material and energy balancing in the process industries 044482409x

Bulsari, Editor Volume 7: Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants V.V... vi Material and Energy Balancing in the Process Indus

MATERIAL A N D E N E R G Y BALAN CIN G

IN THE P R O C E S S I N D U S T R I E S

From Microscopic Balances to Large Plants

COMPUTER-AIDED CHEMICAL ENGINEERING

Advisory Editor: L.M Rose

Volume 1: Distillation Design in Practice (L.M Rose)

Volume 2: The Art of Chemical Process Design (G.L Wells and L.M Rose)

Volume 3: Computer-Programming Examples for Chemical Engineers (G Ross) Volume 4: Analysis and Synthesis of Chemical Process Systems (K Hartmann and

K Kaplick)

Volume 5: Studies in Computer-Aided Modelling, Design and Operation

Part A: Unit Operations (1 Pallai and Z Fony6, Editors)

Part B: Systems (1 Pallai and G.E Veress, Editors)

Volume 6: Neural Networks for Chemical Engineers (A.B Bulsari, Editor)

Volume 7: Material and Energy Balancing in the Process Industries - From Microscopic

Balances to Large Plants (V.V Veverka and F Madron)

COMPUTER-AIDED CHEMICAL ENGINEERING, 7

ELSEVIER SCIENCE B.V

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ISBN: 0-444-82409-x

9 1997 Elsevier Science B.V All rights reserved

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Contents

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Introduction

Balancing in Process Industries

2.1 2.2 2.3 2.4 2.5 2.6 2.7 The P r o b l e m Statement ,

Balancing F l o w s h e e t - Graph Representation

Balancing Variables and Data Preprocessing

Data Reconciliation

Classification of Variables

D a t a Collection and Processing

R e c o m m e n d e d Literature

M a s s 3.1 3.2 3.3 3.4 3.5 3.6 3.7 (Single-Component) B a l ~ c e

Steady-State Mass Balance and its Graph Representation

M o r e on Solvability

3.2.1 Partition of variables and equations

3.2.2 Transformations

Observability and R e d u n d a n c y

3 3.1 Definitions and criteria

3.3.2 Classification methods and transformation of equations

Illustrative E x a m p l e

Just D e t e r m i n e d S y s t e m s

M a i n Results of Chapter 3

R e c o m m e n d e d Literature

M u l t i c o m p o n e n t Balancing

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 C h e m i c a l Reactions and Stoichiometry

C o m p o n e n t Mass Balances

Reaction Invariant Balances

E l e m e n t Balances

Reaction Invariant Balances in a S y s t e m of Units E x a m p l e of M u l t i c o m p o n e n t Balance

U n s t e a d y - S t a t e Balances

M a i n Results of Chapter 4

R e c o m m e n d e d Literature

7

7

12

15

19

20

22

24

25

25

31

31

34

37

37

39

42

49

55

58

59

59

63

70

77

79

83

87

91

96

vi Material and Energy Balancing in the Process Industries

Chapter 5 Energy Balance

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Enthalpy Balance

Graph and Matrix Representation of the Enthalpy Balance

Example of Enthalpy Balance

Heat and Mass Balancing

Remarks on Solvability

Unsteady Energy Balance

Main Results of Chapter 5

Recommended Literature

99 99 108 111 118 123 124 127 132 Chapter 6 Entropy and Exergy Balances

6.1 Exergy and Loss of Exergy

6.2 Simple Examples

6.3 Examples of Exergetic Analysis

6.3.1 Heat exchanger network

6.3.2 Distillation unit

6.3.3 Heating furnace

6.4 Concluding Remarks

6.5 Recommended Literature

135 135 137 170 170 172 173 175 176 Chapter 7 Solvability and Classification of Variables I Linear Systems

7.1 General Analysis

7.2 Canonical Format of the System of Equations

7.3 Commentary and Examples

7.4 Main Results of Chapter 7

7.5 Recommended Literature

177 177 184 191 196 199 Chapter 8 Solvability and Classification of Variables II Nonlinear Systems

8.1 Simple Examples

8.2 Solvability of Multicomponent Balance Equations 8.3 8.2.1 Transformation of the equations

8.2.2 Additional hypotheses

8.2.3 Independent variables

8.2.4 Degrees of freedom

8.2.5 Solution manifold

8.2.6 Remarks

8.2.7 Example

General Solution Manifold

8.3.1 Energy balance equations

201

201

213

213

217

224

228

234

235

238

244

244

vii

8.4 8.5

8.6 8.7

8.3.2 The whole system of balance equations

8.3.3 Heat exchangers

8.3.4 Heat and mass balances

8.3.5 General nonlinear models

Jacobi Matrix and Linearisation

Classification Problems; Observability and Redundancy

8.5.1 Examples

8.5.2 Theoretical analysis

8.5.3 Theoretical classification

8.5.4 Classification in practice

Main Results of Chapter 8

R e c o m m e n d e d Literature

250 251 255 257 258 265 265 270 277 284 288 295 Chapter 9 Balancing Based on Measured D a t a - Reconciliation I Linear Reconciliation

9.1 9.2 9.3 9.4 9.5 9.6 9.7 P r o b l e m Statement

Reconciliation Formulae

Statistical Characteristics of the Solutions; Propagation of Errors

Gross Errors

Systematic Errors

Main Results of Chapter 9

R e c o m m e n d e d Literature

297 297 303 312 329 342 345 350 Chapter 10 Balancing Based on Measured D a t a - Reconciliation II Nonlinear Reconciliation

10.1 Simple Examples and P r o b l e m Statement

10.2 Conditions of M i n i m u m

10.3 Properties of the Solution

10.4 Methods of Solution

10.4.1 Improving the initial guess

10.4.2 Suboptimal reconciliation

10.4.3 Final reconciliation

10.4.4 Short-cut method

10.4.5 Remarks ,

10.5 Gross Errors

10.6 E x a m p l e

10.7 Main Results of Chapter 10

10.8 R e c o m m e n d e d Literature

353 353 364 367 374 374 376 380 384 387 394 399 409 414 Chapter 11 D y n a m i c Balancing

