SELF-INSTRUCTION IN

FRANK M. TILLER University of Houston Houston, Texas 77004 assistant could handle two sections with substantial sav­ ings in time of senior personnel. LTHOUGH SUBSTANTIAL attention has been fo. The author decided to test the self-instruction technique A cused on self-paced instruction in engineering, the with fundamental and presumably difficult topics, (1) first majority of courses continue in the traditional form of law of thermodynamic and (2) . No statistical con­ professorial lectures. A middle path can be followed in trol evaluation was carried out. Informally it was learned which a portion of the material is presented in the form that good students developed excellent comprehension of of "written lectures" akin to, but different from SPI. In the subjects; and poor students performed about as ex­ the self-instruction (not self-paced) technique, standard pected. lectures can be intermixed with written lectures. Students The following material is exactly as presented to stu­ come to scheduled classes and read rather than listen. A dents except for portions which were omitted for editorial monitor is available to ans,ver questions. With the two purposes. lectures described in this article, there were few questions. The students appeared to understand the material. None were able to complete the entire assignment in the 1.5 FIRST LAW OF THERMODYNAMICS hour (30 minutes) second period. The self-instruction process represents a moderate step THIS "LECTURE" IS PART of an experiment away from classical procedures. The advantage is that to see if it is possible to present effectively a students continue to attend class and do not have to break portion of the material in EGR 234 in written abruptly from accustomed practice. At the same time, part of the responsibility for learning as in SPl is transferred rather than oral form. The topics discussed in this to their shoulders. If half of the lectures in a course were "lecture" are equivalent to those which would be given with the self-instruction system, one instructor and given by the lecturer. The material represents the single most important principle in the course of thermodynamics. It is a building block for all sci­ ence and engineering. It is essential that ev~ry stu­ dent has a thorough mastery and understanding of the subject. It is the object to present the underlying basis for the first law of thermodynamics also known as the law of . The concepts which are necessary to an understanding are as follows :

• and • Properties and non-properties, point and path functions • Tests for exact differentials in the form

oP Frank M. Tiller, M. D. Anderson professor of Chemical Engineering a. dz = Pdx + Qdy is exact if ay and Director of International Affairs obtained his bachelor's degree from the Uni versity of Louisville in 1937 and his PhD from the Uni­ versity of qncinnati in 1946. . 1.n 1962 he was awarded a Doutor b. f C dz = 0 Honoris Causa by the University of Brazil. He has been a staff mem­ ber at Cincinnati, Vanderbilt, Lamar Tech, and the lnstituto de Oleos in Note: Line integrals formed a part of prior lectures. Rio de Janeiro. As consultant, adviser and coordinator, his services have been rendered through a variety of organizations including the BACKGROUND MATERIAL Fulbright Commission, Organization of American States, and Agency for International Development. The concept of work and quasiequilibrium processes is important to the first law. (At this point a reading assignment is given.) The authors

