A Flowsheeting

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FLOWSHEETING&

MICROSOFT OFFICE

EXCEL

P. Ditl, J. Skrivánek


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CONTENTS

INTRODUCTION 3Decision-Making 3Working Tools for Decision-Making 3Complete System Characteristic 3

BASIC CONCEPTS 4What Is Flowsheeting? 4What Is a Balance? 4What Is a Component? 5What Is a Balanced System Boundary? 5What Is a Limited Time of Balance? 5What Is an Amount of Component? 6What Is a Continuous Process? 7What Is a Source Stream of Reaction? 7Why Enthalpy Balance and not Heat Balance? 8What is the Difference between Material Balance and Enthalpy Balance? 8Enthalpy Balance 9

Formulation of the Problem Being Solved 9How Many Equations Define the System Behaviour in Full? 9Possibilities of Solving a Model 9

METHODS OF SETTING UP AND SOLVING A MODEL 10Solving a System of Equations – a Model 10General Information 111. Process Basis and Data Available with Respect Thereto 122. Process Flow Sheet and Definition of Balance Blocks, Incidence Matrix 13

Flow Sheet 13Incidence Matrix 15

3. List of Components in Streams and Their Definitions 154. Basis of Calculation and Conversion of Specified Flows and Compositions 17

Reference Stream – Basis of Calculation 18Conversion 19

5. Chemical Reactions, Determination of Number of Independent Chemical Reactions 20Source Stream 20Determination of Number of Independent Chemical Reactions 20

6. Specification Table and Matrix of Coefficients 22Specification Table 22Matrix of Coefficients 23

7. Formulation of Additional Relations 25Types of Additional Relations 25Enthalpy Balance 25

8. Conditions of System Solvability 279. Solving a System of Equations Using Methods of Linear Algebra, with the Advantage 27

of Using Excel10. Table of Solutions Containing Complete System Characteristics 28

CONCLUSION 28

List of Projects Solved by the Authors 29

Enthalpy Data EXCEL fileProblems Solved EXCEL filesProblem 1: Solvay (Ammonia Soda) Process (problem_soda file)Problem 2: Manufacture of Reduced Iron (problem_redukce file)Problem 3: Manufacture of Styrene from Ethylbenzene (problem_styren file)Problem 4: Catalytic Oxidation of SO2 to SO3 (problem_reactor file)


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INTRODUCTION.

Decision-Making

Experience and practice have made us realise that mass and material balances represent effective

tools for decision-making. The concept of decision-making shall include the following spheres ofaction:

• Sizing lines and individual apparatus in designing.• Adjusting a line to the modified composition of materials.• Adjusting a line to modified requirements for quality (composition) of a product.• Diagnosing failures of operation of lines and locating causes of failure.• Checking or substituting measured quantities by calculating those quantities and taking

advantage of this principle to ensure easier control or to carry out process or guaranteemeasurements.

• Optimising operation of a line and searching for alternatives of operation using the method:“What will happen if.....?”

• Training and teaching operating personnel.

This enumeration is certainly not complete, however, it gives us an idea of the content of the conceptof decision-making.

Working Tools for Decision-Making

Decision-making that involves any one process requires a quantitative model. This decision-makingshall mean either proposing a new process, or modifying a process, or controlling an existing process.Such a model should give us an opportunity to study behaviour of the system if the initial conditionschange. In other words, the model should enable one to study changes of the system behaviour aftermodifying initial conditions, that is to search for an answer to the question: What will happen if…. Sucha model constitutes a prerequisite for an objective quantitative economic analysis of the system. It isnecessary to bear in mind at all times that the quantitative model should be exact. That means we

receive one and only solution for the selected values of the designating parameters. When usingempirical coefficients, such as efficiency, yield, or adjustable constants guaranteed by the provider ofthe equipment, maximum attention must be paid in order to define these coefficients accurately. Theuse of empirical coefficients is thus limited to cases where those coefficients include deviations fromthe ideal behaviour, express efficiency of the process, or adjust the theory to the real behaviour of theequipment. These adjustable parameters often constitute the subject of know-how. In any case,however, exact mass and enthalpy balances constitute the basis of the model. Thereby a quantitativemodel of a process establishes a reliable foundation of the system’s economic model, which is thefinal tool for the management in making decisions.

The law of conservation of mass and the law of conservation of energy can be used as an exactmodel for processes accompanied by a chemical reaction and heat exchange. On the basis of thismodel, mass and enthalpy balances can be drawn up for the process in question and subsequentlycomplete system characteristic can be acquired.

Complete System Characteristic

Material balances are used for obtaining a complete characteristic for all streams that are present inthe process flow chart. This information is essential both for a designer and for the operation controlitself. A complete characteristic contains the following details:

• Where the stream comes from and where it goes – that defines the direction of the stream, i.e.its orientation.

• Size of the stream, expressed as weight per time unit (kg.s-1) or moles per time unit (kmol.s-1).• What components the stream contains and their part, expressed as a weight or mole fraction

of the component in the stream in question. The method of expressing the composition of the

stream should be invariant to temperature and pressure.• Temperature and pressure for each stream are added to the above-mentioned in the case ofenthalpy balance.


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BASIC CONCEPTS

If we know the composition, temperature and pressure for a stream, we are able to determine:

• any intensive physical property of the stream – such as density, viscosity, or surface tension,

• whether the stream consists of one or more phases and what these phases consist of.In general we can say that all intensive physical properties represent a function of composition,temperature and pressure. With liquid and solid phases the effect of pressure on the value of intensiveproperties is mostly negligible. As to ideal gases, the change of enthalpy with pressure is null. A morenoticeable dependence of enthalpy on pressure is encountered only with streams the position of whichis in the proximity of a two-phase range. That leads to the following conclusions:

• The composition of the stream must serve the purpose of mass balance.• The temperature of the stream must serve the purpose of enthalpy balance.• The pressure of the stream does not constitute the object of balance calculations. If we need

to know its value (for instance, in order to calculate in what phase the stream is – whether in agaseous or liquid phase), its value must be determined.

What Is Flowsheeting?

Flowsheeting consists of a flow sheet of the process being analysed and the complete systemcharacteristic. It includes neither dimensions and structural design of the apparatus nor the planning ofthe piping system. Flowsheeting consists in the following steps:

• Drawing up a process flow sheet• Defining components in individual streams• Establishing balance equations

By working them out we obtain the missing information for complete system characteristic.Flowsheeting is a working tool for decision-making.

What Is a Balance?

Any extensive property, that is a property, the numerical value of which changes with the size of thesystem, can be balanced. This class includes mass, volume, internal energy, entropy, and so forth. Onthe contrary, intensive properties are properties with their value being independent of the size of thesystem, such as density, temperature, viscosity, specific heat, colour, hardness, and so forth.

Within the defined boundaries of the system being balanced and the limited time for which thebalancing is being performed, the following balance relation holds good for the extensive propertybeing balanced:

Input + Source = Output + Accumulation

• Input is the quantity of the extensive property being balanced that passes into the system

through the balanced system boundaries within the limited time for which the balance is beingperformed.

• Source is the quantity of the extensive property being balanced that is generated within thesystem boundaries in the course of the balanced period. Disappearance of the extensiveproperty is expressed as a source with a negative sign.

• Output is the quantity of the extensive property being balanced that gets out of the systemthrough the balanced system boundaries within the limited time, for which the balance is beingperformed.

• Accumulation is the change in quantity of the extensive property being balanced within thebalanced system boundaries in the course of the balanced period.

The introduction of a source element enables us to balance all extensive properties, even thoseproperties to which the law of conservation does not apply (for instance, entropy, exergy, component

subject to a chemical reaction, and so forth). In order to apply a balance relation it is still necessary todefine in greater detail the concepts involved in the balance relation.


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At the start of each balancing, three concepts must be defined clearly:

• Extensive property being balanced• Balanced system boundaries• Limited time for which the balance is being performed .

What Is a Component?

