(last edited February 5, 2013)
Dr. Larry Bortner

Purpose 


To investigate the thermodynamics of a heat engine cycle. To compare the mechanical work done by a heat engine with the thermodynamic work indicated by a PV diagram. 

Background 

Heat is the flow of energy between two things at different temperatures. A heat engine is an apparatus that transforms heat into mechanical work. We are all quite familiar with such devices in the form of internal combustion engines in our vehicles. 



The theory behind the operation of these engines is basic: a gas in a cylinder with a piston is taken through a series of thermodynamic processes that returns it to its original state. 

The work done in a closed cycle is the area of the cycle represented in a PV diagram. 



What's a PV Diagram? 

A PV diagram is just a plot of pressure vs. volume of a gas. An idealized diagram is shown in Fig. 1, with the cycle consisting of an isometric process (AB, the volume is constant) followed by a isobaric process (BC, the pressure is constant) then another constant volume process (CD), and finally a constant pressure process (DA) that returns the gas to the original pressure and volume. No work is done in the process AB since the piston doesn’t move. The work done by the gas in BC is P_{2}(V_{2} − V_{1}). As in the process AB, no work is done in CD. Work has to be done on the gas in DA to compress it, meaning that the work done by the gas is negative, P_{1}(V_{1} − V_{2}) = −P_{1}(V_{2} − V_{1}). The total work done in the cycle is P_{2}(V_{2} − V_{1}) − P_{1}(V_{2} − V_{1}) = (P_{2} − P_{1})(V_{2} − V_{1}). As stated, this is the area of the cycle, the region bound by the four processes in the graph. 

Figure 1 PV diagram for a closed cycle heat engine. Heat is absorbed in process BC and released in DA. Thermodynamically, the work done is the (pressurevolume) area enclosed by the processes.

Note on calculating area: Using calculus, the general expression for the work done is the integral over the closed loop, which we can break into separate integrals over each process: 
(1)
Ideal Engine Efficiency 
A heat engine utilizes the temperature difference between two objects to do work. The efficiency of the engine is the ratio of the work done to the amount of energy available, which is the difference between the amount of heat put into the system and the amount of heat taken out. Physically, this can never reach 100%. The maximum theoretical efficiency for a heat engine operating between the temperatures T_{hot} and T_{cold} is called the Carnot efficiency and is given by the expression 

(2) 
Engine Description 
The heat engine we investigate today consists mainly of two parts. The first is a tightfitting, lowfriction piston inside a clear cylinder. This is connected via tygon tubing to the second part, a small aluminum can with a rubber stopper plugging up the end. There is a hole in the stopper that the tubing feeds into and allows expanding gas in the can to go through the tube and push up the piston. The gas pushed out of the aluminum can goes into the clear cylinder and lifts a mass on the attached platform. The distance the mass moves is measured and used to calculate the mechanical work done. The work required to vertically raise a mass m a distance h is 

(3) 

The air in the aluminum cylinder produces the thermodynamic work. (See Fig. 2) The metal cylinder starts out submerged in ice water. 

Figure 2 A diagram of the four processes of the heat engine
studied in the lab as the engine lifts a crabapple with a wormhole.
· The first process in the cycle is adiabatic (no heat flows into or out of the system), where a mass is placed on the platform. This action decreases the volume of the air and increases the pressure.
· The second step is isobaric when you move the cylinder from the ice water into boiling water. The air in the aluminum cylinder expands due to the increase in temperature and work is done as the mass rises. For this heat engine, this is the power stroke.
· Removing the mass is an adiabatic process.
· Putting the cylinder back into the ice water is an isobaric step that completes the engine cycle.
A pressure sensor reads the air pressure inside the two cylinders and a motion sensor monitors the platform position. Since the crosssectional area of the cylinder remains the same, a simple calculation gives you the volume. Thus Science Workshop, properly set up, collects the pertinent data as you take the heat engine through the different stages of its cycle. 
Finding the Area of the PV Diagram 
Some of you might be fretting: How do you find the area of a PV diagram without using calculus? 

We assume that since processes BC and DA are isobaric (same pressure), that lines through their data points are parallel. The adiabatic processes AB and CD are approximately linear, so the closed cycle ABCD is a parallelogram in PV space, as in Fig. 3. 
Figure 3 The expected shape of the PV diagram of a heat
engine used in this experiment. The area of the parallelogram is the same as
that of a rectangle that uses average volumes.
Included in Fig. 3 is a rectangle with an equivalent area whose left and right sides have volume coordinates V_{1} and V_{2}. V_{1} is the average of V_{A} and V_{B} , while V_{2} is the average of V_{C} and V_{D}. 

