Thermodynamics defines all of space and all that physically exists it that space to comprise the universe. All of space is infinitely vast. It extends way beyond the localized, vanishingly small, regions of our engineering systems (and their events). Since most of universe is not involved with local events, an approach wherein a portion of mass of the universe, the system, is separated from the universe, is used. To define a system, to select a mass, is a perfunctory task; it cannot be done wrong. However the purpose of the system is to yield some answer or understanding about an event. Some systems, though correctly selected, fail to provde the answers sought. In such cases some other system needs to be addressed.
Though scarcely 300 years old, great efforts at thermodynamics have developed considerable understanding to include a methodology and language. The basic terms of thermodynamics; system, energy, heat, work, temperature (and others) are difficult to describe because they are abstractions. The abstracts comprise a circular, self-supporting structure. Should you become impatient learning the language of thermodynamics, take a moment to define two abstracts you already know: truth and life. As Mark Twain might have said about the language of thermodynamics,Thermodynamics is clear as mud but it covers the ground.
Persons reading these words have done pencil-and-paper analyses of physical events in physics. In each of those past instances, there was "a system." Those investigations were of the physically simplist matter; what we call a "body." To procede from physics, we need to understand the "body" (what we know well). Limitations of "body" in physical reality (as we experience it) coerce us us toward new models of real matter. For now we need four models ( categorizations) of system.
The ideas expressed above and in the previous section are sufficient for us to do elementary calculations similar to what you have done in physics. The problems which provide a taste of thermodynamics, are organized with respect to the principle properties of thermodynamic analysis: mass, momentum and energy.
The first step of any thermodynamic analysis is to identify the system and the event. Once a system/event is well-defined, further considerations focus on system properties and how they change (or do not) with the event. What happens to properties is accounted for by special equations. We call these, Property Equations (of the system/event). They are: the Mass Equation, the Momentum Equation and the Energy Equation. Once developed, these property equations will be seen consistent with the "Conservation Laws" learned in physics.
Considerations of a mass and its event are property-focused: "Does mass leave the system? What forces are assocoiated with momentum changes of the event? What energy changes occur as work, heat or equilibration? Answers to such questions exceed the "Conservation Perspective;" such questions require solution of the Property Equations.
With these words we embark upon a re-construction and explanation of the property equations. The ideas of mass, momentum and energy date back over a thousand years. They believed they had the mass equation. Below we show the simplest forms and how they evolved and were transformed as new ideas arose.
MASS EQUATION: The simplest system is the body and the body has at least one, non-zero property: its mass. The mass of a body is constant. Thus for every event of a body:
The above equation is not the Mass Equation. It is a specialization of the mass equation. Since we must become familiar with equation forms, we start here. Constant mass means the scalar magnitude mass is constant in time. Hence the above expression can be differentiated with respect to time to obtain a mass equation in rate form. Two other forms are the increment form and the difference form. The above equation and each of the three equations below state the same thing; mass is constant. The mass equation is quite simple. The value of the three forms will be realized later when we use the momentum and energy equations.
Events with Constant Mass: The equations above have many "0's." One might ask, "Does anything happen with mass being constant?" Yes... Indeed, lots happens. Look now, cases come later.
MOMENTUM EQUATION AND FORCE: The simplest form of momentum equation applies to the system model,"body." Newton stated, "a body at rest persists in the state of rest until some effect of the surroundings changes that state." Newton created the construct, force, as the mechanism whereby surroundings might cause momentum change. Students generally encounter the momentum equation in physics texts as f = ma, where momentum is included implicitly in mA. By the few algebraic steps below, momentum is made explicit.
The equation (above right) displays momentum of the body explicitly. The equation retains meaning when force equals zero and it retains meaning when velocity is zero. But when mass is set equal to zero, the result is nonsensical. Hence, we deem mass to the domimnant entity and its momentum to to be the equation's principle aspect.
To move toward a consistency in equations, we move the system property, momentum, to be left-of-equality and force, a secondary item in this equation, is placed right-of-equality. Forces by Newton's understanding are either body forces or surface forces. The body force we need is gravity and often more than one surface force applies. The final momentum equation (rate form) is:
The above first order, vector differential equation states Newton's First Law and his Sewcond Law of motion by use of the calculus he invented for the purpose. The equation mathematically describes all manners of "change of motion," everything that might happen to the momentum of an arbitrary body in the 0XYZ space including no change.
Few of us care about "everything," we have specific questions about simple motions like braking of a locomotive, fall-rate of a sky-diver or the best manner to fly-cast a lure. We seek specific solutions. How are such specific solutions obtained? Many solutions begin with the momentum equation for a body.