An extensive property being balanced will be hereinafter referred to as a component. Components canbe

• pure chemical compounds, such as methane, sulphuric acid, water, oxygen, to which aspecific molar mass can be attributed,

• polymers – chemically pure substances, whose molar mass is not clearly defined,• substances representing a group of chemical compounds, such as fats, R2O3, silicates,

hydrocarbons C4+,• substances that cannot be specified chemically, such as water-insoluble residue, substances

with density lower than water, substrate in bioreactors.• Also, a component may be defined as a particle with a size of 50µ up to 100µ, or a particle

with other than white colour.When defining a component it is necessary to define in a precise manner the method through whichthe component is determined. Differences in methods used for determining a component can lead tosignificant discrepancies. For instance, it is not enough to define a component as water content in asolid phase, it being loose material. Water can be on the surface of particles; in the case of porousparticles it can be bound through adsorption to the inner surface of particles; it can be water ofcrystallisation if the particles are of crystalline nature, and in the case of some substances it can bechemically bound water. Based on the method of determining the water content, results can differeven in the order of magnitude.

This applies also to a component defined as a particle size. If we determine the proportion of particlesof a specific dimension by measuring the particle’s largest dimension under the microscope, byseparating on sieves, or by measuring the distribution of the particles using the settling method, we

will arrive at different results.

What Is a Balanced System Boundary?

A boundary of the system being balanced must be defined in order that a component can enter or getout of the system only by crossing the system’s boundaries. System boundaries do not have to beidentical with the apparatus housing. A system can also be an imaginary volume defined within theapparatus. This method is very often used to define differential volume, it being the system with dx, dyand dz dimensions in rectangular coordinates. Then the balance relation for the component leads to apartial differential equation.

The system being balanced does not have to be identical with the individual units of apparatus. Unitsof apparatus can be combined for the purposes of balancing. Admittedly, this reduces the number of

input data, required for the performance of the balance calculation, but the possibility of acquiringfurther information is thereby lost. For instance, if we link, for the purposes of balancing, anevaporating apparatus and crystalliser, we lose the information about the properties of theconcentrated solution entering the crystalliser, which is an information necessary for the structuraldesign of the evaporating apparatus. An opposite approach is possible as well. For instance, insteadof balancing the whole distillation column individual levels can be balanced.

What Is a Limited Time of Balance?

If we divide all elements of the balance relation by the limited time of balance, we will obtain thecomponent’s quantity per unit of time.

This quantity can be termed as stream. Then the balance relation for any one component can be

posted as follows:

Input Stream + Source per Unit of Time = Output Stream + Rate of Accumulation


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Source of component per unit of time (rate of source) can be seen as an imaginary (fictive) stream thatenters into the system; however, it does not cross its boundaries at the same time. It can have bothpositive – a component is formed – and negative - a component disappears - value.

Accumulation per unit of time can be designated as the rate of accumulation; it is the change inquantity of the balanced component in the system with time. Therefore, the final form of the balancerelation reads as follows:

Input Stream of Component + Fictive Stream of Component = Output Stream of Component + Accumulation Rate of Component .

What Is an Amount of Component?

In order to substitute into the above-mentioned relation we need to know the stream of the component,which is the quantity of the component per unit of time. However, we do not usually have theopportunity to measure the component stream directly. In most cases we measure the overall streamand its composition. The component stream can be calculated from these two pieces of information.

If the component is a differential particle size, the size of the stream of this component can becalculated by multiplying the overall flow of particles by the relative frequency of occurrence of the

particles within the monitored interval. When reducing the interval of the particle size to zero we get adifferential and integral description known in literature as population balance.

The numerical value of the flow of the component is required to be invariant to temperature andpressure. This requirement excludes the overall stream to be expressed as volume per unit of time.There remains the possibility of expressing the flows of components as mass or amount (number ofpieces) – kilograms of flour and number of eggs per unit of time.

Furthermore, it is required to have the opportunity to convert mass to amount (number of pieces) andvice versa. For it is common practice to weigh small screws instead of counting our thousands ofpieces. The approach is accurate only if every piece has exactly the same mass as the other ones.This requirement is satisfied strictly only with molecules. Molecules are too small to be counted out.However, we know that one kmol of any substance contains the same number of molecules,corresponding to the Avogadro’s number, which is 6.0221367·10226 kmol-1. If we know the molar mass

for a component (kg·kmol-1

), we can convert the amount of substance (kmol) to mass (kg) and viceversa. If a component cannot be attributed molar mass, its quantity can be expressed only as mass inkg.

Consequently, we shall express the fraction of the component present in the stream only as a weightfraction or a mole fraction, depending on whether the overall stream is expressed in kmol·s-1 or kg·s-1.

It is common practice in industry and business to encounter the definition of composition in volume percent. The following example will show us what ambiguousness this can lead to.

The content of alcohol in rum commonly available in the market is declared on the label as 38 volumeper cent. Let’s assume this denotes a solution created by mixing 38 volume parts of ethanol with 62volume parts of water. If we perform this operation at different temperatures, solutions with variousalcohol content are formed – as the following table shows.

Sample Problem: Declared composition of rum 38 volume per cent of ethanol

Composition of rum with 38 volume per cent in weight fractions at various temperatures

kg/m3 m3/kg weight per cent volume per cent

temperature ethanol water ethanol water ethanol water ethanol

0 806.0 999.8 0.001241 0.001000 33.07 66.93 38.00

10 798.0 999.6 0.001253 0.001000 32.85 67.15 38.00

20 789.0 998.2 0.001267 0.001002 32.64 67.36 38.00

30 781.0 995.6 0.001280 0.001004 32.47 67.53 38.00

40 772.0 992.2 0.001295 0.001008 32.29 67.71 38.00

50 763.0 988.0 0.001311 0.001012 32.13 67.87 38.00

60 754.0 983.2 0.001326 0.001017 31.97 68.03 38.00


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70 745.0 977.7 0.001342 0.001023 31.87 68.17 38.00

Minimum alcohol content

ethanol water

temperature 70°C 0°C

745.0 999.8 0.001342 0.001000 31.35 68.65 38.00

The view of revenue offices on this case is not clear. The relative deviation of the highest and lowestvalue is more than 5%. However, this implies for us that from now on we will use either the amount inkmol·s-1 or in kg s-1 for expressing the size of the stream in balancing. This is in agreement with theexpression of composition by mole fraction of kmol·Σkmol-1 in the first case or weight fraction of kg·Σkg-1 in the second case.

The dimensions of flows selected in that manner agree with the SI system. Energy flows come out inenthalpy balances as kW, that is kJ s-1.

It is necessary to point out that there are also other possibilities of expressing the composition to makeit invariant to temperature and pressure – for instance, a relative mole fraction, molar concentration,

and the like. However, it is not difficult to convert these values to mole or weight fractions.

What Is a Continuous Process?

The accumulation of a component in the system is the change in quantity of the property beingbalanced (hold-up). If a process is run continuously in a steady state, the change in hold-up and itscomposition with time is null, and therefore the rate of accumulation is also null.

The accumulation of mass is usually applied in batch systems. If we choose a limited time of balanceexceeding the time of a batch process cycle by a long way, we can accept the assumption of a zerorate of accumulation. If we choose no matter how long balance period, the system accumulation willbe given by the level of the system hold-up at the end of the balance period minus the system hold-upat the start of the balance period. This can be the difference between the maximum and the minimum

hold-up at the most. If we choose the balance period to be just as long so that the maximumaccumulation possible is negligible compared with the input and output streams for the balance period,we can regard also the periodical batch processes to be continuous and deem the rate ofaccumulation to be zero.

Complete system characteristic does not say anything about the dimensions of individual pieces ofequipment, and therefore the relations we use for balancing do not contain any parameters of theequipment. However, in order to calculate the rate of accumulation it is necessary to know the hold-up,which constitutes a parameter of the equipment. Therefore, our concern will be focused on continuousand periodical batch processes, where the accumulation rate can be ignored.

What Is a Source Stream of Reaction?

Just as in the case of accumulation, it is necessary to know the hold-up of the equipment in which thechemical reaction takes place in order to calculate the source of a component directly. The rate ofgeneration of the component being balanced is the amount of component generated at the given placein the unit of capacity of the system per unit of time. That means that an intensive property is involved.Since the source in balances is an extensive property, we have to integrate the intensive propertythrough the capacity of the system being balanced. In the case of an ideal stirred tank, wherecomposition and temperature do not constitute functions of the place, we simply multiply thementioned intensive property by the capacity of the hold-up, which is, however, a parameter of theequipment. In order to avoid this difficulty, we can substitute the source by a fictive stream of reaction.

For every independent reaction there will be one fictive stream, which will be denominated as nr k forbalances of amount of substance and as mr k for mass balances, where k is the serial number ofreaction in case that more independent reactions are taking place in the block being balanced.