The thermodynamic work done is the area of the rectangle: 

(4) 

Your data set includes points for the entire cycle. To analyze the data, 
· Choose approximations of points A, B, C, and D from the data. Find the average volume V_{1} and the standard error u{V_{1 }} of the volumes in process AB.
· Find the average pressure P_{B} and the standard error u{P_{B}} of the pressures in process BC.
· Find the average volume V_{2 }and the standard error u{V_{2 }} of the volumes in process CD.
· Find the average pressure P_{A} and the standard error u{P_{A}} of the pressures in process DA.
CalculusBased Physics: Quadrature 
A more rigorous analysis of the data would be to fit curves (lines) to each process. Once we have equations for the four lines, we could just plug them into Eq. (1), do the integrations, and add everything up. But by fitting the data to curves and integrating the curves, we may be doing more work than we have to. Might there not be some way to integrate the data directly? 

Yes, there is, and it’s called numerical integration, or quadrature. The simplest technique is called the trapezoid rule. (Refer to Fig. 4.) 
Figure 4 The trapezoid rule in numerical integration.
Suppose we have an arbitrary curve P(V) in PV space (instead of xy space) and two successive points on the curve that we’ll label (V_{i1 }, P_{i1}) and (V_{i }, P_{i}). A good approximation of the area ΔW_{i} under the curve between these two points is the area of the trapezoid bound by the four points (V_{i1 }, 0), (V_{i1 }, P_{i1}), (V_{i }, P_{i}), and (V_{i }, 0). This area is the same as a rectangle of width V_{i} − V_{i}_{1} and a height that is the average of P_{i} and P_{i1}. 

(5) 

Assuming that the measurement uncertainties of the pressure and volume are independent of the index, the square of the uncertainty of this calculation is found by propagating the error: 

(6) 

Say we have N points that describe a curve. Then we can use Eq. 5 to find a reasonable approximation of a bounded integral of that function by summing the areas of the trapezoids determined by the N points. In fact, if, as with your data, the N points describe a closed curve (starting and ending at the same point), we can calculate the entire integral stated in Eq. 1: 

(7) 

The propagated error of Eq. 7 is 

(8) 

When this value is calculated, use Eq. 6 for the uncertainty within the radical. 
You need the following items:
Science Workshop interface
Heat Engine/Gas Law Apparatus
rotary motion sensor, u{y} = 0.005 m
pressure sensor, u{P} = 0.01 kPa
spring balance
2 beakers
Bunsen burner
100 g mass hanger with 100 g locking mass
tongs
Figure 5 Experimental setup of the heat engine used in this experiment.
CAUTION! Note that this experiment requires boiling water. An open gas flame from a Bunsen burner is used to achieve this. Be careful: both the flame and the hot water can cause serious burns. 
1. Fill one metal beaker with water to a depth of at least 3 inches, deep enough that the full height of the aluminum can is covered by water. If the stopper is too much under water, water gets in the can (undesirable). 
a. Place the beaker on the tripod and turn on the Bunsen burner to begin heating the water. This step should be taken before you start the quiz− you want the water to be boiling when you are ready to take data. 
2. Fill the other metal beaker with ice water to the same depth. 
3. Disconnect the tube which runs from the can to the clear cylinder, at the base of the clear cylinder. (The locking connector should release with just an eighth turn counterclockwise.) Place the aluminum can in the ice water. 
4. Adjust the height of the piston to about a centimeter above the bottom of the clear cylinder. If you need it, use the thumbscrew at the top front of the cylinder to temporarily lock the piston in this position. 
5. Reconnect the tubing from the Al cylinder to the base of the clear cylinder. Also, check to make sure that the pressure sensor is attached to the other connector at the base of the clear cylinder. 
6. Loosen the thumbscrew that is holding the piston. The piston should remain in roughly the same position. It should not be resting on the bottom. If the piston moves down, there is a leak in the apparatus and you should have your TA check the equipment for you. 
7. From the Start menu, click on DataStudio Experiments> Third Quarter> Heat Engine. This starts the DataStudio software. 
Heat Engine Cycle 
8. You are now at point A in Figs. 2 and 3. Time to run your little engine through its cycle. For spreadsheet space reasons the elapsed time of this cycle has to be less than 60 seconds. 
a. With the spring balance, pick up the 200g mass (the mass hanger with the 100 g weight added). The spring should be fully extended. 
b. Click on the Start button in DataStudio. This starts the timer. 
c. Process AB: Place the mass on the platform, taking about 10 seconds to uniformly transfer the weight. (The weight indicator should go at a constant speed from 2N to 0.) 
d. Process BC: Transfer the can from the ice water to the hot water. Quickly take the Al cylinder out of the ice water and slowly submerge the cylinder in the boiling water, taking 510 seconds for the submersion. Allow the piston time to stop moving before taking the next step. 
e. Process CD: Slowly, smoothly remove the 200g mass from the platform with the spring balance, taking about 10 seconds. 
f. Process DA: Transfer the aluminum can from the hot water to the ice water. Take 510 seconds to totally submerge the cylinder in the ice water after taking it out of the boiling water. 
g. Click on Stop. 

