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    Design Variable

     Selection

      to

      Simplify

    Process

      Calculations

    WOOYOUNG

      LEE

    JAMES H. CHRISTENSEN and DALE F. RUDD

    University

      of

     Wisconsin

    Madison Wisconsin

    During

      the analysis of a large

      process system

    th e

      vaJues

      of

      cer ta in  design

      variables

      must

    be

      preassigned   to

      allow

      the solution of the design

      equat ions.

      A   careless  select ion of

      these   dis-

    t inguished variables can lead to

      trouble

    magnifying

      c o mp u t a t i o n a l

      prob lems   above

      wha t

      need

    be. A method is   proposed   for determining a set of design

      var iab les

      which could

      minimize

      th e

    computat ional  labor associated with   process  analysis.  This  work   is

      part

      of a larger  project  on

    th e  c om pu te r

      aided

      design of

      process

      systems.

    Process design  is frequently burdened  by   a need  for ex-

    tensive computations; a burden which may not be inherent

    in the

      problem itself

      but

      which

      may

      result  from

      our ac-

    cidental approach to the problem. A careless first step

    may

      lead directly to horrendous computational problems

    which could easily be avoided. This paper deals with the

    strategy

      of making  this  first  step,  the  selection  of  design

    variables.

      A simple algorithm  is  developed which enables

    the  selection of a set of design variables which reduces  the

    computational  labor associated with the process  analysis.

    The process design equations can be expressed in sym-

    bolic  form   as

    fi  to)  = 0

    i =  1, 2,  IV

    / = 1, 2 , . . . . M

     1)

    where N  < M.

    The design equations ft tie together the process  vari-

    ables

      V j

     through material

      and

      energy balances, equipment

    design equations,

      and the

      like.

      The

      equations constitute

    independent sources  of  information  and may  number into

    the hundreds for a typical chemical process. The variables

      flow

      rates, equipment design parameters, temperatures,

    and the like) are even more numerous, the excess vari-

    ables

      affording

      degrees  of freedom for  optimization.

    The

     process analysis

     and

     optimization problem involves:

    the numerical specification o f certain  ke y  design variables;

    the  solution  of the  design equations  for the  unspecified

    state, variables;  the  economic analysis  of the  results;  and

    the

      respecification

      of the

      design variables

     to

      achieve more

    nearly the economic goals.

    We  wish  to determine  a set of  design variables which

    simplifies   the

      solution

      of the

      design equations.

      Our

      atten-

    tion is now  focused  on the first two steps.

    THE

      BIPARTITE  GRAPH

    We represent  the system of equations by a special graph

    which exposes

      the

      structure

      of the

      equations compactly

    and lucidly, and which leads directly to their dissection.

    This  useful notation is the  bipartite  graph of topology  1,

    5).

    Consider

      the

      following system

      of

      equations:

    h   « i >  «2 ,

     v

    z

    )

      = 0

    fa

      to,  0

    4

    , 05) = 0 2)

    /s  o4,

     t>

    5

    , »e 0

      to,»s,  «s) 0

    The corresponding graph is shown in Figure 1.

    Page 11 4

    f-nodes

    v-nodes

    Fig.

      1. The bipartite

      graph

      of

      Equations 2).

    The

     equations form

      the

      / nodes,

      the

      variables  form

      the

    v

      nodes, and an edge  connects the equations to the corre-

    sponding variables.

    The number of edges associated with a node  defines  p,

    the

      local degree

      of the

      node.

      The

      local degree

      of the /

    node /i  is  p f i)  =  3 and the  local degree  of the v  node

    v5

      is

     f > v

    s

    )

      =  2.

    This symbolic notation

      is the key to

      what follows;

     bi-

    partite graphs have certain properties

      of

     extreme interest

    to  the process engineer.

    SELECTION

      OF

      DESIGN VARIABLES

    Generally, it is necessary to  specify  the numerical val-

    ues of certain  of the  variables  in Equation 1) before solu-

    tion

      can

      begin.

      The

      number

      of

      such distinguished vari-

    ables

      is  equal  to the  degrees  of  freedom  of the  system,

    that being

      the

      excess

      of

      variables over equations.