Instead of posting mole and/or weight fractions into the fictive stream column in the specification tablefor k, we


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• enter the component’s stoichiometric coefficient into the component’s row, with a positive signif the component is a product of the reaction (second term on the right side of thestoichiometric equation), or with a negative sign if the component is a reactant (first term onthe left side of the stoichiometric equation) in the case of material balance;

• multiply, in addition, the stoichiometric coefficient (with the appropriate sign) by thecomponent’s molar mass in the case of mass balance.

Sample Problem:

If we take the stoichiometric equation of formation of sodium hydrogen carbonate in manufacturingsoda

NaCl + NH3 +CO2 + H2O = NaHCO3 + NH4Cl

as an example, the column for the fictive stream will read as follows:

for molar balance: for mass balance:

components nr

1 H2O -1

2 NaCl -1

3 CO2 -1

4 NH3 -1

5 NaHCO3 1

6 NH4Cl 1

components mr

1 H2O -18.02

2 NaCl -58.44

3 CO2 -44.01

4 NH3 -17.03

5 NaHCO3 84.01

6 NH4Cl 53.49

Why Enthalpy Balance and Not Heat Balance?

The change in energy of 1 kg of water upon being heated by 0.5°C corresponds to the change in themass flow rate of the same by 65 m·s-1 or raising by 213 m. Such changes in kinetic and potentialenergy can be hardly encountered in a chemical plant, the exception thereof being power-producingmachinery (turbines). For the purposes of balancing we can therefore ignore potential and kinetic

energy of streams and substitute heat balance by enthalpy balance.This has the same advantage as disregarding the accumulation rate. In order to calculate kinetic andpotential energy of a stream at the given place we need to know the diameter of the piping and itsheight above the ground, which are parameters of equipment that do not belong into the systemcharacteristic. We do not need this information to calculate the enthalpy of a stream, and therefore wereplace the overall heat balance by enthalpy balance. The mistake that can be thereby caused isnegligible in the absolute majority of cases.

What Is the Difference between Material Balance and Enthalpy Balance?

For a continuous process or a batch process occurring periodically, material balance of a componenthas the following form:

Input Stream of Component + Fictive Stream of Component = Output Stream of Component +

Accumulation Rate of Component .

That applies to every component, which means that for a balanced system we have as many materialbalances as there are components present in the system.

The dimension of individual parts of the balance equation is kg·s-1, or kmol·s-1.

The system being balanced can interchange energy with the surroundings not only through thematerial streams, but also through the exchange across the system boundaries – heat exchangers, orstirrers as the case may be, and so forth.

In that case, the enthalpy balance for a continuous process or a batch process occurring periodicallywill have the following form:

Input Stream Enthalpy + Efficiency of Source = Output Stream Enthalpy + Interchange of Energy withSurroundings


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Individual parts of the balance equation are expressed in kW.

Removal of energy into the surroundings has a plus sign, while power supply has a negative sign. Thesystem in which the interchange of energy with the surroundings is zero, is called adiabatic .

Only one enthalpy balance holds good for the balanced block.

Enthalpy Balance

To draw up an enthalpy balance physical and chemical data of individual components must beavailable. Various databases can serve as sources of data. Data from individual databases may notbe quite consistent and therefore it is practical to use only one database if possible and not to takedata for various components from different databases. A database adapted for Excel from the book byR.C. Reid, J.M. Prausnitz, and T.K. Sherwood - The Properties of gases and liquids (Mc Graw Hill1982) is used for the purposes of teaching at the Czech Technical University.

The database of the National Bureau of Standards or the DECHEMA database can be recommendedfor industrial applications. The respective Internet addresses are provided at the end.

Formulation of the Problem Being Solved

In order to comply with the law of conservation of mass and enthalpy the following equations musthold true for every block of the flow sheet:

• material balance for every one component (in order words, the number of balance equationsagreeing with the number of different components present in the streams, which pass into or getout of the given block – we take into account both real and fictive streams);

• summation equations for every one stream (in other words, the total of weight or mole fractions ofall components in the given stream must equal one);

• other equations, if any, follow from the laws that determine the relations among the composition,pressure, and temperature of individual streams. These are the laws of thermodynamics, chemicaland phase balance, or laws of kinetics of processes. These relations are almost always non-linear;

• other equations issue from specifications given by technological regulations (such as yield,selectivity) or in data provided by the producer of the equipment (such as efficiency of machines);

• in the case of changes in temperature one enthalpy balance is added.

The first two groups form a system of linear equations.

Enthalpy balance is non-linear, which is given by the fact that molar heats represent functions oftemperature.

This set of equations forms a mathematical model, for which methods of solving must be established.

How Many Equations Define the System Behaviour in Full?

A system is defined in full if the sum of balance equations, summation equations, equations forphysical laws, specification equations, and enthalpy balance is equal to the number of unknowns inthe equations.

Possibilities of Solving a Model

The issue of simplicity of information transmission plays an important role with the model used. Thereis specialised software available for these models, such as CHEMCAD, PRO, or MAX. In addition,advantage can be taken of mathematical software, such as MATLAB, Mathematica, MATHCAD, andso forth.

This software has two big disadvantages – firstly, it is very expensive, and secondly, to become

proficient in the software requires great efforts and a lot of time. It is necessary to bear in mind that amodel set up by means of certain software will be used from a certain level of management upwards. At this level it is possible to encounter the willingness to sacrifice the financial means to buy the


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software, however, it is an absolute illusion to expect that someone would be willing to sacrifice thetime needed in order to become familiar with the software.

However, there is software available with the following advantages:

• It is installed in every computer in an enterprise – one can assume that Microsoft Office isinstalled in every PC.

• Almost everyone, who is able to use a computer, has a good command of it. And if themanager is not willing to communicate with a computer, his secretary is bound to have agood command of this programme.

• Operating records and source data are stored by means of this programme. Transmittingsource data into our model is therefore possible without even one number being retypedusing a keyboard.

• All and any economic and statistical analyses are performed by means of this programme.Transferring data for these analyses is also performed through floppy disks, over the net andthrough other channels, but not by typing the data on the keyboard.

• Tables and diagrams obtained using this programme can be, without the slightest difficulty,transferred into the most widespread Word processor.

We are talking about Excel. Working with Excel may not be as elegant as working with special

programmes, but the above-mentioned advantages are significant to such an extent that for the timebeing Excel cannot be replaced with any other tool – exception perhaps being Lotus II, but it is fullycompatible with Excel.

METHODS OF SETTING UP AND SOLVING A MODEL

The methods of solving have been drawn up to be generally valid and uniform for all types of balancecases. Exactly the same procedure is used for all cases, ranging from the problem of what ratio to useto mix two streams of water with different temperatures so that the resulting stream can have thetemperature required, to systems containing dozens of blocks and streams.

These uniform methods have facilitated the elaboration of an exact method of solving, which isavailable to the engineering public at large and which consists of the following steps:

The following is required for any process we wish to balance:

1. Process Basis and Data Available with Respect Thereto.2. Process Flow Sheet and Definition of Balance Blocks, Incidence Matrix.3. List of Components in Streams and Their Definitions.4. Basis of Calculation and Conversion of Specified Flows and Compositions.5. Chemical Reactions, Determination of Number of Independent Chemical Reactions.6. Specification Table and Matrix of Coefficients.7. Formulation of Additional Relations.8. Conditions of System Solvability.9. Solution of a System of Equations Using Methods of Linear Algebra, with the Advantage of

Using Excel.10. Table of Solutions Containing Complete System Characteristic.

Comments:

Once the balance sheet has been drawn up and once it has been established which components arepresent in individual streams, the rest of the balancing is, more or less, a matter of routine for there isan exact procedure that leads step by step straight to the ultimate goal.

When strictly following the stated procedure the result may be incorrect only if the streams (both realand fictive) and components have not been defined correctly. An operations engineer, who has aperfect command of the process and knows the methods of measuring and analyses in detail, is themost suitable person to set up the balance programme in Excel (or to take part in this task).

Putting together a list of components and defining these components is essential to satisfy thesummation equation. The requirements for perfect acquaintance with the process are focused in thisvery issue.


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Solving a System of Equations – a Model

In general, tasks are not noticeably complicated from the mathematical point of view. Solving a systemof equations, most of which are linear, is always involved.

