Investigating student understanding of heat engine: a case study of Stirling engine
IInvestigating student understanding of heat engine: a case study of Stirling engine
Lilin Zhu and Gang Xiang ∗ Department of Physics, Sichuan University, Chengdu 610064, China (Dated: August 17, 2020)We report on the study of student difficulties regarding heat engine in the context of Stirlingcycle within upper-division undergraduate thermal physics course. An in-class test about a Stirlingengine with a regenerator was taken by three classes, and the students were asked to perform oneof the most basic activities—calculate the efficiency of the heat engine. Our data suggest thatquite a few students have not developed a robust conceptual understanding of basic engineeringknowledge of the heat engine, including the function of the regenerator and the influence of pistonmovements on the heat and work involved in the engine. Most notably, although the science errorratios of the three classes were similar ( ∼ PACS numbers:
I. INTRODUCTION
Any device that transforms heat partly into work ormechanical energy is called a heat engine. All engines ab-sorb heat from a source at a relatively high temperature,perform some mechanical work, and reject some heat ata lower temperature. About two hundred years ago, aFrench engineer Sadi Carnot developed a hypothetical,idealized heat engine that has the maximum possible ef-ficiency consistent with the second law of thermodynam-ics [1]. The cycle of the engine is called Carnot cycle.Almost at the same time (8 years earlier), another cy-cle called Stirling cycle was proposed by Robert Stirling,which is known as a reversible cycle as basic and impor-tant as Carnot cycle in thermodynamics [2–4]. In fact,Stirling engine was one of the earliest heat engines put topractical use and is still used in various facilities such assubmarines and concentrated solar power system. Otherbasic cycles such as Otto cycle [5] and Diesel cycle [6]were proposed later in 19th century and have been usedin the engines of various vehicles such as cars since then.Fig. 1 shows four classical cycles: (a) Carnot cycle, (b)Stirling cycle, (c) Otto cycle and (d) Diesel cycle. Amongthem, both Carnot cycle and Stirling cycle are used in ex-ternal combustion engines, while Otto cycle and Dieselcycle are used in internal combustion engines. Interest-ingly, all the Thermodynamics textbooks describe Carnotcycle in great detail, but many of them just introduce theother three cycles briefly [7] and some of them even don’t ∗ Electronic address: [email protected] cover them at all. Coincidentally, or perhaps correspond-ingly, a number of studies in physics education researchhave shed light on student understanding of basic con-cepts of Carnot cycle [9–13], but very few studies havefocused on the topics related to Stirling cycle, Otto cycleor Diesel cycle [14–17], which are widely used in automo-biles and many other types of machinery. In this sense,evaluating of student understanding of basic knowledgeof different kinds of heat engines, for instance, Carnotengine and Stirling engine, is important and necessaryfor the development of the pedagogical content knowl-edge (PCK) needed to teach the heat engine well, notonly in physics, but also in related applied disciplines.It is worthwhile noting that both Carnot cycle and Stir-ling cycle are reversible cycles working between two heatreservoirs, which means Carnot’s theorem are applicableto both of them.Among various educational evaluation techniques,post-testing is the most widely adopted method in thephysics education research (PER) [18–20]. PER focuseson understanding how students learn physics at all levelsand developing strategies to help students learn physicsmore effectively [21, 22]. Based on these facts, we de-signed an in-class test about Stirling cycle at the end ofthe course and then performed this study. The objec-tives of this study are as follows. First of all, Stirlingcycle is a good example for the students to understandhow to calculate the efficiency of the cycle, since it in-volves a regenerator that alternatively absorbs and re-leases heat, which is unique (and puzzling) and not foundin any other cycles including Carnot cycle. Secondly, ac-cording to Carnot’s theorem, every reversible heat enginebetween a pair of heat reservoirs, such as Carnot engineand Stirling engine, is equally efficient, regardless of the a r X i v : . [ phy s i c s . e d - ph ] A ug OP VQ Q OOP V V V V V Q Q V V V V 𝑉𝑃 Q Q O V V V 𝑃 𝑉Q Q (a) (b)(c) (d) FIG. 1: The P-V diagrams for (a) Carnot cycle, (b) Stirlingcycle, (c) Otto cycle and (d) Diesel cycle. Q and Q representthe quantities of heat absorbed and rejected by the engineduring one cycle, respectively. working substance or the operation