Influence of uncertainties and assessment of significant digits in thermodynamics

Thermodynamics calculations are predominantly deterministic and students are expected to solve a problem for quantities which can be calculated to many significant digits using a calculator or software. Students often report final answers to many more significant digits than justifiable. Textbooks often emphasize that final answers are justifiable to a limited number of digits because of the lack of precision of inputs and/or internal coefficients/models. It is often recommended that final results be expressed to three significant digits. Students are encouraged to keep intermediate digits during intermediate calculations and then round the final result to an appropriate number of digits. It has been observed that significant digits continues to be a difficult concept for some students with the common misconception that an answer with five or six significant digits is equivalent to, if not better than, an answer with fewer digits. This paper reports work where students are required to solve thermodynamic problems with uncertain inputs. Typical problems have input values with no specified uncertainty. For example, the inlet temperature may be specified to be 480°C or the pressure to be 2.0 MPa. A number of problems have been developed that include uncertainties, and the student is expected to report a final answer with the propagated uncertainty. Problems are solved using hand calculations and then with the aid of a spreadsheet. Using software, students can access routines to evaluate thermodynamic properties which are tedious if done by hand. The approach is based on the traditional differential method for uncertainty propagation, yet numerical differentiation in used in the spreadsheet program. Examples show that when uncertainties are considered, there can be relatively large uncertainties in final results. By knowing a result’s uncertainty, students can report final answers to an appropriate number of digits. A student survey was conducted to gauge the effect of the exercises on student learning and attitudes. Students show an increased aptitude for reporting answers to an appropriate number of significant digits and a positive attitude toward the methods covered. Some comments indicate that the methods are inadequately covered in lower-level engineering classes. This is useful information for continuous improvement of the engineering curriculum. Student feedback indicates they understand how to use these techniques in subsequent classes, indicating students have gained a deeper understanding of the concepts.