It is important to solve the essential question of whether the system has been overdetermined, that iswhether also data that should be the result of the calculation are available for the system

characteristic, or whether the system has been determined insufficiently, that means that not enoughinformation is available to enable us to determine the complete system characteristic.

The method of how to proceed from process description to solvable mathematical model has certainlogic, which must be adhered to when computing with a calculator as well as when using programmessuch as PRO, MAX, or CHEMCAD. The mentioned programmes obtain the required information fromthe user using various menus. Formation of and recording in what is known as a specification table is,in fact, making a list of all equations that are available to us to solve the problem.

The methods for EXCEL have exactly the same logical structure as the mentioned programmes, withthe difference consisting in that we know exactly why we take respective steps. Although this methodis more laborious, we know the logic and subject-matter of the procedure, and therefore we are able toeliminate any contingent mistake or error in the specification right at the beginning.

A system can be solved using a calculator only in the cases where it is possible to apply the solutionthrough iterative method by means of substitution.


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The following part demonstrates in detail on sample problems the application of the general methodsdiscussed above. Stress is put on the uniform procedure, which must be unconditionally observed forsuccess of the solution to be guaranteed.

General Information

In describing the methods we will proceed by providing the reader with a standard text and with asolution in EXCEL of the problem being described. This file has been provided with quite a detaileddescription and numerous commentaries, so that the standard text can serve, after the initial workingthrough, only as reference. In addition, series of problems solved in EXCEL have been attached sothat the user can review the methods on other problems as well. This file contains problems rangingfrom the most simple to the larger-scale ones. Commentaries on these problems do not constitute partof the standard text anymore; the commentary, with which the EXCEL file has been provided, shouldbe sufficient for the user.

Let’s discuss the individual phases of the procedure in detail.

1. Process Basis and Data Available with Respect Thereto

It is demonstrated on the Solvay process. The following table contains a typical process description:

Manufacture of Soda – Bicarbonate Column

Description of Technology

A saturated solution of NaCl contains 35.6600 g NaCl/ 100g H2O.

It reacts with the yield of 100.0000 %with a gas mixture consisting of 32.0000 volume % NH3

and 68,0000 volume % CO2.

Solid sodium bicarbonate (NaHCO3) and mother liquorcontaining, on the top of NH4CL, also 7.5300 g NaHCO3/100g H2O

are formed in the reaction.The plant processes 100.0000 tons NaCl / day.

This description, which corresponds to a typical textbook example, is quite insufficient from the point ofview of industrial application.

However, let’s pay attention to the most relevant characteristic of the process description, which arethe cells marked in colour. An essential rule applies to the methods:

The content of the cells can be changed (by entering a value using the keyboard) only in the fields

distinguished with colour .

Let’s take notice of the cell giving the content of carbon dioxide in the entrance gas. If the entrancegas contains only two components and if the content of ammonia has been entered, the content ofcarbon dioxide cannot be entered since the percentage of its content must correspond to 100 – thepercentage of ammonia.

It is very important to obey this rule. If it is disobeyed, the programme will not present a correct resultshould any parameter be changed in a proper manner. It is very difficult to try to find, especially withlarger files, the cell into which a numerical value was entered by mistake instead of allowing theoperation with other cells. The greatest danger arises if there are more persons with access to theprogramme. If another person different from the author uses the programme, the following precautionmust be taken:

To lock all cells apart from the ones marked with colour (i.e. into which data can be entered using

keyboard).


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In describing the technology it is necessary to have the opportunity to check the accuracy of suchdescription. All numerical entries must come from somewhere. The source must be obvious with everynumerical entry. It must be clear whether the figure in the cell has been taken from literature, operatingrecords, whether it is the required value, or an assumption or estimate.

Let’s see to what extent the above-mentioned description meets these requirements:

It has been stated that the saturated solution contains 35.66 g NaCl / 100 g H2O. However, it has beenmentioned neither from which monograph this information was taken nor for which temperature thisinformation holds true.

It reacts with the yield of 100%. It has not been stated whether this is required or assumed. Only onecomponent can actually disappear from the reacting system. If NaCl disappears completely, it meansthat the relief gases will contain a negligible amount of NH3. In this case it is necessary to distinguishprecisely whether those 100 % constitute an assumption or requirement.

The composition of the gas mixture – the information that the entrance gas contains only carbondioxide and ammonia and that there is 38 volume per cent of ammonia – must come from somesource – Encyclopaedia by Kirk-Othmer, operating records from another soda processing plant,assumption of a designer, estimate – but in any case this source must be provided.

As to solubility of NaHCO3, it must be clear for what temperature this solubility holds true, whether theinfluence of NH4Cl being present is considered, or whether a saturated or supersaturated solution istaken into account.

The amount of processed salt can be limited by the sales of the salt, transmission capacity of trafficroads, permissible quantity of salts in a watercourse, investment means available, and so forth.

This determination of the origin of the data is not an end in itself. For purposes of optimising it isnecessary to know which parameters can be changed and to what extent. It is possible to change theproportion of ammonia in the entrance gas to quite a considerable extent, but it is not possible tochange the solubility more than as adequate to the permissible temperature range.

3. Process Flow Sheet and Definition of Balance Blocks, Incidence Matrix

Flow Sheet

A flow sheet is the next step following the process description. This phase gives a lot of freedom to theuser. The nodes being balanced can be drawn as rectangles or as schematic drawings of the piecesof apparatus, which they represent, streams connecting individual nodes can have any thickness,equipment with one input stream and one output stream, with which the amount and composition ofthe stream does not change, can be left out from the flow sheet. The examples of such equipmentinclude, for instance, an intermediate storage tank or a pump. Individual nodes (units of apparatus)can be interconnected for the purposes of balancing into greater units-blocks and these can besubsequently marked in various ways. However, it is necessary to observe certain rules:

The nodes being balanced are designated using alphanumerical signs visibly different from thenumbers assigned to the streams.

Standard types of nodes:

1. Blender - at least two streams entering

2. Divider – output streams are of the same composition and of the same phase – at least twostreams exiting


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3. Separator - output streams are of various composition and possibly of different phase as well

4. Reactor, bioreactor – at least one entering and one exiting stream

Streams are indicated by numbers. In compliance with the practice established in designing the

following indications are recommended:

Streams must be provided with arrows so that it can be unequivocally determined where the stream

proceeds from and where it passes into – orientation of the stream.

Streams cannot be connected or disconnected in any point. Such points must be drawn in the flowsheet as the nodes being balanced (blenders or dividers).

In other respects, all and any tools can be used in order to draw flow sheets, be it a set of rectanglesconnected by streams, represented by connected arrows perpendicular to one another, or moreelegant flow sheets taking advantages of the possibilities offered by AUTOCAD.

The following drawing can serve as an object example of a flow sheet. It was drawn up using EXCEL.One can see the differentiation of the direction of streams and application of the “Divider” balancenodes where the streams split and the “Blender” balance nodes where the streams join.

Example of a Flow Sheet:

With flow sheets that consist of more balance nodes, an incidence matrix must be set up.

2

S

D

K

R

2

6

2


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Incidence Matrix

An incidence matrix provides information about what streams enter into the pieces of apparatus andwhat streams exit from the pieces of apparatus.

The number of columns of an incidence matrix exceeds the number of nodes being balanced by one.

This one extra balance node is the surroundings. The surroundings is formally regarded as a nodebeing balanced. The number of rows agrees with the number of streams.

For every stream in its respective row we enter -1 into the cell in the column corresponding to thenode from which the stream exits and +1 in the column corresponding to the node into which thestream passes. We post zeros into the remaining cells in the row.

If we mark the

Reactor column with RCondenser column with KDivider column with DBlender column with SSurroundings with O,

then the incidence matrix for the given flow sheet will read as follows:

incmat stream R K D S O

1 1 0 0 0 -12 -1 0 0 0 13 -1 1 0 0 04 0 -1 0 0 15 0 -1 1 0 06 0 0 -1 0 17 0 0 -1 1 08 0 0 0 1 -19 1 0 0 -1 010 1 0 0 0 -111 -1 0 0 0 1

The incmati;k element of the incidence matrix is the value for stream i in block k.

We set up a matrix by first filling all cells with zeros. Then we proceed stream by stream – i.e. row byrow – and we mark in each row from which node the stream exits (we enter –1 into the respective cell)and into which node the stream passes (we enter 1 into the respective cell).

