In this chapter, we will define four types of energy: work, heat, kinetic energy, and internal energy. As we shall see, work and heat are forms of communication between two systems or a system and its surroundings. Kinetic energy and internal energy, on the other hand, are forms of energy that a system contains. Work and heat are energy transfers and kinetic energy and internal energy are energy storage.

Sections:

5.1 Definition of Work

5.2 Definition of Heat

5.3 Definition of Kinetic Energy

5.4 Definition of Internal Energy

5.5 The Conservation of Energy Principle

5.6 Definition of Enthalpy

5.7 Chapter Reference

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5.1 Definition of Work

Work is done on a system by the application of a force on the substance or charge that is in motion relative to the reference coordinate frame. The amount of work is given by,

(5-1)

The subscript, F, on the position vector is to remind us that it is the movement of the object upon which the force is acting that contributes to work. For example, it is easy to forget that surface forces acting on a given system are not applied to the system boundary. Rather, these forces are applied to the substance that happens to be at the same location in space as the system boundary. Recall that system boundaries have no real entity; they are mathematical surfaces with no properties except geometrical position and size. It is not physically possible to apply a force on such an object. If the system is a closed system, the substance will move with the boundary, in which case the distinction is not significant. However, if the system is a control volume or an open system with substance (or charge) crossing the boundary, the distinction is extremely important.

The integration of work depends on how the entity that the force is acting on moves as well as the end points of the motion. Such integrals are referred to as path functions and their differentials are referred to as inexact differentials. The bar through the differential symbol will be used to indicate inexact differentials. Exact differentials, on the other hand may be integrated knowing only the end points of the integral. The differential of a property is always an exact differential.

Power is the rate of doing work,

(5-2)

In Chapter 4, we found that there were three types of forces which act on a system -- surface forces, gravitational forces, and electromagnetic forces. The right hand side of Eqn. 4-14 is the sum of these forces. Their application created or destroyed momentum, which is given by the left hand side of Eqn. 4-14. Applying the definition of power to these forces, we have,

(5-3)

These three terms represent surface power, gravitational power, and electromagnetic power, respectively. Each of these types of power will now be explored.

Surface Power. Recall that the material velocity relative to the system boundary was defined in Section 3.4,. Using this relative velocity and the velocity of the system boundary, the surface power becomes,

(5-4)

The first of these terms represents the power done on a system by the surroundings in the process of pushing the substance into the system. As a special, but typical, case, consider a fluid entering a system at an inflow port with negligible shear stress. As we found in Chapter 4, the stress tensor for this case is the negative of the normal stress times the identity tensor. In this case, the power in question is,

(5-5)

Recall that we have defined the specific volume, v, to be the reciprocal of the mass density. The product of specific volume and mass density is everywhere unity, so the above equation may be rewritten,

(5-6)

If we now assume that the pressure and the specific volume are uniform over the inflow port area, we have what we shall call flow power,

(5-7)

There would be a similar flow power out of the system at an outflow port. If the flow is into the system, the dot product of the velocity and area vectors is negative; thus, flow power is positive for inflow and negative for outflow.

The second type of surface power is the power done on the system by a movement of the substance at the system boundary caused by the motion of the boundary itself. Consider a simple case -- a fluid contained in a cylinder and piston arrangement. The system is the fluid inside the cylinder and piston. The external force is acting on the piston, which in turn is acting on the system. The reference coordinate frame is at rest relative to the system and has base vectors as shown.

