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Lexicographic multi-objective optimization of thermoacoustic refrigerator’s stack

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Abstract

This work develops a novel mathematical programming model to optimize the performance of a simple thermoacoustic refrigerator (TAR). This study aims to optimize the geometric parameters namely the stack position, the stack length, the blockage ratio and the plate spacing involved in designing TARs. System parameters and constraints that capture the underlying thermoacoustic dynamics have been used to define the models. The cooling load, the coefficient of performance and the acoustic power loss have been used to measure the performance of the device. The optimization task is formulated as a three-criterion nonlinear programming problem with discontinuous derivatives (DNLP). Since we optimize multiple objectives simultaneously, each objective component has been given a weighting factor to provide appropriate user-defined emphasis. A practical example is given to illustrate the approach. We have determined a design statement of a stack describing how the geometrical parameters describing would change if emphasis is given to one objective in particular. We also considered optimization of multiple objectives components simultaneously and identify global optimal solutions describing the stack geometry using a lexicographic multi-objective optimization scheme. Additionally, this approach illustrates the difference between a design for maximum cooling and best coefficient of performance of a simple TAR.

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Abbreviations

a:

Speed of sound (m/s)

BR:

Blockage ratio

cp :

Isobaric specific heat capacity (J/kgK)

COP:

Coefficient of performance of refrigerator

COPC:

Carnot coefficient of performance

COPR:

Relative coefficient of performance

diri :

Direction of ith objective

DR:

Drive ratio

f:

Frequency (Hz)

K:

Thermal conductivity (W/m K)

l:

Plate half thickness (mm)

LS :

Stack length (mm)

LSn :

Normalised stack length

min:

Minimize

max:

Maximize

pm :

Mean pressure (Pa)

ri :

Range of ith objective function

si :

Surplus of ith objective

Tm :

Mean temperature

Tmn :

Normalized temperature difference

XS :

Stack centre position (mm)

XSn :

Normalised stack position

wi :

Objective function component weight

\(\mathop {{\text{W}}_{2} }\limits^{\text{o}}\) :

Acoustic power loss

yo :

Plate half-gap (mm)

δk :

Gas thermal penetration depth (mm)

δkn :

Normalised thermal penetration depth

δs :

Solid thermal penetration depth (mm)

δv :

Viscous penetration depth

γ:

Isentropic coefficient

εs :

Stack heat capacity correction factor

ω:

Angular frequency (rad/s)

ρm :

Density (kg/m3)

σ:

Prandtl number

θ:

Normalised temperature difference

∆Tm :

Temperature span (K)

Φc :

Normalized cooling load

ΦH :

Normalized heat flow

ΦW :

Normalized acoustic power

ξ:

Objective function

μ:

Dynamic viscosity (kg/m s)

λ:

Wavelength (mm)

εi :

Right hand side of ith objective function

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Authors and Affiliations

  1. Department of Mechanical Engineering, Mangosuthu University of Technology, Box 12363, Durban, 4026, South Africa

    L. K. Tartibu

  2. Department of Mechanical Engineering, Cape Peninsula University of Technology, Box 652, Cape Town, 8000, South Africa

    B. Sun & M. A. E. Kaunda

Authors
  1. L. K. Tartibu
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  2. B. Sun
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  3. M. A. E. Kaunda
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Correspondence to L. K. Tartibu.

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Tartibu, L.K., Sun, B. & Kaunda, M.A.E. Lexicographic multi-objective optimization of thermoacoustic refrigerator’s stack. Heat Mass Transfer 51, 649–660 (2015). https://doi.org/10.1007/s00231-014-1440-z

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