Thereafter we perform a check by establishing in each column the number of streams passing into thenode – the number of cells containing 1 – and the number of streams exiting from the node – thenumber of cells containing -1. This check must be performed carefully because an error may have

fatal consequences.

The incidence matrix will facilitate our further work when setting up a matrix of coefficients.

3. List of Components in Streams and Their Definitions

In creating a list of streams and list of components we proceed in the following manner:

List of Streams:

Column 1 contains the numbers of the streams.

Column 2 contains the working title of the stream - brine, feed, distillate, total solids, and so forth.

Column 3 contains the list of components present in the stream.


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List of Components:

Column 1 contains the component’s serial number.

Column 2 contains the component’s name.

Column 3 contains, in the case of a defined chemical compound, the component’s molar mass.

The total of masses of all components in a sample of the stream must be equal to the total mass of thesample.

Abidance by these definitions is necessary for successful balancing.

Every component is defined using the method through which its amount in the sample of the streamhas been determined. This seemingly simple operation is the most common source of errors. Incomparison with time we will need to look for an error, checking every step three times will actually bea great saving. Remember to take components one after the other and refer to the list of streams tosee through which stream the component passes into the system and through which stream it exitsthe system. It isn’t a waste of time.

Some components present in the system cannot be balanced. To decide which components will bebalanced and which won’t is the most difficult task in balancing.

Molar masses are to be taken from a database. These values are recommended not to be retypedfrom the database, but to be copied using a box. Thereby we protect ourselves against error, whichcould be made by retyping. In retyping we transfer rounded values and thus deprive ourselves of partof the information.

The following examples demonstrate what such a list may look like:

The first example corresponds to the bicarbonate column, the flow sheet of which is shown on p. 12.

List of Streams: List ofComponents:

molar masskg / kmol

1 salt brine NaCl, H2O 1 H2O 18.02

2 relief gases CO2, NH3 2 NaCl 58.44

3 entrance gas NH3, CO2 3 CO2 44.01

4 suspension NH4CL, NaCl, NaHCO3, H2O 4 NH3 17.03

5 product NaHCO3 5 NaHCO3 84.01

6 mother liquor NH4CL, NaCl, NaHCO3, H2O 6 NH4Cl 53.49

The second example is from the balance sheet of a desulphurisation plant. To facilitatecomprehension we provide a flow sheet as well. Notice that one component is defined as gasses fromwhich soluble salts are generated in the solution after absorption. That should serve as a proof of thestatement mentioned above that to define components is not a simple and primitive task. Thedefinition of the composition of streams and the definition of components is, indeed, the most difficult

1 f lue gas in

5 p ro c e s s w a t e r

2 f lue gas ou t

6 l imes tone s lu r ry

4 p ro c e s s s l u r r y 3 ox ida t ion a i r

11 f ic t iv

12 f i c t i v

7 f l y ing ashes

8 g y p s u m

A

B S O R B E R

F

I L T E R

1 0 b l e e d

9 recyc le


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part of the whole work as far as requirements for invention and knowledge of engineers areconcerned.

List of Streams:

absorber components1 flue gas in O2, N2, SO2, CO2, H2O, sgg 1 O2

2 flue gas out O2, N2, SO2, CO2, H2O 2 N23 oxidation air O2, N2, H2O 3 SO24 process slurry gypsum, CaCO3, H2O, solute, solids 4 CO25 process water H2O 5 H2O6 limestone slurry CaCO3, H2O, solids 6 CaCO37 flying ashes solids 7 gypsum8 gypsum product gypsum, CaCO3, H2O, solids 8 solids9 recycle H2O, solute 9 solute10 bleed H2O, solute 10 sgg11 fictive gypsum, CO2, CaCO3, H2O, SO2, O212 fictive solute solute, CO2, H2O, sgg, CaCO3The definition of some components follows

Definition of Components: gypsum CaSO4.2H2Osolids material not soluble in diluted sulphuric acidsolute dissolved materialsgg solute generating gases -sum of NOx and Cl2

In the case of the PRO 2, CHEMCAD, and MAX programmes as well as in the case of the methods forEXCEL, the preparation of a flow sheet, list of streams and list of components makes the biggestrequirements on the skills of an engineer using these tools. Maximum attention must be paid to thesesteps, and it is necessary to advise in strong terms against any haste and superficiality. Creativity andknowledge must be shown in these very points.

In the following procedure we are already bound by strict rules and we have no chance to apply an

individual approach.

4. Basis of Calculation and Conversion of Specified Flows and

Compositions

Balancing mass (kg) is the fundamental method of balancing.

Molar balance (kmol) is practicable only if molar mass can be attributed to all components beingbalanced.

Mass balances are expedient for using the same expression of quantity as economic and productionrecords. We are interested to know the tons of sulphuric acid manufactured and the price for onekilogram of soda. The amount of kilo-moles processed and the price for one kilo-mole are figures thathave no informative value in the world of economy. Economy requires kilograms, not kilo-moles.

Molar balances are expedient in the case where streams are gases. The composition of gases isgiven in volume per cent, which represents mole per cent at the same time. This is the product of thelaw of Avogadro which says that the number of molecules in an identical volume of any gas is, at thegiven temperature and pressure, the same. Measurements of gas flow rate are performed bymeasuring the volume flow rate, with temperature and pressure being measured at the same time, sothe state-of-the-art flow meters indicate the flow in Nm3 (standard cubic meters, which represents thevolume at the temperature of 273.15 °K and pressure of 101.325 kPa). On the assumption of idealbehaviour of the gas, volume can be converted to kmol using the equation of state for ideal gas:

P·V = n·R·T

and vice versa even if we do not know the exact composition of the gas. However, to calculate mass(kg) we must know the exact composition of the gas.

All relations provi ded to us by physical chemistry for calculating intensive properties, phase andchemical balances, as well as relations for kinetics of mass transfer and for kinetics of chemical


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reactions are expressed by mole fractions. Therefore, it is more expedient to carry out all calculationsas molar balances and to convert just the final results to mass.

At the same time, it is necessary to bear in mind that amount of substance may be converted to massat any time while converting mass to amount of substance is possible only if molar mass can beattributed to all components.

The relationship between a mole fraction and a weight fraction is non-linear, as shown in the graph fora two-component mixture of ethanol – water.

weight fractionof ethanol

mole fraction of ethanol

As a non-binding rule we can recommend molar balances (kmol), if possible, with processes withmostly gaseous streams and with chemical reactions in gaseous phase or, as the case may be, withprocesses in which separation processes, such as absorption and distillation, take place.

When taking decisions it is necessary to bear in mind that the flow and composition of gaseousstreams is given in Nm3 and mole fractions, as distinct from kilograms and weight fractions with all theother streams.

Reference Stream – Basis of Calculation

Balancing is practicable only if the magnitude of at least one stream is known in the system. If that isnot the case, we have to select one stream as a reference stream and attribute a numerical value to it.It is completely arbitrary which stream we select as a reference stream and what numerical value weattribute to it. It is sometimes expedient to attribute the value of 1 to the magnitude of the stream andto select such a stream to be the reference stream to make the resulting balance directly indicate thequantity of product made from one kilogram of material or how many kilograms of the respectivematerial are required to produce one kilogram of product. If the magnitude of at least one stream hasbeen specified, a reference stream does not need to be selected. The selection of a reference streamwill be illustrated in the sample problems.

The decision to balance amount of substance, or mass, also indicates the standard onto which thedata specified in the process description must be converted.

First of all, it is necessary to decide on how we shall denote the mass flow rate and the amount ofsubstance.

Notation:

• Amount of substance is usually designated in the same way as in the equation of state forideal gas PV=nRT , i.e. as n. Then the flow should be a dotted n, but EXCEL cannot do that,and therefore we shall indicate the flow in kmol·s-1 as n as well.

• Mass is usually designated as m. Since the same rules apply here as with the flow of amountof substance, we have no other option than to designate the mass flow rate as m.

• The letter x has been introduced in physical chemistry to denote a mole fraction. We shalladhere to this established practice.

• In order to denote a weight fraction we shall choose a letter from the end of the alphabet aswell. However, y could be confused with the symbol introduced for a mole fraction in vapourphase and z clashes with the symbol for compressibility factor. These two properties will be

0,00

0,20

0,40

0,60

0,80

1,00

1,20

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1


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present in our calculations, and therefore it is necessary to look for another sign to denote theweight fraction. The nearest letter from the end of the alphabet is w.