details. However, itis possible that some students would take it for grantedthat the heat engine here only refers to Carnot engine ifother types of heat engines are not taught or just intro-duced briefly. In this sense, our study can estimate towhat extent the students understand Carnot’s theorem.Finally, it is noted that in the last decade or so, a grow-ing number of researchers have shown interest in studentunderstanding of thermodynamics beyond the theoreticallevel, leading to upper-division engineering instruction onheat engines [23–25].Interestingly, Stirling engine with aregenerator in this study is such an exquisite design thatthe more the students work on it, the more the studentswill be stimulated to be curious in thermodynamics andbe interested in integrating theory with practice.This paper is organized as the following: we first in-troduce Stirling cycle in Sec. II. A brief overview of theresearch context and methodology is then presented inSec. III. Then we discuss the student difficulties iden-tified as well as implications for instruction. Finally, asummary is given in Sec. V. II. STIRLING CYCLE
A Stirling cycle is similar to a Carnot cycle, exceptthat the two adiabatic processes are replaced by twoconstant-volume processes. As shown in Fig. 2(a), a Stir-ling cycle involves two isothermal processes → and → , as well as two constant-volume processes → and → . In practice, a Stirling heat engine includesa cylinder containing two opposed pistons and a regen- erator between the pistons. The regenerator serves asa thermodynamic sponge that alternatively absorbs andreleases heat. The cylinder is divided by the regeneratorinto two parts: the volume between the regenerator anda piston maintained at high temperature T max is calledthe expansion space, while the other, maintained at lowtemperature T min the compression space, as illustratedin Fig. 2(b), which shows the piston arrangement at theterminal points of Stirling cycle. At the beginning, all theworking gas is in the compression space with the temper-ature T min , which is corresponding to the point in Fig.2(a). In the constant-volume process → , the work-ing gas is transferred from the compression space to theexpansion one through the regenerator, which is heatedfrom T min to T max . During the final process → ,both pistons move simultaneously, transferring the work-ing gas from the expansion space to the compression one.In the passage through the regenerator, the working gasreleases heat and emerges into the compression space at T min .The question on Stirling cycle, shown in Fig. 2, was de-veloped to assess student understanding of the efficiencycalculation of heat engine. It should be emphasized thatduring the cycle, the heat transferred in the constant-volume processes → and → is contained in theregenerator. Therefore, the regenerator adds an addi-tional layer of complexity. It is pedagogically useful, sinceit can be used to assess or facilitate deeper conceptualunderstanding of important quantitative and qualitativeproperties of heat engines. III. METHODS
This investigation was carried out with three classesof physics undergraduates enrolled in Thermodynamicsand Statistical Mechanics course at Sichuan University(SCU) in 2019, which is a renowned four-year public re-search university in China. This course is typically takenduring the first semester of the junior year. The sizes ofthe classes varied from 14 to 54 students. The studentswere given course credit for participating in in-class testsduring the semester and a final exam at the end. Thethree classes were taught by different instructors, but thelectures, homework assignments and in-class tests werepresented in traditional formats. The only nontraditionalaspect was the addition of voluntary tutorial given by oneof the authors (Xiang), which emphasized not only thetheoretical (science) aspects, but also the experimental(engineering) aspects of the basic knowledge of thermo-dynamics, for instance, how the refrigerator is scientifi-cally and mechanically related to Carnot cycle and howthe engineering improvements of internal combustion en-gines have influenced the scientific and technological ad-vances of our society and impacted our daily life. Thistutorial is new and unusual to the participants, especiallywhen one recognizes that traditional college courses ofphysics in China do not attach much importance to ex- 𝑉 ! 𝑉 " 𝑃 𝑉 𝑇 𝑇 (a) Regenerator 𝑇 !" 𝑇 !$% Compression space Expansion space (b) * Question : The Stirling cycle is shown above: (a) P-V diagram; (b) piston arrangement at the four terminalpoints of the cycle. Assuming that the working material is an ideal diatomic gas and the pistons move withoutfriction or leakage of the gas, calculate the efficiency of the heat engine.
