MATERIAL BALANCE NOTES Revision 3 Irven rinard Department of Chemical Engineering City College of CUNY and Project ECSEL October 1999 C1999 Irven Rinard
MATERIAL BALANCE NOTES Revision 3 Irven Rinard Department of Chemical Engineering City College of CUNY and Project ECSEL October 1999 © 1999 Irven Rinard
CONTENTS INTRODUCTION A. Types of Material Balance Problems B. Historical Perspective CONSERVATION OF MASS A. Control volumes B. Holdup or Inventor C. Material Balance Basis D. Material balances I. PROCESSES The Concept of a process Basic processing functions C. Unit Operatic Modes of process Operations IL. PROCESS MATERIAL BALANCES The Stream Summar B. Equipment Characterization IV. STEADY-STATE PROCESS MODELING 29 Linear Input-Output models Rigorous models V. STEADY-STATE MATERIAL BALANCE CALCULATIONS Sequential Modular BCD Simultaneous Design Specifications Optimization E. Ad Hoc methods VI. RECYCLE STREAMS AND TEAR SETS A. The Node incidence matrix B. Enumeration of tear sets VIL. SOLUTION OF LINEAR MATERIAL BALANCE MODELS A. Use of linear Equation Solvers B. Reduction to the tear set variables C. Design Specifications
i CONTENTS INTRODUCTION 1 A. Types of Material Balance Problems B. Historical Perspective I. CONSERVATION OF MASS 5 A. Control Volumes B. Holdup or Inventory C. Material Balance Basis D. Material Balances II. PROCESSES 13 A. The Concept of a Process B. Basic Processing Functions C. Unit Operations D. Modes of Process Operations III. PROCESS MATERIAL BALANCES 21 A. The Stream Summary B. Equipment Characterization IV. STEADY-STATE PROCESS MODELING 29 A. Linear Input-Output Models B. Rigorous Models V. STEADY-STATE MATERIAL BALANCE CALCULATIONS 33 A. Sequential Modular B. Simultaneous C. Design Specifications D. Optimization E. Ad Hoc Methods VI. RECYCLE STREAMS AND TEAR SETS 37 A. The Node Incidence Matrix B. Enumeration of Tear Sets VII. SOLUTION OF LINEAR MATERIAL BALANCE MODELS 45 A. Use of Linear Equation Solvers B. Reduction to the Tear Set Variables C. Design Specifications
VIIL. SEQUENTIAL MODULAR SOLUTION OF NONLINEAR MATERIAL BALANCE MODELS Convergence by Direct Iteration B. Convergence Acceleration C The method of wegstein MIIXING AND BLENDING PROBLEMS Mixing B. Blending X. PLANT DATA ANALYSIS AND RECONCILIATION 67 Plant data XL THE ELEMENTS OF DYNAMIC PROCESS MODELING 75 Conservation of Mass for Dynamic Systems B. Surge and Mixing Tanks C Gas holders XIL. PROCESS SIMULATORS Steady state Dy ynamIc BILIOGRAPHY APPENDICES Reaction Stoichiometry 91 B. Evaluation of Equipment model Parameters 93 C. Complex Equipment Models D. Linear Material Balance by Spreadsheet- Example E. The Kremser model of gas absorbers
-0- VIII. SEQUENTIAL MODULAR SOLUTION OF NONLINEAR 53 MATERIAL BALANCE MODELS A. Convergence by Direct Iteration B. Convergence Acceleration C. The Method of Wegstein IX. MIXING AND BLENDING PROBLEMS 61 A. Mixing B. Blending X. PLANT DATA ANALYSIS AND RECONCILIATION 67 A. Plant Data B. Data Reconciliation XI. THE ELEMENTS OF DYNAMIC PROCESS MODELING 75 A. Conservation of Mass for Dynamic Systems B. Surge and Mixing Tanks C. Gas Holders XII. PROCESS SIMULATORS 87 A. Steady State B. Dynamic BILIOGRAPHY 89 APPENDICES A. Reaction Stoichiometry 91 B. Evaluation of Equipment Model Parameters 93 C. Complex Equipment Models 96 D. Linear Material Balance by Spreadsheet - Example 97 E. The Kremser Model of Gas Absorbers 98
INTRODUCTION Revised October 2. 1999 The material balance is the fundamental tool of chemical engineering. It is the basis for must thoroughly master its use in the formulation and solution of chemical processing problems uh the analysis and design of chemical processes. So it goes without saying that chemical engineer In chemical processing we deal with the transformation of raw materials of lower value into products of higher value and, in many, cases unwanted byproducts that must be disposed of. In addition many of these chemical compounds may be hazardous. The material balance is the chemical engineer's tool for keeping track of what is entering and leaving the process as well as what goes on internally. Without accurate material balances, it is impossible to design or operate a chemical plant safely and economically The purpose of these notes is to provide a guide to the use of material balances in chemical engineering. Why one might ask? Aren't there already enough books on the subject, books such as those by Felder and Rouseau, by Himmelblau, and by reklaitis? To answer that question we need to look briefly at the history of the problem A. Types of Material Balance Problems First let us look at the types of material balance problems that arise in chemical engineering. There are four basic types of problems (1)Flow sheet material balance models for continuous processes operating in the steady state (2) Mixing and blending material balances (3 Flow sheet material balances for non-steady state processes, either continuous or batch. and (4) Process data analysis and reconciliation A flow sheet is a schematic diagram of a process which shows at various levels of detail the equipment involved and how it is interconnected by the process piping(See, for instance Figures Il-1 and 11-2 in Chapter II). A flow sheet material balance shows the flow rates and compositions of all streams entering and leaving each item of equipment Most of the emphasis on material balance problems has continuous processes operating in the steady state. Again one might ask why. The reason is simple. Of the total tonnage of chemicals produced, the vast majority is produced using continuous steady-state processes. This includes oil refineries as well as chemical plants producing large tonnage products such as sulfuric acid, ethylene, and most of the other commodity chemicals, petrochemicals and