    F =

     M N  3)

    Equation  3)  is valid only when the design equations

    constitute  independent sources of  information, and it gives

    no

     hint

      as to

     which

      F

      variables should

      be

      chosen

      as de-

    sign variables.

    We are faced with a combinatorial problem, for there

    can

      be

     very many different ways

      to

      select

      F

      design vari-

    ables

      from

      N  total variables, not all of which lead to a

    soluble  set of  equations.  A  criterion  is  required  for the

    selection  of design variables.

    Necessary  Conditions

    A

     necessary  and  sufficient  condition  for the  solution  of

    a set of equations is that the Jacobian determinant be

    nonzero

      2,

     4).  While of considerable mathematical sig-

    nificance,

     this condition

     is of

     little direct

     use to the

     process

    engineer, since it is extremely  difficult  to compute the

    Jacobian determinant for a large system. We must look in

    directions other than toward

      the

      classical mathematics.

    A.I.Ch.E.  Journal

    November 1966

  • 8/9/2019 Lee Christensen Rudd 1966

    2/7

    In a  logically consistent  set of IV equations there will  be

    N   unspecified  variables,  and for  each equation  it  will  be

    possible  to  denote  on e  variable  as the  output variable  fo r

    that equation. And, no variable will  be an  output variable

    for  more than  on e equation.  If  such  an  output assignment

    is  possible,  the  equations  ar e  said  to  satisfy  the  diversity

    condition  of Hall  3,

     5),

      a necessary  but not  sufficient  con-

    dition  for the  existence  of a  solution. This condition might

    be   stated

      thus:

      There must  be at  least  K  unspecified  vari

    ables

      associated

      with  each

      an d

      every

      group of  K

      equa-

    tions,  where

      K =

      1, 2 .. . .

     N.

    For

      example,

      the

      bipartite graph

      in

      Figure

      2

      describes

    an insoluble set of equations. Hall's condition

      fails

      at

      K

      =

    2

      where equations

      /i  and fa

      have only

      «i

      associated with

    them.

    desig n vori qbles

    equations

    s ta te

      variables

    Fig

    4.

      Z

      an d

      Vp

    as design

      variables

    The

      fact

      that  an output  set has  been assigned,

      denoted

    by   the  double-headed arrows, indicates  that  the  diversity

    condition

      is

      satisfied.

      In

      addition,

      an

      order

      of

      solution

      is

    suggested

      by the

      direction

      of the

      arrows.

      The

      subgraph

    can be  arranged  as in  Figure 5.

    Fig, 2, The diversity condition  not  satisfied.

    Fig.

      5. The

      precedence order

      of

      equations.

    Armed

      with  the  diversity condition  we  consider  the

    problem  of  selecting design variables which lead  to a

    solvable  set of  equations. Later  we  consider  the  selection

    which leads  to an  easily solvable  set of  equations.

    Assigning an   output set.

      We now

      assign

      a

      direction

      to

    the

      edges

      of the

      bipartite graph,

      a

      step which  identifies

    one of

      many possible orders  of  solution.

    If  a

      variable  Vj is  assigned

      as the

      output variable

      of

    equation

      ft ,

      then

      the  edge

     connecting nodes

      ft and  Vj

     will

    be   oriented

      from

      ft to

      Vj

      and all  other edges associated

    with  f t  will  be  oriented  to ft.

    F =

     7 - 5 = 2

    Fig.  3. The  pipartite  graph  of  Equation

      3).

    Design variables  are not  candidates  for the  output set,

    and

      their edges must

      be

      oriented

      to the

      equations.

      For

    example, consider  the  following  equations  and  those  in

    Figure 3.

    , «

    a

    ) = 0

    , «2 , » s,

     vs)

      = 0

    ,

     «s,

    f a

    fa

    ft  «a,

     «4,

     vs,

    fs   «5 , °6 ,

     07 )

    = 0

    6

      0

    = 0

     4)

    This graph shows  the  precedence order  of the  system

    of  equations  6 ) .  Starting  from  th e  left  an d  preceding  to

    the

      right,

      the set of five

      equations

      can be

      solved

      one at a

    time There  is no  need  for the  simultaneous solution  of

    equations,  a  troublesome task  at

      best.