It makes sense to call Newton's Second Law a momentum equation. To account for the myriad of ways momentum of a body can change (or be changed) the momentum equation contains Newton's construct, force. Force is not a property; it is a mechanism of momentum change, or as some say, of momentum transfer.
Shot Tower, Wythville, Virginia: This engineering structure was built in 1768 before the Revolutionary War to make bullets. The shot tower operated steadily through the War of Revolution than through the Civil War. It produced over a million spherical lead shot for muskets. The manufacture of shot for muskets invlolves the mass equation and momentum equation.
The property mass has no construct. Stated otherwise, there are no means or mechanisms for the creation or destruction of mass. Our next property equation, the energy equation, has two constructs which are work and heat.
ENERGY EQUATION, WORK AND HEAT: Energy and work are ideas that evolve from Newton's Momentum Equation. When that vector equation is scalar multiplied by a vector differential displacement, the energy equation results. A clean derivation of this fact is tedious and a bit sophisticated (LINK to be provided)). Fortunately the energy equation can be applied without a thorough understanding its origin. Below we introduce the energy equation (for a body), then apply it briefly.
The calculus of scalar multiplication of the momentum equation by a differential displacement produces our first energy form, kinetic energy, which arrives left of equality. Force, Newton's construct, is transformed into a new construct - WORK (right of equality). Work is the idea "consequence of a force that acts and moves a body through a displacement". Principally, forces are the gravity force and all surface forces. Work is expressed as two terms.
So, one might ask; where is potential energy? In the language of science, a construct is an "advantageous way of perceiving or considering reality and its events." The atom, body and extended body are models or entities. Force, work, and heat are not models nor properties of matter. They are constructs. Their definitions and relations to reality help us explain (or redict) physical changes of matter.
Returning to potential energy. The work associated with the ever-present force of gravity is indicated above. A detailed inspection of this work reveals a superior perspective. The force of "gravity work" is exerted by Earth. But if the system is taken to include Earth, that is for the body and Earth as system - there is no force. In this new perspective, gravity work is cast as the construct, potential energy which is represented quite conveniently as "mass times change of elevation " of the smaller mass, the body. The new representation, being energy not work, is placed with the other body energy, kinetic energy," left of the equality of the energy equation.
This equation applies to any amount of selected matter (system) approximated as a body. It is independent of the "inner" character of the body. The same equation applies to a tomato, rain drop or planet. Kinetic energy and potential energy are classed as extrinsic energies. Finally, this equation form (where ΔPE means PE2 - PE1, etc) is called the increment form of the extrinsic energy equation. The increment form is readily changed to a rate form. Create a time difference; Δt (being some t2 - t1). Divide the equation by that Δt. The rate form equation arrives in the limit that Δt vanishes.
Terms left-of-equality in the rate-form energy equation , are typical derivatives from calculus. They are instantaneous rates of change of energy of the body in accord with passage of time. The work (and later heat) terms written right-of-equality are not related formally to calculus. These are called "work-rate" which is physical, not mathematical. These terms, created by engineering, are not explained in calculus texts. Work is not a property; it is not energy. It is energy transfer (this distinction is important). The term identifies the rate of energy transfer across the system boundary "from" or "to" the surroundings. (Sign convention is explained in the example below). This rate of energy transfer is called power and is distinguished from work, W. (or W1-2) by being written as W with a dot above, which is pronounced as "work-dot." It is wise to carry a summation sign (σ) with work as a reminder that there might be more than one instance.
One other thing. In the Energy Equation, notice the strike-through of the system-identifying subscript, "and Earth" (shown as and Earth). While it is true that as one falls "down"as we say, Earth falls "up," Earth (of immense mass) falls so little it can be ignored - and is!
Elevator Speed Many events occur as time proceeds and are described by rate-form property equations. In this example electrical power delivered to an elevator results in its steady upward speed.
The energy cahnges of the model, body (or extended body) is limited to change of kinetic, change of potential energy (or both) with work and/or equilibration being the mechanisms of energy change. We are certainly aware that systems experience temperature change. This requires a model superior to the body. The new model must admit a new energy and a new energy transfer mechanism: internal energy and heat. We know what this is about! The next example introduces these concepts by way of fact.
18 - Wheeler Safe-Braking Speeds
Tractor-trailer trucks are driven cautiously to avoid severe braking which can destroy brake drums. In the prompt braking of a truck its kinetic energy is transformed into internal energy of its brakes. There is too little time for there to be heat.
Most of us have some understanding of basic, frictionless motion. Also the idea of "potential energy" and that "water flows downhill" are not new. The next example addresses a simple event of water (a uniformly distributed fluid) that flows under the force of gravity. Though the example is easy, we will use a careful approach to its set-up, event description and calculations. uses the direct relation of temperature of a solid with its internal energy to determine an approximate safe-braking speed.
