Thermodynamic work and efficiency of a heat engine Jared Operaña*, Richmond Crisostomo, Ciara Janer National Institute of Physics, University of the Philippines, Diliman, Quezon City, Philippines *Corresponding author:
[email protected]
Abstract This experiment aims to describe the cyclical operations involved in an ideal heat engine and to quantitatively determine the thermodynamic net work done in a heat engine cycle. The percentage difference calculated for the thermodynamic work done compared with the mechanical work done is 43.47%, while the calculated average efficiency of the heat engine was 3.134 x 10-3. Sources of error might include leaks in the gas chamber, inaccuracy of value measurements. Although the experiment presented a lack of precision, it can be concluded that the experiment was able to describe the behaviour of the operation of a heat engine, with the plotted data points as its supporting detail. Recommendations for future experiments include more careful and precise measurements of data.
Introduction Heat engines convert heat energy into other useful forms such as mechanical or electrical energy. [1] One good example of a heat engine is the one that automobile engines use. They operate in a cyclic manner, using the energy extracted from the burning fuel in the form of heat to do useful work to set the automobile in motion. All heat engines operate by carrying a working substance through a cyclic process involving hot and cold reservoirs. For a heat energy to work, a certain amount of heat is extracted from the hot reservoir, resulting to some mechanical work W done, and the remaining heat that is unable to do useful work is transferred into the cold reservoir. This experiment aims to simulate and investigate the work done by the different processes involved in a heat engine cycle, mainly, the Ericsson cycle. The Ericsson cycle, which will be detailed in the methodology, consists of 4 processes. The total work done by this engine is the sum of the work done by the engine for each individual thermodynamic process, given by the equation: Wone cycle = WA→B + WB→C + WC→D + WD→A
(1)
where the work Wi→f for each state transition i to f can be calculated by: 𝑉
𝑊𝑖→f = ∫𝑉 𝑓 𝑃𝑑𝑉
(2)
𝑖
where P is the pressure and Vf and Vi are the final and initial values of volume, respectively. Assuming that the gas inside the heat engine is an ideal gas, the pressure and volume of the gas are related by: PV = NkT
(3)
where T is the temperature, N is the number of gas particles and k is the Boltzmann constant with value 1.381 J/K. For the Isothermal processes, the work done by the gas is given by: 𝑉
Wi→f isothermal = 𝑃𝑖 𝑉𝑖 𝑙𝑛( 𝑉𝑓) 𝑖
In an Isothermal process, the system is exposed to a constant temperature
(4)
For the Isobaric processes, the work done by the gas is given by: Wi→f isobaric = 𝑃𝑖 (𝑉𝑖 − 𝑉𝑖 )
(5)
The total work Wone cycle of the Ericsson cycle is the sum of all the work down by the gas for each process in the cycle, and is given by: 𝑉
Wone cycle = nR(𝑇𝐻 − 𝑇𝐶 )𝑙𝑛(𝑉𝐴 ) 𝐵
(6)
where TB and TC are the temperatures of the hot and cold reservoirs, respectively. The efficiency 𝜀 of an engine is defined by the amount of work done by the engine in one cycle (Wone cycle) divided by the total heat (Qin) that went inside the engine during that one cycle of work, and can be calculated using the equation: 7
ε = (1 −
𝑇
(1− 𝐶 𝑇𝐶 2 𝑇𝐻 −1 ) [ ] 𝑃 𝑇𝐻 ln( 𝐵 )
(7)
𝑃𝐴
Methodology The Ericsson cycle consists of four stages A to B which is isothermal (constant low temperature), B to C which is isobaric (constant high pressure), C to D which is isothermal (constant high temperature) and D to A which is isobaric (constant low pressure). Following the stages of the Ericsson cycle, the experiment also consisted of four parts. The necessary quantities such as the diameter of the piston, and the mass of the platform were initially recorded from the provided specifications in the heat engine apparatus. The piston was raised to 50 cm and the heat engine apparatus was then connected to an air chamber and the Vernier Lab Quest with the gas pressure sensor using two rubber tubings. A hot bath was prepared by boiling water and a cold bath was prepared by placing ice cubes in a container with water. The temperatures were maintained by adding water and ice cubes in the hot and cold baths respectively. The air chamber was dipped alternately in the hot and cold bath. Stages at which the air chamber is kept at constant hot or constant cold baths were the isothermal stages. A weight of 100g was also placed and removed in the platform alternately in the whole duration of the experiment. Stages at which there is constantly no mass or stages where there constantly has mass are isobaric stages. The changes in the temperature and mass on the platform induced changes in the pressure and volume inside the heat engine apparatus and these changes were recorded. In stage A, the air chamber was placed in the cold bath with no mass on the platform. The pressure, height of piston and the temperature of the cold bath were measured and recorded.