FIG. 2: Question used in this study. Administered after lecture instruction. perimental education or engineering training. It is notedthat neither the structure of Stirling engine, nor the cal-culation of the efficiency of Stirling cycle was covered inthe tutorial. Therefore, the preliminary analysis showedno significant differences in answering the questions ofwhat Stirling cycle is and how it works between tutorialparticipants and nonparticipants.This work focuses on the results from a particular ques-tion from the in-class test, shown in Fig. 2. The studentscompleted the test individually in quiet rooms and werecut off after 30 minutes. In this study, the working mate-rial is assumed to be an ideal diatomic gas for the calcu-lation of the cycle efficiency. The heat transferred in theconstant-volume process → is equal to that in the pro-cess → for ideal Stirling cycle. In this case, one onlyneed to consider the heat supply in the process → and the heat rejection in the process → between theengine and its surroundings. This heat supply and heatrejection at constant temperature satisfies the require-ment of the second law of thermodynamics for maximumthermal efficiency, so that the efficiency of Stirling cycleis the same as Carnot cycle. Any answer that satisfies afew key criteria was considered as a correct one. Thesecriteria include correct calculations of heat rejection inthe isothermal process → , heat supply in the isother-mal process → and efficiency η of the cycle, or clearexpressions of using Carnot’s Theorem to obtain correctefficiency for Stirling cycle.Therefore, Stirling cycle not only gives us opportunityto estimate student understanding of general knowledgeabout the heat, work and efficiency involved in a heatengine, but also provides novelty to the students whichcan be used to explore their understanding of Carnot’s theorem. Although the students have learned that allreversible heat engines between two heat reservoirs havethe same efficiency, regardless of other details, our studywill show that most of the students can only associateCarnot’s theorem with Carnot cycle, but not with Stir-ling cycle.In addition to the collection and analysis of the writ-ten answers by the students, post-test interviews werealso carried out with the students. Each interview lastedan average of 10 minutes. Instead of being constrained tocover all the questions about Stirling cycle, the interviewsfocused on the most commonly discussed or typical ques-tions that were related to the student errors in the in-classtest. During the interviews, the students were required totalk aloud, and the interviewers provided minimal inter-ventions only for reminding the students to keep talkingor asking them to clarify explanations not understood bythe interviewers. IV. RESULTS AND ANALYSIS
The data from the in-class test on Stirling cycle werecollected and analyzed at SCU in 2019 to investigate stu-dent performance. In the following, we discuss the testresults in terms of qualitative and quantitative descrip-tion.
A. Qualitative description of student difficulties
To better understand student thinking behind theirwritten answers, interviews were conducted, which in-volved an undergraduate student and an interviewer eachtime. It was found that the most common error made bythe students was the misunderstanding in the functionof regenerator. As a result, the heat transferred in thetwo constant-volume processes → and → wasincluded when calculating the efficiency of Stirling cycle.Some typical interviews are listed as follows. Interview 1
I: Can you tell me why you consider the heat supplyin the process → and heat rejection in the process → when calculating the efficiency of Stirling cycle?S1: That’s just how I learned. The heat supply andheat rejection of all processes should be included accord-ing to the formula of heat engine’s efficiency.I: You mean the heat transferred in the engine shouldalso be considered ?S1: Yes.I: How about the regenerator? You didn’t consider itduring the calculation.S1: Um... I’m not sure. Maybe it’s designed for im-proving the efficiency. Interview 2
I: Can you explain why you only include the heat sup-ply and the heat rejection in the two isothermal processes → and → when calculating the efficiency?S2: Because in this way, I can get the efficiency of ( T − T ) /T .I: You mean Stirling engine has the same efficiency asCarnot engine?S2: Yes.I: Okay, why?S2: Um... I’m not sure.I: Let me ask in a slightly different way. Where do thetwo constant-volume processes → and → happen?S2: In the engine?I: So you think the heat transferred for the whole cycleis independent of the regenerator?S2: Yes. Actually, I think it is just for the practicalengineering requirement.Both S1 and S2 (and indeed most students inter-viewed) did clearly know that the efficiency of heat engineis defined as η = Q H − Q C Q H . (1)This appeared to be extremely stable in that studentsnever questioned this, even when confronted with otherissues. It should be pointed out that Q H and Q C in Eq.