From the first law of thermodynamics, the energy balance equation can be obtained as
The mass, species, and energy balance equations for all the plant subsystems and a few associated state- and effect-related functions yield a set of independent equations. This set of simultaneous equations is solved by matrix algebra assuming equal temperature intervals for all effects and assuming that all effects have adiabatic walls. The boundary conditions are the specific seawater feed conditions (flow rate, salinity, temperature), the desired distillate production rate, and the specified maximum brine salinity and temperature. The matrix solutions obtained determine the distillation rates in the individual effects, the steam requirements, and hence the performance ratio (Hamed et al., 1996).
The steady-state exergy balance equation may be written as
Total exergy transported into system = Total exergy transported out of system +
Energy destroyed within system (or total irreversibility)
The system overall irreversibility rate can be expressed as the summation of the subsystem irreversibility rate:
j where J is the number of subsystems in the analysis and Ii is the irreversibility rate of subsystem i. The exergy (or second-law) efficiency r|n, given by
% = p (8.22) E Ein is used as a criterion of performance, with Ein and Eout determined by Eqs. (8.19) and (8.20), respectively. The total loss of exergy is obtained from the individual exergy losses of the plant subsystems. The exergy efficiency defect, §i, of each subsystem is defined by (Hamed et al., 1996)
The exergy of the working fluid at each point, calculated from its properties, is given by:
where the subscript o indicates the "dead state" or environment defined in the previous section.
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