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Introduction
On the AP Physics B Exam, most heat engines feature "sealed" pistons, where the number of moles of gas inside the piston remains constant. This simplification makes thermodynamic calculations for the heat engine straightforward. In contrast, many real-world engines, such as gasoline engines, exchange mass with their environment. An additional complication with gasoline engines is that the heat that drives the engine is generated within the piston, rather than from an outside source. Nonetheless, these more complicated systems can still be understood in terms of our familiar pressure-volume (PV) diagram.
In this exercise, we will construct a PV diagram for a gasoline-powered car engine. If we ignore energy loss due to friction, as well as heat loss through the walls of the engine's pistons, we can treat the gasoline engine as an Otto engine (first built by Nikolaus Otto in 1876). It is recommended that student discussion take place after each "step" of the Otto cycle. In order to facilitate this discussion, a separate PV diagram is included for each of the six steps discussed below.
Note: Although not required for the AP Physics B Exam, basic chemistry is included in parts of this problem to increase student understanding. Questions that utilize knowledge of chemistry are marked with an asterisk (*).
There are several good Web sites of the Otto engine: Otto Engine Simulation by Xing Min (Sherman) Wang, Ph.D.
www.rawbw.com/~xmwang/myGUI/OttoG.html
Step-by-Step Explanation of Otto Cycle
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/otto.html
P-V Diagram of the Ideal Otto Cycle
www.grc.nasa.gov/WWW/K-12/airplane/otto.html
The Internal Combustion Engine: Otto Cycle (includes detailed analysis of pressure, temperature, flow, and spray)
www.eng.warwick.ac.uk/~espbc/courses/engine/ic018.htm Step 1: (A –> B) Intake Stroke
During the intake stroke, the piston's valve is opened, and the piston is lowered. Thus, air is drawn into the cylinder. Since the inside of the piston and the atmosphere are in direct contact, the air inside the piston is also at atmospheric pressure. Therefore, step 1 is isobaric.
(A) The engine that we will study is a 3.0 L V6 sports car engine: all six cylinders of the engine taken together will hold three liters of air. How much air is drawn into a single cylinder? Express your answer in both grams and moles. The incoming air has a temperature of 150° C. The atomic mass of air is 0.029 kg/mole.
Note: In many sports cars, the mass of air that enters the cylinder is increased by either increasing the air pressure or cooling the air (which increases its density).
(B) On the PV diagram provided above, show the isobaric expansion from point A to point B. Estimate the amount of work done during this step.
(C) Fuel is sprayed into the cylinder. The optimum air-fuel ratio for the complete combustion of gasoline is 14.7:1 (massair:massfuel). How many grams of fuel were injected into the cylinder?
Step 2: (B –>C) Compression Stroke
During the compression stroke, the valves are sealed, and the piston moves upward. This occurs so rapidly that heat does not have time to escape through the walls of the cylinder. Therefore, step 2 is adiabatic.
(D) A typical gasoline engine has a compression ratio of 8:1 (volumemaximum:volumeminimum). The pressure of the compressed air-fuel mixture is 1.0 x 106 Pa. What is the pre-ignition temperature of the mixture? Assume that the addition of fuel does not significantly change the total number of moles of gas in the cylinder.
(E) The ignition temperature of octane is 280° C. If the temperature in the cylinder exceeds this value (due to compression), the fuel will ignite too early and cause the engine to "knock." Based on your findings, is the compression ratio given in part (D) reasonable?
(F) Diesel engines do not have spark plugs, but rely on the temperature rise during compression to ignite the diesel fuel. What does this imply about the maximum compression inside diesel engines?
(G) On the PV diagram provided above, show the adiabatic expansion from point B to point C. Use your diagram to estimate the amount of work done on the gas during this step.
Step 3: (C —>D) Ignition
During the ignition step, the fuel-air mixture is ignited using a spark plug (not shown). This step occurs so rapidly that the cylinder has not yet had time to move. Therefore, step 3 is isochoric.
