{"version":1,"lectureId":"01M46CT1ZEF8KJE0SG50T7XMW7","attempt":0,"publication":{"slug":"working-principles-of-the-four-stroke-gasoline-internal-combustion-engine","title":"Working Principles of the Four-Stroke Gasoline Internal Combustion Engine","subject":"engineering","summary":"A first look inside a gasoline engine, built for beginning engineering students. One cylinder is drawn to scale and set in motion: the cylinder, piston, connecting rod, crankshaft, valves and spark plug, then the intake, compression, power and exhaust strokes that take two turns of the crank. The lecture then follows the energy, from the chemistry of burning fuel, to gas pressure, to work as the integral of p dV and the area of the Otto cycle loop, and ends with why the compression ratio sets the efficiency and why knock limits it.","metaDescription":"How a four-stroke gasoline engine works: the parts of one cylinder, the four strokes, and how burning fuel becomes work on the crankshaft.","transcript":"Turn the key in a car, and drops of gasoline begin burning inside metal cylinders, each one shoving a piston about twenty five times a second. This lecture follows one cylinder through its four stroke cycle, and tracks how the chemical energy in the fuel becomes work on a turning shaft. Here is where we are going. Energy changes form, from fuel, to heat, to pressure, to work, and the engine does it in four strokes: intake, compression, power, and exhaust. Here is one cylinder, cut open. The cylinder itself is a smooth bore, closed at the top by the cylinder head. Inside it slides the piston, sealed against the wall by rings. Everything happens in the small space above it. Under the piston, a connecting rod links it to the crankshaft. The crank arm swings the bottom of the rod around a circle. In the head sit two valves. The intake valve lets in the mixture of air and fuel, and the exhaust valve lets burned gas out. Between them stands the spark plug. Now turn the crank. The rod changes the crank's rotation into straight up and down sliding of the piston, and it works just as well in reverse. Notice the two places where the piston stops and turns around. The top is top dead center, the bottom is bottom dead center, and one trip between them is a stroke, half a turn of the crank. Bring it back to the top, and we are ready. One full cycle takes four strokes, so the crank turns twice for every single power stroke. Intake. The intake valve opens and the piston slides down. The falling pressure draws a fresh charge of air, with a fine mist of fuel, in through the port. Compression. Both valves are shut, and the piston rises, squeezing the charge to about one ninth of its starting volume. Squeezed that hard, the gas heats up and its pressure climbs. Power. Near the top the spark plug fires. A flame races through the mixture in a few thousandths of a second, and the hot gas, now at many times its earlier pressure, drives the piston down. That is the only stroke that delivers work. The other three are carried through by the spinning crankshaft and its heavy flywheel, or by the other cylinders of the engine. Exhaust. The exhaust valve opens, and the rising piston pushes the spent gas out. Then the exhaust valve closes, the intake valve opens again, and the cycle starts over. Four strokes, two turns of the crank, one push. Next we look inside that push, and follow the energy from the fuel to the crankshaft. So where does the push come from? Gasoline is a blend of hydrocarbons, and octane is a good representative. It burns with oxygen from the air to make carbon dioxide and water, releasing heat. The products hold less chemical energy than the reactants did. The difference comes out as heat, about forty four megajoules for every kilogram of gasoline, so burning a mass m f of fuel releases m f times that. Set free inside a closed cylinder, that heat drives the gas past two thousand kelvin. For a gas, p V equals m R T, so at fixed volume a hotter gas means a higher pressure. And pressure is what pushes the piston. Pressure is force per unit area. On a piston crown of area A, the gas pushes with a force p times A. On the right, I will draw that pressure against the volume above the piston. Let the piston move a small distance d x. The force does work p A d x, and A times d x is exactly the small volume swept, d V. Add those pieces over a whole stroke, and the work is the integral of p d V. Work is the area under a curve of pressure against volume. Now trace one ideal cycle. Intake fills the cylinder at about atmospheric pressure, while the volume grows from the small clearance volume up to the whole cylinder. Compression runs from state one to state two. It is too quick for much heat to leak away, so the pressure climbs steeply as the volume shrinks. Then the spark. The burn is so fast that the piston hardly moves, so the heat arrives at nearly constant volume, and the pressure jumps from state two to state three. The power stroke is the expansion from three to four, the hot gas pushing the piston all the way down. At the bottom the exhaust valve opens and the pressure drops. Then the exhaust stroke sweeps the rest out along the bottom line, back to where we began. Now read the work straight off the picture. During expansion the gas does work on the piston, equal to all the area under the top curve. During compression the piston does work on the gas instead: the area under the bottom curve, which has to be paid back. Take one from the other. The shared part cancels, and what is left is the area enclosed by the loop, the net work one cylinder delivers every cycle. So the chain is complete. Fuel becomes heat, heat becomes pressure, and pressure acting through the swept volume becomes work on the crankshaft. We have the work. Now, what did it cost? Efficiency is the net work out, divided by the heat the fuel put in. Net work is the heat in minus the heat thrown away with the exhaust, so efficiency is one minus heat out over heat in. In the ideal cycle both heats are added and removed at constant volume, so each is proportional to a temperature change: heat in from state two to three, heat out from four back to one. Along the two adiabats, the temperature ratio depends only on the compression ratio r, the full volume over the clearance volume, raised to the power gamma minus one. Gamma is about one point four for air. Those equal ratios make the temperatures cancel, and we are left with the Otto efficiency: one minus one over r to the gamma minus one. It depends only on how hard the engine compresses. Now put in