(1) represent the quantities of heat transferred from thehot and cold reservoirs during one cycle, respectively. Inother words, the heat transferred per cycle in the engine The symbols are used to represent to absolute values of the en-ergy transfers throughout a heat engine cycle. Therefore, theyare inherently positive. should not be contained. But most students misunder-stood the physical meaning of Q H and Q C . S1 was firmerin the belief that all heat transferred during the wholecycle should be considered. Actually, S1 didn’t under-stand the physical meaning of efficiency of heat engine. S2 was trying to speak that all reversible engines havethe same efficiency, but was unable to properly verbal-ize it. Even though Carnot’s theorem is fundamental inthe course of Thermodynamics, S2 was not familiar withit. Therefore, S2 also may not understand that only theheat transferred between the heat engine and hot (cold)reservoir rather than the total heat transferred duringthe whole cycle should be considered when calculatingthe efficiency of heat engine.These interviews were fairly representative of the stu-dents as a whole. Most of the students were not ableto describe how the regenerator works, indicating a lackof understanding of the relevant engineering concepts ofheat engine. As indicated by the interviews, the studentsin general had a more varied mastery of the function ofregenerator. In some cases, the students only treated theregenerator as a required engineering component with-out knowing its function. This appears to stem fromthe students’ lack of engineering knowledge of heat en-gines. In the Thermodynamics course, students majoringin physics usually only learn the theoretical knowledge ofheat engines. Relevant engineering aspects are not in-troduced in the classes, unless the instructor places em-phasis on them. This assertion is supported by students’preformation in the test. Only three students gave cor-rect description of the regenerator, who were from thesame class. Their instructor (Xiang) did give them somegeneral engineering introduction of heat engine. B. Quantitative description of student difficulties
The answers given by the students were analyzed caseby case. Four types of errors were found in the an-swers, i.e., miscalculation, The answers given by the stu-dents were analyzed case by case. Four types of errorswere found in the answers, i.e., miscalculation, misus-ing formulas, misunderstanding cycle process and mis-understanding regenerator. To get a clear picture of thestudent performance, we present the percentages of oc-currence of the four errors for the three classes lecturedby Wu (left panel), Xiang (middle panel) and Zhu (rightpanel) as well as the case of no error in Fig. 3. Firstly,it is remarkable that more students got correct answerin Xiang class (above 40%) than the other two classes( ∼ (c) Zhu A B C D E0.20.40.60.8 (b) Xiang
A B C D E0.20.40.60.8 (a) Wu
A: MiscalculationB: Misusing formulasC: Misunderstanding cycle processD: Misunderstanding regenerator
E: No error
A B C D E00.20.40.60.81 P e r c en t o f S t uden t s FIG. 3: Comparison of student errors in the test on Stirling cycle for (a) Wu class, (b) Xiang class and (c) Zhu class. Somebars are composed of several parts with different colors. The grey part shows the percentage of students who only made oneerror, while the others represent some students made multiple errors, simultaneously. The case of no error is established byempty bar. The detailed explanation can be found in the context. errorpercentage Class Wu Xiang Zhu errorpercentage Class Wu Xiang ZhuMiscalculation 7.6% 0 0Science error 10.2% 8.3% 14.3% Misusing formulas 2.6% 8.3% 14.3%Misunderstanding cycle process 15.4% 2.1% 7.2%Engineering error 71.8% 52.1% 71.4% Misunderstanding regenerator 61.5% 50.0% 64.2%No error 23.1% 45.8% 14.3%
TABLE I: Quantitative results of the errors found in the answers of the three classes lectured by Wu, Xiang and Zhu. Thescience errors include the errors of miscalculation and misusing formulas, and the engineering errors include the errors ofmisunderstanding cycle process and misunderstanding regenerator. magenta part represents the students made the errors ofmiscalculation and misunderstanding cycle process, si-multaneously. Similarly, the green part shows the per-centage of students who made the errors of miscalculationand misunderstanding regenerator, simultaneously. Thesimilar results can be found in the other errors. Generallyspeaking, the results shown in Fig. 3 indicate that mostof students adequately understood Stirling cycle processand knew how to do the calculation with the formulas inthe textbooks. However, almost all students don’t knowthe function of regenerator. This is due to the pedagog-ical techniques we used during the teaching.Since miscalculation and misusing formulas werelargely related to the student ability of understand-ing thermodynamics theory and performing mathemat-ical calculation, they were classified as science errors . And since misunderstanding cycle process and misunder-standing regenerator were basically related to the stu-dents’ ability to understand the mechanical processes inStirling cycle, they were classified as engineering er-rors . In fact, we found that many students who misun-derstood cycle process also misunderstood regenerator,which make sense since both the