    When such

      precedence

      order exists,  the  system does

    no t  possess

      a

      loop;

      the

      structure

      is

      said

      to be

      acyclic.

    Should a  loop appear,  the  simultaneous solution  of  those

    equations within

      the

      loop must

      be

      undertaken  7).

    The

      Effects

      of

      Design

      Variable

      Selection

    The  appearance  of an  acyclic structure  in  Figure  5 was

    not

      accidental;  rather,

      it was the

      direct result

      of the se-

    lection  of   t > an d

      g

      as  design variables.  In  this section  we

    demonstrate  the   effects  of  design variable selection  on

    the  structure  of the  remaining computational problem.

    For  example, consider  the  following system  and  that  in

    Figure  6.

    /i

      « i ,

      t>

    2

    , « s)

      = 0

    fa  »8 , »4, u 5 )  = 0 5)

    fa

      vs,

     «e,  »i) = 0

    If

      «2  and  1 3   are

      selected

      as the  design

      variables,

      the

    subgraph

      in

      Figure

      4  might result.

    F  -

     6 3

     * 3

    Fig.  6. The bipartite graph of  Equations 4).

    Case  1.  Coupling Variables

      vi,

     v3,

      and  u5  are  Design

    Variables:

      Once

      numerical values  are  assigned  to  these

    three variables  the

      bipartite

      graph reduces  to  Figure  7.

    The

      equations

      can be

      solved

     one at a

      time,

      and the

      order

    of  solution  is  immaterial.

    Vol 12 No. 6

    A.I.Ch.E.  Journal

    Page

      11 5

  • 8/9/2019 Lee Christensen Rudd 1966

    3/7

    2^  VM J  V6,

    Fig 7 Vi

    Vs

    an d

      V s

      as   des ign va r iab les

    Case  2;

      Noncoupling

      Variables  t > 2 U 4   an d   €6  are De-

    sign

      Variables: Th e  bipartite graph becomes,  fo r example,

    Figure

      8

    This unfortunate selection

      of

      design variables

    Fig.  8. Vs  V j and  g as  design var iables.

    leads  directly  to a set of  equations which must  be   solved

    simultaneously. We  must  now be  troubled with  the de-

    velopment  of an  appropriate iterative scheme  of  solution,

    with   al l  the  problems  of  convergence  and the  like.  In the

    bipartite graph  th e  order  is

      no t

      even unique, that  is the

    directions may be reversed.

    Case

      3;

      Variables

      Vi, Vt,  an d  v  ar e

      Design Variables:

    Th e  bipartite graph  is now  acyclic  and the  information

    flow

     is

     unique Figure

     9 .

     Equation

     /

    3

     can be

      solved

     first

    for   variable  v ,  which then allows  the  solution  of  /a  fo r

    variable

      t > 3

    which

      in

      turn enables

      the

      solution

      of

      equation

    fi .

      The

      happy situation

     of

      case

      1 has

      appeared again,

     in

    Figure

      9.

    Fig.  9.

     Vj

    V4 and Ve as

     design variables

    W e  temporarily assume that acyclic graphs  are of the

    same

      difficulty  equivalence class, and  that cases  1 and 3

    constitute  equivalent selections  of   design variables. This

    is

      no t  strictly  true however, since certain equations  ca n

    be

     solved more easily

     in one

      direction than

      the

      other.

      For

    example,  if we  have only  a  table  of sin  x,  it is  easier  to

    solve

      y =  sin  x  fo r  y  in  terms  of  x,  than  x  in   terms  of  y.

    The same analogy holds true in  process calculations.

     Later,

    this

      is

      accounted

      for by the

      assignment

      of

      difficulty

      scores

    in  a

      report

      yet to be

      published.

    SELECTION  ALGORITHMS

    It has  been established that  the  careful selection  of a

    set of design variables  can  alter  the   complexion  of  process

    design calculations.

     We now

      busy

      ourselves with altering

    the  problem  in our  favor, toward  a  structure yielding  an

    easily   solved  set of  equations.  A  simple algorithm  for de-

    sign

      variable selection will  be  established.