numbers. Here is that formula plotted against r. At a compression ratio of four, the ideal engine turns about forty three percent of the heat into work. At eight, fifty six percent. At ten, sixty. Squeeze harder still and it keeps climbing, but ever more slowly. So why do gasoline engines stop near ten to thirteen? Squeezing heats the mixture, and too much heat makes it ignite on its own before the spark. That is knock, and it can wreck a piston. A fuel's octane rating measures how well it resists. Real engines also fall below the ideal curve. Heat leaks into the cylinder walls, friction drags on the moving parts, and the burn takes time. A good gasoline engine turns roughly a third of its fuel energy into work, and the rest leaves as heat. Let's gather it up. Four strokes take two turns of the crank, and only one of them delivers work. The valves time the flow of gas in and out, and the spark times the burn. Burning fuel raises the gas pressure, and the work it does is the integral of p d V, the area of the loop. And efficiency climbs with compression ratio, until knock sets the limit. That, in one cylinder, is how a gasoline engine works.","watch":{"version":1,"scenes":[{"title":"One Cylinder, Four Strokes","start":0,"end":162.61085416666663,"objects":{"bdc":"a Line [gray] labelled \"upright(\"BDC\")\" drawn in engine (start=(-1.8, 2.9), end=(-1.1, 2.9), dashed=True)","card":"a Title that says \"Introduction to Thermal Engineering — Working Principles of the Four-Stroke Gasoline Internal Combustion Engine\"","crank":"a VariableNumber","crank_arm":"a Line [blue] drawn in engine (end=((1.2 * sin((crank * 0.017453292519943295))), (1.2 * cos((crank…)","crank_path":"a Circle [gray] drawn in engine (radius=1.2)","engine":"a Figure (x_range=(-3.2, 3.2), y_range=(-1.6, 7.6), aspect=(6.4, 9.2))","flow_in":"a Vector [green] drawn in engine (start=(-2.0, 6.9), end=(-0.8, 5.9))","flow_out":"a Vector [magenta] drawn in engine (start=(0.8, 5.9), end=(2.0, 6.9))","head":"a Line [gray] drawn in engine (start=(-1.0, 5.6), end=(1.0, 5.6))","heading":"a Heading that says \"Inside One Cylinder\"","lift_in":"a VariableNumber","lift_out":"a VariableNumber","name_cyl":"a Point [text] labelled \"upright(\"cylinder\")\" drawn in engine (location=(1.0, 2.3), show_marker=False)","name_in":"a Point [text] labelled \"upright(\"intake valve\")\" drawn in engine (location=(-0.55, 6.5), show_marker=False)","name_out":"a Point [text] labelled \"upright(\"exhaust valve\")\" drawn in engine (location=(0.55, 6.5), show_marker=False)","name_piston":"a Point [text] labelled \"upright(\"piston\")\" drawn in engine (location=(0.95, ((1.2 * cos((crank * 0.017453292519943295))) + sqrt((12.…, show_marker=False)","name_plug":"a Point [text] labelled \"upright(\"spark plug\")\" drawn in engine (location=(0.0, 6.8999999999999995), show_marker=False)","name_rod":"a Point [text] labelled \"upright(\"connecting rod\")\" drawn in engine (location=((((1.2 * sin((crank * 0.017453292519943295))) + 0.0) / 2.0), (…, show_marker=False)","piston":"a Polygon [blue] drawn in engine (vertices=((-0.95, (((1.2 * cos((crank * 0.017453292519943295))) + sqrt((…, fill_opacity=0.6)","plug":"a Line [yellow] drawn in engine (start=(0.0, 5.6), end=(0.0, 6.8999999999999995))","point":"a Point [yellow] drawn in engine (location=(0.0, 5.45))","polygon":"a Polygon [green] drawn in engine (vertices=((-0.95, (((1.2 * cos((crank * 0.017453292519943295))) + sqrt((…, fill_opacity=0.35)","polygon_2":"a Polygon [red] drawn in engine (vertices=((-0.95, (((1.2 * cos((crank * 0.017453292519943295))) + sqrt((…, fill_opacity=0.35)","polygon_3":"a Polygon [gray] drawn in engine (vertices=((-0.95, (((1.2 * cos((crank * 0.017453292519943295))) + sqrt((…, fill_opacity=0.35)","promise":"a Math [text] that says \"$upright(\"fuel\") arrow.r upright(\"heat\") arrow.r upright(\"pressure\") arrow.r upright(\"work\")$\"","rod":"a Line [blue] drawn in engine (start=((1.2 * sin((crank * 0.017453292519943295))), (1.2 * cos((crank…, end=(0.0, ((1.2 * cos((crank * 0.017453292519943295))) + sqrt((12.9…)","shaft":"a Point [yellow] labelled \"upright(\"crankshaft\")\" drawn in engine","spark":"a Point [yellow] drawn in engine (location=(0.0, 5.4799999999999995), marker_radius=0.14)","stem_in":"a Line [green] drawn in engine (start=(-0.55, (5.6 - lift_in)), end=(-0.55, (6.5 - lift_in)))","stem_out":"a Line [magenta] drawn in engine (start=(0.55, (5.6 - lift_out)), end=(0.55, (6.5 - lift_out)))","strokes":"a Table [text] that says \"Stroke Piston Valves Intake down intake open Compression up both closed Power down both closed Exhaust up exhaust open\" (rows=(('Stroke', 'Piston', 'Valves'), ('Intake', 'down', 'intake ope…, header=True)","tdc":"a Line [gray] labelled \"upright(\"TDC\")\" drawn in engine (start=(-1.8, 5.3), end=(-1.1, 5.3), dashed=True)","valve_in":"a Line [green] drawn in engine (start=(-0.85, (5.6 - lift_in)), end=(-0.25, (5.6 - lift_in)))","valve_out":"a Line [magenta] drawn in engine (start=(0.25, (5.6 - lift_out)), end=(0.85, (5.6 - lift_out)))","wall_l":"a Line [gray] drawn in engine (start=(-1.0, 1.9), end=(-1.0, 5.6))","wall_r":"a Line [gray] drawn in engine (start=(1.0, 1.9), end=(1.0, 5.6))"},"beats":[{"start":0,"say":"Turn the key in a car, and drops of gasoline begin burning inside metal cylinders, each one shoving a piston about twenty five times a second. This lecture follows one cylinder through its four stroke cycle, and tracks how the chemical energy in the fuel becomes work on a turning shaft.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.520479156684236,"card: enter:write-left-to-right."],[17.28,"card is hidden from the screen — left the board."]]},{"start":18.48,"say":"Here is where we are going. Energy changes form, from fuel, to heat, to pressure, to work, and the engine does it in four strokes: intake, compression, power, and exhaust.","live":null,"does":[[18.48,"heading is shown on the screen, written out."],[20.6,"promise is shown on the screen, written out."],[26.24,"strokes is shown on the screen, written out."],[27.479999999999997,"strokes is shown on the screen, written out."],[28.4,"strokes is shown on the screen, written out."],[29.24,"strokes is shown on the screen, written out."],[29.839999999999996,"strokes is shown on the screen, written out."],[30.94,"promise moves to a new place on the board."],[30.94,"strokes moves to a new place on the board."]]