Chapter 8 Solvability and classification of variables II

201

Chapter 8 SOLVABILITY

AND CLASSIFICATION

OF VARIABLES

II

N O N L I N E A R SYSTEMS

When also chemical components and/or energy are included in balancing, the system of equations is generally nonlinear. From the mathematical point of view, the analysis then brings numerous problems not completely resolved so far. In practice that means that a method of solution (involving possibly reconciliation of measured data) can fail; the reader perhaps knows from his own experience that this is not an uncommon case in numerical mathematics, when treating nonlinear problems. Along with the (unavoidably incomplete) analysis and classification, our aim is to illustrate what can be the reasons of such failure. The topic is a matter of further research. To the best of the authors' knowledge, the material presented in this chapter is largely quite novel. We have, therefore, at least outlined also certain rather lengthy mathematical proofs. They are fine-printed and the results summarized for a reader not interested in details. The reader is anyway recommended to peruse Sections A.1-A.4 of Appendix A and B.1-B.8 of Appendix B, and of course Chapters 3, 4, 5, and 7.

8.1

SIMPLE EXAMPLES

Let us consider blending of two streams (1 and 2) of mass flowrates mj, mass fractions y~ of k-th component, and temperatures Tj . Possibly, the blending unit includes a heat exchanger; for convenience and in contrast to the general notation in Chapter 5, let Q be the heat withdrawn (Q < 0 if heat is supplied). Stream 3 is the output stream. Q

2

T "- 3 r

-q Fig. 8-1. Blending of streams

202

Material and Energy Balancing in the Process Industries

If we have two components in the mixture, we can determine the composition by the mass fraction of one selected component, thus 3) in stream j. Pressure P is considered constant and known, thus the thermodynamic state variables are yJ and T j in stream j. The set of balance equations reads m~

+ m2

- m3

m l y l + m2y2 _ m 3 y 3

= 0 = 0

(8.1.1)

ml hi + m2h 2 - m 3 h 3 - Q = 0 where hj ( = / T with the notation introduced in Section 5.1) is specific enthalpy of j-th stream, a given function of 3~ and T j, given P. We assume m~ > 0 and m 2 > 0. Then the nonlinear system (8.1.1) in variables mj, 3;, T j (j - 1, 2, 3), and Q is solvable: taking arbitrary values of m~, m 2 , yl, y2, T 1, T 2, Q the solution reads m 3 = m I + m 2 (> 0)

(8.1.2)

thus y3 =

(m 1yl + m2y2 )

(8.1.3)

( m 1 h 1 + m 2 h 2- Q)

(8.1.4)

ml+m2

and with h3 =

m~+m2

the RHS is given function of the above variables, where y3 follows from (8.1.3), and there remains to resolve the equation in unknown T 3 /_~(T3; y3 ) = h 3

(8.1.5)

with (by hypothesis known) specific enthalpy function/4. Here, given y3 the specific enthalpy is increasing function of temperature, thus T 3 is uniquely determined. Let us, however, precise the latter point. The mass flowrates mj are subject to the only condition mj > 0 .

(8.1.6)

For the mass fractions, we suppose 0 < yJ < 1 .

(8.1.7)

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

203

But taking for instance yl = 0 or y~ - 1 the stream 1 is, in fact, a single-component one and the problem is modified; yl is no more a variable of the process. Let us rather adopt the condition 0 < 3~ < 1

(8.1.7a)

thus 3~ has to lie in an open interval. Then, if (8.1.7a) holds for j - 1 and 2, it holds also for j = 3 by (8.1.3), as is readily shown formally. We suppose of course T j > 0 (absolute temperature). But the enthalpy function is continuous within certain limits where the medium does not change its state of aggregation; also the technological process admits some range of temperatures only. When necessary, the heat withdrawn (or supplied) allows one, by (8.1.4), to maintain the h3-value, thus the solution T3 of (8.1.5) in these limits; imagine mixing of water with concentrated sulphuric acid. Generally, in the system of balance equations we suppose a priori that the variables' values are restricted to certain intervals, thus to a multidimensional interval in the whole variables space. The enthalpy function/q is assumed to be infinitely differentiable (imagine a polynomial expression). Then, as can be shown formally, the solutions (8.1.2 and 3) and that of (8.1.5) with (8.1.4) are infinitely differentiable functions of the (chosen) 'independent variables', in number 2 + 2 + 2 + 1 = 7 for 3 'dependent variables' (our solutions). The special partition of variables into 'independent' and 'dependent' as chosen is not critical. The set of equations (8.1.1) can be written as g(z) = 0 (z ~ U)

(8.1.8)

where g is an (infinitely differentiable) M-vector of functions (M = 3), z is the vector of N (= 10) variables mj, 3~, T j, Q, and U is the multidimensional interval of admissible values. The equation is solvable, thus the set M of solutions is nonempty. The fact that the general solution can be found as (infinitely differentiable) function of N-M (= 7) variables (components of z) is expressed, in mathematical language, by saying that M is a 'differentiable manifold', of dimension N-M. It is called the solution manifold of the system (8.1.8). Another example of solution manifold is the linear affine manifold (7.1.8). The dimension (N-M) is, in less formal language, the number of degrees of freedom of the system, thus the 'number of independent variables'. In the example, in accord with the intuition, it equals the number of variables ('dimension of the problem') minus the number of equations ('constraints'). Such set of constraints can be called regular; see later for a precision. It is to be stressed immediately that not any N-M (= 7) variables can be chosen as independent; for example including the three variables ml, m2, m3 among the seven independent ones is erroneous, because they are subject to the condition (8.1.2).

204

Material and Energy Balancing in the Process Industries

The simple picture changes when some of the variables (components of z) are fixed, for example as measured values. Generally, if Zn is the n-th component of z, let Zn be a fixed value. Let us examine the solvability conditions; they read generally z e 914(solution manifold) with certain fixed values Zn" In the system (8.1.1), for example with fixed mass flowrates one of the solvability conditions reads (8.1.9)

/'h 1 -I- /'h 2 - /'~'/3 - - O .

Let us rewrite the system (8.1.1) as m 1

+ m2

= m3

m l (yl_y3)

+ m2 (y2_ y3 )

- 0

m l(hl-h 3)

+ m 2(hz-h 3)

-Q

(8.1.10)

and let us consider less trivial examples.

Example 1 Let us have measured the mass fractions 3: and temperatures 7j, thus also the specific enthalpies hj are known. Let also Q be known, for example having measured the mass flowrate and inlet-outlet temperatures of the cooling (or heating) medium. Then the conditions read, having eliminated the variable m 3 m, Q~l_~3 ) -I- m 2 (~2_~3) _ 0 m, (~,_~3) + m2 (~t2_~t3) _ Q and

(8.1.11)

m~ >O, m 2 > O . The inequalities correspond to the general condition z ~ '// in (8.1.8). For example if it happens that ~ = ~3 then, by the inequalities, we have also ~2 _ ~3 as a solvability condition; precluding this case, the inequalities imply only (.~l_y3) (~2_~3) < 0

(8.1.12)

because m~/m2 > 0, hence the algebraic signs of the differences must be different. Suppose the inequality is satisfied. If now Q , 0 then the solvability condition reads ~'-/93 )(ktz-kt3 ) ~ (.92-~3 )(h'-kt 3 ) for 0 ~ 0

(8.1.13)

Chapter

8 -

Solvability and Classification of Variables H- Nonlinear Systems

205

(linear independence). Let also this condition (inequality) be obeyed. Then the equations are uniquely solvable in the mj. If we had for example m~ = 0 then, according to (8.1.11)~ and (8.1.12) also m 2 - 0 , which would contradict the second equation with ~) g: 0. Hence the solutions m~ g: 0 and m2 g: 0. If it happens that some of the flowrates becomes negative, our data are simply wrong (physically absurd) and such set of measurements has better to be discarded. [By the way, such case could occur as well in the solution of mass balance equations in Chapter 3.] Precluding this case, the given values determine uniquely m I and m2, thus also m 3 ( > 0 ) .

Example 2 Let further the heat exchanger be absent. Then the condition of nonnull flowrates mj reads

Q~l__~3 )(~2__~./3 ) __ Q,~2__~3)(~1__~3 )

for ~) - 0 ;

(8.1.14)

else the equations (8.1.11) have the unique solution m~ = m2 = 0. The absence of the heat exchanger means that ~) does not enter the set of adjustable values. The solvability condition then takes the form of a (nonlinear) equation. With (possibly adjusted) measured values, one of the equations (8.1.11) is eliminated. Assuming again the inequality (8.1.12), we can eliminate the second equation and by (8.1.11)1 with (8.1.2), we have the homogeneous linear system m 1 +

m2

(.91@3)ml + @2@3) m 2

- m 3 =0 =

0

} (8.1.15)

of rank 2, thus the solution manifold is a onedimensional vector space. The equations (8.1.14) and^ (8.1.15) represent a decomposition of the system (8.1.2) and (8.1.11) (with Q - 0) into the solvability condition and those for the unmeasured values.

Example 3 If we add also the variable m 3 to the set of the measured ones, the elimination of Eq.(8.1.2) by (8.1.11) subsists without modification and we arrive again at the condition (8.1.14), whatever be m 3 = rh3 . Then the set of constraints is solvable if and only if the condition (8.1.14) is obeyed, with arbitrary rh3 , and on substituting rh 3 for m 3 in (8.1.15) 1 the variables ml and m2 are uniquely determined.

206

Material and Energy Balancing in the Process Industries

There is a close analogy with the classification of variables in Chapter 7. If m 3 is not measured one would call the variables m~, m 2, and m 3 unobservable; if m 3 = rh 3 is measured then m 1 and m 2 can be regarded as observable, and m3 as nonredundant. The values of measured composition and temperature variables have to belong to the set (say) 914+ determined by the condition (8.1.14). With (8.1.12), the condition can be rewritten

~2 = ~t3 +

~ 2_~3

143

(~t~_~t3)

(8.1.16)

hence applying the same consideration as above, for example temperature ~.2 can be expressed as function of the remaining (3+2 =) 5 variables ~ , ~2, ~3, i,~, ~,2. We conclude again that 914+ is a (differentiable) manifold, of dimension 5. If compared with the linear case there are, however, also essential differences. In Chapter 7, the classification was independent of the variables' values. But now, (a)

The classification is unambiguous only if the admissible flowrates are restricted to nonzero ones and then, in addition

(b)

The solvability condition fails as insufficient when the measured variables take certain special values.

Indeed, concerning point (a), if m 3 is not measured and if we admit zero values for m~ and m2 then Eqs.(8.1.11) with Q - 0 are always solvable by m~ = m 2 = 0 thus also m 3 = 0 , and we have no solvability condition; if m 3 = rn 3 is given then the condition reads either rh 3 = 0, or (8.1.14). Concerning point (b), if for example kt1 = kt2 = kt3 then the condition (8.1.14) is obeyed identically; but if also ~1 _ ~3 then, by the requirement that m2 > 0, we must have also ~2 = ~3. In the latter case, thus admitting ~1 _ ~3, the condition (8.1.14) is insufficient and we, in fact, do not know which of the measured values has to be adjusted and which left fixed, or also according to which criterion. We then can say that admitting ~1 _ ~3 somewhere in the a priori admitted region r variables, the problem of adjustment is 'not well-posed'. In other terms, if it can happen that the mass fractions in the streams according to Fig. 8-1 are equal (or only slightly different) then determining the mass flowrates from measured concentrations is inappropriate. See Section 5.5 for another example of such 'ill-posedness'. In complex systems, less trivial cases than (a) and (b) above can occur.

Chapter 8 - Solvability and Classification of Variables II- Nonlinear Systems

207

Example 4 It is not easy to give simply analyzable examples that should illustrate all mathematically possible cases, and would not be technologically na'fve at the same time. As another example, let us consider again the system (8.1.1), equivalent to (8.1.10). Let us have fixed (measured) the variables yJ = (j = 1, 2, 3), T j - 7"J (j = 1 and 3), and Q = Q; thus T 2 is now unmeasured. Let us, however, admit both positive and negative values of Q; perhaps the heat-transporting medium in the exchanger can be cooler or warmer than stream 3. Then Eqs.(8.1.10) yield ^

~ 2_~3 mI =

-

(8.1.1.7)

m2

from where also m3, and

h2

= ]'/3 q-

q" ,-1-3

(f,/1 f,/3 ) .

(8.1.18)

m2

Here, kt~ and f/2 are given, and h 2 depends only on the unmeasured T 2, given also ~2; cf. (8.1.5). We assume again the condition (inequality) (8.1.12) fulfilled. Then, in the same manner as before and after the formula (8.1.8), we conclude that ml, m3, and T 2 c a n be expressed as (infinitely differentiable) functions of m 2 (> 0), thus that the set of solutions is a (differentiable) manifold, of dimension one. It is thus a curve in the 4-dimensional space of variables ml, m2, m3, T 2. It is generally a curve and not a straight line, because the dependence of T2 on m E by (8.1.1 8) is generally nonlinear. If Q 4: 0, none of the variables is uniquely determined; they are not 'observable'. But if it happens that Q = 0, the mass flowrates are again not uniquely determined, while T 2 (by h 2 ) is; it becomes 'observable', given the measured values. Let ~: be the vector of measured variables, 9V/'(~) the above manifold (curve). Thus in the special case where = 0, 91,f(:~) is a straight line parallel to the (m 1 , m 2 , m 3 )-hyperplane of coordinates. The reader can perhaps imagine variable m 3 eliminated as a 'dependent' variable, thus consider only the system (8.1.17 and 18). Because the coefficient at m 2 in (8.1.17) is positive by (8.1.12), we then have the following picture. ^

208

Material and Energy Balancing in the Process Industries

0>0v~ 0=0vc 0<0% mr

Jm

i

Fig. 8-2. Graphical representation of Eq.(8.1.18)

Approximating the specific heat of stream 2 by a constant, a similar figure is obtained with T 2 in place of h2; the curves in Fig. 8-2 then represent M(i). In the 4-th dimension, also m 3 changes linearly with the parameter m2 of the curve M(:~). We can also eliminate m~ and m 3 as dependent, and imagine only the projection into the (m2, T 2 )-plane. Then, considering also^ ~) as variable, the individual curves in Fig. 8-2 are certain sections at given Q

7~

Fig. 8-3. Curves in Fig. 8-2 with different l)

while connecting the points at given m2-coordinate, according to (8.1.18) we obtain (approximately) a straight line (assuming constant specific heat), with the

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

209

remaining measured variables fixed. The reader can perhaps imagine the whole surface in the (m2 , T 2, {))-space, generated by straight lines parallel to the (T 2, {))-plane and of variable slope 1/m 2 . The reader has, perhaps, already some idea of how different is the analysis of solvability for a nonlinear system. To the observations (a) and (b) above we have added the following

(c)

The classification of unmeasured variables can depend on the values of the measured ones.

In the example in Section 5.5, we have shown that the whole system can be considered observable with the exception of certain special values of the measured variables. On the other hand, in the last example we have shown that an unmeasured variable is unobservable (not uniquely determined) with the exception of certain special measured values. Thus in the first case, a variable is 'almost always observable', in the second case 'almost always unobservable'.

Example 5

In our simple examples according to Fig. 8-1, we have considered the graph of the technological system restricted to one technological unit (node), plus the environment node. The presence of other node balances can change the classification. To Fig. 8-1, let us add a splitter and another heat exchanger. Q

d

T

3

5 1

"1 Q" Fig. 8-4. Nodes added to Fig. 8-1

The stream 4 parameters are unmeasured. Let us have, in addition to Example 4, measured the mass flowrate m 5 = rh5 , temperature T 6 = ~,6, and heat Q' = Q' . Then the splitter balances imply

210

Material and Energy Balancing in the Process Industries

(8.1.19)

m 4 = m 3 - th 5 y4 _ ~3

T 4 __ ~" 3

(8.1.20)

and

=

T 5 _ ~-3

} thus h 5 = ~t3

(8.1.21)

further A

m 6 -- m 5 y6 (= yS) = ~3

(8.1.22)

and we have to satisfy the balance m 6 h 6 + Q' = m 5 h 5

hence the condition /'~/5 (~/3_~./6) = 0 '

(8.1.23)

where ~./6 is known, given y6 = ~3 and the measured ~r,6. The condition (8.1.23) has to be obeyed and if so, in addition the new unmeasured parameters y4, T 4, y5, T 5, m6 ' y6 are uniquely determined (observable), while m 3 and m 4 are subjected to the condition (8.1.19), but not uniquely determined (thus unobservable) because nor the earlier conditions have determined m 3 . The condition (8.1.23) involves the variables rhs, ~3, ~3, ~,-6 (while y6 _ ~3 ), and ~)'; the remaining measured variables (.~1, ~r, 1, ~2) are not affected by the condition, thus can be called 'nonredundant'. The condition as written determines, for example, {)' as function of the four remaining variables. If M § is now the set determined by the condition, by the same consideration as above we conclude that 9V/"§is a (differentiable) manifold, of dimension I-1 where I is the total number of measured variables. The reader may recall Fig. 7-5. Also now, but in a multidimensional space, our M § is 'parallel' to the coordinate axes corresponding to the nonredundant variables (such as was x 3 in Fig. 7-5), and we then have a (5-1 =) 4-dimensional manifold as projection into the subspace of the 'redundant' variables (thus rh5 , ~3, ib3, ~6, ~),), but no more 'linear' (in contrast to the straight line in Fig. 7-5). The solvability problem does not consist in the physical determination of the variables by the balances, but rather in the formal determination of the variables by certain given algebraic conditions. In the energy balance, it can well

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

211

happen that certain thermodynamic data are approximated and do not obey rigorously the conditions of full consistency. Let us assume that our two-component mixture can be regarded as ideal, thus the mixing enthalpy can be neglected; cf. (6.2.47). Then the specific enthalpy depends only weakly on the composition. If the specific heat of stream j is approximated by a constant value Cp.i characteristic of the stream, the variable ~3 (= y5 = y 6 ) is absent in (8.1.23). Then also ~3 is a nonredundant variable.

Example 6 As the last example, let us delete the measurement of m 5 and instead, let us measure m 4 - - t h 4 . Then, with the condition

~./3 ~ ~./6 and 0'~ 0

(8.1.24)

we have

ms=

~./3 ~./6

(8.1.25)

uniquely determined and if the measurement is not subject to gross errors, we have m 5 > 0. Then also m 3 --/~4 + m5 is uniquely determined, hence successively m~ and m 2 by (8.1.2) and (8.1.17), thus also h 2 thus T 2 by (8.1.18), now whatever be Q. Then the whole system is observable (all the variables are uniquely determined), and no variable is redundant. Again somewhat naively from the technological standpoint, let us admit also the value Q' = 0. In addition let Ft3 resp. ~./6 depend on ~,3 resp. ~r,6 only (see the preceding paragraph). Then (replacing rh5 by the unmeasured m 5 , with m 5 > 0) the balance (8.1.23) implies for Q' = 0:~./3 ~./6_ 0

(8.1.26)

hence ~.3 and ~,6 are subject to a solvability condition ('redundant'), the other measured variables are 'nonredundant', and m 5 , thus also m3, m l , and m 2 are unobservable; also T 2 is unobservable with the exception of Q = 0 (attention: the latter Q, the former Q'). It is thus seen that it can also happen that

(d)

Some variable can be (called) 'nonredundant' with the exception of certain special values of some other measured variables, where it appears as 'redundant'.

But such classification is misleading. If we have admitted positive as well as negative values of Q' and found (measured) Q' = 0, we in fact do not know

212

Material and Energy Balancing in the Process Industries

which of the measured values (~.3 and p6, or perhaps Q') has to be corrected; only if it were Q' , we should not know according to which condition, as m 5 is unknown in (8.1.23). Simply, the classification 'redundant'/'nonredundant' can fail. The (adjustment) problem is then again 'not well-posed'. ^

if

Let us summarize. The examples have shown that, according to the selection of measured variables and their admitted values Either (i)

The solvability condition takes the form of a system of algebraic equations (determining a differentiable manifold M § ); generally, the condition does not involve certain variables, which can be called 'nonredundant', while the remaining measured variables are 'redundant'.

(ii)

If the measured values have been adjusted so as to obey the solvability condition (to lie on the manifold M+), the unmeasured variables are subject to a condition taking again the form of a system of algebraic equations (determining a differentiable manifold M(~) where i is the vector of adjusted measured values); generally, given i some unmeasured variables are uniquely determined and can be called 'observable', while the other are not uniquely determined thus 'unobservable'. But the classification can depend on ~,; in the examples, some unobservable variables become observable at some special values of :~

(iii)

There can exist certain values in the admitted region of the measured variables, such that the solvability condition fails as an adjustment condition (the condition does not determine a differentiable manifold); then a meaningful classification of variables is not possible.

Then

Or also

In the latter case, we have called the adjustment problem not well-posed, which can be interpreted as an inappropriate selection of the set of measured variables. Still other cases than illustrated by the examples are possible. It is, alas, generally not easy to analyze the possibilities a priori.

Chapter 8 -

8.2

Solvability and Classification of Variables H- Nonlinear Systems

213

S O L V A B I L I T Y OF M U L T I C O M P O N E N T BALANCE EQUATIONS

Our aim is now to show that, under certain plausible conditions, the whole set of (mass, components, and energy) balance equations is solvable and, in certain sense to be precised, 'regular'. In the first step, the energy balance will not be considered.

8.2.1

Transformation of the equations

The case where only total mass balance is considered has been completely analyzed in Chapter 3. So let us begin with the multicomponent balances, as formulated in Chapter 4. We already know that the balances (4.5.2) with (4.5.1) involve also the total mass balance of the set of nodes as a consequence, hence the latter need not be written explicitly. Generally, in addition to the equations (4.5.2) we have introduced the conditions (4.2.14): certain components are absent in certain streams. If again K is the number of components occurring in the balances we have introduced K sets Ek (k = 1, ..., K); E k is the set of streams in which component Ck can be present. Then instead of (4.5.1), we have the expression (4.5.4). Further, some of the conditions (4.5.2c) are consequences of other conditions. We thus rewrite the set of balance equations in the following manner. We define, for any node n k = 1, . . . , K: n k (n) = - Z

Cniy~mi]M k

(8.2.1)

i~ E k

formally for any k, with some a priori given order. Then our equations read, having again partitioned the set N u of units into S (splitters) and T, = Nu - S, n e T, : D(n)n(n)= 0

(8.2.2a)

with (8.2.1), n(n) being the vector of components n k (n) s u c h t h a t component C k is present in the node n balance, further

s e S"

Z_ C~j mj = 0

(8.2.2b)

J~Js m

where J~ = Js u {j~ } is the set of arcs incident to splitter s, see (4.2.9 and 10), further s e S" y~?- y{ = 0

(j e J~, k = 1, ..., K)

(8.2.2c)

Material and Energy Balancing in the Process Industries

214

and having partitioned the set J into J' (arcs incident to some splitter) and Ju = J - J' according to (5.2.2 and 3), further conditions are K

J e Ju "k__E1YJ = 1

(8.2.2d)

and K

se

S: Z y ~ s = l ;

(8.2.2e)

k=l

the additional conditions read J

~ Ju"

Y~, = 0 for j ff

(k = 1, ..., K)

(8.2.2f)

Ek ( k = 1 , . . . , K )

(8.2.2g)

Ek

and se

S:yJk~ - 0 f o r j ~

Indeed, from (8.2.2e) and (8.2.2c) follows also (4.5.2d) for any j ~ J', and from (8.2.2g) and (8.2.2c) follow also the additional conditions (4.2.14) for j ~ J', because j~ ~ Ek if and only if Js c F_~. Then in (4.5.1), y~ = 0 if i ~ ~ . Recall that each Eq.(8.2.2a) represents K(n) - R o (n) (linearly independent) scalar equations, where R0 (n) is dimension of the reaction space in node n (Ro = 0 in a nonreaction node), and K(n) is the number of components present in the node n balance; see the commentary to (4.3.16), and also Remark (i) at the end of this section. We designate again I MI the number of elements of set M. The number of the scalar equations equals M=

Z

n~T u K

(K(n)-Ro(n))+ IsI + g ~

IJsl + IJul § Izl

K

+ X I J u n ( J - E k)[ + X k=l

s~S

X ]{j~} n(J-gk)]

k=l s~S

;

(8.2.3)

the sophisticated notation in the last sum means, for each k, summation of terms equal to 1 over such splitters s for which j~ ~ J-Ek thus Js ~ Ek, while if j~ ~ then the term equals zero. The number of variables (mass flowrates and mass fractions) equals N = (K + 1) lJ[ .

