Sophomore Engineering Curricula	Introduction	Conservation and Accounting Framework	Curriculum Structure: Texas A&M four-course structure	Curriculum Structure: Texas A&M Five-Course Structure
	Curriculum Structure: Rose-Hulman Institute of Technology Sophomore Engineering Curriculum	Example Problems	Student Performance/Faculty Reactions	Conclusions

Formulating the Physical Laws in the Conservation and Accounting Framework

To help students begin to see the common features of the basic laws of physics and to provide a framework for problem solving it is useful to restate all of the basic laws in terms of the CAF. Answering four questions for each extensive property of interest provides the form of the physical law in the CAF: 

(1)   What is it?

(2)   How can it be stored inside the system?

(3)   How can it be transported across the system boundary? Students need to understand and apply the mechanisms through which an extensive property can cross the boundary. For example if you are counting something like energy, you will need to know all of the ways that energy can cross a boundary and determine which, if any, are applicable to the situation at hand.

(4)   How can it be generated or consumed inside the system? In addition to transport mechanisms, students need to understand and apply knowledge about when and how an extensive property can be created or consumed.  For example if you are counting positive charge, then you need to know that positive charge can be created by ionization processes or consumed by recombination processes. When a quantity can neither be created nor consumed we say that quantity is conserved.  Again, this definition is different from the definition used in many textbooks on physics and engineering science.

Once students have answered these four questions, then they can adapt the accounting principle for each extensive property and demonstrate the underlying similarities between the fundamental principles of physics.  Consider the extensive properties mass and linear momentum.  Table 2 provides answers to each of these questions for mass, and Table 3 provides answers for linear momentum.

Table 4 summarizes the rate-form of the accounting principle for six extensive properties that are commonly used in engineering analysis. As students attempt to solve engineering problems, they are often confronted with relating changes within a system to things that happen to the system.  The accounting equation provides an explicit way to relate these things. It in fact is the only mechanism for relating system interactions that are spatially separated on the boundary of a system. For example, if I consider a compressed spring, how are the forces acting on the ends of the spring related? If we take the stationary spring as the system and apply conservation of linear momentum, we see that the linear momentum of the system is constant (in fact it is zero) and the forces acting on the system boundary must be equal in magnitude and opposite in direction.

This author’s prejudice is to focus on the rate-form of the equations because it is an easy matter to go from the rate-form to the finite-time form by integrating both sides with respect to time.

            Table 4 summarizes the rate-form of the accounting principle for six extensive properties that are commonly used in engineering analysis. As students attempt to solve engineering problems, they are often confronted with relating changes within a system to things that happen to the system.  The accounting equation provides an explicit way to relate these things. It in fact is the only mechanism for relating system interactions that are spatially separated on the boundary of a system. For example, if I consider a compressed spring, how are the forces acting on the ends of the spring related? If we take the stationary spring as the system and apply conservation of linear momentum, we see that the linear momentum of the system is constant (in fact it is zero) and the forces acting on the system boundary must be equal in magnitude and opposite in direction.

Another approach to understanding the CAF is to compare it to the discipline of system dynamics.  In the later discipline, much emphasis is placed on energy storage and transfer by identifying "through" variables and "across" variables. How are the "through" and "across" variables related to the six intensive properties in Table 4?  To facilitate the comparison, consider a simple, linear, ideal spring as a closed system. Application of the conservation of linear momentum and shows that the forces at the two ends of the spring

are opposite direction and differ in magnitude by the rate at which linear momentum is accumulated in the spring.  If the spring is stationary or the spring mass is negligible, the forces (transport rates of momentum) are equal and opposite in direction.  Application of the conservation of energy shows that the rate at which energy accumulates in the system is given by

If the force F₂ is rewritten as the difference between the rate at which linear momentum is accumulating and the force F₁, then the conservation of energy equation may be rewritten.

If the rate at which linear momentum accumulates is zero, e.g., if the spring is stationary or massless, then the rate at which energy accumulates is equal to the force applied to the spring dotted with the difference in the velocities between the two ends of the spring.  In a systems dynamics framework, , the force is the "through" variable for energy transfer.  The through variable exists if and only if the spring does not accumulate linear momentum.  Also, in a systems dynamics framework, the velocity difference between the two ends of the spring is the "across" variable and is related to the rate at which energy accumulates in the spring.  Typically, the "across" variable is really just the difference between the values of an intensive property measured at two points on the system boundary.  So the "through" and "across" variables in a systems dynamics framework can be obtained from the accounting equations for the intensive properties in Table 4.. 

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References

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