• Designations of fundamental properties:Flow of amount of substance n recommended dimension kmol·s*1

Mole fraction xMass flow rate m recommended dimension kg·s-1

Weight fraction w• This notation is not binding; any other notation of fundamental properties can be used as longas it is defined properly.

However, the indexing is binding.

• Every flow designation has an index that is identical with the number of the respective stream.• Every designation of composition has two indexes:• The first index designates the stream• The second index designates the component.

Thus the mole fraction of the component 2 in the stream 3 is x3;2 – x3;2 can be used as notation inorder to increase the speed of posting in EXCEL. Similarly, one can post n3 instead of n3.

Conversion

To convert flows from Nm3·s-1 to kmol·s-1 one needs to know:

Normal temperature 273.15 °KNormal pressure 101.325 kPaGas constant R 8.31451 kJ·kmol-1·K-1

Then we can substitute values in the equation of state for ideal gas:

PV = nRT.

If the flow (production) is given in tons per year, this figure is not converted to the total number ofhours, but to the annual working capacity, which is 8,000 working hours per year as for continuouschemical operations. This is the value recommended in case a more accurate figure cannot beobtained.

In converting the composition in mole fractions to weight fractions we proceed as follows:

1.2. The components present in the stream are entered into the first column.3. The component’s quantity – mole fractions, mole per cent, any numbers designating the ratios

in the numbers of moles of the components - is entered into the second column.4. Molar mass of the components is entered into the third column.5. Scalar product of the second and third column is entered into the fourth column.6. The fifth column is created by dividing the individual parts of the fourth column by the sum of

the fourth column. We get the weight fractions of individual components as a result.

This procedure can be tested on the following table:

[kmol] [kg/kmol] [kg] [kg/kg altogether]NH3 32 17.03 544.96 0.154046 w3;4CO2 68 44.01 2992.68 0.845954 w3;3

[kmol altogether] [kg altogether]100 3537.64

In converting weight fractions to mole fractions we follow the same procedure, only the fourth columnis created by dividing the second column by the third column.

We will demonstrate the procedure in calculating the composition of saturated solution of salt in molefractions. The solubility of salt is specified as 35.66 g NaCl / 100g H2O.

[kg] kg/kmol kmol [kmol/Σkmol ]

NaCl 36,66 58,44 0,6273 0,1016

H2O 100 18,02 5,5494 0,8984

[Σkg ] [Σkmol ]

136,66 6,1767


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Important warning – If properties taken from the cells marked in colour are present in the calculationformulas for converting (for example, molar mass), we have to introduce the value of this property intothe formula by reference to the respective cell and not by retyping the value of such a property.

5. Chemical Reactions, Determination of Number of Independent ChemicalReactions

Source Stream

The next phase is to determine which nodes function as chemical reactors, and to resolve how manyindependent reactions take place therein. For the bicarbonate column, which is part of the Solvayprocess, the flow sheet and description of the reaction will be as follows:

Flow Sheet – Bicarbonate Column

Reaction:

NaCl + H2O + CO2 + NH3 = NaHCO3 + NH4Cl

For every independent reaction we have one stream, which is given by the product of the reaction rate

and the reaction volume, and we shall call it the source stream. The formation or disappearance of

individual components in the reactor through the reaction in kg·s-1

or in kmol s-1

is given by the productof the source stream and the respective stoichiometric coefficient (plus signs for products of reaction,

negative signs for reactants). In the event of more independent reactions the calculated propertiesmust be added up.

Determination of Number of Individual Chemical Reactions

The number of source streams present in the balance agrees with the number of independentreactions taking place in the system. However, there is one problem, namely the word “independent”.With more complicated reaction we are able to estimate, based on the fundamental knowledge ofchemistry, which chemical reactions could take place in a system. Almost all real processes do nottake place as a single reaction, but as a complex of reversible, parallel and consecutive elementary

reactions. However, we still need to establish which reactions are independent. This task is identicalwith the task of determining the rank of a matrix of coefficients. Let’s take the manufacture of nitric

filt

3

2

1

4


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acid by absorption of gases of nitrogen into water as an example. The following reactions may takeplace in the system.

We can set up a matrix of coefficients from the stoichiometric coefficients, which are used formultiplying the formulas of compounds in the individual stoichiometric equations:

NO NO2 O2 H2O HNO2 HNO3

-2 2 -1 0 0 0-1 -1 0 -1 2 02 0 0 1 -3 11 -3 0 -1 0 20 -2 0 -1 1 10 0 -1 0 -2 21 -1 0 0 -1 1We modify this matrix using the method of Gaussian elimination to make the cells below the diagonalshow zero. The Gaussian method of elimination is described for instance by J. Holenda in Linear

Algebra (Holenda J. Lineární algebra. Prague: Czech Technical University 1978).

With matrixes one is allowed to replace any row (column) with a linear combination of the initial row(column) with any other rows (columns) of the matrix. Therefore, if we want to generate zeros belowthe diagonal in the first column, we proceed as follows:

• row 1 – this remains the same,• row 2 – we multiply the second row by –2 and add it up with the first row,

• row 3 – we add the third row up with the first row,• row 4 – we multiply the fourth row by 2 and add it up with the first row,• row 5 and row 6 remain unchanged,• row 7 – we add the row 7 up with the row 2.

We get the following matrix

generation of zeros below the diagonal in the f i rst column

-2 2 -1 0 0 00 4 -1 2 -4 00 2 -1 1 -3 10 -4 0 -2 2 20 -2 0 -1 1 1

0 0 -1 0 -2 20 -2 0 -1 1 1

We proceed in an analogous manner in generating zeros below the diagonal in the second columnand in the third column:

generation of zeros below the diagonal in the second colum n-2 2 -1 0 0 00 4 -1 2 -4 00 0 1 0 2 -20 0 -1 0 -2 20 0 -1 0 -2 2

0 0 -1 0 -2 20 0 -1 0 -2 2

2· NO + O2 = 2· NO2

NO + NO2 + H2O = 2· HNO23· HNO2 = HNO3 + 2 NO + H2O

3· NO2 + H2O = 2· HNO3 + NO2· NO2 + H2O = HNO2 + HNO3

2· HNO2 + O2 = 2· HNO3HNO2 + NO2 = HNO3 + NO


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generation of zeros below the diagonal in the thi rd colum n-2 2 -1 0 0 00 4 -1 2 -4 00 0 1 0 2 -2

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

The elimination for the remaining columns does not take place since they already show zeros belowthe diagonal. Our matrix has the rank 3, and consequently only three of the seven mentionedstoichiometric equations are independent.We can take the liberty to refrain from determining the rank of a matrix only if we are absolutely sure.

An error made by incorrect determination of the number of independent equations is very difficult tofind.

6. Specification Table and Matrix of Coefficients

Specification Table

• A specification table is a method of entering input data to make it capable of being used insetting up a matrix of coefficients of a linear system of balance equations.

• If we have converted all available data to the chosen basis of calculation, we arrange theavailable values of the system characteristic into a specification table.

• The specification table will have as many columns as there are streams present in the system,plus one column for every independent reaction that takes place in the system.

• The first row is reserved for magnitudes of streams. On the top of that, the specification tablewill have as many rows for each block as there are components present in the block.

• Individual fields in the columns of streams contain a weight or mole fraction of the component

corresponding to the respective row.• If we don’t know the value that corresponds to the respective field, we will enter an

alphanumeric sign indicating this property. If we don’t know the mass flow rate of the stream 5,we will enter m5 into the respective field. If we don’t know the mole fraction of the component3 in the stream 4, we will write x4;3 into the respective field.

• If we know the value required, we shall introduce it into the respective field by reference to thecell in the Conversion paragraph – never by retyping!

• If we know that there is only one component in the stream, we will enter 1 into the respectivefield, if we know that there is no component present in the stream, we will enter 0 into therespective field.

• We post stoichiometric coefficients of components of the corresponding reaction into thecolumns for chemical reactions. Stoichiometric coefficients for reactants (components enteringinto the reactions, or components on the left-hand side of the stoichiometric equation) areattributed a negative sign. Stoichiometric coefficients of components, constituting products ofreactions (components on the right-hand side of the stoichiometric equation) will have apositive sign.