(8.2.4)

Observe that we thus formally regard as variables all the mass fractions, even if some of them are put equal to zero by the condition YJk= 0 for j ~ J - ~ , which

Chapter

8 -

Solvability and Classification of Variables II- Nonlinear Systems

215

is a consequence of the conditions (8.2.2c,f,g). The convention facilitates the formal analysis. Let us designate k - 1, "" , K:

mk

(n) = M k n k (n)

--

(8.2.5)

~, CniY~mi i~F~

and i m k --

y~ m

(8.2.6)

i

(mass flowrate of component Ck in stream i). Then, by (8.2.1)

k = 1, ..., K: m k (n) - - E

Cni m ki .

(8.2.7)

i~ E k

The case when J-F_,k ~: O in (8.2.3), thus when some component Ck is absent in certain streams by an a priori technological assumption, complicates the solvability analysis of complex reacting systems. Of course it can be assumed that the technological requirements are formulated in the manner that the system of balance equations is solvable. But it is not easy to find some general criteria of solvability used in the mathematical analysis. We shall not attempt to give a complete answer to the problem. Let us only analyze the case that will be regarded as 'regular'. Let us begin with a simple example. Let the technological system be a sulphuric acid plant as in Section 4.6, G its graph according to Fig. 4-2. Let Ck be sulphur dioxide; the subset Ek of streams is the set E 5 in (4.6.1). Restricting the graph to the arcs j e E k , we have subgraph G k [N, E k ]; N is the set of nodes, node n = 0 (environment) included, thus Gk is

J A2

AI

node 0

G Fig. 8-5. Subgraph Gk for sulphur dioxide

We see that G k contains isolated nodes D, S1, and $3. The remaining nodes and arcs form a connected graph, say G k [N k , E k ]. The subset Juk = Ju n E k is the set of arcs not incident with any splitter and containing component C k . Restricting

216

Material and Energy Balancing in the Process Industries

G k to arcs Juk and deleting the splitters makes the subgraph Guk of G k disconnected.

.-I "1

R nodes Nk'

A2

,,- node 0

nodes Nk~

Fig. 8-6. Subgraph Guk for sulphur dioxide

The subgraph Guk has two connected components; the component Nk 1 contains reaction nodes (B and R), N~k contains the environment node. The reader can himself make the reduction (restriction to E k and deletion of splitters) in the case when C k is sulphur trioxide. According to Fig. 4-2 and the list (4.6.1), he will find node S1 isolated in the first step, in the second step N~ = {B,R} and N O- {D, A1, A2, 0} (stream 13 connects D with A2). If C k = 02 or N 2 , N~ and N Owill be the same as in the preceding example (with different structures of the subgraphs), while if C k = H20, Guk will remain connected and contain the node 0. Finally if C k is elemental sulphur then Guk has nodes 0 and B, and is connected. Alas, the scheme is simplified, for example concerning the flow scheme of water and acids. Adding further streams and splitters, it can happen that the decomposition of Guk will comprise more components. Let us now try to generalize. Let again G [N, J] be the original graph, Ck some chemical component, and ~ the subset of streams as above. S is again the set of splitters, J' the set of streams incident to some s ~ S, and Ju = J - J'. We have the partition m

m

J' = w Js where Js = Js u {Js } s~ S

(8.2.8)

according to (4.2.9 and 10). Let (k = 1, ..., K) Juk = Ju ~ Ek

(8.2.9)

and Sk: set of splitters s ~ S such that j~ ~ E k

(8.2.10)

m

hence also Js c Ek; S k is thus the set of splitters where component C k is present. Then, in (8.2.7)

Chapter 8 - Solvability and Classification of Variables H. Nonlinear Systems

m k (n)

---

E

i Z Cni m k -

ie Juk

Z

_

se Sk je Js

Cnj m~, .

217

(8.2.11)

If j s Js, let us designate (splitting ratio) %

%

(0 < % < 1 )

(8.2.12)

%8 assuming of course mjs > 0. The inequalities follow from the splitter balance mj~ = E mj (mj > 0), and we have j~J~

Z % = 1.

(8.2.13)

J~Js

According to (8.2.6) with the condition y]s - y~, for j e Js, we have for s ~ S k 9 m~, = ctj m~,~

(J ~ Js ) .

(8.2.14)

1 3 j - % i f j ~ Js 9

(8.2.15)

Let us define, for J ~ Js'gj = 1 ifj-Js, Then (8.2.11) reads

n~

]~ Cni mki - s ~ S T u 9, i n k ( n ) --_ i~J~

m~s

(8.2.16)

J~Js

where o f c o u r s e Cnj = 0 if node n is not adjacent to splitter s (not endpoint of any j ~ Js ), and Cnj 4 : 0 for arc j whose n is endpoint; then (see Fig. 4-1) Cnj~ = - 1 if n (~ T, ) is endpoint of Js, Cnj -- 1 if n is endpoint of j ~ J~. We shall now formulate

8.2.2

Additional hypotheses

Let us introduce the factor (subgraph) Gk [N, Ek] of G. It contains generally isolated nodes. If node n is isolated in Gk then component Ck is absent in the adjacent streams; thus mk (n) = 0 if n is isolated in Gk 9

(8.2.17)

218

Material and Energy Balancing in the Process Industries

In fact, such mk (n) does not occur in the balances (8.2.2a). Inthe (full) incidence matrix of Gk, the n-th row equals zero as n is isolated. Let N k be the remaining set of nodes. We shall a s s u m e m

0 ~ Nk (k = 1, ... , K)

(8.2.18)

hence the environment node (denoted by 0) is not isolated in G k , thus any Ck is present at least in one of the input or output streams. The hypothesis is not quite trivial. It can happen that Ck is some intermediate product that arises in some reaction node and is again fully consumed in other reaction nodes. In that case, one would expect that there were an inventory of the intermediate product. But in our steady-state models, wehave included the inventories in the environment node, thus we have still 0 ~ Nk. This assumed, deleting the isolated nodes we have subgraph Gk [Nk, Ek ] of full incidence matrix I

I

graph G,~

I I

I

J,,k

]

J'c~ Ek

:

Nuk

I I I -

Nk

Sk

Fig. 8-7. Full incidence matrix of general Gk

where the void field is zero. The void field in the left lower part of the figure_is due to the fact that the arcs of Ju, thus a fortiori Juk are none of the arcs j ~ J~, whatever be s ~ S. See (8.2.9 and 10). We thus can restrict Gk to the arcs j e Juk and nodes n ~ Nuk = N k - S k , and we have subgraph Guk [Nuk, Juk ] with 0 ~ Nuk. Generally, Guk is not connected. Let Gluk be the/-th connected component, of node set N~ and arc set Jluk. We thus have the partition L k

Nuk = N~uk ~9 l--~N~ where 0 ~ N~k

(8.2.19)

(one and only one of the components contains the environment node); thus Lk+ 1 is the number of connected components (Lk = 0 means that Guk is connected).

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

219

Summation over n e Nk~ in (8.2.16) yields, for

I_>1"(0+) E ( Z

Z CnjBj

se Sk \ je]'s ne N~

m~,s + Z m k(n)-0 neN~

(8.2.20)

because the sum of rows of the incidence matrix of a graph equals zero; here, Cni for n E N~ and i e J~k (C Juk ) are just the elements of the full incidence matrix of G~uk.Conceming the node n = 0, the summation only determines m k (0) by the remaining terms in Eq. (8.2.20), and is of no use (no balance of node 0 is set up). Hence Eq. (8.2.20) is relevant for l > 1, if L k > 1 thus if Guk is not connected. In that case, and in the above manner the integral reaction rates (occurring implicitly in m k (n)), splitting ratios otj (8.2.15), and component mass flowrates m~,~through the splitters are interrelated due to the a priori assumed absence of some components in some streams. Let us precise the point. Recall the observation after formula (8.2.16). In the inner double sum in (8.2.20), given s ~ Sk the summation concerns only nodes n adjacent to node (splitter) s via some arc j e J~. It can happen that for some l > 1, none of the nodes s e Sk is adjacent to any n e N~. This, however, means that even before deleting the splitters, no node n e N~ was connected by a path with any node n' ~ N~, because it is just the arcs j e Js (s e Sk ) that have been deleted. Then also the graph G k [N k , E k ] contains at least two connected components, one of them being a subgraph of node set N~; see Fig. 8-7 and imagine a subset of rows (~ 0) empty in the columns J' n Ek, and with just two nonnull elements in any column of J~,k (incidence matrix of G~k ). Then the sum over Sk in (8.2.20) equals zero and we must have, in case of u

1 > 1, Glukconnected component of Gk" ]~ .mk(n) = 0 . neN~

(8.2.21)

This means that the linear system (8.2.20) in variables m~,, is not generally solvable, unless the condition (8.2.21) is satisfied identically. If so, the condition (8.2.21) is added to the equations (8.2.2a). The case is conceivable but again, one feels thatthere is something wrong in the model. We then cannot have Gu~kas an isolated node in ~ [Nk, Ek ] because in that case, it would have been deleted already by the construction of Gk from Gk [N, Ek ]. Thus Ck is again an intermediate product in a subsystem, as above. We shall not examine this case and instead, we shall

Assume that n

Gk [Nk, Ek] is connected (k = 1, .-., K) .

(8.2.22)

Here, Gk is the subgraph of G k according to Fig. 8-7, comprising all the arcs j e Ek.

220

Material and Energy Balancing in the Process Industries

Then in particular for each 1 > 1 in (8.2.20), some splitter s ~ Sk is adjacent to some node n ~ N~, unless S k is empty. But in the latter case, Gk [Nk, Ek ] = Guk [Nuk, Juk ] (nothing deleted), hence also Guk is connected by hypothesis, hence L k - 0 and we have no condition (8.2.20). Let again Lk > 1. With the above observation, the inner double sum in (8.2.20) equals, for any s ~ S k

s.s .s J--J``

where n s is the non-splitter endpoint of j``, n i that of j e Js, and e(J's ) = 1 if n`` e N~, else zero (8.2.23) e(j)= 1 ifn ie N~,elsezero. Thus recalling again the observation after (8.2.16), with (8.2.15), the sum equals

(say) 7., = - e(j``) + ]~ e(/')% j~J``

(s e

SO

(8.2.24)

given k and I thus N~ in (8.2.23). Thus Eq.(8.2.20) reads

(8.2.24a)

]~ ],``m ~ + ]~ m k (n) = 0 . se Sk ne N~

Let us examine the coefficients (8.2.24), generally dependent on the ratios %. If the splitter s is 'inside' of the subset N~, of nodes, thus if all the adjacent nodes are elements of Nk~ then

(8.2.25a)

7`` = -1 + ] ~ otj = 0 j e Js according to(8.2.13). Else either n`` e N~ but not all nj e N~, then -1 < ~'s < 0

(8.2.25b)

because the incomplete sum of otj over Js is > 0 but < 1 or

n s ~ N~k but some nj e N~, then 0 < ~,``< 1

(8.2.25c)

or

n s ~ N~. and nj ~ N~ for any j, then in the latter case, however, But if it were only the first above, Gk would again not must take place for some s

~,``= 0;

(8.2.25d)

some of the preceding three cases occurs for another s ~ S k . case (all splitters 'inside') then, by the same consideration as be connected. Thus, in fact some of the cases (8.2.25 b or c) ~ Sk, given l > 1" hence some ~'s * 0 for any l > 1.

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

221

i

Assuming L k > 1 and l > 1, let thus S~ be the nonempty subset of S k , of elements s adjacent to some n 9 N~ and some n'~ N~ (thus Ys * 0), and let us generally write ~k for any k = 1 , . . . , K and 1 < l < Lk. We thus have, for m

(8.2.26)

s 9 S ~ ' - I _< ~k < 1 and ~k r 0 m

while ~k = 0 if s 9 S k - S~. For s 9 S~, we have one of the two possibilities (8.2.25 b or c). According to (8.2.24a), Eq. (8.2.20) reads (for k = 1 , - . . , K) Z Y~km~s+E m k ( n ) = 0 seSm, neN~

(8.2.27)

(1 < l < ~ ) .

The coefficients ~,, = ~sk are functions (8_.2.24) of the ratios aj obeying the condition (8.2.13); we have [Js [ > 2. For any l, as S~ ~ ;O, we have some ~sk ~ 0 by (8.2.26).

The Lk equations (8.2.27) are linear in the IS k ] variables m~, and of rank < ~ . They are not homogeneous unless the sums over N~ equal zero. We shall not examine their solvability in general; generally, they impose certain restrictions on all the variables, mk (n) and % included. We shall limit ourselves to the case when they are solvable in the variables mikef o r any k = 1, ..., K such that L k > 1; if L k = 0 (Guk connected), the condition (8.2.27) is absent. More precisely: Recalling (8.2.24) where (given k and l) y~ stands for ~k, let us introduce the matrices k = 1, ..., K" Gk = (~k) with rows 1 = 1, ..., Lk and columns s ~ S~ (8.2.28) (for some given order in each S k ). We then assume that all the matrices are of full row rank, thus rankG k = L k (k = 1,-.., K)

(8.2.29)

whatever be the splitting ratios (xj [obeying (8,2.13)] in some intervals admitted by the technology. This clearly implies

Lk < I Sk [

(8.2.29a)

as a necessary condition for (8.2.29). Denoting by mk: the/_q,-vector of components Z

n~N~

m k (n)

(l = 1, ... , L~ )

(8.2.30)

and by Uk" the [ Ski-vector of components m j~ (s ~ S k)

(8.2.31)

222

Material and Energy Balancing in the Process Industries

the system (8.2.27) reads Gk Uk + mk = 0

(8.2.32)

with (8.2.28) and (.8.2.24) where 7~ stands for ~sk. The hypotheses can be verified as follows. Let k = 1, -.., K (component C k ). (a)

Identify the subgraph Gk [Nk, Ek ], restriction of G to the streams j ~ Ek containing component Ck

(b)

Delete the isolated nodes of Gk" get subgraph Gk [Nk .Ek ]; verify if Gk is connected and if 0 ~ Nk. If so then

(c)

Partition the node set of Gk into the subset Sk of splitters, and the subset Nuk of non-splitters (including the node environment)

(d)

Delete the arcs of E k incident to splitters, and the splitters s ~ Sk; the subgraph Guk is of node set Nuk

(e)

Decompose Guk into Lk+l connected components Gluk (l = 0, ..., Lk ); if Lk = 0 then stop. If Lk > 1 then

(f)

Identify the node sets N~ (1 < l < Lk) of G~k, while N Ois determined as the node set containing node 0 (environment)

(g)

Identify the arcs j~ and j ~ Js incident to splitters s ~ Sk, where Js is the input stream to s, and find the other endpoints n s of Js and nj of j ~ Js

(h)

Identify the coefficients e by (8.2.23) (l = 1,---, Lk ) and

(i)

Taking some admissible values of the splitting ratios % (8.2.12) obeying (8.2.13), compute the coefficients ~k denoted as ~/~ in (8.2.24), for l = 1, -.. ,Lk and s ~ S k Identify matrix Gk (8.2.28)

(k)

Identify the rank of G k (e.g., by Gauss elimination), and verify the condition (8.2.29).

Recall that if, in step (e), Lk = 0 then Guk is connected and there is nothing more to verify; the equation (8.2.32) is absent.

Chapter 8 - Solvability and Classification of Variables II- Nonlinear Systems

223

Because Gk as function of the t~j is continuous, if its rank is maximum at some special values of the % then it is such at least in some neighbourhood of the values.

In certain cases, the hypothesis (8.2.29) can be verified directly. Let for instance L k = 1 as in the above example (sulphuric acid plant). Then Eq.(8.2.32) thus (8.2.27) for 1 = 1 reads Z ]'~km~,~ + Z m k (n) = 0 s~S k n~N~

(L k - 1)

(8.2.33)

where some 7~k, according to (8.2.26), is nonnull whatever be the splitting ratios ctj. Thus clearly, the rank equals 1 - L k and some m~,~ can be expressed as (infinitely differentiable) function of the sum over N~ in (8.2.33), splitting ratios t~j, and the other flowrates mJk~(t ~ S, t ~ S k ), if there are any. Indeed, we then have

mJs

--

_

'~slk

t~S ')(:k mJt + ]~ mk (n)) n~N~ t-4:s k

(8.2.33a)

where the ?-coefficients depend generally on the txj. Going back to Figs. 8-5 and 8-6 (Ck = SO2 ), the whole set Sk consists of one element $2. Denoting by s the splitter $2, according to (8.2.24) We have 3'~k = -1 (+ 0) = -1 and Eq.(8.2.33) reads m~ = m k (B) + m k (R) where, according to Sections 4.1 and 4.2, m k (X) equals the integral production rate of Ck by the chemical reactions in node X; in the present case the sum represents SO2 produced by burning in B minus that consumed by conversion in R. Clearly, it must equal the inlet mass flowrate of SO2 into $2. As an exercise, the reader can set up the balances (8.2.33) for the other chemical components according to the list (4.6.1) and Fig. 4-2. For Ck = N2, one will in addition set mk (n) = 0 in (8.2.33), because nitrogen is (assumed to be) nonreacting. For Ck = H20 (or S), Guk is connected (Lk = 0) and no condition (8.2.32) is formulated.

224

Material and Energy Balancing in the Process Industries

8.2.3

Independent variables

The condition (8.2.29) is not a necessary condition of solvability. Only we have limited ourselves to the cases where it is obeyed as a structural property of the graph, with the choice of the sets E k . If so, our solvability analysis can continue. _

Let us have verified the condition (8.2.29) at some special values of the o~j; then, at least in some neighbourhood of the special values, there exists a subset of columns of Gk that are linearly independent, in number ~ = rankGk. Rearranging appropriately the columns of G k , thus the components of vector Uk (8.2.31) on which the matrix operates, thus in fact the splitters s e Sk, if Uk(S) = m~, Sk = I Skl, s = 1, ..., Sk

(8.2.34)

then the vector Uk is partitioned

uk =

(8.2.35) Vk

} Sk- L k

where the first 4 components correspond to the linearly independent columns; vector Vk is absent if Sk = ~ . Accordingly, matrix Gk is partitioned

Gk = ( G~,, G~ ) Lk

(8.2.36)

Sk-L k

where G~, is L k • ~ regular, and Eq.(8.2.32) is equivalent to G~u~ + G[.' v k + m k = 0 thus to

(8.2.37)

u~ = - G~ l (GkVk + m k ) .

Thus if v k , thus the last Sk- Lk components Uk (S) = m~ are arbitrary, and if the L k components (8.2.30) of mk, thus also if the mk (n) for n e Nuk (8.2.19) are arbitrary then u~, thus also the whole vector Uk (8.2.35) is an (infinitely differentiable) function of the Sk- Lk variables m~,,, of the mk (n), and (via Gk ) of the splitting ratios ~j subject to the conditions (8.2.13) and otherwise arbitrary in a (multidimensional) interval. In addition, by (8.2.14), the remaining m~ (j e Js) are thus also functions of the above variables. Thus in the whole subsystem formed by splitters s e S k and the incident streams j e Js, the vector n

Wk of components mJk (j e Js, s ~ Sk )

(8.2.38)

is an (infinitely differentiable) function of the Sk- ~ components of Vk, of the mk (n) (n ~ Nuk ), and of the otj, say

225

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

(8.2.39)

wk = Fk (Vk, mk, or)

where o~ stands for the composed variable, say vector of components % where j ~ Js and where for each s, one of them has been eliminated by the condition (8.2.13). Recall now Eq.(8.2.16). Given k, we have decomposed graph G,k into connected components G~k (l = 0,-.. , ~ ), of node sets N~. If Lk > 1 then by the summation over n ~ N~, we have eliminated one equation for each N~ (l > 1). For l = 0, instead of such elimination we have simply considered the equations only for n , 0. Thus for each connected component, we have I N~I-1 equations left; they are, with (8.2.14), equivalent to the equations (8.2.11), considered each for n ~ N~ without the eliminated one thus in number ] NLI-1, 1 = 0,--., L k . Restricted to streams (arcs) i ~ Jluk(arc set of Guik), for each I the rows of (Cni) in number I NLI-1 constitute the reduced incidence matrix (say) C~k of connected graph G~uk, hence the matrix is of full row rank. In the same manner as in (8.2.35-37), we can identify some ([ N i l - 1 ) x (] N~ I -1) regular submatrix. Instead of m k in (8.2.37) we now have, for each I, the vector (say) m kI of I NLI 1 component s mk (n), n e N~ (minus the deleted reference node). Let us designate P~ = I N~I - 1

(l = 0, ..., L k)

(8.2.40)

and R~ = [ Jluk ] - P~

(8.2.41)

further

(8.2.42)

i i 6 Jluk Zk, . vector of components mk,

It can, however, happen that some of the G~uk is an isolated node. Then I Jluk I = 0 and I NLI = 1, thus PL = eL = 0 and the subvector z k1 is simply absent. With this exception, there then exists a partition

,

,

Xk

} Rk

0 = o , .-. ,L~)

(8.2.43)

such that the corresponding P~ x P~ submatrix of C~ukis regular, thus such that y~ is an (infinitely differentiable, in fact linear) function y~. = Z~ (x I , w k , m kI )

(8.2.44)

1. recall (8.2.38). Then also (having added the identity x k1 = x k1 ) the whole whatever be x k, vector zki is such function. Now mki is subvector of vector

(8.2.45)

ffak of components m k (n), n e Nuk - {0} where node n = 0 does not occur, and let us introduce the composed vectors

Zk --

and x k l=0,...

