PHYS111Q : Labs

13 A Heat Engine

Introduction

The first law of thermodynamics relates the work done on or by a system and the heat flowing into or out of it to the resulting change in its internal energy as

$$\Delta E^\mathrm{int} = Q - W$$ (62)

This is nothing more than a statement that energy is conserved. The sign conventions are that positive heat $Q$ flows into the system and positive work $W$ is done by the system. Work done by or on a gas can be calculated by finding the area under the graph of its pressure vs. its volume, called a $P$-$V$ diagram. A negative change in volume corresponds to negative work. We can apply the first law to several thermodynamic processes as follows.

• The internal energy of a fixed number of moles of an ideal gas depends only on its temperature. In any process, the change in internal energy is given by
  
  $$\Delta E^\mathrm{int} = n C_V \Delta T$$
  
  
  with
  
  $$C_V = \frac{f}{2} R$$
  
  
  where $f$ is the number of degrees of freedom of the molecules of the gas. We use $f = 3$ for a monatomic gas and $f = 5$ for a diatomic gas (near room temperature).

• An isobaric process takes place at constant pressure. Isobaric processes involve both changes in internal energy and heat flow. For an ideal gas, it is particularly easy to calculate the work done on or by the gas, because the area under the $P$-$V$ diagram for the system is rectangular,
  

  $$W^\mathrm{isobaric} = P \Delta V = nR \Delta T \hspace{1.5cm... ...\right) P \Delta V =\left( \frac{f}{2} + 1 \right) nR \Delta T$$ (63)

  
  

• A constant-volume process involves no work, since $\Delta V=0$, so any heat flowing into or out of the system results entirely in a change in the internal energy of the system,
  

  $$Q^\mathrm{constant V} = \Delta E^\mathrm{int} = n C_V \Delta T = \frac{f}{2} \Delta P V$$ (64)

  
  

• An isothermal process, takes place at constant temperature. If the temperature of a closed system remains constant, so does its internal energy, so according to the first law of thermodynamics, any work done results in heat flowing into or out of the system
  

  $$Q = W^\mathrm{isothermal} = nRT \ln \left(\frac{V_2}{V_1}\right)$$ (65)

  
  

• In an adiabatic process, no heat enters or leaves the system. In this case, according to the first law of thermodynamics, all of the work done goes into changing the internal energy of the system,
  

  $$W^\mathrm{adiabatic} = -\Delta E^\mathrm{int} = - n C_V \Delta T$$ (66)

  
  
  A process in a system that is thermally insulated from its environment is adiabatic. In a system which is not thermally insulated from its environment, an approximately adiabatic process is possible if it occurs rapidly enough that there is not time for a significant quantity of heat flow into or out of the system.

A series of processes which return a system to its initial state is called a cycle. A cycle involves no change in internal energy, since the initial and final states are identical. Hence, the work done results solely in a net flow of heat into or out of the system

$$W^\mathrm{net} = Q^\mathrm{net} = \vert Q^\mathrm{in}\vert - \vert Q^\mathrm{out}\vert$$ (67)

The net work is also given by the area enclosed by the $P$-$V$ diagram of the cycle. It follows from Eq. 67 that a cyclic system designed to do work (a steam engine, for example) must absorb more heat than it gives off. The efficiency of an engine is the ratio

$$\epsilon = \frac{W^\mathrm{net}}{Q^\mathrm{in}}$$ (68)

of the work done to the heat absorbed.

Experiments

Pressure Sensor Calibration

Your pressure sensor is not calibrated well enough to give good results for the heat engine experiment, so you will need to calibrate it.

1. Follow this link to get the current local atmospheric pressure $P_o$ in inches of mercury. Convert it to Pa. (There are 2.54 centimeters in an inch, and 76 cm Hg = 1.01e5 Pa.)

2. Disconnect one of the tubes from the piston chamber, place the piston at a position of 45 mm, and seal the system by re-connecting the tube.

3. Connect the pressure sensor to the port labeled CH 1 on the LabPro interface.

4. Run Logger Pro. It should recognize the pressure sensor and automatically give you a graph of pressure vs. time.

5. Use the Data Collection ... button to set the experiment length to more than 60 seconds so that you can complete the calibration measurements in one run.

