A literature review on latest developments of Harmony Search and its applications to intelligent manufacturing

Jin Yi; Chao Lu; Guomin Li; Jin Yi; Chao Lu; Guomin Li

doi:10.3934/mbe.2019102

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 2086-2117. doi: 10.3934/mbe.2019102

Previous Article Next Article

Research article Special Issues

A literature review on latest developments of Harmony Search and its applications to intelligent manufacturing

Jin Yi ¹,
Chao Lu ^{2
,
,},
Guomin Li ³

1.
Department of Industrial Systems Engineering and Management, National University of Singapore, 119077, Singapore
2.
School of computer science, China University of Geosciences (Wuhan), Wuhan, 430074, China
3.
The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

Received: 14 December 2018 Accepted: 19 February 2019 Published: 11 March 2019

The harmony search (HS) algorithm is one of the most popular meta-heuristic algorithms. The basic idea of HS was inspired by the music improvisation process in which the musicians continuously adjust the pitch of their instruments to generate wonderful harmony. Since its inception in 2001, HS has attracted the attention of many researchers from all over the world, resulting in a lot of improved variants and successful applications. Even for today, the research on improved HS variants design and innovative applications are still hot topics. This paper provides a detailed review of the basic concept of HS and a survey of its latest variants for function optimization. It also provides a survey of the innovative applications of HS in the field of intelligent manufacturing based on about 40 recently published articles. Some potential future research directions for both HS and its applications to intelligent manufacturing are also analyzed and summarized in this paper.

Keywords:

Citation: Jin Yi, Chao Lu, Guomin Li. A literature review on latest developments of Harmony Search and its applications to intelligent manufacturing[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 2086-2117. doi: 10.3934/mbe.2019102

Related Papers:

[1]	Yu Lei, Zhi Su, Xiaotong He, Chao Cheng . Immersive virtual reality application for intelligent manufacturing: Applications and art design. Mathematical Biosciences and Engineering, 2023, 20(3): 4353-4387. doi: 10.3934/mbe.2023202
[2]	Di-Wen Kang, Li-Ping Mo, Fang-Ling Wang, Yun Ou . Adaptive harmony search algorithm utilizing differential evolution and opposition-based learning. Mathematical Biosciences and Engineering, 2021, 18(4): 4226-4246. doi: 10.3934/mbe.2021212
[3]	Qing Wu, Chunjiang Zhang, Mengya Zhang, Fajun Yang, Liang Gao . A modified comprehensive learning particle swarm optimizer and its application in cylindricity error evaluation problem. Mathematical Biosciences and Engineering, 2019, 16(3): 1190-1209. doi: 10.3934/mbe.2019057
[4]	Agustín Halty, Rodrigo Sánchez, Valentín Vázquez, Víctor Viana, Pedro Piñeyro, Daniel Alejandro Rossit . Scheduling in cloud manufacturing systems: Recent systematic literature review. Mathematical Biosciences and Engineering, 2020, 17(6): 7378-7397. doi: 10.3934/mbe.2020377
[5]	Jiang Zhao, Dan Wu . The risk assessment on the security of industrial internet infrastructure under intelligent convergence with the case of G.E.'s intellectual transformation. Mathematical Biosciences and Engineering, 2022, 19(3): 2896-2912. doi: 10.3934/mbe.2022133
[6]	Chao Wang, Jian Li, Haidi Rao, Aiwen Chen, Jun Jiao, Nengfeng Zou, Lichuan Gu . Multi-objective grasshopper optimization algorithm based on multi-group and co-evolution. Mathematical Biosciences and Engineering, 2021, 18(3): 2527-2561. doi: 10.3934/mbe.2021129
[7]	Yun-Fei Fu . Recent advances and future trends in exploring Pareto-optimal topologies and additive manufacturing oriented topology optimization. Mathematical Biosciences and Engineering, 2020, 17(5): 4631-4656. doi: 10.3934/mbe.2020255
[8]	Jiahe Xu, Yuan Tian, Tinghuai Ma, Najla Al-Nabhan . Intelligent manufacturing security model based on improved blockchain. Mathematical Biosciences and Engineering, 2020, 17(5): 5633-5650. doi: 10.3934/mbe.2020303
[9]	Yifan Gu, Hua Xu, Jinfeng Yang, Rui Li . An improved memetic algorithm to solve the energy-efficient distributed flexible job shop scheduling problem with transportation and start-stop constraints. Mathematical Biosciences and Engineering, 2023, 20(12): 21467-21498. doi: 10.3934/mbe.2023950
[10]	Shuang Wang, Heming Jia, Qingxin Liu, Rong Zheng . An improved hybrid Aquila Optimizer and Harris Hawks Optimization for global optimization. Mathematical Biosciences and Engineering, 2021, 18(6): 7076-7109. doi: 10.3934/mbe.2021352

Abstract

1. Introduction

A wide variety of heat-transfer problems applied to Newtonian fluids flow in bounded conduits of cylindrical or parallel-plate geometries with negligible axial conduction has been successfully reduced and known as the Graetz problem ^[1,2]. Multi-stream or multiphase systems, however, are fundamentally different since conjugated boundary conditions must be coupled at the boundaries, referred to as conjugated Graetz problems ^[3,4], which were solved analytically by means of an orthogonal expansion technique ^[5,6] with the eigenfunction expansion in terms of the extended power series. Extension to markedly increased applications of practical processes with recycle-effect concept is possible. It's been widely used in separation, fermentation, and polymerization such as distillation ^[7], extraction ^[8], loop reactors ^[9], air-lift reactor ^[10], draft-tube bubble column ^[11], mass exchanger ^[12], and thermal diffusion column ^[13].

Many materials of food, polymeric systems, biological process, pulp and paper suspensions ^[14] with high molecular weight in processing industries exhibit a range of non-Newtonian fluid behavioral features and display shear-thinning and/or shear thickening behavior ^[15,16]. Those non-Newtonian fluids can be treated as the laminar flow conditions with negligible viscoelastic effects based on their high viscosity levels in the appropriate shear rate range ^[17]. Therefore, the analytical solution could be analogously obtained under the similar mathematical treatment when dealing with Newtonian fluids. A considerable body of literature has shown the practical feasibility of solving the power-law model of non-Newtonian flows by LBM (Lattice Boltzmann Method). It is also devoted to studying the non-Newtonian behavior with shear-thinning and shear-thickening liquids on sedimentation ^[18,19] and flows over a heated cylinder ^[20], an inclined square ^[21], cylinder ^[22] and various shapes ^[23]. Furthermore, the heat-transfer responses to the distributions of the conduit wall and fluid temperature are two major concerns in investigating the heat-transfer efficiency improvement under different kinds of boundary conditions which can be detected at the conduit wall. Two cases of the uniform wall temperature (Dirichlet problem) ^[24,25] and uniform heat flux (Neumann problem) ^[26,27] were processed in the application of engineering field, in general. Recently, a non-trivial amount of research use metal foams of PCMs (phase change materials) and nanoparticles of NEPCMs (nano-encapsulated phase change materials) to enhance heat transfer properties in some applications regarding constant heat load, transient, or cyclic loads. Thermal and energy storage managements of crucial importance of systems were investigated within the thermal performance given variable heat loads ^[28] and non-uniform magnetic sources ^[29], as well as thermal benefit of NEPCMs nanoparticles in microchannels that considers forced convection ^[30] and natural convection flow ^[31,32]. However, the well-known case of sinusoidal wall heat flux distribution ^[33] of non-uniform heating was the simplest model of period heating within the convective heat-transfer problems in the periodic ^[34] and circumferentially ^[35] heating systems. It's been investigated by many researchers to design the cooling tubes in nuclear reactors ^[36].

The present study is an extension of our previous work ^[37] to apply the system of non-Newtonian fluids for the conjugated Graetz problem where the power-law index of the shearing-thinning aqueous polymer solutions were given. Though the phenomenon of heat transfer in the present study could be drawn parallel comparison, in similar sense, with that of heat-transfer mechanism in our previous work ^[37], the manners of non-Newtonian fluids with the convective heat transfer are somehow different. It's actually affected by the velocity profile of shear-thinning fluid. A power-law fluid flowing through a double-pass laminar countercurrent-flow concentric-tube heat exchanger that implemented an impermeable sheet with sinusoidal wall fluxes was investigated for the purpose of examining the heat transfer efficiency and temperature distributions under external recycling, which was solve analytically through the resultant conjugated partial differential equations by the superposition technique. The recycle ratio and impermeable-sheet position are two parameters treated as essential and should be suitably adjusted to an improved design of heat transfer equipment. The comparison of heat-transfer efficiency improvement and Nusselt numbers in both operations (single- and double-pass device) is also discussed.

2. Dimensionless temperature distributions

A double-pass concentric circular heat exchanger was made by inserting an impermeable sheet into a circular tube of inside diameter 2R and length L, as shown in Figure 1. The thickness of the inner (subchannel a) and annular tube (subchannel b) are 2κR and 2(1-κ)R, respectively. Comparing with the radius of outer circular tube R and outer circular tube R₁, the thickness of the impermeable barrier δ is negligible (δ << R). Two flow patterns, flow pattern A and flow pattern B, are proposed by and used in this study. An inlet fluid with volumetric flow rate V and temperature T₁ will enter the subchannel a and then flows reversely into the subchannel b with the aid of a convectional pump at the end of the conduit like flow pattern A, as shown in Figure 1(a). On the other hand, the flow pattern B is that the fluid feeds into subchannel b and exits from the subchannel a, as shown in . The fluid is heated by the outer wall with sinusoidal heat fluxes, ${q}_{w}^{″}\left(z\right) = {q}_{0}^{″}\left[1+\mathit{sin}(\beta z)\right]$ in both flow patterns.

Figure 1. Schematic diagram of a double-pass concentric circular heat exchanger.