~

1

(8.2.46)

Xk

l=O,

... , L k

226

Material and Energy Balancing in the Process Industries

But because the union of subsets J~k is a partition of Juk ,in fact Zk is vector of components mk,i i e Juk"

(8.2.47)

the selection of the x kl depends, however, on the connected component and its incidence matrix. On the other hand, the vector m k (8.2.30) is linear function of the mk (n), n e Nuk - {0}, thus of vector m k . Thus finally, by combination with (8.2.39) we have m

u

(8.2.48)

Zk= Z k(Xk, Vk, (X, m k)

(where Zk is infinitely differentiable). The vector z k is column vector of I Jukl components. The dimension of Xk is

~e~=

/=0

~lJ~kl-

/=0

~ I N ~ I + t k + 1 = IJ~kl - INukl +Zk + 1

/=0

and the dimension of Vk is Sk - ~ ; recall (8.2.35). Hence

xk ) is a (I Juk I +Sk-[ Nuk I +l)-vector

(8.2.49)

Vk of variables (component Ck mass flowrates) m~,, appropriately selected in the set of streams j ~ E k . The result (8.2.48) holds true for any k = 1,-.. , K. Let us now consider the components mk (n) where k = 1, ... , K(n), and n e T, (set of non-splitters). Here, for convenience, we have re-numbered (given node n) the components present in the node as written. Now for any n, the K(n) components mk (n) are subject to the condition (8.2.2a) with (8.2.5), with full row rank (K(n)-Ro(n)) x K(n) matrix D(n). There remains again to select, for each n, certain (K(n)-Ro (n)) linearly independent columns of D(n) and having possibly re-ordered the columns, thus also the components n k (n) of n(n), we have the partitions

n(n) =

n'(n) ] } K(n)-Ro(n) n"(n) } R o (n)

and D(n) = (D'(n), D"(n))

(8.2.50)

from where n' (n) = - D' (n) "1D"(n)n"(n)

(8.2.51)

hence each of the K(n)-Ro (n) components of n'(n) is linear combination of the R 0 (n) remaining components. Having again added the identities n k (n) = nk (n) for the latter R 0 (n) components, we find that each nk (n) (k = 1, ... , K(n)) is linear combination of the R 0 (n) components of n"(n), with constant coefficients. Clearly, the subset of the last R 0 (n) components of n(n) is empty if the node is nonreaction, in which case n'(n) = n(n) = 0. Multiplying each n k by M k according to (8.2.5), we have m k . To the subvector n"(n) thus uniquely corresponds certain m"(n) of R 0 (n) components. Hence each mk (n) is a linear combination of the R 0 (n) components of m"(n), with constant coefficients determined uniquely by matrix D(n) and the choice of the partition. This holds true for any node n ~ T u . Let us introduce the composed vector

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

m" =

227

(8.2.52)

m"(n) n~ T u

Thus in the total, each m k (n) (n ~ T u , k = 1, .-. , K(n)) is a given linear combination of the components of m". Rearranged in another order, the mk(n) constitute vectors m k (k = 1, ... , K) in (8.2.45), where Nuk - {0 } c T,. Hence also the components of each ffak are the above linear combinations and the latter can be substituted for 1~k . Considering the totality of the relations (8.2.48), we first introduce the composed v e c t o r

Z

(8.2.53)

"-

k=l,...,K For each k = 1, .-., K, in z are comprised just the components m ki w h e r e i E Juk = Ju ('~ E k , thus where component Ck can be present. We then introduce the composed vectors

X =

and v =

Xk

Vk

k=l,...K

(8.2.54) k=l,...,K

Further, taking certain ]Js l-1 ratios % in each Js (s E S), the remaining one is determined. Let the former % constitute vector a. Then, finally, we obtain some (by the construction unique) function Z (infinitely differentiable) such that

(8.2.55)

z = Z(x, v, a , m " ) . Here, by (8.2.49)

(x /

is of d i m e n s i o n

V

g

[Juk I

k=l

+ ~: Is~lk=l

~: INu~l +K k=l

(8.2.56)

further

a is of dimension s

[Js[ - I s I

(8.2.57)

R 0 (n) .

(8.2.58)

s~ S

and

m" is of dimension s

n~ T u

228

Material and Energy Balancing in the Process Industries

The whole vector argument in (8.2.55) is thus of dimension K

o:g+

X(IJu~l

k= 1

+

Is~l)- ~ INu~l + x I J s l - I s I + x e0(~) k= 1

s~ S

n~ T,

(8.2.59)

where Juk = J, n Ek, S k is the set of splitters s such that Js e ER, see (8.2.9 and 10). Further Nuk is the node set of subgraph Guk, see the text before (8.2.19). It is thus the subset of nodes n e Tu u {0} incident with streams j e Juk, node 0 included. By (8.2.55), wehave determined the vector z of mk, i e Juk, k = 1, ... , K. In i

addition, also the m~, (j e Js, s e S k ) are determined as w k by (8.2.39), thus finally again

by the variables occurring in (8.2.55). Hence introducing the composed vector

w =

wk

(8.2.60)

k=l,...,K we can writeas well w = W (x, v, o~, m")

(8.2.61)

for some uniquely determined (infinitely differentiable) function W. In the total, the function r

F =

/z 1 W

(8.2.62)

determines the whole vector of c o m p o n e n t mass flowrates.

8.2.4

Degrees of freedom

Our intermediate result can be summarized as follows. W e have, by graph decompositions in the subgraphs G k of G restricted to streams j ~ ~ where c o m p o n e n t Ck can be present, selected certain special parameters (in n u m b e r D), such that the component mass flowrates m~, are determined as (infinitely differentiable) functions of the parameters. It is the vector function denoted as F in (8.2.62) with (8.2.55 and 61). W e have not explicitly written the function, we only k n o w that it is uniquely determined by the structures of the graphs G k , by the assumed reaction stoichiometries, and by the (not necessarily unique, but not arbitrary) selection of the parameters. W e have assumed c e r t a i n c o n d i t i o n s for this kind of solvability. The graphs are successively d e c o m p o s e d and the conditions verified by the procedure (say) P described in points (a)-(k) following after (8.2.32).

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

229

The group (say, vector) of parameters in (8.2.55) is selected in the following manner. Let first k be fixed. Then (a)

One identifies vector Uk of component mass flowrates (8.2.34), thus inlet mass flowrates of Ck into the splitters where Ck occurs, and by the partition (8.2.35), one selects subvector Vk as described, further

(b)

In the ultimate decomposition into subgraphs G~k by P:(e), J~k is the arc set of Gluk. In each (nonempty) J~k, one identifies vector Zkl of component mass flowrates (8.2.42) in streams not incident to splitters and, by the partition (8.2.43), one selects subvector x~ as described; note that for different l, the corresponding sets of flowrates are two-by-two disjoint. Then

(c)

By composition, one identifies vector x (8.2.46).

Now, in the totality of k = 1, .--, K (d)

One forms composed vectors x and v (8.2.54) of selected component mass flowrates; note that by different k, also the components of x and v represent different variables.

Then (e)

One identifies vector (or simply group of variables) ~ of independent splitting ratios % (thus eliminating one for each splitter by (8.2.13)), with the only condition of lying in a (multidimensional) interval, thus in a neighbourhood of the values used in P:(i).

Finally

(f)

For each node n ~ T u one partitions the vector n(n) according to (8.2.50), where n'(n) uniquely corresponds to some selected subset of linearly independent columns of D(n), and n"(n) is the remaining subvector, selected in this manner and further recalculated in terms of mass flowrates to m"(n) according to (8.2.5), and

(g)

One forms composed vector (8.2.52) of selected parameters m k (n).

The selections (a)-(g) determine the whole vector of parameters occurring in (8.2.55 and 61). Their total number is D (8.2.59). Now whatever be the values of the parameters, by (8.2.55 and 61) the vectors z (8.2.53) and w (8.2.60) are uniquely determined. The vector z is

230

Material and Energy Balancing in the Process Industries

composed of subvectors z k (8.2.47), each of them being that of component Ck mass flowrates for streams i ~ Juk, thus for streams not incident to the splitters and such that the presence of component Ck is admitted in the streams. The vector w is composed of subvectors Wk (8.2.38), thus of component mass flowrates through the splitters. Hence the vector composed of z and w is that of all component mass flowrates m~ (j ~ J) that can be nonnull. Given these m~,, recalling (8.2.6) we determine also, for any K

j ~ J: mj =

kZ=lm~,

(8.2.63)

where the summation is, in effect, over such k that j s Ek; we complete formally the definition of m~, by m~ = 0 i f j

~ Ek .

(8.2.63a)

In other terms, we can use the formula (4.5.10) with (4.5.5). Because each j is present at least in one ~ , mj is determined and we can expect mj > 0. We then determine also the mass fractions

yJ=,

(8.2.64)

% obeying clearly K

Z y~ = 1.

(8.2.65)

k=l

In the function F are also hidden the equations (conditions) (8.2.14) with (8.2.13) made use of in the construction. Thus, by (8.2.63 and 64), for J ~ Js" mj = ffj kEmJs = ~j mjs

Y~ _ ctj m~,~ = Y~

(8.2.66)

(8.2.67)

%mjs and

Z% =%s

j~J~

"

(8.2.68)

Chapter 8 - Solvability and Classification of Variables II- Nonlinear Systems

231

The choice of the aj has been subjected to the condition (8.2.12), hence mj~ > 0 makes also mj > 0. In this manner, all the 'primary' process variables occurring in the original set of equations are uniquely determined by the D parameters introduced by (a)-(g) above. Compare with the modified system (4.5.7)-(4.5.11). It is also physically clear that with a realistic choice of the parameters, we shall have indeed mj > 0 for all j ~ J. From the mathematical viewpoint, the case where some mj = 0 is possible; such point in the variables space is then 'singular', as Eq.(8.2.64) is not solvable, or has an infinity of solutions if also m ~ - 0. We shall not attempt specifying formal conditions precluding this possibility. One can say that the D 'independent parameters' represent certain D 'degrees of freedom'. Let us precise the point. Let us conversely have some solution of the system (8.22) with (8.2.1) in the variables mj and y~, (j ~ J, k - 1, .-. , K). Let us assume mj > 0 whatever be j e J

(8.2.69)

(precluding thus possible singular solutions). Then, by (8.2.66) %_

mj

(j E Js)

(8.2.70)

mj~ thus in particular the selected parameters % in (8.2.55 and 61) are uniquely determined. Further~ by (8.2.64) m~ = y~mj (j ~ J, k = 1, ..-, K)

(8.2.71)

thus in particular the selected vector parameters x and v in (8.2.55 and 60), composed of certain component mass flowrates, are uniquely determined. Finally, according to (8.2.7) we find the mk (n), thus in particular the vector m" (8.2.52) in (8.2.55 and 61) is uniquely determined. Consequently, the assignment of the N variables mj and y~, (8.2.4) to the D parameters (8.2.59) by (8.2.55 and 61) with (8.2.63 and 64) is one to one and bicontinuous (continuous along with its inverse; in fact infinitely differentiable). In this manner, the set (say) M of solutions is parametrized. Compare with the parametrisation of a straight line such as (7.3.6), or (in mathematical idealisation) of a territory on Earth's surface by geodetic coordinates (0, q0). Also in the latter case, the parametrisation is conventional (not the unique possible); for instance the zero meridian (q~ = 0) was chosen by convention. In addition, for example North pole is not uniquely assigned the q0-coordinate; also above (8.2.70), the points where mj = 0 (j e Js, thus also mjs = 0) are not uniquely parametrized.

232

Material and Energy Balancing in the Process Industries

If some parametrisation is possible, and if the set is, in certain mathematically precise sense 'smooth' then the set is called (differentiable) manifold; in the cases considered here, it is a 'submanifold' of an 'ambient space'. For the above set M of solutions, the ambient space is R N, phYSically interpreted as the N-space of mass flowrates mj and mass fractions y~,; see (8.2.4). The number D (8.2.59) of parameters is called dimension of Yff, which is a mathematically precise expression for the 'number of degrees of freedom' for the solutions of a set of equations. The proof that the set 9V/chas the required property of'smoothness' is, however, quite nontrivial. The proof is too technical to be reproduced here in detail. One has to show that the matrix of partial derivatives (Jacobi matrix) of F (8.2.62) is of full column rank D. One makes use of the fact that at constant a in (8.2.55 and 61), F is linear, and using the construction of F, that F(x, v, ~, m") = 0 implies x = 0, v = 0, and m" = 0 whatever be in the admissible region; hence the linear map (F restricted to variables x, v, m") is injective, hence the corresponding submatrix of partial derivatives is of full column rank. One then considers, in F, the rows of W (8.2.61) corresponding to the equations (8.2.14), thus m-~ = aj m~s 0" ~ Js ); by the differentiation, one obtains the corresponding rows of the Jacobi matrix. For any such row, thus for any (k, j) where j ~ Js, one multiplies by c~j the m~-row of the Jacobi matrix of W, and subtracts from the former; one considers only I Js l-1 elements j~ Js for each s as corresponds to ] Js 1-1 independent splitting ratios c~j in the subset Js of streams. One finally selects certain rows such that the corresponding submatrix in the a-columns is diagonal, of elements m~s r 0 for an appropriately selected k. This is possible under the condition mj~ > 0 whatever be s ~ S

(8.2.72)

One concludes the first part of the proof by proving that the Jacobi matrix is of full column rank. The second part of the proof makes use of the equations (8.2.63-65) transforming the variables m~ to mj and y~, under the condition (8.2.69), which implies in particular also (8.2.72). In mathematically precise language, the latter transformation is called 'diffeomorphism of manifolds', and it shows that our Yd is, in fact, a submanifold of the intersection of hyperplanes (8.2.65) over j e J.

Let us also observe that all the functions considered above are not only infinitely differentiable, but 'analytic'. Mathematicians know the precise meaning of the concept; for the reader it suffices perhaps to consider functions composed of elementary functions he knows from practice. Thus 914is, in fact, an 'analytic manifold', an expression used in Section 4.5, before-last paragraph. Before summarizing the final results, let us show that the set of equations (8.2.2) with (8.2.1) is, in certain sense, 'regular', on precising the sense; the ultimate precision follows in Section 8.4.

233

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

Let us give another expression to the number D (8.2.59). We have the partitions

(8.2.73)

k = 1, ..., K: Nuk = (N,k n T , ) u {0}

into the set of non-splitters n ~ T u contained in N,k, and the one-element set {0} (node environment), because 0 ~ Nuk for any k according to (8.2.19). Hence

K-Z k=l

IN.~I =-

~(INu~nTul

+ 1)-K)

k=l

and denoting

(8.2.74)

N~,k = Nuk n T u we have K

K-

Z k= 1

I Nu~ I =- ~ I N;k I 9

(8.2.75)

k= 1

Here, N'uk is the set of non-splitters n e T, such that component C k occurs in the node n balance. Let us introduce the numbers 5(n, k) where n ~ T u and k - 1, ..., K, such that 8(n, k) = 1 if Ck occurs in the n-th balance

(8.2.76) = 0 otherwise . In the matrix of elements 5(n, k), the number of nonnull elements equals K

z INu~l

by successive summation over columns:

k=l

Z K(n)

and by successive summation over rows:

n~ T.

where K(n) is the number of nonnull elements in the n-th row, thus of components C k occuring in the n-th balance as introduced in (8.2.3). Hence K k= 1

I N;~I = z K(n).

(8.2.77)

n~ T.

Consequently, by (8.2.59) K

D-- ~ (Ro(n)-g(n))+ Z(IL~I+Is~I)+ ~ n~ T.

k= 1

se S

IJ~l-Is

(8.2.78)

where we recall that Juk "- J, n E k and Sk is the set of splitters s e S such that Js ~ Ek thus E k . The notation is the same as in (8.2.3) Our goal is to compute the sum M + D. Because the sets Juk and Ju n (J-Ek) are disjoint, we have

Js c

234

Material and Energy Balancing in the Process Industries

IJun(J-Ek)J

+ [J,k[ = [ ( J u n ( J - E k ) ) U ( J , n E k ) [ = [ J u n J [

= [Ju[

(8.2.79)

because J, c J. We further have i

x JJ~l + Isl s~]gS [Js[ =sZSlJs u {Js}[ =s~S

because [ Us }1 = 1, hence !

xx

[J~l--KX [J~l-glsJ

s~S

(8.2.80)

seS

and since Sk is the set of s e S such that Js e Ek, while the (disjoint) union of sets {Js } n (J-E k) over s e S is equivalent to the set of s e S such that Js ~ Ek, we have

Is~l + z I {J~} n(J-Ek)J = IsI s~S

thus K

K

(8.2.81)

zls~l + z z I ~ } n(J-gk)[ --KISI"

k=l

k=l se S

n

consequently, summation of the LH-sides of (8.2.80 and 81) gives K Z [Js J, while the s~ S summation in (8.2.79) over k = 1, .-., K gives K IJu I 9 Finally m

s~ S

IJsl + ILl + I s I - - ~

s~ S

I J s u {Js}l + ILl

= x IJ~[ + [Ju[ s~ S

= I JJ

(8.2.82)

Using all the numbered formulae, with summation over k in (8.2.79), according to (8.2.78) and (8.2.3) we obtain

M + D = ( K + 1)lJI .

8.2.5

(8.2.83)

Solution manifold W e can n o w summarize.

T h e set of e q u a t i o n s (8.2.2), in n u m b e r M (8.2.3) is s o l v a b l e in the i v a r i a b l e s ( c o m p o n e n t m a s s flowrates) m k ( k 1, . . . , K; i ~ J), with the substitutions (8.2.6), and w h e r e mj (j ~ J) equals the s u m (8.2.63), u n d e r the c o n d i t i o n s (8.2.18), (8.2.22), and (8.2.29); the conditions are of structural

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

235

character (determined by the graph G and its subgraphs G k ) and can be verified according to the points (a)-(k) following after (8.2.32). The conditions are sufficient, but have not been shown to be necessary; they only appear as plausible and not very restrictive. Basically, the conditions limit the arbitrariness in the choice of the subsets Ek (4.2.14). One then restricts the solution in the primary variables mj (j s J) and y~, (j s J; k = 1, ..., K) to the subset obeying the conditions (8.2.69) of positive mass flowrates. The primary variables are then uniquely determined by the m~, according to (8.2.63, 63a, 64), and vice versa. Then the set M of solutions has D degrees offreedom, where D is the number (8.2.78); in rigorous mathematical terms, M is a differentiable (in fact, analytic) manifold of dimension dimM= D

(8.2.84)

and a submanifold of the N-space of the variables mj and y~, see (8.2.4). We have, by (8.2.83) with (8.2.4) M +D =N

(8.2.85)

hence the number of degrees of freedom equals the number of variables minus the number of equations (constraints), as corresponds to what has been intuitively expected. In this sense, we say that the system of equations (8.2.2) with (8.2.1) obeying the above conditions is regular. The wording will be further precised in Section 8.4.

8.2.6

Remarks

(i)

It is the elimination of 'dependent' equations (constraints) that has made the system regular. Let us for instance replace the number K(n) in the K(n)-R o (n) scalar equations (8.2.2a) for some n ~ T u by the number K of all the components C k occurring in the whole system. According to Chapter 4, the balance will be again correctly written; a species C k not present in the node n balance is nonreacting in the node and does not affect the other component balances. Indeed, let us go back to Section 4.3. For a nonreacting species Ck, we have Vk~ = 0 for r = 1, ... , R, thus the k-th row of matrix S (4.3.8) equals zero. Then, in the matrix S O (4.3.10), having possibly re-ordered the components, the corresponding row equals again zero and is necessarily among the rows of matrix B. Hence n k is one of the components of vector n' in (4.3.12) and by (4.3.13) we have

236

Material and Energy Balancing in the Process Industries

nk = 0 for nonreacting Ck

(8.2.86)

as already stated in Section 4.3. By the way, we have thus shown that in (8.2.50), a nonreacting species cannot be represented by any component of vector n"(n) thus m"(n) in (8.2.52), and its balance does not represent any degree of freedom. It is only the reacting species that can do so, as they represent certain reaction degrees of freedom. If a nonreacting component Ck (such as component C 3 = N 2 in the example 4.6) is present in a reaction node n balance (see n = R in (4.6.2)) then the result (8.2.86) completes the balance. If it is not present (see C, = S in n = R above) then the result (8.2.86) is again correct. But according to (8.2.1) with Cn~ = 0 for all i e Ek, the result (8.2.86) is as well a consequence of the a priori assumption (definition of the subset Ek), as corresponds to (8.2.20. If replacing the K(n)-Ro(n) scalar equations (8.2.2a) by K-R o (n) equations involving also the species Ck not present we thus, in the formal scheme, generate an equation that is a consequence of the other constraints. It can be shown formally (by inspection of the above derivations) that the number D of degrees of freedom will remain unaffected. But the number M of equations will increase to (say) M' by (8.2.87)

M' - M = E ( K - K(n)) ne T~

where possibly, for some n e T u we have K(n) < K, hence the system will no more be regular; we shall have D>N-M'ifK(n)
Tu

(8.2.88)

thus if some component is not present in some node balance. The result (8.2.85) with (8.2.84) has also the following interpretation: The set of equations (8.2.2) with (8.2.1) is a minimal set of equations representing the balance constraints. (ii)

The a priori assumptions (8.2.2f and g) make the corresponding variables y~ 'observable', thus uniquely determined by the constraints. One could also delete a priori the latter variables in the list of all the variables. As can be shown formally, the number D of degrees of freedom would not be altered; only the system (8.2.-2) would be modified. We have preferred including all the equations (constraints) as written for the sake of formal simplicity, leaving possible modifications to the reader.

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems (iii)

237

Let in particular all the nodes be nonreaction (Ro(n) = 0 for any n ~ T u ), and let us admit the presence of all the K c o m p o n e n t s in any stream (E k = J for k = 1, -.., K). Imagine for example a system of distillation units. Then D(n) in (8.2.2a) is unit matrix for any n ~ T u , hence nk

(n) = 0 (n ~ T u ; k = t, . . . , K)

(8.2.89)

stands for (8.2.2a), with E R - J in (8.2.1). Introducing I Tu I x I JI matrix Cu as submatrix of C restricted to rows n ~ T , , further c o l u m n vectors

k = 1, . . . , K: Yk =

(8.2.90)

~mi/Mk

i~J of dimension l J I, and s u b s y s t e m (8.2.2a) reads

I JI x

K matrix Y of column vectors YR, the

m

C u Y = O.

(8.2.91)

The equations (8.2.2b-e) remain unaltered, and because J-E k - Q~, the constraints (8.2.2f and g ) a r e absent. In (8.2.3) we then have K(n) = K and R 0 (n) = 0, and the last two sums over k = 1, -.., K equal zero. In (8.2.78), we have again Ro(n) = 0 and K(n) = K, Juk = Ju n E k = Ju, and Sk = S for any k = 1, . . . , K. (iv)

This book deals with balancing proper, not so m u c h with more general p r o b l e m s of m a t h e m a t i c a l modeling. In Chapter 4, we have eliminated the (integral) reaction rates W~ in the node balances (4.3.2) as a priori u n k n o w n parameters. W h e n necessary, they can be c o m p u t e d according to R e m a r k to Section 4.5; see (4.5.12). In more detailed models, the reaction rates can be given functions of the other process variables and occur explicitly in the model equations. The elimination then brings no advantage and we can return to the original form of balances (4.3.2).

Recall the parametrisation of the solution manifold 5~. The last group of parameters was represented by vector m" (8.2.52), where the subvectors m"(n) were computed from vectors n"(n); see (8.2.50). In each node n ~ Tu , we thus have introduced R0 (n) parameters where Ro (n) is the dimension of the reaction space R(n). Alternatively, we can introduce directly certain R o (n) independent reaction rates Wr(n) (r = 1, ..., R o (n)) in node n e Tu . Indeed, by (4.3.2) the whole vector n(n) is uniquely determined as (linear) function of the Wr(n), instead of the parametrisation using (8.2.50 and 51).

