5.06 Energy Equation

Vector multiplication of Newton's 2'nd Law (with a BODY as system) produced the first and simplest energy equation. Two forms of energy were identified and specified in terms of mass and the outwardly observable properties position, displacement and velocity. Work and/or equilibration were recognized as mechanisms of energy change. Some call this beginning energy equation the Extrinsic Energy Equation and the associated work, extrinsic work. Friction of events, reasonably modeled as extrinsic, is addressed in a limited way.

Our energy equation for a Body, Extended Body (or Bodies) is:

(1) A system as BODY has only kinetic and potential energy forms.

The fact of system energy change and work associated with deformation (with effects of displacement and velocity zero or inconsequential) obliged engineering to studies the intrinsic nature of matter, its energy and work. This is the realm of thermodynamics.

All substances have internal energy, the only intrinsic energy form (enthalpy is a composite property) and all substances have intrinsic work in the compression mode. Simply put, this means the energy of all substances can be changed by compression (by forcing it to occupy a smaller space). Other substances (in esoteric and very important situations have other work modes). Compression work, spring work and shaft work are listed explicitly when writing the intrinsic energy equation. Friction is ever present; at the system boundary and within. Finally, heat is recognized. The intrinsic energy equation we have used is:

(2) Some systems are compressible. These can have energy change
even when potential and kinetic energy changes are zero.

Though the above two equations represent distinct aspects of reality, physical circumstances arise wherein a system event includes matter changes both extrinsic and intrinsic. For such cases, the equations can be used individually or as a convenience, the equations can be written as one that represents both aspects of reality and event. Notice the next equation reduces to either of the previous two upon removal of terms.

(3) Equations (1) and (2) can be combined. This resulting
energy equation is suited for both types of system.

A next and final adjustment of the energy equation requires terms to be added right-of-equality to account for the possibility of energy flow and work attendant with mass flow into and out of a system.

ENERGY AND WORK OF MASS FLOW: The sketch shows a section horizontal pipe which contains two snugly-fitted (but frictionless) pistons with water between them. Our system is the water within the pipe and between the stationary planes labeled "in" and "out."

The system is stationary, its boundary is shown as a dashed line. Initially F_left is equal to F_right and there is no motion. Relative to the stationary system, water will move (flow) to the right. The restriction (orifice) is included to emphasize the fact of pressure decrease in the direction of flow.

We don't need all of the system. What we seek to show will involve only the piston to the left, and the water between the piston face and the "in" plane (the water in the short length of pipe, ΔX).

Imagine the force, F_left to increase slightly and its piston to move to the right at a constant speed. The force continues to displace. That energy rate transmits to the piston at its left face. The piston does not deform; at constant speed its energy does not change. The work rate of our interest passes through the piston to its right face and from that face to the water. We carry the idea of passage of that work the over the short, ΔX (piston-like) expanse of water to fixed place, the entrance of water at the " in " or entrance of the system space of our system.

From previous study we have a work relation that represents what is happening at the left face of the piston. Slightly to the right of the left piston face across the small distance, ΔX, at the stationary, " in " boundary of a system across which a flow of mass occurs. We bring two equations together: the work at the left face of the piston and the mass flow across the " in " plane:

(4) 4

Our plan is to remove the piston and the ΔX of water so as to express the work at the left in terms of properties of water at " in ."

The algebra goes like this. First for W-dot (above left) both vectors have I components: W-dot = |F||V|. Next the speed of the piston, |V|_piston is equal to speed of the water moving across " in ", (|V|_in). The force on the left face of the piston equals the force on the right face which equals the (average) pressure of the water on the right face times the piston area. The speed of the pistion equals the speed of water moving across the plane " in. "

(5) 5

In the last expression above, the density is that of water at the "in" in plane and the pressure is that of water at the piston face. We argue that ΔX is so small that these properties are those of the entrance plane, "in." In a similar fashion the work of flow crossing the "out" plane can be obtained. Both results with the customary change of density, ρ, to specific volume, v.

(6) 6

A next consideration of mass flow into a system is the energy of the mass itself. That energy rate, a sum of the internal, kinetic and potential components of the flow, is written as:

(7) 7

Now, imagine there are many "in" and "out" planes, each with its flow work and energy rate. Let the situation be steady in time. We write the energy equation then collect it. Immediately we see the sum, u + pv appears. This is the second instance when that group was needed. Enthalpy is a very convenient variable. The sum is represented as "h." Our energy equation for this is:

(8) 8

The above development was done for a steady system at no harm to the development but a savings in writing. To obtain our system equation from energy, our last energy equation we simply replace the leading "0" (for steady) with the other familiar term.

(9) 9