is not the work that must be done; ambient pressure contributes, too.

      Let start now with the first cycle described, that corresponds to the well-known Otto cycle, represented in Fig. 1.

Fig. 1. Otto cycle.

      From 2 to 3, Q₂₃=1 kJ is received with V₃=V₂ (i.e. W₂₃=0), and, from the energy balance DE=W+Q=mc_v(T₃-T₂), T₃=T₂+Q₂₃/(mc_v)=723+1000/(0.61∙10^-3∙713)=3020 K, too high for practical combustion engines, that only reach 2000 K, but this is a simple academic exercise. Furthermore, p₃=p₂T₃/T₂=2.5∙10⁶∙3020/723=11 MPa.

      From 3 to 4 there is an isentropic expansion, from V₃=50 cm³ to V₄=500 cm³, so that p₄=p₃(V₃/V₄)^g=11∙10⁶∙(50/500)^1.4=418 kPa, T₄=T₃(V₃/V₄)^g-1=3020∙(50/500)^0.4=1210 K. The work the gas delivers is W₃₄=DE=mc_v(T₄-T₃)=0.61∙10^-3∙713(1210-3020)=-790 J, and the heat is null, Q₃₄=0. Notice again that W₃₄ is not the work delivered to the shaft; ambient pressure resists, too.

      From 4 to 1, heat is released at constant volume (i.e. W₄₁=0), and, from the energy balance DE=W+Q=mc_v(T₁-T₄)=0.61∙10^-3∙713(288-1210)=-400 J. Table 1 shows the summary of energy exchanges.

Table 1. Summary of energy exchanges.

Otto	DE	W	Q
1-2	190	190	0
2-3	1000	0	1000
3-4	-790	-790	0
4-1	-400	0	-400
cycle	0	-600	600

      The energy efficiency of a power cycle is defined as net work delivered divided by heat input, h=W_net/Q_in, that here takes the value:

                                    

      which can be compared with the well-known efficiency of the ideal Otto cycle: h=1-1/r^g-1=1-1/10^0,4=0,60, r being the compression ratio r=V₁/V₂ (with r=10 in this case).

b)      After the compression, 1 kJ of heat is added at constant volume, and then a quick expansion follows, ending with a constant pressure heat release.

Fig. 2. New cycle.

      From 1 to 2 has been worked at the very beginning: p₁=10⁵ Pa, T₁=288 K, p₂=2,5 MPa, T₂=723 K, Q₁₂=0 and W₁₂=190 J.

      From 2 to 3 is the same as in the Otto cycle: p₂=2,5 MPa, T₂=723 K, p₃=11 MPa, T₃=3020 K, Q₂₃=1 kJ and W₂₃=0.

      From 3 to 4 there is an isentropic expansion, from p₃=11 MPa to p₄=10⁵ Pa, so that V₄=V₃(p₃/p₄)^1/g =50∙10^-6∙(11/0,1)^1/1,4=1400 cm³, T₄=T₃(V₃/V₄)^g-1=3020∙(50/1400)^0.4=800 K. The work the gas delivers is W₃₄=DE=mc_v(T₄-T₃)=0.61∙10^-3∙713(800-3020)=-970 J, and the heat is null, Q₃₄=0.

      From 4 to 1, heat is released at constant pressure. The energy balance DE=W+Q takes the form Q=DH=mc_p(T₁-T₄)=0.61∙10^-3∙1000(288-800)=-310 J, whereasW₄₁=-p₁(V₁-V₄)=90 J. Table 2 shows the summary of energy exchanges.

Table 2. Summary of energy exchanges.

	DE	W	Q
1-2	190	190	0
2-3	1000	0	1000
3-4	-970	-970	0
4-1	-220	90	-310
cycle	0	-690	690

      The energy efficiency now takes the value:

                                    

c)      After the compression, 1 kJ of heat is added at constant pressure, and then a quick expansion follows, ending with a constant volume heat release.

            This corresponds to the ideal Diesel cycle, depicted in Fig. 3.

Fig. 3. Diesel cycle.

      From 1 to 2 has been worked at the very beginning: p₁=10⁵ Pa, T₁=288 K, p₂=2,5 MPa, T₂=723 K, Q₁₂=0 and W₁₂=190 J.

      From 2 to 3, Q₂₃=1 kJ is received at constant pressure. The energy balance DE=W+Q takes the form Q=DH=mc_p(T₃-T₂), so thatT₃=T₂+Q₂₃/(mc_p)=723+1000/(0.61∙10^-3∙1000)=2370 K and V₃=V₂T₃/T₂=50∙10^-6∙2370/723=164∙10^-6 m³, with a work (output) of W₂₃=-p₂(V₃-V₂)=-290 J.

      From 3 to 4 there is an isentropic expansion, from V₃=164 cm³ to V₄=500 cm³, so that p₄=p₃(V₃/V₄)^g=2,5∙10⁶∙(164/500)^1.4=525 kPa, T₄=T₃(V₃/V₄)^g-1=2370∙(164/500)^0.4=1520 K. The work the gas delivers is W₃₄=DE=mc_v(T₄-T₃)=0.61∙10^-3∙713(1520-2370)=-370 J, and the heat is null, Q₃₄=0.