238

Material and Energy Balancing in the Process Industries

Let us have, for each node n ~ T u , certain R o (n) independent admissible reactions (R0 (n) = 0 for a nonreaction node). We can extend the set of model variables by the reaction rates W~ (n) where n s Tu, r = 1, ..., R0 (n). The information contained in the equations (8.2.2a) is conserved when replacing each of the equations by n s T u 9 n(n) -R~n)W~ (n)s~ (n)

(8.2.92)

r=l

where s~(n) are the (linearly independent) column vectors of the stoichiometric matrix. For each n e T u, we thus have deleted K(n)-Ro(n) scalar equations and added K(n) scalar equations (8.2.92); recall that the dimension of vector n(n) is K(n). The number of equations has thus been increased by K(n) - (K(n)-R o (n)) = R0 (n), and at the same time, we have R 0 (n) new variables for each n e T u . The difference 'number of variables minus number of equations' remains unaltered and equals again the number of degrees offreedom (dimension D of the solution manifold Yr and the new set of equations is again minimal. The new equations replacing Eqs.(8.2.2a)read more explicitly n~

Tu

Rn (n)

k ~ K(n)" n k ( n ) =

~Z Vkr(n)Wr (n)

r=l

(8.2.93)

where K(n) is the set of chemical components present in the node n balance (K(n) = I K(n) l). The quantities n k (n) are the sums (8.2.1), Vk~(n) are the stoichiomentric coefficients in the R o (n) independent reactions, and W~(n) are the additional variables (integral reaction rates).

8.2.7

Example

The results summarized in Subsections 8.2.5 with 8.2.6 are mainly of theoretical importance in showing that the set of balance equations (8.2.2) h a s D degrees of freedom where D = N-M (8.2.85): the set of constraints is minimal. The D independent parameters specified according to Subsection 8.2.4: points (a)-(g) represent one of the possible choices; they can be found applying a systematic procedure as described. Any mathematical model of the system must involve the balance constraints. The constraints can be used directly when measured data for an existing plant are available and have to be adjusted (reconciled); see further Chapters 9 and 10. Other kind of problems (lying beyond the scope of this book)

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

239

are design and simulation problems. In order to specify some solution, the model is then completed by further equations taking into account the physicochemical processes in the nodes, such as the reaction kinetics. It is then useful to have an a priori idea of what independent parameters have to be fixed, the other being determined by the balance. The parameters are just certain degrees of freedom of the balance constraints; they can occur as independent variables in the additional equations. As an illustration, let us consider the sulphuric acid plant according to Section 4.6. The scheme contains two reaction nodes, B and R. In (8.2.2a), D(n) is unit matrix with the exception of the two reaction nodes; the components nk (n) of n(n) will now be restricted to k ~ K(n) where K(n) is the subset of chemical species actually present in the node n balance. In node n = B, component C 4 (HzO) is absent. In (4.6.5), we thus delete the C4-column and the third row corresponding to the trivial balance n4 (B) = 0. Hence

D(B) =

S

0 2

N2

SO 2 SO 3

C1

C2

C3

C5

-1

1

0

0

1/2

0

0

1

0

0

1

0

0

1

1

C6

(8.2.94)

In node n = R, components C~ (S) and C 4 (H20) are absent. Hence in (4.6.6), we delete columns C~ and C4 and rows 1 and 4 giving

D(R) =

02

N2

SO 2 SO 3

C2

C3

C5

1

0

0

1/2

0

1

0

0

0

0

1

1

C6

(8.2.95) .

From D(B)n(B) = 0 we obtain n 1(B)--

n 5 ( B ) - n 6(B)

3 n2 (B) - - n 5 (B) - 2 n6 (B) n 3 (B) = 0

(8.2.96)

240

Material and Energy Balancing in the Process Industries

and from D(R)n(R) = 0

1 n 2 (R) n 3

-

2

n 6

(R) (8.2.97)

(R) - 0

n 5 (R) =

- n6

(R)

Hence, with n k (n) =

m 5 (B), as

m 6

(B),

m 6

1

mk (n) (8.2.5), we can take (8.2.98)

(R)

independent parameters according to points (f) and (g) of Subsection 8.2.4.

We have three splitters S1, $2, $3 in Fig. 4-2. With splitting ratios ctj (8.2.12) obeying (8.2.13), as independent parameters we can further take for instance cx3 (giving 13~4 "- 1 - cx3), ~7 (giving cx8 = 1 - cx7) , and cx~6, CXl7(giving CXl5= 1

(8.2.99)

- 13~16 - 0~17 )

This is point 8.2.4:(e) Recall that using Figs. 8-5, 8-6, and the analysis that follows after the figures in Subsection 8.2.1 we have verified that the conditions (a)-(k) following after (8.2.32) are fulfilled. Indeed, the subgraphs Guk are either connected or have two connected components ( t h u s / ~ = 1); see then also (8.2.33 and 33a). If G,k is connected, the splitter input flowrates m~,~ (s e Sk" component Ck present in splitter s) are arbitrary, if ~ = 1 then some m~,~is a linear combination (8.2.33a). Let us go through k = 1, ... , 6 according to points 8.2.4: (a)-(d). The reader may himself draw the subgraphs G k and Guk. If k - 1 (C~ = sulphur) then S1 = O (no splitter in G1 ). The subgraph G~ = Gul is connected, Nul = N O = {0, B }, Jux J~ = { 11 }. Vector u 1 (8.2.35) is absent. In (8.2.40-43) we have p0 = 1, R ~ - 0 thus subvector x ~ is absent (no degree of freedom) and subvector y0 is the scalar m 111., it obeys the node B balance n

m~l'

= -

m, (B)

[ = - M, n, (B)]

(8.2.100)

with (8.2.96): elemental sulphur is totally consumed by the two combustion reactions. I l k = 2 (C 2 = 02 ) or k = 3 (C 3 = N 2 ) the subgraphs G k are identical, as well as Guk 9 We have ~ = 1, Sk = {S1, $2 }. The connected component G luk with

Chapter

Solvability and Classification of Variables II- Nonlinear Systems

8 -

241

N~ = {B, R } and J~k -- { 5 } generates a linear dependence between the two input streams m k2 (into S1) and mk6 (into $2), generally according to (8.2.34-37)', for L k = 1 we have more simply (8.2.33). We can take for instance m k2

as

(k-2and3)

(8.2.101)

independent parameters. Then, by (8.2.33a) for

k - 2 and 3"

6

2

mk - mk + mk

(B) +

mk

(8.2.102)

(R)

with (8.2.96 and 97) where again m k (n) = M k n k ( n ) . It is simply the species Ck balance of the subsystem of nodes N~ - {B,R} where m k3 + m k4 - m k2 ., observe that by (4.3.1) rewritten as (4.5.1), mk (n) is the production rate of species Ck in node n, in mass units (zero for the nonreacting species C3). In (8.2.43) we have p1 _ 1 and R 1 - 0, hence subvector Xk is absent (no degree of freedom) and y~ is the scalar mk5 subject to the balance (8.2.16) thus 3

m~ - m k +

mk

(B) (k = 2 and 3) where

3

2

mk -

~3 mk

(8.2.103)

The connected component Gu~ has N o - {0, D, A1, A2} and J ~ k = { 1, 9, 10}. We o is absent (no degree of freedom) and have pO = 3 and R~ = 0 hence subvector Xk 1 mk, 9 m k~0. Because the nodes n e N o are nonreaction we yO is of components mk, have, for k-

2 and

1

2

3" m k = m k ,

9

mk -

~

6

mk,

10

mk

-

6

m k

(8.2.104)

as the reader readily verifies; the splitter outlet streams (j ~ J~) obey m~, - ~j m~?, and for mk6 we have (8.2.102). If k = 4 (C 4 - H20 ) then $4 - {$3} and Gu4 is connected (L 1 - 0), with Nu4 -- N O = {0, D, A1, A2} and Ju4 = J~ - {1, 12, 13, 18} (arcs not incident to $3). Because L~ = 0, the splitter input flowrate m414

(8.2.1.05)

can be taken as an independent parameter. In addition we have p0 _ 3, R ~ = 1, hence one of the flowrates mJ4 (j ~ J~ 4 ) can be chosen arbitrarily. Let for instance 1

m 4

(8.2.106)

be taken as an independent parameter. Then the split output flowrates are determined by the ratios ~j (8.2.99) and m414,and the remaining m~ are determined by the incident (nonreaction) node balances (8.2.16). One finds, after simple transformations

242 13 m4 =

Material and Energy Balancing in the Process Industries 14

m~ + (x15m 4

18 14 m 4 = ~16m4

(8.2.107)

1 m 412 _ (0~16 + ( ~ 17) m 4 I4 - m4

If k = 5 (C 5 = S O 2 ) then S 5 = {$2} and L 5 = 1. In G~5 we have N~ - {B, R} and J~5 = {5}. Here, in (8.2.35) we have S s - L 5 - 0 hence subvector v 5 is absent (no degree of freedom) and u 5, is just the scalar m 65 (input into $2). We already know that, by (8.2.33a) m ~ - m s (B) + m s (R)

(8.2.108)

is determined by the parameters (8.2.98) with (8.2.97). We further have P~ = 1 and R~ - 0. The only stream j - 5 ~ J~5 is determined by the node B balance thus m~ = m 5 (B);

(8.2.109)

the reader can check again that it is the balance (8.2.16). Further, in G~ we have N o = {A1, A2, 0), J~5 = {9, 10}, and the streams are determined by 9

10

m5 = (~7 m6, m5 = m~

(8.2.110)

(no degree of freedom).

Finally if k = 6 (C6 = SO3 ) then $6 = {$2,$3}, L 6 = l , and G~6 = G~5. Hence, in the same manner as above m 6 = m 6 (B) + m 6 (R)

(8.2.111)

and m~ = m 6 (B)

(8.2.112)

are determined, while the input stream to $3, not incident to any node of G]u6, can be taken arbitrarily. The reader may notice that the latter stream cannot be taken as dependent (determined) by (8.2.33a), a s ~ls6 = 0 for s = $3. Thus m~ 4

(8.2.113)

can be taken as independent parameter. In G~ we now have N O = {D, A1, A2, 0} a n d J~ 6 = { 9 , 1 0 , 1 3 , 1 8 } . We have pO = 3 and R ~ = 1, hence one of the streams j ~ J~6 can be taken as independent. Let for instance

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

m~~

243 (8.2.114)

be taken as independent parameter. From the incident (nonreaction) node balances we then obtain, after simple transformations 4 = m~8= m6 - (z17

4-

~

(8.2.115)

m 9 = m~~ - c~8 m 6 + ((Z16 + 0~17) m 14 with (8.2.111). Considering in addition that m~, = ~j m~? for j ~ J~ (output streams from splitter s) we see that all the component mass flowrates m~, are determined by the independent parameters (8.2" 98, 99, 101, 105, 106, 113, 114). Thus 9

m 5 (B), m 6 (B), m 6 (R)

(8.2.116a)

(~3' 0~7' 0~16' (Z17

(8.2.116b)

,4

14

(8.2.116c)

m4 , m6 m 41 ,

m~~

(8.2.116d)

can be taken as independent parameters (degrees of freedom) according to the systematic procedure described. The primary variables mJ

mj = Zk mJk

and

yJk =

(8.2.117)

mj

are then also determined. Formally, we add the assumptions y~, = 0 for j ~ E k thus for the streams j where component Ck is absent. An assiduous reader can verify that the number of scalar equations (8.2.2) is M = 113, the number o f variables (yJk = 0 for j ~ E k included) is N = 126, and correctly D = 13 = N-M is the number of degrees of freedom. The choice of independent parameters is not unique and can be modified. For example the concentration of sulphuric acid in stream 14 is limited by the absorption technology to a rather narrow range (some 98 wt.% of H2804, which corresponds to the SO3 mass fraction y~4 = 0.8). Instead of the parameters m414 and m~ 4 w e can take as independent

14

m14 (-" m 4 + m~4)

and

y 4(m 4) = ~ m14

(8.2.118)

244

Material and Energy Balancing in the Process Industries

thus in (8.2.116c) and

m~4 -- y~4m14

14 14 m4 - m14 - 26 m14

(8.2.118a)

are determined. The nonabsorbed quantities m 9 (from A1) and m~~ (from A2) are both small; the absorbers are designed to make them as small as possible. Considering that, by (8.2.115)3 (~17 m~ 4 -. t~ 8 m6 - (Z16 m~4 -t- m 9 - m~~

we can take, instead of the parameter 0~17 , also the flowrate m 9 as independent and determine

1

1~17 : 0~8 m~ 4 - (~16 q- m-~4 ( m ~ -

m~o

)

(8.2.119)

1 with (8.2.118a). In the group (8.2.116d), we then have m4, m~, m~~ And the like. For instance with the above modifications, the parameters (8.2.116a) are determined by the performance of the reaction nodes B and R (where m 6 (B) thus SO3 production in B can be, according to the conditions in the combustion chamber, assessed as a small or even negligible quantity); the remaining parameters a3, o~7, ~ in (8.2.116b), and also m14 (thus m~4) can be varied as process parameters (with the condition 0 < al7 < 1 in (8.2.119)), while m 41 represents the humidity of air into D. The sum m~+ m 23 represents mass flowrate of dry air into S 1; here, the ratio m~/m~ is, in fact, determined by Nature (thus by the standard composition of air, idealized as mixture of 02 and N 2 ). Having fixed the ratio a priori, we have a further constraint and the number of degrees of freedom decreases by one. - It remains to note that the technological scheme has been simplified; the example was just a simple illustration.

8.3

GENERAL SOLUTION MANIFOLD

8.3.1

Energy balance equations

In addition to the component mass balance equations (total mass balance included as a consequence), let us now consider the energy balance according to Chapter 5, thus (5.2.11). Because the balance (5.2.11b) is consequence of the total mass balance, the additional set of equations reads, by (5.2.11a) Cuhu+

C'h'+

Sfi'

Duq=0

(8.3.1a)

Oeq - 0

(8.3.1b)

= 0

(8.3.1c)

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

245

with (5.2.1-7) and (5.2.12-16). In the first step, let us regard as variables the components of vectors 1~ (5.2.7) and q (components qi )" According to the consideration following after (5.2.4), let us assume rankCu = I T~ I

(8.3.2)

(full row rank) as a simplifying hypothesis; see also the fine-printed remark at the end of this subsection. We now have in addition the subset of nodes D E (energy distributors, see Remark (i) in Section 5.1). Concerning the graph GE with reduced incidence matrix D (5.2.1), let us adopt the following conventions. Given nodes n' and n" of a graph, they are called (mutually) adjacent when there exists an arc of endpoints n' and n". Then our conventions read

(a)

If some node d ~ DE is adjacent to node 0 (environment), but not adjacent to any n ~ T u then it is merged with node 0, thus deleted. [The convention is merely formal.] Any two adjacent nodes d' ~ D E and d" ~ D E are merged. Possible isolated nodes are then deleted [again merely formal convention].

(b) (c)

Consequently, any node d ~ D E is adjacent to some node n ~ T u . Compare with an analogous convention introduced for the splitters (4.2.9-10). As an immediate consequence of the conventions, for any d ~ DE we can find some arc i ~ E (net energy stream) whose other endpoint is in T u , hence with an appropriate arrangement of the set E we have the partition +1 o De

~

D~"

~

(8.3.3)

+1 matrix I' with

I D El

x

I D E[

diagonal

matrix I'

of elements

+1

or -1, and

I DE [ • (I E l - I D E 1) matrix O,". The corresponding partition of vector q reads

q' ] } rDEr q=

q"

} IEI-IDE!

(8.3.4)

and because the inverse of l'is again I', Eq.(8.3.1b) reads q'=-lo

eq .

(8.3.5)

246

Material and Energy Balancing in the Process Industries

Adding the identities qi

=

qi for the components of q", we can write

q = Q(q")

(8.3.6)

where Q is a linear map (function). Thus vector q" uniquely determines the whole q (and vice versa). According to (5.2.12 and 13), Eq.(8.3.1c) represents E I J~ ] scalar equations s~S (8.3.7)

/-~ =/Ts(s ~ S , j ~ Js)

where/T is specific enthalpy of the material in stream j. Adding the identities /~s = ~ we have h' = Hs (fis)

(8.3.8)

with I S I -vector hs of components ~

(s ~ S);

(8.3.9)

the function H s is again linear. Finally with (8.3.2), for an appropriate order of the streams j ~ Ju let us partition Cu=(

Cu~

, Cu2 ) } ITul

(8.3.10)

IWul IJul-lZul where Cul is IT u [ • ]T u I regular; it is the reduced incidence matrix (node 0 deleted) of a spanning tree of (connected) subgraph Gu. The corresponding partition of vector h u reads hu =

hul) } Iwol ha:

(8.3,11)

} IJu[-lTul

and Eq.(8.3.1 a) reads hul = - Cu~ (Cu2 hu2 + C'h' + Duq).

(8.3.12)

To (8.3.10) also corresponds the partition of vectors m u in (5.2.5) and flu in (5.2.7)

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems mu =

(mul)

and flu

mu2

fin2

} IJ.l-lrul

247 (8.3.13)

hence with the notation (5.2.14) hul = (diagmul) llu~

and

hu2 - (diagmu2) flue 9

(8.3.14)

In the same manner as in Section 8.2, we shall limit ourselves to the case when, according to (8.2.69) mj>0foranyj~

J;

hence in particular the matrix diagmul is invertible. Then, by (8.3.12) with (8.3.14), (5.2.16), (8.3.8), and (8.3.6) flu1 = (diagmul)-z hu 1 where hu~

- Cul (CuE (diagmu2)fluE + C (diagm')H s (fi~) + D u Q(q"))

Adding the identity flu2 ^

^

=

flu2 we can write

^

h u - H u (m, huE, h s , q")

(8.3.16)

where m is the whole vector of mass flowrates (5.2.5); H u is an (infinitely differentiable) function of the vector arguments as written. Considering the whole vector 1] (5.2.7) of specific enthalpies/T (j ~ J) and the vector q of net energy flowrates, with (8.3.6), (8.3.8), and (8.3.16) we obtain the (infinitely differentiable) function H determining uniquely fi ) = H ( m , fluE, fis, q"). q

(8.3.17)

In the second step, we shall replace the components /T of 1] by the temperatures 7~ of the streams j ~ J. Generally, /T is function of the thermodynamic state variables

/?]J - f J ( r j, ~; PJ) (j E J)

(8.3.18)

where T is temperature of stream j, yJ represents its composition (independent mass fractions y~ of components Ck present in the stream), and PJ is pressure. We

248

Material and Energy Balancing in the Process Industries

suppose that the state of aggregation of the stream and its pressure are given (known) a priori. [Here, we neglect possible pressure drop along a pipeline through which the material flows.] We have

0f j

- c~ > 0

(8.3.19)

~)Tj J is isobaric specific heat of stream j. To be more precise, we assume that where Cp the function fJ used in the computation (not only the theoretical physical quantity itself) obeys the condition (inequality). Assuming the functions f J infinitely differentiable, conversely T j is uniquely determined by/2/j and yJ (given PJ ) as an infinitely differentiable function. As a result, the variables (8.3.20a)

7~ (j ~ J) in number ]JI and

(8.3.20b)

qi (i ~ E) in number I EI

are uniquely determined by the mass flowrates and mass fractions obeying the equations (8.2.2), and by the additional 'independent' variables Tj

temperatures corresponding to components/T (8.3.21a)

of vector ha2, in number [Ju 1-IT~ I TJs

temperatures corresponding to components/T~ of vector hs, in number

I sI

(8.3.21b)

and qi

"'" components of vector q", in number I E [ - I D E

I

9

(8.3.21c)

In Section 8.2, we have shown that the mass flowrates and mass fractions are uniquely determined by the D parameters (8.2.56-58). In order to conclude that the D former parameters plus the variables (8.3.21) represent a parametrisation of a differentiable manifold, we proceed in an analogous manner as in the fine-printed paragraphs preceding formula (8.2.73). The formal proof is again too technical, though simpler. Basically, it relies upon the fact that the functional relation (8.3.17), with temperatures substituted for specific enthalpies, comprises [Ju 1-]Tu [ identities Tj = Tj by (8.3.21a), [ S I identities Tj~= Ts by (8.3.21b), and IEI-I DE I identities qi = qi by (8.3.21c).

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

249

T h e a d d i t i o n a l e q u a t i o n s (8.3.1) are in n u m b e r

Mh= ITul + IDEI +s slJsl

(8.3.22)

a c c o r d i n g to ( 5 . 2 . 1 ) , ( 5 . 2 . 4 ) , ( 5 . 2 . 1 2 are in n u m b e r

Dh-ILl-ITul

a n d 13). T h e a d d i t i o n a l p a r a m e t e r s ( 8 . 3 . 2 1 )

+ IsI + I E I - I D o l

w h e r e , o f c o u r s e , w e c a n h a v e D E - Q5 thus

(8.3.23) I DE[ = 0. T h u s

Mh + Dh = Nh

(8.3.24)

where

ILl + IsI +

x [Jsl s~ S

+

IEI-

[JI + IEI

(8.3.25)

is the n u m b e r o f the a d d i t i o n a l v a r i a b l e s (8.3.20), a c c o r d i n g to (5.2.2 a n d 3), w i t h (4.2.9). R e c a l l (8.2.85). C o n s e q u e n t l y , for the w h o l e s y s t e m o f b a l a n c e e q u a t i o n s (8.2.2) a n d (8.3.1) w e h a v e (M + Mh) + (D + Dh) = (N + Nh).

(8.3.26)

Although the simplifying hypothesis that Gu is connected has been adopted as plausible, the cases where it does not hold are still conceivable. The reader can imagine for instance two parallelly ranged pre-heaters of a stream (thus preceded by a splitter) followed by another splitter in the direction of the flow. Such cases then require special consideration. So in the case just mentioned, it can be shown that the set of constraints remains minimal, thus (8.3.26) applies. Generally, previous graph analysis can be helpful. We have to examine whether the subgraph G, [N-S, Ju ] obtained by the deletion of splitters (s e S) and incident arcs 0" e J') is connected. If it happens that G, has L+I connected components where L > 1 we can modify the analysis applied to subgraph Guk; see Fig. 8-7 in Subsection 8.2.2. The analysis is simpler; if the reader has perused Section 8.2 in detail he will be able to find the modifications. In Fig. 8-7, we replace Juk by Ju, J' n E k by J', Nuk by T u w {0}, and Sk by S. The connected components (say) G~u (l = 0, ..-, L+l) of Gu will be of node sets N ~and arc sets J~. Now in the equation (8.3.1a), we merge the nodes in each G~o where 1 <_ l < L in the same manner as in (8.2.20)" the vector components m ki are replaced by the components h ~(i e J~) and h '-i (j ~ J') of vectors h, and h', respectively. By the merging, we arrive at equations the components of h', thus in fact h 'j~ (s e S) have to obey, given the splitting ratios ctj. We then obtain a condition analogous to (8.2.29) with L instead of Lk and matrix (say) G in lieu of Gk; it will be a sufficient condition for the solvability. The equation (8.3.6) will subsist. The new solvability condition satisfied, we find the components of vectors hu and h' as functions of vector q" and of components of a composed vector such as (8.2.49), in number

250

Material and Energy Balancing in the Process Industries

IJul + I s ] - I N ~ I + 1= ILl + I s I - I T , I given the splitting ratios aj; adding to this number the number (8.3.21 c) of components of vector q", we have the number D h (8.3.23). As above, we conclude again that the whole set of balance equations is minimal thus obeying (8.3.26), with the same interpretation of

Mh, Dh, Nh.

8.3.2

The whole system of balance equations

Let us summarize the results of Sections 8.2 and 8.3. We have admitted the absence of certain components (chemical species) Ck in certain material streams (arcs)j of the graph G. For any k (= 1, ..., K), Ek is the subset of streams where component Ck can be present. The arbitrariness in the choice of the E k has been limited by the structural conditions (8.2.18), (8.2.22), and (8.2.29); see Section 8.2, first paragraph of Subsection 8.2.5. We have also limited the admissible flowrates to positive ones. [Observe that a system of balance equations that would not admit of such solutions would be technologically absurd.] We have further assumed, as motivated by the consideration following after (5.2.4), that the subgraph G u [N-S, Ju ] of G, obtained by deleting the splitters and the incident arcs, is connected; the hypothesis can be weakened, as remarked at the end of Subsection 8.3.1. Considering the graph GE of net energy streams and the incident nodes (distributors d e D z included), we have adopted the conventions (a)-(c) following after (8.3.2). We have considered an a priori given state of aggregation and pressure PJ in each of the streams j e J. Then the system of balance equations (8.2.2) with (8.2.1), and (8.3.1) has Dto t --

D + Dn

(8.3.27)

degrees offreedom, where D is the number (8.2.78), and D h the number (8.3.23). In rigorous mathematical terms, the s e t ~//'tot o f the solutions is a differentiable

manifold of dimension dimYfftot = Dto t

(8.3.28)

and a submanifold of the space of the Ntot -" N + Nh

(8.3.29)

variables mj, y~ (j e J; k = 1, ..., K), T j (j e J), and q~ (i e E); here, N is the number (8.2.4) and Nh the number (8.3.25).

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

251

We have Mto t -t- Dto t = Nto t

(8.3.30)

where Mto t --

M + Mh

(8.3.31)

is the number of the (scalar) balance equations, with (8.2.3) and (8.3.22). Hence the number of degrees of freedom equals the number of variables minus the number of equations (constraints). We thus call the system of equations regular; see the precision in Section 8.4. See also Remarks (i)-(iv) in Subsection 8.2.6. Further remarks will follow.

8.3.3

Heat exchangers See the last paragraph of Section 5.2. Formally, a heat exchanger is a

couple of nodes {n', n"} c T u

(8.3.32)

(non-splitters) such that there is just one arc (net energy stream) i ~ E of endpoints n' and n", and no arc (material stream)j ~ J connecting directly the two nodes; see Figs. 5-2 and 5-3. Physically, we regard as heat exchangers only such couples of nodes where the energy flowrate qi represents a heat transfer rate. The partition of a heat exchanger (as technological unit) into two nodes (8.3.32) has been motivated by separate (generally multicomponent) mass balances of the 'cold' and 'hot' sides. On the other hand, it can happen that we are not interested in the (unknown) heat transfer rates. Then the latter can be eliminated by merging the nodes (8.3.32). Let us first partition the set E of net energy streams E = E* w ECx (E* n ECx = 0 )

(8.3.33)

where E~x is a subset Of streams i e E corresponding to certain (not necessarily all) heat transfer rates to be eliminated. Each arc i e E~x has precisely two endpoints n'i and n'~ that, according to (8.3.32), represent some heat exchanger, say H i = {n'i , n'~ } (i e Eex) 9 Then the set T u is partitioned

(8.3.34)

252

Material and Energy Balancing in the Process Industries

T u - T~ u u H i

(8.3.35)

i~ Eex

where T~ contains the remaining nodes (units); the environment is not element of any H i (nor element of T~ ). Then merging the two nodes of each H i in the graph G u [N-S, Ju ] yields graph G~ of nodes 0 (environment), n e T~, and (say) h i (i e Eex ) and

(8.3.36)

of arcs j e Ju because no arc j e Ju has been eliminated by the merging. The nodes h i a r e uniquely determined by the couples H i , hence represent the corresponding heat exchangers (technological units) in the reduced graph. With the plausible hypothesis that G u is connected, also G~ is connected. The merging concerns the energy balance only. We can apply the merging as well to the whole graph of nodes n e N and d e D z, and arcs j e J and i e E, thus to the union of graphs G and Gz; see Section 5.2, first paragraph. Then, in the reduced graph, the arcs i ~ E~x are deleted. Recall that merging two nodes (thus n I and n'~, i e Eex ) means summation of the corresponding rows of the incidence matrix. Thus the matrix C (5.2.4) is 'reduced' to C~ C'* ) } T u and nodes hi, i ~ Eex 0

c"

Ju

J'

Is

(8.3.37)

and the matrix D (5.2.1) to D . = / D~ ) D~ E

} T~andnodeshi,i~Eex }D E

(8.3.38)

*

operating on column vector q* of components qi, i ~ E* (= E-Eex );

(8.3.39)

the arcs i e E~x have been deleted. Here, C~ is reduced incidence matrix of graph G~, thus if G~ is connected then

253

C h a p t e r 8 - Solvability and Classification of Variables II- Nonlinear Systems

rankC~ = IT~[ + [Eex[

(8.3.40)

(full row rank). Observe that none of the nodes d e D E has been affected by the merging, and that no d e DE is incident with any i e Eex. Hence the partition of submatrix D; is again of the form (8.3.3), with certain (zero) columns of De deleted. Consequently, instead of the system (8.3.1) we now have Cuhu+

C' h ' + D uq = 0

(8.3.41a)

D: q* - 0

(8.3.4 lb)

Sfi'

= 0.

(8.3.41c)

We can now repeat, step by step, all the considerations of Subsection 8.3.1 leading to the final conclusion, summarized in Subsection 8.3.2. Only in (8.3.22)-(8.2.25), we have the new numbers

~--ITS[

+ IEex[ + IDEI + Z

s~ S

D;= ]Ju]-IT;I-Igexl ~=

[Jsl

+ IsI + IN*l-IDol

(8.3.42) (8.3.43)

[Jul + IsI + ~slJ~l +IE*I

--I J[ + IE*I

(8.3.44)

thus

(8.3.45)

/lfh + Dh = / ~ h . By the way, observe that

IT:I + [Eexi--ITul-IEexl because just one node in each [E*[ = I E [ -

Hi

(8.3.34 an 35) has been deleted, and that

[Eexl

by (8.3.33). Hence, in fact Dh= Dh

(8.3.46)

thus the number of degrees of freedom remains unaltered; indeed, any of the heat transfer rates qi (i E Eex ) is uniquely determined (say) by the node n'i energy

254

Material and Energy Balancing in the Process Industries

balance, given the other values of the variables. The reduction (8.3.41) thus only simplifies the problem, without reducing the information (of course if the heat transfer rates qi, i e Eex a r e not known a priori; else such qi would not be deleted). In the results (8.3.27-31), ~ is substituted for Nh and h/Fhfor M h , Yff~ot stands now for ~tot and we have the number of degrees of freedom d i m M tot = d i m M t o t = Dto t

(8.3.47)

Recall that we have not merged the nodes in the component mass balances. As an example, let us again consider the sulphuric acid plant; see Subsection 8.2.7. In Section 5.3, we have extended the technological scheme according to Figs.5.4-5.7. The component mass balances are extended trivially. By Fig. 5-4 we add the equalities | ~

mk

'

m~ ( k = 2 , 3 , 4 )

and by Fig. 5-7b m k3

m 3' a n d m 6k

m~' ( k = 2 , 3 , 5 , 6 )

In the component mass balances, no new degrees of freedom arise. The subgraph G, is obtained by deleting the splitters (S1, $2, $3) and the incident arcs 2,3, 4, 6', 7, 8, 14, 15, 16, 17; the reader can draw the figure and verify that G u is connected thus condition (8.3.2) is obeyed. The set D E of energy distributors is empty and the matrix D e in (5.2.1) thus (8.3.3) is absent. Thus IDol - 0 and q = q" in (8.3.4). According to the procedure described, all the components of vector q hence qf, qa, q~, qb, qal, qa2 can be taken as independent parameters. According to (8.3.9), the components of subvector hs thus the specific enthalpies

can also be taken as independent. In G u , we can find a spanning tree by the deletion of certain (of course not arbitrary) arcs in number ]Ju [-] T~I = 11-8 = 3; see (8.3.10) The deleted arcs represent independent parameters; let us delete for instance arcs 1, 12, 18 hence

/~,,/~,~,/q,~ can be taken as independent. As independent parameters, instead of the specific enthalpies we can take the temperatures T 2, T 6', T 14, T l, T 12, T 18 "

Observe that by the node B balance mll/~ll + my/~y= m5/2/5 + qb

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

255

where the mass flowrates have been determined according to Subsection 8.2.7, the (so far independent) parameter qb Can be determined when replaced by another independent parameter, viz. /~rll resp. T 11

which is specific enthalpy resp. temperature of input liquid sulphur. The quantity qh represents steam production in energy units and is thus, along with other variables, determined by the energy balance of the plant, given the independent parameters. The new parameters (added to those found in Subsection 8.2.7) represent enthalpies resp. temperatures of input streams 1, 11, 12 and process parameters T 2, T 6', T 14, T 18along with heat withdrawn from the system (qd, qal, qa2), power supply qf to the fan F (Fig. 5-4) and heat qe transferred in the exchanger {E3, E6} (Fig. 5-7b). If the latter two nodes are merged according to (8.3.36), we obtain graph G~ instead of Gu. We can apply the formal procedure as above, or consider simply that the variable qe has been eliminated while, according to (8.3.47), the number of degrees of freedom remains unaltered. Hence a new independent variable has to be added to those found above. We can take for instance the variable T y thus the temperature of dry combustion air into B, according to Fig. 5-7a. Then qe, when necessary, can be computed from the balance of the exchanger where the parameters of the streams are known. Recall finally (cf. Subsection 8.2.7) that the independent parameters are arbitrary only with respect to the set of the balance equations. Eventually, they must obey in addition the equations of the extended model, taking into account the physicochemical processes in the nodes (such as heat transfer in the exchanger) and the construction parameters of the apparatuses.

8.3.4

Heat and mass balances

Recall the introducing paragraphs to Section 5.4, and also the last paragraph before E x a m p l e 6 of Section 8.1. The 'heat and mass balances' (5.4.6) are of the same form as the complete balances examined in this section, only the chemical composition dependencies are suppressed. Instead of the m u l t i c o m p o n e n t mass balance, we have formally a single-component mass balance, and in the energy balance, the (a priori assessed) term s can occur in addition as a constant in (5.4.6)3. The whole analysis holds true as well for the system (5.4.6). W e can write immediately the result. In the single-component mass balance we set (number of components) K = 1 and E k = E~ = J. The only limitation is the condition mj > 0 for a n y j e J

(8.3.48)

(and possibly the condition formulated in the fine-printed paragraphs after (8.3.26), if G u happens to have several connected components). In (8.2.78) we thus have

256

Material and Energy Balancing in the Process Industries

O--ITul + IJu[ + IsI + z [ J s l - I s I s~ S

thus simply

D= IJI-IN~I

;

(8.3.49)

see (3.1.4-6). D is thus the dimension of the null space M = KerC, which is the space of solutions of the mass balance equation (3.1.6). According to (8.2.3) we have formally

M-ITul + IsI +s~slJ~l § IJul + IsI thus M=

INu] + IJI

while in (8.2.4) N=2[JI thus N - I J[ + I J[. The I J[ equations (8.2.2c, d, e) concern I J[ variables y~(j ~ J, with K = 1) and determine only y~ = 1. Deleting the latter trivial equations, the number of equations becomes IN. [ and that of variables mj (j J) equals l J I, as corresponds to the equations (5.4.6)~ and (5.4.6)2. On the other hand, the numbers M h (8.3.22), D h (8.3.23), and N h (8.3.25) remain unaltered. Consequently, if ~M"tot is again the set (manifold) of solutions of (5.4.6), we have

(8.3.50)

dimPl4ftot = D + D h (= Dto t )

with (8.3.49) and (8.3.23). The set of equations is again minimal thus obeying

IJI +N~--INul-I-Mh-I-Otot

;

(8.3.5~)

the number of variables mj, 7~ (j ~ J), and qi (i ~ E) equals the number of scalar equations plus the number of degrees of freedom. Recall that the pressures PJ and the states of aggregation in the streams j ~ J are again regarded as given a priori. By the way and in particular in the simplified heat and mass balances, the pressure dependence of specific enthalpy can often be neglected at least in a condensed (solid or liquid) phase.

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

257

In the same manner as in the previous subsection, we can eliminate the heat transfer rates qi (i E Eex ). The modification is left to the reader.

8.3.5

General nonlinear models . .

The balance equations analyzed in Sections 8.2 and 8.3 are examples of nonlinear models (sets of constraints). We limit ourselves to 'steady-state' models, in the sense explained in Section 4.2, first paragraph. Generally, such model reads g(z;o) = 0

(8.3.52)

where z is an N-vector of unknowns, and p is a vector of a priori given parameters; for instance in the above energy balance equations, p is of components (pressures) PJ (j ~ J). The vector-valued function g is of, say, M components where M is the number of the corresponding scalar equations. Generally, the equations are nonlinear. Introducing the vector p makes sense when the dependence of the solutions on the a priori parameters is analyzed. Among the components of p, we can also have certain thermodynamic data such as specific heats, standard enthalpies, and the like. Some of the components of p can be certain constants of the model specific to the given system, but not known a priori; then the problem reads finding the values of these constants based on certain measured data. We shall not deal with problems of this kind. Hence the model will read more simply g(z) = 0 ;

(8.3.53)

any admissible value of z determines some state of the system (not to be confounded with the narrower concept of a 'thermodynamic state'). Let M b e the set of solutions; we call the model solvable when M~: 0 . A special case is when the model is uniquely solvable, thus when g(z) = 0 if and only if z = z 0

(8.3.54)

where vector z 0 represents the unique solution, thus M = {z 0 }. Generally, the set M of vectors z obeying the condition (8.3.53) is called feasible set (Stanley and Mah, 1981a), with respect to the constraints represented by the M scalar equations in (8.3.53). A meaningful model is clearly solvable. But generally, the set M can have rather strange properties and the analysis is difficult. In the case of balance

258

Material and Energy Balancing in the Process Industries

equations, we have shown that Mis, as the mathematicians put it, a differentiable manifold. We then call M the solution manifold of the system (8.3.53) of constraints. The situation can change when certain components of vector z have been fixed (for example as the measured ones). See the examples in Section 8.1. The analysis is then considerably more difficult. To some extent, it is facilitated when the feasible set is a manifold, as will be shown in the next sections.

8.4

JACOBI MATRIX AND LINEARISATION

In the analysis of nonlinear problems, the first step usually consists in that the problem is linearized. That means that nonlinear functions occurring in the problem are approximated by Taylor expansion up to the first degree. For a vector-valued function of several variables, the coefficients (partial derivatives) are elements of the Jacobi matrix. Let g be an M-vector of scalar functions gm of N scalar variables Zv, thus of N-vector z. Then the Jacobi matrix of g, say G, is the M x N matrix of elements Gmv = ~gm ] ~Zv

(8.4.1)

Let us designate Dg = G

(8.4.2)

the Jacobi matrix Of an arbitrary g. It is again a (matrix) function of z, thus Dg(z) is the value of Dg at point z. We shall suppose that g is (at least) twice continuously differentiable in an N-dimensional (open) interval U; 'open' means that the boundary of U is not included in U. For example in the above balance equations, U is limited by the conditions mj > 0 (j s J), possibly also by other inequalities in addition. Let z 0 s U. For an arbitrary z s U, the Taylor formula (restricted to the first degree) reads g(z) = g(z o ) + Dg(z 0 ) ( z - z o ) + d(z; z 0 )

(8.4.3)

where given Zo, d(z; z 0 ) depends on z in the manner that if the difference Z-Zo tends to zero, d(z; z 0) becomes 'negligibly small'. In fact Taylor's formula gives an exact expression to this d in terms of an integral; if dm is the m-th component then

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

dm =

]~,Dm;vv, (zS~,v )(Z~v, )

259

(8.4.3a)

V, V

where AZv is the v-th component of z-z 0 , and Dm;vv, = Dm,v, v (symmetric) is determined (as an integral expression) by the values of the second derivatives on the line segment between z 0 and z. One thus can say that d m is 'quadratically small' in the differences AZv, if they tend to zero. The linearisation of function g in a neighbourhood of point z 0 (for small AZv ) consists in approximating _

g(z)

--- g(z 0 ) + Dg(z 0 )(z-z 0 ) = Dg(z 0 )z + (g(z 0 )-Dg(z 0 )z 0 ) .

(8.4.4)

In this manner, for example the equation (8.3.53) becomes a linear equation (7.1.1) and can be analyzed by the methods described in Chapter 7. Let us have a model (set of constraints) g(z) = 0

(8.4.5)

as in (8.3.53), where g is the function considered in (8.4.1-4). Let us assume g infinitely differentiable on the N-dimensional interval U, so as to simplify the mathematical terminology and in accord with Sections 8.2 and 8.3. Thus Taylor's formula (8.4.3) applies. Even then, not any such function admits of a meaningful approximation (8.4.4). Imagine a function h, with m-th component h m = (gm)2. Then h(z) = 0 if and only if g(z) = 0, thus the set M of solutions remains unaltered. But because Dh m = 2g mDg m , we have Dh(z 0 ) = 0 for any z 0 ~ M. Although the approximation (8.4.4) with h in lieu of g remains correct up to the first degree, it is of no use when the properties of M have to be examined. Less trivial examples can be invented. Generally, the same set of constraints (representing the same feasible set M) can be formulated by an infinity of equivalent equations. While in linear algebra, the equivalence means simply a regular transformation (multiplying by a regular square matrix), this is not the case when nonlinearity is admitted. Then not any (though equivalent) formulation of the model is equally appropriate for the solvability analysis. Observe that in (7.1.4), we assumed that the matrix C was of full row rank. It is thus natural to require also in the present case that rankDg(z o ) = M for any z o e M

(8.4.6)

thus that Dg(z 0 ) is of full row rank (at least) on the set M of solutions of (8.4.5). Such model (better to say, its formulation using the function g) will be called regular. To be precise, we (of course) suppose that there exists some solution, thus M ~ ~ . Well-known theorems of Analysis then show that the rank remains constant at least in a neighbourhood of the set M (at points 'not very distant'

260

Material and Energy Balancing in the Process Industries

from M), and that the feasible set M is a differentiable manifold, in the rigorous mathematical sense. Moreover, its dimension is .

dimM= N- M ;

(8.4.7)

compare with (7.1.1), (7.1.4), and (7.1.6-8) with the subsequent text. Going back to the approximation (8.4.4) where now z0 ~ M thus g(z 0 ) = 0, and substituting v for z-z 0 we have (approximately) the equation Dg(z 0 )v = 0

(8.4.8)

for any other solution z = z 0 + v ~ M of (8,4.5), 'not too far' from z 0 . Thus any other such solution is (approximately) found when shifting slightly the point z 0 along a vector v obeying (8.4.8), thus belonging to the null space KerDg(z 0 ) of dimension N-M. The null space (generated by the vectors v) is called the tangent space to M at point z o (~ M). In Analysis, it is defined by a more sophisticated construction directly, using the properties of M (independent of the particular g)" the tangent space is generated by tangent vectors to curves traced on M and passing through point z o . Imagine a surface in threedimensional space, and the tangent plane at some point of the surface. In Sections 8.2 and 8.3, we have made use of other ideas leading to the concept of a (differentiable) manifold by way of certain 'degrees of freedom' in a parametrisation of M . The ideas are interrelated and when precised mathematically (which, of course, has not been possible in this book), they lead to the same concept. Although we have stated that the sets M i n (8.2.84) and Mtot in (8.3.50) are differentiable manifolds we have, in fact, still not proved that the models (8.2.2) and (8.3.1) are regular in the above sense. But it is so, indeed. Thus under the structural hypotheses summarized in Subsection 8.3.2 The multcomponent mass balances (8.2.2) with (8.2.1) and the energy balance (8.3.1) constitute a regular model (set o f constraints).

The proof is again quite nontrivial in details; let us only outline the idea of the proof. It follows the same lines as in (8.2.11)-(8.2.72), and in (8.3.2)-(8.3.26). One first transforms the equations (8.2.2) with (8.2.1), using the substitutions (diffeomorphisms) (8.2.70 and 71) with (8.2.69); one proves that at points of M, the rank of the system remains unaltered when Eqs.(8.2.2b) are multiplied by l/mis , Eqs.(8.2.2c) by ~.i mj, Eqs.(8.2.2d and f) by mj, and (8.2.2e and g) by mj~. One then forms the differentials of the transformed equations and puts them equal to zero. The differentials are in terms dm~, d%, and dmk (n), and one finds that all the differentials of the transformed variables dm~ and dotj are linear combinations of the components of dx, dv, dc~, and din" where x, v, c~, and m" are the same as in (8.2.55), thus representing D scalar components. In this manner, one shows that the dimension of the null space is D, thus the rank of the Jacobi matrix equals N-D = M for the system (8.2.2).

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

261

The analysis of subsystem (8.3.1) is easier. One has only to show that the submatrix corresponding to the new variables T -i and qi is of full row rank. At constant mass flowrates and compositions of the streams, by (8.3.19) the transformation (substitution) (8.3.18) is a diffeomorphism, and in terms of t h e / T and qi ~at constant m (vector of mass flowrates) the equations (8.3.1) are linear; the result is then immediate, using the linear transformations (8.3.5)and (8.3.15). Thus the rank of the submatrix equals Mh and is again full. The row rank of the whole system is also full, equal to M + Mh.

In the more detailed analysis of the balance equations that will follow in Section 8.5, we thus can start from the Jacobi matrix Dg where g is represented by the LH-sides of the equations (8.2.2) and (8.3.1). The equations and variables are partitioned into certain groups and not assigned a fixed numeric order; in addition we make use of composed variables (substitutions) such as (8.2.1). So as to find a general expression for the Jacobi matrix, let us rewrite it in terms of differentials. If h is a differentiable function of variable y then its differential is the formal expression dh(y) = Dh(y)dy

(8.4.9)

having the property that with the substitution y = fix) thus k(x) = h(f(x)) we have (8.4.10)

dk(x) = Dh(y)df(x)

(the 'chain rule' of differentiation). The formalism enables us to identify the elements of the Jacobi matrix as coefficients at the corresponding differentials dzv in any row of the differentiated equation, whatever be the order of the equations gm (Z) = 0 and components Zv of z. In the system (8.2.2) with (8.2.1) we thus have the differentials in the rows n ~ T u in number K(n)-R o (n) for each n: D(n)dn(n) where

dn(n) =

dnk'(n) k~ K(n) (8.4.11a)

1 Z Cn i ( y ~ d m i + m i d y ~ ) 9, and dnk (n) = - -Mki~E~

262

Material and Energy Balancing in the Process Industries

here, K(n) is the set of K(n) indices k of components Ck occurring in the node n balance; further, in the rows s~S"

Z_ Csj dmj, thus dmjs - Z dmj

j~js

s~J~

(8.4.11b)

represents the corresponding row of the Jacobi matrix, with element 1 in column mj, -1 in columns mj, j ~ J~, else zero; further

s ~ S , j ~ Js k = 1, ..., K:

dy~s- dye, K Z dy~

J ~ Ju"

k=l K

s e S"

Z dy~

k=l

k=l,...,K J e Ju, J ~ Ek"

(8.4.1 lc) (8.4.11d)

(8.4.11e)

dy~

(8.4.11f)

dye?

