Chapter 3 Mass (single-component) balance

Chapter 3 Mass (single-component) balance

25 Chapter 3 MASS (SINGLE-COMPONENT) BALANCE Single-component balancing means setting-up the balance of one quantity obeying a conservation law; on...

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25

Chapter 3 MASS (SINGLE-COMPONENT)

BALANCE

Single-component balancing means setting-up the balance of one quantity obeying a conservation law; one balance is set up around each node. In chemical technology, it is the total mass (possibly the sum of masses of individual components of a mixture) that remains conserved in any process. Another example is the conservation of electric charge expressed as First Kirchhoff' s Law for (direct-current) electrical networks. It is also possible to consider thebalance of one selected chemical element (irrespective of in what chemical component it occurs), or also of a component not participating in any chemical reaction. This chapter deals with mass balancing. The reader is recommended to peruse Sections A. 1-A.4 of Appendix A. For the necessary notions of linear algebra, see Sections B. 1-B.8 of Appendix B.

3.1

STEADY-STATE MASS BALANCE AND ITS GRAPH REPRESENTATION Generally, the mass balance of a unit (node) reads

Z mj in.Z mj - out

an

(3.1.1)

an .I

h.

node n "k

Fig. 3-1a. Node balance

"~} (mj)~ Jr"

26

Material and Energy Balancing in the Process Industries

where mj is (possibly integral or integral mean) mass flowrate of j-th stream, 'in' means input, 'out' means output streams j; an is increase (per unit time or per period) of accumulation in the (say, n-th) node. A 'steady-state' balance means formally a n = 0. With the above interpretation of an, the change in accumulation (holdup) is then neglected. It is also possible to regard a n as a fictitious rate of mass flow directed outwards; then a n becomes one of the output flowrates mj and the RHS in Eq.(3.1.1) becomes zero.

an (mj)i" {

(mj)out

node n

Fig. 3-lb. Accumulation stream regarded as output stream

The fictitious flowrate a n can be positive or negative. In a system of units n ~ N u (set of units) connected by streams j ~ J (set of streams), the (formally steady-state) balances can be written in the form Z Cnjmj = 0

j~J

(n ~

N u)

(3.1.2)

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

(3.1.3)

Cnj = 0 if stream j is neither input nor output. The notation ~ means sum of terms j over set J, independent of the order in the summatioJ~.JThis kind of notation will be used henceforth for convenience, as in further analysis the sets will be re-arranged in different manners. Also the number of elements of a (general finite) set M is independent of the order and will further be denoted by [ M I . Recall now Appendix A. Clearly, whatever be the index orders the matrix C of elements Cnj is the reduced incidence matrix of an oriented graph (say) G. The set of arcs of G is J and any arc j ~ J is incident to one node n ~ N u at least (Cnj ~ 0), to two,such nodes at most. Let us now introduce the node n o in the

27

Chapter 3 - Mass (Single-component) Balance

following manner: the other endpoint of any arc j ~ J that has only one endpoint in the set N u of units, is no. Thus for example the balancing scheme

I

Fig. 3-2a. Balancing scheme

is completed to a graph

.

-

.

no en

l

-'x..

-•set

Nu of units

Fig. 3-2b. Graph of the balancing scheme

Clearly, the streams incident to node n o , according to the direction, either represent a material supplied to the system, or produced by the system as a whole; hence node n o is called the node environment. But observe that formally, also the fictitious streams such as in Fig. 3-1b (a n) are arcs incident to node n o . We thus have identified the graph G[N, J] whose node set is N = N u u {no} ; {no} m e a n s s e t of one element n 0.

(3.1.4)

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Material and Energy Balancing in the Process Industries

A natural requirement is that G is connected: any node (unit) n attainable by a path, thus also by a sequence of (arbitrarily oriented) arcs at node no; otherwise certain subsystem of units would have no input output, which is technologically absurd. The immediate consequence system of equations (3.1.2) is, by (A.7 and 8) rankC = I Nu I

~ N u is starting and no for the

(3.1.5)

where IN u [ is the number of elements of set Nu; hence the matrix C is of full row rank. If we introduce some orders of elements n s N u and of elements j ~ J, the system (3.1.2) reads Cm = 0 ;

(3.1.6)

here, m is column vector of components mj (j e J), an element of t J l -dimensional space (say) q/'. The set of solutions is the null space KerC of matrix C, a vector subspace of q/', of dimension I J I-INn I. Clearly, the system as a whole has at least one input and one output, and the graph G contains a circuit, hence IJI > INul ;

(3.1.7)

see (A.3), with [Ni - I Nu I + 1. Consequently (and of course), there are always nonnull solutions to Eq.(3.1.6). A correctly written technological scheme (graph) has to satisfy certain other natural requirements. The equation (3.1.6) has, by (3.1.7), always a nonnull solution (m~:0), but it can still happen that certain mi is uniquely determined as mj = 0; such stream is then, in effect, absent. Let us consider first an arc (stream) jl ~ J that lies on a circuit of the graph G; for example

29

Chapter 3 - Mass (Single-component) Balance

Fig. 3-3. A circuit

(recall that the notion of a circuit is independent of orientation). It is then easily shown that if vector m (of j-th component mj) is a solution then if a is arbitrary, there exists always a solution m such that mj, = r~j, + a. The idea of the proof consists in adding a fictitious circulation a along the circuit as indicated in Fig. 3-3; in any of the node balances, the circulation term cancels and all the balance equations are again obeyed. Conversely let arc J2 separate the graph, thus J2 lies on no circuit according to the terminology of Appendix A; then if m is an arbitrary solution, we have uniquely mJ,_= o. The conclusion follows formally from (A.9)-(A.13), where Arcd stands for C (3.1.6). We can also imagine Fig. A-7 as an example. More generally, the graph G is decomposed m

./'2

Fig. 3-4. Arc J2 separates the graph The node n o (environment) belongs to, say, subgraph G1, while G2 is connected by just one stream with G i" when balancing G2 as a whole, the input/output results mi_, = o. Technologically, such case is absurd; the units of G2 would be constantly out of operation. For a correctly written technological scheme, such cases are precluded.

30

Material and Energy Balancing in the Process Industries

In what follows, we shall always suppose that the equation (3.1.6) has some solution in such that mj ~ 0 whatever be j ~ J (and even mj > 0 if j is not a fictitious accumulation stream).

Remark In mass- and energy balancing problems can occur the case when two or more separate plants (systems of units) cooperate via certain streams of energy from one plant into another.

energy stream Fig. 3-5. Cooperating plants

Thus in Fig. 3-5, the input/output streams into/from separate plants G~ and G 2 are incident to environment node no; in addition the plants G~ and G 2 are connected via certain energy stream(s), and some energy stream(s) can connect directly G 2 with n 0. We can obtain connected graph G whose node set consists of Nu~ (units of G1), Nu2 (units of G2), and n 0. The arc set J of G consists of: arcs between the units of G~ , arcs between the units of G 2, and the arcs connecting G~ and G 2 with environment node as drawn. In the set of mass- and energy balance equations as considered later in Chapter 5, we can take the whole G[N,J] as the graph of material streams. On the other hand, the mass balances of G1 and G2 are independent. Indeed, with reduced incident matrices C 1 and C2, respectively, we have the balances (3.1.2) thus (3.1.6) C~m~ = 0

and

Czm 2 = 0

(3.1.8)

w h e r e m I and m 2 are the respective vectors of mass flowrates. This corresponds

to a partition of the (full) incidence matrix of G

Chapter 3- Mass (Single-component) Balance no

~1 ~2

Nul

C1

Nu2

31

(3.1.9) C2

Here, the row vectors c~ resp. o~2 have nonnull elements (_+1 according to the orientation) in the columns i~, j~, k~ resp. i2, J2 according to Fig. 3-5. The void fields are zeros, because no arc incident to some node n ~ Nu~ is incident to any m ~ Nu2 . AS a consequence, we can analyze the two mass balances (3.1.8) separately. The corresponding full incidence matrices are

C1

N,2

/~176 C2

(3.~.1o)

Nu2

The reader can certainly imagine the case where there are more than two cooperating plants interconnected by energy streams only, with arbitrary numbers of streams connecting the plants with the environment. We obtain again a partition such as in (3.1.9), with several matrices C~, C 2 , .-. and row vectors ~ , 0~ 2 , "'" . Each of the matrices (3.1.10) is that of a connected graph (node no is formally split into two or more environment nodes). If we delete node n o and the incident arcs i 1, j~, k~, i 2 , J2 in Fig. 3-5, the graph G becomes disconnected into two separate parts G~ and G2. In graph theory, a node n o having such property is called cutnode of G. Conversely let us have a graph G, with reference (environment) node n 0. We perhaps don't know that the mass balances can be separated. The possibility is easily verified. We delete node no and all the arcs incident to n o. If the graph is disconnected then n o is a cutnode and the partition of the remaining set of nodes determines the cooperating plants; generally, they can be more than two.

3.2

MORE ON SOLVABILITY

3.2.1

Partition of variables and equations

In the set J of arcs j thus of variables mj, let us assume that certain variables have been fixed as (say) mj, +" let J+ be the corresponding set of arcs, j0 _ j _ j+ the remaining subset. Generally, the values m.+ j can be certain measured values, or also certain target values required when designing the plant or otherwise. Making abstraction from the way how the m~ have been determined, we see immediately that the set of equations (3.1.2)

32

Material and Energy Balancing in the Process Industries (n ~ Nu)

jeZ joCn.m. J J + j~Z j+C.m nj + j .-0

(3.2.1)

in variables mj (j e j0) can have generally an infinity of solutions, or also no solution; in a special case, it has just one solution, i.e. a unique subvector of components mj (j e j0) obeying (3.2.1) whatever be the fixed m~. In the first case, the system is undetermined, in the second overdetermined. The special third case where the system is just determined will be examined in more detail in Section 3.5. In the second case, there are certain conditions the variables m~ have to obey so as to make the system (generally not uniquely) solvable. If so, certain variables mi (J e j0) are uniquely determined by the equations, certain other still not in general. Let us analyze the solvability in more detail. When restricting the graph G to arcs j e j0, we obtain a subgraph (say) G~ its reduced incidence matrix is of elements Cnj where n e Nu and j e jo. The subgraph is generally not connected; it can even contain isolated nodes, not incident with any arc j e j0. Example:

Jl

-~ G,

no ~

li .

.

j

i

E: .

a

-:

! ,,,," k

.

J2

b

Fig. 3-6. Fixed and unknown streams j e J+ (fixed) j ~ jo (unknown)

0

G3 O

~t |

1i

G~

! !

O 0 G1

1 | 1i

.,o

1!

O 0 G2

|

1 1!

0

G4 Fig. 3-6a. Graph of unknown streams in G

<)

,, ~o

]

,s"

~,-'""k

Go 5

Chapter

3-

33

Mass (Single-component) Balance

Let G o (k - 1, ..., K) be generally the connected components of G o ( K - l means that G o is connected), with node sets Nk and arc sets jo; in Fig. 3-6a, K - 5, three components are isolated nodes, and for example in G ~ N 5 - {a, b, c} and jo 5 - {i, k}. With a corresponding rearrangement of rows and columns, let C o be the f u l l incidence matrix of G o , C~ a block of the f u l l incidence matrix of G, restricted to nodes (rows) n ~ Nk and arcs ( c o l u m n s ) j ~ J+. Thus for k - 5 in Figs. 3-6 and 3-6a i

k

-1 CO 5 -

a

1 -1

b

1

c

and .....

j,

j2j'

1 C +5 -

l .....

-1

a

b

1 1

c

-1

where the void fields are zeros. Indeed, with the subset N s - {a, b, c } of nodes are incident the unknown streams (arcs) i, k (~ jo) and the fixed streams Jl, J2, j ' , l (~ J+), the other are nonincident. (For convenience, we have ranged the latter streams as written after one another.) The streams Jl, J2, l have the other endpoints in other subsets N k. Observe that the two matrices correspond to the balances of nodes a, b,c mi

- m + j,

+ m .+ Jl

m i - m k

-0 =0

+ m + J2 + m j+, -

mk

m l+ m O

which can be written in the form

c o (mi

+ C~m+-O

mk

where m + is the vector of all the fixed m~.

34

Material and Energy Balancing in the Process Industries

Generally, let us partition the vector of unknown streams into subvectors m k (k - 1, -.-, K). Here, m k is the vector of components mj where j ~ j0; thus in the above example, m 5 = (mi, mOT. Observe that some j0 can be empty; then m k is absent. Let further in + be the vector of fixed components m~ where j e J+. Recall also thatthe balance (3.1.6) thus Cm - 0 is equivalent to the equat_ion Cm = 0 where C is the full incidence matrix of G; indeed, the row n o of C is linear combination of the rows of C. Thus the system (3.2.1) is, with the above partition, equivalent to k - 1, ..- , K:

C ~ m k + C~ m + - 0

(3.2.2)

because nodes n ~ Nk can be incident only with arcs j ~ j0 and some j ~ J+. By the connectedness of each G ~ just one (arbitrary) row of C o is linearly dependent, equal to minus sum of the remaining ones; recall that C o is the full incidence matrix of G ~ If G o is an isolated node then j0 = Q and we put, by convention, C o rnk = 0 thus the equation reads C~ m + - 0.

3.2.2

Transformations

Let us apply the operation 'merging of nodes' to each subset N k of nodes of our graph G. Let G* be the reduced graph obtained in this manner. Thus with Fig. 3-6 we have

Jl

G1 Fig. 3-7 Reduced graph

where for example node ~ represents the three merged nodes a, b, c of G~ observe that arc j ' has been deleted by the merging. The graph G* is connected (as G is), hence its full incidence matrix (of K rows) is of rank K-1. On selecting an arbitrary reference node in G*, let A* be the reduced incidence matrix. The merging of nodes corresponds to the summation of rows in the original full incidence matrix. In particular the arcs j e j0 (k - 1, ..., K) are deleted. But it can

Chapter

3 - Mass (Single-component) Balance

35

happen that in addition certain arcs j ~ J+ are also deleted; for example the arc j ' in Fig. 3-6. Let thus J* ( c J§ be the arc set of G*; then matrix A* operates on columns j ~ J* and let m * be the subvector of m § of components mj , j ~ J* . We thus have A* m* - 0

where

rankA* - K - 1

(3.2.3)

is full row rank; of course if K = 1 then the equation is absent. Consider again the example in Fig. 3-6. According to Fig. 3-6a, for instance the subgraph G O of G O is an isolated node. The balance of the node involves only arcs j e J+ (fixed) and remains unaffected by the merging; it becomes one of the scalar equations (3.2.3). The unknown variables mj (j ~ jo) occur only in the balances of 'nontrivial' subgraphs G Osuch as G Ocontaining the nodes a, b, c; see above. The merging represents summation of the node balances giving m +. J, + m.+ J2 - ml+ - 0 which is the node ~ -balance according to Fig. 3-7; it is also one of the scalar equations (3.2.3), where stream j ' is absent (deleted by the merging). Having replaced the node c -balance by that of node ~ (obtained by the summation), the remaining balances (of nodes a and b) read mi mi- mk

+

m .+ J1

-

+

m .+ J2

m j+ ,

- 0 = 0

which can be written in the form

B5 ( mi mk

+ c5 = 0

where c 5 - A 5m + ;

here, A 5 is the submatrix of C; where the third (c) row has been deleted. Taking c as reference node in G ~ B 5 is the reduced incidence matrix of G ~ Clearly, the original three balances are equivalent to the three new ones; in addition if the first (node g:) balance is satisfied then the remaining two equations are (in the present case even uniquely) solvable in m i and mk. Let us generalize. Let us arrange the subgraphs G Oin the manner that G ~ ---, G ~ (K' < K) are those subgraphs which are not isolated nodes; observe that if G Ois an isolated node, the corresponding (scalar) equation (node balance) in (3.2.2) becomes automatically one of the node balances of the reduced graph G*. Having selected a reference node in each G O for k - 1,-.-, K', let Bk be the reduced incidence matrix of G ~ thus B k is of full row rank; let further A k be the corresponding

36

Material and Energy Balancing in the Process Industries

submatrix of C~, with the reference-node balance deleted. Then the system (3.2.2) is equivalent to k-1, and

..., K'"

Bkmk,+mCk-- 0 - 0

where c k - A k m+ } (equation absent if K = 1)

(3.2.4)

which is a system generated by elementary operations (summations of rows). The subsystem (3.2.4)~ is (generally not uniquely) solvable in variables m k whatever be the e k. Hence the whole system (3.2.4), thus (3.2.1) admits some solution if and only if either K = 1 (G o connected), or the subvector m* obeys Eq.(3.2.4) 2. We also see that the natural requirement is

K-1

< I J*l

if K > 1 ;

'

(3.2.5)

otherwise, by (3.2.4) 2, the only solution in m* would be m* - 0, a case precluded by our convention (see the last paragraph of section 3.1 before Remark). Observe finally that with J' - J+- J*

(3.2.6)

(the set of arcs j e J+ deleted by the graph reduction), the values m.+ for j e J' J are not subject to any condition; they are arbitrary. Let now 1 < k < K' and let m ~ be some solution of the k-th vector e q u a t i o n ( 3 . 2 . 4 ) 1 . If m k is an arbitrary solution, we must have Bk(m k - m ~ = 0 .

(3.2.7)

Now G~ is connected hence denoting y - m k - m ~ the equation in y has the same properties as Eq.(3.1.6). Making use of the same arguments as in the paragraphs below the latter equation (with Figs. 3.3 and 3.4), we conclude" If j e Jk~ then the j-th component yj of y - m k - m 0k is uniquely determined as yj - 0 if and only if arc j sepa.rates the subgraph G ~ This is also a necessary and sufficient condition for the uniqueness of mj in the solution m k. Given the connected graph G and the partition of the arc set J into j0 and J+, also the subgraph G o and its components G o are uniquely determined, as well as the reduced graph G*, hence the subsets J* and J' of J+. By each G ~ also the arcs separating the subgraph are uniquely determined. So the variables (arcs j) can be classified: for j e J+ into j e J* and j e J ' , and for j e j0, thus j e Jk0 for some k - 1,--., K' into those which separate G o and those which lie in some circuit of G o, hence also in some circuit of the whole G~ conversely if an arc lies in some circuit of G ~ it must lie in some circuit of a connected component G ~

Chapter 3

-

Mass (Single-component) Balance

3.3

OBSERVABILITY AND R E D U N D A N C Y

3.3.1

Definitions and criteria

37

In Section 3.2, we have classified the variables by the criteria of solvability. The classification is purely algebraic, but the names given in practice to the respective properties are those introduced in the theory of measurement and control: observability and redundancy. The notions were introduced in a more general context; restricting oneself to steady-state models one considers a system of algebraic equations (constraints) and one assumes that certain variables have been measured. Then an unmeasured variable is called observable when its value is uniquely determined by the measured values and the constraints. A measured variable is called redundant when deleting its measurement, it remains uniquely determined by the other measured variables (and the constraints). As a consequence the measured redundant values, if erroneous, have first to be adjusted so as to obey the constraints; only then the observable unmeasured variables can be computed. Despite of the apparent clarity, this verbal formulation is rather vague in general; see later comments in Chapters 5 and 8. In the present case of mass balance equations, a rigorous mathematical definition is possible. Irrespective of how the values m~ have been fixed, we call the j-th variable m e a s u r e d if j ~ J+, unmeasured if j ~ j0 in (3.2.1). Then an unmeasuredj-th variable is called observable if, whenever there exists a solution to the system (3.2.1), the j-th component mj is uniquely determined by the measured values; thus if m~ is the j-th component of any other solution, we have mi = m]. A measured j-th variable is called r e d u n d a n t if, whatever be the measured vector m § such that the system (3.2.1.) has a solution, the remaining components (say) m~+ where i ~ j determine uniquely mi, +" thus if rh + is another measured vector such that the system is solvable and if mi - + - mi+ f o r a l l i ~ j , we have also rh+ j - m j+ . Using the result following after Eq.(3.2.7) we see immediatly that:

k -

1,

The j-th unmeasured variable where j ~ j0, thus j ~ Jk~ for some 9.., K' is observable if and only if arc j separates subgraph Gk~

or also

The j-th unmeasured variable is observablei f and only if arc j does not lie in any circuit of subgraph G ~ Recall that G O is the subgraph of G restricted to unmeasured streams (arcs), G o its k-th connected component, K' the number of components that are

38

Material and Energy Balancing in the Process Industries

not isolated nodes. The necessary and sufficient conditions give also a precise meaning to unobservability. An unmeasured variable that is not observable is called unobservable. Clearly, we thus have also the following special result.

The whole vector of unmeasured variables is observable if and only if all the connected components Gk~ of subgraph G o are trees (or also isolated nodes). Let us now recall the partition (3.2.6). For the j-th measured variable we have either j ~ J* or j ~ J'. If j ~ J* then mj is a component of vector m* in (3.2.4) 2. So if the necessary and sufficient condition (3.2.4) 2 of solvability is obeyed, at least one row of the vector equation contains variable mj = mj§ (with nonnull coefficient) and the corresponding scalar equation determines uniquely this mi, given the other m]~ (i ~: j). If j ~ J' then mj = m +..~is absent in (3.2.4) 2, hence arbitrary. Consequently, we have the result

The j-th measured variable (j ~ J+) is redundant if and only if j ~ J*, thus if and only if arc j is an arc of the reduced graph G*. A variable that is not redundant is called nonredundant. Thus

The j-th measured variable is nonredundant if and only if arc j is deleted by the graph reduction (merging the nodes connected by unmeasured streams). Clearly, if the arc j closes a circuit with certain unmeasured streams then its both endpoints are merged by the graph reduction, being connected by a sequence of arcs i ~ jo whose endpoints thus lie in one connected component G ~ Conversely if the endpoints of arc j lie in the same G O then there exists a path in G O between the endpoints, hence arc j closes a circuit with some arcs i ~ j0. We thus have the criterion

The j-th measured variable is nonredundant if and only if arc j closes a circuit with certain unmeasured streams (i.e. its endpoints are connected by a sequence of arcs i ~ j0). As an example, recall Fig. 3-6 where the j'-th variable is nonredundant. As drawn, all the unmeasured variables are observable. Note that if we deleted the j'-th measurement, the three unmeasured streams j', i, k forming a circuit would become unobservable. It is thus seen that the observability and redundancy are, in the case of mass balance equations, exact structural properties of the graph G of the system, and of the partition of the streams (arcs) into j0 and J+ corresponding to variables called respectively, by convention, 'unmeasured' and 'measured'. Let us add an observation concerning the redundancy. We call 'redundant' each variable (arc) of J* separately; this does not necessarily imply that also a subset of J* can be regarded as 'redundant' in the sense that the variables mj of the subset would be

Chapter 3

-

Mass (Single-component) Balance

39

uniquely determined by the remaining ones. From (3.2.4)2 it only follows that there exists a subset of (K-1) arcs (linearly independent columns of A*) such that the variables of the subset are uniquely determined by the other ones, and this number is maximum. The number K-1 (full row rank of matrix A*) is called the degree of redundancy. Not any K-1 columns of matrix A* are, however, linearly independent; consider for instance the reduced graph according to Fig. 3-7 where the columns corresponding to the parallel streams j~ and J2 are linearly dependent.

3.3.2

Classification methods and transformation of equations

The formal definitions enable us to classify the variables by graph algorithms. Basically, the classification algorithms consist in eliminating circuits containing unmeasured streams; see for instance Mah (1990), 8-2-2 and 9-1-1. As an example, let us outline a procedure that performs the classification simultaneously with transforming the equations in a manner suitable for adjustment (reconciliation) of measured variables, and computation of the unmeasured observable ones. Given is the connected graph G[N, J] of the system and the partition of the streams (arcs)j ~ J into unmeasured (j0) and measured (J+). Define subgraph G o [N, j0]. Then (a)

Decompose subgraph G o into connected components G o [Nk, Jk~ k = 1, ---, K. If K = 1 then the set of redundant variables is empty. If K > 1 then merge the nodes of each subset N k of nodes in the whole graph G. Get graph G* whose arc set is J* c J+; the arcs j ~ J* determine the set of redundant variables, those of J' = J+- J* the set if nonredundant ones.

(b)

In each subgraph G o that is not an isolated node (thus j0 :r O), find the subset Sk ( c j0) of arcs that separate G ~ Then the union S of the subsets Sk determines the set of unmeasured observable variables.

Let us notice that according to Section A.3 of Appendix A, an arc that separates a connected graph (whose deletion disconnects the graph) must lie in any spanning tree of the graph. So when searching for such arcs (subsets Sk), we can limit ourselves to an (arbitrary) spanning tree (of G~ In particular when decomposing G o into the G o, we can apply the successive procedure according to Section A.4, where the nodes of any connected component are subdivided according to their distance from a selected reference node. In this manner, if the procedure is carried out in all the details we also obtain a spanning tree (say) T k of each G o (that is not an isolated node), and the predecessors of the nodes in T k. Using this information, we can find the inverse (say) R k of the reduced incidence

40

Material and Energy Balancing in the Process Industries

matrix of each Tk, as described in detail in Section A.3; see (A.4)-(A.6) with (A.9) where Are d stands for the reduced incidence matrix B k of G o occurring in (3.2.4). The matrices R k are obtained directly, without numerical inversion. On the other hand, as shown in the last two paragraphs of Section A.3, we thus can find the set Sk of arcs separating G ~ Indeed, having decomposed

Bk = (Bk, Bk)

(3.3.1)

where Bk is the reduced incidence matrix of T k , u s i n g R k = (Bk) -1 We can transform the whole equation (3.2.4)~ (k = l,--., K') into tt

tt

m k + RkBkm k + RkAk m+-

0

(3.3.2)

where the unknown subvector m k is partitioned as corresponds to (3.3._1). The matrix R k B k is the matrix (Cij) occurring in (A.12 and 13) and computed by (A.11) with (A.10); the zero rows of the matrix identify the arcs i ~ S k . In this manner, the classification is complete; the remaining arcs determine the unobservable variables. In addition, we can obtain also the matrix Rk Ak ready for further computation (3.3.2). [Observe that computing the elements of R k Bk, we substitute n k (reference node in G ~ for n o in (A.10). The reader can easily verify that also the elements of R k A k can be computed in this manner; considering G O[N k , j0], in (A.11) we put Rink - 0 as well as Rin - 0 whenever n ~ Nk.] See the example in Section 3.4. Having completed the classification according to steps (a) and (b) above, we can make use of the additional information obtained in the described manner. First, the reduced graph G* determines, having selected a reference node, the reduced incidence matrix A*. It is the matrix occurring in (3.2.4) 2, thus in the constraint equation for the measured vector m + (in fact, only for the subvector m* of redundant variables). The equation is employed for adjusting the given values; if the components of m + have been actually measured then for reconciliation by statistical methods. Having adjusted the components of m + and on introducing the adjusted values into Eqs.(3.3.2), we can compute directly the observable components of vectors m k. In addition we thus obtain certain conditions constraining the remaining (unobservable) components of the unmeasured vector. Though unobservable (thus admitting an infinity of solutions), they are generally not arbitrary. The whole vector of solutions will lie on a 'linear affine manifold' (as the mathematicians put it), thus on a subset of lower dimension than has the whole vector space of unmeasured variables. As a simple example, imagine a straight line or a plane in threedimensional space; for example if the manifold is a straight line in space (x, y, z), parallel to the plane (x, y) in a distance z0 (but not parallel to any of the axes), the variable z - z0 is observable, but x and y are

Chapter 3 - Mass (Single-component) Balance

41

unobservable, nevertheless subjected to the condition of lying on the projection of the straight line into the plane (x, y).

.sr, s ssJ' ..-"

Z0

Fig. 3-8. Observable variable z

Interpreted in terms of balance equations, the figure represents the solutions of a subsystem (3.2.4) k where the connected component G o drawn with incident fixed streams can look like

,

..

2

x t

' k

Fig. 3-8a. Interpretation of Fig. 3-8 unknown; ~ fixed

3

j

42

Material and Energy Balancing in the Process Industries

One of the node balances has been eliminated by graph reduction (as the merged-nodes balance) and the remaining two read m 1

-

m

2

m + i -0

-

+

mj - 0

m 3 -

We set m 1 - x, m2 = y, m3 = z, a n d mj + = z 0 , 9 mi§ is length of the segment on the x-axis in Fig. 3-8. But in spaces of higher dimension, there are more possibilities (not just straight lines or planes) having not individual names and defying any geometric imagination. In a more general frame, the possibilities will be examined in Chapters 7 (linear systems) and 8 (nonlinear systems).

3.4

ILLUSTRATIVE EXAMPLE Let us have graph G of a mass balance system

l; 11

7

e

i13

10 h

y

a

3

b

6/

Fig. 3-9. Example of mass balance system measured; ..... unmeasured

where the reference (environment) node n o has not been drawn. The arc set J is partitioned

43

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

j o _ {3, 4, 6, 7, 8, 12, 13} and J+-

{1, 2, 5, 9, 10, 11, 14, 15}.

So G~ j0] is defined. The simple figure would allow one to solve the problem by mere inspection. But our goal is to illustrate the formal procedure as outlined in Section 3.3. The decomposition of G o starts from some node, say no (environment), giving the connected component G~~ N~ - {no} , j]0 _ O (isolated node). Let us further select node a; we have successively, according to Section A.4 j~l) = -{3 } , j~2) _ {4, 6} , j(3) _ { 7, 8 } ,

N ~l) = {b } N (2) - {c, d} N (3) - {e }

(distance one) (distance two) (distance three)

j ( 4 ) _ Q~ .

We have thus determined G2~ N 2 - {a, b, c, d, e} , J2 ~

{3,4,6,7,8}

with the structure c

O r i i

i i i

"8

4!

b I

| ! |

o

a

3

b

1 1i

6

~:3d

Fig. 3-9a. Component G2~

From the remaining nodes, let us take node f; we find j(1) _ { 12 } ,

j(2)_ { 1 3 } , j(3)_

N (~) - {h) N (2)- {g}

44

Material and Energy Balancing in the Process Industries

hence we have the connected component G3~ N 3 _ {f, g, h} , J3 ~

{12, 13}

with the structure g O I I I

,, |

', 13 ! !

1

f

12

Fig. 3-9b. Component G3~

and no other node remains. We have K - 3 and let us merge the nodes in each N k. The unmeasured arcs are deleted. Let h k be the node obtained by merging N k. Then the incident nondeleted arcs are with

ill" 1, 5, 9, 14, 15 h 2" l, 5, 9, l0 (arc 2 deleted) fi3" 10, 14, 15 (arc 11 deleted).

We thus have J*= {1, 5, 9, 10, 14, 15} and J ' -

{2, ll} (nonredundant).

In this manner graph G* is defined. It has the structure

(3.4.1)

Chapter 3 - Mass (Single-component) Balance

43

14

15

Fig. 3-9c. Reduced graph G*

Observe that the subgraphs Gk~ as well as G, are determined by the original graph G[N,J] with the partition J - jo u J+ and by the prescribed algebraic graph operations; the drawings are only illustrations for the reader. Going back to the components Gk~ of G ~ let us find s p a n n i n g trees T k. G1~ is trivial (isolated node). In G2~ let us start from N~3~= {e }. Again according to Section A.4, from the two arcs j ~ jc3) incident to node e let us s e l e c t for instance in j~3): j ( e ) - 7 thus p ( e ) = c and further, in our simple example uniquely in j(2). j ( c ) = 4 thus p ( c ) - b and

j(d) - 6 thus p(d) - b

finally in J~>" j ( b ) - 3 , while p ( b ) - a is the (chosen)reference node in G2~ Hence the arc set of T 2 is {3, 4, 6, 7}, the

46

Material and Energy Balancing in the Process Industries

node set is N 2. The reader can find the tree T 2 in Fig. 3-9a, having deleted arc 8. Thus c = p(e)

7 =j(e)

()- . . . . . . . . . .

e

t~3

4 =j(c)

Ir

0

---I~O

t~0

a = p(b)

3 =j(b)

b = p(c) = p(d)

6 =j(d)

d

Fig. 3-9d. A spanning tree of G2~

The i n v e r s e R 2 of the reduced incidence matrix is found as in the example at the end of Section A.3: for each node n e N 2, the reference node excluded, we go backwards through the sequence of predecessors. We have b

c

d

e

Ab3

Ab3

Ab3

Ab3

j(b) = 3

Ac4

j(c) = 4

Ac4

R2

J

Ad6

j(d) = 6 Ae7

j(e) = 7

denoting by Anj the elements of the reduced incidence matrix, hence

R2=

b

c

d

e

1

1

1

1

3

0

-1

0

-1

4

0

0

1

0

6

0

0

0

1

7.

According to (3.3.1) we have

(3.4.2)

47

Chapter 3 - Mass (Single-component) Balance B 2 = (B2' , B2" )

} r o w s b, c, d, e

column 8

where

( B 2 ' ) -1 = R 2

and where, with the notation as above

Ab8

0

Ac8

m

B21!

0

Ad8

-1

Ae8

1

.

Recall that generally, the zero rows of R2B2" determine the arcs i ~ $2 that separate G2~ thus denoting by Cij (A.11 and 10) the elements of R2B2", i e $2 if and only if Cij = 0 in all columns j. In our special case, the only column is j = 8. We find C38=

0

G8 = -1 c68 = -1 G8 = 1

(3.4.3)

thus S 2 --{3}

is the set of arcs that separate G2~ Further, as I J3~ - IN 31 - 1, G3~ is a tree, thus T 3 = G3~ See Fig. 3-9b; we find j ( g ) = 1.3 , p ( g ) = .h , j ( h ) = 1 2 , p ( h ) - f

(reference node)

and h

g

R3 / 1 , ) 1 2 0

-1

We have directly S 3 = J3 ~ = { 12, 13 }.

13.

(3.4.4)

48

Material and Energy Balancing in the Process Industries

The classification is complete. The set S=S2uS

(3.4.5)

3= {3, 12, 13}

determines the set of unmeasured observable variables mj (j ~ S), and with (3.4.1), J* (resp. J') determines that of redundant (resp. nonredundant) measured ones. In addition, the incidence matrix of G* determines the conditions the measured variables have to obey in order to have the system solvable. Taking node h~ as reference node, the reduced incidence matrix equals (cf. Fig. 3-9c)

A* -

1

5

9

10

1

-1

1

-1 1

14

15

) -1

-1

h2

(3.4.6)

h3

and the condition (3.2.4) 2 reads m~- m5 + m9- m~0

=0

/~10- /~14- /~15

J

= 0

(3.4.7)

where fil (of components rhj, j ~ J+) is the vector of a priori fixed variables. The conditions (3.4.7) obeyed, we can substitute fil for m + in (3.3.2). We have 1

2

5

1

-1

A2

9

10

11

14

15

c 1

d

-1

e (3.4.8)

(having deleted reference node a), and

A3:/

1

2

5

9

10

11

11

14

15

-1

h

Chapter 3- Mass (Single-component) Balance

49

(having deleted reference node f). With m3 m4

m2

tt

~

m2 - (m8) m6

/ m2)

(3.4.9)

m7

m3

m13

the unmeasured variables are subject to the conditions

-1 m 2 +

m 8

+ R2A2ffl-

0

-1

(3.4.1o) and + R3A31fll -

m 3

0

according to (3.3.2) with (3.4.3), making use of the results (3.4.2),(3.4.4),(3.4.8). The whole space of unmeasured variables (unknowns) is of dimension 7. Given lh obeying (3.4.7), the set of solutions forms a (7-6=) 1-dimensional linear affine manifold (a straight line in 7-dimensional space), with the coordinates m 3, m~2, m~3 uniquely determined by the conditions (3.4.10). We can assign an arbitrary value to any one of the variables m4, m6, mT, m8, then the remaining ones are also uniquely determined by the conditions.

3.5

JUST DETERMINED SYSTEMS

The analysis of solvability applies in particular (and more simply) to systems where the set of a priori given variables' values just determines those of the remaining ones. We consider the graph GIN,J] as in Section 3.1. Denoting again by J+ the set of streams corresponding to the variables to be fixed and j 0 _ j _ j+, the partition determines the subgraph G o [N, jo]. A sufficient and necessary condition for the unique determination reads clearly G O is a tree

(3.5.1)

Material and Energy Balancing in the Process Industries

30

thus a spanning tree of G. [Recall that we assume G connected]. The subgraph G+[N,J § restricted to the arcs i ~ J+ is called the cotree of spanning tree G~ the arcs (called chords) determine certain degrees of freedom of the system (3.1.6). From a pure algebraic point of view, the mass flowrates of streams represented by the chords can be chosen arbitrarily; of course the physical condition is that of positive (or perhaps also null) mass flowrates (with the exception of the fictitious accumulation streams as in Fig. 3-1b). The number of degrees of freedom is thus D-

[JI - ( I N I

- 1)

(3.5.2)

called also the nullity of (connected) graph G; it is also the dimension of the null space KerC of reduced incidence matrix C. The choice of G o, thus also the selection of the chords thus degrees of freedom is by far not unique; only the number D is uniquely determined by G. As shown in Appendix A (A.18), the set Sp(G) of the trees G o has

I Sp(G) I

= det(CCT) "

(3.5.3)

elements; such is also the number of possible choices of the set of chords. To some extent, the selection of the chords can be controlled by our a priori idea of which values have to be fixed preferably. Let P ( c J) be the corresponding set of preferred arcs. In Appendix A, we have given two methods of finding a spanning tree of connected graph G. The selection of any new arc (called branch)j of the tree can be subjected to the condition that, as far as possible, j ~ P. So in the strategy (i)-(iii) according to Section A.5, instead of taking the minimum j ~ S we select, as far as possible, j ~ S but j ~ P (else arbitrary; taking the minimum corresponds to an ordered set with increasing preference). Then the chords remaining as J - R after the final step belong, as far as possible, to the preferred set P. Irrespective of how we have obtained the spanning tree G o, we choose a reference node n o ; in the mass balance problem, it will be clearly the environment node, thus C is the matrix in (3.1.6). We now can classify the nodes (units) n ~ N u = N - {no} according to their distance from n 0. Thus N is

partitioned N - {no} u N(~)u ... u N Cp)

(3.5.4)

where N (p) is the set of nodes of distance p, P is the maximum distance. We put N (~ = {no} and given p >_ 0 and N (p), let A (p) = {no} k.) ... k.) N (p). Then N (p+~) is the set of nodes n ~ N - A (p) that are connected by some arc j ' ~ jo with some n' E N(P); the set of such arcs j ' is j(p+l) and we have the partition

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

31

jo = J(~)u ... u j(r,)

(3.5.5)

of the arcs of G O (not perhaps of all the arcs of G[N,J]). In fact this is the construction according to Fig. A-11 using the fact that G O is a tree. From the theory it then follows that for each n ~ N ~p+~), there exists just one arc j ' having the other endpoint in N Cp), denoted by j(n) (~ J(P+])), and the other endpoint n ' = p(n) (~ N (p)) is called the last predecessor of n ; cf. Fig. A-8. We have j(P+l) = Q~.

3

c

6

1://

"o

',

g

i

no

: L.,

b

i~I

e IF~ ~ I

~

~I

IF ~

i

nodes

N {~

N (-~)

N (3)

Fig. 3-10 Tree G O for example jr2) = {3, 4, 5 } is the set of arcs j(n), n ~ N ~2)

Observe that the distance classification need not represent the distance classification with respect to the whole graph G. Imagine a chord connecting nodes n o and e; then also node e is in distance 1 from no and the whole classification will be different. Then also a tree obtained by the algorithm given in Section A.4 and starting from node n o cannot be G O as drawn in Fig. 3-10.

