
Problem Solving using the Conservation and Accounting Framework
Now that the basic concepts of the conservation and accounting framework have been formulated, we can suggest a problem solving approach that is based on a set of generic questions. As students approach a new physical situation focusing on a recurring set of questions helps students focus on the important issues involved in formulating models of the physics situation and finding the desired quantities.
What is the system? Choose the boundary
This is almost always the starting point for the analysis of every physical situation. In each engineering science, the question of choosing the system is often phrased differently and the focus is on a node, a freebody diagram, a closed system, or a control volume. In fluid mechanics and thermodynamics this is usually accomplished when students are usually asked to sketch the control volume and in dynamics when students are asked to sketch the freebody diagram. It is important for the students to learn that more than one system may be required for the solution of a particular problem. Although terminology differs, the goal is always the same a clear description of exactly what you intend to analyze. Implicit in this question is the need to clearly identify the boundary and the surroundings. The conservation and accounting framework recognizes the interconnectedness of everything. To avoid becoming overwhelmed by the knowledge that almost everything is impacted by almost everything else, answering the first question forces the students to define the system, recognize the surroundings and consider the interactions between the two.
During the process students are required to clearly define a system and its boundary. Then, they can apply the accounting principle by watching the boundary for things entering and leaving. They can also watch the interior of the boundary to determine what changes inside. The boundary can be something physical like a rigid container, or it can be something imaginary. A useful example for introducing concepts about the conservation and accounting frame is a checking account at a bank where the boundary is an imaginary boundary. The bank does not have small areas where they keep money. An account is simply a convenient way to think about your money. A safe deposit box is a physical boundary in a bank and you could use it to apply conservation principles to money you put inside. The method requires students to carefully identify boundaries and interactions between the system and its surroundings.
What should we count?Choose what to count
Next, in order to apply the accounting principle students should determine what to count. There are five quantities that are commonly counted in engineering problems. These quantities are mass, momentum, angular momentum, energy, and entropy. An advantage of the framework is it focuses attention on physical properties and helps students to think about physical processes in terms of these properties. Further, the question focuses attention on what is actually happening in the problem in terms of the extensive properties. Which of the extensive properties mass, charge, linear momentum, angular momentum, energy, or entropy should we be interested in? Which of these properties are changing?
What is the time interval of interest?
This question focuses student attention on the process. What type of process has occurred or will be occurring? This question is basically asking the students to identify whether the rate form of the basic principles or the finite time form is most appropriate.
What are the important interactions?
This question is intimately related to the previous question. For example if linear momentum is to be counted then the student should be on the look out for interactions that transport momentum: external forces and mass flow. Or if a student believes that forces acting on a system are important, then linear and possibly angular momentum must be counted. Although the mechanisms and names vary from property to property, the underlying idea of an exchange of something with the surroundings is a common feature of any engineering system.
Know how to count
The last thing that you need to be able to do is determine how much of a quantity is inside your boundary. For example, if you are counting energy to solve a problem then you will have to determine the amount of energy inside your boundary. The majority of the time spent in the class deals with this concept therefore it is impossible to enumerate all the intricacies here. Essentially these concepts involve relating temperature to internal energy, speed to kinetic energy and vertical height to gravity potential.
Tools for Insight in Analysis
Two additional concepts that are sometimes useful in the analysis and design of a system are the degrees of freedom (dof) and order.
Order is the number of independent storage elements inside the system boundary. The easiest operational definition for independent storage is one that is not dependent on another. Two storage elements are dependent if knowing the quantity stored in one implies the storage in the other is also known. For example, a mass moving in a single direction has the ability to store kinetic energy and linear momentum. If the linear momentum is known, the velocity of the mass is known, once the velocity is known, the kinetic energy is also known therefore the mass has one order. If the mass can move in two directions, the order is two since knowing kinetic energy will not completely specify the two momentums but knowing energy and one momentum will fix the other momentum. Students often find the exercise of determining order helpful in identifying what conservation equations to write, knowing how many equations to expect when they are finished, and for helping to define a proper set of variables to use in the formulation.
Determining the Degree of Freedom (dof) is an exercise in identifying different types of variables. One type of variable is a flow or motion. For example, velocity and current are motion variables. The dof is the minimum number of independent motion variables required for describing the conservation equations. By determining the dof, students are forced to think about the problem formulation before they begin to write equations. In addition, the dof will indicate when extra constraint equations are required. For example, suppose a mass moves in a plane such that it remains a constant distance from a point of rotation. The mass has one dof because a single angle and its derivatives are sufficient to express all the motion related quantities in the conservation equations. If the conservation equations are expressed in terms of two variables (say horizontal and vertical positions) then the conservation equations will have more variables than can be uniquely determined. What is required is a kinematic constraint equation that relates the motion variables together. By counting the dof and the number of motion variables in the conservation equations, a student can determine if constraint equations are required.
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References

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