      From 4 to 1, heat is released at constant volume (i.e. W₄₁=0), and, from the energy balance DE=W+Q=mc_v(T₁-T₄)=0.61∙10^-3∙713(288-1520)=-530 J. Table 3 shows the summary of energy exchanges.

Table 3. Summary of energy exchanges.

Diesel	DE	W	Q
1-2	190	190	0
2-3	710	-290	1000
3-4	-370	-370	0
4-1	-530	0	-530
cycle	0	-470	470

      The energy efficiency here takes the value:

                                    

      which can be compared with the well-known efficiency of the ideal Diesel cycle:

                                   

      r being the compression ratio r=V₁/V₂ as for the Otto cycle (r=10), and r_c being the cut-off ratio, r_c=V₃/V₂=3,3.

d)      After the compression, 1 kJ of heat is added at constant pressure, and then a quick expansion follows, ending with a constant pressure heat release.

      This corresponds to the ideal Brayton cycle, depicted in Fig. 4.

Fig. 4. Brayton cycle.

      From 1 to 2 has been worked at the very beginning: p₁=10⁵ Pa, T₁=288 K, p₂=2,5 MPa, T₂=723 K, Q₁₂=0 and W₁₂=190 J.

      From 2 to 3, Q₂₃=1 kJ is received at constant pressure as in the Diesel cycle above, yielding T₃=2370 K, V₃=164∙10^-6 m³, and W₂₃=-290 J.

      From 3 to 4 there is an isentropic expansion, from p₃=p₂=2,5 MPa to p₄=10⁵ Pa, so that V₄=V₃(p₃/p₄)^1/g =164∙10^-6∙(2,5/0,1)^1/1,4=1640 cm³, T₄=T₃(V₃/V₄)^g-1=2370∙(164/1640)^0.4=940 K. The work the gas delivers is W₃₄=DE=mc_v(T₄-T₃)=0.61∙10^-3∙713(940-2370)=-620 J, and the heat is null, Q₃₄=0.

      From 4 to 1, heat is released at constant pressure. The energy balance DE=W+Q takes the form Q=DH=mc_p(T₁-T₄)=0.61∙10^-3∙1000(288-940)=-400 J, whereasW₄₁=-p₁(V₁-V₄)=114 J. Table 4 shows the summary of energy exchanges.

Table 4. Summary of energy exchanges.

Brayton	DE	W	Q
1-2	189	189	0
2-3	714	-286	1000
3-4	-619	-619	0
4-1	-284	114	-398
cycle	0	-602	602

      The energy efficiency here takes the value:

                                    

      which can be compared with the well-known efficiency of the ideal Brayton cycle:

                                    

      p₁₂ being the pressure ratio p₁₂=p₂/p₁.

Problem2:

A four-cylinder four-stroke 50 kW gasoline engine, running at 3000 rpm, has a compression ratio of 9, 500 cm³ of displacement per cylinder. Evaluate:

a)             The conversion efficiency, deducing it from the basic cycle states.

b)             Maximum pressure and temperature in the cycle, and mean effective pressure.

Solution.                                  

Fig. 1. Sketch of the standard Otto cycle and corresponding piston position.

a)      The conversion efficiency, deducing it from the basic cycle states.

The cold-air-standard model takes as working fluid plain air with constant properties (those at the inlet, i.e. cold), what renders the analysis simple. The energy exchanges for the trapped control mass, m, are W₁₂/m=c_v(T₂-T₁), Q₁₂=0, W₂₃=0, Q₂₃/m=c_v(T₂-T₁), W₃₄/m=c_v(T₄-T₃), Q₃₄=0, W₄₁=0, Q₄₁/m=c_v(T₁-T₄), and, with the isentropic relation,TV^g-1=constant, in terms of the compression ratio r≡V₁/V₂, the energy efficiency becomes:

        

with a numerical values of h_e=1-1/9^0.4=0.58. Taking as reference a Carnot cycle with the same extreme temperatures, the exergy efficiency h_x would be:

   

where T₃ is computed below.

b)             Maximum pressure and temperature in the cycle, and mean effective pressure.

For the isentropic compression T₂=T₁r^g-1=288×9^0.4=694 K, p₂=p₁r^g=100×9^1.4=2.2 MPa. From the mechanical power:

        

where =50 kW, Z=4 cylinders, m=p₁V₁/(RT₁)=100×10³×0.5×10^-3/(287×288)=0.68×10^-3 kg is the control mass per cylinder (neglecting the residual mass), and n=(3000/60)/2=25 s^-1 is the control-mass processing rate (i.e. the charge is renovated 25 times per second; remember that only half of the cycles count, in a 4-stroke engine). The maximum temperature obtained from that is T₃=1060 K, and then the maximum pressure p₃=p₂(T₃/T₂)=2.2×(1060/694)=3.3 MPa. Substitution in the exergy efficiency above, yields h_x=0.80.

The mean effective pressure, p_me, is the unit work divided by the displacement volume, i.e. the pressure difference that produces the same power: