# Closed-form analytical solutions of transient heat conduction in hollow composite cylinders with any number of layers

### Significance Statement

Hollow multilayer cylinders have been applied in a number of engineering applications such as rockets, turbines, heat exchangers, spacecraft, fuel cells, pipelines, and electrochemical reactors in selected industrial processes. In most of these applications, it is necessary to acquire knowledge relating to temperature and heat flow via a cylindrical composite media that eventually translates to a classic problem of transient heat conduction. For a majority of these problems, numerical methods such as finite element and finite difference methods are commonly used.

Although these numerical methods are commonly used, analytical methods remain desirable for efficiency and accuracy in computation and in providing deeper physical insight. In addition, analytical methods can be helpful as a benchmark for the validation of numerical solutions. Several analytical methods are available for transient heat conduction problems including orthogonal and quasi-orthogonal expansion methods, Laplace transform method, finite integral transform method, and their combinations. Generally, analytical methods should be applicable in typical multilayer bodies. Unfortunately, in practice, analytical methods are only limited to composite media composed of two or three layers.

Problems such as complex expressions, formidable derivations, and arithmetic errors in inverse Laplace transform limit typical analytical methods from obtaining transient solutions for composite bodies with several layers. Bingen Yang and Shibing Liu at the University of Southern California demonstrated an effort to obtain closed-form analytical solution for transient heat conduction in hollow composite cylinders. A closed-form analytical solution was described as one that was given by an infinite series with every term presented in an exact mathematical expression with a finite number of terms. They considered a one-dimensional problem, with temperature distribution in the radial direction. Their research work is published in International Journal of Heat and Mass Transfer.

In the development, the authors derived a distributed transfer function formulation as well as a formula for transfer function residues, and eventually yielded transient heat conduction solutions in explicit and closed form. The thrust of the proposed method lied in that it was applicable to composite cylinders of an arbitrary number of layers and subject to typical boundary conditions (including time-dependent boundary excitation). According to the authors, the proposed analytical method was new in the field of heat conduction.

Distributed transfer function method treated multilayer composite cylinders in a systematic way. As opposed to conventional analytical methods, the proposed method did not require different derivations for different cylinder arrangement. For a selected number of layers, boundary conditions and thermal resistance at layer interfaces, the authors only needed to modify the matrices in spatial state equations.

The authors observed that the Laplace transform through the exact residue formula yielded the transient response of a multilayer composite cylinder. Exact transfer function residues were obtained, therefore, no numerical errors were induced in Laplace inversion, and as such, the transient solution was given in an explicit and closed form.

Distributed transfer function method developed by Yang and Liu was applicable to composite cylinders with thermal resistance at layer interfaces. This was realized through simple modification of the matching conditions at layer interfaces. The solution was the same as that of composite cylinders with perfect thermal contact at layer interfaces. Distributed transfer function method-based solution process only entailed two-by-two matrices, irrespective of the number of layers in the composite media. This method made the solution process more efficient in computation.

Bingen “Ben” Yang is Professor of Aerospace and Mechanical Engineering at the University of Southern California. With 20 years of research experience, he has published approximately 150 journal and conference papers in the areas of solid mechanics, vibrations, structural dynamics and controls. His current research interests include modeling, analysis, and deployment control of inflatable space structures, servo control of flexible positioning systems, and development of intelligent biomedical devices via smart materials and micro-electro-mechanical systems. A fellow of the American Society of Mechanical Engineers, Dr. Yang received his Ph. D. in mechanical engineering from the University of California at Berkeley.

He has undertaken various research projects on modelling, dynamic analysis, control, and failure analysis of complex mechanical systems including aircraft and vehicle structures, flexible robot arms, automobile brakes, weapon systems, cable-stayed long span bridges, distributed vibration dampers, intelligent vehicle/highway systems, information storage systems, and machines for material processing and manufacture.

Those projects have been sponsored by various government and industrial funding agencies including the National Science Foundation, the Powell Foundation, the U.S. Army Research Office, Tektronix, Motorola, Caltrans, Structural Research & Analysis Corporation, Ford Motor Company, Hughes, and General Motor Company, Dr. Yang’s research has resulted in over 60 journal publications.

Shibing Liu is a Controls and Dynamics Engineer at the Hyperloop One. He received his Bachelor of Science in Mechanical Engineering Technology from the Southern Polytechnic State University and North China University of Technology, and his MS and PhD in Mechanical Engineering from the University of Southern California (USC), where his thesis work was associated with modeling, analysis, experiments, and optimization of flexible rotor systems with water-lubricated rubber bearings.

He developed an innovative mathematical model to determine dynamic coefficients of water lubricated rubber bearings (WLRBs). Not only this model is the first attempt to consider mixed lubrication in the WLRB analysis, but also enables the capability of deriving an analytical closed-form solution to flexible rotor-WLRB dynamics. Such a closed-form mathematical expression was deployed in optimizing the number and locations of bearings that yield minimum vibration amplitude. His research results have been used in the research and development of new products by top pump manufacturing companies.

His research interests include dynamics and control, vibrations, rotordynamics, heat transfer, structural health monitoring, and optimization of mechanical systems.

### Reference

Bingen Yang and Shibing Liu. Closed-form analytical solutions of transient heat conduction in hollow composite cylinders with any number of layers. International Journal of Heat and Mass Transfer, volume 108 (2017), pages 907–917.