DownLoad: Full-Size Img PowerPoint

The problem of laminar heat transfer at steady state with negligible axial conduction was known as the Graetz problem, and the convective velocity in radial direction is neglected by applying the Navier-stokes relations to obtain the hydro-dynamical equation for laminar flow. The energy balance equations of fluid flowing in the subchannel a and subchannel b in dimensional form with specified velocities are:

$\rho {C}_{p}{v}_{a}\left(r\right)\frac{\partial {T}_{a}(r, z)}{\partial z} = \frac{k}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {T}_{a}\left(r, z\right)}{\partial r}\right)+k\frac{{\partial }^{2}{T}_{a}}{\partial {z}^{2}}$

(1)

$\rho {C}_{p}{v}_{b}\left(r\right)\frac{\partial {T}_{b}(r, z)}{\partial z} = \frac{k}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {T}_{b}\left(r, z\right)}{\partial r}\right)+k\frac{{\partial }^{2}{T}_{b}}{\partial {z}^{2}}$

(2)

The theoretical analysis of double-pass heat exchangers is developed based on the following assumptions: (a) constant physical properties of fluid; (b) fully-developed laminar flow with power law index $\omega$ ( $\tau = -c{\dot{\gamma }}^{\omega }$ ) in each subchannel; (c) neglecting the entrance length and the end effects; (d) ignoring the longitudinal heat conduction and the thermal resistance of the impermeable sheet; (e) well mixed at both inlet and outlet. With these assumptions, the dimensionless energy balance equations and the velocity distributions of a double-pass heat exchanger with sinusoidal heat fluxes were formulated by neglecting the second terms on the right-hand side of Eq. (1) and Eq. (2). The descriptions of two or more contiguous streams of multi-stream (or phases of multi-phase) problems with coupling mutual boundary conditions ^[38,39] becomes

$\left(\frac{{v}_{a}\left(\eta \right){R}^{2}}{\alpha GzL}\right)\frac{\partial {\varphi }_{a}(\eta , \xi )}{\partial \xi } = \frac{1}{\eta }\frac{\partial }{\partial \eta }\left(\eta \frac{\partial {\varphi }_{a}\left(\eta , \xi \right)}{\partial \eta }\right)$

(3)

$\left(\frac{{v}_{b}\left(\eta \right){R}^{2}}{\alpha GzL}\right)\frac{\partial {\varphi }_{b}(\eta , \xi )}{\partial \xi } = \frac{1}{\eta }\frac{\partial }{\partial \eta }\left(\eta \frac{\partial {\varphi }_{b}\left(\eta , \xi \right)}{\partial \eta }\right)$

(4)

where ${v}_{a}$ and ${v}_{b}$ are the velocity distributions in subchannels a and b, respectively, as follows in flow pattern A:

${v}_{a} = \left(\frac{3\omega +1}{\omega +1}\right)\frac{V}{\pi (\kappa R{)}^{2}}\left(1-{\left(\frac{\eta }{\kappa }\right)}^{\frac{\omega +1}{\omega }}\right) = G\left(1-{\left(\frac{\eta }{\kappa }\right)}^{\frac{\omega +1}{\omega }}\right) , 0\le \eta \le \kappa$

(5)

${v}_{b} = -\frac{(3+1/\omega )V}{\pi {R}^{2}\left[{\left(1-{\beta }^{2}\right)}^{1+\frac{1}{\omega }}-{\kappa }^{1-\frac{1}{\omega }}{\left({\beta }^{2}-{\kappa }^{2}\right)}^{1+\frac{1}{\omega }}\right]}{\int }_{\kappa }^{\eta }{\left({\beta }^{2}\frac{1}{\eta }-\eta \right)}^{\frac{1}{\omega }}d\eta = -H{\int }_{\kappa }^{\eta }{\left({\beta }^{2}\frac{1}{\eta }-\eta \right)}^{\frac{1}{\omega }}d\eta , \kappa \le \eta \le \beta$

(6)

${v}_{b} = -\frac{(3+1/\omega )V}{\pi {R}^{2}\left[{\left(1-{\beta }^{2}\right)}^{1+\frac{1}{\omega }}-{\kappa }^{1-\frac{1}{\omega }}{\left({\beta }^{2}-{\kappa }^{2}\right)}^{1+\frac{1}{\omega }}\right]}{\int }_{\eta }^{1}{\left(\eta -{\beta }^{2}\frac{1}{\eta }\right)}^{\frac{1}{\omega }}d\eta = -H{\int }_{\eta }^{1}{\left(\eta -{\beta }^{2}\frac{1}{\eta }\right)}^{\frac{1}{\omega }}d\eta , \beta \le \eta \le 1$

(7)

and in flow pattern B:

${v}_{a} = -\left(\frac{3\omega +1}{\omega +1}\right)\frac{V}{\pi (\kappa R{)}^{2}}\left(1-{\left(\frac{\eta }{\kappa }\right)}^{\frac{\omega +1}{\omega }}\right) = -G\left(1-{\left(\frac{\eta }{\kappa }\right)}^{\frac{\omega +1}{\omega }}\right) , 0\le \eta \le \kappa$

(8)

${v}_{b} = \frac{(3+1/\omega )V}{\pi {R}^{2}\left[{\left(1-{\beta }^{2}\right)}^{1+\frac{1}{\omega }}-{\kappa }^{1-\frac{1}{\omega }}{\left({\beta }^{2}-{\kappa }^{2}\right)}^{1+\frac{1}{\omega }}\right]}{\int }_{\kappa }^{\eta }{\left({\beta }^{2}\frac{1}{\eta }-\eta \right)}^{\frac{1}{\omega }}d\eta = H{\int }_{\kappa }^{\eta }{\left({\beta }^{2}\frac{1}{\eta }-\eta \right)}^{\frac{1}{\omega }}d\eta , \kappa \le \eta \le \beta$

(9)

${v}_{b} = \frac{(3+1/\omega )V}{\pi {R}^{2}\left[{\left(1-{\beta }^{2}\right)}^{1+\frac{1}{\omega }}-{\kappa }^{1-\frac{1}{\omega }}{\left({\beta }^{2}-{\kappa }^{2}\right)}^{1+\frac{1}{\omega }}\right]}{\int }_{\eta }^{1}{\left(\eta -{\beta }^{2}\frac{1}{\eta }\right)}^{\frac{1}{\omega }}d\eta = H{\int }_{\eta }^{1}{\left(\eta -{\beta }^{2}\frac{1}{\eta }\right)}^{\frac{1}{\omega }}d\eta , \beta \le \eta \le 1$

(10)

The values of $\beta$ ( ${v}_{b}\left(\beta \right) = {v}_{b, max}$ ) in Eq. (6) and Eq. (7) for flow pattern A, and Eq. (9) and Eq. (10) for flow pattern B, were obtained via the given $\omega$ and $\kappa$ , as shown in ^[40]. The terms with the power law index $\omega$ on the right-hand side of Eq. (5) and Eq. (6) and (7) for flow pattern A (Eq. (9) and (10) for flow pattern B), respectively, were approximated using the polynomials fitted at the selected points for the acceptable tolerance as follows:

$\left(1-{\left(\frac{{\eta }_{a}}{\kappa }\right)}^{\frac{\omega +1}{\omega }}\right) = {U}_{1}+{U}_{2}\eta +{U}_{3}{\eta }^{2}+{U}_{4}{\eta }^{3}+{U}_{5}{\eta }^{4}$

(11)

${\int }_{\kappa }^{\eta }{\left({\beta }^{2}\frac{1}{\eta }-\eta \right)}^{\frac{1}{\omega }}d\eta +{\int }_{\eta }^{1}{\left(\eta -{\beta }^{2}\frac{1}{\eta }\right)}^{\frac{1}{\omega }}d\eta = {Z}_{1}+{Z}_{2}\eta +{Z}_{3}{\eta }^{2}+{Z}_{4}{\eta }^{3}+{Z}_{5}{\eta }^{4}$

(12)

Table 1. The values of

$\beta$ in Eqs. (4) and (5) (or Eqs. (7) and (8)) for various

$\omega$ and

$\kappa$ .

	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.1	0.3442	0.4687	0.5632	0.6431	0.7140	0.7788	0.8389	0.8954	0.9489
0.2	0.3682	0.4856	0.5749	0.6509	0.7191	0.7818	0.8404	0.896	0.9491
0.3	0.3884	0.4991	0.584	0.657	0.7229	0.784	0.8416	0.8965	0.9492
0.4	0.4052	0.5100	0.5912	0.6617	0.7259	0.7858	0.8426	0.8969	0.9493
0.5	0.4193	0.5189	0.597	0.6655	0.7283	0.7872	0.8433	0.8972	0.9493
0.6	0.4312	0.5262	0.6018	0.6686	0.7303	0.7884	0.8439	0.8975	0.9494
0.7	0.4412	0.5324	0.6059	0.6713	0.7319	0.7893	0.8444	0.8977	0.9495
0.8	0.4498	0.5377	0.6093	0.6735	0.7333	0.7902	0.8449	0.8979	0.9495
0.9	0.4872	0.5422	0.6122	0.6754	0.7345	0.7909	0.8452	0.898	0.9495
1.0	0.4637	0.5461	0.6147	0.6770	0.7355	0.7915	0.8455	0.8981	0.9496

| Show Table

DownLoad: CSV

The coefficients were obtained with the given power law index $\omega = 0.6$ as an illustration, shown in Table 2.

Table 2. The coefficients in Eqs. (3) and (4) (or Eqs. (7) and (8)) for various

$\kappa$ .

coefficients		$\omega =0.6$
$\kappa$	0.3	0.5	0.7	$\kappa$	0.3	0.5	0.7
${U}_{1}$	0.4898	0.8694	0.9467	${Z}_{1}$	-2.4715	-4.3986	-8.5530
${U}_{2}$	3.2650	0.8362	0.3409	${Z}_{2}$	17.0332	23.4914	39.6461
${U}_{3}$	-12.2439	-3.1356	-1.2783	${Z}_{3}$	-39.9047	-46.0023	-68.7476
${U}_{4}$	-16.3251	-4.1808	-1.7045	${Z}_{4}$	39.5577	39.5465	52.9275
${U}_{5}$	1.0203	0.2613	0.1065	${Z}_{5}$	-14.2148	-12.6369	-15.2730

| Show Table

DownLoad: CSV

The corresponding boundary conditions are:

$\frac{\partial {\varphi }_{a}\left(0, \xi \right)}{\partial \eta } = 0$

(13)

$\frac{\partial {\varphi }_{b}(1, \xi )}{\partial \eta } = 1+\mathit{sin}\left(B\xi \right)$

(14)

$\frac{\partial {\varphi }_{a}\left(\kappa , \xi \right)}{\partial \eta } = \frac{\partial {\varphi }_{b}\left(\kappa , \xi \right)}{\partial \eta }$

(15)

${\varphi }_{a}\left(\kappa , \xi \right) = {\varphi }_{b}\left(\kappa , \xi \right)$

(16)

in which

$\eta = \frac{r}{R} , \kappa = \frac{{R}_{1}}{R} , \xi = \frac{z}{LGz} , {\varphi }_{a} = \frac{k\left({T}_{a}-{T}_{i}\right)}{{q}_{0}^{″}R} , {\varphi }_{b} = \frac{k\left({T}_{b}-{T}_{i}\right)}{{q}_{0}^{″}R} , B = \beta GzL = 2\pi Gz , Gz = \frac{4V}{\alpha \pi L}$

(17)

The general form of dimensionless temperature distributions of the laminar double-pass countercurrent-flow concentric tube heat exchangers with sinusoidal wall fluxes can be expressed as follows ^[41]:

${\varphi }_{a}\left(\eta , \xi \right) = {\theta }_{0a}\xi +{\theta }_{1a}\left(\eta \right)+{\theta }_{2a}\left(\eta \right)\mathit{sin}\left(B\xi \right)+{\theta }_{3a}\left(\eta \right)\mathit{cos}\left(B\xi \right)$

(18)

${\varphi }_{b}(\eta , \xi ) = {\theta }_{0b}\left(\frac{1}{Gz}-\xi \right)+{\theta }_{1b}\left(\eta \right)+{\theta }_{2b}\left(\eta \right)\mathit{sin}(B\xi )+{\theta }_{3b}(\eta \left)\mathit{cos}(B\xi \right)$

(19)

in which the ${\theta }_{\text{0a}}$ and ${\theta }_{\text{0b}}$ are the constants yet to be determined, and the ${\theta }_{\text{1a}}\left(\eta \right) , {\theta }_{2a}\left(\eta \right) , {\theta }_{3a}\left(\eta \right) , {\theta }_{\text{1b}}\left(\eta \right) , {\theta }_{2b}\left(\eta \right)$ and ${\theta }_{3b}\left(\eta \right)$ are the functions of $\eta$ to be determined.