-1- INTRODUCTION Revised October 2, 1999 The material balance is the fundamental tool of chemical engineering. It is the basis for the analysis and design of chemical processes. So it goes without saying that chemical engineers must thoroughly master its use in the formulation and solution of chemical processing problems. In chemical processing we deal with the transformation of raw materials of lower value into products of higher value and, in many, cases unwanted byproducts that must be disposed of. In addition many of these chemical compounds may be hazardous. The material balance is the chemical engineer's tool for keeping track of what is entering and leaving the process as well as what goes on internally. Without accurate material balances, it is impossible to design or operate a chemical plant safely and economically. The purpose of these notes is to provide a guide to the use of material balances in chemical engineering. Why one might ask? Aren't there already enough books on the subject, books such as those by Felder and Rouseau, by Himmelblau, and by Reklaitis? To answer that question we need to look briefly at the history of the problem. A. Types of Material Balance Problems First let us look at the types of material balance problems that arise in chemical engineering. There are four basic types of problems: (1) Flow sheet material balance models for continuous processes operating in the steady state, (2) Mixing and blending material balances, (3) Flow sheet material balances for non-steady state processes, either continuous or batch, and (4) Process data analysis and reconciliation A flow sheet is a schematic diagram of a process which shows at various levels of detail the equipment involved and how it is interconnected by the process piping (See, for instance Figures II-1 and II-2 in Chapter II). A flow sheet material balance shows the flow rates and compositions of all streams entering and leaving each item of equipment. Most of the emphasis on material balance problems has been on continuous processes operating in the steady state. Again one might ask why. The reason is simple. Of the total tonnage of chemicals produced, the vast majority is produced using continuous steady-state processes. This includes oil refineries as well as chemical plants producing large tonnage products such as sulfuric acid, ethylene, and most of the other commodity chemicals, petrochemicals and
olymers. It has been found that the most economical and efficient way to produce such chemicals on a large scale is via the continuous process operating in the steady-state. This is the reason for the emphasis on this type of material balance problem Another class of material balance problems is those involving blending and mixing. A sub stantial number of the products produced by the chemical processing industries are blends or mix- tures of various constituents or ingredients. Examples of blends are gasoline and animal feeds of precise mixtures, prescription drugs and polymeric resins Dynamic material balance problems arise in the operation and control of continuous processes. Also, batch processes, by their very nature, are dynamic. In either case we must consider how the state of the process varies as a function of time. In addition to determining the flow rates and compositions of the interconnecting streams, we must also follow the changes in inventory within the process itself. In the three types of problems just discussed, we are interested in predicting the performance of the process or equipment. Our models start by assuming that the law of conservation of mass is obeyed. A fourth type of problem, which encountered by engineers in the plant, starts with actual operating data, generally flow rates and compositions of various streams The problem is to determine the actual performance of the plant from the available data This, in many ways, is a much more difficult problem than the first three. Why? Simple The data may in error for one reason or another. A flow meter may be out of calibration or broken entirely. A composition measurement is not only subject to calibration errors but sampling errors as well. Thus the first thing one must do when dealing with plant data is to determine, if possible, whether or not it is accurate. If it is, then we can proceed to