    A  Selection Criterion

    Generally the  labor  of   solving  a set of  design equations

    increases  with

      the

      number

      of

      equations that must

      be

    solved  simultaneously. W e therefore,  define  th e  following

    criterion  for the  selection  of   design variables:

    A  best  set of  design variables results  in a  structure  in

    which the  largest number of  equations that must be solved

    simultaneously

      is minimized.  A corollary  of  this criterion  is

    that  a  best  set of  design variables results  in an  acyclic

    ordering

      of the

      design equations,

      if

      such

      an

      ordering

      is

    possible Hence,  if the  difficulty  of  solving  any  equation  is

    independent of the  output variable  and an  acyclic assign-

    ment  is  possible, this criterion  may  give  the  minimum

    total

      effort

      fo r  solving  the set of design  equations.*

    An   algorithm  is now  presented which selects  a set of

    design variables,

      yielding

      an   acyclic ordering  of the de-

    sign

      equations,  if  such  an   ordering  is  possible.  The al-

    gorithm

      is based  on the  following observation:

    Th e  graph  of an  acyclic system must contain  at  least

    one  v  node  V j  with  P(VJ)  = 1 and one  /  node

      t  with

    p(fi)

      = 1. This  is so  because every  directed path  in an

    acyclic  graph must terminate,  by  definition,  at a  v  node.

    Such  a  path cannot terminate  at an

      /

      node,  since

      each

    / node

     has one

     outgoing edge; therefore,

      it

     must  terminate

    at

     a

      v  node. Furthermore, since

      a

      v  node

      can

      have only

    «

      In a

      report

      yet to be

      published,

      we

      examine

      the

      validity

      of

      this

    criterion

    an d

      show  that

      in situations

      where

      the

      difficulties

      of

      solving

    parts of the  process design  differ  radically  the   concept  of   difficulty  scores

    must be introduced.

    START

    Y es

    Assign  attached v-node

     as

      output

    of

      f and

      delete both nodes  from

    g r apn .

    Assign  v.  as   output  of   attached

    f - node ;  delete both nodes  from

    graph.

    v.

      is a  design variable;

    1  delete  it.

    f.  is   redundant o r  inconsistent;

      delete it .

    Yes

    System  is

    nonacyclic

    STOP

    Fig 10. Flow   diagram of algorithm I.

    Pa ge   1106

    A I Ch E Journa l

    November 1966

  • 8/9/2019 Lee Christensen Rudd 1966

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    one  incoming edge this

      v

      node must have exactly  one

    edge,  if a  path terminates  at  this

      v

      node.

    If   this

     v

      node and the /  node  of which it is an  output

    are  deleted  f r o m  the  graph the  resulting subgraph must

    also

     be  acyclic; hence it  will also contain  at  least  one  v

    node and one /  node with  one edge each.

    Finally we note that  any /  node connected  to  only  one

    v  node must necessarily have that  v  node  as its  output.

    W e  then have t he  algorithm  fo r output  an d design variable

    selection shown

     in

      Figure

      10.

    Example

      of

      Algorithm

      I

    Suppose  that  we  have  the  system given  in  Figure  11.

    Th e  design equations  of the

      system

      ar e  also given.  W e

    apply algorithm

      I to the

      graph

      of

      Figure

      12. The

      system

    consists of a   mixer a  reactor and a  separator. There  ar e

    five streams in the system each described  by two  variables;

    the  mass flow rate  q  and the  composition  c.  The  other

    variables  in the  system  are the  reactor volume  V, the re-

    action  temperature  T and the  separation ratio  S.

    System:

    ©

    Feed

      »

    ©

    Mixer

    © 

    Reactor

    ®

    Separator Product

    ®

    Fig 11 A simple

      process

      system

    Design Equations:

    Mixer:

    fi

      qi

      +  q & —   < 7 — 0  total mass balance

    /2

     qi

    c

    i

     

    qs

    c

    s

     —

     q z = 0  component balance

    Reactor:

    /s  < 2  —

     qa

      =  0  total mass balance

    ft   q C 2 — fsCa —

     R(CS, T

    V

    =

    component balance

    R   is the  reaction rate relationship

    Separator:

    /5

      c /s  — qi  —  q &  = 0  total

      m a s s

      balance

    f e   73^3 ~

    94^4

      —

     < 7 s C s

      = 0  component balance

    /7

     S =

      Cs/Ci  separation ratio

    .The  variables

      o f

      interest

      ar e

      qfs

      an d

      c/s

    and V, T, and

    S.