11.1 Mass Balance Model with Accumulation

417

417

viii Material and Energy Balancing in the Process Industries

11.2

11.3

11.4

Daily Balancing

Conclusion

R e c o m m e n d e d Literature

422 434 435 Chapter 12 Measurement Planning and Optimisation

12.1 Single-Component Balancing

12.1.1 Finding the first solution

12.1.2 Optimal measurement placement from the standpoint of measurement precision 12.2 The General Case

12.2.1 Problem statement

12.2.2 Objective functions

12.2.3 Theory

12.2.4 Solution of the problem

12.2.5 Results

12.2.6 Example

12.2.7 Optimality of the solution

12.3 Main Results of Chapter 12

12.4 R e c o m m e n d e d Literature

437 437 438 440 441 441 442 443 444 447 449 452 453 454 Chapter 13 Mass and Utilities Balancing with Reconciliation in Large Systems A Case Study 457

13.1 The Problem Statement 457

13.2 Requirements on the Industrial Balancing System 460 13.3 Important Attributes of an Industrial Balancing S y s t e m 4 6 3 13.4 Balancing Errors 465

13.4.1 Mistakes in the balance flowsheet - topology errors 465

13.4.2 Analysis of balance inconsistency - balance differences 467

13.4.3 Identification of the cause of a gross error 468 13.5 Structure of the Balancing System IBS 469

13.6 Configuration of the Balancing System - A Case Study 472

13.7 Examples of Balancing Proper 475

13.7.1 Correct balance - the base case 475

13.7.2 Balancing with a gross error 477

13.7.3 Some quantities unobservable 478

13.7.4 T o p o l o g y e r r o r - some streams missing 479

13.7.5 T o p o l o g y errors - too many fixed streams 480 13.7.6 Reporting 481

ix 13.8

13.9

Experience with C o m p a n y - W i d e Mass and Utilities

Balancing

13.8.1 Balancing in operating plants

13.8.2 Implementation of a c o m p a n y - w i d e balancing system

Conclusions

481 482 482 485 A p p e n d i x A Graph Theory

A.1 Basic Concepts

A.2 Subgraphs and Connectedness, Circuits and Trees A.2.1 Subgraphs and connected c o m p o n e n t s

A.2.2 Connected graphs, circuits, trees

A.3 Properties of the Incidence Matrix

A.3.1 Case when G is a tree

A.3.2 General connected graph, elimination of circuits

A.4 Graph Operations ,

A.4.1 Finding connected components

A.4.2 Derived operations

A.4.3 Merging of nodes

A.5 Costed Graphs

A.6 R e c o m m e n d e d Literature

487 487 491 491 493 497 498 500 503 503 506 506 507 513 Appendix B Vectors and Matrices

B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 B.9 B.10 B.11 Vector Spaces

D i m e n s i o n and Bases

Vector Subspaces

Classical Vector Spaces

Linear Maps

Matrices

M o r e on Matrices; Elimination

Matrix Operations

S y m m e t r i c Matrices

Orthogonality

Determinants

515 515 518 521 523 526 532 538 543 551 554 562 A p p e n d i x C Differential Balances

C.1 C.2 C.3 C.4 C.5 C.6 Material C o n t i n u u m

Interface Balances

T h e r m o d y n a m i c Consistency

Exact Integral Balances

Electrochemical Processes

R e c o m m e n d e d Literature

571

571

575

576

578

582

584

X Material and Energy Balancing in the Process Industries

A p p e n d i x D Differential form of the 2nd Law of Thermodynamics

D.1 Recommended Literature

585

588

Appendix E Probability and Statistics

E.1 Random Variables and their Characteristics

E.2 Special Probability Densities

Chapter 1

I N T R O D U C T I O N

Material and energy balancing belongs to very frequent chemical engineering activities In spite of this fact, a systematic treatment of this topic in the form of a monograph is quite rare Balancing also usually forms one or more introductory chapters in general chemical engineering textbooks, but the limited space available there usually makes not possible to cover this subject in depth The classical treatment of balances in the literature available (which was excellent at the time of publishing) needs to be complemented nowadays for several reasons

Balancing belonged to the first successful use of computers in chemical engineering in early sixties Since then the spread of computers (and especially personal computers) influenced balancing significantly The classical steady state balancing in the stage of process design now represents only a part of balancing problems we can meet in practice

Mass and energy balancing (accounting) now becomes to be a standard feature of plant information systems as a part of efforts to guarantee good house- keeping of operating plants Process plants very often represent complex systems linked to one another to enable functioning in the most efficient way The automatic computerised analysis and synthesis of balancing relations among hundreds or thousands of process streams requires a systematic approach respecting all possible situations which can occur in practice This fact brought the need for new balancing techniques based on graph theory and

statistics

The other important feature of balancing calculations is the danger of

insufficient accuracy of results Unmeasured flows and concentrations calculated

on the basis of imprecise data are in most cases differences of big numbers This fact influences adversely the accuracy of results and in some situation makes them useless The confidence intervals should be an integral part of balancing results based on imprecise data Even more dangerous is the possible occurrence

of gross measurement errors which can devalue results even more seriously The problems of confidence intervals and gross errors detection and elimination can

be solved by data reconciliation

Instrumentation in process plants is very often not sufficient for setting

up detailed and reliable balances Design and maintenance of instrumentation systems is strongly influenced by control engineers with the primary target of process control To improve situation in this area new techniques of instrumenta-

2 Material and Energy Balancing in the Process Industries

tion design and optimisation for balancing purposes are now available (optimum measuring placement)