Substituting Eqs. (18) and (19) into the governing equations, Eqs. (3) and (4), and the boundary conditions, Eqs. (13)–(16), yields

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{1a}\left(\eta \right)}{d\eta }\right)-\frac{{v}_{a}\left(\eta \right){R}^{2}\eta }{GzL\alpha }{\theta }_{0a}+\left[\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{2a}\left(\eta \right)}{d\eta }\right)+\frac{{v}_{a}\left(\eta \right)B{R}^{2}\eta }{GzL\alpha }{\theta }_{3a}\left(\eta \right)\right]\mathit{sin}(B\xi ) \\ +\left[\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{3a}\left(\eta \right)}{d\eta }\right)-\frac{{v}_{a}\left(\eta \right)B{R}^{2}\eta }{GzL\alpha }{\theta }_{2a}\left(\eta \right)\right]\mathit{cos}(B\xi ) = 0$

(20)

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{1b}\left(\eta \right)}{d\eta }\right)+\frac{{v}_{b}\left(\eta \right){R}^{2}\eta }{GzL\alpha }{\theta }_{0b}+\left[\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{2b}\left(\eta \right)}{d\eta }\right)+\frac{{v}_{b}\left(\eta \right)B{R}^{2}\eta }{GzL\alpha }{\theta }_{3b}\left(\eta \right)\right]\mathit{sin}(B\xi ) \\ +\left[\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{3b}\left(\eta \right)}{d\eta }\right)-\frac{{v}_{b}\left(\eta \right)B{R}^{2}\eta }{GzL\alpha }{\theta }_{2b}\left(\eta \right)\right]\mathit{cos}(B\xi ) = 0$

(21)

$\frac{d{\theta }_{1a}\left(0\right)}{d\eta }+\frac{d{\theta }_{2a}\left(0\right)}{d\eta }\mathit{sin}\left(B\xi \right)+\frac{d{\theta }_{3a}\left(0\right)}{d\eta }\mathit{cos}\left(B\xi \right) = 0$

(22)

$\left(\frac{d{\theta }_{1b}\left(1\right)}{d\eta }-1\right)+\left(\frac{d{\theta }_{2b}\left(1\right)}{d\eta }-1\right)\mathit{sin}(B\xi )+\left[\frac{d{\theta }_{3b}\left(1\right)}{d\eta }\right]\mathit{cos}(B\xi ) = 0$

(23)

$\left(\frac{d{\theta }_{1a}\left(\kappa \right)}{d\eta }-\frac{d{\theta }_{1b}\left(\kappa \right)}{d\eta }\right)+\left(\frac{d{\theta }_{2a}\left(\kappa \right)}{d\eta }-\frac{d{\theta }_{2b}\left(\kappa \right)}{d\eta }\right)\mathit{sin}\left(B\xi \right)+\left(\frac{d{\theta }_{3a}\left(\kappa \right)}{d\eta }-\frac{d{\theta }_{3b}\left(\kappa \right)}{d\eta }\right)\mathit{cos}\left(B\xi \right) = 0$

(24)

${\theta }_{0a}\xi +{\theta }_{1a}\left(\kappa \right)+{\theta }_{2a}\left(\kappa \right)\mathit{sin}\left(B\xi \right)+{\theta }_{3a}\left(\kappa \right)\mathit{cos}\left(B\xi \right) \\ = {\theta }_{0b}\left(\frac{1}{Gz}-\xi \right)+{\theta }_{1b}\left(\kappa \right)+{\theta }_{2b}\left(\kappa \right)\mathit{sin}(B\xi )+{\theta }_{3b}(\kappa \left)\mathit{cos}(B\xi \right)$

(25)

2.1. Solving ${\theta }_{2a}\left(\eta \right) , {\theta }_{3a}\left(\eta \right) , {\theta }_{2b}\left(\eta \right)$ and ${\theta }_{3b}\left(\eta \right)$ by using Frobenius method

Multiplying Eqs. (20)–(25) by $\mathit{sin}(B\xi)$ and integrating with respect to ζ in the interval [0, 2π/B] gives:

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{2a}\left(\eta \right)}{d\eta }\right)+\frac{{v}_{a}\left(\eta \right)B{R}^{2}\eta }{\alpha GzL}{\theta }_{3a}\left(\eta \right) = 0$

(26)

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{2b}\left(\eta \right)}{d\eta }\right)+\frac{{v}_{b}\left(\eta \right)B{R}^{2}\eta }{\alpha GzL}{\theta }_{3b}\left(\eta \right) = 0$

(27)

$\frac{d{\theta }_{2a}\left(0\right)}{d\eta } = 0$

(28)

$\frac{d{\theta }_{2b}\left(1\right)}{d\eta } = 1$

(29)

$\frac{d{\theta }_{2a}\left(\kappa \right)}{d\eta } = \frac{d{\theta }_{2b}\left(\kappa \right)}{d\eta }$

(30)

${\theta }_{2a}\left(\kappa \right) = {\theta }_{2b}\left(\kappa \right)$

(31)

Similarly, multiplying Eqs. (20)–(25) by $\mathit{cos}(B\xi)$ and integrating with respect to ζ in the interval [0, 2π//B], one can obtain:

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{3a}\left(\eta \right)}{d\eta }\right)-\frac{{v}_{a}\left(\eta \right)B{R}^{2}\eta }{\alpha GzL}{\theta }_{2a}\left(\eta \right) = 0$

(32)

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{3b}\left(\eta \right)}{d\eta }\right)-\frac{{v}_{b}\left(\eta \right)B{R}^{2}\eta }{\alpha GzL}{\theta }_{2b}\left(\eta \right) = 0$

(33)

$\frac{d{\theta }_{3a}\left(0\right)}{d\eta } = 0$

(34)

$\frac{d{\theta }_{3b}\left(1\right)}{d\eta } = 0$

(35)

$\frac{d{\theta }_{3a}\left(\kappa \right)}{d\eta } = \frac{d{\theta }_{3b}\left(\kappa \right)}{d\eta }$

(36)

${\theta }_{3a}\left(\kappa \right) = {\theta }_{3b}\left(\kappa \right)$

(37)

The complex functions ${\psi }_{a}\left(\eta \right) = {\theta }_{2a}\left(\eta \right)+{\theta }_{3a}\left(\eta \right)i$ and ${\psi }_{b}\left(\eta \right) = {\theta }_{2b}\left(\eta \right)+{\theta }_{3b}\left(\eta \right)i$ were introduced to combine Eqs. (26)–(31) and Eqs. (32)–(37) into a unique one-dimensional boundary value problem according to the prior mathematical treatment ^[41] as follows:

$\frac{d}{d\eta }\left(\eta \frac{\partial {\psi }_{a}\left(\eta \right)}{\partial \eta }\right)-\frac{{v}_{a}\left(\eta \right)B{R}^{2}\eta }{GzL\alpha }{\psi }_{a}\left(\eta \right)i = 0$

(38)

$\frac{d}{d\eta }\left(\eta \frac{\partial {\psi }_{b}\left(\eta \right)}{\partial \eta }\right)-\frac{{v}_{b}\left(\eta \right)B{R}^{2}\eta }{GzL\alpha }{\psi }_{b}\left(\eta \right)i = 0$

(39)

$\frac{d{\psi }_{a}\left(0\right)}{d\eta } = 0$

(40)

$\frac{d{\psi }_{b}\left(1\right)}{d\eta } = 1$

(41)

$\frac{d{\psi }_{a}\left(\kappa \right)}{d\eta } = \frac{d{\psi }_{b}\left(\kappa \right)}{d\eta }$

(42)

${\psi }_{a}\left(\kappa \right) = {\psi }_{b}\left(\kappa \right)$

(43)

We can apply the method of Frobenius which enables one to create a power series solution to solve differential equations. Assuming ${\psi }_{a}\left(\eta \right)$ and ${\psi }_{b}\left(\eta \right)$ have the forms of power series multiplied by unknown powers of $\eta$ , respectively, leads to:

${\psi }_{a}\left(\eta \right) = {\sum }_{n = 0}^{\infty }{a}_{n}{\eta }^{n+{r}_{a}} , n\ge 0$

(44)

${\psi }_{b}\left(\eta \right) = {\sum }_{n = 0}^{\infty }{b}_{n}{\eta }^{n+{r}_{b}} , n\ge 0$

(45)

One can find the constants of ${r}_{a}$ and ${r}_{b}$ are proved to be zero in this system equations. The coefficients ${a}_{n}$ and ${b}_{n}$ are determined by solving Eqs. (38) and (39) and incorporating the boundary conditions of Eqs. (40) to (43), and then comparing with the real and imaginary parts of the complex functions ${\psi }_{a}\left(\eta \right) = {\theta }_{2a}\left(\eta \right)+{\theta }_{3a}\left(\eta \right)i$ and ${\psi }_{b}\left(\eta \right) = {\theta }_{2b}\left(\eta \right)+{\theta }_{3b}\left(\eta \right)i$ , and thus the recursive relations were obtained for flow pattern A, respectively, as follows:

${a}_{0} , {a}_{1} = 0 , {a}_{n} = \frac{BG}{{n}^{2}}({U}_{1}{a}_{n-2}+{U}_{2}{a}_{n-3}+{U}_{3}{a}_{n-4}+{U}_{4}{a}_{n-5}+{U}_{5}{a}_{n-6}) , n\ge 2$

(46)

and

${b}_{0} , {b}_{1} = 0 , {b}_{n} = \frac{-BH}{{n}^{2}}({Z}_{1}{b}_{n-2}+{Z}_{2}{b}_{n-3}+{Z}_{3}{b}_{n-4}+{Z}_{4}{b}_{n-5}+{Z}_{5}{b}_{n-6}) , n\ge 2$

(47)

Similarly, the recursive relations of the coefficients ${a}_{n}$ and ${b}_{n}$ for flow pattern B are

${a}_{0} , {a}_{1} = 0 , {a}_{n} = \frac{-BG}{{n}^{2}}({U}_{1}{a}_{n-2}+{U}_{2}{a}_{n-3}+{U}_{3}{a}_{n-4}+{U}_{4}{a}_{n-5}+{U}_{5}{a}_{n-6}) , n\ge 2$

(48)

and

${b}_{0} , {b}_{1} = 0 , {b}_{n} = \frac{BH}{{n}^{2}}({Z}_{1}{b}_{n-2}+{Z}_{2}{b}_{n-3}+{Z}_{3}{b}_{n-4}+{Z}_{4}{b}_{n-5}+{Z}_{5}{b}_{n-6}) , n\ge 2$

(49)

2.2. Solving ${\theta }_{0a} , {\theta }_{1a}\left(\eta \right) , {\theta }_{0b}$ and ${\theta }_{1b}\left(\eta \right)$ by double integrations