(8.4.11g)

and

k=l,...,K s E S, j~ ~ F_~"

The equations (8.2.2b-g) are linear and the elements of the Jacobi matrix in the rows (8.4.1 lb-g) are constants (+1 or 0). In (8.4.1 la), the elements can be found if the values yik and m i are given. Then, if Dtk (n) (t = 1, --- , K(n)-R o (n); k ~ K(n)) is an element of matrix D(n), the t-th differential in the n-th subset of rows equals

1 - E Dtk (n) ]~ Cni (y~ d m i + m i dyl~ ) . k~ K(n) M k ie F~

Thus for example the element in the column m i (i E J) of the Jacobi matrix equals, in the t-th row of the n-th subset

1 - Y~ Dtk (11) Cn ini k i k~ K(n) M-k Yk

where Bik = 1 if i ~ E k , Bik = 0 if i ~ Ek; cf. (4.5.5). The element in the column y~ (corresponding to dy~ ) equals

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems -Dtk (n)

1

Cni mi

263

for i ~ E k

and zero for i ~ E k . One could also discard directly all the variables y~, i ~ E k , equal to zero by hypothesis, from the list of variables. We leave the modification to the reader. The reader can also set up the Jacobi matrix in the special case where no chemical reaction is admitted, and all components are admitted in all streams; see then Remark (iii) in Subsection 8.2.6. In the system (8.3.1), we shall make use of the substitutions (8.3.18). Consequently, with (5.2.15 and 16) where lau is vector of components/T, j ~ Ju (arcs nonincident to splitters)

(8.4.12)

and fi' is of components/T, j s J' (J' - J-Ju )

(8.4.13)

we have the following differentials. In the subset of rows n ~ Wu: A

C, (diagla u )dm, + C'(diagll')dm' + Cu (diagmu)dllu + C'(diagm')dh' + Dud q (8.4.14a) in the rows d ~ DE:

D e dq

and in the groups of rows s ~ S, in number [Js I for each s" Sdh'

(8.4.14b)

(8.4.14c)

thus -d/-~ + d / ~

(j ~ J s)

represents the corresponding I J~ I rows of the Jacobi matrix, with elements -1 in column/T~, 1 in column/T, else zero. The reader can himself set up the (quite analogous) Jacobi matrix of the reduced system (8.3.41), and also of the (possibly again reduced) system of heat and mass balances (5.4.6). In the latter case, the subsystem of mass balance equations is linear (thus with Jacobi matrix equal to the matrix of the linear subsystem), and the remaining subsystem yields Jacobi (sub)matrix of the same format as (8.4.14). What remains now is to express d/T in terms of the differentials of the variables T and yJ, see (8.3.18), and substitute the differentials for the

264

Material and Energy Balancing in the Process Industries

components of vectors dlau and dl~'. Again, pressure PJ is considered known a priori, not perhaps adjusted according to the measured values, nor computed from the balance; thus dP j does not occur in the differential. By the way, regarding also (some of the) PJ as variables would only increase the number of variables, but not that of the equations (constraints); hence the (full row) rank of the matrix would not change. The expression for d/T (j ~ J) depends on how detailed is our thermodynamic description of the system. Let us begin with the simplest case, that of heat and mass balance equations. Then, according to Section 5.4, the specific enthalpies /T, denoted by hJ~, are functions of temperature only; see (5.4.5). Thus J dT j (j e J)

(8.4.15)

by (8.3.19). [Attention: the subscript s has here nothing in common with any J can still splitter; it is related to the wording 'sensible heat'.] The specific heat Cp depend on temperature (thus inducing a higher degree of nonlinearity), or be assessed as constant. Note that if steam is used as a heating medium, or if it arises in a steam generator then the corresponding items can be included as components of vector q, not necessarily among the heat contents of streams j~J. Generally, thermodynamically consistent are the expressions (C.10) /T

-

Z yJk/4~ k

(8.4.16)

with summation over indices k of components Ck present in the stream;/4~ is partial specific enthalpy of Ck in stream j, thus at T (given PJ ) and 3; (composition of the stream). Since the mass fractions must obey the condition (4.2.2), one of them is not independent and we have E dy~ = 0 .

(8.4.17)

k

Then, by (C.10) and (C.21) we have the thermodynamically exact expression dI)J-c~dTJ+ E /-lJdy~ ke C~j')

(8.4.18)

where C(j) is the set of k such that Ck is present in stream j. But in practice, other expressions for specific enthalpy can be applied directly, for example using tables and interpolation formulae. Recall only again the requirements for

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

265

thermodynamic consistency, in particular if some nodes are reaction nodes, if phase transitions take place, or, in certain cases, also if the heat of mixing is relevant; see the examples in Section 5.3. So the expression for d/T requires an ad hoc consideration. Remark: Complications can arise when stream j is a multiphase mixture. The energy (enthalpy) balance in terms of the specific enthalpies/210 will anyway hold true, as well as the individual component balances, where YJkis the mass fraction of Ck in the whole mixture. The case will, however, require again a special consideration. Generally, new variables, say fie will occur, where fie is mass fraction of phase f in stream j, obeying F. E' ~ f=l

1

(8.4.19)

where Fj (usually - 2) is the number of phases f in stream j. Then

/_Tf= :~J ~/.~ (f) f=l

(8.4.20)

where/T (3') is specific enthalpy of phase f in stream j. We shall not analyze the general case. In particular if the stream j is a vapour-liquid mixture (Fj = 2), an additional condition will be that of thermodynamic phase equilibrium; thus given again PJ, the variables /T, T J, ~ , and the Ck-fractions in the vapour-liquid mixture are interrelated. For example/2p is obtained as function of T j, p~ (mass fraction of the vapour phase), and the chemical composition of the stream as a whole (while the composition variables of individual phases are eliminated by the equilibrium conditions), with parameter PJ; the function can be primarily obtained in molar quantities, and re-calculated to specific enthalpy and mass fractions. In practice, the case can occur as a vapour-liquid feed into a distillation column.

8.5

CLASSIFICATION PROBLEMS; OBSERVABILITY AND REDUNDANCY

8.5.1

Examples

Let us return to the examples in Section 8.1. We have seen that a classification of variables, which was uniquely possible in the linear case, can fail if the equations are nonlinear. Some (though incomplete) information can be obtained when the equations have been linearized. So let us begin with the Jacobi matrix of the system (8.1.1). Let C be the matrix, hence

266

Material and Energy Balancing in the Process Industries m l m2 m3 y l y2 y3

1

C -

1

T 1 T2 T3 Q

-1

yl y2 _y3 ml m2 -m 3

(8.5.1)

h l ha -h3 Pl P2 -P3 q l q2 -q3 -1 where ()h i

i = 1' 2, 3" P i - mi ~~yi ' qi = mi Cpi ,.

(8.5.1a)

Cpi ( > 0 ) is again isobaric specific heat. We have eliminated one of the mass fractions in each of the material streams, say yi1 - 1-y~ where y i = y~, which brings no complication in the present case. We see immediately that if m i > 0

(i = 1, 2, 3) then C is of full row rank even if the variable Q is deleted (heat exchanger absent). But C depends on the state variables (vector z). Let M b e the solution manifold; if z e Mthen z obeys Eqs.(8.1.1). In the sequel, we are referring to the numbered examples of Section 8.1.

Example 1

Let us have measured the variables 3~ and 7~ (j = 1, 2, 3), and Q; we have the equivalent conditions (8.1.11) with (8.1.2) for z e M. Let us partition the matrix C-(

B

,

unmeasured

A

)

(8.5.2)

measured

as in (7.1.10). We shall examine the matrix B; thus ml m2 m3

1

1

-1

B-

(8.5.3) f,2

.

The rank of B remains unaltered when j-th column is multiplied by mj > 0, and the sum of columns 1 and 2 is then added to column 3, giving

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

B' =

m 1

m2

m 1

--I- m 2

- m3

ml ~2

m2~2

ml )1

4. m2 ~2

_ m3 ~3

m, ~1

m2 ~2

m, ~1 + m2 ~./2

_

267

(8.5.4)

m3 ~3

with rankB' = rankB

(8.5.4a)

Let first Q r 0. Then the condition of solvability is (8.1.13) and adding the third column in (8.5.3) to the first and second, one finds immediately that this is also a necessary and sufficient condition for rankB to be full rank. Then, whatever be otherwise the measured values, Eq.(8.1.1) is uniquely solvable. Thus if the admissible region of measured values is restricted to a subset of variables obeying the inequalities (8.1.13) and (say) Q > 0, the system can be called 'observable' in the mj, and at the same time we have rankB = 3. Let us now admit the case that Q = 0; see the text before formula (8.1.17), but let still all the temperatures be measured. The measured values can, nonetheless, obey the condition (8.1.13). But then the only solution is m~ = m 2 = m 3 = 0. Conversely insisting on positive mass flowrates, we must have (8.1.14) for the adjusted values; but then rankB < 2. Or we admit an error in Q; then the mass fractions can be assigned the measured values, but the mass flowrates remain undetermined, as the actual Q is unknown. Simply, we do not know where is the error, and also the adjustment conditions remain undetermined. Observe that having admitted Q - 0, we have rankB < 2 at some points of M, while rankB = 3 at other points. The classification of variables fails, while the problem is 'not well-posed'.

Example 2 Let further the heat exchanger be absent. Then the variable Q is also absent, and we have the condition (8.1.14). Let in addition the condition (8.1.12) be satisfied in the admissible region of measured variables. Then, on M, by (8.1.1) where Q = 0, according to (8.5.4) we have

B'-

m 1

m2

0

mlY 1

m2Y 2

0

m,

m 2 T/2

0

]/1

(8.5.5)

thus by (8.5.4a) rankB = 2 on M; recall that ~1 r ~2 by (8.1.12). Let us suppress the latter condition and admit for instance ~1 = ~3. Then also ~2 _ ~3 and on M,

268

Material and Energy Balancing in the Process Industries

we have points where also ~t~ - ~ / 2 ~./3, and at these points, we have rankB = rankB' = 1. Thus rankB is not constant on M. The classification of measured variables fails. Under the condition (8.1.12) for the admitted region of variables, a classification is theoretically possible, as shown after formula (8.1.15). But some flaw still remains. So as to formulate the solvability condition (8.1.14), a pre-treatment of the equations was necessary, using the condition (8.1.6). In complex systems, such pre-treatment is not always obvious by inspection. Having certain actually measured values (of mass fractions and temperatures in the present example), a condition such as (8.1.14) will most likely not be satisfied, due to measurement errors. Thus in the present case, in the linearized system, when starting from the measured values the matrix B will be of rank 3. The formal degree of redundancy will equal zero and no adjustment (reconciliation) will take place. Without the pre-treatment, the system is linear in m~, m 2 , m 3 and the unique solution is m~ = m 2 = m 3 = 0. Observe that we have here rankB = 3 in the a priori admitted region, with the exception of points z e M where rankB = 2. The problem appears again rather as 'not well-posed'.

Example 3 Let us have again Q = 0 (no heat exchanger), but let us have measured in addition m 3 = m3; let rh3 > 0. Thus our B consists of the columns m~ and m2 in (8.5.3), thus

B'-

1

1

~

~2

i,

(8.5.6) .

It is not difficult to show that if the condition (8.1.12) is fulfilled then ~1 ~: ~2 and rankB = 2 in the whole admissible region, in particular on M. In the linearized system, we thus obtain the degree of redundancy equal to 1. On the other hand, so long as rh3 > 0, we obtain again the condition (8.1.14) and this condition satisfied, we have m~ > 0 and m2 > 0. Further, because the two columns of B are linearly independent and because on M, as shown above, with the third column of (8.5.3) the rank of the matrix equals also 2, the ~ 3 - c o l u m n of the Jacobi matrix is a linear combination of the columns of the new matrix B (8.5.6). According to (7.1.27), at points of M the measured variable rh3 can be qualified as nonredundant, in accord with the tentative qualification before formula (8.1.16). But observe that at points z ~ M, the matrix (8.5.3) is of rank 3, thus in the linearized system, again by (7.1.27), rh3 will be qualified as redundant. If

Chapter

8 -

Solvability and Classification of Variables H- Nonlinear Systems

269

we apply an adjustment (reconciliation) procedure using, in any approximation step, a linearized system then all the temperatures, mass fractions, as well as the variable rh3 will be sucessively adjusted. But in the end, so long as rh 3 remains positive, we must arrive at a solution, thus at a point z ~ 91//where rh3 becomes nonredundant in the sense (7.1.27). A mathematical proof not given here (but see Section 10.3) shows, however, that the final rh 3 will then equal the original one; to be quite precise, under a (plausible) norming condition derived from statistics. Thus the variable/'~/3 c a n be called 'nonadjustable'. We can call the problem 'well-posed'. Nonetheless, under unfavourable circumstances thus when the initial estimates (in particular mass fractions and temperatures) are 'too far' from 914, it can also happen that by the procedure, we arrive at rh3 = 0 and m~ = m 2 -- 0 , along with some incidental values of 53 and ktj. Such possibility can never be avoided; the measured values are then discarded. So in practice, an adjustment procedure can fail either because the problem is 'not well-posed' (it then most likely fails always), or because the initial estimates (measured values) are 'too bad'.

Example 4 Let us now return again to the case when a heat exchanger is present, and let us admit positive as well as negative values of Q; see the text before formula (8.1.17). We now add variable T 2 t o the set of the unmeasured ones. Then, in addition to (8.5.3), we have the T2-column of (8.5.1) in B. Thus ^

B-

m~

m2 m3

T2

1

1

0

-1

)1 )2 _)3

0

~1 h 2 _~3 m2

(8.5.7) 2 Cp

Here, our admissibility conditions read (8.1.6) and (8.1.12). Then, clearly, rankB = 3 is full row rank, and we have no redundancy. Now for observability. Deleting any of the columns m~, m2, or m 3 leaves the rank unaltered. Let us delete the column T 2. We then obtain the matrix (8.5.3). If Q ~ 0, on M w e have (8.1.18), hence the condition (8.1.13) with h 2 instead of Ft2 is fulfilled and as above, B with column T 2 deleted remains with rank 3. According to (7.1.28), all the unmeasured variables will be qualified as unobservable, whatever be an estimated value of m 2 (> 0), and of h 2 such that the inequality (8.1.13) is obeyed; most likely, we shall use such estimates for a classification based on the linearized system. Let now, however, Q = 0. The matrix (8.5.7) is independent of this value and our first estimate of m 2 and h 2 will probably lead to the same

270

Material and Energy Balancing in the Process Industries

classification. But there are certain conditions the unmeasured variables have to obey. If subjecting the values to the conditions, thus estimating m2 and h 2 adequately, with Q - 0 in (8.1.18) we obtain the result (8.1.14) where h 2 stands for the correctly estimated h 2. Then deleting the column T 2 lowers the rank to 2. According to (7.1.28), we qualify T 2 as observable. This is quite in accord with the qualification in the text after formula (8.1.18). The adjustment problem can be called 'well-posed'. In the present case, no adjustment of measured values is necessary. The variable T 2 is 'exceptionally observable', else unobservable. The example was somewhat na'fve, it shall only illustrate the different possibilities. Analyzing the further examples of Section 8.1 in this manner would be annoying. The trickiness of a general and complete classification is perhaps evident. So far, any attempt to find some general criteria of observability/redundancy ended at an incomplete classification. By graph methods, one can arrive at a complete classification only under restrictive hypotheses, not warranted by practice. Else, in the best case one ends up with certain groups ('blocks') of variables that defy further refinement. Let us try to analyze the problem by linearisation, as illustrated by the above examples.

8.5.2

Theoreticalanalysis

Let us have again a general model (8.3.53) thus (8.4.5), with the notation and hypotheses (8.4.1-7). Thus the model reads

(8.5.8)

g(z) = 0 where z ~ U

where U is the admissible region; for simplicity, let U be an open (N-dimensional) interval in the N-space of state variables. Thus the Jacobi matrix Dg is an M • N matrix function of z, and we suppose that it is of full row rank M on the solution manifold M. Recall that M c U by (8.5.8). Let us partition the vector z z_(y

) }J x

} I = N-J

(unmeasured)

(8.5.9)

(measured)

with the corresponding partition

Dg=(B, J

A) }M;

(8.5.10)

I

thus the M x J resp. M x I matrices B resp. A are again functions of z. We suppose that the components of subvector x have been fixed a priori, for example

Chapter 8 - Solvability and Classification of Variables II- Nonlinear Systems

271

as measured. If x § is the vector of the fixed values, we have generally z ~ M if x = x § in (8.5.9), and whatever be y. The adjustment problem then involves finding some ~ such that

/:/

M for some y ;

(8.5.11)

given a priori some x +, we use some criterion for the adjustment ~-x § in the case of reconciliation derived from statistics and minimized by the adjustment. Possibly, we are then also interested in the values of y obeying (8.5.11) with given ~. Generalizing the nomenclature of Section 7.3, let us designate M + the set of ~ obeying (8.5.11)

(8.5.12)

thus the projection of the solution manifold M into the l-subspace of vectors x; cf. (7.3.4). Further, given some ~ obeying the solvability condition ~ ~ M § let us introduce the set M(~,) as in (7.3.10); thus

y e M(~)means

/y/

~ M.

(8.5.13)

Clearly, if ~ e M § then M(~) ~ O, else the set is empty. The adjustment problem thus involves finding ~ e M § 'as close as possible' to an a priori estimated (say, measured) value x +, and then perhaps also identifying the set M(~) of vectors y; in particular if M(~) is a one-element set then this y is unique. If the equation (8.5.8) is linear then the adjustment problem is completely solvable. Generally, because B depends on z, also its rank L(z) = rankB(z)

(8.5.14)

is a function of z; of course the function can take only integer values 0, 1, ..., the maximum being < J and < M. [Basically, it can happen that L(z) = 0 at some z; for instance if admitting also zero mass flowrates in (8.5.1), and if T 1 is the unmeasured variable then the rank of B equals zero if m~ = 0.] Let us designate L: the maximum L(z) for z ~ M .

(8.5.15)

Because the integer L < Min(J, M)

(8.5.15a)

272

Material and Energy Balancing in the Process Industries

the maximum is attained somewhere on M . It can be shown formally that if L(z 0 ) = L at some z 0 ~ M then L(z) = L also in some neighbourhood of z 0 in M (at points z ~ M ' sufficiently close' to z o). Moreover, we then have rankB(z) > L in some N-dimensional interval containing the point z0; but we can have L(z) > L if z ~ M. On the other hand, from the regularity hypothesis (8.4.6) it follows that the rank of Dg(z) equals M also in some N-dimensional interval containing z o , whatever be z0 ~ M . The standard argument reads as follows. If the rank of some matrix M(z) equals K at some z 0 then there exists some K • K submatrix whose determinant is nonnull at z0. Assuming M continuous as function of z, the determinant (as a continuous function) is nonnull also in some neighbourhood of z0, hence the rank is K at least; if K is maximum then rankM(z) = K in this neighbourhood.

The following analysis will be based on the properties of matrix B. In order to preclude 'ill-posed' problems, it would theoretically suffice to assume rankB constant on the manifold M. Motivated by the above examples we restrict ourselves, however, to 'well-posed' problems assuming that the rank equals L also in an N-dimensional neighbourhood of any z ~ M; we can suppose that this neighbourhood is an N-dimensional interval. The assumption rankB = L in some neighbourhood of M

(8.5.16)

means that whatever be z ~ M , there exists an N-dimensional interval (say) U~ such that rankB(z') = L for any z' ~ % .

(8.5.16a)

Such adjustment problem will be called well-posed. Let us designate H = M- L ;

(8.5.17)

we have 0 < H< I.

(8.5.17a)

[Indeed, L < M by (8.5.15a). If we had H > I thus M > L+I then, at z ~ ~ we should have L linearly independent columns of B at most, thus L + I linearly independent columns of Dg(z) at most, which would contradict the hypothesis of regularity.] If the problem is well-posed then also H is constant on any ~ ; it can be regarded as a 'degree of redundancy', as it was in the linear case. The

273

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

theoretical analysis shows that the set M § is again of 'dimension' I-H, but it can happen that it does not have all the properties of a differentiable manifold; only when the solution manifold M i s restricted to some neighbourhood of an arbitrary z o ~ M , its projection into the x-space has the required properties. The following results require a subtle mathematical reasoning and formulation. They are based on the fact that under the condition that rankB(z) = L on M, the mapping (projection) that assigns, to any z e M, its x-component according to (8.5.9) is, in mathematical terms, of constant rank thus a subimmersion of rank I-H. The consequences read. (a)

Locally, thus given an arbitrary z0 ~ M, in some neighbourhood V of z0 the set of x e R I such that

Y )~

VnMforsomey~

RJ

(8.5.18)

x

(projection of V ~ M into the x-subspace R ~) is a differentiable manifold of dimension l-H, a submanifold of R~; compare with (7.3.3)-(7.3.4) where M-L = H.

(b)

Given any i e M § (8.5.12), the set of N-vectors z such that

z=

/y/ i

~ M"

(8.5.19)

(section o f M at constant i) is a differentiable manifold of dimension J-L, a closed submanifold of M ; compare with the fine-printed paragraph following after (7.3.10).

(c)

Under the assumption (8.5.16), given

z0=

I r~ )

e M

(8.5.20)

i0

the neighbourhood V in (8.5.18) can be taken as Cartesian product of J-dimensional interval % and/-dimensional Vx (thus V= Vy • Vx) in the manner that the set

Vy m M(io )

(8.5.21)

is a differentiable manifold of dimension J-L, a closed submanifold of Vy; see (8.5.13) and compare with (7.3.9)-(7.3.10). In particular if L = J then, (at least) in a sufficiently small neighbourhood of Y0, the set (8.5.21) equals {Y0 } thus the solution in y is unique. If we now restrict ourselves to a neighbourhood V = Vy x Vx of some z 0 e M , the result (a) implies that the solvability condition (8.5.18) can be written

2'/4

Material and Energy Balancing in the Process Industries

x ~ Vx:h(x)=0

whereh=

/hi/

and rankDh(x) = H ;

(8.5.22)

(8.5.22a)

hence there exists some function h with full row rank Jacobi matrix, determining the manifold (8.5.18) by the condition (8.5.22). Further, the result (c) says that given some x = i obeying the solvability condition, the set Vy n M(i) of solutions in y at fixed ~ is determined by the condition y e Vy" f(y) = 0

where f =

/'/

and rankDf(y) = L;

(8.5.23)

(8.5.23a)

hence there exists some function f with full row rank Jacobi matrix, determining the manifold of solutions in y (e %). The function f depends on f~ (not written explicitly). Moreover, from the condition (8.5.16a) it follows that V(= Vy• Vx) can be taken such that some L rows of matrix B remain linearly independent in '~, the other M-L rows being their linear combinations at any z ~ 'E. With an appropriate order, let g~, ..., gL be the corresponding components of g (8.5.8). Then the equations

Ye %:gl

/'/ ~

--0, "'", gL

/yl

--0

(8.5.24)

in y, at fixed ~, determine again a closed submanifold of Vy, say M'(x). One has clearly Vy n M (~) c M' (~); tracing a curve on M' (~) starting from a point y e Vy n M (~), one can show that the curve remains on M (:~). One concludes by topological arguments, using the result that Vy n M (~) is a connected component of M '(~). Restricting Vy when necessary, one thus shows that the equations (8.2.24) also determine Vy n M (i), thus the set of solutions in y. Hence the conditions (8.5.23) can take the special form (8.5.24). The beauty of m o d e m Analysis has also certain flaws. One of them is in that m a n y theorems are only existence theorems, and often of local character only. An existence theorem states for example that a solution exists under certain conditions, but the verification of the conditions in practice is frequently not less difficult than finding a solution directly by way of trial. In addition, an existence t h e o r e m need not indicate a practically feasible way of finding the solution. M o r e o v e r , a local theorem states only that (expressed in rather vague terms) the solution exists and is unique only if the 'initial guess' in a numerical procedure is 'not too bad'; else the procedure can fail. So in the best case, a theoretical analysis gives an idea of what can be expected, and perhaps suggests a strategy.