• That holds good in case we balance amount of substance. If we balance mass, we have tomultiply the individual stoichiometric coefficients by molar masses of the respectivecomponents.

• The number of unknowns agrees with the number of different alphanumeric signs in individualcells. Beware of duplications!

• For every block being balanced the number of balance equations available to us agrees withthe number of components present in the streams passing into or out of the blocks.

• Attention: the overall balance of a node or a block is given by the sum of the componentbalances, and therefore it is a dependent equation.

• Moreover, the sum of weight (or mole) fractions in every stream must equal one.For illustration see the sample specification table provided for the case of bicarbonate column.


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Specification Table

stream 1 2 3 4 5 6 reaction

kg·s-1 4,315 1,001157 m3 m4 m5 m6 mrblock component

column 1 H2O 0.732 0 0 w4;1 0 0 -18.02

2 NaCl 0.268 0.00 0 0.02 0 0 -58.443 CO2 0 w2;3 0.846 0 0 0 -44.01

4 NH3 0 w2;4 0.154 0 0 0 -17.03

5 NaHCO3 0 0 0 w4;5 0 0 84.01

6 NH4Cl 0 0 0 w4;6 0 0 53.49

filter 1 H2O 0 0 0 w4;1 0 w6;1 02 NaCl 0 0 0 0,02 0 w6;2 0

5 NaHCO3 0 0 0 w4;5 1 w6;5 0

6 NH4Cl 0 0.00 0 w4;6 0 w6;6 0

Matrix of Coefficients

A specification table can be transformed into a matrix of coefficients and a right-hand-side vector ofthe system of linear algebraic equations in the following manner:

• We open a new clean sheet and transfer the reference to the top left corner of thespecification table into a suitable field (left margin). Only that one cell, not the whole matrix!

• We place the cursor onto the bottom right edge of the cell (+ will appear) and drag it onto thewhole column and then the whole matrix. Thereafter we copy the whole specification table.

Having been copied and slightly adjusted, the specification table on the Sheet 2 looks as follows:

1 2 3 4 5 6 reactionstream kg·s-1 4.315 1.0012 m3 m4 m5 m6 mr

1 H2O 0.732 0 0 w4;1 0 0 -18.022 NaCl 0.268 0 0 0.02 0 0 -58.44

3 CO2 0 w2;3 0.846 0 0 0 -44.01

4 NH3 0 w2;4 0.154 0 0 0 -17.03

5 NaHCO3 0 0 0 w4;5 0 0 84.01

6 NH4Cl 0 0 0 w4;6 0 0 53.49

1 H2O 0 0 0 w4;1 0 w6;1 02 NaCl 0 0 0 0,02 0 w6;2 0

5 NaHCO3 0 0 0 w4;5 1 w6;5 0

6 NH4Cl 0 0 0 w4;6 0 w6;6 0

• First of all, we create a right-hand-side vector. We place it at the end as the last column,designate it, for instance, as “vps” and post zeros into all cells of this column.

• Then we transfer the numerical values from the row of streams into the right-hand-side vector.We proceed in the following manner:

• We create a new end row and post zeros into all fields.• We post 1 in the last row of the column wherefrom we want to transfer the numerical value of

the stream.• We drag and drop the numerical value (using an arrow) in the last row of the column of the

right-hand-side vector.• We enter the alphanumeric designation of the stream for the respective column into the

cleared cell in the row of streams.• We repeat this procedure for every stream for which we know the numerical value. In our case

we start with m1 and go over to m2.


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The alphanumeric designation for concentration of components in streams is removed from the tablein the following manner:

• Behind the column with the alphanumerical designation of the composition of the stream weinsert as many new columns as there are unknown concentrations present in the stream.

• We fill the new columns with zeros.• We enter into the row of streams that the values in the column indicate the product of flow and

composition. Therefore, we will have five columns for stream 6 – m6, mw6;1, mw6;2, mw6;5,and mw6;6.

• We replace the alphanumeric designation of the content of the component in the stream byzero and enter 1 into the respective column in the same row.

After these steps have been taken, the matrix reads as follows:

m1 m2 mw2;3 mw2;4 m3 m4 mw4;1 mw4;5 mw4;6 m5 m6 mw6;1 mw6;2 mw6;5 mw6;6 mr vps

0.73 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -18 0

0.,27 0 0 0 0 -0.02 0 0 0 0 0 0 0 0 0 -58.4 0

0 0 -1 0 0.85 0 0 0 0 0 0 0 0 0 0 -44 0

0 0 0 -1 0.15 0 0 0 0 0 0 0 0 0 0 -17 0

0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 84 0

0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 53.5 0

0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0

0 0 0 0 0 0.02 0 0 0 0 0 0 -1 0 0 0 0

0 0 0 0 0 0 0 1 0 -1 0 0 0 -1 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.31

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Thereby we have created a matrix of coefficients for balance equations. Apart from balances ofcomponents in individual nodes it must hold true that the sum of weight or mole fractions in everystream equals 1.

This relation will be referred to as the “summation relation” and we will incorporate it into the matrix ofcoefficients using the following procedure:

• We create a new end row and insert zeros into all fields.• We choose one stream in which there are some unknown concentrations. We select the

blocks from which this stream exits or into which this stream enters. We enter the value of 1 –the sum of absolute values of the content of the cells in the selected block - into the last row inthe column of this stream. If a known value is present in the stream – for instance in stream 4,a value a will appear in the end row, for which the following holds true:

§ 0


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1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.31

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0 -1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 -0.98 1 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 -1 1 1 1 1 0 0

7. Formulation of Additional Relations

Types of Additional Relations

In addition to balance relations and summation equations there are usually also other relations thatmust be satisfied. They are called additional relations. In most cases they are defined in thedescription of technology. They can include physical and chemical data (phase and chemicalbalances), or adjustable parameters of the process and equipment specified by manufacturers andtechnology support (efficiencies, specific consumption, conversions of chemical reactions). Theserelations can be both linear and non-linear (most relations of equilibrium). Heat balance with non-isothermal processes can also be included in additional relations. Enthalpy balance can serve as anadditional relation as long as it does not contain new unknowns. In that case we actually deal withenthalpy balance and molar balance at the same time. In most cases, molar and enthalpy balances

can be solved independent of each other, with enthalpy balance being based on molar balance.

In our case, the additional relation is the concentration ratio of sodium hydrogen carbonate and waterin mother liquor. The description of technology suggests that this relation will have the following form:

mw6;5 = 0,0753· mw6;1

We create another (this time final) end row for this additional relation and it shall read as follows:

m1 m2 mw2;3 mw2;4 m3 m4 mw4;1 mw4;5 mw4;6 m5 m6 mw6;1 mw6;2 mw6;5 mw6;6 mr vps

· · · · · · · · · · · · · · · · ·

· · · · · · · · · · · · · · · · ·· · · · · · · · · · · · · · · · ·

· · · · · · · · · · · · · · · · ·0 0 0 0 0 0 0 0 0 0 0 0.075 0 -1 0 0 0

Full comprehension of the recommended procedure will best be ensured by the complete Excel filewith commentaries clarifying the particular steps in greater detail. This file is entitled problem_soda.

The linear additional relation as the one stated above is rather an exception than a rule. Additionalrelations arising from phase or chemical balances or from enthalpy balances are non-linear. A problemof a non-linear additional relation is dealt with in the problem_redukce file. The Excel file has beenprovided with abundant commentaries that should be sufficient in order to make one understand theapplied procedure. One non-linear relation can be managed using the Solver function. If there aremore non-linear additional relations, the problem must be dealt with numerically using the Newton-Raphson method. Excel suggests an elegant method of obtaining the value of Jacobian, which isessential for the application of this method. The problem is solved in the problem_reactor file. Thecommentaries provided in this file should be sufficient to make one understand the procedure.

Enthalpy Balance

The number of balance equations of components for each block agrees with the number ofcomponents present in the given block.

We have one enthalpy balance for each block.

For the amount of matter and block l it will read as follows:

∑ ∑∑= ===⋅+⋅⋅⋅

S

1i

N

1k

k k j;i

C

1 j

j;iil;i Qhr nr hxnincmat


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• Index i indicates streams and index j indicates components.• S is the total number of streams; C is the total number of components.• hi;j is the molar enthalpy of component j at the temperature and pressure of stream i.• nr k is the source stream of the k-th reaction and hr k is the reaction enthalpy of this reaction.• N is the total number of independent reactions.