Water Seeks its Level Since water flows, all bodies, amounts or expanses of water naturally attain the equilibrium condition with the least potential energy subject to the physical constraints. This example explores such ideas.
Time is a grand abstract. It is not a property of any system. Time can be made quantitative on a relative scale. It marks and measures sequences of events.. which occured first, second, next and so on. Time does not participate in events, it is an observer. Thermodynamics uses special aspects of matter and of time to describe physical events of that matter.
Commonly, the initial state and its system properties are associated with the time of event commencement, designated with the subscript, "1." All system properties at the final time, the time at which the task is completed, are designated with the subscript, "2." System property differences , pressure for example are written, p2 - p1 or Δp. Work and heat do not exist; they are energy transfers. For a batch event they are written as W1-2 or Q1-2.
TIME and TIME-RELATED NOTATIONS: Time is a dimension; a human perception. Time can be quantified. Our clocks measure the lengths of our lives - as we live. Our clocks tell us where we are in the day, calenders place the day in a larger time frame - the year. We use the idea of time scientifically, to explain the past or predict the future. Some special notations for thermodynamics follow:
The above examples and explanations have refreshed our memories of some aspects of physics. Math, vectors and calculus are the tools of thermodynamic analysis. Here even simple problems, with obvious answers, will be posed mathematically for analysis. The purpose is solve the easy problems correctly so as to learn a method for more difficult problems.
System principles are associated with three system properties: mass, momentum and energy. From the perspective of "property equations," the terminiologies, conservation of mass, conservation of momentum, and conservation of energy are not needed. Such system cases are included in the equation formulations. Events occur in time. Three time-wise descriptions of event are increment, rate and differential. Three equations and three time perspectives are confusing at first. A necessary goal is to attain consistency of equation form. Without consistency, there appear to be hundreds of equations.
We must establish and maintain consistency in writing property equations. While learning we will find it helpful to avoid senseless algebra that mixes up the equations. Notice that in each of the equations below, the system property is written left-of-equality. Terms right-of-equality are "constructs," or constructed mental mechanisms that explain acts of transfer at the system boundary that change the value of the system property.
Mass Equation: For the most elementary system: a body,
"0" right of equality means there are no mechanisms to create or "uncreate" system mass. This mass equation will be "improved" to a better equation which will be called the Mass Equation.
Momentum Equation: Momentum involves the constructed idea (construct), force.
Momentum is a vector property. The system space is noted as 0XYZ. The vector, force is the construct for momentum transfer to (or from) the surroundings. The dimension of force is [F]. And by the dimensional equivalence among force, mass, length and time we have: [1] = [mL]/[Ft·t]. [F] can be written as [F][1] or [F][mL/Ft·t]= [m][L/t]/[t], which is the dimension of d(mV)/dt, so force is the rate of momentum transfer to the system. Momentum change, whether it is positive, an increase of momentum, or negative, is built into equation by its vector nature.
Body forces are categorized as electrostatic, electrodynamic and gravity. Only gravity is needed here. It is written explicitly. Other forces of importance are classed as surface forces.
Energy Equation, System Schematic and Sign Conventions:
A mass, m, moving with relative speed, v, has kinetic energy: KE = (mv²)/2;. When Earth and a mass are both the system, the mass has potential energy proportional to its elevation above Earth: PE = mgo,Earthz. These energies (KE + PE) are subscripted c.m. because they are calculable in terms of the system center of mass.
Systems composed of many particles have KE, PE, density, temperature and internal energy, U. The three energies of a system are [U + (KE + PE)c.m.].
Two mechanisms of energy transport are the constructs work and heat, designated as W and Q, respectively. Signs for work and heat may be arbitrarily assigned.
It is unanimously agreed by investigators and educators that an energy interaction by the mechanism, heat (that occurs in the absence of all other interactions) the result of which is an increase of energy of the system, is defined to be positive." To draw a schematic system with an arrow labeled "Q-dot" (designating heat rate) pointing into the system also means "heat to the system is positive."
Both sign conventions are used for work. This writing adopts the same line of reasoning for work as is unamously used for heat. Work (in the absence of other interactions) that causes an increase of system energy is positive. This is said in the system schematic by the arrow associated with work pointing into the system.
Sign Conventions for Work and Heat: While every thermodynamicist uses a sign convention for work, it might not be the one you use. The clearest way to state your sign conventions is with a quick sketch. The figure below shows three conventions for the purpose of discussion. Selections (1) and (2) are valid. But (3) is inspecific with regard to heat (no "arrow" indicating "positive" direction).
Notice: The sketches above would be valid without writing the summation signs, Σ. Indeed most texts omit that notation in equations. But below the equation they write a reminder that the W and Q are sum, total or net values. Thus the reminders says the work and heat are ΣW and ΣQ.