Take the piston as an auxiliary system. Assuming that the fluid is at or near equilibrium, the stress tensor is just the negative of the pressure times the identity tensor. The force acting on the left hand side of the piston is then given by,

(5-8)

The external force applied on the system is then given by the following in terms of properties of the system,

(5-9)

The motion of the system boundary will be for an increase in the distance, x. Thus the boundary velocity is given by,

(5-10)

The power associated with this external force and motion is then given by,

(5-11)

Because the area of the piston is a constant, it may be taken inside of the derivative. Finally, the product of the area, A, and the displacement, x, is the volume of the system. We then have the final result for this case,

(5-12)

We will refer to this as compressive power. The power is positive when the volume is decreasing with time or the fluid is being compressed. When the volume is increasing, the power is from the system to the surroundings and is frequently called expansion power. The work associated with the compressive power is the time integral of the power,

(5-13)

Gravitational Power. This type of power is the second type given in Eqn 5-3,

(5-14)

Recall that the weight force vector is relative to a coordinate frame attached to the earth, which is an inertial coordinate frame. The material velocity must also be relative to this inertial coordinate frame. For small changes of elevation compared with the mean radius of the earth, we can assume that the acceleration of gravity is approximately constant. When we also assume a uniform velocity throughout the system, we have,

(5-15)

Because the acceleration of gravity, g, is only a week function of elevation, the differential work here is an exact differential. The traditional treatment of work done on a system by the gravitational attraction is to define a potential energy to be the negative of the gravitational work. To keep a consistency between the Force-Momentum Principle and the forthcoming Conservation of Energy Principle, we will retain gravitational work as a work term and not a potential energy term.

Shaft Power Consider a system with a circular shaft crossing the system boundaries, as shown. The shaft has no axial motion relative to the system boundary, so all of the motion is in the plane of the boundary. Use standard cylindrical coordinates with the z-axis along the center of the shaft pointing outward from the system boundary. The shaft is turning with an angular speed in the theta direction. Consider the cross-section of the shaft at the system boundary, as shown. The speed of a circular ring at distance, r, from the center of the shaft and having a width of dr, as shown, is then uniform. The area vector of this ring has the value,

(5-16)

where the base vector, k, is the base vector for the z-axis. The velocity vector of this ring is,

(5-17)

where the base vector, j, is the base vector for the theta-axis. We will not assume anything about the nature of the stress tensor, and include all nine components. The power transmitted by this shaft is then,

(5-18)

This reduces to,

(5-19)

which may be seen to be the angular speed of the shaft times the torque transmitted by the shaft.

Electromagnetic Power. Eqns. 4-5 and 4-6 gave the electromagnetic body force for a continuum. The power associated with this type of force is,

(5-20)

Notice that the cross product of the velocity vector and the magnetic field vector results in a vector perpendicular to the two. When that result is dotted into the velocity vector, the result will be zero. The divergence theorem may be used to derive the following [1],

(5-21)

It is shown in Reference 1 that the volume integral may be interpreted as a storage of electromagnetic energy. For the special case of two cables crossing the system boundary and the storage term is not changing with time, the surface integral term reduces to,

(5-22)

That is, electric power is the product of the electric potential difference between the two wires time the current flowing though them. This result is all that will be needed in this treatise.

Please note that each of these particular types of work represents a communication process between a given system and its surroundings. Each is also a highly organized, mathematical operation. In almost every analysis related to the Conservation of Energy Principle, the first step is to carefully look for forces acting on something that is in motion. For each case found, you must then attempt to evaluate the work or power involved. Look first for body force powers, then look for powers on the system boundary. After all of the works or powers have been identified, you may then look for other types of energy.

In the SI unit system, the units of work and power are the Joule (J) and Watt (W), respectively. They are defined as,

1 J = 1 N-m and 1 W = 1 J/s

In the English unit system, the units of work and power are ft-lbf and Watt (W), respectively. The watt is defined exactly as it is in the SI unit system. By using the definitions of the foot and the lbf, it may be found that,

1 ft-lbf/s = 1.3558 W

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5.2 Definition of Heat

Heat is also an energy transfer process, but unlike work, it is a highly disorganized process. We define heat for a closed system initially and generalize for an open system later. If there is communication between two closed systems that cannot be identified as being work, then it is heat. The amount of heat to the receiving system is equal to the amount of work that would have to be done for the same process in the absence of heat.