Retrofitting and revamping of operating plants is now important part of investment to process industries The new design of the plant relies heavily on underlying process data Material and energy balances needed for retrofitting and revamps must be much more detailed than is common in regular plant operation Special plant tests are run with additional (portable) instrumentation The reliability and completeness of balances influences the success of overall project significantly

Increased interest in energy conservation in large systems requires a systematic approach to consistent problem definition Particularly successful is the use of pinch technology The application of pinch technology to existing plants (revamps) depends on reliable heat and energy balances The Classical energy (or enthalpy) balances are now complemented by balancing of entropy and exergy (availability)

There are also new balancing techniques used in the process control (dynamic balances, instrumentation design and maintenance, etc.) Advanced control techniques incorporated in process computers are very often based not only on directly measured variables but also on secondary variables calculated from balance models

The present book will complement the information present in classical balancing literature by the following topics:

systematic analysis of large systems by Graph theory

comprehensive thermodynamic analysis (entropy and availability) balancing on the basis of redundant raw plant data (reconciliation, propagation of random measurement errors, detection and elimination of gross errors)

measurement design and optimisation (optimal measurement placement) dynamic balancing

experience with plant-wide regular mass and energy balancing as a part

of plant's process information system

The structure of the book is as follows:

The most elementary and unequivocal balance of a technological system

is its mass balance, dealt with in Chapter 3 Although the balance of a node (unit

of the system) is almost trivial (mass in = mass out + possible accumulation), the fact that the individual nodes are connected by streams where output stream from one unit becomes input stream into another unit gives rise to a complex structure represented by the graph of the system The methods of Graph theory enable one

to analyse completely the problems of solvability in a mathematically precise

an algebraic procedure described in Section 4.3, the (usually a priori unknown) source terms can be eliminated giving a set of'reaction invariant balances'; so the number of unknown variables (parameters) is reduced Of main interest are steady-state balances neglecting the terms due to accumulation in the nodes The formulation of an unsteady-state balance brings no theoretical problem (see Section 4.7); specified is the balance of a batch reactor

The exact energy balance (Chapter 5) is only hypothetical In practice, less relevant items (such as kinetic and potential energy) are neglected and neglecting also the accumulation terms, we have the steady-state enthalpy balance Each material stream is assigned its specific enthalpy with the condition that the specific enthalpies are thermodynamically consistent In addition we introduce net energy streams representing heat transfer or also mechanical power supply completing the balance of any node For the case that only total mass is balanced along with energy (enthalpy), a simplified version of the balance is presented in Section 5.4 (heat and mass balancing); an important special case is

a heat exchanger network For completeness, in Section 5.6 we give general form and examples of unsteady energy balance

Of different nature is the entropy balance dealt with in Chapter 6 It is not a balance in the proper sense because no quantity is conserved In fact it is

a method of thermodynamic analysis of the processes in the system More familiar to engineers is the notion of exergy (availability) The (necessarily positive) production of entropy in any node is proportional to the loss of exergy

in the node; the goal is to reduce the loss as far as possible The chapter gives

a number of examples of such thermodynamic analysis, along with critical comments

Chapters 7 and 8 are devoted to the problems of solvability We call a set of equations solvable when there exists some vector of solutions, not necessarily unique In Chapter 3, we have shown that the set of mass balance equations is always solvable if no variable has been fixed a priori With redundant measured variables, the equations need not be (and usually are not) solvable, unless the fixed variables have been adjusted Then certain unknown (unmeasured) variables are uniquely determined (observable), other still not (unobservable variables) Certain measured variables can be nonredundant: they

4 Material and Energy Balancing in the Process Industries

are not affected by the solvability conditions thus their values are arbitrary The

set of mass balance equations is an example of a linear system In Chapter 7 we

analyse the solvability of a linear system in general, using the methods of linear

algebra In particular we transform the equations to (what we call the) canonical format (Section 7.2) and give explicit criteria for the solvability classification The main goal is to give methods that can be applied to the (considerably more difficult) analysis of non-linear systems

The multicomponent chemical species and energy balances are mostly non-linear in the primary variables (mass flowrates, mass fractions, temperatures,

possibly also pressures of the streams) The solvability analysis of non-linear systems in Chapter 8 begins with examples showing that even in simple cases, the terminology introduced for linear systems (in particular observability and redundancy) becomes somewhat vague: certain problems can be 'not well-posed'

We then first analyse the solvability of the whole system of balance equations without a priori fixed variables (Sections 8.2 and 8.3) and show that under

certain plausible structural hypotheses, the system is solvable

If the values of certain variables have been fixed a priori (for example measured), there is no mathematically precise general solution to the observabi- lity and redundancy classification problem We did not attempt to arrive at a complete analysis Still, we have shown that remitting somewhat the mathemati- cal precision, a pragmatically plausible classification is possible if the problem

is linearized

In Chapters 9 and 10, we deal with balancing based on measured data

The measured data are subject to random errors, and are processed by statistical methods A statistically rigorous approach is possible if the balance constraints

are linear; see Chapter 9 The measured data are first adjusted so as to obey the solvability conditions and minimise the generalised sum of squares of the adjustments; the procedure is called reconciliation In particular if the distribution

of measurement errors is Gaussian with zero mean, we thus obtain the maximum likelihood estimates Also the estimates of the unmeasured observable variables can then be computed Even if the distribution is not Gaussian, we can compute

statistical characteristics (in particular standard deviations) of the estimates (and possibly also of other variables dependent linearly on the former) With a