Let us have a spanning tree G o with the partition of J into jo and J+. Having fixed certain values m~+ for i ~ J+, the solution in mj (j ~ jo) is unique. Having partitioned the matrix C C

=

(B

, A)

} rowsn~

Nu=N-

{no}

(3.5.6)

columns j~ jo ie J+ the solution of Eq.(3.1.6) is m~

B-JAm +

(3.5.7)

52

Material and Energy Balancing in the Process Industries

where m ~ is vector of components mj (j ~ j0), m + that of given m + (i ~ J+) In Appendix A, we have given an explicit formula (A.4 and 5) for the inverse B -J Recall that B is reduced incidence matrix of tree G o thus in (A.5), A' stands for B. The computation is based on the knowledge of the predecessors p(n), p(p(n)) etc. and the connecting arcs j(n), j(p(n)) etc.; as an example, see Section 3.4, matrix R 2 (3.4.2). In fact, we need not compute the inverse B -~ and can find the solution directly on rearranging matrix B into upper-triangular form. The procedure is again based on the knowledge of p(n) and j(n). By the one-to-one correspondence, we have p - 1, ..., P: IJr

l -

I N( )I

(3.5.8)

hence in the first step, the partition of B (and also row-partition of A) reads, for example for P = 4 j(p)

///// / /,/,/,/

///// A

Fig. 3-1 l a. Triangularisation of matrix B, first step

where the void fields in B are zeros. Indeed, any n ~ N (p) is incident with (just one) j(n) ~ J~P), and generally with (possibly several)j' ~ J~P+~)(if p < P); each of the j ' is j(n') ~ j~p+l) for some n' ~ N Cp+l) , while n ~ N ~p) is not incident with any j" ~ J~P") where p" < p or p" > p + l , because the arcs incident to n span the distance 1. It can also happen that a branch ends the tree at lower distance from n o than P, thus some n ~ N ~p) can be incident only with one j(n) even if p < P. We can further rearrange arbitrarily the elements in each N Cp), and then re-order

Chapter 3 - Mass (Single-component) Balance

53

the set J(P) in the manner that the n-th column in J(P) is j(n); the (n,j(n))-th element of B is then the unique nonnull (thus _+1) element in row n and column subset J(P) Then each field N(P)x J(P) in B is diagonal j(p) A

r

+_1

N(p) ,

j(n) Fig. 3-11 b. Triangularisation, second step

and in addition, at the RHS of the field we have nonzero field N(P)x j(p+l), unless p = P. Below the diagonal, we have zeros in all rows of B (thus B is upper triangular). The upper-triangular format of matrix B allows one to compute successively the solutions from below. In the last group J(P), the solutions mj where j = fin) with n ~ N (P) are each given as Cnj(n)-times the n-th component of vector (-Am+). Introducing the values of these mj into the equations of rows n N (p-l), w e compute mi for j ~ j(P-l), and so on. Besides, the reader certainly knows how to manage standard operations with triangular matrix B. Example Rearranging the nodes n e N u and arcs j ~ j0 according to Fig. 3-10 the matrix C reads j(1) ,Ik

9

9

1 a N (1)

b c (2)

1 -1

,%

9

2

3 j -1 I j

j(3)

4

9

5

f g

A

9

6

7

-1 1

1

d e

N (3)

j(2)

-1 -1

-1

A

54

Material and Energy Balancing in the Process Industries

Having triangularized the matrix B, it is ready for use in a general analysis of solutions (3.5.7) with arbitrary m +. Otherwise, given m + an even simpler variant not requiring the distance classification is possible. It is based on the following properties of a tree T. Let us call endpoint of tree T any node n incident with just one arc j. Then deleting arc j and node n, we have again a tree. (ii) Any tree has at least two endpoints. [This an easy consequence of (A.3); for (ii), use induction.] Let now n be an endpoint of the tree G ~ n ~ n o, j - j ( n ) the unique arc incident to n in G ~ Then Eq.(3.5.7) with (3.5.6) reads Bm ~ - - Am + and the n-th scalar equation is (i)

Cnimj = - ~ n m + (Cnj - +1)

(3.5.9)

where c~, is the n-th row vector of A; indeed, C n i --- 0 for any i ~ j , i ~ jo. Going through the nodes n ~ N we thus can find, by (ii), an endpoint n ~: no and compute mj (j = j(n)) by (3.5.9); the solution is rhj. We delete the n-th equation and set mj - rhj in the remaining ones. We have thus deleted arc j and node n in G ~ and have again a tree with node set N - {n } and arc set jo _ {j} (unknown variables), while the set of fixed variables is now J§ w {j}. Continue in this manner so long as some unknown variable remains. Of course the former 'classifying method' provides a special sequence of such operations. Conversely, the latter sequence of nodes n and arcs j(n) determines a triangularisation of matrix B.

Remark More generally, let us consider the case where all the unknown (unmeasured) variables are observable. As shown in Subsection 3.3.1, the necessary and sufficient condition is that the connected components Gk~ ( k - 1,-.., K) of G~ j0] are all trees. According to (A.1 and 3), this is equivalent to the condition

IN! - g - - IJ~

.

(3.5.10)

If so and if K > 1, we have the solvability condition (3.2.4) 2. Having adjusted the fixed (measured) values so as to obey the condition, the unknown variables can be computed according to (3.2.4)~ where the K' matrices Bk are reduced incidence matrices of the (nontrivial) trees Gk~ Then the solutions can be computed as above, most simply using the endpoints; see (35.9) and the subsequent paragraph.

Chapter 3 - Mass (Single-component) Balance

3.6

55

MAIN RESULTS OF C H A P T E R 3

The basic idea is that of the oriented graph of a technological system subject to balancing; see Section 3.1 and Appendix A. The graph (G) contains nodes n ~ Nu, which are the individual units of the system, plus one formally introduced node n o representing the (so-called) environment; hence the node set (N) of G is the union (3.1.4). The arc set (J) consists of streams between the units, oriented according to the direction of material flow; arcs whose one endpoint is n o represent inputs/outputs into/from the system. If not stated otherwise, also in the following chapters G[N,J] represents the technological system subject to balancing, with units n ~ Nu and oriented material streams j~J. If only mass is balanced we can formally admit a change in accumulation of mass in a node as a fictitious stream oriented towards the environment; see Fig. 3-1. The conservation of mass is then expressed as Eq.(3.1.6) thus Cm - 0 where m is the (column) vector of mass flowrates and C the reduced incidence matrix of G, thus without row n 0. We assume G connected, hence C is of full row rank (3.1.5) thus INn [ where I MI denotes generally the number of elements of set M. The problems of solvability arise when certain mass flowrates mj are given a priori; see Section 3.2. The graph structure allows one to analyze the problems completely by graph operations. The arc set J is partitioned j = jo u J+

(J+ n jo = 0 )

(3.6.1)

where J§ is the subset of streams (arcs) with a priori given mass flowrates. Restricting graph G to arcs j ~ j0 (thus deleting arcs j ~ J+) we have subgraph G~ see Fig. 3-6. The subgraph is generally no more connected; it can be uniquely decomposed into (say) K connected components Gk~ of node sets Nk and arc sets Jk~ thus G0[N, j0] is union of Gk~

Jk0],

k = 1,--., K.

