## The Problem of the Nonenergetic Production Factors

Toxicity is not the only factor that is not measurable with a purely thermodynamic analysis. We must therefore try to extend our method to include toxicity and the other 'environmental expenses' that must be charged to a certain commodity. It is well known from economic theory that the 'cost' of a product is given by an expression called the 'production function'. Without further justification we can assume the following form for such a function:

where cj is the 'cost' of a unit of the (material or immaterial) commodity, K is the amount of monetary capital (E) required by its production, L the amount of labor (work-hours), E the exergy (J), M the necessary materials (kg), and O the environmental remediation cost (monetary cost of the remedial action necessary to annihilate or reduce the effects of product j on the environment). The five terms inside of the brackets in eqn  are called the 'production factors'.

We are looking for a method in which both cj and the 'production factors' are homogeneously expressed in terms of a thermodynamic function. Exergy 'automatically' accounts for E and M: since the exergy destruction appears to be the most complete indicator we have analyzed so far, it seems proper to use exergy as the common quantifier for the remaining, nonenergetic 'production factors'. To do this, we need a general method to represent the operative structure of complex production systems: this is the topic ofthe following section.

Structural Representation of Energy Conversion Systems

To compute the exergy destruction of a 'complex system' we need to take into proper account the complex interconnection of its components, which makes the response of the system to a variation in the performance of one of its subunits not easily predictable (and, for VLCS, like most of the biological ones, only fuzzily predictable). There may be cases in which an increase in the efficiency ofone component corresponds to an overall performance deterioration. Several methods have been proposed in the past to represent these cross-effects: we describe here in some detail a procedure based on a 'Leontieff input-output model' which results in a matricial representation of the process, similar to the one first developed by Valero and co-workers in 1988-89 for their 'symbolic exergoe-conomic method'.

Consider again the cogeneration process of Figure 3, which we shall here refer to as 'P', and define a 'connectivity matrix' CM constructed according to the following algorithm:

1. the jth column (j = 1.. .N) of CM represents the jth process stream;

2. the ith row of CM (i = 1...M) represents the ith component;

3. the generic element CMj is 0 if flux j does not interact with component' 1 if flux j enters component i, and -1 if flux j exits component i.

With M = 6 and N = 13, Figure 7 represents the CM(6,13) for the plant of Figure 3. This matrix representation of a process is the basis for a compact and elegant computational procedure. By inspection, we see that some of the streams transport mass and exergy (1,2,3,4,5,9,10,11,12) and some only exergy (6,7,8,13). We can correspondingly define a mass vector vm(13), every element of which represents the mass flow rate associated with the corresponding stream, and two energy- and exergy vectors vh(13) and ve(13), which contain the energy and exergy values of each stream, respectively. As a further convention, vm contains a ' 0' in all positions corresponding to a massless exergy carrier' (an energy flux like mechanical work, electricity, etc.). It turns out that the mass and energy balances as well as the exergy accounting of the process are described by the scalar product of the matrix CM with the three vectors vm, vh, and ve, respectively.

This formalism can be manipulated to compute the general transfer function of the process, that is, the matricial function n that premultiplied by any property vector of the input streams (the 'fuels' in Valero's terminology), provides the resulting property vector for the output streams:

where veA is a vector that contains the exergy destruction in each component.

There is a direct relationship between CM and n (neither one of which is necessarily square), the second being a sort of augmented form of the former, but with the '0' and '1' substituted by influence coefficients k, (see below). The transfer function n is itself of course a function of the internal characteristics of each component, in particular of the efficiencies. Once a closed-form expression for n is known, we can study the sensitivity of the process performance to that of each one of its components. It is convenient here to adopt the formalism of the 'influence coefficients', first proposed independently by Beyer and Brodyansky in the 1970s and later systematized by Kotas. For each component, we thus define a 'coefficient of structural bonds aij- and a ' coefficient of influence' rKj: 