* (H) Gasoline is a mixture of different fuels. For purposes of this discussion, we will assume that the fuel is 100 percent octane (and therefore has an octane rating of 100). The formula for octane is C8H18. Write a balanced chemical equation for the complete combustion of octane.
* (I) How many moles of octane were consumed in the reaction?
* (J) Assume that the carbon dioxide and water produced by the complete combustion of the octane is gaseous. How many moles of gas are present in the cylinder after combustion?
* (K) Use the following heats of formation (
) to estimate the energy released by the combustion of the octane in the cylinder.
(L) Students often suggest that gasoline engines work by turning a liquid (fuel) into a gas, which then causes the piston to move. Based on your calculations, which factor is more important in powering a gasoline engine: the production of gas (CO2 and H2O) or the release of heat? Explain.
(M) After combustion, the temperature in the cylinder is 2900° C. What is the pressure in the cylinder at this time?
(N) If the piston in the cylinder has a mass of 8.0 kg and a radius of 5 cm, what is the acceleration of the cylinder?
(O) On the PV diagram provided above, indicate the change in the cylinder during the isochoric combustion (from point C to point D). Use your diagram to estimate the amount of work done on the gas during this step.
Step 4: (D–>E) Power Stroke
During the power stroke, the piston rapidly moves downward, expanding the cylinder volume. This occurs so rapidly that it can be considered adibiatic (the same assumption made in step 2).
(P) If the pressure in the cylinder at the end of step 4 is 2500 kPa, what is the temperature inside the cylinder at this point?
(Q) On the PV diagram provided above, indicate the change that takes place in the cylinder during the adibiatic expansion (from point D to point E). Use your diagram to estimate the amount of work done on the gas during this step.
Step 5: (E–>F) Exhaust Valve Opens
After the power stroke, the exhaust valve is opened, and the air pressure in the cylinder comes into equilibrium with the atmosphere. Since the piston is not moving, this process can be considered isochoric.
(R) Assume that the number of moles of air in the cylinder is now equal to that found in part (A). What is the temperature of the air in the cylinder?
(S) On the PV diagram provided above, indicate the change in the cylinder during this isochoric process (from point E to point F). Use your diagram to estimate the amount of work done on the gas during this step.
Step 6: (F–>G) Exhaust Stroke
Since the air in the cylinder is oxygen poor and has a higher density than the "outside" air (due to the presence of carbon dioxide and water vapor), the exhaust must be forcibly removed from the cylinder. This isobaric process restores the cylinder to its original state.
(T) On the PV diagram provided, indicate the change in the cylinder during this isobaric process (from point F to point G). Use your diagram to estimate the amount of work done on the gas during this step.
Reflection
(U) Combine all your previous PV diagrams to create the PV diagram for the complete Otto cycle.
(V) How much work was done by the piston during one complete cycle?
(W) Based on the energy available due to the combustion of octane, what is the approximate efficiency of this engine?
(X) If this gasoline engine operates at 900 rpm, how many watts does the engine produce?
(Y) James Watt originally defined a horsepower (hp); it corresponds to a lift, performed by a horse, of 100 feet per minute. Today, 1 hp is defined as 746 watts. What is the power rating of this engine in horsepower?
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David Castro taught AP Physics (B and C), AP Calculus (AB and BC), and AP U.S. and European History in a teaching career spanning 14 years, including 5 years as a master AP Physics teacher. In 1997, he received a Special Recognition Teaching Award, and in 2002 his combined AP Physics and AP Calculus syllabus was published in the AP Physics Teacher's Guide. Active as an AP Physics consultant in the Southwest Region since 1995, his areas of expertise include Pre-AP middle school science, AP Vertical Teams, as well as interdisciplinary physics/calculus. He also serves as a Reader for AP Physics. Mr. Castro recently joined the Charles A. Dana Center at the University of Texas, where he continues to focus on providing support for science educators.
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