• Q is the energy exchanged with the surroundings.• All parts are expressed in kW.

Enthalpy balance for mass balance will have the following form:

∑ ∑∑= ==

=⋅+⋅⋅⋅S

1i

N

1k

k k

j

j;iC

1 j

j;iil;i Qhr nr M

hwmincmat

• M j is the molar mass of component j.

Consequently, molar enthalpy of a component at the given temperature and pressure is thefundamental property for enthalpy balance. It is a relative property; its value is related to a specificstate for which we choose the enthalpy of a component to be zero. These conditions shall be referred

to as standard state conditions.

In our calculations this component will behave like ideal gas at the temperature of 298.15°K. Pressuredoes not need to be defined because with ideal gas enthalpy does not constitute a function ofpressure.

A component can be transferred from the standard state conditions into the final state using thefollowing steps:

• By heating component j from the standard state temperature up onto the final temperature Ti (°K). The change in enthalpy is given by the following relation:

j

04

i

j3

i

j2

i

j

i j j HT4

CpD

T3

CpC

T2

CpB

TCpAh −⋅+⋅+⋅+⋅=∆ where

4 j3 j2 j

j j0 15,298

4

CpD15,298

3

CpC15,298

2

CpB15,298CpAH ⋅+⋅+⋅+⋅=

The nomenclature is in compliance with the EnthalpyData database.

• By condensing the component. The database provides evaporation heat Hvap (condensationheat with the opposite sign) for boiling temperature. It can be converted onto any othertemperature using the following equation:

374,0

1

1

·

−

−

=C

B

C

vap

T

T

T

T

Hvaph

This calculation is approximate. For a more accurate calculation access must be found to adatabase containing data for Cp in a liquid phase.

• In order to include heat of admixture, heat of dilution, melting heat and heat of crystallization,one must refer to specialised databases.

Heat of reaction hr can be calculated from standard heats of formation – Hform and stoichiometriccoefficients ? k;j that have a negative sign for reactants and a positive sign for products of reaction. Forone reaction:

j

C

1 j j Hformhr ⋅ ν=∑=


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Our EnthalpyData database defines Hform by stating that a component in a gaseous state is formedfrom elements.

We provide, for information, the internet address of the DECHEMA database www.ddbts.de

and the internet address of the National Bureau of Standards database webbook.nist.gov

An illustration of the application of the methods of calculating enthalpy balances in EXCEL can be

found in the problem_styrene file. In this case, the problem is solved on the basis of both molarbalance and mass balance.

8. Conditions of System Solvability

Using Excel we are able to solve the following problems:

1. We have more equations than unknowns, and the system is linear.2. We have as many equations as unknowns, and the system is linear.3. We have as many equations as unknowns, and one equation is non-linear.4. We have as many equations as unknowns, and more equations are non-linear.5. We have one equation less than unknowns, and the system is linear.

If our problem does not fit into any one of these categories, the problem must be redefined, i.e. otheradditional relations must be found, some blocks in the block diagram must be consolidated, some ofthe components must be disregarded, and so forth.

9. Solving a System of Equations using Methods of Linear Algebra, with theAdvantage of Using Excel

Problem 1

When solving Problem 1 we proceed as follows:

• We multiply the transposed matrix of coefficients by the matrix of coefficients to get a squarematrix of the most expectation values of the matrix coefficients in the sense of the method of

minimum squares. The formula has the following form: PRODUCT OF MATRIXES(TRANSPOSITION(A1:D18);A1:D18), where A1 is the cell in the top left corner of the matrixof coefficients and D18 is the cell in the bottom left corner.

• We multiply the transposed matrix of coefficients by the right-hand-side vector to get acorrected vector with the same dimension as the square matrix.

• We multiply the inversion of the square matrix by the corrected vector to get the vector ofsolutions.

Problem 2

We solve this problem by multiplying the inversion of the square matrix of coefficients by the right-hand-side vector.

The formula has the following form: PRODUCT OF MATRIXES(INVERSION(A1:H8);I1:I8), where A1

is the cell in the top left corner of the matrix of coefficients and H8 is the cell in the bottom right corner.Column I1:I8 would correspond to the right-hand-side vector.

Problem 3.

In solving Problem 3 we proceed as follows:

• We attribute a numerical value to one of the unknowns.• As a consequence of the incorrectness of the value fixed in this manner the non-linear

additional relation is not satisfied.• We keep modifying the fixed value until the non-linear relation is satisfied with the accuracy

required.• This procedure can be performed manually, however, using the Solver function is more

efficient. The procedure is described in detail in the problem_redukce file.

Problem 4


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This Problem is solved using the Newton- Raphson method.

The procedure can be clarified by studying the problem_reactor file. The sheet entitled Directions provides a detailed method of applying this method in the environment of EXCEL. The description ofthe method can be found for instance in the book by M. Kubícek Numerical Algorithms of Solutions toChemical Engineering Problems (Kubícek, M.: Numerické algoritmy rešení chemick o-inženýrských úloh. Prague: SNTL 1983).

Problem 5

In this case we cannot obtain a solution, but functionality. We choose one of the unknowns to be anindependent variable, and then we can attribute some numerical value to it. Now the system issolvable. We can study how other variables (unknowns) change subject to the value of theindependent variable selected, whereby we can obtain valuable information.

10. Table of Solutions Containing Complete System Characteristic.

A table of solutions provides complete characteristic for each stream; that is the flow of components,overall flow, temperature and pressure. The columns represent individual streams and the rows areused for entering the mentioned parameters (see, for instance, the problem_soda file).

CONCLUSION

The proposed methods of balancing using EXCEL enable one:

• To create balance models of lines, nodes and equipment, with the aim to answer the question:“What will happen if ….“

• To serve as a guideline for interventions in the operation of the line, especially if parameterschange unexpectedly.

• To acquire balance sheet values to elaborate a „basic design“.

• To determine the minimum number of measured quantities needed for the purposes ofdetermining the complete system characteristic.• To evaluate and check industrial experiments and operating data.• To balance raw materials, products, and wastes for the purposes of economy and ecology of

an enterprise.• To make the training of the operating personnel easier.


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Address for consultation, training and advisory services: see the main Web page.

List of Selected Projects Solved by the Authors:Project 1: Desulphurisation Unit of the Mitsubishi CompanyProject 2: Co-generation Plant of the Škoda Company

Project 3: Catalytic Reactor - Draslovka KolínProject 4: Co-generation Plant with the Utilisation of a Heat Pump - Brun &Sorensen - DenmarkProject 5: Manufacture of Cole-seed Oil - Sindat EngineeringProject 6: Refining of Crude Spirit - Alcohol DistilleryProject 7: Low-pressure Manufacture of Nitric Acid - Sindat EngineeringProject 8: Manufacture of Pure Salts –OSRAM BruntálProject 9: Enthalpy Balance of a Bakery Factory - Pragosoja StránciceProject 10: Enthalpy Balance of a Boiler for Steam Production - Detmarovice Power Plant - MitsubishiCo.

Problems Solved:Problem 1: Solvay (Ammonia Soda) Process (problem_soda file)Problem 2: Manufacture of Reduced Iron (problem_redukce file)

Problem 3: Manufacture of Styrene from Ethylbenzene (problem_styren file)Problem 4: Catalytic Oxidation of SO2 to SO3 (problem_reaktor file)


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Declared composition of rum 38 volume per cent of ethanol

Composition of rum with 38 volume per cent in weight fractions at various temperatures

kg/m3 m3/kg weight per cent volume per cent

temperature ethanol water ethanol water ethanol water ethanol

0 806.0 999.8 0.001241 0.001000 33.07 66.93 38.00

10 798.0 999.6 0.001253 0.001000 32.85 67.15 38.00

20 789.0 998.2 0.001267 0.001002 32.64 67.36 38.00

30 781.0 995.6 0.001280 0.001004 32.47 67.53 38.00

40 772.0 992.2 0.001295 0.001008 32.29 67.71 38.00

50 763.0 988.0 0.001311 0.001012 32.13 67.87 38.00

60 754.0 983.2 0.001326 0.001017 31.97 68.03 38.00

70 745.0 977.7 0.001342 0.001023 31.87 68.17 38.00

Minimum alcohol content

ethanol water

temperature 70°C 0°C745.0 999.8 0.001342 0.001000 31.35 68.65 38.00

A Flowsheeting

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