Solution manifold

One then restricts the solution in the primary variables (j e J) and yj (/ J = 1, —, K) to the subset obeying the conditions (8.2.69) of positive mass flowrates. The primary variables are then uniquely determined by the ml according to (8.2.63, 63a, 64), and vice versa. Then the set fW of solutions has D degrees of freedom, where D is the number (8.2.78) in rigorous mathematical terms, f is a differentiable (in fact, analytic) manifold of dimension 

Indeed, let us go back to Section 4.3. For a nonreacting species C, , we have = 0 for r = 1, , R, thus the k-th row of matrix S (4.3.8) equals zero. Then, in the matrix (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 ny. is one of the components of vector n in (4.3.12) and by (4.3.13) we have 

If a nonreacting component Q (such as component C3 = N2 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 = R above) then the result (8.2.86) is again correct. But according to (8.2.1) with C = 0 for all / e, the result (8.2.86) is as well a consequence of the a priori assumption (definition of the subset E ), as corresponds to (8.2.20- If replacing the K n)-Rffn) scalar equations (8.2.2a) by K-R ri) equations involving also the species Q 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 

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. 

Recall the parametrisation of the solution manifold 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 ne T , we thus have introduced R(, (n) parameters where / (n) is the dimension of the reaction space ( ). Alternatively, we can introduce directly certain R, (n) independent reaction rates Wfn) (r = 1, , Rain)) in node n e T . Indeed, by (4.3.2) the whole vector n(n) is uniquely determined as (linear) function of the (n), instead of the parametrisation using (8.2.50 and 51). 

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Rheinboldt, W. C. On the Computation of Multi-Dimensional Solution Manifolds of Parameterized Equations. Tech. Report ICMA-86-102, 1986. 

The equations (4.5.2a) with (4.5.1) are bilinear in the variables yi and mj. The whole set of constraints generally does not determine a unique solution, but only what is called an (analytic) solution manifold, a (mathematically smooth, but curved ) subset of the variables space. Only if a sufficiently great number of variables is measured and adjusted so as to make the system solvable, the remaining variables can be computed. We shall not attempt to analyze the problem in the same (mathematically rigorous) manner as in Chapter 3. An example in Section 5.5 will show that such problems may sometimes also be not well-posed . For a different approach, see further Chapter 8. 

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

GENERAL SOLUTION MANIFOLD 8.3.1 Energy balance equations... 

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 Jacobi 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 k (8.5.25) where our hypotheses are expected to hold. Instead of the full solution manifold we limit ourselves to some portion Uof Then ... 

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 " if the sequence converges then the limit value, say z, 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 Vet/ where ll is the admissible region (8.5.8). The interval can be assessed as some neighbourhood of a vector Zq e fSf. A first information can be obtained in the same manner as above, in the linear case. Taking different Zq e we can examine the behaviour of the Jacobi matrix Dg(Zo) on fW (restricted to t thus on r U). [We can also, in the case of balance models, start from different values of the independent parameters representing the degrees of freedom and determining Zq e fW 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 parametrisation of the whole set of solutions is not unique (i.e. the selection of the parameters is not unique), only the number of degrees of freedom (D,ot) is invariant. In mathematical terms, the set is a differentiable manifold of dimension 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. 

Here, according to Chapter 8, is the solution manifold. Because thus g(oo(r)) = 0 we have... 

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