STAGED SEPARATIONS JOHN W. TIERNEY University of Pittsburgh Pittsburgh, Pennsylvania GRADUATE COURSES in calculations for mul- ticomponent distillation and extraction sys­ tems have been a staple part of the chemical engi­ neering curriculum for many years. These systems are industrially important because a large part of John W. Tierney is Professor of Chemical Engineering at the University of Pittsburgh. He teaches courses in the capital investment in a chemical plant may be staged separations, process dynamics and control, and in separation equipment. Many separation proc­ introductory chemical engineering. His research interests esses are carried out in staged equipment, and it include process modelling, digital control, and flow of has been recognized that a common method of dilute polymer solutions. He has also taught at Purdue analysis of all staged processes should be used. University (Assistant Professor, 1953-1956), the Univer­ sidad Tecnica Santa Maria in Valparaiso, Chile (Visiting Nevertheless, it has been our observation that at­ Professor, 1960-62), and the Universidad de Barcelona, tempts to develop a common approach to staged Barcelona, Spain (Fulbright Professor, 1968-69). He has calculations have been only partly successful, and also had industrial experience at the Research Laboratories that this is due primarily to the lack of a truly of the Pure Oil Company and the Research Division of general model for staged processes. We have re­ Remington Rand Univac. rently proposed a model (1-4J which we believe can be the basis for a unified presentation of staged calculations. This article is a description of a grad­ and stripping can be adequately described by these uate course offered at the University of Pittsburgh models. with this end in view. The course is designed for the MS level student but with suitable modification can be used for SCOPE OF COURSE upper level undergraduates or for PhD level students. It is assumed that the student has under­ The course has as a central theme the study of graduate preparation in transport operations, a mathematical model for staged separation proc­ linear algebra, and computer programming. Some esses. The use of models is common in chemical background in numerical analysis, particularly the engineering design and analysis to describe the solution of implicit equations, is highly desirable. variation in temperature, pressure, composition, The first aim of the course is to provide the and other properties of materials as they pass student with the means to solve the mathematical through a chemical process. One very useful type models for staged separations. The essential simi­ of model is based on the assumption that all of the larity of the operations of distillation, extraction, material is contained in a finite number of well­ absorption, and stripping, is emphasized by using mixed chambers or stages, and that flow through the same model for all these processes. A second the process is accomplished by moving from one and equally important aim is to provide the stu­ stage to another. The alternative to this staged dent with an appreciation of the factors involved model is the continuous model, in which the prop­ in designing and selecting separation processes. erties are assumed to change continuously as func­ tions of process dimensions. The term Staged Sep­ SOME UNIQUE FEATURES arations is used to designate those processes which are described by staged models and which have the The course is not a traditional advanced course additional restrictions that no reaction occurs and in distillation or extraction calculations. Some of there is more than one phase in each stage. Many the novel features are summarized here. of the common industrial separation processes • A single model is used for all problems. The such as distillation, liquid extraction, absorption, model is a set of five matrix equations. (More ac- 180 CHEMICAL ENGINEERING EDUCATION curately, there are 2m+3 equations, where m is the computer have an exceptionally large memory the number of components, because two of the or be very rapid as long as the student can obtain equations can be written for each component.) the time he needs. We have used the BASIC pro­ The kind of problem being solved determines the gramming language in the course and found it to method of solving the model. This is very impor­ be very effective for student use. It is easily tant because some of the equations are nonlinear learned; the interactive mode permits rapid cor­ and iterative methods must be used in most prob­ rection of minor errors; and the built-in matrix lems. instructions are especially convenient for the solu­ • The model used is very general. It provides tion of matrix equations. Obviously the students for any number of feeds, any number of products, do not have time to write more than a few pro­ and for flow of both phases between any two grams, so most of the programs needed for the stages. Flow need not be countercurrent, and mul­ course are stored in disc files-about 30 programs tiple column systems are solved as a single prob­ are currently in the files. Students are encouraged lem. Stage inefficiencies are included naturally by to modify these programs for their own use, and adjusting the flow patterns for entrainment or by­ some of the problems which are assigned require passing. modification. During the last few weeks of the • Solution methods are computer oriented. It course, each student has a special design or re­ is assumed that the student is familiar with other search problem assigned which requires that a methods such as the graphical solutions for binary new program be written or an existing program systems, and this can often be used to illustrate a extensively modified. point in discussions, but no attempt is made to Programs available in the library are of two study systematically the graphical and other meth­ types-main programs and function subprograms. ods traditionally used. The main programs solve specific problems such • The student must understand not only the as the constant flow distillation problem. The sub­ general solution methods but the computer pro­ programs are used to obtain physical properties grams as well. There is a danger in computer which are needed in the main programs-equilib­ oriented solution methods that "canned" programs rium ratios, activity coefficients, enthalpies, vapor will be used in which the student does little more pressures, and the like. When solving a problem, a than put the input data into the correct form. To student must first of all obtain the correct main insure that the solution methods are understood program, then the subprograms which are needed, the student solves simple problems by hand­ and then properly combine them. Next he must usually problems with two components and two or enter the data for the problem he is solving. This three stages. He then uses library programs to is done directly at the terminal by changing solve more complex problems, and then, to insure DATA statements in the main program. He then that the computer programs are understood, he runs the program and if necessary makes modifi­ must modify some of the library programs. cations. For example, it may be desirable to sup­ • By the end of the course the student should press some of the output or provide additional out­ understand and be able to solve quite difficult put. Copies of the programs are available in the problems. For example, a multicomponent, multi­ notes for the course and can also be obtained at column distillation with nonideal liquids includ­ the terminals by requesting a program listing. ing heat balancing is an assigned problem. • Methods for obtaining correction algorithms STAGED SEPARATIONS MODEL by vector differentiation of the fundamental equa­ tions are presented. Using these methods the stu­ As noted previously a single model is used for dent can derive correction algorithms rapidly and all separation processes. The model consists of efficiently. five matrix equations. 1. The Overall Material Balance (0MB) equa­ tion. COMPUTER PROGRAMS 2. The Component Material Balance ( CMB) Problem solving is an important part of the equation. One CMB equation can be written for course, and a computer must be used to solve all each component. but the most trivial problems. Good access to com­ 3. The Sum of Compositions (SC) equation. puting facilities is needed. It is not necessary that 4. The interphase transfer equations. For FALL 1973 181 The course has as its central theme the study of a mathematical model for staged separations processes. equilibrium separation, the Equilibrium Relation 3. Constant Flow Distillation. The constant (ER) equation is used. In nonequilibrium separa­ flow (constant molal overflow) problem is shown tions the Mass Transport (MT) equation is used. to be identical to the extraction problem, except Most of the course is devoted to equilibrium sep­ that the flows are fixed and the stage temperatures arations. One equation can be written for each must be varied until a solution is found. The di­ component. rect substitution method must be modified because 5. The Energy Balance (EB) equation. the temperatures do not appear explicitly in the The number of stages must be known in order model equations. The Newton-Raphson correction to write these equations. An unknown number of method is derived and used to solve the single stages cannot be accomodated in the model. This stage, no flow (bubble point) problem. Then it is is different from the graphical methods which extended to the single stage with flow (equilibrium may determine the number of stages needed for a flash) and finally to the multiple stage constant given separation. A problem must normally be flow distillation. Nonderivative methods are also staged in the "operating" form-that is, the in­ used to solve these problems, and the results are puts (feeds, interstage flow connections, and heat compared with the derivative methods.