Chemical source term reaction rate vector

Chemical reactions for which the rank of the reaction coefficient matrix T is equal to the number of reaction rate functions R, (i. e 1,..., I) (i.e., Nr = I), can be expressed in terms of / reaction-progress variables Y, (i. e 1,...,/), in addition to the mixture-fraction vector . For these reactions, the chemical source terms for the reaction-progress variables can be found without resorting to SVD of T. Thus, in this sense, such chemical reactions are simple compared with the general case presented in Section 5.1. 

A particularly simple method of approximating the chemical source term co(Y) is reaction elimination all one does is delete reactions which are numerically insignificant under the current reaction conditions. As shown by Bhattacharjee et al. (2003), at any set of single point reaction condition(s), one can rigorously identify the smallest possible set of reactions which reproduces the rate at which species are made or consumed chemically, and the rate at which heat is released due to chemical reactions, cumulatively >( Y), to within a user-specified tolerance vector tols , Eq. (16). Mathematically, the process of finding the smallest possible model is a constrained interval optimization... 

Ip being the NM-dimensional vector of the unknown scalars, tu( ) the vector of the source terms, including the chemical reaction rates, and P(, V, being the spatial vector... 

It can happen that our information is limited and the detailed chemical components balance not needed. Then only heat and mass balances are set up see Section 5.4. A typical example is a heat exchanger network. In the equations (5.7.11), the quantities (5.7.9) are approximated as functions of temperature only, say h (T ) in stream j. Formally, we can consider in addition certain source terms s n) in some nodes n e T , for example due to heats of reactions, a priori assessed or regarded as unknown p2U ameters to be computed from the set of constraints (given measured values of mass flowrates and temperatures). For example in a heat exchanger network, P ) = (P-Tq) is the sensible heat of stream y, with temperature P, specific heat cj, and reference temperature. We introduce the vectors hs of components hi, j e, and h of components for j 6 J (5.4.8), then the vectors (5.4.9) of components hi (j e J ) and h (5.4.10) of components for j e J the quantity /ij = is the heat flowrate in material stream j, with heat content factor hi. Finally s is the vector of components s(n), n g T . Then the heat and mass balance is represented by the equations (5.4.6). Again, the heat transfer rates through dividing walls can be eliminated by summation of the two scalar equations in (5.7.11), viz. the n]-th... 

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