(3.6.2)

Recall that a connected component can also consist of one (isolated) node, with empty subset of incident arcs. Having merged the nodes of each Nk in G we have the reduced graph G*; see Fig. 3-7. The K nodes of G* correspond uniquely to the K subsets Nk of N. By the graph reduction (merging), we have deleted all the arcs j ~ jo, and some arcs j ~ J+ in addition (subset J ' c J+). The arc set of G*, denoted by J*, consists of the remaining (not deleted) arcs j ~ J+. Thus J§ is partitioned J+ = J* t,.,) J' (J* c3 J' = 0 ) .

(3.6.3)

56

Material and Energy Balancing in the Process Industries

The subgraph G ~ the decomposition (3.6.2) of G ~ the reduced graph G*, and the partition (3.6.3) are uniquely determined by G and by the assumed partition (3.6.1.) of streams. Of course if K = 1 (G o connected) then G* is reduced to one isolated node and J* = O, J' = J+. In the first step, having re-arranged the variables and equations according to the graph decomposition, we have thus rewritten the balance equation Cm = 0 (3.1.6) in equivalent form (3.2.4). The last of the equations (3.2.4) reads A'm* - 0

(3.6.4)

where m* is the subvector of mass flowrates mi, i ~ J*, A* is reduced incidence matrix of graph G*, of full row rank K-l; Eq.(3.6.4) is absent if K = 1. The equation represents the (necessary and sufficient) condition of solvability: the a priori given values of m i (i E J*) must obey Eq.(3.6.4). The vector m* is subvector of vector m +, the latter of components m i , i ~ J+ (D J*); if Eq.(3.6.4) is obeyed and if the remaining a priori given m i (i E J') are arbitrary, the whole set of balance equations is solvable in the components mj, j ~ j0 (3.6.1). Any such solution is that of the subsystem (3.2.4)~. Here, we consider the subgraphs Gk~ (3.6.2) in number K' (< K) that are not isolated nodes (thus such that Jk~ 0 ) . According to the partition of the set jo into nonempty subsets Jk~ we form subvectors mk (k = 1, ..., K') of components mj , j ~ Jk~ then the incidence matrix of Gk~ operates on mk. In each Gk~ we can select a reference node (say nk~ one of them being the environment node no); then Bk is reduced incidence matrix of Gk~ On the other hand, having partitioned the rows of the whole incidence matrix of G according to the node subsets Nk, having deleted nk~ in subset N k we obtain submatrices A k (k = 1, ---, K') operating on subvector m + of m i , i ~ J+. Schematically jo

j+ ,,,~

0 F/k

Nk

0

I I I I------I I

I I I

I

I

I I I

tlk

I

0

Ak

I I I

Then the solutions mj (j ~ j0) are exactly those of the set of equations k = 1, ..., K'"

Bkm k + s

--

0 where c k -

Ak m+

(3.6.5)

Chapter 3- Mass (Single-component)Balance

57

are constant vectors determined by m +, provided Eq.(3.6.4) is satisfied. Here, each Bk is of full row rank, hence the set of equations (3.6.5) is (not necessarily uniquely) solvable. The structure of the graph G along with the partition (3.6.1) allows one to classify all the variables (mass flowrates) with respect to the solvability; see Section 3.3. According to the standard terminology, the variables mi, i ~ J+ are called measured, the remaining mj (j ~ j0) unmeasured. This classification will be used henceforth; any a priori fixed variable of a model will be called 'measured', else 'unmeasured'. The classification of the measured variables j* follows immediately from the partition (3.6.3); if i ~ then m~ is redundant, else (i ~ J') nonredundant. The nonredundant variables are also characterized by the property that i ~ J' if and only if arc i closes a circuit in G with certain unmeasured streams (j ~ j0). The nonredundant variables are unaffected by the solvability condition. The unmeasured variables are subject to the conditions (3.6.5), given the measured values obeying the solvability condition. Some of the mj (j ~ j0) are thus uniquely determined; they are (called) observable. A necessary and sufficient condition for mj to be observable is that arc j separates subgraph GR~ where j JR~ by the decomposition (3.6.2), there exists a unique k such that j ~ Jk~ In particular, the whole subvector of unmeasured variables is observable (uniquely determined) if and only if all the subgraphs Gk~ are trees (possible isolated nodes included). An unmeasured variable is called unobservable when it is not observable (thus lies in some circuit of subgraph G~ In Section 3.3, we give a method of classifying the variables simultaneously with transforming the equations into a form suitable for explicit computation. First, the components of rn* have to be adjusted so as to obey Eq.(3.6.4). Then, by further graph operations the equations (3.6.5) are transformed in the manner that the observable variables are directly expressed as (linear) functions of the measured ones. See also the illustrative example 3.4. If in particular all the unmeasured variables are observable and no measured variable is redundant, the system is called just determined; see Section 3.5. A necessary and sufficient condition is that the subgraph G~ ~ is a tree thus a spanning tree of G[N,J]. This is a special case in Section 3.2 where thus G o is connected, the condition (3.6.4) is absent, and K ' = K - 1 in (3.6.5); thus BR - B (reduced incidence matrix of G ~ and AR = A is the remaining submatrix of C - (B,A). Instead of matrix inversion, the equation can be solved by rearranging matrix B into upper triangular form; see Figs.3-11 a,b. The set of mass balance equations provides an example of a linear model. It will be shown later in Chapter 7 that for any linear model, an analogous transformation of the equations and unambiguous variables classification is possible using the methods of classical linear algebra.

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3.7

Material and Energy Balancing in the Process Industries

R E C O M M E N D E D LITERATURE

A pioneering work in the classification of variables is due to V~iclavek (1969). The application of graph methods to the problems of solvability of balance equations and reconciliation is then due in particular to R.S.H. Mah and his school (Mah et al., 1976); an overview can be found in Mah (1990). See also the literature to Chapter 5. Mah, R.S.H (1990), Chemical Process Structures and Information Flows, Butterworths, Boston Mah, R.S.H, G.M. Stanley, and D.M. Downing (1976), Reconciliation and rectification of process flow and inventory data, I&C Proc. Des. Dev. 15, 175-183 V~iclavek, V. (1969), On the application of the calculus of observations in calculations of chemical engineering balances, Coll. Czech. Chem. Commun. 34, 364-372