     Th e

      number

      of

      design equations

      is

      IV =

      7. The

      number

    of   variables

     is

     M

      = 13.

    Graph:

    F =

      13 7

     

    6

    Fig

    12 The   bipartite graph  far the  system

    Step

     1:

      Look

     for

     nodes with

      the

      local degree

      of 1.

    Step 2:  Assign  c\ V and S to  fa /4,  and  /7,  respec-

    tively.

    Step

     3:

      Eliminate these

      from  the

      graph

      to get

      Figure

    13.

    Repeat steps

      1 to 3:

     Figures

      14 and 15 are

      obtained.

    Design variables:  T c2, c4,  cs  qa  and < 5

    Order

      of

     calculations:

      See

      Figure

      16.

    Fig

    13.

    Fig.  14.

    Fig

    15.

    Fig

    16 The

      ordering  of Figure

      12

    Preferred

      esign

      Variables

    For

     purely

      practical  reasons the  engineer  may

     prefer

      to

    have  close control

      over

      certain variables

    and may

      elect

    to   elevate such preferred variables  to the  level  of a  design

    variable regardless  of the  logical structure  of the  problem.

    Fo r   example if   quality control restrictions  ar e  imposed  o n

    a  product concentration the  engineer  may  wish  to  com-

    pute

      only  designs which meet those restrictions. This

      can

    be  achieved  by  selecting that concentration  as a  design

    variable

    the

      value

     of

     which

      is

      free

      to be

      selected directly.

    This preference

      offers

      no  difficulty.  Such preferred vari-

    ables

      are

      assigned prior

      to the use of

      algorithm

      I. If a re-

    striction exists on the flow rate  into  the  reactor in the pre-

    F -5

    Fig

    17 Preferred

     design variables

    Vol

    12 No. 6

    A.I.Ch.E.

      Journal

    Page

     1107

  • 8/9/2019 Lee Christensen Rudd 1966

    5/7

    vious

      example,  q2  might  be a  preferred variable.  The se-

    lection

      of  7 2   as a

      design variable consumes

      one  degree  of

    freedom   and

      results

      in

      Figure

      17, a

      graph more easily

    reducible by algorithm I.

    Irreducible  Systems

    When algorithm  I  terminated, either  an  acyclic output

    assignment

      was

      produced,

      or

      p A )

      — 2 for any

      node

      A

    in the

      remaining subgraph.

      If an

      acyclic assignment

      is

    possible, then the latter condition is impossible; this can

    be proved by contradiction. If the number of / nodes of

    this subgraph equals

      the

      number

      of  v

      nodes,

      our

      degrees

    of

      freedom  have  been  consumed  and we are not  free  to

    select more design variables.  A  computational scheme  fo r

    this

      subgraph has been already determined and the in-

    formation

      flow can be

      traced

      out.

    Should   the  degrees  of  freedom  not be

      fully

      consumed,

    some  more design variables are to be selected; however,

    an   acyclic structure cannot be

      attained,

      regardless of our

    selection  of the  remaining design variables.  Th e  informa-

    tion  flow can be  changed acyclic only  by  tearing some o f

    state variables. Suppose that if one assigns outputs to  k

      nodes

      an d

      deletes these nodes

      an d

      their outputs  from

    the graph; then an acyclic assignment of the remaining

    subgraph

      may be

      possible

      by

      selecting appropriate

      de-

    sign

      variables, where

      k  is  always

      less than

      the

      number

      of

    /

      nodes

      of the

      graph.

      It is

      said

      that  k

      tears

      are

      supplied

    to  the

      system

      and

      that

      the

      computation might

      be

      per-

    formed

      by a k dimensional iterative  scheme.  We

      wisn

      to

    solve

      the

      system with

      the

      minimum number

      of

      tears  k ,

    since  the  computational burden  of  iterations increases

    with

      the  dimension.