Gaussian distribution, the confidence intervals for the estimates are thus found

Important is the fact that the presence of redundant measured variables generally improves the precision of the results On the other hand a measurement error in

one variable affects also the estimates of other variables (so-called propagation

of errors); in particular the presence of gross errors can spoil the whole result

of reconciliation Methods of detection and elimination of gross errors can be

based on the statistical analysis of data; see Section 9.4

A rigorous formulation of the reconciliation problem is possible even

with non-linear constraints; only the general existence and uniqueness of a

C h a p t e r 1 - Introduction 5

solution is not warranted theoretically Still, with possible exception of considerable gross errors or inadequate selection of the measured variables the problem appears as solvable in practice as shown in Chapter 10 The methods are based on successive approximations (such as successive quadratic programming), where the special character of the criterion to be minimised can be made use of; see Section 10.4 Although not rigorous from the statistical point of view, the approximate statistical characteristics of the estimates can be computed if the constraints are linearised at the final solution The latter linearisation makes also possible to classify the variables according to Subsection 8.5.4 of Chapter 8 Finally the methods of detection and elimination of gross errors according to Chapter 9 can be applied to the linearised system of equations (Section 10.5)

Chapter 11 deals with dynamic (unsteady-state) mass balancing of a technological system involving inventories A straightforward method of reconciliation is presented avoiding accumulation of small systematic errors in the time series of measurements The method has proved successful in practice and can also be supported by theoretical arguments

There is generally a great number of possibilities how to select the set

of values to be measured, thus the measuring points The criteria of optimality and the ways leading towards an optimum are analysed in Chapter 12 A more

or less heuristic procedure is suggested eliminating successively measuring points that involve high costs and/or give unprecise estimates of the required quantities

If theories and methods presented in the book should to be successful in the practice, many prerequisites must be fulfilled Typical example of the complexness of the problem can be regular (e.g daily) balancing in process plants based on measured data Chapter 13 deals with problems which can be encountered in regular balancing of mass and utilities in refinery and petrochemi- cal complexex The analysis of the problem is complemented by the experience gained during development and implementation of mass and utilities balancing with reconciliation in the framework of company's process information system

It is the intention of authors to present balancing techniques in a systematic and rigorous way To make the book at the same time more readable, some parts are included in 5 Appendices The appendices A,B, and E will enable the reader to brush up and possibly extend his knowledge of graph theory, vector and matrix algebra, and statistical theory pertaining to the problems of balancing For the interested reader's convenience, the theory is presented in a largely self- contained manner, with (at least outlined) mathematical proofs In perusing Chapters 3-8, the reader can manage with Appendix A and Sections B.1-8 of Appendix B Sections B.9-11 and Appendix E constitute the theoretical basis of Chapters 9-11

The subtitle of the book reads 'From microscopic balances to large plants' The first part of the promise is fulfilled in Appendices C and D If the reader is more deeply interested in the physics underlying the ('macroscopic') node balances, before perusing Chapters 4 and 5 he can begin with Appendix C,

6 Material and Energy Balancing in the Process Industries

and with Appendix D before Chapter 6 He will then perhaps better understand the complex nature of the processes in the technological units, and the (neces- sary) simplifications adopted in setting-up the balances of a technological system

There are many examples throughout the book which serves for better understanding of theory presented there To gain more practice in solving balancing problems, Appendix F presents several exercises dealing mostly with balancing with reconciliation The exercises are used in reconciliation courses given by one of the authors of the book Unfortunately, the data reconciliation

is out of reach of the power of pocket calculators even for very simple flowsheets If the reader wants to go through the workshop based on Appendix

F, he can obtain (free of charge) mass and heat balancing software RECON suitable for IBM compatible personal computers (see the coupon at the end of the book) This software is sufficient for the complete solution of exercises including the statistical analysis of data

Fig 2-1 depicts a simple scheme for a mass balance Besides the distillation column (1) and a distillate storage tank (2), the scheme comprises also more complex subsystems of apparatuses which are balanced here as "black boxes" Node (5) represents here a distributor of electric energy supply

Fig 2-1 A flowsheet for a single-component balance

Streams are usually classified as process streams (streams 1 to 8) and utilities, such as electricity or cooling water (streams 9 to 11) This classification

is useful, because process streams usually don't mix with utilities Systems of balancing equations of process streams and of individual utilities are often independent and can be solved separately The classification of the stream (12)

8 Material and Energy Balancing in the Process Industries

is problematic In the subsystem (4) arises the process off-gas which is further led to the subsystem (3) to be burned in a furnace; so it becomes here a utility stream

The most simple balance of the system in a certain (usually integral) time interval means the complete information about

exchange of balanced quantities (overall mass, components, different forms of energy, etc.) among separate elements (nodes) of the system or among nodes and the environment

inventories of balanced quantities in nodes at the beginning and at the end of the balancing interval, or how the inventories have changed (accumulation)

At this moment we are interested in the overall mass flows in the case

of mass streams (for example in tons) and electric energy (for example in GWh)

We are not interested in other characteristics of streams as chemical composition, because it can be supposed that the quality of the individual streams is guaranteed by the control system

Balance equations for this problem can be expressed as

sum of inputs = sum of outputs + increase of inventory

for every node in the system It is the case of so-called single component balance

when there is one balancing equation around any node The balance can be also

classified as dynamic because the process need not be at steady state as we admit

the change of inventory in the tank (2) The incentive for setting up the balance

in such system is regular monitoring of yields of main products and of specific consumption of utilities as a part of "good house keeping" of the plant

Let's consider the second flowsheet in Fig 2-2 where is the distillation/absorption train for processing of refinery off-gases Refinery off-gas (1) consists of light hydrocarbons (C1 to C7), hydrogen and inert gases (N2, Ar) The column No 1 serves as absorber/desorber Hydrogen, inerts and C1 and C2 hydrocarbons are boiled up in the bottom part of the column and heavier hydrocarbons (C3, C4) are absorbed in the upper part of the column by heavy hydrocarbons (C6 and C7) The lean gas (2) leaving the head of the column is later burned in the furnace The second column serves for separation of C3 and C4 hydrocarbons from the absorbent (6) which circulates among columns (1) and (2) The middle hydrocarbons (4) consisting mostly of propane and butanes are separated in the third column which produces commercial products propane (7) and butane (8)

