lee christensen rudd 1966
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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.
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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.
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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.
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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
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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
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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
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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
top related