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Thermal and Economical Optimization of a Shell and Tube Evaporator Using Hybrid Backtracking Search—Sine–Cosine Algorithm

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This paper proposes a hybrid optimization algorithm based on the combination of the merits of the backtracking search (BSA) and sine–cosine algorithm (SCA) to achieve the optimal design of a shell and tube evaporator. To the author’s best knowledge, this is the first application of the metaheuristic algorithms over shell and tube evaporator design problems. In order to test the accuracy of the proposed hybrid algorithm, 10 well-known optimization test functions have been solved. Numerical results obtained from the hybrid BSA–SCA have been compared with the literature optimizers including differential search, big bang–big crunch optimization, quantum-behaved particle swarm optimization, bat algorithm, intelligent tuned harmony search algorithm, and backtracking search algorithm. Comparison results reveal that solutions obtained from the BSA–SCA are better than those of the results acquired by the aforementioned optimizers with respect to statistical analysis. Proposed optimization procedure is then utilized to obtain optimum values of the two heat exchanger design objectives including total cost and overall heat transfer coefficient. Six decision variables such as tube outer diameter, shell diameter, baffle spacing, tube length, number of tube passes, and tube bundle configuration are selected to be iteratively optimized. It is found that BSA–SCA provides better results than the compared literature optimizers for both objective functions. In addition, a sensitivity analysis is performed for the design parameters at the optimal point. Results show that variation of the design parameters at the optimum point has considerable effect on the objective function rates.

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\(A_{\mathrm{s}}\) :

Cross-sectional area normal to shell side flow (\(\hbox {m}^{2}\))

\(a_{\mathrm{s}}\) :

Cross-sectional area normal to in-tube flow (\(\hbox {m}^{2}\))


Baffle spacing (m)


Bat algorithm


Big bang–big crunch algorithm


Boiling number


Backtracking search algorithm

\(C_{\mathrm{e}}\) :

Energy cost (€/k Wh)

\(C_{\mathrm{i}}\) :

Capital investment cost (€)

\(C_{\mathrm{o}}\) :

Annual operation cost (€/year)

\(C_{\mathrm{od}}\) :

Total operating cost (€)

\(C_{\mathrm{l}}\) :

Shell side clearance (m)

\(C_{\mathrm{p}}\) :

Specific heat (J/kg K)

D :

Problem dimension


Differential search algorithm

\(D_{\mathrm{e}}\) :

Hydraulic shell diameter (m)

\(D_{\mathrm{s}}\) :

Shell inside diameter (m)

\(d_{\mathrm{i}}\) :

Tube inside diameter (m)

\(d_{\mathrm{o}}\) :

Tube outside diameter (m)

E :

Convection enhancement factor

F :

Correction factor


Firefly algorithm

\(f_\mathrm{s}\) :

Shell side friction factor

\(f_\mathrm{t}\) :

Tube side friction factor

Fr :

Froude number

G :

Mass velocity (\(\hbox {kg/m}^{2}\,\hbox {s}\))

H :

Annual operating time (h/year)

\(h_{\mathrm{s}}\) :

Shell side heat transfer coefficient (\(\hbox {W/m}^{2}\,\hbox {K}\))

\(h_{\mathrm{t}}\) :

Tube side heat transfer coefficient (\(\hbox {W/m}^{2}\,\hbox {K}\))

\(h_{\mathrm{fg}}\) :

Latent heat of vaporization (J/kg)

\(h_{\mathrm{nb}}\) :

Nucleate boiling heat transfer coefficient \((\hbox {W/m}^{2}\,\hbox {K})\)

\(h_{\mathrm{tp}}\) :

Two-phase heat transfer coefficient \((\hbox {W/m}^{2}\,\hbox {K})\)


Intelligent tuned harmony search

i :

Annual discount rate (%)

k :

Heat conductivity (W/m K)

L :

Tube length (m)

M :

Molecular weight (kg/k mol)

\({\dot{m}}_\mathrm{s} \) :

Shell side mass flow rate (kg/s)

\({\dot{m}}_\mathrm{t} \) :

Tube side mass flow rate (kg/s)

N :

Population size


Equipment life (year)

\(N_{\mathrm{b}}\) :

Number of baffles

\(N_{\mathrm{t}}\) :

Tube number

Nu :

Nusselt number

P :

Pumping power (W)

\(P_{\mathrm{t}}\) :

Tube pitch (m)

\(\Delta P_{\mathrm{s} }\) :

Shell side pressure drop (Pa)

\(\Delta P_{\mathrm{t}}\) :

Tube side pressure drop (Pa)

\(Pr_{\mathrm{s}}\) :

Shell side Prandtl number

\(Pr_{\mathrm{t}}\) :

Tube side Prandtl number

\(p_{\mathrm{r}}\) :

Reduced pressure (\(p/p_{\mathrm{sat}}\))

Q :

Heat load (W)


Quantum-behaved particle swarm optimization

\(q{^\prime \prime }\) :

Heat flux \((\hbox {W/m}^{2})\)

\(R_{\mathrm{fs}}\) :

Shell side fouling resistance (\(\hbox {m}^{2} \hbox {K/W}\))

\(R_{\mathrm{ft}}\) :

Tube side fouling resistance (\(\hbox {m}^{2} \hbox {K/W}\))

r :

Uniform random number between 0 and 1

\(Re_{\mathrm{s}}\) :

Shell side Reynold number

\(Re_{\mathrm{t}}\) :

Tube side Reynold number

S :

Empirical boiling suppression

S :

Total heat exchange area (\(\hbox {m}^{2}\))


Sine–cosine algorithm

T :

Temperature (K)

\(\Delta T_{\mathrm{LM}}\) :

Logarithmic mean temperature difference

U :

Overall heat transfer coefficient (\(\hbox {W/m}^{2}\,\hbox {K}\))

\(v_{\mathrm{s}}\) :

Shell side flow velocity (m/s)

x :

Vapor quality

\(\mu \) :

Dynamic viscosity (Pa s)

\(\rho \) :

Density (\(\hbox {kg/m}^{3}\))

\(\eta \) :

Pumping efficiency














Shell side


Tube side


Tube wall


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Correspondence to Oguz Emrah Turgut.

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Turgut, O.E. Thermal and Economical Optimization of a Shell and Tube Evaporator Using Hybrid Backtracking Search—Sine–Cosine Algorithm. Arab J Sci Eng 42, 2105–2123 (2017).

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  • Backtracking search
  • Metaheuristics
  • Numerical modeling
  • Shell and tube evaporator