use it to analyze it to determine process performance. If not, we must try to determine what measurements are in error, by how much, and make the appropriate corrections to the data. This is known as data reconciliation and is possible only if we have redundant measurements B. Historical Perspective The solution of material balance problems for continuous steady-state processes of any complexity used to be very difficult. By its nature, the problem is one of solving a large number of simultaneous algebraic equations, many of which are highly nonlinear. Before the availability of computers and the appropriate software, the solution of the material balance model for a chemical process typically took a team of chemical engineers using slide rules and adding machines days or weeks, if not months. And given the complexity of the problem, errors were ommon The methods used in those days to solve material balance problems days are best described as ad hoc. Typically an engineer started with the process specifications such as the production
-2- polymers. It has been found that the most economical and efficient way to produce such chemicals on a large scale is via the continuous process operating in the steady-state. This is the reason for the emphasis on this type of material balance problem. Another class of material balance problems is those involving blending and mixing. A substantial number of the products produced by the chemical processing industries are blends or mixtures of various constituents or ingredients. Examples of blends are gasoline and animal feeds; of precise mixtures, prescription drugs and polymeric resins. Dynamic material balance problems arise in the operation and control of continuous processes. Also, batch processes, by their very nature, are dynamic. In either case we must consider how the state of the process varies as a function of time. In addition to determining the flow rates and compositions of the interconnecting streams, we must also follow the changes in inventory within the process itself. In the three types of problems just discussed, we are interested in predicting the performance of the process or equipment. Our models start by assuming that the law of conservation of mass is obeyed. A fourth type of problem, which encountered by engineers in the plant, starts with actual operating data, generally flow rates and compositions of various streams. The problem is to determine the actual performance of the plant from the available data. This, in many ways, is a much more difficult problem than the first three. Why? Simple. The data may in error for one reason or another. A flow meter may be out of calibration or broken entirely. A composition measurement is not only subject to calibration errors but sampling errors as well. Thus the first thing one must do when dealing with plant data is to determine, if possible, whether or not it is accurate. If it is, then we can proceed to use it to analyze it to determine process performance. If not, we must try to determine what measurements are in error, by how much, and make the appropriate corrections to the data. This is known as data reconciliation and is possible only if we have redundant measurements. B. Historical Perspective The solution of material balance problems for continuous steady-state processes of any complexity used to be very difficult. By its nature, the problem is one of solving a large number of simultaneous algebraic equations, many of which are highly nonlinear. Before the availability of computers and the appropriate software, the solution of the material balance model for a chemical process typically took a team of chemical engineers using slide rules and adding machines days or weeks, if not months. And given the complexity of the problem, errors were common. The methods used in those days to solve material balance problems days are best described as ad hoc. Typically an engineer started with the process specifications such as the production
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 specifications 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 solution. 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 mathematical 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
engineer can solve significant flow sheeting problems in as little as a day or two and, moreover, do it much more accurately and in much more detail than was previously possible. The process engineer can now concentrate on the process model and the results rather than concocting a scheme to solve the model equations themselves. The simulation program will do that, at least most of the time. However, things do go wrong at times. Either the problem is very difficult for the simulator to solve or a mistake has been made in describing the process to the program. Thu in order to fix what is wrong the engineer does need to know something about how the simula tion program attempts to solve the problem. This is the subject of Sections v, vlll, and IX of these notes. Steady-state simulation programs are