Chapter 2 - Balancing in process industries 9

Such multicomponent balance is important for tracing valuable components in the process The value of commercial products (propane and butane) is much higher than the value of the lean gas Significant economical losses can occur if the absorption in the first column does not operate properly and valuable hydrocarbons are lost in the lean gas

There are also other incentives for setting up such multicomponent balance The balance in a multicomponent system can serve for validation of sampling and analytical methods The analysis of imbalances (inputs minus outputs) can reveal errors caused by improper sampling or erroneous analytical methods

Nowadays, process simulators which can simulate the behaviour of the plant on the computer are more and more used for the optimisation and control

of processes The major problem in using simulators is their tuning to simulate properly all important features of the real process This can be done only by comparison of the result of simulation with the complete component balance set

up on the basis of plant data

10 Material and Energy Balancing in the Process Industries

There is a chlorination reactor in Fig 2-3 Methane is chlorinated in a complex reaction set which can be summarised as

C H 3 C 1 + C12 = CH2C12 + HC1

CH2C12 + C12 = CHC13 + HC1

In this case the individual components are not conserved in the reactor

as they can be consumed or created by reactions The balance of the reactor is governed by the stoichiometry of the reaction system In this case we speak about

the multicomponent balance with chemical reaction The result of balancing is

the complete information about amount of individual components entering and

leaving the reactor and also about reaction characteristics as reaction rates or extents of reaction Balancing of reacting systems is important for detailed

analysis of processes running in the reactor, either in the stage of laboratory or pilot plant studies, or as regular monitoring of industrial reactors behaviour (yields, selectivity) In the last case it is possible to reveal important changes as improper control, ageing of the catalyst, etc., which can cause serious economic losses or even jeopardise the safety of the whole process

Another flowsheet is in Fig 2-4 The distillation column is complemented by two heat exchangers which makes possible preheating of the feed by the contact with hot bottom product In practice, especially in refinery and petrochemical industries, we can meet even with much complicated heat exchange systems consisting of dozens of heat exchangers which make processes economically viable

Fig 2-4 A feed of the column preheat

To analyse such system the heat or enthalpy balance must be set up, which includes the complete information about enthalpy of streams and also

Chapter 2 - Balancing in process industries 11

about heat fluxes occurring among some parts of the system We can meet with this kind of balancing either when designing a new process, or during monitoring or control of the existing process (evaluation of heat transfer coefficients, calculation of unmeasured temperatures, etc.) Such monitoring can reveal, for example a common quiet "thief' in process plants - a heat exchange area fouling

All these examples given above represent a broad class of needs we can meet in practice The information about balanced quantities are obtained by solving the system of balance equations on the basis of measured or otherwise specified values of variables There are two basic types of balance calculations

For the first case is typical that some variables (flows, inventories, etc.) are fixed and the other are calculated by direct solving the system of equations The number of unknowns equals the number of independent equations and the selection of unknowns must guarantee the solvability of the system This situation is typical for example for steady-state balancing in the stage of the

design of the process This type of balancing which can be reduced to equations solving was the main topic in earlier chemical engineering balancing literature The solution is relatively easy (with the use of computers and balancing software which is now available) The one problem can be the selection of the set of independent variables which makes the problem just solvable Such a system of independent variables can differ from the set we would like to specify independently The other problem can be the consistent set of enthalpy data in the case of energy balancing Nowadays process simulators are widely used for designing of new processes which are capable of setting up the complete material and energy balance of even very complicated processes

The other area of mass and energy balancing concerns the analysis of operating processes The advent of process computers brought the huge amount

of process data from operating plants, sometimes of dubious quality At the same time the widely spread personal computers brought an immense computing power

to process engineers

The situation here is much more complicated than in the case of balancing in the design area The occurrence of measured variables in operating processes depends mostly on the instrumentation which is usually dictated by control engineers to guarantee good controllability of the process Less attention

is given to balancing needs, where very often only basic inputs and outputs are measured with sufficient precision (the custody measurement) On the other hand,

in some areas where the control is critical there is more information for balancing than needed, we say that the instrumentation is redundant So we can meet very often with insufficient instrumentation in one part of the plant and with huge amount of redundant data in the rest All this influences the unique solvability

of the system of balancing equations and makes the balancing more difficult

This situation which started in early sixties brought the need for systematic methods for the second type of balancing- the balancing based on

12 Material and Energy Balancing in the Process Industries

measured data Besides problems with redundant or insufficient data also other problems have to be solved Statistical methods were used for the adjustment of controversial data by reconciliation Reconciliation which was developed about two hundred years ago was re-invented for balancing in early sixties Since then many problems typical for the use of reconciliation in process engineering were successfully solved

Results of balancing based on measured data are significantly influenced

by measurement errors The propagation of measurement errors in the process of balancing provides the information about the reliability of results by confidence intervals Very useful are techniques of the statistical analysis of process data aimed at detection and elimination of gross measurement and other gross errors

It is also possible to optimise the instrumentation to improve results of balancing All this theory and techniques are now available to improve the quality of mass and energy balancing in process plants

In the plant documentation we can usually meet with process and instrumentation (P&I) diagrams which contain the process equipment and the instrumentation and control loops P&I diagrams are usually too much detailed for balancing purposes We can simplify them by deleting control loops and a part of instrumentation which is not needed for balancing purposes Very often the flowsheet can be further simplified by ignoring of some parts of equipment

or by merging some pieces into one equipment

For example, in the case of mass balance, pumps can be neglected because they have only one input and one output and their mass balance is a trivial one Even in the case of enthalpy balance, pumps are often neglected as the shaft work exerted on the pump is often negligible in comparison with the enthalpy changes of streams Similarly, if only the multicomponent balance is set

up around a distillation column, the column proper is usually merged with the condenser, boiler, or other supporting equipment

The resulting flowsheet whose complexness is just sufficient for the purpose of the balance is finally redrawn in the graph form The graph representation of the flowsheet for the single-component balance shown in Fig 2-1 is in Fig 2-5 The graph consists of nodes (circles) and streams