},{"start":31.54,"say":"Here is one cylinder, cut open. The cylinder itself is a smooth bore, closed at the top by the cylinder head.","live":["promise","heading"],"does":[[31.54,"engine is shown on the screen, written out."],[34.120000000000005,"wall_l is shown on the screen, written out."],[34.120000000000005,"wall_r is shown on the screen, written out."],[34.519999999999996,"name_cyl is shown on the screen, written out."],[37.72,"head is shown on the screen, written out."]]},{"start":38.98,"say":"Inside it slides the piston, sealed against the wall by rings. Everything happens in the small space above it.","live":["promise","heading","engine","wall_l","wall_r","name_cyl","head"],"does":[[40.31999999999999,"piston is shown on the screen, written out."],[41.07999999999999,"name_piston is shown on the screen, written out."],[44.47999999999999,"point is shown on the screen, grown."]]},{"start":46.44,"say":"Under the piston, a connecting rod links it to the crankshaft. The crank arm swings the bottom of the rod around a circle.","live":["promise","heading","engine","wall_l","wall_r","name_cyl","head","piston","name_piston","point"],"does":[[46.8532269344843,"point is hidden from the screen."],[47.8,"rod is shown on the screen, written out."],[48.44,"name_rod is shown on the screen, written out."],[48.99999999999999,"shaft is shown on the screen, written out."],[50.599999999999994,"crank_arm is shown on the screen, written out."],[52.59999999999999,"crank_path is shown on the screen, written out."]]},{"start":53.919999999999995,"say":"In the head sit two valves. The intake valve lets in the mixture of air and fuel, and the exhaust valve lets burned gas out. Between them stands the spark plug.","live":["promise","heading","engine","wall_l","wall_r","name_cyl","head","piston","name_piston","rod","name_rod","shaft","crank_arm","crank_path"],"does":[[56.47999999999998,"valve_in is shown on the screen, written out."],[56.47999999999998,"stem_in is shown on the screen, written out."],[56.47999999999998,"name_in is shown on the screen, written out."],[59.59999999999998,"valve_out is shown on the screen, written out."],[59.59999999999998,"stem_out is shown on the screen, written out."],[59.59999999999998,"name_out is shown on the screen, written out."],[63.359999999999985,"plug is shown on the screen, written out."],[63.359999999999985,"name_plug is shown on the screen, written out."]]},{"start":64.83999999999999,"say":"Now turn the crank. The rod changes the crank's rotation into straight up and down sliding of the piston, and it works just as well in reverse.","live":["promise","heading","engine","wall_l","wall_r","name_cyl","head","piston","name_piston","rod","name_rod","shaft","crank_arm","crank_path","valve_in","stem_in","name_in","valve_out","stem_out","name_out","plug","name_plug"],"does":[[65.55999999999997,"piston is redrawn as the numbers it depends on change."],[65.55999999999997,"name_piston is redrawn as the numbers it depends on change."],[65.55999999999997,"rod is redrawn as the numbers it depends on change."],[65.55999999999997,"name_rod is redrawn as the numbers it depends on change."],[65.55999999999997,"crank_arm is redrawn as the numbers it depends on change."],[65.55999999999997,"polygon is redrawn as the numbers it depends on change."],[65.55999999999997,"polygon_2 is redrawn as the numbers it depends on change."],[65.55999999999997,"polygon_3 is redrawn as the numbers it depends on change."],[65.55999999999997,"crank ticks to 180.0."],[65.95999999999998,"name_cyl is hidden from the screen."],[65.95999999999998,"name_piston is hidden from the screen."],[65.95999999999998,"name_rod is hidden from the screen."],[65.95999999999998,"name_in is hidden from the screen."],[65.95999999999998,"name_out is hidden from the screen."],[65.95999999999998,"name_plug is hidden from the screen."],[70.03999999999998,"piston is redrawn as the numbers it depends on change."],[70.03999999999998,"rod is redrawn as the numbers it depends on change."],[70.03999999999998,"crank_arm is redrawn as the numbers it depends on change."],[70.03999999999998,"polygon is redrawn as the numbers it depends on change."],[70.03999999999998,"polygon_2 is redrawn as the numbers it depends on change."],[70.03999999999998,"polygon_3 is redrawn as the numbers it depends on change."],[70.03999999999998,"crank ticks to 360.0."]]},{"start":74.32,"say":"Notice the two places where the piston stops and turns around. The top is top dead center, the bottom is bottom dead center, and one trip between them is a stroke, half a turn of the crank.","live":["promise","heading","engine","wall_l","wall_r","head","piston","rod","shaft","crank_arm","crank_path","valve_in","stem_in","valve_out","stem_out","plug"],"does":[[78.96,"tdc is shown on the screen, written out."],[81.03999999999999,"piston is redrawn as the numbers it depends on change."],[81.03999999999999,"rod is redrawn as the numbers it depends on change."],[81.03999999999999,"crank_arm is redrawn as the numbers it depends on change."],[81.03999999999999,"polygon is redrawn as the numbers it depends on change."],[81.03999999999999,"polygon_2 is redrawn as the numbers it depends on change."],[81.03999999999999,"polygon_3 is redrawn as the numbers it depends on change."],[81.03999999999999,"bdc is shown on the screen, written out."],[81.03999999999999,"crank ticks to 540.0."]]},{"start":86.53999999999999,"say":"Bring it back to the top, and we are ready. One full cycle takes four strokes, so the crank turns twice for every single power stroke.","live":["promise","heading","engine","wall_l","wall_r","head","piston","rod","shaft","crank_arm","crank_path","valve_in","stem_in","valve_out","stem_out","plug","tdc","bdc"],"does":[[86.67999999999999,"piston is redrawn as the numbers it depends on change."],[86.67999999999999,"rod is redrawn as the numbers it depends on change."],[86.67999999999999,"crank_arm is redrawn as the numbers it depends on change."],[86.67999999999999,"polygon is redrawn as the numbers it depends on change."],[86.67999999999999,"polygon_2 is redrawn as the numbers it depends on change."],[86.67999999999999,"polygon_3 is redrawn as the numbers it depends on change."],[86.67999999999999,"crank ticks to 720.0."],[93.83999999999999,"strokes (the \"row=4\" part) is indicated — a transient flash."]]