described briefly in Section Xll Simple material balance problems involving only a few variables can still be solved manually. However, it is generally more efficient to use a computer program such as a spread sheet. Both approaches are discussed in Section VIl In order to achieve high levels of mass and energy utilization efficiency, most processes involve the use of recycle. As will be seen, this creates recycle loops within the process which complicate the solution of material balances models for the process. A systematic procedure for identifying recycle loops is presented in Section VI An introduction to problems encountered in determining plant performance from plant is given in Sectionⅹ There are two basic process operating modes that are of interest to chemical engineers, dynamic and steady state. All processes are dynamic in that some or all of the process variables change with time. Many processes are deliberately run dynamically; batch processes being the prime example. However, many large-scale processes such as oil refineries and petrochemical plants are run in what is called the continuous or steady state operation. The appropriate model for dynamic processes are differential equations with respect to time. In general, continuous pro- modeling is discussed briefly in Section XI and dynamic process simulators in Section XI ocess cesses operating in the steady state are modeled by algebraic equations. Dynamic pro Many topics in process modeling are not covered in these notes. The most serious omission is the companion to the material balance, namely, the energy balance. Also, little atten- tion is paid to what are known as first-principle or rigorous equipment models. Such modeling is more properly covered in texts and courses on unit operations and chemical reaction engineering However, a few of the simpler and more useful models are given in Appendix e
-4- engineer can solve significant flow sheeting problems in as little as a day or two and, moreover, do it much more accurately and in much more detail than was previously possible. The process engineer can now concentrate on the process model and the results rather than concocting a scheme to solve the model equations themselves. The simulation program will do that, at least most of the time. However, things do go wrong at times. Either the problem is very difficult for the simulator to solve or a mistake has been made in describing the process to the program. Thus, in order to fix what is wrong, the engineer does need to know something about how the simulation program attempts to solve the problem. This is the subject of Sections V, VIII, and IX of these notes. Steady-state simulation programs are described briefly in Section XII. Simple material balance problems involving only a few variables can still be solved manually. However, it is generally more efficient to use a computer program such as a spreadsheet. Both approaches are discussed in Section VII. In order to achieve high levels of mass and energy utilization efficiency, most processes involve the use of recycle. As will be seen, this creates recycle loops within the process which complicate the solution of material balances models for the process. A systematic procedure for identifying recycle loops is presented in Section VI. An introduction to problems encountered in determining plant performance from plant is given in Section X. There are two basic process operating modes that are of interest to chemical engineers, dynamic and steady state. All processes are dynamic in that some or all of the process variables change with time. Many processes are deliberately run dynamically; batch processes being the prime example. However, many large-scale processes such as oil refineries and petrochemical plants are run in what is called the continuous or steady state operation. The appropriate model for dynamic processes are differential equations with respect to time. In general, continuous processes operating in the steady state are modeled by algebraic equations. Dynamic process modeling is discussed briefly in Section XI and dynamic process simulators in Section XII. Many topics in process modeling are not covered in these notes. The most serious omission is the companion to the material balance, namely, the energy balance. Also, little attention is paid to what are known as first-principle or rigorous equipment models. Such modeling is more properly covered in texts and courses on unit operations and chemical reaction engineering. However, a few of the simpler and more useful models are given in Appendix E