6. Orient the piston chamber vertically, and begin collecting data.

7. Collect data with nothing on the platform and with 50 g, 100 g, 150 g, 200 g, and 250 g loads. Collect data with each load for about 5 seconds. It is important that you work quickly, because the system leaks air slowly when under pressure.

An Engine Cycle

You will make pressure and volume measurements to characterize the behavior of a simple heat engine in which an expanding volume of trapped air lifts a load placed atop a frictionless piston. The engine converts some of the heat flowing into it from a hot water bath into the mechanical work done lifting the load. It then dumps the remaining heat into an ice water bath. The engine cycle consists of processes between the following four states.

A. No load, cold bath.
B. With load, cold bath.
C. With load, hot bath.
D. No load, hot bath.

The engine lifts its load in process $B \rightarrow C$. The other processes return the engine to its initial state so that it can lift another load.

1. Fill one plastic cup 3/4 full of ice water and another 3/4 full of nearly boiling water.

2. Disconnect one of the tubes from the piston chamber, place the piston at a position of 45 mm, and seal the system by re-connecting the tube.

3. Run Logger Pro.

4. Begin collecting data, and take the system through states A, B, C, and D in order. Use a 200 g mass as the load. Leave the system in each state for 5 seconds. Record the piston position corresponding to each state. Also record the approximate elapsed time so that you can determine the pressure corresponding to each state from your Logger Pro data. It is important that you work quickly, because the system leaks air slowly when under pressure.

Analysis

Pressure Sensor Calibration

1. Use the Statistics button to determine the average pressure for each load from your Logger Pro data.

2. Enter your experimental pressures into a column in Excel.

3. In an adjacent column, calculate the corresponding calibrated pressures $P^\mathrm{th}$ using the weight of each load and the area of the piston via
   

   $$P^\mathrm{th} = P_o + \frac{(m^\mathrm{platform} + m^\mathrm{load})g}{\pi r^2}$$ (69)

   
   
   The radius of the piston is $r = 0.01625$ m, and the mass of the piston and platform is $m^\mathrm{platform} = 0.035$ kg. We will assume that these theoretical predictions are more reliable than the pressure sensor readings.

4. A linear fit to a graph of $P^\mathrm{th}$ vs. $P^\mathrm{exp}$ describes a function for correcting the data from your pressure sensor
   

   $$P^\mathrm{cal} = a (P^\mathrm{exp}) + b$$ (70)

   
   
   Use the LINEST() function to find the parameters $a$ and $b$ of your calibration function.

An Engine Cycle

1. In a column in Excel, calculate the volumes for each state you measured from your piston positions.

2. In an adjacent column, enter your measured pressures $P^\mathrm{exp}$. (There should only be two pressures to work with here.)

3. In a third column, calculate calibrated pressures using Eq. 70 and the calibration parameters you determined above.

4. Produce $P$-$V$ diagrams for the two cycles you measured using the calibrated pressures.

5. Calculate the work $W^\mathrm{net}$ done by the engine in each cycle ( $A \rightarrow B \rightarrow C \rightarrow D$) by finding the area inside each $P-V$ diagram. You may use the approximation that each process on the $P-V$ diagram is a straight line segment. You may assume that the uncertainty in your result from this measurement is about 5%.

Before You Leave Lab

Show your work to your instructor and discuss preliminary answers to the questions below.

Group Assignment

Hand in your spreadsheet and answers to the following questions.

1. Calculate the mechanical work ($m g \Delta y$) required to lift the 200 g mass through the vertical distance it traveled between states $A$ and $C$. How does your result compare with your result from step 5 of the analysis?

2. Explain why processes $A \rightarrow B$ and $C \rightarrow D$ are approximately adiabatic.

3. Identify the process in which the heat $Q^\mathrm{in}$ is absorbed by the system. Calculate $Q^\mathrm{in}$ (see the Introduction). Show your work. Assume air is a diatomic ideal gas that can rotate but not vibrate ( $C_V = \frac{5}{2} R$).

4. Use Eq. 68 to calculate the efficiency $\epsilon$ of this engine (ignoring the energy required to heat the water and freeze the ice).

Copyright © 2003-2010, Lewis A. Riley

Updated Fri Aug 27 11:05:11 2010

This work is licensed under a Creative Commons License.