rate and product quality and calculated backwards through the process. Intermediate specifi cations would be used as additional starting points for calculations. As will be seen later in these notes, such an approach goes against the output-from- input structure of the process and can lead to severe numerical instabilities The growing availability of digital computers in the late 1950s led to the development of the first material balance programs such as IBM's GIFS, Dartmouth's PACER and Shells CHEOPS. Almost every major oil and chemical company soon developed in-house programs of which Monsanto's Flowtran is the best-known example. By the 1970,s several companies specializing in flow sheet programs had come into existence. Today companies such as Simulation Sciences, Aspen Technology and Hyprotech provide third-generation versions of eady-state flow sheet simulation programs that provide a wide range of capabilities and are relatively easy to use compared to earlier versions Dynamic simulation is less advanced than steady-state simulation. This is due, in part, to the lack of emphasis until recently on the dynamic aspects of chemical engineering operations his situation is changing rapidly due to demands for improved process control and for simulators for training operating personnel. The companies mentioned in the previous paragraph have all recently added dynamic simulators to their product lines. In addition several companies such as ABB Simcon offer training simulators for the process industries C. Material Balance Methodology There are two major steps involved in applying the principle of conservation of mass to chemical processing problems. The first is the formulation of the problem; the second, its solu- tion By formulation of the problem is meant determining the appropriate mathematical description of the system based on the applicable principles of chemistry and physics. In the case of material balances, the appropriate physical law is the conservation of mass. The resulting set of equations is sometimes referred as a mathematical model of the system What a mathematical model means will be made clearer by the examples contained in these notes. However, some general comments are in order. First, there may be a number of mathe matical models of varying levels of detail that can apply to the same system. Which we use depends upon what aspects of the process we wish to study. This will also become clearer as we proceed. Second, for many systems of practical interest, the number of equations involved in the model can be quite large, on the order of several hundred or even several thousand Thus, the process engineer must have a clear of how to formulate the model to insure that it is a correct and adequate representation of the process for the purposes for which it is intended This is the subject of Sections I-IV of these notes Today, using process simulation program such as PRO-lL, ASPEN, and HYSIM, a single-3- rate and product quality and calculated backwards through the process. Intermediate specifi￾cations would be used as additional starting points for calculations. As will be seen later in these notes, such an approach goes against the output-from-input structure of the process and can lead to severe numerical instabilities. The growing availability of digital computers in the late 1950's led to the development of the first material balance programs such as IBM's GIFS, Dartmouth's PACER and Shell’s CHEOPS. Almost every major oil and chemical company soon developed in-house programs of which Monsanto's Flowtran is the best-known example. By the 1970's several companies specializing in flow sheet programs had come into existence. Today companies such as Simulation Sciences, Aspen Technology and Hyprotech provide third-generation versions of steady-state flow sheet simulation programs that provide a wide range of capabilities and are relatively easy to use compared to earlier versions. Dynamic simulation is less advanced than steady-state simulation. This is due, in part, to the lack of emphasis until recently on the dynamic aspects of chemical engineering operations. This situation is changing rapidly due to demands for improved process control and for simulators for training operating personnel. The companies mentioned in the previous paragraph have all recently added dynamic simulators to their product lines. In addition several companies such as ABB Simcon offer training simulators for the process industries. C. Material Balance Methodology There are two major steps involved in applying the principle of conservation of mass to chemical processing problems. The first is the formulation of the problem; the second, its solu￾tion. By formulation of the problem is meant determining the appropriate mathematical description of the system based on the applicable principles of chemistry and physics. In the case of material balances, the appropriate physical law is the conservation of mass. The resulting set of equations is sometimes referred as a mathematical model of the system. What a mathematical model means will be made clearer by the examples contained in these notes. However, some general comments are in order. First, there may be a number of mathe￾matical models of varying levels of detail that can apply to the same system. Which we use depends upon what aspects of the process we wish to study. This will also become clearer as we proceed. Second, for many systems of practical interest, the number of equations involved in the model can be quite large, on the order of several hundred or even several thousand. Thus, the process engineer must have a clear of how to formulate the model to insure that it is a correct and adequate representation of the process for the purposes for which it is intended. This is the subject of Sections I - IV of these notes. Today, using process simulation program such as PRO-II, ASPEN, and HYSIM, a single