SUMMER 1975 115 ( of the book assigned) imply that a quasiequilib­ and acceleration. Accurate clocks were essential to rium process is inherently frictionless. That is measurement, and no serious progress was possi­ not true. A quasiequilibrium process can be car­ ble prior to the development of accurate measure­ ried out with as well as without friction. A body ment of time. Sun dials were of no value, and can be moved very slowly up a frictional plane. A water clocks offer a dubious means to satisfactory piston moving in a could work against experimentation. The pendulum clock was one of an internal friction generating device. Therefore, the term quasiequilibrium as used by the authors should always be accompanied by the adjective In the self-instruction (n ot self-paced) technique, "frictionless". In general, thermodynamicists use standard lectures can be intermixed w ith the term "reversible process" as equivalent to the w ritten lectures. Students come to scheduled works "frictionless, quasiequilibrium process". I cl asses and read rather than listen. use the expression "frictional quasiequilibrium A monitor is available process" to describe a large group of operations to answer questions. which are not reversible. the first instruments to permit reasonably precise WORK AND HEAT measurement. T HE CONCEPTS OF WORK and heat are es- A study of falling bodies was one of the im­ sential to the development of the first law. portant first steps in modern mechanics with the Heat is a transitory phenomenon which represents recognition of the difference between velocity and the passage of "something" between bodies having acceleration. Out of the Gallileo experiments at different . Work is similarly another the Leaning Tower of Pisa came the basis for the transitory phenomenon which results from me­ first form of the law of conservation of energy in chanical, electrical, magnetic, or gravitational ef­ the form: fects. After establishing the concept of "conserva­ tion" of the effects to be studied, work and heat (2) will be viewed as energy in transitory passage where v is the velocity and z the distance of fall from one body to another. from rest. Equation (2) can be manipulated into You should be clearly aware that d'W is a path a more general form function. Written in the form for a frictionless, 2 2 quasiequilibrium process involving only work of V V 1 2g + Z = 2g + Z 1 (3) expansion, we have Later Bernoulli modified Equation (3) for a d'W=pdv (1) flowing fluid and arrived at Wis given by the area under a curve connecting p v 2 P1 v~ A to B. (Two paths connecting the same endpoints ~ + 2g + Z = ~ + 2g + Z 1 (4) are presented in a figure). As the areas under the curves are different, W is a path function and not where p is the and w the specific weight. a property of the system. Although different language was employed, Equa­ As we do not as yet have variables which give tion ( 4) represented the first law of the conserva- tion of mechanical energy. •· Q in the form Y dX, it is not so easy to show that Q is a path function. However, if we resorted to Simultaneously with development of mechan­ experiment we should discover that heat trans­ ics, experimental studies in heat were taking place. ferred depends upon the path. Just as the clock was basic to mechanics, the thermometer played a similar role in the science of heat. Fahrenheit produced the first reliable HISTORY OF THE FIRST LAW mercury thermometer in the early part of the HISTORICAL APPROACH to the first law is eighteenth century just as the steam engine and A helpful in an analysis of the fundamental un­ industrial revolution began to emerge. The therm­ der-pinnings. Before work could be quantitized ometer permitted studies to be made in a variety and defined, the science of mechanics had to be of areas including irreversible mixing of hot and developed. That required an understanding of mo­ cold bodies. From such experiments came equa­ tion or the inter-relationships of mass, velocity, tions of the form

116 CHEMICAL ENGINEERING EDUCATION It is interesting to note Equation (6) depends upon the straightness of the line. If the data had yielded a curve, where Tis the final and T, and T2 are say a parabola of the form W = KT2 instead of W = KT, the initial temperatures. Equation (5) represents no one would have presumed there was a direct, linear re­ a form of conservation of heat energy. lation between Q and W. If water with dye had been used as a thermometer, the results would have been drastic, The idea of "heat energy" is foreign to modern near 4°C, where water has its maximum density. (A figure thermodynamicists. As pointed out earlier, heat is demonstrates the minimum length of the water column and a transitory phenomenon and therefore not some­ the lack of linearity, The curve is double-valued). We can thing which is indestructible and conserved. philosophize over what might have happended if the length Nevertheless, the of heat evolved of the mercury column had not been approximately pro­ portional to what we call "temperature" today. Even with upon the basis that heat was an imponderable (no mercury, the expansion is not quite linear with tempera­ mass) fluid which could neither be created or ture. There is no question but that the line shown would destroyed. not be straight for any carefully done experiment. Never­ Thus, at the beginning of the nineteenth cen­ theless, a true spark of genius led scientists to assume tury, there were two important theoretical prin­ Equation (6) was valid under all circumstances and to ad­ ciples which might be referred to as : just the specific heat of various substances to force Equa­ tion,., (61,, to remain valid; That has turned out to be a truly-monumental "fudge factor" methodology. • Conservation of mechanical energy • Conservation of heat energy On performing the reverse process of adding heat to a gas or vapor as in a steam engine, it was During the first half of the nineteenth century, a discovered that the work was not all changed into basis was laid for unifying the two principles which at that time appeared to be irreconcilable to many scientists. Nevertheless, a true spark of genius The principle of conservation of energy was led scientists to assume Equation 6 was developed on slim evidence. A highly specilized valid under all circumstances and to adjust case was extrapolated into a great law. The sec­ the specific heat of various substances to remain valid. That has turned out to be ond law of thermodynamics was established be­ a truly monumental "fudge factor" methodology. fore the first law in Carnot's remarkable writings, the numbering of the laws having nothing to do with their chronological development. heat so that we may write A number of people contributed to the develop­ ment of the first law and to the demise of the d'Q - d'W = dz (7) caloric theory. The problem of friction entered where dz simply represents the difference between heavily into the debate over the energy laws. It the heat added d'Q and the work of expansion d'W, was known that mechanical actions such as canon both of which are inexact differentials. It would be boring produced large amounts of heat. It was quite natural to inquire about the properties of z. reasoned that an "imponderable" fluid was not a Is it an exact or inexact differential? If it is exact, reasonable hypothesis. Basically it was found that it is a property of the system. If it is inexact, it is the frictional work as measured by the number of kin to Q and W in that it, is a path function. turns of a frictional device was proportional to the If the process involved only reversible work of increased length of a mercury column in a small expansion, d'W could be written as bore tube (as shown in a figure). The turns were d'W=pdV (8) considered to be proportional to work and the length of the mercury column (temperature) pro­ and (7) would become portional to heat absorbed by the cooling water. dz= d'Q - pdV (9) Stated mathematically We do .not know whether to write dz or d'z. d'W • Jd'Q (6) Before reading further, attempt to answer the following questions. Spend no more than five min­ where the arrow indicates the work was converted utes on the questions, and then turn the page and to heat but the process (by the second law) could read the commentary. not be reversed. It was a one way frictional proc­ ess in which mechanical energy disappeared and 1) What is ·your opinion· about the probability of dz being the cooling water became hotter. · (Continued on page 140.)

