THERMODYNAMICS in the UNDERGRADUATE CURRICULUM in CHEMICAL ENGINEERING* ALBERT B. NEWMAN1 Cooper Union, New York City
I
N THE minds of some people, chemical engineering education has been assumed to be of the practical type, excluding theoretical studies and indirect reasoning. That this is not a true picture of the situation will be evident to anyone who reads the papers in this symposium. The following discussion of thermodynamics for students of chemical engineering shows that modem training in chemical engineering is not concemed with a mass of isolated facts and empirical formulations, but attempts to train the student in mathematical and exact methods of reasoning leadmg to accurate numerical conclusions, thus making him sure-footed when he meets new problems in uncharted territory in industry. In recent years the curriculum-builders in most chemical engineering schools have realized the inadequacy for their students of the usual thermodynamics course as taught to mechanical engineers. They have also come to believe that the course in chemical thermodynamics is not quite what chemical engineering students need, although the content of chemical thermodynamics may be said to be the basis of chemical engineering thermodynamics. What was needed was the combination of the useful material and methods developed by the chemists, physicists, and mechanical engineers and the addition thereto of the graphical mathematics already used by chemical engineers. As no textbook is yet available, few of the smaller schools have yet attempted to offer such a course. After a chemical engineering student has graduated, he usually looks forward to work in equipment design
*Contribution to the Symposium on Chemical Engineering Education, conducted by the Division of Chemical Education at the ninety-fourth meeting of the A. C. S., Rochester, N. Y., September 7. 1937. t present address: College of the City of New York. New York City.
and process development. In this work it is highly important that he be able to calculate the exact thennodynamic relationships involved. Mere heat and material balances are insuflicient in processes where important pressure changes occur; what he needs is energy and material balances, usually followed by the laying out of an economic balance. In such cases, a complete thermodynamic study is necessary before the method and the detailed specifications of equipment giving the lowest annual cost of production can be specified. Obviously i t is not good educational practice to teach students the thermodynamics of the ideal gas and follow i t up by saying that the use of the resulting equations will be twenty per cent. in error if the pressure is 500 atmospheres. If the student is to.use thermodynamics intelligently instead of blindly, he must be able to find out for himself how much emor is introduced by assuming that the gas is ideal. As a starting point, he has pressure-volume-temperature tables, or, what is the same thing in more convenient form, compressibility factor tables. By little-known but extremely useful methods of 'graphical and iiumerical differentiation of these tabulated values,'lie'can obtain cukes or tabulated values of all of the important partial derivatives, sometimes called' partial "differential coefficients, as functionsof pressure, volume; and temperature. These partial derivatives are then useful for substitution in rigorous thermdynamic equations. One successful sequence of courses tied together by thermodynamics gives the student his first exposure to thermodynamics in the first setnester of his junior year as part of his physical chemistry cowse. In that course he will study the first law and the second law and acquire an understanding of such quantities as entropy and enthalpy, and will solve problems with the aid of
these quantities. He will also be introduced to partial ditferentiation of thermodynamic equations and will come to realize how he can convert an equation containing quantities difficult or impossible to measure directly into an equation containing only easily measurable or already known and tabulated quantities, or those obtainable by ditrerentiating tabulated data. The physical chemistry course is followed in the second semester of the junior year with a course called fluid dynamics. In this course no attempt is made to touch on the energy situation in regard to chemical reactions or on the thermodynamics of solutions. The course begins with an exhaustive study of the thermodynamic information which can be extracted from pressurevolume-temperature tables and compressibility factor tables for various gases. Allan Ferguson' discusses the various quantities which can be so extracted and, starting with P-V-T tahles for some gas or liquid, makes the following tangible suggestions, modified and amplified by the present writer: (1) Calculate the compressihility (bV/bP)r, by differentiating a table of volumes down a single temperature column. Then for every place in the column occupied by a volume one can get a numerical value for ( b v / b P ) ~ . These values of (bV/bP)r can then be plotted or tabulated as varying with P a t any single value of T. A complete table or family of curves could be made by repeating the operation on each temperature column of the original P - V-T table. (2) Calculate the thermal dilation (bV/bT)p. This is exactly analogous to the procedure under (1). The table is differentiated horizontally instead of vertically, and a complete table can be worked out without difficulty. (3) The isothermal reversihle work done in compressing the fluid from one pressure to another is given P(bV/bP)dP.