Integrations of Eqs. (20) to (25) with respect to $\xi$ in the interval [0, $2\pi /B$ ], one can obtain

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{1a}\left(\eta \right)}{d\eta }\right)-\frac{{v}_{a}\left(\eta \right){R}^{2}\eta }{GzL\alpha }{\theta }_{0a} = 0$

(50)

$\frac{d}{d\eta }\left(\eta \frac{d{\theta }_{1b}\left(\eta \right)}{d\eta }\right)+\frac{{v}_{b}\left(\eta \right){R}^{2}\eta }{GzL\alpha }{\theta }_{0b} = 0$

(51)

$\frac{d{\theta }_{1a}\left(0\right)}{d\eta } = 0$

(52)

$\frac{d{\theta }_{\text{1b}}\left(1\right)}{d\eta } = 1$

(53)

$\frac{d{\theta }_{1a}\left(\kappa \right)}{d\eta } = \frac{d{\theta }_{1b}\left(\kappa \right)}{d\eta }$

(54)

${\theta }_{0a} = -{\theta }_{0b}$

(55)

${\theta }_{1a}\left(\kappa \right) = \frac{{\theta }_{0b}}{Gz}+{\theta }_{1b}\left(\kappa \right)$

(56)

Furthermore, integrating Eqs. (50) and (51) twice with respect to $\eta$ for ${\theta }_{1a}\left(\eta \right)$ and ${\theta }_{1b}\left(\eta \right)$ , respectively, yields

${\theta }_{1a} = G{\theta }_{0a}\left(\frac{1}{4}{U}_{1}{\eta }^{2}+\frac{1}{9}{U}_{2}{\eta }^{3}+\frac{1}{16}{U}_{3}{\eta }^{4}+\frac{1}{25}{U}_{4}{\eta }^{5}+\frac{1}{36}{U}_{5}{\eta }^{6}\right)+{\gamma }_{1a}\mathit{ln}\eta +{\gamma }_{2a}$

(57)

and

${\theta }_{1b} = -H{\theta }_{0b}\left(\frac{1}{4}{Z}_{1}{\eta }^{2}+\frac{1}{9}{Z}_{2}{\eta }^{3}+\frac{1}{16}{Z}_{3}{\eta }^{4}+\frac{1}{25}{Z}_{4}{\eta }^{5}+\frac{1}{36}{Z}_{5}{\eta }^{6}\right)+{\gamma }_{1b}\mathit{ln}\eta +{\gamma }_{2b}$

(58)

where ${\gamma }_{1a} , {\gamma }_{2a} , {\gamma }_{1b}$ and ${\gamma }_{2b}$ are the integrating constants in Eqs. (57) and (58). Since there are six constants ( ${\theta }_{0a} , {\theta }_{0b} , {\gamma }_{1a} , {\gamma }_{2a} , {\gamma }_{1b}$ and ${\gamma }_{2b}$ ) to be determined given five equations (Eqs. (52) to (56)), it needs one extra equation. The additional one equation of the overall energy balance is required as follows:

$\rho {C}_{p}V({T}_{F}-{T}_{i}) = {\int }_{0}^{L}{q}_{}^{″}\left(z\right)2\pi R dz$

(59)

Eq. (59) can be rewritten as

${\varphi }_{F} = {\int }_{0}^{\frac{1}{Gz}}8[1+\mathit{sin}(B\xi )]d\xi = 8\left[\frac{1}{Gz}-\frac{1}{B}\left(\mathit{cos}\left(\frac{B}{Gz}\right)-1\right)\right]$

(60)

where the ${\varphi }_{F}$ is the average outlet temperature and it is defined as

${\varphi }_{F} = -\frac{1}{V}{\int }_{\kappa }^{1}{v}_{b}2\pi {R}^{2}\eta {\varphi }_{b}\left(\eta , 0\right)d\eta$

(61)

in flow pattern A, and

${\varphi }_{F} = -\frac{1}{V}{\int }_{0}^{\kappa }{v}_{a}2\pi {R}^{2}\eta {\varphi }_{a}\left(\eta , 0\right)d\eta$

(62)

in flow pattern B. Further, Eqs. (61) and (62) should be solved by using Eqs. (6), (7) and (19) and Eqs. (8) and (18) at $\xi = 0$ , respectively. Therefore, the complete solutions of dimensionless temperature distributions in a double-pass concentric circular heat exchanger were obtained by substituting the functions of ${\theta }_{1a} , {\theta }_{1b} , {\theta }_{2a} , {\theta }_{2b} , {\theta }_{3a}$ and ${\theta }_{3b}$ , and constants of ${\theta }_{0a}$ and ${\theta }_{0b}$ into the ${\varphi }_{a}$ and ${\varphi }_{b}$ (say Eqs. (18) and (19)).

3. Heat-transfer efficiency enhancement

The local Nusselt number is usually used to measure the convection heat transfer occurring at the wall surface of double-pass concentric circular heat exchangers in forced convection heat-transfer problems and defined as

$Nu\left(\xi \right) = \frac{hDe}{k}$

(63)

where k is the heat conductivity coefficient of the fluid, $De$ is the equivalent diameter of the conduit, $De = 2R$ , and h is the heat transfer coefficient. The heat-transfer coefficient h is defined as

${q}_{w}^{″}\left(z\right) = h\left({T}_{j}\left(R, z\right)-{T}_{i}\right), \text{j} = \text{a, b}$

(64)

or, in the dimensionless form

$h = \frac{k}{R}\frac{{q}_{w}^{″}\left(\xi \right)}{{q}_{0}^{″}{\varphi }_{j}\left(1, \xi \right)} = \frac{k}{R}\frac{1+\mathit{sin}(B\xi )}{{\varphi }_{j}\left(1, \xi \right)}, \text{j} = \text{a, b}$

(65)

Substituting Eq. (65) into Eq. (63) yields

$Nu\left(\xi \right) = \frac{2\left[1+\mathit{sin}(B\xi )\right]}{{\varphi }_{j}\left(1, \xi \right)}, \text{j} = \text{a, b}$

(66)

Similarly, the local Nusselt number of single-pass heat exchangers is defined as

$N{u}_{0}\left(\xi \right) = \frac{2\left[1+\mathit{sin}(B\xi )\right]}{{\varphi }_{0}\left(1, \xi \right)}$

(67)

where the wall temperature distribution, ${\varphi }_{0}\left(1, \xi \right)$ , of single-pass heat exchangers can be determined, according to the reference ^[41].

Moreover, the average Nusselt numbers of single- and double-pass concentric circular heat exchangers, respectively, were determined by

$\overline{Nu} = Gz{\int }_{0}^{1/Gz}Nu\left(\xi \right)d\xi = Gz{\int }_{0}^{1/Gz}\frac{2\left[1+\mathit{sin}(B\xi )\right]}{{\varphi }_{j}\left(1, \xi \right)}d\xi , \text{j} = \text{a, b}$

(68)

and

$\overline{N{u}_{0}} = Gz{\int }_{0}^{1/Gz}N{u}_{0}\left(\xi \right)d\xi = Gz{\int }_{0}^{1/Gz}\frac{2\left[1+\mathit{sin}(B\xi )\right]}{{\varphi }_{0}\left(1, \xi \right)}d\xi$

(69)

The heat-transfer efficiency enhancement was illustrated by calculating the percentage increase in the device by employing a double-pass operation, based on single-pass device with the same working dimensions and operating parameters

${I}_{h} = \frac{\overline{Nu}-\overline{N{u}_{0}}}{\overline{N{u}_{0}}}\left(\%\right)$

(70)

4. Power consumption increment

The power consumption increment is unavoidable due to inserting an impermeable sheet into a single-pass device to conduct double-pass operations. The power consumption only incurring the friction losses to walls in double-pass operation were significant. And relations to joint, diversion or bending of conduit are neglected for simplicity. It may be obtained by using generalized Bernoulli equation ^[42] with following assumptions: (a) incompressible fluid; (b) no change in average velocity; (c) no change in elevation; (d) no work performed. Hence, the power consumption may determine using Fanning friction factor ${f}_{F}$ ^[43]:

$\ell{w}_{f, j} = \frac{2{f}_{F, j}{\stackrel{̄}{v}}_{j}^{2}L}{D{e}_{j}}, \hspace{1em}j = a, b$

(71)

$P = V\rho \ell{w}_{f, a}+V\rho \ell{w}_{f, b} , {P}_{0} = V\rho \ell{w}_{f, 0}$

(72)

The relative extents ${I}_{P}$ of power consumption increment was illustrated by calculating the percentage increment in the double-pass operation, based on the single-pass device as

${I}_{P} = \frac{{P}_{\text{double}}-{P}_{\text{single}}}{{P}_{\text{single}}}\times 100\%$

(73)

5. Results and discussions

The applications of Frobenius method to solve for differential equations are expanded in terms of an extended power series, say Eqs. (38) and (39). To illustrate, comparison is made to such a power series with terms truncated after $n = 70$ and $n = 75$ with $\kappa = 0.5$ and $\omega = 0.8$ . The accuracy of those comparisons is analyzed and some results are presented in for an extended power series for flow pattern A. It can be observed from that the power series agree reasonably well with the term of $n = 70$ , and hence those power series with $n = 70$ are employed in the calculation procedure.

Table 3. The convergence of power series in Eqs. (44) and (45) with

$n = 70$ and

$n = 75$ for flow pattern A as an illustration.

$Gz$	n	${\theta }_{2a}\left(0.3\right)$	${\theta }_{3a}\left(0.3\right)$	${\theta }_{2b}\left(0.7\right)$	${\theta }_{3b}\left(0.7\right)$	$\overline{Nu}$
1	70	-0.092	0.135	0.136	-1.124	0.07
1	75	-0.092	0.135	0.136	-1.124	0.07
10	70	$-5.2\times 1{0}^{-24}$	$-9.2\times 1{0}^{-23}$	$1.71\times 1{0}^{-13}$	$6.5\times 1{0}^{-15}$	1.87
10	75	$-6.7\times 1{0}^{-25}$	$-2.6\times 1{0}^{-24}$	$2.7\times 1{0}^{-14}$	$1.0\times 1{0}^{-15}$	1.87
50	70	$-1.4\times 1{0}^{-23}$	$-7.7\times 1{0}^{-24}$	$1.8\times 1{0}^{-13}$	$7.2\times 1{0}^{-15}$	6.31
50	75	$-4.4\times 1{0}^{-25}$	$-2.4\times 1{0}^{-25}$	$2.8\times 1{0}^{-14}$	$1.1\times 1{0}^{-15}$	6.31
100	70	$-5.1\times 1{0}^{-24}$	$1.7\times 1{0}^{-24}$	$1.8\times 1{0}^{-13}$	$7.3\times 1{0}^{-15}$	7.92
100	75	$-1.6\times 1{0}^{-25}$	$5.2\times 1{0}^{-26}$	$2.8\times 1{0}^{-14}$	$-1.1\times 1{0}^{-15}$	7.92
1000	70	$-1.3\times 1{0}^{-38}$	$-2.3\times 1{0}^{-38}$	$1.9\times 1{0}^{-13}$	$6.3\times 1{0}^{-15}$	10.06
1000	75	$4.5\times 1{0}^{-40}$	$-4.9\times 1{0}^{-40}$	$3.0\times 1{0}^{-14}$	$1.0\times 1{0}^{-15}$	10.06