      Algorithm

      I cannot possibly find an

    acyclic ordering unless there  is at  least  on e  v  node with  a

    single edge  in the subgraph, This observation leads to the

    following

      algorithm.

    Algorithm  II

    1. Apply algorithm I to the graph of the system.

    2. If  algorithm  I  succeeds  in an  acyclic ordering,

      then

    fc* = 0;  otherwise,  le t  k  min   P (VJ )  — 1, for  t  in the

    vi

    remaining  subgraph.

    3. For

      each combination

      o f

      k f

      nodes giving

      at

      least

    one

      v

      node with just one edge upon their deletion:  1)

    Apply algorithm  I to the  resulting subgraph;  if no  acyclic

    assignment  is

      produced,

      go on to the

      next combination.

      2)

      If an

      acyclic assignment

      is

      obtained,

      an d

      k

      of the

    variables selected as design variables can be assigned as

    outputs

      of the

      deleted  j  nodes, then

      a

      minimum number

    of

      tears  k* = k

      has

      been  obtained; otherwise,

      go to the

    next combination.

    4. If all

      combinations

      of k f

     nodes have been  exhausted,

    increment k

     by one and go to

      step

      3.

    A s

      an example, consider applying algorithm II to the

    simple system given

      in Figure 18.

      Obviously, algorithm

      I

    cannot  reduce the size of the graph. We then have min

    P(VJ )

      —  2;

      hence, start with

      k  1.

      Deletion

      of  fi

      yields

      ig 19

    the  following

      sequence

      of  edge

      assignments

      by

      using

    algorithm   I,

      shown

      in Figure 19.

    Now

      «2

     can be assigned as the output of

     /i

     and fc*

     

    1.

    The

      system

      can  then  be

      arranged

      as in

      Figure

      20

      with

    design variables  V 4  and

      « g.

      In  this case,  a  minimal  tear

    w as   obtained at the first try; however, for large  k the

    number  of combinations  to be  considered  may become  ex -

    cessively large. It is hoped that further investigation may

    yield some way to direct the search in order to reduce the

    number  of  cases which must  be  examined.

    F i g .  20.

    ORDERING THE  PROCESS  FLOW  SHEET

    Algorithms   I and II

     were designed

      to aid the

      computer

    during the

      analysis

      of the design equations. We now ex-

    tend these ideas to aid the engineer in the analysis of the

    process

      flow

      sheet, simplifying

      the

      process calculations.

    Only

      a

      slight extension

      is

     required

      to

      make this transition.

    Figure 21 is the process flow sheet for an example pro-

    cess.  The blocks represent unit operations, distillation,

    compression,  filtration,  heating,  and so  forth,  and the ar-

    rows

     trace  out the  direction  of flow of material  an d energy

    between the blocks. Notice that the direction of physical

    flow does not necessarily indicate the direction of infor-

    mation  flow; the

      specification

     on the

      output product

      from

    stage  M  actually transfers

      information

      back into  th e  sys-

    tem  8).Indeed,  we take advantage of the nonunique-

    ness  of the  connecting arrows  to  reorder  an d  simplify  the

    system. In its present  form,  the specification of the  values

    of  the design variables leads directly to cumbersome

      itera-

    tive recycle calculations, a condition we shall avoid by the

    reassignment of new design variables.

    Fig.  18 Application  of  algorithm  II

    Fig

    21 A   process   flow  diagram.

    A   block  in the  process  flow  sheet  may be  thought  to

    embrace

      a

      number

      of

      design  equations.

      For

      example,

    should

      the

      block

      represent  a

      multicomponent distillation

    tower

      it may

      embrace

      the

     hundreds

      of

     equations

      required

    for   an  adequate description  of  that  operation. To apply

    Page 1108

    A.I.Ch.E.

      Journal November 1966

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    the

      algorithms

      in

      their present  form

      to

      such problems

      is

    unnecessary.

    Notice that once the input variables to a block are

    specified

      the  output variables  can be  solved for; this  is

    the usual convention in flow sheet preparation. Thus, we

    may  think  of a  block  as a  special equation which  can

    possess  several output variables, equal  in  number  to the

    number

      o f output arrows

     f rom

      the  block  in the flow   sheet,

    but not necessarily those particular variables. With this

    interpretation

     we are in a

     position

     to

     apply

      the

     algorithms

    directly.