(arrows) Every stream is incident either with two nodes or with one node and the environment (which can be viewed also as one of the nodes)

Chapter 2 - Balancing in process industries 13

T 13

8

w

Fig 2-5 A graph representation of balancing flowsheet

By comparing Figs 2-1 and 2-5, we can see that the new stream (13) occurred in Fig 2-5 This stream represents the artificial (fictitious) stream - the accumulation of mass in the tank (2) We will meet with artificial streams more

in the next parts of the book

The direct application of conservation laws to the graph in Fig 2-5 in the present form would lead to some problems For example, there are different kinds

of streams incident with the node (3) - mass streams, electric energy streams and the heating gas (12) As we try to set up the single-component balance (only one balancing equation around every node), we must separate streams according to the kind of the stream The final graph representation of the balancing flowsheet

is in Fig 2-6 The original graph breaks into two parts which have the only node common - the environment (the environment node is not shown in the Fig 2-6 explicitly) It is also worth mentioning the stream 12 which is used as the fuel

in the node 3 This stream can't be taken into account in the mass balance of the node 3, so that the direction of this stream is from the node 4 to the environment

14 Material and Energy Balancing in the Process Industries

of nonreaction nodes are so-called splitters with two or more outgoing streams

of the same composition, temperature, etc Some unit operations can be split in two or more fictitious nodes For example, heat exchangers are usually balanced

as tube and shell sides connected by a fictitious stream which represents the heat flux in the exchanger

Reactors are nodes where chemical transformation occurs To maintain the conservation law, the fictitious streams representing sources or sinks of reacting components going to or from the environment can be defined It is also possible to eliminate the fictitious streams by an algebraic procedure expounded

in Chapter 4

Separate units (such as storage tanks) where the material is only stored (and the inventory can be measured) can be called inventory nodes They are each connected by a fictitious stream (accumulation) with the environment

Chapter 2 - Balancing in process industries 15

Process variables which characterise process streams are of different nature (volume flows, temperatures, composition, pH, electric conductivity, etc.)

It is important to transform the primary data to so-called balancing variables

The system of balancing variables must be consistent as concerns the physical dimensions of balancing variables In general, the balance can be set up

in extensive quantities, as mass, or in intensive quantities which are related to a unit of time In the second case the extensive quantities are divided by the length

of the time interval and so-called mean integralflowrates are created In the limit

(for short intervals), they can be regarded as 'instantaneous' and conversely, the former (extensive) as time-integrated

If also energy is being balanced, the streams can comprise 'net energy streams' such as in Fig 2-1 The other streams represent material flow The flow

of energy in a material stream can be expressed as product of the (instantaneous

or integrated) mass flowrate and energy per unit mass The relevant part of the energy associated with the material is its enthalpy; thus the energy factor is specific enthalpy In balancing several chemical components, the factor is component mass/total mass, hence the mass fraction of the component

In this book, we prefer the mass system of variables (mass, specific enthalpy, mass fractions) Nevertheless, also the molar system is worth

mentioning The molar system is widely used in chemistry and classical chemical thermodynamics Based on the elementary idea that matter is composed of molecules, a conventional number of molecules is taken as a unit; traditionally (and later precised for scientific purposes), it is the number of molecules in 32 grammes of oxygen 0 2 , which equals 0.602x1024 (Avogadro's number) This

quantity of matter is called mole, or g-mole more precisely Then 1 0 3 g-moles (1 kg-mole) is denoted as 1 kmol in the system of units consistent with 1 kg as unit of mass

Let generally a chemical species X be assigned chemical formula

E a F b G c where E, F, G, are atoms (atom species) constituting the molecule, with atom masses (atomic weights) M E , MF, MG, "'" , respectively Then the quantity

Mx = aME + bMF + cM C +

represents the number of kilogrammes in 1 kg-mole of species X (thus 32 kg for

X = 02 ) Mx is called the mole mass (molecular weight) of species X or, having been derived from the chemical formula, its formula mass Its physical unit is kg

kmol -~

The molar quantities in chemical thermodynamics are related to unit

mole Thus i f / ) x is specific enthalpy of species X (that of unit mass of the species) then the molar enthalpy equals

16 Material and Energy Balancing in the Process Industries

In a mixture of several components (A, B, C, ), we can introduce the m e a n

or (mechanical) work also electric energy can be considered

Tab 2-1 Consistent systems of balancing variables

kg mass flowrate

mass fraction

mass fraction

specific enthalpy

J kg -I specific enthalpy

J kg -I

a

! heat, work

J

! heat flow rate, power

W

Molar system Generic symbol

C h a p t e r 2 - Balancing in process industries 17

In the case of energy balancing the requirement of consistency means not only the system of units, but also the consistent system of zero-levels for enthalpy, especially for reacting systems All these problems will be dealt with

in next chapters of this book

The transformation of primary variables to balancing variables is a part

of data pre-processing which precedes the balancing proper

Data pre-processing begins with calculating basic statistics, as integration

of flows or averaging of other process variables in the selected time interval After that balancing variables are calculated from pre-processed data This step can range from basic mathematical operations as multiplication of the volume and density to obtain the mass, to calculation of specific enthalpy of streams which can be based on very complex thermodynamic correlations A special sort

of data pre-processing are so-called compensations of instrument readings Typical example here can be the compensation of orifice for the temperature and pressure which depart from values supposed when designing the instrumentation

In practice we can often meet with the situation when underlying data belong to different systems of units which can be even inconsistent The information about composition can be in grammes per litre, the composition of gases can be based on the dry gas, etc The original data must be further re- calculated to the specific consistent set

Consistent composition variables are mass fractions in the mass system and mole fractions in the molar system Quite common are, however, also the component densities expressed per unit volume of the mixture The relations among the composition variables are summarised in Table 2-2 below Let us first summarise the definitions and notation