| Show Table

DownLoad: CSV

The dimensionless temperature distributions of power-law fluids in double-pass concentric circular heat exchanger with sinusoidal wall flux are obtained through solving the energy balance equations with the aid of the linear superposition method. Obtaining the wall temperature distributions in advance of the design of the heat exchanger equipment is important for an engineer to select the appropriate materials and to carefully consider both technical and economic feasibility. It can be observed in Figure 2 for the flow pattern A that the wall temperature is getting lower at the downstream in subchannel b. The wall temperature at the whole part of subchannel b could reach an extremely low value, especially for remarkably high flow rate, say $Gz$ = 100. On the contrary, the wall temperature is getting higher at the downstream in subchannel b for the flow pattern B. The wall temperature profiles are growing up toward the end downstream, as shown in for the flow pattern B. The wall temperatures are found to be close to the inlet temperature at the flow entrance irrespective of the $Gz$ values. The wall temperature becomes a little bit lower for smaller $\omega$ (more apparently shear-thinning), regardless of the flow pattern. It is noteworthy that the wall temperatures are monotonically increasing or decreasing along the heat-exchanger device even with the sinusoidal wall heat flux. Regarding the device performance, the simulated Nusselt number Nu is demonstrated in , and both flow patterns of the double-pass concentric circular heat exchanger show much more effective energy transfer than that from the single-pass device. The performance of device is further improved along with the increasing $Gz$ . The device with the flow pattern B is shown to be more effective than that of the flow pattern A and single-pass operations under the same $\omega$ condtion. For both flow patterns, the Nu is getting lower with the smaller $\omega$ .

Figure 2. Dimensionless wall temperature distribution with and as parameters (flow pattern A).

DownLoad: Full-Size Img PowerPoint

Figure 3. Dimensionless wall temperature distribution with and as parameters (flow pattern B).

DownLoad: Full-Size Img PowerPoint

Figure 4. Average Nusselt number vs.

$Gz$ with

$\omega$ as a parameter for

$\kappa$ = 0.5.

DownLoad: Full-Size Img PowerPoint

The dimensionless wall temperatures are demonstrated in and for the flow patterns A and B, respectively, to illustrate the influence of $\kappa$ value (impermeable-sheet position). The wall temperature shows monotonically decreasing tendency for flow pattern A and increasing tendency for flow pattern B. On the other hand, the wall temperature presented longitudinal fluctuations in accordance with the sinusoidal wall flux. It is also found that the wall temperature increases with small $\kappa$ (relative larger thickness of subchannel b) and decrease with a higher $Gz$ for both flow patterns.

Figure 5. Dimensionless wall temperature vs.

$Gz\xi$ with

$\kappa$ for various

$Gz$ (flow pattern A).

DownLoad: Full-Size Img PowerPoint

Figure 6. Dimensionless wall temperature vs.

$Gz\xi$ with

$\kappa$ for various

$Gz$ (flow pattern B).

DownLoad: Full-Size Img PowerPoint

In , it is found that the Nusselt number Nu of the current double-pass heat exchanger is sensitive to the $\kappa$ values. The Nusselt number Nu increases with the $\kappa$ values which indicated the double-pass device with a narrower subchannel b could accomplish better heat-transfer efficiency. It is also found the device of flow pattern B could have better device performance improvement than that of flow pattern A.

Figure 7. The average Nusselt number vs.

$Gz$ with

$\kappa$ as a parameter as a parameter.

DownLoad: Full-Size Img PowerPoint

When simultaneously considering the heat-transfer improvement enhancement ${I}_{h}$ and the power consumption increment ${I}_{p}$ , its ratio ${I}_{h}/{I}_{p}$ is plotted versus Graetz number $Gz$ with the $\kappa$ value as a parameter in Figure 8 and and . The ratio of ${I}_{h}/{I}_{p}$ rapidly increases with $Gz$ and quickly flatten out. It also rises with the increase in $\kappa$ . The double-pass device of flow pattern B demonstrated to be more beneficial in economic sense than that of flow pattern A.

Figure 8. The ratio of

${I}_{h}/{I}_{p}$ vs.

$Gz$ with

$\kappa$ as a parameter.

DownLoad: Full-Size Img PowerPoint

Table 4. The ratio of

${I}_{h}/{I}_{p}$ with ω,

$Gz$ and

$\kappa$ as parameters for flow pattern A.

	ω = 0.4			ω = 0.6			ω = 0.8			ω = 1.0
	Impermeable-sheet position ( $\kappa$ )
	0.3	0.5	0.7	0.3	0.5	0.7	0.3	0.5	0.7	0.3	0.5	0.7

1	-5.88	-13.68	-11.02	-2.98	-9.34	-7.57	-1.38	-6.38	-5.22	-0.71	-4.31	-3.63
10	-1.36	-3.29	-2.43	-0.60	-2.00	-1.55	-0.25	-1.16	-0.86	-0.1	-0.63	-0.47
100	3.15	13.48	20.48	1.65	9.77	14.74	0.84	6.97	10.58	0.42	4.9	7.55
1000	4.56	19.63	31.99	2.36	14.04	22.98	1.19	9.97	16.29	0.59	6.91	11.56

| Show Table

DownLoad: CSV

Table 5. The ratio of

${I}_{h}/{I}_{p}$ with ω,

$Gz$ and

$\kappa$ as parameters for flow pattern B.

	ω = 0.4			ω = 0.6			ω = 0.8			ω = 1.0
	Impermeable-sheet position ( $\kappa$ )
	0.3	0.5	0.7	0.3	0.5	0.7	0.3	0.5	0.7	0.3	0.5	0.7

1	-0.57	-1.29	-1.02	-2.26	-7.08	-5.65	-1.43	-6.43	-5.27	-0.49	-3.01	-2.45
10	4.71	13.58	13.63	2.59	9.43	9.21	1.24	6.6	6.58	0.62	4.71	4.76
100	4.35	18.59	28.29	2.4	13.34	20.15	1.18	9.37	14.28	0.58	6.52	10.12
1000	4.7	20.29	33.16	2.45	14.51	23.79	1.23	10.12	16.84	0.61	7.12	11.94

| Show Table

DownLoad: CSV

6. Conclusions

A mathematical formulation for a concentric circular double-pass heat exchanger of power-law fluids with sinusoidal wall flux has been formulated, and the analytical solution is obtained using orthogonal expansion technique. The double-pass flow patterns can be achieved by inserting an impermeable sheet into a cylindrical heat exchanger to examine the heat transfer behavior. The double-pass device performance can be significantly enhanced when compared to that of the single-pass one, especially for a narrower annular flow channel (subchannel b). The average Nusselt number of the device of flow pattern B (annular flow in, core flow out) is larger than that of flow pattern A (core flow in, annular flow out). The longitudinal wall temperature profile is able to be more smoothing despite the sinusoidal wall flux. The wall temperature is decreasing in longitudinal direction for flow pattern A and increasing for flow pattern B, and the variations are much more moderate for high flow rates, for example at $Gz$ = 100. A comparison is also made for the heat-transfer improvement enhancement ${I}_{h}$ and the power consumption increment ${I}_{p}$ in the form of ${I}_{h}/{I}_{p}$ . One could find that the flow pattern B always performs better than that of flow pattern A when assessing the economic feasibility of both the flow patterns A and B by ratio ${I}_{h}/{I}_{p}$ .

Acknowledgments

The authors wish to thank the Ministry of Science and Technology (MOST) of the Republic of China for its financial support.

Conflict of interest

The authors have no conflict of interest.

Supplementary

Nomenclature

a_n	[-]	constants
B	[m]	constant B= $\beta GzL$
b_n	[-]	constants
De	[m]	hydraulic diameter
${f}_{F}$	[-]	Fanning friction factor
g_c	[-]	gravity factor
$G$	[-]	constant
$Gz$	[-]	Graetz number
$H$	[-]	constant
$h$	[kW/mK]	heat transfer coefficient
${I}_{h}$	[-]	heat-transfer improvement enhancement
${I}_{p}$	[-]	power consumption increment
$k$	[kW/mK]	thermal conductivity of the fluid
$L$	[m]	conduit length
$\ell{w}_{f}$	[kJ/kg]	friction loss in conduit
$\overline{Nu}$	[-]	the average Nusselt number
P	[(Nm)/s]	power consumption
${q}^{″}$	[kW]	wall heat flux
r	[m]	radius coordinate
R	[m]	outer tube radius
R₁	[m]	inter tube radius
T	[K]	temperature of fluid in conduit
${U}_{i}$	[-]	constants, $i=\mathrm{1, 2}, \mathrm{3, 4}, 5$
$V$	[m³/s]	inlet volumetric flow rate
$v$	[m/s]	velocity distribution of fluid
${Z}_{i}$	[-]	constant, $i=\mathrm{1, 2}, \mathrm{3, 4}, 5$
z	[m]	longitudinal coordinate
$\alpha$	[m²/s]	thermal diffusivity of fluid
β	[1/m]	constant
${\dot{\gamma }}^{}$	[1/s]	shear rate
${\gamma }_{1a} , {\gamma }_{2a}$	[-]	integration constants
${\gamma }_{1b} , {\gamma }_{2b}$	[-]	integration constants
δ	[m]	impermeable sheet thickness
$\theta$	[-]	coefficients
$\eta$	[-]	dimensionless radius coordinate, = r/R
κ	[-]	constant = R₁/R
$\lambda$	[-]	constant
$\xi$	[-]	dimensionless longitudinal coordinate = z/ $GzL$
$\rho$	[kg/m³]	density of the fluid
$\tau$	[Pa]	shear stress
$\varphi$	[-]	dimensionless temperature $k\left(T-{T}_{i}\right)/{q}_{0}^{″}R$
$\omega$	[-]	power-law index
$\psi$	[-]	complex functions of dimensionless temperature

| Show Table

DownLoad: CSV

Subscripts

0	[-] = at the inlet or for the single-pass device
a	[-] = the inner flow channel
b	[-] = the outer flow channel
F	[-] = at the outlet of a double-pass device
i	[-] = at the inlet of a double-pass device
L	[-] = at the end of the channel=
w	[-] = at the wall surface