Figure 1. Stage A of the Ericsson cycle In stage B, the air chamber was kept in the cold bath making A to B stage isothermal. The 100g mass was placed on the platform thus inducing changes in pressure and volume. The pressure, height of piston and the temperature of the cold bath were measured and recorded.
Figure 2. Stage B of the Ericsson cycle During stage C, the 100g mass was kept on the platform making B to C stage isobaric but the air chamber can was transferred into the hot bath. The pressure, height of piston and the temperature of the hot bath were also measured and recorded.
Figure 3. Stage C of the Ericsson cycle In the last part, stage D the 100g mass was removed from the platform but keeping the air chamber hot bath. This made the stages D to A isobaric as well since both had no mass on the platform. The pressure, height of piston and the temperature of the hot bath were measured and recorded.
Figure 4. Stage D of the Ericsson cycle The process was repeated for three trials. Using the data gathered, a Pressure vs. Volume graph was constructed. This diagram was used to compute for the thermodynamic work for each thermodynamic process. The computed net thermodynamic work was then compared with the computed mechanical work in lifting the platform and the mass.
Data and Analysis At the start of the experiment the necessary parameters were recorded as seen below in Table W1. For the sake of convenience, units were adjusted to SI so that converting won’t be necessary anymore. Table W1: Set-up Specifications D-Piston (m) Mass object (kg) V-chamber (m^3) 0.0325 0.135 0.00075 Note that apart from the mass of the metal disk, the mass-contribution of the apparatus platform was also considered (0.035 kg). In the 4 stages of the Heat Engine (A. no weight and cold; B. added weight and cold; C. added weight and hot; D. no weight and hot) the necessary data were gathered as well (pressure, height, and temperature). Table W2 below provides the data for each stage A, B, C, and D of the Heat engine.
Table W2: P, V, T (Trial 1) State P (Pa) h (m) V (m^3) T( 'C) A 100330 0.044 3.65014E-05 0.2 B 101410 0.0405 3.35979E-05 0.2 C 101410 0.065 5.39225E-05 97.5 D 100330 0.067 5.55816E-05 97.5 (Note that all measurements were already converted for the experimenters’ convenience.) The experiment was done for 3 trials. Results of the other 2 follow below.
State A B C D
P (Pa) 100330 101410 101470 100310
Table W2: P, V, T, Trial 2 h (m) V (m^3) 0.044 3.65014E-05 0.0415 3.44274E-05 0.053 4.39676E-05 0.055 4.56267E-05
T( 'C) 0.1 0 97.8 97.8
Table W2: P, V, T, data TRIAL 3 State P (Pa) h (m) V (m^3) T( 'C) A 100280 0.045 3.7331E-05 0.3 B 101420 0.0425 3.5257E-05 0.3 C 101360 0.0675 5.59964E-05 97.5 D 100330 0.0685 5.6826E-05 97.5 As observed, although the numerical values are not equivalent all the time for all the trials, they follow the same trend. A plot of Pressure versus Volume below allows this trend to be seen more clearly. Figures 1-3 below provide a Pressure vs Volume plot for their respective trials. 3 similar quadrilaterals can be seen when the 4 points of each sets are connected to one another. From previous discussion, it was proven that the “area” occupied by the quadrilateral constitutes the thermodynamic work – that is the physical representation of the geometry. Using numerical tools, the thermodynamic work will be discussed further as we go along the discussion (both geometric and analytic means were used in computing the work).
