Hybrid Particle Swarm Optimization and Ant Colony Optimization Technique for the Optimal Design of Shell and Tube Heat Exchangers

Tube side calculations

Number of tubes Nt and tube bundle diameter Db is calculated as follows:

Velocity through tubes is found by

Darcy friction factor ft is calculated as follows:

where Ret is tube side Reynolds number and given by

where di is the inside diameter of tube and for simplicity kept constant as di=0.8do in this work. However, for more rigorous calculation, instead of keeping it constant, variable pipe thickness can be used from tables of tube manufacturers: schedules 40 or 80 or standard tubing gages: B.W.G. and Stub’s gage.

Pr_t is the tube side Prandtl number and given by

According to flow regime, the tube side heat transfer coefficient (h_t) is computed from the following correlation:

Shell side calculation

Clearance between tube bundle diameter and shell diameter is calculated from the figure given in [16] for different head types as follows:

where m and c are empirical constants and assume the following values for different head types: Fixed and U tube (0.01,0.008), outside packed head (0.0,0.038), split ring floating head (0.027,0.0446), pull-through floating head (0.009,0.0862).

Baffle spacing is calculated as

Cross-sectional area normal to flow direction is determined by

where is tube pitch and given by Pt = 1.25 do

Flow velocity for shell side can be obtained from

This flow rate is defined at the equator of the shell where velocity is lowest.

Reynolds number and Prandtl number for shell side can be calculated as

where Des is the shell hydraulic diameter and computed as follows:

For triangular pitch

For square pitch

Kern’s formulation for segmental baffle shell-and-tube exchanger is used for computing shell side heat transfer coefficient h_s

where coefficient j_h is calculated from figure given in [16] for different baffle cuts.

Following approximate equation was used: jh=mResc where coefficients m and c are evaluated from figure given in [16] for different baffle cuts as follows: 15% baffle cut (–7 × 10⁻⁸; 0.100067); 25% baffle cut (–4.5 × 10⁻⁸; 0.070045);35% baffle cut (–4.4 × 10⁻⁸; 0.063044); 45% baffle cut (–3.4 × 10⁻⁸; 0.050034).

Pseudo-code for hybrid PSO and ACO algorithm

• Step 1: Initialize optimization

  • Step 1.1: Initialize constants for PSO and ACO algorithm tmax,P

  • Step 1.2: Initialize randomly all particles positions xti and velocities

  • Step 1.3: Evaluate objective function values as f(xti)

  • Step 1.4: Assign best positions pti=xti with fpti=fxti,i=1,…,P

  • Step 1.5: Find ftbest(ptbest)=min{f(pt1),…,f(pti),….f(ptP)}

And initialize ptg=ptbest and fptg=ftbestptbest

• Step 2: Perform optimization

• Step 2.1: Update particle positions xti and velocities υti according to eqs (47) and (46) of all P particles.

• Step 2.2: Evaluate objective function value as fxti

• Step 2.3: Generate P solutions zti using eq. (49)

• Step 2.4: Evaluate objective function value as fzti and if fzti<fxtithen fxti=fzti and xti=zti

• Step 2.5: Update particle best position if fpti>fxti then pti=xti with fpti=fxti,i=1,…,P

• Step 2.6: Find ftbest(ptbest)=min{f(pt1),…,f(pti),….f(ptP)}

Step 2.7: Increment iteration count t=t + 1

End while

• Step 3: Report best solution p^g of the swarm with objective function value f(p^g)