    Fig. 22. An undirected flow diagram.

    Fig

    23

    Figure 22 is the undirected counterpart of  Figure  21

    with  the  number  of allowed outputs  affixed The  variables

    which appear in only one block are denoted as outputs in

    Figure 23 with the allowed outputs reduced by the ap-

    propriate

      amo unt

    Those blocks with

     no

     more allowed out-

    put are eliminated, and any new variables which now ap-

    pear

      in

      only

      one

      block

      are

      then denoted

      as

      outputs  from

    the reduced graph Figure 24). When there are more

    single  block

      variables than allowed outputs  for a  block,

    one  has a choice which can

      only

      be made rationally by

    physical insight into the type of operation occurring in

    the block.

    Finally,  af ter  all outputs have been assigned this in-

    format ion  is placed  on the original undirected graph.  Any

    variables without assigned direction are design variables

     Figure 25), Notice how this assignment has eliminated

    all  recycle calculations and how the blocks now can be

    handled one at a time in the order prescribed in  Figure

    26,

    Fig.  25 A   reassignment  of   design   variables for

    Figure 21.

    Fig 26 A   precedence ordering

    Fig

    24

    FINAL  COMMENTS

    We

      have demonstrated

      a

      systematic approach

      to the

    selection of design variables based on the criterion that

    acyclicity is desirable. System acyclicity

      will

      be obtained

    by use of  these algorithms  if it is  obtainable merely  by

    selecting

      the

      proper design variables. However,  there

      are

    several points which require

      further

      comment.

    It was implied in this report

      that

     information flow re-

    versals  can be made  at any  equation  or any  system com-

    ponent without  difficulty This is not

      true.

     It is well known

    that  difficulties  are

      often

      encountered should one attempt

    to

      ompute  certain variables, such as the kind of plates to

    use in a distillation tower, and that it is better to  spe ify

    such variables  and  compute  the  consequences.  The  direc-

    tion

     of

     information

     flow in the

      system determines

      the

      ease

    of  component computation. Thus, in certain situations it

    may pay to tolerate cyclic structure in the system to ease

    the computation of a troublesome component.

    However, before

      one can

      begin

      to

      understand

      the im-

    plications of partial acyclicity and the role

     that

      component

    diffi ulty  s ores

     may

      play

      in

      systems analysis,

      the

      limited

    case

      discussed  in  this

      paper

      must  be

     presented. Hence,

     in

    this report the problem of systems analysis with

      completely

    Vol.  12 No. 6

    A I Ch E

    Journal

    Page 1109

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    7/7

    unhindered information flow reversals and equal compo-

    nent

      difficulty

      plays  an  important role  as an  introduction

    to the more complete problem.

    In subsequent reports we will discuss the ordering of

    recycle calculations,

      the  effects  of

      direction dependent

    component  computational  difficulties,  and the  implications

    of

     partial acyclicity. These  topics, along with the methods

    presented in this first report, may lead to a more orderly

    approach  to complex systems calculations.

    Obviously, the  bipartite graph  is

      only

      useful  as a

    method  of  developing  the  algorithms,  and in a practical

    problem the ordering of the equations and the selection

    o f

      design variables would be performed without the use

    of

      the  graph, applying  the  concept  of list processing  to a

    list of

      design equations

      in

      which

      the

      occurrence

      of

      each

    variable

      has

      been noted.

    ACKNOWLEDGMENT

    Th e

      authors  wish

      to

      thank

      Donald

      Engelstad

      fo r

      valuable

    suggestions  and discussions.

      This

      work  was

      supported

      in

      part

    by a

      grant

      from  the National Science

      Foundat ion .