Figure 1
Figure 2
Figure 3 Figure 4 below superimposes all the (3) plots, making their similarities even more evident.
Figure 4 Table W3. Thermodynamic process per transition Transition Thermodynamic Process Isothermal A --> B Isobaric B --> C Isothermal C --> D Isobaric D --> A [2]Transition A to B and C to D exhibited an isothermal thermodynamic process since it was kept constant at low and high temperature respectively and only varying the pressure applied in the system throughout the transition. This means that all the changes in the thermodynamic system during this transition occurred at constant temperature. Transition B to C and D to A however were isobaric thermodynamic processes since both were kept at constant pressure applied (mass on the platform) whilst changing the temperature. This means that all the changes in the thermodynamic system during such transition happened at constant pressure.
Trial 1
Thermodynamic values (Area of the quadrilateral approach) (J) 0.0212785920
Trial 2
0.0104900110
Trial 3
0.0218199029
Average Value
0.017862835
In finding the thermodynamic work, it was mentioned earlier that the geometry of the plot is crucial. This is because connecting the (4) points from the 4 stages (A, B, C, and D) of the heat engine will give rise to a quadrilateral. Now the physical interpretation of its area is the thermodynamic work in joules! In finding the area of the quadrilateral, the liner equations corresponding to the sides of the quadrilateral were obtained. From this, the area of the polygon was calculated. In particular, all the calculations were numerically done (this is to say that the numerical approximations below, as shown in “Thermodynamic values” are close enough, to the actual – with a high degree of certainty). Table W5: Thermodynamic Work and Mechanical Work for Each Cycle Thermodynamic Work Mechanical Work Percent Difference 0.017862835
0.027783
43.47%
Using the eq. (7), the efficiency of the heat engine was calculated for each trial. Trial
Efficiency (𝜀)
1 2 3 Average
3.023 x 10-3 3.024 x 10-3 3.190 x 10-3 3.134 x 10-3
Conclusion From the calculated value of the efficiency, it appears that the heat engine was not very efficient, having only a value 3.134 x10-3. However, We can see that by pressure and volume variance--- using the geometric and the analytic approach --- it is time-reverse symmetric. That is to say, that step-wise reversing the heat engine process from any point in the cycle it must produce the same output, assuming that the heat engine is an ideal one, in other words, heat is fully converted into mechanical work.
References [1] http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heaeng.html. Accessed May 4, 2016. 9:47 PM PST [2] https://www.boundless.com/physics/textbooks/boundless-physics-textbook/thermodynamics14/the-first-law-of-thermodynamics-117/isothermal-processes-408-4347/. Accessed May 4, 2016. 10:30 PM PST
Appendix Table W4: Thermodynamic Work by Gas (Trial 1) Transition Work Done by Gas (J) A --> B -0.303549813 B --> C 2.061120913 C --> D 0.165718136 D --> A -1.914323149 One Cycle 0.008966087
Table W4: Thermodynamic Work by Gas (Trial 2) Transition Work Done by Gas (J) A --> B -0.214223838 B --> C 0.967464918 C --> D 0.165255537 D --> A -0.915363347 One Cycle 0.003133271
Table W4: Thermodynamic Work by Gas (Trial 3) Transition Work Done by Gas (J) A --> B -0.213975283 B --> C 2.103392 C --> D 0.083469131 D --> A -1.95593887 One Cycle 0.016946977 Average Thermodynamic Work 0.009682112 Table W5: Thermodynamic Work and Mechanical Work for Each Cycle Thermodynamic Work Mechanical Work Percent Difference 0.009682112
0.027783
96.62%