The position and pressure data are presented in a table and plotted on a graph. When you finish the cycle, the plot is similar to Fig. 3. You should not have more than 120 data points. You need more than 3 or 4 points in the AB and CD processes. You may want to take several runs to get a cycle with plenty of points in straight lines for each process. 

9. Repeat Step 8 with a 100g mass instead of 200 g. (The spring will extend only half as far.) Make a note of which run # corresponds with which mass. Do not quit DataStudio until you have finished your analysis in Excel. 
Analysis 
Note: do not enter anything into the light orange cells in the template. Numbers will appear as you work through the analysis. 

1. Click on StartTemplatesThird QuarterHeat Engine. 



2. First calculate the radius of the cylinder in meters. Accept the manufacturer’s specifications for the diameter of the cylinder as d = 3.25 ± 0 .01 cm. 



3. You need to transfer the data from Science Workshop into the Excel template. Select the run that gives the best data for 200 g. 



4. From the File menu in DataStudio choose Export Data... 



5. The data measurement you want is under Pressure, Ch A vs Position, Ch 1 & 2. Double click on your chosen Run #. 



6. Save this data under a unique name as a text file in My Documents. 



7. Open the My Documents folder from the desktop. 



8. Double click on the file you just saved. Notepad opens up with your data displayed. 



9. Highlight all the numbers in the two columns (i.e., everything but the first two lines.) 



10. Copy this selection (Ctrl C). 



11. Switch to the Heat Engine template and select cell A8. Paste the data here (Ctrl V). There should be no more than 120 points in the table. 



12. Repeat this process (Steps 311) for your best 100 g data set. To paste the 100g data, select H8. 



For both data sets: 



13. Calculate the pointtopoint change in cylindrical volume from the position data. (V = πr^{2}h, where h is the position. The Excel formula you put in for the 200 g data will work for the 100 g data when you copy it over.) For simplicity, we assume that this change is the volume, since we are only interested in how the volume changes. (E.g., it doesn’t matter if the volume goes from 14.85 to 18.21 or from 0.00 to 3.36, the change is the same.) 



14. Convert the pressure data from kPa to N/m^{2} (1 Pa = 1 N/m^{2}). 



15. P vs. V is automatically plotted when V and P are calculated. You should have a rough parallelogram. 



16. Find the points B, C, and D in your raw data set that corresponds to the vertices of the parallelogram. A is automatically the first point. 



a. Hover the cursor over the best point on the chart; the coordinates appear. 



b. Match these numbers with a data pair in your columns and mark that point with the appropriate letter, in the column to the right of the pressure column. 



17. Find the corners of the approximation rectangle. 



a. Find the average volume V_{1} and the standard error u{V_{1}} of all the volumes in process AB, not just the corners. 



b. Find the average pressure P_{B} and u{P_{B}} of all the pressures in process BC. 



c. Find the average volume V_{2} and u{V_{2}} of all the volumes in process CD. 



d. Find the average pressure P_{A} and u{P_{A}} of all the pressures in process DA. 



18. Using Eq. 4, find the thermodynamic work performed by the heat engine from the area bound by these four lines. Also find the error in this calculation. 



19. Now you need to calculate the mechanical work mgh to compare with the thermodynamic work. The mechanical work occurs during the power stroke, process BC. Find h from the difference between the raw data yvalues associated with B and C from step 16. 



20. Propagate the error for the mechanical work and calculate it. 



21. Compare the thermodynamic work to the mechanical work. 



Quadrature Analysis 



22. Find the thermodynamic work done in the cycle by using numerical integration, for each of your two data sets. Also find the uncertainty in these calculations. 



a. Copy the point A (both pressure and volume) and add it right after the last point. This ensures a closed curve. 



b. Calculate the work done along the curve for each pair of data points, from Eq. (5). 



c. Calculate the error in this number from Eq. (6). Assume 



d. Find the total work done from Eq. (7). 



e. Find the error in this number from Eq. (8). 



23. Compare this thermodynamic work with the mechanical work. 



24. Which technique of finding the thermodynamic work is more accurate? Which is more precise? 


1. What is the maximum theoretical thermal efficiency of this engine? 

2. Discuss any systematic errors that would be introduced by starting with the piston near the top of the cylinder (with enough room to move up without hitting the top during the expansion phase.) rather than near the bottom. 
