An Overview of the Application of Harmony Search for Chemical Engineering Optimization
Table 8
Problem description of the design of heat exchangers.
Application
Type of optimization problem
Objective function
Optimization variables
Constraints
Reference
Shell and tube heat exchanger design
Global optimization
Minimize ,,,, where is the annual total cost, is the annual capital cost, is the annual operating cost, TP is the yearly period of operation time, is the pumping power, is the energy cost per unit, is the capital cost, is the interest rate, and TL is the technical life.
Discrete and continuous variables: inside shell diameter, tube outside diameter, tube pitch, length, baffle cut, central baffle spacing, number of tube passes, tube layout characteristic angle, and number of tubes.
Minimize ,,,,, where is the total cost of the heat exchanger, is the capital investment, is the discounted operation cost, is the operation cost, is the energy cost, represent the heat exchanger coefficients, is the heat exchanger surface area, is the efficiency factor, is the annual number of work hours, ny is the number of years, and is the annual discount rate.
Discrete and continuous variables: the number of tube passes, shell and tube outside diameter, and baffle spacing.
where is the tube side pressure drop, is the major loss pressure drop, and is the minor loss pressure drop.
Minimize  +  + . The first two terms correspond to the annual utility cost, while the rest of the terms represent the capital cost. is the cold utility cost. is the hot utility cost. is the heat exchanged between stream i and cold utility. is the heat exchanged between stream j and hot utility. represent equipment dependent constants. is the area. HP is the index set of hot process stream. CP is the index set of cold process stream. NE is the index set of process heat exchangers.
The stream split fractions and the heat load of the exchangers.
(1) Heat balance:  = ,  = , where is the temperature, is the inlet value is the outlet value, is the heat capacity flow rate, is the heat exchanged, NL is the index set of exchanger levels. (2) Non-negative bounds: ,,,, where is the mole fraction vector. (3) Minimum temperature: ,,, where is the temperature approach and is the minimum temperature approach. (4) Mass conservation: , where SS is the index set of stream splits.
Minimize ,,,,, where is the annual total cost, is the annual capital cost, is the annual operating cost, TP is the yearly period of operation time, ec is the energy cost per unit, is the fan electrical power consumption, is the air cooled heat exchanger capital cost, are the cost factors, is the tube area, is the interest rate, and TL is the technical life.
Discrete and continuous variables: air velocity, tube material, length and outside diameter, fin height, number of tubes per row, and tube passes.
Minimize ,,,, where is the heat transfer area, is the length of side a, is the length of side b, is the number of layers in the hot side, is the fin height, is the fin thickness, and is the fin frequency.
Discrete and continuous variables: ,,,,,, and the lance length .
Constraints were handled using a penalty method. ,,,, where is the heat transfer rate, is the effectiveness, is the minimum heat capacity rate, is the temperature, is the friction factor, is the mass flow velocity, is the density, is the hydraulic diameter, is the non-flow length, and are fluids.
Minimize , where is the pressure drop by friction, is the pressure, are fluids, and max is the maximum value.
Discrete and continuous variables: the length of sides a and b, number of layers in the hot side, fin height, thickness and frequency, and lance length.
Constraints were handled using a penalty method. ,,,, where is the heat transfer rate, is the effectiveness, is the minimum heat capacity rate, is the temperature, is the friction factor, is the length, is the mass flow velocity, is the density, is the hydraulic diameter, and is the non-flow length.
