
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 2: Accounting Principle for Mass
Question

Answer

What is it?

The mass of an object is a measure of the amount of matter in the object.

How can it be stored inside the system?

If there is any matter inside the system, then the system has mass. Given a system of volume _{} and information about the density r of the matter in the system, then the system mass m_{sys} can be calculated from the integral over the system volume
_{}
where r is the mass density of the substance.

How can it be transported across the system boundary?

Mass can only be transported across a system boundary when atoms or molecules physically move across the system boundary between the system and the surroundings. In general, this transport occurs due to either gross fluid motion or through molecular diffusion.
In either case, we can define the mass flow rate to be the rate at which mass crosses a boundary per unit time. The symbol adopted for the mass flow rate will be a dotted lowercase m, _{} .

How can mass be generated or consumed within the system?

Empirical evidence has repeated demonstrated that for the conditions of most engineering applications, mass cannot be created or destroyed within the boundaries of a system. Thus mass is conserved!

Accounting Equation for Mass (Conservation Equation)

Rate form of
Conservation of Mass

_{}
where _{} is the mass flow rate and the summations are over all the inlets and outlets.

Finitetime form of
Conservation of Mass

_{}
where _{} , the amount of mass that flows across the boundary in the time interval.




Table 4 summarizes the rateform 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 authors prejudice is to focus on the rateform of the equations because it is an easy matter to go from the rateform to the finitetime form by integrating both sides with respect to time.
Table 3: Accounting Principle for Linear Momentum
Question

Answer

What is linear momentum?

Linear momentum of a particle is the product of mass and velocity: P = mV

How can linear momentum be stored inside the system?

If there is any matter inside the system and that mass has velocity, then the system has linear momentum.
For a system of n discrete particles, then the linear momentum of the system of particles is
_{}
For a continuous system of volume _{} with density r and velocity V, both functions of position and time, the system linear momentum can be calculated from the integral over the system volume
_{}

How can linear momentum be transported across the system boundary?

Linear momentum can be transported by two mechanisms: forces and masstransport of linear momentum.
For a system, the transport rate of linear momentum by an external force is F_{external}. External forces can be classified as either body forces, like weight, or surface (or contact) forces.
For an open system, every mass that crosses the system boundary carries with it linear momentum due to its velocity. The mass transport rate of linear momentum is the product of the mass flow rate and the velocity, _{} .

How can linear momentum be generated or consumed within the system?

Empirical evidence has repeatedly demonstrated that linear momentum cannot be created or destroyed within the boundaries of a system. Thus linear momentum is conserved!

Accounting Equation for Linear Momentum (Conservation Equation)

Rate form
of
Conservation of Linear Momentum

_{}
where _{} and _{} are the masstransport rates of linear momentum at the boundary and the summations are over all the inlets and outlets.

Finitetime form
of
Conservation of Linear Momentum

_{}
where _{} , the amount of mass that flows across the boundary in the time interval.




Table 4 summarizes the rateform 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.
Table 4  Rateform of the Basic Laws
Mass
(Conserved)

_{}

The rate at which mass is accumulated within the system is equal to the difference between the rate at which mass enters the system and the rate at which mass leaves the system. The symbol _{} is a conventional symbol for mass rate into a system. The symbol _{} is a conventional symbol for mass rate exiting a system.

Charge
(Conserved)

_{}

The rate at which charge is accumulated within the system is equal to the difference between the rate at which change enters the system and the rate at which charge leaves the system.

Linear
Momentum
(Conserved)

_{}

The rate at which linear momentum is accumulated within the system is equal to the sum of the external forces acting upon the system plus the rate at which mass entering the system adds linear momentum minus the rate at which mass leaving the system subtracts linear momentum. For a closed system, i.e., a system that does not exchange mass with its surroundings, the rate law simplifies to a statement of Newtons second law.

Angular
Momentum
(Conserved)

_{}

The rate at which angular momentum is accumulated within the system is equal to the sum of the external moments (or torques) acting upon the system plus the rate at which mass entering the system adds angular momentum minus the rate at which mass leaving the system subtracts angular momentum. For a closed system, i.e., a system that does not exchange mass with its surroundings, the rate law simplifies to a statement that the rate at which angular moment accumulates within a system is equal to the sum of external moments (or torques) acting on the system.

Energy
(Conserved)

_{}

The rate at which energy is accumulated within the system is equal to the net rate of heat flow into the system plus the net work done on the system plus the rate at which mass entering the system adds energy (through either potential, kinetic or internal energy{?}) minus the rate at which mass leaving the system subtracts energy. {Do we need comments about sign conventions?}

Entropy

_{}

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 F_{2} is rewritten as the difference between the rate at which linear momentum is accumulating and the force F_{1}, 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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