t t 

0 if x  is feasible

ΔP  ΔPmax                                                                                                                     (43) 

Penalty(x ) m

               (45) 

s s 

ri gi  (x ) otherwise

The values suggested below can be used as a  general guide, and will normally give designs that are near the optimum。 [15]。 

Liquids。 For fluids having Viscosity <1 mN  s/m2 

maximum pressure drop: 35 kN/m2 [15]。 For fluids having viscosities 1 to 10 mN s/m2, maximum pres- sure drop: 50-70 kN/m2 [15]。 

Gas and vapors。 High vacuum: 0。4-0。8 kN/m2; medium vacuum: 0。1absolute pressure; 1 to 2 bar: 0。5system gauge pressure; above 10 bar: 0。1system gauge pressure。 

For the present case study following constraints were imposed: 

ΔPt 35000 Pa; ΔPs 35000 Pa 

The prerequisite of a good design is to choose the lowest cost exchanger with standard dimensions (as per TEMA standard) while obeying the above constraints。 Attempt has been made in this work to apply SA optimization technique to design a lowest cost heat exchanger with TEMA dimensions and sa- tisfying all of the above constraints。 

However, the value of the constraints of pres- sure drop and velocity is dependent on the detailed design and very much problem specific。 In this work, the values of constraints are selected as per general  guidelines given by Sinnot [15] and the user is not  restricted to adhere this value。 The value of these constraints must be judiciously selected as they  have a big impact on final solution and cost。 In case the user does not have specific restriction on these va- lues, the constraints should be kept as broad as pos- sible。 This will facilitate the lowest cost heat exchanger。 

Handling the constraints 

The original problem can be set as: Minimize Ctot(x) 

Subject to gi(x) ≥ 0 where i = 1, 2, m 

i 1

where ri is a variable penalty coefficient for the ith constraint, ri  varies according to the level of violation。 

Simulated annealing: at a glance [22] 

What Is simulated annealing? 

The simulated annealing method  is  based on the simulation of thermal annealing of critically heated solids。 When a solid (metal) is brought into a molten  state by heating it to a high temperature, the atoms in the molten metal move freely with respect to each other。 However, the movements of atoms get restric- ted as the temperature is reduced。 As  the tempera- ture reduces, the atoms tend to get ordered and fi- nally form crystals having the minimum possible inter- nal energy。 The process of formation of crystals es- sentially depends on the cooling rate。 When the tem- perature of the molten metal is reduced at a very fast rate, it may not be able to achieve the crystalline state; instead, it may attain a polycrystalline state having a higher energy state compared to that of the crystalline state。 In engineering applications, rapid cooling may introduce defects inside the material。 Thus, the tem- perature of the heated solid (molten metal) needs  to  be reduced at a slow and controlled rate to ensure proper solidification with a highly ordered crystalline state that corresponds to the lowest energy state (in- ternal energy)。 This process of cooling at a slow   rate is known as annealing。