. CONSERVATION OF MASS Revised October 2. 1999 6 The principle of conservation of mass is fundamental to all chemical engineering analysis The basic idea is relatively easy to understand since it is fact of our everyday life Let us consider a simple example. Suppose we are required to prepare one kilogram of a solution of ethanol in water such that the solution will contain 40% ethanol by weight. So, we weigh out 400 grams of ethanol and 600 grams of water and mix the two together in a large beaker. If we weigh the resulting mixture(making appropriate allowance for the weight of the aker), experience says it will weigh 1000 grams or one kilogram. And it will. This is manifestation of the conservation of mass That, in the absence of nuclear reactions, mass is conserved is a fundamental law ofnature This law is used throughout these notes and throughout all chemical engineering Suppose we happened to measure the volumes involved in making up our alcohol solution. Assuming that we do this at 20 C. we would find that we added 5989 ml of water to 315.7 ml of ethanol to obtain 935.2 ml of solution. However, the sum of the volumes of the pure components is 914.6 ml. We conclude that volume is not conserved Let us take note of one other fact about our solution. If we were to separate it back into its pure components(something we could do, for instance, by azeotropic distillation) and did this with extreme care to avoid any inadvertent losses, we would obtain 400 gm of ethanol and 600 gm of water. Thus, in this case, not only was total mass conserved but the mass of each of the components was also This is not always true. Suppose that instead of adding ethanol and water, we added ( carefully and slowly) sodium hydroxide to sulfuric acid. Suppose that the H2sO4 solution contains exactly 98.08 pounds of H2SO4 and that we add exactly 80.00 pounds of NaOH. A chemical reaction will take place as follow H2 SO4 +2 Naoh, Na,SO4 2 H,O Notice that the amount of H2so4 in the original solution is 1.0 lb-mol and that the amount of NaoH added is exactly 2.0 lb-mols. What we are left with is 1.0 lb-mol of Na2 SO4 or 142.05 lbs and 2.0 lb-mols of H2O or 36.03 lbs. No individual component is conserved; the H2SO4 and the Naoh have disappeared and in their place we have Na2 SO4 and H2O. However, if we look at the atomic species H, O, S, and Na, we will find that these are all con-served. That is exactly what the reaction equation expresses
-5- I. CONSERVATION OF MASS Revised October 2, 1999 The principle of conservation of mass is fundamental to all chemical engineering analysis. The basic idea is relatively easy to understand since it is fact of our everyday life. Let us consider a simple example. Suppose we are required to prepare one kilogram of a solution of ethanol in water such that the solution will contain 40% ethanol by weight. So, we weigh out 400 grams of ethanol and 600 grams of water and mix the two together in a large beaker. If we weigh the resulting mixture (making appropriate allowance for the weight of the beaker), experience says it will weigh 1000 grams or one kilogram. And it will. This is a manifestation of the conservation of mass. That, in the absence of nuclear reactions, mass is conserved is a fundamental law of nature. This law is used throughout these notes and throughout all chemical engineering. Suppose we happened to measure the volumes involved in making up our alcohol solution. Assuming that we do this at 20 C, we would find that we added 598.9 ml of water to 315.7 ml of ethanol to obtain 935.2 ml of solution. However, the sum of the volumes of the pure components is 914.6 ml. We conclude that volume is not conserved. Let us take note of one other fact about our solution. If we were to separate it back into its pure components (something we could do, for instance, by azeotropic distillation) and did this with extreme care to avoid any inadvertent losses, we would obtain 400 gm of ethanol and 600 gm of water. Thus, in this case, not only was total mass conserved but the mass of each of the components was also. This is not always true. Suppose that instead of adding ethanol and water, we added (carefully and slowly) sodium hydroxide to sulfuric acid. Suppose that the H2SO4 solution contains exactly 98.08 pounds of H2SO4 and that we add exactly 80.00 pounds of NaOH. A chemical reaction will take place as follows: H2SO4 + 2 NaOH ‡ Na2SO4 + 2 H2O. Notice that the amount of H2SO4 in the original solution is 1.0 lb-mol and that the amount of NaOH added is exactly 2.0 lb-mols. What we are left with is 1.0 lb-mol of Na2SO4 or 142.05 lbs and 2.0 lb-mols of H2O or 36.03 lbs. No individual component is conserved; the H2SO4 and the NaOH have disappeared and in their place we have Na2SO4 and H2O. However, if we look at the atomic species H, O, S, and Na, we will find that these are all con-served. That is exactly what the reaction equation expresses