SUMMER 1975 117 SELF-TAUGHT THERMO: Tiller dynamics*, we assume there are two independent Continued from page 117. variables, say p, T, or p, V over which the circuit integral must be performed. In working with d'Q, exact or inexact? A logical approach would revolve around the fact that both d' W and d'Q are inexact. Work with two there will be portions of the cycle when Q and W inexact differentials like the following: ar!;l positive or negative. Let us call the sum of all d'u = ydx + dy the heat transferred to the system Q1 and all d'v = dx + x 2dy transferred from the system Q2 where both quan­ Substract one from the other to obtain some idea of the tities are considered as positive numbers. Then possibility of the difference leading to an exact or inexact differential. !cd'Q = Ql - Q2 = Qnet (17) 2) Test Equation (9) for exactness (if you can) by using the relation op/oy = cQ/ox for an of the where Qnet represents the net heat transferred to form dz = pdx + Qdy. the system. With a similar situation for W, Equa­ Work on these two questions before looking at the answers that follow. tion (16) becomes ANSWERS f cdz = Qnet - Wnet (18) 1) It is demonstrated that no linear combination of certain inexact differentials can be made exact and the following Overwhelming experimental evidence points to­ comment added: It would be judicious (but not necessarily correct) to ward Qnet = Wnet and le dz = 0. This is the es­ think that dz in Equation (9) was inexact. 2) Equation (9) is restricted to a reversible, quasiequilib­ sential point of The Law of Conservation of rium process. Therefore, any conclusions drawn from Equa­ Energy: The Heart of the Subject. tion (9) will apply only to those restrictions. The quantity z now possesses special interest Writing the equation in the form as it is a property of the system. It deserves a dx = d'Q - pdV (10) name. Thermodynamicists generally call it in­ We might say the coefficients could be derived from ternal energy. It represents the storage of heat or dz = Md'Q + NdV (11) work as they are transferred to the system. It is and apply the rule frequently given the symbol U, I, or E. In accord with the textbook, we shall choose U and rewrite ( !~ )Q = (-!~ )v (12) Equation (18) in the form Such a mathematical process would imply that Q is a state variable, i.e., Q=f(p,V). However, Q is a path function; dU = d'Q - d'W (19) and the operations used with properties like p,v,T cannot The integrated form is be used with Q and Q. For example, writing dz= d'Q - d'W (13) llU=U2- U1 =Q - W (20) it could be argued dz is exact because For a frictionless, quasiequilibrium process in o o which there is only work of expansion ~ (1) = 0 and ~ (- 1) = 0 (14) Thus Equation (14) does not have the same status as dU = d'Q-dpV (21) dz=dx - dy (15) Various examples involving the First Law follow. where it is obvious z = x - y. We conclude it is impossible to apply the op/oy = oQ/ ENTROPY ox test. JN THERMODYNAMICS, it is frequently use- INTERNAL-ENERGY ful to define certain quantities which appeal to HERE IS NO PURELY mathematical method the senses like pressure and temperature or may Tto demonstrate the exactness or inexactness of be conceptualized as voltage, current, and re­ dz. It is necessary to resort to experiment, because sistance. In this exercise, a powerful but abstract the first law rests upon the basis of factual evi­ "' In a previous lecture, students were introduced to the dence. Place Equation (9) in integrated form zeroth law and the "half law". The latter which lies be­ tween the zeroth and first laws states that Boyle (or Mar- fcdz= fcd'Q - Jcd'W (16) riot) and Charles showed that C dv=O when p and T were in which the circuit integral starts and stops at f the same point. From the "half" law of thermo- ch;;rnged and, therefore, that v=f (p,T).