| Show Table

DownLoad: CSV

References

[1]	Z.W. Geem, Music-inspired harmony search algorithm: theory and applications, Springer, 2009.
[2]	J. Kennedy, Particle swarm optimization, in Encyclopedia of machine learning, Springer, 2011, 760–766.
[3]	M. Dorigo, V. Maniezzo and A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE T. Syst. Man. Cy. B., 26 (1996), 29–41.
[4]	D. Karaboga and B. Basturk, A powerful and efficient algorithm for numerical function optimization: artificial bee colony (abc) algorithm, J. Global. Optim., 39 (2007), 459–471.
[5]	X. S. Yang and S. Deb, Cuckoo search via lévy flights, in Nature & Biologically Inspired Computing, 2009. NaBIC 2009. World Congress on, IEEE, 2009, 210–214.
[6]	J. H. Holland, Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence, MIT press, 1992.
[7]	R. Storn and K. Price, Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces, J. Global. Optim., 11 (1997), 341–359.
[8]	S. Kirkpatrick, Optimization by simulated annealing: Quantitative studies, J. Stat. Phy., 34 (1984), 975–986.
[9]	A. Y. Lam and V. O. Li, Chemical-reaction-inspired metaheuristic for optimization, IEEE Trans. Evol. Comput., 14 (2010), 381–399.
[10]	Z. W. Geem, J. H. Kim and G. V. Loganathan, A new heuristic optimization algorithm: harmony search, Simulation, 76 (2001), 60–68.
[11]	J. Yi, X. Li, L. Gao, et al., Optimal design of photovoltaic-wind hybrid renewable energy system using a discrete geometric selective harmony search, in Computer Supported Cooperative Work in Design (CSCWD), 2015 IEEE 19th International Conference on, IEEE, (2015), 499–504.
[12]	A. Chauhan and R. Saini, Discrete harmony search based size optimization of integrated renewable energy system for remote rural areas of uttarakhand state in india, Renew. Energy, 94 (2016), 587–604.
[13]	Z. W. Geem and Y. Yoon, Harmony search optimization of renewable energy charging with energy storage system, Int. J. Electr. Power Energy Syst., 86 (2017), 120–126.
[14]	C. Camacho-Gómez, S. Jiménez-Fernández, R. Mallol-Poyato, et al., Optimal design of microgrid's network topology and location of the distributed renewable energy resources using the harmony search algorithm, Soft Comput., 1 (2018), 1–16.
[15]	H. B. Ouyang, L. Q. Gao, S. Li, et al., Improved novel global harmony search with a new relaxation method for reliability optimization problems, Inform. Sci., 305 (2015), 14–55.
[16]	W. Zeng, J. Yi, X. Rao, et al., A two-stage path planning approach for multiple car-like robots based on ph curves and a modified harmony search algorithm, Eng. Optimiz., 49 (2017), 1995– 2012.
[17]	S. Kundu and D. R. Parhi, Navigation of underwater robot based on dynamically adaptive harmony search algorithm, Memet. Comput., 8 (2016), 125–146.
[18]	J. P. Papa, W. Scheirer and D. D. Cox, Fine-tuning deep belief networks using harmony search, Appl. Soft. Comput., 46 (2016), 875–885.
[19]	W. Y. Lee, S. M. Park and K. B. Sim, Optimal hyperparameter tuning of convolutional neural networks based on the parameter-setting-free harmony search algorithm, Optik, 172 (2018), 359– 367.
[20]	S. Kulluk, L. Ozbakir and A. Baykasoglu, Training neural networks with harmony search algorithms for classification problems, Eng. Appl. Artif. Intell., 25 (2012), 11–19.
[21]	S. Mun and Y. H. Cho, Modified harmony search optimization for constrained design problems, Expert. Syst. Appl., 39 (2012), 419–423.
[22]	V. R. Pandi and B. K. Panigrahi, Dynamic economic load dispatch using hybrid swarm intelligence based harmony search algorithm, Expert. Syst. Appl., 38 (2011), 8509–8514.
[23]	A. Kusiak, Intelligent manufacturing systems., Prentice Hall Press, 200 Old Tappan Toad, Old Tappan, NJ 07675, USA, (1990), 448.
[24]	T. Ghosh, S. Sengupta, M. Chattopadhyay, et al., Meta-heuristics in cellular manufacturing: A state-of-the-art review, Int. J. Ind. Eng. Comput., 2 (2011), 87–122.
[25]	M. Alavidoost, M. F. Zarandi, M. Tarimoradi, et al., Modified genetic algorithm for simple straight and u-shaped assembly line balancing with fuzzy processing times, J. Intell. Manuf., 28 (2017), 313–336.
[26]	C. L. Kuo, C. H. Chu, Y. Li, et al., Electromagnetism-like algorithms for optimized tool path planning in 5-axis flank machining, Comput. Ind. Eng., 84 (2015), 70–78.
[27]	X. Li, C. Lu, L. Gao, et al., An effective multi-objective algorithm for energy efficient scheduling in a real-life welding shop, IEEE T. Ind. Inform., 14 (2018), 5400–5409.
[28]	C. Lu, L. Gao, X. Li, et al., A multi-objective approach to welding shop scheduling for makespan, noise pollution and energy consumption, J. Cleaner. Prod., 196 (2018), 773–787.
[29]	C. Lu, L. Gao, Q. Pan, et al., A multi-objective cellular grey wolf optimizer for hybrid flowshop scheduling problem considering noise pollution, Appl. Soft. Comput., 75 (2019), 728–749.
[30]	X. S. Yang, Harmony search as a metaheuristic algorithm, in Music-inspired harmony search algorithm, Springer, (2009), 1–14.
[31]	G. Ingram and T. Zhang, Overview of applications and developments in the harmony search algorithm, in Music-inspired harmony search algorithm, Springer, (2009), 15–37.
[32]	O. Moh'd Alia and R. Mandava, The variants of the harmony search algorithm: an overview, Artif. Intell. Rev., 36 (2011), 49–68.
[33]	X. Wang, X. Z. Gao and K. Zenger, The variations of harmony search and its current research trends, in An Introduction to Harmony Search Optimization Method, Springer, (2015), 21–30.
[34]	A. Askarzadeh, Solving electrical power system problems by harmony search: a review, Artif. Intell. Rev., 47 (2017), 217–251.
[35]	A. Askarzadeh and E. Rashedi, Harmony search algorithm: Basic concepts and engineering applications, in Intelligent Systems: Concepts, Methodologies, Tools, and Applications, 1–30.
[36]	J. Yi, X. Li, C. H. Chu, et al., Parallel chaotic local search enhanced harmony search algorithm for engineering design optimization, J. Intell. Manuf., 30 (2019), 405–428.
[37]	M. Mahdavi, M. Fesanghary and E. Damangir, An improved harmony search algorithm for solving optimization problems, Appl. Math. Comput., 188 (2007), 1567–1579.
[38]	M. G. Omran and M. Mahdavi, Global-best harmony search, Appl. Math. Comput., 198 (2008), 643–656.
[39]	Q. K. Pan, P. N. Suganthan, M. F. Tasgetiren, et al., A self-adaptive global best harmony search algorithm for continuous optimization problems, Appl. Math. Comput., 216 (2010), 830–848.
[40]	D. Zou, L. Gao, J.Wu, et al., Novel global harmony search algorithm for unconstrained problems, Neurocomputing, 73 (2010), 3308–3318.
[41]	J. Chen, Q. K. Pan and J. Q. Li, Harmony search algorithm with dynamic control parameters, Appl. Math. Comput., 219 (2012), 592–604.
[42]	R. Enayatifar, M. Yousefi, A. H. Abdullah, et al., Lahs: a novel harmony search algorithm based on learning automata, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3481–3497.
[43]	A. Kattan and R. Abdullah, A dynamic self-adaptive harmony search algorithm for continuous optimization problems, Appl. Math. Comput., 219 (2013), 8542–8567.
[44]	K. Luo, A novel self-adaptive harmony search algorithm, J. Appl. Math., 2013 (2013), 1–16.
[45]	X.Wang and X. Yan, Global best harmony search algorithm with control parameters co-evolution based on pso and its application to constrained optimal problems, Appl. Math. Comput., 219 (2013), 10059–10072.
[46]	J. Contreras, I. Amaya and R. Correa, An improved variant of the conventional harmony search algorithm, Appl. Math. Comput., 227 (2014), 821–830.
[47]	M. Khalili, R. Kharrat, K. Salahshoor, et al., Global dynamic harmony search algorithm: Gdhs, Appl. Math. Comput., 228 (2014), 195–219.
[48]	V. Kumar, J. K. Chhabra and D. Kumar, Parameter adaptive harmony search algorithm for unimodal and multimodal optimization problems, J. Comput. Sci., 5 (2014), 144–155.
[49]	G. Li and Q. Wang, A cooperative harmony search algorithm for function optimization, Math. Probl. Eng., 2014 (2014), 1–14.
[50]	I. Amaya, J. Cruz and R. Correa, Harmony search algorithm: a variant with self-regulated fretwidth, Appl. Math. Comput., 266 (2015), 1127–1152.
[51]	J. Kalivarapu, S. Jain and S. Bag, An improved harmony search algorithm with dynamically varying bandwidth, Eng. Optimiz., 48 (2016), 1091–1108.
[52]	Y. Wang, Z. Guo and Y. Wang, Enhanced harmony search with dual strategies and adaptive parameters, Soft Comput., 21 (2017), 4431–4445.
[53]	Z. Guo, H. Yang, S. Wang, et al., Adaptive harmony search with best-based search strategy, Soft Comput., 22 (2018), 1335–1349.
[54]	M. A. Al-Betar, I. A. Doush, A. T. Khader, et al., Novel selection schemes for harmony search, Appl. Math. Comput., 218 (2012), 6095–6117.
[55]	M. A. Al-Betar, A. T. Khader, Z. W. Geem, et al., An analysis of selection methods in memory consideration for harmony search, Appl. Math. Comput., 219 (2013), 10753–10767.
[56]	M. Castelli, S. Silva, L. Manzoni, et al., Geometric selective harmony search, Inform. Sci., 279 (2014), 468–482.
[57]	X. Gao, X. Wang, S. Ovaska, et al., A hybrid optimization method of harmony search and opposition-based learning, Eng. Optimiz., 44 (2012), 895–914.
[58]	A. Kaveh and M. Ahangaran, Social harmony search algorithm for continuous optimization, Iran. J. Sci. Technol. Trans. B-Eng., 36 (2012), 121–137.
[59]	P. Yadav, R. Kumar, S. K. Panda, et al., An intelligent tuned harmony search algorithm for optimisation, Inform. Sci., 196 (2012), 47–72.
[60]	M. A. Al-Betar, A. T. Khader, M. A. Awadallah, et al., Cellular harmony search for optimization problems, J. Appl. Math., 2013 (2013), 1–20.
[61]	S. Ashrafi and A. Dariane, Performance evaluation of an improved harmony search algorithm for numerical optimization: Melody search (ms), Eng. Appl. Artif. Intell., 26 (2013), 1301–1321.
[62]	M. El-Abd, An improved global-best harmony search algorithm, Appl. Math. Comput., 222 (2013), 94–106.
[63]	B. H. F. Hasan, I. A. Doush, E. Al Maghayreh, et al., Hybridizing harmony search algorithm with different mutation operators for continuous problems, Appl. Math. Comput., 232 (2014), 1166–1182.
[64]	E. Valian, S. Tavakoli and S. Mohanna, An intelligent global harmony search approach to continuous optimization problems, Appl. Math. Comput., 232 (2014), 670–684.
[65]	A. M. Turky and S. Abdullah, A multi-population harmony search algorithm with external archive for dynamic optimization problems, Inform. Sci., 272 (2014), 84–95.
[66]	M. A. Al-Betar, M. A. Awadallah, A. T. Khader, et al., Island-based harmony search for optimization problems, Expert. Syst. Appl., 42 (2015), 2026–2035.
[67]	J. Yi, L. Gao, X. Li, et al., An efficient modified harmony search algorithm with intersect mutation operator and cellular local search for continuous function optimization problems, Appl. Intell., 44 (2016), 725–753.
[68]	B. Keshtegar and M. O. Sadeq, Gaussian global-best harmony search algorithm for optimization problems, Soft Comput., 21 (2017), 7337–7349.
[69]	H. B. Ouyang, L. Q. Gao, S. Li, et al., Improved harmony search algorithm: Lhs, Appl. Soft. Comput., 53 (2017), 133–167.
[70]	E. A. Portilla-Flores, Á . Sańchez-Maŕquez, L. Flores-Pulido, et al., Enhancing the harmony search algorithm performance on constrained numerical optimization, IEEE Access, 5 (2017), 25759–25780.
[71]	S. Tuo, L. Yong and T. Zhou, An improved harmony search based on teaching-learning strategy for unconstrained optimization problems, Math. Probl. Eng., 2013 (2013), 1–21.
[72]	G. Wang, L. Guo, H. Duan, et al., Hybridizing harmony search with biogeography based optimization for global numerical optimization, J. Comput. Theor. Nanosci., 10 (2013), 2312–2322.
[73]	G. G.Wang, A. H. Gandomi, X. Zhao, et al., Hybridizing harmony search algorithm with cuckoo search for global numerical optimization, Soft Comput., 20 (2016), 273–285.
[74]	X. Yuan, J. Zhao, Y. Yang, et al., Hybrid parallel chaos optimization algorithm with harmony search algorithm, Appl. Soft. Comput., 17 (2014), 12–22.
[75]	F. Zhao, Y. Liu, C. Zhang, et al., A self-adaptive harmony pso search algorithm and its performance analysis, Expert. Syst. Appl., 42 (2015), 7436–7455.
[76]	A. Fouad, D. Boukhetala, F. Boudjema, et al., A novel global harmony search method based on ant colony optimisation algorithm, J. Exp. Theor. Artif. Intell., 28 (2016), 215–238.
[77]	G. Zhang and Y. Li, A memetic algorithm for global optimization of multimodal nonseparable problems, IEEE T. Cy., 46 (2016), 1375–1387.
