Constraints on rates or cumulated quantities

It is necessary to check the data consistency before performing any calculation. In fact, it is difficult to describe accurately the metabolism or to quantify biochemical pathways if some relevant species is omitted or biased [49-5 Ij.The recovery of an entity k (such as carbon or nitrogen atoms) is defined as the ratio between its production rate and its consumption rate. Assuming that represents the amount of k per mole of chemical species /, the recovery is calculated as  

The minus sign indicates that r, 0 for substrate so that / 0. Recovery should equal one, and it is lower than one if a product formation rate is underestimated (or neglected), or if a substrate consumption rate is overestimated. The enthalpy recovery take into account all the products and substrates as well as the heat dissipated and the heat of combustion of the hydroxide ions and the heat of vaporization of gaseous ethanol [32]. 

Another way to judge the correctness of the balance is to compute the balance for the elements. For each entity the net consumption rate in the reactor must be equal to the net production rate. For 5 different species containing the element k the balance for that element can be expressed by rates ( ) or by cumulated quantities () as follows [32,33], on rates  

The constraint is that the balance should equal zero. 

It is possible to investigate the balances individually (each balance should equal zero) or simultaneously. In that case the combined balance is a vector of which each element is one of the balances and the constraint is that each element of the vector should equal zero. We construct the matrix X of element fraction of which each element Xk, is equal to The vector of balances is noted f and the general formula using matrix notation is  

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