140 CHEMICAL ENGINEERING EDUCATION concept which arises out of application of an inte­ different if equivalent conclusion. To simplify the grating factor to d'Q will be developed. problem, we shall assume T is the generalized Our task is to seek a generalized displacement force for heat transfer. which yields heat transferred in a reversible proc­ When a crystal dissolves in a liquid, there is ess. Concepts necessary to this "lecture" are : necessarily a transfer of energy. In fact, any time • First Law of Thermodynamics there is a transfer of mass from one to an­ • Integrating factors which turn inexact into exact dif­ other, there will be an energy transfer. Thus if a ferentials differential mass, dm, passed from one phase to a second, there would be an energy transfer of the GENERALIZED DISPLACEMENT FOR type Xdm. HEAT TRANSFERRED Previous knowledge from non-thermodynamic In general, work is given by sources does not provide us with definitions of the d'W= ± XdY (1) generalized quantities needed for heat and mass where X is a generalized force and Y is a gen­ transfer. We shall seek ari answer for heat trans­ eralized displacement. fer at this time but will postpone seeking a solu­ X and Y are both properties, and they possess tion for mass transfer until we study multicom­ exact differentials. W, is not a property, and its ponent systems and define new thermodynamic differential is inexact. We know from the study functions such as and the work of differentials an inexact differential multiplied function. by an integrating factor yields an exact differ­ There is no general formula for finding new ential. _From Equation 1 we can see (1/ X) serves thermodynamic functions. There is some contro­ as an integrating factor for d'W because dY is versy among thermodynamicists about rigorous exact. Various kinds of generalized quantities are methods for defining our Y in the equation shown in Table 1 along with a description of the d'Q = + TdY (2) type of work involved. The method which will be followed is not rigorous. TABLE 1 It will apply only to a specialized case of an in a frictionless, quasiequilibrium, non-flow GEN- process. Thus the method applies to a highly-re­ ERAL- IZED stricted, non-existent situation. Nevertheless, the DIS- method will produce the same answer which can GENERALIZED PLACE- EQUA- be obtained from more rigorous ( and cont ro­ PROCESS FORCE MENT TION versial) methods. Fluid expansion Pressure P V PdV We shall give the name of entropy to our gen­ Elastic solid Force F length L FdL eralized displacement Y before we find it. Our Electric Voltage E charge Q -EdQ method will involve integrating factors. The proc­ Surface charge Surface tension s area A - sdA ess we are about to consider was commented on by Heat transfer Temperature T unknown Y* + TdY Clausius in an appendix of his book on the me­ Mass transfer unknown X* mass m +Xdm chanical theory of heat. He recognized the lack of Potential energy weight (- mg) distance z -mg dz rigor at that time ( close to 1850) . In Table 1, it is apparent there are two missing Concepts that are needed to understand this quantities marked by * which have not been previ­ development include (1) integrating factors and ously encountered in non-thermodynamics courses. (2) of an ideal gas. If we accept Equation ( 1) as valid for expressing QUESTIONS work done in general terms, there must be a gen­ Take the function eralized displacement Y for transfer of heat. We z = xy + x 2 (3) assume temperature is the generalized force which and fi nd its differential. Then divide the differential by x causes heat to flow. That is not the only assump­ to obtain a new differential du. Show du is inexact by tion which could be made. From elementary ki­ ''' The generalized forces are generally, but not always, netic theory, we know that temperature is pro­ intensive quantities. Pressure, surface tension, and electric portional to the kinetic energy of the molecules. potential are examples of intensive variables. The general­ We might assume the generalized force was the ized displacements are extensive variables in that they are dependent on length, area, volume, mass, or charge. Gravi­ square of the velocity or the velocity of the mole­ tational force mg is an example of a force which is ex­ cules. Such assumptions would lead to a somewhat tensive rather than intensive.