[78]	A. Assad and K. Deep, A hybrid harmony search and simulated annealing algorithm for continuous optimization, Inform. Sci., 450 (2018), 246–266.
[79]	A. Sadollah, H. Sayyaadi, D. G. Yoo, et al., Mine blast harmony search: A new hybrid optimization method for improving exploration and exploitation capabilities, Appl. Soft. Comput., 68 (2018), 548–564.
[80]	L. Wang, H. Hu, R. Liu, et al., An improved differential harmony search algorithm for function optimization problems, Soft Comput., (2018), 1–26.
[81]	B. L. Miller and D. E. Goldberg, Genetic algorithms, selection schemes, and the varying effects of noise, Evol. Comput., 4 (1996), 113–131.
[82]	Y. Shi, H. Liu, L. Gao, et al., Cellular particle swarm optimization, Inform. Sci., 181 (2011), 4460–4493.
[83]	C. Lu, L. Gao and J. Yi, Grey wolf optimizer with cellular topological structure, Expert. Syst. Appl., 107 (2018), 89–114.
[84]	A. Turky, S. Abdullah and A. Dawod, A dual-population multi operators harmony search algorithm for dynamic optimization problems, Comput. Ind. Eng., 117 (2018), 19–28.
[85]	G. Wang and L. Guo, A novel hybrid bat algorithm with harmony search for global numerical optimization, J. Appl. Math., 2013 (2013), 1–21.
[86]	K. Z. Gao, P. N. Suganthan, Q. K. Pan, et al., Pareto-based grouping discrete harmony search algorithm for multi-objective flexible job shop scheduling, Inform. Sci., 289 (2014), 76–90.
[87]	K. Z. Gao, P. N. Suganthan, Q. K. Pan, et al., An effective discrete harmony search algorithm for flexible job shop scheduling problem with fuzzy processing time, Int. J. Prod. Res., 53 (2015), 5896–5911.
[88]	K. Z. Gao, P. N. Suganthan, Q. K. Pan, et al., Discrete harmony search algorithm for flexible job shop scheduling problem with multiple objectives, J. Intell. Manuf., 27 (2016), 363–374.
[89]	K. Gao, L.Wang, J. Luo, et al., Discrete harmony search algorithm for scheduling and rescheduling the reprocessing problems in remanufacturing: a case study, Eng. Optimiz., 50 (2018), 965– 981.
[90]	L. Liu and H. Zhou, Hybridization of harmony search with variable neighborhood search for restrictive single-machine earliness/tardiness problem, Inform. Sci., 226 (2013), 68–92.
[91]	Y. Yuan, H. Xu and J. Yang, A hybrid harmony search algorithm for the flexible job shop scheduling problem, Appl. Soft. Comput., 13 (2013), 3259–3272.
[92]	F. Zammori, M. Braglia and D. Castellano, Harmony search algorithm for single-machine scheduling problem with planned maintenance, Comput. Ind. Eng., 76 (2014), 333–346.
[93]	Y. Li, X. Li and J. N. Gupta, Solving the multi-objective flowline manufacturing cell scheduling problem by hybrid harmony search, Expert. Syst. Appl., 42 (2015), 1409–1417.
[94]	C. Garcia-Santiago, J. Del Ser, C. Upton, et al., A random-key encoded harmony search approach for energy-efficient production scheduling with shared resources, Eng. Optimiz., 47 (2015), 1481–1496.
[95]	A. Maroosi, R. C. Muniyandi, E. Sundararajan, et al., A parallel membrane inspired harmony search for optimization problems: A case study based on a flexible job shop scheduling problem, Appl. Soft. Comput., 49 (2016), 120–136.
[96]	Z. Guo, L. Shi, L. Chen, et al., A harmony search-based memetic optimization model for integrated production and transportation scheduling in mto manufacturing, Omega, 66 (2017), 327– 343.
[97]	F. Zhao, Y. Liu, Y. Zhang, et al., A hybrid harmony search algorithm with efficient job sequence scheme and variable neighborhood search for the permutation flow shop scheduling problems, Eng. Appl. Artif. Intell., 65 (2017), 178–199.
[98]	M. Gaham, B. Bouzouia and N. Achour, An effective operations permutation-based discrete harmony search approach for the flexible job shop scheduling problem with makespan criterion, Appl. Intell., 48 (2018), 1423–1441.
[99]	F. Zhao, S. Qin, G. Yang, et al., A differential-based harmony search algorithm with variable neighborhood search for job shop scheduling problem and its runtime analysis, IEEE Access, 6 (2018), 76313–76330.
[100]	S. M. Lee and S. Y. Han, Topology optimization scheme for dynamic stiffness problems using harmony search method, Int. J. Precis. Eng. Manuf., 17 (2016), 1187–1194.
[101]	S. M. Lee and S. Y. Han, Topology optimization based on the harmony search method, J. Mech. Sci. Technol., 31 (2017), 2875–2882.
[102]	J. Yi, X. Li, M. Xiao, et al., Construction of nested maximin designs based on successive local enumeration and modified novel global harmony search algorithm, Eng. Optimiz., 49 (2017), 161–180.
[103]	B. Keshtegar, P. Hao, Y. Wang, et al., Optimum design of aircraft panels based on adaptive dynamic harmony search, Thin-Walled Struct., 118 (2017), 37–45.
[104]	B. Keshtegar, P. Hao, Y. Wang, et al., An adaptive response surface method and gaussian globalbest harmony search algorithm for optimization of aircraft stiffened panels, Appl. Soft. Comput., 66 (2018), 196–207.
[105]	H. Ouyang, W. Wu, C. Zhang, et al., Improved harmony search with general iteration models for engineering design optimization problems, Soft Comput., 0 (2018), 1–36.
[106]	X. Li, K. Qin, B. Zeng, et al., Assembly sequence planning based on an improved harmony search algorithm, Int. J. Adv. Manuf. Tech., 84 (2016), 2367–2380.
[107]	X. Li, K. Qin, B. Zeng, et al., A dynamic parameter controlled harmony search algorithm for assembly sequence planning, Int. J. Adv. Manuf. Tech., 92 (2017), 3399–3411.
[108]	G. Li, B. Zeng, W. Liao, et al., A new agv scheduling algorithm based on harmony search for material transfer in a real-world manufacturing system, Adv. Mech. Eng., 10 (2018), 1–13.
[109]	G. Li, X. Li, L. Gao, et al., Tasks assigning and sequencing of multiple agvs based on an improved harmony search algorithm, J. Ambient Intell. Humaniz., 0 (2018), 1–14.
[110]	M. B. B. Mahaleh and S. A. Mirroshandel, Harmony search path detection for vision based automated guided vehicle, Robot. Auton. Syst., 107 (2018), 156–166.
[111]	M. Ayyıldız and K. C¸ etinkaya, Comparison of four different heuristic optimization algorithms for the inverse kinematics solution of a real 4-dof serial robot manipulator, Neural Comput. Appl., 27 (2016), 825–836.
[112]	O. Zarei, M. Fesanghary, B. Farshi, et al., Optimization of multi-pass face-milling via harmony search algorithm, J. Mater. Process. Technol., 209 (2009), 2386–2392.
[113]	K. Abhishek, S. Datta and S. S. Mahapatra, Multi-objective optimization in drilling of cfrp (polyester) Measurement, 77 (2016), 222–239.
[114]	S. Kumari, A. Kumar, R. K. Yadav, et al., Optimisation of machining parameters using grey relation analysis integrated with harmony search for turning of aisi d2 steel, Materials Today: Proceedings, 5 (2018), 12750–12756.
[115]	J. Yi, C. H. Chu, C. L. Kuo, et al., Optimized tool path planning for five-axis flank milling of ruled surfaces using geometric decomposition strategy and multi-population harmony search algorithm, Appl. Soft. Comput., 73 (2018), 547–561.
[116]	S. Atta, P. R. S. Mahapatra and A. Mukhopadhyay, Solving tool indexing problem using harmony search algorithm with harmony refinement, Soft Comput., (2018), 1–17.
[117]	C. C. Lin, D. J. Deng, Z. Y. Chen, et al., Key design of driving industry 4.0: Joint energy-efficient deployment and scheduling in group-based industrial wireless sensor networks, IEEE Commun. Mag., 54 (2016), 46–52.
[118]	B. Zeng and Y. Dong, An improved harmony search based energy-efficient routing algorithm for wireless sensor networks, Appl. Soft. Comput., 41 (2016), 135–147.
[119]	L. Wang, L. An, H. Q. Ni, et al., Pareto-based multi-objective node placement of industrial wireless sensor networks using binary differential evolution harmony search, Adv. Manuf., 4 (2016), 66–78.
[120]	O. Moh'd Alia and A. Al-Ajouri, Maximizing wireless sensor network coverage with minimum cost using harmony search algorithm, IEEE Sens. J., 17 (2017), 882–896.
[121]	C. C. Lin, D. J. Deng, J. R. Kang, et al., Forecasting rare faults of critical components in led epitaxy plants using a hybrid grey forecasting and harmony search approach, IEEE Trans. Ind. Inform., 12 (2016), 2228–2235.
[122]	S. Kang and J. Chae, Harmony search for the layout design of an unequal area facility, Expert. Syst. Appl., 79 (2017), 269–281.
[123]	J. Lin, M. Liu, J. Hao, et al., Many-objective harmony search for integrated order planning in steelmaking-continuous casting-hot rolling production of multi-plants, Int. J. Prod. Res., 55 (2017), 4003–4020.
[124]	Y. H. Kim, Y. Yoon and Z. W. Geem, A comparison study of harmony search and genetic algorithm for the max-cut problem, Swarm Evol. Comput., 44 (2018), 130–135.
[125]	B. Naderi, R. Tavakkoli-Moghaddam, et al., Electromagnetism-like mechanism and simulated annealing algorithms for flowshop scheduling problems minimizing the total weighted tardiness and makespan, Knowledge-Based Syst., 23 (2010), 77–85.
[126]	M. R. Garey, D. S. Johnson and R. Sethi, The complexity of flowshop and jobshop scheduling, Math. Oper. Res., 1 (1976), 117–129.
[127]	P. J. Van Laarhoven, E. H. Aarts and J. K. Lenstra, Job shop scheduling by simulated annealing, Oper. Res., 40 (1992), 113–125.
[128]	M. Dell'Amico and M. Trubian, Applying tabu search to the job-shop scheduling problem, Ann. Oper. Res., 41 (1993), 231–252.
[129]	I. Kacem, S. Hammadi and P. Borne, Approach by localization and multiobjective evolutionary optimization for flexible job-shop scheduling problems, IEEE T. Syst. Man. Cy. C., 32 (2002), 1–13.
[130]	W. Xia and Z. Wu, An effective hybrid optimization approach for multi-objective flexible jobshop scheduling problems, Comput. Ind. Eng., 48 (2005), 409–425.
[131]	G. Zhang, X. Shao, P. Li, et al., An effective hybrid particle swarm optimization algorithm for multi-objective flexible job-shop scheduling problem, Comput. Ind. Eng., 56 (2009), 1309–1318.
[132]	G. Zhang, L. Gao and Y. Shi, An effective genetic algorithm for the flexible job-shop scheduling problem, Expert. Syst. Appl., 38 (2011), 3563–3573.
[133]	Q. Lin, L. Gao, X. Li, et al., A hybrid backtracking search algorithm for permutation flow-shop scheduling problem, Comput. Ind. Eng., 85 (2015), 437–446.
[134]	C. Lu, L. Gao, X. Li, et al., Energy-efficient permutation flow shop scheduling problem using a hybrid multi-objective backtracking search algorithm, J. Cleaner. Prod., 144 (2017), 228–238.
[135]	C. Viergutz and S. Knust, Integrated production and distribution scheduling with lifespan constraints, Ann. Oper. Res., 213 (2014), 293–318.
[136]	R. T. Lund, Remanufacturing, Technol. Rev., 87 (1984), 18.
[137]	Y. Liu, H. Dong, N. Lohse, et al., An investigation into minimising total energy consumption and total weighted tardiness in job shops, J. Cleaner. Prod., 65 (2014), 87–96.
[138]	M. Mashayekhi, E. Salajegheh and M. Dehghani, Topology optimization of double and triple layer grid structures using a modified gravitational harmony search algorithm with efficient member grouping strategy, Comput. Struct., 172 (2016), 40–58.
[139]	M. F. F. Rashid, W. Hutabarat and A. Tiwari, A review on assembly sequence planning and assembly line balancing optimisation using soft computing approaches, Int. J. Adv. Manuf. Tech., 59 (2012), 335–349.
[140]	D. Ghosh, A new genetic algorithm for the tool indexing problem, Technical report, Indian Institute of Management Ahmedabad, 2016.
[141]	D. Ghosh, Exploring Lin Kernighan neighborhoods for the indexing problem, Technical report, Indian Institute of Management Ahmedabad, 2016.
[142]	M. Hermann, T. Pentek and B. Otto, Design principles for industrie 4.0 scenarios, in System Sciences (HICSS), 2016 49th Hawaii International Conference on, IEEE, (2016), 3928–3937.
[143]	S. Das, A. Mukhopadhyay, A. Roy, et al., Exploratory power of the harmony search algorithm: analysis and improvements for global numerical optimization, IEEE T. Syst. Man. Cy. B., 41 (2011), 89–106.
[144]	L. Q. Gao, S. Li, X. Kong, et al., On the iterative convergence of harmony search algorithm and a proposed modification, Appl. Math. Comput., 247 (2014), 1064–1095.
[145]	T. G. Dietterich, Ensemble methods in machine learning, in International workshop on multiple classifier systems, Springer, 2000, 1–15.
[146]	S. Mahdavi, M. E. Shiri and S. Rahnamayan, Metaheuristics in large-scale global continues optimization: A survey, Inform. Sci., 295 (2015), 407–428.
[147]	G. Karafotias, M. Hoogendoorn and Á . E. Eiben, Parameter control in evolutionary algorithms: Trends and challenges, IEEE Trans. Evol. Comput., 19 (2015), 167–187.