},{"start":96.22,"say":"Intake. The intake valve opens and the piston slides down. The falling pressure draws a fresh charge of air, with a fine mist of fuel, in through the port.","live":null,"does":[[96.44,"strokes (the \"row=2\" part) is emphasized."],[98.52,"valve_in is redrawn as the numbers it depends on change."],[98.52,"stem_in is redrawn as the numbers it depends on change."],[98.52,"polygon is shown on the screen, written out."],[98.52,"lift_in ticks to 0.2."],[99.48,"piston is redrawn as the numbers it depends on change."],[99.48,"rod is redrawn as the numbers it depends on change."],[99.48,"crank_arm is redrawn as the numbers it depends on change."],[99.48,"polygon is redrawn as the numbers it depends on change."],[99.48,"polygon_2 is redrawn as the numbers it depends on change."],[99.48,"polygon_3 is redrawn as the numbers it depends on change."],[99.48,"crank ticks to 900.0."],[101.67999999999999,"flow_in is shown on the screen, written out."],[105.32,"flow_in is hidden from the screen."]]},{"start":106.56,"say":"Compression. Both valves are shut, and the piston rises, squeezing the charge to about one ninth of its starting volume. Squeezed that hard, the gas heats up and its pressure climbs.","live":["promise","heading","engine","wall_l","wall_r","head","piston","rod","shaft","crank_arm","crank_path","valve_in","stem_in","valve_out","stem_out","plug","tdc","bdc","polygon"],"does":[[106.88,"strokes (the \"row=2\" part) is no longer emphasized."],[106.88,"strokes (the \"row=3\" part) is emphasized."],[108.88,"valve_in is redrawn as the numbers it depends on change."],[108.88,"stem_in is redrawn as the numbers it depends on change."],[108.88,"lift_in ticks to 0.0."],[109.92,"piston is redrawn as the numbers it depends on change."],[109.92,"rod is redrawn as the numbers it depends on change."],[109.92,"crank_arm is redrawn as the numbers it depends on change."],[109.92,"polygon is redrawn as the numbers it depends on change."],[109.92,"polygon_2 is redrawn as the numbers it depends on change."],[109.92,"polygon_3 is redrawn as the numbers it depends on change."],[109.92,"crank ticks to 1080.0."],[115.67999999999999,"polygon is indicated — a transient flash."]]},{"start":118.58,"say":"Power. Near the top the spark plug fires. A flame races through the mixture in a few thousandths of a second, and the hot gas, now at many times its earlier pressure, drives the piston down.","live":null,"does":[[118.75999999999999,"strokes (the \"row=3\" part) is no longer emphasized."],[118.75999999999999,"strokes (the \"row=4\" part) is emphasized."],[121.39999999999999,"spark is shown on the screen, written out."],[122.44,"polygon is hidden from the screen."],[122.44,"polygon_2 is shown on the screen, faded in."],[122.83999999999999,"spark is hidden from the screen."],[129.15999999999997,"piston is redrawn as the numbers it depends on change."],[129.15999999999997,"rod is redrawn as the numbers it depends on change."],[129.15999999999997,"crank_arm is redrawn as the numbers it depends on change."],[129.15999999999997,"polygon_2 is redrawn as the numbers it depends on change."],[129.15999999999997,"polygon_3 is redrawn as the numbers it depends on change."],[129.15999999999997,"crank ticks to 1260.0."]]},{"start":131.35999999999999,"say":"That is the only stroke that delivers work. The other three are carried through by the spinning crankshaft and its heavy flywheel, or by the other cylinders of the engine.","live":["promise","heading","engine","wall_l","wall_r","head","piston","rod","shaft","crank_arm","crank_path","valve_in","stem_in","valve_out","stem_out","plug","tdc","bdc","polygon_2"],"does":[[133.51999999999998,"piston is indicated — a transient flash."],[136.23999999999998,"shaft is indicated — a transient flash."]]},{"start":141.04,"say":"Exhaust. The exhaust valve opens, and the rising piston pushes the spent gas out. Then the exhaust valve closes, the intake valve opens again, and the cycle starts over.","live":null,"does":[[141.4,"strokes (the \"row=4\" part) is no longer emphasized."],[141.4,"strokes (the \"row=5\" part) is emphasized."],[143.4,"valve_out is redrawn as the numbers it depends on change."],[143.4,"stem_out is redrawn as the numbers it depends on change."],[143.4,"polygon_2 is hidden from the screen."],[143.4,"polygon_3 is shown on the screen, written out."],[143.4,"lift_out ticks to 0.2."],[144.32,"piston is redrawn as the numbers it depends on change."],[144.32,"rod is redrawn as the numbers it depends on change."],[144.32,"crank_arm is redrawn as the numbers it depends on change."],[144.32,"polygon_3 is redrawn as the numbers it depends on change."],[144.32,"crank ticks to 1440.0."],[145.07999999999998,"flow_out is shown on the screen, written out."],[148.2,"valve_out is redrawn as the numbers it depends on change."],[148.2,"stem_out is redrawn as the numbers it depends on change."],[148.2,"flow_out is hidden from the screen."],[148.2,"polygon_3 is hidden from the screen."],[148.2,"lift_out ticks to 0.0."],[149.99999999999997,"valve_in is redrawn as the numbers it depends on change."],[149.99999999999997,"stem_in is redrawn as the numbers it depends on change."],[149.99999999999997,"lift_in ticks to 0.2."],[151.88,"strokes (the \"row=5\" part) is no longer emphasized."]]},{"start":152.48,"say":"Four strokes, two turns of the crank, one push. Next we look inside that push, and follow the energy from the fuel to the crankshaft.","live":["promise","heading","engine","wall_l","wall_r","head","piston","rod","shaft","crank_arm","crank_path","valve_in","stem_in","valve_out","stem_out","plug","tdc","bdc"],"does":[[159.11999999999998,"promise is indicated — a transient flash."],[161.5691875,"engine is hidden from the screen — left the board."],[161.5691875,"wall_l is hidden from the screen — engine left the board."],[161.5691875,"wall_r is hidden from the screen — engine left the board."],[161.5691875,"head is hidden from the screen — engine left the board."],[161.5691875,"piston is hidden from the screen — engine left the board."],[161.5691875,"rod is hidden from the screen — engine left the board."],[161.5691875,"shaft is hidden from the screen — engine left the board."],[161.5691875,"crank_arm is hidden from the screen — engine left the board."],[161.5691875,"crank_path is hidden from the screen — engine left the board."],[161.5691875,"valve_in is hidden from the screen — engine left the board."],[161.5691875,"stem_in is hidden from the screen — engine left the board."],[161.5691875,"valve_out is hidden from the screen — engine left the board."],[161.5691875,"stem_out is hidden from the screen — engine left the board."],[161.5691875,"plug is hidden from the screen — engine left the board."],[161.5691875,"tdc is hidden from the screen — engine left the board."],[161.5691875,"bdc is hidden from the screen — engine left the board."],[161.5691875,"heading is hidden from the screen — left the board."],[161.5691875,"promise is hidden from the screen — left the board."],[161.5691875,"strokes is hidden from the screen — left the board."]]