    NOTATION

      i

     

    composition  of  stream

     i

    F   —  degrees  o f  freedom

    /i

      =  z'

    th

     function

    k  =  number

     of

     tears

    k

    f

      =  minimum number  of tears

    M  =  total number

      of

      variables

    N

     

    total number of equations

    qi

      =  m a s s

     flow rate of stream

     i

    R c, T,V)  =  reaction rate relationship

    S

     

    separation ratio

    T

      =  reaction temperature

    V

     

    reactor volume

      j 

    /th  variable

    p ( A )  =  local degree  of node  A

    LITERATURE  CITED

    1. Berge,

      Claude,

      The  Theory of Graphs and Its Applica-

    tions, Wiley, N ew

      York  (1962).

    2.

      Buck,

      R.

      C.,  Advanced  Calculus, McGraw-Hill,

      New

    York  (1956).

    3.

      Hall,

      P.,

      ].

      London

      Math Soc.,

      10, 26

      (1934) .

    4.  Hildebrand,  F. B.,  Advanced  Calculus fo r A pplications,

    Prentice-Hall,

      Englewood  Cliffs,

      N. J.

      (1962).

    5.

      Ore,

     O.,  Graphs  and Their

      Uses, Random   House ,

      New

    York  (1963).

    6.

      R udd ,

      D.

      F.,

      and D. V.  Steward,  paper presented  at

    A.I.Ch.E  La s  Vegas meeting  (Sept., 1964).

    7.  Steward, D.

      V., SJAM

     Rev., 4, 321

     (1962).

    8.

      Rudd,

      D.

     F.,

     and C. C.  Watson,

      Strategy

     in

      Process

      Engi-

    neering, Wiley,  New  York ,  to be

      published.

    Manuscript

      received

      January  28,

      2966;  revi

    1966;

     paper accepted  M ay   3, 1966.

    lision  received April 28,

    Rate ata

      and

      erivatives

    NOEL

      DE

     NEVERS

    University of Utah Salt Lake City Utah

    In

      m o s t

      rate

      process

      p ro b lems the original experimental

      data

      must be di f ferent ia ted to ob-

    tain

      the derivatives

    w h i c h

      are the   i n fo rmat i o n   a c t u a l l y   so ught .  For first  derivatives   all the

    c o m m o n l y  used methods   give  a b o u t  th e  sam e  result . However the  c o m m o n   metho d s o f  calculat-

    ing derivatives can   give  very  d i f ferent  second  derivatives for the

      same

      set o f  starting  data.

    Thus in   problem s   in

      whi c h

      th e   desired

      result

      is a   secon d

      derivative

    th e   an sw er

      obta ined

      is a

    strong

      funct ion

      o f the metho d   used  to

      c a l c u l a t e

      second  derivatives.  This   is   i l lustrated  by

    several examples.

    Churchill  1)  has pointed out one important similarity

    between heat, mass, and momentum

     t ransfer

      and reaction

    kinetics.

      In all of

     these

      fields, the

      engineer

      is

      interested

     in

    determining rates (heat, mass, momentum

      t ransfer ,

      and

    chemical reaction

      rates).

      These  rates  are  generally  ob-

    tained

      by differentiating integral data. For example, the

    momentum  transfer  rate  to a  particle  ( f o r ce )  is  propor-

    tional to the acceleration. The most common way  14)

    to

      obtain such accelerations is to  differentiate  position-

    time  (integral) data twice

      to

      obtain  d?x/dt2.  Similarly,

    chemical  reaction rates are determined by differentiating

    composition-time (integral) data and heat transfer rates

    by differentiating  temperature-time  or  temperature-dis-

    tance data.

    In some cases these rates can be obtained  from  steady

    state experiments,  for  example, steady state heat transfer,

    steady state  mass  transfer, chemical reactions at constant

    composition  (continuous  stirred-tank reactors), or  f rom

    force  measurements on supported bodies. Even these data

    are

      of ten  differentiated

     to

      obtain more interesting informa-

    tion;  for

      example, steady state chemical reaction data

      are

    differentiated  with respect to space velocity to elucidate

    kinetics  (16).

    Thus, in all these fields the desired derivatives are

      ob-

    tained  from  integral data. Given  a table  of x-y  data, there

    are several widely known ways to find

     dy/dx

      and  d2y/dx2.

    As

     will be shown

     here,

     all give practically the same values

    of  dy/dx,  but  they  can

      give very  dif fe rent  values

      of