SUMMER 1975 141 means of the oP/oy = oQ/ ox test. Then assume there is an d'Q = ± TdY (13) integrating factor 11.(x) which is a function of x alone and Basically we want to find a function which is an find "-· Attempt to find another integrating factor which is a function of y alone. integrating factor of d'Q. We do not know if T it­ self will turn out to be the function we are look­ Answer ing for. However, we shall experiment and see The differential what results. du = (z + y/x)dx + dy (4) You are to do the following: is shown to be exact. Assuming 11. is an integrating factor, • Write the first law for a general non-flow process and it is demonstrated that 11.(x) = x and that no integrating solve for d'Q. factor which is a function of y exists. • Assume that work is restricted to a frictionless, quasi- equilibrium expansion process. INTERNAL ENERGY AND ENTROPY • Assume the gas is ideal and replace dU by NC"dT. • Eliminate P by means of PV = NRT. N IDEAL GAS CONSISTS of point masses • Place the differential of Q in the form A having no gravitational, electrical, or mag­ d'Q = AdT + B dv (14) netic interaction. Therefore an ideal gas can only • Find an integrating factor which is a function of T. have translational kinetic energy. From kinetic Do this work before continuing. The answer is on the next theory, it can be shown that sheet. NOTE: Basically the student develops t he concept of 2 entropy in this example by himself. By providing him the PV N ( Mu ) (5) =--{- + appropriate catalyst, a situation is produced in which where u is the square root of the average velocity entropy naturally falls out of the process of finding an integrating factor for d'Q. To the equation of the molecules. Thus the molar kinetic energy is NRT directly related to the ideal gas temperature. In d' Q = NC,.dT + PdV = NC,.dT + - v- dV (15) fact an integrating factor 11.(T) is applied. Then applying the RT= (2/ 3) (molar K.E.). (6) rule for exactness In deriving (5) all but translational kinetic en­ (11.NC .),, = 0 = 11.NRT) (16) _a_ 1 _a_ ( ergy has been neglected. The molecules are as­ av ' oT v v sumed to be incapable of having rotational, vibra­ tional, gravitational, magnetic or electrical energy. The product 11.T must be constant, and 11. = 1/T. Therefore, Therefore, the internal energy of an ideal gas de­ ~ = dY = NC ~ + N~ (17) T V T V pends only on its temperature. We can write It can be seen that dY is exact. We shall call it entropy and use the Jetter S.

~c)V )'L' . = 0; (~OP )T= 0 (7) ( SIGNIFICANCE OF ENTROPY which states that JT HAS BEEN SHOWN that 1/ T is an integrat- u = u (T) (8) ing factor for d'Q. Therefore, the quantity dS The first law for a non-flow, frictionless, quasi­ = d'Q/ T must be an exact differential. It fulfills equilibrium process can be written as our search for a dY to satisfy the relation d'Q = d'Q= dU + pdV (9) ± XdY. The question of sign has been settled; a The generalized specific heat is defined by plus is used. Further as Q is proportional to mass d'Q = NCdT (10) (lbs. or lb-moles), it can be written as Q = Nq. Then where C is the molar specific heat. For a constant d'Q Nd'q volume process, combining (9) and (10) yields dS = ---;_r- = - T- = Nds (18) d'Q = NC,.dT = dU (11) and we see entropy is an extensive variable which Integration gives is proportional to mass. (In the accompanying Q = i,U = NC,.i,T = NC,.T + U0 (12) figure (Three graphs show x (generalized force) At this point, a number of examples involving vs Y (generalized displacement), P vs V, T vs S)) specific and ideal gases are presented. we see that the heat transferred is given by the area under the T-S curve in accord with SEARCHING FOR ENTROPY E ARE LOOKING for the generalized dis­ Q= f TdS (19) Wplacement in the equation There is a direct correlation between the area