}]},{"title":"From Fuel to Work","start":162.61085416666663,"end":320.0233333333333,"objects":{"blow_line":"a Line [gray] drawn in pv (start=(1.0, 2.5), end=(1.0, 1.0))","burn_line":"a Line [yellow] drawn in pv (start=(0.1, 25.118864315095795), end=(0.1, 62.797160787739486))","compression":"a ParametricCurve [blue] drawn in pv (function=<function>)","energy":"a Math [text] that says \"$Q = m_f thin Q_\"LHV\"$\"","expansion":"a ParametricCurve [red] drawn in pv (function=<function>)","gas_law":"a Math [text] that says \"$p V = m R T$\"","h_fuel":"a Heading that says \"Where the Push Comes From\"","h_loop":"a Heading that says \"Work Is the Loop\"","h_work":"a Heading that says \"Pressure Does Work\"","intake_line":"a Line [green] drawn in pv (start=(0.1, 1.0), end=(1.0, 1.0))","lhv":"a Math [text] that says \"$Q_\"LHV\" approx 44 thin upright(\"MJ/kg\")$\"","net":"a Polygon [yellow] drawn in pv (vertices=((0.1, 62.797160787739486), (0.13, 43.49299505075181), (0.16, 3…, fill_opacity=0.45)","point":"a Point [yellow] drawn in pv (location=(0.1, 1.0))","pv":"an Axes (x_range=(0.0, 1.15), y_range=(0.0, 70.0), x_ticks_every=0.25)","reaction":"a Math [text] that says \"$2 upright(C)_8 upright(H)_18 + 25 upright(O)_2 arrow.r 16 upright(C O)_2 + 18 upright(H)_2 upright(O)$\"","s1":"a Point [text] labelled \"1\" drawn in pv (location=(1.0, 1.0))","s2":"a Point [text] labelled \"2\" drawn in pv (location=(0.1, 25.118864315095795))","s3":"a Point [text] labelled \"3\" drawn in pv (location=(0.1, 62.797160787739486))","s4":"a Point [text] labelled \"4\" drawn in pv (location=(1.0, 2.5))","w_in":"a Math [text] that says \"$W_\"in\" = upright(\"area under\") 1 arrow.r 2$\"","w_net":"a Math [text] that says \"$W_\"net\" = W_\"out\" - W_\"in\" = upright(\"loop area\")$\"","w_out":"a Math [text] that says \"$W_\"out\" = upright(\"area under\") 3 arrow.r 4$\"","work":"a Derivation [text] that says \"$F &= p A \\ dif W &= F dif x = p A dif x \\ dif W &= p dif V \\ W &= integral p dif V$\"","work_in":"a Polygon [blue] drawn in pv (vertices=((1.0, 1.0), (0.97, 1.0435651612459236), (0.94, 1.0904882865070…)","work_out":"a Polygon [red] drawn in pv (vertices=((0.1, 62.797160787739486), (0.13, 43.49299505075181), (0.16, 3…)"},"beats":[{"start":162.61085416666663,"say":"So where does the push come from? Gasoline is a blend of hydrocarbons, and octane is a good representative. It burns with oxygen from the air to make carbon dioxide and water, releasing heat.","live":[],"does":[[162.61085416666663,"h_fuel is shown on the screen, written out."],[167.57085416666663,"reaction is shown on the screen, written out."],[169.93085416666662,"reaction (the \"2 upright(C)_8 upright(H)_18\" part) is emphasized."],[170.37085416666662,"reaction (the \"2 upright(C)_8 upright(H)_18\" part) is no longer emphasized."],[170.37085416666662,"reaction (the \"25 upright(O)_2\" part) is emphasized."],[171.81085416666662,"reaction (the \"16 upright(C O)_2\" part) is emphasized."],[171.81085416666662,"reaction (the \"25 upright(O)_2\" part) is no longer emphasized."],[172.77085416666662,"reaction (the \"16 upright(C O)_2\" part) is no longer emphasized."],[172.77085416666662,"reaction (the \"18 upright(H)_2 upright(O)\" part) is emphasized."],[173.33085416666663,"reaction (the \"18 upright(H)_2 upright(O)\" part) is no longer emphasized."]]},{"start":175.05085416666662,"say":"The products hold less chemical energy than the reactants did. The difference comes out as heat, about forty four megajoules for every kilogram of gasoline, so burning a mass m f of fuel releases m f times that.","live":["reaction","h_fuel"],"does":[[180.81085416666662,"lhv is shown on the screen, written out."],[182.57085416666663,"lhv (the \"44\" part) is emphasized."],[184.41085416666664,"lhv (the \"44\" part) is no longer emphasized."],[186.17085416666663,"energy is shown on the screen, written out."]]},{"start":188.63085416666664,"say":"Set free inside a closed cylinder, that heat drives the gas past two thousand kelvin. For a gas, p V equals m R T, so at fixed volume a hotter gas means a higher pressure. And pressure is what pushes the piston.","live":["reaction","lhv","energy","h_fuel"],"does":[[196.05085416666662,"gas_law is shown on the screen, written out."],[199.0108541666666,"gas_law (the \"T\" part) is emphasized."],[200.09085416666662,"gas_law (the \"T\" part) is no longer emphasized."],[200.09085416666662,"gas_law (the \"p\" part) is emphasized."],[202.21085416666662,"gas_law (the \"p\" part) is no longer emphasized."],[203.37085416666662,"energy is hidden from the screen — left the board."],[203.37085416666662,"gas_law is hidden from the screen — left the board."],[203.37085416666662,"h_fuel is hidden from the screen — left the board."],[203.37085416666662,"lhv is hidden from the screen — left the board."],[203.37085416666662,"reaction is hidden from the screen — left the board."]]},{"start":204.57085416666663,"say":"Pressure is force per unit area. On a piston crown of area A, the gas pushes with a force p times A. On the right, I will draw that pressure against the volume above the piston.","live":[],"does":[[204.57085416666663,"h_work is shown on the screen, written out."],[209.73085416666663,"work is shown on the screen, written out."],[212.4508541666666,"pv is shown on the screen, written out."]]},{"start":216.6908541666666,"say":"Let the piston move a small distance d x. The force does work p A d x, and A times d x is exactly the small volume swept, d V.","live":["pv","h_work"],"does":[[220.65085416666662,"work is shown on the screen, written out."],[225.1308541666666,"work is shown on the screen, written out."]]},{"start":227.7508541666666,"say":"Add those pieces over a whole stroke, and the work is the integral of p d V. Work is the area under a curve of pressure against volume.","live":null,"does":[[230.8508541666666,"work is shown on the screen, written out."]]},{"start":237.07085416666663,"say":"Now trace one ideal cycle. Intake fills the cylinder at about atmospheric pressure, while the volume grows from the small clearance volume up to the whole cylinder.","live":null,"does":[[239.6108541666666,"intake_line is shown on the screen, drawn."],[244.3308541666666,"point is shown on the screen, grown."],[245.7308541666666,"s1 is shown on the screen, written out."],[246.03661174825896,"point is hidden from the screen."]]