C h a p t e r 8 - Solvability and Classification of Variables H- Nonlinear Systems

275

In terms more pragmatical, the above theoretical results can be interpreted as follows. We have some idea of where a solution of the problem can be found; it is a neighbourhood of some (assessed) value z 0 of the state variable z, obeying the constraints (8.5.8) thus g(z 0 ) = 0. Let

v = % x Vx

(8.5.25

be this neighbourhood (interval estimate); Vy is a J-dimensional interval of the y-variables, Vx an/-dimensional interval of the x-variables, according to (8.5.9). We assume that the model is regular (8.4.6); this is the case with a set of balance equations, as shown (under certain hypotheses) in Sections 8.2-8.4. We further suppose that the problem is well-posed (8.5.16). We then expect that with our interval estimate, the solvability condition reads x = ~, where :~ is some solution of the set of equations (8.5.22). This means that at least hypothetically, the equation (8.5.22) can be obtained by elimination of vector variable y from the original constraints (8.5.8). The theory also shows that if the interval Vx is sufficiently small, the vector equation (8.5.22) with (8.5.22a) is equivalent to a set of scalar equations x~ = X1 (xH+~, "- , x i ) . . . . . . . . . . . . XH

-- S H(XH+I,

'X• ...

,

(8.5.26)

V x

x I)

for an appropriate selection of the H variables Xl, "'" , XH with the corresponding re-ordering of the x-components. The number H is uniquely determined, while the selection of the 'dependent' variables Xl, " . , xH is generally not unique, but also not arbitrary. Consider a simple example where I = 2 and H = 1. Let the set (manifold) of solutions of (8.5.22) look like

fI I I I I I I

i

I'- . . . . . . .

1

,

I

I_

Fig. 8-8. Possible manifold (curve) of solutions to (8.5.22)

276

Material and Energy Balancing in the Process Industries

In Vx, we can take X1 as dependent (uniquely) on x2, but not conversely. Taking a smaller interval, say Vx, x 1 depends on x 2 , or also conversely. Generally, any set of H variables (say, x~, --. , XH ) that are uniquely determined by the remaining ones according to (8.5.26) can be called 'redundant' in Vx. As shown, however, this concept of a 'redundant variable' can depend on the choice of Vx . Anyway, the (invariant) number H (8.5.17) can be called the degree of

redundancy. Let further some x = ~ obey the solvability condition (8.5.22). We then expect that with our interval estimate (8.5.25), the set of solutions in y will be determined as that of certain L equations (8.5.24) for an appropriate selection of some L components gl, "'", gL of function g. More generally, having eliminated the variable y in certain H scalar equations, some (M-H =) L equations remain. The H scalar equations (8.5.22) and the L scalar equations (8.5.23) represent the result of such (perhaps only hypothetical) elimination; they obey the conditions (8.5.22a) and (8.5.23a). The number L is again uniquely determined. More explicitly, let us write the equations (8.5.23) as dependent on fl (Yl,

"'" , Y J ; I~) = 0

(8.5.27) fL (Yl,

"'" , YJ; IK) -" 0

where the parameter ~ has been fixed; Yl, "'" , YJ are the unknowns. Again, if the interval Vy is sufficiently small, the equations (8.5.27) are equivalent to a set of equations Yl = Y1 (YL+I,

"",

YJ; x) (8.5.28)

YL = YL(YL+I, "'", YJ;s for an appropriate selection of the variables Yl, YL with the corresponding re-ordering of the y -components. The number J-L represents the number of degrees of freedom for the vector variable y, given ~. Observe that if J - L, thus if matrix B is of full column rank then (assuming y e Vy ) the whole vector y is uniquely determined by ~. In that case, for a well-posed problem, rankB(z) = L - J for any solution z e M, hence the conclusion is independent of the special value of ~. More generally, it can happen that some of the equations (8.5.28) is of the form "'"

yj = yj (~,)

,

(8.5.29)

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

2'/'/

thus the j-th variable is uniquely determined by the given ~,, whatever be the other yj. (j'~j). We then can call yj 'observable' (in Vy ) at x = ft (~ V x ).

8.5.3

Theoretical classification

Even for well-posed problems, only an incomplete classification of variables is meaningful, if it has to be independent of the particular choice of point z 0 ~ M. Let us introduce two formal definitions. (i)

The j-th u n m e a s u r e d variable (yj) is called observable when

Yl__ [y =

_

/Y'l e M (~) and y' =

Yj

9

e M (~)] implies yj = yj (8.5.30)

~,Y'j

w h a t e v e r be f~ ~ M§ recall (8.5.12 and 13). Hence given any :~ obeying

the solvability condition, the j-th unmeasured variable is uniquely determined. (ii)

The definition of a nonredundant variable (xi) is trickier. Let us first have some solution

f~ =

/'/

~ M.

(8.5.31)

The idea of nonredundancy is that leaving Xh = 2h constant for all h ~e i and changing x i , the equation (8.5.8) with (8.5.9) remains solvable: the variable x~ is not determined by the remaining 2h, and even arbitrary, with the condition

Xi E ,(I/'i

(8.5.32)

where '~i is the interval of admissible values; see (8.5.8) where the admissible region U is Cartesian product of onedimensional intervals. Moreover, we shall require a 'smooth' solvability when xi varies: There exists a curve on Mpassing through f~ and leaving the coordinates Xh -- 2h constant for h ;e i, while xi takes all the possible values in ,~i; 'possible' means that given xi ~ ,~i, there exists some point on M with i-th measured component xi. We then call the variable xi nonredundant at

278

Material and Energy Balancing in the Process Industries

point ~. The i-th measured variable is called nonredundant when it is nonredundant at any ~ ~ M .

The classification criteria can be based on the properties of the Jacobi matrix (8.5.10). Alas, the analysis requires again a subtle mathematical reasoning. Definition (i) Let us designate S M ( i ) the section of M a t constant ~, thus the set (8.5.19); let us trace an arbitrary curve on the manifold SM(:~). Let

be a parametrisation of the curve, thus x = ~ and y = cp(t), with components yj = q)j (t) (j = 1, ..., J). Denoting by ~0' the derivative, by (8.5.8) with (8.5.10) we have

(8.5.33)

B(z)q~'(t) = 0

for any such curve. If now the variable yj is observable then necessarily q~j is constant, hence cp~ = 0. Thus conversely, whatever be z e 914and v ~ 1~J B(z)v = 0 implies vj = 0

(8.5.34)

because (:~ ~ M § being arbitrary) otherwise we could trace a curve on SM(~) not obeying the condition cpj = const. The condition (8.5.34) is necessary for yj to be observable. It is also sufficient if the manifold (section) SM(:~) is (topologically) connected. More generally, it is sufficient at least for yj to be locally observable at any "2 ~ M . This means that having arbitrary

'2 =

i

~ M(thus S' e M(~))

(8.5.35a)

there exists some J-dimensional open interval Vy' such that ~, ~ ~' and y ~ Vy' n M ( i ) implies yj = ~j.

(8.5.35b)

Observe that Vy' can be different for different '2, in particular for different ~. Moreover, let the condition (8.5.34) be obeyed also in some N-dimensional neighbourhood ~ of an arbitrary z e M . Then, given arbitrary '2 (8.5.35a), there exists some neighbourhood ~ x Vx of ~ (a Cartesian product of intervals) such that whatever be x~Vx [y e % n M('~) and y' e % n M(:~)] implies yj = y] .

(8.5.35c)

This is the condition (8.5.30) restricted to M ~ ( ~ x Vx ); here, ~ is independent of x (~ Vx). The assertion can be proved using the equations (8.5.24) and the implicit function theorem, along with the hypothesis (8.5.16).

279

C h a p t e r 8 - Solvability and Classification of Variqbles H - Nonlinear Systems

Definition (ii) Let us further consider a nonredundant measured variable xi. Let again i ~ M (8.5.31). We have a curve on M , parametrized as tp(t) / 7(0 =

~(t)

with 7(0) = i and ~h (t) = q/h (0) = Xh for h # i

(8.5.36)

q/i ( t ) = xi + t where t is in some interval such that ~i + t ~ :c/~ (8.5.32). Let e~ be the i-th unit vector of R I (in the x-space), and with the partition (8.5.10) let

(8.5.37)

a i = Ae i

be the i-th column vector of matrix A (a function of z). The condition 7(0 r M thus g(7(t)) = 0 reads B0'(t))tp'(t) + ai 0'(t)) = 0

(8.5.38)

thus in particular, at t = 0 a~ (i) = - B(i)tp'(0)

(8.5.39)

where i is arbitrary. Thus the necessary condition reads: a~ (z) is element of the vector space generated by the columns of matrix B(z), whatever be z ~ M . Conversely, let us assume that the condition is fulfilled in some N-dimensional neighbourhood Vz of any i e M . Thus, whatever be z ~ Vz , there exists some vector v(z) ~ R J such that a i (z)

=-

B(z)v(z) .

(8.5.40)

Assuming again that the problem is well-posed, we can take Vz such that there are certain L columns of B that remain linearly independent. We thus can take v(z) as the vector of coordinates of ai (z) relative to these L columns (constituting a basis), the remaining components of v(z) being zero. We thus have a continuous vector field v(z) on Vz. Let us consider an integral curve of the field, parametrized as 9(t), and then a curve

7(0 =

9(0) V(t)

such that 7(0) = i and ~'(t) = e i .

Then, so long as z = 7(0 e d dt g(T(t)) = Dg(z)y'(t) = B(z)v(z) + a i (z) = 0 by (8.5.37) and (8.5.40), thus 0 = g(i) = g0'(0)

(8.5.41)

280

Material and Energy Balancing in the Process Industries

hence T(t) ~ M for any t. We thus have a curve on M passing through i and leaving xh = ~h (t) = ~h(0) = Xh constant for h ~: i, while xi = ~ (t) takes all the values in some neighbourhood of Xi limited by the condition

(8.5.42)

7(0 6 ~ .

This holds true whatever be i ~ M . By the way, replacing i by z 1 = 7(t) ~ M for some t ~: 0, the curve can be continued, and so on. We shall not examine the conditions under which we thus obtain a 'maximal curve' such that finally x~ takes all the possible values in ,c/i (8.5.32). We simply state that the condition (8.5.40) is sufficient for x i to be (at least) locally nonredundant at any i ~ Yv~ thus nonredundant on Vz n M.

The interpretation of the theoretical results, remitting again the mathematical precision, can read as follows. We have a regular model (8.5.8) with full row rank Jac0bi matrix (8.4.6), and a well-posed adjustment problem, thus the partitions (8.5.9 and 10), where matrix B is of constant rank L, thus obeying the condition (8.5.16). We are usually not interested in all the possible values the state variable z can have theoretically, but rather examine a limited region, say an interval V (8.5.25) where our hypotheses are expected to hold. Instead of the full solution manifold M, we limit ourselves to some portion M n V o f M. Then: An observable unmeasured variable (yj) is uniquely determined by the measured variables' values (obeying the condition of solvability), say ~j = Yj (i) where :~ is the vector of (adjusted) measured values; the hypothetical function Yj need not be known explicitly. The necessary condition of observability is (8.5.34). Compare with (7.1.18) where we substitute B(z) for B, with variable z. According to (7.1.28)~, the condition is equivalent to rankB0)(z)=L- 1 (z~ Mn

V)

(8.5.43)

where L = rankB and where Ba) is the M x (J-1) submatrix of B with the j-th column excepted. The necessary condition requires the condition (8.5.43) to be fulfilled at any z e M ~ V. Under plausible mathematical hypotheses, the condition is also sufficient; else, it is sufficient only for yj to be locally observable according to (8.5.35). The local observability is rather a theoretical construction" it does not preclude the existence of another solution (say) y] for given ~, it only states that within some (not a priori known) finite distance from ~j, no other solution exists. [For the uniqueness of the solution in the (a priori assessed) interval V, it is sufficient to suppose that the condition (8.5.43) is fulfilled whatever be z e V, thus not only at points z obeying g(z) = 0; see (8.5.35c). But the latter condition can be too strong.] In particular if L = J thus if matrix B is of full column rank the condition (8.5.43) is obeyed at any z e V, where B is of constant rank by hypothesis. We

Chapter

8 -

Solvability and Classification of Variables II- Nonlinear Systems

281

then call the vector variable y observable (on M n V). Thus the whole vector y is uniquely determined by ~. A n o n r e d u n d a n t measured variable is not affected by the solvability condition; it can be freely varied in some interval leaving the model equations solvable, if the remaining fixed measured variables obey the condition of solvability. The necessary condition reads, according to (8.5.39) a i

(Z) E ImB(z) (z s M m V)

(8.5.44)

where ai is the i-th column vector of matrix A (8.5.10). Compare with (7.1.27) 2. Thus at any z ~ M n V, a i (z) is linear combination of the columns of B(z), thus in fact of certain L (= rankB) linearly independent columns. A sufficient condition assumes that ai (z) ~ ImB(z) at any z ~ V, thus not only at points z obeying g(z) = 0. The theoretical definition following after (8.5.31) is mathematically formal. In the applications (in particular in the statistical reconciliation of measured values), even the fact that the condition (8.5.44) is fulfilled at some z ~ M n V is relevant. Under frequent statistical hypotheses, the variable is then nonadjustable (at point z): by the reconciliation, the measured value can remain unadjusted (at least if the error is considered as uncorrelated with the other errors); see further Chapters 9 and 10. The reader has probably noticed that, in contrast to the linear case, we have not completed the classification by introducing 'unobservable' and 'redundant' variables. The main reason is that we have not considered and classified all the possible cases that can occur theoretically. Still, let us at least examine the cases where some of the above necessary conditions is not satisfied. Going back to (8.5.43), let conversely rankB0~ (z) = L at some z ~ M n V .

(8.5.45)

That means that some set of L (= rankB) columns of B(z), not comprising the j-th column constitutes a basis of ImB(z); the L column vectors are thus linearly independent. By standard arguments, one concludes that the equality (8.5.45) holds true also in some neighbourhood of point z. We thus can state that the variable yj is not observable. It is not even locally observable, because the condition (8.5.43) is necessary even for local observability. The statement (8.5.45) thus 'disqualifies' the j-th unmeasured variable: we cannot expect that with an arbitrary measured (and adjusted) ~, the value of yj will be determined. It can happen that the condition (8.5.43) is fulfilled at certain particular values of z, and even that the yj-value is uniquely determined by some ~; see the example 4 in Section 8.1, Fig. 8-2. But such case is exceptional, due to some coincidence. It is left to the reader's taste, if he then will call the variable 'unobservable' or perhaps 'observable at' some ~.

282

Material and Energy Balancing in the Process Industries

Let us further return to (8.5.44) and let conversely a i (Z) ~ ImB(z) for some z ~ M n V

(8.5.46)

Thus adding the column a i (Z) to those of B(z), the rank of the extended matrix equals L+I. Again by standard arguments, one concludes that a i (Z') ~ I m B ( z ' ) for any z' in a neighbourhood of z. We can say that the variable xi is not nonredundant, or perhaps 'redundant' in some neighbourhood of z. The variable will generally be adjusted by reconciliation, with possible exceptions mentioned above, after formula (8.5.44) (end of the paragraph).

Remarks

(i)

The theory presented in this section applies to any regular model and w e l l - p o s e d adjustment problem, not only to balance equations. As already mentioned, it is in fact sufficient to assume that the function g is twice continuously differentiable. On the other hand, according to Section 8.2, at least in case of component mass balances the set M of solutions is even an 'analytic manifold', and it is such also at least if simple mathematical expressions are used for the enthalpy functions in the whole balance model. In that case, certain stronger conclusions can be drawn from the theory. The parametrisations introduced in Sections 8.2 and 8.3 allow one to suppose that the manifold M is (topologically) connected, and to use the theorem of analytic continuation: if an analytic function equals zero in an open subset of a connected region R then it equals zero in the whole R An analytic function of the state variable z is also an analytic function of the parameters, which can be assumed to lie in a connected region R of dimension D (number of degrees of freedom). By the (analytic) diffeomorphism, to an open subset of M corresponds uniquely an open subset of R . In particular if some determinant of a (sub)matrix (function of z) equals zero in some open subset of M then it equals zero in the whole M. Thus if the rank of some matrix M(z) equals (say) K in M n V where V is open in the z-space 11N then rankM(z) < K on M ; indeed, the determinant of any (K+ 1) x (K+ 1) submatrix (if there is any) equals zero in M ~ V, thus everywhere in M (but not necessarily in the whole admissible region U c 1~y , nor in V). The rank can be lower, however. Then the same theory shows that the points z e Mwhere rankM(z) < K are 'rare' in M; they will be contained in some 'negligible' subset of M, of 'lower dimension' than D = dimM. We shall not precise the latter concepts.

The basic hypothesis adopted in this section is that of 'well-posedness' (8.5.16). Let now M be analytic. The hypothesis then implies rankB(z) < L for any z ~ U

(8.5.47)

Chapter

8 -

Solvability and Classification of Variables H- Nonlinear Systems

283

where U is the admissible region in (8.5.8). Let further the condition (8.5.43) be fulfilled for all z e M n V0 where V0 is an arbitrarily small open N-dimensional interval; then the rank is < L-1 everywhere on M , thus equal to L-1 because rankB(z) = L on Mby hypothesis. Hence if the unmeasured variable yj is (locally) observable in a neighbourhood of some point it is (at least locally) observable on the whole solution manifold M. Conversely if we have (8.5.45) at some arbitrary z e M then (as the rank is < L everywhere on Mby hypothesis) the rank equals L 'almost everywhere' on M. Thus if yj is not (locally) observable at some point, it can be observable at most in some 'exceptional' subsets of M. Thus for well-posed problems (and analytic manifolds), the classification observable / not observable (unobservable) is 'almost complete'. The remaining cases ('exceptional observability') represent mere coincidence. Briefly, an unmeasured variable is either

everywhere 'observable'

or

almost everywhere 'unobservable'

where the quotes recall that mathematical precision has been remitted. Retaining again the well-posedness and analyticity hypotheses, let the condition (8.5.44) be fulfilled for all z e M n V0 where V0 is again an arbitrarily small open N-dimensional interval. Hence the rank of the extended matrix (B(z), a i ( z ) ) equals L on M n V0 , thus it is < L everywhere on M and being also > L (as rankB = L by hypothesis), it equals L on the whole M. The stronger hypothesis, namely that the condition is fulfilled in some V0 open in R N, implies as well that the condition is fulfilled in some neighbourhood of the whole M where rankB = L by hypothesis. Thus if the measured variable is (by the formal definition) nonredundant in a neighbourhood of some point, it is nonredundant on the whole M. According to the terminology introduced in the text following after formula (8.5.44), if the variable is nonadjustable in a neighbourhood of some point then it is nonadjustable on the whole M. Conversely if we have (8.5.46) at some arbitrary z e M then the rank of (B(z), ai (z)) equals L+ 1 thus (being < L+ 1 everywhere on Mby hypothesis), it equals L+ 1 almost everywhere on M. Thus if xi is not nonredundant (in simpler terms, 'redundant') at some point, it can be nonadjustable (or even nonredundant) at most in some 'exceptional' subsets of M. Thus again, for well-posed problems (and analytic manifolds), the classification 'nonadjustable'/'redundant'(not nonredundant) is 'almost

284

Material and Energy Balancing in the Process Industries

complete'. The remaining cases ('exceptional nonadjustability') represent again mere coincidence. Briefly, a measured variable is either

everywhere 'nonadjustable'

or

almost everywhere 'redundant'

where the quotes mean again that mathematical precision has been remitted. But incidentally, it can happen that a variable classified as 'redundant' (almost everywhere) remains unadjusted at some point, though subject to measurement error. Recall also that the expression 'nonadjustable' is imprecise from the statistical point of view (being due to some special, though quite common statistical hypothesis); see further Chapters 9 and 10. (ii)

It is not our aim to introduce a new terminology. The terms '(locally) observable', 'unobservable', ' redundant','nonredundant (nonadjustable)' are common, but (in the nonlinear case) often only vaguely interpreted. A more detailed (and mathematically formal) classification can be found in Stanley and Mah (1981a). We have put up with a simpler classification and shown that and why it remains incomplete.

8.5.4

Classification in practice

In practice, the adjustment problem as formulated in Subsection 8.5.2 (see (8.5.11)) is most frequently a reconciliationproblem: some measured values (vector x + ) are adjusted (reconciled) so as to make the model solvable. The classification of variables gives one an idea of what can be expected from the reconciliation. Thus, first, the degree of redundancy H informs us on the number of independent constraints (scalar equations) the adjusted value ~, has to obey, thus how many measured variables are 'redundant' in the manner that having deleted their measurement, they will be still determined by the remaining measured values. In particular if H = 0 then all the I measured values are necessary (none is redundant). If it happens that H = I then the whole measurement is redundant because the constraints determine the I variables uniquely. Generally, not any H measured variables are determined by the other values, thus redundant; some of them can be nonredundant thus not subject to the constraints (solvability conditions), hence their measurement cannot be deleted. Under frequent hypotheses adopted by the statistical model of measurement, the nonredundant values remain unadjusted by the reconciliation; so they are also called nonadjustable.