In many ways this is a very unsatisfying definition of heat. There are no involved mathematical expressions as there were for work. That is the nature of heat. One important part of the definition is that if we find heat from system 1 to system 2, we can have the same changes occur in system 2 by doing work on it instead of doing heat on it. Work can always replace heat. We will soon see that heat cannot always replace work. For example, if we wish to warm a house in the winter, we can do so by either having heat transfer to the house or by work transfer to the house (e.g., by using an electric heater).

In some analyses, particularly if the details of the heat transfer process are being determined, it is convenient to define the heat flux vector as the vector field such that the heat rate into a system is given by,

(5-23)

for any arbitrary system. The negative sign is included because we wish to have the heat flux vector pointing into the system at those locations where we have heat transfer to the system.

If a system has no heat transfer with its surroundings, it is said to be an adiabatic system.

Because heat is measured by an equivalent amount of work, the units for heat and work are the same. The SI units for heat and heat rate are the Joule and Watt, just as for work and power, respectively. It is traditional to use the British Thermal Unit (Btu) for heat in the English unit system. The International Steam Tables Btu is defined from the following,

1 Btu/lbm = 2326 J/kg

A Btu is also 778.170 ft-lbf. Heat rate is in units of Watts, which is 3.41214 Btu/hr. (There is a slight difference from the definition of the IST Btu and the mean Btu.)

We need to repeat that heat, like work, is an energy transfer process. It is inappropriate to refer to the amount of `heat' something has, because heat can not be stored. What systems do store are kinetic energy and internal energy, which are our next topics.

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5.3 Definition of Kinetic Energy

Kinetic energy is a storable type of mechanical energy. It is defined by the following,

(5-24)

As is the case with work, a system's kinetic energy depends on the condition of the system as well as the coordinate frame used to look at the system. It is also a highly organized form of energy and has a detailed mathematical definition, similar to work. The units for kinetic energy are the same as for work or heat.

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5.4 Definition of Internal Energy

For any process involving any closed system, the change in internal energy, U, may be calculated from the following balance equation:

(5-25)

The subscript, `in,' reminds us that if the energy rate is directed into the system, it enters the balance with a positive sign; otherwise, it enters the equation with a negative sign. Note that internal energy is defined within a time derivative. This means that only changes in its value may be determined. In particular, it is not inferred that internal energy has an absolute zero.

Equation 5-21 may be used to measure changes in internal energy for many different substances in many different situations. In every case, it is found that internal energy is an extensive property of the substance within the system. In particular, the changes are found to be independent of the path taken from an initial state to a final state. This observation is generalized into the Conservation of Energy Principle.

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5.5 The Conservation of Energy Principle

The essence of the Energy Principle is that the internal energy, whose changes are defined by Eqn. 5-21, is an extensive property of the substances. Define the intensive internal energy as a scalar point function of space and time such that the extensive internal energy is given by,

(5-26)

In terms of these properties, the balance equation becomes the following for the general, open system,

(5-27)

The bracket encloses the terms that make up the time rate of change of the two stored energies. This is followed by the net convection of internal and kinetic energies out of the system. On the right hand side are the three forms of power and the heat rate transferred to the system from the surroundings. This equation may thus be cast into the following more concise form,

(5-28)

It is this latter expression, together with the similar expression for the Conservation of Mass Principle (Eqn. 3-6), which will be used in subsequent thermodynamic analyses.

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5.6 Definition of Enthalpy

Whenever we have convection of material across a system boundary, we will have convection of internal energy and flow power. Thus, for any port, there will be two entries into the Conservation of Energy Principle:

Notice that outflows of internal energy are positive on the left hand side of Eqn. 5-24, whereas flow powers from the system enter as negative terms. The result is that when combined the two terms will have the same sign for any given port. It is convenient to define the sum of internal energy and the pv product to be a new intensive property, the enthalpy,

(5-29)

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5.7 Chapter Reference

1. Scott, W. T., The Physics of Electricity and Magnetism, Second Edition, John Wiley & Sons, Inc., New York, 1966, pp. 554ff.

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