},{"start":247.29085416666663,"say":"Compression runs from state one to state two. It is too quick for much heat to leak away, so the pressure climbs steeply as the volume shrinks.","live":["pv","h_work","intake_line","s1"],"does":[[247.41085416666664,"compression is shown on the screen, drawn."],[249.5708541666666,"s2 is shown on the screen, written out."]]},{"start":256.27085416666665,"say":"Then the spark. The burn is so fast that the piston hardly moves, so the heat arrives at nearly constant volume, and the pressure jumps from state two to state three.","live":["pv","h_work","intake_line","s1","compression","s2"],"does":[[264.4908541666666,"burn_line is shown on the screen, drawn."],[266.17085416666663,"s3 is shown on the screen, written out."]]},{"start":267.4308541666666,"say":"The power stroke is the expansion from three to four, the hot gas pushing the piston all the way down.","live":["pv","h_work","intake_line","s1","compression","s2","burn_line","s3"],"does":[[268.6108541666666,"expansion is shown on the screen, drawn."],[270.09085416666665,"s4 is shown on the screen, written out."]]},{"start":274.19085416666667,"say":"At the bottom the exhaust valve opens and the pressure drops. Then the exhaust stroke sweeps the rest out along the bottom line, back to where we began.","live":["pv","h_work","intake_line","s1","compression","s2","burn_line","s3","expansion","s4"],"does":[[276.97085416666664,"blow_line is shown on the screen, drawn."],[279.29085416666663,"intake_line is indicated — a transient flash."],[282.83085416666665,"h_work is hidden from the screen — left the board."],[282.83085416666665,"work is hidden from the screen — left the board."]]},{"start":284.03085416666664,"say":"Now read the work straight off the picture. During expansion the gas does work on the piston, equal to all the area under the top curve.","live":["pv","intake_line","s1","compression","s2","burn_line","s3","expansion","s4","blow_line"],"does":[[284.03085416666664,"h_loop is shown on the screen, written out."],[289.73085416666663,"w_out is shown on the screen, written out."],[290.57085416666666,"work_out is shown on the screen, faded in."]]},{"start":292.77085416666665,"say":"During compression the piston does work on the gas instead: the area under the bottom curve, which has to be paid back.","live":["pv","intake_line","s1","compression","s2","burn_line","s3","expansion","s4","blow_line","w_out","h_loop","work_out"],"does":[[296.69085416666667,"work_in is shown on the screen, faded in."],[297.41085416666664,"w_in is shown on the screen, written out."]]},{"start":300.35085416666664,"say":"Take one from the other. The shared part cancels, and what is left is the area enclosed by the loop, the net work one cylinder delivers every cycle.","live":["pv","intake_line","s1","compression","s2","burn_line","s3","expansion","s4","blow_line","w_out","w_in","h_loop","work_out","work_in"],"does":[[302.73085416666663,"work_out is hidden from the screen."],[302.73085416666663,"work_in is hidden from the screen."],[304.9308541666667,"net is shown on the screen, faded in."],[306.41085416666664,"w_net is shown on the screen, written out."],[309.47085416666664,"A box is drawn around w_net."]]},{"start":310.07085416666666,"say":"So the chain is complete. Fuel becomes heat, heat becomes pressure, and pressure acting through the swept volume becomes work on the crankshaft.","live":["pv","intake_line","s1","compression","s2","burn_line","s3","expansion","s4","blow_line","w_out","w_in","w_net","h_loop","net"],"does":[[317.4108541666666,"net is indicated — a transient flash."],[318.9816666666667,"h_loop is hidden from the screen — left the board."],[318.9816666666667,"pv is hidden from the screen — left the board."],[318.9816666666667,"intake_line is hidden from the screen — pv left the board."],[318.9816666666667,"s1 is hidden from the screen — pv left the board."],[318.9816666666667,"compression is hidden from the screen — pv left the board."],[318.9816666666667,"s2 is hidden from the screen — pv left the board."],[318.9816666666667,"burn_line is hidden from the screen — pv left the board."],[318.9816666666667,"s3 is hidden from the screen — pv left the board."],[318.9816666666667,"expansion is hidden from the screen — pv left the board."],[318.9816666666667,"s4 is hidden from the screen — pv left the board."],[318.9816666666667,"blow_line is hidden from the screen — pv left the board."],[318.9816666666667,"net is hidden from the screen — pv left the board."],[318.9816666666667,"w_in is hidden from the screen — left the board."],[318.9816666666667,"w_net is hidden from the screen — left the board."],[318.9816666666667,"w_out is hidden from the screen — left the board."]]}]},{"title":"How Much of the Heat Becomes Work","start":320.0233333333333,"end":456.7492916666667,"objects":{"deriv":"a Derivation [text] that says \"$eta &= frac(W_\"net\", Q_\"in\") \\ &= 1 - frac(Q_\"out\", Q_\"in\") \\ &= 1 - frac(T_4 - T_1, T_3 - T_2) \\ frac(T_2, T_1) &= frac(T_3, T_4) = r^(gamma - 1) \\ eta &= 1 - frac(1, r^(gamma - 1))$\"","eta_axes":"an Axes (x_range=(1.0, 16.0), y_range=(0.0, 0.8), x_ticks_every=2.0)","eta_curve":"a FunctionPlot [blue] drawn in eta_axes (function=<function>, x_range=(1.0, 16.0))","h_eff":"a Heading that says \"Efficiency of the Ideal Cycle\"","h_recap":"a Heading that says \"What to Carry Away\"","knock":"a Polygon [red] labelled \"upright(\"knock\")\" drawn in eta_axes (vertices=((13.0, 0.0), (16.0, 0.0), (16.0, 0.8), (13.0, 0.8)), fill_opacity=0.2)","line":"a Line [yellow] drawn in eta_axes (start=(4.0, 0.42565082250148245), end=(0.0, 0.42565082250148245), dashed=True)","line_2":"a Line [yellow] drawn in eta_axes (start=(4.0, 0.42565082250148245), end=(4.0, 0.0), dashed=True)","point":"a Point [yellow] drawn in eta_axes (location=(4.0, 0.42565082250148245))","r_live":"a VariableNumber (initial_value=4.0, format_spec='.0f')","real":"a Polygon [gray] labelled \"upright(\"real engines\")\" drawn in eta_axes (vertices=((8.0, 0.28), (13.0, 0.28), (13.0, 0.38), (8.0, 0.38)), fill_opacity=0.4)","recap":"a Block [text] that says \"Four strokes take two crank turns, and only one of them delivers work. The valves time the gas exchange; the spark times the burn. Burning fuel raises gas pressure, and work is the integral of $p dif V$. Efficiency rises with compression r…\"","rider":"a PlotPoint [yellow] labelled \"r = 4\" drawn in eta_axes (target='eta_curve', x=<VariableNumber r_live = 13.0>)"},"beats":[{"start":320.0233333333333,"say":"We have the work. Now, what did it cost? Efficiency is the net work out, divided by the heat the fuel put in.","live":[],"does":[[320.0233333333333,"h_eff is shown on the screen, written out."],[323.4633333333333,"deriv is shown on the screen, written out."]]