Cost equations

The economic factors must be considered in every application. It is important to find a technique that will meet both the technical and economical requirements. In short, pollution control costs depend on the system characteristics and the application. Some cost equations that generalize the economics of the managing systems are available in the literature. Most of these equations give rough estimates and have an accuracy of only about 30% to 50%. For a comprehensive cost comparison of different units, a detailed cost analysis based on the equipment tender proposals and the special characteristics of the project is necessary. 

The cost equations can be developed by considering a 1 metre length of pipe. 

Structural theory facilitates the evaluation of exergy cost and incorporation of thermoeconomics functional analysis (Erlach et al., 1999). It is a common formulation for the various thermoeconomic methods providing the costing equations from a set of modeling equations for the components of a system. The structural theory needs a productive structure displaying how the resource... 

To this point we isolated four variables D, v, Ap, and/, and have introduced three equality constraints—Equations (d (e), and (/)—leaving 1 degree of freedom (one independent variable). To facilitate the solution of the optimization problem, we eliminate three of the four unknown variables (Ap, v, and/) from the objective function using the three equality constraints, leaving D as the single independent variable. Direct substitution yields the cost equation... 

Another source estimates the cost of using two BioNets with four fractures each as ranging from 60,000 to 70,000, while the cost of four BioNets with four fractures each ranged from 80,000 to 90,000. These costs equate to 10 to 20 per ton for remediation of contaminated soil at a typical site (D16920U). 

Use the equationP = R-Cand replace theP with 2,000. Put the revenue equation and cost equation in their respective positions and solve the equation for x. 

The following cost equation shows the economic elements divided into three groups, each affected differently by current density. 

A cost equation may be written to include all the costs, which are expressed in terms of the capacity of the flow-scheme components. The selection of equipment sizes that minimize capital investment is (1) complicated by interrelations between pieces of equipment, (2) limited by the discontinuity in size of standard equipment, (3) fixed by the availability of idle or used equipment, and (4) restricted by the higher cost of custom-made equipment. Writing one equation for a complete plant is a complex task. It is more likely that it may be done for small sections of the plant which can be operated as interrelated trains. 

The cost equations thus written are discontinuous functions of the size of the units which compose the trains. A mathematical minimization of any of these equations may not lead to a practical minimum. It may indicate only the domain where less-expensive solutions may exist. The practical alternative is to draw flow schemes which are equivalent to the process under investigation. The economic analysis of these schemes terminates with the selection of one which requires the minimum capital investment and operating costs. 

Essential relations describing each subsystem are overall equations for mass balance, for energy balance, for performance, and for costing in terms of performance. Presently available cost trends (17) in terms of capacity parameters (e.g. area, mass rate, power,. ..) are suitable costing equations to start with. They may be implemented to include the influence of variables such as pressure, temperature or efficiency whenever sufficient data are available. 

Essential performance variables are less than the design degrees of freedom. In this case, the cost is minimized with respect to the excess degrees of freedom and the envelope of minimum costs is the costing equation. 

These elements of the strategy and the technique of developing the costing equations are illustrated in the following three examples. 

This example treats a simple open-cycle gas turbine for which the cost objective function, equations of constraint and costing equations are all available in analytic form. Figure 3 shows these functions along with the fixed and variable decision variables. Since the set of equations is diagonalized,... 

The heat exchanger costing equation is the costing equation derived in Example 3. The other costing equations were suggested to us by experienced engineers and are illustrative only. 

Examination of costing equation reveals that exchangers in this range should cost about 3.6 k /yr., so that there is obviously an opportunity to improve the system through addition of a heat exchanger. The redistribution of cost given in Table VI shows that the pressure loss penalty is high on the hot... 

Translating a Cost Equation for a Heat Exchanger to an Optimized Costing Equation. 

Table Vll describes the class of heat exchanger chosen and summarizes the design equations and the capital cost equation. The cost of the heat exchanger, expressed in terms of geometric variables, (measured in feet) is given by ...

As indicated in Table Vll, there are 32 variables constrained by 23 equations, leaving 9 degrees of freedom. On the other hand, the cycle analysis, requires a costing equation using four variables, i.e.,... 

Cost Estimation. The capital costing equations used in the cogeneration problem have been designed to yield approximate capital and maintenance expenditures and to reflect the consequence of changing the system s variables on these costs. The form of these equations expresses equipment costs in terms of stream and performance variables. In all cases a capital recovery factor is used to account for the cost of capital (i = 15%) and estimated useful life (n = 40 years). 

The approach taken to develop these costing equations was to single out the most important parameters that influence cost, and use them to yield a base cost, designating them with a prime (i.e. Z ). This base cost is then adjusted by multiplication factors so as to incorporate the influence of other factors. The form of these equations has been suggested in the literature (10-12) and by experienced engineers, then curve fit to available data. However, extreme care must be exercised when applying any of these equations in the field. 

The interested reader should refer to Reference (13) for a detailed description of costing equations for the boiler, turbine, condenser, and pump. The costing equations for this system are listed in Table I. 

The equations of constraint link the cost estimate through the system s thermodynamic performance to fuel costs. The thermodynamic analysis must relate the variables used to describe the system s performance to those used in the cost estimate. In this problem, costing equations are used which are generally in terms of stream and performance variables. Thus the thermodynamic analysis need only be in terms of these variables. Sixteen equations of constraint have been developed from a thermodynamic analysis of the cycle, and are given in Table III. 

The next step is to obtain equations for the solution of the shadow and marginal prices. This requires the evaluation of various derivatives of the constraint equation matrix. However, because not all the constraints are in algebraic form (those constraints that are functions of steam table properties) numerical derivatives must be evaluated. One other note, there are two condenser costing equations. This means that two separate derivatives must be taken and the derivative corresponding to whichever costing equation is valid for that value of condenser area, is the one that should be used. 

In the application of this method to a Rankine cycle cogeneration system, generalized costing equations for the major components have been developed. Also, the utility of the method was extended by relaxing the rule that each state variable (and hence each Lagrange constraint) must correspond to an available-energy flow. The applicability was further extended by the introduction of numerical techniques necessary for the purpose of evaluating partial derivatives of steam table data. 

Optimization Procedure. Given a set of design variables (a working design), capital cost equations as functions of the design variables (9), the unit costs of utilities, and Equations 33-35, the Second-Law based optimization may be performed. 

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