Thus we have to be careful to identify the appropriate conserved species for the system we are analyzing. If no chemical reactions are involved, then each of the molecular species is conserved. If chemical reactions are involved, then only atomic species are conserved. There will be a mass balance for each of the conserved species. In the example above it does not make much difference since there are four conserved atomic species and four molecular species. But, if dditional reactions take place involving, say, Na2S and NaHSO4, then the number of molecular species exceeds the number of conserved atomic species. This will generally be the case A. Control volumes We apply the principle of the conservation of mass to systems to determine changes in the state of the system that result from adding or removing mass from the system or from chemical reactions taking place within the system. The system will generally be the volume contained within a precisely defined section of a piece of equipment. We refer to this precisely defined volume as a control volume may be the entire volume of the equipment. This would be the case if the system is a cylinder containing a gas or gas mixture. Or it may be the volume associated with a particular phase of the material held within the system. For instance, a flash drum is used to allow a mixture of vapor and liquid to separate into separate vapor and liquid phases. The liquid phase will occupy part of the total volume of the drum; the vapor, the remainder of the volume. If we are interested only in what happens to the liquid phase, then we would specify the volume occupied by the liquid as our control volume Note that the control volume can change over the course of an operation. Suppose we are dding liquid to a tank that contains 100 Kg of water to start with and that we add another 50 K The tank would originally contain 100 liters of water but would contain 150 liters after the addition. On the other hand, if our interest is in the entire contents of the tank- both the liquid and the vapor in the space above it-then we would take the volume of the tank itself as our control volume. This volume. of course. will not change B. Holdup or inventory Another concept that we will need to make precise is that of holdup, also known as inventory or accumulation. Holdup refers to the amount of a conserved species contained within a control volume. We can refer to the total holdup as simply the total mass of material contained within the control volume. Or we can refer to the holdup of a particular component, sodium chloride say, which is contained within the control volume. Needless to say, the sum of the holdups of all of the individual components within the control volume must equal the total hold-
-6- Thus we have to be careful to identify the appropriate conserved species for the system we are analyzing. If no chemical reactions are involved, then each of the molecular species is conserved. If chemical reactions are involved, then only atomic species are conserved. There will be a mass balance for each of the conserved species. In the example above it does not make much difference since there are four conserved atomic species and four molecular species. But, if additional reactions take place involving, say, Na2S and NaHSO4, then the number of molecular species exceeds the number of conserved atomic species. This will generally be the case. A. Control Volumes We apply the principle of the conservation of mass to systems to determine changes in the state of the system that result from adding or removing mass from the system or from chemical reactions taking place within the system. The system will generally be the volume contained within a precisely defined section of a piece of equipment. We refer to this precisely defined volume as a control volume. It may be the entire volume of the equipment. This would be the case if the system is a cylinder containing a gas or gas mixture. Or it may be the volume associated with a particular phase of the material held within the system. For instance, a flash drum is used to allow a mixture of vapor and liquid to separate into separate vapor and liquid phases. The liquid phase will occupy part of the total volume of the drum; the vapor, the remainder of the volume. If we are interested only in what happens to the liquid phase, then we would specify the volume occupied by the liquid as our control volume. Note that the control volume can change over the course of an operation. Suppose we are adding liquid to a tank that contains 100 Kg of water to start with and that we add another 50 Kg. The tank would originally contain 100 liters of water but would contain 150 liters after the addition. On the other hand, if our interest is in the entire contents of the tank - both the liquid and the vapor in the space above it - then we would take the volume of the tank itself as our control volume. This volume, of course, will not change. B. Holdup or inventory Another concept that we will need to make precise is that of holdup, also known as inventory or accumulation. Holdup refers to the amount of a conserved species contained within a control volume. We can refer to the total holdup as simply the total mass of material contained within the control volume. Or we can refer to the holdup of a particular component, sodium chloride say, which is contained within the control volume. Needless to say, the sum of the holdups of all of the individual components within the control volume must equal the total holdup