},{"start":328.7233333333333,"say":"Net work is the heat in minus the heat thrown away with the exhaust, so efficiency is one minus heat out over heat in.","live":["h_eff"],"does":[[333.4233333333333,"deriv is shown on the screen, written out."]]},{"start":337.74333333333334,"say":"In the ideal cycle both heats are added and removed at constant volume, so each is proportional to a temperature change: heat in from state two to three, heat out from four back to one.","live":null,"does":[[343.06333333333333,"deriv is shown on the screen, written out."],[345.2633333333333,"deriv (the \"T_3 - T_2\" part) is emphasized."],[347.3033333333333,"deriv (the \"T_3 - T_2\" part) is no longer emphasized."],[347.3033333333333,"deriv (the \"T_4 - T_1\" part) is emphasized."],[349.50333333333333,"deriv (the \"T_4 - T_1\" part) is no longer emphasized."]]},{"start":350.1033333333333,"say":"Along the two adiabats, the temperature ratio depends only on the compression ratio r, the full volume over the clearance volume, raised to the power gamma minus one. Gamma is about one point four for air.","live":null,"does":[[350.9433333333333,"deriv is shown on the screen, written out."],[358.06333333333333,"deriv (the \"r^(gamma - 1)\" part) is emphasized."],[359.0233333333333,"deriv (the \"r^(gamma - 1)\" part) is no longer emphasized."]]},{"start":363.6433333333333,"say":"Those equal ratios make the temperatures cancel, and we are left with the Otto efficiency: one minus one over r to the gamma minus one. It depends only on how hard the engine compresses.","live":null,"does":[[367.7033333333333,"deriv is shown on the screen, written out."],[372.3433333333333,"deriv is indicated — a transient flash."],[375.4033333333333,"deriv moves to a new place on the board."]]},{"start":376.00333333333333,"say":"Now put in numbers. Here is that formula plotted against r. At a compression ratio of four, the ideal engine turns about forty three percent of the heat into work.","live":null,"does":[[376.00333333333333,"eta_axes is shown on the screen, written out."],[378.7833333333333,"eta_curve is shown on the screen, drawn."],[381.6633333333333,"rider is shown on the screen, written out."],[383.8233333333333,"point is shown on the screen, grown."],[383.8233333333333,"line is shown on the screen, drawn."],[383.8233333333333,"line_2 is shown on the screen, drawn."],[385.463898989899,"point is hidden from the screen."],[385.463898989899,"line is hidden from the screen."],[385.463898989899,"line_2 is hidden from the screen."]]},{"start":386.86333333333334,"say":"At eight, fifty six percent. At ten, sixty. Squeeze harder still and it keeps climbing, but ever more slowly.","live":["h_eff","eta_axes","eta_curve","rider"],"does":[[387.22333333333336,"rider is redrawn as the numbers it depends on change."],[387.22333333333336,"r_live ticks to 8.0."],[389.3033333333333,"rider is redrawn as the numbers it depends on change."],[389.3033333333333,"r_live ticks to 10.0."],[390.9433333333333,"rider is redrawn as the numbers it depends on change."],[390.9433333333333,"r_live ticks to 13.0."]]},{"start":395.5833333333333,"say":"So why do gasoline engines stop near ten to thirteen? Squeezing heats the mixture, and too much heat makes it ignite on its own before the spark. That is knock, and it can wreck a piston. A fuel's octane rating measures how well it resists.","live":null,"does":[[405.4233333333333,"knock is shown on the screen, faded in."]]},{"start":411.56333333333333,"say":"Real engines also fall below the ideal curve. Heat leaks into the cylinder walls, friction drags on the moving parts, and the burn takes time. A good gasoline engine turns roughly a third of its fuel energy into work, and the rest leaves as heat.","live":["h_eff","eta_axes","eta_curve","rider","knock"],"does":[[413.62333333333333,"eta_curve is indicated — a transient flash."],[422.9033333333333,"real is shown on the screen, faded in."],[426.6633333333333,"deriv is hidden from the screen — left the board."],[426.6633333333333,"eta_axes is hidden from the screen — left the board."],[426.6633333333333,"eta_curve is hidden from the screen — eta_axes left the board."],[426.6633333333333,"rider is hidden from the screen — eta_axes left the board."],[426.6633333333333,"knock is hidden from the screen — eta_axes left the board."],[426.6633333333333,"real is hidden from the screen — eta_axes left the board."],[426.6633333333333,"h_eff is hidden from the screen — left the board."]]},{"start":427.86333333333334,"say":"Let's gather it up. Four strokes take two turns of the crank, and only one of them delivers work.","live":[],"does":[[427.86333333333334,"h_recap is shown on the screen, written out."],[427.86333333333334,"recap is shown on the screen, written out."],[432.06333333333333,"recap (the \"only one of them delivers work\" part) is emphasized."]]},{"start":434.4233333333333,"say":"The valves time the flow of gas in and out, and the spark times the burn.","live":["recap","h_recap"],"does":[[434.78333333333336,"recap (the \"The valves time the gas exchange\" part) is emphasized."],[434.78333333333336,"recap (the \"only one of them delivers work\" part) is no longer emphasized."]]},{"start":439.7833333333333,"say":"Burning fuel raises the gas pressure, and the work it does is the integral of p d V, the area of the loop.","live":null,"does":[[440.0233333333333,"recap (the \"Burning fuel raises gas pressure\" part) is emphasized."],[440.0233333333333,"recap (the \"The valves time the gas exchange\" part) is no longer emphasized."]]},{"start":447.1433333333333,"say":"And efficiency climbs with compression ratio, until knock sets the limit. That, in one cylinder, is how a gasoline engine works.","live":null,"does":[[450.22333333333336,"recap (the \"Burning fuel raises gas pressure\" part) is no longer emphasized."],[450.22333333333336,"recap (the \"until knock sets the limit\" part) is emphasized."],[455.457625,"recap (the \"until knock sets the limit\" part) is no longer emphasized."],[455.707625,"h_recap is hidden from the screen — left the board."],[455.707625,"recap is hidden from the screen — left the board."]]