Ck "'" k-th component of the mixture, of mole mass Mk

m total mass of the mixture, m k mass of C k in the mixture

n total number of moles, r/k .-" number of moles of Ck

V volume of the mixture (m 3 ), P "'" its (mass) density (kg m -3 )

18 Material and Energy Balancing in the Process Industries

partial mass density of Ck "Ok = m k (kg m -3)

V

n k

partial mole density of Ck :Ck -7_ (kmol m-3)

v

We then have the table

Table 2-2 The most common systems of composition variables

is mole density (kmol m -3) of the mixture

Still other composition variables can occur For example if the mixture can be regarded as solution of components C2, C3, "-" in solvent C~, the concentrations of the former can be expressed as mass (moles) of Ck (k > 2) per unit mass (mole) of C~, say Zk Thus for instance in the mass system

k > 2 : Zk = ~ Yk (Zl= 1)

Yl

Chapter 2 - Balancing in process industries 19

Measurement error is defined as the difference between measured and true (usually unknown) value The errors can be classified as

Gross errors (outliers) are large errors occurring from time to time as a result of inattention, measurement devices failures, unsteady state, etc In data

20 Material and Energy Balancing in the Process Industries

measured in process plants, gross errors are important as a single gross error can invalidate all the results, disable process control systems and the like As gross errors may be quite frequent in plant measurements, their detection and elimination is very important

In some situations we have the information about maximum measurement errors Such maximum errors are provided by vendors of measuring instruments (sometimes as the class of accuracy) As the information about measurement accuracy must be provided in a consistent manner, the basic question concerns the relation between the standard deviation and the maximum error of measured value The rigorous answer to this question probably does not exist In practice

we can recommend to take the standard deviation as one half of the maximum error For more discussions about this controversial topic see (Madron, 1992)

After preparing values of measured or otherwise fixed balancing variables which are available before balancing proper, there remains to specify the set of first guesses of unmeasured variables which are supposed to be the result of balancing In the case of single-component (linear) balancing the situation is simple The solution in this case (if exists) is unique and is obtained in the single calculation step regardless of the first guesses of unmeasured variables

If the balancing model is non-linear (all balancing except the single- component one), the first guesses of unmeasured variables are required There are several reasons to provide the best estimates of unmeasured variables available The solution of a non-linear balance is based on an iterative solution of the system of equations If the starting data set is far from the final solution, the number of iterations can be high or the iteration process may diverge It is also known that there exist multiple solutions of which only one is reasonable The situation is even more complicated in redundant systems with some unobservable variables; such situations will be discussed more throughout the book To summarise, it is worthwhile to spend some effort in providing good estimates of unmeasured variables in non-linear balancing

2.5 C L A S S I F I C A T I O N OF V A R I A B L E S

This section deals with classification of variables which has close relation

to solvability of the balancing problem Let's start with some simple examples

of single-component balance shown in Fig 2-7

C h a p t e r 2 - Balancing in process industries 21

Fig 2-7 Balancing graphs - classification of variables

measured flow; u n m e a s u r e d flow

Graph 2-7a represents one balancing equation around one node There is also one unknown flow which can be calculated from the equation The system

is just solvable and the unmeasured flow is observable The two measured flows are nonredundant

In graph 2-7b all three streams are measured The system is redundant

as one of the streams need not be measured and could be calculated All three streams are redundant but the degree of redundancy is only one (as only one measurement is more than needed to make the system uniquely solvable) The flows in a system with all streams measured will not probably be consistent with the balancing model (the sum of inputs will not equal the sum of outputs) This problem is solved by reconciliation which means adjustment of flows to meet the overall balance

In graph 2-7c there are two unmeasured streams The problem (one equation and two unknowns) is still solvable, but the solution is not unique We say that the two unmeasured flows are not observable (they are unobservable)

The only solution of this obstacle is to complete the measurement (at least one more stream must be measured to make the system fully observable)

In practice we can meet with even more complicated situations - see Fig 2-7d Streams 1, 2, 4 and 5 are measured and redundant (one stream can be calculated from the others) Stream 6 is measured, but nonredundant Streams 3 and 7 are unmeasured and observable Streams 8 and 9 are unmeasured and unobservable The general classification of balancing variables is presented in Fig 2-8 Anyway, we can see that even in a relatively simple flowsheet with the

22 Material and Energy Balancing in the Process Industries

most simple balancing model (single-component balance) the situation starts to

be a little bit difficult to interpret

variable

I measured 1 ( unmeasured ) (c~ (fixed)l

J just determined 1

( redundant I [ observable I (unobservable I

Fig 2-8 Classification of variables

The problem of solvability of balancing systems is usually solved in chemical engineering literature by the concept of degrees of freedom Every balancing node (or unit operation) has its own number of degrees of freedom which is the difference between the number of variables and the number of equations among them Full solvability of such a node is achieved by specification of some variables The number of degrees of freedom of the whole system is the sum of degrees of freedom of individual nodes corrected for variables common to neighbouring nodes

This approach is useful in education of chemical engineers and may play some role in balancing in the stage of the design of new processes For general case with possible redundancy this approach is difficult to apply Throughout the present book the analysis of solvability and classification of variables is based

on more general approach - the detailed analysis of the system of balancing equations or on the analysis of graphs in the case of single-component balance

2.6 DATA C O L L E C T I O N AND PROCESSING

If someone meets with the need of setting up the balance of some process based on one set of data, the problem of data collection seems to be relatively simple and straightforward The major problem usually is obtaining thermodynamic properties of streams in the case of energy balance

A different situation can be met in balancing used in daily operation of process plants Typical applications here are

Chapter 2- Balancing in process industries 23

daily mass and utilities balancing for accounting purposes

monitoring of heat recovery systems

monitoring of operation of chemical reactors

regular balancing of separation systems

balancing as a part of control of processes

If the balancing is done on regular basis, the data collection and processing must be well organised The experience in this area shows that the data collection can be hardly based only on manual inputs The systematic use

of process control systems and process databases provides the way of efficient solving of this problems