C. Material Balance basis will w. Whenever we apply the principle of conservation of mass to define a material balance,we nt to specify the basis for it. Generally, the basis is either the quantity of total mass or the mass of a particular component or conserved species for which the material balances will be defined. Or, for continuous processes, it might be the mass flow rate of a component or Quite often the basis will be set by the specification of the problem to be solved. For instance, if we are told that a tank contains 5,000 pounds of a particular mixture about which certain questions are to be answered, then a natural basis for the problem would the 5,000 pounds of the mixture. Or, if we are looking at a continuous process to make 10,000 Kg/hr of ethanol reasonable choice for a basis would be this production rate Some problems, however, do not have a naturally defined basis so we must choose one For instance, if we are asked what is the mass ratio of Naoh to H2SO+ required to produce neutral solution of NaCl in water, we would have to specify a basis for doing the calculations We might choose 98.08 Kg of H2S04(1.0 Kg-mol) as a basis. Or we could chose 1.0 lb of H2SO4. Either is acceptable. One basis may make the calculations simpler than another, but in this day of personal computers the choice is less critical than it might have been years ago Whatever the choice of basis, it is mandatory that all material balances are defined to be consistent with it D. Material balances We are now in a position to define material balances for some simple systems. (Note Material balances are sometimes referred to as mass balances. There are three basic situations for which we will want to do this a discrete process in which one or more steps are carried out over a finite but definite period of time An example of such a process is the dissolving of a specified quantity of salt in a quantity of water contained in a tank. We are only interested in the concentration in weight of the salt water after it is completely dissolved and not how long it takes for the salt to dissolve A continuous process operating in the steady state interna By definition, continuous process operating in the steady state undergoes no changes in its state variables such as temperatures, pressures, compositions, and liquid levels. In addition, all the flow rates of all streams entering and leaving each item of equipment are constant What this means from the standpoint of material balances is that there is no change in any of the holdups in the system
-7- C. Material Balance Basis Whenever we apply the principle of conservation of mass to define a material balance, we will want to specify the basis for it. Generally, the basis is either the quantity of total mass or the mass of a particular component or conserved species for which the material balances will be defined. Or, for continuous processes, it might be the mass flow rate of a component or conserved species. Quite often the basis will be set by the specification of the problem to be solved. For instance, if we are told that a tank contains 5,000 pounds of a particular mixture about which certain questions are to be answered, then a natural basis for the problem would the 5,000 pounds of the mixture. Or, if we are looking at a continuous process to make 10,000 Kg/hr of ethanol, a reasonable choice for a basis would be this production rate. Some problems, however, do not have a naturally defined basis so we must choose one. For instance, if we are asked what is the mass ratio of NaOH to H2SO4 required to produce a neutral solution of NaCl in water, we would have to specify a basis for doing the calculations. We might choose 98.08 Kg of H2SO4 (1.0 Kg-mol) as a basis. Or we could chose 1.0 lb of H2SO4. Either is acceptable. One basis may make the calculations simpler than another, but in this day of personal computers the choice is less critical than it might have been years ago. Whatever the choice of basis, it is mandatory that all material balances are defined to be consistent with it. D. Material Balances We are now in a position to define material balances for some simple systems. (Note: Material balances are sometimes referred to as mass balances.) There are three basic situations for which we will want to do this: 1) A discrete process in which one or more steps are carried out over a finite but indefinite period of time. An example of such a process is the dissolving of a specified quantity of salt in a quantity of water contained in a tank. We are only interested in the concentration in weight % of the salt in the water after it is completely dissolved and not how long it takes for the salt to dissolve. 2) A continuous process operating in the steady state. By definition, continuous process operating in the steady state undergoes no changes in its internal state variables such as temperatures, pressures, compositions, and liquid levels. In addition, all the flow rates of all streams entering and leaving each item of equipment are constant. What this means from the standpoint of material balances is that there is no change in any of the holdups in the system
