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Why The Balance Principle Should Replace The Reynolds Transport Theorem

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2008 Annual Conference & Exposition


Pittsburgh, Pennsylvania

Publication Date

June 22, 2008

Start Date

June 22, 2008

End Date

June 25, 2008



Conference Session

Thermodynamics, Fluids and Heat Transfer - I

Tagged Division

Mechanical Engineering

Page Count


Page Numbers

13.1401.1 - 13.1401.14



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Paper Authors


Donald Gray West Virginia University

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Donald D. Gray is Professor and Group Coordinator for Environmental and Hydrotechnical Engineering in the Department of Civil and Environmental Engineering at West Virginia University. He has taught introductory fluid mechanics 33 times and wrote A First Course in Fluid Mechanics for Civil Engineers (Water Resources Publications, 2000). He is the author or co-author of more than 175 papers, reports, and conference presentations on fluid mechanics. Dr. Gray is President of the West Virginia Academy of Science and is a Professional Engineer. He graduated from Tulane University (B.S.E.(M.E.)) and Purdue University (M.S.E. and Ph.D.).

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

Why the Balance Principle Should Replace the Reynolds Transport Theorem Abstract

The finite control volume equations for mass, momentum, and energy are important topics in introductory fluids mechanics courses for engineering students. In most contemporary U. S. textbooks these equations are derived by transformation of the corresponding equations for a control mass using the Reynolds Transport Theorem. This paper shows that there is a much simpler path to these equations: the direct application of the balance principle to a control volume. The balance principle is easier to teach, to understand, and to apply in more complex situations. It better prepares students to understand the derivation of the partial differential equations of fluid mechanics and the finite volume equations of computational fluid dynamics. For these reasons the balance principle should replace the Reynolds Transport Theorem in introductory engineering textbooks and courses in fluid mechanics.


Everything should be made as simple as possible, but not simpler. Albert Einstein

The equations for the conservation of mass, momentum, and energy in a finite control volume are among the main topics covered in almost all introductory fluid mechanics courses taken by engineering students in the United States. In today’s most widely used textbooks, the derivation of these equations is performed by using an equation known as the Reynolds Transport Theorem (RTT). This paper will explain how this situation came to be, and will propose that a simpler, yet more powerful, approach based on an intuitively obvious balance principle should be used instead of the RTT.

A control volume is any region of space conceptually isolated from the rest of the universe in order to facilitate the solution of problems involving fluid mechanics, thermodynamics, heat transfer, or other physical phenomena. The boundary of a control volume is called its control surface. In any given situation, the specification of a control volume (a.k.a. an open system) is made by the analyst in order to best solve the problem at hand. In general, the control volume may deform or accelerate in any arbitrary manner and may contain solids as well as fluids. Learning the art of choosing an appropriate control volume is a critical objective for engineering students taking introductory courses in fluid mechanics and other transport sciences.

The control volume is an often viewed as a more convenient alternative to the control mass (a.k.a. control system or closed system) approach in which equations are written for a specified piece of matter conceptually isolated from the rest of the universe. Both the control volume and the control mass can be of finite or infinitesimal size, but in this paper attention will center on the finite case. The derivation of equations governing the matter within a finite control volume is most commonly accomplished by transforming the equations governing the control mass occupying the control volume at some instant by using an equation often called the Reynolds

Gray, D. (2008, June), Why The Balance Principle Should Replace The Reynolds Transport Theorem Paper presented at 2008 Annual Conference & Exposition, Pittsburgh, Pennsylvania. 10.18260/1-2--3852

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