The modem p r o c e s s information system is shown in Fig 2-9 The basis

of the whole system is the core database of process variables Typical variables which are stored in such database are flows, inventories, temperatures, laboratory data, etc The other data needed for balancing are in a databank of physical properties needed for setting up energy balances The system is based on a Local Area Network (LAN) which makes possible the use of the database by other software and plant staff (clients of the information system)

The data input to the core database can be from Distributed Control Systems (DCS), laboratory systems or other process databases In practice also the manual entries must be available as DCS are usually not available in all plants of the company

DATA I N P U T

(communication with ~ ( import from I ( manual )

DCS systems external databases input

e core of a pro / / (Database of process]

Laboratory N, / Labc ] (Physical property~ ~/

24 Material and Energy Balancing in the Process Industries

Above the core database work so-called applications where belong also

balancing systems They usually import data from the core database, make data pre-processing and the balancing calculations proper After that, balancing results are checked by the administrator of the balancing system for correctness and the results are stored in a special database Balancing data are from this moment available via the LAN to all plant personnel and also to other information systems The raw balancing data are usually presented to users in well shaped form of reports highlighting the most important results of the balance

Such an information system must be integrated, which means that the individual users of the database (applications) know where the relevant data are and are able to use them

of one selected chemical element (irrespective of in what chemical component

it occurs), or also of a component not participating in any chemical reaction This chapter deals with mass balancing

The reader is recommended to peruse Sections A 1-A.4 of Appendix A For the necessary notions of linear algebra, see Sections B 1-B.8 of Appendix B

26 Material and Energy Balancing in the Process Industries

where mj is (possibly integral or integral mean) mass flowrate of j-th stream, 'in' means input, 'out' means output streams j; an is increase (per unit time or per period) of accumulation in the (say, n-th) node A 'steady-state' balance means formally a n = 0 With the above interpretation of an, the change in accumulation (holdup) is then neglected

It is also possible to regard a n as a fictitious rate of mass flow directed outwards; then a n becomes one of the output flowrates mj and the RHS in Eq.(3.1.1) becomes zero

an

(mj)out

Fig 3-lb Accumulation stream regarded as output stream

The fictitious flowrate a n can be positive or negative

In a system of units n ~ N u (set of units) connected by streams j ~ J (set

of streams), the (formally steady-state) balances can be written in the form

j~J

where

Cnj = 1 if stream j is input into node n

Cnj = -1 if stream j is output from node n

Cnj = 0 if stream j is neither input nor output

(3.1.3)

The notation ~ means sum of terms j over set J, independent of the order in the summatioJ~.JThis kind of notation will be used henceforth for convenience,

as in further analysis the sets will be re-arranged in different manners Also the

number of elements of a (general finite) set M is independent of the order and

will further be denoted by [ M I

Recall now Appendix A Clearly, whatever be the index orders the matrix

C of elements Cnj is the reduced incidence matrix of an oriented graph (say) G The set of arcs of G is J and any arc j ~ J is incident to one node n ~ N u at least (Cnj ~ 0), to two,such nodes at most Let us now introduce the node n o in the

Chapter 3 - Mass (Single-component) Balance 27 following manner: the other endpoint of any arc j ~ J that has only one endpoint

in the set N u of units, is no Thus for example the balancing scheme

Fig 3-2b Graph of the balancing scheme

Clearly, the streams incident to node n o , according to the direction, either represent a material supplied to the system, or produced by the system as a whole; hence node n o is called the node environment But observe that formally,

also the fictitious streams such as in Fig 3-1b (a n) are arcs incident to node n o

We thus have identified the graph G[N, J] whose node set is

{no} m e a n s s e t of one element n 0

28 Material and Energy Balancing in the Process Industries

A natural requirement is that G is connected: any node (unit) n ~ N u is attainable by a path, thus also by a sequence of (arbitrarily oriented) arcs starting

at node no; otherwise certain subsystem of units would have no input and no output, which is technologically absurd The immediate consequence for the system of equations (3.1.2) is, by (A.7 and 8)

where I N u [ is the number of elements of set Nu; hence the matrix C is of full row rank If we introduce some orders of elements n s N u and of elements j ~ J, the system (3.1.2) reads

here, m is column vector of components mj (j e J), an element of

t J l -dimensional space (say) q/' The set of solutions is the null space KerC of matrix C, a vector subspace of q/', of dimension I J I-INn I Clearly, the system

as a whole has at least one input and one output, and the graph G contains a circuit, hence

is then, in effect, absent

Let us consider first an arc (stream) jl ~ J that lies on a circuit of the graph G; for example

C h a p t e r 3 - Mass (Single-component) Balance 29

Fig 3-3 A circuit

(recall that the notion of a circuit is independent of orientation) It is then easily shown that

m

if vector m (of j-th component mj) is a solution then if a is arbitrary, there exists always

a solution m such that mj, = r~j, + a The idea of the proof consists in adding a fictitious circulation a along the circuit as indicated in Fig 3-3; in any of the node balances, the circulation term cancels and all the balance equations are again obeyed

Conversely let arc J2 separate the graph, thus J2 lies on no circuit according to the terminology of Appendix A; then if m is an arbitrary solution, we have uniquely mJ,_ = o The conclusion follows formally from (A.9)-(A.13), where Arcd stands for C (3.1.6) We can also imagine Fig A-7 as an example More generally, the graph G is decomposed

./'2

Fig 3-4 Arc J2 separates the graph

The node n o (environment) belongs to, say, subgraph G1, while G2 is connected by just one stream with G i" when balancing G2 as a whole, the input/output results mi_, = o Technologically, such case is absurd; the units of G2 would be constantly out of operation For a correctly written technological scheme, such cases are precluded