This article has been cited by:

1.	2021, 9781119624547, 85, 10.1002/9781119624547.ch3
2.	Mohammad Nasir, Ali Sadollah, Jin Hee Yoon, Zong Woo Geem, Comparative Study of Harmony Search Algorithm and its Applications in China, Japan and Korea, 2020, 10, 2076-3417, 3970, 10.3390/app10113970
3.	Yayun Xin, Jin Yi, Kai Zhang, Canfeng Chen, Jian Xiong, Offline Selective Harmonic Elimination With (2N+1) Output Voltage Levels in Modular Multilevel Converter Using a Differential Harmony Search Algorithm, 2020, 8, 2169-3536, 121596, 10.1109/ACCESS.2020.3007022
4.	Liang Guo, Ming Zhou, Yuqian Lu, Tao Yang, Fan Yang, A hybrid 3D feature recognition method based on rule and graph, 2021, 34, 0951-192X, 257, 10.1080/0951192X.2020.1858507
5.	Yuesheng Luo, Chao Lu, Xinyu Li, Ling Wang, Liang Gao, 2019, Green Job Shop Scheduling Problem with Machine at Different Speeds using a multi-objective grey wolf optimization algorithm*, 978-1-7281-0356-3, 573, 10.1109/COASE.2019.8843132
6.	Chao Lu, Liang Gao, Wenyin Gong, Chengyu Hu, Xuesong Yan, Xinyu Li, Sustainable scheduling of distributed permutation flow-shop with non-identical factory using a knowledge-based multi-objective memetic optimization algorithm, 2021, 60, 22106502, 100803, 10.1016/j.swevo.2020.100803
7.	Yong-Woon Jeong, Seung-Min Park, Zong Woo Geem, Kwee-Bo Sim, Advanced Parameter-Setting-Free Harmony Search Algorithm, 2020, 10, 2076-3417, 2586, 10.3390/app10072586
8.	Piotr Nowakowski, Krzysztof Szwarc, Urszula Boryczka, Combining an artificial intelligence algorithm and a novel vehicle for sustainable e-waste collection, 2020, 730, 00489697, 138726, 10.1016/j.scitotenv.2020.138726
9.	Kathiresan Gopal, Lai Soon Lee, Hsin-Vonn Seow, Parameter Estimation of Compartmental Epidemiological Model Using Harmony Search Algorithm and Its Variants, 2021, 11, 2076-3417, 1138, 10.3390/app11031138
10.	Mahima Dubey, Vijay Kumar, Manjit Kaur, Thanh-Phong Dao, Hassène Gritli, A Systematic Review on Harmony Search Algorithm: Theory, Literature, and Applications, 2021, 2021, 1563-5147, 1, 10.1155/2021/5594267
11.	Alireza Sadeghi Hesar, Seyed Reza Kamel, Mahboobeh Houshmand, A quantum multi-objective optimization algorithm based on harmony search method, 2021, 1432-7643, 10.1007/s00500-021-05799-x
12.	Mohsin Nazir, Aneeqa Sabah, Sana Sarwar, Azeema Yaseen, Anca Jurcut, Power and Resource Allocation in Wireless Communication Network, 2021, 0929-6212, 10.1007/s11277-021-08419-x
13.	Jiachen Liang, Shusheng Zhang, Bo Huang, Yajun Zhang, Rui Huang, NC process analysis–based intersecting machining feature recognition and reuse approach, 2022, 123, 0268-3768, 2393, 10.1007/s00170-022-10281-5
14.	Tianqi Liu, Hua Yang, Jing Yu, Kang Zhou, 2022, A Cross-Mutation Global Harmony Search Algorithm Based on Lévy Flight, 978-1-6654-9858-6, 19, 10.1109/MLCCIM55934.2022.00012
15.	Yu Lei, Zhi Su, Xiaotong He, Chao Cheng, Immersive virtual reality application for intelligent manufacturing: Applications and art design, 2022, 20, 1551-0018, 4353, 10.3934/mbe.2023202
16.	Lianyang Zhou, Fei Wang, Edge computing and machinery automation application for intelligent manufacturing equipment, 2021, 87, 01419331, 104389, 10.1016/j.micpro.2021.104389
17.	Wenhui Zeng, Jin Yi, Rongfu Lin, Wenlong Lu, Statistical tolerance–cost–service life optimization of blade bearing of controllable pitch propeller considering the marine environment conditions through meta-heuristic algorithm, 2022, 9, 2288-5048, 689, 10.1093/jcde/qwac023
18.	R. M. Fonseca-Pérez, O. Del-Mazo-Alvarado, A. Meza-de-Luna, A. Bonilla-Petriciolet, Z. W. Geem, Doraiswami Ramkrishna, An Overview of the Application of Harmony Search for Chemical Engineering Optimization, 2022, 2022, 1687-8078, 1, 10.1155/2022/1928343
19.	Mohammed Hadwan, Annealing Harmony Search Algorithm to Solve the Nurse Rostering Problem, 2022, 71, 1546-2226, 5545, 10.32604/cmc.2022.024512
20.	A. Costa, G. Buffa, D. Palmeri, G. Pollara, L. Fratini, Hybrid prediction-optimization approaches for maximizing parts density in SLM of Ti6Al4V titanium alloy, 2022, 33, 0956-5515, 1967, 10.1007/s10845-022-01938-9
21.	Xianwang Li, Zhongxiang Huang, Wenhui Ning, Intelligent manufacturing quality prediction model and evaluation system based on big data machine learning, 2023, 111, 00457906, 108904, 10.1016/j.compeleceng.2023.108904
22.	Jin Hee Yoon, Zong Woo Geem, Empirical Convergence Theory of Harmony Search Algorithm for Box-Constrained Discrete Optimization of Convex Function, 2021, 9, 2227-7390, 545, 10.3390/math9050545
23.	Linchao Yang, Ying Liu, Guanglu Yang, Shi-Tong Peng, Dynamic monitoring and anomaly tracing of the quality in tobacco strip processing based on improved canonical variable analysis and transfer entropy, 2023, 20, 1551-0018, 15309, 10.3934/mbe.2023684

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)