errors were caused bythe poor comprehensive ability of engineering aspects ofStirling cycle. The results of the quantitative statisticsof the errors are listed in Table I.The good news is that the science error ratios in thethree classes were low, which was 8.3% (Xiang), 14.3%(Zhu) and 10.2% (Wu), respectively. The roughly 10%error ratio in all the classes shows that the importance ofthe basic thermodynamics theories was recognized bothin the teaching and learning aspects very well and at theapproximately same level. This is actually a tradition inthe teaching and learning process of most of the physicscourses in China.The bad news is that the engineering error ratios wereextremely high, which were above 50% for all the classes.Detailed analysis show that among the engineering er-rors, the error of misunderstanding regenerator domi-nates. For instance, in Xiang class, 52.1% (25) partic-ipants made engineering errors, 50% (24) participantsmade the error of misunderstanding regenerator, and2.1% (1) participant made the error of misunderstandingcycle process, respectively. The participants who mis-understood regenerator tended to count in the heat ab-sorbed and/or released by the regenerator and obtainedthe wrong answer for the efficiency. The participant whomisunderstood cycle process thought the heat engine didnonzero work during the isovolumic process. Similar re-sults were found in Zhu and Wu classes.Although the engineering errors were high for the threeclasses, there was a big difference between Xiang classand the other two classes: the engineering ratio of Xiangclass was smaller by approximately 20% than those of theother two. As we mentioned before, the only differencebetween the lectures of the three classes was that Xiangclass gave a tutorial, which did not cover anything relatedto Stirling cycle but did introduce general knowledge suchas mechanical counterparts of refrigerator correspondingto the reverse Carnot cycle and the meaning of the engi-neering improvement of heat engines to our society. Thetutorial was very preliminary, but it could help some ofthe students realize that engineering is as important asscience in terms of applications of the thermodynamicsand really inspire their interest in the engineering as-pects of the heat engines. To some extent, the differencebetween not giving any tutorials and even just giving apreliminary tutorial is like the different between “0” and“1” in computer science. Our results demonstrated howthe preliminary tutorial emphasizing engineering aspectsof basic knowledges can influence the teaching results.Another interesting finding is that we can actuallyuse the quantitative results to investigate how the stu-dents understand Carnot’s theorem. Very few students,for instance, two students in Xiang class and one stu-dent in Wu class, made errors in the solution process,but got the right answer for the efficiency of Stirling en-gine. The written answer sheets or the interviews showedthat they used Carnot’s theorem to obtain the answer di-rectly after they got stuck in analyzing the cycle processesand performing efficiency calculations. But this usage ofCarnot’s theorem was kind of passive instead of active, i.e., the students did not realize to use Carnot’s theoremuntil they failed in other ways. In other words, most ofthe students only correlate Carnot’s theorem with Carnotcycle and not with other reversible cycles such as Stirlingcycle. The very poor performance on Carnot’s theoremwas somewhat unexpected, as explicit instruction on itwas included in all classes. Our speculation is that thisis because for a long time traditional Thermodynamicstextbooks only emphasize Carnot cycle and neglect otherimportant cycles. In this sense, it is necessary to empha-size the importance of other cycles besides Carnot cycle.Our results suggest that Stirling cycle could be an im-portant supplement to Carnot cycle.
V. SUMMARY
This paper describes an in-depth investigation of stu-dent understanding of heat engine in the context of Stir-ling engine. Our findings indicate that the majority ofstudents in the three classes studied can comprehendthe scientific theory of heat engine and make the cor-rect mathematical calculations. However, above 50% stu-dents for all the classes don’t understand the basic engi-neering knowledge of Stirling cycle, especially the func-tion of the regenerator. Importantly, the engineering er-ror ratio of the class that was given a tutorial of engineer-ing knowledge of heat engine was smaller by about 20%than those of the other two. The results suggest that it isuseful to include both scientific and engineering aspectsof basic knowledge in instructional approaches, especiallyin the course of Thermodynamics taught in the countriesand regions with a tradition of not paying much attentionto experimental teaching or engineering training. In ad-dition, findings from this investigation also suggest thatit is necessary to attach more importance to other cyclesbesides Carnot cycle in the Thermodynamics textbooksto improve the students’ understanding of Carnot’s the-orem and related knowledge of heat engine.
Acknowledgements
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