On the consistent use of sign convention in thermodynamics

International Journal of Mechanical Engineering Education, Oct 2006 by Boettner, Daisie D, Bailey, Margaret B, Arnas, A Özer

Abstract In thermodynamics, various sign conventions are used for energy transfers in the form of heat and work. Regardless of the sign convention introduced, thermodynamics texts subsequently abandon their established conventions in favor of magnitudes or absolute values. This article illustrates the importance of consistent use of a sign convention throughout a text and applies it to power-producing and power-consuming engineering devices. Additionally, using a selected sign convention, a substantive proof is presented showing why the ratio of energy added/rejected in the form of heat equals the ratio of the absolute temperatures of the energy source/sink, respectively.

Keywords thermodynamics; sign convention; undergraduate education

Notation

c specific heat, kJ/(kg-K)

m mass, kg

M molecular mass, kg/kmol

p pressure, kPa

Q heat transfer, kJ

R universal gas constant, kJ/(kmol-K)

S entropy, kJ/K

T temperature, K

U internal energy, kJ

V volume, m3

W work, kJ

Subscripts

H high-temperature reservoir

L low-temperature reservoir

v constant volume

Greek symbols

η efficiency

β refrigeration cycle coefficient of performance

γ heat pump cycle coefficient of performance

(ProQuest-CSA LLC: ... denotes formulae omitted.)

Introduction

Consistency in presentation and use of concepts in any engineering course is important as it will enhance student learning. Most undergraduate thermodynamics texts introduce a sign convention for energy transfers when presenting the concepts of energy transfers in the forms of heat and work. Adopted by many authors [1-4], the most common sign convention used is: energy added to a system in the form of heat is positive, and that added in the form of work is negative. This is abbreviated as HIP to WIN where Heat In is Positive, and Work In is Negative. Callen [5] and Smith [6] adopt another sign convention, for which all energy in is positive and all energy out is negative. Regardless of the convention adopted, once it is presented, the textbook author should adhere to the convention throughout the text.

The purpose of this article is twofold: to promote consistency in sign convention use; and to encourage preciseness in thermodynamics instruction. First, as indicated by Lewins [7], many individuals find the use of sign conventions perplexing. The engineering textbooks listed in the References and Bibliography sections abandon the established sign convention in favor of magnitudes or absolute values. Çengel and Turner [8] consider abandonment of the sign convention to be a 'relaxed sign convention'. This mid-text change challenges instructors to justify the convention, with which they may or may not agree, and confuses the novice student of thermodynamics. The situation becomes more confusing when the altered sign convention is used in the discussion of performance parameters such as heat engine efficiency and refrigeration and heat pump coefficients of performance.

Second, it is very interesting that Obert [9] says in his preface that 'it was Prof. Joseph Keenan, the teacher, who aroused interest of the author in thermodynamics (and in preciseness)'. This article attempts to illustrate this preciseness to encourage educators to teach correct and precise thermodynamics.

Example applications of sign conventions in use: Carnot and reverse Carnot cycles

Once a text initially establishes a sign convention for energy transfers, it should follow that convention consistently, to avoid any confusion. One example of the application of a sign convention is in the analysis of the Carnot and reverse Carnot (refrigeration and heat pump) cycles to determine the limiting performance of ideal devices. Reversible processes do not exist in nature; however, Hutchinson [10] offers the view that 'Reversible processes form an asymptote to Reality'. By using the established sign convention to analyze the Carnot cycle, the equivalence of the heat transfer ratio to the absolute temperature ratio results. Although Van Ness [11] suggests that 'This is proved in virtually every thermodynamics textbook ever written', the extensive English-language publications listed in the References and Bibliography sections do not support this statement.

This article offers two analyses of the Carnot cycle and an analysis of the reverse Carnot cycle. Each analysis uses a consistent sign convention that does not incorporate the use of absolute values for energy transfers. The aim is to show that the negative ratio of energy transfer in the form of heat associated with a high-temperature reservoir to a low-temperature reservoir is equivalent to the ratio of the absolute temperatures of the high- and low-temperature reservoirs. This result is expressed as:

... (1)

In these analyses, an ideal gas is used as the working substance, since the performance of devices does not depend on the choice of the working substance. Equation 1 also is written in textbooks without the minus sign, which signifies that the absolute values are used since the absolute temperature ratio is always positive whereas one of the heat terms must be negative.