}]}]},"durationSeconds":457,"chapters":[{"title":"One Cylinder, Four Strokes","startSeconds":0,"narration":"Turn the key in a car, and drops of gasoline begin burning inside metal cylinders, each one shoving a piston about twenty five times a second. This lecture follows one cylinder through its four stroke cycle, and tracks how the chemical energy in the fuel becomes work on a turning shaft. Here is where we are going. Energy changes form, from fuel, to heat, to pressure, to work, and the engine does it in four strokes: intake, compression, power, and exhaust. Here is one cylinder, cut open. The cylinder itself is a smooth bore, closed at the top by the cylinder head. Inside it slides the piston, sealed against the wall by rings. Everything happens in the small space above it. Under the piston, a connecting rod links it to the crankshaft. The crank arm swings the bottom of the rod around a circle. In the head sit two valves. The intake valve lets in the mixture of air and fuel, and the exhaust valve lets burned gas out. Between them stands the spark plug. Now turn the crank. The rod changes the crank's rotation into straight up and down sliding of the piston, and it works just as well in reverse. Notice the two places where the piston stops and turns around. The top is top dead center, the bottom is bottom dead center, and one trip between them is a stroke, half a turn of the crank. Bring it back to the top, and we are ready. One full cycle takes four strokes, so the crank turns twice for every single power stroke. Intake. The intake valve opens and the piston slides down. The falling pressure draws a fresh charge of air, with a fine mist of fuel, in through the port. Compression. Both valves are shut, and the piston rises, squeezing the charge to about one ninth of its starting volume. Squeezed that hard, the gas heats up and its pressure climbs. Power. Near the top the spark plug fires. A flame races through the mixture in a few thousandths of a second, and the hot gas, now at many times its earlier pressure, drives the piston down. That is the only stroke that delivers work. The other three are carried through by the spinning crankshaft and its heavy flywheel, or by the other cylinders of the engine. Exhaust. The exhaust valve opens, and the rising piston pushes the spent gas out. Then the exhaust valve closes, the intake valve opens again, and the cycle starts over. Four strokes, two turns of the crank, one push. Next we look inside that push, and follow the energy from the fuel to the crankshaft."},{"title":"From Fuel to Work","startSeconds":162.61085416666663,"narration":"So where does the push come from? Gasoline is a blend of hydrocarbons, and octane is a good representative. It burns with oxygen from the air to make carbon dioxide and water, releasing heat. The products hold less chemical energy than the reactants did. The difference comes out as heat, about forty four megajoules for every kilogram of gasoline, so burning a mass m f of fuel releases m f times that. Set free inside a closed cylinder, that heat drives the gas past two thousand kelvin. For a gas, p V equals m R T, so at fixed volume a hotter gas means a higher pressure. And pressure is what pushes the piston. Pressure is force per unit area. On a piston crown of area A, the gas pushes with a force p times A. On the right, I will draw that pressure against the volume above the piston. Let the piston move a small distance d x. The force does work p A d x, and A times d x is exactly the small volume swept, d V. Add those pieces over a whole stroke, and the work is the integral of p d V. Work is the area under a curve of pressure against volume. Now trace one ideal cycle. Intake fills the cylinder at about atmospheric pressure, while the volume grows from the small clearance volume up to the whole cylinder. Compression runs from state one to state two. It is too quick for much heat to leak away, so the pressure climbs steeply as the volume shrinks. Then the spark. The burn is so fast that the piston hardly moves, so the heat arrives at nearly constant volume, and the pressure jumps from state two to state three. The power stroke is the expansion from three to four, the hot gas pushing the piston all the way down. At the bottom the exhaust valve opens and the pressure drops. Then the exhaust stroke sweeps the rest out along the bottom line, back to where we began. Now read the work straight off the picture. During expansion the gas does work on the piston, equal to all the area under the top curve. During compression the piston does work on the gas instead: the area under the bottom curve, which has to be paid back. Take one from the other. The shared part cancels, and what is left is the area enclosed by the loop, the net work one cylinder delivers every cycle. So the chain is complete. Fuel becomes heat, heat becomes pressure, and pressure acting through the swept volume becomes work on the crankshaft."},{"title":"How Much of the Heat Becomes Work","startSeconds":320.0233333333333,"narration":"We have the work. Now, what did it cost? Efficiency is the net work out, divided by the heat the fuel put in. Net work is the heat in minus the heat thrown away with the exhaust, so efficiency is one minus heat out over heat in. In the ideal cycle both heats are added and removed at constant volume, so each is proportional to a temperature change: heat in from state two to three, heat out from four back to one. Along the two adiabats, the temperature ratio depends only on the compression ratio r, the full volume over the clearance volume, raised to the power gamma minus one. Gamma is about one point four for air. Those equal ratios make the temperatures cancel, and we are left with the Otto efficiency: one minus one over r to the gamma minus one. It depends only on how hard the engine compresses. Now put in numbers. Here is that formula plotted against r. At a compression ratio of four, the ideal engine turns about forty three percent of the heat into work. At eight, fifty six percent. At ten, sixty. Squeeze harder still and it keeps climbing, but ever more slowly. So why do gasoline engines stop near ten to thirteen? Squeezing heats the mixture, and too much heat makes it ignite on its own before the spark. That is knock, and it can wreck a piston. A fuel's octane rating measures how well it resists. Real engines also fall below the ideal curve. Heat leaks into the cylinder walls, friction drags on the moving parts, and the burn takes time. A good gasoline engine turns roughly a third of its fuel energy into work, and the rest leaves as heat. Let's gather it up. Four strokes take two turns of the crank, and only one of them delivers work. The valves time the flow of gas in and out, and the spark times the burn. Burning fuel raises the gas pressure, and the work it does is the integral of p d V, the area of the loop. And efficiency climbs with compression ratio, until knock sets the limit. That, in one cylinder, is how a gasoline engine works."}]}}