142 CHEMICAL ENGINEERING EDUCATION under the X-Y, P-V, and T-S curves. Each area ture. A dial on a voltmeter gives a direct reading corresponds to some kind of work. of voltage. There are an infinite number of integrating If we went deeply into measurement in a so­ factors for d'Q. Each one would lead to an exact phisticated manner, we would find the simple differential with a new X and Y. As an example, concepts must be examined with much more care. vk-i where k = cv/ Cv is an integrating factor for We would discover temperature is difficult to de­ the case of an ideal gas. Thus fine precisely. In the end, we would be asking how vk-l d'Q = dY (20) can we measure the variables and how can we as­ or sure reproducibility and· comparative accuracy. 1 Entropy suffers in comparison with other d'Q = v - k dY (21) physical quantities, because we have no en­ There is no special value attached to this XY com­ tropometer from which we can read values of bination. Further, 1/ T has general application to entropy. However, if we can show how to measure all processes, although the present development entropy, that should enable us to form an intel­ has been restricted to ideal gases. There may be lectual if not a physical notion of what it is. some useful integrating factors other than 1/ T, Returning to the equation for an ideal gas, we but we shall not search for them. have on a unit mass basis WHAT DO WE HAVE? , dT dv wE HAVE FOUND a function, entropy, which ds = d q/ T = Cv-r + R -V (22) is a property of the system. We shall be able This equation can be integrated to give to show that entropy plays a very useful role in thermodynamics. But that comes later. We now s = C,. In T + Rln v + s 0 (23) ask, what do we have? Usually when a physical or quantity is defined, there is some material concept L- s = Cv In T/ T 0 + R In v/ v0 (24) involved which appeals to our senses. In the case These equations tell us how to measure s orL- s for of the other generalized displacements in Table 1, an ideal gas. However, we must make measure­ it is simple to visualize distance, area, and volume. ments of T and v and then calculate s. We could The concept of electric charge is not difficult to develop an instrument which would translate (24) imagine. In the case of magnetic quantities, we into something which could be visibly seen. Such have to use a little more imagination. The gen­ an instrument would not be very useful. eralized forces can be understood in terms of We must be satisfied with a mathematical and measuring instruments. Everyone thinks of a intellectural rather than a familiar physical con­ mercury column in glass as signifying tempera- cept of entropy. •

The organization of the book is excellent. It is [eJ n#I book reviews divided, logically, into two parts: the elements of profitability assessment and the elements of de­ INTRODUCTION TO PROCESS ECONOMICS cision making. Thus, the reader first gets a com­ plete treatment of time value of money calcula­ F. A. Holland, F. A . Watson and J . K . Wilkinson, tions, followed by chapters on profitability esti­ John Wiley & Sons, 290 pages. mates, uncertainties in profitability estimates, cap­ Reviewed by James H. Black, University of Ala­ ital cost estimates, and manufacturing cost esti­ mates. The second part of the book covers such de­ bama, University, Alabama cision making tools and techniques as statistical This book is intended as an undergraduate text analysis, curve fitting and trend analysis, linear for the process engineering disciplines, such as programming, financial and cost accounting, price chemical, metallurgical, and mineral engineering. and cost trends, value engineering, marketing, and It would also give an excellent introduction to some material on risk and insurance. process economics for the practicing engineer or This is a good book, well worth the cost. It serve him as an excellent reference book. It is, as would be of particular interest and value to those the title states, a book covering process economics; who found use in the recent series of articles on but some wider aspects, such as some of the man­ engineering economics, by the same authors, in agement sciences, are also presented. Chemical Engineering magazine.

SUMMER 1975 143