/
obtained directly from W = - -'PdV by the use of a J PI
general law for partial ditrerential equations: dV = (bV/bT)&T (bV/bP)#. The quantity W is obtained by the graphical integration of either W =
+
~d v OI L ; P ( ~ v/bp)&pVand it is agood ideato
Lave the student do both and see if he gets the same value; i t wiU give him confidence in the accuracy of his differentiation. (4) The heat given off during an isothermal reversible P,
compression: Q = r
J
-
(
2
J
T(bZV/bT2)pbP. The term (baV/bTl)p
1
-
[b(bV/bT)p/bT]p and numerical values areobtainedby graphically differentiating a constant pressure line of (bV/bT)p values with varying temperature. Thus if the constant pressure specific heat is known a t one pressure, it can he calculated for any other pressure a t the same temperature provided P-V-T data are available. In this way a whole table of CP values as functions of T and P can he built up. The student may be asked to show that Cp is not a function of pressure in the case of the ideal gas. (8) Having a complete set of values of Cp as calculated under (7), the corresponding complete set of Cv values can he calculated from the results of previous computations by substitution in the rigorous equation: Cv = Cp T(bV/bT)pa/(bV/bP) T. For the ideal gas, the student may be asked to show that T(bV/bT)p2/ ( b v / b P ) ~= R. (9) Entropy tables may he constructed by the use of the previously-gathered tabulations.
+
At T = const..fdS
= -L2(bv/b~)&p.
This convenient form is PR
Lr
is the change in internal energy. For ideal gases, this will be zero, as (bV/bP), = 0. (6) The quantity (bP/bT)v is difficult to obtain directly by graphical differentiation of a tahle of volumes as functions of T and P , because i t involves hunting for identical volumes diagonally across the tahle. However, having sets of values of (bV/bT)p and ( b v / b P ) ~found under (1) and (Z), the quantity is obtained by substitution of these in (dP/bT)v = -(bV/bT)p/(bV/bP)~. (7) From the rigorous thermodynamic equation -
~ = -T d
J SI
( (bV/bTp)dP. ~
J PI
The tabulated quantities found in (2) are substituted here, and a graphical integration performed. The student may be asked to show that for ideal gases the quantities under (3) and (4) are equal. (5) Having determined the values for (3) and (4) for the same gas a t the same temperature, the difference
FsncusoN, ALLAN."The mechanical properties of fluids," 2
Blackie and Son Ltd.. London. 1925, p. 15.
At P
=
const., [d.S
=f
(Cp/T)dT.
By graphical integration, i t is easy to obtain values of relative entropy for each space in the table, the other variables being, as usual, P and T. By plotting curves of these entropy values, constant entropy lines may be drawn, and from the intersections may he read values of P or T for the construction of a table with constant entropy columns. From this table and previously made tahles it is not difficultto get numerical values for (10) and (11) and to make numerical solutions of such problems. (10) The rise of temperature over a specified pressure range in a reversihle adiabatic compression may be calculated by the aid of previously computed quantities by substitution in the rigorous equation(bT/bP)s= r*.
T/Cp(bV/bT)p. From this, a t S=const.,
/ -~T/T=
(11) The difference between reversible adiabatic and
isothermal compressibilities : (b V/bP)s - (b V/bP) T = T/Cp(bV/bT)p2. If compressibility factor tables are used instead of P-V-T tables, it is necessary to know what is meant by the compressibility factor. In "Chemical Engineers' Handbook," edited by Perry, C = PV/PoV, in which PoV, is a t O°C. and 1 atm. If the gas can he assumed to be ideal a t 0% and 1 atm.. C = PV/RTo. To calculatesuch quantities as (bV/bPr) and(bV/bT), for substitution in the previously presented equations, it is only necessary to partially differentiate V = RToC/P, and it turns out that
mechanical engineers is given briefly, but thoroughly, during the first few weeks. The preparation in previous courses enables the students to assimilate these methods rapidly, and to make accurate computations for processes involving steam boilers, turbines, gas engines, and gas compressors. The students are required to construct enthalpy-concentration charts for solutions with one volatile component from thermodynamic data. These are known as Merkel charts and are useful in solving multiple effect evaporation problems in the concurrent unit operations course. They also study the construction of enthalpy-concentration charts for two volatile components. These charts are useful in column distillation problems, but it is unfortunate that little of the fundamental data are reported in the literature. It is planned that the Thus the student may differentiate the compressibility students shall determine some of the missing quantities factors instead of volumes, and make the substitutions in the physical chemistry laboratory course and then as indicated. use the results in the construction of these so-called In addition to the above, tables and curves of en- Ponchon diagrams. Then the students are given a thalpy can be computed. The fugacity is also de- thorough understanding of free energy and related rived, values computed, and uses developed. The quantities and are required to solve problems involving law of straight-line flow of compressible fluids is de- metallurgical and other chemical reactions. rived, and then simplified for the non-compressible Having had continuous exposure to the various case. Then by analogy the corresponding equations phases of thermodynamics for two years, the students for turbulent flow are developed by use of the Reynolds graduate with confidence that they can successfully number. All of this work is accompanied by problem attack any thermodynamic problem for which physical assignments illustrating the use of the equations de- data are in existence, and can approximate solutions veloped. Seldom is there any deviation from the policy even when some of the data are missing. of never developing an equation without giving an ap-. In many chemical engineering curricula there is plication in a numerical problem. much less thermodynamics than outlined above. The The fluid dynamics course is followed in the senior trend, however, is toward very thorough and compreyear by a course called Chemical Engineering Thermo- hensive work in thermodynamics with plenty of pracdynamics. In this course the material usually given to tice in its accurate application.