Chapter 8 - Solvability and Classificationof Variables II- NonlinearSystems

285

Further, if some vector ~ of adjusted measured values obeys the solvability condition then given ~,, certain L independent constraints (scalar equations) are imposed upon the J components of the unmeasured vector y. The number J-L determines the number of degrees of freedom for the unmeasured variables. It can happen that certain unmeasured variables are uniquely determined by the latter constraints (thus by ~); they are called observable. In particular if L = J then there is no degree of freedom and all the y-variables are uniquely determined. If L < J then at least some of the unmeasured variables remain undetermined; they are called unobservable. The above intuitive concepts have a precise meaning when the model is linear. Then the variables can be classified a priori, and the partition of variables into measured and unmeasured (measurement placement) possibly modified. In any case (assuming 0 < H < /), the measured vector x § can be adjusted by standard reconciliation. The situation is not that simple when the model is nonlinear, as the previous analysis has shown. A well-posed adjustment (reconciliation) problem allows one at least to expect that a reconciliation procedure will converge to some adjusted value ~, if the measured x + is not ' too bad'. Then also an a priori classification of variables makes sense, based on the same ideas as in the linear case. The analysis of solvability and classification of variables is not so much a physical or technological problem, but rather a problem of computation relative to the mathematical model adopted. Observe that any model (even if regarded as 'rigorous') represents a mathematical idealisation of the reality. If we put up with a linear approximation, we can make use of the methods described in Chapter 7. We thus assess a 'representative' value z 0 of the state variable (taken for instance from the design of the plant) and in a neighbourhood of point z 0 , linearize the model according to (8.4.4). Then the model (8.5.8) reads By + Ax + c = 0

(8.5.48)

where vector z has been partitioned according to (8.5.9), and Dg(z 0 ) = (B, A)

(8.5.48a)

is the corresponding partition (8.5.10) of the Jacobi matrix at z = z 0 . We then have c = g(z 0 ) - Dg(z 0 )z 0 ;

(8.5.48b)

of course when z 0 obeys the constraints (model equations), we have g(z 0 ) = 0. Then (8.5.48) is the linear model (7.1.1) with (7.1.9 and 10). Transforming the matrix (B, A, c) (7.2.1) according to Section 7.2, the analysis including the variables classification is complete; see the before-last paragraph of Section 7.2

286

Material and Energy Balancing in the Process Industries

before Remarks. The analysis is, of course, complete only if the linearized model (8.5.48) is applied henceforth to any set of measured values to be reconciled. Let now the model be nonlinear. Then the Jacobi matrix depends on the unknown vector z and the reconcilation consists of a number of steps, say of a sequence of approximations z(n); if the sequence converges then the limit value, say ~, represents a point on the solution manifold M, thus an estimate of the actual value of the state vector. So as to have an a priori idea of what can be expected, one can proceed as follows. We suppose that the state vector z can take its values in some N-dimensional interval V c U where U is the admissible region (8.5.8). The interval can be assessed as some neighbourhood of a vector z0 ~ M. A first information can be obtained in the same manner as above, in the linear case. Taking different z0 ~ M, we can examine the behaviour of the Jacobi matrix Dg(z 0 ) on M (restricted to V, thus on M n V ). [We can also, in the case of balance models, start from different values of the independent parameters representing the degrees of freedom and determining z 0 ~ M; see Sections 8.2 and 8.3. But such procedure may be rather tedious.] In the reconciliation, however, also the behaviour of Dg(z) in a neighbourhood of the solution manifold is relevant. The nonlinear reconciliation starts from some initial guess, say z +. The x-component is the measured x +, but also some initial guess y+ of the unmeasured subvector y is necessary; generally, z § ~ M. We also do not know a priori if the vector z will be determined by the reconciled ~, (thus observable). There are methods that allow one to find some ~ ~ M, 'as close as possible' to the initial z § even if y is not observable; see further in Section 10.4. The analysis (Veverka 1992) also shows that if matrix B (8.5.10) is not of constant rank in a neighbourhood of manifold M, it can happen that a sequence of approximations will not converge. The adjustment problem assuming constant rankB (8.5.16) has been called 'well-posed'. In the interval V, let us select randomly a number of initial guesses z +. Starting from any such z +, let us compute ~ as suggested. If the sequence of approximations converges for any z +, it is plausible to assume that the problem is well-posed (in the a priori assessed region V). The divergence (or oscillation) of the approximations signals that the problem may be not well-posed in V. Of course the reason can be also in that some z + is too distant from the solution manifold (a 'bad' initial guess). In the sequence of approximations z (n~,the Jacobi matrix is computed and transformed according to Section 7.2. Thus in particular the rank of B at each z (n~ is known. Observe that it can happen that the rank is not constant and still, the sequence converges, perhaps by lucky chance. But if such case is detected, the problem is still not well-posed, by definition. Conversely (and more likely), it can happen that the sequence does not converge, while the ranks of all the B(z ~n~)

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems

287

are constant. The reason can be a bad initial guess as mentioned above, or again that the problem is not well-posed. The points where the rank of B is lower than the maximum rank are 'rare' in V. Still, as a detailed analysis shows, also in a neighbourhood of such points the matrix B has certain unfavourable properties; see again Veverka (1992). The reader can also imagine the elimination: Certain rows are annulled, but 'what is zero'. Near the 'singular points', the zero becomes uncertain (some subdeterminants are 'nearly zero'). Let us have come to the conclusion that the problem is well-posed. We thus know the constant L (= rankB), thus the number of degrees of freedom J-L for the unmeasured variable y, and also the (constant) degree of redundancy H = M-L (M is the number of scalar equations). So as to draw further conclusions according to Remark (i) to Subsection 8.5.3, let us assume that the manifold M is analytic. That means we consider balance models with simple enthalpy functions, thus algebraic expressions that can be formally extended beyond the limits where they make sense physically (so-called analytic continuation). Further information will be obtained after the last approximation step, thus at points ~ ~ M corresponding to different initial guesses z § We thus know Dg(~) decomposed according to (8.5.10), and transform it to the canonical format according to Section 7.2. See now the before-last paragraph of Section 7.2, before Remarks. According to points (v) and (vi), at each ~, we find the partitions of measured and unmeasured variables, thus their respective classifications. The conclusions will be the following. (a)

If some measured variable x i is classified as nonredundant at each "2 then, almost certainly, it is nonadjustable on the whole V n M (portion of the solution manifold we are interested in)

(b)

If some xi is classified as redundant at some ~ then it is not nonredundant, thus (having simplified the terminology) redundant on the whole V n M, with possible exception of some 'rare' points, due to some coincidence.

['Nonadjustable' means here 'nonadjustable by statistical reconciliation assuming uncorrelated errors'; see Chapters 9 and 10.] Further

(c)

If some unmeasured variable yj is classified as observable at each ~ then, almost certainly, it is (at least locally) observable on the whole V n M

(d)

If some yj is classified as unobservable at some ~ then it is not (even locally) observable, thus (having simplified the terminology)

288

Material and Energy Balancing in the Process Industries

unobservable on the whole V n M, with possible exception of some

'rare' points, due to some coincidence. The wording 'locally' admits of possible other solutions in yj, in some nonnull distance from the estimated value. This is rather a theoretical possibility. In practice, such solutions will most likely represent values not admitted by the technology. The 'rare' ('incidental') points of V n M form certain subsets of 'lower dimension' than D = dimM; imagine isolated points or curves on a surface as a simple illustration. Their presence will, most likely, not be detected by the procedure. It remains, however, at least theoretically. In the case (b) above, in a neighbourhood of the possible 'rare' points, the variable will be 'almost nonadjustable': if an estimate falls into this neighbourhood, the estimated "~i will only slightly differ from the measured x + even if subject to a gross error. Such case can occur in the reconciliation and cannot be avoided by the a priori classification. See later Section 10.5, Remark. Nor can be avoided the cases that for a problem qualified as well-posed, the reconciliation procedure still fails. The only possibility is then to discard the corresponding set of measured data. The possible existence of the 'rare' points, in contrast to the linear case, does not allow one to conclude simply that, for instance, "deleting the measurement of a redundant variable makes it observable"; it can happen that the problem becomes 'not well-posed'. This can also happen when adding the measurement of an unmeasured variable classified as unobservable by the above method (and using the simplified terminology). Compare Examples 4 and 1 in Subsection 8.5.1, where Q - 0 is admitted: if the 'exceptionally observable', else unobservable unmeasured variable T 2 is measured, the originally well-posed problem has no more this property. Thus having changed the measurement placement (having added or, conversely, deleted some measurement), one ought to test if the new adjustment problem is well-posed; one can use again the procedure as above.

8.6

MAIN RESULTS OF C H A P T E R 8

The solvability analysis of a set of nonlinear equations is by far not that straightforward as it was in the case of linear models in Chapter 7. The less straightforward and unambiguous is a solvability classification of variables, which was uniquely determined by the linear model and the partition of variables into 'measured' and 'unmeasured'. This holds even in the case of balance equations having a simple structure; see Section 8.1. Although rather tricky, the structure of the set of balance equations can be analyzed; see Sections 8.2 and 8.3. The complications are mainly due to the

Chapter

8 -

Solvability and Classification of Variables H- Nonlinear Systems

289

presence of reaction nodes, splitters, and to certain a priori technological assumptions precluding the presence of certain chemical components in certain streams; see Section 8.2. We have again (cf. Chapter 4) denoted by Ek (k = 1, .--, K) the set of material streams where component Ck can be present. The arbitrariness in the choice of the E k has been limited by the structural hypotheses (8.2.18), (8.2.22), and (8.2.29) motivated by technological arguments as plausible and not very restrictive. Formally, they can be verified according to the points (a)-(k) following after (8.2.32). Then the set Yd of solutions can be 'parametrized', i.e. the solution z can be expressed as function of certain selected component mass flowrates mki (8.2.6), splitting ratios %, (8.2.12), and component production rates mk(n) (8.2.7) in reaction nodes; see the points (a)-(g) in Subsection 8.2.4. The number D of the parameters represents the 'number of degrees of freedom' for the variables in the set of component mass balance equations. The results are summarized in Subsection 8.2.5, with Remarks 8.2.6 and Example 8.2.7. No further structural complications arise when the constraints are extended by the energy ('enthalpy') balance equations; see Section 8.3. The structural hypothesis (8.3.2) has been motivated in Section 5.2 as plausible and can be weakened, as remarked at the end of Subsection 8.3.1; one adds the conventions (a)-(c) leading to (8.3.3). The specific enthalpies /2p of material streams j are generally considered as functions (8.3.18) of temperature T j, composition 3~(mass fractions of chemical components in stream j), and pressure PJ. Here, we assume that the state of aggregation in each stream is given a priori, and for simplicity, also the pressure PJ (if relevant as a thermodynamic state variable) is assumed to be a priori known thus is not written explicitly as a variable; regarding certain PJ as variables would only increase the number of parameters ('degrees of freedom') without changing the number of equations. The thermodynamic condition of positive specific heat (8.3.19) is formally adopted also for the specific enthalpy functions fJ as given in the database. Then the extended vector z of solutions can again be expressed as function of the D parameters introduced in Section 8.2, plus certain additional D h parameters that are certain selected temperatures of material streams and net energy flowrates q~ (8.3.21). The s u m O t o t -- D + D h is the total number of degrees of freedom for the variables subject to the components mass and energy balance constraints. The results are summarized in Subsection 8.3.2. The parametrisation of the whole set ~ftot of solutions is not unique (i.e. the selection of the parameters is not unique), only the number of degrees of freedom (Dtot) is invariant. In mathematical terms, the set Mtot is a differentiable manifold of dimension Dtot; it is called the solution manifold of the set of balance equations. The main result of the structural analysis is that the set of equations (constraints) (8.2.2) and (8.3.1) is minimal by (8.3.30): the number of degrees of freedom equals the number of variables minus the number of (scalar) constraints.

290

Material and Energy Balancing in the Process Industries

Using the notation introduced in Chapters 4 and 5 (summarized in Sections 4.8 and 5.7), the minimal set o f constraints can be arranged as follows. We designate (8.6.1)

n k (n) - - Z Cniy~mi/Mk i~Ek

which is (4.8.8) with (4.8.5) where y~ - 0 if i ~ Ek. The terms are introduced for any node n ~ T u (non-splitter), and for the K(n) components Ck present in the node n balance thus in some stream incident to n. The K(n)-vector of components nk(n ) is n(n). On the vector n(n) operates the matrix D(n); see the text that follows after (4.8.9), and also (4.8.11). Then the constraints read k-1,...,K J ~ Ju, J ~ Ek"

Y~, = 0

(8.6.2a)

k-1,...,K s ~ S , j~ ~ E k 9

y~,~- 0

(8.6.2b)

y~?- y~,- 0

(8.6.2c)

J ~ Ju"

K Z y~,- 1 - 0 k=l

(8.6.2d)

s ~ S 9

K Z y~s_ 1 - 0 k=l

(8.6.2e)

C"m' = 0

(8.6.2f)

q = 0

(8.6.2g)

Sfi' - 0

(8.6.2h)

D(n)n(n) = 0

(8.6.2i)

Cu (diagmu)hu + C ' ( d i a g m ' ) h ' + Duq = 0

(8.6.2j)

k=l,...,K s ~ S , j ~ J~"

9

De

n e Tu 9

where we recall that for any two vectors a - (a~, ..., a N)T and b = (b~, ..., b N )T we have (diaga)b = (diagb)a = (al b l , "'", aN bN )T. In the graph G[N,J] of material streams, S is again the subset of splitter nodes, T u that of units that are not splitters, J' ( c J) is the subset of material streams incident to some splitter, Ju = J - J'; the set J' is partitioned into subsets J~

Chapter 8 - Solvability and Classification of Variables H- Nonlinear Systems m

291 m

of streams incident to splitters s, j~ (~ J~ ) is the input and J~ ( c Js ) the set of split output streams for splitter s. For the submatrices C u , C', C" of the reduced incidence matrix C of G see (5.7.2a and b); the vector m of mass flowrates mj is again partitioned into m u (j ~ Ju ) and m' (j ~ J'), while hu resp. h' is the corresponding vector of specific enthalpies/T for j ~ Ju resp. j ~ J'. The reduced incidence matrix D of graph GE [NE, E] (5.7.1) of net energy streams is again partitioned into submatrix D u (nodes n ~ Tu) and De (nodes n ~ DE) where D E is the set of energy distributors (possibly empty); q is the vector of net energy flowrates qi (i E E). The equation (8.6.2h) with S according to (5.2.12 and 13) represents the scalar equations - ~ + / T = 0 (s ~ S, j ~ Js ). By (8.6.2c) and assuming also equal pressure PJ = PJ~ (j ~ Js) in streams incident to splitter s, the equation is equivalent to the condition of equality of temperatures s e S, j e Js" Tj~ - Tj - 0

.

(8.6.2h)*

We have thus re-arranged the equations in the manner that Eqs.(8.6.2a-g) are linear, and Eq.(8.6.2h) 'quasilinear', equivalent to the linear subsystem (8.6.2h)*. Observe that the formally written equations (8.6.2a and b) are, in fact, simple instructions (equalities) >~put y~, = 0 ~. The LH-sides of Eqs.(8.6.2) constitute the function g in the model g(z) = 0

(8.6.3)

where z is the vector of model variables (mass flowrates, net energy flowrates, mass fractions, and temperatures) subject to the balance constraints; here, hu and 11' are regarded as functions of the mass fractions and temperatures (given the pressures and states of aggregation). We have included as variables all the mass fractions y~ (j e J, k = 1, 9.. , K), for the sake of formal simplicity. By (8.6.2a-c) however, we have y~ = 0 whenever j ~ Ek; the latter variables are thus always 'observable', i.e. their values are uniquely determined by the model. Recall that a correctly set-up list Ek must not admit the case where, for instance, Js e Ek but j ~ Ek for some J ~ Js. In the solvability analysis, we have adopted the condition of positive mass flowrates mj>0foreachj~

J

(8.6.4)

with the natural assumption that the model admits such solution. Then, with all the structural assumptions fulfilled, the dimension Dto t of solution manifold ~V[tot equals D+D h where D is the number (8.2.78) and D h the number (8.3.23); here, IMI is number of elements of general set M and in addition Juk is the set of streams j ~ Ju ~ Ek, Sk the set of splitters where component C k occurs, K(n) the

292

Material and Energy Balancing in the Process Industries

number of components present in the node n balance, and R o (n) the dimension of the reaction space in node n (R0 (n) = 0 for a nonreaction node). If some unit is a heat exchanger, it has been partitioned into two nodes, say n' and n"; see (8.3.32). In the energy balance, if the heat transfer rate qi between n' and n" is not given a priori, it can be deleted as a variable by merging the two nodes. The procedure is described in detail in Subsection 8.3.3. We introduce the subset Eex ( c E) of net energy streams i to be deleted by the merging, the set of nodes h i (i ~ Eex ) of the corresponding exchangers, and the subset T~ ( c Tu ) of the remaining units. Then the submatrix C u (5.7.2a) is reduced (8.3.37)

(Cu,

c'*)

(8.6.5)

to the matrix of rows (nodes) n ~ T~, and h~(i ~ E~x ). Having merged the nodes also in the graph GE, in the matrix D the submatrix D Ois reduced to the same rows as in (8.6.5), and by the deletion of arcs i ~ Eex to columns i ~ E* = E-Eex ; we thus have D~. Also in the submatrix De, the columns i ~ Eex are deleted and the new submatrix is D~; see (8.3.38). Then the equations (8.6.2a-f) remain, instead of (8.6.2g) we have Deq = 0

(8.6.6a)

Eqs.(8.6.2h and i) remain, and instead of (8.6.2j) we have C~ (diagm u )hu + C'* (diagm')h' + D~q* = 0

(8.6.6b)

where q* is the remaining vector of net energy flowrates qi, i e E*. By the reduction and elimination, the dimension of the solution manifold (say) ~tot (thus the number of degrees of freedom) remains unaltered and is again Dtot (8.3.47). If we consider heat and mass balances only, the set of constraints (5.4.6) is minimal; see Subsection 8.3.4. The dimension of the solution manifold is Dtot = D + D h (8.3.50) where D h is the same as above (8.3.23), and D = I JI - I Nu I (8.3.49) where N uis the set of units; if certain components s(n) of vector s (source terms) are not given a priori, they represent additional degrees of freedom. The set Nu = T Ow S is again set up in the manner that both sides of any heat exchanger are considered separately, so as to take account of separate mass balances. We can again merge the couples of nodes {n',n"} in the energy balance of any heat exchanger, without changing the dimension of the solution manifold. If energy distributors and source terms are absent and if each qi (i ~ E) is a heat transfer rate in a heat exchanger, we have again the set of constraints (5.7.12). The equation (8.6.3) represents a nonlinear model. Generally, g is an M-vector of scalar functions gm of vector variable (N-vector) z. If the function g

Chapter $ -

Solvability and Classification of Variables H- Nonlinear Systems

293

is (at least) twice continuously differentiable, the properties of the solution can be examined using the Jacobi matrix Dg; see Section 8.4. The model is called solvable when there exists some solution z obeying g(z) = 0, uniquely solvable when this solution is a unique point z0 . If M (~ ~ ) is the set of solutions, the model is called regular when the Jacobi matrix Dg(z) is of full row rank (M) at any z ~ M (8.4.6). Then M is a differentiable manifold of dimension N-M (8.4.7). In particular the model (8.6.3) with (8.6.2) is regular, as well as the reduced models following thereafter. The Jacobi matrix of the balance model (8.6.3) has a special structure and its elements can be computed explicitly; see the formulae following after (8.4.9). The columns can be arranged according to the variables mj (j ~ J), y~ (j e J, k = 1, ..., K), TJ(j e J), and qi (i e E); the rows are arranged according to the scalar components gm of g, thus according to the equations (8.6.2). The elements in the rows corresponding to Eqs.(8.6.2a-g) are constants (+1 or 0), and if (8.6.2h) is replaced by (8.6.2h)* then also the elements in the latter rows are constants (+1 or 0). The elements in the rows (8.6.2i and j) can be computed as shown in (8.4.1 la) and illustrated after (8.4.1 lg), and in (8.4.14a); here, we make use of the developments by differentials (8.4.10). A special consideration is necessary if the enthalpy functions involve dependence on the mass fractions (8.4.16-18), and in particular if certain streams are multiphase mixtures; see Remark to Section 8.4. If the values of certain variables are a priori fixed, one can ask again, as in Chapter 7, which of the variables is redundant resp. nonredundant, and which of the unknown ones is observable resp. unobservable. Basically, such solvability classification of variables is possible and follows then the same lines as in Chapter 7, using the Jacobi matrix Dg(z) instead of the constant matrix C of the linear system; see Section 8.5. But since Dg(z) is no more constant, the classification can remain incomplete and need not be unambiguous; see the examples in Subsection 8.5.1. Theoretically, precise definitions and exact theorems can be formulated if the behaviour of the Jacobi matrix Dg is known in the whole admissible region U (8.5.8); in the balance model, the region is anyway limited by the conditions (8.6.4), and perhaps also by other inequalities. But even then, most of the theorems are of local character only (such as "the solution is unique in a neighbourhood of some point"); a global analysis by more advanced topological methods can be hardly thought of. The theory is presented in Subsections 8.5.2 and 3. In every case, the full row rank assumption is adopted for the matrix Dg(z), z e M. It holds true for the minimal set of the balance constraints. The vector z is again partitioned into x (fixed) and y (unknown). As in Chapter 7, x is called the vector of measured variables, y that of the unmeasured ones (8.5.9). Accordingly, matrix Dg is partitioned by (8.5.10) into B (operating on y) and A (operating on x). These conventions and notation are of constant use here and

Material and Energy Balancing in the Process Industries

294

also in the following chapters. The behaviour of matrix B(z) is of key importance for the analysis, and also for the numeric procedures suggested in Chapter 10. Given subvector x = x + in the partition of z, there need not exist any solution z ~ U of g(z) = 0 having this x § as subvector. The adjustment problem consists in finding some ~, such that the equation

g

y:~ / -

0

(8.6.7)

is solvable in y, and further that the difference i-x + minimizes some criterion; a special case is dealt with in Chapter 10 (reconciliation). Irrespective of the criterion, we can say that ~'makes solvable' the equation g(z) = 0. The set of making the equation solvable is denoted by M + (8.5.12). Given ~, ~ M +, we denote by M(~) the set of solutions y of Eq.(8.6.7). In particular, the latter equation can be uniquely solvable, but this property can depend on i. In the most favourable case, also the set M + and the sets M(:~) are differentiable manifolds; M ( i ) can also be a (set of) isolated point(s). Under certain assumptions, the theory is able to determine the properties of M + and M(:~) at least locally: restricting the admissible region to some N-dimensional open interval (say) V (thus formally U = V), the set M + is a manifold and for any ~ ~ M § M(:~) is a manifold whose dimension is independent of ~. One of the assumptions is that of constant rank of matrix B; we have called such problems well-posed. We then designate L = rankB

(8.6.8)

on 'E. Recall that M is the number of scalar equations and N the number of variables thus N = J + I where J is dimension of vector y and I that of x. We designate H = M- L.

(8.6.9)

Also this notation is used henceforth. It corresponds to the notation introduced for the linear systems. The number H can be regarded as the 'number of constraints' the variable x has to obey so as to make the model solvable, thus H is called the degree of redundancy and I-H is the 'number of degrees of freedom' for the variable x. The number L can be regarded as the 'number of constraints' the variable y has to obey, given some ~ ~ M + in (8.6.7), and J-L is the 'number of degrees of freedom' for the variable y; if L = J then the solution in y is (at least locally) unique. In the end, remitting the mathematical precision a 'most likely' classification of variables is possible using the methods of Chapter 7. The main

Chapter 8

-

Solvability and Classification of Variables II- Nonlinear Systems

295

results are summarized in Subsection 8.5.4. Basically, one applies the criteria of Section 7.2 to the model linearized at some point(s) of the solution manifold.

8.7

RECOMMENDED LITERATURE

As already stated in the introduction to this chapter, the present material is largely novel, and only a few relevant references are available. The neceessary mathematics is expounded in university textbooks; we have applied the theorems given in Dieudonn6 (1970). The approach to the observability/redundancy analysis as presented here can be characterized as equation-oriented; cf. Romagnoli and Stephanopoulos (1980), or Crowe (1989). For another (graph-oriented) approach, see Stanley and Mah (1981a and b), Kretsovalis and Mah (1988), summarized in Mah (1990). Crowe, C.M. (1989), Observability and redundancy of process data for steady-state reconciliation, Chem. Eng. Sci. 44, 2909-2917 Dieudonnr,J.(1970), Eldments d'Analyse III: 16.1-8, Gauthier-Villars, Paris Kretsovalis, A. and R.S.H. Mah (1988), Observability and redundancy classification in generalized process networks I and II (Algorithms), Comput. Chem. Engng. 7, 671-703 Mah, R.S.H. (1990), Chemical Process Structures and Information Flows, Butterworths, Boston Romagnoli, J.A. and G. Stephanopoulos (1980), On the rectification of measurement errors for complex chemical plants, Chem. Eng. Sci. 35, 1067-1081 Stanley, G.M. and R.S.H. Mah (198 la), Observability and redundancy in process data estimation, Chem. Eng. Sci. 36, 259-272 Stanley, G.M. and R.S.H. Mah (1981b), Observability and redundancy classification in process networks (theorems and algorithms), Chem. Eng. Sci. 36, 1941-1954 Veverka, V. (1992), A method of reconciliation of measured data with nonlinear constraints, Appl. Math. and Computation 49, 141-176